LMFDB - Embedded newform 5096.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5096,2,Mod(1,5096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5096, 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("5096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5096 = 2^{3} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5096.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:	$40.6917648700$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{3} -3.00000 q^{5} +1.00000 q^{9} +O(q^{10})$

$q-2.00000 q^{3} -3.00000 q^{5} +1.00000 q^{9} +1.00000 q^{13} +6.00000 q^{15} +2.00000 q^{17} -5.00000 q^{19} +1.00000 q^{23} +4.00000 q^{25} +4.00000 q^{27} -5.00000 q^{29} +3.00000 q^{31} -6.00000 q^{37} -2.00000 q^{39} +2.00000 q^{41} +1.00000 q^{43} -3.00000 q^{45} -3.00000 q^{47} -4.00000 q^{51} +11.0000 q^{53} +10.0000 q^{57} +8.00000 q^{59} +10.0000 q^{61} -3.00000 q^{65} +2.00000 q^{67} -2.00000 q^{69} +14.0000 q^{71} +7.00000 q^{73} -8.00000 q^{75} +13.0000 q^{79} -11.0000 q^{81} +1.00000 q^{83} -6.00000 q^{85} +10.0000 q^{87} +11.0000 q^{89} -6.00000 q^{93} +15.0000 q^{95} -5.00000 q^{97} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	6.00000	1.54919
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	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
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.0000	1.32453
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.00000	−0.372104
$66$	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
$69$	−2.00000	−0.240772
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	−8.00000	−0.923760
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.00000	−0.622171
$94$	0	0
$95$	15.0000	1.53897
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	12.0000	1.13899
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−12.0000	−1.03280
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	0	0
$144$	0	0
$145$	15.0000	1.24568
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−9.00000	−0.722897
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−22.0000	−1.74471
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.0000	1.77979	0.889897	−	0.456162i	$-0.150776\pi$
$167$	23.0000	1.77979	0.889897	+	0.456162i	$0.150776\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.00000	−0.382360
$172$	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
$175$	0	0
$176$	0	0
$177$	−16.0000	−1.20263
$178$	0	0
$179$	17.0000	1.27064	0.635320	−	0.772249i	$-0.280868\pi$
$179$	17.0000	1.27064	0.635320	+	0.772249i	$0.280868\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	−20.0000	−1.47844
$184$	0	0
$185$	18.0000	1.32339
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	0	0
$213$	−28.0000	−1.91853
$214$	0	0
$215$	−3.00000	−0.204598
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.00000	−0.589610	−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000	−0.589610	−0.294805	+	0.955557i	$0.595255\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	−26.0000	−1.68888
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.00000	−0.318142
$248$	0	0
$249$	−2.00000	−0.126745
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	12.0000	0.751469
$256$	0	0
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0	0
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	−33.0000	−2.02717
$266$	0	0
$267$	−22.0000	−1.34638
$268$	0	0
$269$	−32.0000	−1.95107	−0.975537	−	0.219834i	$-0.929448\pi$
$269$	−32.0000	−1.95107	−0.975537	+	0.219834i	$0.929448\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	0	0
$285$	−30.0000	−1.77705
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	−23.0000	−1.34367	−0.671837	−	0.740699i	$-0.734495\pi$
$293$	−23.0000	−1.34367	−0.671837	+	0.740699i	$0.734495\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	0	0
$302$	0	0
$303$	24.0000	1.37876
$304$	0	0
$305$	−30.0000	−1.71780
$306$	0	0
$307$	9.00000	0.513657	0.256829	−	0.966457i	$-0.417322\pi$
$307$	9.00000	0.513657	0.256829	+	0.966457i	$0.417322\pi$
$308$	0	0
$309$	32.0000	1.82042
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	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
$315$	0	0
$316$	0	0
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	−10.0000	−0.556415
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	8.00000	0.442401
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	21.0000	1.14394	0.571971	−	0.820274i	$-0.306179\pi$
$337$	21.0000	1.14394	0.571971	+	0.820274i	$0.306179\pi$
$338$	0	0
$339$	30.0000	1.62938
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	6.00000	0.323029
$346$	0	0
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	−1.00000	−0.0535288	−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000	−0.0535288	−0.0267644	+	0.999642i	$0.508520\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	−42.0000	−2.22913
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.0000	0.738892	0.369446	−	0.929252i	$-0.379548\pi$
$359$	14.0000	0.738892	0.369446	+	0.929252i	$0.379548\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	22.0000	1.15470
$364$	0	0
$365$	−21.0000	−1.09919
$366$	0	0
$367$	−36.0000	−1.87918	−0.939592	−	0.342296i	$-0.888796\pi$
