LMFDB - Embedded newform 256.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,4,Mod(1,256)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("256.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,8,0,-12,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.1044889615$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	256.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} -12.0000 q^{5} -32.0000 q^{7} +37.0000 q^{9} -8.00000 q^{11} +20.0000 q^{13} -96.0000 q^{15} -98.0000 q^{17} -88.0000 q^{19} -256.000 q^{21} +32.0000 q^{23} +19.0000 q^{25} +80.0000 q^{27} -172.000 q^{29} +256.000 q^{31} -64.0000 q^{33} +384.000 q^{35} -92.0000 q^{37} +160.000 q^{39} +102.000 q^{41} -296.000 q^{43} -444.000 q^{45} +320.000 q^{47} +681.000 q^{49} -784.000 q^{51} -76.0000 q^{53} +96.0000 q^{55} -704.000 q^{57} +408.000 q^{59} -636.000 q^{61} -1184.00 q^{63} -240.000 q^{65} +552.000 q^{67} +256.000 q^{69} -416.000 q^{71} +138.000 q^{73} +152.000 q^{75} +256.000 q^{77} +64.0000 q^{79} -359.000 q^{81} +392.000 q^{83} +1176.00 q^{85} -1376.00 q^{87} -582.000 q^{89} -640.000 q^{91} +2048.00 q^{93} +1056.00 q^{95} +238.000 q^{97} -296.000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	0	0
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−8.00000	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$11$	−8.00000	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$12$	0	0
$13$	20.0000	0.426692	0.213346	−	0.976977i	$-0.431564\pi$
$13$	20.0000	0.426692	0.213346	+	0.976977i	$0.431564\pi$
$14$	0	0
$15$	−96.0000	−1.65247
$16$	0	0
$17$	−98.0000	−1.39815	−0.699073	−	0.715050i	$-0.746404\pi$
$17$	−98.0000	−1.39815	−0.699073	+	0.715050i	$0.746404\pi$
$18$	0	0
$19$	−88.0000	−1.06256	−0.531279	−	0.847197i	$-0.678288\pi$
$19$	−88.0000	−1.06256	−0.531279	+	0.847197i	$0.678288\pi$
$20$	0	0
$21$	−256.000	−2.66018
$22$	0	0
$23$	32.0000	0.290107	0.145054	−	0.989424i	$-0.453665\pi$
$23$	32.0000	0.290107	0.145054	+	0.989424i	$0.453665\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	−172.000	−1.10137	−0.550683	−	0.834715i	$-0.685633\pi$
$29$	−172.000	−1.10137	−0.550683	+	0.834715i	$0.685633\pi$
$30$	0	0
$31$	256.000	1.48319	0.741596	−	0.670847i	$-0.234069\pi$
$31$	256.000	1.48319	0.741596	+	0.670847i	$0.234069\pi$
$32$	0	0
$33$	−64.0000	−0.337605
$34$	0	0
$35$	384.000	1.85451
$36$	0	0
$37$	−92.0000	−0.408776	−0.204388	−	0.978890i	$-0.565520\pi$
$37$	−92.0000	−0.408776	−0.204388	+	0.978890i	$0.565520\pi$
$38$	0	0
$39$	160.000	0.656936
$40$	0	0
$41$	102.000	0.388530	0.194265	−	0.980949i	$-0.437768\pi$
$41$	102.000	0.388530	0.194265	+	0.980949i	$0.437768\pi$
$42$	0	0
$43$	−296.000	−1.04976	−0.524879	−	0.851177i	$-0.675889\pi$
$43$	−296.000	−1.04976	−0.524879	+	0.851177i	$0.675889\pi$
$44$	0	0
$45$	−444.000	−1.47084
$46$	0	0
$47$	320.000	0.993123	0.496562	−	0.868001i	$-0.334596\pi$
$47$	320.000	0.993123	0.496562	+	0.868001i	$0.334596\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	−784.000	−2.15259
$52$	0	0
$53$	−76.0000	−0.196970	−0.0984849	−	0.995139i	$-0.531400\pi$
$53$	−76.0000	−0.196970	−0.0984849	+	0.995139i	$0.531400\pi$
$54$	0	0
$55$	96.0000	0.235357
$56$	0	0
$57$	−704.000	−1.63591
$58$	0	0
$59$	408.000	0.900289	0.450145	−	0.892956i	$-0.351372\pi$
$59$	408.000	0.900289	0.450145	+	0.892956i	$0.351372\pi$
$60$	0	0
$61$	−636.000	−1.33494	−0.667471	−	0.744636i	$-0.732623\pi$
$61$	−636.000	−1.33494	−0.667471	+	0.744636i	$0.732623\pi$
$62$	0	0
$63$	−1184.00	−2.36778
$64$	0	0
$65$	−240.000	−0.457974
$66$	0	0
$67$	552.000	1.00653	0.503265	−	0.864132i	$-0.332132\pi$
$67$	552.000	1.00653	0.503265	+	0.864132i	$0.332132\pi$
$68$	0	0
$69$	256.000	0.446649
$70$	0	0
$71$	−416.000	−0.695354	−0.347677	−	0.937614i	$-0.613029\pi$
$71$	−416.000	−0.695354	−0.347677	+	0.937614i	$0.613029\pi$
$72$	0	0
$73$	138.000	0.221256	0.110628	−	0.993862i	$-0.464714\pi$
$73$	138.000	0.221256	0.110628	+	0.993862i	$0.464714\pi$
$74$	0	0
$75$	152.000	0.234019
$76$	0	0
$77$	256.000	0.378882
$78$	0	0
$79$	64.0000	0.0911464	0.0455732	−	0.998961i	$-0.485489\pi$
$79$	64.0000	0.0911464	0.0455732	+	0.998961i	$0.485489\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	392.000	0.518405	0.259202	−	0.965823i	$-0.416540\pi$
$83$	392.000	0.518405	0.259202	+	0.965823i	$0.416540\pi$
$84$	0	0
$85$	1176.00	1.50065
$86$	0	0
$87$	−1376.00	−1.69566
$88$	0	0
$89$	−582.000	−0.693167	−0.346584	−	0.938019i	$-0.612658\pi$
