LMFDB - Embedded newform 882.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(1,882)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	882.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$52.0396846251$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+2.00000 q^{2} +4.00000 q^{4} -8.00000 q^{5} +8.00000 q^{8} -16.0000 q^{10} -40.0000 q^{11} +4.00000 q^{13} +16.0000 q^{16} +84.0000 q^{17} +148.000 q^{19} -32.0000 q^{20} -80.0000 q^{22} -84.0000 q^{23} -61.0000 q^{25} +8.00000 q^{26} -58.0000 q^{29} -136.000 q^{31} +32.0000 q^{32} +168.000 q^{34} -222.000 q^{37} +296.000 q^{38} -64.0000 q^{40} -420.000 q^{41} -164.000 q^{43} -160.000 q^{44} -168.000 q^{46} -488.000 q^{47} -122.000 q^{50} +16.0000 q^{52} -478.000 q^{53} +320.000 q^{55} -116.000 q^{58} -548.000 q^{59} +692.000 q^{61} -272.000 q^{62} +64.0000 q^{64} -32.0000 q^{65} -908.000 q^{67} +336.000 q^{68} +524.000 q^{71} +440.000 q^{73} -444.000 q^{74} +592.000 q^{76} +1216.00 q^{79} -128.000 q^{80} -840.000 q^{82} +684.000 q^{83} -672.000 q^{85} -328.000 q^{86} -320.000 q^{88} -604.000 q^{89} -336.000 q^{92} -976.000 q^{94} -1184.00 q^{95} -832.000 q^{97} +O(q^{100})$

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
$5$	−8.00000	−0.715542	−0.357771	−	0.933809i	$-0.616463\pi$
$5$	−8.00000	−0.715542	−0.357771	+	0.933809i	$0.616463\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	0	0
$10$	−16.0000	−0.505964
$11$	−40.0000	−1.09640	−0.548202	−	0.836346i	$-0.684688\pi$
$11$	−40.0000	−1.09640	−0.548202	+	0.836346i	$0.684688\pi$
$12$	0	0
$13$	4.00000	0.0853385	0.0426692	−	0.999089i	$-0.486414\pi$
$13$	4.00000	0.0853385	0.0426692	+	0.999089i	$0.486414\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	84.0000	1.19841	0.599206	−	0.800595i	$-0.295483\pi$
$17$	84.0000	1.19841	0.599206	+	0.800595i	$0.295483\pi$
$18$	0	0
$19$	148.000	1.78703	0.893514	−	0.449036i	$-0.148232\pi$
$19$	148.000	1.78703	0.893514	+	0.449036i	$0.148232\pi$
$20$	−32.0000	−0.357771
$21$	0	0
$22$	−80.0000	−0.775275
$23$	−84.0000	−0.761531	−0.380765	−	0.924672i	$-0.624339\pi$
$23$	−84.0000	−0.761531	−0.380765	+	0.924672i	$0.624339\pi$
$24$	0	0
$25$	−61.0000	−0.488000
$26$	8.00000	0.0603434
$27$	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
$31$	−136.000	−0.787946	−0.393973	−	0.919122i	$-0.628900\pi$
$31$	−136.000	−0.787946	−0.393973	+	0.919122i	$0.628900\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	168.000	0.847405
$35$	0	0
$36$	0	0
$37$	−222.000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−222.000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	296.000	1.26362
$39$	0	0
$40$	−64.0000	−0.252982
$41$	−420.000	−1.59983	−0.799914	−	0.600114i	$-0.795122\pi$
$41$	−420.000	−1.59983	−0.799914	+	0.600114i	$0.795122\pi$
$42$	0	0
$43$	−164.000	−0.581622	−0.290811	−	0.956780i	$-0.593925\pi$
$43$	−164.000	−0.581622	−0.290811	+	0.956780i	$0.593925\pi$
$44$	−160.000	−0.548202
$45$	0	0
$46$	−168.000	−0.538484
$47$	−488.000	−1.51451	−0.757257	−	0.653118i	$-0.773461\pi$
$47$	−488.000	−1.51451	−0.757257	+	0.653118i	$0.773461\pi$
$48$	0	0
$49$	0	0
$50$	−122.000	−0.345068
$51$	0	0
$52$	16.0000	0.0426692
$53$	−478.000	−1.23884	−0.619418	−	0.785061i	$-0.712632\pi$
$53$	−478.000	−1.23884	−0.619418	+	0.785061i	$0.712632\pi$
$54$	0	0
$55$	320.000	0.784523
$56$	0	0
$57$	0	0
$58$	−116.000	−0.262613
$59$	−548.000	−1.20921	−0.604606	−	0.796525i	$-0.706669\pi$
$59$	−548.000	−1.20921	−0.604606	+	0.796525i	$0.706669\pi$
$60$	0	0
$61$	692.000	1.45248	0.726242	−	0.687439i	$-0.241265\pi$
$61$	692.000	1.45248	0.726242	+	0.687439i	$0.241265\pi$
$62$	−272.000	−0.557162
$63$	0	0
$64$	64.0000	0.125000
$65$	−32.0000	−0.0610633
$66$	0	0
$67$	−908.000	−1.65567	−0.827835	−	0.560972i	$-0.810428\pi$
$67$	−908.000	−1.65567	−0.827835	+	0.560972i	$0.810428\pi$
$68$	336.000	0.599206
$69$	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
$73$	440.000	0.705453	0.352727	−	0.935726i	$-0.385255\pi$
$73$	440.000	0.705453	0.352727	+	0.935726i	$0.385255\pi$
$74$	−444.000	−0.697486
$75$	0	0
$76$	592.000	0.893514
$77$	0	0
$78$	0	0
$79$	1216.00	1.73178	0.865890	−	0.500234i	$-0.166753\pi$
$79$	1216.00	1.73178	0.865890	+	0.500234i	$0.166753\pi$
$80$	−128.000	−0.178885
$81$	0	0
$82$	−840.000	−1.13125
$83$	684.000	0.904563	0.452282	−	0.891875i	$-0.350610\pi$
$83$	684.000	0.904563	0.452282	+	0.891875i	$0.350610\pi$
