
# q-expansion of newform 975.2.bp.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 = 975
weight = 2
poly_data = [13, -38, 70, -74, 63, -34, 16, -4, 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, 1, 0, 0, 0, 0, 0, 0], 1], [[-159, 235, -426, 221, -172, 32, -15, -1], 37], [[-70, 298, -390, 256, -146, 22, -8, -3], 37], [[-181, 335, -427, 256, -146, 22, -8, -3], 37], [[-152, 294, -498, 373, -261, 89, -29, 3], 37], [[97, -266, 366, -378, 215, -114, 28, -8], 37], [[39, 38, -84, 276, -184, 159, -41, 17], 37]]

hecke_ring_character_values = [[326, [-1, 0, 0, 0, 0, 0, 0, 0]], [352, [-1, 0, 0, 0, 0, 0, 0, 0]], [301, [0, 0, 0, 0, 1, 0, 0, 0]]]
aps_data = [[0, 0, 1, 0, 0, -1, 0, -1], [1, 0, 0, 0, 0, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, -1, 0], [2, -2, 0, -2, 0, -1, 1, -2], [0, 0, 0, 0, 0, 2, 3, 0], [-1, 3, 3, -1, 2, -3, -2, 1], [2, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -5, 2, -1, 4, 2, 1], [3, 0, 0, 0, 1, 3, -4, 0], [-5, 0, 0, 0, -5, 2, 2, 0], [0, -2, -3, 2, -2, 3, 2, -1], [3, 0, 0, 0, -3, 6, -3, 0], [4, -4, 0, -4, 0, 0, 0, 0], [1, -1, -2, -1, 0, 1, 1, 0], [0, -2, -2, 2, -2, 4, 0, 2], [7, 0, 0, 0, 0, 0, -7, 0], [6, 0, 0, 0, -6, 4, -4, 0], [-2, 2, 2, 2, 0, 0, -2, 2], [-5, 0, 0, 0, 2, -5, 3, 0], [-2, 0, 0, 0, 0, 0, 0, 0], [1, -2, -2, 0, -1, 1, 2, -2], [-6, 6, 2, 6, 0, 1, -3, 4], [-7, 0, 0, 0, -1, -6, 1, 0], [-5, 10, 5, 0, 5, -9, -1, -3], [0, 0, 0, 0, -4, -4, 0, 0], [-8, 16, 8, 0, 8, -10, -6, 4], [5, 0, 0, 0, -11, 5, 6, 0], [-3, 5, -5, 1, 2, 1, 2, 1], [0, 0, 0, 0, 14, -7, 3, 0], [3, -4, 0, -2, -1, -1, 2, -4], [-1, 3, -1, -1, 2, 0, -1, 3], [8, 0, 0, 0, 6, -12, -8, 0], [-4, 5, 3, 3, 1, 0, -4, 5], [2, 0, 0, 0, 6, -2, 4, 0], [3, 0, 0, 0, -10, 10, -6, 0], [0, 0, 0, 0, 8, -8, 8, 0], [0, 8, 6, -8, 8, -12, -2, -2], [0, 6, 6, -6, 6, -6, -6, 6], [4, -8, -4, 0, -4, -3, 11, -18], [0, 0, 0, 0, 3, -3, 0, 0], [5, -8, -8, -2, -3, 8, 3, 2], [-3, 0, 0, 0, -3, -1, -1, 0], [2, -2, 0, -2, 0, -1, 1, -2], [-4, 0, 0, 0, 6, 0, -4, 0], [0, 0, 0, 0, 6, -3, 7, 0], [20, 0, 0, 0, -20, 9, -9, 0], [-2, -8, -12, 12, -10, 20, 2, 8], [9, 0, 0, 0, -5, -9, -14, 0], [6, -6, 0, -6, 0, -6, 6, -12], [1, 2, 6, -4, 3, -9, 0, -6], [2, 0, 0, 0, -7, 9, 7, 0], [1, 4, -4, -6, 5, -6, 5, -6], [-3, 0, -15, 6, -3, 12, 6, 3], [-4, 8, 4, 0, 4, 1, -9, 14], [4, -10, -10, 2, -6, 10, 6, -2], [4, 0, 0, 0, -4, 0, 0, 0], [12, 0, 0, 0, 9, -18, -12, 0], [-6, 7, 12, 5, 1, -1, -12, 12], [0, 0, 0, 0, 18, -9, 9, 0], [4, -1, -7, -7, 3, 0, 4, -1], [8, 0, 0, 0, 4, 8, -12, 0], [-2, -2, -4, 6, -4, 6, 2, 0], [0, 0, 0, 0, -2, 2, 0, 0], [5, -7, -10, -3, -2, 2, 10, -10], [-22, 0, 0, 0, 16, 6, 16, 0], [15, 0, 0, 0, -2, -2, 0, 0], [-6, 10, -10, 2, 4, 2, 4, 2], [-14, 0, 0, 0, 14, 1, -1, 0], [12, -12, -7, -12, 0, 1, 6, -5], [-7, 12, 14, 2, 5, -5, -14, 14], [-12, 0, 0, 0, 0, 20, 6, 0], [-9, 0, 0, 0, -4, 0, -9, 0], [-4, 0, 0, 0, -4, 9, 9, 0], [-8, 0, -16, 16, -8, 16, 8, 0], [1, 