LMFDB - Embedded newform 2576.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 644)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.00000 q^{15} +6.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} +5.00000 q^{27} +1.00000 q^{29} -1.00000 q^{31} -2.00000 q^{33} -2.00000 q^{35} -6.00000 q^{37} +1.00000 q^{39} +3.00000 q^{41} +4.00000 q^{45} +3.00000 q^{47} +1.00000 q^{49} +6.00000 q^{53} -4.00000 q^{55} -6.00000 q^{57} -8.00000 q^{59} -10.0000 q^{61} -2.00000 q^{63} +2.00000 q^{65} +2.00000 q^{67} +1.00000 q^{69} -5.00000 q^{71} -7.00000 q^{73} +1.00000 q^{75} +2.00000 q^{77} -14.0000 q^{79} +1.00000 q^{81} +4.00000 q^{83} -1.00000 q^{87} -12.0000 q^{89} -1.00000 q^{91} +1.00000 q^{93} -12.0000 q^{95} -8.00000 q^{97} -4.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	4.00000	0.596285
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	−4.00000	−0.402015
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−3.00000	−0.270501
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.0000	0.961074	0.480537	−	0.876974i	$-0.340442\pi$
$131$	11.0000	0.961074	0.480537	+	0.876974i	$0.340442\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	−10.0000	−0.860663
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−21.0000	−1.70896	−0.854478	−	0.519488i	$-0.826123\pi$
$151$	−21.0000	−1.70896	−0.854478	+	0.519488i	$0.826123\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−12.0000	−0.917663
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−7.00000	−0.523205	−0.261602	−	0.965176i	$-0.584251\pi$
$179$	−7.00000	−0.523205	−0.261602	+	0.965176i	$0.584251\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−26.0000	−1.88129	−0.940647	−	0.339387i	$-0.889781\pi$
$191$	−26.0000	−1.88129	−0.940647	+	0.339387i	$0.889781\pi$
$192$	0	0
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	1.00000	0.0701862
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	5.00000	0.342594
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.00000	−0.0678844
$218$	0	0
$219$	7.00000	0.473016
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	19.0000	1.24473	0.622366	−	0.782727i	$-0.286172\pi$
$233$	19.0000	1.24473	0.622366	+	0.782727i	$0.286172\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	14.0000	0.909398
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	12.0000	0.734388
$268$	0	0
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	0	0
$285$	12.0000	0.710819
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	0	0
$297$	10.0000	0.580259
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	20.0000	1.14520
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	8.00000	0.442401
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.00000	−0.108306
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	37.0000	1.96931	0.984656	−	0.174509i	$-0.0558337\pi$
$353$	37.0000	1.96931	0.984656	+	0.174509i	$0.0558337\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.0000	1.16112	0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000	1.16112	0.580558	+	0.814219i	$0.302835\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	−38.0000	−1.96757	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000	−1.96757	−0.983783	+	0.179364i	$0.942596\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−11.0000	−0.554877
$394$	0	0
$395$	28.0000	1.40883
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	1.00000	0.0498135
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	8.00000	0.390826	0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000	0.390826	0.195413	+	0.980721i	$0.437395\pi$
$420$	0	0
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−23.0000	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$439$	−23.0000	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−29.0000	−1.37783	−0.688916	−	0.724841i	$-0.741913\pi$
$443$	−29.0000	−1.37783	−0.688916	+	0.724841i	$0.741913\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	6.00000	0.283790
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	21.0000	0.986666
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.00000	−0.326023	−0.163011	−	0.986624i	$-0.552121\pi$
$461$	−7.00000	−0.326023	−0.163011	+	0.986624i	$0.552121\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	34.0000	1.57333	0.786666	−	0.617379i	$-0.211805\pi$
$467$	34.0000	1.57333	0.786666	+	0.617379i	$0.211805\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	25.0000	1.13286	0.566429	−	0.824110i	$-0.308325\pi$
$487$	25.0000	1.13286	0.566429	+	0.824110i	$0.308325\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	8.00000	0.359573
$496$	0	0
$497$	−5.00000	−0.224281
$498$	0	0
$499$	27.0000	1.20869	0.604343	−	0.796724i	$-0.293436\pi$
$499$	27.0000	1.20869	0.604343	+	0.796724i	$0.293436\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	−7.00000	−0.309662
$512$	0	0
$513$	30.0000	1.32453
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.00000	0.263880
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	16.0000	0.694341
$532$	0	0
$533$	−3.00000	−0.129944
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	7.00000	0.302072
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	16.0000	0.685365
$546$	0	0
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	0	0
$549$	20.0000	0.853579
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−14.0000	−0.595341
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	26.0000	1.08617
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	0	0
$579$	−9.00000	−0.374027
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	15.0000	0.619116	0.309558	−	0.950881i	$-0.399819\pi$
$587$	15.0000	0.619116	0.309558	+	0.950881i	$0.399819\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	0	0
$591$	15.0000	0.617018
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.00000	−0.245564
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	17.0000	0.693444	0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000	0.693444	0.346722	+	0.937968i	$0.387295\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	−44.0000	−1.78590	−0.892952	−	0.450151i	$-0.851370\pi$
$607$	−44.0000	−1.78590	−0.892952	+	0.450151i	$0.851370\pi$
$608$	0	0
$609$	−1.00000	−0.0405220