$84$	0	0
$85$	−672.000	−0.857513
$86$	−328.000	−0.411269
$87$	0	0
$88$	−320.000	−0.387638
$89$	−604.000	−0.719369	−0.359685	−	0.933074i	$-0.617116\pi$
$89$	−604.000	−0.719369	−0.359685	+	0.933074i	$0.617116\pi$
$90$	0	0
$91$	0	0
$92$	−336.000	−0.380765
$93$	0	0
$94$	−976.000	−1.07092
$95$	−1184.00	−1.27869
$96$	0	0
$97$	−832.000	−0.870895	−0.435447	−	0.900214i	$-0.643410\pi$
$97$	−832.000	−0.870895	−0.435447	+	0.900214i	$0.643410\pi$
$98$	0	0
$99$	0	0
$100$	−244.000	−0.244000
$101$	−464.000	−0.457126	−0.228563	−	0.973529i	$-0.573403\pi$
$101$	−464.000	−0.457126	−0.228563	+	0.973529i	$0.573403\pi$
$102$	0	0
$103$	−632.000	−0.604590	−0.302295	−	0.953214i	$-0.597753\pi$
$103$	−632.000	−0.604590	−0.302295	+	0.953214i	$0.597753\pi$
$104$	32.0000	0.0301717
$105$	0	0
$106$	−956.000	−0.875990
$107$	160.000	0.144559	0.0722794	−	0.997384i	$-0.476973\pi$
$107$	160.000	0.144559	0.0722794	+	0.997384i	$0.476973\pi$
$108$	0	0
$109$	−2198.00	−1.93147	−0.965735	−	0.259530i	$-0.916432\pi$
$109$	−2198.00	−1.93147	−0.965735	+	0.259530i	$0.916432\pi$
$110$	640.000	0.554742
$111$	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
$115$	672.000	0.544907
$116$	−232.000	−0.185695
$117$	0	0
$118$	−1096.00	−0.855042
$119$	0	0
$120$	0	0
$121$	269.000	0.202104
$122$	1384.00	1.02706
$123$	0	0
$124$	−544.000	−0.393973
$125$	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	$0.520475\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−64.0000	−0.0431782
$131$	1452.00	0.968411	0.484205	−	0.874954i	$-0.339109\pi$
$131$	1452.00	0.968411	0.484205	+	0.874954i	$0.339109\pi$
$132$	0	0
$133$	0	0
$134$	−1816.00	−1.17074
$135$	0	0
$136$	672.000	0.423702
$137$	−646.000	−0.402858	−0.201429	−	0.979503i	$-0.564558\pi$
$137$	−646.000	−0.402858	−0.201429	+	0.979503i	$0.564558\pi$
$138$	0	0
$139$	−3012.00	−1.83795	−0.918973	−	0.394320i	$-0.870980\pi$
$139$	−3012.00	−1.83795	−0.918973	+	0.394320i	$0.870980\pi$
$140$	0	0
$141$	0	0
$142$	1048.00	0.619339
$143$	−160.000	−0.0935655
$144$	0	0
$145$	464.000	0.265746
$146$	880.000	0.498831
$147$	0	0
$148$	−888.000	−0.493197
$149$	3170.00	1.74293	0.871465	−	0.490458i	$-0.163170\pi$
$149$	3170.00	1.74293	0.871465	+	0.490458i	$0.163170\pi$
$150$	0	0
$151$	−1880.00	−1.01319	−0.506597	−	0.862183i	$-0.669097\pi$
$151$	−1880.00	−1.01319	−0.506597	+	0.862183i	$0.669097\pi$
$152$	1184.00	0.631810
$153$	0	0
$154$	0	0
$155$	1088.00	0.563808
$156$	0	0
$157$	604.000	0.307035	0.153517	−	0.988146i	$-0.450940\pi$
$157$	604.000	0.307035	0.153517	+	0.988146i	$0.450940\pi$
$158$	2432.00	1.22455
$159$	0	0
$160$	−256.000	−0.126491
$161$	0	0
$162$	0	0
$163$	1116.00	0.536269	0.268135	−	0.963381i	$-0.413593\pi$
$163$	1116.00	0.536269	0.268135	+	0.963381i	$0.413593\pi$
$164$	−1680.00	−0.799914
$165$	0	0
$166$	1368.00	0.639623
$167$	1784.00	0.826647	0.413324	−	0.910584i	$-0.364368\pi$
$167$	1784.00	0.826647	0.413324	+	0.910584i	$0.364368\pi$
$168$	0	0
$169$	−2181.00	−0.992717
$170$	−1344.00	−0.606353
$171$	0	0
$172$	−656.000	−0.290811
$173$	344.000	0.151178	0.0755891	−	0.997139i	$-0.475916\pi$
$173$	344.000	0.151178	0.0755891	+	0.997139i	$0.475916\pi$
$174$	0	0
$175$	0	0
$176$	−640.000	−0.274101
$177$	0	0
$178$	−1208.00	−0.508671
$179$	−1392.00	−0.581246	−0.290623	−	0.956838i	$-0.593862\pi$
$179$	−1392.00	−0.581246	−0.290623	+	0.956838i	$0.593862\pi$
$180$	0	0
$181$	4052.00	1.66399	0.831997	−	0.554781i	$-0.187198\pi$
$181$	4052.00	1.66399	0.831997	+	0.554781i	$0.187198\pi$
$182$	0	0
$183$	0	0
$184$	−672.000	−0.269242
$185$	1776.00	0.705806
$186$	0	0
$187$	−3360.00	−1.31394
$188$	−1952.00	−0.757257
$189$	0	0
$190$	−2368.00	−0.904173
$191$	3108.00	1.17742	0.588709	−	0.808345i	$-0.299636\pi$
$191$	3108.00	1.17742	0.588709	+	0.808345i	$0.299636\pi$
$192$	0	0
$193$	50.0000	0.0186481	0.00932404	−	0.999957i	$-0.497032\pi$
$193$	50.0000	0.0186481	0.00932404	+	0.999957i	$0.497032\pi$
$194$	−1664.00	−0.615816
$195$	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
$199$	1544.00	0.550006	0.275003	−	0.961443i	$-0.411321\pi$
$199$	1544.00	0.550006	0.275003	+	0.961443i	$0.411321\pi$
$200$	−488.000	−0.172534
$201$	0	0
$202$	−928.000	−0.323237
$203$	0	0
$204$	0	0
$205$	3360.00	1.14474
$206$	−1264.00	−0.427510
$207$	0	0
$208$	64.0000	0.0213346
$209$	−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	$0.562930\pi$
