LMFDB - Embedded newform 9522.2.a.bk.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 = [2,2,0,2,0,0,0,2,0,0,0,0,12,0,0,2,0,0,0,0,0,0,0,0,6,12,0,0,-4] 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:	$0$
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:	no (minimal twist has level 3174)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{8} -2.82843 q^{10} +5.65685 q^{11} +6.00000 q^{13} -2.82843 q^{14} +1.00000 q^{16} +2.82843 q^{17} -8.48528 q^{19} -2.82843 q^{20} +5.65685 q^{22} +3.00000 q^{25} +6.00000 q^{26} -2.82843 q^{28} -2.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.82843 q^{34} +8.00000 q^{35} -8.48528 q^{38} -2.82843 q^{40} +6.00000 q^{41} +2.82843 q^{43} +5.65685 q^{44} -8.00000 q^{47} +1.00000 q^{49} +3.00000 q^{50} +6.00000 q^{52} -8.48528 q^{53} -16.0000 q^{55} -2.82843 q^{56} -2.00000 q^{58} -4.00000 q^{59} +8.00000 q^{62} +1.00000 q^{64} -16.9706 q^{65} +8.48528 q^{67} +2.82843 q^{68} +8.00000 q^{70} -8.00000 q^{71} +6.00000 q^{73} -8.48528 q^{76} -16.0000 q^{77} +2.82843 q^{79} -2.82843 q^{80} +6.00000 q^{82} -5.65685 q^{83} -8.00000 q^{85} +2.82843 q^{86} +5.65685 q^{88} -2.82843 q^{89} -16.9706 q^{91} -8.00000 q^{94} +24.0000 q^{95} +5.65685 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 12 q^{13} + 2 q^{16} + 6 q^{25} + 12 q^{26} - 4 q^{29} + 16 q^{31} + 2 q^{32} + 16 q^{35} + 12 q^{41} - 16 q^{47} + 2 q^{49} + 6 q^{50} + 12 q^{52} - 32 q^{55} - 4 q^{58}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 12 * q^13 + 2 * q^16 + 6 * q^25 + 12 * q^26 - 4 * q^29 + 16 * q^31 + 2 * q^32 + 16 * q^35 + 12 * q^41 - 16 * q^47 + 2 * q^49 + 6 * q^50 + 12 * q^52 - 32 * q^55 - 4 * q^58 - 8 * q^59 + 16 * q^62 + 2 * q^64 + 16 * q^70 - 16 * q^71 + 12 * q^73 - 32 * q^77 + 12 * q^82 - 16 * q^85 - 16 * q^94 + 48 * q^95 + 2 * 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
$5$	−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
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.82843	−0.894427
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	−2.82843	−0.755929
$15$	0	0
$16$	1.00000	0.250000
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−8.48528	−1.94666	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	−8.48528	−1.94666	−0.973329	+	0.229416i	$0.926318\pi$
$20$	−2.82843	−0.632456
$21$	0	0
$22$	5.65685	1.20605
$23$	0	0
$23$	0	0
$24$	0	0
$25$	3.00000	0.600000
$26$	6.00000	1.17670
$27$	0	0
$28$	−2.82843	−0.534522
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.82843	0.485071
$35$	8.00000	1.35225
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−8.48528	−1.37649
$39$	0	0
$40$	−2.82843	−0.447214
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	5.65685	0.852803
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.00000	0.424264
$51$	0	0
$52$	6.00000	0.832050
$53$	−8.48528	−1.16554	−0.582772	−	0.812636i	$-0.698032\pi$
$53$	−8.48528	−1.16554	−0.582772	+	0.812636i	$0.698032\pi$
$54$	0	0
$55$	−16.0000	−2.15744
$56$	−2.82843	−0.377964
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−16.9706	−2.10494
$66$	0	0
$67$	8.48528	1.03664	0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528	1.03664	0.518321	+	0.855186i	$0.326557\pi$
$68$	2.82843	0.342997
$69$	0	0
$70$	8.00000	0.956183
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	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
$76$	−8.48528	−0.973329
$77$	−16.0000	−1.82337
$78$	0	0
$79$	2.82843	0.318223	0.159111	−	0.987261i	$-0.449137\pi$
$79$	2.82843	0.318223	0.159111	+	0.987261i	$0.449137\pi$
$80$	−2.82843	−0.316228
$81$	0	0
$82$	6.00000	0.662589
