LMFDB - Embedded newform 9801.2.a.cq.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$78.2613790211$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 891)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.629509 q^{2} -1.60372 q^{4} -1.36679 q^{5} +0.587994 q^{7} -2.26857 q^{8} -0.860404 q^{10} -3.79991 q^{13} +0.370148 q^{14} +1.77935 q^{16} +0.193493 q^{17} +1.94578 q^{19} +2.19194 q^{20} +0.321232 q^{23} -3.13190 q^{25} -2.39208 q^{26} -0.942977 q^{28} +4.17141 q^{29} +5.63559 q^{31} +5.65726 q^{32} +0.121806 q^{34} -0.803662 q^{35} +10.2849 q^{37} +1.22489 q^{38} +3.10065 q^{40} -6.86081 q^{41} -6.81088 q^{43} +0.202218 q^{46} +5.08784 q^{47} -6.65426 q^{49} -1.97156 q^{50} +6.09398 q^{52} +7.86594 q^{53} -1.33391 q^{56} +2.62594 q^{58} -7.20462 q^{59} +8.63734 q^{61} +3.54766 q^{62} +0.00260672 q^{64} +5.19366 q^{65} -9.09740 q^{67} -0.310308 q^{68} -0.505913 q^{70} -2.86490 q^{71} -0.424593 q^{73} +6.47446 q^{74} -3.12049 q^{76} +4.10547 q^{79} -2.43199 q^{80} -4.31894 q^{82} +7.44009 q^{83} -0.264464 q^{85} -4.28751 q^{86} +3.38915 q^{89} -2.23433 q^{91} -0.515165 q^{92} +3.20284 q^{94} -2.65947 q^{95} +6.46421 q^{97} -4.18892 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 26 q^{4} - 18 q^{7} - 16 q^{10} - 18 q^{13} + 18 q^{16} - 36 q^{19} + 20 q^{25} - 46 q^{28} - 12 q^{31} + 42 q^{34} + 14 q^{37} - 42 q^{40} - 42 q^{43} - 26 q^{46} + 38 q^{49} - 36 q^{52} - 40 q^{58}+ \cdots - 18 q^{97}+O(q^{100}) $ 24 * q + 26 * q^4 - 18 * q^7 - 16 * q^10 - 18 * q^13 + 18 * q^16 - 36 * q^19 + 20 * q^25 - 46 * q^28 - 12 * q^31 + 42 * q^34 + 14 * q^37 - 42 * q^40 - 42 * q^43 - 26 * q^46 + 38 * q^49 - 36 * q^52 - 40 * q^58 - 62 * q^61 - 16 * q^64 + 38 * q^67 + 6 * q^70 - 2 * q^73 - 102 * q^76 - 94 * q^79 - 24 * q^82 - 42 * q^85 - 34 * q^91 - 66 * q^94 - 18 * q^97

Coefficient data

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

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

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

$2$	0.629509	0.445130	0.222565	−	0.974918i	$-0.428557\pi$
$2$	0.629509	0.445130	0.222565	+	0.974918i	$0.428557\pi$
$3$	0	0
$3$	0	0
$4$	−1.60372	−0.801859
$5$	−1.36679	−0.611245	−0.305622	−	0.952153i	$-0.598865\pi$
$5$	−1.36679	−0.611245	−0.305622	+	0.952153i	$0.598865\pi$
$6$	0	0
$7$	0.587994	0.222241	0.111120	−	0.993807i	$-0.464556\pi$
$7$	0.587994	0.222241	0.111120	+	0.993807i	$0.464556\pi$
$8$	−2.26857	−0.802062
$9$	0	0
$10$	−0.860404	−0.272084
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.79991	−1.05391	−0.526953	−	0.849895i	$-0.676666\pi$
$13$	−3.79991	−1.05391	−0.526953	+	0.849895i	$0.676666\pi$
$14$	0.370148	0.0989262
$15$	0	0
$16$	1.77935	0.444837
$17$	0.193493	0.0469290	0.0234645	−	0.999725i	$-0.492530\pi$
$17$	0.193493	0.0469290	0.0234645	+	0.999725i	$0.492530\pi$
$18$	0	0
$19$	1.94578	0.446394	0.223197	−	0.974773i	$-0.428351\pi$
$19$	1.94578	0.446394	0.223197	+	0.974773i	$0.428351\pi$
$20$	2.19194	0.490132
$21$	0	0
$22$	0	0
$23$	0.321232	0.0669815	0.0334907	−	0.999439i	$-0.489338\pi$
$23$	0.321232	0.0669815	0.0334907	+	0.999439i	$0.489338\pi$
$24$	0	0
$25$	−3.13190	−0.626380
$26$	−2.39208	−0.469125
$27$	0	0
$28$	−0.942977	−0.178206
$29$	4.17141	0.774611	0.387306	−	0.921951i	$-0.373406\pi$
$29$	4.17141	0.774611	0.387306	+	0.921951i	$0.373406\pi$
$30$	0	0
$31$	5.63559	1.01218	0.506091	−	0.862480i	$-0.331090\pi$
$31$	5.63559	1.01218	0.506091	+	0.862480i	$0.331090\pi$
$32$	5.65726	1.00007
$33$	0	0
$34$	0.121806	0.0208895
$35$	−0.803662	−0.135844
$36$	0	0
$37$	10.2849	1.69083	0.845417	−	0.534107i	$-0.179352\pi$
$37$	10.2849	1.69083	0.845417	+	0.534107i	$0.179352\pi$
$38$	1.22489	0.198703
$39$	0	0
$40$	3.10065	0.490256
$41$	−6.86081	−1.07148	−0.535739	−	0.844384i	$-0.679967\pi$
$41$	−6.86081	−1.07148	−0.535739	+	0.844384i	$0.679967\pi$
$42$	0	0
$43$	−6.81088	−1.03865	−0.519325	−	0.854577i	$-0.673817\pi$
$43$	−6.81088	−1.03865	−0.519325	+	0.854577i	$0.673817\pi$
$44$	0	0
$45$	0	0
$46$	0.202218	0.0298155
$47$	5.08784	0.742138	0.371069	−	0.928605i	$-0.378991\pi$
$47$	5.08784	0.742138	0.371069	+	0.928605i	$0.378991\pi$
$48$	0	0
$49$	−6.65426	−0.950609
$50$	−1.97156	−0.278821
$51$	0	0
$52$	6.09398	0.845084
$53$	7.86594	1.08047	0.540235	−	0.841514i	$-0.318335\pi$
$53$	7.86594	1.08047	0.540235	+	0.841514i	$0.318335\pi$
$54$	0	0
$55$	0	0
$56$	−1.33391	−0.178251
$57$	0	0
$58$	2.62594	0.344803
$59$	−7.20462	−0.937962	−0.468981	−	0.883208i	$-0.655379\pi$
$59$	−7.20462	−0.937962	−0.468981	+	0.883208i	$0.655379\pi$
$60$	0	0
$61$	8.63734	1.10590	0.552949	−	0.833215i	$-0.313502\pi$
$61$	8.63734	1.10590	0.552949	+	0.833215i	$0.313502\pi$
$62$	3.54766	0.450553
$63$	0	0
$64$	0.00260672	0.000325841 0
$65$	5.19366	0.644194
$66$	0	0
$67$	−9.09740	−1.11142	−0.555712	−	0.831375i	$-0.687554\pi$
$67$	−9.09740	−1.11142	−0.555712	+	0.831375i	$0.687554\pi$
$68$	−0.310308	−0.0376304
$69$	0	0
$70$	−0.505913	−0.0604681
$71$	−2.86490	−0.340001	−0.170000	−	0.985444i	$-0.554377\pi$
