
# q-expansion of newform 475.2.e.f, downloaded from the LMFDB on 19 April 2024.

# 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.

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 = 475
weight = 2
poly_data = [1, 33, 1107, -552, 1009, -147, 484, -93, 109, -18, 17, -3, 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, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-68499350729, -726241735, -24355803798, -38817354653, -15342668397, -4177401796, -3678052623, -801763109, -621660414, 38785859, -46922925, -3959208], 66494854340], [[-6178545, -1547540653, -6717081048, 7520604435, -5508756455, 2957264154, -1701731499, 833713615, -313859676, 76199247, -11701549, 690858], 1955731010], [[-263970961763, -1147600790, -38482449021, 12276068039, -19479845384, -11747270177, 1416907499, -3244495858, 251450977, -731712017, 153015320, -54841341], 66494854340], [[-148724948799, -4114960306, -138016601091, -204345695921, -102294574896, -7077581935, -27189880401, 1833052570, -7500316929, 3072233475, -998274384, 134211789], 66494854340], [[-4335298296, -159117251, -5335902045, -2448832100, -2951210507, -1358444363, -412414770, -345971017, -29949491, -64765518, 9282581, -5051595], 1955731010], [[-612082342, -122568267, -4110047595, -725395450, -2049018039, -1288684281, -259108930, -359601729, 35014563, -93333716, 17844777, -6178545], 1955731010], [[402309009, -680489253663, 432662257370, -647754164847, 212867920667, -294644252580, 86418671699, -67720087321, 16279393858, -9648336987, 1742271771, -469619124], 66494854340], [[-65422139102, -2194621698825, 1145299361381, -2007194188104, 298838370979, -966498199653, 187219927286, -217868445987, 36042149143, -34029515688, 6017448375, -2004496389], 66494854340], [[2372026047, -4859631701974, 2534882432535, -4359372249951, 755327449736, -2086126950865, 426177732417, -469415474758, 81775705089, -72904583371, 12953281828, -4220156417], 66494854340], [[-3616495515, 7195985653273, -3864903739992, 6642795442545, -1144205547385, 3178214179566, -650339598681, 717279166315, -124776050724, 111043957893, -19688524501, 6424307742], 66494854340]]