$83$	−5.65685	−0.620920	−0.310460	−	0.950586i	$-0.600483\pi$
$83$	−5.65685	−0.620920	−0.310460	+	0.950586i	$0.600483\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	2.82843	0.304997
$87$	0	0
$88$	5.65685	0.603023
$89$	−2.82843	−0.299813	−0.149906	−	0.988700i	$-0.547897\pi$
$89$	−2.82843	−0.299813	−0.149906	+	0.988700i	$0.547897\pi$
$90$	0	0
$91$	−16.9706	−1.77900
$92$	0	0
$93$	0	0
$94$	−8.00000	−0.825137
$95$	24.0000	2.46235
$96$	0	0
$97$	5.65685	0.574367	0.287183	−	0.957876i	$-0.407281\pi$
$97$	5.65685	0.574367	0.287183	+	0.957876i	$0.407281\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	3.00000	0.300000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	−8.48528	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$103$	−8.48528	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−8.48528	−0.824163
$107$	5.65685	0.546869	0.273434	−	0.961891i	$-0.411840\pi$
$107$	5.65685	0.546869	0.273434	+	0.961891i	$0.411840\pi$
$108$	0	0
$109$	16.9706	1.62549	0.812743	−	0.582623i	$-0.197974\pi$
$109$	16.9706	1.62549	0.812743	+	0.582623i	$0.197974\pi$
$110$	−16.0000	−1.52554
$111$	0	0
$112$	−2.82843	−0.267261
$113$	8.48528	0.798228	0.399114	−	0.916901i	$-0.369318\pi$
$113$	8.48528	0.798228	0.399114	+	0.916901i	$0.369318\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−4.00000	−0.368230
$119$	−8.00000	−0.733359
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	0	0
$124$	8.00000	0.718421
$125$	5.65685	0.505964
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−16.9706	−1.48842
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	24.0000	2.08106
$134$	8.48528	0.733017
$135$	0	0
$136$	2.82843	0.242536
$137$	−14.1421	−1.20824	−0.604122	−	0.796892i	$-0.706476\pi$
$137$	−14.1421	−1.20824	−0.604122	+	0.796892i	$0.706476\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	−8.00000	−0.671345
$143$	33.9411	2.83830
$144$	0	0
$145$	5.65685	0.469776
$146$	6.00000	0.496564
$147$	0	0
$148$	0	0
$149$	−2.82843	−0.231714	−0.115857	−	0.993266i	$-0.536961\pi$
$149$	−2.82843	−0.231714	−0.115857	+	0.993266i	$0.536961\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−8.48528	−0.688247
$153$	0	0
$154$	−16.0000	−1.28932
$155$	−22.6274	−1.81748
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	2.82843	0.225018
$159$	0	0
$160$	−2.82843	−0.223607
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−5.65685	−0.439057
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−8.00000	−0.613572
$171$	0	0
$172$	2.82843	0.215666
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−8.48528	−0.641427
$176$	5.65685	0.426401
$177$	0	0
$178$	−2.82843	−0.212000
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−5.65685	−0.420471	−0.210235	−	0.977651i	$-0.567423\pi$
$181$	−5.65685	−0.420471	−0.210235	+	0.977651i	$0.567423\pi$
$182$	−16.9706	−1.25794
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.0000	1.17004
$188$	−8.00000	−0.583460
$189$	0	0
$190$	24.0000	1.74114
$191$	5.65685	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	5.65685	0.406138
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	25.4558	1.80452	0.902258	−	0.431196i	$-0.141908\pi$
$199$	25.4558	1.80452	0.902258	+	0.431196i	$0.141908\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	10.0000	0.703598
$203$	5.65685	0.397033
$204$	0	0
$205$	−16.9706	−1.18528
$206$	−8.48528	−0.591198
$207$	0	0
$208$	6.00000	0.416025
$209$	−48.0000	−3.32023
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−8.48528	−0.582772
$213$	0	0
$214$	5.65685	0.386695
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−22.6274	−1.53605
$218$	16.9706	1.14939
$219$	0	0
$220$	−16.0000	−1.07872
