LMFDB - Embedded newform 1386.2.r.c.1277.1

Newspace parameters

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.r (of order $6$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
Defining polynomial:	$x^{8} - x^{4} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1277.1
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1386.1277
Dual form			1386.2.r.c.89.1

$q$-expansion

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.0340742 + 0.0590182i) q^{5} +(-2.19067 + 1.48356i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.0340742 + 0.0590182i) q^{5} +(-2.19067 + 1.48356i) q^{7} +1.00000i q^{8} +(-0.0590182 - 0.0340742i) q^{10} +(-0.866025 - 0.500000i) q^{11} -2.44949i q^{13} +(1.15539 - 2.38014i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.817837 + 1.41654i) q^{17} +(6.28497 - 3.62863i) q^{19} +0.0681483 q^{20} +1.00000 q^{22} +(-0.405083 + 0.233875i) q^{23} +(2.49768 - 4.32611i) q^{25} +(1.22474 + 2.12132i) q^{26} +(0.189469 + 2.63896i) q^{28} +7.23143i q^{29} +(-0.201501 - 0.116337i) q^{31} +(0.866025 + 0.500000i) q^{32} -1.63567i q^{34} +(-0.162203 - 0.0787382i) q^{35} +(2.74496 + 4.75442i) q^{37} +(-3.62863 + 6.28497i) q^{38} +(-0.0590182 + 0.0340742i) q^{40} -5.38891 q^{41} +9.64100 q^{43} +(-0.866025 + 0.500000i) q^{44} +(0.233875 - 0.405083i) q^{46} +(4.31199 + 7.46859i) q^{47} +(2.59808 - 6.50000i) q^{49} +4.99536i q^{50} +(-2.12132 - 1.22474i) q^{52} +(7.34278 + 4.23936i) q^{53} -0.0681483i q^{55} +(-1.48356 - 2.19067i) q^{56} +(-3.61571 - 6.26260i) q^{58} +(-0.439158 + 0.760643i) q^{59} +(-8.63223 + 4.98382i) q^{61} +0.232673 q^{62} -1.00000 q^{64} +(0.144564 - 0.0834643i) q^{65} +(1.52993 - 2.64991i) q^{67} +(0.817837 + 1.41654i) q^{68} +(0.179841 - 0.0129120i) q^{70} -15.8771i q^{71} +(6.76696 + 3.90691i) q^{73} +(-4.75442 - 2.74496i) q^{74} -7.25725i q^{76} +(2.63896 - 0.189469i) q^{77} +(5.76612 + 9.98722i) q^{79} +(0.0340742 - 0.0590182i) q^{80} +(4.66693 - 2.69445i) q^{82} +17.1270 q^{83} -0.111469 q^{85} +(-8.34935 + 4.82050i) q^{86} +(0.500000 - 0.866025i) q^{88} +(5.33814 + 9.24592i) q^{89} +(3.63397 + 5.36603i) q^{91} +0.467750i q^{92} +(-7.46859 - 4.31199i) q^{94} +(0.428310 + 0.247285i) q^{95} -9.99876i q^{97} +(1.00000 + 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8q + 4q^{4} + 8q^{5} + O(q^{10}) $	$ 8q + 4q^{4} + 8q^{5} - 4q^{16} - 4q^{17} + 24q^{19} + 16q^{20} + 8q^{22} + 24q^{23} - 4q^{25} + 12q^{31} + 16q^{35} + 12q^{37} - 8q^{38} + 16q^{41} - 32q^{43} + 8q^{46} + 48q^{53} - 4q^{58} + 16q^{59} + 24q^{62} - 8q^{64} - 12q^{65} - 24q^{67} + 4q^{68} - 20q^{70} - 24q^{73} - 12q^{74} + 40q^{79} + 8q^{80} + 12q^{82} + 72q^{83} - 32q^{85} - 24q^{86} + 4q^{88} - 16q^{89} + 36q^{91} + 24q^{95} + 8q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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 $

$2$	−0.866025	+	0.500000i	−0.612372	+	0.353553i
$2$	−0.866025	+	0.500000i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	0.0340742	+	0.0590182i	0.0152384	+	0.0263937i	0.873544	−	0.486745i	$-0.161816\pi$
$5$	0.0340742	+	0.0590182i	0.0152384	+	0.0263937i	−0.858306	+	0.513139i	$0.828483\pi$
$6$		0			0
$7$	−2.19067	+	1.48356i	−0.827996	+	0.560734i
$7$	−2.19067	+	1.48356i	−0.827996	+	0.560734i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−0.0590182	−	0.0340742i	−0.0186632	−	0.0107752i
$11$	−0.866025	−	0.500000i	−0.261116	−	0.150756i
$11$	−0.866025	−	0.500000i	−0.261116	−	0.150756i
$12$		0			0
$13$		−	2.44949i		−	0.679366i	−0.940540	−	0.339683i	$-0.889680\pi$
$13$		−	2.44949i		−	0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$14$	1.15539	−	2.38014i	0.308792	−	0.636119i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−0.817837	+	1.41654i	−0.198355	+	0.343560i	−0.947995	−	0.318285i	$-0.896893\pi$
$17$	−0.817837	+	1.41654i	−0.198355	+	0.343560i	0.749640	+	0.661845i	$0.230226\pi$
$18$		0			0
$19$	6.28497	−	3.62863i	1.44187	−	0.832464i	0.443895	−	0.896079i	$-0.353596\pi$
$19$	6.28497	−	3.62863i	1.44187	−	0.832464i	0.997975	+	0.0636147i	$0.0202629\pi$
$20$	0.0681483			0.0152384
$21$		0			0
$22$	1.00000			0.213201
$23$	−0.405083	+	0.233875i	−0.0844657	+	0.0487663i	−0.541638	−	0.840612i	$-0.682196\pi$
$23$	−0.405083	+	0.233875i	−0.0844657	+	0.0487663i	0.457172	+	0.889378i	$0.348862\pi$
$24$		0			0
$25$	2.49768	−	4.32611i	0.499536	−	0.865221i
$26$	1.22474	+	2.12132i	0.240192	+	0.416025i
$27$		0			0
$28$	0.189469	+	2.63896i	0.0358062	+	0.498716i
$29$			7.23143i			1.34284i	0.741076	+	0.671421i	$0.234316\pi$
$29$			7.23143i			1.34284i	−0.741076	+	0.671421i	$0.765684\pi$
$30$		0			0
$31$	−0.201501	−	0.116337i	−0.0361906	−	0.0208947i	0.481796	−	0.876284i	$-0.339985\pi$
$31$	−0.201501	−	0.116337i	−0.0361906	−	0.0208947i	−0.517986	+	0.855389i	$0.673318\pi$
$32$	0.866025	+	0.500000i	0.153093	+	0.0883883i
$33$		0			0
$34$		−	1.63567i		−	0.280516i
$35$	−0.162203	−	0.0787382i	−0.0274172	−	0.0133092i
$36$		0			0
$37$	2.74496	+	4.75442i	0.451269	+	0.781621i	0.998465	−	0.0553835i	$-0.0176381\pi$
$37$	2.74496	+	4.75442i	0.451269	+	0.781621i	−0.547196	+	0.837004i	$0.684305\pi$
$38$	−3.62863	+	6.28497i	−0.588641	+	1.01956i
$39$		0			0
$40$	−0.0590182	+	0.0340742i	−0.00933160	+	0.00538760i
$41$	−5.38891			−0.841606			−0.420803	−	0.907152i	$-0.638252\pi$
$41$	−5.38891			−0.841606			−0.420803	+	0.907152i	$0.638252\pi$
$42$		0			0
$43$	9.64100			1.47024			0.735119	−	0.677938i	$-0.237126\pi$
$43$	9.64100			1.47024			0.735119	+	0.677938i	$0.237126\pi$
$44$	−0.866025	+	0.500000i	−0.130558	+	0.0753778i
$45$		0			0
$46$	0.233875	−	0.405083i	0.0344830	−	0.0597263i
