LMFDB - Embedded newform 882.2.g.m.361.2

Newspace parameters

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	882.g (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.04280545828$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
Defining polynomial:	$x^{4} + 2 x^{2} + 4$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	882.361
Dual form			882.2.g.m.667.2

$q$-expansion

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.707107 + 1.22474i) q^{5} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.707107 + 1.22474i) q^{5} -1.00000 q^{8} +(-0.707107 + 1.22474i) q^{10} +(2.00000 - 3.46410i) q^{11} +4.24264 q^{13} +(-0.500000 - 0.866025i) q^{16} +(-3.53553 + 6.12372i) q^{17} +(2.82843 + 4.89898i) q^{19} -1.41421 q^{20} +4.00000 q^{22} +(4.00000 + 6.92820i) q^{23} +(1.50000 - 2.59808i) q^{25} +(2.12132 + 3.67423i) q^{26} -2.00000 q^{29} +(0.500000 - 0.866025i) q^{32} -7.07107 q^{34} +(-2.00000 - 3.46410i) q^{37} +(-2.82843 + 4.89898i) q^{38} +(-0.707107 - 1.22474i) q^{40} -9.89949 q^{41} -4.00000 q^{43} +(2.00000 + 3.46410i) q^{44} +(-4.00000 + 6.92820i) q^{46} +(2.82843 + 4.89898i) q^{47} +3.00000 q^{50} +(-2.12132 + 3.67423i) q^{52} +(2.00000 - 3.46410i) q^{53} +5.65685 q^{55} +(-1.00000 - 1.73205i) q^{58} +(-5.65685 + 9.79796i) q^{59} +(-0.707107 - 1.22474i) q^{61} +1.00000 q^{64} +(3.00000 + 5.19615i) q^{65} +(6.00000 - 10.3923i) q^{67} +(-3.53553 - 6.12372i) q^{68} +(7.77817 - 13.4722i) q^{73} +(2.00000 - 3.46410i) q^{74} -5.65685 q^{76} +(8.00000 + 13.8564i) q^{79} +(0.707107 - 1.22474i) q^{80} +(-4.94975 - 8.57321i) q^{82} +5.65685 q^{83} -10.0000 q^{85} +(-2.00000 - 3.46410i) q^{86} +(-2.00000 + 3.46410i) q^{88} +(3.53553 + 6.12372i) q^{89} -8.00000 q^{92} +(-2.82843 + 4.89898i) q^{94} +(-4.00000 + 6.92820i) q^{95} +7.07107 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4q + 2q^{2} - 2q^{4} - 4q^{8} + O(q^{10}) $	$ 4q + 2q^{2} - 2q^{4} - 4q^{8} + 8q^{11} - 2q^{16} + 16q^{22} + 16q^{23} + 6q^{25} - 8q^{29} + 2q^{32} - 8q^{37} - 16q^{43} + 8q^{44} - 16q^{46} + 12q^{50} + 8q^{53} - 4q^{58} + 4q^{64} + 12q^{65} + 24q^{67} + 8q^{74} + 32q^{79} - 40q^{85} - 8q^{86} - 8q^{88} - 32q^{92} - 16q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	0.707107	+	1.22474i	0.316228	+	0.547723i	0.979698	−	0.200480i	$-0.0642503\pi$
$5$	0.707107	+	1.22474i	0.316228	+	0.547723i	−0.663470	+	0.748203i	$0.730917\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.00000			−0.353553
$9$		0			0
$10$	−0.707107	+	1.22474i	−0.223607	+	0.387298i
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	−0.389338	−	0.921095i	$-0.627296\pi$
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	0.992361	−	0.123371i	$-0.0393705\pi$
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−3.53553	+	6.12372i	−0.857493	+	1.48522i	0.0168199	+	0.999859i	$0.494646\pi$
$17$	−3.53553	+	6.12372i	−0.857493	+	1.48522i	−0.874313	+	0.485363i	$0.838688\pi$
$18$		0			0
$19$	2.82843	+	4.89898i	0.648886	+	1.12390i	0.983389	+	0.181509i	$0.0580980\pi$
$19$	2.82843	+	4.89898i	0.648886	+	1.12390i	−0.334504	+	0.942394i	$0.608569\pi$
$20$	−1.41421			−0.316228
$21$		0			0
$22$	4.00000			0.852803
$23$	4.00000	+	6.92820i	0.834058	+	1.44463i	0.894795	+	0.446476i	$0.147321\pi$
$23$	4.00000	+	6.92820i	0.834058	+	1.44463i	−0.0607377	+	0.998154i	$0.519345\pi$
$24$		0			0
$25$	1.50000	−	2.59808i	0.300000	−	0.519615i
$26$	2.12132	+	3.67423i	0.416025	+	0.720577i
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$	−7.07107			−1.21268
$35$		0			0
$36$		0			0
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	0.653476	−	0.756948i	$-0.273310\pi$
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	−0.982274	+	0.187453i	$0.939977\pi$
$38$	−2.82843	+	4.89898i	−0.458831	+	0.794719i
$39$		0			0
$40$	−0.707107	−	1.22474i	−0.111803	−	0.193649i
$41$	−9.89949			−1.54604			−0.773021	−	0.634381i	$-0.781255\pi$
$41$	−9.89949			−1.54604			−0.773021	+	0.634381i	$0.781255\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$	2.00000	+	3.46410i	0.301511	+	0.522233i
$45$		0			0
$46$	−4.00000	+	6.92820i	−0.589768	+	1.02151i
$47$	2.82843	+	4.89898i	0.412568	+	0.714590i	0.995170	−	0.0981685i	$-0.0312984\pi$
$47$	2.82843	+	4.89898i	0.412568	+	0.714590i	−0.582601	+	0.812758i	$0.697965\pi$
$48$		0			0
$49$		0			0
$50$	3.00000			0.424264
$51$		0			0
$52$	−2.12132	+	3.67423i	−0.294174	+	0.509525i
$53$	2.00000	−	3.46410i	0.274721	−	0.475831i	−0.695344	−	0.718677i	$-0.744748\pi$
$53$	2.00000	−	3.46410i	0.274721	−	0.475831i	0.970065	+	0.242846i	$0.0780811\pi$
$54$		0			0
$55$	5.65685			0.762770
$56$		0			0
$57$		0			0
$58$	−1.00000	−	1.73205i	−0.131306	−	0.227429i
$59$	−5.65685	+	9.79796i	−0.736460	+	1.27559i	0.217620	+	0.976034i	$0.430171\pi$
$59$	−5.65685	+	9.79796i	−0.736460	+	1.27559i	−0.954080	+	0.299552i	$0.903163\pi$
$60$		0			0
$61$	−0.707107	−	1.22474i	−0.0905357	−	0.156813i	0.817201	−	0.576353i	$-0.195525\pi$
