LMFDB - Embedded newform 177.9.c.a.58.8

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$72.1060139808$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.8
Character	$\chi$	$=$	177.58
Dual form			177.9.c.a.58.73

$q$-expansion

$f(q)$	$=$	$q-27.0356i q^{2} +46.7654 q^{3} -474.926 q^{4} +21.6634 q^{5} -1264.33i q^{6} -141.547 q^{7} +5918.81i q^{8} +2187.00 q^{9} +O(q^{10})$	\(q-27.0356i q^{2} +46.7654 q^{3} -474.926 q^{4} +21.6634 q^{5} -1264.33i q^{6} -141.547 q^{7} +5918.81i q^{8} +2187.00 q^{9} -585.684i q^{10} +7994.24i q^{11} -22210.1 q^{12} -55243.6i q^{13} +3826.80i q^{14} +1013.10 q^{15} +38437.8 q^{16} +12422.3 q^{17} -59127.0i q^{18} -130025. q^{19} -10288.5 q^{20} -6619.48 q^{21} +216129. q^{22} +379176. i q^{23} +276795. i q^{24} -390156. q^{25} -1.49355e6 q^{26} +102276. q^{27} +67224.1 q^{28} -149038. q^{29} -27389.7i q^{30} -1.11578e6i q^{31} +476026. i q^{32} +373854. i q^{33} -335844. i q^{34} -3066.38 q^{35} -1.03866e6 q^{36} +2.51615e6i q^{37} +3.51531e6i q^{38} -2.58349e6i q^{39} +128222. i q^{40} +171038. q^{41} +178962. i q^{42} -3.98516e6i q^{43} -3.79667e6i q^{44} +47377.8 q^{45} +1.02513e7 q^{46} +5.38311e6i q^{47} +1.79756e6 q^{48} -5.74477e6 q^{49} +1.05481e7i q^{50} +580931. q^{51} +2.62366e7i q^{52} +6.71423e6 q^{53} -2.76509e6i q^{54} +173182. i q^{55} -837787. i q^{56} -6.08066e6 q^{57} +4.02934e6i q^{58} +(-1.06905e7 + 5.70473e6i) q^{59} -481146. q^{60} +2.35441e7i q^{61} -3.01659e7 q^{62} -309562. q^{63} +2.27097e7 q^{64} -1.19676e6i q^{65} +1.01074e7 q^{66} +3.32648e7i q^{67} -5.89965e6 q^{68} +1.77323e7i q^{69} +82901.5i q^{70} +2.18665e7 q^{71} +1.29444e7i q^{72} -5.20187e6i q^{73} +6.80258e7 q^{74} -1.82458e7 q^{75} +6.17522e7 q^{76} -1.13156e6i q^{77} -6.98462e7 q^{78} -3.31695e7 q^{79} +832693. q^{80} +4.78297e6 q^{81} -4.62413e6i q^{82} +3.11342e7i q^{83} +3.14376e6 q^{84} +269108. q^{85} -1.07741e8 q^{86} -6.96982e6 q^{87} -4.73164e7 q^{88} -8.38935e7i q^{89} -1.28089e6i q^{90} +7.81953e6i q^{91} -1.80081e8i q^{92} -5.21800e7i q^{93} +1.45536e8 q^{94} -2.81678e6 q^{95} +2.22615e7i q^{96} +2.10512e7i q^{97} +1.55313e8i q^{98} +1.74834e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) $	$ 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{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 $

$2$		−	27.0356i		−	1.68973i	−0.534981	−	0.844864i	$-0.679681\pi$
$2$		−	27.0356i		−	1.68973i	0.534981	−	0.844864i	$-0.320319\pi$
$3$	46.7654			0.577350
$3$	46.7654			0.577350
$4$	−474.926			−1.85518
$5$	21.6634			0.0346614			0.0173307	−	0.999850i	$-0.494483\pi$
$5$	21.6634			0.0346614			0.0173307	+	0.999850i	$0.494483\pi$
$6$		−	1264.33i		−	0.975565i
$7$	−141.547			−0.0589531			−0.0294766	−	0.999565i	$-0.509384\pi$
$7$	−141.547			−0.0589531			−0.0294766	+	0.999565i	$0.509384\pi$
$8$			5918.81i			1.44502i
$9$	2187.00			0.333333
$10$		−	585.684i		−	0.0585684i
$11$			7994.24i			0.546017i	0.962012	+	0.273009i	$0.0880188\pi$
$11$			7994.24i			0.546017i	−0.962012	+	0.273009i	$0.911981\pi$
$12$	−22210.1			−1.07109
$13$		−	55243.6i		−	1.93423i	−0.254339	−	0.967115i	$-0.581858\pi$
$13$		−	55243.6i		−	1.93423i	0.254339	−	0.967115i	$-0.418142\pi$
$14$			3826.80i			0.0996148i
$15$	1013.10			0.0200118
$16$	38437.8			0.586514
$17$	12422.3			0.148732			0.0743661	−	0.997231i	$-0.476307\pi$
$17$	12422.3			0.148732			0.0743661	+	0.997231i	$0.476307\pi$
$18$		−	59127.0i		−	0.563243i
$19$	−130025.			−0.997728			−0.498864	−	0.866680i	$-0.666249\pi$
$19$	−130025.			−0.997728			−0.498864	+	0.866680i	$0.666249\pi$
$20$	−10288.5			−0.0643032
$21$	−6619.48			−0.0340366
$22$	216129.			0.922621
$23$			379176.i			1.35497i	0.735536	+	0.677485i	$0.236930\pi$
$23$			379176.i			1.35497i	−0.735536	+	0.677485i	$0.763070\pi$
$24$			276795.i			0.834284i
$25$	−390156.			−0.998799
$26$	−1.49355e6			−3.26832
$27$	102276.			0.192450
$28$	67224.1			0.109369
$29$	−149038.			−0.210720			−0.105360	−	0.994434i	$-0.533599\pi$
$29$	−149038.			−0.210720			−0.105360	+	0.994434i	$0.533599\pi$
$30$		−	27389.7i		−	0.0338145i
$31$		−	1.11578e6i		−	1.20818i	−0.796915	−	0.604092i	$-0.793536\pi$
$31$		−	1.11578e6i		−	1.20818i	0.796915	−	0.604092i	$-0.206464\pi$
$32$			476026.i			0.453973i
$33$			373854.i			0.315243i
$34$		−	335844.i		−	0.251317i
$35$	−3066.38			−0.00204340
$36$	−1.03866e6			−0.618393
$37$			2.51615e6i			1.34255i	0.741209	+	0.671275i	$0.234253\pi$
$37$			2.51615e6i			1.34255i	−0.741209	+	0.671275i	$0.765747\pi$
$38$			3.51531e6i			1.68589i
$39$		−	2.58349e6i		−	1.11673i
$40$			128222.i			0.0500865i
$41$	171038.			0.0605283			0.0302641	−	0.999542i	$-0.490365\pi$
$41$	171038.			0.0605283			0.0302641	+	0.999542i	$0.490365\pi$
$42$			178962.i			0.0575126i
$43$		−	3.98516e6i		−	1.16566i	−0.812594	−	0.582831i	$-0.801945\pi$
$43$		−	3.98516e6i		−	1.16566i	0.812594	−	0.582831i	$-0.198055\pi$
$44$		−	3.79667e6i		−	1.01296i
$45$	47377.8			0.0115538
$46$	1.02513e7			2.28953
$47$			5.38311e6i			1.10317i	0.834119	+	0.551584i	$0.185977\pi$
$47$			5.38311e6i			1.10317i	−0.834119	+	0.551584i	$0.814023\pi$
$48$	1.79756e6			0.338624
$49$	−5.74477e6			−0.996525
$50$			1.05481e7i			1.68770i
$51$	580931.			0.0858705
$52$			2.62366e7i			3.58835i
$53$	6.71423e6			0.850928			0.425464	−	0.904975i	$-0.360111\pi$
$53$	6.71423e6			0.850928			0.425464	+	0.904975i	$0.360111\pi$
$54$		−	2.76509e6i		−	0.325188i
$55$			173182.i			0.0189257i
