LMFDB - Embedded newform 1323.2.g.h.667.12

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:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 441)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			667.12
Character	$\chi$	$=$	1323.667
Dual form			1323.2.g.h.361.12

$q$-expansion

$f(q)$	$=$	$q+(1.08816 + 1.88474i) q^{2} +(-1.36816 + 2.36973i) q^{4} +1.26829 q^{5} -1.60248 q^{8} +O(q^{10})$	\(q+(1.08816 + 1.88474i) q^{2} +(-1.36816 + 2.36973i) q^{4} +1.26829 q^{5} -1.60248 q^{8} +(1.38010 + 2.39040i) q^{10} +5.47733 q^{11} +(-2.37268 - 4.10960i) q^{13} +(0.992580 + 1.71920i) q^{16} +(2.40822 + 4.17116i) q^{17} +(2.69059 - 4.66025i) q^{19} +(-1.73523 + 3.00550i) q^{20} +(5.96019 + 10.3233i) q^{22} +5.17631 q^{23} -3.39144 q^{25} +(5.16368 - 8.94376i) q^{26} +(-2.01656 + 3.49278i) q^{29} +(-0.732093 + 1.26802i) q^{31} +(-3.76264 + 6.51709i) q^{32} +(-5.24103 + 9.07773i) q^{34} +(-0.959170 + 1.66133i) q^{37} +11.7111 q^{38} -2.03241 q^{40} +(-1.94808 - 3.37418i) q^{41} +(-1.66016 + 2.87549i) q^{43} +(-7.49389 + 12.9798i) q^{44} +(5.63263 + 9.75600i) q^{46} +(-1.57773 - 2.73271i) q^{47} +(-3.69042 - 6.39199i) q^{50} +12.9849 q^{52} +(-3.57149 - 6.18601i) q^{53} +6.94684 q^{55} -8.77732 q^{58} +(-0.154341 + 0.267327i) q^{59} +(5.17143 + 8.95719i) q^{61} -3.18652 q^{62} -12.4070 q^{64} +(-3.00924 - 5.21216i) q^{65} +(-2.23655 + 3.87382i) q^{67} -13.1794 q^{68} +1.96688 q^{71} +(-5.27515 - 9.13683i) q^{73} -4.17491 q^{74} +(7.36235 + 12.7520i) q^{76} +(4.50822 + 7.80846i) q^{79} +(1.25888 + 2.18044i) q^{80} +(4.23963 - 7.34326i) q^{82} +(-5.08023 + 8.79921i) q^{83} +(3.05432 + 5.29023i) q^{85} -7.22607 q^{86} -8.77732 q^{88} +(-2.59776 + 4.49945i) q^{89} +(-7.08205 + 12.2665i) q^{92} +(3.43363 - 5.94722i) q^{94} +(3.41245 - 5.91054i) q^{95} +(2.48521 - 4.30451i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24q - 4q^{2} - 12q^{4} + 24q^{8} + O(q^{10}) $	$ 24q - 4q^{2} - 12q^{4} + 24q^{8} + 40q^{11} - 12q^{16} + 64q^{23} + 24q^{25} - 16q^{29} - 48q^{32} - 12q^{37} - 56q^{44} + 24q^{46} + 4q^{50} - 32q^{53} + 96q^{64} - 60q^{65} - 12q^{67} + 112q^{71} + 136q^{74} + 12q^{79} + 12q^{85} + 152q^{86} - 16q^{92} - 64q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	1.08816	+	1.88474i	0.769442	+	1.33271i	0.937866	+	0.346998i	$0.112799\pi$
$2$	1.08816	+	1.88474i	0.769442	+	1.33271i	−0.168424	+	0.985715i	$0.553868\pi$
$3$		0			0
$3$		0			0
$4$	−1.36816	+	2.36973i	−0.684082	+	1.18487i
$5$	1.26829			0.567196			0.283598	−	0.958943i	$-0.408472\pi$
$5$	1.26829			0.567196			0.283598	+	0.958943i	$0.408472\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.60248			−0.566563
$9$		0			0
$10$	1.38010	+	2.39040i	0.436425	+	0.755910i
$11$	5.47733			1.65148			0.825739	−	0.564053i	$-0.190759\pi$
$11$	5.47733			1.65148			0.825739	+	0.564053i	$0.190759\pi$
$12$		0			0
$13$	−2.37268	−	4.10960i	−0.658062	−	1.13980i	−0.981117	−	0.193417i	$-0.938043\pi$
$13$	−2.37268	−	4.10960i	−0.658062	−	1.13980i	0.323054	−	0.946380i	$-0.395290\pi$
$14$		0			0
$15$		0			0
$16$	0.992580	+	1.71920i	0.248145	+	0.429800i
$17$	2.40822	+	4.17116i	0.584079	+	1.01165i	0.994990	+	0.0999785i	$0.0318774\pi$
$17$	2.40822	+	4.17116i	0.584079	+	1.01165i	−0.410911	+	0.911676i	$0.634789\pi$
$18$		0			0
$19$	2.69059	−	4.66025i	0.617265	−	1.06913i	−0.372718	−	0.927945i	$-0.621574\pi$
$19$	2.69059	−	4.66025i	0.617265	−	1.06913i	0.989983	−	0.141189i	$-0.0450925\pi$
$20$	−1.73523	+	3.00550i	−0.388009	+	0.672051i
$21$		0			0
$22$	5.96019	+	10.3233i	1.27072	+	2.20095i
$23$	5.17631			1.07934			0.539668	−	0.841878i	$-0.318550\pi$
$23$	5.17631			1.07934			0.539668	+	0.841878i	$0.318550\pi$
$24$		0			0
$25$	−3.39144			−0.678288
$26$	5.16368	−	8.94376i	1.01268	−	1.75402i
$27$		0			0
$28$		0			0
$29$	−2.01656	+	3.49278i	−0.374466	+	0.648594i	−0.990247	−	0.139324i	$-0.955507\pi$
$29$	−2.01656	+	3.49278i	−0.374466	+	0.648594i	0.615781	+	0.787917i	$0.288841\pi$
$30$		0			0
$31$	−0.732093	+	1.26802i	−0.131488	+	0.227744i	−0.924250	−	0.381787i	$-0.875309\pi$
$31$	−0.732093	+	1.26802i	−0.131488	+	0.227744i	0.792763	+	0.609531i	$0.208642\pi$
$32$	−3.76264	+	6.51709i	−0.665148	+	1.15207i
$33$		0			0
$34$	−5.24103	+	9.07773i	−0.898830	+	1.55682i
$35$		0			0
$36$		0			0
$37$	−0.959170	+	1.66133i	−0.157687	+	0.273121i	−0.934034	−	0.357184i	$-0.883737\pi$
$37$	−0.959170	+	1.66133i	−0.157687	+	0.273121i	0.776347	+	0.630305i	$0.217070\pi$
$38$	11.7111			1.89980
$39$		0			0
$40$	−2.03241			−0.321352
$41$	−1.94808	−	3.37418i	−0.304239	−	0.526958i	0.672852	−	0.739777i	$-0.265069\pi$
$41$	−1.94808	−	3.37418i	−0.304239	−	0.526958i	−0.977092	+	0.212819i	$0.931736\pi$
$42$		0			0
$43$	−1.66016	+	2.87549i	−0.253173	+	0.438508i	−0.964398	−	0.264457i	$-0.914807\pi$
$43$	−1.66016	+	2.87549i	−0.253173	+	0.438508i	0.711225	+	0.702964i	$0.248141\pi$
$44$	−7.49389	+	12.9798i	−1.12975	+	1.95678i
$45$		0			0
$46$	5.63263	+	9.75600i	0.830486	+	1.43844i
$47$	−1.57773	−	2.73271i	−0.230135	−	0.398606i	0.727712	−	0.685882i	$-0.240584\pi$
$47$	−1.57773	−	2.73271i	−0.230135	−	0.398606i	−0.957848	+	0.287276i	$0.907250\pi$
$48$		0			0
$49$		0			0
$50$	−3.69042	−	6.39199i	−0.521904	−	0.903964i
$51$		0			0
$52$	12.9849			1.80068
$53$	−3.57149	−	6.18601i	−0.490582	−	0.849714i	0.509359	−	0.860554i	$-0.329883\pi$
$53$	−3.57149	−	6.18601i	−0.490582	−	0.849714i	−0.999941	+	0.0108405i	$0.996549\pi$
