LMFDB - Embedded newform 690.3.c.a.91.11

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			91.11
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.14

$q$-expansion

$f(q)$	$=$	$q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} +6.98697i q^{11} +3.46410 q^{12} -7.65791 q^{13} +0.581679i q^{14} -3.87298i q^{15} +4.00000 q^{16} +22.0292i q^{17} -4.24264 q^{18} +8.31107i q^{19} -4.47214i q^{20} -0.712409i q^{21} -9.88107i q^{22} +(17.3790 + 15.0656i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +10.8299 q^{26} +5.19615 q^{27} -0.822619i q^{28} +32.6032 q^{29} +5.47723i q^{30} -17.7255 q^{31} -5.65685 q^{32} +12.1018i q^{33} -31.1540i q^{34} -0.919716 q^{35} +6.00000 q^{36} +22.9773i q^{37} -11.7536i q^{38} -13.2639 q^{39} +6.32456i q^{40} +28.8797 q^{41} +1.00750i q^{42} +7.58017i q^{43} +13.9739i q^{44} -6.70820i q^{45} +(-24.5776 - 21.3060i) q^{46} +49.4935 q^{47} +6.92820 q^{48} +48.8308 q^{49} +7.07107 q^{50} +38.1557i q^{51} -15.3158 q^{52} +17.5382i q^{53} -7.34847 q^{54} +15.6233 q^{55} +1.16336i q^{56} +14.3952i q^{57} -46.1078 q^{58} -33.8653 q^{59} -7.74597i q^{60} -39.2567i q^{61} +25.0676 q^{62} -1.23393i q^{63} +8.00000 q^{64} +17.1236i q^{65} -17.1145i q^{66} -70.8924i q^{67} +44.0585i q^{68} +(30.1012 + 26.0944i) q^{69} +1.30067 q^{70} +26.8165 q^{71} -8.48528 q^{72} -62.0148 q^{73} -32.4948i q^{74} -8.66025 q^{75} +16.6221i q^{76} +2.87381 q^{77} +18.7580 q^{78} +35.2805i q^{79} -8.94427i q^{80} +9.00000 q^{81} -40.8420 q^{82} +143.692i q^{83} -1.42482i q^{84} +49.2589 q^{85} -10.7200i q^{86} +56.4704 q^{87} -19.7621i q^{88} +73.4091i q^{89} +9.48683i q^{90} +3.14977i q^{91} +(34.7579 + 30.1312i) q^{92} -30.7014 q^{93} -69.9943 q^{94} +18.5841 q^{95} -9.79796 q^{96} -95.9147i q^{97} -69.0572 q^{98} +20.9609i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	0.411309i		−	0.0587585i	−0.999568	−	0.0293792i	$-0.990647\pi$
$7$		−	0.411309i		−	0.0587585i	0.999568	−	0.0293792i	$-0.00935305\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$			6.98697i			0.635179i	0.948228	+	0.317589i	$0.102873\pi$
$11$			6.98697i			0.635179i	−0.948228	+	0.317589i	$0.897127\pi$
$12$	3.46410			0.288675
$13$	−7.65791			−0.589070			−0.294535	−	0.955641i	$-0.595165\pi$
$13$	−7.65791			−0.589070			−0.294535	+	0.955641i	$0.595165\pi$
$14$			0.581679i			0.0415485i
$15$		−	3.87298i		−	0.258199i
$16$	4.00000			0.250000
$17$			22.0292i			1.29584i	0.761710	+	0.647918i	$0.224360\pi$
$17$			22.0292i			1.29584i	−0.761710	+	0.647918i	$0.775640\pi$
$18$	−4.24264			−0.235702
$19$			8.31107i			0.437425i	0.975789	+	0.218712i	$0.0701857\pi$
$19$			8.31107i			0.437425i	−0.975789	+	0.218712i	$0.929814\pi$
$20$		−	4.47214i		−	0.223607i
$21$		−	0.712409i		−	0.0339242i
$22$		−	9.88107i		−	0.449139i
$23$	17.3790	+	15.0656i	0.755607	+	0.655026i
$23$	17.3790	+	15.0656i	0.755607	+	0.655026i
$24$	−4.89898			−0.204124
$25$	−5.00000			−0.200000
$26$	10.8299			0.416535
$27$	5.19615			0.192450
$28$		−	0.822619i		−	0.0293792i
$29$	32.6032			1.12425			0.562124	−	0.827053i	$-0.309984\pi$
$29$	32.6032			1.12425			0.562124	+	0.827053i	$0.309984\pi$
$30$			5.47723i			0.182574i
$31$	−17.7255			−0.571789			−0.285894	−	0.958261i	$-0.592291\pi$
$31$	−17.7255			−0.571789			−0.285894	+	0.958261i	$0.592291\pi$
$32$	−5.65685			−0.176777
$33$			12.1018i			0.366721i
$34$		−	31.1540i		−	0.916295i
$35$	−0.919716			−0.0262776
$36$	6.00000			0.166667
$37$			22.9773i			0.621008i	0.950572	+	0.310504i	$0.100498\pi$
$37$			22.9773i			0.621008i	−0.950572	+	0.310504i	$0.899502\pi$
$38$		−	11.7536i		−	0.309306i
$39$	−13.2639			−0.340100
$40$			6.32456i			0.158114i
$41$	28.8797			0.704382			0.352191	−	0.935928i	$-0.385437\pi$
$41$	28.8797			0.704382			0.352191	+	0.935928i	$0.385437\pi$
$42$			1.00750i			0.0239881i
$43$			7.58017i			0.176283i	0.996108	+	0.0881415i	$0.0280928\pi$
$43$			7.58017i			0.176283i	−0.996108	+	0.0881415i	$0.971907\pi$
$44$			13.9739i			0.317589i
$45$		−	6.70820i		−	0.149071i
$46$	−24.5776	−	21.3060i	−0.534295	−	0.463173i
$47$	49.4935			1.05305			0.526526	−	0.850159i	$-0.323494\pi$
$47$	49.4935			1.05305			0.526526	+	0.850159i	$0.323494\pi$
$48$	6.92820			0.144338
$49$	48.8308			0.996547
$50$	7.07107			0.141421
$51$			38.1557i			0.748152i
$52$	−15.3158			−0.294535
$53$			17.5382i			0.330910i	0.986217	+	0.165455i	$0.0529093\pi$
$53$			17.5382i			0.330910i	−0.986217	+	0.165455i	$0.947091\pi$
$54$	−7.34847			−0.136083
$55$	15.6233			0.284061
$56$			1.16336i			0.0207743i
$57$			14.3952i			0.252547i
$58$	−46.1078			−0.794963
$59$	−33.8653			−0.573989			−0.286994	−	0.957932i	$-0.592656\pi$
$59$	−33.8653			−0.573989			−0.286994	+	0.957932i	$0.592656\pi$
$60$		−	7.74597i		−	0.129099i
$61$		−	39.2567i		−	0.643552i	−0.946816	−	0.321776i	$-0.895720\pi$
$61$		−	39.2567i		−	0.643552i	0.946816	−	0.321776i	$-0.104280\pi$
$62$	25.0676			0.404316
$63$		−	1.23393i		−	0.0195862i
$64$	8.00000			0.125000
$65$			17.1236i			0.263440i
$66$		−	17.1145i		−	0.259311i
$67$		−	70.8924i		−	1.05810i	−0.848592	−	0.529048i	$-0.822549\pi$
$67$		−	70.8924i		−	1.05810i	0.848592	−	0.529048i	$-0.177451\pi$
