LMFDB - Embedded newform 9409.2.a.n.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9409,2,Mod(1,9409)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9409, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9409.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9409 = 97^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9409.a (trivial)

Newform invariants

comment:select newform

sage:traces = [56,8,16,40,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.1312432618$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 97)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	9409.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.15107 q^{2} -2.03471 q^{3} +2.62711 q^{4} +3.21461 q^{5} +4.37680 q^{6} -4.00609 q^{7} -1.34897 q^{8} +1.14003 q^{9} -6.91487 q^{10} +3.57147 q^{11} -5.34541 q^{12} -2.90782 q^{13} +8.61739 q^{14} -6.54079 q^{15} -2.35250 q^{16} -5.57114 q^{17} -2.45229 q^{18} +4.91272 q^{19} +8.44516 q^{20} +8.15121 q^{21} -7.68249 q^{22} -3.08831 q^{23} +2.74476 q^{24} +5.33374 q^{25} +6.25493 q^{26} +3.78449 q^{27} -10.5245 q^{28} +3.90306 q^{29} +14.0697 q^{30} -1.17331 q^{31} +7.75833 q^{32} -7.26689 q^{33} +11.9839 q^{34} -12.8780 q^{35} +2.99499 q^{36} -9.36027 q^{37} -10.5676 q^{38} +5.91656 q^{39} -4.33642 q^{40} -0.135130 q^{41} -17.5339 q^{42} +0.409118 q^{43} +9.38266 q^{44} +3.66475 q^{45} +6.64317 q^{46} +4.69945 q^{47} +4.78664 q^{48} +9.04875 q^{49} -11.4733 q^{50} +11.3356 q^{51} -7.63917 q^{52} +5.92979 q^{53} -8.14072 q^{54} +11.4809 q^{55} +5.40409 q^{56} -9.99594 q^{57} -8.39578 q^{58} -13.4284 q^{59} -17.1834 q^{60} -6.95895 q^{61} +2.52388 q^{62} -4.56706 q^{63} -11.9837 q^{64} -9.34751 q^{65} +15.6316 q^{66} -2.49921 q^{67} -14.6360 q^{68} +6.28380 q^{69} +27.7016 q^{70} -6.04691 q^{71} -1.53786 q^{72} +12.0089 q^{73} +20.1346 q^{74} -10.8526 q^{75} +12.9063 q^{76} -14.3076 q^{77} -12.7269 q^{78} -0.949556 q^{79} -7.56237 q^{80} -11.1204 q^{81} +0.290674 q^{82} -13.4341 q^{83} +21.4142 q^{84} -17.9091 q^{85} -0.880043 q^{86} -7.94159 q^{87} -4.81781 q^{88} +10.5236 q^{89} -7.88315 q^{90} +11.6490 q^{91} -8.11334 q^{92} +2.38735 q^{93} -10.1089 q^{94} +15.7925 q^{95} -15.7859 q^{96} -19.4645 q^{98} +4.07158 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{2} + 16 q^{3} + 40 q^{4} + 16 q^{6} + 24 q^{8} + 40 q^{9} + 32 q^{11} + 48 q^{12} + 8 q^{16} + 24 q^{18} + 16 q^{22} + 8 q^{25} + 64 q^{27} + 48 q^{31} + 56 q^{32} + 80 q^{33} + 48 q^{35} + 8 q^{36}+ \cdots - 16 q^{99}+O(q^{100}) $ 56 * q + 8 * q^2 + 16 * q^3 + 40 * q^4 + 16 * q^6 + 24 * q^8 + 40 * q^9 + 32 * q^11 + 48 * q^12 + 8 * q^16 + 24 * q^18 + 16 * q^22 + 8 * q^25 + 64 * q^27 + 48 * q^31 + 56 * q^32 + 80 * q^33 + 48 * q^35 + 8 * q^36 + 32 * q^43 + 80 * q^44 + 16 * q^47 + 112 * q^48 + 8 * q^49 + 24 * q^50 + 48 * q^53 + 96 * q^54 + 32 * q^61 + 80 * q^62 - 56 * q^64 + 128 * q^65 + 192 * q^66 + 80 * q^70 - 88 * q^72 + 32 * q^73 + 80 * q^75 + 48 * q^79 + 72 * q^81 + 144 * q^86 + 176 * q^88 + 96 * q^89 + 64 * q^91 + 64 * q^94 + 176 * q^95 + 32 * q^96 - 136 * q^98 - 16 * q^99

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 $

$2$	−2.15107	−1.52104	−0.760519	−	0.649316i	$-0.775055\pi$
$2$	−2.15107	−1.52104	−0.760519	+	0.649316i	$0.775055\pi$
$3$	−2.03471	−1.17474	−0.587369	−	0.809319i	$-0.699836\pi$
$3$	−2.03471	−1.17474	−0.587369	+	0.809319i	$0.699836\pi$
$4$	2.62711	1.31356
$5$	3.21461	1.43762	0.718809	−	0.695207i	$-0.244687\pi$
$5$	3.21461	1.43762	0.718809	+	0.695207i	$0.244687\pi$
$6$	4.37680	1.78682
$7$	−4.00609	−1.51416	−0.757080	−	0.653323i	$-0.773375\pi$
$7$	−4.00609	−1.51416	−0.757080	+	0.653323i	$0.773375\pi$
$8$	−1.34897	−0.476933
$9$	1.14003	0.380010
$10$	−6.91487	−2.18667
$11$	3.57147	1.07684	0.538419	−	0.842677i	$-0.319022\pi$
$11$	3.57147	1.07684	0.538419	+	0.842677i	$0.319022\pi$
$12$	−5.34541	−1.54309
$13$	−2.90782	−0.806484	−0.403242	−	0.915093i	$-0.632117\pi$
