LMFDB - Embedded newform 1216.2.a.r.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,2,Mod(1,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1216 = 2^{6} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1216.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1216.1

$q$ -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^{7} +6.00000 q^{9} +2.00000 q^{11} +1.00000 q^{13} +3.00000 q^{17} -1.00000 q^{19} +3.00000 q^{21} -3.00000 q^{23} -5.00000 q^{25} +9.00000 q^{27} -3.00000 q^{29} -8.00000 q^{31} +6.00000 q^{33} +10.0000 q^{37} +3.00000 q^{39} -12.0000 q^{41} +8.00000 q^{43} +8.00000 q^{47} -6.00000 q^{49} +9.00000 q^{51} +9.00000 q^{53} -3.00000 q^{57} -5.00000 q^{59} -10.0000 q^{61} +6.00000 q^{63} +7.00000 q^{67} -9.00000 q^{69} +10.0000 q^{71} +1.00000 q^{73} -15.0000 q^{75} +2.00000 q^{77} +14.0000 q^{79} +9.00000 q^{81} +6.00000 q^{83} -9.00000 q^{87} -4.00000 q^{89} +1.00000 q^{91} -24.0000 q^{93} -6.00000 q^{97} +12.0000 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	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	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
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	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
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	9.00000	1.26025
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−9.00000	−1.08347
$70$	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	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	−15.0000	−1.73205
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−24.0000	−2.48868
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	12.0000	1.20605
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.00000	0.483368	0.241684	−	0.970355i	$-0.422300\pi$
$107$	5.00000	0.483368	0.241684	+	0.970355i	$0.422300\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	30.0000	2.84747
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−36.0000	−3.24601
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	24.0000	2.11308
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.0000	−1.45241	−0.726204	−	0.687479i	$-0.758717\pi$
$137$	−17.0000	−1.45241	−0.726204	+	0.687479i	$0.758717\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	18.0000	1.45521
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	27.0000	2.14124
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−6.00000	−0.458831
$172$	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
$175$	−5.00000	−0.377964
$176$	0	0
$177$	−15.0000	−1.12747
$178$	0	0
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	−30.0000	−2.21766
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.00000	0.438763
$188$	0	0
$189$	9.00000	0.654654
$190$	0	0
$191$	21.0000	1.51951	0.759753	−	0.650211i	$-0.225320\pi$
$191$	21.0000	1.51951	0.759753	+	0.650211i	$0.225320\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	21.0000	1.48123
$202$	0	0
$203$	−3.00000	−0.210559
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.0000	−1.25109
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	30.0000	2.05557
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	3.00000	0.202721
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	−30.0000	−2.00000
$226$	0	0
$227$	−21.0000	−1.39382	−0.696909	−	0.717159i	$-0.745442\pi$
$227$	−21.0000	−1.39382	−0.696909	+	0.717159i	$0.745442\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	42.0000	2.72819
$238$	0	0
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	−18.0000	−1.11417
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	0	0
$273$	3.00000	0.181568
$274$	0	0
$275$	−10.0000	−0.603023
$276$	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
$279$	−48.0000	−2.87368
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−18.0000	−1.05518
$292$	0	0
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.0000	1.04447
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	−30.0000	−1.72345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−42.0000	−2.38930
$310$	0	0
$311$	5.00000	0.283524	0.141762	−	0.989901i	$-0.454723\pi$
$311$	5.00000	0.283524	0.141762	+	0.989901i	$0.454723\pi$
$312$	0	0
$313$	−3.00000	−0.169570	−0.0847850	−	0.996399i	$-0.527020\pi$
