LMFDB - Embedded newform 690.3.g.a.461.8

Newspace parameters

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	690.g (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.8011382409$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			461.8
Character	$\chi$	$=$	690.461
Dual form			690.3.g.a.461.7

$q$-expansion

$f(q)$	$=$	$q+1.41421i q^{2} +(1.56849 - 2.55731i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(3.61658 + 2.21819i) q^{6} -1.76788 q^{7} -2.82843i q^{8} +(-4.07965 - 8.02225i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(1.56849 - 2.55731i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(3.61658 + 2.21819i) q^{6} -1.76788 q^{7} -2.82843i q^{8} +(-4.07965 - 8.02225i) q^{9} +3.16228 q^{10} +17.7224i q^{11} +(-3.13699 + 5.11462i) q^{12} -20.6890 q^{13} -2.50016i q^{14} +(-5.71832 - 3.50726i) q^{15} +4.00000 q^{16} +0.0288786i q^{17} +(11.3452 - 5.76950i) q^{18} -5.91750 q^{19} +4.47214i q^{20} +(-2.77291 + 4.52102i) q^{21} -25.0633 q^{22} +4.79583i q^{23} +(-7.23316 - 4.43637i) q^{24} -5.00000 q^{25} -29.2586i q^{26} +(-26.9143 - 2.14992i) q^{27} +3.53576 q^{28} +45.1942i q^{29} +(4.96001 - 8.08692i) q^{30} +41.1755 q^{31} +5.65685i q^{32} +(45.3217 + 27.7975i) q^{33} -0.0408405 q^{34} +3.95310i q^{35} +(8.15930 + 16.0445i) q^{36} -70.8841 q^{37} -8.36861i q^{38} +(-32.4505 + 52.9081i) q^{39} -6.32456 q^{40} +18.6543i q^{41} +(-6.39369 - 3.92149i) q^{42} +60.4201 q^{43} -35.4448i q^{44} +(-17.9383 + 9.12238i) q^{45} -6.78233 q^{46} +51.6509i q^{47} +(6.27398 - 10.2292i) q^{48} -45.8746 q^{49} -7.07107i q^{50} +(0.0738515 + 0.0452959i) q^{51} +41.3780 q^{52} +21.7689i q^{53} +(3.04045 - 38.0625i) q^{54} +39.6285 q^{55} +5.00033i q^{56} +(-9.28156 + 15.1329i) q^{57} -63.9142 q^{58} -68.2581i q^{59} +(11.4366 + 7.01452i) q^{60} +1.38027 q^{61} +58.2310i q^{62} +(7.21234 + 14.1824i) q^{63} -8.00000 q^{64} +46.2620i q^{65} +(-39.3116 + 64.0945i) q^{66} -100.715 q^{67} -0.0577572i q^{68} +(12.2644 + 7.52223i) q^{69} -5.59053 q^{70} +56.2569i q^{71} +(-22.6903 + 11.5390i) q^{72} -73.9837 q^{73} -100.245i q^{74} +(-7.84247 + 12.7865i) q^{75} +11.8350 q^{76} -31.3311i q^{77} +(-74.8233 - 45.8920i) q^{78} +4.47024 q^{79} -8.94427i q^{80} +(-47.7129 + 65.4559i) q^{81} -26.3811 q^{82} +21.0356i q^{83} +(5.54583 - 9.04204i) q^{84} +0.0645745 q^{85} +85.4469i q^{86} +(115.575 + 70.8868i) q^{87} +50.1266 q^{88} -28.2078i q^{89} +(-12.9010 - 25.3686i) q^{90} +36.5757 q^{91} -9.59166i q^{92} +(64.5835 - 105.298i) q^{93} -73.0455 q^{94} +13.2319i q^{95} +(14.4663 + 8.87274i) q^{96} +57.7059 q^{97} -64.8765i q^{98} +(142.174 - 72.3013i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56q - 8q^{3} - 112q^{4} + 16q^{6} - 16q^{7} + O(q^{10}) $	$ 56q - 8q^{3} - 112q^{4} + 16q^{6} - 16q^{7} + 16q^{12} + 80q^{13} - 40q^{15} + 224q^{16} - 32q^{18} - 64q^{19} + 56q^{21} - 96q^{22} - 32q^{24} - 280q^{25} + 40q^{27} + 32q^{28} - 80q^{31} + 32q^{33} + 192q^{34} + 240q^{37} - 56q^{39} - 144q^{43} - 32q^{48} + 72q^{49} - 24q^{51} - 160q^{52} + 16q^{54} - 16q^{57} + 80q^{60} + 112q^{61} - 64q^{63} - 448q^{64} + 160q^{66} + 832q^{67} + 64q^{72} - 608q^{73} + 40q^{75} + 128q^{76} - 320q^{78} + 48q^{79} - 32q^{81} - 448q^{82} - 112q^{84} + 240q^{85} + 200q^{87} + 192q^{88} + 80q^{91} - 232q^{93} + 160q^{94} + 64q^{96} - 448q^{97} + 464q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	1.56849	−	2.55731i	0.522831	−	0.852436i
$3$	1.56849	−	2.55731i	0.522831	−	0.852436i
$4$	−2.00000			−0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$	3.61658	+	2.21819i	0.602763	+	0.369698i
$7$	−1.76788			−0.252555			−0.126277	−	0.991995i	$-0.540303\pi$
$7$	−1.76788			−0.252555			−0.126277	+	0.991995i	$0.540303\pi$
$8$		−	2.82843i		−	0.353553i
$9$	−4.07965	−	8.02225i	−0.453295	−	0.891361i
$10$	3.16228			0.316228
$11$			17.7224i			1.61113i	0.592508	+	0.805564i	$0.298138\pi$
$11$			17.7224i			1.61113i	−0.592508	+	0.805564i	$0.701862\pi$
$12$	−3.13699	+	5.11462i	−0.261416	+	0.426218i
$13$	−20.6890			−1.59146			−0.795730	−	0.605652i	$-0.792913\pi$
$13$	−20.6890			−1.59146			−0.795730	+	0.605652i	$0.792913\pi$
$14$		−	2.50016i		−	0.178583i
$15$	−5.71832	−	3.50726i	−0.381221	−	0.233817i
$16$	4.00000			0.250000
$17$			0.0288786i			0.00169874i	1.00000		0.000849371i	$0.000270363\pi$
$17$			0.0288786i			0.00169874i	−1.00000		0.000849371i	$0.999730\pi$
$18$	11.3452	−	5.76950i	0.630287	−	0.320528i
$19$	−5.91750			−0.311447			−0.155724	−	0.987801i	$-0.549771\pi$
$19$	−5.91750			−0.311447			−0.155724	+	0.987801i	$0.549771\pi$
$20$			4.47214i			0.223607i
$21$	−2.77291	+	4.52102i	−0.132043	+	0.215287i
$22$	−25.0633			−1.13924
$23$			4.79583i			0.208514i
$23$			4.79583i			0.208514i
$24$	−7.23316	−	4.43637i	−0.301382	−	0.184849i
$25$	−5.00000			−0.200000
$26$		−	29.2586i		−	1.12533i
$27$	−26.9143	−	2.14992i	−0.996825	−	0.0796268i
$28$	3.53576			0.126277
$29$			45.1942i			1.55842i	0.626763	+	0.779210i	$0.284379\pi$
$29$			45.1942i			1.55842i	−0.626763	+	0.779210i	$0.715621\pi$
$30$	4.96001	−	8.08692i	0.165334	−	0.269564i
$31$	41.1755			1.32824			0.664121	−	0.747625i	$-0.268806\pi$
$31$	41.1755			1.32824			0.664121	+	0.747625i	$0.268806\pi$
$32$			5.65685i			0.176777i
$33$	45.3217	+	27.7975i	1.37338	+	0.842349i
$34$	−0.0408405			−0.00120119
$35$			3.95310i			0.112946i
$36$	8.15930	+	16.0445i	0.226647	+	0.445680i
$37$	−70.8841			−1.91579			−0.957893	−	0.287125i	$-0.907300\pi$
$37$	−70.8841			−1.91579			−0.957893	+	0.287125i	$0.907300\pi$
$38$		−	8.36861i		−	0.220226i
$39$	−32.4505	+	52.9081i	−0.832065	+	1.35662i
$40$	−6.32456			−0.158114
$41$			18.6543i			0.454982i	0.973780	+	0.227491i	$0.0730522\pi$
$41$			18.6543i			0.454982i	−0.973780	+	0.227491i	$0.926948\pi$
$42$	−6.39369	−	3.92149i	−0.152231	−	0.0933688i
$43$	60.4201			1.40512			0.702559	−	0.711625i	$-0.252041\pi$
$43$	60.4201			1.40512			0.702559	+	0.711625i	$0.252041\pi$
$44$		−	35.4448i		−	0.805564i
$45$	−17.9383	+	9.12238i	−0.398629	+	0.202719i
$46$	−6.78233			−0.147442