$47$	4.31199	+	7.46859i	0.628969	+	1.08941i	0.987759	+	0.155987i	$0.0498559\pi$
$47$	4.31199	+	7.46859i	0.628969	+	1.08941i	−0.358791	+	0.933418i	$0.616811\pi$
$48$		0			0
$49$	2.59808	−	6.50000i	0.371154	−	0.928571i
$50$			4.99536i			0.706450i
$51$		0			0
$52$	−2.12132	−	1.22474i	−0.294174	−	0.169842i
$53$	7.34278	+	4.23936i	1.00861	+	0.582320i	0.910784	−	0.412883i	$-0.135478\pi$
$53$	7.34278	+	4.23936i	1.00861	+	0.582320i	0.0978246	+	0.995204i	$0.468812\pi$
$54$		0			0
$55$		−	0.0681483i		−	0.00918912i
$56$	−1.48356	−	2.19067i	−0.198250	−	0.292741i
$57$		0			0
$58$	−3.61571	−	6.26260i	−0.474767	−	0.822320i
$59$	−0.439158	+	0.760643i	−0.0571734	+	0.0990273i	−0.893196	−	0.449668i	$-0.851542\pi$
$59$	−0.439158	+	0.760643i	−0.0571734	+	0.0990273i	0.836022	+	0.548696i	$0.184875\pi$
$60$		0			0
$61$	−8.63223	+	4.98382i	−1.10524	+	0.638113i	−0.937593	−	0.347733i	$-0.886951\pi$
$61$	−8.63223	+	4.98382i	−1.10524	+	0.638113i	−0.167651	+	0.985846i	$0.553618\pi$
$62$	0.232673			0.0295495
$63$		0			0
$64$	−1.00000			−0.125000
$65$	0.144564	−	0.0834643i	0.0179310	−	0.0103525i
$66$		0			0
$67$	1.52993	−	2.64991i	0.186910	−	0.323738i	−0.757308	−	0.653058i	$-0.773486\pi$
$67$	1.52993	−	2.64991i	0.186910	−	0.323738i	0.944219	+	0.329319i	$0.106819\pi$
$68$	0.817837	+	1.41654i	0.0991773	+	0.171780i
$69$		0			0
$70$	0.179841	−	0.0129120i	0.0214951	−	0.00154328i
$71$		−	15.8771i		−	1.88426i	−0.335245	−	0.942131i	$-0.608819\pi$
$71$		−	15.8771i		−	1.88426i	0.335245	−	0.942131i	$-0.391181\pi$
$72$		0			0
$73$	6.76696	+	3.90691i	0.792013	+	0.457269i	0.840671	−	0.541547i	$-0.182161\pi$
$73$	6.76696	+	3.90691i	0.792013	+	0.457269i	−0.0486577	+	0.998816i	$0.515494\pi$
$74$	−4.75442	−	2.74496i	−0.552690	−	0.319095i
$75$		0			0
$76$		−	7.25725i		−	0.832464i
$77$	2.63896	−	0.189469i	0.300737	−	0.0215920i
$78$		0			0
$79$	5.76612	+	9.98722i	0.648740	+	1.12365i	0.983424	+	0.181320i	$0.0580370\pi$
$79$	5.76612	+	9.98722i	0.648740	+	1.12365i	−0.334684	+	0.942330i	$0.608630\pi$
$80$	0.0340742	−	0.0590182i	0.00380961	−	0.00659844i
$81$		0			0
$82$	4.66693	−	2.69445i	0.515376	−	0.297553i
$83$	17.1270			1.87993			0.939967	−	0.341266i	$-0.110856\pi$
$83$	17.1270			1.87993			0.939967	+	0.341266i	$0.110856\pi$
$84$		0			0
$85$	−0.111469			−0.0120905
$86$	−8.34935	+	4.82050i	−0.900333	+	0.519808i
$87$		0			0
$88$	0.500000	−	0.866025i	0.0533002	−	0.0923186i
$89$	5.33814	+	9.24592i	0.565841	+	0.980066i	0.996971	+	0.0777760i	$0.0247819\pi$
$89$	5.33814	+	9.24592i	0.565841	+	0.980066i	−0.431129	+	0.902290i	$0.641885\pi$
$90$		0			0
$91$	3.63397	+	5.36603i	0.380944	+	0.562512i
$92$			0.467750i			0.0487663i
$93$		0			0
$94$	−7.46859	−	4.31199i	−0.770326	−	0.444748i
$95$	0.428310	+	0.247285i	0.0439437	+	0.0253709i
$96$		0			0
$97$		−	9.99876i		−	1.01522i	−0.861587	−	0.507610i	$-0.830529\pi$
$97$		−	9.99876i		−	1.01522i	0.861587	−	0.507610i	$-0.169471\pi$
$98$	1.00000	+	6.92820i	0.101015	+	0.699854i
$99$		0			0
$100$	−2.49768	−	4.32611i	−0.249768	−	0.432611i
$101$	1.76028	−	3.04889i	0.175154	−	0.303376i	−0.765060	−	0.643959i	$-0.777291\pi$
$101$	1.76028	−	3.04889i	0.175154	−	0.303376i	0.940215	+	0.340582i	$0.110624\pi$
$102$		0			0
$103$	13.8471	−	7.99465i	1.36440	−	0.787736i	0.374194	−	0.927351i	$-0.377920\pi$
$103$	13.8471	−	7.99465i	1.36440	−	0.787736i	0.990206	+	0.139614i	$0.0445862\pi$
$104$	2.44949			0.240192
$105$		0			0
$106$	−8.47871			−0.823525
$107$	−5.25201	+	3.03225i	−0.507731	+	0.293139i	−0.731901	−	0.681412i	$-0.761366\pi$
$107$	−5.25201	+	3.03225i	−0.507731	+	0.293139i	0.224169	+	0.974550i	$0.428033\pi$
$108$		0			0
$109$	−9.00877	+	15.6036i	−0.862883	+	1.49456i	0.00625046	+	0.999980i	$0.498010\pi$
$109$	−9.00877	+	15.6036i	−0.862883	+	1.49456i	−0.869134	+	0.494577i	$0.835323\pi$
$110$	0.0340742	+	0.0590182i	0.00324884	+	0.00562716i
$111$		0			0
$112$	2.38014	+	1.15539i	0.224902	+	0.109175i
$113$		−	7.05521i		−	0.663698i	−0.943332	−	0.331849i	$-0.892327\pi$
$113$		−	7.05521i		−	0.663698i	0.943332	−	0.331849i	$-0.107673\pi$
$114$		0			0
$115$	−0.0276058	−	0.0159382i	−0.00257425	−	0.00148624i
$116$	6.26260	+	3.61571i	0.581468	+	0.335711i
$117$		0			0
$118$		−	0.878315i		−	0.0808555i
$119$	−0.309909	−	4.31648i	−0.0284093	−	0.395691i
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$	4.98382	−	8.63223i	0.451214	−	0.781526i
$123$		0			0
$124$	−0.201501	+	0.116337i	−0.0180953	+	0.0104473i
$125$	0.681167			0.0609254
$126$		0			0
$127$	6.83939			0.606898			0.303449	−	0.952848i	$-0.401862\pi$
$127$	6.83939			0.606898			0.303449	+	0.952848i	$0.401862\pi$
$128$	0.866025	−	0.500000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	−0.0834643	+	0.144564i	−0.00732031	+	0.0126791i
$131$	0.757359	+	1.31178i	0.0661708	+	0.114611i	0.897213	−	0.441599i	$-0.145588\pi$
$131$	0.757359	+	1.31178i	0.0661708	+	0.114611i	−0.831042	+	0.556210i	$0.812255\pi$
$132$		0			0
$133$	−8.38499	+	17.2733i	−0.727071	+	1.49778i
$134$			3.05986i			0.264331i
$135$		0			0
$136$	−1.41654	−	0.817837i	−0.121467	−	0.0701290i
$137$	4.27427	+	2.46775i	0.365175	+	0.210834i	0.671349	−	0.741142i	$-0.265715\pi$
$137$	4.27427	+	2.46775i	0.365175	+	0.210834i	−0.306173	+	0.951976i	$0.599049\pi$
$138$		0			0
$139$		−	1.10583i		−	0.0937952i	−0.998900	−	0.0468976i	$-0.985067\pi$
$139$		−	1.10583i		−	0.0937952i	0.998900	−	0.0468976i	$-0.0149334\pi$
$140$	−0.149291	+	0.101102i	−0.0126174	+	0.00854471i
$141$		0			0
$142$	7.93854	+	13.7499i	0.666187	+	1.15387i
$143$	−1.22474	+	2.12132i	−0.102418	+	0.177394i
$144$		0			0
$145$	−0.426786	+	0.246405i	−0.0354426	+	0.0204628i
$146$	−7.81382			−0.646676
$147$		0			0
