LMFDB - Embedded newform 690.3.c.a.91.3

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$18.8011382409$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.3
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.6

$q$-expansion

$f(q)$	$=$	$q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} +1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} +1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -2.20036i q^{11} -3.46410 q^{12} -1.82994 q^{13} -2.15284i q^{14} +3.87298i q^{15} +4.00000 q^{16} +23.2496i q^{17} -4.24264 q^{18} -2.85690i q^{19} -4.47214i q^{20} -2.63668i q^{21} +3.11178i q^{22} +(-14.5452 - 17.8168i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +2.58793 q^{26} -5.19615 q^{27} +3.04458i q^{28} -7.23088 q^{29} -5.47723i q^{30} +49.9104 q^{31} -5.65685 q^{32} +3.81114i q^{33} -32.8800i q^{34} +3.40394 q^{35} +6.00000 q^{36} -9.68006i q^{37} +4.04027i q^{38} +3.16955 q^{39} +6.32456i q^{40} -62.1103 q^{41} +3.72883i q^{42} +76.6308i q^{43} -4.40073i q^{44} -6.70820i q^{45} +(20.5700 + 25.1967i) q^{46} +17.1464 q^{47} -6.92820 q^{48} +46.6826 q^{49} +7.07107 q^{50} -40.2696i q^{51} -3.65988 q^{52} -33.9334i q^{53} +7.34847 q^{54} -4.92016 q^{55} -4.30569i q^{56} +4.94830i q^{57} +10.2260 q^{58} +35.2164 q^{59} +7.74597i q^{60} -8.54346i q^{61} -70.5840 q^{62} +4.56687i q^{63} +8.00000 q^{64} +4.09187i q^{65} -5.38977i q^{66} +78.9586i q^{67} +46.4993i q^{68} +(25.1930 + 30.8596i) q^{69} -4.81390 q^{70} +111.407 q^{71} -8.48528 q^{72} +134.217 q^{73} +13.6897i q^{74} +8.66025 q^{75} -5.71380i q^{76} +3.34959 q^{77} -4.48242 q^{78} +72.4039i q^{79} -8.94427i q^{80} +9.00000 q^{81} +87.8373 q^{82} +119.919i q^{83} -5.27337i q^{84} +51.9878 q^{85} -108.372i q^{86} +12.5243 q^{87} +6.22357i q^{88} -73.9324i q^{89} +9.48683i q^{90} -2.78570i q^{91} +(-29.0904 - 35.6335i) q^{92} -86.4474 q^{93} -24.2486 q^{94} -6.38822 q^{95} +9.79796 q^{96} +178.439i q^{97} -66.0192 q^{98} -6.60109i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32q + 64q^{4} + 96q^{9} + O(q^{10}) $	$ 32q + 64q^{4} + 96q^{9} - 48q^{13} + 128q^{16} - 80q^{23} - 160q^{25} + 120q^{29} + 248q^{31} - 120q^{35} + 192q^{36} - 48q^{39} + 72q^{41} + 160q^{46} + 400q^{47} - 344q^{49} - 96q^{52} - 256q^{58} + 120q^{59} + 160q^{62} + 256q^{64} + 192q^{69} + 104q^{71} + 16q^{73} + 240q^{77} + 192q^{78} + 288q^{81} + 64q^{82} - 120q^{85} + 144q^{87} - 160q^{92} - 192q^{93} + 96q^{94} - 160q^{95} + 64q^{98} + 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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	2.00000			0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$	2.44949			0.408248
$7$			1.52229i			0.217470i	0.994071	+	0.108735i	$0.0346800\pi$
$7$			1.52229i			0.217470i	−0.994071	+	0.108735i	$0.965320\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$		−	2.20036i		−	0.200033i	−0.994986	−	0.100016i	$-0.968110\pi$
$11$		−	2.20036i		−	0.200033i	0.994986	−	0.100016i	$-0.0318896\pi$
$12$	−3.46410			−0.288675
$13$	−1.82994			−0.140765			−0.0703824	−	0.997520i	$-0.522422\pi$
$13$	−1.82994			−0.140765			−0.0703824	+	0.997520i	$0.522422\pi$
$14$		−	2.15284i		−	0.153774i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$			23.2496i			1.36763i	0.729657	+	0.683813i	$0.239680\pi$
$17$			23.2496i			1.36763i	−0.729657	+	0.683813i	$0.760320\pi$
$18$	−4.24264			−0.235702
$19$		−	2.85690i		−	0.150363i	−0.997170	−	0.0751816i	$-0.976046\pi$
$19$		−	2.85690i		−	0.150363i	0.997170	−	0.0751816i	$-0.0239536\pi$
$20$		−	4.47214i		−	0.223607i
$21$		−	2.63668i		−	0.125556i
$22$			3.11178i			0.141445i
$23$	−14.5452	−	17.8168i	−0.632400	−	0.774642i
$23$	−14.5452	−	17.8168i	−0.632400	−	0.774642i
$24$	4.89898			0.204124
$25$	−5.00000			−0.200000
$26$	2.58793			0.0995357
$27$	−5.19615			−0.192450
$28$			3.04458i			0.108735i
$29$	−7.23088			−0.249341			−0.124670	−	0.992198i	$-0.539787\pi$
$29$	−7.23088			−0.249341			−0.124670	+	0.992198i	$0.539787\pi$
$30$		−	5.47723i		−	0.182574i
$31$	49.9104			1.61001			0.805007	−	0.593266i	$-0.202162\pi$
$31$	49.9104			1.61001			0.805007	+	0.593266i	$0.202162\pi$
$32$	−5.65685			−0.176777
$33$			3.81114i			0.115489i
$34$		−	32.8800i		−	0.967058i
$35$	3.40394			0.0972555
$36$	6.00000			0.166667
$37$		−	9.68006i		−	0.261623i	−0.991407	−	0.130812i	$-0.958242\pi$
$37$		−	9.68006i		−	0.261623i	0.991407	−	0.130812i	$-0.0417583\pi$
$38$			4.04027i			0.106323i
$39$	3.16955			0.0812706
$40$			6.32456i			0.158114i
$41$	−62.1103			−1.51489			−0.757443	−	0.652901i	$-0.773552\pi$
$41$	−62.1103			−1.51489			−0.757443	+	0.652901i	$0.773552\pi$
$42$			3.72883i			0.0887817i
$43$			76.6308i			1.78211i	0.453893	+	0.891056i	$0.350035\pi$
$43$			76.6308i			1.78211i	−0.453893	+	0.891056i	$0.649965\pi$
$44$		−	4.40073i		−	0.100016i
$45$		−	6.70820i		−	0.149071i
$46$	20.5700	+	25.1967i	0.447174	+	0.547755i
$47$	17.1464			0.364817			0.182408	−	0.983223i	$-0.441611\pi$
$47$	17.1464			0.364817			0.182408	+	0.983223i	$0.441611\pi$
$48$	−6.92820			−0.144338
$49$	46.6826			0.952707
$50$	7.07107			0.141421
$51$		−	40.2696i		−	0.789599i
$52$	−3.65988			−0.0703824
$53$		−	33.9334i		−	0.640253i	−0.947375	−	0.320127i	$-0.896275\pi$
$53$		−	33.9334i		−	0.640253i	0.947375	−	0.320127i	$-0.103725\pi$
