
# q-expansion of newform 930.2.bg.f, downloaded from the LMFDB on 25 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 = 930
weight = 2
poly_data = [1, 0, 16, 0, 90, 0, 239, 0, 329, 0, 239, 0, 90, 0, 16, 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, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-4, 1, -33, 0, -16, 0, 166, 0, 253, 0, 135, 0, 29, 0, 2, 0], 2], [[-6, 0, -29, 16, 45, 74, 269, 165, 314, 164, 149, 75, 30, 15, 2, 1], 2], [[13, 21, 156, 166, 528, 597, 760, 1092, 539, 956, 194, 404, 33, 77, 2, 5], 2], [[8, 6, 126, 60, 623, 284, 1244, 598, 1119, 553, 479, 239, 92, 46, 6, 3], 2], [[-2, 21, -17, 282, -74, 1151, -165, 2004, -164, 1658, -75, 673, -15, 125, -1, 8], 2], [[-21, 15, -282, 106, -1151, 313, -2004, 494, -1658, 403, -673, 165, -125, 31, -8, 2], 2], [[2, 21, 17, 282, 74, 1151, 165, 2004, 164, 1658, 75, 673, 15, 125, 1, 8], 2], [[6, 2, 60, 97, 284, 668, 598, 1513, 553, 1433, 239, 628, 46, 122, 3, 8], 2], [[22, 1, 317, 0, 1410, 0, 2652, 0, 2313, 0, 973, 0, 185, 0, 12, 0], 2], [[27, -17, 342, -203, 1435, -981, 2602, -2007, 2211, -1836, 912, -793, 171, -153, 11, -10], 2], [[-15, -20, -192, -313, -787, -1480, -1408, -2871, -1194, -2525, -494, -1061, -93, -201, -6, -13], 2], [[27, 17, 342, 203, 1435, 981, 2602, 2007, 2211, 1836, 912, 793, 171, 153, 11, 10], 2], [[-1, 35, 0, 570, 0, 2606, 0, 4925, 0, 4285, 0, 1795, 0, 340, 0, 22], 2], [[26, 48, 342, 547, 1435, 2103, 2602, 3576, 2211, 2927, 912, 1182, 171, 219, 11, 14], 2]]

hecke_ring_character_values = [[311, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [187, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [871, [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, -1, -1, 0, 0, -1, 0, 1, 0, 2, 0, 1, 0, 0, 0], [1, -1, -1, -2, -1, 0, -3, 1, 1, -2, 1, 2, 1, 2, 2, -1], [-1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, -1, -1, -1, 0], [2, 2, 0, 2, 0, 0, 2, -1, -2, 1, -2, -2, -2, -3, 1, 2], [0, -1, -2, -2, 1, -1, -1, 1, 2, 1, 2, 2, 1, -1, 0, -1], [1, 0, -1, -2, -1, -2, -1, 1, 3, -1, 2, 2, 2, 0, 2, 0], [-3, -1, 3, -1, 2, -2, 0, -1, -2, 0, 2, 0, -5, 1, -2, -1], [-1, 1, -4, 0, -1, 3, 0, 1, 