LMFDB - Embedded newform 1323.2.g.g.361.5

Newspace parameters

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.g (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
Defining polynomial:	$x^{12} - 7 x^{10} + 37 x^{8} - 78 x^{6} + 123 x^{4} - 36 x^{2} + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{5} $
Twist minimal:	no (minimal twist has level 441)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.5
Root			$1.82904 - 1.05600i$ of defining polynomial
Character	$\chi$	$=$	1323.361
Dual form			1323.2.g.g.667.5

$q$-expansion

$f(q)$	$=$	$q+(1.23025 - 2.13086i) q^{2} +(-2.02704 - 3.51094i) q^{4} -3.65808 q^{5} -5.05408 q^{8} +O(q^{10})$	\(q+(1.23025 - 2.13086i) q^{2} +(-2.02704 - 3.51094i) q^{4} -3.65808 q^{5} -5.05408 q^{8} +(-4.50036 + 7.79485i) q^{10} -0.406421 q^{11} +(-0.243398 + 0.421578i) q^{13} +(-2.16372 + 3.74766i) q^{16} +(-2.42792 + 4.20528i) q^{17} +(-0.986757 - 1.70911i) q^{19} +(7.41507 + 12.8433i) q^{20} +(-0.500000 + 0.866025i) q^{22} -4.64766 q^{23} +8.38151 q^{25} +(0.598883 + 1.03729i) q^{26} +(3.82383 + 6.62307i) q^{29} +(3.51360 + 6.08573i) q^{31} +(0.269748 + 0.467216i) q^{32} +(5.97391 + 10.3471i) q^{34} +(-1.16372 - 2.01561i) q^{37} -4.85584 q^{38} +18.4882 q^{40} +(-3.75700 + 6.50731i) q^{41} +(1.16372 + 2.01561i) q^{43} +(0.823832 + 1.42692i) q^{44} +(-5.71780 + 9.90352i) q^{46} +(3.15811 - 5.47002i) q^{47} +(10.3114 - 17.8598i) q^{50} +1.97351 q^{52} +(-1.78434 + 3.09056i) q^{53} +1.48672 q^{55} +18.8171 q^{58} +(-3.05919 - 5.29868i) q^{59} +(-4.01356 + 6.95169i) q^{61} +17.2905 q^{62} -7.32743 q^{64} +(0.890369 - 1.54216i) q^{65} +(-1.80039 - 3.11836i) q^{67} +19.6860 q^{68} -8.46050 q^{71} +(0.986757 - 1.70911i) q^{73} -5.72665 q^{74} +(-4.00040 + 6.92889i) q^{76} +(-4.08113 + 7.06872i) q^{79} +(7.91503 - 13.7092i) q^{80} +(9.24411 + 16.0113i) q^{82} +(-6.08600 - 10.5413i) q^{83} +(8.88151 - 15.3832i) q^{85} +5.72665 q^{86} +2.05408 q^{88} +(-7.41507 - 12.8433i) q^{89} +(9.42101 + 16.3177i) q^{92} +(-7.77056 - 13.4590i) q^{94} +(3.60963 + 6.25206i) q^{95} +(-4.74375 - 8.21642i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12q + 2q^{2} - 6q^{4} - 24q^{8} + O(q^{10}) $	$ 12q + 2q^{2} - 6q^{4} - 24q^{8} - 16q^{11} - 6q^{16} - 6q^{22} - 8q^{23} + 24q^{25} + 22q^{29} + 16q^{32} + 6q^{37} - 6q^{43} - 14q^{44} - 12q^{46} + 56q^{50} + 28q^{53} + 36q^{58} - 48q^{64} - 6q^{65} - 76q^{71} - 72q^{74} + 6q^{79} + 30q^{85} + 72q^{86} - 12q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.23025	−	2.13086i	0.869920	−	1.50675i	0.00784213	−	0.999969i	$-0.497504\pi$
$2$	1.23025	−	2.13086i	0.869920	−	1.50675i	0.862078	−	0.506776i	$-0.169163\pi$
$3$		0			0
$3$		0			0
$4$	−2.02704	−	3.51094i	−1.01352	−	1.75547i
$5$	−3.65808			−1.63594			−0.817970	−	0.575260i	$-0.804901\pi$
$5$	−3.65808			−1.63594			−0.817970	+	0.575260i	$0.804901\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−5.05408			−1.78689
$9$		0			0
$10$	−4.50036	+	7.79485i	−1.42314	+	2.46495i
$11$	−0.406421			−0.122540			−0.0612702	−	0.998121i	$-0.519515\pi$
$11$	−0.406421			−0.122540			−0.0612702	+	0.998121i	$0.519515\pi$
$12$		0			0
$13$	−0.243398	+	0.421578i	−0.0675065	+	0.116925i	−0.897803	−	0.440397i	$-0.854838\pi$
$13$	−0.243398	+	0.421578i	−0.0675065	+	0.116925i	0.830297	+	0.557322i	$0.188171\pi$
$14$		0			0
$15$		0			0
$16$	−2.16372	+	3.74766i	−0.540929	+	0.936916i
$17$	−2.42792	+	4.20528i	−0.588857	+	1.01993i	0.405525	+	0.914084i	$0.367089\pi$
$17$	−2.42792	+	4.20528i	−0.588857	+	1.01993i	−0.994382	+	0.105847i	$0.966245\pi$
$18$		0			0
$19$	−0.986757	−	1.70911i	−0.226378	−	0.392097i	0.730354	−	0.683069i	$-0.239355\pi$
$19$	−0.986757	−	1.70911i	−0.226378	−	0.392097i	−0.956732	+	0.290971i	$0.906022\pi$
$20$	7.41507	+	12.8433i	1.65806	+	2.87185i
$21$		0			0
$22$	−0.500000	+	0.866025i	−0.106600	+	0.184637i
$23$	−4.64766			−0.969105			−0.484552	−	0.874762i	$-0.661018\pi$
$23$	−4.64766			−0.969105			−0.484552	+	0.874762i	$0.661018\pi$
$24$		0			0
$25$	8.38151			1.67630
$26$	0.598883	+	1.03729i	0.117451	+	0.203430i
$27$		0			0
$28$		0			0
$29$	3.82383	+	6.62307i	0.710068	+	1.22987i	0.964831	+	0.262870i	$0.0846690\pi$
$29$	3.82383	+	6.62307i	0.710068	+	1.22987i	−0.254764	+	0.967003i	$0.581998\pi$
$30$		0			0
$31$	3.51360	+	6.08573i	0.631061	+	1.09303i	0.987335	+	0.158648i	$0.0507136\pi$
$31$	3.51360	+	6.08573i	0.631061	+	1.09303i	−0.356274	+	0.934381i	$0.615953\pi$
$32$	0.269748	+	0.467216i	0.0476851	+	0.0825930i
$33$		0			0
$34$	5.97391	+	10.3471i	1.02452	+	1.77452i
$35$		0			0
$36$		0			0
$37$	−1.16372	−	2.01561i	−0.191314	−	0.331365i	0.754372	−	0.656447i	$-0.227941\pi$
$37$	−1.16372	−	2.01561i	−0.191314	−	0.331365i	−0.945686	+	0.325082i	$0.894608\pi$
$38$	−4.85584			−0.787721
$39$		0			0
$40$	18.4882			2.92324
$41$	−3.75700	+	6.50731i	−0.586744	+	1.01627i	0.407911	+	0.913022i	$0.366257\pi$
$41$	−3.75700	+	6.50731i	−0.586744	+	1.01627i	−0.994655	+	0.103249i	$0.967076\pi$
$42$		0			0
$43$	1.16372	+	2.01561i	0.177465	+	0.307378i	0.941012	−	0.338374i	$-0.109877\pi$
$43$	1.16372	+	2.01561i	0.177465	+	0.307378i	−0.763547	+	0.645753i	$0.776544\pi$
$44$	0.823832	+	1.42692i	0.124197	+	0.215116i
$45$		0			0
$46$	−5.71780	+	9.90352i	−0.843044	+	1.46019i
$47$	3.15811	−	5.47002i	0.460658	−	0.797884i	−0.538335	−	0.842731i	$-0.680947\pi$
$47$	3.15811	−	5.47002i	0.460658	−	0.797884i	0.998994	+	0.0448469i	$0.0142800\pi$
