# q-expansion of newform 40.11.e.a, downloaded from the LMFDB on 21 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 = 40
weight = 11
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 = [[31, [-1]], [21, [-1]], [17, [-1]]]
aps_data = [[-32], [0], [3125], [26886], [321102], [-682086], [0], [-1288802], [-1772186], [0], [0], [112652586], [-611598], [0], [-222705514], [-552886486], [-646632402], [0], [0], [0], [0], [0], [0], [5417714898], [0], [0], [14044339014], [0], [0], [0], [20318773686], [-72683101698], [0], [-103495919602], [0], [0], [-5400472614], [0], [43509849286], [-184997159686], [-267558407202], [0], [0], [0], [580524950986], [0], [562320155102], [629009357814], [0], [0], [0], [0], [-1618005901598], [-1608098772498], [0], [-2214897910586], [0], [0], [-3089404925814], [-2247119085198], [0], [-1113327388886], [0], [0], [0], [1616308081786], [-3544376403698], [0], [0], [0], [0], [0], [-13297915500714], [14435898778314], [5766608786798], [15442830340214], [0], [-8511052423014], [-20683921279998], [21146642646098], [-21345112988802], [0], [0], [0], [0], [0], [36261574218498], [0], [0], [-29048827604586], [0], [0], [1687217902086], [-51356040682098], [-13729356084002], [-9040014344986], [0], [70425245717202], [0], [0], [0], [103637277435386], [0], [-106986856456302], [91263192214702], [0], [0], [0], [0], [102099889430002], [138364328480886], [71521365211914], [0], [-130417699050802], [0], [-199758798905598], [0], [216797045564486], [-152001204656486], [-139894449714402], [0], [0], [-279816712573814], [0], [310510477583902], [0], [0], [0], [-274031113804314], [-165297764117286], [173644792574398], [197589693396614], [0], [-483909690946614], [292779315175602], [-358734875814302], [299716495082314], [0], [52397951292986], [-23858387137902], [-658837761030898], [0], [-658306425756186], [0], [0], [0], [510601595345514], [0], [697087687375598], [116051258383414], [-356957606711814], [-578569500027198], [0], [-802351357965914], [0], [0], [0], [-1292034043872702], [0], [0], [0], [0], [-901597754454714], [908542780746702], [0], [-1420157971789786], [0], [1734356381950986]]