LMFDB - Embedded newform 8008.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
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	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.00000 q^{15} +4.00000 q^{17} -4.00000 q^{19} +3.00000 q^{21} +1.00000 q^{23} -4.00000 q^{25} +9.00000 q^{27} +4.00000 q^{29} +5.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} +9.00000 q^{37} +3.00000 q^{39} +2.00000 q^{43} +6.00000 q^{45} +1.00000 q^{49} +12.0000 q^{51} -10.0000 q^{53} -1.00000 q^{55} -12.0000 q^{57} +11.0000 q^{59} -8.00000 q^{61} +6.00000 q^{63} +1.00000 q^{65} +1.00000 q^{67} +3.00000 q^{69} +1.00000 q^{71} -2.00000 q^{73} -12.0000 q^{75} -1.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} -6.00000 q^{83} +4.00000 q^{85} +12.0000 q^{87} +7.00000 q^{89} +1.00000 q^{91} +15.0000 q^{93} -4.00000 q^{95} +1.00000 q^{97} -6.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	9.00000	1.47959	0.739795	−	0.672832i	$-0.234922\pi$
$37$	9.00000	1.47959	0.739795	+	0.672832i	$0.234922\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	12.0000	1.68034
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	1.00000	0.122169	0.0610847	−	0.998133i	$-0.480544\pi$
$67$	1.00000	0.122169	0.0610847	+	0.998133i	$0.480544\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−12.0000	−1.38564
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	15.0000	1.55543
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	27.0000	2.56273
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	9.00000	0.774597
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	24.0000	1.94029
$154$	0	0
$155$	5.00000	0.401610
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	−30.0000	−2.37915
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−24.0000	−1.83533
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	33.0000	2.48043
$178$	0	0
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	0	0
$183$	−24.0000	−1.77413
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	9.00000	0.654654
$190$	0	0
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	3.00000	0.214834
$196$	0	0
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	0	0
$213$	3.00000	0.205557
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	5.00000	0.339422
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−23.0000	−1.54019	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−23.0000	−1.54019	−0.770097	+	0.637927i	$0.779792\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	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
$229$	19.0000	1.25556	0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000	1.25556	0.627778	+	0.778393i	$0.283965\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	12.0000	0.751469
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	0	0
$261$	24.0000	1.48556
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	21.0000	1.28518
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	0	0
$279$	30.0000	1.79605
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	0	0
$285$	−12.0000	−0.710819
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	3.00000	0.175863
$292$	0	0
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	11.0000	0.640445
$296$	0	0
$297$	−9.00000	−0.522233
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	24.0000	1.36531
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	5.00000	0.282617	0.141308	−	0.989966i	$-0.454869\pi$
$313$	5.00000	0.282617	0.141308	+	0.989966i	$0.454869\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.00000	−0.494685	−0.247342	−	0.968928i	$-0.579557\pi$
$331$	−9.00000	−0.494685	−0.247342	+	0.968928i	$0.579557\pi$
$332$	0	0
$333$	54.0000	2.95918
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	3.00000	0.162938
$340$	0	0
$341$	−5.00000	−0.270765
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.00000	0.161515
$346$	0	0
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	9.00000	0.480384
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	1.00000	0.0530745
$356$	0	0
$357$	12.0000	0.635107
$358$	0	0
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	3.00000	0.157459
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	11.0000	0.574195	0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000	0.574195	0.287098	+	0.957901i	$0.407310\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	−36.0000	−1.86401	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000	−1.86401	−0.932005	+	0.362446i	$0.881942\pi$
$374$	0	0
$375$	−27.0000	−1.39427
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	54.0000	2.76650
$382$	0	0
$383$	−35.0000	−1.78842	−0.894208	−	0.447651i	$-0.852261\pi$
$383$	−35.0000	−1.78842	−0.894208	+	0.447651i	$0.852261\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	1.00000	0.0507020	0.0253510	−	0.999679i	$-0.491930\pi$
$389$	1.00000	0.0507020	0.0253510	+	0.999679i	$0.491930\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	5.00000	0.249068
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−9.00000	−0.446113
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−15.0000	−0.739895
$412$	0	0
$413$	11.0000	0.541275
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−6.00000	−0.293821
$418$	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
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.0000	−0.776114
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	−3.00000	−0.144841
$430$	0	0
