LMFDB - Embedded newform 177.9.c.a.58.16

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			58.16
Character	$\chi$	$=$	177.58
Dual form			177.9.c.a.58.65

$q$-expansion

$f(q)$	$=$	$q-20.7097i q^{2} +46.7654 q^{3} -172.891 q^{4} -840.056 q^{5} -968.496i q^{6} -4543.72 q^{7} -1721.16i q^{8} +2187.00 q^{9} +O(q^{10})$	\(q-20.7097i q^{2} +46.7654 q^{3} -172.891 q^{4} -840.056 q^{5} -968.496i q^{6} -4543.72 q^{7} -1721.16i q^{8} +2187.00 q^{9} +17397.3i q^{10} -24898.4i q^{11} -8085.30 q^{12} -50820.4i q^{13} +94099.0i q^{14} -39285.5 q^{15} -79904.8 q^{16} -124308. q^{17} -45292.1i q^{18} +132560. q^{19} +145238. q^{20} -212489. q^{21} -515637. q^{22} -269266. i q^{23} -80490.9i q^{24} +315069. q^{25} -1.05247e6 q^{26} +102276. q^{27} +785567. q^{28} -711317. q^{29} +813591. i q^{30} +85871.3i q^{31} +1.21418e6i q^{32} -1.16438e6i q^{33} +2.57438e6i q^{34} +3.81698e6 q^{35} -378112. q^{36} +391679. i q^{37} -2.74527e6i q^{38} -2.37663e6i q^{39} +1.44587e6i q^{40} +4.31923e6 q^{41} +4.40057e6i q^{42} +1.63923e6i q^{43} +4.30470e6i q^{44} -1.83720e6 q^{45} -5.57642e6 q^{46} -4.88539e6i q^{47} -3.73678e6 q^{48} +1.48806e7 q^{49} -6.52498e6i q^{50} -5.81332e6 q^{51} +8.78637e6i q^{52} -4.36472e6 q^{53} -2.11810e6i q^{54} +2.09160e7i q^{55} +7.82049e6i q^{56} +6.19920e6 q^{57} +1.47311e7i q^{58} +(-1.09079e7 + 5.27712e6i) q^{59} +6.79211e6 q^{60} +9.09407e6i q^{61} +1.77837e6 q^{62} -9.93712e6 q^{63} +4.68975e6 q^{64} +4.26919e7i q^{65} -2.41140e7 q^{66} -2.19803e6i q^{67} +2.14917e7 q^{68} -1.25923e7i q^{69} -7.90484e7i q^{70} -1.67746e7 q^{71} -3.76419e6i q^{72} +838268. i q^{73} +8.11154e6 q^{74} +1.47343e7 q^{75} -2.29184e7 q^{76} +1.13131e8i q^{77} -4.92193e7 q^{78} +3.35946e7 q^{79} +6.71245e7 q^{80} +4.78297e6 q^{81} -8.94499e7i q^{82} -7.13611e7i q^{83} +3.67374e7 q^{84} +1.04426e8 q^{85} +3.39480e7 q^{86} -3.32650e7 q^{87} -4.28542e7 q^{88} +2.85634e7i q^{89} +3.80479e7i q^{90} +2.30913e8i q^{91} +4.65537e7i q^{92} +4.01580e6i q^{93} -1.01175e8 q^{94} -1.11358e8 q^{95} +5.67818e7i q^{96} +1.66487e7i q^{97} -3.08172e8i q^{98} -5.44527e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	20.7097i		−	1.29435i	−0.762339	−	0.647177i	$-0.775949\pi$
$2$		−	20.7097i		−	1.29435i	0.762339	−	0.647177i	$-0.224051\pi$
$3$	46.7654			0.577350
$3$	46.7654			0.577350
$4$	−172.891			−0.675355
$5$	−840.056			−1.34409			−0.672045	−	0.740510i	$-0.734584\pi$
$5$	−840.056			−1.34409			−0.672045	+	0.740510i	$0.734584\pi$
$6$		−	968.496i		−	0.747296i
$7$	−4543.72			−1.89243			−0.946214	−	0.323541i	$-0.895127\pi$
$7$	−4543.72			−1.89243			−0.946214	+	0.323541i	$0.895127\pi$
$8$		−	1721.16i		−	0.420206i
$9$	2187.00			0.333333
$10$			17397.3i			1.73973i
$11$		−	24898.4i		−	1.70059i	−0.526305	−	0.850296i	$-0.676423\pi$
$11$		−	24898.4i		−	1.70059i	0.526305	−	0.850296i	$-0.323577\pi$
$12$	−8085.30			−0.389916
$13$		−	50820.4i		−	1.77936i	−0.456583	−	0.889681i	$-0.650927\pi$
$13$		−	50820.4i		−	1.77936i	0.456583	−	0.889681i	$-0.349073\pi$
$14$			94099.0i			2.44947i
$15$	−39285.5			−0.776011
$16$	−79904.8			−1.21925
$17$	−124308.			−1.48835			−0.744173	−	0.667987i	$-0.767156\pi$
$17$	−124308.			−1.48835			−0.744173	+	0.667987i	$0.767156\pi$
$18$		−	45292.1i		−	0.431452i
$19$	132560.			1.01718			0.508589	−	0.861009i	$-0.330167\pi$
$19$	132560.			1.01718			0.508589	+	0.861009i	$0.330167\pi$
$20$	145238.			0.907737
$21$	−212489.			−1.09259
$22$	−515637.			−2.20117
$23$		−	269266.i		−	0.962212i	−0.876662	−	0.481106i	$-0.840235\pi$
$23$		−	269266.i		−	0.962212i	0.876662	−	0.481106i	$-0.159765\pi$
$24$		−	80490.9i		−	0.242606i
$25$	315069.			0.806577
$26$	−1.05247e6			−2.30313
$27$	102276.			0.192450
$28$	785567.			1.27806
$29$	−711317.			−1.00571			−0.502853	−	0.864372i	$-0.667716\pi$
$29$	−711317.			−1.00571			−0.502853	+	0.864372i	$0.667716\pi$
$30$			813591.i			1.00443i
$31$			85871.3i			0.0929825i	0.998919	+	0.0464913i	$0.0148040\pi$
$31$			85871.3i			0.0929825i	−0.998919	+	0.0464913i	$0.985196\pi$
$32$			1.21418e6i			1.15794i
$33$		−	1.16438e6i		−	0.981837i
$34$			2.57438e6i			1.92645i
$35$	3.81698e6			2.54359
$36$	−378112.			−0.225118
$37$			391679.i			0.208989i	0.994525	+	0.104494i	$0.0333225\pi$
$37$			391679.i			0.208989i	−0.994525	+	0.104494i	$0.966678\pi$
$38$		−	2.74527e6i		−	1.31659i
$39$		−	2.37663e6i		−	1.02732i
$40$			1.44587e6i			0.564795i
$41$	4.31923e6			1.52852			0.764260	−	0.644908i	$-0.223104\pi$
$41$	4.31923e6			1.52852			0.764260	+	0.644908i	$0.223104\pi$
$42$			4.40057e6i			1.41420i
$43$			1.63923e6i			0.479476i	0.970838	+	0.239738i	$0.0770616\pi$
$43$			1.63923e6i			0.479476i	−0.970838	+	0.239738i	$0.922938\pi$
$44$			4.30470e6i			1.14850i
$45$	−1.83720e6			−0.448030
$46$	−5.57642e6			−1.24544
$47$		−	4.88539e6i		−	1.00117i	−0.865687	−	0.500585i	$-0.833118\pi$
$47$		−	4.88539e6i		−	1.00117i	0.865687	−	0.500585i	$-0.166882\pi$
$48$	−3.73678e6			−0.703935
$49$	1.48806e7			2.58128
$50$		−	6.52498e6i		−	1.04400i
$51$	−5.81332e6			−0.859297
$52$			8.78637e6i			1.20170i
$53$	−4.36472e6			−0.553163			−0.276581	−	0.960990i	$-0.589202\pi$
$53$	−4.36472e6			−0.553163			−0.276581	+	0.960990i	$0.589202\pi$
$54$		−	2.11810e6i		−	0.249099i
$55$			2.09160e7i			2.28575i
$56$			7.82049e6i			0.795210i
$57$	6.19920e6			0.587268
$58$			1.47311e7i			1.30174i
$59$	−1.09079e7	+	5.27712e6i	−0.900188	+	0.435501i