$47$			51.6509i			1.09896i	0.835508	+	0.549478i	$0.185173\pi$
$47$			51.6509i			1.09896i	−0.835508	+	0.549478i	$0.814827\pi$
$48$	6.27398	−	10.2292i	0.130708	−	0.213109i
$49$	−45.8746			−0.936216
$50$		−	7.07107i		−	0.141421i
$51$	0.0738515	+	0.0452959i	0.00144807	+	0.000888155i
$52$	41.3780			0.795730
$53$			21.7689i			0.410734i	0.978685	+	0.205367i	$0.0658388\pi$
$53$			21.7689i			0.410734i	−0.978685	+	0.205367i	$0.934161\pi$
$54$	3.04045	−	38.0625i	0.0563046	−	0.704862i
$55$	39.6285			0.720519
$56$			5.00033i			0.0892915i
$57$	−9.28156	+	15.1329i	−0.162834	+	0.265489i
$58$	−63.9142			−1.10197
$59$		−	68.2581i		−	1.15692i	−0.815712	−	0.578458i	$-0.803655\pi$
$59$		−	68.2581i		−	1.15692i	0.815712	−	0.578458i	$-0.196345\pi$
$60$	11.4366	+	7.01452i	0.190611	+	0.116909i
$61$	1.38027			0.0226273			0.0113136	−	0.999936i	$-0.496399\pi$
$61$	1.38027			0.0226273			0.0113136	+	0.999936i	$0.496399\pi$
$62$			58.2310i			0.939209i
$63$	7.21234	+	14.1824i	0.114482	+	0.225117i
$64$	−8.00000			−0.125000
$65$			46.2620i			0.711722i
$66$	−39.3116	+	64.0945i	−0.595631	+	0.971129i
$67$	−100.715			−1.50322			−0.751608	−	0.659610i	$-0.770721\pi$
$67$	−100.715			−1.50322			−0.751608	+	0.659610i	$0.770721\pi$
$68$		−	0.0577572i		−	0.000849371i
$69$	12.2644	+	7.52223i	0.177745	+	0.109018i
$70$	−5.59053			−0.0798648
$71$			56.2569i			0.792350i	0.918175	+	0.396175i	$0.129663\pi$
$71$			56.2569i			0.792350i	−0.918175	+	0.396175i	$0.870337\pi$
$72$	−22.6903	+	11.5390i	−0.315144	+	0.160264i
$73$	−73.9837			−1.01348			−0.506738	−	0.862100i	$-0.669149\pi$
$73$	−73.9837			−1.01348			−0.506738	+	0.862100i	$0.669149\pi$
$74$		−	100.245i		−	1.35467i
$75$	−7.84247	+	12.7865i	−0.104566	+	0.170487i
$76$	11.8350			0.155724
$77$		−	31.3311i		−	0.406898i
$78$	−74.8233	−	45.8920i	−0.959274	−	0.588359i
$79$	4.47024			0.0565854			0.0282927	−	0.999600i	$-0.490993\pi$
$79$	4.47024			0.0565854			0.0282927	+	0.999600i	$0.490993\pi$
$80$		−	8.94427i		−	0.111803i
$81$	−47.7129	+	65.4559i	−0.589048	+	0.808098i
$82$	−26.3811			−0.321721
$83$			21.0356i			0.253441i	0.991938	+	0.126721i	$0.0404451\pi$
$83$			21.0356i			0.253441i	−0.991938	+	0.126721i	$0.959555\pi$
$84$	5.54583	−	9.04204i	0.0660217	−	0.107643i
$85$	0.0645745			0.000759700
$86$			85.4469i			0.993569i
$87$	115.575	+	70.8868i	1.32845	+	0.814791i
$88$	50.1266			0.569620
$89$		−	28.2078i		−	0.316942i	−0.987364	−	0.158471i	$-0.949344\pi$
$89$		−	28.2078i		−	0.316942i	0.987364	−	0.158471i	$-0.0506564\pi$
$90$	−12.9010	−	25.3686i	−0.143344	−	0.281873i
$91$	36.5757			0.401930
$92$		−	9.59166i		−	0.104257i
$93$	64.5835	−	105.298i	0.694447	−	1.13224i
$94$	−73.0455			−0.777079
$95$			13.2319i			0.139283i
$96$	14.4663	+	8.87274i	0.150691	+	0.0924244i
$97$	57.7059			0.594906			0.297453	−	0.954736i	$-0.403863\pi$
$97$	57.7059			0.594906			0.297453	+	0.954736i	$0.403863\pi$
$98$		−	64.8765i		−	0.662005i
$99$	142.174	−	72.3013i	1.43610	−	0.730316i
$100$	10.0000			0.100000
$101$		−	68.2488i		−	0.675730i	−0.941195	−	0.337865i	$-0.890295\pi$
$101$		−	68.2488i		−	0.675730i	0.941195	−	0.337865i	$-0.109705\pi$
$102$	−0.0640581	+	0.104442i	−0.000628021	+	0.00102394i
$103$	−61.1415			−0.593607			−0.296803	−	0.954939i	$-0.595921\pi$
$103$	−61.1415			−0.593607			−0.296803	+	0.954939i	$0.595921\pi$
$104$			58.5173i			0.562666i
$105$	10.1093	+	6.20042i	0.0962791	+	0.0590516i
$106$	−30.7859			−0.290433
$107$		−	146.342i		−	1.36769i	−0.729629	−	0.683843i	$-0.760307\pi$
$107$		−	146.342i		−	1.36769i	0.729629	−	0.683843i	$-0.239693\pi$
$108$	53.8285	+	4.29985i	0.498412	+	0.0398134i
$109$	48.8918			0.448549			0.224274	−	0.974526i	$-0.427999\pi$
$109$	48.8918			0.448549			0.224274	+	0.974526i	$0.427999\pi$
$110$			56.0432i			0.509484i
$111$	−111.181	+	181.272i	−1.00163	+	1.63309i
$112$	−7.07153			−0.0631386
$113$			127.489i			1.12822i	0.825698	+	0.564112i	$0.190781\pi$
$113$			127.489i			1.12822i	−0.825698	+	0.564112i	$0.809219\pi$
$114$	−21.4011	−	13.1261i	−0.187729	−	0.115141i
$115$	10.7238			0.0932505
$116$		−	90.3884i		−	0.779210i
$117$	84.4038	+	165.972i	0.721400	+	1.41856i
$118$	96.5315			0.818064
$119$		−	0.0510540i		−	0.000429025i
$120$	−9.92003	+	16.1738i	−0.0826669	+	0.134782i
$121$	−193.084			−1.59574
$122$			1.95199i			0.0159999i
$123$	47.7047	+	29.2591i	0.387843	+	0.237879i
$124$	−82.3510			−0.664121
$125$			11.1803i			0.0894427i
$126$	−20.0569	+	10.1998i	−0.159182	+	0.0809507i
$127$	−230.507			−1.81501			−0.907506	−	0.420038i	$-0.862017\pi$
$127$	−230.507			−1.81501			−0.907506	+	0.420038i	$0.862017\pi$
$128$		−	11.3137i		−	0.0883883i
$129$	94.7685	−	154.513i	0.734640	−	1.19777i
$130$	−65.4243			−0.503264
$131$		−	65.2731i		−	0.498268i	−0.968469	−	0.249134i	$-0.919854\pi$
$131$		−	65.2731i		−	0.498268i	0.968469	−	0.249134i	$-0.0801460\pi$
$132$	−90.6434	−	55.5950i	−0.686692	−	0.421174i
$133$	10.4614			0.0786574
$134$		−	142.433i		−	1.06293i
$135$	−4.80737	+	60.1821i	−0.0356102	+	0.445794i
$136$	0.0816810			0.000600596
$137$		−	123.243i		−	0.899587i	−0.893133	−	0.449793i	$-0.851498\pi$
$137$		−	123.243i		−	0.899587i	0.893133	−	0.449793i	$-0.148502\pi$
$138$	−10.6380	+	17.3445i	−0.0770873	+	0.125685i
$139$	−219.172			−1.57678			−0.788389	−	0.615177i	$-0.789085\pi$
$139$	−219.172			−1.57678			−0.788389	+	0.615177i	$0.789085\pi$
$140$		−	7.90621i		−	0.0564729i
$141$	132.087	+	81.0142i	0.936790	+	0.574569i
$142$	−79.5592			−0.560276
$143$		−	366.659i		−	2.56405i
$144$	−16.3186	−	32.0890i	−0.113324	−	0.222840i
$145$	101.057			0.696947
$146$		−	104.629i		−	0.716635i
$147$	−71.9540	+	117.315i	−0.489483	+	0.798064i
$148$	141.768			0.957893
$149$		−	84.6070i		−	0.567832i	−0.958849	−	0.283916i	$-0.908366\pi$
$149$		−	84.6070i		−	0.567832i	0.958849	−	0.283916i	$-0.0916337\pi$
$150$	−18.0829	−	11.0909i	−0.120553	−	0.0739395i
$151$	−29.2120			−0.193457			−0.0967284	−	0.995311i	$-0.530838\pi$
$151$	−29.2120			−0.193457			−0.0967284	+	0.995311i	$0.530838\pi$
$152$			16.7372i			0.110113i
$153$	0.231671	−	0.117815i	0.00151419	−	0.000770030i
