LMFDB - Embedded newform 1323.2.g.f.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:	$10$
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			361.2
Root			$0.920620 - 1.59456i$ of defining polynomial
Character	$\chi$	$=$	1323.361
Dual form			1323.2.g.f.667.2

$q$-expansion

$f(q)$	$=$	$q+(-0.920620 + 1.59456i) q^{2} +(-0.695084 - 1.20392i) q^{4} +1.33475 q^{5} -1.12285 q^{8} +O(q^{10})$	\(q+(-0.920620 + 1.59456i) q^{2} +(-0.695084 - 1.20392i) q^{4} +1.33475 q^{5} -1.12285 q^{8} +(-1.22880 + 2.12835i) q^{10} -1.51302 q^{11} +(2.58800 - 4.48254i) q^{13} +(2.42388 - 4.19829i) q^{16} +(0.774463 - 1.34141i) q^{17} +(1.25211 + 2.16872i) q^{19} +(-0.927765 - 1.60694i) q^{20} +(1.39291 - 2.41260i) q^{22} +7.36079 q^{23} -3.21843 q^{25} +(4.76513 + 8.25344i) q^{26} +(0.0309713 + 0.0536439i) q^{29} +(-1.92388 - 3.33227i) q^{31} +(3.34011 + 5.78523i) q^{32} +(1.42597 + 2.46986i) q^{34} +(-0.281608 - 0.487760i) q^{37} -4.61087 q^{38} -1.49873 q^{40} +(4.51188 - 7.81481i) q^{41} +(5.09988 + 8.83325i) q^{43} +(1.05167 + 1.82155i) q^{44} +(-6.77649 + 11.7372i) q^{46} +(4.75925 - 8.24327i) q^{47} +(2.96296 - 5.13199i) q^{50} -7.19550 q^{52} +(-0.755374 + 1.30835i) q^{53} -2.01950 q^{55} -0.114051 q^{58} +(4.22166 + 7.31212i) q^{59} +(1.61958 - 2.80520i) q^{61} +7.08467 q^{62} -2.60434 q^{64} +(3.45434 - 5.98309i) q^{65} +(-3.46670 - 6.00449i) q^{67} -2.15327 q^{68} +12.3304 q^{71} +(1.37936 - 2.38912i) q^{73} +1.03702 q^{74} +(1.74064 - 3.01488i) q^{76} +(2.95969 - 5.12633i) q^{79} +(3.23529 - 5.60368i) q^{80} +(8.30746 + 14.3889i) q^{82} +(2.80111 + 4.85167i) q^{83} +(1.03372 - 1.79045i) q^{85} -18.7802 q^{86} +1.69889 q^{88} +(0.703287 + 1.21813i) q^{89} +(-5.11636 - 8.86180i) q^{92} +(8.76293 + 15.1778i) q^{94} +(1.67126 + 2.89470i) q^{95} +(6.09713 + 10.5605i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.920620	+	1.59456i	−0.650977	+	1.12753i	0.331909	+	0.943311i	$0.392307\pi$
$2$	−0.920620	+	1.59456i	−0.650977	+	1.12753i	−0.982886	+	0.184214i	$0.941026\pi$
$3$		0			0
$3$		0			0
$4$	−0.695084	−	1.20392i	−0.347542	−	0.601960i
$5$	1.33475			0.596920			0.298460	−	0.954422i	$-0.403527\pi$
$5$	1.33475			0.596920			0.298460	+	0.954422i	$0.403527\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.12285			−0.396987
$9$		0			0
$10$	−1.22880	+	2.12835i	−0.388581	+	0.673042i
$11$	−1.51302			−0.456192			−0.228096	−	0.973639i	$-0.573250\pi$
$11$	−1.51302			−0.456192			−0.228096	+	0.973639i	$0.573250\pi$
$12$		0			0
$13$	2.58800	−	4.48254i	0.717781	−	1.24323i	−0.244096	−	0.969751i	$-0.578491\pi$
$13$	2.58800	−	4.48254i	0.717781	−	1.24323i	0.961877	−	0.273482i	$-0.0881755\pi$
$14$		0			0
$15$		0			0
$16$	2.42388	−	4.19829i	0.605971	−	1.04957i
$17$	0.774463	−	1.34141i	0.187835	−	0.325340i	−0.756693	−	0.653770i	$-0.773186\pi$
$17$	0.774463	−	1.34141i	0.187835	−	0.325340i	0.944528	+	0.328430i	$0.106520\pi$
$18$		0			0
$19$	1.25211	+	2.16872i	0.287254	+	0.497538i	0.973153	−	0.230158i	$-0.0739244\pi$
$19$	1.25211	+	2.16872i	0.287254	+	0.497538i	−0.685900	+	0.727696i	$0.740591\pi$
$20$	−0.927765	−	1.60694i	−0.207455	−	0.359322i
$21$		0			0
$22$	1.39291	−	2.41260i	0.296970	−	0.514367i
$23$	7.36079			1.53483			0.767415	−	0.641151i	$-0.221543\pi$
$23$	7.36079			1.53483			0.767415	+	0.641151i	$0.221543\pi$
$24$		0			0
$25$	−3.21843			−0.643687
$26$	4.76513	+	8.25344i	0.934518	+	1.61863i
$27$		0			0
$28$		0			0
$29$	0.0309713	+	0.0536439i	0.00575123	+	0.00996143i	0.868887	−	0.495011i	$-0.164836\pi$
$29$	0.0309713	+	0.0536439i	0.00575123	+	0.00996143i	−0.863135	+	0.504972i	$0.831503\pi$
$30$		0			0
$31$	−1.92388	−	3.33227i	−0.345540	−	0.598493i	0.639912	−	0.768448i	$-0.278971\pi$
$31$	−1.92388	−	3.33227i	−0.345540	−	0.598493i	−0.985452	+	0.169956i	$0.945638\pi$
$32$	3.34011	+	5.78523i	0.590453	+	1.02269i
$33$		0			0
$34$	1.42597	+	2.46986i	0.244552	+	0.423577i
$35$		0			0
$36$		0			0
$37$	−0.281608	−	0.487760i	−0.0462961	−	0.0801872i	0.841949	−	0.539557i	$-0.181408\pi$
$37$	−0.281608	−	0.487760i	−0.0462961	−	0.0801872i	−0.888245	+	0.459370i	$0.848075\pi$
$38$	−4.61087			−0.747982
$39$		0			0
$40$	−1.49873			−0.236969
$41$	4.51188	−	7.81481i	0.704638	−	1.22047i	−0.262185	−	0.965018i	$-0.584443\pi$
$41$	4.51188	−	7.81481i	0.704638	−	1.22047i	0.966822	−	0.255450i	$-0.0822237\pi$
$42$		0			0
$43$	5.09988	+	8.83325i	0.777724	+	1.34706i	0.933251	+	0.359226i	$0.116959\pi$
$43$	5.09988	+	8.83325i	0.777724	+	1.34706i	−0.155526	+	0.987832i	$0.549707\pi$
$44$	1.05167	+	1.82155i	0.158546	+	0.274609i
$45$		0			0
$46$	−6.77649	+	11.7372i	−0.999139	+	1.73056i
$47$	4.75925	−	8.24327i	0.694209	−	1.20240i	−0.276238	−	0.961089i	$-0.589088\pi$
$47$	4.75925	−	8.24327i	0.694209	−	1.20240i	0.970447	−	0.241315i	$-0.0775788\pi$
$48$		0			0
$49$		0			0
$50$	2.96296	−	5.13199i	0.419025	−	0.725773i
$51$		0			0
$52$	−7.19550			−0.997836
$53$	−0.755374	+	1.30835i	−0.103759	+	0.179715i	−0.913230	−	0.407444i	$-0.866420\pi$
$53$	−0.755374	+	1.30835i	−0.103759	+	0.179715i	0.809472	+	0.587159i	$0.199754\pi$
$54$		0			0
$55$	−2.01950			−0.272310
$56$		0			0
$57$		0			0
$58$	−0.114051			−0.0149757
$59$	4.22166	+	7.31212i	0.549613	+	0.951957i	0.998301	+	0.0582689i	$0.0185581\pi$
$59$	4.22166	+	7.31212i	0.549613	+	0.951957i	−0.448688	+	0.893688i	$0.648109\pi$
$60$		0			0
