LMFDB - Embedded newform 177.3.c.a.58.8

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.82290067918$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
Defining polynomial:	$x^{20} + 60 x^{18} + 1522 x^{16} + 21286 x^{14} + 179593 x^{12} + 941588 x^{10} + 3057025 x^{8} + 5962230 x^{6} + 6517782 x^{4} + 3428244 x^{2} + 570861$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.8
Root			$-1.39995i$ of defining polynomial
Character	$\chi$	$=$	177.58
Dual form			177.3.c.a.58.13

$q$-expansion

$f(q)$	$=$	$q-1.39995i q^{2} -1.73205 q^{3} +2.04015 q^{4} +0.273484 q^{5} +2.42478i q^{6} -10.8938 q^{7} -8.45589i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.39995i q^{2} -1.73205 q^{3} +2.04015 q^{4} +0.273484 q^{5} +2.42478i q^{6} -10.8938 q^{7} -8.45589i q^{8} +3.00000 q^{9} -0.382863i q^{10} -15.0527i q^{11} -3.53364 q^{12} -11.1132i q^{13} +15.2508i q^{14} -0.473689 q^{15} -3.67721 q^{16} -3.40920 q^{17} -4.19984i q^{18} -15.7121 q^{19} +0.557948 q^{20} +18.8687 q^{21} -21.0729 q^{22} +15.2765i q^{23} +14.6460i q^{24} -24.9252 q^{25} -15.5578 q^{26} -5.19615 q^{27} -22.2251 q^{28} +22.5689 q^{29} +0.663139i q^{30} -59.5188i q^{31} -28.6757i q^{32} +26.0720i q^{33} +4.77271i q^{34} -2.97929 q^{35} +6.12044 q^{36} +42.4500i q^{37} +21.9961i q^{38} +19.2485i q^{39} -2.31255i q^{40} +78.4822 q^{41} -26.4152i q^{42} +78.3986i q^{43} -30.7096i q^{44} +0.820453 q^{45} +21.3862 q^{46} -35.3465i q^{47} +6.36911 q^{48} +69.6759 q^{49} +34.8940i q^{50} +5.90491 q^{51} -22.6725i q^{52} +37.2779 q^{53} +7.27434i q^{54} -4.11666i q^{55} +92.1171i q^{56} +27.2142 q^{57} -31.5952i q^{58} +(52.6472 - 26.6321i) q^{59} -0.966395 q^{60} -44.2464i q^{61} -83.3232 q^{62} -32.6815 q^{63} -54.8532 q^{64} -3.03927i q^{65} +36.4994 q^{66} +17.3224i q^{67} -6.95528 q^{68} -26.4596i q^{69} +4.17086i q^{70} -59.0750 q^{71} -25.3677i q^{72} -85.3122i q^{73} +59.4278 q^{74} +43.1717 q^{75} -32.0550 q^{76} +163.981i q^{77} +26.9470 q^{78} +62.2163 q^{79} -1.00566 q^{80} +9.00000 q^{81} -109.871i q^{82} -98.9653i q^{83} +38.4949 q^{84} -0.932363 q^{85} +109.754 q^{86} -39.0904 q^{87} -127.284 q^{88} +35.9346i q^{89} -1.14859i q^{90} +121.065i q^{91} +31.1663i q^{92} +103.090i q^{93} -49.4833 q^{94} -4.29702 q^{95} +49.6677i q^{96} -11.0179i q^{97} -97.5426i q^{98} -45.1580i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20q - 40q^{4} - 8q^{7} + 60q^{9} + O(q^{10}) $	$ 20q - 40q^{4} - 8q^{7} + 60q^{9} - 24q^{12} + 24q^{15} - 8q^{16} + 16q^{17} - 60q^{19} - 164q^{20} + 40q^{22} + 100q^{25} - 156q^{26} + 200q^{28} - 60q^{29} - 32q^{35} - 120q^{36} + 28q^{41} + 180q^{46} - 96q^{48} + 284q^{49} - 24q^{51} - 8q^{53} + 24q^{57} - 152q^{59} - 72q^{60} - 8q^{62} - 24q^{63} + 204q^{64} - 120q^{66} + 384q^{68} + 92q^{71} + 104q^{74} + 240q^{75} + 120q^{76} - 468q^{78} - 420q^{79} + 376q^{80} + 180q^{81} + 168q^{84} - 348q^{85} + 232q^{86} + 144q^{87} - 212q^{88} + 152q^{94} + 788q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	1.39995i		−	0.699974i	−0.936755	−	0.349987i	$-0.886186\pi$
$2$		−	1.39995i		−	0.699974i	0.936755	−	0.349987i	$-0.113814\pi$
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	2.04015			0.510037
$5$	0.273484			0.0546968			0.0273484	−	0.999626i	$-0.491294\pi$
$5$	0.273484			0.0546968			0.0273484	+	0.999626i	$0.491294\pi$
$6$			2.42478i			0.404130i
$7$	−10.8938			−1.55626			−0.778132	−	0.628101i	$-0.783832\pi$
$7$	−10.8938			−1.55626			−0.778132	+	0.628101i	$0.783832\pi$
$8$		−	8.45589i		−	1.05699i
$9$	3.00000			0.333333
$10$		−	0.382863i		−	0.0382863i
$11$		−	15.0527i		−	1.36842i	−0.729284	−	0.684212i	$-0.760146\pi$
$11$		−	15.0527i		−	1.36842i	0.729284	−	0.684212i	$-0.239854\pi$
$12$	−3.53364			−0.294470
$13$		−	11.1132i		−	0.854858i	−0.904049	−	0.427429i	$-0.859419\pi$
$13$		−	11.1132i		−	0.854858i	0.904049	−	0.427429i	$-0.140581\pi$
$14$			15.2508i			1.08934i
$15$	−0.473689			−0.0315792
$16$	−3.67721			−0.229825
$17$	−3.40920			−0.200541			−0.100271	−	0.994960i	$-0.531971\pi$
$17$	−3.40920			−0.200541			−0.100271	+	0.994960i	$0.531971\pi$
$18$		−	4.19984i		−	0.233325i
$19$	−15.7121			−0.826954			−0.413477	−	0.910515i	$-0.635686\pi$
$19$	−15.7121			−0.826954			−0.413477	+	0.910515i	$0.635686\pi$
$20$	0.557948			0.0278974
$21$	18.8687			0.898509
$22$	−21.0729			−0.957860
$23$			15.2765i			0.664194i	0.943245	+	0.332097i	$0.107756\pi$
$23$			15.2765i			0.664194i	−0.943245	+	0.332097i	$0.892244\pi$
$24$			14.6460i			0.610251i
$25$	−24.9252			−0.997008
$26$	−15.5578			−0.598378
$27$	−5.19615			−0.192450
$28$	−22.2251			−0.793752
$29$	22.5689			0.778237			0.389118	−	0.921188i	$-0.372780\pi$
$29$	22.5689			0.778237			0.389118	+	0.921188i	$0.372780\pi$
$30$			0.663139i			0.0221046i
$31$		−	59.5188i		−	1.91996i	−0.280067	−	0.959981i	$-0.590357\pi$
$31$		−	59.5188i		−	1.91996i	0.280067	−	0.959981i	$-0.409643\pi$
$32$		−	28.6757i		−	0.896114i
$33$			26.0720i			0.790060i
$34$			4.77271i			0.140374i
$35$	−2.97929			−0.0851227
$36$	6.12044			0.170012
$37$			42.4500i			1.14730i	0.819101	+	0.573649i	$0.194473\pi$
$37$			42.4500i			1.14730i	−0.819101	+	0.573649i	$0.805527\pi$
$38$			21.9961i			0.578846i
$39$			19.2485i			0.493553i
$40$		−	2.31255i		−	0.0578138i
$41$	78.4822			1.91420			0.957101	−	0.289756i	$-0.0935741\pi$