$54$		0			0
$55$	6.94684			0.936712
$56$		0			0
$57$		0			0
$58$	−8.77732			−1.15252
$59$	−0.154341	+	0.267327i	−0.0200935	+	0.0348030i	−0.875897	−	0.482498i	$-0.839730\pi$
$59$	−0.154341	+	0.267327i	−0.0200935	+	0.0348030i	0.855804	+	0.517301i	$0.173063\pi$
$60$		0			0
$61$	5.17143	+	8.95719i	0.662134	+	1.14685i	0.980054	+	0.198732i	$0.0636825\pi$
$61$	5.17143	+	8.95719i	0.662134	+	1.14685i	−0.317920	+	0.948118i	$0.602984\pi$
$62$	−3.18652			−0.404689
$63$		0			0
$64$	−12.4070			−1.55088
$65$	−3.00924	−	5.21216i	−0.373250	−	0.646489i
$66$		0			0
$67$	−2.23655	+	3.87382i	−0.273238	+	0.473262i	−0.969689	−	0.244342i	$-0.921428\pi$
$67$	−2.23655	+	3.87382i	−0.273238	+	0.473262i	0.696451	+	0.717604i	$0.254761\pi$
$68$	−13.1794			−1.59823
$69$		0			0
$70$		0			0
$71$	1.96688			0.233426			0.116713	−	0.993166i	$-0.462764\pi$
$71$	1.96688			0.233426			0.116713	+	0.993166i	$0.462764\pi$
$72$		0			0
$73$	−5.27515	−	9.13683i	−0.617409	−	1.06938i	−0.989957	−	0.141371i	$-0.954849\pi$
$73$	−5.27515	−	9.13683i	−0.617409	−	1.06938i	0.372547	−	0.928013i	$-0.378484\pi$
$74$	−4.17491			−0.485323
$75$		0			0
$76$	7.36235	+	12.7520i	0.844520	+	1.46275i
$77$		0			0
$78$		0			0
$79$	4.50822	+	7.80846i	0.507214	+	0.878520i	0.999965	+	0.00835000i	$0.00265792\pi$
$79$	4.50822	+	7.80846i	0.507214	+	0.878520i	−0.492751	+	0.870170i	$0.664009\pi$
$80$	1.25888	+	2.18044i	0.140747	+	0.243781i
$81$		0			0
$82$	4.23963	−	7.34326i	0.468189	−	0.810927i
$83$	−5.08023	+	8.79921i	−0.557627	+	0.965839i	0.440066	+	0.897965i	$0.354955\pi$
$83$	−5.08023	+	8.79921i	−0.557627	+	0.965839i	−0.997694	+	0.0678739i	$0.978378\pi$
$84$		0			0
$85$	3.05432	+	5.29023i	0.331287	+	0.573806i
$86$	−7.22607			−0.779207
$87$		0			0
$88$	−8.77732			−0.935666
$89$	−2.59776	+	4.49945i	−0.275362	+	0.476941i	−0.970226	−	0.242200i	$-0.922131\pi$
$89$	−2.59776	+	4.49945i	−0.275362	+	0.476941i	0.694864	+	0.719141i	$0.255464\pi$
$90$		0			0
$91$		0			0
$92$	−7.08205	+	12.2665i	−0.738354	+	1.27887i
$93$		0			0
$94$	3.43363	−	5.94722i	0.354152	−	0.613409i
$95$	3.41245	−	5.91054i	0.350110	−	0.606409i
$96$		0			0
$97$	2.48521	−	4.30451i	0.252335	−	0.437057i	−0.711833	−	0.702348i	$-0.752135\pi$
$97$	2.48521	−	4.30451i	0.252335	−	0.437057i	0.964168	+	0.265291i	$0.0854682\pi$
$98$		0			0
$99$		0			0
$100$	4.64005	−	8.03680i	0.464005	−	0.803680i
$101$	0.00533808			0.000531159			0.000265580	−	1.00000i	$-0.499915\pi$
$101$	0.00533808			0.000531159			0.000265580		1.00000i	$0.499915\pi$
$102$		0			0
$103$	−13.0348			−1.28436			−0.642180	−	0.766554i	$-0.721970\pi$
$103$	−13.0348			−1.28436			−0.642180	+	0.766554i	$0.721970\pi$
$104$	3.80217	+	6.58555i	0.372834	+	0.645767i
$105$		0			0
$106$	7.77268	−	13.4627i	0.754950	−	1.30761i
$107$	4.71081	−	8.15936i	0.455411	−	0.788795i	−0.543301	−	0.839538i	$-0.682826\pi$
$107$	4.71081	−	8.15936i	0.455411	−	0.788795i	0.998712	+	0.0507430i	$0.0161589\pi$
$108$		0			0
$109$	−8.44513	−	14.6274i	−0.808896	−	1.40105i	−0.913629	−	0.406549i	$-0.866732\pi$
$109$	−8.44513	−	14.6274i	−0.808896	−	1.40105i	0.104732	−	0.994500i	$-0.466601\pi$
$110$	7.55924	+	13.0930i	0.720746	+	1.24837i
$111$		0			0
$112$		0			0
$113$	3.07313	+	5.32281i	0.289095	+	0.500728i	0.973594	−	0.228286i	$-0.0733122\pi$
$113$	3.07313	+	5.32281i	0.289095	+	0.500728i	−0.684499	+	0.729014i	$0.739979\pi$
$114$		0			0
$115$	6.56506			0.612195
$116$	−5.51797	−	9.55741i	−0.512331	−	0.887383i
$117$		0			0
$118$	−0.671790			−0.0618432
$119$		0			0
$120$		0			0
$121$	19.0012			1.72738
$122$	−11.2546	+	19.4936i	−1.01895	+	1.76487i
$123$		0			0
$124$	−2.00325	−	3.46973i	−0.179897	−	0.311591i
$125$	−10.6428			−0.951919
$126$		0			0
$127$	−13.9305			−1.23613			−0.618065	−	0.786127i	$-0.712083\pi$
$127$	−13.9305			−1.23613			−0.618065	+	0.786127i	$0.712083\pi$
$128$	−5.97551	−	10.3499i	−0.528165	−	0.914809i
$129$		0			0
$130$	6.54905	−	11.3433i	0.574389	−	0.994871i
$131$	−0.179156			−0.0156529			−0.00782645	−	0.999969i	$-0.502491\pi$
$131$	−0.179156			−0.0156529			−0.00782645	+	0.999969i	$0.502491\pi$
$132$		0			0
$133$		0			0
$134$	−9.73486			−0.840964
$135$		0			0
$136$	−3.85913	−	6.68420i	−0.330917	−	0.573166i
$137$	−3.15206			−0.269299			−0.134649	−	0.990893i	$-0.542991\pi$
$137$	−3.15206			−0.269299			−0.134649	+	0.990893i	$0.542991\pi$
$138$		0			0
$139$	−9.42857	−	16.3308i	−0.799721	−	1.38516i	−0.919798	−	0.392392i	$-0.871648\pi$
$139$	−9.42857	−	16.3308i	−0.799721	−	1.38516i	0.120077	−	0.992765i	$-0.461686\pi$
$140$		0			0
$141$		0			0
$142$	2.14027	+	3.70706i	0.179608	+	0.311090i
$143$	−12.9959	−	22.5096i	−1.08677	−	1.88235i
$144$		0			0
$145$	−2.55758	+	4.42986i	−0.212396	+	0.367880i
$146$	11.4804	−	19.8846i	0.950122	−	1.64566i
$147$		0			0
$148$	−2.62461	−	4.54595i	−0.215741	−	0.373675i
$149$	21.2740			1.74284			0.871418	−	0.490541i	$-0.163201\pi$
$149$	21.2740			1.74284			0.871418	+	0.490541i	$0.163201\pi$
$150$		0			0
$151$	6.36561			0.518026			0.259013	−	0.965874i	$-0.416603\pi$
$151$	6.36561			0.518026			0.259013	+	0.965874i	$0.416603\pi$
$152$	−4.31163	+	7.46796i	−0.349719	+	0.605731i
$153$		0			0
$154$		0			0
$155$	−0.928506	+	1.60822i	−0.0745794	+	0.129175i
$156$		0			0
$157$	0.697976	−	1.20893i	0.0557045	−	0.0964830i	−0.836828	−	0.547465i	$-0.815593\pi$
$157$	0.697976	−	1.20893i	0.0557045	−	0.0964830i	0.892533	+	0.450982i	$0.148926\pi$
