LMFDB - Embedded newform 1323.2.g.f.361.1

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:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	10.0.991381711347.1
Defining polynomial:	$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$1.19343 - 2.06709i$ of defining polynomial
Character	$\chi$	$=$	1323.361
Dual form			1323.2.g.f.667.1

$q$-expansion

$f(q)$	$=$	$q+(-1.19343 + 2.06709i) q^{2} +(-1.84857 - 3.20182i) q^{4} -2.92087 q^{5} +4.05086 q^{8} +O(q^{10})$	\(q+(-1.19343 + 2.06709i) q^{2} +(-1.84857 - 3.20182i) q^{4} -2.92087 q^{5} +4.05086 q^{8} +(3.48586 - 6.03769i) q^{10} +1.35371 q^{11} +(0.733001 - 1.26960i) q^{13} +(-1.13729 + 1.96984i) q^{16} +(1.65514 - 2.86678i) q^{17} +(1.10329 + 1.91096i) q^{19} +(5.39943 + 9.35209i) q^{20} +(-1.61557 + 2.79825i) q^{22} -2.62830 q^{23} +3.53146 q^{25} +(1.74958 + 3.03036i) q^{26} +(-0.521720 - 0.903646i) q^{29} +(1.63729 + 2.83587i) q^{31} +(1.33629 + 2.31453i) q^{32} +(3.95060 + 6.84263i) q^{34} +(5.43773 + 9.41842i) q^{37} -5.26683 q^{38} -11.8320 q^{40} +(-0.904289 + 1.56627i) q^{41} +(-2.17129 - 3.76078i) q^{43} +(-2.50244 - 4.33435i) q^{44} +(3.13670 - 5.43292i) q^{46} +(-1.98957 + 3.44604i) q^{47} +(-4.21456 + 7.29984i) q^{50} -5.42002 q^{52} +(3.22743 - 5.59008i) q^{53} -3.95402 q^{55} +2.49056 q^{58} +(6.10700 + 10.5776i) q^{59} +(0.279867 - 0.484744i) q^{61} -7.81600 q^{62} -10.9283 q^{64} +(-2.14100 + 3.70832i) q^{65} +(-6.40588 - 11.0953i) q^{67} -12.2386 q^{68} -12.9177 q^{71} +(-5.22772 + 9.05467i) q^{73} -25.9583 q^{74} +(4.07903 - 7.06509i) q^{76} +(-0.383838 + 0.664827i) q^{79} +(3.32187 - 5.75365i) q^{80} +(-2.15842 - 3.73849i) q^{82} +(-0.983707 - 1.70383i) q^{83} +(-4.83443 + 8.37348i) q^{85} +10.3652 q^{86} +5.48371 q^{88} +(3.20356 + 5.54872i) q^{89} +(4.85859 + 8.41533i) q^{92} +(-4.74884 - 8.22524i) q^{94} +(-3.22257 - 5.58166i) q^{95} +(4.14143 + 7.17316i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) $	$ 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + 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 $

$2$	−1.19343	+	2.06709i	−0.843886	+	1.46165i	0.0426999	+	0.999088i	$0.486404\pi$
$2$	−1.19343	+	2.06709i	−0.843886	+	1.46165i	−0.886585	+	0.462565i	$0.846929\pi$
$3$		0			0
$3$		0			0
$4$	−1.84857	−	3.20182i	−0.924286	−	1.60091i
$5$	−2.92087			−1.30625			−0.653125	−	0.757250i	$-0.726543\pi$
$5$	−2.92087			−1.30625			−0.653125	+	0.757250i	$0.726543\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	4.05086			1.43219
$9$		0			0
$10$	3.48586	−	6.03769i	1.10233	−	1.90929i
$11$	1.35371			0.408160			0.204080	−	0.978954i	$-0.434580\pi$
$11$	1.35371			0.408160			0.204080	+	0.978954i	$0.434580\pi$
$12$		0			0
$13$	0.733001	−	1.26960i	0.203298	−	0.352123i	−0.746291	−	0.665620i	$-0.768167\pi$
$13$	0.733001	−	1.26960i	0.203298	−	0.352123i	0.949589	+	0.313497i	$0.101501\pi$
$14$		0			0
$15$		0			0
$16$	−1.13729	+	1.96984i	−0.284323	+	0.492461i
$17$	1.65514	−	2.86678i	0.401430	−	0.695297i	−0.592469	−	0.805593i	$-0.701847\pi$
$17$	1.65514	−	2.86678i	0.401430	−	0.695297i	0.993899	+	0.110297i	$0.0351801\pi$
$18$		0			0
$19$	1.10329	+	1.91096i	0.253113	+	0.438404i	0.964381	−	0.264516i	$-0.0852123\pi$
$19$	1.10329	+	1.91096i	0.253113	+	0.438404i	−0.711268	+	0.702921i	$0.751879\pi$
$20$	5.39943	+	9.35209i	1.20735	+	2.09119i
$21$		0			0
$22$	−1.61557	+	2.79825i	−0.344441	+	0.596589i
$23$	−2.62830			−0.548038			−0.274019	−	0.961724i	$-0.588353\pi$
$23$	−2.62830			−0.548038			−0.274019	+	0.961724i	$0.588353\pi$
$24$		0			0
$25$	3.53146			0.706292
$26$	1.74958	+	3.03036i	0.343121	+	0.594302i
$27$		0			0
$28$		0			0
$29$	−0.521720	−	0.903646i	−0.0968810	−	0.167803i	0.813511	−	0.581549i	$-0.197553\pi$
$29$	−0.521720	−	0.903646i	−0.0968810	−	0.167803i	−0.910392	+	0.413747i	$0.864220\pi$
$30$		0			0
$31$	1.63729	+	2.83587i	0.294066	+	0.509337i	0.974767	−	0.223224i	$-0.0716581\pi$
$31$	1.63729	+	2.83587i	0.294066	+	0.509337i	−0.680701	+	0.732561i	$0.738325\pi$
$32$	1.33629	+	2.31453i	0.236226	+	0.409155i
$33$		0			0
$34$	3.95060	+	6.84263i	0.677521	+	1.17350i
$35$		0			0
$36$		0			0
$37$	5.43773	+	9.41842i	0.893957	+	1.54838i	0.835090	+	0.550113i	$0.185415\pi$
$37$	5.43773	+	9.41842i	0.893957	+	1.54838i	0.0588664	+	0.998266i	$0.481251\pi$
$38$	−5.26683			−0.854393
$39$		0			0
$40$	−11.8320			−1.87081
$41$	−0.904289	+	1.56627i	−0.141226	+	0.244611i	−0.927959	−	0.372683i	$-0.878438\pi$
$41$	−0.904289	+	1.56627i	−0.141226	+	0.244611i	0.786732	+	0.617294i	$0.211771\pi$
$42$		0			0
$43$	−2.17129	−	3.76078i	−0.331118	−	0.573514i	0.651613	−	0.758551i	$-0.274093\pi$
$43$	−2.17129	−	3.76078i	−0.331118	−	0.573514i	−0.982731	+	0.185038i	$0.940759\pi$
$44$	−2.50244	−	4.33435i	−0.377257	−	0.653428i
$45$		0			0
$46$	3.13670	−	5.43292i	0.462481	−	0.801041i
$47$	−1.98957	+	3.44604i	−0.290209	+	0.502656i	−0.973859	−	0.227154i	$-0.927058\pi$
$47$	−1.98957	+	3.44604i	−0.290209	+	0.502656i	0.683650	+	0.729810i	$0.260391\pi$
$48$		0			0
$49$		0			0
$50$	−4.21456	+	7.29984i	−0.596029	+	1.03235i
$51$		0			0
$52$	−5.42002			−0.751622
$53$	3.22743	−	5.59008i	0.443322	−	0.767856i	−0.554612	−	0.832109i	$-0.687133\pi$
$53$	3.22743	−	5.59008i	0.443322	−	0.767856i	0.997934	+	0.0642533i	$0.0204666\pi$
$54$		0			0
$55$	−3.95402			−0.533160
$56$		0			0
$57$		0			0
