
# q-expansion of newform 3120.2.l.m, 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 = 3120
weight = 2
poly_data = [8, -20, 25, -4, 0, 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], 1], [[323, -121, -4, 10, 25, 2], 121], [[-223, -121, 14, -35, -27, -7], 121], [[258, -605, 50, -4, -10, -25], 242], [[574, -1573, 372, 38, -26, -65], 242], [[-774, 1573, -392, 12, 30, 75], 242]]

hecke_ring_character_values = [[1951, [1, 0, 0, 0, 0, 0]], [2341, [1, 0, 0, 0, 0, 0]], [2081, [1, 0, 0, 0, 0, 0]], [2497, [-1, 0, 0, 0, 0, 0]], [2641, [1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0], [2, 0, 0, 0, -1, -1], [0, 0, 0, 1, 0, 0], [0, -1, -1, 2, 1, -1], [0, 1, -1, 0, 1, 1], [0, -1, -1, 4, 1, -1], [2, -2, 2, 0, 0, 0], [-4, 0, 0, 0, 2, 2], [0, -2, -2, -2, 0, 0], [-4, -1, 1, 0, 2, 2], [0, 1, 1, -2, -1, 1], [0, 2, 2, -6, -1, 1], [0, 0, 0, -2, 2, -2], [-2, 0, 0, 0, -3, -3], [0, -1, 1, 0, 1, 1], [0, 2, 2, 0, 2, -2], [-2, 1, -1, 0, 0, 0], [0, -4, -4, -2, 0, 0], [6, -1, 1, 0, -1, -1], [0, 2, 2, 2, 1, -1], [8, 1, -1, 0, -2, -2], [0, 0, 0, 6, 0, 0], [-10, 0, 0, 0, 2, 2], [0, 0, 0, 6, 2, -2], [0, -2, -2, 0, 0, 0], [2, -3, 3, 0, 1, 1], [0, 1, 1, 2, 1, -1], [0, -1, -1, -12, -1, 1], [0, 2, -2, 0, 0, 0], [0, -2, -2, 8, 3, -3], [4, 2, -2, 0, 6, 6], [4, 0, 0, 0, 3, 3], [-4, 2, -2, 0, 4, 4], [0, 5, 5, 12, 1, -1], [0, -2, -2, 4, 0, 0], [0, -2, -2, -10, 1, -1], [0, -4, -4, 2, 4, -4], [8, 2, -2, 0, 2, 2], [2, 2, -2, 0, 6, 6], [0, -4, 4, 0, 0, 0], [0, 2, 2, 2, -4, 4], [0, 6, 6, 8, 3, -3], [-2, -5, 5, 0, -1, -1], [2, -1, 1, 0, -1, -1], [0, -4, -4, 8, 2, -2], [0, 6, 6, 6, -1, 1], [2, -1, 1, 0, -1, -1], [0, 5, 5, 2, 1, -1], [2, 1, -1, 0, 2, 2], [14, -2, 2, 0, 0, 0], [-4, -2, 2, 0, -8, -8], [0, 1, 1, 10, -3, 3], [0, -1, -1, 0, -7, 7], [18, 4, -4, 0, 6, 6], [-4, -6, 6, 0, -2, -2], [0, -3, -3, 0, 1, -1], [12, 5, -5, 0, 2, 2], [0, -4, -4, 4, 4, -4], [0, 4, 4, -4, 3, -3], [0, 0, 0, -20, 0, 0], [16, 4, -4, 0, 2, 2], [0, 3, 3, -10, -1, 1], [0, -2, -2, 0, 3, -3], [-4, 1, -1, 0, 3, 3], [0, -1, -1, 6, -5, 5], [0, -6, -6, 4, -2, 2], [2, 5, -5, 0, 3, 3], [0, -4, -4, -8, -5, 5], [14, 3, -3, 0, 6, 6], [0, -2, -2, -10, -4, 4], [0, -6, -6, -6, 2, -2], [16, 1, -1, 0, 1, 1], [0, 4, 4, -14, -3, 3], [14, 0, 0, 0, 0, 0], [0, -4, -4, 6, 6, -6], [0, 3, -3, 0, -8, -8], [-2, 2, -2, 0, -8, -8], [-4, -4, 4, 0, -6, -6], [18, 1, -1, 0, -3, -3], [6, 5, -5, 0, 0, 0], [0, 5, 5, 18, 1, -1], [16, -4, 4, 0, 4, 4], [0, -8, -8, -4, -2, 2], [-8, -7, 7, 0, 0, 0], [0, 2, 2, 18, -2, 2], [0, 0, 0, 0, 1, 1], [0, 0, 0, 16, 