
# q-expansion of newform 363.3.h.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, 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 = 363
weight = 3
poly_data = [81, -27, -18, 15, 1, 5, -2, -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, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0], 1], [[27, 18, -15, -1, -5, 2, 1, -1], 27], [[0, 16, 0, 0, 0, 0, 1, 0], 3], [[162, -54, -36, -5, 2, 10, -4, -2], 27], [[81, -27, -18, 2, 1, 5, -2, -1], 9], [[16, 0, 0, 0, 0, 1, 0, 0], 1], [[135, 90, -75, -5, 2, 10, 5, -5], 27]]

hecke_ring_character_values = [[122, [-1, 0, 0, 0, 0, 0, 0, 0]], [244, [0, 0, 0, 0, 0, 0, 0, 1]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, -3, -6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-9, 0, 0, 0, 0, 0, 18, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-37, 0, 0, 37, 37, 0, 0, -37], [0, 0, 0, 0, 0, 0, 0, 25], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -48, 0, 24, 0, 0, 0, 0], [-24, 48, -48, -24, -24, -48, 48, -24], [0, 0, -30, 0, 0, 0, 0, -15], [0, 0, 0, 0, 0, 0, 0, 0], [-35, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 15, 30, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [45, 0, 0, 0, 0, 0, -90, 0], [-95, 0, 0, 95, 95, 0, 0, -95], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -190], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -42, 0, 21, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -21, -42, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 215, 0, 0, 0, 0], [70, 0, 0, -70, -70, 0, 0, 70], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 210, 0, -105, 0, 0, 0, 0], [0, 0, 0, 0, -263, 0, 0, 0], [0, 0, -210, 0, 0, 0, 0, -105], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 445, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-433, 0, 0, 433, 433, 0, 0, -433], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-135, 270, -270, -135, -135, -270, 270, -135], [0, 0, -288, 0, 0, 0, 0, -144], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 120, 240, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 240, 0, -120, 0, 0, 0, 0], [0, 0, 0, 0, -265, 0, 0, 0], [189, -378, 378, 189, 189, 378, -378, 189], [563, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [99, 0, 0, 0, 0, 0, -198, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 635], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 133, 0, 0, 0], [-39, 78, -78, -39, -39, -78, 78, -39], [0, 0, 330, 0, 0, 0, 0, 165], [-790, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -240, -480, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [240, 0, 0, 0, 0, 0, -480, 0], [0, 0, 0, -742, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -25], [0, 0, 0, 0, 0, 0, 0, 0], [0, 462, 0, -231, 0, 0, 0, 0], [195, -390, 390, 195, 195, 390, -390, 195], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [35, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -39, -78, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 875, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -602], [0, 0, 0, 0, 0, 0, 0, 0], [0, 90, 0, -45, 0, 0, 0, 0], [0, 0, -390, 0, 0, 0, 0, -195], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1055, 0, 0, 0], [0, 0, 192, 0, 0, 0, 0, 96], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -360, -720, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [336, 0, 0, 0, 0, 0, -672, 0], [0, 0, 0, 1237, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1213], [0, 630, 0, -315, 0, 0, 0, 0], [0, 0, 0, 0, -395, 0, 0, 0], [351, -702, 702, 351, 351, 702, -702, 351], [0, 0, 102, 0, 0, 0, 0, 51], [0, 0, 0, 0, 0, 0, 0, 0], [1153, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-336, 0, 0, 0, 0, 0, 672, 0], [1093, 0, 0, -1093, -1093, 0, 0, 1093], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1057, 0, 0, 0], [0, 0, -510, 0, 0, 0, 0, -255], [-1355, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -973, 0, 0, 0, 0], [-70, 0, 0, 70, 70, 0, 0, -70], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -432, 0, 216, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 51, 102, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-755, 0, 0, 755, 755, 0, 0, -755], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -817], [0, -990, 0, 495, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-757, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 504, 1008, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-285, 0, 0, 0, 0, 0, 570, 0], [0, 0, 0, 1370, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1750, 0, 0, 0], [-120, 240, -240, -120, -120, -240, 240, -120], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 480, 960, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [399, 0, 0, 0, 0, 0, -798, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1170, 0, -585, 0, 0, 0, 0], [-459, 918, -918, -459, -459, -918, 918, -459], [0, 0, -1122, 0, 0, 0, 0, -561], [1582, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0]]
