
# q-expansion of newform 1232.2.q.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 = 1232
weight = 2
poly_data = [4, 0, 2, 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], 1], [[0, 1, 0, 0], 1], [[0, 0, 1, 0], 2], [[0, 0, 0, 1], 2]]

hecke_ring_character_values = [[463, [1, 0, 0, 0]], [309, [1, 0, 0, 0]], [353, [0, 0, 1, 0]], [673, [1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [-1, 1, -1, 0], [0, -1, -2, -1], [0, 1, -1, -1], [1, 0, 1, 0], [-1, 0, 0, -2], [2, 4, 2, 0], [0, 1, 2, 1], [0, -3, 2, -3], [-3, 0, 0, 4], [-4, 0, -4, 0], [0, 1, -8, 1], [-4, 0, 0, -1], [0, 0, 0, 4], [0, -6, 2, -6], [2, 7, 2, 0], [-7, -1, -7, 0], [0, -2, 9, -2], [7, 3, 7, 0], [4, 0, 0, 5], [8, -1, 8, 0], [0, -3, 9, -3], [-2, 0, 0, 10], [0, -6, 4, -6], [-1, 0, 0, -2], [-9, -2, -9, 0], [0, 1, -12, 1], [0, -5, -4, -5], [-8, 6, -8, 0], [-11, 0, 0, 2], [-3, 0, 0, -9], [0, 1, 2, 1], [11, -4, 11, 0], [0, 0, 0, 0], [0, -4, 12, -4], [-3, 9, -3, 0], [-12, 4, -12, 0], [0, 9, 3, 9], [-1, 0, 0, -9], [0, 2, -7, 2], [-9, -7, -9, 0], [-2, 0, 0, -4], [0, 2, 10, 2], [-12, 7, -12, 0], [-9, 0, 0, 6], [10, 7, 10, 0], [-6, 0, 0, -1], [-10, 0, 0, 1], [-26, 2, -26, 0], [0, -8, 12, -8], [0, -1, 0, -1], [1, 0, 0, -15], [-8, 3, -8, 0], [12, 0, 0, 10], [0, 10, -11, 10], [7, 17, 7, 0], [-8, -4, -8, 0], [0, 11, 11, 11], [1, 2, 1, 0], [-4, 0, 0, 9], [-22, 1, -22, 0], [-8, 0, 0, -2], [0, 0, 0, -7], [4, -9, 4, 0], [0, 4, -15, 4], [0, 8, 20, 8], [0, -5, 17, -5], [-24, 0, 0, -3], [16, -1, 16, 0], [0, 0, 0, 0], [14, 8, 14, 0], [0, 3, -15, 3], [2, 9, 2, 0], [0, 12, -3, 12], [-1, 0, 0, -19], [0, -13, -12, -13], [10, -9, 10, 0], [0, 0, 22, 0], [0, -8, 7, -8], [10, 9, 10, 0], [12, 0, 0, 10], [12, 0, 0, -8], [-19, -1, -19, 0], [12, 0, 0, 10], [0, -13, 9, -13], [0, 16, 10, 16], [28, 0, 0, -4], [0, 4, -6, 4], [3, 0, 0, -8], [-10, 0, 0, -2], [0, 0, 34, 0], [-19, -5, -19, 0], [-4, 4, -4, 0], [-22, 0, 0, 2], [0, 10, -8, 10], [17, 0, 0, -15], [0, -8, -2, -8], [0, -20, 0, 0], [0, -9, 0, -9], [0, -10, -27, -10], [-8, 0, 0, -19], [-16, -6, -16, 0], [2, -5, 2, 0], [0, 0, -4, 0], [8, -13, 8, 0], [21, 8, 21, 0], [35, 0, 0, -7], [0, -21, -6, -21], [20, -16, 20, 0], [-2, 0, 0, 24], [0, -12, 26, -12], [-6, 16, -6, 0], [-25, 0, 0, -12], [12, 24, 12, 0], [-36, 0, 0, 9], [13, 20, 13, 0], [-3, 0, 0, 1], [-8, -27, -8, 0], [0, 13, 0, 13], [12, 0, 0, 0], [-2, 12, -2, 0], [-10, 0, 0, 11], [0, 8, -24, 8], [15, 1, 15, 0], [0, 5, -21, 5], [5, 0, 0, -22], [0, 17, 12, 17], [0, 6, 10, 6], [6, 0, 0, -30], [0, 0, -13, 0], [-10, 30, -10, 0], [2, 0, 0, 8], [0, 4, -40, 4], [14, 0, 0, -4], [0, -12, 14, -12], [6, 0, 0, 12], [-12, -24, -12, 0], [-2, -11, -2, 0], [-2, 0, 0, -4], [24, -16, 24, 0], [-4, 0, 0, 20], [0, 6, -21, 6], [-18, 18, -18, 0], [-14, 0, 0, 