LMFDB - Embedded newform 690.3.g.a.461.6

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			461.6
Character	$\chi$	$=$	690.461
Dual form			690.3.g.a.461.5

$q$-expansion

$f(q)$	$=$	$q+1.41421i q^{2} +(-2.73248 + 1.23837i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-1.75132 - 3.86431i) q^{6} +4.14117 q^{7} -2.82843i q^{8} +(5.93289 - 6.76763i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(-2.73248 + 1.23837i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-1.75132 - 3.86431i) q^{6} +4.14117 q^{7} -2.82843i q^{8} +(5.93289 - 6.76763i) q^{9} +3.16228 q^{10} +18.2040i q^{11} +(5.46496 - 2.47674i) q^{12} -8.45273 q^{13} +5.85650i q^{14} +(2.76908 + 6.11001i) q^{15} +4.00000 q^{16} -14.2788i q^{17} +(9.57088 + 8.39037i) q^{18} -25.5703 q^{19} +4.47214i q^{20} +(-11.3157 + 5.12830i) q^{21} -25.7443 q^{22} +4.79583i q^{23} +(3.50263 + 7.72862i) q^{24} -5.00000 q^{25} -11.9540i q^{26} +(-7.83068 + 25.8395i) q^{27} -8.28235 q^{28} -30.0240i q^{29} +(-8.64086 + 3.91606i) q^{30} -20.5971 q^{31} +5.65685i q^{32} +(-22.5432 - 49.7420i) q^{33} +20.1932 q^{34} -9.25994i q^{35} +(-11.8658 + 13.5353i) q^{36} +58.2578 q^{37} -36.1618i q^{38} +(23.0969 - 10.4676i) q^{39} -6.32456 q^{40} -55.4616i q^{41} +(-7.25251 - 16.0028i) q^{42} -61.0293 q^{43} -36.4080i q^{44} +(-15.1329 - 13.2663i) q^{45} -6.78233 q^{46} -13.9735i q^{47} +(-10.9299 + 4.95347i) q^{48} -31.8507 q^{49} -7.07107i q^{50} +(17.6824 + 39.0165i) q^{51} +16.9055 q^{52} +25.4683i q^{53} +(-36.5426 - 11.0743i) q^{54} +40.7054 q^{55} -11.7130i q^{56} +(69.8702 - 31.6654i) q^{57} +42.4604 q^{58} -79.7148i q^{59} +(-5.53815 - 12.2200i) q^{60} +95.2262 q^{61} -29.1286i q^{62} +(24.5691 - 28.0259i) q^{63} -8.00000 q^{64} +18.9009i q^{65} +(70.3459 - 31.8810i) q^{66} +75.8720 q^{67} +28.5576i q^{68} +(-5.93901 - 13.1045i) q^{69} +13.0955 q^{70} -132.680i q^{71} +(-19.1418 - 16.7807i) q^{72} -73.4381 q^{73} +82.3890i q^{74} +(13.6624 - 6.19184i) q^{75} +51.1405 q^{76} +75.3859i q^{77} +(14.8034 + 32.6640i) q^{78} +23.5709 q^{79} -8.94427i q^{80} +(-10.6017 - 80.3032i) q^{81} +78.4345 q^{82} +134.121i q^{83} +(22.6313 - 10.2566i) q^{84} -31.9283 q^{85} -86.3084i q^{86} +(37.1808 + 82.0401i) q^{87} +51.4887 q^{88} -116.975i q^{89} +(18.7614 - 21.4011i) q^{90} -35.0042 q^{91} -9.59166i q^{92} +(56.2810 - 25.5067i) q^{93} +19.7615 q^{94} +57.1768i q^{95} +(-7.00527 - 15.4572i) q^{96} -82.3630 q^{97} -45.0437i q^{98} +(123.198 + 108.002i) 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
$3$	−2.73248	+	1.23837i	−0.910827	+	0.412789i
$3$	−2.73248	+	1.23837i	−0.910827	+	0.412789i
$4$	−2.00000			−0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$	−1.75132	−	3.86431i	−0.291886	−	0.644052i
$7$	4.14117			0.591596			0.295798	−	0.955250i	$-0.404414\pi$
$7$	4.14117			0.591596			0.295798	+	0.955250i	$0.404414\pi$
$8$		−	2.82843i		−	0.353553i
$9$	5.93289	−	6.76763i	0.659210	−	0.751959i
$10$	3.16228			0.316228
$11$			18.2040i			1.65491i	0.561533	+	0.827454i	$0.310212\pi$
$11$			18.2040i			1.65491i	−0.561533	+	0.827454i	$0.689788\pi$
$12$	5.46496	−	2.47674i	0.455413	−	0.206395i
$13$	−8.45273			−0.650210			−0.325105	−	0.945678i	$-0.605400\pi$
$13$	−8.45273			−0.650210			−0.325105	+	0.945678i	$0.605400\pi$
$14$			5.85650i			0.418322i
$15$	2.76908	+	6.11001i	0.184605	+	0.407334i
$16$	4.00000			0.250000
$17$		−	14.2788i		−	0.839928i	−0.907541	−	0.419964i	$-0.862043\pi$
$17$		−	14.2788i		−	0.839928i	0.907541	−	0.419964i	$-0.137957\pi$
$18$	9.57088	+	8.39037i	0.531715	+	0.466132i
$19$	−25.5703			−1.34580			−0.672901	−	0.739732i	$-0.734952\pi$
$19$	−25.5703			−1.34580			−0.672901	+	0.739732i	$0.734952\pi$
$20$			4.47214i			0.223607i
$21$	−11.3157	+	5.12830i	−0.538841	+	0.244205i
$22$	−25.7443			−1.17020
$23$			4.79583i			0.208514i
$23$			4.79583i			0.208514i
$24$	3.50263	+	7.72862i	0.145943	+	0.322026i
$25$	−5.00000			−0.200000
$26$		−	11.9540i		−	0.459768i
$27$	−7.83068	+	25.8395i	−0.290025	+	0.957019i
$28$	−8.28235			−0.295798
$29$		−	30.0240i		−	1.03531i	−0.855589	−	0.517656i	$-0.826805\pi$
$29$		−	30.0240i		−	1.03531i	0.855589	−	0.517656i	$-0.173195\pi$
$30$	−8.64086	+	3.91606i	−0.288029	+	0.130535i
$31$	−20.5971			−0.664421			−0.332211	−	0.943205i	$-0.607794\pi$
$31$	−20.5971			−0.664421			−0.332211	+	0.943205i	$0.607794\pi$
$32$			5.65685i			0.176777i
$33$	−22.5432	−	49.7420i	−0.683129	−	1.50733i
$34$	20.1932			0.593919
$35$		−	9.25994i		−	0.264570i
$36$	−11.8658	+	13.5353i	−0.329605	+	0.375980i
$37$	58.2578			1.57453			0.787267	−	0.616612i	$-0.211495\pi$
$37$	58.2578			1.57453			0.787267	+	0.616612i	$0.211495\pi$
$38$		−	36.1618i		−	0.951626i
$39$	23.0969	−	10.4676i	0.592229	−	0.268400i
$40$	−6.32456			−0.158114
$41$		−	55.4616i		−	1.35272i	−0.736571	−	0.676361i	$-0.763556\pi$
$41$		−	55.4616i		−	1.35272i	0.736571	−	0.676361i	$-0.236444\pi$
$42$	−7.25251	−	16.0028i	−0.172679	−	0.381018i
$43$	−61.0293			−1.41929			−0.709643	−	0.704562i	$-0.751144\pi$
$43$	−61.0293			−1.41929			−0.709643	+	0.704562i	$0.751144\pi$
$44$		−	36.4080i		−	0.827454i
$45$	−15.1329	−	13.2663i	−0.336286	−	0.294808i
$46$	−6.78233			−0.147442
$47$		−	13.9735i		−	0.297308i	−0.988889	−	0.148654i	$-0.952506\pi$
$47$		−	13.9735i		−	0.297308i	0.988889	−	0.148654i	$-0.0474941\pi$
$48$	−10.9299	+	4.95347i	−0.227707	+	0.103197i
$49$	−31.8507			−0.650014
$50$		−	7.07107i		−	0.141421i
$51$	17.6824	+	39.0165i	0.346714	+	0.765029i
$52$	16.9055			0.325105
$53$			25.4683i			0.480533i	0.970707	+	0.240267i	$0.0772348\pi$
$53$			25.4683i			0.480533i	−0.970707	+	0.240267i	$0.922765\pi$
$54$	−36.5426	−	11.0743i	−0.676715	−	0.205079i
$55$	40.7054			0.740098
$56$		−	11.7130i		−	0.209161i
$57$	69.8702	−	31.6654i	1.22579	−	0.555533i
$58$	42.4604			0.732076
