LMFDB - Embedded newform 432.2.be.b.239.4

Newspace parameters

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.be (of order $18$, degree $6$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.44953736732$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$6$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			239.4
Character	$\chi$	$=$	432.239
Dual form			432.2.be.b.47.4

$q$-expansion

$f(q)$	$=$	$q+(0.721706 - 1.57453i) q^{3} +(0.976679 + 0.172215i) q^{5} +(3.12803 + 3.72784i) q^{7} +(-1.95828 - 2.27269i) q^{9} +O(q^{10})$	\(q+(0.721706 - 1.57453i) q^{3} +(0.976679 + 0.172215i) q^{5} +(3.12803 + 3.72784i) q^{7} +(-1.95828 - 2.27269i) q^{9} +(0.961816 + 5.45473i) q^{11} +(2.65607 - 0.966729i) q^{13} +(0.976032 - 1.41352i) q^{15} +(-0.517267 - 0.298644i) q^{17} +(0.141252 - 0.0815519i) q^{19} +(8.12712 - 2.23477i) q^{21} +(-5.35924 - 4.49693i) q^{23} +(-3.77422 - 1.37370i) q^{25} +(-4.99172 + 1.44316i) q^{27} +(0.610528 - 1.67741i) q^{29} +(3.26328 - 3.88903i) q^{31} +(9.28277 + 2.42230i) q^{33} +(2.41309 + 4.17960i) q^{35} +(0.726277 - 1.25795i) q^{37} +(0.394755 - 4.87975i) q^{39} +(-3.84382 - 10.5608i) q^{41} +(6.11082 - 1.07750i) q^{43} +(-1.52122 - 2.55694i) q^{45} +(-6.37666 + 5.35065i) q^{47} +(-2.89670 + 16.4280i) q^{49} +(-0.843539 + 0.598919i) q^{51} -8.40465i q^{53} +5.49316i q^{55} +(-0.0264635 - 0.281262i) q^{57} +(-1.35597 + 7.69008i) q^{59} +(-3.16592 + 2.65652i) q^{61} +(2.34667 - 14.4092i) q^{63} +(2.76061 - 0.486770i) q^{65} +(2.02334 + 5.55908i) q^{67} +(-10.9483 + 5.19281i) q^{69} +(1.86491 - 3.23013i) q^{71} +(3.05997 + 5.30003i) q^{73} +(-4.88681 + 4.95121i) q^{75} +(-17.3258 + 20.6481i) q^{77} +(4.05391 - 11.1380i) q^{79} +(-1.33026 + 8.90115i) q^{81} +(-9.04948 - 3.29374i) q^{83} +(-0.453773 - 0.380761i) q^{85} +(-2.20051 - 2.17189i) q^{87} +(-1.45033 + 0.837348i) q^{89} +(11.9121 + 6.87744i) q^{91} +(-3.76826 - 7.94486i) q^{93} +(0.152002 - 0.0553244i) q^{95} +(-2.34010 - 13.2714i) q^{97} +(10.5134 - 12.8678i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 3 q^{5} + 6 q^{9} + O(q^{10}) $	$ 36 q + 3 q^{5} + 6 q^{9} - 18 q^{11} + 9 q^{15} + 18 q^{21} - 9 q^{25} + 30 q^{29} + 27 q^{31} + 27 q^{33} + 27 q^{35} - 45 q^{39} + 18 q^{41} + 27 q^{45} + 45 q^{47} - 63 q^{51} - 9 q^{57} + 54 q^{59} - 63 q^{63} - 57 q^{65} - 63 q^{69} + 36 q^{71} + 9 q^{73} - 45 q^{75} - 81 q^{77} - 54 q^{81} - 27 q^{83} - 36 q^{85} + 45 q^{87} - 63 q^{89} + 27 q^{91} - 63 q^{93} - 72 q^{95} + 99 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{11}{18}\right)$

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			0
$2$		0			0
$3$	0.721706	−	1.57453i	0.416677	−	0.909055i
$3$	0.721706	−	1.57453i	0.416677	−	0.909055i
$4$		0			0
$5$	0.976679	+	0.172215i	0.436784	+	0.0770169i	0.387717	−	0.921779i	$-0.373264\pi$
$5$	0.976679	+	0.172215i	0.436784	+	0.0770169i	0.0490676	+	0.998795i	$0.484375\pi$
$6$		0			0
$7$	3.12803	+	3.72784i	1.18229	+	1.40899i	0.891990	+	0.452055i	$0.149309\pi$
$7$	3.12803	+	3.72784i	1.18229	+	1.40899i	0.290295	+	0.956937i	$0.406247\pi$
$8$		0			0
$9$	−1.95828	−	2.27269i	−0.652761	−	0.757564i
$10$		0			0
$11$	0.961816	+	5.45473i	0.289998	+	1.64466i	0.686864	+	0.726786i	$0.258987\pi$
$11$	0.961816	+	5.45473i	0.289998	+	1.64466i	−0.396866	+	0.917877i	$0.629902\pi$
$12$		0			0
$13$	2.65607	−	0.966729i	0.736661	−	0.268123i	0.0536791	−	0.998558i	$-0.482905\pi$
$13$	2.65607	−	0.966729i	0.736661	−	0.268123i	0.682981	+	0.730436i	$0.260683\pi$
$14$		0			0
$15$	0.976032	−	1.41352i	0.252010	−	0.364970i
$16$		0			0
$17$	−0.517267	−	0.298644i	−0.125456	−	0.0724319i	0.435959	−	0.899967i	$-0.356409\pi$
$17$	−0.517267	−	0.298644i	−0.125456	−	0.0724319i	−0.561415	+	0.827535i	$0.689743\pi$
$18$		0			0
$19$	0.141252	−	0.0815519i	0.0324055	−	0.0187093i	−0.483710	−	0.875228i	$-0.660711\pi$
$19$	0.141252	−	0.0815519i	0.0324055	−	0.0187093i	0.516115	+	0.856519i	$0.327378\pi$
$20$		0			0
$21$	8.12712	−	2.23477i	1.77348	−	0.487667i
$22$		0			0
$23$	−5.35924	−	4.49693i	−1.11748	−	0.937675i	−0.119004	−	0.992894i	$-0.537970\pi$
$23$	−5.35924	−	4.49693i	−1.11748	−	0.937675i	−0.998474	+	0.0552184i	$0.982415\pi$
$24$		0			0
$25$	−3.77422	−	1.37370i	−0.754844	−	0.274741i
$26$		0			0
$27$	−4.99172	+	1.44316i	−0.960658	+	0.277736i
$28$		0			0
$29$	0.610528	−	1.67741i	0.113372	−	0.311488i	−0.870010	−	0.493034i	$-0.835888\pi$
$29$	0.610528	−	1.67741i	0.113372	−	0.311488i	0.983382	+	0.181546i	$0.0581101\pi$
$30$		0			0
$31$	3.26328	−	3.88903i	0.586102	−	0.698490i	−0.388750	−	0.921343i	$-0.627093\pi$
$31$	3.26328	−	3.88903i	0.586102	−	0.698490i	0.974852	+	0.222854i	$0.0715373\pi$
$32$		0			0
$33$	9.28277	+	2.42230i	1.61592	+	0.421668i
$34$		0			0
$35$	2.41309	+	4.17960i	0.407887	+	0.706482i
$36$		0			0
$37$	0.726277	−	1.25795i	0.119399	−	0.206806i	−0.800131	−	0.599826i	$-0.795236\pi$
$37$	0.726277	−	1.25795i	0.119399	−	0.206806i	0.919530	+	0.393020i	$0.128570\pi$
$38$		0			0
$39$	0.394755	−	4.87975i	0.0632114	−	0.781385i
$40$		0			0
$41$	−3.84382	−	10.5608i	−0.600304	−	1.64932i	−0.750658	−	0.660691i	$-0.770264\pi$
$41$	−3.84382	−	10.5608i	−0.600304	−	1.64932i	0.150354	−	0.988632i	$-0.451959\pi$
$42$		0			0
$43$	6.11082	−	1.07750i	0.931892	−	0.164318i	0.312964	−	0.949765i	$-0.398678\pi$
$43$	6.11082	−	1.07750i	0.931892	−	0.164318i	0.618928	+	0.785447i	$0.287567\pi$
$44$		0			0
$45$	−1.52122	−	2.55694i	−0.226770	−	0.381166i
$46$		0			0
$47$	−6.37666	+	5.35065i	−0.930131	+	0.780473i	−0.975841	−	0.218482i	$-0.929890\pi$
$47$	−6.37666	+	5.35065i	−0.930131	+	0.780473i	0.0457098	+	0.998955i	$0.485445\pi$
$48$		0			0