$367$	−36.0000	−1.87918	−0.939592	+	0.342296i	$0.888796\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	−6.00000	−0.309839
$376$	0	0
$377$	−5.00000	−0.257513
$378$	0	0
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−39.0000	−1.96230
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	0	0
$405$	33.0000	1.63978
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3.00000	−0.147264
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	−30.0000	−1.43839
$436$	0	0
$437$	−5.00000	−0.239182
$438$	0	0
$439$	−38.0000	−1.81364	−0.906821	−	0.421517i	$-0.861498\pi$
$439$	−38.0000	−1.81364	−0.906821	+	0.421517i	$0.861498\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	17.0000	0.807694	0.403847	−	0.914826i	$-0.367673\pi$
$443$	17.0000	0.807694	0.403847	+	0.914826i	$0.367673\pi$
$444$	0	0
$445$	−33.0000	−1.56435
$446$	0	0
$447$	32.0000	1.51355
$448$	0	0
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	0	0
$465$	18.0000	0.834730
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.0000	1.29017
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−20.0000	−0.917663
$476$	0	0
$477$	11.0000	0.503655
$478$	0	0
$479$	−25.0000	−1.14228	−0.571140	−	0.820853i	$-0.693499\pi$
$479$	−25.0000	−1.14228	−0.571140	+	0.820853i	$0.693499\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.0000	0.681115
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	−46.0000	−2.05513
$502$	0	0
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	36.0000	1.60198
$506$	0	0
$507$	−2.00000	−0.0888231
$508$	0	0
$509$	−11.0000	−0.487566	−0.243783	−	0.969830i	$-0.578389\pi$
$509$	−11.0000	−0.487566	−0.243783	+	0.969830i	$0.578389\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	48.0000	2.11513
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−12.0000	−0.526742
$520$	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
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	36.0000	1.55642
$536$	0	0
$537$	−34.0000	−1.46721
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	0	0
$543$	16.0000	0.686626
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	25.0000	1.06504
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−36.0000	−1.52811
$556$	0	0
$557$	−40.0000	−1.69485	−0.847427	−	0.530912i	$-0.821850\pi$
$557$	−40.0000	−1.69485	−0.847427	+	0.530912i	$0.821850\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	0	0
$565$	45.0000	1.89316
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−20.0000	−0.831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−3.00000	−0.124035
$586$	0	0
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	−15.0000	−0.618064
$590$	0	0
$591$	36.0000	1.48084
$592$	0	0
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000	0.818546
$598$	0	0
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	33.0000	1.34164
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	−4.00000	−0.161558	−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000	−0.161558	−0.0807792	+	0.996732i	$0.525741\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	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
$621$	4.00000	0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	0	0
$633$	30.0000	1.19239
$634$	0	0
$635$	−24.0000	−0.952411
$636$	0	0
$637$	0	0
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.00000	0.273096
$658$	0	0
$659$	−5.00000	−0.194772	−0.0973862	−	0.995247i	$-0.531048\pi$
$659$	−5.00000	−0.194772	−0.0973862	+	0.995247i	$0.531048\pi$
$660$	0	0
$661$	−13.0000	−0.505641	−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000	−0.505641	−0.252821	+	0.967513i	$0.581358\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.00000	−0.193601
$668$	0	0
$669$	−34.0000	−1.31452
$670$	0	0
$671$	0	0
$672$	0	0
$673$	45.0000	1.73462	0.867311	−	0.497766i	$-0.165846\pi$
$673$	45.0000	1.73462	0.867311	+	0.497766i	$0.165846\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	0	0
$677$	−32.0000	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.0000	1.53056	0.765279	−	0.643699i	$-0.222601\pi$
$683$	40.0000	1.53056	0.765279	+	0.643699i	$0.222601\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	52.0000	1.98392
$688$	0	0
$689$	11.0000	0.419067
$690$	0	0
$691$	49.0000	1.86405	0.932024	−	0.362397i	$-0.118041\pi$
$691$	49.0000	1.86405	0.932024	+	0.362397i	$0.118041\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	0	0
$703$	30.0000	1.13147
$704$	0	0
$705$	−18.0000	−0.677919
$706$	0	0
$707$	0	0
$708$	0	0
$709$	24.0000	0.901339	0.450669	−	0.892691i	$-0.351185\pi$
$709$	24.0000	0.901339	0.450669	+	0.892691i	$0.351185\pi$
$710$	0	0
$711$	13.0000	0.487538
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−6.00000	−0.222528	−0.111264	−	0.993791i	$-0.535490\pi$
$727$	−6.00000	−0.222528	−0.111264	+	0.993791i	$0.535490\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	2.00000	0.0739727
$732$	0	0
$733$	39.0000	1.44050	0.720249	−	0.693716i	$-0.244028\pi$
$733$	39.0000	1.44050	0.720249	+	0.693716i	$0.244028\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	0	0
$747$	1.00000	0.0365881
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.0000	0.912263	0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000	0.912263	0.456131	+	0.889912i	$0.349235\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	49.0000	1.77625	0.888124	−	0.459603i	$-0.152008\pi$
