LMFDB - Embedded newform 81.9.d.g.53.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,9,Mod(26,81)] 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("81.26"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	81.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,2048] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$32.9976674150$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.1
Character	$\chi$	$=$	81.53
Dual form			81.9.d.g.26.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-25.8080 + 14.9003i) q^{2} +(316.037 - 547.392i) q^{4} +(-153.960 - 88.8891i) q^{5} +(-1695.41 - 2936.53i) q^{7} +11207.2i q^{8} +5297.89 q^{10} +(-19670.7 + 11356.9i) q^{11} +(-24388.6 + 42242.2i) q^{13} +(87510.3 + 50524.1i) q^{14} +(-86085.2 - 149104. i) q^{16} +36661.7i q^{17} -81528.6 q^{19} +(-97314.4 + 56184.5i) q^{20} +(338441. - 586197. i) q^{22} +(38694.0 + 22340.0i) q^{23} +(-179510. - 310920. i) q^{25} -1.45359e6i q^{26} -2.14324e6 q^{28} +(-265895. + 153515. i) q^{29} +(188944. - 327260. i) q^{31} +(1.95871e6 + 1.13086e6i) q^{32} +(-546270. - 946167. i) q^{34} +602813. i q^{35} +1.82114e6 q^{37} +(2.10409e6 - 1.21480e6i) q^{38} +(996199. - 1.72547e6i) q^{40} +(2.22165e6 + 1.28267e6i) q^{41} +(429764. + 744374. i) q^{43} +1.43568e7i q^{44} -1.33149e6 q^{46} +(-2.71933e6 + 1.57000e6i) q^{47} +(-2.86641e6 + 4.96477e6i) q^{49} +(9.26560e6 + 5.34950e6i) q^{50} +(1.54154e7 + 2.67002e7i) q^{52} -1.04587e7i q^{53} +4.03801e6 q^{55} +(3.29103e7 - 1.90008e7i) q^{56} +(4.57482e6 - 7.92382e6i) q^{58} +(-1.74219e7 - 1.00585e7i) q^{59} +(5.33263e6 + 9.23638e6i) q^{61} +1.12613e7i q^{62} -2.33251e7 q^{64} +(7.50975e6 - 4.33575e6i) q^{65} +(1.68746e7 - 2.92277e7i) q^{67} +(2.00683e7 + 1.15865e7i) q^{68} +(-8.98208e6 - 1.55574e7i) q^{70} -387108. i q^{71} +1.74105e7 q^{73} +(-4.70002e7 + 2.71356e7i) q^{74} +(-2.57660e7 + 4.46281e7i) q^{76} +(6.66996e7 + 3.85090e7i) q^{77} +(-6.52931e6 - 1.13091e7i) q^{79} +3.06081e7i q^{80} -7.64487e7 q^{82} +(-5.41503e7 + 3.12637e7i) q^{83} +(3.25883e6 - 5.64445e6i) q^{85} +(-2.21828e7 - 1.28072e7i) q^{86} +(-1.27279e8 - 2.20453e8i) q^{88} -6.33977e7i q^{89} +1.65394e8 q^{91} +(2.44575e7 - 1.41205e7i) q^{92} +(4.67870e7 - 8.10375e7i) q^{94} +(1.25522e7 + 7.24700e6i) q^{95} +(3.26981e7 + 5.66348e7i) q^{97} -1.70841e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40}+ \cdots + 133878688 q^{97}+O(q^{100}) $ 32 * q + 2048 * q^4 - 3692 * q^7 + 21504 * q^10 - 63860 * q^13 - 95116 * q^16 + 370216 * q^19 + 691980 * q^22 + 541712 * q^25 - 1994264 * q^28 - 571136 * q^31 - 1027656 * q^34 + 8708536 * q^37 + 2973768 * q^40 - 5453084 * q^43 - 8468088 * q^46 - 14602560 * q^49 + 25384960 * q^52 - 28905144 * q^55 - 1213068 * q^58 - 31037996 * q^61 + 48506000 * q^64 + 62356828 * q^67 - 11075124 * q^70 - 42547328 * q^73 - 69593876 * q^76 + 151045036 * q^79 - 80402280 * q^82 - 160995564 * q^85 - 366976068 * q^88 + 747361592 * q^91 + 544741584 * q^94 + 133878688 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−25.8080	+	14.9003i	−1.61300	+	0.931268i	−0.624334	+	0.781158i	$0.714629\pi$
$2$	−25.8080	+	14.9003i	−1.61300	+	0.931268i	−0.988669	+	0.150110i	$0.952037\pi$
$3$		0			0
$3$		0			0
$4$	316.037	−	547.392i	1.23452	−	2.13825i
$5$	−153.960	−	88.8891i	−0.246337	−	0.142223i	0.371749	−	0.928333i	$-0.378758\pi$
$5$	−153.960	−	88.8891i	−0.246337	−	0.142223i	−0.618086	+	0.786111i	$0.712092\pi$
$6$		0			0
$7$	−1695.41	−	2936.53i	−0.706125	−	1.22305i	−0.966284	−	0.257479i	$-0.917108\pi$
$7$	−1695.41	−	2936.53i	−0.706125	−	1.22305i	0.260158	−	0.965566i	$-0.416225\pi$
$8$			11207.2i			2.73614i
$9$		0			0
$10$	5297.89			0.529789
$11$	−19670.7	+	11356.9i	−1.34353	+	0.775689i	−0.987324	−	0.158717i	$-0.949264\pi$
$11$	−19670.7	+	11356.9i	−1.34353	+	0.775689i	−0.356209	+	0.934406i	$0.615931\pi$
$12$		0			0
$13$	−24388.6	+	42242.2i	−0.853911	+	1.47902i	0.0237404	+	0.999718i	$0.492442\pi$
$13$	−24388.6	+	42242.2i	−0.853911	+	1.47902i	−0.877652	+	0.479299i	$0.840891\pi$
$14$	87510.3	+	50524.1i	2.27796	+	1.31518i
$15$		0			0
$16$	−86085.2	−	149104.i	−1.31356	−	2.27514i
$17$			36661.7i			0.438952i	0.975618	+	0.219476i	$0.0704348\pi$
$17$			36661.7i			0.438952i	−0.975618	+	0.219476i	$0.929565\pi$
$18$		0			0
$19$	−81528.6			−0.625598			−0.312799	−	0.949819i	$-0.601267\pi$
$19$	−81528.6			−0.625598			−0.312799	+	0.949819i	$0.601267\pi$
$20$	−97314.4	+	56184.5i	−0.608215	+	0.351153i
$21$		0			0
$22$	338441.	−	586197.i	1.44475	−	2.50238i
$23$	38694.0	+	22340.0i	0.138272	+	0.0798311i	0.567540	−	0.823346i	$-0.307895\pi$
$23$	38694.0	+	22340.0i	0.138272	+	0.0798311i	−0.429268	+	0.903177i	$0.641229\pi$
$24$		0			0
$25$	−179510.	−	310920.i	−0.459545	−	0.795956i
$26$		−	1.45359e6i		−	3.18088i
$27$		0			0
$28$	−2.14324e6			−3.48690
$29$	−265895.	+	153515.i	−0.375940	+	0.217049i	−0.676050	−	0.736855i	$-0.736310\pi$
$29$	−265895.	+	153515.i	−0.375940	+	0.217049i	0.300110	+	0.953904i	$0.402976\pi$
$30$		0			0
$31$	188944.	−	327260.i	0.204591	−	0.354362i	−0.745411	−	0.666605i	$-0.767747\pi$
$31$	188944.	−	327260.i	0.204591	−	0.354362i	0.950002	+	0.312243i	$0.101080\pi$
$32$	1.95871e6	+	1.13086e6i	1.86797	+	1.07848i
$33$		0			0
$34$	−546270.	−	946167.i	−0.408782	−	0.708031i
$35$			602813.i			0.401708i
$36$		0			0
$37$	1.82114e6			0.971711			0.485856	−	0.874039i	$-0.338508\pi$
$37$	1.82114e6			0.971711			0.485856	+	0.874039i	$0.338508\pi$
$38$	2.10409e6	−	1.21480e6i	1.00909	−	0.582599i
$39$		0			0
$40$	996199.	−	1.72547e6i	0.389140	−	0.674011i
$41$	2.22165e6	+	1.28267e6i	0.786214	+	0.453921i	0.838628	−	0.544704i	$-0.183358\pi$
$41$	2.22165e6	+	1.28267e6i	0.786214	+	0.453921i	−0.0524137	+	0.998625i	$0.516691\pi$
$42$		0			0
$43$	429764.	+	744374.i	0.125706	+	0.217729i	0.922009	−	0.387169i	$-0.126547\pi$
$43$	429764.	+	744374.i	0.125706	+	0.217729i	−0.796303	+	0.604898i	$0.793214\pi$
$44$			1.43568e7i			3.83041i
$45$		0			0
$46$	−1.33149e6			−0.297377
$47$	−2.71933e6	+	1.57000e6i	−0.557276	+	0.321743i	−0.752051	−	0.659105i	$-0.770935\pi$
$47$	−2.71933e6	+	1.57000e6i	−0.557276	+	0.321743i	0.194776	+	0.980848i	$0.437602\pi$
$48$		0			0
$49$	−2.86641e6	+	4.96477e6i	−0.497226	+	0.861221i
$50$	9.26560e6	+	5.34950e6i	1.48250	+	0.855920i
$51$		0			0
$52$	1.54154e7	+	2.67002e7i	2.10834	+	3.65175i
$53$		−	1.04587e7i		−	1.32548i	−0.748848	−	0.662742i	$-0.769393\pi$
$53$		−	1.04587e7i		−	1.32548i	0.748848	−	0.662742i	$-0.230607\pi$
$54$		0			0
$55$	4.03801e6			0.441282
$56$	3.29103e7	−	1.90008e7i	3.34642	−	1.93205i
$57$		0			0
$58$	4.57482e6	−	7.92382e6i	0.404261	−	0.700201i
$59$	−1.74219e7	−	1.00585e7i	−1.43776	−	0.830093i	−0.440070	−	0.897964i	$-0.645046\pi$
$59$	−1.74219e7	−	1.00585e7i	−1.43776	−	0.830093i	−0.997694	+	0.0678705i	$0.978380\pi$
$60$		0			0
$61$	5.33263e6	+	9.23638e6i	0.385143	+	0.667087i	0.991789	−	0.127885i	$-0.0408188\pi$
$61$	5.33263e6	+	9.23638e6i	0.385143	+	0.667087i	−0.606646	+	0.794972i	$0.707486\pi$
$62$			1.12613e7i			0.762115i
$63$		0			0
$64$	−2.33251e7			−1.39029
$65$	7.50975e6	−	4.33575e6i	0.420699	−	0.242891i
$66$		0			0
$67$	1.68746e7	−	2.92277e7i	0.837404	−	1.45043i	−0.0546543	−	0.998505i	$-0.517406\pi$
$67$	1.68746e7	−	2.92277e7i	0.837404	−	1.45043i	0.892058	−	0.451921i	$-0.149261\pi$
$68$	2.00683e7	+	1.15865e7i	0.938589	+	0.541895i
$69$		0			0
$70$	−8.98208e6	−	1.55574e7i	−0.374098	−	0.647956i
$71$		−	387108.i		−	0.0152335i	−0.999971	−	0.00761673i	$-0.997575\pi$
$71$		−	387108.i		−	0.0152335i	0.999971	−	0.00761673i	$-0.00242450\pi$
$72$		0			0
$73$	1.74105e7			0.613083			0.306542	−	0.951857i	$-0.400828\pi$