$49$	−2.89670	+	16.4280i	−0.413814	+	2.34685i
$50$		0			0
$51$	−0.843539	+	0.598919i	−0.118119	+	0.0838654i
$52$		0			0
$53$		−	8.40465i		−	1.15447i	−0.816579	−	0.577234i	$-0.804132\pi$
$53$		−	8.40465i		−	1.15447i	0.816579	−	0.577234i	$-0.195868\pi$
$54$		0			0
$55$			5.49316i			0.740697i
$56$		0			0
$57$	−0.0264635	−	0.281262i	−0.00350517	−	0.0372541i
$58$		0			0
$59$	−1.35597	+	7.69008i	−0.176532	+	1.00116i	0.759828	+	0.650124i	$0.225283\pi$
$59$	−1.35597	+	7.69008i	−0.176532	+	1.00116i	−0.936361	+	0.351040i	$0.885828\pi$
$60$		0			0
$61$	−3.16592	+	2.65652i	−0.405354	+	0.340133i	−0.822559	−	0.568680i	$-0.807454\pi$
$61$	−3.16592	+	2.65652i	−0.405354	+	0.340133i	0.417204	+	0.908813i	$0.363010\pi$
$62$		0			0
$63$	2.34667	−	14.4092i	0.295653	−	1.81539i
$64$		0			0
$65$	2.76061	−	0.486770i	0.342412	−	0.0603764i
$66$		0			0
$67$	2.02334	+	5.55908i	0.247190	+	0.679150i	0.999786	+	0.0206675i	$0.00657914\pi$
$67$	2.02334	+	5.55908i	0.247190	+	0.679150i	−0.752596	+	0.658483i	$0.771199\pi$
$68$		0			0
$69$	−10.9483	+	5.19281i	−1.31803	+	0.625141i
$70$		0			0
$71$	1.86491	−	3.23013i	0.221325	−	0.383345i	−0.733886	−	0.679273i	$-0.762295\pi$
$71$	1.86491	−	3.23013i	0.221325	−	0.383345i	0.955210	+	0.295927i	$0.0956287\pi$
$72$		0			0
$73$	3.05997	+	5.30003i	0.358143	+	0.620321i	0.987651	−	0.156673i	$-0.0500768\pi$
$73$	3.05997	+	5.30003i	0.358143	+	0.620321i	−0.629508	+	0.776994i	$0.716743\pi$
$74$		0			0
$75$	−4.88681	+	4.95121i	−0.564280	+	0.571716i
$76$		0			0
$77$	−17.3258	+	20.6481i	−1.97446	+	2.35306i
$78$		0			0
$79$	4.05391	−	11.1380i	0.456100	−	1.25312i	−0.472266	−	0.881456i	$-0.656564\pi$
$79$	4.05391	−	11.1380i	0.456100	−	1.25312i	0.928366	−	0.371668i	$-0.121214\pi$
$80$		0			0
$81$	−1.33026	+	8.90115i	−0.147807	+	0.989016i
$82$		0			0
$83$	−9.04948	−	3.29374i	−0.993309	−	0.361535i	−0.206308	−	0.978487i	$-0.566145\pi$
$83$	−9.04948	−	3.29374i	−0.993309	−	0.361535i	−0.787001	+	0.616952i	$0.788367\pi$
$84$		0			0
$85$	−0.453773	−	0.380761i	−0.0492186	−	0.0412993i
$86$		0			0
$87$	−2.20051	−	2.17189i	−0.235920	−	0.232851i
$88$		0			0
$89$	−1.45033	+	0.837348i	−0.153735	+	0.0887587i	−0.574894	−	0.818228i	$-0.694957\pi$
$89$	−1.45033	+	0.837348i	−0.153735	+	0.0887587i	0.421159	+	0.906987i	$0.361623\pi$
$90$		0			0
$91$	11.9121	+	6.87744i	1.24873	+	0.720952i
$92$		0			0
$93$	−3.76826	−	7.94486i	−0.390750	−	0.823844i
$94$		0			0
$95$	0.152002	−	0.0553244i	0.0155951	−	0.00567616i
$96$		0			0
$97$	−2.34010	−	13.2714i	−0.237601	−	1.34750i	−0.837066	−	0.547102i	$-0.815731\pi$
$97$	−2.34010	−	13.2714i	−0.237601	−	1.34750i	0.599465	−	0.800401i	$-0.295380\pi$
$98$		0			0
$99$	10.5134	−	12.8678i	1.05664	−	1.29326i
$100$		0			0
$101$	8.98820	+	10.7117i	0.894360	+	1.06586i	0.997463	+	0.0711857i	$0.0226783\pi$
$101$	8.98820	+	10.7117i	0.894360	+	1.06586i	−0.103103	+	0.994671i	$0.532877\pi$
$102$		0			0
$103$	−5.43335	−	0.958047i	−0.535364	−	0.0943991i	−0.100572	−	0.994930i	$-0.532067\pi$
$103$	−5.43335	−	0.958047i	−0.535364	−	0.0943991i	−0.434792	+	0.900531i	$0.643178\pi$
$104$		0			0
$105$	8.32245	−	0.783045i	0.812188	−	0.0764173i
$106$		0			0
$107$	−11.3996			−1.10204			−0.551021	−	0.834491i	$-0.685762\pi$
$107$	−11.3996			−1.10204			−0.551021	+	0.834491i	$0.685762\pi$
$108$		0			0
$109$	10.4472			1.00066			0.500329	−	0.865835i	$-0.333212\pi$
$109$	10.4472			1.00066			0.500329	+	0.865835i	$0.333212\pi$
$110$		0			0
$111$	−1.45652	−	2.05141i	−0.138247	−	0.194712i
$112$		0			0
$113$	−3.27678	−	0.577786i	−0.308254	−	0.0543535i	0.0173814	−	0.999849i	$-0.494467\pi$
$113$	−3.27678	−	0.577786i	−0.308254	−	0.0543535i	−0.325635	+	0.945495i	$0.605578\pi$
$114$		0			0
$115$	−4.45982	−	5.31500i	−0.415880	−	0.495627i
$116$		0			0
$117$	−7.39841	−	4.14330i	−0.683983	−	0.383048i
$118$		0			0
$119$	−0.504729	−	2.86246i	−0.0462684	−	0.262401i
$120$		0			0
$121$	−18.4923	+	6.73066i	−1.68112	+	0.611878i
$122$		0			0
$123$	−19.4024	−	1.56959i	−1.74946	−	0.141525i
$124$		0			0
$125$	−7.74402	−	4.47101i	−0.692646	−	0.399899i
$126$		0			0
$127$	3.80947	−	2.19940i	0.338036	−	0.195165i	−0.321367	−	0.946955i	$-0.604142\pi$
$127$	3.80947	−	2.19940i	0.338036	−	0.195165i	0.659403	+	0.751790i	$0.270809\pi$
$128$		0			0
$129$	2.71366	−	10.3993i	0.238924	−	0.915608i
$130$		0			0
$131$	−1.04355	−	0.875644i	−0.0911755	−	0.0765053i	0.596061	−	0.802939i	$-0.296732\pi$
$131$	−1.04355	−	0.875644i	−0.0911755	−	0.0765053i	−0.687236	+	0.726434i	$0.741176\pi$
$132$		0			0
$133$	0.745854	+	0.271469i	0.0646737	+	0.0235393i
$134$		0			0
$135$	−5.12385	+	0.549852i	−0.440990	+	0.0473238i
$136$		0			0
$137$	−1.38622	+	3.80861i	−0.118433	+	0.325391i	−0.984718	−	0.174159i	$-0.944279\pi$
$137$	−1.38622	+	3.80861i	−0.118433	+	0.325391i	0.866285	+	0.499551i	$0.166501\pi$
$138$		0			0
$139$	−10.0544	+	11.9823i	−0.852801	+	1.01633i	0.146830	+	0.989162i	$0.453093\pi$
$139$	−10.0544	+	11.9823i	−0.852801	+	1.01633i	−0.999631	+	0.0271664i	$0.991352\pi$
$140$		0			0
$141$	3.82269	+	13.9018i	0.321928	+	1.17075i
$142$		0			0
$143$	7.82789	+	13.5583i	0.654601	+	1.13380i
$144$		0			0
$145$	0.885166	−	1.53315i	0.0735090	−	0.127321i
$146$		0			0
$147$	23.7758	+	16.4171i	1.96099	+	1.35406i
$148$		0			0
$149$	2.36175	+	6.48884i	0.193482	+	0.531587i	0.998060	−	0.0622599i	$-0.0198308\pi$
$149$	2.36175	+	6.48884i	0.193482	+	0.531587i	−0.804578	+	0.593847i	$0.797609\pi$
$150$		0			0
$151$	−5.40054	+	0.952262i	−0.439490	+	0.0774940i	−0.389015	−	0.921231i	$-0.627185\pi$
$151$	−5.40054	+	0.952262i	−0.439490	+	0.0774940i	−0.0504750	+	0.998725i	$0.516074\pi$
$152$		0			0
$153$	0.334228	+	1.76042i	0.0270208	+	0.142321i