$48$		0			0
$49$		0			0
$50$	10.3114	−	17.8598i	1.45825	−	2.52576i
$51$		0			0
$52$	1.97351			0.273677
$53$	−1.78434	+	3.09056i	−0.245097	+	0.424521i	−0.962159	−	0.272489i	$-0.912153\pi$
$53$	−1.78434	+	3.09056i	−0.245097	+	0.424521i	0.717062	+	0.697010i	$0.245487\pi$
$54$		0			0
$55$	1.48672			0.200469
$56$		0			0
$57$		0			0
$58$	18.8171			2.47081
$59$	−3.05919	−	5.29868i	−0.398273	−	0.689829i	0.595240	−	0.803548i	$-0.297057\pi$
$59$	−3.05919	−	5.29868i	−0.398273	−	0.689829i	−0.993513	+	0.113719i	$0.963724\pi$
$60$		0			0
$61$	−4.01356	+	6.95169i	−0.513884	+	0.890073i	0.485987	+	0.873966i	$0.338460\pi$
$61$	−4.01356	+	6.95169i	−0.513884	+	0.890073i	−0.999870	+	0.0161063i	$0.994873\pi$
$62$	17.2905			2.19589
$63$		0			0
$64$	−7.32743			−0.915929
$65$	0.890369	−	1.54216i	0.110437	−	0.191282i
$66$		0			0
$67$	−1.80039	−	3.11836i	−0.219952	−	0.380969i	0.734841	−	0.678240i	$-0.237257\pi$
$67$	−1.80039	−	3.11836i	−0.219952	−	0.380969i	−0.954793	+	0.297271i	$0.903924\pi$
$68$	19.6860			2.38728
$69$		0			0
$70$		0			0
$71$	−8.46050			−1.00408			−0.502039	−	0.864845i	$-0.667416\pi$
$71$	−8.46050			−1.00408			−0.502039	+	0.864845i	$0.667416\pi$
$72$		0			0
$73$	0.986757	−	1.70911i	0.115491	−	0.200037i	−0.802485	−	0.596673i	$-0.796489\pi$
$73$	0.986757	−	1.70911i	0.115491	−	0.200037i	0.917976	+	0.396636i	$0.129822\pi$
$74$	−5.72665			−0.665710
$75$		0			0
$76$	−4.00040	+	6.92889i	−0.458877	+	0.794798i
$77$		0			0
$78$		0			0
$79$	−4.08113	+	7.06872i	−0.459163	+	0.795293i	−0.998917	−	0.0465297i	$-0.985184\pi$
$79$	−4.08113	+	7.06872i	−0.459163	+	0.795293i	0.539754	+	0.841823i	$0.318517\pi$
$80$	7.91503	−	13.7092i	0.884928	−	1.53274i
$81$		0			0
$82$	9.24411	+	16.0113i	1.02084	+	1.76815i
$83$	−6.08600	−	10.5413i	−0.668025	−	1.15705i	−0.978456	−	0.206457i	$-0.933807\pi$
$83$	−6.08600	−	10.5413i	−0.668025	−	1.15705i	0.310431	−	0.950596i	$-0.399527\pi$
$84$		0			0
$85$	8.88151	−	15.3832i	0.963336	−	1.66855i
$86$	5.72665			0.617521
$87$		0			0
$88$	2.05408			0.218966
$89$	−7.41507	−	12.8433i	−0.785996	−	1.36139i	−0.928402	−	0.371577i	$-0.878817\pi$
$89$	−7.41507	−	12.8433i	−0.785996	−	1.36139i	0.142406	−	0.989808i	$-0.454516\pi$
$90$		0			0
$91$		0			0
$92$	9.42101	+	16.3177i	0.982208	+	1.70123i
$93$		0			0
$94$	−7.77056	−	13.4590i	−0.801472	−	1.38819i
$95$	3.60963	+	6.25206i	0.370340	+	0.641448i
$96$		0			0
$97$	−4.74375	−	8.21642i	−0.481655	−	0.834251i	0.518123	−	0.855306i	$-0.326631\pi$
$97$	−4.74375	−	8.21642i	−0.481655	−	0.834251i	−0.999778	+	0.0210547i	$0.993298\pi$
$98$		0			0
$99$		0			0
$100$	−16.9897	−	29.4270i	−1.69897	−	2.94270i
$101$	8.71176			0.866852			0.433426	−	0.901189i	$-0.357304\pi$
$101$	8.71176			0.866852			0.433426	+	0.901189i	$0.357304\pi$
$102$		0			0
$103$	−8.02712			−0.790936			−0.395468	−	0.918480i	$-0.629418\pi$
$103$	−8.02712			−0.790936			−0.395468	+	0.918480i	$0.629418\pi$
$104$	1.23016	−	2.13069i	0.120627	−	0.208931i
$105$		0			0
$106$	4.39037	+	7.60434i	0.426430	+	0.738599i
$107$	6.42101	+	11.1215i	0.620742	+	1.07516i	0.989348	+	0.145571i	$0.0465021\pi$
$107$	6.42101	+	11.1215i	0.620742	+	1.07516i	−0.368605	+	0.929586i	$0.620165\pi$
$108$		0			0
$109$	−1.30039	+	2.25234i	−0.124555	+	0.215735i	−0.921559	−	0.388239i	$-0.873084\pi$
$109$	−1.30039	+	2.25234i	−0.124555	+	0.215735i	0.797004	+	0.603974i	$0.206417\pi$
$110$	1.82904	−	3.16799i	0.174392	−	0.302056i
$111$		0			0
$112$		0			0
$113$	−6.97509	+	12.0812i	−0.656162	+	1.13651i	0.325440	+	0.945563i	$0.394488\pi$
$113$	−6.97509	+	12.0812i	−0.656162	+	1.13651i	−0.981601	+	0.190942i	$0.938846\pi$
$114$		0			0
$115$	17.0015			1.58540
$116$	15.5021	−	26.8505i	1.43934	−	2.49301i
$117$		0			0
$118$	−15.0543			−1.38586
$119$		0			0
$120$		0			0
$121$	−10.8348			−0.984984
$122$	9.87538	+	17.1047i	0.894075	+	1.54858i
$123$		0			0
$124$	14.2444	−	24.6721i	1.27919	−	2.21562i
$125$	−12.3698			−1.10639
$126$		0			0
$127$	−15.5438			−1.37929			−0.689643	−	0.724149i	$-0.742233\pi$
$127$	−15.5438			−1.37929			−0.689643	+	0.724149i	$0.742233\pi$
$128$	−9.55408	+	16.5482i	−0.844470	+	1.46266i
$129$		0			0
$130$	−2.19076	−	3.79450i	−0.192142	−	0.332800i
$131$	−8.51392			−0.743864			−0.371932	−	0.928260i	$-0.621305\pi$
$131$	−8.51392			−0.743864			−0.371932	+	0.928260i	$0.621305\pi$
$132$		0			0
$133$		0			0
$134$	−8.85973			−0.765364
$135$		0			0
$136$	12.2709	−	21.2538i	1.05222	−	1.82250i
$137$	−0.377242			−0.0322300			−0.0161150	−	0.999870i	$-0.505130\pi$
$137$	−0.377242			−0.0322300			−0.0161150	+	0.999870i	$0.505130\pi$
$138$		0			0
$139$	−9.50067	+	16.4556i	−0.805837	+	1.39575i	0.109888	+	0.993944i	$0.464951\pi$
$139$	−9.50067	+	16.4556i	−0.805837	+	1.39575i	−0.915725	+	0.401806i	$0.868383\pi$
$140$		0			0
$141$		0			0
$142$	−10.4086	+	18.0281i	−0.873467	+	1.51289i
$143$	0.0989221	−	0.171338i	0.00827228	−	0.0143280i
$144$		0			0
$145$	−13.9879	−	24.2277i	−1.16163	−	2.01200i
$146$	−2.42792	−	4.20528i	−0.200936	−	0.348032i
$147$		0			0
$148$	−4.71780	+	8.17147i	−0.387801	+	0.671691i
$149$	9.70175			0.794798			0.397399	−	0.917646i	$-0.369913\pi$
$149$	9.70175			0.794798			0.397399	+	0.917646i	$0.369913\pi$
$150$		0			0
$151$	−12.8348			−1.04448			−0.522242	−	0.852798i	$-0.674904\pi$
$151$	−12.8348			−1.04448			−0.522242	+	0.852798i	$0.674904\pi$
$152$	4.98715	+	8.63800i	0.404511	+	0.700634i
$153$		0			0
$154$		0			0
