# q-expansion of newform 198.4.a.b, downloaded from the LMFDB on 25 October 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 = 198
weight = 4
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 = [[-2], [0], [3], [-10], [-11], [-16], [-42], [116], [-189], [120], [-163], [-409], [-468], [110], [-144], [-90], [453], [20], [-97], [465], [848], [-742], [-438], [273], [761], [1578], [1208], [1446], [-1342], [-1761], [2132], [-1782], [-921], [1442], [1806], [-1402], [-2947], [332], [-1920], [-1986], [-111], [899], [1509], [20], [-1254], [4688], [4868], [-4663], [5862], [5051], [720], [522], [-5632], [6345], [-1494], [-1254], [-5442], [2432], [-1666], [3258], [-7804], [-12], [5852], [-2652], [-7153], [-6129], [-823], [5270], [-6576], [-6286], [10701], [-8328], [-3643], [-7942], [-9331], [7731], [-1461], [-6466], [-2010], [-12622], [-10980], [-538], [-5022], [-2587], [704], [-4203], [933], [10316], [-132], [7823], [-723], [7416], [-7171], [12264], [-11860], [-15450], [17181], [2763], [-772], [19388], [1568], [4806], [-9972], [456], [11132], [11657], [15204], [4884], [-22008], [19526], [4934], [-20608], [7038], [14609], [15365], [6753], [5123], [-18075], [-15387], [-24222], [-12967], [17870], [-12978], [28488], [17855], [31674], [371], [-27069], [-20095], [-12220], [-32818], [-24396], [-2485], [-14254], [-17232], [25520], [-14994], [-35224], [-18609], [-16476], [-2722], [18402], [-39763], [1248], [-8659], [41271], [2198], [-27192], [27095], [16008], [22424], [11427], [30020], [-11958], [-13060], [-10188], [-43738], [16470], [23876], [-22218], [9909], [45858], [-33352], [-37665], [35715], [-5949], [-23560], [26786]]