$61$	1.61958	−	2.80520i	0.207367	−	0.359169i	−0.743518	−	0.668716i	$-0.766844\pi$
$61$	1.61958	−	2.80520i	0.207367	−	0.359169i	0.950884	+	0.309547i	$0.100177\pi$
$62$	7.08467			0.899754
$63$		0			0
$64$	−2.60434			−0.325543
$65$	3.45434	−	5.98309i	0.428458	−	0.742111i
$66$		0			0
$67$	−3.46670	−	6.00449i	−0.423524	−	0.733566i	0.572757	−	0.819725i	$-0.305874\pi$
$67$	−3.46670	−	6.00449i	−0.423524	−	0.733566i	−0.996281	+	0.0861595i	$0.972541\pi$
$68$	−2.15327			−0.261122
$69$		0			0
$70$		0			0
$71$	12.3304			1.46335			0.731673	−	0.681656i	$-0.238740\pi$
$71$	12.3304			1.46335			0.731673	+	0.681656i	$0.238740\pi$
$72$		0			0
$73$	1.37936	−	2.38912i	0.161442	−	0.279625i	−0.773944	−	0.633254i	$-0.781719\pi$
$73$	1.37936	−	2.38912i	0.161442	−	0.279625i	0.935386	+	0.353629i	$0.115052\pi$
$74$	1.03702			0.120551
$75$		0			0
$76$	1.74064	−	3.01488i	0.199665	−	0.345830i
$77$		0			0
$78$		0			0
$79$	2.95969	−	5.12633i	0.332991	−	0.576758i	−0.650106	−	0.759844i	$-0.725275\pi$
$79$	2.95969	−	5.12633i	0.332991	−	0.576758i	0.983097	+	0.183086i	$0.0586087\pi$
$80$	3.23529	−	5.60368i	0.361716	−	0.626511i
$81$		0			0
$82$	8.30746	+	14.3889i	0.917406	+	1.58899i
$83$	2.80111	+	4.85167i	0.307462	+	0.532540i	0.977806	−	0.209510i	$-0.0671870\pi$
$83$	2.80111	+	4.85167i	0.307462	+	0.532540i	−0.670344	+	0.742050i	$0.733854\pi$
$84$		0			0
$85$	1.03372	−	1.79045i	0.112122	−	0.194202i
$86$	−18.7802			−2.02512
$87$		0			0
$88$	1.69889			0.181102
$89$	0.703287	+	1.21813i	0.0745483	+	0.129121i	0.900890	−	0.434048i	$-0.142915\pi$
$89$	0.703287	+	1.21813i	0.0745483	+	0.129121i	−0.826341	+	0.563169i	$0.809582\pi$
$90$		0			0
$91$		0			0
$92$	−5.11636	−	8.86180i	−0.533418	−	0.923906i
$93$		0			0
$94$	8.76293	+	15.1778i	0.903827	+	1.56548i
$95$	1.67126	+	2.89470i	0.171467	+	0.296990i
$96$		0			0
$97$	6.09713	+	10.5605i	0.619070	+	1.07226i	0.989656	+	0.143462i	$0.0458236\pi$
$97$	6.09713	+	10.5605i	0.619070	+	1.07226i	−0.370586	+	0.928798i	$0.620843\pi$
$98$		0			0
$99$		0			0
$100$	2.23708	+	3.87474i	0.223708	+	0.387474i
$101$	1.11867			0.111312			0.0556560	−	0.998450i	$-0.482275\pi$
$101$	1.11867			0.111312			0.0556560	+	0.998450i	$0.482275\pi$
$102$		0			0
$103$	−1.93045			−0.190213			−0.0951063	−	0.995467i	$-0.530319\pi$
$103$	−1.93045			−0.190213			−0.0951063	+	0.995467i	$0.530319\pi$
$104$	−2.90593	+	5.03322i	−0.284950	+	0.493548i
$105$		0			0
$106$	−1.39082	−	2.40898i	−0.135089	−	0.233981i
$107$	−2.88969	−	5.00509i	−0.279357	−	0.483860i	0.691868	−	0.722024i	$-0.256788\pi$
$107$	−2.88969	−	5.00509i	−0.279357	−	0.483860i	−0.971225	+	0.238163i	$0.923455\pi$
$108$		0			0
$109$	−4.12106	+	7.13788i	−0.394726	+	0.683685i	−0.993066	−	0.117557i	$-0.962494\pi$
$109$	−4.12106	+	7.13788i	−0.394726	+	0.683685i	0.598340	+	0.801242i	$0.295827\pi$
$110$	1.85920	−	3.22022i	0.177267	−	0.307036i
$111$		0			0
$112$		0			0
$113$	−7.25105	+	12.5592i	−0.682121	+	1.18147i	0.292211	+	0.956354i	$0.405609\pi$
$113$	−7.25105	+	12.5592i	−0.682121	+	1.18147i	−0.974332	+	0.225115i	$0.927724\pi$
$114$		0			0
$115$	9.82483			0.916170
$116$	0.0430553	−	0.0745740i	0.00399759	−	0.00692403i
$117$		0			0
$118$	−15.5462			−1.43114
$119$		0			0
$120$		0			0
$121$	−8.71078			−0.791889
$122$	2.98204	+	5.16505i	0.269982	+	0.467622i
$123$		0			0
$124$	−2.67452	+	4.63241i	−0.240179	+	0.416002i
$125$	−10.9696			−0.981149
$126$		0			0
$127$	8.50004			0.754257			0.377128	−	0.926161i	$-0.376912\pi$
$127$	8.50004			0.754257			0.377128	+	0.926161i	$0.376912\pi$
$128$	−4.28260	+	7.41769i	−0.378532	+	0.655637i
$129$		0			0
$130$	6.36027	+	11.0163i	0.557832	+	0.966194i
$131$	−2.01346			−0.175917			−0.0879585	−	0.996124i	$-0.528034\pi$
$131$	−2.01346			−0.175917			−0.0879585	+	0.996124i	$0.528034\pi$
$132$		0			0
$133$		0			0
$134$	12.7660			1.10282
$135$		0			0
$136$	−0.869605	+	1.50620i	−0.0745680	+	0.129156i
$137$	−2.21740			−0.189445			−0.0947225	−	0.995504i	$-0.530196\pi$
$137$	−2.21740			−0.189445			−0.0947225	+	0.995504i	$0.530196\pi$
$138$		0			0
$139$	−0.377669	+	0.654143i	−0.0320335	+	0.0554836i	−0.881598	−	0.472002i	$-0.843532\pi$
$139$	−0.377669	+	0.654143i	−0.0320335	+	0.0554836i	0.849564	+	0.527485i	$0.176865\pi$
$140$		0			0
$141$		0			0
$142$	−11.3516	+	19.6615i	−0.952604	+	1.64996i
$143$	−3.91568	+	6.78216i	−0.327446	+	0.567153i
$144$		0			0
$145$	0.0413391	+	0.0716014i	0.00343303	+	0.00594618i
$146$	2.53973	+	4.39894i	0.210189	+	0.364059i
$147$		0			0
$148$	−0.391482	+	0.678068i	−0.0321797	+	0.0557368i
$149$	−6.58499			−0.539463			−0.269732	−	0.962936i	$-0.586935\pi$
$149$	−6.58499			−0.539463			−0.269732	+	0.962936i	$0.586935\pi$
$150$		0			0
$151$	12.6671			1.03083			0.515417	−	0.856939i	$-0.327637\pi$
$151$	12.6671			1.03083			0.515417	+	0.856939i	$0.327637\pi$
$152$	−1.40593	−	2.43514i	−0.114036	−	0.197516i
$153$		0			0
$154$		0			0
$155$	−2.56791	−	4.44775i	−0.206260	−	0.357252i
$156$		0			0
$157$	−8.65372	−	14.9887i	−0.690642	−	1.19623i	−0.971628	−	0.236515i	$-0.923995\pi$
$157$	−8.65372	−	14.9887i	−0.690642	−	1.19623i	0.280986	−	0.959712i	$-0.409338\pi$
$158$	5.44950	+	9.43882i	0.433539	+	0.750912i
$159$		0			0
$160$	4.45822	+	7.72186i	0.352453	+	0.610467i
$161$		0			0
$162$		0			0
$163$	6.10963	+	10.5822i	0.478543	+	0.828861i	0.999697	−	0.0246014i	$-0.00783167\pi$
