LMFDB - Embedded newform 1323.2.g.f.361.4

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	10.0.991381711347.1
Defining polynomial:	$x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9$
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.4
Root			$-0.335166 + 0.580525i$ of defining polynomial
Character	$\chi$	$=$	1323.361
Dual form			1323.2.g.f.667.4

$q$-expansion

$f(q)$	$=$	$q+(0.335166 - 0.580525i) q^{2} +(0.775327 + 1.34291i) q^{4} +1.42494 q^{5} +2.38012 q^{8} +O(q^{10})$	\(q+(0.335166 - 0.580525i) q^{2} +(0.775327 + 1.34291i) q^{4} +1.42494 q^{5} +2.38012 q^{8} +(0.477591 - 0.827212i) q^{10} +4.93077 q^{11} +(1.37730 - 2.38556i) q^{13} +(-0.752918 + 1.30409i) q^{16} +(0.559839 - 0.969670i) q^{17} +(2.00752 + 3.47713i) q^{19} +(1.10479 + 1.91356i) q^{20} +(1.65263 - 2.86244i) q^{22} -5.43661 q^{23} -2.96955 q^{25} +(-0.923251 - 1.59912i) q^{26} +(-3.40555 - 5.89858i) q^{29} +(1.25292 + 2.17012i) q^{31} +(2.88483 + 4.99666i) q^{32} +(-0.375279 - 0.650002i) q^{34} +(0.709787 + 1.22939i) q^{37} +2.69142 q^{38} +3.39152 q^{40} +(0.124384 - 0.215440i) q^{41} +(-0.498313 - 0.863104i) q^{43} +(3.82296 + 6.62156i) q^{44} +(-1.82217 + 3.15609i) q^{46} +(4.73790 - 8.20628i) q^{47} +(-0.995294 + 1.72390i) q^{50} +4.27144 q^{52} +(0.410229 - 0.710537i) q^{53} +7.02604 q^{55} -4.56570 q^{58} +(3.29204 + 5.70197i) q^{59} +(0.0376322 - 0.0651809i) q^{61} +1.67974 q^{62} +0.855913 q^{64} +(1.96257 - 3.39927i) q^{65} +(6.29385 + 10.9013i) q^{67} +1.73623 q^{68} -0.0804951 q^{71} +(-5.34551 + 9.25869i) q^{73} +0.951587 q^{74} +(-3.11297 + 5.39183i) q^{76} +(0.922457 - 1.59774i) q^{79} +(-1.07286 + 1.85825i) q^{80} +(-0.0833788 - 0.144416i) q^{82} +(-7.23583 - 12.5328i) q^{83} +(0.797736 - 1.38172i) q^{85} -0.668072 q^{86} +11.7358 q^{88} +(6.76292 + 11.7137i) q^{89} +(-4.21515 - 7.30085i) q^{92} +(-3.17597 - 5.50094i) q^{94} +(2.86059 + 4.95469i) q^{95} +(-2.70160 - 4.67930i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) $	$ 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.335166	−	0.580525i	0.236998	−	0.410493i	−0.722853	−	0.691002i	$-0.757170\pi$
$2$	0.335166	−	0.580525i	0.236998	−	0.410493i	0.959852	+	0.280508i	$0.0905031\pi$
$3$		0			0
$3$		0			0
$4$	0.775327	+	1.34291i	0.387664	+	0.671453i
$5$	1.42494			0.637251			0.318626	−	0.947881i	$-0.396779\pi$
$5$	1.42494			0.637251			0.318626	+	0.947881i	$0.396779\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	2.38012			0.841499
$9$		0			0
$10$	0.477591	−	0.827212i	0.151028	−	0.261587i
$11$	4.93077			1.48668			0.743342	−	0.668911i	$-0.233239\pi$
$11$	4.93077			1.48668			0.743342	+	0.668911i	$0.233239\pi$
$12$		0			0
$13$	1.37730	−	2.38556i	0.381995	−	0.661635i	−0.609352	−	0.792900i	$-0.708571\pi$
$13$	1.37730	−	2.38556i	0.381995	−	0.661635i	0.991347	+	0.131265i	$0.0419038\pi$
$14$		0			0
$15$		0			0
$16$	−0.752918	+	1.30409i	−0.188230	+	0.326023i
$17$	0.559839	−	0.969670i	0.135781	−	0.235180i	−0.790115	−	0.612959i	$-0.789979\pi$
$17$	0.559839	−	0.969670i	0.135781	−	0.235180i	0.925896	+	0.377780i	$0.123312\pi$
$18$		0			0
$19$	2.00752	+	3.47713i	0.460557	+	0.797709i	0.998989	−	0.0449606i	$-0.0143162\pi$
$19$	2.00752	+	3.47713i	0.460557	+	0.797709i	−0.538431	+	0.842669i	$0.680983\pi$
$20$	1.10479	+	1.91356i	0.247039	+	0.427884i
$21$		0			0
$22$	1.65263	−	2.86244i	0.352342	−	0.610274i
$23$	−5.43661			−1.13361			−0.566806	−	0.823851i	$-0.691821\pi$
$23$	−5.43661			−1.13361			−0.566806	+	0.823851i	$0.691821\pi$
$24$		0			0
$25$	−2.96955			−0.593911
$26$	−0.923251	−	1.59912i	−0.181064	−	0.313613i
$27$		0			0
$28$		0			0
$29$	−3.40555	−	5.89858i	−0.632394	−	1.09534i	−0.987061	−	0.160346i	$-0.948739\pi$
$29$	−3.40555	−	5.89858i	−0.632394	−	1.09534i	0.354667	−	0.934993i	$-0.384594\pi$
$30$		0			0
$31$	1.25292	+	2.17012i	0.225031	+	0.389765i	0.956329	−	0.292294i	$-0.0944184\pi$
$31$	1.25292	+	2.17012i	0.225031	+	0.389765i	−0.731298	+	0.682058i	$0.761085\pi$
$32$	2.88483	+	4.99666i	0.509970	+	0.883294i
$33$		0			0
$34$	−0.375279	−	0.650002i	−0.0643597	−	0.111474i
$35$		0			0
$36$		0			0
$37$	0.709787	+	1.22939i	0.116688	+	0.202110i	0.918453	−	0.395529i	$-0.129439\pi$
$37$	0.709787	+	1.22939i	0.116688	+	0.202110i	−0.801765	+	0.597639i	$0.796106\pi$
$38$	2.69142			0.436605
$39$		0			0
$40$	3.39152			0.536247
$41$	0.124384	−	0.215440i	0.0194256	−	0.0336460i	−0.856149	−	0.516729i	$-0.827150\pi$
$41$	0.124384	−	0.215440i	0.0194256	−	0.0336460i	0.875575	+	0.483083i	$0.160483\pi$
$42$		0			0
$43$	−0.498313	−	0.863104i	−0.0759921	−	0.131622i	0.825525	−	0.564365i	$-0.190879\pi$
$43$	−0.498313	−	0.863104i	−0.0759921	−	0.131622i	−0.901517	+	0.432743i	$0.857546\pi$
$44$	3.82296	+	6.62156i	0.576333	+	0.998238i
$45$		0			0
$46$	−1.82217	+	3.15609i	−0.268664	+	0.465340i
$47$	4.73790	−	8.20628i	0.691093	−	1.19701i	−0.280387	−	0.959887i	$-0.590463\pi$
$47$	4.73790	−	8.20628i	0.691093	−	1.19701i	0.971480	−	0.237122i	$-0.0762040\pi$
$48$		0			0
$49$		0			0
$50$	−0.995294	+	1.72390i	−0.140756	+	0.243796i
$51$		0			0
$52$	4.27144			0.592342
$53$	0.410229	−	0.710537i	0.0563493	−	0.0975998i	−0.836475	−	0.548005i	$-0.815387\pi$
$53$	0.410229	−	0.710537i	0.0563493	−	0.0975998i	0.892824	+	0.450406i	$0.148721\pi$
$54$		0			0
$55$	7.02604			0.947392
$56$		0			0
$57$		0			0
$58$	−4.56570			−0.599506
