
# q-expansion of newform 3234.2.a.f, downloaded from the LMFDB on 23 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 = 3234
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], [-1], [0], [0], [1], [4], [-3], [1], [-3], [-9], [-2], [-7], [6], [11], [3], [0], [-9], [10], [-4], [3], [4], [-16], [0], [0], [1], [-15], [4], [-18], [-10], [12], [-1], [-18], [12], [19], [9], [-19], [13], [2], [-12], [6], [-15], [10], [0], [-10], [-21], [-20], [-4], [-8], [-18], [-26], [9], [-24], [28], [-15], [-12], [-18], [-6], [16], [32], [-15], [-20], [-9], [4], [21], [1], [-12], [-10], [-10], [12], [-8], [24], [-18], [4], [-22], [14], [-15], [12], [13], [-6], [-14], [-9], [17], [-12], [-29], [37], [3], [0], [2], [-21], [8], [33], [-6], [-4], [-36], [-10], [-30], [36], [30], [4], [20], [-19], [45], [36], [-21], [29], [22], [-12], [-9], [48], [-38], [-32], [-16], [18], [-14], [-16], [18], [46], [-48], [24], [-18], [-35], [44], [33], [33], [-50], [-27], [35], [-9], [22], [-14], [-4], [6], [2], [-7], [-30], [-50], [-54], [-23], [-42], [-6], [-20], [42], [14], [42], [19], [36], [-20], [57], [40], [24], [32], [-12], [2], [0], [-40], [9], [-7], [-6], [-8], [21], [3], [30], [-37], [0], [-12], [39], [8], [28], [-28], [-6], [-42], [22], [-30], [8], [-32], [-30], [47], [18], [16], [28], [-52], [-18], [29], [33], [-3], [60], [56], [-56], [-4], [-18], [61], [42], [20], [30], [18], [-30], [44], [-37], [42], [-54], [-42], [-5], [-50], [-44], [63], [36], [16], [-6], [30], [58], [2], [48], [44], [12], [0], [-59], [-40], [30], [-12], [66], [-10], [-65], [30], [-70], [36], [-64], [6], [-3], [-17], [-48], [5], [61], [-31], [-57], [58], [72], [-11], [-6], [60], [-48], [33], [64], [-14], [11], [-45], [-36], [16], [-39], [-49], [-36], [-40], [27], [33], [46], [-54], [-18], [62], [37], [42], [-20], [-16], [42], [-32], [-2], [-66], [-38], [-6], [54], [23], [18], [-59], [-52], [34], [-61], [-50], [34], [-33], [-10], [-34], [-6], [42], [8], [-30], [-14], [-23], [39], [62], [15], [64], [75], [-66], [33], [42], [18], [-58], [30], [-20], [18], [66], [25], [-50], [57], [-25], [9], [-82], [-46], [-36], [-41], [-39], [-10], [78], [-54], [45], [-4], [-36], [82], [45], [-42], [13], [-81], [-65], [-67], [3], [-16], [45], [46], [80], [28], [-42], [-6], [44], [48], [-5], [0], [44], [-36], [-61], [-33], [79], [-29], [26], [-24], [-81], [-76], [48], [-12], [13], [26], [42], [-18], [-41], [29], [30], [1], [14], [-39], [0], [-96], [-21], [0], [62], [-45], [-24], [-60], [64], [-55], [51], [20], [-88], [45], [88], [42], [66], [28], [5], [-27], [72], [-56], [78], [-8], [84], [-36], [-16], [-33], [-41], [72], [50], [-41], [95], [-21], [-37], [-30], [-69], [-26], [-3], [68], [79], [12], [38], [-57], [-14], [-36], [32], [102], [99], [-32], [17], [81], [-2], [-33], [10], [-15], [27], [71], [8], [90], [42], [-14], [-18], [-18], [-42], [64], [15], [-90], [61], [84], [72], [21], [16], [6]]
