
# q-expansion of newform 825.2.k.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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = [1, 0, 0, 0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = [[551, [-1, 0, 0, 0]], [727, [0, 0, 1, 0]], [376, [1, 0, 0, 0]]]
aps_data = [[1, 0, -1, -1], [1, -1, 0, -1], [0, 0, 0, 0], [2, -2, 2, 0], [0, 0, -1, 0], [2, 0, -2, 0], [0, 0, 0, 4], [0, 2, 0, 2], [1, 6, 1, 0], [-2, -2, 0, 2], [-6, -2, 0, 2], [3, 4, 3, 0], [0, 4, -2, 4], [4, 0, -4, -6], [3, 0, -3, 6], [-1, 0, -1, 0], [-4, 0, 0, 0], [2, 6, 0, -6], [5, -2, 5, 0], [0, 6, 6, 6], [0, 0, 0, 0], [0, -2, 8, -2], [2, -10, 2, 0], [-4, 4, 0, -4], [-5, 8, -5, 0], [0, -2, -10, -2], [5, 0, -5, 2], [4, 0, -4, -6], [0, 2, 6, 2], [3, -4, 3, 0], [0, 2, 0, 0], [0, -2, 4, -2], [-3, 0, 3, 4], [0, 4, -12, 4], [-10, 0, 0, 0], [-4, -10, 0, 10], [1, -12, 1, 0], [-5, 0, 5, 2], [-2, 0, 2, -18], [0, 12, 0, 0], [10, -10, 0, 10], [0, -4, 0, 4], [0, -8, 0, -8], [-8, 0, 8, 0], [0, 0, 0, 20], [0, 6, -6, 6], [4, 6, 0, -6], [1, 0, -1, 22], [-6, 0, 6, -6], [0, 4, -4, 4], [-14, 8, -14, 0], [-4, 14, 0, -14], [-14, -10, 0, 10], [0, 10, -2, 10], [-1, 0, 1, 4], [6, 6, 6, 0], [-14, 0, 0, 0], [0, -6, 0, 6], [10, -12, 10, 0], [0, -16, 6, -16], [8, 0, -8, 10], [-2, -8, -2, 0], [2, -14, 2, 0], [0, -10, 14, -10], [-7, 0, 7, -8], [9, 0, -9, -16], [2, 6, 0, -6], [8, -4, 8, 0], [8, 0, -8, -14], [0, 0, 30, 0], [-5, 4, -5, 0], [-8, 10, 0, -10], [11, 14, 11, 0], [-12, 0, 12, 12], [0, 10, -14, 10], [-1, -10, -1, 0], [-6, 8, 0, -8], [-9, -4, -9, 0], [0, -4, 30, -4], [0, -6, 10, -6], [-10, -18, 0, 18], [-4, 0, 0, 0], [0, -4, -12, -4], [-11, 0, 11, 16], [0, -8, -4, -8], [25, 6, 25, 0], [-20, -12, 0, 12], [-2, -12, -2, 0], [0, -6, -22, -6], [1, 0, -1, -18], [-1, 0, 1, 2], [-16, 10, 0, -10], [1, 18, 1, 0], [0, 0, -20, 0], [0, 2, -22, 2], [-8, -10, -8, 0], [30, -4, 0, 4], [0, 4, -22, 4], [6, 0, -6, 30], [-22, -8, 0, 8], [-12, -14, -12, 0], [8, 0, -8, -24], [0, -14, 0, 0], [-34, -4, 0, 4], [24, -4, 0, 4], [-17, 0, -17, 0], [-1, 0, 1, 2], [-20, -20, -20, 0], [22, 18, 0, -18], [-2, 18, 0, -18], [12, -14, 12, 0], [16, 0, -16, 0], [-27, 0, 27, -4], [0, 8, 12, 8], [0, 12, 0, -12], [0, 16, -10, 16], [5, 0, -5, 6], [1, 0, -1, -18], [-3, 8, -3, 0], [-8, 12, 0, -12], [14, 0, 0, 0], [28, 0, -28, -8], [-14, 0, 14, -8], [5, 18, 5, 0], [20, 0, 0, 0], [0, -8, -10, -8], [0, 8, -18, 8], [14, 26, 0, -26], [1, 30, 1, 0], [16, 0, -16, -28], [0, -16, -8, -16], [2, -34, 2, 0], [-24, 16, 0, -16], [-3, -4, -3, 0], [0, 6, 10, 6], [0, 16, -30, 16], [-5, 8, -5, 0], [30, 2, 30, 0], [-9, 0, 9, -24], [-30, -10, 0, 10], [-36, 4, 0, -4], [0, -4, -6, -4], [17, 0, -17, 2], [4, 0, -4, 34], [0, -12, -6, -12], [-16, 4, 0, -4], [-4, 0, 4, -20], [8, 0, -8, 12], [0, 0, 36, 0], [3, 22, 3, 0], [16, 32, 16, 0], [0, -12, -4, -12], [5, 0, -5, -2], [14, 0, -14, -6], [-5, 34, -5, 0], [0, 0, -56, 0], [0, 16, 8, 16], [18, 0, 0, 0], [-6, -24, -6, 0], [0, -22, 6, -22], [13, 0, -13, 6], [14, 28, 14, 0], [-20, 26, -20, 0], [0, 0, -20, 0], [7, 0, -7, 44], [-11, 18, -11, 0], [22, -2, 0, 2], [0, 52, 0, 0]]
