LMFDB - Embedded newform 230.3.d.a.91.16

Newspace parameters

Level:	$ N $	$=$	$ 230 = 2 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	230.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.26704608029$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
Defining polynomial:	$ x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 $ x^16 + 78x^14 + 2165x^12 + 28310x^10 + 184804x^8 + 569634x^6 + 696037x^4 + 285578*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.16
Root			$1.00527i$ of defining polynomial
Character	$\chi$	$=$	230.91
Dual form			230.3.d.a.91.15

$q$-expansion

$f(q)$	$=$	$q+1.41421 q^{2} +4.30716 q^{3} +2.00000 q^{4} +2.23607i q^{5} +6.09125 q^{6} -1.47532i q^{7} +2.82843 q^{8} +9.55167 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +4.30716 q^{3} +2.00000 q^{4} +2.23607i q^{5} +6.09125 q^{6} -1.47532i q^{7} +2.82843 q^{8} +9.55167 q^{9} +3.16228i q^{10} -6.04959i q^{11} +8.61433 q^{12} -5.21324 q^{13} -2.08642i q^{14} +9.63111i q^{15} +4.00000 q^{16} +15.7063i q^{17} +13.5081 q^{18} +4.82663i q^{19} +4.47214i q^{20} -6.35447i q^{21} -8.55541i q^{22} +(-2.58013 - 22.8548i) q^{23} +12.1825 q^{24} -5.00000 q^{25} -7.37263 q^{26} +2.37613 q^{27} -2.95065i q^{28} -23.4711 q^{29} +13.6205i q^{30} -20.4887 q^{31} +5.65685 q^{32} -26.0566i q^{33} +22.2121i q^{34} +3.29893 q^{35} +19.1033 q^{36} +15.5248i q^{37} +6.82589i q^{38} -22.4543 q^{39} +6.32456i q^{40} +20.3093 q^{41} -8.98657i q^{42} -38.1696i q^{43} -12.0992i q^{44} +21.3582i q^{45} +(-3.64885 - 32.3216i) q^{46} -13.8273 q^{47} +17.2287 q^{48} +46.8234 q^{49} -7.07107 q^{50} +67.6497i q^{51} -10.4265 q^{52} -38.2742i q^{53} +3.36035 q^{54} +13.5273 q^{55} -4.17285i q^{56} +20.7891i q^{57} -33.1932 q^{58} -33.5696 q^{59} +19.2622i q^{60} +100.567i q^{61} -28.9753 q^{62} -14.0918i q^{63} +8.00000 q^{64} -11.6572i q^{65} -36.8496i q^{66} -32.4469i q^{67} +31.4126i q^{68} +(-11.1130 - 98.4395i) q^{69} +4.66539 q^{70} -24.1306 q^{71} +27.0162 q^{72} +15.1818 q^{73} +21.9554i q^{74} -21.5358 q^{75} +9.65326i q^{76} -8.92511 q^{77} -31.7552 q^{78} -11.2095i q^{79} +8.94427i q^{80} -75.7306 q^{81} +28.7216 q^{82} +44.1310i q^{83} -12.7089i q^{84} -35.1204 q^{85} -53.9800i q^{86} -101.094 q^{87} -17.1108i q^{88} -111.039i q^{89} +30.2050i q^{90} +7.69122i q^{91} +(-5.16026 - 45.7096i) q^{92} -88.2480 q^{93} -19.5547 q^{94} -10.7927 q^{95} +24.3650 q^{96} +154.126i q^{97} +66.2183 q^{98} -57.7837i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9}+O(q^{10}) $ 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9	$ 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + 24 q^{13} + 64 q^{16} - 32 q^{18} + 4 q^{23} - 16 q^{24} - 80 q^{25} + 96 q^{26} - 96 q^{27} - 108 q^{29} - 116 q^{31} + 60 q^{35} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} - 128 q^{47} - 28 q^{49} + 48 q^{52} + 224 q^{54} + 160 q^{58} + 204 q^{59} + 64 q^{62} + 128 q^{64} - 268 q^{69} - 120 q^{70} + 236 q^{71} - 64 q^{72} - 112 q^{73} - 936 q^{77} - 432 q^{78} - 136 q^{81} - 64 q^{82} + 60 q^{85} - 152 q^{87} + 8 q^{92} + 856 q^{93} - 216 q^{94} - 160 q^{95} - 32 q^{96} + 256 q^{98}+O(q^{100}) $ 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9 + 24 * q^13 + 64 * q^16 - 32 * q^18 + 4 * q^23 - 16 * q^24 - 80 * q^25 + 96 * q^26 - 96 * q^27 - 108 * q^29 - 116 * q^31 + 60 * q^35 + 128 * q^36 + 248 * q^39 - 156 * q^41 - 124 * q^46 - 128 * q^47 - 28 * q^49 + 48 * q^52 + 224 * q^54 + 160 * q^58 + 204 * q^59 + 64 * q^62 + 128 * q^64 - 268 * q^69 - 120 * q^70 + 236 * q^71 - 64 * q^72 - 112 * q^73 - 936 * q^77 - 432 * q^78 - 136 * q^81 - 64 * q^82 + 60 * q^85 - 152 * q^87 + 8 * q^92 + 856 * q^93 - 216 * q^94 - 160 * q^95 - 32 * q^96 + 256 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/230\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$	4.30716			1.43572			0.717861	−	0.696187i	$-0.245121\pi$
$3$	4.30716			1.43572			0.717861	+	0.696187i	$0.245121\pi$
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$	6.09125			1.01521
$7$		−	1.47532i		−	0.210761i	−0.994432	−	0.105380i	$-0.966394\pi$
$7$		−	1.47532i		−	0.210761i	0.994432	−	0.105380i	$-0.0336060\pi$
$8$	2.82843			0.353553
$9$	9.55167			1.06130
$10$			3.16228i			0.316228i
$11$		−	6.04959i		−	0.549963i	−0.961450	−	0.274981i	$-0.911328\pi$
$11$		−	6.04959i		−	0.549963i	0.961450	−	0.274981i	$-0.0886717\pi$
$12$	8.61433			0.717861
$13$	−5.21324			−0.401018			−0.200509	−	0.979692i	$-0.564260\pi$
$13$	−5.21324			−0.401018			−0.200509	+	0.979692i	$0.564260\pi$
$14$		−	2.08642i		−	0.149030i
$15$			9.63111i			0.642074i
$16$	4.00000			0.250000
$17$			15.7063i			0.923901i	0.886906	+	0.461951i	$0.152850\pi$
$17$			15.7063i			0.923901i	−0.886906	+	0.461951i	$0.847150\pi$
$18$	13.5081			0.750450
$19$			4.82663i			0.254033i	0.991901	+	0.127017i	$0.0405402\pi$
$19$			4.82663i			0.254033i	−0.991901	+	0.127017i	$0.959460\pi$
$20$			4.47214i			0.223607i
$21$		−	6.35447i		−	0.302594i
$22$		−	8.55541i		−	0.388882i
$23$	−2.58013	−	22.8548i	−0.112180	−	0.993688i
$23$	−2.58013	−	22.8548i	−0.112180	−	0.993688i
$24$	12.1825			0.507604
$25$	−5.00000			−0.200000
$26$	−7.37263			−0.283563
$27$	2.37613			0.0880047
$28$		−	2.95065i		−	0.105380i
$29$	−23.4711			−0.809349			−0.404674	−	0.914461i	$-0.632615\pi$
$29$	−23.4711			−0.809349			−0.404674	+	0.914461i	$0.632615\pi$
$30$			13.6205i			0.454015i
$31$	−20.4887			−0.660924			−0.330462	−	0.943819i	$-0.607205\pi$
$31$	−20.4887			−0.660924			−0.330462	+	0.943819i	$0.607205\pi$