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	27.0000	1.29754	0.648769	−	0.760986i	$-0.275284\pi$
$433$	27.0000	1.29754	0.648769	+	0.760986i	$0.275284\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	−17.0000	−0.807694	−0.403847	−	0.914826i	$-0.632327\pi$
$443$	−17.0000	−0.807694	−0.403847	+	0.914826i	$0.632327\pi$
$444$	0	0
$445$	7.00000	0.331832
$446$	0	0
$447$	−54.0000	−2.55411
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−36.0000	−1.69143
$454$	0	0
$455$	1.00000	0.0468807
$456$	0	0
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	0	0
$459$	36.0000	1.68034
$460$	0	0
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	0	0
$465$	15.0000	0.695608
$466$	0	0
$467$	5.00000	0.231372	0.115686	−	0.993286i	$-0.463093\pi$
$467$	5.00000	0.231372	0.115686	+	0.993286i	$0.463093\pi$
$468$	0	0
$469$	1.00000	0.0461757
$470$	0	0
$471$	21.0000	0.967629
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	16.0000	0.734130
$476$	0	0
$477$	−60.0000	−2.74721
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	9.00000	0.410365
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	1.00000	0.0454077
$486$	0	0
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	1.00000	0.0448561
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−42.0000	−1.87642
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−25.0000	−1.10811	−0.554053	−	0.832482i	$-0.686919\pi$
$509$	−25.0000	−1.10811	−0.554053	+	0.832482i	$0.686919\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	−36.0000	−1.58944
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	23.0000	1.00765	0.503824	−	0.863806i	$-0.331926\pi$
$521$	23.0000	1.00765	0.503824	+	0.863806i	$0.331926\pi$
$522$	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
$525$	−12.0000	−0.523723
$526$	0	0
$527$	20.0000	0.871214
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	66.0000	2.86416
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	45.0000	1.94189
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	−9.00000	−0.386227
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	−48.0000	−2.04859
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	27.0000	1.14609
$556$	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
$559$	2.00000	0.0845910
$560$	0	0
$561$	−12.0000	−0.506640
$562$	0	0
$563$	−10.0000	−0.421450	−0.210725	−	0.977545i	$-0.567582\pi$
$563$	−10.0000	−0.421450	−0.210725	+	0.977545i	$0.567582\pi$
$564$	0	0
$565$	1.00000	0.0420703
$566$	0	0
$567$	9.00000	0.377964
$568$	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
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	−27.0000	−1.12794
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	0	0
$579$	18.0000	0.748054
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	−42.0000	−1.72765
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	−12.0000	−0.491127
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−46.0000	−1.86708	−0.933541	−	0.358470i	$-0.883298\pi$
$607$	−46.0000	−1.86708	−0.933541	+	0.358470i	$0.883298\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	0	0
$619$	−7.00000	−0.281354	−0.140677	−	0.990056i	$-0.544928\pi$
$619$	−7.00000	−0.281354	−0.140677	+	0.990056i	$0.544928\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	0	0
$623$	7.00000	0.280449
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	36.0000	1.43541
$630$	0	0
$631$	−21.0000	−0.835997	−0.417998	−	0.908448i	$-0.637268\pi$
$631$	−21.0000	−0.835997	−0.417998	+	0.908448i	$0.637268\pi$
$632$	0	0
$633$	66.0000	2.62326
$634$	0	0
$635$	18.0000	0.714308
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−5.00000	−0.197488	−0.0987441	−	0.995113i	$-0.531483\pi$
$641$	−5.00000	−0.197488	−0.0987441	+	0.995113i	$0.531483\pi$
$642$	0	0
$643$	−9.00000	−0.354925	−0.177463	−	0.984128i	$-0.556789\pi$
$643$	−9.00000	−0.354925	−0.177463	+	0.984128i	$0.556789\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	5.00000	0.196570	0.0982851	−	0.995158i	$-0.468664\pi$
$647$	5.00000	0.196570	0.0982851	+	0.995158i	$0.468664\pi$
$648$	0	0
$649$	−11.0000	−0.431788
$650$	0	0
$651$	15.0000	0.587896
$652$	0	0
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	−12.0000	−0.468165
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	4.00000	0.154881
$668$	0	0
$669$	−69.0000	−2.66769
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	−36.0000	−1.38564
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	40.0000	1.53056	0.765279	−	0.643699i	$-0.222601\pi$
$683$	40.0000	1.53056	0.765279	+	0.643699i	$0.222601\pi$
$684$	0	0
$685$	−5.00000	−0.191040
$686$	0	0
$687$	57.0000	2.17469
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−13.0000	−0.494543	−0.247272	−	0.968946i	$-0.579534\pi$
$691$	−13.0000	−0.494543	−0.247272	+	0.968946i	$0.579534\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	0	0
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	−36.0000	−1.35777
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	45.0000	1.69001	0.845005	−	0.534758i	$-0.179597\pi$
$709$	45.0000	1.69001	0.845005	+	0.534758i	$0.179597\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	5.00000	0.187251
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	0	0
$717$	−48.0000	−1.79259
$718$	0	0
$719$	7.00000	0.261056	0.130528	−	0.991445i	$-0.458333\pi$
$719$	7.00000	0.261056	0.130528	+	0.991445i	$0.458333\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−90.0000	−3.34714
$724$	0	0
$725$	−16.0000	−0.594225