$59$		−	79.7148i		−	1.35110i	−0.737315	−	0.675549i	$-0.763907\pi$
$59$		−	79.7148i		−	1.35110i	0.737315	−	0.675549i	$-0.236093\pi$
$60$	−5.53815	−	12.2200i	−0.0923025	−	0.203667i
$61$	95.2262			1.56109			0.780543	−	0.625102i	$-0.214943\pi$
$61$	95.2262			1.56109			0.780543	+	0.625102i	$0.214943\pi$
$62$		−	29.1286i		−	0.469817i
$63$	24.5691	−	28.0259i	0.389986	−	0.444856i
$64$	−8.00000			−0.125000
$65$			18.9009i			0.290783i
$66$	70.3459	−	31.8810i	1.06585	−	0.483045i
$67$	75.8720			1.13242			0.566209	−	0.824262i	$-0.308410\pi$
$67$	75.8720			1.13242			0.566209	+	0.824262i	$0.308410\pi$
$68$			28.5576i			0.419964i
$69$	−5.93901	−	13.1045i	−0.0860725	−	0.189920i
$70$	13.0955			0.187079
$71$		−	132.680i		−	1.86873i	−0.356317	−	0.934365i	$-0.615968\pi$
$71$		−	132.680i		−	1.86873i	0.356317	−	0.934365i	$-0.384032\pi$
$72$	−19.1418	−	16.7807i	−0.265858	−	0.233066i
$73$	−73.4381			−1.00600			−0.503000	−	0.864286i	$-0.667771\pi$
$73$	−73.4381			−1.00600			−0.503000	+	0.864286i	$0.667771\pi$
$74$			82.3890i			1.11336i
$75$	13.6624	−	6.19184i	0.182165	−	0.0825579i
$76$	51.1405			0.672901
$77$			75.3859i			0.979038i
$78$	14.8034	+	32.6640i	0.189787	+	0.418769i
$79$	23.5709			0.298366			0.149183	−	0.988810i	$-0.452336\pi$
$79$	23.5709			0.298366			0.149183	+	0.988810i	$0.452336\pi$
$80$		−	8.94427i		−	0.111803i
$81$	−10.6017	−	80.3032i	−0.130885	−	0.991398i
$82$	78.4345			0.956518
$83$			134.121i			1.61591i	0.589243	+	0.807956i	$0.299426\pi$
$83$			134.121i			1.61591i	−0.589243	+	0.807956i	$0.700574\pi$
$84$	22.6313	−	10.2566i	0.269421	−	0.122102i
$85$	−31.9283			−0.375627
$86$		−	86.3084i		−	1.00359i
$87$	37.1808	+	82.0401i	0.427366	+	0.942989i
$88$	51.4887			0.585099
$89$		−	116.975i		−	1.31432i	−0.753751	−	0.657161i	$-0.771757\pi$
$89$		−	116.975i		−	1.31432i	0.753751	−	0.657161i	$-0.228243\pi$
$90$	18.7614	−	21.4011i	0.208460	−	0.237790i
$91$	−35.0042			−0.384662
$92$		−	9.59166i		−	0.104257i
$93$	56.2810	−	25.5067i	0.605172	−	0.274266i
$94$	19.7615			0.210229
$95$			57.1768i			0.601861i
$96$	−7.00527	−	15.4572i	−0.0729715	−	0.161013i
$97$	−82.3630			−0.849104			−0.424552	−	0.905404i	$-0.639568\pi$
$97$	−82.3630			−0.849104			−0.424552	+	0.905404i	$0.639568\pi$
$98$		−	45.0437i		−	0.459629i
$99$	123.198	+	108.002i	1.24442	+	1.09093i
$100$	10.0000			0.100000
$101$		−	133.725i		−	1.32401i	−0.749501	−	0.662003i	$-0.769706\pi$
$101$		−	133.725i		−	1.32401i	0.749501	−	0.662003i	$-0.230294\pi$
$102$	−55.1776	+	25.0067i	−0.540957	+	0.245163i
$103$	−157.226			−1.52647			−0.763233	−	0.646124i	$-0.776389\pi$
$103$	−157.226			−1.52647			−0.763233	+	0.646124i	$0.776389\pi$
$104$			23.9079i			0.229884i
$105$	11.4672	+	25.3026i	0.109212	+	0.240977i
$106$	−36.0175			−0.339788
$107$		−	68.1719i		−	0.637120i	−0.947903	−	0.318560i	$-0.896801\pi$
$107$		−	68.1719i		−	0.637120i	0.947903	−	0.318560i	$-0.103199\pi$
$108$	15.6614	−	51.6790i	0.145013	−	0.478510i
$109$	−3.60776			−0.0330987			−0.0165494	−	0.999863i	$-0.505268\pi$
$109$	−3.60776			−0.0330987			−0.0165494	+	0.999863i	$0.505268\pi$
$110$			57.5661i			0.523328i
$111$	−159.188	+	72.1446i	−1.43413	+	0.649951i
$112$	16.5647			0.147899
$113$		−	82.8721i		−	0.733382i	−0.930343	−	0.366691i	$-0.880491\pi$
$113$		−	82.8721i		−	0.733382i	0.930343	−	0.366691i	$-0.119509\pi$
$114$	44.7816	+	98.8114i	0.392821	+	0.866767i
$115$	10.7238			0.0932505
$116$			60.0481i			0.517656i
$117$	−50.1491	+	57.2050i	−0.428625	+	0.488932i
$118$	112.734			0.955371
$119$		−	59.1309i		−	0.496898i
$120$	17.2817	−	7.83213i	0.144014	−	0.0652677i
$121$	−210.385			−1.73872
$122$			134.670i			1.10385i
$123$	68.6818	+	151.548i	0.558389	+	1.23209i
$124$	41.1941			0.332211
$125$			11.1803i			0.0894427i
$126$	39.6347	+	34.7460i	0.314561	+	0.275762i
$127$	−12.8307			−0.101029			−0.0505145	−	0.998723i	$-0.516086\pi$
$127$	−12.8307			−0.101029			−0.0505145	+	0.998723i	$0.516086\pi$
$128$		−	11.3137i		−	0.0883883i
$129$	166.761	−	75.5767i	1.29272	−	0.585866i
$130$	−26.7299			−0.205615
$131$		−	260.678i		−	1.98991i	−0.100330	−	0.994954i	$-0.531990\pi$
$131$		−	260.678i		−	1.98991i	0.100330	−	0.994954i	$-0.468010\pi$
$132$	45.0865	+	99.4841i	0.341564	+	0.753667i
$133$	−105.891			−0.796172
$134$			107.299i			0.800740i
$135$	57.7789	+	17.5099i	0.427992	+	0.129703i
$136$	−40.3865			−0.296960
$137$			122.563i			0.894619i	0.894379	+	0.447309i	$0.147618\pi$
$137$			122.563i			0.894619i	−0.894379	+	0.447309i	$0.852382\pi$
$138$	18.5326	−	8.39902i	0.134294	−	0.0608625i
$139$	−79.5231			−0.572109			−0.286054	−	0.958213i	$-0.592344\pi$
$139$	−79.5231			−0.572109			−0.286054	+	0.958213i	$0.592344\pi$
$140$			18.5199i			0.132285i
$141$	17.3043	+	38.1822i	0.122726	+	0.270796i
$142$	187.638			1.32139
$143$		−	153.874i		−	1.07604i
$144$	23.7316	−	27.0705i	0.164802	−	0.187990i
$145$	−67.1358			−0.463006
$146$		−	103.857i		−	0.711350i
$147$	87.0314	−	39.4429i	0.592050	−	0.268319i
$148$	−116.516			−0.787267
$149$		−	14.2215i		−	0.0954460i	−0.998861	−	0.0477230i	$-0.984804\pi$
$149$		−	14.2215i		−	0.0954460i	0.998861	−	0.0477230i	$-0.0151965\pi$
$150$	8.75659	+	19.3215i	0.0583772	+	0.128810i
$151$	199.067			1.31832			0.659161	−	0.752002i	$-0.270912\pi$
$151$	199.067			1.31832			0.659161	+	0.752002i	$0.270912\pi$
$152$			72.3236i			0.475813i
$153$	−96.6335	−	84.7144i	−0.631592	−	0.553689i
$154$	−106.612			−0.692284
$155$			46.0564i			0.297138i
$156$	−46.1938	+	20.9352i	−0.296114	+	0.134200i
$157$	−150.983			−0.961676			−0.480838	−	0.876809i	$-0.659668\pi$
$157$	−150.983			−0.961676			−0.480838	+	0.876809i	$0.659668\pi$
$158$			33.3343i			0.210977i
$159$	−31.5391	−	69.5915i	−0.198359	−	0.437682i
$160$	12.6491			0.0790569
$161$			19.8604i			0.123356i
$162$	113.566	−	14.9930i	0.701024	−	0.0925495i
