LMFDB - Embedded newform 936.2.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [936,2,Mod(1,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("936.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.47399762919$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	936.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{5} +0.828427 q^{7} +0.828427 q^{11} +1.00000 q^{13} -4.00000 q^{17} +0.828427 q^{19} -4.00000 q^{23} +3.00000 q^{25} -4.00000 q^{29} -10.4853 q^{31} -2.34315 q^{35} +2.00000 q^{37} -1.17157 q^{41} -5.65685 q^{43} -6.48528 q^{47} -6.31371 q^{49} -2.34315 q^{53} -2.34315 q^{55} -0.828427 q^{59} +9.31371 q^{61} -2.82843 q^{65} -0.828427 q^{67} -14.4853 q^{71} +6.00000 q^{73} +0.686292 q^{77} -4.00000 q^{79} +8.82843 q^{83} +11.3137 q^{85} +4.48528 q^{89} +0.828427 q^{91} -2.34315 q^{95} +17.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 4 q^{11} + 2 q^{13} - 8 q^{17} - 4 q^{19} - 8 q^{23} + 6 q^{25} - 8 q^{29} - 4 q^{31} - 16 q^{35} + 4 q^{37} - 8 q^{41} + 4 q^{47} + 10 q^{49} - 16 q^{53} - 16 q^{55} + 4 q^{59} - 4 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 4 * q^11 + 2 * q^13 - 8 * q^17 - 4 * q^19 - 8 * q^23 + 6 * q^25 - 8 * q^29 - 4 * q^31 - 16 * q^35 + 4 * q^37 - 8 * q^41 + 4 * q^47 + 10 * q^49 - 16 * q^53 - 16 * q^55 + 4 * q^59 - 4 * q^61 + 4 * q^67 - 12 * q^71 + 12 * q^73 + 24 * q^77 - 8 * q^79 + 12 * q^83 - 8 * q^89 - 4 * q^91 - 16 * q^95 + 12 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0.828427	0.313116	0.156558	−	0.987669i	$-0.449960\pi$
$7$	0.828427	0.313116	0.156558	+	0.987669i	$0.449960\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	0.828427	0.190054	0.0950271	−	0.995475i	$-0.469706\pi$
$19$	0.828427	0.190054	0.0950271	+	0.995475i	$0.469706\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	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
$31$	−10.4853	−1.88321	−0.941606	−	0.336717i	$-0.890684\pi$
$31$	−10.4853	−1.88321	−0.941606	+	0.336717i	$0.890684\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.34315	−0.396064
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.17157	−0.182969	−0.0914845	−	0.995807i	$-0.529161\pi$
$41$	−1.17157	−0.182969	−0.0914845	+	0.995807i	$0.529161\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.48528	−0.945976	−0.472988	−	0.881069i	$-0.656825\pi$
$47$	−6.48528	−0.945976	−0.472988	+	0.881069i	$0.656825\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.34315	−0.321856	−0.160928	−	0.986966i	$-0.551449\pi$
$53$	−2.34315	−0.321856	−0.160928	+	0.986966i	$0.551449\pi$
$54$	0	0
$55$	−2.34315	−0.315950
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.828427	−0.107852	−0.0539260	−	0.998545i	$-0.517174\pi$
$59$	−0.828427	−0.107852	−0.0539260	+	0.998545i	$0.517174\pi$
$60$	0	0
$61$	9.31371	1.19250	0.596249	−	0.802799i	$-0.296657\pi$
$61$	9.31371	1.19250	0.596249	+	0.802799i	$0.296657\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.82843	−0.350823
$66$	0	0
$67$	−0.828427	−0.101208	−0.0506042	−	0.998719i	$-0.516115\pi$
$67$	−0.828427	−0.101208	−0.0506042	+	0.998719i	$0.516115\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.4853	−1.71909	−0.859543	−	0.511063i	$-0.829252\pi$
$71$	−14.4853	−1.71909	−0.859543	+	0.511063i	$0.829252\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.686292	0.0782102
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.82843	0.969046	0.484523	−	0.874779i	$-0.338993\pi$
$83$	8.82843	0.969046	0.484523	+	0.874779i	$0.338993\pi$
$84$	0	0
$85$	11.3137	1.22714
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.48528	0.475439	0.237719	−	0.971334i	$-0.423600\pi$
$89$	4.48528	0.475439	0.237719	+	0.971334i	$0.423600\pi$
$90$	0	0
$91$	0.828427	0.0868428
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.34315	−0.240402
$96$	0	0
$97$	17.3137	1.75794	0.878970	−	0.476876i	$-0.158231\pi$
$97$	17.3137	1.75794	0.878970	+	0.476876i	$0.158231\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.3137	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$101$	11.3137	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$102$	0	0
