LMFDB - Embedded newform 1323.2.g.g.361.2

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			361.2
Root			$-0.474636 + 0.274031i$ of defining polynomial
Character	$\chi$	$=$	1323.361
Dual form			1323.2.g.g.667.2

$q$-expansion

$f(q)$	$=$	$q+(-0.849814 + 1.47192i) q^{2} +(-0.444368 - 0.769668i) q^{4} +0.949271 q^{5} -1.88874 q^{8} +O(q^{10})$	\(q+(-0.849814 + 1.47192i) q^{2} +(-0.444368 - 0.769668i) q^{4} +0.949271 q^{5} -1.88874 q^{8} +(-0.806704 + 1.39725i) q^{10} +0.588364 q^{11} +(2.50987 - 4.34722i) q^{13} +(2.49381 - 4.31941i) q^{16} +(-3.79121 + 6.56657i) q^{17} +(2.23061 + 3.86353i) q^{19} +(-0.421826 - 0.730623i) q^{20} +(-0.500000 + 0.866025i) q^{22} -2.47710 q^{23} -4.09888 q^{25} +(4.26584 + 7.38866i) q^{26} +(2.73855 + 4.74331i) q^{29} +(3.03731 + 5.26078i) q^{31} +(2.34981 + 4.07000i) q^{32} +(-6.44364 - 11.1607i) q^{34} +(3.49381 + 6.05146i) q^{37} -7.58242 q^{38} -1.79292 q^{40} +(-0.527445 + 0.913562i) q^{41} +(-3.49381 - 6.05146i) q^{43} +(-0.261450 - 0.452845i) q^{44} +(2.10507 - 3.64610i) q^{46} +(-3.73840 + 6.47510i) q^{47} +(3.48329 - 6.03323i) q^{50} -4.46122 q^{52} +(3.46108 - 5.99476i) q^{53} +0.558517 q^{55} -9.30903 q^{58} +(5.21512 + 9.03284i) q^{59} +(-5.82644 + 10.0917i) q^{61} -10.3246 q^{62} +1.98762 q^{64} +(2.38255 - 4.12669i) q^{65} +(5.93199 + 10.2745i) q^{67} +6.73877 q^{68} -4.30037 q^{71} +(-2.23061 + 3.86353i) q^{73} -11.8764 q^{74} +(1.98242 - 3.43366i) q^{76} +(0.666896 - 1.15510i) q^{79} +(2.36730 - 4.10029i) q^{80} +(-0.896461 - 1.55272i) q^{82} +(-2.84194 - 4.92238i) q^{83} +(-3.59888 + 6.23345i) q^{85} +11.8764 q^{86} -1.11126 q^{88} +(0.421826 + 0.730623i) q^{89} +(1.10074 + 1.90654i) q^{92} +(-6.35389 - 11.0053i) q^{94} +(2.11745 + 3.66754i) q^{95} +(1.70317 + 2.94997i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.849814	+	1.47192i	−0.600909	+	1.04081i	0.391774	+	0.920061i	$0.371861\pi$
$2$	−0.849814	+	1.47192i	−0.600909	+	1.04081i	−0.992684	+	0.120744i	$0.961472\pi$
$3$		0			0
$3$		0			0
$4$	−0.444368	−	0.769668i	−0.222184	−	0.384834i
$5$	0.949271			0.424527			0.212263	−	0.977212i	$-0.431916\pi$
$5$	0.949271			0.424527			0.212263	+	0.977212i	$0.431916\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.88874			−0.667769
$9$		0			0
$10$	−0.806704	+	1.39725i	−0.255102	+	0.441850i
$11$	0.588364			0.177398			0.0886992	−	0.996058i	$-0.471729\pi$
$11$	0.588364			0.177398			0.0886992	+	0.996058i	$0.471729\pi$
$12$		0			0
$13$	2.50987	−	4.34722i	0.696112	−	1.20570i	−0.273692	−	0.961817i	$-0.588245\pi$
$13$	2.50987	−	4.34722i	0.696112	−	1.20570i	0.969804	−	0.243885i	$-0.0784218\pi$
$14$		0			0
$15$		0			0
$16$	2.49381	−	4.31941i	0.623453	−	1.07985i
$17$	−3.79121	+	6.56657i	−0.919503	+	1.59263i	−0.119332	+	0.992854i	$0.538075\pi$
$17$	−3.79121	+	6.56657i	−0.919503	+	1.59263i	−0.800171	+	0.599772i	$0.795258\pi$
$18$		0			0
$19$	2.23061	+	3.86353i	0.511737	+	0.886355i	0.999907	+	0.0136063i	$0.00433116\pi$
$19$	2.23061	+	3.86353i	0.511737	+	0.886355i	−0.488170	+	0.872748i	$0.662336\pi$
$20$	−0.421826	−	0.730623i	−0.0943231	−	0.163372i
$21$		0			0
$22$	−0.500000	+	0.866025i	−0.106600	+	0.184637i
$23$	−2.47710			−0.516511			−0.258256	−	0.966077i	$-0.583148\pi$
$23$	−2.47710			−0.516511			−0.258256	+	0.966077i	$0.583148\pi$
$24$		0			0
$25$	−4.09888			−0.819777
$26$	4.26584	+	7.38866i	0.836601	+	1.44904i
$27$		0			0
$28$		0			0
$29$	2.73855	+	4.74331i	0.508536	+	0.880810i	0.999951	+	0.00988468i	$0.00314644\pi$
$29$	2.73855	+	4.74331i	0.508536	+	0.880810i	−0.491415	+	0.870925i	$0.663520\pi$
$30$		0			0
$31$	3.03731	+	5.26078i	0.545518	+	0.944865i	0.998574	+	0.0533826i	$0.0170003\pi$
$31$	3.03731	+	5.26078i	0.545518	+	0.944865i	−0.453056	+	0.891482i	$0.649666\pi$
$32$	2.34981	+	4.07000i	0.415392	+	0.719481i
$33$		0			0
$34$	−6.44364	−	11.1607i	−1.10508	−	1.91405i
$35$		0			0
$36$		0			0
$37$	3.49381	+	6.05146i	0.574379	+	0.994853i	0.996109	+	0.0881319i	$0.0280897\pi$
$37$	3.49381	+	6.05146i	0.574379	+	0.994853i	−0.421730	+	0.906721i	$0.638577\pi$
$38$	−7.58242			−1.23003
$39$		0			0
$40$	−1.79292			−0.283486
$41$	−0.527445	+	0.913562i	−0.0823731	+	0.142674i	−0.904269	−	0.426964i	$-0.859583\pi$
$41$	−0.527445	+	0.913562i	−0.0823731	+	0.142674i	0.821896	+	0.569638i	$0.192917\pi$
$42$		0			0
$43$	−3.49381	−	6.05146i	−0.532801	−	0.922838i	−0.999266	−	0.0382990i	$-0.987806\pi$
$43$	−3.49381	−	6.05146i	−0.532801	−	0.922838i	0.466465	−	0.884540i	$-0.345527\pi$
$44$	−0.261450	−	0.452845i	−0.0394151	−	0.0682689i
$45$		0			0
$46$	2.10507	−	3.64610i	0.310376	−	0.537587i
$47$	−3.73840	+	6.47510i	−0.545301	+	0.944490i	0.453286	+	0.891365i	$0.350251\pi$
$47$	−3.73840	+	6.47510i	−0.545301	+	0.944490i	−0.998588	+	0.0531249i	$0.983082\pi$
$48$		0			0
$49$		0			0
$50$	3.48329	−	6.03323i	0.492612	−	0.853228i
$51$		0			0
$52$	−4.46122			−0.618660
$53$	3.46108	−	5.99476i	0.475416	−	0.823444i	−0.524188	−	0.851603i	$-0.675631\pi$
$53$	3.46108	−	5.99476i	0.475416	−	0.823444i	0.999603	+	0.0281586i	$0.00896435\pi$
$54$		0			0
$55$	0.558517			0.0753104
$56$		0			0
$57$		0			0
$58$	−9.30903			−1.22234
$59$	5.21512	+	9.03284i	0.678950	+	1.17598i	0.975297	+	0.220896i	$0.0708981\pi$
$59$	5.21512	+	9.03284i	0.678950	+	1.17598i	−0.296347	+	0.955080i	$0.595769\pi$
$60$		0			0
$61$	−5.82644	+	10.0917i	−0.745999	+	1.29211i	0.203727	+	0.979028i	$0.434695\pi$
