LMFDB - Embedded newform 690.2.d.c.139.6

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			139.6
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	690.139
Dual form			690.2.d.c.139.3

$q$-expansion

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.21432 - 0.311108i) q^{5} -1.00000 q^{6} +0.622216i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.21432 - 0.311108i) q^{5} -1.00000 q^{6} +0.622216i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(0.311108 + 2.21432i) q^{10} +4.42864 q^{11} -1.00000i q^{12} -1.37778i q^{13} -0.622216 q^{14} +(0.311108 + 2.21432i) q^{15} +1.00000 q^{16} +6.42864i q^{17} -1.00000i q^{18} +1.37778 q^{19} +(-2.21432 + 0.311108i) q^{20} -0.622216 q^{21} +4.42864i q^{22} -1.00000i q^{23} +1.00000 q^{24} +(4.80642 - 1.37778i) q^{25} +1.37778 q^{26} -1.00000i q^{27} -0.622216i q^{28} +4.23506 q^{29} +(-2.21432 + 0.311108i) q^{30} -4.00000 q^{31} +1.00000i q^{32} +4.42864i q^{33} -6.42864 q^{34} +(0.193576 + 1.37778i) q^{35} +1.00000 q^{36} +11.8064i q^{37} +1.37778i q^{38} +1.37778 q^{39} +(-0.311108 - 2.21432i) q^{40} -10.8573 q^{41} -0.622216i q^{42} +1.05086i q^{43} -4.42864 q^{44} +(-2.21432 + 0.311108i) q^{45} +1.00000 q^{46} -11.6128i q^{47} +1.00000i q^{48} +6.61285 q^{49} +(1.37778 + 4.80642i) q^{50} -6.42864 q^{51} +1.37778i q^{52} -3.37778i q^{53} +1.00000 q^{54} +(9.80642 - 1.37778i) q^{55} +0.622216 q^{56} +1.37778i q^{57} +4.23506i q^{58} -9.61285 q^{59} +(-0.311108 - 2.21432i) q^{60} +8.66370 q^{61} -4.00000i q^{62} -0.622216i q^{63} -1.00000 q^{64} +(-0.428639 - 3.05086i) q^{65} -4.42864 q^{66} +11.8064i q^{67} -6.42864i q^{68} +1.00000 q^{69} +(-1.37778 + 0.193576i) q^{70} -6.99063 q^{71} +1.00000i q^{72} -2.75557i q^{73} -11.8064 q^{74} +(1.37778 + 4.80642i) q^{75} -1.37778 q^{76} +2.75557i q^{77} +1.37778i q^{78} -12.0415 q^{79} +(2.21432 - 0.311108i) q^{80} +1.00000 q^{81} -10.8573i q^{82} -2.19358i q^{83} +0.622216 q^{84} +(2.00000 + 14.2351i) q^{85} -1.05086 q^{86} +4.23506i q^{87} -4.42864i q^{88} -2.75557 q^{89} +(-0.311108 - 2.21432i) q^{90} +0.857279 q^{91} +1.00000i q^{92} -4.00000i q^{93} +11.6128 q^{94} +(3.05086 - 0.428639i) q^{95} -1.00000 q^{96} -2.13335i q^{97} +6.61285i q^{98} -4.42864 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.21432	−	0.311108i	0.990274	−	0.139132i
$5$	2.21432	−	0.311108i	0.990274	−	0.139132i
$6$	−1.00000			−0.408248
$7$			0.622216i			0.235175i	0.993063	+	0.117588i	$0.0375161\pi$
$7$			0.622216i			0.235175i	−0.993063	+	0.117588i	$0.962484\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	0.311108	+	2.21432i	0.0983809	+	0.700229i
$11$	4.42864			1.33529			0.667643	−	0.744482i	$-0.267303\pi$
$11$	4.42864			1.33529			0.667643	+	0.744482i	$0.267303\pi$
$12$		−	1.00000i		−	0.288675i
$13$		−	1.37778i		−	0.382129i	−0.981578	−	0.191064i	$-0.938806\pi$
$13$		−	1.37778i		−	0.382129i	0.981578	−	0.191064i	$-0.0611939\pi$
$14$	−0.622216			−0.166294
$15$	0.311108	+	2.21432i	0.0803277	+	0.571735i
$16$	1.00000			0.250000
$17$			6.42864i			1.55917i	0.626294	+	0.779587i	$0.284571\pi$
$17$			6.42864i			1.55917i	−0.626294	+	0.779587i	$0.715429\pi$
$18$		−	1.00000i		−	0.235702i
$19$	1.37778			0.316085			0.158043	−	0.987432i	$-0.449482\pi$
$19$	1.37778			0.316085			0.158043	+	0.987432i	$0.449482\pi$
$20$	−2.21432	+	0.311108i	−0.495137	+	0.0695658i
$21$	−0.622216			−0.135779
$22$			4.42864i			0.944189i
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$	1.00000			0.204124
$25$	4.80642	−	1.37778i	0.961285	−	0.275557i
$26$	1.37778			0.270206
$27$		−	1.00000i		−	0.192450i
$28$		−	0.622216i		−	0.117588i
$29$	4.23506			0.786432			0.393216	−	0.919446i	$-0.371363\pi$
$29$	4.23506			0.786432			0.393216	+	0.919446i	$0.371363\pi$
$30$	−2.21432	+	0.311108i	−0.404278	+	0.0568003i
$31$	−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
$33$			4.42864i			0.770927i
$34$	−6.42864			−1.10250
$35$	0.193576	+	1.37778i	0.0327203	+	0.232888i
$36$	1.00000			0.166667
$37$			11.8064i			1.94096i	0.241173	+	0.970482i	$0.422468\pi$
$37$			11.8064i			1.94096i	−0.241173	+	0.970482i	$0.577532\pi$
$38$			1.37778i			0.223506i
$39$	1.37778			0.220622
$40$	−0.311108	−	2.21432i	−0.0491905	−	0.350115i
$41$	−10.8573			−1.69562			−0.847811	−	0.530298i	$-0.822080\pi$
$41$	−10.8573			−1.69562			−0.847811	+	0.530298i	$0.822080\pi$
$42$		−	0.622216i		−	0.0960100i
$43$			1.05086i			0.160254i	0.996785	+	0.0801270i	$0.0255326\pi$
$43$			1.05086i			0.160254i	−0.996785	+	0.0801270i	$0.974467\pi$
$44$	−4.42864			−0.667643
$45$	−2.21432	+	0.311108i	−0.330091	+	0.0463772i
$46$	1.00000			0.147442
$47$		−	11.6128i		−	1.69391i	−0.531666	−	0.846954i	$-0.678434\pi$
$47$		−	11.6128i		−	1.69391i	0.531666	−	0.846954i	$-0.321566\pi$
$48$			1.00000i			0.144338i
$49$	6.61285			0.944693
$50$	1.37778	+	4.80642i	0.194848	+	0.679731i
$51$	−6.42864			−0.900190
$52$			1.37778i			0.191064i
$53$		−	3.37778i		−	0.463974i	−0.972719	−	0.231987i	$-0.925477\pi$
$53$		−	3.37778i		−	0.463974i	0.972719	−	0.231987i	$-0.0745227\pi$
$54$	1.00000			0.136083
$55$	9.80642	−	1.37778i	1.32230	−	0.185780i
$56$	0.622216			0.0831471
$57$			1.37778i			0.182492i
$58$			4.23506i			0.556091i
$59$	−9.61285			−1.25149			−0.625743	−	0.780029i	$-0.715204\pi$
$59$	−9.61285			−1.25149			−0.625743	+	0.780029i	$0.715204\pi$
$60$	−0.311108	−	2.21432i	−0.0401638	−	0.285867i
$61$	8.66370			1.10927			0.554637	−	0.832093i	$-0.312857\pi$
$61$	8.66370			1.10927			0.554637	+	0.832093i	$0.312857\pi$
$62$		−	4.00000i		−	0.508001i
$63$		−	0.622216i		−	0.0783918i
$64$	−1.00000			−0.125000
