Eta-products on modular curves \(X_0(N)\) - Modular Forms

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# Eta-products on modular curves \(X_0(N)\)

This package provides a class for representing eta-products, which are meromorphic functions on modular curves of the form

\[\prod_{d | N} \eta(q^d)^{r_d}\]

where \(\eta(q)\) is Dirichlet’s eta function

\[q^{1/24} \prod_{n = 1}^\infty(1-q^n) .\]

These are useful for obtaining explicit models of modular curves.

See Issue #3934 for background.

AUTHOR:

- David Loeffler (2008-08-22): initial version

sage.modular.etaproducts.AllCusps(N)

  Return a list of `CuspFamily` objects corresponding to the cusps of \(X_0(N)\).

  INPUT:

  - `N` – integer; the level

  EXAMPLES:

  Sage

      sage: AllCusps(18)
      [(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]
      sage: AllCusps(0)
      Traceback (most recent call last):
      ...
      ValueError: N must be positive

  Python

      >>> from sage.all import *
      >>> AllCusps(Integer(18))
      [(Inf), (c_{2}), (c_{3,1}), (c_{3,2}), (c_{6,1}), (c_{6,2}), (c_{9}), (0)]
      >>> AllCusps(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: N must be positive

class sage.modular.etaproducts.CuspFamily(N, width, label=None)

  Bases: `SageObject`

  A family of elliptic curves parametrising a region of \(X_0(N)\).

  level()

    Return the level of this cusp.

    EXAMPLES:

    Sage

        sage: e = CuspFamily(10, 1)
        sage: e.level()
        10

    Python

        >>> from sage.all import *
        >>> e = CuspFamily(Integer(10), Integer(1))
        >>> e.level()
        10

  sage_cusp()

    Return the corresponding element of \(\mathbb{P}^1(\QQ)\).

    EXAMPLES:

    Sage

        sage: CuspFamily(10, 1).sage_cusp() # not implemented
        Infinity

    Python

        >>> from sage.all import *
        >>> CuspFamily(Integer(10), Integer(1)).sage_cusp() # not implemented
        Infinity

  width()

    Return the width of this cusp.

    EXAMPLES:

    Sage

        sage: e = CuspFamily(10, 1)
        sage: e.width()
        1

    Python

        >>> from sage.all import *
        >>> e = CuspFamily(Integer(10), Integer(1))
        >>> e.width()
        1

sage.modular.etaproducts.EtaGroup(level)

  Create the group of eta products of the given level.

  EXAMPLES:

  Sage

      sage: EtaGroup(12)
      Group of eta products on X_0(12)
      sage: EtaGroup(1/2)
      Traceback (most recent call last):
      ...
      TypeError: Level (=1/2) must be a positive integer
      sage: EtaGroup(0)
      Traceback (most recent call last):
      ...
      ValueError: Level (=0) must be a positive integer

  Python

      >>> from sage.all import *
      >>> EtaGroup(Integer(12))
      Group of eta products on X_0(12)
      >>> EtaGroup(Integer(1)/Integer(2))
      Traceback (most recent call last):
      ...
      TypeError: Level (=1/2) must be a positive integer
      >>> EtaGroup(Integer(0))
      Traceback (most recent call last):
      ...
      ValueError: Level (=0) must be a positive integer

class sage.modular.etaproducts.EtaGroupElement(parent, rdict)

  Bases: `Element`

  Create an eta product object. Usually called implicitly via EtaGroup_class.__call__ or the EtaProduct factory function.

  EXAMPLES:

  Sage

      sage: EtaProduct(8, {1:24, 2:-24})
      Eta product of level 8 : (eta_1)^24 (eta_2)^-24
      sage: g = _; g == loads(dumps(g))
      True
      sage: TestSuite(g).run()

  Python

      >>> from sage.all import *
      >>> EtaProduct(Integer(8), {Integer(1):Integer(24), Integer(2):-Integer(24)})
      Eta product of level 8 : (eta_1)^24 (eta_2)^-24
      >>> g = _; g == loads(dumps(g))
      True
      >>> TestSuite(g).run()

  degree()

    Return the degree of `self` as a map \(X_0(N) \to \mathbb{P}^1\).

    This is the sum of all the positive coefficients in the divisor of `self`.

