LMFDB - Embedded newform 1323.2.g.h.361.5

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			361.5
Character	$\chi$	$=$	1323.361
Dual form			1323.2.g.h.667.5

$q$-expansion

$f(q)$	$=$	$q+(-0.551407 + 0.955065i) q^{2} +(0.391901 + 0.678793i) q^{4} -0.105466 q^{5} -3.07001 q^{8} +O(q^{10})$	\(q+(-0.551407 + 0.955065i) q^{2} +(0.391901 + 0.678793i) q^{4} -0.105466 q^{5} -3.07001 q^{8} +(0.0581547 - 0.100727i) q^{10} -3.33731 q^{11} +(-1.23997 + 2.14770i) q^{13} +(0.909025 - 1.57448i) q^{16} +(0.806594 - 1.39706i) q^{17} +(3.84133 + 6.65338i) q^{19} +(-0.0413323 - 0.0715896i) q^{20} +(1.84022 - 3.18735i) q^{22} +1.89719 q^{23} -4.98888 q^{25} +(-1.36746 - 2.36851i) q^{26} +(-4.64521 - 8.04574i) q^{29} +(-4.63081 - 8.02080i) q^{31} +(-2.06753 - 3.58107i) q^{32} +(0.889523 + 1.54070i) q^{34} +(0.991268 + 1.71693i) q^{37} -8.47254 q^{38} +0.323782 q^{40} +(-3.74268 + 6.48252i) q^{41} +(-3.77388 - 6.53655i) q^{43} +(-1.30790 - 2.26534i) q^{44} +(-1.04612 + 1.81194i) q^{46} +(-1.59780 + 2.76747i) q^{47} +(2.75090 - 4.76470i) q^{50} -1.94379 q^{52} +(-4.98839 + 8.64015i) q^{53} +0.351974 q^{55} +10.2456 q^{58} +(2.22993 + 3.86235i) q^{59} +(2.83550 - 4.91123i) q^{61} +10.2138 q^{62} +8.19630 q^{64} +(0.130775 - 0.226509i) q^{65} +(-4.98571 - 8.63550i) q^{67} +1.26442 q^{68} -3.29042 q^{71} +(2.36189 - 4.09091i) q^{73} -2.18637 q^{74} +(-3.01084 + 5.21493i) q^{76} +(-3.84705 + 6.66328i) q^{79} +(-0.0958713 + 0.166054i) q^{80} +(-4.12748 - 7.14901i) q^{82} +(0.584428 + 1.01226i) q^{83} +(-0.0850683 + 0.147343i) q^{85} +8.32378 q^{86} +10.2456 q^{88} +(3.01477 + 5.22173i) q^{89} +(0.743509 + 1.28780i) q^{92} +(-1.76208 - 3.05201i) q^{94} +(-0.405130 - 0.701706i) q^{95} +(-1.90127 - 3.29310i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.551407	+	0.955065i	−0.389903	+	0.675333i	−0.992436	−	0.122762i	$-0.960825\pi$
$2$	−0.551407	+	0.955065i	−0.389903	+	0.675333i	0.602533	+	0.798094i	$0.294158\pi$
$3$		0			0
$3$		0			0
$4$	0.391901	+	0.678793i	0.195951	+	0.339396i
$5$	−0.105466			−0.0471659			−0.0235829	−	0.999722i	$-0.507507\pi$
$5$	−0.105466			−0.0471659			−0.0235829	+	0.999722i	$0.507507\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−3.07001			−1.08541
$9$		0			0
$10$	0.0581547	−	0.100727i	0.0183901	−	0.0318527i
$11$	−3.33731			−1.00624			−0.503119	−	0.864217i	$-0.667814\pi$
$11$	−3.33731			−1.00624			−0.503119	+	0.864217i	$0.667814\pi$
$12$		0			0
$13$	−1.23997	+	2.14770i	−0.343907	+	0.595664i	−0.985155	−	0.171670i	$-0.945084\pi$
$13$	−1.23997	+	2.14770i	−0.343907	+	0.595664i	0.641248	+	0.767334i	$0.278417\pi$
$14$		0			0
$15$		0			0
$16$	0.909025	−	1.57448i	0.227256	−	0.393619i
$17$	0.806594	−	1.39706i	0.195628	−	0.338837i	−0.751478	−	0.659758i	$-0.770659\pi$
$17$	0.806594	−	1.39706i	0.195628	−	0.338837i	0.947106	+	0.320921i	$0.103992\pi$
$18$		0			0
$19$	3.84133	+	6.65338i	0.881262	+	1.52639i	0.849939	+	0.526880i	$0.176638\pi$
$19$	3.84133	+	6.65338i	0.881262	+	1.52639i	0.0313221	+	0.999509i	$0.490028\pi$
$20$	−0.0413323	−	0.0715896i	−0.00924218	−	0.0160079i
$21$		0			0
$22$	1.84022	−	3.18735i	0.392336	−	0.679546i
$23$	1.89719			0.395591			0.197795	−	0.980243i	$-0.436622\pi$
$23$	1.89719			0.395591			0.197795	+	0.980243i	$0.436622\pi$
$24$		0			0
$25$	−4.98888			−0.997775
$26$	−1.36746	−	2.36851i	−0.268181	−	0.464503i
$27$		0			0
$28$		0			0
$29$	−4.64521	−	8.04574i	−0.862594	−	1.49406i	−0.869416	−	0.494080i	$-0.835505\pi$
$29$	−4.64521	−	8.04574i	−0.862594	−	1.49406i	0.00682200	−	0.999977i	$-0.497828\pi$
$30$		0			0
$31$	−4.63081	−	8.02080i	−0.831718	−	1.44058i	−0.896675	−	0.442689i	$-0.854024\pi$
$31$	−4.63081	−	8.02080i	−0.831718	−	1.44058i	0.0649574	−	0.997888i	$-0.479309\pi$
$32$	−2.06753	−	3.58107i	−0.365491	−	0.633049i
$33$		0			0
$34$	0.889523	+	1.54070i	0.152552	+	0.264228i
$35$		0			0
$36$		0			0
$37$	0.991268	+	1.71693i	0.162963	+	0.282261i	0.935930	−	0.352186i	$-0.114561\pi$
$37$	0.991268	+	1.71693i	0.162963	+	0.282261i	−0.772967	+	0.634447i	$0.781228\pi$
$38$	−8.47254			−1.37443
$39$		0			0
$40$	0.323782			0.0511945
$41$	−3.74268	+	6.48252i	−0.584509	+	1.01240i	0.410427	+	0.911893i	$0.365379\pi$
$41$	−3.74268	+	6.48252i	−0.584509	+	1.01240i	−0.994936	+	0.100506i	$0.967954\pi$
$42$		0			0
$43$	−3.77388	−	6.53655i	−0.575512	−	0.996815i	−0.995986	−	0.0895108i	$-0.971470\pi$
$43$	−3.77388	−	6.53655i	−0.575512	−	0.996815i	0.420474	−	0.907304i	$-0.361864\pi$
$44$	−1.30790	−	2.26534i	−0.197173	−	0.341514i
$45$		0			0
$46$	−1.04612	+	1.81194i	−0.154242	+	0.267155i
$47$	−1.59780	+	2.76747i	−0.233063	+	0.403677i	−0.958708	−	0.284392i	$-0.908208\pi$
$47$	−1.59780	+	2.76747i	−0.233063	+	0.403677i	0.725645	+	0.688070i	$0.241542\pi$
$48$		0			0
$49$		0			0
$50$	2.75090	−	4.76470i	0.389036	−	0.673830i
$51$		0			0
$52$	−1.94379			−0.269555
$53$	−4.98839	+	8.64015i	−0.685209	+	1.18682i	0.288163	+	0.957581i	$0.406956\pi$
$53$	−4.98839	+	8.64015i	−0.685209	+	1.18682i	−0.973371	+	0.229234i	$0.926378\pi$
$54$		0			0
$55$	0.351974			0.0474601
$56$		0			0
$57$		0			0
$58$	10.2456			1.34531
$59$	2.22993	+	3.86235i	0.290312	+	0.502836i	0.973884	−	0.227048i	$-0.0729075\pi$
$59$	2.22993	+	3.86235i	0.290312	+	0.502836i	−0.683571	+	0.729884i	$0.739574\pi$
$60$		0			0
$61$	2.83550	−	4.91123i	0.363048	−	0.628818i	−0.625413	−	0.780294i	$-0.715069\pi$
