LMFDB - Embedded newform 4368.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -1.00000 q^{13} -3.00000 q^{15} +3.00000 q^{17} +1.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +4.00000 q^{25} +1.00000 q^{27} -5.00000 q^{29} +8.00000 q^{31} +5.00000 q^{33} +3.00000 q^{35} +1.00000 q^{37} -1.00000 q^{39} -12.0000 q^{41} -11.0000 q^{43} -3.00000 q^{45} -8.00000 q^{47} +1.00000 q^{49} +3.00000 q^{51} +10.0000 q^{53} -15.0000 q^{55} +1.00000 q^{57} +8.00000 q^{59} +13.0000 q^{61} -1.00000 q^{63} +3.00000 q^{65} -2.00000 q^{67} -1.00000 q^{69} -8.00000 q^{71} +11.0000 q^{73} +4.00000 q^{75} -5.00000 q^{77} -2.00000 q^{79} +1.00000 q^{81} +18.0000 q^{83} -9.00000 q^{85} -5.00000 q^{87} +18.0000 q^{89} +1.00000 q^{91} +8.00000 q^{93} -3.00000 q^{95} +10.0000 q^{97} +5.00000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	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
$55$	−15.0000	−2.02260
$56$	0	0
$57$	1.00000	0.132453
$58$	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
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	18.0000	1.97576	0.987878	−	0.155230i	$-0.0496119\pi$
$83$	18.0000	1.97576	0.987878	+	0.155230i	$0.0496119\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	−3.00000	−0.307794
$96$	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
$99$	5.00000	0.502519
$100$	0	0
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	0	0
$103$	−5.00000	−0.492665	−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000	−0.492665	−0.246332	+	0.969185i	$0.579225\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	0	0
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	3.00000	0.268328
$126$	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
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−11.0000	−0.961074	−0.480537	−	0.876974i	$-0.659558\pi$
$131$	−11.0000	−0.961074	−0.480537	+	0.876974i	$0.659558\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−3.00000	−0.258199
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−5.00000	−0.418121
$144$	0	0
$145$	15.0000	1.24568
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	3.00000	0.242536
$154$	0	0
$155$	−24.0000	−1.92773
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	−15.0000	−1.16775
$166$	0	0
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	13.0000	0.960988
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	15.0000	1.09691
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−25.0000	−1.80894	−0.904468	−	0.426541i	$-0.859732\pi$
$191$	−25.0000	−1.80894	−0.904468	+	0.426541i	$0.859732\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	16.0000	1.13995	0.569976	−	0.821661i	$-0.306952\pi$
$197$	16.0000	1.13995	0.569976	+	0.821661i	$0.306952\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	5.00000	0.350931
$204$	0	0
$205$	36.0000	2.51435
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	33.0000	2.25058
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	11.0000	0.743311
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	0	0
$255$	−9.00000	−0.563602
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−30.0000	−1.84289
$266$	0	0
$267$	18.0000	1.10158
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	22.0000	1.33640	0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000	1.33640	0.668202	+	0.743980i	$0.267064\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	20.0000	1.20605
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	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
$285$	−3.00000	−0.177705
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000	0.586210
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	11.0000	0.634029
$302$	0	0
$303$	−16.0000	−0.919176
$304$	0	0
$305$	−39.0000	−2.23313
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	−5.00000	−0.284440
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	−25.0000	−1.39973
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	3.00000	0.166924
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	3.00000	0.165900
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	14.0000	0.734809
$364$	0	0
$365$	−33.0000	−1.72730
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	3.00000	0.154919
$376$	0	0
$377$	5.00000	0.257513
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	0	0
$387$	−11.0000	−0.559161
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−11.0000	−0.554877
$394$	0	0
$395$	6.00000	0.301893
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	5.00000	0.247841
$408$	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
$411$	−3.00000	−0.147979
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	−54.0000	−2.65076
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	15.0000	0.732798	0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000	0.732798	0.366399	+	0.930458i	$0.380591\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	−13.0000	−0.629114
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	15.0000	0.719195
$436$	0	0
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	−54.0000	−2.55985
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−60.0000	−2.82529
$452$	0	0
$453$	3.00000	0.140952
