
# q-expansion of newform 171.3.c.f, downloaded from the LMFDB on 15 March 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 = 171
weight = 3
poly_data = [25, -5, -4, -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], [[5, -1, 1, -1], 5], [[5, 9, 1, -1], 5], [[-35, -7, 7, 3], 5]]

hecke_ring_character_values = [[20, [1, 0, 0, 0]], [154, [-1, 0, 0, 0]]]
aps_data = [[0, -1, 0, 0], [0, 0, 0, 0], [3, 0, -1, 0], [9, 0, -1, 0], [3, 0, 1, 0], [0, 3, 0, 1], [-1, 0, 5, 0], [1, -7, -2, 1], [-14, 0, 2, 0], [0, 6, 0, 2], [0, 9, 0, -1], [0, -13, 0, -7], [0, 5, 0, -3], [1, 0, 15, 0], [-17, 0, -9, 0], [0, 3, 0, -9], [0, 9, 0, -7], [-19, 0, 3, 0], [0, 44, 0, -8], [0, 29, 0, 5], [21, 0, 3, 0], [0, -4, 0, 10], [-2, 0, -28, 0], [0, -47, 0, 1], [0, -3, 0, 13], [-118, 0, 10, 0], [0, 18, 0, -20], [0, 44, 0, 8], [0, -28, 0, 10], [0, -24, 0, 20], [0, -38, 0, -8], [35, 0, -47, 0], [127, 0, 17, 0], [1, 0, -13, 0], [147, 0, 23, 0], [0, -29, 0, 5], [66, 0, -28, 0], [10, 0, 16, 0], [0, 10, 0, -34], [0, -8, 0, 0], [0, 22, 0, 2], [0, 6, 0, -12], [143, 0, -29, 0], [0, -54, 0, -22], [-286, 0, -10, 0], [-143, 0, -29, 0], [0, 85, 0, 5], [0, -49, 0, 21], [0, -77, 0, -45], [-27, 0, 67, 0], [151, 0, -23, 0], [111, 0, 55, 0], [0, -9, 0, 35], [-109, 0, -19, 0], [0, 56, 0, 12], [215, 0, 55, 0], [0, -87, 0, 45], [-14, 0, 40, 0], [349, 0, -45, 0], [0, -110, 0, 38], [9, 0, -13, 0], [0, -6, 0, 26], [0, 86, 0, -46], [-73, 0, -21, 0], [90, 0, 76, 0], [0, -29, 0, -49], [0, 8, 0, 28], [0, -127, 0, 9], [299, 0, -15, 0], [-539, 0, 23, 0], [150, 0, 92, 0], [-265, 0, 71, 0], [-398, 0, -40, 0], [0, -22, 0, 4], [0, 78, 0, 54], [0, 65, 0, 1], [-173, 0, -61, 0], [29, 0, -73, 0], [0, -80, 0, -52], [0, -159, 0, 41], [182, 0, 40, 0], [0, 133, 0, -49], [0, 209, 0, -43], [0, 32, 0, 20], [0, 140, 0, 30], [-341, 0, -63, 0], [0, -250, 0, 10], [85, 0, 39, 0], [75, 0, 35, 0], [-175, 0, -5, 0], [427, 0, -75, 0], [-694, 0, 54, 0], [0, -43, 0, -37], [310, 0, -144, 0], [-103, 0, 87, 0], [738, 0, -54, 0], [0, 103, 0, -33], [0, -136, 0, 52], [0, 84, 0, -12], [541, 0, -137, 0], [0, -309, 0, 23], [-37, 0, 27, 0], [0, -267, 0, 13], [0, -203, 0, 5], [162, 0, 100, 0], [325, 0, -161, 0], [-413, 0, 5, 0], [230, 0, -4, 0], [0, 80, 0, 124], [0, -225, 0, 23], [0, -173, 0, -51], [221, 0, -85, 0], [287, 0, 61, 0], [-126, 0, 220, 0], [-175, 0, 191, 0], [0, 89, 0, 57], [489, 0, -221, 0], [-121, 0, -85, 0], [-533, 0, 59, 0], [0, -32, 0, 92], [0, 134, 0, 120], [0, -293, 0, -37], [0, 414, 0, -74], [0, -373, 0, -13], [377, 0, -57, 0], [498, 0, 102, 0], [-54, 0, 80, 0], [63, 0, -33, 0], [9, 0, -77, 0], [426, 0, -232, 0], [41, 0, -213, 0], [0, -220, 0, -36], [0, -154, 0, -180], [-283, 0, -37, 0], [-857, 0, 89, 0], [-227, 0, -93, 0], [0, 361, 0, 17], [0, 250, 0, 98], [0, 234, 0, 22], [-889, 0, -3, 0], [0, 155, 0, 47], [355, 0, -325, 0], [33, 0, 215, 0], [0, -74, 0, 106], [0, 204, 0, 78], [0, -90, 0, -170], [554, 0, 64, 0], [0, -381, 0, 59], [-1207, 0, -41, 0], [0, -401, 0, -5], [0, 178, 0, -4], [-393, 0, 321, 0], [1009, 0, 127, 0], [0, -251, 0, 45], [0, -113, 0, 131], [0, -268, 0, 88], [250, 0, -140, 0], [566, 0, 220, 0], [-227, 0, 19, 0], [0, -212, 0, -216], [-458, 0, -72, 0], [0, -482, 0, 82], [810, 0, -52, 0], [0, 548, 0, 8], [0, -33, 0, 7], [0, -262, 0, 66], [0, 193, 0, 247], [-387, 0, 299, 0]]
