
# q-expansion of newform 3150.2.m.f, downloaded from the LMFDB on 23 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 3150
weight = 2
poly_data = [1, 0, 0, 0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[2801, [-1, 0, 0, 0]], [127, [0, 0, -1, 0]], [451, [1, 0, 0, 0]]]
aps_data = [[0, -1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 2, 2, 2], [2, 2, 2, 0], [-3, 2, -3, 0], [0, 5, 0, 5], [-2, 0, 2, 4], [-6, 2, 0, -2], [-6, -2, 0, 2], [-1, 0, 1, -10], [0, -2, -2, -2], [-5, -2, -5, 0], [-5, -2, -5, 0], [2, 0, -2, -4], [-8, 0, 0, 0], [8, -1, 0, 1], [1, 0, -1, 2], [0, -3, -8, -3], [6, 0, 6, 0], [0, 8, 2, 8], [4, 0, -4, -8], [-2, -8, 0, 8], [4, 0, -4, -6], [0, 4, -10, 4], [-2, -2, -2, 0], [-2, -10, -2, 0], [0, 10, 6, 10], [-4, 0, 4, 2], [-4, 0, 4, -4], [0, 0, 6, 0], [-6, 10, -6, 0], [0, 3, -4, 3], [-6, 8, 0, -8], [-2, -8, 0, 8], [-4, 0, 4, -2], [5, -14, 5, 0], [-3, 14, -3, 0], [-8, 0, 8, 0], [2, 8, 0, -8], [-4, -5, 0, 5], [0, -1, -12, -1], [-6, 0, -6, 0], [-6, 4, -6, 0], [0, 4, -6, 4], [12, -6, 0, 6], [2, -14, 2, 0], [10, -8, 10, 0], [0, 5, -12, 5], [4, 0, -4, -2], [-4, -15, 0, 15], [-2, -4, 0, 4], [0, 12, -10, 12], [3, -14, 3, 0], [-22, 0, 22, 0], [10, 8, 0, -8], [10, -2, 0, 2], [11, 0, -11, -10], [0, -5, 20, -5], [18, 0, 18, 0], [-14, 0, 14, 12], [16, 0, -16, 12], [0, 0, 16, 0], [14, 8, 14, 0], [-8, -6, -8, 0], [-16, 10, 0, -10], [10, 0, -10, 12], [12, 4, 12, 0], [0, 9, -4, 9], [3, 0, -3, 26], [8, 9, 0, -9], [-20, 0, 20, 0], [7, 6, 7, 0], [0, 4, 0, 4], [-1, 0, 1, 6], [-6, 8, 0, -8], [-24, 0, 24, 2], [0, 5, 24, 5], [0, 8, 26, 8], [20, 0, 0, 0], [14, 10, 0, -10], [0, 15, -8, 15], [-20, -6, -20, 0], [0, -14, -6, -14], [-16, 0, 16, 12], [28, 3, 0, -3], [-16, 0, 16, -12], [0, 8, 26, 8], [-14, 12, -14, 0], [4, 8, 4, 0], [-4, -4, 0, 4], [20, 0, -20, 0], [0, -12, -6, -12], [0, 0, 24, 0], [-19, 0, 19, 2], [24, -4, 0, 4], [0, 12, -10, 12], [-2, -40, -2, 0], [18, -6, 0, 6], [11, 0, -11, -18], [-12, 14, -12, 0], [-6, 0, 6, 20], [-8, 13, 0, -13], [-8, 12, 0, -12], [-14, 0, 14, -4], [14, 0, 14, 0], [13, 0, -13, -2], [-12, 5, 0, -5], [-18, 0, 0, 0], [6, 0, -6, 26], [-5, 2, -5, 0], [-26, 2, -26, 0], [0, -19, -8, -19], [-12, 12, 0, -12], [0, 23, 8, 23], [-14, 20, -14, 0], [3, -30, 3, 0], [-4, 0, 4, -2], [-2, 14, 0, -14], [-32, 3, 0, -3], [-6, 28, -6, 0], [-16, 12, -16, 0], [6, 0, -6, 14], [4, -15, 0, 15], [0, -6, -18, -6], [0, -8, 6, -8], [20, -18, 0, 18], [-6, 0, 6, 6], [8, 18, 8, 0], [0, -16, 20, -16], [4, 0, -4, -20], [28, 12, 0, -12], [17, 0, -17, -10], [0, 4, 10, 4], [0, -10, 2, -10], [2, 0, -2, 48], [-24, 0, 24, 16], [-16, 28, -16, 0], [20, -21, 0, 21], [-28, 3, 0, -3], [0, -10, 26, -10], [-26, 16, -26, 0], [8, -4, 8, 0], [0, -19, 8, -19], [24, 2, 0, -2], [-12, 