LMFDB - Embedded newform 9216.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, 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("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9216 = 2^{10} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9216.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:	$73.5901305028$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.27133$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79793 q^{5} +0.158942 q^{7} +O(q^{10})$	$q-1.79793 q^{5} +0.158942 q^{7} +5.37109 q^{11} -5.95687 q^{13} +3.05320 q^{17} +3.05320 q^{19} -2.82843 q^{23} -1.76744 q^{25} +2.96951 q^{29} -4.15894 q^{31} -0.285766 q^{35} -8.46742 q^{37} -2.60365 q^{41} +8.13853 q^{43} +2.82843 q^{47} -6.97474 q^{49} +5.03049 q^{53} -9.65685 q^{55} +5.65685 q^{59} -5.18944 q^{61} +10.7101 q^{65} +1.08532 q^{67} +0.317883 q^{71} +1.33897 q^{73} +0.853690 q^{77} -9.69382 q^{79} +0.163788 q^{83} -5.48946 q^{85} -14.3990 q^{89} -0.946795 q^{91} -5.48946 q^{95} -0.571533 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^7	$ 4 q + 4 q^{5} - 4 q^{7} - 8 q^{13} + 4 q^{25} + 12 q^{29} - 12 q^{31} - 16 q^{37} + 4 q^{49} + 20 q^{53} - 16 q^{55} - 16 q^{61} + 8 q^{65} - 16 q^{67} - 8 q^{71} - 8 q^{73} + 24 q^{77} - 12 q^{79} - 24 q^{85} + 8 q^{89} - 16 q^{91} - 24 q^{95}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 - 8 * q^13 + 4 * q^25 + 12 * q^29 - 12 * q^31 - 16 * q^37 + 4 * q^49 + 20 * q^53 - 16 * q^55 - 16 * q^61 + 8 * q^65 - 16 * q^67 - 8 * q^71 - 8 * q^73 + 24 * q^77 - 12 * q^79 - 24 * q^85 + 8 * q^89 - 16 * q^91 - 24 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.79793	−0.804060	−0.402030	−	0.915627i	$-0.631695\pi$
$5$	−1.79793	−0.804060	−0.402030	+	0.915627i	$0.631695\pi$
$6$	0	0
$7$	0.158942	0.0600743	0.0300371	−	0.999549i	$-0.490437\pi$
$7$	0.158942	0.0600743	0.0300371	+	0.999549i	$0.490437\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.37109	1.61944	0.809722	−	0.586814i	$-0.199618\pi$
$11$	5.37109	1.61944	0.809722	+	0.586814i	$0.199618\pi$
$12$	0	0
$13$	−5.95687	−1.65214	−0.826070	−	0.563568i	$-0.809428\pi$
$13$	−5.95687	−1.65214	−0.826070	+	0.563568i	$0.809428\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.05320	0.740511	0.370255	−	0.928930i	$-0.379270\pi$
$17$	3.05320	0.740511	0.370255	+	0.928930i	$0.379270\pi$
$18$	0	0
$19$	3.05320	0.700453	0.350227	−	0.936665i	$-0.386105\pi$
$19$	3.05320	0.700453	0.350227	+	0.936665i	$0.386105\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−1.76744	−0.353488
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.96951	0.551423	0.275712	−	0.961240i	$-0.411087\pi$
$29$	2.96951	0.551423	0.275712	+	0.961240i	$0.411087\pi$
$30$	0	0
$31$	−4.15894	−0.746968	−0.373484	−	0.927637i	$-0.621837\pi$
$31$	−4.15894	−0.746968	−0.373484	+	0.927637i	$0.621837\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.285766	−0.0483033
$36$	0	0
$37$	−8.46742	−1.39203	−0.696017	−	0.718025i	$-0.745046\pi$
$37$	−8.46742	−1.39203	−0.696017	+	0.718025i	$0.745046\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.60365	−0.406622	−0.203311	−	0.979114i	$-0.565170\pi$
$41$	−2.60365	−0.406622	−0.203311	+	0.979114i	$0.565170\pi$
$42$	0	0
$43$	8.13853	1.24111	0.620557	−	0.784162i	$-0.286907\pi$
$43$	8.13853	1.24111	0.620557	+	0.784162i	$0.286907\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−6.97474	−0.996391
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.03049	0.690992	0.345496	−	0.938420i	$-0.387711\pi$
$53$	5.03049	0.690992	0.345496	+	0.938420i	$0.387711\pi$
$54$	0	0
$55$	−9.65685	−1.30213
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.65685	0.736460	0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685	0.736460	0.368230	+	0.929735i	$0.379964\pi$
$60$	0	0
$61$	−5.18944	−0.664439	−0.332220	−	0.943202i	$-0.607798\pi$
$61$	−5.18944	−0.664439	−0.332220	+	0.943202i	$0.607798\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	10.7101	1.32842
$66$	0	0
$67$	1.08532	0.132593	0.0662966	−	0.997800i	$-0.478882\pi$
$67$	1.08532	0.132593	0.0662966	+	0.997800i	$0.478882\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.317883	0.0377258	0.0188629	−	0.999822i	$-0.493995\pi$
$71$	0.317883	0.0377258	0.0188629	+	0.999822i	$0.493995\pi$
$72$	0	0
$73$	1.33897	0.156715	0.0783573	−	0.996925i	$-0.475032\pi$
$73$	1.33897	0.156715	0.0783573	+	0.996925i	$0.475032\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.853690	0.0972870
$78$	0	0
$79$	−9.69382	−1.09064	−0.545320	−	0.838228i	$-0.683592\pi$
$79$	−9.69382	−1.09064	−0.545320	+	0.838228i	$0.683592\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.163788	0.0179781	0.00898906	−	0.999960i	$-0.497139\pi$