$61$	−5.82644	+	10.0917i	−0.745999	+	1.29211i	−0.949726	+	0.313081i	$0.898639\pi$
$62$	−10.3246			−1.31123
$63$		0			0
$64$	1.98762			0.248453
$65$	2.38255	−	4.12669i	0.295518	−	0.511853i
$66$		0			0
$67$	5.93199	+	10.2745i	0.724708	+	1.25523i	0.959094	+	0.283087i	$0.0913585\pi$
$67$	5.93199	+	10.2745i	0.724708	+	1.25523i	−0.234387	+	0.972143i	$0.575308\pi$
$68$	6.73877			0.817195
$69$		0			0
$70$		0			0
$71$	−4.30037			−0.510360			−0.255180	−	0.966894i	$-0.582135\pi$
$71$	−4.30037			−0.510360			−0.255180	+	0.966894i	$0.582135\pi$
$72$		0			0
$73$	−2.23061	+	3.86353i	−0.261073	+	0.452192i	−0.966527	−	0.256563i	$-0.917410\pi$
$73$	−2.23061	+	3.86353i	−0.261073	+	0.452192i	0.705454	+	0.708756i	$0.250743\pi$
$74$	−11.8764			−1.38060
$75$		0			0
$76$	1.98242	−	3.43366i	0.227400	−	0.393868i
$77$		0			0
$78$		0			0
$79$	0.666896	−	1.15510i	0.0750317	−	0.129959i	−0.826068	−	0.563570i	$-0.809428\pi$
$79$	0.666896	−	1.15510i	0.0750317	−	0.129959i	0.901100	+	0.433611i	$0.142761\pi$
$80$	2.36730	−	4.10029i	0.264672	−	0.458426i
$81$		0			0
$82$	−0.896461	−	1.55272i	−0.0989976	−	0.171469i
$83$	−2.84194	−	4.92238i	−0.311943	−	0.540301i	0.666840	−	0.745201i	$-0.267647\pi$
$83$	−2.84194	−	4.92238i	−0.311943	−	0.540301i	−0.978783	+	0.204900i	$0.934313\pi$
$84$		0			0
$85$	−3.59888	+	6.23345i	−0.390354	+	0.676113i
$86$	11.8764			1.28066
$87$		0			0
$88$	−1.11126			−0.118461
$89$	0.421826	+	0.730623i	0.0447134	+	0.0774459i	0.887516	−	0.460777i	$-0.152429\pi$
$89$	0.421826	+	0.730623i	0.0447134	+	0.0774459i	−0.842803	+	0.538223i	$0.819096\pi$
$90$		0			0
$91$		0			0
$92$	1.10074	+	1.90654i	0.114760	+	0.198771i
$93$		0			0
$94$	−6.35389	−	11.0053i	−0.655353	−	1.13511i
$95$	2.11745	+	3.66754i	0.217246	+	0.376281i
$96$		0			0
$97$	1.70317	+	2.94997i	0.172930	+	0.299524i	0.939443	−	0.342705i	$-0.111343\pi$
$97$	1.70317	+	2.94997i	0.172930	+	0.299524i	−0.766513	+	0.642229i	$0.778010\pi$
$98$		0			0
$99$		0			0
$100$	1.82141	+	3.15478i	0.182141	+	0.315478i
$101$	9.58658			0.953900			0.476950	−	0.878930i	$-0.341742\pi$
$101$	9.58658			0.953900			0.476950	+	0.878930i	$0.341742\pi$
$102$		0			0
$103$	−11.6529			−1.14819			−0.574096	−	0.818788i	$-0.694647\pi$
$103$	−11.6529			−1.14819			−0.574096	+	0.818788i	$0.694647\pi$
$104$	−4.74048	+	8.21075i	−0.464842	+	0.805130i
$105$		0			0
$106$	5.88255	+	10.1889i	0.571363	+	0.989630i
$107$	−1.89926	−	3.28961i	−0.183608	−	0.318018i	0.759499	−	0.650509i	$-0.225444\pi$
$107$	−1.89926	−	3.28961i	−0.183608	−	0.318018i	−0.943107	+	0.332491i	$0.892111\pi$
$108$		0			0
$109$	6.43199	−	11.1405i	0.616073	−	1.06707i	−0.374123	−	0.927379i	$-0.622056\pi$
$109$	6.43199	−	11.1405i	0.616073	−	1.06707i	0.990195	−	0.139690i	$-0.0446106\pi$
$110$	−0.474636	+	0.822093i	−0.0452547	+	0.0783835i
$111$		0			0
$112$		0			0
$113$	4.51052	−	7.81245i	0.424314	−	0.734934i	−0.572042	−	0.820224i	$-0.693849\pi$
$113$	4.51052	−	7.81245i	0.424314	−	0.734934i	0.996356	+	0.0852908i	$0.0271819\pi$
$114$		0			0
$115$	−2.35144			−0.219273
$116$	2.43385	−	4.21555i	0.225977	−	0.391404i
$117$		0			0
$118$	−17.7275			−1.63195
$119$		0			0
$120$		0			0
$121$	−10.6538			−0.968530
$122$	−9.90278	−	17.1521i	−0.896556	−	1.55288i
$123$		0			0
$124$	2.69937	−	4.67545i	0.242411	−	0.419867i
$125$	−8.63731			−0.772544
$126$		0			0
$127$	6.43268			0.570808			0.285404	−	0.958407i	$-0.407872\pi$
$127$	6.43268			0.570808			0.285404	+	0.958407i	$0.407872\pi$
$128$	−6.38874	+	11.0656i	−0.564690	+	0.978071i
$129$		0			0
$130$	4.04944	+	7.01384i	0.355160	+	0.615154i
$131$	−6.63315			−0.579541			−0.289770	−	0.957096i	$-0.593579\pi$
$131$	−6.63315			−0.579541			−0.289770	+	0.957096i	$0.593579\pi$
$132$		0			0
$133$		0			0
$134$	−20.1643			−1.74193
$135$		0			0
$136$	7.16059	−	12.4025i	0.614016	−	1.06351i
$137$	−14.0334			−1.19896			−0.599478	−	0.800391i	$-0.704625\pi$
$137$	−14.0334			−1.19896			−0.599478	+	0.800391i	$0.704625\pi$
$138$		0			0
$139$	−4.40254	+	7.62541i	−0.373418	+	0.646779i	−0.990089	−	0.140442i	$-0.955148\pi$
$139$	−4.40254	+	7.62541i	−0.373418	+	0.646779i	0.616671	+	0.787221i	$0.288481\pi$
$140$		0			0
$141$		0			0
$142$	3.65452	−	6.32981i	0.306680	−	0.531186i
$143$	1.47672	−	2.55775i	0.123489	−	0.213890i
$144$		0			0
$145$	2.59963	+	4.50268i	0.215887	+	0.373928i
$146$	−3.79121	−	6.56657i	−0.313763	−	0.543453i
$147$		0			0
$148$	3.10507	−	5.37815i	0.255236	−	0.442081i
$149$	4.36584			0.357663			0.178832	−	0.983880i	$-0.442768\pi$
$149$	4.36584			0.357663			0.178832	+	0.983880i	$0.442768\pi$
$150$		0			0
$151$	−12.6538			−1.02975			−0.514877	−	0.857264i	$-0.672162\pi$
$151$	−12.6538			−1.02975			−0.514877	+	0.857264i	$0.672162\pi$
$152$	−4.21303	−	7.29719i	−0.341722	−	0.591880i
$153$		0			0
$154$		0			0
$155$	2.88323	+	4.99391i	0.231587	+	0.401120i
$156$		0			0
$157$	−5.63694	−	9.76347i	−0.449877	−	0.779210i	0.548501	−	0.836150i	$-0.315199\pi$
$157$	−5.63694	−	9.76347i	−0.449877	−	0.779210i	−0.998378	+	0.0569405i	$0.981865\pi$
$158$	1.13348	+	1.96324i	0.0901745	+	0.156187i
$159$		0			0
$160$	2.23061	+	3.86353i	0.176345	+	0.305439i
$161$		0			0
$162$		0			0
$163$	0.833104	+	1.44298i	0.0652537	+	0.113023i	0.896807	−	0.442423i	$-0.145881\pi$
$163$	0.833104	+	1.44298i	0.0652537	+	0.113023i	−0.831553	+	0.555446i	$0.812548\pi$
