LMFDB - Embedded newform 912.2.bn.m.449.3

Newspace parameters

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.bn (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.28235666434$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.3
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	912.449
Dual form			912.2.bn.m.65.3

$q$-expansion

$f(q)$	$=$	$q+(0.158919 - 1.72474i) q^{3} +(-1.22474 + 0.707107i) q^{5} +0.267949 q^{7} +(-2.94949 - 0.548188i) q^{9} +O(q^{10})$	\(q+(0.158919 - 1.72474i) q^{3} +(-1.22474 + 0.707107i) q^{5} +0.267949 q^{7} +(-2.94949 - 0.548188i) q^{9} +5.27792i q^{11} +(-0.232051 - 0.133975i) q^{13} +(1.02494 + 2.22474i) q^{15} +(-4.24264 + 2.44949i) q^{17} +(-1.73205 - 4.00000i) q^{19} +(0.0425821 - 0.462144i) q^{21} +(-4.57081 - 2.63896i) q^{23} +(-1.50000 + 2.59808i) q^{25} +(-1.41421 + 5.00000i) q^{27} +(-1.03528 + 1.79315i) q^{29} +2.46410i q^{31} +(9.10306 + 0.838759i) q^{33} +(-0.328169 + 0.189469i) q^{35} +7.73205i q^{37} +(-0.267949 + 0.378937i) q^{39} +(2.82843 + 4.89898i) q^{41} +(2.86603 + 4.96410i) q^{43} +(4.00000 - 1.41421i) q^{45} +(-0.656339 - 0.378937i) q^{47} -6.92820 q^{49} +(3.55051 + 7.70674i) q^{51} +(5.46739 - 9.46979i) q^{53} +(-3.73205 - 6.46410i) q^{55} +(-7.17423 + 2.35167i) q^{57} +(5.60609 + 9.71003i) q^{59} +(5.23205 - 9.06218i) q^{61} +(-0.790313 - 0.146887i) q^{63} +0.378937 q^{65} +(0.866025 + 0.500000i) q^{67} +(-5.27792 + 7.46410i) q^{69} +(-6.69213 - 11.5911i) q^{71} +(1.50000 + 2.59808i) q^{73} +(4.24264 + 3.00000i) q^{75} +1.41421i q^{77} +(-9.06218 + 5.23205i) q^{79} +(8.39898 + 3.23375i) q^{81} -2.07055i q^{83} +(3.46410 - 6.00000i) q^{85} +(2.92820 + 2.07055i) q^{87} +(-3.67423 + 6.36396i) q^{89} +(-0.0621778 - 0.0358984i) q^{91} +(4.24995 + 0.391592i) q^{93} +(4.94975 + 3.67423i) q^{95} +(-0.464102 + 0.267949i) q^{97} +(2.89329 - 15.5672i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} - 4 q^{9}+O(q^{10}) $ 8 * q + 16 * q^7 - 4 * q^9	$ 8 q + 16 q^{7} - 4 q^{9} + 12 q^{13} - 12 q^{21} - 12 q^{25} + 24 q^{33} - 16 q^{39} + 16 q^{43} + 32 q^{45} + 48 q^{51} - 16 q^{55} - 28 q^{57} + 28 q^{61} - 8 q^{63} + 12 q^{73} - 24 q^{79} + 28 q^{81} - 32 q^{87} + 48 q^{91} - 4 q^{93} + 24 q^{97} + 32 q^{99}+O(q^{100}) $ 8 * q + 16 * q^7 - 4 * q^9 + 12 * q^13 - 12 * q^21 - 12 * q^25 + 24 * q^33 - 16 * q^39 + 16 * q^43 + 32 * q^45 + 48 * q^51 - 16 * q^55 - 28 * q^57 + 28 * q^61 - 8 * q^63 + 12 * q^73 - 24 * q^79 + 28 * q^81 - 32 * q^87 + 48 * q^91 - 4 * q^93 + 24 * q^97 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$-1$	$1$

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
$3$	0.158919	−	1.72474i	0.0917517	−	0.995782i
$3$	0.158919	−	1.72474i	0.0917517	−	0.995782i
$4$		0			0
$5$	−1.22474	+	0.707107i	−0.547723	+	0.316228i	−0.748203	−	0.663470i	$-0.769083\pi$
$5$	−1.22474	+	0.707107i	−0.547723	+	0.316228i	0.200480	+	0.979698i	$0.435750\pi$
$6$		0			0
$7$	0.267949			0.101275			0.0506376	−	0.998717i	$-0.483875\pi$
$7$	0.267949			0.101275			0.0506376	+	0.998717i	$0.483875\pi$
$8$		0			0
$9$	−2.94949	−	0.548188i	−0.983163	−	0.182729i
$10$		0			0
$11$			5.27792i			1.59135i	0.605723	+	0.795676i	$0.292884\pi$
$11$			5.27792i			1.59135i	−0.605723	+	0.795676i	$0.707116\pi$
$12$		0			0
$13$	−0.232051	−	0.133975i	−0.0643593	−	0.0371579i	0.467475	−	0.884006i	$-0.345164\pi$
$13$	−0.232051	−	0.133975i	−0.0643593	−	0.0371579i	−0.531834	+	0.846848i	$0.678497\pi$
$14$		0			0
$15$	1.02494	+	2.22474i	0.264639	+	0.574427i
$16$		0			0
$17$	−4.24264	+	2.44949i	−1.02899	+	0.594089i	−0.916696	−	0.399586i	$-0.869154\pi$
$17$	−4.24264	+	2.44949i	−1.02899	+	0.594089i	−0.112296	+	0.993675i	$0.535820\pi$
$18$		0			0
$19$	−1.73205	−	4.00000i	−0.397360	−	0.917663i
$19$	−1.73205	−	4.00000i	−0.397360	−	0.917663i
$20$		0			0
$21$	0.0425821	−	0.462144i	0.00929218	−	0.100848i
$22$		0			0
$23$	−4.57081	−	2.63896i	−0.953080	−	0.550261i	−0.0590435	−	0.998255i	$-0.518805\pi$
$23$	−4.57081	−	2.63896i	−0.953080	−	0.550261i	−0.894036	+	0.447995i	$0.852138\pi$
$24$		0			0
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$	−1.41421	+	5.00000i	−0.272166	+	0.962250i
$28$		0			0
$29$	−1.03528	+	1.79315i	−0.192246	+	0.332980i	−0.945994	−	0.324184i	$-0.894910\pi$
$29$	−1.03528	+	1.79315i	−0.192246	+	0.332980i	0.753748	+	0.657163i	$0.228244\pi$
$30$		0			0
$31$			2.46410i			0.442566i	0.975210	+	0.221283i	$0.0710244\pi$
$31$			2.46410i			0.442566i	−0.975210	+	0.221283i	$0.928976\pi$
$32$		0			0
$33$	9.10306	+	0.838759i	1.58464	+	0.146009i
$34$		0			0
$35$	−0.328169	+	0.189469i	−0.0554708	+	0.0320261i
$36$		0			0
$37$			7.73205i			1.27114i	0.772043	+	0.635571i	$0.219235\pi$
$37$			7.73205i			1.27114i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$	−0.267949	+	0.378937i	−0.0429062	+	0.0606785i
$40$		0			0
$41$	2.82843	+	4.89898i	0.441726	+	0.765092i	0.997818	−	0.0660290i	$-0.0210330\pi$
$41$	2.82843	+	4.89898i	0.441726	+	0.765092i	−0.556092	+	0.831121i	$0.687700\pi$
$42$		0			0
$43$	2.86603	+	4.96410i	0.437065	+	0.757018i	0.997462	−	0.0712058i	$-0.0226847\pi$
$43$	2.86603	+	4.96410i	0.437065	+	0.757018i	−0.560397	+	0.828224i	$0.689351\pi$
$44$		0			0
$45$	4.00000	−	1.41421i	0.596285	−	0.210819i
$46$		0			0
$47$	−0.656339	−	0.378937i	−0.0957369	−	0.0552737i	0.451367	−	0.892338i	$-0.350936\pi$
$47$	−0.656339	−	0.378937i	−0.0957369	−	0.0552737i	−0.547104	+	0.837065i	$0.684270\pi$
$48$		0			0
