LMFDB - Embedded newform 690.2.d.c.139.1

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
Defining polynomial:	$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			139.1
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	690.139
Dual form			690.2.d.c.139.4

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.67513 - 1.48119i) q^{5} -1.00000 q^{6} +2.96239i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.67513 - 1.48119i) q^{5} -1.00000 q^{6} +2.96239i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(-1.48119 + 1.67513i) q^{10} -3.35026 q^{11} +1.00000i q^{12} +4.96239i q^{13} +2.96239 q^{14} +(-1.48119 + 1.67513i) q^{15} +1.00000 q^{16} +1.35026i q^{17} +1.00000i q^{18} +4.96239 q^{19} +(1.67513 + 1.48119i) q^{20} +2.96239 q^{21} +3.35026i q^{22} +1.00000i q^{23} +1.00000 q^{24} +(0.612127 + 4.96239i) q^{25} +4.96239 q^{26} +1.00000i q^{27} -2.96239i q^{28} -7.73813 q^{29} +(1.67513 + 1.48119i) q^{30} -4.00000 q^{31} -1.00000i q^{32} +3.35026i q^{33} +1.35026 q^{34} +(4.38787 - 4.96239i) q^{35} +1.00000 q^{36} -7.61213i q^{37} -4.96239i q^{38} +4.96239 q^{39} +(1.48119 - 1.67513i) q^{40} +4.70052 q^{41} -2.96239i q^{42} +10.3127i q^{43} +3.35026 q^{44} +(1.67513 + 1.48119i) q^{45} +1.00000 q^{46} +3.22425i q^{47} -1.00000i q^{48} -1.77575 q^{49} +(4.96239 - 0.612127i) q^{50} +1.35026 q^{51} -4.96239i q^{52} +6.96239i q^{53} +1.00000 q^{54} +(5.61213 + 4.96239i) q^{55} -2.96239 q^{56} -4.96239i q^{57} +7.73813i q^{58} -1.22425 q^{59} +(1.48119 - 1.67513i) q^{60} -11.0884 q^{61} +4.00000i q^{62} -2.96239i q^{63} -1.00000 q^{64} +(7.35026 - 8.31265i) q^{65} +3.35026 q^{66} -7.61213i q^{67} -1.35026i q^{68} +1.00000 q^{69} +(-4.96239 - 4.38787i) q^{70} -2.18664 q^{71} -1.00000i q^{72} +9.92478i q^{73} -7.61213 q^{74} +(4.96239 - 0.612127i) q^{75} -4.96239 q^{76} -9.92478i q^{77} -4.96239i q^{78} +4.12601 q^{79} +(-1.67513 - 1.48119i) q^{80} +1.00000 q^{81} -4.70052i q^{82} +6.38787i q^{83} -2.96239 q^{84} +(2.00000 - 2.26187i) q^{85} +10.3127 q^{86} +7.73813i q^{87} -3.35026i q^{88} -9.92478 q^{89} +(1.48119 - 1.67513i) q^{90} -14.7005 q^{91} -1.00000i q^{92} +4.00000i q^{93} +3.22425 q^{94} +(-8.31265 - 7.35026i) q^{95} -1.00000 q^{96} +12.8872i q^{97} +1.77575i q^{98} +3.35026 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6q - 6q^{4} - 6q^{6} - 6q^{9} + O(q^{10}) $	$ 6q - 6q^{4} - 6q^{6} - 6q^{9} + 2q^{10} - 4q^{14} + 2q^{15} + 6q^{16} + 8q^{19} - 4q^{21} + 6q^{24} + 2q^{25} + 8q^{26} - 28q^{29} - 24q^{31} - 12q^{34} + 28q^{35} + 6q^{36} + 8q^{39} - 2q^{40} - 12q^{41} + 6q^{46} - 14q^{49} + 8q^{50} - 12q^{51} + 6q^{54} + 32q^{55} + 4q^{56} - 4q^{59} - 2q^{60} - 28q^{61} - 6q^{64} + 24q^{65} + 6q^{69} - 8q^{70} + 12q^{71} - 44q^{74} + 8q^{75} - 8q^{76} + 8q^{79} + 6q^{81} + 4q^{84} + 12q^{85} + 20q^{86} - 16q^{89} - 2q^{90} - 48q^{91} + 16q^{94} - 8q^{95} - 6q^{96} + 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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	−1.00000			−0.500000
$5$	−1.67513	−	1.48119i	−0.749141	−	0.662410i
$5$	−1.67513	−	1.48119i	−0.749141	−	0.662410i
$6$	−1.00000			−0.408248
$7$			2.96239i			1.11968i	0.828602	+	0.559839i	$0.189137\pi$
$7$			2.96239i			1.11968i	−0.828602	+	0.559839i	$0.810863\pi$
$8$			1.00000i			0.353553i
$9$	−1.00000			−0.333333
$10$	−1.48119	+	1.67513i	−0.468395	+	0.529723i
$11$	−3.35026			−1.01014			−0.505071	−	0.863078i	$-0.668534\pi$
$11$	−3.35026			−1.01014			−0.505071	+	0.863078i	$0.668534\pi$
$12$			1.00000i			0.288675i
$13$			4.96239i			1.37632i	0.725559	+	0.688159i	$0.241581\pi$
$13$			4.96239i			1.37632i	−0.725559	+	0.688159i	$0.758419\pi$
$14$	2.96239			0.791732
$15$	−1.48119	+	1.67513i	−0.382443	+	0.432517i
$16$	1.00000			0.250000
$17$			1.35026i			0.327487i	0.986503	+	0.163743i	$0.0523569\pi$
$17$			1.35026i			0.327487i	−0.986503	+	0.163743i	$0.947643\pi$
$18$			1.00000i			0.235702i
$19$	4.96239			1.13845			0.569225	−	0.822182i	$-0.307243\pi$
$19$	4.96239			1.13845			0.569225	+	0.822182i	$0.307243\pi$
$20$	1.67513	+	1.48119i	0.374571	+	0.331205i
$21$	2.96239			0.646446
$22$			3.35026i			0.714278i
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$	1.00000			0.204124
$25$	0.612127	+	4.96239i	0.122425	+	0.992478i
$26$	4.96239			0.973204
$27$			1.00000i			0.192450i
$28$		−	2.96239i		−	0.559839i
$29$	−7.73813			−1.43694			−0.718468	−	0.695560i	$-0.755156\pi$
$29$	−7.73813			−1.43694			−0.718468	+	0.695560i	$0.755156\pi$
$30$	1.67513	+	1.48119i	0.305836	+	0.270428i
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		−	1.00000i		−	0.176777i
$33$			3.35026i			0.583206i
$34$	1.35026			0.231568
$35$	4.38787	−	4.96239i	0.741686	−	0.838797i
$36$	1.00000			0.166667
$37$		−	7.61213i		−	1.25143i	−0.780053	−	0.625713i	$-0.784808\pi$
$37$		−	7.61213i		−	1.25143i	0.780053	−	0.625713i	$-0.215192\pi$
$38$		−	4.96239i		−	0.805006i
$39$	4.96239			0.794618
$40$	1.48119	−	1.67513i	0.234197	−	0.264861i
$41$	4.70052			0.734098			0.367049	−	0.930202i	$-0.380368\pi$
$41$	4.70052			0.734098			0.367049	+	0.930202i	$0.380368\pi$
$42$		−	2.96239i		−	0.457106i
$43$			10.3127i			1.57266i	0.617804	+	0.786332i	$0.288023\pi$
$43$			10.3127i			1.57266i	−0.617804	+	0.786332i	$0.711977\pi$
$44$	3.35026			0.505071
$45$	1.67513	+	1.48119i	0.249714	+	0.220803i
$46$	1.00000			0.147442
$47$			3.22425i			0.470306i	0.971958	+	0.235153i	$0.0755591\pi$
$47$			3.22425i			0.470306i	−0.971958	+	0.235153i	$0.924441\pi$
$48$		−	1.00000i		−	0.144338i
$49$	−1.77575			−0.253678
$50$	4.96239	−	0.612127i	0.701788	−	0.0865678i
$51$	1.35026			0.189074
$52$		−	4.96239i		−	0.688159i
