
# q-expansion of newform 475.3.c.f, downloaded from the LMFDB on 24 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 = 3
poly_data = [28561, 0, -7098, 0, 771, 0, -42, 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], 1], [[0, 4901, 0, -771, 0, 42, 0, -1], 2197], [[0, 9295, 0, -771, 0, 42, 0, -1], 2197], [[0, 10985, 0, -2809, 0, 251, 0, -10], 4394], [[3549, 0, -602, 0, 42, 0, -1, 0], 338], [[-7098, 0, 940, 0, -42, 0, 1, 0], 169], [[0, 93457, 0, -16919, 0, 1259, 0, -34], 4394], [[-32955, 0, 7424, 0, -628, 0, 23, 0], 338]]

hecke_ring_character_values = [[77, [1, 0, 0, 0, 0, 0, 0, 0]], [401, [-1, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 1, 0, 0, 0, 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, 0, 0], [-2, 0, 0, 0, 3, 0, 0, 0], [0, -5, 0, 4, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, -1, 0], [-6, 0, 0, 0, -7, 1, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, -1], [0, 0, 0, 0, 0, -1, 0, 1], [0, 12, 0, -12, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 3], [0, 0, -2, 0, 0, 0, -3, 0], [0, 0, -2, 0, 0, 0, -2, 0], [0, -19, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 4], [22, 0, 0, 0, 1, 0, 0, 0], [0, -8, 0, -19, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 0, -1], [0, 0, -7, 0, 0, 0, -5, 0], [0, 0, 0, 0, 0, 4, 0, -6], [0, 0, 12, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 1, 0, 3], [0, 0, 0, -10, 0, 0, 0, 0], [52, 0, 0, 0, 105, 0, 0, 0], [0, 45, 0, -5, 0, 0, 0, 0], [0, 4, 0, 47, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 3], [0, 35, 0, 17, 0, 0, 0, 0], [0, 38, 0, 30, 0, 0, 0, 0], [-16, 0, 0, 0, 98, 0, 0, 0], [0, 0, 9, 0, 0, 0, -13, 0], [-2, 0, 0, 0, 126, 0, 0, 0], [74, 0, 0, 0, -46, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, 8], [0, 0, 14, 0, 0, 0, 12, 0], [0, 0, -16, 0, 0, 0, 17, 0], [0, -19, 0, 33, 0, 0, 0, 0], [0, -52, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 0, -4], [0, 0, 0, 0, 0, 2, 0, -6], [-131, 0, 0, 0, -29, 0, 0, 0], [0, -118, 0, -16, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 3, 0], [139, 0, 0, 0, -109, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, 12], [0, 24, 0, -48, 0, 0, 0, 0], [0, -2, 0, 89, 0, 0, 0, 0], [-244, 0, 0, 0, 16, 0, 0, 0], [0, 0, -12, 0, 0, 0, -26, 0], [63, 0, 0, 0, 241, 0, 0, 0], [0, 0, 0, 0, 0, -8, 0, -6], [222, 0, 0, 0, -27, 0, 0, 0], [0, 95, 0, -93, 0, 0, 0, 0], [0, 0, -8, 0, 0, 0, 18, 0], [0, 0, 0, 0, 0, -10, 0, 4], [65, 0, 0, 0, 41, 0, 0, 0], [0, 0, -26, 0, 0, 0, 8, 0], [0, 0, 0, 0, 0, -11, 0, 19], [0, 0, -8, 0, 0, 0, 18, 0], [0, 129, 0, -62, 0, 0, 0, 0], [0, -78, 0, -28, 0, 0, 0, 0], [-55, 0, 0, 0, 165, 0, 0, 0], [0, 0, -69, 0, 0, 0, 11, 0], [0, 39, 0, 110, 0, 0, 0, 0], [0, 0, 0, 0, 0, -23, 0, -14], [0, -29, 0, -133, 0, 0, 0, 0], [0, 0, -36, 0, 0, 0, 14, 0], [-2, 0, 0, 0, 28, 0, 0, 0], [0, 0, 45, 0, 0, 0, -11, 0], [-371, 0, 0, 0, 11, 0, 0, 0], [0, 0, 10, 0, 0, 