LMFDB - Embedded newform 9522.2.a.cf.1.2

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:	$0$
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			1.2
Root			$-1.25511$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.66933 q^{5} -3.18696 q^{7} +1.00000 q^{8} -2.66933 q^{10} -5.51998 q^{11} +2.64969 q^{13} -3.18696 q^{14} +1.00000 q^{16} +3.74723 q^{17} -6.37393 q^{19} -2.66933 q^{20} -5.51998 q^{22} +2.12530 q^{25} +2.64969 q^{26} -3.18696 q^{28} -5.50705 q^{29} -10.6638 q^{31} +1.00000 q^{32} +3.74723 q^{34} +8.50705 q^{35} +9.36329 q^{37} -6.37393 q^{38} -2.66933 q^{40} -0.424692 q^{41} -5.38311 q^{43} -5.51998 q^{44} +9.38528 q^{47} +3.15674 q^{49} +2.12530 q^{50} +2.64969 q^{52} -11.6296 q^{53} +14.7346 q^{55} -3.18696 q^{56} -5.50705 q^{58} +5.64969 q^{59} -7.39110 q^{61} -10.6638 q^{62} +1.00000 q^{64} -7.07290 q^{65} -4.30338 q^{67} +3.74723 q^{68} +8.50705 q^{70} +1.91411 q^{71} -5.37377 q^{73} +9.36329 q^{74} -6.37393 q^{76} +17.5920 q^{77} +6.61099 q^{79} -2.66933 q^{80} -0.424692 q^{82} +5.15732 q^{83} -10.0026 q^{85} -5.38311 q^{86} -5.51998 q^{88} +14.9168 q^{89} -8.44448 q^{91} +9.38528 q^{94} +17.0141 q^{95} -12.8109 q^{97} +3.15674 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.66933	−1.19376	−0.596880	−	0.802331i	$-0.703593\pi$
$5$	−2.66933	−1.19376	−0.596880	+	0.802331i	$0.703593\pi$
$6$	0	0
$7$	−3.18696	−1.20456	−0.602280	−	0.798285i	$-0.705741\pi$
$7$	−3.18696	−1.20456	−0.602280	+	0.798285i	$0.705741\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.66933	−0.844115
$11$	−5.51998	−1.66434	−0.832169	−	0.554522i	$-0.812901\pi$
$11$	−5.51998	−1.66434	−0.832169	+	0.554522i	$0.812901\pi$
$12$	0	0
$13$	2.64969	0.734893	0.367446	−	0.930045i	$-0.380232\pi$
$13$	2.64969	0.734893	0.367446	+	0.930045i	$0.380232\pi$
$14$	−3.18696	−0.851752
$15$	0	0
$16$	1.00000	0.250000
$17$	3.74723	0.908838	0.454419	−	0.890788i	$-0.349847\pi$
$17$	3.74723	0.908838	0.454419	+	0.890788i	$0.349847\pi$
$18$	0	0
$19$	−6.37393	−1.46228	−0.731140	−	0.682228i	$-0.761011\pi$
$19$	−6.37393	−1.46228	−0.731140	+	0.682228i	$0.761011\pi$
$20$	−2.66933	−0.596880
$21$	0	0
$22$	−5.51998	−1.17686
$23$	0	0
$23$	0	0
$24$	0	0
$25$	2.12530	0.425061
$26$	2.64969	0.519648
$27$	0	0
$28$	−3.18696	−0.602280
$29$	−5.50705	−1.02263	−0.511317	−	0.859392i	$-0.670842\pi$
$29$	−5.50705	−1.02263	−0.511317	+	0.859392i	$0.670842\pi$
$30$	0	0
$31$	−10.6638	−1.91527	−0.957637	−	0.287979i	$-0.907017\pi$
$31$	−10.6638	−1.91527	−0.957637	+	0.287979i	$0.907017\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.74723	0.642645
$35$	8.50705	1.43795
$36$	0	0
$37$	9.36329	1.53932	0.769658	−	0.638457i	$-0.220427\pi$
$37$	9.36329	1.53932	0.769658	+	0.638457i	$0.220427\pi$
$38$	−6.37393	−1.03399
$39$	0	0
$40$	−2.66933	−0.422058
$41$	−0.424692	−0.0663257	−0.0331629	−	0.999450i	$-0.510558\pi$
$41$	−0.424692	−0.0663257	−0.0331629	+	0.999450i	$0.510558\pi$
$42$	0	0
$43$	−5.38311	−0.820917	−0.410459	−	0.911879i	$-0.634631\pi$
$43$	−5.38311	−0.820917	−0.410459	+	0.911879i	$0.634631\pi$
$44$	−5.51998	−0.832169
$45$	0	0
$46$	0	0
$47$	9.38528	1.36898	0.684492	−	0.729020i	$-0.260024\pi$
$47$	9.38528	1.36898	0.684492	+	0.729020i	$0.260024\pi$
$48$	0	0
$49$	3.15674	0.450963
$50$	2.12530	0.300563
$51$	0	0
$52$	2.64969	0.367446
$53$	−11.6296	−1.59745	−0.798725	−	0.601696i	$-0.794492\pi$
$53$	−11.6296	−1.59745	−0.798725	+	0.601696i	$0.794492\pi$
$54$	0	0
$55$	14.7346	1.98682
$56$	−3.18696	−0.425876
$57$	0	0
$58$	−5.50705	−0.723111
$59$	5.64969	0.735528	0.367764	−	0.929919i	$-0.380124\pi$
$59$	5.64969	0.735528	0.367764	+	0.929919i	$0.380124\pi$
$60$	0	0
$61$	−7.39110	−0.946333	−0.473167	−	0.880973i	$-0.656889\pi$
$61$	−7.39110	−0.946333	−0.473167	+	0.880973i	$0.656889\pi$
$62$	−10.6638	−1.35430
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.07290	−0.877285
$66$	0	0
$67$	−4.30338	−0.525741	−0.262871	−	0.964831i	$-0.584669\pi$
$67$	−4.30338	−0.525741	−0.262871	+	0.964831i	$0.584669\pi$
$68$	3.74723	0.454419
$69$	0	0
$70$	8.50705	1.01679
$71$	1.91411	0.227163	0.113581	−	0.993529i	$-0.463768\pi$
$71$	1.91411	0.227163	0.113581	+	0.993529i	$0.463768\pi$
$72$	0	0
$73$	−5.37377	−0.628953	−0.314476	−	0.949265i	$-0.601829\pi$
$73$	−5.37377	−0.628953	−0.314476	+	0.949265i	$0.601829\pi$
$74$	9.36329	1.08846
$75$	0	0
$76$	−6.37393	−0.731140
$77$	17.5920	2.00479
$78$	0	0
$79$	6.61099	0.743795	0.371897	−	0.928274i	$-0.378707\pi$
$79$	6.61099	0.743795	0.371897	+	0.928274i	$0.378707\pi$
$80$	−2.66933	−0.298440
$81$	0	0
$82$	−0.424692	−0.0468994
$83$	5.15732	0.566090	0.283045	−	0.959107i	$-0.408655\pi$
$83$	5.15732	0.566090	0.283045	+	0.959107i	$0.408655\pi$
$84$	0	0
$85$	−10.0026	−1.08493
$86$	−5.38311	−0.580476
$87$	0	0
$88$	−5.51998	−0.588432
