LMFDB - Embedded newform 177.11.c.a.58.12

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$112.458233723$
Analytic rank:	$0$
Dimension:	$100$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.12
Character	$\chi$	$=$	177.58
Dual form			177.11.c.a.58.89

$q$-expansion

$f(q)$	$=$	$q-51.4146i q^{2} -140.296 q^{3} -1619.46 q^{4} -2372.07 q^{5} +7213.27i q^{6} +15485.2 q^{7} +30615.5i q^{8} +19683.0 q^{9} +O(q^{10})$	\(q-51.4146i q^{2} -140.296 q^{3} -1619.46 q^{4} -2372.07 q^{5} +7213.27i q^{6} +15485.2 q^{7} +30615.5i q^{8} +19683.0 q^{9} +121959. i q^{10} -207798. i q^{11} +227204. q^{12} +9463.66i q^{13} -796164. i q^{14} +332792. q^{15} -84244.9 q^{16} +393426. q^{17} -1.01199e6i q^{18} -3.27325e6 q^{19} +3.84147e6 q^{20} -2.17251e6 q^{21} -1.06839e7 q^{22} -3.87647e6i q^{23} -4.29524e6i q^{24} -4.13893e6 q^{25} +486571. q^{26} -2.76145e6 q^{27} -2.50777e7 q^{28} +1.36358e7 q^{29} -1.71104e7i q^{30} +2.27647e7i q^{31} +3.56817e7i q^{32} +2.91533e7i q^{33} -2.02279e7i q^{34} -3.67318e7 q^{35} -3.18759e7 q^{36} +8.26010e7i q^{37} +1.68293e8i q^{38} -1.32772e6i q^{39} -7.26220e7i q^{40} -1.06754e8 q^{41} +1.11699e8i q^{42} -7.40752e7i q^{43} +3.36521e8i q^{44} -4.66894e7 q^{45} -1.99307e8 q^{46} +2.11955e7i q^{47} +1.18192e7 q^{48} -4.26846e7 q^{49} +2.12802e8i q^{50} -5.51962e7 q^{51} -1.53261e7i q^{52} +2.58725e7 q^{53} +1.41979e8i q^{54} +4.92911e8i q^{55} +4.74087e8i q^{56} +4.59224e8 q^{57} -7.01080e8i q^{58} +(-5.38229e8 - 4.70560e8i) q^{59} -5.38944e8 q^{60} +1.35826e9i q^{61} +1.17044e9 q^{62} +3.04795e8 q^{63} +1.74830e9 q^{64} -2.24484e7i q^{65} +1.49890e9 q^{66} -1.48600e9i q^{67} -6.37140e8 q^{68} +5.43854e8i q^{69} +1.88855e9i q^{70} -8.76816e8 q^{71} +6.02605e8i q^{72} -8.52461e8i q^{73} +4.24690e9 q^{74} +5.80676e8 q^{75} +5.30091e9 q^{76} -3.21779e9i q^{77} -6.82640e7 q^{78} -9.09746e7 q^{79} +1.99834e8 q^{80} +3.87420e8 q^{81} +5.48872e9i q^{82} +1.22363e8i q^{83} +3.51830e9 q^{84} -9.33233e8 q^{85} -3.80855e9 q^{86} -1.91305e9 q^{87} +6.36185e9 q^{88} -5.23788e8i q^{89} +2.40052e9i q^{90} +1.46546e8i q^{91} +6.27780e9i q^{92} -3.19380e9i q^{93} +1.08976e9 q^{94} +7.76436e9 q^{95} -5.00601e9i q^{96} +6.29776e9i q^{97} +2.19462e9i q^{98} -4.09009e9i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + O(q^{10}) $	$ 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + 620136q^{12} + 662904q^{15} + 27925160q^{16} - 2053136q^{17} - 5169828q^{19} - 1076324q^{20} - 1829224q^{22} + 215378180q^{25} - 22082700q^{26} - 102921320q^{28} - 112503588q^{29} - 76491392q^{35} - 1007769600q^{36} + 473464516q^{41} + 1215405588q^{46} - 1676272320q^{48} + 1975297276q^{49} + 733970808q^{51} + 3267506728q^{53} + 591502824q^{57} + 508142200q^{59} + 1264196808q^{60} - 6538206968q^{62} + 362009736q^{63} - 10324137972q^{64} - 2764346616q^{66} + 9997685952q^{68} + 14908523204q^{71} + 4863508712q^{74} + 1890481680q^{75} + 2044437240q^{76} - 758396196q^{78} - 3599839500q^{79} - 23217941144q^{80} + 38742048900q^{81} - 13094894808q^{84} + 23360564412q^{85} + 12186923752q^{86} + 7965322272q^{87} + 32415437996q^{88} - 22098322280q^{94} + 7834510028q^{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 $

$2$		−	51.4146i		−	1.60671i	−0.595502	−	0.803353i	$-0.703047\pi$
$2$		−	51.4146i		−	1.60671i	0.595502	−	0.803353i	$-0.296953\pi$
$3$	−140.296			−0.577350
$3$	−140.296			−0.577350
$4$	−1619.46			−1.58151
$5$	−2372.07			−0.759061			−0.379530	−	0.925179i	$-0.623914\pi$
$5$	−2372.07			−0.759061			−0.379530	+	0.925179i	$0.623914\pi$
$6$			7213.27i			0.927633i
$7$	15485.2			0.921353			0.460676	−	0.887568i	$-0.347607\pi$
$7$	15485.2			0.921353			0.460676	+	0.887568i	$0.347607\pi$
$8$			30615.5i			0.934312i
$9$	19683.0			0.333333
$10$			121959.i			1.21959i
$11$		−	207798.i		−	1.29026i	−0.764072	−	0.645131i	$-0.776803\pi$
$11$		−	207798.i		−	1.29026i	0.764072	−	0.645131i	$-0.223197\pi$
$12$	227204.			0.913084
$13$			9463.66i			0.0254884i	0.999919	+	0.0127442i	$0.00405671\pi$
$13$			9463.66i			0.0254884i	−0.999919	+	0.0127442i	$0.995943\pi$
$14$		−	796164.i		−	1.48034i
$15$	332792.			0.438244
$16$	−84244.9			−0.0803422
$17$	393426.			0.277089			0.138544	−	0.990356i	$-0.455758\pi$
$17$	393426.			0.277089			0.138544	+	0.990356i	$0.455758\pi$
$18$		−	1.01199e6i		−	0.535569i
$19$	−3.27325e6			−1.32194			−0.660969	−	0.750413i	$-0.729855\pi$
$19$	−3.27325e6			−1.32194			−0.660969	+	0.750413i	$0.729855\pi$
$20$	3.84147e6			1.20046
$21$	−2.17251e6			−0.531943
$22$	−1.06839e7			−2.07307
$23$		−	3.87647e6i		−	0.602278i	−0.953580	−	0.301139i	$-0.902633\pi$
$23$		−	3.87647e6i		−	0.602278i	0.953580	−	0.301139i	$-0.0973669\pi$
$24$		−	4.29524e6i		−	0.539425i
$25$	−4.13893e6			−0.423827
$26$	486571.			0.0409524
$27$	−2.76145e6			−0.192450
$28$	−2.50777e7			−1.45713
$29$	1.36358e7			0.664800			0.332400	−	0.943139i	$-0.392142\pi$
$29$	1.36358e7			0.664800			0.332400	+	0.943139i	$0.392142\pi$
$30$		−	1.71104e7i		−	0.704130i
$31$			2.27647e7i			0.795158i	0.917568	+	0.397579i	$0.130150\pi$
$31$			2.27647e7i			0.795158i	−0.917568	+	0.397579i	$0.869850\pi$
$32$			3.56817e7i			1.06340i
$33$			2.91533e7i			0.744934i
$34$		−	2.02279e7i		−	0.445200i
$35$	−3.67318e7			−0.699363
$36$	−3.18759e7			−0.527169
$37$			8.26010e7i			1.19118i	0.803289	+	0.595589i	$0.203081\pi$
$37$			8.26010e7i			1.19118i	−0.803289	+	0.595589i	$0.796919\pi$
$38$			1.68293e8i			2.12397i
$39$		−	1.32772e6i		−	0.0147157i
$40$		−	7.26220e7i		−	0.709199i
$41$	−1.06754e8			−0.921437			−0.460718	−	0.887546i	$-0.652408\pi$
$41$	−1.06754e8			−0.921437			−0.460718	+	0.887546i	$0.652408\pi$
$42$			1.11699e8i			0.854677i
