LMFDB - Embedded newform 5408.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5408 = 2^{5} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 416)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5408.1

$q$ -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} +3.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} -4.00000 q^{11} -9.00000 q^{15} +5.00000 q^{17} -6.00000 q^{19} -3.00000 q^{21} +6.00000 q^{23} +4.00000 q^{25} -9.00000 q^{27} -4.00000 q^{29} +12.0000 q^{33} +3.00000 q^{35} +3.00000 q^{37} -12.0000 q^{41} +3.00000 q^{43} +18.0000 q^{45} -7.00000 q^{47} -6.00000 q^{49} -15.0000 q^{51} -2.00000 q^{53} -12.0000 q^{55} +18.0000 q^{57} +2.00000 q^{59} -12.0000 q^{61} +6.00000 q^{63} -4.00000 q^{67} -18.0000 q^{69} +11.0000 q^{71} -6.00000 q^{73} -12.0000 q^{75} -4.00000 q^{77} +6.00000 q^{79} +9.00000 q^{81} -10.0000 q^{83} +15.0000 q^{85} +12.0000 q^{87} -6.00000 q^{89} -18.0000 q^{95} -24.0000 q^{99} +O(q^{100})

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	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−9.00000	−2.32379
$16$	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
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	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	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	12.0000	2.08893
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	0	0
$45$	18.0000	2.68328
$46$	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
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−15.0000	−2.10042
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	18.0000	2.38416
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	−12.0000	−1.53644	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000	−1.53644	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	0	0
$71$	11.0000	1.30546	0.652730	−	0.757591i	$-0.273624\pi$
$71$	11.0000	1.30546	0.652730	+	0.757591i	$0.273624\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	−12.0000	−1.38564
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	15.0000	1.62698
$86$	0	0
$87$	12.0000	1.28654
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−18.0000	−1.84676
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	−24.0000	−2.41209
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	−9.00000	−0.878310
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	0	0
$111$	−9.00000	−0.854242
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	36.0000	3.24601
$124$	0	0
$125$	−3.00000	−0.268328
$126$	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
$129$	−9.00000	−0.792406
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	−27.0000	−2.32379
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	21.0000	1.76852
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	18.0000	1.48461
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	30.0000	2.42536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	36.0000	2.80260
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−36.0000	−2.75299
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	36.0000	2.66120
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	−20.0000	−1.46254
$188$	0	0
$189$	−9.00000	−0.654654
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−36.0000	−2.51435
$206$	0	0
$207$	36.0000	2.50217
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	3.00000	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$212$	0	0
$213$	−33.0000	−2.26112
$214$	0	0
$215$	9.00000	0.613795
$216$	0	0
$217$	0	0
$218$	0	0
$219$	18.0000	1.21633
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−21.0000	−1.40626	−0.703132	−	0.711059i	$-0.748216\pi$
$223$	−21.0000	−1.40626	−0.703132	+	0.711059i	$0.748216\pi$
$224$	0	0
$225$	24.0000	1.60000
$226$	0	0
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	9.00000	0.594737	0.297368	−	0.954763i	$-0.403891\pi$
$229$	9.00000	0.594737	0.297368	+	0.954763i	$0.403891\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	0	0
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	0	0
