LMFDB - Embedded newform 116.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,2,Mod(1,116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("116.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 116 = 2^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	116.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:	$0.926264663447$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	116.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+2.00000 q^{3} -2.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} +2.00000 q^{13} -4.00000 q^{15} +2.00000 q^{17} -6.00000 q^{19} +8.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} -4.00000 q^{27} -1.00000 q^{29} -6.00000 q^{31} -12.0000 q^{33} -8.00000 q^{35} +2.00000 q^{37} +4.00000 q^{39} +2.00000 q^{41} +10.0000 q^{43} -2.00000 q^{45} -2.00000 q^{47} +9.00000 q^{49} +4.00000 q^{51} +10.0000 q^{53} +12.0000 q^{55} -12.0000 q^{57} +10.0000 q^{61} +4.00000 q^{63} -4.00000 q^{65} -12.0000 q^{67} +8.00000 q^{69} +8.00000 q^{71} +10.0000 q^{73} -2.00000 q^{75} -24.0000 q^{77} -6.00000 q^{79} -11.0000 q^{81} +16.0000 q^{83} -4.00000 q^{85} -2.00000 q^{87} +2.00000 q^{89} +8.00000 q^{91} -12.0000 q^{93} +12.0000 q^{95} +10.0000 q^{97} -6.00000 q^{99} +O(q^{100})$

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	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
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	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
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	−12.0000	−2.08893
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	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
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	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
$63$	4.00000	0.503953
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−24.0000	−2.73505
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	−12.0000	−1.24434
$94$	0	0
$95$	12.0000	1.23117
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−16.0000	−1.56144
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	20.0000	1.76090
$130$	0	0
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	0	0
$135$	8.00000	0.688530
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	18.0000	1.48461
$148$	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
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	20.0000	1.58610
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	24.0000	1.86840
$166$	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
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	20.0000	1.47844
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	−16.0000	−1.16383
$190$	0	0
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	36.0000	2.49017
$210$	0	0
$211$	−18.0000	−1.23917	−0.619586	−	0.784929i	$-0.712699\pi$
$211$	−18.0000	−1.23917	−0.619586	+	0.784929i	$0.712699\pi$
$212$	0	0
$213$	16.0000	1.09630
$214$	0	0
$215$	−20.0000	−1.36399
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	20.0000	1.35147
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−48.0000	−3.15817
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	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
$243$	−10.0000	−0.641500
$244$	0	0
$245$	−18.0000	−1.14998
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	32.0000	2.02792
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−20.0000	−1.22859
$266$	0	0
$267$	4.00000	0.244796
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	16.0000	0.968364
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	24.0000	1.42164
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	20.0000	1.17242
$292$	0	0
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	24.0000	1.39262
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	40.0000	2.30556
$302$	0	0
$303$	−28.0000	−1.60856
$304$	0	0
$305$	−20.0000	−1.14520
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	−20.0000	−1.10600
$328$	0	0
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−12.0000	−0.651751
$340$	0	0
$341$	36.0000	1.94951
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	0	0
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	0	0
$357$	16.0000	0.846810
$358$	0	0
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	50.0000	2.62432
$364$	0	0
$365$	−20.0000	−1.04685
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	40.0000	2.07670
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	24.0000	1.23935
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	48.0000	2.44631
$386$	0	0
$387$	10.0000	0.508329
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−36.0000	−1.81596
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	−48.0000	−2.40301
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	22.0000	1.09319