$59$	3.29204	+	5.70197i	0.428586	+	0.742334i	0.996748	−	0.0805836i	$-0.0256784\pi$
$59$	3.29204	+	5.70197i	0.428586	+	0.742334i	−0.568161	+	0.822917i	$0.692345\pi$
$60$		0			0
$61$	0.0376322	−	0.0651809i	0.00481831	−	0.00834556i	−0.863606	−	0.504167i	$-0.831800\pi$
$61$	0.0376322	−	0.0651809i	0.00481831	−	0.00834556i	0.868425	+	0.495821i	$0.165133\pi$
$62$	1.67974			0.213328
$63$		0			0
$64$	0.855913			0.106989
$65$	1.96257	−	3.39927i	0.243427	−	0.421628i
$66$		0			0
$67$	6.29385	+	10.9013i	0.768916	+	1.33180i	0.938151	+	0.346226i	$0.112537\pi$
$67$	6.29385	+	10.9013i	0.768916	+	1.33180i	−0.169235	+	0.985576i	$0.554130\pi$
$68$	1.73623			0.210549
$69$		0			0
$70$		0			0
$71$	−0.0804951			−0.00955301			−0.00477651	−	0.999989i	$-0.501520\pi$
$71$	−0.0804951			−0.00955301			−0.00477651	+	0.999989i	$0.501520\pi$
$72$		0			0
$73$	−5.34551	+	9.25869i	−0.625644	+	1.08365i	0.362772	+	0.931878i	$0.381830\pi$
$73$	−5.34551	+	9.25869i	−0.625644	+	1.08365i	−0.988416	+	0.151769i	$0.951503\pi$
$74$	0.951587			0.110620
$75$		0			0
$76$	−3.11297	+	5.39183i	−0.357083	+	0.618485i
$77$		0			0
$78$		0			0
$79$	0.922457	−	1.59774i	0.103785	−	0.179760i	−0.809456	−	0.587180i	$-0.800238\pi$
$79$	0.922457	−	1.59774i	0.103785	−	0.179760i	0.913241	+	0.407420i	$0.133571\pi$
$80$	−1.07286	+	1.85825i	−0.119950	+	0.207759i
$81$		0			0
$82$	−0.0833788	−	0.144416i	−0.00920765	−	0.0159481i
$83$	−7.23583	−	12.5328i	−0.794236	−	1.37566i	−0.923323	−	0.384023i	$-0.874538\pi$
$83$	−7.23583	−	12.5328i	−0.794236	−	1.37566i	0.129088	−	0.991633i	$-0.458795\pi$
$84$		0			0
$85$	0.797736	−	1.38172i	0.0865266	−	0.149868i
$86$	−0.668072			−0.0720400
$87$		0			0
$88$	11.7358			1.25104
$89$	6.76292	+	11.7137i	0.716868	+	1.24165i	0.962235	+	0.272222i	$0.0877584\pi$
$89$	6.76292	+	11.7137i	0.716868	+	1.24165i	−0.245366	+	0.969430i	$0.578908\pi$
$90$		0			0
$91$		0			0
$92$	−4.21515	−	7.30085i	−0.439460	−	0.761167i
$93$		0			0
$94$	−3.17597	−	5.50094i	−0.327576	−	0.567378i
$95$	2.86059	+	4.95469i	0.293491	+	0.508341i
$96$		0			0
$97$	−2.70160	−	4.67930i	−0.274306	−	0.475111i	0.695654	−	0.718377i	$-0.255115\pi$
$97$	−2.70160	−	4.67930i	−0.274306	−	0.475111i	−0.969960	+	0.243266i	$0.921781\pi$
$98$		0			0
$99$		0			0
$100$	−2.30238	−	3.98783i	−0.230238	−	0.398783i
$101$	−5.13540			−0.510991			−0.255496	−	0.966810i	$-0.582239\pi$
$101$	−5.13540			−0.510991			−0.255496	+	0.966810i	$0.582239\pi$
$102$		0			0
$103$	14.2112			1.40027			0.700137	−	0.714009i	$-0.253122\pi$
$103$	14.2112			1.40027			0.700137	+	0.714009i	$0.253122\pi$
$104$	3.27814	−	5.67791i	0.321448	−	0.556765i
$105$		0			0
$106$	−0.274990	−	0.476296i	−0.0267094	−	0.0462620i
$107$	−3.83015	−	6.63401i	−0.370274	−	0.641334i	0.619333	−	0.785128i	$-0.287403\pi$
$107$	−3.83015	−	6.63401i	−0.370274	−	0.641334i	−0.989608	+	0.143794i	$0.954070\pi$
$108$		0			0
$109$	−0.849394	+	1.47119i	−0.0813572	+	0.140915i	−0.903833	−	0.427885i	$-0.859259\pi$
$109$	−0.849394	+	1.47119i	−0.0813572	+	0.140915i	0.822476	+	0.568800i	$0.192592\pi$
$110$	2.35489	−	4.07880i	0.224530	−	0.388898i
$111$		0			0
$112$		0			0
$113$	0.300351	−	0.520224i	0.0282547	−	0.0489385i	−0.851552	−	0.524270i	$-0.824338\pi$
$113$	0.300351	−	0.520224i	0.0282547	−	0.0489385i	0.879807	+	0.475331i	$0.157672\pi$
$114$		0			0
$115$	−7.74683			−0.722395
$116$	5.28083	−	9.14666i	0.490312	−	0.849246i
$117$		0			0
$118$	4.41352			0.406297
$119$		0			0
$120$		0			0
$121$	13.3125			1.21023
$122$	−0.0252261	−	0.0436929i	−0.00228386	−	0.00395577i
$123$		0			0
$124$	−1.94284	+	3.36510i	−0.174472	+	0.302195i
$125$	−11.3561			−1.01572
$126$		0			0
$127$	7.25977			0.644200			0.322100	−	0.946706i	$-0.395611\pi$
$127$	7.25977			0.644200			0.322100	+	0.946706i	$0.395611\pi$
$128$	−5.48278	+	9.49645i	−0.484614	+	0.839375i
$129$		0			0
$130$	−1.31557	−	2.27864i	−0.115384	−	0.199850i
$131$	−20.4530			−1.78698			−0.893492	−	0.449079i	$-0.851752\pi$
$131$	−20.4530			−1.78698			−0.893492	+	0.449079i	$0.851752\pi$
$132$		0			0
$133$		0			0
$134$	8.43794			0.728927
$135$		0			0
$136$	1.33248	−	2.30793i	0.114260	−	0.197903i
$137$	−12.2116			−1.04331			−0.521655	−	0.853157i	$-0.674685\pi$
$137$	−12.2116			−1.04331			−0.521655	+	0.853157i	$0.674685\pi$
$138$		0			0
$139$	1.24092	−	2.14933i	0.105253	−	0.182304i	−0.808588	−	0.588375i	$-0.799768\pi$
$139$	1.24092	−	2.14933i	0.105253	−	0.182304i	0.913842	+	0.406071i	$0.133101\pi$
$140$		0			0
$141$		0			0
$142$	−0.0269793	+	0.0467294i	−0.00226405	+	0.00392145i
$143$	6.79117	−	11.7626i	0.567906	−	0.983642i
$144$		0			0
$145$	−4.85269	−	8.40511i	−0.402994	−	0.698006i
$146$	3.58327	+	6.20640i	0.296553	+	0.513645i
$147$		0			0
$148$	−1.10063	+	1.90635i	−0.0904715	+	0.156701i
$149$	8.55593			0.700929			0.350465	−	0.936576i	$-0.386024\pi$
$149$	8.55593			0.700929			0.350465	+	0.936576i	$0.386024\pi$
$150$		0			0
$151$	−17.6592			−1.43709			−0.718544	−	0.695482i	$-0.755191\pi$
$151$	−17.6592			−1.43709			−0.718544	+	0.695482i	$0.755191\pi$
$152$	4.77814	+	8.27599i	0.387559	+	0.671271i
$153$		0			0
$154$		0			0
$155$	1.78533	+	3.09228i	0.143401	+	0.248378i
$156$		0			0
$157$	3.16074	+	5.47457i	0.252255	+	0.436918i	0.964146	−	0.265371i	$-0.0854946\pi$
$157$	3.16074	+	5.47457i	0.252255	+	0.436918i	−0.711891	+	0.702289i	$0.752161\pi$
$158$	−0.618353	−	1.07102i	−0.0491936	−	0.0852057i
$159$		0			0
