# q-expansion of newform 6498.2.a.bn, downloaded from the LMFDB on 20 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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 6498
weight = 2
poly_data = [-1, -3, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0], 1], [[0, 1, 0], 1], [[-2, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[-1, 0, 0], [0, 0, 0], [0, -2, 1], [-1, 1, -1], [0, 0, 1], [2, 0, -1], [-2, 3, -1], [0, 0, 0], [-2, 2, -5], [-2, -5, 2], [3, 3, 0], [3, -4, 6], [-3, 1, 3], [5, 2, 1], [-1, 3, 0], [1, -2, -6], [4, 3, 3], [-11, 1, 0], [4, 1, -4], [-5, 6, -7], [1, -7, 2], [5, -5, -5], [-9, 2, 1], [5, 3, 4], [-4, 0, 1], [1, -4, 8], [1, 6, 1], [7, 3, -7], [13, 1, -2], [-4, 5, 0], [-8, 4, -6], [-2, -2, 5], [-1, 8, -8], [-5, -1, 4], [8, 3, -5], [-8, 9, 5], [-8, -3, -2], [-16, -4, 5], [-4, 5, -5], [1, 2, 3], [-13, -8, 8], [-4, 9, -13], [0, 5, -9], [-5, -4, 5], [-1, -7, -4], [-10, -5, -2], [11, -3, 3], [1, -4, 3], [-3, -2, 5], [-14, 4, 4], [8, 3, 2], [4, -14, 8], [5, 8, 2], [-13, -2, 3], [4, 8, -15], [-1, -10, 7], [-15, -11, 5], [3, -13, 3], [-11, 3, -9], [-5, 3, -3], [-1, 6, -11], [-1, 8, 5], [-13, -5, 3], [9, -8, 5], [-3, 7, 13], [-5, -5, 2], [9, -4, 20], [-2, -11, -5], [-5, 2, -2], [5, 0, -5], [-27, 0, -6], [-8, 0, 4], [-6, 1, 5], [13, -10, -7], [6, -7, 7], [-9, 0, -2], [-1, -3, -8], [0, 4, 7], [5, 17, -2], [-10, -6, 14], [7, 19, -4], [-10, -20, 7], [1, 0, -13], [0, -20, 10], [-23, -8, 3], [-17, -3, -8], [12, -14, 4], [-14, 6, -25], [15, 1, -8], [3, -6, 9], [-26, 3, 6], [-18, -4, 11], [-13, -2, 11], [11, -4, 16], [5, -15, -9], [-1, 15, -3], [-1, -6, -21], [-4, -1, -12], [-3, 21, -8], [7, -1, 19], [-17, 7, -14], [15, -3, 1], [-5, -2, 17], [-3, -8, 16], [21, 2, 9], [-26, 11, -4], [-8, -3, -19], [-11, -14, 3], [-2, -3, 11], [-5, 13, -19], [-8, -7, -10], [-16, 8, -7], [0, -16, 8], [0, 10, 6], [3, 7, -13], [-6, -5, 21], [-29, 2, 2], [7, -2, -13], [-5, 21, -5], [-13, 13, -19], [-10, -5, 7], [-1, -21, 25], [14, -8, 9], [-30, -4, 8], [5, 12, -8], [-13, -20, 7], [6, 25, -19], [-2, 5, 19], [11, 17, -7], [-11, 16, -17], [22, -2, 4], [-21, -5, -5], [6, 0, -2], [-8, -6, 20], [-3, -26, 27], [1, 15, -14], [5, 12, -22], [-3, 23, -21], [21, -9, 4], [-12, -9, -12], [-5, -27, 2], [25, 3, 16], [-13, 15, -26], [-10, 9, 20], [13, 0, 0], [-13, -8, 6], [-22, -6, 14], [8, -19, 31], [18, 10, -3], [-12, -27, 16], [2, 1, 17], [4, 9, 13], [3, 0, -18], [2, 15, -4], [-9, -2, 8], [-11, 17, -21], [14, 1, 5], [-3, -5, 31], [-7, -3, -21], [-12, 3, -4], [-13, -10, -13], [-6, 8, 20], [17, -14, 24], [14, -28, 10], [23, -1, -5], [-34, -3, -4], [1, -22, 10], [20, -11, -17], [17, -15, -3], [3, -8, 25], [-18, -4, -1], [27, -1, 6], [7, 10, -22], [11, -10, -21], [-8, 2, 4], [19, 25, -28], [7, -18, -5], [7, -18, -3], [13, -14, -16], [-3, 14, 10], [-17, -5, 1], [-3, -3, -13], [31, -1, 13], [25, 7, 0], [-23, -10, -4], [5, 1, -15], [30, 16, -6], [24, -6, -11], [7, 25, -27], [27, -7, -14], [19, 9, -5], [24, -18, 18], [-6, -8, 2], [26, -15, 11], [-37, 20, 7], [-10, 13, -36], [3, 4, 21], [-11, -5, 10], [-11, -20, 28], [-10, -43, 28], [-2, 22, 5], [-19, 5, -1], [0, 33, -18], [-16, 1, 13], [-7, -15, 29], [6, -28, 38], [33, 22, -16], [6, -21, -6], [7, -17, 11], [20, 37, -25], [22, 