$212$	−1912.00	−0.619418
$213$	0	0
$214$	320.000	0.102218
$215$	1312.00	0.416175
$216$	0	0
$217$	0	0
$218$	−4396.00	−1.36576
$219$	0	0
$220$	1280.00	0.392262
$221$	336.000	0.102271
$222$	0	0
$223$	2000.00	0.600583	0.300291	−	0.953848i	$-0.402916\pi$
$223$	2000.00	0.600583	0.300291	+	0.953848i	$0.402916\pi$
$224$	0	0
$225$	0	0
$226$	−1540.00	−0.453271
$227$	−388.000	−0.113447	−0.0567235	−	0.998390i	$-0.518065\pi$
$227$	−388.000	−0.113447	−0.0567235	+	0.998390i	$0.518065\pi$
$228$	0	0
$229$	4180.00	1.20621	0.603105	−	0.797662i	$-0.293930\pi$
$229$	4180.00	1.20621	0.603105	+	0.797662i	$0.293930\pi$
$230$	1344.00	0.385308
$231$	0	0
$232$	−464.000	−0.131306
$233$	1322.00	0.371704	0.185852	−	0.982578i	$-0.440495\pi$
$233$	1322.00	0.371704	0.185852	+	0.982578i	$0.440495\pi$
$234$	0	0
$235$	3904.00	1.08370
$236$	−2192.00	−0.604606
$237$	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
$241$	−4336.00	−1.15895	−0.579474	−	0.814991i	$-0.696742\pi$
$241$	−4336.00	−1.15895	−0.579474	+	0.814991i	$0.696742\pi$
$242$	538.000	0.142909
$243$	0	0
$244$	2768.00	0.726242
$245$	0	0
$246$	0	0
$247$	592.000	0.152502
$248$	−1088.00	−0.278581
$249$	0	0
$250$	2976.00	0.752875
$251$	−764.000	−0.192125	−0.0960623	−	0.995375i	$-0.530625\pi$
$251$	−764.000	−0.192125	−0.0960623	+	0.995375i	$0.530625\pi$
$252$	0	0
$253$	3360.00	0.834946
$254$	−368.000	−0.0909070
$255$	0	0
$256$	256.000	0.0625000
$257$	−4300.00	−1.04368	−0.521842	−	0.853042i	$-0.674755\pi$
$257$	−4300.00	−1.04368	−0.521842	+	0.853042i	$0.674755\pi$
$258$	0	0
$259$	0	0
$260$	−128.000	−0.0305316
$261$	0	0
$262$	2904.00	0.684770
$263$	3860.00	0.905011	0.452505	−	0.891762i	$-0.350530\pi$
$263$	3860.00	0.905011	0.452505	+	0.891762i	$0.350530\pi$
$264$	0	0
$265$	3824.00	0.886439
$266$	0	0
$267$	0	0
$268$	−3632.00	−0.827835
$269$	2800.00	0.634643	0.317322	−	0.948318i	$-0.397217\pi$
$269$	2800.00	0.634643	0.317322	+	0.948318i	$0.397217\pi$
$270$	0	0
$271$	−4880.00	−1.09387	−0.546935	−	0.837175i	$-0.684206\pi$
$271$	−4880.00	−1.09387	−0.546935	+	0.837175i	$0.684206\pi$
$272$	1344.00	0.299603
$273$	0	0
$274$	−1292.00	−0.284863
$275$	2440.00	0.535046
$276$	0	0
$277$	−6674.00	−1.44766	−0.723830	−	0.689978i	$-0.757620\pi$
$277$	−6674.00	−1.44766	−0.723830	+	0.689978i	$0.757620\pi$
$278$	−6024.00	−1.29962
$279$	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
$283$	−9100.00	−1.91144	−0.955722	−	0.294270i	$-0.904924\pi$
$283$	−9100.00	−1.91144	−0.955722	+	0.294270i	$0.904924\pi$
$284$	2096.00	0.437939
$285$	0	0
$286$	−320.000	−0.0661608
$287$	0	0
$288$	0	0
$289$	2143.00	0.436190
$290$	928.000	0.187910
$291$	0	0
$292$	1760.00	0.352727
$293$	−5952.00	−1.18676	−0.593378	−	0.804924i	$-0.702206\pi$
$293$	−5952.00	−1.18676	−0.593378	+	0.804924i	$0.702206\pi$
$294$	0	0
$295$	4384.00	0.865242
$296$	−1776.00	−0.348743
$297$	0	0
$298$	6340.00	1.23244
$299$	−336.000	−0.0649879
$300$	0	0
$301$	0	0
$302$	−3760.00	−0.716436
$303$	0	0
$304$	2368.00	0.446757
$305$	−5536.00	−1.03931
$306$	0	0
$307$	−3004.00	−0.558460	−0.279230	−	0.960224i	$-0.590079\pi$
$307$	−3004.00	−0.558460	−0.279230	+	0.960224i	$0.590079\pi$
$308$	0	0
$309$	0	0
$310$	2176.00	0.398673
$311$	−688.000	−0.125443	−0.0627217	−	0.998031i	$-0.519978\pi$
$311$	−688.000	−0.125443	−0.0627217	+	0.998031i	$0.519978\pi$
$312$	0	0
$313$	5592.00	1.00984	0.504918	−	0.863167i	$-0.331523\pi$
$313$	5592.00	1.00984	0.504918	+	0.863167i	$0.331523\pi$
$314$	1208.00	0.217106
$315$	0	0
$316$	4864.00	0.865890
$317$	2922.00	0.517716	0.258858	−	0.965915i	$-0.416654\pi$
$317$	2922.00	0.517716	0.258858	+	0.965915i	$0.416654\pi$
$318$	0	0
$319$	2320.00	0.407195
$320$	−512.000	−0.0894427
$321$	0	0
$322$	0	0
$323$	12432.0	2.14159
$324$	0	0
$325$	−244.000	−0.0416452
$326$	2232.00	0.379200
$327$	0	0
$328$	−3360.00	−0.565625
$329$	0	0
$330$	0	0
$331$	−7492.00	−1.24410	−0.622051	−	0.782977i	$-0.713700\pi$
$331$	−7492.00	−1.24410	−0.622051	+	0.782977i	$0.713700\pi$
$332$	2736.00	0.452282
$333$	0	0
$334$	3568.00	0.584528
$335$	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	$0.164041\pi$
$338$	−4362.00	−0.701957
$339$	0	0
$340$	−2688.00	−0.428757
$341$	5440.00	0.863908
$342$	0	0
$343$	0	0
$344$	−1312.00	−0.205635
$345$	0	0
$346$	688.000	0.106899
$347$	3984.00	0.616347	0.308173	−	0.951330i	$-0.400282\pi$
$347$	3984.00	0.616347	0.308173	+	0.951330i	$0.400282\pi$
