# q-expansion of newform 9675.2.a.s, downloaded from the LMFDB on 06 June 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 = 9675
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], [0], [2], [-6], [4], [-4], [6], [8], [-6], [-2], [1], [4], [-2], [0], [14], [-12], [-8], [-2], [-8], [0], [-14], [14], [18], [8], [0], [-2], [-2], [-16], [4], [-18], [-20], [6], [16], [-14], [4], [-12], [-18], [-20], [22], [-8], [-18], [14], [8], [-4], [16], [-28], [6], [-18], [-28], [-6], [-16], [-18], [24], [10], [8], [2], [-10], [12], [-2], [28], [-12], [-10], [30], [-20], [-18], [-20], [14], [26], [-4], [8], [10], [-28], [-16], [6], [18], [22], [18], [4], [-34], [-36], [-18], [-40], [-16], [-30], [-18], [18], [-16], [-12], [-36], [-24], [12], [4], [0], [-30], [10], [36], [-2], [20], [-18], [16], [-10], [4], [14], [20], [14], [-20], [-22], [16], [-6], [-6], [36], [0], [-6], [-36], [-24], [34], [40], [22], [-50], [-14], [24], [20], [-22], [-10], [28], [-16], [10], [-4], [24], [48], [50], [-14], [18], [-46], [-4], [-18], [-50], [-44], [26], [16], [24], [-10], [16], [-22], [26], [52], [24], [-14], [38], [-52], [8], [28], [16], [32], [-54], [22], [18], [24], [-26], [56], [-32], [34], [-24], [32], [-22], [-54], [14], [36], [-2], [-16], [38], [16], [54], [-36], [-26], [24], [14], [-8], [-64], [-14], [-66], [-24], [6], [-22], [36], [10], [0], [-66], [20], [20], [-2], [-12], [-34], [2], [2], [-6], [-48], [-6], [-8], [50], [-38], [-12], [58], [56], [8], [-14], [12], [14], [66], [48], [0], [24], [-38], [-24], [-30], [48], [54], [-50], [-16], [-54], [64], [36], [22], [34], [-48], [-40], [-52], [58], [28], [-16], [26], [44], [12], [10], [6], [4], [28], [12], [36], [8], [14], [30], [20], [-32], [24], [-4], [-12], [2], [-58], [-20], [2], [34], [-36], [-18], [52], [34], [38], [-40], [-36], [22], [2], [-18], [68], [62], [6], [68], [-42], [46], [-68], [-34], [-32], [-34], [-16], [40], [30], [10], [-4], [-20], [-8], [-52], [-58], [28], [48], [-2], [-62], [24], [78], [-54], [-32], [66], [-28], [-46], [66], [-8], [-18], [40], [28], [54], [-38], [-24], [-80], [12], [-10], [-64], [-58], [-64], [58], [48], [6], [-2], [4], [28], [-70], [64], [-44], [-66], [-30], [-12], [38], [6], [-40], [30], [2], [-20], [-44], [-20], [74], [-10], [-58], [0], [84], [20], [72], [14], [-2], [-30], [32], [-54], [-34], [54], [-80], [-18], [72], [78], [-52], [-24], [-26], [28], [14], [74], [80], [-74], [-58], [64], [-84], [66], [-44], [-22], [-14], [72], [-36], [-12], [-42], [-14], [-56], [-38], [60], [-60], [84], [30], [-24], [26], [36], [-84], [-2], [66], [-34], [10], [2], [-56], [-26], [4], [52], [-72], [-22], [-4], [100], [50], [-54], [-44], [4], [80], [-74], [-40], [-30], [-20], [-10], [-34], [-78], [72], [-14], [30], [32], [90], [-2], [-12], [88], [-34], [-62], [-44], [52], [-98], [18], [-84], [-56], [-54], [24], [70], [26], [8], [0], [54], [18], [68], [74], [12], [-32]]