$235$	−21.0000	−1.36989
$236$	0	0
$237$	−18.0000	−1.16923
$238$	0	0
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−18.0000	−1.14998
$246$	0	0
$247$	0	0
$248$	0	0
$249$	30.0000	1.90117
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	0	0
$255$	−45.0000	−2.81801
$256$	0	0
$257$	5.00000	0.311891	0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000	0.311891	0.155946	+	0.987766i	$0.450158\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	−24.0000	−1.48556
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	18.0000	1.10158
$268$	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
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−16.0000	−0.964836
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	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
$285$	54.0000	3.19868
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	36.0000	2.08893
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	−36.0000	−2.06135
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−18.0000	−1.02398
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	0	0
$315$	18.0000	1.01419
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	−36.0000	−2.00932
$322$	0	0
$323$	−30.0000	−1.66924
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−45.0000	−2.48851
$328$	0	0
$329$	−7.00000	−0.385922
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	0	0
$333$	18.0000	0.986394
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	42.0000	2.28113
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−54.0000	−2.90726
$346$	0	0
$347$	−21.0000	−1.12734	−0.563670	−	0.826000i	$-0.690611\pi$
$347$	−21.0000	−1.12734	−0.563670	+	0.826000i	$0.690611\pi$
$348$	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
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	33.0000	1.75146
$356$	0	0
$357$	−15.0000	−0.793884
$358$	0	0
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−15.0000	−0.787296
$364$	0	0
$365$	−18.0000	−0.942163
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−72.0000	−3.74817
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	12.0000	0.621336	0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000	0.621336	0.310668	+	0.950518i	$0.399447\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	−54.0000	−2.76650
$382$	0	0
$383$	11.0000	0.562074	0.281037	−	0.959697i	$-0.409322\pi$
$383$	11.0000	0.562074	0.281037	+	0.959697i	$0.409322\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	18.0000	0.914991
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	0	0
$393$	27.0000	1.36197
$394$	0	0
$395$	18.0000	0.905678
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	18.0000	0.901127
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	27.0000	1.34164
$406$	0	0
$407$	−12.0000	−0.594818
$408$	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
$411$	−18.0000	−0.887875
$412$	0	0
$413$	2.00000	0.0984136
$414$	0	0
$415$	−30.0000	−1.47264
$416$	0	0
$417$	27.0000	1.32220
$418$	0	0
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−9.00000	−0.438633	−0.219317	−	0.975654i	$-0.570383\pi$
$421$	−9.00000	−0.438633	−0.219317	+	0.975654i	$0.570383\pi$
$422$	0	0
$423$	−42.0000	−2.04211
$424$	0	0
$425$	20.0000	0.970143
$426$	0	0
$427$	−12.0000	−0.580721
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.0000	−0.915198	−0.457599	−	0.889159i	$-0.651290\pi$
$431$	−19.0000	−0.915198	−0.457599	+	0.889159i	$0.651290\pi$
$432$	0	0
$433$	3.00000	0.144171	0.0720854	−	0.997398i	$-0.477035\pi$
$433$	3.00000	0.144171	0.0720854	+	0.997398i	$0.477035\pi$
$434$	0	0
$435$	36.0000	1.72607
$436$	0	0
$437$	−36.0000	−1.72211
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	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
$445$	−18.0000	−0.853282
$446$	0	0
$447$	−18.0000	−0.851371
$448$	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
$451$	48.0000	2.26023
$452$	0	0
$453$	−9.00000	−0.422857
$454$	0	0
$455$	0	0
$456$	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