$65$	−0.428639	−	3.05086i	−0.0531662	−	0.378412i
$66$	−4.42864			−0.545128
$67$			11.8064i			1.44238i	0.692735	+	0.721192i	$0.256406\pi$
$67$			11.8064i			1.44238i	−0.692735	+	0.721192i	$0.743594\pi$
$68$		−	6.42864i		−	0.779587i
$69$	1.00000			0.120386
$70$	−1.37778	+	0.193576i	−0.164677	+	0.0231368i
$71$	−6.99063			−0.829635			−0.414818	−	0.909905i	$-0.636155\pi$
$71$	−6.99063			−0.829635			−0.414818	+	0.909905i	$0.636155\pi$
$72$			1.00000i			0.117851i
$73$		−	2.75557i		−	0.322515i	−0.986912	−	0.161257i	$-0.948445\pi$
$73$		−	2.75557i		−	0.322515i	0.986912	−	0.161257i	$-0.0515550\pi$
$74$	−11.8064			−1.37247
$75$	1.37778	+	4.80642i	0.159093	+	0.554998i
$76$	−1.37778			−0.158043
$77$			2.75557i			0.314026i
$78$			1.37778i			0.156003i
$79$	−12.0415			−1.35477			−0.677387	−	0.735627i	$-0.736888\pi$
$79$	−12.0415			−1.35477			−0.677387	+	0.735627i	$0.736888\pi$
$80$	2.21432	−	0.311108i	0.247568	−	0.0347829i
$81$	1.00000			0.111111
$82$		−	10.8573i		−	1.19899i
$83$		−	2.19358i		−	0.240776i	−0.992727	−	0.120388i	$-0.961586\pi$
$83$		−	2.19358i		−	0.240776i	0.992727	−	0.120388i	$-0.0384139\pi$
$84$	0.622216			0.0678893
$85$	2.00000	+	14.2351i	0.216930	+	1.54401i
$86$	−1.05086			−0.113317
$87$			4.23506i			0.454046i
$88$		−	4.42864i		−	0.472095i
$89$	−2.75557			−0.292090			−0.146045	−	0.989278i	$-0.546654\pi$
$89$	−2.75557			−0.292090			−0.146045	+	0.989278i	$0.546654\pi$
$90$	−0.311108	−	2.21432i	−0.0327936	−	0.233410i
$91$	0.857279			0.0898673
$92$			1.00000i			0.104257i
$93$		−	4.00000i		−	0.414781i
$94$	11.6128			1.19777
$95$	3.05086	−	0.428639i	0.313011	−	0.0439775i
$96$	−1.00000			−0.102062
$97$		−	2.13335i		−	0.216609i	−0.994118	−	0.108305i	$-0.965458\pi$
$97$		−	2.13335i		−	0.216609i	0.994118	−	0.108305i	$-0.0345422\pi$
$98$			6.61285i			0.667998i
$99$	−4.42864			−0.445095
$100$	−4.80642	+	1.37778i	−0.480642	+	0.137778i
$101$	16.2351			1.61545			0.807725	−	0.589560i	$-0.200699\pi$
$101$	16.2351			1.61545			0.807725	+	0.589560i	$0.200699\pi$
$102$		−	6.42864i		−	0.636530i
$103$		−	12.2351i		−	1.20556i	−0.797909	−	0.602778i	$-0.794060\pi$
$103$		−	12.2351i		−	1.20556i	0.797909	−	0.602778i	$-0.205940\pi$
$104$	−1.37778			−0.135103
$105$	−1.37778	+	0.193576i	−0.134458	+	0.0188911i
$106$	3.37778			0.328079
$107$		−	10.6637i		−	1.03090i	−0.856920	−	0.515450i	$-0.827625\pi$
$107$		−	10.6637i		−	1.03090i	0.856920	−	0.515450i	$-0.172375\pi$
$108$			1.00000i			0.0962250i
$109$	15.1526			1.45135			0.725676	−	0.688036i	$-0.241527\pi$
$109$	15.1526			1.45135			0.725676	+	0.688036i	$0.241527\pi$
$110$	1.37778	+	9.80642i	0.131367	+	0.935006i
$111$	−11.8064			−1.12062
$112$			0.622216i			0.0587939i
$113$		−	14.4286i		−	1.35733i	−0.734447	−	0.678666i	$-0.762558\pi$
$113$		−	14.4286i		−	1.35733i	0.734447	−	0.678666i	$-0.237442\pi$
$114$	−1.37778			−0.129041
$115$	−0.311108	−	2.21432i	−0.0290110	−	0.206486i
$116$	−4.23506			−0.393216
$117$			1.37778i			0.127376i
$118$		−	9.61285i		−	0.884934i
$119$	−4.00000			−0.366679
$120$	2.21432	−	0.311108i	0.202139	−	0.0284001i
$121$	8.61285			0.782986
$122$			8.66370i			0.784375i
$123$		−	10.8573i		−	0.978968i
$124$	4.00000			0.359211
$125$	10.2143	−	4.54617i	0.913597	−	0.406622i
$126$	0.622216			0.0554314
$127$		−	4.99063i		−	0.442847i	−0.975178	−	0.221423i	$-0.928930\pi$
$127$		−	4.99063i		−	0.442847i	0.975178	−	0.221423i	$-0.0710703\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−1.05086			−0.0925226
$130$	3.05086	−	0.428639i	0.267578	−	0.0375942i
$131$	0.755569			0.0660143			0.0330072	−	0.999455i	$-0.489492\pi$
$131$	0.755569			0.0660143			0.0330072	+	0.999455i	$0.489492\pi$
$132$		−	4.42864i		−	0.385464i
$133$			0.857279i			0.0743355i
$134$	−11.8064			−1.01992
$135$	−0.311108	−	2.21432i	−0.0267759	−	0.190578i
$136$	6.42864			0.551251
$137$			12.9175i			1.10362i	0.833971	+	0.551808i	$0.186062\pi$
$137$			12.9175i			1.10362i	−0.833971	+	0.551808i	$0.813938\pi$
$138$			1.00000i			0.0851257i
$139$	14.1017			1.19609			0.598046	−	0.801462i	$-0.295944\pi$
$139$	14.1017			1.19609			0.598046	+	0.801462i	$0.295944\pi$
$140$	−0.193576	−	1.37778i	−0.0163602	−	0.116444i
$141$	11.6128			0.977978
$142$		−	6.99063i		−	0.586641i
$143$		−	6.10171i		−	0.510251i
$144$	−1.00000			−0.0833333
$145$	9.37778	−	1.31756i	0.778783	−	0.109418i
$146$	2.75557			0.228052
$147$			6.61285i			0.545418i
$148$		−	11.8064i		−	0.970482i
$149$	−12.4286			−1.01819			−0.509097	−	0.860709i	$-0.670021\pi$
$149$	−12.4286			−1.01819			−0.509097	+	0.860709i	$0.670021\pi$
$150$	−4.80642	+	1.37778i	−0.392443	+	0.112496i
$151$	−4.85728			−0.395280			−0.197640	−	0.980275i	$-0.563328\pi$
$151$	−4.85728			−0.395280			−0.197640	+	0.980275i	$0.563328\pi$
$152$		−	1.37778i		−	0.111753i
$153$		−	6.42864i		−	0.519725i
$154$	−2.75557			−0.222050
$155$	−8.85728	+	1.24443i	−0.711434	+	0.0999551i
$156$	−1.37778			−0.110311
$157$		−	9.90813i		−	0.790755i	−0.918519	−	0.395378i	$-0.870614\pi$
$157$		−	9.90813i		−	0.790755i	0.918519	−	0.395378i	$-0.129386\pi$
$158$		−	12.0415i		−	0.957969i
$159$	3.37778			0.267876
$160$	0.311108	+	2.21432i	0.0245952	+	0.175057i
$161$	0.622216			0.0490375
$162$			1.00000i			0.0785674i
$163$		−	10.1017i		−	0.791227i	−0.918417	−	0.395614i	$-0.870532\pi$
$163$		−	10.1017i		−	0.791227i	0.918417	−	0.395614i	$-0.129468\pi$
$164$	10.8573			0.847811
$165$	1.37778	+	9.80642i	0.107260	+	0.763429i
$166$	2.19358			0.170255
$167$			1.89829i			0.146894i	0.997299	+	0.0734470i	$0.0234000\pi$
$167$			1.89829i			0.146894i	−0.997299	+	0.0734470i	$0.976600\pi$
$168$			0.622216i			0.0480050i
