LMFDB - Embedded newform 3381.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} +3.00000 q^{13} +4.00000 q^{16} -2.00000 q^{17} -3.00000 q^{19} +1.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} +3.00000 q^{31} +2.00000 q^{33} -2.00000 q^{36} -3.00000 q^{37} -3.00000 q^{39} +3.00000 q^{43} +4.00000 q^{44} +10.0000 q^{47} -4.00000 q^{48} +2.00000 q^{51} -6.00000 q^{52} -4.00000 q^{53} +3.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} -8.00000 q^{64} +1.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -6.00000 q^{71} -1.00000 q^{73} +5.00000 q^{75} +6.00000 q^{76} -7.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -6.00000 q^{87} +14.0000 q^{89} -2.00000 q^{92} -3.00000 q^{93} +10.0000 q^{97} -2.00000 q^{99} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	2.00000	0.577350
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	−3.00000	−0.480384
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	−4.00000	−0.577350
$49$	0	0
$50$	0	0
$51$	2.00000	0.280056
$52$	−6.00000	−0.832050
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	6.00000	0.688247
$77$	0	0
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	−2.00000	−0.208514
$93$	−3.00000	−0.311086
$94$	0	0
$95$	0	0
$96$	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
$99$	−2.00000	−0.201008
$100$	10.0000	1.00000
$101$	−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
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	2.00000	0.192450
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	0	0
$116$	−12.0000	−1.11417
$117$	3.00000	0.277350
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	−6.00000	−0.538816
$125$	0	0
$126$	0	0
$127$	1.00000	0.0887357	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	1.00000	0.0887357	0.0443678	+	0.999015i	$0.485873\pi$
$128$	0	0
$129$	−3.00000	−0.264135
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−21.0000	−1.78120	−0.890598	−	0.454791i	$-0.849714\pi$
$139$	−21.0000	−1.78120	−0.890598	+	0.454791i	$0.849714\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	0	0
$143$	−6.00000	−0.501745
$144$	4.00000	0.333333
$145$	0	0
$146$	0	0
$147$	0	0
$148$	6.00000	0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−3.00000	−0.229416
$172$	−6.00000	−0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	−8.00000	−0.603023
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000	0.292509
$188$	−20.0000	−1.45865
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	8.00000	0.577350
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	12.0000	0.832050
$209$	6.00000	0.415029
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	8.00000	0.549442
$213$	6.00000	0.411113
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	−6.00000	−0.397360
$229$	−15.0000	−0.991228	−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000	−0.991228	−0.495614	+	0.868543i	$0.665057\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	7.00000	0.454699
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	−9.00000	−0.572656
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.0000	−0.856786
$268$	−2.00000	−0.122169
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	0	0
$275$	10.0000	0.603023
$276$	2.00000	0.120386
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	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
$283$	−15.0000	−0.891657	−0.445829	−	0.895118i	$-0.647091\pi$
$283$	−15.0000	−0.891657	−0.445829	+	0.895118i	$0.647091\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	3.00000	0.173494
$300$	−10.0000	−0.577350
$301$	0	0
$302$	0	0
$303$	16.0000	0.919176
$304$	−12.0000	−0.688247
$305$	0	0
$306$	0	0
$307$	−27.0000	−1.54097	−0.770486	−	0.637457i	$-0.779986\pi$
$307$	−27.0000	−1.54097	−0.770486	+	0.637457i	$0.779986\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	22.0000	1.24751	0.623753	−	0.781622i	$-0.285607\pi$
$311$	22.0000	1.24751	0.623753	+	0.781622i	$0.285607\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	0	0
$316$	14.0000	0.787562
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	6.00000	0.333849
$324$	−2.00000	−0.111111
$325$	−15.0000	−0.832050
$326$	0	0
$327$	9.00000	0.497701
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	8.00000	0.439057
$333$	−3.00000	−0.164399
$334$	0	0
$335$	0	0
$336$	0	0
$337$	27.0000	1.47078	0.735392	−	0.677642i	$-0.236998\pi$