$13$	−2.90782	−0.806484	−0.403242	+	0.915093i	$0.632117\pi$
$14$	8.61739	2.30309
$15$	−6.54079	−1.68883
$16$	−2.35250	−0.588124
$17$	−5.57114	−1.35120	−0.675600	−	0.737268i	$-0.736115\pi$
$17$	−5.57114	−1.35120	−0.675600	+	0.737268i	$0.736115\pi$
$18$	−2.45229	−0.578009
$19$	4.91272	1.12706	0.563528	−	0.826097i	$-0.309444\pi$
$19$	4.91272	1.12706	0.563528	+	0.826097i	$0.309444\pi$
$20$	8.44516	1.88839
$21$	8.15121	1.77874
$22$	−7.68249	−1.63791
$23$	−3.08831	−0.643956	−0.321978	−	0.946747i	$-0.604348\pi$
$23$	−3.08831	−0.643956	−0.321978	+	0.946747i	$0.604348\pi$
$24$	2.74476	0.560271
$25$	5.33374	1.06675
$26$	6.25493	1.22669
$27$	3.78449	0.728326
$28$	−10.5245	−1.98894
$29$	3.90306	0.724781	0.362390	−	0.932026i	$-0.381961\pi$
$29$	3.90306	0.724781	0.362390	+	0.932026i	$0.381961\pi$
$30$	14.0697	2.56877
$31$	−1.17331	−0.210733	−0.105367	−	0.994433i	$-0.533602\pi$
$31$	−1.17331	−0.210733	−0.105367	+	0.994433i	$0.533602\pi$
$32$	7.75833	1.37149
$33$	−7.26689	−1.26500
$34$	11.9839	2.05523
$35$	−12.8780	−2.17678
$36$	2.99499	0.499165
$37$	−9.36027	−1.53882	−0.769409	−	0.638756i	$-0.779449\pi$
$37$	−9.36027	−1.53882	−0.769409	+	0.638756i	$0.779449\pi$
$38$	−10.5676	−1.71429
$39$	5.91656	0.947407
$40$	−4.33642	−0.685648
$41$	−0.135130	−0.0211037	−0.0105519	−	0.999944i	$-0.503359\pi$
$41$	−0.135130	−0.0211037	−0.0105519	+	0.999944i	$0.503359\pi$
$42$	−17.5339	−2.70553
$43$	0.409118	0.0623899	0.0311950	−	0.999513i	$-0.490069\pi$
$43$	0.409118	0.0623899	0.0311950	+	0.999513i	$0.490069\pi$
$44$	9.38266	1.41449
$45$	3.66475	0.546309
$46$	6.64317	0.979482
$47$	4.69945	0.685485	0.342743	−	0.939429i	$-0.388644\pi$
$47$	4.69945	0.685485	0.342743	+	0.939429i	$0.388644\pi$
$48$	4.78664	0.690892
$49$	9.04875	1.29268
$50$	−11.4733	−1.62256
$51$	11.3356	1.58731
$52$	−7.63917	−1.05936
$53$	5.92979	0.814519	0.407259	−	0.913313i	$-0.366484\pi$
$53$	5.92979	0.814519	0.407259	+	0.913313i	$0.366484\pi$
$54$	−8.14072	−1.10781
$55$	11.4809	1.54808
$56$	5.40409	0.722152
$57$	−9.99594	−1.32399
$58$	−8.39578	−1.10242
$59$	−13.4284	−1.74823	−0.874113	−	0.485723i	$-0.838556\pi$
$59$	−13.4284	−1.74823	−0.874113	+	0.485723i	$0.838556\pi$
$60$	−17.1834	−2.21837
$61$	−6.95895	−0.891002	−0.445501	−	0.895281i	$-0.646974\pi$
$61$	−6.95895	−0.891002	−0.445501	+	0.895281i	$0.646974\pi$
$62$	2.52388	0.320534
$63$	−4.56706	−0.575395
$64$	−11.9837	−1.49797
$65$	−9.34751	−1.15942
$66$	15.6316	1.92412
$67$	−2.49921	−0.305328	−0.152664	−	0.988278i	$-0.548785\pi$
$67$	−2.49921	−0.305328	−0.152664	+	0.988278i	$0.548785\pi$
$68$	−14.6360	−1.77488
$69$	6.28380	0.756480
$70$	27.7016	3.31097
$71$	−6.04691	−0.717636	−0.358818	−	0.933408i	$-0.616820\pi$
$71$	−6.04691	−0.717636	−0.358818	+	0.933408i	$0.616820\pi$
$72$	−1.53786	−0.181239
$73$	12.0089	1.40553	0.702766	−	0.711421i	$-0.251948\pi$
$73$	12.0089	1.40553	0.702766	+	0.711421i	$0.251948\pi$
$74$	20.1346	2.34060
$75$	−10.8526	−1.25315
$76$	12.9063	1.48045
$77$	−14.3076	−1.63051
$78$	−12.7269	−1.44104
$79$	−0.949556	−0.106833	−0.0534167	−	0.998572i	$-0.517011\pi$
$79$	−0.949556	−0.106833	−0.0534167	+	0.998572i	$0.517011\pi$
$80$	−7.56237	−0.845499
$81$	−11.1204	−1.23560
$82$	0.290674	0.0320995
$83$	−13.4341	−1.47459	−0.737295	−	0.675571i	$-0.763897\pi$
$83$	−13.4341	−1.47459	−0.737295	+	0.675571i	$0.763897\pi$
$84$	21.4142	2.33648
$85$	−17.9091	−1.94251
$86$	−0.880043	−0.0948974
$87$	−7.94159	−0.851428
$88$	−4.81781	−0.513580
$89$	10.5236	1.11550	0.557751	−	0.830008i	$-0.311664\pi$
$89$	10.5236	1.11550	0.557751	+	0.830008i	$0.311664\pi$
$90$	−7.88315	−0.830957
$91$	11.6490	1.22114
$92$	−8.11334	−0.845874
$93$	2.38735	0.247557
$94$	−10.1089	−1.04265
$95$	15.7925	1.62028
$96$	−15.7859	−1.61114
$97$	0	0
$97$	0	0
$98$	−19.4645	−1.96621
$99$	4.07158	0.409209
$100$	14.0123	1.40123
$101$	10.4594	1.04075	0.520376	−	0.853937i	$-0.325792\pi$
$101$	10.4594	1.04075	0.520376	+	0.853937i	$0.325792\pi$
$102$	−24.3838	−2.41435
