LMFDB - Embedded newform 882.4.a.i.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 14)
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)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{5} +8.00000 q^{8} +O(q^{10})$

$q+2.00000 q^{2} +4.00000 q^{4} -14.0000 q^{5} +8.00000 q^{8} -28.0000 q^{10} +28.0000 q^{11} -18.0000 q^{13} +16.0000 q^{16} +74.0000 q^{17} -80.0000 q^{19} -56.0000 q^{20} +56.0000 q^{22} +112.000 q^{23} +71.0000 q^{25} -36.0000 q^{26} -190.000 q^{29} -72.0000 q^{31} +32.0000 q^{32} +148.000 q^{34} -346.000 q^{37} -160.000 q^{38} -112.000 q^{40} +162.000 q^{41} -412.000 q^{43} +112.000 q^{44} +224.000 q^{46} +24.0000 q^{47} +142.000 q^{50} -72.0000 q^{52} -318.000 q^{53} -392.000 q^{55} -380.000 q^{58} -200.000 q^{59} +198.000 q^{61} -144.000 q^{62} +64.0000 q^{64} +252.000 q^{65} -716.000 q^{67} +296.000 q^{68} -392.000 q^{71} -538.000 q^{73} -692.000 q^{74} -320.000 q^{76} +240.000 q^{79} -224.000 q^{80} +324.000 q^{82} -1072.00 q^{83} -1036.00 q^{85} -824.000 q^{86} +224.000 q^{88} +810.000 q^{89} +448.000 q^{92} +48.0000 q^{94} +1120.00 q^{95} -1354.00 q^{97} +O(q^{100})$

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
$5$	−14.0000	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$5$	−14.0000	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	0	0
$10$	−28.0000	−0.885438
$11$	28.0000	0.767483	0.383742	−	0.923440i	$-0.374635\pi$
$11$	28.0000	0.767483	0.383742	+	0.923440i	$0.374635\pi$
$12$	0	0
$13$	−18.0000	−0.384023	−0.192012	−	0.981393i	$-0.561501\pi$
$13$	−18.0000	−0.384023	−0.192012	+	0.981393i	$0.561501\pi$
$14$	0	0
$15$	0	0
$16$	16.0000	0.250000
$17$	74.0000	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$17$	74.0000	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$18$	0	0
$19$	−80.0000	−0.965961	−0.482980	−	0.875631i	$-0.660446\pi$
$19$	−80.0000	−0.965961	−0.482980	+	0.875631i	$0.660446\pi$
$20$	−56.0000	−0.626099
$21$	0	0
$22$	56.0000	0.542693
$23$	112.000	1.01537	0.507687	−	0.861541i	$-0.330501\pi$
$23$	112.000	1.01537	0.507687	+	0.861541i	$0.330501\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	−36.0000	−0.271545
$27$	0	0
$28$	0	0
$29$	−190.000	−1.21662	−0.608312	−	0.793698i	$-0.708153\pi$
$29$	−190.000	−1.21662	−0.608312	+	0.793698i	$0.708153\pi$
$30$	0	0
$31$	−72.0000	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$31$	−72.0000	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	148.000	0.746523
$35$	0	0
$36$	0	0
$37$	−346.000	−1.53735	−0.768676	−	0.639638i	$-0.779084\pi$
$37$	−346.000	−1.53735	−0.768676	+	0.639638i	$0.779084\pi$
$38$	−160.000	−0.683038
$39$	0	0
$40$	−112.000	−0.442719
$41$	162.000	0.617077	0.308538	−	0.951212i	$-0.400160\pi$
$41$	162.000	0.617077	0.308538	+	0.951212i	$0.400160\pi$
$42$	0	0
$43$	−412.000	−1.46115	−0.730575	−	0.682833i	$-0.760748\pi$
$43$	−412.000	−1.46115	−0.730575	+	0.682833i	$0.760748\pi$
$44$	112.000	0.383742
$45$	0	0
$46$	224.000	0.717978
$47$	24.0000	0.0744843	0.0372421	−	0.999306i	$-0.488143\pi$
$47$	24.0000	0.0744843	0.0372421	+	0.999306i	$0.488143\pi$
$48$	0	0
$49$	0	0
$50$	142.000	0.401637
$51$	0	0
$52$	−72.0000	−0.192012
$53$	−318.000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−318.000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−392.000	−0.961041
$56$	0	0
$57$	0	0
$58$	−380.000	−0.860284
$59$	−200.000	−0.441318	−0.220659	−	0.975351i	$-0.570821\pi$
$59$	−200.000	−0.441318	−0.220659	+	0.975351i	$0.570821\pi$
$60$	0	0
$61$	198.000	0.415595	0.207798	−	0.978172i	$-0.433370\pi$
$61$	198.000	0.415595	0.207798	+	0.978172i	$0.433370\pi$
$62$	−144.000	−0.294968
$63$	0	0
$64$	64.0000	0.125000
$65$	252.000	0.480873
$66$	0	0
$67$	−716.000	−1.30557	−0.652786	−	0.757542i	$-0.726400\pi$
$67$	−716.000	−1.30557	−0.652786	+	0.757542i	$0.726400\pi$
$68$	296.000	0.527872
$69$	0	0
$70$	0	0
$71$	−392.000	−0.655237	−0.327619	−	0.944810i	$-0.606246\pi$
$71$	−392.000	−0.655237	−0.327619	+	0.944810i	$0.606246\pi$
$72$	0	0
$73$	−538.000	−0.862577	−0.431289	−	0.902214i	$-0.641941\pi$
$73$	−538.000	−0.862577	−0.431289	+	0.902214i	$0.641941\pi$
$74$	−692.000	−1.08707
$75$	0	0
$76$	−320.000	−0.482980
$77$	0	0
$78$	0	0
$79$	240.000	0.341799	0.170899	−	0.985288i	$-0.445333\pi$
$79$	240.000	0.341799	0.170899	+	0.985288i	$0.445333\pi$
$80$	−224.000	−0.313050
$81$	0	0
$82$	324.000	0.436339
$83$	−1072.00	−1.41768	−0.708839	−	0.705370i	$-0.750781\pi$
$83$	−1072.00	−1.41768	−0.708839	+	0.705370i	$0.750781\pi$
$84$	0	0
$85$	−1036.00	−1.32200
$86$	−824.000	−1.03319
$87$	0	0
$88$	224.000	0.271346
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	0	0