$53$			6.96239i			0.956358i	0.878263	+	0.478179i	$0.158703\pi$
$53$			6.96239i			0.956358i	−0.878263	+	0.478179i	$0.841297\pi$
$54$	1.00000			0.136083
$55$	5.61213	+	4.96239i	0.756739	+	0.669128i
$56$	−2.96239			−0.395866
$57$		−	4.96239i		−	0.657284i
$58$			7.73813i			1.01607i
$59$	−1.22425			−0.159384			−0.0796921	−	0.996820i	$-0.525394\pi$
$59$	−1.22425			−0.159384			−0.0796921	+	0.996820i	$0.525394\pi$
$60$	1.48119	−	1.67513i	0.191221	−	0.216258i
$61$	−11.0884			−1.41972			−0.709862	−	0.704341i	$-0.751243\pi$
$61$	−11.0884			−1.41972			−0.709862	+	0.704341i	$0.751243\pi$
$62$			4.00000i			0.508001i
$63$		−	2.96239i		−	0.373226i
$64$	−1.00000			−0.125000
$65$	7.35026	−	8.31265i	0.911688	−	1.03106i
$66$	3.35026			0.412389
$67$		−	7.61213i		−	0.929969i	−0.885319	−	0.464985i	$-0.846060\pi$
$67$		−	7.61213i		−	0.929969i	0.885319	−	0.464985i	$-0.153940\pi$
$68$		−	1.35026i		−	0.163743i
$69$	1.00000			0.120386
$70$	−4.96239	−	4.38787i	−0.593119	−	0.524451i
$71$	−2.18664			−0.259507			−0.129753	−	0.991546i	$-0.541419\pi$
$71$	−2.18664			−0.259507			−0.129753	+	0.991546i	$0.541419\pi$
$72$		−	1.00000i		−	0.117851i
$73$			9.92478i			1.16161i	0.814044	+	0.580804i	$0.197262\pi$
$73$			9.92478i			1.16161i	−0.814044	+	0.580804i	$0.802738\pi$
$74$	−7.61213			−0.884892
$75$	4.96239	−	0.612127i	0.573007	−	0.0706823i
$76$	−4.96239			−0.569225
$77$		−	9.92478i		−	1.13103i
$78$		−	4.96239i		−	0.561880i
$79$	4.12601			0.464212			0.232106	−	0.972690i	$-0.425438\pi$
$79$	4.12601			0.464212			0.232106	+	0.972690i	$0.425438\pi$
$80$	−1.67513	−	1.48119i	−0.187285	−	0.165603i
$81$	1.00000			0.111111
$82$		−	4.70052i		−	0.519086i
$83$			6.38787i			0.701160i	0.936533	+	0.350580i	$0.114016\pi$
$83$			6.38787i			0.701160i	−0.936533	+	0.350580i	$0.885984\pi$
$84$	−2.96239			−0.323223
$85$	2.00000	−	2.26187i	0.216930	−	0.245334i
$86$	10.3127			1.11204
$87$			7.73813i			0.829615i
$88$		−	3.35026i		−	0.357139i
$89$	−9.92478			−1.05202			−0.526012	−	0.850477i	$-0.676313\pi$
$89$	−9.92478			−1.05202			−0.526012	+	0.850477i	$0.676313\pi$
$90$	1.48119	−	1.67513i	0.156132	−	0.176574i
$91$	−14.7005			−1.54103
$92$		−	1.00000i		−	0.104257i
$93$			4.00000i			0.414781i
$94$	3.22425			0.332556
$95$	−8.31265	−	7.35026i	−0.852860	−	0.754121i
$96$	−1.00000			−0.102062
$97$			12.8872i			1.30849i	0.756281	+	0.654247i	$0.227014\pi$
$97$			12.8872i			1.30849i	−0.756281	+	0.654247i	$0.772986\pi$
$98$			1.77575i			0.179377i
$99$	3.35026			0.336714
$100$	−0.612127	−	4.96239i	−0.0612127	−	0.496239i
$101$	4.26187			0.424071			0.212036	−	0.977262i	$-0.431991\pi$
$101$	4.26187			0.424071			0.212036	+	0.977262i	$0.431991\pi$
$102$		−	1.35026i		−	0.133696i
$103$			0.261865i			0.0258023i	0.999917	+	0.0129012i	$0.00410668\pi$
$103$			0.261865i			0.0258023i	−0.999917	+	0.0129012i	$0.995893\pi$
$104$	−4.96239			−0.486602
$105$	−4.96239	−	4.38787i	−0.484280	−	0.428213i
$106$	6.96239			0.676247
$107$		−	9.08840i		−	0.878608i	−0.898338	−	0.439304i	$-0.855225\pi$
$107$		−	9.08840i		−	0.878608i	0.898338	−	0.439304i	$-0.144775\pi$
$108$		−	1.00000i		−	0.0962250i
$109$	−18.9380			−1.81393			−0.906963	−	0.421210i	$-0.861606\pi$
$109$	−18.9380			−1.81393			−0.906963	+	0.421210i	$0.861606\pi$
$110$	4.96239	−	5.61213i	0.473145	−	0.535095i
$111$	−7.61213			−0.722511
$112$			2.96239i			0.279919i
$113$			6.64974i			0.625555i	0.949826	+	0.312777i	$0.101259\pi$
$113$			6.64974i			0.625555i	−0.949826	+	0.312777i	$0.898741\pi$
$114$	−4.96239			−0.464770
$115$	1.48119	−	1.67513i	0.138122	−	0.156207i
$116$	7.73813			0.718468
$117$		−	4.96239i		−	0.458773i
$118$			1.22425i			0.112702i
$119$	−4.00000			−0.366679
$120$	−1.67513	−	1.48119i	−0.152918	−	0.135214i
$121$	0.224254			0.0203867
$122$			11.0884i			1.00390i
$123$		−	4.70052i		−	0.423832i
$124$	4.00000			0.359211
$125$	6.32487	−	9.21933i	0.565713	−	0.824602i
$126$	−2.96239			−0.263911
$127$			0.186642i			0.0165618i	0.999966	+	0.00828091i	$0.00263593\pi$
$127$			0.186642i			0.0165618i	−0.999966	+	0.00828091i	$0.997364\pi$
$128$			1.00000i			0.0883883i
$129$	10.3127			0.907978
$130$	−8.31265	−	7.35026i	−0.729068	−	0.644661i
$131$	7.92478			0.692391			0.346196	−	0.938162i	$-0.387473\pi$
$131$	7.92478			0.692391			0.346196	+	0.938162i	$0.387473\pi$
$132$		−	3.35026i		−	0.291603i
$133$			14.7005i			1.27470i
$134$	−7.61213			−0.657588
$135$	1.48119	−	1.67513i	0.127481	−	0.144172i
$136$	−1.35026			−0.115784
$137$			9.19982i			0.785993i	0.919540	+	0.392997i	$0.128562\pi$
$137$			9.19982i			0.785993i	−0.919540	+	0.392997i	$0.871438\pi$
$138$		−	1.00000i		−	0.0851257i
$139$	−8.62530			−0.731588			−0.365794	−	0.930696i	$-0.619203\pi$
$139$	−8.62530			−0.731588			−0.365794	+	0.930696i	$0.619203\pi$
$140$	−4.38787	+	4.96239i	−0.370843	+	0.419398i
$141$	3.22425			0.271531
$142$			2.18664i			0.183499i
$143$		−	16.6253i		−	1.39028i
$144$	−1.00000			−0.0833333
$145$	12.9624	+	11.4617i	1.07647	+	0.951841i
$146$	9.92478			0.821380
$147$			1.77575i			0.146461i
$148$			7.61213i			0.625713i
$149$	−4.64974			−0.380921			−0.190461	−	0.981695i	$-0.560998\pi$
$149$	−4.64974			−0.380921			−0.190461	+	0.981695i	$0.560998\pi$
$150$	−0.612127	−	4.96239i	−0.0499799	−	0.405177i
$151$	10.7005			0.870796			0.435398	−	0.900238i	$-0.356608\pi$
$151$	10.7005			0.870796			0.435398	+	0.900238i	$0.356608\pi$
$152$			4.96239i			0.402503i
$153$		−	1.35026i		−	0.109162i
$154$	−9.92478			−0.799761
$155$	6.70052	+	5.92478i	0.538199	+	0.475890i
$156$	−4.96239			−0.397309
$157$		−	17.0132i		−	1.35780i	−0.734231	−	0.678900i	$-0.762457\pi$
$157$		−	17.0132i		−	1.35780i	0.734231	−	0.678900i	$-0.237543\pi$