$103$	6.79591	0.669621	0.334810	−	0.942286i	$-0.391328\pi$
$103$	6.79591	0.669621	0.334810	+	0.942286i	$0.391328\pi$
$104$	3.92256	0.384639
$105$	26.2030	2.55715
$106$	−12.7554	−1.23891
$107$	−1.58885	−0.153600	−0.0768002	−	0.997047i	$-0.524470\pi$
$107$	−1.58885	−0.153600	−0.0768002	+	0.997047i	$0.524470\pi$
$108$	9.94230	0.956698
$109$	−7.56134	−0.724245	−0.362123	−	0.932130i	$-0.617948\pi$
$109$	−7.56134	−0.724245	−0.362123	+	0.932130i	$0.617948\pi$
$110$	−24.6962	−2.35469
$111$	19.0454	1.80771
$112$	9.42431	0.890514
$113$	2.87337	0.270304	0.135152	−	0.990825i	$-0.456848\pi$
$113$	2.87337	0.270304	0.135152	+	0.990825i	$0.456848\pi$
$114$	21.5020	2.01385
$115$	−9.92771	−0.925764
$116$	10.2538	0.952041
$117$	−3.31500	−0.306472
$118$	28.8854	2.65912
$119$	22.3185	2.04593
$120$	8.82333	0.805456
$121$	1.75540	0.159582
$122$	14.9692	1.35525
$123$	0.274949	0.0247913
$124$	−3.08243	−0.276810
$125$	1.07285	0.0959583
$126$	9.82408	0.875198
$127$	−2.89578	−0.256959	−0.128479	−	0.991712i	$-0.541010\pi$
$127$	−2.89578	−0.256959	−0.128479	+	0.991712i	$0.541010\pi$
$128$	10.2612	0.906974
$129$	−0.832435	−0.0732918
$130$	20.1072	1.76352
$131$	−15.5710	−1.36045	−0.680223	−	0.733005i	$-0.738117\pi$
$131$	−15.5710	−1.36045	−0.680223	+	0.733005i	$0.738117\pi$
$132$	−19.0910	−1.66165
$133$	−19.6808	−1.70654
$134$	5.37599	0.464415
$135$	12.1657	1.04706
$136$	7.51530	0.644432
$137$	−11.6561	−0.995844	−0.497922	−	0.867222i	$-0.665903\pi$
$137$	−11.6561	−0.995844	−0.497922	+	0.867222i	$0.665903\pi$
$138$	−13.5169	−1.15064
$139$	20.2833	1.72041	0.860205	−	0.509949i	$-0.170336\pi$
$139$	20.2833	1.72041	0.860205	+	0.509949i	$0.170336\pi$
$140$	−33.8321	−2.85933
$141$	−9.56200	−0.805266
$142$	13.0073	1.09155
$143$	−10.3852	−0.868453
$144$	−2.68192	−0.223493
$145$	12.5468	1.04196
$146$	−25.8320	−2.13787
$147$	−18.4115	−1.51856
$148$	−24.5905	−2.02133
$149$	−8.20537	−0.672210	−0.336105	−	0.941824i	$-0.609110\pi$
$149$	−8.20537	−0.672210	−0.336105	+	0.941824i	$0.609110\pi$
$150$	23.3447	1.90609
$151$	12.8962	1.04948	0.524739	−	0.851263i	$-0.324163\pi$
$151$	12.8962	1.04948	0.524739	+	0.851263i	$0.324163\pi$
$152$	−6.62711	−0.537530
$153$	−6.35127	−0.513469
$154$	30.7767	2.48006
$155$	−3.77175	−0.302954
$156$	15.5435	1.24447
$157$	−0.507389	−0.0404941	−0.0202470	−	0.999795i	$-0.506445\pi$
$157$	−0.507389	−0.0404941	−0.0202470	+	0.999795i	$0.506445\pi$
$158$	2.04256	0.162498
$159$	−12.0654	−0.956846
$160$	24.9400	1.97168
$161$	12.3720	0.975053
$162$	23.9208	1.87940
$163$	7.22640	0.566015	0.283008	−	0.959118i	$-0.408668\pi$
$163$	7.22640	0.566015	0.283008	+	0.959118i	$0.408668\pi$
$164$	−0.355001	−0.0277209
$165$	−23.3603	−1.81859
$166$	28.8978	2.24291
$167$	−2.86213	−0.221478	−0.110739	−	0.993850i	$-0.535322\pi$
$167$	−2.86213	−0.221478	−0.110739	+	0.993850i	$0.535322\pi$
$168$	−10.9957	−0.848340
$169$	−4.54459	−0.349584
$170$	38.5237	2.95463
$171$	5.60064	0.428292
$172$	1.07480	0.0819527
$173$	7.17432	0.545454	0.272727	−	0.962092i	$-0.412074\pi$
$173$	7.17432	0.545454	0.272727	+	0.962092i	$0.412074\pi$
$174$	17.0829	1.29505
$175$	−21.3674	−1.61523
$176$	−8.40187	−0.633315
$177$	27.3228	2.05371
$178$	−22.6371	−1.69672
$179$	−3.26534	−0.244063	−0.122031	−	0.992526i	$-0.538941\pi$
$179$	−3.26534	−0.244063	−0.122031	+	0.992526i	$0.538941\pi$
$180$	9.62773	0.717608
$181$	9.70242	0.721175	0.360588	−	0.932725i	$-0.382576\pi$
$181$	9.70242	0.721175	0.360588	+	0.932725i	$0.382576\pi$
$182$	−25.0578	−1.85741
$183$	14.1594	1.04669
$184$	4.16603	0.307124
$185$	−30.0896	−2.21223
$186$	−5.13536	−0.376543
$187$	−19.8972	−1.45503
$188$	12.3460	0.900424
$189$	−15.1610	−1.10280
$190$	−33.9708	−2.46450
$191$	14.5467	1.05256	0.526280	−	0.850311i	$-0.323586\pi$
$191$	14.5467	1.05256	0.526280	+	0.850311i	$0.323586\pi$
$192$	24.3834	1.75972
$193$	6.07319	0.437158	0.218579	−	0.975819i	$-0.429858\pi$
$193$	6.07319	0.437158	0.218579	+	0.975819i	$0.429858\pi$