$154$		0			0
$155$	3.85693	−	3.23635i	0.309796	−	0.259949i
$156$		0			0
$157$	0.626364	−	3.55229i	0.0499893	−	0.283503i	−0.949558	−	0.313592i	$-0.898468\pi$
$157$	0.626364	−	3.55229i	0.0499893	−	0.283503i	0.999547	+	0.0300883i	$0.00957885\pi$
$158$		0			0
$159$	−13.2334	−	6.06568i	−1.04947	−	0.481040i
$160$		0			0
$161$		−	34.0449i		−	2.68312i
$162$		0			0
$163$			5.07293i			0.397343i	0.980066	+	0.198671i	$0.0636627\pi$
$163$			5.07293i			0.397343i	−0.980066	+	0.198671i	$0.936337\pi$
$164$		0			0
$165$	8.64914	+	3.96444i	0.673334	+	0.308631i
$166$		0			0
$167$	1.81056	−	10.2682i	0.140105	−	0.794576i	−0.831062	−	0.556179i	$-0.812267\pi$
$167$	1.81056	−	10.2682i	0.140105	−	0.794576i	0.971168	−	0.238397i	$-0.0766220\pi$
$168$		0			0
$169$	−3.83845	+	3.22084i	−0.295265	+	0.247757i
$170$		0			0
$171$	−0.461954	−	0.161321i	−0.0353265	−	0.0123365i
$172$		0			0
$173$	19.5905	−	3.45433i	1.48944	−	0.262628i	0.631094	−	0.775707i	$-0.282606\pi$
$173$	19.5905	−	3.45433i	1.48944	−	0.262628i	0.858341	+	0.513079i	$0.171495\pi$
$174$		0			0
$175$	−6.68493	−	18.3667i	−0.505333	−	1.38839i
$176$		0			0
$177$	11.1296	+	7.68499i	0.836555	+	0.577639i
$178$		0			0
$179$	7.51483	−	13.0161i	0.561685	−	0.972866i	−0.435665	−	0.900109i	$-0.643487\pi$
$179$	7.51483	−	13.0161i	0.561685	−	0.972866i	0.997350	−	0.0727573i	$-0.0231798\pi$
$180$		0			0
$181$	2.08688	+	3.61458i	0.155116	+	0.268669i	0.933101	−	0.359613i	$-0.117091\pi$
$181$	2.08688	+	3.61458i	0.155116	+	0.268669i	−0.777985	+	0.628283i	$0.783758\pi$
$182$		0			0
$183$	1.89791	+	6.90206i	0.140297	+	0.510215i
$184$		0			0
$185$	0.925978	−	1.10354i	0.0680792	−	0.0811337i
$186$		0			0
$187$	1.13151	−	3.10879i	0.0827440	−	0.227337i
$188$		0			0
$189$	−20.9941	−	14.0941i	−1.52710	−	1.02520i
$190$		0			0
$191$	22.5495	+	8.20734i	1.63162	+	0.593862i	0.985546	−	0.169409i	$-0.0541860\pi$
$191$	22.5495	+	8.20734i	1.63162	+	0.593862i	0.646078	+	0.763272i	$0.276408\pi$
$192$		0			0
$193$	6.52079	+	5.47159i	0.469377	+	0.393854i	0.846567	−	0.532282i	$-0.178665\pi$
$193$	6.52079	+	5.47159i	0.469377	+	0.393854i	−0.377191	+	0.926136i	$0.623110\pi$
$194$		0			0
$195$	1.22591	−	4.69797i	0.0877896	−	0.336428i
$196$		0			0
$197$	3.63484	−	2.09858i	0.258972	−	0.149517i	−0.364894	−	0.931049i	$-0.618895\pi$
$197$	3.63484	−	2.09858i	0.258972	−	0.149517i	0.623865	+	0.781532i	$0.285561\pi$
$198$		0			0
$199$	−4.58250	−	2.64571i	−0.324845	−	0.187549i	0.328705	−	0.944433i	$-0.393388\pi$
$199$	−4.58250	−	2.64571i	−0.324845	−	0.187549i	−0.653550	+	0.756883i	$0.726721\pi$
$200$		0			0
$201$	10.2132	+	0.826213i	0.720383	+	0.0582765i
$202$		0			0
$203$	8.16288	−	2.97105i	0.572922	−	0.208527i
$204$		0			0
$205$	−1.93545	−	10.9765i	−0.135178	−	0.766632i
$206$		0			0
$207$	0.274751	+	20.9862i	0.0190965	+	1.45864i
$208$		0			0
$209$	0.580702	+	0.692054i	0.0401680	+	0.0478704i
$210$		0			0
$211$	−11.1463	−	1.96540i	−0.767344	−	0.135303i	−0.223744	−	0.974648i	$-0.571828\pi$
$211$	−11.1463	−	1.96540i	−0.767344	−	0.135303i	−0.543600	+	0.839345i	$0.682939\pi$
$212$		0			0
$213$	−3.74001	−	5.26756i	−0.256261	−	0.360927i
$214$		0			0
$215$	6.15388			0.419691
$216$		0			0
$217$	24.7053			1.67711
$218$		0			0
$219$	10.5534	−	0.992956i	0.713136	−	0.0670977i
$220$		0			0
$221$	−1.66260	−	0.293162i	−0.111839	−	0.0197202i
$222$		0			0
$223$	11.2504	+	13.4077i	0.753380	+	0.897843i	0.997410	−	0.0719243i	$-0.0229140\pi$
$223$	11.2504	+	13.4077i	0.753380	+	0.897843i	−0.244030	+	0.969768i	$0.578470\pi$
$224$		0			0
$225$	4.26898	+	11.2677i	0.284599	+	0.751182i
$226$		0			0
$227$	0.724677	+	4.10985i	0.0480985	+	0.272780i	0.999367	−	0.0355847i	$-0.0113294\pi$
$227$	0.724677	+	4.10985i	0.0480985	+	0.272780i	−0.951268	+	0.308365i	$0.900218\pi$
$228$		0			0
$229$	−1.78302	+	0.648967i	−0.117825	+	0.0428849i	−0.400260	−	0.916402i	$-0.631080\pi$
$229$	−1.78302	+	0.648967i	−0.117825	+	0.0428849i	0.282435	+	0.959287i	$0.408858\pi$
$230$		0			0
$231$	20.0069	+	42.1818i	1.31635	+	2.77536i
$232$		0			0
$233$	15.7256	+	9.07917i	1.03022	+	0.594796i	0.917047	−	0.398779i	$-0.130566\pi$
$233$	15.7256	+	9.07917i	1.03022	+	0.594796i	0.113171	+	0.993576i	$0.463899\pi$
$234$		0			0
$235$	−7.14941	+	4.12772i	−0.466376	+	0.269262i
$236$		0			0
$237$	−14.6114	−	14.4214i	−0.949112	−	0.936768i
$238$		0			0
$239$	−13.3664	−	11.2158i	−0.864602	−	0.725487i	0.0983526	−	0.995152i	$-0.468643\pi$
$239$	−13.3664	−	11.2158i	−0.864602	−	0.725487i	−0.962954	+	0.269665i	$0.913087\pi$
$240$		0			0
$241$	−25.1799	−	9.16473i	−1.62198	−	0.590352i	−0.638222	−	0.769853i	$-0.720330\pi$
$241$	−25.1799	−	9.16473i	−1.62198	−	0.590352i	−0.983758	+	0.179500i	$0.942552\pi$
$242$		0			0
$243$	13.0551	+	8.51854i	0.837482	+	0.546465i
$244$		0			0
$245$	−5.65828	+	15.5460i	−0.361495	+	0.993198i
$246$		0			0
$247$	0.296336	−	0.353160i	0.0188554	−	0.0224710i
$248$		0			0
$249$	−11.7171	+	11.8716i	−0.742544	+	0.752329i
$250$		0			0
$251$	9.18578	+	15.9102i	0.579801	+	1.00425i	0.995502	+	0.0947437i	$0.0302032\pi$
$251$	9.18578	+	15.9102i	0.579801	+	1.00425i	−0.415700	+	0.909502i	$0.636464\pi$
$252$		0			0
$253$	19.3749	−	33.5584i	1.21809	−	2.10980i
$254$		0			0
$255$	−0.927009	+	0.439682i	−0.0580516	+	0.0275339i
$256$		0			0
$257$	7.13045	+	19.5908i	0.444786	+	1.22204i	0.936310	+	0.351174i	$0.114218\pi$
$257$	7.13045	+	19.5908i	0.444786	+	1.22204i	−0.491525	+	0.870864i	$0.663560\pi$
$258$		0			0
$259$	6.96126	−	1.22746i	0.432551	−	0.0762705i
$260$		0			0
$261$	−5.00783	+	1.89730i	−0.309977	+	0.117440i
$262$		0			0
$263$	−20.5295	+	17.2263i	−1.26590	+	1.06222i	−0.270878	+	0.962614i	$0.587314\pi$
