LMFDB - Embedded newform 8001.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$2$
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	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +O(q^{10})$

$q-2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.00000 q^{10} -1.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} -6.00000 q^{17} -4.00000 q^{19} -2.00000 q^{20} -9.00000 q^{23} -4.00000 q^{25} +2.00000 q^{26} -2.00000 q^{28} -5.00000 q^{29} -3.00000 q^{31} +8.00000 q^{32} +12.0000 q^{34} +1.00000 q^{35} +1.00000 q^{37} +8.00000 q^{38} -10.0000 q^{41} -4.00000 q^{43} +18.0000 q^{46} -12.0000 q^{47} +1.00000 q^{49} +8.00000 q^{50} -2.00000 q^{52} -3.00000 q^{53} +10.0000 q^{58} -3.00000 q^{59} -3.00000 q^{61} +6.00000 q^{62} -8.00000 q^{64} +1.00000 q^{65} +12.0000 q^{67} -12.0000 q^{68} -2.00000 q^{70} -6.00000 q^{71} -7.00000 q^{73} -2.00000 q^{74} -8.00000 q^{76} +4.00000 q^{80} +20.0000 q^{82} +9.00000 q^{83} +6.00000 q^{85} +8.00000 q^{86} -1.00000 q^{89} +1.00000 q^{91} -18.0000 q^{92} +24.0000 q^{94} +4.00000 q^{95} +14.0000 q^{97} -2.00000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	2.00000	0.392232
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	8.00000	1.41421
$33$	0	0
$34$	12.0000	2.05798
$35$	1.00000	0.169031
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	18.0000	2.65396
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.00000	1.13137
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	−8.00000	−1.00000
$65$	1.00000	0.124035
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−12.0000	−1.45521
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−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
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	4.00000	0.447214
$81$	0	0
$82$	20.0000	2.20863
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−18.0000	−1.87663
$93$	0	0
$94$	24.0000	2.47541
$95$	4.00000	0.410391
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	−8.00000	−0.800000
$101$	15.0000	1.49256	0.746278	−	0.665635i	$-0.231839\pi$
$101$	15.0000	1.49256	0.746278	+	0.665635i	$0.231839\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	−10.0000	−0.928477
$117$	0	0
$118$	6.00000	0.552345
$119$	6.00000	0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	6.00000	0.543214
$123$	0	0
$124$	−6.00000	−0.538816
$125$	9.00000	0.804984
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0	0
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−24.0000	−2.07328
$135$	0	0
$136$	0	0
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	12.0000	1.00702
$143$	0	0
$144$	0	0
$145$	5.00000	0.415227
$146$	14.0000	1.15865
$147$	0	0
$148$	2.00000	0.164399
$149$	24.0000	1.96616	0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000	1.96616	0.983078	+	0.183186i	$0.0586410\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	0	0
$159$	0	0
$160$	−8.00000	−0.632456
$161$	9.00000	0.709299
$162$	0	0
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	−20.0000	−1.56174
$165$	0	0
$166$	−18.0000	−1.39707
$167$	−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
$169$	−12.0000	−0.923077
$170$	−12.0000	−0.920358
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	2.00000	0.149906
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	0	0
$188$	−24.0000	−1.75038
$189$	0	0
$190$	−8.00000	−0.580381
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−28.0000	−2.01028
$195$	0	0
$196$	2.00000	0.142857
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	0	0
$202$	−30.0000	−2.11079
$203$	5.00000	0.350931
$204$	0	0
$205$	10.0000	0.698430
$206$	−32.0000	−2.22955