13, 6], [38, -20, -10], [31, 16, -4], [33, 5, 6], [-7, 27, -5], [2, 29, -8], [10, -27, 41], [2, 8, 4], [18, -21, 2], [13, 25, -11], [-15, 3, -19], [1, 1, -6], [-19, 4, -7], [4, 1, -3], [13, 19, -24], [3, -14, -22], [11, -11, 6], [-27, 21, -11], [-17, -5, -10], [-1, 5, 31], [-10, -16, -4], [15, 21, 1], [-8, -2, 7], [10, 16, -19], [-32, 2, 3], [-15, -37, 18], [39, 1, -4], [16, 34, -22], [21, -15, 7], [3, -39, 41], [-8, 29, -25], [-7, 32, -27], [1, 41, -28], [17, -5, -3], [16, -17, 10], [-1, 15, -29], [23, -25, 20], [-8, -36, 19], [19, -16, -1], [-13, 4, -30], [-6, 1, -12], [3, -26, 31], [-12, 17, 29], [-12, 13, -7], [16, -20, -8], [-32, 4, 11], [-4, 20, -17], [-7, -20, 35], [3, 45, -36], [8, 1, 19], [-14, -15, 8], [32, -2, 6], [-16, -24, 8], [-4, -6, 12], [28, 14, 1], [25, -14, -11], [-25, -2, 19], [17, 27, 2], [43, -14, 17], [3, 13, -20], [10, 30, -3], [-22, 5, -11], [18, -23, 21], [36, 13, -6], [9, -32, 30], [26, 19, -23], [51, -12, 28], [-10, 45, -14], [-30, 15, -25], [3, -4, -33], [12, -29, 13], [24, -30, 31], [-4, -1, -44], [16, 27, -26], [2, 14, -45], [27, 45, -24], [8, 44, -43], [26, -12, -3], [5, -13, 18], [-3, -44, 2], [52, -13, 3], [-13, -37, 12], [7, -38, -3], [4, -14, 35], [-1, 9, -21], [-36, -1, -16], [-22, -1, 2], [25, 21, 3], [37, -18, 0], [-51, -6, 1], [10, -16, -22], [-13, -2, -32], [-3, 19, -26], [9, 24, 5], [-21, 27, -40], [-10, 11, -34], [12, -39, 23], [-2, -11, 25], [-6, -18, 3], [-10, -20, 23], [-8, 18, 18], [-66, -2, 6], [48, -11, -5], [-24, -10, -18], [-22, 19, -21], [-12, -26, -4], [42, -4, -20], [-18, 15, 33], [28, -26, 10], [-14, 10, 25], [31, -5, 13], [24, 3, 2], [54, 6, -18], [16, -7, -3], [22, 8, -17], [-30, 22, 5], [30, -3, 10], [28, -14, 9], [-4, 40, -48], [13, -7, 48], [-22, 9, -23], [15, 3, 35], [-36, -11, -8], [11, 60, -25], [18, 35, -16], [9, 19, -5], [69, -6, -3], [-25, 31, -4], [-7, 0, 23], [5, -24, 1], [-79, -1, 2], [3, 25, -33], [11, 21, -43], [13, -17, -10], [-16, -12, 3], [-8, -8, -19], [-14, 11, 2], [-35, 22, 16], [61, 5, -6], [-22, 11, -25], [-36, -32, 29], [3, 5, -6], [27, 29, -25], [-59, 11, 9], [38, -20, 37], [13, 0, 28], [-4, -26, -6], [-2, 3, -20], [29, 11, 8], [-27, 25, 13], [-1, -12, 4], [-40, -4, -11], [-22, -56, 42], [6, 3, 10], [-4, -6, 33], [-21, 4, -11], [3, -4, -25], [35, -9, 34], [48, 41, -28], [34, 24, 2], [4, 22, -11], [17, 6, -24], [-26, -3, -9], [-7, -22, 25], [-35, -51, 31], [-23, -11, 10], [46, 15, -4], [-25, -58, 29], [-25, -43, 14], [39, -27, 42], [-25, 37, -24], [13, -42, 11], [-23, -27, 28], [-70, 3, -2], [-46, -1, -21], [-31, -18, 26], [-28, -14, 16], [-49, 8, 13], [-30, 42, 23], [41, -1, 25], [21, -26, 1], [-43, -24, 29], [31, -26, 14], [-68, 8, 1], [-16, 26, 25], [27, 27, 10], [9, -46, 31], [-21, 0, -7], [59, 4, -10], [24, -10, 5], [-3, -31, 19], [-35, -25, 8], [19, -12, 32], [55, 11, -20], [2, 2, 39], [-10, 24, -21], [-21, -52, 14], [-11, -44, 14], [49, 12, -3], [-36, 37, -10], [10, -16, -16], [-6, 32, 0], [-21, -31, -4], [25, -17, 60], [-11, 26, -54], [-26, 3, -13], [-33, -56, 28], [17, -34, 21], [-15, -3, 32], [-6, -13, -45], [-29, -13, 9], [62, -2, 19], [1, -39, 29], [-17, -1, 38], [65, 17, 2], [53, -35, 0], [10, -6, -23], [-56, -6, -9], [-28, -27, 37], [1, 38, -27]]