$61$	−0.707107	−	1.22474i	−0.0905357	−	0.156813i	−0.907737	+	0.419540i	$0.862191\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			0.125000
$65$	3.00000	+	5.19615i	0.372104	+	0.644503i
$66$		0			0
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	−0.222571	−	0.974916i	$-0.571445\pi$
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	0.955588	−	0.294706i	$-0.0952216\pi$
$68$	−3.53553	−	6.12372i	−0.428746	−	0.742611i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	7.77817	−	13.4722i	0.910366	−	1.57680i	0.0968194	−	0.995302i	$-0.469133\pi$
$73$	7.77817	−	13.4722i	0.910366	−	1.57680i	0.813547	−	0.581499i	$-0.197534\pi$
$74$	2.00000	−	3.46410i	0.232495	−	0.402694i
$75$		0			0
$76$	−5.65685			−0.648886
$77$		0			0
$78$		0			0
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.827401	+	0.561611i	$0.189818\pi$
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.0726692	+	0.997356i	$0.476848\pi$
$80$	0.707107	−	1.22474i	0.0790569	−	0.136931i
$81$		0			0
$82$	−4.94975	−	8.57321i	−0.546608	−	0.946753i
$83$	5.65685			0.620920			0.310460	−	0.950586i	$-0.399517\pi$
$83$	5.65685			0.620920			0.310460	+	0.950586i	$0.399517\pi$
$84$		0			0
$85$	−10.0000			−1.08465
$86$	−2.00000	−	3.46410i	−0.215666	−	0.373544i
$87$		0			0
$88$	−2.00000	+	3.46410i	−0.213201	+	0.369274i
$89$	3.53553	+	6.12372i	0.374766	+	0.649113i	0.990292	−	0.139003i	$-0.0443898\pi$
$89$	3.53553	+	6.12372i	0.374766	+	0.649113i	−0.615526	+	0.788116i	$0.711056\pi$
$90$		0			0
$91$		0			0
$92$	−8.00000			−0.834058
$93$		0			0
$94$	−2.82843	+	4.89898i	−0.291730	+	0.505291i
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	7.07107			0.717958			0.358979	−	0.933346i	$-0.383125\pi$
$97$	7.07107			0.717958			0.358979	+	0.933346i	$0.383125\pi$
$98$		0			0
$99$		0			0
$100$	1.50000	+	2.59808i	0.150000	+	0.259808i
$101$	6.36396	−	11.0227i	0.633238	−	1.09680i	−0.353648	−	0.935379i	$-0.615059\pi$
$101$	6.36396	−	11.0227i	0.633238	−	1.09680i	0.986886	−	0.161421i	$-0.0516078\pi$
$102$		0			0
$103$	−2.82843	−	4.89898i	−0.278693	−	0.482711i	0.692367	−	0.721545i	$-0.256568\pi$
$103$	−2.82843	−	4.89898i	−0.278693	−	0.482711i	−0.971060	+	0.238835i	$0.923235\pi$
$104$	−4.24264			−0.416025
$105$		0			0
$106$	4.00000			0.388514
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	0.753010	−	0.658009i	$-0.228601\pi$
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	−0.946357	+	0.323122i	$0.895268\pi$
$108$		0			0
$109$	−2.00000	+	3.46410i	−0.191565	+	0.331801i	−0.945769	−	0.324840i	$-0.894690\pi$
$109$	−2.00000	+	3.46410i	−0.191565	+	0.331801i	0.754204	+	0.656640i	$0.228023\pi$
$110$	2.82843	+	4.89898i	0.269680	+	0.467099i
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−5.65685	+	9.79796i	−0.527504	+	0.913664i
$116$	1.00000	−	1.73205i	0.0928477	−	0.160817i
$117$		0			0
$118$	−11.3137			−1.04151
$119$		0			0
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$	0.707107	−	1.22474i	0.0640184	−	0.110883i
$123$		0			0
$124$		0			0
$125$	11.3137			1.01193
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$	−3.00000	+	5.19615i	−0.263117	+	0.455733i
$131$	−8.48528	−	14.6969i	−0.741362	−	1.28408i	−0.951875	−	0.306486i	$-0.900847\pi$
$131$	−8.48528	−	14.6969i	−0.741362	−	1.28408i	0.210513	−	0.977591i	$-0.432487\pi$
$132$		0			0
$133$		0			0
$134$	12.0000			1.03664
$135$		0			0
$136$	3.53553	−	6.12372i	0.303170	−	0.525105i
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	−5.65685			−0.479808			−0.239904	−	0.970797i	$-0.577116\pi$
$139$	−5.65685			−0.479808			−0.239904	+	0.970797i	$0.577116\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.48528	−	14.6969i	0.709575	−	1.22902i
$144$		0			0
$145$	−1.41421	−	2.44949i	−0.117444	−	0.203419i
$146$	15.5563			1.28745
$147$		0			0
$148$	4.00000			0.328798
$149$	−10.0000	−	17.3205i	−0.819232	−	1.41895i	−0.906249	−	0.422744i	$-0.861067\pi$
$149$	−10.0000	−	17.3205i	−0.819232	−	1.41895i	0.0870170	−	0.996207i	$-0.472267\pi$
$150$		0			0
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	−0.331842	−	0.943335i	$-0.607670\pi$
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	0.982873	−	0.184284i	$-0.0589965\pi$
$152$	−2.82843	−	4.89898i	−0.229416	−	0.397360i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	9.19239	−	15.9217i	0.733632	−	1.27069i	−0.221688	−	0.975118i	$-0.571157\pi$
$157$	9.19239	−	15.9217i	0.733632	−	1.27069i	0.955321	−	0.295571i	$-0.0955099\pi$
$158$	−8.00000	+	13.8564i	−0.636446	+	1.10236i
$159$		0			0
$160$	1.41421			0.111803
$161$		0			0
$162$		0			0
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$	4.94975	−	8.57321i	0.386510	−	0.669456i
$165$		0			0
$166$	2.82843	+	4.89898i	0.219529	+	0.380235i
$167$	−11.3137			−0.875481			−0.437741	−	0.899101i	$-0.644221\pi$
$167$	−11.3137			−0.875481			−0.437741	+	0.899101i	$0.644221\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$	−5.00000	−	8.66025i	−0.383482	−	0.664211i