$164$	0.937519			0.0732080
$165$		0			0
$166$	9.66047			0.749798
$167$	1.95135	−	3.37984i	0.151000	−	0.261540i	−0.780595	−	0.625037i	$-0.785084\pi$
$167$	1.95135	−	3.37984i	0.151000	−	0.261540i	0.931595	+	0.363497i	$0.118417\pi$
$168$		0			0
$169$	−6.09888	−	10.5636i	−0.469145	−	0.812583i
$170$	−6.11677	−	10.5945i	−0.469134	−	0.812565i
$171$		0			0
$172$	−3.10507	+	5.37815i	−0.236760	+	0.410080i
$173$	8.05705	−	13.9552i	0.612566	−	1.06100i	−0.378240	−	0.925708i	$-0.623471\pi$
$173$	8.05705	−	13.9552i	0.612566	−	1.06100i	0.990806	−	0.135288i	$-0.0431961\pi$
$174$		0			0
$175$		0			0
$176$	1.46727	−	2.54138i	0.110599	−	0.191564i
$177$		0			0
$178$	−1.43389			−0.107475
$179$	7.14400	−	12.3738i	0.533967	−	0.924859i	−0.465245	−	0.885182i	$-0.654034\pi$
$179$	7.14400	−	12.3738i	0.533967	−	0.924859i	0.999213	−	0.0396767i	$-0.0126328\pi$
$180$		0			0
$181$	12.8873			0.957905			0.478952	−	0.877841i	$-0.341017\pi$
$181$	12.8873			0.957905			0.478952	+	0.877841i	$0.341017\pi$
$182$		0			0
$183$		0			0
$184$	4.67859			0.344910
$185$	3.31657	+	5.74447i	0.243839	+	0.422342i
$186$		0			0
$187$	−2.23061	+	3.86353i	−0.163118	+	0.282529i
$188$	6.64490			0.484629
$189$		0			0
$190$	−7.19777			−0.522181
$191$	−1.08217	+	1.87438i	−0.0783034	+	0.135625i	−0.902518	−	0.430652i	$-0.858284\pi$
$191$	−1.08217	+	1.87438i	−0.0783034	+	0.135625i	0.824215	+	0.566277i	$0.191617\pi$
$192$		0			0
$193$	−5.21565	−	9.03377i	−0.375431	−	0.650265i	0.614961	−	0.788558i	$-0.289172\pi$
$193$	−5.21565	−	9.03377i	−0.375431	−	0.650265i	−0.990391	+	0.138293i	$0.955839\pi$
$194$	−5.78949			−0.415661
$195$		0			0
$196$		0			0
$197$	18.7848			1.33836			0.669179	−	0.743101i	$-0.266646\pi$
$197$	18.7848			1.33836			0.669179	+	0.743101i	$0.266646\pi$
$198$		0			0
$199$	−4.21303	+	7.29719i	−0.298654	+	0.517284i	−0.975828	−	0.218539i	$-0.929871\pi$
$199$	−4.21303	+	7.29719i	−0.298654	+	0.517284i	0.677174	+	0.735823i	$0.263204\pi$
$200$	7.74171			0.547422
$201$		0			0
$202$	−8.14681	+	14.1107i	−0.573208	+	0.992825i
$203$		0			0
$204$		0			0
$205$	−0.500689	+	0.867218i	−0.0349696	+	0.0605692i
$206$	9.90278	−	17.1521i	0.689960	−	1.19505i
$207$		0			0
$208$	−12.5183	−	21.6823i	−0.867986	−	1.50340i
$209$	1.31241	+	2.27316i	0.0907814	+	0.157238i
$210$		0			0
$211$	−5.61126	+	9.71899i	−0.386295	+	0.669083i	−0.991948	−	0.126646i	$-0.959579\pi$
$211$	−5.61126	+	9.71899i	−0.386295	+	0.669083i	0.605653	+	0.795729i	$0.292912\pi$
$212$	−6.15197			−0.422519
$213$		0			0
$214$	6.45606			0.441327
$215$	−3.31657	−	5.74447i	−0.226188	−	0.391770i
$216$		0			0
$217$		0			0
$218$	10.9320	+	18.9348i	0.740408	+	1.28242i
$219$		0			0
$220$	−0.248187	−	0.429872i	−0.0167328	−	0.0289820i
$221$	19.0309	+	32.9624i	1.28016	+	2.21729i
$222$		0			0
$223$	10.3774	+	17.9742i	0.694923	+	1.20364i	0.970206	+	0.242279i	$0.0778951\pi$
$223$	10.3774	+	17.9742i	0.694923	+	1.20364i	−0.275283	+	0.961363i	$0.588772\pi$
$224$		0			0
$225$		0			0
$226$	7.66621	+	13.2783i	0.509949	+	0.883257i
$227$	10.4302			0.692279			0.346139	−	0.938183i	$-0.387492\pi$
$227$	10.4302			0.692279			0.346139	+	0.938183i	$0.387492\pi$
$228$		0			0
$229$	15.0592			0.995141			0.497570	−	0.867424i	$-0.334226\pi$
$229$	15.0592			0.995141			0.497570	+	0.867424i	$0.334226\pi$
$230$	1.99829	−	3.46113i	0.131763	−	0.228220i
$231$		0			0
$232$	−5.17240	−	8.95886i	−0.339585	−	0.588178i
$233$	2.19344	+	3.79915i	0.143697	+	0.248890i	0.928886	−	0.370366i	$-0.120768\pi$
$233$	2.19344	+	3.79915i	0.143697	+	0.248890i	−0.785189	+	0.619256i	$0.787434\pi$
$234$		0			0
$235$	−3.54875	+	6.14662i	−0.231495	+	0.400961i
$236$	4.63486	−	8.02781i	0.301704	−	0.522566i
$237$		0			0
$238$		0			0
$239$	−4.77561	+	8.27160i	−0.308909	+	0.535046i	−0.978124	−	0.208023i	$-0.933297\pi$
$239$	−4.77561	+	8.27160i	−0.308909	+	0.535046i	0.669215	+	0.743069i	$0.266630\pi$
$240$		0			0
$241$	10.5358			0.678674			0.339337	−	0.940665i	$-0.389797\pi$
$241$	10.5358			0.678674			0.339337	+	0.940665i	$0.389797\pi$
$242$	9.05377	−	15.6816i	0.581999	−	1.00805i
$243$		0			0
$244$	10.3563			0.662996
$245$		0			0
$246$		0			0
$247$	22.3942			1.42491
$248$	−5.73668	−	9.93623i	−0.364280	−	0.630951i
$249$		0			0
$250$	7.34011	−	12.7134i	0.464229	−	0.804068i
$251$	24.4346			1.54230			0.771148	−	0.636656i	$-0.219683\pi$
$251$	24.4346			1.54230			0.771148	+	0.636656i	$0.219683\pi$
$252$		0			0
$253$	−1.45744			−0.0916282
$254$	−5.46658	+	9.46839i	−0.343004	+	0.594100i
$255$		0			0
$256$	−8.87085	−	15.3648i	−0.554428	−	0.960298i
$257$	−4.00832			−0.250032			−0.125016	−	0.992155i	$-0.539898\pi$
$257$	−4.00832			−0.250032			−0.125016	+	0.992155i	$0.539898\pi$
$258$		0			0
$259$		0			0
$260$	−4.23491			−0.262638
$261$		0			0
$262$	5.63694	−	9.76347i	0.348251	−	0.603189i
$263$	−17.6872			−1.09064			−0.545321	−	0.838227i	$-0.683592\pi$
$263$	−17.6872			−1.09064			−0.545321	+	0.838227i	$0.683592\pi$
$264$		0			0
$265$	3.28550	−	5.69066i	0.201827	−	0.349574i
$266$		0			0
$267$		0			0
$268$	5.27197	−	9.13132i	0.322037	−	0.557784i
$269$	−7.11366	+	12.3212i	−0.433727	+	0.751238i	−0.997191	−	0.0749032i	$-0.976135\pi$
$269$	−7.11366	+	12.3212i	−0.433727	+	0.751238i	0.563463	+	0.826141i	$0.309469\pi$
$270$		0			0
$271$	2.69937	+	4.67545i	0.163975	+	0.284013i	0.936291	−	0.351226i	$-0.114235\pi$
$271$	2.69937	+	4.67545i	0.163975	+	0.284013i	−0.772316	+	0.635239i	$0.780902\pi$
