LMFDB - Embedded newform 9522.2.a.cd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-8,0,8,0,0,0,-8,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,8,-12,0,0, 12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.546984493056.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 20x^{6} + 129x^{4} - 320x^{2} + 256 $ x^8 - 20x^6 + 129x^4 - 320*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.27550$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.68971 q^{5} +1.75786 q^{7} -1.00000 q^{8} +3.68971 q^{10} +3.04470 q^{11} -2.39592 q^{13} -1.75786 q^{14} +1.00000 q^{16} -3.38834 q^{17} +3.51572 q^{19} -3.68971 q^{20} -3.04470 q^{22} +8.61396 q^{25} +2.39592 q^{26} +1.75786 q^{28} +3.48599 q^{29} -1.57605 q^{31} -1.00000 q^{32} +3.38834 q^{34} -6.48599 q^{35} +10.4285 q^{37} -3.51572 q^{38} +3.68971 q^{40} -3.17788 q^{41} -11.7463 q^{43} +3.04470 q^{44} -9.10833 q^{47} -3.90993 q^{49} -8.61396 q^{50} -2.39592 q^{52} -13.4575 q^{53} -11.2341 q^{55} -1.75786 q^{56} -3.48599 q^{58} -0.604079 q^{59} -10.5628 q^{61} +1.57605 q^{62} +1.00000 q^{64} +8.84025 q^{65} +11.2431 q^{67} -3.38834 q^{68} +6.48599 q^{70} +7.90018 q^{71} +13.3656 q^{73} -10.4285 q^{74} +3.51572 q^{76} +5.35215 q^{77} +11.5558 q^{79} -3.68971 q^{80} +3.17788 q^{82} +17.8035 q^{83} +12.5020 q^{85} +11.7463 q^{86} -3.04470 q^{88} -8.55774 q^{89} -4.21169 q^{91} +9.10833 q^{94} -12.9720 q^{95} +10.1656 q^{97} +3.90993 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 8 * q^25 - 12 * q^26 + 12 * q^29 - 12 * q^31 - 8 * q^32 - 36 * q^35 - 24 * q^41 - 48 * q^47 - 16 * q^49 - 8 * q^50 + 12 * q^52 + 12 * q^55 - 12 * q^58 - 36 * q^59 + 12 * q^62 + 8 * q^64 + 36 * q^70 - 24 * q^71 + 12 * q^73 - 12 * q^77 + 24 * q^82 + 12 * q^85 + 48 * q^94 - 72 * q^95 + 16 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.68971	−1.65009	−0.825044	−	0.565068i	$-0.808850\pi$
$5$	−3.68971	−1.65009	−0.825044	+	0.565068i	$0.808850\pi$
$6$	0	0
$7$	1.75786	0.664408	0.332204	−	0.943208i	$-0.392208\pi$
$7$	1.75786	0.664408	0.332204	+	0.943208i	$0.392208\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.68971	1.16679
$11$	3.04470	0.918012	0.459006	−	0.888433i	$-0.348206\pi$
$11$	3.04470	0.918012	0.459006	+	0.888433i	$0.348206\pi$
$12$	0	0
$13$	−2.39592	−0.664509	−0.332255	−	0.943190i	$-0.607809\pi$
$13$	−2.39592	−0.664509	−0.332255	+	0.943190i	$0.607809\pi$
$14$	−1.75786	−0.469807
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.38834	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$17$	−3.38834	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$18$	0	0
$19$	3.51572	0.806561	0.403280	−	0.915076i	$-0.367870\pi$
$19$	3.51572	0.806561	0.403280	+	0.915076i	$0.367870\pi$
$20$	−3.68971	−0.825044
$21$	0	0
$22$	−3.04470	−0.649132
$23$	0	0
$23$	0	0
$24$	0	0
$25$	8.61396	1.72279
$26$	2.39592	0.469879
$27$	0	0
$28$	1.75786	0.332204
$29$	3.48599	0.647332	0.323666	−	0.946171i	$-0.395085\pi$
$29$	3.48599	0.647332	0.323666	+	0.946171i	$0.395085\pi$
$30$	0	0
$31$	−1.57605	−0.283067	−0.141534	−	0.989933i	$-0.545203\pi$
$31$	−1.57605	−0.283067	−0.141534	+	0.989933i	$0.545203\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.38834	0.581096
$35$	−6.48599	−1.09633
$36$	0	0
$37$	10.4285	1.71443	0.857214	−	0.514960i	$-0.172193\pi$
$37$	10.4285	1.71443	0.857214	+	0.514960i	$0.172193\pi$
$38$	−3.51572	−0.570325
$39$	0	0
$40$	3.68971	0.583394
$41$	−3.17788	−0.496302	−0.248151	−	0.968721i	$-0.579823\pi$
$41$	−3.17788	−0.496302	−0.248151	+	0.968721i	$0.579823\pi$
$42$	0	0
$43$	−11.7463	−1.79129	−0.895643	−	0.444773i	$-0.853284\pi$
$43$	−11.7463	−1.79129	−0.895643	+	0.444773i	$0.853284\pi$
$44$	3.04470	0.459006
$45$	0	0
$46$	0	0
$47$	−9.10833	−1.32859	−0.664294	−	0.747472i	$-0.731268\pi$
$47$	−9.10833	−1.32859	−0.664294	+	0.747472i	$0.731268\pi$
$48$	0	0
$49$	−3.90993	−0.558562
$50$	−8.61396	−1.21820
$51$	0	0
$52$	−2.39592	−0.332255
$53$	−13.4575	−1.84853	−0.924264	−	0.381753i	$-0.875320\pi$
$53$	−13.4575	−1.84853	−0.924264	+	0.381753i	$0.875320\pi$
$54$	0	0
$55$	−11.2341	−1.51480
$56$	−1.75786	−0.234904
$57$	0	0
$58$	−3.48599	−0.457733
$59$	−0.604079	−0.0786443	−0.0393222	−	0.999227i	$-0.512520\pi$
$59$	−0.604079	−0.0786443	−0.0393222	+	0.999227i	$0.512520\pi$
$60$	0	0
$61$	−10.5628	−1.35243	−0.676215	−	0.736704i	$-0.736381\pi$
$61$	−10.5628	−1.35243	−0.676215	+	0.736704i	$0.736381\pi$
$62$	1.57605	0.200159
$63$	0	0
$64$	1.00000	0.125000
$65$	8.84025	1.09650
$66$	0	0
$67$	11.2431	1.37357	0.686783	−	0.726862i	$-0.259022\pi$
$67$	11.2431	1.37357	0.686783	+	0.726862i	$0.259022\pi$
$68$	−3.38834	−0.410897
$69$	0	0
$70$	6.48599	0.775224
$71$	7.90018	0.937579	0.468789	−	0.883310i	$-0.344690\pi$
$71$	7.90018	0.937579	0.468789	+	0.883310i	$0.344690\pi$
$72$	0	0
$73$	13.3656	1.56433	0.782165	−	0.623071i	$-0.214115\pi$
$73$	13.3656	1.56433	0.782165	+	0.623071i	$0.214115\pi$