$313$	−3.00000	−0.169570	−0.0847850	+	0.996399i	$0.527020\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	15.0000	0.837218
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−5.00000	−0.277350
$326$	0	0
$327$	−27.0000	−1.49310
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	0	0
$333$	60.0000	3.28798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	9.00000	0.480384
$352$	0	0
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	3.00000	0.158334	0.0791670	−	0.996861i	$-0.474774\pi$
$359$	3.00000	0.158334	0.0791670	+	0.996861i	$0.474774\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−21.0000	−1.10221
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	0	0
$369$	−72.0000	−3.74817
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	−21.0000	−1.08734	−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000	−1.08734	−0.543669	+	0.839299i	$0.682965\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	−54.0000	−2.76650
$382$	0	0
$383$	38.0000	1.94171	0.970855	−	0.239669i	$-0.0770389\pi$
$383$	38.0000	1.94171	0.970855	+	0.239669i	$0.0770389\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	48.0000	2.43998
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	−24.0000	−1.21064
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	8.00000	0.399501	0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000	0.399501	0.199750	+	0.979847i	$0.435987\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	12.0000	0.593362	0.296681	−	0.954977i	$-0.404120\pi$
$409$	12.0000	0.593362	0.296681	+	0.954977i	$0.404120\pi$
$410$	0	0
$411$	−51.0000	−2.51564
$412$	0	0
$413$	−5.00000	−0.246034
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−48.0000	−2.35057
$418$	0	0
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	37.0000	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$421$	37.0000	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$422$	0	0
$423$	48.0000	2.33384
$424$	0	0
$425$	−15.0000	−0.727607
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	−30.0000	−1.40952
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	27.0000	1.26025
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	54.0000	2.47249
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	−9.00000	−0.409514
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	0	0
$489$	24.0000	1.08532
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.0000	0.448561
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	48.0000	2.14448
$502$	0	0
$503$	−27.0000	−1.20387	−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000	−1.20387	−0.601935	+	0.798545i	$0.705603\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−36.0000	−1.59882
$508$	0	0
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	1.00000	0.0442374
$512$	0	0
$513$	−9.00000	−0.397360
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−16.0000	−0.700973	−0.350486	−	0.936568i	$-0.613984\pi$
$521$	−16.0000	−0.700973	−0.350486	+	0.936568i	$0.613984\pi$
$522$	0	0
$523$	1.00000	0.0437269	0.0218635	−	0.999761i	$-0.493040\pi$
$523$	1.00000	0.0437269	0.0218635	+	0.999761i	$0.493040\pi$
$524$	0	0
$525$	−15.0000	−0.654654
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−30.0000	−1.30189
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	72.0000	3.10703
$538$	0	0
$539$	−12.0000	−0.516877
$540$	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
$543$	54.0000	2.31736
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	−60.0000	−2.56074
$550$	0	0
$551$	3.00000	0.127804
$552$	0	0
$553$	14.0000	0.595341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.0000	1.35588	0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000	1.35588	0.677942	+	0.735116i	$0.262872\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	63.0000	2.63186
$574$	0	0
$575$	15.0000	0.625543
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	0	0
$579$	30.0000	1.24676
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	18.0000	0.745484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−33.0000	−1.35060
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	42.0000	1.71037
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	0	0
$609$	−9.00000	−0.364698
$610$	0	0
$611$	8.00000	0.323645
$612$	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
$615$	0	0