25], [-26, -3, -26, 0], [10, 0, 0, 6], [14, 0, 0, 0], [12, -16, 12, 0], [0, -9, -7, -9], [0, 15, -14, 15], [0, -16, -3, -16], [-19, 0, 0, 18], [-3, 0, 0, -9], [0, 7, 13, 7], [22, 8, 22, 0], [22, 0, 0, 6], [0, 10, -8, 10], [0, 18, -15, 18], [-18, 0, 0, -1], [-19, 4, -19, 0], [0, 2, -32, 2], [-10, 0, 0, 11], [8, 0, 0, 20], [0, -9, -35, -9], [-40, 0, -40, 0], [-10, -30, -10, 0], [-20, -13, -20, 0], [-16, -2, -16, 0], [36, 0, 0, -16], [0, 12, -26, 12], [10, 18, 10, 0], [48, 0, 0, -9], [0, 5, 19, 5], [-10, -32, -10, 0], [-2, -11, -2, 0], [-3, 0, 0, -6], [4, 0, 0, 5], [6, -36, 6, 0], [-8, 0, 0, 26], [0, -16, 2, -16], [0, 25, -12, 25], [-14, 0, 0, -14], [-40, 0, 0, 0], [0, -16, -12, -16], [0, 17, 0, 0], [-12, 25, -12, 0], [39, -4, 39, 0], [47, 3, 47, 0], [0, -3, -6, -3], [17, 7, 17, 0], [0, 8, 32, 8], [16, 0, 0, 15], [0, 3, -15, 3], [0, -5, 14, -5], [-36, 12, -36, 0], [-31, 0, -31, 0], [0, 32, 0, 0], [0, -37, 4, -37], [-40, 0, 0, -17], [0, -31, -3, -31], [3, 12, 3, 0], [7, 0, 0, 7], [0, -7, 22, -7], [-20, 27, -20, 0], [-45, 0, 0, 1], [16, 4, 16, 0], [0, -4, -20, -4], [0, 17, 34, 17], [-10, 0, 0, -3], [11, 5, 11, 0], [0, 11, -20, 11], [6, 0, 0, 12], [12, 0, 0, 0], [0, -22, -4, -22], [-3, 33, -3, 0], [0, 8, -31, 8], [-39, -17, -39, 0], [-14, -41, -14, 0], [0, -8, 8, -8], [18, 0, 0, 4], [0, 22, -15, 22], [45, 0, 0, -15], [0, 0, 31, 0], [0, -25, 6, -25], [-2, 0, 0, -18], [7, 0, 0, -4], [0, -2, -33, -2], [24, 11, 24, 0], [0, -3, -3, -3], [0, -2, 26, -2], [-12, -28, -12, 0], [14, -22, 14, 0], [9, 0, 0, 1], [-14, -24, -14, 0], [-33, 0, 0, -3], [-7, -1, -7, 0], [0, -4, -60, -4], [0, -8, -7, -8], [25, 0, 0, -9], [-12, 0, 0, 11], [8, -6, 8, 0], [0, -8, 24, -8], [16, -12, 16, 0], [0, -19, -12, -19], [8, 0, 0, -5], [0, 4, 18, 4], [2, 0, 0, 11], [-9, 31, -9, 0], [22, 1, 22, 0], [15, 0, 0, 27], [3, 0, 0, 16], [0, 24, -32, 24], [22, -27, 22, 0], [-4, 0, 0, -50], [63, 8, 63, 0], [0, 2, 10, 2], [18, -32, 18, 0], [-28, -8, -28, 0], [-16, 0, 0, -11], [0, -41, -10, -41], [-40, -8, -40, 0], [4, 0, 0, -16], [32, -14, 32, 0], [-7, 0, 0, 14], [1, 20, 1, 0], [0, -39, 1, -39], [-16, 0, 0, -36], [-5, 0, 0, 32], [-42, 0, 0, -3], [-32, -2, -32, 0], [0, 20, -2, 20], [4, -12, 4, 0], [-13, 20, -13, 0], [0, -10, -12, -10], [-2, 0, 0, -39], [0, 13, -4, 13], [0, -13, 35, -13], [11, 38, 11, 0], [0, 7, -60, 7], [0, 11, -66, 11], [-11, -49, -11, 0], [-10, 30, -10, 0], [51, 0, 0, -3], [-45, 0, 0, -6], [0, -10, 46, -10], [0, -2, -2, -2], [33, -34, 33, 0], [42, 0, 0, 24], [-2, -39, -2, 0], [8, 0, 0, -12], [-2, 8, -2, 0], [-33, 37, -33, 0], [0, -5, -54, -5], [-6, 0, 0, 2], [1, 0, 0, -2], [-31, 0, -31, 0], [0, 19, -14, 19], [-40, 0, 0, -24], [0, 2, -30, 2], [-14, 0, 0, -7], [0, -15, -58, -15], [0, -16, -17, -16], [-45, -5, -45, 0]]
