LMFDB - Embedded newform 690.3.g.a.461.9

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			461.9
Character	$\chi$	$=$	690.461
Dual form			690.3.g.a.461.10

$q$-expansion

$f(q)$	$=$	$q-1.41421i q^{2} +(2.16025 + 2.08166i) q^{3} -2.00000 q^{4} +2.23607i q^{5} +(2.94392 - 3.05506i) q^{6} +12.7530 q^{7} +2.82843i q^{8} +(0.333364 + 8.99382i) q^{9} +O(q^{10})$	\(q-1.41421i q^{2} +(2.16025 + 2.08166i) q^{3} -2.00000 q^{4} +2.23607i q^{5} +(2.94392 - 3.05506i) q^{6} +12.7530 q^{7} +2.82843i q^{8} +(0.333364 + 8.99382i) q^{9} +3.16228 q^{10} +10.4206i q^{11} +(-4.32050 - 4.16332i) q^{12} -10.0884 q^{13} -18.0355i q^{14} +(-4.65474 + 4.83047i) q^{15} +4.00000 q^{16} -0.278181i q^{17} +(12.7192 - 0.471449i) q^{18} +14.4422 q^{19} -4.47214i q^{20} +(27.5497 + 26.5474i) q^{21} +14.7369 q^{22} +4.79583i q^{23} +(-5.88783 + 6.11011i) q^{24} -5.00000 q^{25} +14.2672i q^{26} +(-18.0020 + 20.1229i) q^{27} -25.5060 q^{28} -1.71142i q^{29} +(6.83131 + 6.58279i) q^{30} -36.9908 q^{31} -5.65685i q^{32} +(-21.6921 + 22.5111i) q^{33} -0.393408 q^{34} +28.5166i q^{35} +(-0.666729 - 17.9876i) q^{36} +17.1960 q^{37} -20.4243i q^{38} +(-21.7936 - 21.0007i) q^{39} -6.32456 q^{40} -36.4515i q^{41} +(37.5437 - 38.9611i) q^{42} -41.5531 q^{43} -20.8412i q^{44} +(-20.1108 + 0.745426i) q^{45} +6.78233 q^{46} +73.4371i q^{47} +(8.64100 + 8.32665i) q^{48} +113.639 q^{49} +7.07107i q^{50} +(0.579080 - 0.600942i) q^{51} +20.1769 q^{52} -20.7239i q^{53} +(28.4580 + 25.4586i) q^{54} -23.3011 q^{55} +36.0709i q^{56} +(31.1987 + 30.0637i) q^{57} -2.42032 q^{58} +74.5843i q^{59} +(9.30948 - 9.66093i) q^{60} +77.3260 q^{61} +52.3129i q^{62} +(4.25139 + 114.698i) q^{63} -8.00000 q^{64} -22.5584i q^{65} +(31.8355 + 30.6773i) q^{66} +103.539 q^{67} +0.556363i q^{68} +(-9.98330 + 10.3602i) q^{69} +40.3285 q^{70} -56.8221i q^{71} +(-25.4384 + 0.942897i) q^{72} -39.9241 q^{73} -24.3188i q^{74} +(-10.8013 - 10.4083i) q^{75} -28.8843 q^{76} +132.894i q^{77} +(-29.6995 + 30.8207i) q^{78} +81.4578 q^{79} +8.94427i q^{80} +(-80.7777 + 5.99644i) q^{81} -51.5502 q^{82} +37.1864i q^{83} +(-55.0993 - 53.0949i) q^{84} +0.622033 q^{85} +58.7650i q^{86} +(3.56261 - 3.69711i) q^{87} -29.4739 q^{88} -129.429i q^{89} +(1.05419 + 28.4410i) q^{90} -128.658 q^{91} -9.59166i q^{92} +(-79.9094 - 77.0024i) q^{93} +103.856 q^{94} +32.2937i q^{95} +(11.7757 - 12.2202i) q^{96} +39.8372 q^{97} -160.710i q^{98} +(-93.7209 + 3.47385i) 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
$3$	2.16025	+	2.08166i	0.720084	+	0.693887i
$3$	2.16025	+	2.08166i	0.720084	+	0.693887i
$4$	−2.00000			−0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$	2.94392	−	3.05506i	0.490653	−	0.509176i
$7$	12.7530			1.82186			0.910928	−	0.412565i	$-0.135367\pi$
$7$	12.7530			1.82186			0.910928	+	0.412565i	$0.135367\pi$
$8$			2.82843i			0.353553i
$9$	0.333364	+	8.99382i	0.0370405	+	0.999314i
$10$	3.16228			0.316228
$11$			10.4206i			0.947326i	0.880706	+	0.473663i	$0.157069\pi$
$11$			10.4206i			0.947326i	−0.880706	+	0.473663i	$0.842931\pi$
$12$	−4.32050	−	4.16332i	−0.360042	−	0.346944i
$13$	−10.0884			−0.776034			−0.388017	−	0.921652i	$-0.626840\pi$
$13$	−10.0884			−0.776034			−0.388017	+	0.921652i	$0.626840\pi$
$14$		−	18.0355i		−	1.28825i
$15$	−4.65474	+	4.83047i	−0.310316	+	0.322031i
$16$	4.00000			0.250000
$17$		−	0.278181i		−	0.0163636i	−0.999967	−	0.00818181i	$-0.997396\pi$
$17$		−	0.278181i		−	0.0163636i	0.999967	−	0.00818181i	$-0.00260438\pi$
$18$	12.7192	−	0.471449i	0.706622	−	0.0261916i
$19$	14.4422			0.760114			0.380057	−	0.924963i	$-0.375904\pi$
$19$	14.4422			0.760114			0.380057	+	0.924963i	$0.375904\pi$
$20$		−	4.47214i		−	0.223607i
$21$	27.5497	+	26.5474i	1.31189	+	1.26416i
$22$	14.7369			0.669861
$23$			4.79583i			0.208514i
$23$			4.79583i			0.208514i
$24$	−5.88783	+	6.11011i	−0.245326	+	0.254588i
$25$	−5.00000			−0.200000
$26$			14.2672i			0.548739i
$27$	−18.0020	+	20.1229i	−0.666739	+	0.745291i
$28$	−25.5060			−0.910928
$29$		−	1.71142i		−	0.0590146i	−0.999565	−	0.0295073i	$-0.990606\pi$
$29$		−	1.71142i		−	0.0590146i	0.999565	−	0.0295073i	$-0.00939383\pi$
$30$	6.83131	+	6.58279i	0.227710	+	0.219426i
$31$	−36.9908			−1.19325			−0.596626	−	0.802520i	$-0.703492\pi$
$31$	−36.9908			−1.19325			−0.596626	+	0.802520i	$0.703492\pi$
$32$		−	5.65685i		−	0.176777i
$33$	−21.6921	+	22.5111i	−0.657338	+	0.682154i
$34$	−0.393408			−0.0115708
$35$			28.5166i			0.814759i
$36$	−0.666729	−	17.9876i	−0.0185202	−	0.499657i
$37$	17.1960			0.464756			0.232378	−	0.972626i	$-0.425349\pi$
$37$	17.1960			0.464756			0.232378	+	0.972626i	$0.425349\pi$
$38$		−	20.4243i		−	0.537482i
$39$	−21.7936	−	21.0007i	−0.558809	−	0.538480i
$40$	−6.32456			−0.158114
$41$		−	36.4515i		−	0.889062i	−0.895764	−	0.444531i	$-0.853370\pi$
$41$		−	36.4515i		−	0.889062i	0.895764	−	0.444531i	$-0.146630\pi$
$42$	37.5437	−	38.9611i	0.893898	−	0.927645i
$43$	−41.5531			−0.966352			−0.483176	−	0.875523i	$-0.660517\pi$
$43$	−41.5531			−0.966352			−0.483176	+	0.875523i	$0.660517\pi$
$44$		−	20.8412i		−	0.473663i
$45$	−20.1108	+	0.745426i	−0.446907	+	0.0165650i
$46$	6.78233			0.147442
$47$			73.4371i			1.56249i	0.624224	+	0.781246i	$0.285415\pi$
$47$			73.4371i			1.56249i	−0.624224	+	0.781246i	$0.714585\pi$
$48$	8.64100	+	8.32665i	0.180021	+	0.173472i
$49$	113.639			2.31916
$50$			7.07107i			0.141421i
$51$	0.579080	−	0.600942i	0.0113545	−	0.0117832i
$52$	20.1769			0.388017
$53$		−	20.7239i		−	0.391017i	−0.980702	−	0.195509i	$-0.937364\pi$
$53$		−	20.7239i		−	0.391017i	0.980702	−	0.195509i	$-0.0626358\pi$
$54$	28.4580	+	25.4586i	0.527001	+	0.471456i
$55$	−23.3011			−0.423657
$56$			36.0709i			0.644123i
$57$	31.1987	+	30.0637i	0.547346	+	0.527434i
$58$	−2.42032			−0.0417297
$59$			74.5843i			1.26414i	0.774911	+	0.632070i	$0.217794\pi$