$56$		−	837787.i		−	0.0851886i
$57$	−6.08066e6			−0.576038
$58$			4.02934e6i			0.356059i
$59$	−1.06905e7	+	5.70473e6i	−0.882246	+	0.470789i
$59$	−1.06905e7	+	5.70473e6i	−0.882246	+	0.470789i
$60$	−481146.			−0.0371255
$61$			2.35441e7i			1.70045i	0.526421	+	0.850224i	$0.323534\pi$
$61$			2.35441e7i			1.70045i	−0.526421	+	0.850224i	$0.676466\pi$
$62$	−3.01659e7			−2.04150
$63$	−309562.			−0.0196510
$64$	2.27097e7			1.35361
$65$		−	1.19676e6i		−	0.0670432i
$66$	1.01074e7			0.532675
$67$			3.32648e7i			1.65077i	0.564571	+	0.825385i	$0.309042\pi$
$67$			3.32648e7i			1.65077i	−0.564571	+	0.825385i	$0.690958\pi$
$68$	−5.89965e6			−0.275925
$69$			1.77323e7i			0.782293i
$70$			82901.5i			0.00345279i
$71$	2.18665e7			0.860490			0.430245	−	0.902712i	$-0.358427\pi$
$71$	2.18665e7			0.860490			0.430245	+	0.902712i	$0.358427\pi$
$72$			1.29444e7i			0.481674i
$73$		−	5.20187e6i		−	0.183176i	−0.995797	−	0.0915880i	$-0.970806\pi$
$73$		−	5.20187e6i		−	0.183176i	0.995797	−	0.0915880i	$-0.0291943\pi$
$74$	6.80258e7			2.26854
$75$	−1.82458e7			−0.576657
$76$	6.17522e7			1.85096
$77$		−	1.13156e6i		−	0.0321894i
$78$	−6.98462e7			−1.88697
$79$	−3.31695e7			−0.851590			−0.425795	−	0.904820i	$-0.640006\pi$
$79$	−3.31695e7			−0.851590			−0.425795	+	0.904820i	$0.640006\pi$
$80$	832693.			0.0203294
$81$	4.78297e6			0.111111
$82$		−	4.62413e6i		−	0.102276i
$83$			3.11342e7i			0.656033i	0.944672	+	0.328017i	$0.106380\pi$
$83$			3.11342e7i			0.656033i	−0.944672	+	0.328017i	$0.893620\pi$
$84$	3.14376e6			0.0631441
$85$	269108.			0.00515527
$86$	−1.07741e8			−1.96965
$87$	−6.96982e6			−0.121659
$88$	−4.73164e7			−0.789007
$89$		−	8.38935e7i		−	1.33711i	−0.743661	−	0.668557i	$-0.766912\pi$
$89$		−	8.38935e7i		−	1.33711i	0.743661	−	0.668557i	$-0.233088\pi$
$90$		−	1.28089e6i		−	0.0195228i
$91$			7.81953e6i			0.114029i
$92$		−	1.80081e8i		−	2.51372i
$93$		−	5.21800e7i		−	0.697545i
$94$	1.45536e8			1.86406
$95$	−2.81678e6			−0.0345827
$96$			2.22615e7i			0.262102i
$97$			2.10512e7i			0.237787i	0.992907	+	0.118894i	$0.0379348\pi$
$97$			2.10512e7i			0.237787i	−0.992907	+	0.118894i	$0.962065\pi$
$98$			1.55313e8i			1.68386i
$99$			1.74834e7i			0.182006i
$100$	1.85295e8			1.85295
$101$		−	1.83997e8i		−	1.76817i	−0.467322	−	0.884087i	$-0.654781\pi$
$101$		−	1.83997e8i		−	1.76817i	0.467322	−	0.884087i	$-0.345219\pi$
$102$		−	1.57059e7i		−	0.145098i
$103$			7.76453e7i			0.689869i	0.938627	+	0.344934i	$0.112099\pi$
$103$			7.76453e7i			0.689869i	−0.938627	+	0.344934i	$0.887901\pi$
$104$	3.26976e8			2.79501
$105$	−143400.			−0.00117976
$106$		−	1.81524e8i		−	1.43784i
$107$	−1.87626e8			−1.43139			−0.715695	−	0.698413i	$-0.753890\pi$
$107$	−1.87626e8			−1.43139			−0.715695	+	0.698413i	$0.753890\pi$
$108$	−4.85735e7			−0.357030
$109$			7.95454e7i			0.563520i	0.959485	+	0.281760i	$0.0909182\pi$
$109$			7.95454e7i			0.563520i	−0.959485	+	0.281760i	$0.909082\pi$
$110$	4.68210e6			0.0319794
$111$			1.17669e8i			0.775121i
$112$	−5.44073e6			−0.0345768
$113$		−	1.61849e8i		−	0.992650i	−0.868137	−	0.496325i	$-0.834682\pi$
$113$		−	1.61849e8i		−	0.992650i	0.868137	−	0.496325i	$-0.165318\pi$
$114$			1.64395e8i			0.973348i
$115$			8.21425e6i			0.0469652i
$116$	7.07821e7			0.390923
$117$		−	1.20818e8i		−	0.644743i
$118$	1.54231e8	+	2.89024e8i	0.795506	+	1.49075i
$119$	−1.75833e6			−0.00876823
$120$			5.99633e6i			0.0289175i
$121$	1.50451e8			0.701865
$122$	6.36531e8			2.87329
$123$	7.99867e6			0.0349460
$124$			5.29915e8i			2.24140i
$125$	−1.69144e7			−0.0692812
$126$			8.36921e6i			0.0332049i
$127$	3.43849e8			1.32176			0.660881	−	0.750491i	$-0.270183\pi$
$127$	3.43849e8			1.32176			0.660881	+	0.750491i	$0.270183\pi$
$128$		−	4.92110e8i		−	1.83325i
$129$		−	1.86368e8i		−	0.672995i
$130$	−3.23553e7			−0.113285
$131$			5.64994e8i			1.91849i	0.282580	+	0.959244i	$0.408810\pi$
$131$			5.64994e8i			1.91849i	−0.282580	+	0.959244i	$0.591190\pi$
$132$		−	1.77553e8i		−	0.584833i
$133$	1.84046e7			0.0588192
$134$	8.99337e8			2.78935
$135$	2.21564e6			0.00667060
$136$			7.35250e7i			0.214921i
$137$	3.35466e7			0.0952283			0.0476142	−	0.998866i	$-0.484838\pi$
$137$	3.35466e7			0.0952283			0.0476142	+	0.998866i	$0.484838\pi$
$138$	4.79405e8			1.32186
$139$	−2.60900e8			−0.698899			−0.349450	−	0.936955i	$-0.613631\pi$
$139$	−2.60900e8			−0.698899			−0.349450	+	0.936955i	$0.613631\pi$
$140$	1.45630e6			0.00379088
$141$			2.51743e8i			0.636915i
$142$		−	5.91175e8i		−	1.45399i
$143$	4.41630e8			1.05612
$144$	8.40634e7			0.195505
$145$	−3.22867e6			−0.00730385
$146$	−1.40636e8			−0.309517
$147$	−2.68656e8			−0.575344
$148$		−	1.19499e9i		−	2.49067i
$149$		−	7.11310e8i		−	1.44316i	−0.692332	−	0.721579i	$-0.743417\pi$
$149$		−	7.11310e8i		−	1.44316i	0.692332	−	0.721579i	$-0.256583\pi$
$150$			4.93286e8i			0.974393i
$151$		−	3.12895e8i		−	0.601853i	−0.953647	−	0.300927i	$-0.902704\pi$
$151$		−	3.12895e8i		−	0.601853i	0.953647	−	0.300927i	$-0.0972959\pi$
$152$		−	7.69593e8i		−	1.44174i
$153$	2.71675e7			0.0495774
$154$	−3.05924e7			−0.0543914
$155$		−	2.41717e7i		−	0.0418774i
$156$			1.22696e9i			2.07173i
$157$			1.07100e9i			1.76275i	0.472420	+	0.881374i	$0.343381\pi$
$157$			1.07100e9i			1.76275i	−0.472420	+	0.881374i	$0.656619\pi$
$158$			8.96759e8i			1.43896i
$159$	3.13994e8			0.491284
$160$			1.03123e7i			0.0157354i
$161$		−	5.36711e7i		−	0.0798798i