$43$		−	7.40752e7i		−	0.503884i	−0.967742	−	0.251942i	$-0.918931\pi$
$43$		−	7.40752e7i		−	0.503884i	0.967742	−	0.251942i	$-0.0810692\pi$
$44$			3.36521e8i			2.04056i
$45$	−4.66894e7			−0.253020
$46$	−1.99307e8			−0.967685
$47$			2.11955e7i			0.0924176i	0.998932	+	0.0462088i	$0.0147140\pi$
$47$			2.11955e7i			0.0924176i	−0.998932	+	0.0462088i	$0.985286\pi$
$48$	1.18192e7			0.0463856
$49$	−4.26846e7			−0.151109
$50$			2.12802e8i			0.680965i
$51$	−5.51962e7			−0.159977
$52$		−	1.53261e7i		−	0.0403101i
$53$	2.58725e7			0.0618671			0.0309335	−	0.999521i	$-0.490152\pi$
$53$	2.58725e7			0.0618671			0.0309335	+	0.999521i	$0.490152\pi$
$54$			1.41979e8i			0.309211i
$55$			4.92911e8i			0.979388i
$56$			4.74087e8i			0.860830i
$57$	4.59224e8			0.763221
$58$		−	7.01080e8i		−	1.06814i
$59$	−5.38229e8	−	4.70560e8i	−0.752847	−	0.658195i
$59$	−5.38229e8	−	4.70560e8i	−0.752847	−	0.658195i
$60$	−5.38944e8			−0.693086
$61$			1.35826e9i			1.60818i	0.594509	+	0.804089i	$0.297347\pi$
$61$			1.35826e9i			1.60818i	−0.594509	+	0.804089i	$0.702653\pi$
$62$	1.17044e9			1.27759
$63$	3.04795e8			0.307118
$64$	1.74830e9			1.62823
$65$		−	2.24484e7i		−	0.0193472i
$66$	1.49890e9			1.19689
$67$		−	1.48600e9i		−	1.10064i	−0.834953	−	0.550321i	$-0.814505\pi$
$67$		−	1.48600e9i		−	1.10064i	0.834953	−	0.550321i	$-0.185495\pi$
$68$	−6.37140e8			−0.438218
$69$			5.43854e8i			0.347726i
$70$			1.88855e9i			1.12367i
$71$	−8.76816e8			−0.485978			−0.242989	−	0.970029i	$-0.578128\pi$
$71$	−8.76816e8			−0.485978			−0.242989	+	0.970029i	$0.578128\pi$
$72$			6.02605e8i			0.311437i
$73$		−	8.52461e8i		−	0.411207i	−0.978635	−	0.205603i	$-0.934084\pi$
$73$		−	8.52461e8i		−	0.411207i	0.978635	−	0.205603i	$-0.0659157\pi$
$74$	4.24690e9			1.91387
$75$	5.80676e8			0.244696
$76$	5.30091e9			2.09065
$77$		−	3.21779e9i		−	1.18879i
$78$	−6.82640e7			−0.0236439
$79$	−9.09746e7			−0.0295655			−0.0147827	−	0.999891i	$-0.504706\pi$
$79$	−9.09746e7			−0.0295655			−0.0147827	+	0.999891i	$0.504706\pi$
$80$	1.99834e8			0.0609846
$81$	3.87420e8			0.111111
$82$			5.48872e9i			1.48048i
$83$			1.22363e8i			0.0310642i	0.999879	+	0.0155321i	$0.00494423\pi$
$83$			1.22363e8i			0.0310642i	−0.999879	+	0.0155321i	$0.995056\pi$
$84$	3.51830e9			0.841272
$85$	−9.33233e8			−0.210327
$86$	−3.80855e9			−0.809594
$87$	−1.91305e9			−0.383822
$88$	6.36185e9			1.20551
$89$		−	5.23788e8i		−	0.0938005i	−0.998900	−	0.0469002i	$-0.985066\pi$
$89$		−	5.23788e8i		−	0.0938005i	0.998900	−	0.0469002i	$-0.0149343\pi$
$90$			2.40052e9i			0.406529i
$91$			1.46546e8i			0.0234838i
$92$			6.27780e9i			0.952508i
$93$		−	3.19380e9i		−	0.459085i
$94$	1.08976e9			0.148488
$95$	7.76436e9			1.00343
$96$		−	5.00601e9i		−	0.613953i
$97$			6.29776e9i			0.733377i	0.930344	+	0.366689i	$0.119509\pi$
$97$			6.29776e9i			0.733377i	−0.930344	+	0.366689i	$0.880491\pi$
$98$			2.19462e9i			0.242788i
$99$		−	4.09009e9i		−	0.430088i
$100$	6.70285e9			0.670285
$101$			9.07488e9i			0.863444i	0.902007	+	0.431722i	$0.142094\pi$
$101$			9.07488e9i			0.863444i	−0.902007	+	0.431722i	$0.857906\pi$
$102$			2.83789e9i			0.257037i
$103$			5.43941e9i			0.469209i	0.972091	+	0.234604i	$0.0753794\pi$
$103$			5.43941e9i			0.469209i	−0.972091	+	0.234604i	$0.924621\pi$
$104$	−2.89735e8			−0.0238141
$105$	5.15333e9			0.403777
$106$		−	1.33023e9i		−	0.0994023i
$107$	6.07134e8			0.0432878			0.0216439	−	0.999766i	$-0.493110\pi$
$107$	6.07134e8			0.0432878			0.0216439	+	0.999766i	$0.493110\pi$
$108$	4.47206e9			0.304361
$109$		−	8.52551e9i		−	0.554100i	−0.960856	−	0.277050i	$-0.910643\pi$
$109$		−	8.52551e9i		−	0.554100i	0.960856	−	0.277050i	$-0.0893568\pi$
$110$	2.53428e10			1.57359
$111$		−	1.15886e10i		−	0.687727i
$112$	−1.30455e9			−0.0740235
$113$			2.98035e10i			1.61762i	0.588072	+	0.808808i	$0.299887\pi$
$113$			2.98035e10i			1.61762i	−0.588072	+	0.808808i	$0.700113\pi$
$114$		−	2.36108e10i		−	1.22627i
$115$			9.19524e9i			0.457166i
$116$	−2.20827e10			−1.05139
$117$			1.86273e8i			0.00849613i
$118$	−2.41937e10	+	2.76728e10i	−1.05753	+	1.20960i
$119$	6.09228e9			0.255296
$120$			1.01886e10i			0.409456i
$121$	−1.72426e10			−0.664779
$122$	6.98345e10			2.58387
$123$	1.49772e10			0.531992
$124$		−	3.68666e10i		−	1.25755i
$125$	3.29825e10			1.08077
$126$		−	1.56709e10i		−	0.493448i
$127$	−8.16922e9			−0.247265			−0.123632	−	0.992328i	$-0.539454\pi$
$127$	−8.16922e9			−0.247265			−0.123632	+	0.992328i	$0.539454\pi$
$128$		−	5.33499e10i		−	1.55269i
$129$			1.03925e10i			0.290918i
$130$	−1.15418e9			−0.0310854
$131$		−	2.12207e10i		−	0.550051i	−0.961437	−	0.275025i	$-0.911314\pi$
$131$		−	2.12207e10i		−	0.550051i	0.961437	−	0.275025i	$-0.0886863\pi$
$132$		−	4.72127e10i		−	1.17812i
$133$	−5.06868e10			−1.21797
$134$	−7.64024e10			−1.76841
$135$	6.55034e9			0.146081
$136$			1.20450e10i			0.258887i
$137$	6.03369e10			1.25020			0.625101	−	0.780544i	$-0.285058\pi$
$137$	6.03369e10			1.25020			0.625101	+	0.780544i	$0.285058\pi$
$138$	2.79620e10			0.558693
$139$	−2.81739e10			−0.542966			−0.271483	−	0.962443i	$-0.587514\pi$
$139$	−2.81739e10			−0.542966			−0.271483	+	0.962443i	$0.587514\pi$
$140$	5.94859e10			1.10605
$141$		−	2.97365e9i		−	0.0533573i
$142$			4.50812e10i			0.780825i
$143$	1.96653e9			0.0328867
$144$	−1.65819e9			−0.0267807
$145$	−3.23450e10			−0.504623
$146$	−4.38290e10			−0.660689
$147$	5.98849e9			0.0872430
$148$		−	1.33769e11i		−	1.88386i
$149$		−	3.53969e10i		−	0.481985i	−0.970527	−	0.240993i	$-0.922527\pi$
$149$		−	3.53969e10i		−	0.481985i	0.970527	−	0.240993i	$-0.0774730\pi$
$150$		−	2.98552e10i		−	0.393155i
$151$		−	7.55219e10i		−	0.962029i	−0.876713	−	0.481015i	$-0.840268\pi$