$49$	−6.92820			−0.989743
$50$		0			0
$51$	3.55051	+	7.70674i	0.497171	+	1.07916i
$52$		0			0
$53$	5.46739	−	9.46979i	0.751003	−	1.30078i	−0.196334	−	0.980537i	$-0.562904\pi$
$53$	5.46739	−	9.46979i	0.751003	−	1.30078i	0.947337	−	0.320239i	$-0.103763\pi$
$54$		0			0
$55$	−3.73205	−	6.46410i	−0.503230	−	0.871619i
$56$		0			0
$57$	−7.17423	+	2.35167i	−0.950251	+	0.311486i
$58$		0			0
$59$	5.60609	+	9.71003i	0.729850	+	1.26414i	0.956946	+	0.290265i	$0.0937436\pi$
$59$	5.60609	+	9.71003i	0.729850	+	1.26414i	−0.227096	+	0.973872i	$0.572923\pi$
$60$		0			0
$61$	5.23205	−	9.06218i	0.669895	−	1.16029i	−0.308038	−	0.951374i	$-0.599672\pi$
$61$	5.23205	−	9.06218i	0.669895	−	1.16029i	0.977933	−	0.208919i	$-0.0669944\pi$
$62$		0			0
$63$	−0.790313	−	0.146887i	−0.0995701	−	0.0185060i
$64$		0			0
$65$	0.378937			0.0470014
$66$		0			0
$67$	0.866025	+	0.500000i	0.105802	+	0.0610847i	0.551967	−	0.833866i	$-0.313877\pi$
$67$	0.866025	+	0.500000i	0.105802	+	0.0610847i	−0.446165	+	0.894951i	$0.647211\pi$
$68$		0			0
$69$	−5.27792	+	7.46410i	−0.635387	+	0.898572i
$70$		0			0
$71$	−6.69213	−	11.5911i	−0.794210	−	1.37561i	−0.923340	−	0.383984i	$-0.874552\pi$
$71$	−6.69213	−	11.5911i	−0.794210	−	1.37561i	0.129130	−	0.991628i	$-0.458782\pi$
$72$		0			0
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	0.940356	−	0.340193i	$-0.110493\pi$
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	−0.764794	+	0.644275i	$0.777159\pi$
$74$		0			0
$75$	4.24264	+	3.00000i	0.489898	+	0.346410i
$76$		0			0
$77$			1.41421i			0.161165i
$78$		0			0
$79$	−9.06218	+	5.23205i	−1.01957	+	0.588652i	−0.913981	−	0.405758i	$-0.867008\pi$
$79$	−9.06218	+	5.23205i	−1.01957	+	0.588652i	−0.105594	+	0.994409i	$0.533674\pi$
$80$		0			0
$81$	8.39898	+	3.23375i	0.933220	+	0.359306i
$82$		0			0
$83$		−	2.07055i		−	0.227273i	−0.993522	−	0.113636i	$-0.963750\pi$
$83$		−	2.07055i		−	0.227273i	0.993522	−	0.113636i	$-0.0362499\pi$
$84$		0			0
$85$	3.46410	−	6.00000i	0.375735	−	0.650791i
$86$		0			0
$87$	2.92820	+	2.07055i	0.313936	+	0.221987i
$88$		0			0
$89$	−3.67423	+	6.36396i	−0.389468	+	0.674579i	−0.992378	−	0.123231i	$-0.960674\pi$
$89$	−3.67423	+	6.36396i	−0.389468	+	0.674579i	0.602910	+	0.797809i	$0.294008\pi$
$90$		0			0
$91$	−0.0621778	−	0.0358984i	−0.00651801	−	0.00376317i
$92$		0			0
$93$	4.24995	+	0.391592i	0.440699	+	0.0406062i
$94$		0			0
$95$	4.94975	+	3.67423i	0.507833	+	0.376969i
$96$		0			0
$97$	−0.464102	+	0.267949i	−0.0471224	+	0.0272061i	−0.523376	−	0.852102i	$-0.675328\pi$
$97$	−0.464102	+	0.267949i	−0.0471224	+	0.0272061i	0.476254	+	0.879308i	$0.341994\pi$
$98$		0			0
$99$	2.89329	−	15.5672i	0.290787	−	1.56456i
$100$		0			0
$101$	1.79315	+	1.03528i	0.178425	+	0.103014i	0.586553	−	0.809911i	$-0.300485\pi$
$101$	1.79315	+	1.03528i	0.178425	+	0.103014i	−0.408127	+	0.912925i	$0.633818\pi$
$102$		0			0
$103$			3.53590i			0.348402i	0.984710	+	0.174201i	$0.0557343\pi$
$103$			3.53590i			0.348402i	−0.984710	+	0.174201i	$0.944266\pi$
$104$		0			0
$105$	0.274633	+	0.596119i	0.0268014	+	0.0581752i
$106$		0			0
$107$	4.89898			0.473602			0.236801	−	0.971558i	$-0.423901\pi$
$107$	4.89898			0.473602			0.236801	+	0.971558i	$0.423901\pi$
$108$		0			0
$109$	−5.53590	+	3.19615i	−0.530243	+	0.306136i	−0.741115	−	0.671378i	$-0.765703\pi$
$109$	−5.53590	+	3.19615i	−0.530243	+	0.306136i	0.210872	+	0.977514i	$0.432370\pi$
$110$		0			0
$111$	13.3358	+	1.22877i	1.26578	+	0.116629i
$112$		0			0
$113$	−3.96524			−0.373018			−0.186509	−	0.982453i	$-0.559717\pi$
$113$	−3.96524			−0.373018			−0.186509	+	0.982453i	$0.559717\pi$
$114$		0			0
$115$	7.46410			0.696031
$116$		0			0
$117$	0.610988	+	0.522364i	0.0564859	+	0.0482926i
$118$		0			0
$119$	−1.13681	+	0.656339i	−0.104211	+	0.0601665i
$120$		0			0
$121$	−16.8564			−1.53240
$122$		0			0
$123$	8.89898	−	4.09978i	0.802394	−	0.369664i
$124$		0			0
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$	−17.1962	−	9.92820i	−1.52591	−	0.880986i	−0.999527	−	0.0307388i	$-0.990214\pi$
$127$	−17.1962	−	9.92820i	−1.52591	−	0.880986i	−0.526384	−	0.850247i	$-0.676453\pi$
$128$		0			0
$129$	9.01727	−	4.15427i	0.793927	−	0.365763i
$130$		0			0
$131$	−10.9348	+	6.31319i	−0.955375	+	0.551586i	−0.894747	−	0.446574i	$-0.852644\pi$
$131$	−10.9348	+	6.31319i	−0.955375	+	0.551586i	−0.0606288	+	0.998160i	$0.519311\pi$
$132$		0			0
$133$	−0.464102	−	1.07180i	−0.0402427	−	0.0929366i
$134$		0			0
$135$	−1.80348	−	7.12372i	−0.155219	−	0.613113i
$136$		0			0
$137$	−9.14162	−	5.27792i	−0.781021	−	0.450923i	0.0557708	−	0.998444i	$-0.482238\pi$
$137$	−9.14162	−	5.27792i	−0.781021	−	0.450923i	−0.836792	+	0.547521i	$0.815572\pi$
$138$		0			0
$139$	1.40192	−	2.42820i	0.118910	−	0.205958i	−0.800426	−	0.599431i	$-0.795393\pi$
$139$	1.40192	−	2.42820i	0.118910	−	0.205958i	0.919336	+	0.393474i	$0.128727\pi$
$140$		0			0
$141$	−0.757875	+	1.07180i	−0.0638246	+	0.0902616i
$142$		0			0
$143$	0.707107	−	1.22474i	0.0591312	−	0.102418i
$144$		0			0
$145$		−	2.92820i		−	0.243174i
$146$		0			0
$147$	−1.10102	+	11.9494i	−0.0908106	+	0.985568i
$148$		0			0
$149$	−2.36156	+	1.36345i	−0.193466	+	0.111698i	−0.593604	−	0.804757i	$-0.702296\pi$
$149$	−2.36156	+	1.36345i	−0.193466	+	0.111698i	0.400138	+	0.916455i	$0.368962\pi$
$150$		0			0
$151$		−	2.00000i		−	0.162758i	−0.996683	−	0.0813788i	$-0.974068\pi$
$151$		−	2.00000i		−	0.162758i	0.996683	−	0.0813788i	$-0.0259324\pi$
$152$		0			0
