# q-expansion of newform 615.2.j.b, downloaded from the LMFDB on 19 June 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 = 615
weight = 2
poly_data = [1, 0, 0, 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 = [[206, [1, 0, 0, 0]], [247, [1, 0, 0, 0]], [211, [0, 0, 1, 0]]]
aps_data = [[0, 1, 0, 1], [0, 0, 0, 1], [0, 0, -1, 0], [2, 0, -2, 0], [0, 0, 0, 3], [-1, 0, 1, -4], [3, -1, 3, 0], [5, 0, 5, 0], [0, 4, 0, -4], [0, 0, 0, 5], [1, 4, 0, -4], [-7, -1, 0, 1], [4, 0, 4, -3], [0, 3, -3, 3], [-7, -1, -7, 0], [2, 0, -2, -2], [-2, -5, 0, 5], [0, -4, 9, -4], [2, -10, 2, 0], [2, 0, -2, 1], [0, -9, 1, -9], [10, 0, -10, -2], [-2, 3, 0, -3], [-2, 0, 2, 8], [2, -2, 2, 0], [4, 5, 4, 0], [0, 3, -11, 3], [10, 6, 0, -6], [-4, 12, -4, 0], [2, -3, 0, 3], [0, -6, 0, 6], [0, 0, 4, 0], [-7, -5, -7, 0], [-10, -8, 0, 8], [-2, -12, -2, 0], [-6, 0, 6, -12], [9, 0, -9, 2], [5, 5, 0, -5], [-2, 0, 2, -8], [0, -6, -2, -6], [-10, -7, -10, 0], [-9, -6, -9, 0], [8, -14, 8, 0], [1, 0, -1, 8], [0, 7, -14, 7], [10, -10, 10, 0], [5, -6, 5, 0], [12, 8, 0, -8], [-15, -7, -15, 0], [0, -6, 0, 0], [-8, 0, 8, 2], [2, 0, -2, -26], [0, 14, 3, 14], [0, 10, -10, 10], [-11, 0, 11, -5], [7, -5, 7, 0], [-16, 4, 0, -4], [-7, -16, 0, 16], [-5, -7, 0, 7], [-8, 11, -8, 0], [-11, -5, 0, 5], [1, 15, 1, 0], [0, 7, 3, 7], [-20, -4, -20, 0], [0, 8, 0, 0], [-7, 0, 7, -9], [0, 0, 0, -34], [0, 13, -9, 13], [-9, 11, -9, 0], [0, 10, 11, 10], [-34, -1, 0, 1], [12, -12, 0, 12], [0, 1, -5, 1], [21, 5, 0, -5], [-10, 4, 0, -4], [3, -3, 3, 0], [0, -4, -6, -4], [-16, 0, 16, 12], [0, -3, -6, -3], [15, 2, 0, -2], [0, 4, 16, 4], [0, 0, 0, -12], [0, 4, -32, 4], [-15, 7, 0, -7], [0, 0, 0, 14], [0, 4, 20, 4], [0, -3, 6, -3], [1, -4, 1, 0], [0, 11, 0, -11], [22, 0, -22, -4], [-2, -9, 0, 9], [2, 0, -2, 7], [0, -7, 19, -7], [8, 13, 0, -13], [13, 0, -13, -10], [7, 0, -7, -21], [-6, 13, -6, 0], [18, 0, -18, -13], [10, 12, 0, -12], [0, 10, 16, 10], [-10, -16, -10, 0], [-23, 7, -23, 0], [-1, 0, 1, -43], [0, 7, 18, 7], [8, 0, -8, 12], [6, 0, -6, 8], [-9, 0, 9, -11], [-9, -23, -9, 0], [10, -13, 0, 13], [20, 6, 20, 0], [0, -16, 2, -16], [0, -10, 30, -10], [0, -3, -10, -3], [-31, 0, 0, 0], [-19, 4, 0, -4], [0, -27, 0, 0], [-11, 0, 11, 0], [0, -5, 20, -5], [-15, 0, 15, 9], [-8, 0, 8, 2], [0, -6, -12, -6], [-18, 14, -18, 0], [0, 6, -10, 6], [-18, 8, -18, 0], [19, -2, 19, 0], [0, -11, 0, 11], [-18, 0, 18, -2], [-2, 29, -2, 0], [-6, 0, 6, -2], [0, 27, 13, 27], [-5, 4, 0, -4], [0, -9, -2, -9], [0, 0, 0, 34], [9, 6, 9, 0], [12, 6, 0, -6], [29, -4, 0, 4], [9, -3, 9, 0], [0, 11, -11, 11], [14, -14, 0, 14], [-2, 0, 2, -20], [0, 32, 11, 32], [-24, 4, 0, -4], [25, 0, -25, -18], [9, 0, -9, -1], [0, -22, -19, -22], [28, 18, 28, 0], [0, -6, -4, -6], [-4, 19, 0, -19], [0, 8, -3, 8], [0, -14, 6, -14], [-15, -7, 0, 7], [0, 10, 20, 10], [10, -30, 10, 0], [21, 9, 21, 0], [0, 36, 6, 36], [0, 11, 34, 11], [-13, 30, -13, 0], [8, 25, 8, 0], [9, -4, 9, 0], [0, 13, -20, 13], [34, -15, 0, 15], [44, 0, 0, 0], [-14, 14, -14, 0], [26, 0, -26, -23], [33, 0, -33, 15], [-6, -1, 0, 1], [20, 0, -20, 28], [-16, 0, 16, 16]]