$41$	78.4822			1.91420			0.957101	+	0.289756i	$0.0935741\pi$
$42$		−	26.4152i		−	0.628933i
$43$			78.3986i			1.82322i	0.411052	+	0.911612i	$0.365162\pi$
$43$			78.3986i			1.82322i	−0.411052	+	0.911612i	$0.634838\pi$
$44$		−	30.7096i		−	0.697946i
$45$	0.820453			0.0182323
$46$	21.3862			0.464918
$47$		−	35.3465i		−	0.752054i	−0.926609	−	0.376027i	$-0.877290\pi$
$47$		−	35.3465i		−	0.752054i	0.926609	−	0.376027i	$-0.122710\pi$
$48$	6.36911			0.132690
$49$	69.6759			1.42196
$50$			34.8940i			0.697879i
$51$	5.90491			0.115783
$52$		−	22.6725i		−	0.436009i
$53$	37.2779			0.703357			0.351678	−	0.936121i	$-0.385611\pi$
$53$	37.2779			0.703357			0.351678	+	0.936121i	$0.385611\pi$
$54$			7.27434i			0.134710i
$55$		−	4.11666i		−	0.0748484i
$56$			92.1171i			1.64495i
$57$	27.2142			0.477442
$58$		−	31.5952i		−	0.544745i
$59$	52.6472	−	26.6321i	0.892326	−	0.451392i
$59$	52.6472	−	26.6321i	0.892326	−	0.451392i
$60$	−0.966395			−0.0161066
$61$		−	44.2464i		−	0.725350i	−0.931916	−	0.362675i	$-0.881864\pi$
$61$		−	44.2464i		−	0.725350i	0.931916	−	0.362675i	$-0.118136\pi$
$62$	−83.3232			−1.34392
$63$	−32.6815			−0.518755
$64$	−54.8532			−0.857082
$65$		−	3.03927i		−	0.0467580i
$66$	36.4994			0.553021
$67$			17.3224i			0.258544i	0.991609	+	0.129272i	$0.0412640\pi$
$67$			17.3224i			0.258544i	−0.991609	+	0.129272i	$0.958736\pi$
$68$	−6.95528			−0.102284
$69$		−	26.4596i		−	0.383473i
$70$			4.17086i			0.0595836i
$71$	−59.0750			−0.832043			−0.416021	−	0.909355i	$-0.636576\pi$
$71$	−59.0750			−0.832043			−0.416021	+	0.909355i	$0.636576\pi$
$72$		−	25.3677i		−	0.352329i
$73$		−	85.3122i		−	1.16866i	−0.811516	−	0.584330i	$-0.801357\pi$
$73$		−	85.3122i		−	1.16866i	0.811516	−	0.584330i	$-0.198643\pi$
$74$	59.4278			0.803079
$75$	43.1717			0.575623
$76$	−32.0550			−0.421777
$77$			163.981i			2.12963i
$78$	26.9470			0.345474
$79$	62.2163			0.787548			0.393774	−	0.919207i	$-0.371169\pi$
$79$	62.2163			0.787548			0.393774	+	0.919207i	$0.371169\pi$
$80$	−1.00566			−0.0125707
$81$	9.00000			0.111111
$82$		−	109.871i		−	1.33989i
$83$		−	98.9653i		−	1.19235i	−0.802853	−	0.596177i	$-0.796686\pi$
$83$		−	98.9653i		−	1.19235i	0.802853	−	0.596177i	$-0.203314\pi$
$84$	38.4949			0.458273
$85$	−0.932363			−0.0109690
$86$	109.754			1.27621
$87$	−39.0904			−0.449315
$88$	−127.284			−1.44640
$89$			35.9346i			0.403759i	0.979410	+	0.201880i	$0.0647050\pi$
$89$			35.9346i			0.403759i	−0.979410	+	0.201880i	$0.935295\pi$
$90$		−	1.14859i		−	0.0127621i
$91$			121.065i			1.33038i
$92$			31.1663i			0.338764i
$93$			103.090i			1.10849i
$94$	−49.4833			−0.526418
$95$	−4.29702			−0.0452318
$96$			49.6677i			0.517372i
$97$		−	11.0179i		−	0.113587i	−0.998386	−	0.0567935i	$-0.981912\pi$
$97$		−	11.0179i		−	0.113587i	0.998386	−	0.0567935i	$-0.0180877\pi$
$98$		−	97.5426i		−	0.995332i
$99$		−	45.1580i		−	0.456141i
$100$	−50.8511			−0.508511
$101$			110.774i			1.09677i	0.836227	+	0.548384i	$0.184757\pi$
$101$			110.774i			1.09677i	−0.836227	+	0.548384i	$0.815243\pi$
$102$		−	8.26657i		−	0.0810448i
$103$			43.0668i			0.418125i	0.977902	+	0.209062i	$0.0670412\pi$
$103$			43.0668i			0.418125i	−0.977902	+	0.209062i	$0.932959\pi$
$104$	−93.9716			−0.903573
$105$	5.16029			0.0491456
$106$		−	52.1871i		−	0.492331i
$107$	−146.405			−1.36827			−0.684137	−	0.729353i	$-0.739821\pi$
$107$	−146.405			−1.36827			−0.684137	+	0.729353i	$0.739821\pi$
$108$	−10.6009			−0.0981567
$109$			141.124i			1.29472i	0.762185	+	0.647359i	$0.224127\pi$
$109$			141.124i			1.29472i	−0.762185	+	0.647359i	$0.775873\pi$
$110$	−5.76311			−0.0523919
$111$		−	73.5256i		−	0.662393i
$112$	40.0589			0.357669
$113$		−	21.1887i		−	0.187511i	−0.995595	−	0.0937555i	$-0.970113\pi$
$113$		−	21.1887i		−	0.187511i	0.995595	−	0.0937555i	$-0.0298872\pi$
$114$		−	38.0984i		−	0.334197i
$115$			4.17787i			0.0363293i
$116$	46.0438			0.396929
$117$		−	33.3395i		−	0.284953i
$118$	−37.2836	−	73.7033i	−0.315963	−	0.624604i
$119$	37.1393			0.312095
$120$			4.00546i			0.0333788i
$121$	−105.582			−0.872582
$122$	−61.9426			−0.507726
$123$	−135.935			−1.10516
$124$		−	121.427i		−	0.979251i
$125$	−13.6538			−0.109230
$126$			45.7524i			0.363115i
$127$	215.084			1.69358			0.846789	−	0.531929i	$-0.178533\pi$
$127$	215.084			1.69358			0.846789	+	0.531929i	$0.178533\pi$
$128$		−	37.9110i		−	0.296180i
$129$		−	135.790i		−	1.05264i
$130$	−4.25482			−0.0327294
$131$		−	79.8202i		−	0.609314i	−0.952462	−	0.304657i	$-0.901458\pi$
$131$		−	79.8202i		−	0.609314i	0.952462	−	0.304657i	$-0.0985419\pi$
$132$			53.1907i			0.402960i
$133$	171.165			1.28696
$134$	24.2505			0.180974
$135$	−1.42107			−0.0105264
$136$			28.8278i			0.211969i
$137$	−138.442			−1.01053			−0.505265	−	0.862965i	$-0.668605\pi$
$137$	−138.442			−1.01053			−0.505265	+	0.862965i	$0.668605\pi$
$138$	−37.0421			−0.268421
$139$	59.8615			0.430658			0.215329	−	0.976542i	$-0.430918\pi$
$139$	59.8615			0.430658			0.215329	+	0.976542i	$0.430918\pi$
$140$	−6.07820			−0.0434157
$141$			61.2220i			0.434198i
$142$			82.7019i			0.582408i
$143$	−167.282			−1.16981
$144$	−11.0316			−0.0766085
$145$	6.17223			0.0425671
$146$	−119.433			−0.818031
$147$	−120.682			−0.820967
$148$			86.6044i			0.585165i
$149$		−	218.369i		−	1.46556i	−0.680464	−	0.732782i	$-0.738222\pi$
$149$		−	218.369i		−	1.46556i	0.680464	−	0.732782i	$-0.261778\pi$
