LMFDB - Embedded newform 432.2.be.b.47.4

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			47.4
Character	$\chi$	$=$	432.47
Dual form			432.2.be.b.239.4

$q$-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 $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$e\left(\frac{7}{18}\right)$

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	0.0490676	−	0.998795i	$-0.484375\pi$
$5$	0.976679	−	0.172215i	0.436784	−	0.0770169i	0.387717	+	0.921779i	$0.373264\pi$
$6$		0			0
$7$	3.12803	−	3.72784i	1.18229	−	1.40899i	0.290295	−	0.956937i	$-0.406247\pi$
$7$	3.12803	−	3.72784i	1.18229	−	1.40899i	0.891990	−	0.452055i	$-0.149309\pi$
$8$		0			0
$9$	−1.95828	+	2.27269i	−0.652761	+	0.757564i
$10$		0			0
$11$	0.961816	−	5.45473i	0.289998	−	1.64466i	−0.396866	−	0.917877i	$-0.629902\pi$
$11$	0.961816	−	5.45473i	0.289998	−	1.64466i	0.686864	−	0.726786i	$-0.258987\pi$
$12$		0			0
$13$	2.65607	+	0.966729i	0.736661	+	0.268123i	0.682981	−	0.730436i	$-0.260683\pi$
$13$	2.65607	+	0.966729i	0.736661	+	0.268123i	0.0536791	+	0.998558i	$0.482905\pi$
$14$		0			0
$15$	0.976032	+	1.41352i	0.252010	+	0.364970i
$16$		0			0
$17$	−0.517267	+	0.298644i	−0.125456	+	0.0724319i	−0.561415	−	0.827535i	$-0.689743\pi$
$17$	−0.517267	+	0.298644i	−0.125456	+	0.0724319i	0.435959	+	0.899967i	$0.356409\pi$
$18$		0			0
$19$	0.141252	+	0.0815519i	0.0324055	+	0.0187093i	0.516115	−	0.856519i	$-0.327378\pi$
$19$	0.141252	+	0.0815519i	0.0324055	+	0.0187093i	−0.483710	+	0.875228i	$0.660711\pi$
$20$		0			0
$21$	8.12712	+	2.23477i	1.77348	+	0.487667i
$22$		0			0
$23$	−5.35924	+	4.49693i	−1.11748	+	0.937675i	−0.998474	−	0.0552184i	$-0.982415\pi$
$23$	−5.35924	+	4.49693i	−1.11748	+	0.937675i	−0.119004	+	0.992894i	$0.537970\pi$
$24$		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	0.983382	−	0.181546i	$-0.0581101\pi$
$29$	0.610528	+	1.67741i	0.113372	+	0.311488i	−0.870010	+	0.493034i	$0.835888\pi$
$30$		0			0
$31$	3.26328	+	3.88903i	0.586102	+	0.698490i	0.974852	−	0.222854i	$-0.0715373\pi$
$31$	3.26328	+	3.88903i	0.586102	+	0.698490i	−0.388750	+	0.921343i	$0.627093\pi$
$32$		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	0.919530	−	0.393020i	$-0.128570\pi$
$37$	0.726277	+	1.25795i	0.119399	+	0.206806i	−0.800131	+	0.599826i	$0.795236\pi$
$38$		0			0
$39$	0.394755	+	4.87975i	0.0632114	+	0.781385i
$40$		0			0
$41$	−3.84382	+	10.5608i	−0.600304	+	1.64932i	0.150354	+	0.988632i	$0.451959\pi$
$41$	−3.84382	+	10.5608i	−0.600304	+	1.64932i	−0.750658	+	0.660691i	$0.770264\pi$
$42$		0			0
$43$	6.11082	+	1.07750i	0.931892	+	0.164318i	0.618928	−	0.785447i	$-0.287567\pi$
$43$	6.11082	+	1.07750i	0.931892	+	0.164318i	0.312964	+	0.949765i	$0.398678\pi$
$44$		0			0
$45$	−1.52122	+	2.55694i	−0.226770	+	0.381166i
$46$		0			0
$47$	−6.37666	−	5.35065i	−0.930131	−	0.780473i	0.0457098	−	0.998955i	$-0.485445\pi$
$47$	−6.37666	−	5.35065i	−0.930131	−	0.780473i	−0.975841	+	0.218482i	$0.929890\pi$
$48$		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	$0.195868\pi$
$53$			8.40465i			1.15447i	−0.816579	+	0.577234i	$0.804132\pi$
$54$		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	−0.936361	−	0.351040i	$-0.885828\pi$