$59$			74.5843i			1.26414i	−0.774911	+	0.632070i	$0.782206\pi$
$60$	9.30948	−	9.66093i	0.155158	−	0.161016i
$61$	77.3260			1.26764			0.633820	−	0.773481i	$-0.281486\pi$
$61$	77.3260			1.26764			0.633820	+	0.773481i	$0.281486\pi$
$62$			52.3129i			0.843756i
$63$	4.25139	+	114.698i	0.0674825	+	1.82061i
$64$	−8.00000			−0.125000
$65$		−	22.5584i		−	0.347053i
$66$	31.8355	+	30.6773i	0.482356	+	0.464808i
$67$	103.539			1.54536			0.772679	−	0.634796i	$-0.218916\pi$
$67$	103.539			1.54536			0.772679	+	0.634796i	$0.218916\pi$
$68$			0.556363i			0.00818181i
$69$	−9.98330	+	10.3602i	−0.144686	+	0.150148i
$70$	40.3285			0.576122
$71$		−	56.8221i		−	0.800311i	−0.916447	−	0.400156i	$-0.868956\pi$
$71$		−	56.8221i		−	0.800311i	0.916447	−	0.400156i	$-0.131044\pi$
$72$	−25.4384	+	0.942897i	−0.353311	+	0.0130958i
$73$	−39.9241			−0.546905			−0.273453	−	0.961886i	$-0.588166\pi$
$73$	−39.9241			−0.546905			−0.273453	+	0.961886i	$0.588166\pi$
$74$		−	24.3188i		−	0.328632i
$75$	−10.8013	−	10.4083i	−0.144017	−	0.138777i
$76$	−28.8843			−0.380057
$77$			132.894i			1.72589i
$78$	−29.6995	+	30.8207i	−0.380763	+	0.395138i
$79$	81.4578			1.03111			0.515556	−	0.856856i	$-0.327586\pi$
$79$	81.4578			1.03111			0.515556	+	0.856856i	$0.327586\pi$
$80$			8.94427i			0.111803i
$81$	−80.7777	+	5.99644i	−0.997256	+	0.0740302i
$82$	−51.5502			−0.628662
$83$			37.1864i			0.448029i	0.974586	+	0.224014i	$0.0719162\pi$
$83$			37.1864i			0.448029i	−0.974586	+	0.224014i	$0.928084\pi$
$84$	−55.0993	−	53.0949i	−0.655944	−	0.632082i
$85$	0.622033			0.00731803
$86$			58.7650i			0.683314i
$87$	3.56261	−	3.69711i	0.0409495	−	0.0424955i
$88$	−29.4739			−0.334930
$89$		−	129.429i		−	1.45426i	−0.686498	−	0.727132i	$-0.740853\pi$
$89$		−	129.429i		−	1.45426i	0.686498	−	0.727132i	$-0.259147\pi$
$90$	1.05419	+	28.4410i	0.0117132	+	0.316011i
$91$	−128.658			−1.41382
$92$		−	9.59166i		−	0.104257i
$93$	−79.9094	−	77.0024i	−0.859241	−	0.827982i
$94$	103.856			1.10485
$95$			32.2937i			0.339933i
$96$	11.7757	−	12.2202i	0.122663	−	0.127294i
$97$	39.8372			0.410693			0.205346	−	0.978689i	$-0.434168\pi$
$97$	39.8372			0.410693			0.205346	+	0.978689i	$0.434168\pi$
$98$		−	160.710i		−	1.63989i
$99$	−93.7209	+	3.47385i	−0.946676	+	0.0350894i
$100$	10.0000			0.100000
$101$			104.192i			1.03161i	0.856707	+	0.515804i	$0.172507\pi$
$101$			104.192i			1.03161i	−0.856707	+	0.515804i	$0.827493\pi$
$102$	−0.849860	−	0.818942i	−0.00833196	−	0.00802885i
$103$	26.0419			0.252834			0.126417	−	0.991977i	$-0.459652\pi$
$103$	26.0419			0.252834			0.126417	+	0.991977i	$0.459652\pi$
$104$		−	28.5344i		−	0.274369i
$105$	−59.3618	+	61.6029i	−0.565351	+	0.586694i
$106$	−29.3080			−0.276491
$107$		−	28.1668i		−	0.263241i	−0.991300	−	0.131621i	$-0.957982\pi$
$107$		−	28.1668i		−	0.263241i	0.991300	−	0.131621i	$-0.0420181\pi$
$108$	36.0039	−	40.2457i	0.333370	−	0.372646i
$109$	138.704			1.27251			0.636257	−	0.771477i	$-0.280482\pi$
$109$	138.704			1.27251			0.636257	+	0.771477i	$0.280482\pi$
$110$			32.9528i			0.299571i
$111$	37.1476	+	35.7962i	0.334663	+	0.322488i
$112$	51.0120			0.455464
$113$		−	59.4090i		−	0.525743i	−0.964831	−	0.262871i	$-0.915330\pi$
$113$		−	59.4090i		−	0.525743i	0.964831	−	0.262871i	$-0.0846695\pi$
$114$	42.5165	−	44.1216i	0.372952	−	0.387032i
$115$	−10.7238			−0.0932505
$116$			3.42285i			0.0295073i
$117$	−3.36313	−	90.7336i	−0.0287447	−	0.775501i
$118$	105.478			0.893882
$119$		−	3.54765i		−	0.0298122i
$120$	−13.6626	−	13.1656i	−0.113855	−	0.109713i
$121$	12.4114			0.102573
$122$		−	109.355i		−	0.896356i
$123$	75.8798	−	78.7444i	0.616909	−	0.640199i
$124$	73.9816			0.596626
$125$		−	11.1803i		−	0.0894427i
$126$	162.208	−	6.01238i	1.28736	−	0.0477173i
$127$	−127.559			−1.00440			−0.502201	−	0.864751i	$-0.667476\pi$
$127$	−127.559			−1.00440			−0.502201	+	0.864751i	$0.667476\pi$
$128$			11.3137i			0.0883883i
$129$	−89.7652	−	86.4996i	−0.695854	−	0.670539i
$130$	−31.9024			−0.245403
$131$		−	24.8275i		−	0.189523i	−0.995500	−	0.0947616i	$-0.969791\pi$
$131$		−	24.8275i		−	0.189523i	0.995500	−	0.0947616i	$-0.0302089\pi$
$132$	43.3843	−	45.0222i	0.328669	−	0.341077i
$133$	184.181			1.38482
$134$		−	146.426i		−	1.09273i
$135$	−44.9961	−	40.2536i	−0.333304	−	0.298175i
$136$	0.786816			0.00578541
$137$			48.0916i			0.351033i	0.984476	+	0.175517i	$0.0561596\pi$
$137$			48.0916i			0.351033i	−0.984476	+	0.175517i	$0.943840\pi$
$138$	14.6515	+	14.1185i	0.106171	+	0.102308i
$139$	−243.193			−1.74959			−0.874795	−	0.484494i	$-0.839004\pi$
$139$	−243.193			−1.74959			−0.874795	+	0.484494i	$0.839004\pi$
$140$		−	57.0331i		−	0.407379i
$141$	−152.871	+	158.642i	−1.08419	+	1.12512i
$142$	−80.3586			−0.565906
$143$		−	105.127i		−	0.735157i
$144$	1.33346	+	35.9753i	0.00926012	+	0.249828i
$145$	3.82686			0.0263921
$146$			56.4612i			0.386720i
$147$	245.488	+	236.558i	1.66999	+	1.60924i
$148$	−34.3920			−0.232378
$149$		−	168.605i		−	1.13158i	−0.824551	−	0.565788i	$-0.808572\pi$
$149$		−	168.605i		−	1.13158i	0.824551	−	0.565788i	$-0.191428\pi$
$150$	−14.7196	+	15.2753i	−0.0981305	+	0.101835i
$151$	−64.6419			−0.428092			−0.214046	−	0.976824i	$-0.568664\pi$
$151$	−64.6419			−0.428092			−0.214046	+	0.976824i	$0.568664\pi$
$152$			40.8486i			0.268741i
$153$	2.50191	−	0.0927358i	0.0163524	−	0.000606116i
$154$	187.940			1.22039
$155$		−	82.7139i		−	0.533638i
$156$	43.5871	+	42.0014i	0.279404	+	0.269240i
$157$	101.508			0.646547			0.323274	−	0.946306i	$-0.395217\pi$
$157$	101.508			0.646547			0.323274	+	0.946306i	$0.395217\pi$
$158$		−	115.199i		−	0.729106i
$159$	43.1402	−	44.7689i	0.271322	−	0.281565i
$160$	12.6491			0.0790569
$161$			61.1612i			0.379883i
$162$	8.48025	+	114.237i	0.0523472	+	0.705166i
$163$	138.601			0.850310			0.425155	−	0.905120i	$-0.360220\pi$