$74$	−10.4285	−1.21228
$75$	0	0
$76$	3.51572	0.403280
$77$	5.35215	0.609934
$78$	0	0
$79$	11.5558	1.30013	0.650066	−	0.759878i	$-0.274741\pi$
$79$	11.5558	1.30013	0.650066	+	0.759878i	$0.274741\pi$
$80$	−3.68971	−0.412522
$81$	0	0
$82$	3.17788	0.350939
$83$	17.8035	1.95419	0.977096	−	0.212798i	$-0.0682577\pi$
$83$	17.8035	1.95419	0.977096	+	0.212798i	$0.0682577\pi$
$84$	0	0
$85$	12.5020	1.35603
$86$	11.7463	1.26663
$87$	0	0
$88$	−3.04470	−0.324566
$89$	−8.55774	−0.907119	−0.453559	−	0.891226i	$-0.649846\pi$
$89$	−8.55774	−0.907119	−0.453559	+	0.891226i	$0.649846\pi$
$90$	0	0
$91$	−4.21169	−0.441505
$92$	0	0
$93$	0	0
$94$	9.10833	0.939453
$95$	−12.9720	−1.33090
$96$	0	0
$97$	10.1656	1.03216	0.516079	−	0.856541i	$-0.327391\pi$
$97$	10.1656	1.03216	0.516079	+	0.856541i	$0.327391\pi$
$98$	3.90993	0.394963
$99$	0	0
$100$	8.61396	0.861396
$101$	−0.760077	−0.0756305	−0.0378152	−	0.999285i	$-0.512040\pi$
$101$	−0.760077	−0.0756305	−0.0378152	+	0.999285i	$0.512040\pi$
$102$	0	0
$103$	−14.2878	−1.40782	−0.703911	−	0.710289i	$-0.748564\pi$
$103$	−14.2878	−1.40782	−0.703911	+	0.710289i	$0.748564\pi$
$104$	2.39592	0.234939
$105$	0	0
$106$	13.4575	1.30711
$107$	6.44057	0.622633	0.311316	−	0.950306i	$-0.399230\pi$
$107$	6.44057	0.622633	0.311316	+	0.950306i	$0.399230\pi$
$108$	0	0
$109$	6.99032	0.669551	0.334775	−	0.942298i	$-0.391340\pi$
$109$	6.99032	0.669551	0.334775	+	0.942298i	$0.391340\pi$
$110$	11.2341	1.07113
$111$	0	0
$112$	1.75786	0.166102
$113$	−5.13170	−0.482749	−0.241375	−	0.970432i	$-0.577598\pi$
$113$	−5.13170	−0.482749	−0.241375	+	0.970432i	$0.577598\pi$
$114$	0	0
$115$	0	0
$116$	3.48599	0.323666
$117$	0	0
$118$	0.604079	0.0556099
$119$	−5.95623	−0.546007
$120$	0	0
$121$	−1.72980	−0.157255
$122$	10.5628	0.956313
$123$	0	0
$124$	−1.57605	−0.141534
$125$	−13.3345	−1.19267
$126$	0	0
$127$	−4.83561	−0.429091	−0.214546	−	0.976714i	$-0.568827\pi$
$127$	−4.83561	−0.429091	−0.214546	+	0.976714i	$0.568827\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.84025	−0.775342
$131$	−12.8101	−1.11922	−0.559612	−	0.828755i	$-0.689050\pi$
$131$	−12.8101	−1.11922	−0.559612	+	0.828755i	$0.689050\pi$
$132$	0	0
$133$	6.18013	0.535885
$134$	−11.2431	−0.971258
$135$	0	0
$136$	3.38834	0.290548
$137$	21.3166	1.82120	0.910601	−	0.413288i	$-0.135620\pi$
$137$	21.3166	1.82120	0.910601	+	0.413288i	$0.135620\pi$
$138$	0	0
$139$	15.7638	1.33707	0.668535	−	0.743681i	$-0.266922\pi$
$139$	15.7638	1.33707	0.668535	+	0.743681i	$0.266922\pi$
$140$	−6.48599	−0.548166
$141$	0	0
$142$	−7.90018	−0.662968
$143$	−7.29486	−0.610027
$144$	0	0
$145$	−12.8623	−1.06815
$146$	−13.3656	−1.10615
$147$	0	0
$148$	10.4285	0.857214
$149$	−4.59761	−0.376651	−0.188326	−	0.982107i	$-0.560306\pi$
$149$	−4.59761	−0.376651	−0.188326	+	0.982107i	$0.560306\pi$
$150$	0	0
$151$	18.4936	1.50499	0.752495	−	0.658598i	$-0.228850\pi$
$151$	18.4936	1.50499	0.752495	+	0.658598i	$0.228850\pi$
$152$	−3.51572	−0.285162
$153$	0	0
$154$	−5.35215	−0.431289
$155$	5.81518	0.467086
$156$	0	0
$157$	1.93619	0.154525	0.0772625	−	0.997011i	$-0.475382\pi$
$157$	1.93619	0.154525	0.0772625	+	0.997011i	$0.475382\pi$
$158$	−11.5558	−0.919331
$159$	0	0
$160$	3.68971	0.291697
$161$	0	0
$162$	0	0
$163$	−0.971974	−0.0761309	−0.0380654	−	0.999275i	$-0.512120\pi$
$163$	−0.971974	−0.0761309	−0.0380654	+	0.999275i	$0.512120\pi$
$164$	−3.17788	−0.248151
$165$	0	0
$166$	−17.8035	−1.38182
$167$	−9.58369	−0.741608	−0.370804	−	0.928711i	$-0.620918\pi$
$167$	−9.58369	−0.741608	−0.370804	+	0.928711i	$0.620918\pi$
$168$	0	0
$169$	−7.25956	−0.558428
$170$	−12.5020	−0.958860
$171$	0	0
$172$	−11.7463	−0.895643
$173$	−11.2438	−0.854852	−0.427426	−	0.904050i	$-0.640579\pi$
$173$	−11.2438	−0.854852	−0.427426	+	0.904050i	$0.640579\pi$
$174$	0	0
$175$	15.1421	1.14464
$176$	3.04470	0.229503
$177$	0	0
$178$	8.55774	0.641430
$179$	−16.8356	−1.25835	−0.629176	−	0.777263i	$-0.716608\pi$
$179$	−16.8356	−1.25835	−0.629176	+	0.777263i	$0.716608\pi$
$180$	0	0
$181$	−3.02638	−0.224949	−0.112475	−	0.993655i	$-0.535878\pi$
$181$	−3.02638	−0.224949	−0.112475	+	0.993655i	$0.535878\pi$
$182$	4.21169	0.312191
$183$	0	0
$184$	0	0
$185$	−38.4780	−2.82896
$186$	0	0
$187$	−10.3165	−0.754417
$188$	−9.10833	−0.664294
$189$	0	0
$190$	12.9720	0.941086
$191$	22.8482	1.65324	0.826619	−	0.562762i	$-0.190261\pi$
$191$	22.8482	1.65324	0.826619	+	0.562762i	$0.190261\pi$
$192$	0	0
$193$	−15.5823	−1.12164	−0.560820	−	0.827938i	$-0.689514\pi$
$193$	−15.5823	−1.12164	−0.560820	+	0.827938i	$0.689514\pi$
$194$	−10.1656	−0.729845
$195$	0	0
$196$	−3.90993	−0.279281
$197$	−25.7481	−1.83447	−0.917237	−	0.398342i	$-0.869586\pi$
$197$	−25.7481	−1.83447	−0.917237	+	0.398342i	$0.869586\pi$
$198$	0	0
$199$	9.79555	0.694388	0.347194	−	0.937793i	$-0.387134\pi$
$199$	9.79555	0.694388	0.347194	+	0.937793i	$0.387134\pi$
$200$	−8.61396	−0.609099
$201$	0	0
$202$	0.760077	0.0534788