$272$	18.9091	+	32.7515i	1.14653	+	1.98585i
$273$		0			0
$274$	11.9258	−	20.6561i	0.720464	−	1.24788i
$275$	−2.41164			−0.145427
$276$		0			0
$277$	7.66621			0.460618			0.230309	−	0.973118i	$-0.426026\pi$
$277$	7.66621			0.460618			0.230309	+	0.973118i	$0.426026\pi$
$278$	−7.48267	−	12.9604i	−0.448781	−	0.777311i
$279$		0			0
$280$		0			0
$281$	−11.3312	−	19.6263i	−0.675965	−	1.17081i	−0.976186	−	0.216936i	$-0.930394\pi$
$281$	−11.3312	−	19.6263i	−0.675965	−	1.17081i	0.300220	−	0.953870i	$-0.402940\pi$
$282$		0			0
$283$	15.9246	+	27.5822i	0.946619	+	1.63959i	0.752476	+	0.658620i	$0.228859\pi$
$283$	15.9246	+	27.5822i	0.946619	+	1.63959i	0.194144	+	0.980973i	$0.437807\pi$
$284$	1.91095	+	3.30986i	0.113394	+	0.196404i
$285$		0			0
$286$	2.50987	+	4.34722i	0.148412	+	0.257057i
$287$		0			0
$288$		0			0
$289$	−20.2465	−	35.0680i	−1.19097	−	2.06282i
$290$	−8.83680			−0.518915
$291$		0			0
$292$	3.96485			0.232025
$293$	13.7468	−	23.8102i	0.803097	−	1.39100i	−0.114472	−	0.993427i	$-0.536517\pi$
$293$	13.7468	−	23.8102i	0.803097	−	1.39100i	0.917568	−	0.397578i	$-0.130149\pi$
$294$		0			0
$295$	4.95056	+	8.57462i	0.288233	+	0.499234i
$296$	−6.59888	−	11.4296i	−0.383552	−	0.664332i
$297$		0			0
$298$	−3.71015	+	6.42617i	−0.214923	+	0.372258i
$299$	−6.21720	+	10.7685i	−0.359550	+	0.622758i
$300$		0			0
$301$		0			0
$302$	10.7534	−	18.6254i	0.618789	−	1.07177i
$303$		0			0
$304$	22.2509			1.27618
$305$	−5.53087	+	9.57975i	−0.316697	+	0.548535i
$306$		0			0
$307$	14.8176			0.845683			0.422841	−	0.906204i	$-0.361033\pi$
$307$	14.8176			0.845683			0.422841	+	0.906204i	$0.361033\pi$
$308$		0			0
$309$		0			0
$310$	−9.80085			−0.556651
$311$	14.5318	+	25.1698i	0.824021	+	1.42725i	0.902665	+	0.430343i	$0.141608\pi$
$311$	14.5318	+	25.1698i	0.824021	+	1.42725i	−0.0786442	+	0.996903i	$0.525059\pi$
$312$		0			0
$313$	12.2390	−	21.1986i	0.691790	−	1.19822i	−0.279461	−	0.960157i	$-0.590156\pi$
$313$	12.2390	−	21.1986i	0.691790	−	1.19822i	0.971251	−	0.238058i	$-0.0765110\pi$
$314$	19.1614			1.08134
$315$		0			0
$316$	−1.18539			−0.0666834
$317$	−3.69344	+	6.39722i	−0.207444	+	0.359304i	−0.950909	−	0.309472i	$-0.899848\pi$
$317$	−3.69344	+	6.39722i	−0.207444	+	0.359304i	0.743465	+	0.668775i	$0.233181\pi$
$318$		0			0
$319$	1.61126	+	2.79079i	0.0902135	+	0.156254i
$320$	1.88679			0.105475
$321$		0			0
$322$		0			0
$323$	−33.8268			−1.88218
$324$		0			0
$325$	−10.2877	+	17.8188i	−0.570657	+	0.988407i
$326$	−2.83193			−0.156846
$327$		0			0
$328$	0.996205	−	1.72548i	0.0550062	−	0.0952736i
$329$		0			0
$330$		0			0
$331$	−10.0309	+	17.3740i	−0.551347	+	0.954960i	0.446831	+	0.894618i	$0.352552\pi$
$331$	−10.0309	+	17.3740i	−0.551347	+	0.954960i	−0.998178	+	0.0603420i	$0.980781\pi$
$332$	−2.52573	+	4.37470i	−0.138618	+	0.240093i
$333$		0			0
$334$	3.31657	+	5.74447i	0.181475	+	0.314324i
$335$	5.63106	+	9.75329i	0.307658	+	0.532879i
$336$		0			0
$337$	−3.20327	+	5.54823i	−0.174493	+	0.302231i	−0.939986	−	0.341214i	$-0.889162\pi$
$337$	−3.20327	+	5.54823i	−0.174493	+	0.302231i	0.765493	+	0.643445i	$0.222495\pi$
$338$	20.7317			1.12765
$339$		0			0
$340$	6.39692			0.346921
$341$	1.78705	+	3.09526i	0.0967740	+	0.167617i
$342$		0			0
$343$		0			0
$344$	6.59888	+	11.4296i	0.355788	+	0.616243i
$345$		0			0
$346$	13.6940	+	23.7187i	0.736194	+	1.27512i
$347$	−14.5963	−	25.2816i	−0.783572	−	1.35719i	−0.929848	−	0.367943i	$-0.880062\pi$
$347$	−14.5963	−	25.2816i	−0.783572	−	1.35719i	0.146276	−	0.989244i	$-0.453271\pi$
$348$		0			0
$349$	2.17192	+	3.76188i	0.116260	+	0.201369i	0.918283	−	0.395925i	$-0.129576\pi$
$349$	2.17192	+	3.76188i	0.116260	+	0.201369i	−0.802022	+	0.597294i	$0.796243\pi$
$350$		0			0
$351$		0			0
$352$	1.38255	+	2.39464i	0.0736899	+	0.127635i
$353$	−25.7007			−1.36791			−0.683955	−	0.729525i	$-0.739741\pi$
$353$	−25.7007			−1.36791			−0.683955	+	0.729525i	$0.739741\pi$
$354$		0			0
$355$	−4.08222			−0.216662
$356$	0.374892	−	0.649331i	0.0198692	−	0.0344145i
$357$		0			0
$358$	12.1421	+	21.0308i	0.641732	+	1.11151i
$359$	10.3436	+	17.9157i	0.545916	+	0.945554i	0.998549	+	0.0538567i	$0.0171514\pi$
$359$	10.3436	+	17.9157i	0.545916	+	0.945554i	−0.452633	+	0.891697i	$0.649515\pi$
$360$		0			0
$361$	−0.451246	+	0.781582i	−0.0237498	+	0.0411359i
$362$	−10.9518	+	18.9691i	−0.575614	+	0.996992i
$363$		0			0
$364$		0			0
$365$	−2.11745	+	3.66754i	−0.110833	+	0.191968i
$366$		0			0
$367$	−2.84781			−0.148655			−0.0743273	−	0.997234i	$-0.523681\pi$
$367$	−2.84781			−0.148655			−0.0743273	+	0.997234i	$0.523681\pi$
$368$	−6.17742	+	10.6996i	−0.322020	+	0.557755i
$369$		0			0
$370$	−11.2739			−0.586101
$371$		0			0
$372$		0			0
$373$	21.4327			1.10974			0.554871	−	0.831936i	$-0.312768\pi$
$373$	21.4327			1.10974			0.554871	+	0.831936i	$0.312768\pi$
$374$	−3.79121	−	6.56657i	−0.196039	−	0.339549i
$375$		0			0
$376$	7.06085	−	12.2297i	0.364135	−	0.630701i
$377$	27.4936			1.41599
$378$		0			0
$379$	27.0494			1.38943			0.694716	−	0.719284i	$-0.255530\pi$
$379$	27.0494			1.38943			0.694716	+	0.719284i	$0.255530\pi$
$380$	1.88186	−	3.25947i	0.0965372	−	0.167207i
$381$		0			0
$382$	−1.83929	−	3.18575i	−0.0941064	−	0.162997i
$383$	14.4268			0.737175			0.368588	−	0.929593i	$-0.379841\pi$
$383$	14.4268			0.737175			0.368588	+	0.929593i	$0.379841\pi$