    EXAMPLES:

    Sage

        sage: e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
        sage: e.degree()
        230

    Python

        >>> from sage.all import *
        >>> e = EtaProduct(Integer(12), {Integer(1):-Integer(336), Integer(2):Integer(576), Integer(3):Integer(696), Integer(4):-Integer(216), Integer(6):-Integer(576), Integer(12):-Integer(144)})
        >>> e.degree()
        230

  divisor()

    Return the divisor of `self`, as a formal sum of CuspFamily objects.

    EXAMPLES:

    Sage

        sage: e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144})
        sage: e.divisor() # FormalSum seems to print things in a random order?
        -131*(Inf) - 50*(c_{2}) + 11*(0) + 50*(c_{6}) + 169*(c_{4}) - 49*(c_{3})
        sage: e = EtaProduct(2^8, {8:1,32:-1})
        sage: e.divisor() # random
        -(c_{2}) - (Inf) - (c_{8,2}) - (c_{8,3}) - (c_{8,4}) - (c_{4,2})
         - (c_{8,1}) - (c_{4,1}) + (c_{32,4}) + (c_{32,3}) + (c_{64,1})
         + (0) + (c_{32,2}) + (c_{64,2}) + (c_{128}) + (c_{32,1})

    Python

        >>> from sage.all import *
        >>> e = EtaProduct(Integer(12), {Integer(1):-Integer(336), Integer(2):Integer(576), Integer(3):Integer(696), Integer(4):-Integer(216), Integer(6):-Integer(576), Integer(12):-Integer(144)})
        >>> e.divisor() # FormalSum seems to print things in a random order?
        -131*(Inf) - 50*(c_{2}) + 11*(0) + 50*(c_{6}) + 169*(c_{4}) - 49*(c_{3})
        >>> e = EtaProduct(Integer(2)**Integer(8), {Integer(8):Integer(1),Integer(32):-Integer(1)})
        >>> e.divisor() # random
        -(c_{2}) - (Inf) - (c_{8,2}) - (c_{8,3}) - (c_{8,4}) - (c_{4,2})
         - (c_{8,1}) - (c_{4,1}) + (c_{32,4}) + (c_{32,3}) + (c_{64,1})
         + (0) + (c_{32,2}) + (c_{64,2}) + (c_{128}) + (c_{32,1})

  is_one()

    Return whether `self` is the one of the monoid.

    EXAMPLES:

    Sage

        sage: e = EtaProduct(3, {3:12, 1:-12})
        sage: e.is_one()
        False
        sage: e.parent().one().is_one()
        True
        sage: ep = EtaProduct(5, {})
        sage: ep.is_one()
        True
        sage: ep.parent().one() == ep
        True

    Python

        >>> from sage.all import *
        >>> e = EtaProduct(Integer(3), {Integer(3):Integer(12), Integer(1):-Integer(12)})
        >>> e.is_one()
        False
        >>> e.parent().one().is_one()
        True
        >>> ep = EtaProduct(Integer(5), {})
        >>> ep.is_one()
        True
        >>> ep.parent().one() == ep
        True

  level()

    Return the level of this eta product.

    EXAMPLES:

    Sage

        sage: e = EtaProduct(3, {3:12, 1:-12})
        sage: e.level()
        3
        sage: EtaProduct(12, {6:6, 2:-6}).level() # not the lcm of the d's
        12
        sage: EtaProduct(36, {6:6, 2:-6}).level() # not minimal
        36

    Python

        >>> from sage.all import *
        >>> e = EtaProduct(Integer(3), {Integer(3):Integer(12), Integer(1):-Integer(12)})
        >>> e.level()
        3
        >>> EtaProduct(Integer(12), {Integer(6):Integer(6), Integer(2):-Integer(6)}).level() # not the lcm of the d's
        12
        >>> EtaProduct(Integer(36), {Integer(6):Integer(6), Integer(2):-Integer(6)}).level() # not minimal
        36

  order_at_cusp(cusp)

    Return the order of vanishing of `self` at the given cusp.

    INPUT:

    - `cusp` – a `CuspFamily` object

    OUTPUT: integer

    EXAMPLES:

    Sage

        sage: e = EtaProduct(2, {2:24, 1:-24})
        sage: e.order_at_cusp(CuspFamily(2, 1)) # cusp at infinity
        1
        sage: e.order_at_cusp(CuspFamily(2, 2)) # cusp 0
        -1

    Python

        >>> from sage.all import *
        >>> e = EtaProduct(Integer(2), {Integer(2):Integer(24), Integer(1):-Integer(24)})
        >>> e.order_at_cusp(CuspFamily(Integer(2), Integer(1))) # cusp at infinity
        1
        >>> e.order_at_cusp(CuspFamily(Integer(2), Integer(2))) # cusp 0
        -1

  q_expansion(n)

    Return the \(q\)-expansion of `self` at the cusp at infinity.

