LMFDB - Embedded newform 162.8.c.q.109.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,8,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.55"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,-32,0,-256,-528] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.6063741284$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 518x^{5} + 53377x^{4} + 11940x^{3} + 3528x^{2} + 1563408x + 346406544 $ x^8 - 2x^7 + 2x^6 + 518x^5 + 53377x^4 + 11940x^3 + 3528x^2 + 1563408*x + 346406544
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.4
Root			$6.43942 - 6.43942i$ of defining polynomial
Character	$\chi$	$=$	162.109
Dual form			162.8.c.q.55.4

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.00000 - 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(140.213 - 242.855i) q^{5} +(82.1617 + 142.308i) q^{7} +512.000 q^{8} -2243.40 q^{10} +(1018.97 + 1764.91i) q^{11} +(839.167 - 1453.48i) q^{13} +(657.293 - 1138.47i) q^{14} +(-2048.00 - 3547.24i) q^{16} +31685.7 q^{17} -12641.8 q^{19} +(8973.61 + 15542.7i) q^{20} +(8151.75 - 14119.3i) q^{22} +(-24733.5 + 42839.6i) q^{23} +(-256.638 - 444.510i) q^{25} -13426.7 q^{26} -10516.7 q^{28} +(3776.40 + 6540.91i) q^{29} +(-78017.6 + 135131. i) q^{31} +(-16384.0 + 28377.9i) q^{32} +(-126743. - 219525. i) q^{34} +46080.4 q^{35} +541440. q^{37} +(50567.2 + 87584.9i) q^{38} +(71788.8 - 124342. i) q^{40} +(-268792. + 465561. i) q^{41} +(-100249. - 173636. i) q^{43} -130428. q^{44} +395735. q^{46} +(205436. + 355825. i) q^{47} +(398270. - 689825. i) q^{49} +(-2053.10 + 3556.08i) q^{50} +(53706.7 + 93022.7i) q^{52} +1.36778e6 q^{53} +571489. q^{55} +(42066.8 + 72861.8i) q^{56} +(30211.2 - 52327.3i) q^{58} +(-399444. + 691858. i) q^{59} +(-284720. - 493149. i) q^{61} +1.24828e6 q^{62} +262144. q^{64} +(-235323. - 407592. i) q^{65} +(2.40226e6 - 4.16084e6i) q^{67} +(-1.01394e6 + 1.75620e6i) q^{68} +(-184322. - 319254. i) q^{70} +2.45714e6 q^{71} +1.60344e6 q^{73} +(-2.16576e6 - 3.75121e6i) q^{74} +(404537. - 700679. i) q^{76} +(-167440. + 290015. i) q^{77} +(-2.79499e6 - 4.84107e6i) q^{79} -1.14862e6 q^{80} +4.30067e6 q^{82} +(-4.91453e6 - 8.51221e6i) q^{83} +(4.44273e6 - 7.69504e6i) q^{85} +(-801990. + 1.38909e6i) q^{86} +(521712. + 903632. i) q^{88} -117249. q^{89} +275789. q^{91} +(-1.58294e6 - 2.74173e6i) q^{92} +(1.64348e6 - 2.84660e6i) q^{94} +(-1.77254e6 + 3.07013e6i) q^{95} +(3.89623e6 + 6.74846e6i) q^{97} -6.37233e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{2} - 256 q^{4} - 528 q^{5} - 560 q^{7} + 4096 q^{8} + 8448 q^{10} - 2160 q^{11} - 13460 q^{13} - 4480 q^{14} - 16384 q^{16} + 45120 q^{17} + 73408 q^{19} - 33792 q^{20} - 17280 q^{22} - 62640 q^{23}+ \cdots + 23190336 q^{98}+O(q^{100}) $ 8 * q - 32 * q^2 - 256 * q^4 - 528 * q^5 - 560 * q^7 + 4096 * q^8 + 8448 * q^10 - 2160 * q^11 - 13460 * q^13 - 4480 * q^14 - 16384 * q^16 + 45120 * q^17 + 73408 * q^19 - 33792 * q^20 - 17280 * q^22 - 62640 * q^23 - 94696 * q^25 + 215360 * q^26 + 71680 * q^28 - 68400 * q^29 - 227504 * q^31 - 131072 * q^32 - 180480 * q^34 - 126048 * q^35 + 1047160 * q^37 - 293632 * q^38 - 270336 * q^40 - 67200 * q^41 - 562640 * q^43 + 276480 * q^44 + 1002240 * q^46 - 515328 * q^47 - 1449396 * q^49 - 757568 * q^50 - 861440 * q^52 + 4996032 * q^53 + 2645568 * q^55 - 286720 * q^56 - 547200 * q^58 - 4155840 * q^59 - 2130764 * q^61 + 3640064 * q^62 + 2097152 * q^64 - 8966496 * q^65 + 1205440 * q^67 - 1443840 * q^68 + 504192 * q^70 + 24972960 * q^71 - 9641720 * q^73 - 4188640 * q^74 - 2349056 * q^76 - 18686592 * q^77 + 11471680 * q^79 + 4325376 * q^80 + 1075200 * q^82 - 16811232 * q^83 + 21006396 * q^85 - 4501120 * q^86 - 1105920 * q^88 + 33715200 * q^89 - 46401088 * q^91 - 4008960 * q^92 - 4122624 * q^94 - 14090160 * q^95 + 27078040 * q^97 + 23190336 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−4.00000	−	6.92820i	−0.353553	−	0.612372i
$2$	−4.00000	−	6.92820i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−32.0000	+	55.4256i	−0.250000	+	0.433013i
$5$	140.213	−	242.855i	0.501640	−	0.868866i	−0.498358	−	0.866971i	$-0.666064\pi$
$5$	140.213	−	242.855i	0.501640	−	0.868866i	0.999998	−	0.00189451i	$-0.000603041\pi$
$6$		0			0
$7$	82.1617	+	142.308i	0.0905370	+	0.156815i	0.907737	−	0.419539i	$-0.137808\pi$
$7$	82.1617	+	142.308i	0.0905370	+	0.156815i	−0.817200	+	0.576354i	$0.804475\pi$
$8$	512.000			0.353553
$9$		0			0
$10$	−2243.40			−0.709426
$11$	1018.97	+	1764.91i	0.230827	+	0.399804i	0.958052	−	0.286595i	$-0.0925235\pi$
$11$	1018.97	+	1764.91i	0.230827	+	0.399804i	−0.727225	+	0.686399i	$0.759190\pi$
$12$		0			0
$13$	839.167	−	1453.48i	0.105937	−	0.183488i	−0.808184	−	0.588930i	$-0.799549\pi$
$13$	839.167	−	1453.48i	0.105937	−	0.183488i	0.914121	+	0.405443i	$0.132883\pi$
$14$	657.293	−	1138.47i	0.0640193	−	0.110885i
$15$		0			0
$16$	−2048.00	−	3547.24i	−0.125000	−	0.216506i
$17$	31685.7			1.56420			0.782099	−	0.623154i	$-0.214149\pi$
$17$	31685.7			1.56420			0.782099	+	0.623154i	$0.214149\pi$
$18$		0			0
$19$	−12641.8			−0.422835			−0.211418	−	0.977396i	$-0.567808\pi$
$19$	−12641.8			−0.422835			−0.211418	+	0.977396i	$0.567808\pi$
$20$	8973.61	+	15542.7i	0.250820	+	0.434433i
$21$		0			0
$22$	8151.75	−	14119.3i	0.163219	−	0.282704i
$23$	−24733.5	+	42839.6i	−0.423874	+	0.734172i	−0.996315	−	0.0857744i	$-0.972664\pi$
$23$	−24733.5	+	42839.6i	−0.423874	+	0.734172i	0.572440	+	0.819947i	$0.305997\pi$
$24$		0			0
$25$	−256.638	−	444.510i	−0.00328497	−	0.00568973i
$26$	−13426.7			−0.149817
$27$		0			0
$28$	−10516.7			−0.0905370
$29$	3776.40	+	6540.91i	0.0287531	+	0.0498018i	0.880044	−	0.474892i	$-0.157513\pi$
$29$	3776.40	+	6540.91i	0.0287531	+	0.0498018i	−0.851291	+	0.524694i	$0.824180\pi$
$30$		0			0
$31$	−78017.6	+	135131.i	−0.470356	+	0.814681i	−0.999425	−	0.0338979i	$-0.989208\pi$
$31$	−78017.6	+	135131.i	−0.470356	+	0.814681i	0.529069	+	0.848579i	$0.322541\pi$
$32$	−16384.0	+	28377.9i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−126743.	−	219525.i	−0.553028	−	0.957872i
$35$	46080.4			0.181668
$36$		0			0
$37$	541440.			1.75729			0.878647	−	0.477472i	$-0.158447\pi$
$37$	541440.			1.75729			0.878647	+	0.477472i	$0.158447\pi$
$38$	50567.2	+	87584.9i	0.149495	+	0.258933i
$39$		0			0
$40$	71788.8	−	124342.i	0.177356	−	0.307190i
$41$	−268792.	+	465561.i	−0.609077	+	1.05495i	0.382316	+	0.924032i	$0.375127\pi$
$41$	−268792.	+	465561.i	−0.609077	+	1.05495i	−0.991393	+	0.130921i	$0.958207\pi$
$42$		0			0
$43$	−100249.	−	173636.i	−0.192282	−	0.333043i	0.753724	−	0.657191i	$-0.228256\pi$
$43$	−100249.	−	173636.i	−0.192282	−	0.333043i	−0.946006	+	0.324148i	$0.894922\pi$
$44$	−130428.			−0.230827
$45$		0			0
$46$	395735.			0.599449
$47$	205436.	+	355825.i	0.288624	+	0.499912i	0.973482	−	0.228765i	$-0.0734688\pi$
$47$	205436.	+	355825.i	0.288624	+	0.499912i	−0.684857	+	0.728677i	$0.740135\pi$
$48$		0			0
$49$	398270.	−	689825.i	0.483606	−	0.837630i
$50$	−2053.10	+	3556.08i	−0.00232282	+	0.00402324i
$51$		0			0
$52$	53706.7	+	93022.7i	0.0529684	+	0.0917439i
$53$	1.36778e6			1.26198			0.630988	−	0.775792i	$-0.282650\pi$
$53$	1.36778e6			1.26198			0.630988	+	0.775792i	$0.282650\pi$
$54$		0			0
$55$	571489.			0.463168
$56$	42066.8	+	72861.8i	0.0320097	+	0.0554424i
$57$		0			0
$58$	30211.2	−	52327.3i	0.0203315	−	0.0352152i
$59$	−399444.	+	691858.i	−0.253206	+	0.438566i	−0.964407	−	0.264423i	$-0.914818\pi$
$59$	−399444.	+	691858.i	−0.253206	+	0.438566i	0.711201	+	0.702989i	$0.248152\pi$
$60$		0			0
$61$	−284720.	−	493149.i	−0.160606	−	0.278179i	0.774480	−	0.632599i	$-0.218012\pi$
$61$	−284720.	−	493149.i	−0.160606	−	0.278179i	−0.935086	+	0.354420i	$0.884678\pi$
