LMFDB - Embedded newform 8002.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
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	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -3.00000 q^{9} -2.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} -3.00000 q^{18} +1.00000 q^{19} -2.00000 q^{22} +3.00000 q^{23} -5.00000 q^{25} +4.00000 q^{26} -4.00000 q^{29} +1.00000 q^{32} -3.00000 q^{36} +6.00000 q^{37} +1.00000 q^{38} -8.00000 q^{41} -12.0000 q^{43} -2.00000 q^{44} +3.00000 q^{46} -10.0000 q^{47} -7.00000 q^{49} -5.00000 q^{50} +4.00000 q^{52} -13.0000 q^{53} -4.00000 q^{58} +7.00000 q^{59} +12.0000 q^{61} +1.00000 q^{64} -16.0000 q^{67} +8.00000 q^{71} -3.00000 q^{72} +4.00000 q^{73} +6.00000 q^{74} +1.00000 q^{76} +10.0000 q^{79} +9.00000 q^{81} -8.00000 q^{82} -4.00000 q^{83} -12.0000 q^{86} -2.00000 q^{88} +15.0000 q^{89} +3.00000 q^{92} -10.0000 q^{94} +10.0000 q^{97} -7.00000 q^{98} +6.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−3.00000	−0.707107
$19$	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
$21$	0	0
$22$	−2.00000	−0.426401
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	4.00000	0.784465
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	−3.00000	−0.500000
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	−8.00000	−1.24939	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000	−1.24939	−0.624695	+	0.780869i	$0.714777\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	3.00000	0.442326
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	−5.00000	−0.707107
$51$	0	0
$52$	4.00000	0.554700
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−4.00000	−0.525226
$59$	7.00000	0.911322	0.455661	−	0.890153i	$-0.349403\pi$
$59$	7.00000	0.911322	0.455661	+	0.890153i	$0.349403\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−3.00000	−0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	−8.00000	−0.883452
$83$	−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
$86$	−12.0000	−1.29399
$87$	0	0
$88$	−2.00000	−0.213201
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	0	0
$92$	3.00000	0.312772
$93$	0	0
$94$	−10.0000	−1.03142
$95$	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	$0.330506\pi$
$98$	−7.00000	−0.707107
$99$	6.00000	0.603023
$100$	−5.00000	−0.500000
$101$	7.00000	0.696526	0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000	0.696526	0.348263	+	0.937397i	$0.386772\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	−13.0000	−1.26267
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−4.00000	−0.371391
$117$	−12.0000	−1.10940
$118$	7.00000	0.644402
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	12.0000	1.08643
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−16.0000	−1.38219
$135$	0	0
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	−8.00000	−0.668994
$144$	−3.00000	−0.250000
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	6.00000	0.493197
$149$	−16.0000	−1.31077	−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000	−1.31077	−0.655386	+	0.755295i	$0.727494\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9.00000	0.707107
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−3.00000	−0.229416
$172$	−12.0000	−0.914991
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	15.0000	1.12430
$179$	−26.0000	−1.94333	−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000	−1.94333	−0.971666	+	0.236360i	$0.924046\pi$
$180$	0	0
$181$	−19.0000	−1.41226	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−19.0000	−1.41226	−0.706129	+	0.708083i	$0.749560\pi$
$182$	0	0
$183$	0	0
$184$	3.00000	0.221163
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−10.0000	−0.729325
$189$	0	0
$190$	0	0
$191$	−27.0000	−1.95365	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−27.0000	−1.95365	−0.976826	+	0.214036i	$0.931339\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−7.00000	−0.500000
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	6.00000	0.426401
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	−5.00000	−0.353553