$83$	0.163788	0.0179781	0.00898906	+	0.999960i	$0.497139\pi$
$84$	0	0
$85$	−5.48946	−0.595415
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.3990	−1.52629	−0.763147	−	0.646225i	$-0.776347\pi$
$89$	−14.3990	−1.52629	−0.763147	+	0.646225i	$0.776347\pi$
$90$	0	0
$91$	−0.946795	−0.0992511
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.48946	−0.563206
$96$	0	0
$97$	−0.571533	−0.0580304	−0.0290152	−	0.999579i	$-0.509237\pi$
$97$	−0.571533	−0.0580304	−0.0290152	+	0.999579i	$0.509237\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.1158	1.00656	0.503281	−	0.864123i	$-0.332126\pi$
$101$	10.1158	1.00656	0.503281	+	0.864123i	$0.332126\pi$
$102$	0	0
$103$	11.3507	1.11841	0.559207	−	0.829028i	$-0.311106\pi$
$103$	11.3507	1.11841	0.559207	+	0.829028i	$0.311106\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.02109	0.0987123	0.0493561	−	0.998781i	$-0.484283\pi$
$107$	1.02109	0.0987123	0.0493561	+	0.998781i	$0.484283\pi$
$108$	0	0
$109$	−2.04313	−0.195696	−0.0978480	−	0.995201i	$-0.531196\pi$
$109$	−2.04313	−0.195696	−0.0978480	+	0.995201i	$0.531196\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.53488	−0.332533	−0.166267	−	0.986081i	$-0.553171\pi$
$113$	−3.53488	−0.332533	−0.166267	+	0.986081i	$0.553171\pi$
$114$	0	0
$115$	5.08532	0.474209
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.485281	0.0444857
$120$	0	0
$121$	17.8486	1.62260
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1674	1.08829
$126$	0	0
$127$	1.49791	0.132918	0.0664591	−	0.997789i	$-0.478830\pi$
$127$	1.49791	0.132918	0.0664591	+	0.997789i	$0.478830\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.7422	−1.28803	−0.644015	−	0.765013i	$-0.722733\pi$
$131$	−14.7422	−1.28803	−0.644015	+	0.765013i	$0.722733\pi$
$132$	0	0
$133$	0.485281	0.0420792
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.7954	−1.17862	−0.589309	−	0.807907i	$-0.700600\pi$
$137$	−13.7954	−1.17862	−0.589309	+	0.807907i	$0.700600\pi$
$138$	0	0
$139$	−3.42847	−0.290799	−0.145399	−	0.989373i	$-0.546447\pi$
$139$	−3.42847	−0.290799	−0.145399	+	0.989373i	$0.546447\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−31.9949	−2.67555
$144$	0	0
$145$	−5.33897	−0.443377
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.14108	0.339250	0.169625	−	0.985509i	$-0.445744\pi$
$149$	4.14108	0.339250	0.169625	+	0.985509i	$0.445744\pi$
$150$	0	0
$151$	22.6644	1.84440	0.922201	−	0.386712i	$-0.126389\pi$
$151$	22.6644	1.84440	0.922201	+	0.386712i	$0.126389\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.47750	0.600607
$156$	0	0
$157$	3.93161	0.313777	0.156888	−	0.987616i	$-0.449854\pi$
$157$	3.93161	0.313777	0.156888	+	0.987616i	$0.449854\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.449555	−0.0354299
$162$	0	0
$163$	−7.68897	−0.602247	−0.301123	−	0.953585i	$-0.597362\pi$
$163$	−7.68897	−0.602247	−0.301123	+	0.953585i	$0.597362\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.95458	−0.306015	−0.153007	−	0.988225i	$-0.548896\pi$
$167$	−3.95458	−0.306015	−0.153007	+	0.988225i	$0.548896\pi$
$168$	0	0
$169$	22.4844	1.72957
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.6011	1.71833	0.859165	−	0.511699i	$-0.170984\pi$
$173$	22.6011	1.71833	0.859165	+	0.511699i	$0.170984\pi$
$174$	0	0
$175$	−0.280920	−0.0212355
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.2981	−1.29292	−0.646462	−	0.762946i	$-0.723752\pi$
$179$	−17.2981	−1.29292	−0.646462	+	0.762946i	$0.723752\pi$
$180$	0	0
$181$	−8.14953	−0.605750	−0.302875	−	0.953030i	$-0.597946\pi$
$181$	−8.14953	−0.605750	−0.302875	+	0.953030i	$0.597946\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15.2238	1.11928
$186$	0	0
$187$	16.3990	1.19922
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.1674	−1.16983	−0.584916	−	0.811094i	$-0.698873\pi$
$191$	−16.1674	−1.16983	−0.584916	+	0.811094i	$0.698873\pi$
$192$	0	0
$193$	−22.1454	−1.59406	−0.797030	−	0.603940i	$-0.793597\pi$
$193$	−22.1454	−1.59406	−0.797030	+	0.603940i	$0.793597\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.2222	1.44077	0.720387	−	0.693572i	$-0.243964\pi$
$197$	20.2222	1.44077	0.720387	+	0.693572i	$0.243964\pi$
$198$	0	0
$199$	−25.0075	−1.77274	−0.886368	−	0.462981i	$-0.846780\pi$
$199$	−25.0075	−1.77274	−0.886368	+	0.462981i	$0.846780\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.471978	0.0331264
$204$	0	0
$205$	4.68119	0.326948
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.3990	1.13434
$210$	0	0
$211$	−26.0559	−1.79376	−0.896881	−	0.442273i	$-0.854172\pi$
$211$	−26.0559	−1.79376	−0.896881	+	0.442273i	$0.854172\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−14.6325	−0.997930
$216$	0	0
$217$	−0.661029	−0.0448736