$61$	2.83550	−	4.91123i	0.363048	−	0.628818i	0.988461	+	0.151476i	$0.0484027\pi$
$62$	10.2138			1.29716
$63$		0			0
$64$	8.19630			1.02454
$65$	0.130775	−	0.226509i	0.0162207	−	0.0280950i
$66$		0			0
$67$	−4.98571	−	8.63550i	−0.609101	−	1.05499i	−0.991389	−	0.130951i	$-0.958197\pi$
$67$	−4.98571	−	8.63550i	−0.609101	−	1.05499i	0.382288	−	0.924043i	$-0.375136\pi$
$68$	1.26442			0.153333
$69$		0			0
$70$		0			0
$71$	−3.29042			−0.390502			−0.195251	−	0.980753i	$-0.562552\pi$
$71$	−3.29042			−0.390502			−0.195251	+	0.980753i	$0.562552\pi$
$72$		0			0
$73$	2.36189	−	4.09091i	0.276438	−	0.478805i	−0.694059	−	0.719919i	$-0.744179\pi$
$73$	2.36189	−	4.09091i	0.276438	−	0.478805i	0.970497	+	0.241113i	$0.0775125\pi$
$74$	−2.18637			−0.254160
$75$		0			0
$76$	−3.01084	+	5.21493i	−0.345367	+	0.598194i
$77$		0			0
$78$		0			0
$79$	−3.84705	+	6.66328i	−0.432827	+	0.749678i	−0.997115	−	0.0758997i	$-0.975817\pi$
$79$	−3.84705	+	6.66328i	−0.432827	+	0.749678i	0.564289	+	0.825577i	$0.309150\pi$
$80$	−0.0958713	+	0.166054i	−0.0107187	+	0.0185654i
$81$		0			0
$82$	−4.12748	−	7.14901i	−0.455804	−	0.789476i
$83$	0.584428	+	1.01226i	0.0641493	+	0.111110i	0.896316	−	0.443415i	$-0.146233\pi$
$83$	0.584428	+	1.01226i	0.0641493	+	0.111110i	−0.832167	+	0.554525i	$0.812900\pi$
$84$		0			0
$85$	−0.0850683	+	0.147343i	−0.00922695	+	0.0159815i
$86$	8.32378			0.897576
$87$		0			0
$88$	10.2456			1.09219
$89$	3.01477	+	5.22173i	0.319565	+	0.553503i	0.980397	−	0.197031i	$-0.0631299\pi$
$89$	3.01477	+	5.22173i	0.319565	+	0.553503i	−0.660832	+	0.750534i	$0.729797\pi$
$90$		0			0
$91$		0			0
$92$	0.743509	+	1.28780i	0.0775162	+	0.134262i
$93$		0			0
$94$	−1.76208	−	3.05201i	−0.181744	−	0.314791i
$95$	−0.405130	−	0.701706i	−0.0415655	−	0.0719935i
$96$		0			0
$97$	−1.90127	−	3.29310i	−0.193045	−	0.334364i	0.753213	−	0.657777i	$-0.228503\pi$
$97$	−1.90127	−	3.29310i	−0.193045	−	0.334364i	−0.946258	+	0.323413i	$0.895170\pi$
$98$		0			0
$99$		0			0
$100$	−1.95515	−	3.38641i	−0.195515	−	0.338641i
$101$	−17.4702			−1.73835			−0.869177	−	0.494501i	$-0.835351\pi$
$101$	−17.4702			−1.73835			−0.869177	+	0.494501i	$0.835351\pi$
$102$		0			0
$103$	−8.73204			−0.860394			−0.430197	−	0.902735i	$-0.641556\pi$
$103$	−8.73204			−0.860394			−0.430197	+	0.902735i	$0.641556\pi$
$104$	3.80674	−	6.59346i	0.373281	−	0.646542i
$105$		0			0
$106$	−5.50127	−	9.52848i	−0.534330	−	0.925487i
$107$	−9.07316	−	15.7152i	−0.877135	−	1.51924i	−0.854471	−	0.519500i	$-0.826118\pi$
$107$	−9.07316	−	15.7152i	−0.877135	−	1.51924i	−0.0226645	−	0.999743i	$-0.507215\pi$
$108$		0			0
$109$	2.11124	−	3.65678i	0.202220	−	0.350256i	−0.747023	−	0.664798i	$-0.768518\pi$
$109$	2.11124	−	3.65678i	0.202220	−	0.350256i	0.949243	+	0.314542i	$0.101851\pi$
$110$	−0.194081	+	0.336157i	−0.0185049	+	0.0320514i
$111$		0			0
$112$		0			0
$113$	−1.02824	+	1.78096i	−0.0967285	+	0.167539i	−0.910329	−	0.413886i	$-0.864171\pi$
$113$	−1.02824	+	1.78096i	−0.0967285	+	0.167539i	0.813600	+	0.581425i	$0.197505\pi$
$114$		0			0
$115$	−0.200089			−0.0186584
$116$	3.64093	−	6.30627i	0.338052	−	0.585523i
$117$		0			0
$118$	−4.91840			−0.452775
$119$		0			0
$120$		0			0
$121$	0.137670			0.0125155
$122$	3.12703	+	5.41617i	0.283108	+	0.490357i
$123$		0			0
$124$	3.62964	−	6.28672i	0.325951	−	0.564564i
$125$	1.05349			0.0942268
$126$		0			0
$127$	0.317159			0.0281433			0.0140717	−	0.999901i	$-0.495521\pi$
$127$	0.317159			0.0281433			0.0140717	+	0.999901i	$0.495521\pi$
$128$	−0.384435	+	0.665862i	−0.0339796	+	0.0588544i
$129$		0			0
$130$	0.144221	+	0.249797i	0.0126490	+	0.0219087i
$131$	14.9563			1.30674			0.653370	−	0.757039i	$-0.273355\pi$
$131$	14.9563			1.30674			0.653370	+	0.757039i	$0.273355\pi$
$132$		0			0
$133$		0			0
$134$	10.9966			0.949962
$135$		0			0
$136$	−2.47625	+	4.28900i	−0.212337	+	0.367779i
$137$	15.2473			1.30267			0.651334	−	0.758791i	$-0.274210\pi$
$137$	15.2473			1.30267			0.651334	+	0.758791i	$0.274210\pi$
$138$		0			0
$139$	−4.05943	+	7.03114i	−0.344316	+	0.596374i	−0.985229	−	0.171240i	$-0.945223\pi$
$139$	−4.05943	+	7.03114i	−0.344316	+	0.596374i	0.640913	+	0.767614i	$0.278556\pi$
$140$		0			0
$141$		0			0
$142$	1.81436	−	3.14257i	0.152258	−	0.263718i
$143$	4.13818	−	7.16754i	0.346052	−	0.599380i
$144$		0			0
$145$	0.489912	+	0.848553i	0.0406850	+	0.0704685i
$146$	2.60473	+	4.51152i	0.215569	+	0.373376i
$147$		0			0
$148$	−0.776958	+	1.34573i	−0.0638656	+	0.110618i
$149$	11.1486			0.913329			0.456664	−	0.889639i	$-0.349044\pi$
$149$	11.1486			0.913329			0.456664	+	0.889639i	$0.349044\pi$
$150$		0			0
$151$	−11.2735			−0.917425			−0.458713	−	0.888585i	$-0.651689\pi$
$151$	−11.2735			−0.917425			−0.458713	+	0.888585i	$0.651689\pi$
$152$	−11.7929	−	20.4260i	−0.956534	−	1.65677i
$153$		0			0
$154$		0			0
$155$	0.488393	+	0.845922i	0.0392287	+	0.0679461i
$156$		0			0
$157$	6.10318	+	10.5710i	0.487087	+	0.843659i	0.999890	−	0.0148476i	$-0.00472630\pi$
$157$	6.10318	+	10.5710i	0.487087	+	0.843659i	−0.512803	+	0.858506i	$0.671393\pi$
$158$	−4.24258	−	7.34836i	−0.337521	−	0.584604i
$159$		0			0
$160$	0.218054	+	0.377681i	0.0172387	+	0.0298583i
$161$		0			0
$162$		0			0
$163$	−4.48132	−	7.76187i	−0.351004	−	0.607957i	0.635422	−	0.772165i	$-0.280826\pi$
$163$	−4.48132	−	7.76187i	−0.351004	−	0.607957i	−0.986426	+	0.164209i	$0.947493\pi$
$164$	−5.86705			−0.458139
$165$		0			0
$166$	−1.28903			−0.100048