$201$	0	0
$202$	7.00000	0.492518
$203$	0	0
$204$	0	0
$205$	0	0
$206$	13.0000	0.905753
$207$	−9.00000	−0.625543
$208$	4.00000	0.277350
$209$	−2.00000	−0.138343
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	−13.0000	−0.892844
$213$	0	0
$214$	11.0000	0.751945
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−18.0000	−1.21911
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	15.0000	1.00000
$226$	−6.00000	−0.399114
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	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
$231$	0	0
$232$	−4.00000	−0.262613
$233$	−16.0000	−1.04819	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000	−1.04819	−0.524097	+	0.851658i	$0.675597\pi$
$234$	−12.0000	−0.784465
$235$	0	0
$236$	7.00000	0.455661
$237$	0	0
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	12.0000	0.768221
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	−18.0000	−1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.0000	1.93373	0.966863	−	0.255294i	$-0.0821723\pi$
$257$	31.0000	1.93373	0.966863	+	0.255294i	$0.0821723\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	12.0000	0.742781
$262$	−12.0000	−0.741362
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−16.0000	−0.977356
$269$	32.0000	1.95107	0.975537	−	0.219834i	$-0.0705517\pi$
$269$	32.0000	1.95107	0.975537	+	0.219834i	$0.0705517\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	−13.0000	−0.785359
$275$	10.0000	0.603023
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	−3.00000	−0.176777
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	4.00000	0.234082
$293$	−15.0000	−0.876309	−0.438155	−	0.898900i	$-0.644368\pi$
$293$	−15.0000	−0.876309	−0.438155	+	0.898900i	$0.644368\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	−16.0000	−0.926855
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	−9.00000	−0.517892
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	23.0000	1.31268	0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000	1.31268	0.656340	+	0.754466i	$0.272104\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	10.0000	0.562544
$317$	31.0000	1.74113	0.870567	−	0.492050i	$-0.163752\pi$
$317$	31.0000	1.74113	0.870567	+	0.492050i	$0.163752\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	9.00000	0.500000
$325$	−20.0000	−1.10940
$326$	8.00000	0.443079
$327$	0	0
$328$	−8.00000	−0.441726
$329$	0	0
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	−4.00000	−0.219529
$333$	−18.0000	−0.986394
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−3.00000	−0.162221
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−24.0000	−1.29025
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	0	0
$356$	15.0000	0.794998
$357$	0	0
$358$	−26.0000	−1.37414
$359$	−33.0000	−1.74167	−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000	−1.74167	−0.870837	+	0.491572i	$0.836422\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−19.0000	−0.998618
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	3.00000	0.156386
$369$	24.0000	1.24939
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	−10.0000	−0.515711
$377$	−16.0000	−0.824042
$378$	0	0
$379$	23.0000	1.18143	0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000	1.18143	0.590715	+	0.806880i	$0.298846\pi$
$380$	0	0
$381$	0	0
$382$	−27.0000	−1.38144
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	0	0
$386$	12.0000	0.610784
$387$	36.0000	1.82998
$388$	10.0000	0.507673
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	0	0
$392$	−7.00000	−0.353553
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	6.00000	0.301511
$397$	−36.0000	−1.80679	−0.903394	−	0.428811i	$-0.858933\pi$
$397$	−36.0000	−1.80679	−0.903394	+	0.428811i	$0.858933\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	7.00000	0.348263
$405$	0	0
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	0	0
$412$	13.0000	0.640464
$413$	0	0
$414$	−9.00000	−0.442326
$415$	0	0
$416$	4.00000	0.196116
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	11.0000	0.535472