$384$		0			0
$385$		0			0
$386$	17.7293			0.902399
$387$		0			0
$388$	1.51366	−	2.62174i	0.0768446	−	0.133099i
$389$	6.10755			0.309665			0.154832	−	0.987941i	$-0.450516\pi$
$389$	6.10755			0.309665			0.154832	+	0.987941i	$0.450516\pi$
$390$		0			0
$391$	9.39120	−	16.2660i	0.474933	−	0.822609i
$392$		0			0
$393$		0			0
$394$	−15.9635	+	27.6497i	−0.804232	+	1.39297i
$395$	0.633065	−	1.09650i	0.0318530	−	0.0551710i
$396$		0			0
$397$	6.44364	+	11.1607i	0.323397	+	0.560140i	0.981187	−	0.193061i	$-0.0618417\pi$
$397$	6.44364	+	11.1607i	0.323397	+	0.560140i	−0.657789	+	0.753202i	$0.728508\pi$
$398$	−7.16059	−	12.4025i	−0.358928	−	0.621682i
$399$		0			0
$400$	−10.2218	+	17.7047i	−0.511092	+	0.885237i
$401$	−8.39060			−0.419006			−0.209503	−	0.977808i	$-0.567185\pi$
$401$	−8.39060			−0.419006			−0.209503	+	0.977808i	$0.567185\pi$
$402$		0			0
$403$	30.4930			1.51897
$404$	−4.25997	−	7.37848i	−0.211941	−	0.367093i
$405$		0			0
$406$		0			0
$407$	2.05563	+	3.56046i	0.101894	+	0.176485i
$408$		0			0
$409$	−3.40633	−	5.89994i	−0.168432	−	0.291733i	0.769437	−	0.638723i	$-0.220537\pi$
$409$	−3.40633	−	5.89994i	−0.168432	−	0.291733i	−0.937869	+	0.346990i	$0.887204\pi$
$410$	−0.850985	−	1.47395i	−0.0420271	−	0.0727931i
$411$		0			0
$412$	5.17817	+	8.96885i	0.255110	+	0.441864i
$413$		0			0
$414$		0			0
$415$	−2.69777	−	4.67267i	−0.132428	−	0.229372i
$416$	23.5909			1.15664
$417$		0			0
$418$	−4.46122			−0.218205
$419$	−5.16231	+	8.94137i	−0.252195	+	0.436815i	−0.964130	−	0.265431i	$-0.914486\pi$
$419$	−5.16231	+	8.94137i	−0.252195	+	0.436815i	0.711935	+	0.702246i	$0.247819\pi$
$420$		0			0
$421$	−1.56801	−	2.71588i	−0.0764202	−	0.132364i	0.825283	−	0.564720i	$-0.191016\pi$
$421$	−1.56801	−	2.71588i	−0.0764202	−	0.132364i	−0.901703	+	0.432356i	$0.857682\pi$
$422$	−9.53706	−	16.5187i	−0.464257	−	0.804117i
$423$		0			0
$424$	−6.53706	+	11.3225i	−0.317468	+	0.549870i
$425$	15.5397	−	26.9156i	0.753787	−	1.30560i
$426$		0			0
$427$		0			0
$428$	−1.68794	+	2.92359i	−0.0815895	+	0.141317i
$429$		0			0
$430$	11.2739			0.543675
$431$	15.9363	−	27.6025i	0.767625	−	1.32957i	−0.171222	−	0.985233i	$-0.554771\pi$
$431$	15.9363	−	27.6025i	0.767625	−	1.32957i	0.938847	−	0.344334i	$-0.111895\pi$
$432$		0			0
$433$	7.48855			0.359877			0.179938	−	0.983678i	$-0.442410\pi$
$433$	7.48855			0.359877			0.179938	+	0.983678i	$0.442410\pi$
$434$		0			0
$435$		0			0
$436$	−11.4327			−0.547526
$437$	−5.52544	−	9.57035i	−0.264318	−	0.457812i
$438$		0			0
$439$	−1.14465	+	1.98259i	−0.0546311	+	0.0946238i	−0.892048	−	0.451941i	$-0.850732\pi$
$439$	−1.14465	+	1.98259i	−0.0546311	+	0.0946238i	0.837417	+	0.546565i	$0.184065\pi$
$440$	−1.05489			−0.0502900
$441$		0			0
$442$	−64.6908			−3.07703
$443$	18.6749	−	32.3458i	0.887270	−	1.53680i	0.0441800	−	0.999024i	$-0.485933\pi$
$443$	18.6749	−	32.3458i	0.887270	−	1.53680i	0.843090	−	0.537773i	$-0.180734\pi$
$444$		0			0
$445$	0.400427	+	0.693560i	0.0189821	+	0.0328779i
$446$	−35.2755			−1.67034
$447$		0			0
$448$		0			0
$449$	6.20286			0.292731			0.146366	−	0.989231i	$-0.453242\pi$
$449$	6.20286			0.292731			0.146366	+	0.989231i	$0.453242\pi$
$450$		0			0
$451$	−0.310330	+	0.537507i	−0.0146129	+	0.0253102i
$452$	−8.01732			−0.377103
$453$		0			0
$454$	−8.86376	+	15.3525i	−0.415997	+	0.720527i
$455$		0			0
$456$		0			0
$457$	−10.0858	+	17.4691i	−0.471795	+	0.817172i	−0.999479	−	0.0322682i	$-0.989727\pi$
$457$	−10.0858	+	17.4691i	−0.471795	+	0.817172i	0.527685	+	0.849440i	$0.323060\pi$
$458$	−12.7975	+	22.1660i	−0.597989	+	1.03575i
$459$		0			0
$460$	1.04490	+	1.80983i	0.0487189	+	0.0843836i
$461$	−11.2680	−	19.5168i	−0.524803	−	0.908986i	−0.999583	−	0.0288813i	$-0.990806\pi$
$461$	−11.2680	−	19.5168i	−0.524803	−	0.908986i	0.474780	−	0.880105i	$-0.342528\pi$
$462$		0			0
$463$	13.8145	−	23.9275i	0.642016	−	1.11200i	−0.342966	−	0.939348i	$-0.611432\pi$
$463$	13.8145	−	23.9275i	0.642016	−	1.11200i	0.984982	−	0.172656i	$-0.0552350\pi$
$464$	27.3177			1.26819
$465$		0			0
$466$	−7.45606			−0.345395
$467$	10.0612	+	17.4265i	0.465577	+	0.806404i	0.999227	−	0.0393016i	$-0.0125133\pi$
$467$	10.0612	+	17.4265i	0.465577	+	0.806404i	−0.533650	+	0.845705i	$0.679180\pi$
$468$		0			0
$469$		0			0
$470$	−6.03156	−	10.4470i	−0.278215	−	0.481883i
$471$		0			0
$472$	−9.84997	−	17.0607i	−0.453382	−	0.785280i
$473$	−2.05563	−	3.56046i	−0.0945181	−	0.163710i
$474$		0			0
$475$	−9.14301	−	15.8362i	−0.419510	−	0.726613i
$476$		0			0
$477$		0			0
$478$	−8.11677	−	14.0586i	−0.371252	−	0.643028i
$479$	9.58658			0.438022			0.219011	−	0.975722i	$-0.429717\pi$
$479$	9.58658			0.438022			0.219011	+	0.975722i	$0.429717\pi$
$480$		0			0
$481$	35.0760			1.59933
$482$	−8.95351	+	15.5079i	−0.407821	+	0.706367i
$483$		0			0
$484$	4.73422	+	8.19991i	0.215192	+	0.372723i
$485$	1.61677	+	2.80032i	0.0734135	+	0.127156i
$486$		0			0
$487$	−6.53706	+	11.3225i	−0.296223	+	0.513073i	−0.975269	−	0.221023i	$-0.929060\pi$
$487$	−6.53706	+	11.3225i	−0.296223	+	0.513073i	0.679046	+	0.734096i	$0.262394\pi$
$488$	11.0046	−	19.0605i	0.498155	−	0.862830i
$489$		0			0
$490$		0			0
$491$	7.67054	−	13.2858i	0.346167	−	0.599578i	−0.639398	−	0.768876i	$-0.720817\pi$
$491$	7.67054	−	13.2858i	0.346167	−	0.599578i	0.985565	+	0.169298i	$0.0541499\pi$
$492$		0			0
$493$	−41.5297			−1.87040