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−18.1876	−1.22343
$222$	0	0
$223$	−18.3465	−1.22857	−0.614286	−	0.789083i	$-0.710556\pi$
$223$	−18.3465	−1.22857	−0.614286	+	0.789083i	$0.710556\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.163788	0.0108710	0.00543551	−	0.999985i	$-0.498270\pi$
$227$	0.163788	0.0108710	0.00543551	+	0.999985i	$0.498270\pi$
$228$	0	0
$229$	4.02756	0.266148	0.133074	−	0.991106i	$-0.457515\pi$
$229$	4.02756	0.266148	0.133074	+	0.991106i	$0.457515\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.7211	0.767874	0.383937	−	0.923359i	$-0.374568\pi$
$233$	11.7211	0.767874	0.383937	+	0.923359i	$0.374568\pi$
$234$	0	0
$235$	−5.08532	−0.331730
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.6517	0.883058	0.441529	−	0.897247i	$-0.354436\pi$
$239$	13.6517	0.883058	0.441529	+	0.897247i	$0.354436\pi$
$240$	0	0
$241$	−2.13167	−0.137313	−0.0686565	−	0.997640i	$-0.521871\pi$
$241$	−2.13167	−0.137313	−0.0686565	+	0.997640i	$0.521871\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.5401	0.801158
$246$	0	0
$247$	−18.1876	−1.15725
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.27020	−0.395771	−0.197886	−	0.980225i	$-0.563407\pi$
$251$	−6.27020	−0.395771	−0.197886	+	0.980225i	$0.563407\pi$
$252$	0	0
$253$	−15.1917	−0.955096
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.0853	−0.940997	−0.470498	−	0.882401i	$-0.655926\pi$
$257$	−15.0853	−0.940997	−0.470498	+	0.882401i	$0.655926\pi$
$258$	0	0
$259$	−1.34583	−0.0836255
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.1706	−1.61375	−0.806875	−	0.590722i	$-0.798843\pi$
$263$	−26.1706	−1.61375	−0.806875	+	0.590722i	$0.798843\pi$
$264$	0	0
$265$	−9.04449	−0.555599
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.1580	−0.741286	−0.370643	−	0.928775i	$-0.620863\pi$
$269$	−12.1580	−0.741286	−0.370643	+	0.928775i	$0.620863\pi$
$270$	0	0
$271$	10.6644	0.647815	0.323907	−	0.946089i	$-0.395003\pi$
$271$	10.6644	0.647815	0.323907	+	0.946089i	$0.395003\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.49307	−0.572453
$276$	0	0
$277$	3.76421	0.226170	0.113085	−	0.993585i	$-0.463927\pi$
$277$	3.76421	0.226170	0.113085	+	0.993585i	$0.463927\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.4496	−0.623368	−0.311684	−	0.950186i	$-0.600893\pi$
$281$	−10.4496	−0.623368	−0.311684	+	0.950186i	$0.600893\pi$
$282$	0	0
$283$	17.6569	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$283$	17.6569	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.413828	−0.0244275
$288$	0	0
$289$	−7.67794	−0.451644
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−30.7465	−1.79623	−0.898114	−	0.439762i	$-0.855063\pi$
$293$	−30.7465	−1.79623	−0.898114	+	0.439762i	$0.855063\pi$
$294$	0	0
$295$	−10.1706	−0.592158
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16.8486	0.974379
$300$	0	0
$301$	1.29355	0.0745590
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.33026	0.534249
$306$	0	0
$307$	−21.2981	−1.21555	−0.607775	−	0.794110i	$-0.707938\pi$
$307$	−21.2981	−1.21555	−0.607775	+	0.794110i	$0.707938\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.77883	0.100868	0.0504342	−	0.998727i	$-0.483939\pi$
$311$	1.77883	0.100868	0.0504342	+	0.998727i	$0.483939\pi$
$312$	0	0
$313$	−2.70320	−0.152794	−0.0763971	−	0.997077i	$-0.524342\pi$
$313$	−2.70320	−0.152794	−0.0763971	+	0.997077i	$0.524342\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0653	1.23931	0.619655	−	0.784874i	$-0.287272\pi$
$317$	22.0653	1.23931	0.619655	+	0.784874i	$0.287272\pi$
$318$	0	0
$319$	15.9495	0.892999
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.32206	0.518693
$324$	0	0
$325$	10.5284	0.584011
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.449555	0.0247848
$330$	0	0
$331$	21.8431	1.20060	0.600302	−	0.799774i	$-0.295047\pi$
$331$	21.8431	1.20060	0.600302	+	0.799774i	$0.295047\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.95133	−0.106613
$336$	0	0
$337$	18.8738	1.02812	0.514062	−	0.857753i	$-0.328140\pi$
$337$	18.8738	1.02812	0.514062	+	0.857753i	$0.328140\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−22.3380	−1.20967
$342$	0	0
$343$	−2.22117	−0.119932
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.0490	1.50575	0.752875	−	0.658163i	$-0.228666\pi$
$347$	28.0490	1.50575	0.752875	+	0.658163i	$0.228666\pi$
$348$	0	0
$349$	−16.9307	−0.906279	−0.453139	−	0.891440i	$-0.649696\pi$
$349$	−16.9307	−0.906279	−0.453139	+	0.891440i	$0.649696\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.6202	0.671705	0.335853	−	0.941915i	$-0.390976\pi$
$353$	12.6202	0.671705	0.335853	+	0.941915i	$0.390976\pi$
$354$	0	0
$355$	−0.571533	−0.0303338