$158$	−9.81128	+	16.9936i	−0.780543	+	1.35194i
$159$		0			0
$160$	−4.77212	+	8.26556i	−0.377269	+	0.653450i
$161$		0			0
$162$		0			0
$163$	9.53086	−	16.5079i	0.746515	−	1.29300i	−0.202969	−	0.979185i	$-0.565059\pi$
$163$	9.53086	−	16.5079i	0.746515	−	1.29300i	0.949484	−	0.313816i	$-0.101608\pi$
$164$	10.6612			0.832499
$165$		0			0
$166$	−22.1123			−1.71625
$167$	0.872003	+	1.51035i	0.0674776	+	0.116875i	0.897790	−	0.440423i	$-0.145172\pi$
$167$	0.872003	+	1.51035i	0.0674776	+	0.116875i	−0.830313	+	0.557298i	$0.811838\pi$
$168$		0			0
$169$	−4.75919	+	8.24317i	−0.366092	+	0.634090i
$170$	−6.64715	+	11.5132i	−0.509813	+	0.883022i
$171$		0			0
$172$	−4.54276	−	7.86828i	−0.346382	−	0.599951i
$173$	5.03794	+	8.72598i	0.383028	+	0.663424i	0.991493	−	0.130157i	$-0.0415480\pi$
$173$	5.03794	+	8.72598i	0.383028	+	0.663424i	−0.608466	+	0.793580i	$0.708215\pi$
$174$		0			0
$175$		0			0
$176$	5.43669	+	9.41662i	0.409806	+	0.709805i
$177$		0			0
$178$	−11.3071			−0.847500
$179$	−9.27118	−	16.0582i	−0.692961	−	1.20024i	−0.970863	−	0.239634i	$-0.922973\pi$
$179$	−9.27118	−	16.0582i	−0.692961	−	1.20024i	0.277902	−	0.960609i	$-0.410361\pi$
$180$		0			0
$181$	8.80982			0.654829			0.327414	−	0.944881i	$-0.393823\pi$
$181$	8.80982			0.654829			0.327414	+	0.944881i	$0.393823\pi$
$182$		0			0
$183$		0			0
$184$	−8.29494			−0.611511
$185$	−1.21651	+	2.10705i	−0.0894393	+	0.154913i
$186$		0			0
$187$	13.1906	+	22.8468i	0.964593	+	1.67072i
$188$	8.63437			0.629726
$189$		0			0
$190$	14.8531			1.07756
$191$	2.45469	+	4.25165i	0.177615	+	0.307639i	0.941063	−	0.338231i	$-0.109828\pi$
$191$	2.45469	+	4.25165i	0.177615	+	0.307639i	−0.763448	+	0.645869i	$0.776495\pi$
$192$		0			0
$193$	4.88380	−	8.45899i	0.351544	−	0.608892i	−0.634976	−	0.772531i	$-0.718990\pi$
$193$	4.88380	−	8.45899i	0.351544	−	0.608892i	0.986520	+	0.163640i	$0.0523235\pi$
$194$	10.8172			0.776629
$195$		0			0
$196$		0			0
$197$	−3.31445			−0.236145			−0.118073	−	0.993005i	$-0.537672\pi$
$197$	−3.31445			−0.236145			−0.118073	+	0.993005i	$0.537672\pi$
$198$		0			0
$199$	5.54432	+	9.60304i	0.393026	+	0.680742i	0.992847	−	0.119393i	$-0.0380948\pi$
$199$	5.54432	+	9.60304i	0.393026	+	0.680742i	−0.599821	+	0.800134i	$0.704761\pi$
$200$	5.43472			0.384293
$201$		0			0
$202$	0.00580866	+	0.0100609i	0.000408696	+	0.000707883i
$203$		0			0
$204$		0			0
$205$	−2.47073	−	4.27943i	−0.172563	−	0.298889i
$206$	−14.1839	−	24.5673i	−0.988240	−	1.71168i
$207$		0			0
$208$	4.71014	−	8.15821i	0.326590	−	0.565670i
$209$	14.7373	−	25.5257i	1.01940	−	1.76565i
$210$		0			0
$211$	−3.66118	−	6.34135i	−0.252046	−	0.436557i	0.712043	−	0.702136i	$-0.247770\pi$
$211$	−3.66118	−	6.34135i	−0.252046	−	0.436557i	−0.964089	+	0.265579i	$0.914437\pi$
$212$	19.5456			1.34240
$213$		0			0
$214$	20.5044			1.40165
$215$	−2.10557	+	3.64695i	−0.143599	+	0.248720i
$216$		0			0
$217$		0			0
$218$	18.3792	−	31.8337i	1.24480	−	2.15605i
$219$		0			0
$220$	−9.50442	+	16.4621i	−0.640788	+	1.10988i
$221$	11.4278	−	19.7936i	0.768720	−	1.33146i
$222$		0			0
$223$	−2.02765	+	3.51199i	−0.135782	+	0.235181i	−0.925896	−	0.377779i	$-0.876688\pi$
$223$	−2.02765	+	3.51199i	−0.135782	+	0.235181i	0.790114	+	0.612960i	$0.210021\pi$
$224$		0			0
$225$		0			0
$226$	−6.68808	+	11.5841i	−0.444884	+	0.770562i
$227$	1.33417			0.0885522			0.0442761	−	0.999019i	$-0.485902\pi$
$227$	1.33417			0.0885522			0.0442761	+	0.999019i	$0.485902\pi$
$228$		0			0
$229$	15.9966			1.05709			0.528544	−	0.848906i	$-0.322738\pi$
$229$	15.9966			1.05709			0.528544	+	0.848906i	$0.322738\pi$
$230$	7.14381	+	12.3734i	0.471049	+	0.815880i
$231$		0			0
$232$	3.23150	−	5.59712i	0.212158	−	0.367469i
$233$	4.06542	−	7.04151i	0.266334	−	0.461305i	−0.701578	−	0.712593i	$-0.747521\pi$
$233$	4.06542	−	7.04151i	0.266334	−	0.461305i	0.967912	+	0.251288i	$0.0808542\pi$
$234$		0			0
$235$	−2.00102	−	3.46586i	−0.130532	−	0.226088i
$236$	−0.422329	−	0.731495i	−0.0274913	−	0.0476163i
$237$		0			0
$238$		0			0
$239$	−11.0509	−	19.1407i	−0.714823	−	1.23811i	−0.963028	−	0.269403i	$-0.913174\pi$
$239$	−11.0509	−	19.1407i	−0.714823	−	1.23811i	0.248204	−	0.968708i	$-0.420160\pi$
$240$		0			0
$241$	−27.5947			−1.77753			−0.888765	−	0.458362i	$-0.848436\pi$
$241$	−27.5947			−1.77753			−0.888765	+	0.458362i	$0.848436\pi$
$242$	20.6762	+	35.8122i	1.32912	+	2.30210i
$243$		0			0
$244$	−28.3015			−1.81182
$245$		0			0
$246$		0			0
$247$	−25.5356			−1.62479
$248$	1.17317	−	2.03198i	0.0744961	−	0.129031i
$249$		0			0
$250$	−11.5810	−	20.0589i	−0.732447	−	1.26863i
$251$	−16.5610			−1.04532			−0.522661	−	0.852541i	$-0.675061\pi$
$251$	−16.5610			−1.04532			−0.522661	+	0.852541i	$0.675061\pi$
$252$		0			0
$253$	28.3524			1.78250
$254$	−15.1585	−	26.2553i	−0.951130	−	1.64741i
$255$		0			0
$256$	0.597516	−	1.03493i	0.0373448	−	0.0646831i
$257$	2.06573			0.128857			0.0644285	−	0.997922i	$-0.479478\pi$
$257$	2.06573			0.128857			0.0644285	+	0.997922i	$0.479478\pi$
$258$		0			0
$259$		0			0
$260$	16.4686			1.02134
$261$		0			0
$262$	−0.194949	−	0.337662i	−0.0120440	−	0.0208608i
$263$	−10.1296			−0.624620			−0.312310	−	0.949980i	$-0.601103\pi$
$263$	−10.1296			−0.624620			−0.312310	+	0.949980i	$0.601103\pi$
$264$		0			0
$265$	−4.52969	−	7.84565i	−0.278257	−	0.481954i
$266$		0			0
$267$		0			0
$268$	−6.11994	−	10.6000i	−0.373835	−	0.647501i
