LMFDB - Embedded newform 2592.2.s.b.863.1

Newspace parameters

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.s (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$1$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			863.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2592.863
Dual form			2592.2.s.b.1727.1

$q$-expansion

$f(q)$	$=$	$q+(-1.67303 - 0.965926i) q^{5} +(-0.633975 + 0.366025i) q^{7} +O(q^{10})$	$q+(-1.67303 - 0.965926i) q^{5} +(-0.633975 + 0.366025i) q^{7} +(2.63896 + 4.57081i) q^{11} +(-2.23205 + 3.86603i) q^{13} -0.896575i q^{17} -1.26795i q^{19} +(0.707107 - 1.22474i) q^{23} +(-0.633975 - 1.09808i) q^{25} +(4.69093 - 2.70831i) q^{29} +(-6.46410 - 3.73205i) q^{31} +1.41421 q^{35} -7.73205 q^{37} +(0.328169 + 0.189469i) q^{41} +(7.56218 - 4.36603i) q^{43} +(-2.31079 - 4.00240i) q^{47} +(-3.23205 + 5.59808i) q^{49} +2.44949i q^{53} -10.1962i q^{55} +(5.41662 - 9.38186i) q^{59} +(0.598076 + 1.03590i) q^{61} +(7.46859 - 4.31199i) q^{65} +(-11.3660 - 6.56218i) q^{67} -13.2827 q^{71} -13.7321 q^{73} +(-3.34607 - 1.93185i) q^{77} +(-14.3660 + 8.29423i) q^{79} +(5.27792 + 9.14162i) q^{83} +(-0.866025 + 1.50000i) q^{85} +14.9372i q^{89} -3.26795i q^{91} +(-1.22474 + 2.12132i) q^{95} +(3.19615 + 5.53590i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{7}+O(q^{10}) $ 8 * q - 12 * q^7	$ 8 q - 12 q^{7} - 4 q^{13} - 12 q^{25} - 24 q^{31} - 48 q^{37} + 12 q^{43} - 12 q^{49} - 16 q^{61} - 84 q^{67} - 96 q^{73} - 108 q^{79} - 16 q^{97}+O(q^{100}) $ 8 * q - 12 * q^7 - 4 * q^13 - 12 * q^25 - 24 * q^31 - 48 * q^37 + 12 * q^43 - 12 * q^49 - 16 * q^61 - 84 * q^67 - 96 * q^73 - 108 * q^79 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$-1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.67303	−	0.965926i	−0.748203	−	0.431975i	0.0768413	−	0.997043i	$-0.475517\pi$
$5$	−1.67303	−	0.965926i	−0.748203	−	0.431975i	−0.825044	+	0.565068i	$0.808850\pi$
$6$		0			0
$7$	−0.633975	+	0.366025i	−0.239620	+	0.138345i	−0.615002	−	0.788526i	$-0.710845\pi$
$7$	−0.633975	+	0.366025i	−0.239620	+	0.138345i	0.375382	+	0.926870i	$0.377511\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.63896	+	4.57081i	0.795676	+	1.37815i	0.922409	+	0.386214i	$0.126217\pi$
$11$	2.63896	+	4.57081i	0.795676	+	1.37815i	−0.126733	+	0.991937i	$0.540449\pi$
$12$		0			0
$13$	−2.23205	+	3.86603i	−0.619060	+	1.07224i	0.370598	+	0.928793i	$0.379153\pi$
$13$	−2.23205	+	3.86603i	−0.619060	+	1.07224i	−0.989658	+	0.143449i	$0.954181\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.896575i		−	0.217451i	−0.994072	−	0.108726i	$-0.965323\pi$
$17$		−	0.896575i		−	0.217451i	0.994072	−	0.108726i	$-0.0346770\pi$
$18$		0			0
$19$		−	1.26795i		−	0.290887i	−0.989367	−	0.145444i	$-0.953539\pi$
$19$		−	1.26795i		−	0.290887i	0.989367	−	0.145444i	$-0.0464610\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.707107	−	1.22474i	0.147442	−	0.255377i	−0.782839	−	0.622224i	$-0.786229\pi$
$23$	0.707107	−	1.22474i	0.147442	−	0.255377i	0.930281	+	0.366847i	$0.119563\pi$
$24$		0			0
$25$	−0.633975	−	1.09808i	−0.126795	−	0.219615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.69093	−	2.70831i	0.871084	−	0.502920i	0.00337538	−	0.999994i	$-0.498926\pi$
$29$	4.69093	−	2.70831i	0.871084	−	0.502920i	0.867708	+	0.497074i	$0.165592\pi$
$30$		0			0
$31$	−6.46410	−	3.73205i	−1.16099	−	0.670296i	−0.209447	−	0.977820i	$-0.567166\pi$
$31$	−6.46410	−	3.73205i	−1.16099	−	0.670296i	−0.951540	+	0.307524i	$0.900500\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.41421			0.239046
$36$		0			0
$37$	−7.73205			−1.27114			−0.635571	−	0.772043i	$-0.719235\pi$
$37$	−7.73205			−1.27114			−0.635571	+	0.772043i	$0.719235\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.328169	+	0.189469i	0.0512514	+	0.0295900i	0.525407	−	0.850851i	$-0.323913\pi$
$41$	0.328169	+	0.189469i	0.0512514	+	0.0295900i	−0.474155	+	0.880441i	$0.657246\pi$
$42$		0			0
$43$	7.56218	−	4.36603i	1.15322	−	0.665813i	0.203551	−	0.979064i	$-0.434752\pi$
$43$	7.56218	−	4.36603i	1.15322	−	0.665813i	0.949670	+	0.313252i	$0.101418\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.31079	−	4.00240i	−0.337063	−	0.583811i	0.646816	−	0.762646i	$-0.276100\pi$
$47$	−2.31079	−	4.00240i	−0.337063	−	0.583811i	−0.983879	+	0.178836i	$0.942767\pi$
$48$		0			0
$49$	−3.23205	+	5.59808i	−0.461722	+	0.799725i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.44949i			0.336463i	0.985747	+	0.168232i	$0.0538057\pi$
$53$			2.44949i			0.336463i	−0.985747	+	0.168232i	$0.946194\pi$
$54$		0			0
$55$		−	10.1962i		−	1.37485i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.41662	−	9.38186i	0.705184	−	1.22141i	−0.261442	−	0.965219i	$-0.584198\pi$