$423$	30.0000	1.45865
$424$	−13.0000	−0.631336
$425$	0	0
$426$	0	0
$427$	0	0
$428$	11.0000	0.531705
$429$	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	$0.593326\pi$
$432$	0	0
$433$	−31.0000	−1.48976	−0.744882	−	0.667196i	$-0.767494\pi$
$433$	−31.0000	−1.48976	−0.744882	+	0.667196i	$0.767494\pi$
$434$	0	0
$435$	0	0
$436$	−18.0000	−0.862044
$437$	3.00000	0.143509
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	−28.0000	−1.32584
$447$	0	0
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	15.0000	0.707107
$451$	16.0000	0.753411
$452$	−6.00000	−0.282216
$453$	0	0
$454$	−2.00000	−0.0938647
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	0	0
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	28.0000	1.30127	0.650635	−	0.759390i	$-0.274503\pi$
$463$	28.0000	1.30127	0.650635	+	0.759390i	$0.274503\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−16.0000	−0.741186
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−12.0000	−0.554700
$469$	0	0
$470$	0	0
$471$	0	0
$472$	7.00000	0.322201
$473$	24.0000	1.10352
$474$	0	0
$475$	−5.00000	−0.229416
$476$	0	0
$477$	39.0000	1.78569
$478$	4.00000	0.182956
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	5.00000	0.227744
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	23.0000	1.04223	0.521115	−	0.853487i	$-0.325516\pi$
$487$	23.0000	1.04223	0.521115	+	0.853487i	$0.325516\pi$
$488$	12.0000	0.543214
$489$	0	0
$490$	0	0
$491$	26.0000	1.17336	0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000	1.17336	0.586682	+	0.809818i	$0.300434\pi$
$492$	0	0
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	0	0
$497$	0	0
$498$	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
$501$	0	0
$502$	12.0000	0.535586
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	0	0
$508$	−18.0000	−0.798621
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	31.0000	1.36735
$515$	0	0
$516$	0	0
$517$	20.0000	0.879599
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	12.0000	0.525226
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−15.0000	−0.654031
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−21.0000	−0.911322
$532$	0	0
$533$	−32.0000	−1.38607
$534$	0	0
$535$	0	0
$536$	−16.0000	−0.691095
$537$	0	0
$538$	32.0000	1.37962
$539$	14.0000	0.603023
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	−13.0000	−0.555332
$549$	−36.0000	−1.53644
$550$	10.0000	0.426401
$551$	−4.00000	−0.170406
$552$	0	0
$553$	0	0
$554$	−20.0000	−0.849719
$555$	0	0
$556$	0	0
$557$	4.00000	0.169485	0.0847427	−	0.996403i	$-0.472993\pi$
$557$	4.00000	0.169485	0.0847427	+	0.996403i	$0.472993\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	0	0
$566$	−6.00000	−0.252199
$567$	0	0
$568$	8.00000	0.335673
$569$	33.0000	1.38343	0.691716	−	0.722170i	$-0.256855\pi$
$569$	33.0000	1.38343	0.691716	+	0.722170i	$0.256855\pi$
$570$	0	0
$571$	3.00000	0.125546	0.0627730	−	0.998028i	$-0.480006\pi$
$571$	3.00000	0.125546	0.0627730	+	0.998028i	$0.480006\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	0	0
$575$	−15.0000	−0.625543
$576$	−3.00000	−0.125000
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	26.0000	1.07681
$584$	4.00000	0.165521
$585$	0	0
$586$	−15.0000	−0.619644
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	6.00000	0.246598
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	0	0
$595$	0	0
$596$	−16.0000	−0.655386
$597$	0	0
$598$	12.0000	0.490716
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	48.0000	1.95471
$604$	−9.00000	−0.366205
$605$	0	0
$606$	0	0
$607$	38.0000	1.54237	0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000	1.54237	0.771186	+	0.636610i	$0.219664\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−40.0000	−1.61823
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	23.0000	0.928204
$615$	0	0
$616$	0	0
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	0	0
$619$	31.0000	1.24600	0.622998	−	0.782224i	$-0.285915\pi$
$619$	31.0000	1.24600	0.622998	+	0.782224i	$0.285915\pi$
$620$	0	0
$621$	0	0