$73$	1.74105e7			0.613083			0.306542	+	0.951857i	$0.400828\pi$
$74$	−4.70002e7	+	2.71356e7i	−1.56737	+	0.904923i
$75$		0			0
$76$	−2.57660e7	+	4.46281e7i	−0.772313	+	1.33768i
$77$	6.66996e7	+	3.85090e7i	1.89741	+	1.09547i
$78$		0			0
$79$	−6.52931e6	−	1.13091e7i	−0.167633	−	0.290349i	0.769954	−	0.638099i	$-0.220279\pi$
$79$	−6.52931e6	−	1.13091e7i	−0.167633	−	0.290349i	−0.937587	+	0.347750i	$0.886946\pi$
$80$			3.06081e7i			0.747269i
$81$		0			0
$82$	−7.64487e7			−1.69089
$83$	−5.41503e7	+	3.12637e7i	−1.14101	+	0.658761i	−0.946680	−	0.322176i	$-0.895586\pi$
$83$	−5.41503e7	+	3.12637e7i	−1.14101	+	0.658761i	−0.194327	+	0.980937i	$0.562252\pi$
$84$		0			0
$85$	3.25883e6	−	5.64445e6i	0.0624289	−	0.108130i
$86$	−2.21828e7	−	1.28072e7i	−0.405529	−	0.234132i
$87$		0			0
$88$	−1.27279e8	−	2.20453e8i	−2.12239	−	3.67609i
$89$		−	6.33977e7i		−	1.01045i	−0.862988	−	0.505224i	$-0.831410\pi$
$89$		−	6.33977e7i		−	1.01045i	0.862988	−	0.505224i	$-0.168590\pi$
$90$		0			0
$91$	1.65394e8			2.41187
$92$	2.44575e7	−	1.41205e7i	0.341398	−	0.197106i
$93$		0			0
$94$	4.67870e7	−	8.10375e7i	0.599258	−	1.03795i
$95$	1.25522e7	+	7.24700e6i	0.154108	+	0.0889742i
$96$		0			0
$97$	3.26981e7	+	5.66348e7i	0.369348	+	0.639730i	0.989464	−	0.144781i	$-0.0462477\pi$
$97$	3.26981e7	+	5.66348e7i	0.369348	+	0.639730i	−0.620116	+	0.784510i	$0.712914\pi$
$98$		−	1.70841e8i		−	1.85220i
$99$		0			0
$100$	−2.26927e8			−2.26927
$101$	−1.16451e7	+	6.72328e6i	−0.111907	+	0.0646094i	−0.554909	−	0.831911i	$-0.687247\pi$
$101$	−1.16451e7	+	6.72328e6i	−0.111907	+	0.0646094i	0.443002	+	0.896521i	$0.353913\pi$
$102$		0			0
$103$	1.43919e7	−	2.49275e7i	0.127870	−	0.221478i	−0.794981	−	0.606634i	$-0.792519\pi$
$103$	1.43919e7	−	2.49275e7i	0.127870	−	0.221478i	0.922851	+	0.385157i	$0.125853\pi$
$104$	−4.73417e8	−	2.73328e8i	−4.04679	−	2.33642i
$105$		0			0
$106$	1.55838e8	+	2.69919e8i	1.23438	+	2.13801i
$107$		−	7.10303e7i		−	0.541886i	−0.962595	−	0.270943i	$-0.912664\pi$
$107$		−	7.10303e7i		−	0.541886i	0.962595	−	0.270943i	$-0.0873356\pi$
$108$		0			0
$109$	5.37568e7			0.380827			0.190413	−	0.981704i	$-0.439017\pi$
$109$	5.37568e7			0.380827			0.190413	+	0.981704i	$0.439017\pi$
$110$	−1.04213e8	+	6.01674e7i	−0.711789	+	0.410952i
$111$		0			0
$112$	−2.91899e8	+	5.05584e8i	−1.85507	+	3.21307i
$113$	1.78818e8	+	1.03241e8i	1.09673	+	0.633195i	0.935359	−	0.353699i	$-0.115076\pi$
$113$	1.78818e8	+	1.03241e8i	1.09673	+	0.633195i	0.161367	+	0.986894i	$0.448410\pi$
$114$		0			0
$115$	−3.97157e6	−	6.87896e6i	−0.0227076	−	0.0393307i
$116$			1.94065e8i			1.07180i
$117$		0			0
$118$	5.99500e8			3.09216
$119$	1.07658e8	−	6.21565e7i	0.536858	−	0.309955i
$120$		0			0
$121$	1.50777e8	−	2.61154e8i	0.703387	−	1.21830i
$122$	−2.75249e8	−	1.58915e8i	−1.24247	−	0.717342i
$123$		0			0
$124$	−1.19426e8	−	2.06853e8i	−0.505143	−	0.874933i
$125$			1.33271e8i			0.545876i
$126$		0			0
$127$	3.63640e8			1.39784			0.698918	−	0.715201i	$-0.253665\pi$
$127$	3.63640e8			1.39784			0.698918	+	0.715201i	$0.253665\pi$
$128$	1.00546e8	−	5.80502e7i	0.374563	−	0.216254i
$129$		0			0
$130$	−1.29208e8	+	2.23795e8i	−0.452393	+	0.783567i
$131$	−1.19720e7	−	6.91203e6i	−0.0406519	−	0.0234704i	0.479536	−	0.877522i	$-0.340805\pi$
$131$	−1.19720e7	−	6.91203e6i	−0.0406519	−	0.0234704i	−0.520188	+	0.854052i	$0.674138\pi$
$132$		0			0
$133$	1.38224e8	+	2.39411e8i	0.441751	+	0.765135i
$134$			1.00575e9i			3.11939i
$135$		0			0
$136$	−4.10876e8			−1.20103
$137$	−1.96816e7	+	1.13632e7i	−0.0558699	+	0.0322565i	−0.527675	−	0.849447i	$-0.676936\pi$
$137$	−1.96816e7	+	1.13632e7i	−0.0558699	+	0.0322565i	0.471805	+	0.881703i	$0.343603\pi$
$138$		0			0
$139$	−2.54007e8	+	4.39952e8i	−0.680434	+	1.17855i	0.294415	+	0.955678i	$0.404875\pi$
$139$	−2.54007e8	+	4.39952e8i	−0.680434	+	1.17855i	−0.974849	+	0.222868i	$0.928458\pi$
$140$	3.29975e8	+	1.90511e8i	0.858952	+	0.495916i
$141$		0			0
$142$	5.76801e6	+	9.99049e6i	0.0141864	+	0.0245716i
$143$		−	1.10791e9i		−	2.64948i
$144$		0			0
$145$	5.45831e7			0.123477
$146$	−4.49331e8	+	2.59421e8i	−0.988905	+	0.570945i
$147$		0			0
$148$	5.75548e8	−	9.96879e8i	1.19960	−	2.07776i
$149$	3.18161e8	+	1.83691e8i	0.645509	+	0.372685i	0.786734	−	0.617293i	$-0.211771\pi$
$149$	3.18161e8	+	1.83691e8i	0.645509	+	0.372685i	−0.141224	+	0.989978i	$0.545104\pi$
$150$		0			0
$151$	−1.07691e7	−	1.86526e7i	−0.0207143	−	0.0358782i	0.855482	−	0.517832i	$-0.173261\pi$
$151$	−1.07691e7	−	1.86526e7i	−0.0207143	−	0.0358782i	−0.876197	+	0.481954i	$0.839927\pi$
$152$		−	9.13708e8i		−	1.71172i
$153$		0			0
$154$	−2.29518e9			−4.08069
$155$	−5.81798e7	+	3.35901e7i	−0.100796	+	0.0581949i
$156$		0			0
$157$	2.58785e8	−	4.48229e8i	0.425933	−	0.737737i	−0.570574	−	0.821246i	$-0.693279\pi$
$157$	2.58785e8	−	4.48229e8i	0.425933	−	0.737737i	0.996507	+	0.0835088i	$0.0266127\pi$
$158$	3.37018e8	+	1.94577e8i	0.540784	+	0.312222i
$159$		0			0
$160$	−2.01043e8	−	3.48217e8i	−0.306767	−	0.531336i
$161$		−	1.51502e8i		−	0.225483i
$162$		0			0
$163$	1.46155e8			0.207045			0.103522	−	0.994627i	$-0.466989\pi$
$163$	1.46155e8			0.207045			0.103522	+	0.994627i	$0.466989\pi$
$164$	1.40425e9	−	8.10744e8i	1.94119	−	1.12075i
$165$		0			0
$166$	9.31675e8	−	1.61371e9i	1.22696	−	2.12517i
$167$	−2.40904e8	−	1.39086e8i	−0.309727	−	0.178821i	0.337077	−	0.941477i	$-0.390562\pi$
$167$	−2.40904e8	−	1.39086e8i	−0.309727	−	0.178821i	−0.646804	+	0.762656i	$0.723895\pi$
$168$		0			0
$169$	−7.81738e8	−	1.35401e9i	−0.958328	−	1.65987i
$170$			1.94230e8i			0.232552i
$171$		0			0
$172$	5.43285e8			0.620747
$173$	1.20763e9	−	6.97225e8i	1.34818	−	0.778374i	0.360192	−	0.932878i	$-0.382711\pi$
$173$	1.20763e9	−	6.97225e8i	1.34818	−	0.778374i	0.987992	+	0.154504i	$0.0493779\pi$
$174$		0			0
$175$	−6.08685e8	+	1.05427e9i	−0.648994	+	1.12409i
$176$	3.38670e9	+	1.95531e9i	3.52961	+	2.03782i
$177$		0			0
$178$	9.44644e8	+	1.63617e9i	0.940997	+	1.62985i
$179$			1.59335e9i			1.55203i	0.630717	+	0.776013i	$0.282761\pi$
$179$			1.59335e9i			1.55203i	−0.630717	+	0.776013i	$0.717239\pi$
$180$		0			0
$181$	4.20532e8			0.391818			0.195909	−	0.980622i	$-0.437234\pi$
$181$	4.20532e8			0.391818			0.195909	+	0.980622i	$0.437234\pi$
$182$	−4.26850e9	+	2.46442e9i	−3.89036	+	2.24610i
$183$		0			0
$184$	−2.50369e8	+	4.33652e8i	−0.218429	+	0.378330i
$185$	−2.80384e8	−	1.61880e8i	−0.239368	−	0.138199i
$186$		0			0
$187$	−4.16362e8	−	7.21160e8i	−0.340490	−	0.589747i
$188$			1.98472e9i			1.58879i
$189$		0			0
$190$	−4.31930e8			−0.331435
$191$	1.06211e9	−	6.13211e8i	0.798063	−	0.460762i	−0.0447302	−	0.998999i	$-0.514243\pi$
$191$	1.06211e9	−	6.13211e8i	0.798063	−	0.460762i	0.842794	+	0.538237i	$0.180909\pi$
$192$		0			0
$193$	−9.80416e8	+	1.69813e9i	−0.706612	+	1.22389i	0.259495	+	0.965745i	$0.416444\pi$
$193$	−9.80416e8	+	1.69813e9i	−0.706612	+	1.22389i	−0.966107	+	0.258143i	$0.916889\pi$
$194$	−1.68775e9	−	9.74422e8i	−1.19152	−	0.687924i
$195$		0			0
$196$	1.81178e9	+	3.13810e9i	1.22767	+	2.12639i
$197$		−	4.53495e7i		−	0.0301098i	−0.999887	−	0.0150549i	$-0.995208\pi$
$197$		−	4.53495e7i		−	0.0301098i	0.999887	−	0.0150549i	$-0.00479231\pi$
$198$		0			0
$199$	−1.93844e9			−1.23606			−0.618029	−	0.786155i	$-0.712069\pi$
$199$	−1.93844e9			−1.23606			−0.618029	+	0.786155i	$0.712069\pi$
$200$	3.48455e9	−	2.01181e9i	2.17784	−	1.25738i
$201$		0			0
$202$	2.00358e8	−	3.47030e8i	0.120337	−	0.208430i
$203$	9.01601e8	+	5.20540e8i	0.530921	+	0.306528i
$204$		0			0
$205$	−2.28031e8	−	3.94962e8i	−0.129116	−	0.223635i
$206$			8.57774e8i			0.476325i
$207$		0			0
$208$	8.39797e9			4.48664