$163$	157.063			0.963577			0.481789	−	0.876287i	$-0.339987\pi$
$163$	157.063			0.963577			0.481789	+	0.876287i	$0.339987\pi$
$164$			110.923i			0.676361i
$165$	−111.227	+	50.4082i	−0.674101	+	0.305504i
$166$	−189.675			−1.14262
$167$		−	140.625i		−	0.842065i	−0.907046	−	0.421032i	$-0.861668\pi$
$167$		−	140.625i		−	0.842065i	0.907046	−	0.421032i	$-0.138332\pi$
$168$	14.5050	+	32.0055i	0.0863394	+	0.190509i
$169$	−97.5513			−0.577226
$170$		−	45.1535i		−	0.265609i
$171$	−151.705	+	173.050i	−0.887167	+	1.01199i
$172$	122.059			0.709643
$173$			129.805i			0.750317i	0.926961	+	0.375158i	$0.122412\pi$
$173$			129.805i			0.750317i	−0.926961	+	0.375158i	$0.877588\pi$
$174$	−116.022	+	52.5816i	−0.666794	+	0.302193i
$175$	−20.7059			−0.118319
$176$			72.8160i			0.413727i
$177$	98.7163	+	217.819i	0.557719	+	1.23062i
$178$	165.427			0.929365
$179$			277.661i			1.55118i	0.631239	+	0.775588i	$0.282547\pi$
$179$			277.661i			1.55118i	−0.631239	+	0.775588i	$0.717453\pi$
$180$	30.2658	+	26.5327i	0.168143	+	0.147404i
$181$	−248.306			−1.37185			−0.685927	−	0.727670i	$-0.740603\pi$
$181$	−248.306			−1.37185			−0.685927	+	0.727670i	$0.740603\pi$
$182$		−	49.5035i		−	0.271997i
$183$	−260.204	+	117.925i	−1.42188	+	0.644400i
$184$	13.5647			0.0737210
$185$		−	130.268i		−	0.704153i
$186$	36.0720	+	79.5934i	0.193935	+	0.427922i
$187$	259.931			1.39000
$188$			27.9470i			0.148654i
$189$	−32.4282	+	107.006i	−0.171578	+	0.566169i
$190$	−80.8602			−0.425580
$191$			114.375i			0.598820i	0.954125	+	0.299410i	$0.0967898\pi$
$191$			114.375i			0.598820i	−0.954125	+	0.299410i	$0.903210\pi$
$192$	21.8598	−	9.90695i	0.113853	−	0.0515987i
$193$	−224.783			−1.16468			−0.582341	−	0.812945i	$-0.697863\pi$
$193$	−224.783			−1.16468			−0.582341	+	0.812945i	$0.697863\pi$
$194$		−	116.479i		−	0.600407i
$195$	−23.4063	−	51.6463i	−0.120032	−	0.264853i
$196$	63.7014			0.325007
$197$		−	237.189i		−	1.20401i	−0.798493	−	0.602004i	$-0.794369\pi$
$197$		−	237.189i		−	1.20401i	0.798493	−	0.602004i	$-0.205631\pi$
$198$	−152.738	+	174.228i	−0.771406	+	0.879940i
$199$	−200.271			−1.00639			−0.503194	−	0.864174i	$-0.667842\pi$
$199$	−200.271			−1.00639			−0.503194	+	0.864174i	$0.667842\pi$
$200$			14.1421i			0.0707107i
$201$	−207.319	+	93.9574i	−1.03144	+	0.467450i
$202$	189.115			0.936214
$203$		−	124.335i		−	0.612486i
$204$	−35.3648	−	78.0330i	−0.173357	−	0.382515i
$205$	−124.016			−0.604955
$206$		−	222.351i		−	1.07937i
$207$	32.4564	+	28.4531i	0.156794	+	0.137455i
$208$	−33.8109			−0.162553
$209$		−	465.481i		−	2.22718i
$210$	−35.7833	+	16.2171i	−0.170397	+	0.0772243i
$211$	−199.810			−0.946967			−0.473483	−	0.880803i	$-0.657004\pi$
$211$	−199.810			−0.946967			−0.473483	+	0.880803i	$0.657004\pi$
$212$		−	50.9365i		−	0.240267i
$213$	164.307	+	362.545i	0.771392	+	1.70209i
$214$	96.4096			0.450512
$215$			136.466i			0.634724i
$216$	73.0852	+	22.1485i	0.338357	+	0.102539i
$217$	−85.2960			−0.393069
$218$		−	5.10215i		−	0.0234043i
$219$	200.668	−	90.9434i	0.916292	−	0.415266i
$220$	−81.4107			−0.370049
$221$			120.695i			0.546130i
$222$	−102.028	−	225.126i	−0.459585	−	1.01408i
$223$	251.698			1.12869			0.564345	−	0.825539i	$-0.309129\pi$
$223$	251.698			1.12869			0.564345	+	0.825539i	$0.309129\pi$
$224$			23.4260i			0.104580i
$225$	−29.6644	+	33.8382i	−0.131842	+	0.150392i
$226$	117.199			0.518579
$227$			209.743i			0.923977i	0.886886	+	0.461988i	$0.152864\pi$
$227$			209.743i			0.923977i	−0.886886	+	0.461988i	$0.847136\pi$
$228$	−139.740	+	63.3308i	−0.612896	+	0.277767i
$229$	310.423			1.35556			0.677781	−	0.735264i	$-0.262942\pi$
$229$	310.423			1.35556			0.677781	+	0.735264i	$0.262942\pi$
$230$			15.1658i			0.0659380i
$231$	−93.3555	−	205.990i	−0.404136	−	0.891733i
$232$	−84.9208			−0.366038
$233$			13.5244i			0.0580446i	0.999579	+	0.0290223i	$0.00923938\pi$
$233$			13.5244i			0.0580446i	−0.999579	+	0.0290223i	$0.990761\pi$
$234$	−80.9001	−	70.9216i	−0.345727	−	0.303084i
$235$	−31.2457			−0.132960
$236$			159.430i			0.675549i
$237$	−64.4070	+	29.1895i	−0.271760	+	0.123162i
$238$	83.6237			0.351360
$239$			162.359i			0.679328i	0.940547	+	0.339664i	$0.110313\pi$
$239$			162.359i			0.679328i	−0.940547	+	0.339664i	$0.889687\pi$
$240$	11.0763	+	24.4400i	0.0461513	+	0.101833i
$241$	114.412			0.474740			0.237370	−	0.971419i	$-0.423715\pi$
$241$	114.412			0.474740			0.237370	+	0.971419i	$0.423715\pi$
$242$		−	297.530i		−	1.22946i
$243$	128.414	+	206.298i	0.528452	+	0.848963i
$244$	−190.452			−0.780543
$245$			71.2203i			0.290695i
$246$	−214.321	+	97.1308i	−0.871222	+	0.394841i
$247$	216.139			0.875055
$248$			58.2573i			0.234908i
$249$	−166.091	−	366.482i	−0.667031	−	1.47181i
$250$	−15.8114			−0.0632456
$251$		−	77.7456i		−	0.309743i	−0.987935	−	0.154872i	$-0.950504\pi$
$251$		−	77.7456i		−	0.309743i	0.987935	−	0.154872i	$-0.0494964\pi$
$252$	−49.1382	+	56.0519i	−0.194993	+	0.222428i
$253$	−87.3033			−0.345072
$254$		−	18.1453i		−	0.0714383i
$255$	87.2435	−	39.5390i	0.342131	−	0.155055i
$256$	16.0000			0.0625000
$257$			252.344i			0.981883i	0.871193	+	0.490941i	$0.163347\pi$
$257$			252.344i			0.981883i	−0.871193	+	0.490941i	$0.836653\pi$
$258$	106.882	+	235.836i	0.414270	+	0.914093i
$259$	241.256			0.931489
$260$		−	37.8018i		−	0.145391i
$261$	−203.192	−	178.129i	−0.778512	−	0.682488i
$262$	368.654			1.40708
$263$		−	192.257i		−	0.731016i	−0.930808	−	0.365508i	$-0.880895\pi$
$263$		−	192.257i		−	0.731016i	0.930808	−	0.365508i	$-0.119105\pi$
$264$	−140.692	+	63.7619i	−0.532923	+	0.241522i
$265$	56.9487			0.214901
$266$		−	149.752i		−	0.562978i
$267$	144.858	+	319.631i	0.542538	+	1.19712i
$268$	−151.744			−0.566209
$269$			334.333i			1.24287i	0.783464	+	0.621437i	$0.213451\pi$
$269$			334.333i			1.24287i	−0.783464	+	0.621437i	$0.786549\pi$