0, 0], [0, 0, 0, -12, -2, 2], [18, 1, -1, 0, 2, 2], [0, -6, -6, 4, 4, -4], [-4, -4, 4, 0, 6, 6], [20, -3, 3, 0, 3, 3], [0, -7, -7, -8, 3, -3], [0, 2, -2, 0, -3, -3], [-2, 0, 0, 0, -6, -6], [0, 5, 5, -2, -9, 9], [6, -3, 3, 0, 1, 1], [0, 11, 11, -10, -7, 7], [0, -2, -2, 12, -9, 9], [0, 2, 2, 20, 2, -2], [14, -2, 2, 0, -8, -8], [2, -1, 1, 0, -13, -13], [0, -4, -4, -18, -2, 2], [0, 10, 10, 10, 1, -1], [0, 6, 6, -20, -3, 3], [0, 4, -4, 0, -6, -6], [8, -3, 3, 0, 7, 7], [0, -2, -2, -22, 0, 0], [0, 4, 4, 6, 2, -2], [0, 6, 6, 8, -3, 3], [12, 3, -3, 0, -7, -7], [-8, -4, 4, 0, -14, -14], [6, 0, 0, 0, 4, 4], [0, 6, 6, 0, 2, -2], [0, 1, 1, 8, 5, -5], [0, 0, 0, -2, 0, 0], [24, -6, 6, 0, 2, 2], [-10, -7, 7, 0, -9, -9], [0, -10, -10, 8, -2, 2], [0, 4, 4, 30, 2, -2], [0, 6, 6, -2, -5, 5], [-40, -1, 1, 0, 1, 1], [-34, 0, 0, 0, 4, 4], [2, -1, 1, 0, 11, 11], [-4, 6, -6, 0, -4, -4], [0, -1, -1, -12, -5, 5], [0, 0, 0, -2, -4, 4], [0, 3, -3, 0, 1, 1], [0, 16, 16, 14, 3, -3], [-16, -4, 4, 0, 0, 0], [0, 0, 0, -10, 8, -8], [-4, -7, 7, 0, 6, 6], [-2, 4, -4, 0, 0, 0], [0, 8, 8, -4, -7, 7], [0, -8, -8, 20, 4, -4], [0, -10, -10, -10, 0, 0], [18, 8, -8, 0, -2, -2], [-20, 3, -3, 0, -1, -1], [12, 2, -2, 0, -11, -11], [0, -8, -8, 2, -6, 6], [0, -6, -6, 14, 3, -3], [24, -3, 3, 0, -1, -1], [2, 3, -3, 0, 4, 4], [0, -4, -4, 10, 2, -2], [0, -1, -1, 26, -1, 1], [18, 7, -7, 0, -5, -5], [0, -2, -2, -10, 5, -5], [0, 2, 2, -10, 4, -4], [-10, -6, 6, 0, -16, -16], [0, 2, 2, -4, -6, 6], [0, 3, 3, 36, -1, 1], [0, 6, 6, 12, 2, -2], [24, 4, -4, 0, -12, -12], [6, -5, 5, 0, -5, -5], [4, -5, 5, 0, -14, -14], [0, -7, -7, -6, 9, -9], [-20, -4, 4, 0, -3, -3], [0, -14, -14, 26, 9, -9], [0, -5, -5, -14, -1, 1], [0, 2, 2, -8, 8, -8], [-12, -2, 2, 0, -2, -2], [0, -4, -4, 4, -7, 7], [0, 4, 4, -18, -1, 1], [8, 4, -4, 0, 0, 0], [0, -9, -9, 8, -1, 1], [-10, 4, -4, 0, 2, 2], [0, 10, 10, 14, -2, 2], [-2, 2, -2, 0, -1, -1], [6, 5, -5, 0, 3, 3], [32, 4, -4, 0, 2, 2], [0, 12, 12, -22, -4, 4], [24, 4, -4, 0, -4, -4], [10, 2, -2, 0, -2, -2], [4, 7, -7, 0, 5, 5], [-20, 8, -8, 0, -3, -3], [0, 2, 2, 10, 0, 0], [-4, -1, 1, 0, 9, 9], [0, -6, -6, -16, -2, 2], [-8, -10, 10, 0, 6, 6], [0, 8, 8, -30, 0, 0], [0, 4, 4, -20, -9, 9], [0, 0, 0, -38, 1, -1], [-10, -10, 10, 0, 6, 6], [0, 5, 5, -12, 1, -1], [0, 2, 2, -4, 8, -8], [-6, 0, 0, 0, -6, -6], [-26, -1, 1, 0, -2, -2], [0, 2, 2, 16, -2, 2], [0, -6, -6, -10, 3, -3], [-4, 8, -8, 0, -8, -8], [-44, -4, 4, 0, 5, 5], [0, 8, 8, -4, -4, 4], [0, -1, -1, -18, -9, 9], [42, 6, -6, 0, 6, 6], [0, 4, 4, -10, -4, 