$348$	0	0
$349$	−180.000	−0.0276080	−0.0138040	−	0.999905i	$-0.504394\pi$
$349$	−180.000	−0.0276080	−0.0138040	+	0.999905i	$0.504394\pi$
$350$	0	0
$351$	0	0
$352$	−1280.00	−0.193819
$353$	10428.0	1.57231	0.786156	−	0.618028i	$-0.212068\pi$
$353$	10428.0	1.57231	0.786156	+	0.618028i	$0.212068\pi$
$354$	0	0
$355$	−4192.00	−0.626727
$356$	−2416.00	−0.359685
$357$	0	0
$358$	−2784.00	−0.411003
$359$	−8684.00	−1.27667	−0.638334	−	0.769759i	$-0.720376\pi$
$359$	−8684.00	−1.27667	−0.638334	+	0.769759i	$0.720376\pi$
$360$	0	0
$361$	15045.0	2.19347
$362$	8104.00	1.17662
$363$	0	0
$364$	0	0
$365$	−3520.00	−0.504781
$366$	0	0
$367$	5648.00	0.803333	0.401666	−	0.915786i	$-0.368431\pi$
$367$	5648.00	0.803333	0.401666	+	0.915786i	$0.368431\pi$
$368$	−1344.00	−0.190383
$369$	0	0
$370$	3552.00	0.499080
$371$	0	0
$372$	0	0
$373$	−2546.00	−0.353423	−0.176712	−	0.984263i	$-0.556546\pi$
$373$	−2546.00	−0.353423	−0.176712	+	0.984263i	$0.556546\pi$
$374$	−6720.00	−0.929099
$375$	0	0
$376$	−3904.00	−0.535461
$377$	−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	$0.310690\pi$
$380$	−4736.00	−0.639347
$381$	0	0
$382$	6216.00	0.832561
$383$	10872.0	1.45048	0.725239	−	0.688497i	$-0.241729\pi$
$383$	10872.0	1.45048	0.725239	+	0.688497i	$0.241729\pi$
$384$	0	0
$385$	0	0
$386$	100.000	0.0131862
$387$	0	0
$388$	−3328.00	−0.435447
$389$	−10434.0	−1.35996	−0.679980	−	0.733230i	$-0.738012\pi$
$389$	−10434.0	−1.35996	−0.679980	+	0.733230i	$0.738012\pi$
$390$	0	0
$391$	−7056.00	−0.912627
$392$	0	0
$393$	0	0
$394$	324.000	0.0414286
$395$	−9728.00	−1.23916
$396$	0	0
$397$	−3044.00	−0.384821	−0.192411	−	0.981315i	$-0.561631\pi$
$397$	−3044.00	−0.384821	−0.192411	+	0.981315i	$0.561631\pi$
$398$	3088.00	0.388913
$399$	0	0
$400$	−976.000	−0.122000
$401$	−8910.00	−1.10959	−0.554793	−	0.831988i	$-0.687203\pi$
$401$	−8910.00	−1.10959	−0.554793	+	0.831988i	$0.687203\pi$
$402$	0	0
$403$	−544.000	−0.0672421
$404$	−1856.00	−0.228563
$405$	0	0
$406$	0	0
$407$	8880.00	1.08149
$408$	0	0
$409$	5616.00	0.678957	0.339478	−	0.940614i	$-0.389749\pi$
$409$	5616.00	0.678957	0.339478	+	0.940614i	$0.389749\pi$
$410$	6720.00	0.809456
$411$	0	0
$412$	−2528.00	−0.302295
$413$	0	0
$414$	0	0
$415$	−5472.00	−0.647253
$416$	128.000	0.0150859
$417$	0	0
$418$	−11840.0	−1.38544
$419$	8932.00	1.04142	0.520712	−	0.853732i	$-0.325666\pi$
$419$	8932.00	1.04142	0.520712	+	0.853732i	$0.325666\pi$
$420$	0	0
$421$	−5538.00	−0.641106	−0.320553	−	0.947231i	$-0.603869\pi$
$421$	−5538.00	−0.641106	−0.320553	+	0.947231i	$0.603869\pi$
$422$	−2408.00	−0.277772
$423$	0	0
$424$	−3824.00	−0.437995
$425$	−5124.00	−0.584825
$426$	0	0
$427$	0	0
$428$	640.000	0.0722794
$429$	0	0
$430$	2624.00	0.294280
$431$	6700.00	0.748788	0.374394	−	0.927270i	$-0.377851\pi$
$431$	6700.00	0.748788	0.374394	+	0.927270i	$0.377851\pi$
$432$	0	0
$433$	−5048.00	−0.560257	−0.280129	−	0.959962i	$-0.590377\pi$
$433$	−5048.00	−0.560257	−0.280129	+	0.959962i	$0.590377\pi$
$434$	0	0
$435$	0	0
$436$	−8792.00	−0.965735
$437$	−12432.0	−1.36088
$438$	0	0
$439$	−1344.00	−0.146118	−0.0730588	−	0.997328i	$-0.523276\pi$
$439$	−1344.00	−0.146118	−0.0730588	+	0.997328i	$0.523276\pi$
$440$	2560.00	0.277371
$441$	0	0
$442$	672.000	0.0723162
$443$	4392.00	0.471039	0.235519	−	0.971870i	$-0.424321\pi$
$443$	4392.00	0.471039	0.235519	+	0.971870i	$0.424321\pi$
$444$	0	0
$445$	4832.00	0.514739
$446$	4000.00	0.424676
$447$	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
$451$	16800.0	1.75406
$452$	−3080.00	−0.320511
$453$	0	0
$454$	−776.000	−0.0802191
$455$	0	0
$456$	0	0
$457$	26.0000	0.00266133	0.00133067	−	0.999999i	$-0.499576\pi$
$457$	26.0000	0.00266133	0.00133067	+	0.999999i	$0.499576\pi$
$458$	8360.00	0.852920
$459$	0	0
$460$	2688.00	0.272454
$461$	−7656.00	−0.773483	−0.386741	−	0.922188i	$-0.626399\pi$
$461$	−7656.00	−0.773483	−0.386741	+	0.922188i	$0.626399\pi$
$462$	0	0
$463$	12608.0	1.26554	0.632768	−	0.774341i	$-0.281919\pi$
$463$	12608.0	1.26554	0.632768	+	0.774341i	$0.281919\pi$
$464$	−928.000	−0.0928477
$465$	0	0
$466$	2644.00	0.262835
$467$	3068.00	0.304005	0.152002	−	0.988380i	$-0.451428\pi$
$467$	3068.00	0.304005	0.152002	+	0.988380i	$0.451428\pi$
$468$	0	0
$469$	0	0
$470$	7808.00	0.766290
$471$	0	0
$472$	−4384.00	−0.427521
$473$	6560.00	0.637694
$474$	0	0
$475$	−9028.00	−0.872070
$476$	0	0
