LMFDB - Embedded newform 1323.2.o.e.440.18

Newspace parameters

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.o (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 441)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			440.18
Character	$\chi$	$=$	1323.440
Dual form			1323.2.o.e.881.18

$q$-expansion

$f(q)$	$=$	$q+(1.02035 + 0.589100i) q^{2} +(-0.305921 - 0.529871i) q^{4} +(2.16601 + 3.75164i) q^{5} -3.07728i q^{8} +O(q^{10})$	\(q+(1.02035 + 0.589100i) q^{2} +(-0.305921 - 0.529871i) q^{4} +(2.16601 + 3.75164i) q^{5} -3.07728i q^{8} +5.10399i q^{10} +(1.87238 + 1.08102i) q^{11} +(2.25256 - 1.30052i) q^{13} +(1.20098 - 2.08016i) q^{16} +1.17115 q^{17} +2.41658i q^{19} +(1.32526 - 2.29541i) q^{20} +(1.27366 + 2.20604i) q^{22} +(3.16186 - 1.82550i) q^{23} +(-6.88321 + 11.9221i) q^{25} +3.06454 q^{26} +(-0.589262 - 0.340210i) q^{29} +(5.67723 - 3.27775i) q^{31} +(-2.87915 + 1.66228i) q^{32} +(1.19499 + 0.689926i) q^{34} -5.10692 q^{37} +(-1.42361 + 2.46576i) q^{38} +(11.5448 - 6.66541i) q^{40} +(3.68473 + 6.38214i) q^{41} +(-2.12577 + 3.68194i) q^{43} -1.32283i q^{44} +4.30162 q^{46} +(-3.57157 + 6.18614i) q^{47} +(-14.0466 + 8.10980i) q^{50} +(-1.37821 - 0.795711i) q^{52} +3.23289i q^{53} +9.36601i q^{55} +(-0.400836 - 0.694269i) q^{58} +(-2.91810 - 5.05430i) q^{59} +(6.21638 + 3.58903i) q^{61} +7.72370 q^{62} -8.72092 q^{64} +(9.75814 + 5.63387i) q^{65} +(-3.32682 - 5.76221i) q^{67} +(-0.358281 - 0.620560i) q^{68} +1.95976i q^{71} -11.9069i q^{73} +(-5.21085 - 3.00849i) q^{74} +(1.28048 - 0.739283i) q^{76} +(4.87702 - 8.44725i) q^{79} +10.4054 q^{80} +8.68270i q^{82} +(-0.796736 + 1.37999i) q^{83} +(2.53673 + 4.39374i) q^{85} +(-4.33806 + 2.50458i) q^{86} +(3.32660 - 5.76184i) q^{88} +6.09921 q^{89} +(-1.93456 - 1.11692i) q^{92} +(-7.28851 + 4.20802i) q^{94} +(-9.06614 + 5.23434i) q^{95} +(-2.36387 - 1.36478i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48q + 24q^{4} + O(q^{10}) $	$ 48q + 24q^{4} + 24q^{11} - 24q^{16} + 48q^{23} - 24q^{25} - 120q^{32} - 48q^{50} - 48q^{64} - 120q^{65} + 168q^{74} - 24q^{79} - 24q^{85} - 24q^{86} + 144q^{92} - 96q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	1.02035	+	0.589100i	0.721498	+	0.416557i	0.815304	−	0.579034i	$-0.196570\pi$
$2$	1.02035	+	0.589100i	0.721498	+	0.416557i	−0.0938059	+	0.995591i	$0.529903\pi$
$3$		0			0
$3$		0			0
$4$	−0.305921	−	0.529871i	−0.152961	−	0.264936i
$5$	2.16601	+	3.75164i	0.968670	+	1.67778i	0.699415	+	0.714716i	$0.253444\pi$
$5$	2.16601	+	3.75164i	0.968670	+	1.67778i	0.269254	+	0.963069i	$0.413223\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		−	3.07728i		−	1.08798i
$9$		0			0
$10$			5.10399i			1.61402i
$11$	1.87238	+	1.08102i	0.564545	+	0.325940i	0.754968	−	0.655762i	$-0.227653\pi$
$11$	1.87238	+	1.08102i	0.564545	+	0.325940i	−0.190423	+	0.981702i	$0.560986\pi$
$12$		0			0
$13$	2.25256	−	1.30052i	0.624748	−	0.360698i	−0.153967	−	0.988076i	$-0.549205\pi$
$13$	2.25256	−	1.30052i	0.624748	−	0.360698i	0.778715	+	0.627378i	$0.215872\pi$
$14$		0			0
$15$		0			0
$16$	1.20098	−	2.08016i	0.300245	−	0.520040i
$17$	1.17115			0.284046			0.142023	−	0.989863i	$-0.454639\pi$
$17$	1.17115			0.284046			0.142023	+	0.989863i	$0.454639\pi$
$18$		0			0
$19$			2.41658i			0.554402i	0.960812	+	0.277201i	$0.0894067\pi$
$19$			2.41658i			0.554402i	−0.960812	+	0.277201i	$0.910593\pi$
$20$	1.32526	−	2.29541i	0.296337	−	0.513270i
$21$		0			0
$22$	1.27366	+	2.20604i	0.271545	+	0.470330i
$23$	3.16186	−	1.82550i	0.659294	−	0.380644i	−0.132714	−	0.991154i	$-0.542369\pi$
$23$	3.16186	−	1.82550i	0.659294	−	0.380644i	0.792008	+	0.610511i	$0.209036\pi$
$24$		0			0
$25$	−6.88321	+	11.9221i	−1.37664	+	2.38441i
$26$	3.06454			0.601006
$27$		0			0
$28$		0			0
$29$	−0.589262	−	0.340210i	−0.109423	−	0.0631755i	0.444290	−	0.895883i	$-0.353456\pi$
$29$	−0.589262	−	0.340210i	−0.109423	−	0.0631755i	−0.553713	+	0.832708i	$0.686789\pi$
$30$		0			0
$31$	5.67723	−	3.27775i	1.01966	−	0.588702i	0.105655	−	0.994403i	$-0.466306\pi$
$31$	5.67723	−	3.27775i	1.01966	−	0.588702i	0.914006	+	0.405701i	$0.132973\pi$
$32$	−2.87915	+	1.66228i	−0.508967	+	0.293852i
$33$		0			0
$34$	1.19499	+	0.689926i	0.204939	+	0.118321i
$35$		0			0
$36$		0			0
$37$	−5.10692			−0.839572			−0.419786	−	0.907623i	$-0.637895\pi$
$37$	−5.10692			−0.839572			−0.419786	+	0.907623i	$0.637895\pi$
$38$	−1.42361	+	2.46576i	−0.230940	+	0.399999i
$39$		0			0
$40$	11.5448	−	6.66541i	1.82540	−	1.05389i
$41$	3.68473	+	6.38214i	0.575458	+	0.996723i	0.995992	+	0.0894458i	$0.0285096\pi$
$41$	3.68473	+	6.38214i	0.575458	+	0.996723i	−0.420534	+	0.907277i	$0.638157\pi$
$42$		0			0
$43$	−2.12577	+	3.68194i	−0.324176	+	0.561490i	−0.981345	−	0.192253i	$-0.938420\pi$
$43$	−2.12577	+	3.68194i	−0.324176	+	0.561490i	0.657169	+	0.753743i	$0.271754\pi$
$44$		−	1.32283i		−	0.199424i
$45$		0			0
$46$	4.30162			0.634239
$47$	−3.57157	+	6.18614i	−0.520967	+	0.902341i	0.478736	+	0.877959i	$0.341095\pi$
$47$	−3.57157	+	6.18614i	−0.520967	+	0.902341i	−0.999703	+	0.0243819i	$0.992238\pi$
$48$		0			0
$49$		0			0
$50$	−14.0466	+	8.10980i	−1.98649	+	1.14690i
$51$		0			0
$52$	−1.37821	−	0.795711i	−0.191124	−	0.110345i
$53$			3.23289i			0.444071i	0.975039	+	0.222036i	$0.0712702\pi$
$53$			3.23289i			0.444071i	−0.975039	+	0.222036i	$0.928730\pi$
$54$		0			0
$55$			9.36601i			1.26291i
$56$		0			0
$57$		0			0