$167$	−8.70833	+	15.0833i	−0.673871	+	1.16718i	0.302927	+	0.953014i	$0.402036\pi$
$167$	−8.70833	+	15.0833i	−0.673871	+	1.16718i	−0.976798	+	0.214165i	$0.931297\pi$
$168$		0			0
$169$	3.42493	+	5.93216i	0.263456	+	0.456320i
$170$	−0.0938145	−	0.162491i	−0.00719524	−	0.0124625i
$171$		0			0
$172$	2.95798	−	5.12337i	0.225544	−	0.390653i
$173$	−1.41466	+	2.45027i	−0.107555	+	0.186291i	−0.914779	−	0.403954i	$-0.867635\pi$
$173$	−1.41466	+	2.45027i	−0.107555	+	0.186291i	0.807224	+	0.590245i	$0.200969\pi$
$174$		0			0
$175$		0			0
$176$	−3.03370	+	5.25453i	−0.228674	+	0.396075i
$177$		0			0
$178$	−6.64946			−0.498398
$179$	−5.08135	+	8.80115i	−0.379798	+	0.657829i	−0.991033	−	0.133620i	$-0.957340\pi$
$179$	−5.08135	+	8.80115i	−0.379798	+	0.657829i	0.611235	+	0.791449i	$0.290673\pi$
$180$		0			0
$181$	17.0870			1.27006			0.635032	−	0.772486i	$-0.280987\pi$
$181$	17.0870			1.27006			0.635032	+	0.772486i	$0.280987\pi$
$182$		0			0
$183$		0			0
$184$	−5.82439			−0.429380
$185$	−0.104545	−	0.181078i	−0.00768631	−	0.0133131i
$186$		0			0
$187$	−2.69186	+	4.66243i	−0.196848	+	0.340951i
$188$	−2.50472			−0.182676
$189$		0			0
$190$	0.893566			0.0648261
$191$	−11.2000	+	19.3990i	−0.810404	+	1.40366i	0.102178	+	0.994766i	$0.467419\pi$
$191$	−11.2000	+	19.3990i	−0.810404	+	1.40366i	−0.912582	+	0.408894i	$0.865914\pi$
$192$		0			0
$193$	0.128393	+	0.222383i	0.00924194	+	0.0160075i	0.870609	−	0.491975i	$-0.163725\pi$
$193$	0.128393	+	0.222383i	0.00924194	+	0.0160075i	−0.861367	+	0.507982i	$0.830391\pi$
$194$	4.19350			0.301076
$195$		0			0
$196$		0			0
$197$	0.763370			0.0543878			0.0271939	−	0.999630i	$-0.491343\pi$
$197$	0.763370			0.0543878			0.0271939	+	0.999630i	$0.491343\pi$
$198$		0			0
$199$	−2.51561	+	4.35716i	−0.178327	+	0.308871i	−0.941307	−	0.337550i	$-0.890402\pi$
$199$	−2.51561	+	4.35716i	−0.178327	+	0.308871i	0.762981	+	0.646421i	$0.223735\pi$
$200$	15.3159			1.08300
$201$		0			0
$202$	9.63321	−	16.6852i	0.677790	−	1.17397i
$203$		0			0
$204$		0			0
$205$	0.394726	−	0.683686i	0.0275689	−	0.0477507i
$206$	4.81491	−	8.33966i	0.335470	−	0.581052i
$207$		0			0
$208$	2.25433	+	3.90462i	0.156310	+	0.270737i
$209$	−12.8197	−	22.2044i	−0.886759	−	1.53591i
$210$		0			0
$211$	−3.60537	+	6.24468i	−0.248204	+	0.429901i	−0.963027	−	0.269403i	$-0.913174\pi$
$211$	−3.60537	+	6.24468i	−0.248204	+	0.429901i	0.714824	+	0.699305i	$0.246507\pi$
$212$	−7.81983			−0.537068
$213$		0			0
$214$	20.0120			1.36799
$215$	0.398017	+	0.689385i	0.0271445	+	0.0470157i
$216$		0			0
$217$		0			0
$218$	2.32831	+	4.03274i	0.157693	+	0.273132i
$219$		0			0
$220$	0.137939	+	0.238917i	0.00929983	+	0.0161078i
$221$	2.00031	+	3.46464i	0.134555	+	0.233057i
$222$		0			0
$223$	−5.59106	−	9.68400i	−0.374405	−	0.648488i	0.615833	−	0.787877i	$-0.288820\pi$
$223$	−5.59106	−	9.68400i	−0.374405	−	0.648488i	−0.990238	+	0.139388i	$0.955486\pi$
$224$		0			0
$225$		0			0
$226$	−1.13395	−	1.96407i	−0.0754295	−	0.130648i
$227$	23.7706			1.57771			0.788857	−	0.614577i	$-0.210673\pi$
$227$	23.7706			1.57771			0.788857	+	0.614577i	$0.210673\pi$
$228$		0			0
$229$	1.90547			0.125917			0.0629586	−	0.998016i	$-0.479946\pi$
$229$	1.90547			0.125917			0.0629586	+	0.998016i	$0.479946\pi$
$230$	0.110330	−	0.191098i	0.00727497	−	0.0126006i
$231$		0			0
$232$	14.2609	+	24.7006i	0.936272	+	1.62167i
$233$	3.27092	+	5.66540i	0.214285	+	0.371153i	0.953051	−	0.302809i	$-0.0979245\pi$
$233$	3.27092	+	5.66540i	0.214285	+	0.371153i	−0.738766	+	0.673962i	$0.764591\pi$
$234$		0			0
$235$	0.168514	−	0.291875i	0.0109926	−	0.0190398i
$236$	−1.74782	+	3.02732i	−0.113774	+	0.197062i
$237$		0			0
$238$		0			0
$239$	−10.6735	+	18.4870i	−0.690409	+	1.19582i	0.281295	+	0.959621i	$0.409236\pi$
$239$	−10.6735	+	18.4870i	−0.690409	+	1.19582i	−0.971704	+	0.236202i	$0.924097\pi$
$240$		0			0
$241$	20.0662			1.29258			0.646288	−	0.763094i	$-0.276321\pi$
$241$	20.0662			1.29258			0.646288	+	0.763094i	$0.276321\pi$
$242$	−0.0759124	+	0.131484i	−0.00487983	+	0.00845212i
$243$		0			0
$244$	4.44494			0.284558
$245$		0			0
$246$		0			0
$247$	−19.0526			−1.21229
$248$	14.2167	+	24.6240i	0.902758	+	1.56362i
$249$		0			0
$250$	−0.580900	+	1.00615i	−0.0367394	+	0.0636344i
$251$	−6.81467			−0.430138			−0.215069	−	0.976599i	$-0.568998\pi$
$251$	−6.81467			−0.430138			−0.215069	+	0.976599i	$0.568998\pi$
$252$		0			0
$253$	−6.33151			−0.398059
$254$	−0.174884	+	0.302907i	−0.0109732	+	0.0190061i
$255$		0			0
$256$	7.77234	+	13.4621i	0.485771	+	0.841380i
$257$	−14.3883			−0.897518			−0.448759	−	0.893653i	$-0.648134\pi$
$257$	−14.3883			−0.897518			−0.448759	+	0.893653i	$0.648134\pi$
$258$		0			0
$259$		0			0
$260$	0.205004			0.0127138
$261$		0			0
$262$	−8.24701	+	14.2842i	−0.509502	+	0.882484i
$263$	1.53901			0.0948992			0.0474496	−	0.998874i	$-0.484891\pi$
$263$	1.53901			0.0948992			0.0474496	+	0.998874i	$0.484891\pi$
$264$		0			0
$265$	0.526106	−	0.911243i	0.0323185	−	0.0559772i
$266$		0			0
$267$		0			0
$268$	3.90781	−	6.76852i	0.238707	−	0.413453i
$269$	13.1285	−	22.7393i	0.800461	−	1.38644i	−0.118852	−	0.992912i	$-0.537921\pi$
$269$	13.1285	−	22.7393i	0.800461	−	1.38644i	0.919313	−	0.393527i	$-0.128745\pi$
$270$		0			0
$271$	8.96673	+	15.5308i	0.544690	+	0.943431i	0.998626	+	0.0523969i	$0.0166861\pi$
$271$	8.96673	+	15.5308i	0.544690	+	0.943431i	−0.453936	+	0.891034i	$0.649981\pi$