$477$	0	0
$478$	−4824.00	−0.461600
$479$	−6456.00	−0.615829	−0.307915	−	0.951414i	$-0.599631\pi$
$479$	−6456.00	−0.615829	−0.307915	+	0.951414i	$0.599631\pi$
$480$	0	0
$481$	−888.000	−0.0841774
$482$	−8672.00	−0.819500
$483$	0	0
$484$	1076.00	0.101052
$485$	6656.00	0.623162
$486$	0	0
$487$	11896.0	1.10690	0.553449	−	0.832883i	$-0.313311\pi$
$487$	11896.0	1.10690	0.553449	+	0.832883i	$0.313311\pi$
$488$	5536.00	0.513531
$489$	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
$493$	−4872.00	−0.445079
$494$	1184.00	0.107835
$495$	0	0
$496$	−2176.00	−0.196986
$497$	0	0
$498$	0	0
$499$	−2628.00	−0.235762	−0.117881	−	0.993028i	$-0.537610\pi$
$499$	−2628.00	−0.235762	−0.117881	+	0.993028i	$0.537610\pi$
$500$	5952.00	0.532363
$501$	0	0
$502$	−1528.00	−0.135853
$503$	13568.0	1.20272	0.601359	−	0.798979i	$-0.294626\pi$
$503$	13568.0	1.20272	0.601359	+	0.798979i	$0.294626\pi$
$504$	0	0
$505$	3712.00	0.327093
$506$	6720.00	0.590396
$507$	0	0
$508$	−736.000	−0.0642809
$509$	−20656.0	−1.79874	−0.899372	−	0.437183i	$-0.855976\pi$
$509$	−20656.0	−1.79874	−0.899372	+	0.437183i	$0.855976\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	0	0
$514$	−8600.00	−0.737996
$515$	5056.00	0.432610
$516$	0	0
$517$	19520.0	1.66052
$518$	0	0
$519$	0	0
$520$	−256.000	−0.0215891
$521$	−3628.00	−0.305078	−0.152539	−	0.988297i	$-0.548745\pi$
$521$	−3628.00	−0.305078	−0.152539	+	0.988297i	$0.548745\pi$
$522$	0	0
$523$	4852.00	0.405666	0.202833	−	0.979213i	$-0.434985\pi$
$523$	4852.00	0.405666	0.202833	+	0.979213i	$0.434985\pi$
$524$	5808.00	0.484205
$525$	0	0
$526$	7720.00	0.639939
$527$	−11424.0	−0.944283
$528$	0	0
$529$	−5111.00	−0.420071
$530$	7648.00	0.626807
$531$	0	0
$532$	0	0
$533$	−1680.00	−0.136527
$534$	0	0
$535$	−1280.00	−0.103438
$536$	−7264.00	−0.585368
$537$	0	0
$538$	5600.00	0.448760
$539$	0	0
$540$	0	0
$541$	−7130.00	−0.566622	−0.283311	−	0.959028i	$-0.591433\pi$
$541$	−7130.00	−0.566622	−0.283311	+	0.959028i	$0.591433\pi$
$542$	−9760.00	−0.773483
$543$	0	0
$544$	2688.00	0.211851
$545$	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	$0.666591\pi$
$548$	−2584.00	−0.201429
$549$	0	0
$550$	4880.00	0.378334
$551$	−8584.00	−0.663685
$552$	0	0
$553$	0	0
$554$	−13348.0	−1.02365
$555$	0	0
$556$	−12048.0	−0.918973
$557$	−2406.00	−0.183026	−0.0915130	−	0.995804i	$-0.529170\pi$
$557$	−2406.00	−0.183026	−0.0915130	+	0.995804i	$0.529170\pi$
$558$	0	0
$559$	−656.000	−0.0496348
$560$	0	0
$561$	0	0
$562$	18804.0	1.41139
$563$	25412.0	1.90229	0.951144	−	0.308748i	$-0.0999099\pi$
$563$	25412.0	1.90229	0.951144	+	0.308748i	$0.0999099\pi$
$564$	0	0
$565$	6160.00	0.458678
$566$	−18200.0	−1.35160
$567$	0	0
$568$	4192.00	0.309670
$569$	9690.00	0.713930	0.356965	−	0.934118i	$-0.383812\pi$
$569$	9690.00	0.713930	0.356965	+	0.934118i	$0.383812\pi$
$570$	0	0
$571$	5604.00	0.410718	0.205359	−	0.978687i	$-0.434164\pi$
$571$	5604.00	0.410718	0.205359	+	0.978687i	$0.434164\pi$
$572$	−640.000	−0.0467828
$573$	0	0
$574$	0	0
$575$	5124.00	0.371627
$576$	0	0
$577$	−21568.0	−1.55613	−0.778066	−	0.628183i	$-0.783799\pi$
$577$	−21568.0	−1.55613	−0.778066	+	0.628183i	$0.783799\pi$
$578$	4286.00	0.308433
$579$	0	0
$580$	1856.00	0.132873
$581$	0	0
$582$	0	0
$583$	19120.0	1.35827
$584$	3520.00	0.249415
$585$	0	0
$586$	−11904.0	−0.839163
$587$	−20300.0	−1.42738	−0.713689	−	0.700463i	$-0.752977\pi$
$587$	−20300.0	−1.42738	−0.713689	+	0.700463i	$0.752977\pi$
$588$	0	0
$589$	−20128.0	−1.40808
$590$	8768.00	0.611818
$591$	0	0
$592$	−3552.00	−0.246598
$593$	−13812.0	−0.956477	−0.478238	−	0.878230i	$-0.658725\pi$
$593$	−13812.0	−0.956477	−0.478238	+	0.878230i	$0.658725\pi$
$594$	0	0
$595$	0	0
$596$	12680.0	0.871465
$597$	0	0
$598$	−672.000	−0.0459534
$599$	21996.0	1.50039	0.750194	−	0.661218i	$-0.229960\pi$
$599$	21996.0	1.50039	0.750194	+	0.661218i	$0.229960\pi$
$600$	0	0
$601$	8368.00	0.567950	0.283975	−	0.958832i	$-0.408347\pi$
$601$	8368.00	0.567950	0.283975	+	0.958832i	$0.408347\pi$
$602$	0	0
$603$	0	0
$604$	−7520.00	−0.506597
$605$	−2152.00	−0.144614
$606$	0	0
$607$	21504.0	1.43792	0.718962	−	0.695049i	$-0.244617\pi$
$607$	21504.0	1.43792	0.718962	+	0.695049i	$0.244617\pi$
$608$	4736.00	0.315905
$609$	0	0
$610$	−11072.0	−0.734905
$611$	−1952.00	−0.129246
$612$	0	0
$613$	−10270.0	−0.676674	−0.338337	−	0.941025i	$-0.609864\pi$