$622$	−27.0000	−1.08260
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	22.0000	0.879297
$627$	0	0
$628$	−10.0000	−0.399043
$629$	0	0
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	31.0000	1.23117
$635$	0	0
$636$	0	0
$637$	−28.0000	−1.10940
$638$	8.00000	0.316723
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	21.0000	0.828159	0.414080	−	0.910241i	$-0.364104\pi$
$643$	21.0000	0.828159	0.414080	+	0.910241i	$0.364104\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.00000	0.157256	0.0786281	−	0.996904i	$-0.474946\pi$
$647$	4.00000	0.157256	0.0786281	+	0.996904i	$0.474946\pi$
$648$	9.00000	0.353553
$649$	−14.0000	−0.549548
$650$	−20.0000	−0.784465
$651$	0	0
$652$	8.00000	0.313304
$653$	25.0000	0.978326	0.489163	−	0.872192i	$-0.337302\pi$
$653$	25.0000	0.978326	0.489163	+	0.872192i	$0.337302\pi$
$654$	0	0
$655$	0	0
$656$	−8.00000	−0.312348
$657$	−12.0000	−0.468165
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	16.0000	0.621858
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−18.0000	−0.697486
$667$	−12.0000	−0.464642
$668$	−8.00000	−0.309529
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	13.0000	0.501113	0.250557	−	0.968102i	$-0.419386\pi$
$673$	13.0000	0.501113	0.250557	+	0.968102i	$0.419386\pi$
$674$	13.0000	0.500741
$675$	0	0
$676$	3.00000	0.115385
$677$	−9.00000	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$677$	−9.00000	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	−3.00000	−0.114708
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−12.0000	−0.457496
$689$	−52.0000	−1.98104
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	−24.0000	−0.912343
$693$	0	0
$694$	3.00000	0.113878
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−25.0000	−0.946264
$699$	0	0
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	2.00000	0.0752710
$707$	0	0
$708$	0	0
$709$	9.00000	0.338002	0.169001	−	0.985616i	$-0.445946\pi$
$709$	9.00000	0.338002	0.169001	+	0.985616i	$0.445946\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	15.0000	0.562149
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−26.0000	−0.971666
$717$	0	0
$718$	−33.0000	−1.23155
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	0	0
$722$	−18.0000	−0.669891
$723$	0	0
$724$	−19.0000	−0.706129
$725$	20.0000	0.742781
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	35.0000	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$733$	35.0000	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$734$	3.00000	0.110732
$735$	0	0
$736$	3.00000	0.110581
$737$	32.0000	1.17874
$738$	24.0000	0.883452
$739$	−24.0000	−0.882854	−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000	−0.882854	−0.441427	+	0.897297i	$0.645528\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	49.0000	1.79764	0.898818	−	0.438322i	$-0.144427\pi$
$743$	49.0000	1.79764	0.898818	+	0.438322i	$0.144427\pi$
$744$	0	0
$745$	0	0
$746$	−22.0000	−0.805477
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	−10.0000	−0.364662
$753$	0	0
$754$	−16.0000	−0.582686
$755$	0	0
$756$	0	0
$757$	−23.0000	−0.835949	−0.417975	−	0.908459i	$-0.637260\pi$
$757$	−23.0000	−0.835949	−0.417975	+	0.908459i	$0.637260\pi$
$758$	23.0000	0.835398
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	0	0
$764$	−27.0000	−0.976826
$765$	0	0
$766$	−4.00000	−0.144526
$767$	28.0000	1.01102
$768$	0	0
$769$	48.0000	1.73092	0.865462	−	0.500974i	$-0.167025\pi$
$769$	48.0000	1.73092	0.865462	+	0.500974i	$0.167025\pi$
$770$	0	0
$771$	0	0
$772$	12.0000	0.431889
$773$	45.0000	1.61854	0.809269	−	0.587439i	$-0.199864\pi$
$773$	45.0000	1.61854	0.809269	+	0.587439i	$0.199864\pi$
$774$	36.0000	1.29399
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	16.0000	0.573628
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	0	0
$784$	−7.00000	−0.250000
$785$	0	0
$786$	0	0
$787$	−3.00000	−0.106938	−0.0534692	−	0.998569i	$-0.517028\pi$
$787$	−3.00000	−0.106938	−0.0534692	+	0.998569i	$0.517028\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	0	0
$791$	0	0
$792$	6.00000	0.213201