$221$	16.9706	1.14156
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−2.82843	−0.188982
$225$	0	0
$226$	8.48528	0.564433
$227$	11.3137	0.750917	0.375459	−	0.926839i	$-0.377485\pi$
$227$	11.3137	0.750917	0.375459	+	0.926839i	$0.377485\pi$
$228$	0	0
$229$	16.9706	1.12145	0.560723	−	0.828003i	$-0.310523\pi$
$229$	16.9706	1.12145	0.560723	+	0.828003i	$0.310523\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	22.6274	1.47605
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−8.00000	−0.518563
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	28.2843	1.82195	0.910975	−	0.412461i	$-0.135331\pi$
$241$	28.2843	1.82195	0.910975	+	0.412461i	$0.135331\pi$
$242$	21.0000	1.34993
$243$	0	0
$244$	0	0
$245$	−2.82843	−0.180702
$246$	0	0
$247$	−50.9117	−3.23943
$248$	8.00000	0.508001
$249$	0	0
$250$	5.65685	0.357771
$251$	−11.3137	−0.714115	−0.357057	−	0.934082i	$-0.616220\pi$
$251$	−11.3137	−0.714115	−0.357057	+	0.934082i	$0.616220\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	0	0
$260$	−16.9706	−1.05247
$261$	0	0
$262$	−4.00000	−0.247121
$263$	−16.9706	−1.04645	−0.523225	−	0.852195i	$-0.675271\pi$
$263$	−16.9706	−1.04645	−0.523225	+	0.852195i	$0.675271\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	24.0000	1.47153
$267$	0	0
$268$	8.48528	0.518321
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.82843	0.171499
$273$	0	0
$274$	−14.1421	−0.854358
$275$	16.9706	1.02336
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	8.00000	0.478091
$281$	−25.4558	−1.51857	−0.759284	−	0.650759i	$-0.774451\pi$
$281$	−25.4558	−1.51857	−0.759284	+	0.650759i	$0.774451\pi$
$282$	0	0
$283$	19.7990	1.17693	0.588464	−	0.808523i	$-0.299733\pi$
$283$	19.7990	1.17693	0.588464	+	0.808523i	$0.299733\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	33.9411	2.00698
$287$	−16.9706	−1.00174
$288$	0	0
$289$	−9.00000	−0.529412
$290$	5.65685	0.332182
$291$	0	0
$292$	6.00000	0.351123
$293$	25.4558	1.48715	0.743573	−	0.668655i	$-0.233130\pi$
$293$	25.4558	1.48715	0.743573	+	0.668655i	$0.233130\pi$
$294$	0	0
$295$	11.3137	0.658710
$296$	0	0
$297$	0	0
$298$	−2.82843	−0.163846
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	−8.48528	−0.486664
$305$	0	0
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−16.0000	−0.911685
$309$	0	0
$310$	−22.6274	−1.28515
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	33.9411	1.91847	0.959233	−	0.282617i	$-0.0912024\pi$
$313$	33.9411	1.91847	0.959233	+	0.282617i	$0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	2.82843	0.159111
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−11.3137	−0.633446
$320$	−2.82843	−0.158114
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	18.0000	0.998460
$326$	−4.00000	−0.221540
$327$	0	0
$328$	6.00000	0.331295
$329$	22.6274	1.24749
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−5.65685	−0.310460
$333$	0	0
$334$	24.0000	1.31322
$335$	−24.0000	−1.31126
$336$	0	0
$337$	−16.9706	−0.924445	−0.462223	−	0.886764i	$-0.652948\pi$
$337$	−16.9706	−0.924445	−0.462223	+	0.886764i	$0.652948\pi$
$338$	23.0000	1.25104
$339$	0	0
$340$	−8.00000	−0.433861
$341$	45.2548	2.45069
$342$	0	0
$343$	16.9706	0.916324
$344$	2.82843	0.152499
$345$	0	0
$346$	14.0000	0.752645
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−8.48528	−0.453557
$351$	0	0
$352$	5.65685	0.301511
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	22.6274	1.20094
$356$	−2.82843	−0.149906
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−5.65685	−0.298557	−0.149279	−	0.988795i	$-0.547695\pi$
$359$	−5.65685	−0.298557	−0.149279	+	0.988795i	$0.547695\pi$