$337$	27.0000	1.47078	0.735392	+	0.677642i	$0.236998\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	12.0000	0.643268
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	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
$355$	0	0
$356$	−28.0000	−1.48400
$357$	0	0
$358$	0	0
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−37.0000	−1.93138	−0.965692	−	0.259690i	$-0.916380\pi$
$367$	−37.0000	−1.93138	−0.965692	+	0.259690i	$0.916380\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	0	0
$372$	6.00000	0.311086
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.0000	0.927047
$378$	0	0
$379$	29.0000	1.48963	0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000	1.48963	0.744815	+	0.667271i	$0.232538\pi$
$380$	0	0
$381$	−1.00000	−0.0512316
$382$	0	0
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.00000	0.152499
$388$	−20.0000	−1.01535
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	4.00000	0.201008
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	0	0
$403$	9.00000	0.448322
$404$	32.0000	1.59206
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−37.0000	−1.82953	−0.914766	−	0.403984i	$-0.867625\pi$
$409$	−37.0000	−1.82953	−0.914766	+	0.403984i	$0.867625\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	−2.00000	−0.0985329
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	21.0000	1.02837
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	0	0
$423$	10.0000	0.486217
$424$	0	0
$425$	10.0000	0.485071
$426$	0	0
$427$	0	0
$428$	32.0000	1.54678
$429$	6.00000	0.289683
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−4.00000	−0.192450
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	36.0000	1.69330
$453$	12.0000	0.563809
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.0000	1.26301	0.631503	−	0.775373i	$-0.282438\pi$
$457$	27.0000	1.26301	0.631503	+	0.775373i	$0.282438\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	−29.0000	−1.34774	−0.673872	−	0.738848i	$-0.735370\pi$
$463$	−29.0000	−1.34774	−0.673872	+	0.738848i	$0.735370\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	0	0
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	15.0000	0.688247
$476$	0	0
$477$	−4.00000	−0.183147
$478$	0	0
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	0	0
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	12.0000	0.538816
$497$	0	0
$498$	0	0
$499$	−35.0000	−1.56682	−0.783408	−	0.621508i	$-0.786520\pi$
$499$	−35.0000	−1.56682	−0.783408	+	0.621508i	$0.786520\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.00000	0.177646
$508$	−2.00000	−0.0887357
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.00000	0.132453
$514$	0	0
$515$	0	0
$516$	6.00000	0.264135
$517$	−20.0000	−0.879599
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−21.0000	−0.918266	−0.459133	−	0.888368i	$-0.651840\pi$
$523$	−21.0000	−0.918266	−0.459133	+	0.888368i	$0.651840\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.00000	−0.261364
$528$	8.00000	0.348155
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	20.0000	0.863064
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	0	0
$543$	−5.00000	−0.214571
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−24.0000	−1.02523
$549$	2.00000	0.0853579
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	42.0000	1.78120
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	20.0000	0.842152
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−40.0000	−1.67689	−0.838444	−	0.544988i	$-0.816534\pi$
$569$	−40.0000	−1.67689	−0.838444	+	0.544988i	$0.816534\pi$
$570$	0	0
$571$	−17.0000	−0.711428	−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000	−0.711428	−0.355714	+	0.934595i	$0.615762\pi$
$572$	12.0000	0.501745
$573$	6.00000	0.250654
$574$	0	0
$575$	−5.00000	−0.208514
$576$	−8.00000	−0.333333
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	0	0
$579$	5.00000	0.207793
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	0	0
$586$	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
$589$	−9.00000	−0.370839
$590$	0	0
$591$	−4.00000	−0.164538
$592$	−12.0000	−0.493197
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	0	0
$596$	−20.0000	−0.819232
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	24.0000	0.976546
$605$	0	0
$606$	0	0
$607$	−39.0000	−1.58296	−0.791481	−	0.611194i	$-0.790689\pi$
$607$	−39.0000	−1.58296	−0.791481	+	0.611194i	$0.790689\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.0000	1.21367