$89$	−582.000	−0.693167	−0.346584	+	0.938019i	$0.612658\pi$
$90$	0	0
$91$	−640.000	−0.737255
$92$	0	0
$93$	2048.00	2.28352
$94$	0	0
$95$	1056.00	1.14046
$96$	0	0
$97$	238.000	0.249126	0.124563	−	0.992212i	$-0.460247\pi$
$97$	238.000	0.249126	0.124563	+	0.992212i	$0.460247\pi$
$98$	0	0
$99$	−296.000	−0.300496
$100$	0	0
$101$	−1468.00	−1.44625	−0.723126	−	0.690716i	$-0.757295\pi$
$101$	−1468.00	−1.44625	−0.723126	+	0.690716i	$0.757295\pi$
$102$	0	0
$103$	992.000	0.948977	0.474489	−	0.880262i	$-0.342633\pi$
$103$	992.000	0.948977	0.474489	+	0.880262i	$0.342633\pi$
$104$	0	0
$105$	3072.00	2.85520
$106$	0	0
$107$	−584.000	−0.527639	−0.263820	−	0.964572i	$-0.584982\pi$
$107$	−584.000	−0.527639	−0.263820	+	0.964572i	$0.584982\pi$
$108$	0	0
$109$	740.000	0.650267	0.325134	−	0.945668i	$-0.394591\pi$
$109$	740.000	0.650267	0.325134	+	0.945668i	$0.394591\pi$
$110$	0	0
$111$	−736.000	−0.629352
$112$	0	0
$113$	−302.000	−0.251414	−0.125707	−	0.992067i	$-0.540120\pi$
$113$	−302.000	−0.251414	−0.125707	+	0.992067i	$0.540120\pi$
$114$	0	0
$115$	−384.000	−0.311376
$116$	0	0
$117$	740.000	0.584727
$118$	0	0
$119$	3136.00	2.41577
$120$	0	0
$121$	−1267.00	−0.951916
$122$	0	0
$123$	816.000	0.598181
$124$	0	0
$125$	1272.00	0.910169
$126$	0	0
$127$	1664.00	1.16265	0.581323	−	0.813673i	$-0.302535\pi$
$127$	1664.00	1.16265	0.581323	+	0.813673i	$0.302535\pi$
$128$	0	0
$129$	−2368.00	−1.61621
$130$	0	0
$131$	−2328.00	−1.55266	−0.776329	−	0.630327i	$-0.782921\pi$
$131$	−2328.00	−1.55266	−0.776329	+	0.630327i	$0.782921\pi$
$132$	0	0
$133$	2816.00	1.83593
$134$	0	0
$135$	−960.000	−0.612027
$136$	0	0
$137$	1734.00	1.08135	0.540677	−	0.841230i	$-0.318168\pi$
$137$	1734.00	1.08135	0.540677	+	0.841230i	$0.318168\pi$
$138$	0	0
$139$	3032.00	1.85015	0.925075	−	0.379784i	$-0.124002\pi$
$139$	3032.00	1.85015	0.925075	+	0.379784i	$0.124002\pi$
$140$	0	0
$141$	2560.00	1.52901
$142$	0	0
$143$	−160.000	−0.0935655
$144$	0	0
$145$	2064.00	1.18211
$146$	0	0
$147$	5448.00	3.05676
$148$	0	0
$149$	−1788.00	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−1788.00	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−480.000	−0.258688	−0.129344	−	0.991600i	$-0.541287\pi$
$151$	−480.000	−0.258688	−0.129344	+	0.991600i	$0.541287\pi$
$152$	0	0
$153$	−3626.00	−1.91598
$154$	0	0
$155$	−3072.00	−1.59193
$156$	0	0
$157$	−2300.00	−1.16917	−0.584586	−	0.811332i	$-0.698743\pi$
$157$	−2300.00	−1.16917	−0.584586	+	0.811332i	$0.698743\pi$
$158$	0	0
$159$	−608.000	−0.303255
$160$	0	0
$161$	−1024.00	−0.501258
$162$	0	0
$163$	−1592.00	−0.765000	−0.382500	−	0.923955i	$-0.624937\pi$
$163$	−1592.00	−0.765000	−0.382500	+	0.923955i	$0.624937\pi$
$164$	0	0
$165$	768.000	0.362356
$166$	0	0
$167$	−2208.00	−1.02311	−0.511557	−	0.859249i	$-0.670931\pi$
$167$	−2208.00	−1.02311	−0.511557	+	0.859249i	$0.670931\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	0	0
$171$	−3256.00	−1.45610
$172$	0	0
$173$	−3948.00	−1.73503	−0.867517	−	0.497408i	$-0.834285\pi$
$173$	−3948.00	−1.73503	−0.867517	+	0.497408i	$0.834285\pi$
$174$	0	0
$175$	−608.000	−0.262631
$176$	0	0
$177$	3264.00	1.38609
$178$	0	0
$179$	−2104.00	−0.878549	−0.439275	−	0.898353i	$-0.644765\pi$
$179$	−2104.00	−0.878549	−0.439275	+	0.898353i	$0.644765\pi$
$180$	0	0
$181$	1412.00	0.579852	0.289926	−	0.957049i	$-0.406369\pi$
$181$	1412.00	0.579852	0.289926	+	0.957049i	$0.406369\pi$
$182$	0	0
$183$	−5088.00	−2.05528
$184$	0	0
$185$	1104.00	0.438744
$186$	0	0
$187$	784.000	0.306587
$188$	0	0
$189$	−2560.00	−0.985252
$190$	0	0
$191$	−3712.00	−1.40624	−0.703118	−	0.711074i	$-0.748209\pi$
$191$	−3712.00	−1.40624	−0.703118	+	0.711074i	$0.748209\pi$
$192$	0	0
$193$	1614.00	0.601960	0.300980	−	0.953630i	$-0.402686\pi$
$193$	1614.00	0.601960	0.300980	+	0.953630i	$0.402686\pi$
$194$	0	0
$195$	−1920.00	−0.705098
$196$	0	0
$197$	−684.000	−0.247376	−0.123688	−	0.992321i	$-0.539472\pi$
$197$	−684.000	−0.247376	−0.123688	+	0.992321i	$0.539472\pi$
$198$	0	0
$199$	4064.00	1.44769	0.723843	−	0.689965i	$-0.242374\pi$
$199$	4064.00	1.44769	0.723843	+	0.689965i	$0.242374\pi$
$200$	0	0
$201$	4416.00	1.54965
$202$	0	0
$203$	5504.00	1.90298
$204$	0	0
$205$	−1224.00	−0.417014
$206$	0	0
$207$	1184.00	0.397554
$208$	0	0
$209$	704.000	0.232999
$210$	0	0
$211$	2120.00	0.691691	0.345846	−	0.938291i	$-0.387592\pi$
$211$	2120.00	0.691691	0.345846	+	0.938291i	$0.387592\pi$
$212$	0	0
$213$	−3328.00	−1.07057
