
# q-expansion of newform 3240.2.a.f, downloaded from the LMFDB on 20 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 = 3240
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 = [[0], [0], [1], [0], [5], [0], [-3], [5], [-6], [10], [-2], [4], [3], [3], [-4], [6], [3], [2], [-11], [14], [-15], [10], [12], [-14], [-13], [12], [0], [17], [-12], [6], [10], [4], [-7], [7], [-4], [22], [0], [16], [22], [2], [0], [-4], [-12], [19], [8], [-24], [12], [2], [-13], [-16], [11], [-6], [-23], [-11], [31], [12], [30], [24], [-14], [-10], [-4], [10], [-7], [-10], [7], [-28], [20], [-31], [3], [32], [-25], [12], [-8], [-26], [-35], [-12], [24], [8], [7], [-7], [0], [38], [-28], [27], [-14], [29], [-9], [-25], [-4], [-4], [11], [-26], [34], [-15], [-23], [-34], [-44], [27], [-4], [24], [-9], [0], [-5], [15], [-3], [-31], [17], [-30], [-16], [-1], [-32], [-2], [-25], [-19], [2], [-33], [-41], [-30], [-34], [-44], [0], [-26], [-28], [13], [-8], [6], [-46], [0], [16], [28], [5], [-34], [20], [-26], [-10], [-54], [30], [28], [-32], [15], [35], [-4], [12], [12], [40], [-14], [32], [2], [-11], [26], [22], [2], [-47], [6], [13], [-6], [14], [-30], [-42], [16], [21], [13], [-6], [-12], [-23], [-52], [28], [-42], [-2], [30], [-11], [28], [18], [17], [-6], [50], [1], [-32], [-16], [-8], [-52], [-37], [4], [-57], [-8], [-18], [-38], [-40], [41], [50], [1], [51], [-39], [16], [-9], [-5], [-25], [-58], [-1], [24], [40], [44], [28], [15], [9], [32], [28], [39], [33], [-17], [7], [-18], [26], [52], [-42], [-15], [-22], [-25], [54], [54], [14], [-26], [-17], [46], [65], [-48], [-42], [6], [58], [-12], [14], [56], [-58], [-59], [-4], [42], [26], [-16], [-61], [-32], [41], [-20], [26], [-10], [-49], [36], [-32], [-3], [-20], [-34], [42], [45], [30], [46], [20], [33], [50], [-52], [-8], [43], [-2], [-33], [22], [66], [-54], [-17], [6], [33], [29], [-24], [-42], [52], [61], [10], [-23], [-34], [-12], [-40], [43], [33], [18], [14], [48], [-10], [-53], [58], [-66], [-12], [-24], [53], [26], [37], [63], [-21], [62], [18], [-26], [-42], [-84], [-20], [7], [6], [0], [84], [52], [-2], [-28], [74], [-54], [14], [-6], [16], [-26], [52], [20], [3], [45], [68], [13], [34], [-27], [3], [-56], [50], [-6], [-35], [-7], [-5], [-52], [-42], [2], [36], [-22], [3], [-83], [-48], [38], [-69], [63], [42], [14], [-62], [0], [36], [84], [11], [-82], [-27], [0], [22], [60], [-37], [-50], [-88], [76], [-34], [-66], [36], [50], [-20], [-92], [57], [-6], [49], [29], [9], [-38], [-92], [45], [-80], [29], [-100], [8], [-56], [32], [-45], [-48], [-70], [-57], [62], [-50], [13], [86], [19], [-61], [-80], [-64], [32], [-52], [22], [-13], [-20], [29], [67], [32], [-50], [52], [87], [-69], [36], [22], [41], [10], [42], [-18], [-28], [-80], [-33], [47], [-28], [-47], [28], [17], [79], [-42], [-76], [-90], [-38], [-23], [-26], [106], [-6], [22], [39], [10], [-24], [-53], [-62], [-16], [68]]
