
# q-expansion of newform 4235.2.a.m, downloaded from the LMFDB on 18 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 = 4235
weight = 2
poly_data = [-4, -1, 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], [-1, 1], [1, 0], [1, 0], [0, 0], [-3, 1], [3, -1], [2, 2], [-2, 2], [1, -3], [0, 0], [6, 0], [0, -2], [-6, 2], [-1, -3], [0, -2], [-4, 0], [0, -6], [0, 4], [8, 0], [2, 4], [5, -1], [-4, 0], [4, -2], [-7, 5], [6, 4], [3, 1], [2, -6], [-13, 3], [-14, 0], [-4, 4], [6, -2], [-12, -2], [10, 2], [-2, -4], [-11, 7], [10, 4], [-2, -2], [-11, 7], [7, -1], [20, 0], [8, -10], [-11, -1], [-4, -6], [4, -2], [-12, 4], [9, -9], [-5, 1], [-19, 3], [16, -2], [0, -2], [-7, -5], [-6, 4], [-14, 2], [10, -8], [18, -2], [10, -12], [16, 0], [-10, 4], [-15, 1], [19, -3], [-5, 3], [27, -3], [6, 10], [11, -13], [-10, 8], [12, 0], [22, -8], [-2, -2], [10, -8], [-7, 5], [-8, 0], [-1, -3], [-20, 6], [4, -8], [-4, 4], [-7, -7], [25, 1], [29, 1], [14, -8], [16, 4], [5, -3], [7, 5], [-2, 4], [6, -6], [-18, 6], [11, -5], [12, 2], [8, 2], [-8, 8], [25, -1], [-10, 2], [2, -2], [13, 11], [13, 11], [-21, -7], [-22, 4], [10, 0], [-20, -8], [25, 5], [-8, 4], [8, 2], [28, 0], [-10, 16], [-20, -8], [-1, -9], [-16, -4], [25, 1], [1, -5], [14, -8], [19, 9], [-10, -12], [22, 4], [-14, 18], [1, -5], [2, 0], [-1, 1], [16, 8], [18, -20], [19, -11], [0, -2], [4, -14], [13, -7], [24, -12], [-4, -8], [-11, -7], [-1, 11], [34, -10], [32, 8], [1, -3], [-17, -7], [-8, 16], [35, -7], [24, 6], [-38, -4], [22, 8], [19, 7], [21, 11], [1, 9], [27, -17], [6, -10], [-17, 15], [-16, -8], [-10, 6], [40, -2], [-34, 2], [38, -12], [26, -4], [4, -8], [-36, 4], [30, 4], [24, -14], [12, -8], [0, -8], [22, -18], [-8, -8], [-7, -13], [-24, 18], [7, 19], [-8, -14], [4, 0], [28, 2], [30, 2], [-12, 24], [-18, 20], [13, 15], [-8, 8], [-5, 3], [7, 11], [-10, 0], [-8, -4], [-18, 16], [3, 1], [-6, 12], [38, 2], [20, -18], [-13, 13], [1, -15], [4, -12], [-24, 2], [-12, 12], [12, 16], [14, -8], [6, 16], [-12, -4], [18, -12], [-4, -18], [23, -23], [51, -9], [-24, 24], [-34, -12], [-14, 10], [11, 5], [52, 2], [-28, 8], [39, -9], [45, -3], [0, -26], [3, 15], [-19, -17], [26, -8], [-30, -10], [-18, 8], [2, -20], [2, 2], [-35, -3], [2, 6], [-4, 16], [-6, 12], [40, -12], [16, -26], [26, 12], [24, 16], [27, -19], [42, -10], [-4, -10], [-14, -10], [24, -2], [26, -10], [-22, 4], [-10, 20], [10, 6], [57, 5], [34, 6], [37, -5], [-38, 16], [17, -27], [-7, 11], [13, 15], [-4, -8], [-4, 10], [-10, -2], [-16, 16], [3, 7], [27, -3], [-51, 7], [-20, 14], [-32, 2], [-37, -3], [-38, -10], [16, 4], [32, -12], [-49, 5], [-2, 16], [-39, 1], [38, -6], [56, -8], [-58, 6], [-15, 23], [-6, -18], [14, 4], [18, -4], [8, -16], [8, -6], [-2, -28], [43, -3], [-2, -8], [12, -24], [3, -9], [-39, -7], [26, -2], [-22, -6], [0, 18], [-15, 13], [-18, 4], [-30, -6], [-47, 9], [-26, 4], [-64, 4], [-10, 20], [36, 2], [-48, 4], [9, 13], [-60, 12], [-42, -8], [-1, 5], [20, 24], [-18, 4], [27, 15], [-22, 26], [5, -1], [-29, 17], [16, -24], [-46, 8], [21, -13], [-7, 27], [-2, 4], [54, 4], [-40, 0], [0, -26], [33, -7], [33, -9], [20, -22], [-8, -20], [30, -28], [36, 6], [18, -10], [59, -17], [-28, 32], [33, -1], [-37, -17], [22, 12], [-21, -15], [-24, -20], [-21, 13], [30, -12], [-42, 14], [-16, 6], [-16, 8], [12, 22], [32, 0], [-53, -1], [2, -20], [-26, -10], [2, 14], [-4, -14], [70, -10], [7, -11], [5, 9], [-1, 19], [6, -18], [-20, -10], [38, 12], [0, -16], [34, -28], [24, -14], [-5, -27], [4, -24], [-12, 12], [-8, -6], [34, -16], [-16, -10], [2, 6], [-13, 21], [-12, 0], [-73, 1], [-3, -3], [2, 12], [42, 0], [-63, 3], [-42, 0], [20, 14], [-2, 12], [-35, 31], [-38, 20], [-68, 0], [-42, 16], [28, -32], [46, -6], [-31, 33], [92, 0], [32, 2], [1, -7], [17, -13], [47, -17], [-2, -24], [26, -18], [-6, 18], [-34, -16], [70, -6], [24, -6], [20, 26], [44, 4], [35, -3], [4, -16], [2, 8], [-9, 15], [14, -14], [31, 11], [-83, -5], [-40, 4], [44, 4], [-3, -7], [6, 26], [42, -28], [0, 4], [3, 9], [-15, -7], [-38, 36], [21, -19], [82, -4], [-30, 8], [40, 16], [-50, 16], [-4, -32], [-45, -15], [25, -37], [-13, 23], [-44, 0], [25, -37], [21, 25], [53, -19], [-1, 25], [-68, 8], [-7, 3], [-48, -2], [36, 12], [-36, 22], [33, -1], [-18, -20], [36, -26], [-8, -30], [-14, 22], [1, -11], [4, -2], [34, -2], [-10, -8], [29, 13], [-20, 24], [12, 24], [-33, 19], [-38, -16], [-54, -6], [68, -8], [-34, 24], [-32, 22], [-65, -3], [4, -28], [-2, 36], [-43, 23], [37, -11], [6, 4], [-30, 6], [-20, -8], [11, 3], [25, 1], [-36, 8], [-18, 24], [52, -24], [-24, 32]]