$169$	11.1017			0.853978
$170$	−14.2351	+	2.00000i	−1.09178	+	0.153393i
$171$	−1.37778			−0.105362
$172$		−	1.05086i		−	0.0801270i
$173$		−	21.2257i		−	1.61376i	−0.590716	−	0.806880i	$-0.701154\pi$
$173$		−	21.2257i		−	1.61376i	0.590716	−	0.806880i	$-0.298846\pi$
$174$	−4.23506			−0.321059
$175$	0.857279	+	2.99063i	0.0648042	+	0.226071i
$176$	4.42864			0.333821
$177$		−	9.61285i		−	0.722546i
$178$		−	2.75557i		−	0.206539i
$179$	−0.488863			−0.0365393			−0.0182697	−	0.999833i	$-0.505816\pi$
$179$	−0.488863			−0.0365393			−0.0182697	+	0.999833i	$0.505816\pi$
$180$	2.21432	−	0.311108i	0.165046	−	0.0231886i
$181$	−20.9304			−1.55575			−0.777873	−	0.628422i	$-0.783701\pi$
$181$	−20.9304			−1.55575			−0.777873	+	0.628422i	$0.783701\pi$
$182$			0.857279i			0.0635457i
$183$			8.66370i			0.640439i
$184$	−1.00000			−0.0737210
$185$	3.67307	+	26.1432i	0.270050	+	1.92209i
$186$	4.00000			0.293294
$187$			28.4701i			2.08194i
$188$			11.6128i			0.846954i
$189$	0.622216			0.0452595
$190$	0.428639	+	3.05086i	0.0310968	+	0.221332i
$191$	18.9590			1.37182			0.685912	−	0.727684i	$-0.259403\pi$
$191$	18.9590			1.37182			0.685912	+	0.727684i	$0.259403\pi$
$192$		−	1.00000i		−	0.0721688i
$193$			1.24443i			0.0895761i	0.998997	+	0.0447881i	$0.0142613\pi$
$193$			1.24443i			0.0895761i	−0.998997	+	0.0447881i	$0.985739\pi$
$194$	2.13335			0.153166
$195$	3.05086	−	0.428639i	0.218476	−	0.0306955i
$196$	−6.61285			−0.472346
$197$		−	15.2444i		−	1.08612i	−0.839694	−	0.543060i	$-0.817265\pi$
$197$		−	15.2444i		−	1.08612i	0.839694	−	0.543060i	$-0.182735\pi$
$198$		−	4.42864i		−	0.314730i
$199$	14.5303			1.03003			0.515015	−	0.857181i	$-0.327786\pi$
$199$	14.5303			1.03003			0.515015	+	0.857181i	$0.327786\pi$
$200$	−1.37778	−	4.80642i	−0.0974241	−	0.339865i
$201$	−11.8064			−0.832761
$202$			16.2351i			1.14230i
$203$			2.63512i			0.184949i
$204$	6.42864			0.450095
$205$	−24.0415	+	3.37778i	−1.67913	+	0.235915i
$206$	12.2351			0.852457
$207$			1.00000i			0.0695048i
$208$		−	1.37778i		−	0.0955322i
$209$	6.10171			0.422064
$210$	−0.193576	−	1.37778i	−0.0133580	−	0.0950762i
$211$	0.266706			0.0183608			0.00918041	−	0.999958i	$-0.497078\pi$
$211$	0.266706			0.0183608			0.00918041	+	0.999958i	$0.497078\pi$
$212$			3.37778i			0.231987i
$213$		−	6.99063i		−	0.478990i
$214$	10.6637			0.728956
$215$	0.326929	+	2.32693i	0.0222964	+	0.158695i
$216$	−1.00000			−0.0680414
$217$		−	2.48886i		−	0.168955i
$218$			15.1526i			1.02626i
$219$	2.75557			0.186204
$220$	−9.80642	+	1.37778i	−0.661149	+	0.0928902i
$221$	8.85728			0.595805
$222$		−	11.8064i		−	0.792395i
$223$			0.990632i			0.0663376i	0.999450	+	0.0331688i	$0.0105599\pi$
$223$			0.990632i			0.0663376i	−0.999450	+	0.0331688i	$0.989440\pi$
$224$	−0.622216			−0.0415735
$225$	−4.80642	+	1.37778i	−0.320428	+	0.0918523i
$226$	14.4286			0.959779
$227$		−	17.1526i		−	1.13846i	−0.822180	−	0.569228i	$-0.807242\pi$
$227$		−	17.1526i		−	1.13846i	0.822180	−	0.569228i	$-0.192758\pi$
$228$		−	1.37778i		−	0.0912460i
$229$	−7.15257			−0.472655			−0.236327	−	0.971673i	$-0.575944\pi$
$229$	−7.15257			−0.472655			−0.236327	+	0.971673i	$0.575944\pi$
$230$	2.21432	−	0.311108i	0.146008	−	0.0205138i
$231$	−2.75557			−0.181303
$232$		−	4.23506i		−	0.278046i
$233$			3.71456i			0.243349i	0.992570	+	0.121674i	$0.0388264\pi$
$233$			3.71456i			0.243349i	−0.992570	+	0.121674i	$0.961174\pi$
$234$	−1.37778			−0.0900686
$235$	−3.61285	−	25.7146i	−0.235676	−	1.67743i
$236$	9.61285			0.625743
$237$		−	12.0415i		−	0.782179i
$238$		−	4.00000i		−	0.259281i
$239$	−27.8479			−1.80133			−0.900666	−	0.434512i	$-0.856921\pi$
$239$	−27.8479			−1.80133			−0.900666	+	0.434512i	$0.856921\pi$
$240$	0.311108	+	2.21432i	0.0200819	+	0.142934i
$241$	−7.12399			−0.458896			−0.229448	−	0.973321i	$-0.573692\pi$
$241$	−7.12399			−0.458896			−0.229448	+	0.973321i	$0.573692\pi$
$242$			8.61285i			0.553655i
$243$			1.00000i			0.0641500i
$244$	−8.66370			−0.554637
$245$	14.6430	−	2.05731i	0.935504	−	0.131437i
$246$	10.8573			0.692235
$247$		−	1.89829i		−	0.120785i
$248$			4.00000i			0.254000i
$249$	2.19358			0.139012
$250$	4.54617	+	10.2143i	0.287525	+	0.646010i
$251$	−22.4099			−1.41450			−0.707250	−	0.706963i	$-0.750065\pi$
$251$	−22.4099			−1.41450			−0.707250	+	0.706963i	$0.750065\pi$
$252$			0.622216i			0.0391959i
$253$		−	4.42864i		−	0.278426i
$254$	4.99063			0.313140
$255$	−14.2351	+	2.00000i	−0.891434	+	0.125245i
$256$	1.00000			0.0625000
$257$			11.7146i			0.730734i	0.930864	+	0.365367i	$0.119057\pi$
$257$			11.7146i			0.730734i	−0.930864	+	0.365367i	$0.880943\pi$
$258$		−	1.05086i		−	0.0654234i
$259$	−7.34614			−0.456467
$260$	0.428639	+	3.05086i	0.0265831	+	0.189206i
$261$	−4.23506			−0.262144
$262$			0.755569i			0.0466792i
$263$		−	8.47013i		−	0.522290i	−0.965300	−	0.261145i	$-0.915900\pi$
$263$		−	8.47013i		−	0.522290i	0.965300	−	0.261145i	$-0.0841001\pi$
$264$	4.42864			0.272564
$265$	−1.05086	−	7.47949i	−0.0645535	−	0.459462i
$266$	−0.857279			−0.0525631
$267$		−	2.75557i		−	0.168638i
$268$		−	11.8064i		−	0.721192i
$269$	−28.8256			−1.75753			−0.878765	−	0.477255i	$-0.841632\pi$
$269$	−28.8256			−1.75753			−0.878765	+	0.477255i	$0.841632\pi$
$270$	2.21432	−	0.311108i	0.134759	−	0.0189334i
$271$	−13.8350			−0.840417			−0.420208	−	0.907428i	$-0.638043\pi$
$271$	−13.8350			−0.840417			−0.420208	+	0.907428i	$0.638043\pi$
$272$			6.42864i			0.389794i
$273$			0.857279i			0.0518849i
$274$	−12.9175			−0.780375
$275$	21.2859	−	6.10171i	1.28359	−	0.367947i
$276$	−1.00000			−0.0601929
