LMFDB - Embedded newform 177.9.c.a.58.11

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			58.11
Character	$\chi$	$=$	177.58
Dual form			177.9.c.a.58.70

$q$-expansion

$f(q)$	$=$	$q-25.1007i q^{2} -46.7654 q^{3} -374.043 q^{4} +458.555 q^{5} +1173.84i q^{6} -1157.18 q^{7} +2962.95i q^{8} +2187.00 q^{9} +O(q^{10})$	\(q-25.1007i q^{2} -46.7654 q^{3} -374.043 q^{4} +458.555 q^{5} +1173.84i q^{6} -1157.18 q^{7} +2962.95i q^{8} +2187.00 q^{9} -11510.0i q^{10} -15446.5i q^{11} +17492.3 q^{12} +19712.2i q^{13} +29046.0i q^{14} -21444.5 q^{15} -21382.9 q^{16} -208.117 q^{17} -54895.1i q^{18} -32459.5 q^{19} -171519. q^{20} +54116.1 q^{21} -387717. q^{22} -244303. i q^{23} -138564. i q^{24} -180353. q^{25} +494789. q^{26} -102276. q^{27} +432836. q^{28} -920152. q^{29} +538270. i q^{30} +725461. i q^{31} +1.29524e6i q^{32} +722362. i q^{33} +5223.87i q^{34} -530631. q^{35} -818032. q^{36} +2.65720e6i q^{37} +814755. i q^{38} -921849. i q^{39} +1.35868e6i q^{40} +3.07208e6 q^{41} -1.35835e6i q^{42} +5.98941e6i q^{43} +5.77766e6i q^{44} +1.00286e6 q^{45} -6.13215e6 q^{46} -8.25605e6i q^{47} +999980. q^{48} -4.42573e6 q^{49} +4.52697e6i q^{50} +9732.67 q^{51} -7.37321e6i q^{52} +7.09542e6 q^{53} +2.56719e6i q^{54} -7.08307e6i q^{55} -3.42868e6i q^{56} +1.51798e6 q^{57} +2.30964e7i q^{58} +(1.16388e7 + 3.37193e6i) q^{59} +8.02115e6 q^{60} +7.65261e6i q^{61} +1.82096e7 q^{62} -2.53076e6 q^{63} +2.70374e7 q^{64} +9.03913e6i q^{65} +1.81318e7 q^{66} +1.47421e7i q^{67} +77844.7 q^{68} +1.14249e7i q^{69} +1.33192e7i q^{70} +3.87749e6 q^{71} +6.47998e6i q^{72} -1.72671e7i q^{73} +6.66974e7 q^{74} +8.43426e6 q^{75} +1.21412e7 q^{76} +1.78744e7i q^{77} -2.31390e7 q^{78} -6.10409e7 q^{79} -9.80523e6 q^{80} +4.78297e6 q^{81} -7.71112e7i q^{82} +7.83794e6i q^{83} -2.02417e7 q^{84} -95433.0 q^{85} +1.50338e8 q^{86} +4.30313e7 q^{87} +4.57673e7 q^{88} -8.95553e7i q^{89} -2.51724e7i q^{90} -2.28106e7i q^{91} +9.13796e7i q^{92} -3.39265e7i q^{93} -2.07232e8 q^{94} -1.48845e7 q^{95} -6.05724e7i q^{96} +1.08068e8i q^{97} +1.11089e8i q^{98} -3.37815e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	25.1007i		−	1.56879i	−0.620261	−	0.784395i	$-0.712973\pi$
$2$		−	25.1007i		−	1.56879i	0.620261	−	0.784395i	$-0.287027\pi$
$3$	−46.7654			−0.577350
$3$	−46.7654			−0.577350
$4$	−374.043			−1.46110
$5$	458.555			0.733687			0.366844	−	0.930283i	$-0.380438\pi$
$5$	458.555			0.733687			0.366844	+	0.930283i	$0.380438\pi$
$6$			1173.84i			0.905742i
$7$	−1157.18			−0.481959			−0.240979	−	0.970530i	$-0.577469\pi$
$7$	−1157.18			−0.481959			−0.240979	+	0.970530i	$0.577469\pi$
$8$			2962.95i			0.723377i
$9$	2187.00			0.333333
$10$		−	11510.0i		−	1.15100i
$11$		−	15446.5i		−	1.05502i	−0.849550	−	0.527509i	$-0.823126\pi$
$11$		−	15446.5i		−	1.05502i	0.849550	−	0.527509i	$-0.176874\pi$
$12$	17492.3			0.843569
$13$			19712.2i			0.690179i	0.938570	+	0.345090i	$0.112151\pi$
$13$			19712.2i			0.690179i	−0.938570	+	0.345090i	$0.887849\pi$
$14$			29046.0i			0.756092i
$15$	−21444.5			−0.423595
$16$	−21382.9			−0.326277
$17$	−208.117			−0.00249179			−0.00124590	−	0.999999i	$-0.500397\pi$
$17$	−208.117			−0.00249179			−0.00124590	+	0.999999i	$0.500397\pi$
$18$		−	54895.1i		−	0.522930i
$19$	−32459.5			−0.249074			−0.124537	−	0.992215i	$-0.539744\pi$
$19$	−32459.5			−0.249074			−0.124537	+	0.992215i	$0.539744\pi$
$20$	−171519.			−1.07199
$21$	54116.1			0.278259
$22$	−387717.			−1.65510
$23$		−	244303.i		−	0.873005i	−0.899703	−	0.436502i	$-0.856217\pi$
$23$		−	244303.i		−	0.873005i	0.899703	−	0.436502i	$-0.143783\pi$
$24$		−	138564.i		−	0.417642i
$25$	−180353.			−0.461703
$26$	494789.			1.08275
$27$	−102276.			−0.192450
$28$	432836.			0.704192
$29$	−920152.			−1.30097			−0.650485	−	0.759519i	$-0.725435\pi$
$29$	−920152.			−1.30097			−0.650485	+	0.759519i	$0.725435\pi$
$30$			538270.i			0.664531i
$31$			725461.i			0.785538i	0.919637	+	0.392769i	$0.128483\pi$
$31$			725461.i			0.785538i	−0.919637	+	0.392769i	$0.871517\pi$
$32$			1.29524e6i			1.23524i
$33$			722362.i			0.609115i
$34$			5223.87i			0.00390910i
$35$	−530631.			−0.353607
$36$	−818032.			−0.487035
$37$			2.65720e6i			1.41781i	0.705306	+	0.708903i	$0.250810\pi$
$37$			2.65720e6i			1.41781i	−0.705306	+	0.708903i	$0.749190\pi$
$38$			814755.i			0.390744i
$39$		−	921849.i		−	0.398475i
$40$			1.35868e6i			0.530733i
$41$	3.07208e6			1.08717			0.543584	−	0.839355i	$-0.317067\pi$
$41$	3.07208e6			1.08717			0.543584	+	0.839355i	$0.317067\pi$
$42$		−	1.35835e6i		−	0.436530i
$43$			5.98941e6i			1.75190i	0.482400	+	0.875951i	$0.339765\pi$
$43$			5.98941e6i			1.75190i	−0.482400	+	0.875951i	$0.660235\pi$
$44$			5.77766e6i			1.54149i
$45$	1.00286e6			0.244562
$46$	−6.13215e6			−1.36956
$47$		−	8.25605e6i		−	1.69192i	−0.533243	−	0.845962i	$-0.679027\pi$
$47$		−	8.25605e6i		−	1.69192i	0.533243	−	0.845962i	$-0.320973\pi$
$48$	999980.			0.188376
$49$	−4.42573e6			−0.767716
$50$			4.52697e6i			0.724315i
$51$	9732.67			0.00143864
$52$		−	7.37321e6i		−	1.00842i
$53$	7.09542e6			0.899238			0.449619	−	0.893220i	$-0.351560\pi$
$53$	7.09542e6			0.899238			0.449619	+	0.893220i	$0.351560\pi$
$54$			2.56719e6i			0.301914i
$55$		−	7.08307e6i		−	0.774053i
$56$		−	3.42868e6i		−	0.348638i
$57$	1.51798e6			0.143803
$58$			2.30964e7i			2.04095i
$59$	1.16388e7	+	3.37193e6i	0.960502	+	0.278273i