$68$			44.0585i			0.647918i
$69$	30.1012	+	26.0944i	0.436250	+	0.378179i
$70$	1.30067			0.0185811
$71$	26.8165			0.377698			0.188849	−	0.982006i	$-0.439524\pi$
$71$	26.8165			0.377698			0.188849	+	0.982006i	$0.439524\pi$
$72$	−8.48528			−0.117851
$73$	−62.0148			−0.849518			−0.424759	−	0.905306i	$-0.639641\pi$
$73$	−62.0148			−0.849518			−0.424759	+	0.905306i	$0.639641\pi$
$74$		−	32.4948i		−	0.439119i
$75$	−8.66025			−0.115470
$76$			16.6221i			0.218712i
$77$	2.87381			0.0373222
$78$	18.7580			0.240487
$79$			35.2805i			0.446589i	0.974751	+	0.223294i	$0.0716811\pi$
$79$			35.2805i			0.446589i	−0.974751	+	0.223294i	$0.928319\pi$
$80$		−	8.94427i		−	0.111803i
$81$	9.00000			0.111111
$82$	−40.8420			−0.498074
$83$			143.692i			1.73123i	0.500712	+	0.865614i	$0.333072\pi$
$83$			143.692i			1.73123i	−0.500712	+	0.865614i	$0.666928\pi$
$84$		−	1.42482i		−	0.0169621i
$85$	49.2589			0.579516
$86$		−	10.7200i		−	0.124651i
$87$	56.4704			0.649084
$88$		−	19.7621i		−	0.224570i
$89$			73.4091i			0.824821i	0.910998	+	0.412411i	$0.135313\pi$
$89$			73.4091i			0.824821i	−0.910998	+	0.412411i	$0.864687\pi$
$90$			9.48683i			0.105409i
$91$			3.14977i			0.0346129i
$92$	34.7579	+	30.1312i	0.377803	+	0.327513i
$93$	−30.7014			−0.330123
$94$	−69.9943			−0.744620
$95$	18.5841			0.195622
$96$	−9.79796			−0.102062
$97$		−	95.9147i		−	0.988811i	−0.869231	−	0.494405i	$-0.835386\pi$
$97$		−	95.9147i		−	0.988811i	0.869231	−	0.494405i	$-0.164614\pi$
$98$	−69.0572			−0.704665
$99$			20.9609i			0.211726i
$100$	−10.0000			−0.100000
$101$	144.998			1.43563			0.717814	−	0.696235i	$-0.245143\pi$
$101$	144.998			1.43563			0.717814	+	0.696235i	$0.245143\pi$
$102$		−	53.9604i		−	0.529023i
$103$			123.567i			1.19968i	0.800121	+	0.599838i	$0.204768\pi$
$103$			123.567i			1.19968i	−0.800121	+	0.599838i	$0.795232\pi$
$104$	21.6598			0.208268
$105$	−1.59299			−0.0151714
$106$		−	24.8028i		−	0.233989i
$107$			92.0932i			0.860684i	0.902666	+	0.430342i	$0.141607\pi$
$107$			92.0932i			0.860684i	−0.902666	+	0.430342i	$0.858393\pi$
$108$	10.3923			0.0962250
$109$			45.0093i			0.412929i	0.978454	+	0.206465i	$0.0661958\pi$
$109$			45.0093i			0.412929i	−0.978454	+	0.206465i	$0.933804\pi$
$110$	−22.0947			−0.200861
$111$			39.7978i			0.358539i
$112$		−	1.64524i		−	0.0146896i
$113$			54.3464i			0.480942i	0.970656	+	0.240471i	$0.0773018\pi$
$113$			54.3464i			0.480942i	−0.970656	+	0.240471i	$0.922698\pi$
$114$		−	20.3579i		−	0.178578i
$115$	33.6877	−	38.8605i	0.292936	−	0.337918i
$116$	65.2063			0.562124
$117$	−22.9737			−0.196357
$118$	47.8928			0.405871
$119$	9.06083			0.0761414
$120$			10.9545i			0.0912871i
$121$	72.1823			0.596548
$122$			55.5173i			0.455060i
$123$	50.0211			0.406675
$124$	−35.4509			−0.285894
$125$			11.1803i			0.0894427i
$126$			1.74504i			0.0138495i
$127$	114.199			0.899203			0.449602	−	0.893229i	$-0.351566\pi$
$127$	114.199			0.899203			0.449602	+	0.893229i	$0.351566\pi$
$128$	−11.3137			−0.0883883
$129$			13.1292i			0.101777i
$130$		−	24.2164i		−	0.186280i
$131$	87.1429			0.665213			0.332606	−	0.943066i	$-0.392072\pi$
$131$	87.1429			0.665213			0.332606	+	0.943066i	$0.392072\pi$
$132$			24.2036i			0.183360i
$133$	3.41842			0.0257024
$134$			100.257i			0.748186i
$135$		−	11.6190i		−	0.0860663i
$136$		−	62.3081i		−	0.458148i
$137$			3.00953i			0.0219674i	0.999940	+	0.0109837i	$0.00349628\pi$
$137$			3.00953i			0.0219674i	−0.999940	+	0.0109837i	$0.996504\pi$
$138$	−42.5696	−	36.9030i	−0.308475	−	0.267413i
$139$	−124.206			−0.893567			−0.446783	−	0.894642i	$-0.647431\pi$
$139$	−124.206			−0.893567			−0.446783	+	0.894642i	$0.647431\pi$
$140$	−1.83943			−0.0131388
$141$	85.7252			0.607980
$142$	−37.9243			−0.267073
$143$		−	53.5056i		−	0.374165i
$144$	12.0000			0.0833333
$145$		−	72.9029i		−	0.502779i
$146$	87.7022			0.600700
$147$	84.5775			0.575357
$148$			45.9546i			0.310504i
$149$		−	11.8453i		−	0.0794988i	−0.999210	−	0.0397494i	$-0.987344\pi$
$149$		−	11.8453i		−	0.0794988i	0.999210	−	0.0397494i	$-0.0126560\pi$
$150$	12.2474			0.0816497
$151$	135.282			0.895908			0.447954	−	0.894057i	$-0.352153\pi$
$151$	135.282			0.895908			0.447954	+	0.894057i	$0.352153\pi$
$152$		−	23.5073i		−	0.154653i
$153$			66.0877i			0.431946i
$154$	−4.06418			−0.0263908
$155$			39.6353i			0.255712i
$156$	−26.5278			−0.170050
$157$			172.890i			1.10121i	0.834765	+	0.550606i	$0.185604\pi$
$157$			172.890i			1.10121i	−0.834765	+	0.550606i	$0.814396\pi$
$158$		−	49.8942i		−	0.315786i
$159$			30.3771i			0.191051i
$160$			12.6491i			0.0790569i
$161$	6.19662	−	7.14813i	0.0384883	−	0.0443983i
$162$	−12.7279			−0.0785674
$163$	−247.308			−1.51723			−0.758613	−	0.651541i	$-0.774123\pi$
$163$	−247.308			−1.51723			−0.758613	+	0.651541i	$0.774123\pi$
$164$	57.7594			0.352191
$165$	27.0604			0.164003
$166$		−	203.211i		−	1.22416i
$167$	−218.390			−1.30773			−0.653863	−	0.756613i	$-0.726853\pi$
$167$	−218.390			−1.30773			−0.653863	+	0.756613i	$0.726853\pi$
$168$			2.01500i			0.0119940i
$169$	−110.356			−0.652997
$170$	−69.6625			−0.409780
$171$			24.9332i			0.145808i
$172$			15.1603i			0.0881415i
$173$	−172.122			−0.994927			−0.497463	−	0.867485i	$-0.665735\pi$