$58$	2.49056			0.327026
$59$	6.10700	+	10.5776i	0.795064	+	1.37709i	0.922799	+	0.385283i	$0.125896\pi$
$59$	6.10700	+	10.5776i	0.795064	+	1.37709i	−0.127735	+	0.991808i	$0.540771\pi$
$60$		0			0
$61$	0.279867	−	0.484744i	0.0358333	−	0.0620651i	−0.847553	−	0.530711i	$-0.821925\pi$
$61$	0.279867	−	0.484744i	0.0358333	−	0.0620651i	0.883386	+	0.468646i	$0.155258\pi$
$62$	−7.81600			−0.992632
$63$		0			0
$64$	−10.9283			−1.36604
$65$	−2.14100	+	3.70832i	−0.265558	+	0.459960i
$66$		0			0
$67$	−6.40588	−	11.0953i	−0.782603	−	1.35551i	−0.930420	−	0.366494i	$-0.880558\pi$
$67$	−6.40588	−	11.0953i	−0.782603	−	1.35551i	0.147817	−	0.989015i	$-0.452775\pi$
$68$	−12.2386			−1.48414
$69$		0			0
$70$		0			0
$71$	−12.9177			−1.53305			−0.766525	−	0.642214i	$-0.778016\pi$
$71$	−12.9177			−1.53305			−0.766525	+	0.642214i	$0.778016\pi$
$72$		0			0
$73$	−5.22772	+	9.05467i	−0.611858	+	1.05977i	0.379069	+	0.925368i	$0.376244\pi$
$73$	−5.22772	+	9.05467i	−0.611858	+	1.05977i	−0.990927	+	0.134401i	$0.957089\pi$
$74$	−25.9583			−3.01759
$75$		0			0
$76$	4.07903	−	7.06509i	0.467897	−	0.810422i
$77$		0			0
$78$		0			0
$79$	−0.383838	+	0.664827i	−0.0431852	+	0.0747989i	−0.886810	−	0.462134i	$-0.847084\pi$
$79$	−0.383838	+	0.664827i	−0.0431852	+	0.0747989i	0.843625	+	0.536933i	$0.180417\pi$
$80$	3.32187	−	5.75365i	0.371397	−	0.643278i
$81$		0			0
$82$	−2.15842	−	3.73849i	−0.238358	−	0.412847i
$83$	−0.983707	−	1.70383i	−0.107976	−	0.187020i	0.806974	−	0.590587i	$-0.201104\pi$
$83$	−0.983707	−	1.70383i	−0.107976	−	0.187020i	−0.914950	+	0.403567i	$0.867770\pi$
$84$		0			0
$85$	−4.83443	+	8.37348i	−0.524368	+	0.908232i
$86$	10.3652			1.11770
$87$		0			0
$88$	5.48371			0.584565
$89$	3.20356	+	5.54872i	0.339576	+	0.588163i	0.984353	−	0.176208i	$-0.0563830\pi$
$89$	3.20356	+	5.54872i	0.339576	+	0.588163i	−0.644777	+	0.764371i	$0.723050\pi$
$90$		0			0
$91$		0			0
$92$	4.85859	+	8.41533i	0.506543	+	0.877359i
$93$		0			0
$94$	−4.74884	−	8.22524i	−0.489806	−	0.848369i
$95$	−3.22257	−	5.58166i	−0.330629	−	0.572666i
$96$		0			0
$97$	4.14143	+	7.17316i	0.420498	+	0.728324i	0.995988	−	0.0894847i	$-0.0285220\pi$
$97$	4.14143	+	7.17316i	0.420498	+	0.728324i	−0.575490	+	0.817809i	$0.695189\pi$
$98$		0			0
$99$		0			0
$100$	−6.52815	−	11.3071i	−0.652815	−	1.13071i
$101$	−16.2266			−1.61461			−0.807305	−	0.590134i	$-0.799075\pi$
$101$	−16.2266			−1.61461			−0.807305	+	0.590134i	$0.799075\pi$
$102$		0			0
$103$	2.22683			0.219416			0.109708	−	0.993964i	$-0.465008\pi$
$103$	2.22683			0.219416			0.109708	+	0.993964i	$0.465008\pi$
$104$	2.96929	−	5.14295i	0.291162	−	0.504308i
$105$		0			0
$106$	7.70346	+	13.3428i	0.748226	+	1.29597i
$107$	8.75403	+	15.1624i	0.846284	+	1.46581i	0.884501	+	0.466537i	$0.154499\pi$
$107$	8.75403	+	15.1624i	0.846284	+	1.46581i	−0.0382175	+	0.999269i	$0.512168\pi$
$108$		0			0
$109$	−7.79917	+	13.5086i	−0.747025	+	1.29388i	0.202218	+	0.979341i	$0.435185\pi$
$109$	−7.79917	+	13.5086i	−0.747025	+	1.29388i	−0.949243	+	0.314544i	$0.898148\pi$
$110$	4.71886	−	8.17331i	0.449926	−	0.779295i
$111$		0			0
$112$		0			0
$113$	0.844555	−	1.46281i	0.0794491	−	0.137610i	−0.823563	−	0.567224i	$-0.808017\pi$
$113$	0.844555	−	1.46281i	0.0794491	−	0.137610i	0.903012	+	0.429615i	$0.141351\pi$
$114$		0			0
$115$	7.67690			0.715875
$116$	−1.92887	+	3.34091i	−0.179092	+	0.310196i
$117$		0			0
$118$	−29.1532			−2.68377
$119$		0			0
$120$		0			0
$121$	−9.16746			−0.833405
$122$	0.668005	+	1.15702i	0.0604784	+	0.104752i
$123$		0			0
$124$	6.05330	−	10.4846i	0.543602	−	0.941546i
$125$	4.28942			0.383657
$126$		0			0
$127$	−3.96918			−0.352208			−0.176104	−	0.984372i	$-0.556350\pi$
$127$	−3.96918			−0.352208			−0.176104	+	0.984372i	$0.556350\pi$
$128$	10.3696	−	17.9607i	0.916552	−	1.58751i
$129$		0			0
$130$	−5.11028	−	8.85127i	−0.448202	−	0.776308i
$131$	5.32863			0.465565			0.232782	−	0.972529i	$-0.425217\pi$
$131$	5.32863			0.465565			0.232782	+	0.972529i	$0.425217\pi$
$132$		0			0
$133$		0			0
$134$	30.5800			2.64171
$135$		0			0
$136$	6.70473	−	11.6129i	0.574925	−	0.995800i
$137$	7.49543			0.640378			0.320189	−	0.947354i	$-0.396254\pi$
$137$	7.49543			0.640378			0.320189	+	0.947354i	$0.396254\pi$
$138$		0			0
$139$	−7.03285	+	12.1812i	−0.596518	+	1.03320i	0.396812	+	0.917900i	$0.370116\pi$
$139$	−7.03285	+	12.1812i	−0.596518	+	1.03320i	−0.993331	+	0.115300i	$0.963217\pi$
$140$		0			0
$141$		0			0
$142$	15.4164	−	26.7021i	1.29372	−	2.24079i
$143$	0.992275	−	1.71867i	0.0829782	−	0.143722i
$144$		0			0
$145$	1.52388	+	2.63943i	0.126551	+	0.219193i
$146$	−12.4779	−	21.6123i	−1.03268	−	1.78865i
$147$		0			0
$148$	20.1041	−	34.8212i	1.65254	−	2.86229i
$149$	−2.17971			−0.178569			−0.0892846	−	0.996006i	$-0.528458\pi$
$149$	−2.17971			−0.178569			−0.0892846	+	0.996006i	$0.528458\pi$
$150$		0			0
$151$	14.0277			1.14156			0.570781	−	0.821102i	$-0.306641\pi$
$151$	14.0277			1.14156			0.570781	+	0.821102i	$0.306641\pi$
$152$	4.46929	+	7.74103i	0.362507	+	0.627880i
$153$		0			0
$154$		0			0
$155$	−4.78231	−	8.28320i	−0.384124	−	0.665322i
$156$		0			0
$157$	1.48312	+	2.56883i	0.118365	+	0.205015i	0.919120	−	0.393978i	$-0.128901\pi$
$157$	1.48312	+	2.56883i	0.118365	+	0.205015i	−0.800755	+	0.598993i	$0.795568\pi$
$158$	−0.916172	−	1.58686i	−0.0728867	−	0.126243i
$159$		0			0
