LMFDB - Embedded newform 882.4.g.c.667.1

Newspace parameters

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.g (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$52.0396846251$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 294)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			667.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	882.667
Dual form			882.4.g.c.361.1

$q$-expansion

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.00000 + 6.92820i) q^{5} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.00000 + 6.92820i) q^{5} +8.00000 q^{8} +(-8.00000 - 13.8564i) q^{10} +(20.0000 + 34.6410i) q^{11} -4.00000 q^{13} +(-8.00000 + 13.8564i) q^{16} +(42.0000 + 72.7461i) q^{17} +(74.0000 - 128.172i) q^{19} +32.0000 q^{20} -80.0000 q^{22} +(42.0000 - 72.7461i) q^{23} +(30.5000 + 52.8275i) q^{25} +(4.00000 - 6.92820i) q^{26} -58.0000 q^{29} +(-68.0000 - 117.779i) q^{31} +(-16.0000 - 27.7128i) q^{32} -168.000 q^{34} +(111.000 - 192.258i) q^{37} +(148.000 + 256.344i) q^{38} +(-32.0000 + 55.4256i) q^{40} +420.000 q^{41} -164.000 q^{43} +(80.0000 - 138.564i) q^{44} +(84.0000 + 145.492i) q^{46} +(-244.000 + 422.620i) q^{47} -122.000 q^{50} +(8.00000 + 13.8564i) q^{52} +(239.000 + 413.960i) q^{53} -320.000 q^{55} +(58.0000 - 100.459i) q^{58} +(-274.000 - 474.582i) q^{59} +(346.000 - 599.290i) q^{61} +272.000 q^{62} +64.0000 q^{64} +(16.0000 - 27.7128i) q^{65} +(454.000 + 786.351i) q^{67} +(168.000 - 290.985i) q^{68} +524.000 q^{71} +(220.000 + 381.051i) q^{73} +(222.000 + 384.515i) q^{74} -592.000 q^{76} +(-608.000 + 1053.09i) q^{79} +(-64.0000 - 110.851i) q^{80} +(-420.000 + 727.461i) q^{82} -684.000 q^{83} -672.000 q^{85} +(164.000 - 284.056i) q^{86} +(160.000 + 277.128i) q^{88} +(-302.000 + 523.079i) q^{89} -336.000 q^{92} +(-488.000 - 845.241i) q^{94} +(592.000 + 1025.37i) q^{95} +832.000 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q - 2q^{2} - 4q^{4} - 8q^{5} + 16q^{8} + O(q^{10}) $	$ 2q - 2q^{2} - 4q^{4} - 8q^{5} + 16q^{8} - 16q^{10} + 40q^{11} - 8q^{13} - 16q^{16} + 84q^{17} + 148q^{19} + 64q^{20} - 160q^{22} + 84q^{23} + 61q^{25} + 8q^{26} - 116q^{29} - 136q^{31} - 32q^{32} - 336q^{34} + 222q^{37} + 296q^{38} - 64q^{40} + 840q^{41} - 328q^{43} + 160q^{44} + 168q^{46} - 488q^{47} - 244q^{50} + 16q^{52} + 478q^{53} - 640q^{55} + 116q^{58} - 548q^{59} + 692q^{61} + 544q^{62} + 128q^{64} + 32q^{65} + 908q^{67} + 336q^{68} + 1048q^{71} + 440q^{73} + 444q^{74} - 1184q^{76} - 1216q^{79} - 128q^{80} - 840q^{82} - 1368q^{83} - 1344q^{85} + 328q^{86} + 320q^{88} - 604q^{89} - 672q^{92} - 976q^{94} + 1184q^{95} + 1664q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	−4.00000	+	6.92820i	−0.357771	+	0.619677i	−0.987588	−	0.157066i	$-0.949796\pi$
$5$	−4.00000	+	6.92820i	−0.357771	+	0.619677i	0.629817	+	0.776743i	$0.283130\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	8.00000			0.353553
$9$		0			0
$10$	−8.00000	−	13.8564i	−0.252982	−	0.438178i
$11$	20.0000	+	34.6410i	0.548202	+	0.949514i	0.998398	+	0.0565844i	$0.0180210\pi$
$11$	20.0000	+	34.6410i	0.548202	+	0.949514i	−0.450195	+	0.892930i	$0.648646\pi$
$12$		0			0
$13$	−4.00000			−0.0853385			−0.0426692	−	0.999089i	$-0.513586\pi$
$13$	−4.00000			−0.0853385			−0.0426692	+	0.999089i	$0.513586\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	42.0000	+	72.7461i	0.599206	+	1.03785i	0.992939	+	0.118630i	$0.0378502\pi$
$17$	42.0000	+	72.7461i	0.599206	+	1.03785i	−0.393733	+	0.919225i	$0.628817\pi$
$18$		0			0
$19$	74.0000	−	128.172i	0.893514	−	1.54761i	0.0578808	−	0.998324i	$-0.481566\pi$
$19$	74.0000	−	128.172i	0.893514	−	1.54761i	0.835633	−	0.549288i	$-0.185101\pi$
$20$	32.0000			0.357771
$21$		0			0
$22$	−80.0000			−0.775275
$23$	42.0000	−	72.7461i	0.380765	−	0.659505i	−0.610406	−	0.792088i	$-0.708994\pi$
$23$	42.0000	−	72.7461i	0.380765	−	0.659505i	0.991172	+	0.132583i	$0.0423272\pi$
$24$		0			0
$25$	30.5000	+	52.8275i	0.244000	+	0.422620i
$26$	4.00000	−	6.92820i	0.0301717	−	0.0522589i
$27$		0			0
$28$		0			0
$29$	−58.0000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−58.0000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−68.0000	−	117.779i	−0.393973	−	0.682381i	0.598997	−	0.800752i	$-0.295566\pi$
$31$	−68.0000	−	117.779i	−0.393973	−	0.682381i	−0.992970	+	0.118370i	$0.962233\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−168.000			−0.847405
$35$		0			0
$36$		0			0
$37$	111.000	−	192.258i	0.493197	−	0.854242i	−0.506772	−	0.862080i	$-0.669162\pi$
$37$	111.000	−	192.258i	0.493197	−	0.854242i	0.999969	+	0.00783774i	$0.00249486\pi$
$38$	148.000	+	256.344i	0.631810	+	1.09433i
$39$		0			0
$40$	−32.0000	+	55.4256i	−0.126491	+	0.219089i
$41$	420.000			1.59983			0.799914	−	0.600114i	$-0.204878\pi$
$41$	420.000			1.59983			0.799914	+	0.600114i	$0.204878\pi$
$42$		0			0
$43$	−164.000			−0.581622			−0.290811	−	0.956780i	$-0.593925\pi$
$43$	−164.000			−0.581622			−0.290811	+	0.956780i	$0.593925\pi$
$44$	80.0000	−	138.564i	0.274101	−	0.474757i
$45$		0			0
$46$	84.0000	+	145.492i	0.269242	+	0.466341i
$47$	−244.000	+	422.620i	−0.757257	+	1.31161i	0.186988	+	0.982362i	$0.440127\pi$
$47$	−244.000	+	422.620i	−0.757257	+	1.31161i	−0.944245	+	0.329245i	$0.893206\pi$
$48$		0			0
$49$		0			0
$50$	−122.000			−0.345068
$51$		0			0
$52$	8.00000	+	13.8564i	0.0213346	+	0.0369527i
$53$	239.000	+	413.960i	0.619418	+	1.07286i	0.989592	+	0.143902i	$0.0459649\pi$