$613$	−10270.0	−0.676674	−0.338337	+	0.941025i	$0.609864\pi$
$614$	−6008.00	−0.394891
$615$	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
$619$	16292.0	1.05788	0.528942	−	0.848658i	$-0.322589\pi$
$619$	16292.0	1.05788	0.528942	+	0.848658i	$0.322589\pi$
$620$	4352.00	0.281904
$621$	0	0
$622$	−1376.00	−0.0887019
$623$	0	0
$624$	0	0
$625$	−4279.00	−0.273856
$626$	11184.0	0.714062
$627$	0	0
$628$	2416.00	0.153517
$629$	−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	$0.384458\pi$
$632$	9728.00	0.612277
$633$	0	0
$634$	5844.00	0.366080
$635$	1472.00	0.0919914
$636$	0	0
$637$	0	0
$638$	4640.00	0.287930
$639$	0	0
$640$	−1024.00	−0.0632456
$641$	−15518.0	−0.956200	−0.478100	−	0.878305i	$-0.658674\pi$
$641$	−15518.0	−0.956200	−0.478100	+	0.878305i	$0.658674\pi$
$642$	0	0
$643$	−10452.0	−0.641037	−0.320518	−	0.947242i	$-0.603857\pi$
$643$	−10452.0	−0.641037	−0.320518	+	0.947242i	$0.603857\pi$
$644$	0	0
$645$	0	0
$646$	24864.0	1.51434
$647$	−72.0000	−0.00437498	−0.00218749	−	0.999998i	$-0.500696\pi$
$647$	−72.0000	−0.00437498	−0.00218749	+	0.999998i	$0.500696\pi$
$648$	0	0
$649$	21920.0	1.32579
$650$	−488.000	−0.0294476
$651$	0	0
$652$	4464.00	0.268135
$653$	−11962.0	−0.716859	−0.358430	−	0.933557i	$-0.616688\pi$
$653$	−11962.0	−0.716859	−0.358430	+	0.933557i	$0.616688\pi$
$654$	0	0
$655$	−11616.0	−0.692938
$656$	−6720.00	−0.399957
$657$	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
$661$	26068.0	1.53393	0.766965	−	0.641689i	$-0.221766\pi$
$661$	26068.0	1.53393	0.766965	+	0.641689i	$0.221766\pi$
$662$	−14984.0	−0.879713
$663$	0	0
$664$	5472.00	0.319811
$665$	0	0
$666$	0	0
$667$	4872.00	0.282825
$668$	7136.00	0.413324
$669$	0	0
$670$	14528.0	0.837710
$671$	−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	$0.700063\pi$
$674$	21532.0	1.23054
$675$	0	0
$676$	−8724.00	−0.496359
$677$	10056.0	0.570877	0.285438	−	0.958397i	$-0.407861\pi$
$677$	10056.0	0.570877	0.285438	+	0.958397i	$0.407861\pi$
$678$	0	0
$679$	0	0
$680$	−5376.00	−0.303177
$681$	0	0
$682$	10880.0	0.610875
$683$	−6152.00	−0.344656	−0.172328	−	0.985040i	$-0.555129\pi$
$683$	−6152.00	−0.344656	−0.172328	+	0.985040i	$0.555129\pi$
$684$	0	0
$685$	5168.00	0.288262
$686$	0	0
$687$	0	0
$688$	−2624.00	−0.145406
$689$	−1912.00	−0.105720
$690$	0	0
$691$	−14716.0	−0.810164	−0.405082	−	0.914280i	$-0.632757\pi$
$691$	−14716.0	−0.810164	−0.405082	+	0.914280i	$0.632757\pi$
$692$	1376.00	0.0755891
$693$	0	0
$694$	7968.00	0.435823
$695$	24096.0	1.31513
$696$	0	0
$697$	−35280.0	−1.91725
$698$	−360.000	−0.0195218
$699$	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
$703$	−32856.0	−1.76271
$704$	−2560.00	−0.137051
$705$	0	0
$706$	20856.0	1.11179
$707$	0	0
$708$	0	0
$709$	22114.0	1.17138	0.585690	−	0.810535i	$-0.300824\pi$
$709$	22114.0	1.17138	0.585690	+	0.810535i	$0.300824\pi$
$710$	−8384.00	−0.443163
$711$	0	0
$712$	−4832.00	−0.254335
$713$	11424.0	0.600045
$714$	0	0
$715$	1280.00	0.0669501
$716$	−5568.00	−0.290623
$717$	0	0
$718$	−17368.0	−0.902741
$719$	9288.00	0.481758	0.240879	−	0.970555i	$-0.422564\pi$
$719$	9288.00	0.481758	0.240879	+	0.970555i	$0.422564\pi$
$720$	0	0
$721$	0	0
$722$	30090.0	1.55102
$723$	0	0
$724$	16208.0	0.831997
$725$	3538.00	0.181239
$726$	0	0
$727$	23848.0	1.21661	0.608304	−	0.793704i	$-0.291850\pi$
$727$	23848.0	1.21661	0.608304	+	0.793704i	$0.291850\pi$
$728$	0	0
$729$	0	0
$730$	−7040.00	−0.356934
$731$	−13776.0	−0.697023
$732$	0	0
$733$	−34756.0	−1.75135	−0.875677	−	0.482898i	$-0.839584\pi$
$733$	−34756.0	−1.75135	−0.875677	+	0.482898i	$0.839584\pi$
$734$	11296.0	0.568042
$735$	0	0
$736$	−2688.00	−0.134621
$737$	36320.0	1.81528
$738$	0	0
$739$	26044.0	1.29641	0.648203	−	0.761468i	$-0.275521\pi$
$739$	26044.0	1.29641	0.648203	+	0.761468i	$0.275521\pi$
$740$	7104.00	0.352903
$741$	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
$745$	−25360.0	−1.24714
$746$	−5092.00	−0.249908
$747$	0	0
$748$	−13440.0	−0.656972
$749$	0	0
$750$	0	0
$751$	−11424.0	−0.555083	−0.277542	−	0.960714i	$-0.589520\pi$
$751$	−11424.0	−0.555083	−0.277542	+	0.960714i	$0.589520\pi$
$752$	−7808.00	−0.378628
$753$	0	0
$754$	−464.000	−0.0224110
$755$	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	$0.630654\pi$
$758$	16536.0	0.792368
$759$	0	0
$760$	−9472.00	−0.452086
$761$	38524.0	1.83508	0.917539	−	0.397646i	$-0.130173\pi$