$163$	138.601			0.850310			0.425155	+	0.905120i	$0.360220\pi$
$164$			72.9031i			0.444531i
$165$	−50.3363	−	48.5051i	−0.305069	−	0.293970i
$166$	52.5895			0.316804
$167$		−	293.071i		−	1.75492i	−0.479651	−	0.877459i	$-0.659237\pi$
$167$		−	293.071i		−	1.75492i	0.479651	−	0.877459i	$-0.340763\pi$
$168$	−75.0875	+	77.9222i	−0.446949	+	0.463823i
$169$	−67.2234			−0.397772
$170$		−	0.879687i		−	0.00517463i
$171$	4.81451	+	129.890i	0.0281550	+	0.759593i
$172$	83.1063			0.483176
$173$		−	139.905i		−	0.808698i	−0.914605	−	0.404349i	$-0.867498\pi$
$173$		−	139.905i		−	0.808698i	0.914605	−	0.404349i	$-0.132502\pi$
$174$	−5.22850	−	5.03829i	−0.0300488	−	0.0289557i
$175$	−63.7650			−0.364371
$176$			41.6823i			0.236832i
$177$	−155.259	+	161.121i	−0.877171	+	0.910286i
$178$	−183.041			−1.02832
$179$		−	77.8984i		−	0.435187i	−0.976040	−	0.217593i	$-0.930179\pi$
$179$		−	77.8984i		−	0.435187i	0.976040	−	0.217593i	$-0.0698206\pi$
$180$	40.2216	−	1.49085i	0.223453	−	0.00828251i
$181$	−39.3128			−0.217198			−0.108599	−	0.994086i	$-0.534636\pi$
$181$	−39.3128			−0.217198			−0.108599	+	0.994086i	$0.534636\pi$
$182$			181.950i			0.999723i
$183$	167.044	+	160.967i	0.912806	+	0.879599i
$184$	−13.5647			−0.0737210
$185$			38.4514i			0.207845i
$186$	−108.898	+	113.009i	−0.585472	+	0.607575i
$187$	2.89881			0.0155017
$188$		−	146.874i		−	0.781246i
$189$	−229.579	+	256.627i	−1.21470	+	1.35781i
$190$	45.6702			0.240369
$191$			112.583i			0.589440i	0.955584	+	0.294720i	$0.0952263\pi$
$191$			112.583i			0.589440i	−0.955584	+	0.294720i	$0.904774\pi$
$192$	−17.2820	−	16.6533i	−0.0900104	−	0.0867359i
$193$	−325.616			−1.68713			−0.843564	−	0.537029i	$-0.819547\pi$
$193$	−325.616			−1.68713			−0.843564	+	0.537029i	$0.819547\pi$
$194$		−	56.3383i		−	0.290404i
$195$	46.9590	−	48.7319i	0.240816	−	0.249907i
$196$	−227.278			−1.15958
$197$		−	232.078i		−	1.17806i	−0.808111	−	0.589031i	$-0.799510\pi$
$197$		−	232.078i		−	1.17806i	0.808111	−	0.589031i	$-0.200490\pi$
$198$	4.91277	+	132.541i	0.0248120	+	0.669401i
$199$	144.995			0.728620			0.364310	−	0.931278i	$-0.381305\pi$
$199$	144.995			0.728620			0.364310	+	0.931278i	$0.381305\pi$
$200$		−	14.1421i		−	0.0707107i
$201$	223.670	+	215.533i	1.11279	+	1.07230i
$202$	147.350			0.729457
$203$		−	21.8258i		−	0.107516i
$204$	−1.15816	+	1.20188i	−0.00567725	+	0.00589158i
$205$	81.5081			0.397600
$206$		−	36.8288i		−	0.178780i
$207$	−43.1329	+	1.59876i	−0.208371	+	0.00772348i
$208$	−40.3537			−0.194008
$209$			150.496i			0.720076i
$210$	87.1197	+	83.9503i	0.414856	+	0.399763i
$211$	152.832			0.724321			0.362160	−	0.932116i	$-0.382039\pi$
$211$	152.832			0.724321			0.362160	+	0.932116i	$0.382039\pi$
$212$			41.4478i			0.195509i
$213$	118.284	−	122.750i	0.555326	−	0.576291i
$214$	−39.8339			−0.186140
$215$		−	92.9156i		−	0.432166i
$216$	−56.9161	−	50.9172i	−0.263500	−	0.235728i
$217$	−471.743			−2.17393
$218$		−	196.157i		−	0.899803i
$219$	−86.2460	−	83.1084i	−0.393817	−	0.379491i
$220$	46.6023			0.211829
$221$			2.80642i			0.0126987i
$222$	50.6235	−	52.5347i	0.228034	−	0.236643i
$223$	275.685			1.23625			0.618127	−	0.786078i	$-0.287892\pi$
$223$	275.685			1.23625			0.618127	+	0.786078i	$0.287892\pi$
$224$		−	72.1418i		−	0.322062i
$225$	−1.66682	−	44.9691i	−0.00740810	−	0.199863i
$226$	−84.0169			−0.371756
$227$		−	372.107i		−	1.63924i	−0.572909	−	0.819619i	$-0.694185\pi$
$227$		−	372.107i		−	1.63924i	0.572909	−	0.819619i	$-0.305815\pi$
$228$	−62.3974	−	60.1274i	−0.273673	−	0.263717i
$229$	−193.596			−0.845397			−0.422699	−	0.906270i	$-0.638917\pi$
$229$	−193.596			−0.845397			−0.422699	+	0.906270i	$0.638917\pi$
$230$			15.1658i			0.0659380i
$231$	−276.640	+	287.084i	−1.19757	+	1.24279i
$232$	4.84064			0.0208648
$233$		−	339.560i		−	1.45734i	−0.684866	−	0.728669i	$-0.740139\pi$
$233$		−	339.560i		−	1.45734i	0.684866	−	0.728669i	$-0.259861\pi$
$234$	−128.317	+	4.75618i	−0.548362	+	0.0203255i
$235$	−164.210			−0.698767
$236$		−	149.169i		−	0.632070i
$237$	175.969	+	169.568i	0.742486	+	0.715475i
$238$	−5.01713			−0.0210804
$239$			190.187i			0.795764i	0.917437	+	0.397882i	$0.130255\pi$
$239$			190.187i			0.795764i	−0.917437	+	0.397882i	$0.869745\pi$
$240$	−18.6190	+	19.3219i	−0.0775790	+	0.0805078i
$241$	−4.45865			−0.0185006			−0.00925031	−	0.999957i	$-0.502945\pi$
$241$	−4.45865			−0.0185006			−0.00925031	+	0.999957i	$0.502945\pi$
$242$		−	17.5523i		−	0.0725302i
$243$	−186.983	−	155.198i	−0.769476	−	0.638676i
$244$	−154.652			−0.633820
$245$			254.104i			1.03716i
$246$	−111.361	−	107.310i	−0.452689	−	0.436220i
$247$	−145.699			−0.589874
$248$		−	104.626i		−	0.421878i
$249$	−77.4095	+	80.3319i	−0.310881	+	0.322618i
$250$	−15.8114			−0.0632456
$251$			32.2001i			0.128287i	0.997941	+	0.0641437i	$0.0204316\pi$
$251$			32.2001i			0.128287i	−0.997941	+	0.0641437i	$0.979568\pi$
$252$	−8.50279	−	229.396i	−0.0337412	−	0.910303i
$253$	−49.9754			−0.197531
$254$			180.396i			0.710220i
$255$	1.34375	+	1.29486i	0.00526959	+	0.00507789i
$256$	16.0000			0.0625000
$257$		−	454.545i		−	1.76866i	−0.466863	−	0.884329i	$-0.654616\pi$
$257$		−	454.545i		−	1.76866i	0.466863	−	0.884329i	$-0.345384\pi$
$258$	−122.329	+	126.947i	−0.474143	+	0.492043i
$259$	219.300			0.846719
$260$			45.1169i			0.173526i
$261$	15.3923	−	0.570528i	0.0589741	−	0.00218593i
$262$	−35.1114			−0.134013
$263$			328.977i			1.25086i	0.780279	+	0.625432i	$0.215077\pi$
$263$			328.977i			1.25086i	−0.780279	+	0.625432i	$0.784923\pi$
$264$	−63.6709	−	61.3546i	−0.241178	−	0.232404i
$265$	46.3401			0.174868
$266$		−	260.471i		−	0.979215i
$267$	269.428	−	279.600i	1.00910	−	1.04719i
$268$	−207.078			−0.772679
$269$		−	35.5555i		−	0.132177i	−0.997814	−	0.0660883i	$-0.978948\pi$
$269$		−	35.5555i		−	0.132177i	0.997814	−	0.0660883i	$-0.0210519\pi$
