LMFDB - Embedded newform 142.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [142,2,Mod(1,142)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(142, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("142.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 142 = 2 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	142.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:	$1.13387570870$
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	$\chi$	$=$	142.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{18} -1.00000 q^{19} -1.00000 q^{21} +3.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +5.00000 q^{31} +1.00000 q^{32} -2.00000 q^{36} -4.00000 q^{37} -1.00000 q^{38} -1.00000 q^{39} -1.00000 q^{42} -1.00000 q^{43} +3.00000 q^{46} +9.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -5.00000 q^{50} -1.00000 q^{52} +6.00000 q^{53} -5.00000 q^{54} -1.00000 q^{56} -1.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} +5.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +8.00000 q^{67} +3.00000 q^{69} -1.00000 q^{71} -2.00000 q^{72} -1.00000 q^{73} -4.00000 q^{74} -5.00000 q^{75} -1.00000 q^{76} -1.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -1.00000 q^{84} -1.00000 q^{86} -3.00000 q^{89} +1.00000 q^{91} +3.00000 q^{92} +5.00000 q^{93} +9.00000 q^{94} +1.00000 q^{96} -16.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−2.00000	−0.471405
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	−1.00000	−0.196116
$27$	−5.00000	−0.962250
$28$	−1.00000	−0.188982
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−1.00000	−0.162221
$39$	−1.00000	−0.160128
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−1.00000	−0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	0	0
$46$	3.00000	0.442326
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−5.00000	−0.707107
$51$	0	0
$52$	−1.00000	−0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−1.00000	−0.132453
$58$	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	$0.459132\pi$
$62$	5.00000	0.635001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	−2.00000	−0.235702
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	−4.00000	−0.464991
$75$	−5.00000	−0.577350
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−1.00000	−0.113228
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−1.00000	−0.107833
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	3.00000	0.312772
$93$	5.00000	0.518476
$94$	9.00000	0.928279
$95$	0	0
$96$	1.00000	0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	−5.00000	−0.500000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	−5.00000	−0.481125
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	2.00000	0.181071
$123$	0	0
$124$	5.00000	0.449013
$125$	0	0
$126$	2.00000	0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−3.00000	−0.262111	−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000	−0.262111	−0.131056	+	0.991375i	$0.541837\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	8.00000	0.691095
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	3.00000	0.255377
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	−1.00000	−0.0839181
$143$	0	0
$144$	−2.00000	−0.166667
$145$	0	0
$146$	−1.00000	−0.0827606
$147$	−6.00000	−0.494872
$148$	−4.00000	−0.328798
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	−5.00000	−0.408248
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000	0.636446
$159$	6.00000	0.475831
$160$	0	0
$161$	−3.00000	−0.236433
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	−1.00000	−0.0771517
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	−1.00000	−0.0762493
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	5.00000	0.377964
$176$	0	0
$177$	6.00000	0.450988
$178$	−3.00000	−0.224860
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	1.00000	0.0741249
$183$	2.00000	0.147844
$184$	3.00000	0.221163
$185$	0	0
$186$	5.00000	0.366618
$187$	0	0
$188$	9.00000	0.656392
$189$	5.00000	0.363696
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	1.00000	0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	−6.00000	−0.428571
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−5.00000	−0.353553
$201$	8.00000	0.564276
$202$	12.0000	0.844317
$203$	0	0
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	−6.00000	−0.417029
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	6.00000	0.412082
$213$	−1.00000	−0.0685189
$214$	−3.00000	−0.205076