hecke_ring_character_values = [[77, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [401, [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0], [1, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 1, -1, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 0, 0, 0, -1, -1, -1, 0], [-1, 1, -1, 1, 0, 1, -1, 0, 0, -1, 0, 0], [-1, 0, 0, 0, -1, -1, 0, -1, 1, 1, -1, 1], [1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 3, 0], [0, 1, 0, 1, 0, 0, 0, 0, -1, 0, -1, -2], [1, 0, 1, 0, 1, -1, 1, 2, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0], [-2, 1, -1, 0, -1, 0, -2, 2, 3, 0, -2, 1], [2, -1, 1, 0, 1, 0, 0, 1, 0, 3, 0, -1], [2, 0, 0, 1, 0, 0, 0, 0, -1, 2, -2, -2], [-1, -4, 0, -1, 0, 0, 0, 0, 1, -1, 2, -1], [4, -2, 2, -1, 2, -1, 3, -3, -5, 4, 2, -2], [-5, 1, 0, -2, 0, 0, 0, 0, 0, -5, 1, 1], [1, -2, 0, 0, 0, 0, 0, 0, -3, 1, 5, 0], [-5, 4, -4, 2, -1, 2, -1, -2, -1, 1, 1, 1], [-1, 1, -1, -1, 0, -1, 3, -4, -4, 1, 2, 0], [-2, 1, -1, 0, -1, 0, 3, 4, 5, -2, 3, 1], [-5, 0, -1, 0, 0, -3, 4, -2, 0, 0, 0, 0], [-4, 2, 0, 1, 0, 0, 0, 0, 4, -4, 3, -1], [-2, 3, -3, -2, 1, -2, -4, 2, 1, -1, -6, -1], [5, 2, 0, -2, 0, 0, 0, 0, -3, 5, 1, 0], [8, 0, 0, 0, -1, -1, -2, 1, 0, 0, 0, 0], [1, 0, -1, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, -1, -8, 1, 0], [2, 0, -3, 0, -1, -2, 2, -4, 0, 0, 0, 0], [-6, 1, 0, 2, 0, 0, 0, 0, 5, -6, -1, -1], [2, -3, 3, 1, -1, 1, 0, -3, -2, 4, 1, 1], [3, -2, 0, -2, 0, 0, 0, 0, -8, 3, 2, -2], [-2, -5, 0, -2, 0, 0, 0, 0, 1, -2, -1, 1], [2, 0, 0, 2, 2, 2, -1, 5, 3, 3, 1, -2], [-4, 0, 2, 0, 0, -1, 7, 1, 0, 0, 0, 0], [0, -3, 3, 0, -3, 0, 1, -1, 2, 10, 1, 3], [2, 0, 4, 0, 2, 3, 3, -3, 0, 0, 0, 0], [-9, 1, 0, 0, 0, 0, 0, 0, 2, -9, 2, 2], [3, 0, 0, -1, 3, -1, 3, 2, -1, 9, 2, -3], [3, 0, -6, 0, -1, -1, -3, -4, 0, 0, 0, 0], [-1, -3, 0, 0, 0, 0, 0, 0, 1, -1, 6, -1], [-1, 0, -3, 0, 4, 2, -7, -1, 0, 0, 0, 0], [-1, 0, 0, -1, -1, -1, -6, -5, -4, 1, -7, 1], [-2, 0, 5, 0, -2, 1, 2, 1, 0, 0, 0, 0], [-6, 4, 0, -3, 0, 0, 0, 0, -7, -6, 2, 1], [3, -5, 5, 0, -2, 0, -4, 0, 2, -5, -4, 2], [-4, 3, -3, 5, -1, 5, -5, 2, 3, -9, 0, 1], [7, 0, 6, 0, 2, -3, 4, 6, 0, 0, 0, 0], [0, 0, 2, 0, 1, -3, -1, -2, 0, 0, 0, 0], [-2, 0, 0, 2, -2, 2, 1, -5, -3, 7, 3, 2], [-2, 0, 0, 0, -3, 0, -1, 0, 0, 0, 0, 0], [-2, 5, 0, 4, 0, 0, 0, 0, -2, -2, 5, 1], [-2, -1, 0, 1, 0, 0, 0, 0, 6, -2, -7, 3], [-3, -1, 0, -3, 0, 0, 0, 0, 0, -3, -4, 2], [2, 2, -2, 1, 4, 1, -4, -4, -8, 9, -3, -4], [-4, 5, -5, 2, 1, 2, 0, 6, 5, -9, 2, -1], [7, -3, 3, 2, 4, 2, 5, 1, -3, 0, 7, -4], [-5, 0, -4, 0, 1, 5, -7, 0, 0, 0, 0, 0], [4, -4, 0, -4, 0, 0, 0, 0, 3, 