$726$	0	0
$727$	7.00000	0.259616	0.129808	−	0.991539i	$-0.458564\pi$
$727$	7.00000	0.259616	0.129808	+	0.991539i	$0.458564\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	8.00000	0.295891
$732$	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
$735$	3.00000	0.110657
$736$	0	0
$737$	−1.00000	−0.0368355
$738$	0	0
$739$	22.0000	0.809283	0.404642	−	0.914475i	$-0.367396\pi$
$739$	22.0000	0.809283	0.404642	+	0.914475i	$0.367396\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	30.0000	1.10059	0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000	1.10059	0.550297	+	0.834969i	$0.314515\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	−36.0000	−1.31717
$748$	0	0
$749$	4.00000	0.146157
$750$	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
$753$	63.0000	2.29585
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	−3.00000	−0.108893
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	11.0000	0.397187
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	27.0000	0.968620
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−1.00000	−0.0357828
$782$	0	0
$783$	36.0000	1.28654
$784$	0	0
$785$	7.00000	0.249841
$786$	0	0
$787$	−18.0000	−0.641631	−0.320815	−	0.947142i	$-0.603957\pi$
$787$	−18.0000	−0.641631	−0.320815	+	0.947142i	$0.603957\pi$
$788$	0	0
$789$	−18.0000	−0.640817
$790$	0	0
$791$	1.00000	0.0355559
$792$	0	0
$793$	−8.00000	−0.284088
$794$	0	0
$795$	−30.0000	−1.06399
$796$	0	0
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	42.0000	1.48400
$802$	0	0
$803$	2.00000	0.0705785
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	−54.0000	−1.90089
$808$	0	0
$809$	34.0000	1.19538	0.597688	−	0.801729i	$-0.296086\pi$
$809$	34.0000	1.19538	0.597688	+	0.801729i	$0.296086\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	−36.0000	−1.26258
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	−32.0000	−1.11681	−0.558404	−	0.829569i	$-0.688586\pi$
$821$	−32.0000	−1.11681	−0.558404	+	0.829569i	$0.688586\pi$
$822$	0	0
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	0	0
$829$	19.0000	0.659897	0.329949	−	0.943999i	$-0.392969\pi$
$829$	19.0000	0.659897	0.329949	+	0.943999i	$0.392969\pi$
$830$	0	0
$831$	12.0000	0.416275
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	−14.0000	−0.484490
$836$	0	0
$837$	45.0000	1.55543
$838$	0	0
$839$	−3.00000	−0.103572	−0.0517858	−	0.998658i	$-0.516491\pi$
$839$	−3.00000	−0.103572	−0.0517858	+	0.998658i	$0.516491\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−36.0000	−1.23991
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	30.0000	1.02960
$850$	0	0
$851$	9.00000	0.308516
$852$	0	0
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	0	0
$857$	−52.0000	−1.77629	−0.888143	−	0.459567i	$-0.848005\pi$
$857$	−52.0000	−1.77629	−0.888143	+	0.459567i	$0.848005\pi$
$858$	0	0
$859$	49.0000	1.67186	0.835929	−	0.548837i	$-0.184929\pi$
$859$	49.0000	1.67186	0.835929	+	0.548837i	$0.184929\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	−4.00000	−0.135691
$870$	0	0
$871$	1.00000	0.0338837
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	0	0
$879$	48.0000	1.61900
$880$	0	0
$881$	−25.0000	−0.842271	−0.421136	−	0.906998i	$-0.638368\pi$
$881$	−25.0000	−0.842271	−0.421136	+	0.906998i	$0.638368\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	33.0000	1.10928
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	−9.00000	−0.301511
$892$	0	0
$893$	0	0
$894$	0	0
$895$	15.0000	0.501395
$896$	0	0
$897$	3.00000	0.100167
$898$	0	0
$899$	20.0000	0.667037
$900$	0	0
$901$	−40.0000	−1.33259
$902$	0	0
$903$	6.00000	0.199667
$904$	0	0
$905$	−3.00000	−0.0997234
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	−24.0000	−0.793416
$916$	0	0
$917$	6.00000	0.198137
$918$	0	0
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	0	0
$921$	−48.0000	−1.58165
$922$	0	0
$923$	1.00000	0.0329154
$924$	0	0
$925$	−36.0000	−1.18367
$926$	0	0
$927$	48.0000	1.57653
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	15.0000	0.489506
$940$	0	0
$941$	60.0000	1.95594	0.977972	−	0.208736i	$-0.0669349\pi$
$941$	60.0000	1.95594	0.977972	+	0.208736i	$0.0669349\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	9.00000	0.292770
$946$	0	0
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	9.00000	0.291845
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	−9.00000	−0.291233
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−5.00000	−0.161458
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	24.0000	0.773389
$964$	0	0
$965$	6.00000	0.193147
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	−48.0000	−1.54198
$970$	0	0
$971$	39.0000	1.25157	0.625785	−	0.779996i	$-0.284779\pi$
$971$	39.0000	1.25157	0.625785	+	0.779996i	$0.284779\pi$
$972$	0	0
$973$	−2.00000	−0.0641171
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	0	0
$979$	−7.00000	−0.223721
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.00000	0.0635963
$990$	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
$993$	−27.0000	−0.856819
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	32.0000	1.01345	0.506725	−	0.862108i	$-0.330856\pi$
$997$	32.0000	1.01345	0.506725	+	0.862108i	$0.330856\pi$
$998$	0	0
$999$	81.0000	2.56273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.e.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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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