$59$	−1.35597	−	7.69008i	−0.176532	−	1.00116i	0.759828	−	0.650124i	$-0.225283\pi$
$60$		0			0
$61$	−3.16592	−	2.65652i	−0.405354	−	0.340133i	0.417204	−	0.908813i	$-0.363010\pi$
$61$	−3.16592	−	2.65652i	−0.405354	−	0.340133i	−0.822559	+	0.568680i	$0.807454\pi$
$62$		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	−0.752596	−	0.658483i	$-0.771199\pi$
$67$	2.02334	−	5.55908i	0.247190	−	0.679150i	0.999786	−	0.0206675i	$-0.00657914\pi$
$68$		0			0
$69$	−10.9483	−	5.19281i	−1.31803	−	0.625141i
$70$		0			0
$71$	1.86491	+	3.23013i	0.221325	+	0.383345i	0.955210	−	0.295927i	$-0.0956287\pi$
$71$	1.86491	+	3.23013i	0.221325	+	0.383345i	−0.733886	+	0.679273i	$0.762295\pi$
$72$		0			0
$73$	3.05997	−	5.30003i	0.358143	−	0.620321i	−0.629508	−	0.776994i	$-0.716743\pi$
$73$	3.05997	−	5.30003i	0.358143	−	0.620321i	0.987651	+	0.156673i	$0.0500768\pi$
$74$		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	0.928366	+	0.371668i	$0.121214\pi$
$79$	4.05391	+	11.1380i	0.456100	+	1.25312i	−0.472266	+	0.881456i	$0.656564\pi$
$80$		0			0
$81$	−1.33026	−	8.90115i	−0.147807	−	0.989016i
$82$		0			0
$83$	−9.04948	+	3.29374i	−0.993309	+	0.361535i	−0.787001	−	0.616952i	$-0.788367\pi$
$83$	−9.04948	+	3.29374i	−0.993309	+	0.361535i	−0.206308	+	0.978487i	$0.566145\pi$
$84$		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	0.421159	−	0.906987i	$-0.361623\pi$
$89$	−1.45033	−	0.837348i	−0.153735	−	0.0887587i	−0.574894	+	0.818228i	$0.694957\pi$
$90$		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	0.599465	+	0.800401i	$0.295380\pi$
$97$	−2.34010	+	13.2714i	−0.237601	+	1.34750i	−0.837066	+	0.547102i	$0.815731\pi$
$98$		0			0
$99$	10.5134	+	12.8678i	1.05664	+	1.29326i
$100$		0			0
$101$	8.98820	−	10.7117i	0.894360	−	1.06586i	−0.103103	−	0.994671i	$-0.532877\pi$
$101$	8.98820	−	10.7117i	0.894360	−	1.06586i	0.997463	−	0.0711857i	$-0.0226783\pi$
$102$		0			0
$103$	−5.43335	+	0.958047i	−0.535364	+	0.0943991i	−0.434792	−	0.900531i	$-0.643178\pi$
$103$	−5.43335	+	0.958047i	−0.535364	+	0.0943991i	−0.100572	+	0.994930i	$0.532067\pi$
$104$		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	−0.325635	−	0.945495i	$-0.605578\pi$
$113$	−3.27678	+	0.577786i	−0.308254	+	0.0543535i	0.0173814	+	0.999849i	$0.494467\pi$
$114$		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	0.659403	−	0.751790i	$-0.270809\pi$
$127$	3.80947	+	2.19940i	0.338036	+	0.195165i	−0.321367	+	0.946955i	$0.604142\pi$
$128$		0			0
$129$	2.71366	+	10.3993i	0.238924	+	0.915608i
$130$		0			0
$131$	−1.04355	+	0.875644i	−0.0911755	+	0.0765053i	−0.687236	−	0.726434i	$-0.741176\pi$
$131$	−1.04355	+	0.875644i	−0.0911755	+	0.0765053i	0.596061	+	0.802939i	$0.296732\pi$
$132$		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	0.866285	−	0.499551i	$-0.166501\pi$
$137$	−1.38622	−	3.80861i	−0.118433	−	0.325391i	−0.984718	+	0.174159i	$0.944279\pi$
$138$		0			0
$139$	−10.0544	−	11.9823i	−0.852801	−	1.01633i	−0.999631	−	0.0271664i	$-0.991352\pi$
$139$	−10.0544	−	11.9823i	−0.852801	−	1.01633i	0.146830	−	0.989162i	$-0.453093\pi$
$140$		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	−0.804578	−	0.593847i	$-0.797609\pi$
