
# q-expansion of newform 361.2.e.f, downloaded from the LMFDB on 04 October 2025.

# 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 = 361
weight = 2
poly_data = [1, 0, 0, -1, 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 = [[2, [0, 0, 0, 0, 0, -1]]]
aps_data = [[1, -1, 1, -1, 1, -1], [0, 1, -1, -1, 0, 1], [1, 0, -1, -1, -1, 0], [0, 1, 0, 0, -1, -1], [0, -1, -1, 0, -1, -1], [3, 3, 0, -1, -2, 0], [0, 2, -1, 1, -2, 0], [0, 0, 0, 0, 0, 0], [2, 0, 0, -2, -2, 2], [1, 0, -5, 0, 1, 0], [3, -1, -2, -3, 3, 3], [0, -1, -1, 0, 3, -2], [-3, 4, 1, 4, 0, -4], [3, 0, -3, 2, 0, 5], [3, -1, -2, -1, 3, 0], [-3, 1, 0, 2, 2, 2], [-2, -7, -2, 2, 7, 2], [4, -4, 0, 0, 0, -3], [-2, 6, 6, 6, -2, 0], [0, 0, 0, -2, -10, -2], [4, 0, 4, 0, 0, 0], [7, 3, 1, -6, 0, 6], [0, 6, -9, 0, 3, 3], [-2, -2, 5, 3, -1, -5], [-4, 5, -2, 2, -5, 4], [1, 1, 5, 3, -4, -5], [0, 1, -4, 6, -4, 1], [0, -3, 6, 0, -3, -3], [-4, 0, 4, 2, 5, -2], [-3, 7, 7, 0, -8, 1], [1, 1, 1, -8, 7, -1], [1, -7, 7, -7, 7, -1], [1, 5, 0, -6, -6, -10], [-5, -5, -6, 0, 5, 6], [3, -8, 7, 4, 0, -4], [-10, -3, -3, 0, 3, 0], [-5, 0, 5, -3, -3, -8], [0, 5, -3, 0, -3, 5], [0, -8, 0, 8, 8, 11], [2, 2, -12, 12, -2, -2], [0, -3, 9, 9, 9, -3], [-2, -8, 2, -8, -2, 0], [9, 4, 4, 0, 7, -11], [2, -9, -3, -5, 0, 5], [6, 1, 10, -6, -11, -11], [8, -8, 12, -8, 8, 0], [6, 3, 7, -7, -3, -6], [1, 0, -1, 3, -9, 4], [-6, -3, -3, 0, -6, 9], [3, 6, 6, 0, -14, 8], [9, 0, -9, -14, -1, -5], [0, 1, 10, 3, 10, 1], [5, 5, -9, -2, -3, 9], [-10, 3, 0, 7, 7, 2], [3, 2, 7, 2, 3, 0], [-13, 1, -4, 9, 0, -9], [-3, 4, 10, 13, 0, -13], [-1, 0, 1, 14, -7, 13], [0, 9, -12, -5, -12, 9], [13, -10, 0, -3, -3, -4], [-7, -7, 1, -1, 7, 7], [0, -7, -1, -3, -1, -7], [5, -11, -8, -13, 0, 13], [-9, -7, 5, 9, 2, 2], [7, 10, -10, 10, 7, 0], [0, 0, -28, -1, 1, 28], [0, 6, 4, -15, 4, 6], [3, 0, -3, 5, 15, 8], [-2, 0, 2, 5, -1, 3], [0, -17, 6, 0, 11, 11], [0, -6, 3, 15, 3, -6], [-1, 14, -6, 6, -14, 1], [14, 14, 9, -8, -6, -9], [-14, -21, 15, 14, 6, 6], [15, 3, 3, 0, -8, 5], [-5, 16, 6, 11, 0, -11], [9, -1, -8, -1, 9, 0], [13, -12, 3, -3, 12, -13], [-3, 5, -7, 7, -5, 3], [-14, -2, -7, -2, -14, 0], [3, -16, -16, 0, 11, 5], [-4, -2, -2, 2, 