7, 14, -9, 8, -15, -7, 0], [-9, 0, 0, 0, 7, 2, 7, 0], [-2, 2, 11, 2, 0, -10, -1, -9], [-15, 0, 0, 0, -2, 17, -2, 0], [-4, 0, -2, 8, -4, 7, -1, 4], [-1, 0, 0, 0, 6, -1, -5, 0], [0, 14, 20, -14, 14, -21, -13, 6], [12, 0, 0, 0, 11, -22, -12, 0], [2, 0, 0, 0, 0, 3, -1, 0], [-8, 2, 4, 14, -6, 4, -2, 0], [-5, 0, -10, 10, -5, 10, 5, 0], [13, 0, 0, 0, -13, 15, -15, 0], [-2, 9, -5, -5, 7, 0, -2, 9], [-7, 0, 0, 0, -15, -7, 22, 0], [9, -18, 0, 0, -9, 9, 0, 0], [10, -10, 0, -10, 0, -5, 5, -10], [15, 0, 0, 0, 7, 8, -7, 0], [12, -24, -12, 0, -12, 15, 9, -6], [-3, 0, 0, 0, -1, -3, 4, 0], [-3, -4, 4, 10, -7, 10, -7, 10], [0, 2, 9, -2, 2, -3, -8, 7], [-3, 11, 0, -5, 8, -13, 5, -10], [-30, 0, 0, 0, 0, 13, 15, 0], [-5, 0, 0, 0, 16, 5, 21, 0], [0, 0, 0, 0, 2, -2, 0, 0], [0, 0, 17, 0, 0, 0, -17, 17], [2, -6, -6, 2, -4, 6, 4, -2], [2, -4, -2, 0, -2, 6, -2, 6], [-11, 0, 0, 0, 21, -21, 22, 0], [-17, 0, 0, 0, -2, 17, 15, 0], [-12, 12, 4, 12, 0, 2, -6, 8], [8, -7, 2, -9, 1, -3, 0, -2], [-9, 16, 0, 2, 7, -5, -2, 4], [0, 0, 0, 0, 18, -9, -8, 0], [-3, 0, 0, 0, -5, 0, -3, 0], [-11, 0, 0, 0, 11, 16, -16, 0], [-1, 15, 13, -13, 14, -28, 1, -15], [28, 0, 0, 0, 18, -28, -10, 0], [-2, 0, 0, 0, -12, 10, 12, 0], [18, -9, 9, -27, 9, -27, 9, -27], [14, 0, 0, 0, 0, 14, 0, 0], [-2, 0, -12, 4, -2, 9, 5, 2], [-4, 8, 4, 0, 4, -14, 6, -16], [10, -22, -22, 2, -12, 22, 12, -2], [-3, 0, 0, 0, 3, -11, 11, 0], [0, 0, 0, 0, 32, -16, -15, 0], [-10, 4, 8, 16, -6, 2, -4, 0], [19, -26, -12, -12, -7, 0, 19, -26], [-10, 0, 0, 0, -10, -17, -17, 0], [-6, 0, 0, 12, -6, 6, 0, 0], [4, 0, 0, 0, -5, 9, 5, 0], [2, -4, 4, 0, -2, 0, -2, 0], [-3, 0, 0, 0, -13, -13, 0, 0], [13, 0, 0, 0, 30, -13, 17, 0], [-16, 0, 0, 0, 16, -30, 30, 0], [-16, 16, 16, 16, 0, -8, -8, 0], [6, 0, 0, 0, 0, 39, -3, 0], [-24, 0, 0, 0, 0, -4, 12, 0], [-3, -4, -10, 10, -7, 14, 3, 4], [18, 0, 0, 0, 18, -23, -23, 0], [9, 4, 22, -22, 13, -26, -9, -4], [11, 0, 0, 0, -9, -2, -9, 0], [-8, 24, 24, -8, 16, -24, -16, 8], [11, 0, 7, -22, 11, -20, 2, -11], [19, 0, 0, 0, -19, 19, 0, 0], [0, 18, 22, -18, 18, -27, -13, 4], [12, 0, 0, 0, -2, 4, -12, 0], [16, -8, 16, -24, 8, -24, 0, -16], [-14, 0, 0, 0, 0, 36, 7, 0], [-1, -4, -6, 6, -5, 10, 1, 4], [-21, 0, 0, 0, 22, -21, -1, 0], [1, -9, 0, 7, -8, 15, -7, 14], [-8, 0, 0, 0, 18, 18, 0, 0], [-7, 12, 14, 2, 5, -5, -14, 14], [-2, 0, 0, 0, 15, -17, -15, 0], [11, -22, -11, 0, -11, 15, 7, -3], [-3, 0, 0, 0, -21, -21, 0, 0], [16, -32, -16, 0, -16, 26, 6, 4], [0, 0, 0, 0, -20, 10, 0, 0], [0, 2, 0, -2, 2, -4, 2, -4], [29, 0, 0, 0, -9, 18, -29, 0], [17, -15, -19, -19, 2, 0, 17, -15], [0, 0, 0, 0, -37, 37, 0, 0], [-7, 16, 18, -2, 9, -27, 0, -18], [0, -14, -2, 14, -14, 21, -5, 12], [10, -12, -12, -8, -2, 12, 2, 8], [-22, 0, 0, 0, -6, 22, 16, 0], [17, -38, -38, 4, -21, 38, 21, -4], [-2, 2, -13, 2, 0, 14, -1, 15], [-13, 4, -18, 22, -9, 27, 0, 18], [0, 0, 0, 0, -48, 24, 16, 0], [0, 0, 0, 0, 7, 0, 0, 0]]