$71$	−2.86490	−0.340001	−0.170000	+	0.985444i	$0.554377\pi$
$72$	0	0
$73$	−0.424593	−0.0496948	−0.0248474	−	0.999691i	$-0.507910\pi$
$73$	−0.424593	−0.0496948	−0.0248474	+	0.999691i	$0.507910\pi$
$74$	6.47446	0.752641
$75$	0	0
$76$	−3.12049	−0.357945
$77$	0	0
$78$	0	0
$79$	4.10547	0.461901	0.230950	−	0.972966i	$-0.425816\pi$
$79$	4.10547	0.461901	0.230950	+	0.972966i	$0.425816\pi$
$80$	−2.43199	−0.271904
$81$	0	0
$82$	−4.31894	−0.476947
$83$	7.44009	0.816656	0.408328	−	0.912835i	$-0.366112\pi$
$83$	7.44009	0.816656	0.408328	+	0.912835i	$0.366112\pi$
$84$	0	0
$85$	−0.264464	−0.0286851
$86$	−4.28751	−0.462335
$87$	0	0
$88$	0	0
$89$	3.38915	0.359249	0.179625	−	0.983735i	$-0.442512\pi$
$89$	3.38915	0.359249	0.179625	+	0.983735i	$0.442512\pi$
$90$	0	0
$91$	−2.23433	−0.234221
$92$	−0.515165	−0.0537097
$93$	0	0
$94$	3.20284	0.330348
$95$	−2.65947	−0.272856
$96$	0	0
$97$	6.46421	0.656341	0.328171	−	0.944618i	$-0.393568\pi$
$97$	6.46421	0.656341	0.328171	+	0.944618i	$0.393568\pi$
$98$	−4.18892	−0.423145
$99$	0	0
$100$	5.02268	0.502268
$101$	18.4974	1.84056	0.920280	−	0.391260i	$-0.127961\pi$
$101$	18.4974	1.84056	0.920280	+	0.391260i	$0.127961\pi$
$102$	0	0
$103$	3.23669	0.318920	0.159460	−	0.987204i	$-0.449025\pi$
$103$	3.23669	0.318920	0.159460	+	0.987204i	$0.449025\pi$
$104$	8.62038	0.845298
$105$	0	0
$106$	4.95168	0.480950
$107$	−16.8700	−1.63088	−0.815441	−	0.578840i	$-0.803506\pi$
$107$	−16.8700	−1.63088	−0.815441	+	0.578840i	$0.803506\pi$
$108$	0	0
$109$	−8.90745	−0.853179	−0.426589	−	0.904445i	$-0.640285\pi$
$109$	−8.90745	−0.853179	−0.426589	+	0.904445i	$0.640285\pi$
$110$	0	0
$111$	0	0
$112$	1.04625	0.0988609
$113$	2.88544	0.271440	0.135720	−	0.990747i	$-0.456665\pi$
$113$	2.88544	0.271440	0.135720	+	0.990747i	$0.456665\pi$
$114$	0	0
$115$	−0.439055	−0.0409421
$116$	−6.68977	−0.621129
$117$	0	0
$118$	−4.53538	−0.417515
$119$	0.113773	0.0104295
$120$	0	0
$121$	0	0
$122$	5.43729	0.492269
$123$	0	0
$124$	−9.03790	−0.811627
$125$	11.1146	0.994116
$126$	0	0
$127$	1.28445	0.113976	0.0569882	−	0.998375i	$-0.481850\pi$
$127$	1.28445	0.113976	0.0569882	+	0.998375i	$0.481850\pi$
$128$	−11.3129	−0.999927
$129$	0	0
$130$	3.26946	0.286750
$131$	6.33798	0.553752	0.276876	−	0.960906i	$-0.410701\pi$
$131$	6.33798	0.553752	0.276876	+	0.960906i	$0.410701\pi$
$132$	0	0
$133$	1.14411	0.0992069
$134$	−5.72690	−0.494729
$135$	0	0
$136$	−0.438953	−0.0376400
$137$	−16.5640	−1.41516	−0.707579	−	0.706634i	$-0.750213\pi$
$137$	−16.5640	−1.41516	−0.707579	+	0.706634i	$0.750213\pi$
$138$	0	0
$139$	−17.4492	−1.48002	−0.740010	−	0.672596i	$-0.765179\pi$
$139$	−17.4492	−1.48002	−0.740010	+	0.672596i	$0.765179\pi$
$140$	1.28885	0.108927
$141$	0	0
$142$	−1.80348	−0.151345
$143$	0	0
$144$	0	0
$145$	−5.70142	−0.473477
$146$	−0.267285	−0.0221207
$147$	0	0
$148$	−16.4941	−1.35581
$149$	6.30584	0.516595	0.258297	−	0.966065i	$-0.416839\pi$
$149$	6.30584	0.516595	0.258297	+	0.966065i	$0.416839\pi$
$150$	0	0
$151$	−16.2316	−1.32091	−0.660453	−	0.750867i	$-0.729636\pi$
$151$	−16.2316	−1.32091	−0.660453	+	0.750867i	$0.729636\pi$
$152$	−4.41416	−0.358035
$153$	0	0
$154$	0	0
$155$	−7.70264	−0.618691
$156$	0	0
$157$	−17.0442	−1.36028	−0.680138	−	0.733084i	$-0.738080\pi$
$157$	−17.0442	−1.36028	−0.680138	+	0.733084i	$0.738080\pi$
$158$	2.58443	0.205606
$159$	0	0
$160$	−7.73226	−0.611289
$161$	0.188882	0.0148860
$162$	0	0
$163$	18.5735	1.45479	0.727396	−	0.686218i	$-0.240730\pi$
$163$	18.5735	1.45479	0.727396	+	0.686218i	$0.240730\pi$
$164$	11.0028	0.859174
$165$	0	0
$166$	4.68360	0.363518
$167$	9.46682	0.732564	0.366282	−	0.930504i	$-0.380630\pi$
$167$	9.46682	0.732564	0.366282	+	0.930504i	$0.380630\pi$
$168$	0	0
$169$	1.43932	0.110717
$170$	−0.166482	−0.0127686
$171$	0	0
$172$	10.9227	0.832851
$173$	17.9680	1.36608	0.683042	−	0.730379i	$-0.260657\pi$
$173$	17.9680	1.36608	0.683042	+	0.730379i	$0.260657\pi$
$174$	0	0
$175$	−1.84154	−0.139207
$176$	0	0
$177$	0	0
$178$	2.13350	0.159913
$179$	−26.4579	−1.97756	−0.988779	−	0.149387i	$-0.952270\pi$
$179$	−26.4579	−1.97756	−0.988779	+	0.149387i	$0.952270\pi$
$180$	0	0
$181$	−0.878966	−0.0653330	−0.0326665	−	0.999466i	$-0.510400\pi$
$181$	−0.878966	−0.0653330	−0.0326665	+	0.999466i	$0.510400\pi$
$182$	−1.40653	−0.104259
$183$	0	0
$184$	−0.728738	−0.0537233
$185$	−14.0573	−1.03351
$186$	0	0
$187$	0	0
$188$	−8.15946	−0.595090
$189$	0	0
$190$	−1.67416	−0.121456
$191$	−15.5215	−1.12309	−0.561547	−	0.827445i	$-0.689794\pi$
$191$	−15.5215	−1.12309	−0.561547	+	0.827445i	$0.689794\pi$
$192$	0	0
$193$	−9.67458	−0.696391	−0.348196	−	0.937422i	$-0.613206\pi$
$193$	−9.67458	−0.696391	−0.348196	+	0.937422i	$0.613206\pi$
$194$	4.06928	0.292157
$195$	0	0
$196$	10.6716	0.762254
$197$	−26.0364	−1.85502	−0.927508	−	0.373803i	$-0.878053\pi$
$197$	−26.0364	−1.85502	−0.927508	+	0.373803i	$0.878053\pi$
$198$	0	0
$199$	−20.4538	−1.44993	−0.724967	−	0.688784i	$-0.758145\pi$
$199$	−20.4538	−1.44993	−0.724967	+	0.688784i	$0.758145\pi$
$200$	7.10494	0.502395
$201$	0	0