$54$	7.34847			0.136083
$55$	−4.92016			−0.0894575
$56$		−	4.30569i		−	0.0768872i
$57$			4.94830i			0.0868122i
$58$	10.2260			0.176311
$59$	35.2164			0.596889			0.298444	−	0.954427i	$-0.403532\pi$
$59$	35.2164			0.596889			0.298444	+	0.954427i	$0.403532\pi$
$60$			7.74597i			0.129099i
$61$		−	8.54346i		−	0.140057i	−0.997545	−	0.0700284i	$-0.977691\pi$
$61$		−	8.54346i		−	0.140057i	0.997545	−	0.0700284i	$-0.0223090\pi$
$62$	−70.5840			−1.13845
$63$			4.56687i			0.0724900i
$64$	8.00000			0.125000
$65$			4.09187i			0.0629519i
$66$		−	5.38977i		−	0.0816631i
$67$			78.9586i			1.17849i	0.807956	+	0.589243i	$0.200574\pi$
$67$			78.9586i			1.17849i	−0.807956	+	0.589243i	$0.799426\pi$
$68$			46.4993i			0.683813i
$69$	25.1930	+	30.8596i	0.365116	+	0.447240i
$70$	−4.81390			−0.0687700
$71$	111.407			1.56911			0.784554	−	0.620061i	$-0.212892\pi$
$71$	111.407			1.56911			0.784554	+	0.620061i	$0.212892\pi$
$72$	−8.48528			−0.117851
$73$	134.217			1.83858			0.919291	−	0.393578i	$-0.128763\pi$
$73$	134.217			1.83858			0.919291	+	0.393578i	$0.128763\pi$
$74$			13.6897i			0.184996i
$75$	8.66025			0.115470
$76$		−	5.71380i		−	0.0751816i
$77$	3.34959			0.0435012
$78$	−4.48242			−0.0574670
$79$			72.4039i			0.916505i	0.888822	+	0.458252i	$0.151524\pi$
$79$			72.4039i			0.916505i	−0.888822	+	0.458252i	$0.848476\pi$
$80$		−	8.94427i		−	0.111803i
$81$	9.00000			0.111111
$82$	87.8373			1.07119
$83$			119.919i			1.44481i	0.691472	+	0.722403i	$0.256963\pi$
$83$			119.919i			1.44481i	−0.691472	+	0.722403i	$0.743037\pi$
$84$		−	5.27337i		−	0.0627782i
$85$	51.9878			0.611621
$86$		−	108.372i		−	1.26014i
$87$	12.5243			0.143957
$88$			6.22357i			0.0707223i
$89$		−	73.9324i		−	0.830701i	−0.909661	−	0.415351i	$-0.863659\pi$
$89$		−	73.9324i		−	0.830701i	0.909661	−	0.415351i	$-0.136341\pi$
$90$			9.48683i			0.105409i
$91$		−	2.78570i		−	0.0306121i
$92$	−29.0904	−	35.6335i	−0.316200	−	0.387321i
$93$	−86.4474			−0.929542
$94$	−24.2486			−0.257964
$95$	−6.38822			−0.0672445
$96$	9.79796			0.102062
$97$			178.439i			1.83958i	0.392408	+	0.919791i	$0.371642\pi$
$97$			178.439i			1.83958i	−0.392408	+	0.919791i	$0.628358\pi$
$98$	−66.0192			−0.673665
$99$		−	6.60109i		−	0.0666777i
$100$	−10.0000			−0.100000
$101$	147.269			1.45811			0.729053	−	0.684457i	$-0.239961\pi$
$101$	147.269			1.45811			0.729053	+	0.684457i	$0.239961\pi$
$102$			56.9498i			0.558331i
$103$			58.9552i			0.572380i	0.958173	+	0.286190i	$0.0923889\pi$
$103$			58.9552i			0.572380i	−0.958173	+	0.286190i	$0.907611\pi$
$104$	5.17586			0.0497679
$105$	−5.89580			−0.0561505
$106$			47.9891i			0.452727i
$107$			111.196i			1.03922i	0.854404	+	0.519609i	$0.173922\pi$
$107$			111.196i			1.03922i	−0.854404	+	0.519609i	$0.826078\pi$
$108$	−10.3923			−0.0962250
$109$			52.7876i			0.484290i	0.970240	+	0.242145i	$0.0778509\pi$
$109$			52.7876i			0.484290i	−0.970240	+	0.242145i	$0.922149\pi$
$110$	6.95816			0.0632560
$111$			16.7664i			0.151048i
$112$			6.08916i			0.0543675i
$113$		−	25.4121i		−	0.224886i	−0.993658	−	0.112443i	$-0.964132\pi$
$113$		−	25.4121i		−	0.224886i	0.993658	−	0.112443i	$-0.0358676\pi$
$114$		−	6.99795i		−	0.0613855i
$115$	−39.8395	+	32.5240i	−0.346431	+	0.282818i
$116$	−14.4618			−0.124670
$117$	−5.48983			−0.0469216
$118$	−49.8035			−0.422064
$119$	−35.3927			−0.297418
$120$		−	10.9545i		−	0.0912871i
$121$	116.158			0.959987
$122$			12.0823i			0.0990350i
$123$	107.578			0.874620
$124$	99.8208			0.805007
$125$			11.1803i			0.0894427i
$126$		−	6.45853i		−	0.0512582i
$127$	100.861			0.794177			0.397089	−	0.917780i	$-0.370021\pi$
$127$	100.861			0.794177			0.397089	+	0.917780i	$0.370021\pi$
$128$	−11.3137			−0.0883883
$129$		−	132.728i		−	1.02890i
$130$		−	5.78678i		−	0.0445137i
$131$	−28.6539			−0.218732			−0.109366	−	0.994002i	$-0.534882\pi$
$131$	−28.6539			−0.218732			−0.109366	+	0.994002i	$0.534882\pi$
$132$			7.62228i			0.0577445i
$133$	4.34903			0.0326995
$134$		−	111.664i		−	0.833316i
$135$			11.6190i			0.0860663i
$136$		−	65.7599i		−	0.483529i
$137$		−	159.048i		−	1.16094i	−0.814282	−	0.580469i	$-0.802869\pi$
$137$		−	159.048i		−	1.16094i	0.814282	−	0.580469i	$-0.197131\pi$
$138$	−35.6283	−	43.6420i	−0.258176	−	0.316246i
$139$	119.112			0.856918			0.428459	−	0.903561i	$-0.359057\pi$
$139$	119.112			0.856918			0.428459	+	0.903561i	$0.359057\pi$
$140$	6.80789			0.0486278
$141$	−29.6984			−0.210627
$142$	−157.553			−1.10953
$143$			4.02654i			0.0281576i
$144$	12.0000			0.0833333
$145$			16.1687i			0.111509i
$146$	−189.811			−1.30007
$147$	−80.8567			−0.550046
$148$		−	19.3601i		−	0.130812i
$149$			245.833i			1.64989i	0.565214	+	0.824945i	$0.308794\pi$
$149$			245.833i			1.64989i	−0.565214	+	0.824945i	$0.691206\pi$
$150$	−12.2474			−0.0816497
$151$	209.449			1.38708			0.693541	−	0.720417i	$-0.256050\pi$
$151$	209.449			1.38708			0.693541	+	0.720417i	$0.256050\pi$
$152$			8.08053i			0.0531614i
$153$			69.7489i			0.455875i
$154$	−4.73704			−0.0307600
$155$		−	111.603i		−	0.720020i
$156$	6.33911			0.0406353
$157$		−	211.566i		−	1.34755i	−0.738935	−	0.673777i	$-0.764671\pi$
$157$		−	211.566i		−	1.34755i	0.738935	−	0.673777i	$-0.235329\pi$
$158$		−	102.395i		−	0.648067i
$159$			58.7744i			0.369650i
$160$			12.6491i			0.0790569i