$62$	1.24828e6			0.665184
$63$		0			0
$64$	262144.			0.125000
$65$	−235323.	−	407592.i	−0.106284	−	0.184090i
$66$		0			0
$67$	2.40226e6	−	4.16084e6i	0.975795	−	1.69013i	0.298510	−	0.954407i	$-0.403510\pi$
$67$	2.40226e6	−	4.16084e6i	0.975795	−	1.69013i	0.677285	−	0.735720i	$-0.263156\pi$
$68$	−1.01394e6	+	1.75620e6i	−0.391050	+	0.677318i
$69$		0			0
$70$	−184322.	−	319254.i	−0.0642293	−	0.111248i
$71$	2.45714e6			0.814751			0.407376	−	0.913261i	$-0.366444\pi$
$71$	2.45714e6			0.814751			0.407376	+	0.913261i	$0.366444\pi$
$72$		0			0
$73$	1.60344e6			0.482418			0.241209	−	0.970473i	$-0.422456\pi$
$73$	1.60344e6			0.482418			0.241209	+	0.970473i	$0.422456\pi$
$74$	−2.16576e6	−	3.75121e6i	−0.621297	−	1.07612i
$75$		0			0
$76$	404537.	−	700679.i	0.105709	−	0.183093i
$77$	−167440.	+	290015.i	−0.0417968	+	0.0723942i
$78$		0			0
$79$	−2.79499e6	−	4.84107e6i	−0.637801	−	1.10470i	−0.985914	−	0.167252i	$-0.946511\pi$
$79$	−2.79499e6	−	4.84107e6i	−0.637801	−	1.10470i	0.348113	−	0.937453i	$-0.386823\pi$
$80$	−1.14862e6			−0.250820
$81$		0			0
$82$	4.30067e6			0.861365
$83$	−4.91453e6	−	8.51221e6i	−0.943427	−	1.63406i	−0.758871	−	0.651241i	$-0.774249\pi$
$83$	−4.91453e6	−	8.51221e6i	−0.943427	−	1.63406i	−0.184556	−	0.982822i	$-0.559085\pi$
$84$		0			0
$85$	4.44273e6	−	7.69504e6i	0.784664	−	1.35908i
$86$	−801990.	+	1.38909e6i	−0.135964	+	0.235497i
$87$		0			0
$88$	521712.	+	903632.i	0.0816097	+	0.141352i
$89$	−117249.			−0.0176296			−0.00881480	−	0.999961i	$-0.502806\pi$
$89$	−117249.			−0.0176296			−0.00881480	+	0.999961i	$0.502806\pi$
$90$		0			0
$91$	275789.			0.0383648
$92$	−1.58294e6	−	2.74173e6i	−0.211937	−	0.367086i
$93$		0			0
$94$	1.64348e6	−	2.84660e6i	0.204088	−	0.353491i
$95$	−1.77254e6	+	3.07013e6i	−0.212111	+	0.367387i
$96$		0			0
$97$	3.89623e6	+	6.74846e6i	0.433454	+	0.750765i	0.997168	−	0.0752054i	$-0.0239613\pi$
$97$	3.89623e6	+	6.74846e6i	0.433454	+	0.750765i	−0.563714	+	0.825970i	$0.690628\pi$
$98$	−6.37233e6			−0.683922
$99$		0			0
$100$	32849.7			0.00328497
$101$	−577424.	−	1.00013e6i	−0.0557661	−	0.0965897i	0.836795	−	0.547517i	$-0.184427\pi$
$101$	−577424.	−	1.00013e6i	−0.0557661	−	0.0965897i	−0.892561	+	0.450927i	$0.851093\pi$
$102$		0			0
$103$	8.49333e6	−	1.47109e7i	0.765857	−	1.32650i	−0.173935	−	0.984757i	$-0.555648\pi$
$103$	8.49333e6	−	1.47109e7i	0.765857	−	1.32650i	0.939792	−	0.341746i	$-0.111018\pi$
$104$	429653.	−	744181.i	0.0374543	−	0.0648727i
$105$		0			0
$106$	−5.47113e6	−	9.47627e6i	−0.446176	−	0.772800i
$107$	2.24189e7			1.76918			0.884589	−	0.466372i	$-0.154439\pi$
$107$	2.24189e7			1.76918			0.884589	+	0.466372i	$0.154439\pi$
$108$		0			0
$109$	1.75728e7			1.29971			0.649857	−	0.760056i	$-0.274829\pi$
$109$	1.75728e7			1.29971			0.649857	+	0.760056i	$0.274829\pi$
$110$	−2.28596e6	−	3.95939e6i	−0.163755	−	0.283631i
$111$		0			0
$112$	336534.	−	582894.i	0.0226343	−	0.0392037i
$113$	3.47819e6	−	6.02441e6i	0.226767	−	0.392772i	−0.730081	−	0.683360i	$-0.760518\pi$
$113$	3.47819e6	−	6.02441e6i	0.226767	−	0.392772i	0.956848	+	0.290589i	$0.0938512\pi$
$114$		0			0
$115$	6.93588e6	+	1.20133e7i	0.425265	+	0.736580i
$116$	−483379.			−0.0287531
$117$		0			0
$118$	6.39111e6			0.358088
$119$	2.60335e6	+	4.50913e6i	0.141618	+	0.245289i
$120$		0			0
$121$	7.66699e6	−	1.32796e7i	0.393438	−	0.681454i
$122$	−2.27776e6	+	3.94519e6i	−0.113566	+	0.196702i
$123$		0			0
$124$	−4.99313e6	−	8.64835e6i	−0.235178	−	0.407340i
$125$	2.17643e7			0.996688
$126$		0			0
$127$	−3.25050e7			−1.40811			−0.704057	−	0.710144i	$-0.748630\pi$
$127$	−3.25050e7			−1.40811			−0.704057	+	0.710144i	$0.748630\pi$
$128$	−1.04858e6	−	1.81619e6i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−1.88259e6	+	3.26074e6i	−0.0751542	+	0.130171i
$131$	−8.15697e6	+	1.41283e7i	−0.317015	+	0.549085i	−0.979864	−	0.199668i	$-0.936014\pi$
$131$	−8.15697e6	+	1.41283e7i	−0.317015	+	0.549085i	0.662849	+	0.748753i	$0.269347\pi$
$132$		0			0
$133$	−1.03867e6	−	1.79903e6i	−0.0382822	−	0.0663068i
$134$	−3.84362e7			−1.37998
$135$		0			0
$136$	1.62231e7			0.553028
$137$	1.80797e7	+	3.13150e7i	0.600716	+	1.04047i	0.992713	+	0.120504i	$0.0384512\pi$
$137$	1.80797e7	+	3.13150e7i	0.600716	+	1.04047i	−0.391997	+	0.919967i	$0.628215\pi$
$138$		0			0
$139$	−2.67592e7	+	4.63482e7i	−0.845125	+	1.46380i	0.0403883	+	0.999184i	$0.487141\pi$
$139$	−2.67592e7	+	4.63482e7i	−0.845125	+	1.46380i	−0.885513	+	0.464615i	$0.846193\pi$
$140$	−1.47457e6	+	2.55404e6i	−0.0454170	+	0.0786645i
$141$		0			0
$142$	−9.82854e6	−	1.70235e7i	−0.288058	−	0.498931i
$143$	3.42034e6			0.0978122
$144$		0			0
$145$	2.11799e6			0.0576948
$146$	−6.41378e6	−	1.11090e7i	−0.170561	−	0.295420i
$147$		0			0
$148$	−1.73261e7	+	3.00097e7i	−0.439323	+	0.760931i
$149$	2.72109e6	−	4.71307e6i	0.0673893	−	0.116722i	−0.830362	−	0.557224i	$-0.811866\pi$
$149$	2.72109e6	−	4.71307e6i	0.0673893	−	0.116722i	0.897751	+	0.440503i	$0.145200\pi$
$150$		0			0
$151$	−1.09681e7	−	1.89972e7i	−0.259245	−	0.449025i	0.706795	−	0.707419i	$-0.250140\pi$
$151$	−1.09681e7	−	1.89972e7i	−0.259245	−	0.449025i	−0.966040	+	0.258393i	$0.916807\pi$
$152$	−6.47260e6			−0.149495
$153$		0			0
$154$	2.67905e6			0.0591096
$155$	2.18781e7	+	3.78940e7i	0.471899	+	0.817353i
$156$		0			0
$157$	−2.49533e6	+	4.32205e6i	−0.0514612	+	0.0891334i	−0.890609	−	0.454771i	$-0.849721\pi$
$157$	−2.49533e6	+	4.32205e6i	−0.0514612	+	0.0891334i	0.839147	+	0.543904i	$0.183055\pi$
$158$	−2.23599e7	+	3.87285e7i	−0.450994	+	0.781144i
$159$		0			0
$160$	4.59449e6	+	7.95788e6i	0.0886782	+	0.153595i
$161$	−8.12857e6			−0.153505
$162$		0			0
$163$	−4.47642e7			−0.809606			−0.404803	−	0.914404i	$-0.632660\pi$
$163$	−4.47642e7			−0.809606			−0.404803	+	0.914404i	$0.632660\pi$
$164$	−1.72027e7	−	2.97959e7i	−0.304539	−	0.527476i
$165$		0			0
$166$	−3.93162e7	+	6.80977e7i	−0.667104	+	1.15546i
$167$	3.07391e7	−	5.32416e7i	0.510720	−	0.884593i	−0.489203	−	0.872170i	$-0.662712\pi$
$167$	3.07391e7	−	5.32416e7i	0.510720	−	0.884593i	0.999923	−	0.0124229i	$-0.00395445\pi$
$168$		0			0
$169$	2.99659e7	+	5.19024e7i	0.477555	+	0.827149i
$170$	−7.10837e7			−1.10968
$171$		0			0
$172$	1.28318e7			0.192282
$173$	5.12455e7	+	8.87599e7i	0.752479	+	1.30333i	0.946618	+	0.322358i	$0.104476\pi$
$173$	5.12455e7	+	8.87599e7i	0.752479	+	1.30333i	−0.194138	+	0.980974i	$0.562191\pi$
$174$		0			0
$175$	42171.6	−	73043.4i	0.000594822	−	0.00103026i
$176$	4.17370e6	−	7.22906e6i	0.0577068	−	0.0999511i
$177$		0			0
$178$	468994.	+	812322.i	0.00623301	+	0.0107959i
$179$	6.08971e7			0.793617			0.396809	−	0.917901i	$-0.370118\pi$
$179$	6.08971e7			0.793617			0.396809	+	0.917901i	$0.370118\pi$
$180$		0			0
$181$	−9.62195e7			−1.20611			−0.603057	−	0.797698i	$-0.706051\pi$
$181$	−9.62195e7			−1.20611			−0.603057	+	0.797698i	$0.706051\pi$
$182$	−1.10316e6	−	1.91072e6i	−0.0135640	−	0.0234935i
$183$		0			0
$184$	−1.26635e7	+	2.19339e7i	−0.149862	+	0.259569i
$185$	7.59167e7	−	1.31492e8i	0.881528	−	1.52685i
$186$		0			0
$187$	3.22867e7	+	5.59223e7i	0.361059	+	0.625373i
$188$	−2.62957e7			−0.288624
$189$		0			0
$190$	2.83606e7			0.299970
$191$	3.69604e7	+	6.40172e7i	0.383813	+	0.664783i	0.991604	−	0.129314i	$-0.0412774\pi$
$191$	3.69604e7	+	6.40172e7i	0.383813	+	0.664783i	−0.607791	+	0.794097i	$0.707944\pi$
$192$		0			0
$193$	−4.07347e7	+	7.05546e7i	−0.407863	+	0.706440i	−0.994650	−	0.103302i	$-0.967059\pi$
$193$	−4.07347e7	+	7.05546e7i	−0.407863	+	0.706440i	0.586787	+	0.809741i	$0.300393\pi$
$194$	3.11698e7	−	5.39877e7i	0.306498	−	0.530871i
$195$		0			0
$196$	2.54893e7	+	4.41488e7i	0.241803	+	0.418815i