$209$	1.60372e9	−	9.25909e8i	0.840512	−	0.485270i
$210$		0			0
$211$	2.82976e8	−	4.90129e8i	0.142764	−	0.247275i	−0.785772	−	0.618516i	$-0.787734\pi$
$211$	2.82976e8	−	4.90129e8i	0.142764	−	0.247275i	0.928537	+	0.371241i	$0.121068\pi$
$212$	−5.72501e9	−	3.30534e9i	−2.83422	−	1.63634i
$213$		0			0
$214$	1.05837e9	+	1.83315e9i	0.504641	+	0.874064i
$215$		−	1.52805e8i		−	0.0715130i
$216$		0			0
$217$	−1.28135e9			−0.577867
$218$	−1.38736e9	+	8.00992e8i	−0.614275	+	0.354652i
$219$		0			0
$220$	1.27616e9	−	2.21037e9i	0.544771	−	0.943571i
$221$	−1.54867e9	−	8.94126e8i	−0.649218	−	0.374826i
$222$		0			0
$223$	9.82951e8	+	1.70252e9i	0.397477	+	0.688451i	0.993414	−	0.114581i	$-0.0365524\pi$
$223$	9.82951e8	+	1.70252e9i	0.397477	+	0.688451i	−0.595937	+	0.803031i	$0.703219\pi$
$224$		−	7.66910e9i		−	3.04616i
$225$		0			0
$226$	−6.15327e9			−2.35870
$227$	2.99796e9	−	1.73087e9i	1.12907	−	0.651870i	0.185371	−	0.982669i	$-0.440651\pi$
$227$	2.99796e9	−	1.73087e9i	1.12907	−	0.651870i	0.943702	+	0.330798i	$0.107318\pi$
$228$		0			0
$229$	−2.54732e9	+	4.41208e9i	−0.926278	+	1.60436i	−0.136784	+	0.990601i	$0.543677\pi$
$229$	−2.54732e9	+	4.41208e9i	−0.926278	+	1.60436i	−0.789493	+	0.613759i	$0.789657\pi$
$230$	2.04997e8	+	1.18355e8i	0.0732548	+	0.0422937i
$231$		0			0
$232$	−1.72047e9	−	2.97994e9i	−0.593875	−	1.02862i
$233$		−	4.76326e9i		−	1.61615i	−0.589083	−	0.808073i	$-0.700511\pi$
$233$		−	4.76326e9i		−	1.61615i	0.589083	−	0.808073i	$-0.299489\pi$
$234$		0			0
$235$	5.58225e8			0.183037
$236$	−1.10119e10	+	6.35774e9i	−3.54989	+	2.04953i
$237$		0			0
$238$	−1.85230e9	+	3.20828e9i	−0.577303	+	0.999917i
$239$	−1.47172e9	−	8.49698e8i	−0.451059	−	0.260419i	0.257218	−	0.966353i	$-0.417194\pi$
$239$	−1.47172e9	−	8.49698e8i	−0.451059	−	0.260419i	−0.708278	+	0.705934i	$0.750527\pi$
$240$		0			0
$241$	−2.74201e9	−	4.74931e9i	−0.812833	−	1.40787i	−0.910874	−	0.412685i	$-0.864591\pi$
$241$	−2.74201e9	−	4.74931e9i	−0.812833	−	1.40787i	0.0980407	−	0.995182i	$-0.468742\pi$
$242$			8.98650e9i			2.62017i
$243$		0			0
$244$	6.74123e9			1.90186
$245$	8.82628e8	−	5.09585e8i	0.244970	−	0.141434i
$246$		0			0
$247$	1.98836e9	−	3.44395e9i	0.534205	−	0.925271i
$248$	3.66768e9	+	2.11753e9i	0.969582	+	0.559788i
$249$		0			0
$250$	−1.98577e9	−	3.43945e9i	−0.508357	−	0.880500i
$251$			2.82808e9i			0.712520i	0.934387	+	0.356260i	$0.115948\pi$
$251$			2.82808e9i			0.712520i	−0.934387	+	0.356260i	$0.884052\pi$
$252$		0			0
$253$	−1.01485e9			−0.247697
$254$	−9.38483e9	+	5.41834e9i	−2.25472	+	1.30176i
$255$		0			0
$256$	1.25569e9	−	2.17492e9i	0.292363	−	0.506388i
$257$	3.34468e9	+	1.93105e9i	0.766693	+	0.442651i	0.831694	−	0.555235i	$-0.187372\pi$
$257$	3.34468e9	+	1.93105e9i	0.766693	+	0.442651i	−0.0650005	+	0.997885i	$0.520705\pi$
$258$		0			0
$259$	−3.08758e9	−	5.34785e9i	−0.686150	−	1.18845i
$260$		−	5.48103e9i		−	1.19941i
$261$		0			0
$262$	4.11965e8			0.0874289
$263$	6.17148e9	−	3.56311e9i	1.28993	−	0.744742i	0.311290	−	0.950315i	$-0.399239\pi$
$263$	6.17148e9	−	3.56311e9i	1.28993	−	0.744742i	0.978642	+	0.205573i	$0.0659057\pi$
$264$		0			0
$265$	−9.29665e8	+	1.61023e9i	−0.188514	+	0.326515i
$266$	−7.13459e9	−	4.11916e9i	−1.42509	−	0.822776i
$267$		0			0
$268$	−1.06660e10	−	1.84741e10i	−2.06758	−	3.58116i
$269$			4.73081e9i			0.903496i	0.892146	+	0.451748i	$0.149199\pi$
$269$			4.73081e9i			0.903496i	−0.892146	+	0.451748i	$0.850801\pi$
$270$		0			0
$271$	−4.16250e9			−0.771751			−0.385875	−	0.922551i	$-0.626100\pi$
$271$	−4.16250e9			−0.771751			−0.385875	+	0.922551i	$0.626100\pi$
$272$	5.46640e9	−	3.15603e9i	0.998679	−	0.576588i
$273$		0			0
$274$	3.38629e8	−	5.86522e8i	0.0600788	−	0.104060i
$275$	7.06216e9	+	4.07734e9i	1.23483	+	0.712929i
$276$		0			0
$277$	4.60026e9	+	7.96788e9i	0.781382	+	1.35339i	0.931137	+	0.364670i	$0.118818\pi$
$277$	4.60026e9	+	7.96788e9i	0.781382	+	1.35339i	−0.149755	+	0.988723i	$0.547848\pi$
$278$		−	1.51391e10i		−	2.53466i
$279$		0			0
$280$	−6.75585e9			−1.09913
$281$	−4.84559e9	+	2.79760e9i	−0.777179	+	0.448705i	−0.835430	−	0.549597i	$-0.814781\pi$
$281$	−4.84559e9	+	2.79760e9i	−0.777179	+	0.448705i	0.0582505	+	0.998302i	$0.481448\pi$
$282$		0			0
$283$	−3.36180e9	+	5.82281e9i	−0.524114	+	0.907793i	0.475491	+	0.879720i	$0.342270\pi$
$283$	−3.36180e9	+	5.82281e9i	−0.524114	+	0.907793i	−0.999606	+	0.0280725i	$0.991063\pi$
$284$	−2.11900e8	−	1.22340e8i	−0.0325729	−	0.0188060i
$285$		0			0
$286$	1.65082e10	+	2.85930e10i	2.46737	+	4.27362i
$287$		−	8.69861e9i		−	1.28210i
$288$		0			0
$289$	5.63168e9			0.807321
$290$	−1.40868e9	+	8.13304e8i	−0.199169	+	0.114990i
$291$		0			0
$292$	5.50235e9	−	9.53036e9i	0.756863	−	1.31092i
$293$	−7.32618e9	−	4.22977e9i	−0.994048	−	0.573914i	−0.0875662	−	0.996159i	$-0.527909\pi$
$293$	−7.32618e9	−	4.22977e9i	−0.994048	−	0.573914i	−0.906482	+	0.422245i	$0.861242\pi$
$294$		0			0
$295$	1.78819e9	+	3.09723e9i	0.236116	+	0.408965i
$296$			2.04099e10i			2.65873i
$297$		0			0
$298$	−1.09482e10			−1.38828
$299$	−1.88738e9	+	1.08968e9i	−0.236143	+	0.136337i
$300$		0			0
$301$	1.45725e9	−	2.52403e9i	0.177529	−	0.307489i
$302$	5.55857e8	+	3.20924e8i	0.0668244	+	0.0385811i
$303$		0			0
$304$	7.01840e9	+	1.21562e10i	0.821758	+	1.42333i
$305$		−	1.89605e9i		−	0.219104i
$306$		0			0
$307$	1.52537e10			1.71721			0.858604	−	0.512639i	$-0.171332\pi$
$307$	1.52537e10			1.71721			0.858604	+	0.512639i	$0.171332\pi$
$308$	4.21590e10	−	2.43405e10i	4.68477	−	2.70475i
$309$		0			0
$310$	1.00100e9	−	1.73379e9i	0.108390	−	0.187737i
$311$	−1.96006e8	−	1.13164e8i	−0.0209521	−	0.0120967i	0.489487	−	0.872010i	$-0.337184\pi$
$311$	−1.96006e8	−	1.13164e8i	−0.0209521	−	0.0120967i	−0.510439	+	0.859914i	$0.670517\pi$
$312$		0			0
$313$	−4.76510e9	−	8.25340e9i	−0.496472	−	0.859915i	0.503520	−	0.863984i	$-0.332038\pi$
$313$	−4.76510e9	−	8.25340e9i	−0.496472	−	0.859915i	−0.999992	+	0.00406912i	$0.998705\pi$
$314$			1.54239e10i			1.58663i
$315$		0			0
$316$	−8.25401e9			−0.827784
$317$	−9.29236e9	+	5.36495e9i	−0.920215	+	0.531286i	−0.883704	−	0.468047i	$-0.844958\pi$
$317$	−9.29236e9	+	5.36495e9i	−0.920215	+	0.531286i	−0.0365111	+	0.999333i	$0.511624\pi$
$318$		0			0
$319$	3.48689e9	−	6.03947e9i	0.336725	−	0.583225i
$320$	3.59115e9	+	2.07335e9i	0.342479	+	0.197730i
$321$		0			0
$322$	2.25742e9	+	3.90996e9i	0.209985	+	0.363705i
$323$		−	2.98898e9i		−	0.274608i
$324$		0			0
$325$	1.75120e10			1.56964
$326$	−3.77199e9	+	2.17776e9i	−0.333964	+	0.192814i
$327$		0			0
$328$	−1.43752e10	+	2.48985e10i	−1.24199	+	2.15119i
$329$	9.22073e9	+	5.32359e9i	0.787013	+	0.454382i
$330$		0			0
$331$	2.29002e9	+	3.96643e9i	0.190778	+	0.330437i	0.945508	−	0.325598i	$-0.105566\pi$
$331$	2.29002e9	+	3.96643e9i	0.190778	+	0.330437i	−0.754731	+	0.656035i	$0.772232\pi$
$332$			3.95219e10i			3.25301i
$333$		0			0
$334$	8.28970e9			0.666120
$335$	−5.19605e9	+	2.99994e9i	−0.412567	+	0.238195i
$336$		0			0
$337$	9.60960e9	−	1.66443e10i	0.745051	−	1.29047i	−0.205120	−	0.978737i	$-0.565759\pi$
$337$	9.60960e9	−	1.66443e10i	0.745051	−	1.29047i	0.950171	−	0.311729i	$-0.100908\pi$
$338$	4.03503e10	+	2.32962e10i	3.09157	+	1.78492i
$339$		0			0
$340$	−2.05982e9	−	3.56771e9i	−0.154139	−	0.266977i
$341$			8.58324e9i			0.634796i
$342$		0			0
$343$	−1.08437e8			−0.00783431
$344$	−8.34235e9	+	4.81646e9i	−0.595737	+	0.343949i
$345$		0			0
$346$	−2.07777e10	+	3.59880e10i	−1.44975	+	2.51104i
$347$	−7.22360e7	−	4.17055e7i	−0.00498237	−	0.00287657i	0.497507	−	0.867460i	$-0.334249\pi$
$347$	−7.22360e7	−	4.17055e7i	−0.00498237	−	0.00287657i	−0.502489	+	0.864584i	$0.667582\pi$
$348$		0			0