$202$	11.6443	0.819289
$203$	2.45277	0.172150
$204$	0	0
$205$	9.37725	0.654936
$206$	2.03752	0.141961
$207$	0	0
$208$	−6.76136	−0.468816
$209$	0	0
$210$	0	0
$211$	15.9168	1.09576	0.547880	−	0.836557i	$-0.315435\pi$
$211$	15.9168	1.09576	0.547880	+	0.836557i	$0.315435\pi$
$212$	−12.6148	−0.866385
$213$	0	0
$214$	−10.6198	−0.725955
$215$	9.30901	0.634869
$216$	0	0
$217$	3.31370	0.224948
$218$	−5.60732	−0.379776
$219$	0	0
$220$	0	0
$221$	−0.735256	−0.0494587
$222$	0	0
$223$	2.50161	0.167520	0.0837602	−	0.996486i	$-0.473307\pi$
$223$	2.50161	0.167520	0.0837602	+	0.996486i	$0.473307\pi$
$224$	3.32644	0.222257
$225$	0	0
$226$	1.81641	0.120826
$227$	−16.1138	−1.06951	−0.534754	−	0.845008i	$-0.679596\pi$
$227$	−16.1138	−1.06951	−0.534754	+	0.845008i	$0.679596\pi$
$228$	0	0
$229$	3.30474	0.218383	0.109192	−	0.994021i	$-0.465174\pi$
$229$	3.30474	0.218383	0.109192	+	0.994021i	$0.465174\pi$
$230$	−0.276389	−0.0182246
$231$	0	0
$232$	−9.46315	−0.621287
$233$	−4.31187	−0.282480	−0.141240	−	0.989975i	$-0.545109\pi$
$233$	−4.31187	−0.282480	−0.141240	+	0.989975i	$0.545109\pi$
$234$	0	0
$235$	−6.95399	−0.453628
$236$	11.5542	0.752113
$237$	0	0
$238$	0.0716211	0.00464250
$239$	15.0401	0.972866	0.486433	−	0.873718i	$-0.338298\pi$
$239$	15.0401	0.972866	0.486433	+	0.873718i	$0.338298\pi$
$240$	0	0
$241$	−13.7709	−0.887064	−0.443532	−	0.896259i	$-0.646275\pi$
$241$	−13.7709	−0.887064	−0.443532	+	0.896259i	$0.646275\pi$
$242$	0	0
$243$	0	0
$244$	−13.8519	−0.886775
$245$	9.09495	0.581055
$246$	0	0
$247$	−7.39381	−0.470457
$248$	−12.7848	−0.811833
$249$	0	0
$250$	6.99672	0.442511
$251$	27.6416	1.74472	0.872361	−	0.488862i	$-0.162588\pi$
$251$	27.6416	1.74472	0.872361	+	0.488862i	$0.162588\pi$
$252$	0	0
$253$	0	0
$254$	0.808572	0.0507343
$255$	0	0
$256$	−7.12678	−0.445424
$257$	−10.9826	−0.685074	−0.342537	−	0.939504i	$-0.611286\pi$
$257$	−10.9826	−0.685074	−0.342537	+	0.939504i	$0.611286\pi$
$258$	0	0
$259$	6.04748	0.375772
$260$	−8.32917	−0.516553
$261$	0	0
$262$	3.98982	0.246492
$263$	17.9503	1.10687	0.553433	−	0.832894i	$-0.313318\pi$
$263$	17.9503	1.10687	0.553433	+	0.832894i	$0.313318\pi$
$264$	0	0
$265$	−10.7511	−0.660432
$266$	0.720228	0.0441600
$267$	0	0
$268$	14.5897	0.891206
$269$	−24.1477	−1.47231	−0.736156	−	0.676812i	$-0.763361\pi$
$269$	−24.1477	−1.47231	−0.736156	+	0.676812i	$0.763361\pi$
$270$	0	0
$271$	14.5492	0.883803	0.441902	−	0.897064i	$-0.354304\pi$
$271$	14.5492	0.883803	0.441902	+	0.897064i	$0.354304\pi$
$272$	0.344291	0.0208757
$273$	0	0
$274$	−10.4272	−0.629930
$275$	0	0
$276$	0	0
$277$	−24.1116	−1.44873	−0.724364	−	0.689418i	$-0.757866\pi$
$277$	−24.1116	−1.44873	−0.724364	+	0.689418i	$0.757866\pi$
$278$	−10.9844	−0.658802
$279$	0	0
$280$	1.82317	0.108955
$281$	1.18791	0.0708648	0.0354324	−	0.999372i	$-0.488719\pi$
$281$	1.18791	0.0708648	0.0354324	+	0.999372i	$0.488719\pi$
$282$	0	0
$283$	−15.3522	−0.912594	−0.456297	−	0.889828i	$-0.650825\pi$
$283$	−15.3522	−0.912594	−0.456297	+	0.889828i	$0.650825\pi$
$284$	4.59448	0.272632
$285$	0	0
$286$	0	0
$287$	−4.03412	−0.238126
$288$	0	0
$289$	−16.9626	−0.997798
$290$	−3.58910	−0.210759
$291$	0	0
$292$	0.680927	0.0398482
$293$	−32.0977	−1.87517	−0.937583	−	0.347761i	$-0.886942\pi$
$293$	−32.0977	−1.87517	−0.937583	+	0.347761i	$0.886942\pi$
$294$	0	0
$295$	9.84717	0.573325
$296$	−23.3321	−1.35615
$297$	0	0
$298$	3.96959	0.229952
$299$	−1.22065	−0.0705921
$300$	0	0
$301$	−4.00476	−0.230830
$302$	−10.2179	−0.587975
$303$	0	0
$304$	3.46223	0.198572
$305$	−11.8054	−0.675975
$306$	0	0
$307$	−33.1199	−1.89025	−0.945126	−	0.326707i	$-0.894061\pi$
$307$	−33.1199	−1.89025	−0.945126	+	0.326707i	$0.894061\pi$
$308$	0	0
$309$	0	0
$310$	−4.84889	−0.275398
$311$	33.9824	1.92697	0.963483	−	0.267769i	$-0.0862865\pi$
$311$	33.9824	1.92697	0.963483	+	0.267769i	$0.0862865\pi$
$312$	0	0
$313$	4.71328	0.266410	0.133205	−	0.991088i	$-0.457473\pi$
$313$	4.71328	0.266410	0.133205	+	0.991088i	$0.457473\pi$
$314$	−10.7295	−0.605500
$315$	0	0
$316$	−6.58401	−0.370379
$317$	−17.4171	−0.978242	−0.489121	−	0.872216i	$-0.662682\pi$
$317$	−17.4171	−0.978242	−0.489121	+	0.872216i	$0.662682\pi$
$318$	0	0
$319$	0	0
$320$	−0.00356283	−0.000199168 0
$321$	0	0
$322$	0.118903	0.00662622
$323$	0.376496	0.0209488
$324$	0	0
$325$	11.9009	0.660145
$326$	11.6922	0.647572
$327$	0	0
$328$	15.5643	0.859392
$329$	2.99162	0.164933
$330$	0	0
$331$	12.9769	0.713275	0.356637	−	0.934243i	$-0.383923\pi$
$331$	12.9769	0.713275	0.356637	+	0.934243i	$0.383923\pi$
$332$	−11.9318	−0.654843
$333$	0	0
$334$	5.95945	0.326087
$335$	12.4342	0.679353
$336$	0	0
$337$	10.3436	0.563452	0.281726	−	0.959495i	$-0.409093\pi$
$337$	10.3436	0.563452	0.281726	+	0.959495i	$0.409093\pi$
$338$	0.906064	0.0492834
$339$	0	0
$340$	0.424125	0.0230014
$341$	0	0
$342$	0	0
$343$	−8.02863	−0.433505
$344$	15.4510	0.833062
$345$	0	0
$346$	11.3110	0.608085
$347$	6.14370	0.329811	0.164905	−	0.986309i	$-0.447268\pi$
$347$	6.14370	0.329811	0.164905	+	0.986309i	$0.447268\pi$