$59$	−1.09079e7	+	5.27712e6i	−0.900188	+	0.435501i
$60$	6.79211e6			0.524082
$61$			9.09407e6i			0.656809i	0.944537	+	0.328405i	$0.106511\pi$
$61$			9.09407e6i			0.656809i	−0.944537	+	0.328405i	$0.893489\pi$
$62$	1.77837e6			0.120352
$63$	−9.93712e6			−0.630809
$64$	4.68975e6			0.279531
$65$			4.26919e7i			2.39162i
$66$	−2.41140e7			−1.27085
$67$		−	2.19803e6i		−	0.109077i	−0.998512	−	0.0545386i	$-0.982631\pi$
$67$		−	2.19803e6i		−	0.109077i	0.998512	−	0.0545386i	$-0.0173688\pi$
$68$	2.14917e7			1.00516
$69$		−	1.25923e7i		−	0.555534i
$70$		−	7.90484e7i		−	3.29231i
$71$	−1.67746e7			−0.660113			−0.330057	−	0.943961i	$-0.607068\pi$
$71$	−1.67746e7			−0.660113			−0.330057	+	0.943961i	$0.607068\pi$
$72$		−	3.76419e6i		−	0.140069i
$73$			838268.i			0.0295183i	0.999891	+	0.0147591i	$0.00469815\pi$
$73$			838268.i			0.0295183i	−0.999891	+	0.0147591i	$0.995302\pi$
$74$	8.11154e6			0.270506
$75$	1.47343e7			0.465677
$76$	−2.29184e7			−0.686956
$77$			1.13131e8i			3.21825i
$78$	−4.92193e7			−1.32971
$79$	3.35946e7			0.862505			0.431253	−	0.902231i	$-0.358072\pi$
$79$	3.35946e7			0.862505			0.431253	+	0.902231i	$0.358072\pi$
$80$	6.71245e7			1.63878
$81$	4.78297e6			0.111111
$82$		−	8.94499e7i		−	1.97845i
$83$		−	7.13611e7i		−	1.50366i	−0.659358	−	0.751829i	$-0.729172\pi$
$83$		−	7.13611e7i		−	1.50366i	0.659358	−	0.751829i	$-0.270828\pi$
$84$	3.67374e7			0.737888
$85$	1.04426e8			2.00047
$86$	3.39480e7			0.620613
$87$	−3.32650e7			−0.580645
$88$	−4.28542e7			−0.714599
$89$			2.85634e7i			0.455249i	0.973749	+	0.227625i	$0.0730959\pi$
$89$			2.85634e7i			0.455249i	−0.973749	+	0.227625i	$0.926904\pi$
$90$			3.80479e7i			0.579910i
$91$			2.30913e8i			3.36731i
$92$			4.65537e7i			0.649835i
$93$			4.01580e6i			0.0536835i
$94$	−1.01175e8			−1.29587
$95$	−1.11358e8			−1.36718
$96$			5.67818e7i			0.668535i
$97$			1.66487e7i			0.188058i	0.995569	+	0.0940292i	$0.0299747\pi$
$97$			1.66487e7i			0.188058i	−0.995569	+	0.0940292i	$0.970025\pi$
$98$		−	3.08172e8i		−	3.34110i
$99$		−	5.44527e7i		−	0.566864i
$100$	−5.44726e7			−0.544726
$101$		−	2.97927e7i		−	0.286302i	−0.989701	−	0.143151i	$-0.954277\pi$
$101$		−	2.97927e7i		−	0.286302i	0.989701	−	0.143151i	$-0.0457235\pi$
$102$			1.20392e8i			1.11224i
$103$		−	1.76307e8i		−	1.56646i	−0.621730	−	0.783232i	$-0.713570\pi$
$103$		−	1.76307e8i		−	1.56646i	0.621730	−	0.783232i	$-0.286430\pi$
$104$	−8.74702e7			−0.747699
$105$	1.78502e8			1.46854
$106$			9.03920e7i			0.715989i
$107$	1.33388e8			1.01761			0.508804	−	0.860883i	$-0.330088\pi$
$107$	1.33388e8			1.01761			0.508804	+	0.860883i	$0.330088\pi$
$108$	−1.76826e7			−0.129972
$109$		−	6.03788e7i		−	0.427738i	−0.976862	−	0.213869i	$-0.931393\pi$
$109$		−	6.03788e7i		−	0.427738i	0.976862	−	0.213869i	$-0.0686066\pi$
$110$	4.33164e8			2.95857
$111$			1.83170e7i			0.120660i
$112$	3.63065e8			2.30734
$113$		−	3.07112e8i		−	1.88358i	−0.336206	−	0.941788i	$-0.609144\pi$
$113$		−	3.07112e8i		−	1.88358i	0.336206	−	0.941788i	$-0.390856\pi$
$114$		−	1.28384e8i		−	0.760133i
$115$			2.26199e8i			1.29330i
$116$	1.22980e8			0.679208
$117$		−	1.11144e8i		−	0.593121i
$118$	1.09288e8	+	2.25899e8i	0.563693	+	1.16516i
$119$	5.64821e8			2.81659
$120$			6.76169e7i			0.326084i
$121$	−4.05570e8			−1.89201
$122$	1.88335e8			0.850144
$123$	2.01991e8			0.882492
$124$		−	1.48464e7i		−	0.0627962i
$125$	6.34711e7			0.259978
$126$			2.05794e8i			0.816491i
$127$	−1.43879e8			−0.553075			−0.276537	−	0.961003i	$-0.589187\pi$
$127$	−1.43879e8			−0.553075			−0.276537	+	0.961003i	$0.589187\pi$
$128$			2.13708e8i			0.796125i
$129$			7.66594e7i			0.276826i
$130$	8.84137e8			3.09561
$131$		−	3.51695e8i		−	1.19421i	−0.802163	−	0.597105i	$-0.796318\pi$
$131$		−	3.51695e8i		−	1.19421i	0.802163	−	0.597105i	$-0.203682\pi$
$132$			2.01311e8i			0.663088i
$133$	−6.02314e8			−1.92494
$134$	−4.55205e7			−0.141185
$135$	−8.59175e7			−0.258670
$136$			2.13955e8i			0.625412i
$137$	5.18425e8			1.47165			0.735823	−	0.677174i	$-0.236795\pi$
$137$	5.18425e8			1.47165			0.735823	+	0.677174i	$0.236795\pi$
$138$	−2.60783e8			−0.719058
$139$	1.33649e8			0.358021			0.179010	−	0.983847i	$-0.442710\pi$
$139$	1.33649e8			0.358021			0.179010	+	0.983847i	$0.442710\pi$
$140$	−6.59921e8			−1.71783
$141$		−	2.28467e8i		−	0.578026i
$142$			3.47396e8i			0.854421i
$143$	−1.26534e9			−3.02597
$144$	−1.74752e8			−0.406417
$145$	5.97546e8			1.35176
$146$	1.73603e7			0.0382072
$147$	6.95896e8			1.49031
$148$		−	6.77177e7i		−	0.141142i
$149$			1.12096e8i			0.227428i	0.993514	+	0.113714i	$0.0362748\pi$
$149$			1.12096e8i			0.227428i	−0.993514	+	0.113714i	$0.963725\pi$
$150$		−	3.05143e8i		−	0.602752i
$151$		−	2.71112e8i		−	0.521484i	−0.965409	−	0.260742i	$-0.916033\pi$
$151$		−	2.71112e8i		−	0.521484i	0.965409	−	0.260742i	$-0.0839672\pi$
$152$		−	2.28157e8i		−	0.427425i
$153$	−2.71862e8			−0.496115
$154$	2.34291e9			4.16556
$155$		−	7.21367e7i		−	0.124977i
$156$			4.10898e8i			0.693802i
$157$		−	8.46853e7i		−	0.139383i	−0.997569	−	0.0696914i	$-0.977799\pi$
$157$		−	8.46853e7i		−	0.139383i	0.997569	−	0.0696914i	$-0.0222015\pi$
$158$		−	6.95734e8i		−	1.11639i
$159$	−2.04118e8			−0.319369
$160$		−	1.01998e9i		−	1.55637i
$161$			1.22347e9i			1.82092i
$162$		−	9.90538e7i		−	0.143817i
$163$	3.76158e8			0.532868			0.266434	−	0.963853i	$-0.414154\pi$