$162$		−	1.29311e8i		−	0.187748i
$163$	−7.32147e8			−1.03717			−0.518583	−	0.855027i	$-0.673540\pi$
$163$	−7.32147e8			−1.03717			−0.518583	+	0.855027i	$0.673540\pi$
$164$	−8.12306e7			−0.112291
$165$			8.09894e6i			0.0109268i
$166$	8.41734e8			1.10852
$167$	−5.61899e8			−0.722424			−0.361212	−	0.932484i	$-0.617637\pi$
$167$	−5.61899e8			−0.722424			−0.361212	+	0.932484i	$0.617637\pi$
$168$		−	3.91794e7i		−	0.0491837i
$169$	−2.23612e9			−2.74125
$170$		−	7.27551e6i		−	0.00871100i
$171$	−2.84364e8			−0.332576
$172$			1.89266e9i			2.16251i
$173$			7.34364e7i			0.0819836i	0.999159	+	0.0409918i	$0.0130517\pi$
$173$			7.34364e7i			0.0819836i	−0.999159	+	0.0409918i	$0.986948\pi$
$174$			1.88434e8i			0.205571i
$175$	5.52252e7			0.0588823
$176$			3.07281e8i			0.320247i
$177$	−4.99945e8	+	2.66784e8i	−0.509365	+	0.271810i
$178$	−2.26811e9			−2.25936
$179$			1.91943e8i			0.186965i	0.995621	+	0.0934824i	$0.0297999\pi$
$179$			1.91943e8i			0.186965i	−0.995621	+	0.0934824i	$0.970200\pi$
$180$	−2.25010e7			−0.0214344
$181$	−1.53677e9			−1.43184			−0.715919	−	0.698183i	$-0.753992\pi$
$181$	−1.53677e9			−1.43184			−0.715919	+	0.698183i	$0.753992\pi$
$182$	2.11406e8			0.192678
$183$			1.10105e9i			0.981754i
$184$	−2.24427e9			−1.95796
$185$			5.45084e7i			0.0465347i
$186$	−1.41072e9			−1.17866
$187$			9.93065e7i			0.0812103i
$188$		−	2.55658e9i		−	2.04658i
$189$	−1.44768e7			−0.0113455
$190$			7.61535e7i			0.0584353i
$191$			3.00969e8i			0.226146i	0.993587	+	0.113073i	$0.0360693\pi$
$191$			3.00969e8i			0.226146i	−0.993587	+	0.113073i	$0.963931\pi$
$192$	1.06203e9			0.781504
$193$	1.92053e9			1.38418			0.692090	−	0.721811i	$-0.256690\pi$
$193$	1.92053e9			1.38418			0.692090	+	0.721811i	$0.256690\pi$
$194$	5.69132e8			0.401796
$195$		−	5.59671e7i		−	0.0387074i
$196$	2.72834e9			1.84873
$197$	2.07571e9			1.37817			0.689084	−	0.724682i	$-0.258013\pi$
$197$	2.07571e9			1.37817			0.689084	+	0.724682i	$0.258013\pi$
$198$	4.72675e8			0.307540
$199$	−2.89818e9			−1.84805			−0.924024	−	0.382334i	$-0.875120\pi$
$199$	−2.89818e9			−1.84805			−0.924024	+	0.382334i	$0.875120\pi$
$200$		−	2.30926e9i		−	1.44329i
$201$			1.55564e9i			0.953072i
$202$	−4.97448e9			−2.98773
$203$	2.10958e7			0.0124226
$204$	−2.75899e8			−0.159305
$205$	3.70527e6			0.00209800
$206$	2.09919e9			1.16569
$207$			8.29259e8i			0.451657i
$208$		−	2.12344e9i		−	1.13445i
$209$		−	1.03945e9i		−	0.544777i
$210$			3.87692e6i			0.00199347i
$211$			5.74536e7i			0.0289859i	0.999895	+	0.0144930i	$0.00461342\pi$
$211$			5.74536e7i			0.0289859i	−0.999895	+	0.0144930i	$0.995387\pi$
$212$	−3.18877e9			−1.57863
$213$	1.02259e9			0.496804
$214$			5.07259e9i			2.41866i
$215$		−	8.63322e7i		−	0.0404035i
$216$			6.05351e8i			0.278095i
$217$			1.57935e8i			0.0712262i
$218$	2.15056e9			0.952195
$219$		−	2.43268e8i		−	0.105757i
$220$		−	8.22488e7i		−	0.0351107i
$221$		−	6.86249e8i		−	0.287682i
$222$	3.18125e9			1.30974
$223$	3.07934e9			1.24520			0.622598	−	0.782542i	$-0.286077\pi$
$223$	3.07934e9			1.24520			0.622598	+	0.782542i	$0.286077\pi$
$224$		−	6.73798e7i		−	0.0267632i
$225$	−8.53271e8			−0.332933
$226$	−4.37569e9			−1.67731
$227$			1.63518e9i			0.615831i	0.951414	+	0.307916i	$0.0996314\pi$
$227$			1.63518e9i			0.615831i	−0.951414	+	0.307916i	$0.900369\pi$
$228$	2.88787e9			1.06865
$229$		−	2.33566e9i		−	0.849314i	−0.905354	−	0.424657i	$-0.860395\pi$
$229$		−	2.33566e9i		−	0.849314i	0.905354	−	0.424657i	$-0.139605\pi$
$230$	2.22077e8			0.0793584
$231$		−	5.29177e7i		−	0.0185846i
$232$		−	8.82128e8i		−	0.304495i
$233$			3.61892e9i			1.22788i	0.789354	+	0.613939i	$0.210416\pi$
$233$			3.61892e9i			1.22788i	−0.789354	+	0.613939i	$0.789584\pi$
$234$	−3.26638e9			−1.08944
$235$			1.16616e8i			0.0382374i
$236$	5.07719e9	−	2.70932e9i	1.63672	−	0.873399i
$237$	−1.55118e9			−0.491666
$238$			4.75375e7i			0.0148159i
$239$	−4.00604e9			−1.22779			−0.613894	−	0.789388i	$-0.710398\pi$
$239$	−4.00604e9			−1.22779			−0.613894	+	0.789388i	$0.710398\pi$
$240$	3.89412e7			0.0117372
$241$	2.30136e9			0.682207			0.341103	−	0.940026i	$-0.389199\pi$
$241$	2.30136e9			0.682207			0.341103	+	0.940026i	$0.389199\pi$
$242$		−	4.06754e9i		−	1.18596i
$243$	2.23677e8			0.0641500
$244$		−	1.11817e10i		−	3.15464i
$245$	−1.24451e8			−0.0345410
$246$		−	2.16249e8i		−	0.0590492i
$247$			7.18303e9i			1.92983i
$248$	6.60411e9			1.74585
$249$			1.45600e9i			0.378761i
$250$			4.57291e8i			0.117066i
$251$	−4.10510e9			−1.03426			−0.517129	−	0.855908i	$-0.672999\pi$
$251$	−4.10510e9			−1.03426			−0.517129	+	0.855908i	$0.672999\pi$
$252$	1.47019e8			0.0364562
$253$	−3.03123e9			−0.739838
$254$		−	9.29619e9i		−	2.23342i
$255$	1.25849e7			0.00297639
$256$	−7.49081e9			−1.74409
$257$	−8.76632e8			−0.200949			−0.100474	−	0.994940i	$-0.532036\pi$
$257$	−8.76632e8			−0.200949			−0.100474	+	0.994940i	$0.532036\pi$
$258$	−5.03857e9			−1.13718
$259$		−	3.56153e8i		−	0.0791475i
$260$			5.68374e8i			0.124377i
$261$	−3.25946e8			−0.0702399
$262$	1.52750e10			3.24172
$263$	−6.03710e9			−1.26184			−0.630922	−	0.775846i	$-0.717323\pi$
$263$	−6.03710e9			−1.26184			−0.630922	+	0.775846i	$0.717323\pi$
$264$	−2.21277e9			−0.455534
$265$	1.45453e8			0.0294944
$266$		−	4.97579e8i		−	0.0993884i
$267$		−	3.92331e9i		−	0.771983i
$268$		−	1.57983e10i		−	3.06247i
$269$			6.22322e9i			1.18852i	0.804273	+	0.594259i	$0.202555\pi$
