
# q-expansion of newform 2240.2.g.m, 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 = 2240
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], [[0, 1, 0, 0, 0, 0], 1], [[100, -121, 10, -25, -2, -5], 121], [[223, 0, -14, 35, 27, 7], 121], [[258, -605, 50, -4, -10, -25], 242], [[574, -1331, 372, 38, -26, -65], 242]]

hecke_ring_character_values = [[1471, [1, 0, 0, 0, 0, 0]], [1541, [1, 0, 0, 0, 0, 0]], [897, [-1, 0, 0, 0, 0, 0]], [1921, [1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1], [0, -1, 0, 0, 0, 1], [0, 0, 0, 0, -1, 0], [2, -1, -1, -1, 0, 0], [0, -2, 2, 0, 2, 1], [0, 1, -1, 0, 0, 1], [-2, -1, -1, -2, 0, 0], [0, 2, -2, 0, -2, 0], [0, -3, -3, -3, 0, 0], [-4, 0, 0, -2, 0, 0], [0, 0, 0, 0, -6, 0], [6, 2, 2, 0, 0, 0], [0, 2, -2, 0, 0, 0], [0, 3, -3, 0, 2, -3], [0, 2, -2, 0, 2, -4], [-2, 3, 3, 0, 0, 0], [-8, 1, 1, 0, 0, 0], [0, -2, 2, 0, -4, -2], [-2, 0, 0, -2, 0, 0], [0, -2, 2, 0, 2, 0], [4, -1, -1, -5, 0, 0], [0, -1, 1, 0, 6, 6], [-2, -2, -2, -6, 0, 0], [0, 1, -1, 0, -4, -3], [0, -3, -3, -4, 0, 0], [0, 1, -1, 0, 2, -1], [0, 0, 0, 0, 0, 0], [4, 3, 3, 7, 0, 0], [0, 0, 0, 0, -6, 4], [0, -6, 6, 0, 2, 8], [-2, -1, -1, -4, 0, 0], [0, -4, 4, 0, 0, 0], [2, 5, 5, 8, 0, 0], [-10, 0, 0, -4, 0, 0], [-2, 1, 1, -1, 0, 0], [0, 3, -3, 0, -4, -2], [0, 0, 0, 0, -4, 2], [0, 3, -3, 0, 14, 1], [0, -2, 2, 0, 14, -1], [-4, -4, -4, 0, 0, 0], [8, 3, 3, 2, 0, 0], [4, -1, -1, 3, 0, 0], [0, -2, 2, 0, 10, 6], [0, -2, 2, 0, -14, -8], [12, 8, 8, 2, 0, 0], [-2, -1, -1, -7, 0, 0], [0, -1, 1, 0, -6, -1], [0, -4, 4, 0, -8, 1], [16, -1, -1, 2, 0, 0], [0, 0, 0, 0, 4, 4], [-14, -1, -1, -3, 0, 0], [2, 2, 2, 0, 0, 0], [-10, 3, 3, -8, 0, 0], [0, 2, -2, 0, 6, 0], [0, 4, -4, 0, -8, 0], [-8, -3, -3, 2, 0, 0], [4, -8, -8, -8, 0, 0], [0, -2, 2, 0, 10, 4], [-16, 3, 3, 5, 0, 0], [0, -2, 2, 0, -4, 9], [0, 4, -4, 0, 10, 3], [0, -2, 2, 0, -28, 3], [12, 4, 4, 4, 0, 0], [0, 5, -5, 0, 8, 5], [0, 0, 0, 0, -22, -2], [-8, -2, -2, 4, 0, 0], [0, -4, 4, 0, 2, 4], [0, 12, -12, 0, -8, -6], [12, 7, 7, 8, 0, 0], [0, 5, -5, 0, 12, -9], [8, 4, 4, 4, 0, 0], [0, -3, 3, 0, 2, 13], [0, 4, -4, 0, 6, 2], [4, 2, 2, 0, 0, 0], [0, 2, -2, 0, 0, 0], [-20, -1, -1, -9, 0, 0], [0, 2, -2, 0, 2, 1], [-6, 3, 3, 1, 0, 0], [-2, 4, 4, 2, 0, 0], [6, -9, -9, -12, 0, 0], [-8, 7, 7, 13, 0, 0], [-6, -9, -9, -11, 0, 0], [0, 4, -4, 0, 22, 0], [-12, 2, 2, -6, 0, 0], [0, 4, -4, 0, 0, -8], [0, 1, 1, 7, 0, 0], [0, 8, -8, 0, 6, -16], [4, 9, 9, 4, 0, 0], [0, 0, 0, 0, 0, -8], [0, 4, -4, 0, 0, -5], [0, 4, 4, -10, 0, 0], [0, 