$163$	3.76158e8			0.532868			0.266434	+	0.963853i	$0.414154\pi$
$164$	−7.46756e8			−1.03229
$165$			9.78146e8i			1.31968i
$166$	−1.47787e9			−1.94627
$167$	−7.52892e8			−0.967982			−0.483991	−	0.875073i	$-0.660813\pi$
$167$	−7.52892e8			−0.967982			−0.483991	+	0.875073i	$0.660813\pi$
$168$			3.65728e8i			0.459115i
$169$	−1.76698e9			−2.16613
$170$		−	2.16262e9i		−	2.58932i
$171$	2.89908e8			0.339059
$172$		−	2.83409e8i		−	0.323817i
$173$		−	1.48929e9i		−	1.66263i	−0.555803	−	0.831314i	$-0.687589\pi$
$173$		−	1.48929e9i		−	1.66263i	0.555803	−	0.831314i	$-0.312411\pi$
$174$			6.88907e8i			0.751560i
$175$	−1.43159e9			−1.52639
$176$			1.98950e9i			2.07345i
$177$	−5.10112e8	+	2.46787e8i	−0.519724	+	0.251437i
$178$	5.91538e8			0.589254
$179$			8.56853e8i			0.834631i	0.908762	+	0.417315i	$0.137029\pi$
$179$			8.56853e8i			0.834631i	−0.908762	+	0.417315i	$0.862971\pi$
$180$	3.17635e8			0.302579
$181$	−1.69536e9			−1.57960			−0.789801	−	0.613363i	$-0.789816\pi$
$181$	−1.69536e9			−1.57960			−0.789801	+	0.613363i	$0.789816\pi$
$182$	4.78214e9			4.35850
$183$			4.25288e8i			0.379209i
$184$	−4.63452e8			−0.404328
$185$		−	3.29032e8i		−	0.280900i
$186$	8.31660e7			0.0694855
$187$			3.09507e9i			2.53107i
$188$			8.44640e8i			0.676146i
$189$	−4.64713e8			−0.364198
$190$			2.30618e9i			1.76961i
$191$		−	9.34661e8i		−	0.702297i	−0.936320	−	0.351149i	$-0.885791\pi$
$191$		−	9.34661e8i		−	0.702297i	0.936320	−	0.351149i	$-0.114209\pi$
$192$	2.19318e8			0.161387
$193$	1.11844e9			0.806092			0.403046	−	0.915180i	$-0.367951\pi$
$193$	1.11844e9			0.806092			0.403046	+	0.915180i	$0.367951\pi$
$194$	3.44789e8			0.243414
$195$			1.99650e9i			1.38080i
$196$	−2.57272e9			−1.74328
$197$	1.06355e9			0.706145			0.353073	−	0.935596i	$-0.385137\pi$
$197$	1.06355e9			0.706145			0.353073	+	0.935596i	$0.385137\pi$
$198$	−1.12770e9			−0.733723
$199$	−4.25436e8			−0.271283			−0.135641	−	0.990758i	$-0.543310\pi$
$199$	−4.25436e8			−0.271283			−0.135641	+	0.990758i	$0.543310\pi$
$200$		−	5.42286e8i		−	0.338929i
$201$		−	1.02792e8i		−	0.0629758i
$202$	−6.16997e8			−0.370576
$203$	3.23203e9			1.90323
$204$	1.00507e9			0.580330
$205$	−3.62840e9			−2.05447
$206$	−3.65126e9			−2.02756
$207$		−	5.88886e8i		−	0.320737i
$208$			4.06079e9i			2.16949i
$209$		−	3.30052e9i		−	1.72981i
$210$		−	3.69673e9i		−	1.90082i
$211$			2.35827e8i			0.118977i	0.998229	+	0.0594887i	$0.0189470\pi$
$211$			2.35827e8i			0.118977i	−0.998229	+	0.0594887i	$0.981053\pi$
$212$	7.54620e8			0.373581
$213$	−7.84470e8			−0.381117
$214$		−	2.76241e9i		−	1.31714i
$215$		−	1.37705e9i		−	0.644459i
$216$		−	1.76034e8i		−	0.0808687i
$217$		−	3.90175e8i		−	0.175963i
$218$	−1.25043e9			−0.553645
$219$			3.92019e7i			0.0170424i
$220$		−	3.61619e9i		−	1.54369i
$221$			6.31738e9i			2.64831i
$222$	3.79339e8			0.156177
$223$	−8.10341e6			−0.00327679			−0.00163839	−	0.999999i	$-0.500522\pi$
$223$	−8.10341e6			−0.00327679			−0.00163839	+	0.999999i	$0.500522\pi$
$224$		−	5.51692e9i		−	2.19131i
$225$	6.89056e8			0.268859
$226$	−6.36020e9			−2.43802
$227$		−	4.59683e9i		−	1.73123i	−0.500710	−	0.865615i	$-0.666928\pi$
$227$		−	4.59683e9i		−	1.73123i	0.500710	−	0.865615i	$-0.333072\pi$
$228$	−1.07179e9			−0.396614
$229$		−	8.36769e8i		−	0.304273i	−0.988359	−	0.152137i	$-0.951385\pi$
$229$		−	8.36769e8i		−	0.304273i	0.988359	−	0.152137i	$-0.0486154\pi$
$230$	4.68451e9			1.67399
$231$			5.29062e9i			1.85806i
$232$			1.22429e9i			0.422604i
$233$			1.78391e9i			0.605269i	0.953107	+	0.302634i	$0.0978661\pi$
$233$			1.78391e9i			0.605269i	−0.953107	+	0.302634i	$0.902134\pi$
$234$	−2.30176e9			−0.767709
$235$			4.10401e9i			1.34566i
$236$	1.88588e9	−	9.12366e8i	0.607946	−	0.294118i
$237$	1.57107e9			0.497968
$238$		−	1.16973e10i		−	3.64566i
$239$	4.17187e8			0.127861			0.0639307	−	0.997954i	$-0.479636\pi$
$239$	4.17187e8			0.127861			0.0639307	+	0.997954i	$0.479636\pi$
$240$	3.13910e9			0.946151
$241$	−2.94836e8			−0.0874003			−0.0437001	−	0.999045i	$-0.513915\pi$
$241$	−2.94836e8			−0.0874003			−0.0437001	+	0.999045i	$0.513915\pi$
$242$			8.39922e9i			2.44894i
$243$	2.23677e8			0.0641500
$244$		−	1.57228e9i		−	0.443579i
$245$	−1.25005e10			−3.46948
$246$		−	4.18316e9i		−	1.14226i
$247$		−	6.73673e9i		−	1.80993i
$248$	1.47799e8			0.0390718
$249$		−	3.33723e9i		−	0.868138i
$250$		−	1.31447e9i		−	0.336503i
$251$	7.08685e8			0.178549			0.0892746	−	0.996007i	$-0.471545\pi$
$251$	7.08685e8			0.178549			0.0892746	+	0.996007i	$0.471545\pi$
$252$	1.71804e9			0.426020
$253$	−6.70429e9			−1.63633
$254$			2.97970e9i			0.715875i
$255$	4.88351e9			1.15497
$256$	5.62640e9			1.31000
$257$	−2.28871e9			−0.524637			−0.262318	−	0.964981i	$-0.584487\pi$
$257$	−2.28871e9			−0.524637			−0.262318	+	0.964981i	$0.584487\pi$
$258$	1.58759e9			0.358311
$259$		−	1.77968e9i		−	0.395497i
$260$		−	7.38105e9i		−	1.61519i
$261$	−1.55565e9			−0.335235
$262$	−7.28349e9			−1.54573
$263$	8.89682e9			1.85957			0.929783	−	0.368108i	$-0.119994\pi$
$263$	8.89682e9			1.85957			0.929783	+	0.368108i	$0.119994\pi$
$264$	−2.00409e9			−0.412574
$265$	3.66661e9			0.743500
$266$			1.24737e10i			2.49155i
$267$			1.33578e9i			0.262838i
$268$			3.80019e8i			0.0736658i
$269$			4.26909e9i			0.815317i	0.913135	+	0.407658i	$0.133655\pi$
$269$			4.26909e9i			0.815317i	−0.913135	+	0.407658i	$0.866345\pi$
$270$			1.77932e9i			0.334811i
