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
where \(\eta(q)\) is Dirichlet’s eta function
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)[source]¶
Return a list of CuspFamily objects corresponding to the cusps of \(X_0(N)\).
INPUT:
N
– integer; the level
EXAMPLES:
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
>>> 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)[source]¶
Bases:
SageObject
A family of elliptic curves parametrising a region of \(X_0(N)\).
- level()[source]¶
Return the level of this cusp.
EXAMPLES:
sage: e = CuspFamily(10, 1) sage: e.level() 10
>>> from sage.all import * >>> e = CuspFamily(Integer(10), Integer(1)) >>> e.level() 10
- sage.modular.etaproducts.EtaGroup(level)[source]¶
Create the group of eta products of the given level.
EXAMPLES:
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
>>> 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)[source]¶
Bases:
Element
Create an eta product object. Usually called implicitly via EtaGroup_class.__call__ or the EtaProduct factory function.
EXAMPLES:
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()
>>> 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()[source]¶
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: e = EtaProduct(12, {1:-336, 2:576, 3:696, 4:-216, 6:-576, 12:-144}) sage: e.degree() 230
>>> 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()[source]¶
Return the divisor of
self
, as a formal sum of CuspFamily objects.EXAMPLES:
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})
>>> 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()[source]¶
Return whether
self
is the one of the monoid.EXAMPLES:
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
>>> 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()[source]¶
Return the level of this eta product.
EXAMPLES:
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
>>> 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)[source]¶
Return the order of vanishing of
self
at the given cusp.INPUT:
cusp
– aCuspFamily
object
OUTPUT: integer
EXAMPLES:
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
>>> 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)[source]¶
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: 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
>>> 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)[source]¶
Alias for
self.q_expansion()
.EXAMPLES:
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
>>> 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)[source]¶
Return the exponent \(r_d\) of \(\eta(q^d)\) in
self
.EXAMPLES:
sage: e = EtaProduct(12, {2:24, 3:-24}) sage: e.r(3) -24 sage: e.r(4) 0
>>> 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)[source]¶
Bases:
UniqueRepresentation
,Parent
The group of eta products of a given level under multiplication.
- Element[source]¶
alias of
EtaGroupElement
- basis(reduce=True)[source]¶
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: 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]
>>> 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()[source]¶
Return the level of
self
.EXAMPLES:
sage: EtaGroup(10).level() 10
>>> from sage.all import * >>> EtaGroup(Integer(10)).level() 10
- one()[source]¶
Return the identity element of
self
.EXAMPLES:
sage: EtaGroup(12).one() Eta product of level 12 : 1
>>> from sage.all import * >>> EtaGroup(Integer(12)).one() Eta product of level 12 : 1
- reduce_basis(long_etas)[source]¶
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: 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]
>>> 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)[source]¶
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, aValueError
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: 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
>>> 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)[source]¶
Find polynomial relations between eta products.
INPUT:
eta_elements
– list; a list of EtaGroupElement objects. Not implemented unless this list has precisely two elements. degreedegree
– integer; the maximal degree of polynomial to look forlabels
– list of strings; labels to use for the polynomial returnedverbose
– boolean (default:False
); ifTrue
, 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: 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]
>>> 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: 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
>>> 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: 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]
>>> 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.num_cusps_of_width(N, d)[source]¶
Return the number of cusps on \(X_0(N)\) of width
d
.INPUT:
N
– integer; the leveld
– integer; an integer dividing \(N\), the cusp width
EXAMPLES:
sage: from sage.modular.etaproducts import num_cusps_of_width sage: [num_cusps_of_width(18,d) for d in divisors(18)] [1, 1, 2, 2, 1, 1] sage: num_cusps_of_width(4,8) Traceback (most recent call last): ... ValueError: N and d must be positive integers with d|N
>>> from sage.all import * >>> from sage.modular.etaproducts import num_cusps_of_width >>> [num_cusps_of_width(Integer(18),d) for d in divisors(Integer(18))] [1, 1, 2, 2, 1, 1] >>> num_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.qexp_eta(ps_ring, prec)[source]¶
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 precisionprec
.INPUT:
ps_ring
– PowerSeriesRing; a power series ringprec
– 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: 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)
>>> 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)