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	−17.0000	−0.791769	−0.395884	−	0.918300i	$-0.629562\pi$
$461$	−17.0000	−0.791769	−0.395884	+	0.918300i	$0.629562\pi$
$462$	0	0
$463$	−5.00000	−0.232370	−0.116185	−	0.993228i	$-0.537067\pi$
$463$	−5.00000	−0.232370	−0.116185	+	0.993228i	$0.537067\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	−31.0000	−1.43451	−0.717254	−	0.696811i	$-0.754601\pi$
$467$	−31.0000	−1.43451	−0.717254	+	0.696811i	$0.754601\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	17.0000	0.783319
$472$	0	0
$473$	−55.0000	−2.52890
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	−27.0000	−1.23366	−0.616831	−	0.787096i	$-0.711584\pi$
$479$	−27.0000	−1.23366	−0.616831	+	0.787096i	$0.711584\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−30.0000	−1.36223
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	0	0
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	0	0
$495$	−15.0000	−0.674200
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	15.0000	0.670151
$502$	0	0
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	48.0000	2.13597
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	7.00000	0.310270	0.155135	−	0.987893i	$-0.450419\pi$
$509$	7.00000	0.310270	0.155135	+	0.987893i	$0.450419\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	15.0000	0.660979
$516$	0	0
$517$	−40.0000	−1.75920
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	24.0000	1.04546
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−30.0000	−1.29701
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	−9.00000	−0.385518
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	13.0000	0.554826
$550$	0	0
$551$	−5.00000	−0.213007
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	8.00000	0.338971	0.169485	−	0.985533i	$-0.445789\pi$
$557$	8.00000	0.338971	0.169485	+	0.985533i	$0.445789\pi$
$558$	0	0
$559$	11.0000	0.465250
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	−54.0000	−2.27180
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	−25.0000	−1.04439
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	8.00000	0.332469
$580$	0	0
$581$	−18.0000	−0.746766
$582$	0	0
$583$	50.0000	2.07079
$584$	0	0
$585$	3.00000	0.124035
$586$	0	0
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	16.0000	0.658152
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	0	0
$597$	17.0000	0.695764
$598$	0	0
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	−42.0000	−1.70754
$606$	0	0
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	0	0
$609$	5.00000	0.202610
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	0	0
$615$	36.0000	1.45166
$616$	0	0
$617$	15.0000	0.603877	0.301939	−	0.953327i	$-0.402366\pi$
$617$	15.0000	0.603877	0.301939	+	0.953327i	$0.402366\pi$
$618$	0	0
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−18.0000	−0.721155
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	5.00000	0.199681
$628$	0	0
$629$	3.00000	0.119618
$630$	0	0
$631$	−9.00000	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$632$	0	0
$633$	15.0000	0.596196
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	−49.0000	−1.93237	−0.966186	−	0.257847i	$-0.916987\pi$
$643$	−49.0000	−1.93237	−0.966186	+	0.257847i	$0.916987\pi$
$644$	0	0
$645$	33.0000	1.29937
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	45.0000	1.76099	0.880493	−	0.474059i	$-0.157212\pi$
$653$	45.0000	1.76099	0.880493	+	0.474059i	$0.157212\pi$
$654$	0	0
$655$	33.0000	1.28942
$656$	0	0
$657$	11.0000	0.429151
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	−3.00000	−0.116510
$664$	0	0
$665$	3.00000	0.116335
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	65.0000	2.50930
$672$	0	0
$673$	31.0000	1.19496	0.597481	−	0.801883i	$-0.296168\pi$
$673$	31.0000	1.19496	0.597481	+	0.801883i	$0.296168\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	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
$679$	−10.0000	−0.383765
$680$	0	0
$681$	10.0000	0.383201
$682$	0	0
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	0	0
$685$	9.00000	0.343872
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	0	0
$693$	−5.00000	−0.189934
$694$	0	0
$695$	−36.0000	−1.36556
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	−8.00000	−0.302588
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	1.00000	0.0377157
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	16.0000	0.601742
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	15.0000	0.560968
$716$	0	0
$717$	−20.0000	−0.746914
$718$	0	0
$719$	−10.0000	−0.372937	−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000	−0.372937	−0.186469	+	0.982461i	$0.559704\pi$
$720$	0	0
$721$	5.00000	0.186210
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−49.0000	−1.81731	−0.908655	−	0.417548i	$-0.862889\pi$
$727$	−49.0000	−1.81731	−0.908655	+	0.417548i	$0.862889\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.0000	−1.22055
$732$	0	0
$733$	44.0000	1.62518	0.812589	−	0.582838i	$-0.198058\pi$
$733$	44.0000	1.62518	0.812589	+	0.582838i	$0.198058\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	−1.00000	−0.0367359
$742$	0	0
$743$	10.0000	0.366864	0.183432	−	0.983032i	$-0.441279\pi$
$743$	10.0000	0.366864	0.183432	+	0.983032i	$0.441279\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	0	0