$360$	0	0
$361$	53.0000	2.78947
$362$	−5.65685	−0.297318
$363$	0	0
$364$	−16.9706	−0.889499
$365$	−16.9706	−0.888280
$366$	0	0
$367$	14.1421	0.738213	0.369107	−	0.929387i	$-0.379664\pi$
$367$	14.1421	0.738213	0.369107	+	0.929387i	$0.379664\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	−22.6274	−1.17160	−0.585802	−	0.810454i	$-0.699220\pi$
$373$	−22.6274	−1.17160	−0.585802	+	0.810454i	$0.699220\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−12.0000	−0.618031
$378$	0	0
$379$	2.82843	0.145287	0.0726433	−	0.997358i	$-0.476857\pi$
$379$	2.82843	0.145287	0.0726433	+	0.997358i	$0.476857\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	5.65685	0.289430
$383$	−5.65685	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	45.2548	2.30640
$386$	−10.0000	−0.508987
$387$	0	0
$388$	5.65685	0.287183
$389$	2.82843	0.143407	0.0717035	−	0.997426i	$-0.477156\pi$
$389$	2.82843	0.143407	0.0717035	+	0.997426i	$0.477156\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	25.4558	1.27599
$399$	0	0
$400$	3.00000	0.150000
$401$	14.1421	0.706225	0.353112	−	0.935581i	$-0.385123\pi$
$401$	14.1421	0.706225	0.353112	+	0.935581i	$0.385123\pi$
$402$	0	0
$403$	48.0000	2.39105
$404$	10.0000	0.497519
$405$	0	0
$406$	5.65685	0.280745
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−16.9706	−0.838116
$411$	0	0
$412$	−8.48528	−0.418040
$413$	11.3137	0.556711
$414$	0	0
$415$	16.0000	0.785409
$416$	6.00000	0.294174
$417$	0	0
$418$	−48.0000	−2.34776
$419$	−28.2843	−1.38178	−0.690889	−	0.722961i	$-0.742780\pi$
$419$	−28.2843	−1.38178	−0.690889	+	0.722961i	$0.742780\pi$
$420$	0	0
$421$	−28.2843	−1.37849	−0.689246	−	0.724528i	$-0.742058\pi$
$421$	−28.2843	−1.37849	−0.689246	+	0.724528i	$0.742058\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	−8.48528	−0.412082
$425$	8.48528	0.411597
$426$	0	0
$427$	0	0
$428$	5.65685	0.273434
$429$	0	0
$430$	−8.00000	−0.385794
$431$	11.3137	0.544962	0.272481	−	0.962161i	$-0.412156\pi$
$431$	11.3137	0.544962	0.272481	+	0.962161i	$0.412156\pi$
$432$	0	0
$433$	11.3137	0.543702	0.271851	−	0.962339i	$-0.412364\pi$
$433$	11.3137	0.543702	0.271851	+	0.962339i	$0.412364\pi$
$434$	−22.6274	−1.08615
$435$	0	0
$436$	16.9706	0.812743
$437$	0	0
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−16.0000	−0.762770
$441$	0	0
$442$	16.9706	0.807207
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	8.00000	0.379236
$446$	−8.00000	−0.378811
$447$	0	0
$448$	−2.82843	−0.133631
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	33.9411	1.59823
$452$	8.48528	0.399114
$453$	0	0
$454$	11.3137	0.530979
$455$	48.0000	2.25027
$456$	0	0
$457$	16.9706	0.793849	0.396925	−	0.917851i	$-0.370077\pi$
$457$	16.9706	0.793849	0.396925	+	0.917851i	$0.370077\pi$
$458$	16.9706	0.792982
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	6.00000	0.277945
$467$	22.6274	1.04707	0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274	1.04707	0.523536	+	0.852004i	$0.324613\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	22.6274	1.04372
$471$	0	0
$472$	−4.00000	−0.184115
$473$	16.0000	0.735681
$474$	0	0
$475$	−25.4558	−1.16799
$476$	−8.00000	−0.366679
$477$	0	0
$478$	−8.00000	−0.365911
$479$	−22.6274	−1.03387	−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274	−1.03387	−0.516937	+	0.856024i	$0.672928\pi$
$480$	0	0
$481$	0	0
$482$	28.2843	1.28831
$483$	0	0
$484$	21.0000	0.954545
$485$	−16.0000	−0.726523
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	0	0
$489$	0	0
$490$	−2.82843	−0.127775
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	−5.65685	−0.254772
$494$	−50.9117	−2.29063
$495$	0	0