$154$	44.3089			0.287720
$155$		−	92.0712i		−	0.594008i
$156$	64.9011	−	105.816i	0.416033	−	0.678309i
$157$	269.251			1.71498			0.857488	−	0.514503i	$-0.172024\pi$
$157$	269.251			1.71498			0.857488	+	0.514503i	$0.172024\pi$
$158$			6.32188i			0.0400119i
$159$	55.6698	+	34.1444i	0.350125	+	0.214745i
$160$	12.6491			0.0790569
$161$		−	8.47846i		−	0.0526613i
$162$	−92.5687	−	67.4762i	−0.571412	−	0.416520i
$163$	−296.191			−1.81712			−0.908561	−	0.417752i	$-0.862818\pi$
$163$	−296.191			−1.81712			−0.908561	+	0.417752i	$0.862818\pi$
$164$		−	37.3085i		−	0.227491i
$165$	62.1571	−	101.342i	0.376710	−	0.614196i
$166$	−29.7488			−0.179210
$167$		−	139.977i		−	0.838188i	−0.907943	−	0.419094i	$-0.862348\pi$
$167$		−	139.977i		−	0.838188i	0.907943	−	0.419094i	$-0.137652\pi$
$168$	12.7874	+	7.84298i	0.0761153	+	0.0466844i
$169$	259.034			1.53274
$170$			0.0913222i			0.000537189i
$171$	24.1413	+	47.4716i	0.141177	+	0.277612i
$172$	−120.840			−0.702559
$173$		−	2.23761i		−	0.0129342i	−0.999979	−	0.00646708i	$-0.997941\pi$
$173$		−	2.23761i		−	0.0129342i	0.999979	−	0.00646708i	$-0.00205855\pi$
$174$	−100.249	+	163.448i	−0.576144	+	0.939359i
$175$	8.83941			0.0505109
$176$			70.8897i			0.402782i
$177$	−174.557	−	107.062i	−0.986198	−	0.604872i
$178$	39.8919			0.224112
$179$			86.0465i			0.480707i	0.970685	+	0.240353i	$0.0772633\pi$
$179$			86.0465i			0.480707i	−0.970685	+	0.240353i	$0.922737\pi$
$180$	35.8766	−	18.2448i	0.199314	−	0.101360i
$181$	43.5809			0.240778			0.120389	−	0.992727i	$-0.461586\pi$
$181$	43.5809			0.240778			0.120389	+	0.992727i	$0.461586\pi$
$182$			51.7258i			0.284208i
$183$	2.16494	−	3.52976i	0.0118303	−	0.0192883i
$184$	13.5647			0.0737210
$185$			158.502i			0.856766i
$186$	148.914	+	91.3349i	0.800616	+	0.491048i
$187$	−0.511799			−0.00273689
$188$		−	103.302i		−	0.549478i
$189$	47.5812	+	3.80081i	0.251753	+	0.0201101i
$190$	−18.7128			−0.0984883
$191$			195.043i			1.02117i	0.859828	+	0.510584i	$0.170571\pi$
$191$			195.043i			1.02117i	−0.859828	+	0.510584i	$0.829429\pi$
$192$	−12.5480	+	20.4585i	−0.0653539	+	0.106555i
$193$	330.591			1.71290			0.856452	−	0.516226i	$-0.172664\pi$
$193$	330.591			1.71290			0.856452	+	0.516226i	$0.172664\pi$
$194$			81.6085i			0.420662i
$195$	118.306	+	72.5616i	0.606698	+	0.372111i
$196$	91.7492			0.468108
$197$		−	91.7002i		−	0.465483i	−0.972539	−	0.232742i	$-0.925230\pi$
$197$		−	91.7002i		−	0.465483i	0.972539	−	0.232742i	$-0.0747697\pi$
$198$	102.249	+	201.064i	0.516411	+	1.01547i
$199$	6.18236			0.0310671			0.0155336	−	0.999879i	$-0.495055\pi$
$199$	6.18236			0.0310671			0.0155336	+	0.999879i	$0.495055\pi$
$200$			14.1421i			0.0707107i
$201$	−157.972	+	257.560i	−0.785928	+	1.28140i
$202$	96.5183			0.477813
$203$		−	79.8980i		−	0.393586i
$204$	−0.147703	−	0.0905919i	−0.000724034	−	0.000444078i
$205$	41.7122			0.203474
$206$		−	86.4671i		−	0.419743i
$207$	38.4733	−	19.5653i	0.185862	−	0.0945185i
$208$	−82.7559			−0.397865
$209$		−	104.872i		−	0.501782i
$210$	−8.76872	+	14.2967i	−0.0417558	+	0.0680796i
$211$	312.639			1.48170			0.740851	−	0.671670i	$-0.234423\pi$
$211$	312.639			1.48170			0.740851	+	0.671670i	$0.234423\pi$
$212$		−	43.5378i		−	0.205367i
$213$	143.866	+	88.2386i	0.675428	+	0.414266i
$214$	206.959			0.967100
$215$		−	135.103i		−	0.628388i
$216$	−6.08090	+	76.1250i	−0.0281523	+	0.352431i
$217$	−72.7934			−0.335454
$218$			69.1435i			0.317172i
$219$	−116.043	+	189.199i	−0.529877	+	0.863923i
$220$	−79.2571			−0.360259
$221$		−	0.597469i		−	0.00270348i
$222$	−256.358	−	157.234i	−1.15477	−	0.708262i
$223$	73.3202			0.328790			0.164395	−	0.986395i	$-0.447433\pi$
$223$	73.3202			0.328790			0.164395	+	0.986395i	$0.447433\pi$
$224$		−	10.0007i		−	0.0446458i
$225$	20.3983	+	40.1112i	0.0906589	+	0.178272i
$226$	−180.297			−0.797775
$227$			54.4085i			0.239685i	0.992793	+	0.119843i	$0.0382390\pi$
$227$			54.4085i			0.239685i	−0.992793	+	0.119843i	$0.961761\pi$
$228$	18.5631	−	30.2657i	0.0814172	−	0.132744i
$229$	−104.576			−0.456662			−0.228331	−	0.973584i	$-0.573327\pi$
$229$	−104.576			−0.456662			−0.228331	+	0.973584i	$0.573327\pi$
$230$			15.1658i			0.0659380i
$231$	−80.1234	−	49.1427i	−0.346854	−	0.212739i
$232$	127.828			0.550985
$233$		−	72.9504i		−	0.313092i	−0.987671	−	0.156546i	$-0.949964\pi$
$233$		−	72.9504i		−	0.313092i	0.987671	−	0.156546i	$-0.0500359\pi$
$234$	−234.720	+	119.365i	−1.00308	+	0.510107i
$235$	115.495			0.491468
$236$			136.516i			0.578458i
$237$	7.01155	−	11.4318i	0.0295846	−	0.0482354i
$238$	0.0722012			0.000303366
$239$			401.263i			1.67892i	0.543418	+	0.839462i	$0.317130\pi$
$239$			401.263i			1.67892i	−0.543418	+	0.839462i	$0.682870\pi$
$240$	−22.8733	−	14.0290i	−0.0953053	−	0.0584543i
$241$	−75.2115			−0.312081			−0.156041	−	0.987751i	$-0.549873\pi$
$241$	−75.2115			−0.312081			−0.156041	+	0.987751i	$0.549873\pi$
$242$		−	273.062i		−	1.12836i
$243$	92.5536	+	224.684i	0.380879	+	0.924625i
$244$	−2.76053			−0.0113136
$245$			102.579i			0.418689i
$246$	−41.3786	+	67.4646i	−0.168206	+	0.274246i
$247$	122.427			0.495656
$248$		−	116.462i		−	0.469604i
$249$	53.7945	+	32.9942i	0.216042	+	0.132507i
$250$	−15.8114			−0.0632456
$251$		−	95.2872i		−	0.379630i	−0.981820	−	0.189815i	$-0.939211\pi$
$251$		−	95.2872i		−	0.379630i	0.981820	−	0.189815i	$-0.0607889\pi$
$252$	−14.4247	−	28.3648i	−0.0572408	−	0.112559i
$253$	−84.9937			−0.335944
$254$		−	325.986i		−	1.28341i
$255$	0.101285	−	0.165137i	0.000397195	−	0.000647596i
$256$	16.0000			0.0625000
$257$			110.044i			0.428188i	0.976813	+	0.214094i	$0.0686799\pi$
$257$			110.044i			0.428188i	−0.976813	+	0.214094i	$0.931320\pi$
$258$	218.514	+	134.023i	0.846954	+	0.519469i
$259$	125.315			0.483840
$260$		−	92.5239i		−	0.355861i
$261$	362.559	−	184.377i	1.38911	−	0.706423i
$262$	92.3101			0.352329
$263$		−	372.584i		−	1.41667i	−0.705876	−	0.708335i	$-0.749446\pi$
$263$		−	372.584i		−	1.41667i	0.705876	−	0.708335i	$-0.250554\pi$
$264$	78.6232	−	128.189i	0.297815	−	0.485565i