$173$	−172.122			−0.994927			−0.497463	+	0.867485i	$0.665735\pi$
$174$	−79.8611			−0.458972
$175$			2.05655i			0.0117517i
$176$			27.9479i			0.158795i
$177$	−58.6565			−0.331392
$178$		−	103.816i		−	0.583237i
$179$	84.0652			0.469638			0.234819	−	0.972039i	$-0.424550\pi$
$179$	84.0652			0.469638			0.234819	+	0.972039i	$0.424550\pi$
$180$		−	13.4164i		−	0.0745356i
$181$		−	277.980i		−	1.53580i	−0.640569	−	0.767901i	$-0.721301\pi$
$181$		−	277.980i		−	1.53580i	0.640569	−	0.767901i	$-0.278699\pi$
$182$		−	4.45445i		−	0.0244750i
$183$		−	67.9946i		−	0.371555i
$184$	−49.1551	−	42.6119i	−0.267147	−	0.231587i
$185$	51.3788			0.277723
$186$	43.4183			0.233432
$187$	−153.918			−0.823088
$188$	98.9869			0.526526
$189$		−	2.13723i		−	0.0113081i
$190$	−26.2819			−0.138326
$191$		−	66.0363i		−	0.345740i	−0.984945	−	0.172870i	$-0.944696\pi$
$191$		−	66.0363i		−	0.345740i	0.984945	−	0.172870i	$-0.0553040\pi$
$192$	13.8564			0.0721688
$193$	−91.4052			−0.473602			−0.236801	−	0.971558i	$-0.576099\pi$
$193$	−91.4052			−0.473602			−0.236801	+	0.971558i	$0.576099\pi$
$194$			135.644i			0.699195i
$195$			29.6590i			0.152097i
$196$	97.6616			0.498274
$197$	34.0865			0.173028			0.0865140	−	0.996251i	$-0.472427\pi$
$197$	34.0865			0.173028			0.0865140	+	0.996251i	$0.472427\pi$
$198$		−	29.6432i		−	0.149713i
$199$		−	276.277i		−	1.38833i	−0.719817	−	0.694164i	$-0.755774\pi$
$199$		−	276.277i		−	1.38833i	0.719817	−	0.694164i	$-0.244226\pi$
$200$	14.1421			0.0707107
$201$		−	122.789i		−	0.610892i
$202$	−205.059			−1.01514
$203$		−	13.4100i		−	0.0660591i
$204$			76.3115i			0.374076i
$205$		−	64.5769i		−	0.315009i
$206$		−	174.750i		−	0.848299i
$207$	52.1369	+	45.1968i	0.251869	+	0.218342i
$208$	−30.6316			−0.147267
$209$	−58.0692			−0.277843
$210$	2.25283			0.0107278
$211$	−159.869			−0.757675			−0.378838	−	0.925463i	$-0.623676\pi$
$211$	−159.869			−0.757675			−0.378838	+	0.925463i	$0.623676\pi$
$212$			35.0765i			0.165455i
$213$	46.4476			0.218064
$214$		−	130.239i		−	0.608596i
$215$	16.9498			0.0788362
$216$	−14.6969			−0.0680414
$217$			7.29065i			0.0335975i
$218$		−	63.6528i		−	0.291985i
$219$	−107.413			−0.490470
$220$	31.2467			0.142030
$221$		−	168.698i		−	0.763339i
$222$		−	56.2826i		−	0.253525i
$223$	−26.6387			−0.119456			−0.0597280	−	0.998215i	$-0.519023\pi$
$223$	−26.6387			−0.119456			−0.0597280	+	0.998215i	$0.519023\pi$
$224$			2.32672i			0.0103871i
$225$	−15.0000			−0.0666667
$226$		−	76.8574i		−	0.340077i
$227$		−	209.504i		−	0.922924i	−0.887160	−	0.461462i	$-0.847325\pi$
$227$		−	209.504i		−	0.922924i	0.887160	−	0.461462i	$-0.152675\pi$
$228$			28.7904i			0.126274i
$229$		−	80.8748i		−	0.353165i	−0.984286	−	0.176583i	$-0.943496\pi$
$229$		−	80.8748i		−	0.353165i	0.984286	−	0.176583i	$-0.0565043\pi$
$230$	−47.6416	+	54.9571i	−0.207137	+	0.238944i
$231$	4.97758			0.0215480
$232$	−92.2157			−0.397481
$233$	−174.793			−0.750184			−0.375092	−	0.926988i	$-0.622389\pi$
$233$	−174.793			−0.750184			−0.375092	+	0.926988i	$0.622389\pi$
$234$	32.4898			0.138845
$235$		−	110.671i		−	0.470939i
$236$	−67.7307			−0.286994
$237$			61.1077i			0.257838i
$238$	−12.8139			−0.0538401
$239$	99.3102			0.415524			0.207762	−	0.978179i	$-0.433382\pi$
$239$	99.3102			0.415524			0.207762	+	0.978179i	$0.433382\pi$
$240$		−	15.4919i		−	0.0645497i
$241$		−	43.9106i		−	0.182202i	−0.995842	−	0.0911008i	$-0.970961\pi$
$241$		−	43.9106i		−	0.182202i	0.995842	−	0.0911008i	$-0.0290386\pi$
$242$	−102.081			−0.421823
$243$	15.5885			0.0641500
$244$		−	78.5134i		−	0.321776i
$245$		−	109.189i		−	0.445670i
$246$	−70.7405			−0.287563
$247$		−	63.6454i		−	0.257674i
$248$	50.1352			0.202158
$249$			248.882i			0.999525i
$250$		−	15.8114i		−	0.0632456i
$251$		−	419.272i		−	1.67041i	−0.549940	−	0.835204i	$-0.685349\pi$
$251$		−	419.272i		−	1.67041i	0.549940	−	0.835204i	$-0.314651\pi$
$252$		−	2.46786i		−	0.00979308i
$253$	−105.263	+	121.426i	−0.416059	+	0.479945i
$254$	−161.501			−0.635833
$255$	85.3188			0.334584
$256$	16.0000			0.0625000
$257$	−207.444			−0.807176			−0.403588	−	0.914941i	$-0.632237\pi$
$257$	−207.444			−0.807176			−0.403588	+	0.914941i	$0.632237\pi$
$258$		−	18.5676i		−	0.0719673i
$259$	9.45077			0.0364895
$260$			34.2472i			0.131720i
$261$	97.8095			0.374749
$262$	−123.239			−0.470376
$263$			342.541i			1.30244i	0.758891	+	0.651218i	$0.225742\pi$
$263$			342.541i			1.30244i	−0.758891	+	0.651218i	$0.774258\pi$
$264$		−	34.2290i		−	0.129655i
$265$	39.2167			0.147987
$266$	−4.83438			−0.0181744
$267$			127.148i			0.476211i
$268$		−	141.785i		−	0.529048i
$269$	−403.239			−1.49903			−0.749514	−	0.661988i	$-0.769713\pi$
$269$	−403.239			−1.49903			−0.749514	+	0.661988i	$0.769713\pi$
$270$			16.4317i			0.0608581i
$271$	45.3290			0.167266			0.0836328	−	0.996497i	$-0.473348\pi$
$271$	45.3290			0.167266			0.0836328	+	0.996497i	$0.473348\pi$
$272$			88.1169i			0.323959i
$273$			5.45556i			0.0199837i
$274$		−	4.25612i		−	0.0155333i
$275$		−	34.9348i		−	0.127036i
$276$	60.2025	+	52.1887i	0.218125	+	0.189090i
$277$	−241.667			−0.872446			−0.436223	−	0.899839i	$-0.643684\pi$
