
# q-expansion of newform 225.4.a.m, downloaded from the LMFDB on 18 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 = 225
weight = 4
poly_data = [-10, 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 = None
aps_data = [[0, 1], [0, 0], [0, 0], [15, 0], [0, 20], [35, 0], [0, 28], [91, 0], [0, -36], [0, -20], [-147, 0], [370, 0], [0, -140], [335, 0], [0, -56], [0, 28], [0, -280], [427, 0], [15, 0], [0, -20], [-70, 0], [-876, 0], [0, -168], [0, 0], [-1085, 0], [0, 420], [1540, 0], [0, 252], [-811, 0], [0, -512], [260, 0], [0, -420], [0, -840], [-784, 0], [0, -680], [-797, 0], [1225, 0], [-275, 0], [0, 84], [0, 1260], [0, 560], [-1687, 0], [0, 700], [3625, 0], [0, -552], [3199, 0], [887, 0], [3535, 0], [0, -28], [-5019, 0], [0, -1140], [0, 140], [-623, 0], [0, 840], [0, 1176], [0, -916], [0, 140], [-308, 0], [-7485, 0], [0, -300], [525, 0], [0, 2016], [-2695, 0], [0, -1680], [-2555, 0], [0, 2360], [-5088, 0], [-11485, 0], [0, -2024], [-9674, 0], [0, -1764], [0, -2820], [8505, 0], [-6775, 0], [9241, 0], [0, -2856], [0, -1380], [-4795, 0], [0, 2220], [6881, 0], [0, 4060], [6418, 0], [0, 420], [-8155, 0], [14749, 0], [0, -3528], [0, -2920], [-1030, 0], [0, 2940], [11940, 0], [0, 3836], [0, -840], [14125, 0], [0, -1280], [1279, 0], [0, 5180], [0, 1400], [0, -3500], [10745, 0], [-9423, 0], [9080, 0], [0, -2640], [0, 2436], [0, -1980], [587, 0], [-15995, 0], [0, -6468], [0, 3668], [0, 6040], [3143, 0], [2660, 0], [-6670, 0], [0, 3492], [5579, 0], [23617, 0], [0, 9960], [-14280, 0], [0, -3920], [0, -3808], [0, 1820], [25438, 0], [15150, 0], [0, 6636], [0, -624], [-20552, 0], [0, -2660], [17281, 0], [0, -2940], [30415, 0], [18550, 0], [24016, 0], [0, -3920], [-7772, 0], [-3865, 0], [0, -5040], [-7441, 0], [0, 8148], [6125, 0], [0, -6328], [0, -860], [28623, 0], [0, -8400], [1135, 0], [0, 316], [-1974, 0], [0, 13300], [36645, 0], [0, -3416], [20104, 0], [0, 680], [-21975, 0], [0, -280], [34915, 0], [0, 15624], [17880, 0], [0, 5480], [-25829, 0], [0, -980], [40005, 0], [0, 9520], [0, 13692], [0, -16976], [-20540, 0], [0, 3360], [0, 8968], [0, 11088], [-18283, 0], [-13230, 0]]