$151$		−	7.55219e10i		−	0.962029i	0.876713	−	0.481015i	$-0.159732\pi$
$152$		−	1.00212e11i		−	1.23510i
$153$	7.74381e9			0.0923629
$154$	−1.65441e11			−1.91003
$155$		−	5.39994e10i		−	0.603574i
$156$			2.15019e9i			0.0232730i
$157$		−	7.86464e10i		−	0.824481i	−0.911075	−	0.412240i	$-0.864746\pi$
$157$		−	7.86464e10i		−	0.824481i	0.911075	−	0.412240i	$-0.135254\pi$
$158$			4.67742e9i			0.0475030i
$159$	−3.62982e9			−0.0357190
$160$		−	8.46394e10i		−	0.807184i
$161$		−	6.00278e10i		−	0.554911i
$162$		−	1.99191e10i		−	0.178523i
$163$	1.75024e11			1.52110			0.760552	−	0.649277i	$-0.224928\pi$
$163$	1.75024e11			1.52110			0.760552	+	0.649277i	$0.224928\pi$
$164$	1.72884e11			1.45726
$165$		−	6.91535e10i		−	0.565450i
$166$	6.29126e9			0.0499111
$167$	−9.22995e9			−0.0710586			−0.0355293	−	0.999369i	$-0.511312\pi$
$167$	−9.22995e9			−0.0710586			−0.0355293	+	0.999369i	$0.511312\pi$
$168$		−	6.65125e10i		−	0.497001i
$169$	1.37769e11			0.999350
$170$			4.79818e10i			0.337934i
$171$	−6.44274e10			−0.440646
$172$			1.19962e11i			0.796896i
$173$		−	5.92302e10i		−	0.382219i	−0.981569	−	0.191110i	$-0.938791\pi$
$173$		−	5.92302e10i		−	0.382219i	0.981569	−	0.191110i	$-0.0612086\pi$
$174$			9.83588e10i			0.616690i
$175$	−6.40921e10			−0.390494
$176$			1.75059e10i			0.103663i
$177$	7.55114e10	+	6.60177e10i	0.434657	+	0.380009i
$178$	−2.69303e10			−0.150710
$179$			2.00252e11i			1.08971i	0.838530	+	0.544855i	$0.183415\pi$
$179$			2.00252e11i			1.08971i	−0.838530	+	0.544855i	$0.816585\pi$
$180$	7.56117e10			0.400153
$181$	−1.09128e11			−0.561750			−0.280875	−	0.959744i	$-0.590625\pi$
$181$	−1.09128e11			−0.561750			−0.280875	+	0.959744i	$0.590625\pi$
$182$	7.53463e9			0.0377316
$183$		−	1.90559e11i		−	0.928482i
$184$	1.18680e11			0.562716
$185$		−	1.95935e11i		−	0.904177i
$186$	−1.64208e11			−0.737615
$187$		−	8.17533e10i		−	0.357517i
$188$		−	3.43254e10i		−	0.146159i
$189$	−4.27615e10			−0.177314
$190$		−	3.99202e11i		−	1.61222i
$191$		−	2.66205e11i		−	1.04725i	−0.851950	−	0.523623i	$-0.824580\pi$
$191$		−	2.66205e11i		−	1.04725i	0.851950	−	0.523623i	$-0.175420\pi$
$192$	−2.45279e11			−0.940057
$193$	1.58248e11			0.590952			0.295476	−	0.955350i	$-0.404522\pi$
$193$	1.58248e11			0.590952			0.295476	+	0.955350i	$0.404522\pi$
$194$	3.23797e11			1.17832
$195$			3.14943e9i			0.0111701i
$196$	6.91262e10			0.238981
$197$	3.48221e11			1.17361			0.586806	−	0.809728i	$-0.300385\pi$
$197$	3.48221e11			1.17361			0.586806	+	0.809728i	$0.300385\pi$
$198$	−2.10290e11			−0.691025
$199$	−3.62970e11			−1.16307			−0.581534	−	0.813522i	$-0.697547\pi$
$199$	−3.62970e11			−1.16307			−0.581534	+	0.813522i	$0.697547\pi$
$200$		−	1.26716e11i		−	0.395986i
$201$			2.08481e11i			0.635456i
$202$	4.66581e11			1.38730
$203$	2.11153e11			0.612515
$204$	8.93882e10			0.253005
$205$	2.53228e11			0.699426
$206$	2.79665e11			0.753881
$207$		−	7.63006e10i		−	0.200759i
$208$		−	7.97266e8i		−	0.00204779i
$209$			6.80175e11i			1.70565i
$210$		−	2.64957e11i		−	0.648752i
$211$		−	3.43143e11i		−	0.820471i	−0.911980	−	0.410236i	$-0.865446\pi$
$211$		−	3.43143e11i		−	0.820471i	0.911980	−	0.410236i	$-0.134554\pi$
$212$	−4.18996e10			−0.0978433
$213$	1.23014e11			0.280580
$214$		−	3.12156e10i		−	0.0695509i
$215$			1.75711e11i			0.382479i
$216$		−	8.45432e10i		−	0.179808i
$217$			3.52516e11i			0.732621i
$218$	−4.38336e11			−0.890276
$219$			1.19597e11i			0.237410i
$220$		−	7.98251e11i		−	1.54891i
$221$			3.72325e9i			0.00706255i
$222$	−5.95824e11			−1.10498
$223$	−1.45253e11			−0.263391			−0.131695	−	0.991290i	$-0.542042\pi$
$223$	−1.45253e11			−0.263391			−0.131695	+	0.991290i	$0.542042\pi$
$224$			5.52538e11i			0.979765i
$225$	−8.14666e10			−0.141276
$226$	1.53234e12			2.59904
$227$			8.25512e11i			1.36960i	0.728731	+	0.684801i	$0.240111\pi$
$227$			8.25512e11i			1.36960i	−0.728731	+	0.684801i	$0.759889\pi$
$228$	−7.43697e11			−1.20704
$229$			2.38084e11i			0.378052i	0.981972	+	0.189026i	$0.0605331\pi$
$229$			2.38084e11i			0.378052i	−0.981972	+	0.189026i	$0.939467\pi$
$230$	4.72770e11			0.734532
$231$			4.51443e11i			0.686347i
$232$			4.17467e11i			0.621130i
$233$			1.15231e12i			1.67799i	0.544140	+	0.838994i	$0.316856\pi$
$233$			1.15231e12i			1.67799i	−0.544140	+	0.838994i	$0.683144\pi$
$234$	9.57717e9			0.0136508
$235$		−	5.02772e10i		−	0.0701506i
$236$	8.71642e11	+	7.62054e11i	1.19063	+	1.04094i
$237$	1.27634e10			0.0170696
$238$		−	3.13232e11i		−	0.410187i
$239$	5.78195e11			0.741455			0.370728	−	0.928742i	$-0.379108\pi$
$239$	5.78195e11			0.741455			0.370728	+	0.928742i	$0.379108\pi$
$240$	−2.80360e10			−0.0352095
$241$	4.61412e11			0.567550			0.283775	−	0.958891i	$-0.408413\pi$
$241$	4.61412e11			0.567550			0.283775	+	0.958891i	$0.408413\pi$
$242$			8.86524e11i			1.06810i
$243$	−5.43536e10			−0.0641500
$244$		−	2.19965e12i		−	2.54335i
$245$	1.01251e11			0.114701
$246$		−	7.70047e11i		−	0.854755i
$247$		−	3.09769e10i		−	0.0336941i
$248$	−6.96954e11			−0.742926
$249$		−	1.71671e10i		−	0.0179349i
$250$		−	1.69578e12i		−	1.73648i
$251$	−7.89068e10			−0.0792038			−0.0396019	−	0.999216i	$-0.512609\pi$
$251$	−7.89068e10			−0.0792038			−0.0396019	+	0.999216i	$0.512609\pi$
$252$	−4.93604e11			−0.485709
$253$	−8.05523e11			−0.777098
$254$			4.20017e11i			0.397282i
$255$	1.30929e11			0.121432
$256$	−9.52708e11			−0.866483
$257$	1.38822e12			1.23821			0.619104	−	0.785309i	$-0.287496\pi$
$257$	1.38822e12			1.23821			0.619104	+	0.785309i	$0.287496\pi$
$258$	5.34325e11			0.467419
$259$			1.27909e12i			1.09750i
$260$			3.63544e10i			0.0305978i
$261$	2.68394e11			0.221600
$262$	−1.09105e12			−0.883771
$263$	2.06493e12			1.64107			0.820535	−	0.571596i	$-0.193676\pi$