$59$	1.16388e7	+	3.37193e6i	0.960502	+	0.278273i
$60$	8.02115e6			0.618916
$61$			7.65261e6i			0.552701i	0.961057	+	0.276351i	$0.0891251\pi$
$61$			7.65261e6i			0.552701i	−0.961057	+	0.276351i	$0.910875\pi$
$62$	1.82096e7			1.23235
$63$	−2.53076e6			−0.160653
$64$	2.70374e7			1.61155
$65$			9.03913e6i			0.506376i
$66$	1.81318e7			0.955573
$67$			1.47421e7i			0.731579i	0.930698	+	0.365790i	$0.119201\pi$
$67$			1.47421e7i			0.731579i	−0.930698	+	0.365790i	$0.880799\pi$
$68$	77844.7			0.00364077
$69$			1.14249e7i			0.504030i
$70$			1.33192e7i			0.554735i
$71$	3.87749e6			0.152587			0.0762935	−	0.997085i	$-0.475691\pi$
$71$	3.87749e6			0.152587			0.0762935	+	0.997085i	$0.475691\pi$
$72$			6.47998e6i			0.241126i
$73$		−	1.72671e7i		−	0.608035i	−0.952666	−	0.304018i	$-0.901672\pi$
$73$		−	1.72671e7i		−	0.608035i	0.952666	−	0.304018i	$-0.0983282\pi$
$74$	6.66974e7			2.22424
$75$	8.43426e6			0.266564
$76$	1.21412e7			0.363923
$77$			1.78744e7i			0.508475i
$78$	−2.31390e7			−0.625124
$79$	−6.10409e7			−1.56716			−0.783579	−	0.621293i	$-0.786608\pi$
$79$	−6.10409e7			−1.56716			−0.783579	+	0.621293i	$0.786608\pi$
$80$	−9.80523e6			−0.239386
$81$	4.78297e6			0.111111
$82$		−	7.71112e7i		−	1.70554i
$83$			7.83794e6i			0.165154i	0.996585	+	0.0825771i	$0.0263151\pi$
$83$			7.83794e6i			0.165154i	−0.996585	+	0.0825771i	$0.973685\pi$
$84$	−2.02417e7			−0.406565
$85$	−95433.0			−0.00182820
$86$	1.50338e8			2.74837
$87$	4.30313e7			0.751116
$88$	4.57673e7			0.763176
$89$		−	8.95553e7i		−	1.42735i	−0.700476	−	0.713676i	$-0.747029\pi$
$89$		−	8.95553e7i		−	1.42735i	0.700476	−	0.713676i	$-0.252971\pi$
$90$		−	2.51724e7i		−	0.383667i
$91$		−	2.28106e7i		−	0.332638i
$92$			9.13796e7i			1.27555i
$93$		−	3.39265e7i		−	0.453531i
$94$	−2.07232e8			−2.65428
$95$	−1.48845e7			−0.182742
$96$		−	6.05724e7i		−	0.713165i
$97$			1.08068e8i			1.22070i	0.792131	+	0.610351i	$0.208971\pi$
$97$			1.08068e8i			1.22070i	−0.792131	+	0.610351i	$0.791029\pi$
$98$			1.11089e8i			1.20439i
$99$		−	3.37815e7i		−	0.351672i
$100$	6.74596e7			0.674596
$101$			4.86906e7i			0.467907i	0.972248	+	0.233954i	$0.0751664\pi$
$101$			4.86906e7i			0.467907i	−0.972248	+	0.233954i	$0.924834\pi$
$102$		−	244296.i		−	0.00225692i
$103$		−	1.01995e8i		−	0.906213i	−0.891456	−	0.453106i	$-0.850316\pi$
$103$		−	1.01995e8i		−	0.906213i	0.891456	−	0.453106i	$-0.149684\pi$
$104$	−5.84064e7			−0.499260
$105$	2.48152e7			0.204155
$106$		−	1.78100e8i		−	1.41072i
$107$	−6.84585e6			−0.0522267			−0.0261133	−	0.999659i	$-0.508313\pi$
$107$	−6.84585e6			−0.0522267			−0.0261133	+	0.999659i	$0.508313\pi$
$108$	3.82556e7			0.281190
$109$			2.20679e8i			1.56334i	0.623690	+	0.781672i	$0.285633\pi$
$109$			2.20679e8i			1.56334i	−0.623690	+	0.781672i	$0.714367\pi$
$110$	−1.77790e8			−1.21433
$111$		−	1.24265e8i		−	0.818571i
$112$	2.47439e7			0.157252
$113$		−	1.71332e8i		−	1.05081i	−0.850851	−	0.525406i	$-0.823913\pi$
$113$		−	1.71332e8i		−	1.05081i	0.850851	−	0.525406i	$-0.176087\pi$
$114$		−	3.81023e7i		−	0.225596i
$115$		−	1.12026e8i		−	0.640513i
$116$	3.44176e8			1.90086
$117$			4.31106e7i			0.230060i
$118$	8.46377e7	−	2.92140e8i	0.436552	−	1.50683i
$119$	240829.			0.00120094
$120$		−	6.35390e7i		−	0.306419i
$121$	−2.42358e7			−0.113062
$122$	1.92086e8			0.867073
$123$	−1.43667e8			−0.627677
$124$		−	2.71354e8i		−	1.14775i
$125$	−2.61824e8			−1.07243
$126$			6.35237e7i			0.252031i
$127$	4.65901e8			1.79093			0.895464	−	0.445133i	$-0.146844\pi$
$127$	4.65901e8			1.79093			0.895464	+	0.445133i	$0.146844\pi$
$128$		−	3.47074e8i		−	1.29295i
$129$		−	2.80097e8i		−	1.01146i
$130$	2.26888e8			0.794398
$131$		−	1.18641e6i		−	0.00402857i	−0.999998	−	0.00201429i	$-0.999359\pi$
$131$		−	1.18641e6i		−	0.00402857i	0.999998	−	0.00201429i	$-0.000641168\pi$
$132$		−	2.70194e8i		−	0.889980i
$133$	3.75616e7			0.120043
$134$	3.70037e8			1.14769
$135$	−4.68991e7			−0.141198
$136$		−	616641.i		−	0.00180251i
$137$	2.77000e8			0.786317			0.393159	−	0.919471i	$-0.371382\pi$
$137$	2.77000e8			0.786317			0.393159	+	0.919471i	$0.371382\pi$
$138$	2.86772e8			0.790717
$139$	1.50839e8			0.404067			0.202034	−	0.979379i	$-0.435245\pi$
$139$	1.50839e8			0.404067			0.202034	+	0.979379i	$0.435245\pi$
$140$	1.98479e8			0.516657
$141$			3.86097e8i			0.976833i
$142$		−	9.73276e7i		−	0.239377i
$143$	3.04485e8			0.728151
$144$	−4.67644e7			−0.108759
$145$	−4.21940e8			−0.954506
$146$	−4.33416e8			−0.953881
$147$	2.06971e8			0.443241
$148$		−	9.93906e8i		−	2.07156i
$149$			4.59699e8i			0.932670i	0.884608	+	0.466335i	$0.154426\pi$
$149$			4.59699e8i			0.932670i	−0.884608	+	0.466335i	$0.845574\pi$
$150$		−	2.11705e8i		−	0.418184i
$151$		−	3.23489e7i		−	0.0622231i	−0.999516	−	0.0311116i	$-0.990095\pi$
$151$		−	3.23489e7i		−	0.0622231i	0.999516	−	0.0311116i	$-0.00990472\pi$
$152$		−	9.61760e7i		−	0.180174i
$153$	−455152.			−0.000830597
$154$	4.48660e8			0.797690
$155$			3.32664e8i			0.576340i
$156$			3.44811e8i			0.582214i
$157$			6.79121e8i			1.11776i	0.829249	+	0.558880i	$0.188769\pi$
$157$			6.79121e8i			1.11776i	−0.829249	+	0.558880i	$0.811231\pi$
$158$			1.53217e9i			2.45854i
$159$	−3.31820e8			−0.519175
$160$			5.93939e8i			0.906279i
$161$			2.82703e8i			0.420752i
$162$		−	1.20056e8i		−	0.174310i
$163$	8.90583e8			1.26161			0.630803	−	0.775943i	$-0.282726\pi$
$163$	8.90583e8			1.26161			0.630803	+	0.775943i	$0.282726\pi$
