
# q-expansion of newform 6042.2.a.f, 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 = 6042
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], [-1], [-1], [0], [4], [-2], [1], [-5], [9], [-3], [-10], [12], [-11], [-2], [-1], [7], [8], [-13], [0], [-6], [-8], [-4], [-1], [10], [11], [-11], [9], [13], [1], [11], [0], [-9], [10], [-12], [7], [-22], [-21], [-14], [-24], [2], [17], [-20], [14], [-16], [-23], [18], [-6], [17], [1], [22], [-16], [-10], [-20], [-20], [-25], [15], [-9], [-20], [-7], [0], [18], [4], [-4], [4], [-10], [-26], [-15], [-24], [-26], [6], [-19], [-16], [35], [-20], [30], [25], [20], [-22], [0], [36], [3], [19], [-30], [26], [9], [-30], [-24], [34], [-4], [42], [25], [-2], [15], [-36], [7], [-30], [10], [-6], [-1], [-8], [-21], [-4], [-36], [-10], [-14], [-24], [20], [-44], [11], [40], [-8], [-38], [7], [22], [-18], [-5], [0], [38], [0], [42], [-28], [18], [24], [-34], [-35], [20], [-49], [-16], [13], [4], [-4], [-42], [19], [10], [-8], [36], [53], [-18], [-27], [24], [-9], [32], [-16], [11], [-9], [34], [-9], [-4], [-17], [22], [-21], [-48], [18], [52], [-21], [-20], [22], [43], [-31], [26], [-5], [24], [23], [-42], [49], [-46], [-3], [-34], [-8], [30], [30], [38], [36], [9], [-40], [4], [-12], [-5], [-39], [20], [12], [53], [46], [36], [-44], [-4], [-40], [46], [42], [-36], [-21], [-33], [-25], [-43], [2], [-20], [6], [-9], [2], [25], [-13], [-11], [63], [10], [63], [19], [57], [40], [45], [-29], [20], [-6], [-10], [25], [42], [-40], [-57], [67], [30], [5], [59], [18], [-74], [-31], [12], [-18], [-12], [2], [-1], [-68], [-16], [32], [-28], [-27], [-27], [-16], [22], [42], [-14], [-44], [-33], [31], [62], [71], [43], [-2], [-38], [26], [44], [66], [64], [54], [21], [-30], [-2], [-28], [-52], [69], [-74], [64], [-31], [-58], [-34], [18], [12], [25], [28], [-31], [-14], [62], [-81], [43], [-56], [32], [48], [36], [8], [-20], [22], [82], [45], [-4], [-8], [46], [18], [-16], [62], [-40], [72], [52], [-42], [24], [-40], [46], [-59], [-50], [-3], [-38], [28], [26], [18], [16], [59], [4], [14], [-51], [34], [-40], [78], [-48], [26], [48], [-14], [-47], [47], [49], [-72], [7], [-48], [-90], [-1], [-48], [-77], [-22], [-21], [-58], [-29], [-10], [14], [48], [-30], [-49], [48], [-3], [47], [10], [27], [76], [-2], [-26], [62], [84], [5], [-41], [-17], [65], [-46], [76], [18], [41], [-83], [-54], [-7], [-54], [86], [27], [-53], [-18], [-21], [-38], [70], [41], [-48], [-31], [6], [-19], [19], [1], [30], [12], [66], [-60], [20], [42], [-36], [-28], [29], [-43], [-14], [18], [54], [26], [0], [20], [51], [-18], [-4], [40], [-23], [-77], [-19], [82], [-53], [-18], [-32], [80], [-60], [-98], [88], [-98], [-57], [-92], [-52], [90], [-70], [-30], [56], [32], [-70], [-24], [8], [-54], [-67], [-2], [12], [-48], [-38], [-59], [-82], [-1], [58], [96], [-33], [84], [-10], [90], [-24], [-34], [-39]]