$161$	27.1223	−	22.1420i	0.168461	−	0.137528i
$162$	−12.7279			−0.0785674
$163$	−14.1494			−0.0868061			−0.0434030	−	0.999058i	$-0.513820\pi$
$163$	−14.1494			−0.0868061			−0.0434030	+	0.999058i	$0.513820\pi$
$164$	−124.221			−0.757443
$165$	8.52197			0.0516483
$166$		−	169.591i		−	1.02163i
$167$	52.0666			0.311776			0.155888	−	0.987775i	$-0.450176\pi$
$167$	52.0666			0.311776			0.155888	+	0.987775i	$0.450176\pi$
$168$			7.45767i			0.0443909i
$169$	−165.651			−0.980185
$170$	−73.5218			−0.432481
$171$		−	8.57070i		−	0.0501211i
$172$			153.262i			0.891056i
$173$	−219.256			−1.26738			−0.633688	−	0.773589i	$-0.718460\pi$
$173$	−219.256			−1.26738			−0.633688	+	0.773589i	$0.718460\pi$
$174$	−17.7120			−0.101793
$175$		−	7.61145i		−	0.0434940i
$176$		−	8.80145i		−	0.0500082i
$177$	−60.9966			−0.344614
$178$			104.556i			0.587395i
$179$	−84.3515			−0.471237			−0.235619	−	0.971846i	$-0.575712\pi$
$179$	−84.3515			−0.471237			−0.235619	+	0.971846i	$0.575712\pi$
$180$		−	13.4164i		−	0.0745356i
$181$		−	272.635i		−	1.50627i	−0.657867	−	0.753134i	$-0.728541\pi$
$181$		−	272.635i		−	1.50627i	0.657867	−	0.753134i	$-0.271459\pi$
$182$			3.93958i			0.0216460i
$183$			14.7977i			0.0808618i
$184$	41.1400	+	50.3934i	0.223587	+	0.273877i
$185$	−21.6453			−0.117001
$186$	122.255			0.657285
$187$	51.1577			0.273570
$188$	34.2928			0.182408
$189$		−	7.91005i		−	0.0418521i
$190$	9.03431			0.0475490
$191$		−	235.645i		−	1.23375i	−0.787063	−	0.616873i	$-0.788399\pi$
$191$		−	235.645i		−	1.23375i	0.787063	−	0.616873i	$-0.211601\pi$
$192$	−13.8564			−0.0721688
$193$	−340.126			−1.76231			−0.881155	−	0.472828i	$-0.843233\pi$
$193$	−340.126			−1.76231			−0.881155	+	0.472828i	$0.843233\pi$
$194$		−	252.352i		−	1.30078i
$195$		−	7.08734i		−	0.0363453i
$196$	93.3653			0.476353
$197$	−221.673			−1.12524			−0.562621	−	0.826715i	$-0.690207\pi$
$197$	−221.673			−1.12524			−0.562621	+	0.826715i	$0.690207\pi$
$198$			9.33535i			0.0471482i
$199$			198.065i			0.995303i	0.867377	+	0.497651i	$0.165804\pi$
$199$			198.065i			0.995303i	−0.867377	+	0.497651i	$0.834196\pi$
$200$	14.1421			0.0707107
$201$		−	136.760i		−	0.680400i
$202$	−208.269			−1.03104
$203$		−	11.0075i		−	0.0542241i
$204$		−	80.5391i		−	0.394800i
$205$			138.883i			0.677478i
$206$		−	83.3752i		−	0.404734i
$207$	−43.6356	−	53.4503i	−0.210800	−	0.258214i
$208$	−7.31977			−0.0351912
$209$	−6.28622			−0.0300776
$210$	8.33792			0.0397044
$211$	−105.845			−0.501636			−0.250818	−	0.968034i	$-0.580700\pi$
$211$	−105.845			−0.501636			−0.250818	+	0.968034i	$0.580700\pi$
$212$		−	67.8668i		−	0.320127i
$213$	−192.962			−0.905925
$214$		−	157.255i		−	0.734838i
$215$	171.352			0.796985
$216$	14.6969			0.0680414
$217$			75.9781i			0.350130i
$218$		−	74.6529i		−	0.342445i
$219$	−232.470			−1.06151
$220$	−9.84032			−0.0447287
$221$		−	42.5455i		−	0.192514i
$222$		−	23.7112i		−	0.106807i
$223$	230.988			1.03582			0.517910	−	0.855435i	$-0.326710\pi$
$223$	230.988			1.03582			0.517910	+	0.855435i	$0.326710\pi$
$224$		−	8.61137i		−	0.0384436i
$225$	−15.0000			−0.0666667
$226$			35.9382i			0.159019i
$227$			146.750i			0.646475i	0.946318	+	0.323237i	$0.104771\pi$
$227$			146.750i			0.646475i	−0.946318	+	0.323237i	$0.895229\pi$
$228$			9.89659i			0.0434061i
$229$			203.857i			0.890203i	0.895480	+	0.445102i	$0.146832\pi$
$229$			203.857i			0.890203i	−0.895480	+	0.445102i	$0.853168\pi$
$230$	56.3416	−	45.9959i	0.244963	−	0.199982i
$231$	−5.80166			−0.0251154
$232$	20.4520			0.0881553
$233$	296.931			1.27438			0.637191	−	0.770706i	$-0.280096\pi$
$233$	296.931			1.27438			0.637191	+	0.770706i	$0.280096\pi$
$234$	7.76379			0.0331786
$235$		−	38.3405i		−	0.163151i
$236$	70.4328			0.298444
$237$		−	125.407i		−	0.529144i
$238$	50.0528			0.210306
$239$	−266.458			−1.11488			−0.557442	−	0.830216i	$-0.688217\pi$
$239$	−266.458			−1.11488			−0.557442	+	0.830216i	$0.688217\pi$
$240$			15.4919i			0.0645497i
$241$		−	327.615i		−	1.35940i	−0.733490	−	0.679700i	$-0.762110\pi$
$241$		−	327.615i		−	1.35940i	0.733490	−	0.679700i	$-0.237890\pi$
$242$	−164.273			−0.678813
$243$	−15.5885			−0.0641500
$244$		−	17.0869i		−	0.0700284i
$245$		−	104.386i		−	0.426063i
$246$	−152.139			−0.618450
$247$			5.22796i			0.0211658i
$248$	−141.168			−0.569226
$249$		−	207.706i		−	0.834160i
$250$		−	15.8114i		−	0.0632456i
$251$		−	5.89337i		−	0.0234796i	−0.999931	−	0.0117398i	$-0.996263\pi$
$251$		−	5.89337i		−	0.0234796i	0.999931	−	0.0117398i	$-0.00373698\pi$
$252$			9.13374i			0.0362450i
$253$	−39.2034	+	32.0047i	−0.154954	+	0.126501i
$254$	−142.638			−0.561568
$255$	−90.0455			−0.353120
$256$	16.0000			0.0625000
$257$	−45.2730			−0.176160			−0.0880798	−	0.996113i	$-0.528073\pi$
$257$	−45.2730			−0.176160			−0.0880798	+	0.996113i	$0.528073\pi$
$258$			187.706i			0.727544i
$259$	14.7359			0.0568952
$260$			8.18375i			0.0314760i
$261$	−21.6926			−0.0831136
$262$	40.5228			0.154667
$263$			211.508i			0.804215i	0.915593	+	0.402107i	$0.131722\pi$
$263$			211.508i			0.804215i	−0.915593	+	0.402107i	$0.868278\pi$
$264$		−	10.7795i		−	0.0408316i
$265$	−75.8774			−0.286330
$266$	−6.15046			−0.0231220
$267$			128.055i			0.479606i
$268$			157.917i			0.589243i
$269$	−133.758			−0.497242			−0.248621	−	0.968601i	$-0.579977\pi$
