
# q-expansion of newform 231.2.j.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 = 231
weight = 2
poly_data = [1, 3, 5, 3, 4, -3, 5, -3, 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], [[-7, -12, -7, -4, 0, -3, 2, -1], 8], [[9, 6, 19, -16, 20, -7, 0, 1], 8], [[-1, -10, 13, -16, 12, -9, 4, -1], 8], [[-5, -8, -13, -4, 8, -17, 10, -3], 8], [[-1, 6, 1, 16, -20, 23, -12, 3], 8], [[-7, -12, -11, -28, 32, -35, 18, -5], 8]]

hecke_ring_character_values = [[155, [1, 0, 0, 0, 0, 0, 0, 0]], [199, [1, 0, 0, 0, 0, 0, 0, 0]], [211, [0, 0, 0, 0, 1, 0, 0, 0]]]
aps_data = [[0, 0, 0, -1, 0, 1, 1, 0], [1, 0, 0, -1, 1, 0, 0, 1], [0, 2, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, -1, 0, 0, 0], [-2, 0, 0, 0, 2, 0, 0, 1], [0, 0, 0, 0, 0, -2, -2, 0], [0, 0, 4, -2, 2, 0, 0, 4], [2, 2, 2, -2, 4, -2, -2, 4], [-3, 0, 0, 3, 0, -4, 0, -3], [-1, 0, 4, 0, 1, -4, 2, -1], [2, 4, 0, 4, 2, -2, -2, 0], [-1, 0, -4, 0, -4, 4, 4, -1], [2, -6, -2, -2, -4, 6, 2, -4], [0, 0, 0, -5, 0, 6, 0, 5], [-2, 2, 4, 2, 4, -2, -4, 4], [-5, 2, 0, 2, -5, 0, 0, 0], [2, 0, -2, 0, -4, 2, 0, 2], [0, -4, -2, 2, -2, 0, 4, 0], [-8, 0, 0, 3, 0, -2, 0, -3], [0, -4, -4, 3, -3, 0, 4, -9], [0, 0, -2, 0, -2, 2, 0, 0], [1, 2, 0, 4, 1, 0, 0, 0], [0, 2, -6, 2, -2, 0, -2, 0], [-2, -6, -6, 4, 0, 2, 0, -4], [6, 0, 0, -8, 6, -4, -4, 0], [4, -2, 0, -6, 4, 0, 0, 0], [6, 0, 0, 0, 12, 0, -4, 6], [5, 8, 8, -5, 4, -8, -8, 4], [-3, 8, 8, -1, 0, -4, 0, 1], [2, 2, 2, -2, 5, -2, -2, 5], [0, 2, 6, -11, 11, 0, -2, -2], [-2, -2, -2, 6, 0, -6, 0, -6], [0, 2, -2, 5, -5, 0, -2, -14], [8, 0, 0, 0, 6, 0, -8, 8], [0, 4, 8, 4, -4, 0, -4, 10], [-1, -12, -10, 1, 2, 12, 10, 2], [4, -6, -4, -4, 4, 6, 4, 4], [1, -8, 0, -3, 1, 2, 2, 0], [-4, 0, 0, -2, -4, 4, 4, 0], [-12, 6, 2, 12, -10, -6, -2, -10], [11, 0, 0, -11, 8, 0, 0, 8], [0, 12, 2, -4, 4, 0, -12, 8], [7, 0, -4, 0, 1, 4, -12, 7], [0, 0, 4, 5, -5, 0, 0, -5], [1, -2, -2, -11, 0, 8, 0, 11], [-10, 14, 14, 2, 0, -12, 0, -2], [11, -12, 0, -1, 11, 2, 2, 0], [-10, 10, 4, 10, -6, -10, -4, -6], [-2, 0, 6, 0, 16, -6, -6, -2], [10, -10, 0, -8, 10, 4, 4, 0], [-8, -8, 0, 2, -8, 0, 0, 0], [-2, -12, -12, 2, -11, 12, 12, -11], [2, -12, -12, 4, 0, 6, 0, -4], [14, 8, 0, 4, 14, -4, -4, 0], [2, 0, 2, 0, 0, -2, 2, 2], [6, -4, -4, 11, 0, 0, 0, -11], [0, 6, -8, 8, -8, 0, -6, 6], [12, 0, 12, 0, 14, -12, -4, 12], [15, 8, 0, 1, 15, -8, -8, 0], [0, 4, 6, -11, 11, 0, -4, 13], [-20, 0, 2, 20, -14, 0, -2, -14], [-8, 0, -2, 0, -16, 2, -4, -8], [-4, -6, -6, -8, 0, 10, 0, 8], [14, 0, 8, -14, 6, 0, -8, 6], [0, 6, 10, -2, 2, 0, -6, 6], [15, -6, 0, -14, 15, 8, 8, 0], [-9, 2, 2, -13, 0, -4, 0, 13], [3, 0, 20, 0, 8, -20, -8, 3], [0, 0, -8, -7, 7, 0, 0, -7], [-12, -2, -8, 12, -6, 2, 8, -6], [-8, 0, 0, 4, 0, 8, 0, -4], [7, 0, -8, 0, -16, 8, 8, 7], [-18, 0, -14, 0, -8, 14, 14, -18], [6, 4, 4, 8, 0, -16, 0, -8], [0, 2, -10, 15, -15, 0, -2, -18], [-2, -6, 0, 6, -2, 0, 0, 0], [19, 0, -2, 0, 14, 2, -2, 19], [22, 0, 0, -6, 