$496$	8.00000	0.359211
$497$	22.6274	1.01498
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	5.65685	0.252982
$501$	0	0
$502$	−11.3137	−0.504956
$503$	22.6274	1.00891	0.504453	−	0.863439i	$-0.331694\pi$
$503$	22.6274	1.00891	0.504453	+	0.863439i	$0.331694\pi$
$504$	0	0
$505$	−28.2843	−1.25863
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−16.9706	−0.750733
$512$	1.00000	0.0441942
$513$	0	0
$514$	14.0000	0.617514
$515$	24.0000	1.05757
$516$	0	0
$517$	−45.2548	−1.99031
$518$	0	0
$519$	0	0
$520$	−16.9706	−0.744208
$521$	−2.82843	−0.123916	−0.0619578	−	0.998079i	$-0.519734\pi$
$521$	−2.82843	−0.123916	−0.0619578	+	0.998079i	$0.519734\pi$
$522$	0	0
$523$	19.7990	0.865749	0.432875	−	0.901454i	$-0.357499\pi$
$523$	19.7990	0.865749	0.432875	+	0.901454i	$0.357499\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−16.9706	−0.739952
$527$	22.6274	0.985666
$528$	0	0
$529$	0	0
$530$	24.0000	1.04249
$531$	0	0
$532$	24.0000	1.04053
$533$	36.0000	1.55933
$534$	0	0
$535$	−16.0000	−0.691740
$536$	8.48528	0.366508
$537$	0	0
$538$	−18.0000	−0.776035
$539$	5.65685	0.243658
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	0	0
$544$	2.82843	0.121268
$545$	−48.0000	−2.05609
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	−14.1421	−0.604122
$549$	0	0
$550$	16.9706	0.723627
$551$	16.9706	0.722970
$552$	0	0
$553$	−8.00000	−0.340195
$554$	10.0000	0.424859
$555$	0	0
$556$	12.0000	0.508913
$557$	14.1421	0.599222	0.299611	−	0.954062i	$-0.403143\pi$
$557$	14.1421	0.599222	0.299611	+	0.954062i	$0.403143\pi$
$558$	0	0
$559$	16.9706	0.717778
$560$	8.00000	0.338062
$561$	0	0
$562$	−25.4558	−1.07379
$563$	11.3137	0.476816	0.238408	−	0.971165i	$-0.423374\pi$
$563$	11.3137	0.476816	0.238408	+	0.971165i	$0.423374\pi$
$564$	0	0
$565$	−24.0000	−1.00969
$566$	19.7990	0.832214
$567$	0	0
$568$	−8.00000	−0.335673
$569$	25.4558	1.06716	0.533582	−	0.845748i	$-0.320845\pi$
$569$	25.4558	1.06716	0.533582	+	0.845748i	$0.320845\pi$
$570$	0	0
$571$	−2.82843	−0.118366	−0.0591830	−	0.998247i	$-0.518850\pi$
$571$	−2.82843	−0.118366	−0.0591830	+	0.998247i	$0.518850\pi$
$572$	33.9411	1.41915
$573$	0	0
$574$	−16.9706	−0.708338
$575$	0	0
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	−9.00000	−0.374351
$579$	0	0
$580$	5.65685	0.234888
$581$	16.0000	0.663792
$582$	0	0
$583$	−48.0000	−1.98796
$584$	6.00000	0.248282
$585$	0	0
$586$	25.4558	1.05157
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−67.8823	−2.79704
$590$	11.3137	0.465778
$591$	0	0
$592$	0	0
$593$	46.0000	1.88899	0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000	1.88899	0.944497	+	0.328521i	$0.106550\pi$
$594$	0	0
$595$	22.6274	0.927634
$596$	−2.82843	−0.115857
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	0	0
$605$	−59.3970	−2.41483
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−8.48528	−0.344124
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	−16.0000	−0.644658
$617$	−14.1421	−0.569341	−0.284670	−	0.958625i	$-0.591884\pi$
$617$	−14.1421	−0.569341	−0.284670	+	0.958625i	$0.591884\pi$
$618$	0	0
$619$	−2.82843	−0.113684	−0.0568420	−	0.998383i	$-0.518103\pi$
$619$	−2.82843	−0.113684	−0.0568420	+	0.998383i	$0.518103\pi$
$620$	−22.6274	−0.908739
$621$	0	0
$622$	−24.0000	−0.962312
$623$	8.00000	0.320513
$624$	0	0
$625$	−31.0000	−1.24000
$626$	33.9411	1.35656
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−8.48528	−0.337794	−0.168897	−	0.985634i	$-0.554020\pi$
$631$	−8.48528	−0.337794	−0.168897	+	0.985634i	$0.554020\pi$
$632$	2.82843	0.112509
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	6.00000	0.237729
$638$	−11.3137	−0.447914