$58$	−0.400836	−	0.694269i	−0.0526324	−	0.0911619i
$59$	−2.91810	−	5.05430i	−0.379905	−	0.658014i	0.611143	−	0.791520i	$-0.290710\pi$
$59$	−2.91810	−	5.05430i	−0.379905	−	0.658014i	−0.991048	+	0.133506i	$0.957377\pi$
$60$		0			0
$61$	6.21638	+	3.58903i	0.795925	+	0.459528i	0.842044	−	0.539408i	$-0.181352\pi$
$61$	6.21638	+	3.58903i	0.795925	+	0.459528i	−0.0461190	+	0.998936i	$0.514685\pi$
$62$	7.72370			0.980911
$63$		0			0
$64$	−8.72092			−1.09012
$65$	9.75814	+	5.63387i	1.21035	+	0.698795i
$66$		0			0
$67$	−3.32682	−	5.76221i	−0.406435	−	0.703966i	0.588052	−	0.808823i	$-0.299895\pi$
$67$	−3.32682	−	5.76221i	−0.406435	−	0.703966i	−0.994487	+	0.104857i	$0.966562\pi$
$68$	−0.358281	−	0.620560i	−0.0434479	−	0.0752540i
$69$		0			0
$70$		0			0
$71$			1.95976i			0.232580i	0.993215	+	0.116290i	$0.0371003\pi$
$71$			1.95976i			0.232580i	−0.993215	+	0.116290i	$0.962900\pi$
$72$		0			0
$73$		−	11.9069i		−	1.39360i	−0.717266	−	0.696799i	$-0.754607\pi$
$73$		−	11.9069i		−	1.39360i	0.717266	−	0.696799i	$-0.245393\pi$
$74$	−5.21085	−	3.00849i	−0.605749	−	0.349729i
$75$		0			0
$76$	1.28048	−	0.739283i	0.146881	−	0.0848016i
$77$		0			0
$78$		0			0
$79$	4.87702	−	8.44725i	0.548708	−	0.950390i	−0.449656	−	0.893202i	$-0.648453\pi$
$79$	4.87702	−	8.44725i	0.548708	−	0.950390i	0.998363	−	0.0571879i	$-0.0182134\pi$
$80$	10.4054			1.16335
$81$		0			0
$82$			8.68270i			0.958844i
$83$	−0.796736	+	1.37999i	−0.0874531	+	0.151473i	−0.906434	−	0.422348i	$-0.861206\pi$
$83$	−0.796736	+	1.37999i	−0.0874531	+	0.151473i	0.818981	+	0.573821i	$0.194539\pi$
$84$		0			0
$85$	2.53673	+	4.39374i	0.275147	+	0.476568i
$86$	−4.33806	+	2.50458i	−0.467785	+	0.270076i
$87$		0			0
$88$	3.32660	−	5.76184i	0.354617	−	0.614214i
$89$	6.09921			0.646515			0.323258	−	0.946311i	$-0.395222\pi$
$89$	6.09921			0.646515			0.323258	+	0.946311i	$0.395222\pi$
$90$		0			0
$91$		0			0
$92$	−1.93456	−	1.11692i	−0.201692	−	0.116447i
$93$		0			0
$94$	−7.28851	+	4.20802i	−0.751753	+	0.434025i
$95$	−9.06614	+	5.23434i	−0.930166	+	0.537032i
$96$		0			0
$97$	−2.36387	−	1.36478i	−0.240014	−	0.138572i	0.375169	−	0.926956i	$-0.377585\pi$
$97$	−2.36387	−	1.36478i	−0.240014	−	0.138572i	−0.615183	+	0.788384i	$0.710918\pi$
$98$		0			0
$99$		0			0
$100$	8.42288			0.842288
$101$	−7.99849	+	13.8538i	−0.795880	+	1.37850i	0.126400	+	0.991979i	$0.459658\pi$
$101$	−7.99849	+	13.8538i	−0.795880	+	1.37850i	−0.922279	+	0.386524i	$0.873676\pi$
$102$		0			0
$103$	−2.61251	+	1.50834i	−0.257419	+	0.148621i	−0.623156	−	0.782097i	$-0.714150\pi$
$103$	−2.61251	+	1.50834i	−0.257419	+	0.148621i	0.365738	+	0.930718i	$0.380817\pi$
$104$	−4.00205	−	6.93175i	−0.392433	−	0.679714i
$105$		0			0
$106$	−1.90450	+	3.29868i	−0.184981	+	0.320397i
$107$		−	11.8484i		−	1.14543i	−0.819754	−	0.572716i	$-0.805890\pi$
$107$		−	11.8484i		−	1.14543i	0.819754	−	0.572716i	$-0.194110\pi$
$108$		0			0
$109$	7.16157			0.685954			0.342977	−	0.939344i	$-0.388565\pi$
$109$	7.16157			0.685954			0.342977	+	0.939344i	$0.388565\pi$
$110$	−5.51752	+	9.55662i	−0.526075	+	0.911188i
$111$		0			0
$112$		0			0
$113$	−2.46102	+	1.42087i	−0.231514	+	0.133664i	−0.611270	−	0.791422i	$-0.709341\pi$
$113$	−2.46102	+	1.42087i	−0.231514	+	0.133664i	0.379756	+	0.925086i	$0.376008\pi$
$114$		0			0
$115$	13.6973	+	7.90812i	1.27728	+	0.737436i
$116$			0.416310i			0.0386535i
$117$		0			0
$118$		−	6.87623i		−	0.633008i
$119$		0			0
$120$		0			0
$121$	−3.16279	−	5.47811i	−0.287526	−	0.498010i
$122$	4.22859	+	7.32414i	0.382839	+	0.663096i
$123$		0			0
$124$	−3.47357	−	2.00547i	−0.311936	−	0.180096i
$125$	−37.9763			−3.39670
$126$		0			0
$127$	18.5344			1.64466			0.822332	−	0.569009i	$-0.192673\pi$
$127$	18.5344			1.64466			0.822332	+	0.569009i	$0.192673\pi$
$128$	−3.14011	−	1.81294i	−0.277549	−	0.160243i
$129$		0			0
$130$	6.63783	+	11.4971i	0.582176	+	1.00836i
$131$	−3.35221	−	5.80619i	−0.292884	−	0.507289i	0.681607	−	0.731719i	$-0.261282\pi$
$131$	−3.35221	−	5.80619i	−0.292884	−	0.507289i	−0.974490	+	0.224429i	$0.927948\pi$
$132$		0			0
$133$		0			0
$134$		−	7.83931i		−	0.677214i
$135$		0			0
$136$		−	3.60396i		−	0.309037i
$137$	−11.8181	−	6.82316i	−1.00969	−	0.582942i	−0.0985856	−	0.995129i	$-0.531432\pi$
$137$	−11.8181	−	6.82316i	−1.00969	−	0.582942i	−0.911099	+	0.412187i	$0.864765\pi$
$138$		0			0
$139$	7.74126	−	4.46942i	0.656605	−	0.379091i	−0.134377	−	0.990930i	$-0.542903\pi$
$139$	7.74126	−	4.46942i	0.656605	−	0.379091i	0.790982	+	0.611839i	$0.209570\pi$
$140$		0			0
$141$		0			0
$142$	−1.15449	+	1.99964i	−0.0968830	+	0.167806i
$143$	5.62354			0.470264
$144$		0			0
$145$		−	2.94760i		−	0.244785i
$146$	7.01436	−	12.1492i	0.580513	−	1.00548i
$147$		0			0
$148$	1.56231	+	2.70601i	0.128421	+	0.222432i
$149$	3.29003	−	1.89950i	0.269530	−	0.155613i	−0.359144	−	0.933282i	$-0.616931\pi$
$149$	3.29003	−	1.89950i	0.269530	−	0.155613i	0.628674	+	0.777669i	$0.283598\pi$
$150$		0			0
$151$	1.91083	−	3.30965i	0.155501	−	0.269336i	−0.777740	−	0.628586i	$-0.783634\pi$
$151$	1.91083	−	3.30965i	0.155501	−	0.269336i	0.933241	+	0.359250i	$0.116967\pi$
$152$	7.43648			0.603178
$153$		0			0
$154$		0			0
$155$	24.5939	+	14.1993i	1.97543	+	1.14051i
$156$		0			0
$157$	−18.6081	+	10.7434i	−1.48509	+	0.857417i	−0.999856	−	0.0169675i	$-0.994599\pi$
$157$	−18.6081	+	10.7434i	−1.48509	+	0.857417i	−0.485234	+	0.874384i	$0.661265\pi$
$158$	9.95256	−	5.74611i	0.791783	−	0.457136i
$159$		0			0