$348$	0	0
$349$	−20.3782	−1.09082	−0.545409	−	0.838170i	$-0.683626\pi$
$349$	−20.3782	−1.09082	−0.545409	+	0.838170i	$0.683626\pi$
$350$	−1.15927	−0.0619653
$351$	0	0
$352$	0	0
$353$	11.9881	0.638064	0.319032	−	0.947744i	$-0.396642\pi$
$353$	11.9881	0.638064	0.319032	+	0.947744i	$0.396642\pi$
$354$	0	0
$355$	3.91570	0.207824
$356$	−5.43524	−0.288067
$357$	0	0
$358$	−16.6555	−0.880271
$359$	17.8820	0.943773	0.471887	−	0.881659i	$-0.343573\pi$
$359$	17.8820	0.943773	0.471887	+	0.881659i	$0.343573\pi$
$360$	0	0
$361$	−15.2139	−0.800733
$362$	−0.553317	−0.0290817
$363$	0	0
$364$	3.58323	0.187812
$365$	0.580327	0.0303757
$366$	0	0
$367$	27.6194	1.44172	0.720860	−	0.693081i	$-0.243747\pi$
$367$	27.6194	1.44172	0.720860	+	0.693081i	$0.243747\pi$
$368$	0.571583	0.0297958
$369$	0	0
$370$	−8.84920	−0.460048
$371$	4.62513	0.240125
$372$	0	0
$373$	−12.3887	−0.641463	−0.320731	−	0.947170i	$-0.603929\pi$
$373$	−12.3887	−0.641463	−0.320731	+	0.947170i	$0.603929\pi$
$374$	0	0
$375$	0	0
$376$	−11.5421	−0.595241
$377$	−15.8510	−0.816367
$378$	0	0
$379$	16.8316	0.864579	0.432290	−	0.901735i	$-0.357706\pi$
$379$	16.8316	0.864579	0.432290	+	0.901735i	$0.357706\pi$
$380$	4.26504	0.218792
$381$	0	0
$382$	−9.77091	−0.499923
$383$	12.4490	0.636114	0.318057	−	0.948072i	$-0.396970\pi$
$383$	12.4490	0.636114	0.318057	+	0.948072i	$0.396970\pi$
$384$	0	0
$385$	0	0
$386$	−6.09024	−0.309985
$387$	0	0
$388$	−10.3668	−0.526293
$389$	−1.22879	−0.0623020	−0.0311510	−	0.999515i	$-0.509917\pi$
$389$	−1.22879	−0.0623020	−0.0311510	+	0.999515i	$0.509917\pi$
$390$	0	0
$391$	0.0621561	0.00314337
$392$	15.0957	0.762447
$393$	0	0
$394$	−16.3902	−0.825724
$395$	−5.61129	−0.282335
$396$	0	0
$397$	20.5917	1.03347	0.516734	−	0.856146i	$-0.327148\pi$
$397$	20.5917	1.03347	0.516734	+	0.856146i	$0.327148\pi$
$398$	−12.8759	−0.645409
$399$	0	0
$400$	−5.57273	−0.278637
$401$	−6.73871	−0.336515	−0.168258	−	0.985743i	$-0.553814\pi$
$401$	−6.73871	−0.336515	−0.168258	+	0.985743i	$0.553814\pi$
$402$	0	0
$403$	−21.4147	−1.06674
$404$	−29.6646	−1.47587
$405$	0	0
$406$	1.54404	0.0766294
$407$	0	0
$408$	0	0
$409$	−29.1705	−1.44239	−0.721194	−	0.692733i	$-0.756407\pi$
$409$	−29.1705	−1.44239	−0.721194	+	0.692733i	$0.756407\pi$
$410$	5.90307	0.291532
$411$	0	0
$412$	−5.19073	−0.255729
$413$	−4.23628	−0.208454
$414$	0	0
$415$	−10.1690	−0.499177
$416$	−21.4971	−1.05398
$417$	0	0
$418$	0	0
$419$	−37.9702	−1.85497	−0.927483	−	0.373865i	$-0.878032\pi$
$419$	−37.9702	−1.85497	−0.927483	+	0.373865i	$0.878032\pi$
$420$	0	0
$421$	−8.10826	−0.395173	−0.197586	−	0.980285i	$-0.563310\pi$
$421$	−8.10826	−0.395173	−0.197586	+	0.980285i	$0.563310\pi$
$422$	10.0198	0.487756
$423$	0	0
$424$	−17.8445	−0.866604
$425$	−0.606001	−0.0293954
$426$	0	0
$427$	5.07871	0.245776
$428$	27.0547	1.30774
$429$	0	0
$430$	5.86011	0.282600
$431$	−30.4491	−1.46668	−0.733342	−	0.679860i	$-0.762041\pi$
$431$	−30.4491	−1.46668	−0.733342	+	0.679860i	$0.762041\pi$
$432$	0	0
$433$	24.0856	1.15748	0.578739	−	0.815513i	$-0.303545\pi$
$433$	24.0856	1.15748	0.578739	+	0.815513i	$0.303545\pi$
$434$	2.08600	0.100131
$435$	0	0
$436$	14.2850	0.684129
$437$	0.625048	0.0299001
$438$	0	0
$439$	−34.3175	−1.63789	−0.818943	−	0.573874i	$-0.805440\pi$
$439$	−34.3175	−1.63789	−0.818943	+	0.573874i	$0.805440\pi$
$440$	0	0
$441$	0	0
$442$	−0.462851	−0.0220156
$443$	8.54020	0.405757	0.202879	−	0.979204i	$-0.434970\pi$
$443$	8.54020	0.405757	0.202879	+	0.979204i	$0.434970\pi$
$444$	0	0
$445$	−4.63224	−0.219589
$446$	1.57479	0.0745684
$447$	0	0
$448$	0.00153274	7.24151e−5 0
$449$	36.8685	1.73993	0.869965	−	0.493113i	$-0.164141\pi$
$449$	36.8685	1.73993	0.869965	+	0.493113i	$0.164141\pi$
$450$	0	0
$451$	0	0
$452$	−4.62744	−0.217656
$453$	0	0
$454$	−10.1438	−0.476071
$455$	3.05384	0.143166
$456$	0	0
$457$	−23.6841	−1.10789	−0.553947	−	0.832552i	$-0.686879\pi$
$457$	−23.6841	−1.10789	−0.553947	+	0.832552i	$0.686879\pi$
$458$	2.08036	0.0972090
$459$	0	0
$460$	0.704120	0.0328298
$461$	28.6862	1.33605	0.668024	−	0.744140i	$-0.267140\pi$
$461$	28.6862	1.33605	0.668024	+	0.744140i	$0.267140\pi$
$462$	0	0
$463$	10.8631	0.504852	0.252426	−	0.967616i	$-0.418772\pi$
$463$	10.8631	0.504852	0.252426	+	0.967616i	$0.418772\pi$
$464$	7.42239	0.344576
$465$	0	0
$466$	−2.71436	−0.125740
$467$	−29.4876	−1.36452	−0.682261	−	0.731108i	$-0.739003\pi$
$467$	−29.4876	−1.36452	−0.682261	+	0.731108i	$0.739003\pi$
$468$	0	0
$469$	−5.34922	−0.247004
$470$	−4.37760	−0.201924
$471$	0	0
$472$	16.3442	0.752304
$473$	0	0
$474$	0	0
$475$	−6.09400	−0.279612
$476$	−0.182460	−0.00836302
$477$	0	0
$478$	9.46791	0.433052
$479$	−21.3570	−0.975828	−0.487914	−	0.872892i	$-0.662242\pi$
$479$	−21.3570	−0.975828	−0.487914	+	0.872892i	$0.662242\pi$
$480$	0	0
$481$	−39.0818	−1.78198
$482$	−8.66894	−0.394859
$483$	0	0
$484$	0	0
$485$	−8.83519	−0.401185
$486$	0	0
$487$	−2.27641	−0.103154	−0.0515769	−	0.998669i	$-0.516425\pi$
$487$	−2.27641	−0.103154	−0.0515769	+	0.998669i	$0.516425\pi$
