LMFDB - Embedded newform 8004.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
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	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -5.00000 q^{13} +2.00000 q^{15} +5.00000 q^{17} +7.00000 q^{19} -2.00000 q^{21} -1.00000 q^{23} -1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{29} +8.00000 q^{31} -4.00000 q^{33} -4.00000 q^{35} -3.00000 q^{37} +5.00000 q^{39} +4.00000 q^{41} +7.00000 q^{43} -2.00000 q^{45} -8.00000 q^{47} -3.00000 q^{49} -5.00000 q^{51} +6.00000 q^{53} -8.00000 q^{55} -7.00000 q^{57} +9.00000 q^{59} -2.00000 q^{61} +2.00000 q^{63} +10.0000 q^{65} +8.00000 q^{67} +1.00000 q^{69} -7.00000 q^{71} -10.0000 q^{73} +1.00000 q^{75} +8.00000 q^{77} +11.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -10.0000 q^{85} +1.00000 q^{87} -3.00000 q^{89} -10.0000 q^{91} -8.00000 q^{93} -14.0000 q^{95} -14.0000 q^{97} +4.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	5.00000	0.800641
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−8.00000	−1.07872
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	10.0000	1.24035
$66$	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
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	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	$0.271155\pi$
$84$	0	0
$85$	−10.0000	−1.08465
$86$	0	0
$87$	1.00000	0.107211
$88$	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
$91$	−10.0000	−1.04828
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	−14.0000	−1.43637
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−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
$109$	−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
$111$	3.00000	0.284747
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	2.00000	0.186501
$116$	0	0
$117$	−5.00000	−0.462250
$118$	0	0
$119$	10.0000	0.916698
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−7.00000	−0.616316
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	14.0000	1.21395
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	−23.0000	−1.96502	−0.982511	−	0.186203i	$-0.940382\pi$
$137$	−23.0000	−1.96502	−0.982511	+	0.186203i	$0.940382\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−20.0000	−1.67248
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	−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
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	15.0000	1.16073	0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000	1.16073	0.580367	+	0.814355i	$0.302909\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	7.00000	0.535303
$172$	0	0
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	11.0000	0.822179	0.411089	−	0.911595i	$-0.365148\pi$
$179$	11.0000	0.822179	0.411089	+	0.911595i	$0.365148\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	20.0000	1.46254
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	25.0000	1.80894	0.904468	−	0.426541i	$-0.140268\pi$
$191$	25.0000	1.80894	0.904468	+	0.426541i	$0.140268\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	−10.0000	−0.716115
$196$	0	0
$197$	7.00000	0.498729	0.249365	−	0.968410i	$-0.419778\pi$
$197$	7.00000	0.498729	0.249365	+	0.968410i	$0.419778\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	28.0000	1.93680
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	7.00000	0.479632
$214$	0	0
$215$	−14.0000	−0.954792
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−25.0000	−1.68168
$222$	0	0
$223$	−29.0000	−1.94198	−0.970992	−	0.239113i	$-0.923143\pi$
$223$	−29.0000	−1.94198	−0.970992	+	0.239113i	$0.923143\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	6.00000	0.398234	0.199117	−	0.979976i	$-0.436193\pi$
$227$	6.00000	0.398234	0.199117	+	0.979976i	$0.436193\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	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
$235$	16.0000	1.04372
$236$	0	0
$237$	−11.0000	−0.714527
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−35.0000	−2.22700
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	10.0000	0.626224
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	3.00000	0.183597
$268$	0	0