$92$	448.000	0.507687
$93$	0	0
$94$	48.0000	0.0526683
$95$	1120.00	1.20957
$96$	0	0
$97$	−1354.00	−1.41730	−0.708649	−	0.705561i	$-0.750695\pi$
$97$	−1354.00	−1.41730	−0.708649	+	0.705561i	$0.750695\pi$
$98$	0	0
$99$	0	0
$100$	284.000	0.284000
$101$	−1358.00	−1.33788	−0.668941	−	0.743316i	$-0.733252\pi$
$101$	−1358.00	−1.33788	−0.668941	+	0.743316i	$0.733252\pi$
$102$	0	0
$103$	832.000	0.795916	0.397958	−	0.917404i	$-0.369719\pi$
$103$	832.000	0.795916	0.397958	+	0.917404i	$0.369719\pi$
$104$	−144.000	−0.135773
$105$	0	0
$106$	−636.000	−0.582772
$107$	−444.000	−0.401150	−0.200575	−	0.979678i	$-0.564281\pi$
$107$	−444.000	−0.401150	−0.200575	+	0.979678i	$0.564281\pi$
$108$	0	0
$109$	1870.00	1.64324	0.821622	−	0.570033i	$-0.193070\pi$
$109$	1870.00	1.64324	0.821622	+	0.570033i	$0.193070\pi$
$110$	−784.000	−0.679559
$111$	0	0
$112$	0	0
$113$	−1378.00	−1.14718	−0.573590	−	0.819143i	$-0.694450\pi$
$113$	−1378.00	−1.14718	−0.573590	+	0.819143i	$0.694450\pi$
$114$	0	0
$115$	−1568.00	−1.27145
$116$	−760.000	−0.608312
$117$	0	0
$118$	−400.000	−0.312059
$119$	0	0
$120$	0	0
$121$	−547.000	−0.410969
$122$	396.000	0.293870
$123$	0	0
$124$	−288.000	−0.208574
$125$	756.000	0.540950
$126$	0	0
$127$	1944.00	1.35828	0.679142	−	0.734007i	$-0.262352\pi$
$127$	1944.00	1.35828	0.679142	+	0.734007i	$0.262352\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	504.000	0.340029
$131$	−848.000	−0.565573	−0.282787	−	0.959183i	$-0.591259\pi$
$131$	−848.000	−0.565573	−0.282787	+	0.959183i	$0.591259\pi$
$132$	0	0
$133$	0	0
$134$	−1432.00	−0.923179
$135$	0	0
$136$	592.000	0.373262
$137$	2966.00	1.84965	0.924827	−	0.380389i	$-0.124210\pi$
$137$	2966.00	1.84965	0.924827	+	0.380389i	$0.124210\pi$
$138$	0	0
$139$	−2800.00	−1.70858	−0.854291	−	0.519795i	$-0.826008\pi$
$139$	−2800.00	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$140$	0	0
$141$	0	0
$142$	−784.000	−0.463323
$143$	−504.000	−0.294731
$144$	0	0
$145$	2660.00	1.52346
$146$	−1076.00	−0.609934
$147$	0	0
$148$	−1384.00	−0.768676
$149$	−510.000	−0.280408	−0.140204	−	0.990123i	$-0.544776\pi$
$149$	−510.000	−0.280408	−0.140204	+	0.990123i	$0.544776\pi$
$150$	0	0
$151$	592.000	0.319048	0.159524	−	0.987194i	$-0.449004\pi$
$151$	592.000	0.319048	0.159524	+	0.987194i	$0.449004\pi$
$152$	−640.000	−0.341519
$153$	0	0
$154$	0	0
$155$	1008.00	0.522352
$156$	0	0
$157$	2686.00	1.36539	0.682695	−	0.730704i	$-0.260808\pi$
$157$	2686.00	1.36539	0.682695	+	0.730704i	$0.260808\pi$
$158$	480.000	0.241688
$159$	0	0
$160$	−448.000	−0.221359
$161$	0	0
$162$	0	0
$163$	−1012.00	−0.486294	−0.243147	−	0.969989i	$-0.578180\pi$
$163$	−1012.00	−0.486294	−0.243147	+	0.969989i	$0.578180\pi$
$164$	648.000	0.308538
$165$	0	0
$166$	−2144.00	−1.00245
$167$	544.000	0.252072	0.126036	−	0.992026i	$-0.459775\pi$
$167$	544.000	0.252072	0.126036	+	0.992026i	$0.459775\pi$
$168$	0	0
$169$	−1873.00	−0.852526
$170$	−2072.00	−0.934795
$171$	0	0
$172$	−1648.00	−0.730575
$173$	1858.00	0.816538	0.408269	−	0.912862i	$-0.366132\pi$
$173$	1858.00	0.816538	0.408269	+	0.912862i	$0.366132\pi$
$174$	0	0
$175$	0	0
$176$	448.000	0.191871
$177$	0	0
$178$	1620.00	0.682158
$179$	300.000	0.125268	0.0626342	−	0.998037i	$-0.480050\pi$
$179$	300.000	0.125268	0.0626342	+	0.998037i	$0.480050\pi$
$180$	0	0
$181$	2358.00	0.968336	0.484168	−	0.874975i	$-0.339122\pi$
$181$	2358.00	0.968336	0.484168	+	0.874975i	$0.339122\pi$
$182$	0	0
$183$	0	0
$184$	896.000	0.358989
$185$	4844.00	1.92507
$186$	0	0
$187$	2072.00	0.810265
$188$	96.0000	0.0372421
$189$	0	0
$190$	2240.00	0.855298
$191$	−1392.00	−0.527338	−0.263669	−	0.964613i	$-0.584933\pi$
$191$	−1392.00	−0.527338	−0.263669	+	0.964613i	$0.584933\pi$
$192$	0	0
$193$	1778.00	0.663126	0.331563	−	0.943433i	$-0.392424\pi$
$193$	1778.00	0.663126	0.331563	+	0.943433i	$0.392424\pi$
$194$	−2708.00	−1.00218
$195$	0	0
$196$	0	0
$197$	−1214.00	−0.439055	−0.219528	−	0.975606i	$-0.570452\pi$
$197$	−1214.00	−0.439055	−0.219528	+	0.975606i	$0.570452\pi$
$198$	0	0
$199$	−1040.00	−0.370471	−0.185235	−	0.982694i	$-0.559305\pi$
$199$	−1040.00	−0.370471	−0.185235	+	0.982694i	$0.559305\pi$
$200$	568.000	0.200818
$201$	0	0
$202$	−2716.00	−0.946025
$203$	0	0
$204$	0	0
$205$	−2268.00	−0.772702
$206$	1664.00	0.562798
$207$	0	0
$208$	−288.000	−0.0960058
$209$	−2240.00	−0.741359
$210$	0	0
$211$	−3868.00	−1.26201	−0.631005	−	0.775779i	$-0.717357\pi$
$211$	−3868.00	−1.26201	−0.631005	+	0.775779i	$0.717357\pi$
$212$	−1272.00	−0.412082
$213$	0	0
$214$	−888.000	−0.283656
$215$	5768.00	1.82965
$216$	0	0
$217$	0	0
$218$	3740.00	1.16195
$219$	0	0
$220$	−1568.00	−0.480521