$171$		0			0
$172$	2.00000	−	3.46410i	0.152499	−	0.264135i
$173$	−6.36396	−	11.0227i	−0.483843	−	0.838041i	0.515985	−	0.856598i	$-0.327426\pi$
$173$	−6.36396	−	11.0227i	−0.483843	−	0.838041i	−0.999828	+	0.0185571i	$0.994093\pi$
$174$		0			0
$175$		0			0
$176$	−4.00000			−0.301511
$177$		0			0
$178$	−3.53553	+	6.12372i	−0.264999	+	0.458993i
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	−0.998286	−	0.0585225i	$-0.981361\pi$
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	0.549825	+	0.835280i	$0.314694\pi$
$180$		0			0
$181$	12.7279			0.946059			0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279			0.946059			0.473029	+	0.881047i	$0.343160\pi$
$182$		0			0
$183$		0			0
$184$	−4.00000	−	6.92820i	−0.294884	−	0.510754i
$185$	2.82843	−	4.89898i	0.207950	−	0.360180i
$186$		0			0
$187$	14.1421	+	24.4949i	1.03418	+	1.79124i
$188$	−5.65685			−0.412568
$189$		0			0
$190$	−8.00000			−0.580381
$191$	−8.00000	−	13.8564i	−0.578860	−	1.00261i	−0.995610	−	0.0935936i	$-0.970165\pi$
$191$	−8.00000	−	13.8564i	−0.578860	−	1.00261i	0.416751	−	0.909021i	$-0.363169\pi$
$192$		0			0
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	0.496119	+	0.868255i	$0.334758\pi$
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$	3.53553	+	6.12372i	0.253837	+	0.439658i
$195$		0			0
$196$		0			0
$197$	4.00000			0.284988			0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000			0.284988			0.142494	+	0.989796i	$0.454488\pi$
$198$		0			0
$199$	−8.48528	+	14.6969i	−0.601506	+	1.04184i	0.391088	+	0.920353i	$0.372099\pi$
$199$	−8.48528	+	14.6969i	−0.601506	+	1.04184i	−0.992593	+	0.121485i	$0.961234\pi$
$200$	−1.50000	+	2.59808i	−0.106066	+	0.183712i
$201$		0			0
$202$	12.7279			0.895533
$203$		0			0
$204$		0			0
$205$	−7.00000	−	12.1244i	−0.488901	−	0.846802i
$206$	2.82843	−	4.89898i	0.197066	−	0.341328i
$207$		0			0
$208$	−2.12132	−	3.67423i	−0.147087	−	0.254762i
$209$	22.6274			1.56517
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	2.00000	+	3.46410i	0.137361	+	0.237915i
$213$		0			0
$214$	2.00000	−	3.46410i	0.136717	−	0.236801i
$215$	−2.82843	−	4.89898i	−0.192897	−	0.334108i
$216$		0			0
$217$		0			0
$218$	−4.00000			−0.270914
$219$		0			0
$220$	−2.82843	+	4.89898i	−0.190693	+	0.330289i
$221$	−15.0000	+	25.9808i	−1.00901	+	1.74766i
$222$		0			0
$223$	−16.9706			−1.13643			−0.568216	−	0.822879i	$-0.692366\pi$
$223$	−16.9706			−1.13643			−0.568216	+	0.822879i	$0.692366\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.48528	−	14.6969i	0.563188	−	0.975470i	−0.434028	−	0.900899i	$-0.642908\pi$
$227$	8.48528	−	14.6969i	0.563188	−	0.975470i	0.997216	−	0.0745706i	$-0.0237586\pi$
$228$		0			0
$229$	−6.36396	−	11.0227i	−0.420542	−	0.728401i	0.575450	−	0.817837i	$-0.304827\pi$
$229$	−6.36396	−	11.0227i	−0.420542	−	0.728401i	−0.995993	+	0.0894361i	$0.971494\pi$
$230$	−11.3137			−0.746004
$231$		0			0
$232$	2.00000			0.131306
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	0.750867	−	0.660454i	$-0.229636\pi$
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	−0.947403	+	0.320043i	$0.896303\pi$
$234$		0			0
$235$	−4.00000	+	6.92820i	−0.260931	+	0.451946i
$236$	−5.65685	−	9.79796i	−0.368230	−	0.637793i
$237$		0			0
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−2.12132	+	3.67423i	−0.136646	+	0.236678i	−0.926225	−	0.376971i	$-0.876966\pi$
$241$	−2.12132	+	3.67423i	−0.136646	+	0.236678i	0.789579	+	0.613649i	$0.210299\pi$
$242$	2.50000	−	4.33013i	0.160706	−	0.278351i
$243$		0			0
$244$	1.41421			0.0905357
$245$		0			0
$246$		0			0
$247$	12.0000	+	20.7846i	0.763542	+	1.32249i
$248$		0			0
$249$		0			0
$250$	5.65685	+	9.79796i	0.357771	+	0.619677i
$251$	5.65685			0.357057			0.178529	−	0.983935i	$-0.442866\pi$
$251$	5.65685			0.357057			0.178529	+	0.983935i	$0.442866\pi$
$252$		0			0
$253$	32.0000			2.01182
$254$	−4.00000	−	6.92820i	−0.250982	−	0.434714i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−6.36396	−	11.0227i	−0.396973	−	0.687577i	0.596378	−	0.802704i	$-0.296606\pi$
$257$	−6.36396	−	11.0227i	−0.396973	−	0.687577i	−0.993351	+	0.115126i	$0.963273\pi$
$258$		0			0
$259$		0			0
$260$	−6.00000			−0.372104
$261$		0			0
$262$	8.48528	−	14.6969i	0.524222	−	0.907980i
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	−0.212565	−	0.977147i	$-0.568182\pi$
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	0.952517	−	0.304487i	$-0.0984850\pi$
$264$		0			0
$265$	5.65685			0.347498
$266$		0			0
$267$		0			0
$268$	6.00000	+	10.3923i	0.366508	+	0.634811i
$269$	−0.707107	+	1.22474i	−0.0431131	+	0.0746740i	−0.886777	−	0.462198i	$-0.847061\pi$
$269$	−0.707107	+	1.22474i	−0.0431131	+	0.0746740i	0.843664	+	0.536872i	$0.180394\pi$
$270$		0			0
$271$	2.82843	+	4.89898i	0.171815	+	0.297592i	0.939054	−	0.343769i	$-0.111704\pi$
