
# q-expansion of newform 4032.2.a.m, downloaded from the LMFDB on 24 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 = 4032
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], [1], [4], [-6], [-2], [-4], [8], [-2], [0], [10], [6], [-4], [0], [6], [-4], [-6], [4], [8], [10], [0], [4], [6], [-14], [-2], [-8], [-12], [2], [14], [0], [20], [-10], [4], [6], [8], [10], [20], [-8], [22], [12], [18], [0], [2], [-10], [-8], [20], [16], [-12], [2], [22], [0], [2], [12], [30], [-24], [22], [0], [10], [-26], [4], [30], [28], [-8], [10], [-18], [-4], [18], [-12], [-22], [30], [-8], [-32], [-22], [-20], [-16], [-26], [-6], [-18], [-22], [36], [-6], [0], [2], [24], [4], [-34], [10], [22], [32], [-28], [-16], [-8], [-12], [-44], [24], [6], [6], [20], [-30], [-12], [-2], [-44], [6], [12], [34], [28], [-18], [24], [26], [-48], [42], [22], [-44], [-8], [-2], [-4], [24], [-18], [28], [2], [2], [-18], [-12], [-4], [-2], [10], [0], [8], [-6], [-12], [24], [-48], [-6], [22], [-14], [-2], [-36], [6], [-10], [-44], [38], [56], [36], [26], [56], [-14], [-42], [20], [-32], [2], [-18], [20], [-24], [12], [0], [-40], [-18], [-22], [-26], [-4], [-26], [-8], [12], [-18], [-24], [16], [-14], [50], [30], [-28], [-22], [-24], [-54], [48], [38], [-36], [38], [40], [10], [0], [4], [-22], [-10], [64], [30], [-30], [44], [-54], [-48], [34], [-12], [-44], [38], [60], [22], [50], [50], [30], [-24], [-66], [-48], [34], [-30], [12], [-42], [-32], [12], [54], [-28], [50], [-50], [-24], [-4], [-24], [26], [32], [-2], [56], [62], [-22], [-40], [30], [-16], [-60], [-38], [70], [16], [72], [-60], [-30], [-20], [-16], [54], [-76], [-48], [-62], [6], [36], [40], [-68], [-28], [-56], [18], [-50], [-24], [48], [52], [-4], [48], [2], [-66], [-24], [10], [38], [-20], [58], [-12], [-2], [74], [-64], [44], [-46], [26], [78], [28], [30], [38], [28], [6], [-6], [-28], [-6], [64], [-14], [24], [52], [66], [10], [36], [0], [-24], [72], [82], [-12], [-64], [18], [38], [-56], [30], [30], [-60], [6], [44], [-14], [6], [-80], [30], [12], [-36], [-38], [78], [-64], [-36], [-52], [2], [-28], [10], [8], [58], [-48], [22], [-34], [-76], [40], [-22], [-12], [64], [2], [-18], [44], [58], [54], [-16], [-10], [2], [20], [4], [80], [-58], [-14], [-34], [0], [4], [76], [92], [-78], [-34], [74], [-16], [10], [6], [-18], [72], [30], [12], [34], [-4], [-48], [-18], [92], [74], [30], [-48], [-22], [-26], [16], [12], [46], [8], [26], [22], [-32], [4], [-20], [-22], [54], [24], [-38], [28], [4], [48], [70], [-40], [2], [-60], [-16], [34], [-2], [74], [-90], [86], [-40], [30], [28], [-56], [-32], [-94], [-68], [-16], [2], [-66], [-44], [-52], [40], [-6], [32], [-26], [12], [-10], [58], [30], [32], [86], [-18], [88], [50], [-18], [44], [4], [82], [-10], [-44], [36], [-22], [86], [32], [-40], [30], [-8], [-34], [-14], [-16], [-84], [26], [70], [-100], [102], [-12], [-88]]
