
# q-expansion of newform 900.3.c.f, downloaded from the LMFDB on 23 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 = 900
weight = 3
poly_data = [1, -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], 1], [[-1, 2], 1]]

hecke_ring_character_values = [[451, [-1, 0]], [101, [1, 0]], [577, [1, 0]]]
aps_data = [[-1, -1], [0, 0], [0, 0], [0, -6], [0, 6], [18, 0], [-10, 0], [0, -8], [0, 4], [36, 0], [0, 4], [54, 0], [-18, 0], [0, -12], [0, 0], [26, 0], [0, -18], [-74, 0], [0, -24], [0, -60], [36, 0], [0, -52], [0, -52], [18, 0], [-72, 0], [-36, 0], [0, -6], [0, -108], [-26, 0], [10, 0], [0, -126], [0, -78], [-110, 0], [0, 108], [-288, 0], [0, 108], [234, 0], [0, -72], [0, 76], [-146, 0], [0, -42], [262, 0], [0, 108], [180, 0], [154, 0], [0, 108], [0, 140], [0, -54], [0, -124], [338, 0], [182, 0], [0, -204], [-106, 0], [0, 186], [-14, 0], [0, 108], [108, 0], [0, -188], [270, 0], [234, 0], [0, 48], [58, 0], [0, 156], [0, 156], [468, 0], [-250, 0], [0, 216], [-468, 0], [0, -324], [-434, 0], [158, 0], [0, 264], [0, -126], [-270, 0], [0, 188], [0, -32], [288, 0], [-306, 0], [450, 0], [-50, 0], [0, 426], [-286, 0], [0, -72], [-36, 0], [0, -452], [0, -124], [-54, 0], [288, 0], [288, 0], [0, -234], [0, 332], [0, -84], [0, 150], [0, 42], [0, 256], [0, 64], [252, 0], [-54, 0], [0, 360], [-650, 0], [0, 396], [-574, 0], [0, 324], [-198, 0], [0, -104], [-504, 0], [0, -236], [-998, 0], [0, -312], [-614, 0], [0, -378], [414, 0], [-58, 0], [0, -108], [0, 476], [-810, 0], [0, 240], [0, 568], [950, 0], [0, -654], [-242, 0], [324, 0], [806, 0], [0, 332], [0, -448], [-756, 0], [-310, 0], [0, 48], [0, 630], [1206, 0], [0, -280], [0, -648], [0, 140], [846, 0], [1458, 0], [1282, 0], [422, 0], [0, 696], [94, 0], [270, 0], [0, 108], [-1188, 0], [0, 222], [0, -260], [718, 0], [0, 528], [-666, 0], [182, 0], [0, 572], [0, 676], [-774, 0], [1602, 0], [0, -240], [0, -212], [0, -828], [0, 624], [0, 108], [-54, 0], [936, 0], [0, 0], [0, 596], [-1550, 0], [0, -702], [0, -1062], [206, 0], [0, -416], [0, 764], [-198, 0]]