$269$			6.22322e9i			1.18852i	−0.804273	+	0.594259i	$0.797445\pi$
$270$		−	5.99013e7i		−	0.0112715i
$271$	9.67398e8			0.179361			0.0896805	−	0.995971i	$-0.471415\pi$
$271$	9.67398e8			0.179361			0.0896805	+	0.995971i	$0.471415\pi$
$272$	4.77484e8			0.0872334
$273$			3.65683e8i			0.0658347i
$274$		−	9.06954e8i		−	0.160910i
$275$		−	3.11900e9i		−	0.545361i
$276$		−	8.42154e9i		−	1.45129i
$277$	4.68317e9			0.795464			0.397732	−	0.917502i	$-0.369797\pi$
$277$	4.68317e9			0.795464			0.397732	+	0.917502i	$0.369797\pi$
$278$			7.05359e9i			1.18095i
$279$		−	2.44022e9i		−	0.402728i
$280$		−	1.81493e7i		−	0.00295276i
$281$	−3.92555e9			−0.629616			−0.314808	−	0.949155i	$-0.601940\pi$
$281$	−3.92555e9			−0.629616			−0.314808	+	0.949155i	$0.601940\pi$
$282$	6.80604e9			1.07621
$283$			2.29831e9i			0.358314i	0.983821	+	0.179157i	$0.0573370\pi$
$283$			2.29831e9i			0.358314i	−0.983821	+	0.179157i	$0.942663\pi$
$284$	−1.03850e10			−1.59636
$285$	−1.31728e8			−0.0199663
$286$		−	1.19398e10i		−	1.78456i
$287$	−2.42099e7			−0.00356833
$288$			1.04107e9i			0.151324i
$289$	−6.82145e9			−0.977879
$290$			8.72892e7i			0.0123415i
$291$			9.84465e8i			0.137287i
$292$			2.47051e9i			0.339824i
$293$	−1.22364e10			−1.66029			−0.830144	−	0.557549i	$-0.811742\pi$
$293$	−1.22364e10			−1.66029			−0.830144	+	0.557549i	$0.811742\pi$
$294$			7.26329e9i			0.972174i
$295$	−2.31592e8	+	1.23584e8i	−0.0305799	+	0.0163182i
$296$	−1.48926e10			−1.94001
$297$			8.17618e8i			0.105081i
$298$	−1.92307e10			−2.43854
$299$	2.09470e10			2.62083
$300$	8.66540e9			1.06980
$301$			5.64086e8i			0.0687194i
$302$	−8.45931e9			−1.01697
$303$		−	8.60469e9i		−	1.02086i
$304$	−4.99787e9			−0.585181
$305$			5.10046e8i			0.0589400i
$306$		−	7.34490e8i		−	0.0837723i
$307$	−1.10109e10			−1.23956			−0.619780	−	0.784776i	$-0.712778\pi$
$307$	−1.10109e10			−1.23956			−0.619780	+	0.784776i	$0.712778\pi$
$308$			5.37406e8i			0.0597172i
$309$			3.63111e9i			0.398296i
$310$	−6.53496e8			−0.0707614
$311$	1.17778e10			1.25899			0.629495	−	0.777005i	$-0.283262\pi$
$311$	1.17778e10			1.25899			0.629495	+	0.777005i	$0.283262\pi$
$312$	1.52912e10			1.61370
$313$		−	1.32404e9i		−	0.137951i	−0.997618	−	0.0689756i	$-0.978027\pi$
$313$		−	1.32404e9i		−	0.137951i	0.997618	−	0.0689756i	$-0.0219731\pi$
$314$	2.89551e10			2.97856
$315$	−6.70617e6			−0.000681133
$316$	1.57531e10			1.57985
$317$	−5.95561e9			−0.589779			−0.294889	−	0.955531i	$-0.595283\pi$
$317$	−5.95561e9			−0.589779			−0.294889	+	0.955531i	$0.595283\pi$
$318$		−	8.48902e9i		−	0.830136i
$319$		−	1.19145e9i		−	0.115057i
$320$	4.91970e8			0.0469179
$321$	−8.77440e9			−0.826413
$322$	−1.45103e9			−0.134975
$323$	−1.61520e9			−0.148394
$324$	−2.27156e9			−0.206131
$325$			2.15536e10i			1.93191i
$326$			1.97941e10i			1.75253i
$327$			3.71997e9i			0.325348i
$328$			1.01234e9i			0.0874647i
$329$		−	7.61961e8i		−	0.0650353i
$330$	2.18960e8			0.0184633
$331$	7.37192e9			0.614142			0.307071	−	0.951687i	$-0.400651\pi$
$331$	7.37192e9			0.614142			0.307071	+	0.951687i	$0.400651\pi$
$332$		−	1.47865e10i		−	1.21706i
$333$			5.50283e9i			0.447516i
$334$			1.51913e10i			1.22070i
$335$			7.20630e8i			0.0572180i
$336$	−2.54438e8			−0.0199629
$337$		−	1.48781e10i		−	1.15353i	−0.816911	−	0.576764i	$-0.804315\pi$
$337$		−	1.48781e10i		−	1.15353i	0.816911	−	0.576764i	$-0.195685\pi$
$338$			6.04549e10i			4.63196i
$339$		−	7.56893e9i		−	0.573107i
$340$	−1.27807e8			−0.00956395
$341$	8.91984e9			0.659689
$342$			7.68797e9i			0.561963i
$343$	1.62914e9			0.117701
$344$	2.35874e10			1.68441
$345$			3.84142e8i			0.0271154i
$346$	1.98540e9			0.138530
$347$		−	6.20527e9i		−	0.427999i	−0.976834	−	0.213999i	$-0.931351\pi$
$347$		−	6.20527e9i		−	0.427999i	0.976834	−	0.213999i	$-0.0686491\pi$
$348$	3.31015e9			0.225700
$349$		−	4.68321e9i		−	0.315676i	−0.987465	−	0.157838i	$-0.949548\pi$
$349$		−	4.68321e9i		−	0.315676i	0.987465	−	0.157838i	$-0.0504524\pi$
$350$		−	1.49305e9i		−	0.0994951i
$351$		−	5.65008e9i		−	0.372243i
$352$	−3.80546e9			−0.247877
$353$		−	1.21392e10i		−	0.781793i	−0.920435	−	0.390897i	$-0.872165\pi$
$353$		−	1.21392e10i		−	0.781793i	0.920435	−	0.390897i	$-0.127835\pi$
$354$	7.21267e9	+	1.35163e10i	0.459286	+	0.860688i
$355$	4.73702e8			0.0298258
$356$			3.98432e10i			2.48059i
$357$	−8.22288e7			−0.00506234
$358$	5.18930e9			0.315920
$359$	−1.60234e10			−0.964665			−0.482333	−	0.875988i	$-0.660210\pi$
$359$	−1.60234e10			−0.964665			−0.482333	+	0.875988i	$0.660210\pi$
$360$			2.80420e8i			0.0166955i
$361$	−7.70990e7			−0.00453962
$362$			4.15475e10i			2.41942i
$363$	7.03590e9			0.405222
$364$		−	3.71370e9i		−	0.211544i
$365$		−	1.12690e8i		−	0.00634914i
$366$	2.97676e10			1.65890
$367$		−	4.52398e8i		−	0.0249377i	−0.999922	−	0.0124688i	$-0.996031\pi$
$367$		−	4.52398e8i		−	0.0249377i	0.999922	−	0.0124688i	$-0.00396906\pi$
$368$			1.45747e10i			0.794709i
$369$	3.74061e8			0.0201761
$370$	1.47367e9			0.0786310
$371$	−9.50377e8			−0.0501649
$372$			2.47817e10i			1.29407i
$373$	−1.13079e10			−0.584182			−0.292091	−	0.956391i	$-0.594351\pi$
$373$	−1.13079e10			−0.584182			−0.292091	+	0.956391i	$0.594351\pi$
$374$	2.68481e9			0.137223
$375$	−7.91006e8			−0.0399995
$376$	−3.18616e10			−1.59410
$377$			8.23339e9i			0.407580i
$378$			3.91389e8i			0.0191709i
$379$	−1.45339e9			−0.0704409			−0.0352205	−	0.999380i	$-0.511213\pi$