$59$	5.41662	−	9.38186i	0.705184	−	1.22141i	0.966625	−	0.256194i	$-0.0824687\pi$
$60$		0			0
$61$	0.598076	+	1.03590i	0.0765758	+	0.132633i	0.901770	−	0.432215i	$-0.142268\pi$
$61$	0.598076	+	1.03590i	0.0765758	+	0.132633i	−0.825195	+	0.564848i	$0.808935\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.46859	−	4.31199i	0.926364	−	0.534837i
$66$		0			0
$67$	−11.3660	−	6.56218i	−1.38858	−	0.801698i	−0.395426	−	0.918498i	$-0.629403\pi$
$67$	−11.3660	−	6.56218i	−1.38858	−	0.801698i	−0.993155	+	0.116800i	$0.962736\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13.2827			−1.57637			−0.788185	−	0.615439i	$-0.788979\pi$
$71$	−13.2827			−1.57637			−0.788185	+	0.615439i	$0.788979\pi$
$72$		0			0
$73$	−13.7321			−1.60721			−0.803607	−	0.595160i	$-0.797089\pi$
$73$	−13.7321			−1.60721			−0.803607	+	0.595160i	$0.797089\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.34607	−	1.93185i	−0.381320	−	0.220155i
$78$		0			0
$79$	−14.3660	+	8.29423i	−1.61630	+	0.933174i	−0.628439	+	0.777859i	$0.716306\pi$
$79$	−14.3660	+	8.29423i	−1.61630	+	0.933174i	−0.987865	+	0.155315i	$0.950361\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.27792	+	9.14162i	0.579327	+	1.00342i	0.995557	+	0.0941638i	$0.0300178\pi$
$83$	5.27792	+	9.14162i	0.579327	+	1.00342i	−0.416230	+	0.909259i	$0.636649\pi$
$84$		0			0
$85$	−0.866025	+	1.50000i	−0.0939336	+	0.162698i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			14.9372i			1.58334i	0.610951	+	0.791669i	$0.290787\pi$
$89$			14.9372i			1.58334i	−0.610951	+	0.791669i	$0.709213\pi$
$90$		0			0
$91$		−	3.26795i		−	0.342574i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.22474	+	2.12132i	−0.125656	+	0.217643i
$96$		0			0
$97$	3.19615	+	5.53590i	0.324520	+	0.562085i	0.981415	−	0.191897i	$-0.0614639\pi$
$97$	3.19615	+	5.53590i	0.324520	+	0.562085i	−0.656895	+	0.753982i	$0.728131\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.46739	+	3.15660i	−0.544025	+	0.314093i	−0.746709	−	0.665151i	$-0.768367\pi$
$101$	−5.46739	+	3.15660i	−0.544025	+	0.314093i	0.202683	+	0.979244i	$0.435034\pi$
$102$		0			0
$103$	3.46410	+	2.00000i	0.341328	+	0.197066i	0.660859	−	0.750510i	$-0.270192\pi$
$103$	3.46410	+	2.00000i	0.341328	+	0.197066i	−0.319531	+	0.947576i	$0.603525\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.79315			−0.173350			−0.0866752	−	0.996237i	$-0.527624\pi$
$107$	−1.79315			−0.173350			−0.0866752	+	0.996237i	$0.527624\pi$
$108$		0			0
$109$	3.92820			0.376254			0.188127	−	0.982145i	$-0.439758\pi$
$109$	3.92820			0.376254			0.188127	+	0.982145i	$0.439758\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.5068	−	10.1075i	−1.64690	−	0.950838i	−0.978294	−	0.207221i	$-0.933558\pi$
$113$	−17.5068	−	10.1075i	−1.64690	−	0.950838i	−0.668605	−	0.743617i	$-0.733108\pi$
$114$		0			0
$115$	−2.36603	+	1.36603i	−0.220633	+	0.127383i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.328169	+	0.568406i	0.0300832	+	0.0521057i
$120$		0			0
$121$	−8.42820	+	14.5981i	−0.766200	+	1.32710i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.1087i			1.08304i
$126$		0			0
$127$		−	21.1244i		−	1.87448i	−0.348680	−	0.937242i	$-0.613370\pi$
$127$		−	21.1244i		−	1.87448i	0.348680	−	0.937242i	$-0.386630\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.0507680	+	0.0879327i	−0.00443562	+	0.00768272i	−0.868235	−	0.496154i	$-0.834745\pi$
$131$	−0.0507680	+	0.0879327i	−0.00443562	+	0.00768272i	0.863799	+	0.503836i	$0.168079\pi$
$132$		0			0
$133$	0.464102	+	0.803848i	0.0402427	+	0.0697024i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.21046	+	2.43091i	−0.359723	+	0.207686i	−0.668959	−	0.743299i	$-0.733260\pi$
$137$	−4.21046	+	2.43091i	−0.359723	+	0.207686i	0.309236	+	0.950985i	$0.399927\pi$
$138$		0			0
$139$	−10.7321	−	6.19615i	−0.910281	−	0.525551i	−0.0297592	−	0.999557i	$-0.509474\pi$
$139$	−10.7321	−	6.19615i	−0.910281	−	0.525551i	−0.880521	+	0.474006i	$0.842807\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−23.5612			−1.97028
$144$		0			0
$145$	−10.4641			−0.868996
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.58510	−	0.915158i	−0.129856	−	0.0749727i	0.433665	−	0.901074i	$-0.357220\pi$
$149$	−1.58510	−	0.915158i	−0.129856	−	0.0749727i	−0.563521	+	0.826102i	$0.690554\pi$
$150$		0			0
$151$	0.928203	−	0.535898i	0.0755361	−	0.0436108i	−0.461756	−	0.887007i	$-0.652781\pi$
$151$	0.928203	−	0.535898i	0.0755361	−	0.0436108i	0.537292	+	0.843396i	$0.319447\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.20977	+	12.4877i	0.579103	+	1.00304i
$156$		0			0
$157$	−5.33013	+	9.23205i	−0.425390	+	0.736798i	−0.996457	−	0.0841060i	$-0.973197\pi$