$488$	−19.5945	−0.886999
$489$	0	0
$490$	5.72535	0.258645
$491$	9.52567	0.429887	0.214944	−	0.976626i	$-0.431043\pi$
$491$	9.52567	0.429887	0.214944	+	0.976626i	$0.431043\pi$
$492$	0	0
$493$	0.807139	0.0363517
$494$	−4.65447	−0.209415
$495$	0	0
$496$	10.0277	0.450256
$497$	−1.68454	−0.0755620
$498$	0	0
$499$	−12.7232	−0.569571	−0.284785	−	0.958591i	$-0.591922\pi$
$499$	−12.7232	−0.569571	−0.284785	+	0.958591i	$0.591922\pi$
$500$	−17.8246	−0.797141
$501$	0	0
$502$	17.4007	0.776629
$503$	5.78240	0.257825	0.128912	−	0.991656i	$-0.458851\pi$
$503$	5.78240	0.257825	0.128912	+	0.991656i	$0.458851\pi$
$504$	0	0
$505$	−25.2820	−1.12503
$506$	0	0
$507$	0	0
$508$	−2.05989	−0.0913930
$509$	−9.24772	−0.409898	−0.204949	−	0.978773i	$-0.565703\pi$
$509$	−9.24772	−0.409898	−0.204949	+	0.978773i	$0.565703\pi$
$510$	0	0
$511$	−0.249658	−0.0110442
$512$	18.1394	0.801656
$513$	0	0
$514$	−6.91363	−0.304947
$515$	−4.42385	−0.194938
$516$	0	0
$517$	0	0
$518$	3.80695	0.167268
$519$	0	0
$520$	−11.7822	−0.516684
$521$	25.3513	1.11066	0.555331	−	0.831629i	$-0.312591\pi$
$521$	25.3513	1.11066	0.555331	+	0.831629i	$0.312591\pi$
$522$	0	0
$523$	−20.9860	−0.917655	−0.458827	−	0.888525i	$-0.651730\pi$
$523$	−20.9860	−0.917655	−0.458827	+	0.888525i	$0.651730\pi$
$524$	−10.1643	−0.444031
$525$	0	0
$526$	11.2999	0.492699
$527$	1.09045	0.0475007
$528$	0	0
$529$	−22.8968	−0.995513
$530$	−6.76789	−0.293978
$531$	0	0
$532$	−1.83483	−0.0795500
$533$	26.0705	1.12924
$534$	0	0
$535$	23.0576	0.996869
$536$	20.6381	0.891432
$537$	0	0
$538$	−15.2012	−0.655371
$539$	0	0
$540$	0	0
$541$	11.1538	0.479540	0.239770	−	0.970830i	$-0.422928\pi$
$541$	11.1538	0.479540	0.239770	+	0.970830i	$0.422928\pi$
$542$	9.15888	0.393408
$543$	0	0
$544$	1.09464	0.0469324
$545$	12.1746	0.521501
$546$	0	0
$547$	−6.09638	−0.260662	−0.130331	−	0.991471i	$-0.541604\pi$
$547$	−6.09638	−0.260662	−0.130331	+	0.991471i	$0.541604\pi$
$548$	26.5640	1.13476
$549$	0	0
$550$	0	0
$551$	8.11667	0.345782
$552$	0	0
$553$	2.41399	0.102653
$554$	−15.1785	−0.644872
$555$	0	0
$556$	27.9836	1.18677
$557$	9.90178	0.419552	0.209776	−	0.977749i	$-0.432727\pi$
$557$	9.90178	0.419552	0.209776	+	0.977749i	$0.432727\pi$
$558$	0	0
$559$	25.8807	1.09464
$560$	−1.42999	−0.0604282
$561$	0	0
$562$	0.747801	0.0315441
$563$	−0.551557	−0.0232453	−0.0116227	−	0.999932i	$-0.503700\pi$
$563$	−0.551557	−0.0232453	−0.0116227	+	0.999932i	$0.503700\pi$
$564$	0	0
$565$	−3.94378	−0.165916
$566$	−9.66436	−0.406223
$567$	0	0
$568$	6.49923	0.272702
$569$	−19.2487	−0.806946	−0.403473	−	0.914991i	$-0.632197\pi$
$569$	−19.2487	−0.806946	−0.403473	+	0.914991i	$0.632197\pi$
$570$	0	0
$571$	−27.4122	−1.14716	−0.573582	−	0.819148i	$-0.694447\pi$
$571$	−27.4122	−1.14716	−0.573582	+	0.819148i	$0.694447\pi$
$572$	0	0
$573$	0	0
$574$	−2.53951	−0.105997
$575$	−1.00607	−0.0419558
$576$	0	0
$577$	−30.0033	−1.24905	−0.624526	−	0.781004i	$-0.714708\pi$
$577$	−30.0033	−1.24905	−0.624526	+	0.781004i	$0.714708\pi$
$578$	−10.6781	−0.444150
$579$	0	0
$580$	9.14347	0.379662
$581$	4.37473	0.181494
$582$	0	0
$583$	0	0
$584$	0.963220	0.0398583
$585$	0	0
$586$	−20.2058	−0.834694
$587$	−5.84198	−0.241124	−0.120562	−	0.992706i	$-0.538470\pi$
$587$	−5.84198	−0.241124	−0.120562	+	0.992706i	$0.538470\pi$
$588$	0	0
$589$	10.9656	0.451832
$590$	6.19889	0.255204
$591$	0	0
$592$	18.3005	0.752145
$593$	−35.2941	−1.44936	−0.724678	−	0.689088i	$-0.758011\pi$
$593$	−35.2941	−1.44936	−0.724678	+	0.689088i	$0.758011\pi$
$594$	0	0
$595$	−0.155503	−0.00637500
$596$	−10.1128	−0.414236
$597$	0	0
$598$	−0.768412	−0.0314227
$599$	22.8350	0.933012	0.466506	−	0.884518i	$-0.345513\pi$
$599$	22.8350	0.933012	0.466506	+	0.884518i	$0.345513\pi$
$600$	0	0
$601$	−11.4829	−0.468399	−0.234200	−	0.972189i	$-0.575247\pi$
$601$	−11.4829	−0.468399	−0.234200	+	0.972189i	$0.575247\pi$
$602$	−2.52103	−0.102750
$603$	0	0
$604$	26.0309	1.05918
$605$	0	0
$606$	0	0
$607$	−12.9207	−0.524434	−0.262217	−	0.965009i	$-0.584454\pi$
$607$	−12.9207	−0.524434	−0.262217	+	0.965009i	$0.584454\pi$
$608$	11.0078	0.446426
$609$	0	0
$610$	−7.43161	−0.300897
$611$	−19.3333	−0.782143
$612$	0	0
$613$	−24.0982	−0.973317	−0.486659	−	0.873592i	$-0.661784\pi$
$613$	−24.0982	−0.973317	−0.486659	+	0.873592i	$0.661784\pi$
$614$	−20.8493	−0.841408
$615$	0	0
$616$	0	0
$617$	22.7911	0.917534	0.458767	−	0.888556i	$-0.348291\pi$
$617$	22.7911	0.917534	0.458767	+	0.888556i	$0.348291\pi$
$618$	0	0
$619$	−20.3174	−0.816624	−0.408312	−	0.912842i	$-0.633883\pi$
$619$	−20.3174	−0.816624	−0.408312	+	0.912842i	$0.633883\pi$
$620$	12.3529	0.496103
$621$	0	0
$622$	21.3923	0.857751
$623$	1.99280	0.0798399
$624$	0	0
$625$	0.468279	0.0187312
$626$	2.96705	0.118587
$627$	0	0
$628$	27.3341	1.09075
$629$	1.99006	0.0793491
$630$	0	0
$631$	15.3567	0.611339	0.305670	−	0.952138i	$-0.401120\pi$
$631$	15.3567	0.611339	0.305670	+	0.952138i	$0.401120\pi$
$632$	−9.31355	−0.370473
$633$	0	0
$634$	−10.9642	−0.435445
$635$	−1.75556	−0.0696675
$636$	0	0