$459$	−45.0000	−2.10042
$460$	0	0
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−42.0000	−1.93526
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	−24.0000	−1.10120
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	37.0000	1.69057	0.845287	−	0.534313i	$-0.179430\pi$
$479$	37.0000	1.69057	0.845287	+	0.534313i	$0.179430\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−18.0000	−0.819028
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	54.0000	2.44196
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	−72.0000	−3.23616
$496$	0	0
$497$	11.0000	0.493417
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	54.0000	2.38416
$514$	0	0
$515$	18.0000	0.793175
$516$	0	0
$517$	28.0000	1.23144
$518$	0	0
$519$	−42.0000	−1.84360
$520$	0	0
$521$	11.0000	0.481919	0.240959	−	0.970535i	$-0.422538\pi$
$521$	11.0000	0.481919	0.240959	+	0.970535i	$0.422538\pi$
$522$	0	0
$523$	−24.0000	−1.04945	−0.524723	−	0.851273i	$-0.675831\pi$
$523$	−24.0000	−1.04945	−0.524723	+	0.851273i	$0.675831\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	0	0
$535$	36.0000	1.55642
$536$	0	0
$537$	63.0000	2.71865
$538$	0	0
$539$	24.0000	1.03375
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	45.0000	1.92759
$546$	0	0
$547$	33.0000	1.41098	0.705489	−	0.708721i	$-0.250727\pi$
$547$	33.0000	1.41098	0.705489	+	0.708721i	$0.250727\pi$
$548$	0	0
$549$	−72.0000	−3.07289
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	6.00000	0.255146
$554$	0	0
$555$	−27.0000	−1.14609
$556$	0	0
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	60.0000	2.53320
$562$	0	0
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	−42.0000	−1.76695
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	0	0
$571$	21.0000	0.878823	0.439411	−	0.898286i	$-0.355187\pi$
$571$	21.0000	0.878823	0.439411	+	0.898286i	$0.355187\pi$
$572$	0	0
$573$	72.0000	3.00784
$574$	0	0
$575$	24.0000	1.00087
$576$	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
$579$	54.0000	2.24416
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	9.00000	0.370211
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	15.0000	0.614940
$596$	0	0
$597$	−36.0000	−1.47338
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−3.00000	−0.122373	−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000	−0.122373	−0.0611863	+	0.998126i	$0.519488\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	0	0
$605$	15.0000	0.609837
$606$	0	0
$607$	−30.0000	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$607$	−30.0000	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	108.000	4.35498
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	−54.0000	−2.16695
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−72.0000	−2.87540
$628$	0	0
$629$	15.0000	0.598089
$630$	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
$633$	−9.00000	−0.357718
$634$	0	0
$635$	54.0000	2.14292
$636$	0	0
$637$	0	0
$638$	0	0
$639$	66.0000	2.61092
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	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
$645$	−27.0000	−1.06312
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.0000	1.09572	0.547862	−	0.836569i	$-0.315442\pi$
$653$	28.0000	1.09572	0.547862	+	0.836569i	$0.315442\pi$
$654$	0	0
$655$	−27.0000	−1.05498
$656$	0	0
$657$	−36.0000	−1.40449
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−18.0000	−0.698010
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	63.0000	2.43572
$670$	0	0
$671$	48.0000	1.85302
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	−36.0000	−1.38564
$676$	0	0
$677$	−14.0000	−0.538064	−0.269032	−	0.963131i	$-0.586704\pi$
$677$	−14.0000	−0.538064	−0.269032	+	0.963131i	$0.586704\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	−27.0000	−1.03011
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	−27.0000	−1.02417
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	57.0000	2.15594
$700$	0	0
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	0	0
$705$	63.0000	2.37272
$706$	0	0
$707$	4.00000	0.150435
