# q-expansion of newform 370.2.m.b, downloaded from the LMFDB on 25 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, 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 = 370
weight = 2
poly_data = [9, -3, -2, -1, 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, 1, 0, 0], 1], [[-3, -2, 2, 1], 6], [[3, 2, 0, -1], 2]]

hecke_ring_character_values = [[297, [-1, 0, 0, 0]], [261, [1, 0, -1, 0]]]
aps_data = [[0, 0, 1, 0], [1, -1, 0, 0], [0, 0, -1, 1], [2, 0, 2, 0], [0, 0, 0, 0], [3, -1, -2, 2], [-1, 2, 2, -1], [0, 0, 0, 0], [2, 2, 2, 2], [-3, -1, 7, 1], [2, 3, -7, -3], [3, 0, -7, 0], [-1, 2, -1, -4], [2, -3, -3, -3], [0, 2, -2, -2], [-3, 0, 1, -1], [-10, 0, 6, 2], [0, -3, -3, 0], [2, -6, -4, 0], [4, -2, -2, 4], [-4, 0, 8, 0], [-2, 6, 4, 0], [8, 0, -6, -4], [-13, 0, 6, -1], [5, -3, -3, -3], [-1, -1, -1, -1], [-6, 4, 4, 4], [1, 7, 8, 0], [3, 0, 0, 3], [-6, 12, 0, -6], [-20, 0, 10, 0], [0, 0, -2, -4], [3, -1, -5, 1], [2, -4, 12, 2], [-13, -1, -1, -1], [-1, -5, 6, 10], [4, 0, 1, 6], [3, -6, 7, 3], [-8, 4, 4, -8], [7, 0, 2, 11], [0, 8, -8, -8], [-12, 5, 7, -10], [-4, -7, 15, 7], [9, 1, 1, 1], [12, 0, -5, 2], [-8, 9, 7, -9], [-6, 4, 4, 4], [2, 6, -10, -6], [-7, -1, 8, 2], [8, -5, -3, 10], [-7, -1, 15, 1], [4, 0, 2, 8], [-12, 0, -12, 0], [2, -10, 6, 10], [5, -10, -16, 5], [0, 4, 4, 0], [-14, 4, 4, 4], [-1, 2, -15, -1], [7, 0, -7, 0], [12, 0, -5, 2], [13, -3, -10, 6], [13, -8, 5, 0], [-16, 3, 29, -3], [-3, 0, -5, -13], [9, -18, -2, 9], [-4, 0, -3, -10], [-32, 0, 16, 0], [2, 3, 5, 0], [-2, -2, -2, -2], [1, -2, -4, 1], [9, -18, -6, 9], [-8, 1, 1, 1], [10, -6, 4, 0], [7, 0, 7, 0], [-2, 4, -4, -2], [-26, 4, 22, -8], [0, -11, -11, 0], [1, 6, -8, -6], [-4, 8, 0, -8], [34, 0, -17, 0], [8, -16, -10, 8], [-7, 9, 5, -9], [13, -11, 2, 0], [-1, 9, -7, -9], [-11, -3, -14, 0], [18, -1, -35, 1], [7, 7, 14, 0], [-4, 7, -3, -14], [18, 0, -14, -10], [18, -4, -14, 8], [20, -1, -1, -1], [17, 0, -15, -13], [2, 0, 0, 0], [4, 10, 10, 10], [8, 0, 8, 0], [-8, 16, 4, -8], [-7, 14, 2, -7], [-1, -1, 2, 2], [7, 3, -10, -6], [5, -21, 11, 21], [-20, 11, 11, 11], [4, -8, 31, 4], [-6, -6, -6, -6], [11, -10, -12, 10], [12, -24, -2, 12], [12, -24, -14, 12], [7, 7, -14, -14], [-1, 5, -3, -5], [17, -1, -16, 2], [-2, 4, -19, -2], [0, 0, -8, 0], [47, 0, -22, 3], [0, 0, 4, 8], [-10, 12, 12, 12], [-31, 0, 11, -9], [0, 0, 27, 0], [4, 5, 5, 5], [-10, -4, 14, 8], [-27, 0, 27, 0], [-28, 8, 20, -16], [22, -15, 7, 0], [-4, 0, -4, 0], [13, -7, -19, 7], [3, -6, -21, 3], [-4, 8, 0, -4], [36, 0, -20, -4], [4, 6, -14, -6], [-9, 18, -9, -9], [-8, 0, 8, 0], [-4, -12, -16, 0], [-2, 8, 8, 8], [10, -14, -4, 0], [2, 9, 9, 9], [6, -12, 7, 6], [-9, 18, -18, -9], [-12, 24, 0, -24], [-11, -14, -25, 0], [-10, 15, 5, -15], [-1, 2, 11, -1], [-13, -5, -18, 0], [10, -6, -4, 12], [12, 6, -18, -12], [-6, 0, 0, -6], [12, -12, 0, 24], [20, 0, -4, 12], [7, -14, 31, 7], [-6, 12, 19, -6], [-7, 5, 5, 5], [-2, 0, 4, 0], [18, 0, -2, 14], [17, -18, -16, 18], [-6, -3, 9, 6], [-29, -9, 38, 18], [2, -10, 6, 10], [-40, 4, 36, -8], [8, -25, 9, 25], [-2, 0, 4, 0], [-11, -8, 19, 16], [63, 0, -30, 3], [5, -10, -4, 5], [5, -10, 23, 5], [0, 0, 4, 8], [-14, -6, 20, 12], [8, -16, -10, 8], [-6, 12, -36, -6], [54, 0, -26, 2], [-8, 3, 13, -3], [-3, 6, -23, -3]]