0, -2], [-4, -4, 17, 13, -9, -17], [-17, 8, 0, 9, 9, 9], [1, -18, 9, -9, 18, -1], [-18, -18, -7, 20, -2, 7], [-15, 11, -16, 15, 5, 5], [0, 10, 10, 0, -13, 3], [-6, 0, 6, 7, -25, 1], [15, 6, -16, -15, 10, 10], [0, 3, -3, -21, -3, 3], [-1, -18, 0, 19, 19, -4], [1, 9, 3, -1, -12, -12], [1, 7, -15, -16, 0, 16], [2, 0, -2, -14, -3, -12], [-17, 0, 1, 0, -17, 0], [1, 20, 0, -21, -21, -1], [0, 16, -8, 18, -8, 16], [19, -21, 0, -21, 19, 0], [-3, -1, 9, -1, -3, 0], [-4, 6, 0, -2, -2, -6], [5, 5, -13, 1, -6, 13], [9, -19, 14, -9, 5, 5], [-12, -9, -9, 0, -3, 12], [0, -10, -10, 0, 6, 4], [7, 6, -3, -7, -3, -3], [1, -28, 7, -7, 28, -1], [2, 0, 0, -2, -2, 8], [-22, 13, -17, 13, -22, 0], [27, 13, -6, -27, -7, -7], [-6, -13, -13, 0, 24, -11], [2, 0, -2, -4, 14, -2], [11, -23, -23, -23, 11, 0], [0, -8, 2, -18, 2, -8], [0, 10, 0, -10, -10, -18], [8, 0, -8, -9, 14, -1], [-4, -33, -9, -5, 0, 5], [-3, -19, -19, 0, 5, 14], [-27, 0, 0, 27, 0, 0], [-6, -6, -1, -19, 25, 1], [-2, -11, 0, 13, 13, 10], [0, -9, 1, 0, 1, -9], [0, 1, -5, 0, 4, 4], [12, 5, 5, 0, -1, -4], [-8, 12, 12, 8, -24, -24], [2, 11, 21, -21, -11, -2], [-5, -5, 21, 12, -7, -21], [29, 13, 8, -21, 0, 21], [17, 0, -17, 4, 27, 21], [0, 0, 6, 25, 6, 0], [-15, 3, -2, 2, -3, 15], [-4, 17, -12, 12, -17, 4], [-7, 13, -3, 13, -7, 0], [15, -18, 11, -4, 0, 4], [-3, -5, -5, 0, 4, 1], [-1, 1, 21, 1, -1, 0], [-13, -13, 2, 13, 0, -2], [0, -13, -9, -12, -9, -13], [9, -22, -22, 0, 11, 11], [0, 4, 19, -27, 19, 4], [7, 7, -2, -8, 1, 2], [-3, 13, 0, -10, -10, 8], [10, 10, -26, -11, 1, 26], [-2, 0, 37, 0, -2, 0], [21, 7, -9, -21, 2, 2], [-12, 19, -20, -8, 0, 8], [-18, -15, -29, 29, 15, 18], [-3, -16, -7, 7, 16, 3], [-1, 9, 0, -8, -8, 3], [0, 11, -28, 6, -28, 11], [-2, 15, 7, 9, 0, -9], [-33, 1, 7, 33, -8, -8], [-18, 17, -15, 17, -18, 0], [17, 17, 23, 4, -21, -23], [-25, 0, 25, 2, -1, -23], [-36, -1, -1, 0, 11, -10], [-3, -13, 22, 3, -9, -9], [7, 11, 19, -19, -11, -7], [7, 7, 9, -6, -1, -9], [-6, 20, -23, 20, -6, 0], [-13, 4, -4, 9, 0, -9], [8, -14, 17, 9, 0, -9], [0, 24, -8, 8, -24, 0], [-12, -22, 8, -8, 22, 12], [0, 6, 18, 3, 18, 6], [-11, 0, 11, 38, -7, 27], [-28, 0, -21, 7, 0, -7], [-3, 14, 12, 14, -3, 0]]