$379$	−1.45339e9			−0.0704409			−0.0352205	+	0.999380i	$0.511213\pi$
$380$	1.33776e9			0.0641571
$381$	1.60802e10			0.763119
$382$	8.13689e9			0.382124
$383$	−6.96464e9			−0.323671			−0.161835	−	0.986818i	$-0.551741\pi$
$383$	−6.96464e9			−0.323671			−0.161835	+	0.986818i	$0.551741\pi$
$384$		−	2.30137e10i		−	1.05843i
$385$		−	2.45134e7i		−	0.00111573i
$386$		−	5.19228e10i		−	2.33889i
$387$		−	8.71555e9i		−	0.388554i
$388$		−	9.99774e9i		−	0.441139i
$389$	2.20334e10			0.962240			0.481120	−	0.876655i	$-0.340230\pi$
$389$	2.20334e10			0.962240			0.481120	+	0.876655i	$0.340230\pi$
$390$	−1.51311e9			−0.0654050
$391$			4.71023e9i			0.201528i
$392$		−	3.40022e10i		−	1.44000i
$393$			2.64222e10i			1.10764i
$394$		−	5.61182e10i		−	2.32873i
$395$	−7.18564e8			−0.0295173
$396$		−	8.30333e9i		−	0.337654i
$397$		−	2.99959e10i		−	1.20754i	−0.797160	−	0.603768i	$-0.793665\pi$
$397$		−	2.99959e10i		−	1.20754i	0.797160	−	0.603768i	$-0.206335\pi$
$398$			7.83542e10i			3.12270i
$399$	8.60696e8			0.0339593
$400$	−1.49967e10			−0.585809
$401$			1.42017e10i			0.549241i	0.961553	+	0.274620i	$0.0885522\pi$
$401$			1.42017e10i			0.549241i	−0.961553	+	0.274620i	$0.911448\pi$
$402$	4.20578e10			1.61043
$403$	−6.16398e10			−2.33691
$404$			8.73850e10i			3.28028i
$405$	1.03615e8			0.00385127
$406$		−	5.70339e8i		−	0.0209908i
$407$	−2.01147e10			−0.733055
$408$			3.43842e9i			0.124085i
$409$		−	1.80649e10i		−	0.645570i	−0.946472	−	0.322785i	$-0.895381\pi$
$409$		−	1.80649e10i		−	0.645570i	0.946472	−	0.322785i	$-0.104619\pi$
$410$		−	1.00174e8i		−	0.00354504i
$411$	1.56882e9			0.0549801
$412$		−	3.68758e10i		−	1.27983i
$413$	1.51320e9	−	8.07484e8i	0.0520112	−	0.0277545i
$414$	2.24195e10			0.763177
$415$			6.74473e8i			0.0227390i
$416$	2.62973e10			0.878089
$417$	−1.22011e10			−0.403510
$418$	−2.81022e10			−0.920524
$419$		−	2.79122e10i		−	0.905603i	−0.891611	−	0.452801i	$-0.850425\pi$
$419$		−	2.79122e10i		−	0.905603i	0.891611	−	0.452801i	$-0.149575\pi$
$420$	6.81046e7			0.00218866
$421$		−	8.74405e9i		−	0.278346i	−0.990268	−	0.139173i	$-0.955556\pi$
$421$		−	8.74405e9i		−	0.278346i	0.990268	−	0.139173i	$-0.0444443\pi$
$422$	1.55329e9			0.0489783
$423$			1.17729e10i			0.367723i
$424$			3.97403e10i			1.22961i
$425$	−4.84661e9			−0.148553
$426$		−	2.76465e10i		−	0.839464i
$427$		−	3.33259e9i		−	0.100247i
$428$	8.91085e10			2.65549
$429$	2.06530e10			0.609753
$430$	−2.33405e9			−0.0682709
$431$			1.69782e10i			0.492021i	0.969267	+	0.246010i	$0.0791197\pi$
$431$			1.69782e10i			0.492021i	−0.969267	+	0.246010i	$0.920880\pi$
$432$	3.93126e9			0.112875
$433$	−7.81317e9			−0.222267			−0.111134	−	0.993805i	$-0.535448\pi$
$433$	−7.81317e9			−0.222267			−0.111134	+	0.993805i	$0.535448\pi$
$434$	4.26988e9			0.120353
$435$	−1.50990e8			−0.00421688
$436$		−	3.77782e10i		−	1.04543i
$437$		−	4.93024e10i		−	1.35189i
$438$	−6.57690e9			−0.178700
$439$	−9.61070e9			−0.258760			−0.129380	−	0.991595i	$-0.541299\pi$
$439$	−9.61070e9			−0.258760			−0.129380	+	0.991595i	$0.541299\pi$
$440$	−1.02503e9			−0.0273481
$441$	−1.25638e10			−0.332175
$442$	−1.85532e10			−0.486105
$443$		−	6.24996e10i		−	1.62279i	−0.584498	−	0.811395i	$-0.698708\pi$
$443$		−	6.24996e10i		−	1.62279i	0.584498	−	0.811395i	$-0.301292\pi$
$444$		−	5.58840e10i		−	1.43799i
$445$		−	1.81742e9i		−	0.0463463i
$446$		−	8.32518e10i		−	2.10404i
$447$		−	3.32647e10i		−	0.833208i
$448$	−3.21448e9			−0.0797993
$449$	−7.16072e10			−1.76186			−0.880930	−	0.473247i	$-0.843082\pi$
$449$	−7.16072e10			−1.76186			−0.880930	+	0.473247i	$0.843082\pi$
$450$			2.30687e10i			0.562566i
$451$			1.36732e9i			0.0330495i
$452$			7.68663e10i			1.84154i
$453$		−	1.46326e10i		−	0.347480i
$454$	4.42081e10			1.04059
$455$			1.69398e8i			0.00395241i
$456$		−	3.59903e10i		−	0.832388i
$457$		−	5.88145e10i		−	1.34840i	−0.738548	−	0.674201i	$-0.764488\pi$
$457$		−	5.88145e10i		−	1.34840i	0.738548	−	0.674201i	$-0.235512\pi$
$458$	−6.31462e10			−1.43511
$459$	1.27050e9			0.0286235
$460$		−	3.90116e9i		−	0.0871290i
$461$	8.90675e10			1.97204			0.986020	−	0.166629i	$-0.0532883\pi$
$461$	8.90675e10			1.97204			0.986020	+	0.166629i	$0.0532883\pi$
$462$	−1.43066e9			−0.0314029
$463$			4.09409e10i			0.890910i	0.895304	+	0.445455i	$0.146958\pi$
$463$			4.09409e10i			0.890910i	−0.895304	+	0.445455i	$0.853042\pi$
$464$	−5.72869e9			−0.123590
$465$		−	1.13040e9i		−	0.0241779i
$466$	9.78397e10			2.07478
$467$		−	9.03824e10i		−	1.90028i	−0.311831	−	0.950138i	$-0.600942\pi$
$467$		−	9.03824e10i		−	1.90028i	0.311831	−	0.950138i	$-0.399058\pi$
$468$			5.73795e10i			1.19612i
$469$		−	4.70852e9i		−	0.0973180i
$470$	3.15280e9			0.0646108
$471$			5.00856e10i			1.01772i
$472$	−3.37652e10	−	6.32750e10i	−0.680301	−	1.27486i
$473$	3.18584e10			0.636471
$474$			4.19373e10i			0.830782i
$475$	5.07299e10			0.996529
$476$	8.35075e8			0.0162666
$477$	1.46840e10			0.283643
$478$			1.08306e11i			2.07463i
$479$	−3.29316e10			−0.625563			−0.312782	−	0.949825i	$-0.601261\pi$
$479$	−3.29316e10			−0.625563			−0.312782	+	0.949825i	$0.601261\pi$
$480$			4.82260e8i			0.00908482i
$481$	1.39001e11			2.59680
$482$		−	6.22187e10i		−	1.15274i
$483$		−	2.50995e9i		−	0.0461186i
$484$	−7.14531e10			−1.30209
$485$			4.56039e8i			0.00824205i
$486$		−	6.04726e9i		−	0.108396i
$487$	−7.35838e10			−1.30818			−0.654089	−	0.756418i	$-0.726948\pi$
$487$	−7.35838e10			−1.30818			−0.654089	+	0.756418i	$0.726948\pi$