$203$	6.12787	0.430092
$204$	0	0
$205$	11.7255	0.818942
$206$	14.2878	0.995480
$207$	0	0
$208$	−2.39592	−0.166127
$209$	10.7043	0.740432
$210$	0	0
$211$	−17.7200	−1.21990	−0.609949	−	0.792441i	$-0.708810\pi$
$211$	−17.7200	−1.21990	−0.609949	+	0.792441i	$0.708810\pi$
$212$	−13.4575	−0.924264
$213$	0	0
$214$	−6.44057	−0.440268
$215$	43.3403	2.95578
$216$	0	0
$217$	−2.77048	−0.188072
$218$	−6.99032	−0.473444
$219$	0	0
$220$	−11.2341	−0.757400
$221$	8.11821	0.546090
$222$	0	0
$223$	−12.8356	−0.859536	−0.429768	−	0.902939i	$-0.641405\pi$
$223$	−12.8356	−0.859536	−0.429768	+	0.902939i	$0.641405\pi$
$224$	−1.75786	−0.117452
$225$	0	0
$226$	5.13170	0.341355
$227$	5.15132	0.341905	0.170952	−	0.985279i	$-0.445316\pi$
$227$	5.15132	0.341905	0.170952	+	0.985279i	$0.445316\pi$
$228$	0	0
$229$	−5.71364	−0.377568	−0.188784	−	0.982019i	$-0.560455\pi$
$229$	−5.71364	−0.377568	−0.188784	+	0.982019i	$0.560455\pi$
$230$	0	0
$231$	0	0
$232$	−3.48599	−0.228866
$233$	2.23017	0.146103	0.0730515	−	0.997328i	$-0.476726\pi$
$233$	2.23017	0.146103	0.0730515	+	0.997328i	$0.476726\pi$
$234$	0	0
$235$	33.6071	2.19229
$236$	−0.604079	−0.0393222
$237$	0	0
$238$	5.95623	0.386085
$239$	−4.12408	−0.266765	−0.133382	−	0.991065i	$-0.542584\pi$
$239$	−4.12408	−0.266765	−0.133382	+	0.991065i	$0.542584\pi$
$240$	0	0
$241$	2.93695	0.189186	0.0945928	−	0.995516i	$-0.469845\pi$
$241$	2.93695	0.189186	0.0945928	+	0.995516i	$0.469845\pi$
$242$	1.72980	0.111196
$243$	0	0
$244$	−10.5628	−0.676215
$245$	14.4265	0.921677
$246$	0	0
$247$	−8.42338	−0.535967
$248$	1.57605	0.100079
$249$	0	0
$250$	13.3345	0.843345
$251$	30.2950	1.91221	0.956103	−	0.293032i	$-0.0946641\pi$
$251$	30.2950	1.91221	0.956103	+	0.293032i	$0.0946641\pi$
$252$	0	0
$253$	0	0
$254$	4.83561	0.303413
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.43383	0.338953	0.169476	−	0.985534i	$-0.445792\pi$
$257$	5.43383	0.338953	0.169476	+	0.985534i	$0.445792\pi$
$258$	0	0
$259$	18.3318	1.13908
$260$	8.84025	0.548249
$261$	0	0
$262$	12.8101	0.791411
$263$	−5.29489	−0.326497	−0.163249	−	0.986585i	$-0.552197\pi$
$263$	−5.29489	−0.326497	−0.163249	+	0.986585i	$0.552197\pi$
$264$	0	0
$265$	49.6542	3.05024
$266$	−6.18013	−0.378928
$267$	0	0
$268$	11.2431	0.686783
$269$	−20.7209	−1.26338	−0.631689	−	0.775222i	$-0.717638\pi$
$269$	−20.7209	−1.26338	−0.631689	+	0.775222i	$0.717638\pi$
$270$	0	0
$271$	14.0183	0.851549	0.425775	−	0.904829i	$-0.360002\pi$
$271$	14.0183	0.851549	0.425775	+	0.904829i	$0.360002\pi$
$272$	−3.38834	−0.205449
$273$	0	0
$274$	−21.3166	−1.28778
$275$	26.2269	1.58154
$276$	0	0
$277$	−18.6920	−1.12309	−0.561547	−	0.827445i	$-0.689794\pi$
$277$	−18.6920	−1.12309	−0.561547	+	0.827445i	$0.689794\pi$
$278$	−15.7638	−0.945451
$279$	0	0
$280$	6.48599	0.387612
$281$	5.45021	0.325132	0.162566	−	0.986698i	$-0.448023\pi$
$281$	5.45021	0.325132	0.162566	+	0.986698i	$0.448023\pi$
$282$	0	0
$283$	−19.0411	−1.13188	−0.565938	−	0.824448i	$-0.691486\pi$
$283$	−19.0411	−1.13188	−0.565938	+	0.824448i	$0.691486\pi$
$284$	7.90018	0.468789
$285$	0	0
$286$	7.29486	0.431354
$287$	−5.58627	−0.329747
$288$	0	0
$289$	−5.51912	−0.324654
$290$	12.8623	0.755299
$291$	0	0
$292$	13.3656	0.782165
$293$	2.28495	0.133488	0.0667442	−	0.997770i	$-0.478739\pi$
$293$	2.28495	0.133488	0.0667442	+	0.997770i	$0.478739\pi$
$294$	0	0
$295$	2.22887	0.129770
$296$	−10.4285	−0.606142
$297$	0	0
$298$	4.59761	0.266333
$299$	0	0
$300$	0	0
$301$	−20.6482	−1.19015
$302$	−18.4936	−1.06419
$303$	0	0
$304$	3.51572	0.201640
$305$	38.9737	2.23163
$306$	0	0
$307$	2.91592	0.166420	0.0832102	−	0.996532i	$-0.473483\pi$
$307$	2.91592	0.166420	0.0832102	+	0.996532i	$0.473483\pi$
$308$	5.35215	0.304967
$309$	0	0
$310$	−5.81518	−0.330280
$311$	0.971974	0.0551156	0.0275578	−	0.999620i	$-0.491227\pi$
$311$	0.971974	0.0551156	0.0275578	+	0.999620i	$0.491227\pi$
$312$	0	0
$313$	−9.84969	−0.556738	−0.278369	−	0.960474i	$-0.589794\pi$
$313$	−9.84969	−0.556738	−0.278369	+	0.960474i	$0.589794\pi$
$314$	−1.93619	−0.109266
$315$	0	0
$316$	11.5558	0.650066
$317$	19.9541	1.12073	0.560367	−	0.828244i	$-0.310660\pi$
$317$	19.9541	1.12073	0.560367	+	0.828244i	$0.310660\pi$
$318$	0	0
$319$	10.6138	0.594258
$320$	−3.68971	−0.206261
$321$	0	0
$322$	0	0
$323$	−11.9125	−0.662827
$324$	0	0
$325$	−20.6384	−1.14481
$326$	0.971974	0.0538327
$327$	0	0
$328$	3.17788	0.175469
$329$	−16.0112	−0.882724
$330$	0	0
$331$	24.7358	1.35960	0.679801	−	0.733397i	$-0.262066\pi$
$331$	24.7358	1.35960	0.679801	+	0.733397i	$0.262066\pi$
$332$	17.8035	0.977096
$333$	0	0
$334$	9.58369	0.524396
$335$	−41.4839	−2.26651
$336$	0	0
$337$	−7.17130	−0.390645	−0.195323	−	0.980739i	$-0.562575\pi$
$337$	−7.17130	−0.390645	−0.195323	+	0.980739i	$0.562575\pi$
$338$	7.25956	0.394868
$339$	0	0
$340$	12.5020	0.678017
$341$	−4.79861	−0.259859
$342$	0	0
$343$	−19.1781	−1.03552
$344$	11.7463	0.633316
$345$	0	0
$346$	11.2438	0.604471
