
# q-expansion of newform 9680.2.a.m, downloaded from the LMFDB on 14 April 2026.

# 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 = 9680
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], [-4], [0], [4], [-4], [-8], [-4], [-8], [4], [6], [-8], [-4], [-12], [-10], [0], [8], [-8], [-12], [12], [8], [-4], [10], [-2], [0], [12], [-4], [-16], [-2], [-12], [-8], [6], [0], [-16], [-8], [2], [8], [-12], [12], [-16], [22], [-12], [-4], [12], [4], [-8], [4], [-12], [22], [-12], [8], [-8], [8], [18], [28], [-2], [24], [-28], [24], [-4], [-28], [28], [-20], [-22], [2], [24], [-4], [12], [-24], [34], [-24], [28], [-4], [16], [4], [10], [-14], [14], [16], [24], [-6], [32], [-34], [-16], [32], [18], [-28], [-32], [-4], [8], [24], [-28], [16], [40], [-44], [-18], [38], [20], [16], [-12], [12], [4], [16], [-32], [-2], [8], [-20], [-12], [8], [-12], [-36], [-42], [16], [4], [2], [32], [-36], [-18], [48], [-6], [-36], [4], [-16], [-24], [-24], [38], [-44], [4], [-4], [16], [-20], [-12], [10], [-16], [-8], [-26], [-20], [-30], [24], [-16], [-24], [-4], [44], [-30], [-4], [44], [-52], [0], [12], [-20], [14], [16], [-28], [-48], [-4], [-16], [30], [-44], [0], [-16], [52], [4], [-8], [-34], [-60], [20], [12], [56], [22], [16], [2], [-24], [-28], [-28], [-26], [16], [54], [20], [-32], [-4], [0], [-38], [36], [-60], [-10], [44], [-56], [-32], [48], [46], [60], [8], [18], [-60], [-6], [-40], [30], [-12], [44], [8], [64], [22], [24], [56], [-30], [4], [4], [8], [56], [4], [22], [-28], [-48], [24], [26], [-4], [-48], [4], [-34], [40], [-8], [46], [-60], [-36], [-32], [22], [12], [4], [24], [-46], [-48], [-16], [-40], [24], [28], [-50], [4], [-40], [-4], [-40], [8], [-4], [50], [-60], [40], [-76], [24], [16], [-36], [36], [-48], [68], [10], [-36], [-48], [10], [-28], [-6], [20], [-52], [52], [-40], [4], [-14], [0], [-18], [-42], [-36], [-12], [34], [-56], [6], [-80], [20], [76], [72], [8], [8], [24], [-4], [-60], [12], [32], [52], [12], [62], [36], [12], [0], [-30], [56], [-76], [24], [20], [-64], [4], [-42], [16], [-36], [36], [12], [-64], [48], [-40], [2], [40], [-62], [-36], [-20], [52], [-6], [48], [-8], [52], [-40], [24], [-56], [-50], [48], [72], [38], [24], [-28], [-60], [-30], [56], [88], [28], [44], [-8], [2], [-8], [-12], [8], [-24], [-14], [4], [-38], [-36], [74], [42], [0], [-52], [-66], [8], [22], [16], [32], [-74], [-80], [-54], [-82], [-92], [-40], [-20], [44], [-56], [-12], [-36], [-44], [32], [-60], [40], [24], [22], [28], [60], [8], [-88], [-32], [-4], [80], [-48], [-62], [80], [-40], [-76], [0], [52], [-78], [-54], [36], [12], [24], [-4], [68], [-22], [-60], [84], [-34], [6], [-16], [-40], [-12], [68], [-16], [90], [64], [24], [48], [78], [92], [-38], [64], [72], [-14], [96], [16], [-48], [20], [52], [-40], [32], [-52], [50], [32], [-76], [-78], [-28], [94], [-28], [-28], [-72], [-54], [-82], [-88], [48], [-80], [72], [-70], [32], [40], [-60], [66], [-18], [-24], [42], [40], [80], [24], [46], [-96], [0], [32], [-108], [-28], [84], [-34]]