$271$	−1.99409e9			−0.369715			−0.184858	−	0.982765i	$-0.559182\pi$
$271$	−1.99409e9			−0.369715			−0.184858	+	0.982765i	$0.559182\pi$
$272$	9.93282e9			1.81467
$273$			1.07988e10i			1.94412i
$274$		−	1.07364e10i		−	1.90483i
$275$		−	7.84471e9i		−	1.37166i
$276$			2.17710e9i			0.375182i
$277$	3.08122e9			0.523363			0.261681	−	0.965154i	$-0.415723\pi$
$277$	3.08122e9			0.523363			0.261681	+	0.965154i	$0.415723\pi$
$278$		−	2.76784e9i		−	0.463406i
$279$			1.87801e8i			0.0309942i
$280$		−	6.56965e9i		−	1.06883i
$281$	7.73056e9			1.23990			0.619948	−	0.784643i	$-0.287153\pi$
$281$	7.73056e9			1.23990			0.619948	+	0.784643i	$0.287153\pi$
$282$	−4.73148e9			−0.748171
$283$		−	1.40153e8i		−	0.0218503i	−0.999940	−	0.0109252i	$-0.996522\pi$
$283$		−	1.40153e8i		−	0.0218503i	0.999940	−	0.0109252i	$-0.00347765\pi$
$284$	2.90017e9			0.445811
$285$	−5.20768e9			−0.789341
$286$			2.62049e10i			3.91668i
$287$	−1.96254e10			−2.89262
$288$			2.65542e9i			0.385979i
$289$	8.47676e9			1.21517
$290$		−	1.23750e10i		−	1.74966i
$291$			7.78581e8i			0.108576i
$292$		−	1.44929e8i		−	0.0199353i
$293$	1.15041e9			0.156092			0.0780462	−	0.996950i	$-0.475132\pi$
$293$	1.15041e9			0.156092			0.0780462	+	0.996950i	$0.475132\pi$
$294$		−	1.44118e10i		−	1.92898i
$295$	9.16325e9	−	4.43308e9i	1.20993	−	0.585352i
$296$	6.74144e8			0.0878185
$297$		−	2.54650e9i		−	0.327279i
$298$	2.32147e9			0.294373
$299$	−1.36842e10			−1.71212
$300$	−2.54743e9			−0.314497
$301$		−	7.44822e9i		−	0.907375i
$302$	−5.61464e9			−0.674985
$303$		−	1.39327e9i		−	0.165297i
$304$	−1.05922e10			−1.24020
$305$		−	7.63953e9i		−	0.882810i
$306$			5.63017e9i			0.642149i
$307$	−6.59310e9			−0.742226			−0.371113	−	0.928588i	$-0.621024\pi$
$307$	−6.59310e9			−0.742226			−0.371113	+	0.928588i	$0.621024\pi$
$308$		−	1.95593e10i		−	2.17346i
$309$		−	8.24506e9i		−	0.904398i
$310$	−1.49393e9			−0.161764
$311$	4.88204e9			0.521867			0.260933	−	0.965357i	$-0.415970\pi$
$311$	4.88204e9			0.521867			0.260933	+	0.965357i	$0.415970\pi$
$312$	−4.09058e9			−0.431684
$313$			1.44353e10i			1.50401i	0.659160	+	0.752003i	$0.270912\pi$
$313$			1.44353e10i			1.50401i	−0.659160	+	0.752003i	$0.729088\pi$
$314$	−1.75381e9			−0.180411
$315$	8.34773e9			0.847864
$316$	−5.80821e9			−0.582497
$317$	2.88442e9			0.285642			0.142821	−	0.989749i	$-0.454383\pi$
$317$	2.88442e9			0.285642			0.142821	+	0.989749i	$0.454383\pi$
$318$			4.22721e9i			0.413376i
$319$			1.77106e10i			1.71030i
$320$	−3.93965e9			−0.375714
$321$	6.23792e9			0.587516
$322$	2.53377e10			2.35691
$323$	−1.64782e10			−1.51391
$324$	−8.26931e8			−0.0750394
$325$		−	1.60119e10i		−	1.43519i
$326$		−	7.79011e9i		−	0.689721i
$327$		−	2.82364e9i		−	0.246955i
$328$		−	7.43411e9i		−	0.642294i
$329$			2.21979e10i			1.89464i
$330$	2.02571e10			1.70813
$331$	−6.73106e9			−0.560753			−0.280377	−	0.959890i	$-0.590459\pi$
$331$	−6.73106e9			−0.560753			−0.280377	+	0.959890i	$0.590459\pi$
$332$			1.23377e10i			1.01550i
$333$			8.56602e8i			0.0696630i
$334$			1.55922e10i			1.25291i
$335$			1.84647e9i			0.146610i
$336$	1.69789e10			1.33215
$337$		−	6.26340e9i		−	0.485614i	−0.970075	−	0.242807i	$-0.921932\pi$
$337$		−	6.26340e9i		−	0.485614i	0.970075	−	0.242807i	$-0.0780681\pi$
$338$			3.65935e10i			2.80374i
$339$		−	1.43622e10i		−	1.08748i
$340$	−1.80543e10			−1.35103
$341$	2.13806e9			0.158125
$342$		−	6.00390e9i		−	0.438863i
$343$	−4.14196e10			−2.99247
$344$	2.82139e9			0.201479
$345$			1.05783e10i			0.746687i
$346$	−3.08427e10			−2.15203
$347$			1.66110e10i			1.14572i	0.819655	+	0.572858i	$0.194165\pi$
$347$			1.66110e10i			1.14572i	−0.819655	+	0.572858i	$0.805835\pi$
$348$	5.75121e9			0.392141
$349$			4.18146e9i			0.281855i	0.990020	+	0.140928i	$0.0450085\pi$
$349$			4.18146e9i			0.281855i	−0.990020	+	0.140928i	$0.954992\pi$
$350$			2.96477e10i			1.97569i
$351$		−	5.19770e9i		−	0.342438i
$352$	3.02312e10			1.96918
$353$			1.36943e10i			0.881945i	0.897521	+	0.440973i	$0.145366\pi$
$353$			1.36943e10i			0.881945i	−0.897521	+	0.440973i	$0.854634\pi$
$354$	5.11087e9	+	1.05643e10i	0.325448	+	0.672707i
$355$	1.40916e10			0.887251
$356$		−	4.93834e9i		−	0.307455i
$357$	2.64141e10			1.62616
$358$	1.77452e10			1.08031
$359$	1.48079e10			0.891486			0.445743	−	0.895161i	$-0.352940\pi$
$359$	1.48079e10			0.891486			0.445743	+	0.895161i	$0.352940\pi$
$360$			3.16213e9i			0.188265i
$361$	5.88508e8			0.0346516
$362$			3.51104e10i			2.04457i
$363$	−1.89666e10			−1.09235
$364$		−	3.99228e10i		−	2.27413i
$365$		−	7.04192e8i		−	0.0396752i
$366$	8.80757e9			0.490831
$367$			1.10191e10i			0.607408i	0.952766	+	0.303704i	$0.0982235\pi$
$367$			1.10191e10i			0.607408i	−0.952766	+	0.303704i	$0.901777\pi$
$368$			2.15157e10i			1.17318i
$369$	9.44616e9			0.509507
$370$	−6.81415e9			−0.363584
$371$	1.98321e10			1.04682
$372$		−	6.94296e8i		−	0.0362554i
$373$	2.84790e10			1.47126			0.735629	−	0.677385i	$-0.236887\pi$
$373$	2.84790e10			1.47126			0.735629	+	0.677385i	$0.236887\pi$
$374$	6.40979e10			3.27610
$375$	2.96825e9			0.150098
$376$	−8.40857e9			−0.420698
$377$			3.61494e10i			1.78952i
$378$			9.62406e9i			0.471401i
$379$	−5.75537e9			−0.278944			−0.139472	−	0.990226i	$-0.544540\pi$
$379$	−5.75537e9			−0.278944			−0.139472	+	0.990226i	$0.544540\pi$
$380$	1.92527e10			0.923331
$381$	−6.72857e9			−0.319318
$382$	−1.93565e10			−0.909022
