LMFDB - Embedded newform 1386.4.a.l.1.1

Newspace parameters

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1386.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$81.7766472680$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1386.1

$q$-expansion

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{5} +7.00000 q^{7} +8.00000 q^{8} +14.0000 q^{10} -11.0000 q^{11} -67.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -30.0000 q^{17} -7.00000 q^{19} +28.0000 q^{20} -22.0000 q^{22} -28.0000 q^{23} -76.0000 q^{25} -134.000 q^{26} +28.0000 q^{28} -121.000 q^{29} -310.000 q^{31} +32.0000 q^{32} -60.0000 q^{34} +49.0000 q^{35} -71.0000 q^{37} -14.0000 q^{38} +56.0000 q^{40} +180.000 q^{41} -108.000 q^{43} -44.0000 q^{44} -56.0000 q^{46} -71.0000 q^{47} +49.0000 q^{49} -152.000 q^{50} -268.000 q^{52} -128.000 q^{53} -77.0000 q^{55} +56.0000 q^{56} -242.000 q^{58} +429.000 q^{59} +22.0000 q^{61} -620.000 q^{62} +64.0000 q^{64} -469.000 q^{65} -803.000 q^{67} -120.000 q^{68} +98.0000 q^{70} -468.000 q^{71} -117.000 q^{73} -142.000 q^{74} -28.0000 q^{76} -77.0000 q^{77} -96.0000 q^{79} +112.000 q^{80} +360.000 q^{82} +1122.00 q^{83} -210.000 q^{85} -216.000 q^{86} -88.0000 q^{88} +1146.00 q^{89} -469.000 q^{91} -112.000 q^{92} -142.000 q^{94} -49.0000 q^{95} -92.0000 q^{97} +98.0000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	7.00000	0.626099	0.313050	−	0.949737i	$-0.398649\pi$
$5$	7.00000	0.626099	0.313050	+	0.949737i	$0.398649\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	14.0000	0.442719
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−67.0000	−1.42942	−0.714710	−	0.699421i	$-0.753441\pi$
$13$	−67.0000	−1.42942	−0.714710	+	0.699421i	$0.753441\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	0	0
$19$	−7.00000	−0.0845216	−0.0422608	−	0.999107i	$-0.513456\pi$
$19$	−7.00000	−0.0845216	−0.0422608	+	0.999107i	$0.513456\pi$
$20$	28.0000	0.313050
$21$	0	0
$22$	−22.0000	−0.213201
$23$	−28.0000	−0.253844	−0.126922	−	0.991913i	$-0.540510\pi$
$23$	−28.0000	−0.253844	−0.126922	+	0.991913i	$0.540510\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	−134.000	−1.01075
$27$	0	0
$28$	28.0000	0.188982
$29$	−121.000	−0.774798	−0.387399	−	0.921912i	$-0.626626\pi$
$29$	−121.000	−0.774798	−0.387399	+	0.921912i	$0.626626\pi$
$30$	0	0
$31$	−310.000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−310.000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−60.0000	−0.302645
$35$	49.0000	0.236643
$36$	0	0
$37$	−71.0000	−0.315468	−0.157734	−	0.987482i	$-0.550419\pi$
$37$	−71.0000	−0.315468	−0.157734	+	0.987482i	$0.550419\pi$
$38$	−14.0000	−0.0597658
$39$	0	0
$40$	56.0000	0.221359
$41$	180.000	0.685641	0.342820	−	0.939401i	$-0.388618\pi$
$41$	180.000	0.685641	0.342820	+	0.939401i	$0.388618\pi$
$42$	0	0
$43$	−108.000	−0.383020	−0.191510	−	0.981491i	$-0.561338\pi$
$43$	−108.000	−0.383020	−0.191510	+	0.981491i	$0.561338\pi$
$44$	−44.0000	−0.150756
$45$	0	0
$46$	−56.0000	−0.179495
$47$	−71.0000	−0.220349	−0.110175	−	0.993912i	$-0.535141\pi$
$47$	−71.0000	−0.220349	−0.110175	+	0.993912i	$0.535141\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−152.000	−0.429921
$51$	0	0
$52$	−268.000	−0.714710
$53$	−128.000	−0.331739	−0.165869	−	0.986148i	$-0.553043\pi$
$53$	−128.000	−0.331739	−0.165869	+	0.986148i	$0.553043\pi$
$54$	0	0
$55$	−77.0000	−0.188776
$56$	56.0000	0.133631
$57$	0	0
$58$	−242.000	−0.547865
$59$	429.000	0.946628	0.473314	−	0.880894i	$-0.343058\pi$
$59$	429.000	0.946628	0.473314	+	0.880894i	$0.343058\pi$
$60$	0	0
$61$	22.0000	0.0461772	0.0230886	−	0.999733i	$-0.492650\pi$
$61$	22.0000	0.0461772	0.0230886	+	0.999733i	$0.492650\pi$
$62$	−620.000	−1.27000
$63$	0	0
$64$	64.0000	0.125000
$65$	−469.000	−0.894958
$66$	0	0
$67$	−803.000	−1.46421	−0.732105	−	0.681192i	$-0.761462\pi$
$67$	−803.000	−1.46421	−0.732105	+	0.681192i	$0.761462\pi$
$68$	−120.000	−0.214002
$69$	0	0
$70$	98.0000	0.167332
$71$	−468.000	−0.782273	−0.391136	−	0.920333i	$-0.627918\pi$
$71$	−468.000	−0.782273	−0.391136	+	0.920333i	$0.627918\pi$
$72$	0	0
$73$	−117.000	−0.187586	−0.0937932	−	0.995592i	$-0.529899\pi$
$73$	−117.000	−0.187586	−0.0937932	+	0.995592i	$0.529899\pi$
$74$	−142.000	−0.223070
$75$	0	0
$76$	−28.0000	−0.0422608
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−96.0000	−0.136720	−0.0683598	−	0.997661i	$-0.521777\pi$
$79$	−96.0000	−0.136720	−0.0683598	+	0.997661i	$0.521777\pi$
$80$	112.000	0.156525
$81$	0	0
$82$	360.000	0.484821
$83$	1122.00	1.48380	0.741901	−	0.670510i	$-0.233925\pi$
$83$	1122.00	1.48380	0.741901	+	0.670510i	$0.233925\pi$
$84$	0	0
$85$	−210.000	−0.267973