$153$	13.8564	−	4.89898i	1.12022	−	0.396059i
$154$		0			0
$155$	−1.74238	−	3.01790i	−0.139952	−	0.242403i
$156$		0			0
$157$	−5.23205	−	9.06218i	−0.417563	−	0.723241i	0.578131	−	0.815944i	$-0.303782\pi$
$157$	−5.23205	−	9.06218i	−0.417563	−	0.723241i	−0.995694	+	0.0927037i	$0.970449\pi$
$158$		0			0
$159$	−15.4641	−	10.9348i	−1.22638	−	0.867184i
$160$		0			0
$161$	−1.22474	−	0.707107i	−0.0965234	−	0.0557278i
$162$		0			0
$163$	5.19615			0.406994			0.203497	−	0.979076i	$-0.434769\pi$
$163$	5.19615			0.406994			0.203497	+	0.979076i	$0.434769\pi$
$164$		0			0
$165$	−11.7420	+	5.40957i	−0.914115	+	0.421134i
$166$		0			0
$167$	10.8840	−	18.8516i	0.842229	−	1.45878i	−0.0457762	−	0.998952i	$-0.514576\pi$
$167$	10.8840	−	18.8516i	0.842229	−	1.45878i	0.888006	−	0.459832i	$-0.152091\pi$
$168$		0			0
$169$	−6.46410	−	11.1962i	−0.497239	−	0.861242i
$170$		0			0
$171$	2.91591	+	12.7474i	0.222985	+	0.974822i
$172$		0			0
$173$	8.76268	+	15.1774i	0.666214	+	1.15392i	0.978955	+	0.204079i	$0.0654198\pi$
$173$	8.76268	+	15.1774i	0.666214	+	1.15392i	−0.312740	+	0.949839i	$0.601247\pi$
$174$		0			0
$175$	−0.401924	+	0.696152i	−0.0303826	+	0.0526242i
$176$		0			0
$177$	17.6382	−	8.12596i	1.32577	−	0.610785i
$178$		0			0
$179$	1.41421			0.105703			0.0528516	−	0.998602i	$-0.483169\pi$
$179$	1.41421			0.105703			0.0528516	+	0.998602i	$0.483169\pi$
$180$		0			0
$181$	−3.00000	−	1.73205i	−0.222988	−	0.128742i	0.384345	−	0.923190i	$-0.374427\pi$
$181$	−3.00000	−	1.73205i	−0.222988	−	0.128742i	−0.607333	+	0.794447i	$0.707761\pi$
$182$		0			0
$183$	−14.7985	−	10.4641i	−1.09393	−	0.773529i
$184$		0			0
$185$	−5.46739	−	9.46979i	−0.401970	−	0.696233i
$186$		0			0
$187$	−12.9282	−	22.3923i	−0.945404	−	1.63749i
$188$		0			0
$189$	−0.378937	+	1.33975i	−0.0275636	+	0.0974522i
$190$		0			0
$191$			13.0053i			0.941032i	0.882391	+	0.470516i	$0.155932\pi$
$191$			13.0053i			0.941032i	−0.882391	+	0.470516i	$0.844068\pi$
$192$		0			0
$193$	−2.42820	+	1.40192i	−0.174786	+	0.100913i	−0.584841	−	0.811148i	$-0.698843\pi$
$193$	−2.42820	+	1.40192i	−0.174786	+	0.100913i	0.410055	+	0.912061i	$0.365510\pi$
$194$		0			0
$195$	0.0602202	−	0.653570i	0.00431246	−	0.0468031i
$196$		0			0
$197$			0.656339i			0.0467622i	0.999727	+	0.0233811i	$0.00744311\pi$
$197$			0.656339i			0.0467622i	−0.999727	+	0.0233811i	$0.992557\pi$
$198$		0			0
$199$	−11.5981	+	20.0885i	−0.822166	+	1.42403i	0.0819004	+	0.996641i	$0.473901\pi$
$199$	−11.5981	+	20.0885i	−0.822166	+	1.42403i	−0.904066	+	0.427392i	$0.859432\pi$
$200$		0			0
$201$	1.00000	−	1.41421i	0.0705346	−	0.0997509i
$202$		0			0
$203$	−0.277401	+	0.480473i	−0.0194698	+	0.0337226i
$204$		0			0
$205$	−6.92820	−	4.00000i	−0.483887	−	0.279372i
$206$		0			0
$207$	12.0349	+	10.2892i	0.836484	+	0.715152i
$208$		0			0
$209$	21.1117	−	9.14162i	1.46032	−	0.632339i
$210$		0			0
$211$	−9.52628	+	5.50000i	−0.655816	+	0.378636i	−0.790681	−	0.612228i	$-0.790273\pi$
$211$	−9.52628	+	5.50000i	−0.655816	+	0.378636i	0.134865	+	0.990864i	$0.456940\pi$
$212$		0			0
$213$	−21.0552	+	9.70017i	−1.44268	+	0.664645i
$214$		0			0
$215$	−7.02030	−	4.05317i	−0.478780	−	0.276424i
$216$		0			0
$217$			0.660254i			0.0448210i
$218$		0			0
$219$	4.71940	−	2.17423i	0.318907	−	0.146921i
$220$		0			0
$221$	1.31268			0.0883003
$222$		0			0
$223$	9.86603	−	5.69615i	0.660678	−	0.381443i	−0.131857	−	0.991269i	$-0.542094\pi$
$223$	9.86603	−	5.69615i	0.660678	−	0.381443i	0.792535	+	0.609826i	$0.208761\pi$
$224$		0			0
$225$	5.84847	−	6.84072i	0.389898	−	0.456048i
$226$		0			0
$227$	4.79744			0.318418			0.159209	−	0.987245i	$-0.449106\pi$
$227$	4.79744			0.318418			0.159209	+	0.987245i	$0.449106\pi$
$228$		0			0
$229$	−11.3923			−0.752825			−0.376412	−	0.926452i	$-0.622842\pi$
$229$	−11.3923			−0.752825			−0.376412	+	0.926452i	$0.622842\pi$
$230$		0			0
$231$	2.43916	+	0.224745i	0.160485	+	0.0147871i
$232$		0			0
$233$	11.5911	−	6.69213i	0.759359	−	0.438416i	−0.0697066	−	0.997568i	$-0.522206\pi$
$233$	11.5911	−	6.69213i	0.759359	−	0.438416i	0.829066	+	0.559151i	$0.188873\pi$
$234$		0			0
$235$	1.07180			0.0699163
$236$		0			0
$237$	7.58380	+	16.4614i	0.492621	+	1.06928i
$238$		0			0
$239$		−	15.2789i		−	0.988313i	−0.869373	−	0.494156i	$-0.835477\pi$
$239$		−	15.2789i		−	0.988313i	0.869373	−	0.494156i	$-0.164523\pi$
$240$		0			0
$241$	9.82051	+	5.66987i	0.632595	+	0.365229i	0.781756	−	0.623584i	$-0.214324\pi$
$241$	9.82051	+	5.66987i	0.632595	+	0.365229i	−0.149162	+	0.988813i	$0.547657\pi$
$242$		0			0
$243$	6.91215	−	13.9722i	0.443415	−	0.896317i
$244$		0			0
$245$	8.48528	−	4.89898i	0.542105	−	0.312984i
$246$		0			0
$247$	−0.133975	+	1.16025i	−0.00852460	+	0.0738252i
$248$		0			0
$249$	−3.57117	−	0.329049i	−0.226314	−	0.0208527i
$250$		0			0
$251$	12.7279	+	7.34847i	0.803379	+	0.463831i	0.844651	−	0.535317i	$-0.179808\pi$
$251$	12.7279	+	7.34847i	0.803379	+	0.463831i	−0.0412721	+	0.999148i	$0.513141\pi$
$252$		0			0
$253$	13.9282	−	24.1244i	0.875659	−	1.51669i
$254$		0			0
$255$	−9.79796	−	6.92820i	−0.613572	−	0.433861i
$256$		0			0
$257$	4.05317	−	7.02030i	0.252830	−	0.437914i	−0.711474	−	0.702713i	$-0.751972\pi$
$257$	4.05317	−	7.02030i	0.252830	−	0.437914i	0.964304	+	0.264798i	$0.0853054\pi$
$258$		0			0
$259$			2.07180i			0.128735i
$260$		0			0
$261$	4.03652	−	4.72135i	0.249854	−	0.292244i
$262$		0			0
$263$	24.9754	−	14.4195i	1.54005	−	0.889147i	0.541213	−	0.840886i	$-0.317965\pi$