$164$	−1.14909e9			−1.58847
$165$			3.31242e8i			0.446900i
$166$	1.96737e8			0.259092
$167$	−1.07002e8			−0.137571			−0.0687853	−	0.997631i	$-0.521912\pi$
$167$	−1.07002e8			−0.137571			−0.0687853	+	0.997631i	$0.521912\pi$
$168$			1.60343e8i			0.201286i
$169$	4.27159e8			0.523652
$170$			2.39543e6i			0.00286806i
$171$	−7.09889e7			−0.0830245
$172$		−	2.24029e9i		−	2.55971i
$173$			1.00080e9i			1.11728i	0.829410	+	0.558640i	$0.188677\pi$
$173$			1.00080e9i			1.11728i	−0.829410	+	0.558640i	$0.811323\pi$
$174$		−	1.08011e9i		−	1.17834i
$175$	2.08701e8			0.222522
$176$			3.30291e8i			0.344228i
$177$	−5.44290e8	−	1.57690e8i	−0.554546	−	0.160661i
$178$	−2.24790e9			−2.23922
$179$			5.84466e8i			0.569308i	0.958630	+	0.284654i	$0.0918787\pi$
$179$			5.84466e8i			0.569308i	−0.958630	+	0.284654i	$0.908121\pi$
$180$	−3.75112e8			−0.357331
$181$	2.10696e9			1.96310			0.981548	−	0.191215i	$-0.0612428\pi$
$181$	2.10696e9			1.96310			0.981548	+	0.191215i	$0.0612428\pi$
$182$	−5.72562e8			−0.521839
$183$		−	3.57877e8i		−	0.319102i
$184$	7.23857e8			0.631512
$185$			1.21847e9i			1.04023i
$186$	−8.51576e8			−0.711495
$187$			3.21468e6i			0.00262888i
$188$			3.08812e9i			2.47208i
$189$	1.18352e8			0.0927530
$190$			3.73610e8i			0.286684i
$191$			4.70580e8i			0.353590i	0.984248	+	0.176795i	$0.0565730\pi$
$191$			4.70580e8i			0.353590i	−0.984248	+	0.176795i	$0.943427\pi$
$192$	−1.26441e9			−0.930431
$193$	−1.59781e9			−1.15158			−0.575791	−	0.817597i	$-0.695306\pi$
$193$	−1.59781e9			−1.15158			−0.575791	+	0.817597i	$0.695306\pi$
$194$	2.71257e9			1.91502
$195$		−	4.22718e8i		−	0.292356i
$196$	1.65541e9			1.12171
$197$	−3.39150e8			−0.225178			−0.112589	−	0.993642i	$-0.535914\pi$
$197$	−3.39150e8			−0.225178			−0.112589	+	0.993642i	$0.535914\pi$
$198$	−8.47938e8			−0.551701
$199$	−2.20463e9			−1.40580			−0.702898	−	0.711290i	$-0.748111\pi$
$199$	−2.20463e9			−1.40580			−0.702898	+	0.711290i	$0.748111\pi$
$200$		−	5.34376e8i		−	0.333985i
$201$		−	6.89422e8i		−	0.422377i
$202$	1.22217e9			0.734048
$203$	1.06478e9			0.627014
$204$	−3.64043e6			−0.00210200
$205$	1.40872e9			0.797642
$206$	−2.56014e9			−1.42166
$207$		−	5.34290e8i		−	0.291002i
$208$		−	4.21504e8i		−	0.225190i
$209$			5.01386e8i			0.262777i
$210$		−	6.22877e8i		−	0.320277i
$211$			1.23787e9i			0.624517i	0.949997	+	0.312259i	$0.101086\pi$
$211$			1.23787e9i			0.624517i	−0.949997	+	0.312259i	$0.898914\pi$
$212$	−2.65399e9			−1.31388
$213$	−1.81332e8			−0.0880961
$214$			1.71835e8i			0.0819327i
$215$			2.74647e9i			1.28535i
$216$		−	3.03039e8i		−	0.139214i
$217$		−	8.39491e8i		−	0.378597i
$218$	5.53918e9			2.45256
$219$			8.07504e8i			0.351049i
$220$			2.64937e9i			1.13097i
$221$		−	4.10245e6i		−	0.00171978i
$222$	−3.11913e9			−1.28417
$223$	3.28091e9			1.32671			0.663353	−	0.748307i	$-0.269133\pi$
$223$	3.28091e9			1.32671			0.663353	+	0.748307i	$0.269133\pi$
$224$		−	1.49883e9i		−	0.595334i
$225$	−3.94431e8			−0.153901
$226$	−4.30055e9			−1.64851
$227$			3.93777e8i			0.148302i	0.997247	+	0.0741509i	$0.0236247\pi$
$227$			3.93777e8i			0.148302i	−0.997247	+	0.0741509i	$0.976375\pi$
$228$	−5.67790e8			−0.210111
$229$			1.21372e9i			0.441344i	0.975348	+	0.220672i	$0.0708250\pi$
$229$			1.21372e9i			0.441344i	−0.975348	+	0.220672i	$0.929175\pi$
$230$	−2.81193e9			−1.00483
$231$		−	8.35904e8i		−	0.293568i
$232$		−	2.72637e9i		−	0.941093i
$233$			3.02094e9i			1.02499i	0.858691	+	0.512493i	$0.171278\pi$
$233$			3.02094e9i			1.02499i	−0.858691	+	0.512493i	$0.828722\pi$
$234$	1.08210e9			0.360916
$235$		−	3.78585e9i		−	1.24134i
$236$	−4.35339e9	−	1.26125e9i	−1.40339	−	0.406586i
$237$	2.85460e9			0.904799
$238$		−	6.04497e6i		−	0.00188402i
$239$	1.27278e8			0.0390087			0.0195043	−	0.999810i	$-0.493791\pi$
$239$	1.27278e8			0.0390087			0.0195043	+	0.999810i	$0.493791\pi$
$240$	4.58545e8			0.138209
$241$	2.47599e9			0.733974			0.366987	−	0.930226i	$-0.380389\pi$
$241$	2.47599e9			0.733974			0.366987	+	0.930226i	$0.380389\pi$
$242$			6.08333e8i			0.177370i
$243$	−2.23677e8			−0.0641500
$244$		−	2.86241e9i		−	0.807555i
$245$	−2.02944e9			−0.563264
$246$			3.60613e9i			0.984694i
$247$		−	6.39849e8i		−	0.171905i
$248$	−2.14951e9			−0.568241
$249$		−	3.66544e8i		−	0.0953518i
$250$			6.57197e9i			1.68242i
$251$	2.75745e9			0.694725			0.347363	−	0.937731i	$-0.387077\pi$
$251$	2.75745e9			0.694725			0.347363	+	0.937731i	$0.387077\pi$
$252$	9.46612e8			0.234731
$253$	−3.77362e9			−0.921035
$254$		−	1.16944e10i		−	2.80959i
$255$	4.46296e6			0.00105551
$256$	−1.79022e9			−0.416818
$257$	−7.67874e9			−1.76018			−0.880091	−	0.474805i	$-0.842518\pi$
$257$	−7.67874e9			−1.76018			−0.880091	+	0.474805i	$0.842518\pi$
$258$	−7.03061e9			−1.58677
$259$		−	3.07486e9i		−	0.683324i
$260$		−	3.38102e9i		−	0.739868i
$261$	−2.01237e9			−0.433657
$262$	−2.97798e7			−0.00631999
$263$	1.52171e9			0.318060			0.159030	−	0.987274i	$-0.449163\pi$
$263$	1.52171e9			0.318060			0.159030	+	0.987274i	$0.449163\pi$
$264$	−2.14032e9			−0.440620
$265$	3.25364e9			0.659760
$266$		−	9.42820e8i		−	0.188323i
$267$			4.18809e9i			0.824083i
$268$		−	5.51419e9i		−	1.06891i
$269$		−	1.90338e9i		−	0.363509i	−0.983344	−	0.181755i	$-0.941822\pi$
$269$		−	1.90338e9i		−	0.363509i	0.983344	−	0.181755i	$-0.0581777\pi$
$270$			1.17720e9i			0.221510i
$271$	−9.32770e9			−1.72941			−0.864704	−	0.502281i	$-0.832494\pi$