$149$	2.36175	−	6.48884i	0.193482	−	0.531587i	0.998060	+	0.0622599i	$0.0198308\pi$
$150$		0			0
$151$	−5.40054	−	0.952262i	−0.439490	−	0.0774940i	−0.0504750	−	0.998725i	$-0.516074\pi$
$151$	−5.40054	−	0.952262i	−0.439490	−	0.0774940i	−0.389015	+	0.921231i	$0.627185\pi$
$152$		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	0.999547	−	0.0300883i	$-0.00957885\pi$
$157$	0.626364	+	3.55229i	0.0499893	+	0.283503i	−0.949558	+	0.313592i	$0.898468\pi$
$158$		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	$-0.936337\pi$
$163$		−	5.07293i		−	0.397343i	0.980066	−	0.198671i	$-0.0636627\pi$
$164$		0			0
$165$	8.64914	−	3.96444i	0.673334	−	0.308631i
$166$		0			0
$167$	1.81056	+	10.2682i	0.140105	+	0.794576i	0.971168	+	0.238397i	$0.0766220\pi$
$167$	1.81056	+	10.2682i	0.140105	+	0.794576i	−0.831062	+	0.556179i	$0.812267\pi$
$168$		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	0.858341	−	0.513079i	$-0.171495\pi$
$173$	19.5905	+	3.45433i	1.48944	+	0.262628i	0.631094	+	0.775707i	$0.282606\pi$
$174$		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	0.997350	+	0.0727573i	$0.0231798\pi$
$179$	7.51483	+	13.0161i	0.561685	+	0.972866i	−0.435665	+	0.900109i	$0.643487\pi$
$180$		0			0
$181$	2.08688	−	3.61458i	0.155116	−	0.268669i	−0.777985	−	0.628283i	$-0.783758\pi$
$181$	2.08688	−	3.61458i	0.155116	−	0.268669i	0.933101	+	0.359613i	$0.117091\pi$
$182$		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	0.646078	−	0.763272i	$-0.276408\pi$
$191$	22.5495	−	8.20734i	1.63162	−	0.593862i	0.985546	+	0.169409i	$0.0541860\pi$
$192$		0			0
$193$	6.52079	−	5.47159i	0.469377	−	0.393854i	−0.377191	−	0.926136i	$-0.623110\pi$
$193$	6.52079	−	5.47159i	0.469377	−	0.393854i	0.846567	+	0.532282i	$0.178665\pi$
$194$		0			0
$195$	1.22591	+	4.69797i	0.0877896	+	0.336428i
$196$		0			0
$197$	3.63484	+	2.09858i	0.258972	+	0.149517i	0.623865	−	0.781532i	$-0.285561\pi$
$197$	3.63484	+	2.09858i	0.258972	+	0.149517i	−0.364894	+	0.931049i	$0.618895\pi$
$198$		0			0
$199$	−4.58250	+	2.64571i	−0.324845	+	0.187549i	−0.653550	−	0.756883i	$-0.726721\pi$
$199$	−4.58250	+	2.64571i	−0.324845	+	0.187549i	0.328705	+	0.944433i	$0.393388\pi$
$200$		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	−0.543600	−	0.839345i	$-0.682939\pi$
$211$	−11.1463	+	1.96540i	−0.767344	+	0.135303i	−0.223744	+	0.974648i	$0.571828\pi$
$212$		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	−0.244030	−	0.969768i	$-0.578470\pi$
$223$	11.2504	−	13.4077i	0.753380	−	0.897843i	0.997410	+	0.0719243i	$0.0229140\pi$
$224$		0			0
$225$	4.26898	−	11.2677i	0.284599	−	0.751182i
$226$		0			0
$227$	0.724677	−	4.10985i	0.0480985	−	0.272780i	−0.951268	−	0.308365i	$-0.900218\pi$
$227$	0.724677	−	4.10985i	0.0480985	−	0.272780i	0.999367	+	0.0355847i	$0.0113294\pi$
$228$		0			0
$229$	−1.78302	−	0.648967i	−0.117825	−	0.0428849i	0.282435	−	0.959287i	$-0.408858\pi$
$229$	−1.78302	−	0.648967i	−0.117825	−	0.0428849i	−0.400260	+	0.916402i	$0.631080\pi$
$230$		0			0
$231$	20.0069	−	42.1818i	1.31635	−	2.77536i
$232$		0			0
$233$	15.7256	−	9.07917i	1.03022	−	0.594796i	0.113171	−	0.993576i	$-0.463899\pi$
$233$	15.7256	−	9.07917i	1.03022	−	0.594796i	0.917047	+	0.398779i	$0.130566\pi$