$197$	−2.44033e7			−0.227414			−0.113707	−	0.993514i	$-0.536272\pi$
$197$	−2.44033e7			−0.227414			−0.113707	+	0.993514i	$0.536272\pi$
$198$		0			0
$199$	−1.52134e8			−1.36848			−0.684241	−	0.729256i	$-0.739867\pi$
$199$	−1.52134e8			−1.36848			−0.684241	+	0.729256i	$0.739867\pi$
$200$	−131399.	−	227589.i	−0.00116141	−	0.00201162i
$201$		0			0
$202$	−4.61940e6	+	8.00103e6i	−0.0394326	+	0.0682993i
$203$	−620550.	+	1.07482e6i	−0.00520644	+	0.00901781i
$204$		0			0
$205$	7.53760e7	+	1.30555e8i	0.611075	+	1.05841i
$206$	−1.35893e8			−1.08309
$207$		0			0
$208$	−6.87445e6			−0.0529684
$209$	−1.28816e7	−	2.23116e7i	−0.0976018	−	0.169051i
$210$		0			0
$211$	2.93912e7	−	5.09071e7i	0.215392	−	0.373069i	−0.738002	−	0.674799i	$-0.764231\pi$
$211$	2.93912e7	−	5.09071e7i	0.215392	−	0.373069i	0.953394	+	0.301729i	$0.0975639\pi$
$212$	−4.37690e7	+	7.58102e7i	−0.315494	+	0.546452i
$213$		0			0
$214$	−8.96757e7	−	1.55323e8i	−0.625499	−	1.08340i
$215$	−5.62246e7			−0.385826
$216$		0			0
$217$	−2.56402e7			−0.170339
$218$	−7.02912e7	−	1.21748e8i	−0.459518	−	0.795909i
$219$		0			0
$220$	−1.82877e7	+	3.16752e7i	−0.115792	+	0.200558i
$221$	2.65896e7	−	4.60545e7i	0.165706	−	0.287011i
$222$		0			0
$223$	1.65907e7	+	2.87360e7i	0.100184	+	0.173524i	0.911760	−	0.410722i	$-0.134723\pi$
$223$	1.65907e7	+	2.87360e7i	0.100184	+	0.173524i	−0.811576	+	0.584246i	$0.801390\pi$
$224$	−5.38455e6			−0.0320097
$225$		0			0
$226$	−5.56511e7			−0.320697
$227$	−6.92959e7	−	1.20024e8i	−0.393203	−	0.681048i	0.599667	−	0.800250i	$-0.295300\pi$
$227$	−6.92959e7	−	1.20024e8i	−0.393203	−	0.681048i	−0.992870	+	0.119202i	$0.961966\pi$
$228$		0			0
$229$	−3.35184e7	+	5.80556e7i	−0.184442	+	0.319463i	−0.943388	−	0.331690i	$-0.892381\pi$
$229$	−3.35184e7	+	5.80556e7i	−0.184442	+	0.319463i	0.758946	+	0.651153i	$0.225714\pi$
$230$	5.54871e7	−	9.61064e7i	0.300707	−	0.520841i
$231$		0			0
$232$	1.93351e6	+	3.34894e6i	0.0101658	+	0.0176076i
$233$	1.30899e8			0.677940			0.338970	−	0.940797i	$-0.389922\pi$
$233$	1.30899e8			0.677940			0.338970	+	0.940797i	$0.389922\pi$
$234$		0			0
$235$	1.15219e8			0.579142
$236$	−2.55644e7	−	4.42789e7i	−0.126603	−	0.219283i
$237$		0			0
$238$	2.08268e7	−	3.60731e7i	0.100139	−	0.173446i
$239$	−9.63707e6	+	1.66919e7i	−0.0456617	+	0.0790884i	−0.887953	−	0.459934i	$-0.847873\pi$
$239$	−9.63707e6	+	1.66919e7i	−0.0456617	+	0.0790884i	0.842291	+	0.539023i	$0.181206\pi$
$240$		0			0
$241$	−3.42495e7	−	5.93219e7i	−0.157614	−	0.272995i	0.776394	−	0.630248i	$-0.217047\pi$
$241$	−3.42495e7	−	5.93219e7i	−0.157614	−	0.272995i	−0.934008	+	0.357253i	$0.883714\pi$
$242$	−1.22672e8			−0.556405
$243$		0			0
$244$	3.64441e7			0.160606
$245$	−1.11685e8	−	1.93444e8i	−0.485192	−	0.840377i
$246$		0			0
$247$	−1.06086e7	+	1.83746e7i	−0.0447938	+	0.0775851i
$248$	−3.99450e7	+	6.91868e7i	−0.166296	+	0.288033i
$249$		0			0
$250$	−8.70571e7	−	1.50787e8i	−0.352382	−	0.610344i
$251$	2.45365e7			0.0979387			0.0489693	−	0.998800i	$-0.484406\pi$
$251$	2.45365e7			0.0979387			0.0489693	+	0.998800i	$0.484406\pi$
$252$		0			0
$253$	−1.00811e8			−0.391367
$254$	1.30020e8	+	2.25202e8i	0.497843	+	0.862290i
$255$		0			0
$256$	−8.38861e6	+	1.45295e7i	−0.0312500	+	0.0541266i
$257$	−1.66362e8	+	2.88148e8i	−0.611349	+	1.05889i	0.379664	+	0.925124i	$0.376039\pi$
$257$	−1.66362e8	+	2.88148e8i	−0.611349	+	1.05889i	−0.991013	+	0.133763i	$0.957294\pi$
$258$		0			0
$259$	4.44856e7	+	7.70514e7i	0.159100	+	0.275570i
$260$	3.01214e7			0.106284
$261$		0			0
$262$	1.30512e8			0.448326
$263$	1.85275e8	+	3.20907e8i	0.628019	+	1.08776i	0.987949	+	0.154781i	$0.0494673\pi$
$263$	1.85275e8	+	3.20907e8i	0.628019	+	1.08776i	−0.359930	+	0.932979i	$0.617199\pi$
$264$		0			0
$265$	1.91780e8	−	3.32173e8i	0.633058	−	1.09649i
$266$	−8.30937e6	+	1.43922e7i	−0.0270696	+	0.0468860i
$267$		0			0
$268$	1.53745e8	+	2.66294e8i	0.487898	+	0.845063i
$269$	4.30980e8			1.34997			0.674985	−	0.737831i	$-0.264150\pi$
$269$	4.30980e8			1.34997			0.674985	+	0.737831i	$0.264150\pi$
$270$		0			0
$271$	3.41000e8			1.04079			0.520394	−	0.853927i	$-0.325785\pi$
$271$	3.41000e8			1.04079			0.520394	+	0.853927i	$0.325785\pi$
$272$	−6.48923e7	−	1.12397e8i	−0.195525	−	0.338659i
$273$		0			0
$274$	1.44638e8	−	2.50520e8i	0.424771	−	0.735724i
$275$	523012.	−	905884.i	0.00151652	−	0.00262669i
$276$		0			0
$277$	−3.41041e8	−	5.90701e8i	−0.964112	−	1.66989i	−0.711982	−	0.702198i	$-0.752202\pi$
$277$	−3.41041e8	−	5.90701e8i	−0.964112	−	1.66989i	−0.252130	−	0.967693i	$-0.581131\pi$
$278$	4.28147e8			1.19519
$279$		0			0
$280$	2.35932e7			0.0642293
$281$	−3.37119e7	−	5.83907e7i	−0.0906382	−	0.156990i	0.817142	−	0.576437i	$-0.195557\pi$
$281$	−3.37119e7	−	5.83907e7i	−0.0906382	−	0.156990i	−0.907780	+	0.419447i	$0.862224\pi$
$282$		0			0
$283$	2.57861e8	−	4.46628e8i	0.676290	−	1.17137i	−0.299800	−	0.954002i	$-0.596920\pi$
$283$	2.57861e8	−	4.46628e8i	0.676290	−	1.17137i	0.976090	−	0.217366i	$-0.0697466\pi$
$284$	−7.86284e7	+	1.36188e8i	−0.203688	+	0.352798i
$285$		0			0
$286$	−1.36814e7	−	2.36968e7i	−0.0345818	−	0.0598975i
$287$	−8.83375e7			−0.220576
$288$		0			0
$289$	5.93644e8			1.44672
$290$	−8.47197e6	−	1.46739e7i	−0.0203982	−	0.0353307i
$291$		0			0
$292$	−5.13102e7	+	8.88719e7i	−0.120605	+	0.208893i
$293$	−4.03009e7	+	6.98032e7i	−0.0936005	+	0.162121i	−0.909024	−	0.416744i	$-0.863171\pi$
$293$	−4.03009e7	+	6.98032e7i	−0.0936005	+	0.162121i	0.815423	+	0.578865i	$0.196504\pi$
$294$		0			0
$295$	1.12014e8	+	1.94014e8i	0.254037	+	0.440004i
$296$	2.77217e8			0.621297
$297$		0			0
$298$	−4.35374e7			−0.0953029
$299$	4.15110e7	+	7.18991e7i	0.0898077	+	0.155552i
$300$		0			0
$301$	1.64732e7	−	2.85324e7i	0.0348173	−	0.0603054i
$302$	−8.77444e7	+	1.51978e8i	−0.183314	+	0.317509i
$303$		0			0
$304$	2.58904e7	+	4.48435e7i	0.0528544	+	0.0915465i
$305$	−1.59685e8			−0.322266
$306$		0			0
$307$	3.87201e8			0.763752			0.381876	−	0.924214i	$-0.375278\pi$
$307$	3.87201e8			0.763752			0.381876	+	0.924214i	$0.375278\pi$
$308$	−1.07162e7	−	1.85610e7i	−0.0208984	−	0.0361971i
$309$		0			0
$310$	1.75025e8	−	3.03152e8i	0.333683	−	0.577956i
$311$	−3.15388e8	+	5.46268e8i	−0.594543	+	1.02978i	0.399068	+	0.916921i	$0.369334\pi$
$311$	−3.15388e8	+	5.46268e8i	−0.594543	+	1.02978i	−0.993611	+	0.112858i	$0.964000\pi$
$312$		0			0
$313$	−2.37475e8	−	4.11320e8i	−0.437737	−	0.758183i	0.559777	−	0.828643i	$-0.310887\pi$
$313$	−2.37475e8	−	4.11320e8i	−0.437737	−	0.758183i	−0.997515	+	0.0704598i	$0.977553\pi$
$314$	3.99254e7			0.0727772
$315$		0			0
$316$	3.57759e8			0.637801
$317$	−1.45155e8	−	2.51415e8i	−0.255931	−	0.443286i	0.709217	−	0.704991i	$-0.249049\pi$
$317$	−1.45155e8	−	2.51415e8i	−0.255931	−	0.443286i	−0.965148	+	0.261704i	$0.915715\pi$
$318$		0			0
$319$	−7.69606e6	+	1.33300e7i	−0.0132740	+	0.0229912i
$320$	3.67559e7	−	6.36631e7i	0.0627050	−	0.108608i
$321$		0			0
$322$	3.25143e7	+	5.63164e7i	0.0542723	+	0.0940024i
$323$	−4.00564e8			−0.661398
$324$		0			0
$325$	−861448.			−0.00139199
$326$	1.79057e8	+	3.10135e8i	0.286239	+	0.495781i
$327$		0			0
$328$	−1.37621e8	+	2.38367e8i	−0.215341	+	0.372982i
$329$	−3.37579e7	+	5.84703e7i	−0.0522624	+	0.0905211i
$330$		0			0
$331$	−5.24913e8	−	9.09175e8i	−0.795589	−	1.37800i	−0.922464	−	0.386083i	$-0.873828\pi$
$331$	−5.24913e8	−	9.09175e8i	−0.795589	−	1.37800i	0.126875	−	0.991919i	$-0.459505\pi$
$332$	6.29059e8			0.943427
$333$		0			0
$334$	−4.91825e8			−0.722267
$335$	−6.73655e8	−	1.16680e9i	−0.978995	−	1.69567i
$336$		0			0
$337$	−2.54630e8	+	4.41032e8i	−0.362414	+	0.627719i	−0.988358	−	0.152149i	$-0.951381\pi$
$337$	−2.54630e8	+	4.41032e8i	−0.362414	+	0.627719i	0.625944	+	0.779868i	$0.284714\pi$