$158$		−	4.12601i		−	0.328248i
$159$	6.96239			0.552153
$160$	−1.48119	+	1.67513i	−0.117099	+	0.132431i
$161$	−2.96239			−0.233469
$162$		−	1.00000i		−	0.0785674i
$163$		−	12.6253i		−	0.988890i	−0.869209	−	0.494445i	$-0.835371\pi$
$163$		−	12.6253i		−	0.988890i	0.869209	−	0.494445i	$-0.164629\pi$
$164$	−4.70052			−0.367049
$165$	4.96239	−	5.61213i	0.386321	−	0.436903i
$166$	6.38787			0.495795
$167$		−	24.6253i		−	1.90556i	−0.303657	−	0.952781i	$-0.598208\pi$
$167$		−	24.6253i		−	1.90556i	0.303657	−	0.952781i	$-0.401792\pi$
$168$			2.96239i			0.228553i
$169$	−11.6253			−0.894254
$170$	−2.26187	−	2.00000i	−0.173477	−	0.153393i
$171$	−4.96239			−0.379483
$172$		−	10.3127i		−	0.786332i
$173$			4.44851i			0.338214i	0.985598	+	0.169107i	$0.0540883\pi$
$173$			4.44851i			0.338214i	−0.985598	+	0.169107i	$0.945912\pi$
$174$	7.73813			0.586626
$175$	−14.7005	+	1.81336i	−1.11126	+	0.137077i
$176$	−3.35026			−0.252535
$177$			1.22425i			0.0920205i
$178$			9.92478i			0.743894i
$179$	13.8496			1.03516			0.517582	−	0.855634i	$-0.326832\pi$
$179$	13.8496			1.03516			0.517582	+	0.855634i	$0.326832\pi$
$180$	−1.67513	−	1.48119i	−0.124857	−	0.110402i
$181$	−22.6859			−1.68623			−0.843116	−	0.537732i	$-0.819281\pi$
$181$	−22.6859			−1.68623			−0.843116	+	0.537732i	$0.819281\pi$
$182$			14.7005i			1.08968i
$183$			11.0884i			0.819678i
$184$	−1.00000			−0.0737210
$185$	−11.2750	+	12.7513i	−0.828957	+	0.937495i
$186$	4.00000			0.293294
$187$		−	4.52373i		−	0.330808i
$188$		−	3.22425i		−	0.235153i
$189$	−2.96239			−0.215482
$190$	−7.35026	+	8.31265i	−0.533244	+	0.603063i
$191$	−19.3258			−1.39837			−0.699184	−	0.714942i	$-0.746453\pi$
$191$	−19.3258			−1.39837			−0.699184	+	0.714942i	$0.746453\pi$
$192$			1.00000i			0.0721688i
$193$			5.92478i			0.426475i	0.977000	+	0.213237i	$0.0684008\pi$
$193$			5.92478i			0.426475i	−0.977000	+	0.213237i	$0.931599\pi$
$194$	12.8872			0.925245
$195$	−8.31265	−	7.35026i	−0.595281	−	0.526363i
$196$	1.77575			0.126839
$197$			8.07522i			0.575336i	0.957730	+	0.287668i	$0.0928799\pi$
$197$			8.07522i			0.575336i	−0.957730	+	0.287668i	$0.907120\pi$
$198$		−	3.35026i		−	0.238093i
$199$	−15.9756			−1.13248			−0.566239	−	0.824241i	$-0.691602\pi$
$199$	−15.9756			−1.13248			−0.566239	+	0.824241i	$0.691602\pi$
$200$	−4.96239	+	0.612127i	−0.350894	+	0.0432839i
$201$	−7.61213			−0.536918
$202$		−	4.26187i		−	0.299864i
$203$		−	22.9234i		−	1.60890i
$204$	−1.35026			−0.0945372
$205$	−7.87399	−	6.96239i	−0.549943	−	0.486274i
$206$	0.261865			0.0182450
$207$		−	1.00000i		−	0.0695048i
$208$			4.96239i			0.344080i
$209$	−16.6253			−1.15000
$210$	−4.38787	+	4.96239i	−0.302792	+	0.342437i
$211$	21.7743			1.49901			0.749503	−	0.662000i	$-0.230292\pi$
$211$	21.7743			1.49901			0.749503	+	0.662000i	$0.230292\pi$
$212$		−	6.96239i		−	0.478179i
$213$			2.18664i			0.149826i
$214$	−9.08840			−0.621270
$215$	15.2750	−	17.2750i	1.04175	−	1.17815i
$216$	−1.00000			−0.0680414
$217$		−	11.8496i		−	0.804400i
$218$			18.9380i			1.28264i
$219$	9.92478			0.670654
$220$	−5.61213	−	4.96239i	−0.378370	−	0.334564i
$221$	−6.70052			−0.450726
$222$			7.61213i			0.510893i
$223$			3.81336i			0.255361i	0.991815	+	0.127681i	$0.0407533\pi$
$223$			3.81336i			0.255361i	−0.991815	+	0.127681i	$0.959247\pi$
$224$	2.96239			0.197933
$225$	−0.612127	−	4.96239i	−0.0408085	−	0.330826i
$226$	6.64974			0.442334
$227$		−	16.9380i		−	1.12421i	−0.827065	−	0.562106i	$-0.809991\pi$
$227$		−	16.9380i		−	1.12421i	0.827065	−	0.562106i	$-0.190009\pi$
$228$			4.96239i			0.328642i
$229$	26.9380			1.78011			0.890055	−	0.455853i	$-0.150666\pi$
$229$	26.9380			1.78011			0.890055	+	0.455853i	$0.150666\pi$
$230$	−1.67513	−	1.48119i	−0.110455	−	0.0976671i
$231$	−9.92478			−0.653002
$232$		−	7.73813i		−	0.508033i
$233$			27.4010i			1.79510i	0.440910	+	0.897551i	$0.354656\pi$
$233$			27.4010i			1.79510i	−0.440910	+	0.897551i	$0.645344\pi$
$234$	−4.96239			−0.324401
$235$	4.77575	−	5.40105i	0.311535	−	0.352325i
$236$	1.22425			0.0796921
$237$		−	4.12601i		−	0.268013i
$238$			4.00000i			0.259281i
$239$	−7.48612			−0.484237			−0.242118	−	0.970247i	$-0.577842\pi$
$239$	−7.48612			−0.484237			−0.242118	+	0.970247i	$0.577842\pi$
$240$	−1.48119	+	1.67513i	−0.0956107	+	0.108129i
$241$	−13.0738			−0.842158			−0.421079	−	0.907024i	$-0.638348\pi$
$241$	−13.0738			−0.842158			−0.421079	+	0.907024i	$0.638348\pi$
$242$		−	0.224254i		−	0.0144156i
$243$		−	1.00000i		−	0.0641500i
$244$	11.0884			0.709862
$245$	2.97461	+	2.63023i	0.190041	+	0.168039i
$246$	−4.70052			−0.299694
$247$			24.6253i			1.56687i
$248$		−	4.00000i		−	0.254000i
$249$	6.38787			0.404815
$250$	−9.21933	−	6.32487i	−0.583082	−	0.400020i
$251$	−5.02302			−0.317050			−0.158525	−	0.987355i	$-0.550674\pi$
$251$	−5.02302			−0.317050			−0.158525	+	0.987355i	$0.550674\pi$
$252$			2.96239i			0.186613i
$253$		−	3.35026i		−	0.210629i
$254$	0.186642			0.0117110
$255$	−2.26187	−	2.00000i	−0.141643	−	0.125245i
$256$	1.00000			0.0625000
$257$			19.4010i			1.21020i	0.796148	+	0.605102i	$0.206868\pi$
$257$			19.4010i			1.21020i	−0.796148	+	0.605102i	$0.793132\pi$
$258$		−	10.3127i		−	0.642038i
$259$	22.5501			1.40119
$260$	−7.35026	+	8.31265i	−0.455844	+	0.515529i
$261$	7.73813			0.478979
$262$		−	7.92478i		−	0.489594i
$263$		−	15.4763i		−	0.954308i	−0.878820	−	0.477154i	$-0.841668\pi$
$263$		−	15.4763i		−	0.954308i	0.878820	−	0.477154i	$-0.158332\pi$
$264$	−3.35026			−0.206194
$265$	10.3127	−	11.6629i	0.633501	−	0.716447i
$266$	14.7005			0.901347
$267$			9.92478i			0.607387i
$268$			7.61213i			0.464985i
$269$	20.2130			1.23241			0.616204	−	0.787587i	$-0.288670\pi$