$271$	2.82843	+	4.89898i	0.171815	+	0.297592i	−0.767240	+	0.641361i	$0.778370\pi$
$272$	7.07107			0.428746
$273$		0			0
$274$	6.00000			0.362473
$275$	−6.00000	−	10.3923i	−0.361814	−	0.626680i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$	−2.82843	−	4.89898i	−0.169638	−	0.293821i
$279$		0			0
$280$		0			0
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$		0			0
$283$	−11.3137	+	19.5959i	−0.672530	+	1.16486i	0.304654	+	0.952463i	$0.401459\pi$
$283$	−11.3137	+	19.5959i	−0.672530	+	1.16486i	−0.977184	+	0.212393i	$0.931874\pi$
$284$		0			0
$285$		0			0
$286$	16.9706			1.00349
$287$		0			0
$288$		0			0
$289$	−16.5000	−	28.5788i	−0.970588	−	1.68111i
$290$	1.41421	−	2.44949i	0.0830455	−	0.143839i
$291$		0			0
$292$	7.77817	+	13.4722i	0.455183	+	0.788400i
$293$	−24.0416			−1.40453			−0.702264	−	0.711917i	$-0.747827\pi$
$293$	−24.0416			−1.40453			−0.702264	+	0.711917i	$0.747827\pi$
$294$		0			0
$295$	−16.0000			−0.931556
$296$	2.00000	+	3.46410i	0.116248	+	0.201347i
$297$		0			0
$298$	10.0000	−	17.3205i	0.579284	−	1.00335i
$299$	16.9706	+	29.3939i	0.981433	+	1.69989i
$300$		0			0
$301$		0			0
$302$	16.0000			0.920697
$303$		0			0
$304$	2.82843	−	4.89898i	0.162221	−	0.280976i
$305$	1.00000	−	1.73205i	0.0572598	−	0.0991769i
$306$		0			0
$307$	5.65685			0.322854			0.161427	−	0.986885i	$-0.448390\pi$
$307$	5.65685			0.322854			0.161427	+	0.986885i	$0.448390\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.82843	+	4.89898i	−0.160385	+	0.277796i	−0.935007	−	0.354629i	$-0.884607\pi$
$311$	−2.82843	+	4.89898i	−0.160385	+	0.277796i	0.774622	+	0.632425i	$0.217940\pi$
$312$		0			0
$313$	−10.6066	−	18.3712i	−0.599521	−	1.03840i	−0.992892	−	0.119020i	$-0.962025\pi$
$313$	−10.6066	−	18.3712i	−0.599521	−	1.03840i	0.393371	−	0.919380i	$-0.371309\pi$
$314$	18.3848			1.03751
$315$		0			0
$316$	−16.0000			−0.900070
$317$	14.0000	+	24.2487i	0.786318	+	1.36194i	0.928208	+	0.372061i	$0.121349\pi$
$317$	14.0000	+	24.2487i	0.786318	+	1.36194i	−0.141890	+	0.989882i	$0.545318\pi$
$318$		0			0
$319$	−4.00000	+	6.92820i	−0.223957	+	0.387905i
$320$	0.707107	+	1.22474i	0.0395285	+	0.0684653i
$321$		0			0
$322$		0			0
$323$	−40.0000			−2.22566
$324$		0			0
$325$	6.36396	−	11.0227i	0.353009	−	0.611430i
$326$	2.00000	−	3.46410i	0.110770	−	0.191859i
$327$		0			0
$328$	9.89949			0.546608
$329$		0			0
$330$		0			0
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	0.998298	+	0.0583130i	$0.0185721\pi$
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	−0.448649	+	0.893708i	$0.648095\pi$
$332$	−2.82843	+	4.89898i	−0.155230	+	0.268866i
$333$		0			0
$334$	−5.65685	−	9.79796i	−0.309529	−	0.536120i
$335$	16.9706			0.927201
$336$		0			0
$337$	−16.0000			−0.871576			−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000			−0.871576			−0.435788	+	0.900049i	$0.643530\pi$
$338$	2.50000	+	4.33013i	0.135982	+	0.235528i
$339$		0			0
$340$	5.00000	−	8.66025i	0.271163	−	0.469668i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	4.00000			0.215666
$345$		0			0
$346$	6.36396	−	11.0227i	0.342129	−	0.592584i
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	−29.6985			−1.58972			−0.794862	−	0.606791i	$-0.792457\pi$
$349$	−29.6985			−1.58972			−0.794862	+	0.606791i	$0.792457\pi$
$350$		0			0
$351$		0			0
$352$	−2.00000	−	3.46410i	−0.106600	−	0.184637i
$353$	0.707107	−	1.22474i	0.0376355	−	0.0651866i	−0.846594	−	0.532239i	$-0.821351\pi$
$353$	0.707107	−	1.22474i	0.0376355	−	0.0651866i	0.884230	+	0.467052i	$0.154684\pi$
$354$		0			0
$355$		0			0
$356$	−7.07107			−0.374766
$357$		0			0
$358$	−12.0000			−0.634220
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	0.996157	−	0.0875892i	$-0.0279163\pi$
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	−0.573933	+	0.818902i	$0.694583\pi$
$360$		0			0
$361$	−6.50000	+	11.2583i	−0.342105	+	0.592544i
$362$	6.36396	+	11.0227i	0.334482	+	0.579340i
$363$		0			0
$364$		0			0
$365$	22.0000			1.15153
$366$		0			0
$367$	−2.82843	+	4.89898i	−0.147643	+	0.255725i	−0.930356	−	0.366658i	$-0.880502\pi$
$367$	−2.82843	+	4.89898i	−0.147643	+	0.255725i	0.782713	+	0.622383i	$0.213835\pi$
$368$	4.00000	−	6.92820i	0.208514	−	0.361158i
$369$		0			0
$370$	5.65685			0.294086
$371$		0			0
$372$		0			0
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	0.707055	−	0.707159i	$-0.250023\pi$
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	−0.965945	+	0.258748i	$0.916690\pi$
$374$	−14.1421	+	24.4949i	−0.731272	+	1.26660i
$375$		0			0
$376$	−2.82843	−	4.89898i	−0.145865	−	0.252646i
$377$	−8.48528			−0.437014
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$	−4.00000	−	6.92820i	−0.205196	−	0.355409i
$381$		0			0
$382$	8.00000	−	13.8564i	0.409316	−	0.708955i
$383$	−2.82843	−	4.89898i	−0.144526	−	0.250326i	0.784670	−	0.619914i	$-0.212832\pi$
$383$	−2.82843	−	4.89898i	−0.144526	−	0.250326i	−0.929196	+	0.369587i	$0.879499\pi$
$384$		0			0