$761$	38524.0	1.83508	0.917539	+	0.397646i	$0.130173\pi$
$762$	0	0
$763$	0	0
$764$	12432.0	0.588709
$765$	0	0
$766$	21744.0	1.02564
$767$	−2192.00	−0.103192
$768$	0	0
$769$	−18440.0	−0.864712	−0.432356	−	0.901703i	$-0.642318\pi$
$769$	−18440.0	−0.864712	−0.432356	+	0.901703i	$0.642318\pi$
$770$	0	0
$771$	0	0
$772$	200.000	0.00932404
$773$	13968.0	0.649928	0.324964	−	0.945726i	$-0.394648\pi$
$773$	13968.0	0.649928	0.324964	+	0.945726i	$0.394648\pi$
$774$	0	0
$775$	8296.00	0.384518
$776$	−6656.00	−0.307908
$777$	0	0
$778$	−20868.0	−0.961638
$779$	−62160.0	−2.85894
$780$	0	0
$781$	−20960.0	−0.960317
$782$	−14112.0	−0.645325
$783$	0	0
$784$	0	0
$785$	−4832.00	−0.219696
$786$	0	0
$787$	10916.0	0.494426	0.247213	−	0.968961i	$-0.420485\pi$
$787$	10916.0	0.494426	0.247213	+	0.968961i	$0.420485\pi$
$788$	648.000	0.0292945
$789$	0	0
$790$	−19456.0	−0.876220
$791$	0	0
$792$	0	0
$793$	2768.00	0.123953
$794$	−6088.00	−0.272110
$795$	0	0
$796$	6176.00	0.275003
$797$	−12360.0	−0.549327	−0.274664	−	0.961540i	$-0.588566\pi$
$797$	−12360.0	−0.549327	−0.274664	+	0.961540i	$0.588566\pi$
$798$	0	0
$799$	−40992.0	−1.81501
$800$	−1952.00	−0.0862670
$801$	0	0
$802$	−17820.0	−0.784596
$803$	−17600.0	−0.773463
$804$	0	0
$805$	0	0
$806$	−1088.00	−0.0475474
$807$	0	0
$808$	−3712.00	−0.161618
$809$	−3402.00	−0.147847	−0.0739233	−	0.997264i	$-0.523552\pi$
$809$	−3402.00	−0.147847	−0.0739233	+	0.997264i	$0.523552\pi$
$810$	0	0
$811$	292.000	0.0126430	0.00632152	−	0.999980i	$-0.497988\pi$
$811$	292.000	0.0126430	0.00632152	+	0.999980i	$0.497988\pi$
$812$	0	0
$813$	0	0
$814$	17760.0	0.764727
$815$	−8928.00	−0.383723
$816$	0	0
$817$	−24272.0	−1.03938
$818$	11232.0	0.480095
$819$	0	0
$820$	13440.0	0.572372
$821$	−6910.00	−0.293740	−0.146870	−	0.989156i	$-0.546920\pi$
$821$	−6910.00	−0.293740	−0.146870	+	0.989156i	$0.546920\pi$
$822$	0	0
$823$	568.000	0.0240574	0.0120287	−	0.999928i	$-0.496171\pi$
$823$	568.000	0.0240574	0.0120287	+	0.999928i	$0.496171\pi$
$824$	−5056.00	−0.213755
$825$	0	0
$826$	0	0
$827$	12144.0	0.510627	0.255313	−	0.966858i	$-0.417821\pi$
$827$	12144.0	0.510627	0.255313	+	0.966858i	$0.417821\pi$
$828$	0	0
$829$	14828.0	0.621228	0.310614	−	0.950536i	$-0.399465\pi$
$829$	14828.0	0.621228	0.310614	+	0.950536i	$0.399465\pi$
$830$	−10944.0	−0.457677
$831$	0	0
$832$	256.000	0.0106673
$833$	0	0
$834$	0	0
$835$	−14272.0	−0.591501
$836$	−23680.0	−0.979653
$837$	0	0
$838$	17864.0	0.736398
$839$	22824.0	0.939180	0.469590	−	0.882885i	$-0.344402\pi$
$839$	22824.0	0.939180	0.469590	+	0.882885i	$0.344402\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	−11076.0	−0.453330
$843$	0	0
$844$	−4816.00	−0.196414
$845$	17448.0	0.710331
$846$	0	0
$847$	0	0
$848$	−7648.00	−0.309709
$849$	0	0
$850$	−10248.0	−0.413534
$851$	18648.0	0.751169
$852$	0	0
$853$	−41780.0	−1.67705	−0.838523	−	0.544866i	$-0.816580\pi$
$853$	−41780.0	−1.67705	−0.838523	+	0.544866i	$0.816580\pi$
$854$	0	0
$855$	0	0
$856$	1280.00	0.0511092
$857$	21420.0	0.853784	0.426892	−	0.904303i	$-0.359608\pi$
$857$	21420.0	0.853784	0.426892	+	0.904303i	$0.359608\pi$
$858$	0	0
$859$	−18132.0	−0.720205	−0.360102	−	0.932913i	$-0.617258\pi$
$859$	−18132.0	−0.720205	−0.360102	+	0.932913i	$0.617258\pi$
$860$	5248.00	0.208088
$861$	0	0
$862$	13400.0	0.529473
$863$	−24036.0	−0.948082	−0.474041	−	0.880503i	$-0.657205\pi$
$863$	−24036.0	−0.948082	−0.474041	+	0.880503i	$0.657205\pi$
$864$	0	0
$865$	−2752.00	−0.108174
$866$	−10096.0	−0.396162
$867$	0	0
$868$	0	0
$869$	−48640.0	−1.89873
$870$	0	0
$871$	−3632.00	−0.141292
$872$	−17584.0	−0.682878
$873$	0	0
$874$	−24864.0	−0.962285
$875$	0	0
$876$	0	0
$877$	−4374.00	−0.168414	−0.0842072	−	0.996448i	$-0.526836\pi$
$877$	−4374.00	−0.168414	−0.0842072	+	0.996448i	$0.526836\pi$
$878$	−2688.00	−0.103321
$879$	0	0
$880$	5120.00	0.196131
$881$	46348.0	1.77242	0.886211	−	0.463282i	$-0.153328\pi$
$881$	46348.0	1.77242	0.886211	+	0.463282i	$0.153328\pi$
$882$	0	0
$883$	−20660.0	−0.787389	−0.393694	−	0.919241i	$-0.628803\pi$
$883$	−20660.0	−0.787389	−0.393694	+	0.919241i	$0.628803\pi$
$884$	1344.00	0.0511353
$885$	0	0
$886$	8784.00	0.333075
$887$	1800.00	0.0681376	0.0340688	−	0.999419i	$-0.489153\pi$
$887$	1800.00	0.0681376	0.0340688	+	0.999419i	$0.489153\pi$
$888$	0	0
$889$	0	0
$890$	9664.00	0.363975
$891$	0	0
$892$	8000.00	0.300291
$893$	−72224.0	−2.70648
$894$	0	0
$895$	11136.0	0.415906