4, -3, -2], [-3, -1, 1, -4, -4, -4, 6, 2, 6, -11, 2, 4], [20, 0, 5, 0, 0, 1, 2, 3, 0, 0, 0, 0], [-1, 4, -4, 4, 3, 4, -1, 6, 3, -3, 3, -3], [15, 0, 6, 0, 0, 1, -2, 8, 0, 0, 0, 0], [13, -6, 0, -5, 0, 0, 0, 0, -5, 13, 0, -3], [-12, -2, 0, -3, 0, 0, 0, 0, 1, -12, 0, 5], [-5, 0, 0, 0, -3, -5, 3, -12, 0, 0, 0, 0], [-6, 4, -4, 0, -2, 0, -3, -9, -7, -2, -3, 2], [1, 3, -3, -2, 4, -2, 2, 2, -2, 5, 0, -4], [-8, 0, 0, 0, 1, 7, 3, 0, 0, 0, 0, 0], [-3, 0, -2, 0, -2, -2, -12, 10, 0, 0, 0, 0], [4, -1, 1, -2, 3, -2, 0, 5, 2, -7, -2, -3], [-7, 1, 0, 4, 0, 0, 0, 0, -2, -7, -4, -2], [11, 0, 7, 0, 2, 1, 1, 14, 0, 0, 0, 0], [10, 0, 10, 0, -3, 3, 0, -1, 0, 0, 0, 0], [5, -8, 8, 0, -3, 0, 14, -9, -6, 14, 14, 3], [-4, 5, 0, 3, 0, 0, 0, 0, 2, -4, -8, 2], [1, 2, -2, -3, 3, -3, -3, 0, -3, -1, -6, -3], [-5, 2, -2, -3, -3, -3, 8, -5, -2, -9, 5, 3], [6, 2, 0, 3, 0, 0, 0, 0, -3, 6, 0, -5], [9, 0, 2, 0, 4, 0, -3, 9, 0, 0, 0, 0], [2, -2, 2, 0, 0, 0, 3, -13, -13, 3, 3, 0], [5, -5, 0, -3, 0, 0, 0, 0, 1, 5, 6, -6], [-10, 0, 0, 4, 0, 0, 0, 0, -5, -10, 5, 2], [1, 2, -2, 4, 3, 4, -12, 1, -2, 18, -8, -3], [-12, 2, 0, -1, 0, 0, 0, 0, 9, -12, -10, 1], [11, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0], [-3, 0, 4, 0, -5, 1, 5, -4, 0, 0, 0, 0], [-13, 9, -9, -5, -4, -5, -6, -3, 1, -2, -11, 4], [-16, 0, -3, 0, 1, -1, 0, 3, 0, 0, 0, 0], [-4, 0, -6, 0, -2, 4, -18, 0, 0, 0, 0, 0], [6, 2, 0, 0, 0, 0, 0, 0, -4, 6, 10, 1], [16, 0, 8, 0, 6, -3, 7, -3, 0, 0, 0, 0], [9, -6, 6, -2, 3, -2, 6, -5, -8, 0, 4, -3], [-5, 2, -2, -2, -3, -2, -7, 6, 9, -5, -9, 3], [5, -2, 0, -1, 0, 0, 0, 0, 0, 5, -3, 4], [3, -2, 0, 3, 0, 0, 0, 0, 0, 3, 13, 3], [0, 0, -6, 0, 8, 0, -6, 8, 0, 0, 0, 0], [2, 0, 0, 4, 0, 0, 0, 0, 2, 2, 6, 1], [4, -7, 0, -3, 0, 0, 0, 0, -15, 4, 5, -6], [5, -2, 0, -1, 0, 0, 0, 0, 6, 5, -7, -5], [8, -3, 0, 2, 0, 0, 0, 0, -2, 8, -3, 1], [4, 0, -9, 0, -4, 0, 2, -2, 0, 0, 0, 0], [2, 0, 6, 0, 5, 7, 4, -11, 0, 0, 0, 0], [5, 0, 4, 0, -1, -1, 9, -10, 0, 0, 0, 0], [7, 0, 0, 0, 4, -4, 5, -7, 0, 0, 0, 0], [1, -6, 6, 5, -5, 5, -7, 0, 5, -8, -2, 5], [4, 4, 0, -2, 0, 0, 0, 0, -15, 4, 7, -2], [16, -5, 0, 1, 0, 0, 0, 0, -8, 16, 9, 1], [8, 0, 6, 0, 0, -2, 6, 12, 0, 0, 0, 0], [-10, 0, -1, 0, 1, 2, -11, -6, 0, 0, 0, 0], [0, 4, -4, 2, 4, 2, 1, -1, -5, 6, 3, -4], [10, 4, 0, -5, 0, 0, 0, 0, 8, 10, -3, -3], [-7, 0, 1, 0, -7, 3, 7, -8, 0, 0, 0, 0], [-8, -1, 0, 6, 0, 