$160$	−12.4725	−	7.20102i	−0.986041	−	0.569291i
$161$		0			0
$162$		0			0
$163$	12.5175			0.980447			0.490223	−	0.871597i	$-0.336915\pi$
$163$	12.5175			0.980447			0.490223	+	0.871597i	$0.336915\pi$
$164$	2.25448	−	3.90487i	0.176045	−	0.304919i
$165$		0			0
$166$	−1.62590	+	0.938715i	−0.126194	+	0.0728584i
$167$	7.70819	+	13.3510i	0.596477	+	1.03313i	0.993337	+	0.115250i	$0.0367668\pi$
$167$	7.70819	+	13.3510i	0.596477	+	1.03313i	−0.396859	+	0.917880i	$0.629900\pi$
$168$		0			0
$169$	−3.11731	+	5.39935i	−0.239793	+	0.415334i
$170$			5.97755i			0.458457i
$171$		0			0
$172$	2.60127			0.198345
$173$	4.30737	−	7.46059i	0.327483	−	0.567218i	−0.654528	−	0.756037i	$-0.727133\pi$
$173$	4.30737	−	7.46059i	0.327483	−	0.567218i	0.982012	+	0.188820i	$0.0604661\pi$
$174$		0			0
$175$		0			0
$176$	4.49739	−	2.59657i	0.339004	−	0.195724i
$177$		0			0
$178$	6.22334	+	3.59305i	0.466459	+	0.269310i
$179$		−	19.1384i		−	1.43047i	−0.698882	−	0.715237i	$-0.746319\pi$
$179$		−	19.1384i		−	1.43047i	0.698882	−	0.715237i	$-0.253681\pi$
$180$		0			0
$181$		−	7.69817i		−	0.572200i	−0.958200	−	0.286100i	$-0.907641\pi$
$181$		−	7.69817i		−	0.572200i	0.958200	−	0.286100i	$-0.0923590\pi$
$182$		0			0
$183$		0			0
$184$	−5.61758	−	9.72993i	−0.414133	−	0.717300i
$185$	−11.0616	−	19.1593i	−0.813268	−	1.40862i
$186$		0			0
$187$	2.19285	+	1.26604i	0.160357	+	0.0925820i
$188$	4.37047			0.318750
$189$		0			0
$190$	−12.3342			−0.894817
$191$	−16.1203	−	9.30704i	−1.16642	−	0.673433i	−0.213587	−	0.976924i	$-0.568515\pi$
$191$	−16.1203	−	9.30704i	−1.16642	−	0.673433i	−0.952834	+	0.303491i	$0.901848\pi$
$192$		0			0
$193$	−9.05721	−	15.6875i	−0.651952	−	1.12921i	−0.982649	−	0.185477i	$-0.940617\pi$
$193$	−9.05721	−	15.6875i	−0.651952	−	1.12921i	0.330696	−	0.943737i	$-0.392716\pi$
$194$	−1.60799	−	2.78511i	−0.115447	−	0.199959i
$195$		0			0
$196$		0			0
$197$		−	16.5945i		−	1.18231i	−0.806559	−	0.591154i	$-0.798672\pi$
$197$		−	16.5945i		−	1.18231i	0.806559	−	0.591154i	$-0.201328\pi$
$198$		0			0
$199$		−	2.71887i		−	0.192735i	−0.995346	−	0.0963677i	$-0.969278\pi$
$199$		−	2.71887i		−	0.192735i	0.995346	−	0.0963677i	$-0.0307225\pi$
$200$	36.6875	+	21.1815i	2.59420	+	1.49776i
$201$		0			0
$202$	−16.3225	+	9.42383i	−1.14845	+	0.663058i
$203$		0			0
$204$		0			0
$205$	−15.9623	+	27.6476i	−1.11486	+	1.93099i
$206$	−3.55425			−0.247636
$207$		0			0
$208$		−	6.24759i		−	0.433192i
$209$	−2.61237	+	4.52476i	−0.180702	+	0.312984i
$210$		0			0
$211$	−13.9445	−	24.1526i	−0.959979	−	1.66273i	−0.722539	−	0.691330i	$-0.757025\pi$
$211$	−13.9445	−	24.1526i	−0.959979	−	1.66273i	−0.237440	−	0.971402i	$-0.576308\pi$
$212$	1.71301	−	0.989010i	0.117650	−	0.0679255i
$213$		0			0
$214$	6.97992	−	12.0896i	0.477138	−	0.826427i
$215$	−18.4177			−1.25608
$216$		0			0
$217$		0			0
$218$	7.30732	+	4.21888i	0.494914	+	0.285739i
$219$		0			0
$220$	4.96278	−	2.86526i	0.334590	−	0.193176i
$221$	2.63809	−	1.52310i	0.177457	−	0.102455i
$222$		0			0
$223$	6.64349	+	3.83562i	0.444881	+	0.256852i	0.705666	−	0.708545i	$-0.250648\pi$
$223$	6.64349	+	3.83562i	0.444881	+	0.256852i	−0.260785	+	0.965397i	$0.583981\pi$
$224$		0			0
$225$		0			0
$226$	−3.34815			−0.222715
$227$	−1.16439	+	2.01677i	−0.0772829	+	0.133858i	−0.902077	−	0.431576i	$-0.857958\pi$
$227$	−1.16439	+	2.01677i	−0.0772829	+	0.133858i	0.824794	+	0.565434i	$0.191291\pi$
$228$		0			0
$229$	−10.3653	+	5.98443i	−0.684961	+	0.395463i	−0.801722	−	0.597698i	$-0.796082\pi$
$229$	−10.3653	+	5.98443i	−0.684961	+	0.395463i	0.116760	+	0.993160i	$0.462749\pi$
$230$	9.31735	+	16.1381i	0.614368	+	1.06412i
$231$		0			0
$232$	−1.04692	+	1.81332i	−0.0687337	+	0.119050i
$233$		−	2.52779i		−	0.165601i	−0.996566	−	0.0828007i	$-0.973614\pi$
$233$		−	2.52779i		−	0.165601i	0.996566	−	0.0828007i	$-0.0263865\pi$
$234$		0			0
$235$	−30.9442			−2.01858
$236$	−1.78542	+	3.09244i	−0.116221	+	0.201301i
$237$		0			0
$238$		0			0
$239$	−17.4587	+	10.0798i	−1.12931	+	0.652006i	−0.943761	−	0.330630i	$-0.892739\pi$
$239$	−17.4587	+	10.0798i	−1.12931	+	0.652006i	−0.185546	+	0.982636i	$0.559405\pi$
$240$		0			0
$241$	−18.1254	−	10.4647i	−1.16756	−	0.674091i	−0.214455	−	0.976734i	$-0.568798\pi$
$241$	−18.1254	−	10.4647i	−1.16756	−	0.674091i	−0.953104	+	0.302643i	$0.902131\pi$
$242$		−	7.45280i		−	0.479084i
$243$		0			0
$244$		−	4.39184i		−	0.281159i
$245$		0			0
$246$		0			0
$247$	3.14280	+	5.44349i	0.199972	+	0.346361i
$248$	−10.0865	−	17.4704i	−0.640496	−	1.10937i
$249$		0			0
$250$	−38.7492	−	22.3718i	−2.45071	−	1.41492i
$251$	25.5747			1.61426			0.807130	−	0.590374i	$-0.201020\pi$
$251$	25.5747			1.61426			0.807130	+	0.590374i	$0.201020\pi$
$252$		0			0
$253$	7.89363			0.496268
$254$	18.9116	+	10.9186i	1.18662	+	0.685096i
$255$		0			0
$256$	6.58491	+	11.4054i	0.411557	+	0.712837i
$257$	−5.93725	−	10.2836i	−0.370355	−	0.641474i	0.619265	−	0.785182i	$-0.287431\pi$
$257$	−5.93725	−	10.2836i	−0.370355	−	0.641474i	−0.989620	+	0.143708i	$0.954097\pi$
$258$		0			0
$259$		0			0
$260$		−	6.89408i		−	0.427553i
$261$		0			0
$262$		−	7.89915i		−	0.488011i
$263$	−19.3705	−	11.1836i	−1.19444	−	0.689608i	−0.235127	−	0.971965i	$-0.575550\pi$
$263$	−19.3705	−	11.1836i	−1.19444	−	0.689608i	−0.959309	+	0.282357i	$0.908884\pi$
$264$		0			0
$265$	−12.1286	+	7.00247i	−0.745056	+	0.430158i
$266$		0			0
$267$		0			0
$268$	−2.03549	+	3.52557i	−0.124337	+	0.215358i