$148$	5.48993			0.451269
$149$	−0.275702	+	0.159176i	−0.0225863	+	0.0130402i	−0.511251	−	0.859432i	$-0.670818\pi$
$149$	−0.275702	+	0.159176i	−0.0225863	+	0.0130402i	0.488664	+	0.872472i	$0.337484\pi$
$150$		0			0
$151$	−1.76941	+	3.06471i	−0.143992	+	0.249402i	−0.928997	−	0.370088i	$-0.879327\pi$
$151$	−1.76941	+	3.06471i	−0.143992	+	0.249402i	0.785004	+	0.619491i	$0.212661\pi$
$152$	3.62863	+	6.28497i	0.294320	+	0.509778i
$153$		0			0
$154$	−2.19067	+	1.48356i	−0.176529	+	0.119549i
$155$		−	0.0158563i		−	0.00127361i
$156$		0			0
$157$	−5.64589	−	3.25966i	−0.450591	−	0.260149i	0.257489	−	0.966281i	$-0.417105\pi$
$157$	−5.64589	−	3.25966i	−0.450591	−	0.260149i	−0.708080	+	0.706132i	$0.750438\pi$
$158$	−9.98722	−	5.76612i	−0.794541	−	0.458728i
$159$		0			0
$160$			0.0681483i			0.00538760i
$161$	0.540436	−	1.11331i	0.0425923	−	0.0877411i
$162$		0			0
$163$	0.553536	+	0.958753i	0.0433563	+	0.0750954i	0.886889	−	0.461982i	$-0.152862\pi$
$163$	0.553536	+	0.958753i	0.0433563	+	0.0750954i	−0.843533	+	0.537078i	$0.819528\pi$
$164$	−2.69445	+	4.66693i	−0.210401	+	0.364426i
$165$		0			0
$166$	−14.8324	+	8.56350i	−1.15122	+	0.664657i
$167$	9.93426			0.768736			0.384368	−	0.923180i	$-0.374419\pi$
$167$	9.93426			0.768736			0.384368	+	0.923180i	$0.374419\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$	0.0965346	−	0.0557343i	0.00740386	−	0.00427462i
$171$		0			0
$172$	4.82050	−	8.34935i	0.367560	−	0.636632i
$173$	6.61107	+	11.4507i	0.502630	+	0.870581i	0.999995	+	0.00303994i	$0.000967644\pi$
$173$	6.61107	+	11.4507i	0.502630	+	0.870581i	−0.497365	+	0.867541i	$0.665699\pi$
$174$		0			0
$175$	0.946464	+	13.1825i	0.0715459	+	0.996506i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$	−9.24592	−	5.33814i	−0.693011	−	0.400110i
$179$	−9.79227	−	5.65357i	−0.731909	−	0.422568i	0.0872113	−	0.996190i	$-0.472204\pi$
$179$	−9.79227	−	5.65357i	−0.731909	−	0.422568i	−0.819120	+	0.573622i	$0.805538\pi$
$180$		0			0
$181$		−	8.26670i		−	0.614459i	−0.951635	−	0.307230i	$-0.900598\pi$
$181$		−	8.26670i		−	0.614459i	0.951635	−	0.307230i	$-0.0994020\pi$
$182$	−5.83013	−	2.83013i	−0.432158	−	0.209783i
$183$		0			0
$184$	−0.233875	−	0.405083i	−0.0172415	−	0.0298631i
$185$	−0.187065	+	0.324006i	−0.0137533	+	0.0238214i
$186$		0			0
$187$	1.41654	−	0.817837i	0.103587	−	0.0598062i
$188$	8.62398			0.628969
$189$		0			0
$190$	−0.494570			−0.0358799
$191$	−10.3133	+	5.95439i	−0.746245	+	0.430845i	−0.824335	−	0.566102i	$-0.808451\pi$
$191$	−10.3133	+	5.95439i	−0.746245	+	0.430845i	0.0780906	+	0.996946i	$0.475118\pi$
$192$		0			0
$193$	−12.7134	+	22.0203i	−0.915132	+	1.58506i	−0.108425	+	0.994105i	$0.534581\pi$
$193$	−12.7134	+	22.0203i	−0.915132	+	1.58506i	−0.806707	+	0.590951i	$0.798753\pi$
$194$	4.99938	+	8.65918i	0.358934	+	0.621693i
$195$		0			0
$196$	−4.33013	−	5.50000i	−0.309295	−	0.392857i
$197$			8.37821i			0.596923i	0.954422	+	0.298461i	$0.0964734\pi$
$197$			8.37821i			0.596923i	−0.954422	+	0.298461i	$0.903527\pi$
$198$		0			0
$199$	17.3736	+	10.0306i	1.23158	+	0.711053i	0.967359	−	0.253410i	$-0.0815521\pi$
$199$	17.3736	+	10.0306i	1.23158	+	0.711053i	0.264220	+	0.964462i	$0.414885\pi$
$200$	4.32611	+	2.49768i	0.305902	+	0.176612i
$201$		0			0
$202$			3.52056i			0.247706i
$203$	−10.7283	−	15.8417i	−0.752978	−	1.11187i
$204$		0			0
$205$	−0.183622	−	0.318043i	−0.0128248	−	0.0222131i
$206$	−7.99465	+	13.8471i	−0.557014	+	0.964776i
$207$		0			0
$208$	−2.12132	+	1.22474i	−0.147087	+	0.0849208i
$209$	−7.25725			−0.501995
$210$		0			0
$211$	2.96399			0.204050			0.102025	−	0.994782i	$-0.467468\pi$
$211$	2.96399			0.204050			0.102025	+	0.994782i	$0.467468\pi$
$212$	7.34278	−	4.23936i	0.504304	−	0.291160i
$213$		0			0
$214$	3.03225	−	5.25201i	0.207280	−	0.359020i
$215$	0.328509	+	0.568994i	0.0224041	+	0.0388051i
$216$		0			0
$217$	0.614014	−	0.0440843i	0.0416820	−	0.00299263i
$218$		−	18.0175i		−	1.22030i
$219$		0			0
$220$	−0.0590182	−	0.0340742i	−0.00397901	−	0.00229728i
$221$	3.46979	+	2.00328i	0.233403	+	0.134755i
$222$		0			0
$223$		−	13.8238i		−	0.925709i	−0.886434	−	0.462854i	$-0.846825\pi$
$223$		−	13.8238i		−	0.925709i	0.886434	−	0.462854i	$-0.153175\pi$
$224$	−2.63896	+	0.189469i	−0.176323	+	0.0126594i
$225$		0			0
$226$	3.52761	+	6.10999i	0.234653	+	0.406431i
$227$	11.1257	−	19.2702i	0.738437	−	1.27901i	−0.214761	−	0.976667i	$-0.568897\pi$
$227$	11.1257	−	19.2702i	0.738437	−	1.27901i	0.953199	−	0.302345i	$-0.0977693\pi$
$228$		0			0
$229$	12.3195	−	7.11269i	0.814098	−	0.470020i	−0.0342790	−	0.999412i	$-0.510913\pi$
$229$	12.3195	−	7.11269i	0.814098	−	0.470020i	0.848377	+	0.529393i	$0.177580\pi$
$230$	0.0318764			0.00210187
$231$		0			0
$232$	−7.23143			−0.474767
$233$	0.445759	−	0.257359i	0.0292027	−	0.0168602i	−0.485328	−	0.874332i	$-0.661300\pi$
$233$	0.445759	−	0.257359i	0.0292027	−	0.0168602i	0.514530	+	0.857472i	$0.327966\pi$
$234$		0			0
$235$	−0.293855	+	0.508972i	−0.0191690	+	0.0332017i
$236$	0.439158	+	0.760643i	0.0285867	+	0.0495137i
$237$		0			0
$238$	2.42663	+	3.58322i	0.157295	+	0.232266i
$239$		−	21.8238i		−	1.41166i	−0.708380	−	0.705832i	$-0.750574\pi$
$239$		−	21.8238i		−	1.41166i	0.708380	−	0.705832i	$-0.249426\pi$
$240$		0			0
$241$	−7.95439	−	4.59247i	−0.512387	−	0.295827i	0.221427	−	0.975177i	$-0.428929\pi$
$241$	−7.95439	−	4.59247i	−0.512387	−	0.295827i	−0.733814	+	0.679350i	$0.762262\pi$
$242$	−0.866025	−	0.500000i	−0.0556702	−	0.0321412i
$243$		0			0
$244$			9.96764i			0.638113i
$245$	0.472146	−	0.0681483i	0.0301643	−	0.00435384i
$246$		0			0
$247$	−8.88828	−	15.3950i	−0.565548	−	0.979558i