$612$	4.00000	0.161690
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	0	0
$616$	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
$619$	7.00000	0.281354	0.140677	−	0.990056i	$-0.455072\pi$
$619$	7.00000	0.281354	0.140677	+	0.990056i	$0.455072\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0	0
$624$	−12.0000	−0.480384
$625$	25.0000	1.00000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	−4.00000	−0.159617
$629$	6.00000	0.239236
$630$	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
$633$	−12.0000	−0.476957
$634$	0	0
$635$	0	0
$636$	−8.00000	−0.317221
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	48.0000	1.89589	0.947943	−	0.318440i	$-0.103159\pi$
$641$	48.0000	1.89589	0.947943	+	0.318440i	$0.103159\pi$
$642$	0	0
$643$	41.0000	1.61688	0.808441	−	0.588577i	$-0.200312\pi$
$643$	41.0000	1.61688	0.808441	+	0.588577i	$0.200312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	40.0000	1.56652
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.00000	−0.0390137
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	0	0
$663$	6.00000	0.233021
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.00000	0.232321
$668$	32.0000	1.23812
$669$	16.0000	0.618596
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	7.00000	0.269830	0.134915	−	0.990857i	$-0.456924\pi$
$673$	7.00000	0.269830	0.134915	+	0.990857i	$0.456924\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	8.00000	0.307692
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.00000	0.0766402
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	0	0
$687$	15.0000	0.572286
$688$	12.0000	0.457496
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−43.0000	−1.63580	−0.817899	−	0.575362i	$-0.804861\pi$
$691$	−43.0000	−1.63580	−0.817899	+	0.575362i	$0.804861\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−2.00000	−0.0756469
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	9.00000	0.339441
$704$	16.0000	0.603023
$705$	0	0
$706$	0	0
$707$	0	0
$708$	12.0000	0.450988
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	−7.00000	−0.262521
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	0	0
$716$	40.0000	1.49487
$717$	24.0000	0.896296
$718$	0	0
$719$	−26.0000	−0.969636	−0.484818	−	0.874615i	$-0.661114\pi$
$719$	−26.0000	−0.969636	−0.484818	+	0.874615i	$0.661114\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−14.0000	−0.520666
$724$	−10.0000	−0.371647
$725$	−30.0000	−1.11417
$726$	0	0
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.00000	−0.221918
$732$	4.00000	0.147844
$733$	−35.0000	−1.29275	−0.646377	−	0.763018i	$-0.723717\pi$
$733$	−35.0000	−1.29275	−0.646377	+	0.763018i	$0.723717\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	29.0000	1.06678	0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000	1.06678	0.533391	+	0.845869i	$0.320917\pi$
$740$	0	0
$741$	9.00000	0.330623
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4.00000	−0.146352
$748$	−8.00000	−0.292509
$749$	0	0
$750$	0	0
$751$	11.0000	0.401396	0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000	0.401396	0.200698	+	0.979653i	$0.435679\pi$
$752$	40.0000	1.45865
$753$	−26.0000	−0.947493
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−46.0000	−1.67190	−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000	−1.67190	−0.835949	+	0.548807i	$0.815082\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	0	0
$767$	18.0000	0.649942
$768$	−16.0000	−0.577350
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	0	0
$771$	−20.0000	−0.720282
$772$	10.0000	0.359908
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	0	0
$775$	−15.0000	−0.538816
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	−8.00000	−0.284988
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	0	0
$799$	−20.0000	−0.707549
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	2.00000	0.0705785
$804$	2.00000	0.0705346
$805$	0	0
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	32.0000	1.12506	0.562530	−	0.826777i	$-0.309828\pi$
$809$	32.0000	1.12506	0.562530	+	0.826777i	$0.309828\pi$
$810$	0	0
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	0	0
$816$	8.00000	0.280056
$817$	−9.00000	−0.314870
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	40.0000	1.39431	0.697156	−	0.716919i	$-0.254448\pi$
$823$	40.0000	1.39431	0.697156	+	0.716919i	$0.254448\pi$
$824$	0	0
$825$	−10.0000	−0.348155