$263$	2.06493e12			1.64107			0.820535	+	0.571596i	$0.193676\pi$
$264$	−8.92543e11			−0.696000
$265$	−6.13714e10			−0.0469609
$266$			2.60604e12i			1.95692i
$267$			7.34854e10i			0.0541557i
$268$			2.40653e12i			1.74067i
$269$			7.98504e10i			0.0566912i	0.999598	+	0.0283456i	$0.00902390\pi$
$269$			7.98504e10i			0.0566912i	−0.999598	+	0.0283456i	$0.990976\pi$
$270$		−	3.36783e11i		−	0.234710i
$271$	8.27816e11			0.566353			0.283177	−	0.959068i	$-0.408612\pi$
$271$	8.27816e11			0.566353			0.283177	+	0.959068i	$0.408612\pi$
$272$	−3.31442e10			−0.0222619
$273$		−	2.05599e10i		−	0.0135584i
$274$		−	3.10220e12i		−	2.00871i
$275$			8.60062e11i			0.546848i
$276$		−	8.80751e11i		−	0.549931i
$277$	2.75212e12			1.68760			0.843799	−	0.536659i	$-0.180314\pi$
$277$	2.75212e12			1.68760			0.843799	+	0.536659i	$0.180314\pi$
$278$			1.44855e12i			0.872387i
$279$			4.48078e11i			0.265053i
$280$		−	1.12456e12i		−	0.653423i
$281$	−9.85832e10			−0.0562693			−0.0281346	−	0.999604i	$-0.508957\pi$
$281$	−9.85832e10			−0.0562693			−0.0281346	+	0.999604i	$0.508957\pi$
$282$	−1.52889e11			−0.0857296
$283$			8.55168e11i			0.471107i	0.971861	+	0.235553i	$0.0756902\pi$
$283$			8.55168e11i			0.471107i	−0.971861	+	0.235553i	$0.924310\pi$
$284$	1.41997e12			0.768578
$285$	−1.08931e12			−0.579331
$286$		−	1.01108e11i		−	0.0528393i
$287$	−1.65311e12			−0.848968
$288$			7.02323e11i			0.354466i
$289$	−1.86121e12			−0.923222
$290$			1.66301e12i			0.810782i
$291$		−	8.83551e11i		−	0.423416i
$292$			1.38053e12i			0.650326i
$293$	3.40558e12			1.57708			0.788538	−	0.614986i	$-0.210838\pi$
$293$	3.40558e12			1.57708			0.788538	+	0.614986i	$0.210838\pi$
$294$		−	3.07896e11i		−	0.140174i
$295$	1.27671e12	+	1.11620e12i	0.571457	+	0.499610i
$296$	−2.52887e12			−1.11293
$297$			5.73824e11i			0.248311i
$298$	−1.81992e12			−0.774409
$299$	3.66856e10			0.0153511
$300$	−9.40384e11			−0.386989
$301$		−	1.14707e12i		−	0.464255i
$302$	−3.88293e12			−1.54570
$303$		−	1.27317e12i		−	0.498509i
$304$	2.75755e11			0.106207
$305$		−	3.22188e12i		−	1.22071i
$306$		−	3.98145e11i		−	0.148400i
$307$	−3.76502e12			−1.38062			−0.690312	−	0.723512i	$-0.742527\pi$
$307$	−3.76502e12			−1.38062			−0.690312	+	0.723512i	$0.742527\pi$
$308$			5.21109e12i			1.88008i
$309$		−	7.63128e11i		−	0.270898i
$310$	−2.77636e12			−0.969766
$311$	−5.66700e12			−1.94783			−0.973915	−	0.226912i	$-0.927137\pi$
$311$	−5.66700e12			−1.94783			−0.973915	+	0.226912i	$0.927137\pi$
$312$	4.06487e10			0.0137491
$313$			1.16286e12i			0.387084i	0.981092	+	0.193542i	$0.0619976\pi$
$313$			1.16286e12i			0.387084i	−0.981092	+	0.193542i	$0.938002\pi$
$314$	−4.04358e12			−1.32470
$315$	−7.22993e11			−0.233121
$316$	1.47330e11			0.0467580
$317$	−1.75108e12			−0.547028			−0.273514	−	0.961868i	$-0.588186\pi$
$317$	−1.75108e12			−0.547028			−0.273514	+	0.961868i	$0.588186\pi$
$318$			1.86626e11i			0.0573899i
$319$		−	2.83350e12i		−	0.857766i
$320$	−4.14707e12			−1.23592
$321$	−8.51786e10			−0.0249922
$322$	−3.08631e12			−0.891579
$323$	−1.28778e12			−0.366294
$324$	−6.27413e11			−0.175723
$325$		−	3.91695e10i		−	0.0108027i
$326$		−	8.99878e12i		−	2.44397i
$327$			1.19610e12i			0.319910i
$328$		−	3.26833e12i		−	0.860909i
$329$			3.28216e11i			0.0851492i
$330$	−3.55550e12			−0.908512
$331$	−6.66533e12			−1.67758			−0.838788	−	0.544459i	$-0.816735\pi$
$331$	−6.66533e12			−1.67758			−0.838788	+	0.544459i	$0.816735\pi$
$332$		−	1.98163e11i		−	0.0491283i
$333$			1.62584e12i			0.397059i
$334$			4.74554e11i			0.114170i
$335$			3.52490e12i			0.835454i
$336$	1.83023e11			0.0427375
$337$			3.31280e12i			0.762160i	0.924542	+	0.381080i	$0.124448\pi$
$337$			3.31280e12i			0.762160i	−0.924542	+	0.381080i	$0.875552\pi$
$338$		−	7.08334e12i		−	1.60566i
$339$		−	4.18132e12i		−	0.933931i
$340$	1.51134e12			0.332634
$341$	4.73046e12			1.02596
$342$			3.31251e12i			0.707989i
$343$	−5.03516e12			−1.06058
$344$	2.26785e12			0.470785
$345$		−	1.29006e12i		−	0.263945i
$346$	−3.04530e12			−0.614114
$347$			4.81033e12i			0.956153i	0.878318	+	0.478076i	$0.158666\pi$
$347$			4.81033e12i			0.956153i	−0.878318	+	0.478076i	$0.841334\pi$
$348$	3.09812e12			0.607018
$349$			1.57464e12i			0.304126i	0.988371	+	0.152063i	$0.0485917\pi$
$349$			1.57464e12i			0.304126i	−0.988371	+	0.152063i	$0.951408\pi$
$350$			3.29527e12i			0.627409i
$351$		−	2.61334e10i		−	0.00490524i
$352$	7.41459e12			1.37206
$353$			2.26537e12i			0.413300i	0.978415	+	0.206650i	$0.0662562\pi$
$353$			2.26537e12i			0.413300i	−0.978415	+	0.206650i	$0.933744\pi$
$354$	3.39428e12	−	3.88239e12i	0.610563	−	0.698366i
$355$	2.07987e12			0.368887
$356$			8.48255e11i			0.148346i
$357$	−8.54723e11			−0.147395
$358$	1.02959e13			1.75084
$359$	6.50960e12			1.09165			0.545824	−	0.837900i	$-0.316217\pi$
$359$	6.50960e12			1.09165			0.545824	+	0.837900i	$0.316217\pi$
$360$		−	1.42942e12i		−	0.236400i
$361$	4.58309e12			0.747520
$362$			5.61077e12i			0.902567i
$363$	2.41908e12			0.383810
$364$		−	2.37327e11i		−	0.0371398i
$365$			2.02209e12i			0.312131i
$366$	−9.79751e12			−1.49180
$367$			6.69893e12i			1.00618i	0.864234	+	0.503090i	$0.167803\pi$
$367$			6.69893e12i			1.00618i	−0.864234	+	0.503090i	$0.832197\pi$
$368$			3.26573e11i			0.0483884i
$369$	−2.10124e12			−0.307146
$370$	−1.00739e13			−1.45275
$371$	4.00641e11			0.0570014
$372$			5.17224e12i			0.726046i
$373$	2.32947e12			0.322635			0.161318	−	0.986903i	$-0.448426\pi$
$373$	2.32947e12			0.322635			0.161318	+	0.986903i	$0.448426\pi$
$374$	−4.20331e12			−0.574426
$375$	−4.62732e12			−0.623983
$376$	−6.48912e11			−0.0863469
$377$			1.29045e11i			0.0169447i
$378$			2.19857e12i			0.284892i