$263$	24.9754	−	14.4195i	1.54005	−	0.889147i	0.998835	−	0.0482609i	$-0.0153679\pi$
$264$		0			0
$265$			15.4641i			0.949952i
$266$		0			0
$267$	10.3923	+	7.34847i	0.635999	+	0.449719i
$268$		0			0
$269$	5.46739	+	9.46979i	0.333352	+	0.577383i	0.983167	−	0.182710i	$-0.0584868\pi$
$269$	5.46739	+	9.46979i	0.333352	+	0.577383i	−0.649815	+	0.760093i	$0.725153\pi$
$270$		0			0
$271$	3.46410	+	6.00000i	0.210429	+	0.364474i	0.951849	−	0.306568i	$-0.0991805\pi$
$271$	3.46410	+	6.00000i	0.210429	+	0.364474i	−0.741420	+	0.671042i	$0.765847\pi$
$272$		0			0
$273$	−0.0717968	+	0.101536i	−0.00434534	+	0.00614524i
$274$		0			0
$275$	−13.7124	−	7.91688i	−0.826891	−	0.477406i
$276$		0			0
$277$	17.8564			1.07289			0.536444	−	0.843936i	$-0.319767\pi$
$277$	17.8564			1.07289			0.536444	+	0.843936i	$0.319767\pi$
$278$		0			0
$279$	1.35079	−	7.26784i	0.0808698	−	0.435114i
$280$		0			0
$281$	4.70951	−	8.15711i	0.280946	−	0.486613i	−0.690672	−	0.723168i	$-0.742685\pi$
$281$	4.70951	−	8.15711i	0.280946	−	0.486613i	0.971618	+	0.236556i	$0.0760185\pi$
$282$		0			0
$283$	2.92820	+	5.07180i	0.174064	+	0.301487i	0.939837	−	0.341624i	$-0.110977\pi$
$283$	2.92820	+	5.07180i	0.174064	+	0.301487i	−0.765773	+	0.643111i	$0.777643\pi$
$284$		0			0
$285$	7.12372	−	7.95315i	0.421973	−	0.471104i
$286$		0			0
$287$	0.757875	+	1.31268i	0.0447359	+	0.0774849i
$288$		0			0
$289$	3.50000	−	6.06218i	0.205882	−	0.356599i
$290$		0			0
$291$	0.388390	+	0.843039i	0.0227678	+	0.0494198i
$292$		0			0
$293$	−13.9391			−0.814329			−0.407164	−	0.913355i	$-0.633482\pi$
$293$	−13.9391			−0.814329			−0.407164	+	0.913355i	$0.633482\pi$
$294$		0			0
$295$	−13.7321	−	7.92820i	−0.799511	−	0.461598i
$296$		0			0
$297$	−26.3896	−	7.46410i	−1.53128	−	0.433111i
$298$		0			0
$299$	0.707107	+	1.22474i	0.0408930	+	0.0708288i
$300$		0			0
$301$	0.767949	+	1.33013i	0.0442639	+	0.0766672i
$302$		0			0
$303$	2.07055	−	2.92820i	0.118950	−	0.168221i
$304$		0			0
$305$			14.7985i			0.847358i
$306$		0			0
$307$	12.0000	−	6.92820i	0.684876	−	0.395413i	−0.116814	−	0.993154i	$-0.537268\pi$
$307$	12.0000	−	6.92820i	0.684876	−	0.395413i	0.801690	+	0.597740i	$0.203935\pi$
$308$		0			0
$309$	6.09852	+	0.561920i	0.346933	+	0.0319665i
$310$		0			0
$311$			12.4505i			0.706004i	0.935623	+	0.353002i	$0.114839\pi$
$311$			12.4505i			0.706004i	−0.935623	+	0.353002i	$0.885161\pi$
$312$		0			0
$313$	−8.39230	+	14.5359i	−0.474361	+	0.821618i	−0.999569	−	0.0293564i	$-0.990654\pi$
$313$	−8.39230	+	14.5359i	−0.474361	+	0.821618i	0.525208	+	0.850974i	$0.323988\pi$
$314$		0			0
$315$	1.07180	−	0.378937i	0.0603889	−	0.0213507i
$316$		0			0
$317$	−4.43211	+	7.67664i	−0.248932	+	0.431163i	−0.963230	−	0.268679i	$-0.913413\pi$
$317$	−4.43211	+	7.67664i	−0.248932	+	0.431163i	0.714298	+	0.699842i	$0.246746\pi$
$318$		0			0
$319$	−9.46410	−	5.46410i	−0.529888	−	0.305931i
$320$		0			0
$321$	0.778539	−	8.44949i	0.0434538	−	0.471605i
$322$		0			0
$323$	17.1464	+	12.7279i	0.954053	+	0.708201i
$324$		0			0
$325$	0.696152	−	0.401924i	0.0386156	−	0.0222947i
$326$		0			0
$327$	4.63279	+	10.0559i	0.256194	+	0.556095i
$328$		0			0
$329$	−0.175865	−	0.101536i	−0.00969578	−	0.00559786i
$330$		0			0
$331$		−	18.0718i		−	0.993316i	−0.867946	−	0.496658i	$-0.834560\pi$
$331$		−	18.0718i		−	0.993316i	0.867946	−	0.496658i	$-0.165440\pi$
$332$		0			0
$333$	4.23862	−	22.8056i	0.232275	−	1.24974i
$334$		0			0
$335$	−1.41421			−0.0772667
$336$		0			0
$337$	−21.3564	+	12.3301i	−1.16336	+	0.671665i	−0.952106	−	0.305767i	$-0.901087\pi$
$337$	−21.3564	+	12.3301i	−1.16336	+	0.671665i	−0.211251	+	0.977432i	$0.567754\pi$
$338$		0			0
$339$	−0.630150	+	6.83903i	−0.0342251	+	0.371445i
$340$		0			0
$341$	−13.0053			−0.704278
$342$		0			0
$343$	−3.73205			−0.201512
$344$		0			0
$345$	1.18618	−	12.8737i	0.0638620	−	0.693095i
$346$		0			0
$347$	−16.6424	+	9.60849i	−0.893410	+	0.515811i	−0.875057	−	0.484021i	$-0.839176\pi$
$347$	−16.6424	+	9.60849i	−0.893410	+	0.515811i	−0.0183540	+	0.999832i	$0.505843\pi$
$348$		0			0
$349$	9.39230			0.502759			0.251379	−	0.967889i	$-0.419116\pi$
$349$	9.39230			0.502759			0.251379	+	0.967889i	$0.419116\pi$
$350$		0			0
$351$	0.998042	−	0.970785i	0.0532716	−	0.0518167i
$352$		0			0
$353$			2.17209i			0.115609i	0.998328	+	0.0578043i	$0.0184099\pi$
$353$			2.17209i			0.115609i	−0.998328	+	0.0578043i	$0.981590\pi$
$354$		0			0
$355$	16.3923	+	9.46410i	0.870013	+	0.502302i
$356$		0			0
$357$	0.951356	+	2.06502i	0.0503511	+	0.109292i
$358$		0			0
$359$	−6.21166	+	3.58630i	−0.327839	+	0.189278i	−0.654881	−	0.755732i	$-0.727281\pi$
$359$	−6.21166	+	3.58630i	−0.327839	+	0.189278i	0.327042	+	0.945010i	$0.393948\pi$
$360$		0			0
$361$	−13.0000	+	13.8564i	−0.684211	+	0.729285i
$362$		0			0
$363$	−2.67880	+	29.0730i	−0.140600	+	1.52594i
$364$		0			0
$365$	−3.67423	−	2.12132i	−0.192318	−	0.111035i
$366$		0			0
$367$	10.5263	−	18.2321i	0.549467	−	0.951705i	−0.448844	−	0.893610i	$-0.648164\pi$
$367$	10.5263	−	18.2321i	0.549467	−	0.951705i	0.998311	−	0.0580950i	$-0.0185026\pi$
$368$		0			0
$369$	−5.65685	−	16.0000i	−0.294484	−	0.832927i
$370$		0			0
$371$	1.46498	−	2.53742i	0.0760581	−	0.131736i
$372$		0			0
$373$			19.4641i			1.00781i	0.863758	+	0.503906i	$0.168104\pi$
$373$			19.4641i			1.00781i	−0.863758	+	0.503906i	$0.831896\pi$
$374$		0			0
$375$	−19.5133	−	1.79796i	−1.00766	−	0.0928462i
$376$		0			0