$194$	0	0
$195$	19.0194	1.36201
$196$	23.7721	1.69801
$197$	−15.0691	−1.07363	−0.536815	−	0.843700i	$-0.680373\pi$
$197$	−15.0691	−1.07363	−0.536815	+	0.843700i	$0.680373\pi$
$198$	−8.75827	−0.622423
$199$	17.1465	1.21548	0.607742	−	0.794135i	$-0.292076\pi$
$199$	17.1465	1.21548	0.607742	+	0.794135i	$0.292076\pi$
$200$	−7.19505	−0.508767
$201$	5.08517	0.358680
$202$	−22.4990	−1.58302
$203$	−15.6360	−1.09743
$204$	29.7800	2.08502
$205$	−0.434390	−0.0303391
$206$	−14.6185	−1.01852
$207$	−3.52076	−0.244710
$208$	6.84064	0.474313
$209$	17.5456	1.21366
$210$	−56.3646	−3.88952
$211$	23.2489	1.60052	0.800261	−	0.599652i	$-0.204694\pi$
$211$	23.2489	1.60052	0.800261	+	0.599652i	$0.204694\pi$
$212$	15.5782	1.06992
$213$	12.3037	0.843034
$214$	3.41774	0.233632
$215$	1.31516	0.0896929
$216$	−5.10517	−0.347363
$217$	4.70040	0.319084
$218$	16.2650	1.10160
$219$	−24.4345	−1.65113
$220$	30.1616	2.03350
$221$	16.1999	1.08972
$222$	−40.9680	−2.74959
$223$	−2.43132	−0.162813	−0.0814067	−	0.996681i	$-0.525941\pi$
$223$	−2.43132	−0.162813	−0.0814067	+	0.996681i	$0.525941\pi$
$224$	−31.0806	−2.07666
$225$	6.08062	0.405375
$226$	−6.18083	−0.411143
$227$	−9.60748	−0.637671	−0.318835	−	0.947810i	$-0.603292\pi$
$227$	−9.60748	−0.637671	−0.318835	+	0.947810i	$0.603292\pi$
$228$	−26.2605	−1.73914
$229$	27.1144	1.79177	0.895884	−	0.444287i	$-0.146543\pi$
$229$	27.1144	1.79177	0.895884	+	0.444287i	$0.146543\pi$
$230$	21.3552	1.40812
$231$	29.1118	1.91542
$232$	−5.26512	−0.345672
$233$	11.8786	0.778193	0.389096	−	0.921197i	$-0.372787\pi$
$233$	11.8786	0.778193	0.389096	+	0.921197i	$0.372787\pi$
$234$	7.13080	0.466155
$235$	15.1069	0.985467
$236$	−35.2779	−2.29639
$237$	1.93207	0.125501
$238$	−48.0087	−3.11194
$239$	20.3682	1.31751	0.658753	−	0.752359i	$-0.271084\pi$
$239$	20.3682	1.31751	0.658753	+	0.752359i	$0.271084\pi$
$240$	15.3872	0.993239
$241$	−4.41257	−0.284238	−0.142119	−	0.989850i	$-0.545392\pi$
$241$	−4.41257	−0.284238	−0.142119	+	0.989850i	$0.545392\pi$
$242$	−3.77599	−0.242730
$243$	11.2733	0.723183
$244$	−18.2820	−1.17038
$245$	29.0882	1.85838
$246$	−0.591436	−0.0377086
$247$	−14.2853	−0.908952
$248$	1.58276	0.100506
$249$	27.3345	1.73226
$250$	−2.30777	−0.145956
$251$	−0.196625	−0.0124108	−0.00620542	−	0.999981i	$-0.501975\pi$
$251$	−0.196625	−0.0124108	−0.00620542	+	0.999981i	$0.501975\pi$
$252$	−11.9982	−0.755815
$253$	−11.0298	−0.693437
$254$	6.22904	0.390844
$255$	36.4397	2.28194
$256$	1.89480	0.118425
$257$	−12.5037	−0.779962	−0.389981	−	0.920823i	$-0.627518\pi$
$257$	−12.5037	−0.779962	−0.389981	+	0.920823i	$0.627518\pi$
$258$	1.79063	0.111480
$259$	37.4981	2.33002
$260$	−24.5570	−1.52296
$261$	4.44961	0.275424
$262$	33.4944	2.06929
$263$	−0.655115	−0.0403962	−0.0201981	−	0.999796i	$-0.506430\pi$
$263$	−0.655115	−0.0403962	−0.0201981	+	0.999796i	$0.506430\pi$
$264$	9.80282	0.603322
$265$	19.0620	1.17097
$266$	42.3348	2.59571
$267$	−21.4125	−1.31042
$268$	−6.56572	−0.401065
$269$	−29.1149	−1.77517	−0.887585	−	0.460645i	$-0.847618\pi$
$269$	−29.1149	−1.77517	−0.887585	+	0.460645i	$0.847618\pi$
$270$	−26.1693	−1.59261
$271$	−26.7765	−1.62656	−0.813279	−	0.581874i	$-0.802320\pi$
$271$	−26.7765	−1.62656	−0.813279	+	0.581874i	$0.802320\pi$
$272$	13.1061	0.794674
$273$	−23.7023	−1.43453
$274$	25.0730	1.51472
$275$	19.0493	1.14872
$276$	16.5083	0.993680
$277$	−10.2741	−0.617311	−0.308656	−	0.951174i	$-0.599879\pi$
$277$	−10.2741	−0.617311	−0.308656	+	0.951174i	$0.599879\pi$
$278$	−43.6309	−2.61681
$279$	−1.33761	−0.0800807
$280$	17.3721	1.03818
$281$	−23.3163	−1.39093	−0.695466	−	0.718559i	$-0.744802\pi$
$281$	−23.3163	−1.39093	−0.695466	+	0.718559i	$0.744802\pi$
$282$	20.5686	1.22484
$283$	3.18662	0.189425	0.0947124	−	0.995505i	$-0.469807\pi$
$283$	3.18662	0.189425	0.0947124	+	0.995505i	$0.469807\pi$
$284$	−15.8859	−0.942656
$285$	−32.1331	−1.90340
$286$	22.3393	1.32095
$287$	0.541341	0.0319544