    INPUT:

    - `n` – integer; number of terms to calculate

    OUTPUT:

    A power series over \(\ZZ\) in the variable \(q\), with a relative precision of \(1 + O(q^n)\).

    ALGORITHM: Calculates eta to (n/m) terms, where m is the smallest integer dividing self.level() such that self.r(m) != 0. Then multiplies.

    EXAMPLES:

    Sage

        sage: EtaProduct(36, {6:6, 2:-6}).q_expansion(10)
        q + 6*q^3 + 27*q^5 + 92*q^7 + 279*q^9 + O(q^11)
        sage: R.<q> = ZZ[[]]
        sage: EtaProduct(2,{2:24,1:-24}).q_expansion(100) == delta_qexp(101)(q^2)/delta_qexp(101)(q)
        True

    Python

        >>> from sage.all import *
        >>> EtaProduct(Integer(36), {Integer(6):Integer(6), Integer(2):-Integer(6)}).q_expansion(Integer(10))
        q + 6*q^3 + 27*q^5 + 92*q^7 + 279*q^9 + O(q^11)
        >>> R = ZZ[['q']]; (q,) = R._first_ngens(1)
        >>> EtaProduct(Integer(2),{Integer(2):Integer(24),Integer(1):-Integer(24)}).q_expansion(Integer(100)) == delta_qexp(Integer(101))(q**Integer(2))/delta_qexp(Integer(101))(q)
        True

  qexp(n)

    Alias for `self.q_expansion()`.

    EXAMPLES:

    Sage

        sage: e = EtaProduct(36, {6:8, 3:-8})
        sage: e.qexp(10)
        q + 8*q^4 + 36*q^7 + O(q^10)
        sage: e.qexp(30) == e.q_expansion(30)
        True

    Python

        >>> from sage.all import *
        >>> e = EtaProduct(Integer(36), {Integer(6):Integer(8), Integer(3):-Integer(8)})
        >>> e.qexp(Integer(10))
        q + 8*q^4 + 36*q^7 + O(q^10)
        >>> e.qexp(Integer(30)) == e.q_expansion(Integer(30))
        True

  r(d)

    Return the exponent \(r_d\) of \(\eta(q^d)\) in `self`.

    EXAMPLES:

    Sage

        sage: e = EtaProduct(12, {2:24, 3:-24})
        sage: e.r(3)
        -24
        sage: e.r(4)
        0

    Python

        >>> from sage.all import *
        >>> e = EtaProduct(Integer(12), {Integer(2):Integer(24), Integer(3):-Integer(24)})
        >>> e.r(Integer(3))
        -24
        >>> e.r(Integer(4))
        0

class sage.modular.etaproducts.EtaGroup_class(level)

  Bases: `UniqueRepresentation`, `Parent`

  The group of eta products of a given level under multiplication.

  Element

    alias of `EtaGroupElement`

  basis(reduce=True)

    Produce a basis for the free abelian group of eta-products of level N (under multiplication), attempting to find basis vectors of the smallest possible degree.

    INPUT:

    - `reduce` – boolean (default: `True`); whether or not to apply LLL-reduction to the calculated basis

    EXAMPLES:

    Sage

        sage: EtaGroup(5).basis()
        [Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
        sage: EtaGroup(12).basis()
        [Eta product of level 12 : (eta_1)^-3 (eta_2)^2 (eta_3)^1 (eta_4)^-1 (eta_6)^-2 (eta_12)^3,
         Eta product of level 12 : (eta_1)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
         Eta product of level 12 : (eta_1)^6 (eta_2)^-9 (eta_3)^-2 (eta_4)^3 (eta_6)^3 (eta_12)^-1,
         Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
         Eta product of level 12 : (eta_1)^3 (eta_3)^-1 (eta_4)^-3 (eta_12)^1]
        sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
        [Eta product of level 12 : (eta_1)^384 (eta_2)^-576 (eta_3)^-696 (eta_4)^216 (eta_6)^576 (eta_12)^96,
         Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
         Eta product of level 12 : (eta_1)^-40 (eta_2)^116 (eta_3)^96 (eta_4)^-30 (eta_6)^-80 (eta_12)^-62,
         Eta product of level 12 : (eta_1)^-4 (eta_2)^-33 (eta_3)^-4 (eta_4)^1 (eta_6)^3 (eta_12)^37,
         Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]