$385$		0			0
$386$	−14.0000			−0.712581
$387$		0			0
$388$	−3.53553	+	6.12372i	−0.179490	+	0.310885i
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	−0.321716	−	0.946836i	$-0.604260\pi$
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	0.980842	−	0.194804i	$-0.0624070\pi$
$390$		0			0
$391$	−56.5685			−2.86079
$392$		0			0
$393$		0			0
$394$	2.00000	+	3.46410i	0.100759	+	0.174519i
$395$	−11.3137	+	19.5959i	−0.569254	+	0.985978i
$396$		0			0
$397$	−3.53553	−	6.12372i	−0.177443	−	0.307341i	0.763561	−	0.645736i	$-0.223449\pi$
$397$	−3.53553	−	6.12372i	−0.177443	−	0.307341i	−0.941004	+	0.338395i	$0.890116\pi$
$398$	−16.9706			−0.850657
$399$		0			0
$400$	−3.00000			−0.150000
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$		0			0
$404$	6.36396	+	11.0227i	0.316619	+	0.548400i
$405$		0			0
$406$		0			0
$407$	−16.0000			−0.793091
$408$		0			0
$409$	10.6066	−	18.3712i	0.524463	−	0.908396i	−0.475132	−	0.879915i	$-0.657600\pi$
$409$	10.6066	−	18.3712i	0.524463	−	0.908396i	0.999594	−	0.0284813i	$-0.00906711\pi$
$410$	7.00000	−	12.1244i	0.345705	−	0.598779i
$411$		0			0
$412$	5.65685			0.278693
$413$		0			0
$414$		0			0
$415$	4.00000	+	6.92820i	0.196352	+	0.340092i
$416$	2.12132	−	3.67423i	0.104006	−	0.180144i
$417$		0			0
$418$	11.3137	+	19.5959i	0.553372	+	0.958468i
$419$	−22.6274			−1.10542			−0.552711	−	0.833373i	$-0.686407\pi$
$419$	−22.6274			−1.10542			−0.552711	+	0.833373i	$0.686407\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$	6.00000	+	10.3923i	0.292075	+	0.505889i
$423$		0			0
$424$	−2.00000	+	3.46410i	−0.0971286	+	0.168232i
$425$	10.6066	+	18.3712i	0.514496	+	0.891133i
$426$		0			0
$427$		0			0
$428$	4.00000			0.193347
$429$		0			0
$430$	2.82843	−	4.89898i	0.136399	−	0.236250i
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	−0.417687	−	0.908591i	$-0.637159\pi$
$431$	12.0000	−	20.7846i	0.578020	−	1.00116i	0.995706	−	0.0925683i	$-0.0295076\pi$
$432$		0			0
$433$	−4.24264			−0.203888			−0.101944	−	0.994790i	$-0.532506\pi$
$433$	−4.24264			−0.203888			−0.101944	+	0.994790i	$0.532506\pi$
$434$		0			0
$435$		0			0
$436$	−2.00000	−	3.46410i	−0.0957826	−	0.165900i
$437$	−22.6274	+	39.1918i	−1.08242	+	1.87480i
$438$		0			0
$439$	−16.9706	−	29.3939i	−0.809961	−	1.40289i	−0.912890	−	0.408205i	$-0.866155\pi$
$439$	−16.9706	−	29.3939i	−0.809961	−	1.40289i	0.102930	−	0.994689i	$-0.467178\pi$
$440$	−5.65685			−0.269680
$441$		0			0
$442$	−30.0000			−1.42695
$443$	10.0000	+	17.3205i	0.475114	+	0.822922i	0.999594	−	0.0285009i	$-0.00907336\pi$
$443$	10.0000	+	17.3205i	0.475114	+	0.822922i	−0.524479	+	0.851423i	$0.675740\pi$
$444$		0			0
$445$	−5.00000	+	8.66025i	−0.237023	+	0.410535i
$446$	−8.48528	−	14.6969i	−0.401790	−	0.695920i
$447$		0			0
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$	−19.7990	+	34.2929i	−0.932298	+	1.61479i
$452$		0			0
$453$		0			0
$454$	16.9706			0.796468
$455$		0			0
$456$		0			0
$457$	3.00000	+	5.19615i	0.140334	+	0.243066i	0.927622	−	0.373519i	$-0.121849\pi$
$457$	3.00000	+	5.19615i	0.140334	+	0.243066i	−0.787288	+	0.616585i	$0.788516\pi$
$458$	6.36396	−	11.0227i	0.297368	−	0.515057i
$459$		0			0
$460$	−5.65685	−	9.79796i	−0.263752	−	0.456832i
$461$	1.41421			0.0658665			0.0329332	−	0.999458i	$-0.489515\pi$
$461$	1.41421			0.0658665			0.0329332	+	0.999458i	$0.489515\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$	1.00000	+	1.73205i	0.0464238	+	0.0804084i
$465$		0			0
$466$	3.00000	−	5.19615i	0.138972	−	0.240707i
$467$	2.82843	+	4.89898i	0.130884	+	0.226698i	0.924018	−	0.382350i	$-0.124885\pi$
$467$	2.82843	+	4.89898i	0.130884	+	0.226698i	−0.793134	+	0.609048i	$0.791552\pi$
$468$		0			0
$469$		0			0
$470$	−8.00000			−0.369012
$471$		0			0
$472$	5.65685	−	9.79796i	0.260378	−	0.450988i
$473$	−8.00000	+	13.8564i	−0.367840	+	0.637118i
$474$		0			0
$475$	16.9706			0.778663
$476$		0			0
$477$		0			0
$478$	12.0000	+	20.7846i	0.548867	+	0.950666i
$479$	−14.1421	+	24.4949i	−0.646171	+	1.11920i	0.337859	+	0.941197i	$0.390297\pi$
$479$	−14.1421	+	24.4949i	−0.646171	+	1.11920i	−0.984030	+	0.178004i	$0.943036\pi$
$480$		0			0
$481$	−8.48528	−	14.6969i	−0.386896	−	0.670123i
$482$	−4.24264			−0.193247
$483$		0			0
$484$	5.00000			0.227273
$485$	5.00000	+	8.66025i	0.227038	+	0.393242i
$486$		0			0
$487$	12.0000	−	20.7846i	0.543772	−	0.941841i	−0.454911	−	0.890537i	$-0.650329\pi$
$487$	12.0000	−	20.7846i	0.543772	−	0.941841i	0.998683	−	0.0513038i	$-0.0163377\pi$
$488$	0.707107	+	1.22474i	0.0320092	+	0.0554416i
$489$		0			0
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	7.07107	−	12.2474i	0.318465	−	0.551597i
$494$	−12.0000	+	20.7846i	−0.539906	+	0.935144i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	0.907314	−	0.420455i	$-0.138129\pi$
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	−0.817781	+	0.575529i	$0.804796\pi$
$500$	−5.65685	+	9.79796i	−0.252982	+	0.438178i