$269$	−4.22669			−0.257706			−0.128853	−	0.991664i	$-0.541130\pi$
$269$	−4.22669			−0.257706			−0.128853	+	0.991664i	$0.541130\pi$
$270$		0			0
$271$		−	22.3943i		−	1.36036i	−0.733046	−	0.680179i	$-0.761902\pi$
$271$		−	22.3943i		−	1.36036i	0.733046	−	0.680179i	$-0.238098\pi$
$272$	1.40653	−	2.43619i	0.0852836	−	0.147715i
$273$		0			0
$274$	−8.03905	−	13.9240i	−0.485657	−	0.841183i
$275$	−25.7760	+	14.8818i	−1.55435	+	0.897405i
$276$		0			0
$277$	−5.69230	+	9.85935i	−0.342017	+	0.592391i	−0.984807	−	0.173651i	$-0.944443\pi$
$277$	−5.69230	+	9.85935i	−0.342017	+	0.592391i	0.642790	+	0.766042i	$0.277777\pi$
$278$	10.5317			0.631652
$279$		0			0
$280$		0			0
$281$	0.702700	+	0.405704i	0.0419196	+	0.0242023i	0.520813	−	0.853671i	$-0.325629\pi$
$281$	0.702700	+	0.405704i	0.0419196	+	0.0242023i	−0.478894	+	0.877873i	$0.658962\pi$
$282$		0			0
$283$	15.8740	−	9.16486i	0.943611	−	0.544794i	0.0525206	−	0.998620i	$-0.483274\pi$
$283$	15.8740	−	9.16486i	0.943611	−	0.544794i	0.891090	+	0.453826i	$0.149941\pi$
$284$	1.03842	−	0.599532i	0.0616188	−	0.0355757i
$285$		0			0
$286$	5.73799	+	3.31283i	0.339294	+	0.195892i
$287$		0			0
$288$		0			0
$289$	−15.6284			−0.919318
$290$	1.73643	−	3.00759i	0.101967	−	0.176612i
$291$		0			0
$292$	−6.30913	+	3.64258i	−0.369214	+	0.213166i
$293$	−6.23639	−	10.8017i	−0.364334	−	0.631044i	0.624335	−	0.781156i	$-0.285370\pi$
$293$	−6.23639	−	10.8017i	−0.364334	−	0.631044i	−0.988669	+	0.150112i	$0.952037\pi$
$294$		0			0
$295$	12.6413	−	21.8954i	0.736004	−	1.27480i
$296$			15.7154i			0.913438i
$297$		0			0
$298$	4.47599			0.259287
$299$	4.74819	−	8.22411i	0.274595	−	0.475613i
$300$		0			0
$301$		0			0
$302$	3.89943	−	2.25134i	0.224387	−	0.129550i
$303$		0			0
$304$	5.02688	+	2.90227i	0.288311	+	0.166457i
$305$			31.0955i			1.78052i
$306$		0			0
$307$			21.3241i			1.21703i	0.793543	+	0.608514i	$0.208234\pi$
$307$			21.3241i			1.21703i	−0.793543	+	0.608514i	$0.791766\pi$
$308$		0			0
$309$		0			0
$310$	16.7296	+	28.9765i	0.950178	+	1.64576i
$311$	3.92094	+	6.79126i	0.222336	+	0.385097i	0.955517	−	0.294936i	$-0.0952985\pi$
$311$	3.92094	+	6.79126i	0.222336	+	0.385097i	−0.733181	+	0.680034i	$0.761965\pi$
$312$		0			0
$313$	8.57593	+	4.95131i	0.484740	+	0.279865i	0.722390	−	0.691486i	$-0.243044\pi$
$313$	8.57593	+	4.95131i	0.484740	+	0.279865i	−0.237650	+	0.971351i	$0.576377\pi$
$314$	−25.3158			−1.42865
$315$		0			0
$316$	−5.96794			−0.335723
$317$	−20.8358	−	12.0296i	−1.17025	−	0.675647i	−0.216515	−	0.976279i	$-0.569469\pi$
$317$	−20.8358	−	12.0296i	−1.17025	−	0.675647i	−0.953740	+	0.300632i	$0.902802\pi$
$318$		0			0
$319$	−0.735549	−	1.27401i	−0.0411828	−	0.0713308i
$320$	−18.8896	−	32.7178i	−1.05596	−	1.82898i
$321$		0			0
$322$		0			0
$323$			2.83018i			0.157476i
$324$		0			0
$325$			35.8069i			1.98621i
$326$	12.7723	+	7.37407i	0.707390	+	0.408412i
$327$		0			0
$328$	19.6396	−	11.3389i	1.08442	−	0.626088i
$329$		0			0
$330$		0			0
$331$	4.53686	−	7.85807i	0.249368	−	0.431918i	−0.713982	−	0.700164i	$-0.753110\pi$
$331$	4.53686	−	7.85807i	0.249368	−	0.431918i	0.963351	+	0.268245i	$0.0864437\pi$
$332$	0.974954			0.0535075
$333$		0			0
$334$			18.1636i			0.993867i
$335$	14.4118	−	24.9620i	0.787403	−	1.36382i
$336$		0			0
$337$	4.02012	+	6.96304i	0.218990	+	0.379301i	0.954499	−	0.298213i	$-0.0963906\pi$
$337$	4.02012	+	6.96304i	0.218990	+	0.379301i	−0.735510	+	0.677514i	$0.763057\pi$
$338$	−6.36152	+	3.67282i	−0.346021	+	0.199775i
$339$		0			0
$340$	1.55208	−	2.68828i	0.0841733	−	0.145792i
$341$	14.1733			0.767525
$342$		0			0
$343$		0			0
$344$	11.3303	+	6.54157i	0.610891	+	0.352698i
$345$		0			0
$346$	8.79007	−	5.07495i	0.472557	−	0.272831i
$347$	30.6345	−	17.6868i	1.64454	−	0.949478i	0.665356	−	0.746526i	$-0.268280\pi$
$347$	30.6345	−	17.6868i	1.64454	−	0.949478i	0.979189	−	0.202952i	$-0.0650536\pi$
$348$		0			0
$349$	21.1868	+	12.2322i	1.13411	+	0.654776i	0.944964	−	0.327174i	$-0.106096\pi$
$349$	21.1868	+	12.2322i	1.13411	+	0.654776i	0.189141	+	0.981950i	$0.439430\pi$
$350$		0			0
$351$		0			0
$352$	−7.18783			−0.383112
$353$	0.485949	−	0.841688i	0.0258644	−	0.0447985i	−0.852803	−	0.522232i	$-0.825100\pi$
$353$	0.485949	−	0.841688i	0.0258644	−	0.0447985i	0.878668	+	0.477433i	$0.158433\pi$
$354$		0			0
$355$	−7.35231	+	4.24486i	−0.390220	+	0.225294i
$356$	−1.86588	−	3.23180i	−0.0988914	−	0.171285i
$357$		0			0
$358$	11.2745	−	19.5279i	0.595874	−	1.03208i
$359$			15.9210i			0.840276i	0.907460	+	0.420138i	$0.138018\pi$
$359$			15.9210i			0.840276i	−0.907460	+	0.420138i	$0.861982\pi$
$360$		0			0
$361$	13.1601			0.692639
$362$	4.53499	−	7.85484i	0.238354	−	0.412841i
$363$		0			0
$364$		0			0
$365$	44.6704	−	25.7905i	2.33816	−	1.34994i
$366$		0			0
$367$	−21.3983	−	12.3543i	−1.11698	−	0.644891i	−0.176355	−	0.984327i	$-0.556431\pi$
$367$	−21.3983	−	12.3543i	−1.11698	−	0.644891i	−0.940629	+	0.339435i	$0.889764\pi$
$368$		−	8.76958i		−	0.457146i
$369$		0			0
$370$		−	26.0657i		−	1.35509i
$371$		0			0
$372$		0			0
$373$	4.71810	+	8.17200i	0.244294	+	0.423130i	0.961933	−	0.273286i	$-0.0881104\pi$
$373$	4.71810	+	8.17200i	0.244294	+	0.423130i	−0.717639	+	0.696416i	$0.754777\pi$
$374$	1.49165	+	2.58361i	0.0771314	+	0.133595i
$375$		0			0
$376$	19.0364	+	10.9907i	0.981730	+	0.566802i
$377$	−1.76980			−0.0911492
$378$		0			0
$379$	20.8031			1.06858			0.534292	−	0.845300i	$-0.320578\pi$
$379$	20.8031			1.06858			0.534292	+	0.845300i	$0.320578\pi$