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	−27.0000	−1.08347
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−6.00000	−0.239617
$628$	0	0
$629$	30.0000	1.19618
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	−3.00000	−0.119239
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	60.0000	2.37356
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	50.0000	1.97181	0.985904	−	0.167313i	$-0.0535092\pi$
$643$	50.0000	1.97181	0.985904	+	0.167313i	$0.0535092\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	−10.0000	−0.392534
$650$	0	0
$651$	−24.0000	−0.940634
$652$	0	0
$653$	4.00000	0.156532	0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000	0.156532	0.0782660	+	0.996933i	$0.475062\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−31.0000	−1.20759	−0.603794	−	0.797140i	$-0.706345\pi$
$659$	−31.0000	−1.20759	−0.603794	+	0.797140i	$0.706345\pi$
$660$	0	0
$661$	39.0000	1.51692	0.758462	−	0.651717i	$-0.225951\pi$
$661$	39.0000	1.51692	0.758462	+	0.651717i	$0.225951\pi$
$662$	0	0
$663$	9.00000	0.349531
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.00000	0.348481
$668$	0	0
$669$	−6.00000	−0.231973
$670$	0	0
$671$	−20.0000	−0.772091
$672$	0	0
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	0	0
$675$	−45.0000	−1.73205
$676$	0	0
$677$	11.0000	0.422764	0.211382	−	0.977403i	$-0.432204\pi$
$677$	11.0000	0.422764	0.211382	+	0.977403i	$0.432204\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	−63.0000	−2.41417
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−66.0000	−2.51806
$688$	0	0
$689$	9.00000	0.342873
$690$	0	0
$691$	18.0000	0.684752	0.342376	−	0.939563i	$-0.388768\pi$
$691$	18.0000	0.684752	0.342376	+	0.939563i	$0.388768\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	78.0000	2.95023
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.0000	−0.376089
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	84.0000	3.15025
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−81.0000	−3.02500
$718$	0	0
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	0	0
$723$	60.0000	2.23142
$724$	0	0
$725$	15.0000	0.557086
$726$	0	0
$727$	−11.0000	−0.407967	−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000	−0.407967	−0.203984	+	0.978974i	$0.565389\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.0000	0.515697
$738$	0	0
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	0	0
$741$	−3.00000	−0.110208
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	36.0000	1.31717
$748$	0	0
$749$	5.00000	0.182696
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	0	0
$759$	−18.0000	−0.653359
$760$	0	0
$761$	−29.0000	−1.05125	−0.525625	−	0.850717i	$-0.676168\pi$
$761$	−29.0000	−1.05125	−0.525625	+	0.850717i	$0.676168\pi$
$762$	0	0
$763$	−9.00000	−0.325822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.00000	−0.180540
$768$	0	0
$769$	13.0000	0.468792	0.234396	−	0.972141i	$-0.424689\pi$
$769$	13.0000	0.468792	0.234396	+	0.972141i	$0.424689\pi$
$770$	0	0
$771$	−36.0000	−1.29651
$772$	0	0
$773$	31.0000	1.11499	0.557496	−	0.830179i	$-0.311762\pi$
$773$	31.0000	1.11499	0.557496	+	0.830179i	$0.311762\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	0	0
$777$	30.0000	1.07624
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	20.0000	0.715656
$782$	0	0
$783$	−27.0000	−0.964901
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−53.0000	−1.88925	−0.944623	−	0.328158i	$-0.893572\pi$
$787$	−53.0000	−1.88925	−0.944623	+	0.328158i	$0.893572\pi$
$788$	0	0
$789$	48.0000	1.70885
$790$	0	0
$791$	2.00000	0.0711118
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.0000	1.31061	0.655304	−	0.755366i	$-0.272541\pi$
$797$	37.0000	1.31061	0.655304	+	0.755366i	$0.272541\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−24.0000	−0.847998
$802$	0	0
$803$	2.00000	0.0705785
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−66.0000	−2.32331
$808$	0	0
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	0	0
$811$	−39.0000	−1.36948	−0.684738	−	0.728790i	$-0.740083\pi$
$811$	−39.0000	−1.36948	−0.684738	+	0.728790i	$0.740083\pi$
$812$	0	0
$813$	−9.00000	−0.315644
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	6.00000	0.209657
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	0	0