    Python

        >>> from sage.all import *
        >>> EtaGroup(Integer(5)).basis()
        [Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
        >>> EtaGroup(Integer(12)).basis()
        [Eta product of level 12 : (eta_1)^-3 (eta_2)^2 (eta_3)^1 (eta_4)^-1 (eta_6)^-2 (eta_12)^3,
         Eta product of level 12 : (eta_1)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
         Eta product of level 12 : (eta_1)^6 (eta_2)^-9 (eta_3)^-2 (eta_4)^3 (eta_6)^3 (eta_12)^-1,
         Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
         Eta product of level 12 : (eta_1)^3 (eta_3)^-1 (eta_4)^-3 (eta_12)^1]
        >>> EtaGroup(Integer(12)).basis(reduce=False) # much bigger coefficients
        [Eta product of level 12 : (eta_1)^384 (eta_2)^-576 (eta_3)^-696 (eta_4)^216 (eta_6)^576 (eta_12)^96,
         Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
         Eta product of level 12 : (eta_1)^-40 (eta_2)^116 (eta_3)^96 (eta_4)^-30 (eta_6)^-80 (eta_12)^-62,
         Eta product of level 12 : (eta_1)^-4 (eta_2)^-33 (eta_3)^-4 (eta_4)^1 (eta_6)^3 (eta_12)^37,
         Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]

    ALGORITHM: An eta product of level \(N\) is uniquely determined by the integers \(r_d\) for \(d | N\) with \(d < N\), since \(\sum_{d | N} r_d = 0\). The valid \(r_d\) are those that satisfy two congruences modulo 24, and one congruence modulo 2 for every prime divisor of N. We beef up the congruences modulo 2 to congruences modulo 24 by multiplying by 12. To calculate the kernel of the ensuing map \(\ZZ^m \to (\ZZ/24\ZZ)^n\) we lift it arbitrarily to an integer matrix and calculate its Smith normal form. This gives a basis for the lattice.

    This lattice typically contains “large” elements, so by default we pass it to the reduce_basis() function which performs LLL-reduction to give a more manageable basis.

  level()

    Return the level of `self`.

    EXAMPLES:

    Sage

        sage: EtaGroup(10).level()
        10

    Python

        >>> from sage.all import *
        >>> EtaGroup(Integer(10)).level()
        10

  one()

    Return the identity element of `self`.

    EXAMPLES:

    Sage

        sage: EtaGroup(12).one()
        Eta product of level 12 : 1

    Python

        >>> from sage.all import *
        >>> EtaGroup(Integer(12)).one()
        Eta product of level 12 : 1

  reduce_basis(long_etas)

    Produce a more manageable basis via LLL-reduction.

    INPUT:

    - `long_etas` – a list of EtaGroupElement objects (which should all be of the same level)

    OUTPUT:

    - a new list of EtaGroupElement objects having hopefully smaller norm

    ALGORITHM: We define the norm of an eta-product to be the \(L^2\) norm of its divisor (as an element of the free \(\ZZ\)-module with the cusps as basis and the standard inner product). Applying LLL-reduction to this gives a basis of hopefully more tractable elements. Of course we’d like to use the \(L^1\) norm as this is just twice the degree, which is a much more natural invariant, but \(L^2\) norm is easier to work with!

    EXAMPLES:

    Sage

        sage: EtaGroup(4).reduce_basis([ EtaProduct(4, {1:8,2:24,4:-32}), EtaProduct(4, {1:8, 4:-8})])
        [Eta product of level 4 : (eta_1)^8 (eta_4)^-8,
         Eta product of level 4 : (eta_1)^-8 (eta_2)^24 (eta_4)^-16]

    Python

        >>> from sage.all import *
        >>> EtaGroup(Integer(4)).reduce_basis([ EtaProduct(Integer(4), {Integer(1):Integer(8),Integer(2):Integer(24),Integer(4):-Integer(32)}), EtaProduct(Integer(4), {Integer(1):Integer(8), Integer(4):-Integer(8)})])
        [Eta product of level 4 : (eta_1)^8 (eta_4)^-8,
         Eta product of level 4 : (eta_1)^-8 (eta_2)^24 (eta_4)^-16]

sage.modular.etaproducts.EtaProduct(level, dic)

  Create an `EtaGroupElement` object representing the function \(\prod_{d | N} \eta(q^d)^{r_d}\).