$271$	−9.32770e9			−1.72941			−0.864704	+	0.502281i	$0.832494\pi$
$272$	4.45015e6			0.000813015
$273$			1.06675e9i			0.192049i
$274$		−	6.95289e9i		−	1.23357i
$275$			2.78582e9i			0.487104i
$276$		−	4.27340e9i		−	0.736440i
$277$	−1.02882e10			−1.74752			−0.873760	−	0.486357i	$-0.838325\pi$
$277$	−1.02882e10			−1.74752			−0.873760	+	0.486357i	$0.838325\pi$
$278$		−	3.78615e9i		−	0.633897i
$279$			1.58658e9i			0.261846i
$280$		−	1.57224e9i		−	0.255791i
$281$	−3.50624e9			−0.562362			−0.281181	−	0.959655i	$-0.590726\pi$
$281$	−3.50624e9			−0.562362			−0.281181	+	0.959655i	$0.590726\pi$
$282$	9.69130e9			1.53245
$283$			6.78254e9i			1.05742i	0.848803	+	0.528709i	$0.177324\pi$
$283$			6.78254e9i			1.05742i	−0.848803	+	0.528709i	$0.822676\pi$
$284$	−1.45035e9			−0.222946
$285$	6.96077e8			0.105506
$286$		−	7.64277e9i		−	1.14232i
$287$	−3.55495e9			−0.523970
$288$			2.83269e9i			0.411746i
$289$	−6.97571e9			−0.999994
$290$			1.05910e10i			1.49742i
$291$		−	5.05383e9i		−	0.704772i
$292$			6.45865e9i			0.888404i
$293$	−1.05002e9			−0.142471			−0.0712353	−	0.997460i	$-0.522694\pi$
$293$	−1.05002e9			−0.142471			−0.0712353	+	0.997460i	$0.522694\pi$
$294$		−	5.19511e9i		−	0.695353i
$295$	5.33700e9	+	1.54622e9i	0.704708	+	0.204165i
$296$	−7.87315e9			−1.02561
$297$			1.57981e9i			0.203038i
$298$	1.15387e10			1.46316
$299$	4.81574e9			0.602530
$300$	−3.15477e9			−0.389478
$301$		−	6.93084e9i		−	0.844344i
$302$	−8.11979e8			−0.0976151
$303$		−	2.27703e9i		−	0.270146i
$304$	6.94079e8			0.0812670
$305$			3.50914e9i			0.405510i
$306$			1.14246e7i			0.00130303i
$307$	−1.07373e9			−0.120876			−0.0604380	−	0.998172i	$-0.519250\pi$
$307$	−1.07373e9			−0.120876			−0.0604380	+	0.998172i	$0.519250\pi$
$308$		−	6.68580e9i		−	0.742935i
$309$			4.76984e9i			0.523202i
$310$	8.35007e9			0.904156
$311$	1.79518e9			0.191896			0.0959480	−	0.995386i	$-0.469412\pi$
$311$	1.79518e9			0.191896			0.0959480	+	0.995386i	$0.469412\pi$
$312$	2.73140e9			0.288248
$313$			1.29198e10i			1.34611i	0.739594	+	0.673053i	$0.235017\pi$
$313$			1.29198e10i			1.34611i	−0.739594	+	0.673053i	$0.764983\pi$
$314$	1.70464e10			1.75353
$315$	−1.16049e9			−0.117869
$316$	2.28319e10			2.28978
$317$	−1.77264e10			−1.75543			−0.877715	−	0.479183i	$-0.840933\pi$
$317$	−1.77264e10			−1.75543			−0.877715	+	0.479183i	$0.840933\pi$
$318$			8.32890e9i			0.814478i
$319$			1.42131e10i			1.37255i
$320$	1.23981e10			1.18238
$321$	3.20149e8			0.0301531
$322$	7.09602e9			0.660072
$323$	6.75537e6			0.000620639
$324$	−1.78904e9			−0.162345
$325$		−	3.55515e9i		−	0.318658i
$326$		−	2.23542e10i		−	1.97920i
$327$		−	1.03201e10i		−	0.902597i
$328$			9.10242e9i			0.786433i
$329$			9.55376e9i			0.815438i
$330$	8.31440e9			0.701092
$331$	1.70745e10			1.42245			0.711223	−	0.702967i	$-0.248142\pi$
$331$	1.70745e10			1.42245			0.711223	+	0.702967i	$0.248142\pi$
$332$		−	2.93173e9i		−	0.241308i
$333$			5.81129e9i			0.472602i
$334$			2.68582e9i			0.215820i
$335$			6.76008e9i			0.536750i
$336$	−1.15716e9			−0.0907895
$337$			5.55679e9i			0.430828i	0.976523	+	0.215414i	$0.0691102\pi$
$337$			5.55679e9i			0.430828i	−0.976523	+	0.215414i	$0.930890\pi$
$338$		−	1.07220e10i		−	0.821501i
$339$			8.01242e9i			0.606687i
$340$	3.56960e7			0.00267119
$341$	1.12058e10			0.828757
$342$			1.78187e9i			0.130248i
$343$	1.17923e10			0.851966
$344$	−1.77463e10			−1.26729
$345$			5.23894e9i			0.369800i
$346$	2.51207e10			1.75278
$347$			6.91524e9i			0.476968i	0.971146	+	0.238484i	$0.0766505\pi$
$347$			6.91524e9i			0.476968i	−0.971146	+	0.238484i	$0.923349\pi$
$348$	−1.60955e10			−1.09746
$349$		−	5.11294e9i		−	0.344643i	−0.985041	−	0.172321i	$-0.944873\pi$
$349$		−	5.11294e9i		−	0.344643i	0.985041	−	0.172321i	$-0.0551268\pi$
$350$		−	5.23853e9i		−	0.349090i
$351$		−	2.01608e9i		−	0.132825i
$352$	2.00070e10			1.30320
$353$		−	6.28088e9i		−	0.404503i	−0.979334	−	0.202251i	$-0.935174\pi$
$353$		−	6.28088e9i		−	0.404503i	0.979334	−	0.202251i	$-0.0648258\pi$
$354$	−3.95811e9	+	1.36620e10i	−0.252043	+	0.869967i
$355$	1.77804e9			0.111951
$356$			3.34975e10i			2.08551i
$357$	−1.12625e7			−0.000693363
$358$	1.46705e10			0.893125
$359$	−4.64931e9			−0.279905			−0.139952	−	0.990158i	$-0.544695\pi$
$359$	−4.64931e9			−0.279905			−0.139952	+	0.990158i	$0.544695\pi$
$360$			2.97142e9i			0.176911i
$361$	−1.59299e10			−0.937962
$362$		−	5.28860e10i		−	3.07969i
$363$	1.13339e9			0.0652761
$364$			8.53215e9i			0.486019i
$365$		−	7.91793e9i		−	0.446108i
$366$	−8.98296e9			−0.500605
$367$			4.38244e9i			0.241575i	0.992678	+	0.120787i	$0.0385419\pi$
$367$			4.38244e9i			0.241575i	−0.992678	+	0.120787i	$0.961458\pi$
$368$			5.22390e9i			0.284842i
$369$	6.71863e9			0.362389
$370$	3.05844e10			1.63190
$371$	−8.21070e9			−0.433395
$372$			1.26900e10i			0.662656i
$373$	6.57158e9			0.339496			0.169748	−	0.985488i	$-0.445705\pi$
$373$	6.57158e9			0.339496			0.169748	+	0.985488i	$0.445705\pi$
$374$	8.06906e7			0.00412417
$375$	1.22443e10			0.619169
$376$	2.44623e10			1.22390
$377$		−	1.81382e10i		−	0.897903i
$378$		−	2.97071e9i		−	0.145510i
$379$	−8.80802e8			−0.0426895			−0.0213448	−	0.999772i	$-0.506795\pi$
$379$	−8.80802e8			−0.0426895			−0.0213448	+	0.999772i	$0.506795\pi$
$380$	5.56743e9			0.267005
$381$	−2.17880e10			−1.03399
$382$	1.18119e10			0.554709
$383$	−2.17036e10			−1.00864			−0.504320	−	0.863517i	$-0.668257\pi$