$160$	−3.90314	−	6.76043i	−0.308570	−	0.534459i
$161$		0			0
$162$		0			0
$163$	−0.194278	−	0.336499i	−0.0152170	−	0.0263566i	0.858317	−	0.513120i	$-0.171511\pi$
$163$	−0.194278	−	0.336499i	−0.0152170	−	0.0263566i	−0.873534	+	0.486764i	$0.838177\pi$
$164$	6.68657			0.522133
$165$		0			0
$166$	4.69596			0.364477
$167$	3.64889	−	6.32006i	0.282360	−	0.489061i	−0.689606	−	0.724185i	$-0.742216\pi$
$167$	3.64889	−	6.32006i	0.282360	−	0.489061i	0.971965	+	0.235124i	$0.0755496\pi$
$168$		0			0
$169$	5.42542	+	9.39710i	0.417340	+	0.722854i
$170$	−11.5392	−	19.9864i	−0.885013	−	1.53289i
$171$		0			0
$172$	−8.02756	+	13.9041i	−0.612096	+	1.06018i
$173$	2.02754	−	3.51181i	0.154151	−	0.266998i	−0.778598	−	0.627522i	$-0.784069\pi$
$173$	2.02754	−	3.51181i	0.154151	−	0.266998i	0.932750	+	0.360525i	$0.117402\pi$
$174$		0			0
$175$		0			0
$176$	−1.53957	+	2.66661i	−0.116049	+	0.201003i
$177$		0			0
$178$	−15.2929			−1.14625
$179$	−5.29243	+	9.16675i	−0.395575	+	0.685155i	−0.993174	−	0.116639i	$-0.962788\pi$
$179$	−5.29243	+	9.16675i	−0.395575	+	0.685155i	0.597600	+	0.801795i	$0.296121\pi$
$180$		0			0
$181$	19.6312			1.45917			0.729586	−	0.683889i	$-0.239713\pi$
$181$	19.6312			1.45917			0.729586	+	0.683889i	$0.239713\pi$
$182$		0			0
$183$		0			0
$184$	−10.6469			−0.784896
$185$	−15.8829	−	27.5099i	−1.16773	−	2.02257i
$186$		0			0
$187$	2.24058	−	3.88081i	0.163848	−	0.283793i
$188$	14.7115			1.07294
$189$		0			0
$190$	15.3837			1.11605
$191$	4.14357	−	7.17688i	0.299818	−	0.519301i	−0.676276	−	0.736648i	$-0.736407\pi$
$191$	4.14357	−	7.17688i	0.299818	−	0.519301i	0.976094	+	0.217348i	$0.0697406\pi$
$192$		0			0
$193$	9.39242	+	16.2682i	0.676082	+	1.17101i	0.976152	+	0.217090i	$0.0696566\pi$
$193$	9.39242	+	16.2682i	0.676082	+	1.17101i	−0.300070	+	0.953917i	$0.597010\pi$
$194$	−19.7701			−1.41941
$195$		0			0
$196$		0			0
$197$	−5.99634			−0.427222			−0.213611	−	0.976919i	$-0.568522\pi$
$197$	−5.99634			−0.427222			−0.213611	+	0.976919i	$0.568522\pi$
$198$		0			0
$199$	−7.20434	+	12.4783i	−0.510702	+	0.884562i	0.489221	+	0.872160i	$0.337281\pi$
$199$	−7.20434	+	12.4783i	−0.510702	+	0.884562i	−0.999923	+	0.0124022i	$0.996052\pi$
$200$	14.3054			1.01155
$201$		0			0
$202$	19.3654	−	33.5419i	1.36255	−	2.36000i
$203$		0			0
$204$		0			0
$205$	2.64131	−	4.57488i	0.184477	−	0.319523i
$206$	−2.65758	+	4.60306i	−0.185162	+	0.320710i
$207$		0			0
$208$	1.66727	+	2.88780i	0.115604	+	0.200233i
$209$	1.49354	+	2.58690i	0.103311	+	0.178939i
$210$		0			0
$211$	−6.92418	+	11.9930i	−0.476680	+	0.825634i	−0.999643	−	0.0267212i	$-0.991493\pi$
$211$	−6.92418	+	11.9930i	−0.476680	+	0.825634i	0.522963	+	0.852356i	$0.324827\pi$
$212$	−23.8646			−1.63902
$213$		0			0
$214$	−41.7894			−2.85667
$215$	6.34204	+	10.9847i	0.432523	+	0.749153i
$216$		0			0
$217$		0			0
$218$	−18.6156	−	32.2431i	−1.26081	−	2.18378i
$219$		0			0
$220$	7.30929	+	12.6601i	0.492792	+	0.853541i
$221$	−2.42644	−	4.20271i	−0.163220	−	0.282705i
$222$		0			0
$223$	−2.33756	−	4.04878i	−0.156535	−	0.271126i	0.777082	−	0.629399i	$-0.216699\pi$
$223$	−2.33756	−	4.04878i	−0.156535	−	0.271126i	−0.933617	+	0.358273i	$0.883366\pi$
$224$		0			0
$225$		0			0
$226$	2.01584	+	3.49154i	0.134092	+	0.232254i
$227$	19.7126			1.30837			0.654187	−	0.756333i	$-0.273011\pi$
$227$	19.7126			1.30837			0.654187	+	0.756333i	$0.273011\pi$
$228$		0			0
$229$	−28.0728			−1.85510			−0.927552	−	0.373694i	$-0.878091\pi$
$229$	−28.0728			−1.85510			−0.927552	+	0.373694i	$0.878091\pi$
$230$	−9.16188	+	15.8688i	−0.604116	+	1.04636i
$231$		0			0
$232$	−2.11342	−	3.66054i	−0.138753	−	0.240326i
$233$	6.90113	+	11.9531i	0.452108	+	0.783074i	0.998517	−	0.0544448i	$-0.0173389\pi$
$233$	6.90113	+	11.9531i	0.452108	+	0.783074i	−0.546409	+	0.837518i	$0.684006\pi$
$234$		0			0
$235$	5.81127	−	10.0654i	0.379085	−	0.656595i
$236$	22.5785	−	39.1070i	1.46973	−	2.54565i
$237$		0			0
$238$		0			0
$239$	−5.53069	+	9.57944i	−0.357751	+	0.619642i	−0.987585	−	0.157087i	$-0.949790\pi$
$239$	−5.53069	+	9.57944i	−0.357751	+	0.619642i	0.629834	+	0.776730i	$0.283123\pi$
$240$		0			0
$241$	23.1697			1.49249			0.746247	−	0.665669i	$-0.231854\pi$
$241$	23.1697			1.49249			0.746247	+	0.665669i	$0.231854\pi$
$242$	10.9408	−	18.9499i	0.703299	−	1.21815i
$243$		0			0
$244$	−2.06942			−0.132481
$245$		0			0
$246$		0			0
$247$	3.23486			0.205829
$248$	6.63243	+	11.4877i	0.421160	+	0.729470i
$249$		0			0
$250$	−5.11914	+	8.86660i	−0.323763	+	0.560773i
$251$	−7.78402			−0.491323			−0.245662	−	0.969356i	$-0.579005\pi$
$251$	−7.78402			−0.491323			−0.245662	+	0.969356i	$0.579005\pi$
$252$		0			0
$253$	−3.55796			−0.223687
$254$	4.73696	−	8.20466i	0.297223	−	0.514806i
$255$		0			0
$256$	13.8226	+	23.9414i	0.863912	+	1.49634i
$257$	10.3760			0.647235			0.323618	−	0.946188i	$-0.395101\pi$
$257$	10.3760			0.647235			0.323618	+	0.946188i	$0.395101\pi$
$258$		0			0
$259$		0			0
$260$	15.8312			0.981807
$261$		0			0
$262$	−6.35937	+	11.0148i	−0.392883	+	0.680494i
$263$	19.1331			1.17980			0.589898	−	0.807478i	$-0.299168\pi$
$263$	19.1331			1.17980			0.589898	+	0.807478i	$0.299168\pi$
$264$		0			0
$265$	−9.42689	+	16.3279i	−0.579090	+	1.00301i
$266$		0			0
$267$		0			0
$268$	−23.6835	+	41.0210i	−1.44670	+	2.50576i
$269$	−4.41840	+	7.65290i	−0.269395	+	0.466605i	−0.968706	−	0.248212i	$-0.920157\pi$