$103$	−6.34315	−0.625009	−0.312504	−	0.949916i	$-0.601168\pi$
$103$	−6.34315	−0.625009	−0.312504	+	0.949916i	$0.601168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−5.31371	−0.508961	−0.254480	−	0.967078i	$-0.581904\pi$
$109$	−5.31371	−0.508961	−0.254480	+	0.967078i	$0.581904\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.65685	−0.908440	−0.454220	−	0.890889i	$-0.650082\pi$
$113$	−9.65685	−0.908440	−0.454220	+	0.890889i	$0.650082\pi$
$114$	0	0
$115$	11.3137	1.05501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.31371	−0.303767
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−13.6569	−1.21185	−0.605925	−	0.795522i	$-0.707197\pi$
$127$	−13.6569	−1.21185	−0.605925	+	0.795522i	$0.707197\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.31371	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$132$	0	0
$133$	0.686292	0.0595090
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.8284	1.95036	0.975182	−	0.221404i	$-0.0710641\pi$
$137$	22.8284	1.95036	0.975182	+	0.221404i	$0.0710641\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.828427	0.0692766
$144$	0	0
$145$	11.3137	0.939552
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.17157	−0.423672	−0.211836	−	0.977305i	$-0.567944\pi$
$149$	−5.17157	−0.423672	−0.211836	+	0.977305i	$0.567944\pi$
$150$	0	0
$151$	−16.8284	−1.36948	−0.684739	−	0.728788i	$-0.740084\pi$
$151$	−16.8284	−1.36948	−0.684739	+	0.728788i	$0.740084\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	29.6569	2.38210
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.31371	−0.261157
$162$	0	0
$163$	−0.828427	−0.0648874	−0.0324437	−	0.999474i	$-0.510329\pi$
$163$	−0.828427	−0.0648874	−0.0324437	+	0.999474i	$0.510329\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.8284	1.61175	0.805876	−	0.592084i	$-0.201695\pi$
$167$	20.8284	1.61175	0.805876	+	0.592084i	$0.201695\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.3137	−0.860165	−0.430083	−	0.902790i	$-0.641516\pi$
$173$	−11.3137	−0.860165	−0.430083	+	0.902790i	$0.641516\pi$
$174$	0	0
$175$	2.48528	0.187870
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.9706	−1.26844	−0.634220	−	0.773153i	$-0.718679\pi$
$179$	−16.9706	−1.26844	−0.634220	+	0.773153i	$0.718679\pi$
$180$	0	0
$181$	2.68629	0.199670	0.0998352	−	0.995004i	$-0.468168\pi$
$181$	2.68629	0.199670	0.0998352	+	0.995004i	$0.468168\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	−3.31371	−0.242322
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.4853	1.17453	0.587264	−	0.809396i	$-0.300205\pi$
$197$	16.4853	1.17453	0.587264	+	0.809396i	$0.300205\pi$
$198$	0	0
$199$	−13.6569	−0.968109	−0.484054	−	0.875038i	$-0.660836\pi$
$199$	−13.6569	−0.968109	−0.484054	+	0.875038i	$0.660836\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.31371	−0.232577
$204$	0	0
$205$	3.31371	0.231439
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.686292	0.0474718
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	−8.68629	−0.589664
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−10.4853	−0.702146	−0.351073	−	0.936348i	$-0.614183\pi$
$223$	−10.4853	−0.702146	−0.351073	+	0.936348i	$0.614183\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.4853	0.695933	0.347966	−	0.937507i	$-0.386872\pi$
$227$	10.4853	0.695933	0.347966	+	0.937507i	$0.386872\pi$
$228$	0	0
$229$	9.31371	0.615467	0.307734	−	0.951473i	$-0.400429\pi$
$229$	9.31371	0.615467	0.307734	+	0.951473i	$0.400429\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.3431	0.677602	0.338801	−	0.940858i	$-0.389979\pi$
$233$	10.3431	0.677602	0.338801	+	0.940858i	$0.389979\pi$
$234$	0	0
$235$	18.3431	1.19657
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.1421	1.04415	0.522074	−	0.852900i	$-0.325158\pi$
$239$	16.1421	1.04415	0.522074	+	0.852900i	$0.325158\pi$
$240$	0	0
$241$	−5.31371	−0.342286	−0.171143	−	0.985246i	$-0.554746\pi$
