# q-expansion of newform 175.6.b.a, downloaded from the LMFDB on 30 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 = 175
weight = 6
poly_data = [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 = [[127, [-1, 0]], [101, [1, 0]]]
aps_data = [[0, 10], [0, -14], [0, 0], [0, 49], [232, 0], [0, -140], [0, 1722], [98, 0], [0, 1824], [-3418, 0], [-7644, 0], [0, 10398], [-17962, 0], [0, 10880], [0, -9324], [0, 2262], [2730, 0], [25648, 0], [0, 48404], [-58560, 0], [0, 68082], [-31784, 0], [0, -20538], [50582, 0], [0, 58506], [38696, 0], [0, 53060], [0, 146324], [-92898, 0], [0, -83354], [0, -60384], [-61586, 0], [0, 204462], [35406, 0], [20226, 0], [70904, 0], [0, -293524], [0, 13192], [0, -493612], [0, 240716], [-294932, 0], [-336980, 0], [358264, 0], [0, -989554], [0, 990050], [840756, 0], [1150732, 0], [0, -824264], [0, -74382], [-1131956, 0], [0, -198726], [-482904, 0], [805910, 0], [430738, 0], [0, 1176910], [0, 1290976], [1277556, 0], [1650544, 0], [0, 1064090], [-22342, 0], [0, -2495738], [0, -1931776], [0, 459074], [667128, 0], [0, -111034], [0, 68778], [-564448, 0], [0, -2077294], [0, 53248], [2272004, 0], [0, 4006450], [-73784, 0], [0, -1404312], [0, -1603234], [4770120, 0], [0, -2230788], [-4840242, 0], [0, -995820], [-3316050, 0], [-3072734, 0], [-2814378, 0], [3058022, 0], [1937496, 0], [0, 3947902], [7417704, 0], [0, 1402692], [590574, 0], [0, 2904842], [-922684, 0], [0, 7182352], [0, 612570], [-2603300, 0], [0, -5463088], [1640900, 0], [-2997964, 0], [0, -6894048], [-2304764, 0], [-12096042, 0], [0, 5484430], [-6717994, 0], [0, 5002352], [0, -9019606], [0, 12405106], [-6488038, 0], [-10228504, 0], [0, -2653378], [0, 14304374], [0, -10026534], [7522920, 0], [3386250, 0], [0, 6908608], [0, -9688958], [0, 7847418], [10197194, 0], [-8362576, 0], [1102830, 0], [0, 17135426], [0, 54964], [0, -485166], [2721360, 0], [-2145248, 0], [0, 2927962], [0, 13499220], [0, -5429724], [20827982, 0], [23514066, 0], [19574670, 0], [26115180, 0], [0, -15412628], [0, -16986816], [-2015112, 0], [0, -15138072], [7214008, 0], [0, 10969726], [19244190, 0], [-8211854, 0], [0, 18618656], [0, -26250070], [0, 10037300], [-14088390, 0], [18143342, 0], [-21366922, 0], [0, 17801736], [0, -16292124], [2084992, 0], [22784972, 0], [0, -22697500], [0, -25289950], [10394734, 0], [0, 43339944], [0, -37165930], [9047850, 0], [0, 32967860], [0, 16109884], [0, 44728564], [-6605184, 0], [30892984, 0], [4872154, 0], [0, 32500398], [-2640400, 0], [0, 40817920], [0, -6719830], [0, 2789792], [33359382, 0], [0, 7600326], [0, -5797596], [12682496, 0], [0, -14440048]]