$150$		−	60.4381i		−	0.402921i
$151$			26.4104i			0.174904i	0.996169	+	0.0874518i	$0.0278724\pi$
$151$			26.4104i			0.174904i	−0.996169	+	0.0874518i	$0.972128\pi$
$152$			132.860i			0.874078i
$153$	−10.2276			−0.0668471
$154$	229.565			1.49068
$155$		−	16.2774i		−	0.105016i
$156$			39.2699i			0.251730i
$157$		−	18.4659i		−	0.117617i	−0.998269	−	0.0588087i	$-0.981270\pi$
$157$		−	18.4659i		−	0.117617i	0.998269	−	0.0588087i	$-0.0187302\pi$
$158$		−	87.0995i		−	0.551263i
$159$	−64.5672			−0.406083
$160$		−	7.84234i		−	0.0490146i
$161$		−	166.419i		−	1.03366i
$162$		−	12.5995i		−	0.0777748i
$163$	−247.452			−1.51811			−0.759055	−	0.651026i	$-0.774339\pi$
$163$	−247.452			−1.51811			−0.759055	+	0.651026i	$0.774339\pi$
$164$	160.115			0.976313
$165$			7.13027i			0.0432138i
$166$	−138.546			−0.834616
$167$	63.0046			0.377273			0.188637	−	0.982047i	$-0.439593\pi$
$167$	63.0046			0.377273			0.188637	+	0.982047i	$0.439593\pi$
$168$		−	159.552i		−	0.949712i
$169$	45.4978			0.269218
$170$			1.30526i			0.00767800i
$171$	−47.1364			−0.275651
$172$			159.945i			0.929912i
$173$		−	163.258i		−	0.943690i	−0.881681	−	0.471845i	$-0.843588\pi$
$173$		−	163.258i		−	0.943690i	0.881681	−	0.471845i	$-0.156412\pi$
$174$			54.7245i			0.314509i
$175$	271.531			1.55161
$176$			55.3517i			0.314498i
$177$	−91.1876	+	46.1282i	−0.515184	+	0.260611i
$178$	50.3065			0.282621
$179$		−	80.5355i		−	0.449919i	−0.974368	−	0.224960i	$-0.927775\pi$
$179$		−	80.5355i		−	0.449919i	0.974368	−	0.224960i	$-0.0722250\pi$
$180$	1.67384			0.00929914
$181$	−130.162			−0.719125			−0.359563	−	0.933121i	$-0.617074\pi$
$181$	−130.162			−0.719125			−0.359563	+	0.933121i	$0.617074\pi$
$182$	169.485			0.931234
$183$			76.6369i			0.418781i
$184$	129.176			0.702044
$185$			11.6094i			0.0627536i
$186$	144.320			0.775914
$187$			51.3176i			0.274426i
$188$		−	72.1121i		−	0.383575i
$189$	56.6061			0.299503
$190$			6.01560i			0.0316610i
$191$			206.275i			1.07997i	0.841674	+	0.539987i	$0.181571\pi$
$191$			206.275i			1.07997i	−0.841674	+	0.539987i	$0.818429\pi$
$192$	95.0086			0.494836
$193$	174.658			0.904964			0.452482	−	0.891773i	$-0.350539\pi$
$193$	174.658			0.904964			0.452482	+	0.891773i	$0.350539\pi$
$194$	−15.4245			−0.0795079
$195$			5.26417i			0.0269958i
$196$	142.149			0.725250
$197$	262.732			1.33367			0.666833	−	0.745207i	$-0.267649\pi$
$197$	262.732			1.33367			0.666833	+	0.745207i	$0.267649\pi$
$198$	−63.2188			−0.319287
$199$	−36.7705			−0.184776			−0.0923882	−	0.995723i	$-0.529450\pi$
$199$	−36.7705			−0.184776			−0.0923882	+	0.995723i	$0.529450\pi$
$200$			210.765i			1.05382i
$201$		−	30.0033i		−	0.149270i
$202$	155.077			0.767709
$203$	−245.862			−1.21114
$204$	12.0469			0.0590534
$205$	21.4637			0.104701
$206$	60.2913			0.292676
$207$			45.8294i			0.221398i
$208$			40.8654i			0.196468i
$209$			236.509i			1.13162i
$210$		−	7.22413i		−	0.0344006i
$211$		−	332.832i		−	1.57740i	−0.614775	−	0.788702i	$-0.710753\pi$
$211$		−	332.832i		−	1.57740i	0.614775	−	0.788702i	$-0.289247\pi$
$212$	76.0525			0.358738
$213$	102.321			0.480380
$214$			204.960i			0.957756i
$215$			21.4408i			0.0997246i
$216$			43.9381i			0.203417i
$217$			648.389i			2.98797i
$218$	197.567			0.906269
$219$			147.765i			0.674726i
$220$		−	8.39860i		−	0.0381755i
$221$			37.8870i			0.171434i
$222$	−102.932			−0.463658
$223$	−68.0441			−0.305131			−0.152565	−	0.988293i	$-0.548753\pi$
$223$	−68.0441			−0.305131			−0.152565	+	0.988293i	$0.548753\pi$
$224$			312.388i			1.39459i
$225$	−74.7756			−0.332336
$226$	−29.6631			−0.131253
$227$		−	297.497i		−	1.31056i	−0.755386	−	0.655280i	$-0.772551\pi$
$227$		−	297.497i		−	1.31056i	0.755386	−	0.655280i	$-0.227449\pi$
$228$	55.5210			0.243513
$229$			27.5954i			0.120504i	0.998183	+	0.0602519i	$0.0191904\pi$
$229$			27.5954i			0.120504i	−0.998183	+	0.0602519i	$0.980810\pi$
$230$	5.84880			0.0254296
$231$		−	284.024i		−	1.22954i
$232$		−	190.840i		−	0.822585i
$233$		−	359.245i		−	1.54182i	−0.636943	−	0.770911i	$-0.719801\pi$
$233$		−	359.245i		−	1.54182i	0.636943	−	0.770911i	$-0.280199\pi$
$234$	−46.6735			−0.199459
$235$		−	9.66672i		−	0.0411350i
$236$	107.408	−	54.3335i	0.455119	−	0.230227i
$237$	−107.762			−0.454691
$238$		−	51.9931i		−	0.218458i
$239$	−265.127			−1.10932			−0.554659	−	0.832078i	$-0.687151\pi$
$239$	−265.127			−1.10932			−0.554659	+	0.832078i	$0.687151\pi$
$240$	1.74185			0.00725771
$241$	−58.8858			−0.244340			−0.122170	−	0.992509i	$-0.538985\pi$
$241$	−58.8858			−0.244340			−0.122170	+	0.992509i	$0.538985\pi$
$242$			147.810i			0.610784i
$243$	−15.5885			−0.0641500
$244$		−	90.2691i		−	0.369955i
$245$	19.0553			0.0777765
$246$			190.302i			0.773586i
$247$			174.611i			0.706928i
$248$	−503.284			−2.02937
$249$			171.413i			0.688406i
$250$			19.1145i			0.0764581i
$251$	1.34813			0.00537104			0.00268552	−	0.999996i	$-0.499145\pi$
$251$	1.34813			0.00537104			0.00268552	+	0.999996i	$0.499145\pi$
$252$	−66.6752			−0.264584
$253$	229.951			0.908899
$254$		−	301.107i		−	1.18546i
$255$	1.61490			0.00633294
$256$	−272.486			−1.06440
$257$	301.120			1.17167			0.585837	−	0.810429i	$-0.300766\pi$
$257$	301.120			1.17167			0.585837	+	0.810429i	$0.300766\pi$
$258$	−190.099			−0.736819
$259$		−	462.444i		−	1.78550i
$260$		−	6.20056i		−	0.0238483i
