# q-expansion of newform 1104.3.o.b, downloaded from the LMFDB on 23 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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 1104
weight = 3
poly_data = [25, 0, -13, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0], 1], [[0, -6, 0, 2], 5], [[0, 58, 0, -6], 5], [[-26, 0, 4, 0], 1]]

hecke_ring_character_values = [[415, [-1, 0, 0, 0]], [277, [1, 0, 0, 0]], [737, [-1, 0, 0, 0]], [97, [-1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [3, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 0], [0, 0, 0, -1], [0, 3, 2, 0], [0, -4, -1, 0], [23, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -5, 4, 0], [0, 0, 0, 5], [0, -2, 7, 0], [0, 0, 0, -7], [0, 0, 0, 0], [0, 10, 9, 0], [-50, 0, 0, 0], [130, 0, 0, 0], [0, -11, 6, 0], [0, 0, 0, 0], [0, -10, -13, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -15, -14, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 8, 17, 0], [0, 0, 0, 0], [-170, 0, 0, 0], [0, 17, -12, 0], [0, 0, 0, 0], [0, -30, -13, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-34, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 17], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 11], [0, 0, 0, 0], [0, -34, 9, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [110, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-550, 0, 0, 0], [0, -25, -42, 0], [0, 40, -21, 0], [0, -13, 38, 0], [0, 0, 0, 0], [0, 0, 0, -3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 25], [0, 0, 0, 15], [0, 0, 0, 0], [0, 0, 0, 0], [0, -25, 44, 0], [0, 0, 0, 0], [0, 39, 56, 0], [0, 0, 0, 0], [0, 15, 58, 0], [-310, 0, 0, 0], [0, -65, 22, 0], [0, 0, 0, -45], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-586, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 57], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -80, 33, 0], [0, 110, 29, 0], [0, 0, 0, -65], [0, 0, 0, 0], [0, -13, -82, 0], [0, 0, 0, 0], [0, -110, -57, 0], [0, -61, -84, 0], [0, 0, 0, -21], [1126, 0, 0, 0], [0, 0, 0, 0], [830, 0, 0, 0], [0, 0, 0, 35], [0, 0, 0, 0], [0, 0, 0, 0], [0, 78, -53, 0], [0, -129, -16, 0], [0, 4, 91, 0], [0, -135, -28, 0], [0, 65, -66, 0], [0, 0, 0, -35], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -21], [0, 132, -7, 0], [934, 0, 0, 0], [0, 0, 0, 0], [0, 95, 98, 0], [0, 0, 0, 0], [0, 0, 0, 37], [0, 145, 66, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 116, 99, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 82, -77, 0], [0, 165, 26, 0], [0, -148, -87, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 95], [0, 0, 0, 0], [-1430, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 5], [730, 0, 0, 0], [0, 45, -112, 0], [0, 0, 0, 0], [0, 0, 0, -35], [0, -170, 21, 0], [0, 0, 0, 0], [0, -146, -119, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -195, -68, 0], [0, 0, 0, 65], [0, 198, 67, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -68, -147, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 119], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [442, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -205, -102, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -220, -21, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1910, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -200, 37, 0], [1130, 0, 0, 0], [0, 155, 156, 0], [-2158, 0, 0, 0], [0, 0, 0, -123], [-2110, 0, 0, 0], [-26, 0, 0, 0], [0, -89, -176, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -38, -177, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -53, -182, 0], [-2354, 0, 0, 0], [0, 95, -142, 0], [0, 0, 0, 0], [1370, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -261, -104, 0], [0, 0, 0, 145], [0, 0, 0, 0], [0, 0, 0, 0], [-110, 0, 0, 0], [0, 0, 0, 0], [0, 215, 164, 0], [0, 0, 0, 0], [1870, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 135], [0, 209, -94, 0], [0, 0, 0, 0], [0, 30, 209, 0], [1654, 0, 0, 0], [0, 0, 0, 55], [0, 228, -83, 0], [0, 0, 0, 53], [0, 190, -119, 0], [1430, 0, 0, 0], [0, 0, 0, 31], [0, 306, 49, 0], [0, 114, 221, 0], [0, 0, 0, 0], [0, 295, 136, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -312, -43, 0], [-2890, 0, 0, 0], [0, 0, 0, -147], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -145], [0, 0, 0, 0], [1390, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 286, -51, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 55, 238, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-2650, 0, 0, 0], [0, 0, 0, 0], [0, -335, -26, 0], [0, 0, 0, 0], [0, 0, 0, -189], [0, 340, 19, 0], [0, 0, 0, 0], [-3062, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 291, -76, 0], [0, 360, 77, 0], [0, 285, -88, 0], [0, -130, -259, 0], [0, 0, 0, 0], [3418, 0, 0, 0], [0, 365, 114, 0], [0, 0, 0, 0], [0, 369, 106, 0], [-1070, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -215], [3322, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 165], [0, -221, -264, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 133], [0, 0, 0, 0], [0, 207, -182, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 370, 187, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -209], [0, 165, 292, 0], [0, 366, 209, 0], [0, 0, 0, 0], [-1930, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -377, 42, 0], [0, 59, 296, 0]]