$53$	239.000	+	413.960i	0.619418	+	1.07286i	−0.370174	+	0.928963i	$0.620702\pi$
$54$		0			0
$55$	−320.000			−0.784523
$56$		0			0
$57$		0			0
$58$	58.0000	−	100.459i	0.131306	−	0.227429i
$59$	−274.000	−	474.582i	−0.604606	−	1.04721i	−0.992114	−	0.125342i	$-0.959997\pi$
$59$	−274.000	−	474.582i	−0.604606	−	1.04721i	0.387507	−	0.921867i	$-0.373336\pi$
$60$		0			0
$61$	346.000	−	599.290i	0.726242	−	1.25789i	−0.232219	−	0.972664i	$-0.574599\pi$
$61$	346.000	−	599.290i	0.726242	−	1.25789i	0.958461	−	0.285224i	$-0.0920681\pi$
$62$	272.000			0.557162
$63$		0			0
$64$	64.0000			0.125000
$65$	16.0000	−	27.7128i	0.0305316	−	0.0528823i
$66$		0			0
$67$	454.000	+	786.351i	0.827835	+	1.43385i	0.899733	+	0.436440i	$0.143761\pi$
$67$	454.000	+	786.351i	0.827835	+	1.43385i	−0.0718987	+	0.997412i	$0.522906\pi$
$68$	168.000	−	290.985i	0.299603	−	0.518927i
$69$		0			0
$70$		0			0
$71$	524.000			0.875878			0.437939	−	0.899005i	$-0.355709\pi$
$71$	524.000			0.875878			0.437939	+	0.899005i	$0.355709\pi$
$72$		0			0
$73$	220.000	+	381.051i	0.352727	+	0.610941i	0.986726	−	0.162393i	$-0.0519212\pi$
$73$	220.000	+	381.051i	0.352727	+	0.610941i	−0.633999	+	0.773334i	$0.718588\pi$
$74$	222.000	+	384.515i	0.348743	+	0.604040i
$75$		0			0
$76$	−592.000			−0.893514
$77$		0			0
$78$		0			0
$79$	−608.000	+	1053.09i	−0.865890	+	1.49977i	0.000269874		1.00000i	$0.499914\pi$
$79$	−608.000	+	1053.09i	−0.865890	+	1.49977i	−0.866160	+	0.499766i	$0.833419\pi$
$80$	−64.0000	−	110.851i	−0.0894427	−	0.154919i
$81$		0			0
$82$	−420.000	+	727.461i	−0.565625	+	0.979691i
$83$	−684.000			−0.904563			−0.452282	−	0.891875i	$-0.649390\pi$
$83$	−684.000			−0.904563			−0.452282	+	0.891875i	$0.649390\pi$
$84$		0			0
$85$	−672.000			−0.857513
$86$	164.000	−	284.056i	0.205635	−	0.356170i
$87$		0			0
$88$	160.000	+	277.128i	0.193819	+	0.335704i
$89$	−302.000	+	523.079i	−0.359685	+	0.622992i	−0.987908	−	0.155041i	$-0.950449\pi$
$89$	−302.000	+	523.079i	−0.359685	+	0.622992i	0.628223	+	0.778033i	$0.283782\pi$
$90$		0			0
$91$		0			0
$92$	−336.000			−0.380765
$93$		0			0
$94$	−488.000	−	845.241i	−0.535461	−	0.927446i
$95$	592.000	+	1025.37i	0.639347	+	1.10738i
$96$		0			0
$97$	832.000			0.870895			0.435447	−	0.900214i	$-0.356590\pi$
$97$	832.000			0.870895			0.435447	+	0.900214i	$0.356590\pi$
$98$		0			0
$99$		0			0
$100$	122.000	−	211.310i	0.122000	−	0.211310i
$101$	−232.000	−	401.836i	−0.228563	−	0.395883i	0.728819	−	0.684706i	$-0.240069\pi$
$101$	−232.000	−	401.836i	−0.228563	−	0.395883i	−0.957382	+	0.288823i	$0.906736\pi$
$102$		0			0
$103$	−316.000	+	547.328i	−0.302295	+	0.523591i	−0.976655	−	0.214812i	$-0.931086\pi$
$103$	−316.000	+	547.328i	−0.302295	+	0.523591i	0.674360	+	0.738402i	$0.264420\pi$
$104$	−32.0000			−0.0301717
$105$		0			0
$106$	−956.000			−0.875990
$107$	−80.0000	+	138.564i	−0.0722794	+	0.125192i	−0.899900	−	0.436096i	$-0.856361\pi$
$107$	−80.0000	+	138.564i	−0.0722794	+	0.125192i	0.827621	+	0.561288i	$0.189694\pi$
$108$		0			0
$109$	1099.00	+	1903.52i	0.965735	+	1.67270i	0.707627	+	0.706586i	$0.249766\pi$
$109$	1099.00	+	1903.52i	0.965735	+	1.67270i	0.258108	+	0.966116i	$0.416901\pi$
$110$	320.000	−	554.256i	0.277371	−	0.480421i
$111$		0			0
$112$		0			0
$113$	−770.000			−0.641022			−0.320511	−	0.947245i	$-0.603855\pi$
$113$	−770.000			−0.641022			−0.320511	+	0.947245i	$0.603855\pi$
$114$		0			0
$115$	336.000	+	581.969i	0.272454	+	0.471903i
$116$	116.000	+	200.918i	0.0928477	+	0.160817i
$117$		0			0
$118$	1096.00			0.855042
$119$		0			0
$120$		0			0
$121$	−134.500	+	232.961i	−0.101052	+	0.175027i
$122$	692.000	+	1198.58i	0.513531	+	0.889461i
$123$		0			0
$124$	−272.000	+	471.118i	−0.196986	+	0.341191i
$125$	−1488.00			−1.06473
$126$		0			0
$127$	−184.000			−0.128562			−0.0642809	−	0.997932i	$-0.520475\pi$
$127$	−184.000			−0.128562			−0.0642809	+	0.997932i	$0.520475\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	32.0000	+	55.4256i	0.0215891	+	0.0373935i
$131$	726.000	−	1257.47i	0.484205	−	0.838668i	−0.515630	−	0.856811i	$-0.672442\pi$
$131$	726.000	−	1257.47i	0.484205	−	0.838668i	0.999835	+	0.0181429i	$0.00577539\pi$
$132$		0			0
$133$		0			0
$134$	−1816.00			−1.17074
$135$		0			0
$136$	336.000	+	581.969i	0.211851	+	0.366937i
$137$	323.000	+	559.452i	0.201429	+	0.348885i	0.948989	−	0.315309i	$-0.102108\pi$
$137$	323.000	+	559.452i	0.201429	+	0.348885i	−0.747560	+	0.664194i	$0.768775\pi$
$138$		0			0
$139$	3012.00			1.83795			0.918973	−	0.394320i	$-0.129020\pi$
$139$	3012.00			1.83795			0.918973	+	0.394320i	$0.129020\pi$
$140$		0			0
$141$		0			0
$142$	−524.000	+	907.595i	−0.309670	+	0.536364i
$143$	−80.0000	−	138.564i	−0.0467828	−	0.0810301i
$144$		0			0
$145$	232.000	−	401.836i	0.132873	−	0.230142i
$146$	−880.000			−0.498831
$147$		0			0
$148$	−888.000			−0.493197
$149$	−1585.00	+	2745.30i	−0.871465	+	1.50942i	−0.0109833	+	0.999940i	$0.503496\pi$
$149$	−1585.00	+	2745.30i	−0.871465	+	1.50942i	−0.860482	+	0.509482i	$0.829837\pi$
$150$		0			0
$151$	940.000	+	1628.13i	0.506597	+	0.877451i	0.999971	+	0.00763414i	$0.00243005\pi$
$151$	940.000	+	1628.13i	0.506597	+	0.877451i	−0.493374	+	0.869817i	$0.664237\pi$
$152$	592.000	−	1025.37i	0.315905	−	0.547163i
$153$		0			0
$154$		0			0
$155$	1088.00			0.563808
$156$		0			0
$157$	302.000	+	523.079i	0.153517	+	0.265900i	0.932518	−	0.361123i	$-0.117607\pi$