$270$	−56.9272	+	63.6341i	−0.210841	+	0.235682i
$271$	−253.129			−0.934054			−0.467027	−	0.884243i	$-0.654675\pi$
$271$	−253.129			−0.934054			−0.467027	+	0.884243i	$0.654675\pi$
$272$		−	1.11273i		−	0.00409090i
$273$	−277.933	−	267.822i	−1.01807	−	0.981033i
$274$	68.0118			0.248218
$275$		−	52.1029i		−	0.189465i
$276$	19.9666	−	20.7204i	0.0723428	−	0.0750739i
$277$	−330.147			−1.19187			−0.595933	−	0.803034i	$-0.703218\pi$
$277$	−330.147			−1.19187			−0.595933	+	0.803034i	$0.703218\pi$
$278$			343.927i			1.23715i
$279$	−12.3314	−	332.689i	−0.0441986	−	1.19243i
$280$	−80.6570			−0.288061
$281$			151.413i			0.538838i	0.963023	+	0.269419i	$0.0868316\pi$
$281$			151.413i			0.538838i	−0.963023	+	0.269419i	$0.913168\pi$
$282$	224.354	+	216.193i	0.795583	+	0.766640i
$283$	−214.644			−0.758458			−0.379229	−	0.925303i	$-0.623811\pi$
$283$	−214.644			−0.758458			−0.379229	+	0.925303i	$0.623811\pi$
$284$			113.644i			0.400156i
$285$	−67.2245	+	69.7624i	−0.235876	+	0.244780i
$286$	−148.673			−0.519834
$287$		−	464.866i		−	1.61974i
$288$	50.8768	−	1.88579i	0.176655	−	0.00654790i
$289$	288.923			0.999732
$290$		−	5.41200i		−	0.0186621i
$291$	86.0583	+	82.9276i	0.295733	+	0.284975i
$292$	79.8481			0.273453
$293$		−	220.365i		−	0.752099i	−0.926600	−	0.376050i	$-0.877282\pi$
$293$		−	220.365i		−	0.752099i	0.926600	−	0.376050i	$-0.122718\pi$
$294$	334.543	−	347.173i	1.13790	−	1.18086i
$295$	−166.775			−0.565341
$296$			48.6376i			0.164316i
$297$	−209.692	−	187.591i	−0.706034	−	0.631619i
$298$	−238.443			−0.800145
$299$		−	48.3824i		−	0.161814i
$300$	21.6025	+	20.8166i	0.0720084	+	0.0693887i
$301$	−529.927			−1.76055
$302$			91.4174i			0.302707i
$303$	−216.893	+	225.082i	−0.715820	+	0.742844i
$304$	57.7687			0.190029
$305$			172.906i			0.566906i
$306$	−0.131148	−	3.53824i	−0.000428589	−	0.0115629i
$307$	559.586			1.82276			0.911378	−	0.411571i	$-0.135020\pi$
$307$	559.586			1.82276			0.911378	+	0.411571i	$0.135020\pi$
$308$		−	265.787i		−	0.862946i
$309$	56.2569	+	54.2104i	0.182061	+	0.175438i
$310$	−116.975			−0.377339
$311$			326.410i			1.04955i	0.851241	+	0.524775i	$0.175851\pi$
$311$			326.410i			1.04955i	−0.851241	+	0.524775i	$0.824149\pi$
$312$	59.3990	−	61.6415i	0.190381	−	0.197569i
$313$	−450.011			−1.43773			−0.718867	−	0.695148i	$-0.755339\pi$
$313$	−450.011			−1.43773			−0.718867	+	0.695148i	$0.755339\pi$
$314$		−	143.554i		−	0.457178i
$315$	−256.473	+	9.50641i	−0.814200	+	0.0301791i
$316$	−162.916			−0.515556
$317$		−	153.743i		−	0.484994i	−0.970152	−	0.242497i	$-0.922033\pi$
$317$		−	153.743i		−	0.484994i	0.970152	−	0.242497i	$-0.0779665\pi$
$318$	−63.3127	−	61.0095i	−0.199097	−	0.191854i
$319$	17.8340			0.0559061
$320$		−	17.8885i		−	0.0559017i
$321$	58.6338	−	60.8474i	0.182660	−	0.189556i
$322$	86.4950			0.268618
$323$		−	4.01754i		−	0.0124382i
$324$	161.555	−	11.9929i	0.498628	−	0.0370151i
$325$	50.4422			0.155207
$326$		−	196.011i		−	0.601260i
$327$	299.635	+	288.735i	0.916316	+	0.882982i
$328$	103.100			0.314331
$329$			936.543i			2.84663i
$330$	−68.5966	+	71.1863i	−0.207868	+	0.215716i
$331$	498.766			1.50685			0.753423	−	0.657536i	$-0.228401\pi$
$331$	498.766			1.50685			0.753423	+	0.657536i	$0.228401\pi$
$332$		−	74.3727i		−	0.224014i
$333$	5.73253	+	154.658i	0.0172148	+	0.464437i
$334$	−414.466			−1.24092
$335$			231.520i			0.691105i
$336$	110.199	+	106.190i	0.327972	+	0.316041i
$337$	−238.449			−0.707564			−0.353782	−	0.935328i	$-0.615104\pi$
$337$	−238.449			−0.707564			−0.353782	+	0.935328i	$0.615104\pi$
$338$			95.0683i			0.281267i
$339$	123.669	−	128.338i	0.364806	−	0.378579i
$340$	−1.24407			−0.00365902
$341$		−	385.466i		−	1.13040i
$342$	183.693	−	6.80874i	0.537113	−	0.0199086i
$343$	824.339			2.40332
$344$		−	117.530i		−	0.341657i
$345$	−23.1661	−	22.3233i	−0.0671481	−	0.0647053i
$346$	−197.855			−0.571836
$347$		−	225.681i		−	0.650377i	−0.945649	−	0.325189i	$-0.894572\pi$
$347$		−	225.681i		−	0.650377i	0.945649	−	0.325189i	$-0.105428\pi$
$348$	−7.12522	+	7.39421i	−0.0204748	+	0.0212477i
$349$	100.694			0.288521			0.144261	−	0.989540i	$-0.453920\pi$
$349$	100.694			0.288521			0.144261	+	0.989540i	$0.453920\pi$
$350$			90.1773i			0.257649i
$351$	181.612	−	203.008i	0.517412	−	0.578371i
$352$	58.9477			0.167465
$353$			443.896i			1.25749i	0.777610	+	0.628747i	$0.216432\pi$
$353$			443.896i			1.25749i	−0.777610	+	0.628747i	$0.783568\pi$
$354$	227.859	+	219.570i	0.643670	+	0.620253i
$355$	127.058			0.357910
$356$			258.859i			0.727132i
$357$	7.38500	−	7.66380i	0.0206863	−	0.0214672i
$358$	−110.165			−0.307723
$359$			39.2939i			0.109454i	0.998501	+	0.0547269i	$0.0174288\pi$
$359$			39.2939i			0.109454i	−0.998501	+	0.0547269i	$0.982571\pi$
$360$	−2.10838	−	56.8819i	−0.00585662	−	0.158005i
$361$	−152.424			−0.422226
$362$			55.5968i			0.153582i
$363$	26.8116	+	25.8363i	0.0738613	+	0.0711743i
$364$	257.316			0.706911
$365$		−	89.2729i		−	0.244583i
$366$	227.641	−	236.235i	0.621970	−	0.645451i
$367$	9.93672			0.0270755			0.0135378	−	0.999908i	$-0.495691\pi$
$367$	9.93672			0.0270755			0.0135378	+	0.999908i	$0.495691\pi$
$368$			19.1833i			0.0521286i
$369$	327.839	−	12.1516i	0.888452	−	0.0329313i
$370$	54.3784			0.146969
$371$		−	264.292i		−	0.712377i
$372$	159.819	+	154.005i	0.429620	+	0.413991i
$373$	680.417			1.82417			0.912087	−	0.409997i	$-0.134470\pi$
$373$	680.417			1.82417			0.912087	+	0.409997i	$0.134470\pi$
$374$		−	4.09954i		−	0.0109613i
$375$	23.2737	−	24.1523i	0.0620632	−	0.0644062i
$376$	−207.711			−0.552424
$377$			17.2656i			0.0457973i
$378$	362.925	+	324.673i	0.960119	+	0.858924i
$379$	−602.442			−1.58956			−0.794778	−	0.606900i	$-0.792413\pi$
$379$	−602.442			−1.58956			−0.794778	+	0.606900i	$0.792413\pi$
$380$		−	64.5874i		−	0.169967i