$269$	20.2130			1.23241			0.616204	+	0.787587i	$0.288670\pi$
$270$	−1.67513	−	1.48119i	−0.101945	−	0.0901426i
$271$	30.3996			1.84665			0.923323	−	0.384024i	$-0.125462\pi$
$271$	30.3996			1.84665			0.923323	+	0.384024i	$0.125462\pi$
$272$			1.35026i			0.0818716i
$273$			14.7005i			0.889716i
$274$	9.19982			0.555781
$275$	−2.05079	−	16.6253i	−0.123667	−	1.00254i
$276$	−1.00000			−0.0601929
$277$			20.8119i			1.25047i	0.780437	+	0.625234i	$0.214997\pi$
$277$			20.8119i			1.25047i	−0.780437	+	0.625234i	$0.785003\pi$
$278$			8.62530i			0.517311i
$279$	4.00000			0.239474
$280$	4.96239	+	4.38787i	0.296559	+	0.262226i
$281$	19.8496			1.18413			0.592063	−	0.805892i	$-0.298314\pi$
$281$	19.8496			1.18413			0.592063	+	0.805892i	$0.298314\pi$
$282$		−	3.22425i		−	0.192002i
$283$		−	12.3879i		−	0.736383i	−0.929750	−	0.368191i	$-0.879977\pi$
$283$		−	12.3879i		−	0.736383i	0.929750	−	0.368191i	$-0.120023\pi$
$284$	2.18664			0.129753
$285$	−7.35026	+	8.31265i	−0.435392	+	0.492399i
$286$	−16.6253			−0.983075
$287$			13.9248i			0.821954i
$288$			1.00000i			0.0589256i
$289$	15.1768			0.892753
$290$	11.4617	−	12.9624i	0.673053	−	0.761178i
$291$	12.8872			0.755459
$292$		−	9.92478i		−	0.580804i
$293$		−	9.03761i		−	0.527983i	−0.964525	−	0.263991i	$-0.914961\pi$
$293$		−	9.03761i		−	0.527983i	0.964525	−	0.263991i	$-0.0850391\pi$
$294$	1.77575			0.103564
$295$	2.05079	+	1.81336i	0.119401	+	0.105578i
$296$	7.61213			0.442446
$297$		−	3.35026i		−	0.194402i
$298$			4.64974i			0.269352i
$299$	−4.96239			−0.286982
$300$	−4.96239	+	0.612127i	−0.286504	+	0.0353412i
$301$	−30.5501			−1.76088
$302$		−	10.7005i		−	0.615746i
$303$		−	4.26187i		−	0.244838i
$304$	4.96239			0.284613
$305$	18.5745	+	16.4241i	1.06357	+	0.940440i
$306$	−1.35026			−0.0771893
$307$		−	30.5501i		−	1.74359i	−0.489875	−	0.871793i	$-0.662958\pi$
$307$		−	30.5501i		−	1.74359i	0.489875	−	0.871793i	$-0.337042\pi$
$308$			9.92478i			0.565517i
$309$	0.261865			0.0148970
$310$	5.92478	−	6.70052i	0.336505	−	0.380564i
$311$	14.4387			0.818741			0.409371	−	0.912368i	$-0.365748\pi$
$311$	14.4387			0.818741			0.409371	+	0.912368i	$0.365748\pi$
$312$			4.96239i			0.280940i
$313$		−	27.9610i		−	1.58045i	−0.612818	−	0.790224i	$-0.709964\pi$
$313$		−	27.9610i		−	1.58045i	0.612818	−	0.790224i	$-0.290036\pi$
$314$	−17.0132			−0.960109
$315$	−4.38787	+	4.96239i	−0.247229	+	0.279599i
$316$	−4.12601			−0.232106
$317$			1.47627i			0.0829156i	0.999140	+	0.0414578i	$0.0132002\pi$
$317$			1.47627i			0.0829156i	−0.999140	+	0.0414578i	$0.986800\pi$
$318$		−	6.96239i		−	0.390431i
$319$	25.9248			1.45151
$320$	1.67513	+	1.48119i	0.0936427	+	0.0828013i
$321$	−9.08840			−0.507265
$322$			2.96239i			0.165087i
$323$			6.70052i			0.372827i
$324$	−1.00000			−0.0555556
$325$	−24.6253	+	3.03761i	−1.36597	+	0.168496i
$326$	−12.6253			−0.699251
$327$			18.9380i			1.04727i
$328$			4.70052i			0.259543i
$329$	−9.55149			−0.526591
$330$	−5.61213	−	4.96239i	−0.308937	−	0.273171i
$331$	23.1754			1.27383			0.636917	−	0.770932i	$-0.280209\pi$
$331$	23.1754			1.27383			0.636917	+	0.770932i	$0.280209\pi$
$332$		−	6.38787i		−	0.350580i
$333$			7.61213i			0.417142i
$334$	−24.6253			−1.34744
$335$	−11.2750	+	12.7513i	−0.616021	+	0.696678i
$336$	2.96239			0.161612
$337$		−	21.0376i		−	1.14599i	−0.819558	−	0.572996i	$-0.805781\pi$
$337$		−	21.0376i		−	1.14599i	0.819558	−	0.572996i	$-0.194219\pi$
$338$			11.6253i			0.632333i
$339$	6.64974			0.361164
$340$	−2.00000	+	2.26187i	−0.108465	+	0.122667i
$341$	13.4010			0.725707
$342$			4.96239i			0.268335i
$343$			15.4763i			0.835640i
$344$	−10.3127			−0.556021
$345$	−1.67513	−	1.48119i	−0.0901860	−	0.0797448i
$346$	4.44851			0.239153
$347$			3.37470i			0.181163i	0.995889	+	0.0905817i	$0.0288726\pi$
$347$			3.37470i			0.181163i	−0.995889	+	0.0905817i	$0.971127\pi$
$348$		−	7.73813i		−	0.414808i
$349$	4.44851			0.238123			0.119062	−	0.992887i	$-0.462011\pi$
$349$	4.44851			0.238123			0.119062	+	0.992887i	$0.462011\pi$
$350$	1.81336	+	14.7005i	0.0969280	+	0.785776i
$351$	−4.96239			−0.264873
$352$			3.35026i			0.178570i
$353$			35.4010i			1.88421i	0.335321	+	0.942104i	$0.391155\pi$
$353$			35.4010i			1.88421i	−0.335321	+	0.942104i	$0.608845\pi$
$354$	1.22425			0.0650684
$355$	3.66291	+	3.23884i	0.194407	+	0.171900i
$356$	9.92478			0.526012
$357$			4.00000i			0.211702i
$358$		−	13.8496i		−	0.731972i
$359$	34.5501			1.82348			0.911742	−	0.410764i	$-0.134738\pi$
$359$	34.5501			1.82348			0.911742	+	0.410764i	$0.134738\pi$
$360$	−1.48119	+	1.67513i	−0.0780658	+	0.0882871i
$361$	5.62530			0.296068
$362$			22.6859i			1.19235i
$363$		−	0.224254i		−	0.0117703i
$364$	14.7005			0.770517
$365$	14.7005	−	16.6253i	0.769461	−	0.870208i
$366$	11.0884			0.579600
$367$			22.8119i			1.19077i	0.803439	+	0.595387i	$0.203001\pi$
$367$			22.8119i			1.19077i	−0.803439	+	0.595387i	$0.796999\pi$
$368$			1.00000i			0.0521286i
$369$	−4.70052			−0.244699
$370$	12.7513	+	11.2750i	0.662909	+	0.586161i
$371$	−20.6253			−1.07081
$372$		−	4.00000i		−	0.207390i
$373$		−	10.8364i		−	0.561087i	−0.959841	−	0.280543i	$-0.909485\pi$
$373$		−	10.8364i		−	0.561087i	0.959841	−	0.280543i	$-0.0905146\pi$
$374$	−4.52373			−0.233917
$375$	−9.21933	−	6.32487i	−0.476084	−	0.326615i
$376$	−3.22425			−0.166278
$377$		−	38.3996i		−	1.97768i
$378$			2.96239i			0.152369i
$379$	32.4387			1.66626			0.833131	−	0.553076i	$-0.186546\pi$
$379$	32.4387			1.66626			0.833131	+	0.553076i	$0.186546\pi$
$380$	8.31265	+	7.35026i	0.426430	+	0.377060i
$381$	0.186642			0.00956198
$382$			19.3258i			0.988795i
$383$			32.9986i			1.68615i	0.537797	+	0.843074i	$0.319257\pi$