$248$	0.116337	−	0.201501i	0.00738738	−	0.0127953i
$249$		0			0
$250$	−0.589908	+	0.340583i	−0.0373091	+	0.0215404i
$251$	−31.3044			−1.97592			−0.987959	−	0.154718i	$-0.950553\pi$
$251$	−31.3044			−1.97592			−0.987959	+	0.154718i	$0.950553\pi$
$252$		0			0
$253$	0.467750			0.0294072
$254$	−5.92309	+	3.41970i	−0.371647	+	0.214571i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	2.01033	+	3.48200i	0.125401	+	0.217201i	0.921890	−	0.387453i	$-0.126645\pi$
$257$	2.01033	+	3.48200i	0.125401	+	0.217201i	−0.796489	+	0.604654i	$0.793312\pi$
$258$		0			0
$259$	−13.0668	−	6.34303i	−0.811931	−	0.394137i
$260$		−	0.166929i		−	0.0103525i
$261$		0			0
$262$	−1.31178	−	0.757359i	−0.0810423	−	0.0467898i
$263$	11.6746	+	6.74032i	0.719885	+	0.415626i	0.814710	−	0.579868i	$-0.196896\pi$
$263$	11.6746	+	6.74032i	0.719885	+	0.415626i	−0.0948253	+	0.995494i	$0.530229\pi$
$264$		0			0
$265$			0.577810i			0.0354946i
$266$	−1.37502	−	19.1516i	−0.0843080	−	1.17426i
$267$		0			0
$268$	−1.52993	−	2.64991i	−0.0934552	−	0.161869i
$269$	15.7399	−	27.2624i	0.959681	−	1.66222i	0.236408	−	0.971654i	$-0.424030\pi$
$269$	15.7399	−	27.2624i	0.959681	−	1.66222i	0.723273	−	0.690562i	$-0.242637\pi$
$270$		0			0
$271$	−4.89157	+	2.82415i	−0.297142	+	0.171555i	−0.641158	−	0.767409i	$-0.721546\pi$
$271$	−4.89157	+	2.82415i	−0.297142	+	0.171555i	0.344017	+	0.938964i	$0.388212\pi$
$272$	1.63567			0.0991773
$273$		0			0
$274$	−4.93550			−0.298164
$275$	−4.32611	+	2.49768i	−0.260874	+	0.150616i
$276$		0			0
$277$	−3.45982	+	5.99259i	−0.207881	+	0.360060i	−0.951047	−	0.309047i	$-0.899990\pi$
$277$	−3.45982	+	5.99259i	−0.207881	+	0.360060i	0.743166	+	0.669107i	$0.233323\pi$
$278$	0.552914	+	0.957676i	0.0331616	+	0.0574376i
$279$		0			0
$280$	0.0787382	−	0.162203i	0.00470551	−	0.00969346i
$281$		−	30.4582i		−	1.81698i	−0.417902	−	0.908492i	$-0.637234\pi$
$281$		−	30.4582i		−	1.81698i	0.417902	−	0.908492i	$-0.362766\pi$
$282$		0			0
$283$	−22.8490	−	13.1919i	−1.35823	−	0.784175i	−0.368846	−	0.929490i	$-0.620247\pi$
$283$	−22.8490	−	13.1919i	−1.35823	−	0.784175i	−0.989385	+	0.145315i	$0.953580\pi$
$284$	−13.7499	−	7.93854i	−0.815909	−	0.471065i
$285$		0			0
$286$		−	2.44949i		−	0.144841i
$287$	11.8053	−	7.99479i	0.696846	−	0.471917i
$288$		0			0
$289$	7.16228	+	12.4054i	0.421311	+	0.729732i
$290$	0.246405	−	0.426786i	0.0144694	−	0.0250617i
$291$		0			0
$292$	6.76696	−	3.90691i	0.396007	−	0.228634i
$293$	−15.0095			−0.876862			−0.438431	−	0.898765i	$-0.644466\pi$
$293$	−15.0095			−0.876862			−0.438431	+	0.898765i	$0.644466\pi$
$294$		0			0
$295$	−0.0598557			−0.00348494
$296$	−4.75442	+	2.74496i	−0.276345	+	0.159548i
$297$		0			0
$298$	0.159176	−	0.275702i	0.00922084	−	0.0159710i
$299$	0.572874	+	0.992248i	0.0331302	+	0.0573832i
$300$		0			0
$301$	−21.1203	+	14.3030i	−1.21735	+	0.824413i
$302$		−	3.53882i		−	0.203636i
$303$		0			0
$304$	−6.28497	−	3.62863i	−0.360467	−	0.208116i
$305$	−0.588272	−	0.339639i	−0.0336844	−	0.0194477i
$306$		0			0
$307$			15.0711i			0.860151i	0.902793	+	0.430076i	$0.141513\pi$
$307$			15.0711i			0.860151i	−0.902793	+	0.430076i	$0.858487\pi$
$308$	1.15539	−	2.38014i	0.0658347	−	0.135621i
$309$		0			0
$310$	0.00792814	+	0.0137319i	0.000450288	+	0.000779922i
$311$	9.93645	−	17.2104i	0.563445	−	0.975915i	−0.433748	−	0.901034i	$-0.642809\pi$
$311$	9.93645	−	17.2104i	0.563445	−	0.975915i	0.997193	−	0.0748804i	$-0.0238575\pi$
$312$		0			0
$313$	−7.81257	+	4.51059i	−0.441593	+	0.254954i	−0.704273	−	0.709929i	$-0.748727\pi$
$313$	−7.81257	+	4.51059i	−0.441593	+	0.254954i	0.262680	+	0.964883i	$0.415394\pi$
$314$	6.51931			0.367906
$315$		0			0
$316$	11.5322			0.648740
$317$	−10.2958	+	5.94427i	−0.578268	+	0.333863i	−0.760445	−	0.649403i	$-0.775019\pi$
$317$	−10.2958	+	5.94427i	−0.578268	+	0.333863i	0.182177	+	0.983266i	$0.441686\pi$
$318$		0			0
$319$	3.61571	−	6.26260i	0.202441	−	0.350638i
$320$	−0.0340742	−	0.0590182i	−0.00190480	−	0.00329922i
$321$		0			0
$322$	0.0886240	+	1.23437i	0.00493882	+	0.0687889i
$323$			11.8705i			0.660492i
$324$		0			0
$325$	−10.5967	−	6.11804i	−0.587802	−	0.339368i
$326$	−0.958753	−	0.553536i	−0.0531004	−	0.0306575i
$327$		0			0
$328$		−	5.38891i		−	0.297553i
$329$	−20.5263	−	9.96410i	−1.13165	−	0.549339i
$330$		0			0
$331$	−11.5175	−	19.9489i	−0.633061	−	1.09649i	−0.986922	−	0.161196i	$-0.948465\pi$
$331$	−11.5175	−	19.9489i	−0.633061	−	1.09649i	0.353862	−	0.935298i	$-0.384868\pi$
$332$	8.56350	−	14.8324i	0.469983	−	0.814035i
$333$		0			0
$334$	−8.60332	+	4.96713i	−0.470753	+	0.271789i
$335$	0.208524			0.0113929
$336$		0			0
$337$	−20.8191			−1.13409			−0.567045	−	0.823687i	$-0.691914\pi$
$337$	−20.8191			−1.13409			−0.567045	+	0.823687i	$0.691914\pi$
$338$	−6.06218	+	3.50000i	−0.329739	+	0.190375i
$339$		0			0
$340$	−0.0557343	+	0.0965346i	−0.00302261	+	0.00523532i
$341$	0.116337	+	0.201501i	0.00629998	+	0.0109119i
$342$		0			0
$343$	3.95164	+	18.0938i	0.213368	+	0.976972i
$344$			9.64100i			0.519808i
$345$		0			0
$346$	−11.4507	−	6.61107i	−0.615594	−	0.355413i
$347$	6.30751	+	3.64165i	0.338605	+	0.195494i	0.659655	−	0.751569i	$-0.270702\pi$
$347$	6.30751	+	3.64165i	0.338605	+	0.195494i	−0.321050	+	0.947062i	$0.604036\pi$
$348$		0			0
$349$		−	33.7366i		−	1.80588i	−0.429767	−	0.902940i	$-0.641404\pi$
$349$		−	33.7366i		−	1.80588i	0.429767	−	0.902940i	$-0.358596\pi$
$350$	−7.41093	−	10.9432i	−0.396131	−	0.584938i
$351$		0			0
$352$	−0.500000	−	0.866025i	−0.0266501	−	0.0461593i
$353$	8.71467	−	15.0943i	0.463835	−	0.803386i	−0.535313	−	0.844654i	$-0.679806\pi$
$353$	8.71467	−	15.0943i	0.463835	−	0.803386i	0.999148	+	0.0412678i	$0.0131397\pi$