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.0867	−1.42958	−0.714790	−	0.699339i	$-0.753478\pi$
$359$	−27.0867	−1.42958	−0.714790	+	0.699339i	$0.753478\pi$
$360$	0	0
$361$	−9.67794	−0.509365
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.40738	−0.126008
$366$	0	0
$367$	−20.4937	−1.06976	−0.534882	−	0.844927i	$-0.679644\pi$
$367$	−20.4937	−1.06976	−0.534882	+	0.844927i	$0.679644\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.799555	0.0415108
$372$	0	0
$373$	−1.46190	−0.0756943	−0.0378471	−	0.999284i	$-0.512050\pi$
$373$	−1.46190	−0.0756943	−0.0378471	+	0.999284i	$0.512050\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−17.6890	−0.911028
$378$	0	0
$379$	−24.9871	−1.28350	−0.641751	−	0.766913i	$-0.721792\pi$
$379$	−24.9871	−1.28350	−0.641751	+	0.766913i	$0.721792\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.0958	−1.58892	−0.794460	−	0.607316i	$-0.792246\pi$
$383$	−31.0958	−1.58892	−0.794460	+	0.607316i	$0.792246\pi$
$384$	0	0
$385$	−1.53488	−0.0782245
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.62218	0.183652	0.0918260	−	0.995775i	$-0.470730\pi$
$389$	3.62218	0.183652	0.0918260	+	0.995775i	$0.470730\pi$
$390$	0	0
$391$	−8.63577	−0.436729
$392$	0	0
$393$	0	0
$394$	0	0
$395$	17.4288	0.876940
$396$	0	0
$397$	−7.20959	−0.361839	−0.180920	−	0.983498i	$-0.557907\pi$
$397$	−7.20959	−0.361839	−0.180920	+	0.983498i	$0.557907\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−15.2660	−0.762349	−0.381174	−	0.924503i	$-0.624480\pi$
$401$	−15.2660	−0.762349	−0.381174	+	0.924503i	$0.624480\pi$
$402$	0	0
$403$	24.7743	1.23410
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−45.4792	−2.25432
$408$	0	0
$409$	−11.3779	−0.562603	−0.281302	−	0.959619i	$-0.590766\pi$
$409$	−11.3779	−0.562603	−0.281302	+	0.959619i	$0.590766\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.899110	0.0442423
$414$	0	0
$415$	−0.294481	−0.0144555
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.9618	−1.61029	−0.805146	−	0.593077i	$-0.797913\pi$
$419$	−32.9618	−1.61029	−0.805146	+	0.593077i	$0.797913\pi$
$420$	0	0
$421$	−24.9119	−1.21413	−0.607065	−	0.794652i	$-0.707653\pi$
$421$	−24.9119	−1.21413	−0.607065	+	0.794652i	$0.707653\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.39635	−0.261761
$426$	0	0
$427$	−0.824818	−0.0399157
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.3211	−0.497151	−0.248576	−	0.968612i	$-0.579962\pi$
$431$	−10.3211	−0.497151	−0.248576	+	0.968612i	$0.579962\pi$
$432$	0	0
$433$	15.3137	0.735930	0.367965	−	0.929840i	$-0.380055\pi$
$433$	15.3137	0.735930	0.367965	+	0.929840i	$0.380055\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.63577	−0.413105
$438$	0	0
$439$	−22.5735	−1.07738	−0.538688	−	0.842505i	$-0.681080\pi$
$439$	−22.5735	−1.07738	−0.538688	+	0.842505i	$0.681080\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.5334	1.59322	0.796610	−	0.604494i	$-0.206625\pi$
$443$	33.5334	1.59322	0.796610	+	0.604494i	$0.206625\pi$
$444$	0	0
$445$	25.8885	1.22723
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.75506	0.0828266	0.0414133	−	0.999142i	$-0.486814\pi$
$449$	1.75506	0.0828266	0.0414133	+	0.999142i	$0.486814\pi$
$450$	0	0
$451$	−13.9844	−0.658501
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.70227	0.0798039
$456$	0	0
$457$	−26.7422	−1.25095	−0.625473	−	0.780246i	$-0.715094\pi$
$457$	−26.7422	−1.25095	−0.625473	+	0.780246i	$0.715094\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.0662	−0.608555	−0.304277	−	0.952584i	$-0.598415\pi$
$461$	−13.0662	−0.608555	−0.304277	+	0.952584i	$0.598415\pi$
$462$	0	0
$463$	29.4474	1.36854	0.684268	−	0.729231i	$-0.260122\pi$
$463$	29.4474	1.36854	0.684268	+	0.729231i	$0.260122\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.7040	−1.28199	−0.640995	−	0.767545i	$-0.721478\pi$
$467$	−27.7040	−1.28199	−0.640995	+	0.767545i	$0.721478\pi$
$468$	0	0
$469$	0.172503	0.00796544
$470$	0	0
$471$	0	0
$472$	0	0
$473$	43.7127	2.00991
$474$	0	0
$475$	−5.39635	−0.247602
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.5499	1.62432	0.812159	−	0.583436i	$-0.198292\pi$
$479$	35.5499	1.62432	0.812159	+	0.583436i	$0.198292\pi$
$480$	0	0
$481$	50.4393	2.29984
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.02758	0.0466599
$486$	0	0
$487$	−9.86632	−0.447086	−0.223543	−	0.974694i	$-0.571762\pi$
$487$	−9.86632	−0.447086	−0.223543	+	0.974694i	$0.571762\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.635767	0.0286917	0.0143459	−	0.999897i	$-0.495433\pi$
$491$	0.635767	0.0286917	0.0143459	+	0.999897i	$0.495433\pi$
$492$	0	0
$493$	9.06651	0.408335
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.0505249	0.00226635
$498$	0	0
$499$	−3.82750	−0.171342	−0.0856712	−	0.996323i	$-0.527303\pi$