$241$	−5.31371	−0.342286	−0.171143	+	0.985246i	$0.554746\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	17.8579	1.14090
$246$	0	0
$247$	0.828427	0.0527116
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.6569	−1.61945	−0.809723	−	0.586812i	$-0.800383\pi$
$251$	−25.6569	−1.61945	−0.809723	+	0.586812i	$0.800383\pi$
$252$	0	0
$253$	−3.31371	−0.208331
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.65685	0.352865	0.176432	−	0.984313i	$-0.443544\pi$
$257$	5.65685	0.352865	0.176432	+	0.984313i	$0.443544\pi$
$258$	0	0
$259$	1.65685	0.102952
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.34315	−0.391135	−0.195568	−	0.980690i	$-0.562655\pi$
$263$	−6.34315	−0.391135	−0.195568	+	0.980690i	$0.562655\pi$
$264$	0	0
$265$	6.62742	0.407119
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−23.3137	−1.42146	−0.710731	−	0.703464i	$-0.751636\pi$
$269$	−23.3137	−1.42146	−0.710731	+	0.703464i	$0.751636\pi$
$270$	0	0
$271$	20.1421	1.22355	0.611774	−	0.791033i	$-0.290456\pi$
$271$	20.1421	1.22355	0.611774	+	0.791033i	$0.290456\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.48528	0.149868
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.4558	1.27995	0.639974	−	0.768396i	$-0.278945\pi$
$281$	21.4558	1.27995	0.639974	+	0.768396i	$0.278945\pi$
$282$	0	0
$283$	25.6569	1.52514	0.762571	−	0.646905i	$-0.223937\pi$
$283$	25.6569	1.52514	0.762571	+	0.646905i	$0.223937\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.970563	−0.0572905
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.1421	1.29356	0.646779	−	0.762678i	$-0.276116\pi$
$293$	22.1421	1.29356	0.646779	+	0.762678i	$0.276116\pi$
$294$	0	0
$295$	2.34315	0.136423
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−4.68629	−0.270113
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−26.3431	−1.50840
$306$	0	0
$307$	8.82843	0.503865	0.251932	−	0.967745i	$-0.418934\pi$
$307$	8.82843	0.503865	0.251932	+	0.967745i	$0.418934\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	34.6274	1.96354	0.981770	−	0.190071i	$-0.0608718\pi$
$311$	34.6274	1.96354	0.981770	+	0.190071i	$0.0608718\pi$
$312$	0	0
$313$	−6.68629	−0.377932	−0.188966	−	0.981984i	$-0.560514\pi$
$313$	−6.68629	−0.377932	−0.188966	+	0.981984i	$0.560514\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.1716	0.739789	0.369895	−	0.929074i	$-0.379394\pi$
$317$	13.1716	0.739789	0.369895	+	0.929074i	$0.379394\pi$
$318$	0	0
$319$	−3.31371	−0.185532
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.31371	−0.184380
$324$	0	0
$325$	3.00000	0.166410
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.37258	−0.296200
$330$	0	0
$331$	−8.82843	−0.485254	−0.242627	−	0.970120i	$-0.578009\pi$
$331$	−8.82843	−0.485254	−0.242627	+	0.970120i	$0.578009\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.34315	0.128020
$336$	0	0
$337$	13.3137	0.725244	0.362622	−	0.931936i	$-0.381882\pi$
$337$	13.3137	0.725244	0.362622	+	0.931936i	$0.381882\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.68629	−0.470389
$342$	0	0
$343$	−11.0294	−0.595534
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−33.9411	−1.82206	−0.911028	−	0.412346i	$-0.864710\pi$
$347$	−33.9411	−1.82206	−0.911028	+	0.412346i	$0.864710\pi$
$348$	0	0
$349$	−33.3137	−1.78324	−0.891621	−	0.452783i	$-0.850431\pi$
$349$	−33.3137	−1.78324	−0.891621	+	0.452783i	$0.850431\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.51472	0.187070	0.0935348	−	0.995616i	$-0.470183\pi$
$353$	3.51472	0.187070	0.0935348	+	0.995616i	$0.470183\pi$
$354$	0	0
$355$	40.9706	2.17449
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.1716	−1.43406	−0.717030	−	0.697042i	$-0.754499\pi$
$359$	−27.1716	−1.43406	−0.717030	+	0.697042i	$0.754499\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.9706	−0.888280
$366$	0	0
$367$	−19.3137	−1.00817	−0.504084	−	0.863655i	$-0.668170\pi$
$367$	−19.3137	−1.00817	−0.504084	+	0.863655i	$0.668170\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.94113	−0.100778
$372$	0	0