$157$	302.000	+	523.079i	0.153517	+	0.265900i	−0.779001	+	0.627023i	$0.784273\pi$
$158$	−1216.00	−	2106.17i	−0.612277	−	1.06049i
$159$		0			0
$160$	256.000			0.126491
$161$		0			0
$162$		0			0
$163$	−558.000	+	966.484i	−0.268135	+	0.464423i	−0.968380	−	0.249479i	$-0.919741\pi$
$163$	−558.000	+	966.484i	−0.268135	+	0.464423i	0.700246	+	0.713902i	$0.253074\pi$
$164$	−840.000	−	1454.92i	−0.399957	−	0.692746i
$165$		0			0
$166$	684.000	−	1184.72i	0.319811	−	0.553930i
$167$	−1784.00			−0.826647			−0.413324	−	0.910584i	$-0.635632\pi$
$167$	−1784.00			−0.826647			−0.413324	+	0.910584i	$0.635632\pi$
$168$		0			0
$169$	−2181.00			−0.992717
$170$	672.000	−	1163.94i	0.303177	−	0.525118i
$171$		0			0
$172$	328.000	+	568.113i	0.145406	+	0.251850i
$173$	172.000	−	297.913i	0.0755891	−	0.130924i	−0.825753	−	0.564032i	$-0.809250\pi$
$173$	172.000	−	297.913i	0.0755891	−	0.130924i	0.901342	+	0.433107i	$0.142583\pi$
$174$		0			0
$175$		0			0
$176$	−640.000			−0.274101
$177$		0			0
$178$	−604.000	−	1046.16i	−0.254335	−	0.440522i
$179$	696.000	+	1205.51i	0.290623	+	0.503373i	0.973957	−	0.226732i	$-0.0728042\pi$
$179$	696.000	+	1205.51i	0.290623	+	0.503373i	−0.683334	+	0.730106i	$0.739471\pi$
$180$		0			0
$181$	−4052.00			−1.66399			−0.831997	−	0.554781i	$-0.812802\pi$
$181$	−4052.00			−1.66399			−0.831997	+	0.554781i	$0.812802\pi$
$182$		0			0
$183$		0			0
$184$	336.000	−	581.969i	0.134621	−	0.233170i
$185$	888.000	+	1538.06i	0.352903	+	0.611246i
$186$		0			0
$187$	−1680.00	+	2909.85i	−0.656972	+	1.13791i
$188$	1952.00			0.757257
$189$		0			0
$190$	−2368.00			−0.904173
$191$	−1554.00	+	2691.61i	−0.588709	+	1.01967i	0.405692	+	0.914010i	$0.367030\pi$
$191$	−1554.00	+	2691.61i	−0.588709	+	1.01967i	−0.994402	+	0.105665i	$0.966303\pi$
$192$		0			0
$193$	−25.0000	−	43.3013i	−0.00932404	−	0.0161497i	0.861326	−	0.508053i	$-0.169635\pi$
$193$	−25.0000	−	43.3013i	−0.00932404	−	0.0161497i	−0.870650	+	0.491903i	$0.836301\pi$
$194$	−832.000	+	1441.07i	−0.307908	+	0.533312i
$195$		0			0
$196$		0			0
$197$	162.000			0.0585889			0.0292945	−	0.999571i	$-0.490674\pi$
$197$	162.000			0.0585889			0.0292945	+	0.999571i	$0.490674\pi$
$198$		0			0
$199$	772.000	+	1337.14i	0.275003	+	0.476319i	0.970136	−	0.242562i	$-0.0779878\pi$
$199$	772.000	+	1337.14i	0.275003	+	0.476319i	−0.695133	+	0.718881i	$0.744654\pi$
$200$	244.000	+	422.620i	0.0862670	+	0.149419i
$201$		0			0
$202$	928.000			0.323237
$203$		0			0
$204$		0			0
$205$	−1680.00	+	2909.85i	−0.572372	+	0.991378i
$206$	−632.000	−	1094.66i	−0.213755	−	0.370234i
$207$		0			0
$208$	32.0000	−	55.4256i	0.0106673	−	0.0184763i
$209$	5920.00			1.95931
$210$		0			0
$211$	−1204.00			−0.392828			−0.196414	−	0.980521i	$-0.562930\pi$
$211$	−1204.00			−0.392828			−0.196414	+	0.980521i	$0.562930\pi$
$212$	956.000	−	1655.84i	0.309709	−	0.536432i
$213$		0			0
$214$	−160.000	−	277.128i	−0.0511092	−	0.0885238i
$215$	656.000	−	1136.23i	0.208088	−	0.360418i
$216$		0			0
$217$		0			0
$218$	−4396.00			−1.36576
$219$		0			0
$220$	640.000	+	1108.51i	0.196131	+	0.339709i
$221$	−168.000	−	290.985i	−0.0511353	−	0.0885690i
$222$		0			0
$223$	−2000.00			−0.600583			−0.300291	−	0.953848i	$-0.597084\pi$
$223$	−2000.00			−0.600583			−0.300291	+	0.953848i	$0.597084\pi$
$224$		0			0
$225$		0			0
$226$	770.000	−	1333.68i	0.226636	−	0.392544i
$227$	−194.000	−	336.018i	−0.0567235	−	0.0982480i	0.836269	−	0.548319i	$-0.184732\pi$
$227$	−194.000	−	336.018i	−0.0567235	−	0.0982480i	−0.892993	+	0.450071i	$0.851399\pi$
$228$		0			0
$229$	2090.00	−	3619.99i	0.603105	−	1.04461i	−0.389243	−	0.921135i	$-0.627263\pi$
$229$	2090.00	−	3619.99i	0.603105	−	1.04461i	0.992348	−	0.123474i	$-0.0394034\pi$
$230$	−1344.00			−0.385308
$231$		0			0
$232$	−464.000			−0.131306
$233$	−661.000	+	1144.89i	−0.185852	+	0.321905i	−0.943863	−	0.330336i	$-0.892838\pi$
$233$	−661.000	+	1144.89i	−0.185852	+	0.321905i	0.758011	+	0.652242i	$0.226171\pi$
$234$		0			0
$235$	−1952.00	−	3380.96i	−0.541849	−	0.938509i
$236$	−1096.00	+	1898.33i	−0.302303	+	0.523604i
$237$		0			0
$238$		0			0
$239$	−2412.00			−0.652800			−0.326400	−	0.945232i	$-0.605836\pi$
$239$	−2412.00			−0.652800			−0.326400	+	0.945232i	$0.605836\pi$
$240$		0			0
$241$	−2168.00	−	3755.09i	−0.579474	−	1.00368i	−0.995540	−	0.0943434i	$-0.969925\pi$
$241$	−2168.00	−	3755.09i	−0.579474	−	1.00368i	0.416066	−	0.909334i	$-0.363409\pi$
$242$	−269.000	−	465.922i	−0.0714544	−	0.123763i
$243$		0			0
$244$	−2768.00			−0.726242
$245$		0			0
$246$		0			0
$247$	−296.000	+	512.687i	−0.0762511	+	0.132071i
$248$	−544.000	−	942.236i	−0.139290	−	0.241258i
$249$		0			0
$250$	1488.00	−	2577.29i	0.376438	−	0.652009i
$251$	764.000			0.192125			0.0960623	−	0.995375i	$-0.469375\pi$
$251$	764.000			0.192125			0.0960623	+	0.995375i	$0.469375\pi$
$252$		0			0
$253$	3360.00			0.834946
$254$	184.000	−	318.697i	0.0454535	−	0.0787278i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−2150.00	+	3723.91i	−0.521842	+	0.903856i	0.477835	+	0.878449i	$0.341421\pi$
$257$	−2150.00	+	3723.91i	−0.521842	+	0.903856i	−0.999677	+	0.0254070i	$0.991912\pi$
$258$		0			0
$259$		0			0
$260$	−128.000			−0.0305316
$261$		0			0
$262$	1452.00	+	2514.94i	0.342385	+	0.593028i
$263$	−1930.00	−	3342.86i	−0.452505	−	0.783762i	0.546036	−	0.837762i	$-0.316136\pi$
$263$	−1930.00	−	3342.86i	−0.452505	−	0.783762i	−0.998541	+	0.0539998i	$0.982803\pi$
$264$		0			0
$265$	−3824.00			−0.886439
$266$		0			0