$265$	48.6767			0.183686
$266$			14.7947i			0.0556192i
$267$	−72.1361	−	44.2438i	−0.270173	−	0.165707i
$268$	201.431			0.751608
$269$			61.0243i			0.226856i	0.993546	+	0.113428i	$0.0361831\pi$
$269$			61.0243i			0.226856i	−0.993546	+	0.113428i	$0.963817\pi$
$270$	−85.1104	−	6.79865i	−0.315224	−	0.0251802i
$271$	489.007			1.80445			0.902227	−	0.431262i	$-0.141932\pi$
$271$	489.007			1.80445			0.902227	+	0.431262i	$0.141932\pi$
$272$			0.115514i			0.000424685i
$273$	57.3687	−	93.5353i	0.210142	−	0.342620i
$274$	174.292			0.636104
$275$		−	88.6121i		−	0.322226i
$276$	−24.5288	−	15.0445i	−0.0888726	−	0.0545089i
$277$	319.748			1.15432			0.577162	−	0.816630i	$-0.304160\pi$
$277$	319.748			1.15432			0.577162	+	0.816630i	$0.304160\pi$
$278$		−	309.956i		−	1.11495i
$279$	−167.982	−	330.320i	−0.602085	−	1.18394i
$280$	11.1811			0.0399324
$281$			244.012i			0.868371i	0.900824	+	0.434185i	$0.142964\pi$
$281$			244.012i			0.868371i	−0.900824	+	0.434185i	$0.857036\pi$
$282$	−114.571	+	186.800i	−0.406282	+	0.662410i
$283$	−312.536			−1.10437			−0.552183	−	0.833723i	$-0.686205\pi$
$283$	−312.536			−1.10437			−0.552183	+	0.833723i	$0.686205\pi$
$284$		−	112.514i		−	0.396175i
$285$	33.8381	+	20.7542i	0.118730	+	0.0728218i
$286$	518.534			1.81305
$287$		−	32.9785i		−	0.114908i
$288$	45.3807	−	23.0780i	0.157572	−	0.0801319i
$289$	288.999			0.999997
$290$			142.917i			0.492816i
$291$	90.5114	−	147.572i	0.311036	−	0.507120i
$292$	147.967			0.506738
$293$		−	439.469i		−	1.49989i	−0.661498	−	0.749947i	$-0.730079\pi$
$293$		−	439.469i		−	1.49989i	0.661498	−	0.749947i	$-0.269921\pi$
$294$	−165.909	−	101.758i	−0.564317	−	0.346117i
$295$	−152.630			−0.517389
$296$			200.490i			0.677333i
$297$	38.1018	−	476.986i	0.128289	−	1.60601i
$298$	119.652			0.401518
$299$		−	99.2209i		−	0.331842i
$300$	15.6849	−	25.5731i	0.0522831	−	0.0852436i
$301$	−106.816			−0.354869
$302$		−	41.3120i		−	0.136795i
$303$	−174.533	−	107.048i	−0.576017	−	0.353293i
$304$	−23.6700			−0.0778618
$305$		−	3.08637i		−	0.0101192i
$306$	0.166615	+	0.327633i	0.000544494	+	0.00107070i
$307$	−147.088			−0.479115			−0.239557	−	0.970882i	$-0.577002\pi$
$307$	−147.088			−0.479115			−0.239557	+	0.970882i	$0.577002\pi$
$308$			62.6623i			0.203449i
$309$	−95.9001	+	156.358i	−0.310356	+	0.506012i
$310$	130.208			0.420027
$311$			601.613i			1.93445i	0.253930	+	0.967223i	$0.418277\pi$
$311$			601.613i			1.93445i	−0.253930	+	0.967223i	$0.581723\pi$
$312$	149.647	+	91.7840i	0.479637	+	0.294179i
$313$	−88.4556			−0.282606			−0.141303	−	0.989966i	$-0.545129\pi$
$313$	−88.4556			−0.282606			−0.141303	+	0.989966i	$0.545129\pi$
$314$			380.779i			1.21267i
$315$	31.7128	−	16.1273i	0.100675	−	0.0511977i
$316$	−8.94049			−0.0282927
$317$			435.939i			1.37520i	0.726089	+	0.687601i	$0.241336\pi$
$317$			435.939i			1.37520i	−0.726089	+	0.687601i	$0.758664\pi$
$318$	−48.2875	+	78.7290i	−0.151847	+	0.247575i
$319$	−800.950			−2.51082
$320$			17.8885i			0.0559017i
$321$	−374.243	−	229.537i	−1.16586	−	0.715069i
$322$	11.9904			0.0372371
$323$		−	0.170889i		−	0.000529068i
$324$	95.4258	−	130.912i	0.294524	−	0.404049i
$325$	103.445			0.318292
$326$		−	418.877i		−	1.28490i
$327$	76.6865	−	125.031i	0.234515	−	0.382359i
$328$	52.7622			0.160860
$329$		−	91.3128i		−	0.277546i
$330$	143.320	+	87.9034i	0.434302	+	0.266374i
$331$	305.202			0.922060			0.461030	−	0.887384i	$-0.347480\pi$
$331$	305.202			0.922060			0.461030	+	0.887384i	$0.347480\pi$
$332$		−	42.0712i		−	0.126721i
$333$	289.182	+	568.650i	0.868415	+	1.70766i
$334$	197.958			0.592688
$335$			225.207i			0.672258i
$336$	−11.0917	+	18.0841i	−0.0330109	+	0.0538217i
$337$	222.147			0.659189			0.329594	−	0.944123i	$-0.393088\pi$
$337$	222.147			0.659189			0.329594	+	0.944123i	$0.393088\pi$
$338$			366.329i			1.08381i
$339$	326.029	+	199.966i	0.961739	+	0.589871i
$340$	−0.129149			−0.000379850
$341$			729.729i			2.13997i
$342$	−67.1350	+	34.1410i	−0.196301	+	0.0998275i
$343$	167.727			0.489000
$344$		−	170.894i		−	0.496784i
$345$	16.8202	−	27.4241i	0.0487543	−	0.0794901i
$346$	3.16446			0.00914583
$347$		−	656.982i		−	1.89332i	−0.322235	−	0.946660i	$-0.604434\pi$
$347$		−	656.982i		−	1.89332i	0.322235	−	0.946660i	$-0.395566\pi$
$348$	−231.151	−	141.774i	−0.664227	−	0.407396i
$349$	−409.550			−1.17350			−0.586748	−	0.809769i	$-0.699592\pi$
$349$	−409.550			−1.17350			−0.586748	+	0.809769i	$0.699592\pi$
$350$			12.5008i			0.0357166i
$351$	556.829	+	44.4797i	1.58641	+	0.126723i
$352$	−100.253			−0.284810
$353$		−	249.433i		−	0.706609i	−0.935508	−	0.353304i	$-0.885058\pi$
$353$		−	249.433i		−	0.706609i	0.935508	−	0.353304i	$-0.114942\pi$
$354$	151.409	−	246.861i	0.427709	−	0.697347i
$355$	125.794			0.354350
$356$			56.4156i			0.158471i
$357$	−0.130561	−	0.0800778i	−0.000365716	−	0.000224308i
$358$	−121.688			−0.339911
$359$			333.458i			0.928851i	0.885612	+	0.464426i	$0.153739\pi$
$359$			333.458i			0.928851i	−0.885612	+	0.464426i	$0.846261\pi$
$360$	25.8020	+	50.7371i	0.0716722	+	0.140937i
$361$	−325.983			−0.903001
$362$			61.6327i			0.170256i
$363$	−302.851	+	493.775i	−0.834301	+	1.36026i
$364$	−73.1513			−0.200965
$365$			165.433i			0.453240i
$366$	4.99184	+	3.06168i	0.0136389	+	0.00836526i
$367$	−144.077			−0.392581			−0.196290	−	0.980546i	$-0.562889\pi$
$367$	−144.077			−0.392581			−0.196290	+	0.980546i	$0.562889\pi$
$368$			19.1833i			0.0521286i
$369$	149.649	−	76.1029i	0.405553	−	0.206241i
$370$	−224.155			−0.605825
$371$		−	38.4849i		−	0.103733i
$372$	−129.167	+	210.597i	−0.347223	+	0.566121i
$373$	−494.074			−1.32459			−0.662297	−	0.749241i	$-0.730418\pi$
$373$	−494.074			−1.32459			−0.662297	+	0.749241i	$0.730418\pi$
$374$		−	0.723793i		−	0.00193527i
$375$	28.5916	+	17.5363i	0.0762442	+	0.0467635i
$376$	146.091			0.388540
$377$		−	935.021i		−	2.48016i
$378$	−5.37516	+	67.2900i	−0.0142200	+	0.178016i
$379$	−533.662			−1.40808			−0.704040	−	0.710160i	$-0.748622\pi$
$379$	−533.662			−1.40808			−0.704040	+	0.710160i	$0.748622\pi$