$380$	5.54705	+	3.20259i	0.284558	+	0.164289i
$381$		0			0
$382$	−10.9656	−	18.9929i	−0.561047	−	0.971761i
$383$	3.23008	+	5.59467i	0.165050	+	0.285874i	0.936673	−	0.350205i	$-0.113888\pi$
$383$	3.23008	+	5.59467i	0.165050	+	0.285874i	−0.771623	+	0.636080i	$0.780555\pi$
$384$		0			0
$385$		0			0
$386$		−	21.3424i		−	1.08630i
$387$		0			0
$388$			1.67006i			0.0847845i
$389$	0.0445846	+	0.0257409i	0.00226053	+	0.00130512i	0.501130	−	0.865372i	$-0.332918\pi$
$389$	0.0445846	+	0.0257409i	0.00226053	+	0.00130512i	−0.498869	+	0.866677i	$0.666251\pi$
$390$		0			0
$391$	3.70303	−	2.13794i	0.187270	−	0.108120i
$392$		0			0
$393$		0			0
$394$	9.77582	−	16.9322i	0.492499	−	0.853033i
$395$	42.2547			2.12607
$396$		0			0
$397$			12.7131i			0.638052i	0.947746	+	0.319026i	$0.103356\pi$
$397$			12.7131i			0.638052i	−0.947746	+	0.319026i	$0.896644\pi$
$398$	1.60169	−	2.77420i	0.0802853	−	0.139058i
$399$		0			0
$400$	16.5332	+	28.6364i	0.826660	+	1.43182i
$401$	2.19725	−	1.26858i	0.109725	−	0.0633500i	−0.444133	−	0.895961i	$-0.646488\pi$
$401$	2.19725	−	1.26858i	0.109725	−	0.0633500i	0.553858	+	0.832611i	$0.313155\pi$
$402$		0			0
$403$	8.52554	−	14.7667i	0.424687	−	0.735580i
$404$	9.78764			0.486953
$405$		0			0
$406$		0			0
$407$	−9.56210	−	5.52068i	−0.473976	−	0.273650i
$408$		0			0
$409$	−0.0495655	+	0.0286167i	−0.00245086	+	0.00141500i	−0.501225	−	0.865317i	$-0.667117\pi$
$409$	−0.0495655	+	0.0286167i	−0.00245086	+	0.00141500i	0.498774	+	0.866732i	$0.333784\pi$
$410$	−32.5744	+	18.8068i	−1.60873	+	0.928803i
$411$		0			0
$412$	1.59845	+	0.922864i	0.0787499	+	0.0454663i
$413$		0			0
$414$		0			0
$415$	−6.90295			−0.338853
$416$	−4.32364	+	7.48876i	−0.211984	+	0.367167i
$417$		0			0
$418$	−5.33108	+	3.07790i	−0.260752	+	0.150545i
$419$	−3.08007	−	5.33484i	−0.150471	−	0.260624i	0.780930	−	0.624619i	$-0.214746\pi$
$419$	−3.08007	−	5.33484i	−0.150471	−	0.260624i	−0.931401	+	0.363995i	$0.881412\pi$
$420$		0			0
$421$	15.0693	−	26.1007i	0.734431	−	1.27207i	−0.220542	−	0.975378i	$-0.570783\pi$
$421$	15.0693	−	26.1007i	0.734431	−	1.27207i	0.954973	−	0.296694i	$-0.0958842\pi$
$422$		−	32.8588i		−	1.59954i
$423$		0			0
$424$	9.94849			0.483141
$425$	−8.06128	+	13.9626i	−0.391030	+	0.677283i
$426$		0			0
$427$		0			0
$428$	−6.27815	+	3.62469i	−0.303466	+	0.175206i
$429$		0			0
$430$	−18.7926	−	10.8499i	−0.906258	−	0.523228i
$431$			8.07140i			0.388785i	0.980924	+	0.194393i	$0.0622736\pi$
$431$			8.07140i			0.388785i	−0.980924	+	0.194393i	$0.937726\pi$
$432$		0			0
$433$			28.4938i			1.36933i	0.728860	+	0.684663i	$0.240051\pi$
$433$			28.4938i			1.36933i	−0.728860	+	0.684663i	$0.759949\pi$
$434$		0			0
$435$		0			0
$436$	−2.19088	−	3.79471i	−0.104924	−	0.181734i
$437$	4.41147	+	7.64090i	0.211029	+	0.365514i
$438$		0			0
$439$	1.77067	+	1.02230i	0.0845096	+	0.0487916i	0.541659	−	0.840598i	$-0.317796\pi$
$439$	1.77067	+	1.02230i	0.0845096	+	0.0487916i	−0.457150	+	0.889390i	$0.651130\pi$
$440$	28.8218			1.37402
$441$		0			0
$442$	3.58904			0.170713
$443$	21.1324	+	12.2008i	1.00403	+	0.579677i	0.909438	−	0.415839i	$-0.136512\pi$
$443$	21.1324	+	12.2008i	1.00403	+	0.579677i	0.0945924	+	0.995516i	$0.469845\pi$
$444$		0			0
$445$	13.2110	+	22.8821i	0.626260	+	1.08471i
$446$	4.51913	+	7.82737i	0.213987	+	0.370637i
$447$		0			0
$448$		0			0
$449$			0.293539i			0.0138529i	0.999976	+	0.00692647i	$0.00220478\pi$
$449$			0.293539i			0.0138529i	−0.999976	+	0.00692647i	$0.997795\pi$
$450$		0			0
$451$			15.9331i			0.750259i
$452$	1.50576	+	0.869351i	0.0708250	+	0.0408908i
$453$		0			0
$454$	−2.37616	+	1.37188i	−0.111519	+	0.0643855i
$455$		0			0
$456$		0			0
$457$	−8.27470	+	14.3322i	−0.387074	+	0.670432i	−0.992055	−	0.125808i	$-0.959848\pi$
$457$	−8.27470	+	14.3322i	−0.387074	+	0.670432i	0.604981	+	0.796240i	$0.293181\pi$
$458$	−14.1017			−0.658931
$459$		0			0
$460$		−	9.67705i		−	0.451195i
$461$	−10.0560	+	17.4175i	−0.468354	+	0.811213i	−0.999346	−	0.0361638i	$-0.988486\pi$
$461$	−10.0560	+	17.4175i	−0.468354	+	0.811213i	0.530992	+	0.847377i	$0.321820\pi$
$462$		0			0
$463$	9.34602	+	16.1878i	0.434346	+	0.752310i	0.997242	−	0.0742181i	$-0.0236461\pi$
$463$	9.34602	+	16.1878i	0.434346	+	0.752310i	−0.562896	+	0.826528i	$0.690313\pi$
$464$	−1.41538	+	0.817173i	−0.0657076	+	0.0379363i
$465$		0			0
$466$	1.48912	−	2.57924i	0.0689824	−	0.119481i
$467$	−29.3605			−1.35864			−0.679322	−	0.733841i	$-0.737726\pi$
$467$	−29.3605			−1.35864			−0.679322	+	0.733841i	$0.737726\pi$
$468$		0			0
$469$		0			0
$470$	−31.5740	−	18.2293i	−1.45640	−	0.840853i
$471$		0			0
$472$	−15.5535	+	8.97981i	−0.715907	+	0.413329i
$473$	−7.96050	+	4.59599i	−0.366024	+	0.211324i
$474$		0			0
$475$	−28.8106	−	16.6338i	−1.32192	−	0.763212i
$476$		0			0
$477$		0			0
$478$	−23.7520			−1.08639
$479$	−10.9660	+	18.9938i	−0.501051	+	0.867847i	0.498948	+	0.866632i	$0.333720\pi$
$479$	−10.9660	+	18.9938i	−0.501051	+	0.867847i	−0.999999	+	0.00121455i	$0.999613\pi$
$480$		0			0
$481$	−11.5036	+	6.64163i	−0.524521	+	0.302832i
$482$	−12.3295	−	21.3554i	−0.561594	−	0.972710i
$483$		0			0
$484$	−1.93513	+	3.35174i	−0.0879604	+	0.152352i
$485$		−	11.8245i		−	0.536923i
$486$		0			0
$487$	1.07779			0.0488394			0.0244197	−	0.999702i	$-0.492226\pi$
$487$	1.07779			0.0488394			0.0244197	+	0.999702i	$0.492226\pi$
$488$	11.0444	−	19.1295i	0.499957	−	0.865952i
$489$		0			0
