LMFDB - Embedded newform 8008.2.a.b.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-2.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} -2.00000 q^{17} -5.00000 q^{19} +2.00000 q^{21} +9.00000 q^{23} -4.00000 q^{25} +4.00000 q^{27} +7.00000 q^{29} -1.00000 q^{31} +2.00000 q^{33} -1.00000 q^{35} -6.00000 q^{37} -2.00000 q^{39} +10.0000 q^{41} -11.0000 q^{43} +1.00000 q^{45} -7.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} +3.00000 q^{53} -1.00000 q^{55} +10.0000 q^{57} +14.0000 q^{61} -1.00000 q^{63} +1.00000 q^{65} -2.00000 q^{67} -18.0000 q^{69} +10.0000 q^{71} -1.00000 q^{73} +8.00000 q^{75} +1.00000 q^{77} -7.00000 q^{79} -11.0000 q^{81} +9.00000 q^{83} -2.00000 q^{85} -14.0000 q^{87} +7.00000 q^{89} -1.00000 q^{91} +2.00000 q^{93} -5.00000 q^{95} -9.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	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
$9$	1.00000	0.333333
$10$	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
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	10.0000	1.32453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	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
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−14.0000	−1.50096
$88$	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
$93$	2.00000	0.207390
$94$	0	0
$95$	−5.00000	−0.512989
$96$	0	0
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	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
$109$	−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
$111$	12.0000	1.13899
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−20.0000	−1.80334
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	22.0000	1.93699
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	14.0000	1.17901
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	7.00000	0.581318
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−9.00000	−0.709299
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−17.0000	−1.31550	−0.657750	−	0.753237i	$-0.728492\pi$
$167$	−17.0000	−1.31550	−0.657750	+	0.753237i	$0.728492\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.00000	−0.382360
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	−28.0000	−2.06982
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−7.00000	−0.491304
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	9.00000	0.625543
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	−20.0000	−1.37038
$214$	0	0
$215$	−11.0000	−0.750194
$216$	0	0
$217$	1.00000	0.0678844
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	29.0000	1.94198	0.970992	−	0.239113i	$-0.0768565\pi$
$223$	29.0000	1.94198	0.970992	+	0.239113i	$0.0768565\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	−7.00000	−0.456630
$236$	0	0
$237$	14.0000	0.909398
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	27.0000	1.73922	0.869611	−	0.493737i	$-0.164369\pi$
$241$	27.0000	1.73922	0.869611	+	0.493737i	$0.164369\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−5.00000	−0.318142
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	7.00000	0.433289
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	3.00000	0.184289
$266$	0	0
$267$	−14.0000	−0.856786
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	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
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	10.0000	0.592349
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	18.0000	1.05518
$292$	0	0
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	9.00000	0.520483
$300$	0	0
$301$	11.0000	0.634029
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.0000	0.801638
$306$	0	0
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	0	0
$319$	−7.00000	−0.391925
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	10.0000	0.556415
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	32.0000	1.76960
$328$	0	0
$329$	7.00000	0.385922
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	−15.0000	−0.817102	−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000	−0.817102	−0.408551	+	0.912735i	$0.633966\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−18.0000	−0.969087
$346$	0	0
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	0	0
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−1.00000	−0.0523424
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	−3.00000	−0.155752
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	18.0000	0.929516
$376$	0	0
$377$	7.00000	0.360518
$378$	0	0
$379$	−36.0000	−1.84920	−0.924598	−	0.380945i	$-0.875599\pi$
$379$	−36.0000	−1.84920	−0.924598	+	0.380945i	$0.875599\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−11.0000	−0.559161
$388$	0	0
$389$	−38.0000	−1.92668	−0.963338	−	0.268290i	$-0.913542\pi$
$389$	−38.0000	−1.92668	−0.963338	+	0.268290i	$0.913542\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.00000	−0.352208
$396$	0	0
$397$	−9.00000	−0.451697	−0.225849	−	0.974162i	$-0.572515\pi$
$397$	−9.00000	−0.451697	−0.225849	+	0.974162i	$0.572515\pi$
$398$	0	0
$399$	−10.0000	−0.500626
$400$	0	0
$401$	−32.0000	−1.59800	−0.799002	−	0.601329i	$-0.794638\pi$
$401$	−32.0000	−1.59800	−0.799002	+	0.601329i	$0.794638\pi$