$261$	67.7066			0.259412
$262$	−111.744			−0.426504
$263$	82.6739			0.314350			0.157175	−	0.987571i	$-0.449761\pi$
$263$	82.6739			0.314350			0.157175	+	0.987571i	$0.449761\pi$
$264$	220.462			0.835082
$265$	10.1949			0.0384714
$266$		−	239.623i		−	0.900837i
$267$		−	62.2405i		−	0.233111i
$268$			35.3403i			0.131867i
$269$			208.155i			0.773810i	0.922120	+	0.386905i	$0.126456\pi$
$269$			208.155i			0.773810i	−0.922120	+	0.386905i	$0.873544\pi$
$270$			1.98942i			0.00736821i
$271$	38.7081			0.142834			0.0714171	−	0.997447i	$-0.477248\pi$
$271$	38.7081			0.142834			0.0714171	+	0.997447i	$0.477248\pi$
$272$	12.5363			0.0460895
$273$		−	209.691i		−	0.768098i
$274$			193.812i			0.707344i
$275$			375.191i			1.36433i
$276$		−	53.9815i		−	0.195585i
$277$	209.375			0.755866			0.377933	−	0.925833i	$-0.376635\pi$
$277$	209.375			0.755866			0.377933	+	0.925833i	$0.376635\pi$
$278$		−	83.8030i		−	0.301450i
$279$		−	178.556i		−	0.639987i
$280$			25.1926i			0.0899735i
$281$	307.082			1.09282			0.546409	−	0.837519i	$-0.315995\pi$
$281$	307.082			1.09282			0.546409	+	0.837519i	$0.315995\pi$
$282$	85.7075			0.303927
$283$			229.159i			0.809749i	0.914372	+	0.404875i	$0.132685\pi$
$283$			229.159i			0.809749i	−0.914372	+	0.404875i	$0.867315\pi$
$284$	−120.522			−0.424373
$285$	7.44265			0.0261146
$286$			234.187i			0.818834i
$287$	−854.974			−2.97900
$288$		−	86.0270i		−	0.298705i
$289$	−277.377			−0.959783
$290$		−	8.64079i		−	0.0297958i
$291$			19.0836i			0.0655795i
$292$		−	174.049i		−	0.596060i
$293$	56.5578			0.193030			0.0965150	−	0.995332i	$-0.469230\pi$
$293$	56.5578			0.193030			0.0965150	+	0.995332i	$0.469230\pi$
$294$			168.949i			0.574655i
$295$	14.3982	−	7.28347i	0.0488074	−	0.0246897i
$296$	358.953			1.21268
$297$			78.2159i			0.263353i
$298$	−305.705			−1.02586
$299$	169.770			0.567792
$300$	88.0767			0.293589
$301$		−	854.063i		−	2.83742i
$302$	36.9732			0.122428
$303$		−	191.865i		−	0.633219i
$304$	57.7767			0.190055
$305$		−	12.1007i		−	0.0396744i
$306$			14.3181i			0.0467912i
$307$	−38.1615			−0.124305			−0.0621523	−	0.998067i	$-0.519796\pi$
$307$	−38.1615			−0.124305			−0.0621523	+	0.998067i	$0.519796\pi$
$308$			334.546i			1.08619i
$309$		−	74.5940i		−	0.241404i
$310$	−22.7876			−0.0735083
$311$	−237.381			−0.763282			−0.381641	−	0.924311i	$-0.624641\pi$
$311$	−237.381			−0.763282			−0.381641	+	0.924311i	$0.624641\pi$
$312$	162.764			0.521678
$313$		−	468.363i		−	1.49637i	−0.663491	−	0.748184i	$-0.730926\pi$
$313$		−	468.363i		−	1.49637i	0.663491	−	0.748184i	$-0.269074\pi$
$314$	−25.8513			−0.0823291
$315$	−8.93788			−0.0283742
$316$	126.930			0.401679
$317$	−97.7779			−0.308447			−0.154224	−	0.988036i	$-0.549288\pi$
$317$	−97.7779			−0.308447			−0.154224	+	0.988036i	$0.549288\pi$
$318$			90.3907i			0.284248i
$319$		−	339.721i		−	1.06496i
$320$	−15.0015			−0.0468797
$321$	253.582			0.789974
$322$	−232.978			−0.723536
$323$	53.5658			0.165838
$324$	18.3613			0.0566708
$325$			276.998i			0.852300i
$326$			346.420i			1.06264i
$327$		−	244.434i		−	0.747506i
$328$		−	663.637i		−	2.02328i
$329$			385.060i			1.17039i
$330$	9.98200			0.0302485
$331$	−248.132			−0.749644			−0.374822	−	0.927097i	$-0.622296\pi$
$331$	−248.132			−0.749644			−0.374822	+	0.927097i	$0.622296\pi$
$332$		−	201.904i		−	0.608144i
$333$			127.350i			0.382433i
$334$		−	88.2032i		−	0.264081i
$335$			4.73741i			0.0141415i
$336$	−69.3841			−0.206500
$337$			84.1105i			0.249586i	0.992183	+	0.124793i	$0.0398267\pi$
$337$			84.1105i			0.249586i	−0.992183	+	0.124793i	$0.960173\pi$
$338$		−	63.6945i		−	0.188445i
$339$			36.7000i			0.108260i
$340$	−1.90216			−0.00559459
$341$	−895.916			−2.62732
$342$			65.9884i			0.192949i
$343$	−225.240			−0.656676
$344$	662.930			1.92712
$345$		−	7.23629i		−	0.0209747i
$346$	−228.553			−0.660558
$347$		−	521.687i		−	1.50342i	−0.659493	−	0.751711i	$-0.729229\pi$
$347$		−	521.687i		−	1.50342i	0.659493	−	0.751711i	$-0.270771\pi$
$348$	−79.7502			−0.229167
$349$			299.873i			0.859234i	0.903011	+	0.429617i	$0.141351\pi$
$349$			299.873i			0.859234i	−0.903011	+	0.429617i	$0.858649\pi$
$350$		−	380.130i		−	1.08608i
$351$			57.7456i			0.164518i
$352$	−431.645			−1.22626
$353$			633.470i			1.79453i	0.441491	+	0.897266i	$0.354450\pi$
$353$			633.470i			1.79453i	−0.441491	+	0.897266i	$0.645550\pi$
$354$	64.5771	+	127.658i	0.182421	+	0.360616i
$355$	−16.1561			−0.0455101
$356$			73.3119i			0.205932i
$357$	−64.3272			−0.180188
$358$	−112.745			−0.314932
$359$	456.186			1.27071			0.635356	−	0.772219i	$-0.280853\pi$
$359$	456.186			1.27071			0.635356	+	0.772219i	$0.280853\pi$
$360$		−	6.93765i		−	0.0192713i
$361$	−114.129			−0.316148
$362$			182.220i			0.503369i
$363$	182.874			0.503786
$364$			246.990i			0.678545i
$365$		−	23.3315i		−	0.0639220i
$366$	107.288			0.293136
$367$			505.395i			1.37710i	0.725190	+	0.688549i	$0.241752\pi$
$367$			505.395i			1.37710i	−0.725190	+	0.688549i	$0.758248\pi$
$368$		−	56.1747i		−	0.152649i
$369$	235.447			0.638067
$370$	16.2526			0.0439259
$371$	−406.100			−1.09461
$372$			210.318i			0.565371i
$373$	−262.292			−0.703195			−0.351598	−	0.936151i	$-0.614361\pi$
$373$	−262.292			−0.703195			−0.351598	+	0.936151i	$0.614361\pi$
$374$	71.8419			0.192091
$375$	23.6490			0.0630640
$376$	−298.886			−0.794910