$234$		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	−0.962954	−	0.269665i	$-0.913087\pi$
$239$	−13.3664	+	11.2158i	−0.864602	+	0.725487i	0.0983526	+	0.995152i	$0.468643\pi$
$240$		0			0
$241$	−25.1799	+	9.16473i	−1.62198	+	0.590352i	−0.983758	−	0.179500i	$-0.942552\pi$
$241$	−25.1799	+	9.16473i	−1.62198	+	0.590352i	−0.638222	+	0.769853i	$0.720330\pi$
$242$		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	−0.415700	−	0.909502i	$-0.636464\pi$
$251$	9.18578	−	15.9102i	0.579801	−	1.00425i	0.995502	−	0.0947437i	$-0.0302032\pi$
$252$		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	−0.491525	−	0.870864i	$-0.663560\pi$
$257$	7.13045	−	19.5908i	0.444786	−	1.22204i	0.936310	−	0.351174i	$-0.114218\pi$
$258$		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	−0.995027	−	0.0996068i	$-0.968242\pi$
$263$	−20.5295	−	17.2263i	−1.26590	−	1.06222i	−0.270878	−	0.962614i	$-0.587314\pi$
$264$		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	$-0.940396\pi$
$269$		−	6.10651i		−	0.372321i	0.982519	−	0.186160i	$-0.0596044\pi$
$270$		0			0
$271$		−	5.24989i		−	0.318908i	−0.987205	−	0.159454i	$-0.949027\pi$
$271$		−	5.24989i		−	0.318908i	0.987205	−	0.159454i	$-0.0509734\pi$
$272$		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	0.988282	+	0.152640i	$0.0487775\pi$
$277$	21.8063	+	18.2977i	1.31022	+	1.09940i	0.321934	+	0.946762i	$0.395667\pi$
$278$		0			0
$279$	−15.2290	−	0.199378i	−0.911735	−	0.0119364i
$280$		0			0
$281$	2.99467	+	0.528040i	0.178647	+	0.0315002i	0.262256	−	0.964998i	$-0.415534\pi$
$281$	2.99467	+	0.528040i	0.178647	+	0.0315002i	−0.0836090	+	0.996499i	$0.526645\pi$
$282$		0			0
$283$	−5.13245	+	14.1013i	−0.305092	+	0.838235i	0.688502	+	0.725234i	$0.258268\pi$
$283$	−5.13245	+	14.1013i	−0.305092	+	0.838235i	−0.993595	+	0.113001i	$0.963954\pi$
$284$		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	0.994645	−	0.103347i	$-0.0329552\pi$
$293$	12.3270	+	14.6907i	0.720150	+	0.858241i	−0.274495	+	0.961588i	$0.588511\pi$
$294$		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	−0.842567	−	0.538592i	$-0.818957\pi$
$307$	−13.9719	+	8.06666i	−0.797416	+	0.460388i	0.0451507	+	0.998980i	$0.485623\pi$
$308$		0			0
$309$	−5.42975	−	7.86354i	−0.308888	−	0.447341i
$310$		0			0
$311$	−17.9054	−	6.51704i	−1.01532	−	0.369548i	−0.219849	−	0.975534i	$-0.570556\pi$
$311$	−17.9054	−	6.51704i	−1.01532	−	0.369548i	−0.795475	+	0.605986i	$0.792779\pi$
$312$		0			0
$313$	1.68661	−	9.56523i	0.0953328	−	0.540659i	−0.899312	−	0.437307i	$-0.855932\pi$
$313$	1.68661	−	9.56523i	0.0953328	−	0.540659i	0.994645	−	0.103352i	$-0.0329567\pi$
$314$		0			0
$315$	4.77343	+	13.6691i	0.268952	+	0.770164i
$316$		0			0
$317$	12.4921	−	14.8875i	0.701628	−	0.836167i	−0.291082	−	0.956698i	$-0.594015\pi$
$317$	12.4921	−	14.8875i	0.701628	−	0.836167i	0.992710	+	0.120531i	$0.0384596\pi$
$318$		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	−0.254020	+	0.967199i	$0.581753\pi$
$331$	−21.1483	+	25.2035i	−1.16241	+	1.38531i	−0.908395	+	0.418113i	$0.862692\pi$
$332$		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	0.244673	−	0.969606i	$-0.421319\pi$