$347$	−1.99401	−0.107044	−0.0535221	−	0.998567i	$-0.517045\pi$
$347$	−1.99401	−0.107044	−0.0535221	+	0.998567i	$0.517045\pi$
$348$	0	0
$349$	−32.1516	−1.72104	−0.860519	−	0.509419i	$-0.829860\pi$
$349$	−32.1516	−1.72104	−0.860519	+	0.509419i	$0.829860\pi$
$350$	−15.1421	−0.809380
$351$	0	0
$352$	−3.04470	−0.162283
$353$	−9.64785	−0.513503	−0.256752	−	0.966477i	$-0.582652\pi$
$353$	−9.64785	−0.513503	−0.256752	+	0.966477i	$0.582652\pi$
$354$	0	0
$355$	−29.1494	−1.54709
$356$	−8.55774	−0.453559
$357$	0	0
$358$	16.8356	0.889790
$359$	31.4833	1.66163	0.830814	−	0.556551i	$-0.187876\pi$
$359$	31.4833	1.66163	0.830814	+	0.556551i	$0.187876\pi$
$360$	0	0
$361$	−6.63974	−0.349460
$362$	3.02638	0.159063
$363$	0	0
$364$	−4.21169	−0.220753
$365$	−49.3154	−2.58128
$366$	0	0
$367$	−10.7015	−0.558615	−0.279307	−	0.960202i	$-0.590105\pi$
$367$	−10.7015	−0.558615	−0.279307	+	0.960202i	$0.590105\pi$
$368$	0	0
$369$	0	0
$370$	38.4780	2.00038
$371$	−23.6564	−1.22818
$372$	0	0
$373$	−17.6354	−0.913125	−0.456562	−	0.889691i	$-0.650919\pi$
$373$	−17.6354	−0.913125	−0.456562	+	0.889691i	$0.650919\pi$
$374$	10.3165	0.533453
$375$	0	0
$376$	9.10833	0.469726
$377$	−8.35215	−0.430158
$378$	0	0
$379$	5.82597	0.299260	0.149630	−	0.988742i	$-0.452192\pi$
$379$	5.82597	0.299260	0.149630	+	0.988742i	$0.452192\pi$
$380$	−12.9720	−0.665448
$381$	0	0
$382$	−22.8482	−1.16902
$383$	−12.0097	−0.613666	−0.306833	−	0.951763i	$-0.599269\pi$
$383$	−12.0097	−0.613666	−0.306833	+	0.951763i	$0.599269\pi$
$384$	0	0
$385$	−19.7479	−1.00645
$386$	15.5823	0.793119
$387$	0	0
$388$	10.1656	0.516079
$389$	−2.66258	−0.134998	−0.0674992	−	0.997719i	$-0.521502\pi$
$389$	−2.66258	−0.134998	−0.0674992	+	0.997719i	$0.521502\pi$
$390$	0	0
$391$	0	0
$392$	3.90993	0.197482
$393$	0	0
$394$	25.7481	1.29717
$395$	−42.6376	−2.14533
$396$	0	0
$397$	−11.4920	−0.576766	−0.288383	−	0.957515i	$-0.593118\pi$
$397$	−11.4920	−0.576766	−0.288383	+	0.957515i	$0.593118\pi$
$398$	−9.79555	−0.491007
$399$	0	0
$400$	8.61396	0.430698
$401$	−12.1214	−0.605313	−0.302656	−	0.953100i	$-0.597873\pi$
$401$	−12.1214	−0.605313	−0.302656	+	0.953100i	$0.597873\pi$
$402$	0	0
$403$	3.77610	0.188101
$404$	−0.760077	−0.0378152
$405$	0	0
$406$	−6.12787	−0.304121
$407$	31.7515	1.57387
$408$	0	0
$409$	4.95535	0.245026	0.122513	−	0.992467i	$-0.460905\pi$
$409$	4.95535	0.245026	0.122513	+	0.992467i	$0.460905\pi$
$410$	−11.7255	−0.579080
$411$	0	0
$412$	−14.2878	−0.703911
$413$	−1.06188	−0.0522519
$414$	0	0
$415$	−65.6899	−3.22459
$416$	2.39592	0.117470
$417$	0	0
$418$	−10.7043	−0.523564
$419$	−26.9590	−1.31703	−0.658516	−	0.752567i	$-0.728815\pi$
$419$	−26.9590	−1.31703	−0.658516	+	0.752567i	$0.728815\pi$
$420$	0	0
$421$	−11.1158	−0.541749	−0.270874	−	0.962615i	$-0.587313\pi$
$421$	−11.1158	−0.541749	−0.270874	+	0.962615i	$0.587313\pi$
$422$	17.7200	0.862598
$423$	0	0
$424$	13.4575	0.653554
$425$	−29.1871	−1.41578
$426$	0	0
$427$	−18.5679	−0.898566
$428$	6.44057	0.311316
$429$	0	0
$430$	−43.3403	−2.09005
$431$	4.96088	0.238957	0.119479	−	0.992837i	$-0.461878\pi$
$431$	4.96088	0.238957	0.119479	+	0.992837i	$0.461878\pi$
$432$	0	0
$433$	−6.98481	−0.335669	−0.167834	−	0.985815i	$-0.553677\pi$
$433$	−6.98481	−0.335669	−0.167834	+	0.985815i	$0.553677\pi$
$434$	2.77048	0.132987
$435$	0	0
$436$	6.99032	0.334775
$437$	0	0
$438$	0	0
$439$	−19.4325	−0.927461	−0.463730	−	0.885976i	$-0.653489\pi$
$439$	−19.4325	−0.927461	−0.463730	+	0.885976i	$0.653489\pi$
$440$	11.2341	0.535563
$441$	0	0
$442$	−8.11821	−0.386144
$443$	21.3887	1.01621	0.508104	−	0.861296i	$-0.330347\pi$
$443$	21.3887	1.01621	0.508104	+	0.861296i	$0.330347\pi$
$444$	0	0
$445$	31.5756	1.49683
$446$	12.8356	0.607784
$447$	0	0
$448$	1.75786	0.0830510
$449$	−26.5422	−1.25260	−0.626301	−	0.779581i	$-0.715432\pi$
$449$	−26.5422	−1.25260	−0.626301	+	0.779581i	$0.715432\pi$
$450$	0	0
$451$	−9.67570	−0.455611
$452$	−5.13170	−0.241375
$453$	0	0
$454$	−5.15132	−0.241763
$455$	15.5399	0.728522
$456$	0	0
$457$	−27.2862	−1.27639	−0.638196	−	0.769874i	$-0.720319\pi$
$457$	−27.2862	−1.27639	−0.638196	+	0.769874i	$0.720319\pi$
$458$	5.71364	0.266981
$459$	0	0
$460$	0	0
$461$	−19.9647	−0.929851	−0.464925	−	0.885350i	$-0.653919\pi$
$461$	−19.9647	−0.929851	−0.464925	+	0.885350i	$0.653919\pi$
$462$	0	0
$463$	−19.8713	−0.923496	−0.461748	−	0.887011i	$-0.652778\pi$
$463$	−19.8713	−0.923496	−0.461748	+	0.887011i	$0.652778\pi$
$464$	3.48599	0.161833
$465$	0	0
$466$	−2.23017	−0.103310
$467$	29.3272	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$467$	29.3272	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$468$	0	0
$469$	19.7638	0.912608
$470$	−33.6071	−1.55018
$471$	0	0
$472$	0.604079	0.0278050
$473$	−35.7638	−1.64442
$474$	0	0
$475$	30.2842	1.38954
$476$	−5.95623	−0.273003
$477$	0	0
$478$	4.12408	0.188631
$479$	19.9768	0.912763	0.456381	−	0.889784i	$-0.349145\pi$