$373$	−28.6274	−1.48227	−0.741136	−	0.671355i	$-0.765713\pi$
$373$	−28.6274	−1.48227	−0.741136	+	0.671355i	$0.765713\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	16.8284	0.864418	0.432209	−	0.901773i	$-0.357734\pi$
$379$	16.8284	0.864418	0.432209	+	0.901773i	$0.357734\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−36.8284	−1.88184	−0.940922	−	0.338622i	$-0.890039\pi$
$383$	−36.8284	−1.88184	−0.940922	+	0.338622i	$0.890039\pi$
$384$	0	0
$385$	−1.94113	−0.0989289
$386$	0	0
$387$	0	0
$388$	0	0
$389$	20.9706	1.06325	0.531625	−	0.846980i	$-0.321582\pi$
$389$	20.9706	1.06325	0.531625	+	0.846980i	$0.321582\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.85786	0.292528	0.146264	−	0.989246i	$-0.453275\pi$
$401$	5.85786	0.292528	0.146264	+	0.989246i	$0.453275\pi$
$402$	0	0
$403$	−10.4853	−0.522309
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.65685	0.0821272
$408$	0	0
$409$	−21.3137	−1.05390	−0.526948	−	0.849898i	$-0.676664\pi$
$409$	−21.3137	−1.05390	−0.526948	+	0.849898i	$0.676664\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.686292	−0.0337702
$414$	0	0
$415$	−24.9706	−1.22576
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−31.3137	−1.52977	−0.764887	−	0.644164i	$-0.777205\pi$
$419$	−31.3137	−1.52977	−0.764887	+	0.644164i	$0.777205\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	7.71573	0.373390
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.8284	−1.00327	−0.501635	−	0.865079i	$-0.667268\pi$
$431$	−20.8284	−1.00327	−0.501635	+	0.865079i	$0.667268\pi$
$432$	0	0
$433$	16.6274	0.799063	0.399531	−	0.916720i	$-0.369173\pi$
$433$	16.6274	0.799063	0.399531	+	0.916720i	$0.369173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.31371	−0.158516
$438$	0	0
$439$	24.2843	1.15903	0.579513	−	0.814963i	$-0.303243\pi$
$439$	24.2843	1.15903	0.579513	+	0.814963i	$0.303243\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.6274	0.504924	0.252462	−	0.967607i	$-0.418760\pi$
$443$	10.6274	0.504924	0.252462	+	0.967607i	$0.418760\pi$
$444$	0	0
$445$	−12.6863	−0.601388
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.4853	0.589217	0.294608	−	0.955618i	$-0.404811\pi$
$449$	12.4853	0.589217	0.294608	+	0.955618i	$0.404811\pi$
$450$	0	0
$451$	−0.970563	−0.0457020
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.34315	−0.109848
$456$	0	0
$457$	−27.9411	−1.30703	−0.653515	−	0.756913i	$-0.726707\pi$
$457$	−27.9411	−1.30703	−0.653515	+	0.756913i	$0.726707\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.1421	−1.40386	−0.701930	−	0.712246i	$-0.747678\pi$
$461$	−30.1421	−1.40386	−0.701930	+	0.712246i	$0.747678\pi$
$462$	0	0
$463$	−3.85786	−0.179290	−0.0896451	−	0.995974i	$-0.528573\pi$
$463$	−3.85786	−0.179290	−0.0896451	+	0.995974i	$0.528573\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.6569	0.817062	0.408531	−	0.912744i	$-0.366041\pi$
$467$	17.6569	0.817062	0.408531	+	0.912744i	$0.366041\pi$
$468$	0	0
$469$	−0.686292	−0.0316900
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.68629	−0.215476
$474$	0	0
$475$	2.48528	0.114033
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.85786	0.359035	0.179517	−	0.983755i	$-0.442546\pi$
$479$	7.85786	0.359035	0.179517	+	0.983755i	$0.442546\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−48.9706	−2.22364
$486$	0	0
$487$	−33.1127	−1.50048	−0.750240	−	0.661166i	$-0.770062\pi$
$487$	−33.1127	−1.50048	−0.750240	+	0.661166i	$0.770062\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−28.9706	−1.30742	−0.653712	−	0.756744i	$-0.726789\pi$
$491$	−28.9706	−1.30742	−0.653712	+	0.756744i	$0.726789\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	37.7990	1.69212	0.846058	−	0.533092i	$-0.178970\pi$
$499$	37.7990	1.69212	0.846058	+	0.533092i	$0.178970\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.02944	−0.135076	−0.0675380	−	0.997717i	$-0.521514\pi$
$503$	−3.02944	−0.135076	−0.0675380	+	0.997717i	$0.521514\pi$
$504$	0	0
$505$	−32.0000	−1.42398