$708$	0	0
$709$	42.0000	1.57734	0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000	1.57734	0.788672	+	0.614815i	$0.210769\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15.0000	−0.560185
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	18.0000	0.669427
$724$	0	0
$725$	−16.0000	−0.594225
$726$	0	0
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	15.0000	0.554795
$732$	0	0
$733$	33.0000	1.21888	0.609441	−	0.792831i	$-0.291394\pi$
$733$	33.0000	1.21888	0.609441	+	0.792831i	$0.291394\pi$
$734$	0	0
$735$	54.0000	1.99182
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.0000	−0.403551	−0.201775	−	0.979432i	$-0.564671\pi$
$743$	−11.0000	−0.403551	−0.201775	+	0.979432i	$0.564671\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	0	0
$747$	−60.0000	−2.19529
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−36.0000	−1.31366	−0.656829	−	0.754039i	$-0.728103\pi$
$751$	−36.0000	−1.31366	−0.656829	+	0.754039i	$0.728103\pi$
$752$	0	0
$753$	36.0000	1.31191
$754$	0	0
$755$	9.00000	0.327544
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	72.0000	2.61343
$760$	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
$763$	15.0000	0.543036
$764$	0	0
$765$	90.0000	3.25396
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−48.0000	−1.73092	−0.865462	−	0.500974i	$-0.832975\pi$
$769$	−48.0000	−1.73092	−0.865462	+	0.500974i	$0.832975\pi$
$770$	0	0
$771$	−15.0000	−0.540212
$772$	0	0
$773$	27.0000	0.971123	0.485561	−	0.874203i	$-0.338615\pi$
$773$	27.0000	0.971123	0.485561	+	0.874203i	$0.338615\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.00000	−0.322873
$778$	0	0
$779$	72.0000	2.57967
$780$	0	0
$781$	−44.0000	−1.57444
$782$	0	0
$783$	36.0000	1.28654
$784$	0	0
$785$	42.0000	1.49904
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	0	0
$789$	−36.0000	−1.28163
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	0	0
$794$	0	0
$795$	18.0000	0.638394
$796$	0	0
$797$	−50.0000	−1.77109	−0.885545	−	0.464553i	$-0.846215\pi$
$797$	−50.0000	−1.77109	−0.885545	+	0.464553i	$0.846215\pi$
$798$	0	0
$799$	−35.0000	−1.23821
$800$	0	0
$801$	−36.0000	−1.27200
$802$	0	0
$803$	24.0000	0.846942
$804$	0	0
$805$	18.0000	0.634417
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	−25.0000	−0.878953	−0.439477	−	0.898254i	$-0.644836\pi$
$809$	−25.0000	−0.878953	−0.439477	+	0.898254i	$0.644836\pi$
$810$	0	0
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	0	0
$813$	33.0000	1.15736
$814$	0	0
$815$	−54.0000	−1.89154
$816$	0	0
$817$	−18.0000	−0.629740
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	−30.0000	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$823$	−30.0000	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$824$	0	0
$825$	48.0000	1.67115
$826$	0	0
$827$	14.0000	0.486828	0.243414	−	0.969923i	$-0.421733\pi$
$827$	14.0000	0.486828	0.243414	+	0.969923i	$0.421733\pi$
$828$	0	0
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	0	0
$831$	36.0000	1.24883
$832$	0	0
$833$	−30.0000	−1.03944
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−54.0000	−1.85986
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	36.0000	1.23552
$850$	0	0
$851$	18.0000	0.617032
$852$	0	0
$853$	33.0000	1.12990	0.564949	−	0.825126i	$-0.308896\pi$
$853$	33.0000	1.12990	0.564949	+	0.825126i	$0.308896\pi$
$854$	0	0
$855$	−108.000	−3.69352
$856$	0	0
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	48.0000	1.63774	0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000	1.63774	0.818869	+	0.573980i	$0.194601\pi$
$860$	0	0
$861$	36.0000	1.22688
$862$	0	0
$863$	29.0000	0.987171	0.493586	−	0.869697i	$-0.335686\pi$
$863$	29.0000	0.987171	0.493586	+	0.869697i	$0.335686\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	0	0
$867$	−24.0000	−0.815083
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−27.0000	−0.911725	−0.455863	−	0.890050i	$-0.650669\pi$
$877$	−27.0000	−0.911725	−0.455863	+	0.890050i	$0.650669\pi$
$878$	0	0
$879$	27.0000	0.910687
$880$	0	0
$881$	41.0000	1.38133	0.690663	−	0.723177i	$-0.257319\pi$