$155$	−12.8530	−	22.2621i	−1.03238	−	1.78813i
$156$		0			0
$157$	10.4743	+	18.1420i	0.835937	+	1.44789i	0.893265	+	0.449531i	$0.148409\pi$
$157$	10.4743	+	18.1420i	0.835937	+	1.44789i	−0.0573276	+	0.998355i	$0.518258\pi$
$158$	10.0416	+	17.3926i	0.798869	+	1.38368i
$159$		0			0
$160$	−0.986757	−	1.70911i	−0.0780100	−	0.135117i
$161$		0			0
$162$		0			0
$163$	5.58113	+	9.66679i	0.437148	+	0.757162i	0.997468	−	0.0711140i	$-0.0226554\pi$
$163$	5.58113	+	9.66679i	0.437148	+	0.757162i	−0.560321	+	0.828276i	$0.689322\pi$
$164$	30.4624			2.37871
$165$		0			0
$166$	−29.9492			−2.32451
$167$	−1.73012	+	2.99665i	−0.133880	+	0.231888i	−0.925169	−	0.379555i	$-0.876077\pi$
$167$	−1.73012	+	2.99665i	−0.133880	+	0.231888i	0.791289	+	0.611443i	$0.209410\pi$
$168$		0			0
$169$	6.38151	+	11.0531i	0.490886	+	0.850239i
$170$	−21.8530	−	37.8505i	−1.67605	−	2.90300i
$171$		0			0
$172$	4.71780	−	8.17147i	0.359729	−	0.623069i
$173$	3.02680	−	5.24258i	0.230124	−	0.398586i	−0.727721	−	0.685874i	$-0.759420\pi$
$173$	3.02680	−	5.24258i	0.230124	−	0.398586i	0.957844	+	0.287288i	$0.0927536\pi$
$174$		0			0
$175$		0			0
$176$	0.879379	−	1.52313i	0.0662857	−	0.114810i
$177$		0			0
$178$	−36.4896			−2.73501
$179$	4.56654	−	7.90947i	0.341319	−	0.591182i	−0.643359	−	0.765565i	$-0.722460\pi$
$179$	4.56654	−	7.90947i	0.341319	−	0.591182i	0.984678	+	0.174383i	$0.0557930\pi$
$180$		0			0
$181$	−11.9478			−0.888074			−0.444037	−	0.896008i	$-0.646454\pi$
$181$	−11.9478			−0.888074			−0.444037	+	0.896008i	$0.646454\pi$
$182$		0			0
$183$		0			0
$184$	23.4897			1.73168
$185$	4.25696	+	7.37327i	0.312978	+	0.542093i
$186$		0			0
$187$	0.986757	−	1.70911i	0.0721588	−	0.124983i
$188$	−25.6065			−1.86755
$189$		0			0
$190$	17.7630			1.28867
$191$	4.57014	−	7.91571i	0.330683	−	0.572760i	−0.651963	−	0.758251i	$-0.726054\pi$
$191$	4.57014	−	7.91571i	0.330683	−	0.572760i	0.982646	+	0.185491i	$0.0593874\pi$
$192$		0			0
$193$	−8.47150	−	14.6731i	−0.609792	−	1.05619i	−0.991274	−	0.131814i	$-0.957920\pi$
$193$	−8.47150	−	14.6731i	−0.609792	−	1.05619i	0.381483	−	0.924376i	$-0.375414\pi$
$194$	−23.3441			−1.67601
$195$		0			0
$196$		0			0
$197$	21.3173			1.51880			0.759398	−	0.650627i	$-0.225494\pi$
$197$	21.3173			1.51880			0.759398	+	0.650627i	$0.225494\pi$
$198$		0			0
$199$	4.98715	−	8.63800i	0.353530	−	0.612332i	−0.633335	−	0.773877i	$-0.718315\pi$
$199$	4.98715	−	8.63800i	0.353530	−	0.612332i	0.986865	+	0.161546i	$0.0516479\pi$
$200$	−42.3609			−2.99537
$201$		0			0
$202$	10.7177	−	18.5635i	0.754092	−	1.30613i
$203$		0			0
$204$		0			0
$205$	13.7434	−	23.8042i	0.959879	−	1.66256i
$206$	−9.87538	+	17.1047i	−0.688051	+	1.19174i
$207$		0			0
$208$	−1.05329	−	1.82435i	−0.0730324	−	0.126496i
$209$	0.401038	+	0.694619i	0.0277404	+	0.0480478i
$210$		0			0
$211$	−2.44592	+	4.23645i	−0.168384	+	0.291649i	−0.937852	−	0.347036i	$-0.887188\pi$
$211$	−2.44592	+	4.23645i	−0.168384	+	0.291649i	0.769468	+	0.638685i	$0.220521\pi$
$212$	14.4677			0.993646
$213$		0			0
$214$	31.5979			2.15998
$215$	−4.25696	−	7.37327i	−0.290322	−	0.502853i
$216$		0			0
$217$		0			0
$218$	3.19961	+	5.54189i	0.216705	+	0.375344i
$219$		0			0
$220$	−3.01364	−	5.21978i	−0.203179	−	0.351917i
$221$	−1.18190	−	2.04712i	−0.0795034	−	0.137704i
$222$		0			0
$223$	−11.7044	−	20.2727i	−0.783786	−	1.35756i	−0.929722	−	0.368263i	$-0.879953\pi$
$223$	−11.7044	−	20.2727i	−0.783786	−	1.35756i	0.145936	−	0.989294i	$-0.453381\pi$
$224$		0			0
$225$		0			0
$226$	17.1623	+	29.7259i	1.14162	+	1.97734i
$227$	−6.11839			−0.406092			−0.203046	−	0.979169i	$-0.565084\pi$
$227$	−6.11839			−0.406092			−0.203046	+	0.979169i	$0.565084\pi$
$228$		0			0
$229$	−1.46039			−0.0965052			−0.0482526	−	0.998835i	$-0.515365\pi$
$229$	−1.46039			−0.0965052			−0.0482526	+	0.998835i	$0.515365\pi$
$230$	20.9161	−	36.2278i	1.37917	−	2.38879i
$231$		0			0
$232$	−19.3260	−	33.4736i	−1.26881	−	2.19765i
$233$	−6.62422	−	11.4735i	−0.433967	−	0.751653i	0.563244	−	0.826291i	$-0.309553\pi$
$233$	−6.62422	−	11.4735i	−0.433967	−	0.751653i	−0.997211	+	0.0746378i	$0.976220\pi$
$234$		0			0
$235$	−11.5526	+	20.0097i	−0.753610	+	1.30529i
$236$	−12.4022	+	21.4813i	−0.807316	+	1.39831i
$237$		0			0
$238$		0			0
$239$	9.69436	−	16.7911i	0.627076	−	1.08613i	−0.361060	−	0.932543i	$-0.617585\pi$
$239$	9.69436	−	16.7911i	0.627076	−	1.08613i	0.988136	−	0.153584i	$-0.0490817\pi$
$240$		0			0
$241$	5.05368			0.325536			0.162768	−	0.986664i	$-0.447958\pi$
$241$	5.05368			0.325536			0.162768	+	0.986664i	$0.447958\pi$
$242$	−13.3296	+	23.0875i	−0.856857	+	1.48412i
$243$		0			0
$244$	32.5426			2.08333
$245$		0			0
$246$		0			0
$247$	0.960699			0.0611278
$248$	−17.7580	−	30.7578i	−1.12764	−	1.95312i
$249$		0			0
$250$	−15.2180	+	26.3584i	−0.962472	+	1.66705i
$251$	−15.0928			−0.952647			−0.476324	−	0.879270i	$-0.658031\pi$
$251$	−15.0928			−0.952647			−0.476324	+	0.879270i	$0.658031\pi$
$252$		0			0
$253$	1.88891			0.118755
$254$	−19.1228	+	33.1216i	−1.19987	+	2.07823i
$255$		0			0
$256$	16.1804	+	28.0253i	1.01128	+	1.75158i
$257$	−7.71184			−0.481051			−0.240526	−	0.970643i	$-0.577320\pi$
$257$	−7.71184			−0.481051			−0.240526	+	0.970643i	$0.577320\pi$
$258$		0			0
$259$		0			0
$260$	−7.21926			−0.447720
$261$		0			0
$262$	−10.4743	+	18.1420i	−0.647102	+	1.12081i
$263$	−4.21206			−0.259727			−0.129864	−	0.991532i	$-0.541454\pi$
$263$	−4.21206			−0.259727			−0.129864	+	0.991532i	$0.541454\pi$