$354$		0			0
$355$	0.937036	−	0.540998i	0.0497327	−	0.0287132i
$356$	10.6763			0.565841
$357$		0			0
$358$	11.3071			0.597601
$359$	−22.6722	+	13.0898i	−1.19659	+	0.690852i	−0.959794	−	0.280707i	$-0.909431\pi$
$359$	−22.6722	+	13.0898i	−1.19659	+	0.690852i	−0.236798	+	0.971559i	$0.576098\pi$
$360$		0			0
$361$	16.8339	−	29.1571i	0.885992	−	1.53458i
$362$	4.13335	+	7.15918i	0.217244	+	0.376278i
$363$		0			0
$364$	6.46410	−	0.464102i	0.338811	−	0.0243255i
$365$			0.532499i			0.0278723i
$366$		0			0
$367$	9.91510	+	5.72449i	0.517564	+	0.298816i	0.735937	−	0.677050i	$-0.236742\pi$
$367$	9.91510	+	5.72449i	0.517564	+	0.298816i	−0.218373	+	0.975865i	$0.570075\pi$
$368$	0.405083	+	0.233875i	0.0211164	+	0.0121916i
$369$		0			0
$370$		−	0.374129i		−	0.0194501i
$371$	−22.3750	+	1.60645i	−1.16165	+	0.0834028i
$372$		0			0
$373$	14.1189	+	24.4546i	0.731048	+	1.26621i	0.956436	+	0.291943i	$0.0943016\pi$
$373$	14.1189	+	24.4546i	0.731048	+	1.26621i	−0.225388	+	0.974269i	$0.572365\pi$
$374$	−0.817837	+	1.41654i	−0.0422894	+	0.0732473i
$375$		0			0
$376$	−7.46859	+	4.31199i	−0.385163	+	0.222374i
$377$	17.7133			0.912282
$378$		0			0
$379$	9.93550			0.510352			0.255176	−	0.966895i	$-0.417867\pi$
$379$	9.93550			0.510352			0.255176	+	0.966895i	$0.417867\pi$
$380$	0.428310	−	0.247285i	0.0219718	−	0.0126854i
$381$		0			0
$382$	5.95439	−	10.3133i	0.304653	−	0.527675i
$383$	−6.97381	−	12.0790i	−0.356345	−	0.617208i	0.631002	−	0.775781i	$-0.282644\pi$
$383$	−6.97381	−	12.0790i	−0.356345	−	0.617208i	−0.987347	+	0.158573i	$0.949311\pi$
$384$		0			0
$385$	0.101102	+	0.149291i	0.00515266	+	0.00760855i
$386$		−	25.4269i		−	1.29419i
$387$		0			0
$388$	−8.65918	−	4.99938i	−0.439603	−	0.253805i
$389$	1.53141	+	0.884161i	0.0776457	+	0.0448288i	0.538320	−	0.842740i	$-0.319059\pi$
$389$	1.53141	+	0.884161i	0.0776457	+	0.0448288i	−0.460674	+	0.887569i	$0.652392\pi$
$390$		0			0
$391$		−	0.765087i		−	0.0386921i
$392$	6.50000	+	2.59808i	0.328300	+	0.131223i
$393$		0			0
$394$	−4.18910	−	7.25574i	−0.211044	−	0.365539i
$395$	−0.392952	+	0.680613i	−0.0197716	+	0.0342453i
$396$		0			0
$397$	−1.04114	+	0.601102i	−0.0522533	+	0.0301685i	−0.525899	−	0.850547i	$-0.676271\pi$
$397$	−1.04114	+	0.601102i	−0.0522533	+	0.0301685i	0.473646	+	0.880715i	$0.342938\pi$
$398$	−20.0613			−1.00558
$399$		0			0
$400$	−4.99536			−0.249768
$401$	−15.4934	+	8.94510i	−0.773702	+	0.446697i	−0.834194	−	0.551472i	$-0.814067\pi$
$401$	−15.4934	+	8.94510i	−0.773702	+	0.446697i	0.0604915	+	0.998169i	$0.480733\pi$
$402$		0			0
$403$	−0.284965	+	0.493574i	−0.0141951	+	0.0245867i
$404$	−1.76028	−	3.04889i	−0.0875771	−	0.151688i
$405$		0			0
$406$	17.2118	+	8.35515i	0.854208	+	0.414659i
$407$		−	5.48993i		−	0.272126i
$408$		0			0
$409$	−1.35436	−	0.781939i	−0.0669687	−	0.0386644i	0.466142	−	0.884710i	$-0.345644\pi$
$409$	−1.35436	−	0.781939i	−0.0669687	−	0.0386644i	−0.533110	+	0.846046i	$0.678977\pi$
$410$	0.318043	+	0.183622i	0.0157070	+	0.00906847i
$411$		0			0
$412$		−	15.9893i		−	0.787736i
$413$	−0.166413	−	2.31784i	−0.00818866	−	0.114053i
$414$		0			0
$415$	0.583589	+	1.01081i	0.0286472	+	0.0496185i
$416$	1.22474	−	2.12132i	0.0600481	−	0.104006i
$417$		0			0
$418$	6.28497	−	3.62863i	0.307408	−	0.177482i
$419$	4.99020			0.243787			0.121894	−	0.992543i	$-0.461103\pi$
$419$	4.99020			0.243787			0.121894	+	0.992543i	$0.461103\pi$
$420$		0			0
$421$	−37.0235			−1.80441			−0.902207	−	0.431302i	$-0.858054\pi$
$421$	−37.0235			−1.80441			−0.902207	+	0.431302i	$0.858054\pi$
$422$	−2.56689	+	1.48200i	−0.124954	+	0.0721425i
$423$		0			0
$424$	−4.23936	+	7.34278i	−0.205881	+	0.356597i
$425$	4.08539	+	7.07610i	0.198170	+	0.343241i
$426$		0			0
$427$	11.5166	−	23.7244i	0.557325	−	1.14810i
$428$			6.06450i			0.293139i
$429$		0			0
$430$	−0.568994	−	0.328509i	−0.0274393	−	0.0158421i
$431$	11.4567	+	6.61453i	0.551850	+	0.318611i	0.749868	−	0.661588i	$-0.230117\pi$
$431$	11.4567	+	6.61453i	0.551850	+	0.318611i	−0.198018	+	0.980198i	$0.563450\pi$
$432$		0			0
$433$			12.5794i			0.604527i	0.953224	+	0.302263i	$0.0977422\pi$
$433$			12.5794i			0.604527i	−0.953224	+	0.302263i	$0.902258\pi$
$434$	−0.509710	+	0.345185i	−0.0244669	+	0.0165694i
$435$		0			0
$436$	9.00877	+	15.6036i	0.431442	+	0.747279i
$437$	−1.69729	+	2.93979i	−0.0811924	+	0.140629i
$438$		0			0
$439$	−15.2358	+	8.79639i	−0.727165	+	0.419829i	−0.817384	−	0.576093i	$-0.804577\pi$
$439$	−15.2358	+	8.79639i	−0.727165	+	0.419829i	0.0902191	+	0.995922i	$0.471243\pi$
$440$	0.0681483			0.00324884
$441$		0			0
$442$	−4.00657			−0.190573
$443$	−23.4746	+	13.5531i	−1.11531	+	0.643926i	−0.940200	−	0.340623i	$-0.889362\pi$
$443$	−23.4746	+	13.5531i	−1.11531	+	0.643926i	−0.175112	+	0.984549i	$0.556029\pi$
$444$		0			0
$445$	−0.363785	+	0.630094i	−0.0172451	+	0.0298693i
$446$	6.91189	+	11.9717i	0.327288	+	0.566879i
$447$		0			0
$448$	2.19067	−	1.48356i	0.103499	−	0.0700918i
$449$			14.9950i			0.707658i	0.935310	+	0.353829i	$0.115121\pi$
$449$			14.9950i			0.707658i	−0.935310	+	0.353829i	$0.884879\pi$
$450$		0			0
$451$	4.66693	+	2.69445i	0.219757	+	0.126877i
$452$	−6.10999	−	3.52761i	−0.287390	−	0.165925i
$453$		0			0
$454$			22.2514i			1.04431i
$455$	−0.192868	+	0.397314i	−0.00904181	+	0.0186263i
$456$		0			0
$457$	0.475430	+	0.823468i	0.0222397	+	0.0385202i	0.876931	−	0.480616i	$-0.159587\pi$
$457$	0.475430	+	0.823468i	0.0222397	+	0.0385202i	−0.854691	+	0.519136i	$0.826254\pi$
$458$	−7.11269	+	12.3195i	−0.332354	+	0.575654i
$459$		0			0
$460$	−0.0276058	+	0.0159382i	−0.00128713	+	0.000743122i
$461$	17.7698			0.827621			0.413810	−	0.910363i	$-0.364198\pi$
