
# q-expansion of newform 825.2.c.f, downloaded from the LMFDB on 16 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 = 825
weight = 2
poly_data = [8, -20, 25, -4, 0, 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, 0, 0], 1], [[323, 0, -4, 10, 25, 2], 121], [[-223, 0, 14, -35, -27, -7], 121], [[258, -605, 50, -4, -10, -25], 242], [[574, -1331, 372, 38, -26, -65], 242], [[-774, 1815, -392, 12, 30, 75], 242]]

hecke_ring_character_values = [[551, [1, 0, 0, 0, 0, 0]], [727, [-1, 0, 0, 0, 0, 0]], [376, [1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 1, 1, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 1, 1], [1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -1], [0, 0, 0, 1, 1, 2], [1, 1, -1, 0, 0, 0], [0, 0, 0, 1, 2, -1], [0, 0, -2, 0, 0, 0], [6, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, -3], [1, 0, 1, 0, 0, 0], [0, 0, 0, 4, -2, 3], [0, 0, 0, 3, 0, 1], [0, 0, 0, -2, 4, 4], [1, -1, 2, 0, 0, 0], [0, -1, -4, 0, 0, 0], [0, 0, 0, -4, -6, -1], [9, 1, 0, 0, 0, 0], [0, 0, 0, -2, 2, -2], [-1, 2, -1, 0, 0, 0], [0, 0, 0, 4, 6, 0], [-2, 0, 4, 0, 0, 0], [0, 0, 0, -11, -6, -2], [-1, 2, 3, 0, 0, 0], [0, 0, 0, -4, 4, 0], [0, 0, 0, -2, 0, -4], [0, 1, -2, 0, 0, 0], [0, 0, 0, -6, 0, 0], [0, 0, 0, -3, 1, -4], [4, -4, -6, 0, 0, 0], [0, 0, 0, -6, 6, 4], [2, -6, -4, 0, 0, 0], [3, -4, -1, 0, 0, 0], [8, 5, 6, 0, 0, 0], [0, 0, 0, -4, 2, 1], [0, 0, 0, 6, 4, -1], [0, 0, 0, 2, -2, -4], [0, 0, 0, -15, -1, -2], [-13, 1, 0, 0, 0, 0], [-1, -6, -8, 0, 0, 0], [5, 1, -4, 0, 0, 0], [0, 0, 0, 6, -2, -1], [0, 0, 0, 5, 1, 4], [2, 5, 4, 0, 0, 0], [-8, 1, 6, 0, 0, 0], [0, 0, 0, -2, -4, 5], [0, 0, 0, -6, 6, 4], [3, -2, -6, 0, 0, 0], [0, 0, 0, 11, 1, -6], [-10, 0, 8, 0, 0, 0], [-6, 3, 4, 0, 0, 0], [-4, -2, 0, 0, 0, 0], [0, 0, 0, 6, -4, -10], [0, 0, 0, 2, -4, -4], [4, -4, 6, 0, 0, 0], [15, -4, -3, 0, 0, 0], [0, 0, 0, -12, -4, -5], [-5, 0, 1, 0, 0, 0], [0, 0, 0, 15, -5, -1], [0, 0, 0, -27, 1, 2], [0, 0, 0, 0, 10, 7], [-4, -6, 8, 0, 0, 0], [0, 0, 0, -1, -10, -4], [0, 0, 0, -8, -6, -4], [26, 0, 2, 0, 0, 0], [0, 0, 0, -16, 4, -1], [0, 0, 0, 16, -6, 2], [18, 2, 2, 0, 0, 0], [0, 0, 0, -16, -2, 0], [-20, -10, -6, 0, 