$86$	−216.000	−0.270836
$87$	0	0
$88$	−88.0000	−0.106600
$89$	1146.00	1.36490	0.682448	−	0.730934i	$-0.260915\pi$
$89$	1146.00	1.36490	0.682448	+	0.730934i	$0.260915\pi$
$90$	0	0
$91$	−469.000	−0.540270
$92$	−112.000	−0.126922
$93$	0	0
$94$	−142.000	−0.155810
$95$	−49.0000	−0.0529189
$96$	0	0
$97$	−92.0000	−0.0963009	−0.0481504	−	0.998840i	$-0.515333\pi$
$97$	−92.0000	−0.0963009	−0.0481504	+	0.998840i	$0.515333\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−304.000	−0.304000
$101$	202.000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	202.000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	−1798.00	−1.72002	−0.860011	−	0.510276i	$-0.829543\pi$
$103$	−1798.00	−1.72002	−0.860011	+	0.510276i	$0.829543\pi$
$104$	−536.000	−0.505376
$105$	0	0
$106$	−256.000	−0.234575
$107$	1431.00	1.29290	0.646449	−	0.762958i	$-0.276253\pi$
$107$	1431.00	1.29290	0.646449	+	0.762958i	$0.276253\pi$
$108$	0	0
$109$	−278.000	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$109$	−278.000	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$110$	−154.000	−0.133485
$111$	0	0
$112$	112.000	0.0944911
$113$	2322.00	1.93306	0.966528	−	0.256560i	$-0.0825892\pi$
$113$	2322.00	1.93306	0.966528	+	0.256560i	$0.0825892\pi$
$114$	0	0
$115$	−196.000	−0.158931
$116$	−484.000	−0.387399
$117$	0	0
$118$	858.000	0.669367
$119$	−210.000	−0.161770
$120$	0	0
$121$	121.000	0.0909091
$122$	44.0000	0.0326522
$123$	0	0
$124$	−1240.00	−0.898027
$125$	−1407.00	−1.00677
$126$	0	0
$127$	2254.00	1.57488	0.787442	−	0.616389i	$-0.211405\pi$
$127$	2254.00	1.57488	0.787442	+	0.616389i	$0.211405\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−938.000	−0.632831
$131$	−2196.00	−1.46462	−0.732311	−	0.680971i	$-0.761558\pi$
$131$	−2196.00	−1.46462	−0.732311	+	0.680971i	$0.761558\pi$
$132$	0	0
$133$	−49.0000	−0.0319462
$134$	−1606.00	−1.03535
$135$	0	0
$136$	−240.000	−0.151322
$137$	−766.000	−0.477692	−0.238846	−	0.971057i	$-0.576769\pi$
$137$	−766.000	−0.477692	−0.238846	+	0.971057i	$0.576769\pi$
$138$	0	0
$139$	−1080.00	−0.659024	−0.329512	−	0.944151i	$-0.606884\pi$
$139$	−1080.00	−0.659024	−0.329512	+	0.944151i	$0.606884\pi$
$140$	196.000	0.118322
$141$	0	0
$142$	−936.000	−0.553151
$143$	737.000	0.430986
$144$	0	0
$145$	−847.000	−0.485100
$146$	−234.000	−0.132644
$147$	0	0
$148$	−284.000	−0.157734
$149$	−1373.00	−0.754903	−0.377451	−	0.926029i	$-0.623200\pi$
$149$	−1373.00	−0.754903	−0.377451	+	0.926029i	$0.623200\pi$
$150$	0	0
$151$	−3210.00	−1.72997	−0.864987	−	0.501794i	$-0.832673\pi$
$151$	−3210.00	−1.72997	−0.864987	+	0.501794i	$0.832673\pi$
$152$	−56.0000	−0.0298829
$153$	0	0
$154$	−154.000	−0.0805823
$155$	−2170.00	−1.12451
$156$	0	0
$157$	452.000	0.229768	0.114884	−	0.993379i	$-0.463350\pi$
$157$	452.000	0.229768	0.114884	+	0.993379i	$0.463350\pi$
$158$	−192.000	−0.0966753
$159$	0	0
$160$	224.000	0.110680
$161$	−196.000	−0.0959439
$162$	0	0
$163$	−11.0000	−0.00528581	−0.00264290	−	0.999997i	$-0.500841\pi$
$163$	−11.0000	−0.00528581	−0.00264290	+	0.999997i	$0.500841\pi$
$164$	720.000	0.342820
$165$	0	0
$166$	2244.00	1.04921
$167$	−504.000	−0.233537	−0.116769	−	0.993159i	$-0.537254\pi$
$167$	−504.000	−0.233537	−0.116769	+	0.993159i	$0.537254\pi$
$168$	0	0
$169$	2292.00	1.04324
$170$	−420.000	−0.189485
$171$	0	0
$172$	−432.000	−0.191510
$173$	−1688.00	−0.741828	−0.370914	−	0.928667i	$-0.620956\pi$
$173$	−1688.00	−0.741828	−0.370914	+	0.928667i	$0.620956\pi$
$174$	0	0
$175$	−532.000	−0.229802
$176$	−176.000	−0.0753778
$177$	0	0
$178$	2292.00	0.965127
$179$	−460.000	−0.192078	−0.0960391	−	0.995378i	$-0.530617\pi$
$179$	−460.000	−0.192078	−0.0960391	+	0.995378i	$0.530617\pi$
$180$	0	0
$181$	2296.00	0.942875	0.471437	−	0.881900i	$-0.343735\pi$
$181$	2296.00	0.942875	0.471437	+	0.881900i	$0.343735\pi$
$182$	−938.000	−0.382028
$183$	0	0
$184$	−224.000	−0.0897473
$185$	−497.000	−0.197514
$186$	0	0
$187$	330.000	0.129048
$188$	−284.000	−0.110175
$189$	0	0
$190$	−98.0000	−0.0374193
$191$	−878.000	−0.332617	−0.166309	−	0.986074i	$-0.553185\pi$
$191$	−878.000	−0.332617	−0.166309	+	0.986074i	$0.553185\pi$
$192$	0	0
$193$	−2268.00	−0.845877	−0.422938	−	0.906158i	$-0.639001\pi$
$193$	−2268.00	−0.845877	−0.422938	+	0.906158i	$0.639001\pi$
$194$	−184.000	−0.0680950
$195$	0	0
$196$	196.000	0.0714286
$197$	−5298.00	−1.91608	−0.958038	−	0.286642i	$-0.907461\pi$
$197$	−5298.00	−1.91608	−0.958038	+	0.286642i	$0.907461\pi$
$198$	0	0
$199$	1044.00	0.371895	0.185948	−	0.982560i	$-0.440464\pi$
$199$	1044.00	0.371895	0.185948	+	0.982560i	$0.440464\pi$
$200$	−608.000	−0.214960
$201$	0	0
$202$	404.000	0.140720
$203$	−847.000	−0.292846
$204$	0	0
$205$	1260.00	0.429279