$157$	−5.33013	+	9.23205i	−0.425390	+	0.736798i	0.571066	+	0.820904i	$0.306530\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.03528i			0.0815912i
$162$		0			0
$163$		−	1.60770i		−	0.125924i	−0.998016	−	0.0629622i	$-0.979945\pi$
$163$		−	1.60770i		−	0.125924i	0.998016	−	0.0629622i	$-0.0200548\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.32868	−	9.22955i	0.412346	−	0.714204i	−0.582800	−	0.812616i	$-0.698043\pi$
$167$	5.32868	−	9.22955i	0.412346	−	0.714204i	0.995146	+	0.0984115i	$0.0313761\pi$
$168$		0			0
$169$	−3.46410	−	6.00000i	−0.266469	−	0.461538i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	9.58991	−	5.53674i	0.729107	−	0.420950i	−0.0889883	−	0.996033i	$-0.528363\pi$
$173$	9.58991	−	5.53674i	0.729107	−	0.420950i	0.818095	+	0.575082i	$0.195030\pi$
$174$		0			0
$175$	0.803848	+	0.464102i	0.0607652	+	0.0350828i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.8695			1.63461			0.817303	−	0.576208i	$-0.195468\pi$
$179$	21.8695			1.63461			0.817303	+	0.576208i	$0.195468\pi$
$180$		0			0
$181$	−23.3205			−1.73340			−0.866700	−	0.498830i	$-0.833763\pi$
$181$	−23.3205			−1.73340			−0.866700	+	0.498830i	$0.833763\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.9360	+	7.46859i	0.951072	+	0.549101i
$186$		0			0
$187$	4.09808	−	2.36603i	0.299681	−	0.173021i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.2277	−	17.7148i	−0.740048	−	1.28180i	−0.952473	−	0.304623i	$-0.901470\pi$
$191$	−10.2277	−	17.7148i	−0.740048	−	1.28180i	0.212425	−	0.977177i	$-0.431864\pi$
$192$		0			0
$193$	4.23205	−	7.33013i	0.304630	−	0.527634i	−0.672549	−	0.740052i	$-0.734801\pi$
$193$	4.23205	−	7.33013i	0.304630	−	0.527634i	0.977179	+	0.212418i	$0.0681340\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.76079i		−	0.695428i	−0.937601	−	0.347714i	$-0.886958\pi$
$197$		−	9.76079i		−	0.695428i	0.937601	−	0.347714i	$-0.113042\pi$
$198$		0			0
$199$		−	8.00000i		−	0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$		−	8.00000i		−	0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.98262	+	3.43400i	−0.139153	+	0.241019i
$204$		0			0
$205$	−0.366025	−	0.633975i	−0.0255643	−	0.0442787i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.79555	−	3.34607i	0.400887	−	0.231452i
$210$		0			0
$211$	−16.5622	−	9.56218i	−1.14019	−	0.658287i	−0.193710	−	0.981059i	$-0.562052\pi$
$211$	−16.5622	−	9.56218i	−1.14019	−	0.658287i	−0.946477	+	0.322771i	$0.895386\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−16.8690			−1.15046
$216$		0			0
$217$	5.46410			0.370927
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.46618	+	2.00120i	0.233161	+	0.134615i
$222$		0			0
$223$	14.9545	−	8.63397i	1.00143	−	0.578174i	0.0927563	−	0.995689i	$-0.470432\pi$
$223$	14.9545	−	8.63397i	1.00143	−	0.578174i	0.908670	+	0.417515i	$0.137099\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.67475	+	15.0251i	0.575763	+	0.997251i	0.995958	+	0.0898175i	$0.0286284\pi$
$227$	8.67475	+	15.0251i	0.575763	+	0.997251i	−0.420195	+	0.907434i	$0.638038\pi$
$228$		0			0
$229$	−5.50000	+	9.52628i	−0.363450	+	0.629514i	−0.988526	−	0.151050i	$-0.951735\pi$
$229$	−5.50000	+	9.52628i	−0.363450	+	0.629514i	0.625076	+	0.780564i	$0.285068\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	18.8009i		−	1.23169i	−0.787869	−	0.615843i	$-0.788815\pi$
$233$		−	18.8009i		−	1.23169i	0.787869	−	0.615843i	$-0.211185\pi$
$234$		0			0
$235$			8.92820i			0.582412i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.52056	−	16.4901i	0.615834	−	1.06666i	−0.374404	−	0.927266i	$-0.622153\pi$
$239$	9.52056	−	16.4901i	0.615834	−	1.06666i	0.990238	−	0.139390i	$-0.0445140\pi$
$240$		0			0
$241$	8.06218	+	13.9641i	0.519331	+	0.899507i	0.999748	+	0.0224667i	$0.00715197\pi$
$241$	8.06218	+	13.9641i	0.519331	+	0.899507i	−0.480417	+	0.877040i	$0.659515\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.8147	−	6.24384i	0.690923	−	0.398904i
$246$		0			0
$247$	4.90192	+	2.83013i	0.311902	+	0.180077i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	23.8386			1.50468			0.752338	−	0.658777i	$-0.228926\pi$
$251$	23.8386			1.50468			0.752338	+	0.658777i	$0.228926\pi$
$252$		0			0
$253$	7.46410			0.469264
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.688524	−	0.397520i	−0.0429490	−	0.0247966i	0.478372	−	0.878157i	$-0.341227\pi$
$257$	−0.688524	−	0.397520i	−0.0429490	−	0.0247966i	−0.521321	+	0.853361i	$0.674560\pi$
$258$		0			0
$259$	4.90192	−	2.83013i	0.304591	−	0.175856i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.72500	−	6.45189i	−0.229694	−	0.397841i	0.728024	−	0.685552i	$-0.240439\pi$
$263$	−3.72500	−	6.45189i	−0.229694	−	0.397841i	−0.957717	+	0.287711i	$0.907106\pi$
$264$		0			0