$383$	−8.57438e8			−0.0398481			−0.0199240	−	0.999801i	$-0.506342\pi$
$383$	−8.57438e8			−0.0398481			−0.0199240	+	0.999801i	$0.506342\pi$
$384$			9.99414e9i			0.459643i
$385$		−	9.50366e10i		−	4.32561i
$386$		−	2.31626e10i		−	1.04337i
$387$			3.58501e9i			0.159825i
$388$		−	2.87840e9i		−	0.127006i
$389$	−1.61501e10			−0.705306			−0.352653	−	0.935754i	$-0.614720\pi$
$389$	−1.61501e10			−0.705306			−0.352653	+	0.935754i	$0.614720\pi$
$390$	4.13470e10			1.78725
$391$			3.34720e10i			1.43210i
$392$		−	2.56120e10i		−	1.08467i
$393$		−	1.64471e10i		−	0.689477i
$394$		−	2.20258e10i		−	0.914003i
$395$	−2.82214e10			−1.15928
$396$			9.41438e9i			0.382834i
$397$			3.60875e10i			1.45276i	0.687292	+	0.726381i	$0.258799\pi$
$397$			3.60875e10i			1.45276i	−0.687292	+	0.726381i	$0.741201\pi$
$398$			8.81065e9i			0.351136i
$399$	−2.81674e10			−1.11136
$400$	−2.51755e10			−0.983420
$401$			9.33954e8i			0.0361200i	0.999837	+	0.0180600i	$0.00574899\pi$
$401$			9.33954e8i			0.0361200i	−0.999837	+	0.0180600i	$0.994251\pi$
$402$	−2.12878e9			−0.0815130
$403$	4.36401e9			0.165450
$404$			5.15088e9i			0.193355i
$405$	−4.01796e9			−0.149343
$406$		−	6.69342e10i		−	2.46345i
$407$	9.75216e9			0.355405
$408$			1.00057e10i			0.361082i
$409$			3.19361e10i			1.14127i	0.821204	+	0.570635i	$0.193303\pi$
$409$			3.19361e10i			1.14127i	−0.821204	+	0.570635i	$0.806697\pi$
$410$			7.51430e10i			2.65921i
$411$	2.42443e10			0.849655
$412$			3.04818e10i			1.05792i
$413$	4.95625e10	−	2.39778e10i	1.70354	−	0.824154i
$414$	−1.21956e10			−0.415148
$415$			5.99473e10i			2.02105i
$416$	6.17053e10			2.06039
$417$	6.25017e9			0.206703
$418$	−6.83527e10			−2.23898
$419$		−	4.81100e10i		−	1.56091i	−0.625210	−	0.780457i	$-0.714987\pi$
$419$		−	4.81100e10i		−	1.56091i	0.625210	−	0.780457i	$-0.285013\pi$
$420$	−3.08614e10			−0.991788
$421$		−	5.57972e10i		−	1.77617i	−0.459682	−	0.888084i	$-0.652036\pi$
$421$		−	5.57972e10i		−	1.77617i	0.459682	−	0.888084i	$-0.347964\pi$
$422$	4.88391e9			0.153999
$423$		−	1.06844e10i		−	0.333724i
$424$			7.51240e9i			0.232442i
$425$	−3.91657e10			−1.20047
$426$			1.62461e10i			0.493300i
$427$		−	4.13209e10i		−	1.24296i
$428$	−2.30615e10			−0.687246
$429$	−5.91743e10			−1.74704
$430$	−2.85182e10			−0.834159
$431$			2.56565e10i			0.743514i	0.928330	+	0.371757i	$0.121244\pi$
$431$			2.56565e10i			0.743514i	−0.928330	+	0.371757i	$0.878756\pi$
$432$	−8.17233e9			−0.234645
$433$	−3.15184e10			−0.896628			−0.448314	−	0.893876i	$-0.647975\pi$
$433$	−3.15184e10			−0.896628			−0.448314	+	0.893876i	$0.647975\pi$
$434$	−8.08040e9			−0.227758
$435$	2.79445e10			0.780439
$436$			1.04389e10i			0.288875i
$437$		−	3.56939e10i		−	0.978741i
$438$	8.11859e8			0.0220589
$439$	−4.77667e10			−1.28608			−0.643039	−	0.765834i	$-0.722327\pi$
$439$	−4.77667e10			−1.28608			−0.643039	+	0.765834i	$0.722327\pi$
$440$	3.59999e10			0.960486
$441$	3.25439e10			0.860428
$442$	1.30831e11			3.42785
$443$		−	6.06381e10i		−	1.57446i	−0.616662	−	0.787228i	$-0.711516\pi$
$443$		−	6.06381e10i		−	1.57446i	0.616662	−	0.787228i	$-0.288484\pi$
$444$		−	3.16684e9i		−	0.0814882i
$445$		−	2.39948e10i		−	0.611896i
$446$			1.67819e8i			0.00424133i
$447$			5.24220e9i			0.131306i
$448$	−2.13089e10			−0.528992
$449$	−3.18003e10			−0.782431			−0.391216	−	0.920299i	$-0.627945\pi$
$449$	−3.18003e10			−0.782431			−0.391216	+	0.920299i	$0.627945\pi$
$450$		−	1.42701e10i		−	0.347999i
$451$		−	1.07542e11i		−	2.59939i
$452$			5.30969e10i			1.27208i
$453$		−	1.26787e10i		−	0.301079i
$454$	−9.51988e10			−2.24083
$455$		−	1.93980e11i		−	4.52597i
$456$		−	1.06698e10i		−	0.246774i
$457$		−	2.02468e8i		−	0.00464186i	−0.999997	−	0.00232093i	$-0.999261\pi$
$457$		−	2.02468e8i		−	0.00464186i	0.999997	−	0.00232093i	$-0.000738776\pi$
$458$	−1.73292e10			−0.393838
$459$	−1.27137e10			−0.286432
$460$		−	3.91077e10i		−	0.873436i
$461$	1.84288e10			0.408031			0.204015	−	0.978968i	$-0.434601\pi$
$461$	1.84288e10			0.408031			0.204015	+	0.978968i	$0.434601\pi$
$462$	1.09567e11			2.40498
$463$			5.51189e10i			1.19943i	0.800212	+	0.599717i	$0.204720\pi$
$463$			5.51189e10i			1.19943i	−0.800212	+	0.599717i	$0.795280\pi$
$464$	5.68376e10			1.22621
$465$		−	3.37350e9i		−	0.0721554i
$466$	3.69441e10			0.783433
$467$			2.34956e10i			0.493991i	0.969017	+	0.246996i	$0.0794433\pi$
$467$			2.34956e10i			0.493991i	−0.969017	+	0.246996i	$0.920557\pi$
$468$			1.92158e10i			0.400567i
$469$			9.98723e9i			0.206421i
$470$	8.49926e10			1.74177
$471$		−	3.96034e9i		−	0.0804727i
$472$	9.08280e9	+	1.87743e10i	0.183000	+	0.378265i
$473$	4.08143e10			0.815394
$474$		−	3.25363e10i		−	0.644547i
$475$	4.17655e10			0.820433
$476$	−9.76524e10			−1.90220
$477$	−9.54564e9			−0.184388
$478$		−	8.63981e9i		−	0.165498i
$479$	−3.87685e10			−0.736438			−0.368219	−	0.929739i	$-0.620032\pi$
$479$	−3.87685e10			−0.736438			−0.368219	+	0.929739i	$0.620032\pi$
$480$		−	4.76999e10i		−	0.898571i
$481$	1.99053e10			0.371867
$482$			6.10597e9i			0.113127i
$483$			5.72161e10i			1.05131i
$484$	7.01193e10			1.27778
$485$		−	1.39858e10i		−	0.252767i
$486$		−	4.63229e9i		−	0.0830329i
$487$	−1.06085e11			−1.88598			−0.942990	−	0.332820i	$-0.892000\pi$
$487$	−1.06085e11			−1.88598			−0.942990	+	0.332820i	$0.892000\pi$
$488$	1.56524e10			0.275995
$489$	1.75912e10			0.307652
$490$			2.58882e11i			4.49074i