$89$	14.9168	1.58117	0.790587	−	0.612349i	$-0.209775\pi$
$89$	14.9168	1.58117	0.790587	+	0.612349i	$0.209775\pi$
$90$	0	0
$91$	−8.44448	−0.885222
$92$	0	0
$93$	0	0
$94$	9.38528	0.968018
$95$	17.0141	1.74561
$96$	0	0
$97$	−12.8109	−1.30075	−0.650374	−	0.759614i	$-0.725388\pi$
$97$	−12.8109	−1.30075	−0.650374	+	0.759614i	$0.725388\pi$
$98$	3.15674	0.318879
$99$	0	0
$100$	2.12530	0.212530
$101$	−6.74615	−0.671267	−0.335633	−	0.941993i	$-0.608950\pi$
$101$	−6.74615	−0.671267	−0.335633	+	0.941993i	$0.608950\pi$
$102$	0	0
$103$	−1.21661	−0.119876	−0.0599380	−	0.998202i	$-0.519090\pi$
$103$	−1.21661	−0.119876	−0.0599380	+	0.998202i	$0.519090\pi$
$104$	2.64969	0.259824
$105$	0	0
$106$	−11.6296	−1.12957
$107$	6.63640	0.641565	0.320782	−	0.947153i	$-0.396054\pi$
$107$	6.63640	0.641565	0.320782	+	0.947153i	$0.396054\pi$
$108$	0	0
$109$	16.8759	1.61641	0.808207	−	0.588898i	$-0.200438\pi$
$109$	16.8759	1.61641	0.808207	+	0.588898i	$0.200438\pi$
$110$	14.7346	1.40489
$111$	0	0
$112$	−3.18696	−0.301140
$113$	8.31491	0.782201	0.391100	−	0.920348i	$-0.372095\pi$
$113$	8.31491	0.782201	0.391100	+	0.920348i	$0.372095\pi$
$114$	0	0
$115$	0	0
$116$	−5.50705	−0.511317
$117$	0	0
$118$	5.64969	0.520097
$119$	−11.9423	−1.09475
$120$	0	0
$121$	19.4702	1.77002
$122$	−7.39110	−0.669159
$123$	0	0
$124$	−10.6638	−0.957637
$125$	7.67350	0.686339
$126$	0	0
$127$	−12.6429	−1.12188	−0.560938	−	0.827857i	$-0.689560\pi$
$127$	−12.6429	−1.12188	−0.560938	+	0.827857i	$0.689560\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−7.07290	−0.620334
$131$	−4.07085	−0.355672	−0.177836	−	0.984060i	$-0.556910\pi$
$131$	−4.07085	−0.355672	−0.177836	+	0.984060i	$0.556910\pi$
$132$	0	0
$133$	20.3135	1.76140
$134$	−4.30338	−0.371755
$135$	0	0
$136$	3.74723	0.321323
$137$	10.4983	0.896928	0.448464	−	0.893801i	$-0.351971\pi$
$137$	10.4983	0.896928	0.448464	+	0.893801i	$0.351971\pi$
$138$	0	0
$139$	9.71471	0.823991	0.411995	−	0.911186i	$-0.364832\pi$
$139$	9.71471	0.823991	0.411995	+	0.911186i	$0.364832\pi$
$140$	8.50705	0.718977
$141$	0	0
$142$	1.91411	0.160628
$143$	−14.6263	−1.22311
$144$	0	0
$145$	14.7001	1.22078
$146$	−5.37377	−0.444737
$147$	0	0
$148$	9.36329	0.769658
$149$	11.3155	0.927005	0.463503	−	0.886096i	$-0.346592\pi$
$149$	11.3155	0.927005	0.463503	+	0.886096i	$0.346592\pi$
$150$	0	0
$151$	−8.75552	−0.712514	−0.356257	−	0.934388i	$-0.615947\pi$
$151$	−8.75552	−0.712514	−0.356257	+	0.934388i	$0.615947\pi$
$152$	−6.37393	−0.516994
$153$	0	0
$154$	17.5920	1.41760
$155$	28.4651	2.28637
$156$	0	0
$157$	9.69959	0.774112	0.387056	−	0.922056i	$-0.373492\pi$
$157$	9.69959	0.774112	0.387056	+	0.922056i	$0.373492\pi$
$158$	6.61099	0.525942
$159$	0	0
$160$	−2.66933	−0.211029
$161$	0	0
$162$	0	0
$163$	−5.01410	−0.392734	−0.196367	−	0.980530i	$-0.562914\pi$
$163$	−5.01410	−0.392734	−0.196367	+	0.980530i	$0.562914\pi$
$164$	−0.424692	−0.0331629
$165$	0	0
$166$	5.15732	0.400286
$167$	−10.5988	−0.820158	−0.410079	−	0.912050i	$-0.634499\pi$
$167$	−10.5988	−0.820158	−0.410079	+	0.912050i	$0.634499\pi$
$168$	0	0
$169$	−5.97912	−0.459932
$170$	−10.0026	−0.767164
$171$	0	0
$172$	−5.38311	−0.410459
$173$	−11.9773	−0.910615	−0.455308	−	0.890334i	$-0.650471\pi$
$173$	−11.9773	−0.910615	−0.455308	+	0.890334i	$0.650471\pi$
$174$	0	0
$175$	−6.77327	−0.512011
$176$	−5.51998	−0.416084
$177$	0	0
$178$	14.9168	1.11806
$179$	24.6429	1.84190	0.920949	−	0.389683i	$-0.127416\pi$
$179$	24.6429	1.84190	0.920949	+	0.389683i	$0.127416\pi$
$180$	0	0
$181$	19.9599	1.48361	0.741805	−	0.670616i	$-0.233970\pi$
$181$	19.9599	1.48361	0.741805	+	0.670616i	$0.233970\pi$
$182$	−8.44448	−0.625947
$183$	0	0
$184$	0	0
$185$	−24.9937	−1.83757
$186$	0	0
$187$	−20.6847	−1.51261
$188$	9.38528	0.684492
$189$	0	0
$190$	17.0141	1.23433
$191$	−15.1004	−1.09263	−0.546313	−	0.837581i	$-0.683969\pi$
$191$	−15.1004	−1.09263	−0.546313	+	0.837581i	$0.683969\pi$
$192$	0	0
$193$	2.60321	0.187383	0.0936916	−	0.995601i	$-0.470133\pi$
$193$	2.60321	0.187383	0.0936916	+	0.995601i	$0.470133\pi$
$194$	−12.8109	−0.919768
$195$	0	0
$196$	3.15674	0.225482
$197$	−2.24169	−0.159714	−0.0798568	−	0.996806i	$-0.525446\pi$
$197$	−2.24169	−0.159714	−0.0798568	+	0.996806i	$0.525446\pi$
$198$	0	0
$199$	3.43816	0.243725	0.121862	−	0.992547i	$-0.461113\pi$
$199$	3.43816	0.243725	0.121862	+	0.992547i	$0.461113\pi$
$200$	2.12530	0.150282
$201$	0	0
$202$	−6.74615	−0.474657
$203$	17.5508	1.23182
$204$	0	0
$205$	1.13364	0.0791769
$206$	−1.21661	−0.0847651
$207$	0	0
$208$	2.64969	0.183723
$209$	35.1840	2.43373
$210$	0	0
$211$	6.22759	0.428725	0.214363	−	0.976754i	$-0.431233\pi$
$211$	6.22759	0.428725	0.214363	+	0.976754i	$0.431233\pi$
$212$	−11.6296	−0.798725
$213$	0	0
$214$	6.63640	0.453655
$215$	14.3693	0.979977