$221$	−1332.00	−0.405430
$222$	0	0
$223$	−3968.00	−1.19156	−0.595778	−	0.803149i	$-0.703156\pi$
$223$	−3968.00	−1.19156	−0.595778	+	0.803149i	$0.703156\pi$
$224$	0	0
$225$	0	0
$226$	−2756.00	−0.811179
$227$	−3936.00	−1.15084	−0.575422	−	0.817857i	$-0.695162\pi$
$227$	−3936.00	−1.15084	−0.575422	+	0.817857i	$0.695162\pi$
$228$	0	0
$229$	−4810.00	−1.38801	−0.694004	−	0.719971i	$-0.744155\pi$
$229$	−4810.00	−1.38801	−0.694004	+	0.719971i	$0.744155\pi$
$230$	−3136.00	−0.899051
$231$	0	0
$232$	−1520.00	−0.430142
$233$	2182.00	0.613509	0.306754	−	0.951789i	$-0.400757\pi$
$233$	2182.00	0.613509	0.306754	+	0.951789i	$0.400757\pi$
$234$	0	0
$235$	−336.000	−0.0932690
$236$	−800.000	−0.220659
$237$	0	0
$238$	0	0
$239$	3000.00	0.811941	0.405970	−	0.913886i	$-0.366934\pi$
$239$	3000.00	0.811941	0.405970	+	0.913886i	$0.366934\pi$
$240$	0	0
$241$	−2042.00	−0.545796	−0.272898	−	0.962043i	$-0.587982\pi$
$241$	−2042.00	−0.545796	−0.272898	+	0.962043i	$0.587982\pi$
$242$	−1094.00	−0.290599
$243$	0	0
$244$	792.000	0.207798
$245$	0	0
$246$	0	0
$247$	1440.00	0.370951
$248$	−576.000	−0.147484
$249$	0	0
$250$	1512.00	0.382509
$251$	−528.000	−0.132777	−0.0663886	−	0.997794i	$-0.521148\pi$
$251$	−528.000	−0.132777	−0.0663886	+	0.997794i	$0.521148\pi$
$252$	0	0
$253$	3136.00	0.779283
$254$	3888.00	0.960452
$255$	0	0
$256$	256.000	0.0625000
$257$	5634.00	1.36747	0.683734	−	0.729731i	$-0.260355\pi$
$257$	5634.00	1.36747	0.683734	+	0.729731i	$0.260355\pi$
$258$	0	0
$259$	0	0
$260$	1008.00	0.240437
$261$	0	0
$262$	−1696.00	−0.399921
$263$	−168.000	−0.0393891	−0.0196945	−	0.999806i	$-0.506269\pi$
$263$	−168.000	−0.0393891	−0.0196945	+	0.999806i	$0.506269\pi$
$264$	0	0
$265$	4452.00	1.03202
$266$	0	0
$267$	0	0
$268$	−2864.00	−0.652786
$269$	−1310.00	−0.296922	−0.148461	−	0.988918i	$-0.547432\pi$
$269$	−1310.00	−0.296922	−0.148461	+	0.988918i	$0.547432\pi$
$270$	0	0
$271$	2208.00	0.494932	0.247466	−	0.968897i	$-0.420402\pi$
$271$	2208.00	0.494932	0.247466	+	0.968897i	$0.420402\pi$
$272$	1184.00	0.263936
$273$	0	0
$274$	5932.00	1.30790
$275$	1988.00	0.435931
$276$	0	0
$277$	5294.00	1.14832	0.574162	−	0.818742i	$-0.305328\pi$
$277$	5294.00	1.14832	0.574162	+	0.818742i	$0.305328\pi$
$278$	−5600.00	−1.20815
$279$	0	0
$280$	0	0
$281$	−3242.00	−0.688262	−0.344131	−	0.938922i	$-0.611826\pi$
$281$	−3242.00	−0.688262	−0.344131	+	0.938922i	$0.611826\pi$
$282$	0	0
$283$	1592.00	0.334398	0.167199	−	0.985923i	$-0.446528\pi$
$283$	1592.00	0.334398	0.167199	+	0.985923i	$0.446528\pi$
$284$	−1568.00	−0.327619
$285$	0	0
$286$	−1008.00	−0.208407
$287$	0	0
$288$	0	0
$289$	563.000	0.114594
$290$	5320.00	1.07725
$291$	0	0
$292$	−2152.00	−0.431289
$293$	−5022.00	−1.00133	−0.500663	−	0.865642i	$-0.666910\pi$
$293$	−5022.00	−1.00133	−0.500663	+	0.865642i	$0.666910\pi$
$294$	0	0
$295$	2800.00	0.552618
$296$	−2768.00	−0.543536
$297$	0	0
$298$	−1020.00	−0.198279
$299$	−2016.00	−0.389927
$300$	0	0
$301$	0	0
$302$	1184.00	0.225601
$303$	0	0
$304$	−1280.00	−0.241490
$305$	−2772.00	−0.520407
$306$	0	0
$307$	9536.00	1.77280	0.886398	−	0.462924i	$-0.153200\pi$
$307$	9536.00	1.77280	0.886398	+	0.462924i	$0.153200\pi$
$308$	0	0
$309$	0	0
$310$	2016.00	0.369358
$311$	−968.000	−0.176496	−0.0882480	−	0.996099i	$-0.528127\pi$
$311$	−968.000	−0.176496	−0.0882480	+	0.996099i	$0.528127\pi$
$312$	0	0
$313$	−3058.00	−0.552231	−0.276116	−	0.961124i	$-0.589047\pi$
$313$	−3058.00	−0.552231	−0.276116	+	0.961124i	$0.589047\pi$
$314$	5372.00	0.965476
$315$	0	0
$316$	960.000	0.170899
$317$	4986.00	0.883412	0.441706	−	0.897160i	$-0.354373\pi$
$317$	4986.00	0.883412	0.441706	+	0.897160i	$0.354373\pi$
$318$	0	0
$319$	−5320.00	−0.933739
$320$	−896.000	−0.156525
$321$	0	0
$322$	0	0
$323$	−5920.00	−1.01981
$324$	0	0
$325$	−1278.00	−0.218125
$326$	−2024.00	−0.343862
$327$	0	0
$328$	1296.00	0.218170
$329$	0	0
$330$	0	0
$331$	8612.00	1.43009	0.715043	−	0.699081i	$-0.246407\pi$
$331$	8612.00	1.43009	0.715043	+	0.699081i	$0.246407\pi$
$332$	−4288.00	−0.708839
$333$	0	0
$334$	1088.00	0.178242
$335$	10024.0	1.63483
$336$	0	0
$337$	−10206.0	−1.64972	−0.824861	−	0.565336i	$-0.808747\pi$
$337$	−10206.0	−1.64972	−0.824861	+	0.565336i	$0.808747\pi$
$338$	−3746.00	−0.602827
$339$	0	0
$340$	−4144.00	−0.661000
$341$	−2016.00	−0.320154
$342$	0	0
$343$	0	0
$344$	−3296.00	−0.516594
$345$	0	0
$346$	3716.00	0.577380
$347$	−2004.00	−0.310030	−0.155015	−	0.987912i	$-0.549543\pi$
$347$	−2004.00	−0.310030	−0.155015	+	0.987912i	$0.549543\pi$
$348$	0	0
$349$	−1330.00	−0.203992	−0.101996	−	0.994785i	$-0.532523\pi$
$349$	−1330.00	−0.203992	−0.101996	+	0.994785i	$0.532523\pi$