$214$	0	0
$215$	3552.00	1.12672
$216$	0	0
$217$	−8192.00	−2.56272
$218$	0	0
$219$	1104.00	0.340646
$220$	0	0
$221$	−1960.00	−0.596579
$222$	0	0
$223$	−2816.00	−0.845620	−0.422810	−	0.906218i	$-0.638956\pi$
$223$	−2816.00	−0.845620	−0.422810	+	0.906218i	$0.638956\pi$
$224$	0	0
$225$	703.000	0.208296
$226$	0	0
$227$	3848.00	1.12511	0.562557	−	0.826759i	$-0.309818\pi$
$227$	3848.00	1.12511	0.562557	+	0.826759i	$0.309818\pi$
$228$	0	0
$229$	−652.000	−0.188146	−0.0940729	−	0.995565i	$-0.529989\pi$
$229$	−652.000	−0.188146	−0.0940729	+	0.995565i	$0.529989\pi$
$230$	0	0
$231$	2048.00	0.583327
$232$	0	0
$233$	3050.00	0.857563	0.428781	−	0.903408i	$-0.358943\pi$
$233$	3050.00	0.857563	0.428781	+	0.903408i	$0.358943\pi$
$234$	0	0
$235$	−3840.00	−1.06593
$236$	0	0
$237$	512.000	0.140329
$238$	0	0
$239$	6336.00	1.71482	0.857410	−	0.514635i	$-0.172072\pi$
$239$	6336.00	1.71482	0.857410	+	0.514635i	$0.172072\pi$
$240$	0	0
$241$	−4610.00	−1.23218	−0.616092	−	0.787674i	$-0.711285\pi$
$241$	−4610.00	−1.23218	−0.616092	+	0.787674i	$0.711285\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	−8172.00	−2.13098
$246$	0	0
$247$	−1760.00	−0.453385
$248$	0	0
$249$	3136.00	0.798136
$250$	0	0
$251$	792.000	0.199166	0.0995829	−	0.995029i	$-0.468249\pi$
$251$	792.000	0.199166	0.0995829	+	0.995029i	$0.468249\pi$
$252$	0	0
$253$	−256.000	−0.0636149
$254$	0	0
$255$	9408.00	2.31040
$256$	0	0
$257$	−5374.00	−1.30436	−0.652181	−	0.758063i	$-0.726146\pi$
$257$	−5374.00	−1.30436	−0.652181	+	0.758063i	$0.726146\pi$
$258$	0	0
$259$	2944.00	0.706298
$260$	0	0
$261$	−6364.00	−1.50928
$262$	0	0
$263$	−3488.00	−0.817792	−0.408896	−	0.912581i	$-0.634086\pi$
$263$	−3488.00	−0.817792	−0.408896	+	0.912581i	$0.634086\pi$
$264$	0	0
$265$	912.000	0.211410
$266$	0	0
$267$	−4656.00	−1.06720
$268$	0	0
$269$	−4764.00	−1.07980	−0.539900	−	0.841729i	$-0.681538\pi$
$269$	−4764.00	−1.07980	−0.539900	+	0.841729i	$0.681538\pi$
$270$	0	0
$271$	−1344.00	−0.301263	−0.150631	−	0.988590i	$-0.548131\pi$
$271$	−1344.00	−0.301263	−0.150631	+	0.988590i	$0.548131\pi$
$272$	0	0
$273$	−5120.00	−1.13508
$274$	0	0
$275$	−152.000	−0.0333307
$276$	0	0
$277$	8596.00	1.86456	0.932281	−	0.361735i	$-0.117816\pi$
$277$	8596.00	1.86456	0.932281	+	0.361735i	$0.117816\pi$
$278$	0	0
$279$	9472.00	2.03252
$280$	0	0
$281$	2874.00	0.610137	0.305068	−	0.952330i	$-0.401321\pi$
$281$	2874.00	0.610137	0.305068	+	0.952330i	$0.401321\pi$
$282$	0	0
$283$	−2888.00	−0.606621	−0.303311	−	0.952892i	$-0.598092\pi$
$283$	−2888.00	−0.606621	−0.303311	+	0.952892i	$0.598092\pi$
$284$	0	0
$285$	8448.00	1.75585
$286$	0	0
$287$	−3264.00	−0.671316
$288$	0	0
$289$	4691.00	0.954814
$290$	0	0
$291$	1904.00	0.383555
$292$	0	0
$293$	−6540.00	−1.30400	−0.651998	−	0.758221i	$-0.726069\pi$
$293$	−6540.00	−1.30400	−0.651998	+	0.758221i	$0.726069\pi$
$294$	0	0
$295$	−4896.00	−0.966292
$296$	0	0
$297$	−640.000	−0.125039
$298$	0	0
$299$	640.000	0.123786
$300$	0	0
$301$	9472.00	1.81381
$302$	0	0
$303$	−11744.0	−2.22665
$304$	0	0
$305$	7632.00	1.43281
$306$	0	0
$307$	−10584.0	−1.96762	−0.983812	−	0.179202i	$-0.942649\pi$
$307$	−10584.0	−1.96762	−0.983812	+	0.179202i	$0.942649\pi$
$308$	0	0
$309$	7936.00	1.46105
$310$	0	0
$311$	−6368.00	−1.16108	−0.580540	−	0.814231i	$-0.697159\pi$
$311$	−6368.00	−1.16108	−0.580540	+	0.814231i	$0.697159\pi$
$312$	0	0
$313$	8758.00	1.58157	0.790785	−	0.612094i	$-0.209673\pi$
$313$	8758.00	1.58157	0.790785	+	0.612094i	$0.209673\pi$
$314$	0	0
$315$	14208.0	2.54137
$316$	0	0
$317$	−716.000	−0.126860	−0.0634299	−	0.997986i	$-0.520204\pi$
$317$	−716.000	−0.126860	−0.0634299	+	0.997986i	$0.520204\pi$
$318$	0	0
$319$	1376.00	0.241508
$320$	0	0
$321$	−4672.00	−0.812354
$322$	0	0
$323$	8624.00	1.48561
$324$	0	0
$325$	380.000	0.0648573
$326$	0	0
$327$	5920.00	1.00115
$328$	0	0
$329$	−10240.0	−1.71596
$330$	0	0
$331$	4408.00	0.731981	0.365990	−	0.930619i	$-0.380730\pi$
$331$	4408.00	0.731981	0.365990	+	0.930619i	$0.380730\pi$
$332$	0	0
$333$	−3404.00	−0.560174
$334$	0	0
$335$	−6624.00	−1.08032
$336$	0	0
$337$	1202.00	0.194294	0.0971471	−	0.995270i	$-0.469028\pi$
$337$	1202.00	0.194294	0.0971471	+	0.995270i	$0.469028\pi$
$338$	0	0
$339$	−2416.00	−0.387077
$340$	0	0
$341$	−2048.00	−0.325236
$342$	0	0
$343$	−10816.0	−1.70265
$344$	0	0
$345$	−3072.00	−0.479394
$346$	0	0
$347$	−5160.00	−0.798280	−0.399140	−	0.916890i	$-0.630691\pi$