$337$	−3.50899	−	1.27717i	−0.191147	−	0.0695717i	−0.435820	+	0.900034i	$0.643542\pi$
$338$		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	−0.0479028	−	0.998852i	$-0.515254\pi$
$347$	17.2766	−	14.4968i	0.927456	−	0.778228i	0.975359	+	0.220624i	$0.0708093\pi$
$348$		0			0
$349$	−7.89430	+	2.87329i	−0.422572	+	0.153804i	−0.544548	−	0.838729i	$-0.683299\pi$
$349$	−7.89430	+	2.87329i	−0.422572	+	0.153804i	0.121977	+	0.992533i	$0.461077\pi$
$350$		0			0
$351$	−11.8632	−	8.65877i	−0.633211	−	0.462171i
$352$		0			0
$353$	9.50025	+	26.1017i	0.505648	+	1.38926i	0.885686	+	0.464285i	$0.153688\pi$
$353$	9.50025	+	26.1017i	0.505648	+	1.38926i	−0.380039	+	0.924971i	$0.624089\pi$
$354$		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	−0.407762	−	0.913088i	$-0.633691\pi$
$359$	11.1197	−	19.2599i	0.586877	−	1.01650i	0.994639	−	0.103412i	$-0.0329759\pi$
$360$		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	−0.0120154	−	0.999928i	$-0.503825\pi$
$367$	−6.99817	−	1.23397i	−0.365302	−	0.0644125i	−0.353286	+	0.935515i	$0.614936\pi$
$368$		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	−0.954543	−	0.298075i	$-0.903656\pi$
$373$	−3.08072	−	17.4716i	−0.159514	−	0.904646i	0.795029	−	0.606571i	$-0.207456\pi$
$374$		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	$0.00569479\pi$
$379$			0.696553i			0.0357795i	−0.999840	+	0.0178898i	$0.994305\pi$
$380$		0			0
$381$	−0.713701	+	7.58544i	−0.0365640	+	0.388614i
$382$		0			0
$383$	2.97272	+	16.8591i	0.151899	+	0.861462i	0.961566	+	0.274573i	$0.0885365\pi$
$383$	2.97272	+	16.8591i	0.151899	+	0.861462i	−0.809667	+	0.586889i	$0.800352\pi$
$384$		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	−0.273925	−	0.961751i	$-0.588322\pi$
$389$	−16.9672	−	2.99177i	−0.860269	−	0.151689i	−0.586344	+	0.810062i	$0.699433\pi$
$390$		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	−0.739632	−	0.673012i	$-0.765000\pi$
$397$	4.24458	−	7.35182i	0.213029	−	0.368977i	0.952661	+	0.304034i	$0.0983337\pi$
$398$		0			0
$399$	0.965722	+	0.978448i	0.0483466	+	0.0489837i
$400$		0			0
$401$	6.54824	+	7.80389i	0.327004	+	0.389708i	0.904350	−	0.426792i	$-0.140356\pi$
$401$	6.54824	+	7.80389i	0.327004	+	0.389708i	−0.577346	+	0.816499i	$0.695912\pi$
$402$		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	−0.974039	−	0.226380i	$-0.927311\pi$
$409$	−18.6107	+	15.6162i	−0.920238	+	0.772171i	0.0538011	+	0.998552i	$0.482866\pi$
$410$		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	0.340483	−	0.940251i	$-0.389410\pi$
$419$	−0.0629252	−	0.0229029i	−0.00307410	−	0.00111888i	−0.343557	+	0.939132i	$0.611632\pi$
$420$		0			0
$421$	−4.84853	+	27.4974i	−0.236303	+	1.34014i	0.603550	+	0.797325i	$0.293752\pi$
$421$	−4.84853	+	27.4974i	−0.236303	+	1.34014i	−0.839853	+	0.542814i	$0.817359\pi$
$422$		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	−0.518039	−	0.855357i	$-0.673338\pi$
$439$	8.68012	−	10.3446i	0.414280	−	0.493719i	0.932319	+	0.361638i	$0.117782\pi$
$440$		0			0
$441$	43.0083	+	25.5873i	2.04801	+	1.21844i
$442$		0			0
$443$	−1.24107	+	7.03844i	−0.0589649	+	0.334406i	−0.999992	−	0.00392386i	$-0.998751\pi$