$277$	−241.667			−0.872446			−0.436223	+	0.899839i	$0.643684\pi$
$278$	175.654			0.631847
$279$	−53.1764			−0.190596
$280$	2.60135			0.00929053
$281$		−	231.673i		−	0.824459i	−0.911080	−	0.412229i	$-0.864750\pi$
$281$		−	231.673i		−	0.824459i	0.911080	−	0.412229i	$-0.135250\pi$
$282$	−121.234			−0.429907
$283$			175.141i			0.618874i	0.950920	+	0.309437i	$0.100141\pi$
$283$			175.141i			0.618874i	−0.950920	+	0.309437i	$0.899859\pi$
$284$	53.6331			0.188849
$285$	32.1886			0.112943
$286$			75.6683i			0.264575i
$287$		−	11.8785i		−	0.0413884i
$288$	−16.9706			−0.0589256
$289$	−196.287			−0.679194
$290$			103.100i			0.355518i
$291$		−	166.129i		−	0.570890i
$292$	−124.030			−0.424759
$293$		−	306.727i		−	1.04685i	−0.852072	−	0.523424i	$-0.824654\pi$
$293$		−	306.727i		−	1.04685i	0.852072	−	0.523424i	$-0.175346\pi$
$294$	−119.611			−0.406839
$295$			75.7252i			0.256696i
$296$		−	64.9896i		−	0.219559i
$297$			36.3054i			0.122240i
$298$			16.7518i			0.0562141i
$299$	−133.086	−	115.371i	−0.445105	−	0.385856i
$300$	−17.3205			−0.0577350
$301$	3.11780			0.0103581
$302$	−191.318			−0.633503
$303$	251.145			0.828860
$304$			33.2443i			0.109356i
$305$	−87.7806			−0.287805
$306$		−	93.4621i		−	0.305432i
$307$	106.431			0.346682			0.173341	−	0.984862i	$-0.444544\pi$
$307$	106.431			0.346682			0.173341	+	0.984862i	$0.444544\pi$
$308$	5.74761			0.0186611
$309$			214.024i			0.692633i
$310$		−	56.0528i		−	0.180816i
$311$	604.165			1.94265			0.971327	−	0.237749i	$-0.0764097\pi$
$311$	604.165			1.94265			0.971327	+	0.237749i	$0.0764097\pi$
$312$	37.5159			0.120243
$313$		−	505.257i		−	1.61424i	−0.590388	−	0.807120i	$-0.701025\pi$
$313$		−	505.257i		−	1.61424i	0.590388	−	0.807120i	$-0.298975\pi$
$314$		−	244.504i		−	0.778675i
$315$	−2.75915			−0.00875920
$316$			70.5611i			0.223294i
$317$	104.465			0.329544			0.164772	−	0.986332i	$-0.447311\pi$
$317$	104.465			0.329544			0.164772	+	0.986332i	$0.447311\pi$
$318$		−	42.9597i		−	0.135093i
$319$			227.797i			0.714098i
$320$		−	17.8885i		−	0.0559017i
$321$			159.510i			0.496916i
$322$	−8.76334	+	10.1090i	−0.0272154	+	0.0313943i
$323$	−183.086			−0.566831
$324$	18.0000			0.0555556
$325$	38.2895			0.117814
$326$	349.746			1.07284
$327$			77.9584i			0.238405i
$328$	−81.6841			−0.249037
$329$		−	20.3571i		−	0.0618758i
$330$	−38.2692			−0.115967
$331$	−41.3629			−0.124963			−0.0624817	−	0.998046i	$-0.519902\pi$
$331$	−41.3629			−0.124963			−0.0624817	+	0.998046i	$0.519902\pi$
$332$			287.384i			0.865614i
$333$			68.9319i			0.207003i
$334$	308.851			0.924702
$335$	−158.520			−0.473195
$336$		−	2.84964i		−	0.00848106i
$337$		−	181.417i		−	0.538331i	−0.963094	−	0.269165i	$-0.913252\pi$
$337$		−	181.417i		−	0.538331i	0.963094	−	0.269165i	$-0.0867478\pi$
$338$	156.068			0.461738
$339$			94.1307i			0.277672i
$340$	98.5177			0.289758
$341$		−	123.847i		−	0.363188i
$342$		−	35.2609i		−	0.103102i
$343$		−	40.2387i		−	0.117314i
$344$		−	21.4400i		−	0.0623255i
$345$	58.3488	−	67.3084i	0.169127	−	0.195097i
$346$	243.418			0.703519
$347$	202.642			0.583984			0.291992	−	0.956421i	$-0.405682\pi$
$347$	202.642			0.583984			0.291992	+	0.956421i	$0.405682\pi$
$348$	112.941			0.324542
$349$	319.984			0.916859			0.458430	−	0.888731i	$-0.348412\pi$
$349$	319.984			0.916859			0.458430	+	0.888731i	$0.348412\pi$
$350$		−	2.90840i		−	0.00830971i
$351$	−39.7917			−0.113367
$352$		−	39.5243i		−	0.112285i
$353$	−308.167			−0.872994			−0.436497	−	0.899706i	$-0.643781\pi$
$353$	−308.167			−0.872994			−0.436497	+	0.899706i	$0.643781\pi$
$354$	82.9528			0.234330
$355$		−	59.9636i		−	0.168912i
$356$			146.818i			0.412411i
$357$	15.6938			0.0439603
$358$	−118.886			−0.332084
$359$			301.630i			0.840196i	0.907479	+	0.420098i	$0.138004\pi$
$359$			301.630i			0.840196i	−0.907479	+	0.420098i	$0.861996\pi$
$360$			18.9737i			0.0527046i
$361$	291.926			0.808660
$362$			393.123i			1.08598i
$363$	125.023			0.344417
$364$			6.29954i			0.0173064i
$365$			138.669i			0.379916i
$366$			96.1588i			0.262729i
$367$			470.453i			1.28189i	0.767587	+	0.640945i	$0.221457\pi$
$367$			470.453i			1.28189i	−0.767587	+	0.640945i	$0.778543\pi$
$368$	69.5158	+	60.2624i	0.188902	+	0.163756i
$369$	86.6390			0.234794
$370$	−72.6606			−0.196380
$371$	7.21364			0.0194438
$372$	−61.4028			−0.165061
$373$			249.106i			0.667844i	0.942601	+	0.333922i	$0.108372\pi$
$373$			249.106i			0.667844i	−0.942601	+	0.333922i	$0.891628\pi$
$374$	217.672			0.582011
$375$			19.3649i			0.0516398i
$376$	−139.989			−0.372310
$377$	−249.672			−0.662260
$378$			3.02249i			0.00799602i
$379$			141.700i			0.373879i	0.982371	+	0.186940i	$0.0598569\pi$
$379$			141.700i			0.373879i	−0.982371	+	0.186940i	$0.940143\pi$
$380$	37.1682			0.0978112
$381$	197.798			0.519155
$382$			93.3895i			0.244475i
$383$		−	110.480i		−	0.288460i	−0.989544	−	0.144230i	$-0.953929\pi$
$383$		−	110.480i		−	0.288460i	0.989544	−	0.144230i	$-0.0460705\pi$
$384$	−19.5959			−0.0510310
$385$		−	6.42603i		−	0.0166910i
$386$	129.266			0.334887
$387$			22.7405i			0.0587610i
$388$		−	191.829i		−	0.494405i
$389$		−	705.447i		−	1.81349i	−0.421681	−	0.906744i	$-0.638560\pi$