$163$	6.10963	+	10.5822i	0.478543	+	0.828861i	−0.521154	+	0.853463i	$0.674498\pi$
$164$	−12.5445			−0.979564
$165$		0			0
$166$	−10.3150			−0.800602
$167$	1.76248	−	3.05270i	0.136385	−	0.236225i	−0.789741	−	0.613440i	$-0.789785\pi$
$167$	1.76248	−	3.05270i	0.136385	−	0.236225i	0.926126	+	0.377215i	$0.123118\pi$
$168$		0			0
$169$	−6.89546	−	11.9433i	−0.530420	−	0.918714i
$170$	1.90332	+	3.29665i	0.145978	+	0.252842i
$171$		0			0
$172$	7.08968	−	12.2797i	0.540583	−	0.936318i
$173$	−5.07046	+	8.78229i	−0.385500	+	0.667705i	−0.991838	−	0.127502i	$-0.959304\pi$
$173$	−5.07046	+	8.78229i	−0.385500	+	0.667705i	0.606339	+	0.795206i	$0.292638\pi$
$174$		0			0
$175$		0			0
$176$	−3.66738	+	6.35208i	−0.276439	+	0.478806i
$177$		0			0
$178$	−2.58984			−0.194117
$179$	−0.850579	+	1.47325i	−0.0635752	+	0.110116i	−0.896061	−	0.443931i	$-0.853584\pi$
$179$	−0.850579	+	1.47325i	−0.0635752	+	0.110116i	0.832486	+	0.554046i	$0.186917\pi$
$180$		0			0
$181$	16.9941			1.26316			0.631581	−	0.775310i	$-0.282406\pi$
$181$	16.9941			1.26316			0.631581	+	0.775310i	$0.282406\pi$
$182$		0			0
$183$		0			0
$184$	−8.26505			−0.609308
$185$	−0.375877	−	0.651039i	−0.0276351	−	0.0478653i
$186$		0			0
$187$	−1.17178	+	2.02957i	−0.0856887	+	0.148417i
$188$	−13.2323			−0.965066
$189$		0			0
$190$	−6.15437			−0.446485
$191$	11.3470	−	19.6535i	0.821038	−	1.42208i	−0.0838717	−	0.996477i	$-0.526729\pi$
$191$	11.3470	−	19.6535i	0.821038	−	1.42208i	0.904910	−	0.425603i	$-0.139938\pi$
$192$		0			0
$193$	−3.09349	−	5.35808i	−0.222674	−	0.385683i	0.732945	−	0.680288i	$-0.238145\pi$
$193$	−3.09349	−	5.35808i	−0.222674	−	0.385683i	−0.955619	+	0.294605i	$0.904812\pi$
$194$	−22.4526			−1.61200
$195$		0			0
$196$		0			0
$197$	−9.77010			−0.696091			−0.348045	−	0.937478i	$-0.613154\pi$
$197$	−9.77010			−0.696091			−0.348045	+	0.937478i	$0.613154\pi$
$198$		0			0
$199$	4.33973	−	7.51664i	0.307636	−	0.532840i	−0.670209	−	0.742172i	$-0.733796\pi$
$199$	4.33973	−	7.51664i	0.307636	−	0.532840i	0.977845	+	0.209332i	$0.0671289\pi$
$200$	3.61381			0.255535
$201$		0			0
$202$	−1.02987	+	1.78379i	−0.0724615	+	0.125507i
$203$		0			0
$204$		0			0
$205$	6.02225	−	10.4308i	0.420612	−	0.728522i
$206$	1.77721	−	3.07822i	0.123824	−	0.214470i
$207$		0			0
$208$	−12.5460	−	21.7303i	−0.869909	−	1.50673i
$209$	−1.89446	−	3.28130i	−0.131043	−	0.226973i
$210$		0			0
$211$	−2.84219	+	4.92283i	−0.195665	+	0.338901i	−0.947118	−	0.320885i	$-0.896020\pi$
$211$	−2.84219	+	4.92283i	−0.195665	+	0.338901i	0.751453	+	0.659786i	$0.229353\pi$
$212$	2.10019			0.144242
$213$		0			0
$214$	10.6412			0.727420
$215$	6.80708	+	11.7902i	0.464239	+	0.804086i
$216$		0			0
$217$		0			0
$218$	−7.58786	−	13.1426i	−0.513915	−	0.890126i
$219$		0			0
$220$	1.40372	+	2.43132i	0.0946390	+	0.163920i
$221$	−4.00862	−	6.94313i	−0.269649	−	0.467045i
$222$		0			0
$223$	−5.86133	−	10.1521i	−0.392503	−	0.679836i	0.600276	−	0.799793i	$-0.295058\pi$
$223$	−5.86133	−	10.1521i	−0.392503	−	0.679836i	−0.992779	+	0.119957i	$0.961724\pi$
$224$		0			0
$225$		0			0
$226$	−13.3509	−	23.1245i	−0.888091	−	1.53822i
$227$	11.1831			0.742247			0.371123	−	0.928584i	$-0.378973\pi$
$227$	11.1831			0.742247			0.371123	+	0.928584i	$0.378973\pi$
$228$		0			0
$229$	9.65647			0.638118			0.319059	−	0.947735i	$-0.396633\pi$
$229$	9.65647			0.638118			0.319059	+	0.947735i	$0.396633\pi$
$230$	−9.04494	+	15.6663i	−0.596406	+	1.03301i
$231$		0			0
$232$	−0.0347761	−	0.0602340i	−0.00228317	−	0.00395456i
$233$	9.64492	+	16.7055i	0.631860	+	1.09441i	0.987171	+	0.159666i	$0.0510416\pi$
$233$	9.64492	+	16.7055i	0.631860	+	1.09441i	−0.355311	+	0.934748i	$0.615625\pi$
$234$		0			0
$235$	6.35243	−	11.0027i	0.414387	−	0.717739i
$236$	5.86881	−	10.1651i	0.382027	−	0.661690i
$237$		0			0
$238$		0			0
$239$	0.194641	−	0.337128i	0.0125903	−	0.0218070i	−0.859662	−	0.510864i	$-0.829326\pi$
$239$	0.194641	−	0.337128i	0.0125903	−	0.0218070i	0.872252	+	0.489057i	$0.162659\pi$
$240$		0			0
$241$	−10.6361			−0.685134			−0.342567	−	0.939493i	$-0.611296\pi$
$241$	−10.6361			−0.685134			−0.342567	+	0.939493i	$0.611296\pi$
$242$	8.01932	−	13.8899i	0.515502	−	0.892875i
$243$		0			0
$244$	−4.50299			−0.288274
$245$		0			0
$246$		0			0
$247$	12.9618			0.824741
$248$	2.16023	+	3.74163i	0.137175	+	0.237594i
$249$		0			0
$250$	10.0988	−	17.4917i	0.638705	−	1.10627i
$251$	−3.26628			−0.206166			−0.103083	−	0.994673i	$-0.532871\pi$
$251$	−3.26628			−0.206166			−0.103083	+	0.994673i	$0.532871\pi$
$252$		0			0
$253$	−11.1370			−0.700176
$254$	−7.82531	+	13.5538i	−0.491004	+	0.850443i
$255$		0			0
$256$	−10.4896	−	18.1686i	−0.655603	−	1.13554i
$257$	−4.69573			−0.292912			−0.146456	−	0.989217i	$-0.546787\pi$
$257$	−4.69573			−0.292912			−0.146456	+	0.989217i	$0.546787\pi$
$258$		0			0
$259$		0			0
$260$	−9.60421			−0.595628
$261$		0			0
$262$	1.85363	−	3.21059i	0.114518	−	0.198351i
$263$	−19.5498			−1.20549			−0.602747	−	0.797932i	$-0.705927\pi$
$263$	−19.5498			−1.20549			−0.602747	+	0.797932i	$0.705927\pi$
$264$		0			0
$265$	−1.00824	+	1.74632i	−0.0619355	+	0.107276i
$266$		0			0
$267$		0			0
$268$	−4.81929	+	8.34725i	−0.294385	+	0.509890i
$269$	7.88365	−	13.6549i	0.480675	−	0.832553i	−0.519079	−	0.854726i	$-0.673725\pi$
$269$	7.88365	−	13.6549i	0.480675	−	0.832553i	0.999754	+	0.0221730i	$0.00705846\pi$