$637$	25.2856	1.00185
$638$	0	0
$639$	0	0
$640$	15.4623	0.611201
$641$	13.5329	0.534517	0.267259	−	0.963625i	$-0.413882\pi$
$641$	13.5329	0.534517	0.267259	+	0.963625i	$0.413882\pi$
$642$	0	0
$643$	12.6487	0.498816	0.249408	−	0.968399i	$-0.419764\pi$
$643$	12.6487	0.498816	0.249408	+	0.968399i	$0.419764\pi$
$644$	−0.302914	−0.0119365
$645$	0	0
$646$	0.237008	0.00932495
$647$	−1.62146	−0.0637461	−0.0318730	−	0.999492i	$-0.510147\pi$
$647$	−1.62146	−0.0637461	−0.0318730	+	0.999492i	$0.510147\pi$
$648$	0	0
$649$	0	0
$650$	7.49175	0.293851
$651$	0	0
$652$	−29.7867	−1.16654
$653$	33.0538	1.29350	0.646748	−	0.762704i	$-0.276129\pi$
$653$	33.0538	1.29350	0.646748	+	0.762704i	$0.276129\pi$
$654$	0	0
$655$	−8.66266	−0.338478
$656$	−12.2078	−0.476633
$657$	0	0
$658$	1.88325	0.0734169
$659$	47.9012	1.86597	0.932983	−	0.359922i	$-0.117197\pi$
$659$	47.9012	1.86597	0.932983	+	0.359922i	$0.117197\pi$
$660$	0	0
$661$	−18.5052	−0.719768	−0.359884	−	0.932997i	$-0.617184\pi$
$661$	−18.5052	−0.719768	−0.359884	+	0.932997i	$0.617184\pi$
$662$	8.16908	0.317500
$663$	0	0
$664$	−16.8784	−0.655009
$665$	−1.56375	−0.0606397
$666$	0	0
$667$	1.33999	0.0518846
$668$	−15.1821	−0.587413
$669$	0	0
$670$	7.82744	0.302401
$671$	0	0
$672$	0	0
$673$	−39.1351	−1.50855	−0.754273	−	0.656560i	$-0.772011\pi$
$673$	−39.1351	−1.50855	−0.754273	+	0.656560i	$0.772011\pi$
$674$	6.51139	0.250809
$675$	0	0
$676$	−2.30826	−0.0887792
$677$	12.4835	0.479780	0.239890	−	0.970800i	$-0.422889\pi$
$677$	12.4835	0.479780	0.239890	+	0.970800i	$0.422889\pi$
$678$	0	0
$679$	3.80092	0.145866
$680$	0.599955	0.0230072
$681$	0	0
$682$	0	0
$683$	17.3888	0.665363	0.332681	−	0.943039i	$-0.392047\pi$
$683$	17.3888	0.665363	0.332681	+	0.943039i	$0.392047\pi$
$684$	0	0
$685$	22.6394	0.865008
$686$	−5.05410	−0.192966
$687$	0	0
$688$	−12.1189	−0.462030
$689$	−29.8899	−1.13871
$690$	0	0
$691$	3.65485	0.139037	0.0695185	−	0.997581i	$-0.477854\pi$
$691$	3.65485	0.139037	0.0695185	+	0.997581i	$0.477854\pi$
$692$	−28.8156	−1.09541
$693$	0	0
$694$	3.86751	0.146809
$695$	23.8493	0.904655
$696$	0	0
$697$	−1.32752	−0.0502834
$698$	−12.8282	−0.485556
$699$	0	0
$700$	2.95331	0.111625
$701$	−5.58621	−0.210988	−0.105494	−	0.994420i	$-0.533642\pi$
$701$	−5.58621	−0.210988	−0.105494	+	0.994420i	$0.533642\pi$
$702$	0	0
$703$	20.0123	0.754777
$704$	0	0
$705$	0	0
$706$	7.54665	0.284022
$707$	10.8764	0.409048
$708$	0	0
$709$	13.8571	0.520413	0.260207	−	0.965553i	$-0.416209\pi$
$709$	13.8571	0.520413	0.260207	+	0.965553i	$0.416209\pi$
$710$	2.46497	0.0925086
$711$	0	0
$712$	−7.68854	−0.288140
$713$	1.81033	0.0677974
$714$	0	0
$715$	0	0
$716$	42.4310	1.58572
$717$	0	0
$718$	11.2569	0.420102
$719$	26.6724	0.994712	0.497356	−	0.867547i	$-0.334304\pi$
$719$	26.6724	0.994712	0.497356	+	0.867547i	$0.334304\pi$
$720$	0	0
$721$	1.90315	0.0708771
$722$	−9.57731	−0.356430
$723$	0	0
$724$	1.40961	0.0523879
$725$	−13.0644	−0.485201
$726$	0	0
$727$	−22.6397	−0.839659	−0.419830	−	0.907603i	$-0.637910\pi$
$727$	−22.6397	−0.839659	−0.419830	+	0.907603i	$0.637910\pi$
$728$	5.06873	0.187860
$729$	0	0
$730$	0.365321	0.0135211
$731$	−1.31786	−0.0487428
$732$	0	0
$733$	−22.0364	−0.813934	−0.406967	−	0.913443i	$-0.633414\pi$
$733$	−22.0364	−0.813934	−0.406967	+	0.913443i	$0.633414\pi$
$734$	17.3867	0.641753
$735$	0	0
$736$	1.81729	0.0669863
$737$	0	0
$738$	0	0
$739$	−48.6055	−1.78798	−0.893990	−	0.448086i	$-0.852106\pi$
$739$	−48.6055	−1.78798	−0.893990	+	0.448086i	$0.852106\pi$
$740$	22.5439	0.828732
$741$	0	0
$742$	2.91156	0.106887
$743$	−5.50420	−0.201929	−0.100965	−	0.994890i	$-0.532193\pi$
$743$	−5.50420	−0.201929	−0.100965	+	0.994890i	$0.532193\pi$
$744$	0	0
$745$	−8.61873	−0.315766
$746$	−7.79881	−0.285535
$747$	0	0
$748$	0	0
$749$	−9.91945	−0.362449
$750$	0	0
$751$	−3.72066	−0.135769	−0.0678843	−	0.997693i	$-0.521625\pi$
$751$	−3.72066	−0.135769	−0.0678843	+	0.997693i	$0.521625\pi$
$752$	9.05304	0.330130
$753$	0	0
$754$	−9.97834	−0.363390
$755$	22.1851	0.807397
$756$	0	0
$757$	−30.4331	−1.10611	−0.553055	−	0.833145i	$-0.686538\pi$
$757$	−30.4331	−1.10611	−0.553055	+	0.833145i	$0.686538\pi$
$758$	10.5956	0.384851
$759$	0	0
$760$	6.03321	0.218847
$761$	22.4600	0.814175	0.407087	−	0.913389i	$-0.366544\pi$
$761$	22.4600	0.814175	0.407087	+	0.913389i	$0.366544\pi$
$762$	0	0
$763$	−5.23753	−0.189611
$764$	24.8921	0.900563
$765$	0	0
$766$	7.83676	0.283154
$767$	27.3769	0.988523
$768$	0	0
$769$	−38.9595	−1.40492	−0.702458	−	0.711725i	$-0.747914\pi$
$769$	−38.9595	−1.40492	−0.702458	+	0.711725i	$0.747914\pi$
$770$	0	0
$771$	0	0
$772$	15.5153	0.558408
$773$	−53.3792	−1.91992	−0.959959	−	0.280141i	$-0.909619\pi$
$773$	−53.3792	−1.91992	−0.959959	+	0.280141i	$0.909619\pi$
$774$	0	0
$775$	−17.6501	−0.634010
$776$	−14.6645	−0.526426
$777$	0	0
$778$	−0.773534	−0.0277325
$779$	−13.3497	−0.478301
$780$	0	0
$781$	0	0
$782$	0.0391279	0.00139921
$783$	0	0
$784$	−11.8402	−0.422866
$785$	23.2958	0.831462
$786$	0	0
$787$	−28.8324	−1.02776	−0.513882	−	0.857861i	$-0.671793\pi$
