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67.3.b.b.qexp.sage
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67.3.b.b.qexp.sage
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# q-expansion of newform 67.3.b.b, downloaded from the LMFDB on 14 February 2022.
# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.
# To create the q-expansion as a power series, type "qexp = make_data()"
def make_data():
from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR
def discrete_log(elts, gens, mod):
# algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
def table_gens(gens, mod):
T = [1]
n = len(gens)
r = [None]*n
s = [None]*n
for i in range(n):
beta = gens[i]
r[i] = 1
N = len(T)
while beta not in T:
for Tj in T[:N]:
T.append((beta*Tj) % mod)
beta = (beta*gens[i]) % mod
r[i] += 1
s[i] = T.index(beta)
return T, r, s
T, r, s = table_gens(gens, mod)
n = len(gens)
N = [ prod(r[:j]) for j in range(n) ]
Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
return [Z(T.index(elt % mod)) for elt in elts]
def extend_multiplicatively(an):
for pp in prime_powers(len(an)-1):
for k in range(1, (len(an) - 1)//pp + 1):
if gcd(k, pp) == 1:
an[pp*k] = an[pp]*an[k]
from sage.all import PolynomialRing, NumberField, ZZ
R = PolynomialRing(QQ, "x")
f = R(poly_data)
K = NumberField(f, "a")
betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
# convert aps to K elements
primes = primes_first_n(len(aps_data))
good_primes = [p for p in primes if not p.divides(level)]
aps = map(convert_elt_to_field, aps_data)
if not hecke_ring_character_values:
# trivial character
char_values = dict(zip(good_primes, [1]*len(good_primes)))
else:
gens = [elt[0] for elt in hecke_ring_character_values]
gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
char_values = dict([(
p,prod(g**k for g, k in zip(gens_values, elt)))
for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
])
an_list_bound = next_prime(primes[-1])
an = [0]*an_list_bound
an[1] = 1
from sage.all import PowerSeriesRing
PS = PowerSeriesRing(K, "q")
for p, ap in zip(primes, aps):
if p.divides(level):
euler_factor = [1, -ap]
else:
euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
k = RR(an_list_bound).log(p).floor() + 1
foo = (1/PS(euler_factor)).padded_list(k)
for i in range(1, k):
an[p**i] = foo[i]
extend_multiplicatively(an)
return PS(an)
level = 67
weight = 3
poly_data = [1519, 0, 3366, 0, 1725, 0, 357, 0, 32, 0, 1]
# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data = [\
[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1],
[[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], 1],
[[-311, 0, -375, 0, -150, 0, -23, 0, -1, 0], 80],
[[0, 951, 0, 455, 0, 150, 0, 23, 0, 1], 80],
[[-291, 0, -615, 0, -222, 0, -27, 0, -1, 0], 16],
[[1123, 0, 1765, 0, 630, 0, 79, 0, 3, 0], 40],
[[0, -1163, 0, -1765, 0, -630, 0, -79, 0, -3], 40],
[[0, -247, 0, -910, 0, -380, 0, -51, 0, -2], 10],
[[0, 347, 0, 920, 0, 380, 0, 51, 0, 2], 10],
[[2551, 0, 5445, 0, 2150, 0, 283, 0, 11, 0], 40]]
hecke_ring_character_values = [\
[2, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [\
[0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, -1, 0, 0, 0],
[0, 0, 0, -1, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0, 0, -1, 0],
[0, 1, 0, 1, 0, 0, 1, -1, 0, 0],
[0, -2, 0, 1, 0, 0, 1, 0, 1, 0],
[9, 0, 3, 0, 0, 1, 0, 0, 0, -3],
[5, 0, -1, 0, 5, 0, 0, 0, 0, 5],
[2, 0, -1, 0, -5, 3, 0, 0, 0, 1],
[2, 0, -4, 0, -4, -3, 0, 0, 0, -5],
[0, -6, 0, 4, 0, 0, 4, 1, 0, 0],
[-9, 0, -3, 0, 5, -11, 0, 0, 0, -2],
[0, 1, 0, -4, 0, 0, 6, -1, -5, 0],
[0, -1, 0, 3, 0, 0, 8, 2, 5, 0],
[-29, 0, 7, 0, -10, -1, 0, 0, 0, -2],
[0, -7, 0, 2, 0, 0, -3, -2, -1, 0],
[-34, 0, -4, 0, -2, 6, 0, 0, 0, 8],
[0, -16, 0, -6, 0, 0, -1, 1, 0, 0],
[9, 7, -7, -4, -15, 1, -8, 3, 0, 2],
[17, 0, -7, 0, 0, -22, 0, 0, 0, -4],
[-46, 0, -2, 0, 5, 16, 0, 0, 0, 12],
[0, -21, 0, 4, 0, 0, -6, 1, -5, 0],
[-21, 0, 13, 0, 5, -4, 0, 0, 0, 17],
[-48, 0, 1, 0, 31, 1, 0, 0, 0, 5],
[0, -16, 0, -2, 0, 0, -22, -3, 0, 0],
[0, 42, 0, 2, 0, 0, 27, -2, 0, 0],
[11, 0, 12, 0, 0, 19, 0, 0, 0, -32],
[-32, 0, -4, 0, -10, -3, 0, 0, 0, -21],
[0, 21, 0, 1, 0, 0, -14, 4, 5, 0],
[0, 25, 0, -10, 0, 0, -16, 0, 5, 0],
[-52, 0, -4, 0, -15, -8, 0, 0, 0, 9],
[18, 0, -22, 0, 36, 21, 0, 0, 0, 13],
[0, -16, 0, 2, 0, 0, -18, 9, 6, 0],
[0, -28, 0, 12, 0, 0, 22, -7, 0, 0],
[-25, 0, 9, 0, -17, -28, 0, 0, 0, -35],
[-66, 0, 2, 0, -17, -7, 0, 0, 0, 17],
[-19, 0, -33, 0, 15, -1, 0, 0, 0, 8],
[-9, 0, -13, 0, 5, 14, 0, 0, 0, -27],
[4, 0, 18, 0, -5, 26, 0, 0, 0, 2],
[-5, 0, -15, 0, -75, -20, 0, 0, 0, -15],
[0, 41, 0, -4, 0, 0, -9, -1, -5, 0],
[-3, 0, -17, 0, -17, 50, 0, 0, 0, -9],
[0, -26, 0, 4, 0, 0, -36, 11, 20, 0],
[-25, 0, 40, 0, 10, -10, 0, 0, 0, -35],
[0, -7, 0, 3, 0, 0, 27, 7, 0, 0],
[-4, 0, -26, 0, 21, -13, 0, 0, 0, 52],
[65, 0, -3, 0, 1, 23, 0, 0, 0, 14],
[37, 0, -21, 0, -40, 13, 0, 0, 0, 11],