$270$		0			0
$271$	−7.39882	−	12.8151i	−0.449446	−	0.778464i	0.548904	−	0.835886i	$-0.315045\pi$
$271$	−7.39882	−	12.8151i	−0.449446	−	0.778464i	−0.998350	+	0.0574218i	$0.981712\pi$
$272$	−3.75442	−	6.50285i	−0.227645	−	0.394293i
$273$		0			0
$274$	2.04138	−	3.53578i	0.123324	−	0.213604i
$275$	4.86954			0.293644
$276$		0			0
$277$	−7.45122			−0.447701			−0.223850	−	0.974624i	$-0.571863\pi$
$277$	−7.45122			−0.447701			−0.223850	+	0.974624i	$0.571863\pi$
$278$	−0.695380	−	1.20443i	−0.0417061	−	0.0722371i
$279$		0			0
$280$		0			0
$281$	12.9938	+	22.5060i	0.775146	+	1.34259i	0.934712	+	0.355406i	$0.115657\pi$
$281$	12.9938	+	22.5060i	0.775146	+	1.34259i	−0.159566	+	0.987187i	$0.551009\pi$
$282$		0			0
$283$	9.37768	+	16.2426i	0.557445	+	0.965524i	0.997709	+	0.0676550i	$0.0215517\pi$
$283$	9.37768	+	16.2426i	0.557445	+	0.965524i	−0.440263	+	0.897869i	$0.645115\pi$
$284$	−8.57064	−	14.8448i	−0.508574	−	0.880876i
$285$		0			0
$286$	−7.20971	−	12.4876i	−0.426319	−	0.738406i
$287$		0			0
$288$		0			0
$289$	7.30041	+	12.6447i	0.429436	+	0.743805i
$290$	−0.152230			−0.00893928
$291$		0			0
$292$	−3.83507			−0.224431
$293$	−1.23089	+	2.13196i	−0.0719093	+	0.124551i	−0.899738	−	0.436430i	$-0.856243\pi$
$293$	−1.23089	+	2.13196i	−0.0719093	+	0.124551i	0.827829	+	0.560981i	$0.189576\pi$
$294$		0			0
$295$	5.63487	+	9.75988i	0.328075	+	0.568242i
$296$	0.316203	+	0.547680i	0.0183790	+	0.0318333i
$297$		0			0
$298$	6.06227	−	10.5002i	0.351178	−	0.608258i
$299$	19.0497	−	32.9950i	1.10167	−	1.90815i
$300$		0			0
$301$		0			0
$302$	−11.6616	+	20.1985i	−0.671050	+	1.16229i
$303$		0			0
$304$	12.1399			0.696270
$305$	2.16175	−	3.74425i	0.123781	−	0.214395i
$306$		0			0
$307$	4.66277			0.266118			0.133059	−	0.991108i	$-0.457520\pi$
$307$	4.66277			0.266118			0.133059	+	0.991108i	$0.457520\pi$
$308$		0			0
$309$		0			0
$310$	9.45629			0.537081
$311$	−13.7410	−	23.8002i	−0.779183	−	1.34958i	−0.932413	−	0.361393i	$-0.882301\pi$
$311$	−13.7410	−	23.8002i	−0.779183	−	1.34958i	0.153231	−	0.988190i	$-0.451032\pi$
$312$		0			0
$313$	2.74666	−	4.75735i	0.155250	−	0.268901i	−0.777900	−	0.628388i	$-0.783715\pi$
$313$	2.74666	−	4.75735i	0.155250	−	0.268901i	0.933150	+	0.359487i	$0.117048\pi$
$314$	31.8671			1.79837
$315$		0			0
$316$	−8.22893			−0.462914
$317$	4.93879	−	8.55424i	0.277390	−	0.480454i	−0.693345	−	0.720606i	$-0.743864\pi$
$317$	4.93879	−	8.55424i	0.277390	−	0.480454i	0.970735	+	0.240152i	$0.0771972\pi$
$318$		0			0
$319$	−0.0468601	−	0.0811641i	−0.00262366	−	0.00454432i
$320$	−3.47615			−0.194323
$321$		0			0
$322$		0			0
$323$	3.87885			0.215825
$324$		0			0
$325$	−8.32930	+	14.4268i	−0.462026	+	0.800253i
$326$	−22.4986			−1.24608
$327$		0			0
$328$	−5.06616	+	8.77485i	−0.279732	+	0.484510i
$329$		0			0
$330$		0			0
$331$	10.3471	−	17.9217i	0.568729	−	0.985067i	−0.427963	−	0.903796i	$-0.640769\pi$
$331$	10.3471	−	17.9217i	0.568729	−	0.985067i	0.996692	−	0.0812710i	$-0.0258979\pi$
$332$	3.89401	−	6.74463i	0.213712	−	0.370160i
$333$		0			0
$334$	3.24514	+	5.62076i	0.177566	+	0.307554i
$335$	−4.62718	−	8.01452i	−0.252810	−	0.437880i
$336$		0			0
$337$	0.748747	−	1.29687i	0.0407869	−	0.0706449i	−0.844911	−	0.534906i	$-0.820347\pi$
$337$	0.748747	−	1.29687i	0.0407869	−	0.0706449i	0.885698	+	0.464261i	$0.153680\pi$
$338$	25.3924			1.38116
$339$		0			0
$340$	−2.87408			−0.155869
$341$	2.91087	+	5.04177i	0.157632	+	0.273027i
$342$		0			0
$343$		0			0
$344$	−5.72639	−	9.91840i	−0.308746	−	0.534764i
$345$		0			0
$346$	−9.33593	−	16.1703i	−0.501903	−	0.869321i
$347$	−14.7694	−	25.5813i	−0.792862	−	1.37328i	−0.924188	−	0.381938i	$-0.875257\pi$
$347$	−14.7694	−	25.5813i	−0.792862	−	1.37328i	0.131326	−	0.991339i	$-0.458077\pi$
$348$		0			0
$349$	−18.0006	−	31.1780i	−0.963551	−	1.66892i	−0.713458	−	0.700698i	$-0.752872\pi$
$349$	−18.0006	−	31.1780i	−0.963551	−	1.66892i	−0.250094	−	0.968222i	$-0.580461\pi$
$350$		0			0
$351$		0			0
$352$	−5.05363	−	8.75315i	−0.269360	−	0.466545i
$353$	−29.4930			−1.56975			−0.784877	−	0.619652i	$-0.787274\pi$
$353$	−29.4930			−1.56975			−0.784877	+	0.619652i	$0.787274\pi$
$354$		0			0
$355$	16.4580			0.873500
$356$	0.977687	−	1.69340i	0.0518173	−	0.0897502i
$357$		0			0
$358$	−1.56612	−	2.71260i	−0.0827720	−	0.143365i
$359$	−2.70535	−	4.68580i	−0.142783	−	0.247307i	0.785761	−	0.618531i	$-0.212272\pi$
$359$	−2.70535	−	4.68580i	−0.142783	−	0.247307i	−0.928544	+	0.371224i	$0.878938\pi$
$360$		0			0
$361$	6.36444	−	11.0235i	0.334971	−	0.580186i
$362$	−15.6451	+	27.0981i	−0.822289	+	1.42425i
$363$		0			0
$364$		0			0
$365$	1.84110	−	3.18888i	0.0963676	−	0.166914i
$366$		0			0
$367$	23.0843			1.20499			0.602496	−	0.798122i	$-0.294173\pi$
$367$	23.0843			1.20499			0.602496	+	0.798122i	$0.294173\pi$
$368$	17.8417	−	30.9027i	0.930063	−	1.61092i
$369$		0			0
$370$	1.38416			0.0719591
$371$		0			0
$372$		0			0
$373$	21.5030			1.11338			0.556692	−	0.830719i	$-0.312070\pi$
$373$	21.5030			1.11338			0.556692	+	0.830719i	$0.312070\pi$
$374$	−2.15752	−	3.73694i	−0.111563	−	0.193232i
$375$		0			0
$376$	−5.34392	+	9.25595i	−0.275592	+	0.477339i
$377$	0.320615			0.0165125
$378$		0			0
$379$	5.72168			0.293903			0.146952	−	0.989144i	$-0.453054\pi$
$379$	5.72168			0.293903			0.146952	+	0.989144i	$0.453054\pi$