$350$	0	0
$351$	0	0
$352$	896.000	0.135673
$353$	978.000	0.147461	0.0737304	−	0.997278i	$-0.476510\pi$
$353$	978.000	0.147461	0.0737304	+	0.997278i	$0.476510\pi$
$354$	0	0
$355$	5488.00	0.820487
$356$	3240.00	0.482359
$357$	0	0
$358$	600.000	0.0885782
$359$	9680.00	1.42309	0.711547	−	0.702638i	$-0.247995\pi$
$359$	9680.00	1.42309	0.711547	+	0.702638i	$0.247995\pi$
$360$	0	0
$361$	−459.000	−0.0669194
$362$	4716.00	0.684717
$363$	0	0
$364$	0	0
$365$	7532.00	1.08012
$366$	0	0
$367$	8656.00	1.23117	0.615585	−	0.788070i	$-0.288920\pi$
$367$	8656.00	1.23117	0.615585	+	0.788070i	$0.288920\pi$
$368$	1792.00	0.253844
$369$	0	0
$370$	9688.00	1.36123
$371$	0	0
$372$	0	0
$373$	5278.00	0.732666	0.366333	−	0.930484i	$-0.380613\pi$
$373$	5278.00	0.732666	0.366333	+	0.930484i	$0.380613\pi$
$374$	4144.00	0.572944
$375$	0	0
$376$	192.000	0.0263342
$377$	3420.00	0.467212
$378$	0	0
$379$	6340.00	0.859272	0.429636	−	0.903002i	$-0.358642\pi$
$379$	6340.00	0.859272	0.429636	+	0.903002i	$0.358642\pi$
$380$	4480.00	0.604787
$381$	0	0
$382$	−2784.00	−0.372884
$383$	−6232.00	−0.831437	−0.415718	−	0.909493i	$-0.636470\pi$
$383$	−6232.00	−0.831437	−0.415718	+	0.909493i	$0.636470\pi$
$384$	0	0
$385$	0	0
$386$	3556.00	0.468901
$387$	0	0
$388$	−5416.00	−0.708649
$389$	14810.0	1.93033	0.965163	−	0.261649i	$-0.0842664\pi$
$389$	14810.0	1.93033	0.965163	+	0.261649i	$0.0842664\pi$
$390$	0	0
$391$	8288.00	1.07197
$392$	0	0
$393$	0	0
$394$	−2428.00	−0.310459
$395$	−3360.00	−0.428000
$396$	0	0
$397$	−5154.00	−0.651566	−0.325783	−	0.945445i	$-0.605628\pi$
$397$	−5154.00	−0.651566	−0.325783	+	0.945445i	$0.605628\pi$
$398$	−2080.00	−0.261962
$399$	0	0
$400$	1136.00	0.142000
$401$	−3282.00	−0.408716	−0.204358	−	0.978896i	$-0.565511\pi$
$401$	−3282.00	−0.408716	−0.204358	+	0.978896i	$0.565511\pi$
$402$	0	0
$403$	1296.00	0.160194
$404$	−5432.00	−0.668941
$405$	0	0
$406$	0	0
$407$	−9688.00	−1.17989
$408$	0	0
$409$	−5810.00	−0.702411	−0.351205	−	0.936298i	$-0.614228\pi$
$409$	−5810.00	−0.702411	−0.351205	+	0.936298i	$0.614228\pi$
$410$	−4536.00	−0.546383
$411$	0	0
$412$	3328.00	0.397958
$413$	0	0
$414$	0	0
$415$	15008.0	1.77521
$416$	−576.000	−0.0678864
$417$	0	0
$418$	−4480.00	−0.524220
$419$	13560.0	1.58102	0.790512	−	0.612446i	$-0.209814\pi$
$419$	13560.0	1.58102	0.790512	+	0.612446i	$0.209814\pi$
$420$	0	0
$421$	−738.000	−0.0854345	−0.0427172	−	0.999087i	$-0.513601\pi$
$421$	−738.000	−0.0854345	−0.0427172	+	0.999087i	$0.513601\pi$
$422$	−7736.00	−0.892376
$423$	0	0
$424$	−2544.00	−0.291386
$425$	5254.00	0.599662
$426$	0	0
$427$	0	0
$428$	−1776.00	−0.200575
$429$	0	0
$430$	11536.0	1.29376
$431$	−1272.00	−0.142158	−0.0710790	−	0.997471i	$-0.522644\pi$
$431$	−1272.00	−0.142158	−0.0710790	+	0.997471i	$0.522644\pi$
$432$	0	0
$433$	5062.00	0.561811	0.280906	−	0.959735i	$-0.409365\pi$
$433$	5062.00	0.561811	0.280906	+	0.959735i	$0.409365\pi$
$434$	0	0
$435$	0	0
$436$	7480.00	0.821622
$437$	−8960.00	−0.980812
$438$	0	0
$439$	−5640.00	−0.613172	−0.306586	−	0.951843i	$-0.599187\pi$
$439$	−5640.00	−0.613172	−0.306586	+	0.951843i	$0.599187\pi$
$440$	−3136.00	−0.339779
$441$	0	0
$442$	−2664.00	−0.286682
$443$	−13388.0	−1.43585	−0.717927	−	0.696119i	$-0.754909\pi$
$443$	−13388.0	−1.43585	−0.717927	+	0.696119i	$0.754909\pi$
$444$	0	0
$445$	−11340.0	−1.20802
$446$	−7936.00	−0.842557
$447$	0	0
$448$	0	0
$449$	3230.00	0.339495	0.169747	−	0.985488i	$-0.445705\pi$
$449$	3230.00	0.339495	0.169747	+	0.985488i	$0.445705\pi$
$450$	0	0
$451$	4536.00	0.473596
$452$	−5512.00	−0.573590
$453$	0	0
$454$	−7872.00	−0.813769
$455$	0	0
$456$	0	0
$457$	−10646.0	−1.08971	−0.544857	−	0.838529i	$-0.683416\pi$
$457$	−10646.0	−1.08971	−0.544857	+	0.838529i	$0.683416\pi$
$458$	−9620.00	−0.981470
$459$	0	0
$460$	−6272.00	−0.635725
$461$	7282.00	0.735698	0.367849	−	0.929886i	$-0.380094\pi$
$461$	7282.00	0.735698	0.367849	+	0.929886i	$0.380094\pi$
$462$	0	0
$463$	12688.0	1.27357	0.636783	−	0.771043i	$-0.280265\pi$
$463$	12688.0	1.27357	0.636783	+	0.771043i	$0.280265\pi$
$464$	−3040.00	−0.304156
$465$	0	0
$466$	4364.00	0.433816
$467$	−2816.00	−0.279034	−0.139517	−	0.990220i	$-0.544555\pi$
$467$	−2816.00	−0.279034	−0.139517	+	0.990220i	$0.544555\pi$
$468$	0	0
$469$	0	0
$470$	−672.000	−0.0659512
$471$	0	0
$472$	−1600.00	−0.156030
$473$	−11536.0	−1.12141
$474$	0	0
$475$	−5680.00	−0.548666
$476$	0	0
$477$	0	0
$478$	6000.00	0.574129
$479$	−3160.00	−0.301428	−0.150714	−	0.988577i	$-0.548157\pi$
$479$	−3160.00	−0.301428	−0.150714	+	0.988577i	$0.548157\pi$
$480$	0	0
$481$	6228.00	0.590379
$482$	−4084.00	−0.385936
$483$	0	0