$383$			32.9986i			1.68615i	−0.537797	+	0.843074i	$0.680743\pi$
$384$	1.00000			0.0510310
$385$	−14.7005	+	16.6253i	−0.749208	+	0.847304i
$386$	5.92478			0.301563
$387$		−	10.3127i		−	0.524221i
$388$		−	12.8872i		−	0.654247i
$389$	−29.1246			−1.47668			−0.738338	−	0.674431i	$-0.764389\pi$
$389$	−29.1246			−1.47668			−0.738338	+	0.674431i	$0.764389\pi$
$390$	−7.35026	+	8.31265i	−0.372195	+	0.420927i
$391$	−1.35026			−0.0682857
$392$		−	1.77575i		−	0.0896887i
$393$		−	7.92478i		−	0.399752i
$394$	8.07522			0.406824
$395$	−6.91160	−	6.11142i	−0.347761	−	0.307499i
$396$	−3.35026			−0.168357
$397$			13.7381i			0.689497i	0.938695	+	0.344749i	$0.112036\pi$
$397$			13.7381i			0.689497i	−0.938695	+	0.344749i	$0.887964\pi$
$398$			15.9756i			0.800783i
$399$	14.7005			0.735947
$400$	0.612127	+	4.96239i	0.0306063	+	0.248119i
$401$	−1.44992			−0.0724057			−0.0362028	−	0.999344i	$-0.511526\pi$
$401$	−1.44992			−0.0724057			−0.0362028	+	0.999344i	$0.511526\pi$
$402$			7.61213i			0.379658i
$403$		−	19.8496i		−	0.988777i
$404$	−4.26187			−0.212036
$405$	−1.67513	−	1.48119i	−0.0832379	−	0.0736011i
$406$	−22.9234			−1.13767
$407$			25.5026i			1.26412i
$408$			1.35026i			0.0668479i
$409$	17.8496			0.882604			0.441302	−	0.897359i	$-0.354517\pi$
$409$	17.8496			0.882604			0.441302	+	0.897359i	$0.354517\pi$
$410$	−6.96239	+	7.87399i	−0.343848	+	0.388869i
$411$	9.19982			0.453793
$412$		−	0.261865i		−	0.0129012i
$413$		−	3.62672i		−	0.178459i
$414$	−1.00000			−0.0491473
$415$	9.46168	−	10.7005i	0.464456	−	0.525268i
$416$	4.96239			0.243301
$417$			8.62530i			0.422383i
$418$			16.6253i			0.813170i
$419$	−17.9003			−0.874489			−0.437244	−	0.899343i	$-0.644046\pi$
$419$	−17.9003			−0.874489			−0.437244	+	0.899343i	$0.644046\pi$
$420$	4.96239	+	4.38787i	0.242140	+	0.214106i
$421$	7.61213			0.370992			0.185496	−	0.982645i	$-0.440611\pi$
$421$	7.61213			0.370992			0.185496	+	0.982645i	$0.440611\pi$
$422$		−	21.7743i		−	1.05996i
$423$		−	3.22425i		−	0.156769i
$424$	−6.96239			−0.338123
$425$	−6.70052	+	0.826531i	−0.325023	+	0.0400927i
$426$	2.18664			0.105943
$427$		−	32.8481i		−	1.58963i
$428$			9.08840i			0.439304i
$429$	−16.6253			−0.802677
$430$	−17.2750	−	15.2750i	−0.833076	−	0.736628i
$431$	−22.9525			−1.10558			−0.552792	−	0.833319i	$-0.686438\pi$
$431$	−22.9525			−1.10558			−0.552792	+	0.833319i	$0.686438\pi$
$432$			1.00000i			0.0481125i
$433$		−	33.7645i		−	1.62262i	−0.584618	−	0.811309i	$-0.698756\pi$
$433$		−	33.7645i		−	1.62262i	0.584618	−	0.811309i	$-0.301244\pi$
$434$	−11.8496			−0.568797
$435$	11.4617	−	12.9624i	0.549546	−	0.621499i
$436$	18.9380			0.906963
$437$			4.96239i			0.237383i
$438$		−	9.92478i		−	0.474224i
$439$	23.4763			1.12046			0.560231	−	0.828337i	$-0.310713\pi$
$439$	23.4763			1.12046			0.560231	+	0.828337i	$0.310713\pi$
$440$	−4.96239	+	5.61213i	−0.236573	+	0.267548i
$441$	1.77575			0.0845593
$442$			6.70052i			0.318711i
$443$			30.5501i			1.45148i	0.687970	+	0.725739i	$0.258502\pi$
$443$			30.5501i			1.45148i	−0.687970	+	0.725739i	$0.741498\pi$
$444$	7.61213			0.361256
$445$	16.6253	+	14.7005i	0.788115	+	0.696872i
$446$	3.81336			0.180568
$447$			4.64974i			0.219925i
$448$		−	2.96239i		−	0.139960i
$449$	−32.7005			−1.54323			−0.771617	−	0.636088i	$-0.780552\pi$
$449$	−32.7005			−1.54323			−0.771617	+	0.636088i	$0.780552\pi$
$450$	−4.96239	+	0.612127i	−0.233929	+	0.0288559i
$451$	−15.7480			−0.741544
$452$		−	6.64974i		−	0.312777i
$453$		−	10.7005i		−	0.502754i
$454$	−16.9380			−0.794937
$455$	24.6253	+	21.7743i	1.15445	+	1.02080i
$456$	4.96239			0.232385
$457$			5.81336i			0.271937i	0.990713	+	0.135969i	$0.0434147\pi$
$457$			5.81336i			0.271937i	−0.990713	+	0.135969i	$0.956585\pi$
$458$		−	26.9380i		−	1.25873i
$459$	−1.35026			−0.0630248
$460$	−1.48119	+	1.67513i	−0.0690610	+	0.0781034i
$461$	−23.9902			−1.11733			−0.558666	−	0.829392i	$-0.688687\pi$
$461$	−23.9902			−1.11733			−0.558666	+	0.829392i	$0.688687\pi$
$462$			9.92478i			0.461742i
$463$			35.3620i			1.64341i	0.569911	+	0.821706i	$0.306978\pi$
$463$			35.3620i			1.64341i	−0.569911	+	0.821706i	$0.693022\pi$
$464$	−7.73813			−0.359234
$465$	5.92478	−	6.70052i	0.274755	−	0.310729i
$466$	27.4010			1.26933
$467$		−	1.98541i		−	0.0918739i	−0.998944	−	0.0459369i	$-0.985373\pi$
$467$		−	1.98541i		−	0.0918739i	0.998944	−	0.0459369i	$-0.0146273\pi$
$468$			4.96239i			0.229386i
$469$	22.5501			1.04127
$470$	−5.40105	−	4.77575i	−0.249132	−	0.220289i
$471$	−17.0132			−0.783926
$472$		−	1.22425i		−	0.0563508i
$473$		−	34.5501i		−	1.58861i
$474$	−4.12601			−0.189514
$475$	3.03761	+	24.6253i	0.139375	+	1.12989i
$476$	4.00000			0.183340
$477$		−	6.96239i		−	0.318786i
$478$			7.48612i			0.342407i
$479$	−11.0738			−0.505975			−0.252988	−	0.967470i	$-0.581413\pi$
$479$	−11.0738			−0.505975			−0.252988	+	0.967470i	$0.581413\pi$
$480$	1.67513	+	1.48119i	0.0764589	+	0.0676070i
$481$	37.7743			1.72236
$482$			13.0738i			0.595496i
$483$			2.96239i			0.134793i
$484$	−0.224254			−0.0101934
$485$	19.0884	−	21.5877i	0.866759	−	0.980246i
$486$	−1.00000			−0.0453609
$487$			17.4372i			0.790157i	0.918647	+	0.395078i	$0.129283\pi$
$487$			17.4372i			0.790157i	−0.918647	+	0.395078i	$0.870717\pi$
$488$		−	11.0884i		−	0.501948i
$489$	−12.6253			−0.570936
$490$	2.63023	−	2.97461i	0.118821	−	0.134379i
$491$	14.8773			0.671404			0.335702	−	0.941968i	$-0.391027\pi$
$491$	14.8773			0.671404			0.335702	+	0.941968i	$0.391027\pi$
$492$			4.70052i			0.211916i
$493$		−	10.4485i		−	0.470577i
$494$	24.6253			1.10794
$495$	−5.61213	−	4.96239i	−0.252246	−	0.223043i
$496$	−4.00000			−0.179605
$497$		−	6.47768i		−	0.290564i