$277$		−	2.88892i		−	0.173578i	−0.996227	−	0.0867892i	$-0.972339\pi$
$277$		−	2.88892i		−	0.173578i	0.996227	−	0.0867892i	$-0.0276607\pi$
$278$			14.1017i			0.845764i
$279$	4.00000			0.239474
$280$	1.37778	−	0.193576i	0.0823384	−	0.0115684i
$281$	5.51114			0.328767			0.164383	−	0.986397i	$-0.447437\pi$
$281$	5.51114			0.328767			0.164383	+	0.986397i	$0.447437\pi$
$282$			11.6128i			0.691535i
$283$			8.19358i			0.487058i	0.969894	+	0.243529i	$0.0783050\pi$
$283$			8.19358i			0.487058i	−0.969894	+	0.243529i	$0.921695\pi$
$284$	6.99063			0.414818
$285$	0.428639	+	3.05086i	0.0253904	+	0.180717i
$286$	6.10171			0.360802
$287$		−	6.75557i		−	0.398769i
$288$		−	1.00000i		−	0.0589256i
$289$	−24.3274			−1.43102
$290$	1.31756	+	9.37778i	0.0773699	+	0.550682i
$291$	2.13335			0.125059
$292$			2.75557i			0.161257i
$293$			12.6222i			0.737398i	0.929549	+	0.368699i	$0.120197\pi$
$293$			12.6222i			0.737398i	−0.929549	+	0.368699i	$0.879803\pi$
$294$	−6.61285			−0.385669
$295$	−21.2859	+	2.99063i	−1.23931	+	0.174121i
$296$	11.8064			0.686234
$297$		−	4.42864i		−	0.256976i
$298$		−	12.4286i		−	0.719972i
$299$	−1.37778			−0.0796793
$300$	−1.37778	−	4.80642i	−0.0795464	−	0.277499i
$301$	−0.653858			−0.0376878
$302$		−	4.85728i		−	0.279505i
$303$			16.2351i			0.932680i
$304$	1.37778			0.0790214
$305$	19.1842	−	2.69535i	1.09848	−	0.154335i
$306$	6.42864			0.367501
$307$			0.653858i			0.0373177i	0.999826	+	0.0186588i	$0.00593964\pi$
$307$			0.653858i			0.0373177i	−0.999826	+	0.0186588i	$0.994060\pi$
$308$		−	2.75557i		−	0.157013i
$309$	12.2351			0.696028
$310$	−1.24443	−	8.85728i	−0.0706789	−	0.503060i
$311$	−13.0923			−0.742399			−0.371199	−	0.928553i	$-0.621053\pi$
$311$	−13.0923			−0.742399			−0.371199	+	0.928553i	$0.621053\pi$
$312$		−	1.37778i		−	0.0780017i
$313$			11.2573i			0.636302i	0.948040	+	0.318151i	$0.103062\pi$
$313$			11.2573i			0.636302i	−0.948040	+	0.318151i	$0.896938\pi$
$314$	9.90813			0.559148
$315$	−0.193576	−	1.37778i	−0.0109068	−	0.0776294i
$316$	12.0415			0.677387
$317$			22.4701i			1.26205i	0.775763	+	0.631024i	$0.217365\pi$
$317$			22.4701i			1.26205i	−0.775763	+	0.631024i	$0.782635\pi$
$318$			3.37778i			0.189417i
$319$	18.7556			1.05011
$320$	−2.21432	+	0.311108i	−0.123784	+	0.0173915i
$321$	10.6637			0.595190
$322$			0.622216i			0.0346747i
$323$			8.85728i			0.492832i
$324$	−1.00000			−0.0555556
$325$	−1.89829	−	6.62222i	−0.105298	−	0.367334i
$326$	10.1017			0.559482
$327$			15.1526i			0.837939i
$328$			10.8573i			0.599493i
$329$	7.22570			0.398365
$330$	−9.80642	+	1.37778i	−0.539826	+	0.0758445i
$331$	−29.4479			−1.61860			−0.809300	−	0.587395i	$-0.800153\pi$
$331$	−29.4479			−1.61860			−0.809300	+	0.587395i	$0.800153\pi$
$332$			2.19358i			0.120388i
$333$		−	11.8064i		−	0.646988i
$334$	−1.89829			−0.103870
$335$	3.67307	+	26.1432i	0.200681	+	1.42836i
$336$	−0.622216			−0.0339446
$337$			24.6222i			1.34126i	0.741793	+	0.670629i	$0.233976\pi$
$337$			24.6222i			1.34126i	−0.741793	+	0.670629i	$0.766024\pi$
$338$			11.1017i			0.603853i
$339$	14.4286			0.783656
$340$	−2.00000	−	14.2351i	−0.108465	−	0.772005i
$341$	−17.7146			−0.959297
$342$		−	1.37778i		−	0.0745020i
$343$			8.47013i			0.457344i
$344$	1.05086			0.0566583
$345$	2.21432	−	0.311108i	0.119215	−	0.0167495i
$346$	21.2257			1.14110
$347$		−	26.1017i		−	1.40121i	−0.713548	−	0.700607i	$-0.752913\pi$
$347$		−	26.1017i		−	1.40121i	0.713548	−	0.700607i	$-0.247087\pi$
$348$		−	4.23506i		−	0.227023i
$349$	21.2257			1.13619			0.568093	−	0.822965i	$-0.307681\pi$
$349$	21.2257			1.13619			0.568093	+	0.822965i	$0.307681\pi$
$350$	−2.99063	+	0.857279i	−0.159856	+	0.0458235i
$351$	−1.37778			−0.0735407
$352$			4.42864i			0.236047i
$353$		−	4.28544i		−	0.228091i	−0.993476	−	0.114046i	$-0.963619\pi$
$353$		−	4.28544i		−	0.228091i	0.993476	−	0.114046i	$-0.0363810\pi$
$354$	9.61285			0.510917
$355$	−15.4795	+	2.17484i	−0.821566	+	0.115429i
$356$	2.75557			0.146045
$357$		−	4.00000i		−	0.211702i
$358$		−	0.488863i		−	0.0258372i
$359$	4.65386			0.245621			0.122811	−	0.992430i	$-0.460809\pi$
$359$	4.65386			0.245621			0.122811	+	0.992430i	$0.460809\pi$
$360$	0.311108	+	2.21432i	0.0163968	+	0.116705i
$361$	−17.1017			−0.900090
$362$		−	20.9304i		−	1.10008i
$363$			8.61285i			0.452057i
$364$	−0.857279			−0.0449336
$365$	−0.857279	−	6.10171i	−0.0448720	−	0.319378i
$366$	−8.66370			−0.452859
$367$		−	4.88892i		−	0.255200i	−0.991826	−	0.127600i	$-0.959273\pi$
$367$		−	4.88892i		−	0.255200i	0.991826	−	0.127600i	$-0.0407273\pi$
$368$		−	1.00000i		−	0.0521286i
$369$	10.8573			0.565207
$370$	−26.1432	+	3.67307i	−1.35912	+	0.190954i
$371$	2.10171			0.109115
$372$			4.00000i			0.207390i
$373$			23.4193i			1.21260i	0.795235	+	0.606302i	$0.207348\pi$
$373$			23.4193i			1.21260i	−0.795235	+	0.606302i	$0.792652\pi$
$374$	−28.4701			−1.47216
$375$	4.54617	+	10.2143i	0.234763	+	0.527465i
$376$	−11.6128			−0.598887
$377$		−	5.83500i		−	0.300518i
$378$			0.622216i			0.0320033i
$379$	4.90766			0.252089			0.126045	−	0.992025i	$-0.459772\pi$
$379$	4.90766			0.252089			0.126045	+	0.992025i	$0.459772\pi$
$380$	−3.05086	+	0.428639i	−0.156506	+	0.0219887i
$381$	4.99063			0.255678
$382$			18.9590i			0.970026i
$383$		−	19.8796i		−	1.01580i	−0.861417	−	0.507899i	$-0.830422\pi$
$383$		−	19.8796i		−	1.01580i	0.861417	−	0.507899i	$-0.169578\pi$
$384$	1.00000			0.0510310
$385$	0.857279	+	6.10171i	0.0436910	+	0.310972i
$386$	−1.24443			−0.0633399
$387$		−	1.05086i		−	0.0534180i
$388$			2.13335i			0.108305i
$389$	0.161933			0.00821034			0.00410517	−	0.999992i	$-0.498693\pi$