$32$	5.65685			0.176777
$33$		−	26.0566i		−	0.789593i
$34$			22.2121i			0.653297i
$35$	3.29893			0.0942550
$36$	19.1033			0.530648
$37$			15.5248i			0.419589i	0.977746	+	0.209794i	$0.0672795\pi$
$37$			15.5248i			0.419589i	−0.977746	+	0.209794i	$0.932721\pi$
$38$			6.82589i			0.179629i
$39$	−22.4543			−0.575751
$40$			6.32456i			0.158114i
$41$	20.3093			0.495348			0.247674	−	0.968843i	$-0.420334\pi$
$41$	20.3093			0.495348			0.247674	+	0.968843i	$0.420334\pi$
$42$		−	8.98657i		−	0.213966i
$43$		−	38.1696i		−	0.887666i	−0.896109	−	0.443833i	$-0.853618\pi$
$43$		−	38.1696i		−	0.887666i	0.896109	−	0.443833i	$-0.146382\pi$
$44$		−	12.0992i		−	0.274981i
$45$			21.3582i			0.474626i
$46$	−3.64885	−	32.3216i	−0.0793229	−	0.702643i
$47$	−13.8273			−0.294198			−0.147099	−	0.989122i	$-0.546994\pi$
$47$	−13.8273			−0.294198			−0.147099	+	0.989122i	$0.546994\pi$
$48$	17.2287			0.358930
$49$	46.8234			0.955580
$50$	−7.07107			−0.141421
$51$			67.6497i			1.32647i
$52$	−10.4265			−0.200509
$53$		−	38.2742i		−	0.722154i	−0.932536	−	0.361077i	$-0.882409\pi$
$53$		−	38.2742i		−	0.722154i	0.932536	−	0.361077i	$-0.117591\pi$
$54$	3.36035			0.0622287
$55$	13.5273			0.245951
$56$		−	4.17285i		−	0.0745152i
$57$			20.7891i			0.364721i
$58$	−33.1932			−0.572296
$59$	−33.5696			−0.568977			−0.284488	−	0.958679i	$-0.591824\pi$
$59$	−33.5696			−0.568977			−0.284488	+	0.958679i	$0.591824\pi$
$60$			19.2622i			0.321037i
$61$			100.567i			1.64864i	0.566124	+	0.824320i	$0.308442\pi$
$61$			100.567i			1.64864i	−0.566124	+	0.824320i	$0.691558\pi$
$62$	−28.9753			−0.467344
$63$		−	14.0918i		−	0.223680i
$64$	8.00000			0.125000
$65$		−	11.6572i		−	0.179341i
$66$		−	36.8496i		−	0.558327i
$67$		−	32.4469i		−	0.484282i	−0.970241	−	0.242141i	$-0.922150\pi$
$67$		−	32.4469i		−	0.484282i	0.970241	−	0.242141i	$-0.0778497\pi$
$68$			31.4126i			0.461951i
$69$	−11.1130	−	98.4395i	−0.161059	−	1.42666i
$70$	4.66539			0.0666484
$71$	−24.1306			−0.339868			−0.169934	−	0.985455i	$-0.554355\pi$
$71$	−24.1306			−0.339868			−0.169934	+	0.985455i	$0.554355\pi$
$72$	27.0162			0.375225
$73$	15.1818			0.207969			0.103985	−	0.994579i	$-0.466841\pi$
$73$	15.1818			0.207969			0.103985	+	0.994579i	$0.466841\pi$
$74$			21.9554i			0.296694i
$75$	−21.5358			−0.287144
$76$			9.65326i			0.127017i
$77$	−8.92511			−0.115910
$78$	−31.7552			−0.407117
$79$		−	11.2095i		−	0.141893i	−0.997480	−	0.0709463i	$-0.977398\pi$
$79$		−	11.2095i		−	0.141893i	0.997480	−	0.0709463i	$-0.0226019\pi$
$80$			8.94427i			0.111803i
$81$	−75.7306			−0.934946
$82$	28.7216			0.350264
$83$			44.1310i			0.531698i	0.964015	+	0.265849i	$0.0856523\pi$
$83$			44.1310i			0.531698i	−0.964015	+	0.265849i	$0.914348\pi$
$84$		−	12.7089i		−	0.151297i
$85$	−35.1204			−0.413181
$86$		−	53.9800i		−	0.627675i
$87$	−101.094			−1.16200
$88$		−	17.1108i		−	0.194441i
$89$		−	111.039i		−	1.24763i	−0.781572	−	0.623815i	$-0.785582\pi$
$89$		−	111.039i		−	1.24763i	0.781572	−	0.623815i	$-0.214418\pi$
$90$			30.2050i			0.335611i
$91$			7.69122i			0.0845189i
$92$	−5.16026	−	45.7096i	−0.0560898	−	0.496844i
$93$	−88.2480			−0.948903
$94$	−19.5547			−0.208029
$95$	−10.7927			−0.113607
$96$	24.3650			0.253802
$97$			154.126i			1.58893i	0.607312	+	0.794463i	$0.292248\pi$
$97$			154.126i			1.58893i	−0.607312	+	0.794463i	$0.707752\pi$
$98$	66.2183			0.675697
$99$		−	57.7837i		−	0.583673i
$100$	−10.0000			−0.100000
$101$	−58.8607			−0.582779			−0.291390	−	0.956604i	$-0.594118\pi$
$101$	−58.8607			−0.582779			−0.291390	+	0.956604i	$0.594118\pi$
$102$			95.6712i			0.937952i
$103$		−	54.2662i		−	0.526856i	−0.964679	−	0.263428i	$-0.915147\pi$
$103$		−	54.2662i		−	0.526856i	0.964679	−	0.263428i	$-0.0848531\pi$
$104$	−14.7453			−0.141781
$105$	14.2090			0.135324
$106$		−	54.1279i		−	0.510640i
$107$			119.124i			1.11331i	0.830745	+	0.556653i	$0.187915\pi$
$107$			119.124i			1.11331i	−0.830745	+	0.556653i	$0.812085\pi$
$108$	4.75225			0.0440023
$109$		−	149.223i		−	1.36902i	−0.729003	−	0.684510i	$-0.760016\pi$
$109$		−	149.223i		−	1.36902i	0.729003	−	0.684510i	$-0.239984\pi$
$110$	19.1305			0.173913
$111$			66.8678i			0.602413i
$112$		−	5.90130i		−	0.0526902i
$113$			35.3339i			0.312690i	0.987703	+	0.156345i	$0.0499711\pi$
$113$			35.3339i			0.312690i	−0.987703	+	0.156345i	$0.950029\pi$
$114$			29.4002i			0.257897i
$115$	51.1049	−	5.76935i	0.444391	−	0.0501682i
$116$	−46.9422			−0.404674
$117$	−49.7951			−0.425599
$118$	−47.4746			−0.402327
$119$	23.1719			0.194722
$120$			27.2409i			0.227008i
$121$	84.4025			0.697541
$122$			142.223i			1.16576i
$123$	87.4753			0.711181
$124$	−40.9773			−0.330462
$125$		−	11.1803i		−	0.0894427i
$126$		−	19.9288i		−	0.158165i
$127$	92.8398			0.731022			0.365511	−	0.930807i	$-0.380894\pi$
$127$	92.8398			0.731022			0.365511	+	0.930807i	$0.380894\pi$
$128$	11.3137			0.0883883
$129$		−	164.403i		−	1.27444i
$130$		−	16.4857i		−	0.126813i
$131$	151.709			1.15808			0.579042	−	0.815297i	$-0.303427\pi$
$131$	151.709			1.15808			0.579042	+	0.815297i	$0.303427\pi$
$132$		−	52.1131i		−	0.394797i
$133$	7.12085			0.0535402
$134$		−	45.8869i		−	0.342439i
$135$			5.31318i			0.0393569i
$136$			44.4242i			0.326648i
$137$			267.464i			1.95229i	0.217121	+	0.976145i	$0.430333\pi$
$137$			267.464i			1.95229i	−0.217121	+	0.976145i	$0.569667\pi$
$138$	−15.7162	−	139.214i	−0.113886	−	1.00880i
$139$	239.779			1.72503			0.862516	−	0.506030i	$-0.168888\pi$
$139$	239.779			1.72503			0.862516	+	0.506030i	$0.168888\pi$
$140$	6.59785			0.0471275
$141$	−59.5564			−0.422386