$501$		0			0
$502$	2.82843	+	4.89898i	0.126239	+	0.218652i
$503$	−28.2843			−1.26113			−0.630567	−	0.776135i	$-0.717177\pi$
$503$	−28.2843			−1.26113			−0.630567	+	0.776135i	$0.717177\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$	16.0000	+	27.7128i	0.711287	+	1.23198i
$507$		0			0
$508$	4.00000	−	6.92820i	0.177471	−	0.307389i
$509$	16.2635	+	28.1691i	0.720865	+	1.24857i	0.960653	+	0.277750i	$0.0895886\pi$
$509$	16.2635	+	28.1691i	0.720865	+	1.24857i	−0.239788	+	0.970825i	$0.577078\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	6.36396	−	11.0227i	0.280702	−	0.486191i
$515$	4.00000	−	6.92820i	0.176261	−	0.305293i
$516$		0			0
$517$	22.6274			0.995153
$518$		0			0
$519$		0			0
$520$	−3.00000	−	5.19615i	−0.131559	−	0.227866i
$521$	0.707107	−	1.22474i	0.0309789	−	0.0536570i	−0.850120	−	0.526589i	$-0.823471\pi$
$521$	0.707107	−	1.22474i	0.0309789	−	0.0536570i	0.881099	+	0.472931i	$0.156804\pi$
$522$		0			0
$523$	16.9706	+	29.3939i	0.742071	+	1.28530i	0.951551	+	0.307492i	$0.0994896\pi$
$523$	16.9706	+	29.3939i	0.742071	+	1.28530i	−0.209480	+	0.977813i	$0.567177\pi$
$524$	16.9706			0.741362
$525$		0			0
$526$	24.0000			1.04645
$527$		0			0
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$	2.82843	+	4.89898i	0.122859	+	0.212798i
$531$		0			0
$532$		0			0
$533$	−42.0000			−1.81922
$534$		0			0
$535$	2.82843	−	4.89898i	0.122284	−	0.211801i
$536$	−6.00000	+	10.3923i	−0.259161	+	0.448879i
$537$		0			0
$538$	−1.41421			−0.0609711
$539$		0			0
$540$		0			0
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	0.886721	−	0.462304i	$-0.152977\pi$
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	−0.843728	+	0.536771i	$0.819644\pi$
$542$	−2.82843	+	4.89898i	−0.121491	+	0.210429i
$543$		0			0
$544$	3.53553	+	6.12372i	0.151585	+	0.262553i
$545$	−5.65685			−0.242313
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$	3.00000	+	5.19615i	0.128154	+	0.221969i
$549$		0			0
$550$	6.00000	−	10.3923i	0.255841	−	0.443129i
$551$	−5.65685	−	9.79796i	−0.240990	−	0.417407i
$552$		0			0
$553$		0			0
$554$	−10.0000			−0.424859
$555$		0			0
$556$	2.82843	−	4.89898i	0.119952	−	0.207763i
$557$	18.0000	−	31.1769i	0.762684	−	1.32101i	−0.178778	−	0.983890i	$-0.557214\pi$
$557$	18.0000	−	31.1769i	0.762684	−	1.32101i	0.941462	−	0.337119i	$-0.109452\pi$
$558$		0			0
$559$	−16.9706			−0.717778
$560$		0			0
$561$		0			0
$562$	−5.00000	−	8.66025i	−0.210912	−	0.365311i
$563$	−5.65685	+	9.79796i	−0.238408	+	0.412935i	−0.960258	−	0.279115i	$-0.909959\pi$
$563$	−5.65685	+	9.79796i	−0.238408	+	0.412935i	0.721850	+	0.692050i	$0.243292\pi$
$564$		0			0
$565$		0			0
$566$	−22.6274			−0.951101
$567$		0			0
$568$		0			0
$569$	−13.0000	−	22.5167i	−0.544988	−	0.943948i	−0.998608	−	0.0527519i	$-0.983201\pi$
$569$	−13.0000	−	22.5167i	−0.544988	−	0.943948i	0.453619	−	0.891196i	$-0.350133\pi$
$570$		0			0
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	−0.904835	−	0.425762i	$-0.860006\pi$
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	0.821138	+	0.570730i	$0.193340\pi$
$572$	8.48528	+	14.6969i	0.354787	+	0.614510i
$573$		0			0
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$	−6.36396	+	11.0227i	−0.264935	+	0.458881i	−0.967547	−	0.252693i	$-0.918684\pi$
$577$	−6.36396	+	11.0227i	−0.264935	+	0.458881i	0.702611	+	0.711574i	$0.252017\pi$
$578$	16.5000	−	28.5788i	0.686310	−	1.18872i
$579$		0			0
$580$	2.82843			0.117444
$581$		0			0
$582$		0			0
$583$	−8.00000	−	13.8564i	−0.331326	−	0.573874i
$584$	−7.77817	+	13.4722i	−0.321863	+	0.557483i
$585$		0			0
$586$	−12.0208	−	20.8207i	−0.496575	−	0.860094i
$587$	16.9706			0.700450			0.350225	−	0.936666i	$-0.386105\pi$
$587$	16.9706			0.700450			0.350225	+	0.936666i	$0.386105\pi$
$588$		0			0
$589$		0			0
$590$	−8.00000	−	13.8564i	−0.329355	−	0.570459i
$591$		0			0
$592$	−2.00000	+	3.46410i	−0.0821995	+	0.142374i
$593$	−17.6777	−	30.6186i	−0.725935	−	1.25736i	−0.958588	−	0.284796i	$-0.908074\pi$
$593$	−17.6777	−	30.6186i	−0.725935	−	1.25736i	0.232653	−	0.972560i	$-0.425259\pi$
$594$		0			0
$595$		0			0
$596$	20.0000			0.819232
$597$		0			0
$598$	−16.9706	+	29.3939i	−0.693978	+	1.20201i
$599$	−8.00000	+	13.8564i	−0.326871	+	0.566157i	−0.981889	−	0.189456i	$-0.939328\pi$
$599$	−8.00000	+	13.8564i	−0.326871	+	0.566157i	0.655018	+	0.755613i	$0.272661\pi$
$600$		0			0
$601$	4.24264			0.173061			0.0865305	−	0.996249i	$-0.472422\pi$
$601$	4.24264			0.173061			0.0865305	+	0.996249i	$0.472422\pi$
$602$		0			0
$603$		0			0
$604$	8.00000	+	13.8564i	0.325515	+	0.563809i
$605$	3.53553	−	6.12372i	0.143740	−	0.248965i
$606$		0			0
$607$	−8.48528	−	14.6969i	−0.344407	−	0.596530i	0.640839	−	0.767675i	$-0.278587\pi$
$607$	−8.48528	−	14.6969i	−0.344407	−	0.596530i	−0.985246	+	0.171145i	$0.945253\pi$
$608$	5.65685			0.229416
$609$		0			0
$610$	2.00000			0.0809776
$611$	12.0000	+	20.7846i	0.485468	+	0.840855i