$270$	−24.7628	+	81.7117i	−0.0917140	+	0.302636i
$271$	−415.940			−1.53483			−0.767416	−	0.641149i	$-0.778458\pi$
$271$	−415.940			−1.53483			−0.767416	+	0.641149i	$0.778458\pi$
$272$		−	57.1151i		−	0.209982i
$273$	95.6484	−	43.3481i	0.350360	−	0.158784i
$274$	−173.330			−0.632591
$275$		−	91.0200i		−	0.330982i
$276$	11.8780	+	26.2090i	0.0430363	+	0.0949602i
$277$	95.1282			0.343423			0.171712	−	0.985147i	$-0.445070\pi$
$277$	95.1282			0.343423			0.171712	+	0.985147i	$0.445070\pi$
$278$		−	112.463i		−	0.404542i
$279$	−122.200	+	139.393i	−0.437993	+	0.499617i
$280$	−26.1911			−0.0935396
$281$		−	62.8658i		−	0.223722i	−0.993724	−	0.111861i	$-0.964319\pi$
$281$		−	62.8658i		−	0.223722i	0.993724	−	0.111861i	$-0.0356811\pi$
$282$	−53.9979	+	24.4720i	−0.191482	+	0.0867801i
$283$	469.458			1.65886			0.829431	−	0.558609i	$-0.188665\pi$
$283$	469.458			1.65886			0.829431	+	0.558609i	$0.188665\pi$
$284$			265.360i			0.934365i
$285$	−70.8060	−	156.235i	−0.248442	−	0.548191i
$286$	217.610			0.760874
$287$		−	229.676i		−	0.800265i
$288$	38.2835	+	33.5615i	0.132929	+	0.116533i
$289$	85.1164			0.294520
$290$		−	94.9444i		−	0.327394i
$291$	225.055	−	101.996i	0.773386	−	0.350501i
$292$	146.876			0.503000
$293$			288.659i			0.985184i	0.870260	+	0.492592i	$0.163951\pi$
$293$			288.659i			0.985184i	−0.870260	+	0.492592i	$0.836049\pi$
$294$	55.7807	+	123.081i	0.189730	+	0.418643i
$295$	−178.248			−0.604229
$296$		−	164.778i		−	0.556682i
$297$	−470.382	−	142.550i	−1.58378	−	0.479965i
$298$	20.1122			0.0674905
$299$		−	40.5379i		−	0.135578i
$300$	−27.3248	+	12.3837i	−0.0910827	+	0.0412789i
$301$	−252.733			−0.839644
$302$			281.523i			0.932194i
$303$	165.600	+	365.400i	0.546536	+	1.20594i
$304$	−102.281			−0.336451
$305$		−	212.932i		−	0.698139i
$306$	119.804	−	136.660i	0.391517	−	0.446603i
$307$	−453.095			−1.47588			−0.737940	−	0.674867i	$-0.764201\pi$
$307$	−453.095			−1.47588			−0.737940	+	0.674867i	$0.764201\pi$
$308$		−	150.772i		−	0.489519i
$309$	429.617	−	194.704i	1.39035	−	0.630109i
$310$	−65.1336			−0.210108
$311$			445.022i			1.43094i	0.698644	+	0.715470i	$0.253787\pi$
$311$			445.022i			1.43094i	−0.698644	+	0.715470i	$0.746213\pi$
$312$	−29.6068	−	65.3280i	−0.0948937	−	0.209385i
$313$	−548.133			−1.75122			−0.875612	−	0.483016i	$-0.839541\pi$
$313$	−548.133			−1.75122			−0.875612	+	0.483016i	$0.839541\pi$
$314$		−	213.522i		−	0.680008i
$315$	−62.6679	−	54.9382i	−0.198946	−	0.174407i
$316$	−47.1418			−0.149183
$317$			1.18880i			0.00375016i	0.999998	+	0.00187508i	$0.000596858\pi$
$317$			1.18880i			0.00375016i	−0.999998	+	0.00187508i	$0.999403\pi$
$318$	98.4172	−	44.6030i	0.309488	−	0.140261i
$319$	546.558			1.71335
$320$			17.8885i			0.0559017i
$321$	84.4219	+	186.278i	0.262996	+	0.580306i
$322$	−28.0868			−0.0872261
$323$			365.112i			1.13038i
$324$	21.2033	+	160.606i	0.0654424	+	0.495699i
$325$	42.2637			0.130042
$326$			222.121i			0.681352i
$327$	9.85814	−	4.46774i	0.0301472	−	0.0136628i
$328$	−156.869			−0.478259
$329$		−	57.8666i		−	0.175886i
$330$	−71.2880	−	157.298i	−0.216024	−	0.476661i
$331$	−84.6998			−0.255891			−0.127945	−	0.991781i	$-0.540838\pi$
$331$	−84.6998			−0.255891			−0.127945	+	0.991781i	$0.540838\pi$
$332$		−	268.241i		−	0.807956i
$333$	345.637	−	394.267i	1.03795	−	1.18399i
$334$	198.874			0.595430
$335$		−	169.655i		−	0.506432i
$336$	−45.2627	+	20.5132i	−0.134710	+	0.0610511i
$337$	301.194			0.893752			0.446876	−	0.894596i	$-0.352537\pi$
$337$	301.194			0.893752			0.446876	+	0.894596i	$0.352537\pi$
$338$		−	137.958i		−	0.408161i
$339$	102.626	+	226.446i	0.302732	+	0.667983i
$340$	63.8567			0.187814
$341$		−	374.949i		−	1.09956i
$342$	−244.730	−	214.544i	−0.715584	−	0.627321i
$343$	−334.817			−0.976142
$344$			172.617i			0.501793i
$345$	−29.3026	+	13.2800i	−0.0849350	+	0.0384928i
$346$	−183.572			−0.530554
$347$			74.3987i			0.214405i	0.994237	+	0.107203i	$0.0341894\pi$
$347$			74.3987i			0.214405i	−0.994237	+	0.107203i	$0.965811\pi$
$348$	−74.3616	−	164.080i	−0.213683	−	0.471495i
$349$	122.217			0.350193			0.175096	−	0.984551i	$-0.443976\pi$
$349$	122.217			0.350193			0.175096	+	0.984551i	$0.443976\pi$
$350$		−	29.2825i		−	0.0836643i
$351$	66.1906	−	218.415i	0.188577	−	0.622264i
$352$	−102.977			−0.292549
$353$		−	257.047i		−	0.728179i	−0.931364	−	0.364089i	$-0.881380\pi$
$353$		−	257.047i		−	0.728179i	0.931364	−	0.364089i	$-0.118620\pi$
$354$	−308.043	+	139.606i	−0.870177	+	0.394367i
$355$	−296.681			−0.835722
$356$			233.949i			0.657161i
$357$	73.2258	+	161.574i	0.205114	+	0.452588i
$358$	−392.671			−1.09685
$359$			411.848i			1.14721i	0.819133	+	0.573604i	$0.194455\pi$
$359$			411.848i			1.14721i	−0.819133	+	0.573604i	$0.805545\pi$
$360$	−37.5229	+	42.8023i	−0.104230	+	0.118895i
$361$	292.838			0.811185
$362$		−	351.157i		−	0.970047i
$363$	574.874	−	260.535i	1.58368	−	0.717726i
$364$	70.0085			0.192331
$365$			164.213i			0.449897i
$366$	−166.771	−	367.984i	−0.455659	−	1.00542i
$367$	390.609			1.06433			0.532165	−	0.846640i	$-0.321378\pi$
$367$	390.609			1.06433			0.532165	+	0.846640i	$0.321378\pi$
$368$			19.1833i			0.0521286i
$369$	−375.343	−	329.047i	−1.01719	−	0.891727i
$370$	184.227			0.497912
$371$			105.468i			0.284281i
$372$	−112.562	+	51.0135i	−0.302586	+	0.137133i
$373$	−358.049			−0.959916			−0.479958	−	0.877292i	$-0.659348\pi$
$373$	−358.049			−0.959916			−0.479958	+	0.877292i	$0.659348\pi$
$374$			367.598i			0.982882i
$375$	−13.8454	−	30.5500i	−0.0369210	−	0.0814668i
$376$	−39.5230			−0.105114
$377$			253.785i			0.673170i
$378$	−151.329	−	45.8604i	−0.400342	−	0.121324i
$379$	−292.500			−0.771769			−0.385884	−	0.922547i	$-0.626104\pi$
$379$	−292.500			−0.771769			−0.385884	+	0.922547i	$0.626104\pi$
$380$		−	114.354i		−	0.300931i
$381$	35.0596	−	15.8891i	0.0920199	−	0.0417037i