$269$	−4.41840	+	7.65290i	−0.269395	+	0.466605i	0.699311	+	0.714818i	$0.253490\pi$
$270$		0			0
$271$	9.16955	+	15.8821i	0.557010	+	0.964770i	0.997744	+	0.0671321i	$0.0213849\pi$
$271$	9.16955	+	15.8821i	0.557010	+	0.964770i	−0.440734	+	0.897638i	$0.645282\pi$
$272$	3.76474	+	6.52073i	0.228271	+	0.395377i
$273$		0			0
$274$	−8.94531	+	15.4937i	−0.540406	+	0.936010i
$275$	4.78059			0.288280
$276$		0			0
$277$	5.10482			0.306719			0.153360	−	0.988170i	$-0.450991\pi$
$277$	5.10482			0.306719			0.153360	+	0.988170i	$0.450991\pi$
$278$	−16.7865	−	29.0750i	−1.00679	−	1.74381i
$279$		0			0
$280$		0			0
$281$	0.853180	+	1.47775i	0.0508964	+	0.0881552i	0.890351	−	0.455274i	$-0.150459\pi$
$281$	0.853180	+	1.47775i	0.0508964	+	0.0881552i	−0.839455	+	0.543430i	$0.817125\pi$
$282$		0			0
$283$	−6.24415	−	10.8152i	−0.371176	−	0.642896i	0.618571	−	0.785729i	$-0.287712\pi$
$283$	−6.24415	−	10.8152i	−0.371176	−	0.642896i	−0.989747	+	0.142833i	$0.954379\pi$
$284$	23.8793	+	41.3602i	1.41698	+	2.45428i
$285$		0			0
$286$	2.36843	+	4.10224i	0.140048	+	0.242571i
$287$		0			0
$288$		0			0
$289$	3.02104	+	5.23260i	0.177708	+	0.307800i
$290$	−7.27458			−0.427178
$291$		0			0
$292$	38.6552			2.26213
$293$	−2.60202	+	4.50684i	−0.152012	+	0.263292i	−0.931967	−	0.362543i	$-0.881909\pi$
$293$	−2.60202	+	4.50684i	−0.152012	+	0.263292i	0.779955	+	0.625835i	$0.215242\pi$
$294$		0			0
$295$	−17.8377	−	30.8959i	−1.03855	−	1.79883i
$296$	22.0275	+	38.1527i	1.28032	+	2.21758i
$297$		0			0
$298$	2.60135	−	4.50566i	0.150692	−	0.261006i
$299$	−1.92654	+	3.33687i	−0.111415	+	0.192976i
$300$		0			0
$301$		0			0
$302$	−16.7412	+	28.9966i	−0.963347	+	1.66857i
$303$		0			0
$304$	−5.01906			−0.287863
$305$	−0.817453	+	1.41587i	−0.0468072	+	0.0810725i
$306$		0			0
$307$	−5.00136			−0.285442			−0.142721	−	0.989763i	$-0.545585\pi$
$307$	−5.00136			−0.285442			−0.142721	+	0.989763i	$0.545585\pi$
$308$		0			0
$309$		0			0
$310$	22.8295			1.29663
$311$	16.1984	+	28.0565i	0.918528	+	1.59094i	0.801652	+	0.597791i	$0.203955\pi$
$311$	16.1984	+	28.0565i	0.918528	+	1.59094i	0.116876	+	0.993146i	$0.462712\pi$
$312$		0			0
$313$	0.759535	−	1.31555i	0.0429315	−	0.0743595i	−0.843761	−	0.536719i	$-0.819664\pi$
$313$	0.759535	−	1.31555i	0.0429315	−	0.0743595i	0.886693	+	0.462359i	$0.152997\pi$
$314$	−7.08000			−0.399548
$315$		0			0
$316$	2.83821			0.159662
$317$	−10.7544	+	18.6272i	−0.604029	+	1.04621i	0.388175	+	0.921586i	$0.373106\pi$
$317$	−10.7544	+	18.6272i	−0.604029	+	1.04621i	−0.992204	+	0.124623i	$0.960228\pi$
$318$		0			0
$319$	−0.706261	−	1.22328i	−0.0395430	−	0.0684905i
$320$	31.9200			1.78439
$321$		0			0
$322$		0			0
$323$	7.30441			0.406428
$324$		0			0
$325$	2.58856	−	4.48352i	0.143588	−	0.248701i
$326$	0.927430			0.0513656
$327$		0			0
$328$	−3.66315	+	6.34476i	−0.202263	+	0.350330i
$329$		0			0
$330$		0			0
$331$	−9.73902	+	16.8685i	−0.535305	+	0.927175i	0.463844	+	0.885917i	$0.346470\pi$
$331$	−9.73902	+	16.8685i	−0.535305	+	0.927175i	−0.999149	+	0.0412580i	$0.986863\pi$
$332$	−3.63691	+	6.29931i	−0.199601	+	0.345719i
$333$		0			0
$334$	8.70942	+	15.0852i	0.476558	+	0.825423i
$335$	18.7107	+	32.4079i	1.02228	+	1.77063i
$336$		0			0
$337$	4.84742	−	8.39598i	0.264056	−	0.457358i	−0.703260	−	0.710933i	$-0.748273\pi$
$337$	4.84742	−	8.39598i	0.264056	−	0.457358i	0.967316	+	0.253575i	$0.0816063\pi$
$338$	−25.8995			−1.40875
$339$		0			0
$340$	35.7472			1.93866
$341$	2.21642	+	3.83896i	0.120026	+	0.207891i
$342$		0			0
$343$		0			0
$344$	−8.79558	−	15.2344i	−0.474226	−	0.821383i
$345$		0			0
$346$	4.83948	+	8.38222i	0.260172	+	0.450631i
$347$	1.01302	+	1.75460i	0.0543817	+	0.0941919i	0.891935	−	0.452164i	$-0.149348\pi$
$347$	1.01302	+	1.75460i	0.0543817	+	0.0941919i	−0.837553	+	0.546356i	$0.816015\pi$
$348$		0			0
$349$	−8.14577	−	14.1089i	−0.436033	−	0.755231i	0.561346	−	0.827581i	$-0.310284\pi$
$349$	−8.14577	−	14.1089i	−0.436033	−	0.755231i	−0.997379	+	0.0723497i	$0.976950\pi$
$350$		0			0
$351$		0			0
$352$	1.80896	+	3.13321i	0.0964180	+	0.167001i
$353$	17.0614			0.908089			0.454045	−	0.890979i	$-0.349981\pi$
$353$	17.0614			0.908089			0.454045	+	0.890979i	$0.349981\pi$
$354$		0			0
$355$	37.7309			2.00255
$356$	11.8440	−	20.5144i	0.627731	−	1.08726i
$357$		0			0
$358$	−12.6323	−	21.8798i	−0.667639	−	1.15639i
$359$	−1.48363	−	2.56972i	−0.0783030	−	0.135625i	0.824215	−	0.566277i	$-0.191617\pi$
$359$	−1.48363	−	2.56972i	−0.0783030	−	0.135625i	−0.902518	+	0.430652i	$0.858283\pi$
$360$		0			0
$361$	7.06549	−	12.2378i	0.371868	−	0.644094i
$362$	−23.4285	+	40.5794i	−1.23137	+	2.13280i
$363$		0			0
$364$		0			0
$365$	15.2695	−	26.4475i	0.799240	−	1.38432i
$366$		0			0
$367$	10.1575			0.530216			0.265108	−	0.964219i	$-0.414592\pi$
$367$	10.1575			0.530216			0.265108	+	0.964219i	$0.414592\pi$
$368$	2.98914	−	5.17733i	0.155819	−	0.269887i
$369$		0			0
$370$	75.8207			3.94173
$371$		0			0
$372$		0			0
$373$	−25.4846			−1.31954			−0.659771	−	0.751467i	$-0.729347\pi$
$373$	−25.4846			−1.31954			−0.659771	+	0.751467i	$0.729347\pi$
$374$	5.34798	+	9.26297i	0.276537	+	0.478977i
$375$		0			0
$376$	−8.05947	+	13.9594i	−0.415635	+	0.719902i
$377$	−1.52969			−0.0787829
$378$		0			0
$379$	9.85497			0.506216			0.253108	−	0.967438i	$-0.418547\pi$
