# q-expansion of newform 370.2.h.d, downloaded from the LMFDB on 02 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 = [160, -152, 113, 166, 24, 12, -26, -8, 3, 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, 0, 0, 0, 0, 0, 0], 1], [[-881880, 840473, -211635, 120257, 73205, -123604, 34822, -17854, 3868, 105], 887851], [[-1481360957, 5881796255, 4194704142, -51534513, -422935188, -248412067, 36230609, 39766871, 27279422, 4830421], 5161077863], [[1481360957, -720718392, -4194704142, 51534513, 422935188, 248412067, -36230609, -39766871, -27279422, -4830421], 5161077863], [[-18775197608, -20817063063, -9133030804, 731490770, 3464752928, 577196282, 26574182, -195227485, -79772050, -12470333], 10322155726], [[1358343716, 18569839845, -11350242598, 3432795828, -3136611584, -2353824822, 1711086296, -506483273, 281415368, -40939147], 20644311452], [[10748, 1644771, 4140830, 821948, -229852, -433934, -247792, 81613, 420, 22047], 3551404], [[-13883295960, -23630326501, -17048329972, 1196070998, 1991880712, 1835553924, 565223046, -432258509, -53270874, -78590417], 10322155726], [[27078850384, -28243314479, -19246167516, -3262671660, -910662862, 3952005236, 621147016, -234376097, -59569470, -106312323], 10322155726], [[25888295120, -6371750919, -22346975474, -5878785400, -2322899252, 3598338044, 1324404682, -412250275, 32728460, -151550159], 10322155726]]

hecke_ring_character_values = [[297, [0, 0, 0, 0, 0, 0, 1, 0, 0, 0]], [261, [0, 0, 0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 1, -1, 1, -1, 0, 0, -1], [0, -1, 0, 0, 1, 0, 0, 1, -1, 1], [0, 1, -2, -2, 0, 0, 0, 1, -1, 2], [-1, -1, 1, 2, -1, 1, -1, 1, 0, -1], [0, -1, 0, 0, -1, 0, -2, 1, 1, -1], [2, 1, 3, 1, 1, 0, 0, 0, 1, 0], [1, -1, -1, -2, 1, -1, 1, 1, -2, 1], [-3, -1, 1, 1, -1, 2, -5, -1, 0, -1], [-2, 0, 0, 0, 0, 0, 2, 0, -1, -1], [-1, 1, -1, -1, 1, 0, 2, -2, 3, 0], [1, -1, 3, 2, 1, 1, -1, -1, 0, -1], [0, -1, -1, -1, -1, 0, 0, 0, -1, 2], [4, 0, -1, -1, 2, -2, 6, -1, 0, 1], [-1, 1, -4, -4, 1, 0, 1, -1, -1, 3], [-2, 4, -3, -5, 4, 0, 4, -3, 0, 5], [-1, 0, 1, -1, 0, 0, 3, 1, -1, 0], [-2, -2, 0, 0, -2, 0, 2, 2, 1, 1], [-2, 2, -2, -4, 2, -2, 2, -2, 0, 2], [1, -1, 1, 1, -1, 2, -1, 2, -4, 3], [2, 1, 4, 2, 1, 0, 0, 0, 0, -2], [4, 2, 1, -5, 2, 0, 2, 1, -2, 3], [-1, 2, -2, -2, 2, -4, 3, 0, 2, 0], [0, -1, 1, 1, 1, 0, 0, 0, 1, 0], [2, 1, 2, 0, -1, 2, 0, -1, -3, -1], [-4, 2, -4, 0, 1, -4, -4, -2, 2, 1], [0, 0, 1, 1, -2, 2, -2, 6, 1, -2], [-1, -4, 0, 8, -4, 0, -7, 0, 2, -6], [2, 1, 1, -1, -1, 2, 4, 0, -3, 0], [-4, -4, 1, 1, 2, 2, -6, 3, -2, 1], [-6, 2, 1, 7, 2, 0, 0, -5, 