$377$	0.480473	−	0.277401i	0.0247456	−	0.0142869i
$378$		0			0
$379$			23.7846i			1.22173i	0.791733	+	0.610867i	$0.209179\pi$
$379$			23.7846i			1.22173i	−0.791733	+	0.610867i	$0.790821\pi$
$380$		0			0
$381$	−19.8564	+	28.0812i	−1.01727	+	1.43864i
$382$		0			0
$383$	6.26243	+	10.8468i	0.319995	+	0.554248i	0.980487	−	0.196586i	$-0.0629854\pi$
$383$	6.26243	+	10.8468i	0.319995	+	0.554248i	−0.660492	+	0.750833i	$0.729652\pi$
$384$		0			0
$385$	−1.00000	−	1.73205i	−0.0509647	−	0.0882735i
$386$		0			0
$387$	−5.73205	−	16.2127i	−0.291377	−	0.824137i
$388$		0			0
$389$	−12.8159	−	7.39924i	−0.649790	−	0.375156i	0.138586	−	0.990350i	$-0.455744\pi$
$389$	−12.8159	−	7.39924i	−0.649790	−	0.375156i	−0.788376	+	0.615194i	$0.789078\pi$
$390$		0			0
$391$	25.8564			1.30761
$392$		0			0
$393$	9.15091	+	19.8630i	0.461602	+	1.00195i
$394$		0			0
$395$	7.39924	−	12.8159i	0.372296	−	0.644836i
$396$		0			0
$397$	13.1603	+	22.7942i	0.660494	+	1.14401i	0.980486	+	0.196589i	$0.0629865\pi$
$397$	13.1603	+	22.7942i	0.660494	+	1.14401i	−0.319992	+	0.947420i	$0.603680\pi$
$398$		0			0
$399$	−1.92233	+	0.630128i	−0.0962369	+	0.0315459i
$400$		0			0
$401$	11.4016	+	19.7482i	0.569371	+	0.986179i	0.996628	+	0.0820492i	$0.0261464\pi$
$401$	11.4016	+	19.7482i	0.569371	+	0.986179i	−0.427257	+	0.904130i	$0.640520\pi$
$402$		0			0
$403$	0.330127	−	0.571797i	0.0164448	−	0.0284832i
$404$		0			0
$405$	−12.5732	+	1.97846i	−0.624768	+	0.0983103i
$406$		0			0
$407$	−40.8091			−2.02283
$408$		0			0
$409$	17.5359	+	10.1244i	0.867094	+	0.500617i	0.866382	−	0.499383i	$-0.166440\pi$
$409$	17.5359	+	10.1244i	0.867094	+	0.500617i	0.000712791		1.00000i	$0.499773\pi$
$410$		0			0
$411$	−10.5558	+	14.9282i	−0.520681	+	0.736354i
$412$		0			0
$413$	1.50215	+	2.60179i	0.0739158	+	0.128026i
$414$		0			0
$415$	1.46410	+	2.53590i	0.0718699	+	0.124482i
$416$		0			0
$417$	−3.96524	−	2.80385i	−0.194179	−	0.137305i
$418$		0			0
$419$		−	7.55154i		−	0.368917i	−0.982840	−	0.184458i	$-0.940947\pi$
$419$		−	7.55154i		−	0.368917i	0.982840	−	0.184458i	$-0.0590531\pi$
$420$		0			0
$421$	24.7128	−	14.2679i	1.20443	−	0.695377i	0.242892	−	0.970053i	$-0.421904\pi$
$421$	24.7128	−	14.2679i	1.20443	−	0.695377i	0.961537	+	0.274676i	$0.0885706\pi$
$422$		0			0
$423$	1.72814	+	1.47747i	0.0840248	+	0.0718370i
$424$		0			0
$425$		−	14.6969i		−	0.712906i
$426$		0			0
$427$	1.40192	−	2.42820i	0.0678438	−	0.117509i
$428$		0			0
$429$	−2.00000	−	1.41421i	−0.0965609	−	0.0682789i
$430$		0			0
$431$	−12.7279	+	22.0454i	−0.613082	+	1.06189i	0.377635	+	0.925954i	$0.376737\pi$
$431$	−12.7279	+	22.0454i	−0.613082	+	1.06189i	−0.990718	+	0.135935i	$0.956596\pi$
$432$		0			0
$433$	−17.8923	−	10.3301i	−0.859849	−	0.496434i	0.00411252	−	0.999992i	$-0.498691\pi$
$433$	−17.8923	−	10.3301i	−0.859849	−	0.496434i	−0.863962	+	0.503557i	$0.832024\pi$
$434$		0			0
$435$	−5.05040	−	0.465346i	−0.242148	−	0.0223116i
$436$		0			0
$437$	−2.63896	+	22.8541i	−0.126239	+	1.09326i
$438$		0			0
$439$	13.4545	−	7.76795i	0.642147	−	0.370744i	−0.143294	−	0.989680i	$-0.545769\pi$
$439$	13.4545	−	7.76795i	0.642147	−	0.370744i	0.785441	+	0.618936i	$0.212436\pi$
$440$		0			0
$441$	20.4347	+	3.79796i	0.973079	+	0.180855i
$442$		0			0
$443$	8.96575	+	5.17638i	0.425976	+	0.245937i	0.697631	−	0.716457i	$-0.254238\pi$
$443$	8.96575	+	5.17638i	0.425976	+	0.245937i	−0.271655	+	0.962395i	$0.587571\pi$
$444$		0			0
$445$		−	10.3923i		−	0.492642i
$446$		0			0
$447$	1.97630	+	4.28976i	0.0934758	+	0.202899i
$448$		0			0
$449$	−5.10205			−0.240781			−0.120390	−	0.992727i	$-0.538415\pi$
$449$	−5.10205			−0.240781			−0.120390	+	0.992727i	$0.538415\pi$
$450$		0			0
$451$	−25.8564	+	14.9282i	−1.21753	+	0.702942i
$452$		0			0
$453$	−3.44949	−	0.317837i	−0.162071	−	0.0149333i
$454$		0			0
$455$	0.101536			0.00476008
$456$		0			0
$457$	34.7128			1.62380			0.811898	−	0.583799i	$-0.198434\pi$
$457$	34.7128			1.62380			0.811898	+	0.583799i	$0.198434\pi$
$458$		0			0
$459$	−6.24745	−	24.6773i	−0.291606	−	1.15184i
$460$		0			0
$461$	−30.9232	+	17.8535i	−1.44024	+	0.831522i	−0.997865	−	0.0653090i	$-0.979197\pi$
$461$	−30.9232	+	17.8535i	−1.44024	+	0.831522i	−0.442373	+	0.896831i	$0.645863\pi$
$462$		0			0
$463$	−1.58846			−0.0738219			−0.0369109	−	0.999319i	$-0.511752\pi$
$463$	−1.58846			−0.0738219			−0.0369109	+	0.999319i	$0.511752\pi$
$464$		0			0
$465$	−5.48200	+	2.52557i	−0.254222	+	0.117120i
$466$		0			0
$467$			22.6274i			1.04707i	0.852004	+	0.523536i	$0.175387\pi$
$467$			22.6274i			1.04707i	−0.852004	+	0.523536i	$0.824613\pi$
$468$		0			0
$469$	0.232051	+	0.133975i	0.0107151	+	0.00618637i
$470$		0			0
$471$	−16.4614	+	7.58380i	−0.758502	+	0.349443i
$472$		0			0
$473$	−26.2001	+	15.1266i	−1.20468	+	0.695524i
$474$		0			0
$475$	12.9904	+	1.50000i	0.596040	+	0.0688247i
$476$		0			0
$477$	−21.3172	+	24.9339i	−0.976049	+	1.14164i
$478$		0			0
$479$	−15.1774	−	8.76268i	−0.693474	−	0.400377i	0.111438	−	0.993771i	$-0.464454\pi$
$479$	−15.1774	−	8.76268i	−0.693474	−	0.400377i	−0.804912	+	0.593394i	$0.797788\pi$
$480$		0			0
$481$	1.03590	−	1.79423i	0.0472329	−	0.0818098i
$482$		0			0
$483$	−1.41421	+	2.00000i	−0.0643489	+	0.0910032i
$484$		0			0
$485$	0.378937	−	0.656339i	0.0172067	−	0.0298028i
$486$		0			0
$487$			11.0718i			0.501711i	0.968025	+	0.250856i	$0.0807119\pi$
$487$			11.0718i			0.501711i	−0.968025	+	0.250856i	$0.919288\pi$
$488$		0			0
$489$	0.825765	−	8.96204i	0.0373424	−	0.405277i