$461$	17.7698			0.827621			0.413810	+	0.910363i	$0.364198\pi$
$462$		0			0
$463$	1.21592			0.0565088			0.0282544	−	0.999601i	$-0.491005\pi$
$463$	1.21592			0.0565088			0.0282544	+	0.999601i	$0.491005\pi$
$464$	6.26260	−	3.61571i	0.290734	−	0.167855i
$465$		0			0
$466$	−0.257359	+	0.445759i	−0.0119219	+	0.0206494i
$467$	−4.45982	−	7.72464i	−0.206376	−	0.357454i	0.744194	−	0.667963i	$-0.232834\pi$
$467$	−4.45982	−	7.72464i	−0.206376	−	0.357454i	−0.950570	+	0.310510i	$0.899500\pi$
$468$		0			0
$469$	0.579747	+	8.07483i	0.0267702	+	0.372861i
$470$		−	0.587710i		−	0.0271090i
$471$		0			0
$472$	−0.760643	−	0.439158i	−0.0350114	−	0.0202139i
$473$	−8.34935	−	4.82050i	−0.383903	−	0.221647i
$474$		0			0
$475$		−	36.2526i		−	1.66338i
$476$	−3.89313	−	1.88985i	−0.178441	−	0.0866211i
$477$		0			0
$478$	10.9119	+	18.9000i	0.499098	+	0.864464i
$479$	10.3161	−	17.8680i	0.471355	−	0.816411i	−0.528108	−	0.849177i	$-0.677098\pi$
$479$	10.3161	−	17.8680i	0.471355	−	0.816411i	0.999463	+	0.0327661i	$0.0104317\pi$
$480$		0			0
$481$	11.6459	−	6.72376i	0.531007	−	0.306577i
$482$	9.18494			0.418363
$483$		0			0
$484$	1.00000			0.0454545
$485$	0.590109	−	0.340699i	0.0267954	−	0.0154704i
$486$		0			0
$487$	−0.0189644	+	0.0328473i	−0.000859360	+	0.00148845i	−0.866455	−	0.499256i	$-0.833607\pi$
$487$	−0.0189644	+	0.0328473i	−0.000859360	+	0.00148845i	0.865595	+	0.500744i	$0.166940\pi$
$488$	−4.98382	−	8.63223i	−0.225607	−	0.390763i
$489$		0			0
$490$	−0.374816	+	0.295091i	−0.0169325	+	0.0133309i
$491$		−	34.6919i		−	1.56562i	−0.622258	−	0.782812i	$-0.713785\pi$
$491$		−	34.6919i		−	1.56562i	0.622258	−	0.782812i	$-0.286215\pi$
$492$		0			0
$493$	−10.2436	−	5.91413i	−0.461347	−	0.266359i
$494$	15.3950	+	8.88828i	0.692652	+	0.399903i
$495$		0			0
$496$			0.232673i			0.0104473i
$497$	23.5546	+	34.7814i	1.05657	+	1.56016i
$498$		0			0
$499$	10.2268	+	17.7133i	0.457814	+	0.792957i	0.998845	−	0.0480457i	$-0.0152993\pi$
$499$	10.2268	+	17.7133i	0.457814	+	0.792957i	−0.541031	+	0.841002i	$0.681966\pi$
$500$	0.340583	−	0.589908i	0.0152314	−	0.0263815i
$501$		0			0
$502$	27.1104	−	15.6522i	1.21000	−	0.698592i
$503$	−31.4660			−1.40300			−0.701501	−	0.712669i	$-0.747486\pi$
$503$	−31.4660			−1.40300			−0.701501	+	0.712669i	$0.747486\pi$
$504$		0			0
$505$	0.239920			0.0106763
$506$	−0.405083	+	0.233875i	−0.0180082	+	0.0103970i
$507$		0			0
$508$	3.41970	−	5.92309i	0.151724	−	0.262794i
$509$	−4.97798	−	8.62211i	−0.220645	−	0.382168i	0.734359	−	0.678761i	$-0.237483\pi$
$509$	−4.97798	−	8.62211i	−0.220645	−	0.382168i	−0.955004	+	0.296593i	$0.904150\pi$
$510$		0			0
$511$	−20.6203	+	1.48047i	−0.912190	+	0.0654923i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−3.48200	−	2.01033i	−0.153584	−	0.0886720i
$515$	0.943660	+	0.544822i	0.0415826	+	0.0240077i
$516$		0			0
$517$		−	8.62398i		−	0.379282i
$518$	14.4877	−	1.04017i	0.636552	−	0.0457024i
$519$		0			0
$520$	0.0834643	+	0.144564i	0.00366015	+	0.00633957i
$521$	−20.4947	+	35.4979i	−0.897891	+	1.55519i	−0.0677047	+	0.997705i	$0.521568\pi$
$521$	−20.4947	+	35.4979i	−0.897891	+	1.55519i	−0.830186	+	0.557487i	$0.811766\pi$
$522$		0			0
$523$	32.0347	−	18.4952i	1.40078	−	0.808741i	0.406307	−	0.913736i	$-0.366816\pi$
$523$	32.0347	−	18.4952i	1.40078	−	0.808741i	0.994473	+	0.104996i	$0.0334829\pi$
$524$	1.51472			0.0661708
$525$		0			0
$526$	−13.4806			−0.587784
$527$	0.329590	−	0.190289i	0.0143571	−	0.00828910i
$528$		0			0
$529$	−11.3906	+	19.7291i	−0.495244	+	0.857787i
$530$	−0.288905	−	0.500398i	−0.0125492	−	0.0217359i
$531$		0			0
$532$	10.7666	+	15.8983i	0.466791	+	0.689277i
$533$			13.2001i			0.571758i
$534$		0			0
$535$	−0.357916	−	0.206643i	−0.0154741	−	0.00893395i
$536$	2.64991	+	1.52993i	0.114459	+	0.0660828i
$537$		0			0
$538$			31.4799i			1.35719i
$539$	−5.50000	+	4.33013i	−0.236902	+	0.186512i
$540$		0			0
$541$	17.7301	+	30.7095i	0.762277	+	1.32030i	0.941674	+	0.336526i	$0.109252\pi$
$541$	17.7301	+	30.7095i	0.762277	+	1.32030i	−0.179397	+	0.983777i	$0.557415\pi$
$542$	2.82415	−	4.89157i	0.121308	−	0.210111i
$543$		0			0
$544$	−1.41654	+	0.817837i	−0.0607335	+	0.0350645i
$545$	−1.22786			−0.0525960
$546$		0			0
$547$	33.2066			1.41981			0.709907	−	0.704296i	$-0.248737\pi$
$547$	33.2066			1.41981			0.709907	+	0.704296i	$0.248737\pi$
$548$	4.27427	−	2.46775i	0.182588	−	0.105417i
$549$		0			0
$550$	2.49768	−	4.32611i	0.106501	−	0.184466i
$551$	26.2402	+	45.4493i	1.11787	+	1.93620i
$552$		0			0
$553$	−27.4484	−	13.3243i	−1.16722	−	0.566607i
$554$		−	6.91964i		−	0.293987i
$555$		0			0
$556$	−0.957676	−	0.552914i	−0.0406145	−	0.0234488i
$557$	1.95592	+	1.12925i	0.0828748	+	0.0478478i	0.540865	−	0.841110i	$-0.318097\pi$
$557$	1.95592	+	1.12925i	0.0828748	+	0.0478478i	−0.457990	+	0.888957i	$0.651430\pi$
$558$		0			0
$559$		−	23.6155i		−	0.998830i
$560$	0.0129120	+	0.179841i	0.000545631	+	0.00759965i
$561$		0			0
$562$	15.2291	+	26.3776i	0.642401	+	1.11267i
$563$	4.02066	−	6.96399i	0.169451	−	0.293497i	−0.768776	−	0.639518i	$-0.779134\pi$
$563$	4.02066	−	6.96399i	0.169451	−	0.293497i	0.938227	+	0.346021i	$0.112467\pi$
$564$		0			0
$565$	0.416386	−	0.240401i	0.0175175	−	0.0101137i
$566$	26.3837			1.10899
$567$		0			0
$568$	15.8771			0.666187
$569$	−15.6841	+	9.05521i	−0.657511	+	0.379614i	−0.791328	−	0.611392i	$-0.790610\pi$
$569$	−15.6841	+	9.05521i	−0.657511	+	0.379614i	0.133817	+	0.991006i	$0.457277\pi$
$570$		0			0
$571$	−7.15006	+	12.3843i	−0.299221	+	0.518266i	−0.975958	−	0.217959i	$-0.930060\pi$
$571$	−7.15006	+	12.3843i	−0.299221	+	0.518266i	0.676737	+	0.736225i	$0.263393\pi$