$881$	41.0000	1.38133	0.690663	+	0.723177i	$0.257319\pi$
$882$	0	0
$883$	9.00000	0.302874	0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000	0.302874	0.151437	+	0.988467i	$0.451610\pi$
$884$	0	0
$885$	−18.0000	−0.605063
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	−36.0000	−1.20605
$892$	0	0
$893$	42.0000	1.40548
$894$	0	0
$895$	−63.0000	−2.10586
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−10.0000	−0.333148
$902$	0	0
$903$	−9.00000	−0.299501
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−21.0000	−0.697294	−0.348647	−	0.937254i	$-0.613359\pi$
$907$	−21.0000	−0.697294	−0.348647	+	0.937254i	$0.613359\pi$
$908$	0	0
$909$	24.0000	0.796030
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	40.0000	1.32381
$914$	0	0
$915$	108.000	3.57037
$916$	0	0
$917$	−9.00000	−0.297206
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−6.00000	−0.197707
$922$	0	0
$923$	0	0
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	36.0000	1.18240
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	−18.0000	−0.589294
$934$	0	0
$935$	−60.0000	−1.96221
$936$	0	0
$937$	54.0000	1.76410	0.882052	−	0.471153i	$-0.156162\pi$
$937$	54.0000	1.76410	0.882052	+	0.471153i	$0.156162\pi$
$938$	0	0
$939$	63.0000	2.05593
$940$	0	0
$941$	−3.00000	−0.0977972	−0.0488986	−	0.998804i	$-0.515571\pi$
$941$	−3.00000	−0.0977972	−0.0488986	+	0.998804i	$0.515571\pi$
$942$	0	0
$943$	−72.0000	−2.34464
$944$	0	0
$945$	−27.0000	−0.878310
$946$	0	0
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−41.0000	−1.32812	−0.664060	−	0.747679i	$-0.731168\pi$
$953$	−41.0000	−1.32812	−0.664060	+	0.747679i	$0.731168\pi$
$954$	0	0
$955$	−72.0000	−2.32987
$956$	0	0
$957$	−48.0000	−1.55162
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	72.0000	2.32017
$964$	0	0
$965$	−54.0000	−1.73832
$966$	0	0
$967$	−33.0000	−1.06121	−0.530604	−	0.847620i	$-0.678035\pi$
$967$	−33.0000	−1.06121	−0.530604	+	0.847620i	$0.678035\pi$
$968$	0	0
$969$	90.0000	2.89122
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	−9.00000	−0.288527
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	90.0000	2.87348
$982$	0	0
$983$	−35.0000	−1.11633	−0.558163	−	0.829731i	$-0.688494\pi$
$983$	−35.0000	−1.11633	−0.558163	+	0.829731i	$0.688494\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	0	0
$987$	21.0000	0.668437
$988$	0	0
$989$	18.0000	0.572367
$990$	0	0
$991$	54.0000	1.71537	0.857683	−	0.514178i	$-0.171903\pi$
$991$	54.0000	1.71537	0.857683	+	0.514178i	$0.171903\pi$
$992$	0	0
$993$	78.0000	2.47526
$994$	0	0
$995$	36.0000	1.14128
$996$	0	0
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	0	0
$999$	−27.0000	−0.854242

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.b.1.1		1
4.3	odd	2		5408.2.a.m.1.1		1
13.5	odd	4		416.2.f.a.129.1	✓	2
13.8	odd	4		416.2.f.a.129.2	yes	2
13.12	even	2		5408.2.a.a.1.1		1
39.5	even	4		3744.2.c.c.3457.2		2
39.8	even	4		3744.2.c.c.3457.1		2
52.31	even	4		416.2.f.c.129.1	yes	2
52.47	even	4		416.2.f.c.129.2	yes	2
52.51	odd	2		5408.2.a.l.1.1		1
104.5	odd	4		832.2.f.e.129.2		2
104.21	odd	4		832.2.f.e.129.1		2
104.83	even	4		832.2.f.a.129.2		2
104.99	even	4		832.2.f.a.129.1		2
156.47	odd	4		3744.2.c.b.3457.1		2
156.83	odd	4		3744.2.c.b.3457.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.f.a.129.1	✓	2	13.5	odd	4
416.2.f.a.129.2	yes	2	13.8	odd	4
416.2.f.c.129.1	yes	2	52.31	even	4
416.2.f.c.129.2	yes	2	52.47	even	4
832.2.f.a.129.1		2	104.99	even	4
832.2.f.a.129.2		2	104.83	even	4
832.2.f.e.129.1		2	104.21	odd	4
832.2.f.e.129.2		2	104.5	odd	4
3744.2.c.b.3457.1		2	156.47	odd	4
3744.2.c.b.3457.2		2	156.83	odd	4
3744.2.c.c.3457.1		2	39.8	even	4
3744.2.c.c.3457.2		2	39.5	even	4
5408.2.a.a.1.1		1	13.12	even	2
5408.2.a.b.1.1		1	1.1	even	1	trivial
5408.2.a.l.1.1		1	52.51	odd	2
5408.2.a.m.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5408.2.a.b.1.1

Level	$5408$
Weight	$2$
Character	5408.1
Self dual	yes
Analytic conductor	$43.183$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$