$787$	−28.8324	−1.02776	−0.513882	+	0.857861i	$0.671793\pi$
$788$	41.7550	1.48746
$789$	0	0
$790$	−3.53236	−0.125676
$791$	1.69662	0.0603250
$792$	0	0
$793$	−32.8211	−1.16551
$794$	12.9627	0.460028
$795$	0	0
$796$	32.8022	1.16264
$797$	33.1727	1.17504	0.587519	−	0.809211i	$-0.300105\pi$
$797$	33.1727	1.17504	0.587519	+	0.809211i	$0.300105\pi$
$798$	0	0
$799$	0.984462	0.0348278
$800$	−17.7180	−0.626425
$801$	0	0
$802$	−4.24208	−0.149793
$803$	0	0
$804$	0	0
$805$	−0.258162	−0.00909900
$806$	−13.4808	−0.474840
$807$	0	0
$808$	−41.9627	−1.47624
$809$	−24.5695	−0.863816	−0.431908	−	0.901918i	$-0.642160\pi$
$809$	−24.5695	−0.863816	−0.431908	+	0.901918i	$0.642160\pi$
$810$	0	0
$811$	−37.3592	−1.31186	−0.655929	−	0.754822i	$-0.727723\pi$
$811$	−37.3592	−1.31186	−0.655929	+	0.754822i	$0.727723\pi$
$812$	−3.93354	−0.138040
$813$	0	0
$814$	0	0
$815$	−25.3860	−0.889234
$816$	0	0
$817$	−13.2525	−0.463647
$818$	−18.3631	−0.642051
$819$	0	0
$820$	−15.0385	−0.525166
$821$	−12.2011	−0.425822	−0.212911	−	0.977072i	$-0.568294\pi$
$821$	−12.2011	−0.425822	−0.212911	+	0.977072i	$0.568294\pi$
$822$	0	0
$823$	−49.4786	−1.72471	−0.862357	−	0.506300i	$-0.831013\pi$
$823$	−49.4786	−1.72471	−0.862357	+	0.506300i	$0.831013\pi$
$824$	−7.34266	−0.255794
$825$	0	0
$826$	−2.66678	−0.0927890
$827$	−3.03613	−0.105577	−0.0527884	−	0.998606i	$-0.516811\pi$
$827$	−3.03613	−0.105577	−0.0527884	+	0.998606i	$0.516811\pi$
$828$	0	0
$829$	−48.1573	−1.67257	−0.836286	−	0.548294i	$-0.815278\pi$
$829$	−48.1573	−1.67257	−0.836286	+	0.548294i	$0.815278\pi$
$830$	−6.40148	−0.222199
$831$	0	0
$832$	−0.00990532	−0.000343405 0
$833$	−1.28755	−0.0446111
$834$	0	0
$835$	−12.9391	−0.447776
$836$	0	0
$837$	0	0
$838$	−23.9026	−0.825702
$839$	−38.9382	−1.34430	−0.672148	−	0.740417i	$-0.734628\pi$
$839$	−38.9382	−1.34430	−0.672148	+	0.740417i	$0.734628\pi$
$840$	0	0
$841$	−11.5993	−0.399977
$842$	−5.10423	−0.175903
$843$	0	0
$844$	−25.5261	−0.878646
$845$	−1.96724	−0.0676750
$846$	0	0
$847$	0	0
$848$	13.9962	0.480633
$849$	0	0
$850$	−0.381483	−0.0130848
$851$	3.30385	0.113254
$852$	0	0
$853$	37.0104	1.26721	0.633606	−	0.773656i	$-0.281574\pi$
$853$	37.0104	1.26721	0.633606	+	0.773656i	$0.281574\pi$
$854$	3.19709	0.109402
$855$	0	0
$856$	38.2708	1.30807
$857$	−28.5426	−0.974997	−0.487499	−	0.873124i	$-0.662091\pi$
$857$	−28.5426	−0.974997	−0.487499	+	0.873124i	$0.662091\pi$
$858$	0	0
$859$	38.3413	1.30819	0.654094	−	0.756413i	$-0.273050\pi$
$859$	38.3413	1.30819	0.654094	+	0.756413i	$0.273050\pi$
$860$	−14.9290	−0.509076
$861$	0	0
$862$	−19.1680	−0.652865
$863$	27.9656	0.951961	0.475981	−	0.879456i	$-0.342093\pi$
$863$	27.9656	0.951961	0.475981	+	0.879456i	$0.342093\pi$
$864$	0	0
$865$	−24.5584	−0.835012
$866$	15.1621	0.515229
$867$	0	0
$868$	−5.31423	−0.180377
$869$	0	0
$870$	0	0
$871$	34.5693	1.17134
$872$	20.2072	0.684302
$873$	0	0
$874$	0.393474	0.0133094
$875$	6.53530	0.220933
$876$	0	0
$877$	21.0397	0.710459	0.355229	−	0.934779i	$-0.384403\pi$
$877$	21.0397	0.710459	0.355229	+	0.934779i	$0.384403\pi$
$878$	−21.6032	−0.729073
$879$	0	0
$880$	0	0
$881$	17.0076	0.573000	0.286500	−	0.958080i	$-0.407508\pi$
$881$	17.0076	0.573000	0.286500	+	0.958080i	$0.407508\pi$
$882$	0	0
$883$	6.75447	0.227306	0.113653	−	0.993521i	$-0.463745\pi$
$883$	6.75447	0.227306	0.113653	+	0.993521i	$0.463745\pi$
$884$	1.17914	0.0396589
$885$	0	0
$886$	5.37614	0.180615
$887$	−33.8492	−1.13654	−0.568272	−	0.822841i	$-0.692388\pi$
$887$	−33.8492	−1.13654	−0.568272	+	0.822841i	$0.692388\pi$
$888$	0	0
$889$	0.755248	0.0253302
$890$	−2.91604	−0.0977459
$891$	0	0
$892$	−4.01188	−0.134328
$893$	9.89985	0.331286
$894$	0	0
$895$	36.1623	1.20877
$896$	−6.65191	−0.222225
$897$	0	0
$898$	23.2090	0.774496
$899$	23.5084	0.784048
$900$	0	0
$901$	1.52201	0.0507054
$902$	0	0
$903$	0	0
$904$	−6.54585	−0.217712
$905$	1.20136	0.0399345
$906$	0	0
$907$	28.1467	0.934595	0.467297	−	0.884100i	$-0.345228\pi$
$907$	28.1467	0.934595	0.467297	+	0.884100i	$0.345228\pi$
$908$	25.8419	0.857595
$909$	0	0
$910$	1.92242	0.0637277
$911$	8.18259	0.271101	0.135551	−	0.990770i	$-0.456720\pi$
$911$	8.18259	0.271101	0.135551	+	0.990770i	$0.456720\pi$
$912$	0	0
$913$	0	0
$914$	−14.9093	−0.493157
$915$	0	0
$916$	−5.29987	−0.175113
$917$	3.72670	0.123066
$918$	0	0
$919$	−11.6564	−0.384509	−0.192255	−	0.981345i	$-0.561580\pi$
$919$	−11.6564	−0.384509	−0.192255	+	0.981345i	$0.561580\pi$
$920$	0.996028	0.0328381
$921$	0	0
$922$	18.0582	0.594716
$923$	10.8863	0.358328
$924$	0	0
$925$	−32.2114	−1.05910
$926$	6.83844	0.224725
$927$	0	0
$928$	23.5988	0.774668
$929$	31.7731	1.04244	0.521221	−	0.853422i	$-0.325477\pi$
$929$	31.7731	1.04244	0.521221	+	0.853422i	$0.325477\pi$
$930$	0	0
$931$	−12.9478	−0.424346
$932$	6.91502	0.226509
$933$	0	0
$934$	−18.5627	−0.607391
$935$	0	0
$936$	0	0
$937$	43.2510	1.41295	0.706474	−	0.707739i	$-0.250285\pi$
$937$	43.2510	1.41295	0.706474	+	0.707739i	$0.250285\pi$
$938$	−3.36738	−0.109949
$939$	0	0