$338$	2.39727e8	−	4.15219e8i	0.337682	−	0.584883i
$339$		0			0
$340$	2.84335e8	+	4.92482e8i	0.392332	+	0.679539i
$341$	−3.17990e8			−0.434284
$342$		0			0
$343$	2.66218e8			0.356211
$344$	−5.13274e7	−	8.89016e7i	−0.0679821	−	0.117748i
$345$		0			0
$346$	4.09964e8	−	7.10079e8i	0.532083	−	0.921595i
$347$	−2.34537e8	+	4.06230e8i	−0.301341	+	0.521938i	−0.976440	−	0.215789i	$-0.930768\pi$
$347$	−2.34537e8	+	4.06230e8i	−0.301341	+	0.521938i	0.675099	+	0.737727i	$0.264101\pi$
$348$		0			0
$349$	1.09391e8	+	1.89470e8i	0.137750	+	0.238590i	0.926645	−	0.375938i	$-0.122680\pi$
$349$	1.09391e8	+	1.89470e8i	0.137750	+	0.238590i	−0.788895	+	0.614529i	$0.789346\pi$
$350$	−674746.			−0.000841205
$351$		0			0
$352$	−6.67792e7			−0.0816097
$353$	−2.21516e8	−	3.83677e8i	−0.268036	−	0.464252i	0.700318	−	0.713831i	$-0.253041\pi$
$353$	−2.21516e8	−	3.83677e8i	−0.268036	−	0.464252i	−0.968355	+	0.249578i	$0.919708\pi$
$354$		0			0
$355$	3.44521e8	−	5.96729e8i	0.408712	−	0.707909i
$356$	3.75195e6	−	6.49857e6i	0.00440740	−	0.00763384i
$357$		0			0
$358$	−2.43588e8	−	4.21908e8i	−0.280586	−	0.485989i
$359$	−4.34672e8			−0.495828			−0.247914	−	0.968782i	$-0.579745\pi$
$359$	−4.34672e8			−0.495828			−0.247914	+	0.968782i	$0.579745\pi$
$360$		0			0
$361$	−7.34057e8			−0.821210
$362$	3.84878e8	+	6.66628e8i	0.426426	+	0.738591i
$363$		0			0
$364$	−8.82526e6	+	1.52858e7i	−0.00959119	+	0.0166124i
$365$	2.24823e8	−	3.89405e8i	0.242000	−	0.419157i
$366$		0			0
$367$	8.23673e8	+	1.42664e9i	0.869809	+	1.50655i	0.862192	+	0.506582i	$0.169091\pi$
$367$	8.23673e8	+	1.42664e9i	0.869809	+	1.50655i	0.00761697	+	0.999971i	$0.497575\pi$
$368$	2.02616e8			0.211937
$369$		0			0
$370$	−1.21467e9			−1.24667
$371$	1.12379e8	+	1.94647e8i	0.114256	+	0.197896i
$372$		0			0
$373$	−2.85414e8	+	4.94351e8i	−0.284770	+	0.493235i	−0.972553	−	0.232680i	$-0.925250\pi$
$373$	−2.85414e8	+	4.94351e8i	−0.284770	+	0.493235i	0.687784	+	0.725916i	$0.258584\pi$
$374$	2.58294e8	−	4.47378e8i	0.255308	−	0.442206i
$375$		0			0
$376$	1.05183e8	+	1.82182e8i	0.102044	+	0.176746i
$377$	1.26761e7			0.0121840
$378$		0			0
$379$	−1.71431e8			−0.161753			−0.0808763	−	0.996724i	$-0.525772\pi$
$379$	−1.71431e8			−0.161753			−0.0808763	+	0.996724i	$0.525772\pi$
$380$	−1.13442e8	−	1.96488e8i	−0.106055	−	0.183693i
$381$		0			0
$382$	2.95683e8	−	5.12138e8i	0.271397	−	0.470073i
$383$	3.33432e8	−	5.77522e8i	0.303258	−	0.525258i	−0.673614	−	0.739083i	$-0.735259\pi$
$383$	3.33432e8	−	5.77522e8i	0.303258	−	0.525258i	0.976872	+	0.213825i	$0.0685924\pi$
$384$		0			0
$385$	4.69545e7	+	8.13276e7i	0.0419339	+	0.0726316i
$386$	6.51756e8			0.576806
$387$		0			0
$388$	−4.98717e8			−0.433454
$389$	−2.54407e8	−	4.40646e8i	−0.219132	−	0.379547i	0.735411	−	0.677621i	$-0.236989\pi$
$389$	−2.54407e8	−	4.40646e8i	−0.219132	−	0.379547i	−0.954543	+	0.298074i	$0.903656\pi$
$390$		0			0
$391$	−7.83696e8	+	1.35740e9i	−0.663024	+	1.14839i
$392$	2.03914e8	−	3.53190e8i	0.170981	−	0.296147i
$393$		0			0
$394$	9.76133e7	+	1.69071e8i	0.0804030	+	0.139262i
$395$	−1.56757e9			−1.27979
$396$		0			0
$397$	9.18742e8			0.736931			0.368466	−	0.929641i	$-0.379883\pi$
$397$	9.18742e8			0.736931			0.368466	+	0.929641i	$0.379883\pi$
$398$	6.08534e8	+	1.05401e9i	0.483832	+	0.838021i
$399$		0			0
$400$	−1.05119e6	+	1.82071e6i	−0.000821241	+	0.00142243i
$401$	9.88625e8	−	1.71235e9i	0.765643	−	1.32613i	−0.174264	−	0.984699i	$-0.555754\pi$
$401$	9.88625e8	−	1.71235e9i	0.765643	−	1.32613i	0.939906	−	0.341433i	$-0.110912\pi$
$402$		0			0
$403$	1.30940e8	+	2.26794e8i	0.0996560	+	0.172609i
$404$	7.39103e7			0.0557661
$405$		0			0
$406$	9.92880e6			0.00736301
$407$	5.51711e8	+	9.55591e8i	0.405631	+	0.702573i
$408$		0			0
$409$	−1.24460e9	+	2.15572e9i	−0.899497	+	1.55797i	−0.0713582	+	0.997451i	$0.522733\pi$
$409$	−1.24460e9	+	2.15572e9i	−0.899497	+	1.55797i	−0.828139	+	0.560523i	$0.810600\pi$
$410$	6.03008e8	−	1.04444e9i	0.432095	−	0.748411i
$411$		0			0
$412$	5.43573e8	+	9.41496e8i	0.382928	+	0.663252i
$413$	−1.31276e8			−0.0916981
$414$		0			0
$415$	−2.75631e9			−1.89304
$416$	2.74978e7	+	4.76276e7i	0.0187271	+	0.0324364i
$417$		0			0
$418$	−1.03053e8	+	1.78493e8i	−0.0690149	+	0.119537i
$419$	−8.95795e8	+	1.55156e9i	−0.594921	+	1.03043i	0.398637	+	0.917109i	$0.369483\pi$
$419$	−8.95795e8	+	1.55156e9i	−0.594921	+	1.03043i	−0.993558	+	0.113325i	$0.963850\pi$
$420$		0			0
$421$	6.98308e6	+	1.20950e7i	0.00456099	+	0.00789987i	0.868297	−	0.496045i	$-0.165215\pi$
$421$	6.98308e6	+	1.20950e7i	0.00456099	+	0.00789987i	−0.863736	+	0.503945i	$0.831882\pi$
$422$	−4.70259e8			−0.304610
$423$		0			0
$424$	7.00304e8			0.446176
$425$	−8.13175e6	−	1.40846e7i	−0.00513834	−	0.00889986i
$426$		0			0
$427$	4.67861e7	−	8.10359e7i	0.0290817	−	0.0503709i
$428$	−7.17405e8	+	1.24258e9i	−0.442294	+	0.766076i
$429$		0			0
$430$	2.24898e8	+	3.89535e8i	0.136410	+	0.236269i
$431$	−2.57849e9			−1.55129			−0.775647	−	0.631166i	$-0.782577\pi$
$431$	−2.57849e9			−1.55129			−0.775647	+	0.631166i	$0.782577\pi$
$432$		0			0
$433$	1.82232e9			1.07874			0.539372	−	0.842068i	$-0.318662\pi$
$433$	1.82232e9			1.07874			0.539372	+	0.842068i	$0.318662\pi$
$434$	1.02561e8	+	1.77641e8i	0.0602238	+	0.104311i
$435$		0			0
$436$	−5.62329e8	+	9.73983e8i	−0.324929	+	0.562793i
$437$	3.12675e8	−	5.41569e8i	0.179229	−	0.310434i
$438$		0			0
$439$	−6.65655e8	−	1.15295e9i	−0.375512	−	0.650405i	0.614892	−	0.788611i	$-0.289200\pi$
$439$	−6.65655e8	−	1.15295e9i	−0.375512	−	0.650405i	−0.990403	+	0.138206i	$0.955866\pi$
$440$	2.92603e8			0.163755
$441$		0			0
$442$	−4.25433e8			−0.234344
$443$	1.85680e7	+	3.21607e7i	0.0101473	+	0.0175757i	0.871054	−	0.491186i	$-0.163437\pi$
$443$	1.85680e7	+	3.21607e7i	0.0101473	+	0.0175757i	−0.860907	+	0.508762i	$0.830103\pi$
$444$		0			0
$445$	−1.64397e7	+	2.84744e7i	−0.00884371	+	0.0153178i
$446$	1.32726e8	−	2.29888e8i	0.0708409	−	0.122700i
$447$		0			0
$448$	2.15382e7	+	3.73052e7i	0.0113171	+	0.0196018i
$449$	−1.95772e9			−1.02068			−0.510338	−	0.859974i	$-0.670480\pi$
$449$	−1.95772e9			−1.02068			−0.510338	+	0.859974i	$0.670480\pi$
$450$		0			0
$451$	−1.09556e9			−0.562366
$452$	2.22604e8	+	3.85562e8i	0.113383	+	0.196386i
$453$		0			0
$454$	−5.54367e8	+	9.60192e8i	−0.278037	+	0.481574i
$455$	3.86691e7	−	6.69769e7i	0.0192453	−	0.0333338i
$456$		0			0
$457$	−4.99589e8	−	8.65314e8i	−0.244854	−	0.424099i	0.717237	−	0.696829i	$-0.245407\pi$
$457$	−4.99589e8	−	8.65314e8i	−0.244854	−	0.424099i	−0.962090	+	0.272731i	$0.912073\pi$
$458$	5.36295e8			0.260840
$459$		0			0
$460$	−8.87793e8			−0.425265
$461$	8.24033e8	+	1.42727e9i	0.391734	+	0.678503i	0.992678	−	0.120788i	$-0.0385420\pi$
$461$	8.24033e8	+	1.42727e9i	0.391734	+	0.678503i	−0.600944	+	0.799291i	$0.705209\pi$
$462$		0			0
$463$	−1.73531e9	+	3.00564e9i	−0.812538	+	1.40736i	0.0985450	+	0.995133i	$0.468581\pi$
$463$	−1.73531e9	+	3.00564e9i	−0.812538	+	1.40736i	−0.911083	+	0.412224i	$0.864752\pi$
$464$	1.54681e7	−	2.67916e7i	0.00718827	−	0.0124505i
$465$		0			0
$466$	−5.23597e8	−	9.06896e8i	−0.239688	−	0.415152i
$467$	2.49926e9			1.13554			0.567769	−	0.823188i	$-0.307807\pi$
$467$	2.49926e9			1.13554			0.567769	+	0.823188i	$0.307807\pi$
$468$		0			0
$469$	7.89496e8			0.353382
$470$	−4.60874e8	−	7.98258e8i	−0.204758	−	0.354651i
$471$		0			0
$472$	−2.04516e8	+	3.54231e8i	−0.0895219	+	0.155056i
$473$	2.04301e8	−	3.53859e8i	0.0887680	−	0.153751i
$474$		0			0
$475$	3.24436e6	+	5.61940e6i	0.00138900	+	0.00240582i
$476$	−3.33229e8			−0.141618
$477$		0			0
$478$	1.54193e8			0.0645754
$479$	2.19805e9	+	3.80714e9i	0.913827	+	1.58280i	0.808609	+	0.588347i	$0.200221\pi$