$267$		0			0
$268$	1816.00	−	3145.40i	0.413917	−	0.716926i
$269$	1400.00	+	2424.87i	0.317322	+	0.549617i	0.979928	−	0.199350i	$-0.0638832\pi$
$269$	1400.00	+	2424.87i	0.317322	+	0.549617i	−0.662607	+	0.748968i	$0.730550\pi$
$270$		0			0
$271$	−2440.00	+	4226.20i	−0.546935	+	0.947320i	0.451547	+	0.892247i	$0.350872\pi$
$271$	−2440.00	+	4226.20i	−0.546935	+	0.947320i	−0.998482	+	0.0550723i	$0.982461\pi$
$272$	−1344.00			−0.299603
$273$		0			0
$274$	−1292.00			−0.284863
$275$	−1220.00	+	2113.10i	−0.267523	+	0.463363i
$276$		0			0
$277$	3337.00	+	5779.85i	0.723830	+	1.25371i	0.959454	+	0.281866i	$0.0909533\pi$
$277$	3337.00	+	5779.85i	0.723830	+	1.25371i	−0.235624	+	0.971844i	$0.575713\pi$
$278$	−3012.00	+	5216.94i	−0.649812	+	1.12551i
$279$		0			0
$280$		0			0
$281$	9402.00			1.99600			0.998001	−	0.0632056i	$-0.0201324\pi$
$281$	9402.00			1.99600			0.998001	+	0.0632056i	$0.0201324\pi$
$282$		0			0
$283$	−4550.00	−	7880.83i	−0.955722	−	1.65536i	−0.732706	−	0.680545i	$-0.761743\pi$
$283$	−4550.00	−	7880.83i	−0.955722	−	1.65536i	−0.223016	−	0.974815i	$-0.571590\pi$
$284$	−1048.00	−	1815.19i	−0.218970	−	0.379266i
$285$		0			0
$286$	320.000			0.0661608
$287$		0			0
$288$		0			0
$289$	−1071.50	+	1855.89i	−0.218095	+	0.377751i
$290$	464.000	+	803.672i	0.0939552	+	0.162735i
$291$		0			0
$292$	880.000	−	1524.20i	0.176363	−	0.305470i
$293$	5952.00			1.18676			0.593378	−	0.804924i	$-0.297794\pi$
$293$	5952.00			1.18676			0.593378	+	0.804924i	$0.297794\pi$
$294$		0			0
$295$	4384.00			0.865242
$296$	888.000	−	1538.06i	0.174371	−	0.302020i
$297$		0			0
$298$	−3170.00	−	5490.60i	−0.616219	−	1.06732i
$299$	−168.000	+	290.985i	−0.0324939	+	0.0562812i
$300$		0			0
$301$		0			0
$302$	−3760.00			−0.716436
$303$		0			0
$304$	1184.00	+	2050.75i	0.223378	+	0.386903i
$305$	2768.00	+	4794.32i	0.519656	+	0.900071i
$306$		0			0
$307$	3004.00			0.558460			0.279230	−	0.960224i	$-0.409921\pi$
$307$	3004.00			0.558460			0.279230	+	0.960224i	$0.409921\pi$
$308$		0			0
$309$		0			0
$310$	−1088.00	+	1884.47i	−0.199336	+	0.345261i
$311$	−344.000	−	595.825i	−0.0627217	−	0.108637i	0.832959	−	0.553334i	$-0.186645\pi$
$311$	−344.000	−	595.825i	−0.0627217	−	0.108637i	−0.895681	+	0.444697i	$0.853311\pi$
$312$		0			0
$313$	2796.00	−	4842.81i	0.504918	−	0.874543i	−0.495066	−	0.868855i	$-0.664856\pi$
$313$	2796.00	−	4842.81i	0.504918	−	0.874543i	0.999984	−	0.00568790i	$-0.00181052\pi$
$314$	−1208.00			−0.217106
$315$		0			0
$316$	4864.00			0.865890
$317$	−1461.00	+	2530.53i	−0.258858	+	0.448355i	−0.965936	−	0.258780i	$-0.916679\pi$
$317$	−1461.00	+	2530.53i	−0.258858	+	0.448355i	0.707078	+	0.707135i	$0.250013\pi$
$318$		0			0
$319$	−1160.00	−	2009.18i	−0.203597	−	0.352641i
$320$	−256.000	+	443.405i	−0.0447214	+	0.0774597i
$321$		0			0
$322$		0			0
$323$	12432.0			2.14159
$324$		0			0
$325$	−122.000	−	211.310i	−0.0208226	−	0.0360658i
$326$	−1116.00	−	1932.97i	−0.189600	−	0.328396i
$327$		0			0
$328$	3360.00			0.565625
$329$		0			0
$330$		0			0
$331$	3746.00	−	6488.26i	0.622051	−	1.07742i	−0.367053	−	0.930200i	$-0.619633\pi$
$331$	3746.00	−	6488.26i	0.622051	−	1.07742i	0.989103	−	0.147223i	$-0.0470336\pi$
$332$	1368.00	+	2369.45i	0.226141	+	0.391687i
$333$		0			0
$334$	1784.00	−	3089.98i	0.292264	−	0.506216i
$335$	−7264.00			−1.18470
$336$		0			0
$337$	10766.0			1.74024			0.870121	−	0.492839i	$-0.164041\pi$
$337$	10766.0			1.74024			0.870121	+	0.492839i	$0.164041\pi$
$338$	2181.00	−	3777.60i	0.350979	−	0.607913i
$339$		0			0
$340$	1344.00	+	2327.88i	0.214378	+	0.371314i
$341$	2720.00	−	4711.18i	0.431954	−	0.748166i
$342$		0			0
$343$		0			0
$344$	−1312.00			−0.205635
$345$		0			0
$346$	344.000	+	595.825i	0.0534496	+	0.0925774i
$347$	−1992.00	−	3450.25i	−0.308173	−	0.533772i	0.669789	−	0.742551i	$-0.266384\pi$
$347$	−1992.00	−	3450.25i	−0.308173	−	0.533772i	−0.977963	+	0.208779i	$0.933051\pi$
$348$		0			0
$349$	180.000			0.0276080			0.0138040	−	0.999905i	$-0.495606\pi$
$349$	180.000			0.0276080			0.0138040	+	0.999905i	$0.495606\pi$
$350$		0			0
$351$		0			0
$352$	640.000	−	1108.51i	0.0969094	−	0.167852i
$353$	5214.00	+	9030.91i	0.786156	+	1.36166i	0.928306	+	0.371818i	$0.121265\pi$
$353$	5214.00	+	9030.91i	0.786156	+	1.36166i	−0.142149	+	0.989845i	$0.545401\pi$
$354$		0			0
$355$	−2096.00	+	3630.38i	−0.313364	+	0.542762i
$356$	2416.00			0.359685
$357$		0			0
$358$	−2784.00			−0.411003
$359$	4342.00	−	7520.56i	0.638334	−	1.10563i	−0.347464	−	0.937693i	$-0.612957\pi$
$359$	4342.00	−	7520.56i	0.638334	−	1.10563i	0.985798	−	0.167934i	$-0.0537096\pi$
$360$		0			0
$361$	−7522.50	−	13029.4i	−1.09673	−	1.89960i
$362$	4052.00	−	7018.27i	0.588310	−	1.01898i
$363$		0			0
$364$		0			0
$365$	−3520.00			−0.504781
$366$		0			0
$367$	2824.00	+	4891.31i	0.401666	+	0.695707i	0.993927	−	0.110040i	$-0.0350978\pi$
$367$	2824.00	+	4891.31i	0.401666	+	0.695707i	−0.592261	+	0.805746i	$0.701764\pi$
$368$	672.000	+	1163.94i	0.0951914	+	0.164876i
$369$		0			0
$370$	−3552.00			−0.499080
$371$		0			0
$372$		0			0
$373$	1273.00	−	2204.90i	0.176712	−	0.306074i	−0.764041	−	0.645168i	$-0.776787\pi$
$373$	1273.00	−	2204.90i	0.176712	−	0.306074i	0.940752	+	0.339095i	$0.110121\pi$
$374$	−3360.00	−	5819.69i	−0.464549	−	0.804623i
$375$		0			0
$376$	−1952.00	+	3380.96i	−0.267731	+	0.463723i
$377$	232.000			0.0316939
$378$		0			0
$379$	8268.00			1.12058			0.560288	−	0.828298i	$-0.310690\pi$