$488$	−1.39353e11			−2.45718
$489$	−3.42391e10			−0.598808
$490$			3.36462e9i			0.0583648i
$491$	−1.62672e10			−0.279889			−0.139944	−	0.990159i	$-0.544692\pi$
$491$	−1.62672e10			−0.279889			−0.139944	+	0.990159i	$0.544692\pi$
$492$	−3.79878e9			−0.0648311
$493$	−1.85139e9			−0.0313408
$494$	1.94198e11			3.26090
$495$			3.78750e8i			0.00630858i
$496$		−	4.28882e10i		−	0.708617i
$497$	−3.09513e9			−0.0507286
$498$	3.93640e10			0.640003
$499$	−2.65185e10			−0.427707			−0.213853	−	0.976866i	$-0.568601\pi$
$499$	−2.65185e10			−0.427707			−0.213853	+	0.976866i	$0.568601\pi$
$500$	8.03307e9			0.128529
$501$	−2.62774e10			−0.417092
$502$			1.10984e11i			1.74761i
$503$			4.34884e10i			0.679362i	0.940541	+	0.339681i	$0.110319\pi$
$503$			4.34884e10i			0.679362i	−0.940541	+	0.339681i	$0.889681\pi$
$504$		−	1.83224e9i		−	0.0283962i
$505$		−	3.98600e9i		−	0.0612875i
$506$			8.19512e10i			1.25012i
$507$	−1.04573e11			−1.58266
$508$	−1.63303e11			−2.45211
$509$		−	9.17675e10i		−	1.36715i	−0.729878	−	0.683577i	$-0.760423\pi$
$509$		−	9.17675e10i		−	1.36715i	0.729878	−	0.683577i	$-0.239577\pi$
$510$		−	3.40242e8i		−	0.00502930i
$511$			7.36307e8i			0.0107988i
$512$			7.65389e10i			1.11379i
$513$	−1.32984e10			−0.192013
$514$			2.37003e10i			0.339548i
$515$			1.68206e9i			0.0239118i
$516$			8.85109e10i			1.24853i
$517$	−4.30339e10			−0.602349
$518$	−9.62882e9			−0.133738
$519$			3.43428e9i			0.0473332i
$520$	7.08341e9			0.0968789
$521$	4.07268e10			0.552751			0.276376	−	0.961050i	$-0.410867\pi$
$521$	4.07268e10			0.552751			0.276376	+	0.961050i	$0.410867\pi$
$522$			8.81217e9i			0.118686i
$523$	8.55616e10			1.14359			0.571797	−	0.820395i	$-0.306246\pi$
$523$	8.55616e10			1.14359			0.571797	+	0.820395i	$0.306246\pi$
$524$		−	2.68331e11i		−	3.55914i
$525$	2.58263e9			0.0339957
$526$			1.63217e11i			2.13217i
$527$		−	1.38605e10i		−	0.179696i
$528$			1.43701e10i			0.184895i
$529$	−6.54637e10			−0.835946
$530$		−	3.93242e9i		−	0.0498375i
$531$	−2.33801e10	+	1.24762e10i	−0.294082	+	0.156930i
$532$	−8.74081e9			−0.109120
$533$		−	9.44877e9i		−	0.117076i
$534$	−1.06069e11			−1.30444
$535$	−4.06462e9			−0.0496140
$536$	−1.96888e11			−2.38540
$537$			8.97628e9i			0.107944i
$538$	1.68249e11			2.00827
$539$		−	4.59250e10i		−	0.544120i
$540$	−1.05227e9			−0.0123752
$541$		−	1.41371e11i		−	1.65033i	−0.564894	−	0.825164i	$-0.691083\pi$
$541$		−	1.41371e11i		−	1.65033i	0.564894	−	0.825164i	$-0.308917\pi$
$542$		−	2.61542e10i		−	0.303071i
$543$	−7.18675e10			−0.826673
$544$			5.91331e9i			0.0675204i
$545$			1.72322e9i			0.0195324i
$546$	9.88648e9			0.111243
$547$	2.82315e10			0.315344			0.157672	−	0.987492i	$-0.449601\pi$
$547$	2.82315e10			0.315344			0.157672	+	0.987492i	$0.449601\pi$
$548$	−1.59322e10			−0.176666
$549$			5.14910e10i			0.566816i
$550$	−8.43241e10			−0.921512
$551$	1.93787e10			0.210241
$552$	−1.04954e11			−1.13043
$553$	4.69503e9			0.0502039
$554$		−	1.26613e11i		−	1.34412i
$555$			2.54911e9i			0.0268668i
$556$	1.23908e11			1.29658
$557$	−7.13671e10			−0.741442			−0.370721	−	0.928744i	$-0.620889\pi$
$557$	−7.13671e10			−0.741442			−0.370721	+	0.928744i	$0.620889\pi$
$558$	−6.59729e10			−0.680501
$559$	−2.20155e11			−2.25466
$560$	−1.17865e8			−0.00119848
$561$			4.64410e9i			0.0468868i
$562$			1.06130e11i			1.06388i
$563$			2.46340e10i			0.245189i	0.992457	+	0.122595i	$0.0391215\pi$
$563$			2.46340e10i			0.245189i	−0.992457	+	0.122595i	$0.960879\pi$
$564$		−	1.19559e11i		−	1.18159i
$565$		−	3.50620e9i		−	0.0344067i
$566$	6.21364e10			0.605453
$567$	−6.77013e8			−0.00655035
$568$			1.29424e11i			1.24343i
$569$		−	1.72401e11i		−	1.64472i	−0.568970	−	0.822358i	$-0.692658\pi$
$569$		−	1.72401e11i		−	1.64472i	0.568970	−	0.822358i	$-0.307342\pi$
$570$			3.56134e9i			0.0337376i
$571$			8.92475e9i			0.0839560i	0.999119	+	0.0419780i	$0.0133659\pi$
$571$			8.92475e9i			0.0839560i	−0.999119	+	0.0419780i	$0.986634\pi$
$572$	−2.09742e11			−1.95930
$573$			1.40749e10i			0.130565i
$574$			6.54530e8i			0.00602951i
$575$		−	1.47938e11i		−	1.35334i
$576$	4.96662e10			0.451202
$577$	−5.89305e10			−0.531663			−0.265832	−	0.964019i	$-0.585647\pi$
$577$	−5.89305e10			−0.531663			−0.265832	+	0.964019i	$0.585647\pi$
$578$			1.84422e11i			1.65235i
$579$	8.98144e10			0.799156
$580$	1.53338e9			0.0135500
$581$		−	4.40694e9i		−	0.0386752i
$582$	2.66156e10			0.231977
$583$			5.36752e10i			0.464622i
$584$	3.07889e10			0.264693
$585$		−	2.61732e9i		−	0.0223477i
$586$			3.30819e11i			2.80544i
$587$		−	1.15907e11i		−	0.976243i	−0.872776	−	0.488122i	$-0.837682\pi$
$587$		−	1.15907e11i		−	0.976243i	0.872776	−	0.488122i	$-0.162318\pi$
$588$	1.27592e11			1.06737
$589$			1.45080e11i			1.20544i
$590$	3.34117e9	+	6.26125e9i	0.0275734	+	0.0516717i
$591$	9.70714e10			0.795685
$592$			9.67153e10i			0.787424i
$593$	−5.41875e10			−0.438208			−0.219104	−	0.975701i	$-0.570313\pi$
$593$	−5.41875e10			−0.438208			−0.219104	+	0.975701i	$0.570313\pi$
$594$	2.21048e10			0.177558
$595$	−3.80913e7			−0.000303919
$596$			3.37820e11i			2.67732i
$597$	−1.35535e11			−1.06697
$598$		−	5.66317e11i		−	4.42848i
$599$	−1.14987e11			−0.893188			−0.446594	−	0.894737i	$-0.647363\pi$
$599$	−1.14987e11			−0.893188			−0.446594	+	0.894737i	$0.647363\pi$
$600$		−	1.07993e11i		−	0.833282i
$601$			1.86707e11i			1.43108i	0.698573	+	0.715539i	$0.253819\pi$