$265$	2.36603	−	4.09808i	0.145344	−	0.251743i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			22.7661i			1.38807i	0.719939	+	0.694037i	$0.244170\pi$
$269$			22.7661i			1.38807i	−0.719939	+	0.694037i	$0.755830\pi$
$270$		0			0
$271$			13.2679i			0.805971i	0.915206	+	0.402985i	$0.132027\pi$
$271$			13.2679i			0.805971i	−0.915206	+	0.402985i	$0.867973\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.34607	−	5.79555i	0.201775	−	0.349485i
$276$		0			0
$277$	−0.535898	−	0.928203i	−0.0321990	−	0.0557703i	0.849477	−	0.527626i	$-0.176918\pi$
$277$	−0.535898	−	0.928203i	−0.0321990	−	0.0557703i	−0.881676	+	0.471856i	$0.843584\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.91808	−	5.72620i	0.591663	−	0.341597i	−0.174092	−	0.984729i	$-0.555699\pi$
$281$	9.91808	−	5.72620i	0.591663	−	0.341597i	0.765755	+	0.643133i	$0.222366\pi$
$282$		0			0
$283$	−16.7321	−	9.66025i	−0.994617	−	0.574242i	−0.0879660	−	0.996123i	$-0.528037\pi$
$283$	−16.7321	−	9.66025i	−0.994617	−	0.574242i	−0.906651	+	0.421881i	$0.861370\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.277401			−0.0163745
$288$		0			0
$289$	16.1962			0.952715
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−20.6126	−	11.9007i	−1.20420	−	0.695246i	−0.242715	−	0.970098i	$-0.578038\pi$
$293$	−20.6126	−	11.9007i	−1.20420	−	0.695246i	−0.961487	+	0.274852i	$0.911371\pi$
$294$		0			0
$295$	−18.1244	+	10.4641i	−1.05524	+	0.609244i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.15660	+	5.46739i	0.182551	+	0.316187i
$300$		0			0
$301$	−3.19615	+	5.53590i	−0.184223	+	0.319084i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	2.31079i		−	0.132315i
$306$		0			0
$307$			18.0000i			1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$			18.0000i			1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.1087	+	20.9730i	−0.686624	+	1.18927i	0.286299	+	0.958140i	$0.407575\pi$
$311$	−12.1087	+	20.9730i	−0.686624	+	1.18927i	−0.972923	+	0.231128i	$0.925759\pi$
$312$		0			0
$313$	−14.1603	−	24.5263i	−0.800385	−	1.38631i	−0.919363	−	0.393410i	$-0.871295\pi$
$313$	−14.1603	−	24.5263i	−0.800385	−	1.38631i	0.118978	−	0.992897i	$-0.462038\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.33178	+	3.65565i	−0.355628	+	0.205322i	−0.667161	−	0.744913i	$-0.732491\pi$
$317$	−6.33178	+	3.65565i	−0.355628	+	0.205322i	0.311533	+	0.950235i	$0.399157\pi$
$318$		0			0
$319$	24.7583	+	14.2942i	1.38620	+	0.800323i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.13681			−0.0632539
$324$		0			0
$325$	5.66025			0.313974
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.92996	+	1.69161i	0.161534	+	0.0932618i
$330$		0			0
$331$	−6.75833	+	3.90192i	−0.371471	+	0.214469i	−0.674101	−	0.738639i	$-0.735469\pi$
$331$	−6.75833	+	3.90192i	−0.371471	+	0.214469i	0.302630	+	0.953108i	$0.402135\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.6772	+	21.9575i	0.692627	+	1.19967i
$336$		0			0
$337$	2.46410	−	4.26795i	0.134228	−	0.232490i	−0.791074	−	0.611720i	$-0.790478\pi$
$337$	2.46410	−	4.26795i	0.134228	−	0.232490i	0.925302	+	0.379230i	$0.123811\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	39.3949i		−	2.13335i
$342$		0			0
$343$		−	9.85641i		−	0.532196i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.88160	+	11.9193i	−0.369424	+	0.639860i	−0.989476	−	0.144700i	$-0.953778\pi$
$347$	−6.88160	+	11.9193i	−0.369424	+	0.639860i	0.620052	+	0.784561i	$0.287112\pi$
$348$		0			0
$349$	1.07180	+	1.85641i	0.0573720	+	0.0993712i	0.893285	−	0.449491i	$-0.148395\pi$
$349$	1.07180	+	1.85641i	0.0573720	+	0.0993712i	−0.835913	+	0.548862i	$0.815061\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24.2311	+	13.9898i	−1.28969	+	0.744604i	−0.978599	−	0.205776i	$-0.934028\pi$
$353$	−24.2311	+	13.9898i	−1.28969	+	0.744604i	−0.311092	+	0.950380i	$0.600695\pi$
$354$		0			0
$355$	22.2224	+	12.8301i	1.17944	+	0.680952i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.44949			−0.129279			−0.0646396	−	0.997909i	$-0.520590\pi$
$359$	−2.44949			−0.129279			−0.0646396	+	0.997909i	$0.520590\pi$
$360$		0			0
$361$	17.3923			0.915384
$362$		0			0
$363$		0			0
$364$		0			0
$365$	22.9742	+	13.2641i	1.20252	+	0.694277i
$366$		0			0
$367$	−10.8564	+	6.26795i	−0.566700	+	0.327184i	−0.755830	−	0.654768i	$-0.772766\pi$
$367$	−10.8564	+	6.26795i	−0.566700	+	0.327184i	0.189130	+	0.981952i	$0.439433\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.896575	−	1.55291i	−0.0465479	−	0.0806233i
$372$		0			0
$373$	16.1244	−	27.9282i	0.834887	−	1.44607i	−0.0592345	−	0.998244i	$-0.518866\pi$
$373$	16.1244	−	27.9282i	0.834887	−	1.44607i	0.894122	−	0.447823i	$-0.147801\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			24.1803i			1.24535i
