
# q-expansion of newform 1296.2.a.f, downloaded from the LMFDB on 17 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 = 1296
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], [0], [-2], [-3], [2], [3], [1], [-6], [-6], [4], [-4], [-9], [1], [-6], [-12], [3], [8], [-5], [-12], [11], [4], [12], [-6], [5], [0], [-14], [3], [-16], [-6], [-2], [0], [3], [19], [6], [10], [-4], [4], [-12], [6], [12], [14], [-18], [5], [12], [10], [-20], [-26], [21], [14], [-3], [6], [-7], [21], [21], [18], [24], [-20], [-10], [-6], [4], [-30], [7], [-18], [29], [18], [4], [-1], [33], [-16], [21], [-18], [28], [-34], [-23], [0], [-18], [20], [27], [17], [-12], [20], [-30], [-7], [-8], [3], [-9], [17], [30], [-20], [-15], [-42], [-26], [-15], [13], [0], [0], [3], [-20], [-4], [1], [30], [-39], [-45], [37], [11], [-9], [-6], [12], [-37], [28], [-16], [-27], [-35], [40], [3], [-23], [18], [6], [-36], [-4], [-22], [-36], [-9], [-8], [-30], [-4], [-36], [-26], [14], [-47], [6], [-8], [2], [-42], [2], [-18], [4], [12], [-33], [7], [18], [16], [-12], [-10], [12], [26], [42], [-35], [24], [8], [42], [19], [0], [31], [48], [-38], [6], [14], [60], [3], [-9], [22], [-36], [51], [0], [16], [-28], [2], [0], [-3], [-40], [36], [11], [-38], [-30], [1], [-54], [28], [8], [-38], [-9], [-22], [-27], [54], [0], [-16], [-56], [29], [60], [-1], [-15], [-59], [-36], [33], [-3], [47], [-4], [9], [6], [30], [58], [56], [35], [21], [66], [64], [-27], [-27], [-17], [-67], [-30], [-62], [36], [24], [-55], [-68], [-21], [66], [-48], [-34], [40], [27], [-38], [51], [-22], [6], [12], [34], [24], [26], [52], [22], [9], [16], [42], [-70], [18], [-57], [-42], [-45], [-20], [4], [38], [33], [-48], [28], [45], [-68], [-54], [-22], [-27], [12], [26], [42], [-39], [-16], [28], [0], [-43], [-32], [-15], [-4], [14], [18], [55], [12], [-75], [-41], [30], [-28], [-68], [35], [-50], [23], [-20], [36], [-10], [-25], [-27], [-18], [58], [-30], [26], [49], [-18], [26], [-48], [28], [-63], [24], [39], [69], [51], [-64], [-66], [-50], [-54], [-36], [-68], [-7], [18], [16], [-36], [-8], [2], [60], [-40], [-6], [80], [-48], [72], [-66], [16], [-12], [11], [33], [12], [-37], [-6], [-35], [29], [-84], [-2], [-18], [41], [25], [-71], [-60], [-30], [80], [-12], [16], [21], [-47], [84], [2], [-63], [23], [-20], [32], [-6], [12], [-20], [18], [-33], [-52], [-17], [-72], [78], [28], [89], [78], [28], [-76], [42], [-84], [60], [6], [6], [-22], [-15], [-72], [45], [-53], [-25], [72], [-26], [-19], [60], [85], [36], [36], [-38], [-46], [87], [-54], [-10], [-69], [26], [-42], [15], [40], [45], [-65], [60], [10], [80], [-80], [-12], [-1], [-36], [33], [97], [66], [-58], [-32], [51], [-41], [72], [14], [39], [76], [-42], [-18], [-20], [-94], [-69], [-59], [36], [71], [102], [-33], [-17], [62], [-6], [66], [-14], [-21], [0], [-60], [20], [-36], [51], [-70], [60], [-39], [54], [52], [0]]