$216$	0	0
$217$	33.9851	2.30706
$218$	16.8759	1.14298
$219$	0	0
$220$	14.7346	0.993409
$221$	9.92902	0.667898
$222$	0	0
$223$	−20.6429	−1.38235	−0.691176	−	0.722687i	$-0.742907\pi$
$223$	−20.6429	−1.38235	−0.691176	+	0.722687i	$0.742907\pi$
$224$	−3.18696	−0.212938
$225$	0	0
$226$	8.31491	0.553099
$227$	21.7031	1.44049	0.720244	−	0.693721i	$-0.244030\pi$
$227$	21.7031	1.44049	0.720244	+	0.693721i	$0.244030\pi$
$228$	0	0
$229$	8.76768	0.579385	0.289692	−	0.957120i	$-0.406447\pi$
$229$	8.76768	0.579385	0.289692	+	0.957120i	$0.406447\pi$
$230$	0	0
$231$	0	0
$232$	−5.50705	−0.361555
$233$	−6.98878	−0.457850	−0.228925	−	0.973444i	$-0.573521\pi$
$233$	−6.98878	−0.457850	−0.228925	+	0.973444i	$0.573521\pi$
$234$	0	0
$235$	−25.0524	−1.63424
$236$	5.64969	0.367764
$237$	0	0
$238$	−11.9423	−0.774104
$239$	26.3417	1.70390	0.851951	−	0.523622i	$-0.175420\pi$
$239$	26.3417	1.70390	0.851951	+	0.523622i	$0.175420\pi$
$240$	0	0
$241$	−14.3769	−0.926098	−0.463049	−	0.886333i	$-0.653245\pi$
$241$	−14.3769	−0.926098	−0.463049	+	0.886333i	$0.653245\pi$
$242$	19.4702	1.25159
$243$	0	0
$244$	−7.39110	−0.473167
$245$	−8.42638	−0.538341
$246$	0	0
$247$	−16.8890	−1.07462
$248$	−10.6638	−0.677151
$249$	0	0
$250$	7.67350	0.485315
$251$	−27.8370	−1.75706	−0.878528	−	0.477692i	$-0.841474\pi$
$251$	−27.8370	−1.75706	−0.878528	+	0.477692i	$0.841474\pi$
$252$	0	0
$253$	0	0
$254$	−12.6429	−0.793287
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.1882	0.947413	0.473706	−	0.880683i	$-0.342916\pi$
$257$	15.1882	0.947413	0.473706	+	0.880683i	$0.342916\pi$
$258$	0	0
$259$	−29.8405	−1.85420
$260$	−7.07290	−0.438643
$261$	0	0
$262$	−4.07085	−0.251498
$263$	−29.3640	−1.81066	−0.905331	−	0.424706i	$-0.860377\pi$
$263$	−29.3640	−1.81066	−0.905331	+	0.424706i	$0.860377\pi$
$264$	0	0
$265$	31.0432	1.90697
$266$	20.3135	1.24550
$267$	0	0
$268$	−4.30338	−0.262871
$269$	16.5062	1.00640	0.503199	−	0.864171i	$-0.332156\pi$
$269$	16.5062	1.00640	0.503199	+	0.864171i	$0.332156\pi$
$270$	0	0
$271$	7.22854	0.439102	0.219551	−	0.975601i	$-0.429541\pi$
$271$	7.22854	0.439102	0.219551	+	0.975601i	$0.429541\pi$
$272$	3.74723	0.227209
$273$	0	0
$274$	10.4983	0.634224
$275$	−11.7316	−0.707445
$276$	0	0
$277$	1.21349	0.0729118	0.0364559	−	0.999335i	$-0.488393\pi$
$277$	1.21349	0.0729118	0.0364559	+	0.999335i	$0.488393\pi$
$278$	9.71471	0.582649
$279$	0	0
$280$	8.50705	0.508393
$281$	6.88927	0.410980	0.205490	−	0.978659i	$-0.434121\pi$
$281$	6.88927	0.410980	0.205490	+	0.978659i	$0.434121\pi$
$282$	0	0
$283$	9.24316	0.549449	0.274724	−	0.961523i	$-0.411413\pi$
$283$	9.24316	0.549449	0.274724	+	0.961523i	$0.411413\pi$
$284$	1.91411	0.113581
$285$	0	0
$286$	−14.6263	−0.864870
$287$	1.35348	0.0798933
$288$	0	0
$289$	−2.95824	−0.174014
$290$	14.7001	0.863220
$291$	0	0
$292$	−5.37377	−0.314476
$293$	14.3366	0.837551	0.418776	−	0.908090i	$-0.362459\pi$
$293$	14.3366	0.837551	0.418776	+	0.908090i	$0.362459\pi$
$294$	0	0
$295$	−15.0809	−0.878043
$296$	9.36329	0.544230
$297$	0	0
$298$	11.3155	0.655492
$299$	0	0
$300$	0	0
$301$	17.1558	0.988844
$302$	−8.75552	−0.503824
$303$	0	0
$304$	−6.37393	−0.365570
$305$	19.7293	1.12969
$306$	0	0
$307$	15.0423	0.858509	0.429254	−	0.903184i	$-0.358776\pi$
$307$	15.0423	0.858509	0.429254	+	0.903184i	$0.358776\pi$
$308$	17.5920	1.00240
$309$	0	0
$310$	28.4651	1.61671
$311$	−5.01410	−0.284323	−0.142162	−	0.989843i	$-0.545405\pi$
$311$	−5.01410	−0.284323	−0.142162	+	0.989843i	$0.545405\pi$
$312$	0	0
$313$	−1.36031	−0.0768893	−0.0384446	−	0.999261i	$-0.512240\pi$
$313$	−1.36031	−0.0768893	−0.0384446	+	0.999261i	$0.512240\pi$
$314$	9.69959	0.547380
$315$	0	0
$316$	6.61099	0.371897
$317$	29.9622	1.68285	0.841423	−	0.540377i	$-0.181718\pi$
$317$	29.9622	1.68285	0.841423	+	0.540377i	$0.181718\pi$
$318$	0	0
$319$	30.3988	1.70201
$320$	−2.66933	−0.149220
$321$	0	0
$322$	0	0
$323$	−23.8846	−1.32897
$324$	0	0
$325$	5.63140	0.312374
$326$	−5.01410	−0.277705
$327$	0	0
$328$	−0.424692	−0.0234497
$329$	−29.9106	−1.64902
$330$	0	0
$331$	22.7288	1.24929	0.624644	−	0.780910i	$-0.285244\pi$
$331$	22.7288	1.24929	0.624644	+	0.780910i	$0.285244\pi$
$332$	5.15732	0.283045
$333$	0	0
$334$	−10.5988	−0.579939
$335$	11.4871	0.627608
$336$	0	0
$337$	−28.7294	−1.56499	−0.782496	−	0.622656i	$-0.786054\pi$
$337$	−28.7294	−1.56499	−0.782496	+	0.622656i	$0.786054\pi$
$338$	−5.97912	−0.325221
$339$	0	0
$340$	−10.0026	−0.542467
$341$	58.8640	3.18766
$342$	0	0
$343$	12.2483	0.661347
$344$	−5.38311	−0.290238
$345$	0	0
$346$	−11.9773	−0.643902
$347$	−17.1990	−0.923292	−0.461646	−	0.887064i	$-0.652741\pi$
$347$	−17.1990	−0.923292	−0.461646	+	0.887064i	$0.652741\pi$
$348$	0	0
$349$	30.1540	1.61410	0.807052	−	0.590480i	$-0.201062\pi$