$378$		0			0
$379$			14.3923i			0.739283i	0.929174	+	0.369642i	$0.120520\pi$
$379$			14.3923i			0.739283i	−0.929174	+	0.369642i	$0.879480\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.60849	+	16.6424i	−0.490971	+	0.850387i	−0.999946	−	0.0103947i	$-0.996691\pi$
$383$	−9.60849	+	16.6424i	−0.490971	+	0.850387i	0.508975	+	0.860781i	$0.330025\pi$
$384$		0			0
$385$	3.73205	+	6.46410i	0.190203	+	0.329441i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.29392	+	5.36585i	−0.471221	+	0.272059i	−0.716751	−	0.697330i	$-0.754371\pi$
$389$	−9.29392	+	5.36585i	−0.471221	+	0.272059i	0.245530	+	0.969389i	$0.421038\pi$
$390$		0			0
$391$	−1.09808	−	0.633975i	−0.0555321	−	0.0320615i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	32.0464			1.61243
$396$		0			0
$397$	21.5885			1.08349			0.541747	−	0.840542i	$-0.317763\pi$
$397$	21.5885			1.08349			0.541747	+	0.840542i	$0.317763\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−16.2820	−	9.40044i	−0.813086	−	0.469436i	0.0349403	−	0.999389i	$-0.488876\pi$
$401$	−16.2820	−	9.40044i	−0.813086	−	0.469436i	−0.848026	+	0.529954i	$0.822209\pi$
$402$		0			0
$403$	28.8564	−	16.6603i	1.43744	−	0.829906i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.4046	−	35.3417i	−1.01142	−	1.75182i
$408$		0			0
$409$	−7.40192	+	12.8205i	−0.366002	+	0.633933i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−7.40192	+	12.8205i	−0.366002	+	0.633933i	0.622935	+	0.782274i	$0.285940\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			7.93048i			0.390233i
$414$		0			0
$415$		−	20.3923i		−	1.00102i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.795040	−	1.37705i	0.0388402	−	0.0672732i	−0.845952	−	0.533259i	$-0.820967\pi$
$419$	0.795040	−	1.37705i	0.0388402	−	0.0672732i	0.884792	+	0.465986i	$0.154300\pi$
$420$		0			0
$421$	−4.42820	−	7.66987i	−0.215817	−	0.373807i	0.737708	−	0.675120i	$-0.235908\pi$
$421$	−4.42820	−	7.66987i	−0.215817	−	0.373807i	−0.953525	+	0.301313i	$0.902575\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.984508	+	0.568406i	−0.0477557	+	0.0275717i
$426$		0			0
$427$	−0.758330	−	0.437822i	−0.0366982	−	0.0211877i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.48528			−0.408722			−0.204361	−	0.978896i	$-0.565512\pi$
$431$	−8.48528			−0.408722			−0.204361	+	0.978896i	$0.565512\pi$
$432$		0			0
$433$	−34.1769			−1.64244			−0.821219	−	0.570613i	$-0.806706\pi$
$433$	−34.1769			−1.64244			−0.821219	+	0.570613i	$0.806706\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.55291	−	0.896575i	−0.0742860	−	0.0428890i
$438$		0			0
$439$	−6.46410	+	3.73205i	−0.308515	+	0.178121i	−0.646262	−	0.763116i	$-0.723669\pi$
$439$	−6.46410	+	3.73205i	−0.308515	+	0.178121i	0.337747	+	0.941237i	$0.390335\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.52245	−	14.7613i	−0.404914	−	0.701331i	0.589398	−	0.807843i	$-0.299365\pi$
$443$	−8.52245	−	14.7613i	−0.404914	−	0.701331i	−0.994311	+	0.106512i	$0.966032\pi$
$444$		0			0
$445$	14.4282	−	24.9904i	0.683962	−	1.18466i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	23.0064i		−	1.08574i	−0.839818	−	0.542868i	$-0.817338\pi$
$449$		−	23.0064i		−	1.08574i	0.839818	−	0.542868i	$-0.182662\pi$
$450$		0			0
$451$			2.00000i			0.0941763i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.15660	+	5.46739i	−0.147984	+	0.256315i
$456$		0			0
$457$	1.40192	+	2.42820i	0.0655792	+	0.113587i	0.896951	−	0.442130i	$-0.145777\pi$
$457$	1.40192	+	2.42820i	0.0655792	+	0.113587i	−0.831372	+	0.555717i	$0.812444\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−30.8589	+	17.8164i	−1.43724	+	0.829791i	−0.997657	−	0.0684108i	$-0.978207\pi$
$461$	−30.8589	+	17.8164i	−1.43724	+	0.829791i	−0.439583	+	0.898202i	$0.644874\pi$
$462$		0			0
$463$	15.9282	+	9.19615i	0.740246	+	0.427381i	0.822159	−	0.569258i	$-0.192769\pi$
$463$	15.9282	+	9.19615i	0.740246	+	0.427381i	−0.0819125	+	0.996640i	$0.526103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0716			−0.558606			−0.279303	−	0.960203i	$-0.590103\pi$
$467$	−12.0716			−0.558606			−0.279303	+	0.960203i	$0.590103\pi$
$468$		0			0
$469$	9.60770			0.443642
$470$		0			0
$471$		0			0
$472$		0			0
$473$	39.9125	+	23.0435i	1.83518	+	1.05954i
$474$		0			0
$475$	−1.39230	+	0.803848i	−0.0638833	+	0.0368831i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.6443	−	27.0967i	−0.714805	−	1.23808i	−0.963035	−	0.269378i	$-0.913182\pi$
$479$	−15.6443	−	27.0967i	−0.714805	−	1.23808i	0.248229	−	0.968701i	$-0.420151\pi$
$480$		0			0
$481$	17.2583	−	29.8923i	0.786912	−	1.36297i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	12.3490i		−	0.560739i
$486$		0			0
$487$			36.9282i			1.67338i	0.547679	+	0.836688i	$0.315511\pi$
