# q-expansion of newform 370.2.n.a, downloaded from the LMFDB on 26 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 = 370
weight = 2
poly_data = [1, 0, -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 = [[297, [-1, 0, 0, 0]], [261, [0, 0, -1, 0]]]
aps_data = [[0, 1, 0, 0], [0, 0, 0, 0], [-2, 0, 0, 1], [-2, -1, 1, 1], [3, -2, 0, 1], [-4, 2, 2, -2], [1, 0, 1, 0], [0, -1, 3, -1], [-2, 0, 4, -2], [0, 2, 0, -1], [-1, 10, 0, -5], [0, -3, 0, 7], [0, -4, 1, -4], [1, 0, -2, -7], [-1, 0, 2, 5], [6, -2, 6, 0], [-6, -2, 6, 4], [0, -1, -6, -1], [2, 7, -1, -7], [0, 7, -3, 7], [8, 0, -16, 2], [0, 5, 7, 5], [-2, 6, -2, 0], [-3, -2, 3, 4], [3, 0, -6, -2], [2, -2, 0, 1], [-3, 0, 6, -9], [6, 1, -3, -1], [-8, 3, 8, -6], [4, 10, 4, 0], [4, 4, 4, 0], [4, -4, -4, 8], [-1, 0, 2, 0], [4, 4, -4, -8], [4, -18, 0, 9], [0, -2, 14, -2], [0, -17, 0, 0], [7, -3, 7, 0], [-16, -4, 8, 4], [-4, 7, -4, 0], [8, 8, 0, -4], [0, -9, 4, -9], [-2, -20, 0, 10], [5, 0, -10, -8], [4, -13, 4, 0], [-7, 6, 0, -3], [-1, 22, 0, -11], [4, 0, -8, 8], [-10, 13, 5, -13], [0, -9, -10, -9], [-1, 0, 2, 12], [1, -9, -1, 18], [0, 2, 14, 2], [17, -6, 0, 3], [3, -18, 3, 0], [-8, 16, 4, -16], [4, 16, 0, -8], [-7, -1, 7, 2], [12, -9, -6, 9], [3, 0, -3, 0], [6, -11, -3, 11], [16, 15, -8, -15], [-7, 0, 14, -19], [-3, -1, 3, 2], [-3, 8, -3, 0], [10, -3, 10, 0], [10, 6, -10, -12], [-6, 4, 3, -4], [16, 0, -32, 0], [18, 9, -18, -18], [1, 12, 1, 0], [14, 4, 0, -2], [16, 20, -8, -20], [0, -1, 0, 1], [21, 9, -21, -18], [26, -7, -13, 7], [0, 1, -10, 1], [4, 0, -8, 7], [-2, 20, 0, -10], [17, -12, -17, 24], [23, 3, -23, -6], [6, -26, 0, 13], [0, -1, -11, -1], [-21, 0, 42, -4], [0, -8, -20, -8], [-12, 0, 24, 4], [0, 14, 18, 14], [6, -14, -3, 14], [-6, -16, 6, 32], [-30, 5, 15, -5], [-2, 0, 4, -18], [-6, 2, 6, -4], [-2, 0, 4, 30], [-21, -10, 0, 5], [0, -13, 15, -13], [-11, 5, -11, 0], [18, 13, -18, -26], [0, 14, 10, 14], [-12, -2, 6, 2], [26, -2, 0, 1], [-21, 0, 42, 7], [-4, -9, -4, 0], [8, 0, -16, 8], [-5, -44, 0, 22], [2, 14, -2, -28], [0, -2, 0, 0], [-8, -12, 4, 12], [-1, 0, 2, 18], [0, -11, 19, -11], [-13, -14, 13, 28], [-17, 3, -17, 0], [10, 15, 10, 0], [-12, -12, -12, 0], [-1, -34, 0, 17], [8, -12, -8, 24], [1, 16, -1, -32], [-5, 0, 10, 31], [-6, 13, 3, -13], [-20, -25, 10, 25], [0, 2, -18, 2], [0, -1, 40, -1], [24, 18, -12, -18], [0, 0, 0, -39], [-5, -7, -5, 0], [17, -3, -17, 6], [10, 12, -10, -24], [-8, 8, 0, -4], [2, 6, -2, -12], [28, 14, -14, -14], [-4, -14, 2, 14], [-24, -8, 0, 4], [10, 17, -5, -17], [25, 6, 0, -3], [-2, 5, -2, 0], [-1, 0, 1, 0], [14, -12, 0, 6], [-16, -27, 8, 27], [24, 0, -48, -4], [2, -30, 2, 0], [0, 10, -6, 10], [0, 6, 30, 6], [0, 0, -28, 0], [-10, -34, -10, 0], [32, 24, -16, -24], [-38, -4, 38, 8], [-42, -6, 42, 12], [8, 39, 8, 0], [13, 0, -26, -10], [-19, -22, 0, 11], [14, -6, 14, 0], [-22, 0, 44, -11], [0, 2, -21, 2], [-12, -10, 6, 10], [-2, 0, 4, 2], [-2, 5, 1, -5], [12, 8, 0, -4], [10, 52, 0, -26], [0, -2, 33, -2], [5, 28, 5, 0], [10, 5, -10, -10], [12, -8, 12, 0], [-4, -6, -4, 0], [-32, 16, 16, -16], [-8, 20, 8, -40], [8, -42, 8, 0], [-21, -1, -21, 0], [21, -2, 0, 1], [-6, -2, -6, 0]]