[55, 0, 35, 0, 25, 45, 0, 0, 0, -50],
[0, -67, 0, 8, 0, 0, -32, -3, -15, 0],
[0, 40, 0, -16, 0, 0, -26, -4, -6, 0],
[0, 75, 0, -30, 0, 0, -20, 10, 5, 0],
[15, 0, 17, 0, 26, 3, 0, 0, 0, 44],
[0, -6, 0, 39, 0, 0, 44, 1, -10, 0],
[94, 0, -12, 0, 65, -29, 0, 0, 0, 37],
[133, 0, 56, 0, 0, -8, 0, 0, 0, -31],
[-27, 0, -17, 0, 61, 25, 0, 0, 0, -2],
[0, -24, 0, -44, 0, 0, 16, -1, -20, 0],
[-30, 0, 0, 0, -40, 20, 0, 0, 0, -10],
[0, -75, 0, -20, 0, 0, -10, -10, 15, 0],
[76, 0, 62, 0, 40, -61, 0, 0, 0, 23],
[-2, 0, 6, 0, -70, -53, 0, 0, 0, -11],
[150, 0, 50, 0, 110, 50, 0, 0, 0, 40],
[0, -36, 0, 14, 0, 0, -6, 16, 20, 0],
[0, -17, 0, -14, 0, 0, 88, 12, 5, 0],
[48, 0, -14, 0, 70, -43, 0, 0, 0, 59],
[0, -36, 0, -21, 0, 0, -16, 1, 0, 0],
[0, 25, 0, 30, 0, 0, -36, 25, 35, 0],
[0, 16, 0, -25, 0, 0, 50, 10, -26, 0],
[218, 0, -12, 0, -74, -34, 0, 0, 0, -62],
[0, 85, 0, 8, 0, 0, 18, -7, -9, 0],
[78, 0, 48, 0, 76, 36, 0, 0, 0, 88],
[0, 92, 0, -34, 0, 0, 26, -16, -20, 0],
[0, 52, 0, 27, 0, 0, 21, -27, -5, 0],
[0, -97, 0, 33, 0, 0, -57, 22, 40, 0],
[0, -55, 0, -16, 0, 0, -26, 6, -35, 0],
[320, 0, -6, 0, -10, -40, 0, 0, 0, 0],
[49, 0, 43, 0, 55, 111, 0, 0, 0, 42],
[0, -47, 0, 38, 0, 0, 58, -8, 45, 0],
[0, 32, 0, -18, 0, 0, -38, -42, 10, 0],
[-216, 0, -38, 0, 100, -74, 0, 0, 0, 2],
[-219, 0, -11, 0, -49, 32, 0, 0, 0, -33],
[-216, 0, 0, 0, -89, -21, 0, 0, 0, 1],
[0, -15, 0, 14, 0, 0, -46, 36, 9, 0],
[-115, 0, -38, 0, 6, -41, 0, 0, 0, -26],
[0, -55, 0, 9, 0, 0, -51, -9, 20, 0],
[-128, 0, 6, 0, 11, 117, 0, 0, 0, 10],
[353, 0, 61, 0, 0, 2, 0, 0, 0, -66],
[1, 0, -99, 0, -137, -39, 0, 0, 0, 18],
[0, 135, 0, 40, 0, 0, 74, -10, -15, 0],
[48, 0, 66, 0, -10, -123, 0, 0, 0, -91],
[-24, 0, -22, 0, -115, -104, 0, 0, 0, 12],
[0, -10, 0, -70, 0, 0, -26, -15, -20, 0],
[52, 0, 98, 0, 140, 93, 0, 0, 0, -49],
[0, -29, 0, -64, 0, 0, 41, -16, -45, 0],
[0, 8, 0, -32, 0, 0, -8, -33, 0, 0],
[78, 0, 80, 0, 0, -88, 0, 0, 0, 44],
[0, -78, 0, -18, 0, 0, 2, -12, -20, 0],
[29, 0, -107, 0, 55, -99, 0, 0, 0, -58],
[0, 41, 0, -39, 0, 0, -29, -16, 20, 0],
[0, -13, 0, -15, 0, 0, 120, -4, 21, 0],
[325, 0, -75, 0, 175, 50, 0, 0, 0, 135],
[0, 21, 0, 15, 0, 0, -35, 30, 24, 0],
[-256, 0, 35, 0, -59, 4, 0, 0, 0, -54],
[217, 0, -89, 0, -169, 52, 0, 0, 0, 25],
[0, 70, 0, -16, 0, 0, -126, 21, 14, 0],
[0, 128, 0, 52, 0, 0, -53, -7, -30, 0],
[0, -148, 0, -28, 0, 0, -54, -2, -40, 0],
[0, -154, 0, -14, 0, 0, 36, 24, -30, 0],