$263$	−20.5295	+	17.2263i	−1.26590	+	1.06222i	−0.995027	+	0.0996068i	$0.968242\pi$
$264$		0			0
$265$	1.44741	−	8.20865i	0.0889135	−	0.504253i
$266$		0			0
$267$	0.271718	+	2.88790i	0.0166289	+	0.176737i
$268$		0			0
$269$			6.10651i			0.372321i	0.982519	+	0.186160i	$0.0596044\pi$
$269$			6.10651i			0.372321i	−0.982519	+	0.186160i	$0.940396\pi$
$270$		0			0
$271$			5.24989i			0.318908i	0.987205	+	0.159454i	$0.0509734\pi$
$271$			5.24989i			0.318908i	−0.987205	+	0.159454i	$0.949027\pi$
$272$		0			0
$273$	19.4257	−	13.7924i	1.17570	−	0.834756i
$274$		0			0
$275$	3.86307	−	21.9086i	0.232952	−	1.32114i
$276$		0			0
$277$	21.8063	−	18.2977i	1.31022	−	1.09940i	0.321934	−	0.946762i	$-0.395667\pi$
$277$	21.8063	−	18.2977i	1.31022	−	1.09940i	0.988282	−	0.152640i	$-0.0487775\pi$
$278$		0			0
$279$	−15.2290	+	0.199378i	−0.911735	+	0.0119364i
$280$		0			0
$281$	2.99467	−	0.528040i	0.178647	−	0.0315002i	−0.0836090	−	0.996499i	$-0.526645\pi$
$281$	2.99467	−	0.528040i	0.178647	−	0.0315002i	0.262256	+	0.964998i	$0.415534\pi$
$282$		0			0
$283$	−5.13245	−	14.1013i	−0.305092	−	0.838235i	−0.993595	−	0.113001i	$-0.963954\pi$
$283$	−5.13245	−	14.1013i	−0.305092	−	0.838235i	0.688502	−	0.725234i	$-0.258268\pi$
$284$		0			0
$285$	0.0225912	−	0.279260i	0.00133819	−	0.0165419i
$286$		0			0
$287$	27.3455	−	47.3638i	1.61415	−	2.79579i
$288$		0			0
$289$	−8.32162	−	14.4135i	−0.489507	−	0.847851i
$290$		0			0
$291$	−22.5850	−	5.89346i	−1.32396	−	0.345481i
$292$		0			0
$293$	12.3270	−	14.6907i	0.720150	−	0.858241i	−0.274495	−	0.961588i	$-0.588511\pi$
$293$	12.3270	−	14.6907i	0.720150	−	0.858241i	0.994645	+	0.103347i	$0.0329552\pi$
$294$		0			0
$295$	−2.64869	+	7.27723i	−0.154213	+	0.423696i
$296$		0			0
$297$	−12.6731	−	25.8404i	−0.735370	−	1.49941i
$298$		0			0
$299$	−18.5818	−	6.76323i	−1.07461	−	0.391127i
$300$		0			0
$301$	23.1316	+	19.4097i	1.33328	+	1.11876i
$302$		0			0
$303$	23.3528	−	6.42147i	1.34158	−	0.368904i
$304$		0			0
$305$	−3.54958	+	2.04935i	−0.203248	+	0.117346i
$306$		0			0
$307$	−13.9719	−	8.06666i	−0.797416	−	0.460388i	0.0451507	−	0.998980i	$-0.485623\pi$
$307$	−13.9719	−	8.06666i	−0.797416	−	0.460388i	−0.842567	+	0.538592i	$0.818957\pi$
$308$		0			0
$309$	−5.42975	+	7.86354i	−0.308888	+	0.447341i
$310$		0			0
$311$	−17.9054	+	6.51704i	−1.01532	+	0.369548i	−0.795475	−	0.605986i	$-0.792779\pi$
$311$	−17.9054	+	6.51704i	−1.01532	+	0.369548i	−0.219849	+	0.975534i	$0.570556\pi$
$312$		0			0
$313$	1.68661	+	9.56523i	0.0953328	+	0.540659i	0.994645	+	0.103352i	$0.0329567\pi$
$313$	1.68661	+	9.56523i	0.0953328	+	0.540659i	−0.899312	+	0.437307i	$0.855932\pi$
$314$		0			0
$315$	4.77343	−	13.6691i	0.268952	−	0.770164i
$316$		0			0
$317$	12.4921	+	14.8875i	0.701628	+	0.836167i	0.992710	−	0.120531i	$-0.0384596\pi$
$317$	12.4921	+	14.8875i	0.701628	+	0.836167i	−0.291082	+	0.956698i	$0.594015\pi$
$318$		0			0
$319$	9.73704	+	1.71690i	0.545170	+	0.0961281i
$320$		0			0
$321$	−8.22716	+	17.9490i	−0.459195	+	1.00182i
$322$		0			0
$323$	−0.0974201			−0.00542060
$324$		0			0
$325$	−11.3526			−0.629728
$326$		0			0
$327$	7.53979	−	16.4494i	0.416951	−	0.909653i
$328$		0			0
$329$	−39.8928	−	7.03417i	−2.19936	−	0.387807i
$330$		0			0
$331$	−21.1483	−	25.2035i	−1.16241	−	1.38531i	−0.908395	−	0.418113i	$-0.862692\pi$
$331$	−21.1483	−	25.2035i	−1.16241	−	1.38531i	−0.254020	−	0.967199i	$-0.581753\pi$
$332$		0			0
$333$	−4.28119	+	0.812814i	−0.234608	+	0.0445420i
$334$		0			0
$335$	1.01880	+	5.77789i	0.0556629	+	0.315680i
$336$		0			0
$337$	−3.50899	+	1.27717i	−0.191147	+	0.0695717i	−0.435820	−	0.900034i	$-0.643542\pi$
$337$	−3.50899	+	1.27717i	−0.191147	+	0.0695717i	0.244673	+	0.969606i	$0.421319\pi$
$338$		0			0
$339$	−3.27461	+	4.74240i	−0.177853	+	0.257572i
$340$		0			0
$341$	24.3522	+	14.0598i	1.31875	+	0.761380i
$342$		0			0
$343$	−40.8012	+	23.5566i	−2.20306	+	1.27193i
$344$		0			0
$345$	−11.5873	+	3.18624i	−0.623839	+	0.171542i
$346$		0			0
$347$	17.2766	+	14.4968i	0.927456	+	0.778228i	0.975359	−	0.220624i	$-0.0708093\pi$
$347$	17.2766	+	14.4968i	0.927456	+	0.778228i	−0.0479028	+	0.998852i	$0.515254\pi$
$348$		0			0
$349$	−7.89430	−	2.87329i	−0.422572	−	0.153804i	0.121977	−	0.992533i	$-0.461077\pi$
$349$	−7.89430	−	2.87329i	−0.422572	−	0.153804i	−0.544548	+	0.838729i	$0.683299\pi$
$350$		0			0
$351$	−11.8632	+	8.65877i	−0.633211	+	0.462171i
$352$		0			0
$353$	9.50025	−	26.1017i	0.505648	−	1.38926i	−0.380039	−	0.924971i	$-0.624089\pi$
$353$	9.50025	−	26.1017i	0.505648	−	1.38926i	0.885686	−	0.464285i	$-0.153688\pi$
$354$		0			0
$355$	2.37770	−	2.83363i	0.126195	−	0.150394i
$356$		0			0
$357$	−4.87129	−	1.27114i	−0.257816	−	0.0672760i
$358$		0			0
$359$	11.1197	+	19.2599i	0.586877	+	1.01650i	0.994639	+	0.103412i	$0.0329759\pi$
$359$	11.1197	+	19.2599i	0.586877	+	1.01650i	−0.407762	+	0.913088i	$0.633691\pi$
$360$		0			0
$361$	−9.48670	+	16.4314i	−0.499300	+	0.864813i
$362$		0			0
$363$	−2.74840	+	33.9743i	−0.144254	+	1.78319i
$364$		0			0
$365$	2.07587	+	5.70340i	0.108656	+	0.298530i
$366$		0			0
$367$	−6.99817	+	1.23397i	−0.365302	+	0.0644125i	−0.353286	−	0.935515i	$-0.614936\pi$
$367$	−6.99817	+	1.23397i	−0.365302	+	0.0644125i	−0.0120154	+	0.999928i	$0.503825\pi$
$368$		0			0
$369$	−16.4742	+	29.4169i	−0.857613	+	1.53138i
$370$		0			0
$371$	31.3312	−	26.2900i	1.62664	−	1.36491i
$372$		0			0
$373$	−3.08072	+	17.4716i	−0.159514	+	0.904646i	0.795029	+	0.606571i	$0.207456\pi$
$373$	−3.08072	+	17.4716i	−0.159514	+	0.904646i	−0.954543	+	0.298075i	$0.903656\pi$
$374$		0			0
$375$	−12.6286	+	8.96642i	−0.652140	+	0.463024i
$376$		0			0
$377$		−	5.04554i		−	0.259858i
$378$		0			0