$382$	−161.750			−0.423430
$383$			67.9007i			0.177286i	0.996063	+	0.0886432i	$0.0282531\pi$
$383$			67.9007i			0.177286i	−0.996063	+	0.0886432i	$0.971747\pi$
$384$	14.0105	+	30.9145i	0.0364858	+	0.0805064i
$385$	168.568			0.437839
$386$		−	317.892i		−	0.823554i
$387$	−362.080	+	413.024i	−0.935607	+	1.06724i
$388$	164.726			0.424552
$389$			332.519i			0.854806i	0.904061	+	0.427403i	$0.140571\pi$
$389$			332.519i			0.854806i	−0.904061	+	0.427403i	$0.859429\pi$
$390$	73.0389	−	33.1015i	0.187279	−	0.0848755i
$391$	68.4786			0.175137
$392$			90.0874i			0.229815i
$393$	322.815	+	712.297i	0.821413	+	1.81246i
$394$	335.437			0.851362
$395$		−	52.7061i		−	0.133433i
$396$	−246.396	−	216.005i	−0.622212	−	0.545466i
$397$	185.010			0.466020			0.233010	−	0.972474i	$-0.425142\pi$
$397$	185.010			0.466020			0.233010	+	0.972474i	$0.425142\pi$
$398$		−	283.226i		−	0.711624i
$399$	289.345	−	131.132i	0.725174	−	0.328651i
$400$	−20.0000			−0.0500000
$401$		−	28.7486i		−	0.0716924i	−0.999357	−	0.0358462i	$-0.988587\pi$
$401$		−	28.7486i		−	0.0716924i	0.999357	−	0.0358462i	$-0.0114126\pi$
$402$	−132.876	−	293.193i	−0.330537	−	0.729335i
$403$	174.101			0.432014
$404$			267.449i			0.662003i
$405$	−179.563	+	23.7060i	−0.443366	+	0.0585334i
$406$	175.836			0.433093
$407$			1060.52i			2.60571i
$408$	110.355	−	50.0134i	0.270479	−	0.122582i
$409$	307.733			0.752402			0.376201	−	0.926538i	$-0.377230\pi$
$409$	307.733			0.752402			0.376201	+	0.926538i	$0.377230\pi$
$410$		−	175.385i		−	0.427768i
$411$	−151.778	−	334.900i	−0.369289	−	0.814842i
$412$	314.452			0.763233
$413$		−	330.113i		−	0.799304i
$414$	−40.2388	+	45.9003i	−0.0971952	+	0.110870i
$415$	299.903			0.722657
$416$		−	47.8159i		−	0.114942i
$417$	217.295	−	98.4789i	0.521092	−	0.236161i
$418$	658.289			1.57485
$419$		−	346.936i		−	0.828010i	−0.910275	−	0.414005i	$-0.864130\pi$
$419$		−	346.936i		−	0.828010i	0.910275	−	0.414005i	$-0.135870\pi$
$420$	−22.9344	−	50.6052i	−0.0546058	−	0.120489i
$421$	216.953			0.515329			0.257664	−	0.966234i	$-0.417047\pi$
$421$	216.953			0.515329			0.257664	+	0.966234i	$0.417047\pi$
$422$		−	282.574i		−	0.669607i
$423$	−94.5674	−	82.9031i	−0.223563	−	0.195988i
$424$	72.0351			0.169894
$425$			71.3939i			0.167986i
$426$	−512.716	+	232.364i	−1.20356	+	0.545457i
$427$	394.348			0.923532
$428$			136.344i			0.318560i
$429$	190.552	+	420.456i	0.444177	+	0.980085i
$430$	−192.991			−0.448817
$431$		−	595.252i		−	1.38109i	−0.723287	−	0.690547i	$-0.757370\pi$
$431$		−	595.252i		−	1.38109i	0.723287	−	0.690547i	$-0.242630\pi$
$432$	−31.3227	+	103.358i	−0.0725063	+	0.239255i
$433$	−162.757			−0.375882			−0.187941	−	0.982180i	$-0.560181\pi$
$433$	−162.757			−0.375882			−0.187941	+	0.982180i	$0.560181\pi$
$434$		−	120.627i		−	0.277942i
$435$	183.447	−	83.1388i	0.421718	−	0.191124i
$436$	7.21553			0.0165494
$437$		−	122.631i		−	0.280619i
$438$	128.613	+	283.787i	0.293638	+	0.647916i
$439$	−137.285			−0.312721			−0.156361	−	0.987700i	$-0.549976\pi$
$439$	−137.285			−0.312721			−0.156361	+	0.987700i	$0.549976\pi$
$440$		−	115.132i		−	0.261664i
$441$	−188.967	+	215.554i	−0.428496	+	0.488784i
$442$	−170.688			−0.386172
$443$		−	309.185i		−	0.697935i	−0.937135	−	0.348967i	$-0.886532\pi$
$443$		−	309.185i		−	0.697935i	0.937135	−	0.348967i	$-0.113468\pi$
$444$	318.376	−	144.289i	0.717064	−	0.324976i
$445$	−261.563			−0.587782
$446$			355.954i			0.798104i
$447$	17.6114	+	38.8598i	0.0393991	+	0.0869348i
$448$	−33.1294			−0.0739495
$449$			61.1567i			0.136206i	0.997678	+	0.0681032i	$0.0216947\pi$
$449$			61.1567i			0.136206i	−0.997678	+	0.0681032i	$0.978305\pi$
$450$	−47.8544	−	41.9519i	−0.106343	−	0.0932264i
$451$	1009.62			2.23863
$452$			165.744i			0.366691i
$453$	−543.945	+	246.518i	−1.20076	+	0.544189i
$454$	−296.621			−0.653350
$455$			78.2718i			0.172026i
$456$	−89.5632	−	197.623i	−0.196411	−	0.433383i
$457$	325.253			0.711714			0.355857	−	0.934540i	$-0.384189\pi$
$457$	325.253			0.711714			0.355857	+	0.934540i	$0.384189\pi$
$458$			439.005i			0.958526i
$459$	368.957	+	111.813i	0.803827	+	0.243600i
$460$	−21.4476			−0.0466252
$461$		−	178.646i		−	0.387518i	−0.981049	−	0.193759i	$-0.937932\pi$
$461$		−	178.646i		−	0.387518i	0.981049	−	0.193759i	$-0.0620679\pi$
$462$	291.314	−	132.025i	0.630551	−	0.285768i
$463$	753.662			1.62778			0.813890	−	0.581019i	$-0.197346\pi$
$463$	753.662			1.62778			0.813890	+	0.581019i	$0.197346\pi$
$464$		−	120.096i		−	0.258828i
$465$	−57.0348	−	125.848i	−0.122655	−	0.270641i
$466$	−19.1264			−0.0410437
$467$			341.612i			0.731504i	0.930712	+	0.365752i	$0.119188\pi$
$467$			341.612i			0.731504i	−0.930712	+	0.365752i	$0.880812\pi$
$468$	100.298	−	114.410i	0.214313	−	0.244466i
$469$	314.199			0.669934
$470$		−	44.1880i		−	0.0940171i
$471$	412.558	−	186.973i	0.875920	−	0.396970i
$472$	−225.467			−0.477685
$473$		−	1110.98i		−	2.34879i
$474$	−41.2801	−	91.0853i	−0.0870889	−	0.192163i
$475$	127.851			0.269161
$476$			118.262i			0.248449i
$477$	172.360	+	151.100i	0.361341	+	0.316772i
$478$	−229.611			−0.480358
$479$		−	129.264i		−	0.269862i	−0.990855	−	0.134931i	$-0.956919\pi$
$479$		−	129.264i		−	0.269862i	0.990855	−	0.134931i	$-0.0430813\pi$
$480$	−34.5634	+	15.6643i	−0.0720072	+	0.0326339i
$481$	−492.438			−1.02378
$482$			161.803i			0.335692i
$483$	−24.5944	−	54.2680i	−0.0509202	−	0.112356i
$484$	420.771			0.869362
$485$			184.169i			0.379731i
$486$	−291.750	+	181.604i	−0.600308	+	0.373672i
$487$	−356.900			−0.732855			−0.366428	−	0.930447i	$-0.619419\pi$
$487$	−356.900			−0.732855			−0.366428	+	0.930447i	$0.619419\pi$
$488$		−	269.340i		−	0.551927i
$489$	−429.172	+	194.502i	−0.877652	+	0.397755i
$490$	−100.721			−0.205552
$491$		−	576.525i		−	1.17419i	−0.809520	−	0.587093i	$-0.800272\pi$