$264$		0			0
$265$	6.52724	−	11.3055i	0.400965	−	0.694492i
$266$		0			0
$267$		0			0
$268$	−7.29893	+	12.6421i	−0.445853	+	0.772240i
$269$	10.3753	−	17.9706i	0.632596	−	1.09569i	−0.354423	−	0.935085i	$-0.615323\pi$
$269$	10.3753	−	17.9706i	0.632596	−	1.09569i	0.987019	−	0.160603i	$-0.0513439\pi$
$270$		0			0
$271$	14.2444	+	24.6721i	0.865287	+	1.49872i	0.866762	+	0.498723i	$0.166197\pi$
$271$	14.2444	+	24.6721i	0.865287	+	1.49872i	−0.00147433	+	0.999999i	$0.500469\pi$
$272$	−10.5067	−	18.1981i	−0.637060	−	1.10342i
$273$		0			0
$274$	−0.464103	+	0.803851i	−0.0280375	+	0.0485624i
$275$	−3.40642			−0.205415
$276$		0			0
$277$	17.1623			1.03118			0.515590	−	0.856835i	$-0.327573\pi$
$277$	17.1623			1.03118			0.515590	+	0.856835i	$0.327573\pi$
$278$	23.3765	+	40.4892i	1.40203	+	2.42838i
$279$		0			0
$280$		0			0
$281$	4.72140	+	8.17770i	0.281655	+	0.487841i	0.971793	−	0.235837i	$-0.0757833\pi$
$281$	4.72140	+	8.17770i	0.281655	+	0.487841i	−0.690138	+	0.723678i	$0.742450\pi$
$282$		0			0
$283$	−8.43422	−	14.6085i	−0.501362	−	0.868385i	−0.999999	−	0.00157378i	$-0.999499\pi$
$283$	−8.43422	−	14.6085i	−0.501362	−	0.868385i	0.498636	−	0.866811i	$-0.333834\pi$
$284$	17.1498	+	29.7043i	1.01765	+	1.76263i
$285$		0			0
$286$	−0.243398	−	0.421578i	−0.0143924	−	0.0249284i
$287$		0			0
$288$		0			0
$289$	−3.28959	−	5.69774i	−0.193505	−	0.335161i
$290$	−68.8344			−4.04210
$291$		0			0
$292$	−8.00079			−0.468211
$293$	−1.86143	+	3.22409i	−0.108746	+	0.188353i	−0.915262	−	0.402858i	$-0.868017\pi$
$293$	−1.86143	+	3.22409i	−0.108746	+	0.188353i	0.806517	+	0.591211i	$0.201350\pi$
$294$		0			0
$295$	11.1908	+	19.3830i	0.651551	+	1.12852i
$296$	5.88151	+	10.1871i	0.341856	+	0.592112i
$297$		0			0
$298$	11.9356	−	20.6731i	0.691411	−	1.19756i
$299$	1.13123	−	1.95935i	0.0654209	−	0.113312i
$300$		0			0
$301$		0			0
$302$	−15.7901	+	27.3492i	−0.908617	+	1.57377i
$303$		0			0
$304$	8.54024			0.489817
$305$	14.6819	−	25.4298i	0.840683	−	1.45611i
$306$		0			0
$307$	30.5691			1.74467			0.872335	−	0.488908i	$-0.162605\pi$
$307$	30.5691			1.74467			0.872335	+	0.488908i	$0.162605\pi$
$308$		0			0
$309$		0			0
$310$	−63.2498			−3.59235
$311$	−5.21739	−	9.03678i	−0.295851	−	0.512429i	0.679332	−	0.733831i	$-0.262270\pi$
$311$	−5.21739	−	9.03678i	−0.295851	−	0.512429i	−0.975182	+	0.221403i	$0.928936\pi$
$312$		0			0
$313$	0.309930	−	0.536815i	0.0175183	−	0.0303426i	−0.857133	−	0.515095i	$-0.827757\pi$
$313$	0.309930	−	0.536815i	0.0175183	−	0.0303426i	0.874652	+	0.484752i	$0.161090\pi$
$314$	51.5440			2.90879
$315$		0			0
$316$	33.0905			1.86148
$317$	5.12422	−	8.87541i	0.287805	−	0.498493i	−0.685481	−	0.728091i	$-0.740408\pi$
$317$	5.12422	−	8.87541i	0.287805	−	0.498493i	0.973285	+	0.229598i	$0.0737412\pi$
$318$		0			0
$319$	−1.55408	−	2.69175i	−0.0870120	−	0.150709i
$320$	26.8043			1.49841
$321$		0			0
$322$		0			0
$323$	9.58307			0.533216
$324$		0			0
$325$	−2.04005	+	3.53346i	−0.113161	+	0.196001i
$326$	27.4648			1.52113
$327$		0			0
$328$	18.9882	−	32.8885i	1.04845	−	1.81596i
$329$		0			0
$330$		0			0
$331$	10.1819	−	17.6356i	0.559648	−	0.969339i	−0.437878	−	0.899035i	$-0.644270\pi$
$331$	10.1819	−	17.6356i	0.559648	−	0.969339i	0.997526	−	0.0703042i	$-0.0223970\pi$
$332$	−24.6731	+	42.7351i	−1.35411	+	2.34539i
$333$		0			0
$334$	4.25696	+	7.37327i	0.232930	+	0.403447i
$335$	6.58596	+	11.4072i	0.359829	+	0.623242i
$336$		0			0
$337$	2.85594	−	4.94662i	0.155573	−	0.269460i	−0.777695	−	0.628642i	$-0.783611\pi$
$337$	2.85594	−	4.94662i	0.155573	−	0.269460i	0.933267	+	0.359182i	$0.116944\pi$
$338$	31.4035			1.70812
$339$		0			0
$340$	−72.0128			−3.90544
$341$	−1.42800	−	2.47337i	−0.0773305	−	0.133940i
$342$		0			0
$343$		0			0
$344$	−5.88151	−	10.1871i	−0.317110	−	0.549251i
$345$		0			0
$346$	−7.44746	−	12.8994i	−0.400378	−	0.693475i
$347$	4.44066	+	7.69145i	0.238387	+	0.412899i	0.960252	−	0.279136i	$-0.0900480\pi$
$347$	4.44066	+	7.69145i	0.238387	+	0.412899i	−0.721865	+	0.692034i	$0.756715\pi$
$348$		0			0
$349$	10.4874	+	18.1648i	0.561379	+	0.972337i	0.997376	+	0.0723893i	$0.0230624\pi$
$349$	10.4874	+	18.1648i	0.561379	+	0.972337i	−0.435997	+	0.899948i	$0.643604\pi$
$350$		0			0
$351$		0			0
$352$	−0.109631	−	0.189886i	−0.00584335	−	0.0101210i
$353$	−14.7654			−0.785881			−0.392941	−	0.919564i	$-0.628542\pi$
$353$	−14.7654			−0.785881			−0.392941	+	0.919564i	$0.628542\pi$
$354$		0			0
$355$	30.9492			1.64261
$356$	−30.0613	+	52.0677i	−1.59325	+	2.75959i
$357$		0			0
$358$	−11.2360	−	19.4613i	−0.593840	−	1.02856i
$359$	3.60603	+	6.24583i	0.190319	+	0.329642i	0.945356	−	0.326040i	$-0.105714\pi$
$359$	3.60603	+	6.24583i	0.190319	+	0.329642i	−0.755037	+	0.655682i	$0.772381\pi$
$360$		0			0
$361$	7.55262	−	13.0815i	0.397506	−	0.688501i
$362$	−14.6988	+	25.4591i	−0.772554	+	1.33810i
$363$		0			0
$364$		0			0
$365$	−3.60963	+	6.25206i	−0.188937	+	0.327248i
$366$		0			0
$367$	10.9742			0.572850			0.286425	−	0.958103i	$-0.407533\pi$
$367$	10.9742			0.572850			0.286425	+	0.958103i	$0.407533\pi$
$368$	10.0562	−	17.4179i	0.524217	−	0.907970i
$369$		0			0
$370$	20.9485			1.08906
$371$		0			0
$372$		0			0
$373$	−0.543767			−0.0281552			−0.0140776	−	0.999901i	$-0.504481\pi$
$373$	−0.543767			−0.0281552			−0.0140776	+	0.999901i	$0.504481\pi$
$374$	−2.42792	−	4.20528i	−0.125545	−	0.217450i
$375$		0			0
