
# q-expansion of newform 3234.2.a.m, 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 = 3234
weight = 2
poly_data = [0, 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], [0], [0], [1], [4], [-4], [0], [0], [2], [4], [10], [12], [4], [-4], [-10], [4], [-4], [4], [-8], [-4], [8], [16], [-8], [8], [12], [-4], [-12], [18], [2], [0], [8], [6], [16], [-10], [-8], [-8], [12], [0], [-4], [-4], [0], [-16], [-14], [6], [12], [-12], [12], [8], [16], [6], [8], [20], [-4], [24], [-8], [8], [24], [-10], [-10], [24], [4], [8], [-4], [-32], [-18], [20], [-2], [12], [28], [-24], [-32], [12], [22], [4], [4], [10], [8], [-34], [-20], [36], [22], [-40], [-40], [40], [4], [18], [26], [-28], [-40], [-36], [-24], [-8], [20], [20], [24], [24], [0], [-8], [-34], [-36], [14], [0], [6], [-12], [-40], [-36], [36], [-24], [-28], [-16], [26], [22], [-36], [-16], [-2], [-20], [20], [-30], [4], [-40], [-2], [-4], [4], [-28], [2], [26], [36], [-12], [-4], [28], [32], [32], [-22], [-44], [-20], [-40], [-32], [-16], [10], [8], [54], [0], [-28], [24], [36], [4], [4], [-52], [-32], [-46], [-16], [4], [8], [12], [-40], [-16], [-48], [-36], [-28], [-12], [-54], [-8], [-20], [-50], [60], [56], [-28], [-30], [-40], [-12], [-40], [24], [-10], [28], [-16], [-20], [58], [40], [20], [32], [0], [-6], [28], [24], [24], [-2], [-52], [-10], [64], [-40], [4], [20], [32], [-4], [-24], [-34], [-14], [-12], [-8], [-18], [-48], [-48], [-4], [60], [-32], [-36], [-44], [-22], [36], [-34], [-40], [-8], [-28], [56], [48], [-8], [20], [24], [-46], [-38], [-40], [18], [64], [24], [10], [-48], [24], [-8], [60], [2], [56], [-56], [-6], [60], [-8], [40], [54], [20], [-60], [-60], [-16], [20], [-34], [12], [64], [20], [52], [-76], [-16], [-34], [20], [-8], [-16], [36], [60], [-38], [64], [16], [-60], [16], [36], [28], [-76], [72], [-68], [46], [-32], [-4], [58], [-8], [76], [8], [48], [44], [-68], [-36], [-34], [-70], [48], [64], [-32], [-24], [-28], [16], [8], [2], [-74], [28], [-20], [46], [60], [-26], [0], [-18], [-12], [68], [56], [-16], [-64], [-36], [-14], [56], [4], [52], [78], [60], [56], [-48], [6], [-48], [58], [46], [-52], [-8], [-20], [60], [-48], [-24], [30], [40], [38], [36], [88], [74], [16], [-36], [-68], [24], [-6], [14], [66], [48], [-64], [12], [12], [82], [-52], [-32], [88], [-74], [6], [-52], [-48], [50], [-60], [80], [84], [-48], [-24], [-32], [-38], [-2], [24], [-74], [-52], [60], [8], [66], [-64], [6], [60], [-8], [84], [-52], [-10], [12], [8], [74], [92], [-28], [-48], [54], [-32], [-50], [-52], [-72], [-44], [60], [-44], [-48], [-54], [48], [82], [48], [-84], [-72], [8], [-52], [-20], [-2], [-8], [52], [28], [0], [-90], [-56], [-24], [60], [-54], [92], [34], [104], [56], [-36], [64], [-82], [-78], [-60], [-60], [-4], [42], [-4], [20], [-74], [48], [80], [84], [-8], [16], [-78], [12], [96], [-40], [64], [40], [-108], [-10], [-76], [24]]
