# q-expansion of newform 7650.2.a.bx, downloaded from the LMFDB on 24 August 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 = 7650
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], [0], [0], [0], [-4], [2], [1], [4], [0], [2], [8], [-6], [6], [4], [0], [-10], [4], [-2], [-4], [0], [6], [8], [-12], [6], [14], [10], [8], [4], [14], [-14], [0], [20], [10], [20], [-6], [8], [18], [-12], [0], [14], [12], [22], [0], [-18], [6], [-16], [12], [0], [12], [6], [-22], [-16], [18], [-12], [2], [-24], [18], [-16], [10], [6], [-4], [6], [-20], [-16], [22], [30], [-4], [30], [4], [-2], [34], [-24], [-8], [-6], [4], [-32], [-22], [18], [30], [-38], [36], [6], [-24], [-18], [-32], [12], [-18], [22], [-30], [0], [-12], [40], [-32], [20], [-20], [0], [18], [6], [-44], [-2], [-28], [14], [-44], [6], [-44], [30], [12], [18], [40], [26], [-8], [-22], [-22], [4], [40], [-18], [-28], [24], [14], [-4], [-10], [-50], [6], [4], [12], [18], [-26], [24], [-8], [-14], [20], [0], [24], [-38], [6], [2], [38], [-12], [-2], [6], [20], [-54], [-16], [4], [30], [-16], [10], [-6], [-20], [48], [-14], [30], [-52], [-48], [12], [40], [-40], [-18], [-10], [-46], [-20], [58], [24], [20], [-46], [48], [-56], [-38], [18], [-10], [-44], [30], [-48], [-42], [0], [-10], [4], [-6], [-8], [14], [-48], [52], [26], [10], [-16], [-38], [34], [60], [10], [-24], [-50], [-44], [-60], [50], [-4], [-54], [-46], [34], [-62], [24], [-46], [40], [58], [34], [-28], [-66], [-64], [4], [38], [44], [-2], [-6], [16], [60], [-16], [10], [32], [-50], [32], [62], [38], [64], [-66], [-24], [4], [-10], [-6], [56], [-8], [-4], [2], [28], [-48], [54], [-12], [-48], [-46], [-10], [-36], [8], [-36], [-36], [8], [-50], [-14], [-16], [-40], [-60], [28], [-16], [50], [46], [72], [74], [-18], [60], [-10], [12], [6], [38], [24], [-12], [-58], [-62], [2], [-28], [-30], [70], [-36], [-58], [46], [-52], [70], [-16], [-18], [-56], [-4], [62], [10], [-20], [48], [-16], [32], [70], [-68], [-80], [-34], [-42], [40], [30], [50], [44], [-6], [-4], [50], [34], [-64], [-10], [44], [-52], [-74], [-18], [-56], [76], [36], [46], [-36], [-82], [-56], [42], [-8], [42], [-18], [76], [24], [-22], [-20], [-56], [14], [-18], [44], [22], [50], [16], [-54], [50], [-20], [-68], [24], [-58], [78], [30], [72], [20], [36], [-28], [30], [66], [-54], [-48], [26], [-6], [-6], [8], [-66], [36], [38], [-76], [-40], [54], [-76], [70], [2], [-24], [-42], [90], [-80], [-84], [18], [24], [-6], [-58], [0], [-20], [44], [-10], [78], [-88], [-6], [60], [4], [24], [-6], [88], [34], [-28], [-72], [-34], [14], [38], [34], [74], [-32], [2], [92], [-48], [-32], [-38], [-20], [-80], [34], [-58], [4], [12], [-8], [-42], [-64], [-42], [-60], [-6], [30], [2], [96], [-38], [10], [16], [-94], [78], [28], [52], [62], [22], [44], [12], [22], [-78], [-72], [80], [-14], [8], [82], [90], [-24], [68], [70], [-82], [12], [54], [-68], [0], [-6], [-36], [36], [-24], [66], [-2], [10], [-74], [28], [-40], [20], [94], [-58], [-64], [-46], [66], [68], [-40], [-94], [14], [60], [-48], [-84], [-10], [14], [58], [-34]]