
# q-expansion of newform 592.2.i.f, downloaded from the LMFDB on 30 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 = 592
weight = 2
poly_data = [4, -12, 38, 2, 7, -1, 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], [[-84, 12, -38, 49, -7, 1], 254], [[1096, -84, 266, -89, 49, -7], 254], [[14, 760, 91, 140, -20, 21], 254], [[1044, -3306, -72, -609, 87, -85], 254]]

hecke_ring_character_values = [[223, [1, 0, 0, 0, 0, 0]], [149, [1, 0, 0, 0, 0, 0]], [113, [-1, 0, 0, 0, 1, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0], [0, -1, -1, 0, 0, 0], [1, 0, 0, 0, -1, 0], [0, -1, -1, 0, 0, -1], [0, 0, 0, -1, 0, 0], [-2, 1, 1, 0, 2, -1], [0, 0, 0, -2, 1, 2], [-2, 1, 1, 0, 2, 0], [2, 0, 2, 1, 0, 0], [-1, 0, -3, -1, 0, 0], [0, 0, 0, 1, 0, 0], [-1, -3, -2, 1, 3, -1], [-5, 2, 2, 0, 5, 0], [2, 0, -2, -2, 0, 0], [4, 0, 4, 1, 0, 0], [0, -1, 0, 1, -6, -1], [0, -3, 0, -1, -2, 1], [3, 2, 2, 0, -3, 0], [2, -3, -3, 0, -2, 2], [-4, 3, 3, 0, 4, -1], [8, 0, 2, 0, 0, 0], [-6, -3, -3, 0, 6, 0], [0, 3, 0, 2, 4, -2], [0, 4, 0, 0, -7, 0], [-3, 0, -5, 1, 0, 0], [7, 0, 1, 1, 0, 0], [10, 0, 2, -2, 0, 0], [4, 3, 3, 0, -4, -1], [0, -2, 0, 2, -5, -2], [0, 3, 0, -1, -6, 1], [0, -1, 0, 4, 4, -4], [0, 3, 0, -2, 0, 2], [3, 0, 1, 1, 0, 0], [0, -1, 0, 2, -12, -2], [-3, 0, -1, -1, 0, 0], [-4, -1, -1, 0, 4, 4], [0, 2, 0, 0, -5, 0], [0, 3, 0, 1, -4, -1], [-6, 5, 5, 0, 6, 1], [0, 0, 0, -2, 9, 2], [-4, 0, 0, -4, 0, 0], [-5, -2, -2, 0, 5, -2], [-10, 0, -2, 1, 0, 0], [1, 0, -1, 1, 0, 0], [0, 6, 0, 2, -1, -2], [-10, 0, -2, -2, 0, 0], [-14, 0, -2, 2, 0, 0], [2, 0, 2, -3, 0, 0], [-14, 1, 1, 0, 14, 2], [-3, -8, -8, 0, 3, 2], [11, 0, 9, 3, 0, 0], [0, -1, 0, 5, -4, -5], [2, 5, 5, 0, -2, 3], [2, 0, 2, 6, 0, 0], [0, -2, 0, -4, -1, 4], [14, 1, 1, 0, -14, -3], [4, 0, 2, -2, 0, 0], [0, -7, 0, 4, 6, -4], [1, 0, 0, 0, -1, -2], [0, -6, 0, 8, -1, -8], [4, -1, -1, 0, -4, -1], [-1, -2, -2, 0, 1, -2], [14, 0, -2, 2, 0, 0], [0, 11, 0, 1, 0, -1], [0, 10, 0, 0, 3, 0], [0, -4, 0, 6, -11, -6], [0, -3, 0, 3, 10, -3], [-3, -8, -8, 0, 3, -2], [-14, 0, 6, 1, 0, 0], [0, 6, 0, -2, -9, 2], [0, -10, 0, 2, 7, -2], [-20, 0, -8, 0, 0, 0], [-10, -3, -3, 0, 10, -3], [9, -4, -4, 0, -9, 4], [0, 