$494$	−19.0309	+	32.9624i	−0.856239	+	1.48305i
$495$		0			0
$496$	30.2979			1.36042
$497$		0			0
$498$		0			0
$499$	4.86535			0.217803			0.108902	−	0.994053i	$-0.465267\pi$
$499$	4.86535			0.217803			0.108902	+	0.994053i	$0.465267\pi$
$500$	3.83814	+	6.64786i	0.171647	+	0.297301i
$501$		0			0
$502$	−20.7648	+	35.9657i	−0.926780	+	1.60523i
$503$	−16.0085			−0.713783			−0.356892	−	0.934146i	$-0.616163\pi$
$503$	−16.0085			−0.713783			−0.356892	+	0.934146i	$0.616163\pi$
$504$		0			0
$505$	9.10026			0.404956
$506$	1.23855	−	2.14523i	0.0550603	−	0.0953672i
$507$		0			0
$508$	−2.85848	−	4.95102i	−0.126824	−	0.219666i
$509$	31.1851			1.38225			0.691127	−	0.722733i	$-0.257115\pi$
$509$	31.1851			1.38225			0.691127	+	0.722733i	$0.257115\pi$
$510$		0			0
$511$		0			0
$512$	4.59937			0.203265
$513$		0			0
$514$	3.40633	−	5.89994i	0.150247	−	0.260235i
$515$	−11.0617			−0.487439
$516$		0			0
$517$	−2.19954	+	3.80971i	−0.0967356	+	0.167551i
$518$		0			0
$519$		0			0
$520$	−4.50000	+	7.79423i	−0.197338	+	0.341800i
$521$	10.4830	−	18.1572i	0.459270	−	0.795480i	−0.539652	−	0.841888i	$-0.681444\pi$
$521$	10.4830	−	18.1572i	0.459270	−	0.795480i	0.998923	+	0.0464085i	$0.0147776\pi$
$522$		0			0
$523$	−21.7821	−	37.7277i	−0.952465	−	1.64972i	−0.740064	−	0.672536i	$-0.765205\pi$
$523$	−21.7821	−	37.7277i	−0.952465	−	1.64972i	−0.212401	−	0.977183i	$-0.568128\pi$
$524$	2.94756	+	5.10532i	0.128765	+	0.223027i
$525$		0			0
$526$	15.0309	−	26.0342i	0.655377	−	1.13515i
$527$	−46.0604			−2.00642
$528$		0			0
$529$	−16.8640			−0.733216
$530$	5.58413	+	9.67200i	0.242559	+	0.420125i
$531$		0			0
$532$		0			0
$533$	2.64764	+	4.58584i	0.114682	+	0.198635i
$534$		0			0
$535$	−1.80291	−	3.12273i	−0.0779466	−	0.135007i
$536$	−11.2040	−	19.4058i	−0.483937	−	0.838204i
$537$		0			0
$538$	−12.0906	−	20.9415i	−0.521262	−	0.902852i
$539$		0			0
$540$		0			0
$541$	−4.93268	−	8.54365i	−0.212072	−	0.367320i	0.740291	−	0.672287i	$-0.234688\pi$
$541$	−4.93268	−	8.54365i	−0.212072	−	0.367320i	−0.952363	+	0.304967i	$0.901355\pi$
$542$	−9.17585			−0.394137
$543$		0			0
$544$	−35.6345			−1.52782
$545$	6.10570	−	10.5754i	0.261539	−	0.453000i
$546$		0			0
$547$	−0.284350	−	0.492509i	−0.0121579	−	0.0210582i	0.859882	−	0.510492i	$-0.170537\pi$
$547$	−0.284350	−	0.492509i	−0.0121579	−	0.0210582i	−0.872040	+	0.489434i	$0.837203\pi$
$548$	6.23600	+	10.8011i	0.266389	+	0.461399i
$549$		0			0
$550$	2.04944	−	3.54974i	0.0873885	−	0.151361i
$551$	−12.2173	+	21.1609i	−0.520473	+	0.901487i
$552$		0			0
$553$		0			0
$554$	−6.51485	+	11.2841i	−0.276789	+	0.479413i
$555$		0			0
$556$	7.82538			0.331870
$557$	−1.29349	+	2.24040i	−0.0548071	+	0.0949286i	−0.892127	−	0.451784i	$-0.850788\pi$
$557$	−1.29349	+	2.24040i	−0.0548071	+	0.0949286i	0.837320	+	0.546713i	$0.184121\pi$
$558$		0			0
$559$	−35.0760			−1.48356
$560$		0			0
$561$		0			0
$562$	38.5178			1.62478
$563$	−16.6416	−	28.8240i	−0.701358	−	1.21479i	−0.967990	−	0.250989i	$-0.919244\pi$
$563$	−16.6416	−	28.8240i	−0.701358	−	1.21479i	0.266632	−	0.963798i	$-0.414089\pi$
$564$		0			0
$565$	4.28171	−	7.41613i	0.180133	−	0.311999i
$566$	−54.1318			−2.27533
$567$		0			0
$568$	8.12227			0.340803
$569$	−2.67673	+	4.63623i	−0.112214	+	0.194361i	−0.916663	−	0.399662i	$-0.869128\pi$
$569$	−2.67673	+	4.63623i	−0.112214	+	0.194361i	0.804448	+	0.594022i	$0.202461\pi$
$570$		0			0
$571$	−2.45056	−	4.24449i	−0.102553	−	0.177626i	0.810183	−	0.586177i	$-0.199368\pi$
$571$	−2.45056	−	4.24449i	−0.102553	−	0.177626i	−0.912736	+	0.408551i	$0.866034\pi$
$572$	−2.62482			−0.109749
$573$		0			0
$574$		0			0
$575$	10.1533			0.423424
$576$		0			0
$577$	18.0378	−	31.2425i	0.750925	−	1.30064i	−0.196450	−	0.980514i	$-0.562941\pi$
$577$	18.0378	−	31.2425i	0.750925	−	1.30064i	0.947375	−	0.320127i	$-0.103725\pi$
$578$	68.8231			2.86266
$579$		0			0
$580$	2.31038	−	4.00170i	0.0959333	−	0.166161i
$581$		0			0
$582$		0			0
$583$	2.03637	−	3.52710i	0.0843380	−	0.146078i
$584$	4.21303	−	7.29719i	0.174337	−	0.301960i
$585$		0			0
$586$	23.3645	+	40.4684i	0.965177	+	1.67174i
$587$	0.527445	+	0.913562i	0.0217700	+	0.0377068i	0.876705	−	0.481028i	$-0.159737\pi$
$587$	0.527445	+	0.913562i	0.0217700	+	0.0377068i	−0.854935	+	0.518735i	$0.826403\pi$
$588$		0			0
$589$	−13.5501	+	23.4695i	−0.558323	+	0.967045i
$590$	−16.8282			−0.692807
$591$		0			0
$592$	34.8516			1.43239
$593$	−7.53548	−	13.0518i	−0.309445	−	0.535975i	0.668796	−	0.743446i	$-0.266810\pi$
$593$	−7.53548	−	13.0518i	−0.309445	−	0.535975i	−0.978241	+	0.207471i	$0.933477\pi$
$594$		0			0
$595$		0			0
$596$	−1.94004	−	3.36024i	−0.0794670	−	0.137641i
$597$		0			0
$598$	−10.5669	−	18.3024i	−0.432114	−	0.748443i
$599$	21.0283	+	36.4221i	0.859194	+	1.48817i	0.872699	+	0.488259i	$0.162368\pi$
$599$	21.0283	+	36.4221i	0.859194	+	1.48817i	−0.0135047	+	0.999909i	$0.504299\pi$
$600$		0			0
$601$	−9.44989	−	16.3677i	−0.385469	−	0.667652i	0.606365	−	0.795186i	$-0.292627\pi$
$601$	−9.44989	−	16.3677i	−0.385469	−	0.667652i	−0.991834	+	0.127534i	$0.959294\pi$
$602$		0			0
$603$		0			0
$604$	5.62296	+	9.73924i	0.228795	+	0.396284i
$605$	−10.1134			−0.411167
$606$		0			0
$607$	−29.4425			−1.19504			−0.597518	−	0.801856i	$-0.703846\pi$
$607$	−29.4425			−1.19504			−0.597518	+	0.801856i	$0.703846\pi$
$608$	−10.4830	+	18.1572i	−0.425143	+	0.736370i