$612$		0			0
$613$	−6.00000	+	10.3923i	−0.242338	+	0.419741i	−0.961380	−	0.275225i	$-0.911248\pi$
$613$	−6.00000	+	10.3923i	−0.242338	+	0.419741i	0.719042	+	0.694967i	$0.244581\pi$
$614$	2.82843	+	4.89898i	0.114146	+	0.197707i
$615$		0			0
$616$		0			0
$617$	−22.0000			−0.885687			−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000			−0.885687			−0.442843	+	0.896599i	$0.646030\pi$
$618$		0			0
$619$	−5.65685	+	9.79796i	−0.227368	+	0.393813i	−0.957027	−	0.289998i	$-0.906345\pi$
$619$	−5.65685	+	9.79796i	−0.227368	+	0.393813i	0.729659	+	0.683811i	$0.239679\pi$
$620$		0			0
$621$		0			0
$622$	−5.65685			−0.226819
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.0200000	+	0.0346410i
$626$	10.6066	−	18.3712i	0.423925	−	0.734260i
$627$		0			0
$628$	9.19239	+	15.9217i	0.366816	+	0.635344i
$629$	28.2843			1.12777
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	−8.00000	−	13.8564i	−0.318223	−	0.551178i
$633$		0			0
$634$	−14.0000	+	24.2487i	−0.556011	+	0.963039i
$635$	−5.65685	−	9.79796i	−0.224485	−	0.388820i
$636$		0			0
$637$		0			0
$638$	−8.00000			−0.316723
$639$		0			0
$640$	−0.707107	+	1.22474i	−0.0279508	+	0.0484123i
$641$	1.00000	−	1.73205i	0.0394976	−	0.0684119i	−0.845601	−	0.533816i	$-0.820758\pi$
$641$	1.00000	−	1.73205i	0.0394976	−	0.0684119i	0.885098	+	0.465404i	$0.154091\pi$
$642$		0			0
$643$	11.3137			0.446169			0.223085	−	0.974799i	$-0.428387\pi$
$643$	11.3137			0.446169			0.223085	+	0.974799i	$0.428387\pi$
$644$		0			0
$645$		0			0
$646$	−20.0000	−	34.6410i	−0.786889	−	1.36293i
$647$	16.9706	−	29.3939i	0.667182	−	1.15559i	−0.311507	−	0.950244i	$-0.600834\pi$
$647$	16.9706	−	29.3939i	0.667182	−	1.15559i	0.978689	−	0.205349i	$-0.0658329\pi$
$648$		0			0
$649$	22.6274	+	39.1918i	0.888204	+	1.53841i
$650$	12.7279			0.499230
$651$		0			0
$652$	4.00000			0.156652
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$	12.0000	−	20.7846i	0.468879	−	0.812122i
$656$	4.94975	+	8.57321i	0.193255	+	0.334728i
$657$		0			0
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	−2.12132	+	3.67423i	−0.0825098	+	0.142911i	−0.904327	−	0.426840i	$-0.859627\pi$
$661$	−2.12132	+	3.67423i	−0.0825098	+	0.142911i	0.821818	+	0.569751i	$0.192960\pi$
$662$	−10.0000	+	17.3205i	−0.388661	+	0.673181i
$663$		0			0
$664$	−5.65685			−0.219529
$665$		0			0
$666$		0			0
$667$	−8.00000	−	13.8564i	−0.309761	−	0.536522i
$668$	5.65685	−	9.79796i	0.218870	−	0.379094i
$669$		0			0
$670$	8.48528	+	14.6969i	0.327815	+	0.567792i
$671$	−5.65685			−0.218380
$672$		0			0
$673$	24.0000			0.925132			0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000			0.925132			0.462566	+	0.886585i	$0.346929\pi$
$674$	−8.00000	−	13.8564i	−0.308148	−	0.533729i
$675$		0			0
$676$	−2.50000	+	4.33013i	−0.0961538	+	0.166543i
$677$	2.12132	+	3.67423i	0.0815290	+	0.141212i	0.903907	−	0.427729i	$-0.140686\pi$
$677$	2.12132	+	3.67423i	0.0815290	+	0.141212i	−0.822378	+	0.568942i	$0.807353\pi$
$678$		0			0
$679$		0			0
$680$	10.0000			0.383482
$681$		0			0
$682$		0			0
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	−0.957685	−	0.287819i	$-0.907070\pi$
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	0.728101	+	0.685470i	$0.240403\pi$
$684$		0			0
$685$	8.48528			0.324206
$686$		0			0
$687$		0			0
$688$	2.00000	+	3.46410i	0.0762493	+	0.132068i
$689$	8.48528	−	14.6969i	0.323263	−	0.559909i
$690$		0			0
$691$	25.4558	+	44.0908i	0.968386	+	1.67729i	0.700229	+	0.713918i	$0.253081\pi$
$691$	25.4558	+	44.0908i	0.968386	+	1.67729i	0.268157	+	0.963375i	$0.413585\pi$
$692$	12.7279			0.483843
$693$		0			0
$694$	−12.0000			−0.455514
$695$	−4.00000	−	6.92820i	−0.151729	−	0.262802i
$696$		0			0
$697$	35.0000	−	60.6218i	1.32572	−	2.29621i
$698$	−14.8492	−	25.7196i	−0.562052	−	0.973503i
$699$		0			0
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	11.3137	−	19.5959i	0.426705	−	0.739074i
$704$	2.00000	−	3.46410i	0.0753778	−	0.130558i
$705$		0			0
$706$	1.41421			0.0532246
$707$		0			0
$708$		0			0
$709$	−14.0000	−	24.2487i	−0.525781	−	0.910679i	−0.999549	−	0.0300298i	$-0.990440\pi$
$709$	−14.0000	−	24.2487i	−0.525781	−	0.910679i	0.473768	−	0.880650i	$-0.342894\pi$
$710$		0			0
$711$		0			0
$712$	−3.53553	−	6.12372i	−0.132500	−	0.229496i
$713$		0			0
$714$		0			0
$715$	24.0000			0.897549
$716$	−6.00000	−	10.3923i	−0.224231	−	0.388379i
$717$		0			0
$718$	−8.00000	+	13.8564i	−0.298557	+	0.517116i
$719$	19.7990	+	34.2929i	0.738378	+	1.27891i	0.953225	+	0.302260i	$0.0977411\pi$
$719$	19.7990	+	34.2929i	0.738378	+	1.27891i	−0.214848	+	0.976648i	$0.568926\pi$
$720$		0			0
$721$		0			0
$722$	−13.0000			−0.483810
$723$		0			0
$724$	−6.36396	+	11.0227i	−0.236515	+	0.409656i
$725$	−3.00000	+	5.19615i	−0.111417	+	0.192980i
$726$		0			0
$727$	28.2843			1.04901			0.524503	−	0.851409i	$-0.324251\pi$
$727$	28.2843			1.04901			0.524503	+	0.851409i	$0.324251\pi$
$728$		0			0