$347$	−5160.00	−0.798280	−0.399140	+	0.916890i	$0.630691\pi$
$348$	0	0
$349$	−4876.00	−0.747869	−0.373935	−	0.927455i	$-0.621992\pi$
$349$	−4876.00	−0.747869	−0.373935	+	0.927455i	$0.621992\pi$
$350$	0	0
$351$	1600.00	0.243310
$352$	0	0
$353$	4834.00	0.728861	0.364430	−	0.931231i	$-0.381264\pi$
$353$	4834.00	0.728861	0.364430	+	0.931231i	$0.381264\pi$
$354$	0	0
$355$	4992.00	0.746332
$356$	0	0
$357$	25088.0	3.71932
$358$	0	0
$359$	−4128.00	−0.606873	−0.303437	−	0.952852i	$-0.598134\pi$
$359$	−4128.00	−0.606873	−0.303437	+	0.952852i	$0.598134\pi$
$360$	0	0
$361$	885.000	0.129028
$362$	0	0
$363$	−10136.0	−1.46557
$364$	0	0
$365$	−1656.00	−0.237477
$366$	0	0
$367$	4416.00	0.628102	0.314051	−	0.949406i	$-0.398314\pi$
$367$	4416.00	0.628102	0.314051	+	0.949406i	$0.398314\pi$
$368$	0	0
$369$	3774.00	0.532430
$370$	0	0
$371$	2432.00	0.340332
$372$	0	0
$373$	4180.00	0.580247	0.290124	−	0.956989i	$-0.406304\pi$
$373$	4180.00	0.580247	0.290124	+	0.956989i	$0.406304\pi$
$374$	0	0
$375$	10176.0	1.40130
$376$	0	0
$377$	−3440.00	−0.469944
$378$	0	0
$379$	−13736.0	−1.86166	−0.930832	−	0.365446i	$-0.880916\pi$
$379$	−13736.0	−1.86166	−0.930832	+	0.365446i	$0.880916\pi$
$380$	0	0
$381$	13312.0	1.79001
$382$	0	0
$383$	−512.000	−0.0683080	−0.0341540	−	0.999417i	$-0.510874\pi$
$383$	−512.000	−0.0683080	−0.0341540	+	0.999417i	$0.510874\pi$
$384$	0	0
$385$	−3072.00	−0.406659
$386$	0	0
$387$	−10952.0	−1.43856
$388$	0	0
$389$	1732.00	0.225748	0.112874	−	0.993609i	$-0.463994\pi$
$389$	1732.00	0.225748	0.112874	+	0.993609i	$0.463994\pi$
$390$	0	0
$391$	−3136.00	−0.405612
$392$	0	0
$393$	−18624.0	−2.39047
$394$	0	0
$395$	−768.000	−0.0978285
$396$	0	0
$397$	10436.0	1.31931	0.659657	−	0.751567i	$-0.270701\pi$
$397$	10436.0	1.31931	0.659657	+	0.751567i	$0.270701\pi$
$398$	0	0
$399$	22528.0	2.82659
$400$	0	0
$401$	−12130.0	−1.51058	−0.755291	−	0.655390i	$-0.772504\pi$
$401$	−12130.0	−1.51058	−0.755291	+	0.655390i	$0.772504\pi$
$402$	0	0
$403$	5120.00	0.632867
$404$	0	0
$405$	4308.00	0.528559
$406$	0	0
$407$	736.000	0.0896368
$408$	0	0
$409$	5014.00	0.606177	0.303088	−	0.952962i	$-0.401982\pi$
$409$	5014.00	0.606177	0.303088	+	0.952962i	$0.401982\pi$
$410$	0	0
$411$	13872.0	1.66485
$412$	0	0
$413$	−13056.0	−1.55555
$414$	0	0
$415$	−4704.00	−0.556410
$416$	0	0
$417$	24256.0	2.84849
$418$	0	0
$419$	−8024.00	−0.935556	−0.467778	−	0.883846i	$-0.654945\pi$
$419$	−8024.00	−0.935556	−0.467778	+	0.883846i	$0.654945\pi$
$420$	0	0
$421$	−2348.00	−0.271816	−0.135908	−	0.990721i	$-0.543395\pi$
$421$	−2348.00	−0.271816	−0.135908	+	0.990721i	$0.543395\pi$
$422$	0	0
$423$	11840.0	1.36095
$424$	0	0
$425$	−1862.00	−0.212518
$426$	0	0
$427$	20352.0	2.30656
$428$	0	0
$429$	−1280.00	−0.144054
$430$	0	0
$431$	1728.00	0.193120	0.0965601	−	0.995327i	$-0.469216\pi$
$431$	1728.00	0.193120	0.0965601	+	0.995327i	$0.469216\pi$
$432$	0	0
$433$	62.0000	0.00688113	0.00344057	−	0.999994i	$-0.498905\pi$
$433$	62.0000	0.00688113	0.00344057	+	0.999994i	$0.498905\pi$
$434$	0	0
$435$	16512.0	1.81998
$436$	0	0
$437$	−2816.00	−0.308255
$438$	0	0
$439$	14112.0	1.53423	0.767117	−	0.641507i	$-0.221690\pi$
$439$	14112.0	1.53423	0.767117	+	0.641507i	$0.221690\pi$
$440$	0	0
$441$	25197.0	2.72076
$442$	0	0
$443$	−4488.00	−0.481335	−0.240667	−	0.970608i	$-0.577366\pi$
$443$	−4488.00	−0.481335	−0.240667	+	0.970608i	$0.577366\pi$
$444$	0	0
$445$	6984.00	0.743985
$446$	0	0
$447$	−14304.0	−1.51355
$448$	0	0
$449$	−2482.00	−0.260875	−0.130437	−	0.991457i	$-0.541638\pi$
$449$	−2482.00	−0.260875	−0.130437	+	0.991457i	$0.541638\pi$
$450$	0	0
$451$	−816.000	−0.0851972
$452$	0	0
$453$	−3840.00	−0.398276
$454$	0	0
$455$	7680.00	0.791305
$456$	0	0
$457$	5894.00	0.603303	0.301652	−	0.953418i	$-0.402462\pi$
$457$	5894.00	0.603303	0.301652	+	0.953418i	$0.402462\pi$
$458$	0	0
$459$	−7840.00	−0.797255
$460$	0	0
$461$	−7068.00	−0.714077	−0.357039	−	0.934090i	$-0.616214\pi$
$461$	−7068.00	−0.714077	−0.357039	+	0.934090i	$0.616214\pi$
$462$	0	0
$463$	−7616.00	−0.764461	−0.382231	−	0.924067i	$-0.624844\pi$
$463$	−7616.00	−0.764461	−0.382231	+	0.924067i	$0.624844\pi$
$464$	0	0
$465$	−24576.0	−2.45093
$466$	0	0
$467$	−13080.0	−1.29608	−0.648041	−	0.761606i	$-0.724411\pi$
$467$	−13080.0	−1.29608	−0.648041	+	0.761606i	$0.724411\pi$
$468$	0	0
$469$	−17664.0	−1.73912
$470$	0	0
$471$	−18400.0	−1.80006
$472$	0	0
$473$	2368.00	0.230192
$474$	0	0
$475$	−1672.00	−0.161509
$476$	0	0