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.5147	−0.687678	−0.343839	−	0.939029i	$-0.611727\pi$
$509$	−15.5147	−0.687678	−0.343839	+	0.939029i	$0.611727\pi$
$510$	0	0
$511$	4.97056	0.219885
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.9411	0.790580
$516$	0	0
$517$	−5.37258	−0.236286
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.9706	−0.918737	−0.459369	−	0.888246i	$-0.651924\pi$
$521$	−20.9706	−0.918737	−0.459369	+	0.888246i	$0.651924\pi$
$522$	0	0
$523$	26.6274	1.16434	0.582168	−	0.813069i	$-0.302205\pi$
$523$	26.6274	1.16434	0.582168	+	0.813069i	$0.302205\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	41.9411	1.82698
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.17157	−0.0507465
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.23045	−0.225291
$540$	0	0
$541$	23.9411	1.02931	0.514655	−	0.857398i	$-0.327920\pi$
$541$	23.9411	1.02931	0.514655	+	0.857398i	$0.327920\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.0294	0.643790
$546$	0	0
$547$	34.6274	1.48056	0.740281	−	0.672298i	$-0.234693\pi$
$547$	34.6274	1.48056	0.740281	+	0.672298i	$0.234693\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.31371	−0.141169
$552$	0	0
$553$	−3.31371	−0.140913
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.82843	0.119844	0.0599222	−	0.998203i	$-0.480915\pi$
$557$	2.82843	0.119844	0.0599222	+	0.998203i	$0.480915\pi$
$558$	0	0
$559$	−5.65685	−0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.9706	−1.38954	−0.694772	−	0.719230i	$-0.744495\pi$
$563$	−32.9706	−1.38954	−0.694772	+	0.719230i	$0.744495\pi$
$564$	0	0
$565$	27.3137	1.14910
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.9706	−1.04682	−0.523410	−	0.852081i	$-0.675340\pi$
$569$	−24.9706	−1.04682	−0.523410	+	0.852081i	$0.675340\pi$
$570$	0	0
$571$	−9.65685	−0.404127	−0.202063	−	0.979372i	$-0.564765\pi$
$571$	−9.65685	−0.404127	−0.202063	+	0.979372i	$0.564765\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	37.3137	1.55339	0.776695	−	0.629877i	$-0.216895\pi$
$577$	37.3137	1.55339	0.776695	+	0.629877i	$0.216895\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.31371	0.303424
$582$	0	0
$583$	−1.94113	−0.0803932
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.1421	1.49175	0.745873	−	0.666088i	$-0.232033\pi$
$587$	36.1421	1.49175	0.745873	+	0.666088i	$0.232033\pi$
$588$	0	0
$589$	−8.68629	−0.357912
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.8284	−1.26597	−0.632986	−	0.774163i	$-0.718171\pi$
$593$	−30.8284	−1.26597	−0.632986	+	0.774163i	$0.718171\pi$
$594$	0	0
$595$	9.37258	0.384238
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.9706	−1.51058	−0.755288	−	0.655393i	$-0.772503\pi$
$599$	−36.9706	−1.51058	−0.755288	+	0.655393i	$0.772503\pi$
$600$	0	0
$601$	9.31371	0.379914	0.189957	−	0.981792i	$-0.439165\pi$
$601$	9.31371	0.379914	0.189957	+	0.981792i	$0.439165\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29.1716	1.18599
$606$	0	0
$607$	24.0000	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.48528	−0.262366
$612$	0	0
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.1421	0.730375	0.365187	−	0.930934i	$-0.381005\pi$
$617$	18.1421	0.730375	0.365187	+	0.930934i	$0.381005\pi$
$618$	0	0
$619$	−45.7990	−1.84082	−0.920408	−	0.390958i	$-0.872144\pi$
$619$	−45.7990	−1.84082	−0.920408	+	0.390958i	$0.872144\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.71573	0.148868
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−13.5147	−0.538012	−0.269006	−	0.963138i	$-0.586695\pi$
$631$	−13.5147	−0.538012	−0.269006	+	0.963138i	$0.586695\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	38.6274	1.53288
$636$	0	0
$637$	−6.31371	−0.250158
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.3137	−0.920836	−0.460418	−	0.887702i	$-0.652300\pi$
$641$	−23.3137	−0.920836	−0.460418	+	0.887702i	$0.652300\pi$
$642$	0	0
$643$	−28.1421	−1.10982	−0.554909	−	0.831911i	$-0.687247\pi$
$643$	−28.1421	−1.10982	−0.554909	+	0.831911i	$0.687247\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.6274	1.20409	0.602044	−	0.798463i	$-0.294353\pi$