$206$	−3596.00	−1.21624
$207$	0	0
$208$	−1072.00	−0.357355
$209$	77.0000	0.0254842
$210$	0	0
$211$	4566.00	1.48975	0.744873	−	0.667206i	$-0.232510\pi$
$211$	4566.00	1.48975	0.744873	+	0.667206i	$0.232510\pi$
$212$	−512.000	−0.165869
$213$	0	0
$214$	2862.00	0.914216
$215$	−756.000	−0.239808
$216$	0	0
$217$	−2170.00	−0.678844
$218$	−556.000	−0.172739
$219$	0	0
$220$	−308.000	−0.0943880
$221$	2010.00	0.611797
$222$	0	0
$223$	198.000	0.0594577	0.0297288	−	0.999558i	$-0.490536\pi$
$223$	198.000	0.0594577	0.0297288	+	0.999558i	$0.490536\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	4644.00	1.36688
$227$	−3762.00	−1.09997	−0.549984	−	0.835175i	$-0.685366\pi$
$227$	−3762.00	−1.09997	−0.549984	+	0.835175i	$0.685366\pi$
$228$	0	0
$229$	−704.000	−0.203151	−0.101576	−	0.994828i	$-0.532388\pi$
$229$	−704.000	−0.203151	−0.101576	+	0.994828i	$0.532388\pi$
$230$	−392.000	−0.112381
$231$	0	0
$232$	−968.000	−0.273932
$233$	−2210.00	−0.621382	−0.310691	−	0.950511i	$-0.600560\pi$
$233$	−2210.00	−0.621382	−0.310691	+	0.950511i	$0.600560\pi$
$234$	0	0
$235$	−497.000	−0.137960
$236$	1716.00	0.473314
$237$	0	0
$238$	−420.000	−0.114389
$239$	−2451.00	−0.663356	−0.331678	−	0.943393i	$-0.607615\pi$
$239$	−2451.00	−0.663356	−0.331678	+	0.943393i	$0.607615\pi$
$240$	0	0
$241$	25.0000	0.00668212	0.00334106	−	0.999994i	$-0.498937\pi$
$241$	25.0000	0.00668212	0.00334106	+	0.999994i	$0.498937\pi$
$242$	242.000	0.0642824
$243$	0	0
$244$	88.0000	0.0230886
$245$	343.000	0.0894427
$246$	0	0
$247$	469.000	0.120817
$248$	−2480.00	−0.635001
$249$	0	0
$250$	−2814.00	−0.711892
$251$	5267.00	1.32450	0.662251	−	0.749282i	$-0.269601\pi$
$251$	5267.00	1.32450	0.662251	+	0.749282i	$0.269601\pi$
$252$	0	0
$253$	308.000	0.0765367
$254$	4508.00	1.11361
$255$	0	0
$256$	256.000	0.0625000
$257$	1623.00	0.393930	0.196965	−	0.980411i	$-0.436891\pi$
$257$	1623.00	0.393930	0.196965	+	0.980411i	$0.436891\pi$
$258$	0	0
$259$	−497.000	−0.119236
$260$	−1876.00	−0.447479
$261$	0	0
$262$	−4392.00	−1.03564
$263$	2475.00	0.580285	0.290143	−	0.956983i	$-0.406297\pi$
$263$	2475.00	0.580285	0.290143	+	0.956983i	$0.406297\pi$
$264$	0	0
$265$	−896.000	−0.207701
$266$	−98.0000	−0.0225893
$267$	0	0
$268$	−3212.00	−0.732105
$269$	−1746.00	−0.395745	−0.197873	−	0.980228i	$-0.563403\pi$
$269$	−1746.00	−0.395745	−0.197873	+	0.980228i	$0.563403\pi$
$270$	0	0
$271$	1653.00	0.370526	0.185263	−	0.982689i	$-0.440686\pi$
$271$	1653.00	0.370526	0.185263	+	0.982689i	$0.440686\pi$
$272$	−480.000	−0.107001
$273$	0	0
$274$	−1532.00	−0.337779
$275$	836.000	0.183319
$276$	0	0
$277$	−2056.00	−0.445968	−0.222984	−	0.974822i	$-0.571580\pi$
$277$	−2056.00	−0.445968	−0.222984	+	0.974822i	$0.571580\pi$
$278$	−2160.00	−0.466001
$279$	0	0
$280$	392.000	0.0836660
$281$	−837.000	−0.177691	−0.0888456	−	0.996045i	$-0.528318\pi$
$281$	−837.000	−0.177691	−0.0888456	+	0.996045i	$0.528318\pi$
$282$	0	0
$283$	8969.00	1.88393	0.941964	−	0.335713i	$-0.108977\pi$
$283$	8969.00	1.88393	0.941964	+	0.335713i	$0.108977\pi$
$284$	−1872.00	−0.391136
$285$	0	0
$286$	1474.00	0.304753
$287$	1260.00	0.259148
$288$	0	0
$289$	−4013.00	−0.816813
$290$	−1694.00	−0.343018
$291$	0	0
$292$	−468.000	−0.0937932
$293$	−5120.00	−1.02087	−0.510433	−	0.859918i	$-0.670515\pi$
$293$	−5120.00	−1.02087	−0.510433	+	0.859918i	$0.670515\pi$
$294$	0	0
$295$	3003.00	0.592683
$296$	−568.000	−0.111535
$297$	0	0
$298$	−2746.00	−0.533797
$299$	1876.00	0.362849
$300$	0	0
$301$	−756.000	−0.144768
$302$	−6420.00	−1.22328
$303$	0	0
$304$	−112.000	−0.0211304
$305$	154.000	0.0289115
$306$	0	0
$307$	−2892.00	−0.537639	−0.268819	−	0.963191i	$-0.586634\pi$
$307$	−2892.00	−0.537639	−0.268819	+	0.963191i	$0.586634\pi$
$308$	−308.000	−0.0569803
$309$	0	0
$310$	−4340.00	−0.795147
$311$	−3208.00	−0.584916	−0.292458	−	0.956278i	$-0.594473\pi$
$311$	−3208.00	−0.584916	−0.292458	+	0.956278i	$0.594473\pi$
$312$	0	0
$313$	3550.00	0.641079	0.320540	−	0.947235i	$-0.396136\pi$
$313$	3550.00	0.641079	0.320540	+	0.947235i	$0.396136\pi$
$314$	904.000	0.162470
$315$	0	0
$316$	−384.000	−0.0683598
$317$	5524.00	0.978734	0.489367	−	0.872078i	$-0.337228\pi$
$317$	5524.00	0.978734	0.489367	+	0.872078i	$0.337228\pi$
$318$	0	0
$319$	1331.00	0.233610
$320$	448.000	0.0782624
$321$	0	0
$322$	−392.000	−0.0678426
$323$	210.000	0.0361756
$324$	0	0
$325$	5092.00	0.869087
$326$	−22.0000	−0.00373763
$327$	0	0
$328$	1440.00	0.242411
$329$	−497.000	−0.0832842
$330$	0	0
$331$	−2144.00	−0.356027	−0.178013	−	0.984028i	$-0.556967\pi$
$331$	−2144.00	−0.356027	−0.178013	+	0.984028i	$0.556967\pi$
$332$	4488.00	0.741901
$333$	0	0
$334$	−1008.00	−0.165136
$335$	−5621.00	−0.916740