4], [0, 16, 16, 4, -5, 5], [0, 15, 15, 8, 7, -7], [24, -8, 8, 0, -1, -1], [-24, -4, 4, 0, 0, 0], [0, 4, 4, -26, -4, 4], [2, 2, -2, 0, 2, 2], [-10, 0, 0, 0, -3, -3], [0, 14, 14, 26, 4, -4], [12, -8, 8, 0, 6, 6], [0, -18, -18, 0, 4, -4], [-24, -3, 3, 0, 8, 8], [-34, -5, 5, 0, 3, 3], [0, 7, 7, -18, 3, -3], [-14, 10, -10, 0, -2, -2], [0, -5, -5, -20, 7, -7], [0, -10, -10, 26, 1, -1], [-6, 9, -9, 0, 10, 10], [22, 6, -6, 0, -2, -2], [0, -3, -3, -28, -3, 3], [38, -4, 4, 0, -4, -4], [0, 0, 0, -6, 1, -1], [0, 14, 14, -12, 1, -1], [-30, 0, 0, 0, -8, -8], [-24, -2, 2, 0, -2, -2], [24, 7, -7, 0, 0, 0], [0, -4, -4, 24, -2, 2], [0, -2, -2, 12, 8, -8], [6, 8, -8, 0, 8, 8], [0, -9, -9, -26, 5, -5], [8, -8, 8, 0, 2, 2], [0, -9, -9, -36, 3, -3], [6, -14, 14, 0, 3, 3], [0, -5, -5, 28, -1, 1], [-12, 8, -8, 0, 4, 4], [-8, 0, 0, 0, 8, 8], [-2, 2, -2, 0, -2, -2], [0, 7, 7, -10, -11, 11], [0, -2, -2, 26, -1, 1], [-14, 4, -4, 0, 10, 10], [0, -12, -12, 12, 5, -5], [-24, -2, 2, 0, 14, 14], [20, -10, 10, 0, -2, -2], [0, -6, -6, -14, 5, -5], [14, 7, -7, 0, 3, 3], [0, 1, 1, -20, 9, -9], [-18, 9, -9, 0, -9, -9], [0, 14, 14, 44, 3, -3], [-8, -12, 12, 0, 2, 2], [0, -10, -10, 24, -6, 6], [38, 4, -4, 0, 9, 9], [0, 9, -9, 0, -3, -3], [0, -15, -15, -12, -5, 5], [0, 6, 6, 6, -6, 6], [52, -5, 5, 0, 0, 0], [0, -12, -12, -6, 1, -1], [-24, -9, 9, 0, -3, -3], [0, 4, 4, -22, 0, 0], [46, 4, -4, 0, 1, 1], [8, -13, 13, 0, 9, 9], [0, 0, 0, 24, 6, -6], [0, 10, 10, -10, 4, -4], [0, 6, 6, 34, 8, -8], [0, 3, 3, 20, -5, 5], [0, -12, -12, 4, -12, 12], [-22, -13, 13, 0, -9, -9], [0, 7, 7, -8, 15, -15], [0, -14, -14, 12, 3, -3], [26, 3, -3, 0, -17, -17], [16, 4, -4, 0, 17, 17], [4, -5, 5, 0, 4, 4], [0, 2, 2, -64, -2, 2], [0, -12, -12, 22, 0, 0], [-4, 5, -5, 0, -13, -13], [0, -6, -6, 0, -2, 2], [0, 2, 2, 6, 10, -10], [8, -12, 12, 0, 8, 8], [0, 5, 5, -34, 1, -1], [0, -2, -2, 36, 2, -2], [0, 2, 2, 42, -3, 3], [2, -1, 1, 0, -19, -19], [66, 2, -2, 0, 2, 2], [-16, 0, 0, 0, 10, 10], [0, -9, -9, -4, 7, -7], [68, 0, 0, 0, -4, -4], [0, -7, -7, -8, 5, -5], [10, 11, -11, 0, -7, -7], [0, 24, 24, -12, -4, 4], [20, 2, -2, 0, 6, 6], [0, 13, 13, -10, -15, 15], [0, 2, 2, -36, -5, 5], [8, 2, -2, 0, 12, 12], [18, 6, -6, 0, 12, 12], [-10, 0, 0, 0, -20, -20], [0, -4, -4, 48, 8, -8], [0, 0, 0, -44, -1, 1], [-6, 6, -6, 0, 1, 1], [0, 1, 1, 36, -3, 3], [42, 6, -6, 0, 2, 2], [8, 16, -16, 0, 4, 4], [0, 8, 8, 34, -2, 2], [-12, -2, 2, 0, -10, -10], [0, 8, 8, 4, 8, -8], [0, -2, -2, -12, -6, 6], [0, -4, -4, 16, 9, -9], [32, 4, -4, 0, -24, -24]]