$491$	−1.01600e11			−1.74810			−0.874049	−	0.485838i	$-0.838514\pi$
$491$	−1.01600e11			−1.74810			−0.874049	+	0.485838i	$0.838514\pi$
$492$	−3.49223e10			−0.595995
$493$	8.84225e10			1.49684
$494$	−1.39516e11			−2.34269
$495$			4.57433e10i			0.761916i
$496$		−	6.86153e9i		−	0.113369i
$497$	7.62190e10			1.24922
$498$	−6.91129e10			−1.12368
$499$	−4.47378e10			−0.721560			−0.360780	−	0.932651i	$-0.617490\pi$
$499$	−4.47378e10			−0.721560			−0.360780	+	0.932651i	$0.617490\pi$
$500$	−1.09736e10			−0.175577
$501$	−3.52093e10			−0.558864
$502$		−	1.46766e10i		−	0.231106i
$503$			7.34878e9i			0.114800i	0.998351	+	0.0574002i	$0.0182811\pi$
$503$			7.34878e9i			0.114800i	−0.998351	+	0.0574002i	$0.981719\pi$
$504$			1.71034e10i			0.265070i
$505$			2.50275e10i			0.384816i
$506$			1.38844e11i			2.11799i
$507$	−8.26334e10			−1.25062
$508$	2.48754e10			0.373522
$509$			5.25488e10i			0.782874i	0.920205	+	0.391437i	$0.128022\pi$
$509$			5.25488e10i			0.782874i	−0.920205	+	0.391437i	$0.871978\pi$
$510$		−	1.01136e11i		−	1.49494i
$511$		−	3.80885e9i		−	0.0558613i
$512$		−	6.18117e10i		−	0.899479i
$513$	1.35577e10			0.195756
$514$			4.73985e10i			0.679066i
$515$			1.48108e11i			2.10547i
$516$		−	1.32537e10i		−	0.186956i
$517$	−1.21638e11			−1.70258
$518$	−3.68566e10			−0.511913
$519$		−	6.96472e10i		−	0.959919i
$520$	7.34799e10			1.00497
$521$	3.02402e10			0.410426			0.205213	−	0.978717i	$-0.434211\pi$
$521$	3.02402e10			0.410426			0.205213	+	0.978717i	$0.434211\pi$
$522$			3.22170e10i			0.433914i
$523$	9.02188e10			1.20584			0.602921	−	0.797801i	$-0.294003\pi$
$523$	9.02188e10			1.20584			0.602921	+	0.797801i	$0.294003\pi$
$524$			6.08048e10i			0.806515i
$525$	−6.69487e10			−0.881261
$526$		−	1.84250e11i		−	2.40694i
$527$		−	1.06745e10i		−	0.138390i
$528$			9.30397e10i			1.19711i
$529$	5.80656e9			0.0741475
$530$		−	7.59343e10i		−	0.962353i
$531$	−2.38556e10	+	1.15411e10i	−0.300063	+	0.145167i
$532$	1.04135e11			1.30002
$533$		−	2.19505e11i		−	2.71979i
$534$	2.76635e10			0.340206
$535$	−1.12053e11			−1.36776
$536$	−3.78317e9			−0.0458349
$537$			4.00711e10i			0.481874i
$538$	8.84115e10			1.05531
$539$		−	3.70502e11i		−	4.38971i
$540$	1.48543e10			0.174694
$541$		−	2.53245e9i		−	0.0295632i	−0.999891	−	0.0147816i	$-0.995295\pi$
$541$		−	2.53245e9i		−	0.0295632i	0.999891	−	0.0147816i	$-0.00470530\pi$
$542$			4.12969e10i			0.478543i
$543$	−7.92842e10			−0.911984
$544$		−	1.50933e11i		−	1.72341i
$545$			5.07216e10i			0.574919i
$546$	2.23639e11			2.51638
$547$	−2.40448e10			−0.268579			−0.134289	−	0.990942i	$-0.542875\pi$
$547$	−2.40448e10			−0.268579			−0.134289	+	0.990942i	$0.542875\pi$
$548$	−8.96309e10			−0.993883
$549$			1.98887e10i			0.218936i
$550$	−1.62461e11			−1.77541
$551$	−9.42920e10			−1.02298
$552$	−2.16735e10			−0.233439
$553$	−1.52645e11			−1.63223
$554$		−	6.38110e10i		−	0.677417i
$555$		−	1.53873e10i		−	0.162178i
$556$	−2.31068e10			−0.241791
$557$	6.66433e10			0.692366			0.346183	−	0.938167i	$-0.387478\pi$
$557$	6.66433e10			0.692366			0.346183	+	0.938167i	$0.387478\pi$
$558$	3.88929e9			0.0401175
$559$	8.33065e10			0.853162
$560$	−3.04995e11			−3.10128
$561$			1.44742e11i			1.46131i
$562$		−	1.60097e11i		−	1.60487i
$563$		−	1.64430e11i		−	1.63661i	−0.574781	−	0.818307i	$-0.694913\pi$
$563$		−	1.64430e11i		−	1.63661i	0.574781	−	0.818307i	$-0.305087\pi$
$564$			3.94999e10i			0.390373i
$565$			2.57991e11i			2.53170i
$566$	−2.90253e9			−0.0282820
$567$	−2.17325e10			−0.210270
$568$			2.88718e10i			0.277384i
$569$			1.01693e11i			0.970162i	0.874469	+	0.485081i	$0.161210\pi$
$569$			1.01693e11i			0.970162i	−0.874469	+	0.485081i	$0.838790\pi$
$570$			1.07849e11i			1.02169i
$571$		−	1.81949e10i		−	0.171161i	−0.996331	−	0.0855806i	$-0.972725\pi$
$571$		−	1.81949e10i		−	0.171161i	0.996331	−	0.0855806i	$-0.0272745\pi$
$572$	2.18766e11			2.04360
$573$		−	4.37098e10i		−	0.405471i
$574$			4.06435e11i			3.74407i
$575$		−	8.48376e10i		−	0.776098i
$576$	1.02565e10			0.0931769
$577$	1.56217e11			1.40937			0.704684	−	0.709521i	$-0.251089\pi$
$577$	1.56217e11			1.40937			0.704684	+	0.709521i	$0.251089\pi$
$578$		−	1.75551e11i		−	1.57287i
$579$	5.23044e10			0.465397
$580$	−1.03310e11			−0.912917
$581$			3.24245e11i			2.84557i
$582$	1.61242e10			0.140535
$583$			1.08674e11i			0.940704i
$584$	1.44280e9			0.0124038
$585$			9.33673e10i			0.797207i
$586$		−	2.38246e10i		−	0.202039i
$587$			1.71108e11i			1.44118i	0.693361	+	0.720591i	$0.256129\pi$
$587$			1.71108e11i			1.44118i	−0.693361	+	0.720591i	$0.743871\pi$
$588$	−1.20314e11			−1.00648
$589$			1.13831e10i			0.0945798i
$590$	−9.18076e10	−	1.89768e11i	−0.757654	−	1.56608i
$591$	4.97374e10			0.407693
$592$		−	3.12970e10i		−	0.254810i
$593$	−7.33238e10			−0.592961			−0.296481	−	0.955039i	$-0.595813\pi$
$593$	−7.33238e10			−0.592961			−0.296481	+	0.955039i	$0.595813\pi$
$594$	−5.27372e10			−0.423615
$595$	−4.74482e11			−3.78575
$596$		−	1.93803e10i		−	0.153595i
$597$	−1.98957e10			−0.156625
$598$			2.83396e11i			2.21610i
$599$	1.65450e11			1.28516			0.642582	−	0.766217i	$-0.277863\pi$
$599$	1.65450e11			1.28516			0.642582	+	0.766217i	$0.277863\pi$
$600$		−	2.53602e10i		−	0.195681i
$601$		−	6.47113e10i		−	0.496001i	−0.968760	−	0.248000i	$-0.920227\pi$
$601$		−	6.47113e10i		−	0.496001i	0.968760	−	0.248000i	$-0.0797734\pi$
$602$	−1.54250e11			−1.17446
$603$		−	4.80709e9i		−	0.0363591i