$336$	0	0
$337$	−884.000	−0.142892	−0.0714459	−	0.997444i	$-0.522761\pi$
$337$	−884.000	−0.142892	−0.0714459	+	0.997444i	$0.522761\pi$
$338$	4584.00	0.737683
$339$	0	0
$340$	−840.000	−0.133986
$341$	3410.00	0.541530
$342$	0	0
$343$	343.000	0.0539949
$344$	−864.000	−0.135418
$345$	0	0
$346$	−3376.00	−0.524552
$347$	4040.00	0.625010	0.312505	−	0.949916i	$-0.398832\pi$
$347$	4040.00	0.625010	0.312505	+	0.949916i	$0.398832\pi$
$348$	0	0
$349$	1225.00	0.187888	0.0939438	−	0.995578i	$-0.470053\pi$
$349$	1225.00	0.187888	0.0939438	+	0.995578i	$0.470053\pi$
$350$	−1064.00	−0.162495
$351$	0	0
$352$	−352.000	−0.0533002
$353$	−2483.00	−0.374382	−0.187191	−	0.982324i	$-0.559938\pi$
$353$	−2483.00	−0.374382	−0.187191	+	0.982324i	$0.559938\pi$
$354$	0	0
$355$	−3276.00	−0.489780
$356$	4584.00	0.682448
$357$	0	0
$358$	−920.000	−0.135820
$359$	−7804.00	−1.14730	−0.573648	−	0.819102i	$-0.694472\pi$
$359$	−7804.00	−1.14730	−0.573648	+	0.819102i	$0.694472\pi$
$360$	0	0
$361$	−6810.00	−0.992856
$362$	4592.00	0.666713
$363$	0	0
$364$	−1876.00	−0.270135
$365$	−819.000	−0.117448
$366$	0	0
$367$	3074.00	0.437225	0.218612	−	0.975812i	$-0.429847\pi$
$367$	3074.00	0.437225	0.218612	+	0.975812i	$0.429847\pi$
$368$	−448.000	−0.0634609
$369$	0	0
$370$	−994.000	−0.139664
$371$	−896.000	−0.125385
$372$	0	0
$373$	674.000	0.0935614	0.0467807	−	0.998905i	$-0.485104\pi$
$373$	674.000	0.0935614	0.0467807	+	0.998905i	$0.485104\pi$
$374$	660.000	0.0912508
$375$	0	0
$376$	−568.000	−0.0779052
$377$	8107.00	1.10751
$378$	0	0
$379$	12051.0	1.63329	0.816647	−	0.577138i	$-0.195830\pi$
$379$	12051.0	1.63329	0.816647	+	0.577138i	$0.195830\pi$
$380$	−196.000	−0.0264594
$381$	0	0
$382$	−1756.00	−0.235196
$383$	12664.0	1.68956	0.844778	−	0.535116i	$-0.179732\pi$
$383$	12664.0	1.68956	0.844778	+	0.535116i	$0.179732\pi$
$384$	0	0
$385$	−539.000	−0.0713506
$386$	−4536.00	−0.598125
$387$	0	0
$388$	−368.000	−0.0481504
$389$	−264.000	−0.0344096	−0.0172048	−	0.999852i	$-0.505477\pi$
$389$	−264.000	−0.0344096	−0.0172048	+	0.999852i	$0.505477\pi$
$390$	0	0
$391$	840.000	0.108646
$392$	392.000	0.0505076
$393$	0	0
$394$	−10596.0	−1.35487
$395$	−672.000	−0.0856000
$396$	0	0
$397$	−14394.0	−1.81968	−0.909842	−	0.414956i	$-0.863797\pi$
$397$	−14394.0	−1.81968	−0.909842	+	0.414956i	$0.863797\pi$
$398$	2088.00	0.262970
$399$	0	0
$400$	−1216.00	−0.152000
$401$	12304.0	1.53225	0.766125	−	0.642691i	$-0.222182\pi$
$401$	12304.0	1.53225	0.766125	+	0.642691i	$0.222182\pi$
$402$	0	0
$403$	20770.0	2.56731
$404$	808.000	0.0995037
$405$	0	0
$406$	−1694.00	−0.207073
$407$	781.000	0.0951173
$408$	0	0
$409$	−3154.00	−0.381309	−0.190654	−	0.981657i	$-0.561061\pi$
$409$	−3154.00	−0.381309	−0.190654	+	0.981657i	$0.561061\pi$
$410$	2520.00	0.303546
$411$	0	0
$412$	−7192.00	−0.860011
$413$	3003.00	0.357792
$414$	0	0
$415$	7854.00	0.929006
$416$	−2144.00	−0.252688
$417$	0	0
$418$	154.000	0.0180201
$419$	4635.00	0.540417	0.270208	−	0.962802i	$-0.412907\pi$
$419$	4635.00	0.540417	0.270208	+	0.962802i	$0.412907\pi$
$420$	0	0
$421$	−7265.00	−0.841032	−0.420516	−	0.907285i	$-0.638151\pi$
$421$	−7265.00	−0.841032	−0.420516	+	0.907285i	$0.638151\pi$
$422$	9132.00	1.05341
$423$	0	0
$424$	−1024.00	−0.117287
$425$	2280.00	0.260226
$426$	0	0
$427$	154.000	0.0174534
$428$	5724.00	0.646449
$429$	0	0
$430$	−1512.00	−0.169570
$431$	−7589.00	−0.848142	−0.424071	−	0.905629i	$-0.639399\pi$
$431$	−7589.00	−0.848142	−0.424071	+	0.905629i	$0.639399\pi$
$432$	0	0
$433$	5164.00	0.573132	0.286566	−	0.958061i	$-0.407486\pi$
$433$	5164.00	0.573132	0.286566	+	0.958061i	$0.407486\pi$
$434$	−4340.00	−0.480015
$435$	0	0
$436$	−1112.00	−0.122145
$437$	196.000	0.0214553
$438$	0	0
$439$	14221.0	1.54608	0.773042	−	0.634354i	$-0.218734\pi$
$439$	14221.0	1.54608	0.773042	+	0.634354i	$0.218734\pi$
$440$	−616.000	−0.0667424
$441$	0	0
$442$	4020.00	0.432606
$443$	−6924.00	−0.742594	−0.371297	−	0.928514i	$-0.621087\pi$
$443$	−6924.00	−0.742594	−0.371297	+	0.928514i	$0.621087\pi$
$444$	0	0
$445$	8022.00	0.854560
$446$	396.000	0.0420429
$447$	0	0
$448$	448.000	0.0472456
$449$	−2524.00	−0.265289	−0.132645	−	0.991164i	$-0.542347\pi$
$449$	−2524.00	−0.265289	−0.132645	+	0.991164i	$0.542347\pi$
$450$	0	0
$451$	−1980.00	−0.206729
$452$	9288.00	0.966528
$453$	0	0
$454$	−7524.00	−0.777795
$455$	−3283.00	−0.338262
$456$	0	0
$457$	1928.00	0.197348	0.0986740	−	0.995120i	$-0.468540\pi$
$457$	1928.00	0.197348	0.0986740	+	0.995120i	$0.468540\pi$
$458$	−1408.00	−0.143650
$459$	0	0
$460$	−784.000	−0.0794656
$461$	12852.0	1.29843	0.649216	−	0.760604i	$-0.275097\pi$
$461$	12852.0	1.29843	0.649216	+	0.760604i	$0.275097\pi$