$383$	−2.17036e10			−1.00864			−0.504320	+	0.863517i	$0.668257\pi$
$384$			1.62310e10i			0.746486i
$385$			8.19640e9i			0.373061i
$386$			4.01060e10i			1.80659i
$387$			1.30988e10i			0.583968i
$388$		−	4.04220e10i		−	1.78357i
$389$	−1.91308e10			−0.835478			−0.417739	−	0.908567i	$-0.637177\pi$
$389$	−1.91308e10			−0.835478			−0.417739	+	0.908567i	$0.637177\pi$
$390$	−1.06105e10			−0.458646
$391$			5.08435e7i			0.00217535i
$392$		−	1.31132e10i		−	0.555348i
$393$			5.54831e7i			0.00232590i
$394$			8.51288e9i			0.353258i
$395$	−2.79906e10			−1.14980
$396$			1.26357e10i			0.513830i
$397$		−	2.75869e10i		−	1.11056i	−0.831664	−	0.555279i	$-0.812611\pi$
$397$		−	2.75869e10i		−	1.11056i	0.831664	−	0.555279i	$-0.187389\pi$
$398$			5.53375e10i			2.20540i
$399$	−1.75658e9			−0.0693069
$400$	3.85646e9			0.150643
$401$		−	4.26888e10i		−	1.65096i	−0.564433	−	0.825479i	$-0.690905\pi$
$401$		−	4.26888e10i		−	1.65096i	0.564433	−	0.825479i	$-0.309095\pi$
$402$	−1.73049e10			−0.662622
$403$	−1.43004e10			−0.542162
$404$		−	1.82124e10i		−	0.683661i
$405$	2.19325e9			0.0815208
$406$		−	2.67268e10i		−	0.983654i
$407$	4.10444e10			1.49581
$408$			2.88374e7i			0.00104068i
$409$		−	1.58000e9i		−	0.0564631i	−0.999601	−	0.0282315i	$-0.991012\pi$
$409$		−	1.58000e9i		−	0.0564631i	0.999601	−	0.0282315i	$-0.00898757\pi$
$410$		−	3.53597e10i		−	1.25133i
$411$	−1.29540e10			−0.453980
$412$			3.81505e10i			1.32407i
$413$	−1.34682e10	−	3.90194e9i	−0.462922	−	0.134116i
$414$	−1.34110e10			−0.456521
$415$			3.59412e9i			0.121172i
$416$	−2.55321e10			−0.852536
$417$	−7.05403e9			−0.233288
$418$	1.25851e10			0.412242
$419$			1.53991e10i			0.499620i	0.968295	+	0.249810i	$0.0803682\pi$
$419$			1.53991e10i			0.499620i	−0.968295	+	0.249810i	$0.919632\pi$
$420$	−9.28194e9			−0.298292
$421$			4.73339e10i			1.50676i	0.657586	+	0.753380i	$0.271578\pi$
$421$			4.73339e10i			1.50676i	−0.657586	+	0.753380i	$0.728422\pi$
$422$	3.10713e10			0.979737
$423$		−	1.80560e10i		−	0.563975i
$424$			2.10234e10i			0.650488i
$425$	3.75344e7			0.00115047
$426$			4.55156e9i			0.138204i
$427$		−	8.85547e9i		−	0.266379i
$428$	2.56064e9			0.0763087
$429$	−1.42393e10			−0.420398
$430$	6.89382e10			2.01644
$431$			1.05757e10i			0.306478i	0.988189	+	0.153239i	$0.0489704\pi$
$431$			1.05757e10i			0.306478i	−0.988189	+	0.153239i	$0.951030\pi$
$432$	2.18696e9			0.0627921
$433$	1.93675e9			0.0550964			0.0275482	−	0.999620i	$-0.491230\pi$
$433$	1.93675e9			0.0550964			0.0275482	+	0.999620i	$0.491230\pi$
$434$	−2.10718e10			−0.593939
$435$	1.97322e10			0.551084
$436$		−	8.25433e10i		−	2.28421i
$437$			7.92994e9i			0.217442i
$438$	2.02689e10			0.550723
$439$	1.52316e9			0.0410097			0.0205048	−	0.999790i	$-0.493473\pi$
$439$	1.52316e9			0.0410097			0.0205048	+	0.999790i	$0.493473\pi$
$440$	2.09868e10			0.559932
$441$	−9.67907e9			−0.255905
$442$	−1.02974e8			−0.00269798
$443$		−	5.44692e10i		−	1.41428i	−0.707072	−	0.707142i	$-0.749984\pi$
$443$		−	5.44692e10i		−	1.41428i	0.707072	−	0.707142i	$-0.250016\pi$
$444$			4.64804e10i			1.19602i
$445$		−	4.10660e10i		−	1.04723i
$446$		−	8.23530e10i		−	2.08132i
$447$		−	2.14980e10i		−	0.538478i
$448$	−3.12872e10			−0.776702
$449$	−3.04171e10			−0.748398			−0.374199	−	0.927348i	$-0.622082\pi$
$449$	−3.04171e10			−0.748398			−0.374199	+	0.927348i	$0.622082\pi$
$450$			9.90048e9i			0.241438i
$451$		−	4.74529e10i		−	1.14698i
$452$			6.40856e10i			1.53535i
$453$			1.51281e9i			0.0359245i
$454$	9.88405e9			0.232655
$455$		−	1.04599e10i		−	0.244052i
$456$			4.49771e9i			0.104024i
$457$			6.39260e8i			0.0146559i	0.999973	+	0.00732795i	$0.00233258\pi$
$457$			6.39260e8i			0.0146559i	−0.999973	+	0.00732795i	$0.997667\pi$
$458$	3.04652e10			0.692376
$459$	2.12853e7			0.000479546
$460$			4.19026e10i			0.935856i
$461$	−1.63617e10			−0.362263			−0.181132	−	0.983459i	$-0.557976\pi$
$461$	−1.63617e10			−0.362263			−0.181132	+	0.983459i	$0.557976\pi$
$462$	−2.09817e10			−0.460547
$463$		−	8.06345e10i		−	1.75468i	−0.479874	−	0.877338i	$-0.659317\pi$
$463$		−	8.06345e10i		−	1.75468i	0.479874	−	0.877338i	$-0.340683\pi$
$464$	1.96755e10			0.424477
$465$		−	1.55571e10i		−	0.332750i
$466$	7.58275e10			1.60799
$467$			1.34566e10i			0.282924i	0.989944	+	0.141462i	$0.0451803\pi$
$467$			1.34566e10i			0.282924i	−0.989944	+	0.141462i	$0.954820\pi$
$468$		−	1.61252e10i		−	0.336141i
$469$		−	1.70593e10i		−	0.352591i
$470$	−9.50274e10			−1.94741
$471$		−	3.17593e10i		−	0.645339i
$472$	−9.99088e9	+	3.44851e10i	−0.201296	+	0.694805i
$473$	9.25154e10			1.84829
$474$		−	7.16524e10i		−	1.41944i
$475$	5.85416e9			0.114998
$476$	−9.00805e7			−0.00175470
$477$	1.55177e10			0.299746
$478$		−	3.19476e9i		−	0.0611964i
$479$	1.40052e10			0.266041			0.133021	−	0.991113i	$-0.457532\pi$
$479$	1.40052e10			0.266041			0.133021	+	0.991113i	$0.457532\pi$
$480$		−	2.77758e10i		−	0.523240i
$481$	−5.23793e10			−0.978541
$482$		−	6.21490e10i		−	1.15145i
$483$		−	1.32207e10i		−	0.242921i
$484$	9.06521e9			0.165195
$485$			4.95550e10i			0.895613i
$486$			5.61445e9i			0.100638i
$487$	1.25982e10			0.223972			0.111986	−	0.993710i	$-0.464279\pi$
$487$	1.25982e10			0.223972			0.111986	+	0.993710i	$0.464279\pi$
$488$	−2.26743e10			−0.399812
$489$	−4.16484e10			−0.728389
$490$			5.09402e10i			0.883643i
$491$	−6.06960e9			−0.104432			−0.0522161	−	0.998636i	$-0.516628\pi$