$484$	−2188.00	−0.205485
$485$	18956.0	1.77474
$486$	0	0
$487$	−14176.0	−1.31905	−0.659523	−	0.751684i	$-0.729242\pi$
$487$	−14176.0	−1.31905	−0.659523	+	0.751684i	$0.729242\pi$
$488$	1584.00	0.146935
$489$	0	0
$490$	0	0
$491$	11268.0	1.03568	0.517839	−	0.855478i	$-0.326737\pi$
$491$	11268.0	1.03568	0.517839	+	0.855478i	$0.326737\pi$
$492$	0	0
$493$	−14060.0	−1.28444
$494$	2880.00	0.262302
$495$	0	0
$496$	−1152.00	−0.104287
$497$	0	0
$498$	0	0
$499$	−4460.00	−0.400114	−0.200057	−	0.979784i	$-0.564113\pi$
$499$	−4460.00	−0.400114	−0.200057	+	0.979784i	$0.564113\pi$
$500$	3024.00	0.270475
$501$	0	0
$502$	−1056.00	−0.0938876
$503$	−1512.00	−0.134029	−0.0670147	−	0.997752i	$-0.521347\pi$
$503$	−1512.00	−0.134029	−0.0670147	+	0.997752i	$0.521347\pi$
$504$	0	0
$505$	19012.0	1.67529
$506$	6272.00	0.551036
$507$	0	0
$508$	7776.00	0.679142
$509$	−11790.0	−1.02668	−0.513342	−	0.858184i	$-0.671593\pi$
$509$	−11790.0	−1.02668	−0.513342	+	0.858184i	$0.671593\pi$
$510$	0	0
$511$	0	0
$512$	512.000	0.0441942
$513$	0	0
$514$	11268.0	0.966946
$515$	−11648.0	−0.996645
$516$	0	0
$517$	672.000	0.0571654
$518$	0	0
$519$	0	0
$520$	2016.00	0.170014
$521$	1362.00	0.114530	0.0572652	−	0.998359i	$-0.481762\pi$
$521$	1362.00	0.114530	0.0572652	+	0.998359i	$0.481762\pi$
$522$	0	0
$523$	−6968.00	−0.582580	−0.291290	−	0.956635i	$-0.594084\pi$
$523$	−6968.00	−0.582580	−0.291290	+	0.956635i	$0.594084\pi$
$524$	−3392.00	−0.282787
$525$	0	0
$526$	−336.000	−0.0278523
$527$	−5328.00	−0.440401
$528$	0	0
$529$	377.000	0.0309855
$530$	8904.00	0.729745
$531$	0	0
$532$	0	0
$533$	−2916.00	−0.236972
$534$	0	0
$535$	6216.00	0.502320
$536$	−5728.00	−0.461589
$537$	0	0
$538$	−2620.00	−0.209956
$539$	0	0
$540$	0	0
$541$	7062.00	0.561218	0.280609	−	0.959822i	$-0.409464\pi$
$541$	7062.00	0.561218	0.280609	+	0.959822i	$0.409464\pi$
$542$	4416.00	0.349969
$543$	0	0
$544$	2368.00	0.186631
$545$	−26180.0	−2.05767
$546$	0	0
$547$	−8196.00	−0.640650	−0.320325	−	0.947308i	$-0.603792\pi$
$547$	−8196.00	−0.640650	−0.320325	+	0.947308i	$0.603792\pi$
$548$	11864.0	0.924827
$549$	0	0
$550$	3976.00	0.308249
$551$	15200.0	1.17521
$552$	0	0
$553$	0	0
$554$	10588.0	0.811987
$555$	0	0
$556$	−11200.0	−0.854291
$557$	7466.00	0.567944	0.283972	−	0.958833i	$-0.408348\pi$
$557$	7466.00	0.567944	0.283972	+	0.958833i	$0.408348\pi$
$558$	0	0
$559$	7416.00	0.561115
$560$	0	0
$561$	0	0
$562$	−6484.00	−0.486674
$563$	24968.0	1.86905	0.934526	−	0.355896i	$-0.115824\pi$
$563$	24968.0	1.86905	0.934526	+	0.355896i	$0.115824\pi$
$564$	0	0
$565$	19292.0	1.43650
$566$	3184.00	0.236455
$567$	0	0
$568$	−3136.00	−0.231661
$569$	−14250.0	−1.04990	−0.524948	−	0.851134i	$-0.675915\pi$
$569$	−14250.0	−1.04990	−0.524948	+	0.851134i	$0.675915\pi$
$570$	0	0
$571$	6372.00	0.467005	0.233503	−	0.972356i	$-0.424981\pi$
$571$	6372.00	0.467005	0.233503	+	0.972356i	$0.424981\pi$
$572$	−2016.00	−0.147366
$573$	0	0
$574$	0	0
$575$	7952.00	0.576733
$576$	0	0
$577$	8366.00	0.603607	0.301803	−	0.953370i	$-0.402411\pi$
$577$	8366.00	0.603607	0.301803	+	0.953370i	$0.402411\pi$
$578$	1126.00	0.0810301
$579$	0	0
$580$	10640.0	0.761728
$581$	0	0
$582$	0	0
$583$	−8904.00	−0.632532
$584$	−4304.00	−0.304967
$585$	0	0
$586$	−10044.0	−0.708044
$587$	20384.0	1.43328	0.716642	−	0.697441i	$-0.245678\pi$
$587$	20384.0	1.43328	0.716642	+	0.697441i	$0.245678\pi$
$588$	0	0
$589$	5760.00	0.402948
$590$	5600.00	0.390760
$591$	0	0
$592$	−5536.00	−0.384338
$593$	9378.00	0.649424	0.324712	−	0.945813i	$-0.394733\pi$
$593$	9378.00	0.649424	0.324712	+	0.945813i	$0.394733\pi$
$594$	0	0
$595$	0	0
$596$	−2040.00	−0.140204
$597$	0	0
$598$	−4032.00	−0.275720
$599$	9000.00	0.613907	0.306953	−	0.951725i	$-0.400690\pi$
$599$	9000.00	0.613907	0.306953	+	0.951725i	$0.400690\pi$
$600$	0	0
$601$	−7562.00	−0.513245	−0.256623	−	0.966512i	$-0.582610\pi$
$601$	−7562.00	−0.513245	−0.256623	+	0.966512i	$0.582610\pi$
$602$	0	0
$603$	0	0
$604$	2368.00	0.159524
$605$	7658.00	0.514615
$606$	0	0
$607$	2976.00	0.198999	0.0994993	−	0.995038i	$-0.468276\pi$
$607$	2976.00	0.198999	0.0994993	+	0.995038i	$0.468276\pi$
$608$	−2560.00	−0.170759
$609$	0	0
$610$	−5544.00	−0.367984
$611$	−432.000	−0.0286037
$612$	0	0
$613$	4278.00	0.281871	0.140935	−	0.990019i	$-0.454989\pi$
$613$	4278.00	0.281871	0.140935	+	0.990019i	$0.454989\pi$
$614$	19072.0	1.25356
$615$	0	0
$616$	0	0
$617$	−18794.0	−1.22629	−0.613143	−	0.789972i	$-0.710095\pi$
$617$	−18794.0	−1.22629	−0.613143	+	0.789972i	$0.710095\pi$
$618$	0	0
$619$	−18040.0	−1.17139	−0.585694	−	0.810532i	$-0.699178\pi$
$619$	−18040.0	−1.17139	−0.585694	+	0.810532i	$0.699178\pi$