$379$		−	0.696553i		−	0.0357795i	−0.999840	−	0.0178898i	$-0.994305\pi$
$379$		−	0.696553i		−	0.0357795i	0.999840	−	0.0178898i	$-0.00569479\pi$
$380$		0			0
$381$	−0.713701	−	7.58544i	−0.0365640	−	0.388614i
$382$		0			0
$383$	2.97272	−	16.8591i	0.151899	−	0.861462i	−0.809667	−	0.586889i	$-0.800352\pi$
$383$	2.97272	−	16.8591i	0.151899	−	0.861462i	0.961566	−	0.274573i	$-0.0885365\pi$
$384$		0			0
$385$	−20.4776	+	17.1828i	−1.04364	+	0.875715i
$386$		0			0
$387$	−14.4156	−	11.7780i	−0.732784	−	0.598708i
$388$		0			0
$389$	−16.9672	+	2.99177i	−0.860269	+	0.151689i	−0.586344	−	0.810062i	$-0.699433\pi$
$389$	−16.9672	+	2.99177i	−0.860269	+	0.151689i	−0.273925	+	0.961751i	$0.588322\pi$
$390$		0			0
$391$	1.42917	+	3.92662i	0.0722764	+	0.198578i
$392$		0			0
$393$	−2.13186	+	1.01114i	−0.107538	+	0.0510055i
$394$		0			0
$395$	5.87750	−	10.1801i	0.295729	−	0.512218i
$396$		0			0
$397$	4.24458	+	7.35182i	0.213029	+	0.368977i	0.952661	−	0.304034i	$-0.0983337\pi$
$397$	4.24458	+	7.35182i	0.213029	+	0.368977i	−0.739632	+	0.673012i	$0.765000\pi$
$398$		0			0
$399$	0.965722	−	0.978448i	0.0483466	−	0.0489837i
$400$		0			0
$401$	6.54824	−	7.80389i	0.327004	−	0.389708i	−0.577346	−	0.816499i	$-0.695912\pi$
$401$	6.54824	−	7.80389i	0.327004	−	0.389708i	0.904350	+	0.426792i	$0.140356\pi$
$402$		0			0
$403$	4.90786	−	13.4842i	0.244478	−	0.671697i
$404$		0			0
$405$	−2.83215	+	8.46447i	−0.140731	+	0.420603i
$406$		0			0
$407$	7.56031	+	2.75173i	0.374751	+	0.136398i
$408$		0			0
$409$	−18.6107	−	15.6162i	−0.920238	−	0.772171i	0.0538011	−	0.998552i	$-0.482866\pi$
$409$	−18.6107	−	15.6162i	−0.920238	−	0.772171i	−0.974039	+	0.226380i	$0.927311\pi$
$410$		0			0
$411$	4.99632	+	4.93134i	0.246450	+	0.243245i
$412$		0			0
$413$	−32.9089	+	19.0000i	−1.61934	+	0.934928i
$414$		0			0
$415$	−8.27121	−	4.77538i	−0.406018	−	0.234414i
$416$		0			0
$417$	11.6102	+	24.4786i	0.568556	+	1.19872i
$418$		0			0
$419$	−0.0629252	+	0.0229029i	−0.00307410	+	0.00111888i	−0.343557	−	0.939132i	$-0.611632\pi$
$419$	−0.0629252	+	0.0229029i	−0.00307410	+	0.00111888i	0.340483	+	0.940251i	$0.389410\pi$
$420$		0			0
$421$	−4.84853	−	27.4974i	−0.236303	−	1.34014i	−0.839853	−	0.542814i	$-0.817359\pi$
$421$	−4.84853	−	27.4974i	−0.236303	−	1.34014i	0.603550	−	0.797325i	$-0.293752\pi$
$422$		0			0
$423$	24.6477	+	4.01410i	1.19841	+	0.195172i
$424$		0			0
$425$	1.54203	+	1.83772i	0.0747995	+	0.0891425i
$426$		0			0
$427$	−19.8062	−	3.49237i	−0.958489	−	0.169007i
$428$		0			0
$429$	26.9974	−	2.54014i	1.30345	−	0.122639i
$430$		0			0
$431$	−28.9982			−1.39679			−0.698397	−	0.715710i	$-0.746103\pi$
$431$	−28.9982			−1.39679			−0.698397	+	0.715710i	$0.746103\pi$
$432$		0			0
$433$	−25.6912			−1.23464			−0.617320	−	0.786712i	$-0.711782\pi$
$433$	−25.6912			−1.23464			−0.617320	+	0.786712i	$0.711782\pi$
$434$		0			0
$435$	−1.77516	−	2.50020i	−0.0851126	−	0.119876i
$436$		0			0
$437$	−1.12374	−	0.198145i	−0.0537556	−	0.00947857i
$438$		0			0
$439$	8.68012	+	10.3446i	0.414280	+	0.493719i	0.932319	−	0.361638i	$-0.117782\pi$
$439$	8.68012	+	10.3446i	0.414280	+	0.493719i	−0.518039	+	0.855357i	$0.673338\pi$
$440$		0			0
$441$	43.0083	−	25.5873i	2.04801	−	1.21844i
$442$		0			0
$443$	−1.24107	−	7.03844i	−0.0589649	−	0.334406i	0.941027	−	0.338330i	$-0.109862\pi$
$443$	−1.24107	−	7.03844i	−0.0589649	−	0.334406i	−0.999992	+	0.00392386i	$0.998751\pi$
$444$		0			0
$445$	−1.56071	+	0.568052i	−0.0739848	+	0.0269283i
$446$		0			0
$447$	11.9214	+	0.964397i	0.563861	+	0.0456144i
$448$		0			0
$449$	−18.7841	−	10.8450i	−0.886477	−	0.511807i	−0.0136882	−	0.999906i	$-0.504357\pi$
$449$	−18.7841	−	10.8450i	−0.886477	−	0.511807i	−0.872788	+	0.488099i	$0.837691\pi$
$450$		0			0
$451$	53.9093	−	31.1246i	2.53849	−	1.46560i
$452$		0			0
$453$	−2.39824	+	9.19057i	−0.112679	+	0.431810i
$454$		0			0
$455$	10.4499	+	8.76849i	0.489898	+	0.411073i
$456$		0			0
$457$	34.0401	+	12.3896i	1.59233	+	0.579561i	0.977838	−	0.209363i	$-0.0671390\pi$
$457$	34.0401	+	12.3896i	1.59233	+	0.579561i	0.614492	+	0.788923i	$0.289361\pi$
$458$		0			0
$459$	3.01304	+	0.744252i	0.140637	+	0.0347387i
$460$		0			0
$461$	−4.10359	+	11.2745i	−0.191123	+	0.525107i	−0.997830	−	0.0658453i	$-0.979026\pi$
$461$	−4.10359	+	11.2745i	−0.191123	+	0.525107i	0.806707	+	0.590952i	$0.201248\pi$
$462$		0			0
$463$	0.669388	−	0.797745i	0.0311091	−	0.0370744i	−0.750266	−	0.661137i	$-0.770074\pi$
$463$	0.669388	−	0.797745i	0.0311091	−	0.0370744i	0.781375	+	0.624062i	$0.214519\pi$
$464$		0			0
$465$	−2.31215	−	8.40853i	−0.107224	−	0.389936i
$466$		0			0
$467$	−2.21810	−	3.84186i	−0.102641	−	0.177780i	0.810131	−	0.586249i	$-0.199396\pi$
$467$	−2.21810	−	3.84186i	−0.102641	−	0.177780i	−0.912772	+	0.408469i	$0.866063\pi$
$468$		0			0
$469$	−14.3943	+	24.9317i	−0.664668	+	1.15124i
$470$		0			0
$471$	−5.14113	−	3.54993i	−0.236891	−	0.163572i
$472$		0			0
$473$	11.7550	+	32.2965i	0.540494	+	1.48500i
$474$		0			0
$475$	−0.645144	+	0.113756i	−0.0296013	+	0.00521950i
$476$		0			0
$477$	−19.1012	+	16.4587i	−0.874583	+	0.753591i
$478$		0			0
$479$	−12.7569	+	10.7043i	−0.582876	+	0.489091i	−0.885890	−	0.463895i	$-0.846452\pi$
$479$	−12.7569	+	10.7043i	−0.582876	+	0.489091i	0.303014	+	0.952986i	$0.402007\pi$
$480$		0			0
$481$	0.712945	−	4.04331i	0.0325075	−	0.184359i
$482$		0			0
$483$	−53.6048	−	24.5704i	−2.43910	−	1.11799i
$484$		0			0
$485$		−	13.3649i		−	0.606867i
$486$		0			0
$487$		−	15.4161i		−	0.698570i	−0.937016	−	0.349285i	$-0.886424\pi$
$487$		−	15.4161i		−	0.698570i	0.937016	−	0.349285i	$-0.113576\pi$
$488$		0			0
$489$	7.98748	+	3.66117i	0.361206	+	0.165564i
