# q-expansion of newform 1408.2.e.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, 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 = 1408
weight = 2
poly_data = [73, -114, 65, -32, 115, 34, -176, 98, 19, -40, 21, -6, 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, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-1365130309773, -1106157351050, 345017292419, 152552118594, 1360237467448, -1324524483670, -109128530976, 459263240134, -263317478264, 68405568646, -12088176341, -1526954786], 1073263342573], [[-1525720033354, 2275238933214, -257669849217, 2743674542514, -2521096478461, -518540415664, 2158116738159, -1348724548266, 452378117190, -71192108680, -187476460, 4235588970], 1073263342573], [[-1162129902116, 719896835892, 1859077966861, -3551699778076, 84259472836, 2395914932848, -1226709485811, -121409436348, 369739830397, -182663219884, 51896829036, -6707558652], 1073263342573], [[-1432167708553, 2840411794222, -1676461214281, 397362273754, 1892071789518, -2950349029262, 1060289059809, 651425204654, -752223782575, 350610870646, -94655597176, 14652726490], 1073263342573], [[-194601194056, -4953607751021, -839914590849, 315798958462, 897127192705, 1451909036750, -1537042494911, -89689572634, 569933496771, -336293808425, 104700990672, -19486544494], 1073263342573], [[129635798, -70531440, -276604964, -234125046, 198022230, 200613438, -217944716, -4782762, 68672502, -42298230, 12929344, -2429272], 113440793], [[-3073128488976, -1311183144306, -2366122165858, 4954086980328, 425893489986, -4585530225030, 2607626444187, 333735321458, -908223889445, 487243576432, -138890739065, 23049450630], 1073263342573], [[-1522477601173, -160386958868, 869054774163, -1647711870972, 454478275842, 2101303385444, -2186580916814, 66393314848, 602422910600, -400542655764, 122939596861, -25569521988], 1073263342573], [[-1979790531563, 2283818208512, 619030598518, 2049873980500, -1594489756775, -2842576164636, 2685843357030, -138237078958, -698078423367, 440856661502, -138664711514, 25907161020], 1073263342573], [[-5483510658246, 2544267114413, -1039148821397, 4637773508306, 1025231858015, -7751575714110, 4508422571445, 894306058570, -1903124838975, 1009780387641, -290213463740, 49488046390], 1073263342573], [[3866130617820, -5720927840810, -334158903471, -3675002285156, -736281824173, 6836387426076, -4376785035355, -863677430794, 1888496518668, -1023554888328, 296621918924, -52457270576], 1073263342573]]