$477$	−2812.00	−0.269922
$478$	0	0
$479$	13568.0	1.29423	0.647117	−	0.762391i	$-0.275975\pi$
$479$	13568.0	1.29423	0.647117	+	0.762391i	$0.275975\pi$
$480$	0	0
$481$	−1840.00	−0.174422
$482$	0	0
$483$	−8192.00	−0.771737
$484$	0	0
$485$	−2856.00	−0.267390
$486$	0	0
$487$	−1696.00	−0.157809	−0.0789046	−	0.996882i	$-0.525142\pi$
$487$	−1696.00	−0.157809	−0.0789046	+	0.996882i	$0.525142\pi$
$488$	0	0
$489$	−12736.0	−1.17780
$490$	0	0
$491$	3096.00	0.284563	0.142282	−	0.989826i	$-0.454556\pi$
$491$	3096.00	0.284563	0.142282	+	0.989826i	$0.454556\pi$
$492$	0	0
$493$	16856.0	1.53987
$494$	0	0
$495$	3552.00	0.322526
$496$	0	0
$497$	13312.0	1.20146
$498$	0	0
$499$	19208.0	1.72318	0.861591	−	0.507603i	$-0.169468\pi$
$499$	19208.0	1.72318	0.861591	+	0.507603i	$0.169468\pi$
$500$	0	0
$501$	−17664.0	−1.57519
$502$	0	0
$503$	−16224.0	−1.43816	−0.719078	−	0.694929i	$-0.755436\pi$
$503$	−16224.0	−1.43816	−0.719078	+	0.694929i	$0.755436\pi$
$504$	0	0
$505$	17616.0	1.55228
$506$	0	0
$507$	−14376.0	−1.25929
$508$	0	0
$509$	−11292.0	−0.983318	−0.491659	−	0.870788i	$-0.663609\pi$
$509$	−11292.0	−0.983318	−0.491659	+	0.870788i	$0.663609\pi$
$510$	0	0
$511$	−4416.00	−0.382294
$512$	0	0
$513$	−7040.00	−0.605894
$514$	0	0
$515$	−11904.0	−1.01855
$516$	0	0
$517$	−2560.00	−0.217773
$518$	0	0
$519$	−31584.0	−2.67126
$520$	0	0
$521$	−5178.00	−0.435417	−0.217709	−	0.976014i	$-0.569858\pi$
$521$	−5178.00	−0.435417	−0.217709	+	0.976014i	$0.569858\pi$
$522$	0	0
$523$	−6856.00	−0.573216	−0.286608	−	0.958048i	$-0.592528\pi$
$523$	−6856.00	−0.573216	−0.286608	+	0.958048i	$0.592528\pi$
$524$	0	0
$525$	−4864.00	−0.404347
$526$	0	0
$527$	−25088.0	−2.07372
$528$	0	0
$529$	−11143.0	−0.915838
$530$	0	0
$531$	15096.0	1.23373
$532$	0	0
$533$	2040.00	0.165783
$534$	0	0
$535$	7008.00	0.566322
$536$	0	0
$537$	−16832.0	−1.35262
$538$	0	0
$539$	−5448.00	−0.435365
$540$	0	0
$541$	13732.0	1.09128	0.545642	−	0.838018i	$-0.316286\pi$
$541$	13732.0	1.09128	0.545642	+	0.838018i	$0.316286\pi$
$542$	0	0
$543$	11296.0	0.892740
$544$	0	0
$545$	−8880.00	−0.697940
$546$	0	0
$547$	−10968.0	−0.857327	−0.428663	−	0.903464i	$-0.641015\pi$
$547$	−10968.0	−0.857327	−0.428663	+	0.903464i	$0.641015\pi$
$548$	0	0
$549$	−23532.0	−1.82936
$550$	0	0
$551$	15136.0	1.17026
$552$	0	0
$553$	−2048.00	−0.157486
$554$	0	0
$555$	8832.00	0.675491
$556$	0	0
$557$	−25612.0	−1.94832	−0.974161	−	0.225855i	$-0.927482\pi$
$557$	−25612.0	−1.94832	−0.974161	+	0.225855i	$0.927482\pi$
$558$	0	0
$559$	−5920.00	−0.447924
$560$	0	0
$561$	6272.00	0.472021
$562$	0	0
$563$	9768.00	0.731212	0.365606	−	0.930770i	$-0.380862\pi$
$563$	9768.00	0.731212	0.365606	+	0.930770i	$0.380862\pi$
$564$	0	0
$565$	3624.00	0.269846
$566$	0	0
$567$	11488.0	0.850883
$568$	0	0
$569$	22838.0	1.68263	0.841317	−	0.540542i	$-0.181781\pi$
$569$	22838.0	1.68263	0.841317	+	0.540542i	$0.181781\pi$
$570$	0	0
$571$	9208.00	0.674856	0.337428	−	0.941351i	$-0.390443\pi$
$571$	9208.00	0.674856	0.337428	+	0.941351i	$0.390443\pi$
$572$	0	0
$573$	−29696.0	−2.16504
$574$	0	0
$575$	608.000	0.0440963
$576$	0	0
$577$	−10878.0	−0.784848	−0.392424	−	0.919785i	$-0.628363\pi$
$577$	−10878.0	−0.784848	−0.392424	+	0.919785i	$0.628363\pi$
$578$	0	0
$579$	12912.0	0.926778
$580$	0	0
$581$	−12544.0	−0.895719
$582$	0	0
$583$	608.000	0.0431917
$584$	0	0
$585$	−8880.00	−0.627595
$586$	0	0
$587$	18040.0	1.26847	0.634234	−	0.773141i	$-0.281316\pi$
$587$	18040.0	1.26847	0.634234	+	0.773141i	$0.281316\pi$
$588$	0	0
$589$	−22528.0	−1.57598
$590$	0	0
$591$	−5472.00	−0.380860
$592$	0	0
$593$	26994.0	1.86933	0.934663	−	0.355534i	$-0.115701\pi$
$593$	26994.0	1.86933	0.934663	+	0.355534i	$0.115701\pi$
$594$	0	0
$595$	−37632.0	−2.59288
$596$	0	0
$597$	32512.0	2.22886
$598$	0	0
$599$	18336.0	1.25073	0.625366	−	0.780331i	$-0.284950\pi$
$599$	18336.0	1.25073	0.625366	+	0.780331i	$0.284950\pi$
$600$	0	0
$601$	−9286.00	−0.630256	−0.315128	−	0.949049i	$-0.602047\pi$
$601$	−9286.00	−0.630256	−0.315128	+	0.949049i	$0.602047\pi$
$602$	0	0
$603$	20424.0	1.37932
$604$	0	0
$605$	15204.0	1.02170
$606$	0	0
$607$	17536.0	1.17259	0.586297	−	0.810096i	$-0.300585\pi$
$607$	17536.0	1.17259	0.586297	+	0.810096i	$0.300585\pi$
$608$	0	0
$609$	44032.0	2.92983
$610$	0	0
$611$	6400.00	0.423758
$612$	0	0
$613$	−5868.00	−0.386633	−0.193317	−	0.981136i	$-0.561924\pi$
$613$	−5868.00	−0.386633	−0.193317	+	0.981136i	$0.561924\pi$
$614$	0	0