0, 4, 0, 6], [0, 2, 10, -7, 7, 0, -2, -6], [2, -10, 0, -6, 2, 4, 4, 0], [-10, -8, -8, 8, 0, -2, 0, -8], [-3, 4, -8, 3, -4, -4, 8, -4], [5, -16, 0, -9, 5, 8, 8, 0], [0, 0, 6, 0, 14, -6, -8, 0], [-2, 2, 2, 20, 0, -18, 0, -20], [-6, 4, -8, 6, -11, -4, 8, -11], [-13, 4, 0, 15, -13, -2, -2, 0], [0, -4, -4, 3, -3, 0, 4, -32], [-24, 2, 2, 18, 0, -8, 0, -18], [-4, 0, 0, -7, 0, -2, 0, 7], [0, 18, 4, -8, 8, 0, -18, 24], [0, 2, 12, -6, 6, 0, -2, 6], [-20, 14, 2, 20, -7, -14, -2, -7], [-8, 0, -4, 0, 8, 4, 4, -8], [-11, 0, 12, 0, -19, -12, 10, -11], [0, 14, 14, 0, 14, -14, -14, 14], [4, 22, 8, -4, 12, -22, -8, 12], [18, 0, 10, 0, 18, -10, -8, 18], [0, 6, -12, -2, 2, 0, -6, 10], [-1, 0, 0, 9, -1, -4, -4, 0], [-12, -12, -8, 12, -8, 12, 8, -8], [9, 0, 8, 0, 3, -8, -2, 9], [-14, 4, 0, 12, -14, -10, -10, 0], [-18, 4, 8, 18, -14, -4, -8, -14], [-1, 2, 2, -17, 0, 8, 0, 17], [0, -2, -18, 10, -10, 0, 2, -22], [12, 0, -8, 0, -10, 8, -6, 12], [-8, 12, 12, 10, 0, -16, 0, -10], [0, 0, 16, 1, -1, 0, 0, -12], [-14, 0, 2, 0, -12, -2, 26, -14], [10, -6, 0, -16, 10, 8, 8, 0], [-12, -16, -24, 12, -25, 16, 24, -25], [-2, 8, 8, 15, 0, -22, 0, -15], [34, -8, 4, -34, 30, 8, -4, 30], [5, 0, 18, 0, -2, -18, -10, 5], [2, 26, 26, -2, 19, -26, -26, 19], [0, 4, 12, -10, 10, 0, -4, 16], [-8, -18, 0, 4, -8, 2, 2, 0], [11, 0, 2, 0, 10, -2, -6, 11], [28, -4, -4, -24, 0, 8, 0, 24], [16, 14, 14, -24, 0, -2, 0, 24], [3, 8, 0, 16, 3, -4, -4, 0], [0, -6, 16, -6, 6, 0, 6, 14], [1, 0, 0, 3, 0, -12, 0, -3], [-32, 10, 0, 22, -32, 12, 12, 0], [17, -8, -6, -17, 22, 8, 6, 22], [0, 8, 16, -21, 21, 0, -8, 17], [-8, 0, 6, 0, -24, -6, -2, -8], [-4, -4, -4, -12, 0, 4, 0, 12], [-20, 0, -10, 0, -2, 10, 8, -20], [-15, 22, 0, -12, -15, -12, -12, 0], [0, 4, 0, 7, -7, 0, -4, -33], [17, -4, 14, -17, 26, 4, -14, 26], [15, -8, 0, 4, 15, -16, -16, 0], [22, -4, 0, -20, 22, 2, 2, 0], [-14, 2, 2, 14, 0, 4, 0, -14], [-8, -12, -12, 8, 8, 12, 12, 8], [0, -22, -14, 4, -4, 0, 22, -24], [0, -8, -6, 16, -16, 0, 8, -20], [0, -22, -22, 11, -11, 0, 22, -18], [-4, 2, 2, 4, 4, -2, -2, 4], [4, 0, 12, 0, 14, -12, 8, 4], [-5, 2, 0, 16, -5, 4, 4, 0], [4, 24, 0, 24, 4, -36, -36, 0], [-6, 0, -24, 0, -26, 24, 10, -6], [32, 0, -2, -32, 10, 0, 2, 10], [0, 12, -2, 16, -16, 0, -12, 14], [-18, 22, 22, -6, 0, -18, 0, 6], [2, -14, -14, -22, 0, 30, 0, 22], [0, 12, 0, -24, 24, 0, -12, 20], [8, -24, -8, -8, 3, 24, 8, 3], [-22, 2, 2, -2, 0, 12, 0, 2], [0, 0, -12, 0, -24, 0, 12, -24], [16, 0, 8, 0, 2, -8, 6, 16], [0, 0, 6, -9, 9, 0, 0, 5], [-4, 24, 0, -4, -4, -24, -24, 0], [0, -8, -26, 11, -11, 0, 8, -23], [0, -10, 4, -16, 16, 0, 10, -16], [8, 16, 0, -4, 8, 0, 0, 0], [0, -4, -8, 18, -18, 0, 4, -6], [-24, 16, 16, 8, 0, -12, 0, -8], [-7, 0, -14, 0, -22, 14, -14, -7], [-23, 10, 10, 25, 0, -24, 0, -25], [18, -22, -8, -18, 36, 22, 8, 36], [5, -8, 0, 9, 5, 10, 10, 0], [2, 0, -10, 0, -10, 10, -16, 2], [12, -4, -4, -8, 0, 16, 0, 8], [16, 0, 14, 0, 8, -14, -2, 16]]