$940$	11.1522	0.363746
$941$	12.2021	0.397776	0.198888	−	0.980022i	$-0.436267\pi$
$941$	12.2021	0.397776	0.198888	+	0.980022i	$0.436267\pi$
$942$	0	0
$943$	−2.20391	−0.0717692
$944$	−12.8195	−0.417240
$945$	0	0
$946$	0	0
$947$	31.1861	1.01341	0.506706	−	0.862119i	$-0.330863\pi$
$947$	31.1861	1.01341	0.506706	+	0.862119i	$0.330863\pi$
$948$	0	0
$949$	1.61341	0.0523736
$950$	−3.83623	−0.124464
$951$	0	0
$952$	−0.258102	−0.00836514
$953$	18.3282	0.593707	0.296854	−	0.954923i	$-0.404063\pi$
$953$	18.3282	0.593707	0.296854	+	0.954923i	$0.404063\pi$
$954$	0	0
$955$	21.2145	0.686485
$956$	−24.1201	−0.780101
$957$	0	0
$958$	−13.4445	−0.434371
$959$	−9.73954	−0.314506
$960$	0	0
$961$	0.759889	0.0245126
$962$	−24.6024	−0.793213
$963$	0	0
$964$	22.0847	0.711300
$965$	13.2231	0.425666
$966$	0	0
$967$	−20.8508	−0.670515	−0.335258	−	0.942127i	$-0.608823\pi$
$967$	−20.8508	−0.670515	−0.335258	+	0.942127i	$0.608823\pi$
$968$	0	0
$969$	0	0
$970$	−5.56183	−0.178580
$971$	−28.4589	−0.913289	−0.456644	−	0.889649i	$-0.650949\pi$
$971$	−28.4589	−0.913289	−0.456644	+	0.889649i	$0.650949\pi$
$972$	0	0
$973$	−10.2600	−0.328921
$974$	−1.43302	−0.0459169
$975$	0	0
$976$	15.3688	0.491944
$977$	−22.2591	−0.712131	−0.356066	−	0.934461i	$-0.615882\pi$
$977$	−22.2591	−0.712131	−0.356066	+	0.934461i	$0.615882\pi$
$978$	0	0
$979$	0	0
$980$	−14.5857	−0.465924
$981$	0	0
$982$	5.99650	0.191356
$983$	−23.0934	−0.736566	−0.368283	−	0.929714i	$-0.620054\pi$
$983$	−23.0934	−0.736566	−0.368283	+	0.929714i	$0.620054\pi$
$984$	0	0
$985$	35.5862	1.13387
$986$	0.508102	0.0161813
$987$	0	0
$988$	11.8576	0.377240
$989$	−2.18787	−0.0695703
$990$	0	0
$991$	−6.16408	−0.195809	−0.0979043	−	0.995196i	$-0.531214\pi$
$991$	−6.16408	−0.195809	−0.0979043	+	0.995196i	$0.531214\pi$
$992$	31.8820	1.01226
$993$	0	0
$994$	−1.06044	−0.0336350
$995$	27.9560	0.886264
$996$	0	0
$997$	−25.3794	−0.803774	−0.401887	−	0.915689i	$-0.631646\pi$
$997$	−25.3794	−0.803774	−0.401887	+	0.915689i	$0.631646\pi$
$998$	−8.00940	−0.253533
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.cq.1.15		24
3.2	odd	2	inner	9801.2.a.cq.1.10		24
11.2	odd	10		891.2.f.g.730.7	yes	48
11.6	odd	10		891.2.f.g.487.7	yes	48
11.10	odd	2		9801.2.a.cr.1.10		24
33.2	even	10		891.2.f.g.730.6	yes	48
33.17	even	10		891.2.f.g.487.6	✓	48
33.32	even	2		9801.2.a.cr.1.15		24
99.2	even	30		891.2.n.l.433.6		96
99.13	odd	30		891.2.n.l.136.6		96
99.50	even	30		891.2.n.l.784.6		96
99.61	odd	30		891.2.n.l.190.6		96
99.68	even	30		891.2.n.l.136.7		96
99.79	odd	30		891.2.n.l.433.7		96
99.83	even	30		891.2.n.l.190.7		96
99.94	odd	30		891.2.n.l.784.7		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
891.2.f.g.487.6	✓	48	33.17	even	10
891.2.f.g.487.7	yes	48	11.6	odd	10
891.2.f.g.730.6	yes	48	33.2	even	10
891.2.f.g.730.7	yes	48	11.2	odd	10
891.2.n.l.136.6		96	99.13	odd	30
891.2.n.l.136.7		96	99.68	even	30
891.2.n.l.190.6		96	99.61	odd	30
891.2.n.l.190.7		96	99.83	even	30
891.2.n.l.433.6		96	99.2	even	30
891.2.n.l.433.7		96	99.79	odd	30
891.2.n.l.784.6		96	99.50	even	30
891.2.n.l.784.7		96	99.94	odd	30
9801.2.a.cq.1.10		24	3.2	odd	2	inner
9801.2.a.cq.1.15		24	1.1	even	1	trivial
9801.2.a.cr.1.10		24	11.10	odd	2
9801.2.a.cr.1.15		24	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q+0.629509 q^{2} -1.60372 q^{4} -1.36679 q^{5} +0.587994 q^{7} -2.26857 q^{8} -0.860404 q^{10} -3.79991 q^{13} +0.370148 q^{14} +1.77935 q^{16} +0.193493 q^{17} +1.94578 q^{19} +2.19194 q^{20} +0.321232 q^{23} -3.13190 q^{25} -2.39208 q^{26} -0.942977 q^{28} +4.17141 q^{29} +5.63559 q^{31} +5.65726 q^{32} +0.121806 q^{34} -0.803662 q^{35} +10.2849 q^{37} +1.22489 q^{38} +3.10065 q^{40} -6.86081 q^{41} -6.81088 q^{43} +0.202218 q^{46} +5.08784 q^{47} -6.65426 q^{49} -1.97156 q^{50} +6.09398 q^{52} +7.86594 q^{53} -1.33391 q^{56} +2.62594 q^{58} -7.20462 q^{59} +8.63734 q^{61} +3.54766 q^{62} +0.00260672 q^{64} +5.19366 q^{65} -9.09740 q^{67} -0.310308 q^{68} -0.505913 q^{70} -2.86490 q^{71} -0.424593 q^{73} +6.47446 q^{74} -3.12049 q^{76} +4.10547 q^{79} -2.43199 q^{80} -4.31894 q^{82} +7.44009 q^{83} -0.264464 q^{85} -4.28751 q^{86} +3.38915 q^{89} -2.23433 q^{91} -0.515165 q^{92} +3.20284 q^{94} -2.65947 q^{95} +6.46421 q^{97} -4.18892 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 26 q^{4} - 18 q^{7} - 16 q^{10} - 18 q^{13} + 18 q^{16} - 36 q^{19} + 20 q^{25} - 46 q^{28} - 12 q^{31} + 42 q^{34} + 14 q^{37} - 42 q^{40} - 42 q^{43} - 26 q^{46} + 38 q^{49} - 36 q^{52} - 40 q^{58}+ \cdots - 18 q^{97}+O(q^{100}) \) 24 * q + 26 * q^4 - 18 * q^7 - 16 * q^10 - 18 * q^13 + 18 * q^16 - 36 * q^19 + 20 * q^25 - 46 * q^28 - 12 * q^31 + 42 * q^34 + 14 * q^37 - 42 * q^40 - 42 * q^43 - 26 * q^46 + 38 * q^49 - 36 * q^52 - 40 * q^58 - 62 * q^61 - 16 * q^64 + 38 * q^67 + 6 * q^70 - 2 * q^73 - 102 * q^76 - 94 * q^79 - 24 * q^82 - 42 * q^85 - 34 * q^91 - 66 * q^94 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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