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−32.0000	−1.57082
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−2.00000	−0.0972433
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	40.0000	1.93574
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	−24.0000	−1.14808
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−2.00000	−0.0950229	−0.0475114	−	0.998871i	$-0.515129\pi$
$443$	−2.00000	−0.0950229	−0.0475114	+	0.998871i	$0.515129\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	−32.0000	−1.50349
$454$	0	0
$455$	−16.0000	−0.750092
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	0	0
$465$	24.0000	1.11297
$466$	0	0
$467$	42.0000	1.94353	0.971764	−	0.235954i	$-0.0758216\pi$
$467$	42.0000	1.94353	0.971764	+	0.235954i	$0.0758216\pi$
$468$	0	0
$469$	−48.0000	−2.21643
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	−60.0000	−2.75880
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	32.0000	1.45605
$484$	0	0
$485$	−20.0000	−0.908153
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	−36.0000	−1.62798
$490$	0	0
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	−18.0000	−0.799408
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	24.0000	1.05963
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	36.0000	1.58022
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	40.0000	1.72613
$538$	0	0
$539$	−54.0000	−2.32594
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	44.0000	1.88822
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	20.0000	0.845910
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	22.0000	0.927189	0.463595	−	0.886047i	$-0.346559\pi$
$563$	22.0000	0.927189	0.463595	+	0.886047i	$0.346559\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	0	0
$567$	−44.0000	−1.84783
$568$	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
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	0	0
$573$	20.0000	0.835512
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	0	0
$579$	4.00000	0.166234
$580$	0	0
$581$	64.0000	2.65517
$582$	0	0
$583$	−60.0000	−2.48495
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−8.00000	−0.330195	−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000	−0.330195	−0.165098	+	0.986277i	$0.552794\pi$
$588$	0	0
$589$	36.0000	1.48335
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	40.0000	1.63709
$598$	0	0
$599$	−38.0000	−1.55264	−0.776319	−	0.630340i	$-0.782915\pi$
$599$	−38.0000	−1.55264	−0.776319	+	0.630340i	$0.782915\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	−50.0000	−2.03279
$606$	0	0
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	−8.00000	−0.322591
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	−16.0000	−0.642058
$622$	0	0
$623$	8.00000	0.320513
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	72.0000	2.87540
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−36.0000	−1.43087
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−38.0000	−1.50091	−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000	−1.50091	−0.750455	+	0.660922i	$0.770166\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	−40.0000	−1.57500
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−48.0000	−1.88127
$652$	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
$655$	36.0000	1.40664
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	48.0000	1.86136
$666$	0	0
$667$	−4.00000	−0.154881
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	−60.0000	−2.31627
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	0	0
$679$	40.0000	1.53506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	20.0000	0.763048
$688$	0	0
$689$	20.0000	0.761939
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	16.0000	0.606915
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−44.0000	−1.66423
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	−56.0000	−2.10610
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	−44.0000	−1.64092	−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000	−1.64092	−0.820462	+	0.571702i	$0.806283\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	4.00000	0.148762
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	−36.0000	−1.32788
$736$	0	0
$737$	72.0000	2.65215
$738$	0	0
$739$	−14.0000	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$739$	−14.0000	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$740$	0	0
$741$	−24.0000	−0.881662
$742$	0	0
$743$	38.0000	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$743$	38.0000	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	16.0000	0.585409
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	32.0000	1.16460
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	−48.0000	−1.74229
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−40.0000	−1.44810
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−4.00000	−0.144056