$604$			4.68728e10i			0.352187i
$605$	3.40701e11			2.54304
$606$	−2.88541e10			−0.213952
$607$	1.06456e11			0.784178			0.392089	−	0.919927i	$-0.371752\pi$
$607$	1.06456e11			0.784178			0.392089	+	0.919927i	$0.371752\pi$
$608$			1.60952e11i			1.17783i
$609$	1.51147e11			1.09883
$610$	−1.58212e11			−1.14267
$611$	−2.48278e11			−1.78145
$612$	4.70024e10			0.335054
$613$		−	1.71688e11i		−	1.21590i	−0.793976	−	0.607950i	$-0.791992\pi$
$613$		−	1.71688e11i		−	1.21590i	0.793976	−	0.607950i	$-0.208008\pi$
$614$			1.36541e11i			0.960704i
$615$	−1.69683e11			−1.18615
$616$	1.94717e11			1.35233
$617$	6.22771e8			0.00429722			0.00214861	−	0.999998i	$-0.499316\pi$
$617$	6.22771e8			0.00429722			0.00214861	+	0.999998i	$0.499316\pi$
$618$	−1.70752e11			−1.17061
$619$	1.40781e11			0.958921			0.479460	−	0.877564i	$-0.340832\pi$
$619$	1.40781e11			0.958921			0.479460	+	0.877564i	$0.340832\pi$
$620$			1.24718e10i			0.0844037i
$621$		−	2.75395e10i		−	0.185178i
$622$		−	1.01105e11i		−	0.675481i
$623$		−	1.29784e11i		−	0.861526i
$624$			1.89904e11i			1.25255i
$625$	−1.76393e11			−1.15601
$626$	2.98951e11			1.94672
$627$		−	1.54350e11i		−	0.998703i
$628$			1.46413e10i			0.0941329i
$629$		−	4.86889e10i		−	0.311048i
$630$		−	1.72879e11i		−	1.09744i
$631$	3.02486e11			1.90804			0.954022	−	0.299736i	$-0.0968987\pi$
$631$	3.02486e11			1.90804			0.954022	+	0.299736i	$0.0968987\pi$
$632$		−	5.78219e10i		−	0.362430i
$633$			1.10286e10i			0.0686916i
$634$		−	5.97355e10i		−	0.369722i
$635$	1.20867e11			0.743382
$636$	3.52901e10			0.215687
$637$		−	7.56237e11i		−	4.59304i
$638$	3.66781e11			2.21373
$639$	−3.66860e10			−0.220038
$640$		−	1.79527e11i		−	1.07006i
$641$	2.62634e11			1.55567			0.777837	−	0.628466i	$-0.216317\pi$
$641$	2.62634e11			1.55567			0.777837	+	0.628466i	$0.216317\pi$
$642$		−	1.29185e11i		−	0.760454i
$643$	1.74145e11			1.01875			0.509374	−	0.860545i	$-0.329877\pi$
$643$	1.74145e11			1.01875			0.509374	+	0.860545i	$0.329877\pi$
$644$		−	2.11527e11i		−	1.22977i
$645$		−	6.43982e10i		−	0.372079i
$646$			3.41259e11i			1.95954i
$647$	6.34406e9			0.0362035			0.0181017	−	0.999836i	$-0.494238\pi$
$647$	6.34406e9			0.0362035			0.0181017	+	0.999836i	$0.494238\pi$
$648$		−	8.23228e9i		−	0.0466896i
$649$	1.31392e11	+	2.71589e11i	0.740609	+	1.53085i
$650$	−3.31602e11			−1.85765
$651$		−	1.82467e10i		−	0.101592i
$652$	−6.50343e10			−0.359875
$653$	−3.28080e11			−1.80437			−0.902187	−	0.431346i	$-0.858039\pi$
$653$	−3.28080e11			−1.80437			−0.902187	+	0.431346i	$0.858039\pi$
$654$	−5.84766e10			−0.319647
$655$			2.95443e11i			1.60513i
$656$	−3.45128e11			−1.86365
$657$			1.83329e9i			0.00983943i
$658$	4.59711e11			2.45234
$659$		−	3.44431e11i		−	1.82625i	−0.407678	−	0.913126i	$-0.633661\pi$
$659$		−	3.44431e11i		−	1.82625i	0.407678	−	0.913126i	$-0.366339\pi$
$660$		−	1.69112e11i		−	0.891250i
$661$	1.08781e11			0.569833			0.284916	−	0.958552i	$-0.408034\pi$
$661$	1.08781e11			0.569833			0.284916	+	0.958552i	$0.408034\pi$
$662$			1.39398e11i			0.725813i
$663$			2.95435e11i			1.52900i
$664$	−1.22824e11			−0.631847
$665$	5.05978e11			2.58729
$666$	1.77399e10			0.0901686
$667$			1.91534e11i			0.967703i
$668$	1.30168e11			0.653731
$669$	−3.78959e8			−0.00189185
$670$	3.82397e10			0.189765
$671$	2.26428e11			1.11696
$672$		−	2.58001e11i		−	1.26515i
$673$		−	3.75875e9i		−	0.0183224i	−0.999958	−	0.00916122i	$-0.997084\pi$
$673$		−	3.75875e9i		−	0.0183224i	0.999958	−	0.00916122i	$-0.00291615\pi$
$674$	−1.29713e11			−0.628556
$675$	3.22240e10			0.155226
$676$	3.05494e11			1.46291
$677$	−8.68097e10			−0.413251			−0.206625	−	0.978420i	$-0.566248\pi$
$677$	−8.68097e10			−0.413251			−0.206625	+	0.978420i	$0.566248\pi$
$678$	−2.97437e11			−1.40759
$679$		−	7.56469e10i		−	0.355887i
$680$		−	1.79734e11i		−	0.840610i
$681$		−	2.14972e11i		−	0.999526i
$682$		−	4.42784e10i		−	0.204670i
$683$		−	1.53291e11i		−	0.704425i	−0.935920	−	0.352213i	$-0.885429\pi$
$683$		−	1.53291e11i		−	0.704425i	0.935920	−	0.352213i	$-0.114571\pi$
$684$	−5.01224e10			−0.228985
$685$	−4.35506e11			−1.97802
$686$			8.57787e11i			3.87332i
$687$		−	3.91318e10i		−	0.175672i
$688$		−	1.30983e11i		−	0.584602i
$689$			2.21817e11i			0.984277i
$690$	2.19073e11			0.966478
$691$			4.99163e10i			0.218942i	0.993990	+	0.109471i	$0.0349157\pi$
$691$			4.99163e10i			0.218942i	−0.993990	+	0.109471i	$0.965084\pi$
$692$			2.57485e11i			1.12286i
$693$			2.47418e11i			1.07275i
$694$	3.44008e11			1.48296
$695$	−1.12273e11			−0.481212
$696$			5.72545e10i			0.243991i
$697$	−5.36916e11			−2.27497
$698$	8.65966e10			0.364821
$699$			8.34250e10i			0.349452i
$700$	2.47508e11			1.03085
$701$		−	7.98312e10i		−	0.330598i	−0.986243	−	0.165299i	$-0.947141\pi$
$701$		−	7.98312e10i		−	0.330598i	0.986243	−	0.165299i	$-0.0528590\pi$
$702$	−1.07643e11			−0.443237
$703$			5.19208e10i			0.212579i
$704$		−	1.16767e11i		−	0.475368i
$705$			1.91925e11i			0.776919i
$706$	2.83605e11			1.14155
$707$			1.35370e11i			0.541806i
$708$	8.81937e10	−	4.26671e10i	0.350998	−	0.169809i
$709$	−1.26648e11			−0.501202			−0.250601	−	0.968090i	$-0.580628\pi$
$709$	−1.26648e11			−0.501202			−0.250601	+	0.968090i	$0.580628\pi$
$710$		−	2.91832e11i		−	1.14842i
$711$	7.34715e10			0.287502
$712$	4.91622e10			0.191299
$713$	2.31223e10			0.0894689
$714$		−	5.47027e11i		−	2.10483i
$715$	1.06296e12			4.06717