$376$	−15.9614	+	27.6459i	−0.823145	+	1.42573i
$377$	−3.72286			−0.191737
$378$		0			0
$379$	−22.6912			−1.16557			−0.582785	−	0.812626i	$-0.698037\pi$
$379$	−22.6912			−1.16557			−0.582785	+	0.812626i	$0.698037\pi$
$380$	14.6337	−	25.3464i	0.750695	−	1.30024i
$381$		0			0
$382$	−11.2448	−	19.4766i	−0.575336	−	0.996511i
$383$	35.7139			1.82489			0.912447	−	0.409194i	$-0.134190\pi$
$383$	35.7139			1.82489			0.912447	+	0.409194i	$0.134190\pi$
$384$		0			0
$385$		0			0
$386$	−41.6883			−2.12188
$387$		0			0
$388$	−19.2316	+	33.3101i	−0.976336	+	1.69106i
$389$	−38.6591			−1.96010			−0.980048	−	0.198761i	$-0.936308\pi$
$389$	−38.6591			−1.96010			−0.980048	+	0.198761i	$0.936308\pi$
$390$		0			0
$391$	11.2842	−	19.5447i	0.570664	−	0.988420i
$392$		0			0
$393$		0			0
$394$	26.2257	−	45.4242i	1.32123	−	2.28844i
$395$	14.9291	−	25.8579i	0.751163	−	1.30105i
$396$		0			0
$397$	−5.97391	−	10.3471i	−0.299822	−	0.519307i	0.676273	−	0.736651i	$-0.263594\pi$
$397$	−5.97391	−	10.3471i	−0.299822	−	0.519307i	−0.976095	+	0.217344i	$0.930261\pi$
$398$	−12.2709	−	21.2538i	−0.615085	−	1.06536i
$399$		0			0
$400$	−18.1352	+	31.4111i	−0.906761	+	1.57056i
$401$	−32.3566			−1.61581			−0.807906	−	0.589311i	$-0.799399\pi$
$401$	−32.3566			−1.61581			−0.807906	+	0.589311i	$0.799399\pi$
$402$		0			0
$403$	−3.42082			−0.170403
$404$	−17.6591	−	30.5865i	−0.878573	−	1.52173i
$405$		0			0
$406$		0			0
$407$	0.472958	+	0.819187i	0.0234437	+	0.0406056i
$408$		0			0
$409$	9.48751	+	16.4328i	0.469127	+	0.812552i	0.999377	−	0.0352893i	$-0.0112353\pi$
$409$	9.48751	+	16.4328i	0.469127	+	0.812552i	−0.530250	+	0.847841i	$0.677902\pi$
$410$	−33.8157	−	58.5704i	−1.67004	−	2.89259i
$411$		0			0
$412$	16.2713	+	28.1827i	0.801630	+	1.38846i
$413$		0			0
$414$		0			0
$415$	22.2630	+	38.5607i	1.09285	+	1.89287i
$416$	−0.262624			−0.0128762
$417$		0			0
$418$	1.97351			0.0965277
$419$	8.64523	−	14.9740i	0.422347	−	0.731526i	−0.573822	−	0.818980i	$-0.694540\pi$
$419$	8.64523	−	14.9740i	0.422347	−	0.731526i	0.996169	+	0.0874539i	$0.0278730\pi$
$420$		0			0
$421$	−9.30039	−	16.1087i	−0.453273	−	0.785092i	0.545314	−	0.838232i	$-0.316410\pi$
$421$	−9.30039	−	16.1087i	−0.453273	−	0.785092i	−0.998587	+	0.0531397i	$0.983077\pi$
$422$	6.01819	+	10.4238i	0.292961	+	0.507423i
$423$		0			0
$424$	9.01819	−	15.6200i	0.437962	−	0.758572i
$425$	−20.3496	+	35.2466i	−0.987103	+	1.70971i
$426$		0			0
$427$		0			0
$428$	26.0313	−	45.0876i	1.25827	−	2.17939i
$429$		0			0
$430$	−20.9485			−1.01023
$431$	−7.93920	+	13.7511i	−0.382418	+	0.662367i	−0.991407	−	0.130811i	$-0.958242\pi$
$431$	−7.93920	+	13.7511i	−0.382418	+	0.662367i	0.608990	+	0.793178i	$0.291575\pi$
$432$		0			0
$433$	−40.4367			−1.94326			−0.971631	−	0.236501i	$-0.923999\pi$
$433$	−40.4367			−1.94326			−0.971631	+	0.236501i	$0.923999\pi$
$434$		0			0
$435$		0			0
$436$	10.5438			0.504955
$437$	4.58611	+	7.94338i	0.219384	+	0.379984i
$438$		0			0
$439$	6.23047	−	10.7915i	0.297364	−	0.515050i	−0.678168	−	0.734907i	$-0.737226\pi$
$439$	6.23047	−	10.7915i	0.297364	−	0.515050i	0.975532	+	0.219857i	$0.0705591\pi$
$440$	−7.51399			−0.358216
$441$		0			0
$442$	−5.81616			−0.276646
$443$	−4.11537	+	7.12802i	−0.195527	+	0.338663i	−0.947073	−	0.321018i	$-0.895975\pi$
$443$	−4.11537	+	7.12802i	−0.195527	+	0.338663i	0.751546	+	0.659680i	$0.229308\pi$
$444$		0			0
$445$	27.1249	+	46.9817i	1.28584	+	2.22715i
$446$	−57.5976			−2.72732
$447$		0			0
$448$		0			0
$449$	−5.64474			−0.266392			−0.133196	−	0.991090i	$-0.542524\pi$
$449$	−5.64474			−0.266392			−0.133196	+	0.991090i	$0.542524\pi$
$450$		0			0
$451$	1.52692	−	2.64471i	0.0718999	−	0.124534i
$452$	56.5552			2.66013
$453$		0			0
$454$	−7.52716	+	13.0374i	−0.353267	+	0.611876i
$455$		0			0
$456$		0			0
$457$	−2.53443	+	4.38977i	−0.118556	+	0.205345i	−0.919196	−	0.393801i	$-0.871160\pi$
$457$	−2.53443	+	4.38977i	−0.118556	+	0.205345i	0.800640	+	0.599146i	$0.204493\pi$
$458$	−1.79665	+	3.11188i	−0.0839518	+	0.145409i
$459$		0			0
$460$	−34.4628	−	59.6913i	−1.60683	−	2.78312i
$461$	3.88831	+	6.73475i	0.181097	+	0.313669i	0.942254	−	0.334898i	$-0.108702\pi$
$461$	3.88831	+	6.73475i	0.181097	+	0.313669i	−0.761158	+	0.648567i	$0.775369\pi$
$462$		0			0
$463$	4.58998	−	7.95008i	0.213314	−	0.369472i	−0.739435	−	0.673228i	$-0.764907\pi$
$463$	4.58998	−	7.95008i	0.213314	−	0.369472i	0.952750	+	0.303756i	$0.0982408\pi$
$464$	−33.0947			−1.53638
$465$		0			0
$466$	−32.5979			−1.51007
$467$	6.88272	+	11.9212i	0.318494	+	0.551648i	0.980174	−	0.198138i	$-0.0634895\pi$
$467$	6.88272	+	11.9212i	0.318494	+	0.551648i	−0.661680	+	0.749787i	$0.730156\pi$
$468$		0			0
$469$		0			0
$470$	28.4253	+	49.2340i	1.31116	+	2.27100i
$471$		0			0
$472$	15.4614	+	26.7800i	0.711669	+	1.23265i
$473$	−0.472958	−	0.819187i	−0.0217466	−	0.0376663i
$474$		0			0
$475$	−8.27052	−	14.3250i	−0.379477	−	0.657274i
$476$		0			0
$477$		0			0
$478$	−23.8530	−	41.3146i	−1.09101	−	1.88969i
$479$	8.71176			0.398050			0.199025	−	0.979994i	$-0.436222\pi$
$479$	8.71176			0.398050			0.199025	+	0.979994i	$0.436222\pi$
$480$		0			0
$481$	1.13298			0.0516597
$482$	6.21731	−	10.7687i	0.283191	−	0.490501i
$483$		0			0
$484$	21.9626	+	38.0404i	0.998302	+	1.72911i
$485$	17.3530	+	30.0563i	0.787960	+	1.36479i
$486$		0			0
$487$	9.01819	−	15.6200i	0.408653	−	0.707808i	−0.586086	−	0.810249i	$-0.699332\pi$