$349$	30.1540	1.61410	0.807052	+	0.590480i	$0.201062\pi$
$350$	−6.77327	−0.362046
$351$	0	0
$352$	−5.51998	−0.294216
$353$	32.5920	1.73470	0.867348	−	0.497701i	$-0.165822\pi$
$353$	32.5920	1.73470	0.867348	+	0.497701i	$0.165822\pi$
$354$	0	0
$355$	−5.10937	−0.271177
$356$	14.9168	0.790587
$357$	0	0
$358$	24.6429	1.30242
$359$	−10.9584	−0.578361	−0.289180	−	0.957275i	$-0.593383\pi$
$359$	−10.9584	−0.578361	−0.289180	+	0.957275i	$0.593383\pi$
$360$	0	0
$361$	21.6270	1.13826
$362$	19.9599	1.04907
$363$	0	0
$364$	−8.44448	−0.442611
$365$	14.3444	0.750818
$366$	0	0
$367$	−14.6009	−0.762159	−0.381080	−	0.924542i	$-0.624448\pi$
$367$	−14.6009	−0.762159	−0.381080	+	0.924542i	$0.624448\pi$
$368$	0	0
$369$	0	0
$370$	−24.9937	−1.29936
$371$	37.0632	1.92422
$372$	0	0
$373$	34.9805	1.81122	0.905610	−	0.424111i	$-0.139413\pi$
$373$	34.9805	1.81122	0.905610	+	0.424111i	$0.139413\pi$
$374$	−20.6847	−1.06958
$375$	0	0
$376$	9.38528	0.484009
$377$	−14.5920	−0.751526
$378$	0	0
$379$	12.9184	0.663573	0.331786	−	0.943355i	$-0.392349\pi$
$379$	12.9184	0.663573	0.331786	+	0.943355i	$0.392349\pi$
$380$	17.0141	0.872805
$381$	0	0
$382$	−15.1004	−0.772604
$383$	3.50470	0.179082	0.0895409	−	0.995983i	$-0.471460\pi$
$383$	3.50470	0.179082	0.0895409	+	0.995983i	$0.471460\pi$
$384$	0	0
$385$	−46.9588	−2.39324
$386$	2.60321	0.132500
$387$	0	0
$388$	−12.8109	−0.650374
$389$	35.8835	1.81936	0.909682	−	0.415305i	$-0.136325\pi$
$389$	35.8835	1.81936	0.909682	+	0.415305i	$0.136325\pi$
$390$	0	0
$391$	0	0
$392$	3.15674	0.159440
$393$	0	0
$394$	−2.24169	−0.112935
$395$	−17.6469	−0.887912
$396$	0	0
$397$	−32.7061	−1.64147	−0.820736	−	0.571308i	$-0.806436\pi$
$397$	−32.7061	−1.64147	−0.820736	+	0.571308i	$0.806436\pi$
$398$	3.43816	0.172339
$399$	0	0
$400$	2.12530	0.106265
$401$	−33.1293	−1.65440	−0.827200	−	0.561907i	$-0.810068\pi$
$401$	−33.1293	−1.65440	−0.827200	+	0.561907i	$0.810068\pi$
$402$	0	0
$403$	−28.2558	−1.40752
$404$	−6.74615	−0.335633
$405$	0	0
$406$	17.5508	0.871030
$407$	−51.6852	−2.56194
$408$	0	0
$409$	−32.6760	−1.61573	−0.807863	−	0.589370i	$-0.799376\pi$
$409$	−32.6760	−1.61573	−0.807863	+	0.589370i	$0.799376\pi$
$410$	1.13364	0.0559865
$411$	0	0
$412$	−1.21661	−0.0599380
$413$	−18.0054	−0.885987
$414$	0	0
$415$	−13.7666	−0.675775
$416$	2.64969	0.129912
$417$	0	0
$418$	35.1840	1.72091
$419$	−8.40048	−0.410390	−0.205195	−	0.978721i	$-0.565783\pi$
$419$	−8.40048	−0.410390	−0.205195	+	0.978721i	$0.565783\pi$
$420$	0	0
$421$	−5.81779	−0.283542	−0.141771	−	0.989900i	$-0.545280\pi$
$421$	−5.81779	−0.283542	−0.141771	+	0.989900i	$0.545280\pi$
$422$	6.22759	0.303154
$423$	0	0
$424$	−11.6296	−0.564784
$425$	7.96401	0.386311
$426$	0	0
$427$	23.5552	1.13991
$428$	6.63640	0.320782
$429$	0	0
$430$	14.3693	0.692949
$431$	20.4753	0.986259	0.493129	−	0.869956i	$-0.335853\pi$
$431$	20.4753	0.986259	0.493129	+	0.869956i	$0.335853\pi$
$432$	0	0
$433$	26.5736	1.27705	0.638523	−	0.769602i	$-0.279546\pi$
$433$	26.5736	1.27705	0.638523	+	0.769602i	$0.279546\pi$
$434$	33.9851	1.63134
$435$	0	0
$436$	16.8759	0.808207
$437$	0	0
$438$	0	0
$439$	−0.807385	−0.0385344	−0.0192672	−	0.999814i	$-0.506133\pi$
$439$	−0.807385	−0.0385344	−0.0192672	+	0.999814i	$0.506133\pi$
$440$	14.7346	0.702446
$441$	0	0
$442$	9.92902	0.472275
$443$	15.1349	0.719082	0.359541	−	0.933129i	$-0.382933\pi$
$443$	15.1349	0.719082	0.359541	+	0.933129i	$0.382933\pi$
$444$	0	0
$445$	−39.8177	−1.88754
$446$	−20.6429	−0.977470
$447$	0	0
$448$	−3.18696	−0.150570
$449$	6.19710	0.292459	0.146230	−	0.989251i	$-0.453286\pi$
$449$	6.19710	0.292459	0.146230	+	0.989251i	$0.453286\pi$
$450$	0	0
$451$	2.34429	0.110388
$452$	8.31491	0.391100
$453$	0	0
$454$	21.7031	1.01858
$455$	22.5411	1.05674
$456$	0	0
$457$	5.24075	0.245152	0.122576	−	0.992459i	$-0.460884\pi$
$457$	5.24075	0.245152	0.122576	+	0.992459i	$0.460884\pi$
$458$	8.76768	0.409687
$459$	0	0
$460$	0	0
$461$	−7.47112	−0.347965	−0.173982	−	0.984749i	$-0.555664\pi$
$461$	−7.47112	−0.347965	−0.173982	+	0.984749i	$0.555664\pi$
$462$	0	0
$463$	5.63375	0.261823	0.130911	−	0.991394i	$-0.458210\pi$
$463$	5.63375	0.261823	0.130911	+	0.991394i	$0.458210\pi$
$464$	−5.50705	−0.255658
$465$	0	0
$466$	−6.98878	−0.323749
$467$	20.1267	0.931354	0.465677	−	0.884955i	$-0.345811\pi$
$467$	20.1267	0.931354	0.465677	+	0.884955i	$0.345811\pi$
$468$	0	0
$469$	13.7147	0.633286
$470$	−25.0524	−1.15558
$471$	0	0
$472$	5.64969	0.260048
$473$	29.7147	1.36628
$474$	0	0
$475$	−13.5465	−0.621558
$476$	−11.9423	−0.547374
$477$	0	0
$478$	26.3417	1.20484
$479$	−17.1402	−0.783154	−0.391577	−	0.920145i	$-0.628070\pi$
$479$	−17.1402	−0.783154	−0.391577	+	0.920145i	$0.628070\pi$
$480$	0	0
$481$	24.8098	1.13123
$482$	−14.3769	−0.654850
$483$	0	0
$484$	19.4702	0.885010
$485$	34.1964	1.55278