$647$	30.6274	1.20409	0.602044	+	0.798463i	$0.294353\pi$
$648$	0	0
$649$	−0.686292	−0.0269393
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.3137	−0.912336	−0.456168	−	0.889894i	$-0.650778\pi$
$653$	−23.3137	−0.912336	−0.456168	+	0.889894i	$0.650778\pi$
$654$	0	0
$655$	−20.6863	−0.808280
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−35.3137	−1.37563	−0.687813	−	0.725888i	$-0.741429\pi$
$659$	−35.3137	−1.37563	−0.687813	+	0.725888i	$0.741429\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.94113	−0.0752736
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.71573	0.297862
$672$	0	0
$673$	41.3137	1.59253	0.796263	−	0.604950i	$-0.206807\pi$
$673$	41.3137	1.59253	0.796263	+	0.604950i	$0.206807\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.9706	1.26716	0.633581	−	0.773677i	$-0.281584\pi$
$677$	32.9706	1.26716	0.633581	+	0.773677i	$0.281584\pi$
$678$	0	0
$679$	14.3431	0.550439
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2.48528	−0.0950966	−0.0475483	−	0.998869i	$-0.515141\pi$
$683$	−2.48528	−0.0950966	−0.0475483	+	0.998869i	$0.515141\pi$
$684$	0	0
$685$	−64.5685	−2.46704
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.34315	−0.0892667
$690$	0	0
$691$	−24.8284	−0.944518	−0.472259	−	0.881460i	$-0.656561\pi$
$691$	−24.8284	−0.944518	−0.472259	+	0.881460i	$0.656561\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−33.9411	−1.28746
$696$	0	0
$697$	4.68629	0.177506
$698$	0	0
$699$	0	0
$700$	0	0
$701$	44.9706	1.69851	0.849257	−	0.527979i	$-0.177050\pi$
$701$	44.9706	1.69851	0.849257	+	0.527979i	$0.177050\pi$
$702$	0	0
$703$	1.65685	0.0624894
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.37258	0.352492
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	41.9411	1.57071
$714$	0	0
$715$	−2.34315	−0.0876287
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.6863	−0.920643	−0.460322	−	0.887752i	$-0.652266\pi$
$719$	−24.6863	−0.920643	−0.460322	+	0.887752i	$0.652266\pi$
$720$	0	0
$721$	−5.25483	−0.195700
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.0000	−0.445669
$726$	0	0
$727$	−7.02944	−0.260707	−0.130354	−	0.991468i	$-0.541611\pi$
$727$	−7.02944	−0.260707	−0.130354	+	0.991468i	$0.541611\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22.6274	0.836905
$732$	0	0
$733$	2.68629	0.0992204	0.0496102	−	0.998769i	$-0.484202\pi$
$733$	2.68629	0.0992204	0.0496102	+	0.998769i	$0.484202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.686292	−0.0252799
$738$	0	0
$739$	24.8284	0.913328	0.456664	−	0.889639i	$-0.349044\pi$
$739$	24.8284	0.913328	0.456664	+	0.889639i	$0.349044\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.79899	−0.359490	−0.179745	−	0.983713i	$-0.557527\pi$
$743$	−9.79899	−0.359490	−0.179745	+	0.983713i	$0.557527\pi$
$744$	0	0
$745$	14.6274	0.535907
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.31371	−0.121080
$750$	0	0
$751$	−24.2843	−0.886146	−0.443073	−	0.896486i	$-0.646112\pi$
$751$	−24.2843	−0.886146	−0.443073	+	0.896486i	$0.646112\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	47.5980	1.73227
$756$	0	0
$757$	37.3137	1.35619	0.678095	−	0.734974i	$-0.262806\pi$
$757$	37.3137	1.35619	0.678095	+	0.734974i	$0.262806\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.79899	−0.282713	−0.141357	−	0.989959i	$-0.545146\pi$
$761$	−7.79899	−0.282713	−0.141357	+	0.989959i	$0.545146\pi$
$762$	0	0
$763$	−4.40202	−0.159364
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.828427	−0.0299128
$768$	0	0
$769$	−4.62742	−0.166869	−0.0834345	−	0.996513i	$-0.526589\pi$
$769$	−4.62742	−0.166869	−0.0834345	+	0.996513i	$0.526589\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11.7990	−0.424380	−0.212190	−	0.977228i	$-0.568060\pi$
$773$	−11.7990	−0.424380	−0.212190	+	0.977228i	$0.568060\pi$
$774$	0	0
$775$	−31.4558	−1.12993
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.970563	−0.0347740
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.9706	1.74783
