
# q-expansion of newform 6498.2.a.f, 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], [0], [-1], [6], [-5], [-3], [0], [-3], [9], [4], [-2], [0], [8], [0], [-3], [9], [-10], [-5], [-6], [-7], [10], [6], [-12], [10], [-18], [-14], [-9], [-11], [6], [-2], [0], [9], [-4], [0], [10], [-22], [20], [12], [6], [0], [-2], [-3], [-14], [0], [11], [-5], [-26], [-15], [-22], [6], [21], [-8], [-6], [12], [-24], [-6], [11], [8], [0], [-22], [-21], [-20], [21], [-19], [-9], [1], [4], [-18], [-10], [15], [-21], [-28], [-23], [7], [18], [-18], [20], [0], [-32], [12], [-17], [6], [-2], [28], [18], [-18], [17], [12], [-4], [-18], [-36], [-2], [36], [-4], [21], [30], [-36], [-11], [2], [-44], [-24], [-12], [-24], [-4], [11], [12], [30], [-24], [28], [22], [2], [6], [-10], [-16], [6], [-22], [-27], [-24], [-45], [13], [-44], [-33], [36], [-10], [-12], [-10], [-39], [-37], [32], [-16], [-36], [40], [2], [21], [5], [51], [31], [-39], [-9], [-11], [0], [41], [33], [-11], [-48], [-46], [-12], [14], [-18], [-23], [18], [-34], [-42], [37], [48], [-7], [-33], [-7], [21], [48], [30], [32], [12], [-12], [-30], [-20], [8], [-20], [24], [39], [-5], [0], [-1], [-32], [-30], [8], [-30], [-56], [8], [53], [-33], [22], [48], [-51], [6], [61], [-8], [58], [-27], [34], [36], [-56], [51], [12], [12], [41], [44], [-3], [27], [30], [-8], [-5], [28], [6], [48], [47], [-15], [3], [67], [29], [24], [-28], [27], [60], [46], [23], [30], [18], [-18], [58], [-14], [18], [-61], [27], [62], [66], [-42], [10], [-66], [14], [-29], [40], [-30], [26], [39], [-58], [-72], [-18], [-12], [-12], [74], [-31], [43], [36], [60], [-16], [-60], [4], [57], [-58], [69], [57], [-62], [12], [30], [2], [28], [-42], [38], [28], [60], [32], [7], [-66], [-17], [-45], [-3], [13], [-66], [-5], [-47], [-55], [5], [-14], [29], [-6], [-53], [-20], [0], [-66], [23], [-27], [-17], [-4], [-24], [-79], [-6], [-19], [-18], [-66], [-36], [66], [60], [46], [12], [-26], [-6], [3], [-46], [14], [39], [-25], [69], [-4], [34], [81], [-23], [24], [20], [-48], [-30], [90], [4], [24], [16], [54], [-18], [-37], [-78], [37], [23], [-39], [70], [9], [-2], [7], [73], [24], [12], [80], [-9], [-1], [-30], [74], [48], [-47], [24], [-49], [-43], [-71], [57], [-15], [10], [-27], [84], [2], [37], [-66], [72], [40], [-2], [-48], [-8], [22], [72], [9], [-48], [33], [-54], [8], [90], [42], [-15], [74], [-44], [42], [-74], [-38], [-90], [61], [9], [-39], [-4], [-16], [-45], [-72], [38], [3], [-44], [-78], [30], [65], [66], [-5], [-42], [17], [-4], [-34], [24], [82], [87], [54], [-10], [36], [-14], [-74], [48], [-68], [72], [-14], [-66], [94], [-72], [6], [11], [-82], [84], [-23], [24], [64], [-54], [-27], [62], [77], [-36], [6], [28], [-51], [48], [42], [-83], [15], [-69], [-2], [87], [21], [-3], [-88], [21]]