$486$	0	0
$487$	−12.2644	−0.555754	−0.277877	−	0.960617i	$-0.589631\pi$
$487$	−12.2644	−0.555754	−0.277877	+	0.960617i	$0.589631\pi$
$488$	−7.39110	−0.334579
$489$	0	0
$490$	−8.42638	−0.380665
$491$	−6.37029	−0.287487	−0.143744	−	0.989615i	$-0.545914\pi$
$491$	−6.37029	−0.287487	−0.143744	+	0.989615i	$0.545914\pi$
$492$	0	0
$493$	−20.6362	−0.929408
$494$	−16.8890	−0.759870
$495$	0	0
$496$	−10.6638	−0.478818
$497$	−6.10019	−0.273631
$498$	0	0
$499$	−21.9981	−0.984770	−0.492385	−	0.870377i	$-0.663875\pi$
$499$	−21.9981	−0.984770	−0.492385	+	0.870377i	$0.663875\pi$
$500$	7.67350	0.343170
$501$	0	0
$502$	−27.8370	−1.24243
$503$	19.2333	0.857568	0.428784	−	0.903407i	$-0.358942\pi$
$503$	19.2333	0.857568	0.428784	+	0.903407i	$0.358942\pi$
$504$	0	0
$505$	18.0077	0.801331
$506$	0	0
$507$	0	0
$508$	−12.6429	−0.560938
$509$	23.2616	1.03105	0.515527	−	0.856874i	$-0.327596\pi$
$509$	23.2616	1.03105	0.515527	+	0.856874i	$0.327596\pi$
$510$	0	0
$511$	17.1260	0.757611
$512$	1.00000	0.0441942
$513$	0	0
$514$	15.1882	0.669922
$515$	3.24752	0.143103
$516$	0	0
$517$	−51.8066	−2.27845
$518$	−29.8405	−1.31111
$519$	0	0
$520$	−7.07290	−0.310167
$521$	−27.6565	−1.21165	−0.605827	−	0.795596i	$-0.707158\pi$
$521$	−27.6565	−1.21165	−0.605827	+	0.795596i	$0.707158\pi$
$522$	0	0
$523$	−26.8574	−1.17439	−0.587197	−	0.809444i	$-0.699769\pi$
$523$	−26.8574	−1.17439	−0.587197	+	0.809444i	$0.699769\pi$
$524$	−4.07085	−0.177836
$525$	0	0
$526$	−29.3640	−1.28033
$527$	−39.9597	−1.74067
$528$	0	0
$529$	0	0
$530$	31.0432	1.34843
$531$	0	0
$532$	20.3135	0.880701
$533$	−1.12530	−0.0487423
$534$	0	0
$535$	−17.7147	−0.765874
$536$	−4.30338	−0.185878
$537$	0	0
$538$	16.5062	0.711631
$539$	−17.4252	−0.750555
$540$	0	0
$541$	−35.9222	−1.54441	−0.772207	−	0.635371i	$-0.780847\pi$
$541$	−35.9222	−1.54441	−0.772207	+	0.635371i	$0.780847\pi$
$542$	7.22854	0.310492
$543$	0	0
$544$	3.74723	0.160661
$545$	−45.0472	−1.92961
$546$	0	0
$547$	9.51531	0.406845	0.203423	−	0.979091i	$-0.434793\pi$
$547$	9.51531	0.406845	0.203423	+	0.979091i	$0.434793\pi$
$548$	10.4983	0.448464
$549$	0	0
$550$	−11.7316	−0.500239
$551$	35.1015	1.49538
$552$	0	0
$553$	−21.0690	−0.895945
$554$	1.21349	0.0515565
$555$	0	0
$556$	9.71471	0.411995
$557$	−39.2924	−1.66487	−0.832437	−	0.554120i	$-0.813055\pi$
$557$	−39.2924	−1.66487	−0.832437	+	0.554120i	$0.813055\pi$
$558$	0	0
$559$	−14.2636	−0.603286
$560$	8.50705	0.359488
$561$	0	0
$562$	6.88927	0.290606
$563$	1.85603	0.0782224	0.0391112	−	0.999235i	$-0.487547\pi$
$563$	1.85603	0.0782224	0.0391112	+	0.999235i	$0.487547\pi$
$564$	0	0
$565$	−22.1952	−0.933759
$566$	9.24316	0.388519
$567$	0	0
$568$	1.91411	0.0803141
$569$	−30.4311	−1.27574	−0.637868	−	0.770146i	$-0.720184\pi$
$569$	−30.4311	−1.27574	−0.637868	+	0.770146i	$0.720184\pi$
$570$	0	0
$571$	33.2231	1.39034	0.695172	−	0.718843i	$-0.255328\pi$
$571$	33.2231	1.39034	0.695172	+	0.718843i	$0.255328\pi$
$572$	−14.6263	−0.611555
$573$	0	0
$574$	1.35348	0.0564931
$575$	0	0
$576$	0	0
$577$	−9.00095	−0.374714	−0.187357	−	0.982292i	$-0.559992\pi$
$577$	−9.00095	−0.374714	−0.187357	+	0.982292i	$0.559992\pi$
$578$	−2.95824	−0.123047
$579$	0	0
$580$	14.7001	0.610389
$581$	−16.4362	−0.681888
$582$	0	0
$583$	64.1953	2.65870
$584$	−5.37377	−0.222368
$585$	0	0
$586$	14.3366	0.592238
$587$	35.0840	1.44807	0.724037	−	0.689761i	$-0.242285\pi$
$587$	35.0840	1.44807	0.724037	+	0.689761i	$0.242285\pi$
$588$	0	0
$589$	67.9702	2.80067
$590$	−15.0809	−0.620870
$591$	0	0
$592$	9.36329	0.384829
$593$	−0.355244	−0.0145881	−0.00729406	−	0.999973i	$-0.502322\pi$
$593$	−0.355244	−0.0145881	−0.00729406	+	0.999973i	$0.502322\pi$
$594$	0	0
$595$	31.8779	1.30687
$596$	11.3155	0.463503
$597$	0	0
$598$	0	0
$599$	−2.61472	−0.106834	−0.0534172	−	0.998572i	$-0.517011\pi$
$599$	−2.61472	−0.106834	−0.0534172	+	0.998572i	$0.517011\pi$
$600$	0	0
$601$	−18.9109	−0.771390	−0.385695	−	0.922626i	$-0.626038\pi$
$601$	−18.9109	−0.771390	−0.385695	+	0.922626i	$0.626038\pi$
$602$	17.1558	0.699218
$603$	0	0
$604$	−8.75552	−0.356257
$605$	−51.9724	−2.11298
$606$	0	0
$607$	−7.40033	−0.300370	−0.150185	−	0.988658i	$-0.547987\pi$
$607$	−7.40033	−0.300370	−0.150185	+	0.988658i	$0.547987\pi$
$608$	−6.37393	−0.258497
$609$	0	0
$610$	19.7293	0.798814
$611$	24.8681	1.00606
$612$	0	0
$613$	−21.1742	−0.855219	−0.427610	−	0.903963i	$-0.640644\pi$
$613$	−21.1742	−0.855219	−0.427610	+	0.903963i	$0.640644\pi$
$614$	15.0423	0.607057
$615$	0	0
$616$	17.5920	0.708802
$617$	−23.1202	−0.930785	−0.465392	−	0.885104i	$-0.654087\pi$
$617$	−23.1202	−0.930785	−0.465392	+	0.885104i	$0.654087\pi$
$618$	0	0
$619$	−13.2953	−0.534385	−0.267192	−	0.963643i	$-0.586096\pi$
$619$	−13.2953	−0.534385	−0.267192	+	0.963643i	$0.586096\pi$