22, -12, 0], [-3, -34, -3, 0], [0, 9, -28, 9], [4, 0, -4, 32], [-7, 0, 7, 18], [0, -6, -34, -6], [-29, -14, -29, 0], [19, -26, 19, 0], [-7, 0, 7, -6], [0, -7, 20, -7], [0, 8, 20, 8], [-18, 4, 0, -4], [-24, 0, 24, 10], [0, 8, -2, 8], [-10, -2, -10, 0], [22, 0, -22, -26], [24, 0, -24, 24], [0, -4, -30, -4], [20, 26, 20, 0], [23, 0, -23, 2], [-12, 8, 0, -8], [-4, 0, 4, -46], [0, 8, -32, 8], [8, 0, -8, 36], [22, 6, 0, -6], [-44, -7, 0, 7], [0, -5, 12, -5], [-18, 32, -18, 0], [0, -6, -26, -6], [-34, -8, 0, 8], [-8, 2, 0, -2], [0, 0, -18, 0], [-6, -42, -6, 0], [0, 1, 8, 1], [-6, 0, 6, 20], [0, 24, 8, 24], [-23, -22, -23, 0], [-9, 6, -9, 0], [-14, 0, 14, -44], [-48, -4, 0, 4], [9, 0, -9, -14], [14, 44, 14, 0], [0, 0, 4, 0], [0, 10, -44, 10], [0, 34, 0, 0], [-22, 0, 22, 6], [24, -24, 0, 24], [0, -8, 38, -8], [18, 10, 18, 0], [15, 0, -15, 34], [-8, -32, 0, 32], [17, -22, 17, 0], [-1, -46, -1, 0], [-11, 0, 11, -14], [-14, -24, 0, 24], [6, 12, 0, -12], [6, 0, -6, -18], [0, 38, -2, 38], [-46, -4, 0, 4], [14, -12, 14, 0], [0, -6, 46, -6], [16, 0, -16, -12], [40, -1, 0, 1], [-4, -23, 0, 23], [26, 0, -26, -4], [0, 40, -2, 40], [-4, -16, -4, 0], [32, 12, 32, 0], [-12, -26, 0, 26], [-10, -6, 0, 6], [16, 0, -16, 40], [0, 6, 42, 6], [34, 8, 34, 0], [0, 0, 0, -6], [-42, 0, 0, 0], [0, 20, 2, 20], [-20, 21, 0, -21], [-10, -24, -10, 0], [14, -8, 14, 0], [0, 0, 26, 0], [41, 0, -41, -14], [12, 17, 0, -17], [8, 0, -8, 16], [0, -24, 38, -24], [-7, -6, -7, 0], [0, 29, 24, 29], [24, -20, 0, 20], [0, -17, 8, -17], [-12, 48, -12, 0], [7, -50, 7, 0], [0, 8, 18, 8], [-6, 0, 6, 8], [-38, 24, 0, -24], [0, 26, -24, 26], [30, 0, -30, -2], [16, -3, 0, 3], [8, -56, 8, 0], [0, 30, 14, 30], [25, 0, -25, 18], [-8, 10, 0, -10], [6, 0, -6, -38], [0, 8, -34, 8], [0, -10, 0, -10], [12, 0, -12, 12], [-25, 0, 25, -18], [0, 2, -42, 2], [8, 36, 8, 0], [0, 26, -18, 26], [-30, 0, 30, -4], [-2, -24, 0, 24], [-10, -22, 0, 22], [-10, 0, 10, 52], [2, 16, 2, 0], [8, 0, -8, 54], [-34, 12, -34, 0], [-4, 60, -4, 0], [0, -21, -24, -21], [-10, -14, -10, 0], [-13, -10, -13, 0], [0, 39, 4, 39], [-66, -4, 0, 4], [0, 24, -22, 24], [21, 30, 21, 0], [-30, 0, 30, 32], [12, -27, 0, 27], [7, 0, -7, 2], [14, -16, 14, 0], [0, 4, -42, 4], [14, 0, -14, 24], [16, 40, 16, 0], [18, -46, 18, 0], [0, -36, -22, -36], [56, 20, 0, -20], [0, 32, 0, 32], [-3, 0, 3, 30], [-42, 0, 0, 0], [7, -58, 7, 0], [24, -15, 0, 15], [-14, 0, 14, -12], [0, -39, -8, -39], [0, 32, 0, 0], [22, -8, 22, 0], [0, 2, -54, 2], [70, 0, 0, 0], [0, -20, 10, -20], [-10, -16, -10, 0], [8, 0, -8, -58], [0, 16, 46, 16], [-25, 14, -25, 0], [-20, 40, 0, -40], [-14, 14, 0, -14], [-26, 0, 26, -16], [-2, 0, 0, 0], [20, 0, -20, -36], [8, -22, 8, 0], [-40, -18, -40, 0], [0, -4, -40, -4]]