$462$	0	0
$463$	4741.00	0.475881	0.237941	−	0.971280i	$-0.423528\pi$
$463$	4741.00	0.475881	0.237941	+	0.971280i	$0.423528\pi$
$464$	−1936.00	−0.193699
$465$	0	0
$466$	−4420.00	−0.439383
$467$	6809.00	0.674696	0.337348	−	0.941380i	$-0.390470\pi$
$467$	6809.00	0.674696	0.337348	+	0.941380i	$0.390470\pi$
$468$	0	0
$469$	−5621.00	−0.553419
$470$	−994.000	−0.0975528
$471$	0	0
$472$	3432.00	0.334683
$473$	1188.00	0.115485
$474$	0	0
$475$	532.000	0.0513891
$476$	−840.000	−0.0808852
$477$	0	0
$478$	−4902.00	−0.469063
$479$	−1734.00	−0.165404	−0.0827020	−	0.996574i	$-0.526355\pi$
$479$	−1734.00	−0.165404	−0.0827020	+	0.996574i	$0.526355\pi$
$480$	0	0
$481$	4757.00	0.450937
$482$	50.0000	0.00472497
$483$	0	0
$484$	484.000	0.0454545
$485$	−644.000	−0.0602939
$486$	0	0
$487$	−7368.00	−0.685577	−0.342788	−	0.939413i	$-0.611371\pi$
$487$	−7368.00	−0.685577	−0.342788	+	0.939413i	$0.611371\pi$
$488$	176.000	0.0163261
$489$	0	0
$490$	686.000	0.0632456
$491$	−13201.0	−1.21335	−0.606673	−	0.794952i	$-0.707496\pi$
$491$	−13201.0	−1.21335	−0.606673	+	0.794952i	$0.707496\pi$
$492$	0	0
$493$	3630.00	0.331617
$494$	938.000	0.0854304
$495$	0	0
$496$	−4960.00	−0.449013
$497$	−3276.00	−0.295671
$498$	0	0
$499$	16539.0	1.48374	0.741871	−	0.670543i	$-0.233939\pi$
$499$	16539.0	1.48374	0.741871	+	0.670543i	$0.233939\pi$
$500$	−5628.00	−0.503384
$501$	0	0
$502$	10534.0	0.936565
$503$	14780.0	1.31015	0.655077	−	0.755562i	$-0.272636\pi$
$503$	14780.0	1.31015	0.655077	+	0.755562i	$0.272636\pi$
$504$	0	0
$505$	1414.00	0.124598
$506$	616.000	0.0541196
$507$	0	0
$508$	9016.00	0.787442
$509$	11766.0	1.02459	0.512297	−	0.858808i	$-0.328795\pi$
$509$	11766.0	1.02459	0.512297	+	0.858808i	$0.328795\pi$
$510$	0	0
$511$	−819.000	−0.0709010
$512$	512.000	0.0441942
$513$	0	0
$514$	3246.00	0.278550
$515$	−12586.0	−1.07690
$516$	0	0
$517$	781.000	0.0664378
$518$	−994.000	−0.0843125
$519$	0	0
$520$	−3752.00	−0.316416
$521$	13503.0	1.13546	0.567732	−	0.823213i	$-0.307821\pi$
$521$	13503.0	1.13546	0.567732	+	0.823213i	$0.307821\pi$
$522$	0	0
$523$	−18681.0	−1.56188	−0.780940	−	0.624606i	$-0.785259\pi$
$523$	−18681.0	−1.56188	−0.780940	+	0.624606i	$0.785259\pi$
$524$	−8784.00	−0.732311
$525$	0	0
$526$	4950.00	0.410324
$527$	9300.00	0.768718
$528$	0	0
$529$	−11383.0	−0.935563
$530$	−1792.00	−0.146867
$531$	0	0
$532$	−196.000	−0.0159731
$533$	−12060.0	−0.980069
$534$	0	0
$535$	10017.0	0.809482
$536$	−6424.00	−0.517676
$537$	0	0
$538$	−3492.00	−0.279834
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−2000.00	−0.158940	−0.0794702	−	0.996837i	$-0.525323\pi$
$541$	−2000.00	−0.158940	−0.0794702	+	0.996837i	$0.525323\pi$
$542$	3306.00	0.262002
$543$	0	0
$544$	−960.000	−0.0756611
$545$	−1946.00	−0.152950
$546$	0	0
$547$	19368.0	1.51392	0.756961	−	0.653459i	$-0.226683\pi$
$547$	19368.0	1.51392	0.756961	+	0.653459i	$0.226683\pi$
$548$	−3064.00	−0.238846
$549$	0	0
$550$	1672.00	0.129626
$551$	847.000	0.0654871
$552$	0	0
$553$	−672.000	−0.0516751
$554$	−4112.00	−0.315347
$555$	0	0
$556$	−4320.00	−0.329512
$557$	−4669.00	−0.355174	−0.177587	−	0.984105i	$-0.556829\pi$
$557$	−4669.00	−0.355174	−0.177587	+	0.984105i	$0.556829\pi$
$558$	0	0
$559$	7236.00	0.547496
$560$	784.000	0.0591608
$561$	0	0
$562$	−1674.00	−0.125647
$563$	9324.00	0.697975	0.348987	−	0.937127i	$-0.386526\pi$
$563$	9324.00	0.697975	0.348987	+	0.937127i	$0.386526\pi$
$564$	0	0
$565$	16254.0	1.21028
$566$	17938.0	1.33214
$567$	0	0
$568$	−3744.00	−0.276575
$569$	−25646.0	−1.88952	−0.944759	−	0.327765i	$-0.893705\pi$
$569$	−25646.0	−1.88952	−0.944759	+	0.327765i	$0.893705\pi$
$570$	0	0
$571$	−12928.0	−0.947496	−0.473748	−	0.880661i	$-0.657099\pi$
$571$	−12928.0	−0.947496	−0.473748	+	0.880661i	$0.657099\pi$
$572$	2948.00	0.215493
$573$	0	0
$574$	2520.00	0.183245
$575$	2128.00	0.154337
$576$	0	0
$577$	−14862.0	−1.07229	−0.536147	−	0.844125i	$-0.680121\pi$
$577$	−14862.0	−1.07229	−0.536147	+	0.844125i	$0.680121\pi$
$578$	−8026.00	−0.577574
$579$	0	0
$580$	−3388.00	−0.242550
$581$	7854.00	0.560824
$582$	0	0
$583$	1408.00	0.100023
$584$	−936.000	−0.0663218
$585$	0	0
$586$	−10240.0	−0.721861
$587$	13383.0	0.941015	0.470507	−	0.882396i	$-0.344071\pi$
$587$	13383.0	0.941015	0.470507	+	0.882396i	$0.344071\pi$
$588$	0	0
$589$	2170.00	0.151805
$590$	6006.00	0.419090
$591$	0	0
$592$	−1136.00	−0.0788671
$593$	10152.0	0.703023	0.351512	−	0.936184i	$-0.385668\pi$
$593$	10152.0	0.703023	0.351512	+	0.936184i	$0.385668\pi$
$594$	0	0
$595$	−1470.00	−0.101284
$596$	−5492.00	−0.377451
$597$	0	0
$598$	3752.00	0.256573
$599$	−16592.0	−1.13177	−0.565885	−	0.824484i	$-0.691466\pi$
$599$	−16592.0	−1.13177	−0.565885	+	0.824484i	$0.691466\pi$