$142$	−34.1258			−0.240323
$143$			31.5380i			0.220545i
$144$	38.2067			0.265324
$145$		−	52.4830i		−	0.361952i
$146$	21.4703			0.147057
$147$	201.676			1.37195
$148$			31.0496i			0.209794i
$149$		−	201.647i		−	1.35333i	−0.736290	−	0.676667i	$-0.763424\pi$
$149$		−	201.647i		−	1.35333i	0.736290	−	0.676667i	$-0.236576\pi$
$150$	−30.4563			−0.203042
$151$	120.787			0.799912			0.399956	−	0.916534i	$-0.369025\pi$
$151$	120.787			0.799912			0.399956	+	0.916534i	$0.369025\pi$
$152$			13.6518i			0.0898143i
$153$			150.022i			0.980533i
$154$	−12.6220			−0.0819611
$155$		−	45.8140i		−	0.295574i
$156$	−44.9086			−0.287875
$157$			194.897i			1.24138i	0.784055	+	0.620692i	$0.213148\pi$
$157$			194.897i			1.24138i	−0.784055	+	0.620692i	$0.786852\pi$
$158$		−	15.8526i		−	0.100333i
$159$		−	164.853i		−	1.03681i
$160$			12.6491i			0.0790569i
$161$	−33.7183	+	3.80653i	−0.209430	+	0.0236430i
$162$	−107.099			−0.661107
$163$	153.813			0.943635			0.471817	−	0.881696i	$-0.343598\pi$
$163$	153.813			0.943635			0.471817	+	0.881696i	$0.343598\pi$
$164$	40.6185			0.247674
$165$	58.2643			0.353117
$166$			62.4106i			0.375967i
$167$	48.4949			0.290389			0.145194	−	0.989403i	$-0.453619\pi$
$167$	48.4949			0.290389			0.145194	+	0.989403i	$0.453619\pi$
$168$		−	17.9731i		−	0.106983i
$169$	−141.822			−0.839184
$170$	−49.6678			−0.292163
$171$			46.1024i			0.269605i
$172$		−	76.3393i		−	0.443833i
$173$	−111.269			−0.643174			−0.321587	−	0.946880i	$-0.604216\pi$
$173$	−111.269			−0.643174			−0.321587	+	0.946880i	$0.604216\pi$
$174$	−142.968			−0.821658
$175$			7.37662i			0.0421521i
$176$		−	24.1984i		−	0.137491i
$177$	−144.590			−0.816892
$178$		−	157.033i		−	0.882207i
$179$	262.590			1.46698			0.733492	−	0.679698i	$-0.237889\pi$
$179$	262.590			1.46698			0.733492	+	0.679698i	$0.237889\pi$
$180$			42.7164i			0.237313i
$181$		−	211.167i		−	1.16667i	−0.812231	−	0.583335i	$-0.801747\pi$
$181$		−	211.167i		−	1.16667i	0.812231	−	0.583335i	$-0.198253\pi$
$182$			10.8770i			0.0597639i
$183$			433.159i			2.36699i
$184$	−7.29771	−	64.6432i	−0.0396615	−	0.351322i
$185$	−34.7145			−0.187646
$186$	−124.802			−0.670976
$187$	95.0168			0.508111
$188$	−27.6546			−0.147099
$189$		−	3.50556i		−	0.0185479i
$190$	−15.2631			−0.0803324
$191$		−	72.8209i		−	0.381261i	−0.981662	−	0.190631i	$-0.938947\pi$
$191$		−	72.8209i		−	0.381261i	0.981662	−	0.190631i	$-0.0610533\pi$
$192$	34.4573			0.179465
$193$	198.550			1.02875			0.514377	−	0.857564i	$-0.328023\pi$
$193$	198.550			1.02875			0.514377	+	0.857564i	$0.328023\pi$
$194$			217.967i			1.12354i
$195$		−	50.2093i		−	0.257484i
$196$	93.6468			0.477790
$197$	28.2055			0.143175			0.0715876	−	0.997434i	$-0.477193\pi$
$197$	28.2055			0.143175			0.0715876	+	0.997434i	$0.477193\pi$
$198$		−	81.7184i		−	0.412719i
$199$			214.499i			1.07788i	0.842343	+	0.538941i	$0.181175\pi$
$199$			214.499i			1.07788i	−0.842343	+	0.538941i	$0.818825\pi$
$200$	−14.1421			−0.0707107
$201$		−	139.754i		−	0.695295i
$202$	−83.2416			−0.412087
$203$			34.6275i			0.170579i
$204$			135.299i			0.663233i
$205$			45.4129i			0.221526i
$206$		−	76.7439i		−	0.372543i
$207$	−24.6445	−	218.302i	−0.119056	−	1.05460i
$208$	−20.8530			−0.100255
$209$	29.1991			0.139709
$210$	20.0946			0.0956885
$211$	240.262			1.13868			0.569340	−	0.822102i	$-0.307199\pi$
$211$	240.262			1.13868			0.569340	+	0.822102i	$0.307199\pi$
$212$		−	76.5483i		−	0.361077i
$213$	−103.935			−0.487956
$214$			168.466i			0.787226i
$215$	85.3499			0.396976
$216$	6.72070			0.0311144
$217$			30.2274i			0.139297i
$218$		−	211.034i		−	0.968044i
$219$	65.3904			0.298586
$220$	27.0546			0.122975
$221$		−	81.8808i		−	0.370501i
$222$			94.5654i			0.425970i
$223$	257.402			1.15427			0.577136	−	0.816648i	$-0.304170\pi$
$223$	257.402			1.15427			0.577136	+	0.816648i	$0.304170\pi$
$224$		−	8.34570i		−	0.0372576i
$225$	−47.7583			−0.212259
$226$			49.9697i			0.221105i
$227$		−	106.401i		−	0.468727i	−0.972149	−	0.234363i	$-0.924699\pi$
$227$		−	106.401i		−	0.468727i	0.972149	−	0.234363i	$-0.0753005\pi$
$228$			41.5782i			0.182360i
$229$			328.507i			1.43453i	0.696801	+	0.717265i	$0.254606\pi$
$229$			328.507i			1.43453i	−0.696801	+	0.717265i	$0.745394\pi$
$230$	72.2733	−	8.15909i	0.314232	−	0.0354743i
$231$	−38.4419			−0.166415
$232$	−66.3863			−0.286148
$233$	−149.213			−0.640400			−0.320200	−	0.947350i	$-0.603750\pi$
$233$	−149.213			−0.640400			−0.320200	+	0.947350i	$0.603750\pi$
$234$	−70.4210			−0.300944
$235$		−	30.9187i		−	0.131569i
$236$	−67.1393			−0.284488
$237$		−	48.2812i		−	0.203718i
$238$	32.7701			0.137689
$239$	−343.168			−1.43585			−0.717925	−	0.696121i	$-0.754908\pi$
$239$	−343.168			−1.43585			−0.717925	+	0.696121i	$0.754908\pi$
$240$			38.5245i			0.160519i
$241$		−	348.748i		−	1.44709i	−0.690278	−	0.723544i	$-0.742512\pi$
$241$		−	348.748i		−	1.44709i	0.690278	−	0.723544i	$-0.257488\pi$
$242$	119.363			0.493236
$243$	−347.570			−1.43033
$244$			201.134i			0.824320i
$245$			104.700i			0.427348i
$246$	123.709			0.502881
$247$		−	25.1624i		−	0.101872i
$248$	−57.9507			−0.233672
$249$			190.079i			0.763371i
$250$		−	15.8114i		−	0.0632456i
$251$		−	225.099i		−	0.896809i	−0.893831	−	0.448405i	$-0.851992\pi$
$251$		−	225.099i		−	0.896809i	0.893831	−	0.448405i	$-0.148008\pi$
$252$		−	28.1836i		−	0.111840i
$253$	−138.262	+	15.6087i	−0.546491	+	0.0616946i
$254$	131.295			0.516911
$255$	−151.269			−0.593213
$256$	16.0000			0.0625000
$257$	−393.632			−1.53164			−0.765821	−	0.643054i	$-0.777667\pi$