5, -1, 1, 3, 1, 2, -1, 2], [1, -1, 1, -1, 4, 0, 4, 2, 3, 4, 0, -1, -1, -3, 2, 1], [-1, 0, 4, -3, 3, -4, 1, 0, -2, 4, -1, -3, -1, -3, -1, -3], [2, 4, 1, -1, -4, 3, -1, 2, -1, -2, -5, 1, -3, 4, 0, -2], [1, 1, -1, -3, 1, 2, 3, 3, 1, -5, -1, 2, -2, 2, 3, -3], [0, -1, 2, -4, 1, -1, 0, 2, -1, 0, -1, 0, 0, -4, 0, -3], [3, 5, 0, -3, 0, -4, 0, 1, -5, -2, -4, 0, 0, -2, 2, -1], [-1, 2, -5, -6, -3, 3, -2, 3, 2, -6, 1, 5, 0, 8, 3, -3], [8, 0, 0, 1, 1, 5, 5, 3, 3, 0, -1, 1, -1, -4, 7, 2], [0, 3, -2, 1, 2, -2, 0, 0, 1, 3, -1, 0, 5, -2, 0, 0], [4, 6, -4, -5, -3, 0, 0, 1, 0, -4, -6, 2, -2, 4, 6, 0], [0, 0, 1, 1, 0, -3, -4, -5, -1, 2, 5, -5, -1, -1, 0, 2], [-2, 1, 6, 3, 1, -6, 0, -7, -6, 3, -1, -4, -3, 2, -5, -4], [-6, 5, -1, 1, 0, -8, -5, -4, -3, 2, 0, -2, 5, 0, -3, -4], [-2, 3, 0, 4, -4, 0, 1, -4, -3, -2, 0, -2, -3, 8, 0, 0], [3, 3, 4, -2, 1, 0, 1, 3, -7, -3, -7, 3, -7, 5, 6, -5], [-2, -2, 0, -4, 3, 0, 1, 2, 4, 5, 1, -2, 2, -5, 0, -2], [0, -3, 2, -3, 0, -2, 2, -6, -4, -2, 0, -6, 1, 3, 0, -2], [0, 0, 0, -3, 0, 0, -3, 5, 0, -6, 0, 3, 3, 3, 1, -4], [1, -1, 5, 2, 4, 0, 7, -3, -1, -3, 1, -6, -3, -7, -3, 2], [5, -1, -3, 4, 3, 4, 6, 3, 4, -3, 1, -4, 6, -9, 2, 10], [3, 0, 2, 4, -1, 4, 2, 0, 1, -3, -5, 0, 1, 3, -1, 1], [6, 2, 0, 0, -5, 1, 0, 4, 3, -3, -3, -3, 5, 1, 5, 2], [-4, -2, -2, -4, 2, 0, -3, 11, 7, -6, 7, 8, 5, 2, -3, 0], [3, 1, 5, -3, 0, -2, 2, -3, -5, 1, -3, -4, -5, -6, -5, 3], [-3, -6, 0, 10, 1, 7, 5, -12, 6, 7, 0, -6, -1, -6, -2, 3], [-11, -3, 0, 0, 0, 0, -3, 0, 0, -1, 3, 2, 0, 11, -1, -2], [-4, -1, 0, 8, 1, -2, 1, -4, 3, 1, 4, -6, 4, -2, -8, 5], [2, 1, 1, 11, -1, 0, 5, -3, -1, 5, 4, -11, 0, -3, -4, 3], [-4, 6, 1, -1, -2, -7, 1, -6, -1, 0, 0, -6, 2, -2, -2, 2], [-6, 0, -8, -1, -5, 6, -5, -6, 3, -10, 5, 6, 3, 16, 1, -7], [9, 1, 0, 2, -1, 3, 3, 7, 0, -1, -3, 5, -4, -8, 7, 4], [2, -1, 0, -1, -3, 0, 0, 8, 3, -8, -1, 4, 0, 1, 3, -2], [-2, 0, -4, -3, 5, 1, 3, -3, -4, 8, -1, 3, -4, -5, 4, -7], [1, 0, 2, 6, 1, 0, 2, -7, -4, 5, 0, -14, 2, -8, 1, 14], [4, 3, -3, -3, -4, 4, -3, 4, 7, -4, -4, 10, 4, 4, -6, 0], [3, -9, -1, -3, 1, -1, 2, -6, 2, 8, 5, -10, 1, -10, 1, 3], [-2, -1, 15, 2, 3, -3, 6, 6, -12, -2, -7, 