$443$	−1.24107	+	7.03844i	−0.0589649	+	0.334406i	0.941027	+	0.338330i	$0.109862\pi$
$444$		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	−0.872788	−	0.488099i	$-0.837691\pi$
$449$	−18.7841	+	10.8450i	−0.886477	+	0.511807i	−0.0136882	+	0.999906i	$0.504357\pi$
$450$		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	0.614492	−	0.788923i	$-0.289361\pi$
$457$	34.0401	−	12.3896i	1.59233	−	0.579561i	0.977838	+	0.209363i	$0.0671390\pi$
$458$		0			0
$459$	3.01304	−	0.744252i	0.140637	−	0.0347387i
$460$		0			0
$461$	−4.10359	−	11.2745i	−0.191123	−	0.525107i	0.806707	−	0.590952i	$-0.201248\pi$
$461$	−4.10359	−	11.2745i	−0.191123	−	0.525107i	−0.997830	+	0.0658453i	$0.979026\pi$
$462$		0			0
$463$	0.669388	+	0.797745i	0.0311091	+	0.0370744i	0.781375	−	0.624062i	$-0.214519\pi$
$463$	0.669388	+	0.797745i	0.0311091	+	0.0370744i	−0.750266	+	0.661137i	$0.770074\pi$
$464$		0			0
$465$	−2.31215	+	8.40853i	−0.107224	+	0.389936i
$466$		0			0
$467$	−2.21810	+	3.84186i	−0.102641	+	0.177780i	−0.912772	−	0.408469i	$-0.866063\pi$
$467$	−2.21810	+	3.84186i	−0.102641	+	0.177780i	0.810131	+	0.586249i	$0.199396\pi$
$468$		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	0.303014	−	0.952986i	$-0.402007\pi$
$479$	−12.7569	−	10.7043i	−0.582876	−	0.489091i	−0.885890	+	0.463895i	$0.846452\pi$
$480$		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	$0.113576\pi$
$487$			15.4161i			0.698570i	−0.937016	+	0.349285i	$0.886424\pi$
$488$		0			0
$489$	7.98748	−	3.66117i	0.361206	−	0.165564i
$490$		0			0
$491$	−1.16739	−	6.62060i	−0.0526836	−	0.298784i	0.947069	−	0.321030i	$-0.104029\pi$
$491$	−1.16739	−	6.62060i	−0.0526836	−	0.298784i	−0.999753	+	0.0222468i	$0.992918\pi$
$492$		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	−0.331346	−	0.943509i	$-0.607503\pi$
$499$	11.8160	−	32.4641i	0.528956	−	1.45329i	0.860302	−	0.509785i	$-0.170275\pi$
$500$		0			0
$501$	−14.8609	+	10.2614i	−0.663935	+	0.458445i
$502$		0			0
$503$	−7.55590	−	13.0872i	−0.336901	−	0.583530i	0.646947	−	0.762535i	$-0.276045\pi$
$503$	−7.55590	−	13.0872i	−0.336901	−	0.583530i	−0.983848	+	0.179005i	$0.942712\pi$
$504$		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	0.636337	−	0.771411i	$-0.280449\pi$
$509$	−5.27599	−	6.28768i	−0.233854	−	0.278696i	−0.870191	+	0.492715i	$0.836004\pi$
$510$		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	0.445215	−	0.895424i	$-0.353127\pi$
$521$	−2.45686	−	1.41847i	−0.107637	−	0.0621444i	−0.552852	+	0.833279i	$0.686461\pi$
$522$		0			0
$523$	26.4086	−	15.2470i	1.15477	−	0.666705i	0.204722	−	0.978820i	$-0.434371\pi$
$523$	26.4086	−	15.2470i	1.15477	−	0.666705i	0.950044	+	0.312116i	$0.101038\pi$
$524$		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	−0.969179	−	0.246356i	$-0.920767\pi$
$547$	−13.0568	+	15.5605i	−0.558270	+	0.665320i	0.410909	+	0.911676i	$0.365211\pi$
$548$		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	0.181733	−	0.983348i	$-0.441830\pi$
$557$	26.5321	−	15.3183i	1.12420	−	0.649059i	0.942471	+	0.334289i	$0.108496\pi$
$558$		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	−0.262138	−	0.965030i	$-0.584428\pi$
$563$	15.2500	−	12.7963i	0.642712	−	0.539299i	0.904850	+	0.425731i	$0.139983\pi$
