# q-expansion of newform 110.6.a.b, downloaded from the LMFDB on 22 September 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 = 110
weight = 6
poly_data = [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 = None
aps_data = [[-4], [12], [-25], [54], [-121], [-540], [340], [-952], [1092], [-62], [-7560], [-9186], [-6818], [-13310], [-22420], [19654], [48292], [17530], [-35344], [-22912], [47852], [52396], [7890], [41958], [-37602], [57406], [-50528], [-78350], [59990], [-60194], [115466], [302896], [-131658], [129052], [494814], [27044], [-252414], [-325724], [-134214], [-359936], [413652], [-112030], [-317280], [-494996], [202032], [471896], [171136], [832572], [1206998], [-1025018], [722808], [-51556], [-955890], [-1809476], [-977806], [1394550], [1189310], [510412], [-1998276], [-2614050], [-1800366], [867228], [-735734], [1554048], [2047434], [-1110662], [-1297348], [-146684], [-3182358], [-2146134], [-3656086], [751316], [-2307092], [4955928], [3645972], [3822008], [-2368074], [890626], [3582766], [-4077494], [-20820], [4935146], [-1278808], [3047294], [3553040], [3876884], [-2294202], [391368], [869814], [-5750852], [-4564448], [3156036], [-3141204], [-7233876], [-7844020], [-5946654], [-6209714], [-8890858], [2634254], [505174], [-2531434], [-7027392], [-2215002], [3700770], [1183188], [13302278], [-1839240], [-13522484], [-169296], [-5300706], [-26534], [-8549096], [1800982], [7279820], [-5312976], [-1037442], [-14851436], [-10095360], [10154470], [18179388], [11311454], [-7562836], [7847276], [-13851200], [-5211788], [14547890], [25102634], [-16576936], [16210280], [-24870240], [26791804], [-5528634], [20122160], [20578322], [-9531198], [21374318], [12748046], [-31439102], [-17799510], [-1995498], [9092956], [-24567894], [-6203080], [28837402], [20705794], [-16802232], [20620240], [-23298572], [26480796], [7808784], [23096912], [24852058], [33930756], [-39987926], [-4522296], [29095152], [-25131488], [-19330238], [39593144], [44973406], [35769480], [-38310696], [-11948766], [27443364], [-27426858], [13219176], [32018032], [47822296]]