$269$	−133.758			−0.497242			−0.248621	+	0.968601i	$0.579977\pi$
$270$		−	16.4317i		−	0.0608581i
$271$	186.815			0.689353			0.344676	−	0.938722i	$-0.387989\pi$
$271$	186.815			0.689353			0.344676	+	0.938722i	$0.387989\pi$
$272$			92.9986i			0.341907i
$273$			4.82498i			0.0176739i
$274$			224.929i			0.820907i
$275$			11.0018i			0.0400066i
$276$	50.3860	+	61.7191i	0.182558	+	0.223620i
$277$	−128.144			−0.462615			−0.231308	−	0.972881i	$-0.574300\pi$
$277$	−128.144			−0.462615			−0.231308	+	0.972881i	$0.574300\pi$
$278$	−168.449			−0.605933
$279$	149.731			0.536671
$280$	−9.62781			−0.0343850
$281$			183.268i			0.652200i	0.945335	+	0.326100i	$0.105735\pi$
$281$			183.268i			0.652200i	−0.945335	+	0.326100i	$0.894265\pi$
$282$	41.9999			0.148936
$283$		−	436.731i		−	1.54322i	−0.636097	−	0.771609i	$-0.719452\pi$
$283$		−	436.731i		−	1.54322i	0.636097	−	0.771609i	$-0.280548\pi$
$284$	222.813			0.784554
$285$	11.0647			0.0388236
$286$		−	5.69438i		−	0.0199104i
$287$		−	94.5499i		−	0.329442i
$288$	−16.9706			−0.0589256
$289$	−251.546			−0.870401
$290$		−	22.8661i		−	0.0788485i
$291$		−	309.066i		−	1.06208i
$292$	268.433			0.919291
$293$			34.4017i			0.117412i	0.998275	+	0.0587060i	$0.0186974\pi$
$293$			34.4017i			0.117412i	−0.998275	+	0.0587060i	$0.981303\pi$
$294$	114.349			0.388941
$295$		−	78.7463i		−	0.266937i
$296$			27.3793i			0.0924978i
$297$			11.4334i			0.0384964i
$298$		−	347.661i		−	1.16665i
$299$	26.6169	+	32.6037i	0.0890196	+	0.109042i
$300$	17.3205			0.0577350
$301$	−116.654			−0.387556
$302$	−296.206			−0.980815
$303$	−255.077			−0.841838
$304$		−	11.4276i		−	0.0375908i
$305$	−19.1038			−0.0626353
$306$		−	98.6399i		−	0.322353i
$307$	−263.872			−0.859518			−0.429759	−	0.902944i	$-0.641401\pi$
$307$	−263.872			−0.859518			−0.429759	+	0.902944i	$0.641401\pi$
$308$	6.69918			0.0217506
$309$		−	102.113i		−	0.330464i
$310$			157.831i			0.509131i
$311$	12.9685			0.0416992			0.0208496	−	0.999783i	$-0.493363\pi$
$311$	12.9685			0.0416992			0.0208496	+	0.999783i	$0.493363\pi$
$312$	−8.96485			−0.0287335
$313$			71.7752i			0.229314i	0.993405	+	0.114657i	$0.0365769\pi$
$313$			71.7752i			0.229314i	−0.993405	+	0.114657i	$0.963423\pi$
$314$			299.199i			0.952864i
$315$	10.2118			0.0324185
$316$			144.808i			0.458252i
$317$	193.888			0.611633			0.305816	−	0.952090i	$-0.401071\pi$
$317$	193.888			0.611633			0.305816	+	0.952090i	$0.401071\pi$
$318$		−	83.1196i		−	0.261382i
$319$			15.9106i			0.0498764i
$320$		−	17.8885i		−	0.0559017i
$321$		−	192.598i		−	0.599993i
$322$	−38.3567	+	31.3135i	−0.119120	+	0.0972469i
$323$	66.4219			0.205641
$324$	18.0000			0.0555556
$325$	9.14971			0.0281530
$326$	20.0103			0.0613812
$327$		−	91.4308i		−	0.279605i
$328$	175.675			0.535593
$329$			26.1018i			0.0793367i
$330$	−12.0519			−0.0365209
$331$	393.325			1.18829			0.594147	−	0.804356i	$-0.297490\pi$
$331$	393.325			1.18829			0.594147	+	0.804356i	$0.297490\pi$
$332$			239.838i			0.722403i
$333$		−	29.0402i		−	0.0872078i
$334$	−73.6333			−0.220459
$335$	176.557			0.527035
$336$		−	10.5467i		−	0.0313891i
$337$			39.3542i			0.116778i	0.998294	+	0.0583891i	$0.0185964\pi$
$337$			39.3542i			0.116778i	−0.998294	+	0.0583891i	$0.981404\pi$
$338$	234.266			0.693096
$339$			44.0151i			0.129838i
$340$	103.976			0.305811
$341$		−	109.821i		−	0.322056i
$342$			12.1208i			0.0354409i
$343$			145.657i			0.424655i
$344$		−	216.745i		−	0.630072i
$345$	69.0041	−	56.3333i	0.200012	−	0.163285i
$346$	310.075			0.896170
$347$	309.497			0.891922			0.445961	−	0.895052i	$-0.352862\pi$
$347$	309.497			0.891922			0.445961	+	0.895052i	$0.352862\pi$
$348$	25.0485			0.0719785
$349$	−328.191			−0.940375			−0.470188	−	0.882567i	$-0.655814\pi$
$349$	−328.191			−0.940375			−0.470188	+	0.882567i	$0.655814\pi$
$350$			10.7642i			0.0307549i
$351$	9.50866			0.0270902
$352$			12.4471i			0.0353612i
$353$	−111.495			−0.315851			−0.157925	−	0.987451i	$-0.550481\pi$
$353$	−111.495			−0.315851			−0.157925	+	0.987451i	$0.550481\pi$
$354$	86.2623			0.243679
$355$		−	249.113i		−	0.701726i
$356$		−	147.865i		−	0.415351i
$357$	61.3020			0.171714
$358$	119.291			0.333215
$359$			516.911i			1.43986i	0.694045	+	0.719932i	$0.255827\pi$
$359$			516.911i			1.43986i	−0.694045	+	0.719932i	$0.744173\pi$
$360$			18.9737i			0.0527046i
$361$	352.838			0.977391
$362$			385.564i			1.06509i
$363$	−201.192			−0.554249
$364$		−	5.57140i		−	0.0153061i
$365$		−	300.117i		−	0.822239i
$366$		−	20.9271i		−	0.0571779i
$367$			32.4661i			0.0884635i	0.999021	+	0.0442318i	$0.0140840\pi$
$367$			32.4661i			0.0884635i	−0.999021	+	0.0442318i	$0.985916\pi$
$368$	−58.1808	−	71.2671i	−0.158100	−	0.193661i
$369$	−186.331			−0.504962
$370$	30.6110			0.0827325
$371$	51.6565			0.139236
$372$	−172.895			−0.464771
$373$		−	153.444i		−	0.411377i	−0.978617	−	0.205689i	$-0.934057\pi$
$373$		−	153.444i		−	0.411377i	0.978617	−	0.205689i	$-0.0659434\pi$
$374$	−72.3478			−0.193443
$375$		−	19.3649i		−	0.0516398i
$376$	−48.4973			−0.128982
$377$	13.2321			0.0350984
$378$			11.1865i			0.0295939i
$379$			597.537i			1.57662i	0.615281	+	0.788308i	$0.289042\pi$
$379$			597.537i			1.57662i	−0.615281	+	0.788308i	$0.710958\pi$
$380$	−12.7764			−0.0336222
$381$	−174.696			−0.458518