$288$	8.84473	0.521180
$289$	14.0376	0.825743
$290$	−26.9892	−1.58486
$291$	0	0
$292$	31.5487	1.84625
$293$	−11.0409	−0.645018	−0.322509	−	0.946566i	$-0.604526\pi$
$293$	−11.0409	−0.645018	−0.322509	+	0.946566i	$0.604526\pi$
$294$	39.6046	2.30979
$295$	−43.1670	−2.51328
$296$	12.6267	0.733913
$297$	13.5162	0.784290
$298$	17.6504	1.02246
$299$	8.98024	0.519340
$300$	−28.5110	−1.64608
$301$	−1.63896	−0.0944683
$302$	−27.7407	−1.59630
$303$	−21.2819	−1.22261
$304$	−11.5572	−0.662849
$305$	−22.3703	−1.28092
$306$	13.6620	0.781007
$307$	−5.98627	−0.341655	−0.170827	−	0.985301i	$-0.554644\pi$
$307$	−5.98627	−0.341655	−0.170827	+	0.985301i	$0.554644\pi$
$308$	−37.5878	−2.14176
$309$	−13.8277	−0.786629
$310$	8.11331	0.460805
$311$	3.96138	0.224629	0.112315	−	0.993673i	$-0.464174\pi$
$311$	3.96138	0.224629	0.112315	+	0.993673i	$0.464174\pi$
$312$	−7.98126	−0.451850
$313$	−25.3354	−1.43204	−0.716021	−	0.698079i	$-0.754038\pi$
$313$	−25.3354	−1.43204	−0.716021	+	0.698079i	$0.754038\pi$
$314$	1.09143	0.0615930
$315$	−14.6813	−0.827199
$316$	−2.49459	−0.140332
$317$	−12.1586	−0.682894	−0.341447	−	0.939901i	$-0.610917\pi$
$317$	−12.1586	−0.682894	−0.341447	+	0.939901i	$0.610917\pi$
$318$	25.9535	1.45540
$319$	13.9397	0.780472
$320$	−38.5231	−2.15351
$321$	3.23285	0.180440
$322$	−26.6131	−1.48309
$323$	−27.3695	−1.52288
$324$	−29.2146	−1.62303
$325$	−15.5096	−0.860315
$326$	−15.5445	−0.860931
$327$	15.3851	0.850798
$328$	0.182286	0.0100651
$329$	−18.8264	−1.03793
$330$	50.2496	2.76615
$331$	22.0905	1.21420	0.607101	−	0.794625i	$-0.292333\pi$
$331$	22.0905	1.21420	0.607101	+	0.794625i	$0.292333\pi$
$332$	−35.2930	−1.93696
$333$	−10.6710	−0.584766
$334$	6.15664	0.336876
$335$	−8.03401	−0.438945
$336$	−19.1757	−1.04612
$337$	−15.8190	−0.861714	−0.430857	−	0.902420i	$-0.641789\pi$
$337$	−15.8190	−0.861714	−0.430857	+	0.902420i	$0.641789\pi$
$338$	9.77574	0.531730
$339$	−5.84647	−0.317537
$340$	−47.0492	−2.55160
$341$	−4.19046	−0.226926
$342$	−12.0474	−0.651449
$343$	−8.20747	−0.443161
$344$	−0.551888	−0.0297558
$345$	20.2000	1.08753
$346$	−15.4325	−0.829656
$347$	32.3071	1.73434	0.867169	−	0.498015i	$-0.165937\pi$
$347$	32.3071	1.73434	0.867169	+	0.498015i	$0.165937\pi$
$348$	−20.8635	−1.11840
$349$	−10.1458	−0.543092	−0.271546	−	0.962425i	$-0.587535\pi$
$349$	−10.1458	−0.543092	−0.271546	+	0.962425i	$0.587535\pi$
$350$	45.9629	2.45682
$351$	−11.0046	−0.587383
$352$	27.7087	1.47688
$353$	−6.67676	−0.355368	−0.177684	−	0.984088i	$-0.556861\pi$
$353$	−6.67676	−0.355368	−0.177684	+	0.984088i	$0.556861\pi$
$354$	−58.7733	−3.12377
$355$	−19.4385	−1.03169
$356$	27.6468	1.46528
$357$	−45.4116	−2.40344
$358$	7.02399	0.371229
$359$	17.0326	0.898944	0.449472	−	0.893294i	$-0.351612\pi$
$359$	17.0326	0.898944	0.449472	+	0.893294i	$0.351612\pi$
$360$	−4.94364	−0.260553
$361$	5.13482	0.270254
$362$	−20.8706	−1.09693
$363$	−3.57172	−0.187467
$364$	30.6032	1.60404
$365$	38.6039	2.02062
$366$	−30.4579	−1.59206
$367$	8.08227	0.421891	0.210946	−	0.977498i	$-0.432346\pi$
$367$	8.08227	0.421891	0.210946	+	0.977498i	$0.432346\pi$
$368$	7.26523	0.378726
$369$	−0.154052	−0.00801961
$370$	64.7250	3.36489
$371$	−23.7553	−1.23331
$372$	6.27184	0.325180
$373$	−5.36766	−0.277927	−0.138963	−	0.990298i	$-0.544377\pi$
$373$	−5.36766	−0.277927	−0.138963	+	0.990298i	$0.544377\pi$
$374$	42.8003	2.21315
$375$	−2.18293	−0.112726
$376$	−6.33942	−0.326931
$377$	−11.3494	−0.584524
$378$	32.6125	1.67740
$379$	−15.8362	−0.813451	−0.406726	−	0.913550i	$-0.633329\pi$
$379$	−15.8362	−0.813451	−0.406726	+	0.913550i	$0.633329\pi$
$380$	41.4887	2.12833
$381$	5.89206	0.301860
$382$	−31.2909	−1.60098
$383$	−9.60522	−0.490804	−0.245402	−	0.969421i	$-0.578920\pi$
$383$	−9.60522	−0.490804	−0.245402	+	0.969421i	$0.578920\pi$
$384$	−20.8786	−1.06546
$385$	−45.9935	−2.34405
$386$	−13.0639	−0.664934
$387$	0.466406	0.0237088
$388$	0	0
$389$	−2.34108	−0.118697	−0.0593486	−	0.998237i	$-0.518902\pi$