$490$		0			0
$491$	−16.3708	+	9.45168i	−0.738804	+	0.426549i	−0.821634	−	0.570015i	$-0.806937\pi$
$491$	−16.3708	+	9.45168i	−0.738804	+	0.426549i	0.0828305	+	0.996564i	$0.473604\pi$
$492$		0			0
$493$	−0.690115	−	0.398438i	−0.0310812	−	0.0179448i
$494$			7.40570i			0.333198i
$495$		0			0
$496$		−	15.7461i		−	0.707020i
$497$		0			0
$498$		0			0
$499$	8.34290	+	14.4503i	0.373479	+	0.646885i	0.990098	−	0.140377i	$-0.0448314\pi$
$499$	8.34290	+	14.4503i	0.373479	+	0.646885i	−0.616619	+	0.787262i	$0.711498\pi$
$500$	11.6178	+	20.1225i	0.519562	+	0.899908i
$501$		0			0
$502$	26.0952	+	15.0661i	1.16468	+	0.672431i
$503$	−21.2386			−0.946981			−0.473491	−	0.880799i	$-0.657006\pi$
$503$	−21.2386			−0.946981			−0.473491	+	0.880799i	$0.657006\pi$
$504$		0			0
$505$	−69.2993			−3.08378
$506$	8.05428	+	4.65014i	0.358056	+	0.206724i
$507$		0			0
$508$	−5.67007	−	9.82085i	−0.251569	−	0.435730i
$509$	5.72252	+	9.91170i	0.253646	+	0.439328i	0.964527	−	0.263984i	$-0.0850367\pi$
$509$	5.72252	+	9.91170i	0.253646	+	0.439328i	−0.710881	+	0.703313i	$0.751703\pi$
$510$		0			0
$511$		0			0
$512$			22.7685i			1.00623i
$513$		0			0
$514$		−	13.9905i		−	0.617096i
$515$	−11.3175	−	6.53414i	−0.498707	−	0.287929i
$516$		0			0
$517$	−13.3747	+	7.72188i	−0.588218	+	0.339608i
$518$		0			0
$519$		0			0
$520$	17.3370	−	30.0285i	0.760276	−	1.31684i
$521$	−20.7998			−0.911254			−0.455627	−	0.890171i	$-0.650585\pi$
$521$	−20.7998			−0.911254			−0.455627	+	0.890171i	$0.650585\pi$
$522$		0			0
$523$		−	14.9338i		−	0.653009i	−0.945196	−	0.326505i	$-0.894129\pi$
$523$		−	14.9338i		−	0.653009i	0.945196	−	0.326505i	$-0.105871\pi$
$524$	−2.05102	+	3.55248i	−0.0895994	+	0.155191i
$525$		0			0
$526$	−13.1765	−	22.8223i	−0.574522	−	0.995101i
$527$	6.64890	−	3.83875i	0.289631	−	0.167218i
$528$		0			0
$529$	−4.83508	+	8.37460i	−0.210221	+	0.364113i
$530$	−16.5006			−0.716742
$531$		0			0
$532$		0			0
$533$	16.6002	+	9.58410i	0.719033	+	0.415134i
$534$		0			0
$535$	44.4511	−	25.6639i	1.92179	−	1.10955i
$536$	−17.7319	+	10.2375i	−0.765902	+	0.442194i
$537$		0			0
$538$	−4.31272	−	2.48995i	−0.185934	−	0.107349i
$539$		0			0
$540$		0			0
$541$	31.1677			1.34000			0.670002	−	0.742360i	$-0.266293\pi$
$541$	31.1677			1.34000			0.670002	+	0.742360i	$0.266293\pi$
$542$	13.1925	−	22.8501i	0.566667	−	0.981496i
$543$		0			0
$544$	−3.37192	+	1.94678i	−0.144570	+	0.0834675i
$545$	15.5120	+	26.8676i	0.664462	+	1.15088i
$546$		0			0
$547$	−15.7410	+	27.2642i	−0.673035	+	1.16573i	0.304004	+	0.952671i	$0.401677\pi$
$547$	−15.7410	+	27.2642i	−0.673035	+	1.16573i	−0.977039	+	0.213061i	$0.931657\pi$
$548$			8.34940i			0.356669i
$549$		0			0
$550$	−35.0674			−1.49528
$551$	0.822146	−	1.42400i	0.0350246	−	0.0606644i
$552$		0			0
$553$		0			0
$554$	−11.6163	+	6.70667i	−0.493529	+	0.284939i
$555$		0			0
$556$	−4.73643	−	2.73458i	−0.200870	−	0.115972i
$557$		−	27.2389i		−	1.15415i	−0.816692	−	0.577074i	$-0.804194\pi$
$557$		−	27.2389i		−	1.15415i	0.816692	−	0.577074i	$-0.195806\pi$
$558$		0			0
$559$			11.0584i			0.467720i
$560$		0			0
$561$		0			0
$562$	0.478001	+	0.827922i	0.0201633	+	0.0349238i
$563$	−14.1871	−	24.5728i	−0.597916	−	1.03562i	−0.993128	−	0.117031i	$-0.962662\pi$
$563$	−14.1871	−	24.5728i	−0.597916	−	1.03562i	0.395212	−	0.918590i	$-0.370671\pi$
$564$		0			0
$565$	−10.6612	−	6.15525i	−0.448521	−	0.258953i
$566$	21.5961			0.907751
$567$		0			0
$568$	6.03071			0.253043
$569$	−29.4616	−	17.0097i	−1.23509	−	0.713082i	−0.267007	−	0.963695i	$-0.586035\pi$
$569$	−29.4616	−	17.0097i	−1.23509	−	0.713082i	−0.968087	+	0.250613i	$0.919368\pi$
$570$		0			0
$571$	22.3455	+	38.7035i	0.935130	+	1.61969i	0.774402	+	0.632693i	$0.218051\pi$
$571$	22.3455	+	38.7035i	0.935130	+	1.61969i	0.160727	+	0.986999i	$0.448616\pi$
$572$	−1.72036	−	2.97975i	−0.0719319	−	0.124590i
$573$		0			0
$574$		0			0
$575$			50.2613i			2.09604i
$576$		0			0
$577$			7.34738i			0.305875i	0.988236	+	0.152938i	$0.0488734\pi$
$577$			7.34738i			0.305875i	−0.988236	+	0.152938i	$0.951127\pi$
$578$	−15.9465	−	9.20670i	−0.663286	−	0.382948i
$579$		0			0
$580$	−1.56185	+	0.901733i	−0.0648522	+	0.0374424i
$581$		0			0
$582$		0			0
$583$	−3.49482	+	6.05320i	−0.144741	+	0.250698i
$584$	−36.6408			−1.51621
$585$		0			0
$586$		−	14.6954i		−	0.607063i
$587$	−13.1328	+	22.7466i	−0.542048	+	0.938855i	0.456738	+	0.889601i	$0.349018\pi$
$587$	−13.1328	+	22.7466i	−0.542048	+	0.938855i	−0.998786	+	0.0492535i	$0.984316\pi$
$588$		0			0
$589$	7.92095	+	13.7195i	0.326377	+	0.565302i
$590$	25.7971	−	14.8940i	1.06205	−	0.613175i
$591$		0			0
$592$	−6.13331	+	10.6232i	−0.252078	+	0.436611i
$593$	−9.12418			−0.374685			−0.187343	−	0.982295i	$-0.559987\pi$
$593$	−9.12418			−0.374685			−0.187343	+	0.982295i	$0.559987\pi$
$594$		0			0
$595$		0			0
$596$	−2.01298	−	1.16220i	−0.0824550	−	0.0476054i
$597$		0			0
$598$	9.68966	−	5.59433i	0.396240	−	0.228769i
$599$	20.0987	−	11.6040i	0.821210	−	0.474126i	−0.0296234	−	0.999561i	$-0.509431\pi$
$599$	20.0987	−	11.6040i	0.821210	−	0.474126i	0.850834	+	0.525435i	$0.176097\pi$
$600$		0			0
$601$	19.0021	+	10.9709i	0.775111	+	0.447510i	0.834695	−	0.550713i	$-0.185644\pi$
$601$	19.0021	+	10.9709i	0.775111	+	0.447510i	−0.0595840	+	0.998223i	$0.518977\pi$
$602$		0			0
$603$		0			0
$604$	−2.33825			−0.0951421
$605$	13.7013	−	23.7313i	0.557036	−	0.964815i
$606$		0			0
$607$	38.6289	−	22.3024i	1.56790	−	0.905226i	0.571484	−	0.820613i	$-0.306368\pi$