$389$	0.161933			0.00821034			0.00410517	+	0.999992i	$0.498693\pi$
$390$	0.428639	+	3.05086i	0.0217050	+	0.154486i
$391$	6.42864			0.325110
$392$		−	6.61285i		−	0.333999i
$393$			0.755569i			0.0381134i
$394$	15.2444			0.768003
$395$	−26.6637	+	3.74620i	−1.34160	+	0.188492i
$396$	4.42864			0.222548
$397$		−	1.76494i		−	0.0885796i	−0.999019	−	0.0442898i	$-0.985898\pi$
$397$		−	1.76494i		−	0.0885796i	0.999019	−	0.0442898i	$-0.0141025\pi$
$398$			14.5303i			0.728341i
$399$	−0.857279			−0.0429176
$400$	4.80642	−	1.37778i	0.240321	−	0.0688892i
$401$	−31.3461			−1.56535			−0.782676	−	0.622430i	$-0.786146\pi$
$401$	−31.3461			−1.56535			−0.782676	+	0.622430i	$0.786146\pi$
$402$		−	11.8064i		−	0.588851i
$403$			5.51114i			0.274529i
$404$	−16.2351			−0.807725
$405$	2.21432	−	0.311108i	0.110030	−	0.0154591i
$406$	−2.63512			−0.130779
$407$			52.2864i			2.59174i
$408$			6.42864i			0.318265i
$409$	3.51114			0.173615			0.0868073	−	0.996225i	$-0.472334\pi$
$409$	3.51114			0.173615			0.0868073	+	0.996225i	$0.472334\pi$
$410$	−3.37778	−	24.0415i	−0.166817	−	1.18732i
$411$	−12.9175			−0.637173
$412$			12.2351i			0.602778i
$413$		−	5.98126i		−	0.294319i
$414$	−1.00000			−0.0491473
$415$	−0.682439	−	4.85728i	−0.0334996	−	0.238434i
$416$	1.37778			0.0675514
$417$			14.1017i			0.690564i
$418$			6.10171i			0.298444i
$419$	19.7748			0.966061			0.483031	−	0.875603i	$-0.339536\pi$
$419$	19.7748			0.966061			0.483031	+	0.875603i	$0.339536\pi$
$420$	1.37778	−	0.193576i	0.0672290	−	0.00944555i
$421$	11.8064			0.575410			0.287705	−	0.957719i	$-0.407108\pi$
$421$	11.8064			0.575410			0.287705	+	0.957719i	$0.407108\pi$
$422$			0.266706i			0.0129831i
$423$			11.6128i			0.564636i
$424$	−3.37778			−0.164040
$425$	8.85728	+	30.8988i	0.429641	+	1.49881i
$426$	6.99063			0.338697
$427$			5.39069i			0.260874i
$428$			10.6637i			0.515450i
$429$	6.10171			0.294593
$430$	−2.32693	+	0.326929i	−0.112214	+	0.0157659i
$431$	24.9403			1.20133			0.600665	−	0.799501i	$-0.294903\pi$
$431$	24.9403			1.20133			0.600665	+	0.799501i	$0.294903\pi$
$432$		−	1.00000i		−	0.0481125i
$433$		−	32.0513i		−	1.54029i	−0.637870	−	0.770144i	$-0.720184\pi$
$433$		−	32.0513i		−	1.54029i	0.637870	−	0.770144i	$-0.279816\pi$
$434$	2.48886			0.119469
$435$	1.31756	+	9.37778i	0.0631722	+	0.449630i
$436$	−15.1526			−0.725676
$437$		−	1.37778i		−	0.0659084i
$438$			2.75557i			0.131666i
$439$	−0.470127			−0.0224379			−0.0112190	−	0.999937i	$-0.503571\pi$
$439$	−0.470127			−0.0224379			−0.0112190	+	0.999937i	$0.503571\pi$
$440$	−1.37778	−	9.80642i	−0.0656833	−	0.467503i
$441$	−6.61285			−0.314898
$442$			8.85728i			0.421298i
$443$		−	0.653858i		−	0.0310658i	−0.999879	−	0.0155329i	$-0.995056\pi$
$443$		−	0.653858i		−	0.0310658i	0.999879	−	0.0155329i	$-0.00494447\pi$
$444$	11.8064			0.560308
$445$	−6.10171	+	0.857279i	−0.289249	+	0.0406389i
$446$	−0.990632			−0.0469078
$447$		−	12.4286i		−	0.587854i
$448$		−	0.622216i		−	0.0293969i
$449$	−17.1427			−0.809015			−0.404508	−	0.914535i	$-0.632557\pi$
$449$	−17.1427			−0.809015			−0.404508	+	0.914535i	$0.632557\pi$
$450$	−1.37778	−	4.80642i	−0.0649494	−	0.226577i
$451$	−48.0830			−2.26414
$452$			14.4286i			0.678666i
$453$		−	4.85728i		−	0.228215i
$454$	17.1526			0.805010
$455$	1.89829	−	0.266706i	0.0889932	−	0.0125034i
$456$	1.37778			0.0645207
$457$		−	1.00937i		−	0.0472162i	−0.999721	−	0.0236081i	$-0.992485\pi$
$457$		−	1.00937i		−	0.0472162i	0.999721	−	0.0236081i	$-0.00751540\pi$
$458$		−	7.15257i		−	0.334217i
$459$	6.42864			0.300063
$460$	0.311108	+	2.21432i	0.0145055	+	0.103243i
$461$	20.3180			0.946305			0.473153	−	0.880980i	$-0.343116\pi$
$461$	20.3180			0.946305			0.473153	+	0.880980i	$0.343116\pi$
$462$		−	2.75557i		−	0.128201i
$463$			12.4572i			0.578936i	0.957188	+	0.289468i	$0.0934784\pi$
$463$			12.4572i			0.578936i	−0.957188	+	0.289468i	$0.906522\pi$
$464$	4.23506			0.196608
$465$	−1.24443	−	8.85728i	−0.0577091	−	0.410746i
$466$	−3.71456			−0.172074
$467$			15.7877i			0.730567i	0.930896	+	0.365284i	$0.119028\pi$
$467$			15.7877i			0.730567i	−0.930896	+	0.365284i	$0.880972\pi$
$468$		−	1.37778i		−	0.0636881i
$469$	−7.34614			−0.339213
$470$	25.7146	−	3.61285i	1.18612	−	0.166648i
$471$	9.90813			0.456543
$472$			9.61285i			0.442467i
$473$			4.65386i			0.213985i
$474$	12.0415			0.553084
$475$	6.62222	−	1.89829i	0.303848	−	0.0870995i
$476$	4.00000			0.183340
$477$			3.37778i			0.154658i
$478$		−	27.8479i		−	1.27373i
$479$	−5.12399			−0.234121			−0.117060	−	0.993125i	$-0.537347\pi$
$479$	−5.12399			−0.234121			−0.117060	+	0.993125i	$0.537347\pi$
$480$	−2.21432	+	0.311108i	−0.101069	+	0.0142001i
$481$	16.2667			0.741698
$482$		−	7.12399i		−	0.324489i
$483$			0.622216i			0.0283118i
$484$	−8.61285			−0.391493
$485$	−0.663703	−	4.72393i	−0.0301372	−	0.214502i
$486$	−1.00000			−0.0453609
$487$			23.2128i			1.05187i	0.850524	+	0.525936i	$0.176285\pi$
$487$			23.2128i			1.05187i	−0.850524	+	0.525936i	$0.823715\pi$
$488$		−	8.66370i		−	0.392187i
$489$	10.1017			0.456815
$490$	2.05731	+	14.6430i	0.0929397	+	0.661501i
$491$	−40.1847			−1.81351			−0.906755	−	0.421658i	$-0.861448\pi$
$491$	−40.1847			−1.81351			−0.906755	+	0.421658i	$0.861448\pi$
$492$			10.8573i			0.489484i
$493$			27.2257i			1.22618i
$494$	1.89829			0.0854081
$495$	−9.80642	+	1.37778i	−0.440766	+	0.0619268i
$496$	−4.00000			−0.179605
$497$		−	4.34968i		−	0.195110i
$498$			2.19358i			0.0982965i
$499$	−30.5718			−1.36858			−0.684292	−	0.729208i	$-0.739888\pi$
$499$	−30.5718			−1.36858			−0.684292	+	0.729208i	$0.739888\pi$