$479$	2.19805e9	+	3.80714e9i	0.913827	+	1.58280i	0.105219	+	0.994449i	$0.466446\pi$
$480$		0			0
$481$	4.54358e8	−	7.86972e8i	0.186162	−	0.322442i
$482$	−2.73996e8	+	4.74575e8i	−0.111450	+	0.193037i
$483$		0			0
$484$	4.90687e8	+	8.49895e8i	0.196719	+	0.340727i
$485$	2.18520e9			0.869752
$486$		0			0
$487$	−2.82989e9			−1.11025			−0.555123	−	0.831769i	$-0.687329\pi$
$487$	−2.82989e9			−1.11025			−0.555123	+	0.831769i	$0.687329\pi$
$488$	−1.45776e8	−	2.52492e8i	−0.0567830	−	0.0983510i
$489$		0			0
$490$	−8.93480e8	+	1.54755e9i	−0.343083	+	0.594237i
$491$	−1.22921e9	+	2.12905e9i	−0.468641	+	0.811710i	−0.999358	−	0.0358390i	$-0.988590\pi$
$491$	−1.22921e9	+	2.12905e9i	−0.468641	+	0.811710i	0.530716	+	0.847550i	$0.321923\pi$
$492$		0			0
$493$	1.19658e8	+	2.07253e8i	0.0449755	+	0.0778999i
$494$	1.69737e8			0.0633480
$495$		0			0
$496$	6.39121e8			0.235178
$497$	2.01882e8	+	3.49671e8i	0.0737651	+	0.127765i
$498$		0			0
$499$	1.50302e9	−	2.60331e9i	0.541518	−	0.937938i	−0.457299	−	0.889313i	$-0.651183\pi$
$499$	1.50302e9	−	2.60331e9i	0.541518	−	0.937938i	0.998817	−	0.0486243i	$-0.0154837\pi$
$500$	−6.96457e8	+	1.20630e9i	−0.249172	+	0.431579i
$501$		0			0
$502$	−9.81459e7	−	1.69994e8i	−0.0346266	−	0.0599750i
$503$	−2.43279e8			−0.0852348			−0.0426174	−	0.999091i	$-0.513570\pi$
$503$	−2.43279e8			−0.0852348			−0.0426174	+	0.999091i	$0.513570\pi$
$504$		0			0
$505$	−3.23849e8			−0.111898
$506$	4.03242e8	+	6.98436e8i	0.138369	+	0.239662i
$507$		0			0
$508$	1.04016e9	−	1.80161e9i	0.352028	−	0.609731i
$509$	2.00216e9	−	3.46785e9i	0.672956	−	1.16559i	−0.304105	−	0.952638i	$-0.598357\pi$
$509$	2.00216e9	−	3.46785e9i	0.672956	−	1.16559i	0.977062	−	0.212956i	$-0.0683092\pi$
$510$		0			0
$511$	1.31742e8	+	2.28183e8i	0.0436767	+	0.0756503i
$512$	1.34218e8			0.0441942
$513$		0			0
$514$	2.66180e9			0.864578
$515$	−2.38174e9	−	4.12530e9i	−0.768369	−	1.33085i
$516$		0			0
$517$	−4.18665e8	+	7.25149e8i	−0.133245	+	0.230786i
$518$	3.55885e8	−	6.16411e8i	0.112501	−	0.194857i
$519$		0			0
$520$	−1.20486e8	−	2.08687e8i	−0.0375771	−	0.0650855i
$521$	5.72614e9			1.77390			0.886951	−	0.461864i	$-0.152819\pi$
$521$	5.72614e9			1.77390			0.886951	+	0.461864i	$0.152819\pi$
$522$		0			0
$523$	1.53753e9			0.469966			0.234983	−	0.971999i	$-0.424497\pi$
$523$	1.53753e9			0.469966			0.234983	+	0.971999i	$0.424497\pi$
$524$	−5.22046e8	−	9.04210e8i	−0.158507	−	0.274543i
$525$		0			0
$526$	1.48220e9	−	2.56725e9i	0.444077	−	0.769163i
$527$	−2.47204e9	+	4.28170e9i	−0.735731	+	1.27432i
$528$		0			0
$529$	4.78926e8	+	8.29523e8i	0.140661	+	0.243632i
$530$	−3.06848e9			−0.895279
$531$		0			0
$532$	1.32950e8			0.0382822
$533$	4.51122e8	+	7.81366e8i	0.129047	+	0.223516i
$534$		0			0
$535$	3.14341e9	−	5.44455e9i	0.887490	−	1.53718i
$536$	1.22996e9	−	2.13035e9i	0.344996	−	0.597550i
$537$		0			0
$538$	−1.72392e9	−	2.98592e9i	−0.477287	−	0.826685i
$539$	1.62330e9			0.446518
$540$		0			0
$541$	3.90611e9			1.06061			0.530303	−	0.847808i	$-0.322078\pi$
$541$	3.90611e9			1.06061			0.530303	+	0.847808i	$0.322078\pi$
$542$	−1.36400e9	−	2.36252e9i	−0.367974	−	0.637349i
$543$		0			0
$544$	−5.19138e8	+	8.99174e8i	−0.138257	+	0.239468i
$545$	2.46393e9	−	4.26764e9i	0.651988	−	1.12928i
$546$		0			0
$547$	2.50427e9	+	4.33752e9i	0.654222	+	1.13315i	0.982088	+	0.188422i	$0.0603372\pi$
$547$	2.50427e9	+	4.33752e9i	0.654222	+	1.13315i	−0.327866	+	0.944724i	$0.606330\pi$
$548$	−2.31420e9			−0.600716
$549$		0			0
$550$	−8.36820e6			−0.00214468
$551$	−4.77404e7	−	8.26888e7i	−0.0121578	−	0.0210580i
$552$		0			0
$553$	4.59282e8	−	7.95500e8i	0.115489	−	0.200033i
$554$	−2.72833e9	+	4.72560e9i	−0.681730	+	1.18079i
$555$		0			0
$556$	−1.71259e9	−	2.96629e9i	−0.422562	−	0.731899i
$557$	−1.45387e9			−0.356479			−0.178239	−	0.983987i	$-0.557040\pi$
$557$	−1.45387e9			−0.356479			−0.178239	+	0.983987i	$0.557040\pi$
$558$		0			0
$559$	−3.36502e8			−0.0814790
$560$	−9.43727e7	−	1.63458e8i	−0.0227085	−	0.0393323i
$561$		0			0
$562$	−2.69695e8	+	4.67126e8i	−0.0640909	+	0.111009i
$563$	−2.95600e9	+	5.11994e9i	−0.698111	+	1.20916i	0.271009	+	0.962577i	$0.412643\pi$
$563$	−2.95600e9	+	5.11994e9i	−0.698111	+	1.20916i	−0.969121	+	0.246587i	$0.920691\pi$
$564$		0			0
$565$	−9.75373e8	−	1.68940e9i	−0.227510	−	0.394060i
$566$	−4.12577e9			−0.956418
$567$		0			0
$568$	1.25805e9			0.288058
$569$	−3.81277e9	−	6.60392e9i	−0.867657	−	1.50283i	−0.864384	−	0.502831i	$-0.832292\pi$
$569$	−3.81277e9	−	6.60392e9i	−0.867657	−	1.50283i	−0.00327263	−	0.999995i	$-0.501042\pi$
$570$		0			0
$571$	3.68647e9	−	6.38515e9i	0.828675	−	1.43531i	−0.0704037	−	0.997519i	$-0.522429\pi$
$571$	3.68647e9	−	6.38515e9i	0.828675	−	1.43531i	0.899078	−	0.437788i	$-0.144238\pi$
$572$	−1.09451e8	+	1.89574e8i	−0.0244531	+	0.0423539i
$573$		0			0
$574$	3.53350e8	+	6.12020e8i	0.0779854	+	0.135075i
$575$	2.53902e7			0.00556965
$576$		0			0
$577$	1.95790e9			0.424302			0.212151	−	0.977237i	$-0.431953\pi$
$577$	1.95790e9			0.424302			0.212151	+	0.977237i	$0.431953\pi$
$578$	−2.37458e9	−	4.11289e9i	−0.511492	−	0.885930i
$579$		0			0
$580$	−6.77758e7	+	1.17391e8i	−0.0144237	+	0.0249826i
$581$	8.07571e8	−	1.39875e9i	0.170830	−	0.295886i
$582$		0			0
$583$	1.39373e9	+	2.41401e9i	0.291298	+	0.504543i
$584$	8.20964e8			0.170561
$585$		0			0
$586$	6.44815e8			0.132371
$587$	1.09928e9	+	1.90401e9i	0.224324	+	0.388541i	0.956117	−	0.292987i	$-0.0946492\pi$
$587$	1.09928e9	+	1.90401e9i	0.224324	+	0.388541i	−0.731792	+	0.681528i	$0.761316\pi$
$588$		0			0
$589$	9.86283e8	−	1.70829e9i	0.198883	−	0.344476i
$590$	8.96114e8	−	1.55212e9i	0.179631	−	0.311130i
$591$		0			0
$592$	−1.10887e9	−	1.92062e9i	−0.219662	−	0.380465i
$593$	6.54769e9			1.28943			0.644713	−	0.764425i	$-0.276977\pi$
$593$	6.54769e9			1.28943			0.644713	+	0.764425i	$0.276977\pi$
$594$		0			0
$595$	1.46009e9			0.284165
$596$	1.74150e8	+	3.01636e8i	0.0336946	+	0.0583608i
$597$		0			0
$598$	3.32088e8	−	5.75193e8i	0.0635037	−	0.109992i
$599$	2.63642e9	−	4.56641e9i	0.501210	−	0.868122i	−0.498789	−	0.866724i	$-0.666222\pi$
$599$	2.63642e9	−	4.56641e9i	0.501210	−	0.868122i	0.999999	−	0.00139803i	$-0.000445007\pi$
$600$		0			0
$601$	−2.59635e9	−	4.49701e9i	−0.487869	−	0.845013i	0.512034	−	0.858965i	$-0.328892\pi$
$601$	−2.59635e9	−	4.49701e9i	−0.487869	−	0.845013i	−0.999903	+	0.0139518i	$0.995559\pi$
$602$	−2.63571e8			−0.0492392
$603$		0			0
$604$	1.40391e9			0.259245
$605$	−2.15002e9	−	3.72394e9i	−0.394728	−	0.683689i
$606$		0			0
$607$	1.21854e9	−	2.11058e9i	0.221147	−	0.383037i	−0.734010	−	0.679139i	$-0.762353\pi$
$607$	1.21854e9	−	2.11058e9i	0.221147	−	0.383037i	0.955156	+	0.296102i	$0.0956868\pi$
$608$	2.07123e8	−	3.58748e8i	0.0373737	−	0.0647332i
$609$		0			0
$610$	6.38741e8	+	1.10633e9i	0.113938	+	0.197347i
$611$	6.89579e8			0.122304
$612$		0			0
$613$	−3.31215e9			−0.580763			−0.290382	−	0.956911i	$-0.593782\pi$
$613$	−3.31215e9			−0.580763			−0.290382	+	0.956911i	$0.593782\pi$
$614$	−1.54880e9	−	2.68261e9i	−0.270027	−	0.467700i
$615$		0			0
$616$	−8.57295e7	+	1.48488e8i	−0.0147774	+	0.0255952i
$617$	−1.63565e9	+	2.83303e9i	−0.280345	+	0.485572i	−0.971470	−	0.237164i	$-0.923782\pi$
$617$	−1.63565e9	+	2.83303e9i	−0.280345	+	0.485572i	0.691125	+	0.722736i	$0.257116\pi$
$618$		0			0
$619$	7.18024e8	+	1.24365e9i	0.121681	+	0.210757i	0.920431	−	0.390906i	$-0.127838\pi$
$619$	7.18024e8	+	1.24365e9i	0.121681	+	0.210757i	−0.798750	+	0.601663i	$0.794505\pi$
$620$	−2.80040e9			−0.471899
$621$		0			0
$622$	5.04620e9			0.840811
$623$	−9.63334e6	−	1.66854e7i	−0.00159613	−	0.00276458i
$624$		0			0
