# q-expansion of newform 6534.2.a.u, downloaded from the LMFDB on 22 September 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 = 6534
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], [-2], [1], [0], [-6], [5], [8], [-1], [9], [0], [-3], [-9], [5], [-9], [4], [-3], [2], [4], [0], [12], [7], [-2], [2], [19], [-7], [6], [-14], [18], [4], [12], [10], [16], [19], [2], [8], [11], [10], [6], [-10], [1], [19], [-27], [-10], [1], [8], [-23], [-12], [-18], [-1], [22], [20], [-10], [-5], [22], [30], [-8], [25], [-8], [11], [-4], [-14], [-19], [-31], [-34], [-18], [20], [-18], [32], [-30], [6], [-18], [-28], [8], [-10], [16], [-6], [10], [2], [32], [12], [-7], [-18], [29], [29], [8], [-8], [26], [26], [12], [-27], [-36], [-16], [26], [14], [-22], [10], [6], [1], [-10], [-23], [3], [-38], [22], [-1], [-7], [-3], [-33], [-27], [20], [8], [-6], [28], [32], [10], [-18], [-34], [-20], [-36], [36], [39], [34], [-47], [39], [-36], [-9], [-21], [36], [-36], [-8], [1], [28], [26], [-43], [42], [-34], [-48], [-52], [-36], [-23], [13], [-11], [-16], [20], [-23], [9], [0], [-7], [40], [-11], [-18], [-46], [22], [-12], [46], [-9], [31], [48], [14], [-27], [-24], [21], [-13], [-12], [-2], [-24], [-28], [8], [14], [4], [-40], [-13], [30], [4], [54], [14], [4], [-54], [3], [4], [10], [42], [9], [9], [27], [40], [-8], [-42], [-24], [-56], [-13], [12], [42], [18], [-64], [52], [12], [2], [57], [-6], [-29], [20], [-38], [-6], [-9], [-68], [20], [-26], [45], [-10], [12], [-42], [-50], [9], [24], [-10], [8], [-2], [8], [14], [-4], [-31], [-4], [-20], [38], [-18], [-46], [36], [17], [72], [-46], [-31], [55], [58], [-10], [30], [-1], [-33], [-36], [-64], [-15], [19], [-44], [-46], [14], [-28], [72], [17], [31], [14], [-34], [18], [65], [57], [35], [48], [-7], [17], [54], [30], [71], [6], [78], [-54], [42], [62], [-58], [6], [9], [41], [2], [-6], [-77], [-4], [-4], [-34], [-61], [-38], [-22], [-58], [60], [-8], [-60], [34], [27], [-57], [19], [57], [66], [-50], [52], [-15], [-45], [-36], [-10], [74], [-68], [-74], [-12], [19], [-68], [43], [-40], [24], [-32], [-3], [-60], [-34], [-39], [-74], [84], [54], [-45], [-12], [-38], [20], [61], [38], [39], [-75], [-87], [13], [7], [16], [31], [-43], [-36], [-34], [-42], [37], [6], [-6], [-77], [-4], [36], [84], [-33], [-37], [-39], [56], [19], [-48], [-53], [58], [-42], [-44], [-41], [8], [38], [14], [-65], [41], [-24], [-9], [-84], [-38], [40], [-88], [33], [32], [34], [-74], [-19], [20], [-46], [-43], [-73], [81], [52], [39], [60], [-30], [-3], [36], [38], [-11], [16], [-86], [49], [-68], [4], [30], [88], [-67], [79], [9], [82], [46], [36], [-3], [22], [0], [-91], [-24], [28], [46], [21], [-70], [48], [78], [44], [30], [5], [-4], [-65], [-47], [86], [87], [16], [29], [-14], [-77], [-84], [-9], [-46], [52], [62], [-32], [-14], [-54], [-96], [-42], [55], [-42], [-73], [12], [51], [11], [16], [-18]]