$269$	7.54972	+	13.0765i	0.460315	+	0.797289i	0.998976	−	0.0452336i	$-0.0144032\pi$
$269$	7.54972	+	13.0765i	0.460315	+	0.797289i	−0.538662	+	0.842522i	$0.681070\pi$
$270$		0			0
$271$	−14.4026	+	24.9459i	−0.874893	+	1.51536i	−0.0180156	+	0.999838i	$0.505735\pi$
$271$	−14.4026	+	24.9459i	−0.874893	+	1.51536i	−0.856877	+	0.515521i	$0.827598\pi$
$272$	−4.78070	+	8.28041i	−0.289872	+	0.502074i
$273$		0			0
$274$	−3.42993	−	5.94082i	−0.207210	−	0.358898i
$275$	−18.5760			−1.12018
$276$		0			0
$277$	−2.69963			−0.162205			−0.0811026	−	0.996706i	$-0.525844\pi$
$277$	−2.69963			−0.162205			−0.0811026	+	0.996706i	$0.525844\pi$
$278$	20.5195	−	35.5408i	1.23068	−	2.13160i
$279$		0			0
$280$		0			0
$281$	2.46312	−	4.26626i	0.146938	−	0.254503i	−0.783157	−	0.621825i	$-0.786392\pi$
$281$	2.46312	−	4.26626i	0.146938	−	0.254503i	0.930094	+	0.367321i	$0.119725\pi$
$282$		0			0
$283$	1.79079	−	3.10173i	0.106451	−	0.184379i	−0.807879	−	0.589348i	$-0.799385\pi$
$283$	1.79079	−	3.10173i	0.106451	−	0.184379i	0.914330	+	0.404969i	$0.132718\pi$
$284$	−2.69102	+	4.66098i	−0.159682	+	0.276578i
$285$		0			0
$286$	28.2832	−	48.9879i	1.67242	−	2.89672i
$287$		0			0
$288$		0			0
$289$	−3.09903	+	5.36768i	−0.182296	+	0.315746i
$290$	−11.1322			−0.653704
$291$		0			0
$292$	28.8691			1.68944
$293$	12.1955	+	21.1232i	0.712469	+	1.23403i	0.963928	+	0.266164i	$0.0857563\pi$
$293$	12.1955	+	21.1232i	0.712469	+	1.23403i	−0.251459	+	0.967868i	$0.580910\pi$
$294$		0			0
$295$	−0.195750	+	0.339048i	−0.0113970	+	0.0197401i
$296$	1.53705	−	2.66225i	0.0893394	−	0.154740i
$297$		0			0
$298$	23.1494	+	40.0960i	1.34101	+	2.32270i
$299$	−12.2817	−	21.2726i	−0.710270	−	1.23022i
$300$		0			0
$301$		0			0
$302$	6.92678	+	11.9975i	0.398591	+	0.690380i
$303$		0			0
$304$	10.6825			0.612685
$305$	6.55887	+	11.3603i	0.375560	+	0.650489i
$306$		0			0
$307$	23.9025			1.36419			0.682094	−	0.731265i	$-0.261070\pi$
$307$	23.9025			1.36419			0.682094	+	0.731265i	$0.261070\pi$
$308$		0			0
$309$		0			0
$310$	−4.04143			−0.229538
$311$	6.47082	−	11.2078i	0.366926	−	0.635535i	−0.622157	−	0.782893i	$-0.713743\pi$
$311$	6.47082	−	11.2078i	0.366926	−	0.635535i	0.989083	+	0.147357i	$0.0470768\pi$
$312$		0			0
$313$	−13.4340	−	23.2684i	−0.759336	−	1.31521i	−0.943189	−	0.332255i	$-0.892190\pi$
$313$	−13.4340	−	23.2684i	−0.759336	−	1.31521i	0.183853	−	0.982954i	$-0.441143\pi$
$314$	3.03802			0.171446
$315$		0			0
$316$	−24.6719			−1.38790
$317$	4.15584	+	7.19813i	0.233415	+	0.404287i	0.958811	−	0.284045i	$-0.0916765\pi$
$317$	4.15584	+	7.19813i	0.233415	+	0.404287i	−0.725396	+	0.688332i	$0.758343\pi$
$318$		0			0
$319$	−11.0454	+	19.1311i	−0.618422	+	1.07114i
$320$	−15.7357			−0.879654
$321$		0			0
$322$		0			0
$323$	25.9182			1.44212
$324$		0			0
$325$	8.04680	+	13.9375i	0.446356	+	0.773111i
$326$	41.4842			2.29760
$327$		0			0
$328$	3.12177	+	5.40706i	0.172371	+	0.298555i
$329$		0			0
$330$		0			0
$331$	−6.19889	−	10.7368i	−0.340722	−	0.590147i	0.643845	−	0.765156i	$-0.277338\pi$
$331$	−6.19889	−	10.7368i	−0.340722	−	0.590147i	−0.984567	+	0.175009i	$0.944005\pi$
$332$	−13.9012	−	24.0775i	−0.762926	−	1.32143i
$333$		0			0
$334$	−1.89775	+	3.28700i	−0.103840	+	0.179857i
$335$	−2.83659	+	4.91312i	−0.154980	+	0.268433i
$336$		0			0
$337$	−12.9588	−	22.4454i	−0.705913	−	1.22268i	−0.966361	−	0.257189i	$-0.917204\pi$
$337$	−12.9588	−	22.4454i	−0.705913	−	1.22268i	0.260448	−	0.965488i	$-0.416130\pi$
$338$	−20.7150			−1.12675
$339$		0			0
$340$	−16.7152			−0.906511
$341$	−4.00992	+	6.94538i	−0.217149	+	0.376113i
$342$		0			0
$343$		0			0
$344$	2.66038	−	4.60792i	0.143438	−	0.248442i
$345$		0			0
$346$	−10.9641	+	18.9904i	−0.589435	+	1.02093i
$347$	−8.42415	+	14.5911i	−0.452232	+	0.783289i	−0.998524	−	0.0543058i	$-0.982705\pi$
$347$	−8.42415	+	14.5911i	−0.452232	+	0.783289i	0.546292	+	0.837595i	$0.316039\pi$
$348$		0			0
$349$	−15.5503	+	26.9340i	−0.832390	+	1.44174i	0.0637477	+	0.997966i	$0.479695\pi$
$349$	−15.5503	+	26.9340i	−0.832390	+	1.44174i	−0.896138	+	0.443776i	$0.853639\pi$
$350$		0			0
$351$		0			0
$352$	−20.6092	+	35.6963i	−1.09848	+	1.90262i
$353$	2.65938			0.141544			0.0707722	−	0.997493i	$-0.477454\pi$
$353$	2.65938			0.141544			0.0707722	+	0.997493i	$0.477454\pi$
$354$		0			0
$355$	2.49457			0.132398
$356$	−7.10833	−	12.3120i	−0.376741	−	0.652534i
$357$		0			0
$358$	20.1770	−	34.9476i	1.06639	−	1.84704i
$359$	−16.2715	+	28.1830i	−0.858775	+	1.48744i	0.0143230	+	0.999897i	$0.495441\pi$
$359$	−16.2715	+	28.1830i	−0.858775	+	1.48744i	−0.873098	+	0.487545i	$0.837893\pi$
$360$		0			0
$361$	−4.97859	−	8.62318i	−0.262031	−	0.453852i
$362$	9.58646	+	16.6042i	0.503853	+	0.872699i
$363$		0			0
$364$		0			0
$365$	−6.69042	−	11.5881i	−0.350192	−	0.606551i
$366$		0			0
$367$	−14.1536			−0.738809			−0.369405	−	0.929269i	$-0.620438\pi$
$367$	−14.1536			−0.738809			−0.369405	+	0.929269i	$0.620438\pi$
$368$	5.13790	+	8.89911i	0.267832	+	0.463898i
$369$		0			0
$370$	−5.29499			−0.275273
$371$		0			0
$372$		0			0
$373$	2.67628			0.138573			0.0692863	−	0.997597i	$-0.477928\pi$
$373$	2.67628			0.138573			0.0692863	+	0.997597i	$0.477928\pi$
$374$	−28.7069	+	49.7217i	−1.48440	+	2.57105i
$375$		0			0
$376$	2.52828	+	4.37911i	0.130386	+	0.225835i
$377$	19.1386			0.985687
$378$		0			0
$379$	−0.312929			−0.0160741			−0.00803705	−	0.999968i	$-0.502558\pi$
$379$	−0.312929			−0.0160741			−0.00803705	+	0.999968i	$0.502558\pi$
