
# q-expansion of newform 6498.2.a.m, downloaded from the LMFDB on 19 April 2024.

# 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 = 6498
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], [-3], [2], [3], [1], [0], [-5], [-3], [6], [-6], [12], [-10], [8], [-3], [3], [0], [-15], [0], [-11], [12], [-2], [6], [-12], [-10], [-6], [-3], [-3], [12], [-12], [-14], [-19], [6], [8], [18], [0], [-6], [-12], [-18], [-12], [-18], [11], [-6], [-4], [-7], [-3], [-18], [3], [-12], [-14], [-1], [-24], [20], [-18], [8], [-6], [-11], [30], [12], [-14], [-9], [-12], [11], [21], [-33], [-9], [-18], [-16], [28], [31], [-19], [8], [21], [-3], [18], [-26], [2], [36], [-6], [14], [27], [24], [-30], [0], [22], [6], [1], [-4], [32], [-8], [40], [18], [-8], [18], [-1], [18], [-12], [-9], [2], [36], [-22], [-24], [-24], [-32], [15], [28], [2], [-36], [6], [12], [18], [-10], [-24], [0], [18], [32], [23], [10], [-15], [-15], [48], [3], [-36], [-8], [40], [-8], [43], [-35], [-22], [-12], [6], [-12], [12], [13], [-15], [15], [9], [33], [11], [-33], [-2], [3], [15], [-51], [-18], [46], [24], [54], [48], [-27], [-34], [-24], [-24], [15], [-30], [15], [-41], [-47], [-27], [46], [-30], [-24], [-48], [-30], [-24], [-6], [6], [6], [-62], [-3], [39], [-16], [-9], [0], [-26], [30], [10], [-42], [8], [35], [45], [54], [-18], [-29], [2], [45], [-12], [24], [11], [6], [28], [12], [15], [-50], [54], [13], [-44], [-11], [61], [-66], [18], [27], [-24], [-38], [-68], [-7], [51], [-5], [-3], [-27], [38], [44], [45], [-66], [42], [49], [-18], [-48], [58], [-18], [30], [-24], [45], [33], [-42], [0], [-42], [-36], [-32], [-72], [57], [36], [-36], [-12], [-7], [-58], [4], [-26], [60], [36], [-16], [-49], [15], [36], [-32], [24], [-60], [60], [31], [42], [-55], [-25], [30], [-64], [26], [-20], [-48], [54], [-42], [6], [24], [-40], [-21], [38], [45], [-45], [17], [-33], [58], [3], [15], [9], [-21], [-48], [3], [-70], [-69], [-18], [10], [72], [69], [-61], [-81], [22], [28], [-3], [78], [51], [18], [52], [52], [-54], [36], [42], [-14], [-48], [-56], [-39], [36], [-42], [-81], [-79], [-9], [60], [48], [75], [-15], [52], [-2], [16], [44], [36], [-36], [-40], [48], [-20], [30], [55], [70], [-33], [-77], [45], [6], [49], [0], [-15], [-21], [60], [-56], [78], [15], [-15], [16], [-70], [62], [-51], [-30], [3], [79], [-15], [41], [21], [84], [45], [-60], [12], [-9], [90], [-76], [-60], [-18], [-80], [-30], [-18], [-96], [-25], [74], [-83], [-6], [-8], [38], [18], [-45], [72], [24], [-8], [-84], [-42], [62], [15], [95], [57], [40], [-52], [21], [56], [-2], [-53], [-12], [30], [-50], [43], [-70], [-75], [-6], [-1], [-30], [-6], [-36], [-90], [-9], [-4], [-24], [-84], [90], [-66], [66], [-48], [40], [30], [38], [12], [-84], [-78], [61], [-10], [-72], [-69], [2], [6], [-64], [75], [8], [-23], [-46], [90], [-30], [37], [36], [-66], [-39], [5], [-15], [-12], [63], [15], [25], [-48], [59]]