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	0	0
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	14.0000	0.555573
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	6.00000	0.236617	0.118308	−	0.992977i	$-0.462253\pi$
$643$	6.00000	0.236617	0.118308	+	0.992977i	$0.462253\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.0000	−0.825595	−0.412798	−	0.910823i	$-0.635448\pi$
$647$	−21.0000	−0.825595	−0.412798	+	0.910823i	$0.635448\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	1.00000	0.0391931
$652$	0	0
$653$	−13.0000	−0.508729	−0.254365	−	0.967108i	$-0.581866\pi$
$653$	−13.0000	−0.508729	−0.254365	+	0.967108i	$0.581866\pi$
$654$	0	0
$655$	−22.0000	−0.859611
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	−30.0000	−1.16686	−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000	−1.16686	−0.583432	+	0.812162i	$0.698291\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	−20.0000	−0.772091
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	33.0000	1.26271	0.631355	−	0.775494i	$-0.282499\pi$
$683$	33.0000	1.26271	0.631355	+	0.775494i	$0.282499\pi$
$684$	0	0
$685$	16.0000	0.611329
$686$	0	0
$687$	24.0000	0.915657
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	−10.0000	−0.379322
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−19.0000	−0.718646
$700$	0	0
$701$	−28.0000	−1.05755	−0.528773	−	0.848763i	$-0.677348\pi$
$701$	−28.0000	−1.05755	−0.528773	+	0.848763i	$0.677348\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	10.0000	0.376089
$708$	0	0
$709$	−36.0000	−1.35201	−0.676004	−	0.736898i	$-0.736290\pi$
$709$	−36.0000	−1.35201	−0.676004	+	0.736898i	$0.736290\pi$
$710$	0	0
$711$	28.0000	1.05008
$712$	0	0
$713$	1.00000	0.0374503
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	5.00000	0.186728
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	6.00000	0.220416
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	−24.0000	−0.874609
$754$	0	0
$755$	42.0000	1.52854
$756$	0	0
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	21.0000	0.756297
$772$	0	0
$773$	16.0000	0.575480	0.287740	−	0.957709i	$-0.407096\pi$
$773$	16.0000	0.575480	0.287740	+	0.957709i	$0.407096\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	−10.0000	−0.357828
$782$	0	0
$783$	5.00000	0.178685
$784$	0	0
$785$	0	0
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	−14.0000	−0.494049
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	15.0000	0.528025
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−49.0000	−1.72062	−0.860311	−	0.509769i	$-0.829731\pi$
$811$	−49.0000	−1.72062	−0.860311	+	0.509769i	$0.829731\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	22.0000	0.770626
$816$	0	0
$817$	0	0
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	0	0
$823$	−5.00000	−0.174289	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	−5.00000	−0.174289	−0.0871445	+	0.996196i	$0.527774\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	0	0
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	0	0
$831$	−5.00000	−0.173448
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−5.00000	−0.172825
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	28.0000	0.964371
$844$	0	0
$845$	24.0000	0.825625
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−6.00000	−0.205919
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	0	0
$857$	−49.0000	−1.67381	−0.836904	−	0.547350i	$-0.815637\pi$
$857$	−49.0000	−1.67381	−0.836904	+	0.547350i	$0.815637\pi$
$858$	0	0
$859$	−9.00000	−0.307076	−0.153538	−	0.988143i	$-0.549067\pi$
$859$	−9.00000	−0.307076	−0.153538	+	0.988143i	$0.549067\pi$
$860$	0	0
$861$	−3.00000	−0.102240
$862$	0	0
$863$	17.0000	0.578687	0.289343	−	0.957225i	$-0.406563\pi$
$863$	17.0000	0.578687	0.289343	+	0.957225i	$0.406563\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	−28.0000	−0.949835
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	12.0000	0.405674
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	4.00000	0.134917
$880$	0	0
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	0	0
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	0	0
$885$	−16.0000	−0.537834
$886$	0	0
$887$	−37.0000	−1.24234	−0.621169	−	0.783676i	$-0.713342\pi$
$887$	−37.0000	−1.24234	−0.621169	+	0.783676i	$0.713342\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	18.0000	0.602347
$894$	0	0
$895$	14.0000	0.467968
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	0	0
$899$	−1.00000	−0.0333519
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	0	0
$909$	−20.0000	−0.663358
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	0	0
$915$	−20.0000	−0.661180
$916$	0	0
$917$	11.0000	0.363252
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	0	0
$923$	5.00000	0.164577
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	11.0000	0.360898	0.180449	−	0.983584i	$-0.442245\pi$
$929$	11.0000	0.360898	0.180449	+	0.983584i	$0.442245\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	−15.0000	−0.491078
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	28.0000	0.913745
$940$	0	0
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	0	0
$943$	−3.00000	−0.0976934
$944$	0	0
$945$	−10.0000	−0.325300
$946$	0	0
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	52.0000	1.68268
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−18.0000	−0.579441
$966$	0	0
$967$	−27.0000	−0.868261	−0.434131	−	0.900850i	$-0.642944\pi$
$967$	−27.0000	−0.868261	−0.434131	+	0.900850i	$0.642944\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	16.0000	0.510841
$982$	0	0
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	0	0
$985$	30.0000	0.955879
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	0	0
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	−5.00000	−0.158670
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	0	0
$999$	−30.0000	−0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.g.1.1		1
4.3	odd	2		644.2.a.b.1.1	✓	1
12.11	even	2		5796.2.a.f.1.1		1
28.27	even	2		4508.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.b.1.1	✓	1	4.3	odd	2
2576.2.a.g.1.1		1	1.1	even	1	trivial
4508.2.a.b.1.1		1	28.27	even	2
5796.2.a.f.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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