$572$	1.22474	+	2.12132i	0.0512092	+	0.0886969i
$573$		0			0
$574$	−6.22631	+	12.8263i	−0.259881	+	0.535361i
$575$			2.33658i			0.0974420i
$576$		0			0
$577$	−27.3960	−	15.8171i	−1.14051	−	0.658475i	−0.193955	−	0.981011i	$-0.562131\pi$
$577$	−27.3960	−	15.8171i	−1.14051	−	0.658475i	−0.946557	+	0.322536i	$0.895465\pi$
$578$	−12.4054	−	7.16228i	−0.515998	−	0.297912i
$579$		0			0
$580$			0.492810i			0.0204628i
$581$	−37.5196	+	25.4090i	−1.55658	+	1.05414i
$582$		0			0
$583$	−4.23936	−	7.34278i	−0.175576	−	0.304107i
$584$	−3.90691	+	6.76696i	−0.161669	+	0.280019i
$585$		0			0
$586$	12.9986	−	7.50473i	0.536966	−	0.310017i
$587$	25.9909			1.07276			0.536381	−	0.843976i	$-0.319791\pi$
$587$	25.9909			1.07276			0.536381	+	0.843976i	$0.319791\pi$
$588$		0			0
$589$	−1.68857			−0.0695762
$590$	0.0518366	−	0.0299279i	0.00213408	−	0.00123211i
$591$		0			0
$592$	2.74496	−	4.75442i	0.112817	−	0.195405i
$593$	15.0260	+	26.0258i	0.617043	+	1.06875i	0.990022	+	0.140911i	$0.0450030\pi$
$593$	15.0260	+	26.0258i	0.617043	+	1.06875i	−0.372979	+	0.927840i	$0.621664\pi$
$594$		0			0
$595$	0.244191	−	0.165371i	0.0100108	−	0.00677954i
$596$			0.318353i			0.0130402i
$597$		0			0
$598$	−0.992248	−	0.572874i	−0.0405760	−	0.0234266i
$599$	−8.18991	−	4.72844i	−0.334631	−	0.193199i	0.323265	−	0.946309i	$-0.395220\pi$
$599$	−8.18991	−	4.72844i	−0.334631	−	0.193199i	−0.657895	+	0.753110i	$0.728553\pi$
$600$		0			0
$601$		−	2.30182i		−	0.0938931i	−0.998897	−	0.0469465i	$-0.985051\pi$
$601$		−	2.30182i		−	0.0938931i	0.998897	−	0.0469465i	$-0.0149490\pi$
$602$	11.1392	−	22.9469i	0.453998	−	0.935247i
$603$		0			0
$604$	1.76941	+	3.06471i	0.0719962	+	0.124701i
$605$	−0.0340742	+	0.0590182i	−0.00138531	+	0.00239943i
$606$		0			0
$607$	−3.13710	+	1.81121i	−0.127331	+	0.0735145i	−0.562313	−	0.826925i	$-0.690088\pi$
$607$	−3.13710	+	1.81121i	−0.127331	+	0.0735145i	0.434982	+	0.900439i	$0.356755\pi$
$608$	7.25725			0.294320
$609$		0			0
$610$	0.679278			0.0275032
$611$	18.2942	−	10.5622i	0.740105	−	0.427300i
$612$		0			0
$613$	4.97209	−	8.61192i	0.200821	−	0.347832i	−0.747972	−	0.663730i	$-0.768972\pi$
$613$	4.97209	−	8.61192i	0.200821	−	0.347832i	0.948793	+	0.315898i	$0.102306\pi$
$614$	−7.53553	−	13.0519i	−0.304109	−	0.526733i
$615$		0			0
$616$	0.189469	+	2.63896i	0.00763391	+	0.106327i
$617$			5.20936i			0.209721i	0.994487	+	0.104860i	$0.0334396\pi$
$617$			5.20936i			0.209721i	−0.994487	+	0.104860i	$0.966560\pi$
$618$		0			0
$619$	10.6324	+	6.13859i	0.427350	+	0.246731i	0.698217	−	0.715886i	$-0.253977\pi$
$619$	10.6324	+	6.13859i	0.427350	+	0.246731i	−0.270867	+	0.962617i	$0.587310\pi$
$620$	−0.0137319	−	0.00792814i	−0.000551488	−	0.000318402i
$621$		0			0
$622$			19.8729i			0.796831i
$623$	−25.4110	−	12.3353i	−1.01807	−	0.494204i
$624$		0			0
$625$	−12.4652	−	21.5903i	−0.498607	−	0.863613i
$626$	4.51059	−	7.81257i	0.180279	−	0.312253i
$627$		0			0
$628$	−5.64589	+	3.25966i	−0.225296	+	0.130074i
$629$	−8.97973			−0.358045
$630$		0			0
$631$	1.18708			0.0472568			0.0236284	−	0.999721i	$-0.492478\pi$
$631$	1.18708			0.0472568			0.0236284	+	0.999721i	$0.492478\pi$
$632$	−9.98722	+	5.76612i	−0.397270	+	0.229364i
$633$		0			0
$634$	5.94427	−	10.2958i	0.236077	−	0.408897i
$635$	0.233047	+	0.403649i	0.00924817	+	0.0160183i
$636$		0			0
$637$	−15.9217	−	6.36396i	−0.630840	−	0.252149i
$638$			7.23143i			0.286295i
$639$		0			0
$640$	0.0590182	+	0.0340742i	0.00233290	+	0.00134690i
$641$	32.2791	+	18.6364i	1.27495	+	0.736092i	0.975915	−	0.218151i	$-0.0700025\pi$
$641$	32.2791	+	18.6364i	1.27495	+	0.736092i	0.299033	+	0.954243i	$0.403336\pi$
$642$		0			0
$643$		−	2.75684i		−	0.108719i	−0.998521	−	0.0543597i	$-0.982688\pi$
$643$		−	2.75684i		−	0.108719i	0.998521	−	0.0543597i	$-0.0173117\pi$
$644$	−0.693937	−	1.02469i	−0.0273450	−	0.0403783i
$645$		0			0
$646$	−5.93525	−	10.2802i	−0.233519	−	0.404467i
$647$	15.2508	−	26.4151i	0.599570	−	1.03849i	−0.393315	−	0.919404i	$-0.628672\pi$
$647$	15.2508	−	26.4151i	0.599570	−	1.03849i	0.992884	−	0.119081i	$-0.0379950\pi$
$648$		0			0
$649$	0.760643	−	0.439158i	0.0298579	−	0.0172384i
$650$	12.2361			0.479938
$651$		0			0
$652$	1.10707			0.0433563
$653$	−9.77473	+	5.64344i	−0.382515	+	0.220845i	−0.678912	−	0.734220i	$-0.737548\pi$
$653$	−9.77473	+	5.64344i	−0.382515	+	0.220845i	0.296397	+	0.955065i	$0.404215\pi$
$654$		0			0
$655$	−0.0516128	+	0.0893960i	−0.00201668	+	0.00349299i
$656$	2.69445	+	4.66693i	0.105201	+	0.182213i
$657$		0			0
$658$	22.7583	−	1.63397i	0.887212	−	0.0636990i
$659$		−	10.2338i		−	0.398652i	−0.979933	−	0.199326i	$-0.936125\pi$
$659$		−	10.2338i		−	0.398652i	0.979933	−	0.199326i	$-0.0638753\pi$
$660$		0			0
$661$	−19.6739	−	11.3587i	−0.765226	−	0.441803i	0.0659431	−	0.997823i	$-0.478994\pi$
$661$	−19.6739	−	11.3587i	−0.765226	−	0.441803i	−0.831169	+	0.556020i	$0.812328\pi$
$662$	19.9489	+	11.5175i	0.775338	+	0.447642i
$663$		0			0
$664$			17.1270i			0.664657i
$665$	−1.30515	+	0.0937055i	−0.0506115	+	0.00363374i
$666$		0			0
$667$	−1.69125	−	2.92933i	−0.0654855	−	0.113424i
$668$	4.96713	−	8.60332i	0.192184	−	0.332872i
$669$		0			0
$670$	−0.180587	+	0.104262i	−0.00697669	+	0.00402799i
$671$	9.96764			0.384797
$672$		0			0
$673$	−28.2268			−1.08806			−0.544031	−	0.839065i	$-0.683103\pi$
$673$	−28.2268			−1.08806			−0.544031	+	0.839065i	$0.683103\pi$
$674$	18.0299	−	10.4096i	0.694486	−	0.400962i
$675$		0			0
$676$	3.50000	−	6.06218i	0.134615	−	0.233161i
$677$	8.40665	+	14.5607i	0.323094	+	0.559615i	0.981125	−	0.193376i	$-0.0619437\pi$
$677$	8.40665	+	14.5607i	0.323094	+	0.559615i	−0.658031	+	0.752991i	$0.728610\pi$