$257$	−393.632			−1.53164			−0.765821	+	0.643054i	$0.777667\pi$
$258$		−	232.501i		−	0.901166i
$259$	22.9041			0.0884328
$260$		−	23.3143i		−	0.0896704i
$261$	−224.188			−0.858959
$262$	214.549			0.818890
$263$			132.720i			0.504640i	0.967644	+	0.252320i	$0.0811935\pi$
$263$			132.720i			0.504640i	−0.967644	+	0.252320i	$0.918806\pi$
$264$		−	73.6991i		−	0.279163i
$265$	85.5836			0.322957
$266$	10.0704			0.0378586
$267$		−	478.263i		−	1.79125i
$268$		−	64.8938i		−	0.242141i
$269$	−438.116			−1.62868			−0.814341	−	0.580386i	$-0.802902\pi$
$269$	−438.116			−1.62868			−0.814341	+	0.580386i	$0.802902\pi$
$270$			7.51397i			0.0278295i
$271$	−204.698			−0.755342			−0.377671	−	0.925940i	$-0.623275\pi$
$271$	−204.698			−0.755342			−0.377671	+	0.925940i	$0.623275\pi$
$272$			62.8253i			0.230975i
$273$			33.1274i			0.121346i
$274$			378.251i			1.38048i
$275$			30.2479i			0.109993i
$276$	−22.2261	−	196.879i	−0.0805293	−	0.713330i
$277$	173.804			0.627452			0.313726	−	0.949514i	$-0.398423\pi$
$277$	173.804			0.627452			0.313726	+	0.949514i	$0.398423\pi$
$278$	339.099			1.21978
$279$	−195.701			−0.701437
$280$	9.33077			0.0333242
$281$		−	143.605i		−	0.511051i	−0.966802	−	0.255526i	$-0.917752\pi$
$281$		−	143.605i		−	0.511051i	0.966802	−	0.255526i	$-0.0822485\pi$
$282$	−84.2255			−0.298672
$283$			72.1714i			0.255023i	0.991837	+	0.127511i	$0.0406989\pi$
$283$			72.1714i			0.255023i	−0.991837	+	0.127511i	$0.959301\pi$
$284$	−48.2612			−0.169934
$285$	−46.4858			−0.163108
$286$			44.6014i			0.155949i
$287$		−	29.9627i		−	0.104400i
$288$	54.0324			0.187612
$289$	42.3114			0.146406
$290$		−	74.2222i		−	0.255939i
$291$			663.845i			2.28126i
$292$	30.3635			0.103985
$293$			124.199i			0.423889i	0.977282	+	0.211944i	$0.0679796\pi$
$293$			124.199i			0.423889i	−0.977282	+	0.211944i	$0.932020\pi$
$294$	285.213			0.970113
$295$		−	75.0640i		−	0.254454i
$296$			43.9107i			0.148347i
$297$		−	14.3746i		−	0.0483993i
$298$		−	285.171i		−	0.956951i
$299$	13.4508	+	119.148i	0.0449861	+	0.398487i
$300$	−43.0716			−0.143572
$301$	−56.3126			−0.187085
$302$	170.818			0.565623
$303$	−253.523			−0.836709
$304$			19.3065i			0.0635083i
$305$	−224.875			−0.737294
$306$			212.163i			0.693342i
$307$	219.717			0.715690			0.357845	−	0.933781i	$-0.383512\pi$
$307$	219.717			0.715690			0.357845	+	0.933781i	$0.383512\pi$
$308$	−17.8502			−0.0579552
$309$		−	233.733i		−	0.756418i
$310$		−	64.7908i		−	0.209003i
$311$	−317.069			−1.01951			−0.509757	−	0.860319i	$-0.670265\pi$
$311$	−317.069			−1.01951			−0.509757	+	0.860319i	$0.670265\pi$
$312$	−63.5103			−0.203559
$313$		−	484.654i		−	1.54841i	−0.632932	−	0.774207i	$-0.718149\pi$
$313$		−	484.654i		−	1.54841i	0.632932	−	0.774207i	$-0.281851\pi$
$314$			275.626i			0.877791i
$315$	31.5103			0.100033
$316$		−	22.4190i		−	0.0709463i
$317$	−521.910			−1.64640			−0.823202	−	0.567748i	$-0.807815\pi$
$317$	−521.910			−1.64640			−0.823202	+	0.567748i	$0.807815\pi$
$318$		−	233.138i		−	0.733137i
$319$			141.991i			0.445111i
$320$			17.8885i			0.0559017i
$321$			513.086i			1.59840i
$322$	−47.6849	+	5.38325i	−0.148090	+	0.0167182i
$323$	−75.8086			−0.234702
$324$	−151.461			−0.467473
$325$	26.0662			0.0802037
$326$	217.524			0.667251
$327$		−	642.729i		−	1.96553i
$328$	57.4432			0.175132
$329$			20.3997i			0.0620053i
$330$	82.3981			0.249691
$331$	−318.270			−0.961540			−0.480770	−	0.876847i	$-0.659643\pi$
$331$	−318.270			−0.961540			−0.480770	+	0.876847i	$0.659643\pi$
$332$			88.2619i			0.265849i
$333$			148.288i			0.445308i
$334$	68.5822			0.205336
$335$	72.5535			0.216578
$336$		−	25.4179i		−	0.0756484i
$337$			116.539i			0.345812i	0.984938	+	0.172906i	$0.0553156\pi$
$337$			116.539i			0.345812i	−0.984938	+	0.172906i	$0.944684\pi$
$338$	−200.567			−0.593393
$339$			152.189i			0.448935i
$340$	−70.2408			−0.206591
$341$			123.948i			0.363484i
$342$			65.1986i			0.190639i
$343$		−	141.371i		−	0.412159i
$344$		−	107.960i		−	0.313837i
$345$	220.117	−	24.8495i	0.638021	−	0.0720276i
$346$	−157.358			−0.454793
$347$	−100.715			−0.290245			−0.145122	−	0.989414i	$-0.546358\pi$
$347$	−100.715			−0.290245			−0.145122	+	0.989414i	$0.546358\pi$
$348$	−202.188			−0.581000
$349$	143.476			0.411106			0.205553	−	0.978646i	$-0.434101\pi$
$349$	143.476			0.411106			0.205553	+	0.978646i	$0.434101\pi$
$350$			10.4321i			0.0298061i
$351$	−12.3873			−0.0352915
$352$		−	34.2216i		−	0.0972206i
$353$	−274.394			−0.777320			−0.388660	−	0.921381i	$-0.627062\pi$
$353$	−274.394			−0.777320			−0.388660	+	0.921381i	$0.627062\pi$
$354$	−204.481			−0.577630
$355$		−	53.9577i		−	0.151994i
$356$		−	222.078i		−	0.623815i
$357$	99.8053			0.279567
$358$	371.358			1.03731
$359$		−	476.871i		−	1.32833i	−0.747586	−	0.664166i	$-0.768787\pi$
$359$		−	476.871i		−	1.32833i	0.747586	−	0.664166i	$-0.231213\pi$
$360$			60.4101i			0.167806i
$361$	337.704			0.935467
$362$		−	298.636i		−	0.824961i
$363$	363.535			1.00147
$364$			15.3824i			0.0422595i
$365$			33.9475i			0.0930067i
$366$			612.579i			1.67371i
$367$		−	14.0319i		−	0.0382340i	−0.999817	−	0.0191170i	$-0.993914\pi$
$367$		−	14.0319i		−	0.0382340i	0.999817	−	0.0191170i	$-0.00608551\pi$
$368$	−10.3205	−	91.4193i	−0.0280449	−	0.248422i
$369$	193.987			0.525711
$370$	−49.0937			−0.132686
$371$	−56.4668			−0.152202
$372$	−176.496			−0.474452
$373$		−	105.662i		−	0.283277i	−0.989918	−	0.141639i	$-0.954763\pi$
$373$		−	105.662i		−	0.283277i	0.989918	−	0.141639i	$-0.0452371\pi$
$374$	134.374			0.359289
$375$		−	48.1556i		−	0.128415i