$381$	−275.560	−	265.535i	−0.723254	−	0.696942i
$382$	159.216			0.416797
$383$			92.5380i			0.241613i	0.992676	+	0.120807i	$0.0385481\pi$
$383$			92.5380i			0.241613i	−0.992676	+	0.120807i	$0.961452\pi$
$384$	−23.5513	+	24.4404i	−0.0613316	+	0.0636470i
$385$	−297.159			−0.771842
$386$			460.490i			1.19298i
$387$	−13.8523	−	373.722i	−0.0357942	−	0.965689i
$388$	−79.6744			−0.205346
$389$			512.539i			1.31758i	0.752326	+	0.658791i	$0.228932\pi$
$389$			512.539i			1.31758i	−0.752326	+	0.658791i	$0.771068\pi$
$390$	−68.9173	−	66.4101i	−0.176711	−	0.170282i
$391$	1.33411			0.00341205
$392$			321.419i			0.819947i
$393$	51.6825	−	53.6337i	0.131508	−	0.136473i
$394$	−328.208			−0.833015
$395$			182.145i			0.461127i
$396$	187.442	−	6.94771i	0.473338	−	0.0175447i
$397$	734.655			1.85052			0.925258	−	0.379337i	$-0.123848\pi$
$397$	734.655			1.85052			0.925258	+	0.379337i	$0.123848\pi$
$398$		−	205.054i		−	0.515212i
$399$	397.877	+	383.402i	0.997185	+	0.960908i
$400$	−20.0000			−0.0500000
$401$			264.580i			0.659799i	0.944016	+	0.329900i	$0.107015\pi$
$401$			264.580i			0.659799i	−0.944016	+	0.329900i	$0.892985\pi$
$402$	304.810	−	316.318i	0.758234	−	0.786859i
$403$	373.179			0.926003
$404$		−	208.385i		−	0.515804i
$405$	−13.4085	−	180.625i	−0.0331073	−	0.445986i
$406$	−30.8663			−0.0760254
$407$			179.192i			0.440276i
$408$	1.69972	+	1.63788i	0.00416598	+	0.00401442i
$409$	−13.6447			−0.0333612			−0.0166806	−	0.999861i	$-0.505310\pi$
$409$	−13.6447			−0.0333612			−0.0166806	+	0.999861i	$0.505310\pi$
$410$		−	115.270i		−	0.281146i
$411$	−100.110	+	103.890i	−0.243578	+	0.252773i
$412$	−52.0837			−0.126417
$413$			951.173i			2.30308i
$414$	2.26099	+	60.9991i	0.00546132	+	0.147341i
$415$	−83.1513			−0.200364
$416$			57.0688i			0.137185i
$417$	−525.358	−	506.246i	−1.25985	−	1.21402i
$418$	212.833			0.509171
$419$		−	664.827i		−	1.58670i	−0.608766	−	0.793350i	$-0.708335\pi$
$419$		−	664.827i		−	1.58670i	0.608766	−	0.793350i	$-0.291665\pi$
$420$	118.724	−	123.206i	0.282675	−	0.293347i
$421$	−55.6544			−0.132196			−0.0660978	−	0.997813i	$-0.521055\pi$
$421$	−55.6544			−0.132196			−0.0660978	+	0.997813i	$0.521055\pi$
$422$		−	216.137i		−	0.512172i
$423$	−660.480	+	24.4813i	−1.56142	+	0.0578754i
$424$	58.6161			0.138246
$425$			1.39091i			0.00327272i
$426$	−173.595	−	167.279i	−0.407499	−	0.392675i
$427$	986.138			2.30946
$428$			56.3336i			0.131621i
$429$	218.840	−	227.102i	0.510116	−	0.529374i
$430$	−131.403			−0.305587
$431$		−	771.728i		−	1.79055i	−0.445513	−	0.895276i	$-0.646979\pi$
$431$		−	771.728i		−	1.79055i	0.445513	−	0.895276i	$-0.353021\pi$
$432$	−72.0078	+	80.4915i	−0.166685	+	0.186323i
$433$	853.202			1.97044			0.985222	−	0.171281i	$-0.0547905\pi$
$433$	853.202			1.97044			0.985222	+	0.171281i	$0.0547905\pi$
$434$			667.146i			1.53720i
$435$	8.26698	+	7.96623i	0.0190046	+	0.0183132i
$436$	−277.408			−0.636257
$437$			69.2622i			0.158495i
$438$	−117.533	+	121.970i	−0.268340	+	0.278471i
$439$	−593.263			−1.35140			−0.675698	−	0.737179i	$-0.736158\pi$
$439$	−593.263			−1.35140			−0.675698	+	0.737179i	$0.736158\pi$
$440$		−	65.9056i		−	0.149785i
$441$	37.8832	+	1022.05i	0.0859028	+	2.31757i
$442$	3.96887			0.00897935
$443$			467.644i			1.05563i	0.849359	+	0.527815i	$0.176989\pi$
$443$			467.644i			1.05563i	−0.849359	+	0.527815i	$0.823011\pi$
$444$	−74.2952	−	71.5924i	−0.167332	−	0.161244i
$445$	289.413			0.650367
$446$		−	389.877i		−	0.874164i
$447$	350.978	−	364.228i	0.785186	−	0.814829i
$448$	−102.024			−0.227732
$449$		−	863.590i		−	1.92336i	−0.274168	−	0.961682i	$-0.588403\pi$
$449$		−	863.590i		−	1.92336i	0.274168	−	0.961682i	$-0.411597\pi$
$450$	−63.5959	+	2.35724i	−0.141324	+	0.00523832i
$451$	379.846			0.842231
$452$			118.818i			0.262871i
$453$	−139.643	−	134.563i	−0.308262	−	0.297048i
$454$	−526.239			−1.15912
$455$		−	287.688i		−	0.632280i
$456$	−85.0330	+	88.2433i	−0.186476	+	0.193516i
$457$	−352.546			−0.771435			−0.385717	−	0.922617i	$-0.626046\pi$
$457$	−352.546			−0.771435			−0.385717	+	0.922617i	$0.626046\pi$
$458$			273.786i			0.597786i
$459$	5.59781	+	5.00781i	0.0121957	+	0.0109103i
$460$	21.4476			0.0466252
$461$			595.807i			1.29242i	0.763158	+	0.646212i	$0.223648\pi$
$461$			595.807i			1.29242i	−0.763158	+	0.646212i	$0.776352\pi$
$462$	405.998	+	391.228i	0.878783	+	0.846813i
$463$	671.390			1.45009			0.725043	−	0.688704i	$-0.241820\pi$
$463$	671.390			1.45009			0.725043	+	0.688704i	$0.241820\pi$
$464$		−	6.84570i		−	0.0147537i
$465$	172.182	−	178.683i	0.370285	−	0.384264i
$466$	−480.210			−1.03049
$467$		−	797.745i		−	1.70823i	−0.520082	−	0.854116i	$-0.674099\pi$
$467$		−	797.745i		−	1.70823i	0.520082	−	0.854116i	$-0.325901\pi$
$468$	6.72625	+	181.467i	0.0143723	+	0.387751i
$469$	1320.43			2.81542
$470$			232.228i			0.494103i
$471$	219.282	+	211.305i	0.465568	+	0.448631i
$472$	−210.956			−0.446941
$473$		−	433.008i		−	0.915450i
$474$	239.805	−	248.858i	0.505917	−	0.525017i
$475$	−72.2109			−0.152023
$476$			7.09529i			0.0149061i
$477$	186.387	−	6.90862i	0.390749	−	0.0144835i
$478$	268.966			0.562690
$479$		−	706.387i		−	1.47471i	−0.675505	−	0.737356i	$-0.736074\pi$
$479$		−	706.387i		−	1.47471i	0.675505	−	0.737356i	$-0.263926\pi$
$480$	27.3252	+	26.3312i	0.0569276	+	0.0548566i
$481$	−173.481			−0.360666
$482$			6.30548i			0.0130819i
$483$	−127.317	+	132.124i	−0.263596	+	0.273548i
$484$	−24.8227			−0.0512866
$485$			89.0787i			0.183667i
$486$	−219.483	+	264.433i	−0.451612	+	0.544102i
$487$	747.631			1.53518			0.767589	−	0.640943i	$-0.221456\pi$
$487$	747.631			1.53518			0.767589	+	0.640943i	$0.221456\pi$
$488$			218.711i			0.448178i
$489$	299.412	+	288.520i	0.612294	+	0.590020i
$490$	359.358			0.733383
$491$			526.765i			1.07284i	0.843951	+	0.536420i	$0.180224\pi$