$490$		0			0
$491$	−25.6317	+	14.7985i	−1.15674	+	0.667846i	−0.950521	−	0.310660i	$-0.899450\pi$
$491$	−25.6317	+	14.7985i	−1.15674	+	0.667846i	−0.206222	+	0.978505i	$0.566117\pi$
$492$		0			0
$493$		−	10.1436i		−	0.456844i
$494$		0			0
$495$	7.46410	+	21.1117i	0.335486	+	0.948899i
$496$		0			0
$497$	−1.79315	−	3.10583i	−0.0804338	−	0.139315i
$498$		0			0
$499$	4.06218	+	7.03590i	0.181848	+	0.314970i	0.942510	−	0.334178i	$-0.108459\pi$
$499$	4.06218	+	7.03590i	0.181848	+	0.314970i	−0.760662	+	0.649148i	$0.775125\pi$
$500$		0			0
$501$	−30.7846	−	21.7680i	−1.37535	−	0.972523i
$502$		0			0
$503$	−1.13681	−	0.656339i	−0.0506879	−	0.0292647i	0.474442	−	0.880287i	$-0.342650\pi$
$503$	−1.13681	−	0.656339i	−0.0506879	−	0.0292647i	−0.525130	+	0.851022i	$0.675983\pi$
$504$		0			0
$505$	−2.92820			−0.130303
$506$		0			0
$507$	−20.3378	+	9.36965i	−0.903232	+	0.416121i
$508$		0			0
$509$	21.4906	−	37.2228i	0.952554	−	1.64987i	0.212686	−	0.977121i	$-0.431779\pi$
$509$	21.4906	−	37.2228i	0.952554	−	1.64987i	0.739868	−	0.672752i	$-0.234888\pi$
$510$		0			0
$511$	0.401924	+	0.696152i	0.0177801	+	0.0307960i
$512$		0			0
$513$	22.4495	−	3.00340i	0.991169	−	0.132603i
$514$		0			0
$515$	−2.50026	−	4.33057i	−0.110175	−	0.190828i
$516$		0			0
$517$	2.00000	−	3.46410i	0.0879599	−	0.152351i
$518$		0			0
$519$	27.5697	−	12.7014i	1.21018	−	0.557530i
$520$		0			0
$521$	−10.1769			−0.445858			−0.222929	−	0.974835i	$-0.571562\pi$
$521$	−10.1769			−0.445858			−0.222929	+	0.974835i	$0.571562\pi$
$522$		0			0
$523$	9.40192	+	5.42820i	0.411117	+	0.237359i	0.691270	−	0.722597i	$-0.257052\pi$
$523$	9.40192	+	5.42820i	0.411117	+	0.237359i	−0.280152	+	0.959956i	$0.590385\pi$
$524$		0			0
$525$	1.13681	+	0.803848i	0.0496145	+	0.0350828i
$526$		0			0
$527$	−6.03579	−	10.4543i	−0.262923	−	0.455396i
$528$		0			0
$529$	2.42820	+	4.20577i	0.105574	+	0.182860i
$530$		0			0
$531$	−11.2122	−	31.7128i	−0.486567	−	1.37622i
$532$		0			0
$533$		−	1.51575i		−	0.0656544i
$534$		0			0
$535$	−6.00000	+	3.46410i	−0.259403	+	0.149766i
$536$		0			0
$537$	0.224745	−	2.43916i	0.00969846	−	0.105257i
$538$		0			0
$539$		−	36.5665i		−	1.57503i
$540$		0			0
$541$	12.2321	−	21.1865i	0.525897	−	0.910880i	−0.473648	−	0.880714i	$-0.657063\pi$
$541$	12.2321	−	21.1865i	0.525897	−	0.910880i	0.999545	−	0.0301660i	$-0.00960358\pi$
$542$		0			0
$543$	−3.46410	+	4.89898i	−0.148659	+	0.210235i
$544$		0			0
$545$	4.52004	−	7.82894i	0.193617	−	0.335355i
$546$		0			0
$547$	12.8660	+	7.42820i	0.550112	+	0.317607i	0.749167	−	0.662381i	$-0.230454\pi$
$547$	12.8660	+	7.42820i	0.550112	+	0.317607i	−0.199055	+	0.979988i	$0.563787\pi$
$548$		0			0
$549$	−20.3997	+	23.8607i	−0.870636	+	1.01835i
$550$		0			0
$551$	8.96575	+	1.03528i	0.381954	+	0.0441042i
$552$		0			0
$553$	−2.42820	+	1.40192i	−0.103258	+	0.0596159i
$554$		0			0
$555$	−17.2018	+	7.92492i	−0.730177	+	0.336394i
$556$		0			0
$557$	22.5259	+	13.0053i	0.954452	+	0.551053i	0.894461	−	0.447146i	$-0.147559\pi$
$557$	22.5259	+	13.0053i	0.954452	+	0.551053i	0.0599911	+	0.998199i	$0.480893\pi$
$558$		0			0
$559$		−	1.53590i		−	0.0649616i
$560$		0			0
$561$	−40.6755	+	18.7393i	−1.71732	+	0.791174i
$562$		0			0
$563$	−26.0106			−1.09622			−0.548109	−	0.836407i	$-0.684652\pi$
$563$	−26.0106			−1.09622			−0.548109	+	0.836407i	$0.684652\pi$
$564$		0			0
$565$	4.85641	−	2.80385i	0.204311	−	0.117959i
$566$		0			0
$567$	2.25050	+	0.866481i	0.0945121	+	0.0363888i
$568$		0			0
$569$	−25.2528			−1.05865			−0.529326	−	0.848419i	$-0.677555\pi$
$569$	−25.2528			−1.05865			−0.529326	+	0.848419i	$0.677555\pi$
$570$		0			0
$571$	−14.8038			−0.619522			−0.309761	−	0.950814i	$-0.600249\pi$
$571$	−14.8038			−0.619522			−0.309761	+	0.950814i	$0.600249\pi$
$572$		0			0
$573$	22.4309	+	2.06679i	0.937063	+	0.0863413i
$574$		0			0
$575$	13.7124	−	7.91688i	0.571848	−	0.330157i
$576$		0			0
$577$	29.0718			1.21027			0.605137	−	0.796121i	$-0.293118\pi$
$577$	29.0718			1.21027			0.605137	+	0.796121i	$0.293118\pi$
$578$		0			0
$579$	2.03207	+	4.41082i	0.0844501	+	0.183308i
$580$		0			0
$581$		−	0.554803i		−	0.0230171i
$582$		0			0
$583$	49.9808	+	28.8564i	2.06999	+	1.19511i
$584$		0			0
$585$	−1.11767	−	0.207729i	−0.0462100	−	0.00858854i
$586$		0			0
$587$	28.8898	−	16.6796i	1.19241	−	0.688439i	0.233559	−	0.972343i	$-0.424963\pi$
$587$	28.8898	−	16.6796i	1.19241	−	0.688439i	0.958853	+	0.283904i	$0.0916296\pi$
$588$		0			0
$589$	9.85641	−	4.26795i	0.406126	−	0.175858i
$590$		0			0
$591$	1.13202	+	0.104304i	0.0465650	+	0.00429051i
$592$		0			0
$593$	29.1301	+	16.8183i	1.19623	+	0.690643i	0.959712	−	0.280984i	$-0.0906609\pi$
$593$	29.1301	+	16.8183i	1.19623	+	0.690643i	0.236517	+	0.971627i	$0.423994\pi$
$594$		0			0
$595$	0.928203	−	1.60770i	0.0380526	−	0.0659091i
$596$		0			0
$597$	32.8043	+	23.1962i	1.34259	+	0.949355i
$598$		0			0
$599$	1.36345	−	2.36156i	0.0557089	−	0.0964906i	−0.836826	−	0.547469i	$-0.815591\pi$
$599$	1.36345	−	2.36156i	0.0557089	−	0.0964906i	0.892535	+	0.450978i	$0.148925\pi$
$600$		0			0
$601$			28.3731i			1.15736i	0.815554	+	0.578681i	$0.196432\pi$
$601$			28.3731i			1.15736i	−0.815554	+	0.578681i	$0.803568\pi$
$602$		0			0
$603$	−2.28024	−	1.94949i	−0.0928585	−	0.0793894i
$604$		0			0
$605$	20.6448	−	11.9193i	0.839330	−	0.484588i
$606$		0			0
$607$			35.2487i			1.43070i	0.698766	+	0.715351i	$0.253733\pi$
$607$			35.2487i			1.43070i	−0.698766	+	0.715351i	$0.746267\pi$