$620$	28.4651	1.14319
$621$	0	0
$622$	−5.01410	−0.201047
$623$	−47.5392	−1.90462
$624$	0	0
$625$	−31.1096	−1.24438
$626$	−1.36031	−0.0543689
$627$	0	0
$628$	9.69959	0.387056
$629$	35.0864	1.39899
$630$	0	0
$631$	25.6775	1.02221	0.511103	−	0.859520i	$-0.329237\pi$
$631$	25.6775	1.02221	0.511103	+	0.859520i	$0.329237\pi$
$632$	6.61099	0.262971
$633$	0	0
$634$	29.9622	1.18995
$635$	33.7481	1.33925
$636$	0	0
$637$	8.36440	0.331410
$638$	30.3988	1.20350
$639$	0	0
$640$	−2.66933	−0.105514
$641$	6.19355	0.244631	0.122315	−	0.992491i	$-0.460968\pi$
$641$	6.19355	0.244631	0.122315	+	0.992491i	$0.460968\pi$
$642$	0	0
$643$	−25.2528	−0.995872	−0.497936	−	0.867214i	$-0.665909\pi$
$643$	−25.2528	−0.995872	−0.497936	+	0.867214i	$0.665909\pi$
$644$	0	0
$645$	0	0
$646$	−23.8846	−0.939727
$647$	−11.5969	−0.455922	−0.227961	−	0.973670i	$-0.573206\pi$
$647$	−11.5969	−0.455922	−0.227961	+	0.973670i	$0.573206\pi$
$648$	0	0
$649$	−31.1862	−1.22417
$650$	5.63140	0.220882
$651$	0	0
$652$	−5.01410	−0.196367
$653$	−33.4707	−1.30981	−0.654904	−	0.755712i	$-0.727291\pi$
$653$	−33.4707	−1.30981	−0.654904	+	0.755712i	$0.727291\pi$
$654$	0	0
$655$	10.8664	0.424586
$656$	−0.424692	−0.0165814
$657$	0	0
$658$	−29.9106	−1.16604
$659$	−37.9004	−1.47639	−0.738196	−	0.674587i	$-0.764322\pi$
$659$	−37.9004	−1.47639	−0.738196	+	0.674587i	$0.764322\pi$
$660$	0	0
$661$	−10.6084	−0.412619	−0.206309	−	0.978487i	$-0.566145\pi$
$661$	−10.6084	−0.412619	−0.206309	+	0.978487i	$0.566145\pi$
$662$	22.7288	0.883380
$663$	0	0
$664$	5.15732	0.200143
$665$	−54.2233	−2.10269
$666$	0	0
$667$	0	0
$668$	−10.5988	−0.410079
$669$	0	0
$670$	11.4871	0.443786
$671$	40.7988	1.57502
$672$	0	0
$673$	37.9337	1.46224	0.731118	−	0.682251i	$-0.238999\pi$
$673$	37.9337	1.46224	0.731118	+	0.682251i	$0.238999\pi$
$674$	−28.7294	−1.10662
$675$	0	0
$676$	−5.97912	−0.229966
$677$	−3.28167	−0.126125	−0.0630624	−	0.998010i	$-0.520087\pi$
$677$	−3.28167	−0.126125	−0.0630624	+	0.998010i	$0.520087\pi$
$678$	0	0
$679$	40.8278	1.56683
$680$	−10.0026	−0.383582
$681$	0	0
$682$	58.8640	2.25402
$683$	−3.48285	−0.133267	−0.0666337	−	0.997778i	$-0.521226\pi$
$683$	−3.48285	−0.133267	−0.0666337	+	0.997778i	$0.521226\pi$
$684$	0	0
$685$	−28.0233	−1.07072
$686$	12.2483	0.467643
$687$	0	0
$688$	−5.38311	−0.205229
$689$	−30.8149	−1.17396
$690$	0	0
$691$	−13.2135	−0.502665	−0.251333	−	0.967901i	$-0.580869\pi$
$691$	−13.2135	−0.502665	−0.251333	+	0.967901i	$0.580869\pi$
$692$	−11.9773	−0.455308
$693$	0	0
$694$	−17.1990	−0.652866
$695$	−25.9317	−0.983646
$696$	0	0
$697$	−1.59142	−0.0602793
$698$	30.1540	1.14134
$699$	0	0
$700$	−6.77327	−0.256005
$701$	−0.552017	−0.0208494	−0.0104247	−	0.999946i	$-0.503318\pi$
$701$	−0.552017	−0.0208494	−0.0104247	+	0.999946i	$0.503318\pi$
$702$	0	0
$703$	−59.6809	−2.25091
$704$	−5.51998	−0.208042
$705$	0	0
$706$	32.5920	1.22662
$707$	21.4997	0.808581
$708$	0	0
$709$	7.35489	0.276219	0.138109	−	0.990417i	$-0.455897\pi$
$709$	7.35489	0.276219	0.138109	+	0.990417i	$0.455897\pi$
$710$	−5.10937	−0.191751
$711$	0	0
$712$	14.9168	0.559030
$713$	0	0
$714$	0	0
$715$	39.0423	1.46010
$716$	24.6429	0.920949
$717$	0	0
$718$	−10.9584	−0.408963
$719$	−28.9564	−1.07989	−0.539946	−	0.841700i	$-0.681555\pi$
$719$	−28.9564	−1.07989	−0.539946	+	0.841700i	$0.681555\pi$
$720$	0	0
$721$	3.87729	0.144398
$722$	21.6270	0.804872
$723$	0	0
$724$	19.9599	0.741805
$725$	−11.7041	−0.434681
$726$	0	0
$727$	−30.6754	−1.13769	−0.568844	−	0.822445i	$-0.692609\pi$
$727$	−30.6754	−1.13769	−0.568844	+	0.822445i	$0.692609\pi$
$728$	−8.44448	−0.312973
$729$	0	0
$730$	14.3444	0.530909
$731$	−20.1718	−0.746080
$732$	0	0
$733$	−21.9225	−0.809727	−0.404863	−	0.914377i	$-0.632681\pi$
$733$	−21.9225	−0.809727	−0.404863	+	0.914377i	$0.632681\pi$
$734$	−14.6009	−0.538928
$735$	0	0
$736$	0	0
$737$	23.7546	0.875011
$738$	0	0
$739$	−27.0840	−0.996303	−0.498151	−	0.867090i	$-0.665988\pi$
$739$	−27.0840	−0.996303	−0.498151	+	0.867090i	$0.665988\pi$
$740$	−24.9937	−0.918786
$741$	0	0
$742$	37.0632	1.36063
$743$	−39.3316	−1.44294	−0.721468	−	0.692448i	$-0.756532\pi$
$743$	−39.3316	−1.44294	−0.721468	+	0.692448i	$0.756532\pi$
$744$	0	0
$745$	−30.2049	−1.10662
$746$	34.9805	1.28073
$747$	0	0
$748$	−20.6847	−0.756306
$749$	−21.1500	−0.772803
$750$	0	0
$751$	7.33934	0.267816	0.133908	−	0.990994i	$-0.457247\pi$
$751$	7.33934	0.267816	0.133908	+	0.990994i	$0.457247\pi$
$752$	9.38528	0.342246
$753$	0	0
$754$	−14.5920	−0.531409
$755$	23.3713	0.850570
$756$	0	0
$757$	8.00496	0.290945	0.145473	−	0.989362i	$-0.453530\pi$
$757$	8.00496	0.290945	0.145473	+	0.989362i	$0.453530\pi$
$758$	12.9184	0.469217
$759$	0	0
$760$	17.0141	0.617166
$761$	48.5601	1.76030	0.880151	−	0.474693i	$-0.157441\pi$
$761$	48.5601	1.76030	0.880151	+	0.474693i	$0.157441\pi$