$491$		−	576.525i		−	1.17419i	0.809520	−	0.587093i	$-0.199728\pi$
$492$	−137.364	−	303.095i	−0.279194	−	0.616047i
$493$	−428.707			−0.869588
$494$			305.666i			0.618757i
$495$	241.500	−	275.479i	0.487880	−	0.556523i
$496$	−82.3882			−0.166105
$497$		−	549.450i		−	1.10553i
$498$	518.284	−	234.888i	1.04073	−	0.471662i
$499$	−622.821			−1.24814			−0.624069	−	0.781369i	$-0.714522\pi$
$499$	−622.821			−1.24814			−0.624069	+	0.781369i	$0.714522\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	174.145	+	384.255i	0.347595	+	0.766975i
$502$	109.949			0.219022
$503$		−	407.531i		−	0.810200i	−0.914272	−	0.405100i	$-0.867237\pi$
$503$		−	407.531i		−	0.810200i	0.914272	−	0.405100i	$-0.132763\pi$
$504$	−79.2693	−	69.4920i	−0.157280	−	0.137881i
$505$	−299.018			−0.592114
$506$		−	123.466i		−	0.244003i
$507$	266.557	−	120.804i	0.525753	−	0.238273i
$508$	25.6614			0.0505145
$509$		−	274.372i		−	0.539041i	−0.962995	−	0.269520i	$-0.913135\pi$
$509$		−	274.372i		−	0.539041i	0.962995	−	0.269520i	$-0.0868651\pi$
$510$	55.9166	+	123.381i	0.109640	+	0.241923i
$511$	−304.120			−0.595146
$512$			22.6274i			0.0441942i
$513$	200.232	−	660.723i	0.390317	−	1.28796i
$514$	−356.868			−0.694296
$515$			351.568i			0.682656i
$516$	−333.522	+	151.153i	−0.646361	+	0.292933i
$517$	254.373			0.492018
$518$			341.187i			0.658662i
$519$	−160.746	−	354.689i	−0.309723	−	0.683409i
$520$	53.4598			0.102807
$521$		−	227.719i		−	0.437080i	−0.975828	−	0.218540i	$-0.929871\pi$
$521$		−	227.719i		−	0.437080i	0.975828	−	0.218540i	$-0.0701295\pi$
$522$	251.913	−	287.356i	0.482592	−	0.550491i
$523$	178.583			0.341460			0.170730	−	0.985318i	$-0.445387\pi$
$523$	178.583			0.341460			0.170730	+	0.985318i	$0.445387\pi$
$524$			521.356i			0.994954i
$525$	56.5783	−	25.6415i	0.107768	−	0.0488409i
$526$	271.893			0.516907
$527$			294.101i			0.558066i
$528$	−90.1730	−	198.968i	−0.170782	−	0.376834i
$529$	−23.0000			−0.0434783
$530$			80.5377i			0.151958i
$531$	−539.480	−	472.939i	−1.01597	−	0.890657i
$532$	211.782			0.398086
$533$			468.802i			0.879553i
$534$	−452.026	+	204.860i	−0.846491	+	0.383632i
$535$	−152.437			−0.284929
$536$		−	214.598i		−	0.400370i
$537$	−343.846	−	758.702i	−0.640309	−	1.41285i
$538$	−472.818			−0.878845
$539$		−	579.810i		−	1.07571i
$540$	−115.558	−	35.0199i	−0.213996	−	0.0648516i
$541$	−406.572			−0.751519			−0.375760	−	0.926717i	$-0.622618\pi$
$541$	−406.572			−0.751519			−0.375760	+	0.926717i	$0.622618\pi$
$542$		−	588.228i		−	1.08529i
$543$	678.490	−	307.494i	1.24952	−	0.566287i
$544$	80.7730			0.148480
$545$			8.06720i			0.0148022i
$546$	61.3035	+	135.267i	0.112277	+	0.247742i
$547$	−762.540			−1.39404			−0.697020	−	0.717051i	$-0.745491\pi$
$547$	−762.540			−1.39404			−0.697020	+	0.717051i	$0.745491\pi$
$548$		−	245.126i		−	0.447309i
$549$	564.967	−	644.456i	1.02908	−	1.17387i
$550$	128.722			0.234039
$551$			767.722i			1.39333i
$552$	−37.0652	+	16.7980i	−0.0671470	+	0.0304312i
$553$	97.6112			0.176512
$554$			134.532i			0.242837i
$555$	161.320	+	355.956i	0.290667	+	0.641362i
$556$	159.046			0.286054
$557$		−	868.898i		−	1.55996i	−0.625804	−	0.779980i	$-0.715229\pi$
$557$		−	868.898i		−	1.55996i	0.625804	−	0.779980i	$-0.284771\pi$
$558$	−197.132	−	172.817i	−0.353283	−	0.309708i
$559$	515.864			0.922834
$560$		−	37.0398i		−	0.0661425i
$561$	−710.256	+	321.890i	−1.26605	+	0.573779i
$562$	88.9056			0.158195
$563$		−	703.584i		−	1.24971i	−0.780743	−	0.624853i	$-0.785159\pi$
$563$		−	703.584i		−	1.24971i	0.780743	−	0.624853i	$-0.214841\pi$
$564$	−34.6086	−	76.3645i	−0.0613628	−	0.135398i
$565$	−185.308			−0.327978
$566$			663.914i			1.17299i
$567$	−43.9033	−	332.549i	−0.0774309	−	0.586507i
$568$	−375.275			−0.660696
$569$			594.365i			1.04458i	0.852768	+	0.522289i	$0.174922\pi$
$569$			594.365i			1.04458i	−0.852768	+	0.522289i	$0.825078\pi$
$570$	220.949	−	100.135i	0.387630	−	0.175675i
$571$	526.041			0.921263			0.460631	−	0.887592i	$-0.347623\pi$
$571$	526.041			0.921263			0.460631	+	0.887592i	$0.347623\pi$
$572$			307.747i			0.538019i
$573$	−141.638	−	312.526i	−0.247186	−	0.545421i
$574$	324.811			0.565872
$575$		−	23.9792i		−	0.0417029i
$576$	−47.4631	+	54.1410i	−0.0824012	+	0.0939949i
$577$	−146.152			−0.253296			−0.126648	−	0.991948i	$-0.540422\pi$
$577$	−146.152			−0.253296			−0.126648	+	0.991948i	$0.540422\pi$
$578$			120.373i			0.208257i
$579$	614.216	−	278.365i	1.06082	−	0.480768i
$580$	134.272			0.231503
$581$			555.417i			0.955967i
$582$	144.244	+	318.276i	0.247842	+	0.546866i
$583$	−463.624			−0.795238
$584$			207.714i			0.355675i
$585$	127.914	+	112.137i	0.218657	+	0.191687i
$586$	−408.226			−0.696631
$587$		−	478.018i		−	0.814340i	−0.913352	−	0.407170i	$-0.866516\pi$
$587$		−	478.018i		−	0.814340i	0.913352	−	0.407170i	$-0.133484\pi$
$588$	−174.063	+	78.8858i	−0.296025	+	0.134159i
$589$	526.672			0.894180
$590$		−	252.080i		−	0.427255i
$591$	293.728	+	648.115i	0.497002	+	1.09664i
$592$	233.031			0.393634
$593$			793.839i			1.33868i	0.742955	+	0.669341i	$0.233424\pi$
$593$			793.839i			1.33868i	−0.742955	+	0.669341i	$0.766576\pi$
$594$	201.596	−	665.221i	0.339387	−	1.11990i
$595$	−132.221			−0.222220
$596$			28.4429i			0.0477230i
$597$	547.237	−	248.009i	0.916645	−	0.415426i
$598$	57.3292			0.0958683
$599$			329.048i			0.549329i	0.961540	+	0.274664i	$0.0885667\pi$
$599$			329.048i			0.549329i	−0.961540	+	0.274664i	$0.911433\pi$
$600$	−17.5132	−	38.6431i	−0.0291886	−	0.0644052i
$601$	−98.7656			−0.164335			−0.0821677	−	0.996619i	$-0.526184\pi$
$601$	−98.7656			−0.164335			−0.0821677	+	0.996619i	$0.526184\pi$
$602$		−	357.418i		−	0.593718i
$603$	450.140	−	513.473i	0.746501	−	0.851531i
$604$	−398.133			−0.659161
$605$			470.436i			0.777581i
$606$	−516.754	+	234.194i	−0.852729	+	0.386459i