$615$	−9792.00	−0.642035
$616$	0	0
$617$	−19286.0	−1.25839	−0.629194	−	0.777248i	$-0.716615\pi$
$617$	−19286.0	−1.25839	−0.629194	+	0.777248i	$0.716615\pi$
$618$	0	0
$619$	5240.00	0.340248	0.170124	−	0.985423i	$-0.445583\pi$
$619$	5240.00	0.340248	0.170124	+	0.985423i	$0.445583\pi$
$620$	0	0
$621$	2560.00	0.165426
$622$	0	0
$623$	18624.0	1.19768
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	5632.00	0.358725
$628$	0	0
$629$	9016.00	0.571529
$630$	0	0
$631$	15520.0	0.979147	0.489573	−	0.871962i	$-0.337153\pi$
$631$	15520.0	0.979147	0.489573	+	0.871962i	$0.337153\pi$
$632$	0	0
$633$	16960.0	1.06493
$634$	0	0
$635$	−19968.0	−1.24788
$636$	0	0
$637$	13620.0	0.847165
$638$	0	0
$639$	−15392.0	−0.952892
$640$	0	0
$641$	654.000	0.0402987	0.0201493	−	0.999797i	$-0.493586\pi$
$641$	654.000	0.0402987	0.0201493	+	0.999797i	$0.493586\pi$
$642$	0	0
$643$	8232.00	0.504881	0.252440	−	0.967612i	$-0.418767\pi$
$643$	8232.00	0.504881	0.252440	+	0.967612i	$0.418767\pi$
$644$	0	0
$645$	28416.0	1.73470
$646$	0	0
$647$	24672.0	1.49916	0.749580	−	0.661914i	$-0.230256\pi$
$647$	24672.0	1.49916	0.749580	+	0.661914i	$0.230256\pi$
$648$	0	0
$649$	−3264.00	−0.197416
$650$	0	0
$651$	−65536.0	−3.94556
$652$	0	0
$653$	22052.0	1.32153	0.660767	−	0.750591i	$-0.270231\pi$
$653$	22052.0	1.32153	0.660767	+	0.750591i	$0.270231\pi$
$654$	0	0
$655$	27936.0	1.66649
$656$	0	0
$657$	5106.00	0.303202
$658$	0	0
$659$	−12024.0	−0.710757	−0.355378	−	0.934723i	$-0.615648\pi$
$659$	−12024.0	−0.710757	−0.355378	+	0.934723i	$0.615648\pi$
$660$	0	0
$661$	−19100.0	−1.12391	−0.561955	−	0.827168i	$-0.689950\pi$
$661$	−19100.0	−1.12391	−0.561955	+	0.827168i	$0.689950\pi$
$662$	0	0
$663$	−15680.0	−0.918493
$664$	0	0
$665$	−33792.0	−1.97052
$666$	0	0
$667$	−5504.00	−0.319514
$668$	0	0
$669$	−22528.0	−1.30192
$670$	0	0
$671$	5088.00	0.292727
$672$	0	0
$673$	9902.00	0.567153	0.283577	−	0.958950i	$-0.408479\pi$
$673$	9902.00	0.567153	0.283577	+	0.958950i	$0.408479\pi$
$674$	0	0
$675$	1520.00	0.0866738
$676$	0	0
$677$	19684.0	1.11746	0.558728	−	0.829351i	$-0.311290\pi$
$677$	19684.0	1.11746	0.558728	+	0.829351i	$0.311290\pi$
$678$	0	0
$679$	−7616.00	−0.430450
$680$	0	0
$681$	30784.0	1.73223
$682$	0	0
$683$	19864.0	1.11285	0.556424	−	0.830899i	$-0.312173\pi$
$683$	19864.0	1.11285	0.556424	+	0.830899i	$0.312173\pi$
$684$	0	0
$685$	−20808.0	−1.16063
$686$	0	0
$687$	−5216.00	−0.289669
$688$	0	0
$689$	−1520.00	−0.0840456
$690$	0	0
$691$	−3256.00	−0.179253	−0.0896267	−	0.995975i	$-0.528567\pi$
$691$	−3256.00	−0.179253	−0.0896267	+	0.995975i	$0.528567\pi$
$692$	0	0
$693$	9472.00	0.519209
$694$	0	0
$695$	−36384.0	−1.98579
$696$	0	0
$697$	−9996.00	−0.543222
$698$	0	0
$699$	24400.0	1.32030
$700$	0	0
$701$	−2876.00	−0.154957	−0.0774786	−	0.996994i	$-0.524687\pi$
$701$	−2876.00	−0.154957	−0.0774786	+	0.996994i	$0.524687\pi$
$702$	0	0
$703$	8096.00	0.434348
$704$	0	0
$705$	−30720.0	−1.64111
$706$	0	0
$707$	46976.0	2.49889
$708$	0	0
$709$	7300.00	0.386682	0.193341	−	0.981132i	$-0.438068\pi$
$709$	7300.00	0.386682	0.193341	+	0.981132i	$0.438068\pi$
$710$	0	0
$711$	2368.00	0.124904
$712$	0	0
$713$	8192.00	0.430284
$714$	0	0
$715$	1920.00	0.100425
$716$	0	0
$717$	50688.0	2.64014
$718$	0	0
$719$	2880.00	0.149382	0.0746912	−	0.997207i	$-0.476203\pi$
$719$	2880.00	0.149382	0.0746912	+	0.997207i	$0.476203\pi$
$720$	0	0
$721$	−31744.0	−1.63968
$722$	0	0
$723$	−36880.0	−1.89707
$724$	0	0
$725$	−3268.00	−0.167408
$726$	0	0
$727$	−8800.00	−0.448933	−0.224466	−	0.974482i	$-0.572064\pi$
$727$	−8800.00	−0.448933	−0.224466	+	0.974482i	$0.572064\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	29008.0	1.46771
$732$	0	0
$733$	21076.0	1.06202	0.531009	−	0.847366i	$-0.321813\pi$
$733$	21076.0	1.06202	0.531009	+	0.847366i	$0.321813\pi$
$734$	0	0
$735$	−65376.0	−3.28086
$736$	0	0
$737$	−4416.00	−0.220713
$738$	0	0
$739$	19336.0	0.962498	0.481249	−	0.876584i	$-0.340183\pi$
$739$	19336.0	0.962498	0.481249	+	0.876584i	$0.340183\pi$
$740$	0	0
$741$	−14080.0	−0.698032
$742$	0	0
$743$	13664.0	0.674675	0.337338	−	0.941384i	$-0.390474\pi$
$743$	13664.0	0.674675	0.337338	+	0.941384i	$0.390474\pi$
$744$	0	0
$745$	21456.0	1.05515
$746$	0	0
$747$	14504.0	0.710406
$748$	0	0
$749$	18688.0	0.911675
$750$	0	0
$751$	−19520.0	−0.948462	−0.474231	−	0.880400i	$-0.657274\pi$
$751$	−19520.0	−0.948462	−0.474231	+	0.880400i	$0.657274\pi$
$752$	0	0
$753$	6336.00	0.306636