$349$	1.07373e10	+	1.85976e10i	0.723759	+	1.25359i	0.959483	+	0.281767i	$0.0909204\pi$
$349$	1.07373e10	+	1.85976e10i	0.723759	+	1.25359i	−0.235724	+	0.971820i	$0.575746\pi$
$350$		−	3.62783e10i		−	2.41755i
$351$		0			0
$352$	−5.13723e10			−3.34625
$353$	1.63038e10	−	9.41302e9i	1.05000	−	0.606220i	0.127353	−	0.991857i	$-0.459352\pi$
$353$	1.63038e10	−	9.41302e9i	1.05000	−	0.606220i	0.922650	+	0.385638i	$0.126019\pi$
$354$		0			0
$355$	−3.44097e7	+	5.95993e7i	−0.00216654	+	0.00375256i
$356$	−3.47034e10	−	2.00360e10i	−2.16059	−	1.24742i
$357$		0			0
$358$	−2.37414e10	−	4.11213e10i	−1.44535	−	2.50342i
$359$		−	1.82528e10i		−	1.09888i	−0.835532	−	0.549442i	$-0.814840\pi$
$359$		−	1.82528e10i		−	1.09888i	0.835532	−	0.549442i	$-0.185160\pi$
$360$		0			0
$361$	−1.03367e10			−0.608627
$362$	−1.08531e10	+	6.26604e9i	−0.632004	+	0.364888i
$363$		0			0
$364$	5.22706e10	−	9.05354e10i	2.97750	−	5.15719i
$365$	−2.68053e9	−	1.54760e9i	−0.151025	−	0.0871943i
$366$		0			0
$367$	2.38301e9	+	4.12749e9i	0.131359	+	0.227521i	0.924201	−	0.381907i	$-0.124733\pi$
$367$	2.38301e9	+	4.12749e9i	0.131359	+	0.227521i	−0.792842	+	0.609428i	$0.791399\pi$
$368$		−	7.69258e9i		−	0.419450i
$369$		0			0
$370$	9.64822e9			0.514802
$371$	−3.07123e10	+	1.77318e10i	−1.62113	+	0.935958i
$372$		0			0
$373$	6.02586e9	−	1.04371e10i	0.311303	−	0.539193i	−0.667341	−	0.744752i	$-0.732568\pi$
$373$	6.02586e9	−	1.04371e10i	0.311303	−	0.539193i	0.978645	+	0.205559i	$0.0659011\pi$
$374$	2.14910e10	+	1.24078e10i	1.09842	+	0.634175i
$375$		0			0
$376$	−1.75954e10	−	3.04761e10i	−0.880333	−	1.52478i
$377$		−	1.49760e10i		−	0.741362i
$378$		0			0
$379$	−1.49538e10			−0.724759			−0.362380	−	0.932031i	$-0.618036\pi$
$379$	−1.49538e10			−0.724759			−0.362380	+	0.932031i	$0.618036\pi$
$380$	7.93390e9	−	4.58064e9i	0.380498	−	0.219681i
$381$		0			0
$382$	−1.82740e10	+	3.16516e10i	−0.858186	+	1.48642i
$383$	−1.25661e10	−	7.25502e9i	−0.583988	−	0.337166i	0.178728	−	0.983898i	$-0.442802\pi$
$383$	−1.25661e10	−	7.25502e9i	−0.583988	−	0.337166i	−0.762717	+	0.646733i	$0.776135\pi$
$384$		0			0
$385$	−6.84607e9	−	1.18577e10i	−0.311600	−	0.539708i
$386$		−	5.84339e10i		−	2.63218i
$387$		0			0
$388$	4.13352e10			1.82387
$389$	−4.32932e9	+	2.49954e9i	−0.189070	+	0.109159i	−0.591547	−	0.806271i	$-0.701483\pi$
$389$	−4.32932e9	+	2.49954e9i	−0.189070	+	0.109159i	0.402477	+	0.915430i	$0.368149\pi$
$390$		0			0
$391$	−8.19023e8	+	1.41859e9i	−0.0350420	+	0.0606946i
$392$	−5.56412e10	−	3.21245e10i	−2.35642	−	1.36048i
$393$		0			0
$394$	6.75721e8	+	1.17038e9i	0.0280403	+	0.0485672i
$395$			2.32154e9i			0.0953647i
$396$		0			0
$397$	−4.71797e9			−0.189930			−0.0949649	−	0.995481i	$-0.530274\pi$
$397$	−4.71797e9			−0.189930			−0.0949649	+	0.995481i	$0.530274\pi$
$398$	5.00272e10	−	2.88832e10i	1.99377	−	1.15110i
$399$		0			0
$400$	−3.09063e10	+	5.35313e10i	−1.20728	+	2.09106i
$401$	3.75165e10	+	2.16602e10i	1.45093	+	0.837692i	0.998534	−	0.0541274i	$-0.0172377\pi$
$401$	3.75165e10	+	2.16602e10i	1.45093	+	0.837692i	0.452391	+	0.891820i	$0.350571\pi$
$402$		0			0
$403$	9.21614e9	+	1.59628e10i	0.349405	+	0.605187i
$404$			8.49922e9i			0.319046i
$405$		0			0
$406$	−3.10247e10			−1.14184
$407$	−3.58231e10	+	2.06825e10i	−1.30553	+	0.753746i
$408$		0			0
$409$	−8.93484e9	+	1.54756e10i	−0.319296	+	0.553037i	−0.980341	−	0.197309i	$-0.936780\pi$
$409$	−8.93484e9	+	1.54756e10i	−0.319296	+	0.553037i	0.661045	+	0.750346i	$0.270113\pi$
$410$	1.17701e10	+	6.79546e9i	0.416528	+	0.240483i
$411$		0			0
$412$	−9.09674e9	−	1.57560e10i	−0.315716	−	0.546837i
$413$			6.82133e10i			2.34460i
$414$		0			0
$415$	1.11160e10			0.374763
$416$	−9.55404e10	+	5.51603e10i	−3.19017	+	1.84184i
$417$		0			0
$418$	−2.75926e10	+	4.77918e10i	−0.903832	+	1.56548i
$419$	−3.81615e10	−	2.20326e10i	−1.23814	−	0.714840i	−0.269425	−	0.963021i	$-0.586834\pi$
$419$	−3.81615e10	−	2.20326e10i	−1.23814	−	0.714840i	−0.968714	+	0.248182i	$0.920167\pi$
$420$		0			0
$421$	9.23335e9	+	1.59926e10i	0.293921	+	0.509086i	0.974733	−	0.223372i	$-0.0717064\pi$
$421$	9.23335e9	+	1.59926e10i	0.293921	+	0.509086i	−0.680812	+	0.732458i	$0.738373\pi$
$422$			1.68657e10i			0.531807i
$423$		0			0
$424$	1.17213e11			3.62670
$425$	1.13989e10	−	6.58114e9i	0.349387	−	0.201718i
$426$		0			0
$427$	1.80819e10	−	3.13189e10i	0.543918	−	0.942094i
$428$	−3.88814e10	−	2.24482e10i	−1.15869	−	0.668969i
$429$		0			0
$430$	2.27685e9	+	3.94361e9i	0.0665978	+	0.115351i
$431$			3.36109e10i			0.974026i	0.873395	+	0.487013i	$0.161914\pi$
$431$			3.36109e10i			0.974026i	−0.873395	+	0.487013i	$0.838086\pi$
$432$		0			0
$433$	1.68004e10			0.477933			0.238966	−	0.971028i	$-0.423191\pi$
$433$	1.68004e10			0.477933			0.238966	+	0.971028i	$0.423191\pi$
$434$	3.30691e10	−	1.90924e10i	0.932101	−	0.538149i
$435$		0			0
$436$	1.69891e10	−	2.94260e10i	0.470138	−	0.814303i
$437$	−3.15467e9	−	1.82135e9i	−0.0865024	−	0.0499422i
$438$		0			0
$439$	−1.20317e10	−	2.08395e10i	−0.323944	−	0.561087i	0.657354	−	0.753582i	$-0.271675\pi$
$439$	−1.20317e10	−	2.08395e10i	−0.323944	−	0.561087i	−0.981298	+	0.192495i	$0.938342\pi$
$440$			4.52548e10i			1.20741i
$441$		0			0
$442$	5.32909e10			1.39625
$443$	3.52382e10	−	2.03448e10i	0.914954	−	0.528249i	0.0329322	−	0.999458i	$-0.489515\pi$
$443$	3.52382e10	−	2.03448e10i	0.914954	−	0.528249i	0.882022	+	0.471209i	$0.156182\pi$
$444$		0			0
$445$	−5.63537e9	+	9.76074e9i	−0.143708	+	0.248910i
$446$	−5.07361e10	−	2.92925e10i	−1.28226	−	0.740315i
$447$		0			0
$448$	3.95456e10	+	6.84950e10i	0.981717	+	1.70038i
$449$			2.01989e10i			0.496983i	0.968634	+	0.248492i	$0.0799349\pi$
$449$			2.01989e10i			0.496983i	−0.968634	+	0.248492i	$0.920065\pi$
$450$		0			0
$451$	−5.82686e10			−1.40841
$452$	1.13026e11	−	6.52558e10i	2.70786	−	1.56338i
$453$		0			0
$454$	−5.15809e10	+	8.93408e10i	−1.21413	+	2.10294i
$455$	−2.54642e10	−	1.47017e10i	−0.594133	−	0.343023i
$456$		0			0
$457$	−1.44679e9	−	2.50591e9i	−0.0331695	−	0.0574513i	0.848964	−	0.528451i	$-0.177227\pi$
$457$	−1.44679e9	−	2.50591e9i	−0.0331695	−	0.0574513i	−0.882134	+	0.470999i	$0.843893\pi$
$458$		−	1.51823e11i		−	3.45045i
$459$		0			0
$460$	−5.02065e9			−0.112132
$461$	−7.80009e10	+	4.50338e10i	−1.72701	+	0.997092i	−0.825405	+	0.564541i	$0.809053\pi$
$461$	−7.80009e10	+	4.50338e10i	−1.72701	+	0.997092i	−0.901609	+	0.432551i	$0.857613\pi$
$462$		0			0
$463$	−2.82355e9	+	4.89054e9i	−0.0614430	+	0.106422i	−0.895111	−	0.445844i	$-0.852904\pi$
$463$	−2.82355e9	+	4.89054e9i	−0.0614430	+	0.106422i	0.833668	+	0.552266i	$0.186237\pi$
$464$	4.57792e10	+	2.64307e10i	0.987635	+	0.570212i
$465$		0			0
$466$	7.09739e10	+	1.22930e11i	1.50506	+	2.60685i
$467$		−	3.24590e10i		−	0.682445i	−0.939983	−	0.341223i	$-0.889159\pi$
$467$		−	3.24590e10i		−	0.682445i	0.939983	−	0.341223i	$-0.110841\pi$
$468$		0			0
$469$	−1.14437e11			−2.36525
$470$	−1.44067e10	+	8.31771e9i	−0.295239	+	0.170456i
$471$		0			0
$472$	1.12728e11	−	1.95251e11i	2.27125	−	3.93392i
$473$	−1.69075e10	−	9.76155e9i	−0.337781	−	0.195018i
$474$		0			0
$475$	1.46352e10	+	2.53489e10i	0.287491	+	0.497949i
$476$		−	7.85750e10i		−	1.53058i
$477$		0			0
$478$	5.06430e10			0.970080
$479$	8.46906e10	−	4.88961e10i	1.60877	−	0.928822i	0.619121	−	0.785296i	$-0.287489\pi$
$479$	8.46906e10	−	4.88961e10i	1.60877	−	0.928822i	0.989646	−	0.143526i	$-0.0458442\pi$
$480$		0			0
$481$	−4.44151e10	+	7.69291e10i	−0.829755	+	1.43718i
$482$	1.41532e11	+	8.17135e10i	2.62220	+	1.51393i
$483$		0			0
$484$	−9.53024e10	−	1.65069e11i	−1.73669	−	3.00803i
$485$		−	1.16260e10i		−	0.210119i
$486$		0			0