$215$	0	0
$216$	−5.00000	−0.340207
$217$	−5.00000	−0.339422
$218$	−16.0000	−1.08366
$219$	−1.00000	−0.0675737
$220$	0	0
$221$	0	0
$222$	−4.00000	−0.268462
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	−1.00000	−0.0668153
$225$	10.0000	0.666667
$226$	6.00000	0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	−1.00000	−0.0662266
$229$	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
$231$	0	0
$232$	0	0
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	6.00000	0.390567
$237$	8.00000	0.519656
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−11.0000	−0.707107
$243$	16.0000	1.02640
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	1.00000	0.0636285
$248$	5.00000	0.317500
$249$	12.0000	0.760469
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−1.00000	−0.0622573
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	−3.00000	−0.185341
$263$	−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
$265$	0	0
$266$	1.00000	0.0613139
$267$	−3.00000	−0.183597
$268$	8.00000	0.488678
$269$	−27.0000	−1.64622	−0.823110	−	0.567883i	$-0.807763\pi$
$269$	−27.0000	−1.64622	−0.823110	+	0.567883i	$0.807763\pi$
$270$	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
$273$	1.00000	0.0605228
$274$	−12.0000	−0.724947
$275$	0	0
$276$	3.00000	0.180579
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−16.0000	−0.959616
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	9.00000	0.535942
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−1.00000	−0.0593391
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−16.0000	−0.937937
$292$	−1.00000	−0.0585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−15.0000	−0.868927
$299$	−3.00000	−0.173494
$300$	−5.00000	−0.288675
$301$	1.00000	0.0576390
$302$	−16.0000	−0.920697
$303$	12.0000	0.689382
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	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
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	−1.00000	−0.0566139
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	0	0
$320$	0	0
$321$	−3.00000	−0.167444
$322$	−3.00000	−0.167183
$323$	0	0
$324$	1.00000	0.0555556
$325$	5.00000	0.277350
$326$	−4.00000	−0.221540
$327$	−16.0000	−0.884802
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	12.0000	0.658586
$333$	8.00000	0.438397
$334$	−18.0000	−0.984916
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	−12.0000	−0.652714
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	2.00000	0.108148
$343$	13.0000	0.701934
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	21.0000	1.12897
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−13.0000	−0.695874	−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000	−0.695874	−0.347937	+	0.937518i	$0.613118\pi$
$350$	5.00000	0.267261
$351$	5.00000	0.266880
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	−3.00000	−0.159000
$357$	0	0
$358$	−9.00000	−0.475665
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	11.0000	0.578147
$363$	−11.0000	−0.577350
$364$	1.00000	0.0524142
$365$	0	0
$366$	2.00000	0.104542
$367$	38.0000	1.98358	0.991792	−	0.127862i	$-0.0408116\pi$
$367$	38.0000	1.98358	0.991792	+	0.127862i	$0.0408116\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	5.00000	0.259238
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	9.00000	0.464140
$377$	0	0
$378$	5.00000	0.257172
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	6.00000	0.306987
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−4.00000	−0.203595
$387$	2.00000	0.101666
$388$	−16.0000	−0.812277
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	0	0
$392$	−6.00000	−0.303046
$393$	−3.00000	−0.151330
$394$	21.0000	1.05796
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	8.00000	0.401004
$399$	1.00000	0.0500626
$400$	−5.00000	−0.250000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	8.00000	0.399004
$403$	−5.00000	−0.249068
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	0	0
$408$	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
$411$	−12.0000	−0.591916
$412$	8.00000	0.394132
$413$	−6.00000	−0.295241
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	−16.0000	−0.783523
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	14.0000	0.681509
$423$	−18.0000	−0.875190
$424$	6.00000	0.291386
$425$	0	0
$426$	−1.00000	−0.0484502
$427$	−2.00000	−0.0967868
$428$	−3.00000	−0.145010
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−5.00000	−0.240563