$625$	3.07168e9	−	5.32030e9i	0.503263	−	0.871678i
$626$	−1.89980e9	+	3.29056e9i	−0.309527	+	0.536117i
$627$		0			0
$628$	−1.59701e8	−	2.76611e8i	−0.0257306	−	0.0445667i
$629$	1.71559e10			2.74876
$630$		0			0
$631$	−7.26152e9			−1.15060			−0.575301	−	0.817942i	$-0.695115\pi$
$631$	−7.26152e9			−1.15060			−0.575301	+	0.817942i	$0.695115\pi$
$632$	−1.43104e9	−	2.47863e9i	−0.225497	−	0.390572i
$633$		0			0
$634$	−1.16124e9	+	2.01132e9i	−0.180971	+	0.313451i
$635$	−4.55762e9	+	7.89402e9i	−0.706366	+	1.22346i
$636$		0			0
$637$	−6.68430e8	−	1.15776e9i	−0.102463	−	0.177472i
$638$	1.23137e8			0.0187722
$639$		0			0
$640$	−5.88094e8			−0.0886782
$641$	4.47848e9	+	7.75695e9i	0.671625	+	1.16329i	0.977443	+	0.211199i	$0.0677369\pi$
$641$	4.47848e9	+	7.75695e9i	0.671625	+	1.16329i	−0.305818	+	0.952090i	$0.598930\pi$
$642$		0			0
$643$	1.29774e9	−	2.24774e9i	0.192508	−	0.333433i	−0.753573	−	0.657364i	$-0.771671\pi$
$643$	1.29774e9	−	2.24774e9i	0.192508	−	0.333433i	0.946081	+	0.323931i	$0.105005\pi$
$644$	2.60114e8	−	4.50531e8i	0.0383763	−	0.0664698i
$645$		0			0
$646$	1.60226e9	+	2.77519e9i	0.233840	+	0.405022i
$647$	−9.11067e9			−1.32247			−0.661234	−	0.750180i	$-0.729967\pi$
$647$	−9.11067e9			−1.32247			−0.661234	+	0.750180i	$0.729967\pi$
$648$		0			0
$649$	−1.62809e9			−0.233787
$650$	3.44579e6	+	5.96829e6i	0.000492144	+	0.000852418i
$651$		0			0
$652$	1.43245e9	−	2.48108e9i	0.202402	−	0.350570i
$653$	5.11670e9	−	8.86238e9i	0.719107	−	1.24553i	−0.242247	−	0.970215i	$-0.577884\pi$
$653$	5.11670e9	−	8.86238e9i	0.719107	−	1.24553i	0.961354	−	0.275315i	$-0.0887823\pi$
$654$		0			0
$655$	2.28742e9	+	3.96193e9i	0.318054	+	0.550886i
$656$	2.20194e9			0.304539
$657$		0			0
$658$	5.40126e8			0.0739102
$659$	5.37910e9	+	9.31688e9i	0.732168	+	1.26815i	0.955955	+	0.293515i	$0.0948249\pi$
$659$	5.37910e9	+	9.31688e9i	0.732168	+	1.26815i	−0.223786	+	0.974638i	$0.571842\pi$
$660$		0			0
$661$	−3.02911e9	+	5.24658e9i	−0.407953	+	0.706596i	−0.994660	−	0.103204i	$-0.967091\pi$
$661$	−3.02911e9	+	5.24658e9i	−0.407953	+	0.706596i	0.586707	+	0.809799i	$0.300424\pi$
$662$	−4.19930e9	+	7.27340e9i	−0.562567	+	0.974394i
$663$		0			0
$664$	−2.51624e9	−	4.35825e9i	−0.333552	−	0.577729i
$665$	−5.82539e8			−0.0768156
$666$		0			0
$667$	−3.73613e8			−0.0487508
$668$	1.96730e9	+	3.40746e9i	0.255360	+	0.442296i
$669$		0			0
$670$	−5.38924e9	+	9.33444e9i	−0.692254	+	1.19902i
$671$	5.80241e8	−	1.00501e9i	0.0741446	−	0.128422i
$672$		0			0
$673$	−2.11743e9	−	3.66750e9i	−0.267767	−	0.463786i	0.700518	−	0.713635i	$-0.252952\pi$
$673$	−2.11743e9	−	3.66750e9i	−0.267767	−	0.463786i	−0.968285	+	0.249849i	$0.919619\pi$
$674$	4.07408e9			0.512530
$675$		0			0
$676$	−3.83563e9			−0.477555
$677$	−9.92582e8	−	1.71920e9i	−0.122944	−	0.212945i	0.797984	−	0.602679i	$-0.205900\pi$
$677$	−9.92582e8	−	1.71920e9i	−0.122944	−	0.212945i	−0.920927	+	0.389735i	$0.872567\pi$
$678$		0			0
$679$	−6.40241e8	+	1.10893e9i	−0.0784873	+	0.135944i
$680$	2.27468e9	−	3.93986e9i	0.277421	−	0.480507i
$681$		0			0
$682$	1.27196e9	+	2.20310e9i	0.153543	+	0.265943i
$683$	−2.07576e9			−0.249290			−0.124645	−	0.992201i	$-0.539779\pi$
$683$	−2.07576e9			−0.249290			−0.124645	+	0.992201i	$0.539779\pi$
$684$		0			0
$685$	1.01400e10			1.20537
$686$	−1.06487e9	−	1.84441e9i	−0.125940	−	0.218134i
$687$		0			0
$688$	−4.10619e8	+	7.11213e8i	−0.0480706	+	0.0832607i
$689$	1.14780e9	−	1.98804e9i	0.133690	−	0.231557i
$690$		0			0
$691$	−7.28221e8	−	1.26132e9i	−0.0839634	−	0.145429i	0.820986	−	0.570949i	$-0.193425\pi$
$691$	−7.28221e8	−	1.26132e9i	−0.0839634	−	0.145429i	−0.904949	+	0.425520i	$0.860091\pi$
$692$	−6.55943e9			−0.752479
$693$		0			0
$694$	3.75259e9			0.426160
$695$	7.50394e9	+	1.29972e10i	0.847896	+	1.46860i
$696$		0			0
$697$	−8.51685e9	+	1.47516e10i	−0.952718	+	1.65016i
$698$	8.75127e8	−	1.51576e9i	0.0974040	−	0.168709i
$699$		0			0
$700$	2.69898e6	+	4.67478e6i	0.000297411	+	0.000515131i
$701$	−6.55669e9			−0.718905			−0.359452	−	0.933163i	$-0.617036\pi$
$701$	−6.55669e9			−0.718905			−0.359452	+	0.933163i	$0.617036\pi$
$702$		0			0
$703$	−6.84477e9			−0.743046
$704$	2.67117e8	+	4.62660e8i	0.0288534	+	0.0499755i
$705$		0			0
$706$	−1.77213e9	+	3.06941e9i	−0.189530	+	0.328276i
$707$	9.48843e7	−	1.64344e8i	0.0100978	−	0.0174899i
$708$		0			0
$709$	−3.20151e9	−	5.54518e9i	−0.337360	−	0.584324i	0.646576	−	0.762850i	$-0.276200\pi$
$709$	−3.20151e9	−	5.54518e9i	−0.337360	−	0.584324i	−0.983935	+	0.178526i	$0.942867\pi$
$710$	−5.51234e9			−0.578005
$711$		0			0
$712$	−6.00313e7			−0.00623301
$713$	−3.85929e9	−	6.68449e9i	−0.398744	−	0.690645i
$714$		0			0
$715$	4.79575e8	−	8.30648e8i	0.0490665	−	0.0849857i
$716$	−1.94871e9	+	3.37526e9i	−0.198404	+	0.343646i
$717$		0			0
$718$	1.73869e9	+	3.01150e9i	0.175302	+	0.303631i
$719$	−1.33984e10			−1.34432			−0.672158	−	0.740408i	$-0.734632\pi$
$719$	−1.33984e10			−1.34432			−0.672158	+	0.740408i	$0.734632\pi$
$720$		0			0
$721$	2.79130e9			0.277354
$722$	2.93623e9	+	5.08569e9i	0.290342	+	0.502887i
$723$		0			0
$724$	3.07902e9	−	5.33303e9i	0.301528	−	0.522262i
$725$	1.93833e6	−	3.35729e6i	0.000188906	−	0.000327194i
$726$		0			0
$727$	7.20401e9	+	1.24777e10i	0.695351	+	1.20438i	0.970062	+	0.242856i	$0.0780844\pi$
$727$	7.20401e9	+	1.24777e10i	0.695351	+	1.20438i	−0.274711	+	0.961527i	$0.588582\pi$
$728$	1.41204e8			0.0135640
$729$		0			0
$730$	−3.59717e9			−0.342240
$731$	−3.17645e9	−	5.50178e9i	−0.300768	−	0.520945i
$732$		0			0
$733$	−5.60638e9	+	9.71053e9i	−0.525797	+	0.910708i	0.473751	+	0.880659i	$0.342900\pi$
$733$	−5.60638e9	+	9.71053e9i	−0.525797	+	0.910708i	−0.999548	+	0.0300489i	$0.990434\pi$
$734$	6.58939e9	−	1.14132e10i	0.615048	−	1.06529i
$735$		0			0
$736$	−8.10466e8	−	1.40377e9i	−0.0749311	−	0.129785i
$737$	9.79133e9			0.900960
$738$		0			0
$739$	7.79784e9			0.710753			0.355377	−	0.934723i	$-0.384353\pi$
$739$	7.79784e9			0.710753			0.355377	+	0.934723i	$0.384353\pi$
$740$	4.85867e9	+	8.41546e9i	0.440764	+	0.763426i
$741$		0			0
$742$	8.99034e8	−	1.55717e9i	0.0807909	−	0.139934i
$743$	−7.80793e9	+	1.35237e10i	−0.698353	+	1.20958i	0.270684	+	0.962668i	$0.412750\pi$
$743$	−7.80793e9	+	1.35237e10i	−0.698353	+	1.20958i	−0.969037	+	0.246915i	$0.920583\pi$
$744$		0			0
$745$	−7.63062e8	−	1.32166e9i	−0.0676103	−	0.117104i
$746$	4.56662e9			0.402725
$747$		0			0
$748$	−4.13270e9			−0.361059
$749$	1.84198e9	+	3.19040e9i	0.160176	+	0.277433i
$750$		0			0
$751$	−6.37283e9	+	1.10381e10i	−0.549025	+	0.950940i	0.449316	+	0.893373i	$0.351668\pi$
$751$	−6.37283e9	+	1.10381e10i	−0.549025	+	0.950940i	−0.998342	+	0.0575671i	$0.981666\pi$
$752$	8.41464e8	−	1.45746e9i	0.0721561	−	0.124978i
$753$		0			0
$754$	−5.07044e7	−	8.78226e7i	−0.00430770	−	0.00746116i
$755$	−6.15144e9			−0.520190
$756$		0			0
$757$	−2.31819e10			−1.94229			−0.971144	−	0.238492i	$-0.923347\pi$
$757$	−2.31819e10			−1.94229			−0.971144	+	0.238492i	$0.923347\pi$
$758$	6.85722e8	+	1.18771e9i	0.0571882	+	0.0990528i
$759$		0			0
$760$	−9.07540e8	+	1.57191e9i	−0.0749926	+	0.129891i
$761$	1.11016e10	−	1.92286e10i	0.913148	−	1.58162i	0.103557	−	0.994624i	$-0.466978\pi$
$761$	1.11016e10	−	1.92286e10i	0.913148	−	1.58162i	0.809591	−	0.586995i	$-0.199689\pi$
$762$		0			0
$763$	1.44381e9	+	2.50075e9i	0.117672	+	0.203814i
$764$	−4.73093e9			−0.383813
$765$		0			0
$766$	−5.33492e9			−0.428871
$767$	6.70401e8	+	1.16117e9i	0.0536477	+	0.0929205i
$768$		0			0
$769$	−1.09956e10	+	1.90449e10i	−0.871916	+	1.51020i	−0.0119049	+	0.999929i	$0.503790\pi$
$769$	−1.09956e10	+	1.90449e10i	−0.871916	+	1.51020i	−0.860012	+	0.510275i	$0.829544\pi$
$770$	3.75636e8	−	6.50621e8i	0.0296517	−	0.0513583i