$380$	9.33759	+	16.1732i	0.479008	+	0.829667i
$381$		0			0
$382$	−5.34218	+	9.25292i	−0.273330	+	0.473421i
$383$	−8.98880			−0.459306			−0.229653	−	0.973273i	$-0.573759\pi$
$383$	−8.98880			−0.459306			−0.229653	+	0.973273i	$0.573759\pi$
$384$		0			0
$385$		0			0
$386$	21.2573			1.08197
$387$		0			0
$388$	6.80036	+	11.7786i	0.345236	+	0.597966i
$389$	26.9869			1.36829			0.684144	−	0.729347i	$-0.260176\pi$
$389$	26.9869			1.36829			0.684144	+	0.729347i	$0.260176\pi$
$390$		0			0
$391$	12.4657	+	21.5912i	0.630417	+	1.09191i
$392$		0			0
$393$		0			0
$394$	−3.60664	−	6.24689i	−0.181700	−	0.314714i
$395$	5.71772	+	9.90339i	0.287690	+	0.498293i
$396$		0			0
$397$	14.7503	−	25.5482i	0.740295	−	1.28223i	−0.212066	−	0.977255i	$-0.568019\pi$
$397$	14.7503	−	25.5482i	0.740295	−	1.28223i	0.952361	−	0.304973i	$-0.0986475\pi$
$398$	−12.0662	+	20.8992i	−0.604822	+	1.04758i
$399$		0			0
$400$	−3.36628	−	5.83056i	−0.168314	−	0.291528i
$401$	34.2784			1.71178			0.855891	−	0.517156i	$-0.173009\pi$
$401$	34.2784			1.71178			0.855891	+	0.517156i	$0.173009\pi$
$402$		0			0
$403$	6.94808			0.346109
$404$	−0.00730338	+	0.0126498i	−0.000363357	+	0.000629352i
$405$		0			0
$406$		0			0
$407$	−5.25369	+	9.09966i	−0.260416	+	0.451054i
$408$		0			0
$409$	5.49225	−	9.51286i	0.271574	−	0.470381i	−0.697691	−	0.716399i	$-0.745789\pi$
$409$	5.49225	−	9.51286i	0.271574	−	0.470381i	0.969265	+	0.246018i	$0.0791224\pi$
$410$	5.37708	−	9.31338i	0.265555	−	0.459955i
$411$		0			0
$412$	17.8338	−	30.8890i	0.878608	−	1.52179i
$413$		0			0
$414$		0			0
$415$	−6.44320	+	11.1599i	−0.316284	+	0.547820i
$416$	35.7102			1.75083
$417$		0			0
$418$	64.1458			3.13747
$419$	−3.33207	−	5.77132i	−0.162782	−	0.281947i	0.773083	−	0.634305i	$-0.218714\pi$
$419$	−3.33207	−	5.77132i	−0.162782	−	0.281947i	−0.935866	+	0.352357i	$0.885380\pi$
$420$		0			0
$421$	−17.0430	+	29.5193i	−0.830625	+	1.43868i	0.0669186	+	0.997758i	$0.478683\pi$
$421$	−17.0430	+	29.5193i	−0.830625	+	1.43868i	−0.897543	+	0.440926i	$0.854650\pi$
$422$	7.96787	−	13.8008i	0.387870	−	0.671810i
$423$		0			0
$424$	5.72325	+	9.91297i	0.277946	+	0.481416i
$425$	−8.16733	−	14.1462i	−0.396174	−	0.686193i
$426$		0			0
$427$		0			0
$428$	12.8903	+	22.3267i	0.623077	+	1.07920i
$429$		0			0
$430$	−9.16474			−0.441963
$431$	1.12969	+	1.95669i	0.0544155	+	0.0942504i	0.891950	−	0.452134i	$-0.149337\pi$
$431$	1.12969	+	1.95669i	0.0544155	+	0.0942504i	−0.837535	+	0.546384i	$0.816004\pi$
$432$		0			0
$433$	−34.3904			−1.65270			−0.826348	−	0.563160i	$-0.809585\pi$
$433$	−34.3904			−1.65270			−0.826348	+	0.563160i	$0.809585\pi$
$434$		0			0
$435$		0			0
$436$	46.2173			2.21341
$437$	13.9274	−	24.1229i	0.666236	−	1.15395i
$438$		0			0
$439$	2.99569	+	5.18869i	0.142977	+	0.247643i	0.928616	−	0.371042i	$-0.120999\pi$
$439$	2.99569	+	5.18869i	0.142977	+	0.247643i	−0.785640	+	0.618684i	$0.787666\pi$
$440$	−11.1322			−0.530706
$441$		0			0
$442$	49.7411			2.36594
$443$	−19.7190	−	34.1543i	−0.936879	−	1.62272i	−0.771249	−	0.636534i	$-0.780368\pi$
$443$	−19.7190	−	34.1543i	−0.936879	−	1.62272i	−0.165630	−	0.986188i	$-0.552966\pi$
$444$		0			0
$445$	−3.29471	+	5.70661i	−0.156184	+	0.270519i
$446$	−8.82560			−0.417904
$447$		0			0
$448$		0			0
$449$	−2.45092			−0.115666			−0.0578330	−	0.998326i	$-0.518419\pi$
$449$	−2.45092			−0.115666			−0.0578330	+	0.998326i	$0.518419\pi$
$450$		0			0
$451$	−10.6703	−	18.4815i	−0.502444	−	0.870259i
$452$	−16.8182			−0.791060
$453$		0			0
$454$	1.45179	+	2.51457i	0.0681358	+	0.118015i
$455$		0			0
$456$		0			0
$457$	−5.51058	−	9.54461i	−0.257774	−	0.446478i	0.707871	−	0.706342i	$-0.249656\pi$
$457$	−5.51058	−	9.54461i	−0.257774	−	0.446478i	−0.965645	+	0.259864i	$0.916322\pi$
$458$	17.4068	+	30.1495i	0.813368	+	1.40879i
$459$		0			0
$460$	−8.98208	+	15.5574i	−0.418792	+	0.725369i
$461$	14.6540	−	25.3814i	0.682503	−	1.18213i	−0.291711	−	0.956506i	$-0.594225\pi$
$461$	14.6540	−	25.3814i	0.682503	−	1.18213i	0.974215	−	0.225624i	$-0.0724420\pi$
$462$		0			0
$463$	0.593566	+	1.02809i	0.0275853	+	0.0477792i	0.879489	−	0.475920i	$-0.157885\pi$
$463$	0.593566	+	1.02809i	0.0275853	+	0.0477792i	−0.851903	+	0.523699i	$0.824552\pi$
$464$	−8.00639			−0.371687
$465$		0			0
$466$	17.6952			0.819715
$467$	−11.0573	+	19.1519i	−0.511673	+	0.886243i	0.488236	+	0.872712i	$0.337641\pi$
$467$	−11.0573	+	19.1519i	−0.511673	+	0.886243i	−0.999908	+	0.0135313i	$0.995693\pi$
$468$		0			0
$469$		0			0
$470$	4.35483	−	7.54280i	0.200874	−	0.347923i
$471$		0			0
$472$	0.247329	−	0.428387i	0.0113842	−	0.0197181i
$473$	−9.09327	+	15.7500i	−0.418109	+	0.724186i
$474$		0			0
$475$	−9.12499	+	15.8050i	−0.418683	+	0.725181i
$476$		0			0
$477$		0			0
$478$	24.0502	−	41.6562i	1.10003	−	1.90531i
$479$	−25.1428			−1.14880			−0.574402	−	0.818573i	$-0.694766\pi$
$479$	−25.1428			−1.14880			−0.574402	+	0.818573i	$0.694766\pi$
$480$		0			0
$481$	9.10321			0.415071
$482$	−30.0273	−	52.0088i	−1.36771	−	2.36894i
$483$		0			0
$484$	−25.9967	+	45.0276i	−1.18167	+	2.04671i
$485$	3.15197	−	5.45937i	0.143123	−	0.247897i
$486$		0			0
$487$	−6.78904	−	11.7590i	−0.307641	−	0.532849i	0.670205	−	0.742176i	$-0.266206\pi$
$487$	−6.78904	−	11.7590i	−0.307641	−	0.532849i	−0.977846	+	0.209327i	$0.932873\pi$
$488$	−8.28713	−	14.3537i	−0.375141	−	0.649763i
$489$		0			0
$490$		0			0