$754$	0	0
$755$	5760.00	0.277653
$756$	0	0
$757$	20004.0	0.960446	0.480223	−	0.877146i	$-0.340556\pi$
$757$	20004.0	0.960446	0.480223	+	0.877146i	$0.340556\pi$
$758$	0	0
$759$	−2048.00	−0.0979416
$760$	0	0
$761$	31478.0	1.49944	0.749722	−	0.661753i	$-0.230187\pi$
$761$	31478.0	1.49944	0.749722	+	0.661753i	$0.230187\pi$
$762$	0	0
$763$	−23680.0	−1.12356
$764$	0	0
$765$	43512.0	2.05644
$766$	0	0
$767$	8160.00	0.384147
$768$	0	0
$769$	7054.00	0.330785	0.165393	−	0.986228i	$-0.447111\pi$
$769$	7054.00	0.330785	0.165393	+	0.986228i	$0.447111\pi$
$770$	0	0
$771$	−42992.0	−2.00820
$772$	0	0
$773$	9604.00	0.446872	0.223436	−	0.974719i	$-0.428273\pi$
$773$	9604.00	0.446872	0.223436	+	0.974719i	$0.428273\pi$
$774$	0	0
$775$	4864.00	0.225445
$776$	0	0
$777$	23552.0	1.08742
$778$	0	0
$779$	−8976.00	−0.412835
$780$	0	0
$781$	3328.00	0.152478
$782$	0	0
$783$	−13760.0	−0.628023
$784$	0	0
$785$	27600.0	1.25489
$786$	0	0
$787$	3144.00	0.142403	0.0712017	−	0.997462i	$-0.477317\pi$
$787$	3144.00	0.142403	0.0712017	+	0.997462i	$0.477317\pi$
$788$	0	0
$789$	−27904.0	−1.25907
$790$	0	0
$791$	9664.00	0.434402
$792$	0	0
$793$	−12720.0	−0.569610
$794$	0	0
$795$	7296.00	0.325487
$796$	0	0
$797$	22084.0	0.981500	0.490750	−	0.871300i	$-0.336723\pi$
$797$	22084.0	0.981500	0.490750	+	0.871300i	$0.336723\pi$
$798$	0	0
$799$	−31360.0	−1.38853
$800$	0	0
$801$	−21534.0	−0.949896
$802$	0	0
$803$	−1104.00	−0.0485172
$804$	0	0
$805$	12288.0	0.538006
$806$	0	0
$807$	−38112.0	−1.66246
$808$	0	0
$809$	14950.0	0.649708	0.324854	−	0.945764i	$-0.394685\pi$
$809$	14950.0	0.649708	0.324854	+	0.945764i	$0.394685\pi$
$810$	0	0
$811$	−23432.0	−1.01456	−0.507280	−	0.861781i	$-0.669349\pi$
$811$	−23432.0	−1.01456	−0.507280	+	0.861781i	$0.669349\pi$
$812$	0	0
$813$	−10752.0	−0.463824
$814$	0	0
$815$	19104.0	0.821085
$816$	0	0
$817$	26048.0	1.11543
$818$	0	0
$819$	−23680.0	−1.01031
$820$	0	0
$821$	1044.00	0.0443798	0.0221899	−	0.999754i	$-0.492936\pi$
$821$	1044.00	0.0443798	0.0221899	+	0.999754i	$0.492936\pi$
$822$	0	0
$823$	18208.0	0.771192	0.385596	−	0.922668i	$-0.373996\pi$
$823$	18208.0	0.771192	0.385596	+	0.922668i	$0.373996\pi$
$824$	0	0
$825$	−1216.00	−0.0513160
$826$	0	0
$827$	−12488.0	−0.525091	−0.262546	−	0.964920i	$-0.584562\pi$
$827$	−12488.0	−0.525091	−0.262546	+	0.964920i	$0.584562\pi$
$828$	0	0
$829$	−30172.0	−1.26407	−0.632037	−	0.774938i	$-0.717781\pi$
$829$	−30172.0	−1.26407	−0.632037	+	0.774938i	$0.717781\pi$
$830$	0	0
$831$	68768.0	2.87068
$832$	0	0
$833$	−66738.0	−2.77591
$834$	0	0
$835$	26496.0	1.09812
$836$	0	0
$837$	20480.0	0.845750
$838$	0	0
$839$	−32544.0	−1.33915	−0.669573	−	0.742746i	$-0.733523\pi$
$839$	−32544.0	−1.33915	−0.669573	+	0.742746i	$0.733523\pi$
$840$	0	0
$841$	5195.00	0.213006
$842$	0	0
$843$	22992.0	0.939367
$844$	0	0
$845$	21564.0	0.877898
$846$	0	0
$847$	40544.0	1.64476
$848$	0	0
$849$	−23104.0	−0.933954
$850$	0	0
$851$	−2944.00	−0.118589
$852$	0	0
$853$	−26156.0	−1.04990	−0.524950	−	0.851133i	$-0.675916\pi$
$853$	−26156.0	−1.04990	−0.524950	+	0.851133i	$0.675916\pi$
$854$	0	0
$855$	39072.0	1.56285
$856$	0	0
$857$	18646.0	0.743215	0.371607	−	0.928390i	$-0.378807\pi$
$857$	18646.0	0.743215	0.371607	+	0.928390i	$0.378807\pi$
$858$	0	0
$859$	−5800.00	−0.230377	−0.115188	−	0.993344i	$-0.536747\pi$
$859$	−5800.00	−0.230377	−0.115188	+	0.993344i	$0.536747\pi$
$860$	0	0
$861$	−26112.0	−1.03356
$862$	0	0
$863$	25088.0	0.989578	0.494789	−	0.869013i	$-0.335245\pi$
$863$	25088.0	0.989578	0.494789	+	0.869013i	$0.335245\pi$
$864$	0	0
$865$	47376.0	1.86223
$866$	0	0
$867$	37528.0	1.47003
$868$	0	0
$869$	−512.000	−0.0199867
$870$	0	0
$871$	11040.0	0.429479
$872$	0	0
$873$	8806.00	0.341395
$874$	0	0
$875$	−40704.0	−1.57262
$876$	0	0
$877$	−3004.00	−0.115665	−0.0578323	−	0.998326i	$-0.518419\pi$
$877$	−3004.00	−0.115665	−0.0578323	+	0.998326i	$0.518419\pi$
$878$	0	0
$879$	−52320.0	−2.00763
$880$	0	0
$881$	43282.0	1.65517	0.827587	−	0.561338i	$-0.189713\pi$
$881$	43282.0	1.65517	0.827587	+	0.561338i	$0.189713\pi$
$882$	0	0
$883$	27880.0	1.06256	0.531278	−	0.847198i	$-0.321712\pi$
$883$	27880.0	1.06256	0.531278	+	0.847198i	$0.321712\pi$
$884$	0	0
$885$	−39168.0	−1.48770
$886$	0	0
$887$	−7392.00	−0.279819	−0.139909	−	0.990164i	$-0.544681\pi$
$887$	−7392.00	−0.279819	−0.139909	+	0.990164i	$0.544681\pi$
$888$	0	0
$889$	−53248.0	−2.00886
$890$	0	0
$891$	2872.00	0.107986
$892$	0	0
