
# q-expansion of newform 637.2.a.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 = 637
weight = 2
poly_data = [-1, -1, 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 = [[-1, -1], [1, -2], [-1, 2], [0, 0], [-3, 0], [-1, 0], [-5, 4], [3, 0], [-5, -2], [-2, 4], [5, 0], [-5, 6], [2, -4], [-8, 0], [-1, -4], [-1, -4], [5, -4], [3, 0], [-3, 0], [4, -8], [7, -6], [7, -6], [0, 0], [-1, 2], [-10, 12], [-9, 0], [7, -6], [-13, 2], [7, -6], [-10, 8], [8, -12], [-5, -2], [-5, -2], [-4, -12], [-9, 6], [1, 12], [7, 0], [3, -12], [16, 4], [3, -12], [9, -18], [-6, -12], [-5, 10], [-9, 6], [-22, 8], [-11, -6], [4, 0], [-4, 0], [-7, 8], [-5, 18], [-1, 8], [0, 12], [-5, 6], [8, -4], [-13, 8], [-3, 18], [7, 4], [-1, -12], [19, -12], [14, 8], [-3, 18], [-2, -8], [0, 12], [-31, 2], [-25, 12], [-5, -2], [-21, 12], [18, 0], [-19, 26], [-10, -12], [27, -6], [-1, 8], [-3, -6], [-21, 12], [4, -12], [15, 0], [-5, 4], [3, -18], [11, 2], [-1, 6], [20, 8], [-6, 12], [1, -20], [10, 12], [-5, 18], [-1, 2], [-2, -20], [23, -18], [2, 20], [-24, 0], [-13, 14], [-15, 24], [17, -24], [-32, 4], [7, 12], [8, 8], [-17, -2], [-5, -8], [23, 6], [11, -6], [-24, 12], [-9, -6], [-23, 22], [19, 4], [-7, -18], [-29, 6], [-32, 16], [31, -2], [25, -26], [-22, 0], [-5, 18], [11, -18], [2, -4], [17, 0], [-16, 24], [-1, 8], [-4, 24], [17, 2], [11, -4], [4, -8], [3, -6], [10, -12], [23, -16], [-7, -4], [-13, -24], [10, -20], [-21, -18], [15, 6], [0, -24], [19, 6], [21, -24], [20, 8], [19, -18], [-14, -24], [15, -30], [46, 0], [-5, -26], [1, 12], [22, -8], [3, 12], [24, -24], [35, 2], [5, 18], [-12, 24], [-25, 0], [20, -16], [6, -36], [31, -20], [11, 6], [11, 8], [3, 18], [-2, 28], [20, 0], [-17, -2], [9, 6], [24, -36], [-9, 18], [-41, 10], [6, 12], [47, 2], [-29, 4], [10, -32], [36, -12], [-21, 6], [-25, 14], [1, 4], [21, 6], [7, 12]]