$380$		−	26.4639i		−	0.0696417i
$381$	−361.548	+	589.476i	−0.948946	+	1.54718i
$382$	−275.833			−0.722075
$383$		−	245.335i		−	0.640563i	−0.947322	−	0.320281i	$-0.896223\pi$
$383$		−	245.335i		−	0.640563i	0.947322	−	0.320281i	$-0.103777\pi$
$384$	−28.9326	−	17.7455i	−0.0753454	−	0.0462122i
$385$	−70.0586			−0.181970
$386$			467.526i			1.21121i
$387$	−246.493	−	484.705i	−0.636932	−	1.25247i
$388$	−115.412			−0.297453
$389$			119.365i			0.306852i	0.988160	+	0.153426i	$0.0490306\pi$
$389$			119.365i			0.306852i	−0.988160	+	0.153426i	$0.950969\pi$
$390$	−102.618	+	167.310i	−0.263122	+	0.429000i
$391$	−0.138497			−0.000354212
$392$			129.753i			0.331002i
$393$	−166.923	−	102.380i	−0.424742	−	0.260510i
$394$	129.684			0.329147
$395$		−	9.99577i		−	0.0253057i
$396$	−284.347	+	144.603i	−0.718049	+	0.365158i
$397$	424.026			1.06808			0.534038	−	0.845461i	$-0.320674\pi$
$397$	424.026			1.06808			0.534038	+	0.845461i	$0.320674\pi$
$398$			8.74317i			0.0219678i
$399$	16.4087	−	26.7531i	0.0411246	−	0.0670504i
$400$	−20.0000			−0.0500000
$401$		−	787.894i		−	1.96482i	−0.186731	−	0.982411i	$-0.559789\pi$
$401$		−	787.894i		−	1.96482i	0.186731	−	0.982411i	$-0.440211\pi$
$402$	−364.245	−	223.406i	−0.906083	−	0.555735i
$403$	−851.879			−2.11384
$404$			136.498i			0.337865i
$405$	146.364	+	106.689i	0.361392	+	0.263430i
$406$	112.993			0.278307
$407$		−	1256.24i		−	3.08658i
$408$	0.128116	−	0.208884i	0.000314010	−	0.000511970i
$409$	−9.78449			−0.0239230			−0.0119615	−	0.999928i	$-0.503808\pi$
$409$	−9.78449			−0.0239230			−0.0119615	+	0.999928i	$0.503808\pi$
$410$			58.9900i			0.143878i
$411$	−315.171	−	193.307i	−0.766840	−	0.470332i
$412$	122.283			0.296803
$413$			120.672i			0.292185i
$414$	27.6695	+	54.4095i	0.0668346	+	0.131424i
$415$	47.0371			0.113342
$416$		−	117.035i		−	0.281333i
$417$	−343.770	+	560.491i	−0.824389	+	1.34410i
$418$	148.312			0.354813
$419$		−	542.817i		−	1.29551i	−0.761850	−	0.647753i	$-0.775709\pi$
$419$		−	542.817i		−	1.29551i	0.761850	−	0.647753i	$-0.224291\pi$
$420$	−20.2186	−	12.4008i	−0.0481396	−	0.0295258i
$421$	−675.751			−1.60511			−0.802555	−	0.596578i	$-0.796527\pi$
$421$	−675.751			−1.60511			−0.802555	+	0.596578i	$0.796527\pi$
$422$			442.138i			1.04772i
$423$	414.357	−	210.718i	0.979566	−	0.498151i
$424$	61.5718			0.145216
$425$		−	0.144393i		−	0.000339748i
$426$	−124.788	+	203.457i	−0.292930	+	0.477600i
$427$	−2.44015			−0.00571463
$428$			292.685i			0.683843i
$429$	−937.659	−	575.102i	−2.18569	−	1.34056i
$430$	191.065			0.444337
$431$			357.378i			0.829184i	0.910007	+	0.414592i	$0.136076\pi$
$431$			357.378i			0.829184i	−0.910007	+	0.414592i	$0.863924\pi$
$432$	−107.657	−	8.59969i	−0.249206	−	0.0199067i
$433$	−59.5104			−0.137437			−0.0687187	−	0.997636i	$-0.521891\pi$
$433$	−59.5104			−0.137437			−0.0687187	+	0.997636i	$0.521891\pi$
$434$		−	102.945i		−	0.237201i
$435$	158.508	−	258.435i	0.364386	−	0.594102i
$436$	−97.7836			−0.224274
$437$		−	28.3793i		−	0.0649412i
$438$	−267.568	−	164.110i	−0.610886	−	0.374679i
$439$	292.816			0.667006			0.333503	−	0.942749i	$-0.391769\pi$
$439$	292.816			0.667006			0.333503	+	0.942749i	$0.391769\pi$
$440$		−	112.086i		−	0.254742i
$441$	187.152	+	368.017i	0.424382	+	0.834506i
$442$	0.844948			0.00191165
$443$		−	70.1379i		−	0.158325i	−0.996862	−	0.0791624i	$-0.974775\pi$
$443$		−	70.1379i		−	0.158325i	0.996862	−	0.0791624i	$-0.0252246\pi$
$444$	222.363	−	362.545i	0.500817	−	0.816543i
$445$	−63.0746			−0.141741
$446$			103.690i			0.232490i
$447$	−216.366	−	132.706i	−0.484040	−	0.296880i
$448$	14.1431			0.0315693
$449$			751.278i			1.67323i	0.547795	+	0.836613i	$0.315468\pi$
$449$			751.278i			1.67323i	−0.547795	+	0.836613i	$0.684532\pi$
$450$	−56.7259	+	28.8475i	−0.126057	+	0.0641055i
$451$	−330.599			−0.733035
$452$		−	254.979i		−	0.564112i
$453$	−45.8188	+	74.7040i	−0.101145	+	0.164910i
$454$	−76.9453			−0.169483
$455$		−	81.7857i		−	0.179749i
$456$	42.8022	+	26.2522i	0.0938645	+	0.0575707i
$457$	−296.898			−0.649666			−0.324833	−	0.945771i	$-0.605308\pi$
$457$	−296.898			−0.649666			−0.324833	+	0.945771i	$0.605308\pi$
$458$		−	147.892i		−	0.322909i
$459$	0.0620868	−	0.777247i	0.000135265	−	0.00169335i
$460$	−21.4476			−0.0466252
$461$			44.6191i			0.0967875i	0.998828	+	0.0483938i	$0.0154102\pi$
$461$			44.6191i			0.0967875i	−0.998828	+	0.0483938i	$0.984590\pi$
$462$	69.4983	−	113.312i	0.150429	−	0.245263i
$463$	−652.545			−1.40938			−0.704692	−	0.709513i	$-0.748915\pi$
$463$	−652.545			−1.40938			−0.704692	+	0.709513i	$0.748915\pi$
$464$			180.777i			0.389605i
$465$	−235.454	−	144.413i	−0.506354	−	0.310566i
$466$	103.167			0.221389
$467$		−	614.730i		−	1.31634i	−0.752870	−	0.658169i	$-0.771331\pi$
$467$		−	614.730i		−	1.31634i	0.752870	−	0.658169i	$-0.228669\pi$
$468$	−168.808	−	331.944i	−0.360700	−	0.709282i
$469$	178.053			0.379644
$470$			163.335i			0.347520i
$471$	422.319	−	688.559i	0.896644	−	1.46191i
$472$	−193.063			−0.409032
$473$			1070.79i			2.26383i
$474$	16.1670	+	9.91583i	0.0341076	+	0.0209195i
$475$	29.5875			0.0622895
$476$			0.102108i			0.000214512i
$477$	174.636	−	88.8095i	0.366112	−	0.186184i
$478$	−567.471			−1.18718
$479$			633.184i			1.32189i	0.750436	+	0.660944i	$0.229844\pi$
$479$			633.184i			1.32189i	−0.750436	+	0.660944i	$0.770156\pi$
$480$	19.8401	−	32.3477i	0.0413335	−	0.0673910i
$481$	1466.52			3.04890
$482$		−	106.365i		−	0.220675i
$483$	−21.6820	−	13.2984i	−0.0448904	−	0.0275330i
$484$	386.168			0.797868
$485$		−	129.034i		−	0.266050i
$486$	−317.751	+	130.891i	−0.653809	+	0.269322i
$487$	−169.555			−0.348163			−0.174081	−	0.984731i	$-0.555696\pi$
$487$	−169.555			−0.348163			−0.174081	+	0.984731i	$0.555696\pi$
$488$		−	3.90398i		−	0.00799996i
$489$	−464.574	+	757.451i	−0.950049	+	1.54898i
$490$	−145.068			−0.296058
$491$			694.164i			1.41378i	0.707325	+	0.706888i	$0.249902\pi$
$491$			694.164i			1.41378i	−0.707325	+	0.706888i	$0.750098\pi$
$492$	−95.4094	−	58.5182i	−0.193922	−	0.118939i
$493$	−1.30514			−0.00264735