$380$	2.32333	−	4.02412i	0.119184	−	0.206433i
$381$		0			0
$382$	20.8925	+	36.1869i	1.06895	+	1.85148i
$383$	−34.9209			−1.78437			−0.892187	−	0.451666i	$-0.850830\pi$
$383$	−34.9209			−1.78437			−0.892187	+	0.451666i	$0.850830\pi$
$384$		0			0
$385$		0			0
$386$	11.3917			0.579823
$387$		0			0
$388$	8.47603	−	14.6809i	0.430305	−	0.745311i
$389$	28.8822			1.46438			0.732192	−	0.681098i	$-0.238497\pi$
$389$	28.8822			1.46438			0.732192	+	0.681098i	$0.238497\pi$
$390$		0			0
$391$	5.70066	−	9.87383i	0.288295	−	0.499341i
$392$		0			0
$393$		0			0
$394$	8.99455	−	15.5790i	0.453139	−	0.784860i
$395$	3.95046	−	6.84239i	0.198769	−	0.344278i
$396$		0			0
$397$	−5.59226	−	9.68607i	−0.280667	−	0.486130i	0.690882	−	0.722968i	$-0.257222\pi$
$397$	−5.59226	−	9.68607i	−0.280667	−	0.486130i	−0.971549	+	0.236838i	$0.923889\pi$
$398$	7.99049	+	13.8399i	0.400527	+	0.693734i
$399$		0			0
$400$	−7.80111	+	13.5119i	−0.390056	+	0.675596i
$401$	1.08212			0.0540386			0.0270193	−	0.999635i	$-0.491398\pi$
$401$	1.08212			0.0540386			0.0270193	+	0.999635i	$0.491398\pi$
$402$		0			0
$403$	−19.9160			−0.992088
$404$	−0.777570	−	1.34679i	−0.0386856	−	0.0670054i
$405$		0			0
$406$		0			0
$407$	0.426078	+	0.737988i	0.0211199	+	0.0365807i
$408$		0			0
$409$	−10.8674	−	18.8229i	−0.537360	−	0.930735i	−0.999045	−	0.0436908i	$-0.986088\pi$
$409$	−10.8674	−	18.8229i	−0.537360	−	0.930735i	0.461685	−	0.887044i	$-0.347245\pi$
$410$	11.0884	+	19.2057i	0.547618	+	0.948501i
$411$		0			0
$412$	1.34182	+	2.32410i	0.0661069	+	0.114500i
$413$		0			0
$414$		0			0
$415$	3.73879	+	6.47578i	0.183530	+	0.317884i
$416$	34.5767			1.69526
$417$		0			0
$418$	6.97632			0.341223
$419$	12.5906	−	21.8075i	0.615090	−	1.06537i	−0.375279	−	0.926912i	$-0.622453\pi$
$419$	12.5906	−	21.8075i	0.615090	−	1.06537i	0.990369	−	0.138455i	$-0.0442135\pi$
$420$		0			0
$421$	−14.8304	−	25.6869i	−0.722788	−	1.25191i	−0.959878	−	0.280418i	$-0.909527\pi$
$421$	−14.8304	−	25.6869i	−0.722788	−	1.25191i	0.237090	−	0.971488i	$-0.423806\pi$
$422$	−5.23316	−	9.06411i	−0.254746	−	0.441234i
$423$		0			0
$424$	0.848171	−	1.46907i	0.0411908	−	0.0713446i
$425$	−2.49256	+	4.31724i	−0.120907	+	0.209417i
$426$		0			0
$427$		0			0
$428$	−4.01715	+	6.95791i	−0.194176	+	0.336323i
$429$		0			0
$430$	−25.0669			−1.20884
$431$	−2.44517	+	4.23516i	−0.117780	+	0.204000i	−0.918887	−	0.394520i	$-0.870911\pi$
$431$	−2.44517	+	4.23516i	−0.117780	+	0.204000i	0.801108	+	0.598520i	$0.204244\pi$
$432$		0			0
$433$	−9.71430			−0.466839			−0.233420	−	0.972376i	$-0.574992\pi$
$433$	−9.71430			−0.466839			−0.233420	+	0.972376i	$0.574992\pi$
$434$		0			0
$435$		0			0
$436$	11.4579			0.548735
$437$	9.21651	+	15.9635i	0.440885	+	0.763636i
$438$		0			0
$439$	−7.41176	+	12.8375i	−0.353744	+	0.612703i	−0.986902	−	0.161320i	$-0.948425\pi$
$439$	−7.41176	+	12.8375i	−0.353744	+	0.612703i	0.633158	+	0.774022i	$0.281758\pi$
$440$	2.26760			0.108103
$441$		0			0
$442$	14.7617			0.702141
$443$	−10.9510	+	18.9676i	−0.520297	+	0.901180i	0.479425	+	0.877583i	$0.340845\pi$
$443$	−10.9510	+	18.9676i	−0.520297	+	0.901180i	−0.999722	+	0.0235972i	$0.992488\pi$
$444$		0			0
$445$	0.938715	+	1.62590i	0.0444994	+	0.0770751i
$446$	21.5842			1.02204
$447$		0			0
$448$		0			0
$449$	−21.4952			−1.01442			−0.507212	−	0.861822i	$-0.669324\pi$
$449$	−21.4952			−1.01442			−0.507212	+	0.861822i	$0.669324\pi$
$450$		0			0
$451$	−6.82655	+	11.8239i	−0.321450	+	0.556767i
$452$	20.1603			0.948263
$453$		0			0
$454$	−10.2954	+	17.8321i	−0.483185	+	0.836902i
$455$		0			0
$456$		0			0
$457$	−20.3128	+	35.1827i	−0.950190	+	1.64578i	−0.205181	+	0.978724i	$0.565778\pi$
$457$	−20.3128	+	35.1827i	−0.950190	+	1.64578i	−0.745009	+	0.667054i	$0.767555\pi$
$458$	−8.88995	+	15.3978i	−0.415400	+	0.719494i
$459$		0			0
$460$	−6.82908	−	11.8283i	−0.318408	−	0.551498i
$461$	1.41541	+	2.45155i	0.0659220	+	0.114180i	0.897103	−	0.441822i	$-0.145668\pi$
$461$	1.41541	+	2.45155i	0.0659220	+	0.114180i	−0.831181	+	0.556003i	$0.812334\pi$
$462$		0			0
$463$	−13.9324	+	24.1317i	−0.647494	+	1.12149i	0.336225	+	0.941782i	$0.390850\pi$
$463$	−13.9324	+	24.1317i	−0.647494	+	1.12149i	−0.983719	+	0.179711i	$0.942484\pi$
$464$	0.300284			0.0139403
$465$		0			0
$466$	−35.5173			−1.64531
$467$	−13.3219	−	23.0742i	−0.616464	−	1.06775i	−0.990126	−	0.140182i	$-0.955231\pi$
$467$	−13.3219	−	23.0742i	−0.616464	−	1.06775i	0.373661	−	0.927565i	$-0.378102\pi$
$468$		0			0
$469$		0			0
$470$	11.6964	+	20.2587i	0.539513	+	0.934463i
$471$		0			0
$472$	−4.74028	−	8.21041i	−0.218189	−	0.377915i
$473$	−7.71620	−	13.3648i	−0.354791	−	0.614516i
$474$		0			0
$475$	−4.02983	−	6.97987i	−0.184901	−	0.320258i
$476$		0			0
$477$		0			0
$478$	0.358381	+	0.620734i	0.0163920	+	0.0283917i
$479$	−31.5791			−1.44289			−0.721443	−	0.692474i	$-0.756521\pi$
$479$	−31.5791			−1.44289			−0.721443	+	0.692474i	$0.756521\pi$
$480$		0			0
$481$	−2.91520			−0.132922
$482$	9.79185	−	16.9600i	0.446007	−	0.772506i
$483$		0			0
$484$	6.05472	+	10.4871i	0.275215	+	0.476686i
$485$	8.13817	+	14.0957i	0.369535	+	0.640054i
$486$		0			0
$487$	−0.153087	+	0.265154i	−0.00693703	+	0.0120153i	−0.869473	−	0.493980i	$-0.835541\pi$
$487$	−0.153087	+	0.265154i	−0.00693703	+	0.0120153i	0.862536	+	0.505996i	$0.168875\pi$
$488$	−1.81855	+	3.14982i	−0.0823218	+	0.142586i