$487$	−5.47177e10			−0.972774			−0.486387	−	0.873743i	$-0.661685\pi$
$487$	−5.47177e10			−0.972774			−0.486387	+	0.873743i	$0.661685\pi$
$488$	−1.03514e11	+	5.97639e10i	−1.82524	+	1.05380i
$489$		0			0
$490$	−1.51859e10	+	2.63028e10i	−0.263425	+	0.456266i
$491$	−8.48185e9	−	4.89700e9i	−0.145937	−	0.0842566i	0.425254	−	0.905074i	$-0.360185\pi$
$491$	−8.48185e9	−	4.89700e9i	−0.145937	−	0.0842566i	−0.571190	+	0.820818i	$0.693518\pi$
$492$		0			0
$493$	−5.62811e9	−	9.74817e9i	−0.0952741	−	0.165020i
$494$			1.18509e11i			1.98995i
$495$		0			0
$496$	−6.50611e10			−1.07497
$497$	−1.13675e9	+	6.56305e8i	−0.0186312	+	0.0107567i
$498$		0			0
$499$	1.30629e10	−	2.26256e10i	0.210687	−	0.364920i	−0.741243	−	0.671237i	$-0.765763\pi$
$499$	1.30629e10	−	2.26256e10i	0.210687	−	0.364920i	0.951930	+	0.306317i	$0.0990967\pi$
$500$	7.29512e10	+	4.21184e10i	1.16722	+	0.673894i
$501$		0			0
$502$	−4.21392e10	−	7.29873e10i	−0.663547	−	1.14930i
$503$			7.49906e10i			1.17148i	0.810499	+	0.585740i	$0.199196\pi$
$503$			7.49906e10i			1.17148i	−0.810499	+	0.585740i	$0.800804\pi$
$504$		0			0
$505$	2.39051e9			0.0367557
$506$	2.61913e10	−	1.51216e10i	0.399535	−	0.230672i
$507$		0			0
$508$	1.14924e11	−	1.99053e11i	1.72566	−	2.98892i
$509$	−8.79610e10	−	5.07843e10i	−1.31045	−	0.756586i	−0.328276	−	0.944582i	$-0.606468\pi$
$509$	−8.79610e10	−	5.07843e10i	−1.31045	−	0.756586i	−0.982170	+	0.187996i	$0.939801\pi$
$510$		0			0
$511$	−2.95179e10	−	5.11264e10i	−0.432914	−	0.749828i
$512$			1.04562e11i			1.52158i
$513$		0			0
$514$	−1.15093e11			−1.64890
$515$	−4.43157e9	+	2.55857e9i	−0.0629983	+	0.0363721i
$516$		0			0
$517$	3.56607e10	−	6.17661e10i	0.499145	−	0.864545i
$518$	1.59369e11	+	9.20116e10i	2.21352	+	1.27798i
$519$		0			0
$520$	4.85917e10	+	8.41633e10i	0.664582	+	1.15109i
$521$			3.32430e10i			0.451179i	0.974222	+	0.225590i	$0.0724309\pi$
$521$			3.32430e10i			0.451179i	−0.974222	+	0.225590i	$0.927569\pi$
$522$		0			0
$523$	1.23700e10			0.165334			0.0826669	−	0.996577i	$-0.473656\pi$
$523$	1.23700e10			0.165334			0.0826669	+	0.996577i	$0.473656\pi$
$524$	−7.56718e9	+	4.36891e9i	−0.100371	+	0.0579493i
$525$		0			0
$526$	−1.06183e11	+	1.83914e11i	−1.38711	+	2.40254i
$527$	1.19979e10	+	6.92701e9i	0.155548	+	0.0898056i
$528$		0			0
$529$	−3.81573e10	−	6.60905e10i	−0.487254	−	0.843949i
$530$		−	5.54091e10i		−	0.702227i
$531$		0			0
$532$	1.74736e11			2.18140
$533$	−1.08366e11	+	6.25651e10i	−1.34271	+	0.775217i
$534$		0			0
$535$	−6.31382e9	+	1.09359e10i	−0.0770685	+	0.133487i
$536$	3.27561e11	+	1.89117e11i	3.96856	+	2.29125i
$537$		0			0
$538$	−7.04904e10	−	1.22093e11i	−0.841397	−	1.45734i
$539$		−	1.30214e11i		−	1.54277i
$540$		0			0
$541$	6.00939e10			0.701522			0.350761	−	0.936465i	$-0.385923\pi$
$541$	6.00939e10			0.701522			0.350761	+	0.936465i	$0.385923\pi$
$542$	1.07426e11	−	6.20224e10i	1.24484	−	0.718706i
$543$		0			0
$544$	−4.14594e10	+	7.18098e10i	−0.473399	+	0.819951i
$545$	−8.27643e9	−	4.77840e9i	−0.0938117	−	0.0541622i
$546$		0			0
$547$	3.65825e10	+	6.33627e10i	0.408624	+	0.707757i	0.994736	−	0.102473i	$-0.0326755\pi$
$547$	3.65825e10	+	6.33627e10i	0.408624	+	0.707757i	−0.586112	+	0.810230i	$0.699342\pi$
$548$			1.43647e10i			0.159285i
$549$		0			0
$550$	−2.43014e11			−2.65571
$551$	2.16780e10	−	1.25158e10i	0.235187	−	0.135785i
$552$		0			0
$553$	−2.21397e10	+	3.83471e10i	−0.236740	+	0.410045i
$554$	−2.37447e11	−	1.37090e11i	−2.52074	−	1.45535i
$555$		0			0
$556$	1.60551e11	+	2.78082e11i	1.68002	+	2.90987i
$557$			1.00228e11i			1.04128i	0.853775	+	0.520642i	$0.174308\pi$
$557$			1.00228e11i			1.04128i	−0.853775	+	0.520642i	$0.825692\pi$
$558$		0			0
$559$	−4.19253e10			−0.429368
$560$	8.98817e10	−	5.18933e10i	0.913944	−	0.527666i
$561$		0			0
$562$	8.33701e10	−	1.44401e11i	0.835728	−	1.44752i
$563$	−9.12464e9	−	5.26811e9i	−0.0908201	−	0.0524350i	0.453902	−	0.891052i	$-0.350032\pi$
$563$	−9.12464e9	−	5.26811e9i	−0.0908201	−	0.0524350i	−0.544722	+	0.838617i	$0.683365\pi$
$564$		0			0
$565$	−1.83540e10	−	3.17900e10i	−0.180109	−	0.311959i
$566$		−	2.00367e11i		−	1.95236i
$567$		0			0
$568$	4.33840e9			0.0416808
$569$	9.91931e10	−	5.72692e10i	0.946308	−	0.546351i	0.0543757	−	0.998521i	$-0.482683\pi$
$569$	9.91931e10	−	5.72692e10i	0.946308	−	0.546351i	0.891932	+	0.452170i	$0.149350\pi$
$570$		0			0
$571$	1.68644e10	−	2.92100e10i	0.158645	−	0.274782i	−0.775735	−	0.631059i	$-0.782621\pi$
$571$	1.68644e10	−	2.92100e10i	0.158645	−	0.274782i	0.934380	+	0.356277i	$0.115954\pi$
$572$	−6.06461e11	−	3.50140e11i	−5.66525	−	3.27083i
$573$		0			0
$574$	1.29612e11	+	2.24494e11i	1.19398	+	2.06803i
$575$		−	1.60410e10i		−	0.146744i
$576$		0			0
$577$	1.77096e11			1.59773			0.798867	−	0.601508i	$-0.205433\pi$
$577$	1.77096e11			1.59773			0.798867	+	0.601508i	$0.205433\pi$
$578$	−1.45343e11	+	8.39136e10i	−1.30221	+	0.751832i
$579$		0			0
$580$	1.72503e10	−	2.98784e10i	0.152435	−	0.264025i
$581$	1.83613e11	+	1.06009e11i	1.61139	+	0.930335i
$582$		0			0
$583$	1.18778e11	+	2.05730e11i	1.02816	+	1.78083i
$584$			1.95123e11i			1.67748i
$585$		0			0
$586$	2.52099e11			2.13787
$587$	8.73817e10	−	5.04498e10i	0.735984	−	0.424920i	−0.0846236	−	0.996413i	$-0.526969\pi$
$587$	8.73817e10	−	5.04498e10i	0.735984	−	0.424920i	0.820607	+	0.571493i	$0.193635\pi$
$588$		0			0
$589$	−1.54043e10	+	2.66811e10i	−0.127992	+	0.221688i
$590$	−9.22994e10	−	5.32891e10i	−0.761712	−	0.439774i
$591$		0			0
$592$	−1.56773e11	−	2.71540e11i	−1.27640	−	2.21078i
$593$		−	2.10071e10i		−	0.169882i	−0.996386	−	0.0849410i	$-0.972930\pi$
$593$		−	2.10071e10i		−	0.169882i	0.996386	−	0.0849410i	$-0.0270702\pi$
$594$		0			0
$595$	−2.21002e10			−0.176331
$596$	2.01101e11	−	1.16106e11i	1.59379	−	0.920173i
$597$		0			0
$598$	3.24731e10	−	5.62451e10i	0.253933	−	0.439825i
$599$	1.62889e11	+	9.40438e10i	1.26527	+	0.730505i	0.974089	−	0.226163i	$-0.0726182\pi$
$599$	1.62889e11	+	9.40438e10i	1.26527	+	0.730505i	0.291182	+	0.956668i	$0.405952\pi$
$600$		0			0
$601$	−4.83946e10	−	8.38218e10i	−0.370936	−	0.642479i	0.618774	−	0.785569i	$-0.287630\pi$
$601$	−4.83946e10	−	8.38218e10i	−0.370936	−	0.642479i	−0.989710	+	0.143090i	$0.954296\pi$
$602$			8.68538e10i			0.661307i
$603$		0			0
$604$	−1.36137e10			−0.102289
$605$	−4.64275e10	+	2.68049e10i	−0.346540	+	0.200075i
$606$		0			0
$607$	−3.28298e10	+	5.68630e10i	−0.241832	+	0.418866i	−0.961236	−	0.275726i	$-0.911082\pi$
$607$	−3.28298e10	+	5.68630e10i	−0.241832	+	0.418866i	0.719404	+	0.694592i	$0.244415\pi$
$608$	−1.59691e11	−	9.21977e10i	−1.16860	−	0.674692i
$609$		0			0
$610$	2.82517e10	+	4.89333e10i	0.204045	+	0.353415i
$611$		−	1.53161e11i		−	1.09896i
$612$		0			0
$613$	1.95329e10			0.138333			0.0691665	−	0.997605i	$-0.477966\pi$
$613$	1.95329e10			0.138333			0.0691665	+	0.997605i	$0.477966\pi$
$614$	−3.93669e11	+	2.27285e11i	−2.76986	+	1.59918i
$615$		0			0
$616$	−4.31579e11	+	7.47516e11i	−2.99735	+	5.19156i
$617$	−7.24452e10	−	4.18262e10i	−0.499883	−	0.288608i	0.228782	−	0.973478i	$-0.426526\pi$
$617$	−7.24452e10	−	4.18262e10i	−0.499883	−	0.288608i	−0.728665	+	0.684870i	$0.759859\pi$
$618$		0			0
$619$	1.55055e10	+	2.68563e10i	0.105614	+	0.182929i	0.913989	−	0.405739i	$-0.132986\pi$
$619$	1.55055e10	+	2.68563e10i	0.105614	+	0.182929i	−0.808375	+	0.588668i	$0.799652\pi$
$620$			4.24629e10i			0.287371i
$621$		0			0
$622$	6.74470e9			0.0450610
$623$	−1.86169e11	+	1.07485e11i	−1.23582	+	0.713503i
$624$		0			0
$625$	−5.82748e10	+	1.00935e11i	−0.381910	+	0.661487i
$626$	2.45956e11	+	1.42003e11i	1.60162	+	0.924696i
$627$		0			0
$628$	−1.63571e11	−	2.83314e11i	−1.05164	−	1.82150i
$629$			6.67662e10i			0.426535i