$601$			1.86707e11i			1.43108i	−0.698573	+	0.715539i	$0.746181\pi$
$602$	1.52504e10			0.116117
$603$			7.27502e10i			0.550256i
$604$			1.48602e11i			1.11655i
$605$	3.25928e9			0.0243276
$606$	−2.32633e11			−1.72497
$607$	1.25763e11			0.926398			0.463199	−	0.886254i	$-0.346702\pi$
$607$	1.25763e11			0.926398			0.463199	+	0.886254i	$0.346702\pi$
$608$		−	6.18952e10i		−	0.452942i
$609$	9.86554e8			0.00717219
$610$	1.37894e10			0.0995925
$611$	2.97382e11			2.13378
$612$	−1.29025e10			−0.0919750
$613$			1.57522e11i			1.11558i	0.829983	+	0.557788i	$0.188350\pi$
$613$			1.57522e11i			1.11558i	−0.829983	+	0.557788i	$0.811650\pi$
$614$			2.97686e11i			2.09452i
$615$	1.73278e8			0.00121128
$616$	6.69747e9			0.0465145
$617$	−1.27615e11			−0.880566			−0.440283	−	0.897859i	$-0.645122\pi$
$617$	−1.27615e11			−0.880566			−0.440283	+	0.897859i	$0.645122\pi$
$618$	9.81695e10			0.673011
$619$	5.27837e10			0.359532			0.179766	−	0.983709i	$-0.442466\pi$
$619$	5.27837e10			0.359532			0.179766	+	0.983709i	$0.442466\pi$
$620$			1.14798e10i			0.0776901i
$621$			3.87806e10i			0.260764i
$622$		−	3.18420e11i		−	2.12735i
$623$			1.18748e10i			0.0788270i
$624$		−	9.93034e10i		−	0.654977i
$625$	1.52038e11			0.996397
$626$	−3.57964e10			−0.233100
$627$		−	4.86103e10i		−	0.314527i
$628$		−	5.08645e11i		−	3.27021i
$629$			3.12563e10i			0.199680i
$630$			1.81306e8i			0.00115093i
$631$	−1.41100e11			−0.890038			−0.445019	−	0.895521i	$-0.646803\pi$
$631$	−1.41100e11			−0.890038			−0.445019	+	0.895521i	$0.646803\pi$
$632$		−	1.96324e11i		−	1.23057i
$633$			2.68684e9i			0.0167350i
$634$			1.61014e11i			0.996565i
$635$	7.44894e9			0.0458142
$636$	−1.49124e11			−0.911420
$637$			3.17361e11i			1.92751i
$638$	−3.22115e10			−0.194414
$639$	4.78220e10			0.286830
$640$		−	1.06608e10i		−	0.0635431i
$641$	1.96649e11			1.16482			0.582411	−	0.812894i	$-0.302109\pi$
$641$	1.96649e11			1.16482			0.582411	+	0.812894i	$0.302109\pi$
$642$			2.37222e11i			1.39641i
$643$	−7.88497e10			−0.461271			−0.230635	−	0.973040i	$-0.574080\pi$
$643$	−7.88497e10			−0.461271			−0.230635	+	0.973040i	$0.574080\pi$
$644$			2.54898e10i			0.148191i
$645$		−	4.03736e9i		−	0.0233270i
$646$			4.36680e10i			0.250746i
$647$	2.18154e11			1.24493			0.622467	−	0.782646i	$-0.286130\pi$
$647$	2.18154e11			1.24493			0.622467	+	0.782646i	$0.286130\pi$
$648$			2.83095e10i			0.160558i
$649$	−4.56050e10	−	8.54623e10i	−0.257059	−	0.481721i
$650$	5.82715e11			3.26440
$651$			7.38590e9i			0.0411225i
$652$	3.47716e11			1.92413
$653$	−2.06412e11			−1.13522			−0.567612	−	0.823296i	$-0.692133\pi$
$653$	−2.06412e11			−1.13522			−0.567612	+	0.823296i	$0.692133\pi$
$654$	1.00572e11			0.549750
$655$			1.22397e10i			0.0664975i
$656$	6.57433e9			0.0355007
$657$		−	1.13765e10i		−	0.0610586i
$658$	−2.06001e10			−0.109892
$659$		−	5.94177e8i		−	0.00315046i	−0.999999	−	0.00157523i	$-0.999499\pi$
$659$		−	5.94177e8i		−	0.00315046i	0.999999	−	0.00157523i	$-0.000501412\pi$
$660$		−	3.84640e9i		−	0.0202712i
$661$	−9.83164e10			−0.515015			−0.257508	−	0.966276i	$-0.582901\pi$
$661$	−9.83164e10			−0.515015			−0.257508	+	0.966276i	$0.582901\pi$
$662$		−	1.99305e11i		−	1.03773i
$663$		−	3.20927e10i		−	0.166093i
$664$	−1.84278e11			−0.947983
$665$	3.98705e8			0.00203876
$666$	1.48773e11			0.756181
$667$		−	5.65117e10i		−	0.285519i
$668$	2.66860e11			1.34023
$669$	1.44006e11			0.718914
$670$	1.94827e10			0.0966829
$671$	−1.88217e11			−0.928474
$672$		−	3.15104e9i		−	0.0154517i
$673$		−	8.62581e10i		−	0.420475i	−0.977650	−	0.210237i	$-0.932576\pi$
$673$		−	8.62581e10i		−	0.420475i	0.977650	−	0.210237i	$-0.0674236\pi$
$674$	−4.02239e11			−1.94915
$675$	−3.99035e10			−0.192219
$676$	1.06199e12			5.08551
$677$	1.61084e11			0.766828			0.383414	−	0.923577i	$-0.374748\pi$
$677$	1.61084e11			0.766828			0.383414	+	0.923577i	$0.374748\pi$
$678$	−2.04631e11			−0.968394
$679$		−	2.97972e9i		−	0.0140183i
$680$			1.59280e9i			0.00744948i
$681$			7.64697e10i			0.355550i
$682$		−	2.41154e11i		−	1.11470i
$683$		−	2.63470e11i		−	1.21073i	−0.795947	−	0.605366i	$-0.793027\pi$
$683$		−	2.63470e11i		−	1.21073i	0.795947	−	0.605366i	$-0.206973\pi$
$684$	1.35052e11			0.616988
$685$	7.26733e8			0.00330075
$686$		−	4.40448e10i		−	0.198883i
$687$		−	1.09228e11i		−	0.490352i
$688$		−	1.53181e11i		−	0.683676i
$689$		−	3.70918e11i		−	1.64589i
$690$	1.03855e10			0.0458176
$691$		−	1.11539e11i		−	0.489231i	−0.969620	−	0.244615i	$-0.921338\pi$
$691$		−	1.11539e11i		−	0.489231i	0.969620	−	0.244615i	$-0.0786617\pi$
$692$		−	3.48768e10i		−	0.152094i
$693$		−	2.47471e9i		−	0.0107298i
$694$	−1.67763e11			−0.723201
$695$	−5.65198e9			−0.0242248
$696$		−	4.12530e10i		−	0.175800i
$697$	2.12468e9			0.00900249
$698$	−1.26614e11			−0.533407
$699$			1.69240e11i			0.708915i
$700$	−2.62279e10			−0.109237
$701$			2.29404e11i			0.950011i	0.879983	+	0.475005i	$0.157554\pi$
$701$			2.29404e11i			0.950011i	−0.879983	+	0.475005i	$0.842446\pi$
$702$	−1.52754e11			−0.628989
$703$		−	3.27163e11i		−	1.33950i
$704$			1.81547e11i			0.739092i
$705$			5.45361e9i			0.0220764i
$706$	−3.28192e11			−1.32102
$707$			2.60441e10i			0.104239i
$708$	2.37437e11	−	1.26703e11i	0.944963	−	0.504257i
$709$	7.40603e10			0.293090			0.146545	−	0.989204i	$-0.453185\pi$
$709$	7.40603e10			0.293090			0.146545	+	0.989204i	$0.453185\pi$
$710$		−	1.28069e10i		−	0.0503975i
$711$	−7.25417e10			−0.283863