$377$		−	250.811i		−	0.665282i
$378$		−	79.2455i		−	0.209644i
$379$	532.499			1.40501			0.702505	−	0.711679i	$-0.252065\pi$
$379$	532.499			1.40501			0.702505	+	0.711679i	$0.252065\pi$
$380$	−8.76655			−0.0230699
$381$	−372.537			−0.977787
$382$	288.774			0.755953
$383$	286.541			0.748150			0.374075	−	0.927398i	$-0.377960\pi$
$383$	286.541			0.748150			0.374075	+	0.927398i	$0.377960\pi$
$384$			65.6638i			0.170999i
$385$			44.8463i			0.116484i
$386$		−	244.512i		−	0.633451i
$387$			235.196i			0.607741i
$388$		−	22.4782i		−	0.0579336i
$389$	620.394			1.59484			0.797421	−	0.603423i	$-0.206197\pi$
$389$	620.394			1.59484			0.797421	+	0.603423i	$0.206197\pi$
$390$	7.36956			0.0188963
$391$		−	52.0806i		−	0.133198i
$392$		−	589.171i		−	1.50299i
$393$			138.253i			0.351788i
$394$		−	367.811i		−	0.933531i
$395$	17.0152			0.0430764
$396$		−	92.1289i		−	0.232649i
$397$		−	434.013i		−	1.09323i	−0.837383	−	0.546616i	$-0.815916\pi$
$397$		−	434.013i		−	1.09323i	0.837383	−	0.546616i	$-0.184084\pi$
$398$			51.4767i			0.129339i
$399$	−296.467			−0.743026
$400$	91.6551			0.229138
$401$			107.207i			0.267350i	0.991025	+	0.133675i	$0.0426778\pi$
$401$			107.207i			0.267350i	−0.991025	+	0.133675i	$0.957322\pi$
$402$	−42.0031			−0.104485
$403$	−661.442			−1.64129
$404$			225.994i			0.559392i
$405$	2.46136			0.00607743
$406$			344.193i			0.847767i
$407$	638.986			1.56999
$408$		−	49.9313i		−	0.122381i
$409$			550.503i			1.34597i	0.739654	+	0.672987i	$0.234989\pi$
$409$			550.503i			1.34597i	−0.739654	+	0.672987i	$0.765011\pi$
$410$		−	30.0480i		−	0.0732878i
$411$	239.789			0.583429
$412$			87.8627i			0.213259i
$413$	−573.531	+	290.126i	−1.38869	+	0.702485i
$414$	64.1587			0.154973
$415$		−	27.0655i		−	0.0652180i
$416$	−318.677			−0.766050
$417$	−103.683			−0.248641
$418$	331.100			0.792106
$419$			274.431i			0.654967i	0.944857	+	0.327483i	$0.106201\pi$
$419$			274.431i			0.654967i	−0.944857	+	0.327483i	$0.893799\pi$
$420$	10.5278			0.0250661
$421$			46.0928i			0.109484i	0.998501	+	0.0547420i	$0.0174336\pi$
$421$			46.0928i			0.109484i	−0.998501	+	0.0547420i	$0.982566\pi$
$422$	−465.948			−1.10414
$423$		−	106.040i		−	0.250685i
$424$		−	315.218i		−	0.743438i
$425$	84.9751			0.199941
$426$		−	143.244i		−	0.336253i
$427$			482.013i			1.12884i
$428$	−298.689			−0.697871
$429$	289.742			0.675389
$430$	30.0160			0.0698046
$431$			256.182i			0.594389i	0.954817	+	0.297195i	$0.0960510\pi$
$431$			256.182i			0.594389i	−0.954817	+	0.297195i	$0.903949\pi$
$432$	19.1073			0.0442299
$433$	76.7056			0.177149			0.0885746	−	0.996070i	$-0.471769\pi$
$433$	76.7056			0.177149			0.0885746	+	0.996070i	$0.471769\pi$
$434$	907.710			2.09150
$435$	−10.6906			−0.0245761
$436$			287.914i			0.660354i
$437$		−	240.026i		−	0.549258i
$438$	206.863			0.472291
$439$	622.390			1.41775			0.708873	−	0.705336i	$-0.249204\pi$
$439$	622.390			1.41775			0.708873	+	0.705336i	$0.249204\pi$
$440$	−34.8100			−0.0791137
$441$	209.028			0.473986
$442$	53.0398			0.120000
$443$			682.698i			1.54108i	0.637392	+	0.770540i	$0.280013\pi$
$443$			682.698i			1.54108i	−0.637392	+	0.770540i	$0.719987\pi$
$444$		−	150.003i		−	0.337845i
$445$			9.82754i			0.0220844i
$446$			95.2582i			0.213583i
$447$			378.226i			0.846144i
$448$	597.563			1.33385
$449$	−144.491			−0.321806			−0.160903	−	0.986970i	$-0.551441\pi$
$449$	−144.491			−0.321806			−0.160903	+	0.986970i	$0.551441\pi$
$450$			104.682i			0.232626i
$451$		−	1181.37i		−	2.61944i
$452$		−	43.2282i		−	0.0956375i
$453$		−	45.7442i		−	0.100981i
$454$	−416.480			−0.917358
$455$			33.1094i			0.0727678i
$456$		−	230.120i		−	0.504649i
$457$		−	293.853i		−	0.643004i	−0.946909	−	0.321502i	$-0.895812\pi$
$457$		−	293.853i		−	0.643004i	0.946909	−	0.321502i	$-0.104188\pi$
$458$	38.6321			0.0843495
$459$	17.7147			0.0385942
$460$			8.52348i			0.0185293i
$461$	457.634			0.992698			0.496349	−	0.868123i	$-0.334674\pi$
$461$	457.634			0.992698			0.496349	+	0.868123i	$0.334674\pi$
$462$	−397.619			−0.860646
$463$			671.285i			1.44986i	0.688822	+	0.724930i	$0.258128\pi$
$463$			671.285i			1.44986i	−0.688822	+	0.724930i	$0.741872\pi$
$464$	−82.9904			−0.178859
$465$			28.1934i			0.0606309i
$466$	−502.923			−1.07923
$467$		−	358.075i		−	0.766756i	−0.923592	−	0.383378i	$-0.874761\pi$
$467$		−	358.075i		−	0.766756i	0.923592	−	0.383378i	$-0.125239\pi$
$468$		−	68.0174i		−	0.145336i
$469$		−	188.708i		−	0.402362i
$470$	−13.5329			−0.0287934
$471$			31.9839i			0.0679065i
$472$	−225.198	−	445.179i	−0.477115	−	0.943176i
$473$	1180.11			2.49494
$474$			150.861i			0.318272i
$475$	391.628			0.824480
$476$	75.7697			0.159180
$477$	111.834			0.234452
$478$			371.164i			0.776493i
$479$	−23.3301			−0.0487059			−0.0243530	−	0.999703i	$-0.507753\pi$
$479$	−23.3301			−0.0487059			−0.0243530	+	0.999703i	$0.507753\pi$
$480$			13.5833i			0.0282986i
$481$	471.754			0.980777
$482$			82.4371i			0.171031i
$483$			288.247i			0.596785i
$484$	−215.404			−0.445049
$485$		−	3.01323i		−	0.00621285i
$486$			21.8230i			0.0449033i
$487$	718.517			1.47539			0.737697	−	0.675132i	$-0.235913\pi$
$487$	718.517			1.47539			0.737697	+	0.675132i	$0.235913\pi$
$488$	−374.142			−0.766685
$489$	428.600			0.876482
$490$		−	26.6763i		−	0.0544415i
$491$	406.760			0.828431			0.414215	−	0.910179i	$-0.364056\pi$