$630$		0			0
$631$	9.80750e10			0.618644			0.309322	−	0.950957i	$-0.399898\pi$
$631$	9.80750e10			0.618644			0.309322	+	0.950957i	$0.399898\pi$
$632$	1.26743e11	−	7.31754e10i	0.794433	−	0.458666i
$633$		0			0
$634$	1.59879e11	−	2.76918e11i	0.989539	−	1.71393i
$635$	−5.59862e10	−	3.23236e10i	−0.344339	−	0.198804i
$636$		0			0
$637$	−1.39815e11	−	2.42167e11i	−0.849174	−	1.47081i
$638$			2.07823e11i			1.25432i
$639$		0			0
$640$	−2.06401e10			−0.123025
$641$	1.15177e11	−	6.64973e10i	0.682232	−	0.393887i	−0.118463	−	0.992958i	$-0.537797\pi$
$641$	1.15177e11	−	6.64973e10i	0.682232	−	0.393887i	0.800696	+	0.599071i	$0.204463\pi$
$642$		0			0
$643$	1.24503e11	−	2.15646e11i	0.728346	−	1.26153i	−0.229236	−	0.973371i	$-0.573623\pi$
$643$	1.24503e11	−	2.15646e11i	0.728346	−	1.26153i	0.957582	−	0.288161i	$-0.0930439\pi$
$644$	−8.29308e10	−	4.78801e10i	−0.482139	−	0.278363i
$645$		0			0
$646$	4.45366e10	+	7.71397e10i	0.255733	+	0.442943i
$647$		−	2.38080e11i		−	1.35864i	−0.733840	−	0.679322i	$-0.762274\pi$
$647$		−	2.38080e11i		−	1.35864i	0.733840	−	0.679322i	$-0.237726\pi$
$648$		0			0
$649$	4.56934e11			2.57558
$650$	−4.51949e11	+	2.60933e11i	−2.53184	+	1.46176i
$651$		0			0
$652$	4.61905e10	−	8.00043e10i	0.255601	−	0.442714i
$653$	1.62795e11	+	9.39897e10i	0.895340	+	0.516925i	0.875685	−	0.482882i	$-0.160410\pi$
$653$	1.62795e11	+	9.39897e10i	0.895340	+	0.516925i	0.0196547	+	0.999807i	$0.493743\pi$
$654$		0			0
$655$	1.22881e9	+	2.12836e9i	0.00667604	+	0.0115632i
$656$		−	4.41676e11i		−	2.38500i
$657$		0			0
$658$	−3.17292e11			−1.69261
$659$	1.79165e11	−	1.03441e11i	0.949972	−	0.548466i	0.0568994	−	0.998380i	$-0.481879\pi$
$659$	1.79165e11	−	1.03441e11i	0.949972	−	0.548466i	0.893072	+	0.449914i	$0.148545\pi$
$660$		0			0
$661$	8.44372e10	−	1.46250e11i	0.442312	−	0.766106i	−0.555549	−	0.831484i	$-0.687492\pi$
$661$	8.44372e10	−	1.46250e11i	0.442312	−	0.766106i	0.997861	+	0.0653777i	$0.0208252\pi$
$662$	−1.18202e11	−	6.82439e10i	−0.615450	−	0.355330i
$663$		0			0
$664$	−3.50378e11	−	6.06873e11i	−1.80246	−	3.12195i
$665$		−	4.91465e10i		−	0.251308i
$666$		0			0
$667$	−1.37181e10			−0.0693090
$668$	−1.52269e11	+	8.79128e10i	−0.764727	+	0.441516i
$669$		0			0
$670$	8.93999e10	−	1.54845e11i	0.443648	−	0.768420i
$671$	−2.09793e11	−	1.21124e11i	−1.03490	−	0.597502i
$672$		0			0
$673$	1.63036e11	+	2.82387e11i	0.794736	+	1.37652i	0.923006	+	0.384785i	$0.125724\pi$
$673$	1.63036e11	+	2.82387e11i	0.794736	+	1.37652i	−0.128270	+	0.991739i	$0.540942\pi$
$674$			5.72743e11i			2.77537i
$675$		0			0
$676$	−9.88232e11			−4.73230
$677$	−3.13972e9	+	1.81272e9i	−0.0149464	+	0.00862930i	−0.507455	−	0.861679i	$-0.669413\pi$
$677$	−3.13972e9	+	1.81272e9i	−0.0149464	+	0.00862930i	0.492508	+	0.870308i	$0.336080\pi$
$678$		0			0
$679$	1.10873e11	−	1.92038e11i	0.521612	−	0.903459i
$680$	6.32586e10	+	3.65224e10i	0.295858	+	0.170814i
$681$		0			0
$682$	−1.27893e11	−	2.21517e11i	−0.591165	−	1.02393i
$683$		−	2.50569e11i		−	1.15145i	−0.817644	−	0.575724i	$-0.804720\pi$
$683$		−	2.50569e11i		−	1.15145i	0.817644	−	0.575724i	$-0.195280\pi$
$684$		0			0
$685$	4.04025e9			0.0183504
$686$	2.79854e9	−	1.61574e9i	0.0126368	−	0.00729584i
$687$		0			0
$688$	7.39927e10	−	1.28159e11i	0.330244	−	0.571999i
$689$	4.41799e11	+	2.55073e11i	1.96041	+	1.13185i
$690$		0			0
$691$	3.95297e10	+	6.84675e10i	0.173385	+	0.300312i	0.939601	−	0.342271i	$-0.111196\pi$
$691$	3.95297e10	+	6.84675e10i	0.173385	+	0.300312i	−0.766216	+	0.642583i	$0.777863\pi$
$692$		−	8.81395e11i		−	3.84367i
$693$		0			0
$694$	2.48569e9			0.0107154
$695$	7.82140e10	−	4.51569e10i	0.335232	−	0.193546i
$696$		0			0
$697$	−4.70250e10	+	8.14497e10i	−0.199250	+	0.345110i
$698$	−5.54218e11	−	3.19978e11i	−2.33485	−	1.34803i
$699$		0			0
$700$	3.84734e11	+	6.66378e11i	1.60239	+	2.77542i
$701$		−	1.92497e11i		−	0.797173i	−0.917131	−	0.398587i	$-0.869501\pi$
$701$		−	1.92497e11i		−	0.797173i	0.917131	−	0.398587i	$-0.130499\pi$
$702$		0			0
$703$	−1.48475e11			−0.607901
$704$	4.58821e11	−	2.64900e11i	1.86790	−	1.07843i
$705$		0			0
$706$	−2.80513e11	+	4.85863e11i	−1.12911	+	1.95567i
$707$	3.94863e10	+	2.27974e10i	0.158041	+	0.0912447i
$708$		0			0
$709$	−4.01574e10	−	6.95547e10i	−0.158921	−	0.275259i	0.775559	−	0.631275i	$-0.217468\pi$
$709$	−4.01574e10	−	6.95547e10i	−0.158921	−	0.275259i	−0.934480	+	0.356016i	$0.884135\pi$
$710$		−	2.05085e9i		−	0.00807052i
$711$		0			0
$712$	7.10512e11			2.76472
$713$	1.46220e10	−	8.44202e9i	0.0565782	−	0.0326654i
$714$		0			0
$715$	−9.84812e10	+	1.70574e11i	−0.376816	+	0.652664i
$716$	8.72187e11	+	5.03557e11i	3.31862	+	1.91601i
$717$		0			0
$718$	2.71972e11	+	4.71069e11i	1.02335	+	1.77250i
$719$		−	2.23610e11i		−	0.836712i	−0.908283	−	0.418356i	$-0.862607\pi$
$719$		−	2.23610e11i		−	0.836712i	0.908283	−	0.418356i	$-0.137393\pi$
$720$		0			0
$721$	−9.76005e10			−0.361170
$722$	2.66769e11	−	1.54019e11i	0.981717	−	0.566795i
$723$		0			0
$724$	1.32904e11	−	2.30196e11i	0.483707	−	0.837805i
$725$	9.54616e10	+	5.51148e10i	0.345523	+	0.199488i
$726$		0			0
$727$	−1.83712e11	−	3.18198e11i	−0.657656	−	1.13909i	−0.981221	−	0.192888i	$-0.938215\pi$
$727$	−1.83712e11	−	3.18198e11i	−0.657656	−	1.13909i	0.323565	−	0.946206i	$-0.395119\pi$
$728$			1.85361e12i			6.59921i
$729$		0			0
$730$	9.22389e10			0.324805
$731$	−2.72900e10	+	1.57559e10i	−0.0955728	+	0.0551790i
$732$		0			0
$733$	−2.14763e11	+	3.71981e11i	−0.743950	+	1.28856i	0.206733	+	0.978397i	$0.433717\pi$
$733$	−2.14763e11	+	3.71981e11i	−0.743950	+	1.28856i	−0.950684	+	0.310163i	$0.899617\pi$
$734$	−1.23001e11	−	7.10149e10i	−0.423766	−	0.244661i
$735$		0			0
$736$	5.05270e10	+	8.75154e10i	0.172192	+	0.298245i
$737$			7.66571e11i			2.59826i
$738$		0			0
$739$	5.45040e11			1.82747			0.913736	−	0.406309i	$-0.133184\pi$
$739$	5.45040e11			1.82747			0.913736	+	0.406309i	$0.133184\pi$
$740$	−1.77223e11	+	1.02320e11i	−0.591009	+	0.341219i
$741$		0			0
$742$	5.28417e11	−	9.15245e11i	1.74326	−	3.01941i
$743$	−5.16331e11	−	2.98104e11i	−1.69423	−	0.978166i	−0.951029	−	0.309103i	$-0.899971\pi$
$743$	−5.16331e11	−	2.98104e11i	−1.69423	−	0.978166i	−0.743205	−	0.669064i	$-0.766695\pi$
$744$		0			0
$745$	−3.26562e10	−	5.65622e10i	−0.106008	−	0.183612i
$746$			3.59148e11i			1.15963i
$747$		0			0
$748$	−5.26343e11			−1.68137
$749$	−2.08583e11	+	1.20425e11i	−0.662752	+	0.382640i
$750$		0			0
$751$	−1.31908e11	+	2.28471e11i	−0.414678	+	0.718243i	−0.995395	−	0.0958625i	$-0.969439\pi$
$751$	−1.31908e11	+	2.28471e11i	−0.414678	+	0.718243i	0.580717	+	0.814106i	$0.302772\pi$
$752$	4.68187e11	+	2.70308e11i	1.46402	+	0.845255i
$753$		0			0
$754$	2.23147e11	+	3.86501e11i	0.690407	+	1.19582i
$755$			3.82901e9i			0.0117842i
$756$		0			0
$757$	−6.45369e10			−0.196528			−0.0982641	−	0.995160i	$-0.531329\pi$
$757$	−6.45369e10			−0.196528			−0.0982641	+	0.995160i	$0.531329\pi$
$758$	3.85927e11	−	2.22815e11i	1.16904	−	0.674945i
$759$		0			0
$760$	−8.12187e10	+	1.40675e11i	−0.243445	+	0.421660i
$761$	3.81798e10	+	2.20431e10i	0.113840	+	0.0657256i	0.555839	−	0.831290i	$-0.312397\pi$
$761$	3.81798e10	+	2.20431e10i	0.113840	+	0.0657256i	−0.441999	+	0.897016i	$0.645730\pi$
$762$		0			0
$763$	−9.11397e10	−	1.57859e11i	−0.268912	−	0.465768i
$764$		−	7.75190e11i		−	2.27528i
$765$		0			0
$766$	4.32407e11			1.25597
$767$	8.49790e11	−	4.90626e11i	2.45544	−	1.41765i
$768$		0			0
$769$	−8.20480e10	+	1.42111e11i	−0.234619	+	0.406372i	−0.959162	−	0.282858i	$-0.908718\pi$
$769$	−8.20480e10	+	1.42111e11i	−0.234619	+	0.406372i	0.724543	+	0.689230i	$0.242051\pi$