$771$		0			0
$772$	−2.60702e9	−	4.51550e9i	−0.203932	−	0.353220i
$773$	−1.12771e10			−0.878151			−0.439075	−	0.898450i	$-0.644694\pi$
$773$	−1.12771e10			−0.878151			−0.439075	+	0.898450i	$0.644694\pi$
$774$		0			0
$775$	8.00891e7			0.00618042
$776$	1.99487e9	+	3.45521e9i	0.153249	+	0.265435i
$777$		0			0
$778$	−2.03525e9	+	3.52516e9i	−0.154950	+	0.268380i
$779$	3.39801e9	−	5.88553e9i	0.257539	−	0.446071i
$780$		0			0
$781$	2.50375e9	+	4.33662e9i	0.188067	+	0.325741i
$782$	1.25391e10			0.937657
$783$		0			0
$784$	−3.26263e9			−0.241803
$785$	6.99755e8	+	1.21201e9i	0.0516300	+	0.0894258i
$786$		0			0
$787$	9.17090e9	−	1.58845e10i	0.670657	−	1.16161i	−0.307061	−	0.951690i	$-0.599346\pi$
$787$	9.17090e9	−	1.58845e10i	0.670657	−	1.16161i	0.977718	−	0.209922i	$-0.0673210\pi$
$788$	7.80906e8	−	1.35257e9i	0.0568535	−	0.0984731i
$789$		0			0
$790$	6.27029e9	+	1.08605e10i	0.452473	+	0.783706i
$791$	1.14310e9			0.0821231
$792$		0			0
$793$	−9.55709e8			−0.0680565
$794$	−3.67497e9	−	6.36523e9i	−0.260544	−	0.451276i
$795$		0			0
$796$	4.86827e9	−	8.43210e9i	0.342121	−	0.592570i
$797$	−7.85176e8	+	1.35997e9i	−0.0549368	+	0.0951533i	−0.892186	−	0.451668i	$-0.850829\pi$
$797$	−7.85176e8	+	1.35997e9i	−0.0549368	+	0.0951533i	0.837249	+	0.546822i	$0.184162\pi$
$798$		0			0
$799$	6.50937e9	+	1.12746e10i	0.451466	+	0.781962i
$800$	1.68190e7			0.00116141
$801$		0			0
$802$	−1.58180e10			−1.08278
$803$	1.63386e9	+	2.82993e9i	0.111355	+	0.192873i
$804$		0			0
$805$	−1.13973e9	+	1.97407e9i	−0.0770044	+	0.133376i
$806$	1.04752e9	−	1.81435e9i	0.0704674	−	0.122053i
$807$		0			0
$808$	−2.95641e8	−	5.12066e8i	−0.0197163	−	0.0341496i
$809$	−3.66110e9			−0.243104			−0.121552	−	0.992585i	$-0.538787\pi$
$809$	−3.66110e9			−0.243104			−0.121552	+	0.992585i	$0.538787\pi$
$810$		0			0
$811$	−2.32307e10			−1.52929			−0.764645	−	0.644451i	$-0.777086\pi$
$811$	−2.32307e10			−1.52929			−0.764645	+	0.644451i	$0.777086\pi$
$812$	−3.97152e7	−	6.87887e7i	−0.00260322	−	0.00450891i
$813$		0			0
$814$	4.41369e9	−	7.64473e9i	0.286824	−	0.496794i
$815$	−6.27650e9	+	1.08712e10i	−0.406131	+	0.703439i
$816$		0			0
$817$	1.26732e9	+	2.19507e9i	0.0813038	+	0.140822i
$818$	1.99137e10			1.27208
$819$		0			0
$820$	−9.64812e9			−0.611075
$821$	−1.26531e10	−	2.19157e10i	−0.797984	−	1.38215i	−0.920927	−	0.389736i	$-0.872566\pi$
$821$	−1.26531e10	−	2.19157e10i	−0.797984	−	1.38215i	0.122943	−	0.992414i	$-0.460767\pi$
$822$		0			0
$823$	−4.14339e9	+	7.17655e9i	−0.259093	+	0.448763i	−0.965999	−	0.258545i	$-0.916757\pi$
$823$	−4.14339e9	+	7.17655e9i	−0.259093	+	0.448763i	0.706906	+	0.707307i	$0.250090\pi$
$824$	4.34858e9	−	7.53197e9i	0.270771	−	0.468990i
$825$		0			0
$826$	5.25104e8	+	9.09507e8i	0.0324202	+	0.0561534i
$827$	−3.06921e10			−1.88694			−0.943470	−	0.331459i	$-0.892459\pi$
$827$	−3.06921e10			−1.88694			−0.943470	+	0.331459i	$0.892459\pi$
$828$		0			0
$829$	1.83150e10			1.11652			0.558260	−	0.829666i	$-0.311469\pi$
$829$	1.83150e10			1.11652			0.558260	+	0.829666i	$0.311469\pi$
$830$	1.10253e10	+	1.90963e10i	0.669291	+	1.15925i
$831$		0			0
$832$	2.19982e8	−	3.81021e8i	0.0132421	−	0.0229360i
$833$	1.26195e10	−	2.18576e10i	0.756456	−	1.31022i
$834$		0			0
$835$	−8.62000e9	−	1.49303e10i	−0.512395	−	0.887494i
$836$	1.64885e9			0.0976018
$837$		0			0
$838$	1.43327e10			0.841346
$839$	−1.50106e10	−	2.59991e10i	−0.877466	−	1.51982i	−0.854113	−	0.520088i	$-0.825899\pi$
$839$	−1.50106e10	−	2.59991e10i	−0.877466	−	1.51982i	−0.0233529	−	0.999727i	$-0.507434\pi$
$840$		0			0
$841$	8.59642e9	−	1.48894e10i	0.498347	−	0.863161i
$842$	5.58646e7	−	9.67603e7i	0.00322511	−	0.00558605i
$843$		0			0
$844$	1.88104e9	+	3.25805e9i	0.107696	+	0.186535i
$845$	1.68064e10			0.958242
$846$		0			0
$847$	2.51973e9			0.142483
$848$	−2.80122e9	−	4.85185e9i	−0.157747	−	0.273226i
$849$		0			0
$850$	−6.50540e7	+	1.12677e8i	−0.00363335	+	0.00629315i
$851$	−1.33917e10	+	2.31951e10i	−0.744872	+	1.29016i
$852$		0			0
$853$	−1.39325e10	−	2.41319e10i	−0.768615	−	1.33128i	−0.938314	−	0.345784i	$-0.887613\pi$
$853$	−1.39325e10	−	2.41319e10i	−0.768615	−	1.33128i	0.169699	−	0.985496i	$-0.445720\pi$
$854$	−7.48578e8			−0.0411277
$855$		0			0
$856$	1.14785e10			0.625499
$857$	1.70531e9	+	2.95369e9i	0.0925488	+	0.160299i	0.908583	−	0.417704i	$-0.137165\pi$
$857$	1.70531e9	+	2.95369e9i	0.0925488	+	0.160299i	−0.816034	+	0.578004i	$0.803832\pi$
$858$		0			0
$859$	−4.50766e9	+	7.80749e9i	−0.242647	+	0.420277i	−0.961467	−	0.274919i	$-0.911349\pi$
$859$	−4.50766e9	+	7.80749e9i	−0.242647	+	0.420277i	0.718820	+	0.695196i	$0.244682\pi$
$860$	1.79919e9	−	3.11628e9i	0.0964565	−	0.167068i
$861$		0			0
$862$	1.03139e10	+	1.78643e10i	0.548466	+	0.949970i
$863$	−2.43096e9			−0.128748			−0.0643739	−	0.997926i	$-0.520505\pi$
$863$	−2.43096e9			−0.128748			−0.0643739	+	0.997926i	$0.520505\pi$
$864$		0			0
$865$	2.87411e10			1.50989
$866$	−7.28930e9	−	1.26254e10i	−0.381393	−	0.660593i
$867$		0			0
$868$	8.20488e8	−	1.42113e9i	0.0425847	−	0.0737588i
$869$	5.69602e9	−	9.86579e9i	0.294444	−	0.509991i
$870$		0			0
$871$	−4.03180e9	−	6.98328e9i	−0.206745	−	0.358093i
$872$	8.99727e9			0.459518
$873$		0			0
$874$	−5.00280e9			−0.253468
$875$	1.78819e9	+	3.09724e9i	0.0902372	+	0.156295i
$876$		0			0
$877$	4.99117e9	−	8.64495e9i	0.249864	−	0.432777i	−0.713624	−	0.700529i	$-0.752947\pi$
$877$	4.99117e9	−	8.64495e9i	0.249864	−	0.432777i	0.963488	+	0.267752i	$0.0862808\pi$
$878$	−5.32524e9	+	9.22359e9i	−0.265527	+	0.459906i
$879$		0			0
$880$	−1.17041e9	−	2.02721e9i	−0.0578960	−	0.100279i
$881$	−2.89122e10			−1.42451			−0.712256	−	0.701920i	$-0.752327\pi$
$881$	−2.89122e10			−1.42451			−0.712256	+	0.701920i	$0.752327\pi$
$882$		0			0
$883$	−8.55345e9			−0.418099			−0.209049	−	0.977905i	$-0.567037\pi$
$883$	−8.55345e9			−0.418099			−0.209049	+	0.977905i	$0.567037\pi$
$884$	1.70173e9	+	2.94749e9i	0.0828530	+	0.143506i
$885$		0			0
$886$	1.48544e8	−	2.57286e8i	0.00717525	−	0.0124279i
$887$	−8.91592e9	+	1.54428e10i	−0.428977	+	0.743009i	−0.996783	−	0.0801531i	$-0.974459\pi$
$887$	−8.91592e9	+	1.54428e10i	−0.428977	+	0.743009i	0.567806	+	0.823162i	$0.307792\pi$
$888$		0			0
$889$	−2.67067e9	−	4.62573e9i	−0.127486	−	0.220813i
$890$	2.63036e8			0.0125069
$891$		0			0
$892$	−2.12362e9			−0.100184
$893$	−2.59707e9	−	4.49826e9i	−0.122041	−	0.211380i
$894$		0			0
$895$	8.53854e9	−	1.47892e10i	0.398110	−	0.689547i
$896$	1.72306e8	−	2.98442e8i	0.00800242	−	0.0138606i
$897$		0			0
$898$	7.83088e9	+	1.35635e10i	0.360864	+	0.625034i
$899$	−1.17850e9			−0.0540968
$900$		0			0
$901$	4.33391e10			1.97398
$902$	4.38225e9	+	7.59028e9i	0.198826	+	0.344377i
$903$		0			0
$904$	1.78084e9	−	3.08450e9i	0.0801742	−	0.138866i
$905$	−1.34912e10	+	2.33674e10i	−0.605035	+	1.04795i
$906$		0			0
$907$	3.21315e8	+	5.56534e8i	0.0142990	+	0.0247666i	0.873086	−	0.487566i	$-0.162115\pi$
$907$	3.21315e8	+	5.56534e8i	0.0142990	+	0.0247666i	−0.858787	+	0.512332i	$0.828782\pi$
$908$	8.86988e9			0.393203
$909$		0			0
$910$	−6.18706e8			−0.0272170
$911$	4.61488e9	+	7.99321e9i	0.202230	+	0.350273i	0.949247	−	0.314533i	$-0.101848\pi$
$911$	4.61488e9	+	7.99321e9i	0.202230	+	0.350273i	−0.747016	+	0.664806i	$0.768514\pi$
$912$		0			0
$913$	1.00155e10	−	1.73474e10i	0.435537	−	0.754372i
$914$	−3.99671e9	+	6.92251e9i	−0.173138	+	0.299883i
$915$		0			0
$916$	−2.14518e9	−	3.71556e9i	−0.0922209	−	0.159731i
$917$	−2.68076e9			−0.114806
$918$		0			0
$919$	1.03928e10			0.441701			0.220850	−	0.975308i	$-0.429117\pi$
$919$	1.03928e10			0.441701			0.220850	+	0.975308i	$0.429117\pi$
$920$	3.55117e9	+	6.15081e9i	0.150354	+	0.260420i