$491$	406.760			0.828431			0.414215	+	0.910179i	$0.364056\pi$
$492$	−277.328			−0.563675
$493$	−76.9419			−0.156069
$494$	244.446			0.494831
$495$		−	12.3500i		−	0.0249495i
$496$			218.863i			0.441256i
$497$	643.554			1.29488
$498$	239.969			0.481866
$499$	−828.694			−1.66071			−0.830355	−	0.557235i	$-0.811862\pi$
$499$	−828.694			−1.66071			−0.830355	+	0.557235i	$0.811862\pi$
$500$	−27.8557			−0.0557114
$501$	−109.127			−0.217819
$502$		−	1.88731i		−	0.00375959i
$503$		−	491.016i		−	0.976175i	−0.872795	−	0.488088i	$-0.837695\pi$
$503$		−	491.016i		−	0.976175i	0.872795	−	0.488088i	$-0.162305\pi$
$504$			276.351i			0.548316i
$505$			30.2948i			0.0599897i
$506$		−	321.920i		−	0.636205i
$507$	−78.8045			−0.155433
$508$	438.804			0.863787
$509$		−	299.875i		−	0.589146i	−0.955629	−	0.294573i	$-0.904823\pi$
$509$		−	299.875i		−	0.589146i	0.955629	−	0.294573i	$-0.0951774\pi$
$510$		−	2.26078i		−	0.00443289i
$511$			929.378i			1.81874i
$512$			229.822i			0.448872i
$513$	81.6426			0.159147
$514$		−	421.553i		−	0.820141i
$515$			11.7781i			0.0228701i
$516$		−	277.033i		−	0.536885i
$517$	−532.059			−1.02913
$518$	−647.398			−1.24980
$519$			282.772i			0.544840i
$520$	−25.6997			−0.0494226
$521$	−793.658			−1.52334			−0.761668	−	0.647967i	$-0.775619\pi$
$521$	−793.658			−1.52334			−0.761668	+	0.647967i	$0.775619\pi$
$522$		−	94.7857i		−	0.181582i
$523$	−991.881			−1.89652			−0.948261	−	0.317493i	$-0.897159\pi$
$523$	−991.881			−1.89652			−0.948261	+	0.317493i	$0.897159\pi$
$524$		−	162.845i		−	0.310773i
$525$	−470.306			−0.895821
$526$		−	115.739i		−	0.220036i
$527$			202.912i			0.385032i
$528$		−	95.8720i		−	0.181576i
$529$	295.630			0.558846
$530$		−	14.2723i		−	0.0269290i
$531$	157.942	−	79.8964i	0.297442	−	0.150464i
$532$	349.203			0.656396
$533$		−	872.185i		−	1.63637i
$534$	−87.1335			−0.163171
$535$	−40.0396			−0.0748403
$536$	146.477			0.273277
$537$			139.492i			0.259761i
$538$	291.406			0.541646
$539$		−	1048.81i		−	1.94584i
$540$	−2.89918			−0.00536886
$541$		−	200.875i		−	0.371303i	−0.982616	−	0.185652i	$-0.940560\pi$
$541$		−	200.875i		−	0.371303i	0.982616	−	0.185652i	$-0.0594396\pi$
$542$		−	54.1893i		−	0.0999802i
$543$	225.447			0.415187
$544$			97.7612i			0.179708i
$545$			38.5953i			0.0708170i
$546$	−293.556			−0.537648
$547$	−412.389			−0.753911			−0.376955	−	0.926231i	$-0.623029\pi$
$547$	−412.389			−0.753911			−0.376955	+	0.926231i	$0.623029\pi$
$548$	−282.443			−0.515407
$549$		−	132.739i		−	0.241783i
$550$	525.247			0.954994
$551$	−354.605			−0.643566
$552$	−223.740			−0.405325
$553$	−677.775			−1.22563
$554$		−	293.114i		−	0.529086i
$555$		−	20.1081i		−	0.0362308i
$556$	122.126			0.219652
$557$	−214.797			−0.385632			−0.192816	−	0.981235i	$-0.561762\pi$
$557$	−214.797			−0.385632			−0.192816	+	0.981235i	$0.561762\pi$
$558$	−249.970			−0.447974
$559$	871.256			1.55860
$560$	10.9555			0.0195634
$561$		−	88.8846i		−	0.158440i
$562$		−	429.898i		−	0.764943i
$563$		−	104.763i		−	0.186079i	−0.995662	−	0.0930396i	$-0.970342\pi$
$563$		−	104.763i		−	0.186079i	0.995662	−	0.0930396i	$-0.0296583\pi$
$564$			124.902i			0.221457i
$565$		−	5.79479i		−	0.0102563i
$566$	320.810			0.566803
$567$	−98.0446			−0.172918
$568$			499.532i			0.879458i
$569$			765.436i			1.34523i	0.739992	+	0.672615i	$0.234829\pi$
$569$			765.436i			1.34523i	−0.739992	+	0.672615i	$0.765171\pi$
$570$		−	10.4193i		−	0.0182795i
$571$			249.624i			0.437170i	0.975818	+	0.218585i	$0.0701441\pi$
$571$			249.624i			0.437170i	−0.975818	+	0.218585i	$0.929856\pi$
$572$	−341.281			−0.596645
$573$		−	357.279i		−	0.623523i
$574$			1196.92i			2.08522i
$575$		−	380.769i		−	0.662207i
$576$	−164.560			−0.285694
$577$	−268.842			−0.465930			−0.232965	−	0.972485i	$-0.574843\pi$
$577$	−268.842			−0.465930			−0.232965	+	0.972485i	$0.574843\pi$
$578$			388.314i			0.671823i
$579$	−302.517			−0.522481
$580$	12.5923			0.0217108
$581$			1078.11i			1.85562i
$582$	26.7161			0.0459039
$583$		−	561.132i		−	0.962490i
$584$	−721.390			−1.23526
$585$		−	9.11782i		−	0.0155860i
$586$		−	79.1779i		−	0.135116i
$587$			793.085i			1.35108i	0.737322	+	0.675541i	$0.236090\pi$
$587$			793.085i			1.35108i	−0.737322	+	0.675541i	$0.763910\pi$
$588$	−246.209			−0.418724
$589$			935.166i			1.58772i
$590$	−10.1965	−	20.1567i	−0.0172822	−	0.0341639i
$591$	−455.066			−0.769993
$592$		−	156.098i		−	0.263678i
$593$	−108.147			−0.182373			−0.0911866	−	0.995834i	$-0.529066\pi$
$593$	−108.147			−0.182373			−0.0911866	+	0.995834i	$0.529066\pi$
$594$	109.498			0.184340
$595$	10.1570			0.0170706
$596$		−	445.505i		−	0.747492i
$597$	63.6884			0.106681
$598$		−	237.669i		−	0.397439i
$599$	21.0152			0.0350837			0.0175419	−	0.999846i	$-0.494416\pi$
$599$	21.0152			0.0350837			0.0175419	+	0.999846i	$0.494416\pi$
$600$		−	365.055i		−	0.608425i
$601$		−	792.854i		−	1.31922i	−0.751606	−	0.659612i	$-0.770721\pi$
$601$		−	792.854i		−	1.31922i	0.751606	−	0.659612i	$-0.229279\pi$
$602$	−1195.64			−1.98612
$603$			51.9673i			0.0861813i
$604$			53.8812i			0.0892073i
$605$	−28.8751			−0.0477275
$606$	−268.602			−0.443237
$607$	848.008			1.39705			0.698524	−	0.715587i	$-0.253841\pi$
$607$	848.008			1.39705			0.698524	+	0.715587i	$0.253841\pi$
$608$			450.555i			0.741045i
$609$	425.845			0.699253
$610$	−16.9403			−0.0277710