hecke_ring_character_values = [[639, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [133, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1025, [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1], [0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 1, 1, 1, 0, 0, 0, -1, -1], [0, 0, 0, 0, -2, -1, 0, 0, 0, 2, 1, 0], [0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 1, 0, -1, 0, -1, -1, 0, -1, 0], [0, 0, -1, 0, -2, 0, -1, 0, 0, 1, 0, 0], [2, -2, 0, 0, 0, -1, 0, 2, 0, 0, -1, 0], [0, 0, 1, 0, 2, 0, 0, 0, 0, -2, 0, -1], [-1, 0, 0, 3, 0, -1, 0, 1, -1, 0, -1, 0], [0, 0, 1, 0, -2, -1, -1, 0, 0, 1, 1, 1], [0, 0, 2, 0, 2, -1, -2, 0, 0, 0, 1, 0], [0, 0, 2, 0, 2, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, -1, 1, 0, 0, 2, 1, 1], [0, 0, -1, 0, -2, 0, 1, 0, 0, -1, 0, 2], [0, -2, 0, 0, 0, -2, 0, 2, 0, 0, -2, 0], [-4, -4, 0, -2, 0, 1, 0, 0, 0, 0, 1, 0], [1, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, 0], [2, 0, 0, -4, 0, 2, 0, 0, 3, 0, 2, 0], [0, 0, -2, 0, 0, 2, 2, 0, 0, -3, -2, -1], [0, 0, 2, 0, 0, -1, -1, 0, 0, -2, 1, 0], [-4, 2, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0], [0, 0, 2, 0, 0, -1, 2, 0, 0, -4, 1, 0], [0, 2, 0, 2, 0, -2, 0, 2, -1, 0, -2, 0], [7, 2, 0, -1, 0, 2, 0, 0, 2, 0, 2, 0], [4, 0, 0, 4, 0, -2, 0, 1, -3, 0, -2, 0], [-3, 2, 0, 1, 0, -1, 0, -1, -2, 0, -1, 0], [6, 0, 0, -2, 0, 2, 0, -1, 3, 0, 2, 0], [0, 0, 2, 0, -2, -2, -6, 0, 0, 1, 2, 3], [8, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0], [-3, -1, 0, -1, 0, -3, 0, 5, -1, 0, -3, 0], [0, 0, 2, 0, -4, -3, -6, 0, 0, 0, 3, 2], [-1, -4, 0, 3, 0, -2, 0, 2, -2, 0, -2, 0], [0, 0, 0, 0, -4, 1, -4, 0, 0, 0, -1, -2], [0, 0, -1, 0, -2, -3, -3, 0, 0, 1, 3, 3], [0, 1, 0, 4, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, -4, 0, -1, 3, 3, 0, 0, 2, -3, 1], [0, 0, 2, 0, -4, -2, -4, 0, 0, -2, 2, 0], [0, 0, 0, 0, 0, 1, 4, 0, 0, -6, -1, 0], [0, 0, -1, 0, 2, 2, 4, 0, 0, -6, -2, -3], [0, 0, 0, 6, 0, -1, 0, 2, -2, 0, -1, 0], [0, 0, 0, 0, 3, 3, 3, 0, 0, 2, -3, 1], [4, -2, 0, 0, 0, 2, 0, -3, -1, 0, 2, 0], [1, -1, 0, 3, 0, -3, 0, 1, -5, 0, -3, 0], [0, 0, -2, 0, -2, -2, 0, 0, 0, 4, 2, -2], [4, 0, 0, 0, 0, -2, 0, -2, -2, 0, -2, 0], [0, 0, 2, 0, 2, -2, 4, 0, 0, 2, 2, 0], [0, 0, -3, 0, 4, 1, 5, 0, 0, -1, -1, 3], [1, -6, 0, -3, 0, 1, 0, -1, 3, 0, 1, 0], [-13, -2, 0, 3, 0, -2, 0, -2, -4, 0, -2, 0], [3, 0, 0, -1, 0, 3, 0, -3, -1, 0, 3, 0], [-7, -2, 0, -3, 0, 0, 0, -2, 4, 0, 0, 0], [0, 0, 0, 0, -2, -1, -4, 0, 0, -4, 1, 2], [0, 0, 0, 0, -2, -4, -6, 0, 0, 0, 4, 0], [-18, 2, 0, 0, 0, -1, 0, 0, -2, 0, -1, 0], [0, 0, 4, 0, 4, 2, 0, 0, 0, -1, -2, -1], [6, -2, 0, -2, 0, -4, 0, 3, -3, 0, -4, 0], [0, 0, -3, 0, -2, 0, -2, 0, 0, -6, 0, 3], [-2, -4, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-3, 1, 0, -5, 0, 5, 0, -3, 1, 0, 5, 0], [0, 0, -2, 0, 0, 2, -2, 0, 0, 1, -2, 2], [0, 0, 1, 0, 6, 4, 5, 0, 0, -3, -4, -2], [-6, -2, 