$729$		0			0
$730$	11.0000	+	19.0526i	0.407128	+	0.705167i
$731$	14.1421	−	24.4949i	0.523066	−	0.905977i
$732$		0			0
$733$	−6.36396	−	11.0227i	−0.235058	−	0.407133i	0.724231	−	0.689557i	$-0.242195\pi$
$733$	−6.36396	−	11.0227i	−0.235058	−	0.407133i	−0.959290	+	0.282424i	$0.908861\pi$
$734$	−5.65685			−0.208798
$735$		0			0
$736$	8.00000			0.294884
$737$	−24.0000	−	41.5692i	−0.884051	−	1.53122i
$738$		0			0
$739$	−6.00000	+	10.3923i	−0.220714	+	0.382287i	−0.955025	−	0.296526i	$-0.904172\pi$
$739$	−6.00000	+	10.3923i	−0.220714	+	0.382287i	0.734311	+	0.678813i	$0.237505\pi$
$740$	2.82843	+	4.89898i	0.103975	+	0.180090i
$741$		0			0
$742$		0			0
$743$	−16.0000			−0.586983			−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000			−0.586983			−0.293492	+	0.955962i	$0.594817\pi$
$744$		0			0
$745$	14.1421	−	24.4949i	0.518128	−	0.897424i
$746$	5.00000	−	8.66025i	0.183063	−	0.317074i
$747$		0			0
$748$	−28.2843			−1.03418
$749$		0			0
$750$		0			0
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	0.956963	+	0.290209i	$0.0937250\pi$
$751$	20.0000	+	34.6410i	0.729810	+	1.26407i	−0.227153	+	0.973859i	$0.572942\pi$
$752$	2.82843	−	4.89898i	0.103142	−	0.178647i
$753$		0			0
$754$	−4.24264	−	7.34847i	−0.154508	−	0.267615i
$755$	22.6274			0.823496
$756$		0			0
$757$	−28.0000			−1.01768			−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000			−1.01768			−0.508839	+	0.860862i	$0.669925\pi$
$758$	14.0000	+	24.2487i	0.508503	+	0.880753i
$759$		0			0
$760$	4.00000	−	6.92820i	0.145095	−	0.251312i
$761$	−4.94975	−	8.57321i	−0.179428	−	0.310779i	0.762257	−	0.647275i	$-0.224091\pi$
$761$	−4.94975	−	8.57321i	−0.179428	−	0.310779i	−0.941685	+	0.336496i	$0.890758\pi$
$762$		0			0
$763$		0			0
$764$	16.0000			0.578860
$765$		0			0
$766$	2.82843	−	4.89898i	0.102195	−	0.177007i
$767$	−24.0000	+	41.5692i	−0.866590	+	1.50098i
$768$		0			0
$769$	−4.24264			−0.152994			−0.0764968	−	0.997070i	$-0.524373\pi$
$769$	−4.24264			−0.152994			−0.0764968	+	0.997070i	$0.524373\pi$
$770$		0			0
$771$		0			0
$772$	−7.00000	−	12.1244i	−0.251936	−	0.436365i
$773$	16.2635	−	28.1691i	0.584956	−	1.01317i	−0.409925	−	0.912119i	$-0.634445\pi$
$773$	16.2635	−	28.1691i	0.584956	−	1.01317i	0.994881	−	0.101054i	$-0.0322215\pi$
$774$		0			0
$775$		0			0
$776$	−7.07107			−0.253837
$777$		0			0
$778$	26.0000			0.932145
$779$	−28.0000	−	48.4974i	−1.00320	−	1.73760i
$780$		0			0
$781$		0			0
$782$	−28.2843	−	48.9898i	−1.01144	−	1.75187i
$783$		0			0
$784$		0			0
$785$	26.0000			0.927980
$786$		0			0
$787$	−2.82843	+	4.89898i	−0.100823	+	0.174630i	−0.912024	−	0.410137i	$-0.865481\pi$
$787$	−2.82843	+	4.89898i	−0.100823	+	0.174630i	0.811201	+	0.584767i	$0.198814\pi$
$788$	−2.00000	+	3.46410i	−0.0712470	+	0.123404i
$789$		0			0
$790$	−22.6274			−0.805047
$791$		0			0
$792$		0			0
$793$	−3.00000	−	5.19615i	−0.106533	−	0.184521i
$794$	3.53553	−	6.12372i	0.125471	−	0.217323i
$795$		0			0
$796$	−8.48528	−	14.6969i	−0.300753	−	0.520919i

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	882.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.707107 + 1.22474i) q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.707107 + 1.22474i) q^{5} -1.00000 q^{8} +(-0.707107 + 1.22474i) q^{10} +(2.00000 - 3.46410i) q^{11} +4.24264 q^{13} +(-0.500000 - 0.866025i) q^{16} +(-3.53553 + 6.12372i) q^{17} +(2.82843 + 4.89898i) q^{19} -1.41421 q^{20} +4.00000 q^{22} +(4.00000 + 6.92820i) q^{23} +(1.50000 - 2.59808i) q^{25} +(2.12132 + 3.67423i) q^{26} -2.00000 q^{29} +(0.500000 - 0.866025i) q^{32} -7.07107 q^{34} +(-2.00000 - 3.46410i) q^{37} +(-2.82843 + 4.89898i) q^{38} +(-0.707107 - 1.22474i) q^{40} -9.89949 q^{41} -4.00000 q^{43} +(2.00000 + 3.46410i) q^{44} +(-4.00000 + 6.92820i) q^{46} +(2.82843 + 4.89898i) q^{47} +3.00000 q^{50} +(-2.12132 + 3.67423i) q^{52} +(2.00000 - 3.46410i) q^{53} +5.65685 q^{55} +(-1.00000 - 1.73205i) q^{58} +(-5.65685 + 9.79796i) q^{59} +(-0.707107 - 1.22474i) q^{61} +1.00000 q^{64} +(3.00000 + 5.19615i) q^{65} +(6.00000 - 10.3923i) q^{67} +(-3.53553 - 6.12372i) q^{68} +(7.77817 - 13.4722i) q^{73} +(2.00000 - 3.46410i) q^{74} -5.65685 q^{76} +(8.00000 + 13.8564i) q^{79} +(0.707107 - 1.22474i) q^{80} +(-4.94975 - 8.57321i) q^{82} +5.65685 q^{83} -10.0000 q^{85} +(-2.00000 - 3.46410i) q^{86} +(-2.00000 + 3.46410i) q^{88} +(3.53553 + 6.12372i) q^{89} -8.00000 q^{92} +(-2.82843 + 4.89898i) q^{94} +(-4.00000 + 6.92820i) q^{95} +7.07107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q + 2q^{2} - 2q^{4} - 4q^{8} + O(q^{10}) \)	\( 4q + 2q^{2} - 2q^{4} - 4q^{8} + 8q^{11} - 2q^{16} + 16q^{22} + 16q^{23} + 6q^{25} - 8q^{29} + 2q^{32} - 8q^{37} - 16q^{43} + 8q^{44} - 16q^{46} + 12q^{50} + 8q^{53} - 4q^{58} + 4q^{64} + 12q^{65} + 24q^{67} + 8q^{74} + 32q^{79} - 40q^{85} - 8q^{86} - 8q^{88} - 32q^{92} - 16q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more