$479$	19.9768	0.912763	0.456381	+	0.889784i	$0.349145\pi$
$480$	0	0
$481$	−24.9858	−1.13925
$482$	−2.93695	−0.133774
$483$	0	0
$484$	−1.72980	−0.0786274
$485$	−37.5080	−1.70315
$486$	0	0
$487$	−7.49574	−0.339665	−0.169832	−	0.985473i	$-0.554323\pi$
$487$	−7.49574	−0.339665	−0.169832	+	0.985473i	$0.554323\pi$
$488$	10.5628	0.478156
$489$	0	0
$490$	−14.4265	−0.651724
$491$	16.3713	0.738826	0.369413	−	0.929265i	$-0.379559\pi$
$491$	16.3713	0.738826	0.369413	+	0.929265i	$0.379559\pi$
$492$	0	0
$493$	−11.8117	−0.531973
$494$	8.42338	0.378986
$495$	0	0
$496$	−1.57605	−0.0707669
$497$	13.8874	0.622935
$498$	0	0
$499$	39.4766	1.76722	0.883608	−	0.468227i	$-0.155107\pi$
$499$	39.4766	1.76722	0.883608	+	0.468227i	$0.155107\pi$
$500$	−13.3345	−0.596335
$501$	0	0
$502$	−30.2950	−1.35213
$503$	1.25232	0.0558382	0.0279191	−	0.999610i	$-0.491112\pi$
$503$	1.25232	0.0558382	0.0279191	+	0.999610i	$0.491112\pi$
$504$	0	0
$505$	2.80446	0.124797
$506$	0	0
$507$	0	0
$508$	−4.83561	−0.214546
$509$	36.7459	1.62874	0.814368	−	0.580349i	$-0.197084\pi$
$509$	36.7459	1.62874	0.814368	+	0.580349i	$0.197084\pi$
$510$	0	0
$511$	23.4949	1.03935
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−5.43383	−0.239676
$515$	52.7179	2.32303
$516$	0	0
$517$	−27.7321	−1.21966
$518$	−18.3318	−0.805451
$519$	0	0
$520$	−8.84025	−0.387671
$521$	−11.5547	−0.506219	−0.253109	−	0.967438i	$-0.581453\pi$
$521$	−11.5547	−0.506219	−0.253109	+	0.967438i	$0.581453\pi$
$522$	0	0
$523$	−31.0787	−1.35898	−0.679489	−	0.733685i	$-0.737799\pi$
$523$	−31.0787	−1.35898	−0.679489	+	0.733685i	$0.737799\pi$
$524$	−12.8101	−0.559612
$525$	0	0
$526$	5.29489	0.230868
$527$	5.34021	0.232623
$528$	0	0
$529$	0	0
$530$	−49.6542	−2.15684
$531$	0	0
$532$	6.18013	0.267943
$533$	7.61396	0.329797
$534$	0	0
$535$	−23.7638	−1.02740
$536$	−11.2431	−0.485629
$537$	0	0
$538$	20.7209	0.893342
$539$	−11.9046	−0.512766
$540$	0	0
$541$	27.7021	1.19100	0.595502	−	0.803354i	$-0.296953\pi$
$541$	27.7021	1.19100	0.595502	+	0.803354i	$0.296953\pi$
$542$	−14.0183	−0.602136
$543$	0	0
$544$	3.38834	0.145274
$545$	−25.7922	−1.10482
$546$	0	0
$547$	31.4278	1.34376	0.671878	−	0.740662i	$-0.265488\pi$
$547$	31.4278	1.34376	0.671878	+	0.740662i	$0.265488\pi$
$548$	21.3166	0.910601
$549$	0	0
$550$	−26.2269	−1.11832
$551$	12.2557	0.522112
$552$	0	0
$553$	20.3135	0.863817
$554$	18.6920	0.794148
$555$	0	0
$556$	15.7638	0.668535
$557$	−27.8425	−1.17972	−0.589862	−	0.807504i	$-0.700818\pi$
$557$	−27.8425	−1.17972	−0.589862	+	0.807504i	$0.700818\pi$
$558$	0	0
$559$	28.1431	1.19033
$560$	−6.48599	−0.274083
$561$	0	0
$562$	−5.45021	−0.229903
$563$	36.3006	1.52989	0.764945	−	0.644096i	$-0.222766\pi$
$563$	36.3006	1.52989	0.764945	+	0.644096i	$0.222766\pi$
$564$	0	0
$565$	18.9345	0.796579
$566$	19.0411	0.800358
$567$	0	0
$568$	−7.90018	−0.331484
$569$	−3.11917	−0.130763	−0.0653813	−	0.997860i	$-0.520826\pi$
$569$	−3.11917	−0.130763	−0.0653813	+	0.997860i	$0.520826\pi$
$570$	0	0
$571$	−11.9923	−0.501863	−0.250931	−	0.968005i	$-0.580737\pi$
$571$	−11.9923	−0.501863	−0.250931	+	0.968005i	$0.580737\pi$
$572$	−7.29486	−0.305013
$573$	0	0
$574$	5.58627	0.233166
$575$	0	0
$576$	0	0
$577$	−39.7383	−1.65433	−0.827164	−	0.561961i	$-0.810047\pi$
$577$	−39.7383	−1.65433	−0.827164	+	0.561961i	$0.810047\pi$
$578$	5.51912	0.229565
$579$	0	0
$580$	−12.8623	−0.534077
$581$	31.2961	1.29838
$582$	0	0
$583$	−40.9740	−1.69697
$584$	−13.3656	−0.553074
$585$	0	0
$586$	−2.28495	−0.0943905
$587$	−20.3968	−0.841866	−0.420933	−	0.907092i	$-0.638297\pi$
$587$	−20.3968	−0.841866	−0.420933	+	0.907092i	$0.638297\pi$
$588$	0	0
$589$	−5.54095	−0.228311
$590$	−2.22887	−0.0917613
$591$	0	0
$592$	10.4285	0.428607
$593$	−16.3390	−0.670962	−0.335481	−	0.942047i	$-0.608899\pi$
$593$	−16.3390	−0.670962	−0.335481	+	0.942047i	$0.608899\pi$
$594$	0	0
$595$	21.9768	0.900959
$596$	−4.59761	−0.188326
$597$	0	0
$598$	0	0
$599$	2.89167	0.118150	0.0590751	−	0.998254i	$-0.481185\pi$
$599$	2.89167	0.118150	0.0590751	+	0.998254i	$0.481185\pi$
$600$	0	0
$601$	−14.5677	−0.594227	−0.297114	−	0.954842i	$-0.596024\pi$
$601$	−14.5677	−0.594227	−0.297114	+	0.954842i	$0.596024\pi$
$602$	20.6482	0.841560
$603$	0	0
$604$	18.4936	0.752495
$605$	6.38247	0.259484
$606$	0	0
$607$	−33.8186	−1.37266	−0.686328	−	0.727292i	$-0.740778\pi$
$607$	−33.8186	−1.37266	−0.686328	+	0.727292i	$0.740778\pi$
$608$	−3.51572	−0.142581
$609$	0	0
$610$	−38.9737	−1.57800
$611$	21.8229	0.882858
$612$	0	0
$613$	−7.39509	−0.298685	−0.149342	−	0.988786i	$-0.547716\pi$
$613$	−7.39509	−0.298685	−0.149342	+	0.988786i	$0.547716\pi$
$614$	−2.91592	−0.117677
$615$	0	0
$616$	−5.35215	−0.215644
$617$	4.39461	0.176920	0.0884601	−	0.996080i	$-0.471805\pi$
$617$	4.39461	0.176920	0.0884601	+	0.996080i	$0.471805\pi$
$618$	0	0
$619$	41.8376	1.68160	0.840799	−	0.541348i	$-0.182086\pi$
$619$	41.8376	1.68160	0.840799	+	0.541348i	$0.182086\pi$
