
# q-expansion of newform 3136.2.a.f, 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 = 3136
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], [-2], [0], [0], [4], [-4], [2], [-6], [8], [-2], [4], [-10], [10], [-4], [-4], [2], [10], [-8], [8], [0], [6], [-16], [2], [-18], [2], [0], [4], [16], [-10], [6], [-16], [-6], [-14], [-10], [-6], [-16], [12], [-4], [-20], [-12], [0], [-12], [0], [14], [-6], [-4], [-8], [-24], [-22], [-20], [-6], [24], [-14], [-10], [-2], [-16], [-12], [16], [-14], [-6], [10], [8], [10], [24], [26], [34], [12], [-18], [12], [20], [-18], [-16], [-8], [-6], [-28], [-20], [6], [-28], [-2], [2], [6], [18], [8], [-22], [-24], [24], [-30], [22], [28], [24], [18], [-12], [24], [-8], [8], [16], [20], [-38], [-14], [-30], [20], [-30], [46], [6], [4], [14], [-18], [6], [24], [46], [-48], [22], [22], [-22], [-16], [-34], [-34], [12], [38], [-12], [24], [-6], [36], [24], [-22], [6], [-26], [36], [-28], [8], [-4], [16], [24], [-10], [-30], [-6], [-24], [-46], [20], [-26], [34], [2], [8], [0], [24], [52], [-36], [18], [-26], [48], [14], [38], [16], [-28], [-32], [16], [-40], [18], [22], [-40], [4], [-22], [-8], [42], [26], [36], [-40], [8], [-2], [-4], [-24], [-36], [-48], [-6], [-28], [-26], [16], [54], [32], [-8], [-8], [-18], [14], [-34], [0], [-60], [-22], [-6], [-14], [12], [-2], [24], [32], [-20], [-32], [6], [-18], [-2], [6], [8], [-54], [28], [-56], [22], [66], [40], [-4], [12], [30], [-2], [-50], [32], [-16], [-54], [-40], [2], [40], [-54], [24], [-10], [-50], [-40], [-30], [24], [26], [62], [-50], [40], [0], [4], [6], [-42], [8], [-22], [2], [32], [-30], [42], [8], [32], [36], [18], [48], [10], [-26], [28], [-16], [-6], [60], [24], [2], [-34], [-16], [-50], [-56], [-64], [14], [62], [64], [18], [-24], [28], [16], [-56], [50], [-10], [-14], [-18], [48], [-66], [36], [-76], [38], [-64], [14], [16], [60], [50], [-2], [50], [-56], [40], [20], [-8], [2], [0], [-46], [-38], [52], [30], [-70], [18], [6], [-46], [-22], [-44], [-80], [-12], [38], [46], [-46], [6], [-56], [44], [76], [58], [-4], [-32], [40], [-30], [-24], [54], [10], [-68], [32], [-2], [-2], [-24], [70], [34], [-42], [-78], [-88], [32], [54], [-22], [-28], [34], [-64], [46], [50], [6], [60], [-74], [-32], [-14], [-74], [78], [14], [92], [-38], [-42], [-28], [48], [-74], [84], [52], [-12], [-72], [56], [-50], [26], [-54], [12], [58], [-42], [84], [42], [66], [72], [-22], [6], [-48], [-48], [10], [-26], [20], [-56], [-26], [80], [22], [40], [-86], [24], [2], [-70], [40], [-10], [50], [66], [20], [38], [40], [82], [94], [24], [-16], [-16], [-88], [92], [-34], [16], [-96], [22], [48], [34], [4], [-54], [80], [22], [72], [66], [24], [-14], [76], [52], [10], [-66], [38], [14], [-90], [46], [-92], [88], [74], [-84], [-96], [-12], [66], [0], [-26], [56], [24], [30], [-54], [-64], [-76], [-86], [-18], [12]]