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	24.0000	1.64061
$215$	4.00000	0.272798
$216$	0	0
$217$	3.00000	0.203653
$218$	−8.00000	−0.541828
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	20.0000	1.33038
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	0	0
$233$	29.0000	1.89985	0.949927	−	0.312473i	$-0.101157\pi$
$233$	29.0000	1.89985	0.949927	+	0.312473i	$0.101157\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−12.0000	−0.777844
$239$	7.00000	0.452792	0.226396	−	0.974035i	$-0.427306\pi$
$239$	7.00000	0.452792	0.226396	+	0.974035i	$0.427306\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	22.0000	1.41421
$243$	0	0
$244$	−6.00000	−0.384111
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	−18.0000	−1.13842
$251$	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
$253$	0	0
$254$	−2.00000	−0.125491
$255$	0	0
$256$	16.0000	1.00000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	2.00000	0.124035
$261$	0	0
$262$	12.0000	0.741362
$263$	−26.0000	−1.60323	−0.801614	−	0.597841i	$-0.796025\pi$
$263$	−26.0000	−1.60323	−0.801614	+	0.597841i	$0.796025\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	−8.00000	−0.490511
$267$	0	0
$268$	24.0000	1.46603
$269$	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
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	42.0000	2.53731
$275$	0	0
$276$	0	0
$277$	−24.0000	−1.44202	−0.721010	−	0.692925i	$-0.756322\pi$
$277$	−24.0000	−1.44202	−0.721010	+	0.692925i	$0.756322\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	10.0000	0.590281
$288$	0	0
$289$	19.0000	1.11765
$290$	−10.0000	−0.587220
$291$	0	0
$292$	−14.0000	−0.819288
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	0	0
$297$	0	0
$298$	−48.0000	−2.78057
$299$	9.00000	0.520483
$300$	0	0
$301$	4.00000	0.230556
$302$	8.00000	0.460348
$303$	0	0
$304$	16.0000	0.917663
$305$	3.00000	0.171780
$306$	0	0
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	0	0
$309$	0	0
$310$	−6.00000	−0.340777
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	−44.0000	−2.48306
$315$	0	0
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	0	0
$320$	8.00000	0.447214
$321$	0	0
$322$	−18.0000	−1.00310
$323$	24.0000	1.33540
$324$	0	0
$325$	4.00000	0.221880
$326$	26.0000	1.44001
$327$	0	0
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	18.0000	0.987878
$333$	0	0
$334$	16.0000	0.875481
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	24.0000	1.30543
$339$	0	0
$340$	12.0000	0.650791
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	28.0000	1.50529
$347$	23.0000	1.23470	0.617352	−	0.786687i	$-0.288205\pi$
$347$	23.0000	1.23470	0.617352	+	0.786687i	$0.288205\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	0	0
$353$	−12.0000	−0.638696	−0.319348	−	0.947638i	$-0.603464\pi$
$353$	−12.0000	−0.638696	−0.319348	+	0.947638i	$0.603464\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−32.0000	−1.69125
$359$	−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
$361$	−3.00000	−0.157895
$362$	16.0000	0.840941
$363$	0	0
$364$	2.00000	0.104828
$365$	7.00000	0.366397
$366$	0	0
$367$	−7.00000	−0.365397	−0.182699	−	0.983169i	$-0.558483\pi$
$367$	−7.00000	−0.365397	−0.182699	+	0.983169i	$0.558483\pi$
$368$	36.0000	1.87663
$369$	0	0
$370$	2.00000	0.103975
$371$	3.00000	0.155752
$372$	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
$376$	0	0
$377$	5.00000	0.257513
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	12.0000	0.613973
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	0	0
$388$	28.0000	1.42148
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	54.0000	2.73090
$392$	0	0
$393$	0	0
$394$	24.0000	1.20910
