
# q-expansion of newform 180.3.f.f, 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 = 180
weight = 3
poly_data = [16, 0, -7, 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], 1], [[0, 1, 0, 0], 1], [[0, -3, 0, 1], 4], [[-4, 0, 1, 0], 1]]

hecke_ring_character_values = [[91, [-1, 0, 0, 0]], [101, [1, 0, 0, 0]], [37, [-1, 0, 0, 0]]]
aps_data = [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 5, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 14, 0], [-8, 0, 0, -16], [0, 16, -8, 0], [0, 0, 0, 0], [-16, 0, 0, -32], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -48, 24, 0], [0, 0, -86, 0], [0, 0, 0, 0], [118, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [32, 0, 0, 64], [0, 32, -16, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -96, 48, 0], [-22, 0, 0, 0], [0, 0, 206, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 226, 0], [24, 0, 0, 48], [0, 0, 0, 0], [48, 0, 0, 96], [0, 0, 0, 0], [0, 0, 0, 0], [0, -112, 56, 0], [0, 0, -154, 0], [0, 0, 0, 0], [-122, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -374, 0], [-96, 0, 0, -192], [-56, 0, 0, -112], [0, 0, 0, 0], [0, -224, 112, 0], [-218, 0, 0, 0], [0, 0, 34, 0], [0, 0, 0, 0], [-478, 0, 0, 0], [0, 0, 0, 0], [0, 0, 466, 0], [0, 144, -72, 0], [0, 0, 0, 0], [-64, 0, 0, -128], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -394, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 134, 0], [168, 0, 0, 336], [0, 0, 0, 0], [0, 192, -96, 0], [458, 0, 0, 0], [0, 0, 274, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-40, 0, 0, -80], [0, -176, 88, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-142, 0, 0, 0], [0, 0, 0, 0], [602, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [160, 0, 0, 320], [0, -352, 176, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 448, -224, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [88, 0, 0, 176], [0, -80, 40, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1078, 0, 0, 0], [0, 0, 0, 0], [0, 0, 614, 0], [0, -576, 288, 0], [0, 0, 0, 0], [-280, 0, 0, -560], [0, 0, 0, 0], [0, 416, -208, 0], [0, 0, 1166, 0], [0, 0, 0, 0], [-242, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1186, 0], [-264, 0, 0, -528], [-320, 0, 0, -640], [0, 0, 0, 0], [0, 0, 0, 0], [0, 560, -280, 0], [0, 0, 1114, 0], [0, 0, 0, 0], [838, 0, 0, 0], [0, 0, 0, 0], [0, 0, 374, 0], [0, 704, -352, 0], [-104, 0, 0, -208], [0, 0, 0, 0], [742, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [56, 0, 0, 112], [0, -720, 360, 0], [112, 0, 0, 224], [0, 0, 0, 0], [0, 0, 0, 0], [578, 0, 0, 0], [0, 0, -1526, 0], [0, 0, 0, 0], [0, 0, 826, 0], [0, 0, 0, 0], [312, 0, 0, 624], [0, 0, 0, 0], [0, 0, 0, 0], [0, -832, 416, 0], [-502, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1666, 0], [440, 0, 0, 880], [0, 880, -440, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 272, -136, 0], [0, 0, 0, 0], [0, 0, 0, 0], [336, 0, 0, 672], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 544, -272, 0], [0, 0, -1474, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1934, 0], [0, 112, -56, 0], [448, 0, 0, 896], [0, 0, 0, 0]]