$389$		−	705.447i		−	1.81349i	0.421681	−	0.906744i	$-0.361440\pi$
$390$		−	41.9441i		−	0.107549i
$391$	−331.883	+	382.845i	−0.848807	+	0.979143i
$392$	−138.114			−0.352333
$393$	150.936			0.384061
$394$	−48.2056			−0.122349
$395$	78.8897			0.199721
$396$			41.9218i			0.105863i
$397$	10.3766			0.0261376			0.0130688	−	0.999915i	$-0.495840\pi$
$397$	10.3766			0.0261376			0.0130688	+	0.999915i	$0.495840\pi$
$398$			390.715i			0.981696i
$399$	5.92088			0.0148393
$400$	−20.0000			−0.0500000
$401$			50.4144i			0.125722i	0.998022	+	0.0628609i	$0.0200224\pi$
$401$			50.4144i			0.125722i	−0.998022	+	0.0628609i	$0.979978\pi$
$402$			173.650i			0.431966i
$403$	135.740			0.336824
$404$	289.997			0.717814
$405$		−	20.1246i		−	0.0496904i
$406$			18.9646i			0.0467108i
$407$	−160.542			−0.394451
$408$		−	107.921i		−	0.264512i
$409$	71.8300			0.175624			0.0878118	−	0.996137i	$-0.472013\pi$
$409$	71.8300			0.175624			0.0878118	+	0.996137i	$0.472013\pi$
$410$			91.3256i			0.222745i
$411$			5.21266i			0.0126829i
$412$			247.133i			0.599838i
$413$			13.9291i			0.0337267i
$414$	−73.7327	−	63.9179i	−0.178098	−	0.154391i
$415$	321.305			0.774229
$416$	43.3197			0.104134
$417$	−215.131			−0.515901
$418$	82.1222			0.196465
$419$			355.682i			0.848884i	0.905455	+	0.424442i	$0.139530\pi$
$419$			355.682i			0.848884i	−0.905455	+	0.424442i	$0.860470\pi$
$420$	−3.18599			−0.00758569
$421$			277.750i			0.659738i	0.944027	+	0.329869i	$0.107005\pi$
$421$			277.750i			0.659738i	−0.944027	+	0.329869i	$0.892995\pi$
$422$	226.090			0.535757
$423$	148.480			0.351017
$424$		−	49.6056i		−	0.116994i
$425$		−	110.146i		−	0.259167i
$426$	−65.6868			−0.154194
$427$	−16.1466			−0.0378141
$428$			184.186i			0.430342i
$429$		−	92.6744i		−	0.216024i
$430$	−23.9706			−0.0557456
$431$		−	219.586i		−	0.509481i	−0.967009	−	0.254741i	$-0.918010\pi$
$431$		−	219.586i		−	0.509481i	0.967009	−	0.254741i	$-0.0819901\pi$
$432$	20.7846			0.0481125
$433$		−	284.730i		−	0.657574i	−0.944404	−	0.328787i	$-0.893360\pi$
$433$		−	284.730i		−	0.657574i	0.944404	−	0.328787i	$-0.106640\pi$
$434$		−	10.3105i		−	0.0237570i
$435$		−	126.272i		−	0.290279i
$436$			90.0186i			0.206465i
$437$	−125.211	+	144.438i	−0.286524	+	0.330521i
$438$	151.905			0.346814
$439$	−80.6153			−0.183634			−0.0918170	−	0.995776i	$-0.529267\pi$
$439$	−80.6153			−0.183634			−0.0918170	+	0.995776i	$0.529267\pi$
$440$	−44.1895			−0.100431
$441$	146.492			0.332182
$442$			238.575i			0.539762i
$443$	208.114			0.469784			0.234892	−	0.972021i	$-0.424526\pi$
$443$	208.114			0.469784			0.234892	+	0.972021i	$0.424526\pi$
$444$			79.5956i			0.179269i
$445$	164.148			0.368871
$446$	37.6728			0.0844681
$447$		−	20.5167i		−	0.0458986i
$448$		−	3.29048i		−	0.00734481i
$449$	−409.241			−0.911450			−0.455725	−	0.890121i	$-0.650620\pi$
$449$	−409.241			−0.911450			−0.455725	+	0.890121i	$0.650620\pi$
$450$	21.2132			0.0471405
$451$			201.781i			0.447409i
$452$			108.693i			0.240471i
$453$	234.316			0.517253
$454$			296.283i			0.652606i
$455$	7.04310			0.0154793
$456$		−	40.7158i		−	0.0892890i
$457$		−	310.013i		−	0.678365i	−0.940721	−	0.339183i	$-0.889849\pi$
$457$		−	310.013i		−	0.678365i	0.940721	−	0.339183i	$-0.110151\pi$
$458$			114.374i			0.249726i
$459$			114.467i			0.249384i
$460$	67.3754	−	77.7210i	0.146468	−	0.168959i
$461$	463.484			1.00539			0.502694	−	0.864465i	$-0.332342\pi$
$461$	463.484			1.00539			0.502694	+	0.864465i	$0.332342\pi$
$462$	−7.03936			−0.0152367
$463$	604.690			1.30603			0.653013	−	0.757347i	$-0.273505\pi$
$463$	604.690			1.30603			0.653013	+	0.757347i	$0.273505\pi$
$464$	130.413			0.281062
$465$			68.6504i			0.147635i
$466$	247.194			0.530460
$467$			271.585i			0.581551i	0.956791	+	0.290776i	$0.0939134\pi$
$467$			271.585i			0.581551i	−0.956791	+	0.290776i	$0.906087\pi$
$468$	−45.9475			−0.0981783
$469$	−29.1587			−0.0621721
$470$			156.512i			0.333004i
$471$			299.455i			0.635786i
$472$	95.7856			0.202936
$473$	−52.9624			−0.111971
$474$		−	86.4193i		−	0.182319i
$475$		−	41.5554i		−	0.0874850i
$476$	18.1217			0.0380707
$477$			52.6147i			0.110303i
$478$	−140.446			−0.293820
$479$			793.694i			1.65698i	0.560003	+	0.828491i	$0.310800\pi$
$479$			793.694i			1.65698i	−0.560003	+	0.828491i	$0.689200\pi$
$480$			21.9089i			0.0456435i
$481$		−	175.958i		−	0.365817i
$482$			62.0990i			0.128836i
$483$	10.7329	−	12.3809i	0.0222212	−	0.0256334i
$484$	144.365			0.298274
$485$	−214.472			−0.442210
$486$	−22.0454			−0.0453609
$487$	−434.574			−0.892350			−0.446175	−	0.894946i	$-0.647214\pi$
$487$	−434.574			−0.892350			−0.446175	+	0.894946i	$0.647214\pi$
$488$			111.035i			0.227530i
$489$	−428.350			−0.875971
$490$			154.417i			0.315136i
$491$	95.0833			0.193652			0.0968261	−	0.995301i	$-0.469131\pi$
$491$	95.0833			0.193652			0.0968261	+	0.995301i	$0.469131\pi$
$492$	100.042			0.203338
$493$			718.223i			1.45684i
$494$			90.0082i			0.182203i
$495$	46.8700			0.0946869
$496$	−70.9018			−0.142947
$497$		−	11.0299i		−	0.0221929i
$498$		−	351.972i		−	0.706771i
$499$	826.993			1.65730			0.828650	−	0.559767i	$-0.189109\pi$
$499$	826.993			1.65730			0.828650	+	0.559767i	$0.189109\pi$