$609$		0			0
$610$	−9.40043	−	16.2820i	−0.380612	−	0.659240i
$611$	18.7658	+	32.5033i	0.759182	+	1.31494i
$612$		0			0
$613$	5.83379	−	10.1044i	0.235625	−	0.408114i	−0.723829	−	0.689979i	$-0.757620\pi$
$613$	5.83379	−	10.1044i	0.235625	−	0.408114i	0.959454	+	0.281865i	$0.0909531\pi$
$614$	−12.5922	+	21.8103i	−0.508179	+	0.880191i
$615$		0			0
$616$		0			0
$617$	−16.4054	+	28.4151i	−0.660458	+	1.14395i	0.320037	+	0.947405i	$0.396305\pi$
$617$	−16.4054	+	28.4151i	−0.660458	+	1.14395i	−0.980495	+	0.196542i	$0.937029\pi$
$618$		0			0
$619$	−24.1612			−0.971119			−0.485560	−	0.874204i	$-0.661384\pi$
$619$	−24.1612			−0.971119			−0.485560	+	0.874204i	$0.661384\pi$
$620$	2.56243	−	4.43827i	0.102910	−	0.178245i
$621$		0			0
$622$	−49.3972			−1.98065
$623$		0			0
$624$		0			0
$625$	12.2953			0.491811
$626$	20.8018	+	36.0297i	0.831406	+	1.44004i
$627$		0			0
$628$	−5.00975	+	8.67714i	−0.199911	+	0.346256i
$629$	−52.9830			−2.11257
$630$		0			0
$631$	−11.1003			−0.441894			−0.220947	−	0.975286i	$-0.570915\pi$
$631$	−11.1003			−0.441894			−0.220947	+	0.975286i	$0.570915\pi$
$632$	−1.25959	+	2.18168i	−0.0501038	+	0.0867824i
$633$		0			0
$634$	−6.27747	−	10.8729i	−0.249310	−	0.431818i
$635$	6.10635			0.242323
$636$		0			0
$637$		0			0
$638$	−5.47710			−0.216840
$639$		0			0
$640$	−6.06464	+	10.5043i	−0.239726	+	0.415218i
$641$	−7.30037			−0.288347			−0.144174	−	0.989552i	$-0.546052\pi$
$641$	−7.30037			−0.288347			−0.144174	+	0.989552i	$0.546052\pi$
$642$		0			0
$643$	−10.6256	+	18.4041i	−0.419033	+	0.725787i	−0.995842	−	0.0910922i	$-0.970964\pi$
$643$	−10.6256	+	18.4041i	−0.419033	+	0.725787i	0.576809	+	0.816879i	$0.304298\pi$
$644$		0			0
$645$		0			0
$646$	28.7465	−	49.7904i	1.13102	−	1.95898i
$647$	8.47300	−	14.6757i	0.333108	−	0.576960i	−0.650011	−	0.759924i	$-0.725236\pi$
$647$	8.47300	−	14.6757i	0.333108	−	0.576960i	0.983120	+	0.182964i	$0.0585693\pi$
$648$		0			0
$649$	3.06839	+	5.31460i	0.120445	+	0.208616i
$650$	−17.4852	−	30.2853i	−0.685826	−	1.18789i
$651$		0			0
$652$	0.740409	−	1.28243i	0.0289967	−	0.0502237i
$653$	3.73305			0.146085			0.0730427	−	0.997329i	$-0.476729\pi$
$653$	3.73305			0.146085			0.0730427	+	0.997329i	$0.476729\pi$
$654$		0			0
$655$	−6.29665			−0.246031
$656$	2.63070	+	4.55650i	0.102711	+	0.177902i
$657$		0			0
$658$		0			0
$659$	−11.7992	−	20.4368i	−0.459632	−	0.796105i	0.539310	−	0.842107i	$-0.318685\pi$
$659$	−11.7992	−	20.4368i	−0.459632	−	0.796105i	−0.998941	+	0.0460022i	$0.985352\pi$
$660$		0			0
$661$	−17.2588	−	29.8930i	−0.671288	−	1.16270i	−0.977539	−	0.210754i	$-0.932408\pi$
$661$	−17.2588	−	29.8930i	−0.671288	−	1.16270i	0.306252	−	0.951951i	$-0.400925\pi$
$662$	−17.0488	−	29.5293i	−0.662619	−	1.14769i
$663$		0			0
$664$	5.36767	+	9.29708i	0.208306	+	0.360796i
$665$		0			0
$666$		0			0
$667$	−6.78366	−	11.7496i	−0.262664	−	0.454948i
$668$	−3.46847			−0.134199
$669$		0			0
$670$	−19.1414			−0.739498
$671$	−3.42807	+	5.93759i	−0.132339	+	0.229218i
$672$		0			0
$673$	12.2287	+	21.1808i	0.471382	+	0.816458i	0.999464	−	0.0327353i	$-0.0104218\pi$
$673$	12.2287	+	21.1808i	0.471382	+	0.816458i	−0.528082	+	0.849194i	$0.677088\pi$
$674$	−5.44437	−	9.42992i	−0.209709	−	0.363227i
$675$		0			0
$676$	−5.42030	+	9.38823i	−0.208473	+	0.361086i
$677$	−4.16022	+	7.20572i	−0.159890	+	0.276938i	−0.934829	−	0.355098i	$-0.884447\pi$
$677$	−4.16022	+	7.20572i	−0.159890	+	0.276938i	0.774939	+	0.632037i	$0.217781\pi$
$678$		0			0
$679$		0			0
$680$	6.79734	−	11.7733i	0.260666	−	0.451487i
$681$		0			0
$682$	−6.07463			−0.232610
$683$	−21.2312	+	36.7735i	−0.812389	+	1.40710i	0.0987988	+	0.995107i	$0.468500\pi$
$683$	−21.2312	+	36.7735i	−0.812389	+	1.40710i	−0.911188	+	0.411991i	$0.864833\pi$
$684$		0			0
$685$	−13.3215			−0.508989
$686$		0			0
$687$		0			0
$688$	−34.8516			−1.32870
$689$	−17.3737	−	30.0921i	−0.661885	−	1.14642i
$690$		0			0
$691$	−17.6964	+	30.6511i	−0.673204	+	1.16602i	0.303786	+	0.952740i	$0.401749\pi$
$691$	−17.6964	+	30.6511i	−0.673204	+	1.16602i	−0.976990	+	0.213284i	$0.931584\pi$
$692$	−14.3212			−0.544410
$693$		0			0
$694$	49.6167			1.88342
$695$	−4.17920	+	7.23859i	−0.158526	+	0.274575i
$696$		0			0
$697$	−3.99931	−	6.92701i	−0.151485	−	0.262379i
$698$	−7.38293			−0.279448
$699$		0			0
$700$		0			0
$701$	7.00372			0.264527			0.132263	−	0.991215i	$-0.457776\pi$
$701$	7.00372			0.264527			0.132263	+	0.991215i	$0.457776\pi$
$702$		0			0
$703$	−15.5867	+	26.9969i	−0.587862	+	1.01821i
$704$	1.16944			0.0440751
$705$		0			0
$706$	21.8408	−	37.8294i	0.821989	−	1.42373i
$707$		0			0
$708$		0			0
$709$	1.11126	−	1.92477i	0.0417344	−	0.0722861i	−0.844404	−	0.535707i	$-0.820045\pi$
$709$	1.11126	−	1.92477i	0.0417344	−	0.0722861i	0.886138	+	0.463421i	$0.153378\pi$
$710$	3.46913	−	6.00870i	0.130194	−	0.225503i
$711$		0			0
$712$	−0.796717	−	1.37995i	−0.0298582	−	0.0517160i
$713$	−7.52373	−	13.0315i	−0.281766	−	0.488033i
$714$		0			0
$715$	1.40180	−	2.42800i	0.0524245	−	0.0908019i
$716$	−12.6983			−0.474556
$717$		0			0
$718$	−35.1606			−1.31218
$719$	13.0088	+	22.5319i	0.485145	+	0.840296i	0.999854	−	0.0170686i	$-0.00543337\pi$
$719$	13.0088	+	22.5319i	0.485145	+	0.840296i	−0.514709	+	0.857365i	$0.672100\pi$
$720$		0			0
$721$		0			0
$722$	−0.766951	−	1.32840i	−0.0285430	−	0.0494379i
$723$		0			0