$382$			333.253i			0.872390i
$383$		−	309.730i		−	0.808694i	−0.914606	−	0.404347i	$-0.867499\pi$
$383$		−	309.730i		−	0.808694i	0.914606	−	0.404347i	$-0.132501\pi$
$384$	19.5959			0.0510310
$385$		−	7.48991i		−	0.0194543i
$386$	481.010			1.24614
$387$			229.892i			0.594037i
$388$			356.879i			0.919791i
$389$			295.042i			0.758461i	0.925302	+	0.379231i	$0.123811\pi$
$389$			295.042i			0.758461i	−0.925302	+	0.379231i	$0.876189\pi$
$390$			10.0230i			0.0257000i
$391$	414.234	−	338.171i	1.05942	−	0.864886i
$392$	−132.038			−0.336833
$393$	49.6301			0.126285
$394$	313.492			0.795666
$395$	161.900			0.409873
$396$		−	13.2022i		−	0.0333388i
$397$	45.4461			0.114474			0.0572368	−	0.998361i	$-0.481771\pi$
$397$	45.4461			0.114474			0.0572368	+	0.998361i	$0.481771\pi$
$398$		−	280.107i		−	0.703785i
$399$	−7.53274			−0.0188791
$400$	−20.0000			−0.0500000
$401$		−	51.4990i		−	0.128427i	−0.997936	−	0.0642133i	$-0.979546\pi$
$401$		−	51.4990i		−	0.128427i	0.997936	−	0.0642133i	$-0.0204538\pi$
$402$			193.408i			0.481115i
$403$	−91.3332			−0.226633
$404$	294.537			0.729053
$405$		−	20.1246i		−	0.0496904i
$406$			15.5669i			0.0383422i
$407$	−21.2996			−0.0523333
$408$			113.900i			0.279166i
$409$	−507.527			−1.24090			−0.620449	−	0.784247i	$-0.713050\pi$
$409$	−507.527			−1.24090			−0.620449	+	0.784247i	$0.713050\pi$
$410$		−	196.410i		−	0.479049i
$411$			275.480i			0.670268i
$412$			117.910i			0.286190i
$413$			53.6096i			0.129805i
$414$	61.7100	+	75.5902i	0.149058	+	0.182585i
$415$	268.147			0.646137
$416$	10.3517			0.0248839
$417$	−206.307			−0.494742
$418$	8.89005			0.0212681
$419$			471.677i			1.12572i	0.826552	+	0.562861i	$0.190299\pi$
$419$			471.677i			1.12572i	−0.826552	+	0.562861i	$0.809701\pi$
$420$	−11.7916			−0.0280753
$421$		−	760.247i		−	1.80581i	−0.429838	−	0.902906i	$-0.641430\pi$
$421$		−	760.247i		−	1.80581i	0.429838	−	0.902906i	$-0.358570\pi$
$422$	149.688			0.354711
$423$	51.4391			0.121606
$424$			95.9782i			0.226364i
$425$		−	116.248i		−	0.273525i
$426$	272.889			0.640585
$427$	13.0056			0.0304581
$428$			222.393i			0.519609i
$429$		−	6.97417i		−	0.0162568i
$430$	−242.328			−0.563553
$431$			601.309i			1.39515i	0.716512	+	0.697575i	$0.245738\pi$
$431$			601.309i			1.39515i	−0.716512	+	0.697575i	$0.754262\pi$
$432$	−20.7846			−0.0481125
$433$			466.438i			1.07722i	0.842554	+	0.538612i	$0.181051\pi$
$433$			466.438i			1.07722i	−0.842554	+	0.538612i	$0.818949\pi$
$434$		−	107.449i		−	0.247579i
$435$		−	28.0051i		−	0.0643795i
$436$			105.575i			0.242145i
$437$	−50.9007	+	41.5542i	−0.116478	+	0.0950896i
$438$	328.762			0.750598
$439$	−294.320			−0.670432			−0.335216	−	0.942141i	$-0.608809\pi$
$439$	−294.320			−0.670432			−0.335216	+	0.942141i	$0.608809\pi$
$440$	13.9163			0.0316280
$441$	140.048			0.317569
$442$			60.1684i			0.136128i
$443$	642.170			1.44959			0.724797	−	0.688962i	$-0.241934\pi$
$443$	642.170			1.44959			0.724797	+	0.688962i	$0.241934\pi$
$444$			33.5327i			0.0755241i
$445$	−165.318			−0.371501
$446$	−326.666			−0.732435
$447$		−	425.796i		−	0.952564i
$448$			12.1783i			0.0271837i
$449$	663.924			1.47867			0.739337	−	0.673336i	$-0.235139\pi$
$449$	663.924			1.47867			0.739337	+	0.673336i	$0.235139\pi$
$450$	21.2132			0.0471405
$451$			136.665i			0.303027i
$452$		−	50.8243i		−	0.112443i
$453$	−362.777			−0.800832
$454$		−	207.536i		−	0.457127i
$455$	−6.22902			−0.0136902
$456$		−	13.9959i		−	0.0306928i
$457$		−	22.1190i		−	0.0484005i	−0.999707	−	0.0242002i	$-0.992296\pi$
$457$		−	22.1190i		−	0.0484005i	0.999707	−	0.0242002i	$-0.00770393\pi$
$458$		−	288.297i		−	0.629469i
$459$		−	120.809i		−	0.263200i
$460$	−79.6790	+	65.0481i	−0.173215	+	0.141409i
$461$	−332.992			−0.722325			−0.361162	−	0.932503i	$-0.617620\pi$
$461$	−332.992			−0.722325			−0.361162	+	0.932503i	$0.617620\pi$
$462$	8.20479			0.0177593
$463$	−58.4021			−0.126139			−0.0630693	−	0.998009i	$-0.520089\pi$
$463$	−58.4021			−0.126139			−0.0630693	+	0.998009i	$0.520089\pi$
$464$	−28.9235			−0.0623352
$465$			193.302i			0.415704i
$466$	−419.924			−0.901124
$467$		−	609.288i		−	1.30469i	−0.757924	−	0.652343i	$-0.773786\pi$
$467$		−	609.288i		−	1.30469i	0.757924	−	0.652343i	$-0.226214\pi$
$468$	−10.9797			−0.0234608
$469$	−120.198			−0.256285
$470$			54.2216i			0.115365i
$471$			366.443i			0.778010i
$472$	−99.6071			−0.211032
$473$	168.616			0.356481
$474$			177.353i			0.374161i
$475$			14.2845i			0.0300726i
$476$	−70.7854			−0.148709
$477$		−	101.800i		−	0.213418i
$478$	376.828			0.788343
$479$		−	200.294i		−	0.418150i	−0.977900	−	0.209075i	$-0.932955\pi$
$479$		−	200.294i		−	0.418150i	0.977900	−	0.209075i	$-0.0670453\pi$
$480$		−	21.9089i		−	0.0456435i
$481$			17.7140i			0.0368273i
$482$			463.318i			0.961241i
$483$	−46.9772	+	38.3511i	−0.0972613	+	0.0794018i
$484$	232.317			0.479993
$485$	399.003			0.822686
$486$	22.0454			0.0453609
$487$	136.133			0.279534			0.139767	−	0.990184i	$-0.455365\pi$
$487$	136.133			0.279534			0.139767	+	0.990184i	$0.455365\pi$
$488$			24.1646i			0.0495175i
$489$	24.5075			0.0501175
$490$			147.623i			0.301272i
$491$	−217.319			−0.442604			−0.221302	−	0.975205i	$-0.571031\pi$
$491$	−217.319			−0.442604			−0.221302	+	0.975205i	$0.571031\pi$
$492$	215.156			0.437310