0, 0], [0, 0, 0, 0, 2, 9], [0, 0, 0, 0, 6, 5], [-10, 3, 12, 0, 0, 0], [0, 0, 0, 24, 4, 2], [-4, 6, 8, 0, 0, 0], [0, 0, 0, -4, -6, 1], [-16, 0, 6, 0, 0, 0], [16, -1, -4, 0, 0, 0], [-3, 9, 4, 0, 0, 0], [-9, -13, -2, 0, 0, 0], [4, 0, 6, 0, 0, 0], [0, 0, 0, -4, -2, 11], [-21, -1, 1, 0, 0, 0], [0, 0, 0, 1, -2, 7], [14, 12, 6, 0, 0, 0], [0, 0, 0, -2, -6, -2], [0, -4, 2, 0, 0, 0], [0, 0, 0, -6, -8, -6], [0, 0, 0, 16, -4, 2], [0, 2, -10, 0, 0, 0], [0, 0, 0, 20, -4, 7], [2, 6, -2, 0, 0, 0], [8, -5, -12, 0, 0, 0], [0, 0, 0, 2, -8, 0], [-6, -6, -4, 0, 0, 0], [2, -6, 2, 0, 0, 0], [0, 0, 0, 19, 9, -3], [6, -3, -6, 0, 0, 0], [0, 0, 0, 23, 9, -2], [0, 0, 0, -8, -6, -10], [0, 0, 0, 4, 0, -2], [-5, -2, -3, 0, 0, 0], [0, 9, 10, 0, 0, 0], [0, 0, 0, -3, 0, 2], [0, 0, 0, 7, 10, 5], [0, 0, 0, -4, -2, 12], [3, 9, -2, 0, 0, 0], [6, 5, 0, 0, 0, 0], [0, 0, 0, -6, 4, 2], [0, 0, 0, 2, -4, -4], [0, 0, 0, -4, 6, 14], [-10, -1, 6, 0, 0, 0], [10, -1, 6, 0, 0, 0], [2, 14, 6, 0, 0, 0], [0, 0, 0, 4, -4, -8], [0, 0, 0, 5, 10, -7], [0, 0, 0, -4, -6, -10], [0, 2, 2, 0, 0, 0], [1, -7, -10, 0, 0, 0], [0, 0, 0, 2, 0, -12], [0, 0, 0, 20, 2, 4], [0, 0, 0, 5, 8, 9], [-6, -12, -10, 0, 0, 0], [-21, 8, 9, 0, 0, 0], [7, -2, 6, 0, 0, 0], [0, -14, 0, 0, 0, 0], [0, 0, 0, -6, -10, -11], [0, 0, 0, 22, 18, 6], [-11, -8, 3, 0, 0, 0], [0, 0, 0, 12, 6, 0], [8, -4, 4, 0, 0, 0], [0, 0, 0, -12, 6, 11], [-12, 0, -6, 0, 0, 0], [24, 3, -2, 0, 0, 0], [0, 0, 0, 6, -14, -2], [0, 0, 0, 33, -5, -5], [0, 0, 0, -20, 4, -8], [23, 0, -11, 0, 0, 0], [-17, -1, 1, 0, 0, 0], [-4, -10, 10, 0, 0, 0], [0, 0, 0, 12, -2, -3], [0, 0, 0, 36, -4, -6], [14, 4, -8, 0, 0, 0], [-8, 6, 4, 0, 0, 0], [0, 0, 0, 36, 8, -1], [0, 0, 0, 1, -19, -6], [12, 2, -14, 0, 0, 0], [0, 0, 0, -24, 8, 12], [0, 0, 0, 6, -4, -5], [12, -14, 4, 0, 0, 0], [0, 0, 0, -6, -10, -1], [0, 0, 0, -28, -18, -4], [0, 0, 0, 18, -14, -12], [13, 7, -8, 0, 0, 0], [15, 9, 13, 0, 0, 0], [2, -4, -6, 0, 0, 0], [0, 0, 0, -34, -2, 1], [-7, 4, -13, 0, 0, 0], [0, 0, 0, 1, -4, 1], [0, 0, 0, -11, -9, 0], [0, 0, 0, 26, 8, -6], [17, 9, 6, 0, 0, 0], [0, 0, 0, -28, 4, -2], [0, 0, 0, -37, -10, -1], [32, -3, -2, 0, 0, 0], [0, 0, 0, -30, 0, 4]]