$498$		−	6.38787i		−	0.286247i
$499$	16.1016			0.720805			0.360403	−	0.932797i	$-0.382639\pi$
$499$	16.1016			0.720805			0.360403	+	0.932797i	$0.382639\pi$
$500$	−6.32487	+	9.21933i	−0.282857	+	0.412301i
$501$	−24.6253			−1.10018
$502$			5.02302i			0.224188i
$503$		−	38.8021i		−	1.73010i	−0.501686	−	0.865050i	$-0.667287\pi$
$503$		−	38.8021i		−	1.73010i	0.501686	−	0.865050i	$-0.332713\pi$
$504$	2.96239			0.131955
$505$	−7.13918	−	6.31265i	−0.317689	−	0.280909i
$506$	−3.35026			−0.148937
$507$			11.6253i			0.516298i
$508$		−	0.186642i		−	0.00828091i
$509$	−23.4372			−1.03884			−0.519419	−	0.854520i	$-0.673852\pi$
$509$	−23.4372			−1.03884			−0.519419	+	0.854520i	$0.673852\pi$
$510$	−2.00000	+	2.26187i	−0.0885615	+	0.100157i
$511$	−29.4010			−1.30063
$512$		−	1.00000i		−	0.0441942i
$513$			4.96239i			0.219095i
$514$	19.4010			0.855743
$515$	0.387873	−	0.438658i	0.0170917	−	0.0193296i
$516$	−10.3127			−0.453989
$517$		−	10.8021i		−	0.475076i
$518$		−	22.5501i		−	0.990794i
$519$	4.44851			0.195268
$520$	8.31265	+	7.35026i	0.364534	+	0.322330i
$521$	21.1490			0.926556			0.463278	−	0.886213i	$-0.346673\pi$
$521$	21.1490			0.926556			0.463278	+	0.886213i	$0.346673\pi$
$522$		−	7.73813i		−	0.338689i
$523$		−	11.7626i		−	0.514341i	−0.966366	−	0.257171i	$-0.917210\pi$
$523$		−	11.7626i		−	0.514341i	0.966366	−	0.257171i	$-0.0827903\pi$
$524$	−7.92478			−0.346196
$525$	1.81336	+	14.7005i	0.0791414	+	0.641583i
$526$	−15.4763			−0.674797
$527$		−	5.40105i		−	0.235273i
$528$			3.35026i			0.145801i
$529$	−1.00000			−0.0434783
$530$	−11.6629	−	10.3127i	−0.506605	−	0.447953i
$531$	1.22425			0.0531281
$532$		−	14.7005i		−	0.637349i
$533$			23.3258i			1.01035i
$534$	9.92478			0.429487
$535$	−13.4617	+	15.2243i	−0.581999	+	0.658202i
$536$	7.61213			0.328794
$537$		−	13.8496i		−	0.597652i
$538$		−	20.2130i		−	0.871444i
$539$	5.94921			0.256251
$540$	−1.48119	+	1.67513i	−0.0637405	+	0.0720862i
$541$	−21.1754			−0.910401			−0.455200	−	0.890389i	$-0.650432\pi$
$541$	−21.1754			−0.910401			−0.455200	+	0.890389i	$0.650432\pi$
$542$		−	30.3996i		−	1.30578i
$543$			22.6859i			0.973547i
$544$	1.35026			0.0578920
$545$	31.7235	+	28.0508i	1.35889	+	1.20156i
$546$	14.7005			0.629124
$547$			5.29948i			0.226589i	0.993561	+	0.113295i	$0.0361404\pi$
$547$			5.29948i			0.226589i	−0.993561	+	0.113295i	$0.963860\pi$
$548$		−	9.19982i		−	0.392997i
$549$	11.0884			0.473241
$550$	−16.6253	+	2.05079i	−0.708905	+	0.0874458i
$551$	−38.3996			−1.63588
$552$			1.00000i			0.0425628i
$553$			12.2228i			0.519768i
$554$	20.8119			0.884215
$555$	12.7513	+	11.2750i	0.541263	+	0.478599i
$556$	8.62530			0.365794
$557$		−	7.99015i		−	0.338554i	−0.985569	−	0.169277i	$-0.945857\pi$
$557$		−	7.99015i		−	0.338554i	0.985569	−	0.169277i	$-0.0541432\pi$
$558$		−	4.00000i		−	0.169334i
$559$	−51.1754			−2.16449
$560$	4.38787	−	4.96239i	0.185421	−	0.209699i
$561$	−4.52373			−0.190992
$562$		−	19.8496i		−	0.837303i
$563$		−	15.1636i		−	0.639070i	−0.947574	−	0.319535i	$-0.896473\pi$
$563$		−	15.1636i		−	0.639070i	0.947574	−	0.319535i	$-0.103527\pi$
$564$	−3.22425			−0.135766
$565$	9.84955	−	11.1392i	0.414374	−	0.468629i
$566$	−12.3879			−0.520701
$567$			2.96239i			0.124409i
$568$		−	2.18664i		−	0.0917495i
$569$	25.5223			1.06995			0.534976	−	0.844868i	$-0.320321\pi$
$569$	25.5223			1.06995			0.534976	+	0.844868i	$0.320321\pi$
$570$	8.31265	+	7.35026i	0.348179	+	0.307869i
$571$	6.76590			0.283144			0.141572	−	0.989928i	$-0.454784\pi$
$571$	6.76590			0.283144			0.141572	+	0.989928i	$0.454784\pi$
$572$			16.6253i			0.695139i
$573$			19.3258i			0.807348i
$574$	13.9248			0.581209
$575$	−4.96239	+	0.612127i	−0.206946	+	0.0255275i
$576$	1.00000			0.0416667
$577$			4.00000i			0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$			4.00000i			0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$		−	15.1768i		−	0.631271i
$579$	5.92478			0.246225
$580$	−12.9624	−	11.4617i	−0.538234	−	0.475920i
$581$	−18.9234			−0.785073
$582$		−	12.8872i		−	0.534190i
$583$		−	23.3258i		−	0.966057i
$584$	−9.92478			−0.410690
$585$	−7.35026	+	8.31265i	−0.303896	+	0.343686i
$586$	−9.03761			−0.373340
$587$		−	12.8773i		−	0.531504i	−0.964041	−	0.265752i	$-0.914380\pi$
$587$		−	12.8773i		−	0.531504i	0.964041	−	0.265752i	$-0.0856202\pi$
$588$		−	1.77575i		−	0.0732305i
$589$	−19.8496			−0.817887
$590$	1.81336	−	2.05079i	0.0746548	−	0.0844295i
$591$	8.07522			0.332170
$592$		−	7.61213i		−	0.312856i
$593$		−	16.2981i		−	0.669281i	−0.942346	−	0.334641i	$-0.891385\pi$
$593$		−	16.2981i		−	0.669281i	0.942346	−	0.334641i	$-0.108615\pi$
$594$	−3.35026			−0.137463
$595$	6.70052	+	5.92478i	0.274695	+	0.242892i
$596$	4.64974			0.190461
$597$			15.9756i			0.653836i
$598$			4.96239i			0.202927i
$599$	−9.91493			−0.405113			−0.202556	−	0.979271i	$-0.564925\pi$
$599$	−9.91493			−0.405113			−0.202556	+	0.979271i	$0.564925\pi$
$600$	0.612127	+	4.96239i	0.0249900	+	0.202589i
$601$	−7.40105			−0.301895			−0.150948	−	0.988542i	$-0.548232\pi$
$601$	−7.40105			−0.301895			−0.150948	+	0.988542i	$0.548232\pi$
$602$			30.5501i			1.24513i
$603$			7.61213i			0.309990i
$604$	−10.7005			−0.435398
$605$	−0.375654	−	0.332163i	−0.0152725	−	0.0135044i
$606$	−4.26187			−0.173126
$607$		−	13.9610i		−	0.566658i	−0.959023	−	0.283329i	$-0.908561\pi$
$607$		−	13.9610i		−	0.566658i	0.959023	−	0.283329i	$-0.0914389\pi$
$608$		−	4.96239i		−	0.201251i
$609$	−22.9234			−0.928902
$610$	16.4241	−	18.5745i	0.664991	−	0.752060i
$611$	−16.0000			−0.647291
$612$			1.35026i			0.0545811i
$613$			41.3865i			1.67158i	0.549047	+	0.835792i	$0.314991\pi$