0, 2, 0, 2, 0, -3, 3, 0, 2, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, -1, 0, -1], [-8, 0, 0, 2, 0, -4, 0, 2, -3, 0, -4, 0], [-7, 4, 0, 1, 0, 0, 0, -4, 0, 0, 0, 0], [0, 0, -1, 0, 3, -2, -2, 0, 0, 0, 2, -2], [0, 0, 0, 0, 2, 0, 8, 0, 0, -3, 0, 1], [0, 0, 1, 0, -6, -1, -9, 0, 0, 1, 1, -3], [0, 0, 1, 0, 5, 2, 4, 0, 0, -2, -2, 2], [12, -3, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0], [1, 0, 0, 1, 0, -3, 0, 3, 5, 0, -3, 0], [-5, -6, 0, -1, 0, -1, 0, -3, 1, 0, -1, 0], [0, 0, 2, 0, 0, -2, -6, 0, 0, 6, 2, 4], [0, 0, 0, 0, 0, -1, 6, 0, 0, 0, 1, 4], [-3, -4, 0, -3, 0, 5, 0, -1, 5, 0, 5, 0], [-11, 4, 0, 5, 0, -4, 0, -2, -6, 0, -4, 0], [9, 8, 0, 5, 0, 3, 0, -1, 4, 0, 3, 0], [-10, -2, 0, -2, 0, 2, 0, 0, 6, 0, 2, 0], [0, 0, -3, 0, -4, 1, 5, 0, 0, 3, -1, 3], [3, 6, 0, 3, 0, 3, 0, -7, 0, 0, 3, 0], [0, 0, -4, 0, -2, 2, 0, 0, 0, 4, -2, 0], [0, 0, -4, 0, 2, 5, 8, 0, 0, 0, -5, -2], [0, 0, 1, 0, 1, -2, -4, 0, 0, 2, 2, -2], [0, 0, 1, 0, 2, 3, 7, 0, 0, -3, -3, -3], [13, -4, 0, 1, 0, 2, 0, 0, -4, 0, 2, 0], [0, 0, 3, 0, 1, 0, -4, 0, 0, 10, 0, 0], [-2, 0, 0, 0, 0, -3, 0, -2, -6, 0, -3, 0], [0, 0, 0, 0, 4, -1, 2, 0, 0, 4, 1, 0], [16, -2, 0, 0, 0, 2, 0, 0, -2, 0, 2, 0], [-1, 5, 0, 9, 0, -5, 0, -1, -9, 0, -5, 0], [0, -2, 0, -2, 0, -4, 0, 2, 3, 0, -4, 0], [-4, 2, 0, -4, 0, 0, 0, 5, 3, 0, 0, 0], [0, 0, 2, 0, -4, -4, -6, 0, 0, 1, 4, 3], [-10, -2, 0, -8, 0, 3, 0, -2, 8, 0, 3, 0], [0, 0, 0, 0, 2, -2, 0, 0, 0, -7, 2, 1], [4, 2, 0, 6, 0, -3, 0, -2, 0, 0, -3, 0], [0, 0, 2, 0, -2, 0, -6, 0, 0, 0, 0, -2], [24, 4, 0, 2, 0, 3, 0, -2, 2, 0, 3, 0], [-1, 6, 0, -5, 0, 3, 0, 1, 0, 0, 3, 0], [0, 0, 3, 0, 0, 2, -1, 0, 0, 2, -2, 1], [0, 0, -6, 0, -4, 4, 10, 0, 0, -6, -4, -4], [0, 0, 1, 0, 6, 4, 10, 0, 0, -10, -4, -1], [0, 0, -1, 0, 4, 2, 5, 0, 0, 1, -2, -4], [14, 6, 0, -6, 0, 0, 0, 4, 2, 0, 0, 0], [0, 0, 2, 0, -8, -7, -14, 0, 0, 4, 7, 4], [1, -2, 0, -3, 0, 3, 0, 3, 3, 0, 3, 0], [0, 0, 1, 0, -2, -5, 3, 0, 0, 1, 5, 1], [0, 0, -1, 0, 3, 2, 4, 0, 0, 4, -2, -6], [-6, -6, 0, 4, 0, -4, 0, 0, -3, 0, -4, 0], [0, 0, 5, 0, 4, 1, 9, 0, 0, -9, -1, -5], [0, 0, 2, 0, 3, -3, -9, 0, 0, 2, 3, 3], [-3, 1, 0, 3, 0, 1, 0, -3, -7, 0, 1, 0], [10, 6, 0, -8, 0, 5, 0, 0, 2, 0, 5, 0], [10, -1, 0, -2, 0, 2, 0, -6, 2, 0, 2, 0], [0, 0, 4, 0, 0, -2, -10, 0, 0, 0, 2, 4], [0, 0, 0, 0, 6, -2, -8, 0, 0, -1, 2, -1], [0, 0, 0, 0, 4, -1, -10, 0, 0, 0, 1, 0], [0, 0, 3, 0, 0, -5, -13, 0, 0, -5, 5, 1], [0, 0, -4, 0, -6, -2, -8, 0, 0, 9, 2, 5], [5, -5, 0, 7, 0, -3, 0, 5, -1, 0, -3, 0], [0, 0, 1, 0, -1, 0, 4, 0, 0, 4, 0, 0], [0, 0, 4, 0, -5, -5, 1, 0, 0, 14, 5, 1], [0, 0, -2, 0, 2, -1, 6, 0, 0, -8, 1, -4], [-20, -4, 0, -6, 0, -1, 0, 6, -2, 0, -1, 0], [18, 0, 0, 2, 0, -4, 