$272$	−1.46643	−	2.53993i	−0.0889152	−	0.154006i
$273$		0			0
$274$	−8.40748	+	14.5622i	−0.507915	+	0.879734i
$275$	16.6495			1.00400
$276$		0			0
$277$	−18.8713			−1.13386			−0.566932	−	0.823764i	$-0.691870\pi$
$277$	−18.8713			−1.13386			−0.566932	+	0.823764i	$0.691870\pi$
$278$	−4.47680	−	7.75404i	−0.268500	−	0.465056i
$279$		0			0
$280$		0			0
$281$	2.49578	+	4.32283i	0.148886	+	0.257878i	0.930816	−	0.365488i	$-0.119098\pi$
$281$	2.49578	+	4.32283i	0.148886	+	0.257878i	−0.781930	+	0.623366i	$0.785765\pi$
$282$		0			0
$283$	7.69634	+	13.3304i	0.457500	+	0.792413i	0.998828	−	0.0483984i	$-0.0154117\pi$
$283$	7.69634	+	13.3304i	0.457500	+	0.792413i	−0.541328	+	0.840811i	$0.682078\pi$
$284$	−1.28952	−	2.23352i	−0.0765190	−	0.132535i
$285$		0			0
$286$	4.56364	+	7.90446i	0.269854	+	0.467401i
$287$		0			0
$288$		0			0
$289$	7.19881	+	12.4687i	0.423460	+	0.733454i
$290$	−1.08056			−0.0634529
$291$		0			0
$292$	3.70251			0.216673
$293$	12.9013	−	22.3456i	0.753700	−	1.30545i	−0.192318	−	0.981333i	$-0.561601\pi$
$293$	12.9013	−	22.3456i	0.753700	−	1.30545i	0.946018	−	0.324114i	$-0.105066\pi$
$294$		0			0
$295$	−0.235182	−	0.407347i	−0.0136928	−	0.0237167i
$296$	−3.04321	−	5.27099i	−0.176883	−	0.306370i
$297$		0			0
$298$	−6.14741	+	10.6476i	−0.356110	+	0.616801i
$299$	−2.35246	+	4.07458i	−0.136046	+	0.235639i
$300$		0			0
$301$		0			0
$302$	6.21629	−	10.7669i	0.357707	−	0.619567i
$303$		0			0
$304$	13.9675			0.801089
$305$	−0.299049	+	0.517968i	−0.0171235	+	0.0296588i
$306$		0			0
$307$	−22.2914			−1.27224			−0.636120	−	0.771590i	$-0.719462\pi$
$307$	−22.2914			−1.27224			−0.636120	+	0.771590i	$0.719462\pi$
$308$		0			0
$309$		0			0
$310$	−1.07721			−0.0611816
$311$	−0.654931	−	1.13437i	−0.0371377	−	0.0643245i	0.846859	−	0.531817i	$-0.178491\pi$
$311$	−0.654931	−	1.13437i	−0.0371377	−	0.0643245i	−0.883997	+	0.467493i	$0.845157\pi$
$312$		0			0
$313$	−10.7885	+	18.6862i	−0.609802	+	1.05621i	0.381471	+	0.924381i	$0.375418\pi$
$313$	−10.7885	+	18.6862i	−0.609802	+	1.05621i	−0.991273	+	0.131827i	$0.957916\pi$
$314$	−13.4613			−0.759667
$315$		0			0
$316$	−6.03065			−0.339250
$317$	−12.3910	+	21.4618i	−0.695946	+	1.20541i	0.273915	+	0.961754i	$0.411681\pi$
$317$	−12.3910	+	21.4618i	−0.695946	+	1.20541i	−0.969861	+	0.243660i	$0.921652\pi$
$318$		0			0
$319$	15.5025	+	26.8512i	0.867975	+	1.50338i
$320$	−0.864432			−0.0483232
$321$		0			0
$322$		0			0
$323$	12.3936			0.689597
$324$		0			0
$325$	6.18608	−	10.7146i	0.343142	−	0.594339i
$326$	9.88412			0.547431
$327$		0			0
$328$	11.4901	−	19.9014i	0.634434	−	1.09887i
$329$		0			0
$330$		0			0
$331$	−6.92256	+	11.9902i	−0.380498	+	0.659042i	−0.991133	−	0.132870i	$-0.957581\pi$
$331$	−6.92256	+	11.9902i	−0.380498	+	0.659042i	0.610635	+	0.791912i	$0.290914\pi$
$332$	−0.458076	+	0.793410i	−0.0251402	+	0.0435440i
$333$		0			0
$334$	−9.60367	−	16.6340i	−0.525489	−	0.910174i
$335$	0.525823	+	0.910752i	0.0287288	+	0.0497597i
$336$		0			0
$337$	1.69444	−	2.93485i	0.0923018	−	0.159871i	−0.816178	−	0.577801i	$-0.803911\pi$
$337$	1.69444	−	2.93485i	0.0923018	−	0.159871i	0.908479	+	0.417930i	$0.137244\pi$
$338$	−7.55412			−0.410890
$339$		0			0
$340$	−0.133353			−0.00723210
$341$	15.4545	+	26.7679i	0.836906	+	1.44956i
$342$		0			0
$343$		0			0
$344$	11.5859	+	20.0673i	0.624668	+	1.08196i
$345$		0			0
$346$	−1.56011	−	2.70219i	−0.0838720	−	0.145271i
$347$	−7.25739	−	12.5702i	−0.389597	−	0.674802i	0.602798	−	0.797894i	$-0.294052\pi$
$347$	−7.25739	−	12.5702i	−0.389597	−	0.674802i	−0.992395	+	0.123091i	$0.960719\pi$
$348$		0			0
$349$	7.86412	+	13.6211i	0.420957	+	0.729119i	0.996033	−	0.0889810i	$-0.0283610\pi$
$349$	7.86412	+	13.6211i	0.420957	+	0.729119i	−0.575076	+	0.818100i	$0.695028\pi$
$350$		0			0
$351$		0			0
$352$	6.90000	+	11.9511i	0.367771	+	0.636998i
$353$	4.14423			0.220575			0.110287	−	0.993900i	$-0.464823\pi$
$353$	4.14423			0.220575			0.110287	+	0.993900i	$0.464823\pi$
$354$		0			0
$355$	0.347028			0.0184183
$356$	−2.36298	+	4.09281i	−0.125238	+	0.216918i
$357$		0			0
$358$	−5.60378	−	9.70603i	−0.296169	−	0.512979i
$359$	3.96994	+	6.87614i	0.209525	+	0.362909i	0.951565	−	0.307447i	$-0.0994748\pi$
$359$	3.96994	+	6.87614i	0.209525	+	0.362909i	−0.742040	+	0.670356i	$0.766141\pi$
$360$		0			0
$361$	−20.0116	+	34.6612i	−1.05324	+	1.82427i
$362$	−9.42187	+	16.3192i	−0.495202	+	0.857716i
$363$		0			0
$364$		0			0
$365$	−0.249099	+	0.431453i	−0.0130385	+	0.0225833i
$366$		0			0
$367$	−13.1491			−0.686377			−0.343189	−	0.939266i	$-0.611507\pi$
$367$	−13.1491			−0.686377			−0.343189	+	0.939266i	$0.611507\pi$
$368$	1.72459	−	2.98708i	0.0899004	−	0.155712i
$369$		0			0
$370$	0.230588			0.0119877
$371$		0			0
$372$		0			0
$373$	7.81086			0.404431			0.202216	−	0.979341i	$-0.435186\pi$
$373$	7.81086			0.404431			0.202216	+	0.979341i	$0.435186\pi$
$374$	−2.96862	−	5.14180i	−0.153504	−	0.265876i
$375$		0			0
$376$	4.90527	−	8.49618i	0.252970	−	0.438157i
$377$	23.0398			1.18661
$378$		0			0
$379$	−31.6147			−1.62394			−0.811968	−	0.583702i	$-0.801604\pi$
$379$	−31.6147			−1.62394			−0.811968	+	0.583702i	$0.801604\pi$
$380$	0.317542	−	0.549999i	0.0162896	−	0.0282143i
$381$		0			0
$382$	−12.3515	−	21.3934i	−0.631958	−	1.09458i
$383$	10.7319			0.548373			0.274186	−	0.961677i	$-0.411592\pi$
$383$	10.7319			0.548373			0.274186	+	0.961677i	$0.411592\pi$