$160$	4.11070	+	7.11993i	0.324979	+	0.562880i
$161$		0			0
$162$		0			0
$163$	−4.01134	−	6.94784i	−0.314192	−	0.544197i	0.665073	−	0.746778i	$-0.268400\pi$
$163$	−4.01134	−	6.94784i	−0.314192	−	0.544197i	−0.979265	+	0.202581i	$0.935067\pi$
$164$	0.385754			0.0301223
$165$		0			0
$166$	−9.70083			−0.752930
$167$	1.06038	−	1.83663i	0.0820545	−	0.142123i	−0.822078	−	0.569375i	$-0.807185\pi$
$167$	1.06038	−	1.83663i	0.0820545	−	0.142123i	0.904132	+	0.427253i	$0.140518\pi$
$168$		0			0
$169$	2.70608	+	4.68706i	0.208160	+	0.360543i
$170$	−0.534749	−	0.926212i	−0.0410133	−	0.0710372i
$171$		0			0
$172$	0.772712	−	1.33838i	0.0589187	−	0.102050i
$173$	9.14404	−	15.8379i	0.695208	−	1.20414i	−0.274902	−	0.961472i	$-0.588646\pi$
$173$	9.14404	−	15.8379i	0.695208	−	1.20414i	0.970110	−	0.242664i	$-0.0780212\pi$
$174$		0			0
$175$		0			0
$176$	−3.71247	+	6.43018i	−0.279838	+	0.484693i
$177$		0			0
$178$	9.06681			0.679586
$179$	−3.81276	+	6.60389i	−0.284979	+	0.493598i	−0.972604	−	0.232468i	$-0.925320\pi$
$179$	−3.81276	+	6.60389i	−0.284979	+	0.493598i	0.687625	+	0.726066i	$0.258653\pi$
$180$		0			0
$181$	−15.5305			−1.15438			−0.577188	−	0.816611i	$-0.695850\pi$
$181$	−15.5305			−1.15438			−0.577188	+	0.816611i	$0.695850\pi$
$182$		0			0
$183$		0			0
$184$	−12.9398			−0.953933
$185$	1.01140	+	1.75180i	0.0743597	+	0.128795i
$186$		0			0
$187$	2.76044	−	4.78122i	0.201863	−	0.349638i
$188$	14.6937			1.07165
$189$		0			0
$190$	3.83510			0.278227
$191$	7.41624	−	12.8453i	0.536620	−	0.929454i	−0.462463	−	0.886639i	$-0.653034\pi$
$191$	7.41624	−	12.8453i	0.536620	−	0.929454i	0.999083	−	0.0428150i	$-0.0136326\pi$
$192$		0			0
$193$	−8.28387	−	14.3481i	−0.596286	−	1.03280i	−0.993364	−	0.115013i	$-0.963309\pi$
$193$	−8.28387	−	14.3481i	−0.596286	−	1.03280i	0.397078	−	0.917785i	$-0.370024\pi$
$194$	−3.62194			−0.260040
$195$		0			0
$196$		0			0
$197$	4.03740			0.287653			0.143826	−	0.989603i	$-0.454059\pi$
$197$	4.03740			0.287653			0.143826	+	0.989603i	$0.454059\pi$
$198$		0			0
$199$	12.6407	−	21.8943i	0.896076	−	1.55205i	0.0636081	−	0.997975i	$-0.479739\pi$
$199$	12.6407	−	21.8943i	0.896076	−	1.55205i	0.832468	−	0.554074i	$-0.186927\pi$
$200$	−7.06789			−0.499775
$201$		0			0
$202$	−1.72121	+	2.98123i	−0.121104	+	0.209758i
$203$		0			0
$204$		0			0
$205$	0.177240	−	0.306988i	0.0123790	−	0.0214410i
$206$	4.76312	−	8.24997i	0.331862	−	0.574803i
$207$		0			0
$208$	2.07399	+	3.59226i	0.143805	+	0.249078i
$209$	9.89864	+	17.1449i	0.684703	+	1.18594i
$210$		0			0
$211$	−3.76246	+	6.51678i	−0.259019	+	0.448634i	−0.965979	−	0.258619i	$-0.916732\pi$
$211$	−3.76246	+	6.51678i	−0.259019	+	0.448634i	0.706961	+	0.707253i	$0.250066\pi$
$212$	1.27225			0.0873782
$213$		0			0
$214$	−5.13495			−0.351018
$215$	−0.710065	−	1.22987i	−0.0484261	−	0.0838764i
$216$		0			0
$217$		0			0
$218$	0.569377	+	0.986190i	0.0385631	+	0.0667932i
$219$		0			0
$220$	5.44748	+	9.43531i	0.367269	+	0.636129i
$221$	−1.54214	−	2.67106i	−0.103735	−	0.179675i
$222$		0			0
$223$	−6.49230	−	11.2450i	−0.434757	−	0.753020i	0.562519	−	0.826784i	$-0.309832\pi$
$223$	−6.49230	−	11.2450i	−0.434757	−	0.753020i	−0.997276	+	0.0737638i	$0.976499\pi$
$224$		0			0
$225$		0			0
$226$	−0.201335	−	0.348723i	−0.0133926	−	0.0231967i
$227$	−28.9665			−1.92257			−0.961286	−	0.275551i	$-0.911140\pi$
$227$	−28.9665			−1.92257			−0.961286	+	0.275551i	$0.911140\pi$
$228$		0			0
$229$	−15.4358			−1.02003			−0.510013	−	0.860167i	$-0.670360\pi$
$229$	−15.4358			−1.02003			−0.510013	+	0.860167i	$0.670360\pi$
$230$	−2.59648	+	4.49723i	−0.171207	+	0.296538i
$231$		0			0
$232$	−8.10561	−	14.0393i	−0.532159	−	0.921727i
$233$	2.47324	+	4.28378i	0.162027	+	0.280640i	0.935596	−	0.353073i	$-0.114863\pi$
$233$	2.47324	+	4.28378i	0.162027	+	0.280640i	−0.773568	+	0.633713i	$0.781530\pi$
$234$		0			0
$235$	6.75121	−	11.6934i	0.440400	−	0.762795i
$236$	−5.10481	+	8.84179i	−0.332295	+	0.575551i
$237$		0			0
$238$		0			0
$239$	−6.51732	+	11.2883i	−0.421571	+	0.730182i	−0.996093	−	0.0883069i	$-0.971854\pi$
$239$	−6.51732	+	11.2883i	−0.421571	+	0.730182i	0.574523	+	0.818489i	$0.305188\pi$
$240$		0			0
$241$	−14.5825			−0.939339			−0.469670	−	0.882842i	$-0.655627\pi$
$241$	−14.5825			−0.939339			−0.469670	+	0.882842i	$0.655627\pi$
$242$	4.46191	−	7.72826i	0.286823	−	0.496791i
$243$		0			0
$244$	0.116709			0.00747154
$245$		0			0
$246$		0			0
$247$	11.0599			0.703722
$248$	2.98209	+	5.16514i	0.189363	+	0.327987i
$249$		0			0
$250$	−3.80619	+	6.59251i	−0.240724	+	0.416947i
$251$	−14.0715			−0.888187			−0.444094	−	0.895980i	$-0.646474\pi$
$251$	−14.0715			−0.888187			−0.444094	+	0.895980i	$0.646474\pi$
$252$		0			0
$253$	−26.8067			−1.68532
$254$	2.43323	−	4.21448i	0.152674	−	0.264440i
$255$		0			0
$256$	4.53120	+	7.84826i	0.283200	+	0.490517i
$257$	−8.36215			−0.521617			−0.260808	−	0.965391i	$-0.583989\pi$
$257$	−8.36215			−0.521617			−0.260808	+	0.965391i	$0.583989\pi$
$258$		0			0
$259$		0			0
$260$	6.08653			0.377471
$261$		0			0
$262$	−6.85515	+	11.8735i	−0.423512	+	0.733545i
$263$	−3.27066			−0.201678			−0.100839	−	0.994903i	$-0.532153\pi$
$263$	−3.27066			−0.201678			−0.100839	+	0.994903i	$0.532153\pi$
$264$		0			0
$265$	0.584551	−	1.01247i	0.0359087	−	0.0621956i
$266$		0			0
$267$		0			0
$268$	−9.75958	+	16.9041i	−0.596161	+	1.03258i