$786$	0	0
$787$	−18.4853	−0.658929	−0.329465	−	0.944168i	$-0.606868\pi$
$787$	−18.4853	−0.658929	−0.329465	+	0.944168i	$0.606868\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	9.31371	0.330739
$794$	0	0
$795$	0	0
$796$	0	0
$797$	51.5980	1.82769	0.913847	−	0.406058i	$-0.133097\pi$
$797$	51.5980	1.82769	0.913847	+	0.406058i	$0.133097\pi$
$798$	0	0
$799$	25.9411	0.917731
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.97056	0.175407
$804$	0	0
$805$	9.37258	0.330340
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17.6569	−0.620782	−0.310391	−	0.950609i	$-0.600460\pi$
$809$	−17.6569	−0.620782	−0.310391	+	0.950609i	$0.600460\pi$
$810$	0	0
$811$	15.4558	0.542728	0.271364	−	0.962477i	$-0.412525\pi$
$811$	15.4558	0.542728	0.271364	+	0.962477i	$0.412525\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.34315	0.0820768
$816$	0	0
$817$	−4.68629	−0.163953
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.85786	−0.0648399	−0.0324200	−	0.999474i	$-0.510321\pi$
$821$	−1.85786	−0.0648399	−0.0324200	+	0.999474i	$0.510321\pi$
$822$	0	0
$823$	−33.6569	−1.17320	−0.586602	−	0.809875i	$-0.699535\pi$
$823$	−33.6569	−1.17320	−0.586602	+	0.809875i	$0.699535\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.8284	0.863369	0.431685	−	0.902025i	$-0.357919\pi$
$827$	24.8284	0.863369	0.431685	+	0.902025i	$0.357919\pi$
$828$	0	0
$829$	−10.6863	−0.371150	−0.185575	−	0.982630i	$-0.559415\pi$
$829$	−10.6863	−0.371150	−0.185575	+	0.982630i	$0.559415\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	25.2548	0.875028
$834$	0	0
$835$	−58.9117	−2.03872
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.4853	1.05247	0.526234	−	0.850340i	$-0.323603\pi$
$839$	30.4853	1.05247	0.526234	+	0.850340i	$0.323603\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.82843	−0.0973009
$846$	0	0
$847$	−8.54416	−0.293581
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	50.0000	1.71197	0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000	1.71197	0.855984	+	0.517003i	$0.172952\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.9706	1.67280	0.836401	−	0.548118i	$-0.184655\pi$
$857$	48.9706	1.67280	0.836401	+	0.548118i	$0.184655\pi$
$858$	0	0
$859$	28.6863	0.978764	0.489382	−	0.872070i	$-0.337222\pi$
$859$	28.6863	0.978764	0.489382	+	0.872070i	$0.337222\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.17157	0.107962	0.0539808	−	0.998542i	$-0.482809\pi$
$863$	3.17157	0.107962	0.0539808	+	0.998542i	$0.482809\pi$
$864$	0	0
$865$	32.0000	1.08803
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.31371	−0.112410
$870$	0	0
$871$	−0.828427	−0.0280702
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.68629	0.158426
$876$	0	0
$877$	27.9411	0.943505	0.471752	−	0.881731i	$-0.343622\pi$
$877$	27.9411	0.943505	0.471752	+	0.881731i	$0.343622\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34.3431	1.15705	0.578525	−	0.815665i	$-0.303629\pi$
$881$	34.3431	1.15705	0.578525	+	0.815665i	$0.303629\pi$
$882$	0	0
$883$	−44.9706	−1.51338	−0.756690	−	0.653774i	$-0.773185\pi$
$883$	−44.9706	−1.51338	−0.756690	+	0.653774i	$0.773185\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.97056	−0.301202	−0.150601	−	0.988595i	$-0.548121\pi$
$887$	−8.97056	−0.301202	−0.150601	+	0.988595i	$0.548121\pi$
$888$	0	0
$889$	−11.3137	−0.379450
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.37258	−0.179787
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41.9411	1.39881
$900$	0	0
$901$	9.37258	0.312246
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.59798	−0.252565
$906$	0	0
$907$	27.5980	0.916376	0.458188	−	0.888855i	$-0.348499\pi$
$907$	27.5980	0.916376	0.458188	+	0.888855i	$0.348499\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−50.9117	−1.68678	−0.843390	−	0.537302i	$-0.819443\pi$
$911$	−50.9117	−1.68678	−0.843390	+	0.537302i	$0.819443\pi$
$912$	0	0
$913$	7.31371	0.242048
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.05887	0.200082
$918$	0	0