$793$	48.0000	1.70453
$794$	−36.0000	−1.27759
$795$	0	0
$796$	24.0000	0.850657
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	−45.0000	−1.59000
$802$	−18.0000	−0.635602
$803$	−8.00000	−0.282314
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	7.00000	0.246259
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−38.0000	−1.33436	−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000	−1.33436	−0.667180	+	0.744896i	$0.732499\pi$
$812$	0	0
$813$	0	0
$814$	−12.0000	−0.420600
$815$	0	0
$816$	0	0
$817$	−12.0000	−0.419827
$818$	−23.0000	−0.804176
$819$	0	0
$820$	0	0
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	0	0
$823$	48.0000	1.67317	0.836587	−	0.547833i	$-0.184547\pi$
$823$	48.0000	1.67317	0.836587	+	0.547833i	$0.184547\pi$
$824$	13.0000	0.452876
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	−9.00000	−0.312772
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	4.00000	0.138675
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−2.00000	−0.0691714
$837$	0	0
$838$	−14.0000	−0.483622
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	13.0000	0.448010
$843$	0	0
$844$	11.0000	0.378636
$845$	0	0
$846$	30.0000	1.03142
$847$	0	0
$848$	−13.0000	−0.446422
$849$	0	0
$850$	0	0
$851$	18.0000	0.617032
$852$	0	0
$853$	36.0000	1.23262	0.616308	−	0.787505i	$-0.288628\pi$
$853$	36.0000	1.23262	0.616308	+	0.787505i	$0.288628\pi$
$854$	0	0
$855$	0	0
$856$	11.0000	0.375972
$857$	49.0000	1.67381	0.836904	−	0.547350i	$-0.184363\pi$
$857$	49.0000	1.67381	0.836904	+	0.547350i	$0.184363\pi$
$858$	0	0
$859$	42.0000	1.43302	0.716511	−	0.697576i	$-0.245738\pi$
$859$	42.0000	1.43302	0.716511	+	0.697576i	$0.245738\pi$
$860$	0	0
$861$	0	0
$862$	−12.0000	−0.408722
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	0	0
$866$	−31.0000	−1.05342
$867$	0	0
$868$	0	0
$869$	−20.0000	−0.678454
$870$	0	0
$871$	−64.0000	−2.16856
$872$	−18.0000	−0.609557
$873$	−30.0000	−1.01535
$874$	3.00000	0.101477
$875$	0	0
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	22.0000	0.742464
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	21.0000	0.707107
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−18.0000	−0.603023
$892$	−28.0000	−0.937509
$893$	−10.0000	−0.334637
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−9.00000	−0.300334
$899$	0	0
$900$	15.0000	0.500000
$901$	0	0
$902$	16.0000	0.532742
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−2.00000	−0.0663723
$909$	−21.0000	−0.696526
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	−22.0000	−0.727695
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	33.0000	1.08680
$923$	32.0000	1.05329
$924$	0	0
$925$	−30.0000	−0.986394
$926$	28.0000	0.920137
$927$	−39.0000	−1.28093
$928$	−4.00000	−0.131306
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	−16.0000	−0.524097
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	−12.0000	−0.392232
$937$	33.0000	1.07806	0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000	1.07806	0.539032	+	0.842286i	$0.318790\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	7.00000	0.227831
$945$	0	0
$946$	24.0000	0.780307
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	−5.00000	−0.162221
$951$	0	0
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	39.0000	1.26267
$955$	0	0
$956$	4.00000	0.129369
$957$	0	0
$958$	−16.0000	−0.516937
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	24.0000	0.773791
$963$	−33.0000	−1.06341
$964$	5.00000	0.161039
$965$	0	0
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	0	0
$973$	0	0
$974$	23.0000	0.736968
$975$	0	0
$976$	12.0000	0.384111
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	0	0
$981$	54.0000	1.72409
$982$	26.0000	0.829693
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	4.00000	0.127257
$989$	−36.0000	−1.14473
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	36.0000	1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.b.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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