$823$	−41.0000	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$823$	−41.0000	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$824$	0	0
$825$	−30.0000	−1.04447
$826$	0	0
$827$	51.0000	1.77344	0.886722	−	0.462303i	$-0.152977\pi$
$827$	51.0000	1.77344	0.886722	+	0.462303i	$0.152977\pi$
$828$	0	0
$829$	23.0000	0.798823	0.399412	−	0.916772i	$-0.369214\pi$
$829$	23.0000	0.798823	0.399412	+	0.916772i	$0.369214\pi$
$830$	0	0
$831$	−36.0000	−1.24883
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−72.0000	−2.48868
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	18.0000	0.617758
$850$	0	0
$851$	−30.0000	−1.02839
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.0000	0.683187	0.341593	−	0.939848i	$-0.389033\pi$
$857$	20.0000	0.683187	0.341593	+	0.939848i	$0.389033\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	0	0
$863$	−38.0000	−1.29354	−0.646768	−	0.762687i	$-0.723880\pi$
$863$	−38.0000	−1.29354	−0.646768	+	0.762687i	$0.723880\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−24.0000	−0.815083
$868$	0	0
$869$	28.0000	0.949835
$870$	0	0
$871$	7.00000	0.237186
$872$	0	0
$873$	−36.0000	−1.21842
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	0	0
$879$	−3.00000	−0.101187
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.0000	0.335767	0.167884	−	0.985807i	$-0.446307\pi$
$887$	10.0000	0.335767	0.167884	+	0.985807i	$0.446307\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	18.0000	0.603023
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.00000	−0.300501
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	27.0000	0.899500
$902$	0	0
$903$	24.0000	0.798670
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33.0000	1.09575	0.547874	−	0.836561i	$-0.315438\pi$
$907$	33.0000	1.09575	0.547874	+	0.836561i	$0.315438\pi$
$908$	0	0
$909$	−60.0000	−1.99007
$910$	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
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.00000	−0.264183
$918$	0	0
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	0	0
$921$	36.0000	1.18624
$922$	0	0
$923$	10.0000	0.329154
$924$	0	0
$925$	−50.0000	−1.64399
$926$	0	0
$927$	−84.0000	−2.75892
$928$	0	0
$929$	−39.0000	−1.27955	−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000	−1.27955	−0.639774	+	0.768563i	$0.720972\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	15.0000	0.491078
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	0	0
$939$	−9.00000	−0.293704
$940$	0	0
$941$	33.0000	1.07577	0.537885	−	0.843018i	$-0.319224\pi$
$941$	33.0000	1.07577	0.537885	+	0.843018i	$0.319224\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.0000	−0.519930	−0.259965	−	0.965618i	$-0.583711\pi$
$947$	−16.0000	−0.519930	−0.259965	+	0.965618i	$0.583711\pi$
$948$	0	0
$949$	1.00000	0.0324614
$950$	0	0
$951$	−63.0000	−2.04291
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−18.0000	−0.581857
$958$	0	0
$959$	−17.0000	−0.548959
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	30.0000	0.966736
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	−9.00000	−0.289122
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	−16.0000	−0.512936
$974$	0	0
$975$	−15.0000	−0.480384
$976$	0	0
$977$	16.0000	0.511885	0.255943	−	0.966692i	$-0.417614\pi$
$977$	16.0000	0.511885	0.255943	+	0.966692i	$0.417614\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−54.0000	−1.72409
$982$	0	0
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	0	0
$993$	39.0000	1.23763
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	0	0
$999$	90.0000	2.84747

Display

a_p

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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	1216.2.a.r.1.1		1
4.3	odd	2		1216.2.a.a.1.1		1
8.3	odd	2		608.2.a.f.1.1	yes	1
8.5	even	2		608.2.a.a.1.1	✓	1
24.5	odd	2		5472.2.a.l.1.1		1
24.11	even	2		5472.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.2.a.a.1.1	✓	1	8.5	even	2
608.2.a.f.1.1	yes	1	8.3	odd	2
1216.2.a.a.1.1		1	4.3	odd	2
1216.2.a.r.1.1		1	1.1	even	1	trivial
5472.2.a.i.1.1		1	24.11	even	2
5472.2.a.l.1.1		1	24.5	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1216.2.a.r.1.1

Level	$1216$
Weight	$2$
Character	1216.1
Self dual	yes
Analytic conductor	$9.710$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$