$607$	131.584			0.216777			0.108389	−	0.994109i	$-0.465431\pi$
$607$	131.584			0.216777			0.108389	+	0.994109i	$0.465431\pi$
$608$		−	144.647i		−	0.237907i
$609$	153.972	+	339.742i	0.252828	+	0.557869i
$610$	301.132			0.493659
$611$			118.114i			0.193313i
$612$	193.267	+	169.429i	0.315796	+	0.276845i
$613$	537.185			0.876322			0.438161	−	0.898897i	$-0.355630\pi$
$613$	537.185			0.876322			0.438161	+	0.898897i	$0.355630\pi$
$614$		−	640.773i		−	1.04360i
$615$	338.871	−	153.577i	0.551009	−	0.249719i
$616$	213.224			0.346142
$617$			997.204i			1.61621i	0.589036	+	0.808107i	$0.299508\pi$
$617$			997.204i			1.61621i	−0.589036	+	0.808107i	$0.700492\pi$
$618$	275.352	+	607.570i	0.445554	+	0.983123i
$619$	−146.256			−0.236277			−0.118139	−	0.992997i	$-0.537693\pi$
$619$	−146.256			−0.236277			−0.118139	+	0.992997i	$0.537693\pi$
$620$		−	92.1128i		−	0.148569i
$621$	−123.922	−	37.5546i	−0.199552	−	0.0604744i
$622$	−629.356			−1.01183
$623$		−	484.412i		−	0.777547i
$624$	92.3877	−	41.8704i	0.148057	−	0.0671000i
$625$	25.0000			0.0400000
$626$		−	775.177i		−	1.23830i
$627$	576.437	+	1271.92i	0.919357	+	2.02858i
$628$	301.966			0.480838
$629$		−	831.850i		−	1.32250i
$630$	77.6944	−	88.6258i	0.123324	−	0.140676i
$631$	−520.059			−0.824182			−0.412091	−	0.911143i	$-0.635201\pi$
$631$	−520.059			−0.824182			−0.412091	+	0.911143i	$0.635201\pi$
$632$		−	66.6686i		−	0.105488i
$633$	545.977	−	247.438i	0.862522	−	0.390898i
$634$	−1.68122			−0.00265177
$635$			28.6903i			0.0451815i
$636$	63.0781	+	139.183i	0.0991795	+	0.218841i
$637$	269.225			0.422646
$638$			772.949i			1.21152i
$639$	−897.928	−	787.175i	−1.40521	−	1.23189i
$640$	−25.2982			−0.0395285
$641$		−	343.613i		−	0.536058i	−0.963411	−	0.268029i	$-0.913628\pi$
$641$		−	343.613i		−	0.536058i	0.963411	−	0.268029i	$-0.0863723\pi$
$642$	−263.437	+	119.391i	−0.410338	+	0.185967i
$643$	−972.075			−1.51178			−0.755890	−	0.654698i	$-0.772796\pi$
$643$	−972.075			−1.51178			−0.755890	+	0.654698i	$0.772796\pi$
$644$		−	39.7207i		−	0.0616782i
$645$	−168.995	−	372.889i	−0.262007	−	0.578123i
$646$	−516.347			−0.799298
$647$		−	356.832i		−	0.551518i	−0.961227	−	0.275759i	$-0.911071\pi$
$647$		−	356.832i		−	0.551518i	0.961227	−	0.275759i	$-0.0889292\pi$
$648$	−227.132	+	29.9860i	−0.350512	+	0.0462747i
$649$	1451.13			2.23594
$650$			59.7699i			0.0919536i
$651$	233.069	−	105.628i	0.358018	−	0.162255i
$652$	−314.126			−0.481789
$653$			374.245i			0.573116i	0.958063	+	0.286558i	$0.0925111\pi$
$653$			374.245i			0.573116i	−0.958063	+	0.286558i	$0.907489\pi$
$654$	6.31834	+	13.9415i	0.00966107	+	0.0213173i
$655$	−582.894			−0.889914
$656$		−	221.846i		−	0.338180i
$657$	−435.700	+	497.002i	−0.663166	+	0.756471i
$658$	81.8357			0.124370
$659$		−	30.2813i		−	0.0459504i	−0.999736	−	0.0229752i	$-0.992686\pi$
$659$		−	30.2813i		−	0.0459504i	0.999736	−	0.0229752i	$-0.00731387\pi$
$660$	222.453	−	100.816i	0.337050	−	0.152752i
$661$	−1230.40			−1.86143			−0.930714	−	0.365749i	$-0.880813\pi$
$661$	−1230.40			−1.86143			−0.930714	+	0.365749i	$0.880813\pi$
$662$		−	119.784i		−	0.180942i
$663$	−149.465	−	329.796i	−0.225437	−	0.497430i
$664$	379.350			0.571311
$665$			236.779i			0.356059i
$666$	557.578	+	488.805i	0.837204	+	0.733941i
$667$	143.990			0.215877
$668$			281.250i			0.421032i
$669$	−687.759	+	311.695i	−1.02804	+	0.465911i
$670$	239.928			0.358102
$671$			1733.50i			2.58345i
$672$	−29.0100	−	64.0111i	−0.0431697	−	0.0952546i
$673$	746.022			1.10850			0.554251	−	0.832350i	$-0.313005\pi$
$673$	746.022			1.10850			0.554251	+	0.832350i	$0.313005\pi$
$674$			425.953i			0.631978i
$675$	39.1534	−	129.198i	0.0580050	−	0.191404i
$676$	195.103			0.288613
$677$			491.931i			0.726634i	0.931666	+	0.363317i	$0.118356\pi$
$677$			491.931i			0.726634i	−0.931666	+	0.363317i	$0.881644\pi$
$678$	−320.243	+	145.135i	−0.472336	+	0.214064i
$679$	−341.080			−0.502326
$680$			90.3070i			0.132804i
$681$	−259.739	−	573.118i	−0.381408	−	0.841583i
$682$	530.258			0.777504
$683$		−	399.771i		−	0.585316i	−0.956217	−	0.292658i	$-0.905460\pi$
$683$		−	399.771i		−	0.585316i	0.956217	−	0.292658i	$-0.0945397\pi$
$684$	303.411	−	346.100i	0.443583	−	0.505994i
$685$	274.059			0.400086
$686$		−	473.502i		−	0.690237i
$687$	−848.226	+	384.419i	−1.23468	+	0.559561i
$688$	−244.117			−0.354821
$689$		−	215.276i		−	0.312448i
$690$	−18.7808	−	41.4401i	−0.0272185	−	0.0600581i
$691$	205.782			0.297804			0.148902	−	0.988852i	$-0.452426\pi$
$691$	205.782			0.297804			0.148902	+	0.988852i	$0.452426\pi$
$692$		−	259.610i		−	0.375158i
$693$	510.184	+	447.256i	0.736196	+	0.645391i
$694$	−105.216			−0.151608
$695$			177.819i			0.255855i
$696$	232.044	−	105.163i	0.333397	−	0.151097i
$697$	−791.924			−1.13619
$698$			172.841i			0.247624i
$699$	−16.7482	−	36.9551i	−0.0239602	−	0.0528686i
$700$	41.4117			0.0591596
$701$		−	95.2339i		−	0.135854i	−0.997690	−	0.0679272i	$-0.978361\pi$
$701$		−	95.2339i		−	0.135854i	0.997690	−	0.0679272i	$-0.0216385\pi$
$702$	308.885	+	93.6077i	0.440007	+	0.133344i
$703$	−1489.67			−2.11901
$704$		−	145.632i		−	0.206864i
$705$	85.3781	−	38.6936i	0.121104	−	0.0548846i
$706$	363.519			0.514900
$707$		−	553.777i		−	0.783277i
$708$	−197.433	−	435.638i	−0.278859	−	0.615308i
$709$	−1225.99			−1.72918			−0.864589	−	0.502479i	$-0.832421\pi$
$709$	−1225.99			−1.72918			−0.864589	+	0.502479i	$0.832421\pi$
$710$		−	419.571i		−	0.590944i
$711$	139.844	−	159.519i	0.196686	−	0.224359i
$712$	−330.854			−0.464683
$713$		−	98.7800i		−	0.138541i
$714$	−228.500	+	103.557i	−0.320028	+	0.145038i
$715$	−344.072			−0.481219
$716$		−	555.321i		−	0.775588i
$717$	−201.061	−	443.644i	−0.280420	−	0.618750i
$718$	−582.441			−0.811199
$719$			1015.78i			1.41277i	0.707826	+	0.706387i	$0.249676\pi$