$639$	0	0
$640$	−2.82843	−0.111803
$641$	14.1421	0.558581	0.279290	−	0.960207i	$-0.409901\pi$
$641$	14.1421	0.558581	0.279290	+	0.960207i	$0.409901\pi$
$642$	0	0
$643$	36.7696	1.45005	0.725025	−	0.688723i	$-0.241828\pi$
$643$	36.7696	1.45005	0.725025	+	0.688723i	$0.241828\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−22.6274	−0.888204
$650$	18.0000	0.706018
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	11.3137	0.442063
$656$	6.00000	0.234261
$657$	0	0
$658$	22.6274	0.882109
$659$	45.2548	1.76288	0.881439	−	0.472298i	$-0.156575\pi$
$659$	45.2548	1.76288	0.881439	+	0.472298i	$0.156575\pi$
$660$	0	0
$661$	−11.3137	−0.440052	−0.220026	−	0.975494i	$-0.570614\pi$
$661$	−11.3137	−0.440052	−0.220026	+	0.975494i	$0.570614\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−5.65685	−0.219529
$665$	−67.8823	−2.63236
$666$	0	0
$667$	0	0
$668$	24.0000	0.928588
$669$	0	0
$670$	−24.0000	−0.927201
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−16.9706	−0.653682
$675$	0	0
$676$	23.0000	0.884615
$677$	−36.7696	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$677$	−36.7696	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	−8.00000	−0.306786
$681$	0	0
$682$	45.2548	1.73290
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	40.0000	1.52832
$686$	16.9706	0.647939
$687$	0	0
$688$	2.82843	0.107833
$689$	−50.9117	−1.93958
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−33.9411	−1.28746
$696$	0	0
$697$	16.9706	0.642806
$698$	26.0000	0.984115
$699$	0	0
$700$	−8.48528	−0.320713
$701$	19.7990	0.747798	0.373899	−	0.927470i	$-0.378021\pi$
$701$	19.7990	0.747798	0.373899	+	0.927470i	$0.378021\pi$
$702$	0	0
$703$	0	0
$704$	5.65685	0.213201
$705$	0	0
$706$	34.0000	1.27961
$707$	−28.2843	−1.06374
$708$	0	0
$709$	28.2843	1.06224	0.531119	−	0.847297i	$-0.321772\pi$
$709$	28.2843	1.06224	0.531119	+	0.847297i	$0.321772\pi$
$710$	22.6274	0.849192
$711$	0	0
$712$	−2.82843	−0.106000
$713$	0	0
$714$	0	0
$715$	−96.0000	−3.59020
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−5.65685	−0.211112
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	53.0000	1.97246
$723$	0	0
$724$	−5.65685	−0.210235
$725$	−6.00000	−0.222834
$726$	0	0
$727$	−14.1421	−0.524503	−0.262251	−	0.965000i	$-0.584465\pi$
$727$	−14.1421	−0.524503	−0.262251	+	0.965000i	$0.584465\pi$
$728$	−16.9706	−0.628971
$729$	0	0
$730$	−16.9706	−0.628109
$731$	8.00000	0.295891
$732$	0	0
$733$	−39.5980	−1.46258	−0.731292	−	0.682064i	$-0.761082\pi$
$733$	−39.5980	−1.46258	−0.731292	+	0.682064i	$0.761082\pi$
$734$	14.1421	0.521996
$735$	0	0
$736$	0	0
$737$	48.0000	1.76810
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	24.0000	0.881068
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	−22.6274	−0.828449
$747$	0	0
$748$	16.0000	0.585018
$749$	−16.0000	−0.584627
$750$	0	0
$751$	8.48528	0.309632	0.154816	−	0.987943i	$-0.450521\pi$
$751$	8.48528	0.309632	0.154816	+	0.987943i	$0.450521\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	−28.2843	−1.02801	−0.514005	−	0.857787i	$-0.671839\pi$
$757$	−28.2843	−1.02801	−0.514005	+	0.857787i	$0.671839\pi$
$758$	2.82843	0.102733
$759$	0	0
$760$	24.0000	0.870572
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	−48.0000	−1.73772
$764$	5.65685	0.204658
$765$	0	0
$766$	−5.65685	−0.204390
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−33.9411	−1.22395	−0.611974	−	0.790878i	$-0.709624\pi$
$769$	−33.9411	−1.22395	−0.611974	+	0.790878i	$0.709624\pi$
$770$	45.2548	1.63087
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−48.0833	−1.72943	−0.864717	−	0.502259i	$-0.832502\pi$