$379$	8268.00			1.12058			0.560288	+	0.828298i	$0.310690\pi$
$380$	2368.00	−	4101.50i	0.319673	−	0.553690i
$381$		0			0
$382$	−3108.00	−	5383.21i	−0.416280	−	0.721019i
$383$	5436.00	−	9415.43i	0.725239	−	1.25615i	−0.233636	−	0.972324i	$-0.575062\pi$
$383$	5436.00	−	9415.43i	0.725239	−	1.25615i	0.958875	−	0.283827i	$-0.0916042\pi$
$384$		0			0
$385$		0			0
$386$	100.000			0.0131862
$387$		0			0
$388$	−1664.00	−	2882.13i	−0.217724	−	0.377109i
$389$	5217.00	+	9036.11i	0.679980	+	1.17776i	0.974986	+	0.222265i	$0.0713451\pi$
$389$	5217.00	+	9036.11i	0.679980	+	1.17776i	−0.295006	+	0.955495i	$0.595322\pi$
$390$		0			0
$391$	7056.00			0.912627
$392$		0			0
$393$		0			0
$394$	−162.000	+	280.592i	−0.0207143	+	0.0358783i
$395$	−4864.00	−	8424.70i	−0.619581	−	1.07315i
$396$		0			0
$397$	−1522.00	+	2636.18i	−0.192411	+	0.333265i	−0.946049	−	0.324025i	$-0.894964\pi$
$397$	−1522.00	+	2636.18i	−0.192411	+	0.333265i	0.753638	+	0.657290i	$0.228297\pi$
$398$	−3088.00			−0.388913
$399$		0			0
$400$	−976.000			−0.122000
$401$	4455.00	−	7716.29i	0.554793	−	0.960930i	−0.443126	−	0.896459i	$-0.646131\pi$
$401$	4455.00	−	7716.29i	0.554793	−	0.960930i	0.997920	−	0.0644709i	$-0.0205360\pi$
$402$		0			0
$403$	272.000	+	471.118i	0.0336211	+	0.0582334i
$404$	−928.000	+	1607.34i	−0.114281	+	0.197941i
$405$		0			0
$406$		0			0
$407$	8880.00			1.08149
$408$		0			0
$409$	2808.00	+	4863.60i	0.339478	+	0.587994i	0.984335	−	0.176310i	$-0.0564161\pi$
$409$	2808.00	+	4863.60i	0.339478	+	0.587994i	−0.644856	+	0.764304i	$0.723083\pi$
$410$	−3360.00	−	5819.69i	−0.404728	−	0.701010i
$411$		0			0
$412$	2528.00			0.302295
$413$		0			0
$414$		0			0
$415$	2736.00	−	4738.89i	0.323626	−	0.560537i
$416$	64.0000	+	110.851i	0.00754293	+	0.0130647i
$417$		0			0
$418$	−5920.00	+	10253.7i	−0.692719	+	1.19983i
$419$	−8932.00			−1.04142			−0.520712	−	0.853732i	$-0.674334\pi$
$419$	−8932.00			−1.04142			−0.520712	+	0.853732i	$0.674334\pi$
$420$		0			0
$421$	−5538.00			−0.641106			−0.320553	−	0.947231i	$-0.603869\pi$
$421$	−5538.00			−0.641106			−0.320553	+	0.947231i	$0.603869\pi$
$422$	1204.00	−	2085.39i	0.138886	−	0.240557i
$423$		0			0
$424$	1912.00	+	3311.68i	0.218997	+	0.379315i
$425$	−2562.00	+	4437.51i	−0.292412	+	0.506473i
$426$		0			0
$427$		0			0
$428$	640.000			0.0722794
$429$		0			0
$430$	1312.00	+	2272.45i	0.147140	+	0.254854i
$431$	−3350.00	−	5802.37i	−0.374394	−	0.648469i	0.615842	−	0.787870i	$-0.288816\pi$
$431$	−3350.00	−	5802.37i	−0.374394	−	0.648469i	−0.990236	+	0.139400i	$0.955483\pi$
$432$		0			0
$433$	5048.00			0.560257			0.280129	−	0.959962i	$-0.409623\pi$
$433$	5048.00			0.560257			0.280129	+	0.959962i	$0.409623\pi$
$434$		0			0
$435$		0			0
$436$	4396.00	−	7614.10i	0.482867	−	0.836351i
$437$	−6216.00	−	10766.4i	−0.680438	−	1.17855i
$438$		0			0
$439$	−672.000	+	1163.94i	−0.0730588	+	0.126542i	−0.900240	−	0.435393i	$-0.856609\pi$
$439$	−672.000	+	1163.94i	−0.0730588	+	0.126542i	0.827182	+	0.561935i	$0.189943\pi$
$440$	−2560.00			−0.277371
$441$		0			0
$442$	672.000			0.0723162
$443$	−2196.00	+	3803.58i	−0.235519	+	0.407932i	−0.959424	−	0.281969i	$-0.909012\pi$
$443$	−2196.00	+	3803.58i	−0.235519	+	0.407932i	0.723904	+	0.689901i	$0.242346\pi$
$444$		0			0
$445$	−2416.00	−	4184.63i	−0.257369	−	0.445777i
$446$	2000.00	−	3464.10i	0.212338	−	0.367780i
$447$		0			0
$448$		0			0
$449$	−3666.00			−0.385321			−0.192661	−	0.981265i	$-0.561712\pi$
$449$	−3666.00			−0.385321			−0.192661	+	0.981265i	$0.561712\pi$
$450$		0			0
$451$	8400.00	+	14549.2i	0.877030	+	1.51906i
$452$	1540.00	+	2667.36i	0.160256	+	0.277571i
$453$		0			0
$454$	776.000			0.0802191
$455$		0			0
$456$		0			0
$457$	−13.0000	+	22.5167i	−0.00133067	+	0.00230478i	−0.866690	−	0.498847i	$-0.833757\pi$
$457$	−13.0000	+	22.5167i	−0.00133067	+	0.00230478i	0.865359	+	0.501152i	$0.167090\pi$
$458$	4180.00	+	7239.97i	0.426460	+	0.738650i
$459$		0			0
$460$	1344.00	−	2327.88i	0.136227	−	0.235952i
$461$	7656.00			0.773483			0.386741	−	0.922188i	$-0.373601\pi$
$461$	7656.00			0.773483			0.386741	+	0.922188i	$0.373601\pi$
$462$		0			0
$463$	12608.0			1.26554			0.632768	−	0.774341i	$-0.281919\pi$
$463$	12608.0			1.26554			0.632768	+	0.774341i	$0.281919\pi$
$464$	464.000	−	803.672i	0.0464238	−	0.0804084i
$465$		0			0
$466$	−1322.00	−	2289.77i	−0.131417	−	0.227621i
$467$	1534.00	−	2656.97i	0.152002	−	0.263276i	−0.779961	−	0.625828i	$-0.784761\pi$
$467$	1534.00	−	2656.97i	0.152002	−	0.263276i	0.931963	+	0.362552i	$0.118095\pi$
$468$		0			0
$469$		0			0
$470$	7808.00			0.766290
$471$		0			0
$472$	−2192.00	−	3796.66i	−0.213761	−	0.370244i
$473$	−3280.00	−	5681.13i	−0.318847	−	0.552259i
$474$		0			0
$475$	9028.00			0.872070
$476$		0			0
$477$		0			0
$478$	2412.00	−	4177.71i	0.230800	−	0.399757i
$479$	−3228.00	−	5591.06i	−0.307915	−	0.533324i	0.669991	−	0.742369i	$-0.266298\pi$
$479$	−3228.00	−	5591.06i	−0.307915	−	0.533324i	−0.977906	+	0.209045i	$0.932964\pi$
$480$		0			0
$481$	−444.000	+	769.031i	−0.0420887	+	0.0728997i
$482$	8672.00			0.819500
$483$		0			0
$484$	1076.00			0.101052
$485$	−3328.00	+	5764.27i	−0.311581	+	0.539674i
$486$		0			0
$487$	−5948.00	−	10302.2i	−0.553449	−	0.958602i	−0.998022	−	0.0628592i	$-0.979978\pi$
$487$	−5948.00	−	10302.2i	−0.553449	−	0.958602i	0.444574	−	0.895742i	$-0.353355\pi$
$488$	2768.00	−	4794.32i	0.256765	−	0.444731i
$489$		0			0
$490$		0			0
$491$	264.000			0.0242651			0.0121325	−	0.999926i	$-0.496138\pi$