$770$	3.53367e11	+	2.04017e11i	1.00523	+	0.580367i
$771$		0			0
$772$	6.19695e11	+	1.07334e12i	1.74465	+	3.02183i
$773$			6.54118e11i			1.83205i	0.401116	+	0.916027i	$0.368622\pi$
$773$			6.54118e11i			1.83205i	−0.401116	+	0.916027i	$0.631378\pi$
$774$		0			0
$775$	−1.35669e11			−0.376075
$776$	−6.34718e11	+	3.66455e11i	−1.75039	+	1.01059i
$777$		0			0
$778$	7.44876e10	−	1.29016e11i	0.203313	−	0.352149i
$779$	−1.81128e11	−	1.04574e11i	−0.491854	−	0.283972i
$780$		0			0
$781$	4.39633e9	+	7.61466e9i	0.0118164	+	0.0204666i
$782$		−	4.88147e10i		−	0.130534i
$783$		0			0
$784$	9.87022e11			2.61254
$785$	−7.96854e10	+	4.60064e10i	−0.209846	+	0.121155i
$786$		0			0
$787$	5.89916e10	−	1.02176e11i	0.153777	−	0.266350i	−0.778836	−	0.627228i	$-0.784190\pi$
$787$	5.89916e10	−	1.02176e11i	0.153777	−	0.266350i	0.932613	+	0.360878i	$0.117523\pi$
$788$	−2.48240e10	−	1.43321e10i	−0.0643823	−	0.0371711i
$789$		0			0
$790$	−3.45916e10	−	5.99144e10i	−0.0888101	−	0.153824i
$791$		−	7.00141e11i		−	1.78846i
$792$		0			0
$793$	−5.20220e11			−1.31551
$794$	1.21762e11	−	7.02991e10i	0.306357	−	0.176876i
$795$		0			0
$796$	−6.12617e11	+	1.06108e12i	−1.52594	+	2.64300i
$797$	4.23761e10	+	2.44659e10i	0.105024	+	0.0606356i	0.551592	−	0.834114i	$-0.314021\pi$
$797$	4.23761e10	+	2.44659e10i	0.105024	+	0.0606356i	−0.446568	+	0.894750i	$0.647354\pi$
$798$		0			0
$799$	−5.75591e10	−	9.96952e10i	−0.141230	−	0.244617i
$800$		−	8.12005e11i		−	1.98243i
$801$		0			0
$802$	−1.29097e12			−3.12046
$803$	−3.42476e11	+	1.97729e11i	−0.823697	+	0.475562i
$804$		0			0
$805$	−1.34669e10	+	2.33253e10i	−0.0320688	+	0.0555448i
$806$	−4.75701e11	−	2.74646e11i	−1.12718	−	0.650779i
$807$		0			0
$808$	−7.53492e10	−	1.30509e11i	−0.176780	−	0.306192i
$809$		−	8.81881e10i		−	0.205881i	−0.994688	−	0.102940i	$-0.967175\pi$
$809$		−	8.81881e10i		−	0.205881i	0.994688	−	0.102940i	$-0.0328251\pi$
$810$		0			0
$811$	−7.58870e11			−1.75422			−0.877110	−	0.480290i	$-0.840532\pi$
$811$	−7.58870e11			−1.75422			−0.877110	+	0.480290i	$0.840532\pi$
$812$	5.69878e11	−	3.29019e11i	1.31087	−	0.756828i
$813$		0			0
$814$	6.16350e11	−	1.06755e12i	1.40388	−	2.43159i
$815$	−2.25022e10	−	1.29916e10i	−0.0510028	−	0.0294465i
$816$		0			0
$817$	−3.50381e10	−	6.06877e10i	−0.0786415	−	0.136211i
$818$		−	5.32527e11i		−	1.18940i
$819$		0			0
$820$	−2.88265e11			−0.637583
$821$	3.71778e11	−	2.14646e11i	0.818297	−	0.472444i	−0.0315316	−	0.999503i	$-0.510038\pi$
$821$	3.71778e11	−	2.14646e11i	0.818297	−	0.472444i	0.849829	+	0.527059i	$0.176705\pi$
$822$		0			0
$823$	2.81758e11	−	4.88019e11i	0.614154	−	1.06375i	−0.376379	−	0.926466i	$-0.622831\pi$
$823$	2.81758e11	−	4.88019e11i	0.614154	−	1.06375i	0.990532	−	0.137279i	$-0.0438358\pi$
$824$	2.79368e11	+	1.61293e11i	0.605993	+	0.349870i
$825$		0			0
$826$	−1.01640e12	−	1.76045e12i	−2.18345	−	3.78185i
$827$			2.13022e11i			0.455410i	0.973730	+	0.227705i	$0.0731221\pi$
$827$			2.13022e11i			0.455410i	−0.973730	+	0.227705i	$0.926878\pi$
$828$		0			0
$829$	5.16375e11			1.09332			0.546660	−	0.837355i	$-0.315899\pi$
$829$	5.16375e11			1.09332			0.546660	+	0.837355i	$0.315899\pi$
$830$	−2.86882e11	+	1.65632e11i	−0.604493	+	0.349004i
$831$		0			0
$832$	5.68867e11	−	9.85306e11i	1.18718	−	2.05626i
$833$	−1.82017e11	−	1.05088e11i	−0.378035	−	0.218259i
$834$		0			0
$835$	2.47265e10	+	4.28276e10i	0.0508647	+	0.0881003i
$836$		−	1.17049e12i		−	2.39630i
$837$		0			0
$838$	1.31317e12			2.66283
$839$	−5.95758e10	+	3.43961e10i	−0.120233	+	0.0694163i	−0.558910	−	0.829228i	$-0.688780\pi$
$839$	−5.95758e10	+	3.43961e10i	−0.120233	+	0.0694163i	0.438678	+	0.898644i	$0.355447\pi$
$840$		0			0
$841$	−2.02990e11	+	3.51589e11i	−0.405779	+	0.702831i
$842$	−4.76589e11	−	2.75159e11i	−0.948191	−	0.547438i
$843$		0			0
$844$	−1.78862e11	−	3.09797e11i	−0.352491	−	0.610532i
$845$			2.77952e11i			0.545184i
$846$		0			0
$847$	−1.02252e12			−1.98672
$848$	−1.55943e12	+	9.00340e11i	−3.01567	+	1.74110i
$849$		0			0
$850$	−1.96122e11	+	3.39693e11i	−0.375708	+	0.650745i
$851$	7.04674e10	+	4.06844e10i	0.134360	+	0.0775728i
$852$		0			0
$853$	−1.16992e11	−	2.02636e11i	−0.220984	−	0.382755i	0.734123	−	0.679016i	$-0.237593\pi$
$853$	−1.16992e11	−	2.02636e11i	−0.220984	−	0.382755i	−0.955107	+	0.296261i	$0.904260\pi$
$854$			1.07770e12i			2.02613i
$855$		0			0
$856$	7.96051e11			1.48267
$857$	−7.35165e11	+	4.24448e11i	−1.36289	+	0.786867i	−0.990008	−	0.141011i	$-0.954965\pi$
$857$	−7.35165e11	+	4.24448e11i	−1.36289	+	0.786867i	−0.372885	+	0.927878i	$0.621631\pi$
$858$		0			0
$859$	−5.83644e10	+	1.01090e11i	−0.107195	+	0.185668i	−0.914633	−	0.404285i	$-0.867520\pi$
$859$	−5.83644e10	+	1.01090e11i	−0.107195	+	0.185668i	0.807438	+	0.589953i	$0.200854\pi$
$860$	−8.36445e10	−	4.82922e10i	−0.152913	−	0.0882842i
$861$		0			0
$862$	−5.00811e11	−	8.67431e11i	−0.907079	−	1.57111i
$863$		−	5.73876e11i		−	1.03461i	−0.855803	−	0.517303i	$-0.826936\pi$
$863$		−	5.73876e11i		−	1.03461i	0.855803	−	0.517303i	$-0.173064\pi$
$864$		0			0
$865$	−2.47903e11			−0.442810
$866$	−4.33584e11	+	2.50330e11i	−0.770907	+	0.445083i
$867$		0			0
$868$	−4.04953e11	+	7.01399e11i	−0.713388	+	1.23562i
$869$	2.56872e11	+	1.48305e11i	0.450440	+	0.260062i
$870$		0			0
$871$	8.23096e11	+	1.42564e12i	1.43014	+	2.47707i
$872$			6.02464e11i			1.04199i
$873$		0			0
$874$	1.08555e11			0.186038
$875$	3.91353e11	−	2.25948e11i	0.667631	−	0.385457i
$876$		0			0
$877$	−3.66765e11	+	6.35255e11i	−0.619996	+	1.07387i	0.369489	+	0.929235i	$0.379533\pi$
$877$	−3.66765e11	+	6.35255e11i	−0.619996	+	1.07387i	−0.989486	+	0.144630i	$0.953801\pi$
$878$	6.21030e11	+	3.58552e11i	1.04504	+	0.603356i
$879$		0			0
$880$	−3.47612e11	−	6.02082e11i	−0.579648	−	1.00398i
$881$			1.09657e12i			1.82026i	0.414327	+	0.910128i	$0.364017\pi$
$881$			1.09657e12i			1.82026i	−0.414327	+	0.910128i	$0.635983\pi$
$882$		0			0
$883$	−3.36234e11			−0.553093			−0.276547	−	0.961001i	$-0.589190\pi$
$883$	−3.36234e11			−0.553093			−0.276547	+	0.961001i	$0.589190\pi$
$884$	−9.78875e11	+	5.65154e11i	−1.60294	+	0.925460i
$885$		0			0
$886$	−6.06287e11	+	1.05012e12i	−0.983882	+	1.70413i
$887$	−5.13551e11	−	2.96499e11i	−0.829639	−	0.478993i	0.0240898	−	0.999710i	$-0.492331\pi$
$887$	−5.13551e11	−	2.96499e11i	−0.829639	−	0.478993i	−0.853729	+	0.520717i	$0.825665\pi$
$888$		0			0
$889$	−6.16518e11	−	1.06784e12i	−0.987048	−	1.70962i
$890$		−	3.35874e11i		−	0.535324i
$891$		0			0
$892$	1.24259e12			1.96277
$893$	2.21703e11	−	1.28000e11i	0.348631	−	0.201282i
$894$		0			0
$895$	1.41631e11	−	2.45313e11i	0.220733	−	0.382321i
$896$	−3.40932e11	−	1.96837e11i	−0.528976	−	0.305405i
$897$		0			0
$898$	−3.00969e11	−	5.21294e11i	−0.462825	−	0.801636i
$899$			1.16023e11i			0.177625i
$900$		0			0
$901$	3.83434e11			0.581824
$902$	1.50380e12	−	8.68218e11i	2.27176	−	1.31160i
$903$		0			0
$904$	−1.15704e12	+	2.00405e12i	−1.73251	+	3.00079i
$905$	−6.47453e10	−	3.73807e10i	−0.0965192	−	0.0557254i
$906$		0			0
$907$	1.62762e11	+	2.81912e11i	0.240505	+	0.416567i	0.960858	−	0.277041i	$-0.0893537\pi$
$907$	1.62762e11	+	2.81912e11i	0.240505	+	0.416567i	−0.720353	+	0.693607i	$0.756020\pi$
$908$		−	2.18808e12i		−	3.21899i
$909$		0			0
$910$	8.76240e11			1.27778
$911$	9.90701e10	−	5.71982e10i	0.143837	−	0.0830441i	−0.426355	−	0.904556i	$-0.640202\pi$
$911$	9.90701e10	−	5.71982e10i	0.143837	−	0.0830441i	0.570191	+	0.821512i	$0.306869\pi$
$912$		0			0
$913$	7.10114e11	−	1.22995e12i	1.02199	−	1.77013i
$914$	7.46774e10	+	4.31150e10i	0.107005	+	0.0617794i
$915$		0			0
$916$	1.61009e12	+	2.78876e12i	2.28701	+	3.96123i
$917$			4.68748e10i			0.0662922i
$918$		0			0