$500$	−10.2143	+	4.54617i	−0.456798	+	0.203311i
$501$	−1.89829			−0.0848093
$502$		−	22.4099i		−	1.00020i
$503$		−	23.4291i		−	1.04465i	−0.852746	−	0.522326i	$-0.825064\pi$
$503$		−	23.4291i		−	1.04465i	0.852746	−	0.522326i	$-0.174936\pi$
$504$	−0.622216			−0.0277157
$505$	35.9496	−	5.05086i	1.59974	−	0.224760i
$506$	4.42864			0.196877
$507$			11.1017i			0.493044i
$508$			4.99063i			0.221423i
$509$	17.2128			0.762943			0.381472	−	0.924381i	$-0.375417\pi$
$509$	17.2128			0.762943			0.381472	+	0.924381i	$0.375417\pi$
$510$	−2.00000	−	14.2351i	−0.0885615	−	0.630339i
$511$	1.71456			0.0758476
$512$			1.00000i			0.0441942i
$513$		−	1.37778i		−	0.0608307i
$514$	−11.7146			−0.516707
$515$	−3.80642	−	27.0923i	−0.167731	−	1.19383i
$516$	1.05086			0.0462613
$517$		−	51.4291i		−	2.26185i
$518$		−	7.34614i		−	0.322771i
$519$	21.2257			0.931705
$520$	−3.05086	+	0.428639i	−0.133789	+	0.0187971i
$521$	22.3684			0.979978			0.489989	−	0.871729i	$-0.337001\pi$
$521$	22.3684			0.979978			0.489989	+	0.871729i	$0.337001\pi$
$522$		−	4.23506i		−	0.185364i
$523$			30.2953i			1.32472i	0.749186	+	0.662360i	$0.230445\pi$
$523$			30.2953i			1.32472i	−0.749186	+	0.662360i	$0.769555\pi$
$524$	−0.755569			−0.0330072
$525$	−2.99063	+	0.857279i	−0.130522	+	0.0374147i
$526$	8.47013			0.369315
$527$		−	25.7146i		−	1.12014i
$528$			4.42864i			0.192732i
$529$	−1.00000			−0.0434783
$530$	7.47949	−	1.05086i	0.324888	−	0.0456462i
$531$	9.61285			0.417162
$532$		−	0.857279i		−	0.0371678i
$533$			14.9590i			0.647946i
$534$	2.75557			0.119245
$535$	−3.31756	−	23.6128i	−0.143431	−	1.02087i
$536$	11.8064			0.509960
$537$		−	0.488863i		−	0.0210960i
$538$		−	28.8256i		−	1.24276i
$539$	29.2859			1.26143
$540$	0.311108	+	2.21432i	0.0133879	+	0.0952891i
$541$	31.4479			1.35205			0.676024	−	0.736879i	$-0.263701\pi$
$541$	31.4479			1.35205			0.676024	+	0.736879i	$0.263701\pi$
$542$		−	13.8350i		−	0.594264i
$543$		−	20.9304i		−	0.898210i
$544$	−6.42864			−0.275626
$545$	33.5526	−	4.71408i	1.43724	−	0.201929i
$546$	−0.857279			−0.0366882
$547$		−	20.8573i		−	0.891793i	−0.895085	−	0.445896i	$-0.852885\pi$
$547$		−	20.8573i		−	0.891793i	0.895085	−	0.445896i	$-0.147115\pi$
$548$		−	12.9175i		−	0.551808i
$549$	−8.66370			−0.369758
$550$	6.10171	+	21.2859i	0.260178	+	0.907635i
$551$	5.83500			0.248580
$552$		−	1.00000i		−	0.0425628i
$553$		−	7.49240i		−	0.318609i
$554$	2.88892			0.122739
$555$	−26.1432	+	3.67307i	−1.10972	+	0.155913i
$556$	−14.1017			−0.598046
$557$		−	36.3180i		−	1.53884i	−0.638740	−	0.769422i	$-0.720544\pi$
$557$		−	36.3180i		−	1.53884i	0.638740	−	0.769422i	$-0.279456\pi$
$558$			4.00000i			0.169334i
$559$	1.44785			0.0612376
$560$	0.193576	+	1.37778i	0.00818009	+	0.0582220i
$561$	−28.4701			−1.20201
$562$			5.51114i			0.232473i
$563$			2.58073i			0.108765i	0.998520	+	0.0543824i	$0.0173190\pi$
$563$			2.58073i			0.108765i	−0.998520	+	0.0543824i	$0.982681\pi$
$564$	−11.6128			−0.488989
$565$	−4.48886	−	31.9496i	−0.188848	−	1.34413i
$566$	−8.19358			−0.344402
$567$			0.622216i			0.0261306i
$568$			6.99063i			0.293320i
$569$	36.3497			1.52386			0.761929	−	0.647661i	$-0.224253\pi$
$569$	36.3497			1.52386			0.761929	+	0.647661i	$0.224253\pi$
$570$	−3.05086	+	0.428639i	−0.127786	+	0.0179537i
$571$	−45.9309			−1.92215			−0.961074	−	0.276292i	$-0.910894\pi$
$571$	−45.9309			−1.92215			−0.961074	+	0.276292i	$0.910894\pi$
$572$			6.10171i			0.255125i
$573$			18.9590i			0.792023i
$574$	6.75557			0.281972
$575$	−1.37778	−	4.80642i	−0.0574576	−	0.200442i
$576$	1.00000			0.0416667
$577$		−	4.00000i		−	0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$		−	4.00000i		−	0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$		−	24.3274i		−	1.01189i
$579$	−1.24443			−0.0517168
$580$	−9.37778	+	1.31756i	−0.389391	+	0.0547088i
$581$	1.36488			0.0566247
$582$			2.13335i			0.0884303i
$583$		−	14.9590i		−	0.619538i
$584$	−2.75557			−0.114026
$585$	0.428639	+	3.05086i	0.0177221	+	0.126137i
$586$	−12.6222			−0.521419
$587$		−	42.1847i		−	1.74115i	−0.492037	−	0.870574i	$-0.663748\pi$
$587$		−	42.1847i		−	1.74115i	0.492037	−	0.870574i	$-0.336252\pi$
$588$		−	6.61285i		−	0.272709i
$589$	−5.51114			−0.227082
$590$	−2.99063	−	21.2859i	−0.123122	−	0.876327i
$591$	15.2444			0.627072
$592$			11.8064i			0.485241i
$593$			18.7368i			0.769430i	0.923036	+	0.384715i	$0.125700\pi$
$593$			18.7368i			0.769430i	−0.923036	+	0.384715i	$0.874300\pi$
$594$	4.42864			0.181709
$595$	−8.85728	+	1.24443i	−0.363113	+	0.0510167i
$596$	12.4286			0.509097
$597$			14.5303i			0.594688i
$598$		−	1.37778i		−	0.0563418i
$599$	41.5625			1.69820			0.849098	−	0.528235i	$-0.177146\pi$
$599$	41.5625			1.69820			0.849098	+	0.528235i	$0.177146\pi$
$600$	4.80642	−	1.37778i	0.196221	−	0.0562478i
$601$	23.7146			0.967337			0.483668	−	0.875251i	$-0.339304\pi$
$601$	23.7146			0.967337			0.483668	+	0.875251i	$0.339304\pi$
$602$		−	0.653858i		−	0.0266493i
$603$		−	11.8064i		−	0.480795i
$604$	4.85728			0.197640
$605$	19.0716	−	2.67952i	0.775371	−	0.108938i
$606$	−16.2351			−0.659504
$607$		−	2.74266i		−	0.111321i	−0.998450	−	0.0556606i	$-0.982274\pi$
$607$		−	2.74266i		−	0.111321i	0.998450	−	0.0556606i	$-0.0177265\pi$
$608$			1.37778i			0.0558765i
$609$	−2.63512			−0.106781
$610$	2.69535	+	19.1842i	0.109131	+	0.776746i
$611$	−16.0000			−0.647291
$612$			6.42864i			0.259862i
$613$		−	24.0731i		−	0.972305i	−0.873874	−	0.486152i	$-0.838400\pi$
$613$		−	24.0731i		−	0.972305i	0.873874	−	0.486152i	$-0.161600\pi$
$614$	−0.653858			−0.0263876
$615$	−3.37778	−	24.0415i	−0.136205	−	0.969446i