$491$	−6.06960e9			−0.104432			−0.0522161	+	0.998636i	$0.516628\pi$
$492$	5.37376e10			0.917102
$493$	1.91499e8			0.00324175
$494$	−1.60606e10			−0.269684
$495$		−	1.54907e10i		−	0.258018i
$496$		−	1.55125e10i		−	0.256303i
$497$	−4.48697e9			−0.0735406
$498$	−9.20050e9			−0.149587
$499$	−9.92585e10			−1.60090			−0.800452	−	0.599397i	$-0.795407\pi$
$499$	−9.92585e10			−1.60090			−0.800452	+	0.599397i	$0.795407\pi$
$500$	9.79336e10			1.56694
$501$	5.00398e9			0.0794264
$502$		−	6.92139e10i		−	1.08988i
$503$			2.23408e10i			0.349001i	0.984657	+	0.174500i	$0.0558310\pi$
$503$			2.23408e10i			0.349001i	−0.984657	+	0.174500i	$0.944169\pi$
$504$		−	7.49852e9i		−	0.116213i
$505$			2.23273e10i			0.343298i
$506$			9.47204e10i			1.44491i
$507$	−1.99763e10			−0.302331
$508$	−1.74267e11			−2.61674
$509$			1.18747e11i			1.76909i	0.466456	+	0.884545i	$0.345531\pi$
$509$			1.18747e11i			1.76909i	−0.466456	+	0.884545i	$0.654469\pi$
$510$		−	1.12023e8i		−	0.00165587i
$511$			1.99812e10i			0.293048i
$512$		−	4.39153e10i		−	0.639052i
$513$	3.31982e9			0.0479342
$514$			1.92741e11i			2.76136i
$515$		−	4.67703e10i		−	0.664877i
$516$			1.04768e11i			1.47785i
$517$	−1.27527e11			−1.78501
$518$	−7.71811e10			−1.07199
$519$		−	4.68027e10i		−	0.645062i
$520$	−2.67825e10			−0.366301
$521$	−5.20004e10			−0.705758			−0.352879	−	0.935669i	$-0.614797\pi$
$521$	−5.20004e10			−0.705758			−0.352879	+	0.935669i	$0.614797\pi$
$522$			5.05119e10i			0.680317i
$523$	−1.48520e10			−0.198509			−0.0992543	−	0.995062i	$-0.531646\pi$
$523$	−1.48520e10			−0.198509			−0.0992543	+	0.995062i	$0.531646\pi$
$524$			4.43770e8i			0.00588617i
$525$	−9.75998e9			−0.128473
$526$		−	3.81959e10i		−	0.498969i
$527$		−	1.50981e8i		−	0.00195740i
$528$		−	1.54462e10i		−	0.198740i
$529$	1.86272e10			0.237862
$530$		−	8.16685e10i		−	1.03503i
$531$	2.54539e10	+	7.37442e9i	0.320167	+	0.0927576i
$532$	−1.40496e10			−0.175396
$533$			6.05575e10i			0.750341i
$534$	1.05124e11			1.29281
$535$	−3.13920e9			−0.0383181
$536$	−4.36803e10			−0.529208
$537$		−	2.73328e10i		−	0.328690i
$538$	−4.77760e10			−0.570270
$539$			6.83621e10i			0.809954i
$540$	1.75423e10			0.206305
$541$		−	6.49767e10i		−	0.758522i	−0.925290	−	0.379261i	$-0.876178\pi$
$541$		−	6.49767e10i		−	0.758522i	0.925290	−	0.379261i	$-0.123822\pi$
$542$			2.34131e11i			2.71308i
$543$	−9.85327e10			−1.13339
$544$		−	2.69562e8i		−	0.00307796i
$545$			1.01193e11i			1.14701i
$546$	2.67761e10			0.301284
$547$	−4.70937e10			−0.526034			−0.263017	−	0.964791i	$-0.584717\pi$
$547$	−4.70937e10			−0.526034			−0.263017	+	0.964791i	$0.584717\pi$
$548$	−1.03610e11			−1.14889
$549$			1.67363e10i			0.184234i
$550$	6.99259e10			0.764165
$551$	2.98677e10			0.324037
$552$	−3.38514e10			−0.364604
$553$	7.06355e10			0.755305
$554$			2.58242e11i			2.74149i
$555$		−	5.69822e10i		−	0.600575i
$556$	−5.64201e10			−0.590385
$557$	−4.72589e10			−0.490979			−0.245490	−	0.969399i	$-0.578949\pi$
$557$	−4.72589e10			−0.490979			−0.245490	+	0.969399i	$0.578949\pi$
$558$	3.98243e10			0.410782
$559$	−1.18064e11			−1.20913
$560$	1.13464e10			0.115374
$561$		−	1.50336e8i		−	0.00151779i
$562$			8.80089e10i			0.882229i
$563$			4.73136e9i			0.0470925i	0.999723	+	0.0235463i	$0.00749570\pi$
$563$			4.73136e9i			0.0470925i	−0.999723	+	0.0235463i	$0.992504\pi$
$564$		−	1.44417e11i		−	1.42726i
$565$		−	7.85652e10i		−	0.770968i
$566$	1.70246e11			1.65887
$567$	−5.53477e9			−0.0535509
$568$			1.14888e10i			0.110378i
$569$		−	1.46316e9i		−	0.0139586i	−0.999976	−	0.00697932i	$-0.997778\pi$
$569$		−	1.46316e9i		−	0.0139586i	0.999976	−	0.00697932i	$-0.00222160\pi$
$570$		−	1.74720e10i		−	0.165517i
$571$			1.36252e11i			1.28173i	0.767652	+	0.640867i	$0.221425\pi$
$571$			1.36252e11i			1.28173i	−0.767652	+	0.640867i	$0.778575\pi$
$572$	−1.13890e11			−1.06391
$573$		−	2.20069e10i		−	0.204145i
$574$			8.92317e10i			0.821999i
$575$			4.40606e10i			0.403069i
$576$	5.91307e10			0.537184
$577$	−2.11877e11			−1.91152			−0.955762	−	0.294142i	$-0.904966\pi$
$577$	−2.11877e11			−1.91152			−0.955762	+	0.294142i	$0.904966\pi$
$578$			1.75095e11i			1.56878i
$579$	7.47220e10			0.664866
$580$	1.57824e11			1.39463
$581$		−	9.06993e9i		−	0.0795975i
$582$	−1.26854e11			−1.10564
$583$		−	1.09599e11i		−	0.948712i
$584$	5.11617e10			0.439839
$585$			1.97686e10i			0.168792i
$586$			2.63561e10i			0.223507i
$587$			8.76572e10i			0.738304i	0.929369	+	0.369152i	$0.120352\pi$
$587$			8.76572e10i			0.738304i	−0.929369	+	0.369152i	$0.879648\pi$
$588$	−7.74160e10			−0.647622
$589$		−	2.35481e10i		−	0.195657i
$590$	3.88110e10	−	1.33962e11i	0.320293	−	1.10554i
$591$	1.58605e10			0.130007
$592$		−	5.68186e10i		−	0.462598i
$593$	−3.49038e10			−0.282263			−0.141132	−	0.989991i	$-0.545074\pi$
$593$	−3.49038e10			−0.282263			−0.141132	+	0.989991i	$0.545074\pi$
$594$	3.96541e10			0.318524
$595$	1.10433e8			0.000881115
$596$		−	1.71947e11i		−	1.36273i
$597$	1.03100e11			0.811637
$598$		−	1.20878e11i		−	0.945244i
$599$	1.94116e10			0.150784			0.0753918	−	0.997154i	$-0.475979\pi$
$599$	1.94116e10			0.150784			0.0753918	+	0.997154i	$0.475979\pi$
$600$			2.49903e10i			0.192826i
$601$			6.35319e10i			0.486960i	0.969906	+	0.243480i	$0.0782891\pi$
$601$			6.35319e10i			0.486960i	−0.969906	+	0.243480i	$0.921711\pi$
$602$	−1.73969e11			−1.32460
$603$			3.22411e10i			0.243860i