$493$		−	168.115i		−	0.341005i
$494$		−	7.39345i		−	0.0149665i
$495$	−14.7605			−0.0298192
$496$	199.642			0.402503
$497$			169.593i			0.341234i
$498$			293.740i			0.589840i
$499$	−620.672			−1.24383			−0.621916	−	0.783084i	$-0.713646\pi$
$499$	−620.672			−1.24383			−0.621916	+	0.783084i	$0.713646\pi$
$500$			22.3607i			0.0447214i
$501$	−90.1821			−0.180004
$502$			8.33449i			0.0166026i
$503$			165.943i			0.329906i	0.986301	+	0.164953i	$0.0527473\pi$
$503$			165.943i			0.329906i	−0.986301	+	0.164953i	$0.947253\pi$
$504$		−	12.9171i		−	0.0256291i
$505$		−	329.303i		−	0.652085i
$506$	55.4419	−	45.2615i	0.109569	−	0.0894496i
$507$	286.916			0.565910
$508$	201.721			0.397089
$509$	−376.623			−0.739928			−0.369964	−	0.929046i	$-0.620630\pi$
$509$	−376.623			−0.739928			−0.369964	+	0.929046i	$0.620630\pi$
$510$	127.344			0.249693
$511$			204.316i			0.399836i
$512$	−22.6274			−0.0441942
$513$			14.8449i			0.0289374i
$514$	64.0257			0.124564
$515$	131.828			0.255976
$516$		−	265.457i		−	0.514452i
$517$		−	37.7283i		−	0.0729754i
$518$	−20.8397			−0.0402310
$519$	379.762			0.731720
$520$		−	11.5736i		−	0.0222569i
$521$			675.726i			1.29698i	0.761223	+	0.648490i	$0.224599\pi$
$521$			675.726i			1.29698i	−0.761223	+	0.648490i	$0.775401\pi$
$522$	30.6780			0.0587702
$523$			53.9295i			0.103116i	0.998670	+	0.0515578i	$0.0164186\pi$
$523$			53.9295i			0.103116i	−0.998670	+	0.0515578i	$0.983581\pi$
$524$	−57.3079			−0.109366
$525$			13.1834i			0.0251113i
$526$		−	299.118i		−	0.568666i
$527$			1160.40i			2.20190i
$528$			15.2446i			0.0288723i
$529$	−105.875	+	518.297i	−0.200141	+	0.979767i
$530$	107.307			0.202466
$531$	105.649			0.198963
$532$	8.69806			0.0163497
$533$	113.658			0.213243
$534$		−	181.097i		−	0.339132i
$535$	248.643			0.464752
$536$		−	223.329i		−	0.416658i
$537$	146.101			0.272069
$538$	189.163			0.351604
$539$		−	102.719i		−	0.190573i
$540$			23.2379i			0.0430331i
$541$	32.6654			0.0603797			0.0301899	−	0.999544i	$-0.490389\pi$
$541$	32.6654			0.0603797			0.0301899	+	0.999544i	$0.490389\pi$
$542$	−264.196			−0.487446
$543$			472.217i			0.869645i
$544$		−	131.520i		−	0.241764i
$545$	118.037			0.216581
$546$		−	6.82355i		−	0.0124973i
$547$	125.227			0.228934			0.114467	−	0.993427i	$-0.463484\pi$
$547$	125.227			0.228934			0.114467	+	0.993427i	$0.463484\pi$
$548$		−	318.097i		−	0.580469i
$549$		−	25.6304i		−	0.0466856i
$550$		−	15.5589i		−	0.0282889i
$551$			20.6579i			0.0374917i
$552$	−71.2566	−	87.2840i	−0.129088	−	0.158123i
$553$	−110.220			−0.199312
$554$	181.224			0.327118
$555$	37.4907			0.0675508
$556$	238.223			0.428459
$557$			18.9083i			0.0339466i	0.999856	+	0.0169733i	$0.00540303\pi$
$557$			18.9083i			0.0339466i	−0.999856	+	0.0169733i	$0.994597\pi$
$558$	−211.752			−0.379484
$559$		−	140.230i		−	0.250859i
$560$	13.6158			0.0243139
$561$	−88.6077			−0.157946
$562$		−	259.181i		−	0.461175i
$563$		−	663.769i		−	1.17899i	−0.807774	−	0.589493i	$-0.799328\pi$
$563$		−	663.769i		−	1.17899i	0.807774	−	0.589493i	$-0.200672\pi$
$564$	−59.3968			−0.105313
$565$	−56.8233			−0.100572
$566$			617.630i			1.09122i
$567$			13.7006i			0.0241633i
$568$	−315.105			−0.554763
$569$		−	491.805i		−	0.864332i	−0.901794	−	0.432166i	$-0.857749\pi$
$569$		−	491.805i		−	0.864332i	0.901794	−	0.432166i	$-0.142251\pi$
$570$	−15.6479			−0.0274524
$571$		−	880.062i		−	1.54126i	−0.637280	−	0.770632i	$-0.719940\pi$
$571$		−	880.062i		−	1.54126i	0.637280	−	0.770632i	$-0.280060\pi$
$572$			8.05307i			0.0140788i
$573$			408.150i			0.712303i
$574$			133.714i			0.232951i
$575$	72.7260	+	89.0839i	0.126480	+	0.154928i
$576$	24.0000			0.0416667
$577$	−4.02697			−0.00697914			−0.00348957	−	0.999994i	$-0.501111\pi$
$577$	−4.02697			−0.00697914			−0.00348957	+	0.999994i	$0.501111\pi$
$578$	355.740			0.615467
$579$	589.115			1.01747
$580$			32.3375i			0.0557543i
$581$	−182.551			−0.314202
$582$			437.086i			0.751006i
$583$	−74.6658			−0.128072
$584$	−379.622			−0.650037
$585$			12.2756i			0.0209840i
$586$		−	48.6513i		−	0.0830228i
$587$	−313.115			−0.533415			−0.266708	−	0.963778i	$-0.585936\pi$
$587$	−313.115			−0.533415			−0.266708	+	0.963778i	$0.585936\pi$
$588$	−161.713			−0.275023
$589$		−	142.589i		−	0.242087i
$590$			111.364i			0.188753i
$591$	383.948			0.649659
$592$		−	38.7202i		−	0.0654058i
$593$	643.656			1.08542			0.542712	−	0.839919i	$-0.317398\pi$
$593$	643.656			1.08542			0.542712	+	0.839919i	$0.317398\pi$
$594$		−	16.1693i		−	0.0272210i
$595$			79.1405i			0.133009i
$596$			491.667i			0.824945i
$597$		−	343.059i		−	0.574638i
$598$	−37.6419	−	46.1085i	−0.0629464	−	0.0771046i
$599$	482.724			0.805884			0.402942	−	0.915226i	$-0.367988\pi$
$599$	482.724			0.805884			0.402942	+	0.915226i	$0.367988\pi$
$600$	−24.4949			−0.0408248
$601$	−510.476			−0.849377			−0.424689	−	0.905340i	$-0.639616\pi$
$601$	−510.476			−0.849377			−0.424689	+	0.905340i	$0.639616\pi$
$602$	164.974			0.274043
$603$			236.876i			0.392829i
$604$	418.899			0.693541
$605$		−	259.738i		−	0.429319i
$606$	360.733			0.595269
$607$	−719.184			−1.18482			−0.592409	−	0.805638i	$-0.701823\pi$
$607$	−719.184			−1.18482			−0.592409	+	0.805638i	$0.701823\pi$
$608$			16.1611i			0.0265807i
$609$			19.0655i			0.0313063i
$610$	27.0168			0.0442898