$762$	0	0
$763$	−53.7828	−1.94707
$764$	−15.1004	−0.546313
$765$	0	0
$766$	3.50470	0.126630
$767$	14.9700	0.540534
$768$	0	0
$769$	1.81228	0.0653526	0.0326763	−	0.999466i	$-0.489597\pi$
$769$	1.81228	0.0653526	0.0326763	+	0.999466i	$0.489597\pi$
$770$	−46.9588	−1.69228
$771$	0	0
$772$	2.60321	0.0936916
$773$	−8.27640	−0.297682	−0.148841	−	0.988861i	$-0.547554\pi$
$773$	−8.27640	−0.297682	−0.148841	+	0.988861i	$0.547554\pi$
$774$	0	0
$775$	−22.6638	−0.814107
$776$	−12.8109	−0.459884
$777$	0	0
$778$	35.8835	1.28648
$779$	2.70696	0.0969867
$780$	0	0
$781$	−10.5658	−0.378075
$782$	0	0
$783$	0	0
$784$	3.15674	0.112741
$785$	−25.8914	−0.924103
$786$	0	0
$787$	−3.23096	−0.115171	−0.0575857	−	0.998341i	$-0.518340\pi$
$787$	−3.23096	−0.115171	−0.0575857	+	0.998341i	$0.518340\pi$
$788$	−2.24169	−0.0798568
$789$	0	0
$790$	−17.6469	−0.627849
$791$	−26.4993	−0.942207
$792$	0	0
$793$	−19.5842	−0.695454
$794$	−32.7061	−1.16070
$795$	0	0
$796$	3.43816	0.121862
$797$	14.6545	0.519088	0.259544	−	0.965731i	$-0.416428\pi$
$797$	14.6545	0.519088	0.259544	+	0.965731i	$0.416428\pi$
$798$	0	0
$799$	35.1688	1.24418
$800$	2.12530	0.0751408
$801$	0	0
$802$	−33.1293	−1.16984
$803$	29.6632	1.04679
$804$	0	0
$805$	0	0
$806$	−28.2558	−0.995268
$807$	0	0
$808$	−6.74615	−0.237329
$809$	37.2257	1.30879	0.654394	−	0.756154i	$-0.272924\pi$
$809$	37.2257	1.30879	0.654394	+	0.756154i	$0.272924\pi$
$810$	0	0
$811$	10.5135	0.369178	0.184589	−	0.982816i	$-0.440905\pi$
$811$	10.5135	0.369178	0.184589	+	0.982816i	$0.440905\pi$
$812$	17.5508	0.615911
$813$	0	0
$814$	−51.6852	−1.81157
$815$	13.3843	0.468830
$816$	0	0
$817$	34.3116	1.20041
$818$	−32.6760	−1.14249
$819$	0	0
$820$	1.13364	0.0395885
$821$	2.43266	0.0849005	0.0424503	−	0.999099i	$-0.486484\pi$
$821$	2.43266	0.0849005	0.0424503	+	0.999099i	$0.486484\pi$
$822$	0	0
$823$	−27.2576	−0.950141	−0.475071	−	0.879948i	$-0.657577\pi$
$823$	−27.2576	−0.950141	−0.475071	+	0.879948i	$0.657577\pi$
$824$	−1.21661	−0.0423825
$825$	0	0
$826$	−18.0054	−0.626487
$827$	50.7147	1.76352	0.881761	−	0.471697i	$-0.156358\pi$
$827$	50.7147	1.76352	0.881761	+	0.471697i	$0.156358\pi$
$828$	0	0
$829$	34.5670	1.20056	0.600280	−	0.799790i	$-0.295056\pi$
$829$	34.5670	1.20056	0.600280	+	0.799790i	$0.295056\pi$
$830$	−13.7666	−0.477845
$831$	0	0
$832$	2.64969	0.0918616
$833$	11.8291	0.409852
$834$	0	0
$835$	28.2916	0.979071
$836$	35.1840	1.21686
$837$	0	0
$838$	−8.40048	−0.290190
$839$	−7.91222	−0.273160	−0.136580	−	0.990629i	$-0.543611\pi$
$839$	−7.91222	−0.273160	−0.136580	+	0.990629i	$0.543611\pi$
$840$	0	0
$841$	1.32758	0.0457787
$842$	−5.81779	−0.200494
$843$	0	0
$844$	6.22759	0.214363
$845$	15.9602	0.549048
$846$	0	0
$847$	−62.0509	−2.13209
$848$	−11.6296	−0.399363
$849$	0	0
$850$	7.96401	0.273163
$851$	0	0
$852$	0	0
$853$	15.4693	0.529658	0.264829	−	0.964295i	$-0.414684\pi$
$853$	15.4693	0.529658	0.264829	+	0.964295i	$0.414684\pi$
$854$	23.5552	0.806041
$855$	0	0
$856$	6.63640	0.226827
$857$	17.0835	0.583563	0.291781	−	0.956485i	$-0.405752\pi$
$857$	17.0835	0.583563	0.291781	+	0.956485i	$0.405752\pi$
$858$	0	0
$859$	15.5987	0.532222	0.266111	−	0.963942i	$-0.414261\pi$
$859$	15.5987	0.532222	0.266111	+	0.963942i	$0.414261\pi$
$860$	14.3693	0.489989
$861$	0	0
$862$	20.4753	0.697390
$863$	43.1680	1.46946	0.734729	−	0.678361i	$-0.237309\pi$
$863$	43.1680	1.46946	0.734729	+	0.678361i	$0.237309\pi$
$864$	0	0
$865$	31.9713	1.08706
$866$	26.5736	0.903008
$867$	0	0
$868$	33.9851	1.15353
$869$	−36.4926	−1.23793
$870$	0	0
$871$	−11.4026	−0.386363
$872$	16.8759	0.571489
$873$	0	0
$874$	0	0
$875$	−24.4552	−0.826736
$876$	0	0
$877$	0.673001	0.0227256	0.0113628	−	0.999935i	$-0.496383\pi$
$877$	0.673001	0.0227256	0.0113628	+	0.999935i	$0.496383\pi$
$878$	−0.807385	−0.0272479
$879$	0	0
$880$	14.7346	0.496705
$881$	23.1728	0.780711	0.390355	−	0.920664i	$-0.372352\pi$
$881$	23.1728	0.780711	0.390355	+	0.920664i	$0.372352\pi$
$882$	0	0
$883$	−43.9841	−1.48018	−0.740091	−	0.672507i	$-0.765217\pi$
$883$	−43.9841	−1.48018	−0.740091	+	0.672507i	$0.765217\pi$
$884$	9.92902	0.333949
$885$	0	0
$886$	15.1349	0.508467
$887$	−7.47118	−0.250858	−0.125429	−	0.992103i	$-0.540031\pi$
$887$	−7.47118	−0.250858	−0.125429	+	0.992103i	$0.540031\pi$
$888$	0	0
$889$	40.2925	1.35137
$890$	−39.8177	−1.33469
$891$	0	0
$892$	−20.6429	−0.691176
$893$	−59.8211	−2.00184
$894$	0	0
$895$	−65.7800	−2.19878
$896$	−3.18696	−0.106469
$897$	0	0
$898$	6.19710	0.206800
$899$	58.7260	1.95862
$900$	0	0
$901$	−43.5789	−1.45182
$902$	2.34429	0.0780564
$903$	0	0
$904$	8.31491	0.276550
$905$	−53.2795	−1.77107
$906$	0	0
$907$	15.6253	0.518831	0.259415	−	0.965766i	$-0.416470\pi$
$907$	15.6253	0.518831	0.259415	+	0.965766i	$0.416470\pi$