$600$	0	0
$601$	−1001.00	−0.0679395	−0.0339698	−	0.999423i	$-0.510815\pi$
$601$	−1001.00	−0.0679395	−0.0339698	+	0.999423i	$0.510815\pi$
$602$	−1512.00	−0.102366
$603$	0	0
$604$	−12840.0	−0.864987
$605$	847.000	0.0569181
$606$	0	0
$607$	−18427.0	−1.23217	−0.616086	−	0.787679i	$-0.711283\pi$
$607$	−18427.0	−1.23217	−0.616086	+	0.787679i	$0.711283\pi$
$608$	−224.000	−0.0149414
$609$	0	0
$610$	308.000	0.0204435
$611$	4757.00	0.314972
$612$	0	0
$613$	5606.00	0.369371	0.184685	−	0.982798i	$-0.440873\pi$
$613$	5606.00	0.369371	0.184685	+	0.982798i	$0.440873\pi$
$614$	−5784.00	−0.380168
$615$	0	0
$616$	−616.000	−0.0402911
$617$	−6378.00	−0.416157	−0.208078	−	0.978112i	$-0.566721\pi$
$617$	−6378.00	−0.416157	−0.208078	+	0.978112i	$0.566721\pi$
$618$	0	0
$619$	−4070.00	−0.264276	−0.132138	−	0.991231i	$-0.542184\pi$
$619$	−4070.00	−0.264276	−0.132138	+	0.991231i	$0.542184\pi$
$620$	−8680.00	−0.562254
$621$	0	0
$622$	−6416.00	−0.413598
$623$	8022.00	0.515882
$624$	0	0
$625$	−349.000	−0.0223360
$626$	7100.00	0.453312
$627$	0	0
$628$	1808.00	0.114884
$629$	2130.00	0.135022
$630$	0	0
$631$	23252.0	1.46695	0.733477	−	0.679715i	$-0.237896\pi$
$631$	23252.0	1.46695	0.733477	+	0.679715i	$0.237896\pi$
$632$	−768.000	−0.0483377
$633$	0	0
$634$	11048.0	0.692070
$635$	15778.0	0.986033
$636$	0	0
$637$	−3283.00	−0.204203
$638$	2662.00	0.165187
$639$	0	0
$640$	896.000	0.0553399
$641$	−3758.00	−0.231563	−0.115782	−	0.993275i	$-0.536937\pi$
$641$	−3758.00	−0.231563	−0.115782	+	0.993275i	$0.536937\pi$
$642$	0	0
$643$	13068.0	0.801480	0.400740	−	0.916192i	$-0.368753\pi$
$643$	13068.0	0.801480	0.400740	+	0.916192i	$0.368753\pi$
$644$	−784.000	−0.0479719
$645$	0	0
$646$	420.000	0.0255800
$647$	31027.0	1.88531	0.942656	−	0.333765i	$-0.108319\pi$
$647$	31027.0	1.88531	0.942656	+	0.333765i	$0.108319\pi$
$648$	0	0
$649$	−4719.00	−0.285419
$650$	10184.0	0.614537
$651$	0	0
$652$	−44.0000	−0.00264290
$653$	18456.0	1.10603	0.553016	−	0.833171i	$-0.313477\pi$
$653$	18456.0	1.10603	0.553016	+	0.833171i	$0.313477\pi$
$654$	0	0
$655$	−15372.0	−0.916998
$656$	2880.00	0.171410
$657$	0	0
$658$	−994.000	−0.0588908
$659$	−11141.0	−0.658561	−0.329281	−	0.944232i	$-0.606806\pi$
$659$	−11141.0	−0.658561	−0.329281	+	0.944232i	$0.606806\pi$
$660$	0	0
$661$	5172.00	0.304338	0.152169	−	0.988354i	$-0.451374\pi$
$661$	5172.00	0.304338	0.152169	+	0.988354i	$0.451374\pi$
$662$	−4288.00	−0.251749
$663$	0	0
$664$	8976.00	0.524603
$665$	−343.000	−0.0200015
$666$	0	0
$667$	3388.00	0.196677
$668$	−2016.00	−0.116769
$669$	0	0
$670$	−11242.0	−0.648233
$671$	−242.000	−0.0139230
$672$	0	0
$673$	−10460.0	−0.599113	−0.299557	−	0.954078i	$-0.596839\pi$
$673$	−10460.0	−0.599113	−0.299557	+	0.954078i	$0.596839\pi$
$674$	−1768.00	−0.101040
$675$	0	0
$676$	9168.00	0.521620
$677$	15546.0	0.882543	0.441271	−	0.897374i	$-0.354528\pi$
$677$	15546.0	0.882543	0.441271	+	0.897374i	$0.354528\pi$
$678$	0	0
$679$	−644.000	−0.0363983
$680$	−1680.00	−0.0947427
$681$	0	0
$682$	6820.00	0.382920
$683$	−31138.0	−1.74445	−0.872227	−	0.489101i	$-0.837325\pi$
$683$	−31138.0	−1.74445	−0.872227	+	0.489101i	$0.837325\pi$
$684$	0	0
$685$	−5362.00	−0.299082
$686$	686.000	0.0381802
$687$	0	0
$688$	−1728.00	−0.0957549
$689$	8576.00	0.474194
$690$	0	0
$691$	21574.0	1.18772	0.593859	−	0.804569i	$-0.297604\pi$
$691$	21574.0	1.18772	0.593859	+	0.804569i	$0.297604\pi$
$692$	−6752.00	−0.370914
$693$	0	0
$694$	8080.00	0.441949
$695$	−7560.00	−0.412615
$696$	0	0
$697$	−5400.00	−0.293457
$698$	2450.00	0.132857
$699$	0	0
$700$	−2128.00	−0.114901
$701$	−2074.00	−0.111746	−0.0558730	−	0.998438i	$-0.517794\pi$
$701$	−2074.00	−0.111746	−0.0558730	+	0.998438i	$0.517794\pi$
$702$	0	0
$703$	497.000	0.0266639
$704$	−704.000	−0.0376889
$705$	0	0
$706$	−4966.00	−0.264728
$707$	1414.00	0.0752177
$708$	0	0
$709$	−6829.00	−0.361733	−0.180866	−	0.983508i	$-0.557890\pi$
$709$	−6829.00	−0.361733	−0.180866	+	0.983508i	$0.557890\pi$
$710$	−6552.00	−0.346327
$711$	0	0
$712$	9168.00	0.482564
$713$	8680.00	0.455917
$714$	0	0
$715$	5159.00	0.269840
$716$	−1840.00	−0.0960391
$717$	0	0
$718$	−15608.0	−0.811261
$719$	−6921.00	−0.358984	−0.179492	−	0.983759i	$-0.557445\pi$
$719$	−6921.00	−0.358984	−0.179492	+	0.983759i	$0.557445\pi$
$720$	0	0
$721$	−12586.0	−0.650107
$722$	−13620.0	−0.702055
$723$	0	0
$724$	9184.00	0.471437
$725$	9196.00	0.471077
$726$	0	0
$727$	24064.0	1.22763	0.613813	−	0.789451i	$-0.289635\pi$
$727$	24064.0	1.22763	0.613813	+	0.789451i	$0.289635\pi$
$728$	−3752.00	−0.191014
$729$	0	0
$730$	−1638.00	−0.0830481
$731$	3240.00	0.163934
$732$	0	0
$733$	30262.0	1.52490	0.762451	−	0.647047i	$-0.223996\pi$