$389$	−2.34108	−0.118697	−0.0593486	+	0.998237i	$0.518902\pi$
$390$	−40.9122	−2.07167
$391$	17.2054	0.870114
$392$	−12.2065	−0.616521
$393$	31.6824	1.59817
$394$	32.4148	1.63303
$395$	−3.05245	−0.153586
$396$	10.6965	0.537520
$397$	−3.72400	−0.186902	−0.0934510	−	0.995624i	$-0.529790\pi$
$397$	−3.72400	−0.186902	−0.0934510	+	0.995624i	$0.529790\pi$
$398$	−36.8834	−1.84880
$399$	40.0446	2.00474
$400$	−12.5476	−0.627380
$401$	20.4389	1.02067	0.510336	−	0.859975i	$-0.329521\pi$
$401$	20.4389	1.02067	0.510336	+	0.859975i	$0.329521\pi$
$402$	−10.9386	−0.545566
$403$	3.41178	0.169953
$404$	27.4781	1.36709
$405$	−35.7479	−1.77633
$406$	33.6342	1.66924
$407$	−33.4299	−1.65706
$408$	−15.2914	−0.757039
$409$	−25.0610	−1.23919	−0.619593	−	0.784923i	$-0.712702\pi$
$409$	−25.0610	−1.23919	−0.619593	+	0.784923i	$0.712702\pi$
$410$	0.934404	0.0461469
$411$	23.7166	1.16986
$412$	17.8536	0.879585
$413$	53.7953	2.64709
$414$	7.57341	0.372213
$415$	−43.1856	−2.11990
$416$	−22.5598	−1.10609
$417$	−41.2706	−2.02103
$418$	−37.7419	−1.84602
$419$	−27.8182	−1.35901	−0.679504	−	0.733672i	$-0.737805\pi$
$419$	−27.8182	−1.35901	−0.679504	+	0.733672i	$0.737805\pi$
$420$	68.8383	3.35896
$421$	7.25610	0.353641	0.176820	−	0.984243i	$-0.443419\pi$
$421$	7.25610	0.353641	0.176820	+	0.984243i	$0.443419\pi$
$422$	−50.0101	−2.43445
$423$	5.35751	0.260491
$424$	−7.99910	−0.388471
$425$	−29.7150	−1.44139
$426$	−26.4661	−1.28229
$427$	27.8782	1.34912
$428$	−4.17410	−0.201763
$429$	21.1308	1.02020
$430$	−2.82900	−0.136426
$431$	11.7593	0.566423	0.283212	−	0.959057i	$-0.408600\pi$
$431$	11.7593	0.566423	0.283212	+	0.959057i	$0.408600\pi$
$432$	−8.90301	−0.428346
$433$	−21.5649	−1.03634	−0.518171	−	0.855277i	$-0.673387\pi$
$433$	−21.5649	−1.03634	−0.518171	+	0.855277i	$0.673387\pi$
$434$	−10.1109	−0.485339
$435$	−25.5291	−1.22403
$436$	−19.8645	−0.951338
$437$	−15.1720	−0.725774
$438$	52.5604	2.51144
$439$	25.1054	1.19821	0.599107	−	0.800669i	$-0.295522\pi$
$439$	25.1054	1.19821	0.599107	+	0.800669i	$0.295522\pi$
$440$	−15.4874	−0.738332
$441$	10.3158	0.491230
$442$	−34.8471	−1.65751
$443$	29.8209	1.41683	0.708417	−	0.705794i	$-0.249410\pi$
$443$	29.8209	1.41683	0.708417	+	0.705794i	$0.249410\pi$
$444$	50.0344	2.37453
$445$	33.8294	1.60367
$446$	5.22995	0.247645
$447$	16.6955	0.789671
$448$	48.0079	2.26816
$449$	15.3931	0.726445	0.363222	−	0.931702i	$-0.381677\pi$
$449$	15.3931	0.726445	0.363222	+	0.931702i	$0.381677\pi$
$450$	−13.0799	−0.616590
$451$	−0.482612	−0.0227253
$452$	7.54868	0.355060
$453$	−26.2400	−1.23286
$454$	20.6664	0.969922
$455$	37.4470	1.75554
$456$	13.4842	0.631457
$457$	−41.5865	−1.94533	−0.972666	−	0.232208i	$-0.925405\pi$
$457$	−41.5865	−1.94533	−0.972666	+	0.232208i	$0.925405\pi$
$458$	−58.3250	−2.72535
$459$	−21.0840	−0.984115
$460$	−26.0812	−1.21604
$461$	11.4548	0.533505	0.266753	−	0.963765i	$-0.414049\pi$
$461$	11.4548	0.533505	0.266753	+	0.963765i	$0.414049\pi$
$462$	−62.6216	−2.91342
$463$	14.3559	0.667174	0.333587	−	0.942719i	$-0.391741\pi$
$463$	14.3559	0.667174	0.333587	+	0.942719i	$0.391741\pi$
$464$	−9.18195	−0.426261
$465$	7.67440	0.355892
$466$	−25.5517	−1.18366
$467$	21.8383	1.01056	0.505278	−	0.862957i	$-0.331390\pi$
$467$	21.8383	1.01056	0.505278	+	0.862957i	$0.331390\pi$
$468$	−8.70888	−0.402568
$469$	10.0121	0.462315
$470$	−32.4961	−1.49893
$471$	1.03239	0.0475699
$472$	18.1145	0.833786
$473$	1.46115	0.0671839
$474$	−4.15602	−0.190892
$475$	26.2032	1.20228
$476$	58.6332	2.68745
$477$	6.76013	0.309525
$478$	−43.8134	−2.00398
$479$	3.87460	0.177035	0.0885176	−	0.996075i	$-0.471787\pi$
$479$	3.87460	0.177035	0.0885176	+	0.996075i	$0.471787\pi$
$480$	−50.7457	−2.31621
$481$	27.2180	1.24103
$482$	9.49176	0.432338
$483$	−25.1734	−1.14543
$484$	4.61163	0.209620
$485$	0	0
$486$	−24.2497	−1.09999
$487$	−9.88112	−0.447757	−0.223878	−	0.974617i	$-0.571872\pi$
$487$	−9.88112	−0.447757	−0.223878	+	0.974617i	$0.571872\pi$