$773$	−48.0833	−1.72943	−0.864717	+	0.502259i	$0.832502\pi$
$774$	0	0
$775$	24.0000	0.862105
$776$	5.65685	0.203069
$777$	0	0
$778$	2.82843	0.101404
$779$	−50.9117	−1.82410
$780$	0	0
$781$	−45.2548	−1.61935
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	14.1421	0.504113	0.252056	−	0.967713i	$-0.418893\pi$
$787$	14.1421	0.504113	0.252056	+	0.967713i	$0.418893\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−24.0000	−0.853342
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	0	0
$796$	25.4558	0.902258
$797$	−31.1127	−1.10207	−0.551034	−	0.834483i	$-0.685767\pi$
$797$	−31.1127	−1.10207	−0.551034	+	0.834483i	$0.685767\pi$
$798$	0	0
$799$	−22.6274	−0.800500
$800$	3.00000	0.106066
$801$	0	0
$802$	14.1421	0.499376
$803$	33.9411	1.19776
$804$	0	0
$805$	0	0
$806$	48.0000	1.69073
$807$	0	0
$808$	10.0000	0.351799
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	5.65685	0.198517
$813$	0	0
$814$	0	0
$815$	11.3137	0.396302
$816$	0	0
$817$	−24.0000	−0.839654
$818$	14.0000	0.489499
$819$	0	0
$820$	−16.9706	−0.592638
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−8.48528	−0.295599
$825$	0	0
$826$	11.3137	0.393654
$827$	−22.6274	−0.786832	−0.393416	−	0.919360i	$-0.628707\pi$
$827$	−22.6274	−0.786832	−0.393416	+	0.919360i	$0.628707\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	16.0000	0.555368
$831$	0	0
$832$	6.00000	0.208013
$833$	2.82843	0.0979992
$834$	0	0
$835$	−67.8823	−2.34916
$836$	−48.0000	−1.66011
$837$	0	0
$838$	−28.2843	−0.977064
$839$	−50.9117	−1.75767	−0.878833	−	0.477129i	$-0.841677\pi$
$839$	−50.9117	−1.75767	−0.878833	+	0.477129i	$0.841677\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−28.2843	−0.974740
$843$	0	0
$844$	20.0000	0.688428
$845$	−65.0538	−2.23792
$846$	0	0
$847$	−59.3970	−2.04090
$848$	−8.48528	−0.291386
$849$	0	0
$850$	8.48528	0.291043
$851$	0	0
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	0	0
$856$	5.65685	0.193347
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	11.3137	0.385346
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	−39.5980	−1.34637
$866$	11.3137	0.384455
$867$	0	0
$868$	−22.6274	−0.768025
$869$	16.0000	0.542763
$870$	0	0
$871$	50.9117	1.72508
$872$	16.9706	0.574696
$873$	0	0
$874$	0	0
$875$	−16.0000	−0.540899
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	−16.0000	−0.539360
$881$	19.7990	0.667045	0.333522	−	0.942742i	$-0.391763\pi$
$881$	19.7990	0.667045	0.333522	+	0.942742i	$0.391763\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	16.9706	0.570782
$885$	0	0
$886$	−4.00000	−0.134383
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	8.00000	0.268161
$891$	0	0
$892$	−8.00000	−0.267860
$893$	67.8823	2.27159
$894$	0	0
$895$	11.3137	0.378176
$896$	−2.82843	−0.0944911
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	−16.0000	−0.533630
$900$	0	0
$901$	−24.0000	−0.799556
$902$	33.9411	1.13012
$903$	0	0
$904$	8.48528	0.282216
$905$	16.0000	0.531858
$906$	0	0
$907$	42.4264	1.40875	0.704373	−	0.709830i	$-0.251228\pi$
$907$	42.4264	1.40875	0.704373	+	0.709830i	$0.251228\pi$
$908$	11.3137	0.375459
$909$	0	0
$910$	48.0000	1.59118
$911$	−22.6274	−0.749680	−0.374840	−	0.927090i	$-0.622302\pi$
$911$	−22.6274	−0.749680	−0.374840	+	0.927090i	$0.622302\pi$
$912$	0	0
$913$	−32.0000	−1.05905
$914$	16.9706	0.561336
$915$	0	0
$916$	16.9706	0.560723
$917$	11.3137	0.373612
$918$	0	0
$919$	25.4558	0.839711	0.419855	−	0.907591i	$-0.362081\pi$
$919$	25.4558	0.839711	0.419855	+	0.907591i	$0.362081\pi$
$920$	0	0
$921$	0	0
$922$	30.0000	0.987997
$923$	−48.0000	−1.57994
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−8.48528	−0.278094