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	−16.0000	−0.766261
$437$	−3.00000	−0.143509
$438$	−1.00000	−0.0477818
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	8.00000	0.378811
$447$	−15.0000	−0.709476
$448$	−1.00000	−0.0472456
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	10.0000	0.471405
$451$	0	0
$452$	6.00000	0.282216
$453$	−16.0000	−0.751746
$454$	0	0
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	0	0
$465$	0	0
$466$	15.0000	0.694862
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	2.00000	0.0924500
$469$	−8.00000	−0.369406
$470$	0	0
$471$	14.0000	0.645086
$472$	6.00000	0.276172
$473$	0	0
$474$	8.00000	0.367452
$475$	5.00000	0.229416
$476$	0	0
$477$	−12.0000	−0.549442
$478$	−15.0000	−0.686084
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	14.0000	0.637683
$483$	−3.00000	−0.136505
$484$	−11.0000	−0.500000
$485$	0	0
$486$	16.0000	0.725775
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	2.00000	0.0905357
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	0	0
$494$	1.00000	0.0449921
$495$	0	0
$496$	5.00000	0.224507
$497$	1.00000	0.0448561
$498$	12.0000	0.537733
$499$	41.0000	1.83541	0.917706	−	0.397260i	$-0.130039\pi$
$499$	41.0000	1.83541	0.917706	+	0.397260i	$0.130039\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	−15.0000	−0.669483
$503$	−18.0000	−0.802580	−0.401290	−	0.915951i	$-0.631438\pi$
$503$	−18.0000	−0.802580	−0.401290	+	0.915951i	$0.631438\pi$
$504$	2.00000	0.0890871
$505$	0	0
$506$	0	0
$507$	−12.0000	−0.532939
$508$	−16.0000	−0.709885
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	1.00000	0.0441942
$513$	5.00000	0.220755
$514$	6.00000	0.264649
$515$	0	0
$516$	−1.00000	−0.0440225
$517$	0	0
$518$	4.00000	0.175750
$519$	21.0000	0.921798
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−10.0000	−0.437269	−0.218635	−	0.975807i	$-0.570160\pi$
$523$	−10.0000	−0.437269	−0.218635	+	0.975807i	$0.570160\pi$
$524$	−3.00000	−0.131056
$525$	5.00000	0.218218
$526$	−12.0000	−0.523225
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−12.0000	−0.520756
$532$	1.00000	0.0433555
$533$	0	0
$534$	−3.00000	−0.129823
$535$	0	0
$536$	8.00000	0.345547
$537$	−9.00000	−0.388379
$538$	−27.0000	−1.16405
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−10.0000	−0.429537
$543$	11.0000	0.472055
$544$	0	0
$545$	0	0
$546$	1.00000	0.0427960
$547$	−31.0000	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$547$	−31.0000	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$548$	−12.0000	−0.512615
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	3.00000	0.127688
$553$	−8.00000	−0.340195
$554$	8.00000	0.339887
$555$	0	0
$556$	−16.0000	−0.678551
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	−10.0000	−0.423334
$559$	1.00000	0.0422955
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	9.00000	0.378968
$565$	0	0
$566$	14.0000	0.588464
$567$	−1.00000	−0.0419961
$568$	−1.00000	−0.0419591
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	−15.0000	−0.625543
$576$	−2.00000	−0.0833333
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−17.0000	−0.707107
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−12.0000	−0.497844
$582$	−16.0000	−0.663221
$583$	0	0
$584$	−1.00000	−0.0413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−6.00000	−0.247436
$589$	−5.00000	−0.206021
$590$	0	0
$591$	21.0000	0.863825
$592$	−4.00000	−0.164399
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	0	0
$596$	−15.0000	−0.614424
$597$	8.00000	0.327418
$598$	−3.00000	−0.122679
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	−5.00000	−0.204124
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	1.00000	0.0407570
$603$	−16.0000	−0.651570
$604$	−16.0000	−0.651031
$605$	0	0
$606$	12.0000	0.487467
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	0	0
$617$	39.0000	1.57008	0.785040	−	0.619445i	$-0.212642\pi$
$617$	39.0000	1.57008	0.785040	+	0.619445i	$0.212642\pi$
$618$	8.00000	0.321807
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	−12.0000	−0.481156
$623$	3.00000	0.120192
$624$	−1.00000	−0.0400320
$625$	25.0000	1.00000
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	8.00000	0.318223
$633$	14.0000	0.556450
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	6.00000	0.237729
$638$	0	0
$639$	2.00000	0.0791188
$640$	0	0
$641$	45.0000	1.77739	0.888697	−	0.458496i	$-0.151612\pi$
$641$	45.0000	1.77739	0.888697	+	0.458496i	$0.151612\pi$
$642$	−3.00000	−0.118401