$487$	9.01819	−	15.6200i	0.408653	−	0.707808i	0.994739	+	0.102441i	$0.0326653\pi$
$488$	20.2849	−	35.1344i	0.918253	−	1.59046i
$489$		0			0
$490$		0			0
$491$	1.02344	−	1.77266i	0.0461874	−	0.0799989i	−0.842007	−	0.539466i	$-0.818626\pi$
$491$	1.02344	−	1.77266i	0.0461874	−	0.0799989i	0.888195	+	0.459467i	$0.151960\pi$
$492$		0			0
$493$	−37.1358			−1.67251
$494$	1.18190	−	2.04712i	0.0531763	−	0.0921041i
$495$		0			0
$496$	−30.4097			−1.36544
$497$		0			0
$498$		0			0
$499$	−39.0875			−1.74980			−0.874899	−	0.484305i	$-0.839072\pi$
$499$	−39.0875			−1.74980			−0.874899	+	0.484305i	$0.839072\pi$
$500$	25.0742	+	43.4297i	1.12135	+	1.94224i
$501$		0			0
$502$	−18.5679	+	32.1606i	−0.828727	+	1.43540i
$503$	5.11846			0.228221			0.114111	−	0.993468i	$-0.463598\pi$
$503$	5.11846			0.228221			0.114111	+	0.993468i	$0.463598\pi$
$504$		0			0
$505$	−31.8683			−1.41812
$506$	2.32383	−	4.02499i	0.103307	−	0.178933i
$507$		0			0
$508$	31.5079	+	54.5732i	1.39794	+	2.42130i
$509$	−29.5272			−1.30877			−0.654386	−	0.756161i	$-0.727073\pi$
$509$	−29.5272			−1.30877			−0.654386	+	0.756161i	$0.727073\pi$
$510$		0			0
$511$		0			0
$512$	41.4078			1.82998
$513$		0			0
$514$	−9.48751	+	16.4328i	−0.418476	+	0.724822i
$515$	29.3638			1.29392
$516$		0			0
$517$	−1.28352	+	2.22313i	−0.0564493	+	0.0977730i
$518$		0			0
$519$		0			0
$520$	−4.50000	+	7.79423i	−0.197338	+	0.341800i
$521$	−0.532351	+	0.922058i	−0.0233227	+	0.0403961i	−0.877451	−	0.479666i	$-0.840758\pi$
$521$	−0.532351	+	0.922058i	−0.0233227	+	0.0403961i	0.854128	+	0.520062i	$0.174091\pi$
$522$		0			0
$523$	6.69094	+	11.5890i	0.292574	+	0.506754i	0.974418	−	0.224745i	$-0.0721548\pi$
$523$	6.69094	+	11.5890i	0.292574	+	0.506754i	−0.681843	+	0.731498i	$0.738821\pi$
$524$	17.2581	+	29.8918i	0.753922	+	1.30583i
$525$		0			0
$526$	−5.18190	+	8.97532i	−0.225942	+	0.391343i
$527$	−34.1230			−1.48642
$528$		0			0
$529$	−1.39922			−0.0608358
$530$	−16.0603	−	27.8173i	−0.697615	−	1.20830i
$531$		0			0
$532$		0			0
$533$	−1.82889	−	3.16774i	−0.0792181	−	0.137210i
$534$		0			0
$535$	−23.4885	−	40.6833i	−1.01550	−	1.75889i
$536$	9.09931	+	15.7605i	0.393031	+	0.680749i
$537$		0			0
$538$	−25.5286	−	44.2168i	−1.10062	−	1.90632i
$539$		0			0
$540$		0			0
$541$	17.0438	+	29.5207i	0.732769	+	1.26919i	0.955695	+	0.294358i	$0.0951056\pi$
$541$	17.0438	+	29.5207i	0.732769	+	1.26919i	−0.222927	+	0.974835i	$0.571561\pi$
$542$	70.0970			3.01092
$543$		0			0
$544$	−2.61970			−0.112319
$545$	4.75692	−	8.23922i	0.203764	−	0.352930i
$546$		0			0
$547$	2.97150	+	5.14678i	0.127052	+	0.220060i	0.922533	−	0.385918i	$-0.126115\pi$
$547$	2.97150	+	5.14678i	0.127052	+	0.220060i	−0.795481	+	0.605978i	$0.792782\pi$
$548$	0.764686	+	1.32448i	0.0326658	+	0.0565788i
$549$		0			0
$550$	−4.19076	+	7.25860i	−0.178694	+	0.309508i
$551$	7.54638	−	13.0707i	0.321487	−	0.556831i
$552$		0			0
$553$		0			0
$554$	21.1139	−	36.5704i	0.897044	−	1.55373i
$555$		0			0
$556$	77.0331			3.26693
$557$	−15.0402	+	26.0503i	−0.637272	+	1.10379i	0.348756	+	0.937213i	$0.386604\pi$
$557$	−15.0402	+	26.0503i	−0.637272	+	1.10379i	−0.986029	+	0.166575i	$0.946729\pi$
$558$		0			0
$559$	−1.13298			−0.0479202
$560$		0			0
$561$		0			0
$562$	23.2340			0.980069
$563$	−9.81060	−	16.9925i	−0.413468	−	0.716147i	0.581799	−	0.813333i	$-0.302349\pi$
$563$	−9.81060	−	16.9925i	−0.413468	−	0.716147i	−0.995266	+	0.0971860i	$0.969016\pi$
$564$		0			0
$565$	25.5154	−	44.1940i	1.07344	−	1.85926i
$566$	−41.5049			−1.74458
$567$		0			0
$568$	42.7601			1.79417
$569$	−0.687159	+	1.19019i	−0.0288072	+	0.0498955i	−0.880070	−	0.474845i	$-0.842504\pi$
$569$	−0.687159	+	1.19019i	−0.0288072	+	0.0498955i	0.851262	+	0.524740i	$0.175838\pi$
$570$		0			0
$571$	−8.69076	−	15.0528i	−0.363697	−	0.629941i	0.624869	−	0.780729i	$-0.285152\pi$
$571$	−8.69076	−	15.0528i	−0.363697	−	0.629941i	−0.988566	+	0.150788i	$0.951819\pi$
$572$	−0.802077			−0.0335365
$573$		0			0
$574$		0			0
$575$	−38.9545			−1.62451
$576$		0			0
$577$	13.5274	−	23.4301i	0.563153	−	0.975409i	−0.434066	−	0.900881i	$-0.642922\pi$
$577$	13.5274	−	23.4301i	0.563153	−	0.975409i	0.997219	−	0.0745283i	$-0.0237451\pi$
$578$	−16.1881			−0.673337
$579$		0			0
$580$	−56.7080	+	98.2211i	−2.35467	+	4.07841i
$581$		0			0
$582$		0			0
$583$	0.725191	−	1.25607i	0.0300344	−	0.0520210i
$584$	−4.98715	+	8.63800i	−0.206370	+	0.357443i
$585$		0			0
$586$	4.58005	+	7.93288i	0.189200	+	0.327704i
$587$	3.75700	+	6.50731i	0.155068	+	0.268585i	0.933084	−	0.359659i	$-0.117107\pi$
$587$	3.75700	+	6.50731i	0.155068	+	0.268585i	−0.778016	+	0.628245i	$0.783774\pi$
$588$		0			0
$589$	6.93414	−	12.0103i	0.285716	−	0.494875i
$590$	55.0698			2.26719
$591$		0			0
$592$	10.0718			0.413948
$593$	17.7904	+	30.8139i	0.730565	+	1.26538i	0.956642	+	0.291266i	$0.0940765\pi$
$593$	17.7904	+	30.8139i	0.730565	+	1.26538i	−0.226077	+	0.974109i	$0.572590\pi$
$594$		0			0
$595$		0			0
$596$	−19.6659	−	34.0623i	−0.805545	−	1.39524i
$597$		0			0
$598$	−2.78340	−	4.82100i	−0.113822	−	0.197145i
$599$	−5.74105	−	9.94379i	−0.234573	−	0.406292i	0.724576	−	0.689195i	$-0.242036\pi$
$599$	−5.74105	−	9.94379i	−0.234573	−	0.406292i	−0.959148	+	0.282903i	$0.908702\pi$
$600$		0			0
$601$	0.190030	+	0.329142i	0.00775150	+	0.0134260i	0.869875	−	0.493272i	$-0.164199\pi$
$601$	0.190030	+	0.329142i	0.00775150	+	0.0134260i	−0.862124	+	0.506698i	$0.830866\pi$
$602$		0			0
$603$		0			0
