
# q-expansion of newform 1134.2.a.f, downloaded from the LMFDB on 25 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 = 1134
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], [-1], [-1], [-6], [5], [-7], [-4], [4], [-6], [2], [-3], [-1], [0], [-12], [7], [-12], [13], [8], [1], [-6], [-16], [6], [-5], [4], [14], [-3], [-2], [10], [-12], [4], [19], [-5], [24], [10], [2], [-4], [-20], [-2], [-24], [0], [-12], [17], [-10], [14], [-16], [-4], [-3], [-26], [29], [-6], [23], [-3], [15], [-18], [-20], [-6], [2], [-22], [4], [0], [7], [-2], [-17], [6], [8], [-9], [3], [-14], [15], [-2], [-22], [22], [-17], [-4], [-8], [18], [-9], [11], [12], [-12], [0], [-25], [-24], [-7], [-17], [1], [-14], [-8], [13], [20], [-10], [-33], [-29], [-6], [-30], [9], [-28], [-24], [-21], [28], [-31], [15], [-33], [-35], [47], [6], [-24], [-19], [-24], [42], [17], [37], [8], [-1], [-7], [-12], [6], [-16], [28], [14], [-30], [-39], [32], [-8], [-4], [-6], [-14], [-18], [-33], [6], [-18], [-48], [-10], [22], [52], [-12], [12], [43], [31], [-46], [34], [-12], [4], [20], [44], [-30], [29], [-38], [16], [-42], [53], [6], [-27], [0], [16], [18], [2], [-20], [37], [35], [14], [12], [15], [60], [-28], [2], [26], [-48], [55], [-14], [30], [-9], [-24], [10], [-21], [-10], [20], [38], [12], [23], [22], [55], [64], [30], [60], [44], [-39], [48], [37], [-21], [-65], [38], [-33], [47], [23], [-50], [-9], [-18], [-54], [18], [56], [9], [21], [30], [8], [27], [21], [-13], [-55], [40], [6], [16], [-54], [-25], [34], [29], [14], [-22], [22], [-38], [39], [-38], [15], [-10], [26], [-56], [24], [-60], [-6], [-44], [-50], [57], [-4], [-6], [-2], [-42], [57], [4], [-55], [-28], [32], [24], [35], [6], [-54], [-19], [20], [-42], [-8], [-57], [-12], [10], [-22], [47], [-58], [-60], [-38], [47], [-14], [43], [-62], [70], [-6], [55], [-18], [31], [-11], [-54], [-42], [-64], [21], [-52], [-23], [-24], [-4], [0], [-29], [-15], [0], [18], [78], [50], [-27], [-52], [-54], [2], [-28], [63], [18], [63], [-39], [-77], [38], [-24], [8], [-42], [72], [-60], [67], [48], [58], [24], [-52], [2], [-32], [8], [20], [-4], [54], [-80], [70], [8], [34], [-55], [-15], [-44], [-7], [78], [-7], [-71], [-72], [-8], [-26], [-37], [71], [-47], [-58], [14], [-70], [80], [-60], [41], [55], [4], [6], [7], [77], [64], [26], [78], [0], [18], [-42], [9], [2], [53], [12], [6], [20], [17], [-30], [60], [52], [-10], [-26], [-84], [-58], [56], [88], [-73], [4], [-73], [79], [-61], [12], [24], [-83], [-52], [25], [-54], [-32], [-50], [-92], [-69], [-40], [-2], [-43], [38], [-12], [-29], [16], [-51], [31], [-62], [68], [-56], [52], [2], [35], [-66], [71], [-55], [34], [18], [-86], [-67], [45], [74], [-76], [39], [64], [-38], [-102], [-16], [52], [51], [-7], [104], [-83], [-10], [-19], [-27], [62], [-4], [20], [52], [-39], [76], [92], [-58], [-10], [63], [14], [-32], [-53], [18], [12], [-40]]