$919$	50.6274	1.67004	0.835022	−	0.550216i	$-0.185455\pi$
$919$	50.6274	1.67004	0.835022	+	0.550216i	$0.185455\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.4853	−0.476789
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.1127	−1.15201	−0.576005	−	0.817446i	$-0.695389\pi$
$929$	−35.1127	−1.15201	−0.576005	+	0.817446i	$0.695389\pi$
$930$	0	0
$931$	−5.23045	−0.171421
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.37258	0.306516
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31.5147	1.02735	0.513675	−	0.857985i	$-0.328284\pi$
$941$	31.5147	1.02735	0.513675	+	0.857985i	$0.328284\pi$
$942$	0	0
$943$	4.68629	0.152607
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.4853	1.12062	0.560311	−	0.828283i	$-0.310682\pi$
$947$	34.4853	1.12062	0.560311	+	0.828283i	$0.310682\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−10.3431	−0.335047	−0.167524	−	0.985868i	$-0.553577\pi$
$953$	−10.3431	−0.335047	−0.167524	+	0.985868i	$0.553577\pi$
$954$	0	0
$955$	−48.0000	−1.55324
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.9117	0.610690
$960$	0	0
$961$	78.9411	2.54649
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.9706	0.546302
$966$	0	0
$967$	−53.7990	−1.73006	−0.865029	−	0.501721i	$-0.832700\pi$
$967$	−53.7990	−1.73006	−0.865029	+	0.501721i	$0.832700\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	9.94113	0.318698
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.1127	0.355527	0.177763	−	0.984073i	$-0.443114\pi$
$977$	11.1127	0.355527	0.177763	+	0.984073i	$0.443114\pi$
$978$	0	0
$979$	3.71573	0.118755
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.7990	0.567700	0.283850	−	0.958869i	$-0.408388\pi$
$983$	17.7990	0.567700	0.283850	+	0.958869i	$0.408388\pi$
$984$	0	0
$985$	−46.6274	−1.48567
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.6274	0.719510
$990$	0	0
$991$	6.62742	0.210527	0.105263	−	0.994444i	$-0.466431\pi$
$991$	6.62742	0.210527	0.105263	+	0.994444i	$0.466431\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	38.6274	1.22457
$996$	0	0
$997$	44.6274	1.41336	0.706682	−	0.707531i	$-0.250191\pi$
$997$	44.6274	1.41336	0.706682	+	0.707531i	$0.250191\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.2.a.k.1.1	✓	2
3.2	odd	2		936.2.a.l.1.2	yes	2
4.3	odd	2		1872.2.a.y.1.1		2
8.3	odd	2		7488.2.a.cp.1.2		2
8.5	even	2		7488.2.a.ck.1.2		2
12.11	even	2		1872.2.a.x.1.2		2
24.5	odd	2		7488.2.a.ci.1.1		2
24.11	even	2		7488.2.a.cr.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.a.k.1.1	✓	2	1.1	even	1	trivial
936.2.a.l.1.2	yes	2	3.2	odd	2
1872.2.a.x.1.2		2	12.11	even	2
1872.2.a.y.1.1		2	4.3	odd	2
7488.2.a.ci.1.1		2	24.5	odd	2
7488.2.a.ck.1.2		2	8.5	even	2
7488.2.a.cp.1.2		2	8.3	odd	2
7488.2.a.cr.1.1		2	24.11	even	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

Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{5} +0.828427 q^{7} +0.828427 q^{11} +1.00000 q^{13} -4.00000 q^{17} +0.828427 q^{19} -4.00000 q^{23} +3.00000 q^{25} -4.00000 q^{29} -10.4853 q^{31} -2.34315 q^{35} +2.00000 q^{37} -1.17157 q^{41} -5.65685 q^{43} -6.48528 q^{47} -6.31371 q^{49} -2.34315 q^{53} -2.34315 q^{55} -0.828427 q^{59} +9.31371 q^{61} -2.82843 q^{65} -0.828427 q^{67} -14.4853 q^{71} +6.00000 q^{73} +0.686292 q^{77} -4.00000 q^{79} +8.82843 q^{83} +11.3137 q^{85} +4.48528 q^{89} +0.828427 q^{91} -2.34315 q^{95} +17.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 4 q^{11} + 2 q^{13} - 8 q^{17} - 4 q^{19} - 8 q^{23} + 6 q^{25} - 8 q^{29} - 4 q^{31} - 16 q^{35} + 4 q^{37} - 8 q^{41} + 4 q^{47} + 10 q^{49} - 16 q^{53} - 16 q^{55} + 4 q^{59} - 4 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 4 * q^11 + 2 * q^13 - 8 * q^17 - 4 * q^19 - 8 * q^23 + 6 * q^25 - 8 * q^29 - 4 * q^31 - 16 * q^35 + 4 * q^37 - 8 * q^41 + 4 * q^47 + 10 * q^49 - 16 * q^53 - 16 * q^55 + 4 * q^59 - 4 * q^61 + 4 * q^67 - 12 * q^71 + 12 * q^73 + 24 * q^77 - 8 * q^79 + 12 * q^83 - 8 * q^89 - 4 * q^91 - 16 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more