$908$	21.7031	0.720244
$909$	0	0
$910$	22.5411	0.747229
$911$	21.0891	0.698714	0.349357	−	0.936990i	$-0.386400\pi$
$911$	21.0891	0.698714	0.349357	+	0.936990i	$0.386400\pi$
$912$	0	0
$913$	−28.4683	−0.942164
$914$	5.24075	0.173349
$915$	0	0
$916$	8.76768	0.289692
$917$	12.9737	0.428428
$918$	0	0
$919$	−17.4325	−0.575045	−0.287522	−	0.957774i	$-0.592832\pi$
$919$	−17.4325	−0.575045	−0.287522	+	0.957774i	$0.592832\pi$
$920$	0	0
$921$	0	0
$922$	−7.47112	−0.246048
$923$	5.07180	0.166940
$924$	0	0
$925$	19.8998	0.654302
$926$	5.63375	0.185137
$927$	0	0
$928$	−5.50705	−0.180778
$929$	−25.9963	−0.852911	−0.426456	−	0.904508i	$-0.640238\pi$
$929$	−25.9963	−0.852911	−0.426456	+	0.904508i	$0.640238\pi$
$930$	0	0
$931$	−20.1209	−0.659434
$932$	−6.98878	−0.228925
$933$	0	0
$934$	20.1267	0.658567
$935$	55.2141	1.80570
$936$	0	0
$937$	−16.1080	−0.526224	−0.263112	−	0.964765i	$-0.584749\pi$
$937$	−16.1080	−0.526224	−0.263112	+	0.964765i	$0.584749\pi$
$938$	13.7147	0.447801
$939$	0	0
$940$	−25.0524	−0.817119
$941$	24.7832	0.807909	0.403954	−	0.914779i	$-0.367635\pi$
$941$	24.7832	0.807909	0.403954	+	0.914779i	$0.367635\pi$
$942$	0	0
$943$	0	0
$944$	5.64969	0.183882
$945$	0	0
$946$	29.7147	0.966109
$947$	33.7002	1.09511	0.547555	−	0.836769i	$-0.315559\pi$
$947$	33.7002	1.09511	0.547555	+	0.836769i	$0.315559\pi$
$948$	0	0
$949$	−14.2389	−0.462213
$950$	−13.5465	−0.439508
$951$	0	0
$952$	−11.9423	−0.387052
$953$	−26.7114	−0.865266	−0.432633	−	0.901570i	$-0.642415\pi$
$953$	−26.7114	−0.865266	−0.432633	+	0.901570i	$0.642415\pi$
$954$	0	0
$955$	40.3079	1.30433
$956$	26.3417	0.851951
$957$	0	0
$958$	−17.1402	−0.553773
$959$	−33.4576	−1.08040
$960$	0	0
$961$	82.7164	2.66827
$962$	24.8098	0.799902
$963$	0	0
$964$	−14.3769	−0.463049
$965$	−6.94882	−0.223690
$966$	0	0
$967$	−18.4502	−0.593320	−0.296660	−	0.954983i	$-0.595873\pi$
$967$	−18.4502	−0.593320	−0.296660	+	0.954983i	$0.595873\pi$
$968$	19.4702	0.625797
$969$	0	0
$970$	34.1964	1.09798
$971$	43.0914	1.38287	0.691435	−	0.722439i	$-0.256979\pi$
$971$	43.0914	1.38287	0.691435	+	0.722439i	$0.256979\pi$
$972$	0	0
$973$	−30.9604	−0.992546
$974$	−12.2644	−0.392977
$975$	0	0
$976$	−7.39110	−0.236583
$977$	54.0937	1.73061	0.865306	−	0.501244i	$-0.167124\pi$
$977$	54.0937	1.73061	0.865306	+	0.501244i	$0.167124\pi$
$978$	0	0
$979$	−82.3404	−2.63161
$980$	−8.42638	−0.269171
$981$	0	0
$982$	−6.37029	−0.203284
$983$	1.46495	0.0467245	0.0233623	−	0.999727i	$-0.492563\pi$
$983$	1.46495	0.0467245	0.0233623	+	0.999727i	$0.492563\pi$
$984$	0	0
$985$	5.98380	0.190660
$986$	−20.6362	−0.657190
$987$	0	0
$988$	−16.8890	−0.537309
$989$	0	0
$990$	0	0
$991$	23.2490	0.738529	0.369264	−	0.929324i	$-0.379610\pi$
$991$	23.2490	0.738529	0.369264	+	0.929324i	$0.379610\pi$
$992$	−10.6638	−0.338576
$993$	0	0
$994$	−6.10019	−0.193486
$995$	−9.17757	−0.290949
$996$	0	0
$997$	−37.4158	−1.18497	−0.592485	−	0.805582i	$-0.701853\pi$
$997$	−37.4158	−1.18497	−0.592485	+	0.805582i	$0.701853\pi$
$998$	−21.9981	−0.696338
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.cf.1.2	yes	8
3.2	odd	2		9522.2.a.cd.1.7	yes	8
23.22	odd	2	inner	9522.2.a.cf.1.7	yes	8
69.68	even	2		9522.2.a.cd.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9522.2.a.cd.1.2	✓	8	69.68	even	2
9522.2.a.cd.1.7	yes	8	3.2	odd	2
9522.2.a.cf.1.2	yes	8	1.1	even	1	trivial
9522.2.a.cf.1.7	yes	8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.66933 q^{5} -3.18696 q^{7} +1.00000 q^{8} -2.66933 q^{10} -5.51998 q^{11} +2.64969 q^{13} -3.18696 q^{14} +1.00000 q^{16} +3.74723 q^{17} -6.37393 q^{19} -2.66933 q^{20} -5.51998 q^{22} +2.12530 q^{25} +2.64969 q^{26} -3.18696 q^{28} -5.50705 q^{29} -10.6638 q^{31} +1.00000 q^{32} +3.74723 q^{34} +8.50705 q^{35} +9.36329 q^{37} -6.37393 q^{38} -2.66933 q^{40} -0.424692 q^{41} -5.38311 q^{43} -5.51998 q^{44} +9.38528 q^{47} +3.15674 q^{49} +2.12530 q^{50} +2.64969 q^{52} -11.6296 q^{53} +14.7346 q^{55} -3.18696 q^{56} -5.50705 q^{58} +5.64969 q^{59} -7.39110 q^{61} -10.6638 q^{62} +1.00000 q^{64} -7.07290 q^{65} -4.30338 q^{67} +3.74723 q^{68} +8.50705 q^{70} +1.91411 q^{71} -5.37377 q^{73} +9.36329 q^{74} -6.37393 q^{76} +17.5920 q^{77} +6.61099 q^{79} -2.66933 q^{80} -0.424692 q^{82} +5.15732 q^{83} -10.0026 q^{85} -5.38311 q^{86} -5.51998 q^{88} +14.9168 q^{89} -8.44448 q^{91} +9.38528 q^{94} +17.0141 q^{95} -12.8109 q^{97} +3.15674 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 12 q^{13} + 8 q^{16} + 8 q^{25} + 12 q^{26} - 12 q^{29} - 12 q^{31} + 8 q^{32} + 36 q^{35} + 24 q^{41} + 48 q^{47} - 16 q^{49} + 8 q^{50} + 12 q^{52} + 12 q^{55} - 12 q^{58}+ \cdots - 16 q^{98}+O(q^{100}) \) 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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