$379$	1.03290e13			1.32087			0.660436	−	0.750882i	$-0.270371\pi$
$379$	1.03290e13			1.32087			0.660436	+	0.750882i	$0.270371\pi$
$380$	−1.25741e13			−1.58693
$381$	1.14611e12			0.142758
$382$	−1.36868e13			−1.68262
$383$	1.21823e13			1.47820			0.739102	−	0.673594i	$-0.235250\pi$
$383$	1.21823e13			1.47820			0.739102	+	0.673594i	$0.235250\pi$
$384$			7.48478e12i			0.896443i
$385$			7.63281e12i			0.902362i
$386$		−	8.13627e12i		−	0.949486i
$387$		−	1.45802e12i		−	0.167961i
$388$		−	1.01990e13i		−	1.15984i
$389$	1.17149e13			1.31520			0.657599	−	0.753368i	$-0.271572\pi$
$389$	1.17149e13			1.31520			0.657599	+	0.753368i	$0.271572\pi$
$390$	1.61927e11			0.0179471
$391$		−	1.52511e12i		−	0.166885i
$392$		−	1.30681e12i		−	0.141183i
$393$			2.97718e12i			0.317572i
$394$		−	1.79037e13i		−	1.88565i
$395$	2.15798e11			0.0224420
$396$			6.62375e12i			0.680187i
$397$		−	8.14478e12i		−	0.825899i	−0.910754	−	0.412950i	$-0.864499\pi$
$397$		−	8.14478e12i		−	0.825899i	0.910754	−	0.412950i	$-0.135501\pi$
$398$			1.86620e13i			1.86871i
$399$	7.11117e12			0.703196
$400$	3.48684e11			0.0340512
$401$		−	9.47598e12i		−	0.913907i	−0.889491	−	0.456954i	$-0.848941\pi$
$401$		−	9.47598e12i		−	0.913907i	0.889491	−	0.456954i	$-0.151059\pi$
$402$	1.07190e13			1.02099
$403$	−2.15438e11			−0.0202673
$404$		−	1.46964e13i		−	1.36554i
$405$	−9.18987e11			−0.0843401
$406$		−	1.08563e13i		−	0.984132i
$407$	1.71643e13			1.53693
$408$		−	1.68986e12i		−	0.149469i
$409$			1.12656e12i			0.0984326i	0.998788	+	0.0492163i	$0.0156724\pi$
$409$			1.12656e12i			0.0984326i	−0.998788	+	0.0492163i	$0.984328\pi$
$410$		−	1.30196e13i		−	1.12377i
$411$	−8.46503e12			−0.721804
$412$		−	8.80893e12i		−	0.742057i
$413$	−8.33457e12	−	7.28670e12i	−0.693638	−	0.606430i
$414$	−3.92297e12			−0.322562
$415$		−	2.90254e11i		−	0.0235796i
$416$	−3.37680e11			−0.0271043
$417$	3.95268e12			0.313481
$418$	3.49709e13			2.74048
$419$			2.01771e13i			1.56239i	0.624289	+	0.781193i	$0.285389\pi$
$419$			2.01771e13i			1.56239i	−0.624289	+	0.781193i	$0.714611\pi$
$420$	−8.34564e12			−0.638577
$421$			1.04111e13i			0.787199i	0.919282	+	0.393600i	$0.128770\pi$
$421$			1.04111e13i			0.787199i	−0.919282	+	0.393600i	$0.871230\pi$
$422$	−1.76426e13			−1.31826
$423$			4.17191e11i			0.0308059i
$424$			7.92101e11i			0.0578031i
$425$	−1.62836e12			−0.117438
$426$		−	6.32471e12i		−	0.450809i
$427$			2.10329e13i			1.48170i
$428$	−9.83232e11			−0.0684600
$429$	−2.75897e11			−0.0189872
$430$	9.03413e12			0.614531
$431$			1.69638e13i			1.14061i	0.821433	+	0.570305i	$0.193175\pi$
$431$			1.69638e13i			1.14061i	−0.821433	+	0.570305i	$0.806825\pi$
$432$	2.32638e11			0.0154619
$433$	1.53769e13			1.01025			0.505127	−	0.863045i	$-0.331446\pi$
$433$	1.53769e13			1.01025			0.505127	+	0.863045i	$0.331446\pi$
$434$	1.81245e13			1.17711
$435$	4.53788e12			0.291344
$436$			1.38068e13i			0.876313i
$437$			1.26887e13i			0.796175i
$438$	6.14903e12			0.381449
$439$	−1.65688e13			−1.01618			−0.508088	−	0.861305i	$-0.669648\pi$
$439$	−1.65688e13			−1.01618			−0.508088	+	0.861305i	$0.669648\pi$
$440$	−1.50907e13			−0.915054
$441$	−8.40162e11			−0.0503698
$442$	1.91430e11			0.0113474
$443$		−	8.43036e12i		−	0.494114i	−0.969001	−	0.247057i	$-0.920536\pi$
$443$		−	8.43036e12i		−	0.494114i	0.969001	−	0.247057i	$-0.0794636\pi$
$444$			1.87673e13i			1.08765i
$445$			1.24246e12i			0.0712003i
$446$			7.46813e12i			0.423192i
$447$			4.96605e12i			0.278274i
$448$	2.70727e13			1.50017
$449$	−2.91559e13			−1.59770			−0.798849	−	0.601531i	$-0.794558\pi$
$449$	−2.91559e13			−1.59770			−0.798849	+	0.601531i	$0.794558\pi$
$450$			4.18857e12i			0.226988i
$451$			2.21833e13i			1.18890i
$452$		−	4.82657e13i		−	2.55827i
$453$			1.05954e13i			0.555428i
$454$	4.24434e13			2.20055
$455$		−	3.47618e11i		−	0.0178256i
$456$			1.40594e13i			0.713086i
$457$			1.45565e13i			0.730259i	0.930957	+	0.365129i	$0.118975\pi$
$457$			1.45565e13i			0.730259i	−0.930957	+	0.365129i	$0.881025\pi$
$458$	1.22410e13			0.607420
$459$	−1.08643e12			−0.0533257
$460$		−	1.48914e13i		−	0.723011i
$461$	7.73613e12			0.371552			0.185776	−	0.982592i	$-0.440520\pi$
$461$	7.73613e12			0.371552			0.185776	+	0.982592i	$0.440520\pi$
$462$	2.32108e13			1.10276
$463$			1.09254e13i			0.513490i	0.966479	+	0.256745i	$0.0826501\pi$
$463$			1.09254e13i			0.513490i	−0.966479	+	0.256745i	$0.917350\pi$
$464$	−1.14875e12			−0.0534115
$465$			7.57590e12i			0.348473i
$466$	5.92455e13			2.69604
$467$		−	1.60922e13i		−	0.724488i	−0.932083	−	0.362244i	$-0.882011\pi$
$467$		−	1.60922e13i		−	0.724488i	0.932083	−	0.362244i	$-0.117989\pi$
$468$		−	3.01663e11i		−	0.0134367i
$469$		−	2.30110e13i		−	1.01408i
$470$	−2.58498e12			−0.112711
$471$			1.10338e13i			0.476014i
$472$	1.44064e13	−	1.64782e13i	0.614959	−	0.703394i
$473$	−1.53927e13			−0.650143
$474$		−	6.56224e11i		−	0.0274259i
$475$	1.35478e13			0.560272
$476$	−9.86622e12			−0.403753
$477$	5.09249e11			0.0206224
$478$		−	2.97277e13i		−	1.19130i
$479$	−3.75735e13			−1.49006			−0.745031	−	0.667030i	$-0.767565\pi$
$479$	−3.75735e13			−1.49006			−0.745031	+	0.667030i	$0.767565\pi$
$480$			1.18746e13i			0.466028i
$481$	−7.81708e11			−0.0303612
$482$		−	2.37233e13i		−	0.911886i
$483$			8.42167e12i			0.320378i
$484$	2.79238e13			1.05135
$485$		−	1.49387e13i		−	0.556678i
$486$			2.79457e12i			0.103070i
$487$	−3.48485e13			−1.27215			−0.636076	−	0.771626i	$-0.719444\pi$
$487$	−3.48485e13			−1.27215			−0.636076	+	0.771626i	$0.719444\pi$
$488$	−4.15839e13			−1.50254
$489$	−2.45552e13			−0.878210
$490$		−	5.20577e12i		−	0.184291i
$491$	−2.22606e13			−0.780061			−0.390031	−	0.920802i	$-0.627536\pi$