$620$	5.81518	0.233543
$621$	0	0
$622$	−0.971974	−0.0389726
$623$	−15.0433	−0.602697
$624$	0	0
$625$	6.13050	0.245220
$626$	9.84969	0.393673
$627$	0	0
$628$	1.93619	0.0772625
$629$	−35.3352	−1.40891
$630$	0	0
$631$	−15.6837	−0.624359	−0.312180	−	0.950023i	$-0.601059\pi$
$631$	−15.6837	−0.624359	−0.312180	+	0.950023i	$0.601059\pi$
$632$	−11.5558	−0.459666
$633$	0	0
$634$	−19.9541	−0.792479
$635$	17.8420	0.708039
$636$	0	0
$637$	9.36790	0.371170
$638$	−10.6138	−0.420204
$639$	0	0
$640$	3.68971	0.145849
$641$	−33.3971	−1.31910	−0.659552	−	0.751659i	$-0.729254\pi$
$641$	−33.3971	−1.31910	−0.659552	+	0.751659i	$0.729254\pi$
$642$	0	0
$643$	−13.9391	−0.549703	−0.274852	−	0.961487i	$-0.588629\pi$
$643$	−13.9391	−0.549703	−0.274852	+	0.961487i	$0.588629\pi$
$644$	0	0
$645$	0	0
$646$	11.9125	0.468689
$647$	−33.0871	−1.30079	−0.650393	−	0.759598i	$-0.725396\pi$
$647$	−33.0871	−1.30079	−0.650393	+	0.759598i	$0.725396\pi$
$648$	0	0
$649$	−1.83924	−0.0721964
$650$	20.6384	0.809503
$651$	0	0
$652$	−0.971974	−0.0380654
$653$	−48.4393	−1.89558	−0.947789	−	0.318899i	$-0.896687\pi$
$653$	−48.4393	−1.89558	−0.947789	+	0.318899i	$0.896687\pi$
$654$	0	0
$655$	47.2656	1.84682
$656$	−3.17788	−0.124076
$657$	0	0
$658$	16.0112	0.624180
$659$	−16.8655	−0.656985	−0.328492	−	0.944507i	$-0.606541\pi$
$659$	−16.8655	−0.656985	−0.328492	+	0.944507i	$0.606541\pi$
$660$	0	0
$661$	−31.4488	−1.22321	−0.611607	−	0.791161i	$-0.709477\pi$
$661$	−31.4488	−1.22321	−0.611607	+	0.791161i	$0.709477\pi$
$662$	−24.7358	−0.961384
$663$	0	0
$664$	−17.8035	−0.690911
$665$	−22.8029	−0.884258
$666$	0	0
$667$	0	0
$668$	−9.58369	−0.370804
$669$	0	0
$670$	41.4839	1.60266
$671$	−32.1606	−1.24155
$672$	0	0
$673$	−7.22807	−0.278622	−0.139311	−	0.990249i	$-0.544489\pi$
$673$	−7.22807	−0.278622	−0.139311	+	0.990249i	$0.544489\pi$
$674$	7.17130	0.276228
$675$	0	0
$676$	−7.25956	−0.279214
$677$	−25.7187	−0.988451	−0.494226	−	0.869334i	$-0.664548\pi$
$677$	−25.7187	−0.988451	−0.494226	+	0.869334i	$0.664548\pi$
$678$	0	0
$679$	17.8696	0.685773
$680$	−12.5020	−0.479430
$681$	0	0
$682$	4.79861	0.183748
$683$	−42.9312	−1.64272	−0.821358	−	0.570413i	$-0.806783\pi$
$683$	−42.9312	−1.64272	−0.821358	+	0.570413i	$0.806783\pi$
$684$	0	0
$685$	−78.6521	−3.00514
$686$	19.1781	0.732224
$687$	0	0
$688$	−11.7463	−0.447822
$689$	32.2431	1.22836
$690$	0	0
$691$	6.69202	0.254576	0.127288	−	0.991866i	$-0.459373\pi$
$691$	6.69202	0.254576	0.127288	+	0.991866i	$0.459373\pi$
$692$	−11.2438	−0.427426
$693$	0	0
$694$	1.99401	0.0756917
$695$	−58.1639	−2.20628
$696$	0	0
$697$	10.7678	0.407858
$698$	32.1516	1.21696
$699$	0	0
$700$	15.1421	0.572318
$701$	−34.4091	−1.29962	−0.649808	−	0.760099i	$-0.725151\pi$
$701$	−34.4091	−1.29962	−0.649808	+	0.760099i	$0.725151\pi$
$702$	0	0
$703$	36.6635	1.38279
$704$	3.04470	0.114751
$705$	0	0
$706$	9.64785	0.363102
$707$	−1.33611	−0.0502495
$708$	0	0
$709$	30.9925	1.16395	0.581974	−	0.813208i	$-0.302281\pi$
$709$	30.9925	1.16395	0.581974	+	0.813208i	$0.302281\pi$
$710$	29.1494	1.09396
$711$	0	0
$712$	8.55774	0.320715
$713$	0	0
$714$	0	0
$715$	26.9159	1.00660
$716$	−16.8356	−0.629176
$717$	0	0
$718$	−31.4833	−1.17495
$719$	7.01574	0.261643	0.130822	−	0.991406i	$-0.458239\pi$
$719$	7.01574	0.261643	0.130822	+	0.991406i	$0.458239\pi$
$720$	0	0
$721$	−25.1160	−0.935368
$722$	6.63974	0.247105
$723$	0	0
$724$	−3.02638	−0.112475
$725$	30.0281	1.11522
$726$	0	0
$727$	−4.17081	−0.154687	−0.0773434	−	0.997005i	$-0.524644\pi$
$727$	−4.17081	−0.154687	−0.0773434	+	0.997005i	$0.524644\pi$
$728$	4.21169	0.156096
$729$	0	0
$730$	49.3154	1.82524
$731$	39.8004	1.47207
$732$	0	0
$733$	−0.754458	−0.0278665	−0.0139333	−	0.999903i	$-0.504435\pi$
$733$	−0.754458	−0.0278665	−0.0139333	+	0.999903i	$0.504435\pi$
$734$	10.7015	0.395000
$735$	0	0
$736$	0	0
$737$	34.2319	1.26095
$738$	0	0
$739$	−12.3968	−0.456024	−0.228012	−	0.973658i	$-0.573222\pi$
$739$	−12.3968	−0.456024	−0.228012	+	0.973658i	$0.573222\pi$
$740$	−38.4780	−1.41448
$741$	0	0
$742$	23.6564	0.868452
$743$	41.4504	1.52067	0.760334	−	0.649532i	$-0.225035\pi$
$743$	41.4504	1.52067	0.760334	+	0.649532i	$0.225035\pi$
$744$	0	0
$745$	16.9639	0.621508
$746$	17.6354	0.645677
$747$	0	0
$748$	−10.3165	−0.377208
$749$	11.3216	0.413682
$750$	0	0
$751$	−0.781305	−0.0285102	−0.0142551	−	0.999898i	$-0.504538\pi$
$751$	−0.781305	−0.0285102	−0.0142551	+	0.999898i	$0.504538\pi$
$752$	−9.10833	−0.332147
$753$	0	0
$754$	8.35215	0.304167
$755$	−68.2361	−2.48337
$756$	0	0
$757$	42.9645	1.56157	0.780785	−	0.624800i	$-0.214819\pi$
$757$	42.9645	1.56157	0.780785	+	0.624800i	$0.214819\pi$
$758$	−5.82597	−0.211609
$759$	0	0
$760$	12.9720	0.470543
$761$	15.3029	0.554728	0.277364	−	0.960765i	$-0.410539\pi$
$761$	15.3029	0.554728	0.277364	+	0.960765i	$0.410539\pi$
$762$	0	0
$763$	12.2880	0.444855
$764$	22.8482	0.826619
$765$	0	0
$766$	12.0097	0.433927
