# q-expansion of newform 144.16.a.t, downloaded from the LMFDB on 23 October 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 = 144
weight = 16
poly_data = [-370, 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], 1], [[0, 288], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0], [0, 0], [0, 29], [2591260, 0], [0, -6820], [-77911990, 0], [0, 13578], [976566184, 0], [0, 3761416], [0, -20546645], [-162227927108, 0], [-606605347330, 0], [0, -409799290], [-1764832689560, 0], [0, -691427224], [0, 1055009763], [0, -4971118520], [31978275004262, 0], [-126767977040, 0], [0, 2620593120], [101136696626630, 0], [-163797938010884, 0], [0, 55904217908], [0, 12236279700], [-1025848103778610, 0], [0, 365294390095], [309640145289460, 0], [0, -308018020392], [867906802624394, 0], [0, -364286613292], [4203785278051540, 0], [0, 449132151520], [0, -498078193190], [-12880788652661456, 0], [0, 550927836545], [-23114621242583588, 0], [37849846209126950, 0], [-18164275028413400, 0], [0, 4463758405096], [0, 37611476485], [0, 1649417559240], [70445933806207538, 0], [0, 37421507240600], [-116276873798305150, 0], [0, -6527429301957], [-29134015054698044, 0], [-58023917253126992, 0], [-505447812461156060, 0], [0, -51633998714932], [-497421675578884894, 0], [0, 89292814023660], [0, 201786627635560], [241627780925903342, 0], [0, -67584086225940], [0, -310398098812024], [0, 378577947018176], [0, 169357415673015], [-2327208965191166492, 0], [-3517713462637824910, 0], [0, 250862977123000], [-2298167203006920800, 0], [0, -439581063077389], [-689944193108947280, 0], [0, 800449845281680], [-8092129081195733830, 0], [0, -899655574119965], [-1161945411485282912, 0], [2486597263362539390, 0], [0, 2951310803643604], [-1967156606538594394, 0], [0, 709493448205876], [0, -3019085260222080], [-7925169507059118980, 0], [31797512754770219150, 0], [-23229771901311034136, 0], [0, -3969104942747144], [0, 791184365228545], [24130042494777092630, 0], [0, -4809229710370930], [35231426129385257366, 0], [0, 6322766531876140], [-18233476044635708062, 0], [0, -1859415220132920], [76705387233041393570, 0], [54372471676053071236, 0], [0, -2580249084936172], [0, 5116457102610930], [38057987342154755270, 0], [0, 27467641506488665], [48933507044661751060, 0], [0, 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