$724$	−5.72670	−	9.91893i	−0.212831	−	0.368634i
$725$	−11.2250	−	19.4423i	−0.416886	−	0.722068i
$726$		0			0
$727$	0.685875	+	1.18797i	0.0254377	+	0.0440594i	0.878464	−	0.477809i	$-0.158569\pi$
$727$	0.685875	+	1.18797i	0.0254377	+	0.0440594i	−0.853026	+	0.521868i	$0.825235\pi$
$728$		0			0
$729$		0			0
$730$	−3.59888	−	6.23345i	−0.133201	−	0.230710i
$731$	52.9830			1.95965
$732$		0			0
$733$	−0.800174			−0.0295551			−0.0147776	−	0.999891i	$-0.504704\pi$
$733$	−0.800174			−0.0295551			−0.0147776	+	0.999891i	$0.504704\pi$
$734$	2.42011	−	4.19176i	0.0893280	−	0.154721i
$735$		0			0
$736$	−5.82072	−	10.0818i	−0.214555	−	0.371620i
$737$	3.49017	+	6.04515i	0.128562	+	0.222676i
$738$		0			0
$739$	−2.68547	+	4.65136i	−0.0987865	+	0.171103i	−0.911183	−	0.412003i	$-0.864829\pi$
$739$	−2.68547	+	4.65136i	−0.0987865	+	0.171103i	0.812396	+	0.583106i	$0.198163\pi$
$740$	2.94756	−	5.10532i	0.108354	−	0.187675i
$741$		0			0
$742$		0			0
$743$	−6.63162	+	11.4863i	−0.243290	+	0.421391i	−0.961650	−	0.274281i	$-0.911560\pi$
$743$	−6.63162	+	11.4863i	−0.243290	+	0.421391i	0.718359	+	0.695672i	$0.244893\pi$
$744$		0			0
$745$	4.14436			0.151838
$746$	−18.2138	+	31.5472i	−0.666854	+	1.15503i
$747$		0			0
$748$	3.96485			0.144969
$749$		0			0
$750$		0			0
$751$	5.55632			0.202753			0.101377	−	0.994848i	$-0.467675\pi$
$751$	5.55632			0.202753			0.101377	+	0.994848i	$0.467675\pi$
$752$	18.6457	+	32.2953i	0.679939	+	1.17769i
$753$		0			0
$754$	−23.3645	+	40.4684i	−0.850883	+	1.47377i
$755$	−12.0119			−0.437158
$756$		0			0
$757$	−13.3942			−0.486819			−0.243410	−	0.969924i	$-0.578266\pi$
$757$	−13.3942			−0.486819			−0.243410	+	0.969924i	$0.578266\pi$
$758$	−22.9869	+	39.8145i	−0.834923	+	1.44613i
$759$		0			0
$760$	−3.99931	−	6.92701i	−0.145070	−	0.251269i
$761$	12.8438			0.465588			0.232794	−	0.972526i	$-0.425213\pi$
$761$	12.8438			0.465588			0.232794	+	0.972526i	$0.425213\pi$
$762$		0			0
$763$		0			0
$764$	1.92353			0.0695910
$765$		0			0
$766$	−12.2601	+	21.2351i	−0.442975	+	0.767256i
$767$	52.3570			1.89050
$768$		0			0
$769$	−1.48259	+	2.56793i	−0.0534636	+	0.0926018i	−0.891519	−	0.452984i	$-0.850359\pi$
$769$	−1.48259	+	2.56793i	−0.0534636	+	0.0926018i	0.838055	+	0.545586i	$0.183693\pi$
$770$		0			0
$771$		0			0
$772$	−4.63533	+	8.02864i	−0.166829	+	0.288957i
$773$	−9.63939	+	16.6959i	−0.346705	+	0.600510i	−0.985662	−	0.168732i	$-0.946033\pi$
$773$	−9.63939	+	16.6959i	−0.346705	+	0.600510i	0.638957	+	0.769242i	$0.279366\pi$
$774$		0			0
$775$	−12.4496	−	21.5633i	−0.447203	−	0.774578i
$776$	−3.21683	−	5.57171i	−0.115477	−	0.200013i
$777$		0			0
$778$	−5.19028	+	8.98983i	−0.186080	+	0.322301i
$779$	−4.70610			−0.168614
$780$		0			0
$781$	−2.53018			−0.0905371
$782$	15.9616	+	27.6462i	0.570784	+	0.988627i
$783$		0			0
$784$		0			0
$785$	−5.35098	−	9.26818i	−0.190985	−	0.330795i
$786$		0			0
$787$	−6.82265	−	11.8172i	−0.243201	−	0.421237i	0.718423	−	0.695606i	$-0.244864\pi$
$787$	−6.82265	−	11.8172i	−0.243201	−	0.421237i	−0.961624	+	0.274370i	$0.911531\pi$
$788$	−8.34734	−	14.4580i	−0.297362	−	0.515046i
$789$		0			0
$790$	1.07598	+	1.86364i	0.0382815	+	0.0663055i
$791$		0			0
$792$		0			0
$793$	29.2472	+	50.6577i	1.03860	+	1.79891i
$794$	−21.9036			−0.777330
$795$		0			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)

\(f(q)\)	\(=\)	\(q+(-0.849814 + 1.47192i) q^{2} +(-0.444368 - 0.769668i) q^{4} +0.949271 q^{5} -1.88874 q^{8} +O(q^{10})\)	\(q+(-0.849814 + 1.47192i) q^{2} +(-0.444368 - 0.769668i) q^{4} +0.949271 q^{5} -1.88874 q^{8} +(-0.806704 + 1.39725i) q^{10} +0.588364 q^{11} +(2.50987 - 4.34722i) q^{13} +(2.49381 - 4.31941i) q^{16} +(-3.79121 + 6.56657i) q^{17} +(2.23061 + 3.86353i) q^{19} +(-0.421826 - 0.730623i) q^{20} +(-0.500000 + 0.866025i) q^{22} -2.47710 q^{23} -4.09888 q^{25} +(4.26584 + 7.38866i) q^{26} +(2.73855 + 4.74331i) q^{29} +(3.03731 + 5.26078i) q^{31} +(2.34981 + 4.07000i) q^{32} +(-6.44364 - 11.1607i) q^{34} +(3.49381 + 6.05146i) q^{37} -7.58242 q^{38} -1.79292 q^{40} +(-0.527445 + 0.913562i) q^{41} +(-3.49381 - 6.05146i) q^{43} +(-0.261450 - 0.452845i) q^{44} +(2.10507 - 3.64610i) q^{46} +(-3.73840 + 6.47510i) q^{47} +(3.48329 - 6.03323i) q^{50} -4.46122 q^{52} +(3.46108 - 5.99476i) q^{53} +0.558517 q^{55} -9.30903 q^{58} +(5.21512 + 9.03284i) q^{59} +(-5.82644 + 10.0917i) q^{61} -10.3246 q^{62} +1.98762 q^{64} +(2.38255 - 4.12669i) q^{65} +(5.93199 + 10.2745i) q^{67} +6.73877 q^{68} -4.30037 q^{71} +(-2.23061 + 3.86353i) q^{73} -11.8764 q^{74} +(1.98242 - 3.43366i) q^{76} +(0.666896 - 1.15510i) q^{79} +(2.36730 - 4.10029i) q^{80} +(-0.896461 - 1.55272i) q^{82} +(-2.84194 - 4.92238i) q^{83} +(-3.59888 + 6.23345i) q^{85} +11.8764 q^{86} -1.11126 q^{88} +(0.421826 + 0.730623i) q^{89} +(1.10074 + 1.90654i) q^{92} +(-6.35389 - 11.0053i) q^{94} +(2.11745 + 3.66754i) q^{95} +(1.70317 + 2.94997i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12q + 2q^{2} - 6q^{4} - 24q^{8} + O(q^{10}) \)	\( 12q + 2q^{2} - 6q^{4} - 24q^{8} - 16q^{11} - 6q^{16} - 6q^{22} - 8q^{23} + 24q^{25} + 22q^{29} + 16q^{32} + 6q^{37} - 6q^{43} - 14q^{44} - 12q^{46} + 56q^{50} + 28q^{53} + 36q^{58} - 48q^{64} - 6q^{65} - 76q^{71} - 72q^{74} + 6q^{79} + 30q^{85} + 72q^{86} - 12q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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