$919$	−5.33897e11			−0.748506			−0.374253	−	0.927327i	$-0.622101\pi$
$919$	−5.33897e11			−0.748506			−0.374253	+	0.927327i	$0.622101\pi$
$920$	7.70939e10	−	4.45102e10i	0.107614	−	0.0621310i
$921$		0			0
$922$	1.34203e12	−	2.32447e12i	1.85712	−	3.21663i
$923$	1.63523e10	+	9.44100e9i	0.0225305	+	0.0130080i
$924$		0			0
$925$	−3.26913e11	−	5.66231e11i	−0.446546	−	0.773440i
$926$		−	1.68287e11i		−	0.228879i
$927$		0			0
$928$	−6.94416e11			−0.936328
$929$	−2.42386e11	+	1.39942e11i	−0.325421	+	0.187882i	−0.653806	−	0.756662i	$-0.726829\pi$
$929$	−2.42386e11	+	1.39942e11i	−0.325421	+	0.187882i	0.328386	+	0.944544i	$0.393495\pi$
$930$		0			0
$931$	2.33694e11	−	4.04771e11i	0.311064	−	0.538778i
$932$	−2.60737e12	−	1.50537e12i	−3.45572	−	1.99516i
$933$		0			0
$934$	4.83649e11	+	8.37704e11i	0.635539	+	1.10079i
$935$			1.48040e11i			0.193702i
$936$		0			0
$937$	9.25694e11			1.20091			0.600453	−	0.799660i	$-0.294987\pi$
$937$	9.25694e11			1.20091			0.600453	+	0.799660i	$0.294987\pi$
$938$	2.95341e12	−	1.70515e12i	3.81515	−	2.20268i
$939$		0			0
$940$	1.76420e11	−	3.05568e11i	0.225962	−	0.391378i
$941$	−1.07534e12	−	6.20848e11i	−1.37147	−	0.791820i	−0.380359	−	0.924839i	$-0.624200\pi$
$941$	−1.07534e12	−	6.20848e11i	−1.37147	−	0.791820i	−0.991114	+	0.133018i	$0.957533\pi$
$942$		0			0
$943$	5.73099e10	+	9.92636e10i	0.0724741	+	0.125529i
$944$			3.46356e12i			4.36149i
$945$		0			0
$946$	5.81799e11			0.726455
$947$	−1.16909e12	+	6.74975e11i	−1.45361	+	0.839243i	−0.998684	−	0.0512859i	$-0.983668\pi$
$947$	−1.16909e12	+	6.74975e11i	−1.45361	+	0.839243i	−0.454927	+	0.890529i	$0.650335\pi$
$948$		0			0
$949$	−4.24617e11	+	7.35457e11i	−0.523519	+	0.906761i
$950$	−7.55411e11	−	4.36137e11i	−0.927447	−	0.535462i
$951$		0			0
$952$	6.96601e11	+	1.20655e12i	0.848079	+	1.46892i
$953$			6.09272e11i			0.738651i	0.929300	+	0.369326i	$0.120411\pi$
$953$			6.09272e11i			0.738651i	−0.929300	+	0.369326i	$0.879589\pi$
$954$		0			0
$955$	−2.18031e11			−0.262123
$956$	−9.30236e11	+	5.37072e11i	−1.11368	+	0.642985i
$957$		0			0
$958$	−1.45713e12	+	2.52383e12i	−1.72996	+	2.99639i
$959$	6.67366e10	+	3.85304e10i	0.0789023	+	0.0455542i
$960$		0			0
$961$	3.55046e11	+	6.14958e11i	0.416285	+	0.721027i
$962$		−	2.64719e12i		−	3.09090i
$963$		0			0
$964$	−3.46631e12			−4.01383
$965$	3.01891e11	−	1.74297e11i	0.348129	−	0.200992i
$966$		0			0
$967$	1.71762e11	−	2.97500e11i	0.196436	−	0.340237i	−0.750934	−	0.660377i	$-0.770397\pi$
$967$	1.71762e11	−	2.97500e11i	0.196436	−	0.340237i	0.947370	+	0.320140i	$0.103730\pi$
$968$	2.92681e12	+	1.68979e12i	3.33344	+	1.92456i
$969$		0			0
$970$	1.73231e11	+	3.00045e11i	0.195677	+	0.338922i
$971$		−	8.09302e11i		−	0.910403i	−0.890388	−	0.455202i	$-0.849567\pi$
$971$		−	8.09302e11i		−	0.910403i	0.890388	−	0.455202i	$-0.150433\pi$
$972$		0			0
$973$	1.72258e12			1.92189
$974$	1.41216e12	−	8.15309e11i	1.56909	−	0.905913i
$975$		0			0
$976$	9.18120e11	−	1.59023e12i	1.01181	−	1.75251i
$977$	9.76462e11	+	5.63761e11i	1.07171	+	0.618752i	0.928648	−	0.370962i	$-0.120972\pi$
$977$	9.76462e11	+	5.63761e11i	1.07171	+	0.618752i	0.143062	+	0.989714i	$0.454305\pi$
$978$		0			0
$979$	7.19999e11	+	1.24708e12i	0.783793	+	1.35757i
$980$		−	6.44191e11i		−	0.698410i
$981$		0			0
$982$	2.91867e11			0.313862
$983$	−5.69453e11	+	3.28774e11i	−0.609879	+	0.352114i	−0.772918	−	0.634506i	$-0.781204\pi$
$983$	−5.69453e11	+	3.28774e11i	−0.609879	+	0.352114i	0.163039	+	0.986620i	$0.447870\pi$
$984$		0			0
$985$	−4.03108e9	+	6.98204e9i	−0.00428230	+	0.00741715i
$986$	2.90501e11	+	1.67721e11i	0.307355	+	0.177451i
$987$		0			0
$988$	−1.25679e12	−	2.17683e12i	−1.31897	−	2.28453i
$989$			3.84038e10i			0.0401411i
$990$		0			0
$991$	3.09933e11			0.321347			0.160673	−	0.987008i	$-0.448633\pi$
$991$	3.09933e11			0.321347			0.160673	+	0.987008i	$0.448633\pi$
$992$	7.40174e11	−	4.27340e11i	0.764341	−	0.441292i
$993$		0			0
$994$	1.95583e10	−	3.38759e10i	0.0200348	−	0.0347013i
$995$	2.98442e11	+	1.72306e11i	0.304487	+	0.175795i
$996$		0			0
$997$	4.13046e11	+	7.15417e11i	0.418040	+	0.724067i	0.995742	−	0.0921807i	$-0.0293837\pi$
$997$	4.13046e11	+	7.15417e11i	0.418040	+	0.724067i	−0.577702	+	0.816248i	$0.696050\pi$
$998$			7.78563e11i			0.784823i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.9.d.g.53.1		32
3.2	odd	2	inner	81.9.d.g.53.16		32
9.2	odd	6	inner	81.9.d.g.26.1		32
9.4	even	3		81.9.b.b.80.1	✓	16
9.5	odd	6		81.9.b.b.80.16	yes	16
9.7	even	3	inner	81.9.d.g.26.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.9.b.b.80.1	✓	16	9.4	even	3
81.9.b.b.80.16	yes	16	9.5	odd	6
81.9.d.g.26.1		32	9.2	odd	6	inner
81.9.d.g.26.16		32	9.7	even	3	inner
81.9.d.g.53.1		32	1.1	even	1	trivial
81.9.d.g.53.16		32	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	81.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-25.8080 + 14.9003i) q^{2} +(316.037 - 547.392i) q^{4} +(-153.960 - 88.8891i) q^{5} +(-1695.41 - 2936.53i) q^{7} +11207.2i q^{8} +5297.89 q^{10} +(-19670.7 + 11356.9i) q^{11} +(-24388.6 + 42242.2i) q^{13} +(87510.3 + 50524.1i) q^{14} +(-86085.2 - 149104. i) q^{16} +36661.7i q^{17} -81528.6 q^{19} +(-97314.4 + 56184.5i) q^{20} +(338441. - 586197. i) q^{22} +(38694.0 + 22340.0i) q^{23} +(-179510. - 310920. i) q^{25} -1.45359e6i q^{26} -2.14324e6 q^{28} +(-265895. + 153515. i) q^{29} +(188944. - 327260. i) q^{31} +(1.95871e6 + 1.13086e6i) q^{32} +(-546270. - 946167. i) q^{34} +602813. i q^{35} +1.82114e6 q^{37} +(2.10409e6 - 1.21480e6i) q^{38} +(996199. - 1.72547e6i) q^{40} +(2.22165e6 + 1.28267e6i) q^{41} +(429764. + 744374. i) q^{43} +1.43568e7i q^{44} -1.33149e6 q^{46} +(-2.71933e6 + 1.57000e6i) q^{47} +(-2.86641e6 + 4.96477e6i) q^{49} +(9.26560e6 + 5.34950e6i) q^{50} +(1.54154e7 + 2.67002e7i) q^{52} -1.04587e7i q^{53} +4.03801e6 q^{55} +(3.29103e7 - 1.90008e7i) q^{56} +(4.57482e6 - 7.92382e6i) q^{58} +(-1.74219e7 - 1.00585e7i) q^{59} +(5.33263e6 + 9.23638e6i) q^{61} +1.12613e7i q^{62} -2.33251e7 q^{64} +(7.50975e6 - 4.33575e6i) q^{65} +(1.68746e7 - 2.92277e7i) q^{67} +(2.00683e7 + 1.15865e7i) q^{68} +(-8.98208e6 - 1.55574e7i) q^{70} -387108. i q^{71} +1.74105e7 q^{73} +(-4.70002e7 + 2.71356e7i) q^{74} +(-2.57660e7 + 4.46281e7i) q^{76} +(6.66996e7 + 3.85090e7i) q^{77} +(-6.52931e6 - 1.13091e7i) q^{79} +3.06081e7i q^{80} -7.64487e7 q^{82} +(-5.41503e7 + 3.12637e7i) q^{83} +(3.25883e6 - 5.64445e6i) q^{85} +(-2.21828e7 - 1.28072e7i) q^{86} +(-1.27279e8 - 2.20453e8i) q^{88} -6.33977e7i q^{89} +1.65394e8 q^{91} +(2.44575e7 - 1.41205e7i) q^{92} +(4.67870e7 - 8.10375e7i) q^{94} +(1.25522e7 + 7.24700e6i) q^{95} +(3.26981e7 + 5.66348e7i) q^{97} -1.70841e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2048 q^{4} - 3692 q^{7} + 21504 q^{10} - 63860 q^{13} - 95116 q^{16} + 370216 q^{19} + 691980 q^{22} + 541712 q^{25} - 1994264 q^{28} - 571136 q^{31} - 1027656 q^{34} + 8708536 q^{37} + 2973768 q^{40}+ \cdots + 133878688 q^{97}+O(q^{100}) \) 32 * q + 2048 * q^4 - 3692 * q^7 + 21504 * q^10 - 63860 * q^13 - 95116 * q^16 + 370216 * q^19 + 691980 * q^22 + 541712 * q^25 - 1994264 * q^28 - 571136 * q^31 - 1027656 * q^34 + 8708536 * q^37 + 2973768 * q^40 - 5453084 * q^43 - 8468088 * q^46 - 14602560 * q^49 + 25384960 * q^52 - 28905144 * q^55 - 1213068 * q^58 - 31037996 * q^61 + 48506000 * q^64 + 62356828 * q^67 - 11075124 * q^70 - 42547328 * q^73 - 69593876 * q^76 + 151045036 * q^79 - 80402280 * q^82 - 160995564 * q^85 - 366976068 * q^88 + 747361592 * q^91 + 544741584 * q^94 + 133878688 * q^97

Introduction

L-functions

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