$395$	0	0
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	22.0000	1.10276
$399$	0	0
$400$	16.0000	0.800000
$401$	13.0000	0.649189	0.324595	−	0.945853i	$-0.394772\pi$
$401$	13.0000	0.649189	0.324595	+	0.945853i	$0.394772\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	30.0000	1.49256
$405$	0	0
$406$	−10.0000	−0.496292
$407$	0	0
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	−20.0000	−0.987730
$411$	0	0
$412$	32.0000	1.57653
$413$	3.00000	0.147620
$414$	0	0
$415$	−9.00000	−0.441793
$416$	−8.00000	−0.392232
$417$	0	0
$418$	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
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	−30.0000	−1.46038
$423$	0	0
$424$	0	0
$425$	24.0000	1.16417
$426$	0	0
$427$	3.00000	0.145180
$428$	−24.0000	−1.16008
$429$	0	0
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	8.00000	0.383131
$437$	36.0000	1.72211
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	20.0000	0.947027
$447$	0	0
$448$	8.00000	0.377964
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	0	0
$452$	−20.0000	−0.940721
$453$	0	0
$454$	36.0000	1.68956
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	40.0000	1.86908
$459$	0	0
$460$	18.0000	0.839254
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	20.0000	0.928477
$465$	0	0
$466$	−58.0000	−2.68680
$467$	−9.00000	−0.416470	−0.208235	−	0.978079i	$-0.566772\pi$
$467$	−9.00000	−0.416470	−0.208235	+	0.978079i	$0.566772\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	−24.0000	−1.10704
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	16.0000	0.734130
$476$	12.0000	0.550019
$477$	0	0
$478$	−14.0000	−0.640345
$479$	34.0000	1.55350	0.776750	−	0.629809i	$-0.216867\pi$
$479$	34.0000	1.55350	0.776750	+	0.629809i	$0.216867\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	−36.0000	−1.63976
$483$	0	0
$484$	−22.0000	−1.00000
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−14.0000	−0.634401	−0.317200	−	0.948359i	$-0.602743\pi$
$487$	−14.0000	−0.634401	−0.317200	+	0.948359i	$0.602743\pi$
$488$	0	0
$489$	0	0
$490$	2.00000	0.0903508
$491$	−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
$493$	30.0000	1.35113
$494$	−8.00000	−0.359937
$495$	0	0
$496$	12.0000	0.538816
$497$	6.00000	0.269137
$498$	0	0
$499$	−26.0000	−1.16392	−0.581960	−	0.813217i	$-0.697714\pi$
$499$	−26.0000	−1.16392	−0.581960	+	0.813217i	$0.697714\pi$
$500$	18.0000	0.804984
$501$	0	0
$502$	−24.0000	−1.07117
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−15.0000	−0.667491
$506$	0	0
$507$	0	0
$508$	2.00000	0.0887357
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	−32.0000	−1.41421
$513$	0	0
$514$	6.00000	0.264649
$515$	−16.0000	−0.705044
$516$	0	0
$517$	0	0
$518$	2.00000	0.0878750
$519$	0	0
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	33.0000	1.44299	0.721495	−	0.692420i	$-0.243455\pi$
$523$	33.0000	1.44299	0.721495	+	0.692420i	$0.243455\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	52.0000	2.26731
$527$	18.0000	0.784092
$528$	0	0
$529$	58.0000	2.52174
$530$	−6.00000	−0.260623
$531$	0	0
$532$	8.00000	0.346844
$533$	10.0000	0.433148
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	0	0
$538$	−48.0000	−2.06943
$539$	0	0
$540$	0	0
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	−22.0000	−0.944981
$543$	0	0
$544$	−48.0000	−2.05798
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	−42.0000	−1.79415
$549$	0	0
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	0	0
$554$	48.0000	2.03932
$555$	0	0
$556$	4.00000	0.169638
$557$	20.0000	0.847427	0.423714	−	0.905796i	$-0.360726\pi$
