
# q-expansion of newform 9126.2.a.m, downloaded from the LMFDB on 12 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 = 9126
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], [2], [-3], [0], [0], [2], [-3], [-6], [-4], [2], [0], [8], [9], [-6], [-9], [2], [8], [0], [5], [14], [15], [6], [5], [-12], [-4], [9], [5], [6], [-4], [15], [-12], [-22], [18], [8], [23], [14], [3], [18], [-15], [-7], [3], [23], [-6], [8], [2], [-10], [-3], [5], [-6], [12], [17], [-27], [18], [12], [30], [20], [17], [-6], [-22], [-30], [-10], [-15], [-1], [-12], [32], [-22], [-3], [35], [30], [33], [2], [11], [20], [-12], [-30], [23], [-18], [-19], [-27], [-34], [-9], [14], [-22], [-15], [12], [-7], [12], [-34], [36], [0], [-16], [-15], [14], [9], [-12], [24], [2], [38], [38], [-18], [0], [24], [-22], [2], [27], [36], [39], [35], [-4], [23], [-42], [44], [-16], [12], [2], [-24], [-36], [21], [-13], [23], [48], [36], [-10], [42], [-31], [-24], [14], [14], [20], [39], [-40], [-7], [-48], [14], [36], [8], [48], [54], [-22], [-36], [14], [-3], [-19], [-24], [-37], [42], [26], [-9], [-22], [30], [-46], [-9], [44], [51], [2], [-6], [-25], [12], [-57], [-24], [-22], [27], [54], [-51], [2], [-13], [2], [-6], [-12], [23], [45], [53], [14], [-18], [-10], [48], [26], [26], [-10], [12], [-43], [-6], [-24], [6], [-43], [26], [2], [63], [-19], [-24], [-40], [0], [-3], [48], [11], [14], [66], [-21], [-30], [32], [-43], [29], [21], [42], [62], [27], [36], [-58], [-13], [36], [2], [36], [-45], [-37], [-46], [-12], [39], [-18], [-34], [2], [-18], [-70], [12], [59], [12], [-33], [-34], [-36], [41], [-58], [50], [18], [2], [21], [-43], [60], [21], [48], [27], [56], [50], [2], [-18], [-60], [-28], [21], [8], [-51], [38], [-36], [-69], [-7], [24], [-21], [-43], [-64], [72], [-25], [-10], [-75], [26], [-13], [-54], [20], [60], [-30], [-76], [6], [35], [2], [-55], [-4], [41], [-16], [36], [-46], [-22], [-45], [-27], [-58], [-9], [-10], [32], [3], [-55], [0], [-34], [-24], [30], [21], [-60], [-45], [59], [0], [-28], [84], [-15], [14], [11], [66], [2], [81], [8], [2], [-27], [-58], [-3], [11], [-24], [-36], [-66], [2], [45], [77], [-36], [63], [-43], [12], [26], [38], [-30], [2], [6], [2], [-64], [44], [-87], [36], [-46], [-42], [-16], [-21], [-10], [-3], [50], [6], [-58], [8], [26], [-18], [30], [-28], [24], [27], [-55], [-10], [-21], [48], [32], [-79], [-36], [-76], [95], [-84], [-39], [3], [-42], [0], [-73], [-24], [-48], [39], [-64], [-61], [18], [-28], [-61], [12], [56], [21], [-36], [68], [-79], [60], [-48], [-73], [60], [26], [-48], [84], [56], [84], [-34], [57], [44], [-22], [98], [-48], [53], [-36], [-45], [44], [-33], [-19], [-34], [-54], [-58], [90], [26], [42], [-82], [-6], [66], [-76], [-67], [18], [68], [0], [38], [-36], [-27], [-64], [77], [-12], [-45], [32], [66], [-27], [42], [2], [-48], [-75], [86], [-12], [36], [-96], [44], [-45], [26], [-48], [-58], [-96], [47], [6], [74], [83], [-46], [-4], [60], [66], [-13], [48], [41], [60], [-10], [-72], [-61], [38], [2], [-3], [33], [-78], [41], [96], [-19], [69], [-58], [-36], [74], [-40]]