$712$	4.96550e11			1.93216
$713$	4.23079e11			1.63705
$714$			2.22311e9i			0.00855397i
$715$	9.56721e9			0.0366067
$716$		−	9.11587e10i		−	0.346853i
$717$	−1.87344e11			−0.708864
$718$			4.33203e11i			1.63002i
$719$			1.90365e11i			0.712315i	0.934426	+	0.356158i	$0.115913\pi$
$719$			1.90365e11i			0.712315i	−0.934426	+	0.356158i	$0.884087\pi$
$720$	1.82110e9			0.00677647
$721$		−	1.09904e10i		−	0.0406699i
$722$			2.08442e9i			0.00767073i
$723$	1.07624e11			0.393872
$724$	7.29852e11			2.65632
$725$	5.81480e10			0.210467
$726$		−	1.90220e11i		−	0.684715i
$727$	−2.66371e11			−0.953562			−0.476781	−	0.879022i	$-0.658197\pi$
$727$	−2.66371e11			−0.953562			−0.476781	+	0.879022i	$0.658197\pi$
$728$	−4.62823e10			−0.164774
$729$	1.04604e10			0.0370370
$730$	−3.04665e9			−0.0107283
$731$		−	4.95047e10i		−	0.173371i
$732$		−	5.22917e11i		−	1.82133i
$733$	−2.05423e11			−0.711596			−0.355798	−	0.934563i	$-0.615791\pi$
$733$	−2.05423e11			−0.711596			−0.355798	+	0.934563i	$0.615791\pi$
$734$	−1.22309e10			−0.0421379
$735$	−5.82000e9			−0.0199422
$736$	−1.80498e11			−0.615121
$737$	−2.65927e11			−0.901349
$738$		−	1.01130e10i		−	0.0340921i
$739$			4.75269e11i			1.59353i	0.604287	+	0.796767i	$0.293458\pi$
$739$			4.75269e11i			1.59353i	−0.604287	+	0.796767i	$0.706542\pi$
$740$		−	2.58875e10i		−	0.0863302i
$741$			3.35917e11i			1.11419i
$742$			2.56940e10i			0.0847651i
$743$	−4.87871e11			−1.60085			−0.800423	−	0.599436i	$-0.795392\pi$
$743$	−4.87871e11			−1.60085			−0.800423	+	0.599436i	$0.795392\pi$
$744$	3.08844e11			1.00797
$745$		−	1.54094e10i		−	0.0500219i
$746$			3.05717e11i			0.987109i
$747$			6.80906e10i			0.218678i
$748$		−	4.71632e10i		−	0.150660i
$749$	2.65578e10			0.0843849
$750$			2.13854e10i			0.0675883i
$751$			1.72924e11i			0.543620i	0.962351	+	0.271810i	$0.0876223\pi$
$751$			1.72924e11i			0.543620i	−0.962351	+	0.271810i	$0.912378\pi$
$752$			2.06915e11i			0.647024i
$753$	−1.91976e11			−0.597129
$754$	2.22595e11			0.688700
$755$		−	6.77836e9i		−	0.0208611i
$756$	6.87541e9			0.0210480
$757$	−2.66320e11			−0.811000			−0.405500	−	0.914095i	$-0.632903\pi$
$757$	−2.66320e11			−0.811000			−0.405500	+	0.914095i	$0.632903\pi$
$758$			3.92933e10i			0.119026i
$759$	−1.41756e11			−0.427145
$760$		−	1.66720e10i		−	0.0499727i
$761$	2.54534e11			0.758938			0.379469	−	0.925204i	$-0.376107\pi$
$761$	2.54534e11			0.758938			0.379469	+	0.925204i	$0.376107\pi$
$762$		−	4.34740e11i		−	1.28946i
$763$		−	1.12594e10i		−	0.0332213i
$764$		−	1.42938e11i		−	0.419541i
$765$	5.88540e8			0.00171842
$766$			1.88293e11i			0.546915i
$767$	3.15149e11	+	5.90581e11i	0.910615	+	1.70647i
$768$	−3.50311e11			−1.00695
$769$			4.99936e11i			1.42958i	0.699339	+	0.714791i	$0.253478\pi$
$769$			4.99936e11i			1.42958i	−0.699339	+	0.714791i	$0.746522\pi$
$770$	−6.62735e8			−0.00188528
$771$	−4.09960e10			−0.116018
$772$	−9.12111e11			−2.56790
$773$			5.95523e11i			1.66794i	0.551810	+	0.833970i	$0.313937\pi$
$773$			5.95523e11i			1.66794i	−0.551810	+	0.833970i	$0.686063\pi$
$774$	−2.35631e11			−0.656550
$775$			4.35329e11i			1.20673i
$776$	−1.24598e11			−0.343608
$777$		−	1.66556e10i		−	0.0456958i
$778$		−	5.95687e11i		−	1.62592i
$779$	−2.22392e10			−0.0603907
$780$			2.65802e10i			0.0718092i
$781$			1.74806e11i			0.469842i
$782$	1.27344e11			0.340527

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-27.0356i q^{2} +46.7654 q^{3} -474.926 q^{4} +21.6634 q^{5} -1264.33i q^{6} -141.547 q^{7} +5918.81i q^{8} +2187.00 q^{9} +O(q^{10})\)	\(q-27.0356i q^{2} +46.7654 q^{3} -474.926 q^{4} +21.6634 q^{5} -1264.33i q^{6} -141.547 q^{7} +5918.81i q^{8} +2187.00 q^{9} -585.684i q^{10} +7994.24i q^{11} -22210.1 q^{12} -55243.6i q^{13} +3826.80i q^{14} +1013.10 q^{15} +38437.8 q^{16} +12422.3 q^{17} -59127.0i q^{18} -130025. q^{19} -10288.5 q^{20} -6619.48 q^{21} +216129. q^{22} +379176. i q^{23} +276795. i q^{24} -390156. q^{25} -1.49355e6 q^{26} +102276. q^{27} +67224.1 q^{28} -149038. q^{29} -27389.7i q^{30} -1.11578e6i q^{31} +476026. i q^{32} +373854. i q^{33} -335844. i q^{34} -3066.38 q^{35} -1.03866e6 q^{36} +2.51615e6i q^{37} +3.51531e6i q^{38} -2.58349e6i q^{39} +128222. i q^{40} +171038. q^{41} +178962. i q^{42} -3.98516e6i q^{43} -3.79667e6i q^{44} +47377.8 q^{45} +1.02513e7 q^{46} +5.38311e6i q^{47} +1.79756e6 q^{48} -5.74477e6 q^{49} +1.05481e7i q^{50} +580931. q^{51} +2.62366e7i q^{52} +6.71423e6 q^{53} -2.76509e6i q^{54} +173182. i q^{55} -837787. i q^{56} -6.08066e6 q^{57} +4.02934e6i q^{58} +(-1.06905e7 + 5.70473e6i) q^{59} -481146. q^{60} +2.35441e7i q^{61} -3.01659e7 q^{62} -309562. q^{63} +2.27097e7 q^{64} -1.19676e6i q^{65} +1.01074e7 q^{66} +3.32648e7i q^{67} -5.89965e6 q^{68} +1.77323e7i q^{69} +82901.5i q^{70} +2.18665e7 q^{71} +1.29444e7i q^{72} -5.20187e6i q^{73} +6.80258e7 q^{74} -1.82458e7 q^{75} +6.17522e7 q^{76} -1.13156e6i q^{77} -6.98462e7 q^{78} -3.31695e7 q^{79} +832693. q^{80} +4.78297e6 q^{81} -4.62413e6i q^{82} +3.11342e7i q^{83} +3.14376e6 q^{84} +269108. q^{85} -1.07741e8 q^{86} -6.96982e6 q^{87} -4.73164e7 q^{88} -8.38935e7i q^{89} -1.28089e6i q^{90} +7.81953e6i q^{91} -1.80081e8i q^{92} -5.21800e7i q^{93} +1.45536e8 q^{94} -2.81678e6 q^{95} +2.22615e7i q^{96} +2.10512e7i q^{97} +1.55313e8i q^{98} +1.74834e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) \)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about