$604$			1.20999e10i			0.0909145i
$605$	−1.11134e10			−0.0829519
$606$	−5.71550e10			−0.423803
$607$	8.61311e10			0.634462			0.317231	−	0.948348i	$-0.397247\pi$
$607$	8.61311e10			0.634462			0.317231	+	0.948348i	$0.397247\pi$
$608$		−	4.20429e10i		−	0.307665i
$609$	−4.97950e10			−0.362007
$610$	8.80818e10			0.636160
$611$	1.62745e11			1.16773
$612$	1.70246e8			0.00121359
$613$		−	5.93916e10i		−	0.420613i	−0.977635	−	0.210307i	$-0.932554\pi$
$613$		−	5.93916e10i		−	0.420613i	0.977635	−	0.210307i	$-0.0674463\pi$
$614$			2.69512e10i			0.189629i
$615$	−6.58791e10			−0.460519
$616$	−5.29611e10			−0.367819
$617$	2.34982e11			1.62142			0.810708	−	0.585450i	$-0.199082\pi$
$617$	2.34982e11			1.62142			0.810708	+	0.585450i	$0.199082\pi$
$618$	1.19726e11			0.820795
$619$	1.33083e10			0.0906486			0.0453243	−	0.998972i	$-0.485568\pi$
$619$	1.33083e10			0.0906486			0.0453243	+	0.998972i	$0.485568\pi$
$620$		−	1.24430e11i		−	0.842093i
$621$			2.49863e10i			0.168010i
$622$		−	4.50601e10i		−	0.301045i
$623$			1.03632e11i			0.687925i
$624$			1.97118e10i			0.130013i
$625$	−4.96106e10			−0.325128
$626$	3.24296e11			2.11176
$627$		−	2.34475e10i		−	0.151714i
$628$		−	2.54020e11i		−	1.63316i
$629$		−	5.53008e8i		−	0.00353288i
$630$			2.91291e10i			0.184912i
$631$	1.02889e10			0.0649009			0.0324505	−	0.999473i	$-0.489669\pi$
$631$	1.02889e10			0.0649009			0.0324505	+	0.999473i	$0.489669\pi$
$632$		−	1.80861e11i		−	1.13365i
$633$		−	5.78893e10i		−	0.360565i
$634$			4.44944e11i			2.75390i
$635$	2.13641e11			1.31398
$636$	1.24115e11			0.758570
$637$		−	8.72409e10i		−	0.529862i
$638$	3.56759e11			2.15324
$639$	8.48007e9			0.0508623
$640$		−	1.59152e11i		−	0.948622i
$641$	1.09157e11			0.646574			0.323287	−	0.946301i	$-0.395212\pi$
$641$	1.09157e11			0.646574			0.323287	+	0.946301i	$0.395212\pi$
$642$		−	8.03595e9i		−	0.0473039i
$643$	4.76191e10			0.278572			0.139286	−	0.990252i	$-0.455519\pi$
$643$	4.76191e10			0.278572			0.139286	+	0.990252i	$0.455519\pi$
$644$		−	1.05743e11i		−	0.614763i
$645$		−	1.28440e11i		−	0.742097i
$646$		−	1.69564e8i		−	0.000973654i
$647$	−2.74577e11			−1.56692			−0.783461	−	0.621441i	$-0.786548\pi$
$647$	−2.74577e11			−1.56692			−0.783461	+	0.621441i	$0.786548\pi$
$648$			1.41717e10i			0.0803753i
$649$	5.20846e10	−	1.79778e11i	0.293583	−	1.01335i
$650$	−8.92366e10			−0.499907
$651$			3.92591e10i			0.218583i
$652$	−3.33116e11			−1.84334
$653$	−2.39958e11			−1.31972			−0.659862	−	0.751387i	$-0.729385\pi$
$653$	−2.39958e11			−1.31972			−0.659862	+	0.751387i	$0.729385\pi$
$654$	−2.59042e11			−1.41599
$655$		−	5.44036e8i		−	0.00295571i
$656$	−6.56900e10			−0.354718
$657$		−	3.77632e10i		−	0.202678i
$658$	2.39806e11			1.27925
$659$			2.41605e11i			1.28105i	0.767939	+	0.640523i	$0.221282\pi$
$659$			2.41605e11i			1.28105i	−0.767939	+	0.640523i	$0.778718\pi$
$660$		−	1.23899e11i		−	0.652967i
$661$	4.97939e10			0.260838			0.130419	−	0.991459i	$-0.458368\pi$
$661$	4.97939e10			0.260838			0.130419	+	0.991459i	$0.458368\pi$
$662$		−	4.28581e11i		−	2.23152i
$663$			1.91852e8i			0.000992917i
$664$	−2.32235e10			−0.119469
$665$	1.72240e10			0.0880741
$666$	1.45867e11			0.741414
$667$			2.24796e11i			1.13575i
$668$	4.00233e10			0.201005
$669$	−1.53433e11			−0.765974
$670$	1.69682e11			0.842049
$671$	1.18206e11			0.583109
$672$			7.00933e10i			0.343716i
$673$			4.04861e11i			1.97354i	0.162128	+	0.986770i	$0.448164\pi$
$673$			4.04861e11i			1.97354i	−0.162128	+	0.986770i	$0.551836\pi$
$674$	1.39479e11			0.675880
$675$	1.84457e10			0.0888547
$676$	−1.59776e11			−0.765111
$677$	−4.01952e11			−1.91346			−0.956731	−	0.290973i	$-0.906021\pi$
$677$	−4.01952e11			−1.91346			−0.956731	+	0.290973i	$0.906021\pi$
$678$	2.01117e11			0.951765
$679$		−	1.25054e11i		−	0.588327i
$680$		−	2.82764e8i		−	0.00132248i
$681$		−	1.84151e10i		−	0.0856221i
$682$		−	2.81274e11i		−	1.30015i
$683$			1.28178e11i			0.589019i	0.955648	+	0.294510i	$0.0951563\pi$
$683$			1.28178e11i			0.589019i	−0.955648	+	0.294510i	$0.904844\pi$
$684$	2.65529e10			0.121308
$685$	1.27020e11			0.576911
$686$		−	2.95995e11i		−	1.33656i
$687$		−	5.67601e10i		−	0.254810i
$688$		−	1.28071e11i		−	0.571606i
$689$			1.39866e11i			0.620636i
$690$	1.31501e11			0.580139
$691$		−	1.72925e11i		−	0.758483i	−0.925298	−	0.379241i	$-0.876185\pi$
$691$		−	1.72925e11i		−	0.758483i	0.925298	−	0.379241i	$-0.123815\pi$
$692$		−	3.74342e11i		−	1.63246i
$693$			3.90914e10i			0.169492i
$694$	1.73577e11			0.748264
$695$	6.91678e10			0.296459
$696$			1.27500e11i			0.543340i
$697$	−6.39352e8			−0.00270900
$698$	−1.28338e11			−0.540673
$699$		−	1.41275e11i		−	0.591776i
$700$	−7.80631e10			−0.325127
$701$		−	5.06019e10i		−	0.209554i	−0.994496	−	0.104777i	$-0.966587\pi$
$701$		−	5.06019e10i		−	0.209554i	0.994496	−	0.104777i	$-0.0334128\pi$
$702$	−5.06050e10			−0.208375
$703$		−	8.62514e10i		−	0.353138i
$704$		−	4.17633e11i		−	1.70022i
$705$			1.77047e11i			0.716690i
$706$	−1.57654e11			−0.634580
$707$		−	5.63439e10i		−	0.225512i
$708$	2.03588e11	+	5.89827e10i	0.810250	+	0.234742i
$709$	2.68732e11			1.06349			0.531746	−	0.846904i	$-0.321536\pi$
$709$	2.68732e11			1.06349			0.531746	+	0.846904i	$0.321536\pi$
$710$		−	4.46300e10i		−	0.175628i
$711$	−1.33496e11			−0.522386
$712$	2.65348e11			1.03251
$713$	1.77232e11			0.685779
$714$			2.82695e8i			0.00108774i