$500$			22.3607i			0.0447214i
$501$	−378.263			−0.755016
$502$			592.941i			1.18116i
$503$		−	424.858i		−	0.844648i	−0.906445	−	0.422324i	$-0.861215\pi$
$503$		−	424.858i		−	0.844648i	0.906445	−	0.422324i	$-0.138785\pi$
$504$			3.49008i			0.00692475i
$505$		−	324.226i		−	0.642032i
$506$	148.864	−	171.723i	0.294198	−	0.339373i
$507$	−191.143			−0.377008
$508$	228.398			0.449602
$509$	−786.116			−1.54443			−0.772216	−	0.635360i	$-0.780852\pi$
$509$	−786.116			−1.54443			−0.772216	+	0.635360i	$0.780852\pi$
$510$	−120.659			−0.236586
$511$			25.5073i			0.0499164i
$512$	−22.6274			−0.0441942
$513$			43.1856i			0.0841824i
$514$	293.371			0.570760
$515$	276.303			0.536511
$516$			26.2585i			0.0508885i
$517$			345.809i			0.668877i
$518$	−13.3654			−0.0258020
$519$	−298.125			−0.574421
$520$		−	48.4329i		−	0.0931401i
$521$		−	792.323i		−	1.52077i	−0.649471	−	0.760387i	$-0.725009\pi$
$521$		−	792.323i		−	1.52077i	0.649471	−	0.760387i	$-0.274991\pi$
$522$	−138.324			−0.264988
$523$			123.543i			0.236220i	0.993001	+	0.118110i	$0.0376835\pi$
$523$			123.543i			0.236220i	−0.993001	+	0.118110i	$0.962316\pi$
$524$	174.286			0.332606
$525$			3.56204i			0.00678485i
$526$		−	484.425i		−	0.920961i
$527$		−	390.478i		−	0.740945i
$528$			48.4071i			0.0916802i
$529$	75.0560	+	523.648i	0.141883	+	0.989883i
$530$	−55.4608			−0.104643
$531$	−101.596			−0.191330
$532$	6.83684			0.0128512
$533$	−221.158			−0.414930
$534$		−	179.815i		−	0.336732i
$535$	205.927			0.384910
$536$			200.514i			0.374093i
$537$	145.605			0.271145
$538$	570.266			1.05997
$539$			341.179i			0.632986i
$540$		−	23.2379i		−	0.0430331i
$541$	−746.830			−1.38046			−0.690231	−	0.723589i	$-0.742491\pi$
$541$	−746.830			−1.38046			−0.690231	+	0.723589i	$0.742491\pi$
$542$	−64.1048			−0.118275
$543$		−	481.476i		−	0.886696i
$544$		−	124.616i		−	0.229074i
$545$	100.644			0.184668
$546$		−	7.71533i		−	0.0141306i
$547$	760.793			1.39085			0.695423	−	0.718600i	$-0.255217\pi$
$547$	760.793			1.39085			0.695423	+	0.718600i	$0.255217\pi$
$548$			6.01906i			0.0109837i
$549$		−	117.770i		−	0.214517i
$550$			49.4053i			0.0898279i
$551$			270.967i			0.491774i
$552$	−85.1391	−	73.8060i	−0.154238	−	0.133707i
$553$	14.5112			0.0262409
$554$	341.769			0.616912
$555$	88.9906			0.160343
$556$	−248.412			−0.446783
$557$		−	619.940i		−	1.11300i	−0.830848	−	0.556499i	$-0.812144\pi$
$557$		−	619.940i		−	1.11300i	0.830848	−	0.556499i	$-0.187856\pi$
$558$	75.2028			0.134772
$559$		−	58.0483i		−	0.103843i
$560$	−3.67886			−0.00656940
$561$	−266.593			−0.475210
$562$			327.635i			0.582980i
$563$			674.480i			1.19801i	0.800745	+	0.599006i	$0.204437\pi$
$563$			674.480i			1.19801i	−0.800745	+	0.599006i	$0.795563\pi$
$564$	171.450			0.303990
$565$	121.522			0.215084
$566$		−	247.687i		−	0.437610i
$567$		−	3.70178i		−	0.00652872i
$568$	−75.8486			−0.133536
$569$		−	456.740i		−	0.802706i	−0.915923	−	0.401353i	$-0.868540\pi$
$569$		−	456.740i		−	0.802706i	0.915923	−	0.401353i	$-0.131460\pi$
$570$	−45.5216			−0.0798625
$571$			484.210i			0.848004i	0.905661	+	0.424002i	$0.139375\pi$
$571$			484.210i			0.848004i	−0.905661	+	0.424002i	$0.860625\pi$
$572$		−	107.011i		−	0.187082i
$573$		−	114.378i		−	0.199613i
$574$			16.7987i			0.0292660i
$575$	−86.8948	−	75.3280i	−0.151121	−	0.131005i
$576$	24.0000			0.0416667
$577$	72.0514			0.124872			0.0624362	−	0.998049i	$-0.480113\pi$
$577$	72.0514			0.124872			0.0624362	+	0.998049i	$0.480113\pi$
$578$	277.592			0.480262
$579$	−158.318			−0.273434
$580$		−	145.806i		−	0.251389i
$581$	59.1018			0.101724
$582$			234.942i			0.403680i
$583$	−122.539			−0.210187
$584$	175.404			0.300350
$585$			51.3708i			0.0878134i
$586$			433.777i			0.740234i
$587$	−119.743			−0.203991			−0.101995	−	0.994785i	$-0.532523\pi$
$587$	−119.743			−0.203991			−0.101995	+	0.994785i	$0.532523\pi$
$588$	169.155			0.287678
$589$		−	147.318i		−	0.250115i
$590$		−	107.092i		−	0.181511i
$591$	59.0396			0.0998978
$592$			91.9091i			0.155252i
$593$	−648.197			−1.09308			−0.546540	−	0.837433i	$-0.684055\pi$
$593$	−648.197			−1.09308			−0.546540	+	0.837433i	$0.684055\pi$
$594$		−	51.3435i		−	0.0864369i
$595$		−	20.2606i		−	0.0340515i
$596$		−	23.6906i		−	0.0397494i
$597$		−	478.526i		−	0.801551i
$598$	188.213	+	163.159i	0.314737	+	0.272841i
$599$	269.000			0.449082			0.224541	−	0.974465i	$-0.427912\pi$
$599$	269.000			0.449082			0.224541	+	0.974465i	$0.427912\pi$
$600$	24.4949			0.0408248
$601$	−339.169			−0.564340			−0.282170	−	0.959364i	$-0.591054\pi$
$601$	−339.169			−0.564340			−0.282170	+	0.959364i	$0.591054\pi$
$602$	−4.40923			−0.00732430
$603$		−	212.677i		−	0.352698i
$604$	270.564			0.447954
$605$		−	161.404i		−	0.266784i
$606$	−355.172			−0.586093
$607$	372.696			0.613997			0.306998	−	0.951710i	$-0.400675\pi$
$607$	372.696			0.613997			0.306998	+	0.951710i	$0.400675\pi$
$608$		−	47.0145i		−	0.0773265i
$609$		−	23.2268i		−	0.0381392i
$610$	124.141			0.203509
$611$	−379.016			−0.620322
$612$			132.175i			0.215973i
$613$			125.657i			0.204987i	0.994734	+	0.102493i	$0.0326820\pi$
$613$			125.657i			0.204987i	−0.994734	+	0.102493i	$0.967318\pi$