$379$	9.85497			0.506216			0.253108	+	0.967438i	$0.418547\pi$
$380$	−11.9143	+	20.6362i	−0.611191	+	1.05861i
$381$		0			0
$382$	9.89016	+	17.1303i	0.506025	+	0.876460i
$383$	−27.3127			−1.39561			−0.697806	−	0.716286i	$-0.745840\pi$
$383$	−27.3127			−1.39561			−0.697806	+	0.716286i	$0.745840\pi$
$384$		0			0
$385$		0			0
$386$	−44.8370			−2.28214
$387$		0			0
$388$	15.3114	−	26.5202i	0.777321	−	1.34636i
$389$	−4.18446			−0.212161			−0.106080	−	0.994358i	$-0.533830\pi$
$389$	−4.18446			−0.212161			−0.106080	+	0.994358i	$0.533830\pi$
$390$		0			0
$391$	−4.35019	+	7.53475i	−0.219999	+	0.381049i
$392$		0			0
$393$		0			0
$394$	7.15624	−	12.3950i	0.360526	−	0.624450i
$395$	1.12114	−	1.94187i	0.0564107	−	0.0977062i
$396$		0			0
$397$	−15.3354	−	26.5618i	−0.769664	−	1.33310i	−0.937745	−	0.347323i	$-0.887091\pi$
$397$	−15.3354	−	26.5618i	−0.769664	−	1.33310i	0.168082	−	0.985773i	$-0.446243\pi$
$398$	−17.1958	−	29.7840i	−0.861948	−	1.49294i
$399$		0			0
$400$	−4.01629	+	6.95642i	−0.200815	+	0.347821i
$401$	6.84803			0.341974			0.170987	−	0.985273i	$-0.445304\pi$
$401$	6.84803			0.341974			0.170987	+	0.985273i	$0.445304\pi$
$402$		0			0
$403$	4.80055			0.239132
$404$	29.9961	+	51.9547i	1.49236	+	2.58485i
$405$		0			0
$406$		0			0
$407$	7.36113	+	12.7499i	0.364878	+	0.631987i
$408$		0			0
$409$	−9.13490	−	15.8221i	−0.451692	−	0.782353i	0.546799	−	0.837264i	$-0.315846\pi$
$409$	−9.13490	−	15.8221i	−0.451692	−	0.782353i	−0.998491	+	0.0549104i	$0.982513\pi$
$410$	6.30445	+	10.9196i	0.311355	+	0.539282i
$411$		0			0
$412$	−4.11646	−	7.12991i	−0.202803	−	0.351265i
$413$		0			0
$414$		0			0
$415$	2.87328	+	4.97666i	0.141044	+	0.244295i
$416$	3.91802			0.192097
$417$		0			0
$418$	−7.12979			−0.348729
$419$	11.2310	−	19.4526i	0.548669	−	0.950322i	−0.449698	−	0.893181i	$-0.648468\pi$
$419$	11.2310	−	19.4526i	0.548669	−	0.950322i	0.998366	−	0.0571410i	$-0.0181984\pi$
$420$		0			0
$421$	10.4177	+	18.0440i	0.507728	+	0.879411i	0.999960	+	0.00894684i	$0.00284791\pi$
$421$	10.4177	+	18.0440i	0.507728	+	0.879411i	−0.492232	+	0.870464i	$0.663819\pi$
$422$	−16.5271	−	28.6258i	−0.804527	−	1.39348i
$423$		0			0
$424$	13.0739	−	22.6446i	0.634923	−	1.09972i
$425$	5.84505	−	10.1239i	0.283526	−	0.491082i
$426$		0			0
$427$		0			0
$428$	32.3649	−	56.0577i	1.56442	−	2.70965i
$429$		0			0
$430$	−30.2752			−1.46000
$431$	10.1213	−	17.5307i	0.487527	−	0.844422i	−0.512370	−	0.858765i	$-0.671232\pi$
$431$	10.1213	−	17.5307i	0.487527	−	0.844422i	0.999897	+	0.0143427i	$0.00456557\pi$
$432$		0			0
$433$	21.6764			1.04170			0.520851	−	0.853648i	$-0.325615\pi$
$433$	21.6764			1.04170			0.520851	+	0.853648i	$0.325615\pi$
$434$		0			0
$435$		0			0
$436$	57.6693			2.76186
$437$	−2.89978	−	5.02257i	−0.138715	−	0.240262i
$438$		0			0
$439$	−17.7390	+	30.7249i	−0.846639	+	1.46642i	0.0375520	+	0.999295i	$0.488044\pi$
$439$	−17.7390	+	30.7249i	−0.846639	+	1.46642i	−0.884191	+	0.467126i	$0.845289\pi$
$440$	−16.0172			−0.763589
$441$		0			0
$442$	11.5832			0.550955
$443$	−9.60313	+	16.6331i	−0.456258	+	0.790263i	−0.998760	−	0.0497923i	$-0.984144\pi$
$443$	−9.60313	+	16.6331i	−0.456258	+	0.790263i	0.542501	+	0.840055i	$0.317477\pi$
$444$		0			0
$445$	−9.35716	−	16.2071i	−0.443572	−	0.768289i
$446$	11.1589			0.528390
$447$		0			0
$448$		0			0
$449$	29.6082			1.39730			0.698648	−	0.715465i	$-0.253785\pi$
$449$	29.6082			1.39730			0.698648	+	0.715465i	$0.253785\pi$
$450$		0			0
$451$	−1.22415	+	2.12029i	−0.0576429	+	0.0998405i
$452$	−6.24488			−0.293735
$453$		0			0
$454$	−23.5257	+	40.7478i	−1.10412	+	1.91239i
$455$		0			0
$456$		0			0
$457$	4.78098	−	8.28090i	0.223645	−	0.387364i	−0.732267	−	0.681017i	$-0.761538\pi$
$457$	4.78098	−	8.28090i	0.223645	−	0.387364i	0.955912	+	0.293653i	$0.0948711\pi$
$458$	33.5031	−	58.0290i	1.56550	−	2.71152i
$459$		0			0
$460$	−14.1913	−	24.5800i	−0.661673	−	1.14605i
$461$	10.9187	+	18.9118i	0.508536	+	0.880809i	0.999951	+	0.00988416i	$0.00314628\pi$
$461$	10.9187	+	18.9118i	0.508536	+	0.880809i	−0.491416	+	0.870925i	$0.663520\pi$
$462$		0			0
$463$	13.0744	−	22.6456i	0.607621	−	1.05243i	−0.384010	−	0.923329i	$-0.625457\pi$
$463$	13.0744	−	22.6456i	0.607621	−	1.05243i	0.991631	−	0.129102i	$-0.0412094\pi$
$464$	2.37339			0.110182
$465$		0			0
$466$	−32.9442			−1.52611
$467$	−17.4764	−	30.2699i	−0.808709	−	1.40073i	−0.913758	−	0.406258i	$-0.866833\pi$
$467$	−17.4764	−	30.2699i	−0.808709	−	1.40073i	0.105049	−	0.994467i	$-0.466500\pi$
$468$		0			0
$469$		0			0
$470$	13.8707	+	24.0248i	0.639809	+	1.10818i
$471$		0			0
$472$	24.7386	+	42.8485i	1.13869	+	1.97226i
$473$	−2.93930	−	5.09102i	−0.135149	−	0.234086i
$474$		0			0
$475$	3.89623	+	6.74848i	0.178771	+	0.309641i
$476$		0			0
$477$		0			0
$478$	−13.2010	−	22.8649i	−0.603801	−	1.04581i
$479$	−29.8109			−1.36209			−0.681047	−	0.732240i	$-0.738475\pi$
$479$	−29.8109			−1.36209			−0.681047	+	0.732240i	$0.738475\pi$
$480$		0			0
$481$	15.9434			0.726959
$482$	−27.6516	+	47.8939i	−1.25949	+	2.18151i
$483$		0			0
$484$	16.9467	+	29.3525i	0.770304	+	1.33421i
$485$	−12.0965	−	20.9518i	−0.549276	−	0.951374i
$486$		0			0
$487$	−11.2253	+	19.4428i	−0.508667	+	0.881037i	0.491283	+	0.871000i	$0.336528\pi$
$487$	−11.2253	+	19.4428i	−0.508667	+	0.881037i	−0.999950	+	0.0100365i	$0.996805\pi$