$488$	9.38741	0.424948
$489$	−14.7036	−0.664920
$490$	−62.5709	−2.82667
$491$	−6.94620	−0.313478	−0.156739	−	0.987640i	$-0.550098\pi$
$491$	−6.94620	−0.313478	−0.156739	+	0.987640i	$0.550098\pi$
$492$	0.722323	0.0325648
$493$	−21.7445	−0.979324
$494$	30.7287	1.38255
$495$	13.0886	0.588287
$496$	2.76022	0.123937
$497$	24.2244	1.08661
$498$	−58.7986	−2.63483
$499$	−19.2605	−0.862220	−0.431110	−	0.902299i	$-0.641878\pi$
$499$	−19.2605	−0.862220	−0.431110	+	0.902299i	$0.641878\pi$
$500$	2.81849	0.126047
$501$	5.82359	0.260179
$502$	0.422954	0.0188774
$503$	−5.78591	−0.257981	−0.128990	−	0.991646i	$-0.541174\pi$
$503$	−5.78591	−0.257981	−0.128990	+	0.991646i	$0.541174\pi$
$504$	6.16082	0.274425
$505$	33.6230	1.49621
$506$	23.7259	1.05474
$507$	9.24691	0.410669
$508$	−7.60755	−0.337530
$509$	−3.11498	−0.138069	−0.0690345	−	0.997614i	$-0.521992\pi$
$509$	−3.11498	−0.138069	−0.0690345	+	0.997614i	$0.521992\pi$
$510$	−78.3845	−3.47092
$511$	−48.1086	−2.12820
$512$	−24.5984	−1.08710
$513$	18.5922	0.820864
$514$	26.8965	1.18635
$515$	21.8462	0.962659
$516$	−2.18690	−0.0962730
$517$	16.7839	0.738157
$518$	−80.6611	−3.54404
$519$	−14.5976	−0.640765
$520$	12.6095	0.552964
$521$	−26.5454	−1.16297	−0.581487	−	0.813556i	$-0.697529\pi$
$521$	−26.5454	−1.16297	−0.581487	+	0.813556i	$0.697529\pi$
$522$	−9.57143	−0.418930
$523$	26.3606	1.15267	0.576335	−	0.817214i	$-0.304482\pi$
$523$	26.3606	1.15267	0.576335	+	0.817214i	$0.304482\pi$
$524$	−40.9068	−1.78702
$525$	43.4765	1.89747
$526$	1.40920	0.0614441
$527$	6.53670	0.284743
$528$	17.0953	0.743979
$529$	−13.4624	−0.585320
$530$	−41.0037	−1.78109
$531$	−15.3087	−0.664343
$532$	−51.7037	−2.24164
$533$	0.392933	0.0170198
$534$	46.0598	1.99320
$535$	−5.10755	−0.220819
$536$	3.37137	0.145621
$537$	6.64401	0.286710
$538$	62.6284	2.70010
$539$	32.3173	1.39201
$540$	31.9607	1.37537
$541$	−23.4530	−1.00832	−0.504162	−	0.863609i	$-0.668199\pi$
$541$	−23.4530	−1.00832	−0.504162	+	0.863609i	$0.668199\pi$
$542$	57.5983	2.47406
$543$	−19.7416	−0.847192
$544$	−43.2228	−1.85316
$545$	−24.3068	−1.04119
$546$	50.9853	2.18197
$547$	27.6167	1.18080	0.590402	−	0.807110i	$-0.298969\pi$
$547$	27.6167	1.18080	0.590402	+	0.807110i	$0.298969\pi$
$548$	−30.6218	−1.30810
$549$	−7.93341	−0.338590
$550$	−40.9764	−1.74724
$551$	19.1747	0.816868
$552$	−8.47665	−0.360790
$553$	3.80400	0.161763
$554$	22.1004	0.938954
$555$	61.2236	2.59880
$556$	53.2866	2.25986
$557$	−21.2847	−0.901860	−0.450930	−	0.892559i	$-0.648908\pi$
$557$	−21.2847	−0.901860	−0.450930	+	0.892559i	$0.648908\pi$
$558$	2.87730	0.121806
$559$	−1.18964	−0.0503164
$560$	30.2955	1.28022
$561$	40.4849	1.70927
$562$	50.1550	2.11566
$563$	43.2082	1.82101	0.910504	−	0.413501i	$-0.135694\pi$
$563$	43.2082	1.82101	0.910504	+	0.413501i	$0.135694\pi$
$564$	−25.1205	−1.05776
$565$	9.23678	0.388594
$566$	−6.85465	−0.288122
$567$	44.5494	1.87090
$568$	8.15710	0.342264
$569$	22.5398	0.944916	0.472458	−	0.881353i	$-0.343367\pi$
$569$	22.5398	0.944916	0.472458	+	0.881353i	$0.343367\pi$
$570$	69.1206	2.89514
$571$	14.1563	0.592421	0.296211	−	0.955123i	$-0.404277\pi$
$571$	14.1563	0.592421	0.296211	+	0.955123i	$0.404277\pi$
$572$	−27.2831	−1.14076
$573$	−29.5982	−1.23648
$574$	−1.16446	−0.0486038
$575$	−16.4722	−0.686939
$576$	−13.6618	−0.569242
$577$	15.4920	0.644942	0.322471	−	0.946579i	$-0.395487\pi$
$577$	15.4920	0.644942	0.322471	+	0.946579i	$0.395487\pi$
$578$	−30.1960	−1.25599
$579$	−12.3572	−0.513546
$580$	32.9620	1.36867
$581$	53.8184	2.23276
$582$	0	0
$583$	21.1781	0.877105
$584$	−16.1996	−0.670344
$585$	−10.6564	−0.440590
$586$	23.7499	0.981097
$587$	−29.7163	−1.22652	−0.613261	−	0.789881i	$-0.710143\pi$
$587$	−29.7163	−1.22652	−0.613261	+	0.789881i	$0.710143\pi$
$588$	−48.3692	−1.99471
$589$	−5.76416	−0.237508
$590$	92.8554	3.82280
$591$	30.6612	1.26123
$592$	22.0200	0.905017
$593$	−7.40153	−0.303944	−0.151972	−	0.988385i	$-0.548562\pi$