$269$	−7.69349	+	13.3255i	−0.469081	+	0.812471i	−0.999375	−	0.0353420i	$-0.988748\pi$
$269$	−7.69349	+	13.3255i	−0.469081	+	0.812471i	0.530295	+	0.847813i	$0.322081\pi$
$270$		0			0
$271$	−4.06308	−	7.03747i	−0.246815	−	0.427496i	0.715825	−	0.698279i	$-0.246051\pi$
$271$	−4.06308	−	7.03747i	−0.246815	−	0.427496i	−0.962640	+	0.270783i	$0.912717\pi$
$272$	0.843026	+	1.46016i	0.0511160	+	0.0885355i
$273$		0			0
$274$	−4.09293	+	7.08915i	−0.247263	+	0.428271i
$275$	−14.6422			−0.882958
$276$		0			0
$277$	12.8457			0.771826			0.385913	−	0.922535i	$-0.373887\pi$
$277$	12.8457			0.771826			0.385913	+	0.922535i	$0.373887\pi$
$278$	−0.831826	−	1.44077i	−0.0498896	−	0.0864114i
$279$		0			0
$280$		0			0
$281$	0.724081	+	1.25415i	0.0431951	+	0.0748161i	0.886815	−	0.462125i	$-0.152913\pi$
$281$	0.724081	+	1.25415i	0.0431951	+	0.0748161i	−0.843620	+	0.536941i	$0.819580\pi$
$282$		0			0
$283$	−8.71926	−	15.1022i	−0.518306	−	0.897732i	−0.999774	−	0.0212686i	$-0.993229\pi$
$283$	−8.71926	−	15.1022i	−0.518306	−	0.897732i	0.481468	−	0.876464i	$-0.340104\pi$
$284$	−0.0624100	−	0.108097i	−0.00370335	−	0.00641440i
$285$		0			0
$286$	−4.55234	−	7.88489i	−0.269186	−	0.466243i
$287$		0			0
$288$		0			0
$289$	7.87316	+	13.6367i	0.463127	+	0.802160i
$290$	−6.50584			−0.382036
$291$		0			0
$292$	−16.5781			−0.970158
$293$	−0.900048	+	1.55893i	−0.0525814	+	0.0910736i	−0.891118	−	0.453772i	$-0.850078\pi$
$293$	−0.900048	+	1.55893i	−0.0525814	+	0.0910736i	0.838537	+	0.544845i	$0.183412\pi$
$294$		0			0
$295$	4.69094	+	8.12495i	0.273117	+	0.473053i
$296$	1.68938	+	2.92609i	0.0981931	+	0.170075i
$297$		0			0
$298$	2.86766	−	4.96693i	0.166119	−	0.287727i
$299$	−7.48786	+	12.9693i	−0.433034	+	0.750037i
$300$		0			0
$301$		0			0
$302$	−5.91878	+	10.2516i	−0.340588	+	0.589915i
$303$		0			0
$304$	−6.04600			−0.346762
$305$	0.0536236	−	0.0928787i	0.00307048	−	0.00531822i
$306$		0			0
$307$	−1.06478			−0.0607699			−0.0303850	−	0.999538i	$-0.509673\pi$
$307$	−1.06478			−0.0607699			−0.0303850	+	0.999538i	$0.509673\pi$
$308$		0			0
$309$		0			0
$310$	2.39353			0.135943
$311$	8.46463	+	14.6612i	0.479985	+	0.831359i	0.999736	−	0.0229591i	$-0.00730874\pi$
$311$	8.46463	+	14.6612i	0.479985	+	0.831359i	−0.519751	+	0.854318i	$0.673975\pi$
$312$		0			0
$313$	−4.13928	+	7.16944i	−0.233966	+	0.405241i	−0.958972	−	0.283502i	$-0.908504\pi$
$313$	−4.13928	+	7.16944i	−0.233966	+	0.405241i	0.725006	+	0.688743i	$0.241837\pi$
$314$	4.23750			0.239136
$315$		0			0
$316$	2.86082			0.160934
$317$	3.27371	−	5.67023i	0.183870	−	0.318472i	−0.759325	−	0.650711i	$-0.774471\pi$
$317$	3.27371	−	5.67023i	0.183870	−	0.318472i	0.943195	+	0.332239i	$0.107804\pi$
$318$		0			0
$319$	−16.7920	−	29.0846i	−0.940171	−	1.62842i
$320$	1.21962			0.0681790
$321$		0			0
$322$		0			0
$323$	4.49556			0.250140
$324$		0			0
$325$	−4.08997	+	7.08404i	−0.226871	+	0.392952i
$326$	−5.37786			−0.297852
$327$		0			0
$328$	0.296049	−	0.512773i	0.0163466	−	0.0283131i
$329$		0			0
$330$		0			0
$331$	13.3629	−	23.1453i	0.734493	−	1.27218i	−0.220453	−	0.975398i	$-0.570754\pi$
$331$	13.3629	−	23.1453i	0.734493	−	1.27218i	0.954946	−	0.296781i	$-0.0959131\pi$
$332$	11.2203	−	19.4341i	0.615792	−	1.06658i
$333$		0			0
$334$	−0.710806	−	1.23115i	−0.0388936	−	0.0673657i
$335$	8.96834	+	15.5336i	0.489993	+	0.848692i
$336$		0			0
$337$	−4.76164	+	8.24740i	−0.259383	+	0.449264i	−0.966077	−	0.258255i	$-0.916853\pi$
$337$	−4.76164	+	8.24740i	−0.259383	+	0.449264i	0.706694	+	0.707520i	$0.250186\pi$
$338$	3.62794			0.197334
$339$		0			0
$340$	2.47403			0.134173
$341$	6.17786	+	10.7004i	0.334550	+	0.579457i
$342$		0			0
$343$		0			0
$344$	−1.18605	−	2.05429i	−0.0639473	−	0.110760i
$345$		0			0
$346$	−6.12955	−	10.6167i	−0.329526	−	0.570757i
$347$	−9.35156	−	16.1974i	−0.502018	−	0.869521i	−0.999997	−	0.00233189i	$-0.999258\pi$
$347$	−9.35156	−	16.1974i	−0.502018	−	0.869521i	0.497979	−	0.867189i	$-0.334076\pi$
$348$		0			0
$349$	15.0542	+	26.0747i	0.805834	+	1.39574i	0.915727	+	0.401801i	$0.131616\pi$
$349$	15.0542	+	26.0747i	0.805834	+	1.39574i	−0.109893	+	0.993943i	$0.535051\pi$
$350$		0			0
$351$		0			0
$352$	14.2244	+	24.6374i	0.758164	+	1.31318i
$353$	6.25933			0.333150			0.166575	−	0.986029i	$-0.446729\pi$
$353$	6.25933			0.333150			0.166575	+	0.986029i	$0.446729\pi$
$354$		0			0
$355$	−0.114700			−0.00608767
$356$	−10.4870	+	18.1639i	−0.555807	+	0.962686i
$357$		0			0
$358$	2.55582	+	4.42680i	0.135079	+	0.233964i
$359$	5.09755	+	8.82921i	0.269038	+	0.465988i	0.968614	−	0.248571i	$-0.0799608\pi$
$359$	5.09755	+	8.82921i	0.269038	+	0.465988i	−0.699575	+	0.714559i	$0.746628\pi$
$360$		0			0
$361$	1.43970	−	2.49364i	0.0757739	−	0.131244i
$362$	−5.20532	+	9.01587i	−0.273585	+	0.473864i
$363$		0			0
$364$		0			0
$365$	−7.61701	+	13.1931i	−0.398693	+	0.690556i
$366$		0			0
$367$	28.6557			1.49581			0.747906	−	0.663804i	$-0.231059\pi$
$367$	28.6557			1.49581			0.747906	+	0.663804i	$0.231059\pi$
$368$	4.09332	−	7.08984i	0.213379	−	0.369584i
$369$		0			0
$370$	1.35595			0.0704926
$371$		0			0
$372$		0			0
$373$	−16.0734			−0.832249			−0.416124	−	0.909308i	$-0.636612\pi$
$373$	−16.0734			−0.832249			−0.416124	+	0.909308i	$0.636612\pi$
$374$	−1.85041	−	3.20501i	−0.0956826	−	0.165727i
$375$		0			0
$376$	11.2768	−	19.5319i	0.581555	−	1.00728i
$377$	−18.7619			−0.966286