$376$	−39.1095			−0.104015
$377$	122.361			0.324564
$378$		−	4.95761i		−	0.0131154i
$379$			339.983i			0.897053i	0.893770	+	0.448527i	$0.148051\pi$
$379$			339.983i			0.897053i	−0.893770	+	0.448527i	$0.851949\pi$
$380$	−21.5854			−0.0568036
$381$	399.876			1.04954
$382$		−	102.984i		−	0.269592i
$383$			699.796i			1.82714i	0.406676	+	0.913572i	$0.366688\pi$
$383$			699.796i			1.82714i	−0.406676	+	0.913572i	$0.633312\pi$
$384$	48.7300			0.126901
$385$		−	19.9571i		−	0.0518367i
$386$	280.792			0.727439
$387$		−	364.584i		−	0.942077i
$388$			308.252i			0.794463i
$389$		−	124.817i		−	0.320867i	−0.987047	−	0.160433i	$-0.948711\pi$
$389$		−	124.817i		−	0.320867i	0.987047	−	0.160433i	$-0.0512892\pi$
$390$		−	71.0067i		−	0.182068i
$391$	358.965	−	40.5244i	0.918070	−	0.103643i
$392$	132.437			0.337849
$393$	653.436			1.66269
$394$	39.8886			0.101240
$395$	25.0652			0.0634563
$396$		−	115.567i		−	0.291837i
$397$	−149.557			−0.376719			−0.188360	−	0.982100i	$-0.560317\pi$
$397$	−149.557			−0.376719			−0.188360	+	0.982100i	$0.560317\pi$
$398$			303.347i			0.762178i
$399$	30.6707			0.0768688
$400$	−20.0000			−0.0500000
$401$		−	426.696i		−	1.06408i	−0.846719	−	0.532040i	$-0.821426\pi$
$401$		−	426.696i		−	1.06408i	0.846719	−	0.532040i	$-0.178574\pi$
$402$		−	197.642i		−	0.491647i
$403$	106.812			0.265043
$404$	−117.721			−0.291390
$405$		−	169.339i		−	0.418121i
$406$			48.9707i			0.120617i
$407$	93.9186			0.230758
$408$			191.342i			0.468976i
$409$	346.263			0.846610			0.423305	−	0.905987i	$-0.360870\pi$
$409$	346.263			0.846610			0.423305	+	0.905987i	$0.360870\pi$
$410$			64.2235i			0.156643i
$411$			1152.01i			2.80294i
$412$		−	108.532i		−	0.263428i
$413$			49.5261i			0.119918i
$414$	−34.8527	−	308.725i	−0.0841852	−	0.745713i
$415$	−98.6798			−0.237783
$416$	−29.4905			−0.0708907
$417$	1032.77			2.47667
$418$	41.2938			0.0987890
$419$		−	158.433i		−	0.378121i	−0.981965	−	0.189060i	$-0.939456\pi$
$419$		−	158.433i		−	0.378121i	0.981965	−	0.189060i	$-0.0605442\pi$
$420$	28.4180			0.0676620
$421$		−	18.0568i		−	0.0428902i	−0.999770	−	0.0214451i	$-0.993173\pi$
$421$		−	18.0568i		−	0.0428902i	0.999770	−	0.0214451i	$-0.00682671\pi$
$422$	339.781			0.805169
$423$	−132.074			−0.312231
$424$		−	108.256i		−	0.255320i
$425$		−	78.5316i		−	0.184780i
$426$	−146.986			−0.345037
$427$	148.369			0.347469
$428$			238.247i			0.556653i
$429$			135.839i			0.316641i
$430$	120.703			0.280705
$431$		−	151.730i		−	0.352041i	−0.984387	−	0.176020i	$-0.943678\pi$
$431$		−	151.730i		−	0.352041i	0.984387	−	0.176020i	$-0.0563225\pi$
$432$	9.50451			0.0220012
$433$			625.471i			1.44451i	0.691629	+	0.722253i	$0.256894\pi$
$433$			625.471i			1.44451i	−0.691629	+	0.722253i	$0.743106\pi$
$434$			42.7480i			0.0984978i
$435$		−	226.053i		−	0.519662i
$436$		−	298.447i		−	0.684510i
$437$	110.312	−	12.4533i	0.252430	−	0.0284973i
$438$	92.4759			0.211132
$439$	272.842			0.621509			0.310755	−	0.950490i	$-0.399418\pi$
$439$	272.842			0.621509			0.310755	+	0.950490i	$0.399418\pi$
$440$	38.2610			0.0869567
$441$	447.242			1.01415
$442$		−	115.797i		−	0.261984i
$443$	5.57528			0.0125853			0.00629264	−	0.999980i	$-0.497997\pi$
$443$	5.57528			0.0125853			0.00629264	+	0.999980i	$0.497997\pi$
$444$			133.736i			0.301206i
$445$	248.291			0.557957
$446$	364.022			0.816193
$447$		−	868.525i		−	1.94301i
$448$		−	11.8026i		−	0.0263451i
$449$	455.331			1.01410			0.507050	−	0.861916i	$-0.330736\pi$
$449$	455.331			1.01410			0.507050	+	0.861916i	$0.330736\pi$
$450$	−67.5405			−0.150090
$451$		−	122.863i		−	0.272423i
$452$			70.6679i			0.156345i
$453$	520.248			1.14845
$454$		−	150.474i		−	0.331440i
$455$	−17.1981			−0.0377980
$456$			58.8004i			0.128948i
$457$			597.785i			1.30806i	0.756467	+	0.654032i	$0.226924\pi$
$457$			597.785i			1.30806i	−0.756467	+	0.654032i	$0.773076\pi$
$458$			464.579i			1.01437i
$459$			37.3202i			0.0813077i
$460$	102.210	−	11.5387i	0.222195	−	0.0250841i
$461$	−340.556			−0.738734			−0.369367	−	0.929284i	$-0.620425\pi$
$461$	−340.556			−0.738734			−0.369367	+	0.929284i	$0.620425\pi$
$462$	−54.3651			−0.117673
$463$	−471.395			−1.01813			−0.509066	−	0.860727i	$-0.670009\pi$
$463$	−471.395			−1.01813			−0.509066	+	0.860727i	$0.670009\pi$
$464$	−93.8844			−0.202337
$465$		−	197.329i		−	0.424362i
$466$	−211.019			−0.452831
$467$		−	245.681i		−	0.526084i	−0.964784	−	0.263042i	$-0.915274\pi$
$467$		−	245.681i		−	0.526084i	0.964784	−	0.263042i	$-0.0847257\pi$
$468$	−99.5903			−0.212800
$469$	−47.8697			−0.102068
$470$		−	43.7257i		−	0.0930334i
$471$			839.455i			1.78228i
$472$	−94.9493			−0.201164
$473$	−230.911			−0.488183
$474$		−	68.2800i		−	0.144051i
$475$		−	24.1332i		−	0.0508066i
$476$	46.3439			0.0973610
$477$		−	365.582i		−	0.766420i
$478$	−485.313			−1.01530
$479$			203.940i			0.425763i	0.977078	+	0.212881i	$0.0682848\pi$
$479$			203.940i			0.425763i	−0.977078	+	0.212881i	$0.931715\pi$
$480$			54.4818i			0.113504i
$481$		−	80.9345i		−	0.168263i
$482$		−	493.205i		−	1.02325i
$483$	−145.230	+	16.3954i	−0.300684	+	0.0339448i
$484$	168.805			0.348771
$485$	−344.636			−0.710589
$486$	−491.538			−1.01139
$487$	−681.456			−1.39929			−0.699647	−	0.714489i	$-0.746659\pi$
$487$	−681.456			−1.39929			−0.699647	+	0.714489i	$0.746659\pi$
$488$			284.447i			0.582882i
$489$	662.496			1.35480
$490$			148.069i			0.302181i
$491$	−329.666			−0.671417			−0.335709	−	0.941966i	$-0.608976\pi$
$491$	−329.666			−0.671417			−0.335709	+	0.941966i	$0.608976\pi$