4, -3], [5, -3, 3, 3, -3, 6, -1, 6, -4, 1], [3, 2, -4, -3, 2, 1, -1, -7, 3, 1], [2, 2, -4, -6, 2, 2, 0, 4, -4, 8], [-4, 0, -3, 1, 0, -4, -8, -1, 4, -1], [1, 6, -3, -3, 0, -6, 7, 1, 0, 3], [0, 3, 0, 0, 5, 0, 4, -3, -3, 5], [4, 2, 0, -4, 1, 4, 4, 2, -6, 5], [-1, -2, 2, 2, -2, 4, -5, 4, -4, 2], [0, -6, 0, 0, 6, 0, 0, -4, 0, 0], [6, -5, 4, 6, -3, 2, -2, 7, -3, -3], [-2, 4, -2, -2, 0, -4, 2, -8, 7, -5], [4, 2, 6, 8, -6, 2, -2, 2, 4, -2], [3, -1, 0, 0, 1, 0, 3, -3, 7, -7], [2, 6, -1, -1, -4, -2, 4, -5, 4, -3], [-2, -1, 2, 8, -2, 6, -6, -3, 5, -6], [2, 1, 5, -1, 1, 0, 4, 2, 1, 2], [2, 4, -3, -5, -6, 2, -6, 1, -6, -1], [0, 2, 3, 3, -2, 0, -12, -5, -2, -5], [3, -5, -1, -1, 7, -2, 5, -5, 2, -1], [6, 2, 0, -8, 2, 0, 2, 2, -5, 3], [-3, -1, 2, 2, -3, 4, -7, 1, 1, -3], [2, 1, -7, -1, 1, 0, -8, -4, -3, -2], [-8, 3, -7, 1, -3, -8, -10, -4, 5, -4], [-4, -3, 2, 2, -1, 4, -8, 7, 3, -5], [2, 1, 2, 0, 1, 2, -4, -1, -3, 1], [-8, 0, -3, -3, -2, 0, 0, -3, 0, 5], [-4, 6, -4, -8, 3, -4, 4, -6, 2, 5], [5, 0, -1, -1, 2, -2, 7, 1, 4, -3], [-2, 1, -2, 0, 1, -2, 6, -1, 1, 1], [5, -7, 6, 2, -7, 0, -1, 9, -1, -3], [6, -1, 4, 10, -1, 0, -12, -2, 4, -6], [0, -4, -1, -5, -4, 0, 4, 6, -3, 2], [-14, 5, -4, -8, 7, -4, 4, -5, 1, 1], [-5, 1, -11, -1, 1, 0, -5, -6, 3, 4], [4, 5, -4, -4, 3, -8, 12, -5, -1, 5], [-1, 3, -7, -9, 3, 0, 3, -2, -8, 1], [0, -4, -7, -7, 4, 0, 0, -3, -4, 3], [0, -7, 5, 5, 7, 0, 10, 2, 7, 2], [-6, 6, -12, -6, -2, -6, 2, 0, 0, 4], [4, -4, 7, 3, 4, 4, 4, 1, 0, 1], [-12, 12, -8, -8, 4, -16, 4, -8, 6, 2], [-3, -2, -3, -3, 8, -6, 3, -5, -4, 7], [-1, 2, -6, -5, -2, -1, -11, 3, -1, 3], [14, 0, 4, 2, 2, -2, 2, 6, -2, -4], [-15, -3, -3, 1, -3, 0, 11, 1, 2, 1], [-9, 0, -2, 10, 0, 0, -3, -6, 8, -2], [-3, -6, 6, 2, -6, 0, 7, 8, -4, -6], [-1, -1, -3, -3, 7, -6, 5, -6, 1, 2], [5, 0, -2, -7, -2, 5, -9, 7, -5, 5], [-7, -5, 3, 3, -5, 0, 7, 5, 6, 3], [16, 4, 5, -9, 4, 0, -2, 3, -2, 7], [7, 8, 0, -6, 8, 0, -1, -5, 1, 7], [-8, 13, -2, -2, -9, -4, -4, -2, 0, 2], [6, 7, -2, -2, -3, -4, 10, 5, 0, 2], [-7, -6, 3, 1, -6, 0, 9, 7, 0, -1], [4, 1, -1, -5, -1, 4, -4, 4, -5, 4], [5, -8, 6, 6, -4, 12, -7, 0, -4, -2], [11, -3, 5, 10, -5, 5, -5, 3, 2, -5], [-4, -1, 3, 7, -1, -4, -20, -6, 5, -8], [-12, 0, -5, 7, 0, 0, 0, -6, 3, -4], [0, 4, 0, 0, 2, 0, -12, -4, -4, 2], [13, -1, -3, -4, 11, -1, 1, -1, -2, -7], [0, 2, 2, 2, -6, 4, -4, -8, 8, -10], [14, -4, -4, -6, 7, -2, 2, 2, -6, -1], [-4, 0, 8, 10, -12, 2, -2, 6, 2, 2], [-8, 2, -6, 2, -2, -8, -2, -4, 6, -4], [2, -3, 11, 13, -9, 2, -2, 12, -1, -4], [11, 2, 1, 1, -4, 2, 9, 1, 1, -2], [6, -20, 3, 17, -6, 14, -14, 9, -6, -11], [10, 11, 