$767$	1.44732	0.0522599
$768$	0	0
$769$	−21.8581	−0.788225	−0.394113	−	0.919062i	$-0.628948\pi$
$769$	−21.8581	−0.788225	−0.394113	+	0.919062i	$0.628948\pi$
$770$	19.7479	0.711664
$771$	0	0
$772$	−15.5823	−0.560820
$773$	21.6950	0.780315	0.390158	−	0.920748i	$-0.372420\pi$
$773$	21.6950	0.780315	0.390158	+	0.920748i	$0.372420\pi$
$774$	0	0
$775$	−13.5761	−0.487666
$776$	−10.1656	−0.364923
$777$	0	0
$778$	2.66258	0.0954583
$779$	−11.1725	−0.400298
$780$	0	0
$781$	24.0537	0.860708
$782$	0	0
$783$	0	0
$784$	−3.90993	−0.139641
$785$	−7.14399	−0.254980
$786$	0	0
$787$	−29.4128	−1.04845	−0.524226	−	0.851579i	$-0.675645\pi$
$787$	−29.4128	−1.04845	−0.524226	+	0.851579i	$0.675645\pi$
$788$	−25.7481	−0.917237
$789$	0	0
$790$	42.6376	1.51698
$791$	−9.02079	−0.320742
$792$	0	0
$793$	25.3077	0.898702
$794$	11.4920	0.407835
$795$	0	0
$796$	9.79555	0.347194
$797$	−37.9570	−1.34450	−0.672252	−	0.740322i	$-0.734673\pi$
$797$	−37.9570	−1.34450	−0.672252	+	0.740322i	$0.734673\pi$
$798$	0	0
$799$	30.8622	1.09183
$800$	−8.61396	−0.304549
$801$	0	0
$802$	12.1214	0.428021
$803$	40.6944	1.43607
$804$	0	0
$805$	0	0
$806$	−3.77610	−0.133007
$807$	0	0
$808$	0.760077	0.0267394
$809$	11.2234	0.394595	0.197297	−	0.980344i	$-0.436784\pi$
$809$	11.2234	0.394595	0.197297	+	0.980344i	$0.436784\pi$
$810$	0	0
$811$	7.92443	0.278264	0.139132	−	0.990274i	$-0.455569\pi$
$811$	7.92443	0.278264	0.139132	+	0.990274i	$0.455569\pi$
$812$	6.12787	0.215046
$813$	0	0
$814$	−31.7515	−1.11289
$815$	3.58630	0.125623
$816$	0	0
$817$	−41.2965	−1.44478
$818$	−4.95535	−0.173260
$819$	0	0
$820$	11.7255	0.409471
$821$	−9.05979	−0.316189	−0.158094	−	0.987424i	$-0.550535\pi$
$821$	−9.05979	−0.316189	−0.158094	+	0.987424i	$0.550535\pi$
$822$	0	0
$823$	−19.7273	−0.687650	−0.343825	−	0.939034i	$-0.611723\pi$
$823$	−19.7273	−0.687650	−0.343825	+	0.939034i	$0.611723\pi$
$824$	14.2878	0.497740
$825$	0	0
$826$	1.06188	0.0369477
$827$	−43.3110	−1.50607	−0.753035	−	0.657980i	$-0.771411\pi$
$827$	−43.3110	−1.50607	−0.753035	+	0.657980i	$0.771411\pi$
$828$	0	0
$829$	24.6102	0.854747	0.427374	−	0.904075i	$-0.359439\pi$
$829$	24.6102	0.854747	0.427374	+	0.904075i	$0.359439\pi$
$830$	65.6899	2.28013
$831$	0	0
$832$	−2.39592	−0.0830636
$833$	13.2482	0.459023
$834$	0	0
$835$	35.3610	1.22372
$836$	10.7043	0.370216
$837$	0	0
$838$	26.9590	0.931282
$839$	−53.5839	−1.84992	−0.924961	−	0.380062i	$-0.875903\pi$
$839$	−53.5839	−1.84992	−0.924961	+	0.380062i	$0.875903\pi$
$840$	0	0
$841$	−16.8479	−0.580962
$842$	11.1158	0.383074
$843$	0	0
$844$	−17.7200	−0.609949
$845$	26.7857	0.921455
$846$	0	0
$847$	−3.04075	−0.104481
$848$	−13.4575	−0.462132
$849$	0	0
$850$	29.1871	1.00111
$851$	0	0
$852$	0	0
$853$	−36.4681	−1.24864	−0.624322	−	0.781167i	$-0.714625\pi$
$853$	−36.4681	−1.24864	−0.624322	+	0.781167i	$0.714625\pi$
$854$	18.5679	0.635382
$855$	0	0
$856$	−6.44057	−0.220134
$857$	−26.1331	−0.892689	−0.446345	−	0.894861i	$-0.647274\pi$
$857$	−26.1331	−0.892689	−0.446345	+	0.894861i	$0.647274\pi$
$858$	0	0
$859$	−41.5569	−1.41790	−0.708952	−	0.705257i	$-0.750832\pi$
$859$	−41.5569	−1.41790	−0.708952	+	0.705257i	$0.750832\pi$
$860$	43.3403	1.47789
$861$	0	0
$862$	−4.96088	−0.168968
$863$	23.1797	0.789045	0.394522	−	0.918886i	$-0.370910\pi$
$863$	23.1797	0.789045	0.394522	+	0.918886i	$0.370910\pi$
$864$	0	0
$865$	41.4864	1.41058
$866$	6.98481	0.237354
$867$	0	0
$868$	−2.77048	−0.0940361
$869$	35.1840	1.19354
$870$	0	0
$871$	−26.9376	−0.912747
$872$	−6.99032	−0.236722
$873$	0	0
$874$	0	0
$875$	−23.4401	−0.792420
$876$	0	0
$877$	46.2562	1.56196	0.780980	−	0.624556i	$-0.214720\pi$
$877$	46.2562	1.56196	0.780980	+	0.624556i	$0.214720\pi$
$878$	19.4325	0.655814
$879$	0	0
$880$	−11.2341	−0.378700
$881$	−36.6012	−1.23313	−0.616564	−	0.787305i	$-0.711476\pi$
$881$	−36.6012	−1.23313	−0.616564	+	0.787305i	$0.711476\pi$
$882$	0	0
$883$	−23.5246	−0.791667	−0.395834	−	0.918322i	$-0.629544\pi$
$883$	−23.5246	−0.791667	−0.395834	+	0.918322i	$0.629544\pi$
$884$	8.11821	0.273045
$885$	0	0
$886$	−21.3887	−0.718567
$887$	17.0085	0.571090	0.285545	−	0.958365i	$-0.407825\pi$
$887$	17.0085	0.571090	0.285545	+	0.958365i	$0.407825\pi$
$888$	0	0
$889$	−8.50032	−0.285092
$890$	−31.5756	−1.05842
$891$	0	0
$892$	−12.8356	−0.429768
$893$	−32.0223	−1.07159
$894$	0	0
$895$	62.1185	2.07639
$896$	−1.75786	−0.0587259
$897$	0	0
$898$	26.5422	0.885723
$899$	−5.49410	−0.183238
$900$	0	0
$901$	45.5986	1.51911
$902$	9.67570	0.322166
$903$	0	0
$904$	5.13170	0.170678
$905$	11.1665	0.371186
$906$	0	0
$907$	16.9985	0.564426	0.282213	−	0.959352i	$-0.408931\pi$
$907$	16.9985	0.564426	0.282213	+	0.959352i	$0.408931\pi$
$908$	5.15132	0.170952
$909$	0	0
$910$	−15.5399	−0.515143
$911$	−27.4408	−0.909153	−0.454577	−	0.890708i	$-0.650209\pi$
$911$	−27.4408	−0.909153	−0.454577	+	0.890708i	$0.650209\pi$
$912$	0	0
$913$	54.2064	1.79397
$914$	27.2862	0.902546