$384$		0			0
$385$		0			0
$386$	−0.283187			−0.0144139
$387$		0			0
$388$	1.49022	−	2.58114i	0.0756546	−	0.131038i
$389$	−24.1468			−1.22429			−0.612147	−	0.790744i	$-0.709694\pi$
$389$	−24.1468			−1.22429			−0.612147	+	0.790744i	$0.709694\pi$
$390$		0			0
$391$	1.53026	−	2.65049i	0.0773885	−	0.134041i
$392$		0			0
$393$		0			0
$394$	−0.420927	+	0.729067i	−0.0212060	+	0.0367299i
$395$	0.405733	−	0.702750i	0.0204146	−	0.0353592i
$396$		0			0
$397$	−12.0285	−	20.8339i	−0.603691	−	1.04562i	−0.992257	−	0.124203i	$-0.960363\pi$
$397$	−12.0285	−	20.8339i	−0.603691	−	1.04562i	0.388566	−	0.921421i	$-0.372971\pi$
$398$	−2.77424	−	4.80513i	−0.139060	−	0.240860i
$399$		0			0
$400$	−4.53501	+	7.85487i	−0.226751	+	0.392744i
$401$	1.56232			0.0780183			0.0390092	−	0.999239i	$-0.487580\pi$
$401$	1.56232			0.0780183			0.0390092	+	0.999239i	$0.487580\pi$
$402$		0			0
$403$	22.9683			1.14413
$404$	−6.84661	−	11.8587i	−0.340631	−	0.589991i
$405$		0			0
$406$		0			0
$407$	−3.30817	−	5.72992i	−0.163980	−	0.284022i
$408$		0			0
$409$	11.1728	+	19.3519i	0.552460	+	0.956889i	0.998096	+	0.0616748i	$0.0196442\pi$
$409$	11.1728	+	19.3519i	0.552460	+	0.956889i	−0.445636	+	0.895214i	$0.647023\pi$
$410$	0.435309	+	0.753978i	0.0214984	+	0.0372363i
$411$		0			0
$412$	−3.42210	−	5.92725i	−0.168595	−	0.292014i
$413$		0			0
$414$		0			0
$415$	−0.0616373	−	0.106759i	−0.00302566	−	0.00524059i
$416$	10.2547			0.502779
$417$		0			0
$418$	28.2755			1.38300
$419$	−2.98648	+	5.17273i	−0.145899	+	0.252704i	−0.929708	−	0.368298i	$-0.879941\pi$
$419$	−2.98648	+	5.17273i	−0.145899	+	0.252704i	0.783809	+	0.621002i	$0.213274\pi$
$420$		0			0
$421$	7.31594	+	12.6716i	0.356557	+	0.617575i	0.987383	−	0.158349i	$-0.0506172\pi$
$421$	7.31594	+	12.6716i	0.356557	+	0.617575i	−0.630826	+	0.775924i	$0.717284\pi$
$422$	−3.97605	−	6.88672i	−0.193551	−	0.335240i
$423$		0			0
$424$	15.3144	−	26.5254i	0.743735	−	1.28819i
$425$	−4.02400	+	6.96977i	−0.195193	+	0.338083i
$426$		0			0
$427$		0			0
$428$	7.11156	−	12.3176i	0.343750	−	0.595393i
$429$		0			0
$430$	−0.877876			−0.0423349
$431$	−9.70169	+	16.8038i	−0.467314	+	0.809411i	−0.999303	−	0.0373401i	$-0.988112\pi$
$431$	−9.70169	+	16.8038i	−0.467314	+	0.809411i	0.531989	+	0.846751i	$0.321445\pi$
$432$		0			0
$433$	1.35217			0.0649810			0.0324905	−	0.999472i	$-0.489656\pi$
$433$	1.35217			0.0649810			0.0324905	+	0.999472i	$0.489656\pi$
$434$		0			0
$435$		0			0
$436$	3.30959			0.158501
$437$	7.28772	+	12.6227i	0.348619	+	0.603826i
$438$		0			0
$439$	−8.67059	+	15.0179i	−0.413825	+	0.716766i	−0.995304	−	0.0967954i	$-0.969141\pi$
$439$	−8.67059	+	15.0179i	−0.413825	+	0.716766i	0.581479	+	0.813561i	$0.302474\pi$
$440$	−1.08056			−0.0515139
$441$		0			0
$442$	−4.41194			−0.209854
$443$	9.80499	−	16.9827i	0.465849	−	0.806874i	−0.533390	−	0.845869i	$-0.679082\pi$
$443$	9.80499	−	16.9827i	0.465849	−	0.806874i	0.999239	+	0.0389949i	$0.0124156\pi$
$444$		0			0
$445$	−0.317956	−	0.550716i	−0.0150726	−	0.0261064i
$446$	12.3318			0.583927
$447$		0			0
$448$		0			0
$449$	17.7345			0.836942			0.418471	−	0.908230i	$-0.362566\pi$
$449$	17.7345			0.836942			0.418471	+	0.908230i	$0.362566\pi$
$450$		0			0
$451$	12.4905	−	21.6342i	0.588155	−	1.01871i
$452$	−1.61187			−0.0758160
$453$		0			0
$454$	−13.1073	+	22.7025i	−0.615156	+	1.06548i
$455$		0			0
$456$		0			0
$457$	−0.242725	+	0.420413i	−0.0113542	+	0.0196661i	−0.871647	−	0.490135i	$-0.836948\pi$
$457$	−0.242725	+	0.420413i	−0.0113542	+	0.0196661i	0.860292	+	0.509801i	$0.170281\pi$
$458$	−1.05069	+	1.81985i	−0.0490956	+	0.0850361i
$459$		0			0
$460$	−0.0784150	−	0.135819i	−0.00365612	−	0.00633259i
$461$	−3.99687	−	6.92279i	−0.186153	−	0.322426i	0.757811	−	0.652474i	$-0.226269\pi$
$461$	−3.99687	−	6.92279i	−0.186153	−	0.322426i	−0.943964	+	0.330047i	$0.892935\pi$
$462$		0			0
$463$	5.24280	−	9.08080i	0.243654	−	0.422021i	−0.718098	−	0.695942i	$-0.754987\pi$
$463$	5.24280	−	9.08080i	0.243654	−	0.422021i	0.961752	+	0.273921i	$0.0883206\pi$
$464$	−16.8905			−0.784120
$465$		0			0
$466$	−7.21443			−0.334202
$467$	10.9489	+	18.9640i	0.506653	+	0.877549i	0.999970	+	0.00769944i	$0.00245083\pi$
$467$	10.9489	+	18.9640i	0.506653	+	0.877549i	−0.493317	+	0.869849i	$0.664216\pi$
$468$		0			0
$469$		0			0
$470$	0.185839	+	0.321883i	0.00857213	+	0.0148474i
$471$		0			0
$472$	−6.84592	−	11.8575i	−0.315109	−	0.545785i
$473$	12.5946	+	21.8145i	0.579102	+	1.00303i
$474$		0			0
$475$	−19.1639	−	33.1929i	−0.879301	−	1.52299i
$476$		0			0
$477$		0			0
$478$	−11.7708	−	20.3877i	−0.538386	−	0.932512i
$479$	−4.00169			−0.182842			−0.0914210	−	0.995812i	$-0.529141\pi$
$479$	−4.00169			−0.182842			−0.0914210	+	0.995812i	$0.529141\pi$
$480$		0			0
$481$	−4.91658			−0.224177
$482$	−11.0646	+	19.1645i	−0.503980	+	0.872918i
$483$		0			0
$484$	0.0539532	+	0.0934496i	0.00245242	+	0.00424771i
$485$	0.200520	+	0.347311i	0.00910514	+	0.0157706i
$486$		0			0
$487$	13.2377	−	22.9284i	0.599859	−	1.03899i	−0.392982	−	0.919546i	$-0.628557\pi$
$487$	13.2377	−	22.9284i	0.599859	−	1.03899i	0.992841	−	0.119440i	$-0.0381100\pi$
$488$	−8.70502	+	15.0775i	−0.394058	+	0.682528i
$489$		0			0
$490$		0			0
$491$	−14.2149	+	24.6210i	−0.641511	+	1.11113i	0.343584	+	0.939122i	$0.388359\pi$
$491$	−14.2149	+	24.6210i	−0.641511	+	1.11113i	−0.985096	+	0.172008i	$0.944975\pi$
$492$		0			0
$493$	−14.9872			−0.674989
