
# q-expansion of newform 4368.2.a.m, downloaded from the LMFDB on 24 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 = 4368
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], [1], [-3], [-1], [-6], [-1], [-8], [1], [-1], [-5], [-3], [12], [10], [11], [3], [-1], [8], [2], [-2], [-8], [11], [-13], [7], [7], [-1], [6], [-16], [-12], [14], [-15], [-20], [0], [8], [-10], [-10], [-8], [-16], [12], [-7], [24], [7], [14], [8], [-14], [-6], [-16], [15], [-1], [-12], [14], [3], [24], [-1], [18], [30], [21], [-4], [0], [-11], [22], [4], [5], [27], [-30], [10], [-28], [18], [9], [16], [-17], [-10], [-8], [-6], [22], [-34], [4], [-26], [7], [12], [-29], [-18], [-8], [-2], [16], [10], [-29], [20], [0], [-6], [6], [2], [17], [-14], [-28], [-8], [30], [29], [-28], [42], [36], [13], [8], [32], [-21], [-37], [18], [15], [33], [31], [26], [-34], [8], [-18], [28], [2], [45], [-16], [-10], [34], [-31], [43], [9], [14], [-16], [7], [-11], [-18], [34], [6], [-33], [10], [-12], [23], [5], [-43], [5], [30], [17], [-18], [5], [4], [42], [40], [12], [42], [24], [-7], [26], [42], [-12], [-8], [-30], [36], [-8], [-37], [29], [40], [33], [-12], [15], [-50], [37], [-22], [42], [-58], [-51], [36], [-8], [-10], [36], [-34], [30], [-11], [-8], [-12], [-8], [12], [-6], [2], [-38], [-32], [-6], [-58], [13], [-42], [-58], [3], [17], [-14], [-19], [32], [46], [3], [-19], [13], [-60], [-34], [-39], [-33], [-30], [60], [32], [3], [28], [15], [48], [1], [31], [-40], [58], [59], [0], [-16], [-4], [3], [9], [59], [8], [28], [-24], [-23], [56], [44], [-46], [42], [67], [36], [-15], [26], [-18], [18], [-52], [-56], [-42], [-22], [3], [62], [36], [-63], [-54], [-36], [46], [-44], [-78], [-21], [-28], [23], [-55], [44], [73], [32], [6], [-14], [-50], [14], [0], [27], [52], [56], [51], [43], [61], [31], [-62], [-18], [-70], [-6], [3], [-6], [74], [-14], [16], [-70], [5], [-8], [16], [64], [4], [-72], [-24], [6], [-26], [-16], [-17], [76], [-45], [-37], [-84], [40], [52], [-21], [52], [46], [5], [1], [20], [-78], [-36], [20], [61], [-14], [-12], [-57], [-1], [-35], [-54], [4], [28], [50], [-13], [80], [74], [38], [-59], [12], [36], [-15], [-10], [-63], [-15], [-40], [4], [58], [56], [60], [-8], [-8], [47], [-68], [-10], [-30], [23], [26], [-33], [46], [57], [4], [-46], [50], [18], [-24], [70], [26], [-24], [96], [24], [42], [-72], [-27], [46], [-11], [-10], [-68], [46], [-66], [-14], [51], [7], [-6], [26], [34], [-78], [-21], [-16], [-98], [-17], [73], [-14], [-25], [-12], [-38], [36], [-93], [-68], [-25], [28], [-96], [22], [16], [86], [89], [26], [44], [-72], [-21], [91], [20], [-80], [64], [12], [-34], [51], [-48], [-36], [-12], [-23], [-80], [66], [-23], [-8], [-71], [79], [58], [39], [49], [-60], [-12], [52], [-61], [-45], [-52], [17], [-87], [-28], [71], [36], [68], [-68], [50], [68], [-17], [38], [-12], [-36], [-22], [33], [91], [84], [59], [-70]]