$921$		0			0
$922$	6.59226e9	−	1.14181e10i	0.276998	−	0.479774i
$923$	2.06195e9	−	3.57140e9i	0.0863120	−	0.149497i
$924$		0			0
$925$	−1.38954e8	−	2.40675e8i	−0.00577265	−	0.00999852i
$926$	2.77650e10			1.14910
$927$		0			0
$928$	−2.47490e8			−0.0101658
$929$	−8.38051e9	−	1.45155e10i	−0.342938	−	0.593986i	0.642039	−	0.766672i	$-0.278089\pi$
$929$	−8.38051e9	−	1.45155e10i	−0.342938	−	0.593986i	−0.984977	+	0.172686i	$0.944755\pi$
$930$		0			0
$931$	−5.03485e9	+	8.72062e9i	−0.204486	+	0.354180i
$932$	−4.18877e9	+	7.25517e9i	−0.169485	+	0.293557i
$933$		0			0
$934$	−9.99702e9	−	1.73153e10i	−0.401473	−	0.695372i
$935$	1.81080e10			0.724487
$936$		0			0
$937$	2.27134e9			0.0901973			0.0450987	−	0.998983i	$-0.485640\pi$
$937$	2.27134e9			0.0901973			0.0450987	+	0.998983i	$0.485640\pi$
$938$	−3.15798e9	−	5.46979e9i	−0.124940	−	0.216402i
$939$		0			0
$940$	−3.68699e9	+	6.38606e9i	−0.144785	+	0.250776i
$941$	9.34719e8	−	1.61898e9i	0.0365694	−	0.0633400i	−0.847161	−	0.531336i	$-0.821690\pi$
$941$	9.34719e8	−	1.61898e9i	0.0365694	−	0.0633400i	0.883731	+	0.467996i	$0.155024\pi$
$942$		0			0
$943$	−1.32963e10	−	2.30299e10i	−0.516345	−	0.894335i
$944$	3.27225e9			0.126603
$945$		0			0
$946$	−3.26881e9			−0.125537
$947$	2.49897e9	+	4.32834e9i	0.0956171	+	0.165614i	0.909866	−	0.414902i	$-0.136184\pi$
$947$	2.49897e9	+	4.32834e9i	0.0956171	+	0.165614i	−0.814249	+	0.580516i	$0.802851\pi$
$948$		0			0
$949$	1.34556e9	−	2.33057e9i	0.0511058	−	0.0885179i
$950$	2.59549e7	−	4.49552e7i	0.000982171	−	0.00170117i
$951$		0			0
$952$	1.33291e9	+	2.30868e9i	0.0500695	+	0.0867229i
$953$	4.30036e10			1.60946			0.804730	−	0.593641i	$-0.202310\pi$
$953$	4.30036e10			1.60946			0.804730	+	0.593641i	$0.202310\pi$
$954$		0			0
$955$	2.07292e10			0.770143
$956$	−6.16772e8	−	1.06828e9i	−0.0228309	−	0.0395442i
$957$		0			0
$958$	1.75844e10	−	3.04571e10i	0.646174	−	1.11921i
$959$	−2.97092e9	+	5.14578e9i	−0.108774	+	0.188402i
$960$		0			0
$961$	1.58280e9	+	2.74149e9i	0.0575300	+	0.0996449i
$962$	−7.26973e9			−0.263273
$963$		0			0
$964$	4.38394e9			0.157614
$965$	1.14230e10	+	1.97853e10i	0.409201	+	0.708756i
$966$		0			0
$967$	2.30180e10	−	3.98684e10i	0.818607	−	1.41787i	−0.0881016	−	0.996111i	$-0.528080\pi$
$967$	2.30180e10	−	3.98684e10i	0.818607	−	1.41787i	0.906709	−	0.421758i	$-0.138587\pi$
$968$	3.92550e9	−	6.79916e9i	0.139101	−	0.240930i
$969$		0			0
$970$	−8.74080e9	−	1.51395e10i	−0.307504	−	0.532612i
$971$	3.33640e10			1.16953			0.584765	−	0.811203i	$-0.301187\pi$
$971$	3.33640e10			1.16953			0.584765	+	0.811203i	$0.301187\pi$
$972$		0			0
$973$	−8.79431e9			−0.306060
$974$	1.13196e10	+	1.96061e10i	0.392531	+	0.679884i
$975$		0			0
$976$	−1.16621e9	+	2.01994e9i	−0.0401516	+	0.0695446i
$977$	−1.39320e10	+	2.41309e10i	−0.477950	+	0.827833i	−0.999680	−	0.0252770i	$-0.991953\pi$
$977$	−1.39320e10	+	2.41309e10i	−0.477950	+	0.827833i	0.521731	+	0.853110i	$0.325287\pi$
$978$		0			0
$979$	−1.19473e8	−	2.06933e8i	−0.00406939	−	0.00704839i
$980$	1.42957e10			0.485192
$981$		0			0
$982$	1.96673e10			0.662759
$983$	−1.54698e10	−	2.67945e10i	−0.519455	−	0.899722i	−0.999744	−	0.0226119i	$-0.992802\pi$
$983$	−1.54698e10	−	2.67945e10i	−0.519455	−	0.899722i	0.480290	−	0.877110i	$-0.340532\pi$
$984$		0			0
$985$	−3.42165e9	+	5.92647e9i	−0.114080	+	0.197592i
$986$	9.57261e8	−	1.65803e9i	0.0318025	−	0.0550836i
$987$		0			0
$988$	−6.78949e8	−	1.17597e9i	−0.0223969	−	0.0387925i
$989$	9.91799e9			0.326014
$990$		0			0
$991$	−1.80515e10			−0.589192			−0.294596	−	0.955622i	$-0.595185\pi$
$991$	−1.80515e10			−0.589192			−0.294596	+	0.955622i	$0.595185\pi$
$992$	−2.55648e9	−	4.42796e9i	−0.0831480	−	0.144017i
$993$		0			0
$994$	1.61506e9	−	2.79737e9i	0.0521598	−	0.0903435i
$995$	−2.13310e10	+	3.69465e10i	−0.686485	+	1.18903i
$996$		0			0
$997$	−1.31421e9	−	2.27627e9i	−0.0419982	−	0.0727429i	0.844262	−	0.535930i	$-0.180039\pi$
$997$	−1.31421e9	−	2.27627e9i	−0.0419982	−	0.0727429i	−0.886260	+	0.463187i	$0.846706\pi$
$998$	−2.40483e10			−0.765823
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.c.q.109.4		8
3.2	odd	2		162.8.c.r.109.1		8
9.2	odd	6		162.8.c.r.55.1		8
9.4	even	3		162.8.a.j.1.1	yes	4
9.5	odd	6		162.8.a.g.1.4	✓	4
9.7	even	3	inner	162.8.c.q.55.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.8.a.g.1.4	✓	4	9.5	odd	6
162.8.a.j.1.1	yes	4	9.4	even	3
162.8.c.q.55.4		8	9.7	even	3	inner
162.8.c.q.109.4		8	1.1	even	1	trivial
162.8.c.r.55.1		8	9.2	odd	6
162.8.c.r.109.1		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(140.213 - 242.855i) q^{5} +(82.1617 + 142.308i) q^{7} +512.000 q^{8} -2243.40 q^{10} +(1018.97 + 1764.91i) q^{11} +(839.167 - 1453.48i) q^{13} +(657.293 - 1138.47i) q^{14} +(-2048.00 - 3547.24i) q^{16} +31685.7 q^{17} -12641.8 q^{19} +(8973.61 + 15542.7i) q^{20} +(8151.75 - 14119.3i) q^{22} +(-24733.5 + 42839.6i) q^{23} +(-256.638 - 444.510i) q^{25} -13426.7 q^{26} -10516.7 q^{28} +(3776.40 + 6540.91i) q^{29} +(-78017.6 + 135131. i) q^{31} +(-16384.0 + 28377.9i) q^{32} +(-126743. - 219525. i) q^{34} +46080.4 q^{35} +541440. q^{37} +(50567.2 + 87584.9i) q^{38} +(71788.8 - 124342. i) q^{40} +(-268792. + 465561. i) q^{41} +(-100249. - 173636. i) q^{43} -130428. q^{44} +395735. q^{46} +(205436. + 355825. i) q^{47} +(398270. - 689825. i) q^{49} +(-2053.10 + 3556.08i) q^{50} +(53706.7 + 93022.7i) q^{52} +1.36778e6 q^{53} +571489. q^{55} +(42066.8 + 72861.8i) q^{56} +(30211.2 - 52327.3i) q^{58} +(-399444. + 691858. i) q^{59} +(-284720. - 493149. i) q^{61} +1.24828e6 q^{62} +262144. q^{64} +(-235323. - 407592. i) q^{65} +(2.40226e6 - 4.16084e6i) q^{67} +(-1.01394e6 + 1.75620e6i) q^{68} +(-184322. - 319254. i) q^{70} +2.45714e6 q^{71} +1.60344e6 q^{73} +(-2.16576e6 - 3.75121e6i) q^{74} +(404537. - 700679. i) q^{76} +(-167440. + 290015. i) q^{77} +(-2.79499e6 - 4.84107e6i) q^{79} -1.14862e6 q^{80} +4.30067e6 q^{82} +(-4.91453e6 - 8.51221e6i) q^{83} +(4.44273e6 - 7.69504e6i) q^{85} +(-801990. + 1.38909e6i) q^{86} +(521712. + 903632. i) q^{88} -117249. q^{89} +275789. q^{91} +(-1.58294e6 - 2.74173e6i) q^{92} +(1.64348e6 - 2.84660e6i) q^{94} +(-1.77254e6 + 3.07013e6i) q^{95} +(3.89623e6 + 6.74846e6i) q^{97} -6.37233e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 256 q^{4} - 528 q^{5} - 560 q^{7} + 4096 q^{8} + 8448 q^{10} - 2160 q^{11} - 13460 q^{13} - 4480 q^{14} - 16384 q^{16} + 45120 q^{17} + 73408 q^{19} - 33792 q^{20} - 17280 q^{22} - 62640 q^{23}+ \cdots + 23190336 q^{98}+O(q^{100}) \) 8 * q - 32 * q^2 - 256 * q^4 - 528 * q^5 - 560 * q^7 + 4096 * q^8 + 8448 * q^10 - 2160 * q^11 - 13460 * q^13 - 4480 * q^14 - 16384 * q^16 + 45120 * q^17 + 73408 * q^19 - 33792 * q^20 - 17280 * q^22 - 62640 * q^23 - 94696 * q^25 + 215360 * q^26 + 71680 * q^28 - 68400 * q^29 - 227504 * q^31 - 131072 * q^32 - 180480 * q^34 - 126048 * q^35 + 1047160 * q^37 - 293632 * q^38 - 270336 * q^40 - 67200 * q^41 - 562640 * q^43 + 276480 * q^44 + 1002240 * q^46 - 515328 * q^47 - 1449396 * q^49 - 757568 * q^50 - 861440 * q^52 + 4996032 * q^53 + 2645568 * q^55 - 286720 * q^56 - 547200 * q^58 - 4155840 * q^59 - 2130764 * q^61 + 3640064 * q^62 + 2097152 * q^64 - 8966496 * q^65 + 1205440 * q^67 - 1443840 * q^68 + 504192 * q^70 + 24972960 * q^71 - 9641720 * q^73 - 4188640 * q^74 - 2349056 * q^76 - 18686592 * q^77 + 11471680 * q^79 + 4325376 * q^80 + 1075200 * q^82 - 16811232 * q^83 + 21006396 * q^85 - 4501120 * q^86 - 1105920 * q^88 + 33715200 * q^89 - 46401088 * q^91 - 4008960 * q^92 - 4122624 * q^94 - 14090160 * q^95 + 27078040 * q^97 + 23190336 * q^98

Introduction

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