$487$			36.9282i			1.67338i	−0.547679	+	0.836688i	$0.684489\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.12184	−	12.3354i	0.321404	−	0.556688i	−0.659374	−	0.751815i	$-0.729179\pi$
$491$	7.12184	−	12.3354i	0.321404	−	0.556688i	0.980778	+	0.195127i	$0.0625119\pi$
$492$		0			0
$493$	−2.42820	−	4.20577i	−0.109361	−	0.189418i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.42091	−	4.86181i	0.377729	−	0.218082i
$498$		0			0
$499$	31.5622	+	18.2224i	1.41292	+	0.815748i	0.995662	−	0.0930414i	$-0.0296589\pi$
$499$	31.5622	+	18.2224i	1.41292	+	0.815748i	0.417255	+	0.908790i	$0.362992\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	29.1165			1.29824			0.649120	−	0.760686i	$-0.275137\pi$
$503$	29.1165			1.29824			0.649120	+	0.760686i	$0.275137\pi$
$504$		0			0
$505$	12.1962			0.542722
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.9957	+	18.4727i	1.41818	+	0.818788i	0.996139	−	0.0877870i	$-0.0279795\pi$
$509$	31.9957	+	18.4727i	1.41818	+	0.818788i	0.422044	+	0.906575i	$0.361313\pi$
$510$		0			0
$511$	8.70577	−	5.02628i	0.385121	−	0.222350i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.86370	−	6.69213i	−0.170255	−	0.294891i
$516$		0			0
$517$	12.1962	−	21.1244i	0.536386	−	0.929048i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15.2789i			0.669383i	0.942328	+	0.334691i	$0.108632\pi$
$521$			15.2789i			0.669383i	−0.942328	+	0.334691i	$0.891368\pi$
$522$		0			0
$523$			16.1962i			0.708208i	0.935206	+	0.354104i	$0.115214\pi$
$523$			16.1962i			0.708208i	−0.935206	+	0.354104i	$0.884786\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.34607	+	5.79555i	−0.145757	+	0.252458i
$528$		0			0
$529$	10.5000	+	18.1865i	0.456522	+	0.790719i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.46498	+	0.845807i	−0.0634554	+	0.0366360i
$534$		0			0
$535$	3.00000	+	1.73205i	0.129701	+	0.0748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−34.1170			−1.46952
$540$		0			0
$541$	7.00000			0.300954			0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000			0.300954			0.150477	+	0.988614i	$0.451919\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.57201	−	3.79435i	−0.281514	−	0.162532i
$546$		0			0
$547$	1.14359	−	0.660254i	0.0488965	−	0.0282304i	−0.475352	−	0.879795i	$-0.657679\pi$
$547$	1.14359	−	0.660254i	0.0488965	−	0.0282304i	0.524249	+	0.851565i	$0.324346\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3.43400	−	5.94786i	−0.146293	−	0.253387i
$552$		0			0
$553$	6.07180	−	10.5167i	0.258199	−	0.447214i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	2.03339i		−	0.0861574i	−0.999072	−	0.0430787i	$-0.986283\pi$
$557$		−	2.03339i		−	0.0861574i	0.999072	−	0.0430787i	$-0.0137166\pi$
$558$		0			0
$559$			38.9808i			1.64871i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.27981	+	7.41284i	−0.180372	+	0.312414i	−0.942007	−	0.335592i	$-0.891064\pi$
$563$	−4.27981	+	7.41284i	−0.180372	+	0.312414i	0.761635	+	0.648006i	$0.224397\pi$
$564$		0			0
$565$	19.5263	+	33.8205i	0.821477	+	1.42284i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.09154	+	3.51695i	−0.255371	+	0.147438i	−0.622221	−	0.782842i	$-0.713769\pi$
$569$	−6.09154	+	3.51695i	−0.255371	+	0.147438i	0.366850	+	0.930280i	$0.380436\pi$
$570$		0			0
$571$	34.5167	+	19.9282i	1.44448	+	0.833969i	0.998144	−	0.0608975i	$-0.0193963\pi$
$571$	34.5167	+	19.9282i	1.44448	+	0.833969i	0.446333	+	0.894867i	$0.352730\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.79315			−0.0747796
$576$		0			0
$577$	18.4641			0.768671			0.384335	−	0.923194i	$-0.374431\pi$
$577$	18.4641			0.768671			0.384335	+	0.923194i	$0.374431\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.69213	−	3.86370i	−0.277636	−	0.160293i
$582$		0			0
$583$	−11.1962	+	6.46410i	−0.463697	+	0.267716i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.845807	−	1.46498i	−0.0349102	−	0.0604663i	0.848042	−	0.529928i	$-0.177781\pi$
$587$	−0.845807	−	1.46498i	−0.0349102	−	0.0604663i	−0.882953	+	0.469462i	$0.844448\pi$
$588$		0			0
$589$	−4.73205	+	8.19615i	−0.194981	+	0.337717i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	21.1488i		−	0.868478i	−0.900798	−	0.434239i	$-0.857017\pi$
$593$		−	21.1488i		−	0.868478i	0.900798	−	0.434239i	$-0.142983\pi$
$594$		0			0
$595$		−	1.26795i		−	0.0519808i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.20977	−	12.4877i	0.294583	−	0.510233i	−0.680305	−	0.732929i	$-0.738153\pi$
$599$	7.20977	−	12.4877i	0.294583	−	0.510233i	0.974888	+	0.222696i	$0.0714859\pi$
$600$		0			0
$601$	−1.89230	−	3.27757i	−0.0771887	−	0.133695i	0.824847	−	0.565356i	$-0.191261\pi$