$494$	10.5057	−	18.1965i	0.472675	−	0.818697i
$495$		0			0
$496$	−16.8381			−0.756052
$497$		0			0
$498$		0			0
$499$	−7.43118			−0.332665			−0.166333	−	0.986070i	$-0.553193\pi$
$499$	−7.43118			−0.332665			−0.166333	+	0.986070i	$0.553193\pi$
$500$	0.412863	+	0.715100i	0.0184638	+	0.0319802i
$501$		0			0
$502$	3.75765	−	6.50845i	0.167712	−	0.290486i
$503$	10.1610			0.453057			0.226529	−	0.974004i	$-0.427262\pi$
$503$	10.1610			0.453057			0.226529	+	0.974004i	$0.427262\pi$
$504$		0			0
$505$	1.84252			0.0819910
$506$	3.49124	−	6.04700i	0.155204	−	0.268822i
$507$		0			0
$508$	0.124295	+	0.215285i	0.00551470	+	0.00955174i
$509$	−28.9063			−1.28125			−0.640625	−	0.767854i	$-0.721325\pi$
$509$	−28.9063			−1.28125			−0.640625	+	0.767854i	$0.721325\pi$
$510$		0			0
$511$		0			0
$512$	−18.6806			−0.825575
$513$		0			0
$514$	7.93381	−	13.7418i	0.349945	−	0.606123i
$515$	0.920934			0.0405812
$516$		0			0
$517$	5.33237	−	9.23593i	0.234517	−	0.406196i
$518$		0			0
$519$		0			0
$520$	−0.401482	+	0.695387i	−0.0176061	+	0.0304947i
$521$	−16.8995	+	29.2708i	−0.740381	+	1.28238i	0.211941	+	0.977283i	$0.432022\pi$
$521$	−16.8995	+	29.2708i	−0.740381	+	1.28238i	−0.952322	+	0.305095i	$0.901312\pi$
$522$		0			0
$523$	−7.18895	−	12.4516i	−0.314351	−	0.544471i	0.664949	−	0.746889i	$-0.268453\pi$
$523$	−7.18895	−	12.4516i	−0.314351	−	0.544471i	−0.979299	+	0.202418i	$0.935120\pi$
$524$	5.86140	+	10.1522i	0.256056	+	0.443502i
$525$		0			0
$526$	−0.848618	+	1.46985i	−0.0370015	+	0.0640885i
$527$	−14.9407			−0.650828
$528$		0			0
$529$	−19.4007			−0.843508
$530$	0.580197	+	1.00493i	0.0252022	+	0.0436514i
$531$		0			0
$532$		0			0
$533$	−9.28166	−	16.0763i	−0.402033	−	0.696342i
$534$		0			0
$535$	0.956910	+	1.65742i	0.0413708	+	0.0716564i
$536$	15.3062	+	26.5111i	0.661127	+	1.14511i
$537$		0			0
$538$	14.4783	+	25.0772i	0.624205	+	1.08116i
$539$		0			0
$540$		0			0
$541$	12.5882	+	21.8034i	0.541210	+	0.937403i	0.998835	+	0.0482577i	$0.0153669\pi$
$541$	12.5882	+	21.8034i	0.541210	+	0.937403i	−0.457625	+	0.889145i	$0.651300\pi$
$542$	−19.7773			−0.849506
$543$		0			0
$544$	−6.67063			−0.286001
$545$	−0.222664	+	0.385666i	−0.00953789	+	0.0165201i
$546$		0			0
$547$	1.59011	+	2.75416i	0.0679883	+	0.117759i	0.898016	−	0.439963i	$-0.145009\pi$
$547$	1.59011	+	2.75416i	0.0679883	+	0.117759i	−0.830027	+	0.557723i	$0.811675\pi$
$548$	5.97545	+	10.3498i	0.255258	+	0.442121i
$549$		0			0
$550$	−9.18062	+	15.9013i	−0.391463	+	0.678034i
$551$	35.6876	−	61.8127i	1.52034	−	2.63331i
$552$		0			0
$553$		0			0
$554$	10.4057	−	18.0233i	0.442098	−	0.765736i
$555$		0			0
$556$	−6.36358			−0.269876
$557$	10.0229	−	17.3602i	0.424686	−	0.735577i	−0.571705	−	0.820459i	$-0.693718\pi$
$557$	10.0229	−	17.3602i	0.424686	−	0.735577i	0.996391	+	0.0848820i	$0.0270513\pi$
$558$		0			0
$559$	18.7181			0.791689
$560$		0			0
$561$		0			0
$562$	−5.50477			−0.232205
$563$	−19.9007	−	34.4690i	−0.838713	−	1.45269i	−0.890971	−	0.454060i	$-0.849975\pi$
$563$	−19.9007	−	34.4690i	−0.838713	−	1.45269i	0.0522584	−	0.998634i	$-0.483358\pi$
$564$		0			0
$565$	0.108444	−	0.187831i	0.00456228	−	0.00790211i
$566$	−16.9753			−0.713523
$567$		0			0
$568$	10.1017			0.423856
$569$	6.90797	−	11.9649i	0.289597	−	0.501597i	−0.684117	−	0.729373i	$-0.739812\pi$
$569$	6.90797	−	11.9649i	0.289597	−	0.501597i	0.973714	+	0.227776i	$0.0731454\pi$
$570$		0			0
$571$	−5.21935	−	9.04019i	−0.218423	−	0.378320i	0.735903	−	0.677087i	$-0.236758\pi$
$571$	−5.21935	−	9.04019i	−0.218423	−	0.378320i	−0.954326	+	0.298767i	$0.903425\pi$
$572$	6.48703			0.271236
$573$		0			0
$574$		0			0
$575$	−9.46483			−0.394711
$576$		0			0
$577$	−12.7461	+	22.0769i	−0.530628	+	0.919075i	0.468733	+	0.883340i	$0.344711\pi$
$577$	−12.7461	+	22.0769i	−0.530628	+	0.919075i	−0.999361	+	0.0357353i	$0.988623\pi$
$578$	−15.8779			−0.660433
$579$		0			0
$580$	−0.383994	+	0.665098i	−0.0159445	+	0.0276167i
$581$		0			0
$582$		0			0
$583$	16.6478	−	28.8349i	0.689483	−	1.19422i
$584$	−7.25104	+	12.5592i	−0.300050	+	0.519702i
$585$		0			0
$586$	14.2277	+	24.6431i	0.587740	+	1.01800i
$587$	17.5168	+	30.3401i	0.722998	+	1.25227i	0.959793	+	0.280709i	$0.0905697\pi$
$587$	17.5168	+	30.3401i	0.722998	+	1.25227i	−0.236795	+	0.971560i	$0.576097\pi$
$588$		0			0
$589$	35.5769	−	61.6210i	1.46592	−	2.53905i
$590$	0.518724			0.0213555
$591$		0			0
$592$	3.60435			0.148138
$593$	−18.0646	−	31.2888i	−0.741824	−	1.28488i	−0.951664	−	0.307141i	$-0.900628\pi$
$593$	−18.0646	−	31.2888i	−0.741824	−	1.28488i	0.209840	−	0.977736i	$-0.432706\pi$
$594$		0			0
$595$		0			0
$596$	4.36915	+	7.56759i	0.178967	+	0.309980i
$597$		0			0
$598$	−2.59433	−	4.49350i	−0.106090	−	0.183753i
$599$	−20.4742	−	35.4623i	−0.836552	−	1.44895i	−0.892760	−	0.450532i	$-0.851234\pi$
$599$	−20.4742	−	35.4623i	−0.836552	−	1.44895i	0.0562080	−	0.998419i	$-0.482099\pi$
$600$		0			0
$601$	12.8547	+	22.2650i	0.524354	+	0.908207i	0.999598	+	0.0283533i	$0.00902635\pi$
$601$	12.8547	+	22.2650i	0.524354	+	0.908207i	−0.475244	+	0.879854i	$0.657640\pi$
$602$		0			0
$603$		0			0
$604$	−4.41810	−	7.65238i	−0.179770	−	0.311371i
$605$	−0.0145196			−0.000590304
$606$		0			0
$607$	6.84516			0.277836			0.138918	−	0.990304i	$-0.455638\pi$
$607$	6.84516			0.277836			0.138918	+	0.990304i	$0.455638\pi$
$608$	15.8841	−	27.5121i	0.644187	−	1.11576i
$609$		0			0