$492$	174.951			0.355591
$493$		−	368.645i		−	0.747758i
$494$		−	35.5850i		−	0.0720344i
$495$	129.208			0.261027
$496$	−81.9546			−0.165231
$497$			35.6005i			0.0716308i
$498$			268.813i			0.539785i
$499$	−993.708			−1.99140			−0.995700	−	0.0926377i	$-0.970470\pi$
$499$	−993.708			−1.99140			−0.995700	+	0.0926377i	$0.970470\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	208.876			0.416918
$502$		−	318.338i		−	0.634140i
$503$			880.515i			1.75053i	0.483646	+	0.875264i	$0.339312\pi$
$503$			880.515i			1.75053i	−0.483646	+	0.875264i	$0.660688\pi$
$504$		−	39.8577i		−	0.0790827i
$505$		−	131.617i		−	0.260627i
$506$	−195.532	+	22.0741i	−0.386428	+	0.0436246i
$507$	−610.851			−1.20483
$508$	185.680			0.365511
$509$	−689.907			−1.35542			−0.677708	−	0.735331i	$-0.737027\pi$
$509$	−689.907			−1.35542			−0.677708	+	0.735331i	$0.737027\pi$
$510$	−213.927			−0.419465
$511$		−	22.3980i		−	0.0438318i
$512$	22.6274			0.0441942
$513$			11.4687i			0.0223561i
$514$	−556.680			−1.08303
$515$	121.343			0.235617
$516$		−	328.806i		−	0.637221i
$517$			83.6494i			0.161798i
$518$	32.3913			0.0625315
$519$	−479.255			−0.923419
$520$		−	32.9714i		−	0.0634066i
$521$		−	408.559i		−	0.784183i	−0.919926	−	0.392092i	$-0.871752\pi$
$521$		−	408.559i		−	0.784183i	0.919926	−	0.392092i	$-0.128248\pi$
$522$	−317.050			−0.607376
$523$		−	516.837i		−	0.988216i	−0.869400	−	0.494108i	$-0.835495\pi$
$523$		−	516.837i		−	0.988216i	0.869400	−	0.494108i	$-0.164505\pi$
$524$	303.418			0.579042
$525$			31.7723i			0.0605187i
$526$			187.695i			0.356834i
$527$		−	321.801i		−	0.610629i
$528$		−	104.226i		−	0.197398i
$529$	−515.686	+	117.937i	−0.974831	+	0.222943i
$530$	121.034			0.228365
$531$	−320.646			−0.603853
$532$	14.2417			0.0267701
$533$	−105.877			−0.198644
$534$		−	676.367i		−	1.26660i
$535$	−266.369			−0.497886
$536$		−	91.7737i		−	0.171220i
$537$	1131.02			2.10618
$538$	−619.589			−1.15165
$539$		−	283.262i		−	0.525533i
$540$			10.6264i			0.0196784i
$541$	−825.026			−1.52500			−0.762501	−	0.646987i	$-0.776029\pi$
$541$	−825.026			−1.52500			−0.762501	+	0.646987i	$0.776029\pi$
$542$	−289.486			−0.534107
$543$		−	909.533i		−	1.67501i
$544$			88.8484i			0.163324i
$545$	333.673			0.612245
$546$			46.8492i			0.0858043i
$547$	−749.566			−1.37032			−0.685161	−	0.728392i	$-0.740268\pi$
$547$	−749.566			−1.37032			−0.685161	+	0.728392i	$0.740268\pi$
$548$			534.927i			0.976145i
$549$			960.583i			1.74970i
$550$			42.7770i			0.0777765i
$551$		−	113.286i		−	0.205601i
$552$	−31.4324	−	278.429i	−0.0569428	−	0.504400i
$553$	−16.5377			−0.0299054
$554$	245.796			0.443675
$555$	−149.521			−0.269407
$556$	479.559			0.862516
$557$			406.149i			0.729173i	0.931170	+	0.364586i	$0.118790\pi$
$557$			406.149i			0.729173i	−0.931170	+	0.364586i	$0.881210\pi$
$558$	−276.763			−0.495991
$559$			198.988i			0.355971i
$560$	13.1957			0.0235638
$561$	409.253			0.729506
$562$		−	203.089i		−	0.361368i
$563$		−	285.771i		−	0.507587i	−0.967258	−	0.253793i	$-0.918322\pi$
$563$		−	285.771i		−	0.507587i	0.967258	−	0.253793i	$-0.0816784\pi$
$564$	−119.113			−0.211193
$565$	−79.0091			−0.139839
$566$			102.066i			0.180328i
$567$			111.727i			0.197050i
$568$	−68.2517			−0.120161
$569$		−	315.624i		−	0.554699i	−0.960769	−	0.277349i	$-0.910544\pi$
$569$		−	315.624i		−	0.554699i	0.960769	−	0.277349i	$-0.0894560\pi$
$570$	−65.7409			−0.115335
$571$			383.907i			0.672342i	0.941801	+	0.336171i	$0.109132\pi$
$571$			383.907i			0.672342i	−0.941801	+	0.336171i	$0.890868\pi$
$572$			63.0759i			0.110273i
$573$		−	313.652i		−	0.547385i
$574$		−	42.3737i		−	0.0738218i
$575$	12.9007	+	114.274i	0.0224359	+	0.198738i
$576$	76.4133			0.132662
$577$	−717.832			−1.24408			−0.622038	−	0.782987i	$-0.713695\pi$
$577$	−717.832			−1.24408			−0.622038	+	0.782987i	$0.713695\pi$
$578$	59.8374			0.103525
$579$	855.186			1.47701
$580$		−	104.966i		−	0.180976i
$581$	65.1075			0.112061
$582$			938.819i			1.61309i
$583$	−231.543			−0.397158
$584$	42.9405			0.0735283
$585$		−	111.345i		−	0.190334i
$586$			175.644i			0.299735i
$587$	−296.393			−0.504928			−0.252464	−	0.967606i	$-0.581241\pi$
$587$	−296.393			−0.504928			−0.252464	+	0.967606i	$0.581241\pi$
$588$	403.352			0.685973
$589$		−	98.8912i		−	0.167897i
$590$		−	106.157i		−	0.179926i
$591$	121.486			0.205560
$592$			62.0992i			0.104897i
$593$	654.259			1.10330			0.551652	−	0.834075i	$-0.313998\pi$
$593$	654.259			1.10330			0.551652	+	0.834075i	$0.313998\pi$
$594$		−	20.3287i		−	0.0342235i
$595$			51.8140i			0.0870824i
$596$		−	403.293i		−	0.676667i
$597$			923.881i			1.54754i
$598$	19.0224	+	168.500i	0.0318100	+	0.281773i
$599$	368.673			0.615480			0.307740	−	0.951470i	$-0.400427\pi$
$599$	368.673			0.615480			0.307740	+	0.951470i	$0.400427\pi$
$600$	−60.9125			−0.101521
$601$	148.316			0.246782			0.123391	−	0.992358i	$-0.460623\pi$
$601$	148.316			0.246782			0.123391	+	0.992358i	$0.460623\pi$
$602$	−79.6381			−0.132289
$603$		−	309.922i		−	0.513967i
$604$	241.573			0.399956
$605$			188.730i			0.311950i
$606$	−358.535			−0.591643
$607$	939.932			1.54849			0.774244	−	0.632887i	$-0.218130\pi$
$607$	939.932			1.54849			0.774244	+	0.632887i	$0.218130\pi$
$608$			27.3036i			0.0449072i
$609$			149.146i			0.244904i
$610$	−318.021			−0.521346
$611$	72.0850			0.117979
$612$			300.043i			0.490267i
$613$		−	602.963i		−	0.983626i	−0.870701	−	0.491813i	$-0.836334\pi$
$613$		−	602.963i		−	0.983626i	0.870701	−	0.491813i	$-0.163666\pi$
$614$	310.727			0.506069
$615$			195.601i			0.318050i
