
# q-expansion of newform 700.2.c.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 = 700
weight = 2
poly_data = [1, 0, -1, 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, 1, 1], 1], [[0, 1, -1, -1], 1], [[-1, 1, 1, -1], 1]]

hecke_ring_character_values = [[351, [-1, 0, 0, 0]], [477, [-1, 0, 0, 0]], [101, [-1, 0, 0, 0]]]
aps_data = [[1, 0, 0, 1], [0, 0, 1, -1], [0, 0, 0, 0], [2, 0, -1, 1], [1, -1, -1, 2], [-2, 1, 2, 1], [-4, -1, -2, -1], [6, 0, 0, 0], [1, -1, -2, -1], [-3, -2, -4, -2], [6, 0, 0, 0], [3, -3, 2, 1], [0, 0, -2, 2], [-2, 0, 0, 0], [0, 0, 1, -1], [1, -1, 0, 1], [-1, -1, -2, -1], [3, -3, 2, 1], [-1, -1, -2, -1], [2, -2, 2, 0], [-8, -2, -4, -2], [-3, 3, -5, 2], [-6, 6, -2, -4], [-3, 3, 2, -5], [-6, -3, -6, -3], [-6, 6, -2, -4], [6, -6, -3, 9], [5, -3, -6, -3], [-7, 2, 4, 2], [1, -1, 2, -1], [-13, -1, -2, -1], [-5, 1, 2, 1], [-5, 5, -6, 1], [2, 2, 4, 2], [-8, 2, 4, 2], [5, -5, 3, 2], [-2, 4, 8, 4], [6, 6, 12, 6], [0, 0, -3, 3], [2, 5, 10, 5], [-2, 2, 6, -8], [3, -3, -4, 7], [1, -1, 3, -2], [-3, 3, -2, -1], [-2, 2, 10, -12], [1, 1, 2, 1], [-1, 1, 3, -4], [6, -6, 1, 5], [6, -6, 5, 1], [6, -6, -2, 8], [-8, 8, 4, -12], [-1, 1, 15, -16], [3, -3, 6, -3], [16, 4, 8, 4], [6, 0, 0, 0], [9, 1, 2, 1], [6, -6, 0, 6], [-7, -1, -2, -1], [2, -2, -12, 14], [-3, -2, -4, -2], [0, 0, -7, 7], [-8, -5, -10, -5], [0, 0, 1, -1], [2, -4, -8, -4], [20, -1, -2, -1], [5, -5, -4, 9], [-8, 8, -6, -2], [0, 0, 0, 0], [17, 7, 14, 7], [-6, 6, 10, -16], [0, 3, 6, 3], [-4, 4, -10, 6], [12, -12, -7, 19], [-7, 7, 6, -13], [-8, 8, 6, -14], [-12, 12, -2, -10], [11, 4, 8, 4], [10, 1, 2, 1], [-15, 2, 4, 2], [-9, 9, 8, -17], [-7, -7, -14, -7], [19, 0, 0, 0], [1, -1, -9, 10], [22, 4, 8, 4], [-20, -8, -16, -8], [26, 0, 0, 0], [-7, -2, -4, -2], [-12, 12, 2, -14], [0, 0, 16, -16], [9, 3, 6, 3], [0, 0, 13, -13], [3, 9, 18, 9], [2, -6, -12, -6], [-11, 11, 7, -18], [-5, 5, -9, 4], [0, 0, 9, -9], [-6, 6, -8, 2], [12, -12, -6, 18], [0, 0, 14, -14], [-7, 6, 12, 6], [19, 1, 2, 1], [4, -4, 8, -4], [6, -6, -18, 24], [-12, 2, 4, 2], [-12, 12, 10, -22], [12, 3, 6, 3], [-12, 12, 2, -14], [-28, -1, -2, -1], [-1, 1, 7, -8], [3, -3, -12, 15], [-12, 12, -3, -9], [-5, 5, 0, -5], [8, -8, 4, 4], [13, -5, -10, -5], [3, -3, 19, -16], [4, 2, 4, 2], [6, -6, -11, 17], [0, 0, -6, 6], [14, -14, -6, 20], [13, -13, 3, 10], [6, -6, 16, -10], [9, -9, -18, 27], [-18, -3, -6, -3], [5, -5, -10, -5], [-2, 10, 20, 10], [-23, -6, -12, -6], [-21, 0, 0, 0], [41, -1, -2, -1], [-12, 12, 6, -18], [-14, 1, 2, 1], [3, -3, -1, 4], [-23, -3, -6, -3], [-5, 5, -9, 4], [-12, 12, -8, -4], [-9, 9, -14, 5], [-9, 9, 0, -9], [8, -1, -2, -1], [12, -12, 5, 7], [0, -9, -18, -9], [-3, -2, -4, -2], [-34, 2, 4, 2], [13, 6, 12, 6], [-16, 2, 4, 2], [24, -4, -8, -4], [12, -12, -14, 26], [-3, 3, 6, 3], [-12, 6, 12, 6], [38, 8, 16, 8], [20, 8, 16, 8], [30, 4, 8, 4], [4, -4, 6, -2], [-18, 18, 4, -22], [31, -3, -6, -3], [-6, 6, 2, -8], [-25, -3, -6, -3], [2, -2, -14, 16], [11, -11, -9, 20], [3, -3, 28, -25], [-18, -3, -6, -3], [-3, 3, 26, -29], [8, 6, 12, 6], [2, -2, 10, -8], [9, 9, 18, 9], [0, -6, -12, -6], [-10, 10, 14, -24], [6, -6, 9, -3], [8, -8, 18, -10], [8, -1, -2, -1]]