$489$		0			0
$490$		0			0
$491$	9.06981	−	15.7094i	0.409315	−	0.708954i	−0.585498	−	0.810674i	$-0.699101\pi$
$491$	9.06981	−	15.7094i	0.409315	−	0.708954i	0.994813	+	0.101720i	$0.0324345\pi$
$492$		0			0
$493$	0.0959447			0.00432113
$494$	−11.9329	+	20.6684i	−0.536887	+	0.929916i
$495$		0			0
$496$	−18.6531			−0.837549
$497$		0			0
$498$		0			0
$499$	−21.3091			−0.953928			−0.476964	−	0.878923i	$-0.658263\pi$
$499$	−21.3091			−0.953928			−0.476964	+	0.878923i	$0.658263\pi$
$500$	7.62478	+	13.2065i	0.340990	+	0.590613i
$501$		0			0
$502$	3.00701	−	5.20829i	0.134209	−	0.232457i
$503$	−17.0738			−0.761285			−0.380642	−	0.924722i	$-0.624297\pi$
$503$	−17.0738			−0.761285			−0.380642	+	0.924722i	$0.624297\pi$
$504$		0			0
$505$	1.49315			0.0664443
$506$	10.2529	−	17.7586i	0.455799	−	0.789466i
$507$		0			0
$508$	−5.90824	−	10.2334i	−0.262136	−	0.454032i
$509$	36.7735			1.62996			0.814979	−	0.579490i	$-0.196748\pi$
$509$	36.7735			1.62996			0.814979	+	0.579490i	$0.196748\pi$
$510$		0			0
$511$		0			0
$512$	21.4975			0.950065
$513$		0			0
$514$	4.32299	−	7.48764i	0.190679	−	0.330265i
$515$	−2.57667			−0.113542
$516$		0			0
$517$	−7.20083	+	12.4722i	−0.316692	+	0.548527i
$518$		0			0
$519$		0			0
$520$	−3.87870	+	6.71810i	−0.170092	+	0.294608i
$521$	−9.57535	+	16.5850i	−0.419504	+	0.726602i	−0.995890	−	0.0905758i	$-0.971129\pi$
$521$	−9.57535	+	16.5850i	−0.419504	+	0.726602i	0.576386	+	0.817178i	$0.304463\pi$
$522$		0			0
$523$	20.9715	+	36.3236i	0.917018	+	1.58832i	0.803920	+	0.594737i	$0.202744\pi$
$523$	20.9715	+	36.3236i	0.917018	+	1.58832i	0.113097	+	0.993584i	$0.463923\pi$
$524$	1.39952	+	2.42405i	0.0611385	+	0.105895i
$525$		0			0
$526$	17.9980	−	31.1734i	0.784749	−	1.35922i
$527$	−5.95991			−0.259618
$528$		0			0
$529$	31.1812			1.35570
$530$	−1.85641	−	3.21539i	−0.0806372	−	0.139668i
$531$		0			0
$532$		0			0
$533$	−23.3535	−	40.4494i	−1.01155	−	1.75206i
$534$		0			0
$535$	−3.85702	−	6.68056i	−0.166754	−	0.288826i
$536$	3.89258	+	6.74214i	0.168134	+	0.291216i
$537$		0			0
$538$	14.5157	+	25.1419i	0.625816	+	1.08395i
$539$		0			0
$540$		0			0
$541$	−1.44272	−	2.49886i	−0.0620273	−	0.107434i	0.833344	−	0.552754i	$-0.186423\pi$
$541$	−1.44272	−	2.49886i	−0.0620273	−	0.107434i	−0.895371	+	0.445320i	$0.853090\pi$
$542$	27.2460			1.17032
$543$		0			0
$544$	10.3472			0.443631
$545$	−5.50059	+	9.52731i	−0.235620	+	0.408105i
$546$		0			0
$547$	1.38738	+	2.40301i	0.0593201	+	0.102745i	0.894160	−	0.447747i	$-0.147773\pi$
$547$	1.38738	+	2.40301i	0.0593201	+	0.102745i	−0.834840	+	0.550492i	$0.814440\pi$
$548$	1.54128	+	2.66957i	0.0658401	+	0.114038i
$549$		0			0
$550$	−4.48300	+	7.76478i	−0.191156	+	0.331091i
$551$	−0.0775590	+	0.134336i	−0.00330413	+	0.00572291i
$552$		0			0
$553$		0			0
$554$	6.85975	−	11.8814i	0.291443	−	0.504794i
$555$		0			0
$556$	1.05005			0.0445319
$557$	−15.5344	+	26.9064i	−0.658214	+	1.14006i	0.322864	+	0.946445i	$0.395354\pi$
$557$	−15.5344	+	26.9064i	−0.658214	+	1.14006i	−0.981078	+	0.193614i	$0.937979\pi$
$558$		0			0
$559$	52.7939			2.23294
$560$		0			0
$561$		0			0
$562$	−47.8495			−2.01841
$563$	−0.144020	−	0.249451i	−0.00606973	−	0.0105131i	0.862975	−	0.505247i	$-0.168599\pi$
$563$	−0.144020	−	0.249451i	−0.00606973	−	0.0105131i	−0.869044	+	0.494734i	$0.835265\pi$
$564$		0			0
$565$	−9.67836	+	16.7634i	−0.407172	+	0.705242i
$566$	−34.5331			−1.45154
$567$		0			0
$568$	−13.8451			−0.580929
$569$	−8.04004	+	13.9258i	−0.337056	+	0.583798i	−0.983878	−	0.178843i	$-0.942765\pi$
$569$	−8.04004	+	13.9258i	−0.337056	+	0.583798i	0.646821	+	0.762641i	$0.276098\pi$
$570$		0			0
$571$	7.64289	+	13.2379i	0.319845	+	0.553988i	0.980456	−	0.196741i	$-0.0630358\pi$
$571$	7.64289	+	13.2379i	0.319845	+	0.553988i	−0.660610	+	0.750729i	$0.729702\pi$
$572$	10.8869			0.455204
$573$		0			0
$574$		0			0
$575$	−23.6902			−0.987950
$576$		0			0
$577$	−12.0812	+	20.9253i	−0.502949	+	0.871133i	0.497045	+	0.867725i	$0.334418\pi$
$577$	−12.0812	+	20.9253i	−0.502949	+	0.871133i	−0.999994	+	0.00340833i	$0.998915\pi$
$578$	−26.8836			−1.11821
$579$		0			0
$580$	0.0574683	−	0.0995380i	0.00238624	−	0.00413309i
$581$		0			0
$582$		0			0
$583$	1.14289	−	1.97955i	0.0473338	−	0.0819845i
$584$	−1.54881	+	2.68262i	−0.0640902	+	0.111007i
$585$		0			0
$586$	−2.26636	−	3.92546i	−0.0936226	−	0.162159i
$587$	18.0145	+	31.2020i	0.743537	+	1.28784i	0.950875	+	0.309574i	$0.100186\pi$
$587$	18.0145	+	31.2020i	0.743537	+	1.28784i	−0.207339	+	0.978269i	$0.566480\pi$
$588$		0			0
$589$	4.81783	−	8.34472i	0.198515	−	0.343838i
$590$	−20.7503			−0.854276
$591$		0			0
$592$	−2.73034			−0.112216
$593$	12.4668	+	21.5932i	0.511951	+	0.886726i	0.999904	+	0.0138558i	$0.00441057\pi$
$593$	12.4668	+	21.5932i	0.511951	+	0.886726i	−0.487953	+	0.872870i	$0.662256\pi$
$594$		0			0
$595$		0			0
$596$	4.57712	+	7.92780i	0.187486	+	0.324735i
$597$		0			0
$598$	35.0751	+	60.7518i	1.43433	+	2.48433i
$599$	19.7642	+	34.2325i	0.807542	+	1.39870i	0.914561	+	0.404447i	$0.132536\pi$
$599$	19.7642	+	34.2325i	0.807542	+	1.39870i	−0.107019	+	0.994257i	$0.534131\pi$
$600$		0			0
$601$	−1.86447	−	3.22936i	−0.0760534	−	0.131728i	0.825490	−	0.564416i	$-0.190899\pi$
$601$	−1.86447	−	3.22936i	−0.0760534	−	0.131728i	−0.901544	+	0.432688i	$0.857565\pi$
$602$		0			0