$611$	−31.3769			−0.0513533
$612$			139.498i			0.227938i
$613$		−	672.054i		−	1.09634i	−0.836368	−	0.548168i	$-0.815325\pi$
$613$		−	672.054i		−	1.09634i	0.836368	−	0.548168i	$-0.184675\pi$
$614$	373.171			0.607771
$615$		−	240.552i		−	0.391142i
$616$	−9.47407			−0.0153800
$617$		−	295.731i		−	0.479305i	−0.970859	−	0.239653i	$-0.922966\pi$
$617$		−	295.731i		−	0.479305i	0.970859	−	0.239653i	$-0.0770336\pi$
$618$			144.410i			0.233673i
$619$			823.442i			1.33028i	0.746720	+	0.665139i	$0.231628\pi$
$619$			823.442i			1.33028i	−0.746720	+	0.665139i	$0.768372\pi$
$620$		−	223.206i		−	0.360010i
$621$	75.5790	+	92.5787i	0.121705	+	0.149080i
$622$	−18.3402			−0.0294858
$623$	112.547			0.180653
$624$	12.6782			0.0203176
$625$	25.0000			0.0400000
$626$		−	101.505i		−	0.162149i
$627$	10.8880			0.0173653
$628$		−	423.132i		−	0.673777i
$629$	225.058			0.357803
$630$	−14.4417			−0.0229233
$631$		−	471.460i		−	0.747163i	−0.927597	−	0.373582i	$-0.878130\pi$
$631$		−	471.460i		−	0.747163i	0.927597	−	0.373582i	$-0.121870\pi$
$632$		−	204.789i		−	0.324033i
$633$	183.329			0.289620
$634$	−274.199			−0.432490
$635$		−	225.531i		−	0.355167i
$636$			117.549i			0.184825i
$637$	−85.4265			−0.134108
$638$		−	22.5009i		−	0.0352679i
$639$	334.220			0.523036
$640$			25.2982i			0.0395285i
$641$			1114.03i			1.73796i	0.494850	+	0.868979i	$0.335223\pi$
$641$			1114.03i			1.73796i	−0.494850	+	0.868979i	$0.664777\pi$
$642$			272.374i			0.424259i
$643$			182.160i			0.283298i	0.989917	+	0.141649i	$0.0452404\pi$
$643$			182.160i			0.283298i	−0.989917	+	0.141649i	$0.954760\pi$
$644$	54.2446	−	44.2840i	0.0842307	−	0.0687640i
$645$	−296.790			−0.460139
$646$	−93.9348			−0.145410
$647$	231.496			0.357799			0.178899	−	0.983867i	$-0.442746\pi$
$647$	231.496			0.357799			0.178899	+	0.983867i	$0.442746\pi$
$648$	−25.4558			−0.0392837
$649$		−	77.4889i		−	0.119397i
$650$	−12.9396			−0.0199071
$651$		−	131.598i		−	0.202147i
$652$	−28.2988			−0.0434030
$653$	961.708			1.47275			0.736377	−	0.676572i	$-0.236535\pi$
$653$	961.708			1.47275			0.736377	+	0.676572i	$0.236535\pi$
$654$			129.303i			0.197710i
$655$			64.0722i			0.0978201i
$656$	−248.441			−0.378721
$657$	402.650			0.612861
$658$		−	36.9135i		−	0.0560995i
$659$		−	482.196i		−	0.731709i	−0.930672	−	0.365854i	$-0.880777\pi$
$659$		−	482.196i		−	0.731709i	0.930672	−	0.365854i	$-0.119223\pi$
$660$	17.0439			0.0258241
$661$			514.781i			0.778792i	0.921070	+	0.389396i	$0.127316\pi$
$661$			514.781i			0.778792i	−0.921070	+	0.389396i	$0.872684\pi$
$662$	−556.246			−0.840251
$663$			73.6910i			0.111148i
$664$		−	339.182i		−	0.510816i
$665$		−	9.72473i		−	0.0146236i
$666$			41.0690i			0.0616652i
$667$	105.175	+	128.831i	0.157683	+	0.193150i
$668$	104.133			0.155888
$669$	−400.083			−0.598031
$670$	−249.689			−0.372670
$671$	−18.7987			−0.0280160
$672$			14.9153i			0.0221954i
$673$	−1100.79			−1.63565			−0.817824	−	0.575468i	$-0.804820\pi$
$673$	−1100.79			−1.63565			−0.817824	+	0.575468i	$0.804820\pi$
$674$		−	55.6553i		−	0.0825746i
$675$	25.9808			0.0384900
$676$	−331.303			−0.490093
$677$			155.504i			0.229695i	0.993383	+	0.114848i	$0.0366380\pi$
$677$			155.504i			0.229695i	−0.993383	+	0.114848i	$0.963362\pi$
$678$		−	62.2468i		−	0.0918094i
$679$	−271.637			−0.400054
$680$	−147.044			−0.216241
$681$		−	254.178i		−	0.373242i
$682$			155.310i			0.227728i
$683$	−1077.79			−1.57803			−0.789014	−	0.614376i	$-0.789408\pi$
$683$	−1077.79			−1.57803			−0.789014	+	0.614376i	$0.789408\pi$
$684$		−	17.1414i		−	0.0250605i
$685$	−355.643			−0.519187
$686$		−	205.990i		−	0.300276i
$687$		−	353.090i		−	0.513959i
$688$			306.523i			0.445528i
$689$			62.0962i			0.0901251i
$690$	−97.5865	+	79.6673i	−0.141430	+	0.115460i
$691$	−884.748			−1.28039			−0.640194	−	0.768214i	$-0.721146\pi$
$691$	−884.748			−1.28039			−0.640194	+	0.768214i	$0.721146\pi$
$692$	−438.512			−0.633688
$693$	10.0488			0.0145004
$694$	−437.695			−0.630684
$695$		−	266.342i		−	0.383225i
$696$	−35.4239			−0.0508965
$697$		−	1444.04i		−	2.07180i
$698$	464.132			0.664946
$699$	−514.299			−0.735765
$700$		−	15.2229i		−	0.0217470i
$701$			666.703i			0.951074i	0.879696	+	0.475537i	$0.157746\pi$
$701$			666.703i			0.951074i	−0.879696	+	0.475537i	$0.842254\pi$
$702$	−13.4473			−0.0191557
$703$	−27.6550			−0.0393385
$704$		−	17.6029i		−	0.0250041i
$705$			66.4076i			0.0941952i
$706$	157.678			0.223340
$707$			224.186i			0.317094i
$708$	−121.993			−0.172307
$709$		−	967.418i		−	1.36448i	−0.731127	−	0.682241i	$-0.761005\pi$
$709$		−	967.418i		−	1.36448i	0.731127	−	0.682241i	$-0.238995\pi$
$710$			352.299i			0.496195i
$711$			217.212i			0.305502i
$712$			209.112i			0.293697i
$713$	−725.957	−	889.242i	−1.01817	−	1.24718i
$714$	−86.6941			−0.121420
$715$	9.00361			0.0125925
$716$	−168.703			−0.235619
$717$	461.518			0.643679
$718$		−	731.023i		−	1.01814i
$719$	−216.266			−0.300787			−0.150393	−	0.988626i	$-0.548054\pi$
$719$	−216.266			−0.300787			−0.150393	+	0.988626i	$0.548054\pi$
$720$		−	26.8328i		−	0.0372678i
$721$	−89.7469			−0.124476
$722$	−498.988			−0.691120
$723$			567.447i			0.784850i
$724$		−	545.269i		−	0.753134i
$725$	36.1544			0.0498681
$726$	284.529			0.391913
$727$			113.587i			0.156240i	0.996944	+	0.0781200i	$0.0248917\pi$