$499$	−3.82750	−0.171342	−0.0856712	+	0.996323i	$0.527303\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.6719	1.05548	0.527739	−	0.849407i	$-0.323040\pi$
$503$	23.6719	1.05548	0.527739	+	0.849407i	$0.323040\pi$
$504$	0	0
$505$	−18.1876	−0.809336
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.7970	−1.54235	−0.771175	−	0.636623i	$-0.780331\pi$
$509$	−34.7970	−1.54235	−0.771175	+	0.636623i	$0.780331\pi$
$510$	0	0
$511$	0.212818	0.00941453
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−20.4077	−0.899273
$516$	0	0
$517$	15.1917	0.668132
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.4889	0.634770	0.317385	−	0.948297i	$-0.397195\pi$
$521$	14.4889	0.634770	0.317385	+	0.948297i	$0.397195\pi$
$522$	0	0
$523$	−27.5742	−1.20574	−0.602868	−	0.797841i	$-0.705975\pi$
$523$	−27.5742	−1.20574	−0.602868	+	0.797841i	$0.705975\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.6981	−0.553138
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.5096	0.671796
$534$	0	0
$535$	−1.83585	−0.0793706
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−37.4619	−1.61360
$540$	0	0
$541$	14.1982	0.610428	0.305214	−	0.952284i	$-0.401272\pi$
$541$	14.1982	0.610428	0.305214	+	0.952284i	$0.401272\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.67340	0.157351
$546$	0	0
$547$	−10.1807	−0.435295	−0.217648	−	0.976027i	$-0.569838\pi$
$547$	−10.1807	−0.435295	−0.217648	+	0.976027i	$0.569838\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.06651	0.386246
$552$	0	0
$553$	−1.54075	−0.0655194
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.44432	−0.0611979	−0.0305990	−	0.999532i	$-0.509741\pi$
$557$	−1.44432	−0.0611979	−0.0305990	+	0.999532i	$0.509741\pi$
$558$	0	0
$559$	−48.4802	−2.05049
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.48585	0.399781	0.199890	−	0.979818i	$-0.435941\pi$
$563$	9.48585	0.399781	0.199890	+	0.979818i	$0.435941\pi$
$564$	0	0
$565$	6.35547	0.267377
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.98711	−0.376759	−0.188380	−	0.982096i	$-0.560324\pi$
$569$	−8.98711	−0.376759	−0.188380	+	0.982096i	$0.560324\pi$
$570$	0	0
$571$	12.9706	0.542801	0.271401	−	0.962466i	$-0.412513\pi$
$571$	12.9706	0.542801	0.271401	+	0.962466i	$0.412513\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.99907	0.208476
$576$	0	0
$577$	29.5013	1.22815	0.614077	−	0.789246i	$-0.289528\pi$
$577$	29.5013	1.22815	0.614077	+	0.789246i	$0.289528\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0.0260328	0.00108002
$582$	0	0
$583$	27.0192	1.11902
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.57988	−0.106483	−0.0532416	−	0.998582i	$-0.516955\pi$
$587$	−2.57988	−0.106483	−0.0532416	+	0.998582i	$0.516955\pi$
$588$	0	0
$589$	−12.6981	−0.523216
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−35.4338	−1.45509	−0.727546	−	0.686058i	$-0.759339\pi$
$593$	−35.4338	−1.45509	−0.727546	+	0.686058i	$0.759339\pi$
$594$	0	0
$595$	−0.872503	−0.0357691
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.1632	−1.10986	−0.554930	−	0.831897i	$-0.687255\pi$
$599$	−27.1632	−1.10986	−0.554930	+	0.831897i	$0.687255\pi$
$600$	0	0
$601$	−5.33897	−0.217781	−0.108891	−	0.994054i	$-0.534730\pi$
$601$	−5.33897	−0.217781	−0.108891	+	0.994054i	$0.534730\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−32.0906	−1.30467
$606$	0	0
$607$	−16.1084	−0.653820	−0.326910	−	0.945055i	$-0.606007\pi$
$607$	−16.1084	−0.653820	−0.326910	+	0.945055i	$0.606007\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.8486	−0.681621
$612$	0	0
$613$	0.617903	0.0249569	0.0124784	−	0.999922i	$-0.496028\pi$
$613$	0.617903	0.0249569	0.0124784	+	0.999922i	$0.496028\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.80641	−0.354533	−0.177266	−	0.984163i	$-0.556725\pi$
$617$	−8.80641	−0.354533	−0.177266	+	0.984163i	$0.556725\pi$
$618$	0	0
$619$	2.72847	0.109666	0.0548332	−	0.998496i	$-0.482537\pi$
$619$	2.72847	0.109666	0.0548332	+	0.998496i	$0.482537\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.28861	−0.0916910
$624$	0	0
$625$	−13.0390	−0.521559
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.8528	−1.03082
$630$	0	0
$631$	−38.7864	−1.54406	−0.772030	−	0.635586i	$-0.780759\pi$
$631$	−38.7864	−1.54406	−0.772030	+	0.635586i	$0.780759\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.69315	−0.106874
$636$	0	0
$637$	41.5476	1.64618
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−33.1091	−1.30773	−0.653865	−	0.756611i	$-0.726854\pi$
$641$	−33.1091	−1.30773	−0.653865	+	0.756611i	$0.726854\pi$
$642$	0	0
$643$	−27.2797	−1.07581	−0.537904	−	0.843006i	$-0.680784\pi$