$733$	30262.0	1.52490	0.762451	+	0.647047i	$0.223996\pi$
$734$	6148.00	0.309165
$735$	0	0
$736$	−896.000	−0.0448736
$737$	8833.00	0.441476
$738$	0	0
$739$	24024.0	1.19586	0.597928	−	0.801550i	$-0.295991\pi$
$739$	24024.0	1.19586	0.597928	+	0.801550i	$0.295991\pi$
$740$	−1988.00	−0.0987572
$741$	0	0
$742$	−1792.00	−0.0886609
$743$	227.000	0.0112084	0.00560419	−	0.999984i	$-0.498216\pi$
$743$	227.000	0.0112084	0.00560419	+	0.999984i	$0.498216\pi$
$744$	0	0
$745$	−9611.00	−0.472644
$746$	1348.00	0.0661579
$747$	0	0
$748$	1320.00	0.0645240
$749$	10017.0	0.488669
$750$	0	0
$751$	−23435.0	−1.13869	−0.569344	−	0.822099i	$-0.692803\pi$
$751$	−23435.0	−1.13869	−0.569344	+	0.822099i	$0.692803\pi$
$752$	−1136.00	−0.0550873
$753$	0	0
$754$	16214.0	0.783129
$755$	−22470.0	−1.08314
$756$	0	0
$757$	1609.00	0.0772524	0.0386262	−	0.999254i	$-0.487702\pi$
$757$	1609.00	0.0772524	0.0386262	+	0.999254i	$0.487702\pi$
$758$	24102.0	1.15491
$759$	0	0
$760$	−392.000	−0.0187097
$761$	−6976.00	−0.332299	−0.166150	−	0.986101i	$-0.553133\pi$
$761$	−6976.00	−0.332299	−0.166150	+	0.986101i	$0.553133\pi$
$762$	0	0
$763$	−1946.00	−0.0923328
$764$	−3512.00	−0.166309
$765$	0	0
$766$	25328.0	1.19470
$767$	−28743.0	−1.35313
$768$	0	0
$769$	−36079.0	−1.69186	−0.845931	−	0.533292i	$-0.820955\pi$
$769$	−36079.0	−1.69186	−0.845931	+	0.533292i	$0.820955\pi$
$770$	−1078.00	−0.0504525
$771$	0	0
$772$	−9072.00	−0.422938
$773$	23507.0	1.09377	0.546887	−	0.837206i	$-0.315813\pi$
$773$	23507.0	1.09377	0.546887	+	0.837206i	$0.315813\pi$
$774$	0	0
$775$	23560.0	1.09200
$776$	−736.000	−0.0340475
$777$	0	0
$778$	−528.000	−0.0243313
$779$	−1260.00	−0.0579515
$780$	0	0
$781$	5148.00	0.235864
$782$	1680.00	0.0768244
$783$	0	0
$784$	784.000	0.0357143
$785$	3164.00	0.143857
$786$	0	0
$787$	−35145.0	−1.59185	−0.795924	−	0.605397i	$-0.793014\pi$
$787$	−35145.0	−1.59185	−0.795924	+	0.605397i	$0.793014\pi$
$788$	−21192.0	−0.958038
$789$	0	0
$790$	−1344.00	−0.0605283
$791$	16254.0	0.730627
$792$	0	0
$793$	−1474.00	−0.0660067
$794$	−28788.0	−1.28671
$795$	0	0
$796$	4176.00	0.185948
$797$	−34155.0	−1.51798	−0.758991	−	0.651101i	$-0.774308\pi$
$797$	−34155.0	−1.51798	−0.758991	+	0.651101i	$0.774308\pi$
$798$	0	0
$799$	2130.00	0.0943104
$800$	−2432.00	−0.107480
$801$	0	0
$802$	24608.0	1.08346
$803$	1287.00	0.0565595
$804$	0	0
$805$	−1372.00	−0.0600704
$806$	41540.0	1.81536
$807$	0	0
$808$	1616.00	0.0703598
$809$	3439.00	0.149455	0.0747273	−	0.997204i	$-0.476191\pi$
$809$	3439.00	0.149455	0.0747273	+	0.997204i	$0.476191\pi$
$810$	0	0
$811$	−1625.00	−0.0703594	−0.0351797	−	0.999381i	$-0.511200\pi$
$811$	−1625.00	−0.0703594	−0.0351797	+	0.999381i	$0.511200\pi$
$812$	−3388.00	−0.146423
$813$	0	0
$814$	1562.00	0.0672581
$815$	−77.0000	−0.00330944
$816$	0	0
$817$	756.000	0.0323734
$818$	−6308.00	−0.269626
$819$	0	0
$820$	5040.00	0.214640
$821$	−31531.0	−1.34036	−0.670182	−	0.742196i	$-0.733784\pi$
$821$	−31531.0	−1.34036	−0.670182	+	0.742196i	$0.733784\pi$
$822$	0	0
$823$	−31003.0	−1.31312	−0.656559	−	0.754274i	$-0.727989\pi$
$823$	−31003.0	−1.31312	−0.656559	+	0.754274i	$0.727989\pi$
$824$	−14384.0	−0.608119
$825$	0	0
$826$	6006.00	0.252997
$827$	15015.0	0.631345	0.315673	−	0.948868i	$-0.397770\pi$
$827$	15015.0	0.631345	0.315673	+	0.948868i	$0.397770\pi$
$828$	0	0
$829$	12280.0	0.514478	0.257239	−	0.966348i	$-0.417187\pi$
$829$	12280.0	0.514478	0.257239	+	0.966348i	$0.417187\pi$
$830$	15708.0	0.656907
$831$	0	0
$832$	−4288.00	−0.178677
$833$	−1470.00	−0.0611434
$834$	0	0
$835$	−3528.00	−0.146217
$836$	308.000	0.0127421
$837$	0	0
$838$	9270.00	0.382132
$839$	−26599.0	−1.09452	−0.547258	−	0.836964i	$-0.684328\pi$
$839$	−26599.0	−1.09452	−0.547258	+	0.836964i	$0.684328\pi$
$840$	0	0
$841$	−9748.00	−0.399688
$842$	−14530.0	−0.594699
$843$	0	0
$844$	18264.0	0.744873
$845$	16044.0	0.653172
$846$	0	0
$847$	847.000	0.0343604
$848$	−2048.00	−0.0829347
$849$	0	0
$850$	4560.00	0.184008
$851$	1988.00	0.0800796
$852$	0	0
$853$	−3798.00	−0.152451	−0.0762257	−	0.997091i	$-0.524287\pi$
$853$	−3798.00	−0.152451	−0.0762257	+	0.997091i	$0.524287\pi$
$854$	308.000	0.0123414
$855$	0	0
$856$	11448.0	0.457108
$857$	38726.0	1.54359	0.771794	−	0.635873i	$-0.219360\pi$
$857$	38726.0	1.54359	0.771794	+	0.635873i	$0.219360\pi$
$858$	0	0
$859$	3296.00	0.130917	0.0654587	−	0.997855i	$-0.479149\pi$
$859$	3296.00	0.130917	0.0654587	+	0.997855i	$0.479149\pi$
$860$	−3024.00	−0.119904
$861$	0	0
$862$	−15178.0	−0.599727
$863$	−13488.0	−0.532024	−0.266012	−	0.963970i	$-0.585706\pi$
$863$	−13488.0	−0.532024	−0.266012	+	0.963970i	$0.585706\pi$
$864$	0	0
$865$	−11816.0	−0.464458