0, -36, 0], [0, -41, 0, -118, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, 0, -12], [0, -131, 0, -65, 0, 0, 0, 0], [-288, 0, 0, 0, 3, 0, 0, 0], [0, 0, -58, 0, 0, 0, -16, 0], [0, 0, 0, 0, 0, 5, 0, -21], [0, 0, 0, 0, 0, -39, 0, -7], [376, 0, 0, 0, -105, 0, 0, 0], [0, 0, 0, 0, 0, -26, 0, 15], [0, 0, 0, 0, 0, 1, 0, 9], [0, -239, 0, -15, 0, 0, 0, 0], [0, 0, 0, 0, 0, 23, 0, 7], [0, 0, -8, 0, 0, 0, 43, 0], [0, 0, 0, 0, 0, -10, 0, 16], [0, 0, 3, 0, 0, 0, 9, 0], [182, 0, 0, 0, 361, 0, 0, 0], [0, 0, 14, 0, 0, 0, -56, 0], [0, 0, 64, 0, 0, 0, 13, 0], [302, 0, 0, 0, -8, 0, 0, 0], [0, -46, 0, -94, 0, 0, 0, 0], [-168, 0, 0, 0, -283, 0, 0, 0], [-144, 0, 0, 0, 481, 0, 0, 0], [0, 0, 13, 0, 0, 0, -26, 0], [0, 0, 0, 0, 0, -26, 0, -14], [0, 0, 0, 0, 0, -24, 0, 42], [0, 98, 0, 39, 0, 0, 0, 0], [-182, 0, 0, 0, -81, 0, 0, 0], [0, 204, 0, -188, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -72, 0], [0, -246, 0, 116, 0, 0, 0, 0], [0, 0, 0, 0, 0, -14, 0, 20], [-770, 0, 0, 0, 237, 0, 0, 0], [0, 0, 13, 0, 0, 0, 91, 0], [0, 0, -50, 0, 0, 0, -5, 0], [0, 0, -54, 0, 0, 0, -26, 0], [0, 0, 0, 0, 0, 49, 0, 9], [0, 0, 0, 0, 0, -51, 0, 13], [0, -86, 0, -12, 0, 0, 0, 0], [0, 0, 40, 0, 0, 0, -15, 0], [0, 0, -24, 0, 0, 0, 6, 0], [840, 0, 0, 0, 1, 0, 0, 0], [216, 0, 0, 0, 342, 0, 0, 0], [0, 0, 0, 0, 0, -37, 0, -31], [0, 0, -30, 0, 0, 0, 18, 0], [0, 0, -5, 0, 0, 0, 36, 0], [0, 0, 2, 0, 0, 0, -26, 0], [0, 0, 0, 0, 0, -15, 0, -32], [0, 0, 0, 0, 0, 48, 0, -13], [0, -112, 0, 290, 0, 0, 0, 0], [0, -349, 0, 210, 0, 0, 0, 0], [0, 30, 0, -12, 0, 0, 0, 0], [-420, 0, 0, 0, -116, 0, 0, 0], [-576, 0, 0, 0, -202, 0, 0, 0], [-280, 0, 0, 0, -374, 0, 0, 0], [-799, 0, 0, 0, 307, 0, 0, 0], [0, 0, -69, 0, 0, 0, -30, 0], [0, 0, 52, 0, 0, 0, -84, 0], [32, 0, 0, 0, -371, 0, 0, 0], [0, 5, 0, 121, 0, 0, 0, 0], [0, 0, 0, 0, 0, -35, 0, -13], [0, 0, 100, 0, 0, 0, -32, 0], [-175, 0, 0, 0, -606, 0, 0, 0], [-421, 0, 0, 0, -100, 0, 0, 0], [0, 211, 0, 54, 0, 0, 0, 0], [0, -88, 0, 445, 0, 0, 0, 0], [0, 227, 0, -306, 0, 0, 0, 0], [603, 0, 0, 0, -386, 0, 0, 0], [0, 0, 0, 0, 0, 43, 0, 18], [262, 0, 0, 0, -556, 0, 0, 0], [0, 0, 89, 0, 0, 0, 48, 0], [0, 42, 0, 137, 0, 0, 0, 0], [0, 0, 0, 0, 0, -24, 0, -17], [0, 0, 0, 0, 0, -68, 0, 36], [0, 0, 32, 0, 0, 0, 90, 0], [0, 274, 0, 346, 0, 0, 0, 0], [-408, 0, 0, 0, 519, 0, 0, 0], [0, -213, 0, 181, 0, 0, 0, 0], [0, -311, 0, 12, 0, 0, 0, 0], [-1380, 0, 0, 0, -22, 0, 0, 0], [0, 0, 132, 0, 0, 0, -12, 0], [0, -317, 0, -19, 0, 0, 0, 0], [0, -248, 0, 281, 0, 0, 0, 0], [0, 0, 0, 0, 0, 36, 0, -52], [-285, 0, 0, 0, 623, 0, 0, 0], [-385, 0, 0, 0, -402, 0, 0, 0], [0, 0, -3, 0, 0, 0, 27, 0], [0, 0, 0, 0, 0, -4, 0, 31], [0, 0, -28, 0, 0, 0, -6, 0], [0, -83, 0, 105, 0, 0, 0, 0], [0, 0, -78, 0, 0, 0, -96, 0], [0, 0, 0, 0, 0, -24, 0, -18], [0, -129, 0, -83, 0, 0, 0, 0], [0, 16, 0, -292, 0, 0, 0, 0], [0, 0, 0, 0, 0, 79, 0, -29], [0, 0, -150, 0, 0, 0, -3, 0]]