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	10.0000	0.605228
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	14.0000	0.829288
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	5.00000	0.289157
$300$	0	0
$301$	14.0000	0.806947
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	35.0000	1.94745
$324$	0	0
$325$	5.00000	0.277350
$326$	0	0
$327$	16.0000	0.884802
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	0	0
$333$	−3.00000	−0.164399
$334$	0	0
$335$	−16.0000	−0.874173
$336$	0	0
$337$	−3.00000	−0.163420	−0.0817102	−	0.996656i	$-0.526038\pi$
$337$	−3.00000	−0.163420	−0.0817102	+	0.996656i	$0.526038\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	21.0000	1.12734	0.563670	−	0.826000i	$-0.309389\pi$
$347$	21.0000	1.12734	0.563670	+	0.826000i	$0.309389\pi$
$348$	0	0
$349$	−21.0000	−1.12410	−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000	−1.12410	−0.562052	+	0.827102i	$0.689988\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	0	0
$357$	−10.0000	−0.529256
$358$	0	0
$359$	−23.0000	−1.21389	−0.606947	−	0.794742i	$-0.707606\pi$
$359$	−23.0000	−1.21389	−0.606947	+	0.794742i	$0.707606\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	20.0000	1.04685
$366$	0	0
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	36.0000	1.86401	0.932005	−	0.362446i	$-0.118058\pi$
$373$	36.0000	1.86401	0.932005	+	0.362446i	$0.118058\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	5.00000	0.257513
$378$	0	0
$379$	21.0000	1.07870	0.539349	−	0.842082i	$-0.318670\pi$
$379$	21.0000	1.07870	0.539349	+	0.842082i	$0.318670\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	7.00000	0.355830
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	−5.00000	−0.252861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.0000	−1.10694
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	−14.0000	−0.700877
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	−40.0000	−1.99254
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	23.0000	1.13451
$412$	0	0
$413$	18.0000	0.885722
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−5.00000	−0.242536
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	20.0000	0.965609
$430$	0	0
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	−7.00000	−0.334855
$438$	0	0
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	16.0000	0.756774
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	−17.0000	−0.798730
$454$	0	0
$455$	20.0000	0.937614
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	0	0
$467$	7.00000	0.323921	0.161961	−	0.986797i	$-0.448218\pi$
$467$	7.00000	0.323921	0.161961	+	0.986797i	$0.448218\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	7.00000	0.322543
$472$	0	0
$473$	28.0000	1.28744
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	28.0000	1.27141
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−5.00000	−0.225189
$494$	0	0
$495$	−8.00000	−0.359573
$496$	0	0
$497$	−14.0000	−0.627986
$498$	0	0
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	0	0
$501$	−15.0000	−0.670151
$502$	0	0
$503$	41.0000	1.82810	0.914050	−	0.405602i	$-0.132938\pi$
$503$	41.0000	1.82810	0.914050	+	0.405602i	$0.132938\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	−12.0000	−0.532939
$508$	0	0
$509$	5.00000	0.221621	0.110811	−	0.993842i	$-0.464655\pi$
$509$	5.00000	0.221621	0.110811	+	0.993842i	$0.464655\pi$
$510$	0	0
$511$	−20.0000	−0.884748
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−32.0000	−1.40736
$518$	0	0
$519$	1.00000	0.0438951
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	40.0000	1.74243
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.00000	0.390567
$532$	0	0
$533$	−20.0000	−0.866296
$534$	0	0
$535$	24.0000	1.03761
$536$	0	0
$537$	−11.0000	−0.474685
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	44.0000	1.89171	0.945854	−	0.324593i	$-0.105227\pi$
$541$	44.0000	1.89171	0.945854	+	0.324593i	$0.105227\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	32.0000	1.37073
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−7.00000	−0.298210
$552$	0	0
$553$	22.0000	0.935535
$554$	0	0
$555$	−6.00000	−0.254686