$378$		0			0
$379$	−1.01893			−0.0523388			−0.0261694	−	0.999658i	$-0.508331\pi$
$379$	−1.01893			−0.0523388			−0.0261694	+	0.999658i	$0.508331\pi$
$380$	−4.43579	+	7.68302i	−0.227551	+	0.394130i
$381$		0			0
$382$	−4.97135	−	8.61063i	−0.254356	−	0.440558i
$383$	−11.5865			−0.592044			−0.296022	−	0.955181i	$-0.595660\pi$
$383$	−11.5865			−0.592044			−0.296022	+	0.955181i	$0.595660\pi$
$384$		0			0
$385$		0			0
$386$	−11.1059			−0.565275
$387$		0			0
$388$	4.18924	−	7.25598i	0.212677	−	0.368367i
$389$	−17.8135			−0.903181			−0.451590	−	0.892225i	$-0.649143\pi$
$389$	−17.8135			−0.903181			−0.451590	+	0.892225i	$0.649143\pi$
$390$		0			0
$391$	−3.04363	+	5.27172i	−0.153923	+	0.266602i
$392$		0			0
$393$		0			0
$394$	1.35320	−	2.34381i	0.0681732	−	0.118079i
$395$	1.31444	−	2.27668i	0.0661369	−	0.114552i
$396$		0			0
$397$	6.54229	+	11.3316i	0.328348	+	0.568715i	0.982184	−	0.187921i	$-0.0601748\pi$
$397$	6.54229	+	11.3316i	0.328348	+	0.568715i	−0.653836	+	0.756636i	$0.726841\pi$
$398$	−8.47348	−	14.6765i	−0.424737	−	0.735666i
$399$		0			0
$400$	2.23583	−	3.87257i	0.111792	−	0.193629i
$401$	−14.1033			−0.704285			−0.352143	−	0.935946i	$-0.614547\pi$
$401$	−14.1033			−0.704285			−0.352143	+	0.935946i	$0.614547\pi$
$402$		0			0
$403$	6.90259			0.343842
$404$	−3.98161	−	6.89636i	−0.198093	−	0.343107i
$405$		0			0
$406$		0			0
$407$	3.49980	+	6.06183i	0.173479	+	0.300474i
$408$		0			0
$409$	−1.32300	−	2.29150i	−0.0654179	−	0.113307i	0.831461	−	0.555583i	$-0.187505\pi$
$409$	−1.32300	−	2.29150i	−0.0654179	−	0.113307i	−0.896879	+	0.442275i	$0.854171\pi$
$410$	−0.118810	−	0.205784i	−0.00586759	−	0.0101630i
$411$		0			0
$412$	11.0183	+	19.0843i	0.542835	+	0.940217i
$413$		0			0
$414$		0			0
$415$	−10.3106	−	17.8585i	−0.506128	−	0.876639i
$416$	15.8931			0.779224
$417$		0			0
$418$	13.2708			0.649094
$419$	16.7567	−	29.0235i	0.818619	−	1.41789i	−0.0880816	−	0.996113i	$-0.528074\pi$
$419$	16.7567	−	29.0235i	0.818619	−	1.41789i	0.906700	−	0.421776i	$-0.138593\pi$
$420$		0			0
$421$	−2.41950	−	4.19071i	−0.117919	−	0.204242i	0.801024	−	0.598633i	$-0.204289\pi$
$421$	−2.41950	−	4.19071i	−0.117919	−	0.204242i	−0.918943	+	0.394390i	$0.870956\pi$
$422$	2.52210	+	4.36841i	0.122774	+	0.212651i
$423$		0			0
$424$	0.976394	−	1.69116i	0.0474179	−	0.0821302i
$425$	−1.66247	+	2.87949i	−0.0806418	+	0.139676i
$426$		0			0
$427$		0			0
$428$	5.93923	−	10.2871i	0.287084	−	0.497244i
$429$		0			0
$430$	−0.951960			−0.0459076
$431$	−17.6643	+	30.5954i	−0.850858	+	1.47373i	0.0295774	+	0.999562i	$0.490584\pi$
$431$	−17.6643	+	30.5954i	−0.850858	+	1.47373i	−0.880435	+	0.474166i	$0.842749\pi$
$432$		0			0
$433$	−5.47404			−0.263066			−0.131533	−	0.991312i	$-0.541990\pi$
$433$	−5.47404			−0.263066			−0.131533	+	0.991312i	$0.541990\pi$
$434$		0			0
$435$		0			0
$436$	−2.63423			−0.126157
$437$	−10.9141	−	18.9038i	−0.522093	−	0.904292i
$438$		0			0
$439$	3.19906	−	5.54093i	0.152683	−	0.264454i	−0.779530	−	0.626365i	$-0.784542\pi$
$439$	3.19906	−	5.54093i	0.152683	−	0.264454i	0.932213	+	0.361911i	$0.117875\pi$
$440$	16.7228			0.797229
$441$		0			0
$442$	−2.06749			−0.0983404
$443$	−3.19341	+	5.53115i	−0.151723	+	0.262793i	−0.931861	−	0.362815i	$-0.881816\pi$
$443$	−3.19341	+	5.53115i	−0.151723	+	0.262793i	0.780138	+	0.625608i	$0.215149\pi$
$444$		0			0
$445$	9.63674	+	16.6913i	0.456825	+	0.791245i
$446$	−8.70400			−0.412146
$447$		0			0
$448$		0			0
$449$	11.7460			0.554327			0.277163	−	0.960823i	$-0.410606\pi$
$449$	11.7460			0.554327			0.277163	+	0.960823i	$0.410606\pi$
$450$		0			0
$451$	0.613311	−	1.06229i	0.0288797	−	0.0500210i
$452$	0.931482			0.0438132
$453$		0			0
$454$	−9.70859	+	16.8158i	−0.455647	+	0.789203i
$455$		0			0
$456$		0			0
$457$	−5.26120	+	9.11266i	−0.246108	+	0.426272i	−0.962443	−	0.271485i	$-0.912485\pi$
$457$	−5.26120	+	9.11266i	−0.246108	+	0.426272i	0.716334	+	0.697757i	$0.245819\pi$
$458$	−5.17356	+	8.96087i	−0.241745	+	0.418714i
$459$		0			0
$460$	−6.00633	−	10.4033i	−0.280046	−	0.485055i
$461$	−3.54278	−	6.13627i	−0.165004	−	0.285794i	0.771653	−	0.636044i	$-0.219430\pi$
$461$	−3.54278	−	6.13627i	−0.165004	−	0.285794i	−0.936657	+	0.350249i	$0.886097\pi$
$462$		0			0
$463$	16.3760	−	28.3641i	0.761059	−	1.31819i	−0.181246	−	0.983438i	$-0.558013\pi$
$463$	16.3760	−	28.3641i	0.761059	−	1.31819i	0.942305	−	0.334755i	$-0.108654\pi$
$464$	10.2564			0.476141
$465$		0			0
$466$	3.31579			0.153601
$467$	1.96216	+	3.39856i	0.0907978	+	0.157266i	0.907847	−	0.419301i	$-0.137725\pi$
$467$	1.96216	+	3.39856i	0.0907978	+	0.157266i	−0.817049	+	0.576568i	$0.804392\pi$
$468$		0			0
$469$		0			0
$470$	−4.52555	−	7.83849i	−0.208748	−	0.361563i
$471$		0			0
$472$	7.83544	+	13.5714i	0.360655	+	0.624673i
$473$	−2.45707	−	4.25577i	−0.112976	−	0.195681i
$474$		0			0
$475$	−5.96145	−	10.3255i	−0.273530	−	0.473768i
$476$		0			0
$477$		0			0
$478$	4.36878	+	7.56694i	0.199823	+	0.346104i
$479$	16.0865			0.735010			0.367505	−	0.930022i	$-0.380212\pi$
$479$	16.0865			0.735010			0.367505	+	0.930022i	$0.380212\pi$
$480$		0			0
$481$	3.91036			0.178297
$482$	−4.88755	+	8.46549i	−0.222622	+	0.385592i
$483$		0			0
$484$	10.3216	+	17.8775i	0.469162	+	0.812612i
$485$	−3.84961	−	6.66771i	−0.174802	−	0.302765i
$486$		0			0
$487$	−1.75172	+	3.03407i	−0.0793781	+	0.137487i	−0.902982	−	0.429679i	$-0.858627\pi$