6, -6, 0, -18, -4], [18, 7, 7, 9, 0, 0], [-14, 7, 7, -11, 0, 0], [0, -3, 3, 0, -10, 5], [16, -1, -1, 2, 0, 0], [10, -4, -4, -6, 0, 0], [0, 9, -9, 0, -6, -14], [-8, 5, 5, -5, 0, 0], [0, -10, 10, 0, 8, 12], [0, 4, -4, 0, -2, 2], [0, -5, 5, 0, 22, 2], [4, -2, -2, -10, 0, 0], [20, 2, 2, 8, 0, 0], [0, -3, 3, 0, -12, 7], [0, -7, 7, 0, 10, 2], [0, -5, 5, 0, 0, -7], [-4, -15, -15, -7, 0, 0], [2, 2, 2, 10, 0, 0], [0, -5, 5, 0, 2, -9], [0, 6, -6, 0, 6, -6], [0, 16, -16, 0, -10, -8], [-34, -1, -1, -4, 0, 0], [4, -1, -1, 3, 0, 0], [-4, -10, -10, 6, 0, 0], [0, -8, 8, 0, 8, 7], [0, -2, 2, 0, 0, -8], [0, -2, 2, 0, 18, 10], [-6, 5, 5, 9, 0, 0], [-8, -5, -5, 8, 0, 0], [0, 8, -8, 0, 24, -12], [0, 0, 0, 0, -10, 11], [0, 4, -4, 0, 16, -4], [10, -11, -11, -2, 0, 0], [-8, -5, -5, -13, 0, 0], [12, 1, 1, 9, 0, 0], [-12, 8, 8, -2, 0, 0], [0, -6, 6, 0, -4, 8], [0, 10, -10, 0, -18, -11], [-14, -15, -15, -11, 0, 0], [0, -2, 2, 0, 2, 4], [-2, -7, -7, 3, 0, 0], [0, -2, 2, 0, -6, -6], [26, -2, -2, 4, 0, 0], [-30, 4, 4, 10, 0, 0], [0, -8, 8, 0, 14, 9], [0, -2, 2, 0, -4, 5], [0, -4, 4, 0, -6, -3], [-6, -11, -11, -13, 0, 0], [-2, 3, 3, 4, 0, 0], [0, -5, -5, 1, 0, 0], [0, 10, -10, 0, 20, -6], [0, 6, -6, 0, 16, -2], [-12, -3, -3, 2, 0, 0], [-16, 8, 8, 2, 0, 0], [0, -11, 11, 0, -4, 6], [0, 4, -4, 0, -2, 4], [-6, -11, -11, 6, 0, 0], [0, 6, -6, 0, -36, 2], [0, -6, 6, 0, 2, -2], [-10, -2, -2, -18, 0, 0], [0, -20, 20, 0, 16, 8], [0, -2, 2, 0, -16, -4], [0, -2, 2, 0, 4, -14], [16, -12, -12, 4, 0, 0], [36, -5, -5, -5, 0, 0], [-10, -6, -6, -20, 0, 0], [0, 11, -11, 0, -12, -15], [-24, -9, -9, -16, 0, 0], [0, 0, 0, 0, -24, -2], [0, -2, 2, 0, 8, -6], [0, 12, -12, 0, -18, -2], [10, 5, 5, 8, 0, 0], [0, -4, 4, 0, -8, 8], [0, -13, 13, 0, -6, -3], [-14, 12, 12, 10, 0, 0], [0, -6, 6, 0, -2, 9], [-22, 9, 9, 7, 0, 0], [0, 3, -3, 0, 28, -6], [-4, 4, 4, -16, 0, 0], [-12, -17, -17, -4, 0, 0], [32, 7, 7, 7, 0, 0], [0, -2, 2, 0, -24, -14], [0, -14, -14, -8, 0, 0], [-10, -6, -6, -2, 0, 0], [-2, 9, 9, -1, 0, 0], [-8, -3, -3, 5, 0, 0], [0, -4, 4, 0, 4, 0], [40, -3, -3, -4, 0, 0], [0, 8, -8, 0, 26, -10], [-34, 3, 3, 6, 0, 0], [0, 6, -6, 0, -22, 6], [0, -22, 22, 0, 14, 12], [0, 8, -8, 0, -18, -18], [0, 1, 1, -6, 0, 0], [0, 10, -10, 0, -14, 0], [0, 0, 0, 0, -24, -5], [12, -13, -13, -7, 0, 0], [12, -10, -10, -4, 0, 0], [0, -2, 2, 0, 38, -4], [0, 8, -8, 0, 28, 4], [30, 15, 15, 13, 0, 0], [40, -5, -5, 2, 0, 0], [0, 12, -12, 0, -12, -8], [0, -1, 1, 0, -24, 13], [-30, -3, -3, 3, 0, 0], [0, -8, 8, 0, 6, 28], [0, -9, 9, 0, 