$488$	1.13370	−	1.96363i	0.0513202	−	0.0888892i
$489$		0			0
$490$		0			0
$491$	−17.5222	+	30.3494i	−0.790767	+	1.36965i	0.134726	+	0.990883i	$0.456984\pi$
$491$	−17.5222	+	30.3494i	−0.790767	+	1.36965i	−0.925493	+	0.378765i	$0.876349\pi$
$492$		0			0
$493$	−3.45407			−0.155564
$494$	−3.86060	+	6.68675i	−0.173696	+	0.300851i
$495$		0			0
$496$	−7.44830			−0.334438
$497$		0			0
$498$		0			0
$499$	−8.93520			−0.399994			−0.199997	−	0.979796i	$-0.564093\pi$
$499$	−8.93520			−0.399994			−0.199997	+	0.979796i	$0.564093\pi$
$500$	−7.92929	−	13.7339i	−0.354609	−	0.614200i
$501$		0			0
$502$	9.28972	−	16.0903i	0.414621	−	0.718144i
$503$	−12.6403			−0.563603			−0.281802	−	0.959473i	$-0.590932\pi$
$503$	−12.6403			−0.563603			−0.281802	+	0.959473i	$0.590932\pi$
$504$		0			0
$505$	47.3958			2.10909
$506$	4.24620	−	7.35463i	0.188766	−	0.326953i
$507$		0			0
$508$	7.33732	+	12.7086i	0.325541	+	0.563854i
$509$	−28.1110			−1.24600			−0.623000	−	0.782222i	$-0.714086\pi$
$509$	−28.1110			−1.24600			−0.623000	+	0.782222i	$0.714086\pi$
$510$		0			0
$511$		0			0
$512$	−24.5070			−1.08307
$513$		0			0
$514$	−12.3830	+	21.4480i	−0.546192	+	0.946033i
$515$	−6.50427			−0.286613
$516$		0			0
$517$	−2.69331	+	4.66495i	−0.118452	+	0.205164i
$518$		0			0
$519$		0			0
$520$	−8.67288	+	15.0219i	−0.380331	+	0.658753i
$521$	4.23768	−	7.33988i	0.185656	−	0.321566i	−0.758141	−	0.652090i	$-0.773892\pi$
$521$	4.23768	−	7.33988i	0.185656	−	0.321566i	0.943797	+	0.330524i	$0.107226\pi$
$522$		0			0
$523$	−16.7236	−	28.9662i	−0.731273	−	1.26660i	−0.956339	−	0.292259i	$-0.905593\pi$
$523$	−16.7236	−	28.9662i	−0.731273	−	1.26660i	0.225066	−	0.974344i	$-0.427740\pi$
$524$	−9.85035	−	17.0613i	−0.430315	−	0.745327i
$525$		0			0
$526$	−22.8341	+	39.5498i	−0.995613	+	1.72445i
$527$	10.8398			0.472187
$528$		0			0
$529$	−16.0921			−0.699655
$530$	−22.5008	−	38.9725i	−0.977371	−	1.69286i
$531$		0			0
$532$		0			0
$533$	1.32569	+	2.29616i	0.0574220	+	0.0994579i
$534$		0			0
$535$	−25.5693	−	44.2874i	−1.10546	−	1.91471i
$536$	−25.9493	−	44.9456i	−1.12084	−	1.94135i
$537$		0			0
$538$	−10.5461	−	18.2665i	−0.454677	−	0.787523i
$539$		0			0
$540$		0			0
$541$	−9.12929	−	15.8124i	−0.392499	−	0.679828i	0.600280	−	0.799790i	$-0.295056\pi$
$541$	−9.12929	−	15.8124i	−0.392499	−	0.679828i	−0.992778	+	0.119962i	$0.961723\pi$
$542$	−43.7730			−1.88021
$543$		0			0
$544$	8.84701			0.379312
$545$	22.7803	−	39.4567i	0.975802	−	1.69014i
$546$		0			0
$547$	−2.88599	−	4.99869i	−0.123396	−	0.213728i	0.797709	−	0.603043i	$-0.206045\pi$
$547$	−2.88599	−	4.99869i	−0.123396	−	0.213728i	−0.921105	+	0.389315i	$0.872712\pi$
$548$	−13.8558	−	23.9990i	−0.591892	−	1.02519i
$549$		0			0
$550$	−5.70532	+	9.88190i	−0.243276	+	0.421366i
$551$	1.15122	−	1.99397i	0.0490437	−	0.0849461i
$552$		0			0
$553$		0			0
$554$	−6.09227	+	10.5521i	−0.258836	+	0.448317i
$555$		0			0
$556$	52.0029			2.20541
$557$	−16.6911	+	28.9098i	−0.707223	+	1.22495i	0.258661	+	0.965968i	$0.416719\pi$
$557$	−16.6911	+	28.9098i	−0.707223	+	1.22495i	−0.965883	+	0.258977i	$0.916614\pi$
$558$		0			0
$559$	−6.36623			−0.269263
$560$		0			0
$561$		0			0
$562$	−4.07286			−0.171803
$563$	−1.09566	−	1.89773i	−0.0461764	−	0.0799799i	0.842013	−	0.539457i	$-0.181370\pi$
$563$	−1.09566	−	1.89773i	−0.0461764	−	0.0799799i	−0.888190	+	0.459477i	$0.848037\pi$
$564$		0			0
$565$	−2.46683	+	4.27268i	−0.103780	+	0.179753i
$566$	29.8079			1.25292
$567$		0			0
$568$	−52.3278			−2.19563
$569$	9.49302	−	16.4424i	0.397968	−	0.689301i	−0.595507	−	0.803350i	$-0.703049\pi$
$569$	9.49302	−	16.4424i	0.397968	−	0.689301i	0.993475	+	0.114049i	$0.0363822\pi$
$570$		0			0
$571$	10.8690	+	18.8257i	0.454854	+	0.787831i	0.998680	−	0.0513674i	$-0.0163580\pi$
$571$	10.8690	+	18.8257i	0.454854	+	0.787831i	−0.543825	+	0.839198i	$0.683025\pi$
$572$	−7.33717			−0.306782
$573$		0			0
$574$		0			0
$575$	−9.28172			−0.387074
$576$		0			0
$577$	15.4516	−	26.7629i	0.643258	−	1.11416i	−0.341443	−	0.939903i	$-0.610916\pi$
$577$	15.4516	−	26.7629i	0.643258	−	1.11416i	0.984701	−	0.174253i	$-0.0557511\pi$
$578$	−14.4217			−0.599862
$579$		0			0
$580$	5.63398	−	9.75835i	0.233938	−	0.405193i
$581$		0			0
$582$		0			0
$583$	4.36902	−	7.56737i	0.180946	−	0.313408i
$584$	−21.1767	+	36.6792i	−0.876299	+	1.51780i
$585$		0			0
$586$	−6.21069	−	10.7572i	−0.256561	−	0.444377i
$587$	−9.18332	−	15.9060i	−0.379036	−	0.656510i	0.611886	−	0.790946i	$-0.290411\pi$
$587$	−9.18332	−	15.9060i	−0.379036	−	0.656510i	−0.990922	+	0.134436i	$0.957078\pi$
$588$		0			0
$589$	−3.61282	+	6.25759i	−0.148864	+	0.257840i
$590$	85.1527			3.50568
$591$		0			0
$592$	−24.7371			−1.01669
$593$	13.8775	+	24.0365i	0.569880	+	0.987061i	0.996577	+	0.0826662i	$0.0263435\pi$
$593$	13.8775	+	24.0365i	0.569880	+	0.987061i	−0.426698	+	0.904394i	$0.640323\pi$
$594$		0			0
$595$		0			0
$596$	4.02936	+	6.97905i	0.165049	+	0.285873i
$597$		0			0
$598$	−4.59841	−	7.96468i	−0.188043	−	0.325700i
$599$	0.201412	+	0.348855i	0.00822945	+	0.0142538i	0.870111	−	0.492856i	$-0.164047\pi$
$599$	0.201412	+	0.348855i	0.00822945	+	0.0142538i	−0.861881	+	0.507110i	$0.830714\pi$
$600$		0			0
$601$	−12.3733	−	21.4312i	−0.504717	−	0.874196i	−0.999985	−	0.00545577i	$-0.998263\pi$
$601$	−12.3733	−	21.4312i	−0.504717	−	0.874196i	0.495268	−	0.868740i	$-0.335070\pi$