$643$	−27.2797	−1.07581	−0.537904	+	0.843006i	$0.680784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.8477	1.64520	0.822601	−	0.568620i	$-0.192522\pi$
$647$	41.8477	1.64520	0.822601	+	0.568620i	$0.192522\pi$
$648$	0	0
$649$	30.3835	1.19266
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.8937	−0.817634	−0.408817	−	0.912616i	$-0.634059\pi$
$653$	−20.8937	−0.817634	−0.408817	+	0.912616i	$0.634059\pi$
$654$	0	0
$655$	26.5054	1.03565
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.15142	−0.122762	−0.0613809	−	0.998114i	$-0.519550\pi$
$659$	−3.15142	−0.122762	−0.0613809	+	0.998114i	$0.519550\pi$
$660$	0	0
$661$	25.5527	0.993886	0.496943	−	0.867783i	$-0.334456\pi$
$661$	25.5527	0.993886	0.496943	+	0.867783i	$0.334456\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.872503	−0.0338342
$666$	0	0
$667$	−8.39903	−0.325212
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−27.8729	−1.07602
$672$	0	0
$673$	20.7981	0.801706	0.400853	−	0.916142i	$-0.368714\pi$
$673$	20.7981	0.801706	0.400853	+	0.916142i	$0.368714\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−41.0423	−1.57738	−0.788692	−	0.614789i	$-0.789241\pi$
$677$	−41.0423	−1.57738	−0.788692	+	0.614789i	$0.789241\pi$
$678$	0	0
$679$	−0.0908404	−0.00348613
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.5784	1.01699	0.508497	−	0.861064i	$-0.330201\pi$
$683$	26.5784	1.01699	0.508497	+	0.861064i	$0.330201\pi$
$684$	0	0
$685$	24.8032	0.947680
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−29.9660	−1.14161
$690$	0	0
$691$	−14.7899	−0.562633	−0.281316	−	0.959615i	$-0.590771\pi$
$691$	−14.7899	−0.562633	−0.281316	+	0.959615i	$0.590771\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.16415	0.233820
$696$	0	0
$697$	−7.94948	−0.301108
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25.9245	0.979155	0.489577	−	0.871960i	$-0.337151\pi$
$701$	25.9245	0.979155	0.489577	+	0.871960i	$0.337151\pi$
$702$	0	0
$703$	−25.8528	−0.975055
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.60782	0.0604685
$708$	0	0
$709$	20.6082	0.773958	0.386979	−	0.922089i	$-0.373519\pi$
$709$	20.6082	0.773958	0.386979	+	0.922089i	$0.373519\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.7633	0.440538
$714$	0	0
$715$	57.5247	2.15130
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−44.0949	−1.64446	−0.822230	−	0.569155i	$-0.807270\pi$
$719$	−44.0949	−1.64446	−0.822230	+	0.569155i	$0.807270\pi$
$720$	0	0
$721$	1.80409	0.0671880
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.24842	−0.194921
$726$	0	0
$727$	9.23457	0.342491	0.171246	−	0.985228i	$-0.445221\pi$
$727$	9.23457	0.342491	0.171246	+	0.985228i	$0.445221\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24.8486	0.919058
$732$	0	0
$733$	−25.8467	−0.954670	−0.477335	−	0.878721i	$-0.658397\pi$
$733$	−25.8467	−0.954670	−0.477335	+	0.878721i	$0.658397\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.82936	0.214727
$738$	0	0
$739$	24.0403	0.884337	0.442169	−	0.896932i	$-0.354209\pi$
$739$	24.0403	0.884337	0.442169	+	0.896932i	$0.354209\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.8748	−0.655762	−0.327881	−	0.944719i	$-0.606335\pi$
$743$	−17.8748	−0.655762	−0.327881	+	0.944719i	$0.606335\pi$
$744$	0	0
$745$	−7.44538	−0.272778
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.162293	0.00593007
$750$	0	0
$751$	−35.0731	−1.27984	−0.639918	−	0.768443i	$-0.721032\pi$
$751$	−35.0731	−1.27984	−0.639918	+	0.768443i	$0.721032\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−40.7490	−1.48301
$756$	0	0
$757$	46.3962	1.68630	0.843150	−	0.537679i	$-0.180699\pi$
$757$	46.3962	1.68630	0.843150	+	0.537679i	$0.180699\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.5531	0.382550	0.191275	−	0.981536i	$-0.438738\pi$
$761$	10.5531	0.382550	0.191275	+	0.981536i	$0.438738\pi$
$762$	0	0
$763$	−0.324738	−0.0117563
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−33.6972	−1.21673
$768$	0	0
$769$	−35.2068	−1.26959	−0.634795	−	0.772681i	$-0.718915\pi$
$769$	−35.2068	−1.26959	−0.634795	+	0.772681i	$0.718915\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	27.4212	0.986271	0.493136	−	0.869952i	$-0.335851\pi$
$773$	27.4212	0.986271	0.493136	+	0.869952i	$0.335851\pi$
$774$	0	0
$775$	7.35067	0.264044
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.94948	−0.284820
$780$	0	0
$781$	1.70738	0.0610948
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.06877	−0.252295
$786$	0	0
$787$	9.46058	0.337233	0.168617	−	0.985682i	$-0.446070\pi$
$787$	9.46058	0.337233	0.168617	+	0.985682i	$0.446070\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.561839	−0.0199767
$792$	0	0
$793$	30.9128	1.09775
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.1791	0.679359	0.339680	−	0.940541i	$-0.389681\pi$