$402$	0	0
$403$	−1.00000	−0.0498135
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	−36.0000	−1.77575
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9.00000	0.441793
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	−7.00000	−0.340352
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	−14.0000	−0.677507
$428$	0	0
$429$	2.00000	0.0965609
$430$	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
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	−14.0000	−0.671249
$436$	0	0
$437$	−45.0000	−2.15264
$438$	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
$441$	1.00000	0.0476190
$442$	0	0
$443$	−39.0000	−1.85295	−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000	−1.85295	−0.926473	+	0.376361i	$0.877175\pi$
$444$	0	0
$445$	7.00000	0.331832
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−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
$451$	−10.0000	−0.470882
$452$	0	0
$453$	36.0000	1.69143
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	2.00000	0.0927478
$466$	0	0
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	−36.0000	−1.65879
$472$	0	0
$473$	11.0000	0.505781
$474$	0	0
$475$	20.0000	0.917663
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	23.0000	1.05090	0.525448	−	0.850825i	$-0.323898\pi$
$479$	23.0000	1.05090	0.525448	+	0.850825i	$0.323898\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	18.0000	0.819028
$484$	0	0
$485$	−9.00000	−0.408669
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	0	0
$489$	40.0000	1.80886
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	−14.0000	−0.630528
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−10.0000	−0.448561
$498$	0	0
$499$	−18.0000	−0.805791	−0.402895	−	0.915246i	$-0.631996\pi$
$499$	−18.0000	−0.805791	−0.402895	+	0.915246i	$0.631996\pi$
$500$	0	0
$501$	34.0000	1.51901
$502$	0	0
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.00000	−0.0888231
$508$	0	0
$509$	−7.00000	−0.310270	−0.155135	−	0.987893i	$-0.549581\pi$
$509$	−7.00000	−0.310270	−0.155135	+	0.987893i	$0.549581\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	7.00000	0.307860
$518$	0	0
$519$	20.0000	0.877903
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	2.00000	0.0871214
$528$	0	0
$529$	58.0000	2.52174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.0000	0.433148
$534$	0	0
$535$	4.00000	0.172935
$536$	0	0
$537$	46.0000	1.98505
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	24.0000	1.02994
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	−35.0000	−1.49105
$552$	0	0
$553$	7.00000	0.297670
$554$	0	0
$555$	12.0000	0.509372
$556$	0	0
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	0	0
$567$	11.0000	0.461957
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−3.00000	−0.125546	−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000	−0.125546	−0.0627730	+	0.998028i	$0.519994\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−36.0000	−1.50130
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	−52.0000	−2.16105
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	−3.00000	−0.124247
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	9.00000	0.371470	0.185735	−	0.982600i	$-0.440533\pi$
$587$	9.00000	0.371470	0.185735	+	0.982600i	$0.440533\pi$
$588$	0	0
$589$	5.00000	0.206021
$590$	0	0
$591$	−4.00000	−0.164538
$592$	0	0
$593$	−43.0000	−1.76580	−0.882899	−	0.469563i	$-0.844412\pi$
$593$	−43.0000	−1.76580	−0.882899	+	0.469563i	$0.844412\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	28.0000	1.14596
$598$	0	0
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	14.0000	0.567309
$610$	0	0
$611$	−7.00000	−0.283190
$612$	0	0
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	0	0
$615$	−20.0000	−0.806478
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	36.0000	1.44463
$622$	0	0
$623$	−7.00000	−0.280449
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−10.0000	−0.399362
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	0	0
$633$	−26.0000	−1.03341
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	−20.0000	−0.788723	−0.394362	−	0.918955i	$-0.629034\pi$
$643$	−20.0000	−0.788723	−0.394362	+	0.918955i	$0.629034\pi$
$644$	0	0
$645$	22.0000	0.866249
$646$	0	0
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.00000	−0.0390137
$658$	0	0
$659$	−1.00000	−0.0389545	−0.0194772	−	0.999810i	$-0.506200\pi$
$659$	−1.00000	−0.0389545	−0.0194772	+	0.999810i	$0.506200\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	5.00000	0.193892
$666$	0	0
$667$	63.0000	2.43937
$668$	0	0
$669$	−58.0000	−2.24241
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	−23.0000	−0.886585	−0.443292	−	0.896377i	$-0.646190\pi$
$673$	−23.0000	−0.886585	−0.443292	+	0.896377i	$0.646190\pi$
$674$	0	0
$675$	−16.0000	−0.615840
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	9.00000	0.345388
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	20.0000	0.763048
$688$	0	0
$689$	3.00000	0.114291
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−39.0000	−1.47301	−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000	−1.47301	−0.736505	+	0.676432i	$0.763525\pi$
$702$	0	0