$491$			526.765i			1.07284i	−0.843951	+	0.536420i	$0.819776\pi$
$492$	−151.760	+	157.489i	−0.308454	+	0.320099i
$493$	−0.476087			−0.000965693
$494$			206.049i			0.417104i
$495$	−7.76777	−	209.566i	−0.0156925	−	0.423366i
$496$	−147.963			−0.298313
$497$		−	724.652i		−	1.45805i
$498$	113.606	+	109.474i	0.228125	+	0.219826i
$499$	836.489			1.67633			0.838165	−	0.545417i	$-0.183629\pi$
$499$	836.489			1.67633			0.838165	+	0.545417i	$0.183629\pi$
$500$			22.3607i			0.0447214i
$501$	610.076	−	633.108i	1.21772	−	1.26369i
$502$	45.5378			0.0907128
$503$		−	378.646i		−	0.752775i	−0.926462	−	0.376387i	$-0.877166\pi$
$503$		−	378.646i		−	0.752775i	0.926462	−	0.376387i	$-0.122834\pi$
$504$	−324.415	+	12.0248i	−0.643681	+	0.0238587i
$505$	−232.981			−0.461349
$506$			70.6759i			0.139676i
$507$	−145.219	−	139.937i	−0.286429	−	0.276009i
$508$	255.118			0.502201
$509$			449.590i			0.883280i	0.897192	+	0.441640i	$0.145603\pi$
$509$			449.590i			0.883280i	−0.897192	+	0.441640i	$0.854397\pi$
$510$	1.83121	−	1.90034i	0.00359061	−	0.00372616i
$511$	−509.151			−0.996382
$512$		−	22.6274i		−	0.0441942i
$513$	−259.987	+	290.618i	−0.506798	+	0.566507i
$514$	−642.824			−1.25063
$515$			58.2314i			0.113071i
$516$	179.530	+	172.999i	0.347927	+	0.335270i
$517$	−765.258			−1.48019
$518$		−	310.137i		−	0.598721i
$519$	291.234	−	302.229i	0.561145	−	0.582330i
$520$	63.8049			0.122702
$521$			644.286i			1.23663i	0.785929	+	0.618317i	$0.212185\pi$
$521$			644.286i			1.23663i	−0.785929	+	0.618317i	$0.787815\pi$
$522$	−0.806849	−	21.7679i	−0.00154569	−	0.0417010i
$523$	312.889			0.598257			0.299129	−	0.954213i	$-0.403304\pi$
$523$	312.889			0.598257			0.299129	+	0.954213i	$0.403304\pi$
$524$			49.6551i			0.0947616i
$525$	−137.748	−	132.737i	−0.262378	−	0.252833i
$526$	465.244			0.884494
$527$			10.2902i			0.0195259i
$528$	−86.7686	+	90.0443i	−0.164334	+	0.170538i
$529$	−23.0000			−0.0434783
$530$		−	65.5348i		−	0.123651i
$531$	−670.798	+	24.8637i	−1.26327	+	0.0468244i
$532$	−368.362			−0.692409
$533$			367.739i			0.689942i
$534$	−395.414	−	381.029i	−0.740476	−	0.713538i
$535$	62.9829			0.117725
$536$			292.853i			0.546367i
$537$	162.158	−	168.280i	0.301971	−	0.313371i
$538$	−50.2831			−0.0934629
$539$			1184.18i			2.19700i
$540$	89.9922	+	80.5072i	0.166652	+	0.149087i
$541$	−25.0281			−0.0462627			−0.0231314	−	0.999732i	$-0.507364\pi$
$541$	−25.0281			−0.0462627			−0.0231314	+	0.999732i	$0.507364\pi$
$542$			357.978i			0.660476i
$543$	−84.9256	−	81.8361i	−0.156401	−	0.150711i
$544$	−1.57363			−0.00289271
$545$			310.152i			0.569086i
$546$	−378.758	+	393.057i	−0.693695	+	0.719884i
$547$	−285.873			−0.522619			−0.261310	−	0.965255i	$-0.584154\pi$
$547$	−285.873			−0.522619			−0.261310	+	0.965255i	$0.584154\pi$
$548$		−	96.1832i		−	0.175517i
$549$	25.7777	+	695.456i	0.0469540	+	1.26677i
$550$	−73.6847			−0.133972
$551$		−	24.7167i		−	0.0448579i
$552$	−29.3031	−	28.2370i	−0.0530853	−	0.0511541i
$553$	1038.83			1.87854
$554$			466.898i			0.842776i
$555$	−80.0428	+	83.0646i	−0.144221	+	0.149666i
$556$	486.386			0.874795
$557$			768.565i			1.37983i	0.723890	+	0.689915i	$0.242352\pi$
$557$			768.565i			1.37983i	−0.723890	+	0.689915i	$0.757648\pi$
$558$	−470.493	+	17.4393i	−0.843177	+	0.0312531i
$559$	419.206			0.749922
$560$			114.066i			0.203690i
$561$	6.26216	+	6.03435i	0.0111625	+	0.0107564i
$562$	214.131			0.381016
$563$			472.921i			0.840002i	0.907524	+	0.420001i	$0.137970\pi$
$563$			472.921i			0.840002i	−0.907524	+	0.420001i	$0.862030\pi$
$564$	305.742	−	317.285i	0.542096	−	0.562562i
$565$	132.842			0.235119
$566$			303.552i			0.536311i
$567$	−1030.16	+	76.4726i	−1.81686	+	0.134872i
$568$	160.717			0.282953
$569$		−	44.6673i		−	0.0785013i	−0.999229	−	0.0392507i	$-0.987503\pi$
$569$		−	44.6673i		−	0.0785013i	0.999229	−	0.0392507i	$-0.0124971\pi$
$570$	98.6590	+	95.0698i	0.173086	+	0.166789i
$571$	−502.895			−0.880726			−0.440363	−	0.897820i	$-0.645150\pi$
$571$	−502.895			−0.880726			−0.440363	+	0.897820i	$0.645150\pi$
$572$			210.255i			0.367578i
$573$	−234.360	+	243.207i	−0.409005	+	0.424446i
$574$	−657.420			−1.14533
$575$		−	23.9792i		−	0.0417029i
$576$	−2.66692	−	71.9506i	−0.00463006	−	0.124914i
$577$	−94.3235			−0.163472			−0.0817361	−	0.996654i	$-0.526046\pi$
$577$	−94.3235			−0.163472			−0.0817361	+	0.996654i	$0.526046\pi$
$578$		−	408.598i		−	0.706917i
$579$	−703.411	−	677.822i	−1.21487	−	1.17068i
$580$	−7.65372			−0.0131961
$581$			474.238i			0.816244i
$582$	117.277	−	121.705i	0.201507	−	0.209115i
$583$	215.955			0.370421
$584$		−	112.922i		−	0.193360i
$585$	202.887	−	7.52018i	0.346815	−	0.0128550i
$586$	−311.643			−0.531814
$587$		−	883.355i		−	1.50486i	−0.658670	−	0.752432i	$-0.728881\pi$
$587$		−	883.355i		−	1.50486i	0.658670	−	0.752432i	$-0.271119\pi$
$588$	−490.977	−	473.115i	−0.834994	−	0.804618i
$589$	−534.227			−0.907008
$590$			235.856i			0.399756i
$591$	483.108	−	501.347i	0.817442	−	0.848303i
$592$	68.7839			0.116189
$593$		−	951.840i		−	1.60513i	−0.596567	−	0.802563i	$-0.703469\pi$
$593$		−	951.840i		−	1.60513i	0.596567	−	0.802563i	$-0.296531\pi$
$594$	−265.294	+	296.549i	−0.446622	+	0.499241i
$595$	7.93278			0.0133324
$596$			337.209i			0.565788i
$597$	313.226	+	301.831i	0.524667	+	0.505580i
$598$	−68.4231			−0.114420
$599$		−	118.761i		−	0.198265i	−0.995074	−	0.0991326i	$-0.968393\pi$
$599$		−	118.761i		−	0.198265i	0.995074	−	0.0991326i	$-0.0316068\pi$
$600$	29.4392	−	30.5506i	0.0490653	−	0.0509176i
$601$	40.5976			0.0675500			0.0337750	−	0.999429i	$-0.489247\pi$
$601$	40.5976			0.0675500			0.0337750	+	0.999429i	$0.489247\pi$
$602$			749.430i			1.24490i
$603$	34.5162	+	931.212i	0.0572409	+	1.54430i
$604$	129.284			0.214046
$605$			27.7526i			0.0458721i
$606$	318.314	+	306.734i	0.525270	+	0.506161i