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	−2.00000	−0.0695048
$829$	−29.0000	−1.00721	−0.503606	−	0.863934i	$-0.667994\pi$
$829$	−29.0000	−1.00721	−0.503606	+	0.863934i	$0.667994\pi$
$830$	0	0
$831$	7.00000	0.242827
$832$	−24.0000	−0.832050
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	−3.00000	−0.103695
$838$	0	0
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	22.0000	0.757720
$844$	−24.0000	−0.826114
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−16.0000	−0.549442
$849$	15.0000	0.514799
$850$	0	0
$851$	−3.00000	−0.102839
$852$	−12.0000	−0.411113
$853$	17.0000	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$853$	17.0000	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	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
$861$	0	0
$862$	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
$865$	0	0
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	14.0000	0.474917
$870$	0	0
$871$	3.00000	0.101651
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	−9.00000	−0.302874	−0.151437	−	0.988467i	$-0.548390\pi$
$883$	−9.00000	−0.302874	−0.151437	+	0.988467i	$0.548390\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	0	0
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	32.0000	1.07144
$893$	−30.0000	−1.00391
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.00000	−0.100167
$898$	0	0
$899$	18.0000	0.600334
$900$	10.0000	0.333333
$901$	8.00000	0.266519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25.0000	0.830111	0.415056	−	0.909796i	$-0.363762\pi$
$907$	25.0000	0.830111	0.415056	+	0.909796i	$0.363762\pi$
$908$	4.00000	0.132745
$909$	−16.0000	−0.530687
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	12.0000	0.397360
$913$	8.00000	0.264761
$914$	0	0
$915$	0	0
$916$	30.0000	0.991228
$917$	0	0
$918$	0	0
$919$	−39.0000	−1.28649	−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000	−1.28649	−0.643246	+	0.765660i	$0.722413\pi$
$920$	0	0
$921$	27.0000	0.889680
$922$	0	0
$923$	−18.0000	−0.592477
$924$	0	0
$925$	15.0000	0.493197
$926$	0	0
$927$	1.00000	0.0328443
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	0	0
$932$	−4.00000	−0.131024
$933$	−22.0000	−0.720248
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−25.0000	−0.816714	−0.408357	−	0.912822i	$-0.633898\pi$
$937$	−25.0000	−0.816714	−0.408357	+	0.912822i	$0.633898\pi$
$938$	0	0
$939$	1.00000	0.0326338
$940$	0	0
$941$	36.0000	1.17357	0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000	1.17357	0.586783	+	0.809744i	$0.300394\pi$
$942$	0	0
$943$	0	0
$944$	24.0000	0.781133
$945$	0	0
$946$	0	0
$947$	−10.0000	−0.324956	−0.162478	−	0.986712i	$-0.551949\pi$
$947$	−10.0000	−0.324956	−0.162478	+	0.986712i	$0.551949\pi$
$948$	−14.0000	−0.454699
$949$	−3.00000	−0.0973841
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	0	0
$955$	0	0
$956$	48.0000	1.55243
$957$	12.0000	0.387905
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−16.0000	−0.515593
$964$	−28.0000	−0.901819
$965$	0	0
$966$	0	0
$967$	19.0000	0.610999	0.305499	−	0.952192i	$-0.401177\pi$
$967$	19.0000	0.610999	0.305499	+	0.952192i	$0.401177\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−30.0000	−0.962746	−0.481373	−	0.876516i	$-0.659862\pi$
$971$	−30.0000	−0.962746	−0.481373	+	0.876516i	$0.659862\pi$
$972$	2.00000	0.0641500
$973$	0	0
$974$	0	0
$975$	15.0000	0.480384
$976$	8.00000	0.256074
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	0	0
$979$	−28.0000	−0.894884
$980$	0	0
$981$	−9.00000	−0.287348
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	18.0000	0.572656
$989$	3.00000	0.0953945
$990$	0	0
$991$	9.00000	0.285894	0.142947	−	0.989730i	$-0.454342\pi$
$991$	9.00000	0.285894	0.142947	+	0.989730i	$0.454342\pi$
$992$	0	0
$993$	13.0000	0.412543
$994$	0	0
$995$	0	0
$996$	−8.00000	−0.253490
$997$	1.00000	0.0316703	0.0158352	−	0.999875i	$-0.494959\pi$
$997$	1.00000	0.0316703	0.0158352	+	0.999875i	$0.494959\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.e.1.1		1
7.3	odd	6		483.2.i.a.415.1	yes	2
7.5	odd	6		483.2.i.a.277.1	✓	2
7.6	odd	2		3381.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.a.277.1	✓	2	7.5	odd	6
483.2.i.a.415.1	yes	2	7.3	odd	6
3381.2.a.e.1.1		1	1.1	even	1	trivial
3381.2.a.h.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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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