$797$	19.1791	0.679359	0.339680	+	0.940541i	$0.389681\pi$
$798$	0	0
$799$	8.63577	0.305511
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.19173	0.253791
$804$	0	0
$805$	0.808269	0.0284878
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43.1578	1.51735	0.758673	−	0.651472i	$-0.225848\pi$
$809$	43.1578	1.51735	0.758673	+	0.651472i	$0.225848\pi$
$810$	0	0
$811$	3.87518	0.136076	0.0680380	−	0.997683i	$-0.478326\pi$
$811$	3.87518	0.136076	0.0680380	+	0.997683i	$0.478326\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.8243	0.484242
$816$	0	0
$817$	24.8486	0.869342
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.62084	0.196169	0.0980843	−	0.995178i	$-0.468729\pi$
$821$	5.62084	0.196169	0.0980843	+	0.995178i	$0.468729\pi$
$822$	0	0
$823$	38.5255	1.34291	0.671457	−	0.741043i	$-0.265669\pi$
$823$	38.5255	1.34291	0.671457	+	0.741043i	$0.265669\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.23674	0.147326	0.0736629	−	0.997283i	$-0.476531\pi$
$827$	4.23674	0.147326	0.0736629	+	0.997283i	$0.476531\pi$
$828$	0	0
$829$	35.3128	1.22646	0.613231	−	0.789903i	$-0.289869\pi$
$829$	35.3128	1.22646	0.613231	+	0.789903i	$0.289869\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−21.2953	−0.737838
$834$	0	0
$835$	7.11007	0.246054
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.6005	−1.36716	−0.683580	−	0.729876i	$-0.739578\pi$
$839$	−39.6005	−1.36716	−0.683580	+	0.729876i	$0.739578\pi$
$840$	0	0
$841$	−20.1820	−0.695932
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−40.4253	−1.39067
$846$	0	0
$847$	2.83688	0.0974765
$848$	0	0
$849$	0	0
$850$	0	0
$851$	23.9495	0.820977
$852$	0	0
$853$	−10.8683	−0.372124	−0.186062	−	0.982538i	$-0.559572\pi$
$853$	−10.8683	−0.372124	−0.186062	+	0.982538i	$0.559572\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.0082	1.16170	0.580849	−	0.814011i	$-0.302721\pi$
$857$	34.0082	1.16170	0.580849	+	0.814011i	$0.302721\pi$
$858$	0	0
$859$	−12.9973	−0.443463	−0.221731	−	0.975108i	$-0.571171\pi$
$859$	−12.9973	−0.443463	−0.221731	+	0.975108i	$0.571171\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.8307	−0.879289	−0.439644	−	0.898172i	$-0.644896\pi$
$863$	−25.8307	−0.879289	−0.439644	+	0.898172i	$0.644896\pi$
$864$	0	0
$865$	−40.6353	−1.38164
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−52.0663	−1.76623
$870$	0	0
$871$	−6.46512	−0.219062
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.93391	0.0653780
$876$	0	0
$877$	45.2150	1.52680	0.763400	−	0.645926i	$-0.223528\pi$
$877$	45.2150	1.52680	0.763400	+	0.645926i	$0.223528\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	34.7403	1.17043	0.585215	−	0.810878i	$-0.301010\pi$
$881$	34.7403	1.17043	0.585215	+	0.810878i	$0.301010\pi$
$882$	0	0
$883$	48.9366	1.64685	0.823424	−	0.567427i	$-0.192061\pi$
$883$	48.9366	1.64685	0.823424	+	0.567427i	$0.192061\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.9284	0.769860	0.384930	−	0.922946i	$-0.374226\pi$
$887$	22.9284	0.769860	0.384930	+	0.922946i	$0.374226\pi$
$888$	0	0
$889$	0.238081	0.00798497
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.63577	0.288985
$894$	0	0
$895$	31.1009	1.03959
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.3500	−0.411896
$900$	0	0
$901$	15.3591	0.511687
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.6523	0.487059
$906$	0	0
$907$	−23.1680	−0.769280	−0.384640	−	0.923067i	$-0.625674\pi$
$907$	−23.1680	−0.769280	−0.384640	+	0.923067i	$0.625674\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.4078	0.974324	0.487162	−	0.873312i	$-0.338032\pi$
$911$	29.4078	0.974324	0.487162	+	0.873312i	$0.338032\pi$
$912$	0	0
$913$	0.879722	0.0291146
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.34315	−0.0773775
$918$	0	0
$919$	6.86029	0.226300	0.113150	−	0.993578i	$-0.463906\pi$
$919$	6.86029	0.226300	0.113150	+	0.993578i	$0.463906\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.89359	−0.0623283
$924$	0	0
$925$	14.9656	0.492067
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.1385	−0.529488	−0.264744	−	0.964319i	$-0.585287\pi$
$929$	−16.1385	−0.529488	−0.264744	+	0.964319i	$0.585287\pi$
$930$	0	0
$931$	−21.2953	−0.697925
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29.4844	−0.964241
$936$	0	0
$937$	−34.7669	−1.13579	−0.567893	−	0.823102i	$-0.692241\pi$
$937$	−34.7669	−1.13579	−0.567893	+	0.823102i	$0.692241\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52.7024	1.71805	0.859025	−	0.511934i	$-0.171071\pi$
$941$	52.7024	1.71805	0.859025	+	0.511934i	$0.171071\pi$
$942$	0	0
$943$	7.36423	0.239812
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.1972	0.851296	0.425648	−	0.904889i	$-0.360046\pi$