$896$	0	0
$897$	0	0
$898$	−7332.00	−0.272463
$899$	7888.00	0.292636
$900$	0	0
$901$	−40152.0	−1.48464
$902$	33600.0	1.24031
$903$	0	0
$904$	−6160.00	−0.226636
$905$	−32416.0	−1.19066
$906$	0	0
$907$	−41996.0	−1.53744	−0.768718	−	0.639588i	$-0.779105\pi$
$907$	−41996.0	−1.53744	−0.768718	+	0.639588i	$0.779105\pi$
$908$	−1552.00	−0.0567235
$909$	0	0
$910$	0	0
$911$	41308.0	1.50230	0.751150	−	0.660132i	$-0.229500\pi$
$911$	41308.0	1.50230	0.751150	+	0.660132i	$0.229500\pi$
$912$	0	0
$913$	−27360.0	−0.991768
$914$	52.0000	0.00188185
$915$	0	0
$916$	16720.0	0.603105
$917$	0	0
$918$	0	0
$919$	3936.00	0.141280	0.0706402	−	0.997502i	$-0.477496\pi$
$919$	3936.00	0.141280	0.0706402	+	0.997502i	$0.477496\pi$
$920$	5376.00	0.192654
$921$	0	0
$922$	−15312.0	−0.546935
$923$	2096.00	0.0747461
$924$	0	0
$925$	13542.0	0.481360
$926$	25216.0	0.894870
$927$	0	0
$928$	−1856.00	−0.0656532
$929$	7212.00	0.254702	0.127351	−	0.991858i	$-0.459353\pi$
$929$	7212.00	0.254702	0.127351	+	0.991858i	$0.459353\pi$
$930$	0	0
$931$	0	0
$932$	5288.00	0.185852
$933$	0	0
$934$	6136.00	0.214964
$935$	26880.0	0.940182
$936$	0	0
$937$	−38976.0	−1.35890	−0.679451	−	0.733721i	$-0.737782\pi$
$937$	−38976.0	−1.35890	−0.679451	+	0.733721i	$0.737782\pi$
$938$	0	0
$939$	0	0
$940$	15616.0	0.541849
$941$	−53544.0	−1.85493	−0.927463	−	0.373916i	$-0.878015\pi$
$941$	−53544.0	−1.85493	−0.927463	+	0.373916i	$0.878015\pi$
$942$	0	0
$943$	35280.0	1.21832
$944$	−8768.00	−0.302303
$945$	0	0
$946$	13120.0	0.450918
$947$	21392.0	0.734051	0.367026	−	0.930211i	$-0.380376\pi$
$947$	21392.0	0.734051	0.367026	+	0.930211i	$0.380376\pi$
$948$	0	0
$949$	1760.00	0.0602023
$950$	−18056.0	−0.616646
$951$	0	0
$952$	0	0
$953$	−21162.0	−0.719312	−0.359656	−	0.933085i	$-0.617106\pi$
$953$	−21162.0	−0.719312	−0.359656	+	0.933085i	$0.617106\pi$
$954$	0	0
$955$	−24864.0	−0.842492
$956$	−9648.00	−0.326400
$957$	0	0
$958$	−12912.0	−0.435457
$959$	0	0
$960$	0	0
$961$	−11295.0	−0.379141
$962$	−1776.00	−0.0595224
$963$	0	0
$964$	−17344.0	−0.579474
$965$	−400.000	−0.0133435
$966$	0	0
$967$	8224.00	0.273491	0.136746	−	0.990606i	$-0.456336\pi$
$967$	8224.00	0.273491	0.136746	+	0.990606i	$0.456336\pi$
$968$	2152.00	0.0714544
$969$	0	0
$970$	13312.0	0.440642
$971$	8140.00	0.269027	0.134513	−	0.990912i	$-0.457053\pi$
$971$	8140.00	0.269027	0.134513	+	0.990912i	$0.457053\pi$
$972$	0	0
$973$	0	0
$974$	23792.0	0.782695
$975$	0	0
$976$	11072.0	0.363121
$977$	−32158.0	−1.05305	−0.526523	−	0.850161i	$-0.676505\pi$
$977$	−32158.0	−1.05305	−0.526523	+	0.850161i	$0.676505\pi$
$978$	0	0
$979$	24160.0	0.788720
$980$	0	0
$981$	0	0
$982$	528.000	0.0171580
$983$	41416.0	1.34381	0.671905	−	0.740637i	$-0.265476\pi$
$983$	41416.0	1.34381	0.671905	+	0.740637i	$0.265476\pi$
$984$	0	0
$985$	−1296.00	−0.0419228
$986$	−9744.00	−0.314718
$987$	0	0
$988$	2368.00	0.0762511
$989$	13776.0	0.442923
$990$	0	0
$991$	12296.0	0.394143	0.197071	−	0.980389i	$-0.436857\pi$
$991$	12296.0	0.394143	0.197071	+	0.980389i	$0.436857\pi$
$992$	−4352.00	−0.139290
$993$	0	0
$994$	0	0
$995$	−12352.0	−0.393552
$996$	0	0
$997$	57652.0	1.83135	0.915676	−	0.401918i	$-0.131656\pi$
$997$	57652.0	1.83135	0.915676	+	0.401918i	$0.131656\pi$
$998$	−5256.00	−0.166709
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.j.1.1		1
3.2	odd	2		294.4.a.f.1.1	yes	1
7.2	even	3		882.4.g.j.361.1		2
7.3	odd	6		882.4.g.c.667.1		2
7.4	even	3		882.4.g.j.667.1		2
7.5	odd	6		882.4.g.c.361.1		2
7.6	odd	2		882.4.a.q.1.1		1
12.11	even	2		2352.4.a.m.1.1		1
21.2	odd	6		294.4.e.f.67.1		2
21.5	even	6		294.4.e.j.67.1		2
21.11	odd	6		294.4.e.f.79.1		2
21.17	even	6		294.4.e.j.79.1		2
21.20	even	2		294.4.a.b.1.1	✓	1
84.83	odd	2		2352.4.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.4.a.b.1.1	✓	1	21.20	even	2
294.4.a.f.1.1	yes	1	3.2	odd	2
294.4.e.f.67.1		2	21.2	odd	6
294.4.e.f.79.1		2	21.11	odd	6
294.4.e.j.67.1		2	21.5	even	6
294.4.e.j.79.1		2	21.17	even	6
882.4.a.j.1.1		1	1.1	even	1	trivial
882.4.a.q.1.1		1	7.6	odd	2
882.4.g.c.361.1		2	7.5	odd	6
882.4.g.c.667.1		2	7.3	odd	6
882.4.g.j.361.1		2	7.2	even	3
882.4.g.j.667.1		2	7.4	even	3
2352.4.a.m.1.1		1	12.11	even	2
2352.4.a.z.1.1		1	84.83	odd	2

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.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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