$772$	0	0
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	4.00000	0.142948
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	36.0000	1.28163
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	20.0000	0.710221
$794$	0	0
$795$	−40.0000	−1.41865
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−60.0000	−2.11735
$804$	0	0
$805$	−32.0000	−1.12785
$806$	0	0
$807$	−12.0000	−0.422420
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	0	0
$815$	36.0000	1.26102
$816$	0	0
$817$	−60.0000	−2.09913
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	2.00000	0.0697156	0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000	0.0697156	0.0348578	+	0.999392i	$0.488902\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	22.0000	0.765015	0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000	0.765015	0.382507	+	0.923952i	$0.375061\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	24.0000	0.829561
$838$	0	0
$839$	54.0000	1.86429	0.932144	−	0.362089i	$-0.117936\pi$
$839$	54.0000	1.86429	0.932144	+	0.362089i	$0.117936\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−12.0000	−0.413302
$844$	0	0
$845$	18.0000	0.619219
$846$	0	0
$847$	100.000	3.43604
$848$	0	0
$849$	8.00000	0.274559
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	16.0000	0.545279
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	−36.0000	−1.22404
$866$	0	0
$867$	−26.0000	−0.883006
$868$	0	0
$869$	36.0000	1.22122
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	48.0000	1.62270
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	4.00000	0.134917
$880$	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
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.0000	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$887$	−30.0000	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	66.0000	2.21108
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	−40.0000	−1.33705
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	20.0000	0.666297
$902$	0	0
$903$	80.0000	2.66223
$904$	0	0
$905$	−44.0000	−1.46261
$906$	0	0
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−34.0000	−1.12647	−0.563235	−	0.826297i	$-0.690443\pi$
$911$	−34.0000	−1.12647	−0.563235	+	0.826297i	$0.690443\pi$
$912$	0	0
$913$	−96.0000	−3.17714
$914$	0	0
$915$	−40.0000	−1.32236
$916$	0	0
$917$	−72.0000	−2.37765
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	0	0
$933$	20.0000	0.654771
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	22.0000	0.717180	0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000	0.717180	0.358590	+	0.933495i	$0.383258\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	32.0000	1.04096
$946$	0	0
$947$	42.0000	1.36482	0.682408	−	0.730971i	$-0.260933\pi$
$947$	42.0000	1.36482	0.682408	+	0.730971i	$0.260933\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	0	0
$951$	36.0000	1.16738
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	−20.0000	−0.647185
$956$	0	0
$957$	12.0000	0.387905
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−8.00000	−0.257796
$964$	0	0
$965$	−4.00000	−0.128765
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	26.0000	0.834380	0.417190	−	0.908819i	$-0.363015\pi$
$971$	26.0000	0.834380	0.417190	+	0.908819i	$0.363015\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−50.0000	−1.59964	−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000	−1.59964	−0.799821	+	0.600239i	$0.795072\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−50.0000	−1.59475	−0.797376	−	0.603483i	$-0.793779\pi$
$983$	−50.0000	−1.59475	−0.797376	+	0.603483i	$0.793779\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	−16.0000	−0.509286
$988$	0	0
$989$	40.0000	1.27193
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−40.0000	−1.26809
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.2.a.c.1.1	✓	1
3.2	odd	2		1044.2.a.i.1.1		1
4.3	odd	2		464.2.a.a.1.1		1
5.2	odd	4		2900.2.c.b.349.1		2
5.3	odd	4		2900.2.c.b.349.2		2
5.4	even	2		2900.2.a.a.1.1		1
7.6	odd	2		5684.2.a.c.1.1		1
8.3	odd	2		1856.2.a.n.1.1		1
8.5	even	2		1856.2.a.c.1.1		1
12.11	even	2		4176.2.a.x.1.1		1
29.12	odd	4		3364.2.c.c.1681.2		2
29.17	odd	4		3364.2.c.c.1681.1		2
29.28	even	2		3364.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.a.c.1.1	✓	1	1.1	even	1	trivial
464.2.a.a.1.1		1	4.3	odd	2
1044.2.a.i.1.1		1	3.2	odd	2
1856.2.a.c.1.1		1	8.5	even	2
1856.2.a.n.1.1		1	8.3	odd	2
2900.2.a.a.1.1		1	5.4	even	2
2900.2.c.b.349.1		2	5.2	odd	4
2900.2.c.b.349.2		2	5.3	odd	4
3364.2.a.a.1.1		1	29.28	even	2
3364.2.c.c.1681.1		2	29.17	odd	4
3364.2.c.c.1681.2		2	29.12	odd	4
4176.2.a.x.1.1		1	12.11	even	2
5684.2.a.c.1.1		1	7.6	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:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	116.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more