$491$	−7.25177	−	12.5604i	−0.327268	−	0.566844i	0.654701	−	0.755888i	$-0.272795\pi$
$491$	−7.25177	−	12.5604i	−0.327268	−	0.566844i	−0.981969	+	0.189044i	$0.939461\pi$
$492$		0			0
$493$	−19.4253			−0.874870
$494$	−27.7868	−	48.1281i	−1.25019	−	2.16538i
$495$		0			0
$496$	−2.90664			−0.130512
$497$		0			0
$498$		0			0
$499$	13.9915			0.626345			0.313172	−	0.949696i	$-0.398608\pi$
$499$	13.9915			0.626345			0.313172	+	0.949696i	$0.398608\pi$
$500$	14.5611	−	25.2205i	0.651191	−	1.12790i
$501$		0			0
$502$	−18.0209	−	31.2132i	−0.804314	−	1.39311i
$503$	28.4011			1.26634			0.633171	−	0.774012i	$-0.281753\pi$
$503$	28.4011			1.26634			0.633171	+	0.774012i	$0.281753\pi$
$504$		0			0
$505$	0.00677023			0.000301271
$506$	30.8518	+	53.4369i	1.37153	+	2.37556i
$507$		0			0
$508$	19.0592	−	33.0115i	0.845615	−	1.46465i
$509$	−3.45993			−0.153359			−0.0766794	−	0.997056i	$-0.524432\pi$
$509$	−3.45993			−0.153359			−0.0766794	+	0.997056i	$0.524432\pi$
$510$		0			0
$511$		0			0
$512$	−21.3013			−0.941392
$513$		0			0
$514$	2.24784	+	3.89337i	0.0991480	+	0.171729i
$515$	−16.5319			−0.728484
$516$		0			0
$517$	−8.64174	−	14.9679i	−0.380063	−	0.658289i
$518$		0			0
$519$		0			0
$520$	4.82225	+	8.35239i	0.211470	+	0.366276i
$521$	−3.56797	−	6.17991i	−0.156316	−	0.270747i	0.777222	−	0.629227i	$-0.216628\pi$
$521$	−3.56797	−	6.17991i	−0.156316	−	0.270747i	−0.933537	+	0.358480i	$0.883295\pi$
$522$		0			0
$523$	−6.53235	+	11.3144i	−0.285640	+	0.494743i	−0.972764	−	0.231797i	$-0.925539\pi$
$523$	−6.53235	+	11.3144i	−0.285640	+	0.494743i	0.687124	+	0.726540i	$0.258873\pi$
$524$	0.245114	−	0.424551i	0.0107079	−	0.0185466i
$525$		0			0
$526$	−11.0226	−	19.0917i	−0.480609	−	0.832439i
$527$	−7.05216			−0.307197
$528$		0			0
$529$	3.79420			0.164965
$530$	9.85801	−	17.0746i	0.428205	−	0.741672i
$531$		0			0
$532$		0			0
$533$	−9.24434	+	16.0117i	−0.400417	+	0.693542i
$534$		0			0
$535$	5.97467	−	10.3484i	0.258308	−	0.447402i
$536$	3.58403	−	6.20772i	0.154807	−	0.268133i
$537$		0			0
$538$	−16.4305	+	28.4585i	−0.708371	+	1.22693i
$539$		0			0
$540$		0			0
$541$	−2.46788	+	4.27450i	−0.106103	+	0.183775i	−0.914188	−	0.405290i	$-0.867171\pi$
$541$	−2.46788	+	4.27450i	−0.106103	+	0.183775i	0.808086	+	0.589065i	$0.200504\pi$
$542$	−62.6889			−2.69272
$543$		0			0
$544$	−36.2451			−1.55399
$545$	−10.7109	−	18.5518i	−0.458803	−	0.794670i
$546$		0			0
$547$	0.559964	−	0.969887i	0.0239423	−	0.0414694i	−0.853806	−	0.520591i	$-0.825712\pi$
$547$	0.559964	−	0.969887i	0.0239423	−	0.0414694i	0.877748	+	0.479122i	$0.159045\pi$
$548$	4.31254	−	7.46954i	0.184223	−	0.319083i
$549$		0			0
$550$	−20.2136	−	35.0110i	−0.861912	−	1.49288i
$551$	10.8515	+	18.7953i	0.462289	+	0.800708i
$552$		0			0
$553$		0			0
$554$	−2.93762	−	5.08811i	−0.124808	−	0.216173i
$555$		0			0
$556$	51.5993			2.18830
$557$	−5.47832	−	9.48873i	−0.232124	−	0.402050i	0.726309	−	0.687368i	$-0.241234\pi$
$557$	−5.47832	−	9.48873i	−0.232124	−	0.402050i	−0.958433	+	0.285318i	$0.907901\pi$
$558$		0			0
$559$	15.7561			0.666413
$560$		0			0
$561$		0			0
$562$	10.7210			0.452240
$563$	2.38048	−	4.12311i	0.100325	−	0.173768i	−0.811493	−	0.584361i	$-0.801345\pi$
$563$	2.38048	−	4.12311i	0.100325	−	0.173768i	0.911819	+	0.410593i	$0.134678\pi$
$564$		0			0
$565$	3.89761	+	6.75087i	0.163974	+	0.284011i
$566$	7.79462			0.327632
$567$		0			0
$568$	−3.15189			−0.132250
$569$	1.74988	+	3.03088i	0.0733588	+	0.127061i	0.900371	−	0.435122i	$-0.143295\pi$
$569$	1.74988	+	3.03088i	0.0733588	+	0.127061i	−0.827013	+	0.562183i	$0.809962\pi$
$570$		0			0
$571$	−3.53051	+	6.11501i	−0.147747	+	0.255905i	−0.930394	−	0.366560i	$-0.880535\pi$
$571$	−3.53051	+	6.11501i	−0.147747	+	0.255905i	0.782647	+	0.622465i	$0.213869\pi$
$572$	71.1223			2.97377
$573$		0			0
$574$		0			0
$575$	−17.5552			−0.732101
$576$		0			0
$577$	6.44149	+	11.1570i	0.268163	+	0.464472i	0.968387	−	0.249451i	$-0.0802502\pi$
$577$	6.44149	+	11.1570i	0.268163	+	0.464472i	−0.700225	+	0.713923i	$0.746917\pi$
$578$	−13.4889			−0.561064
$579$		0			0
$580$	−6.99838	−	12.1216i	−0.290592	−	0.503320i
$581$		0			0
$582$		0			0
$583$	−19.5623	−	33.8828i	−0.810186	−	1.40328i
$584$	8.45333	+	14.6416i	0.349801	+	0.605874i
$585$		0			0
$586$	−26.5412	+	45.9707i	−1.09641	+	1.89903i
$587$	−19.5044	+	33.7826i	−0.805034	+	1.39436i	0.111235	+	0.993794i	$0.464519\pi$
$587$	−19.5044	+	33.7826i	−0.805034	+	1.39436i	−0.916268	+	0.400565i	$0.868814\pi$
$588$		0			0
$589$	3.93953	+	6.82347i	0.162326	+	0.281156i
$590$	−0.852024			−0.0350773
$591$		0			0
$592$	−3.80821			−0.156517
$593$	−20.1513	+	34.9031i	−0.827515	+	1.43330i	0.0724676	+	0.997371i	$0.476913\pi$
$593$	−20.1513	+	34.9031i	−0.827515	+	1.43330i	−0.899982	+	0.435927i	$0.856421\pi$
$594$		0			0
$595$		0			0
$596$	−29.1064	+	50.4137i	−1.19224	+	2.06503i
$597$		0			0
$598$	26.7288	−	46.2957i	1.09302	−	1.89317i
$599$	−6.39103	+	11.0696i	−0.261130	+	0.452291i	−0.966543	−	0.256506i	$-0.917429\pi$
$599$	−6.39103	+	11.0696i	−0.261130	+	0.452291i	0.705412	+	0.708797i	$0.250762\pi$
$600$		0			0
$601$	4.86311	−	8.42316i	0.198371	−	0.343588i	−0.749630	−	0.661858i	$-0.769768\pi$
$601$	4.86311	−	8.42316i	0.198371	−	0.343588i	0.948000	+	0.318270i	$0.103102\pi$
$602$		0			0
$603$		0			0
$604$	−8.70921	+	15.0848i	−0.354373	+	0.613792i
$605$	24.0990			0.979762
$606$		0			0
$607$	−41.4873			−1.68392			−0.841959	−	0.539541i	$-0.818598\pi$