$616$	−25.2440			−0.0409805
$617$			885.697i			1.43549i	0.696306	+	0.717745i	$0.254826\pi$
$617$			885.697i			1.43549i	−0.696306	+	0.717745i	$0.745174\pi$
$618$		−	330.549i		−	0.534869i
$619$			85.9838i			0.138908i	0.997585	+	0.0694538i	$0.0221257\pi$
$619$			85.9838i			0.138908i	−0.997585	+	0.0694538i	$0.977874\pi$
$620$		−	91.6280i		−	0.147787i
$621$	−6.13072	−	54.3060i	−0.00987233	−	0.0874492i
$622$	−448.403			−0.720905
$623$	−163.819			−0.262951
$624$	−89.8171			−0.143938
$625$	25.0000			0.0400000
$626$		−	685.404i		−	1.09489i
$627$	125.765			0.200583
$628$			389.794i			0.620692i
$629$	−243.837			−0.387659
$630$	44.5622			0.0707337
$631$		−	645.385i		−	1.02280i	−0.859344	−	0.511398i	$-0.829128\pi$
$631$		−	645.385i		−	1.02280i	0.859344	−	0.511398i	$-0.170872\pi$
$632$		−	31.7053i		−	0.0501666i
$633$	1034.85			1.63483
$634$	−738.092			−1.16418
$635$			207.596i			0.326923i
$636$		−	329.706i		−	0.518406i
$637$	−244.102			−0.383205
$638$			200.805i			0.314741i
$639$	−230.488			−0.360701
$640$			25.2982i			0.0395285i
$641$		−	140.119i		−	0.218594i	−0.994009	−	0.109297i	$-0.965140\pi$
$641$		−	140.119i		−	0.218594i	0.994009	−	0.109297i	$-0.0348599\pi$
$642$			725.613i			1.13024i
$643$		−	1028.11i		−	1.59892i	−0.600718	−	0.799461i	$-0.705118\pi$
$643$		−	1028.11i		−	1.59892i	0.600718	−	0.799461i	$-0.294882\pi$
$644$	−67.4366	+	7.61306i	−0.104715	+	0.0118215i
$645$	367.616			0.569948
$646$	−107.210			−0.165959
$647$	1248.29			1.92934			0.964672	−	0.263455i	$-0.0848621\pi$
$647$	1248.29			1.92934			0.964672	+	0.263455i	$0.0848621\pi$
$648$	−214.199			−0.330553
$649$			203.082i			0.312916i
$650$	36.8632			0.0567126
$651$			130.194i			0.199992i
$652$	307.625			0.471817
$653$	−457.504			−0.700618			−0.350309	−	0.936634i	$-0.613923\pi$
$653$	−457.504			−0.700618			−0.350309	+	0.936634i	$0.613923\pi$
$654$		−	908.956i		−	1.38984i
$655$			339.232i			0.517911i
$656$	81.2370			0.123837
$657$	145.011			0.220717
$658$			28.8496i			0.0438444i
$659$			173.014i			0.262540i	0.991347	+	0.131270i	$0.0419055\pi$
$659$			173.014i			0.262540i	−0.991347	+	0.131270i	$0.958095\pi$
$660$	116.529			0.176558
$661$		−	582.673i		−	0.881502i	−0.897629	−	0.440751i	$-0.854712\pi$
$661$		−	582.673i		−	0.881502i	0.897629	−	0.440751i	$-0.145288\pi$
$662$	−450.101			−0.679911
$663$		−	352.674i		−	0.531937i
$664$			124.821i			0.187984i
$665$			15.9227i			0.0239439i
$666$			209.710i			0.314880i
$667$	60.5585	+	536.428i	0.0907924	+	0.804240i
$668$	96.9899			0.145194
$669$	1108.67			1.65721
$670$	102.606			0.153144
$671$	608.389			0.906690
$672$		−	35.9463i		−	0.0534915i
$673$	−390.678			−0.580502			−0.290251	−	0.956951i	$-0.593739\pi$
$673$	−390.678			−0.580502			−0.290251	+	0.956951i	$0.593739\pi$
$674$			164.810i			0.244526i
$675$	−11.8806			−0.0176009
$676$	−283.644			−0.419592
$677$		−	952.947i		−	1.40760i	−0.710397	−	0.703801i	$-0.751485\pi$
$677$		−	952.947i		−	1.40760i	0.710397	−	0.703801i	$-0.248515\pi$
$678$			215.228i			0.317445i
$679$	227.386			0.334883
$680$	−99.3355			−0.146082
$681$		−	458.286i		−	0.672961i
$682$			175.289i			0.257022i
$683$	299.732			0.438847			0.219423	−	0.975630i	$-0.429582\pi$
$683$	299.732			0.438847			0.219423	+	0.975630i	$0.429582\pi$
$684$			92.2048i			0.134802i
$685$	−598.067			−0.873090
$686$		−	199.928i		−	0.291441i
$687$			1414.93i			2.05958i
$688$		−	152.679i		−	0.221917i
$689$			199.532i			0.289597i
$690$	311.293	−	35.1425i	0.451149	−	0.0509312i
$691$	165.585			0.239631			0.119815	−	0.992796i	$-0.461770\pi$
$691$	165.585			0.239631			0.119815	+	0.992796i	$0.461770\pi$
$692$	−222.538			−0.321587
$693$	−85.2497			−0.123015
$694$	−142.432			−0.205234
$695$			536.163i			0.771458i
$696$	−285.937			−0.410829
$697$			318.984i			0.457652i
$698$	202.906			0.290696
$699$	−642.685			−0.919436
$700$			14.7532i			0.0210761i
$701$		−	831.750i		−	1.18652i	−0.805011	−	0.593260i	$-0.797841\pi$
$701$		−	831.750i		−	1.18652i	0.805011	−	0.593260i	$-0.202159\pi$
$702$	−17.5183			−0.0249549
$703$	−74.9324			−0.106590
$704$		−	48.3967i		−	0.0687453i
$705$		−	133.172i		−	0.188897i
$706$	−388.052			−0.549649
$707$			86.8387i			0.122827i
$708$	−289.180			−0.408446
$709$			835.784i			1.17882i	0.807834	+	0.589410i	$0.200640\pi$
$709$			835.784i			1.17882i	−0.807834	+	0.589410i	$0.799360\pi$
$710$		−	76.3077i		−	0.107476i
$711$		−	107.070i		−	0.150590i
$712$		−	314.066i		−	0.441104i
$713$	52.8634	+	468.265i	0.0741422	+	0.656753i
$714$	141.146			0.197683
$715$	−70.5210			−0.0986308
$716$	525.180			0.733492
$717$	−1478.08			−2.06148
$718$		−	674.397i		−	0.939272i
$719$	881.656			1.22623			0.613113	−	0.789996i	$-0.289917\pi$
$719$	881.656			1.22623			0.613113	+	0.789996i	$0.289917\pi$
$720$			85.4327i			0.118657i
$721$	−80.0602			−0.111041
$722$	477.585			0.661475
$723$		−	1502.12i		−	2.07762i
$724$		−	422.335i		−	0.583335i
$725$	117.356			0.161870
$726$	514.117			0.708150
$727$		−	531.577i		−	0.731193i	−0.930774	−	0.365596i	$-0.880865\pi$
$727$		−	531.577i		−	0.731193i	0.930774	−	0.365596i	$-0.119135\pi$
$728$			21.7541i			0.0298820i
$729$	−815.463			−1.11861
$730$			48.0090i			0.0657657i
$731$	599.505			0.820116
$732$			866.318i			1.18349i
$733$		−	18.4847i		−	0.0252178i	−0.999921	−	0.0126089i	$-0.995986\pi$
$733$		−	18.4847i		−	0.0252178i	0.999921	−	0.0126089i	$-0.00401364\pi$
$734$		−	19.8441i		−	0.0270356i
$735$			450.962i			0.613553i
$736$	−14.5954	−	129.286i	−0.0198307	−	0.175661i
$737$	−196.290			−0.266337
$738$	274.339			0.371734