$607$	−1154.28			−1.90162			−0.950809	−	0.309777i	$-0.899746\pi$
$607$	−1154.28			−1.90162			−0.950809	+	0.309777i	$0.899746\pi$
$608$		−	81.6973i		−	0.134370i
$609$	45.4339	−	47.1492i	0.0746041	−	0.0774206i
$610$	244.526			0.400863
$611$		−	740.865i		−	1.21255i
$612$	−5.00383	+	0.185472i	−0.00817619	+	0.000303058i
$613$	−673.217			−1.09823			−0.549117	−	0.835746i	$-0.685036\pi$
$613$	−673.217			−1.09823			−0.549117	+	0.835746i	$0.685036\pi$
$614$		−	791.374i		−	1.28888i
$615$	176.078	+	169.672i	0.286306	+	0.275890i
$616$	−375.880			−0.610195
$617$			376.318i			0.609916i	0.952366	+	0.304958i	$0.0986424\pi$
$617$			376.318i			0.609916i	−0.952366	+	0.304958i	$0.901358\pi$
$618$	76.6650	−	79.5593i	0.124053	−	0.128737i
$619$	276.885			0.447310			0.223655	−	0.974668i	$-0.428201\pi$
$619$	276.885			0.447310			0.223655	+	0.974668i	$0.428201\pi$
$620$			165.428i			0.266819i
$621$	−96.5059	−	86.3343i	−0.155404	−	0.139025i
$622$	461.613			0.742144
$623$		−	1650.61i		−	2.64946i
$624$	−87.1742	−	84.0029i	−0.139702	−	0.134620i
$625$	25.0000			0.0400000
$626$			636.411i			1.01663i
$627$	−313.282	+	325.109i	−0.499652	+	0.518515i
$628$	−203.016			−0.323274
$629$		−	4.78360i		−	0.00760509i
$630$	13.4441	+	362.707i	0.0213398	+	0.575726i
$631$	−760.606			−1.20540			−0.602699	−	0.797969i	$-0.705908\pi$
$631$	−760.606			−1.20540			−0.602699	+	0.797969i	$0.705908\pi$
$632$			230.397i			0.364553i
$633$	330.155	+	318.144i	0.521572	+	0.502597i
$634$	−217.426			−0.342943
$635$		−	285.231i		−	0.449183i
$636$	−86.2804	+	89.5377i	−0.135661	+	0.140783i
$637$	−1146.44			−1.79975
$638$		−	25.2212i		−	0.0395316i
$639$	511.048	−	18.9425i	0.799762	−	0.0296439i
$640$	−25.2982			−0.0395285
$641$		−	292.499i		−	0.456316i	−0.973624	−	0.228158i	$-0.926730\pi$
$641$		−	292.499i		−	0.456316i	0.973624	−	0.228158i	$-0.0732703\pi$
$642$	−86.0512	−	82.9207i	−0.134036	−	0.129160i
$643$	487.514			0.758187			0.379094	−	0.925358i	$-0.376236\pi$
$643$	487.514			0.758187			0.379094	+	0.925358i	$0.376236\pi$
$644$		−	122.322i		−	0.189942i
$645$	193.419	−	200.721i	0.299874	−	0.311195i
$646$	−5.68166			−0.00879515
$647$			764.731i			1.18196i	0.806685	+	0.590982i	$0.201260\pi$
$647$			764.731i			1.18196i	−0.806685	+	0.590982i	$0.798740\pi$
$648$	−16.9605	−	228.474i	−0.0261736	−	0.352583i
$649$	−777.212			−1.19755
$650$		−	71.3360i		−	0.109748i
$651$	−1019.08	−	982.010i	−1.56541	−	1.50846i
$652$	−277.201			−0.425155
$653$		−	398.531i		−	0.610308i	−0.952303	−	0.305154i	$-0.901292\pi$
$653$		−	398.531i		−	0.610308i	0.952303	−	0.305154i	$-0.0987079\pi$
$654$	408.333	−	423.749i	0.624362	−	0.647934i
$655$	55.5161			0.0847573
$656$		−	145.806i		−	0.222265i
$657$	−13.3093	−	359.070i	−0.0202576	−	0.546530i
$658$	1324.47			2.01287
$659$			828.825i			1.25770i	0.777526	+	0.628850i	$0.216474\pi$
$659$			828.825i			1.25770i	−0.777526	+	0.628850i	$0.783526\pi$
$660$	100.673	+	97.0102i	0.152534	+	0.146985i
$661$	−1248.13			−1.88825			−0.944124	−	0.329591i	$-0.893089\pi$
$661$	−1248.13			−1.88825			−0.944124	+	0.329591i	$0.893089\pi$
$662$		−	705.362i		−	1.06550i
$663$	−5.84201	+	6.06256i	−0.00881148	+	0.00914413i
$664$	−105.179			−0.158402
$665$			411.841i			0.619310i
$666$	218.719	−	8.10702i	0.328407	−	0.0121727i
$667$	8.20770			0.0123054
$668$			586.143i			0.877459i
$669$	595.548	+	573.883i	0.890206	+	0.857821i
$670$	327.419			0.488685
$671$			805.782i			1.20087i
$672$	150.175	−	155.844i	0.223475	−	0.231911i
$673$	−1228.15			−1.82489			−0.912447	−	0.409195i	$-0.865810\pi$
$673$	−1228.15			−1.82489			−0.912447	+	0.409195i	$0.865810\pi$
$674$			337.218i			0.500323i
$675$	90.0098	−	100.614i	0.133348	−	0.149058i
$676$	134.447			0.198886
$677$		−	197.067i		−	0.291089i	−0.989352	−	0.145544i	$-0.953507\pi$
$677$		−	197.067i		−	0.291089i	0.989352	−	0.145544i	$-0.0464934\pi$
$678$	−181.498	−	174.895i	−0.267696	−	0.257957i
$679$	508.044			0.748223
$680$			1.75937i			0.00258731i
$681$	774.601	−	803.844i	1.13745	−	1.18039i
$682$	−545.131			−0.799312
$683$			650.986i			0.953127i	0.879140	+	0.476564i	$0.158118\pi$
$683$			650.986i			0.953127i	−0.879140	+	0.476564i	$0.841882\pi$
$684$	−9.62901	−	259.781i	−0.0140775	−	0.379796i
$685$	−107.536			−0.156987
$686$		−	1165.79i		−	1.69940i
$687$	−418.216	−	403.001i	−0.608757	−	0.586610i
$688$	−166.213			−0.241588
$689$			209.072i			0.303443i
$690$	−31.5700	+	32.7618i	−0.0457536	+	0.0474809i
$691$	485.418			0.702486			0.351243	−	0.936284i	$-0.385759\pi$
$691$	485.418			0.702486			0.351243	+	0.936284i	$0.385759\pi$
$692$			279.809i			0.404349i
$693$	−1195.22	+	44.3020i	−1.72471	+	0.0639279i
$694$	−319.161			−0.459886
$695$		−	543.796i		−	0.782440i
$696$	10.4570	+	10.0766i	0.0150244	+	0.0144778i
$697$	−10.1401			−0.0145483
$698$		−	142.403i		−	0.204015i
$699$	706.848	−	733.534i	1.01123	−	1.04940i
$700$	127.530			0.182186
$701$		−	1073.24i		−	1.53101i	−0.643431	−	0.765504i	$-0.722490\pi$
$701$		−	1073.24i		−	1.53101i	0.643431	−	0.765504i	$-0.277510\pi$
$702$	−287.097	−	256.838i	−0.408970	−	0.365865i
$703$	248.347			0.353268
$704$		−	83.3647i		−	0.118416i
$705$	−354.735	−	341.830i	−0.503171	−	0.484866i
$706$	627.763			0.889183
$707$			1328.77i			1.87944i
$708$	310.518	−	322.241i	0.438585	−	0.455143i
$709$	−83.9792			−0.118447			−0.0592237	−	0.998245i	$-0.518863\pi$
$709$	−83.9792			−0.118447			−0.0592237	+	0.998245i	$0.518863\pi$
$710$		−	179.687i		−	0.253081i
$711$	27.1551	+	732.617i	0.0381929	+	1.03040i
$712$	366.082			0.514160
$713$		−	177.402i		−	0.248810i
$714$	−10.8383	−	10.4440i	−0.0151796	−	0.0146274i
$715$	235.072			0.328772
$716$			155.797i			0.217593i
$717$	−395.906	+	410.853i	−0.552170	+	0.573016i
$718$	55.5700			0.0773955
$719$			638.283i			0.887737i	0.896092	+	0.443868i	$0.146394\pi$