$487$	−1.75172	+	3.03407i	−0.0793781	+	0.137487i	0.823604	+	0.567166i	$0.191960\pi$
$488$	0.0895692	−	0.155138i	0.00405461	−	0.00702279i
$489$		0			0
$490$		0			0
$491$	20.5546	−	35.6017i	0.927618	−	1.60668i	0.140321	−	0.990106i	$-0.455186\pi$
$491$	20.5546	−	35.6017i	0.927618	−	1.60668i	0.787296	−	0.616575i	$-0.211480\pi$
$492$		0			0
$493$	−7.62624			−0.343468
$494$	3.70689	−	6.42053i	0.166781	−	0.288873i
$495$		0			0
$496$	−3.77338			−0.169430
$497$		0			0
$498$		0			0
$499$	11.8297			0.529571			0.264785	−	0.964307i	$-0.414699\pi$
$499$	11.8297			0.529571			0.264785	+	0.964307i	$0.414699\pi$
$500$	−8.80470	−	15.2502i	−0.393758	−	0.682009i
$501$		0			0
$502$	−4.71631	+	8.16888i	−0.210499	+	0.364595i
$503$	21.8595			0.974665			0.487332	−	0.873217i	$-0.337970\pi$
$503$	21.8595			0.974665			0.487332	+	0.873217i	$0.337970\pi$
$504$		0			0
$505$	−7.31762			−0.325630
$506$	−8.98470	+	15.5620i	−0.399419	+	0.691813i
$507$		0			0
$508$	5.62869	+	9.74918i	0.249733	+	0.432550i
$509$	16.8966			0.748930			0.374465	−	0.927241i	$-0.377826\pi$
$509$	16.8966			0.748930			0.374465	+	0.927241i	$0.377826\pi$
$510$		0			0
$511$		0			0
$512$	−15.8563			−0.700756
$513$		0			0
$514$	−2.80271	+	4.85444i	−0.123622	+	0.214120i
$515$	20.2501			0.892326
$516$		0			0
$517$	23.3615	−	40.4633i	1.02744	−	1.77957i
$518$		0			0
$519$		0			0
$520$	4.67115	−	8.09067i	0.204843	−	0.354799i
$521$	−17.2466	+	29.8720i	−0.755587	+	1.30872i	0.189495	+	0.981882i	$0.439315\pi$
$521$	−17.2466	+	29.8720i	−0.755587	+	1.30872i	−0.945082	+	0.326834i	$0.894018\pi$
$522$		0			0
$523$	−0.995615	−	1.72445i	−0.0435352	−	0.0754051i	0.843437	−	0.537229i	$-0.180529\pi$
$523$	−0.995615	−	1.72445i	−0.0435352	−	0.0754051i	−0.886972	+	0.461823i	$0.847195\pi$
$524$	−15.8577	−	27.4664i	−0.692749	−	1.19988i
$525$		0			0
$526$	−1.09622	+	1.89870i	−0.0477972	+	0.0827873i
$527$	2.80573			0.122220
$528$		0			0
$529$	6.55673			0.285075
$530$	−0.391843	−	0.678693i	−0.0170206	−	0.0294805i
$531$		0			0
$532$		0			0
$533$	−0.342629	−	0.593452i	−0.0148409	−	0.0257052i
$534$		0			0
$535$	−5.45772	−	9.45305i	−0.235958	−	0.408691i
$536$	14.9801	+	25.9463i	0.647042	+	1.12071i
$537$		0			0
$538$	5.15720	+	8.93253i	0.222343	+	0.385109i
$539$		0			0
$540$		0			0
$541$	−15.0681	−	26.0988i	−0.647830	−	1.12207i	−0.983640	−	0.180145i	$-0.942343\pi$
$541$	−15.0681	−	26.0988i	−0.647830	−	1.12207i	0.335810	−	0.941930i	$-0.390990\pi$
$542$	−5.44724			−0.233979
$543$		0			0
$544$	6.46015			0.276977
$545$	−1.21033	+	2.09636i	−0.0518450	+	0.0897982i
$546$		0			0
$547$	7.68070	+	13.3034i	0.328403	+	0.568810i	0.982195	−	0.187864i	$-0.0601563\pi$
$547$	7.68070	+	13.3034i	0.328403	+	0.568810i	−0.653792	+	0.756674i	$0.726823\pi$
$548$	−9.46800	−	16.3991i	−0.404453	−	0.700533i
$549$		0			0
$550$	−4.90757	+	8.50016i	−0.209260	+	0.362448i
$551$	13.6734	−	23.6831i	0.582508	−	1.00893i
$552$		0			0
$553$		0			0
$554$	4.30546	−	7.45728i	0.182921	−	0.316829i
$555$		0			0
$556$	3.84846			0.163211
$557$	11.6412	−	20.1631i	0.493252	−	0.854338i	−0.506718	−	0.862112i	$-0.669141\pi$
$557$	11.6412	−	20.1631i	0.493252	−	0.854338i	0.999970	+	0.00777438i	$0.00247469\pi$
$558$		0			0
$559$	−2.74531			−0.116114
$560$		0			0
$561$		0			0
$562$	0.970751			0.0409487
$563$	−2.27942	−	3.94808i	−0.0960663	−	0.166392i	0.813987	−	0.580883i	$-0.197293\pi$
$563$	−2.27942	−	3.94808i	−0.0960663	−	0.166392i	−0.910053	+	0.414492i	$0.863959\pi$
$564$		0			0
$565$	0.427982	−	0.741286i	0.0180053	−	0.0311861i
$566$	−11.6896			−0.491351
$567$		0			0
$568$	−0.191588			−0.00803885
$569$	9.09976	−	15.7612i	0.381482	−	0.660746i	−0.609793	−	0.792561i	$-0.708747\pi$
$569$	9.09976	−	15.7612i	0.381482	−	0.660746i	0.991274	+	0.131815i	$0.0420806\pi$
$570$		0			0
$571$	8.52275	+	14.7618i	0.356666	+	0.617763i	0.987402	−	0.158234i	$-0.0505801\pi$
$571$	8.52275	+	14.7618i	0.356666	+	0.617763i	−0.630736	+	0.775998i	$0.717247\pi$
$572$	21.0615			0.880625
$573$		0			0
$574$		0			0
$575$	16.1443			0.673264
$576$		0			0
$577$	5.70473	−	9.88088i	0.237491	−	0.411346i	−0.722503	−	0.691368i	$-0.757008\pi$
$577$	5.70473	−	9.88088i	0.237491	−	0.411346i	0.959994	+	0.280022i	$0.0903417\pi$
$578$	10.5553			0.439041
$579$		0			0
$580$	7.52485	−	13.0334i	0.312452	−	0.541183i
$581$		0			0
$582$		0			0
$583$	2.02275	−	3.50350i	0.0837736	−	0.145100i
$584$	−12.7229	+	22.0368i	−0.526479	+	0.911889i
$585$		0			0
$586$	0.603332	+	1.04500i	0.0249234	+	0.0431686i
$587$	2.52544	+	4.37420i	0.104236	+	0.180543i	0.913426	−	0.407005i	$-0.133427\pi$
$587$	2.52544	+	4.37420i	0.104236	+	0.180543i	−0.809190	+	0.587548i	$0.800094\pi$
$588$		0			0
$589$	−5.03052	+	8.71312i	−0.207279	+	0.359018i
$590$	6.28899			0.258913
$591$		0			0
$592$	−2.13765			−0.0878567
$593$	−9.98892	−	17.3013i	−0.410196	−	0.710480i	0.584715	−	0.811239i	$-0.301206\pi$
$593$	−9.98892	−	17.3013i	−0.410196	−	0.710480i	−0.994911	+	0.100759i	$0.967873\pi$
$594$		0			0
$595$		0			0
$596$	6.63365	+	11.4898i	0.271725	+	0.470641i
$597$		0			0
$598$	5.01935	+	8.69378i	0.205257	+	0.355515i
$599$	2.19660	+	3.80463i	0.0897508	+	0.155453i	0.907406	−	0.420256i	$-0.138060\pi$
$599$	2.19660	+	3.80463i	0.0897508	+	0.155453i	−0.817655	+	0.575709i	$0.804726\pi$
$600$		0			0
$601$	−12.1778	−	21.0926i	−0.496743	−	0.860385i	0.503250	−	0.864141i	$-0.332138\pi$