$604$	26.0167	+	45.0623i	1.05861	+	1.83356i
$605$	39.6346			1.61138
$606$		0			0
$607$	−18.5409			−0.752551			−0.376275	−	0.926508i	$-0.622795\pi$
$607$	−18.5409			−0.752551			−0.376275	+	0.926508i	$0.622795\pi$
$608$	0.532351	−	0.922058i	0.0215897	−	0.0373944i
$609$		0			0
$610$	−36.1249	−	62.5702i	−1.46265	−	2.53339i
$611$	1.53736	+	2.66278i	0.0621949	+	0.107725i
$612$		0			0
$613$	−3.66225	+	6.34321i	−0.147917	+	0.256200i	−0.930457	−	0.366400i	$-0.880590\pi$
$613$	−3.66225	+	6.34321i	−0.147917	+	0.256200i	0.782540	+	0.622600i	$0.213924\pi$
$614$	37.6077	−	65.1385i	1.51772	−	2.62877i
$615$		0			0
$616$		0			0
$617$	−12.7427	+	22.0710i	−0.513002	+	0.888546i	0.486884	+	0.873466i	$0.338133\pi$
$617$	−12.7427	+	22.0710i	−0.513002	+	0.888546i	−0.999886	+	0.0150791i	$0.995200\pi$
$618$		0			0
$619$	32.8963			1.32222			0.661108	−	0.750291i	$-0.270087\pi$
$619$	32.8963			1.32222			0.661108	+	0.750291i	$0.270087\pi$
$620$	−52.1072	+	90.2523i	−2.09267	+	3.62462i
$621$		0			0
$622$	−25.6748			−1.02947
$623$		0			0
$624$		0			0
$625$	3.34221			0.133689
$626$	−0.762585	−	1.32084i	−0.0304790	−	0.0527912i
$627$		0			0
$628$	42.4636	−	73.5490i	1.69448	−	2.93493i
$629$	11.3016			0.450626
$630$		0			0
$631$	29.8683			1.18904			0.594519	−	0.804082i	$-0.297343\pi$
$631$	29.8683			1.18904			0.594519	+	0.804082i	$0.297343\pi$
$632$	20.6264	−	35.7259i	0.820472	−	1.42110i
$633$		0			0
$634$	−12.6082	−	21.8380i	−0.500734	−	0.867297i
$635$	56.8603			2.25643
$636$		0			0
$637$		0			0
$638$	−7.64766			−0.302774
$639$		0			0
$640$	34.9496	−	60.5344i	1.38150	−	2.39283i
$641$	−11.4605			−0.452663			−0.226331	−	0.974050i	$-0.572673\pi$
$641$	−11.4605			−0.452663			−0.226331	+	0.974050i	$0.572673\pi$
$642$		0			0
$643$	8.69078	−	15.0529i	0.342731	−	0.593627i	−0.642208	−	0.766531i	$-0.721981\pi$
$643$	8.69078	−	15.0529i	0.342731	−	0.593627i	0.984939	+	0.172903i	$0.0553147\pi$
$644$		0			0
$645$		0			0
$646$	11.7896	−	20.4202i	0.463855	−	0.803421i
$647$	12.6720	−	21.9485i	0.498186	−	0.862883i	−0.501812	−	0.864977i	$-0.667333\pi$
$647$	12.6720	−	21.9485i	0.498186	−	0.862883i	0.999998	+	0.00209358i	$0.000666408\pi$
$648$		0			0
$649$	1.24332	+	2.15349i	0.0488045	+	0.0845320i
$650$	5.01954	+	8.69410i	0.196883	+	0.341011i
$651$		0			0
$652$	22.6264	−	39.1900i	0.886116	−	1.53480i
$653$	−14.0833			−0.551121			−0.275560	−	0.961284i	$-0.588863\pi$
$653$	−14.0833			−0.551121			−0.275560	+	0.961284i	$0.588863\pi$
$654$		0			0
$655$	31.1445			1.21692
$656$	−16.2581	−	28.1599i	−0.634774	−	1.09946i
$657$		0			0
$658$		0			0
$659$	19.0854	+	33.0569i	0.743462	+	1.28771i	0.950910	+	0.309467i	$0.100151\pi$
$659$	19.0854	+	33.0569i	0.743462	+	1.28771i	−0.207449	+	0.978246i	$0.566516\pi$
$660$		0			0
$661$	0.176866	+	0.306341i	0.00687930	+	0.0119153i	0.869445	−	0.494031i	$-0.164477\pi$
$661$	0.176866	+	0.306341i	0.00687930	+	0.0119153i	−0.862565	+	0.505946i	$0.831144\pi$
$662$	−25.0526	−	43.3924i	−0.973698	−	1.68649i
$663$		0			0
$664$	30.7591	+	53.2764i	1.19369	+	2.06752i
$665$		0			0
$666$		0			0
$667$	−17.7719	−	30.7818i	−0.688130	−	1.19188i
$668$	14.0281			0.542762
$669$		0			0
$670$	32.4096			1.25209
$671$	1.63119	−	2.82531i	0.0629715	−	0.109070i
$672$		0			0
$673$	10.5555	+	18.2827i	0.406886	+	0.704748i	0.994539	−	0.104365i	$-0.0332811\pi$
$673$	10.5555	+	18.2827i	0.406886	+	0.704748i	−0.587653	+	0.809113i	$0.699948\pi$
$674$	−7.02704	−	12.1712i	−0.270672	−	0.468817i
$675$		0			0
$676$	25.8712	−	44.8102i	0.995046	−	1.72347i
$677$	10.5732	−	18.3133i	0.406361	−	0.703837i	−0.588118	−	0.808775i	$-0.700131\pi$
$677$	10.5732	−	18.3133i	0.406361	−	0.703837i	0.994479	+	0.104938i	$0.0334643\pi$
$678$		0			0
$679$		0			0
$680$	−44.8879	+	77.7482i	−1.72137	+	2.98151i
$681$		0			0
$682$	−7.02720			−0.269085
$683$	17.3858	−	30.1131i	0.665249	−	1.15224i	−0.313969	−	0.949433i	$-0.601659\pi$
$683$	17.3858	−	30.1131i	0.665249	−	1.15224i	0.979218	−	0.202811i	$-0.0650078\pi$
$684$		0			0
$685$	1.37998			0.0527264
$686$		0			0
$687$		0			0
$688$	−10.0718			−0.383984
$689$	−0.868609	−	1.50447i	−0.0330914	−	0.0573159i
$690$		0			0
$691$	−17.3246	+	30.0071i	−0.659059	+	1.14152i	0.321801	+	0.946807i	$0.395712\pi$
$691$	−17.3246	+	30.0071i	−0.659059	+	1.14152i	−0.980860	+	0.194716i	$0.937622\pi$
$692$	−24.5418			−0.932940
$693$		0			0
$694$	21.8525			0.829511
$695$	34.7542	−	60.1960i	1.31830	−	2.28336i
$696$		0			0
$697$	−18.2434	−	31.5985i	−0.691017	−	1.19688i
$698$	51.6087			1.95342
$699$		0			0
$700$		0			0
$701$	48.6050			1.83579			0.917894	−	0.396826i	$-0.129889\pi$
$701$	48.6050			1.83579			0.917894	+	0.396826i	$0.129889\pi$
$702$		0			0
$703$	−2.29661	+	3.97784i	−0.0866182	+	0.150027i
$704$	2.97802			0.112238
$705$		0			0
$706$	−18.1651	+	31.4629i	−0.683654	+	1.18412i
$707$		0			0
$708$		0			0
$709$	−2.05408	+	3.55778i	−0.0771428	+	0.133615i	−0.902016	−	0.431702i	$-0.857913\pi$
$709$	−2.05408	+	3.55778i	−0.0771428	+	0.133615i	0.824873	+	0.565318i	$0.191246\pi$
$710$	38.0753	−	65.9483i	1.42894	−	2.47500i
$711$		0			0
$712$	37.4764	+	64.9110i	1.40449	+	2.43264i
$713$	−16.3300	−	28.2844i	−0.611564	−	1.05926i
$714$		0			0
$715$	−0.361864	+	0.626767i	−0.0135330	+	0.0234398i
$716$	−37.0263			−1.38374
$717$		0			0
$718$	17.7453			0.662249
$719$	24.1408	+	41.8131i	0.900299	+	1.55936i	0.827106	+	0.562047i	$0.189986\pi$
$719$	24.1408	+	41.8131i	0.900299	+	1.55936i	0.0731939	+	0.997318i	$0.476681\pi$