$494$			173.138i			0.350482i
$495$	−161.671	−	317.910i	−0.326607	−	0.642242i
$496$	164.702			0.332060
$497$		−	99.4555i		−	0.200112i
$498$	−46.6609	+	76.0770i	−0.0936966	+	0.152765i
$499$	155.122			0.310865			0.155433	−	0.987847i	$-0.450323\pi$
$499$	155.122			0.310865			0.155433	+	0.987847i	$0.450323\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	−357.965	−	219.554i	−0.714502	−	0.438231i
$502$	134.756			0.268439
$503$			460.543i			0.915593i	0.889057	+	0.457796i	$0.151361\pi$
$503$			460.543i			0.915593i	−0.889057	+	0.457796i	$0.848639\pi$
$504$	40.1138	−	20.3996i	0.0795910	−	0.0404754i
$505$	−152.609			−0.302196
$506$		−	120.199i		−	0.237548i
$507$	406.293	−	662.429i	0.801367	−	1.30657i
$508$	461.013			0.907506
$509$			720.090i			1.41472i	0.706856	+	0.707358i	$0.250113\pi$
$509$			720.090i			1.41472i	−0.706856	+	0.707358i	$0.749887\pi$
$510$	0.233539	+	0.143238i	0.000457919	+	0.000280859i
$511$	130.794			0.255958
$512$			22.6274i			0.0441942i
$513$	159.265	+	12.7222i	0.310458	+	0.0247995i
$514$	−155.626			−0.302775
$515$			136.717i			0.265469i
$516$	−189.537	+	309.026i	−0.367320	+	0.598887i
$517$	−915.379			−1.77056
$518$			177.222i			0.342127i
$519$	−5.72226	−	3.50968i	−0.0110255	−	0.00676238i
$520$	130.849			0.251632
$521$		−	139.098i		−	0.266983i	−0.991050	−	0.133491i	$-0.957381\pi$
$521$		−	139.098i		−	0.266983i	0.991050	−	0.133491i	$-0.0426188\pi$
$522$	260.748	+	512.736i	0.499517	+	0.982252i
$523$	−665.486			−1.27244			−0.636220	−	0.771508i	$-0.719503\pi$
$523$	−665.486			−1.27244			−0.636220	+	0.771508i	$0.719503\pi$
$524$			130.546i			0.249134i
$525$	13.8646	−	22.6051i	0.0264087	−	0.0430573i
$526$	526.914			1.00174
$527$			1.18909i			0.00225634i
$528$	181.287	+	111.190i	0.343346	+	0.210587i
$529$	−23.0000			−0.0434783
$530$			68.8393i			0.129886i
$531$	−547.583	+	278.469i	−1.03123	+	0.524424i
$532$	−20.9229			−0.0393287
$533$		−	385.938i		−	0.724086i
$534$	62.5702	−	102.016i	0.117173	−	0.191041i
$535$	−327.232			−0.611648
$536$			284.866i			0.531467i
$537$	220.048	+	134.963i	0.409772	+	0.251329i
$538$	−86.3014			−0.160412
$539$		−	813.009i		−	1.50836i
$540$	9.61475	−	120.364i	0.0178051	−	0.222897i
$541$	673.074			1.24413			0.622065	−	0.782966i	$-0.286294\pi$
$541$	673.074			1.24413			0.622065	+	0.782966i	$0.286294\pi$
$542$			691.560i			1.27594i
$543$	68.3564	−	111.450i	0.125887	−	0.205248i
$544$	−0.163362			−0.000300298
$545$		−	109.325i		−	0.200597i
$546$	132.279	+	81.1316i	0.242269	+	0.148593i
$547$	−298.530			−0.545759			−0.272880	−	0.962048i	$-0.587976\pi$
$547$	−298.530			−0.545759			−0.272880	+	0.962048i	$0.587976\pi$
$548$			246.487i			0.449793i
$549$	−5.63100	−	11.0728i	−0.0102568	−	0.0201691i
$550$	125.316			0.227848
$551$		−	267.437i		−	0.485366i
$552$	21.2761	−	34.6890i	0.0385436	−	0.0628424i
$553$	−7.90286			−0.0142909
$554$			452.191i			0.816230i
$555$	405.338	+	248.609i	0.730338	+	0.447944i
$556$	438.344			0.788389
$557$			152.164i			0.273185i	0.990627	+	0.136592i	$0.0436150\pi$
$557$			152.164i			0.273185i	−0.990627	+	0.136592i	$0.956385\pi$
$558$	467.143	−	237.562i	0.837174	−	0.425738i
$559$	−1250.03			−2.23619
$560$			15.8124i			0.0282365i
$561$	−0.802753	+	1.30883i	−0.00143093	+	0.00233302i
$562$	−345.085			−0.614031
$563$			216.421i			0.384407i	0.981355	+	0.192203i	$0.0615633\pi$
$563$			216.421i			0.384407i	−0.981355	+	0.192203i	$0.938437\pi$
$564$	−264.175	−	162.028i	−0.468395	−	0.287284i
$565$	285.075			0.504557
$566$		−	441.992i		−	0.780905i
$567$	84.3508	−	115.718i	0.148767	−	0.204089i
$568$	159.118			0.280138
$569$		−	168.117i		−	0.295460i	−0.989028	−	0.147730i	$-0.952803\pi$
$569$		−	168.117i		−	0.295460i	0.989028	−	0.147730i	$-0.0471966\pi$
$570$	−29.3509	+	47.8543i	−0.0514928	+	0.0839550i
$571$	17.0074			0.0297853			0.0148926	−	0.999889i	$-0.495259\pi$
$571$	17.0074			0.0297853			0.0148926	+	0.999889i	$0.495259\pi$
$572$			733.317i			1.28202i
$573$	498.785	+	305.924i	0.870480	+	0.533899i
$574$	46.6387			0.0812521
$575$		−	23.9792i		−	0.0417029i
$576$	32.6372	+	64.1780i	0.0566618	+	0.111420i
$577$	955.512			1.65600			0.828000	−	0.560728i	$-0.189478\pi$
$577$	955.512			1.65600			0.828000	+	0.560728i	$0.189478\pi$
$578$			408.707i			0.707105i
$579$	518.530	−	845.422i	0.895560	−	1.46014i
$580$	−202.115			−0.348473
$581$		−	37.1885i		−	0.0640077i
$582$	208.698	+	128.002i	0.358588	+	0.219935i
$583$	−385.798			−0.661745
$584$			209.257i			0.358318i
$585$	371.125	−	188.733i	0.634401	−	0.322620i
$586$	621.503			1.06059
$587$			727.196i			1.23883i	0.785062	+	0.619417i	$0.212631\pi$
$587$			727.196i			1.23883i	−0.785062	+	0.619417i	$0.787369\pi$
$588$	143.908	−	234.631i	0.244742	−	0.399032i
$589$	−243.656			−0.413677
$590$		−	215.851i		−	0.365849i
$591$	−234.506	−	143.831i	−0.396795	−	0.243369i
$592$	−283.536			−0.478947
$593$			335.046i			0.565002i	0.959267	+	0.282501i	$0.0911641\pi$
$593$			335.046i			0.565002i	−0.959267	+	0.282501i	$0.908836\pi$
$594$	674.560	+	53.8841i	1.13562	+	0.0907140i
$595$	−0.114160			−0.000191866
$596$			169.214i			0.283916i
$597$	9.69699	−	15.8102i	0.0162429	−	0.0264827i
$598$	140.319			0.234648
$599$		−	228.357i		−	0.381230i	−0.981665	−	0.190615i	$-0.938952\pi$
$599$		−	228.357i		−	0.381230i	0.981665	−	0.190615i	$-0.0610482\pi$
$600$	36.1658	+	22.1819i	0.0602763	+	0.0369698i
$601$	710.722			1.18257			0.591283	−	0.806464i	$-0.298622\pi$
$601$	710.722			1.18257			0.591283	+	0.806464i	$0.298622\pi$
$602$		−	151.060i		−	0.250930i
$603$	410.884	+	807.964i	0.681399	+	1.33991i
$604$	58.4239			0.0967284
$605$			431.749i			0.713635i
$606$	151.388	−	246.827i	0.249816	−	0.407305i
$607$	690.106			1.13691			0.568456	−	0.822713i	$-0.307541\pi$
$607$	690.106			1.13691			0.568456	+	0.822713i	$0.307541\pi$
$608$		−	33.4744i		−	0.0550566i
$609$	−204.324	−	125.320i	−0.335507	−	0.205779i
$610$	4.36478			0.00715538
$611$		−	1068.61i		−	1.74894i
$612$	−0.463343	+	0.235629i	−0.000757096	+	0.000385015i
$613$	−812.071			−1.32475			−0.662374	−	0.749173i	$-0.730451\pi$