$491$	−2.22606e13			−0.780061			−0.390031	+	0.920802i	$0.627536\pi$
$492$	−2.42550e13			−0.841349
$493$	5.36469e12			0.184208
$494$	−1.59267e12			−0.0541365
$495$			9.70196e12i			0.326463i
$496$		−	1.91781e12i		−	0.0638848i
$497$	−1.35777e13			−0.447757
$498$	−8.82640e11			−0.0288162
$499$	4.69879e13			1.51874			0.759370	−	0.650659i	$-0.225507\pi$
$499$	4.69879e13			1.51874			0.759370	+	0.650659i	$0.225507\pi$
$500$	−5.34140e13			−1.70925
$501$	1.29493e12			0.0410257
$502$			4.05697e12i			0.127257i
$503$			1.27426e13i			0.395748i	0.980227	+	0.197874i	$0.0634037\pi$
$503$			1.27426e13i			0.395748i	−0.980227	+	0.197874i	$0.936596\pi$
$504$			9.33145e12i			0.286943i
$505$		−	2.15262e13i		−	0.655406i
$506$			4.14157e13i			1.24857i
$507$	−1.93284e13			−0.576975
$508$	1.32298e13			0.391051
$509$		−	5.19116e13i		−	1.51941i	−0.650267	−	0.759706i	$-0.725343\pi$
$509$		−	5.19116e13i		−	1.51941i	0.650267	−	0.759706i	$-0.274657\pi$
$510$		−	6.73166e12i		−	0.195106i
$511$		−	1.32005e13i		−	0.378866i
$512$		−	5.64711e12i		−	0.160500i
$513$	9.03891e12			0.254407
$514$		−	7.13750e13i		−	1.98944i
$515$		−	1.29026e13i		−	0.356158i
$516$		−	1.68302e13i		−	0.460088i
$517$	4.40439e12			0.119243
$518$	6.57640e13			1.76335
$519$			8.30976e12i			0.220674i
$520$	6.87270e11			0.0180764
$521$	−1.40221e13			−0.365278			−0.182639	−	0.983180i	$-0.558464\pi$
$521$	−1.40221e13			−0.365278			−0.182639	+	0.983180i	$0.558464\pi$
$522$		−	1.37994e13i		−	0.356046i
$523$	−5.23571e13			−1.33803			−0.669017	−	0.743247i	$-0.733285\pi$
$523$	−5.23571e13			−1.33803			−0.669017	+	0.743247i	$0.733285\pi$
$524$			3.43661e13i			0.869909i
$525$	8.99187e12			0.225452
$526$		−	1.06168e14i		−	2.63672i
$527$			8.95624e12i			0.220329i
$528$		−	2.45602e12i		−	0.0598496i
$529$	2.63995e13			0.637261
$530$			3.15538e12i			0.0754524i
$531$	−1.05940e13	−	9.26203e12i	−0.250949	−	0.219398i
$532$	8.20855e13			1.92623
$533$		−	1.01029e12i		−	0.0234859i
$534$	3.77822e12			0.0870124
$535$	−1.44016e12			−0.0328581
$536$	4.54948e13			1.02834
$537$		−	2.80945e13i		−	0.629144i
$538$	4.10548e12			0.0910862
$539$			8.86979e12i			0.194971i
$540$	−1.06080e13			−0.231029
$541$		−	7.10673e13i		−	1.53350i	−0.641947	−	0.766749i	$-0.721873\pi$
$541$		−	7.10673e13i		−	1.53350i	0.641947	−	0.766749i	$-0.278127\pi$
$542$		−	4.25618e13i		−	0.909963i
$543$	1.53102e13			0.324326
$544$			1.40381e13i			0.294656i
$545$			2.02231e13i			0.420596i
$546$	−1.05708e12			−0.0217843
$547$	−1.72935e13			−0.353140			−0.176570	−	0.984288i	$-0.556500\pi$
$547$	−1.72935e13			−0.353140			−0.176570	+	0.984288i	$0.556500\pi$
$548$	−9.77134e13			−1.97720
$549$			2.67347e13i			0.536059i
$550$	4.42198e13			0.878624
$551$	−4.46334e13			−0.878824
$552$	−1.66504e13			−0.324884
$553$	−1.40876e12			−0.0272402
$554$		−	1.41499e14i		−	2.71148i
$555$			2.74889e13i			0.522027i
$556$	4.56266e13			0.858704
$557$	−3.54388e13			−0.661003			−0.330502	−	0.943805i	$-0.607218\pi$
$557$	−3.54388e13			−0.661003			−0.330502	+	0.943805i	$0.607218\pi$
$558$	2.30378e13			0.425862
$559$	7.01023e11			0.0128432
$560$	3.09447e12			0.0561884
$561$			1.14697e13i			0.206413i
$562$			5.06862e12i			0.0904082i
$563$		−	3.72684e13i		−	0.658869i	−0.944178	−	0.329435i	$-0.893142\pi$
$563$		−	3.72684e13i		−	0.658869i	0.944178	−	0.329435i	$-0.106858\pi$
$564$			4.81572e12i			0.0843850i
$565$		−	7.06959e13i		−	1.22787i
$566$	4.39681e13			0.756930
$567$	5.99927e12			0.102373
$568$		−	2.68442e13i		−	0.454055i
$569$			3.76671e13i			0.631541i	0.948836	+	0.315770i	$0.102263\pi$
$569$			3.76671e13i			0.631541i	−0.948836	+	0.315770i	$0.897737\pi$
$570$			5.60064e13i			0.930816i
$571$		−	3.06504e13i		−	0.504959i	−0.967602	−	0.252479i	$-0.918754\pi$
$571$		−	3.06504e13i		−	0.504959i	0.967602	−	0.252479i	$-0.0812460\pi$
$572$	−3.18473e12			−0.0520106
$573$			3.73475e13i			0.604628i
$574$			8.49938e13i			1.36404i
$575$			1.60444e13i			0.255262i
$576$	3.44117e13			0.542742
$577$	2.72045e11			0.00425364			0.00212682	−	0.999998i	$-0.499323\pi$
$577$	2.72045e11			0.00425364			0.00212682	+	0.999998i	$0.499323\pi$
$578$			9.56934e13i			1.48335i
$579$	−2.22016e13			−0.341186
$580$	5.23816e13			0.798066
$581$			1.89482e12i			0.0286211i
$582$	−4.54275e13			−0.680305
$583$		−	5.37627e12i		−	0.0798248i
$584$	2.60985e13			0.384195
$585$		−	4.41852e11i		−	0.00644908i
$586$		−	1.75097e14i		−	2.53390i
$587$			6.02247e13i			0.864139i	0.901840	+	0.432070i	$0.142217\pi$
$587$			6.02247e13i			0.864139i	−0.901840	+	0.432070i	$0.857783\pi$
$588$	−9.69814e12			−0.137975
$589$		−	7.45146e13i		−	1.05115i
$590$	5.73889e13	−	6.56418e13i	0.802727	−	0.918164i
$591$	−4.88541e13			−0.677585
$592$		−	6.95872e12i		−	0.0957019i
$593$	−4.32391e13			−0.589662			−0.294831	−	0.955549i	$-0.595263\pi$
$593$	−4.32391e13			−0.589662			−0.294831	+	0.955549i	$0.595263\pi$
$594$	2.95029e13			0.398963
$595$	−1.44513e13			−0.193786
$596$			5.73240e13i			0.762263i
$597$	5.09233e13			0.671498
$598$		−	1.88618e12i		−	0.0246647i
$599$	−5.02386e13			−0.651483			−0.325742	−	0.945459i	$-0.605614\pi$
$599$	−5.02386e13			−0.651483			−0.325742	+	0.945459i	$0.605614\pi$
$600$			1.77777e13i			0.228623i
$601$			4.09214e13i			0.521889i	0.965354	+	0.260945i	$0.0840340\pi$
$601$			4.09214e13i			0.521889i	−0.965354	+	0.260945i	$0.915966\pi$
$602$	−5.89760e13			−0.745922
$603$		−	2.92490e13i		−	0.366881i
$604$			1.22305e14i			1.52146i
$605$	4.09007e13			0.504607
$606$	−6.54596e13			−0.800958
$607$	−6.92676e13			−0.840595			−0.420298	−	0.907386i	$-0.638074\pi$
$607$	−6.92676e13			−0.840595			−0.420298	+	0.907386i	$0.638074\pi$
$608$		−	1.16795e14i		−	1.40575i
$609$	−2.96239e13			−0.353636
