
# q-expansion of newform 2366.2.a.f, downloaded from the LMFDB on 20 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 = 2366
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], [3], [1], [0], [0], [6], [-4], [3], [6], [-10], [8], [0], [8], [6], [12], [3], [11], [2], [-3], [2], [-4], [0], [-6], [2], [-6], [14], [-18], [-16], [-18], [11], [15], [3], [-16], [0], [-1], [2], [20], [-12], [9], [6], [5], [-24], [5], [12], [14], [-4], [8], [-15], [-22], [-3], [21], [-28], [3], [-6], [-27], [9], [2], [26], [18], [-31], [-6], [-13], [-6], [8], [-12], [-10], [14], [-24], [29], [-6], [-9], [-10], [32], [-28], [-6], [-30], [-7], [6], [-22], [0], [26], [-33], [-4], [14], [18], [9], [29], [21], [-31], [-3], [30], [-13], [12], [-22], [42], [-15], [-36], [-1], [38], [20], [24], [-36], [-45], [8], [-28], [3], [0], [3], [26], [-10], [-22], [-3], [-37], [5], [-9], [41], [-36], [30], [6], [23], [-10], [-51], [-24], [-1], [-12], [-10], [-48], [-52], [35], [-16], [48], [17], [-52], [12], [-10], [-30], [53], [-15], [-42], [-19], [12], [-13], [24], [-25], [-54], [17], [-18], [-4], [-57], [-22], [-6], [20], [12], [14], [21], [-7], [18], [8], [15], [-42], [-15], [-16], [3], [39], [-30], [17], [35], [-7], [-3], [-48], [-31], [-27], [17], [56], [0], [38], [-12], [38], [-1], [-7], [12], [-28], [-42], [33], [21], [-52], [11], [11], [48], [-16], [-12], [2], [-18], [48], [18], [50], [44], [60], [24], [-36], [-46], [50], [2], [12], [-45], [-22], [24], [-9], [-13], [11], [-6], [-16], [-39], [6], [32], [-61], [60], [-27], [-42], [32], [14], [42], [8], [-33], [74], [-24], [21], [44], [18], [-4], [44], [68], [27], [-1], [-30], [14], [36], [30], [18], [66], [-4], [8], [-16], [42], [-42], [-52], [-15], [-34], [27], [32], [6], [9], [-22], [66], [12], [-28], [-37], [-33], [20], [11], [-18], [-70], [2], [-12], [-1], [18], [30], [-4], [-30], [5], [32], [8], [-7], [2], [-70], [-42], [-10], [59], [-48], [-24], [-40], [-24], [-37], [41], [0], [65], [42], [-4], [-18], [-66], [0], [18], [-12], [26], [-63], [-58], [9], [27], [-31], [-40], [36], [11], [36], [14], [-58], [-48], [35], [81], [-4], [-48], [-48], [39], [80], [57], [32], [72], [-9], [-34], [-39], [-73], [59], [3], [65], [81], [-28], [-88], [71], [72], [-78], [-22], [-60], [62], [12], [-34], [33], [-58], [0], [14], [-58], [-4], [-33], [-42], [-37], [60], [-60], [5], [44], [54], [18], [44], [65], [-12], [-22], [2], [-60], [60], [57], [18], [51], [-10], [54], [-96], [-36], [44], [-10], [45], [53], [65], [72], [-1], [15], [60], [-28], [62], [84], [24], [-4], [12], [-10], [-66], [45], [29], [-33], [5], [-12], [53], [-73], [-16], [6], [89], [39], [-42], [-85], [-24], [-1], [-34], [6], [86], [-42], [83], [57], [-43], [24], [-18], [-4], [-28], [-3], [-4], [39], [-58], [-42], [-6], [-10], [-43], [21], [-21], [8], [102], [-48], [72], [-97], [48], [-33], [-82], [-33], [84], [-39], [35], [-24]]