$727$			113.587i			0.156240i	−0.996944	+	0.0781200i	$0.975108\pi$
$728$			7.87916i			0.0108230i
$729$	27.0000			0.0370370
$730$			424.430i			0.581411i
$731$	−1781.64			−2.43726
$732$			29.5954i			0.0404309i
$733$			104.468i			0.142522i	0.997458	+	0.0712608i	$0.0227023\pi$
$733$			104.468i			0.142522i	−0.997458	+	0.0712608i	$0.977298\pi$
$734$		−	45.9140i		−	0.0625531i
$735$			180.801i			0.245988i
$736$	82.2800	+	100.787i	0.111794	+	0.136939i
$737$	173.738			0.235736
$738$	263.512			0.357062
$739$	828.795			1.12151			0.560755	−	0.827982i	$-0.310511\pi$
$739$	828.795			1.12151			0.560755	+	0.827982i	$0.310511\pi$
$740$	−43.2905			−0.0585007
$741$		−	9.05510i		−	0.0122201i
$742$	−73.0533			−0.0984546
$743$		−	1067.78i		−	1.43712i	−0.695465	−	0.718560i	$-0.744802\pi$
$743$		−	1067.78i		−	1.43712i	0.695465	−	0.718560i	$-0.255198\pi$
$744$	244.510			0.328643
$745$	549.700			0.737853
$746$			217.002i			0.290888i
$747$			359.757i			0.481602i
$748$	102.315			0.136785
$749$	−169.273			−0.225999
$750$			27.3861i			0.0365148i
$751$			116.437i			0.155043i	0.996991	+	0.0775213i	$0.0247006\pi$
$751$			116.437i			0.155043i	−0.996991	+	0.0775213i	$0.975299\pi$
$752$	68.5855			0.0912042
$753$			10.2076i			0.0135559i
$754$	−18.7130			−0.0248183
$755$		−	468.343i		−	0.620322i
$756$		−	15.8201i		−	0.0209261i
$757$		−	893.537i		−	1.18037i	−0.807269	−	0.590183i	$-0.799056\pi$
$757$		−	893.537i		−	1.18037i	0.807269	−	0.590183i	$-0.200944\pi$
$758$		−	845.045i		−	1.11484i
$759$	67.9022	−	55.4338i	0.0894627	−	0.0730353i
$760$	18.0686			0.0237745
$761$	−80.0115			−0.105140			−0.0525700	−	0.998617i	$-0.516741\pi$
$761$	−80.0115			−0.105140			−0.0525700	+	0.998617i	$0.516741\pi$
$762$	247.057			0.324221
$763$	−80.3580			−0.105318
$764$		−	471.291i		−	0.616873i
$765$	155.963			0.203874
$766$			438.024i			0.571833i
$767$	−64.4440			−0.0840209
$768$	−27.7128			−0.0360844
$769$		−	890.267i		−	1.15769i	−0.815436	−	0.578847i	$-0.803503\pi$
$769$		−	890.267i		−	1.15769i	0.815436	−	0.578847i	$-0.196497\pi$
$770$			10.5923i			0.0137563i
$771$	78.4151			0.101706
$772$	−680.251			−0.881155
$773$			619.222i			0.801063i	0.916283	+	0.400531i	$0.131174\pi$
$773$			619.222i			0.801063i	−0.916283	+	0.400531i	$0.868826\pi$
$774$		−	325.117i		−	0.420048i
$775$	−249.552			−0.322003
$776$		−	504.703i		−	0.650391i
$777$	−25.5233			−0.0328485
$778$		−	417.252i		−	0.536313i
$779$			177.443i			0.227783i
$780$		−	14.1747i		−	0.0181727i
$781$		−	245.135i		−	0.313873i
$782$	−585.815	+	478.245i	−0.749124	+	0.611567i
$783$	37.5728			0.0479856
$784$	186.731			0.238177
$785$	−473.076			−0.602644
$786$	−70.1875			−0.0892971
$787$			375.043i			0.476548i	0.971198	+	0.238274i	$0.0765816\pi$
$787$			375.043i			0.476548i	−0.971198	+	0.238274i	$0.923418\pi$
$788$	−443.345			−0.562621
$789$		−	366.343i		−	0.464314i
$790$	−228.961			−0.289824
$791$	38.6846			0.0489060

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} +1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} +1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -2.20036i q^{11} -3.46410 q^{12} -1.82994 q^{13} -2.15284i q^{14} +3.87298i q^{15} +4.00000 q^{16} +23.2496i q^{17} -4.24264 q^{18} -2.85690i q^{19} -4.47214i q^{20} -2.63668i q^{21} +3.11178i q^{22} +(-14.5452 - 17.8168i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +2.58793 q^{26} -5.19615 q^{27} +3.04458i q^{28} -7.23088 q^{29} -5.47723i q^{30} +49.9104 q^{31} -5.65685 q^{32} +3.81114i q^{33} -32.8800i q^{34} +3.40394 q^{35} +6.00000 q^{36} -9.68006i q^{37} +4.04027i q^{38} +3.16955 q^{39} +6.32456i q^{40} -62.1103 q^{41} +3.72883i q^{42} +76.6308i q^{43} -4.40073i q^{44} -6.70820i q^{45} +(20.5700 + 25.1967i) q^{46} +17.1464 q^{47} -6.92820 q^{48} +46.6826 q^{49} +7.07107 q^{50} -40.2696i q^{51} -3.65988 q^{52} -33.9334i q^{53} +7.34847 q^{54} -4.92016 q^{55} -4.30569i q^{56} +4.94830i q^{57} +10.2260 q^{58} +35.2164 q^{59} +7.74597i q^{60} -8.54346i q^{61} -70.5840 q^{62} +4.56687i q^{63} +8.00000 q^{64} +4.09187i q^{65} -5.38977i q^{66} +78.9586i q^{67} +46.4993i q^{68} +(25.1930 + 30.8596i) q^{69} -4.81390 q^{70} +111.407 q^{71} -8.48528 q^{72} +134.217 q^{73} +13.6897i q^{74} +8.66025 q^{75} -5.71380i q^{76} +3.34959 q^{77} -4.48242 q^{78} +72.4039i q^{79} -8.94427i q^{80} +9.00000 q^{81} +87.8373 q^{82} +119.919i q^{83} -5.27337i q^{84} +51.9878 q^{85} -108.372i q^{86} +12.5243 q^{87} +6.22357i q^{88} -73.9324i q^{89} +9.48683i q^{90} -2.78570i q^{91} +(-29.0904 - 35.6335i) q^{92} -86.4474 q^{93} -24.2486 q^{94} -6.38822 q^{95} +9.79796 q^{96} +178.439i q^{97} -66.0192 q^{98} -6.60109i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32q + 64q^{4} + 96q^{9} + O(q^{10}) \)	\( 32q + 64q^{4} + 96q^{9} - 48q^{13} + 128q^{16} - 80q^{23} - 160q^{25} + 120q^{29} + 248q^{31} - 120q^{35} + 192q^{36} - 48q^{39} + 72q^{41} + 160q^{46} + 400q^{47} - 344q^{49} - 96q^{52} - 256q^{58} + 120q^{59} + 160q^{62} + 256q^{64} + 192q^{69} + 104q^{71} + 16q^{73} + 240q^{77} + 192q^{78} + 288q^{81} + 64q^{82} - 120q^{85} + 144q^{87} - 160q^{92} - 192q^{93} + 96q^{94} - 160q^{95} + 64q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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