$616$	2.75557			0.111025
$617$			39.4893i			1.58978i	0.606753	+	0.794890i	$0.292472\pi$
$617$			39.4893i			1.58978i	−0.606753	+	0.794890i	$0.707528\pi$
$618$			12.2351i			0.492166i
$619$	22.6222			0.909264			0.454632	−	0.890679i	$-0.349771\pi$
$619$	22.6222			0.909264			0.454632	+	0.890679i	$0.349771\pi$
$620$	8.85728	−	1.24443i	0.355717	−	0.0499776i
$621$	−1.00000			−0.0401286
$622$		−	13.0923i		−	0.524955i
$623$		−	1.71456i		−	0.0686923i
$624$	1.37778			0.0551555
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$	−11.2573			−0.449934
$627$			6.10171i			0.243679i
$628$			9.90813i			0.395378i
$629$	−75.8992			−3.02630
$630$	1.37778	−	0.193576i	0.0548922	−	0.00771226i
$631$	5.93978			0.236459			0.118229	−	0.992986i	$-0.462278\pi$
$631$	5.93978			0.236459			0.118229	+	0.992986i	$0.462278\pi$
$632$			12.0415i			0.478985i
$633$			0.266706i			0.0106006i
$634$	−22.4701			−0.892403
$635$	−1.55262	−	11.0509i	−0.0616140	−	0.438540i
$636$	−3.37778			−0.133938
$637$		−	9.11108i		−	0.360994i
$638$			18.7556i			0.742540i
$639$	6.99063			0.276545
$640$	−0.311108	−	2.21432i	−0.0122976	−	0.0875287i
$641$	19.8163			0.782696			0.391348	−	0.920243i	$-0.372009\pi$
$641$	19.8163			0.782696			0.391348	+	0.920243i	$0.372009\pi$
$642$			10.6637i			0.420863i
$643$			44.7467i			1.76464i	0.470653	+	0.882318i	$0.344018\pi$
$643$			44.7467i			1.76464i	−0.470653	+	0.882318i	$0.655982\pi$
$644$	−0.622216			−0.0245187
$645$	−2.32693	+	0.326929i	−0.0916227	+	0.0128728i
$646$	−8.85728			−0.348485
$647$		−	32.9403i		−	1.29501i	−0.762059	−	0.647507i	$-0.775811\pi$
$647$		−	32.9403i		−	1.29501i	0.762059	−	0.647507i	$-0.224189\pi$
$648$		−	1.00000i		−	0.0392837i
$649$	−42.5718			−1.67109
$650$	6.62222	−	1.89829i	0.259745	−	0.0744571i
$651$	2.48886			0.0975462
$652$			10.1017i			0.395614i
$653$			37.2257i			1.45675i	0.685177	+	0.728377i	$0.259725\pi$
$653$			37.2257i			1.45675i	−0.685177	+	0.728377i	$0.740275\pi$
$654$	−15.1526			−0.592512
$655$	1.67307	−	0.235063i	0.0653723	−	0.00918468i
$656$	−10.8573			−0.423906
$657$			2.75557i			0.107505i
$658$			7.22570i			0.281687i
$659$	−24.6953			−0.961994			−0.480997	−	0.876722i	$-0.659725\pi$
$659$	−24.6953			−0.961994			−0.480997	+	0.876722i	$0.659725\pi$
$660$	−1.37778	−	9.80642i	−0.0536302	−	0.381715i
$661$	−14.8287			−0.576770			−0.288385	−	0.957515i	$-0.593118\pi$
$661$	−14.8287			−0.576770			−0.288385	+	0.957515i	$0.593118\pi$
$662$		−	29.4479i		−	1.14452i
$663$			8.85728i			0.343988i
$664$	−2.19358			−0.0851273
$665$	0.266706	+	1.89829i	0.0103424	+	0.0736125i
$666$	11.8064			0.457490
$667$		−	4.23506i		−	0.163982i
$668$		−	1.89829i		−	0.0734470i
$669$	−0.990632			−0.0383000
$670$	−26.1432	+	3.67307i	−1.01000	+	0.141903i
$671$	38.3684			1.48120
$672$		−	0.622216i		−	0.0240025i
$673$			8.77430i			0.338225i	0.985597	+	0.169112i	$0.0540901\pi$
$673$			8.77430i			0.338225i	−0.985597	+	0.169112i	$0.945910\pi$
$674$	−24.6222			−0.948412
$675$	−1.37778	−	4.80642i	−0.0530309	−	0.184999i
$676$	−11.1017			−0.426989
$677$			12.8256i			0.492929i	0.969152	+	0.246465i	$0.0792689\pi$
$677$			12.8256i			0.492929i	−0.969152	+	0.246465i	$0.920731\pi$
$678$			14.4286i			0.554129i
$679$	1.32741			0.0509412
$680$	14.2351	−	2.00000i	0.545890	−	0.0766965i
$681$	17.1526			0.657288
$682$		−	17.7146i		−	0.678325i
$683$			8.47013i			0.324100i	0.986783	+	0.162050i	$0.0518106\pi$
$683$			8.47013i			0.324100i	−0.986783	+	0.162050i	$0.948189\pi$
$684$	1.37778			0.0526809
$685$	4.01874	+	28.6035i	0.153548	+	1.09288i
$686$	−8.47013			−0.323391
$687$		−	7.15257i		−	0.272887i
$688$			1.05086i			0.0400635i
$689$	−4.65386			−0.177298
$690$	0.311108	+	2.21432i	0.0118437	+	0.0842977i
$691$	25.7975			0.981384			0.490692	−	0.871333i	$-0.336744\pi$
$691$	25.7975			0.981384			0.490692	+	0.871333i	$0.336744\pi$
$692$			21.2257i			0.806880i
$693$		−	2.75557i		−	0.104675i
$694$	26.1017			0.990807
$695$	31.2257	−	4.38715i	1.18446	−	0.166414i
$696$	4.23506			0.160530
$697$		−	69.7975i		−	2.64377i
$698$			21.2257i			0.803404i
$699$	−3.71456			−0.140497
$700$	−0.857279	−	2.99063i	−0.0324021	−	0.113035i
$701$	7.45091			0.281417			0.140709	−	0.990051i	$-0.455062\pi$
$701$	7.45091			0.281417			0.140709	+	0.990051i	$0.455062\pi$
$702$		−	1.37778i		−	0.0520011i
$703$			16.2667i			0.613510i
$704$	−4.42864			−0.166911
$705$	25.7146	−	3.61285i	0.968466	−	0.136068i
$706$	4.28544			0.161285
$707$			10.1017i			0.379914i
$708$			9.61285i			0.361273i
$709$	−13.2543			−0.497775			−0.248887	−	0.968532i	$-0.580065\pi$
$709$	−13.2543			−0.497775			−0.248887	+	0.968532i	$0.580065\pi$
$710$	−2.17484	−	15.4795i	−0.0816203	−	0.580935i
$711$	12.0415			0.451591
$712$			2.75557i			0.103269i
$713$			4.00000i			0.149801i
$714$	4.00000			0.149696
$715$	−1.89829	−	13.5111i	−0.0709920	−	0.505288i
$716$	0.488863			0.0182697
$717$		−	27.8479i		−	1.04000i
$718$			4.65386i			0.173680i
$719$	2.33677			0.0871469			0.0435735	−	0.999050i	$-0.486126\pi$
$719$	2.33677			0.0871469			0.0435735	+	0.999050i	$0.486126\pi$
$720$	−2.21432	+	0.311108i	−0.0825228	+	0.0115943i
$721$	7.61285			0.283517
$722$		−	17.1017i		−	0.636460i
$723$		−	7.12399i		−	0.264944i
$724$	20.9304			0.777873
$725$	20.3555	−	5.83500i	0.755985	−	0.216707i
$726$	−8.61285			−0.319653
$727$			3.37778i			0.125275i	0.998036	+	0.0626375i	$0.0199512\pi$
$727$			3.37778i			0.125275i	−0.998036	+	0.0626375i	$0.980049\pi$
$728$		−	0.857279i		−	0.0317729i
$729$	−1.00000			−0.0370370
$730$	6.10171	−	0.857279i	0.225834	−	0.0317293i
$731$	−6.75557			−0.249864
$732$		−	8.66370i		−	0.320220i