$614$	−150.517			−0.245141
$615$		−	111.851i		−	0.181871i
$616$	−8.12835			−0.0131954
$617$		−	702.136i		−	1.13798i	−0.822343	−	0.568992i	$-0.807333\pi$
$617$		−	702.136i		−	1.13798i	0.822343	−	0.568992i	$-0.192667\pi$
$618$		−	302.675i		−	0.489766i
$619$		−	888.543i		−	1.43545i	−0.696327	−	0.717724i	$-0.745184\pi$
$619$		−	888.543i		−	1.43545i	0.696327	−	0.717724i	$-0.254816\pi$
$620$			79.2707i			0.127856i
$621$	90.3037	+	78.2831i	0.145417	+	0.126060i
$622$	−854.418			−1.37366
$623$	30.1938			0.0484652
$624$	−53.0556			−0.0850249
$625$	25.0000			0.0400000
$626$			714.541i			1.14144i
$627$	−100.579			−0.160413
$628$			345.781i			0.550606i
$629$	−506.172			−0.804725
$630$	3.90202			0.00619369
$631$			189.781i			0.300763i	0.988628	+	0.150381i	$0.0480501\pi$
$631$			189.781i			0.300763i	−0.988628	+	0.150381i	$0.951950\pi$
$632$		−	99.7884i		−	0.157893i
$633$	−276.902			−0.437444
$634$	−147.736			−0.233023
$635$		−	255.356i		−	0.402136i
$636$			60.7542i			0.0955255i
$637$	−373.942			−0.587036
$638$		−	322.154i		−	0.504944i
$639$	80.4496			0.125899
$640$			25.2982i			0.0395285i
$641$		−	682.549i		−	1.06482i	−0.846487	−	0.532410i	$-0.821287\pi$
$641$		−	682.549i		−	1.06482i	0.846487	−	0.532410i	$-0.178713\pi$
$642$		−	225.581i		−	0.351373i
$643$		−	436.832i		−	0.679365i	−0.940540	−	0.339682i	$-0.889680\pi$
$643$		−	436.832i		−	0.679365i	0.940540	−	0.339682i	$-0.110320\pi$
$644$	12.3932	−	14.2963i	0.0192442	−	0.0221992i
$645$	29.3579			0.0455161
$646$	258.923			0.400810
$647$	853.291			1.31884			0.659421	−	0.751774i	$-0.270801\pi$
$647$	853.291			1.31884			0.659421	+	0.751774i	$0.270801\pi$
$648$	−25.4558			−0.0392837
$649$		−	236.616i		−	0.364586i
$650$	−54.1496			−0.0833071
$651$			12.6278i			0.0193975i
$652$	−494.616			−0.758613
$653$	−502.550			−0.769601			−0.384801	−	0.923000i	$-0.625730\pi$
$653$	−502.550			−0.769601			−0.384801	+	0.923000i	$0.625730\pi$
$654$		−	110.250i		−	0.168578i
$655$		−	194.857i		−	0.297492i
$656$	115.519			0.176096
$657$	−186.044			−0.283173
$658$			28.7893i			0.0437528i
$659$			325.726i			0.494273i	0.968981	+	0.247136i	$0.0794895\pi$
$659$			325.726i			0.494273i	−0.968981	+	0.247136i	$0.920510\pi$
$660$	54.1208			0.0820013
$661$			578.734i			0.875543i	0.899086	+	0.437771i	$0.144232\pi$
$661$			578.734i			0.875543i	−0.899086	+	0.437771i	$0.855768\pi$
$662$	58.4960			0.0883625
$663$		−	292.193i		−	0.440714i
$664$		−	406.422i		−	0.612081i
$665$		−	7.64382i		−	0.0114945i
$666$		−	97.4844i		−	0.146373i
$667$	566.609	+	491.186i	0.849489	+	0.736411i
$668$	−436.781			−0.653863
$669$	−46.1396			−0.0689679
$670$	224.181			0.334599
$671$	274.285			0.408771
$672$			4.02999i			0.00599701i
$673$	787.850			1.17065			0.585327	−	0.810798i	$-0.300966\pi$
$673$	787.850			1.17065			0.585327	+	0.810798i	$0.300966\pi$
$674$			256.563i			0.380657i
$675$	−25.9808			−0.0384900
$676$	−220.713			−0.326498
$677$			1053.03i			1.55543i	0.628616	+	0.777716i	$0.283622\pi$
$677$			1053.03i			1.55543i	−0.628616	+	0.777716i	$0.716378\pi$
$678$		−	133.121i		−	0.196344i
$679$	−39.4506			−0.0581010
$680$	−139.325			−0.204890
$681$		−	362.871i		−	0.532850i
$682$			175.146i			0.256813i
$683$	−416.716			−0.610127			−0.305063	−	0.952332i	$-0.598678\pi$
$683$	−416.716			−0.610127			−0.305063	+	0.952332i	$0.598678\pi$
$684$			49.8664i			0.0729041i
$685$	6.72951			0.00982410
$686$			56.9062i			0.0829536i
$687$		−	140.079i		−	0.203900i
$688$			30.3207i			0.0440708i
$689$		−	134.306i		−	0.194929i
$690$	−82.5176	+	95.1884i	−0.119591	+	0.137954i
$691$	144.765			0.209500			0.104750	−	0.994499i	$-0.466596\pi$
$691$	144.765			0.209500			0.104750	+	0.994499i	$0.466596\pi$
$692$	−344.245			−0.497463
$693$	8.62142			0.0124407
$694$	−286.580			−0.412939
$695$			277.733i			0.399615i
$696$	−159.722			−0.229486
$697$			636.197i			0.912765i
$698$	−452.525			−0.648317
$699$	−302.750			−0.433119
$700$			4.11309i			0.00587585i
$701$		−	723.630i		−	1.03228i	−0.856503	−	0.516142i	$-0.827368\pi$
$701$		−	723.630i		−	1.03228i	0.856503	−	0.516142i	$-0.172632\pi$
$702$	56.2739			0.0801623
$703$	−190.966			−0.271644
$704$			55.8958i			0.0793974i
$705$		−	191.687i		−	0.271897i
$706$	435.814			0.617300
$707$		−	59.6392i		−	0.0843553i
$708$	−117.313			−0.165696
$709$			4.77044i			0.00672841i	0.999994	+	0.00336420i	$0.00107086\pi$
$709$			4.77044i			0.00672841i	−0.999994	+	0.00336420i	$0.998929\pi$
$710$			84.8013i			0.119438i
$711$			105.842i			0.148863i
$712$		−	207.632i		−	0.291618i
$713$	−308.050	−	267.044i	−0.432048	−	0.374536i
$714$	−22.1944			−0.0310846
$715$	−119.642			−0.167332
$716$	168.130			0.234819
$717$	172.010			0.239903
$718$		−	426.570i		−	0.594108i
$719$	−163.295			−0.227115			−0.113557	−	0.993531i	$-0.536225\pi$
$719$	−163.295			−0.227115			−0.113557	+	0.993531i	$0.536225\pi$
$720$		−	26.8328i		−	0.0372678i
$721$	50.8241			0.0704911
$722$	−412.846			−0.571809
$723$		−	76.0554i		−	0.105194i
$724$		−	555.960i		−	0.767901i
$725$	−163.016			−0.224849
$726$	−176.810			−0.243540
$727$		−	1100.34i		−	1.51354i	−0.653683	−	0.756769i	$-0.726777\pi$
$727$		−	1100.34i		−	1.51354i	0.653683	−	0.756769i	$-0.273223\pi$