$610$	−0.329795	−	0.571222i	−0.0133530	−	0.0231281i
$611$	−3.96246	−	6.86319i	−0.160304	−	0.277655i
$612$		0			0
$613$	14.5648	−	25.2271i	0.588269	−	1.01891i	−0.406191	−	0.913788i	$-0.633143\pi$
$613$	14.5648	−	25.2271i	0.588269	−	1.01891i	0.994459	−	0.105123i	$-0.0335235\pi$
$614$	12.2917	−	21.2898i	0.496051	−	0.859185i
$615$		0			0
$616$		0			0
$617$	10.3395	−	17.9085i	0.416252	−	0.720969i	−0.579307	−	0.815109i	$-0.696677\pi$
$617$	10.3395	−	17.9085i	0.416252	−	0.720969i	0.995559	+	0.0941404i	$0.0300102\pi$
$618$		0			0
$619$	−8.86355			−0.356256			−0.178128	−	0.984007i	$-0.557004\pi$
$619$	−8.86355			−0.356256			−0.178128	+	0.984007i	$0.557004\pi$
$620$	−0.382804	+	0.663035i	−0.0153738	+	0.0266281i
$621$		0			0
$622$	1.44453			0.0579205
$623$		0			0
$624$		0			0
$625$	24.8333			0.993331
$626$	−11.8977	−	20.6074i	−0.475528	−	0.823638i
$627$		0			0
$628$	−4.78368	+	8.28558i	−0.190890	+	0.330631i
$629$	3.19820			0.127521
$630$		0			0
$631$	26.4661			1.05360			0.526799	−	0.849990i	$-0.323392\pi$
$631$	26.4661			1.05360			0.526799	+	0.849990i	$0.323392\pi$
$632$	11.8105	−	20.4564i	0.469796	−	0.813711i
$633$		0			0
$634$	−13.6649	−	23.6683i	−0.542704	−	0.939990i
$635$	−0.0334495			−0.00132740
$636$		0			0
$637$		0			0
$638$	−34.1928			−1.35371
$639$		0			0
$640$	0.0405449	−	0.0702258i	0.00160268	−	0.00277592i
$641$	16.5319			0.652971			0.326486	−	0.945202i	$-0.394136\pi$
$641$	16.5319			0.652971			0.326486	+	0.945202i	$0.394136\pi$
$642$		0			0
$643$	−15.4460	+	26.7532i	−0.609130	+	1.05504i	0.382254	+	0.924057i	$0.375148\pi$
$643$	−15.4460	+	26.7532i	−0.609130	+	1.05504i	−0.991384	+	0.130987i	$0.958185\pi$
$644$		0			0
$645$		0			0
$646$	−6.83390	+	11.8367i	−0.268876	+	0.465707i
$647$	−0.649903	+	1.12567i	−0.0255503	+	0.0442545i	−0.878518	−	0.477710i	$-0.841467\pi$
$647$	−0.649903	+	1.12567i	−0.0255503	+	0.0442545i	0.852968	+	0.521964i	$0.174800\pi$
$648$		0			0
$649$	−7.44198	−	12.8899i	−0.292123	−	0.505972i
$650$	6.82209	+	11.8162i	0.267584	+	0.463470i
$651$		0			0
$652$	3.51247	−	6.08377i	0.137559	−	0.238259i
$653$	44.8870			1.75656			0.878281	−	0.478144i	$-0.158690\pi$
$653$	44.8870			1.75656			0.878281	+	0.478144i	$0.158690\pi$
$654$		0			0
$655$	−1.57738			−0.0616335
$656$	6.80438	+	11.7855i	0.265667	+	0.460148i
$657$		0			0
$658$		0			0
$659$	−8.96167	−	15.5221i	−0.349097	−	0.604654i	0.636992	−	0.770870i	$-0.280178\pi$
$659$	−8.96167	−	15.5221i	−0.349097	−	0.604654i	−0.986089	+	0.166216i	$0.946845\pi$
$660$		0			0
$661$	−16.5128	−	28.6010i	−0.642274	−	1.11245i	−0.984924	−	0.172989i	$-0.944658\pi$
$661$	−16.5128	−	28.6010i	−0.642274	−	1.11245i	0.342649	−	0.939463i	$-0.388676\pi$
$662$	−7.63429	−	13.2230i	−0.296715	−	0.513925i
$663$		0			0
$664$	−1.79420	−	3.10765i	−0.0696285	−	0.120600i
$665$		0			0
$666$		0			0
$667$	−8.81283	−	15.2643i	−0.341234	−	0.591035i
$668$	−13.6512			−0.528182
$669$		0			0
$670$	−1.15977			−0.0448058
$671$	−9.46295	+	16.3903i	−0.365313	+	0.632741i
$672$		0			0
$673$	−10.6758	−	18.4909i	−0.411520	−	0.712774i	0.583536	−	0.812087i	$-0.301669\pi$
$673$	−10.6758	−	18.4909i	−0.411520	−	0.712774i	−0.995056	+	0.0993135i	$0.968335\pi$
$674$	1.86865	+	3.23659i	0.0719776	+	0.124669i
$675$		0			0
$676$	−2.68447	+	4.64964i	−0.103249	+	0.178832i
$677$	4.15084	−	7.18946i	0.159530	−	0.276313i	−0.775170	−	0.631753i	$-0.782336\pi$
$677$	4.15084	−	7.18946i	0.159530	−	0.276313i	0.934699	+	0.355440i	$0.115669\pi$
$678$		0			0
$679$		0			0
$680$	0.261161	−	0.452344i	0.0100151	−	0.0173466i
$681$		0			0
$682$	−34.0868			−1.30525
$683$	1.24728	−	2.16036i	0.0477259	−	0.0826637i	−0.841176	−	0.540762i	$-0.818136\pi$
$683$	1.24728	−	2.16036i	0.0477259	−	0.0826637i	0.888902	+	0.458098i	$0.151469\pi$
$684$		0			0
$685$	−1.60808			−0.0614414
$686$		0			0
$687$		0			0
$688$	−13.7222			−0.523154
$689$	−12.3710	−	21.4271i	−0.471296	−	0.816308i
$690$		0			0
$691$	8.43455	−	14.6091i	0.320865	−	0.555755i	−0.659801	−	0.751440i	$-0.729360\pi$
$691$	8.43455	−	14.6091i	0.320865	−	0.555755i	0.980667	+	0.195685i	$0.0626930\pi$
$692$	−2.21763			−0.0843017
$693$		0			0
$694$	16.0071			0.607621
$695$	0.428132	−	0.741547i	0.0162400	−	0.0281285i
$696$		0			0
$697$	6.03765	+	10.4575i	0.228692	+	0.396107i
$698$	−17.3453			−0.656530
$699$		0			0
$700$		0			0
$701$	−16.4806			−0.622465			−0.311232	−	0.950334i	$-0.600742\pi$
$701$	−16.4806			−0.622465			−0.311232	+	0.950334i	$0.600742\pi$
$702$		0			0
$703$	−7.61558	+	13.1906i	−0.287227	+	0.497492i
$704$	−27.3536			−1.03093
$705$		0			0
$706$	−2.28515	+	3.95800i	−0.0860029	+	0.148961i
$707$		0			0
$708$		0			0
$709$	14.7462	−	25.5412i	0.553807	−	0.959222i	−0.444188	−	0.895933i	$-0.646508\pi$
$709$	14.7462	−	25.5412i	0.553807	−	0.959222i	0.997995	−	0.0632882i	$-0.0201587\pi$
$710$	−0.191354	+	0.331434i	−0.00718138	+	0.0124385i
$711$		0			0
$712$	−9.25539	−	16.0308i	−0.346860	−	0.600780i
$713$	−8.78551	−	15.2169i	−0.329020	−	0.569879i
$714$		0			0
$715$	−0.436438	+	0.755933i	−0.0163219	+	0.0282703i
$716$	−7.96554			−0.297686
$717$		0			0
$718$	−8.75620			−0.326779
$719$	−0.217311	−	0.376394i	−0.00810433	−	0.0140371i	0.861945	−	0.507002i	$-0.169246\pi$
$719$	−0.217311	−	0.376394i	−0.00810433	−	0.0140371i	−0.870049	+	0.492965i	$0.835913\pi$
$720$		0			0
$721$		0			0
$722$	−22.0691	−	38.2248i	−0.821327	−	1.42258i
$723$		0			0
$724$	6.69640	+	11.5985i	0.248870	+	0.431055i