$608$		0			0
$609$	0.784610	+	0.554803i	0.0317940	+	0.0224817i
$610$		0			0
$611$	0.101536	+	0.175865i	0.00410771	+	0.00711475i
$612$		0			0
$613$	23.3205	+	40.3923i	0.941906	+	1.63143i	0.761830	+	0.647777i	$0.224301\pi$
$613$	23.3205	+	40.3923i	0.941906	+	1.63143i	0.180077	+	0.983653i	$0.442365\pi$
$614$		0			0
$615$	−8.00000	+	11.3137i	−0.322591	+	0.456213i
$616$		0			0
$617$	−19.9885	−	11.5403i	−0.804705	−	0.464597i	0.0404087	−	0.999183i	$-0.487134\pi$
$617$	−19.9885	−	11.5403i	−0.804705	−	0.464597i	−0.845114	+	0.534587i	$0.820467\pi$
$618$		0			0
$619$	−13.1962			−0.530398			−0.265199	−	0.964194i	$-0.585438\pi$
$619$	−13.1962			−0.530398			−0.265199	+	0.964194i	$0.585438\pi$
$620$		0			0
$621$	19.6589	−	19.1220i	0.788884	−	0.767339i
$622$		0			0
$623$	−0.984508	+	1.70522i	−0.0394435	+	0.0683181i
$624$		0			0
$625$	0.500000	+	0.866025i	0.0200000	+	0.0346410i
$626$		0			0
$627$	−12.4119	−	37.8650i	−0.495685	−	1.51218i
$628$		0			0
$629$	−18.9396	−	32.8043i	−0.755170	−	1.30799i
$630$		0			0
$631$	5.47372	−	9.48076i	0.217905	−	0.377423i	−0.736262	−	0.676697i	$-0.763411\pi$
$631$	5.47372	−	9.48076i	0.217905	−	0.377423i	0.954167	+	0.299273i	$0.0967442\pi$
$632$		0			0
$633$	7.97219	+	17.3045i	0.316866	+	0.687790i
$634$		0			0
$635$	28.0812			1.11437
$636$		0			0
$637$	1.60770	+	0.928203i	0.0636992	+	0.0367768i
$638$		0			0
$639$	13.3843	+	37.8564i	0.529473	+	1.49758i
$640$		0			0
$641$	16.2127	+	28.0812i	0.640363	+	1.10914i	0.985352	+	0.170534i	$0.0545493\pi$
$641$	16.2127	+	28.0812i	0.640363	+	1.10914i	−0.344989	+	0.938607i	$0.612117\pi$
$642$		0			0
$643$	−6.20577	−	10.7487i	−0.244732	−	0.423888i	0.717324	−	0.696739i	$-0.245367\pi$
$643$	−6.20577	−	10.7487i	−0.244732	−	0.423888i	−0.962056	+	0.272852i	$0.912033\pi$
$644$		0			0
$645$	−8.10634	+	11.4641i	−0.319187	+	0.451399i
$646$		0			0
$647$			30.3548i			1.19337i	0.802475	+	0.596686i	$0.203516\pi$
$647$			30.3548i			1.19337i	−0.802475	+	0.596686i	$0.796484\pi$
$648$		0			0
$649$	−51.2487	+	29.5885i	−2.01169	+	1.16145i
$650$		0			0
$651$	1.13877	+	0.104927i	0.0446319	+	0.00411240i
$652$		0			0
$653$		−	1.86748i		−	0.0730802i	−0.999332	−	0.0365401i	$-0.988366\pi$
$653$		−	1.86748i		−	0.0730802i	0.999332	−	0.0365401i	$-0.0116337\pi$
$654$		0			0
$655$	8.92820	−	15.4641i	0.348854	−	0.604232i
$656$		0			0
$657$	−3.00000	−	8.48528i	−0.117041	−	0.331042i
$658$		0			0
$659$	−1.84392	+	3.19376i	−0.0718289	+	0.124411i	−0.899703	−	0.436503i	$-0.856217\pi$
$659$	−1.84392	+	3.19376i	−0.0718289	+	0.124411i	0.827874	+	0.560914i	$0.189550\pi$
$660$		0			0
$661$	−29.3205	−	16.9282i	−1.14044	−	0.658431i	−0.193897	−	0.981022i	$-0.562113\pi$
$661$	−29.3205	−	16.9282i	−1.14044	−	0.658431i	−0.946538	+	0.322591i	$0.895446\pi$
$662$		0			0
$663$	0.208609	−	2.26403i	0.00810170	−	0.0879278i
$664$		0			0
$665$	1.32628	+	0.984508i	0.0514310	+	0.0381776i
$666$		0			0
$667$	9.46410	−	5.46410i	0.366451	−	0.211571i
$668$		0			0
$669$	−8.25651	−	17.9216i	−0.319215	−	0.692889i
$670$		0			0
$671$	47.8294	+	27.6143i	1.84643	+	1.06604i
$672$		0			0
$673$		−	29.9808i		−	1.15567i	−0.816152	−	0.577837i	$-0.803897\pi$
$673$		−	29.9808i		−	1.15567i	0.816152	−	0.577837i	$-0.196103\pi$
$674$		0			0
$675$	−10.8691	−	11.1742i	−0.418350	−	0.430096i
$676$		0			0
$677$	−30.1518			−1.15883			−0.579413	−	0.815034i	$-0.696718\pi$
$677$	−30.1518			−1.15883			−0.579413	+	0.815034i	$0.696718\pi$
$678$		0			0
$679$	−0.124356	+	0.0717968i	−0.00477233	+	0.00275531i
$680$		0			0
$681$	0.762403	−	8.27437i	0.0292154	−	0.317074i
$682$		0			0
$683$	−26.1122			−0.999155			−0.499577	−	0.866269i	$-0.666511\pi$
$683$	−26.1122			−0.999155			−0.499577	+	0.866269i	$0.666511\pi$
$684$		0			0
$685$	14.9282			0.570377
$686$		0			0
$687$	−1.81045	+	19.6488i	−0.0690730	+	0.749649i
$688$		0			0
$689$	−2.53742	+	1.46498i	−0.0966681	+	0.0558114i
$690$		0			0
$691$	17.8564			0.679290			0.339645	−	0.940554i	$-0.389693\pi$
$691$	17.8564			0.679290			0.339645	+	0.940554i	$0.389693\pi$
$692$		0			0
$693$	0.775255	−	4.17121i	0.0294495	−	0.158451i
$694$		0			0
$695$			3.96524i			0.150410i
$696$		0			0
$697$	−24.0000	−	13.8564i	−0.909065	−	0.524849i
$698$		0			0
$699$	−9.70017	−	21.0552i	−0.366894	−	0.796381i
$700$		0			0
$701$	34.2049	−	19.7482i	1.29190	−	0.745880i	0.312911	−	0.949782i	$-0.398696\pi$
$701$	34.2049	−	19.7482i	1.29190	−	0.745880i	0.978991	+	0.203902i	$0.0653625\pi$
$702$		0			0
$703$	30.9282	−	13.3923i	1.16648	−	0.505100i
$704$		0			0
$705$	0.170328	−	1.84858i	0.00641494	−	0.0696214i
$706$		0			0
$707$	0.480473	+	0.277401i	0.0180701	+	0.0104328i
$708$		0			0
$709$	−3.83975	+	6.65064i	−0.144205	+	0.249770i	−0.929076	−	0.369889i	$-0.879396\pi$
$709$	−3.83975	+	6.65064i	−0.144205	+	0.249770i	0.784871	+	0.619659i	$0.212729\pi$
$710$		0			0
$711$	29.5969	−	10.4641i	1.10997	−	0.392434i
$712$		0			0
$713$	6.50266	−	11.2629i	0.243527	−	0.421800i
$714$		0			0
$715$			2.00000i			0.0747958i
$716$		0			0
$717$	−26.3523	−	2.42811i	−0.984144	−	0.0906794i
$718$		0			0
$719$	9.46979	−	5.46739i	0.353164	−	0.203899i	−0.312914	−	0.949781i	$-0.601305\pi$
$719$	9.46979	−	5.46739i	0.353164	−	0.203899i	0.666078	+	0.745882i	$0.267972\pi$
$720$		0			0
$721$			0.947441i			0.0352846i
$722$		0			0
$723$	11.3397	−	16.0368i	0.421730	−	0.596416i
$724$		0			0
$725$	−3.10583	−	5.37945i	−0.115348	−	0.199788i
$726$		0			0