4, -6, 2, -10, 10, 3, 1, 4], [4, 1, 6, 10, -9, 4, -4, 1, 5, -1], [-3, -8, 0, 0, 8, 0, -3, 0, -2, 2], [17, -4, 8, 11, -4, 3, -3, 9, -1, -7], [12, -4, 17, 5, 2, 12, -14, -1, -8, -3], [8, 5, 1, -7, 5, -8, 8, 4, -3, 2], [5, -8, -5, 1, -8, 0, -11, 5, -5, -6], [-10, 2, -9, 1, 4, -10, 6, -3, 8, 3], [14, -1, 2, 0, -2, -2, 2, 5, -3, 2], [1, 9, -7, -16, 7, -9, 9, -7, 0, 9], [1, 7, -7, -7, 7, -14, 15, -12, -3, 10], [-5, 1, -1, 3, 1, 0, 1, -3, 0, -3], [4, -3, 6, 14, -4, 8, -8, 1, 5, -10], [-4, 8, -3, -7, 8, 0, 8, -6, 1, 8], [7, 13, 7, 4, -5, -3, 3, -3, 10, 1], [2, 5, -8, -10, -7, 2, -10, 5, -7, 3], [-11, -12, -4, -1, -4, 3, -3, 5, -9, 5], [-3, -4, 0, 3, -4, -3, -13, 1, 7, -7], [25, -2, 2, -3, 4, -5, 5, 9, -7, -1], [-9, -1, -7, -11, -1, 0, 13, 3, -10, 1], [-13, -7, -9, -1, -7, 0, 5, 3, 2, 3], [-9, -3, -7, 11, -3, 0, -9, -6, 7, -4], [-8, 9, 9, 9, -7, 0, 0, 0, 9, -2], [0, -2, 2, 2, 8, 0, -16, 0, 2, 6], [-21, 16, -13, -13, 10, -26, 5, -9, 13, 0], [3, 1, -25, -31, 1, 0, 3, 2, -10, 21], [8, -12, 13, 5, 0, 8, -4, 7, 4, -5], [-26, 13, -2, -8, 2, -6, 6, -9, 7, 6], [5, 6, -3, -3, 0, -6, 11, 7, -2, 5], [-10, 6, -4, -2, 7, 2, -2, -12, 8, -5], [8, -3, 0, 0, 3, 0, 8, 3, -1, 1], [6, 2, 9, 3, 2, 6, -14, -5, -8, -1], [14, -6, 8, -6, -11, 14, -6, 12, -8, -5], [-5, 0, 6, 11, 14, -5, -7, -11, 5, 3], [5, -12, 7, 7, -2, 14, -9, -1, 2, -9], [13, 7, 11, -11, 7, 0, 9, 4, -3, 8], [6, 12, -1, -1, -10, -2, 8, 11, 2, -1], [-3, 2, 4, 3, -2, -1, 1, 3, 1, -1], [-23, 4, -3, -3, 2, -6, -17, -3, 0, 3], [-6, 6, -12, -22, 9, -10, 10, -8, -4, 13], [-8, 0, -1, -3, -2, -2, 2, 1, -2, 5], [-20, -4, 6, 18, -4, 0, 8, -2, 2, -16], [8, 3, 6, -2, 3, 8, 16, -1, -11, 5], [3, 13, -5, -5, -3, -10, 13, -13, 8, -3], [8, -15, 11, 15, -9, 4, -4, 22, -11, -6], [0, -4, 8, 8, 11, 0, 0, -4, 4, 3], [-6, 4, -6, 0, 4, -6, -6, -4, 2, 4], [4, 10, 0, 0, 10, 0, -4, -10, -2, -2], [8, 9, 4, -20, 9, 0, 16, 3, -13, 7], [13, -11, 2, 2, 7, 4, 9, 15, -7, 5], [10, -9, 10, 0, -8, 10, -16, 9, -1, -8], [2, -7, 2, 0, 1, 2, 22, 7, 5, 1], [12, -4, 0, -4, -4, 0, -8, 6, -10, -6], [-2, -2, 7, 9, 8, -2, 22, -7, 4, -1], [-10, 12, -3, -3, -6, -6, -4, 11, -6, 9], [26, -8, 1, 1, 6, 2, 24, 7, -2, 1], [-6, 6, 18, 26, -13, 8, -8, 4, 14, -13], [21, -8, 2, 2, 4, 4, 17, 19, -7, 5], [20, 1, -10, -6, 5, 4, -4, -15, 5, 1], [2, -9, 3, 1, -15, 2, 6, 8, 7, -16], [-3, 6, -5, -5, 4, -10, 7, -17, 11, -6], [-6, 3, -8, -2, 0, -6, -22, -1, 3, 2], [-8, 7, -6, -10, 4, -4, 4, -9, 3, 6], [-14, 3, 12, 12, -13, 0, 0, 9, 3, 1], [0, -8, 10, 10, -12, 20, -20, -2, -2, -8], [-20, 0, -1, 19, 0, 0, 0, -10, 13, -6], [-4, -2, -2, 2, -4, -4, -8, 0, 6, -6]]