$602$		0			0
$603$		0			0
$604$	−25.9313	−	44.9143i	−1.05513	−	1.82754i
$605$	26.7769			1.08864
$606$		0			0
$607$	−24.0697			−0.976957			−0.488479	−	0.872576i	$-0.662448\pi$
$607$	−24.0697			−0.976957			−0.488479	+	0.872576i	$0.662448\pi$
$608$	−2.94865	+	5.10721i	−0.119584	+	0.207125i
$609$		0			0
$610$	−1.95115	−	3.37950i	−0.0789999	−	0.136832i
$611$	2.91672	+	5.05190i	0.117998	+	0.204378i
$612$		0			0
$613$	10.1907	−	17.6509i	0.411600	−	0.712912i	−0.583465	−	0.812138i	$-0.698303\pi$
$613$	10.1907	−	17.6509i	0.411600	−	0.712912i	0.995065	+	0.0992261i	$0.0316367\pi$
$614$	5.96879	−	10.3382i	0.240881	−	0.417218i
$615$		0			0
$616$		0			0
$617$	20.9315	−	36.2544i	0.842669	−	1.45955i	−0.0449604	−	0.998989i	$-0.514316\pi$
$617$	20.9315	−	36.2544i	0.842669	−	1.45955i	0.887630	−	0.460558i	$-0.152350\pi$
$618$		0			0
$619$	−14.8219			−0.595743			−0.297871	−	0.954606i	$-0.596277\pi$
$619$	−14.8219			−0.595743			−0.297871	+	0.954606i	$0.596277\pi$
$620$	−17.6809	+	30.6242i	−0.710081	+	1.22990i
$621$		0			0
$622$	−77.3270			−3.10053
$623$		0			0
$624$		0			0
$625$	−30.1861			−1.20744
$626$	1.81291	+	3.14005i	0.0724585	+	0.125502i
$627$		0			0
$628$	5.48329	−	9.49734i	0.218807	−	0.378985i
$629$	36.0007			1.43544
$630$		0			0
$631$	−21.0294			−0.837169			−0.418585	−	0.908178i	$-0.637474\pi$
$631$	−21.0294			−0.837169			−0.418585	+	0.908178i	$0.637474\pi$
$632$	−1.55487	+	2.69312i	−0.0618496	+	0.107127i
$633$		0			0
$634$	−25.6694	−	44.4607i	−1.01946	−	1.76576i
$635$	11.5935			0.460072
$636$		0			0
$637$		0			0
$638$	3.37150			0.133479
$639$		0			0
$640$	−30.2882	+	52.4607i	−1.19725	+	2.07369i
$641$	−11.9318			−0.471279			−0.235640	−	0.971840i	$-0.575719\pi$
$641$	−11.9318			−0.471279			−0.235640	+	0.971840i	$0.575719\pi$
$642$		0			0
$643$	19.9678	−	34.5852i	0.787452	−	1.36391i	−0.140072	−	0.990141i	$-0.544733\pi$
$643$	19.9678	−	34.5852i	0.787452	−	1.36391i	0.927524	−	0.373765i	$-0.121933\pi$
$644$		0			0
$645$		0			0
$646$	−8.71733	+	15.0989i	−0.342979	+	0.594057i
$647$	0.494477	−	0.856459i	0.0194399	−	0.0336709i	−0.856142	−	0.516741i	$-0.827145\pi$
$647$	0.494477	−	0.856459i	0.0194399	−	0.0336709i	0.875582	+	0.483070i	$0.160478\pi$
$648$		0			0
$649$	8.26714	+	14.3191i	0.324514	+	0.562074i
$650$	6.17856	+	10.7016i	0.242343	+	0.419751i
$651$		0			0
$652$	−0.718272	+	1.24408i	−0.0281297	+	0.0487221i
$653$	−22.7147			−0.888894			−0.444447	−	0.895805i	$-0.646600\pi$
$653$	−22.7147			−0.888894			−0.444447	+	0.895805i	$0.646600\pi$
$654$		0			0
$655$	−15.5642			−0.608144
$656$	−2.05688	−	3.56262i	−0.0803076	−	0.139097i
$657$		0			0
$658$		0			0
$659$	19.1943	+	33.2454i	0.747702	+	1.29506i	0.948922	+	0.315512i	$0.102176\pi$
$659$	19.1943	+	33.2454i	0.747702	+	1.29506i	−0.201220	+	0.979546i	$0.564491\pi$
$660$		0			0
$661$	16.9629	+	29.3806i	0.659780	+	1.14277i	0.980672	+	0.195657i	$0.0626839\pi$
$661$	16.9629	+	29.3806i	0.659780	+	1.14277i	−0.320892	+	0.947116i	$0.603983\pi$
$662$	−23.2458	−	40.2628i	−0.903472	−	1.56486i
$663$		0			0
$664$	−3.98486	−	6.90198i	−0.154642	−	0.267849i
$665$		0			0
$666$		0			0
$667$	1.37124	+	2.37505i	0.0530944	+	0.0919623i
$668$	−26.9809			−1.04392
$669$		0			0
$670$	−89.3201			−3.45074
$671$	0.378860	−	0.656205i	0.0146257	−	0.0253325i
$672$		0			0
$673$	−16.1030	−	27.8912i	−0.620725	−	1.07513i	−0.989351	−	0.145549i	$-0.953505\pi$
$673$	−16.1030	−	27.8912i	−0.620725	−	1.07513i	0.368626	−	0.929578i	$-0.379828\pi$
$674$	11.5702	+	20.0401i	0.445666	+	0.771916i
$675$		0			0
$676$	20.0585	−	34.7424i	0.771483	−	1.33625i
$677$	18.9842	−	32.8816i	0.729622	−	1.26374i	−0.227421	−	0.973797i	$-0.573029\pi$
$677$	18.9842	−	32.8816i	0.729622	−	1.26374i	0.957043	−	0.289946i	$-0.0936375\pi$
$678$		0			0
$679$		0			0
$680$	−19.5836	+	33.9198i	−0.750997	+	1.30076i
$681$		0			0
$682$	−10.5806			−0.405153
$683$	−7.59357	+	13.1525i	−0.290560	+	0.503265i	−0.973942	−	0.226796i	$-0.927175\pi$
$683$	−7.59357	+	13.1525i	−0.290560	+	0.503265i	0.683382	+	0.730061i	$0.260508\pi$
$684$		0			0
$685$	−21.8932			−0.836495
$686$		0			0
$687$		0			0
$688$	9.87754			0.376578
$689$	−4.73142	−	8.19507i	−0.180253	−	0.312207i
$690$		0			0
$691$	1.34574	−	2.33089i	0.0511943	−	0.0886711i	−0.839293	−	0.543680i	$-0.817031\pi$
$691$	1.34574	−	2.33089i	0.0511943	−	0.0886711i	0.890487	+	0.455009i	$0.150364\pi$
$692$	−14.9922			−0.569919
$693$		0			0
$694$	−4.83589			−0.183568
$695$	20.5420	−	35.5798i	0.779203	−	1.34962i
$696$		0			0
$697$	2.99344	+	5.18480i	0.113385	+	0.196388i
$698$	38.8858			1.47185
$699$		0			0
$700$		0			0
$701$	11.8515			0.447625			0.223813	−	0.974632i	$-0.428150\pi$
$701$	11.8515			0.447625			0.223813	+	0.974632i	$0.428150\pi$
$702$		0			0
$703$	−11.9988	+	20.7826i	−0.452544	+	0.783829i
$704$	−14.7938			−0.557562
$705$		0			0
$706$	−20.3617	+	35.2675i	−0.766323	+	1.32731i
$707$		0			0
$708$		0			0
$709$	20.5167	−	35.5359i	0.770520	−	1.33458i	−0.166759	−	0.985998i	$-0.553330\pi$
$709$	20.5167	−	35.5359i	0.770520	−	1.33458i	0.937278	−	0.348582i	$-0.113337\pi$
$710$	−45.0294	+	77.9931i	−1.68992	+	2.92703i
$711$		0			0
$712$	12.9772	+	22.4771i	0.486339	+	0.842364i
$713$	−4.30328	−	7.45351i	−0.161159	−	0.279136i
$714$		0			0
$715$	−2.89830	+	5.02001i	−0.108390	+	0.187738i
$716$	39.1337			1.46250
$717$		0			0
