
# q-expansion of newform 8820.2.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 = 8820
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], [-1], [0], [4], [-4], [-2], [0], [4], [0], [-8], [2], [2], [4], [-4], [-4], [-4], [8], [-4], [-12], [4], [4], [8], [-14], [12], [-10], [-8], [-12], [-2], [20], [0], [-12], [-4], [8], [-8], [-12], [20], [-4], [12], [-18], [-4], [-16], [-12], [18], [-12], [16], [-16], [16], [-24], [-8], [20], [-4], [-8], [20], [-30], [-12], [-14], [-8], [2], [-8], [-8], [6], [-8], [-16], [28], [12], [8], [-14], [28], [-8], [-18], [20], [-16], [-18], [-16], [-20], [24], [-4], [-16], [-32], [36], [-22], [-12], [-4], [16], [-4], [0], [6], [-18], [16], [-8], [-40], [-32], [12], [16], [12], [-30], [-18], [-16], [2], [-20], [-36], [0], [-16], [-24], [-20], [16], [-34], [-44], [40], [32], [34], [-12], [24], [-20], [-24], [-24], [-28], [-44], [4], [40], [46], [-26], [12], [8], [32], [6], [24], [8], [-36], [0], [-36], [28], [-2], [-30], [40], [6], [0], [-2], [40], [56], [48], [56], [52], [-16], [0], [-4], [-6], [32], [-12], [42], [-26], [52], [20], [-20], [20], [-4], [-6], [-12], [34], [-12], [-44], [8], [20], [-12], [-44], [-60], [-36], [-2], [-10], [-12], [-16], [36], [42], [-32], [-18], [-40], [40], [8], [-32], [-32], [12], [46], [6], [4], [6], [-22], [-32], [-6], [-24], [60], [-12], [24], [30], [-28], [54], [-50], [-26], [-46], [60], [-48], [-64], [20], [40], [12], [30], [16], [-12], [56], [-32], [46], [-6], [40], [40], [40], [-8], [-40], [22], [-60], [52], [6], [56], [-24], [-56], [56], [10], [42], [60], [-40], [-36], [54], [-16], [-28], [-56], [56], [-52], [16], [-12], [-36], [-8], [-36], [40], [0], [-34], [2], [64], [72], [76], [40], [36], [-54], [6], [44], [-32], [14], [20], [-42], [72], [-58], [12], [72], [-52], [56], [36], [62], [-72], [-16], [-66], [60], [-60], [32], [-44], [44], [20], [28], [-24], [52], [-50], [10], [68], [36], [-60], [-28], [32], [-64], [-28], [46], [-52], [-8], [-38], [80], [72], [68], [44], [70], [78], [0], [-54], [44], [-16], [-44], [-44], [44], [36], [32], [2], [12], [24], [-60], [-34], [44], [32], [-32], [60], [-12], [8], [36], [28], [-52], [-16], [-80], [38], [-18], [8], [44], [-8], [56], [40], [44], [44], [50], [52], [8], [24], [-40], [48], [18], [-30], [-8], [88], [46], [36], [26], [-12], [-44], [36], [-40], [-52], [-72], [-10], [48], [22], [-96], [-32], [-74], [-42], [64], [36], [-60], [92], [-66], [18], [-84], [-44], [-64], [-10], [-78], [88], [-38], [-36], [-48], [-52], [-72], [88], [90], [68], [-20], [-68], [-10], [-28], [-2], [-28], [56], [-20], [-40], [-60], [-28], [-20], [20], [12], [82], [26], [-44], [88], [-12], [42], [16], [-34], [-40], [64], [96], [36], [-16], [10], [-22], [-48], [42], [64], [80], [-36], [-76], [-20], [-36], [-8], [-22], [2], [28], [24], [-2], [-20], [-24], [-68], [28], [84], [60], [46], [-20], [-32], [88], [-96], [-80], [84], [0], [12], [20], [-22], [-38], [42], [-84], [32], [96], [-56], [-22], [68], [48], [44], [16], [-44], [-24], [8], [-4], [0], [36], [14], [-2], [0], [30], [-60], [100], [4], [64], [44], [60], [-26], [0], [-46], [-4], [-64], [-72], [56], [32], [-76], [-88], [-34], [-68], [20], [64], [40], [-60], [-20], [-76], [30], [-60], [-10], [104], [20], [-18], [-12], [80], [-12], [-60], [-68], [-54], [-14], [100], [24], [-56], [84], [-24], [-80], [-104], [-40], [-20], [-104], [22], [-98], [68], [-8], [62], [24], [-68], [-48], [-84], [84], [-32], [-62], [-38], [-32], [60], [-80], [-50], [8], [-56], [-20], [-48], [20], [-44], [-118], [-108], [30], [-48], [52], [-32], [60], [76], [-44], [28], [-46], [-2], [36], [-56], [20], [-64], [48], [8], [-32], [-16], [40], [-8], [-22], [0], [-72], [-84], [-68], [68], [-64], [-92]]