$643$	−31.0000	−1.22252	−0.611260	−	0.791430i	$-0.709337\pi$
$643$	−31.0000	−1.22252	−0.611260	+	0.791430i	$0.709337\pi$
$644$	−3.00000	−0.118217
$645$	0	0
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	5.00000	0.196116
$651$	−5.00000	−0.195965
$652$	−4.00000	−0.156652
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−16.0000	−0.625650
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	−9.00000	−0.350857
$659$	27.0000	1.05177	0.525885	−	0.850555i	$-0.323734\pi$
$659$	27.0000	1.05177	0.525885	+	0.850555i	$0.323734\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	8.00000	0.309994
$667$	0	0
$668$	−18.0000	−0.696441
$669$	8.00000	0.309298
$670$	0	0
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	8.00000	0.308148
$675$	25.0000	0.962250
$676$	−12.0000	−0.461538
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	6.00000	0.230429
$679$	16.0000	0.614024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	13.0000	0.496342
$687$	14.0000	0.534133
$688$	−1.00000	−0.0381246
$689$	−6.00000	−0.228582
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	21.0000	0.798300
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−13.0000	−0.492057
$699$	15.0000	0.567352
$700$	5.00000	0.188982
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	5.00000	0.188713
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	30.0000	1.12906
$707$	−12.0000	−0.451306
$708$	6.00000	0.225494
$709$	−13.0000	−0.488225	−0.244113	−	0.969747i	$-0.578497\pi$
$709$	−13.0000	−0.488225	−0.244113	+	0.969747i	$0.578497\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	−3.00000	−0.112430
$713$	15.0000	0.561754
$714$	0	0
$715$	0	0
$716$	−9.00000	−0.336346
$717$	−15.0000	−0.560185
$718$	−24.0000	−0.895672
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	−18.0000	−0.669891
$723$	14.0000	0.520666
$724$	11.0000	0.408812
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	1.00000	0.0370625
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	2.00000	0.0739221
$733$	−49.0000	−1.80986	−0.904928	−	0.425564i	$-0.860076\pi$
$733$	−49.0000	−1.80986	−0.904928	+	0.425564i	$0.860076\pi$
$734$	38.0000	1.40261
$735$	0	0
$736$	3.00000	0.110581
$737$	0	0
$738$	0	0
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	0	0
$741$	1.00000	0.0367359
$742$	−6.00000	−0.220267
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	5.00000	0.183309
$745$	0	0
$746$	26.0000	0.951928
$747$	−24.0000	−0.878114
$748$	0	0
$749$	3.00000	0.109618
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	9.00000	0.328196
$753$	−15.0000	−0.546630
$754$	0	0
$755$	0	0
$756$	5.00000	0.181848
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	−16.0000	−0.579619
$763$	16.0000	0.579239
$764$	6.00000	0.217072
$765$	0	0
$766$	21.0000	0.758761
$767$	−6.00000	−0.216647
$768$	1.00000	0.0360844
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−4.00000	−0.143963
$773$	−33.0000	−1.18693	−0.593464	−	0.804861i	$-0.702240\pi$
$773$	−33.0000	−1.18693	−0.593464	+	0.804861i	$0.702240\pi$
$774$	2.00000	0.0718885
$775$	−25.0000	−0.898027
$776$	−16.0000	−0.574367
$777$	4.00000	0.143499
$778$	21.0000	0.752886
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−3.00000	−0.107006
$787$	5.00000	0.178231	0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000	0.178231	0.0891154	+	0.996021i	$0.471596\pi$
$788$	21.0000	0.748094
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	2.00000	0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	1.00000	0.0353996
$799$	0	0
$800$	−5.00000	−0.176777
$801$	6.00000	0.212000
$802$	−12.0000	−0.423735
$803$	0	0
$804$	8.00000	0.282138
$805$	0	0
$806$	−5.00000	−0.176117
$807$	−27.0000	−0.950445
$808$	12.0000	0.422159
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.00000	0.0349856
$818$	−7.00000	−0.244749
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	−12.0000	−0.418548
$823$	5.00000	0.174289	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	5.00000	0.174289	0.0871445	+	0.996196i	$0.472226\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−30.0000	−1.04320	−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000	−1.04320	−0.521601	+	0.853189i	$0.674665\pi$
$828$	−6.00000	−0.208514
$829$	−40.0000	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	−16.0000	−0.554035
$835$	0	0
$836$	0	0
$837$	−25.0000	−0.864126
$838$	36.0000	1.24360
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	11.0000	0.379085
$843$	−30.0000	−1.03325