$607$	38.6289	−	22.3024i	1.56790	−	0.905226i	0.996414	−	0.0846136i	$-0.0269656\pi$
$608$	−4.01703	−	6.95770i	−0.162912	−	0.282172i
$609$		0			0
$610$	−18.3184	+	31.7283i	−0.741689	+	1.28464i
$611$			18.5795i			0.751647i
$612$		0			0
$613$	11.6560			0.470780			0.235390	−	0.971901i	$-0.424363\pi$
$613$	11.6560			0.470780			0.235390	+	0.971901i	$0.424363\pi$
$614$	−12.5620	+	21.7580i	−0.506961	+	0.878083i
$615$		0			0
$616$		0			0
$617$	−36.6143	+	21.1393i	−1.47403	+	0.851034i	−0.999572	−	0.0292416i	$-0.990691\pi$
$617$	−36.6143	+	21.1393i	−1.47403	+	0.851034i	−0.474462	+	0.880276i	$0.657357\pi$
$618$		0			0
$619$	30.0633	+	17.3571i	1.20835	+	0.697640i	0.962398	−	0.271643i	$-0.0875670\pi$
$619$	30.0633	+	17.3571i	1.20835	+	0.697640i	0.245949	+	0.969283i	$0.420900\pi$
$620$		−	17.3755i		−	0.697815i
$621$		0			0
$622$			9.23930i			0.370462i
$623$		0			0
$624$		0			0
$625$	−47.8410	−	82.8631i	−1.91364	−	3.31452i
$626$	5.83364	+	10.1042i	0.233159	+	0.403844i
$627$		0			0
$628$	11.3852	+	6.57327i	0.454321	+	0.262302i
$629$	−5.98098			−0.238477
$630$		0			0
$631$	−12.8860			−0.512982			−0.256491	−	0.966547i	$-0.582566\pi$
$631$	−12.8860			−0.512982			−0.256491	+	0.966547i	$0.582566\pi$
$632$	−25.9945	−	15.0079i	−1.03401	−	0.596984i
$633$		0			0
$634$	−14.1732	−	24.5488i	−0.562891	−	0.974956i
$635$	40.1457	+	69.5345i	1.59314	+	2.75939i
$636$		0			0
$637$		0			0
$638$		−	1.73325i		−	0.0686200i
$639$		0			0
$640$		−	15.7074i		−	0.620890i
$641$	16.5666	+	9.56474i	0.654342	+	0.377785i	0.790118	−	0.612955i	$-0.210019\pi$
$641$	16.5666	+	9.56474i	0.654342	+	0.377785i	−0.135776	+	0.990740i	$0.543353\pi$
$642$		0			0
$643$	9.77521	−	5.64372i	0.385497	−	0.222567i	−0.294710	−	0.955587i	$-0.595223\pi$
$643$	9.77521	−	5.64372i	0.385497	−	0.222567i	0.680207	+	0.733020i	$0.261890\pi$
$644$		0			0
$645$		0			0
$646$	−1.66726	+	2.88778i	−0.0655976	+	0.113618i
$647$	−5.08677			−0.199982			−0.0999909	−	0.994988i	$-0.531881\pi$
$647$	−5.08677			−0.199982			−0.0999909	+	0.994988i	$0.531881\pi$
$648$		0			0
$649$		−	12.6181i		−	0.495305i
$650$	−21.0939	+	36.5356i	−0.827369	+	1.43305i
$651$		0			0
$652$	−3.82937	−	6.63267i	−0.149970	−	0.259755i
$653$	32.9044	−	18.9974i	1.28765	−	0.743424i	0.309414	−	0.950927i	$-0.399867\pi$
$653$	32.9044	−	18.9974i	1.28765	−	0.743424i	0.978234	+	0.207503i	$0.0665338\pi$
$654$		0			0
$655$	14.5218	−	25.1526i	0.567415	−	0.982792i
$656$	17.7012			0.691115
$657$		0			0
$658$		0			0
$659$	−9.97949	−	5.76166i	−0.388746	−	0.224442i	0.292871	−	0.956152i	$-0.405389\pi$
$659$	−9.97949	−	5.76166i	−0.388746	−	0.224442i	−0.681617	+	0.731710i	$0.738723\pi$
$660$		0			0
$661$	−38.0928	+	21.9929i	−1.48164	+	0.855424i	−0.999783	−	0.0208274i	$-0.993370\pi$
$661$	−38.0928	+	21.9929i	−1.48164	+	0.855424i	−0.481854	+	0.876251i	$0.660037\pi$
$662$	9.25838	−	5.34533i	0.359837	−	0.207752i
$663$		0			0
$664$	4.24660	+	2.45178i	0.164800	+	0.0951473i
$665$		0			0
$666$		0			0
$667$	−2.48422			−0.0961894
$668$	4.71620	−	8.16869i	0.182475	−	0.316056i
$669$		0			0
$670$	29.4103	−	16.9800i	1.13622	−	0.655996i
$671$	7.75962	+	13.4401i	0.299557	+	0.518848i
$672$		0			0
$673$	−21.9316	+	37.9866i	−0.845400	+	1.46428i	0.0398735	+	0.999205i	$0.487305\pi$
$673$	−21.9316	+	37.9866i	−0.845400	+	1.46428i	−0.885273	+	0.465071i	$0.846029\pi$
$674$			9.47301i			0.364887i
$675$		0			0
$676$	3.81461			0.146716
$677$	−0.738999	+	1.27998i	−0.0284020	+	0.0491938i	−0.879877	−	0.475201i	$-0.842375\pi$
$677$	−0.738999	+	1.27998i	−0.0284020	+	0.0491938i	0.851475	+	0.524395i	$0.175709\pi$
$678$		0			0
$679$		0			0
$680$	13.5208	−	7.80621i	0.518497	−	0.299355i
$681$		0			0
$682$	14.4617	+	8.34948i	0.553768	+	0.319718i
$683$			10.3259i			0.395111i	0.980292	+	0.197555i	$0.0633002\pi$
$683$			10.3259i			0.395111i	−0.980292	+	0.197555i	$0.936700\pi$
$684$		0			0
$685$		−	59.1162i		−	2.25871i
$686$		0			0
$687$		0			0
$688$	5.10601	+	8.84388i	0.194665	+	0.337170i
$689$	4.20442	+	7.28228i	0.160176	+	0.277433i
$690$		0			0
$691$	6.58166	+	3.79992i	0.250378	+	0.144556i	0.619937	−	0.784651i	$-0.287158\pi$
$691$	6.58166	+	3.79992i	0.250378	+	0.144556i	−0.369559	+	0.929207i	$0.620491\pi$
$692$	−5.27087			−0.200368
$693$		0			0
$694$	41.6773			1.58205
$695$	33.5353	+	19.3616i	1.27207	+	0.734428i
$696$		0			0
$697$	4.31538	+	7.47446i	0.163457	+	0.283115i
$698$	14.4120	+	24.9624i	0.545503	+	0.944839i
$699$		0			0
$700$		0			0
$701$		−	6.35907i		−	0.240179i	−0.992763	−	0.120089i	$-0.961682\pi$
$701$		−	6.35907i		−	0.240179i	0.992763	−	0.120089i	$-0.0383181\pi$
$702$		0			0
$703$		−	12.3413i		−	0.465460i
$704$	−16.3289	−	9.42749i	−0.615419	−	0.355312i
$705$		0			0
$706$	0.991677	−	0.572545i	0.0373223	−	0.0215480i
$707$		0			0
$708$		0			0
$709$	23.8048	−	41.2311i	0.894007	−	1.54847i	0.0589776	−	0.998259i	$-0.481216\pi$
$709$	23.8048	−	41.2311i	0.894007	−	1.54847i	0.835029	−	0.550206i	$-0.185451\pi$
$710$	−10.0026			−0.375390
$711$		0			0
$712$		−	18.7690i		−	0.703396i
$713$	11.9671	−	20.7276i	0.448171	−	0.776255i
$714$		0			0
$715$	12.1806	+	21.0975i	0.455530	+	0.789002i
$716$	−10.1409	+	5.85486i	−0.378983	+	0.218806i
$717$		0			0
$718$	−9.37904	+	16.2450i	−0.350023	+	0.606257i
$719$	14.1470			0.527594			0.263797	−	0.964578i	$-0.415025\pi$
$719$	14.1470			0.527594			0.263797	+	0.964578i	$0.415025\pi$
$720$		0			0
$721$		0			0