$557$	20.0000	0.847427	0.423714	+	0.905796i	$0.360726\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	−4.00000	−0.169031
$561$	0	0
$562$	20.0000	0.843649
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	−32.0000	−1.34506
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	0	0
$573$	0	0
$574$	−20.0000	−0.834784
$575$	36.0000	1.50130
$576$	0	0
$577$	−37.0000	−1.54033	−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000	−1.54033	−0.770165	+	0.637845i	$0.779826\pi$
$578$	−38.0000	−1.58059
$579$	0	0
$580$	10.0000	0.415227
$581$	−9.00000	−0.373383
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	36.0000	1.48715
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	−6.00000	−0.247016
$591$	0	0
$592$	−4.00000	−0.164399
$593$	−23.0000	−0.944497	−0.472248	−	0.881466i	$-0.656557\pi$
$593$	−23.0000	−0.944497	−0.472248	+	0.881466i	$0.656557\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	48.0000	1.96616
$597$	0	0
$598$	−18.0000	−0.736075
$599$	−19.0000	−0.776319	−0.388159	−	0.921592i	$-0.626889\pi$
$599$	−19.0000	−0.776319	−0.388159	+	0.921592i	$0.626889\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	−8.00000	−0.325515
$605$	11.0000	0.447214
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−32.0000	−1.29777
$609$	0	0
$610$	−6.00000	−0.242933
$611$	12.0000	0.485468
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−44.0000	−1.77570
$615$	0	0
$616$	0	0
$617$	−13.0000	−0.523360	−0.261680	−	0.965155i	$-0.584277\pi$
$617$	−13.0000	−0.523360	−0.261680	+	0.965155i	$0.584277\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	30.0000	1.20289
$623$	1.00000	0.0400642
$624$	0	0
$625$	11.0000	0.440000
$626$	32.0000	1.27898
$627$	0	0
$628$	44.0000	1.75579
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	0	0
$634$	−20.0000	−0.794301
$635$	−1.00000	−0.0396838
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	18.0000	0.709299
$645$	0	0
$646$	−48.0000	−1.88853
$647$	9.00000	0.353827	0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000	0.353827	0.176913	+	0.984226i	$0.443389\pi$
$648$	0	0
$649$	0	0
$650$	−8.00000	−0.313786
$651$	0	0
$652$	−26.0000	−1.01824
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	40.0000	1.56174
$657$	0	0
$658$	−24.0000	−0.935617
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	49.0000	1.90588	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	49.0000	1.90588	0.952940	+	0.303160i	$0.0980418\pi$
$662$	40.0000	1.55464
$663$	0	0
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	45.0000	1.74241
$668$	−16.0000	−0.619059
$669$	0	0
$670$	24.0000	0.927201
$671$	0	0
$672$	0	0
$673$	3.00000	0.115642	0.0578208	−	0.998327i	$-0.481585\pi$
$673$	3.00000	0.115642	0.0578208	+	0.998327i	$0.481585\pi$
$674$	36.0000	1.38667
$675$	0	0
$676$	−24.0000	−0.923077
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	21.0000	0.802369
$686$	2.00000	0.0763604
$687$	0	0
$688$	16.0000	0.609994
$689$	3.00000	0.114291
$690$	0	0
$691$	−6.00000	−0.228251	−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000	−0.228251	−0.114125	+	0.993466i	$0.536407\pi$
$692$	−28.0000	−1.06440
$693$	0	0
$694$	−46.0000	−1.74614
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	60.0000	2.27266
$698$	8.00000	0.302804
$699$	0	0
$700$	8.00000	0.302372
$701$	39.0000	1.47301	0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000	1.47301	0.736505	+	0.676432i	$0.236475\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	0	0
$706$	24.0000	0.903252
$707$	−15.0000	−0.564133
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	−12.0000	−0.450352
$711$	0	0
$712$	0	0
$713$	27.0000	1.01116
$714$	0	0
$715$	0	0
$716$	32.0000	1.19590
$717$	0	0
$718$	48.0000	1.79134