$613$	−812.071			−1.32475			−0.662374	+	0.749173i	$0.730451\pi$
$614$		−	208.014i		−	0.338785i
$615$	65.4253	−	106.671i	0.106383	−	0.173449i
$616$	−88.6178			−0.143860
$617$			1026.40i			1.66354i	0.555120	+	0.831770i	$0.312672\pi$
$617$			1026.40i			1.66354i	−0.555120	+	0.831770i	$0.687328\pi$
$618$	−221.123	−	135.623i	−0.357804	−	0.219455i
$619$	−682.233			−1.10215			−0.551077	−	0.834455i	$-0.685783\pi$
$619$	−682.233			−1.10215			−0.551077	+	0.834455i	$0.685783\pi$
$620$			184.142i			0.297004i
$621$	10.3107	−	129.076i	0.0166033	−	0.207852i
$622$	−850.809			−1.36786
$623$			49.8681i			0.0800451i
$624$	−129.802	+	211.632i	−0.208016	+	0.339154i
$625$	25.0000			0.0400000
$626$		−	125.095i		−	0.199833i
$627$	−268.191	−	164.492i	−0.427737	−	0.262347i
$628$	−538.503			−0.857488
$629$		−	2.04703i		−	0.00325443i
$630$	22.8074	+	44.8486i	0.0362023	+	0.0711883i
$631$	19.9854			0.0316726			0.0158363	−	0.999875i	$-0.494959\pi$
$631$	19.9854			0.0316726			0.0158363	+	0.999875i	$0.494959\pi$
$632$		−	12.6438i		−	0.0200059i
$633$	490.373	−	799.514i	0.774680	−	1.26306i
$634$	−616.511			−0.972415
$635$			515.428i			0.811698i
$636$	−111.340	−	68.2888i	−0.175062	−	0.107372i
$637$	949.098			1.48995
$638$		−	1132.71i		−	1.77541i
$639$	451.306	−	229.508i	0.706270	−	0.359168i
$640$	−25.2982			−0.0395285
$641$			78.5394i			0.122526i	0.998122	+	0.0612632i	$0.0195129\pi$
$641$			78.5394i			0.122526i	−0.998122	+	0.0612632i	$0.980487\pi$
$642$	324.615	−	529.259i	0.505630	−	0.824391i
$643$	947.424			1.47344			0.736722	−	0.676196i	$-0.236373\pi$
$643$	947.424			1.47344			0.736722	+	0.676196i	$0.236373\pi$
$644$			16.9569i			0.0263306i
$645$	−345.501	−	211.909i	−0.535661	−	0.328541i
$646$	0.241674			0.000374108
$647$		−	405.423i		−	0.626619i	−0.949651	−	0.313310i	$-0.898562\pi$
$647$		−	405.423i		−	0.626619i	0.949651	−	0.313310i	$-0.101438\pi$
$648$	185.137	+	134.952i	0.285706	+	0.208260i
$649$	1209.70			1.86394
$650$			146.293i			0.225066i
$651$	−114.176	+	186.155i	−0.175386	+	0.285953i
$652$	592.382			0.908561
$653$			953.530i			1.46023i	0.683325	+	0.730115i	$0.260533\pi$
$653$			953.530i			1.46023i	−0.683325	+	0.730115i	$0.739467\pi$
$654$	176.821	+	108.451i	0.270369	+	0.165827i
$655$	−145.955			−0.222832
$656$			74.6170i			0.113746i
$657$	301.828	+	593.515i	0.459403	+	0.903372i
$658$	129.136			0.196255
$659$		−	784.567i		−	1.19054i	−0.803525	−	0.595271i	$-0.797045\pi$
$659$		−	784.567i		−	1.19054i	0.803525	−	0.595271i	$-0.202955\pi$
$660$	−124.314	+	202.685i	−0.188355	+	0.307098i
$661$	−525.618			−0.795186			−0.397593	−	0.917562i	$-0.630154\pi$
$661$	−525.618			−0.795186			−0.397593	+	0.917562i	$0.630154\pi$
$662$			431.621i			0.651995i
$663$	−1.52791	−	0.937126i	−0.00230454	−	0.00141346i
$664$	59.4977			0.0896050
$665$		−	23.3925i		−	0.0351767i
$666$	−804.192	+	408.966i	−1.20750	+	0.614062i
$667$	−216.744			−0.324953
$668$			279.955i			0.419094i
$669$	115.002	−	187.502i	0.171902	−	0.280273i
$670$	−318.490			−0.475358
$671$			24.4616i			0.0364555i
$672$	−25.5747	−	15.6860i	−0.0380577	−	0.0233422i
$673$	124.971			0.185693			0.0928465	−	0.995680i	$-0.470403\pi$
$673$	124.971			0.185693			0.0928465	+	0.995680i	$0.470403\pi$
$674$			314.163i			0.466117i
$675$	134.571	+	10.7496i	0.199365	+	0.0159254i
$676$	−518.068			−0.766372
$677$			867.241i			1.28101i	0.767956	+	0.640503i	$0.221274\pi$
$677$			867.241i			1.28101i	−0.767956	+	0.640503i	$0.778726\pi$
$678$	−282.795	+	461.075i	−0.417102	+	0.680052i
$679$	−102.017			−0.150246
$680$		−	0.182644i		−	0.000268595i
$681$	139.139	+	85.3395i	0.204316	+	0.125315i
$682$	−1031.99			−1.51319
$683$			765.086i			1.12018i	0.828430	+	0.560092i	$0.189234\pi$
$683$			765.086i			1.12018i	−0.828430	+	0.560092i	$0.810766\pi$
$684$	−48.2827	−	94.9433i	−0.0705887	−	0.138806i
$685$	−275.581			−0.402307
$686$			237.202i			0.345775i
$687$	−164.026	+	267.432i	−0.238757	+	0.389275i
$688$	241.680			0.351280
$689$		−	450.376i		−	0.653667i
$690$	38.7835	+	23.7874i	0.0562080	+	0.0344745i
$691$	896.181			1.29693			0.648467	−	0.761243i	$-0.275411\pi$
$691$	896.181			1.29693			0.648467	+	0.761243i	$0.275411\pi$
$692$			4.47522i			0.00646708i
$693$	−251.346	+	127.820i	−0.362693	+	0.184445i
$694$	929.113			1.33878
$695$			490.084i			0.705157i
$696$	200.498	−	326.897i	0.288072	−	0.469679i
$697$	−0.538709			−0.000772897
$698$		−	579.191i		−	0.829787i
$699$	−186.557	−	114.422i	−0.266891	−	0.163694i
$700$	−17.6788			−0.0252555
$701$		−	90.6113i		−	0.129260i	−0.997909	−	0.0646300i	$-0.979413\pi$
$701$		−	90.6113i		−	0.129260i	0.997909	−	0.0646300i	$-0.0205867\pi$
$702$	−62.9038	+	787.475i	−0.0896066	+	1.12176i
$703$	419.456			0.596666
$704$		−	141.779i		−	0.201391i
$705$	181.153	−	295.356i	0.256955	−	0.418945i
$706$	352.751			0.499648
$707$			120.656i			0.170659i
$708$	349.114	+	214.125i	0.493099	+	0.302436i
$709$	−393.144			−0.554505			−0.277253	−	0.960797i	$-0.589424\pi$
$709$	−393.144			−0.554505			−0.277253	+	0.960797i	$0.589424\pi$
$710$			177.900i			0.250563i
$711$	−18.2370	−	35.8614i	−0.0256498	−	0.0504380i
$712$	−79.7838			−0.112056
$713$			197.471i			0.276958i
$714$	0.113247	−	0.184641i	0.000158609	−	0.000258600i
$715$	−819.874			−1.14668
$716$		−	172.093i		−	0.240353i
$717$	1026.15	+	629.379i	1.43118	+	0.877794i
$718$	−471.580			−0.656797
$719$		−	924.192i		−	1.28539i	−0.766124	−	0.642693i	$-0.777817\pi$
$719$		−	924.192i		−	1.28539i	0.766124	−	0.642693i	$-0.222183\pi$
$720$	−71.7532	+	36.4895i	−0.0996572	+	0.0506799i
$721$	108.091			0.149918
$722$		−	461.010i		−	0.638518i
$723$	−117.969	+	192.339i	−0.163166	+	0.266029i
$724$	−87.1618			−0.120389
$725$		−	225.971i		−	0.311684i
$726$	−698.304	−	428.296i	−0.961851	−	0.589940i
$727$	1152.02			1.58463			0.792314	−	0.610114i	$-0.208876\pi$
$727$	1152.02			1.58463			0.792314	+	0.610114i	$0.208876\pi$
$728$		−	103.452i		−	0.142104i
$729$	719.756	+	115.727i	0.987319	+	0.158748i
$730$	−233.957			−0.320489
$731$			1.74485i			0.00238693i