$739$	−960.371			−1.29955			−0.649777	−	0.760125i	$-0.725138\pi$
$739$	−960.371			−1.29955			−0.649777	+	0.760125i	$0.725138\pi$
$740$	−69.4290			−0.0938229
$741$		−	108.379i		−	0.146260i
$742$	−79.8562			−0.107623
$743$			248.966i			0.335082i	0.985865	+	0.167541i	$0.0535826\pi$
$743$			248.966i			0.335082i	−0.985865	+	0.167541i	$0.946417\pi$
$744$	−249.603			−0.335488
$745$	450.896			0.605229
$746$		−	149.429i		−	0.200307i
$747$			421.524i			0.564289i
$748$	190.034			0.254056
$749$	175.746			0.234641
$750$		−	68.1023i		−	0.0908030i
$751$			24.0757i			0.0320582i	0.999872	+	0.0160291i	$0.00510244\pi$
$751$			24.0757i			0.0320582i	−0.999872	+	0.0160291i	$0.994898\pi$
$752$	−55.3091			−0.0735494
$753$		−	969.539i		−	1.28757i
$754$	173.044			0.229501
$755$			270.087i			0.357731i
$756$		−	7.01112i		−	0.00927396i
$757$			914.334i			1.20784i	0.797045	+	0.603920i	$0.206395\pi$
$757$			914.334i			1.20784i	−0.797045	+	0.603920i	$0.793605\pi$
$758$			480.809i			0.634312i
$759$	−595.518	+	67.2294i	−0.784609	+	0.0885762i
$760$	−30.5263			−0.0401662
$761$	313.770			0.412312			0.206156	−	0.978519i	$-0.433905\pi$
$761$	313.770			0.412312			0.206156	+	0.978519i	$0.433905\pi$
$762$	565.511			0.742140
$763$	−220.153			−0.288536
$764$		−	145.642i		−	0.190631i
$765$	−335.458			−0.438508
$766$			989.662i			1.29199i
$767$	175.007			0.228170
$768$	68.9146			0.0897326
$769$		−	309.335i		−	0.402257i	−0.979565	−	0.201128i	$-0.935539\pi$
$769$		−	309.335i		−	0.402257i	0.979565	−	0.201128i	$-0.0644608\pi$
$770$		−	28.2237i		−	0.0366541i
$771$	−1695.44			−2.19901
$772$	397.099			0.514377
$773$			233.356i			0.301884i	0.988543	+	0.150942i	$0.0482306\pi$
$773$			233.356i			0.301884i	−0.988543	+	0.150942i	$0.951769\pi$
$774$		−	515.599i		−	0.666149i
$775$	102.443			0.132185
$776$			435.934i			0.561770i
$777$	98.6518			0.126965
$778$		−	176.518i		−	0.226887i
$779$			98.0253i			0.125835i
$780$		−	100.419i		−	0.128742i
$781$			145.980i			0.186915i
$782$	507.653	−	57.3101i	0.649173	−	0.0732866i
$783$	−55.7703			−0.0712265
$784$	187.294			0.238895
$785$	−435.803			−0.555164
$786$	924.098			1.17570
$787$			690.751i			0.877702i	0.898560	+	0.438851i	$0.144614\pi$

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:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	230.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +4.30716 q^{3} +2.00000 q^{4} +2.23607i q^{5} +6.09125 q^{6} -1.47532i q^{7} +2.82843 q^{8} +9.55167 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +4.30716 q^{3} +2.00000 q^{4} +2.23607i q^{5} +6.09125 q^{6} -1.47532i q^{7} +2.82843 q^{8} +9.55167 q^{9} +3.16228i q^{10} -6.04959i q^{11} +8.61433 q^{12} -5.21324 q^{13} -2.08642i q^{14} +9.63111i q^{15} +4.00000 q^{16} +15.7063i q^{17} +13.5081 q^{18} +4.82663i q^{19} +4.47214i q^{20} -6.35447i q^{21} -8.55541i q^{22} +(-2.58013 - 22.8548i) q^{23} +12.1825 q^{24} -5.00000 q^{25} -7.37263 q^{26} +2.37613 q^{27} -2.95065i q^{28} -23.4711 q^{29} +13.6205i q^{30} -20.4887 q^{31} +5.65685 q^{32} -26.0566i q^{33} +22.2121i q^{34} +3.29893 q^{35} +19.1033 q^{36} +15.5248i q^{37} +6.82589i q^{38} -22.4543 q^{39} +6.32456i q^{40} +20.3093 q^{41} -8.98657i q^{42} -38.1696i q^{43} -12.0992i q^{44} +21.3582i q^{45} +(-3.64885 - 32.3216i) q^{46} -13.8273 q^{47} +17.2287 q^{48} +46.8234 q^{49} -7.07107 q^{50} +67.6497i q^{51} -10.4265 q^{52} -38.2742i q^{53} +3.36035 q^{54} +13.5273 q^{55} -4.17285i q^{56} +20.7891i q^{57} -33.1932 q^{58} -33.5696 q^{59} +19.2622i q^{60} +100.567i q^{61} -28.9753 q^{62} -14.0918i q^{63} +8.00000 q^{64} -11.6572i q^{65} -36.8496i q^{66} -32.4469i q^{67} +31.4126i q^{68} +(-11.1130 - 98.4395i) q^{69} +4.66539 q^{70} -24.1306 q^{71} +27.0162 q^{72} +15.1818 q^{73} +21.9554i q^{74} -21.5358 q^{75} +9.65326i q^{76} -8.92511 q^{77} -31.7552 q^{78} -11.2095i q^{79} +8.94427i q^{80} -75.7306 q^{81} +28.7216 q^{82} +44.1310i q^{83} -12.7089i q^{84} -35.1204 q^{85} -53.9800i q^{86} -101.094 q^{87} -17.1108i q^{88} -111.039i q^{89} +30.2050i q^{90} +7.69122i q^{91} +(-5.16026 - 45.7096i) q^{92} -88.2480 q^{93} -19.5547 q^{94} -10.7927 q^{95} +24.3650 q^{96} +154.126i q^{97} +66.2183 q^{98} -57.7837i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9}+O(q^{10}) \) 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9	\( 16 q + 32 q^{4} - 8 q^{6} + 64 q^{9} + 24 q^{13} + 64 q^{16} - 32 q^{18} + 4 q^{23} - 16 q^{24} - 80 q^{25} + 96 q^{26} - 96 q^{27} - 108 q^{29} - 116 q^{31} + 60 q^{35} + 128 q^{36} + 248 q^{39} - 156 q^{41} - 124 q^{46} - 128 q^{47} - 28 q^{49} + 48 q^{52} + 224 q^{54} + 160 q^{58} + 204 q^{59} + 64 q^{62} + 128 q^{64} - 268 q^{69} - 120 q^{70} + 236 q^{71} - 64 q^{72} - 112 q^{73} - 936 q^{77} - 432 q^{78} - 136 q^{81} - 64 q^{82} + 60 q^{85} - 152 q^{87} + 8 q^{92} + 856 q^{93} - 216 q^{94} - 160 q^{95} - 32 q^{96} + 256 q^{98}+O(q^{100}) \) 16 * q + 32 * q^4 - 8 * q^6 + 64 * q^9 + 24 * q^13 + 64 * q^16 - 32 * q^18 + 4 * q^23 - 16 * q^24 - 80 * q^25 + 96 * q^26 - 96 * q^27 - 108 * q^29 - 116 * q^31 + 60 * q^35 + 128 * q^36 + 248 * q^39 - 156 * q^41 - 124 * q^46 - 128 * q^47 - 28 * q^49 + 48 * q^52 + 224 * q^54 + 160 * q^58 + 204 * q^59 + 64 * q^62 + 128 * q^64 - 268 * q^69 - 120 * q^70 + 236 * q^71 - 64 * q^72 - 112 * q^73 - 936 * q^77 - 432 * q^78 - 136 * q^81 - 64 * q^82 + 60 * q^85 - 152 * q^87 + 8 * q^92 + 856 * q^93 - 216 * q^94 - 160 * q^95 - 32 * q^96 + 256 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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