[216, 0, 171, 0, 113, 26, 0, 0, 0, -32],
[265, 0, 135, 0, 130, 150, 0, 0, 0, 65],
[24, 0, 8, 0, -50, -189, 0, 0, 0, -23],
[-187, 0, 81, 0, -35, 197, 0, 0, 0, -36],
[414, 0, 60, 0, 86, 124, 0, 0, 0, 66],
[0, 27, 0, 42, 0, 0, 22, 28, 15, 0],
[0, 82, 0, 112, 0, 0, 82, -47, 20, 0],
[-505, 0, -85, 0, -75, 110, 0, 0, 0, 65],
[0, 114, 0, -88, 0, 0, -28, 24, 26, 0],
[0, -40, 0, -16, 0, 0, 24, 56, -10, 0],
[368, 0, 12, 0, -60, 9, 0, 0, 0, -177],
[0, 233, 0, -82, 0, 0, -27, -8, -25, 0],
[0, -199, 0, 4, 0, 0, 76, -31, -35, 0],
[0, 15, 0, 39, 0, 0, -21, 71, 50, 0],
[0, 141, 0, -21, 0, 0, -34, -21, -35, 0],
[178, 0, -22, 0, -44, -4, 0, 0, 0, 18],
[0, 250, 0, -20, 0, 0, 120, -10, -60, 0],
[-194, 0, 36, 0, 8, 16, 0, 0, 0, -112],
[-299, 0, -159, 0, -50, -126, 0, 0, 0, 58],
[0, -78, 0, -8, 0, 0, -144, -22, 10, 0],
[0, -208, 0, 1, 0, 0, 86, -1, 54, 0],
[0, 84, 0, 4, 0, 0, -71, -9, -70, 0],
[-618, 0, 34, 0, 135, 113, 0, 0, 0, 61],
[92, 0, -157, 0, -185, 3, 0, 0, 0, -129],
[0, -120, 0, -25, 0, 0, -36, 20, 40, 0],
[242, 0, -152, 0, -140, -22, 0, 0, 0, 46],
[0, 26, 0, -64, 0, 0, -34, -21, 50, 0],
[176, 0, 42, 0, -130, 99, 0, 0, 0, -197],
[0, 251, 0, 19, 0, 0, 151, -31, 40, 0],
[-225, 0, 5, 0, 65, -80, 0, 0, 0, -95],
[0, 191, 0, -74, 0, 0, 96, -21, 35, 0],
[0, 91, 0, -24, 0, 0, 31, -66, -105, 0],
[-124, 0, 34, 0, 36, 57, 0, 0, 0, 207],
[191, 0, -23, 0, 10, -156, 0, 0, 0, -147],
[310, 0, -90, 0, -30, -80, 0, 0, 0, -170],
[-306, 0, 90, 0, 206, -95, 0, 0, 0, 31],
[-146, 0, 65, 0, 221, 130, 0, 0, 0, -54],
[33, 0, -89, 0, 105, -28, 0, 0, 0, 109],
[0, 168, 0, 66, 0, 0, 146, -35, -74, 0],
[-15, 0, 49, 0, -137, -108, 0, 0, 0, 55],
[195, 0, 115, 0, -45, 65, 0, 0, 0, -50],
[-68, 0, 154, 0, 320, 13, 0, 0, 0, 231],
[278, 0, -115, 0, -65, 187, 0, 0, 0, 169],
[0, -209, 0, 159, 0, 0, -1, 37, 56, 0],
[-255, 0, -175, 0, -20, -60, 0, 0, 0, -50],
[-335, 0, -95, 0, -145, -110, 0, 0, 0, 225],
[-73, 0, 160, 0, -200, -30, 0, 0, 0, -35],
[0, 2, 0, 32, 0, 0, 122, -7, -70, 0],
[0, -10, 0, 30, 0, 0, -40, -35, -80, 0],
[0, -156, 0, 82, 0, 0, -148, -11, 26, 0],
[0, -210, 0, 120, 0, 0, 90, -5, 130, 0],
[547, 0, 49, 0, 225, -67, 0, 0, 0, 106],
[-84, 0, -78, 0, -95, 239, 0, 0, 0, -57],
[-701, 0, -97, 0, -130, -84, 0, 0, 0, 32],
[-516, 0, 110, 0, -154, 45, 0, 0, 0, 41],
[-92, 0, -104, 0, -20, -213, 0, 0, 0, 79],
[0, -228, 0, -24, 0, 0, -214, -11, -66, 0],
[0, -275, 0, -50, 0, 0, -270, 35, 65, 0],
[-668, 0, 104, 0, -60, 223, 0, 0, 0, -59]]