$732$	−4.32988	+	7.05953i	−0.00591513	+	0.00964416i
$733$	−624.542			−0.852035			−0.426017	−	0.904715i	$-0.640084\pi$
$733$	−624.542			−0.852035			−0.426017	+	0.904715i	$0.640084\pi$
$734$		−	203.756i		−	0.277596i
$735$	262.325	+	160.894i	0.356905	+	0.218904i
$736$	−27.1293			−0.0368605
$737$		−	1784.92i		−	2.42187i
$738$	107.626	+	211.636i	0.145834	+	0.286769i
$739$	−858.733			−1.16202			−0.581010	−	0.813896i	$-0.697342\pi$
$739$	−858.733			−1.16202			−0.581010	+	0.813896i	$0.697342\pi$
$740$		−	317.003i		−	0.428383i
$741$	192.026	−	313.084i	0.259144	−	0.422515i
$742$	54.4258			0.0733501
$743$		−	1376.70i		−	1.85289i	−0.376429	−	0.926445i	$-0.622848\pi$
$743$		−	1376.70i		−	1.85289i	0.376429	−	0.926445i	$-0.377152\pi$
$744$	−297.829	−	182.670i	−0.400308	−	0.245524i
$745$	−189.187			−0.253942
$746$		−	698.726i		−	0.936630i
$747$	168.753	−	85.8180i	0.225907	−	0.114883i
$748$	1.02360			0.00136845
$749$			258.716i			0.345415i
$750$	−24.8001	+	40.4346i	−0.0330668	+	0.0539128i
$751$	−841.206			−1.12011			−0.560057	−	0.828454i	$-0.689221\pi$
$751$	−841.206			−1.12011			−0.560057	+	0.828454i	$0.689221\pi$
$752$			206.604i			0.274739i
$753$	−243.679	−	149.457i	−0.323610	−	0.198483i
$754$	1322.32			1.75374
$755$			65.3199i			0.0865165i
$756$	−95.1625	−	7.60162i	−0.125876	−	0.0100551i
$757$	21.0572			0.0278167			0.0139083	−	0.999903i	$-0.495573\pi$
$757$	21.0572			0.0278167			0.0139083	+	0.999903i	$0.495573\pi$
$758$		−	754.713i		−	0.995663i
$759$	−133.312	+	217.355i	−0.175642	+	0.286370i
$760$	37.4255			0.0492441
$761$			209.778i			0.275661i	0.990456	+	0.137830i	$0.0440129\pi$
$761$			209.778i			0.275661i	−0.990456	+	0.137830i	$0.955987\pi$
$762$	−833.646	−	511.307i	−1.09402	−	0.671006i
$763$	−86.4350			−0.113283
$764$		−	390.086i		−	0.510584i
$765$	−0.263442	−	0.518033i	−0.000344368	−	0.000677167i
$766$	346.957			0.452946
$767$			1412.19i			1.84119i
$768$	25.0959	−	40.9169i	0.0326770	−	0.0532773i
$769$	−217.912			−0.283370			−0.141685	−	0.989912i	$-0.545252\pi$
$769$	−217.912			−0.283370			−0.141685	+	0.989912i	$0.545252\pi$
$770$		−	99.0778i		−	0.128672i
$771$	281.417	+	172.604i	0.365003	+	0.223870i
$772$	−661.181			−0.856452
$773$			1256.63i			1.62565i	0.582507	+	0.812826i	$0.302072\pi$
$773$			1256.63i			1.62565i	−0.582507	+	0.812826i	$0.697928\pi$
$774$	685.476	−	348.594i	0.885628	−	0.450379i
$775$	−205.878			−0.265648
$776$		−	163.217i		−	0.210331i
$777$	196.555	−	320.468i	0.252967	−	0.412443i
$778$	−168.808			−0.216977
$779$		−	110.387i		−	0.141703i
$780$	−236.612	−	145.123i	−0.303349	−	0.186055i
$781$	−997.007			−1.27658
$782$		−	0.195864i		−	0.000250466i
$783$	97.1640	−	1216.37i	0.124092	−	1.55347i
$784$	−183.498			−0.234054
$785$		−	602.064i		−	0.766961i
$786$	144.788

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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	690.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(1.56849 - 2.55731i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(3.61658 + 2.21819i) q^{6} -1.76788 q^{7} -2.82843i q^{8} +(-4.07965 - 8.02225i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(1.56849 - 2.55731i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(3.61658 + 2.21819i) q^{6} -1.76788 q^{7} -2.82843i q^{8} +(-4.07965 - 8.02225i) q^{9} +3.16228 q^{10} +17.7224i q^{11} +(-3.13699 + 5.11462i) q^{12} -20.6890 q^{13} -2.50016i q^{14} +(-5.71832 - 3.50726i) q^{15} +4.00000 q^{16} +0.0288786i q^{17} +(11.3452 - 5.76950i) q^{18} -5.91750 q^{19} +4.47214i q^{20} +(-2.77291 + 4.52102i) q^{21} -25.0633 q^{22} +4.79583i q^{23} +(-7.23316 - 4.43637i) q^{24} -5.00000 q^{25} -29.2586i q^{26} +(-26.9143 - 2.14992i) q^{27} +3.53576 q^{28} +45.1942i q^{29} +(4.96001 - 8.08692i) q^{30} +41.1755 q^{31} +5.65685i q^{32} +(45.3217 + 27.7975i) q^{33} -0.0408405 q^{34} +3.95310i q^{35} +(8.15930 + 16.0445i) q^{36} -70.8841 q^{37} -8.36861i q^{38} +(-32.4505 + 52.9081i) q^{39} -6.32456 q^{40} +18.6543i q^{41} +(-6.39369 - 3.92149i) q^{42} +60.4201 q^{43} -35.4448i q^{44} +(-17.9383 + 9.12238i) q^{45} -6.78233 q^{46} +51.6509i q^{47} +(6.27398 - 10.2292i) q^{48} -45.8746 q^{49} -7.07107i q^{50} +(0.0738515 + 0.0452959i) q^{51} +41.3780 q^{52} +21.7689i q^{53} +(3.04045 - 38.0625i) q^{54} +39.6285 q^{55} +5.00033i q^{56} +(-9.28156 + 15.1329i) q^{57} -63.9142 q^{58} -68.2581i q^{59} +(11.4366 + 7.01452i) q^{60} +1.38027 q^{61} +58.2310i q^{62} +(7.21234 + 14.1824i) q^{63} -8.00000 q^{64} +46.2620i q^{65} +(-39.3116 + 64.0945i) q^{66} -100.715 q^{67} -0.0577572i q^{68} +(12.2644 + 7.52223i) q^{69} -5.59053 q^{70} +56.2569i q^{71} +(-22.6903 + 11.5390i) q^{72} -73.9837 q^{73} -100.245i q^{74} +(-7.84247 + 12.7865i) q^{75} +11.8350 q^{76} -31.3311i q^{77} +(-74.8233 - 45.8920i) q^{78} +4.47024 q^{79} -8.94427i q^{80} +(-47.7129 + 65.4559i) q^{81} -26.3811 q^{82} +21.0356i q^{83} +(5.54583 - 9.04204i) q^{84} +0.0645745 q^{85} +85.4469i q^{86} +(115.575 + 70.8868i) q^{87} +50.1266 q^{88} -28.2078i q^{89} +(-12.9010 - 25.3686i) q^{90} +36.5757 q^{91} -9.59166i q^{92} +(64.5835 - 105.298i) q^{93} -73.0455 q^{94} +13.2319i q^{95} +(14.4663 + 8.87274i) q^{96} +57.7059 q^{97} -64.8765i q^{98} +(142.174 - 72.3013i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56q - 8q^{3} - 112q^{4} + 16q^{6} - 16q^{7} + O(q^{10}) \)	\( 56q - 8q^{3} - 112q^{4} + 16q^{6} - 16q^{7} + 16q^{12} + 80q^{13} - 40q^{15} + 224q^{16} - 32q^{18} - 64q^{19} + 56q^{21} - 96q^{22} - 32q^{24} - 280q^{25} + 40q^{27} + 32q^{28} - 80q^{31} + 32q^{33} + 192q^{34} + 240q^{37} - 56q^{39} - 144q^{43} - 32q^{48} + 72q^{49} - 24q^{51} - 160q^{52} + 16q^{54} - 16q^{57} + 80q^{60} + 112q^{61} - 64q^{63} - 448q^{64} + 160q^{66} + 832q^{67} + 64q^{72} - 608q^{73} + 40q^{75} + 128q^{76} - 320q^{78} + 48q^{79} - 32q^{81} - 448q^{82} - 112q^{84} + 240q^{85} + 200q^{87} + 192q^{88} + 80q^{91} - 232q^{93} + 160q^{94} + 64q^{96} - 448q^{97} + 464q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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