$607$	−41.4873			−1.68392			−0.841959	+	0.539541i	$0.818598\pi$
$608$	20.2475	+	35.0697i	0.821145	+	1.42226i
$609$		0			0
$610$	−14.2742	+	24.7236i	−0.577943	+	1.00103i
$611$	−7.48688	+	12.9677i	−0.302887	+	0.524615i
$612$		0			0
$613$	−7.64783	−	13.2464i	−0.308893	−	0.535018i	0.669228	−	0.743057i	$-0.266625\pi$
$613$	−7.64783	−	13.2464i	−0.308893	−	0.535018i	−0.978120	+	0.208039i	$0.933292\pi$
$614$	26.0096	+	45.0500i	1.04966	+	1.81807i
$615$		0			0
$616$		0			0
$617$	2.66563	+	4.61700i	0.107314	+	0.185873i	0.914681	−	0.404176i	$-0.132442\pi$
$617$	2.66563	+	4.61700i	0.107314	+	0.185873i	−0.807367	+	0.590049i	$0.799108\pi$
$618$		0			0
$619$	12.6841			0.509817			0.254908	−	0.966965i	$-0.417955\pi$
$619$	12.6841			0.509817			0.254908	+	0.966965i	$0.417955\pi$
$620$	−2.54070	−	4.40062i	−0.102037	−	0.176733i
$621$		0			0
$622$	28.1650			1.12931
$623$		0			0
$624$		0			0
$625$	3.45909			0.138363
$626$	29.2366	−	50.6393i	1.16853	−	2.02395i
$627$		0			0
$628$	1.90989	+	3.30803i	0.0762129	+	0.132005i
$629$	−9.23957			−0.368406
$630$		0			0
$631$	0.123764			0.00492698			0.00246349	−	0.999997i	$-0.499216\pi$
$631$	0.123764			0.00492698			0.00246349	+	0.999997i	$0.499216\pi$
$632$	−7.22433	−	12.5129i	−0.287369	−	0.497737i
$633$		0			0
$634$	−9.04441	+	15.6654i	−0.359199	+	0.622151i
$635$	−17.6679			−0.701128
$636$		0			0
$637$		0			0
$638$	−48.0763			−1.90336
$639$		0			0
$640$	−7.57868	−	13.1267i	−0.299573	−	0.518876i
$641$	−5.93177			−0.234291			−0.117145	−	0.993115i	$-0.537374\pi$
$641$	−5.93177			−0.234291			−0.117145	+	0.993115i	$0.537374\pi$
$642$		0			0
$643$	23.4140	+	40.5542i	0.923358	+	1.59930i	0.794180	+	0.607682i	$0.207900\pi$
$643$	23.4140	+	40.5542i	0.923358	+	1.59930i	0.129178	+	0.991621i	$0.458766\pi$
$644$		0			0
$645$		0			0
$646$	28.2030	+	48.8490i	1.10963	+	1.92194i
$647$	19.5701	+	33.8964i	0.769379	+	1.33260i	0.937900	+	0.346905i	$0.112767\pi$
$647$	19.5701	+	33.8964i	0.769379	+	1.33260i	−0.168521	+	0.985698i	$0.553899\pi$
$648$		0			0
$649$	−0.845379	+	1.46424i	−0.0331840	+	0.0574764i
$650$	−17.5123	+	30.3322i	−0.686890	+	1.18973i
$651$		0			0
$652$	26.0796	+	45.1711i	1.02135	+	1.76904i
$653$	−43.3281			−1.69556			−0.847779	−	0.530350i	$-0.822061\pi$
$653$	−43.3281			−1.69556			−0.847779	+	0.530350i	$0.822061\pi$
$654$		0			0
$655$	−0.227221			−0.00887827
$656$	3.86726	−	6.69828i	0.150991	−	0.261524i
$657$		0			0
$658$		0			0
$659$	−3.43895	+	5.95643i	−0.133962	+	0.232030i	−0.925201	−	0.379478i	$-0.876103\pi$
$659$	−3.43895	+	5.95643i	−0.133962	+	0.232030i	0.791238	+	0.611508i	$0.209437\pi$
$660$		0			0
$661$	19.3835	−	33.5733i	0.753932	−	1.30585i	−0.191971	−	0.981401i	$-0.561488\pi$
$661$	19.3835	−	33.5733i	0.753932	−	1.30585i	0.945903	−	0.324449i	$-0.105179\pi$
$662$	13.4907	−	23.3666i	0.524331	−	0.908168i
$663$		0			0
$664$	8.14097	−	14.1006i	0.315931	−	0.547209i
$665$		0			0
$666$		0			0
$667$	−10.4383	+	18.0797i	−0.404174	+	0.700050i
$668$	−4.77217			−0.184641
$669$		0			0
$670$	−12.3466			−0.476991
$671$	28.3257	+	49.0615i	1.09350	+	1.89400i
$672$		0			0
$673$	17.9897	−	31.1591i	0.693452	−	1.20109i	−0.277248	−	0.960798i	$-0.589422\pi$
$673$	17.9897	−	31.1591i	0.693452	−	1.20109i	0.970700	−	0.240295i	$-0.0772443\pi$
$674$	28.2025	−	48.8481i	1.08632	−	1.88156i
$675$		0			0
$676$	−13.0227	−	22.5560i	−0.500874	−	0.867539i
$677$	−2.23329	−	3.86817i	−0.0858322	−	0.148666i	0.819913	−	0.572488i	$-0.194022\pi$
$677$	−2.23329	−	3.86817i	−0.0858322	−	0.148666i	−0.905746	+	0.423822i	$0.860688\pi$
$678$		0			0
$679$		0			0
$680$	−4.89449	−	8.47750i	−0.187695	−	0.325097i
$681$		0			0
$682$	−17.4536			−0.668335
$683$	−13.3356	−	23.0980i	−0.510274	−	0.883821i	−0.999929	−	0.0119046i	$-0.996211\pi$
$683$	−13.3356	−	23.0980i	−0.510274	−	0.883821i	0.489655	−	0.871916i	$-0.337123\pi$
$684$		0			0
$685$	−3.99773			−0.152745
$686$		0			0
$687$		0			0
$688$	−6.59138			−0.251294
$689$	−16.9480	+	29.3548i	−0.645668	+	1.11833i
$690$		0			0
$691$	−20.5220	−	35.5452i	−0.780694	−	1.35220i	−0.931538	−	0.363644i	$-0.881532\pi$
$691$	−20.5220	−	35.5452i	−0.780694	−	1.35220i	0.150844	−	0.988558i	$-0.451801\pi$
$692$	−27.5709			−1.04809
$693$		0			0
$694$	−36.6671			−1.39187
$695$	−11.9582	−	20.7121i	−0.453599	−	0.785656i
$696$		0			0
$697$	9.38281	−	16.2515i	0.355399	−	0.615570i
$698$	−67.6847			−2.56190
$699$		0			0
$700$		0			0
$701$	−9.63355			−0.363854			−0.181927	−	0.983312i	$-0.558233\pi$
$701$	−9.63355			−0.363854			−0.181927	+	0.983312i	$0.558233\pi$
$702$		0			0
$703$	5.16148	+	8.93994i	0.194669	+	0.337176i
$704$	−67.9575			−2.56125
$705$		0			0
$706$	2.89382	+	5.01224i	0.108910	+	0.188638i
$707$		0			0
$708$		0			0
$709$	5.07131	+	8.78376i	0.190457	+	0.329881i	0.945402	−	0.325907i	$-0.105670\pi$
$709$	5.07131	+	8.78376i	0.190457	+	0.329881i	−0.754945	+	0.655788i	$0.772336\pi$
$710$	2.71448	+	4.70162i	0.101873	+	0.176449i
$711$		0			0
$712$	4.16286	−	7.21029i	0.156010	−	0.270217i
$713$	−3.78954	+	6.56368i	−0.141919	+	0.245812i
$714$		0			0
$715$	−16.4826	−	28.5487i	−0.616415	−	1.06766i
$716$	50.7380			1.89617
$717$		0			0
$718$	−70.8235			−2.64311
$719$	20.6844	−	35.8264i	0.771397	−	1.33610i	−0.165400	−	0.986227i	$-0.552891\pi$
$719$	20.6844	−	35.8264i	0.771397	−	1.33610i	0.936797	−	0.349873i	$-0.113775\pi$
$720$		0			0
$721$		0			0
$722$	10.8350	−	18.7667i	0.403236	−	0.698425i