$611$	−392.811			−0.642899
$612$	−20.8658			−0.0340945
$613$			355.689i			0.580243i	0.956990	+	0.290121i	$0.0936957\pi$
$613$			355.689i			0.580243i	−0.956990	+	0.290121i	$0.906304\pi$
$614$			53.4241i			0.0870100i
$615$	−37.1761			−0.0604490
$616$	1386.61			2.25099
$617$	120.858			0.195880			0.0979402	−	0.995192i	$-0.468775\pi$
$617$	120.858			0.195880			0.0979402	+	0.995192i	$0.468775\pi$
$618$	−104.428			−0.168977
$619$	−112.377			−0.181546			−0.0907732	−	0.995872i	$-0.528934\pi$
$619$	−112.377			−0.181546			−0.0907732	+	0.995872i	$0.528934\pi$
$620$		−	33.2084i		−	0.0535619i
$621$		−	79.3789i		−	0.127824i
$622$			332.320i			0.534277i
$623$		−	391.466i		−	0.628356i
$624$		−	70.7809i		−	0.113431i
$625$	619.396			0.991034
$626$	−655.684			−1.04742
$627$		−	409.646i		−	0.653343i
$628$		−	37.6732i		−	0.0599892i
$629$		−	144.721i		−	0.230081i
$630$			12.5126i			0.0198612i
$631$	467.803			0.741368			0.370684	−	0.928759i	$-0.379123\pi$
$631$	467.803			0.741368			0.370684	+	0.928759i	$0.379123\pi$
$632$		−	526.094i		−	0.832427i
$633$			576.483i			0.910715i
$634$			136.884i			0.215905i
$635$	58.8222			0.0926333
$636$	−131.727			−0.207117
$637$		−	774.319i		−	1.21557i
$638$	−475.592			−0.745442
$639$	−177.225			−0.277348
$640$		−	10.3681i		−	0.0162001i
$641$	−1006.77			−1.57062			−0.785311	−	0.619102i	$-0.787497\pi$
$641$	−1006.77			−1.57062			−0.785311	+	0.619102i	$0.787497\pi$
$642$		−	355.001i		−	0.552961i
$643$	65.4231			0.101747			0.0508734	−	0.998705i	$-0.483800\pi$
$643$	65.4231			0.101747			0.0508734	+	0.998705i	$0.483800\pi$
$644$		−	339.520i		−	0.527205i
$645$		−	37.1365i		−	0.0575760i
$646$		−	74.9893i		−	0.116083i
$647$	1000.16			1.54584			0.772920	−	0.634503i	$-0.218795\pi$
$647$	1000.16			1.54584			0.772920	+	0.634503i	$0.218795\pi$
$648$		−	76.1030i		−	0.117443i
$649$	−400.884	−	792.480i	−0.617696	−	1.22108i
$650$	387.782			0.596588
$651$		−	1123.04i		−	1.72510i
$652$	−504.839			−0.774293
$653$	−635.037			−0.972492			−0.486246	−	0.873822i	$-0.661634\pi$
$653$	−635.037			−0.972492			−0.486246	+	0.873822i	$0.661634\pi$
$654$	−342.195			−0.523234
$655$		−	21.8296i		−	0.0333276i
$656$	−288.595			−0.439932
$657$		−	255.937i		−	0.389553i
$658$	539.063			0.819245
$659$			445.774i			0.676440i	0.941067	+	0.338220i	$0.109825\pi$
$659$			445.774i			0.676440i	−0.941067	+	0.338220i	$0.890175\pi$
$660$			14.5468i			0.0220406i
$661$	4.13501			0.00625568			0.00312784	−	0.999995i	$-0.499004\pi$
$661$	4.13501			0.00625568			0.00312784	+	0.999995i	$0.499004\pi$
$662$			347.372i			0.524731i
$663$		−	65.6222i		−	0.0989777i
$664$	−836.840			−1.26030
$665$	46.8110			0.0703925
$666$	178.283			0.267693
$667$			344.773i			0.516900i
$668$	128.539			0.192423
$669$	117.856			0.176167
$670$	6.63213			0.00989870
$671$	−666.025			−0.992586
$672$		−	541.072i		−	0.805167i
$673$		−	144.454i		−	0.214641i	−0.994224	−	0.107321i	$-0.965773\pi$
$673$		−	144.454i		−	0.214641i	0.994224	−	0.107321i	$-0.0342271\pi$
$674$	117.750			0.174704
$675$	129.515			0.191874
$676$	92.8222			0.137311
$677$	567.757			0.838637			0.419318	−	0.907839i	$-0.362269\pi$
$677$	567.757			0.838637			0.419318	+	0.907839i	$0.362269\pi$
$678$	51.3780			0.0757788
$679$			120.028i			0.176771i
$680$			7.88396i			0.0115941i
$681$			515.280i			0.756652i
$682$			1254.24i			1.83905i
$683$			505.717i			0.740435i	0.928945	+	0.370218i	$0.120717\pi$
$683$			505.717i			0.740435i	−0.928945	+	0.370218i	$0.879283\pi$
$684$	−96.1651			−0.140592
$685$	−37.8618			−0.0552727
$686$			315.324i			0.459656i
$687$		−	47.7966i		−	0.0695729i
$688$		−	288.288i		−	0.419023i
$689$		−	414.275i		−	0.601270i
$690$	−10.1304			−0.0146818
$691$			843.080i			1.22009i	0.792368	+	0.610044i	$0.208848\pi$
$691$			843.080i			1.22009i	−0.792368	+	0.610044i	$0.791152\pi$
$692$		−	333.071i		−	0.481317i
$693$			491.944i			0.709876i
$694$	−730.334			−1.05236
$695$	16.3712			0.0235557
$696$			330.544i			0.474920i
$697$	−267.562			−0.383877
$698$	419.806			0.601441
$699$			622.230i			0.890171i
$700$	553.964			0.791377
$701$			873.826i			1.24654i	0.782006	+	0.623271i	$0.214197\pi$
$701$			873.826i			1.24654i	−0.782006	+	0.623271i	$0.785803\pi$
$702$	80.8409			0.115158
$703$		−	666.980i		−	0.948763i
$704$			825.687i			1.17285i
$705$			16.7432i			0.0237493i
$706$	886.824			1.25612
$707$		−	1206.75i		−	1.70686i
$708$	−186.036	+	94.1084i	−0.262763	+	0.132921i
$709$	159.514			0.224985			0.112492	−	0.993653i	$-0.464117\pi$
$709$	159.514			0.224985			0.112492	+	0.993653i	$0.464117\pi$
$710$			22.6177i			0.0318559i
$711$	186.649			0.262516
$712$	303.859			0.426768
$713$	909.237			1.27523
$714$			90.0547i			0.126127i
$715$	−45.7491			−0.0639848
$716$		−	164.304i		−	0.229475i
$717$	459.213			0.640465
$718$		−	638.636i		−	0.889465i
$719$		−	35.0202i		−	0.0487068i	−0.999703	−	0.0243534i	$-0.992247\pi$
$719$		−	35.0202i		−	0.0487068i	0.999703	−	0.0243534i	$-0.00775270\pi$
$720$	−3.01697			−0.00419024
$721$		−	469.164i		−	0.650712i
$722$			159.775i			0.221295i
$723$	101.993			0.141070
$724$	−265.549			−0.366781
$725$	−562.534			−0.775908
$726$		−	256.014i		−	0.352637i
$727$	1123.28			1.54509			0.772544	−	0.634961i	$-0.218984\pi$
$727$	1123.28			1.54509			0.772544	+	0.634961i	$0.218984\pi$
$728$	1023.71			1.40620
$729$	27.0000			0.0370370