$844$	14.0000	0.481900
$845$	0	0
$846$	−18.0000	−0.618853
$847$	11.0000	0.377964
$848$	6.00000	0.206041
$849$	14.0000	0.480479
$850$	0	0
$851$	−12.0000	−0.411355
$852$	−1.00000	−0.0342594
$853$	−40.0000	−1.36957	−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000	−1.36957	−0.684787	+	0.728743i	$0.740105\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−3.00000	−0.102538
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−3.00000	−0.102121	−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000	−0.102121	−0.0510606	+	0.998696i	$0.516260\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−34.0000	−1.15537
$867$	−17.0000	−0.577350
$868$	−5.00000	−0.169711
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−16.0000	−0.541828
$873$	32.0000	1.08304
$874$	−3.00000	−0.101477
$875$	0	0
$876$	−1.00000	−0.0337869
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	8.00000	0.269987
$879$	−18.0000	−0.607125
$880$	0	0
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	12.0000	0.404061
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	0	0
$885$	0	0
$886$	18.0000	0.604722
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−4.00000	−0.134231
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	8.00000	0.267860
$893$	−9.00000	−0.301174
$894$	−15.0000	−0.501675
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−3.00000	−0.100167
$898$	−12.0000	−0.400445
$899$	0	0
$900$	10.0000	0.333333
$901$	0	0
$902$	0	0
$903$	1.00000	0.0332779
$904$	6.00000	0.199557
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	0	0
$909$	−24.0000	−0.796030
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	−1.00000	−0.0331133
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	14.0000	0.462573
$917$	3.00000	0.0990687
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	−21.0000	−0.691598
$923$	1.00000	0.0329154
$924$	0	0
$925$	20.0000	0.657596
$926$	−22.0000	−0.722965
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	15.0000	0.491341
$933$	−12.0000	−0.392862
$934$	30.0000	0.981630
$935$	0	0
$936$	2.00000	0.0653720
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	−8.00000	−0.261209
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	14.0000	0.456145
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	8.00000	0.259828
$949$	1.00000	0.0324614
$950$	5.00000	0.162221
$951$	18.0000	0.583690
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−15.0000	−0.485135
$957$	0	0
$958$	15.0000	0.484628
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	4.00000	0.128965
$963$	6.00000	0.193347
$964$	14.0000	0.450910
$965$	0	0
$966$	−3.00000	−0.0965234
$967$	−13.0000	−0.418052	−0.209026	−	0.977910i	$-0.567029\pi$
$967$	−13.0000	−0.418052	−0.209026	+	0.977910i	$0.567029\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	0	0
$971$	−33.0000	−1.05902	−0.529510	−	0.848304i	$-0.677624\pi$
$971$	−33.0000	−1.05902	−0.529510	+	0.848304i	$0.677624\pi$
$972$	16.0000	0.513200
$973$	16.0000	0.512936
$974$	8.00000	0.256337
$975$	5.00000	0.160128
$976$	2.00000	0.0640184
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	−4.00000	−0.127906
$979$	0	0
$980$	0	0
$981$	32.0000	1.02168
$982$	−24.0000	−0.765871
$983$	−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
$987$	−9.00000	−0.286473
$988$	1.00000	0.0318142
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	29.0000	0.921215	0.460608	−	0.887604i	$-0.347632\pi$
$991$	29.0000	0.921215	0.460608	+	0.887604i	$0.347632\pi$
$992$	5.00000	0.158750
$993$	20.0000	0.634681
$994$	1.00000	0.0317181
$995$	0	0
$996$	12.0000	0.380235
$997$	44.0000	1.39349	0.696747	−	0.717317i	$-0.254630\pi$
$997$	44.0000	1.39349	0.696747	+	0.717317i	$0.254630\pi$
$998$	41.0000	1.29783
$999$	20.0000	0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	142.2.a.e.1.1	✓	1
3.2	odd	2		1278.2.a.b.1.1		1
4.3	odd	2		1136.2.a.b.1.1		1
5.4	even	2		3550.2.a.d.1.1		1
7.6	odd	2		6958.2.a.j.1.1		1
8.3	odd	2		4544.2.a.k.1.1		1
8.5	even	2		4544.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
142.2.a.e.1.1	✓	1	1.1	even	1	trivial
1136.2.a.b.1.1		1	4.3	odd	2
1278.2.a.b.1.1		1	3.2	odd	2
3550.2.a.d.1.1		1	5.4	even	2
4544.2.a.f.1.1		1	8.5	even	2
4544.2.a.k.1.1		1	8.3	odd	2
6958.2.a.j.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:	\( N \)	\(=\)	\( 142 = 2 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	142.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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