def hecke_operator_on_qexp(f, n, k, eps = None, prec=None, check=True, _return_list=False): r""" Given the `q`-expansion `f` of a modular form with character `\varepsilon`, this function computes the image of `f` under the Hecke operator `T_{n,k}` of weight `k`. EXAMPLES:: 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) An example on a formal power series:: sage: R.<q> = QQ[[]] sage: f = q + q^2 + q^3 + q^7 + O(q^8) sage: hecke_operator_on_qexp(f, 3, 12) q + O(q^3) sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec() 8 sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec() 9 An example of computing `T_{p,k}` in characteristic `p`:: sage: p = 199 sage: fp = delta_qexp(prec=p^2+1, K=GF(p)) sage: tfp = hecke_operator_on_qexp(fp, p, 12) sage: tfp == fp[p] * fp True sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p)) sage: tfp == tf True """ if eps is None: # Need to have base_ring=ZZ to work over finite fields, since # ZZ can coerce to GF(p), but QQ can't. eps = DirichletGroup(1, base_ring=ZZ)[0] if check: if not (is_PowerSeries(f) or is_ModularFormElement(f)): raise TypeError("f (=%s) must be a power series or modular form"%f) if not is_DirichletCharacter(eps): raise TypeError("eps (=%s) must be a Dirichlet character"%eps) k = Integer(k) n = Integer(n) v = [] if prec is None: if is_ModularFormElement(f): # always want at least three coefficients, but not too many, unless # requested pr = max(f.prec(), f.parent().prec(), (n+1)*3) pr = min(pr, 100*(n+1)) prec = pr // n + 1 else: prec = (f.prec() / ZZ(n)).ceil() if prec == Infinity: prec = f.parent().default_prec() // n + 1 if f.prec() < prec: f._compute_q_expansion(prec) p = Integer(f.base_ring().characteristic()) if k != 1 and p.is_prime() and n.is_power_of(p): # if computing T_{p^a} in characteristic p, use the simpler (and faster) # formula v = [f[m*n] for m in range(prec)] else: l = k-1 for m in range(prec): am = sum([eps(d) * d**l * f[m*n//(d*d)] for \ d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0]) v.append(am) if _return_list: return v if is_ModularFormElement(f): R = f.parent()._q_expansion_ring() else: R = f.parent() return R(v, prec)
def hecke_operator_on_qexp(f, n, k, eps=None, prec=None, check=True, _return_list=False): r""" Given the `q`-expansion `f` of a modular form with character `\varepsilon`, this function computes the image of `f` under the Hecke operator `T_{n,k}` of weight `k`. EXAMPLES:: 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) An example on a formal power series:: sage: R.<q> = QQ[[]] sage: f = q + q^2 + q^3 + q^7 + O(q^8) sage: hecke_operator_on_qexp(f, 3, 12) q + O(q^3) sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec() 8 sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec() 9 An example of computing `T_{p,k}` in characteristic `p`:: sage: p = 199 sage: fp = delta_qexp(prec=p^2+1, K=GF(p)) sage: tfp = hecke_operator_on_qexp(fp, p, 12) sage: tfp == fp[p] * fp True sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p)) sage: tfp == tf True """ if eps is None: # Need to have base_ring=ZZ to work over finite fields, since # ZZ can coerce to GF(p), but QQ can't. eps = DirichletGroup(1, base_ring=ZZ).gen(0) if check: if not (is_PowerSeries(f) or is_ModularFormElement(f)): raise TypeError, "f (=%s) must be a power series or modular form" % f if not is_DirichletCharacter(eps): raise TypeError, "eps (=%s) must be a Dirichlet character" % eps k = Integer(k) n = Integer(n) v = [] if prec is None: if is_ModularFormElement(f): # always want at least three coefficients, but not too many, unless # requested pr = max(f.prec(), f.parent().prec(), (n + 1) * 3) pr = min(pr, 100 * (n + 1)) prec = pr // n + 1 else: prec = (f.prec() / ZZ(n)).ceil() if prec == Infinity: prec = f.parent().default_prec() // n + 1 if f.prec() < prec: f._compute_q_expansion(prec) p = Integer(f.base_ring().characteristic()) if k != 1 and p.is_prime() and n.is_power_of(p): # if computing T_{p^a} in characteristic p, use the simpler (and faster) # formula v = [f[m * n] for m in range(prec)] else: l = k - 1 for m in range(prec): am = sum([eps(d) * d**l * f[m*n//(d*d)] for \ d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0]) v.append(am) if _return_list: return v if is_ModularFormElement(f): R = f.parent()._q_expansion_ring() else: R = f.parent() return R(v, prec)