
# q-expansion of newform 525.2.r.f, downloaded from the LMFDB on 20 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 = 525
weight = 2
poly_data = [1, 0, -1, 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 = [[176, [1, 0, 0, 0]], [127, [-1, 0, 0, 0]], [451, [0, 0, -1, 0]]]
aps_data = [[2, -1, -1, 1], [0, -1, 0, 0], [0, 0, 0, 0], [-2, 0, -1, 0], [1, 1, -1, -2], [1, 0, -2, -4], [-1, -5, -1, 0], [0, -2, 1, -2], [-2, 3, 1, -3], [-1, -6, 0, 3], [-3, -2, 3, 4], [6, 2, -3, -2], [1, 2, 0, -1], [-3, 0, 6, 2], [0, 2, 0, -2], [-6, 2, -6, 0], [5, 3, -5, -6], [0, 0, -4, 0], [-5, 6, -5, 0], [1, -6, 0, 3], [-5, 4, -5, 0], [0, 6, 3, 6], [-7, 0, 14, -3], [0, -7, 3, -7], [4, 0, -8, 8], [9, 1, -9, -2], [-6, 4, 3, -4], [-6, -3, 3, 3], [11, 0, -11, 0], [-4, 0, 8, -2], [-3, 0, 6, 10], [0, 2, -12, 2], [3, 3, 3, 0], [1, 8, 0, -4], [0, -8, -8, -8], [2, -4, -2, 8], [6, 4, 6, 0], [16, -8, -8, 8], [-5, 0, 10, -9], [-4, -18, 2, 18], [-10, 0, 10, 0], [-7, 20, 0, -10], [0, -4, 2, -4], [-3, -4, -3, 0], [5, 0, -10, -9], [22, 0, -22, 0], [14, 8, 0, -4], [6, 0, -12, 10], [-5, -7, -5, 0], [0, 0, -3, 0], [20, 0, -10, 0], [-14, -8, 0, 4], [-10, 2, 10, -4], [-9, 18, 0, -9], [10, 3, -5, -3], [-6, -2, -6, 0], [-16, -2, 16, 4], [0, -4, -10, -4], [-5, 6, -5, 0], [0, -16, 0, 8], [7, -12, 7, 0], [-4, 0, 8, -12], [-7, 0, 14, -20], [3, 7, -3, -14], [-10, 4, 5, -4], [34, 1, -17, -1], [0, -4, -15, -4], [15, 0, -30, -8], [4, 28, 4, 0], [-22, 0, 0, 0], [7, 9, 7, 0], [0, -1, -3, -1], [3, -6, 3, 0], [26, -4, -13, 4], [11, 20, 0, -10], [-20, 6, 10, -6], [21, 9, -21, -18], [6, -26, -3, 26], [0, -6, 6, -6], [17, 8, -17, -16], [32, -4, 0, 2], [7, 8, 0, -4], [0, 10, 0, -20], [-3, 0, 6, -10], [0, 2, 4, 2], [4, -6, -2, 6], [22, -16, 0, 8], [10, -8, -5, 8], [-9, -30, 0, 15], [-1, 0, 2, -24], [-16, -14, 8, 14], [-12, -12, 12, 24], [9, 16, 9, 0], [-14, 28, 0, -14], [0, -2, -17, -2], [-6, 0, 12, -4], [0, 2, 8, 2], [2, -2, -2, 4], [-2, 26, 1, -26], [0, 12, 15, 12], [-14, 0, 28, -2], [12, 6, 12, 0], [0, -18, 0, 0], [0, -17, 3, -17], [-29, -6, 29, 12], [5, -20, 5, 0], [1, 0, -2, 39], [38, 5, -19, -5], [-14, 14, 14, -28], [-7, -8, 0, 4], [-6, -2, 3, 2], [4, -20, 4, 0], [-14, 0, 28, -12], [-37, -4, 37, 8], [28, -24, 0, 12], [-9, -3, 9, 6], [-13, 0, 26, 18], [-19, -5, -19, 0], [-18, 29, 9, -29], [4, 8, 0, -4], [-7, 2, 7, -4], [-5, 0, 10, 36], [14, -21, -7, 21], [11, -9, 11, 0], [0, 8, 5, 8], [7, 18, 0, -9], [0, -4, -8, -4], [0, -2, -24, -2], [5, 0, -10, -22], [10, 10, -5, -10], [7, -12, -7, 24], [19, 0, -38, -17], [0, -4, -25, -4], [-8, 0, 16, -14], [0, 1, -45, 1], [-35, 20, 0, -10], [-13, 21, -13, 0], [2, -10, 2, 0], [11, 0, -22, -23], [-2, -16, 2, 32], [0, 4, 0, -2], [0, -13, 3, -13], [-18, 8, -18, 0], [14, 0, -28, -28], [-15, -6, 15, 12], [11, 34, 0, -17], [15, 0, -30, 6], [7, -17, 7, 0], [0, -2, 4, -2], [12, -4, -6, 4], [-16, -18, 8, 18], [-4, -16, 0, 8], [-9, 0, 18, -38], [-2, -27, 1, 27], [9, -18, 9, 0], [-13, 2, 0, -1], [0, 8, -17, 8], [0, 17, -23, 17], [1, 0, -2, 30], [11, -11, -11, 22], [10, 3, -5, -3], [22, 0, -44, -2], [-7, 0, 14, 2], [0, 0, 24, 0], [-9, 1, -9, 0], [3, 15, 3, 0], [-17, -22, 17, 44], [-15, 30, -15, 0]]