$718$	7.08246			0.264315
$719$	10.4555	+	18.1094i	0.389923	+	0.675366i	0.992439	−	0.122741i	$-0.0391685\pi$
$719$	10.4555	+	18.1094i	0.389923	+	0.675366i	−0.602516	+	0.798107i	$0.705835\pi$
$720$		0			0
$721$		0			0
$722$	16.8644	+	29.2100i	0.627628	+	1.08708i
$723$		0			0
$724$	−36.2896	−	62.8554i	−1.34869	−	2.33600i
$725$	−1.84243	−	3.19119i	−0.0684263	−	0.118518i
$726$		0			0
$727$	−1.32165	−	2.28917i	−0.0490173	−	0.0849005i	0.840476	−	0.541849i	$-0.182276\pi$
$727$	−1.32165	−	2.28917i	−0.0490173	−	0.0849005i	−0.889493	+	0.456949i	$0.848942\pi$
$728$		0			0
$729$		0			0
$730$	36.4462	+	63.1267i	1.34893	+	2.33642i
$731$	−14.3751			−0.531683
$732$		0			0
$733$	−14.1489			−0.522602			−0.261301	−	0.965257i	$-0.584152\pi$
$733$	−14.1489			−0.522602			−0.261301	+	0.965257i	$0.584152\pi$
$734$	−12.1223	+	20.9964i	−0.447442	+	0.774992i
$735$		0			0
$736$	−3.51218	−	6.08327i	−0.129461	−	0.224232i
$737$	−8.67174	−	15.0199i	−0.319428	−	0.553265i
$738$		0			0
$739$	−7.85905	+	13.6123i	−0.289100	+	0.500736i	−0.973595	−	0.228282i	$-0.926689\pi$
$739$	−7.85905	+	13.6123i	−0.289100	+	0.500736i	0.684495	+	0.729017i	$0.260023\pi$
$740$	−58.7212	+	101.708i	−2.15864	+	3.73887i
$741$		0			0
$742$		0			0
$743$	−10.5496	+	18.2724i	−0.387026	+	0.670348i	−0.992048	−	0.125861i	$-0.959831\pi$
$743$	−10.5496	+	18.2724i	−0.387026	+	0.670348i	0.605022	+	0.796208i	$0.293164\pi$
$744$		0			0
$745$	6.36665			0.233256
$746$	30.4142	−	52.6789i	1.11354	−	1.92871i
$747$		0			0
$748$	−16.5675			−0.605768
$749$		0			0
$750$		0			0
$751$	13.0370			0.475725			0.237863	−	0.971299i	$-0.423553\pi$
$751$	13.0370			0.475725			0.237863	+	0.971299i	$0.423553\pi$
$752$	−4.52544	−	7.83829i	−0.165026	−	0.285833i
$753$		0			0
$754$	1.82558	−	3.16200i	0.0664838	−	0.115153i
$755$	−40.9732			−1.49117
$756$		0			0
$757$	−12.6856			−0.461065			−0.230532	−	0.973065i	$-0.574047\pi$
$757$	−12.6856			−0.461065			−0.230532	+	0.973065i	$0.574047\pi$
$758$	−11.7613	+	20.3711i	−0.427188	+	0.739912i
$759$		0			0
$760$	−13.0542	−	22.6105i	−0.473525	−	0.820169i
$761$	−6.04077			−0.218978			−0.109489	−	0.993988i	$-0.534921\pi$
$761$	−6.04077			−0.218978			−0.109489	+	0.993988i	$0.534921\pi$
$762$		0			0
$763$		0			0
$764$	−30.6388			−1.10847
$765$		0			0
$766$	32.5959	−	56.4577i	1.17774	−	2.03990i
$767$	17.9058			0.646540
$768$		0			0
$769$	−0.108129	+	0.187285i	−0.00389924	+	0.00675368i	−0.867968	−	0.496619i	$-0.834575\pi$
$769$	−0.108129	+	0.187285i	−0.00389924	+	0.00675368i	0.864069	+	0.503373i	$0.167908\pi$
$770$		0			0
$771$		0			0
$772$	34.7251	−	60.1457i	1.24979	−	2.16469i
$773$	18.8132	−	32.5854i	0.676663	−	1.17202i	−0.299316	−	0.954154i	$-0.596759\pi$
$773$	18.8132	−	32.5854i	0.676663	−	1.17202i	0.975980	−	0.217861i	$-0.0699081\pi$
$774$		0			0
$775$	5.78202	+	10.0148i	0.207696	+	0.359741i
$776$	16.7763	+	29.0575i	0.602235	+	1.04310i
$777$		0			0
$778$	4.99388	−	8.64965i	0.179039	−	0.310105i
$779$	−3.99078			−0.142985
$780$		0			0
$781$	−17.4869			−0.625730
$782$	−10.3833	−	17.9845i	−0.371307	−	0.643123i
$783$		0			0
$784$		0			0
$785$	−4.33198	−	7.50321i	−0.154615	−	0.267801i
$786$		0			0
$787$	15.4067	+	26.6853i	0.549191	+	0.951226i	0.998330	+	0.0577648i	$0.0183973\pi$
$787$	15.4067	+	26.6853i	0.549191	+	0.951226i	−0.449139	+	0.893462i	$0.648269\pi$
$788$	11.0847	+	19.1992i	0.394875	+	0.683943i
$789$		0			0
$790$	2.67601	+	4.63499i	0.0952083	+	0.164906i
$791$		0			0
$792$		0			0
$793$	−0.410286	−	0.710636i	−0.0145697	−	0.0252354i
$794$	73.2074			2.59803
$795$		0			0
$796$	53.2

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)

\(f(q)\)	\(=\)	\(q+(-1.19343 + 2.06709i) q^{2} +(-1.84857 - 3.20182i) q^{4} -2.92087 q^{5} +4.05086 q^{8} +O(q^{10})\)	\(q+(-1.19343 + 2.06709i) q^{2} +(-1.84857 - 3.20182i) q^{4} -2.92087 q^{5} +4.05086 q^{8} +(3.48586 - 6.03769i) q^{10} +1.35371 q^{11} +(0.733001 - 1.26960i) q^{13} +(-1.13729 + 1.96984i) q^{16} +(1.65514 - 2.86678i) q^{17} +(1.10329 + 1.91096i) q^{19} +(5.39943 + 9.35209i) q^{20} +(-1.61557 + 2.79825i) q^{22} -2.62830 q^{23} +3.53146 q^{25} +(1.74958 + 3.03036i) q^{26} +(-0.521720 - 0.903646i) q^{29} +(1.63729 + 2.83587i) q^{31} +(1.33629 + 2.31453i) q^{32} +(3.95060 + 6.84263i) q^{34} +(5.43773 + 9.41842i) q^{37} -5.26683 q^{38} -11.8320 q^{40} +(-0.904289 + 1.56627i) q^{41} +(-2.17129 - 3.76078i) q^{43} +(-2.50244 - 4.33435i) q^{44} +(3.13670 - 5.43292i) q^{46} +(-1.98957 + 3.44604i) q^{47} +(-4.21456 + 7.29984i) q^{50} -5.42002 q^{52} +(3.22743 - 5.59008i) q^{53} -3.95402 q^{55} +2.49056 q^{58} +(6.10700 + 10.5776i) q^{59} +(0.279867 - 0.484744i) q^{61} -7.81600 q^{62} -10.9283 q^{64} +(-2.14100 + 3.70832i) q^{65} +(-6.40588 - 11.0953i) q^{67} -12.2386 q^{68} -12.9177 q^{71} +(-5.22772 + 9.05467i) q^{73} -25.9583 q^{74} +(4.07903 - 7.06509i) q^{76} +(-0.383838 + 0.664827i) q^{79} +(3.32187 - 5.75365i) q^{80} +(-2.15842 - 3.73849i) q^{82} +(-0.983707 - 1.70383i) q^{83} +(-4.83443 + 8.37348i) q^{85} +10.3652 q^{86} +5.48371 q^{88} +(3.20356 + 5.54872i) q^{89} +(4.85859 + 8.41533i) q^{92} +(-4.74884 - 8.22524i) q^{94} +(-3.22257 - 5.58166i) q^{95} +(4.14143 + 7.17316i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) \)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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