$947$	26.1972	0.851296	0.425648	+	0.904889i	$0.360046\pi$
$948$	0	0
$949$	−7.97608	−0.258915
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.1752	0.362000	0.181000	−	0.983483i	$-0.442067\pi$
$953$	11.1752	0.362000	0.181000	+	0.983483i	$0.442067\pi$
$954$	0	0
$955$	29.0679	0.940615
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.19266	−0.0708047
$960$	0	0
$961$	−13.7032	−0.442039
$962$	0	0
$963$	0	0
$964$	0	0
$965$	39.8159	1.28172
$966$	0	0
$967$	12.8452	0.413075	0.206537	−	0.978439i	$-0.433780\pi$
$967$	12.8452	0.413075	0.206537	+	0.978439i	$0.433780\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.40774	−0.269817	−0.134909	−	0.990858i	$-0.543074\pi$
$971$	−8.40774	−0.269817	−0.134909	+	0.990858i	$0.543074\pi$
$972$	0	0
$973$	−0.544926	−0.0174695
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.4156	1.29301	0.646504	−	0.762910i	$-0.276230\pi$
$977$	40.4156	1.29301	0.646504	+	0.762910i	$0.276230\pi$
$978$	0	0
$979$	−77.3385	−2.47175
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.9202	−0.443985	−0.221993	−	0.975048i	$-0.571256\pi$
$983$	−13.9202	−0.443985	−0.221993	+	0.975048i	$0.571256\pi$
$984$	0	0
$985$	−36.3582	−1.15847
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23.0192	−0.731969
$990$	0	0
$991$	−41.0309	−1.30339	−0.651695	−	0.758481i	$-0.725942\pi$
$991$	−41.0309	−1.30339	−0.651695	+	0.758481i	$0.725942\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	44.9618	1.42539
$996$	0	0
$997$	38.6427	1.22383	0.611913	−	0.790925i	$-0.290400\pi$
$997$	38.6427	1.22383	0.611913	+	0.790925i	$0.290400\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.bn.1.1		4
3.2	odd	2		3072.2.a.o.1.4		4
4.3	odd	2		9216.2.a.bo.1.1		4
8.3	odd	2		9216.2.a.y.1.4		4
8.5	even	2		9216.2.a.x.1.4		4
12.11	even	2		3072.2.a.i.1.4		4
24.5	odd	2		3072.2.a.n.1.1		4
24.11	even	2		3072.2.a.t.1.1		4
32.3	odd	8		1152.2.k.c.289.3		8
32.5	even	8		576.2.k.b.433.2		8
32.11	odd	8		1152.2.k.c.865.3		8
32.13	even	8		576.2.k.b.145.2		8
32.19	odd	8		144.2.k.b.109.4		8
32.21	even	8		1152.2.k.f.865.3		8
32.27	odd	8		144.2.k.b.37.4		8
32.29	even	8		1152.2.k.f.289.3		8
48.5	odd	4		3072.2.d.i.1537.4		8
48.11	even	4		3072.2.d.f.1537.8		8
48.29	odd	4		3072.2.d.i.1537.5		8
48.35	even	4		3072.2.d.f.1537.1		8
96.5	odd	8		192.2.j.a.49.2		8
96.11	even	8		384.2.j.b.97.1		8
96.29	odd	8		384.2.j.a.289.3		8
96.35	even	8		384.2.j.b.289.1		8
96.53	odd	8		384.2.j.a.97.3		8
96.59	even	8		48.2.j.a.37.1	yes	8
96.77	odd	8		192.2.j.a.145.2		8
96.83	even	8		48.2.j.a.13.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.1	✓	8	96.83	even	8
48.2.j.a.37.1	yes	8	96.59	even	8
144.2.k.b.37.4		8	32.27	odd	8
144.2.k.b.109.4		8	32.19	odd	8
192.2.j.a.49.2		8	96.5	odd	8
192.2.j.a.145.2		8	96.77	odd	8
384.2.j.a.97.3		8	96.53	odd	8
384.2.j.a.289.3		8	96.29	odd	8
384.2.j.b.97.1		8	96.11	even	8
384.2.j.b.289.1		8	96.35	even	8
576.2.k.b.145.2		8	32.13	even	8
576.2.k.b.433.2		8	32.5	even	8
1152.2.k.c.289.3		8	32.3	odd	8
1152.2.k.c.865.3		8	32.11	odd	8
1152.2.k.f.289.3		8	32.29	even	8
1152.2.k.f.865.3		8	32.21	even	8
3072.2.a.i.1.4		4	12.11	even	2
3072.2.a.n.1.1		4	24.5	odd	2
3072.2.a.o.1.4		4	3.2	odd	2
3072.2.a.t.1.1		4	24.11	even	2
3072.2.d.f.1537.1		8	48.35	even	4
3072.2.d.f.1537.8		8	48.11	even	4
3072.2.d.i.1537.4		8	48.5	odd	4
3072.2.d.i.1537.5		8	48.29	odd	4
9216.2.a.x.1.4		4	8.5	even	2
9216.2.a.y.1.4		4	8.3	odd	2
9216.2.a.bn.1.1		4	1.1	even	1	trivial
9216.2.a.bo.1.1		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79793 q^{5} +0.158942 q^{7} +O(q^{10})\)	\(q-1.79793 q^{5} +0.158942 q^{7} +5.37109 q^{11} -5.95687 q^{13} +3.05320 q^{17} +3.05320 q^{19} -2.82843 q^{23} -1.76744 q^{25} +2.96951 q^{29} -4.15894 q^{31} -0.285766 q^{35} -8.46742 q^{37} -2.60365 q^{41} +8.13853 q^{43} +2.82843 q^{47} -6.97474 q^{49} +5.03049 q^{53} -9.65685 q^{55} +5.65685 q^{59} -5.18944 q^{61} +10.7101 q^{65} +1.08532 q^{67} +0.317883 q^{71} +1.33897 q^{73} +0.853690 q^{77} -9.69382 q^{79} +0.163788 q^{83} -5.48946 q^{85} -14.3990 q^{89} -0.946795 q^{91} -5.48946 q^{95} -0.571533 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^7	\( 4 q + 4 q^{5} - 4 q^{7} - 8 q^{13} + 4 q^{25} + 12 q^{29} - 12 q^{31} - 16 q^{37} + 4 q^{49} + 20 q^{53} - 16 q^{55} - 16 q^{61} + 8 q^{65} - 16 q^{67} - 8 q^{71} - 8 q^{73} + 24 q^{77} - 12 q^{79} - 24 q^{85} + 8 q^{89} - 16 q^{91} - 24 q^{95}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 - 8 * q^13 + 4 * q^25 + 12 * q^29 - 12 * q^31 - 16 * q^37 + 4 * q^49 + 20 * q^53 - 16 * q^55 - 16 * q^61 + 8 * q^65 - 16 * q^67 - 8 * q^71 - 8 * q^73 + 24 * q^77 - 12 * q^79 - 24 * q^85 + 8 * q^89 - 16 * q^91 - 24 * q^95

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