$678$		0			0
$679$	14.8338	+	21.9040i	0.569269	+	0.840598i
$680$		−	0.111469i		−	0.00427462i
$681$		0			0
$682$	−0.201501	−	0.116337i	−0.00771586	−	0.00445476i
$683$	17.9178	+	10.3448i	0.685604	+	0.395834i	0.801963	−	0.597374i	$-0.203789\pi$
$683$	17.9178	+	10.3448i	0.685604	+	0.395834i	−0.116359	+	0.993207i	$0.537122\pi$
$684$		0			0
$685$			0.336346i			0.0128511i
$686$	−12.4691	−	13.6938i	−0.476073	−	0.522834i
$687$		0			0
$688$	−4.82050	−	8.34935i	−0.183780	−	0.318316i
$689$	10.3843	−	17.9861i	0.395609	−	0.685215i
$690$		0			0
$691$	40.3537	−	23.2982i	1.53513	−	0.886306i	0.536014	−	0.844209i	$-0.319929\pi$
$691$	40.3537	−	23.2982i	1.53513	−	0.886306i	0.999114	−	0.0420972i	$-0.0134039\pi$
$692$	13.2221			0.502630
$693$		0			0
$694$	−7.28329			−0.276470
$695$	0.0652640	−	0.0376802i	0.00247561	−	0.00142929i
$696$		0			0
$697$	4.40725	−	7.63358i	0.166936	−	0.289142i
$698$	16.8683	+	29.2168i	0.638475	+	1.10587i
$699$		0			0
$700$	11.8896	+	5.77161i	0.449386	+	0.218146i
$701$			25.8612i			0.976765i	0.872630	+	0.488382i	$0.162413\pi$
$701$			25.8612i			0.976765i	−0.872630	+	0.488382i	$0.837587\pi$
$702$		0			0
$703$	34.5040	+	19.9209i	1.30134	+	0.751331i
$704$	0.866025	+	0.500000i	0.0326396	+	0.0188445i
$705$		0			0
$706$			17.4293i			0.655962i
$707$	0.667035	+	9.29061i	0.0250864	+	0.349409i
$708$		0			0
$709$	−10.3750	−	17.9700i	−0.389641	−	0.674878i	0.602760	−	0.797923i	$-0.294068\pi$
$709$	−10.3750	−	17.9700i	−0.389641	−	0.674878i	−0.992401	+	0.123044i	$0.960734\pi$
$710$	−0.540998	+	0.937036i	−0.0203033	+	0.0351663i
$711$		0			0
$712$	−9.24592	+	5.33814i	−0.346506	+	0.200055i
$713$	0.108833			0.00407582
$714$		0			0
$715$	−0.166929			−0.00624278
$716$	−9.79227	+	5.65357i	−0.365954	+	0.211284i
$717$		0			0
$718$	13.0898	−	22.6722i	0.488506	−	0.846118i
$719$	12.4722	+	21.6025i	0.465136	+	0.805639i	0.999208	−	0.0398001i	$-0.0126721\pi$
$719$	12.4722	+	21.6025i	0.465136	+	0.805639i	−0.534072	+	0.845439i	$0.679339\pi$
$720$		0			0
$721$	−18.4740	+	38.0568i	−0.688006	+	1.41731i
$722$			33.6677i			1.25298i
$723$		0			0
$724$	−7.15918	−	4.13335i	−0.266069	−	0.153615i
$725$	31.2839	+	18.0618i	1.16186	+	0.670798i
$726$		0			0
$727$			19.4534i			0.721487i	0.932665	+	0.360743i	$0.117477\pi$
$727$			19.4534i			0.721487i	−0.932665	+	0.360743i	$0.882523\pi$
$728$	−5.36603	+	3.63397i	−0.198878	+	0.134684i
$729$		0			0
$730$	−0.266249	−	0.461157i	−0.00985433	−	0.0170682i
$731$	−7.88477	+	13.6568i	−0.291629	+	0.505116i
$732$		0			0
$733$	2.02326	−	1.16813i	0.0747309	−	0.0431459i	−0.462169	−	0.886792i	$-0.652929\pi$
$733$	2.02326	−	1.16813i	0.0747309	−	0.0431459i	0.536900	+	0.843646i	$0.319595\pi$
$734$	−11.4490			−0.422589
$735$		0			0
$736$	−0.467750			−0.0172415
$737$	−2.64991	+	1.52993i	−0.0976108	+	0.0563556i
$738$		0			0
$739$	−18.2066	+	31.5348i	−0.669742	+	1.16003i	0.308234	+	0.951310i	$0.400262\pi$
$739$	−18.2066	+	31.5348i	−0.669742	+	1.16003i	−0.977976	+	0.208717i	$0.933071\pi$
$740$	0.187065	+	0.324006i	0.00687663	+	0.0119107i
$741$		0			0
$742$	18.5741	−	12.5787i	0.681876	−	0.461779i
$743$			52.1166i			1.91197i	0.293409	+	0.955987i	$0.405210\pi$
$743$			52.1166i			1.91197i	−0.293409	+	0.955987i	$0.594790\pi$
$744$		0			0
$745$	−0.0187886	−	0.0108476i	−0.000688361	−	0.000397425i
$746$	−24.4546	−	14.1189i	−0.895347	−	0.516929i
$747$		0			0
$748$		−	1.63567i		−	0.0598062i
$749$	7.00689	−	14.4344i	0.256026	−	0.527420i
$750$		0			0
$751$	4.88661	+	8.46385i	0.178315	+	0.308850i	0.941303	−	0.337562i	$-0.109602\pi$
$751$	4.88661	+	8.46385i	0.178315	+	0.308850i	−0.762989	+	0.646412i	$0.776269\pi$
$752$	4.31199	−	7.46859i	0.157242	−	0.272351i
$753$		0			0
$754$	−15.3402	+	8.85666i	−0.558656	+	0.322540i
$755$	−0.241165			−0.00877688
$756$		0			0
$757$	27.6251			1.00405			0.502026	−	0.864852i	$-0.332588\pi$
$757$	27.6251			1.00405			0.502026	+	0.864852i	$0.332588\pi$
$758$	−8.60440	+	4.96775i	−0.312526	+	0.180437i
$759$		0			0
$760$	−0.247285	+	0.428310i	−0.00896997	+	0.0155364i
$761$	10.0318	+	17.3755i	0.363651	+	0.629863i	0.988559	−	0.150836i	$-0.0481966\pi$
$761$	10.0318	+	17.3755i	0.363651	+	0.629863i	−0.624907	+	0.780699i	$0.714863\pi$
$762$		0			0
$763$	−3.41376	−	47.5475i	−0.123586	−	1.72134i
$764$			11.9088i			0.430845i
$765$		0			0
$766$	12.0790	+	6.97381i	0.436432	+	0.251974i
$767$	1.86319	+	1.07571i	0.0672758	+	0.0388417i
$768$		0			0
$769$			31.4568i			1.13436i	0.823593	+	0.567181i	$0.191966\pi$
$769$			31.4568i			1.13436i	−0.823593	+	0.567181i	$0.808034\pi$
$770$	−0.162203	−	0.0787382i	−0.00584537	−	0.00283753i
$771$		0			0
$772$	12.7134	+	22.0203i	0.457566	+	0.792528i
$773$	−3.56548	+	6.17559i	−0.128241	+	0.222121i	−0.922995	−	0.384811i	$-0.874267\pi$
$773$	−3.56548	+	6.17559i	−0.128241	+	0.222121i	0.794754	+	0.606932i	$0.207600\pi$
$774$		0			0
$775$	−1.00657	+	0.581142i	−0.0361570	+	0.0208752i
$776$	9.99876			0.358934
$777$		0			0
$778$	−1.76832			−0.0633974
$779$	−33.8691	+	19.5543i	−1.21349	+	0.700606i
$780$		0			0
$781$	−7.93854	+	13.7499i	−0.284063	+	0.492012i
$782$	0.382543	+	0.662585i	0.0136797	+	0.0236940i
$783$		0			0
$784$	−6.92820	+	1.00000i	−0.247436	+	0.0357143i
$785$		−	0.444280i		−	0.0158570i
$786$		0			0
$787$	20.7613	+	11.9865i	0.740061	+	0.427274i	0.822091	−	0.569356i	$-0.192807\pi$
$787$	20.7613	+	11.9865i	0.740061	+	0.427274i	−0.0820306	+	0.996630i	$0.526141\pi$
$788$	7.25574	+	4.18910i	0.258475	+	0.149231i
$789$		0

Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.r (of order \(6\), degree \(2\), minimal)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about