7, 0, -3, 8, 3], [16, 7, 7, 0, -16, 3], [-13, 10, 10, 0, 13, -4], [13, 0, 3, -1, 0, 0], [-8, 0, 2, 0, 0, 0], [0, 6, 0, 4, -5, -4], [0, -5, 0, 1, 28, -1], [-7, 0, -5, 7, 0, 0], [-2, 1, 1, 0, 2, 4], [-5, 0, 9, 7, 0, 0], [2, -7, -7, 0, -2, -3], [-12, 0, 0, 0, 0, 0], [10, -7, -7, 0, -10, -1], [7, 6, 6, 0, -7, 4], [0, 7, 0, 3, -10, -3], [-20, 3, 3, 0, 20, 3], [16, 0, -4, 0, 0, 0], [0, 1, 0, -5, -14, 5], [-10, 0, -2, 7, 0, 0], [12, 0, 8, 5, 0, 0], [-10, 5, 5, 0, 10, 4], [0, 5, 0, 4, -10, -4], [0, 4, 0, -2, -19, 2], [18, -3, -3, 0, -18, -9], [28, -1, -1, 0, -28, -6], [1, 0, 11, 3, 0, 0], [16, 0, -4, -3, 0, 0], [0, 0, 0, -6, 13, 6], [18, 0, 2, -3, 0, 0], [-9, 0, -3, 1, 0, 0], [0, -1, 0, -4, -20, 4], [0, 3, 0, 1, 6, -1], [16, -1, -1, 0, -16, 0], [3, 0, -7, 7, 0, 0], [-10, 9, 9, 0, 10, -2], [0, 0, 0, -6, -19, 6], [0, 9, 0, -8, -10, 8], [0, -6, 0, -4, 3, 4], [0, 3, 0, 1, 2, -1], [-6, 0, -2, 2, 0, 0], [0, -5, 0, 4, 0, -4], [0, 0, 0, -8, -3, 8], [20, 0, 0, 7, 0, 0], [16, 7, 7, 0, -16, 7], [-19, 8, 8, 0, 19, -2], [-30, -11, -11, 0, 30, -3], [9, 4, 4, 0, -9, 4], [2, 9, 9, 0, -2, -9], [21, 0, -1, -9, 0, 0], [0, -5, 0, 1, -16, -1], [0, -7, 0, 4, 14, -4], [0, 5, 0, -1, 12, 1], [-4, 0, -2, -6, 0, 0], [0, -5, 0, -8, -16, 8], [-10, -3, -3, 0, 10, -3], [-10, 9, 9, 0, 10, 7], [10, 0, -2, -9, 0, 0], [-6, -7, -7, 0, 6, 2], [14, 0, -6, -2, 0, 0], [0, -10, 0, -4, -9, 4], [0, -2, 0, -2, -9, 2], [16, 0, 10, 0, 0, 0], [-27, -4, -4, 0, 27, 4], [-4, 0, 4, -8, 0, 0], [0, -7, 0, 11, 8, -11], [32, -1, -1, 0, -32, -5], [2, 17, 17, 0, -2, 1], [-26, -3, -3, 0, 26, -3], [0, 7, 0, 2, -24, -2], [-38, -11, -11, 0, 38, -3], [0, -5, 0, -1, 22, 1], [0, -5, 0, 8, 4, -8], [0, -4, 0, 4, 13, -4], [23, 0, 5, -3, 0, 0], [2, 0, -18, 6, 0, 0], [0, 9, 0, 3, 30, -3], [-29, 0, -15, -3, 0, 0], [11, 6, 6, 0, -11, 2], [0, 15, 15, 0, 0, 8], [10, 0, -6, 5, 0, 0], [18, -7, -7, 0, -18, 6], [6, 0, 6, -13, 0, 0], [10, 0, -2, -3, 0, 0], [-1, -18, -18, 0, 1, -2], [0, -14, 0, 6, 35, -6], [0, -12, 0, -8, 1, 8], [0, -7, 0, -1, 10, 1], [0, -7, 0, -5, -28, 5], [18, 13, 13, 0, -18, 5], [0, 3, 0, -8, -8, 8], [0, 3, 0, -5, -38, 5], [0, 17, 0, -6, -22, 6], [6, 0, 22, 2, 0, 0], [0, 3, 0, 11, -10, -11]]
