
# q-expansion of newform 6042.2.a.m, 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
    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 = 6042
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], [2], [4], [0], [4], [-6], [-1], [0], [6], [4], [-4], [10], [-4], [-8], [1], [-14], [-4], [4], [0], [-10], [12], [-6], [0], [18], [14], [12], [-10], [-6], [12], [-4], [-12], [-12], [16], [-22], [0], [4], [12], [-8], [-14], [-24], [-22], [16], [-18], [6], [-8], [-8], [2], [2], [-22], [-4], [24], [10], [6], [-18], [-12], [6], [-20], [28], [28], [16], [22], [8], [8], [22], [-10], [0], [18], [12], [-12], [24], [-24], [20], [-6], [-36], [-24], [-6], [12], [18], [18], [30], [30], [-32], [-10], [26], [34], [-40], [-22], [-14], [-22], [0], [-28], [14], [30], [12], [-36], [30], [-32], [40], [-2], [24], [18], [-28], [30], [-16], [-2], [0], [-42], [-24], [-2], [-6], [8], [-12], [-36], [10], [42], [44], [-32], [50], [0], [40], [6], [26], [-26], [36], [30], [-36], [4], [40], [-30], [-40], [-36], [-42], [22], [12], [22], [-46], [4], [18], [48], [-20], [-26], [52], [-16], [6], [-44], [-36], [-36], [4], [-36], [-44], [0], [-8], [-48], [-44], [0], [-10], [-30], [26], [-22], [32], [12], [44], [18], [-14], [0], [-46], [-42], [14], [6], [56], [-42], [-24], [6], [40], [-6], [-12], [-34], [48], [-34], [-26], [-28], [50], [-48], [64], [30], [-44], [-4], [42], [-56], [62], [-14], [-60], [-22], [-14], [62], [50], [2], [-8], [-48], [42], [0], [-30], [58], [16], [-10], [-8], [10], [26], [-60], [-2], [30], [-26], [-12], [36], [46], [-22], [12], [-60], [-6], [-30], [-4], [18], [-46], [-30], [56], [-18], [24], [22], [-50], [28], [-32], [22], [-6], [36], [28], [-6], [6], [-24], [64], [24], [28], [36], [46], [-16], [32], [6], [-28], [-72], [8], [-18], [-6], [56], [70], [-18], [-16], [64], [-24], [-2], [14], [24], [66], [-28], [14], [62], [-28], [-30], [-30], [-12], [-2], [-50], [-20], [66], [-24], [38], [38], [-20], [-4], [66], [-20], [-8], [76], [-60], [80], [40], [16], [-38], [-18], [-4], [-46], [18], [-56], [-30], [-28], [28], [66], [10], [30], [-60], [76], [-18], [-22], [44], [-78], [-80], [-30], [54], [72], [24], [28], [8], [-42], [66], [32], [-8], [-46], [42], [-32], [-18], [10], [-28], [6], [66], [-28], [44], [18], [-20], [44], [24], [-6], [28], [26], [4], [20], [36], [-30], [-32], [88], [30], [42], [46], [-36], [-30], [-92], [70], [30], [26], [-52], [-40], [6], [-60], [58], [78], [40], [-56], [-6], [24], [-6], [48], [48], [-10], [40], [12], [48], [28], [54], [46], [24], [-78], [-88], [-12], [0], [-14], [88], [-10], [-24], [-24], [14], [-56], [-38], [90], [38], [-14], [6], [0], [84], [18], [8], [80], [8], [42], [-30], [-32], [-36], [-24], [-70], [-70], [-86], [68], [54], [24], [-36], [-22], [-70], [18], [-62], [-40], [42], [-56], [24], [-22], [-54], [-68], [60], [-10], [66], [-20], [-26], [48], [72], [2], [38], [-92], [-90], [70], [-18], [-32], [22], [68], [-80]]