$733$			21.0509i			0.777531i	0.921337	+	0.388766i	$0.127098\pi$
$733$			21.0509i			0.777531i	−0.921337	+	0.388766i	$0.872902\pi$
$734$	4.88892			0.180453
$735$	2.05731	+	14.6430i	0.0758850	+	0.540114i
$736$	1.00000			0.0368605
$737$			52.2864i			1.92599i
$738$			10.8573i			0.399662i
$739$	22.6351			0.832646			0.416323	−	0.909217i	$-0.363318\pi$
$739$	22.6351			0.832646			0.416323	+	0.909217i	$0.363318\pi$
$740$	−3.67307	−	26.1432i	−0.135025	−	0.961043i
$741$	1.89829			0.0697354
$742$			2.10171i			0.0771562i
$743$		−	3.87955i		−	0.142327i	−0.997465	−	0.0711635i	$-0.977329\pi$
$743$		−	3.87955i		−	0.142327i	0.997465	−	0.0711635i	$-0.0226712\pi$
$744$	−4.00000			−0.146647
$745$	−27.5210	+	3.86665i	−1.00829	+	0.141663i
$746$	−23.4193			−0.857440
$747$			2.19358i			0.0802588i
$748$		−	28.4701i		−	1.04097i
$749$	6.63512			0.242442
$750$	−10.2143	+	4.54617i	−0.372974	+	0.166003i
$751$	19.7748			0.721592			0.360796	−	0.932645i	$-0.382505\pi$
$751$	19.7748			0.721592			0.360796	+	0.932645i	$0.382505\pi$
$752$		−	11.6128i		−	0.423477i
$753$		−	22.4099i		−	0.816662i
$754$	5.83500			0.212498
$755$	−10.7556	+	1.51114i	−0.391435	+	0.0549959i
$756$	−0.622216			−0.0226298
$757$		−	8.19358i		−	0.297801i	−0.988852	−	0.148900i	$-0.952427\pi$
$757$		−	8.19358i		−	0.297801i	0.988852	−	0.148900i	$-0.0475733\pi$
$758$			4.90766i			0.178254i
$759$	4.42864			0.160749
$760$	−0.428639	−	3.05086i	−0.0155484	−	0.110666i
$761$	−45.1624			−1.63714			−0.818568	−	0.574410i	$-0.805232\pi$
$761$	−45.1624			−1.63714			−0.818568	+	0.574410i	$0.805232\pi$
$762$			4.99063i			0.180792i
$763$			9.42816i			0.341322i
$764$	−18.9590			−0.685912
$765$	−2.00000	−	14.2351i	−0.0723102	−	0.514670i
$766$	19.8796			0.718277
$767$			13.2444i			0.478229i
$768$			1.00000i			0.0360844i
$769$	−29.4924			−1.06352			−0.531762	−	0.846894i	$-0.678470\pi$
$769$	−29.4924			−1.06352			−0.531762	+	0.846894i	$0.678470\pi$
$770$	−6.10171	+	0.857279i	−0.219890	+	0.0308942i
$771$	−11.7146			−0.421889
$772$		−	1.24443i		−	0.0447881i
$773$		−	41.0923i		−	1.47799i	−0.673712	−	0.738994i	$-0.735301\pi$
$773$		−	41.0923i		−	1.47799i	0.673712	−	0.738994i	$-0.264699\pi$
$774$	1.05086			0.0377722
$775$	−19.2257	+	5.51114i	−0.690607	+	0.197966i
$776$	−2.13335			−0.0765829
$777$		−	7.34614i		−	0.263541i
$778$			0.161933i			0.00580559i
$779$	−14.9590			−0.535961
$780$	−3.05086	+	0.428639i	−0.109238	+	0.0153478i
$781$	−30.9590			−1.10780
$782$			6.42864i			0.229888i
$783$		−	4.23506i		−	0.151349i
$784$	6.61285			0.236173
$785$	−3.08250	−	21.9398i	−0.110019	−	0.783064i
$786$	−0.755569			−0.0269502
$787$		−	10.0919i		−	0.359736i	−0.983691	−	0.179868i	$-0.942433\pi$
$787$		−	10.0919i		−	0.359736i	0.983691	−	0.179868i	$-0.0575671\pi$
$788$			15.2444i

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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.21432 - 0.311108i) q^{5} -1.00000 q^{6} +0.622216i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(2.21432 - 0.311108i) q^{5} -1.00000 q^{6} +0.622216i q^{7} -1.00000i q^{8} -1.00000 q^{9} +(0.311108 + 2.21432i) q^{10} +4.42864 q^{11} -1.00000i q^{12} -1.37778i q^{13} -0.622216 q^{14} +(0.311108 + 2.21432i) q^{15} +1.00000 q^{16} +6.42864i q^{17} -1.00000i q^{18} +1.37778 q^{19} +(-2.21432 + 0.311108i) q^{20} -0.622216 q^{21} +4.42864i q^{22} -1.00000i q^{23} +1.00000 q^{24} +(4.80642 - 1.37778i) q^{25} +1.37778 q^{26} -1.00000i q^{27} -0.622216i q^{28} +4.23506 q^{29} +(-2.21432 + 0.311108i) q^{30} -4.00000 q^{31} +1.00000i q^{32} +4.42864i q^{33} -6.42864 q^{34} +(0.193576 + 1.37778i) q^{35} +1.00000 q^{36} +11.8064i q^{37} +1.37778i q^{38} +1.37778 q^{39} +(-0.311108 - 2.21432i) q^{40} -10.8573 q^{41} -0.622216i q^{42} +1.05086i q^{43} -4.42864 q^{44} +(-2.21432 + 0.311108i) q^{45} +1.00000 q^{46} -11.6128i q^{47} +1.00000i q^{48} +6.61285 q^{49} +(1.37778 + 4.80642i) q^{50} -6.42864 q^{51} +1.37778i q^{52} -3.37778i q^{53} +1.00000 q^{54} +(9.80642 - 1.37778i) q^{55} +0.622216 q^{56} +1.37778i q^{57} +4.23506i q^{58} -9.61285 q^{59} +(-0.311108 - 2.21432i) q^{60} +8.66370 q^{61} -4.00000i q^{62} -0.622216i q^{63} -1.00000 q^{64} +(-0.428639 - 3.05086i) q^{65} -4.42864 q^{66} +11.8064i q^{67} -6.42864i q^{68} +1.00000 q^{69} +(-1.37778 + 0.193576i) q^{70} -6.99063 q^{71} +1.00000i q^{72} -2.75557i q^{73} -11.8064 q^{74} +(1.37778 + 4.80642i) q^{75} -1.37778 q^{76} +2.75557i q^{77} +1.37778i q^{78} -12.0415 q^{79} +(2.21432 - 0.311108i) q^{80} +1.00000 q^{81} -10.8573i q^{82} -2.19358i q^{83} +0.622216 q^{84} +(2.00000 + 14.2351i) q^{85} -1.05086 q^{86} +4.23506i q^{87} -4.42864i q^{88} -2.75557 q^{89} +(-0.311108 - 2.21432i) q^{90} +0.857279 q^{91} +1.00000i q^{92} -4.00000i q^{93} +11.6128 q^{94} +(3.05086 - 0.428639i) q^{95} -1.00000 q^{96} -2.13335i q^{97} +6.61285i q^{98} -4.42864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6q - 6q^{4} - 6q^{6} - 6q^{9} + O(q^{10}) \)	\( 6q - 6q^{4} - 6q^{6} - 6q^{9} + 2q^{10} - 4q^{14} + 2q^{15} + 6q^{16} + 8q^{19} - 4q^{21} + 6q^{24} + 2q^{25} + 8q^{26} - 28q^{29} - 24q^{31} - 12q^{34} + 28q^{35} + 6q^{36} + 8q^{39} - 2q^{40} - 12q^{41} + 6q^{46} - 14q^{49} + 8q^{50} - 12q^{51} + 6q^{54} + 32q^{55} + 4q^{56} - 4q^{59} - 2q^{60} - 28q^{61} - 6q^{64} + 24q^{65} + 6q^{69} - 8q^{70} + 12q^{71} - 44q^{74} + 8q^{75} - 8q^{76} + 8q^{79} + 6q^{81} + 4q^{84} + 12q^{85} + 20q^{86} - 16q^{89} - 2q^{90} - 48q^{91} + 16q^{94} - 8q^{95} - 6q^{96} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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