$727$	8.79423	+	15.2321i	0.326160	+	0.564925i	0.981746	−	0.190195i	$-0.0609119\pi$
$727$	8.79423	+	15.2321i	0.326160	+	0.564925i	−0.655587	+	0.755120i	$0.727579\pi$
$728$		0			0
$729$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$730$		0			0
$731$	−24.3190	−	14.0406i	−0.899472	−	0.519310i
$732$		0			0
$733$	19.7128			0.728109			0.364055	−	0.931378i	$-0.381392\pi$
$733$	19.7128			0.728109			0.364055	+	0.931378i	$0.381392\pi$
$734$		0			0
$735$	−7.10102	−	15.4135i	−0.261925	−	0.568535i
$736$		0			0
$737$	−2.63896	+	4.57081i	−0.0972073	+	0.168368i
$738$		0			0
$739$	−22.5981	−	39.1410i	−0.831284	−	1.43983i	−0.897021	−	0.441989i	$-0.854273\pi$
$739$	−22.5981	−	39.1410i	−0.831284	−	1.43983i	0.0657370	−	0.997837i	$-0.479060\pi$
$740$		0			0
$741$	1.97985	+	0.415458i	0.0727316	+	0.0152622i
$742$		0			0
$743$	3.53553	+	6.12372i	0.129706	+	0.224658i	0.923563	−	0.383447i	$-0.125263\pi$
$743$	3.53553	+	6.12372i	0.129706	+	0.224658i	−0.793857	+	0.608105i	$0.791930\pi$
$744$		0			0
$745$	1.92820	−	3.33975i	0.0706439	−	0.122359i
$746$		0			0
$747$	−1.13505	+	6.10707i	−0.0415294	+	0.223446i
$748$		0			0
$749$	1.31268			0.0479642
$750$		0			0
$751$	−25.4545	−	14.6962i	−0.928847	−	0.536270i	−0.0424005	−	0.999101i	$-0.513501\pi$
$751$	−25.4545	−	14.6962i	−0.928847	−	0.536270i	−0.886447	+	0.462830i	$0.846834\pi$
$752$		0			0
$753$	14.6969	−	20.7846i	0.535586	−	0.757433i
$754$		0			0
$755$	1.41421	+	2.44949i	0.0514685	+	0.0891461i
$756$		0			0
$757$	−12.6962	−	21.9904i	−0.461450	−	0.799254i	0.537584	−	0.843210i	$-0.319337\pi$
$757$	−12.6962	−	21.9904i	−0.461450	−	0.799254i	−0.999033	+	0.0439562i	$0.986004\pi$
$758$		0			0
$759$	−39.3949	−	27.8564i	−1.42994	−	1.01112i
$760$		0			0
$761$			32.1208i			1.16438i	0.813054	+	0.582188i	$0.197803\pi$
$761$			32.1208i			1.16438i	−0.813054	+	0.582188i	$0.802197\pi$
$762$		0			0
$763$	−1.48334	+	0.856406i	−0.0537005	+	0.0310040i
$764$		0			0
$765$	−13.5065	+	15.7980i	−0.488327	+	0.571176i
$766$		0			0
$767$		−	3.00429i		−	0.108479i
$768$		0			0
$769$	−13.4282	+	23.2583i	−0.484233	+	0.838717i	−0.999836	−	0.0181110i	$-0.994235\pi$
$769$	−13.4282	+	23.2583i	−0.484233	+	0.838717i	0.515603	+	0.856828i	$0.327568\pi$
$770$		0			0
$771$	−11.4641	−	8.10634i	−0.412870	−	0.291943i
$772$		0			0
$773$	−20.3538	+	35.2538i	−0.732075	+	1.26799i	0.223920	+	0.974608i	$0.428115\pi$
$773$	−20.3538	+	35.2538i	−0.732075	+	1.26799i	−0.955995	+	0.293384i	$0.905219\pi$
$774$		0			0
$775$	−6.40192	−	3.69615i	−0.229964	−	0.132770i
$776$		0			0
$777$	3.57332	+	0.329247i	0.128192	+	0.0118117i
$778$		0			0
$779$	14.6969	−	19.7990i	0.526572	−	0.709372i
$780$		0			0
$781$	61.1769	−	35.3205i	2.18908	−	1.26387i
$782$		0			0
$783$	−7.50165	−	7.71228i	−0.268087	−	0.275614i
$784$		0			0
$785$	12.8159	+	7.39924i	0.457417	+	0.264090i
$786$		0			0
$787$			37.9282i			1.35199i	0.736904	+	0.675997i	$0.236287\pi$
$787$			37.9282i			1.35199i	−0.736904	+	0.675997i	$0.763713\pi$
$788$		0			0
$789$	−20.9010	−	45.3677i	−0.744094	−	1.61513i
$790$		0			0
$791$	−1.06248			−0.0377775
$792$		0			0
$793$	−2.42820	+	1.40192i	−0.0862280	+

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.bn (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.158919 - 1.72474i) q^{3} +(-1.22474 + 0.707107i) q^{5} +0.267949 q^{7} +(-2.94949 - 0.548188i) q^{9} +O(q^{10})\)	\(q+(0.158919 - 1.72474i) q^{3} +(-1.22474 + 0.707107i) q^{5} +0.267949 q^{7} +(-2.94949 - 0.548188i) q^{9} +5.27792i q^{11} +(-0.232051 - 0.133975i) q^{13} +(1.02494 + 2.22474i) q^{15} +(-4.24264 + 2.44949i) q^{17} +(-1.73205 - 4.00000i) q^{19} +(0.0425821 - 0.462144i) q^{21} +(-4.57081 - 2.63896i) q^{23} +(-1.50000 + 2.59808i) q^{25} +(-1.41421 + 5.00000i) q^{27} +(-1.03528 + 1.79315i) q^{29} +2.46410i q^{31} +(9.10306 + 0.838759i) q^{33} +(-0.328169 + 0.189469i) q^{35} +7.73205i q^{37} +(-0.267949 + 0.378937i) q^{39} +(2.82843 + 4.89898i) q^{41} +(2.86603 + 4.96410i) q^{43} +(4.00000 - 1.41421i) q^{45} +(-0.656339 - 0.378937i) q^{47} -6.92820 q^{49} +(3.55051 + 7.70674i) q^{51} +(5.46739 - 9.46979i) q^{53} +(-3.73205 - 6.46410i) q^{55} +(-7.17423 + 2.35167i) q^{57} +(5.60609 + 9.71003i) q^{59} +(5.23205 - 9.06218i) q^{61} +(-0.790313 - 0.146887i) q^{63} +0.378937 q^{65} +(0.866025 + 0.500000i) q^{67} +(-5.27792 + 7.46410i) q^{69} +(-6.69213 - 11.5911i) q^{71} +(1.50000 + 2.59808i) q^{73} +(4.24264 + 3.00000i) q^{75} +1.41421i q^{77} +(-9.06218 + 5.23205i) q^{79} +(8.39898 + 3.23375i) q^{81} -2.07055i q^{83} +(3.46410 - 6.00000i) q^{85} +(2.92820 + 2.07055i) q^{87} +(-3.67423 + 6.36396i) q^{89} +(-0.0621778 - 0.0358984i) q^{91} +(4.24995 + 0.391592i) q^{93} +(4.94975 + 3.67423i) q^{95} +(-0.464102 + 0.267949i) q^{97} +(2.89329 - 15.5672i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} - 4 q^{9}+O(q^{10}) \) 8 * q + 16 * q^7 - 4 * q^9	\( 8 q + 16 q^{7} - 4 q^{9} + 12 q^{13} - 12 q^{21} - 12 q^{25} + 24 q^{33} - 16 q^{39} + 16 q^{43} + 32 q^{45} + 48 q^{51} - 16 q^{55} - 28 q^{57} + 28 q^{61} - 8 q^{63} + 12 q^{73} - 24 q^{79} + 28 q^{81} - 32 q^{87} + 48 q^{91} - 4 q^{93} + 24 q^{97} + 32 q^{99}+O(q^{100}) \) 8 * q + 16 * q^7 - 4 * q^9 + 12 * q^13 - 12 * q^21 - 12 * q^25 + 24 * q^33 - 16 * q^39 + 16 * q^43 + 32 * q^45 + 48 * q^51 - 16 * q^55 - 28 * q^57 + 28 * q^61 - 8 * q^63 + 12 * q^73 - 24 * q^79 + 28 * q^81 - 32 * q^87 + 48 * q^91 - 4 * q^93 + 24 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more