$490$		0			0
$491$	−1.16739	+	6.62060i	−0.0526836	+	0.298784i	−0.999753	−	0.0222468i	$-0.992918\pi$
$491$	−1.16739	+	6.62060i	−0.0526836	+	0.298784i	0.947069	+	0.321030i	$0.104029\pi$
$492$		0			0
$493$	−0.816756	+	0.685340i	−0.0367848	+	0.0308661i
$494$		0			0
$495$	12.4843	−	10.7572i	0.561126	−	0.483498i
$496$		0			0
$497$	17.8749	−	3.15183i	0.801799	−	0.141379i
$498$		0			0
$499$	11.8160	+	32.4641i	0.528956	+	1.45329i	0.860302	+	0.509785i	$0.170275\pi$
$499$	11.8160	+	32.4641i	0.528956	+	1.45329i	−0.331346	+	0.943509i	$0.607503\pi$
$500$		0			0
$501$	−14.8609	−	10.2614i	−0.663935	−	0.458445i
$502$		0			0
$503$	−7.55590	+	13.0872i	−0.336901	+	0.583530i	−0.983848	−	0.179005i	$-0.942712\pi$
$503$	−7.55590	+	13.0872i	−0.336901	+	0.583530i	0.646947	+	0.762535i	$0.276045\pi$
$504$		0			0
$505$	6.93387	+	12.0098i	0.308553	+	0.534430i
$506$		0			0
$507$	2.30108	+	8.36825i	0.102194	+	0.371647i
$508$		0			0
$509$	−5.27599	+	6.28768i	−0.233854	+	0.278696i	−0.870191	−	0.492715i	$-0.836004\pi$
$509$	−5.27599	+	6.28768i	−0.233854	+	0.278696i	0.636337	+	0.771411i	$0.280449\pi$
$510$		0			0
$511$	−10.1860	+	27.9858i	−0.450601	+	1.23802i
$512$		0			0
$513$	−0.587399	+	0.610934i	−0.0259343	+	0.0269734i
$514$		0			0
$515$	−5.14165	−	1.87141i	−0.226568	−	0.0824641i
$516$		0			0
$517$	−35.3195	−	29.6366i	−1.55335	−	1.30342i
$518$		0			0
$519$	8.69961	−	33.3388i	0.381870	−	1.46341i
$520$		0			0
$521$	−2.45686	+	1.41847i	−0.107637	+	0.0621444i	−0.552852	−	0.833279i	$-0.686461\pi$
$521$	−2.45686	+	1.41847i	−0.107637	+	0.0621444i	0.445215	+	0.895424i	$0.353127\pi$
$522$		0			0
$523$	26.4086	+	15.2470i	1.15477	+	0.666705i	0.950044	−	0.312116i	$-0.101038\pi$
$523$	26.4086	+	15.2470i	1.15477	+	0.666705i	0.204722	+	0.978820i	$0.434371\pi$
$524$		0			0
$525$	−33.7434	−	2.72973i	−1.47268	−	0.119135i
$526$		0			0
$527$	−2.84942	+	1.03710i	−0.124123	+	0.0451770i
$528$		0			0
$529$	4.50510	+	25.5497i	0.195874	+	1.11086i
$530$		0			0
$531$	20.1326	−	11.9776i	0.873679	−	0.519786i
$532$		0			0
$533$	−20.4189	−	24.3343i	−0.884441	−	1.05404i
$534$		0			0
$535$	−11.1338	−	1.96318i	−0.481355	−	0.0848758i
$536$		0			0
$537$	−15.0707	−	21.2261i	−0.650348	−	0.915973i
$538$		0			0
$539$	−92.3962			−3.97979
$540$		0			0
$541$	35.3176			1.51842			0.759212	−	0.650844i	$-0.225585\pi$
$541$	35.3176			1.51842			0.759212	+	0.650844i	$0.225585\pi$
$542$		0			0
$543$	7.19737	−	0.677188i	0.308869	−	0.0290609i
$544$		0			0
$545$	10.2035	+	1.79916i	0.437072	+	0.0770676i
$546$		0			0
$547$	−13.0568	−	15.5605i	−0.558270	−	0.665320i	0.410909	−	0.911676i	$-0.365211\pi$
$547$	−13.0568	−	15.5605i	−0.558270	−	0.665320i	−0.969179	+	0.246356i	$0.920767\pi$
$548$		0			0
$549$	12.2372	+	1.99294i	0.522272	+	0.0850567i
$550$		0			0
$551$	−0.0505578	−	0.286728i	−0.00215384	−	0.0122150i
$552$		0			0
$553$	54.2015	−	19.7277i	2.30488	−	0.838909i
$554$		0			0
$555$	−1.06927	−	2.25441i	−0.0453879	−	0.0956943i
$556$		0			0
$557$	26.5321	+	15.3183i	1.12420	+	0.649059i	0.942471	−	0.334289i	$-0.108496\pi$
$557$	26.5321	+	15.3183i	1.12420	+	0.649059i	0.181733	+	0.983348i	$0.441830\pi$
$558$		0			0
$559$	15.1891	−	8.76943i	0.642431	−	0.370908i
$560$		0			0
$561$	−4.07827	−	4.02522i	−0.172185	−	0.169945i
$562$		0			0
$563$	15.2500	+	12.7963i	0.642712	+	0.539299i	0.904850	−	0.425731i	$-0.139983\pi$
$563$	15.2500	+	12.7963i	0.642712	+	0.539299i	−0.262138	+	0.965030i	$0.584428\pi$
$564$		0			0
$565$	−3.10086	−	1.12862i	−0.130454	−	0.0474815i
$566$		0			0
$567$	−37.3432	+	22.8841i	−1.56827	+	0.961040i
$568$		0			0
$569$	−2.77000	+	7.61051i	−0.116124	+	0.319049i	−0.984115	−	0.177531i	$-0.943189\pi$
$569$	−2.77000	+	7.61051i	−0.116124	+	0.319049i	0.867991	+	0.496580i	$0.165411\pi$
$570$		0			0
$571$	−13.7051	+	16.3331i	−0.573540	+	0.683518i	−0.972353	−	0.233514i	$-0.924978\pi$
$571$	−13.7051	+	16.3331i	−0.573540	+	0.683518i	0.398814	+	0.917032i	$0.369422\pi$
$572$		0			0
$573$	29.1968	−	29.5815i	1.21971	−	1.23579i
$574$		0			0
$575$	14.0495	+	24.3344i	0.585904	+	1.01482i
$576$		0			0
$577$	0.0682767	−	0.118259i	0.00284239	−	0.00492317i	−0.864601	−	0.502460i	$-0.832429\pi$
$577$	0.0682767	−	0.118259i	0.00284239	−	0.00492317i	0.867443	+	0.497536i	$0.165762\pi$
$578$		0			0
$579$	13.3213	−	6.31829i	0.553613	−	0.262579i
$580$		0			0
$581$	−16.0285	−	44.0380i	−0.664975	−	1.82700i
$582$		0			0
$583$	45.8451	−	8.08372i	1.89871	−	0.334794i
$584$		0			0
$585$	−6.51233	−	5.32079i	−0.269252	−	0.219987i
$586$		0			0
$587$	21.3306	−	17.8985i	0.880407	−	0.738749i	−0.0858556	−	0.996308i	$-0.527362\pi$
$587$	21.3306	−	17.8985i	0.880407	−	0.738749i	0.966263	+	0.257558i	$0.0829179\pi$
$588$		0			0
$589$	0.143788	−	0.815460i	0.00592466	−	0.0336004i
$590$		0			0
$591$	−0.680984	−	7.23771i	−0.0280119	−	0.297720i
$592$		0			0
$593$		−	16.6379i		−	0.683238i	−0.939839	−	0.341619i	$-0.889025\pi$
$593$		−	16.6379i		−	0.683238i	0.939839	−	0.341619i	$-0.110975\pi$
$594$		0			0
$595$		−	2.88263i		−	0.118176i
$596$		0			0
$597$	−7.47297	+	5.30586i	−0.305848	+	0.217154i
$598$		0			0
$599$	−4.24539	+	24.0768i	−0.173462	+	0.983752i	0.766442	+	0.642314i	$0.222025\pi$
$599$	−4.24539	+	24.0768i	−0.173462	+	0.983752i	−0.939904	+	0.341439i	$0.889086\pi$
$600$		0			0
$601$	30.7637	−	25.8138i	1.25488	−	1.05297i	0.258671	−	0.965965i	$-0.416715\pi$
$601$	30.7637	−	25.8138i	1.25488	−	1.05297i	0.996208	−	0.0870032i	$-0.0277290\pi$
$602$		0			0
$603$	8.67181	−	15.4847i	0.353144	−	0.630585i
$604$		0			0
$605$	−19.2202	+	3.38904i	−0.781413	+	0.137784i
$606$		0			0
$607$	5.65409	+	15.5345i	0.229493	+	0.630526i	0.999976	−	0.00693444i	$-0.00220732\pi$