$703$	30.0000	1.13147
$704$	0	0
$705$	14.0000	0.527271
$706$	0	0
$707$	0	0
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	−7.00000	−0.262521
$712$	0	0
$713$	−9.00000	−0.337053
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	0	0
$717$	−48.0000	−1.79259
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	−54.0000	−2.00828
$724$	0	0
$725$	−28.0000	−1.03989
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	22.0000	0.813699
$732$	0	0
$733$	−9.00000	−0.332423	−0.166211	−	0.986090i	$-0.553153\pi$
$733$	−9.00000	−0.332423	−0.166211	+	0.986090i	$0.553153\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	2.00000	0.0736709
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.00000	0.329293
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−15.0000	−0.547358	−0.273679	−	0.961821i	$-0.588241\pi$
$751$	−15.0000	−0.547358	−0.273679	+	0.961821i	$0.588241\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	−18.0000	−0.655087
$756$	0	0
$757$	−31.0000	−1.12671	−0.563357	−	0.826214i	$-0.690490\pi$
$757$	−31.0000	−1.12671	−0.563357	+	0.826214i	$0.690490\pi$
$758$	0	0
$759$	18.0000	0.653359
$760$	0	0
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	−12.0000	−0.430498
$778$	0	0
$779$	−50.0000	−1.79144
$780$	0	0
$781$	−10.0000	−0.357828
$782$	0	0
$783$	28.0000	1.00064
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	−1.00000	−0.0356462	−0.0178231	−	0.999841i	$-0.505674\pi$
$787$	−1.00000	−0.0356462	−0.0178231	+	0.999841i	$0.505674\pi$
$788$	0	0
$789$	18.0000	0.640817
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	14.0000	0.497155
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	14.0000	0.495284
$800$	0	0
$801$	7.00000	0.247333
$802$	0	0
$803$	1.00000	0.0352892
$804$	0	0
$805$	−9.00000	−0.317208
$806$	0	0
$807$	8.00000	0.281613
$808$	0	0
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	55.0000	1.92421
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−8.00000	−0.278524
$826$	0	0
$827$	24.0000	0.834562	0.417281	−	0.908778i	$-0.362983\pi$
$827$	24.0000	0.834562	0.417281	+	0.908778i	$0.362983\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	38.0000	1.31821
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−17.0000	−0.588309
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	−32.0000	−1.10214
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	24.0000	0.823678
$850$	0	0
$851$	−54.0000	−1.85110
$852$	0	0
$853$	1.00000	0.0342393	0.0171197	−	0.999853i	$-0.494550\pi$
$853$	1.00000	0.0342393	0.0171197	+	0.999853i	$0.494550\pi$
$854$	0	0
$855$	−5.00000	−0.170996
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	26.0000	0.883006
$868$	0	0
$869$	7.00000	0.237459
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	−9.00000	−0.304604
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	11.0000	0.368514
$892$	0	0
$893$	35.0000	1.17123
$894$	0	0
$895$	−23.0000	−0.768805
$896$	0	0
$897$	−18.0000	−0.601003
$898$	0	0
$899$	−7.00000	−0.233463
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−22.0000	−0.732114
$904$	0	0
$905$	−12.0000	−0.398893
$906$	0	0
$907$	−51.0000	−1.69343	−0.846714	−	0.532049i	$-0.821422\pi$
$907$	−51.0000	−1.69343	−0.846714	+	0.532049i	$0.821422\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.00000	−0.165657	−0.0828287	−	0.996564i	$-0.526395\pi$
$911$	−5.00000	−0.165657	−0.0828287	+	0.996564i	$0.526395\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	0	0
$915$	−28.0000	−0.925651
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−50.0000	−1.64756
$922$	0	0
$923$	10.0000	0.329154
$924$	0	0
$925$	24.0000	0.789115
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	13.0000	0.426516	0.213258	−	0.976996i	$-0.431592\pi$
$929$	13.0000	0.426516	0.213258	+	0.976996i	$0.431592\pi$
$930$	0	0
$931$	−5.00000	−0.163868
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	2.00000	0.0654070
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	32.0000	1.04428
$940$	0	0
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	0	0
$943$	90.0000	2.93080
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	22.0000	0.714904	0.357452	−	0.933932i	$-0.383646\pi$
$947$	22.0000	0.714904	0.357452	+	0.933932i	$0.383646\pi$
$948$	0	0
$949$	−1.00000	−0.0324614
$950$	0	0
$951$	−16.0000	−0.518836
$952$	0	0
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	14.0000	0.452556
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	26.0000	0.836970
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	−20.0000	−0.642493
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8.00000	0.256205
$976$	0	0
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	−7.00000	−0.223721
$980$	0	0
$981$	−16.0000	−0.510841
$982$	0	0
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	−14.0000	−0.445625
$988$	0	0
$989$	−99.0000	−3.14802
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	−14.0000	−0.443830
$996$	0	0
$997$	12.0000	0.380044	0.190022	−	0.981780i	$-0.439144\pi$
$997$	12.0000	0.380044	0.190022	+	0.981780i	$0.439144\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
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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