$722$	13.4280	+	7.75264i	0.499737	+	0.288524i
$723$		0			0
$724$	−4.07904	+	2.35503i	−0.151596	+	0.0875241i
$725$	8.11202	−	4.68348i	0.301273	−	0.173940i
$726$		0			0
$727$	−40.1828	−	23.1996i	−1.49030	−	0.860424i	−0.490360	−	0.871520i	$-0.663135\pi$
$727$	−40.1828	−	23.1996i	−1.49030	−	0.860424i	−0.999938	+	0.0110955i	$0.996468\pi$
$728$		0			0
$729$		0			0
$730$	60.7728			2.24930
$731$	−2.48960	+	4.31211i	−0.0920811	+	0.159489i
$732$		0			0
$733$	−22.8893	+	13.2151i	−0.845436	+	0.488112i	−0.859108	−	0.511794i	$-0.828981\pi$
$733$	−22.8893	+	13.2151i	−0.845436	+	0.488112i	0.0136726	+	0.999907i	$0.495648\pi$
$734$	−14.5559	−	25.2115i	−0.537268	−	0.930575i
$735$		0			0
$736$	−6.06899	+	10.5118i	−0.223706	+	0.387470i
$737$		−	14.3854i		−	0.529894i
$738$		0			0
$739$	−46.2670			−1.70196			−0.850979	−	0.525200i	$-0.823991\pi$
$739$	−46.2670			−1.70196			−0.850979	+	0.525200i	$0.823991\pi$
$740$	−6.76798	+	11.7225i	−0.248796	+	0.430927i
$741$		0			0
$742$		0			0
$743$	36.5640	−	21.1102i	1.34140	−	0.774458i	0.354388	−	0.935098i	$-0.384689\pi$
$743$	36.5640	−	21.1102i	1.34140	−	0.774458i	0.987013	+	0.160640i	$0.0513558\pi$
$744$		0			0
$745$	14.2525	+	8.22868i	0.522171	+	0.301476i
$746$			11.1177i			0.407050i
$747$		0			0
$748$		−	1.54923i		−	0.0566456i
$749$		0			0
$750$		0			0
$751$	8.02320	+	13.8966i	0.292771	+	0.507094i	0.974464	−	0.224544i	$-0.0720894\pi$
$751$	8.02320	+	13.8966i	0.292771	+	0.507094i	−0.681693	+	0.731638i	$0.738756\pi$
$752$	8.57877	+	14.8589i	0.312836	+	0.541847i
$753$		0			0
$754$	−1.80582	−	1.04259i	−0.0657639	−	0.0379688i
$755$	16.5555			0.602516
$756$		0			0
$757$	25.0149			0.909183			0.454591	−	0.890700i	$-0.349785\pi$
$757$	25.0149			0.909183			0.454591	+	0.890700i	$0.349785\pi$
$758$	21.2265	+	12.2551i	0.770981	+	0.445126i
$759$		0			0
$760$	16.1075	+	27.8990i	0.584281	+	1.01200i
$761$	3.00365	+	5.20247i	0.108882	+	0.188589i	0.915318	−	0.402733i	$-0.131940\pi$
$761$	3.00365	+	5.20247i	0.108882	+	0.188589i	−0.806436	+	0.591322i	$0.798606\pi$
$762$		0			0
$763$		0			0
$764$			11.3889i			0.412035i
$765$		0			0
$766$			7.61138i			0.275010i
$767$	−13.1464	−	7.59008i	−0.474689	−	0.274062i
$768$		0			0
$769$	−28.9946	+	16.7400i	−1.04557	+	0.603661i	−0.921406	−	0.388600i	$-0.872959\pi$
$769$	−28.9946	+	16.7400i	−1.04557	+	0.603661i	−0.124166	+	0.992262i	$0.539625\pi$
$770$		0			0
$771$		0			0
$772$	−5.54159	+	9.59831i	−0.199446	+	0.345451i
$773$	36.2016			1.30208			0.651040	−	0.759043i	$-0.274333\pi$
$773$	36.2016			1.30208			0.651040	+	0.759043i	$0.274333\pi$
$774$		0			0
$775$			90.2458i			3.24172i
$776$	−4.19980	+	7.27427i	−0.150764	+	0.261131i
$777$		0			0
$778$	0.0303280	+	0.0525296i	0.00108731	+	0.00188328i
$779$	−15.4230	+	8.90445i	−0.552585	+	0.319035i
$780$		0			0
$781$	−2.11854	+	3.66942i	−0.0758073	+	0.131302i
$782$	5.03785			0.180153
$783$		0			0
$784$		0			0
$785$	−80.6108	−	46.5407i	−2.87712	−	1.66111i
$786$		0			0
$787$	−14.1930	+	8.19433i	−0.505926	+	0.292096i	−0.731157	−	0.682209i	$-0.761019\pi$
$787$	−14.1930	+	8.19433i	−0.505926	+	0.292096i	0.225232	+	0.974305i	$0.427686\pi$
$788$	−8.79294	+	5.07661i	−0.313236	+	0.180847i
$789$		0			0
$790$	43.1147	+	24.8923i	1.53395	+	0.885628i
$791$		0			0
$792$		0			0
$793$	18.6703			0.663004
$794$	−7.48929	+	12.9718i	−0.265785	+	0.460353i
$795$		0			0
$796$	−1.44065	+	0.831760i	−0.0510625	+	0.0294809i

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.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.02035 + 0.589100i) q^{2} +(-0.305921 - 0.529871i) q^{4} +(2.16601 + 3.75164i) q^{5} -3.07728i q^{8} +O(q^{10})\)	\(q+(1.02035 + 0.589100i) q^{2} +(-0.305921 - 0.529871i) q^{4} +(2.16601 + 3.75164i) q^{5} -3.07728i q^{8} +5.10399i q^{10} +(1.87238 + 1.08102i) q^{11} +(2.25256 - 1.30052i) q^{13} +(1.20098 - 2.08016i) q^{16} +1.17115 q^{17} +2.41658i q^{19} +(1.32526 - 2.29541i) q^{20} +(1.27366 + 2.20604i) q^{22} +(3.16186 - 1.82550i) q^{23} +(-6.88321 + 11.9221i) q^{25} +3.06454 q^{26} +(-0.589262 - 0.340210i) q^{29} +(5.67723 - 3.27775i) q^{31} +(-2.87915 + 1.66228i) q^{32} +(1.19499 + 0.689926i) q^{34} -5.10692 q^{37} +(-1.42361 + 2.46576i) q^{38} +(11.5448 - 6.66541i) q^{40} +(3.68473 + 6.38214i) q^{41} +(-2.12577 + 3.68194i) q^{43} -1.32283i q^{44} +4.30162 q^{46} +(-3.57157 + 6.18614i) q^{47} +(-14.0466 + 8.10980i) q^{50} +(-1.37821 - 0.795711i) q^{52} +3.23289i q^{53} +9.36601i q^{55} +(-0.400836 - 0.694269i) q^{58} +(-2.91810 - 5.05430i) q^{59} +(6.21638 + 3.58903i) q^{61} +7.72370 q^{62} -8.72092 q^{64} +(9.75814 + 5.63387i) q^{65} +(-3.32682 - 5.76221i) q^{67} +(-0.358281 - 0.620560i) q^{68} +1.95976i q^{71} -11.9069i q^{73} +(-5.21085 - 3.00849i) q^{74} +(1.28048 - 0.739283i) q^{76} +(4.87702 - 8.44725i) q^{79} +10.4054 q^{80} +8.68270i q^{82} +(-0.796736 + 1.37999i) q^{83} +(2.53673 + 4.39374i) q^{85} +(-4.33806 + 2.50458i) q^{86} +(3.32660 - 5.76184i) q^{88} +6.09921 q^{89} +(-1.93456 - 1.11692i) q^{92} +(-7.28851 + 4.20802i) q^{94} +(-9.06614 + 5.23434i) q^{95} +(-2.36387 - 1.36478i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48q + 24q^{4} + O(q^{10}) \)	\( 48q + 24q^{4} + 24q^{11} - 24q^{16} + 48q^{23} - 24q^{25} - 120q^{32} - 48q^{50} - 48q^{64} - 120q^{65} + 168q^{74} - 24q^{79} - 24q^{85} - 24q^{86} + 144q^{92} - 96q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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