$607$	5.65409	+	15.5345i	0.229493	+	0.630526i	−0.770483	+	0.637460i	$0.779985\pi$
$608$		0			0
$609$	1.21320	−	14.9969i	0.0491613	−	0.607706i
$610$		0			0
$611$	−11.7642	+	20.3762i	−0.475929	+	0.824333i
$612$		0			0
$613$	−1.00864	−	1.74701i	−0.0407385	−	0.0705611i	0.844937	−	0.534865i	$-0.179638\pi$
$613$	−1.00864	−	1.74701i	−0.0407385	−	0.0705611i	−0.885676	+	0.464304i	$0.846304\pi$
$614$		0			0
$615$	−18.6796	−	4.87437i	−0.753236	−	0.196554i
$616$		0			0
$617$	20.7709	−	24.7538i	0.836204	−	0.996549i	−0.163746	−	0.986503i	$-0.552358\pi$
$617$	20.7709	−	24.7538i	0.836204	−	0.996549i	0.999950	−	0.0100465i	$-0.00319796\pi$
$618$		0			0
$619$	−4.84499	+	13.3115i	−0.194737	+	0.535035i	−0.998177	−	0.0603502i	$-0.980778\pi$
$619$	−4.84499	+	13.3115i	−0.194737	+	0.535035i	0.803441	+	0.595385i	$0.203000\pi$
$620$		0			0
$621$	33.2416	+	14.7132i	1.33394	+	0.590421i
$622$		0			0
$623$	−7.65818	−	2.78735i	−0.306818	−	0.111673i
$624$		0			0
$625$	8.59041	+	7.20821i	0.343616	+	0.288328i
$626$		0			0
$627$	1.50875	−	0.414873i	0.0602538	−	0.0165684i
$628$		0			0
$629$	−0.751359	+	0.433797i	−0.0299586	+	0.0172966i
$630$		0			0
$631$	13.0965	+	7.56128i	0.521365	+	0.301010i	0.737493	−	0.675355i	$-0.236010\pi$
$631$	13.0965	+	7.56128i	0.521365	+	0.301010i	−0.216128	+	0.976365i	$0.569343\pi$
$632$		0			0
$633$	−11.1389	+	16.1318i	−0.442733	+	0.641180i
$634$		0			0
$635$	4.09940	−	1.49206i	0.162680	−	0.0592106i
$636$		0			0
$637$	8.18759	+	46.4341i	0.324404	+	1.83979i
$638$		0			0
$639$	−10.9931	+	2.08712i	−0.434881	+	0.0825653i
$640$		0			0
$641$	2.66991	+	3.18187i	0.105455	+	0.125676i	0.816190	−	0.577783i	$-0.196082\pi$
$641$	2.66991	+	3.18187i	0.105455	+	0.125676i	−0.710735	+	0.703460i	$0.751638\pi$
$642$		0			0
$643$	43.0780	+	7.59582i	1.69883	+	0.299550i	0.937286	−	0.348561i	$-0.113329\pi$
$643$	43.0780	+	7.59582i	1.69883	+	0.299550i	0.761546	+	0.648111i	$0.224441\pi$
$644$		0			0
$645$	4.44129	−	9.68946i	0.174876	−	0.381522i
$646$		0			0
$647$	12.1823			0.478936			0.239468	−	0.970904i	$-0.423027\pi$
$647$	12.1823			0.478936			0.239468	+	0.970904i	$0.423027\pi$
$648$		0			0
$649$	−43.2515			−1.69777
$650$		0			0
$651$	17.8300	−	38.8992i	0.698812	−	1.52458i
$652$		0			0
$653$	36.7762	+	6.48464i	1.43916	+	0.253763i	0.838135	−	0.545463i	$-0.183646\pi$
$653$	36.7762	+	6.48464i	1.43916	+	0.253763i	0.601030	+	0.799227i	$0.294757\pi$
$654$		0			0
$655$	−0.868416	−	1.03494i	−0.0339318	−	0.0404384i
$656$		0			0
$657$	6.05305	−	17.3333i	0.236152	−	0.676238i
$658$		0			0
$659$	−1.48344	−	8.41298i	−0.0577865	−	0.327723i	0.942186	−	0.335089i	$-0.108766\pi$
$659$	−1.48344	−	8.41298i	−0.0577865	−	0.327723i	−0.999973	+	0.00736587i	$0.997655\pi$
$660$		0			0
$661$	0.439161	−	0.159841i	0.0170814	−	0.00621711i	−0.333465	−	0.942762i	$-0.608218\pi$
$661$	0.439161	−	0.159841i	0.0170814	−	0.00621711i	0.350547	+	0.936545i	$0.385996\pi$
$662$		0			0
$663$	−1.66150	+	2.40624i	−0.0645274	+	0.0934507i
$664$		0			0
$665$	0.681709	+	0.393585i	0.0264355	+	0.0152626i
$666$		0			0
$667$	−10.8152	+	6.24415i	−0.418765	+	0.241774i
$668$		0			0
$669$	29.2302	−	8.03764i	1.13010	−	0.310753i
$670$		0			0
$671$	−17.5356	−	14.7141i	−0.676956	−	0.568033i
$672$		0			0
$673$	4.43038	+	1.61253i	0.170779	+	0.0621584i	0.425994	−	0.904726i	$-0.359924\pi$
$673$	4.43038	+	1.61253i	0.170779	+	0.0621584i	−0.255216	+	0.966884i	$0.582147\pi$
$674$		0			0
$675$	20.8223	+	1.41036i	0.801452	+	0.0542846i
$676$		0			0
$677$	9.86735	−	27.1103i	0.379233	−	1.04193i	−0.592442	−	0.805613i	$-0.701836\pi$
$677$	9.86735	−	27.1103i	0.379233	−	1.04193i	0.971675	−	0.236321i	$-0.0759417\pi$
$678$		0			0
$679$	42.1537	−	50.2368i	1.61771	−	1.92791i
$680$		0			0
$681$	6.99407	+	1.82507i	0.268014	+	0.0699370i
$682$		0			0
$683$	22.0332	+	38.1626i	0.843076	+	1.46025i	0.887282	+	0.461228i	$0.152591\pi$
$683$	22.0332	+	38.1626i	0.843076	+	1.46025i	−0.0442060	+	0.999022i	$0.514076\pi$
$684$		0			0
$685$	−2.00979	+	3.48106i	−0.0767902	+	0.133004i
$686$		0			0
$687$	−0.265000	+	3.27578i	−0.0101104	+	0.124979i
$688$		0			0
$689$	−8.12502	−	22.3233i	−0.309539	−	0.850451i
$690$		0			0
$691$	−41.2765	+	7.27817i	−1.57023	+	0.276875i	−0.889944	−	0.456069i	$-0.849257\pi$
$691$	−41.2765	+	7.27817i	−1.57023	+	0.276875i	−0.680289	+	0.732944i	$0.738146\pi$
$692$		0			0
$693$	80.8555	−	1.05856i	3.07144	−	0.0402114i
$694$		0			0
$695$	−11.8834	+	9.97138i	−0.450764	+	0.378236i
$696$		0			0
$697$	−1.16564	+	6.61070i	−0.0441519	+	0.250398i
$698$		0			0
$699$	25.6447	−	18.2079i	0.969970	−	0.688686i
$700$		0			0
$701$			0.277648i			0.0104866i	0.999986	+	0.00524332i	$0.00166901\pi$
$701$			0.277648i			0.0104866i	−0.999986	+	0.00524332i	$0.998331\pi$
$702$		0			0
$703$		−	0.236917i		−	0.00893550i
$704$		0			0
$705$	1.33944	+	14.2360i	0.0504461	+	0.536157i
$706$		0			0
$707$	−11.8162	+	67.0132i	−0.444395	+	2.52029i
$708$		0			0
$709$	−26.9969	+	22.6531i	−1.01389	+	0.850754i	−0.988847	−	0.148933i	$-0.952416\pi$
$709$	−26.9969	+	22.6531i	−1.01389	+	0.850754i	−0.0250415	+	0.999686i	$0.507972\pi$
$710$		0			0
$711$	−33.2520	+	12.5981i	−1.24705	+	0.472465i
$712$		0			0
$713$	−34.9774	+	6.16746i	−1.30991	+	0.230973i
$714$		0			0
$715$	5.31040	+	14.5902i	0.198598	+	0.545642i
$716$		0			0
$717$	−27.3061	+	12.9513i	−1.01977	+	0.483677i
$718$		0			0
$719$	25.5033	−	44.1730i	0.951113	−	1.64738i	0.208091	−	0.978109i	$-0.433275\pi$
$719$	25.5033	−	44.1730i	0.951113	−	1.64738i	0.743022	−	0.669267i	$-0.233392\pi$
$720$		0			0
$721$	−13.4243	−	23.2515i	−0.499945	−	0.865931i
$722$		0			0
$723$	−32.6026	+	33.0322i	−1.21250	+	1.22848i
$724$		0			0
