
# q-expansion of newform 196.4.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 = 196
weight = 4
poly_data = [-2, 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, 0], [0, 1], [0, -2], [0, 0], [-26, 0], [0, -24], [0, 73], [0, -67], [-148, 0], [-118, 0], [0, 210], [-254, 0], [0, -65], [122, 0], [0, -218], [-170, 0], [0, -215], [0, 430], [420, 0], [420, 0], [0, -575], [1052, 0], [0, 1025], [0, 725], [0, -223], [0, -1014], [0, -26], [884, 0], [598, 0], [-1548, 0], [2080, 0], [0, -1521], [-2100, 0], [0, -247], [-2990, 0], [656, 0], [0, 1520], [-1742, 0], [0, 998], [0, -876], [-4620, 0], [0, -1560], [-676, 0], [800, 0], [-1270, 0], [0, 1890], [-1236, 0], [0, -3456], [0, -3657], [0, 2340], [840, 0], [-2028, 0], [0, -1401], [0, 5369], [0, -3705], [2172, 0], [0, -3952], [0, 4160], [4246, 0], [-1664, 0], [0, -4199], [0, 3806], [0, 2863], [0, -3016], [0, 351], [3874, 0], [-1790, 0], [9794, 0], [-10086, 0], [0, -1992], [0, 1729], [-7040, 0], [0, 7852], [-2990, 0], [-11570, 0], [0, 4082], [-9838, 0], [0, -10312], [-9906, 0], [0, 2013], [0, 1495], [1686, 0], [8844, 0], [0, -195], [0, 8172], [-5212, 0], [-3042, 0], [-360, 0], [0, -1300], [-4160, 0], [0, -7007], [0, -2642], [11964, 0], [-9636, 0], [-12172, 0], [0, -2716], [0, 12248], [0, 2041], [0, -4785], [11746, 0], [16990, 0], [12090, 0], [0, -14495], [-8390, 0], [-6170, 0], [0, -9633], [0, 12309], [0, 15701], [25232, 0], [0, 5019], [0, -10452], [-11622, 0], [16246, 0], [0, -4693], [21404, 0], [-3622, 0], [0, 8489], [0, 13650], [-19578, 0], [-1962, 0], [0, -1830], [-26460, 0], [0, -14508], [5460, 0], [0, -10545], [-15306, 0], [-29822, 0], [0, 7462], [0, 3830], [0, 17106], [-35958, 0], [23920, 0], [-30724, 0], [22010, 0], [0, -6211], [0, -11373], [0, 25454], [0, 9097], [0, -5208], [-15952, 0], [0, 8541], [-7674, 0], [-5720, 0], [41964, 0], [0, -19370], [0, -16022], [0, -1752], [0, 25355], [0, 18355], [39260, 0], [46874, 0], [0, -3765], [-10540, 0], [0, -4030], [-9524, 0], [15560, 0], [22048, 0], [0, 14545], [0, 31433], [0, -5930], [9990, 0], [7990, 0], [-26364, 0], [0, 34945], [13884, 0], [0, -43066], [16640, 0], [0, 598]]