$719$			1015.78i			1.41277i	−0.707826	+	0.706387i	$0.750324\pi$
$720$	−60.5315	−	53.0654i	−0.0840716	−	0.0737019i
$721$	−651.100			−0.903051
$722$			414.135i			0.573595i
$723$	−312.629	+	141.685i	−0.432406	+	0.195968i
$724$	496.611			0.685927
$725$			150.120i			0.207062i
$726$	368.452	+	812.995i	0.507509	+	1.11983i
$727$	−794.186			−1.09242			−0.546208	−	0.837650i	$-0.683929\pi$
$727$	−794.186			−1.09242			−0.546208	+	0.837650i	$0.683929\pi$
$728$			99.0069i			0.135999i
$729$	−606.361	−	404.682i	−0.831771	−	0.555119i
$730$	−232.232			−0.318125
$731$			871.424i			1.19210i
$732$	520.407	−	235.850i	0.710939	−	0.322200i
$733$	502.321			0.685295			0.342648	−	0.939464i	$-0.388676\pi$
$733$	502.321			0.685295			0.342648	+	0.939464i	$0.388676\pi$
$734$			552.405i			0.752595i
$735$	−88.1970	−	194.608i	−0.119996	−	0.264773i
$736$	−27.1293			−0.0368605
$737$			1381.17i			1.87405i
$738$	465.343	−	530.816i	0.630546	−	0.719263i
$739$	505.526			0.684067			0.342034	−	0.939688i	$-0.388884\pi$
$739$	505.526			0.684067			0.342034	+	0.939688i	$0.388884\pi$
$740$			260.537i			0.352077i
$741$	−590.594	+	267.659i	−0.797023	+	0.361213i
$742$	−149.155			−0.201017
$743$			345.072i			0.464430i	0.972664	+	0.232215i	$0.0745973\pi$
$743$			345.072i			0.464430i	−0.972664	+	0.232215i	$0.925403\pi$
$744$	−72.1439	−	159.187i	−0.0969677	−	0.213961i
$745$	−31.8001			−0.0426848
$746$		−	506.357i		−	0.678763i
$747$	907.679	+	795.723i	1.21510	+	1.06522i
$748$	−519.862			−0.695002
$749$		−	282.311i		−	0.376918i
$750$	43.2043	−	19.5803i	0.0576057	−	0.0261071i
$751$	691.345			0.920566			0.460283	−	0.887772i	$-0.347748\pi$
$751$	691.345			0.920566			0.460283	+	0.887772i	$0.347748\pi$
$752$		−	55.8939i		−	0.0743270i
$753$	96.2776	+	212.438i	0.127859	+	0.282122i
$754$	−358.907			−0.476003
$755$		−	445.126i		−	0.589571i
$756$	64.8564	−	214.012i	0.0857889	−	0.283084i
$757$	−127.590			−0.168546			−0.0842732	−	0.996443i	$-0.526857\pi$
$757$	−127.590			−0.168546			−0.0842732	+	0.996443i	$0.526857\pi$
$758$		−	413.658i		−	0.545723i
$759$	238.554	−	108.114i	0.314301	−	0.142442i
$760$	161.720			0.212790
$761$			183.936i			0.241703i	0.992671	+	0.120852i	$0.0385625\pi$
$761$			183.936i			0.241703i	−0.992671	+	0.120852i	$0.961438\pi$
$762$	22.4706	+	49.5817i	0.0294890	+	0.0650679i
$763$	−14.9404			−0.0195811
$764$		−	228.749i		−	0.299410i
$765$	−189.427	+	216.079i	−0.247617	+	0.282456i
$766$	−96.0261			−0.125360
$767$			673.808i			0.878498i
$768$	−43.7197	+	19.8139i	−0.0569267	+	0.0257993i
$769$	−1436.65			−1.86821			−0.934104	−	0.357002i	$-0.883799\pi$
$769$	−1436.65			−1.86821			−0.934104	+	0.357002i	$0.883799\pi$
$770$			238.391i			0.309599i
$771$	−312.495	−	689.524i	−0.405311	−	0.894325i
$772$	449.567			0.582341
$773$			666.904i			0.862748i	0.902173	+	0.431374i	$0.141971\pi$
$773$			666.904i			0.862748i	−0.902173	+	0.431374i	$0.858029\pi$
$774$	−584.103	−	512.058i	−0.754656	−	0.661574i
$775$	102.985			0.132884
$776$			232.958i			0.300203i
$777$	−659.226	+	298.763i	−0.848425	+	0.384509i
$778$	−470.254			−0.604439
$779$			1418.17i			1.82050i
$780$	46.8125	+	103.293i	0.0600161	+	0.132426i
$781$	2415.30			3.09258
$782$			96.8434i			0.123841i
$783$	775.807	+	235.109i	0.990813	+	0.300266i
$784$	−127.403			−0.162504
$785$			337.609i			0.430075i
$786$	−1007.34

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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-2.73248 + 1.23837i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-1.75132 - 3.86431i) q^{6} +4.14117 q^{7} -2.82843i q^{8} +(5.93289 - 6.76763i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(-2.73248 + 1.23837i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-1.75132 - 3.86431i) q^{6} +4.14117 q^{7} -2.82843i q^{8} +(5.93289 - 6.76763i) q^{9} +3.16228 q^{10} +18.2040i q^{11} +(5.46496 - 2.47674i) q^{12} -8.45273 q^{13} +5.85650i q^{14} +(2.76908 + 6.11001i) q^{15} +4.00000 q^{16} -14.2788i q^{17} +(9.57088 + 8.39037i) q^{18} -25.5703 q^{19} +4.47214i q^{20} +(-11.3157 + 5.12830i) q^{21} -25.7443 q^{22} +4.79583i q^{23} +(3.50263 + 7.72862i) q^{24} -5.00000 q^{25} -11.9540i q^{26} +(-7.83068 + 25.8395i) q^{27} -8.28235 q^{28} -30.0240i q^{29} +(-8.64086 + 3.91606i) q^{30} -20.5971 q^{31} +5.65685i q^{32} +(-22.5432 - 49.7420i) q^{33} +20.1932 q^{34} -9.25994i q^{35} +(-11.8658 + 13.5353i) q^{36} +58.2578 q^{37} -36.1618i q^{38} +(23.0969 - 10.4676i) q^{39} -6.32456 q^{40} -55.4616i q^{41} +(-7.25251 - 16.0028i) q^{42} -61.0293 q^{43} -36.4080i q^{44} +(-15.1329 - 13.2663i) q^{45} -6.78233 q^{46} -13.9735i q^{47} +(-10.9299 + 4.95347i) q^{48} -31.8507 q^{49} -7.07107i q^{50} +(17.6824 + 39.0165i) q^{51} +16.9055 q^{52} +25.4683i q^{53} +(-36.5426 - 11.0743i) q^{54} +40.7054 q^{55} -11.7130i q^{56} +(69.8702 - 31.6654i) q^{57} +42.4604 q^{58} -79.7148i q^{59} +(-5.53815 - 12.2200i) q^{60} +95.2262 q^{61} -29.1286i q^{62} +(24.5691 - 28.0259i) q^{63} -8.00000 q^{64} +18.9009i q^{65} +(70.3459 - 31.8810i) q^{66} +75.8720 q^{67} +28.5576i q^{68} +(-5.93901 - 13.1045i) q^{69} +13.0955 q^{70} -132.680i q^{71} +(-19.1418 - 16.7807i) q^{72} -73.4381 q^{73} +82.3890i q^{74} +(13.6624 - 6.19184i) q^{75} +51.1405 q^{76} +75.3859i q^{77} +(14.8034 + 32.6640i) q^{78} +23.5709 q^{79} -8.94427i q^{80} +(-10.6017 - 80.3032i) q^{81} +78.4345 q^{82} +134.121i q^{83} +(22.6313 - 10.2566i) q^{84} -31.9283 q^{85} -86.3084i q^{86} +(37.1808 + 82.0401i) q^{87} +51.4887 q^{88} -116.975i q^{89} +(18.7614 - 21.4011i) q^{90} -35.0042 q^{91} -9.59166i q^{92} +(56.2810 - 25.5067i) q^{93} +19.7615 q^{94} +57.1768i q^{95} +(-7.00527 - 15.4572i) q^{96} -82.3630 q^{97} -45.0437i q^{98} +(123.198 + 108.002i) 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

Related objects

Downloads

Learn more about