$491$	264.000			0.0242651			0.0121325	+	0.999926i	$0.496138\pi$
$492$		0			0
$493$	−2436.00	−	4219.28i	−0.222539	−	0.385450i
$494$	−592.000	−	1025.37i	−0.0539177	−	0.0933882i
$495$		0			0
$496$	2176.00			0.196986
$497$		0			0
$498$		0			0
$499$	1314.00	−	2275.91i	0.117881	−	0.204176i	−0.801047	−	0.598602i	$-0.795723\pi$
$499$	1314.00	−	2275.91i	0.117881	−	0.204176i	0.918928	+	0.394426i	$0.129056\pi$
$500$	2976.00	+	5154.58i	0.266182	+	0.461040i
$501$		0			0
$502$	−764.000	+	1323.29i	−0.0679263	+	0.117652i
$503$	−13568.0			−1.20272			−0.601359	−	0.798979i	$-0.705374\pi$
$503$	−13568.0			−1.20272			−0.601359	+	0.798979i	$0.705374\pi$
$504$		0			0
$505$	3712.00			0.327093
$506$	−3360.00	+	5819.69i	−0.295198	+	0.511298i
$507$		0			0
$508$	368.000	+	637.395i	0.0321405	+	0.0556689i
$509$	−10328.0	+	17888.6i	−0.899372	+	1.55776i	−0.0710743	+	0.997471i	$0.522643\pi$
$509$	−10328.0	+	17888.6i	−0.899372	+	1.55776i	−0.828298	+	0.560288i	$0.810691\pi$
$510$		0			0
$511$		0			0
$512$	512.000			0.0441942
$513$		0			0
$514$	−4300.00	−	7447.82i	−0.368998	−	0.639123i
$515$	−2528.00	−	4378.62i	−0.216305	−	0.374651i
$516$		0			0
$517$	−19520.0			−1.66052
$518$		0			0
$519$		0			0
$520$	128.000	−	221.703i	0.0107946	−	0.0186967i
$521$	−1814.00	−	3141.94i	−0.152539	−	0.264205i	0.779621	−	0.626251i	$-0.215412\pi$
$521$	−1814.00	−	3141.94i	−0.152539	−	0.264205i	−0.932160	+	0.362046i	$0.882078\pi$
$522$		0			0
$523$	2426.00	−	4201.96i	0.202833	−	0.351317i	−0.746607	−	0.665265i	$-0.768319\pi$
$523$	2426.00	−	4201.96i	0.202833	−	0.351317i	0.949440	+	0.313948i	$0.101652\pi$
$524$	−5808.00			−0.484205
$525$		0			0
$526$	7720.00			0.639939
$527$	5712.00	−	9893.47i	0.472142	−	0.817773i
$528$		0			0
$529$	2555.50	+	4426.26i	0.210035	+	0.363792i
$530$	3824.00	−	6623.36i	0.313404	−	0.542831i
$531$		0			0
$532$		0			0
$533$	−1680.00			−0.136527
$534$		0			0
$535$	−640.000	−	1108.51i	−0.0517189	−	0.0895798i
$536$	3632.00	+	6290.81i	0.292684	+	0.506943i
$537$		0			0
$538$	−5600.00			−0.448760
$539$		0			0
$540$		0			0
$541$	3565.00	−	6174.76i	0.283311	−	0.490709i	−0.688887	−	0.724869i	$-0.741900\pi$
$541$	3565.00	−	6174.76i	0.283311	−	0.490709i	0.972198	+	0.234159i	$0.0752338\pi$
$542$	−4880.00	−	8452.41i	−0.386742	−	0.669856i
$543$		0			0
$544$	1344.00	−	2327.88i	0.105926	−	0.183469i
$545$	−17584.0			−1.38205
$546$		0			0
$547$	−12788.0			−0.999589			−0.499795	−	0.866144i	$-0.666591\pi$
$547$	−12788.0			−0.999589			−0.499795	+	0.866144i	$0.666591\pi$
$548$	1292.00	−	2237.81i	0.100714	−	0.174443i
$549$		0			0
$550$	−2440.00	−	4226.20i	−0.189167	−	0.327647i
$551$	−4292.00	+	7433.96i	−0.331843	+	0.574768i
$552$		0			0
$553$		0			0
$554$	−13348.0			−1.02365
$555$		0			0
$556$	−6024.00	−	10433.9i	−0.459487	−	0.795854i
$557$	1203.00	+	2083.66i	0.0915130	+	0.158505i	0.908148	−	0.418649i	$-0.137496\pi$
$557$	1203.00	+	2083.66i	0.0915130	+	0.158505i	−0.816635	+	0.577155i	$0.804163\pi$
$558$		0			0
$559$	656.000			0.0496348
$560$		0			0
$561$		0			0
$562$	−9402.00	+	16284.7i	−0.705693	+	1.22230i
$563$	12706.0	+	22007.4i	0.951144	+	1.64743i	0.742955	+	0.669341i	$0.233423\pi$
$563$	12706.0	+	22007.4i	0.951144	+	1.64743i	0.208189	+	0.978089i	$0.433243\pi$
$564$		0			0
$565$	3080.00	−	5334.72i	0.229339	−	0.397227i
$566$	18200.0			1.35160
$567$		0			0
$568$	4192.00			0.309670
$569$	−4845.00	+	8391.79i	−0.356965	+	0.618281i	−0.987452	−	0.157918i	$-0.949522\pi$
$569$	−4845.00	+	8391.79i	−0.356965	+	0.618281i	0.630487	+	0.776199i	$0.282855\pi$
$570$		0			0
$571$	−2802.00	−	4853.21i	−0.205359	−	0.355692i	0.744888	−	0.667190i	$-0.232503\pi$
$571$	−2802.00	−	4853.21i	−0.205359	−	0.355692i	−0.950247	+	0.311497i	$0.899170\pi$
$572$	−320.000	+	554.256i	−0.0233914	+	0.0405151i
$573$		0			0
$574$		0			0
$575$	5124.00			0.371627
$576$		0			0
$577$	−10784.0	−	18678.4i	−0.778066	−	1.34765i	−0.933055	−	0.359733i	$-0.882868\pi$
$577$	−10784.0	−	18678.4i	−0.778066	−	1.34765i	0.154990	−	0.987916i	$-0.450466\pi$
$578$	−2143.00	−	3711.78i	−0.154216	−	0.267111i
$579$		0			0
$580$	−1856.00			−0.132873
$581$		0			0
$582$		0			0
$583$	−9560.00	+	16558.4i	−0.679133	+	1.17629i
$584$	1760.00	+	3048.41i	0.124708	+	0.216000i
$585$		0			0
$586$	−5952.00	+	10309.2i	−0.419582	+	0.726737i
$587$	20300.0			1.42738			0.713689	−	0.700463i	$-0.247023\pi$
$587$	20300.0			1.42738			0.713689	+	0.700463i	$0.247023\pi$
$588$		0			0
$589$	−20128.0			−1.40808
$590$	−4384.00	+	7593.31i	−0.305909	+	0.529850i
$591$		0			0
$592$	1776.00	+	3076.12i	0.123299	+	0.213561i
$593$	−6906.00	+	11961.5i	−0.478238	+	0.828333i	−0.999689	−	0.0249483i	$-0.992058\pi$
$593$	−6906.00	+	11961.5i	−0.478238	+	0.828333i	0.521450	+	0.853282i	$0.325391\pi$
$594$		0			0
$595$		0			0
$596$	12680.0			0.871465
$597$		0			0
$598$	−336.000	−	581.969i	−0.0229767	−	0.0397968i
$599$	−10998.0	−	19049.1i	−0.750194	−	1.29937i	−0.947728	−	0.319078i	$-0.896627\pi$
$599$	−10998.0	−	19049.1i	−0.750194	−	1.29937i	0.197535	−	0.980296i	$-0.436706\pi$
$600$		0			0
$601$	−8368.00			−0.567950			−0.283975	−	0.958832i	$-0.591653\pi$
$601$	−8368.00			−0.567950			−0.283975	+	0.958832i	$0.591653\pi$
$602$		0			0
$603$		0			0
$604$	3760.00	−	6512.51i	0.253298	−	0.438726i
$605$	−1076.00	−	1863.69i	−0.0723068	−	0.125239i
$606$		0			0
$607$	10752.0	−	18623.0i	0.718962	−	1.24528i	−0.242449	−	0.970164i	$-0.577951\pi$