$601$	−1.89230	−	3.27757i	−0.0771887	−	0.133695i	−0.902036	+	0.431661i	$0.857928\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	28.2013	−	16.2820i	1.14655	−	0.661959i
$606$		0			0
$607$	−5.83013	−	3.36603i	−0.236638	−	0.136623i	0.376993	−	0.926216i	$-0.376958\pi$
$607$	−5.83013	−	3.36603i	−0.236638	−	0.136623i	−0.613630	+	0.789593i	$0.710291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.6312			0.834649
$612$		0			0
$613$	13.6077			0.549610			0.274805	−	0.961500i	$-0.411387\pi$
$613$	13.6077			0.549610			0.274805	+	0.961500i	$0.411387\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.03699	+	4.64016i	0.323557	+	0.186806i	0.652977	−	0.757378i	$-0.273520\pi$
$617$	8.03699	+	4.64016i	0.323557	+	0.186806i	−0.329420	+	0.944184i	$0.606853\pi$
$618$		0			0
$619$	−4.26795	+	2.46410i	−0.171543	+	0.0990406i	−0.583313	−	0.812247i	$-0.698244\pi$
$619$	−4.26795	+	2.46410i	−0.171543	+	0.0990406i	0.411770	+	0.911288i	$0.364911\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.46739	−	9.46979i	−0.219046	−	0.379399i
$624$		0			0
$625$	8.52628	−	14.7679i	0.341051	−	0.590718i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			6.93237i			0.276412i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20.4046	+	35.3417i	−0.809730	+	1.40249i
$636$		0			0
$637$	−14.4282	−	24.9904i	−0.571666	−	0.990155i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−32.1158	+	18.5421i	−1.26850	+	0.732367i	−0.974704	−	0.223502i	$-0.928251\pi$
$641$	−32.1158	+	18.5421i	−1.26850	+	0.732367i	−0.293794	+	0.955869i	$0.594918\pi$
$642$		0			0
$643$	−3.46410	−	2.00000i	−0.136611	−	0.0788723i	0.430137	−	0.902764i	$-0.358465\pi$
$643$	−3.46410	−	2.00000i	−0.136611	−	0.0788723i	−0.566748	+	0.823891i	$0.691799\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.6660			0.655206			0.327603	−	0.944815i	$-0.393759\pi$
$647$	16.6660			0.655206			0.327603	+	0.944815i	$0.393759\pi$
$648$		0			0
$649$	57.1769			2.24439
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.64566	−	5.56892i	−0.377464	−	0.217929i	0.299251	−	0.954175i	$-0.403263\pi$
$653$	−9.64566	−	5.56892i	−0.377464	−	0.217929i	−0.676714	+	0.736246i	$0.736597\pi$
$654$		0			0
$655$	0.169873	−	0.0980762i	0.00663749	−	0.00383215i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.8840	+	18.8516i	0.423981	+	0.734356i	0.996325	−	0.0856577i	$-0.0272991\pi$
$659$	10.8840	+	18.8516i	0.423981	+	0.734356i	−0.572344	+	0.820014i	$0.693966\pi$
$660$		0			0
$661$	4.25833	−	7.37564i	0.165630	−	0.286879i	−0.771249	−	0.636534i	$-0.780368\pi$
$661$	4.25833	−	7.37564i	0.165630	−	0.286879i	0.936879	+	0.349654i	$0.113701\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.79315i		−	0.0695354i
$666$		0			0
$667$		−	7.66025i		−	0.296606i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.15660	+	5.46739i	−0.121859	+	0.211066i
$672$		0			0
$673$	−15.1603	−	26.2583i	−0.584385	−	1.01218i	−0.994952	−	0.100354i	$-0.968003\pi$
$673$	−15.1603	−	26.2583i	−0.584385	−	1.01218i	0.410567	−	0.911830i	$-0.365331\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−33.6130	+	19.4064i	−1.29185	+	0.745850i	−0.978982	−	0.203946i	$-0.934623\pi$
$677$	−33.6130	+	19.4064i	−1.29185	+	0.745850i	−0.312869	+	0.949796i	$0.601290\pi$
$678$		0			0
$679$	−4.05256	−	2.33975i	−0.155523	−	0.0897912i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.37945			0.205839			0.102920	−	0.994690i	$-0.467182\pi$
$683$	5.37945			0.205839			0.102920	+	0.994690i	$0.467182\pi$
$684$		0			0
$685$	9.39230			0.358862
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.46979	−	5.46739i	−0.360770	−	0.208291i
$690$		0			0
$691$	−2.24167	+	1.29423i	−0.0852771	+	0.0492348i	−0.542032	−	0.840358i	$-0.682345\pi$
$691$	−2.24167	+	1.29423i	−0.0852771	+	0.0492348i	0.456755	+	0.889593i	$0.349012\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.9700	+	20.7327i	0.454050	+	0.786437i
$696$		0			0
$697$	0.169873	−	0.294229i	0.00643440	−	0.0111447i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			8.06918i			0.304769i	0.988321	+	0.152384i	$0.0486952\pi$
$701$			8.06918i			0.304769i	−0.988321	+	0.152384i	$0.951305\pi$
$702$		0			0
$703$			9.80385i			0.369759i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.31079	−	4.00240i	0.0869062	−	0.150526i
$708$		0			0
$709$	6.50000	+	11.2583i	0.244113	+	0.422815i	0.961882	−	0.273466i	$-0.0881700\pi$
$709$	6.50000	+	11.2583i	0.244113	+	0.422815i	−0.717769	+	0.696281i	$0.754837\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.14162	+	5.27792i	−0.342356	+	0.197660i
$714$		0			0
$715$	39.4186	+	22.7583i	1.47417	+	0.851113i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	36.3906			1.35714			0.678570	−	0.734535i	$-0.262600\pi$
$719$	36.3906			1.35714			0.678570	+	0.734535i	$0.262600\pi$