$603$		0			0
$604$	−8.80470	−	15.2502i	−0.358258	−	0.620521i
$605$	−11.6267			−0.472694
$606$		0			0
$607$	−23.6528			−0.960036			−0.480018	−	0.877259i	$-0.659370\pi$
$607$	−23.6528			−0.960036			−0.480018	+	0.877259i	$0.659370\pi$
$608$	−8.36436	+	14.4875i	−0.339219	+	0.587545i
$609$		0			0
$610$	3.98029	+	6.89407i	0.161157	+	0.279133i
$611$	−24.6339	−	42.6671i	−0.996580	−	1.72613i
$612$		0			0
$613$	1.89952	−	3.29006i	0.0767208	−	0.132884i	−0.825113	−	0.564968i	$-0.808888\pi$
$613$	1.89952	−	3.29006i	0.0767208	−	0.132884i	0.901833	+	0.432084i	$0.142222\pi$
$614$	−4.29264	+	7.43507i	−0.173237	+	0.300055i
$615$		0			0
$616$		0			0
$617$	17.5615	−	30.4174i	0.706999	−	1.22456i	−0.258966	−	0.965886i	$-0.583382\pi$
$617$	17.5615	−	30.4174i	0.706999	−	1.22456i	0.965965	−	0.258672i	$-0.0832849\pi$
$618$		0			0
$619$	21.1632			0.850622			0.425311	−	0.905047i	$-0.360165\pi$
$619$	21.1632			0.850622			0.425311	+	0.905047i	$0.360165\pi$
$620$	−3.56983	+	6.18312i	−0.143368	+	0.248320i
$621$		0			0
$622$	50.6011			2.02892
$623$		0			0
$624$		0			0
$625$	1.45048			0.0580192
$626$	5.05726	+	8.75943i	0.202129	+	0.350097i
$627$		0			0
$628$	−12.0301	+	20.8368i	−0.480054	+	0.831477i
$629$	−0.872381			−0.0347841
$630$		0			0
$631$	4.74845			0.189033			0.0945164	−	0.995523i	$-0.469870\pi$
$631$	4.74845			0.189033			0.0945164	+	0.995523i	$0.469870\pi$
$632$	−3.32329	+	5.75610i	−0.132193	+	0.228965i
$633$		0			0
$634$	9.09350	+	15.7504i	0.361149	+	0.625529i
$635$	11.3455			0.450231
$636$		0			0
$637$		0			0
$638$	0.172562			0.00683178
$639$		0			0
$640$	−5.71622	+	9.90078i	−0.225953	+	0.391363i
$641$	9.87469			0.390027			0.195013	−	0.980801i	$-0.437525\pi$
$641$	9.87469			0.390027			0.195013	+	0.980801i	$0.437525\pi$
$642$		0			0
$643$	−21.9748	+	38.0615i	−0.866602	+	1.50100i	−0.00115462	+	0.999999i	$0.500368\pi$
$643$	−21.9748	+	38.0615i	−0.866602	+	1.50100i	−0.865448	+	0.501000i	$0.832966\pi$
$644$		0			0
$645$		0			0
$646$	−3.57095	+	6.18507i	−0.140497	+	0.243348i
$647$	22.1936	−	38.4404i	0.872521	−	1.51125i	0.0131398	−	0.999914i	$-0.495817\pi$
$647$	22.1936	−	38.4404i	0.872521	−	1.51125i	0.859381	−	0.511336i	$-0.170849\pi$
$648$		0			0
$649$	−6.38743	−	11.0634i	−0.250729	−	0.434275i
$650$	−15.3362	−	26.5631i	−0.601537	−	1.04189i
$651$		0			0
$652$	8.49341	−	14.7110i	0.332628	−	0.576128i
$653$	−41.9912			−1.64324			−0.821622	−	0.570033i	$-0.806930\pi$
$653$	−41.9912			−1.64324			−0.821622	+	0.570033i	$0.806930\pi$
$654$		0			0
$655$	−2.68748			−0.105008
$656$	−21.8726	−	37.8844i	−0.853980	−	1.47914i
$657$		0			0
$658$		0			0
$659$	19.6365	+	34.0114i	0.764928	+	1.32489i	0.940284	+	0.340390i	$0.110559\pi$
$659$	19.6365	+	34.0114i	0.764928	+	1.32489i	−0.175356	+	0.984505i	$0.556108\pi$
$660$		0			0
$661$	−0.0933694	−	0.161721i	−0.00363165	−	0.00629020i	0.864204	−	0.503142i	$-0.167823\pi$
$661$	−0.0933694	−	0.161721i	−0.00363165	−	0.00629020i	−0.867836	+	0.496852i	$0.834489\pi$
$662$	19.0515	+	32.9982i	0.740459	+	1.28251i
$663$		0			0
$664$	−3.14522	−	5.44769i	−0.122058	−	0.211411i
$665$		0			0
$666$		0			0
$667$	0.227973	+	0.394862i	0.00882717	+	0.0152891i
$668$	−4.90028			−0.189597
$669$		0			0
$670$	17.0395			0.658294
$671$	−2.45046	+	4.24432i	−0.0945989	+	0.163850i
$672$		0			0
$673$	−5.43382	−	9.41166i	−0.209458	−	0.362793i	0.742086	−	0.670305i	$-0.233837\pi$
$673$	−5.43382	−	9.41166i	−0.209458	−	0.362793i	−0.951544	+	0.307512i	$0.900503\pi$
$674$	1.37862	+	2.38785i	0.0531026	+	0.0919764i
$675$		0			0
$676$	−9.58584	+	16.6032i	−0.368686	+	0.638583i
$677$	−14.1950	+	24.5865i	−0.545560	+	0.944937i	0.453012	+	0.891505i	$0.350350\pi$
$677$	−14.1950	+	24.5865i	−0.545560	+	0.944937i	−0.998571	+	0.0534326i	$0.982984\pi$
$678$		0			0
$679$		0			0
$680$	−1.16071	+	2.01041i	−0.0445111	+	0.0770956i
$681$		0			0
$682$	−10.7192			−0.410460
$683$	−5.92034	+	10.2543i	−0.226536	+	0.392371i	−0.956779	−	0.290816i	$-0.906073\pi$
$683$	−5.92034	+	10.2543i	−0.226536	+	0.392371i	0.730243	+	0.683187i	$0.239407\pi$
$684$		0			0
$685$	−2.95968			−0.113083
$686$		0			0
$687$		0			0
$688$	49.4461			1.88511
$689$	3.90981	+	6.77199i	0.148952	+	0.257992i
$690$		0			0
$691$	5.95416	−	10.3129i	0.226507	−	0.392321i	−0.730264	−	0.683165i	$-0.760603\pi$
$691$	5.95416	−	10.3129i	0.226507	−	0.392321i	0.956770	+	0.290844i	$0.0939361\pi$
$692$	14.0976			0.535909
$693$		0			0
$694$	54.3880			2.06454
$695$	−0.504096	+	0.873119i	−0.0191214	+	0.0331193i
$696$		0			0
$697$	−6.98857	−	12.1046i	−0.264711	−	0.458493i
$698$	66.2870			2.50900
$699$		0			0
$700$		0			0
$701$	31.3902			1.18559			0.592795	−	0.805353i	$-0.298024\pi$
$701$	31.3902			1.18559			0.592795	+	0.805353i	$0.298024\pi$
$702$		0			0
$703$	0.705208	−	1.22146i	0.0265974	−	0.0460681i
$704$	3.94041			0.148510
$705$		0			0
$706$	27.1518	−	47.0284i	1.02187	−	1.76994i
$707$		0			0
$708$		0			0
$709$	−0.312609	+	0.541455i	−0.0117403	+	0.0203348i	−0.871836	−	0.489798i	$-0.837070\pi$
$709$	−0.312609	+	0.541455i	−0.0117403	+	0.0203348i	0.860096	+	0.510133i	$0.170404\pi$
$710$	−15.1516	+	26.2433i	−0.568628	+	0.984893i
$711$		0			0
$712$	−0.789685	−	1.36777i	−0.0295947	−	0.0512595i
$713$	−14.1613	−	24.5281i	−0.530345	−	0.918584i
$714$		0			0
$715$	−5.22647	+	9.05251i	−0.195459	+	0.338545i
$716$	2.36489			0.0883802
$717$		0			0