$730$	−32.6629			−0.0447437
$731$		−	267.277i		−	0.365632i
$732$			156.351i			0.213594i
$733$	−1337.07			−1.82411			−0.912053	−	0.410073i	$-0.865503\pi$
$733$	−1337.07			−1.82411			−0.912053	+	0.410073i	$0.865503\pi$
$734$	707.526			0.963932
$735$	−33.0047			−0.0449043
$736$	438.063			0.595194
$737$	260.749			0.353797
$738$		−	329.613i		−	0.446630i
$739$			393.835i			0.532930i	0.963845	+	0.266465i	$0.0858556\pi$
$739$			393.835i			0.532930i	−0.963845	+	0.266465i	$0.914144\pi$
$740$			23.6849i			0.0320067i
$741$		−	302.435i		−	0.408145i
$742$			568.518i			0.766197i
$743$	−42.0432			−0.0565857			−0.0282929	−	0.999600i	$-0.509007\pi$
$743$	−42.0432			−0.0565857			−0.0282929	+	0.999600i	$0.509007\pi$
$744$	871.714			1.17166
$745$		−	59.7205i		−	0.0801617i
$746$			367.195i			0.492218i
$747$		−	296.896i		−	0.397451i
$748$			104.695i			0.139967i
$749$	1594.92			2.12940
$750$		−	33.1073i		−	0.0441431i
$751$		−	1078.71i		−	1.43636i	−0.695858	−	0.718180i	$-0.744975\pi$
$751$		−	1078.71i		−	1.43636i	0.695858	−	0.718180i	$-0.255025\pi$
$752$			129.976i			0.172841i
$753$	−2.33503			−0.00310097
$754$	−351.123			−0.465680
$755$			7.22284i			0.00956667i
$756$	115.485			0.152758
$757$	−247.277			−0.326654			−0.163327	−	0.986572i	$-0.552223\pi$
$757$	−247.277			−0.326654			−0.163327	+	0.986572i	$0.552223\pi$
$758$		−	745.470i		−	0.983470i
$759$	−398.288			−0.524753
$760$			36.3351i			0.0478093i
$761$	681.030			0.894914			0.447457	−	0.894305i	$-0.352330\pi$
$761$	681.030			0.894914			0.447457	+	0.894305i	$0.352330\pi$
$762$			521.532i			0.684425i
$763$		−	1537.39i		−	2.01492i
$764$			420.831i			0.550826i
$765$	−2.79709			−0.00365633
$766$		−	401.143i		−	0.523685i
$767$	−295.967	−	585.077i	−0.385876	−	0.762812i
$768$	471.960			0.614531
$769$			1018.02i			1.32382i	0.749582	+	0.661912i	$0.230255\pi$
$769$			1018.02i			1.32382i	−0.749582	+	0.661912i	$0.769745\pi$
$770$	62.7824			0.0815356
$771$	−521.556			−0.676466
$772$	356.328			0.461565
$773$		−	1082.79i		−	1.40076i	−0.713770	−	0.700380i	$-0.753014\pi$
$773$		−	1082.79i		−	1.40076i	0.713770	−	0.700380i	$-0.246986\pi$
$774$	329.262			0.425403
$775$			1483.52i			1.91422i
$776$	−93.1664			−0.120060
$777$			800.977i			1.03086i
$778$		−	868.518i		−	1.11635i
$779$	−1233.12			−1.58296
$780$			10.7397i			0.0137688i
$781$			889.236i			1.13859i
$782$	−72.9101			−0.0932354
$783$	−117.271			−0.149772
$784$	−256.213			−0.326802
$785$		−	5.05014i		−	0.00643330i
$786$	193.546			0.246242
$787$	302.623			0.384527			0.192263	−	0.981343i	$-0.438417\pi$
$787$	302.623			0.384527			0.192263	+	0.981343i	$0.438417\pi$
$788$	536.013			0.680219
$789$	−143.195			−0.181490
$790$		−	23.8203i		−	0.0301523i
$791$			230.827i			0.291817i

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.39995i q^{2} -1.73205 q^{3} +2.04015 q^{4} +0.273484 q^{5} +2.42478i q^{6} -10.8938 q^{7} -8.45589i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.39995i q^{2} -1.73205 q^{3} +2.04015 q^{4} +0.273484 q^{5} +2.42478i q^{6} -10.8938 q^{7} -8.45589i q^{8} +3.00000 q^{9} -0.382863i q^{10} -15.0527i q^{11} -3.53364 q^{12} -11.1132i q^{13} +15.2508i q^{14} -0.473689 q^{15} -3.67721 q^{16} -3.40920 q^{17} -4.19984i q^{18} -15.7121 q^{19} +0.557948 q^{20} +18.8687 q^{21} -21.0729 q^{22} +15.2765i q^{23} +14.6460i q^{24} -24.9252 q^{25} -15.5578 q^{26} -5.19615 q^{27} -22.2251 q^{28} +22.5689 q^{29} +0.663139i q^{30} -59.5188i q^{31} -28.6757i q^{32} +26.0720i q^{33} +4.77271i q^{34} -2.97929 q^{35} +6.12044 q^{36} +42.4500i q^{37} +21.9961i q^{38} +19.2485i q^{39} -2.31255i q^{40} +78.4822 q^{41} -26.4152i q^{42} +78.3986i q^{43} -30.7096i q^{44} +0.820453 q^{45} +21.3862 q^{46} -35.3465i q^{47} +6.36911 q^{48} +69.6759 q^{49} +34.8940i q^{50} +5.90491 q^{51} -22.6725i q^{52} +37.2779 q^{53} +7.27434i q^{54} -4.11666i q^{55} +92.1171i q^{56} +27.2142 q^{57} -31.5952i q^{58} +(52.6472 - 26.6321i) q^{59} -0.966395 q^{60} -44.2464i q^{61} -83.3232 q^{62} -32.6815 q^{63} -54.8532 q^{64} -3.03927i q^{65} +36.4994 q^{66} +17.3224i q^{67} -6.95528 q^{68} -26.4596i q^{69} +4.17086i q^{70} -59.0750 q^{71} -25.3677i q^{72} -85.3122i q^{73} +59.4278 q^{74} +43.1717 q^{75} -32.0550 q^{76} +163.981i q^{77} +26.9470 q^{78} +62.2163 q^{79} -1.00566 q^{80} +9.00000 q^{81} -109.871i q^{82} -98.9653i q^{83} +38.4949 q^{84} -0.932363 q^{85} +109.754 q^{86} -39.0904 q^{87} -127.284 q^{88} +35.9346i q^{89} -1.14859i q^{90} +121.065i q^{91} +31.1663i q^{92} +103.090i q^{93} -49.4833 q^{94} -4.29702 q^{95} +49.6677i q^{96} -11.0179i q^{97} -97.5426i q^{98} -45.1580i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20q - 40q^{4} - 8q^{7} + 60q^{9} + O(q^{10}) \)	\( 20q - 40q^{4} - 8q^{7} + 60q^{9} - 24q^{12} + 24q^{15} - 8q^{16} + 16q^{17} - 60q^{19} - 164q^{20} + 40q^{22} + 100q^{25} - 156q^{26} + 200q^{28} - 60q^{29} - 32q^{35} - 120q^{36} + 28q^{41} + 180q^{46} - 96q^{48} + 284q^{49} - 24q^{51} - 8q^{53} + 24q^{57} - 152q^{59} - 72q^{60} - 8q^{62} - 24q^{63} + 204q^{64} - 120q^{66} + 384q^{68} + 92q^{71} + 104q^{74} + 240q^{75} + 120q^{76} - 468q^{78} - 420q^{79} + 376q^{80} + 180q^{81} + 168q^{84} - 348q^{85} + 232q^{86} + 144q^{87} - 212q^{88} + 152q^{94} + 788q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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