$761$	49.0000	1.77625	0.888124	+	0.459603i	$0.152008\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	−48.0000	−1.72868
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	12.0000	0.431053
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10.0000	−0.358287
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−20.0000	−0.714742
$784$	0	0
$785$	42.0000	1.49904
$786$	0	0
$787$	−1.00000	−0.0356462	−0.0178231	−	0.999841i	$-0.505674\pi$
$787$	−1.00000	−0.0356462	−0.0178231	+	0.999841i	$0.505674\pi$
$788$	0	0
$789$	42.0000	1.49524
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	66.0000	2.34078
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	11.0000	0.388666
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	64.0000	2.25291
$808$	0	0
$809$	−43.0000	−1.51180	−0.755900	−	0.654687i	$-0.772800\pi$
$809$	−43.0000	−1.51180	−0.755900	+	0.654687i	$0.772800\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	−48.0000	−1.68343
$814$	0	0
$815$	36.0000	1.26102
$816$	0	0
$817$	−5.00000	−0.174928
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	36.0000	1.25033	0.625166	−	0.780492i	$-0.285031\pi$
$829$	36.0000	1.25033	0.625166	+	0.780492i	$0.285031\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−69.0000	−2.38784
$836$	0	0
$837$	12.0000	0.414781
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	40.0000	1.37767
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	0	0
$848$	0	0
$849$	64.0000	2.19647
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	25.0000	0.855984	0.427992	−	0.903783i	$-0.359221\pi$
$853$	25.0000	0.855984	0.427992	+	0.903783i	$0.359221\pi$
$854$	0	0
$855$	15.0000	0.512989
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	−42.0000	−1.43302	−0.716511	−	0.697576i	$-0.754262\pi$
$859$	−42.0000	−1.43302	−0.716511	+	0.697576i	$0.754262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	26.0000	0.883006
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	−5.00000	−0.169224
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.0000	−0.877958	−0.438979	−	0.898497i	$-0.644660\pi$
$877$	−26.0000	−0.877958	−0.438979	+	0.898497i	$0.644660\pi$
$878$	0	0
$879$	46.0000	1.55154
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−48.0000	−1.61533	−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000	−1.61533	−0.807664	+	0.589643i	$0.799269\pi$
$884$	0	0
$885$	48.0000	1.61350
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.0000	0.501956
$894$	0	0
$895$	−51.0000	−1.70474
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	−15.0000	−0.500278
$900$	0	0
$901$	22.0000	0.732926
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.0000	0.797787
$906$	0	0
$907$	−35.0000	−1.16216	−0.581078	−	0.813848i	$-0.697369\pi$
$907$	−35.0000	−1.16216	−0.581078	+	0.813848i	$0.697369\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−29.0000	−0.960813	−0.480406	−	0.877046i	$-0.659511\pi$
$911$	−29.0000	−0.960813	−0.480406	+	0.877046i	$0.659511\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	60.0000	1.98354
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	0	0
$923$	14.0000	0.460816
$924$	0	0
$925$	−24.0000	−0.789115
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	9.00000	0.295280	0.147640	−	0.989041i	$-0.452832\pi$
$929$	9.00000	0.295280	0.147640	+	0.989041i	$0.452832\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−48.0000	−1.57145
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	56.0000	1.82749
$940$	0	0
$941$	−9.00000	−0.293392	−0.146696	−	0.989182i	$-0.546864\pi$
$941$	−9.00000	−0.293392	−0.146696	+	0.989182i	$0.546864\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−58.0000	−1.88475	−0.942373	−	0.334563i	$-0.891411\pi$
$947$	−58.0000	−1.88475	−0.942373	+	0.334563i	$0.891411\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	0	0
$951$	−40.0000	−1.29709
$952$	0	0
$953$	−11.0000	−0.356325	−0.178162	−	0.984001i	$-0.557015\pi$
$953$	−11.0000	−0.356325	−0.178162	+	0.984001i	$0.557015\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	−30.0000	−0.965734
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−8.00000	−0.256205
$976$	0	0
$977$	44.0000	1.40768	0.703842	−	0.710356i	$-0.251466\pi$
$977$	44.0000	1.40768	0.703842	+	0.710356i	$0.251466\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.00000	0.0317982
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	30.0000	0.951064
$996$	0	0
$997$	−52.0000	−1.64686	−0.823428	−	0.567420i	$-0.807941\pi$
$997$	−52.0000	−1.64686	−0.823428	+	0.567420i	$0.807941\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5096.2.a.a.1.1		1
7.6	odd	2		728.2.a.d.1.1	✓	1
21.20	even	2		6552.2.a.c.1.1		1
28.27	even	2		1456.2.a.c.1.1		1
56.13	odd	2		5824.2.a.c.1.1		1
56.27	even	2		5824.2.a.ba.1.1		1
91.90	odd	2		9464.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.d.1.1	✓	1	7.6	odd	2
1456.2.a.c.1.1		1	28.27	even	2
5096.2.a.a.1.1		1	1.1	even	1	trivial
5824.2.a.c.1.1		1	56.13	odd	2
5824.2.a.ba.1.1		1	56.27	even	2
6552.2.a.c.1.1		1	21.20	even	2
9464.2.a.e.1.1		1	91.90	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:	\( N \)	\(=\)	\( 5096 = 2^{3} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5096.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more