
# q-expansion of newform 6336.2.a.m, downloaded from the LMFDB on 23 September 2025.

# 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 = 6336
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 = [[0], [0], [-2], [-4], [-1], [-4], [-6], [-2], [-6], [-2], [-8], [-6], [-2], [6], [2], [-6], [8], [12], [-4], [14], [-10], [-8], [4], [-8], [-10], [-6], [-16], [-20], [-4], [4], [-8], [20], [-16], [14], [-6], [-8], [-10], [-4], [0], [14], [4], [-26], [18], [-18], [10], [-24], [-6], [8], [12], [6], [6], [24], [2], [-28], [24], [-28], [-22], [32], [8], [-18], [14], [14], [14], [10], [-26], [-6], [4], [6], [28], [-8], [0], [-4], [8], [16], [20], [18], [-34], [14], [0], [-30], [12], [-22], [4], [-34], [4], [-8], [-12], [-6], [-18], [8], [-4], [20], [-32], [-20], [-40], [24], [2], [40], [-42], [4], [-22], [-34], [-12], [-46], [-22], [-2], [-40], [-42], [-6], [26], [-8], [4], [20], [-8], [-8], [44], [-8], [-38], [-42], [-12], [-26], [-2], [-30], [-12], [16], [22], [-34], [6], [8], [-12], [-50], [4], [40], [-14], [18], [-26], [34], [14], [-2], [18], [-22], [42], [24], [-44], [2], [-18], [44], [-22], [-4], [-42], [48], [-36], [4], [-12], [24], [-42], [32], [40], [-22], [42], [-24], [42], [40], [8], [-32], [34], [-8], [40], [-46], [46], [44], [-2], [-60], [22], [8], [4], [-14], [30], [4], [-64], [-24], [-12], [58], [-46], [14], [-26], [-16], [48], [34], [-52], [34], [-44], [-24], [30], [-52], [-16], [-38], [62], [34], [-48], [18], [-12], [-42], [38], [-12], [18], [48], [-44], [-30], [-52], [-18], [-42], [-48], [20], [44], [-22], [-52], [-42], [-42], [-66], [-32], [0], [-48], [64], [28], [-8], [12], [-18], [-4], [12], [-22], [50], [44], [42], [64], [-8], [-42], [54], [-60], [-26], [36], [46], [-40], [30], [-34], [-12], [-32], [0], [-54], [-36], [-8], [46], [-78], [-26], [-14], [60], [10], [18], [-62], [38], [16], [-20], [20], [-24], [12], [-20], [6], [20], [-82], [-26], [70], [72], [-70], [-44], [70], [-32], [-8], [36], [58], [28], [68], [-56], [-12], [64], [58], [-14], [54], [-42], [32], [-2], [70], [-64], [70], [68], [24], [18], [-8], [-46], [-4], [2], [6], [62], [20], [84], [-68], [-62], [32], [-6], [-34], [-48], [48], [-14], [-66], [32], [-28], [-10], [-44], [-48], [14], [58], [-34], [38], [-30], [-16], [22], [-14], [12], [64], [-72], [70], [-32], [-26], [-44], [84], [6], [4], [-74], [-66], [22], [64], [-50], [36], [62], [80], [-6], [52], [-2], [92], [68], [-6], [-14], [54], [-62], [-72], [-20], [66], [-34], [-76], [-6], [74], [60], [14], [18], [-12], [-20], [26], [18], [-8], [-90], [40], [92], [-48], [42], [-4], [-86], [0], [8], [34], [-30], [-54], [-2], [-12], [32], [-30], [30], [-54], [64], [-22], [-50], [18], [-82], [54], [-36], [8], [86], [2], [-12], [-60], [-68], [54], [-12], [-72], [-12], [84], [-82], [-36], [42], [-26], [28], [104], [-62], [-6], [56], [-94], [46], [-70], [24], [-80], [-12], [56], [78], [-28], [-66], [52], [22], [-70], [72], [22], [-92], [36]]