$620$	4032.00	0.261176
$621$	0	0
$622$	−1936.00	−0.124801
$623$	0	0
$624$	0	0
$625$	−19459.0	−1.24538
$626$	−6116.00	−0.390486
$627$	0	0
$628$	10744.0	0.682695
$629$	−25604.0	−1.62305
$630$	0	0
$631$	−21688.0	−1.36828	−0.684141	−	0.729350i	$-0.739823\pi$
$631$	−21688.0	−1.36828	−0.684141	+	0.729350i	$0.739823\pi$
$632$	1920.00	0.120844
$633$	0	0
$634$	9972.00	0.624667
$635$	−27216.0	−1.70084
$636$	0	0
$637$	0	0
$638$	−10640.0	−0.660253
$639$	0	0
$640$	−1792.00	−0.110680
$641$	10558.0	0.650571	0.325285	−	0.945616i	$-0.394540\pi$
$641$	10558.0	0.650571	0.325285	+	0.945616i	$0.394540\pi$
$642$	0	0
$643$	26152.0	1.60394	0.801971	−	0.597363i	$-0.203785\pi$
$643$	26152.0	1.60394	0.801971	+	0.597363i	$0.203785\pi$
$644$	0	0
$645$	0	0
$646$	−11840.0	−0.721112
$647$	25584.0	1.55458	0.777288	−	0.629145i	$-0.216595\pi$
$647$	25584.0	1.55458	0.777288	+	0.629145i	$0.216595\pi$
$648$	0	0
$649$	−5600.00	−0.338705
$650$	−2556.00	−0.154238
$651$	0	0
$652$	−4048.00	−0.243147
$653$	−15198.0	−0.910787	−0.455393	−	0.890290i	$-0.650501\pi$
$653$	−15198.0	−0.910787	−0.455393	+	0.890290i	$0.650501\pi$
$654$	0	0
$655$	11872.0	0.708210
$656$	2592.00	0.154269
$657$	0	0
$658$	0	0
$659$	6100.00	0.360580	0.180290	−	0.983613i	$-0.442296\pi$
$659$	6100.00	0.360580	0.180290	+	0.983613i	$0.442296\pi$
$660$	0	0
$661$	2318.00	0.136399	0.0681995	−	0.997672i	$-0.478275\pi$
$661$	2318.00	0.136399	0.0681995	+	0.997672i	$0.478275\pi$
$662$	17224.0	1.01122
$663$	0	0
$664$	−8576.00	−0.501225
$665$	0	0
$666$	0	0
$667$	−21280.0	−1.23533
$668$	2176.00	0.126036
$669$	0	0
$670$	20048.0	1.15600
$671$	5544.00	0.318962
$672$	0	0
$673$	−10222.0	−0.585482	−0.292741	−	0.956192i	$-0.594567\pi$
$673$	−10222.0	−0.585482	−0.292741	+	0.956192i	$0.594567\pi$
$674$	−20412.0	−1.16653
$675$	0	0
$676$	−7492.00	−0.426263
$677$	25434.0	1.44388	0.721941	−	0.691955i	$-0.243250\pi$
$677$	25434.0	1.44388	0.721941	+	0.691955i	$0.243250\pi$
$678$	0	0
$679$	0	0
$680$	−8288.00	−0.467397
$681$	0	0
$682$	−4032.00	−0.226383
$683$	8532.00	0.477991	0.238996	−	0.971021i	$-0.423182\pi$
$683$	8532.00	0.477991	0.238996	+	0.971021i	$0.423182\pi$
$684$	0	0
$685$	−41524.0	−2.31613
$686$	0	0
$687$	0	0
$688$	−6592.00	−0.365287
$689$	5724.00	0.316498
$690$	0	0
$691$	−20672.0	−1.13806	−0.569030	−	0.822317i	$-0.692681\pi$
$691$	−20672.0	−1.13806	−0.569030	+	0.822317i	$0.692681\pi$
$692$	7432.00	0.408269
$693$	0	0
$694$	−4008.00	−0.219224
$695$	39200.0	2.13948
$696$	0	0
$697$	11988.0	0.651475
$698$	−2660.00	−0.144244
$699$	0	0
$700$	0	0
$701$	21458.0	1.15614	0.578072	−	0.815985i	$-0.303805\pi$
$701$	21458.0	1.15614	0.578072	+	0.815985i	$0.303805\pi$
$702$	0	0
$703$	27680.0	1.48502
$704$	1792.00	0.0959354
$705$	0	0
$706$	1956.00	0.104271
$707$	0	0
$708$	0	0
$709$	−9850.00	−0.521755	−0.260878	−	0.965372i	$-0.584012\pi$
$709$	−9850.00	−0.521755	−0.260878	+	0.965372i	$0.584012\pi$
$710$	10976.0	0.580172
$711$	0	0
$712$	6480.00	0.341079
$713$	−8064.00	−0.423561
$714$	0	0
$715$	7056.00	0.369062
$716$	1200.00	0.0626342
$717$	0	0
$718$	19360.0	1.00628
$719$	−18840.0	−0.977209	−0.488605	−	0.872505i	$-0.662494\pi$
$719$	−18840.0	−0.977209	−0.488605	+	0.872505i	$0.662494\pi$
$720$	0	0
$721$	0	0
$722$	−918.000	−0.0473191
$723$	0	0
$724$	9432.00	0.484168
$725$	−13490.0	−0.691043
$726$	0	0
$727$	−37504.0	−1.91327	−0.956634	−	0.291291i	$-0.905915\pi$
$727$	−37504.0	−1.91327	−0.956634	+	0.291291i	$0.905915\pi$
$728$	0	0
$729$	0	0
$730$	15064.0	0.763758
$731$	−30488.0	−1.54260
$732$	0	0
$733$	−13338.0	−0.672101	−0.336051	−	0.941844i	$-0.609091\pi$
$733$	−13338.0	−0.672101	−0.336051	+	0.941844i	$0.609091\pi$
$734$	17312.0	0.870569
$735$	0	0
$736$	3584.00	0.179495
$737$	−20048.0	−1.00200
$738$	0	0
$739$	17100.0	0.851196	0.425598	−	0.904912i	$-0.360064\pi$
$739$	17100.0	0.851196	0.425598	+	0.904912i	$0.360064\pi$
$740$	19376.0	0.962535
$741$	0	0
$742$	0	0
$743$	19632.0	0.969352	0.484676	−	0.874694i	$-0.338938\pi$
$743$	19632.0	0.969352	0.484676	+	0.874694i	$0.338938\pi$
$744$	0	0
$745$	7140.00	0.351127
$746$	10556.0	0.518073
$747$	0	0
$748$	8288.00	0.405133
$749$	0	0
$750$	0	0
$751$	33912.0	1.64776	0.823879	−	0.566766i	$-0.191805\pi$
$751$	33912.0	1.64776	0.823879	+	0.566766i	$0.191805\pi$
$752$	384.000	0.0186211
$753$	0	0
$754$	6840.00	0.330369
$755$	−8288.00	−0.399512
$756$	0	0
$757$	−31386.0	−1.50693	−0.753463	−	0.657490i	$-0.771618\pi$
$757$	−31386.0	−1.50693	−0.753463	+	0.657490i	$0.771618\pi$
$758$	12680.0	0.607597
$759$	0	0
$760$	8960.00	0.427649
$761$	−34558.0	−1.64616	−0.823079	−	0.567927i	$-0.807746\pi$
$761$	−34558.0	−1.64616	−0.823079	+	0.567927i	$0.807746\pi$
