
# q-expansion of newform 5610.2.a.m, downloaded from the LMFDB on 23 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 = 5610
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], [-1], [0], [-1], [4], [1], [-8], [4], [0], [-6], [-8], [-2], [4], [-6], [8], [4], [-10], [-4], [8], [6], [-8], [-12], [-6], [10], [18], [-8], [0], [-2], [-6], [-14], [16], [18], [-12], [2], [-10], [-14], [12], [-10], [-20], [-24], [16], [6], [-14], [-16], [18], [-20], [-28], [-12], [18], [14], [0], [10], [-16], [-18], [-12], [-6], [-14], [-22], [-22], [28], [18], [-8], [4], [18], [-6], [-20], [-22], [-8], [0], [-26], [28], [14], [-12], [24], [-10], [-8], [-20], [-18], [14], [-20], [-14], [-10], [-2], [-20], [36], [-18], [10], [22], [40], [40], [34], [18], [4], [-20], [-2], [40], [38], [-16], [-6], [-36], [-42], [12], [-42], [28], [46], [-28], [-34], [-10], [38], [-32], [-44], [-42], [-44], [-4], [2], [-40], [46], [30], [-12], [-50], [34], [24], [-4], [-40], [-18], [28], [0], [-20], [-44], [-36], [6], [-42], [22], [-10], [22], [-4], [28], [16], [14], [-20], [-44], [38], [4], [38], [36], [42], [-6], [44], [-18], [22], [54], [-12], [26], [-4], [8], [-6], [42], [-10], [-4], [36], [30], [-10], [20], [34], [16], [-58], [18], [14], [18], [-4], [22], [30], [-34], [32], [6], [36], [-38], [-46], [44], [-20], [-24], [-16], [-6], [30], [24], [-50], [-20], [38], [38], [-34], [-40], [-4], [0], [-60], [50], [34], [4], [6], [-48], [16], [20], [50], [-22], [12], [-48], [16], [-4], [-22], [52], [-18], [28], [50], [-48], [-18], [70], [62], [-18], [0], [60], [-32], [16], [30], [46], [-36], [24], [18], [-8], [34], [0], [34], [28], [-10], [58], [-12], [0], [14], [-36], [-20], [2], [44], [40], [-24], [58], [-18], [-22], [2], [-44], [44], [-36], [4], [10], [-14], [-46], [-46], [28], [12], [12], [18], [-22], [44], [68], [26], [78], [46], [28], [32], [42], [-28], [14], [60], [-4], [6], [50], [-26], [-8], [12], [4], [-46], [-12], [-68], [-10], [42], [16], [-52], [38], [-6], [-48], [20], [2], [42], [36], [70], [28], [82], [84], [-48], [80], [-20], [-68], [-70], [-6], [56], [-60], [-52], [-22], [-36], [76], [34], [-80], [70], [54], [-90], [-28], [0], [-6], [-52], [-66], [-82], [-50], [28], [82], [18], [24], [-86], [-34], [-56], [80], [58], [0], [10], [-26], [0], [-12], [12], [-28], [30], [46], [-38], [-34], [-34], [-54], [0], [52], [-16], [4], [-56], [-12], [-30], [54], [-48], [50], [0], [0], [48], [-86], [-18], [24], [42], [-14], [-50], [-46], [54], [-72], [-68], [6], [-64], [-54], [-10], [84], [-36], [-54], [-14], [-90], [-20], [-60], [22], [38], [46], [26], [-22], [10], [68], [54], [-36], [-60], [-32], [-42], [12], [102], [-62], [-30], [-20], [-68], [38], [-26], [-22], [-10], [-20], [42], [14], [10], [-6], [10], [-30], [-28], [-2], [-54], [-68], [-60], [-6], [-62], [-36], [44], [46], [-62], [-70], [-46], [38], [56], [-48], [-70], [72], [20], [66], [-52], [-76], [50], [-28], [-70]]