$613$			41.3865i			1.67158i	−0.549047	+	0.835792i	$0.685009\pi$
$614$	−30.5501			−1.23290
$615$	−6.96239	+	7.87399i	−0.280751	+	0.317510i
$616$	9.92478			0.399881
$617$			29.3014i			1.17963i	0.807539	+	0.589815i	$0.200799\pi$
$617$			29.3014i			1.17963i	−0.807539	+	0.589815i	$0.799201\pi$
$618$		−	0.261865i		−	0.0105338i
$619$	19.0376			0.765186			0.382593	−	0.923917i	$-0.375031\pi$
$619$	19.0376			0.765186			0.382593	+	0.923917i	$0.375031\pi$
$620$	−6.70052	−	5.92478i	−0.269099	−	0.237945i
$621$	−1.00000			−0.0401286
$622$		−	14.4387i		−	0.578937i
$623$		−	29.4010i		−	1.17793i
$624$	4.96239			0.198655
$625$	−24.2506	+	6.07522i	−0.970024	+	0.243009i
$626$	−27.9610			−1.11755
$627$			16.6253i			0.663951i
$628$			17.0132i			0.678900i
$629$	10.2784			0.409825
$630$	4.96239	+	4.38787i	0.197706	+	0.174817i
$631$	12.4993			0.497589			0.248794	−	0.968556i	$-0.419966\pi$
$631$	12.4993			0.497589			0.248794	+	0.968556i	$0.419966\pi$
$632$			4.12601i			0.164124i
$633$		−	21.7743i		−	0.865452i
$634$	1.47627			0.0586302
$635$	0.276454	−	0.312650i	0.0109707	−	0.0124072i
$636$	−6.96239			−0.276077
$637$		−	8.81194i		−	0.349142i
$638$		−	25.9248i		−	1.02637i
$639$	2.18664			0.0865022
$640$	1.48119	−	1.67513i	0.0585493	−	0.0662154i
$641$	−34.0263			−1.34396			−0.671980	−	0.740569i	$-0.734556\pi$
$641$	−34.0263			−1.34396			−0.671980	+	0.740569i	$0.734556\pi$
$642$			9.08840i			0.358690i
$643$			7.34041i			0.289478i	0.989470	+	0.144739i	$0.0462342\pi$
$643$			7.34041i			0.289478i	−0.989470	+	0.144739i	$0.953766\pi$
$644$	2.96239			0.116734
$645$	−17.2750	−	15.2750i	−0.680204	−	0.601454i
$646$	6.70052			0.263629
$647$		−	14.9525i		−	0.587845i	−0.955829	−	0.293922i	$-0.905039\pi$
$647$		−	14.9525i		−	0.587845i	0.955829	−	0.293922i	$-0.0949608\pi$
$648$			1.00000i			0.0392837i
$649$	4.10157			0.161001
$650$	3.03761	+	24.6253i	0.119145	+	0.965884i
$651$	−11.8496			−0.464421
$652$			12.6253i			0.494445i
$653$		−	20.4485i		−	0.800212i	−0.916469	−	0.400106i	$-0.868973\pi$
$653$		−	20.4485i		−	0.800212i	0.916469	−	0.400106i	$-0.131027\pi$
$654$	18.9380			0.740532
$655$	−13.2750	−	11.7381i	−0.518699	−	0.458647i
$656$	4.70052			0.183525
$657$		−	9.92478i		−	0.387202i
$658$			9.55149i			0.372356i
$659$	−38.4241			−1.49679			−0.748395	−	0.663254i	$-0.769175\pi$
$659$	−38.4241			−1.49679			−0.748395	+	0.663254i	$0.769175\pi$
$660$	−4.96239	+	5.61213i	−0.193161	+	0.218452i
$661$	−39.3112			−1.52903			−0.764515	−	0.644606i	$-0.777021\pi$
$661$	−39.3112			−1.52903			−0.764515	+	0.644606i	$0.777021\pi$
$662$		−	23.1754i		−	0.900737i
$663$			6.70052i			0.260227i
$664$	−6.38787			−0.247898
$665$	21.7743	−	24.6253i	0.844372	−	0.954928i
$666$	7.61213			0.294964
$667$		−	7.73813i		−	0.299622i
$668$			24.6253i			0.952781i
$669$	3.81336			0.147433
$670$	12.7513	+	11.2750i	0.492626	+	0.435593i
$671$	37.1490			1.43412
$672$		−	2.96239i		−	0.114277i
$673$		−	25.5515i		−	0.984938i	−0.870330	−	0.492469i	$-0.836095\pi$
$673$		−	25.5515i		−	0.984938i	0.870330	−	0.492469i	$-0.163905\pi$
$674$	−21.0376			−0.810339
$675$	−4.96239	+	0.612127i	−0.191002	+	0.0235608i
$676$	11.6253			0.447127
$677$			36.2130i			1.39178i	0.718149	+	0.695889i	$0.244990\pi$
$677$			36.2130i			1.39178i	−0.718149	+	0.695889i	$0.755010\pi$
$678$		−	6.64974i		−	0.255382i
$679$	−38.1768			−1.46509
$680$	2.26187	+	2.00000i	0.0867386	+	0.0766965i
$681$	−16.9380			−0.649064
$682$		−	13.4010i		−	0.513153i
$683$			15.4763i			0.592183i	0.955160	+	0.296092i	$0.0956833\pi$
$683$			15.4763i			0.592183i	−0.955160	+	0.296092i	$0.904317\pi$
$684$	4.96239			0.189742
$685$	13.6267	−	15.4109i	0.520650	−	0.588820i
$686$	15.4763			0.590887
$687$		−	26.9380i		−	1.02775i
$688$			10.3127i			0.393166i
$689$	−34.5501			−1.31625
$690$	−1.48119	+	1.67513i	−0.0563881	+	0.0637711i
$691$	−37.6531			−1.43239			−0.716195	−	0.697900i	$-0.754118\pi$
$691$	−37.6531			−1.43239			−0.716195	+	0.697900i	$0.754118\pi$
$692$		−	4.44851i		−	0.169107i
$693$			9.92478i			0.377011i
$694$	3.37470			0.128102
$695$	14.4485	+	12.7757i	0.548063	+	0.484612i
$696$	−7.73813			−0.293313
$697$			6.34694i			0.240407i
$698$		−	4.44851i		−	0.168378i
$699$	27.4010			1.03640
$700$	14.7005	−	1.81336i	0.555628	−	0.0685385i
$701$	28.3488			1.07072			0.535361	−	0.844624i	$-0.320176\pi$
$701$	28.3488			1.07072			0.535361	+	0.844624i	$0.320176\pi$
$702$			4.96239i			0.187293i
$703$		−	37.7743i		−	1.42469i
$704$	3.35026			0.126268
$705$	−5.40105	−	4.77575i	−0.203415	−	0.179865i
$706$	35.4010			1.33234
$707$			12.6253i			0.474823i
$708$		−	1.22425i		−	0.0460103i
$709$	43.5633			1.63605			0.818026	−	0.575181i	$-0.195068\pi$
$709$	43.5633			1.63605			0.818026	+	0.575181i	$0.195068\pi$
$710$	3.23884	−	3.66291i	0.121552	−	0.137467i
$711$	−4.12601			−0.154737
$712$		−	9.92478i		−	0.371947i
$713$		−	4.00000i		−	0.149801i
$714$	4.00000			0.149696
$715$	−24.6253	+	27.8496i	−0.920934	+	1.04151i
$716$	−13.8496			−0.517582
$717$			7.48612i			0.279574i
$718$		−	34.5501i		−	1.28940i
$719$	−32.3634			−1.20695			−0.603476	−	0.797381i	$-0.706218\pi$
$719$	−32.3634			−1.20695			−0.603476	+	0.797381i	$0.706218\pi$
$720$	1.67513	+	1.48119i	0.0624284	+	0.0552009i
$721$	−0.775746			−0.0288903
$722$		−	5.62530i		−	0.209352i
$723$			13.0738i			0.486220i
$724$	22.6859			0.843116
$725$	−4.73672	−	38.3996i	−0.175917	−	1.42613i
$726$	−0.224254			−0.00832284
$727$		−	6.96239i		−	0.258221i	−0.991630	−	0.129110i	$-0.958788\pi$
$727$		−	6.96239i		−	0.258221i	0.991630	−	0.129110i	$-0.0412121\pi$
$728$		−	14.7005i		−	0.544838i
$729$	−1.00000			−0.0370370
$730$	−16.6253	−	14.7005i	−0.615330	−	0.544091i
$731$	−13.9248			−0.515026