$725$	−4.60853	+	5.49224i	−0.171157	+	0.203977i
$726$		0			0
$727$	3.74866	−	10.2994i	0.139030	−	0.381982i	−0.850564	−	0.525872i	$-0.823739\pi$
$727$	3.74866	−	10.2994i	0.139030	−	0.381982i	0.989594	+	0.143890i	$0.0459612\pi$
$728$		0			0
$729$	22.8346	−	14.4077i	0.845726	−	0.533618i
$730$		0			0
$731$	−3.48272	−	1.26761i	−0.128813	−	0.0468841i
$732$		0			0
$733$	−5.12425	−	4.29976i	−0.189269	−	0.158815i	0.543230	−	0.839584i	$-0.317201\pi$
$733$	−5.12425	−	4.29976i	−0.189269	−	0.158815i	−0.732498	+	0.680769i	$0.761646\pi$
$734$		0			0
$735$	20.3940	+	20.1288i	0.752245	+	0.742461i
$736$		0			0
$737$	−28.3772	+	16.3836i	−1.04529	+	0.603497i
$738$		0			0
$739$	14.5944	+	8.42609i	0.536864	+	0.309959i	0.743807	−	0.668394i	$-0.233018\pi$
$739$	14.5944	+	8.42609i	0.536864	+	0.309959i	−0.206943	+	0.978353i	$0.566351\pi$
$740$		0			0
$741$	−0.342193	−	0.721468i	−0.0125708	−	0.0265038i
$742$		0			0
$743$	10.5585	−	3.84299i	0.387355	−	0.140986i	−0.140999	−	0.990010i	$-0.545031\pi$
$743$	10.5585	−	3.84299i	0.387355	−	0.140986i	0.528354	+	0.849024i	$0.322809\pi$
$744$		0			0
$745$	1.18919	+	6.74425i	0.0435687	+	0.247090i
$746$		0			0
$747$	10.2358	+	27.0168i	0.374507	+	0.988491i
$748$		0			0
$749$	−35.6584	−	42.4960i	−1.30293	−	1.55277i
$750$		0			0
$751$	46.3569	+	8.17398i	1.69159	+	0.298273i	0.934744	−	0.355322i	$-0.115629\pi$
$751$	46.3569	+	8.17398i	1.69159	+	0.298273i	0.756844	+	0.653595i	$0.226740\pi$
$752$		0			0
$753$	31.6806	−	2.98077i	1.15450	−	0.108625i
$754$		0			0
$755$	−5.43859			−0.197931
$756$		0			0
$757$	−34.8478			−1.26656			−0.633282	−	0.773921i	$-0.718292\pi$
$757$	−34.8478			−1.26656			−0.633282	+	0.773921i	$0.718292\pi$
$758$		0			0
$759$	−38.8557	−	54.7257i	−1.41037	−	1.98642i
$760$		0			0
$761$	8.01971	+	1.41409i	0.290714	+	0.0512608i	0.317103	−	0.948391i	$-0.397290\pi$
$761$	8.01971	+	1.41409i	0.290714	+	0.0512608i	−0.0263892	+	0.999652i	$0.508401\pi$
$762$		0			0
$763$	32.6791	+	38.9455i	1.18306	+	1.40992i
$764$		0			0
$765$	0.0232635	+	1.77692i	0.000841093	+	0.0642448i
$766$		0			0
$767$	3.83268	+	21.7362i	0.138390	+	0.784850i
$768$		0			0
$769$	−33.8466	+	12.3191i	−1.22054	+	0.444240i	−0.870347	−	0.492438i	$-0.836106\pi$
$769$	−33.8466	+	12.3191i	−1.22054	+	0.444240i	−0.350191	+	0.936678i	$0.613883\pi$
$770$		0			0
$771$	35.9923	+	2.91166i	1.29623	+	0.104861i
$772$		0			0
$773$	22.6182	+	13.0586i	0.813521	+	0.469687i	0.848177	−	0.529713i	$-0.177700\pi$
$773$	22.6182	+	13.0586i	0.813521	+	0.469687i	−0.0346560	+	0.999399i	$0.511034\pi$
$774$		0			0
$775$	−17.6587	+	10.1953i	−0.634319	+	0.366224i
$776$		0			0
$777$	3.09131	−	11.8466i	0.110900	−	0.424993i
$778$		0			0
$779$	−1.40420	−	1.17827i	−0.0503108	−	0.0422158i
$780$		0			0
$781$	19.4132	+	7.06581i	0.694657	+	0.252835i
$782$		0			0
$783$	−0.626818	+	9.25427i	−0.0224006	+	0.330721i
$784$		0			0
$785$	1.22351	−	3.36157i	0.0436690	−	0.119980i
$786$		0			0
$787$	−6.45027	+	7.68713i	−0.229927	+	0.274017i	−0.868656	−	0.495415i	$-0.835016\pi$
$787$	−6.45027	+	7.68713i	−0.229927	+	0.274017i	0.638729	+	0.769432i	$0.279460\pi$
$788$		0			0
$789$	12.3071	+	44.7567i	0.438143	+	1.59338i
$790$		0			0
$791$	−8.09599	−	14.0227i	−0.287860	−	0.498589i
$792$		0			0
$793$	−5.84076	+	10.1165i	−0.207411	+	0.359247i
$794$		0			0
$795$	−11.8802	−	8.20321i	−0.421346	−	0.290938i

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.be (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(0.721706 - 1.57453i) q^{3} +(0.976679 + 0.172215i) q^{5} +(3.12803 + 3.72784i) q^{7} +(-1.95828 - 2.27269i) q^{9} +O(q^{10})\)	\(q+(0.721706 - 1.57453i) q^{3} +(0.976679 + 0.172215i) q^{5} +(3.12803 + 3.72784i) q^{7} +(-1.95828 - 2.27269i) q^{9} +(0.961816 + 5.45473i) q^{11} +(2.65607 - 0.966729i) q^{13} +(0.976032 - 1.41352i) q^{15} +(-0.517267 - 0.298644i) q^{17} +(0.141252 - 0.0815519i) q^{19} +(8.12712 - 2.23477i) q^{21} +(-5.35924 - 4.49693i) q^{23} +(-3.77422 - 1.37370i) q^{25} +(-4.99172 + 1.44316i) q^{27} +(0.610528 - 1.67741i) q^{29} +(3.26328 - 3.88903i) q^{31} +(9.28277 + 2.42230i) q^{33} +(2.41309 + 4.17960i) q^{35} +(0.726277 - 1.25795i) q^{37} +(0.394755 - 4.87975i) q^{39} +(-3.84382 - 10.5608i) q^{41} +(6.11082 - 1.07750i) q^{43} +(-1.52122 - 2.55694i) q^{45} +(-6.37666 + 5.35065i) q^{47} +(-2.89670 + 16.4280i) q^{49} +(-0.843539 + 0.598919i) q^{51} -8.40465i q^{53} +5.49316i q^{55} +(-0.0264635 - 0.281262i) q^{57} +(-1.35597 + 7.69008i) q^{59} +(-3.16592 + 2.65652i) q^{61} +(2.34667 - 14.4092i) q^{63} +(2.76061 - 0.486770i) q^{65} +(2.02334 + 5.55908i) q^{67} +(-10.9483 + 5.19281i) q^{69} +(1.86491 - 3.23013i) q^{71} +(3.05997 + 5.30003i) q^{73} +(-4.88681 + 4.95121i) q^{75} +(-17.3258 + 20.6481i) q^{77} +(4.05391 - 11.1380i) q^{79} +(-1.33026 + 8.90115i) q^{81} +(-9.04948 - 3.29374i) q^{83} +(-0.453773 - 0.380761i) q^{85} +(-2.20051 - 2.17189i) q^{87} +(-1.45033 + 0.837348i) q^{89} +(11.9121 + 6.87744i) q^{91} +(-3.76826 - 7.94486i) q^{93} +(0.152002 - 0.0553244i) q^{95} +(-2.34010 - 13.2714i) q^{97} +(10.5134 - 12.8678i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{5} + 6 q^{9} + O(q^{10}) \)	\( 36 q + 3 q^{5} + 6 q^{9} - 18 q^{11} + 9 q^{15} + 18 q^{21} - 9 q^{25} + 30 q^{29} + 27 q^{31} + 27 q^{33} + 27 q^{35} - 45 q^{39} + 18 q^{41} + 27 q^{45} + 45 q^{47} - 63 q^{51} - 9 q^{57} + 54 q^{59} - 63 q^{63} - 57 q^{65} - 63 q^{69} + 36 q^{71} + 9 q^{73} - 45 q^{75} - 81 q^{77} - 54 q^{81} - 27 q^{83} - 36 q^{85} + 45 q^{87} - 63 q^{89} + 27 q^{91} - 63 q^{93} - 72 q^{95} + 99 q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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