$607$	10752.0	−	18623.0i	0.718962	−	1.24528i	0.961411	−	0.275115i	$-0.0887160\pi$
$608$	−4736.00			−0.315905
$609$		0			0
$610$	−11072.0			−0.734905
$611$	976.000	−	1690.48i	0.0646231	−	0.111931i
$612$		0			0
$613$	5135.00	+	8894.08i	0.338337	+	0.586017i	0.984120	−	0.177504i	$-0.0568022\pi$
$613$	5135.00	+	8894.08i	0.338337	+	0.586017i	−0.645783	+	0.763521i	$0.723469\pi$
$614$	−3004.00	+	5203.08i	−0.197446	+	0.341986i
$615$		0			0
$616$		0			0
$617$	−28358.0			−1.85032			−0.925162	−	0.379572i	$-0.876071\pi$
$617$	−28358.0			−1.85032			−0.925162	+	0.379572i	$0.876071\pi$
$618$		0			0
$619$	8146.00	+	14109.3i	0.528942	+	0.916155i	0.999430	+	0.0337488i	$0.0107446\pi$
$619$	8146.00	+	14109.3i	0.528942	+	0.916155i	−0.470488	+	0.882406i	$0.655922\pi$
$620$	−2176.00	−	3768.94i	−0.140952	−	0.244136i
$621$		0			0
$622$	1376.00			0.0887019
$623$		0			0
$624$		0			0
$625$	2139.50	−	3705.72i	0.136928	−	0.237166i
$626$	5592.00	+	9685.63i	0.357031	+	0.618395i
$627$		0			0
$628$	1208.00	−	2092.32i	0.0767587	−	0.132950i
$629$	18648.0			1.18211
$630$		0			0
$631$	11256.0			0.710134			0.355067	−	0.934841i	$-0.384458\pi$
$631$	11256.0			0.710134			0.355067	+	0.934841i	$0.384458\pi$
$632$	−4864.00	+	8424.70i	−0.306138	+	0.530247i
$633$		0			0
$634$	−2922.00	−	5061.05i	−0.183040	−	0.317035i
$635$	736.000	−	1274.79i	0.0459957	−	0.0796669i
$636$		0			0
$637$		0			0
$638$	4640.00			0.287930
$639$		0			0
$640$	−512.000	−	886.810i	−0.0316228	−	0.0547723i
$641$	7759.00	+	13439.0i	0.478100	+	0.828093i	0.999685	−	0.0251060i	$-0.00799233\pi$
$641$	7759.00	+	13439.0i	0.478100	+	0.828093i	−0.521585	+	0.853199i	$0.674659\pi$
$642$		0			0
$643$	10452.0			0.641037			0.320518	−	0.947242i	$-0.396143\pi$
$643$	10452.0			0.641037			0.320518	+	0.947242i	$0.396143\pi$
$644$		0			0
$645$		0			0
$646$	−12432.0	+	21532.9i	−0.757168	+	1.31145i
$647$	−36.0000	−	62.3538i	−0.00218749	−	0.00378884i	0.864930	−	0.501893i	$-0.167363\pi$
$647$	−36.0000	−	62.3538i	−0.00218749	−	0.00378884i	−0.867117	+	0.498104i	$0.834030\pi$
$648$		0			0
$649$	10960.0	−	18983.3i	0.662893	−	1.14816i
$650$	488.000			0.0294476
$651$		0			0
$652$	4464.00			0.268135
$653$	5981.00	−	10359.4i	0.358430	−	0.620819i	−0.629269	−	0.777188i	$-0.716646\pi$
$653$	5981.00	−	10359.4i	0.358430	−	0.620819i	0.987699	+	0.156369i	$0.0499789\pi$
$654$		0			0
$655$	5808.00	+	10059.8i	0.346469	+	0.600102i
$656$	−3360.00	+	5819.69i	−0.199979	+	0.346373i
$657$		0			0
$658$		0			0
$659$	6016.00			0.355615			0.177807	−	0.984065i	$-0.443100\pi$
$659$	6016.00			0.355615			0.177807	+	0.984065i	$0.443100\pi$
$660$		0			0
$661$	13034.0	+	22575.6i	0.766965	+	1.32842i	0.939201	+	0.343367i	$0.111567\pi$
$661$	13034.0	+	22575.6i	0.766965	+	1.32842i	−0.172236	+	0.985056i	$0.555099\pi$
$662$	7492.00	+	12976.5i	0.439856	+	0.761853i
$663$		0			0
$664$	−5472.00			−0.319811
$665$		0			0
$666$		0			0
$667$	−2436.00	+	4219.28i	−0.141413	+	0.244934i
$668$	3568.00	+	6179.96i	0.206662	+	0.357949i
$669$		0			0
$670$	7264.00	−	12581.6i	0.418855	−	0.725478i
$671$	27680.0			1.59251
$672$		0			0
$673$	−20530.0			−1.17589			−0.587945	−	0.808901i	$-0.700063\pi$
$673$	−20530.0			−1.17589			−0.587945	+	0.808901i	$0.700063\pi$
$674$	−10766.0	+	18647.3i	−0.615268	+	1.06568i
$675$		0			0
$676$	4362.00	+	7555.21i	0.248179	+	0.429859i
$677$	5028.00	−	8708.75i	0.285438	−	0.494394i	−0.687277	−	0.726395i	$-0.741194\pi$
$677$	5028.00	−	8708.75i	0.285438	−	0.494394i	0.972715	+	0.232002i	$0.0745275\pi$
$678$		0			0
$679$		0			0
$680$	−5376.00			−0.303177
$681$		0			0
$682$	5440.00	+	9422.36i	0.305437	+	0.529033i
$683$	3076.00	+	5327.79i	0.172328	+	0.298480i	0.939233	−	0.343280i	$-0.111538\pi$
$683$	3076.00	+	5327.79i	0.172328	+	0.298480i	−0.766905	+	0.641760i	$0.778205\pi$
$684$		0			0
$685$	−5168.00			−0.288262
$686$		0			0
$687$		0			0
$688$	1312.00	−	2272.45i	0.0727028	−	0.125925i
$689$	−956.000	−	1655.84i	−0.0528602	−	0.0915566i
$690$		0			0
$691$	−7358.00	+	12744.4i	−0.405082	+	0.701622i	−0.994331	−	0.106329i	$-0.966090\pi$
$691$	−7358.00	+	12744.4i	−0.405082	+	0.701622i	0.589249	+	0.807951i	$0.299424\pi$
$692$	−1376.00			−0.0755891
$693$		0			0
$694$	7968.00			0.435823
$695$	−12048.0	+	20867.7i	−0.657564	+	1.13893i
$696$		0			0
$697$	17640.0	+	30553.4i	0.958626	+	1.66039i
$698$	−180.000	+	311.769i	−0.00976089	+	0.0169064i
$699$		0			0
$700$		0			0
$701$	−28202.0			−1.51951			−0.759754	−	0.650211i	$-0.774681\pi$
$701$	−28202.0			−1.51951			−0.759754	+	0.650211i	$0.774681\pi$
$702$		0			0
$703$	−16428.0	−	28454.1i	−0.881357	−	1.52655i
$704$	1280.00	+	2217.03i	0.0685253	+	0.118689i
$705$		0			0
$706$	−20856.0			−1.11179
$707$		0			0
$708$		0			0
$709$	−11057.0	+	19151.3i	−0.585690	+	1.01445i	0.409099	+	0.912490i	$0.365843\pi$
$709$	−11057.0	+	19151.3i	−0.585690	+	1.01445i	−0.994789	+	0.101955i	$0.967490\pi$
$710$	−4192.00	−	7260.76i	−0.221582	−	0.383791i
$711$		0			0
$712$	−2416.00	+	4184.63i	−0.127168	+	0.220261i
$713$	−11424.0			−0.600045
$714$		0			0
$715$	1280.00			0.0669501
$716$	2784.00	−	4822.03i	0.145311	−	0.251687i
$717$		0			0
$718$	8684.00	+	15041.1i	0.451370	+	0.781797i
$719$	4644.00	−	8043.64i	0.240879	−	0.417215i	−0.720086	−	0.693885i	$-0.755898\pi$
$719$	4644.00	−	8043.64i	0.240879	−	0.417215i	0.960965	+	0.276670i	$0.0892310\pi$
$720$		0			0
$721$		0			0
$722$	30090.0			1.55102
$723$		0			0
$724$	8104.00	+	14036.5i	0.415998	+	0.720530i
$725$	−1769.00	−	3064.00i	−0.0906193	−	0.156957i