$610$	−1.65652e14			−1.96132
$611$	−2.00587e11			−0.00235558
$612$	−1.25408e13			−0.146073
$613$			8.23233e13i			0.951087i	0.879692	+	0.475543i	$0.157749\pi$
$613$			8.23233e13i			0.951087i	−0.879692	+	0.475543i	$0.842251\pi$
$614$			1.93577e14i			2.21826i
$615$	−3.55269e13			−0.403814
$616$	9.85143e13			1.11070
$617$	−7.21436e12			−0.0806811			−0.0403406	−	0.999186i	$-0.512844\pi$
$617$	−7.21436e12			−0.0806811			−0.0403406	+	0.999186i	$0.512844\pi$
$618$	−3.92360e13			−0.435253
$619$	−4.97946e13			−0.547935			−0.273968	−	0.961739i	$-0.588336\pi$
$619$	−4.97946e13			−0.547935			−0.273968	+	0.961739i	$0.588336\pi$
$620$			8.74500e13i			0.954556i
$621$			1.07047e13i			0.115909i
$622$			2.91367e14i			3.12959i
$623$		−	8.11094e12i		−	0.0864233i
$624$			1.11853e11i			0.00118229i
$625$	−3.78174e13			−0.396544
$626$	5.97880e13			0.621931
$627$		−	9.54259e13i		−	0.984756i
$628$			1.27365e14i			1.30392i
$629$			3.24974e13i			0.330062i
$630$			3.71724e13i			0.374557i
$631$	−8.82637e13			−0.882339			−0.441169	−	0.897424i	$-0.645436\pi$
$631$	−8.82637e13			−0.882339			−0.441169	+	0.897424i	$0.645436\pi$
$632$		−	2.78523e12i		−	0.0276233i
$633$			4.81417e13i			0.473699i
$634$			9.00311e13i			0.878914i
$635$	1.93779e13			0.187689
$636$	5.87836e12			0.0564898
$637$		−	4.03953e11i		−	0.00385153i
$638$	−1.45683e14			−1.37818
$639$	−1.72584e13			−0.161993
$640$			1.26549e14i			1.17858i
$641$	−6.88079e13			−0.635840			−0.317920	−	0.948117i	$-0.602984\pi$
$641$	−6.88079e13			−0.635840			−0.317920	+	0.948117i	$0.602984\pi$
$642$			4.37943e12i			0.0401552i
$643$	1.12758e14			1.02587			0.512937	−	0.858426i	$-0.328557\pi$
$643$	1.12758e14			1.02587			0.512937	+	0.858426i	$0.328557\pi$
$644$			9.72129e13i			0.877595i
$645$		−	2.46516e13i		−	0.220824i
$646$			6.62108e13i			0.588527i
$647$	−8.96543e13			−0.790769			−0.395385	−	0.918516i	$-0.629389\pi$
$647$	−8.96543e13			−0.790769			−0.395385	+	0.918516i	$0.629389\pi$
$648$			1.18611e13i			0.103812i
$649$	−9.77815e13	+	1.11843e14i	−0.849245	+	0.971371i
$650$	−2.01388e12			−0.0173567
$651$		−	4.94566e13i		−	0.422979i
$652$	−2.83445e14			−2.40564
$653$	8.89596e13			0.749251			0.374625	−	0.927176i	$-0.377771\pi$
$653$	8.89596e13			0.749251			0.374625	+	0.927176i	$0.377771\pi$
$654$	6.14969e13			0.514001
$655$			5.03368e13i			0.417522i
$656$	8.99349e12			0.0740303
$657$		−	1.67790e13i		−	0.137069i
$658$	1.68751e13			0.136810
$659$		−	3.64044e13i		−	0.292905i	−0.989218	−	0.146453i	$-0.953214\pi$
$659$		−	3.64044e13i		−	0.292905i	0.989218	−	0.146453i	$-0.0467856\pi$
$660$			1.11991e14i			0.894263i
$661$	−4.09725e13			−0.324702			−0.162351	−	0.986733i	$-0.551908\pi$
$661$	−4.09725e13			−0.324702			−0.162351	+	0.986733i	$0.551908\pi$
$662$			3.42696e14i			2.69537i
$663$		−	5.22358e11i		−	0.00407756i
$664$	−3.74622e12			−0.0290237
$665$	1.20232e14			0.924514
$666$	8.35917e13			0.637958
$667$		−	5.28588e13i		−	0.400395i
$668$	1.49476e13			0.112380
$669$	2.03784e13			0.152069
$670$	1.81231e14			1.34233
$671$	2.82244e14			2.07497
$672$		−	7.75189e13i		−	0.565667i
$673$		−	4.87480e12i		−	0.0353087i	−0.999844	−	0.0176544i	$-0.994380\pi$
$673$		−	4.87480e12i		−	0.0353087i	0.999844	−	0.0176544i	$-0.00561985\pi$
$674$	1.70326e14			1.22457
$675$	1.14294e13			0.0815655
$676$	−2.23112e14			−1.58048
$677$	9.64158e13			0.677961			0.338981	−	0.940793i	$-0.389918\pi$
$677$	9.64158e13			0.677961			0.338981	+	0.940793i	$0.389918\pi$
$678$	−2.14981e14			−1.50055
$679$			9.75219e13i			0.675699i
$680$		−	2.85714e13i		−	0.196511i
$681$		−	1.15816e14i		−	0.790740i
$682$		−	2.43215e14i		−	1.64842i
$683$			7.87699e13i			0.529977i	0.964252	+	0.264988i	$0.0853682\pi$
$683$			7.87699e13i			0.529977i	−0.964252	+	0.264988i	$0.914632\pi$
$684$	1.04338e14			0.696885
$685$	−1.43123e14			−0.948979
$686$			2.58881e14i			1.70404i
$687$		−	3.34022e13i		−	0.218269i
$688$			6.24046e12i			0.0404832i
$689$			2.44849e11i			0.00157689i
$690$	−6.63278e13			−0.424082
$691$		−	6.25229e13i		−	0.396871i	−0.980114	−	0.198435i	$-0.936414\pi$
$691$		−	6.25229e13i		−	0.396871i	0.980114	−	0.198435i	$-0.0635860\pi$
$692$			9.59211e13i			0.604482i
$693$		−	6.33358e13i		−	0.396262i
$694$	2.47321e14			1.53626
$695$	6.68303e13			0.412144
$696$		−	5.85690e13i		−	0.358610i
$697$	−4.19999e13			−0.255320
$698$	8.09595e13			0.488642
$699$		−	1.61664e14i		−	0.968787i
$700$	1.03795e14			0.617569
$701$			2.41001e14i			1.42373i	0.702316	+	0.711865i	$0.252149\pi$
$701$			2.41001e14i			1.42373i	−0.702316	+	0.711865i	$0.747851\pi$
$702$	−1.34364e12			−0.00788129
$703$		−	2.70374e14i		−	1.57466i
$704$		−	3.63292e14i		−	2.10084i
$705$			7.05369e12i			0.0405015i
$706$	1.16473e14			0.664052
$707$			1.40526e14i			0.795536i
$708$	−1.22288e14	−	1.06913e14i	−0.687412	−	0.600987i
$709$	−2.57817e14			−1.43906			−0.719532	−	0.694459i	$-0.755644\pi$
$709$	−2.57817e14			−1.43906			−0.719532	+	0.694459i	$0.755644\pi$
$710$		−	1.06935e14i		−	0.592693i
$711$	−1.79065e12			−0.00985515
$712$	1.60360e13			0.0876389
$713$	8.82467e13			0.478907
$714$			4.39452e13i			0.236821i
$715$	−4.66474e12			−0.0249630
$716$		−	3.24300e14i		−	1.72338i
$717$	−8.11185e13			−0.428079
$718$		−	3.34689e14i		−	1.75396i
$719$			1.74694e14i			0.909148i	0.890709	+	0.454574i	$0.150209\pi$
$719$			1.74694e14i			0.909148i	−0.890709	+	0.454574i	$0.849791\pi$
$720$	3.93334e12			0.0203282
$721$			8.42302e13i			0.432306i
$722$		−	2.35638e14i		−	1.20105i
$723$	−6.47344e13			−0.327675
$724$	1.76729e14			0.888411
$725$	−5.64377e13			−0.281760
$726$		−	1.24376e14i		−	0.616670i
$727$	−1.26239e13			−0.0621617			−0.0310809	−	0.999517i	$-0.509895\pi$