$601$	−12.1778	−	21.0926i	−0.496743	−	0.860385i	−0.999993	+	0.00375637i	$0.998804\pi$
$602$		0			0
$603$		0			0
$604$	−13.6917	−	23.7147i	−0.557107	−	0.964937i
$605$	18.9695			0.771221
$606$		0			0
$607$	−13.1256			−0.532752			−0.266376	−	0.963869i	$-0.585826\pi$
$607$	−13.1256			−0.532752			−0.266376	+	0.963869i	$0.585826\pi$
$608$	−11.5827	+	20.0618i	−0.469741	+	0.813615i
$609$		0			0
$610$	−0.0359456	−	0.0622597i	−0.00145540	−	0.00252082i
$611$	−13.0510	−	22.6051i	−0.527988	−	0.914502i
$612$		0			0
$613$	−23.2403	+	40.2534i	−0.938667	+	1.62582i	−0.170707	+	0.985322i	$0.554605\pi$
$613$	−23.2403	+	40.2534i	−0.938667	+	1.62582i	−0.767960	+	0.640497i	$0.778728\pi$
$614$	−0.356877	+	0.618129i	−0.0144024	+	0.0249456i
$615$		0			0
$616$		0			0
$617$	−14.1948	+	24.5862i	−0.571463	+	0.989803i	0.424953	+	0.905215i	$0.360291\pi$
$617$	−14.1948	+	24.5862i	−0.571463	+	0.989803i	−0.996416	+	0.0845873i	$0.973043\pi$
$618$		0			0
$619$	−31.9212			−1.28302			−0.641511	−	0.767114i	$-0.721692\pi$
$619$	−31.9212			−1.28302			−0.641511	+	0.767114i	$0.721692\pi$
$620$	−2.76843	+	4.79506i	−0.111183	+	0.192574i
$621$		0			0
$622$	11.3482			0.455023
$623$		0			0
$624$		0			0
$625$	−1.33399			−0.0533594
$626$	2.77469	+	4.80591i	0.110899	+	0.192083i
$627$		0			0
$628$	−4.90122	+	8.48916i	−0.195580	+	0.338754i
$629$	1.58947			0.0633762
$630$		0			0
$631$	38.7184			1.54135			0.770677	−	0.637226i	$-0.219918\pi$
$631$	38.7184			1.54135			0.770677	+	0.637226i	$0.219918\pi$
$632$	2.19556	−	3.80282i	0.0873346	−	0.151268i
$633$		0			0
$634$	−2.19447	−	3.80094i	−0.0871537	−	0.150955i
$635$	10.3447			0.410517
$636$		0			0
$637$		0			0
$638$	−22.5124			−0.891276
$639$		0			0
$640$	−7.81261	+	13.5318i	−0.308821	+	0.534893i
$641$	40.4001			1.59571			0.797854	−	0.602851i	$-0.205968\pi$
$641$	40.4001			1.59571			0.797854	+	0.602851i	$0.205968\pi$
$642$		0			0
$643$	−6.27355	+	10.8661i	−0.247405	+	0.428517i	−0.962805	−	0.270198i	$-0.912911\pi$
$643$	−6.27355	+	10.8661i	−0.247405	+	0.428517i	0.715400	+	0.698715i	$0.246244\pi$
$644$		0			0
$645$		0			0
$646$	1.50676	−	2.60979i	0.0592827	−	0.102681i
$647$	17.2774	−	29.9253i	0.679245	−	1.17649i	−0.295964	−	0.955199i	$-0.595641\pi$
$647$	17.2774	−	29.9253i	0.679245	−	1.17649i	0.975209	−	0.221287i	$-0.0710258\pi$
$648$		0			0
$649$	16.2323	+	28.1151i	0.637173	+	1.10362i
$650$	2.74164	+	4.74866i	0.107536	+	0.186258i
$651$		0			0
$652$	6.22019	−	10.7737i	0.243602	−	0.421930i
$653$	22.2944			0.872446			0.436223	−	0.899839i	$-0.356316\pi$
$653$	22.2944			0.872446			0.436223	+	0.899839i	$0.356316\pi$
$654$		0			0
$655$	−29.1442			−1.13876
$656$	0.187302	+	0.324417i	0.00731293	+	0.0126664i
$657$		0			0
$658$		0			0
$659$	−3.57493	−	6.19196i	−0.139259	−	0.241204i	0.787957	−	0.615730i	$-0.211139\pi$
$659$	−3.57493	−	6.19196i	−0.139259	−	0.241204i	−0.927217	+	0.374526i	$0.877806\pi$
$660$		0			0
$661$	21.4530	+	37.1577i	0.834425	+	1.44527i	0.894498	+	0.447072i	$0.147533\pi$
$661$	21.4530	+	37.1577i	0.834425	+	1.44527i	−0.0600736	+	0.998194i	$0.519134\pi$
$662$	−8.95760	−	15.5150i	−0.348147	−	0.603008i
$663$		0			0
$664$	−17.2221	−	29.8296i	−0.668349	−	1.15761i
$665$		0			0
$666$		0			0
$667$	18.5146	+	32.0683i	0.716889	+	1.24169i
$668$	3.28856			0.127238
$669$		0			0
$670$	12.0235			0.464510
$671$	0.185556	−	0.321392i	0.00716331	−	0.0124072i
$672$		0			0
$673$	−18.8270	−	32.6094i	−0.725729	−	1.25700i	−0.958673	−	0.284510i	$-0.908169\pi$
$673$	−18.8270	−	32.6094i	−0.725729	−	1.25700i	0.232944	−	0.972490i	$-0.425164\pi$
$674$	3.19188	+	5.52850i	0.122947	+	0.212950i
$675$		0			0
$676$	−4.19619	+	7.26801i	−0.161392	+	0.279539i
$677$	13.1808	−	22.8298i	0.506580	−	0.877422i	−0.493391	−	0.869808i	$-0.664243\pi$
$677$	13.1808	−	22.8298i	0.506580	−	0.877422i	0.999971	−	0.00761453i	$-0.00242380\pi$
$678$		0			0
$679$		0			0
$680$	1.89871	−	3.28866i	0.0728121	−	0.126114i
$681$		0			0
$682$	8.28244			0.317151
$683$	−1.96588	+	3.40500i	−0.0752222	+	0.130289i	−0.901183	−	0.433439i	$-0.857300\pi$
$683$	−1.96588	+	3.40500i	−0.0752222	+	0.130289i	0.825961	+	0.563728i	$0.190633\pi$
$684$		0			0
$685$	−17.4008			−0.664850
$686$		0			0
$687$		0			0
$688$	1.50076			0.0572158
$689$	−1.13002	−	1.95725i	−0.0430503	−	0.0745653i
$690$		0			0
$691$	9.95052	−	17.2348i	0.378536	−	0.655643i	−0.612314	−	0.790615i	$-0.709761\pi$
$691$	9.95052	−	17.2348i	0.378536	−	0.655643i	0.990849	+	0.134972i	$0.0430944\pi$
$692$	28.3585			1.07803
$693$		0			0
$694$	−12.5373			−0.475910
$695$	1.76823	−	3.06266i	0.0670727	−	0.116173i
$696$		0			0
$697$	−0.139270	−	0.241223i	−0.00527524	−	0.00913699i
$698$	20.1827			0.763925
$699$		0			0
$700$		0			0
$701$	−43.7908			−1.65396			−0.826979	−	0.562234i	$-0.809942\pi$
$701$	−43.7908			−1.65396			−0.826979	+	0.562234i	$0.809942\pi$
$702$		0			0
$703$	−2.84983	+	4.93604i	−0.107483	+	0.186166i
$704$	4.22031			0.159059
$705$		0			0
$706$	2.09792	−	3.63370i	0.0789561	−	0.136756i
$707$		0			0
$708$		0			0
$709$	−22.3172	+	38.6545i	−0.838139	+	1.45170i	0.0533097	+	0.998578i	$0.483023\pi$
$709$	−22.3172	+	38.6545i	−0.838139	+	1.45170i	−0.891449	+	0.453121i	$0.850310\pi$
$710$	−0.0384437	+	0.0665865i	−0.00144277	+	0.00249895i
$711$		0			0
$712$	16.0966	+	27.8801i	0.603244	+	1.04485i
$713$	−6.81163	−	11.7981i	−0.255097	−	0.441842i
$714$		0			0
$715$	9.67699	−	16.7610i	0.361899	−	0.626827i