$564$		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	0.867991	−	0.496580i	$-0.165411\pi$
$569$	−2.77000	−	7.61051i	−0.116124	−	0.319049i	−0.984115	+	0.177531i	$0.943189\pi$
$570$		0			0
$571$	−13.7051	−	16.3331i	−0.573540	−	0.683518i	0.398814	−	0.917032i	$-0.369422\pi$
$571$	−13.7051	−	16.3331i	−0.573540	−	0.683518i	−0.972353	+	0.233514i	$0.924978\pi$
$572$		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	0.867443	−	0.497536i	$-0.165762\pi$
$577$	0.0682767	+	0.118259i	0.00284239	+	0.00492317i	−0.864601	+	0.502460i	$0.832429\pi$
$578$		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	0.966263	−	0.257558i	$-0.0829179\pi$
$587$	21.3306	+	17.8985i	0.880407	+	0.738749i	−0.0858556	+	0.996308i	$0.527362\pi$
$588$		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	$0.110975\pi$
$593$			16.6379i			0.683238i	−0.939839	+	0.341619i	$0.889025\pi$
$594$		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	−0.939904	−	0.341439i	$-0.889086\pi$
$599$	−4.24539	−	24.0768i	−0.173462	−	0.983752i	0.766442	−	0.642314i	$-0.222025\pi$
$600$		0			0
$601$	30.7637	+	25.8138i	1.25488	+	1.05297i	0.996208	+	0.0870032i	$0.0277290\pi$
$601$	30.7637	+	25.8138i	1.25488	+	1.05297i	0.258671	+	0.965965i	$0.416715\pi$
$602$		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	−0.770483	−	0.637460i	$-0.779985\pi$
$607$	5.65409	−	15.5345i	0.229493	−	0.630526i	0.999976	+	0.00693444i	$0.00220732\pi$
$608$		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	−0.885676	−	0.464304i	$-0.846304\pi$
$613$	−1.00864	+	1.74701i	−0.0407385	+	0.0705611i	0.844937	+	0.534865i	$0.179638\pi$
$614$		0			0
$615$	−18.6796	+	4.87437i	−0.753236	+	0.196554i
$616$		0			0
$617$	20.7709	+	24.7538i	0.836204	+	0.996549i	0.999950	+	0.0100465i	$0.00319796\pi$
$617$	20.7709	+	24.7538i	0.836204	+	0.996549i	−0.163746	+	0.986503i	$0.552358\pi$
$618$		0			0
$619$	−4.84499	−	13.3115i	−0.194737	−	0.535035i	0.803441	−	0.595385i	$-0.203000\pi$
$619$	−4.84499	−	13.3115i	−0.194737	−	0.535035i	−0.998177	+	0.0603502i	$0.980778\pi$
$620$		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	−0.216128	−	0.976365i	$-0.569343\pi$
$631$	13.0965	−	7.56128i	0.521365	−	0.301010i	0.737493	+	0.675355i	$0.236010\pi$
$632$		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	−0.710735	−	0.703460i	$-0.751638\pi$
$641$	2.66991	−	3.18187i	0.105455	−	0.125676i	0.816190	+	0.577783i	$0.196082\pi$
$642$		0			0
$643$	43.0780	−	7.59582i	1.69883	−	0.299550i	0.761546	−	0.648111i	$-0.224441\pi$
$643$	43.0780	−	7.59582i	1.69883	−	0.299550i	0.937286	+	0.348561i	$0.113329\pi$
$644$		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	0.601030	−	0.799227i	$-0.294757\pi$
$653$	36.7762	−	6.48464i	1.43916	−	0.253763i	0.838135	+	0.545463i	$0.183646\pi$
$654$		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	−0.999973	−	0.00736587i	$-0.997655\pi$
$659$	−1.48344	+	8.41298i	−0.0577865	+	0.327723i	0.942186	+	0.335089i	$0.108766\pi$
$660$		0			0
$661$	0.439161	+	0.159841i	0.0170814	+	0.00621711i	0.350547	−	0.936545i	$-0.385996\pi$
$661$	0.439161	+	0.159841i	0.0170814	+	0.00621711i	−0.333465	+	0.942762i	$0.608218\pi$