$726$		0			0
$727$	−23848.0			−1.21661			−0.608304	−	0.793704i	$-0.708150\pi$
$727$	−23848.0			−1.21661			−0.608304	+	0.793704i	$0.708150\pi$
$728$		0			0
$729$		0			0
$730$	3520.00	−	6096.82i	0.178467	−	0.309114i
$731$	−6888.00	−	11930.4i	−0.348511	−	0.603640i
$732$		0			0
$733$	−17378.0	+	30099.6i	−0.875677	+	1.51672i	−0.0196367	+	0.999807i	$0.506251\pi$
$733$	−17378.0	+	30099.6i	−0.875677	+	1.51672i	−0.856040	+	0.516909i	$0.827082\pi$
$734$	−11296.0			−0.568042
$735$		0			0
$736$	−2688.00			−0.134621
$737$	−18160.0	+	31454.0i	−0.907642	+	1.57208i
$738$		0			0
$739$	−13022.0	−	22554.8i	−0.648203	−	1.12272i	−0.983552	−	0.180626i	$-0.942187\pi$
$739$	−13022.0	−	22554.8i	−0.648203	−	1.12272i	0.335349	−	0.942094i	$-0.391146\pi$
$740$	3552.00	−	6152.24i	0.176452	−	0.305623i
$741$		0			0
$742$		0			0
$743$	−36204.0			−1.78761			−0.893806	−	0.448454i	$-0.851975\pi$
$743$	−36204.0			−1.78761			−0.893806	+	0.448454i	$0.851975\pi$
$744$		0			0
$745$	−12680.0	−	21962.4i	−0.623569	−	1.08005i
$746$	2546.00	+	4409.80i	0.124954	+	0.216427i
$747$		0			0
$748$	13440.0			0.656972
$749$		0			0
$750$		0			0
$751$	5712.00	−	9893.47i	0.277542	−	0.480716i	−0.693232	−	0.720715i	$-0.743814\pi$
$751$	5712.00	−	9893.47i	0.277542	−	0.480716i	0.970773	+	0.239999i	$0.0771470\pi$
$752$	−3904.00	−	6761.93i	−0.189314	−	0.327902i
$753$		0			0
$754$	−232.000	+	401.836i	−0.0112055	+	0.0194085i
$755$	−15040.0			−0.724982
$756$		0			0
$757$	−16622.0			−0.798067			−0.399034	−	0.916936i	$-0.630654\pi$
$757$	−16622.0			−0.798067			−0.399034	+	0.916936i	$0.630654\pi$
$758$	−8268.00	+	14320.6i	−0.396184	+	0.686210i
$759$		0			0
$760$	4736.00	+	8202.99i	0.226043	+	0.391518i
$761$	19262.0	−	33362.8i	0.917539	−	1.58922i	0.114397	−	0.993435i	$-0.463506\pi$
$761$	19262.0	−	33362.8i	0.917539	−	1.58922i	0.803141	−	0.595789i	$-0.203160\pi$
$762$		0			0
$763$		0			0
$764$	12432.0			0.588709
$765$		0			0
$766$	10872.0	+	18830.9i	0.512822	+	0.888233i
$767$	1096.00	+	1898.33i	0.0515962	+	0.0893672i
$768$		0			0
$769$	18440.0			0.864712			0.432356	−	0.901703i	$-0.357682\pi$
$769$	18440.0			0.864712			0.432356	+	0.901703i	$0.357682\pi$
$770$		0			0
$771$		0			0
$772$	−100.000	+	173.205i	−0.00466202	+	0.00807485i
$773$	6984.00	+	12096.6i	0.324964	+	0.562854i	0.981505	−	0.191436i	$-0.0613145\pi$
$773$	6984.00	+	12096.6i	0.324964	+	0.562854i	−0.656541	+	0.754290i	$0.727981\pi$
$774$		0			0
$775$	4148.00	−	7184.55i	0.192259	−	0.333002i
$776$	6656.00			0.307908
$777$		0			0
$778$	−20868.0			−0.961638
$779$	31080.0	−	53832.1i	1.42947	−	2.47591i
$780$		0			0
$781$	10480.0	+	18151.9i	0.480159	+	0.831659i
$782$	−7056.00	+	12221.4i	−0.322662	+	0.558868i
$783$		0			0
$784$		0			0
$785$	−4832.00			−0.219696
$786$		0			0
$787$	5458.00	+	9453.53i	0.247213	+	0.428186i	0.962752	−	0.270388i	$-0.0871519\pi$
$787$	5458.00	+	9453.53i	0.247213	+	0.428186i	−0.715538	+	0.698573i	$0.753819\pi$
$788$	−324.000	−	561.184i	−0.0146472	−	0.0253698i
$789$		0			0
$790$	19456.0			0.876220

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	882.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.00000 + 6.92820i) q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(-4.00000 + 6.92820i) q^{5} +8.00000 q^{8} +(-8.00000 - 13.8564i) q^{10} +(20.0000 + 34.6410i) q^{11} -4.00000 q^{13} +(-8.00000 + 13.8564i) q^{16} +(42.0000 + 72.7461i) q^{17} +(74.0000 - 128.172i) q^{19} +32.0000 q^{20} -80.0000 q^{22} +(42.0000 - 72.7461i) q^{23} +(30.5000 + 52.8275i) q^{25} +(4.00000 - 6.92820i) q^{26} -58.0000 q^{29} +(-68.0000 - 117.779i) q^{31} +(-16.0000 - 27.7128i) q^{32} -168.000 q^{34} +(111.000 - 192.258i) q^{37} +(148.000 + 256.344i) q^{38} +(-32.0000 + 55.4256i) q^{40} +420.000 q^{41} -164.000 q^{43} +(80.0000 - 138.564i) q^{44} +(84.0000 + 145.492i) q^{46} +(-244.000 + 422.620i) q^{47} -122.000 q^{50} +(8.00000 + 13.8564i) q^{52} +(239.000 + 413.960i) q^{53} -320.000 q^{55} +(58.0000 - 100.459i) q^{58} +(-274.000 - 474.582i) q^{59} +(346.000 - 599.290i) q^{61} +272.000 q^{62} +64.0000 q^{64} +(16.0000 - 27.7128i) q^{65} +(454.000 + 786.351i) q^{67} +(168.000 - 290.985i) q^{68} +524.000 q^{71} +(220.000 + 381.051i) q^{73} +(222.000 + 384.515i) q^{74} -592.000 q^{76} +(-608.000 + 1053.09i) q^{79} +(-64.0000 - 110.851i) q^{80} +(-420.000 + 727.461i) q^{82} -684.000 q^{83} -672.000 q^{85} +(164.000 - 284.056i) q^{86} +(160.000 + 277.128i) q^{88} +(-302.000 + 523.079i) q^{89} -336.000 q^{92} +(-488.000 - 845.241i) q^{94} +(592.000 + 1025.37i) q^{95} +832.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - 2q^{2} - 4q^{4} - 8q^{5} + 16q^{8} + O(q^{10}) \)	\( 2q - 2q^{2} - 4q^{4} - 8q^{5} + 16q^{8} - 16q^{10} + 40q^{11} - 8q^{13} - 16q^{16} + 84q^{17} + 148q^{19} + 64q^{20} - 160q^{22} + 84q^{23} + 61q^{25} + 8q^{26} - 116q^{29} - 136q^{31} - 32q^{32} - 336q^{34} + 222q^{37} + 296q^{38} - 64q^{40} + 840q^{41} - 328q^{43} + 160q^{44} + 168q^{46} - 488q^{47} - 244q^{50} + 16q^{52} + 478q^{53} - 640q^{55} + 116q^{58} - 548q^{59} + 692q^{61} + 544q^{62} + 128q^{64} + 32q^{65} + 908q^{67} + 336q^{68} + 1048q^{71} + 440q^{73} + 444q^{74} - 1184q^{76} - 1216q^{79} - 128q^{80} - 840q^{82} - 1368q^{83} - 1344q^{85} + 328q^{86} + 320q^{88} - 604q^{89} - 672q^{92} - 976q^{94} + 1184q^{95} + 1664q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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