$747$	18.0000	0.658586
$748$	0	0
$749$	−10.0000	−0.365392
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	−15.0000	−0.546630
$754$	0	0
$755$	−9.00000	−0.327544
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	0	0
$759$	−5.00000	−0.181489
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	−3.00000	−0.108607
$764$	0	0
$765$	−9.00000	−0.325396
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	49.0000	1.76699	0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000	1.76699	0.883493	+	0.468445i	$0.155186\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	−25.0000	−0.899188	−0.449594	−	0.893233i	$-0.648431\pi$
$773$	−25.0000	−0.899188	−0.449594	+	0.893233i	$0.648431\pi$
$774$	0	0
$775$	32.0000	1.14947
$776$	0	0
$777$	−1.00000	−0.0358748
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−40.0000	−1.43131
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	−51.0000	−1.82027
$786$	0	0
$787$	17.0000	0.605985	0.302992	−	0.952993i	$-0.402014\pi$
$787$	17.0000	0.605985	0.302992	+	0.952993i	$0.402014\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−18.0000	−0.640006
$792$	0	0
$793$	−13.0000	−0.461644
$794$	0	0
$795$	−30.0000	−1.06399
$796$	0	0
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	55.0000	1.94091
$804$	0	0
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−26.0000	−0.915243
$808$	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
$811$	−51.0000	−1.79085	−0.895426	−	0.445210i	$-0.853129\pi$
$811$	−51.0000	−1.79085	−0.895426	+	0.445210i	$0.853129\pi$
$812$	0	0
$813$	22.0000	0.771574
$814$	0	0
$815$	30.0000	1.05085
$816$	0	0
$817$	−11.0000	−0.384841
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	20.0000	0.696311
$826$	0	0
$827$	45.0000	1.56480	0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000	1.56480	0.782402	+	0.622774i	$0.213994\pi$
$828$	0	0
$829$	53.0000	1.84077	0.920383	−	0.391018i	$-0.127877\pi$
$829$	53.0000	1.84077	0.920383	+	0.391018i	$0.127877\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	−45.0000	−1.55729
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	−22.0000	−0.757720
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	−1.00000	−0.0342796
$852$	0	0
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	10.0000	0.340404	0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000	0.340404	0.170202	+	0.985409i	$0.445558\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	−10.0000	−0.339227
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	3.00000	0.101073	0.0505363	−	0.998722i	$-0.483907\pi$
$881$	3.00000	0.101073	0.0505363	+	0.998722i	$0.483907\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	1.00000	0.0333890
$898$	0	0
$899$	−40.0000	−1.33407
$900$	0	0
$901$	30.0000	0.999445
$902$	0	0
$903$	11.0000	0.366057
$904$	0	0
$905$	−42.0000	−1.39613
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	−16.0000	−0.530687
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	90.0000	2.97857
$914$	0	0
$915$	−39.0000	−1.28930
$916$	0	0
$917$	11.0000	0.363252
$918$	0	0
$919$	18.0000	0.593765	0.296883	−	0.954914i	$-0.404053\pi$
$919$	18.0000	0.593765	0.296883	+	0.954914i	$0.404053\pi$
$920$	0	0
$921$	16.0000	0.527218
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	4.00000	0.131519
$926$	0	0
$927$	−5.00000	−0.164222
$928$	0	0
$929$	44.0000	1.44359	0.721797	−	0.692105i	$-0.243317\pi$
$929$	44.0000	1.44359	0.721797	+	0.692105i	$0.243317\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−30.0000	−0.982156
$934$	0	0
$935$	−45.0000	−1.47166
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	0	0
$945$	3.00000	0.0975900
$946$	0	0
$947$	−39.0000	−1.26733	−0.633665	−	0.773608i	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	0	0
$949$	−11.0000	−0.357075
$950$	0	0
$951$	−28.0000	−0.907962
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	75.0000	2.42694
$956$	0	0
$957$	−25.0000	−0.808135
$958$	0	0
$959$	3.00000	0.0968751
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	10.0000	0.322245
$964$	0	0
$965$	−24.0000	−0.772587
$966$	0	0
$967$	33.0000	1.06121	0.530604	−	0.847620i	$-0.321965\pi$
$967$	33.0000	1.06121	0.530604	+	0.847620i	$0.321965\pi$
$968$	0	0
$969$	3.00000	0.0963739
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	19.0000	0.607864	0.303932	−	0.952694i	$-0.401700\pi$
$977$	19.0000	0.607864	0.303932	+	0.952694i	$0.401700\pi$
$978$	0	0
$979$	90.0000	2.87641
$980$	0	0
$981$	3.00000	0.0957826
$982$	0	0
$983$	−51.0000	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$983$	−51.0000	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$984$	0	0
$985$	−48.0000	−1.52941
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	11.0000	0.349780
$990$	0	0
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	0	0
$993$	−26.0000	−0.825085
$994$	0	0
$995$	−51.0000	−1.61681
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.n.1.1		1
4.3	odd	2		2184.2.a.a.1.1	✓	1
12.11	even	2		6552.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.a.1.1	✓	1	4.3	odd	2
4368.2.a.n.1.1		1	1.1	even	1	trivial
6552.2.a.z.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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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