$728$		−	8.90890i		−	0.0122375i
$729$	27.0000			0.0370370
$730$		−	196.108i		−	0.268641i
$731$	−166.985			−0.228434
$732$		−	135.989i		−	0.185777i
$733$		−	362.979i		−	0.495197i	−0.968863	−	0.247599i	$-0.920359\pi$
$733$		−	362.979i		−	0.495197i	0.968863	−	0.247599i	$-0.0796414\pi$
$734$		−	665.321i		−	0.906432i
$735$		−	189.121i		−	0.257307i
$736$	−98.3102	−	85.2238i	−0.133574	−	0.115793i
$737$	495.323			0.672080
$738$	−122.526			−0.166025
$739$	−475.365			−0.643254			−0.321627	−	0.946866i	$-0.604230\pi$
$739$	−475.365			−0.643254			−0.321627	+	0.946866i	$0.604230\pi$
$740$	102.758			0.138862
$741$		−	110.237i		−	0.148768i
$742$	−10.2016			−0.0137488
$743$			691.352i			0.930487i	0.885183	+	0.465244i	$0.154033\pi$
$743$			691.352i			0.930487i	−0.885183	+	0.465244i	$0.845967\pi$
$744$	86.8367			0.116716
$745$	−26.4869			−0.0355529
$746$		−	352.289i		−	0.472237i
$747$			431.076i			0.577076i
$748$	−307.835			−0.411544
$749$	37.8788			0.0505725
$750$		−	27.3861i		−	0.0365148i
$751$			971.631i			1.29378i	0.762582	+	0.646892i	$0.223931\pi$
$751$			971.631i			1.29378i	−0.762582	+	0.646892i	$0.776069\pi$
$752$	197.974			0.263263
$753$		−	726.201i		−	0.964411i
$754$	353.090			0.468289
$755$		−	302.500i		−	0.400662i
$756$		−	4.27445i		−	0.00565404i
$757$		−	983.792i		−	1.29959i	−0.760108	−	0.649797i	$-0.774854\pi$
$757$		−	983.792i		−	1.29959i	0.760108	−	0.649797i	$-0.225146\pi$
$758$		−	200.394i		−	0.264373i
$759$	−182.321	+	210.316i	−0.240212	+	0.277097i
$760$	−52.5638			−0.0691629
$761$	−73.1433			−0.0961147			−0.0480574	−	0.998845i	$-0.515303\pi$
$761$	−73.1433			−0.0961147			−0.0480574	+	0.998845i	$0.515303\pi$
$762$	−279.729			−0.367098
$763$	18.5127			0.0242631
$764$		−	132.073i		−	0.172870i
$765$	147.777			0.193172
$766$			156.243i			0.203972i
$767$	259.338			0.338119
$768$	27.7128			0.0360844
$769$			687.166i			0.893584i	0.894638	+	0.446792i	$0.147434\pi$
$769$			687.166i			0.893584i	−0.894638	+	0.446792i	$0.852566\pi$
$770$			9.08777i			0.0118023i
$771$	−359.304			−0.466023
$772$	−182.810			−0.236801
$773$		−	56.9245i		−	0.0736411i	−0.999322	−	0.0368205i	$-0.988277\pi$
$773$		−	56.9245i		−	0.0736411i	0.999322	−	0.0368205i	$-0.0117230\pi$
$774$		−	32.1599i		−	0.0415503i
$775$	88.6273			0.114358
$776$			271.288i			0.349597i
$777$	16.3692			0.0210672
$778$			997.653i			1.28233i
$779$			240.021i			0.308114i
$780$			59.3179i			0.0760486i
$781$			187.366i			0.239906i
$782$	469.354	−	541.424i	0.600197	−	0.692359i
$783$	169.411			0.216361
$784$	195.323			0.249137
$785$	386.595			0.492477
$786$	−213.456			−0.271572
$787$			339.738i			0.431688i	0.976428	+	0.215844i	$0.0692502\pi$
$787$			339.738i			0.431688i	−0.976428	+	0.215844i	$0.930750\pi$
$788$	68.1730			0.0865140
$789$			593.298i			0.751961i
$790$	−111.567			−0.141224
$791$	22.3532			0.0282594
$792$		−	59.2864i

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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} +6.98697i q^{11} +3.46410 q^{12} -7.65791 q^{13} +0.581679i q^{14} -3.87298i q^{15} +4.00000 q^{16} +22.0292i q^{17} -4.24264 q^{18} +8.31107i q^{19} -4.47214i q^{20} -0.712409i q^{21} -9.88107i q^{22} +(17.3790 + 15.0656i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +10.8299 q^{26} +5.19615 q^{27} -0.822619i q^{28} +32.6032 q^{29} +5.47723i q^{30} -17.7255 q^{31} -5.65685 q^{32} +12.1018i q^{33} -31.1540i q^{34} -0.919716 q^{35} +6.00000 q^{36} +22.9773i q^{37} -11.7536i q^{38} -13.2639 q^{39} +6.32456i q^{40} +28.8797 q^{41} +1.00750i q^{42} +7.58017i q^{43} +13.9739i q^{44} -6.70820i q^{45} +(-24.5776 - 21.3060i) q^{46} +49.4935 q^{47} +6.92820 q^{48} +48.8308 q^{49} +7.07107 q^{50} +38.1557i q^{51} -15.3158 q^{52} +17.5382i q^{53} -7.34847 q^{54} +15.6233 q^{55} +1.16336i q^{56} +14.3952i q^{57} -46.1078 q^{58} -33.8653 q^{59} -7.74597i q^{60} -39.2567i q^{61} +25.0676 q^{62} -1.23393i q^{63} +8.00000 q^{64} +17.1236i q^{65} -17.1145i q^{66} -70.8924i q^{67} +44.0585i q^{68} +(30.1012 + 26.0944i) q^{69} +1.30067 q^{70} +26.8165 q^{71} -8.48528 q^{72} -62.0148 q^{73} -32.4948i q^{74} -8.66025 q^{75} +16.6221i q^{76} +2.87381 q^{77} +18.7580 q^{78} +35.2805i q^{79} -8.94427i q^{80} +9.00000 q^{81} -40.8420 q^{82} +143.692i q^{83} -1.42482i q^{84} +49.2589 q^{85} -10.7200i q^{86} +56.4704 q^{87} -19.7621i q^{88} +73.4091i q^{89} +9.48683i q^{90} +3.14977i q^{91} +(34.7579 + 30.1312i) q^{92} -30.7014 q^{93} -69.9943 q^{94} +18.5841 q^{95} -9.79796 q^{96} -95.9147i q^{97} -69.0572 q^{98} +20.9609i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32q + 64q^{4} + 96q^{9} + O(q^{10}) \)	\( 32q + 64q^{4} + 96q^{9} - 48q^{13} + 128q^{16} - 80q^{23} - 160q^{25} + 120q^{29} + 248q^{31} - 120q^{35} + 192q^{36} - 48q^{39} + 72q^{41} + 160q^{46} + 400q^{47} - 344q^{49} - 96q^{52} - 256q^{58} + 120q^{59} + 160q^{62} + 256q^{64} + 192q^{69} + 104q^{71} + 16q^{73} + 240q^{77} + 192q^{78} + 288q^{81} + 64q^{82} - 120q^{85} + 144q^{87} - 160q^{92} - 192q^{93} + 96q^{94} - 160q^{95} + 64q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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