$662$		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	−0.255216	−	0.966884i	$-0.582147\pi$
$673$	4.43038	−	1.61253i	0.170779	−	0.0621584i	0.425994	+	0.904726i	$0.359924\pi$
$674$		0			0
$675$	20.8223	−	1.41036i	0.801452	−	0.0542846i
$676$		0			0
$677$	9.86735	+	27.1103i	0.379233	+	1.04193i	0.971675	+	0.236321i	$0.0759417\pi$
$677$	9.86735	+	27.1103i	0.379233	+	1.04193i	−0.592442	+	0.805613i	$0.701836\pi$
$678$		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	−0.0442060	−	0.999022i	$-0.514076\pi$
$683$	22.0332	−	38.1626i	0.843076	−	1.46025i	0.887282	−	0.461228i	$-0.152591\pi$
$684$		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	−0.680289	−	0.732944i	$-0.738146\pi$
$691$	−41.2765	−	7.27817i	−1.57023	−	0.276875i	−0.889944	+	0.456069i	$0.849257\pi$
$692$		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	$-0.998331\pi$
$701$		−	0.277648i		−	0.0104866i	0.999986	−	0.00524332i	$-0.00166901\pi$
$702$		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	−0.0250415	−	0.999686i	$-0.507972\pi$
$709$	−26.9969	−	22.6531i	−1.01389	−	0.850754i	−0.988847	+	0.148933i	$0.952416\pi$
$710$		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	0.743022	+	0.669267i	$0.233392\pi$
$719$	25.5033	+	44.1730i	0.951113	+	1.64738i	0.208091	+	0.978109i	$0.433275\pi$
$720$		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	0.989594	−	0.143890i	$-0.0459612\pi$
$727$	3.74866	+	10.2994i	0.139030	+	0.381982i	−0.850564	+	0.525872i	$0.823739\pi$
$728$		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	−0.732498	−	0.680769i	$-0.761646\pi$
$733$	−5.12425	+	4.29976i	−0.189269	+	0.158815i	0.543230	+	0.839584i	$0.317201\pi$
$734$		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	−0.206943	−	0.978353i	$-0.566351\pi$
$739$	14.5944	−	8.42609i	0.536864	−	0.309959i	0.743807	+	0.668394i	$0.233018\pi$
$740$		0			0
$741$	−0.342193	+	0.721468i	−0.0125708	+	0.0265038i
$742$		0			0
$743$	10.5585	+	3.84299i	0.387355	+	0.140986i	0.528354	−	0.849024i	$-0.322809\pi$
$743$	10.5585	+	3.84299i	0.387355	+	0.140986i	−0.140999	+	0.990010i	$0.545031\pi$
$744$		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	0.756844	−	0.653595i	$-0.226740\pi$
$751$	46.3569	−	8.17398i	1.69159	−	0.298273i	0.934744	+	0.355322i	$0.115629\pi$
$752$		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	−0.0263892	−	0.999652i	$-0.508401\pi$
$761$	8.01971	−	1.41409i	0.290714	−	0.0512608i	0.317103	+	0.948391i	$0.397290\pi$
$762$		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	−0.350191	−	0.936678i	$-0.613883\pi$
$769$	−33.8466	−	12.3191i	−1.22054	−	0.444240i	−0.870347	+	0.492438i	$0.836106\pi$
$770$		0			0
$771$	35.9923	−	2.91166i	1.29623	−	0.104861i
$772$		0			0
$773$	22.6182	−	13.0586i	0.813521	−	0.469687i	−0.0346560	−	0.999399i	$-0.511034\pi$
$773$	22.6182	−	13.0586i	0.813521	−	0.469687i	0.848177	+	0.529713i	$0.177700\pi$
$774$		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	0.638729	−	0.769432i	$-0.279460\pi$
$787$	−6.45027	−	7.68713i	−0.229927	−	0.274017i	−0.868656	+	0.495415i	$0.835016\pi$
$788$		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

Related objects

Downloads

Learn more