Module: sage.modular.modform.hecke_operator_on_qexp
Hecke Operators on
-expansions.
Module-level Functions
B, n, k, [eps=None], [already_echelonized=False]) |
Given a basis
of
-expansions for a space of modular forms
with character
to precision at least
,
this function computes the matrix of
relative to
.
NOTE: If the elements of B are not known to sufficient precision, this function will report that the vectors are linearly dependent (since they are to the specified precision).
Input:
sage: sage.modular.modform.constructor.ModularForms_clear_cache() sage: ModularForms(1,12).q_expansion_basis() [ q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6), 1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + 3199218815520/691*q^5 + O(q^6) ] sage: hecke_operator_on_basis(ModularForms(1,12).q_expansion_basis(), 3, 12) Traceback (most recent call last): ... ValueError: basis vectors must be linearly independent.
sage: hecke_operator_on_basis(ModularForms(1,12).q_expansion_basis(30), 3, 12) [ 252 0] [ 0 177148]
f, n, k, [eps=None], [prec=None], [check=True], [_return_list=False]) |
Given the
-expansion
of a modular form with character
, this function computes the Hecke operator
of weight
on
.
sage: M = ModularForms(1,12) sage: hecke_operator_on_qexp(M.basis()[0], 3, 12) 252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5) sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7) q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7) sage: hecke_operator_on_qexp(M.basis()[0], 1, 12) q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20) 20 sage: hecke_operator_on_qexp(M.basis()[0], 3, 12) 252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7) sage: hecke_operator_on_qexp(M.basis()[0], 1, 12) q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7) 252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12) -6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)