$762$	0	0
$763$	0	0
$764$	−5568.00	−0.263669
$765$	0	0
$766$	−12464.0	−0.587915
$767$	3600.00	0.169476
$768$	0	0
$769$	−39130.0	−1.83493	−0.917467	−	0.397812i	$-0.869769\pi$
$769$	−39130.0	−1.83493	−0.917467	+	0.397812i	$0.869769\pi$
$770$	0	0
$771$	0	0
$772$	7112.00	0.331563
$773$	−25982.0	−1.20894	−0.604468	−	0.796629i	$-0.706614\pi$
$773$	−25982.0	−1.20894	−0.604468	+	0.796629i	$0.706614\pi$
$774$	0	0
$775$	−5112.00	−0.236940
$776$	−10832.0	−0.501090
$777$	0	0
$778$	29620.0	1.36495
$779$	−12960.0	−0.596072
$780$	0	0
$781$	−10976.0	−0.502884
$782$	16576.0	0.758001
$783$	0	0
$784$	0	0
$785$	−37604.0	−1.70974
$786$	0	0
$787$	−35424.0	−1.60448	−0.802242	−	0.596999i	$-0.796360\pi$
$787$	−35424.0	−1.60448	−0.802242	+	0.596999i	$0.796360\pi$
$788$	−4856.00	−0.219528
$789$	0	0
$790$	−6720.00	−0.302642
$791$	0	0
$792$	0	0
$793$	−3564.00	−0.159598
$794$	−10308.0	−0.460727
$795$	0	0
$796$	−4160.00	−0.185235
$797$	−30606.0	−1.36025	−0.680126	−	0.733096i	$-0.738075\pi$
$797$	−30606.0	−1.36025	−0.680126	+	0.733096i	$0.738075\pi$
$798$	0	0
$799$	1776.00	0.0786362
$800$	2272.00	0.100409
$801$	0	0
$802$	−6564.00	−0.289006
$803$	−15064.0	−0.662014
$804$	0	0
$805$	0	0
$806$	2592.00	0.113275
$807$	0	0
$808$	−10864.0	−0.473013
$809$	−16810.0	−0.730542	−0.365271	−	0.930901i	$-0.619024\pi$
$809$	−16810.0	−0.730542	−0.365271	+	0.930901i	$0.619024\pi$
$810$	0	0
$811$	9368.00	0.405616	0.202808	−	0.979218i	$-0.434993\pi$
$811$	9368.00	0.405616	0.202808	+	0.979218i	$0.434993\pi$
$812$	0	0
$813$	0	0
$814$	−19376.0	−0.834310
$815$	14168.0	0.608937
$816$	0	0
$817$	32960.0	1.41141
$818$	−11620.0	−0.496679
$819$	0	0
$820$	−9072.00	−0.386351
$821$	−34382.0	−1.46156	−0.730780	−	0.682614i	$-0.760843\pi$
$821$	−34382.0	−1.46156	−0.730780	+	0.682614i	$0.760843\pi$
$822$	0	0
$823$	−4472.00	−0.189410	−0.0947048	−	0.995505i	$-0.530191\pi$
$823$	−4472.00	−0.189410	−0.0947048	+	0.995505i	$0.530191\pi$
$824$	6656.00	0.281399
$825$	0	0
$826$	0	0
$827$	1716.00	0.0721538	0.0360769	−	0.999349i	$-0.488514\pi$
$827$	1716.00	0.0721538	0.0360769	+	0.999349i	$0.488514\pi$
$828$	0	0
$829$	7910.00	0.331394	0.165697	−	0.986177i	$-0.447013\pi$
$829$	7910.00	0.331394	0.165697	+	0.986177i	$0.447013\pi$
$830$	30016.0	1.25527
$831$	0	0
$832$	−1152.00	−0.0480029
$833$	0	0
$834$	0	0
$835$	−7616.00	−0.315644
$836$	−8960.00	−0.370680
$837$	0	0
$838$	27120.0	1.11795
$839$	−19360.0	−0.796641	−0.398320	−	0.917246i	$-0.630407\pi$
$839$	−19360.0	−0.796641	−0.398320	+	0.917246i	$0.630407\pi$
$840$	0	0
$841$	11711.0	0.480175
$842$	−1476.00	−0.0604113
$843$	0	0
$844$	−15472.0	−0.631005
$845$	26222.0	1.06753
$846$	0	0
$847$	0	0
$848$	−5088.00	−0.206041
$849$	0	0
$850$	10508.0	0.424025
$851$	−38752.0	−1.56099
$852$	0	0
$853$	−698.000	−0.0280177	−0.0140088	−	0.999902i	$-0.504459\pi$
$853$	−698.000	−0.0280177	−0.0140088	+	0.999902i	$0.504459\pi$
$854$	0	0
$855$	0	0
$856$	−3552.00	−0.141828
$857$	−23406.0	−0.932945	−0.466472	−	0.884536i	$-0.654475\pi$
$857$	−23406.0	−0.932945	−0.466472	+	0.884536i	$0.654475\pi$
$858$	0	0
$859$	−7280.00	−0.289162	−0.144581	−	0.989493i	$-0.546183\pi$
$859$	−7280.00	−0.289162	−0.144581	+	0.989493i	$0.546183\pi$
$860$	23072.0	0.914824
$861$	0	0
$862$	−2544.00	−0.100521
$863$	−9808.00	−0.386869	−0.193435	−	0.981113i	$-0.561963\pi$
$863$	−9808.00	−0.386869	−0.193435	+	0.981113i	$0.561963\pi$
$864$	0	0
$865$	−26012.0	−1.02247
$866$	10124.0	0.397260
$867$	0	0
$868$	0	0
$869$	6720.00	0.262325
$870$	0	0
$871$	12888.0	0.501370
$872$	14960.0	0.580974
$873$	0	0
$874$	−17920.0	−0.693539
$875$	0	0
$876$	0	0
$877$	−8066.00	−0.310570	−0.155285	−	0.987870i	$-0.549630\pi$
$877$	−8066.00	−0.310570	−0.155285	+	0.987870i	$0.549630\pi$
$878$	−11280.0	−0.433578
$879$	0	0
$880$	−6272.00	−0.240260
$881$	25842.0	0.988240	0.494120	−	0.869394i	$-0.335490\pi$
$881$	25842.0	0.988240	0.494120	+	0.869394i	$0.335490\pi$
$882$	0	0
$883$	−5692.00	−0.216932	−0.108466	−	0.994100i	$-0.534594\pi$
$883$	−5692.00	−0.216932	−0.108466	+	0.994100i	$0.534594\pi$
$884$	−5328.00	−0.202715
$885$	0	0
$886$	−26776.0	−1.01530
$887$	−13536.0	−0.512395	−0.256198	−	0.966624i	$-0.582470\pi$
$887$	−13536.0	−0.512395	−0.256198	+	0.966624i	$0.582470\pi$
$888$	0	0
$889$	0	0
$890$	−22680.0	−0.854197
$891$	0	0
$892$	−15872.0	−0.595778
$893$	−1920.00	−0.0719489
$894$	0	0
$895$	−4200.00	−0.156861
$896$	0	0
$897$	0	0
$898$	6460.00	0.240059
$899$	13680.0	0.507512
$900$	0	0
$901$	−23532.0	−0.870105
$902$	9072.00	0.334883
$903$	0	0
$904$	−11024.0	−0.405589
$905$	−33012.0	−1.21255
$906$	0	0
$907$	17004.0	0.622501	0.311251	−	0.950328i	$-0.399252\pi$