$715$	1.39623e11			0.534235
$716$		−	2.18615e11i		−	0.831819i
$717$	−5.95219e9			−0.0225217
$718$			1.16701e11i			0.439112i
$719$			2.69631e11i			1.00891i	0.863437	+	0.504457i	$0.168307\pi$
$719$			2.69631e11i			1.00891i	−0.863437	+	0.504457i	$0.831693\pi$
$720$	−2.14440e10			−0.0797952
$721$			1.18027e11i			0.436757i
$722$			3.99852e11i			1.47147i
$723$	−1.15791e11			−0.423760
$724$	−7.88093e11			−2.86829
$725$	1.65952e11			0.600662
$726$		−	2.84489e10i		−	0.102405i
$727$	−3.93515e11			−1.40872			−0.704358	−	0.709845i	$-0.748765\pi$
$727$	−3.93515e11			−1.40872			−0.704358	+	0.709845i	$0.748765\pi$
$728$	6.75868e10			0.240623
$729$	1.04604e10			0.0370370
$730$	−1.98745e11			−0.699850
$731$		−	1.24650e9i		−	0.00436538i
$732$			1.33861e11i			0.466242i
$733$	−2.39451e11			−0.829471			−0.414736	−	0.909942i	$-0.636126\pi$
$733$	−2.39451e11			−0.829471			−0.414736	+	0.909942i	$0.636126\pi$
$734$	1.10002e11			0.378980
$735$	9.49075e10			0.325200
$736$	3.16431e11			1.07837
$737$	2.27715e11			0.771829
$738$		−	1.68642e11i		−	0.568513i
$739$		−	3.89964e11i		−	1.30751i	−0.756704	−	0.653757i	$-0.773192\pi$
$739$		−	3.89964e11i		−	1.30751i	0.756704	−	0.653757i	$-0.226808\pi$
$740$		−	4.55760e11i		−	1.51988i
$741$			2.99228e10i			0.0992496i
$742$			2.06094e11i			0.679907i
$743$	3.43743e11			1.12792			0.563960	−	0.825802i	$-0.309277\pi$
$743$	3.43743e11			1.12792			0.563960	+	0.825802i	$0.309277\pi$
$744$	1.00523e11			0.328074
$745$			2.10797e11i			0.684289i
$746$		−	1.64951e11i		−	0.532598i
$747$			1.71416e10i			0.0550514i
$748$		−	1.20243e9i		−	0.00384108i
$749$	7.92190e9			0.0251711
$750$		−	3.07340e11i		−	0.971347i
$751$			1.75253e10i			0.0550942i	0.999621	+	0.0275471i	$0.00876962\pi$
$751$			1.75253e10i			0.0550942i	−0.999621	+	0.0275471i	$0.991230\pi$
$752$			1.76538e11i			0.552037i
$753$	−1.28953e11			−0.401100
$754$	−4.55282e11			−1.40862
$755$		−	1.48337e10i		−	0.0456523i
$756$	−4.42687e10			−0.135522
$757$	−4.13727e11			−1.25988			−0.629942	−	0.776642i	$-0.716921\pi$
$757$	−4.13727e11			−1.25988			−0.629942	+	0.776642i	$0.716921\pi$
$758$			2.21087e10i			0.0669709i
$759$	1.76475e11			0.531760
$760$		−	4.41020e10i		−	0.132191i
$761$	−2.30836e10			−0.0688281			−0.0344141	−	0.999408i	$-0.510956\pi$
$761$	−2.30836e10			−0.0688281			−0.0344141	+	0.999408i	$0.510956\pi$
$762$			5.46893e11i			1.62212i
$763$		−	2.55366e11i		−	0.753467i
$764$		−	1.76017e11i		−	0.516632i
$765$	−2.08712e8			−0.000609399
$766$			5.44774e11i			1.58235i
$767$	−6.64682e10	+	2.29426e11i	−0.192058	+	0.662919i
$768$	8.37203e10			0.240650
$769$			4.05959e11i			1.16085i	0.814313	+	0.580426i	$0.197114\pi$
$769$			4.05959e11i			1.16085i	−0.814313	+	0.580426i	$0.802886\pi$
$770$	2.05735e11			0.585255
$771$	3.59099e11			1.01624
$772$	5.97648e11			1.68258
$773$		−	6.15256e11i		−	1.72321i	−0.507580	−	0.861604i	$-0.669460\pi$
$773$		−	6.15256e11i		−	1.72321i	0.507580	−	0.861604i	$-0.330540\pi$
$774$	3.28789e11			0.916123
$775$		−	1.30839e11i		−	0.362685i
$776$	−3.20200e11			−0.883027
$777$			1.43797e11i			0.394517i
$778$			4.80196e11i			1.31069i
$779$	−9.97181e10			−0.270785
$780$			1.58115e11i			0.427163i
$781$		−	5.98937e10i		−	0.160982i
$782$	1.27621e9			0.00341266

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)

\(f(q)\)	\(=\)	\(q-25.1007i q^{2} -46.7654 q^{3} -374.043 q^{4} +458.555 q^{5} +1173.84i q^{6} -1157.18 q^{7} +2962.95i q^{8} +2187.00 q^{9} +O(q^{10})\)	\(q-25.1007i q^{2} -46.7654 q^{3} -374.043 q^{4} +458.555 q^{5} +1173.84i q^{6} -1157.18 q^{7} +2962.95i q^{8} +2187.00 q^{9} -11510.0i q^{10} -15446.5i q^{11} +17492.3 q^{12} +19712.2i q^{13} +29046.0i q^{14} -21444.5 q^{15} -21382.9 q^{16} -208.117 q^{17} -54895.1i q^{18} -32459.5 q^{19} -171519. q^{20} +54116.1 q^{21} -387717. q^{22} -244303. i q^{23} -138564. i q^{24} -180353. q^{25} +494789. q^{26} -102276. q^{27} +432836. q^{28} -920152. q^{29} +538270. i q^{30} +725461. i q^{31} +1.29524e6i q^{32} +722362. i q^{33} +5223.87i q^{34} -530631. q^{35} -818032. q^{36} +2.65720e6i q^{37} +814755. i q^{38} -921849. i q^{39} +1.35868e6i q^{40} +3.07208e6 q^{41} -1.35835e6i q^{42} +5.98941e6i q^{43} +5.77766e6i q^{44} +1.00286e6 q^{45} -6.13215e6 q^{46} -8.25605e6i q^{47} +999980. q^{48} -4.42573e6 q^{49} +4.52697e6i q^{50} +9732.67 q^{51} -7.37321e6i q^{52} +7.09542e6 q^{53} +2.56719e6i q^{54} -7.08307e6i q^{55} -3.42868e6i q^{56} +1.51798e6 q^{57} +2.30964e7i q^{58} +(1.16388e7 + 3.37193e6i) q^{59} +8.02115e6 q^{60} +7.65261e6i q^{61} +1.82096e7 q^{62} -2.53076e6 q^{63} +2.70374e7 q^{64} +9.03913e6i q^{65} +1.81318e7 q^{66} +1.47421e7i q^{67} +77844.7 q^{68} +1.14249e7i q^{69} +1.33192e7i q^{70} +3.87749e6 q^{71} +6.47998e6i q^{72} -1.72671e7i q^{73} +6.66974e7 q^{74} +8.43426e6 q^{75} +1.21412e7 q^{76} +1.78744e7i q^{77} -2.31390e7 q^{78} -6.10409e7 q^{79} -9.80523e6 q^{80} +4.78297e6 q^{81} -7.71112e7i q^{82} +7.83794e6i q^{83} -2.02417e7 q^{84} -95433.0 q^{85} +1.50338e8 q^{86} +4.30313e7 q^{87} +4.57673e7 q^{88} -8.95553e7i q^{89} -2.51724e7i q^{90} -2.28106e7i q^{91} +9.13796e7i q^{92} -3.39265e7i q^{93} -2.07232e8 q^{94} -1.48845e7 q^{95} -6.05724e7i q^{96} +1.08068e8i q^{97} +1.11089e8i q^{98} -3.37815e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) \)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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