$932$	6.00000	0.196537
$933$	0	0
$934$	22.6274	0.740392
$935$	−45.2548	−1.47999
$936$	0	0
$937$	−33.9411	−1.10881	−0.554404	−	0.832248i	$-0.687054\pi$
$937$	−33.9411	−1.10881	−0.554404	+	0.832248i	$0.687054\pi$
$938$	−24.0000	−0.783628
$939$	0	0
$940$	22.6274	0.738025
$941$	19.7990	0.645429	0.322714	−	0.946496i	$-0.395405\pi$
$941$	19.7990	0.645429	0.322714	+	0.946496i	$0.395405\pi$
$942$	0	0
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	−25.4558	−0.825897
$951$	0	0
$952$	−8.00000	−0.259281
$953$	−19.7990	−0.641352	−0.320676	−	0.947189i	$-0.603910\pi$
$953$	−19.7990	−0.641352	−0.320676	+	0.947189i	$0.603910\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	−8.00000	−0.258738
$957$	0	0
$958$	−22.6274	−0.731059
$959$	40.0000	1.29167
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	28.2843	0.910975
$965$	28.2843	0.910503
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	−16.0000	−0.513729
$971$	45.2548	1.45230	0.726148	−	0.687538i	$-0.241309\pi$
$971$	45.2548	1.45230	0.726148	+	0.687538i	$0.241309\pi$
$972$	0	0
$973$	−33.9411	−1.08810
$974$	−8.00000	−0.256337
$975$	0	0
$976$	0	0
$977$	−2.82843	−0.0904894	−0.0452447	−	0.998976i	$-0.514407\pi$
$977$	−2.82843	−0.0904894	−0.0452447	+	0.998976i	$0.514407\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	−2.82843	−0.0903508
$981$	0	0
$982$	36.0000	1.14881
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−16.9706	−0.540727
$986$	−5.65685	−0.180151
$987$	0	0
$988$	−50.9117	−1.61972
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	22.6274	0.717698
$995$	−72.0000	−2.28255
$996$	0	0
$997$	50.0000	1.58352	0.791758	−	0.610835i	$-0.209166\pi$
$997$	50.0000	1.58352	0.791758	+	0.610835i	$0.209166\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bk.1.1		2
3.2	odd	2		3174.2.a.i.1.2	yes	2
23.22	odd	2	inner	9522.2.a.bk.1.2		2
69.68	even	2		3174.2.a.i.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3174.2.a.i.1.1	✓	2	69.68	even	2
3174.2.a.i.1.2	yes	2	3.2	odd	2
9522.2.a.bk.1.1		2	1.1	even	1	trivial
9522.2.a.bk.1.2		2	23.22	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{8} -2.82843 q^{10} +5.65685 q^{11} +6.00000 q^{13} -2.82843 q^{14} +1.00000 q^{16} +2.82843 q^{17} -8.48528 q^{19} -2.82843 q^{20} +5.65685 q^{22} +3.00000 q^{25} +6.00000 q^{26} -2.82843 q^{28} -2.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.82843 q^{34} +8.00000 q^{35} -8.48528 q^{38} -2.82843 q^{40} +6.00000 q^{41} +2.82843 q^{43} +5.65685 q^{44} -8.00000 q^{47} +1.00000 q^{49} +3.00000 q^{50} +6.00000 q^{52} -8.48528 q^{53} -16.0000 q^{55} -2.82843 q^{56} -2.00000 q^{58} -4.00000 q^{59} +8.00000 q^{62} +1.00000 q^{64} -16.9706 q^{65} +8.48528 q^{67} +2.82843 q^{68} +8.00000 q^{70} -8.00000 q^{71} +6.00000 q^{73} -8.48528 q^{76} -16.0000 q^{77} +2.82843 q^{79} -2.82843 q^{80} +6.00000 q^{82} -5.65685 q^{83} -8.00000 q^{85} +2.82843 q^{86} +5.65685 q^{88} -2.82843 q^{89} -16.9706 q^{91} -8.00000 q^{94} +24.0000 q^{95} +5.65685 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 12 q^{13} + 2 q^{16} + 6 q^{25} + 12 q^{26} - 4 q^{29} + 16 q^{31} + 2 q^{32} + 16 q^{35} + 12 q^{41} - 16 q^{47} + 2 q^{49} + 6 q^{50} + 12 q^{52} - 32 q^{55} - 4 q^{58}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 12 * q^13 + 2 * q^16 + 6 * q^25 + 12 * q^26 - 4 * q^29 + 16 * q^31 + 2 * q^32 + 16 * q^35 + 12 * q^41 - 16 * q^47 + 2 * q^49 + 6 * q^50 + 12 * q^52 - 32 * q^55 - 4 * q^58 - 8 * q^59 + 16 * q^62 + 2 * q^64 + 16 * q^70 - 16 * q^71 + 12 * q^73 - 32 * q^77 + 12 * q^82 - 16 * q^85 - 16 * q^94 + 48 * q^95 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more