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	6.00000	0.223297
$723$	0	0
$724$	−16.0000	−0.594635
$725$	20.0000	0.742781
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	0	0
$730$	−14.0000	−0.518163
$731$	24.0000	0.887672
$732$	0	0
$733$	−5.00000	−0.184679	−0.0923396	−	0.995728i	$-0.529435\pi$
$733$	−5.00000	−0.184679	−0.0923396	+	0.995728i	$0.529435\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	−72.0000	−2.65396
$737$	0	0
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	−6.00000	−0.220267
$743$	41.0000	1.50414	0.752072	−	0.659081i	$-0.229055\pi$
$743$	41.0000	1.50414	0.752072	+	0.659081i	$0.229055\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	44.0000	1.61095
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	48.0000	1.75038
$753$	0	0
$754$	−10.0000	−0.364179
$755$	4.00000	0.145575
$756$	0	0
$757$	−7.00000	−0.254419	−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000	−0.254419	−0.127210	+	0.991876i	$0.540602\pi$
$758$	−24.0000	−0.871719
$759$	0	0
$760$	0	0
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	−12.0000	−0.434145
$765$	0	0
$766$	48.0000	1.73431
$767$	3.00000	0.108324
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	0	0
$772$	8.00000	0.287926
$773$	48.0000	1.72644	0.863220	−	0.504828i	$-0.168444\pi$
$773$	48.0000	1.72644	0.863220	+	0.504828i	$0.168444\pi$
$774$	0	0
$775$	12.0000	0.431053
$776$	0	0
$777$	0	0
$778$	−24.0000	−0.860442
$779$	40.0000	1.43315
$780$	0	0
$781$	0	0
$782$	−108.000	−3.86207
$783$	0	0
$784$	−4.00000	−0.142857
$785$	−22.0000	−0.785214
$786$	0	0
$787$	39.0000	1.39020	0.695100	−	0.718913i	$-0.255360\pi$
$787$	39.0000	1.39020	0.695100	+	0.718913i	$0.255360\pi$
$788$	−24.0000	−0.854965
$789$	0	0
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	3.00000	0.106533
$794$	26.0000	0.922705
$795$	0	0
$796$	−22.0000	−0.779769
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	−32.0000	−1.13137
$801$	0	0
$802$	−26.0000	−0.918092
$803$	0	0
$804$	0	0
$805$	−9.00000	−0.317208
$806$	−6.00000	−0.211341
$807$	0	0
$808$	0	0
$809$	−44.0000	−1.54696	−0.773479	−	0.633822i	$-0.781485\pi$
$809$	−44.0000	−1.54696	−0.773479	+	0.633822i	$0.781485\pi$
$810$	0	0
$811$	19.0000	0.667180	0.333590	−	0.942718i	$-0.391740\pi$
$811$	19.0000	0.667180	0.333590	+	0.942718i	$0.391740\pi$
$812$	10.0000	0.350931
$813$	0	0
$814$	0	0
$815$	13.0000	0.455370
$816$	0	0
$817$	16.0000	0.559769
$818$	60.0000	2.09785
$819$	0	0
$820$	20.0000	0.698430
$821$	−31.0000	−1.08191	−0.540954	−	0.841052i	$-0.681937\pi$
$821$	−31.0000	−1.08191	−0.540954	+	0.841052i	$0.681937\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	−6.00000	−0.208767
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	0	0
$829$	−8.00000	−0.277851	−0.138926	−	0.990303i	$-0.544365\pi$
$829$	−8.00000	−0.277851	−0.138926	+	0.990303i	$0.544365\pi$
$830$	18.0000	0.624789
$831$	0	0
$832$	8.00000	0.277350
$833$	−6.00000	−0.207888
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	−72.0000	−2.48720
$839$	25.0000	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$839$	25.0000	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−36.0000	−1.24064
$843$	0	0
$844$	30.0000	1.03264
$845$	12.0000	0.412813
$846$	0	0
$847$	11.0000	0.377964
$848$	12.0000	0.412082
$849$	0	0
$850$	−48.0000	−1.64639
$851$	−9.00000	−0.308516
$852$	0	0
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	0	0
$857$	−51.0000	−1.74213	−0.871063	−	0.491171i	$-0.836569\pi$
$857$	−51.0000	−1.74213	−0.871063	+	0.491171i	$0.836569\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	0	0
$863$	−31.0000	−1.05525	−0.527626	−	0.849477i	$-0.676918\pi$