$719$			638.283i			0.887737i	−0.896092	+	0.443868i	$0.853606\pi$
$720$	−80.4432	+	2.98170i	−0.111727	+	0.00414125i
$721$	332.112			0.460626
$722$			215.560i			0.298559i
$723$	−9.63180	−	9.28140i	−0.0133220	−	0.0128374i
$724$	78.6257			0.108599
$725$			8.55712i			0.0118029i
$726$	36.5380	−	37.9174i	0.0503278	−	0.0522278i
$727$	344.419			0.473754			0.236877	−	0.971540i	$-0.423876\pi$
$727$	344.419			0.473754			0.236877	+	0.971540i	$0.423876\pi$
$728$		−	363.899i		−	0.499861i
$729$	−80.8594	−	724.502i	−0.110918	−	0.993830i
$730$	−126.251			−0.172947
$731$			11.5593i			0.0158130i
$732$	−334.087	−	321.933i	−0.456403	−	0.439799i
$733$	−511.714			−0.698109			−0.349054	−	0.937103i	$-0.613497\pi$
$733$	−511.714			−0.698109			−0.349054	+	0.937103i	$0.613497\pi$
$734$		−	14.0526i		−	0.0191453i
$735$	−528.959	+	548.929i	−0.719672	+	0.746842i
$736$	27.1293			0.0368605
$737$			1078.94i			1.46396i
$738$	−17.1850	−	463.634i	−0.0232859	−	0.628230i
$739$	256.144			0.346608			0.173304	−	0.984868i	$-0.444556\pi$
$739$	256.144			0.346608			0.173304	+	0.984868i	$0.444556\pi$
$740$		−	76.9027i		−	0.103923i
$741$	−314.746	−	303.296i	−0.424759	−	0.409306i
$742$	−373.765			−0.503727
$743$			74.9836i			0.100920i	0.998726	+	0.0504600i	$0.0160687\pi$
$743$			74.9836i			0.100920i	−0.998726	+	0.0504600i	$0.983931\pi$
$744$	217.796	−	226.018i	0.292736	−	0.303787i
$745$	377.012			0.506056
$746$		−	962.255i		−	1.28989i
$747$	−334.448	+	12.3966i	−0.447721	+	0.0165952i
$748$	−5.79763			−0.00775084
$749$		−	359.211i		−	0.479588i
$750$	−34.1566	−	32.9140i	−0.0455421	−	0.0438853i
$751$	−920.046			−1.22509			−0.612547	−	0.790434i	$-0.709855\pi$
$751$	−920.046			−1.22509			−0.612547	+	0.790434i	$0.709855\pi$
$752$			293.748i			0.390623i
$753$	−67.0298	+	69.5603i	−0.0890170	+	0.0923776i
$754$	24.4172			0.0323836
$755$		−	144.544i		−	0.191449i
$756$	459.158	−	513.254i	0.607351	−	0.678907i
$757$	1040.64			1.37469			0.687344	−	0.726332i	$-0.258777\pi$
$757$	1040.64			1.37469			0.687344	+	0.726332i	$0.258777\pi$
$758$			851.982i			1.12399i
$759$	−107.959	−	104.032i	−0.142239	−	0.137064i
$760$	−91.3403			−0.120185
$761$			1047.60i			1.37662i	0.725419	+	0.688308i	$0.241646\pi$
$761$			1047.60i			1.37662i	−0.725419	+	0.688308i	$0.758354\pi$
$762$	−375.523	+	389.700i	−0.492813	+	0.511418i
$763$	1768.89			2.31834
$764$		−	225.166i		−	0.294720i
$765$	0.207364	+	5.59445i	0.000271063	+	0.00731301i
$766$	130.868			0.170847
$767$		−	752.439i		−	0.981015i
$768$	34.5640	+	33.3066i	0.0450052	+	0.0433680i
$769$	650.393			0.845764			0.422882	−	0.906185i	$-0.361018\pi$
$769$	650.393			0.845764			0.422882	+	0.906185i	$0.361018\pi$
$770$			420.247i			0.545775i
$771$	946.210	−	981.932i	1.22725	−	1.27358i
$772$	651.231			0.843564
$773$			1302.06i			1.68442i	0.539149	+	0.842210i	$0.318746\pi$
$773$			1302.06i			1.68442i	−0.539149	+	0.842210i	$0.681254\pi$
$774$	−528.522	+	19.5902i	−0.682845	+	0.0253103i
$775$	184.954			0.238650
$776$			112.677i			0.145202i
$777$	473.743	+	456.509i	0.609708	+	0.587528i
$778$	724.840			0.931671
$779$		−	526.439i		−	0.675788i
$780$	−93.9181	+	97.4637i	−0.120408	+	0.124953i
$781$	592.120			0.758156
$782$		−	1.88672i		−	0.00241268i
$783$	34.4388	+	30.8090i	0.0439831	+	0.0393474i
$784$	454.555			0.579790
$785$			226.979i			0.289145i
$786$	−75.8495	−	73.0902i

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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(2.16025 + 2.08166i) q^{3} -2.00000 q^{4} +2.23607i q^{5} +(2.94392 - 3.05506i) q^{6} +12.7530 q^{7} +2.82843i q^{8} +(0.333364 + 8.99382i) q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +(2.16025 + 2.08166i) q^{3} -2.00000 q^{4} +2.23607i q^{5} +(2.94392 - 3.05506i) q^{6} +12.7530 q^{7} +2.82843i q^{8} +(0.333364 + 8.99382i) q^{9} +3.16228 q^{10} +10.4206i q^{11} +(-4.32050 - 4.16332i) q^{12} -10.0884 q^{13} -18.0355i q^{14} +(-4.65474 + 4.83047i) q^{15} +4.00000 q^{16} -0.278181i q^{17} +(12.7192 - 0.471449i) q^{18} +14.4422 q^{19} -4.47214i q^{20} +(27.5497 + 26.5474i) q^{21} +14.7369 q^{22} +4.79583i q^{23} +(-5.88783 + 6.11011i) q^{24} -5.00000 q^{25} +14.2672i q^{26} +(-18.0020 + 20.1229i) q^{27} -25.5060 q^{28} -1.71142i q^{29} +(6.83131 + 6.58279i) q^{30} -36.9908 q^{31} -5.65685i q^{32} +(-21.6921 + 22.5111i) q^{33} -0.393408 q^{34} +28.5166i q^{35} +(-0.666729 - 17.9876i) q^{36} +17.1960 q^{37} -20.4243i q^{38} +(-21.7936 - 21.0007i) q^{39} -6.32456 q^{40} -36.4515i q^{41} +(37.5437 - 38.9611i) q^{42} -41.5531 q^{43} -20.8412i q^{44} +(-20.1108 + 0.745426i) q^{45} +6.78233 q^{46} +73.4371i q^{47} +(8.64100 + 8.32665i) q^{48} +113.639 q^{49} +7.07107i q^{50} +(0.579080 - 0.600942i) q^{51} +20.1769 q^{52} -20.7239i q^{53} +(28.4580 + 25.4586i) q^{54} -23.3011 q^{55} +36.0709i q^{56} +(31.1987 + 30.0637i) q^{57} -2.42032 q^{58} +74.5843i q^{59} +(9.30948 - 9.66093i) q^{60} +77.3260 q^{61} +52.3129i q^{62} +(4.25139 + 114.698i) q^{63} -8.00000 q^{64} -22.5584i q^{65} +(31.8355 + 30.6773i) q^{66} +103.539 q^{67} +0.556363i q^{68} +(-9.98330 + 10.3602i) q^{69} +40.3285 q^{70} -56.8221i q^{71} +(-25.4384 + 0.942897i) q^{72} -39.9241 q^{73} -24.3188i q^{74} +(-10.8013 - 10.4083i) q^{75} -28.8843 q^{76} +132.894i q^{77} +(-29.6995 + 30.8207i) q^{78} +81.4578 q^{79} +8.94427i q^{80} +(-80.7777 + 5.99644i) q^{81} -51.5502 q^{82} +37.1864i q^{83} +(-55.0993 - 53.0949i) q^{84} +0.622033 q^{85} +58.7650i q^{86} +(3.56261 - 3.69711i) q^{87} -29.4739 q^{88} -129.429i q^{89} +(1.05419 + 28.4410i) q^{90} -128.658 q^{91} -9.59166i q^{92} +(-79.9094 - 77.0024i) q^{93} +103.856 q^{94} +32.2937i q^{95} +(11.7757 - 12.2202i) q^{96} +39.8372 q^{97} -160.710i q^{98} +(-93.7209 + 3.47385i) 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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