  This checks the criteria of Ligozat to ensure that this product really is the \(q\)-expansion of a meromorphic function on \(X_0(N)\).

  INPUT:

  - `level` – integer; the N such that this eta product is a function on \(X_0(N)\)

  - `dic` – a dictionary indexed by divisors of N such that the coefficient of \(\eta(q^d)\) is r[d]. Only nonzero coefficients need be specified. If Ligozat’s criteria are not satisfied, a `ValueError` will be raised.

  OUTPUT:

  An EtaGroupElement object, whose parent is the EtaGroup of level N and whose coefficients are the given dictionary.

  Note

  The dictionary `dic` does not uniquely specify N. It is possible for two EtaGroupElements with different \(N\)’s to be created with the same dictionary, and these represent different objects (although they will have the same \(q\)-expansion at the cusp \(\infty\)).

  EXAMPLES:

  Sage

      sage: EtaProduct(3, {3:12, 1:-12})
      Eta product of level 3 : (eta_1)^-12 (eta_3)^12
      sage: EtaProduct(3, {3:6, 1:-6})
      Traceback (most recent call last):
      ...
      ValueError: sum d r_d (=12) is not 0 mod 24
      sage: EtaProduct(3, {4:6, 1:-6})
      Traceback (most recent call last):
      ...
      ValueError: 4 does not divide 3

  Python

      >>> from sage.all import *
      >>> EtaProduct(Integer(3), {Integer(3):Integer(12), Integer(1):-Integer(12)})
      Eta product of level 3 : (eta_1)^-12 (eta_3)^12
      >>> EtaProduct(Integer(3), {Integer(3):Integer(6), Integer(1):-Integer(6)})
      Traceback (most recent call last):
      ...
      ValueError: sum d r_d (=12) is not 0 mod 24
      >>> EtaProduct(Integer(3), {Integer(4):Integer(6), Integer(1):-Integer(6)})
      Traceback (most recent call last):
      ...
      ValueError: 4 does not divide 3

sage.modular.etaproducts.eta_poly_relations(eta_elements, degree, labels=['x1', 'x2'], verbose=False)

  Find polynomial relations between eta products.

  INPUT:

  - `eta_elements` – list; a list of EtaGroupElement objects. Not implemented unless this list has precisely two elements. degree

  - `degree` – integer; the maximal degree of polynomial to look for

  - `labels` – list of strings; labels to use for the polynomial returned

  - `verbose` – boolean (default: `False`); if `True`, prints information as it goes

  OUTPUT: list of polynomials which is a Groebner basis for the part of the ideal of relations between eta_elements which is generated by elements up to the given degree; or None, if no relations were found.

  ALGORITHM: An expression of the form \(\sum_{0 \le i,j \le d} a_{ij} x^i y^j\) is zero if and only if it vanishes at the cusp infinity to degree at least \(v = d(deg(x) + deg(y))\). For all terms up to \(q^v\) in the \(q\)-expansion of this expression to be zero is a system of \(v + k\) linear equations in \(d^2\) coefficients, where \(k\) is the number of nonzero negative coefficients that can appear.

  Solving these equations and calculating a basis for the solution space gives us a set of polynomial relations, but this is generally far from a minimal generating set for the ideal, so we calculate a Groebner basis.

  As a test, we calculate five extra terms of \(q\)-expansion and check that this doesn’t change the answer.

  EXAMPLES:

  Sage

      sage: from sage.modular.etaproducts import eta_poly_relations
      sage: t = EtaProduct(26, {2:2,13:2,26:-2,1:-2})
      sage: u = EtaProduct(26, {2:4,13:2,26:-4,1:-2})
      sage: eta_poly_relations([t, u], 3)
      sage: eta_poly_relations([t, u], 4)
      [x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

  Python

      >>> from sage.all import *
      >>> from sage.modular.etaproducts import eta_poly_relations
      >>> t = EtaProduct(Integer(26), {Integer(2):Integer(2),Integer(13):Integer(2),Integer(26):-Integer(2),Integer(1):-Integer(2)})
      >>> u = EtaProduct(Integer(26), {Integer(2):Integer(4),Integer(13):Integer(2),Integer(26):-Integer(4),Integer(1):-Integer(2)})
      >>> eta_poly_relations([t, u], Integer(3))
      >>> eta_poly_relations([t, u], Integer(4))
      [x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

  Use `verbose=True` to see the details of the computation:

  Sage

      sage: eta_poly_relations([t, u], 3, verbose=True)
      Trying to find a relation of degree 3
      Lowest order of a term at infinity = -12
      Highest possible degree of a term = 15
      Trying all coefficients from q^-12 to q^15 inclusive
      No polynomial relation of order 3 valid for 28 terms
      Check:
      Trying all coefficients from q^-12 to q^20 inclusive
      No polynomial relation of order 3 valid for 33 terms

  [Python]

      >>> from sage.all import *
      >>> eta_poly_relations([t, u], Integer(3), verbose=True)
      Trying to find a relation of degree 3
      Lowest order of a term at infinity = -12
      Highest possible degree of a term = 15
      Trying all coefficients from q^-12 to q^15 inclusive
      No polynomial relation of order 3 valid for 28 terms
      Check:
      Trying all coefficients from q^-12 to q^20 inclusive
      No polynomial relation of order 3 valid for 33 terms

  Sage

      sage: eta_poly_relations([t, u], 4, verbose=True)
      Trying to find a relation of degree 4
      Lowest order of a term at infinity = -16
      Highest possible degree of a term = 20
      Trying all coefficients from q^-16 to q^20 inclusive
      Check:
      Trying all coefficients from q^-16 to q^25 inclusive
      [x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

  Python

      >>> from sage.all import *
      >>> eta_poly_relations([t, u], Integer(4), verbose=True)
      Trying to find a relation of degree 4
      Lowest order of a term at infinity = -16
      Highest possible degree of a term = 20
      Trying all coefficients from q^-16 to q^20 inclusive
      Check:
      Trying all coefficients from q^-16 to q^25 inclusive
      [x1^3*x2 - 13*x1^3 - 4*x1^2*x2 - 4*x1*x2 - x2^2 + x2]

sage.modular.etaproducts.n_cusps_of_width(N, d)

  Return the number of cusps on \(X_0(N)\) of width `d`.

  INPUT:

  - `N` – integer; the level

  - `d` – integer; an integer dividing \(N\), the cusp width

  EXAMPLES:

  Sage

      sage: from sage.modular.etaproducts import n_cusps_of_width
      sage: [n_cusps_of_width(18,d) for d in divisors(18)]
      [1, 1, 2, 2, 1, 1]
      sage: n_cusps_of_width(4,8)
      Traceback (most recent call last):
      ...
      ValueError: N and d must be positive integers with d|N

  Python

      >>> from sage.all import *
      >>> from sage.modular.etaproducts import n_cusps_of_width
      >>> [n_cusps_of_width(Integer(18),d) for d in divisors(Integer(18))]
      [1, 1, 2, 2, 1, 1]
      >>> n_cusps_of_width(Integer(4),Integer(8))
      Traceback (most recent call last):
      ...
      ValueError: N and d must be positive integers with d|N

sage.modular.etaproducts.num_cusps_of_width(N, d)

  alias of `n_cusps_of_width()`.

sage.modular.etaproducts.qexp_eta(ps_ring, prec)

  Return the \(q\)-expansion of \(\eta(q) / q^{1/24}\).

  Here \(\eta(q)\) is Dedekind’s function

  \[\eta(q) = q^{1/24}\prod_{n=1}^\infty (1-q^n).\]

  The result is an element of `ps_ring`, with precision `prec`.

  INPUT:

  - `ps_ring` – PowerSeriesRing; a power series ring

  - `prec` – integer; the number of terms to compute

  OUTPUT: an element of `ps_ring` which is the \(q\)-expansion of \(\eta(q)/q^{1/24}\) truncated to prec terms.

  ALGORITHM: We use the Euler identity

  \[\eta(q) = q^{1/24}( 1 + \sum_{n \ge 1} (-1)^n (q^{n(3n+1)/2} + q^{n(3n-1)/2})\]

  to compute the expansion.

  EXAMPLES:

  Sage

      sage: from sage.modular.etaproducts import qexp_eta
      sage: qexp_eta(ZZ[['q']], 100)
      1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + q^22 + q^26 - q^35 - q^40 + q^51 + q^57 - q^70 - q^77 + q^92 + O(q^100)

  Python

      >>> from sage.all import *
      >>> from sage.modular.etaproducts import qexp_eta
      >>> qexp_eta(ZZ[['q']], Integer(100))
      1 - q - q^2 + q^5 + q^7 - q^12 - q^15 + q^22 + q^26 - q^35 - q^40 + q^51 + q^57 - q^70 - q^77 + q^92 + O(q^100)