$720$		0			0
$721$	−2.92820			−0.109052
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−5.94786	−	3.43400i	−0.220898	−	0.127535i
$726$		0			0
$727$	22.4378	−	12.9545i	0.832173	−	0.480455i	−0.0224233	−	0.999749i	$-0.507138\pi$
$727$	22.4378	−	12.9545i	0.832173	−	0.480455i	0.854596	+	0.519293i	$0.173805\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.91447	−	6.78006i	−0.144782	−	0.250770i
$732$		0			0
$733$	−19.6603	+	34.0526i	−0.726168	+	1.25776i	0.232323	+	0.972639i	$0.425367\pi$
$733$	−19.6603	+	34.0526i	−0.726168	+	1.25776i	−0.958491	+	0.285121i	$0.907966\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	69.2693i		−	2.55157i
$738$		0			0
$739$			32.0000i			1.17714i	0.808447	+	0.588570i	$0.200309\pi$
$739$			32.0000i			1.17714i	−0.808447	+	0.588570i	$0.799691\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14.9000	+	25.8076i	−0.546628	+	0.946788i	0.451874	+	0.892082i	$0.350756\pi$
$743$	−14.9000	+	25.8076i	−0.546628	+	0.946788i	−0.998502	+	0.0547064i	$0.982578\pi$
$744$		0			0
$745$	1.76795	+	3.06218i	0.0647726	+	0.112190i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.13681	−	0.656339i	0.0415382	−	0.0239821i
$750$		0			0
$751$	8.15064	+	4.70577i	0.297421	+	0.171716i	0.641284	−	0.767304i	$-0.278402\pi$
$751$	8.15064	+	4.70577i	0.297421	+	0.171716i	−0.343863	+	0.939020i	$0.611736\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.07055			−0.0753551
$756$		0			0
$757$	−24.7846			−0.900812			−0.450406	−	0.892824i	$-0.648721\pi$
$757$	−24.7846			−0.900812			−0.450406	+	0.892824i	$0.648721\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.82534	+	1.05386i	0.0661684	+	0.0382023i	0.532719	−	0.846292i	$-0.321170\pi$
$761$	1.82534	+	1.05386i	0.0661684	+	0.0382023i	−0.466551	+	0.884494i	$0.654504\pi$
$762$		0			0
$763$	−2.49038	+	1.43782i	−0.0901578	+	0.0520527i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.1803	+	41.8816i	0.873101	+	1.51226i
$768$		0			0
$769$	20.0167	−	34.6699i	0.721819	−	1.25023i	−0.238451	−	0.971155i	$-0.576640\pi$
$769$	20.0167	−	34.6699i	0.721819	−	1.25023i	0.960270	−	0.279073i	$-0.0900271\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	35.9745i		−	1.29391i	−0.762527	−	0.646957i	$-0.776041\pi$
$773$		−	35.9745i		−	1.29391i	0.762527	−	0.646957i	$-0.223959\pi$
$774$		0			0
$775$			9.46410i			0.339961i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.240237	−	0.416102i	0.00860737	−	0.0149084i
$780$		0			0
$781$	−35.0526	−	60.7128i	−1.25428	−	2.17248i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.8350	−	10.2970i	0.636557	−	0.367516i
$786$		0			0
$787$	−21.4186	−	12.3660i	−0.763490	−	0.440801i	0.0670573	−	0.997749i	$-0.478639\pi$
$787$	−21.4186	−	12.3660i	−0.763490	−	0.440801i	−0.830547	+	0.556948i	$0.811972\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	14.7985			0.526173
$792$		0			0
$793$	−5.33975			−0.189620
$794$		0			0
$795$		0			0
$796$		0			0
$797$	44.3632	+	25.6131i	1.57143	+	0.907264i	0.995995	+	0.0894138i	$0.0284994\pi$
$797$	44.3632	+	25.6131i	1.57143	+	0.907264i	0.575432	+	0.817850i	$0.304834\pi$
$798$		0			0
$799$	−3.58846	+	2.07180i	−0.126950	+	0.0732949i
$800$		0			0
$801$		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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.67303 - 0.965926i) q^{5} +(-0.633975 + 0.366025i) q^{7} +O(q^{10})\)	\(q+(-1.67303 - 0.965926i) q^{5} +(-0.633975 + 0.366025i) q^{7} +(2.63896 + 4.57081i) q^{11} +(-2.23205 + 3.86603i) q^{13} -0.896575i q^{17} -1.26795i q^{19} +(0.707107 - 1.22474i) q^{23} +(-0.633975 - 1.09808i) q^{25} +(4.69093 - 2.70831i) q^{29} +(-6.46410 - 3.73205i) q^{31} +1.41421 q^{35} -7.73205 q^{37} +(0.328169 + 0.189469i) q^{41} +(7.56218 - 4.36603i) q^{43} +(-2.31079 - 4.00240i) q^{47} +(-3.23205 + 5.59808i) q^{49} +2.44949i q^{53} -10.1962i q^{55} +(5.41662 - 9.38186i) q^{59} +(0.598076 + 1.03590i) q^{61} +(7.46859 - 4.31199i) q^{65} +(-11.3660 - 6.56218i) q^{67} -13.2827 q^{71} -13.7321 q^{73} +(-3.34607 - 1.93185i) q^{77} +(-14.3660 + 8.29423i) q^{79} +(5.27792 + 9.14162i) q^{83} +(-0.866025 + 1.50000i) q^{85} +14.9372i q^{89} -3.26795i q^{91} +(-1.22474 + 2.12132i) q^{95} +(3.19615 + 5.53590i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{7}+O(q^{10}) \) 8 * q - 12 * q^7	\( 8 q - 12 q^{7} - 4 q^{13} - 12 q^{25} - 24 q^{31} - 48 q^{37} + 12 q^{43} - 12 q^{49} - 16 q^{61} - 84 q^{67} - 96 q^{73} - 108 q^{79} - 16 q^{97}+O(q^{100}) \) 8 * q - 12 * q^7 - 4 * q^13 - 12 * q^25 - 24 * q^31 - 48 * q^37 + 12 * q^43 - 12 * q^49 - 16 * q^61 - 84 * q^67 - 96 * q^73 - 108 * q^79 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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