$723$		0			0
$724$	−12.0533	+	20.8769i	−0.447957	+	0.775884i
$725$	6.83904	−	11.8456i	0.253996	−	0.439934i
$726$		0			0
$727$	4.86372	−	8.42422i	0.180386	−	0.312437i	−0.761626	−	0.648016i	$-0.775599\pi$
$727$	4.86372	−	8.42422i	0.180386	−	0.312437i	0.942012	+	0.335580i	$0.108932\pi$
$728$		0			0
$729$		0			0
$730$	14.5604	−	25.2194i	0.538906	−	0.933412i
$731$	−15.9921			−0.591491
$732$		0			0
$733$	−28.9108			−1.06784			−0.533922	−	0.845534i	$-0.679282\pi$
$733$	−28.9108			−1.06784			−0.533922	+	0.845534i	$0.679282\pi$
$734$	−15.4013	−	26.6758i	−0.568471	−	0.984621i
$735$		0			0
$736$	−19.4766	+	33.7345i	−0.717918	+	1.24347i
$737$	−12.2503	+	21.2182i	−0.451247	+	0.781582i
$738$		0			0
$739$	6.67467	+	11.5609i	0.245532	+	0.425273i	0.962281	−	0.272058i	$-0.0877041\pi$
$739$	6.67467	+	11.5609i	0.245532	+	0.425273i	−0.716749	+	0.697331i	$0.754371\pi$
$740$	−3.32876	−	5.76558i	−0.122368	−	0.211947i
$741$		0			0
$742$		0			0
$743$	−19.9100	−	34.4851i	−0.730425	−	1.26513i	−0.956702	−	0.291071i	$-0.905988\pi$
$743$	−19.9100	−	34.4851i	−0.730425	−	1.26513i	0.226276	−	0.974063i	$-0.427345\pi$
$744$		0			0
$745$	26.9816			0.988530
$746$	2.91221	+	5.04410i	0.106624	+	0.184678i
$747$		0			0
$748$	−72.1877			−2.63944
$749$		0			0
$750$		0			0
$751$	−38.4345			−1.40250			−0.701248	−	0.712917i	$-0.747374\pi$
$751$	−38.4345			−1.40250			−0.701248	+	0.712917i	$0.747374\pi$
$752$	3.13204	−	5.42486i	0.114214	−	0.197824i
$753$		0			0
$754$	20.8258	+	36.0713i	0.758429	+	1.31364i
$755$	8.07344			0.293823
$756$		0			0
$757$	−5.66698			−0.205970			−0.102985	−	0.994683i	$-0.532839\pi$
$757$	−5.66698			−0.205970			−0.102985	+	0.994683i	$0.532839\pi$
$758$	−0.340516	−	0.589791i	−0.0123681	−	0.0214222i
$759$		0			0
$760$	−5.46839	+	9.47153i	−0.198359	+	0.343569i
$761$	52.3321			1.89704			0.948519	−	0.316719i	$-0.102581\pi$
$761$	52.3321			1.89704			0.948519	+	0.316719i	$0.102581\pi$
$762$		0			0
$763$		0			0
$764$	−13.4337			−0.486014
$765$		0			0
$766$	−9.78121	−	16.9416i	−0.353410	−	0.612123i
$767$	1.46481			0.0528912
$768$		0			0
$769$	−1.17360	−	2.03274i	−0.0423212	−	0.0733025i	0.844089	−	0.536203i	$-0.180142\pi$
$769$	−1.17360	−	2.03274i	−0.0423212	−	0.0733025i	−0.886410	+	0.462901i	$0.846809\pi$
$770$		0			0
$771$		0			0
$772$	13.3637	+	23.1466i	0.480970	+	0.833064i
$773$	−18.1814	−	31.4912i	−0.653941	−	1.13266i	−0.982158	−	0.188057i	$-0.939781\pi$
$773$	−18.1814	−	31.4912i	−0.653941	−	1.13266i	0.328217	−	0.944602i	$-0.393552\pi$
$774$		0			0
$775$	2.48285	−	4.30042i	0.0891866	−	0.154476i
$776$	−3.98251	+	6.89790i	−0.142964	+	0.247620i
$777$		0			0
$778$	29.3659	+	50.8633i	1.05282	+	1.82354i
$779$	−20.9660			−0.751185
$780$		0			0
$781$	10.7733			0.385497
$782$	−27.1292	+	46.9892i	−0.970139	+	1.68033i
$783$		0			0
$784$		0			0
$785$	0.885235	−	1.53327i	0.0315954	−	0.0547248i
$786$		0			0
$787$	−15.8846	+	27.5129i	−0.566224	+	0.980729i	0.430711	+	0.902490i	$0.358263\pi$
$787$	−15.8846	+	27.5129i	−0.566224	+	0.980729i	−0.996935	+	0.0782386i	$0.975070\pi$
$788$	4.53472	−	7.85437i	0.161543	−	0.279800i
$789$		0			0
$790$	−12.4435	+	21.5528i	−0.442721	+	0.766816i
$791$		0			0
$792$		0			0
$793$	24.5403	−	42.5050i	0.871451	−	1.50940i
$794$	64.2024			2.27846
$795$		0			0
$796$	−30.3422			−1.07545

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)

\(f(q)\)	\(=\)	\(q+(1.08816 + 1.88474i) q^{2} +(-1.36816 + 2.36973i) q^{4} +1.26829 q^{5} -1.60248 q^{8} +O(q^{10})\)	\(q+(1.08816 + 1.88474i) q^{2} +(-1.36816 + 2.36973i) q^{4} +1.26829 q^{5} -1.60248 q^{8} +(1.38010 + 2.39040i) q^{10} +5.47733 q^{11} +(-2.37268 - 4.10960i) q^{13} +(0.992580 + 1.71920i) q^{16} +(2.40822 + 4.17116i) q^{17} +(2.69059 - 4.66025i) q^{19} +(-1.73523 + 3.00550i) q^{20} +(5.96019 + 10.3233i) q^{22} +5.17631 q^{23} -3.39144 q^{25} +(5.16368 - 8.94376i) q^{26} +(-2.01656 + 3.49278i) q^{29} +(-0.732093 + 1.26802i) q^{31} +(-3.76264 + 6.51709i) q^{32} +(-5.24103 + 9.07773i) q^{34} +(-0.959170 + 1.66133i) q^{37} +11.7111 q^{38} -2.03241 q^{40} +(-1.94808 - 3.37418i) q^{41} +(-1.66016 + 2.87549i) q^{43} +(-7.49389 + 12.9798i) q^{44} +(5.63263 + 9.75600i) q^{46} +(-1.57773 - 2.73271i) q^{47} +(-3.69042 - 6.39199i) q^{50} +12.9849 q^{52} +(-3.57149 - 6.18601i) q^{53} +6.94684 q^{55} -8.77732 q^{58} +(-0.154341 + 0.267327i) q^{59} +(5.17143 + 8.95719i) q^{61} -3.18652 q^{62} -12.4070 q^{64} +(-3.00924 - 5.21216i) q^{65} +(-2.23655 + 3.87382i) q^{67} -13.1794 q^{68} +1.96688 q^{71} +(-5.27515 - 9.13683i) q^{73} -4.17491 q^{74} +(7.36235 + 12.7520i) q^{76} +(4.50822 + 7.80846i) q^{79} +(1.25888 + 2.18044i) q^{80} +(4.23963 - 7.34326i) q^{82} +(-5.08023 + 8.79921i) q^{83} +(3.05432 + 5.29023i) q^{85} -7.22607 q^{86} -8.77732 q^{88} +(-2.59776 + 4.49945i) q^{89} +(-7.08205 + 12.2665i) q^{92} +(3.43363 - 5.94722i) q^{94} +(3.41245 - 5.91054i) q^{95} +(2.48521 - 4.30451i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24q - 4q^{2} - 12q^{4} + 24q^{8} + O(q^{10}) \)	\( 24q - 4q^{2} - 12q^{4} + 24q^{8} + 40q^{11} - 12q^{16} + 64q^{23} + 24q^{25} - 16q^{29} - 48q^{32} - 12q^{37} - 56q^{44} + 24q^{46} + 4q^{50} - 32q^{53} + 96q^{64} - 60q^{65} - 12q^{67} + 112q^{71} + 136q^{74} + 12q^{79} + 12q^{85} + 152q^{86} - 16q^{92} - 64q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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