7, -15, 5, 3, -2], [3, 1, -7, 1, -8, 7, 2, -7, -4, -11, -3, -6, -2, 11, 0, -6], [2, 1, 0, -12, -1, 1, -2, 10, 0, -1, -3, 12, -1, 3, 2, -10], [-2, -1, 2, -8, -2, -3, -1, 8, 2, -18, 5, 10, -1, 10, 8, -9], [1, -4, 1, -7, -1, 4, 0, 3, 1, -3, 3, -1, -5, -5, -1, -4], [-5, 0, -4, 4, -7, -6, -9, -5, 1, -2, 11, 5, 1, 7, -4, -1], [-6, 2, 5, 5, 2, -9, -3, 3, -7, 3, -4, -1, -3, -1, -2, 4], [-1, -2, 4, 8, -6, 10, 0, 0, 0, 6, -2, 1, -8, 3, 1, -5], [-1, -9, -8, -6, 8, -1, 7, 7, 2, -4, 3, 7, 1, -2, 5, -2], [-3, 1, 0, 4, -3, 0, -10, -3, 6, 0, 9, 1, 3, 4, -4, 1], [1, -2, 7, -7, 9, -8, 9, 1, 3, 14, 0, 5, 0, -14, 2, -1], [2, -5, -5, 10, -4, 8, -2, 10, 8, -3, -2, 3, 13, 2, 1, 7], [-10, 9, 4, 0, 4, -2, 11, 0, -2, 2, -7, -3, -6, 6, -2, 3], [1, -8, 2, 1, 2, -3, 6, -4, 4, 4, 0, -2, 7, -12, -3, 3], [4, 6, -9, -7, 0, 7, 2, -3, 3, -3, -11, -5, 3, 5, 2, -1], [-2, -11, -1, -6, 6, 8, 7, 8, 12, 3, 7, 0, 7, -9, 3, -3], [-3, 5, 1, -14, 7, -7, 4, -2, -4, -9, -11, -2, 0, 0, 7, -7], [2, -6, 0, 10, 2, 0, 2, 5, 6, 2, 0, 7, 2, -2, 7, 10], [-10, 0, 6, -5, -1, -7, 5, -8, -12, 8, -11, 8, -12, 0, 3, -11], [-8, 0, 0, 12, -13, 0, 1, -14, -14, -5, -1, -2, -7, 10, -5, -2], [8, 2, -7, -7, 2, 9, -4, 24, 12, -10, 0, 12, 3, 4, 17, -4], [-1, 4, -12, 1, -1, 6, -7, 9, 11, -8, 4, 8, 12, 11, 5, -1], [-3, -2, 4, 8, 6, 5, 6, -3, -7, 4, -4, -5, -4, -8, -6, 7], [8, -3, 2, 8, 3, -1, -1, 0, -2, 16, -2, -8, 1, -8, -5, 0], [-1, 4, -9, -6, -2, 0, -9, -3, 4, 0, 5, 2, 2, -2, -7, 2], [-15, 0, 0, 1, 12, 4, 4, 0, -4, 8, 4, -5, 4, -9, -16, 2], [-22, -7, 6, 2, 3, -3, 0, -7, 0, -2, 8, -5, 0, -2, -1, 1], [-5, 0, -9, -4, -1, 3, -10, 3, 13, -4, 19, -3, 13, 1, -4, 7], [-4, -19, -5, 2, 5, 0, -9, 2, 15, -6, 19, 4, 9, -3, 2, 2], [10, 16, -10, -10, 0, 10, 3, 8, 7, 1, -13, 3, -6, 5, 5, 6], [4, 2, -10, -4, 4, 9, -5, 14, 14, -2, 4, 2, 10, 6, -1, 4], [-6, 1, -4, -6, -2, 0, -3, -6, 0, -1, 1, 4, 4, 8, 1, -10], [-3, 8, -6, 3, -11, -1, -10, 9, -5, -12, -1, 4, 6, 23, -6, -3], [1, -4, -2, -4, 2, -2, 2, -8, -3, -6, -1, 3, 0, 2, 16, 4], [-2, 1, 1, -3, 8, -4, 1, -8, -3, 14, 2, -12, -8, -20, -2, 6], [-5, -9, 11, -6, 8, -1, -3, 12, -2, 8, 10, 7, -3, -8, 7, -6], [-6, 