0, 0, 0, 7, -8, 1, 1], [0, 1, -1, 6, 1, 6, -1, -5, -6, 15, 5, -1], [6, -8, 8, 2, -2, 2, 5, -3, -1, 3, 7, 2], [-4, 0, -7, 0, 1, 7, -2, -1, 0, 0, 0, 0], [18, 0, 2, 0, 9, 7, 1, 10, 0, 0, 0, 0], [11, 2, 0, 7, 0, 0, 0, 0, -10, 11, -5, -8], [11, 1, 0, -1, 0, 0, 0, 0, -9, 11, 3, 6], [9, 0, 7, 0, -3, -2, -4, -8, 0, 0, 0, 0], [-9, 0, 6, 0, -2, 2, 8, 0, 0, 0, 0, 0], [-3, 0, -4, 0, -8, 1, 2, -12, 0, 0, 0, 0], [13, 0, 15, 0, 0, -1, 10, 6, 0, 0, 0, 0], [4, -1, 1, -8, 3, -8, 3, -5, -8, -11, -5, -3], [2, -6, 0, 1, 0, 0, 0, 0, 7, 2, 10, -1], [-2, 0, 0, -3, -2, -3, -5, -7, -5, 0, -8, 2], [-7, 10, -10, 2, 3, 2, -6, 5, 2, 8, -4, -3], [0, 0, -2, 0, -5, 3, 6, -17, 0, 0, 0, 0], [0, -2, 2, -3, -2, -3, 11, -9, -7, 4, 8, 2], [-1, -6, 6, -2, -7, -2, 9, -2, 5, 15, 7, 7], [-9, 2, 0, -5, 0, 0, 0, 0, 6, -9, 1, 3], [14, -10, 10, 4, 4, 4, 5, 1, -3, -8, 9, -4], [3, 0, 8, 0, -4, 0, 11, 7, 0, 0, 0, 0], [1, -6, 0, 6, 0, 0, 0, 0, -9, 1, 17, 2], [-1, 2, 0, 6, 0, 0, 0, 0, 14, -1, -2, 4], [8, 0, 0, 0, -7, -4, 7, 16, 0, 0, 0, 0], [4, 0, 1, 0, 7, 0, 0, 10, 0, 0, 0, 0], [-1, 0, -5, 0, 3, -2, -7, 11, 0, 0, 0, 0], [14, -11, 0, 0, 0, 0, 0, 0, -1, 14, 3, 2], [-4, 6, 0, 2, 0, 0, 0, 0, 6, -4, 0, -10], [-2, 2, 0, 3, 0, 0, 0, 0, -7, -2, -2, -7], [-9, -2, 0, -3, 0, 0, 0, 0, -8, -9, 3, -4], [-12, 0, -5, 0, -2, -1, -5, 4, 0, 0, 0, 0], [-4, 3, -3, -5, -1, -5, 9, -10, -9, -11, 4, 1], [5, 3, -3, 1, 8, 1, 8, 11, 3, 5, 9, -8], [4, -6, 6, -4, -2, -4, -5, 1, 3, 4, -9, 2], [-17, -3, 0, -3, 0, 0, 0, 0, 21, -17, -2, 1], [10, 0, 10, 0, 2, -6, 8, 4, 0, 0, 0, 0], [-12, 14, -14, 0, 2, 0, -8, 0, -2, -9, -8, -2], [-9, 0, -9, 0, -1, 6, -24, -2, 0, 0, 0, 0], [4, -2, 0, 3, 0, 0, 0, 0, -15, 4, -2, -1], [-30, 5, 0, -4, 0, 0, 0, 0, -5, -30, -9, -1], [0, 1, -1, -7, 1, -7, 17, 0, -1, 3, 10, -1], [-6, 0, 0, 0, -3, 1, -18, -5, 0, 0, 0, 0], [9, 0, 1, 0, 8, -8, -1, 7, 0, 0, 0, 0], [7, -1, 1, -1, 6, -1, 3, -2, -8, 25, 2, -6], [4, -17, 0, 5, 0, 0, 0, 0, -6, 4, 16, -4], [-14, -6, 0, 1, 0, 0, 0, 0, -8, -14, 11, -1], [-6, -4, 4, -3, -10, -3, 11, -17, -7, 10, 8, 10], [-9, 9, -9, 5, 0, 5, 7, -6, -6, -4, 12, 0], [5, -10, 10, -3, -5, -3, -1, -18, -13, 12, -4, 5], [9, -6, 6, -2, 3, -2, -10, 13, 10, 10, -12, -3], [2, 0, -9, 0, -4, 3, 1, -12, 0, 0, 0, 0], [0, 0, 0, -4, 0, -4, -11, 9, 9, 25, -15, 0], [-7, 9, -9, 1, 2, 1, -11, -13, -15, -9, -10, -2], [5, 0, 0, 2, 0, 0, 0, 0, 14, 5, 2, 2]]