$915$	0	0
$916$	−5.71364	−0.188784
$917$	−22.5184	−0.743622
$918$	0	0
$919$	−51.7407	−1.70677	−0.853384	−	0.521282i	$-0.825454\pi$
$919$	−51.7407	−1.70677	−0.853384	+	0.521282i	$0.825454\pi$
$920$	0	0
$921$	0	0
$922$	19.9647	0.657504
$923$	−18.9282	−0.623029
$924$	0	0
$925$	89.8303	2.95360
$926$	19.8713	0.653011
$927$	0	0
$928$	−3.48599	−0.114433
$929$	−23.0068	−0.754827	−0.377414	−	0.926045i	$-0.623186\pi$
$929$	−23.0068	−0.754827	−0.377414	+	0.926045i	$0.623186\pi$
$930$	0	0
$931$	−13.7462	−0.450514
$932$	2.23017	0.0730515
$933$	0	0
$934$	−29.3272	−0.959617
$935$	38.0649	1.24485
$936$	0	0
$937$	30.5849	0.999165	0.499582	−	0.866266i	$-0.333487\pi$
$937$	30.5849	0.999165	0.499582	+	0.866266i	$0.333487\pi$
$938$	−19.7638	−0.645311
$939$	0	0
$940$	33.6071	1.09614
$941$	15.8069	0.515291	0.257646	−	0.966240i	$-0.417053\pi$
$941$	15.8069	0.515291	0.257646	+	0.966240i	$0.417053\pi$
$942$	0	0
$943$	0	0
$944$	−0.604079	−0.0196611
$945$	0	0
$946$	35.7638	1.16278
$947$	−40.4618	−1.31483	−0.657416	−	0.753528i	$-0.728351\pi$
$947$	−40.4618	−1.31483	−0.657416	+	0.753528i	$0.728351\pi$
$948$	0	0
$949$	−32.0230	−1.03951
$950$	−30.2842	−0.982550
$951$	0	0
$952$	5.95623	0.193043
$953$	−47.1119	−1.52611	−0.763053	−	0.646336i	$-0.776300\pi$
$953$	−47.1119	−1.52611	−0.763053	+	0.646336i	$0.776300\pi$
$954$	0	0
$955$	−84.3033	−2.72799
$956$	−4.12408	−0.133382
$957$	0	0
$958$	−19.9768	−0.645421
$959$	37.4716	1.21002
$960$	0	0
$961$	−28.5161	−0.919873
$962$	24.9858	0.805574
$963$	0	0
$964$	2.93695	0.0945928
$965$	57.4942	1.85080
$966$	0	0
$967$	7.70518	0.247782	0.123891	−	0.992296i	$-0.460463\pi$
$967$	7.70518	0.247782	0.123891	+	0.992296i	$0.460463\pi$
$968$	1.72980	0.0555980
$969$	0	0
$970$	37.5080	1.20431
$971$	13.1100	0.420721	0.210361	−	0.977624i	$-0.432536\pi$
$971$	13.1100	0.420721	0.210361	+	0.977624i	$0.432536\pi$
$972$	0	0
$973$	27.7106	0.888360
$974$	7.49574	0.240179
$975$	0	0
$976$	−10.5628	−0.338108
$977$	21.8829	0.700096	0.350048	−	0.936732i	$-0.386165\pi$
$977$	21.8829	0.700096	0.350048	+	0.936732i	$0.386165\pi$
$978$	0	0
$979$	−26.0557	−0.832745
$980$	14.4265	0.460838
$981$	0	0
$982$	−16.3713	−0.522429
$983$	−14.8811	−0.474633	−0.237317	−	0.971432i	$-0.576268\pi$
$983$	−14.8811	−0.474633	−0.237317	+	0.971432i	$0.576268\pi$
$984$	0	0
$985$	95.0029	3.02704
$986$	11.8117	0.376162
$987$	0	0
$988$	−8.42338	−0.267983
$989$	0	0
$990$	0	0
$991$	−11.5446	−0.366725	−0.183363	−	0.983045i	$-0.558698\pi$
$991$	−11.5446	−0.366725	−0.183363	+	0.983045i	$0.558698\pi$
$992$	1.57605	0.0500397
$993$	0	0
$994$	−13.8874	−0.440481
$995$	−36.1427	−1.14580
$996$	0	0
$997$	7.15387	0.226565	0.113283	−	0.993563i	$-0.463863\pi$
$997$	7.15387	0.226565	0.113283	+	0.993563i	$0.463863\pi$
$998$	−39.4766	−1.24961
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cd.1.1	✓	8
3.2	odd	2		9522.2.a.cf.1.8	yes	8
23.22	odd	2	inner	9522.2.a.cd.1.8	yes	8
69.68	even	2		9522.2.a.cf.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.cd.1.1	✓	8	1.1	even	1	trivial
9522.2.a.cd.1.8	yes	8	23.22	odd	2	inner
9522.2.a.cf.1.1	yes	8	69.68	even	2
9522.2.a.cf.1.8	yes	8	3.2	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

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.68971 q^{5} +1.75786 q^{7} -1.00000 q^{8} +3.68971 q^{10} +3.04470 q^{11} -2.39592 q^{13} -1.75786 q^{14} +1.00000 q^{16} -3.38834 q^{17} +3.51572 q^{19} -3.68971 q^{20} -3.04470 q^{22} +8.61396 q^{25} +2.39592 q^{26} +1.75786 q^{28} +3.48599 q^{29} -1.57605 q^{31} -1.00000 q^{32} +3.38834 q^{34} -6.48599 q^{35} +10.4285 q^{37} -3.51572 q^{38} +3.68971 q^{40} -3.17788 q^{41} -11.7463 q^{43} +3.04470 q^{44} -9.10833 q^{47} -3.90993 q^{49} -8.61396 q^{50} -2.39592 q^{52} -13.4575 q^{53} -11.2341 q^{55} -1.75786 q^{56} -3.48599 q^{58} -0.604079 q^{59} -10.5628 q^{61} +1.57605 q^{62} +1.00000 q^{64} +8.84025 q^{65} +11.2431 q^{67} -3.38834 q^{68} +6.48599 q^{70} +7.90018 q^{71} +13.3656 q^{73} -10.4285 q^{74} +3.51572 q^{76} +5.35215 q^{77} +11.5558 q^{79} -3.68971 q^{80} +3.17788 q^{82} +17.8035 q^{83} +12.5020 q^{85} +11.7463 q^{86} -3.04470 q^{88} -8.55774 q^{89} -4.21169 q^{91} +9.10833 q^{94} -12.9720 q^{95} +10.1656 q^{97} +3.90993 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} - 12 q^{26} + 12 q^{29} - 12 q^{31} - 8 q^{32} - 36 q^{35} - 24 q^{41} - 48 q^{47} - 16 q^{49} - 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots + 16 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^8 + 12 * q^13 + 8 * q^16 + 8 * q^25 - 12 * q^26 + 12 * q^29 - 12 * q^31 - 8 * q^32 - 36 * q^35 - 24 * q^41 - 48 * q^47 - 16 * q^49 - 8 * q^50 + 12 * q^52 + 12 * q^55 - 12 * q^58 - 36 * q^59 + 12 * q^62 + 8 * q^64 + 36 * q^70 - 24 * q^71 + 12 * q^73 - 12 * q^77 + 24 * q^82 + 12 * q^85 + 48 * q^94 - 72 * q^95 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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