$863$	−31.0000	−1.05525	−0.527626	+	0.849477i	$0.676918\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	−76.0000	−2.58259
$867$	0	0
$868$	6.00000	0.203653
$869$	0	0
$870$	0	0
$871$	−12.0000	−0.406604
$872$	0	0
$873$	0	0
$874$	−72.0000	−2.43544
$875$	−9.00000	−0.304256
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	0	0
$881$	−3.00000	−0.101073	−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000	−0.101073	−0.0505363	+	0.998722i	$0.516093\pi$
$882$	0	0
$883$	−43.0000	−1.44707	−0.723533	−	0.690290i	$-0.757483\pi$
$883$	−43.0000	−1.44707	−0.723533	+	0.690290i	$0.757483\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	28.0000	0.940678
$887$	−47.0000	−1.57811	−0.789053	−	0.614325i	$-0.789428\pi$
$887$	−47.0000	−1.57811	−0.789053	+	0.614325i	$0.789428\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	−20.0000	−0.669650
$893$	48.0000	1.60626
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	0	0
$898$	56.0000	1.86874
$899$	15.0000	0.500278
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.00000	0.265929
$906$	0	0
$907$	−13.0000	−0.431658	−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−13.0000	−0.431658	−0.215829	+	0.976431i	$0.569245\pi$
$908$	−36.0000	−1.19470
$909$	0	0
$910$	2.00000	0.0662994
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	58.0000	1.91847
$915$	0	0
$916$	−40.0000	−1.32164
$917$	6.00000	0.198137
$918$	0	0
$919$	45.0000	1.48441	0.742207	−	0.670171i	$-0.233779\pi$
$919$	45.0000	1.48441	0.742207	+	0.670171i	$0.233779\pi$
$920$	0	0
$921$	0	0
$922$	28.0000	0.922131
$923$	6.00000	0.197492
$924$	0	0
$925$	−4.00000	−0.131519
$926$	38.0000	1.24876
$927$	0	0
$928$	−40.0000	−1.31306
$929$	−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
$931$	−4.00000	−0.131095
$932$	58.0000	1.89985
$933$	0	0
$934$	18.0000	0.588978
$935$	0	0
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	24.0000	0.783628
$939$	0	0
$940$	24.0000	0.782794
$941$	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
$943$	90.0000	2.93080
$944$	12.0000	0.390567
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	−32.0000	−1.03822
$951$	0	0
$952$	0	0
$953$	−56.0000	−1.81402	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000	−1.81402	−0.907009	+	0.421111i	$0.861640\pi$
$954$	0	0
$955$	6.00000	0.194155
$956$	14.0000	0.452792
$957$	0	0
$958$	−68.0000	−2.19698
$959$	21.0000	0.678125
$960$	0	0
$961$	−22.0000	−0.709677
$962$	2.00000	0.0644826
$963$	0	0
$964$	36.0000	1.15948
$965$	−4.00000	−0.128765
$966$	0	0
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	0	0
$969$	0	0
$970$	28.0000	0.899026
$971$	16.0000	0.513464	0.256732	−	0.966483i	$-0.417354\pi$
$971$	16.0000	0.513464	0.256732	+	0.966483i	$0.417354\pi$
$972$	0	0
$973$	−2.00000	−0.0641171
$974$	28.0000	0.897178
$975$	0	0
$976$	12.0000	0.384111
$977$	−16.0000	−0.511885	−0.255943	−	0.966692i	$-0.582386\pi$
$977$	−16.0000	−0.511885	−0.255943	+	0.966692i	$0.582386\pi$
$978$	0	0
$979$	0	0
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	48.0000	1.53174
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	−60.0000	−1.91079
$987$	0	0
$988$	8.00000	0.254514
$989$	36.0000	1.14473
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	−24.0000	−0.762001
$993$	0	0
$994$	−12.0000	−0.380617
$995$	11.0000	0.348723
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	52.0000	1.64603
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.b.1.1	✓	1
3.2	odd	2		8001.2.a.g.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.b.1.1	✓	1	1.1	even	1	trivial
8001.2.a.g.1.1	yes	1	3.2	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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