$907$	17004.0	0.622501	0.311251	+	0.950328i	$0.399252\pi$
$908$	−15744.0	−0.575422
$909$	0	0
$910$	0	0
$911$	14568.0	0.529813	0.264906	−	0.964274i	$-0.414659\pi$
$911$	14568.0	0.529813	0.264906	+	0.964274i	$0.414659\pi$
$912$	0	0
$913$	−30016.0	−1.08804
$914$	−21292.0	−0.770544
$915$	0	0
$916$	−19240.0	−0.694004
$917$	0	0
$918$	0	0
$919$	−1400.00	−0.0502522	−0.0251261	−	0.999684i	$-0.507999\pi$
$919$	−1400.00	−0.0502522	−0.0251261	+	0.999684i	$0.507999\pi$
$920$	−12544.0	−0.449525
$921$	0	0
$922$	14564.0	0.520217
$923$	7056.00	0.251626
$924$	0	0
$925$	−24566.0	−0.873216
$926$	25376.0	0.900548
$927$	0	0
$928$	−6080.00	−0.215071
$929$	−13830.0	−0.488426	−0.244213	−	0.969722i	$-0.578530\pi$
$929$	−13830.0	−0.488426	−0.244213	+	0.969722i	$0.578530\pi$
$930$	0	0
$931$	0	0
$932$	8728.00	0.306754
$933$	0	0
$934$	−5632.00	−0.197307
$935$	−29008.0	−1.01461
$936$	0	0
$937$	24166.0	0.842549	0.421275	−	0.906933i	$-0.361583\pi$
$937$	24166.0	0.842549	0.421275	+	0.906933i	$0.361583\pi$
$938$	0	0
$939$	0	0
$940$	−1344.00	−0.0466345
$941$	−10838.0	−0.375461	−0.187730	−	0.982221i	$-0.560113\pi$
$941$	−10838.0	−0.375461	−0.187730	+	0.982221i	$0.560113\pi$
$942$	0	0
$943$	18144.0	0.626564
$944$	−3200.00	−0.110330
$945$	0	0
$946$	−23072.0	−0.792955
$947$	40916.0	1.40400	0.702002	−	0.712175i	$-0.252290\pi$
$947$	40916.0	1.40400	0.702002	+	0.712175i	$0.252290\pi$
$948$	0	0
$949$	9684.00	0.331250
$950$	−11360.0	−0.387965
$951$	0	0
$952$	0	0
$953$	−56618.0	−1.92449	−0.962244	−	0.272189i	$-0.912253\pi$
$953$	−56618.0	−1.92449	−0.962244	+	0.272189i	$0.912253\pi$
$954$	0	0
$955$	19488.0	0.660332
$956$	12000.0	0.405970
$957$	0	0
$958$	−6320.00	−0.213142
$959$	0	0
$960$	0	0
$961$	−24607.0	−0.825988
$962$	12456.0	0.417461
$963$	0	0
$964$	−8168.00	−0.272898
$965$	−24892.0	−0.830365
$966$	0	0
$967$	17504.0	0.582100	0.291050	−	0.956708i	$-0.405995\pi$
$967$	17504.0	0.582100	0.291050	+	0.956708i	$0.405995\pi$
$968$	−4376.00	−0.145300
$969$	0	0
$970$	37912.0	1.25493
$971$	23112.0	0.763851	0.381926	−	0.924193i	$-0.375261\pi$
$971$	23112.0	0.763851	0.381926	+	0.924193i	$0.375261\pi$
$972$	0	0
$973$	0	0
$974$	−28352.0	−0.932707
$975$	0	0
$976$	3168.00	0.103899
$977$	−23874.0	−0.781778	−0.390889	−	0.920438i	$-0.627832\pi$
$977$	−23874.0	−0.781778	−0.390889	+	0.920438i	$0.627832\pi$
$978$	0	0
$979$	22680.0	0.740404
$980$	0	0
$981$	0	0
$982$	22536.0	0.732335
$983$	−15312.0	−0.496823	−0.248411	−	0.968655i	$-0.579909\pi$
$983$	−15312.0	−0.496823	−0.248411	+	0.968655i	$0.579909\pi$
$984$	0	0
$985$	16996.0	0.549784
$986$	−28120.0	−0.908239
$987$	0	0
$988$	5760.00	0.185476
$989$	−46144.0	−1.48361
$990$	0	0
$991$	−16528.0	−0.529797	−0.264899	−	0.964276i	$-0.585339\pi$
$991$	−16528.0	−0.529797	−0.264899	+	0.964276i	$0.585339\pi$
$992$	−2304.00	−0.0737420
$993$	0	0
$994$	0	0
$995$	14560.0	0.463903
$996$	0	0
$997$	28606.0	0.908687	0.454344	−	0.890827i	$-0.349874\pi$
$997$	28606.0	0.908687	0.454344	+	0.890827i	$0.349874\pi$
$998$	−8920.00	−0.282924
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.4.a.i.1.1		1
3.2	odd	2		98.4.a.a.1.1		1
7.2	even	3		882.4.g.k.361.1		2
7.3	odd	6		882.4.g.b.667.1		2
7.4	even	3		882.4.g.k.667.1		2
7.5	odd	6		882.4.g.b.361.1		2
7.6	odd	2		126.4.a.h.1.1		1
12.11	even	2		784.4.a.s.1.1		1
15.14	odd	2		2450.4.a.bo.1.1		1
21.2	odd	6		98.4.c.f.67.1		2
21.5	even	6		98.4.c.d.67.1		2
21.11	odd	6		98.4.c.f.79.1		2
21.17	even	6		98.4.c.d.79.1		2
21.20	even	2		14.4.a.a.1.1	✓	1
28.27	even	2		1008.4.a.s.1.1		1
84.83	odd	2		112.4.a.a.1.1		1
105.62	odd	4		350.4.c.b.99.1		2
105.83	odd	4		350.4.c.b.99.2		2
105.104	even	2		350.4.a.l.1.1		1
168.83	odd	2		448.4.a.o.1.1		1
168.125	even	2		448.4.a.b.1.1		1
231.230	odd	2		1694.4.a.g.1.1		1
273.272	even	2		2366.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.a.a.1.1	✓	1	21.20	even	2
98.4.a.a.1.1		1	3.2	odd	2
98.4.c.d.67.1		2	21.5	even	6
98.4.c.d.79.1		2	21.17	even	6
98.4.c.f.67.1		2	21.2	odd	6
98.4.c.f.79.1		2	21.11	odd	6
112.4.a.a.1.1		1	84.83	odd	2
126.4.a.h.1.1		1	7.6	odd	2
350.4.a.l.1.1		1	105.104	even	2
350.4.c.b.99.1		2	105.62	odd	4
350.4.c.b.99.2		2	105.83	odd	4
448.4.a.b.1.1		1	168.125	even	2
448.4.a.o.1.1		1	168.83	odd	2
784.4.a.s.1.1		1	12.11	even	2
882.4.a.i.1.1		1	1.1	even	1	trivial
882.4.g.b.361.1		2	7.5	odd	6
882.4.g.b.667.1		2	7.3	odd	6
882.4.g.k.361.1		2	7.2	even	3
882.4.g.k.667.1		2	7.4	even	3
1008.4.a.s.1.1		1	28.27	even	2
1694.4.a.g.1.1		1	231.230	odd	2
2366.4.a.h.1.1		1	273.272	even	2
2450.4.a.bo.1.1		1	15.14	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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