$866$	10328.0	0.405265
$867$	0	0
$868$	−8680.00	−0.339422
$869$	1056.00	0.0412225
$870$	0	0
$871$	53801.0	2.09297
$872$	−2224.00	−0.0863694
$873$	0	0
$874$	392.000	0.0151712
$875$	−9849.00	−0.380522
$876$	0	0
$877$	−17144.0	−0.660105	−0.330052	−	0.943963i	$-0.607066\pi$
$877$	−17144.0	−0.660105	−0.330052	+	0.943963i	$0.607066\pi$
$878$	28442.0	1.09325
$879$	0	0
$880$	−1232.00	−0.0471940
$881$	−11993.0	−0.458632	−0.229316	−	0.973352i	$-0.573649\pi$
$881$	−11993.0	−0.458632	−0.229316	+	0.973352i	$0.573649\pi$
$882$	0	0
$883$	10483.0	0.399526	0.199763	−	0.979844i	$-0.435983\pi$
$883$	10483.0	0.399526	0.199763	+	0.979844i	$0.435983\pi$
$884$	8040.00	0.305899
$885$	0	0
$886$	−13848.0	−0.525093
$887$	−4562.00	−0.172691	−0.0863455	−	0.996265i	$-0.527519\pi$
$887$	−4562.00	−0.172691	−0.0863455	+	0.996265i	$0.527519\pi$
$888$	0	0
$889$	15778.0	0.595250
$890$	16044.0	0.604265
$891$	0	0
$892$	792.000	0.0297288
$893$	497.000	0.0186243
$894$	0	0
$895$	−3220.00	−0.120260
$896$	896.000	0.0334077
$897$	0	0
$898$	−5048.00	−0.187588
$899$	37510.0	1.39158
$900$	0	0
$901$	3840.00	0.141986
$902$	−3960.00	−0.146179
$903$	0	0
$904$	18576.0	0.683439
$905$	16072.0	0.590333
$906$	0	0
$907$	18556.0	0.679318	0.339659	−	0.940549i	$-0.389688\pi$
$907$	18556.0	0.679318	0.339659	+	0.940549i	$0.389688\pi$
$908$	−15048.0	−0.549984
$909$	0	0
$910$	−6566.00	−0.239188
$911$	−5826.00	−0.211881	−0.105941	−	0.994372i	$-0.533785\pi$
$911$	−5826.00	−0.211881	−0.105941	+	0.994372i	$0.533785\pi$
$912$	0	0
$913$	−12342.0	−0.447383
$914$	3856.00	0.139546
$915$	0	0
$916$	−2816.00	−0.101576
$917$	−15372.0	−0.553575
$918$	0	0
$919$	41492.0	1.48933	0.744665	−	0.667438i	$-0.232609\pi$
$919$	41492.0	1.48933	0.744665	+	0.667438i	$0.232609\pi$
$920$	−1568.00	−0.0561907
$921$	0	0
$922$	25704.0	0.918130
$923$	31356.0	1.11820
$924$	0	0
$925$	5396.00	0.191805
$926$	9482.00	0.336499
$927$	0	0
$928$	−3872.00	−0.136966
$929$	15381.0	0.543202	0.271601	−	0.962410i	$-0.412447\pi$
$929$	15381.0	0.543202	0.271601	+	0.962410i	$0.412447\pi$
$930$	0	0
$931$	−343.000	−0.0120745
$932$	−8840.00	−0.310691
$933$	0	0
$934$	13618.0	0.477082
$935$	2310.00	0.0807969
$936$	0	0
$937$	−55054.0	−1.91946	−0.959731	−	0.280921i	$-0.909360\pi$
$937$	−55054.0	−1.91946	−0.959731	+	0.280921i	$0.909360\pi$
$938$	−11242.0	−0.391327
$939$	0	0
$940$	−1988.00	−0.0689802
$941$	3892.00	0.134831	0.0674153	−	0.997725i	$-0.478525\pi$
$941$	3892.00	0.134831	0.0674153	+	0.997725i	$0.478525\pi$
$942$	0	0
$943$	−5040.00	−0.174046
$944$	6864.00	0.236657
$945$	0	0
$946$	2376.00	0.0816601
$947$	36158.0	1.24074	0.620368	−	0.784311i	$-0.286983\pi$
$947$	36158.0	1.24074	0.620368	+	0.784311i	$0.286983\pi$
$948$	0	0
$949$	7839.00	0.268140
$950$	1064.00	0.0363376
$951$	0	0
$952$	−1680.00	−0.0571944
$953$	23959.0	0.814384	0.407192	−	0.913343i	$-0.366508\pi$
$953$	23959.0	0.814384	0.407192	+	0.913343i	$0.366508\pi$
$954$	0	0
$955$	−6146.00	−0.208251
$956$	−9804.00	−0.331678
$957$	0	0
$958$	−3468.00	−0.116958
$959$	−5362.00	−0.180551
$960$	0	0
$961$	66309.0	2.22581
$962$	9514.00	0.318860
$963$	0	0
$964$	100.000	0.00334106
$965$	−15876.0	−0.529603
$966$	0	0
$967$	−56106.0	−1.86582	−0.932910	−	0.360110i	$-0.882739\pi$
$967$	−56106.0	−1.86582	−0.932910	+	0.360110i	$0.882739\pi$
$968$	968.000	0.0321412
$969$	0	0
$970$	−1288.00	−0.0426342
$971$	35079.0	1.15936	0.579680	−	0.814844i	$-0.303178\pi$
$971$	35079.0	1.15936	0.579680	+	0.814844i	$0.303178\pi$
$972$	0	0
$973$	−7560.00	−0.249088
$974$	−14736.0	−0.484776
$975$	0	0
$976$	352.000	0.0115443
$977$	26184.0	0.857421	0.428711	−	0.903442i	$-0.358968\pi$
$977$	26184.0	0.857421	0.428711	+	0.903442i	$0.358968\pi$
$978$	0	0
$979$	−12606.0	−0.411532
$980$	1372.00	0.0447214
$981$	0	0
$982$	−26402.0	−0.857965
$983$	8584.00	0.278522	0.139261	−	0.990256i	$-0.455527\pi$
$983$	8584.00	0.278522	0.139261	+	0.990256i	$0.455527\pi$
$984$	0	0
$985$	−37086.0	−1.19965
$986$	7260.00	0.234488
$987$	0	0
$988$	1876.00	0.0604084
$989$	3024.00	0.0972271
$990$	0	0
$991$	−51331.0	−1.64539	−0.822696	−	0.568482i	$-0.807531\pi$
$991$	−51331.0	−1.64539	−0.822696	+	0.568482i	$0.807531\pi$
$992$	−9920.00	−0.317500
$993$	0	0
$994$	−6552.00	−0.209071
$995$	7308.00	0.232843
$996$	0	0
$997$	−38542.0	−1.22431	−0.612155	−	0.790738i	$-0.709697\pi$
$997$	−38542.0	−1.22431	−0.612155	+	0.790738i	$0.709697\pi$
$998$	33078.0	1.04916
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.4.a.l.1.1		1
3.2	odd	2		462.4.a.d.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.d.1.1	✓	1	3.2	odd	2
1386.4.a.l.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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