2, 7, -20, 13, -13, 5, 14, -5, 7, -11, 18, 0, 1, 10, -10], [1, -4, 0, -6, 6, 1, 1, 8, -6, 7, -5, 15, -1, -9, 7, -12], [-3, 2, 1, -5, 1, -6, 0, 2, -3, 4, -3, 3, 3, -6, 3, -11], [-10, -1, -2, -6, 2, 1, -1, 21, -1, -10, -2, 16, 3, 4, -9, 0], [1, 3, 8, -8, 5, 0, 5, 1, -11, 11, -14, 14, -14, 4, 2, -16], [-14, -10, 7, 10, 5, 3, 5, -6, -4, 8, 0, -3, -7, -6, 4, -11], [1, 19, -5, -8, 5, -8, 6, -8, -8, 2, -1, -14, -3, -6, -3, 6], [-14, -17, -1, 5, 4, 9, -9, 5, 9, -5, 13, -6, 10, 10, -9, 11], [-6, -5, -1, -4, 0, 0, -18, 8, 17, -2, 17, 11, 6, 7, -3, -5], [1, 5, -5, -12, 14, -11, 0, 15, 4, -4, 0, -1, 19, -9, -1, 7], [-8, 4, 0, -3, -6, -2, -2, 1, 0, 4, 0, 14, -4, -2, -5, -6], [3, 7, 0, 17, -1, 2, -1, -13, -7, -4, 2, -16, -1, -2, -13, 20], [-13, 0, 7, 8, 2, 1, 3, -11, 4, 15, -4, 11, 4, -4, -10, -1], [1, 1, 3, 5, 6, 2, 3, -1, 3, 8, -6, 5, -6, -10, -8, 2], [11, 8, 2, -11, -3, -1, -2, -2, -5, 4, -5, -18, -2, -10, -5, 11], [9, -4, 2, -10, 2, -2, 2, 20, 0, -8, 0, 10, 2, -9, 10, -1], [6, 4, 9, -8, 0, -7, 7, -12, -12, -7, 2, -15, -13, 5, 3, -4], [15, 2, 2, -13, -6, 0, -10, -2, -10, -8, -2, -4, 6, 4, 4, -7], [-2, 5, -1, 24, -5, 5, 0, -9, 0, 10, -9, 6, 0, 8, -12, 12], [7, -4, 0, 5, -4, -5, -5, -9, -1, 9, -4, -1, 0, -5, 2, 10], [4, 9, 10, -1, -3, 0, 3, -3, -6, 0, -9, -6, -13, 0, -3, -1], [-5, -5, 10, 14, 13, -1, 10, -13, -6, 21, -4, -12, -13, -29, -9, 8], [-6, -1, 7, -7, 1, 8, 2, 10, -7, -12, -13, 7, -8, 6, 7, -10], [0, 9, 7, -3, -4, -7, 9, -3, -7, -9, -13, 0, -14, 4, -5, -3], [-10, -3, 1, -13, 1, 6, 2, 2, 1, 0, 0, 1, -5, 13, -11, -11], [2, -13, -16, 1, -11, 16, -4, 13, 22, -13, 17, 14, 11, 11, 2, 14], [-3, 6, -14, 6, -6, -3, 1, -9, 14, -8, 18, -6, 3, 5, -1, 9], [-3, 0, -1, -12, 3, -1, -1, -2, -1, 2, 2, -9, -3, -1, 8, -1], [10, 0, 4, 15, 6, 6, 5, -2, 6, 6, -4, 2, 6, -4, -3, 1], [10, -14, -4, 2, 0, 11, 4, 5, 18, -5, 7, 5, 4, -5, 10, 7], [-21, -5, 11, 26, 0, 0, 7, -15, -7, 11, -1, -9, 0, -7, -13, 13], [23, -2, 7, -23, 10, 0, 18, 0, 6, -11, 2, 0, -11, -10, 25, 2], [15, 1, -13, -7, 6, 13, 4, 12, 14, -9, -5, 5, 7, 4, 5, 5], [8, -8, -5, -3, 3, 9, 3, 8, 4, -10, 3, 7, 3, 5, 16, -3], [5, 2, 2, 10, -6, -9, 4, 