$727$	−1.26239e13			−0.0621617			−0.0310809	+	0.999517i	$0.509895\pi$
$728$	−4.48660e12			−0.0219412
$729$	7.62560e12			0.0370370
$730$	1.03965e14			0.501503
$731$		−	2.91431e13i		−	0.139621i
$732$			3.08603e14i			1.46840i
$733$	−2.14382e13			−0.101314			−0.0506569	−	0.998716i	$-0.516131\pi$
$733$	−2.14382e13			−0.101314			−0.0506569	+	0.998716i	$0.516131\pi$
$734$	3.44423e14			1.61664
$735$	−1.42051e13			−0.0662228
$736$	1.38319e14			0.640462
$737$	−3.08789e14			−1.42012
$738$			1.08035e14i			0.493493i
$739$		−	1.33153e14i		−	0.604127i	−0.953288	−	0.302063i	$-0.902325\pi$
$739$		−	1.33153e14i		−	0.604127i	0.953288	−	0.302063i	$-0.0976754\pi$
$740$			3.17310e14i			1.42996i
$741$			4.34594e12i			0.0194533i
$742$		−	2.05988e13i		−	0.0915846i
$743$	1.42151e14			0.627777			0.313889	−	0.949460i	$-0.398368\pi$
$743$	1.42151e14			0.627777			0.313889	+	0.949460i	$0.398368\pi$
$744$	9.77799e13			0.428928
$745$			8.39638e13i			0.365856i
$746$		−	1.19769e14i		−	0.518381i
$747$			2.40848e12i			0.0103547i
$748$			1.32396e14i			0.565416i
$749$	9.40158e12			0.0398834
$750$			2.37912e14i			1.00256i
$751$		−	2.91415e14i		−	1.21986i	−0.792453	−	0.609932i	$-0.791197\pi$
$751$		−	2.91415e14i		−	1.21986i	0.792453	−	0.609932i	$-0.208803\pi$
$752$		−	1.78562e12i		−	0.00742504i
$753$	1.10703e13			0.0457283
$754$	6.63478e12			0.0272251
$755$			1.79143e14i			0.730239i
$756$	6.92507e13			0.280424
$757$	−1.57836e14			−0.634933			−0.317466	−	0.948269i	$-0.602832\pi$
$757$	−1.57836e14			−0.634933			−0.317466	+	0.948269i	$0.602832\pi$
$758$		−	5.31060e14i		−	2.12226i
$759$	1.13012e14			0.448657
$760$			2.37710e14i			0.937518i
$761$	−2.73383e14			−1.07115			−0.535573	−	0.844489i	$-0.679904\pi$
$761$	−2.73383e14			−1.07115			−0.535573	+	0.844489i	$0.679904\pi$
$762$		−	5.89268e13i		−	0.229371i
$763$		−	1.32019e14i		−	0.510521i
$764$			4.31109e14i			1.65623i
$765$	−1.83688e13			−0.0701091
$766$		−	6.26346e14i		−	2.37504i
$767$	4.45322e12	−	5.09362e12i	0.0167763	−	0.0191889i
$768$	1.33661e14			0.500264
$769$			4.46828e14i			1.66153i	0.556622	+	0.830766i	$0.312097\pi$
$769$			4.46828e14i			1.66153i	−0.556622	+	0.830766i	$0.687903\pi$
$770$	3.92438e14			1.44983
$771$	−1.94762e14			−0.714880
$772$	−2.56277e14			−0.934594
$773$		−	2.97425e12i		−	0.0107766i	−0.999985	−	0.00538828i	$-0.998285\pi$
$773$		−	2.97425e12i		−	0.0107766i	0.999985	−	0.00538828i	$-0.00171515\pi$
$774$	−7.49637e13			−0.269865
$775$		−	9.42216e13i		−	0.337009i
$776$	−1.92809e14			−0.685203
$777$		−	1.79451e14i		−	0.633639i
$778$		−	6.02318e14i		−	2.11314i
$779$	3.49433e14			1.21808

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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-51.4146i q^{2} -140.296 q^{3} -1619.46 q^{4} -2372.07 q^{5} +7213.27i q^{6} +15485.2 q^{7} +30615.5i q^{8} +19683.0 q^{9} +O(q^{10})\)	\(q-51.4146i q^{2} -140.296 q^{3} -1619.46 q^{4} -2372.07 q^{5} +7213.27i q^{6} +15485.2 q^{7} +30615.5i q^{8} +19683.0 q^{9} +121959. i q^{10} -207798. i q^{11} +227204. q^{12} +9463.66i q^{13} -796164. i q^{14} +332792. q^{15} -84244.9 q^{16} +393426. q^{17} -1.01199e6i q^{18} -3.27325e6 q^{19} +3.84147e6 q^{20} -2.17251e6 q^{21} -1.06839e7 q^{22} -3.87647e6i q^{23} -4.29524e6i q^{24} -4.13893e6 q^{25} +486571. q^{26} -2.76145e6 q^{27} -2.50777e7 q^{28} +1.36358e7 q^{29} -1.71104e7i q^{30} +2.27647e7i q^{31} +3.56817e7i q^{32} +2.91533e7i q^{33} -2.02279e7i q^{34} -3.67318e7 q^{35} -3.18759e7 q^{36} +8.26010e7i q^{37} +1.68293e8i q^{38} -1.32772e6i q^{39} -7.26220e7i q^{40} -1.06754e8 q^{41} +1.11699e8i q^{42} -7.40752e7i q^{43} +3.36521e8i q^{44} -4.66894e7 q^{45} -1.99307e8 q^{46} +2.11955e7i q^{47} +1.18192e7 q^{48} -4.26846e7 q^{49} +2.12802e8i q^{50} -5.51962e7 q^{51} -1.53261e7i q^{52} +2.58725e7 q^{53} +1.41979e8i q^{54} +4.92911e8i q^{55} +4.74087e8i q^{56} +4.59224e8 q^{57} -7.01080e8i q^{58} +(-5.38229e8 - 4.70560e8i) q^{59} -5.38944e8 q^{60} +1.35826e9i q^{61} +1.17044e9 q^{62} +3.04795e8 q^{63} +1.74830e9 q^{64} -2.24484e7i q^{65} +1.49890e9 q^{66} -1.48600e9i q^{67} -6.37140e8 q^{68} +5.43854e8i q^{69} +1.88855e9i q^{70} -8.76816e8 q^{71} +6.02605e8i q^{72} -8.52461e8i q^{73} +4.24690e9 q^{74} +5.80676e8 q^{75} +5.30091e9 q^{76} -3.21779e9i q^{77} -6.82640e7 q^{78} -9.09746e7 q^{79} +1.99834e8 q^{80} +3.87420e8 q^{81} +5.48872e9i q^{82} +1.22363e8i q^{83} +3.51830e9 q^{84} -9.33233e8 q^{85} -3.80855e9 q^{86} -1.91305e9 q^{87} +6.36185e9 q^{88} -5.23788e8i q^{89} +2.40052e9i q^{90} +1.46546e8i q^{91} +6.27780e9i q^{92} -3.19380e9i q^{93} +1.08976e9 q^{94} +7.76436e9 q^{95} -5.00601e9i q^{96} +6.29776e9i q^{97} +2.19462e9i q^{98} -4.09009e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + O(q^{10}) \)	\( 100q - 51200q^{4} + 18392q^{7} + 1968300q^{9} + 620136q^{12} + 662904q^{15} + 27925160q^{16} - 2053136q^{17} - 5169828q^{19} - 1076324q^{20} - 1829224q^{22} + 215378180q^{25} - 22082700q^{26} - 102921320q^{28} - 112503588q^{29} - 76491392q^{35} - 1007769600q^{36} + 473464516q^{41} + 1215405588q^{46} - 1676272320q^{48} + 1975297276q^{49} + 733970808q^{51} + 3267506728q^{53} + 591502824q^{57} + 508142200q^{59} + 1264196808q^{60} - 6538206968q^{62} + 362009736q^{63} - 10324137972q^{64} - 2764346616q^{66} + 9997685952q^{68} + 14908523204q^{71} + 4863508712q^{74} + 1890481680q^{75} + 2044437240q^{76} - 758396196q^{78} - 3599839500q^{79} - 23217941144q^{80} + 38742048900q^{81} - 13094894808q^{84} + 23360564412q^{85} + 12186923752q^{86} + 7965322272q^{87} + 32415437996q^{88} - 22098322280q^{94} + 7834510028q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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