# q-expansion of newform 100.4.e.c, downloaded from the LMFDB on 01 May 2026.

# 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 = 100
weight = 4
poly_data = [1, 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 = [[51, [-1, 0]], [77, [0, 1]]]
aps_data = [[2, -2], [-7, -7], [0, 0], [9, -9], [0, 0], [0, 0], [0, 0], [0, 0], [-67, -67], [0, -306], [0, 0], [0, 0], [252, 0], [-297, -297], [-301, 301], [0, 0], [0, 0], [952, 0], [549, -549], [0, 0], [0, 0], [0, 0], [-77, -77], [0, -1386], [0, 0], [-378, 0], [1323, 1323], [-221, 221], [0, -1136], [0, 0], [1089, -1089], [0, 0], [0, 0], [0, 0], [0, -3096], [0, 0], [0, 0], [2943, 2943], [1309, -1309], [0, 0], [0, 0], [-1078, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [4473, 4473], [-4081, 4081], [0, 6874], [0, 0], [0, 0], [-1708, 0], [0, 0], [0, 0], [-6017, -6017], [0, 5544], [0, 0], [0, 0], [-8388, 0], [693, 693], [0, 0], [2709, -2709], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-7871, 7871], [0, -9646], [0, 0], [0, 0], [9639, -9639], [0, 0], [0, 0], [8113, 8113], [0, 14184], [0, 0], [15822, 0], [0, 4844], [0, 0], [9592, 0], [0, 0], [0, 0], [0, 0], [-10237, -10237], [0, -1836], [0, 0], [16002, 0], [11043, 11043], [7889, -7889], [0, 0], [-81, 81], [0, 0], [0, 0], [12803, 12803], [0, -8946], [8442, 0], [-14427, -14427], [6802, 0], [-17721, 17721], [0, 0], [-1687, -1687], [0, -14796], [0, 0], [0, 0], [-11641, 11641], [0, 0], [0, 0], [28532, 0], [-11781, 11781], [0, 0], [0, 0], [0, 0], [0, 0], [-21348, 0], [-21987, -21987], [20839, -20839], [0, 0], [0, 0], [-30688, 0], [0, 0], [0, 0], [-3937, -3937], [0, 0], [-9648, 0], [0, -506], [0, 0], [6489, -6489], [0, 0], [0, 0], [28633, 28633], [0, 0], [0, 0], [-21798, 0], [0, 29554], [0, 0], [-24381, 24381], [0, 0], [0, -26406], [0, 0], [7632, 0], [-22347, -22347], [32729, -32729], [0, 36344], [0, 0], [0, 0], [0, 0], [0, 0], [31693, 31693], [0, 0], [-29988, 0], [-15327, -15327], [2849, -2849], [8109, -8109], [0, 0], [0, 0], [0, -53676], [0, 0], [-44478, 0], [-36851, 36851], [0, 0], [23859, -23859], [0, 0], [0, 0], [36743, 36743], [0, 0], [0, 0]]