2, -6, -7, 0, 1, 3, -18, 12, 4], [21, 6, -9, 8, -5, 0, -3, 8, 0, 1, -2, -18, 9, -16, 6, 26], [-14, -2, 0, 12, 2, 5, 2, -14, -3, 18, 6, -12, 6, -9, -28, 5], [12, 10, -12, 3, -11, 12, -11, -5, 2, -23, 1, -2, 1, 12, 11, -2], [-10, 4, -4, -1, -13, -11, 0, -6, -10, -15, 0, 10, -9, 24, 10, -9], [-11, -6, 0, -2, -3, -18, -18, -5, -4, 9, 5, 9, 11, -2, -9, -4], [-5, 9, -12, 8, -10, 10, -2, -5, 2, -7, -6, -13, 0, -2, -4, 4], [-2, 15, -11, -9, -21, 4, 0, -11, -3, -25, -4, 2, -6, 18, 2, -7], [1, 2, -1, 18, -2, -1, 5, -28, 1, 9, 7, -18, 1, -8, -10, 28], [1, 5, 3, 20, -9, 2, 4, 0, -11, -9, 0, 0, -13, 11, 20, 1], [3, -11, -9, -3, 8, 8, 0, 3, 17, 3, 10, 2, 9, -8, 4, 3], [0, 3, 1, -10, 0, -1, -15, -5, 14, 2, 16, -5, -4, 10, 0, 2], [7, 15, -5, 3, -11, 0, -7, -10, -7, -2, -15, -20, -3, 1, -14, -1], [13, -1, 0, 5, 3, 19, 19, 9, 14, -11, -3, -10, -2, -3, 8, 10], [-13, 1, -5, 22, -3, 3, -1, -26, 1, -7, 3, -15, 0, 3, -11, 11], [3, 0, 18, 3, 7, -12, 8, -1, -3, 22, -9, 1, -3, -21, 1, -2], [-2, 5, -13, 1, -3, -4, -13, -7, 1, 2, 12, 1, 3, -7, -13, 5], [36, -13, 7, -3, 0, -4, 0, 3, -1, -2, 7, -15, 6, -15, 18, 16], [-4, 6, -4, 4, -12, -5, -7, -11, -8, -4, -1, -4, 4, 5, -6, -4], [7, -15, 15, 1, 6, 5, 5, 1, 5, 12, 1, -5, -10, -13, 6, 6], [-2, -4, 3, 1, 4, -9, 0, 10, -3, -12, 6, -1, 9, 10, 14, -10], [6, 0, 11, -15, -4, 3, 0, 8, -6, 3, -11, -6, -4, -24, -6, -16], [-3, -8, -1, 15, 7, -2, 8, 6, 9, 5, -3, -1, 8, -9, 1, 17], [-17, 3, 0, -18, -9, -3, -3, 11, 0, -16, -3, 22, -6, 23, 1, -36], [8, 9, 3, -2, -3, 6, -3, 10, 9, -7, 3, -6, 0, 5, -6, 1], [8, -1, -4, -5, 0, 0, -14, -14, 10, 8, 10, -18, 5, 1, 4, 12], [7, 2, 0, 1, -4, 0, 4, -13, -10, -11, -6, -12, -5, 24, -13, -12], [10, -1, -5, -3, -4, 8, -4, 10, -2, -21, 2, 9, 2, 20, 20, -9], [-9, 15, 0, -4, 3, -15, 0, -4, -15, 4, 3, -18, -15, 6, 1, 14], [12, 5, -1, -2, 1, 6, 5, 0, -9, -3, -15, -10, -4, 1, 4, -14], [12, -1, -15, -28, -8, 0, -17, 16, 8, -30, 1, 32, 8, 24, 17, -27], [7, 6, 7, 26, -17, 20, 0, 9, -2, -10, -13, -7, -11, 29, -7, 26], [-12, 14, 0, 6, -6, 8, 8, -17, -9, 14, -9, -3, -23, -3, -18, 12], [-1, 15, 4, -1, 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