$725$	23.1744	+	40.1392i	0.860675	+	1.49073i
$726$		0			0
$727$	13.5839	+	23.5280i	0.503799	+	0.872605i	0.999990	+	0.00439187i	$0.00139798\pi$
$727$	13.5839	+	23.5280i	0.503799	+	0.872605i	−0.496192	+	0.868213i	$0.665269\pi$
$728$		0			0
$729$		0			0
$730$	−0.274710	−	0.475812i	−0.0101675	−	0.0176106i
$731$	−12.1760			−0.450344
$732$		0			0
$733$	−5.66614			−0.209284			−0.104642	−	0.994510i	$-0.533370\pi$
$733$	−5.66614			−0.209284			−0.104642	+	0.994510i	$0.533370\pi$
$734$	7.25050	−	12.5582i	0.267621	−	0.463533i
$735$		0			0
$736$	−3.92249	−	6.79395i	−0.144585	−	0.250428i
$737$	16.6389	+	28.8194i	0.612901	+	1.06158i
$738$		0			0
$739$	6.80540	−	11.7873i	0.250341	−	0.433603i	−0.713279	−	0.700880i	$-0.752791\pi$
$739$	6.80540	−	11.7873i	0.250341	−	0.433603i	0.963620	+	0.267278i	$0.0861241\pi$
$740$	0.0819427	−	0.141929i	0.00301227	−	0.00521741i
$741$		0			0
$742$		0			0
$743$	6.33421	−	10.9712i	0.232380	−	0.402493i	−0.726128	−	0.687559i	$-0.758682\pi$
$743$	6.33421	−	10.9712i	0.232380	−	0.402493i	0.958508	+	0.285066i	$0.0920155\pi$
$744$		0			0
$745$	−1.17580			−0.0430779
$746$	−4.30696	+	7.45988i	−0.157689	+	0.273126i
$747$		0			0
$748$	−4.21977			−0.154290
$749$		0			0
$750$		0			0
$751$	−7.14538			−0.260739			−0.130369	−	0.991465i	$-0.541616\pi$
$751$	−7.14538			−0.260739			−0.130369	+	0.991465i	$0.541616\pi$
$752$	2.90488	+	5.03140i	0.105930	+	0.183476i
$753$		0			0
$754$	−12.7043	+	22.0045i	−0.462663	+	0.801355i
$755$	1.18897			0.0432712
$756$		0			0
$757$	37.6446			1.36822			0.684108	−	0.729381i	$-0.260192\pi$
$757$	37.6446			1.36822			0.684108	+	0.729381i	$0.260192\pi$
$758$	17.4325	−	30.1940i	0.633178	−	1.09670i
$759$		0			0
$760$	1.24376	+	2.15425i	0.0451157	+	0.0781428i
$761$	−10.0472			−0.364209			−0.182104	−	0.983279i	$-0.558291\pi$
$761$	−10.0472			−0.364209			−0.182104	+	0.983279i	$0.558291\pi$
$762$		0			0
$763$		0			0
$764$	−17.5572			−0.635196
$765$		0			0
$766$	−5.91762	+	10.2496i	−0.213812	+	0.370334i
$767$	−11.0602			−0.399361
$768$		0			0
$769$	16.1463	−	27.9663i	0.582252	−	1.00849i	−0.412960	−	0.910749i	$-0.635505\pi$
$769$	16.1463	−	27.9663i	0.582252	−	1.00849i	0.995212	−	0.0977407i	$-0.0311616\pi$
$770$		0			0
$771$		0			0
$772$	−0.100635	+	0.174305i	−0.00362192	+	0.00627336i
$773$	−24.2939	+	42.0783i	−0.873792	+	1.51345i	−0.0157473	+	0.999876i	$0.505013\pi$
$773$	−24.2939	+	42.0783i	−0.873792	+	1.51345i	−0.858044	+	0.513576i	$0.828321\pi$
$774$		0			0
$775$	23.1025	+	40.0148i	0.829867	+	1.43737i
$776$	5.83694	+	10.1099i	0.209534	+	0.362923i
$777$		0			0
$778$	13.3147	−	23.0618i	0.477356	−	0.826806i
$779$	−57.5075			−2.06042
$780$		0			0
$781$	10.9812			0.392938
$782$	1.68759	+	2.92299i	0.0603481	+	0.104526i
$783$		0			0
$784$		0			0
$785$	−0.643678	−	1.11488i	−0.0229739	−	0.0397919i
$786$		0			0
$787$	−24.4776	−	42.3964i	−0.872531	−	1.51127i	−0.859370	−	0.511354i	$-0.829144\pi$
$787$	−24.4776	−	42.3964i	−0.872531	−	1.51127i	−0.0131602	−	0.999913i	$-0.504189\pi$
$788$	0.299165	+	0.518170i	0.0106573	+	0.0184590i
$789$		0			0
$790$	0.447448	+	0.775003i	0.0159195	+	0.0275734i
$791$		0			0
$792$		0			0
$793$	7.03188	+	12.1796i	0.249710	+	0.432510i
$794$	26.5303			0.941525
$795$		0			0
$796$	−3.94348			−0.139773

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)

\(f(q)\)	\(=\)	\(q+(-0.551407 + 0.955065i) q^{2} +(0.391901 + 0.678793i) q^{4} -0.105466 q^{5} -3.07001 q^{8} +O(q^{10})\)	\(q+(-0.551407 + 0.955065i) q^{2} +(0.391901 + 0.678793i) q^{4} -0.105466 q^{5} -3.07001 q^{8} +(0.0581547 - 0.100727i) q^{10} -3.33731 q^{11} +(-1.23997 + 2.14770i) q^{13} +(0.909025 - 1.57448i) q^{16} +(0.806594 - 1.39706i) q^{17} +(3.84133 + 6.65338i) q^{19} +(-0.0413323 - 0.0715896i) q^{20} +(1.84022 - 3.18735i) q^{22} +1.89719 q^{23} -4.98888 q^{25} +(-1.36746 - 2.36851i) q^{26} +(-4.64521 - 8.04574i) q^{29} +(-4.63081 - 8.02080i) q^{31} +(-2.06753 - 3.58107i) q^{32} +(0.889523 + 1.54070i) q^{34} +(0.991268 + 1.71693i) q^{37} -8.47254 q^{38} +0.323782 q^{40} +(-3.74268 + 6.48252i) q^{41} +(-3.77388 - 6.53655i) q^{43} +(-1.30790 - 2.26534i) q^{44} +(-1.04612 + 1.81194i) q^{46} +(-1.59780 + 2.76747i) q^{47} +(2.75090 - 4.76470i) q^{50} -1.94379 q^{52} +(-4.98839 + 8.64015i) q^{53} +0.351974 q^{55} +10.2456 q^{58} +(2.22993 + 3.86235i) q^{59} +(2.83550 - 4.91123i) q^{61} +10.2138 q^{62} +8.19630 q^{64} +(0.130775 - 0.226509i) q^{65} +(-4.98571 - 8.63550i) q^{67} +1.26442 q^{68} -3.29042 q^{71} +(2.36189 - 4.09091i) q^{73} -2.18637 q^{74} +(-3.01084 + 5.21493i) q^{76} +(-3.84705 + 6.66328i) q^{79} +(-0.0958713 + 0.166054i) q^{80} +(-4.12748 - 7.14901i) q^{82} +(0.584428 + 1.01226i) q^{83} +(-0.0850683 + 0.147343i) q^{85} +8.32378 q^{86} +10.2456 q^{88} +(3.01477 + 5.22173i) q^{89} +(0.743509 + 1.28780i) q^{92} +(-1.76208 - 3.05201i) q^{94} +(-0.405130 - 0.701706i) q^{95} +(-1.90127 - 3.29310i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24q - 4q^{2} - 12q^{4} + 24q^{8} + O(q^{10}) \)	\( 24q - 4q^{2} - 12q^{4} + 24q^{8} + 40q^{11} - 12q^{16} + 64q^{23} + 24q^{25} - 16q^{29} - 48q^{32} - 12q^{37} - 56q^{44} + 24q^{46} + 4q^{50} - 32q^{53} + 96q^{64} - 60q^{65} - 12q^{67} + 112q^{71} + 136q^{74} + 12q^{79} + 12q^{85} + 152q^{86} - 16q^{92} - 64q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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