20, 9], [0, 3, -3, 0, 6, -9], [-26, -12, -12, 0, 0, 0], [-12, 2, 2, 0, 0, 0], [0, -1, 1, 0, 40, -2], [26, -10, -10, 2, 0, 0], [6, 15, 15, 10, 0, 0], [0, -8, 8, 0, -18, 5], [-36, -4, -4, 2, 0, 0], [0, 14, -14, 0, 8, -12], [36, 10, 10, -2, 0, 0], [-14, -7, -7, 6, 0, 0], [0, 12, -12, 0, -2, -20], [12, -7, -7, 14, 0, 0], [0, -14, 14, 0, 10, 24], [0, -12, 12, 0, -8, 25], [16, 8, 8, 20, 0, 0], [2, -8, -8, 10, 0, 0], [0, -10, 10, 0, 4, -2], [-10, -6, -6, -24, 0, 0], [0, -16, 16, 0, 16, 0], [0, -4, 4, 0, -10, -4], [2, -2, -2, 16, 0, 0], [-8, 8, 8, 8, 0, 0], [8, 15, 15, 9, 0, 0], [0, -8, 8, 0, 30, -6], [0, -20, 20, 0, 16, 17], [-6, 10, 10, -8, 0, 0], [0, -3, 3, 0, -24, -17], [30, 5, 5, 7, 0, 0], [0, 1, -1, 0, 30, 9], [-16, -6, -6, -12, 0, 0], [0, -14, 14, 0, 10, -2], [6, -13, -13, -14, 0, 0], [30, 4, 4, 6, 0, 0], [-40, 9, 9, 15, 0, 0], [0, 0, 0, 0, -8, -17], [0, -3, 3, 0, 22, -3], [14, 6, 6, 16, 0, 0], [0, 12, -12, 0, 10, -18], [-2, -15, -15, -1, 0, 0], [-48, -6, -6, -2, 0, 0], [0, -2, 2, 0, 40, 12], [-18, -21, -21, -10, 0, 0], [0, -1, 1, 0, 22, 23], [-18, 8, 8, -24, 0, 0], [0, 7, -7, 0, -12, 5], [8, -14, -14, -4, 0, 0], [0, 12, -12, 0, 12, -28], [-2, 15, 15, -10, 0, 0], [18, 15, 15, -1, 0, 0], [0, -4, 4, 0, 0, -12], [0, 0, 0, 0, 30, -6], [34, -2, -2, -6, 0, 0], [0, 2, -2, 0, 4, 14], [10, -2, -2, 6, 0, 0], [0, 19, -19, 0, 0, -10], [-14, 7, 7, 21, 0, 0], [-18, -10, -10, -4, 0, 0], [0, 13, -13, 0, 10, -2], [0, 0, 0, 0, -46, -5], [0, 11, -11, 0, 12, -1], [0, -12, 12, 0, 14, -2], [0, 8, -8, 0, -36, -14], [28, -9, -9, 2, 0, 0], [0, 4, -4, 0, -2, 9], [0, 8, -8, 0, -2, -4], [2, 1, 1, -4, 0, 0], [24, 9, 9, 3, 0, 0], [-10, -14, -14, -6, 0, 0], [0, -14, 14, 0, 4, 26], [0, -6, 6, 0, -10, 6], [-8, -21, -21, -10, 0, 0], [0, 4, -4, 0, -16, -10], [0, -23, 23, 0, -8, 27], [-32, -8, -8, 0, 0, 0], [0, 6, -6, 0, 14, -16], [0, 7, -7, 0, -34, -17], [0, -14, 14, 0, 4, -8], [-10, -8, -8, 8, 0, 0], [-22, 3, 3, 13, 0, 0], [-18, 3, 3, 22, 0, 0], [0, 3, -3, 0, 18, -13], [-4, 1, 1, -23, 0, 0], [0, -10, 10, 0, -32, 0], [4, 19, 19, 10, 0, 0], [0, 6, -6, 0, 12, -3], [18, 7, 7, -7, 0, 0], [0, 0, 0, 0, -36, -4], [0, 24, -24, 0, -10, -14], [-20, 0, 0, -10, 0, 0], [18, 0, 0, -16, 0, 0], [12, 3, 3, -15, 0, 0], [0, 4, -4, 0, -8, -21], [0, 16, -16, 0, -18, -16], [-6, 5, 5, -20, 0, 0], [0, 8, -8, 0, -46, -6], [8, 13, 13, 10, 0, 0], [20, -14, -14, -30, 0, 0], [0, -2, 2, 0, 2, -15], [-46, -7, -7, -14, 0, 0], [0, 12, -12, 0, 32, -3], [0, -5, 5, 0, 28, -9], [0, 8, -8, 0, -2, 20], [-4, 1, 1, -15, 0, 0]]