$893$	−28160.0	−1.05525
$894$	0	0
$895$	25248.0	0.942958
$896$	0	0
$897$	5120.00	0.190582
$898$	0	0
$899$	−44032.0	−1.63354
$900$	0	0
$901$	7448.00	0.275393
$902$	0	0
$903$	75776.0	2.79254
$904$	0	0
$905$	−16944.0	−0.622362
$906$	0	0
$907$	29080.0	1.06459	0.532296	−	0.846558i	$-0.321329\pi$
$907$	29080.0	1.06459	0.532296	+	0.846558i	$0.321329\pi$
$908$	0	0
$909$	−54316.0	−1.98190
$910$	0	0
$911$	−26688.0	−0.970596	−0.485298	−	0.874349i	$-0.661289\pi$
$911$	−26688.0	−0.970596	−0.485298	+	0.874349i	$0.661289\pi$
$912$	0	0
$913$	−3136.00	−0.113676
$914$	0	0
$915$	61056.0	2.20596
$916$	0	0
$917$	74496.0	2.68274
$918$	0	0
$919$	−19680.0	−0.706402	−0.353201	−	0.935547i	$-0.614907\pi$
$919$	−19680.0	−0.706402	−0.353201	+	0.935547i	$0.614907\pi$
$920$	0	0
$921$	−84672.0	−3.02936
$922$	0	0
$923$	−8320.00	−0.296702
$924$	0	0
$925$	−1748.00	−0.0621339
$926$	0	0
$927$	36704.0	1.30045
$928$	0	0
$929$	−48466.0	−1.71164	−0.855822	−	0.517270i	$-0.826948\pi$
$929$	−48466.0	−1.71164	−0.855822	+	0.517270i	$0.826948\pi$
$930$	0	0
$931$	−59928.0	−2.10962
$932$	0	0
$933$	−50944.0	−1.78760
$934$	0	0
$935$	−9408.00	−0.329064
$936$	0	0
$937$	13610.0	0.474514	0.237257	−	0.971447i	$-0.423752\pi$
$937$	13610.0	0.474514	0.237257	+	0.971447i	$0.423752\pi$
$938$	0	0
$939$	70064.0	2.43499
$940$	0	0
$941$	30692.0	1.06326	0.531632	−	0.846976i	$-0.321579\pi$
$941$	30692.0	1.06326	0.531632	+	0.846976i	$0.321579\pi$
$942$	0	0
$943$	3264.00	0.112715
$944$	0	0
$945$	30720.0	1.05748
$946$	0	0
$947$	−4824.00	−0.165532	−0.0827661	−	0.996569i	$-0.526375\pi$
$947$	−4824.00	−0.165532	−0.0827661	+	0.996569i	$0.526375\pi$
$948$	0	0
$949$	2760.00	0.0944082
$950$	0	0
$951$	−5728.00	−0.195313
$952$	0	0
$953$	−22986.0	−0.781311	−0.390656	−	0.920537i	$-0.627752\pi$
$953$	−22986.0	−0.781311	−0.390656	+	0.920537i	$0.627752\pi$
$954$	0	0
$955$	44544.0	1.50933
$956$	0	0
$957$	11008.0	0.371827
$958$	0	0
$959$	−55488.0	−1.86841
$960$	0	0
$961$	35745.0	1.19986
$962$	0	0
$963$	−21608.0	−0.723061
$964$	0	0
$965$	−19368.0	−0.646091
$966$	0	0
$967$	−17184.0	−0.571458	−0.285729	−	0.958310i	$-0.592236\pi$
$967$	−17184.0	−0.571458	−0.285729	+	0.958310i	$0.592236\pi$
$968$	0	0
$969$	68992.0	2.28725
$970$	0	0
$971$	−2920.00	−0.0965059	−0.0482530	−	0.998835i	$-0.515365\pi$
$971$	−2920.00	−0.0965059	−0.0482530	+	0.998835i	$0.515365\pi$
$972$	0	0
$973$	−97024.0	−3.19676
$974$	0	0
$975$	3040.00	0.0998543
$976$	0	0
$977$	−27042.0	−0.885517	−0.442759	−	0.896641i	$-0.646000\pi$
$977$	−27042.0	−0.885517	−0.442759	+	0.896641i	$0.646000\pi$
$978$	0	0
$979$	4656.00	0.151998
$980$	0	0
$981$	27380.0	0.891107
$982$	0	0
$983$	44192.0	1.43388	0.716941	−	0.697134i	$-0.245542\pi$
$983$	44192.0	1.43388	0.716941	+	0.697134i	$0.245542\pi$
$984$	0	0
$985$	8208.00	0.265511
$986$	0	0
$987$	−81920.0	−2.64189
$988$	0	0
$989$	−9472.00	−0.304542
$990$	0	0
$991$	−29824.0	−0.955995	−0.477997	−	0.878361i	$-0.658637\pi$
$991$	−29824.0	−0.955995	−0.477997	+	0.878361i	$0.658637\pi$
$992$	0	0
$993$	35264.0	1.12696
$994$	0	0
$995$	−48768.0	−1.55382
$996$	0	0
$997$	−11612.0	−0.368862	−0.184431	−	0.982845i	$-0.559044\pi$
$997$	−11612.0	−0.368862	−0.184431	+	0.982845i	$0.559044\pi$
$998$	0	0
$999$	−7360.00	−0.233093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.4.a.f.1.1		1
3.2	odd	2		2304.4.a.m.1.1		1
4.3	odd	2		256.4.a.b.1.1		1
8.3	odd	2		256.4.a.g.1.1		1
8.5	even	2		256.4.a.c.1.1		1
12.11	even	2		2304.4.a.n.1.1		1
16.3	odd	4		128.4.b.a.65.1	✓	2
16.5	even	4		128.4.b.d.65.1	yes	2
16.11	odd	4		128.4.b.a.65.2	yes	2
16.13	even	4		128.4.b.d.65.2	yes	2
24.5	odd	2		2304.4.a.c.1.1		1
24.11	even	2		2304.4.a.d.1.1		1
48.5	odd	4		1152.4.d.h.577.2		2
48.11	even	4		1152.4.d.a.577.2		2
48.29	odd	4		1152.4.d.h.577.1		2
48.35	even	4		1152.4.d.a.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.b.a.65.1	✓	2	16.3	odd	4
128.4.b.a.65.2	yes	2	16.11	odd	4
128.4.b.d.65.1	yes	2	16.5	even	4
128.4.b.d.65.2	yes	2	16.13	even	4
256.4.a.b.1.1		1	4.3	odd	2
256.4.a.c.1.1		1	8.5	even	2
256.4.a.f.1.1		1	1.1	even	1	trivial
256.4.a.g.1.1		1	8.3	odd	2
1152.4.d.a.577.1		2	48.35	even	4
1152.4.d.a.577.2		2	48.11	even	4
1152.4.d.h.577.1		2	48.29	odd	4
1152.4.d.h.577.2		2	48.5	odd	4
2304.4.a.c.1.1		1	24.5	odd	2
2304.4.a.d.1.1		1	24.11	even	2
2304.4.a.m.1.1		1	3.2	odd	2
2304.4.a.n.1.1		1	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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