$716$	−11.8245			−0.441904
$717$		0			0
$718$	6.83411			0.255047
$719$	−19.5096	−	33.7917i	−0.727586	−	1.26022i	−0.957901	−	0.287100i	$-0.907309\pi$
$719$	−19.5096	−	33.7917i	−0.727586	−	1.26022i	0.230315	−	0.973116i	$-0.426024\pi$
$720$		0			0
$721$		0			0
$722$	−0.965081	−	1.67157i	−0.0359166	−	0.0622094i
$723$		0			0
$724$	−12.0413	−	20.8561i	−0.447510	−	0.775109i
$725$	10.1130	+	17.5162i	0.375586	+	0.650534i
$726$		0			0
$727$	11.2554	+	19.4949i	0.417439	+	0.723025i	0.995681	−	0.0928402i	$-0.0295946\pi$
$727$	11.2554	+	19.4949i	0.417439	+	0.723025i	−0.578242	+	0.815865i	$0.696261\pi$
$728$		0			0
$729$		0			0
$730$	5.10593	+	8.84373i	0.188979	+	0.327321i
$731$	−1.11590			−0.0412731
$732$		0			0
$733$	0.897039			0.0331329			0.0165664	−	0.999863i	$-0.494726\pi$
$733$	0.897039			0.0331329			0.0165664	+	0.999863i	$0.494726\pi$
$734$	9.60441	−	16.6353i	0.354505	−	0.614021i
$735$		0			0
$736$	−15.6837	−	27.1649i	−0.578108	−	1.00131i
$737$	31.0335	+	53.7517i	1.14314	+	1.97997i
$738$		0			0
$739$	1.79032	−	3.10092i	0.0658578	−	0.114069i	−0.831216	−	0.555949i	$-0.812355\pi$
$739$	1.79032	−	3.10092i	0.0658578	−	0.114069i	0.897074	+	0.441880i	$0.145688\pi$
$740$	−1.56833	+	2.71643i	−0.0576531	+	0.0998581i
$741$		0			0
$742$		0			0
$743$	24.7964	−	42.9486i	0.909691	−	1.57563i	0.0951977	−	0.995458i	$-0.469652\pi$
$743$	24.7964	−	42.9486i	0.909691	−	1.57563i	0.814493	−	0.580173i	$-0.197015\pi$
$744$		0			0
$745$	12.1917			0.446668
$746$	−5.38726	+	9.33101i	−0.197242	+	0.341633i
$747$		0			0
$748$	8.56098			0.313020
$749$		0			0
$750$		0			0
$751$	−42.9030			−1.56555			−0.782776	−	0.622304i	$-0.786197\pi$
$751$	−42.9030			−1.56555			−0.782776	+	0.622304i	$0.786197\pi$
$752$	7.13450	+	12.3573i	0.260168	+	0.450625i
$753$		0			0
$754$	−6.28835	+	10.8917i	−0.229008	+	0.396654i
$755$	−25.1633			−0.915786
$756$		0			0
$757$	13.8029			0.501677			0.250838	−	0.968029i	$-0.419294\pi$
$757$	13.8029			0.501677			0.250838	+	0.968029i	$0.419294\pi$
$758$	−0.341510	+	0.591513i	−0.0124042	+	0.0214847i
$759$		0			0
$760$	6.80856	+	11.7928i	0.246972	+	0.427769i
$761$	40.7197			1.47609			0.738044	−	0.674752i	$-0.235749\pi$
$761$	40.7197			1.47609			0.738044	+	0.674752i	$0.235749\pi$
$762$		0			0
$763$		0			0
$764$	23.0001			0.832113
$765$		0			0
$766$	−3.88342	+	6.72627i	−0.140313	+	0.243030i
$767$	18.1365			0.654871
$768$		0			0
$769$	−5.57381	+	9.65413i	−0.200997	+	0.348137i	−0.948850	−	0.315728i	$-0.897751\pi$
$769$	−5.57381	+	9.65413i	−0.200997	+	0.348137i	0.747853	+	0.663864i	$0.231085\pi$
$770$		0			0
$771$		0			0
$772$	12.8454	−	22.2489i	0.462317	−	0.800756i
$773$	−0.462831	+	0.801647i	−0.0166469	+	0.0288332i	−0.874229	−	0.485514i	$-0.838632\pi$
$773$	−0.462831	+	0.801647i	−0.0166469	+	0.0288332i	0.857582	+	0.514347i	$0.171966\pi$
$774$		0			0
$775$	−3.72061	−	6.44428i	−0.133648	−	0.231485i
$776$	−6.43012	−	11.1373i	−0.230828	−	0.399806i
$777$		0			0
$778$	−5.97049	+	10.3412i	−0.214052	+	0.370750i
$779$	0.998817			0.0357863
$780$		0			0
$781$	−0.396903			−0.0142023
$782$	2.04024	+	3.53381i	0.0729590	+	0.126369i
$783$		0			0
$784$		0			0
$785$	4.50386	+	7.80092i	0.160750	+	0.278427i
$786$		0			0
$787$	11.5120	+	19.9393i	0.410358	+	0.710761i	0.994929	−	0.100582i	$-0.0320704\pi$
$787$	11.5120	+	19.9393i	0.410358	+	0.710761i	−0.584571	+	0.811343i	$0.698737\pi$
$788$	3.13030	+	5.42184i	0.111512	+	0.193145i
$789$		0			0
$790$	−0.881115	−	1.52614i	−0.0313487	−	0.0542975i
$791$		0			0
$792$		0			0
$793$	−0.103662	−	0.179548i	−0.00368114	−	0.00637593i
$794$	8.77102			0.311272
$795$		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)

\(f(q)\)	\(=\)	\(q+(0.335166 - 0.580525i) q^{2} +(0.775327 + 1.34291i) q^{4} +1.42494 q^{5} +2.38012 q^{8} +O(q^{10})\)	\(q+(0.335166 - 0.580525i) q^{2} +(0.775327 + 1.34291i) q^{4} +1.42494 q^{5} +2.38012 q^{8} +(0.477591 - 0.827212i) q^{10} +4.93077 q^{11} +(1.37730 - 2.38556i) q^{13} +(-0.752918 + 1.30409i) q^{16} +(0.559839 - 0.969670i) q^{17} +(2.00752 + 3.47713i) q^{19} +(1.10479 + 1.91356i) q^{20} +(1.65263 - 2.86244i) q^{22} -5.43661 q^{23} -2.96955 q^{25} +(-0.923251 - 1.59912i) q^{26} +(-3.40555 - 5.89858i) q^{29} +(1.25292 + 2.17012i) q^{31} +(2.88483 + 4.99666i) q^{32} +(-0.375279 - 0.650002i) q^{34} +(0.709787 + 1.22939i) q^{37} +2.69142 q^{38} +3.39152 q^{40} +(0.124384 - 0.215440i) q^{41} +(-0.498313 - 0.863104i) q^{43} +(3.82296 + 6.62156i) q^{44} +(-1.82217 + 3.15609i) q^{46} +(4.73790 - 8.20628i) q^{47} +(-0.995294 + 1.72390i) q^{50} +4.27144 q^{52} +(0.410229 - 0.710537i) q^{53} +7.02604 q^{55} -4.56570 q^{58} +(3.29204 + 5.70197i) q^{59} +(0.0376322 - 0.0651809i) q^{61} +1.67974 q^{62} +0.855913 q^{64} +(1.96257 - 3.39927i) q^{65} +(6.29385 + 10.9013i) q^{67} +1.73623 q^{68} -0.0804951 q^{71} +(-5.34551 + 9.25869i) q^{73} +0.951587 q^{74} +(-3.11297 + 5.39183i) q^{76} +(0.922457 - 1.59774i) q^{79} +(-1.07286 + 1.85825i) q^{80} +(-0.0833788 - 0.144416i) q^{82} +(-7.23583 - 12.5328i) q^{83} +(0.797736 - 1.38172i) q^{85} -0.668072 q^{86} +11.7358 q^{88} +(6.76292 + 11.7137i) q^{89} +(-4.21515 - 7.30085i) q^{92} +(-3.17597 - 5.50094i) q^{94} +(2.86059 + 4.95469i) q^{95} +(-2.70160 - 4.67930i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) \)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about