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	−35.0000	−1.48034
$560$	0	0
$561$	−20.0000	−0.844401
$562$	0	0
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	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
$573$	−25.0000	−1.04439
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	0	0
$579$	−10.0000	−0.415586
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	10.0000	0.413449
$586$	0	0
$587$	−23.0000	−0.949312	−0.474656	−	0.880172i	$-0.657427\pi$
$587$	−23.0000	−0.949312	−0.474656	+	0.880172i	$0.657427\pi$
$588$	0	0
$589$	56.0000	2.30744
$590$	0	0
$591$	−7.00000	−0.287942
$592$	0	0
$593$	17.0000	0.698106	0.349053	−	0.937103i	$-0.386503\pi$
$593$	17.0000	0.698106	0.349053	+	0.937103i	$0.386503\pi$
$594$	0	0
$595$	−20.0000	−0.819920
$596$	0	0
$597$	12.0000	0.491127
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	4.00000	0.163163	0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000	0.163163	0.0815817	+	0.996667i	$0.474003\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	15.0000	0.602901	0.301450	−	0.953482i	$-0.402529\pi$
$619$	15.0000	0.602901	0.301450	+	0.953482i	$0.402529\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−28.0000	−1.11821
$628$	0	0
$629$	−15.0000	−0.598089
$630$	0	0
$631$	42.0000	1.67199	0.835997	−	0.548734i	$-0.184890\pi$
$631$	42.0000	1.67199	0.835997	+	0.548734i	$0.184890\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	15.0000	0.594322
$638$	0	0
$639$	−7.00000	−0.276916
$640$	0	0
$641$	−13.0000	−0.513469	−0.256735	−	0.966482i	$-0.582647\pi$
$641$	−13.0000	−0.513469	−0.256735	+	0.966482i	$0.582647\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	14.0000	0.551249
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	33.0000	1.28550	0.642749	−	0.766077i	$-0.277794\pi$
$659$	33.0000	1.28550	0.642749	+	0.766077i	$0.277794\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	0	0
$663$	25.0000	0.970920
$664$	0	0
$665$	−28.0000	−1.08579
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	29.0000	1.12120
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−15.0000	−0.578208	−0.289104	−	0.957298i	$-0.593357\pi$
$673$	−15.0000	−0.578208	−0.289104	+	0.957298i	$0.593357\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−28.0000	−1.07454
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	46.0000	1.75757
$686$	0	0
$687$	1.00000	0.0381524
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	36.0000	1.36950	0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000	1.36950	0.684752	+	0.728776i	$0.259910\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	−40.0000	−1.51729
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	−29.0000	−1.09688
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	−21.0000	−0.792030
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	−4.00000	−0.150435
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	40.0000	1.49592
$716$	0	0
$717$	−15.0000	−0.560185
$718$	0	0
$719$	−9.00000	−0.335643	−0.167822	−	0.985817i	$-0.553673\pi$
$719$	−9.00000	−0.335643	−0.167822	+	0.985817i	$0.553673\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	22.0000	0.818189
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.0000	1.29452
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	−50.0000	−1.83928	−0.919640	−	0.392763i	$-0.871519\pi$
$739$	−50.0000	−1.83928	−0.919640	+	0.392763i	$0.871519\pi$
$740$	0	0
$741$	35.0000	1.28576
$742$	0	0
$743$	−27.0000	−0.990534	−0.495267	−	0.868741i	$-0.664930\pi$
$743$	−27.0000	−0.990534	−0.495267	+	0.868741i	$0.664930\pi$
$744$	0	0
$745$	32.0000	1.17239
$746$	0	0
$747$	12.0000	0.439057
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	0	0
$753$	−9.00000	−0.327978
$754$	0	0
$755$	−34.0000	−1.23739
$756$	0	0
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	25.0000	0.906249	0.453125	−	0.891447i	$-0.350309\pi$
$761$	25.0000	0.906249	0.453125	+	0.891447i	$0.350309\pi$
$762$	0	0
$763$	−32.0000	−1.15848
$764$	0	0
$765$	−10.0000	−0.361551
$766$	0	0
$767$	−45.0000	−1.62486
$768$	0	0
$769$	31.0000	1.11789	0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000	1.11789	0.558944	+	0.829205i	$0.311207\pi$
$770$	0	0
$771$	−3.00000	−0.108042