$732$		−	11.0884i		−	0.409839i
$733$		−	9.68735i		−	0.357810i	−0.983866	−	0.178905i	$-0.942744\pi$
$733$		−	9.68735i		−	0.357810i	0.983866	−	0.178905i	$-0.0572555\pi$
$734$	22.8119			0.842004
$735$	2.63023	−	2.97461i	0.0970173	−	0.109720i
$736$	1.00000			0.0368605
$737$			25.5026i			0.939401i
$738$			4.70052i			0.173029i
$739$	42.9234			1.57896			0.789481	−	0.613775i	$-0.210350\pi$
$739$	42.9234			1.57896			0.789481	+	0.613775i	$0.210350\pi$
$740$	11.2750	−	12.7513i	0.414479	−	0.468747i
$741$	24.6253			0.904633
$742$			20.6253i			0.757179i
$743$			16.9986i			0.623618i	0.950145	+	0.311809i	$0.100935\pi$
$743$			16.9986i			0.623618i	−0.950145	+	0.311809i	$0.899065\pi$
$744$	−4.00000			−0.146647
$745$	7.78892	+	6.88717i	0.285364	+	0.252326i
$746$	−10.8364			−0.396748
$747$		−	6.38787i		−	0.233720i
$748$			4.52373i			0.165404i
$749$	26.9234			0.983758
$750$	−6.32487	+	9.21933i	−0.230952	+	0.336642i
$751$	−17.9003			−0.653193			−0.326596	−	0.945164i	$-0.605902\pi$
$751$	−17.9003			−0.653193			−0.326596	+	0.945164i	$0.605902\pi$
$752$			3.22425i			0.117576i
$753$			5.02302i			0.183049i
$754$	−38.3996			−1.39843
$755$	−17.9248	−	15.8496i	−0.652349	−	0.576824i
$756$	2.96239			0.107741
$757$			12.3879i			0.450245i	0.974330	+	0.225122i	$0.0722782\pi$
$757$			12.3879i			0.450245i	−0.974330	+	0.225122i	$0.927722\pi$
$758$		−	32.4387i		−	1.17823i
$759$	−3.35026			−0.121607
$760$	7.35026	−	8.31265i	0.266622	−	0.301532i
$761$	38.5764			1.39839			0.699197	−	0.714929i	$-0.253541\pi$
$761$	38.5764			1.39839			0.699197	+	0.714929i	$0.253541\pi$
$762$		−	0.186642i		−	0.00676134i
$763$		−	56.1016i		−	2.03101i
$764$	19.3258			0.699184
$765$	−2.00000	+	2.26187i	−0.0723102	+	0.0817779i
$766$	32.9986			1.19229
$767$		−	6.07522i		−	0.219364i
$768$		−	1.00000i		−	0.0360844i
$769$	−34.2228			−1.23411			−0.617054	−	0.786921i	$-0.711674\pi$
$769$	−34.2228			−1.23411			−0.617054	+	0.786921i	$0.711674\pi$
$770$	16.6253	+	14.7005i	0.599134	+	0.529770i
$771$	19.4010			0.698712
$772$		−	5.92478i		−	0.213237i
$773$			13.5613i			0.487768i	0.969804	+	0.243884i	$0.0784215\pi$
$773$			13.5613i			0.487768i	−0.969804	+	0.243884i	$0.921578\pi$
$774$	−10.3127			−0.370681
$775$	−2.44851	−	19.8496i	−0.0879530	−	0.713017i
$776$	−12.8872			−0.462622
$777$		−	22.5501i		−	0.808980i
$778$			29.1246i			1.04417i
$779$	23.3258			0.835734
$780$	8.31265	+	7.35026i	0.297641	+	0.263182i
$781$	7.32582			0.262139
$782$			1.35026i			0.0482853i
$783$		−	7.73813i		−	0.276538i
$784$	−1.77575			−0.0634195
$785$	−25.1998	+	28.4993i	−0.899420	+	1.01718i
$786$	−7.92478			−0.282667
$787$			37.0132i			1.31938i	0.751539	+	0.659689i	$0.229312\pi$
$787$			37.0132i			1.31938i	−0.751539	+	0.659689i	$0.770688\pi$
$788$		−	8.07522i		−	0.287668i

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.67513 - 1.48119i) q^{5} -1.00000 q^{6} +2.96239i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +(-1.67513 - 1.48119i) q^{5} -1.00000 q^{6} +2.96239i q^{7} +1.00000i q^{8} -1.00000 q^{9} +(-1.48119 + 1.67513i) q^{10} -3.35026 q^{11} +1.00000i q^{12} +4.96239i q^{13} +2.96239 q^{14} +(-1.48119 + 1.67513i) q^{15} +1.00000 q^{16} +1.35026i q^{17} +1.00000i q^{18} +4.96239 q^{19} +(1.67513 + 1.48119i) q^{20} +2.96239 q^{21} +3.35026i q^{22} +1.00000i q^{23} +1.00000 q^{24} +(0.612127 + 4.96239i) q^{25} +4.96239 q^{26} +1.00000i q^{27} -2.96239i q^{28} -7.73813 q^{29} +(1.67513 + 1.48119i) q^{30} -4.00000 q^{31} -1.00000i q^{32} +3.35026i q^{33} +1.35026 q^{34} +(4.38787 - 4.96239i) q^{35} +1.00000 q^{36} -7.61213i q^{37} -4.96239i q^{38} +4.96239 q^{39} +(1.48119 - 1.67513i) q^{40} +4.70052 q^{41} -2.96239i q^{42} +10.3127i q^{43} +3.35026 q^{44} +(1.67513 + 1.48119i) q^{45} +1.00000 q^{46} +3.22425i q^{47} -1.00000i q^{48} -1.77575 q^{49} +(4.96239 - 0.612127i) q^{50} +1.35026 q^{51} -4.96239i q^{52} +6.96239i q^{53} +1.00000 q^{54} +(5.61213 + 4.96239i) q^{55} -2.96239 q^{56} -4.96239i q^{57} +7.73813i q^{58} -1.22425 q^{59} +(1.48119 - 1.67513i) q^{60} -11.0884 q^{61} +4.00000i q^{62} -2.96239i q^{63} -1.00000 q^{64} +(7.35026 - 8.31265i) q^{65} +3.35026 q^{66} -7.61213i q^{67} -1.35026i q^{68} +1.00000 q^{69} +(-4.96239 - 4.38787i) q^{70} -2.18664 q^{71} -1.00000i q^{72} +9.92478i q^{73} -7.61213 q^{74} +(4.96239 - 0.612127i) q^{75} -4.96239 q^{76} -9.92478i q^{77} -4.96239i q^{78} +4.12601 q^{79} +(-1.67513 - 1.48119i) q^{80} +1.00000 q^{81} -4.70052i q^{82} +6.38787i q^{83} -2.96239 q^{84} +(2.00000 - 2.26187i) q^{85} +10.3127 q^{86} +7.73813i q^{87} -3.35026i q^{88} -9.92478 q^{89} +(1.48119 - 1.67513i) q^{90} -14.7005 q^{91} -1.00000i q^{92} +4.00000i q^{93} +3.22425 q^{94} +(-8.31265 - 7.35026i) q^{95} -1.00000 q^{96} +12.8872i q^{97} +1.77575i q^{98} +3.35026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6q - 6q^{4} - 6q^{6} - 6q^{9} + O(q^{10}) \)	\( 6q - 6q^{4} - 6q^{6} - 6q^{9} + 2q^{10} - 4q^{14} + 2q^{15} + 6q^{16} + 8q^{19} - 4q^{21} + 6q^{24} + 2q^{25} + 8q^{26} - 28q^{29} - 24q^{31} - 12q^{34} + 28q^{35} + 6q^{36} + 8q^{39} - 2q^{40} - 12q^{41} + 6q^{46} - 14q^{49} + 8q^{50} - 12q^{51} + 6q^{54} + 32q^{55} + 4q^{56} - 4q^{59} - 2q^{60} - 28q^{61} - 6q^{64} + 24q^{65} + 6q^{69} - 8q^{70} + 12q^{71} - 44q^{74} + 8q^{75} - 8q^{76} + 8q^{79} + 6q^{81} + 4q^{84} + 12q^{85} + 20q^{86} - 16q^{89} - 2q^{90} - 48q^{91} + 16q^{94} - 8q^{95} - 6q^{96} + O(q^{100}) \)

Introduction

L-functions

Modular forms

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Fields

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Groups

Properties

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