
# q-expansion of newform 825.2.a.m, downloaded from the LMFDB on 19 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 = [-2, -6, -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], 1], [[0, 1, 0], 1], [[-4, -2, 1], 1]]

hecke_ring_character_values = None
aps_data = [[1, -1, 0], [1, 0, 0], [0, 0, 0], [-1, -1, -1], [1, 0, 0], [-2, 0, -1], [1, -1, -2], [-1, 1, -1], [-1, 2, -1], [0, 2, 0], [6, 0, 1], [1, 0, 3], [1, 1, 0], [-4, -2, 3], [3, 0, -1], [2, 4, 4], [-1, -2, 1], [0, -4, -1], [-4, 6, 1], [9, 0, 1], [2, 2, -2], [1, 1, -2], [-4, 6, 0], [2, -4, 0], [-11, 6, 2], [-1, 3, 2], [4, 4, 0], [-2, 0, 4], [0, 2, -1], [6, 0, 0], [-3, -1, 4], [4, -6, -4], [-6, -6, -4], [-2, 4, 6], [-3, 1, 4], [8, 6, 5], [-4, -2, -1], [-6, 4, -1], [2, 2, 4], [15, -1, -2], [13, 0, -1], [-1, -8, -6], [5, -4, 1], [-6, -2, -1], [5, -1, -4], [-2, -4, -5], [-8, 6, 1], [2, -4, 5], [-6, -6, -4], [-3, 6, 2], [-11, 1, -6], [10, -8, 0], [-6, 4, 3], [-4, 0, -2], [6, 4, 10], [-2, -4, -4], [-4, -6, 4], [15, -3, -4], [-12, 4, 5], [-5, 1, 0], [-15, -5, -1], [27, 1, 2], [0, -10, -7], [-4, 8, -6], [1, -10, -4], [-8, 6, 4], [26, 2, 0], [-16, -4, 1], [16, 6, -2], [-18, -2, -2], [16, -2, 0], [20, 6, 10], [0, -2, -9], [0, 6, 5], [10, -12, -3], [-24, 4, 2], [4, -8, -6], [-4, 6, -1], [-16, 6, 0], [-16, 4, 1], [3, -4, -9], [-9, -2, -13], [4, 6, 0], [4, -2, 11], [21, -1, 1], [-1, -2, 7], [-14, -6, -12], [-2, 6, 2], [0, 2, -4], [6, -8, -6], [16, 4, -2], [0, 10, -2], [20, 4, -7], [2, -2, 6], [-8, 12, 5], [-2, -8, 0], [6, 4, 6], [2, 2, -6], [-19, 9, -3], [6, -6, -3], [23, -9, 2], [-8, 6, 10], [-4, 0, -2], [5, 3, 2], [0, 10, 9], [-3, 0, -2], [7, -10, -5], [4, -2, 12], [-3, 2, -9], [6, 0, 5], [-6, -4, -2], [-2, -4, -4], [-4, -6, -14], [10, -6, 1], [10, 6, -1], [2, 6, 14], [-4, -4, -8], [5, -10, 7], [4, -6, -10], [0, -2, -2], [1, -10, -7], [-2, 0, -12], [20, -2, -4], [-5, 8, 9], [-6, -10, -12], [-21, 9, 8], [-7, -6, 2], [0, 0, 14], [-6, 10, 11], [-22, 18, 6], [11, -3, 8], [-12, 6, 0], [8, 4, -4], [-12, -6, -11], [-12, -6, 0], [-24, 2, -3], [-6, -14, -2], [33, 5, 5], [-20, -4, 8], [-23, 11, 0], [-17, 1, -1], [-4, 10, -10], [-12, -2, -3], [36, 4, 6], [-14, 8, -4], [8, -4, -6], [-36, 8, -1], [1, 19, 6], [-12, 14, -2], [24, 8, 12], [6, 4, 5], [12, 4, -14], [6, -10, -1], [-28, 18, 4], [18, 14, 12], [13, -8, 7], [-15, -13, -9], [-2, 6, 4], [-34, 2, -1], [-7, -13, 4], [1, 4, -1], [11, -9, 0], [26, -8, 6], [17, 6, 9], [-28, -4, 2], [37, -10, -1], [32, -2, -3], [-30, 0, -4]]