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	28.0000	1.00320
$780$	0	0
$781$	−28.0000	−1.00192
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	14.0000	0.499681
$786$	0	0
$787$	−42.0000	−1.49714	−0.748569	−	0.663057i	$-0.769259\pi$
$787$	−42.0000	−1.49714	−0.748569	+	0.663057i	$0.769259\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	−3.00000	−0.106000
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	−4.00000	−0.140807
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	−4.00000	−0.140286
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	49.0000	1.71429
$818$	0	0
$819$	−10.0000	−0.349428
$820$	0	0
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	−54.0000	−1.88232	−0.941161	−	0.337959i	$-0.890263\pi$
$823$	−54.0000	−1.88232	−0.941161	+	0.337959i	$0.890263\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	45.0000	1.56480	0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000	1.56480	0.782402	+	0.622774i	$0.213994\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	−11.0000	−0.381586
$832$	0	0
$833$	−15.0000	−0.519719
$834$	0	0
$835$	−30.0000	−1.03819
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−28.0000	−0.966667	−0.483334	−	0.875436i	$-0.660574\pi$
$839$	−28.0000	−0.966667	−0.483334	+	0.875436i	$0.660574\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.0000	−0.551069
$844$	0	0
$845$	−24.0000	−0.825625
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	−14.0000	−0.478790
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	44.0000	1.49260
$870$	0	0
$871$	−40.0000	−1.35535
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	−25.0000	−0.844190	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	−25.0000	−0.844190	−0.422095	+	0.906552i	$0.638705\pi$
$878$	0	0
$879$	−19.0000	−0.640854
$880$	0	0
$881$	−35.0000	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$881$	−35.0000	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$882$	0	0
$883$	−11.0000	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	0	0
$887$	34.0000	1.14161	0.570804	−	0.821086i	$-0.306632\pi$
$887$	34.0000	1.14161	0.570804	+	0.821086i	$0.306632\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−56.0000	−1.87397
$894$	0	0
$895$	−22.0000	−0.735379
$896$	0	0
$897$	−5.00000	−0.166945
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	30.0000	0.999445
$902$	0	0
$903$	−14.0000	−0.465891
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	29.0000	0.962929	0.481465	−	0.876466i	$-0.340105\pi$
$907$	29.0000	0.962929	0.481465	+	0.876466i	$0.340105\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	48.0000	1.58857
$914$	0	0
$915$	−4.00000	−0.132236
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	0	0
$923$	35.0000	1.15204
$924$	0	0
$925$	3.00000	0.0986394
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	0	0
$931$	−21.0000	−0.688247
$932$	0	0
$933$	−28.0000	−0.916679
$934$	0	0
$935$	−40.0000	−1.30814
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	0	0
$943$	−4.00000	−0.130258
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	50.0000	1.62307
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	−50.0000	−1.61796
$956$	0	0
$957$	4.00000	0.129302
$958$	0	0
$959$	−46.0000	−1.48542
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	−20.0000	−0.643823
$966$	0	0
$967$	52.0000	1.67221	0.836104	−	0.548572i	$-0.184828\pi$
$967$	52.0000	1.67221	0.836104	+	0.548572i	$0.184828\pi$
$968$	0	0
$969$	−35.0000	−1.12436
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	40.0000	1.28234
$974$	0	0
$975$	−5.00000	−0.160128
$976$	0	0
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−16.0000	−0.510841
$982$	0	0
$983$	21.0000	0.669796	0.334898	−	0.942254i	$-0.391298\pi$
$983$	21.0000	0.669796	0.334898	+	0.942254i	$0.391298\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	−7.00000	−0.222587
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−6.00000	−0.190404
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	−40.0000	−1.26681	−0.633406	−	0.773819i	$-0.718344\pi$
$997$	−40.0000	−1.26681	−0.633406	+	0.773819i	$0.718344\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more