$716$		−	1.48142e11i		−	0.563672i
$717$	1.95099e10			0.0738208
$718$		−	3.06666e11i		−	1.15390i
$719$			5.88819e10i			0.220326i	0.993914	+	0.110163i	$0.0351373\pi$
$719$			5.88819e10i			0.220326i	−0.993914	+	0.110163i	$0.964863\pi$
$720$	1.46801e11			0.546261
$721$			8.01089e11i			2.96442i
$722$		−	1.21878e10i		−	0.0448515i
$723$	−1.37881e10			−0.0504606
$724$	2.93112e11			1.06679
$725$	−2.24114e11			−0.811180
$726$			3.92793e11i			1.41389i
$727$	2.41969e10			0.0866209			0.0433104	−	0.999062i	$-0.486210\pi$
$727$	2.41969e10			0.0866209			0.0433104	+	0.999062i	$0.486210\pi$
$728$	3.97440e11			1.41497
$729$	1.04604e10			0.0370370
$730$	−1.45836e10			−0.0513538
$731$		−	2.03770e11i		−	0.713627i
$732$		−	7.35283e10i		−	0.256100i
$733$	−3.00481e11			−1.04088			−0.520441	−	0.853897i	$-0.674233\pi$
$733$	−3.00481e11			−1.04088			−0.520441	+	0.853897i	$0.674233\pi$
$734$	2.28201e11			0.786202
$735$	−5.84592e11			−2.00310
$736$	3.26939e11			1.11418
$737$	−5.47273e10			−0.185496
$738$		−	1.95627e11i		−	0.659483i
$739$			2.67704e11i			0.897587i	0.893636	+	0.448793i	$0.148146\pi$
$739$			2.67704e11i			0.897587i	−0.893636	+	0.448793i	$0.851854\pi$
$740$			5.68866e10i			0.189707i
$741$		−	3.15046e11i		−	1.04496i
$742$		−	4.10716e11i		−	1.35496i
$743$	−1.01469e11			−0.332948			−0.166474	−	0.986046i	$-0.553238\pi$
$743$	−1.01469e11			−0.332948			−0.166474	+	0.986046i	$0.553238\pi$
$744$	6.91186e9			0.0225581
$745$		−	9.41668e10i		−	0.305684i
$746$		−	5.89790e11i		−	1.90433i
$747$		−	1.56067e11i		−	0.501219i
$748$		−	5.35109e11i		−	1.70937i
$749$	−6.06076e11			−1.92575
$750$		−	6.14715e10i		−	0.194280i
$751$		−	3.88339e11i		−	1.22082i	−0.792086	−	0.610409i	$-0.791005\pi$
$751$		−	3.88339e11i		−	1.22082i	0.792086	−	0.610409i	$-0.208995\pi$
$752$			3.90367e11i			1.22068i
$753$	3.31419e10			0.103085
$754$	7.48642e11			2.31627
$755$			2.27749e11i			0.700921i
$756$	8.03446e10			0.245963
$757$	−4.92875e10			−0.150091			−0.0750453	−	0.997180i	$-0.523910\pi$
$757$	−4.92875e10			−0.150091			−0.0750453	+	0.997180i	$0.523910\pi$
$758$			1.19192e11i			0.361052i
$759$	−3.13529e11			−0.944736
$760$			1.91665e11i			0.574497i
$761$	−4.76705e11			−1.42138			−0.710692	−	0.703504i	$-0.751618\pi$
$761$	−4.76705e11			−1.42138			−0.710692	+	0.703504i	$0.751618\pi$
$762$			1.39347e11i			0.413311i
$763$			2.74344e11i			0.809464i
$764$			1.61594e11i			0.474300i
$765$	2.28379e11			0.666823
$766$			1.77573e10i			0.0515776i
$767$	2.68185e11	+	5.54344e11i	0.774914	+	1.60176i
$768$	2.63121e11			0.756328
$769$		−	3.24575e11i		−	0.928132i	−0.885801	−	0.464066i	$-0.846390\pi$
$769$		−	3.24575e11i		−	0.928132i	0.885801	−	0.464066i	$-0.153610\pi$
$770$	−1.96818e12			−5.59888
$771$	−1.07033e11			−0.302899
$772$	−1.93368e11			−0.544398
$773$		−	5.12558e10i		−	0.143557i	−0.997421	−	0.0717786i	$-0.977132\pi$
$773$		−	5.12558e10i		−	0.143557i	0.997421	−	0.0717786i	$-0.0228675\pi$
$774$	7.42443e10			0.206871
$775$			2.70554e10i			0.0749976i
$776$	2.86551e10			0.0790233
$777$		−	8.32274e10i		−	0.228340i
$778$			3.34464e11i			0.912916i
$779$	5.72556e11			1.55478
$780$		−	3.45177e11i		−	0.932532i
$781$			4.17660e11i			1.12258i
$782$	6.93195e11			1.85365

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)

\(f(q)\)	\(=\)	\(q-20.7097i q^{2} +46.7654 q^{3} -172.891 q^{4} -840.056 q^{5} -968.496i q^{6} -4543.72 q^{7} -1721.16i q^{8} +2187.00 q^{9} +O(q^{10})\)	\(q-20.7097i q^{2} +46.7654 q^{3} -172.891 q^{4} -840.056 q^{5} -968.496i q^{6} -4543.72 q^{7} -1721.16i q^{8} +2187.00 q^{9} +17397.3i q^{10} -24898.4i q^{11} -8085.30 q^{12} -50820.4i q^{13} +94099.0i q^{14} -39285.5 q^{15} -79904.8 q^{16} -124308. q^{17} -45292.1i q^{18} +132560. q^{19} +145238. q^{20} -212489. q^{21} -515637. q^{22} -269266. i q^{23} -80490.9i q^{24} +315069. q^{25} -1.05247e6 q^{26} +102276. q^{27} +785567. q^{28} -711317. q^{29} +813591. i q^{30} +85871.3i q^{31} +1.21418e6i q^{32} -1.16438e6i q^{33} +2.57438e6i q^{34} +3.81698e6 q^{35} -378112. q^{36} +391679. i q^{37} -2.74527e6i q^{38} -2.37663e6i q^{39} +1.44587e6i q^{40} +4.31923e6 q^{41} +4.40057e6i q^{42} +1.63923e6i q^{43} +4.30470e6i q^{44} -1.83720e6 q^{45} -5.57642e6 q^{46} -4.88539e6i q^{47} -3.73678e6 q^{48} +1.48806e7 q^{49} -6.52498e6i q^{50} -5.81332e6 q^{51} +8.78637e6i q^{52} -4.36472e6 q^{53} -2.11810e6i q^{54} +2.09160e7i q^{55} +7.82049e6i q^{56} +6.19920e6 q^{57} +1.47311e7i q^{58} +(-1.09079e7 + 5.27712e6i) q^{59} +6.79211e6 q^{60} +9.09407e6i q^{61} +1.77837e6 q^{62} -9.93712e6 q^{63} +4.68975e6 q^{64} +4.26919e7i q^{65} -2.41140e7 q^{66} -2.19803e6i q^{67} +2.14917e7 q^{68} -1.25923e7i q^{69} -7.90484e7i q^{70} -1.67746e7 q^{71} -3.76419e6i q^{72} +838268. i q^{73} +8.11154e6 q^{74} +1.47343e7 q^{75} -2.29184e7 q^{76} +1.13131e8i q^{77} -4.92193e7 q^{78} +3.35946e7 q^{79} +6.71245e7 q^{80} +4.78297e6 q^{81} -8.94499e7i q^{82} -7.13611e7i q^{83} +3.67374e7 q^{84} +1.04426e8 q^{85} +3.39480e7 q^{86} -3.32650e7 q^{87} -4.28542e7 q^{88} +2.85634e7i q^{89} +3.80479e7i q^{90} +2.30913e8i q^{91} +4.65537e7i q^{92} +4.01580e6i q^{93} -1.01175e8 q^{94} -1.11358e8 q^{95} +5.67818e7i q^{96} +1.66487e7i q^{97} -3.08172e8i q^{98} -5.44527e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) \)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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