$718$	9.96239			0.371793
$719$	12.1969	+	21.1257i	0.454869	+	0.787857i	0.998681	−	0.0513506i	$-0.0163526\pi$
$719$	12.1969	+	21.1257i	0.454869	+	0.787857i	−0.543811	+	0.839208i	$0.683019\pi$
$720$		0			0
$721$		0			0
$722$	11.7185	+	20.2970i	0.436116	+	0.755376i
$723$		0			0
$724$	−11.8123	−	20.4595i	−0.439002	−	0.760373i
$725$	−0.0996792	−	0.172649i	−0.00370199	−	0.00641204i
$726$		0			0
$727$	18.9253	+	32.7796i	0.701900	+	1.21573i	0.967799	+	0.251726i	$0.0809980\pi$
$727$	18.9253	+	32.7796i	0.701900	+	1.21573i	−0.265899	+	0.964001i	$0.585669\pi$
$728$		0			0
$729$		0			0
$730$	3.38991	+	5.87150i	0.125466	+	0.217314i
$731$	15.7987			0.584335
$732$		0			0
$733$	−2.40155			−0.0887033			−0.0443516	−	0.999016i	$-0.514122\pi$
$733$	−2.40155			−0.0887033			−0.0443516	+	0.999016i	$0.514122\pi$
$734$	−21.2519	+	36.8093i	−0.784421	+	1.35866i
$735$		0			0
$736$	24.5858	+	42.5839i	0.906245	+	1.56966i
$737$	5.24517	+	9.08490i	0.193208	+	0.334646i
$738$		0			0
$739$	−15.1940	+	26.3167i	−0.558920	+	0.968077i	0.438667	+	0.898650i	$0.355451\pi$
$739$	−15.1940	+	26.3167i	−0.558920	+	0.968077i	−0.997587	+	0.0694277i	$0.977883\pi$
$740$	−0.522533	+	0.905053i	−0.0192087	+	0.0332704i
$741$		0			0
$742$		0			0
$743$	2.54785	−	4.41300i	0.0934715	−	0.161897i	−0.815498	−	0.578760i	$-0.803537\pi$
$743$	2.54785	−	4.41300i	0.0934715	−	0.161897i	0.908970	+	0.416862i	$0.136870\pi$
$744$		0			0
$745$	−8.78934			−0.322016
$746$	−19.7961	+	34.2879i	−0.724787	+	1.25537i
$747$		0			0
$748$	3.25793			0.119122
$749$		0			0
$750$		0			0
$751$	−0.975011			−0.0355787			−0.0177893	−	0.999842i	$-0.505663\pi$
$751$	−0.975011			−0.0355787			−0.0177893	+	0.999842i	$0.505663\pi$
$752$	−23.0718	−	39.9615i	−0.841341	−	1.45724i
$753$		0			0
$754$	−0.295165	+	0.511240i	−0.0107493	+	0.0186183i
$755$	16.9075			0.615326
$756$		0			0
$757$	11.6346			0.422865			0.211433	−	0.977393i	$-0.432187\pi$
$757$	11.6346			0.422865			0.211433	+	0.977393i	$0.432187\pi$
$758$	−5.26750	+	9.12357i	−0.191324	+	0.331383i
$759$		0			0
$760$	−1.87657	−	3.25031i	−0.0680703	−	0.117901i
$761$	−54.1749			−1.96384			−0.981920	−	0.189298i	$-0.939379\pi$
$761$	−54.1749			−1.96384			−0.981920	+	0.189298i	$0.939379\pi$
$762$		0			0
$763$		0			0
$764$	−31.5484			−1.14138
$765$		0			0
$766$	32.1489	−	55.6835i	1.16159	−	2.01193i
$767$	43.7025			1.57801
$768$		0			0
$769$	10.4326	−	18.0698i	0.376208	−	0.651612i	−0.614299	−	0.789074i	$-0.710561\pi$
$769$	10.4326	−	18.0698i	0.376208	−	0.651612i	0.990507	+	0.137462i	$0.0438943\pi$
$770$		0			0
$771$		0			0
$772$	−4.30047	+	7.44863i	−0.154777	+	0.268082i
$773$	−27.4972	+	47.6266i	−0.989007	+	1.71301i	−0.366447	+	0.930439i	$0.619426\pi$
$773$	−27.4972	+	47.6266i	−0.989007	+	1.71301i	−0.622561	+	0.782572i	$0.713908\pi$
$774$		0			0
$775$	6.19189	+	10.7247i	0.222419	+	0.385242i
$776$	−6.84616	−	11.8579i	−0.245763	−	0.425674i
$777$		0			0
$778$	−26.5895	+	46.0544i	−0.953281	+	1.65113i
$779$	22.5975			0.809639
$780$		0			0
$781$	−18.6560			−0.667566
$782$	10.4963	+	18.1801i	0.375346	+	0.650119i
$783$		0			0
$784$		0			0
$785$	−11.5506	−	20.0062i	−0.412258	−	0.714051i
$786$		0			0
$787$	4.59475	+	7.95833i	0.163785	+	0.283684i	0.936223	−	0.351406i	$-0.114296\pi$
$787$	4.59475	+	7.95833i	0.163785	+	0.283684i	−0.772438	+	0.635090i	$0.780963\pi$
$788$	6.79103	+	11.7624i	0.241921	+	0.419019i
$789$		0			0
$790$	7.27374	+	12.5985i	0.258788	+	0.448234i
$791$		0			0
$792$		0			0
$793$	−8.38296	−	14.5197i	−0.297688	−	0.515610i
$794$	20.5934			0.730832
$795$		0			0
$796$

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)

\(f(q)\)	\(=\)	\(q+(-0.920620 + 1.59456i) q^{2} +(-0.695084 - 1.20392i) q^{4} +1.33475 q^{5} -1.12285 q^{8} +O(q^{10})\)	\(q+(-0.920620 + 1.59456i) q^{2} +(-0.695084 - 1.20392i) q^{4} +1.33475 q^{5} -1.12285 q^{8} +(-1.22880 + 2.12835i) q^{10} -1.51302 q^{11} +(2.58800 - 4.48254i) q^{13} +(2.42388 - 4.19829i) q^{16} +(0.774463 - 1.34141i) q^{17} +(1.25211 + 2.16872i) q^{19} +(-0.927765 - 1.60694i) q^{20} +(1.39291 - 2.41260i) q^{22} +7.36079 q^{23} -3.21843 q^{25} +(4.76513 + 8.25344i) q^{26} +(0.0309713 + 0.0536439i) q^{29} +(-1.92388 - 3.33227i) q^{31} +(3.34011 + 5.78523i) q^{32} +(1.42597 + 2.46986i) q^{34} +(-0.281608 - 0.487760i) q^{37} -4.61087 q^{38} -1.49873 q^{40} +(4.51188 - 7.81481i) q^{41} +(5.09988 + 8.83325i) q^{43} +(1.05167 + 1.82155i) q^{44} +(-6.77649 + 11.7372i) q^{46} +(4.75925 - 8.24327i) q^{47} +(2.96296 - 5.13199i) q^{50} -7.19550 q^{52} +(-0.755374 + 1.30835i) q^{53} -2.01950 q^{55} -0.114051 q^{58} +(4.22166 + 7.31212i) q^{59} +(1.61958 - 2.80520i) q^{61} +7.08467 q^{62} -2.60434 q^{64} +(3.45434 - 5.98309i) q^{65} +(-3.46670 - 6.00449i) q^{67} -2.15327 q^{68} +12.3304 q^{71} +(1.37936 - 2.38912i) q^{73} +1.03702 q^{74} +(1.74064 - 3.01488i) q^{76} +(2.95969 - 5.12633i) q^{79} +(3.23529 - 5.60368i) q^{80} +(8.30746 + 14.3889i) q^{82} +(2.80111 + 4.85167i) q^{83} +(1.03372 - 1.79045i) q^{85} -18.7802 q^{86} +1.69889 q^{88} +(0.703287 + 1.21813i) q^{89} +(-5.11636 - 8.86180i) q^{92} +(8.76293 + 15.1778i) q^{94} +(1.67126 + 2.89470i) q^{95} +(6.09713 + 10.5605i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) \)	\( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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