0, 2, 2, 0, -4, 0], [0, 0, -5, 0, 5, 0, 10, 0, 0, -2, 0, 0], [-25, -4, 0, 3, 0, -5, 0, 1, -1, 0, -5, 0], [0, 0, 1, 0, -4, 0, 5, 0, 0, 3, 0, 0], [0, 0, 3, 0, 6, 4, 9, 0, 0, -15, -4, 0], [-10, 4, 0, 0, 0, 2, 0, 0, 4, 0, 2, 0], [0, 2, 0, -10, 0, 3, 0, 0, 2, 0, 3, 0], [-4, 9, 0, 0, 0, 0, 0, -4, 2, 0, 0, 0], [0, 0, 5, 0, -10, -2, -11, 0, 0, 5, 2, -2], [-20, 2, 0, -2, 0, 1, 0, -4, 6, 0, 1, 0], [0, 0, 2, 0, -8, 1, 6, 0, 0, -4, -1, 4], [0, 0, -2, 0, -11, -1, -9, 0, 0, 0, 1, -3], [-2, -4, 0, -6, 0, 4, 0, 1, 1, 0, 4, 0], [-5, -3, 0, -3, 0, -1, 0, 3, 7, 0, -1, 0], [0, 0, -1, 0, 1, 0, -4, 0, 0, 14, 0, -4], [0, 0, -2, 0, 8, 3, 6, 0, 0, -6, -3, 2], [0, 0, -4, 0, 8, 4, 8, 0, 0, -6, -4, -2], [0, 0, 6, 0, 2, -4, -8, 0, 0, 7, 4, 4], [0, 0, -3, 0, 2, 0, -8, 0, 0, 10, 0, 3], [0, 0, 0, 0, 2, 0, -4, 0, 0, 15, 0, -5], [-16, 4, 0, 2, 0, -4, 0, 2, -3, 0, -4, 0], [0, 0, 1, 0, 8, 4, -1, 0, 0, -6, -4, -9], [-4, 2, 0, 4, 0, -6, 0, 8, -6, 0, -6, 0], [0, 0, -4, 0, -8, 3, -6, 0, 0, 8, -3, 0], [0, 0, -5, 0, 2, 2, 10, 0, 0, -2, -2, -3], [-6, -2, 0, 2, 0, -6, 0, 8, -2, 0, -6, 0], [11, 10, 0, 3, 0, 5, 0, -3, 5, 0, 5, 0], [0, 0, -3, 0, 4, 4, -5, 0, 0, -3, -4, 2], [0, 0, 2, 0, -2, 5, -6, 0, 0, -2, -5, 2], [0, 0, -4, 0, 0, 0, 16, 0, 0, -7, 0, -4], [0, 0, 3, 0, 8, 0, 3, 0, 0, 9, 0, 0], [-12, 2, 0, 0, 0, 4, 0, 2, 4, 0, 4, 0], [0, 0, -2, 0, 4, 6, 6, 0, 0, 1, -6, -2], [1, -8, 0, -3, 0, 3, 0, -1, 4, 0, 3, 0], [0, 0, -1, 0, -7, 0, -4, 0, 0, 10, 0, 4], [0, 0, -5, 0, 2, 6, -2, 0, 0, -6, -6, -3], [0, 0, -8, 0, -2, 0, 4, 0, 0, -7, 0, 5], [0, 0, 0, 0, -6, 2, -6, 0, 0, 4, -2, 0], [7, 5, 0, -7, 0, -1, 0, -1, 3, 0, -1, 0], [8, -4, 0, 10, 0, -1, 0, 2, -2, 0, -1, 0], [-7, 8, 0, 1, 0, 1, 0, 3, 1, 0, 1, 0], [11, 4, 0, 7, 0, 2, 0, -10, -6, 0, 2, 0], [0, 0, -3, 0, -2, -2, 11, 0, 0, -9, 2, 4], [0, 0, 2, 0, -9, -7, -19, 0, 0, -6, 7, 3], [-7, 0, 0, -7, 0, 1, 0, 7, 4, 0, 1, 0], [-2, -2, 0, 6, 0, -4, 0, 1, -5, 0, -4, 0], [17, 5, 0, 7, 0, 5, 0, 1, 1, 0, 5, 0], [6, -8, 0, 10, 0, -2, 0, 8, 0, 0, -2, 0], [0, 0, 4, 0, 4, 1, -8, 0, 0, -4, -1, -2], [0, 0, 1, 0, 2, 0, 12, 0, 0, -2, 0, 7], [2, 8, 0, -4, 0, 1, 0, -8, -4, 0, 1, 0], [13, 3, 0, -1, 0, 1, 0, -3, -1, 0, 1, 0], [0, 0, -2, 0, -2, 0, 0, 0, 0, -10, 0, 4], [0, 0, 0, 0, -3, -3, 3, 0, 0, -4, 3, 7], [-15, 4, 0, 1, 0, 5, 0, -5, 5, 0, 5, 0], [0, 0, 6, 0, -2, -2, 2, 0, 0, 3, 2, 1], [0, 0, -6, 0, 8, 5, 10, 0, 0, -8, -5, -2], [0, 0, -5, 0, -2, -2, 3, 0, 0, 11, 2, 2], [-10, 8, 0, 14, 0, -4, 0, 0, -8, 0, -4, 0], [-3, 4, 0, -11, 0, 3, 0, -1, 0, 0, 3, 0], [8, 6, 0, 0, 0, 4, 0, -7, 3, 0, 4, 0], [0, 0, 1, 0, -4, 1, 9, 0, 0, -5, -1, 7], [6, -4, 0, 8, 0, -2, 0, 8, 0, 0, -2, 0], [6, 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