$720$		0			0
$721$		0			0
$722$	−18.5833	−	32.1872i	−0.691597	−	1.19788i
$723$		0			0
$724$	24.2187	+	41.9481i	0.900082	+	1.55899i
$725$	32.0495	+	55.5114i	1.19029	+	2.06164i
$726$		0			0
$727$	20.5151	+	35.5332i	0.760863	+	1.31785i	0.942406	+	0.334470i	$0.108557\pi$
$727$	20.5151	+	35.5332i	0.760863	+	1.31785i	−0.181543	+	0.983383i	$0.558109\pi$
$728$		0			0
$729$		0			0
$730$	8.88151	+	15.3832i	0.328720	+	0.569359i
$731$	−11.3016			−0.418006
$732$		0			0
$733$	30.5428			1.12812			0.564062	−	0.825733i	$-0.309238\pi$
$733$	30.5428			1.12812			0.564062	+	0.825733i	$0.309238\pi$
$734$	13.5011	−	23.3845i	0.498334	−	0.863139i
$735$		0			0
$736$	−1.25370	−	2.17147i	−0.0462118	−	0.0800413i
$737$	0.731715	+	1.26737i	0.0269531	+	0.0466841i
$738$		0			0
$739$	−11.9100	+	20.6288i	−0.438117	+	0.758841i	−0.997544	−	0.0700384i	$-0.977688\pi$
$739$	−11.9100	+	20.6288i	−0.438117	+	0.758841i	0.559427	+	0.828880i	$0.311021\pi$
$740$	17.2581	−	29.8918i	0.634419	−	1.09885i
$741$		0			0
$742$		0			0
$743$	5.26089	−	9.11213i	0.193003	−	0.334292i	−0.753241	−	0.657745i	$-0.771510\pi$
$743$	5.26089	−	9.11213i	0.193003	−	0.334292i	0.946244	+	0.323453i	$0.104844\pi$
$744$		0			0
$745$	−35.4897			−1.30024
$746$	−0.668971	+	1.15869i	−0.0244928	+	0.0424227i
$747$		0			0
$748$	−8.00079			−0.292538
$749$		0			0
$750$		0			0
$751$	−10.2704			−0.374773			−0.187386	−	0.982286i	$-0.560002\pi$
$751$	−10.2704			−0.374773			−0.187386	+	0.982286i	$0.560002\pi$
$752$	13.6665	+	23.6711i	0.498367	+	0.863197i
$753$		0			0
$754$	−4.58005	+	7.93288i	−0.166796	+	0.288899i
$755$	46.9507			1.70871
$756$		0			0
$757$	8.03930			0.292193			0.146097	−	0.989270i	$-0.453329\pi$
$757$	8.03930			0.292193			0.146097	+	0.989270i	$0.453329\pi$
$758$	−27.9159	+	48.3518i	−1.01395	+	1.75622i
$759$		0			0
$760$	−18.2434	−	31.5985i	−0.661757	−	1.14620i
$761$	−27.6604			−1.00269			−0.501345	−	0.865247i	$-0.667161\pi$
$761$	−27.6604			−1.00269			−0.501345	+	0.865247i	$0.667161\pi$
$762$		0			0
$763$		0			0
$764$	−37.0554			−1.34062
$765$		0			0
$766$	43.9371	−	76.1013i	1.58751	−	2.74965i
$767$	2.97841			0.107544
$768$		0			0
$769$	16.9613	−	29.3778i	0.611640	−	1.05939i	−0.379324	−	0.925264i	$-0.623843\pi$
$769$	16.9613	−	29.3778i	0.611640	−	1.05939i	0.990964	−	0.134128i	$-0.0428233\pi$
$770$		0			0
$771$		0			0
$772$	−34.3442	+	59.4858i	−1.23607	+	2.14094i
$773$	−14.2978	+	24.7645i	−0.514256	+	0.890717i	0.485607	+	0.874177i	$0.338599\pi$
$773$	−14.2978	+	24.7645i	−0.514256	+	0.890717i	−0.999863	+	0.0165403i	$0.994735\pi$
$774$		0			0
$775$	29.4493	+	51.0077i	1.05785	+	1.83225i
$776$	23.9753	+	41.5265i	0.860664	+	1.49071i
$777$		0			0
$778$	−47.5605	+	82.3772i	−1.70513	+	2.95337i
$779$	14.8290			0.531303
$780$		0			0
$781$	3.43852			0.123040
$782$	−27.7647	−	48.0899i	−0.992864	−	1.71969i
$783$		0			0
$784$		0			0
$785$	−38.3157	−	66.3647i	−1.36754	−	2.36866i
$786$		0			0
$787$	−23.0017	−	39.8402i	−0.819923	−	1.42015i	−0.905738	−	0.423838i	$-0.860683\pi$
$787$	−23.0017	−	39.8402i	−0.819923	−	1.42015i	0.0858145	−	0.996311i	$-0.472651\pi$
$788$	−43.2111	−	74.8438i	−1.53933	−	2.66620i
$789$		0			0
$790$	−36.7330	−	63.6235i	−1.30690	−	2.26362i
$791$		0			0
$792$		0			0
$793$	−1.95379	−	3.38406i	−0.0693810	−	0.120171i
$794$	−29.3977			−1.04328
$795$		0

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.23025 - 2.13086i) q^{2} +(-2.02704 - 3.51094i) q^{4} -3.65808 q^{5} -5.05408 q^{8} +O(q^{10})\)	\(q+(1.23025 - 2.13086i) q^{2} +(-2.02704 - 3.51094i) q^{4} -3.65808 q^{5} -5.05408 q^{8} +(-4.50036 + 7.79485i) q^{10} -0.406421 q^{11} +(-0.243398 + 0.421578i) q^{13} +(-2.16372 + 3.74766i) q^{16} +(-2.42792 + 4.20528i) q^{17} +(-0.986757 - 1.70911i) q^{19} +(7.41507 + 12.8433i) q^{20} +(-0.500000 + 0.866025i) q^{22} -4.64766 q^{23} +8.38151 q^{25} +(0.598883 + 1.03729i) q^{26} +(3.82383 + 6.62307i) q^{29} +(3.51360 + 6.08573i) q^{31} +(0.269748 + 0.467216i) q^{32} +(5.97391 + 10.3471i) q^{34} +(-1.16372 - 2.01561i) q^{37} -4.85584 q^{38} +18.4882 q^{40} +(-3.75700 + 6.50731i) q^{41} +(1.16372 + 2.01561i) q^{43} +(0.823832 + 1.42692i) q^{44} +(-5.71780 + 9.90352i) q^{46} +(3.15811 - 5.47002i) q^{47} +(10.3114 - 17.8598i) q^{50} +1.97351 q^{52} +(-1.78434 + 3.09056i) q^{53} +1.48672 q^{55} +18.8171 q^{58} +(-3.05919 - 5.29868i) q^{59} +(-4.01356 + 6.95169i) q^{61} +17.2905 q^{62} -7.32743 q^{64} +(0.890369 - 1.54216i) q^{65} +(-1.80039 - 3.11836i) q^{67} +19.6860 q^{68} -8.46050 q^{71} +(0.986757 - 1.70911i) q^{73} -5.72665 q^{74} +(-4.00040 + 6.92889i) q^{76} +(-4.08113 + 7.06872i) q^{79} +(7.91503 - 13.7092i) q^{80} +(9.24411 + 16.0113i) q^{82} +(-6.08600 - 10.5413i) q^{83} +(8.88151 - 15.3832i) q^{85} +5.72665 q^{86} +2.05408 q^{88} +(-7.41507 - 12.8433i) q^{89} +(9.42101 + 16.3177i) q^{92} +(-7.77056 - 13.4590i) q^{94} +(3.60963 + 6.25206i) q^{95} +(-4.74375 - 8.21642i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12q + 2q^{2} - 6q^{4} - 24q^{8} + O(q^{10}) \)	\( 12q + 2q^{2} - 6q^{4} - 24q^{8} - 16q^{11} - 6q^{16} - 6q^{22} - 8q^{23} + 24q^{25} + 22q^{29} + 16q^{32} + 6q^{37} - 6q^{43} - 14q^{44} - 12q^{46} + 56q^{50} + 28q^{53} + 36q^{58} - 48q^{64} - 6q^{65} - 76q^{71} - 72q^{74} + 6q^{79} + 30q^{85} + 72q^{86} - 12q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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