$593$	−7.40153	−0.303944	−0.151972	+	0.988385i	$0.548562\pi$
$594$	−29.0743	−1.19293
$595$	71.7453	2.94127
$596$	−21.5565	−0.882987
$597$	−34.8881	−1.42787
$598$	−19.3171	−0.789937
$599$	−28.6118	−1.16905	−0.584523	−	0.811377i	$-0.698718\pi$
$599$	−28.6118	−1.16905	−0.584523	+	0.811377i	$0.698718\pi$
$600$	14.6398	0.597668
$601$	33.6887	1.37419	0.687095	−	0.726568i	$-0.258886\pi$
$601$	33.6887	1.37419	0.687095	+	0.726568i	$0.258886\pi$
$602$	3.52553	0.143690
$603$	−2.84918	−0.116027
$604$	33.8798	1.37855
$605$	5.64293	0.229418
$606$	45.7789	1.85964
$607$	31.1618	1.26482	0.632408	−	0.774635i	$-0.282067\pi$
$607$	31.1618	1.26482	0.632408	+	0.774635i	$0.282067\pi$
$608$	38.1145	1.54575
$609$	31.8147	1.28920
$610$	48.1202	1.94833
$611$	−13.6652	−0.552833
$612$	−16.6855	−0.674472
$613$	−31.2640	−1.26274	−0.631371	−	0.775481i	$-0.717507\pi$
$613$	−31.2640	−1.26274	−0.631371	+	0.775481i	$0.717507\pi$
$614$	12.8769	0.519670
$615$	0.883855	0.0356405
$616$	19.3006	0.777642
$617$	28.7359	1.15686	0.578431	−	0.815731i	$-0.303665\pi$
$617$	28.7359	1.15686	0.578431	+	0.815731i	$0.303665\pi$
$618$	29.7443	1.19649
$619$	30.6917	1.23360	0.616802	−	0.787118i	$-0.288428\pi$
$619$	30.6917	1.23360	0.616802	+	0.787118i	$0.288428\pi$
$620$	−9.90882	−0.397948
$621$	−11.6877	−0.469010
$622$	−8.52122	−0.341670
$623$	−42.1586	−1.68905
$624$	−13.9187	−0.557193
$625$	−23.2199	−0.928797
$626$	54.4983	2.17819
$627$	−35.7002	−1.42573
$628$	−1.33297	−0.0531913
$629$	52.1474	2.07925
$630$	31.5806	1.25820
$631$	10.8372	0.431421	0.215710	−	0.976457i	$-0.430793\pi$
$631$	10.8372	0.431421	0.215710	+	0.976457i	$0.430793\pi$
$632$	1.28092	0.0509523
$633$	−47.3047	−1.88019
$634$	26.1540	1.03871
$635$	−9.30882	−0.369409
$636$	−31.6971	−1.25687
$637$	−26.3121	−1.04252
$638$	−29.9853	−1.18713
$639$	−6.89365	−0.272709
$640$	32.9859	1.30388
$641$	22.1959	0.876687	0.438344	−	0.898807i	$-0.355565\pi$
$641$	22.1959	0.876687	0.438344	+	0.898807i	$0.355565\pi$
$642$	−6.95410	−0.274456
$643$	−4.00127	−0.157795	−0.0788974	−	0.996883i	$-0.525140\pi$
$643$	−4.00127	−0.157795	−0.0788974	+	0.996883i	$0.525140\pi$
$644$	32.5027	1.28079
$645$	−2.67596	−0.105366
$646$	58.8737	2.31636
$647$	19.0433	0.748671	0.374335	−	0.927293i	$-0.377871\pi$
$647$	19.0433	0.748671	0.374335	+	0.927293i	$0.377871\pi$
$648$	15.0011	0.589299
$649$	−47.9590	−1.88256
$650$	33.3622	1.30857
$651$	−9.56393	−0.374840
$652$	18.9846	0.743494
$653$	27.3882	1.07178	0.535891	−	0.844287i	$-0.319976\pi$
$653$	27.3882	1.07178	0.535891	+	0.844287i	$0.319976\pi$
$654$	−33.0945	−1.29410
$655$	−50.0548	−1.95580
$656$	0.317892	0.0124116
$657$	13.6905	0.534116
$658$	40.4970	1.57874
$659$	43.1146	1.67951	0.839753	−	0.542969i	$-0.182700\pi$
$659$	43.1146	1.67951	0.839753	+	0.542969i	$0.182700\pi$
$660$	−61.3701	−2.38883
$661$	−18.4012	−0.715724	−0.357862	−	0.933775i	$-0.616494\pi$
$661$	−18.4012	−0.715724	−0.357862	+	0.933775i	$0.616494\pi$
$662$	−47.5182	−1.84685
$663$	−32.9620	−1.28014
$664$	18.1223	0.703280
$665$	−63.2661	−2.45336
$666$	22.9541	0.889452
$667$	−12.0539	−0.466727
$668$	−7.51913	−0.290924
$669$	4.94703	0.191263
$670$	17.2817	0.667652
$671$	−24.8537	−0.959466
$672$	63.2398	2.43953
$673$	38.2676	1.47511	0.737553	−	0.675289i	$-0.235981\pi$
$673$	38.2676	1.47511	0.737553	+	0.675289i	$0.235981\pi$
$674$	34.0278	1.31070
$675$	20.1855	0.776941
$676$	−11.9392	−0.459198
$677$	1.71199	0.0657973	0.0328987	−	0.999459i	$-0.489526\pi$
$677$	1.71199	0.0657973	0.0328987	+	0.999459i	$0.489526\pi$
$678$	12.5762	0.482985
$679$	0	0
$680$	24.1588	0.926448
$681$	19.5484	0.749096
$682$	9.01398	0.345163
$683$	5.06609	0.193848	0.0969242	−	0.995292i	$-0.469100\pi$
$683$	5.06609	0.193848	0.0969242	+	0.995292i	$0.469100\pi$
$684$	14.7135	0.562586
$685$	−37.4697	−1.43164
$686$	17.6549	0.674066
$687$	−55.1698	−2.10486
$688$	−0.962449	−0.0366930
$689$	−17.2427	−0.656896
$690$	−43.4516	−1.65417
$691$	−28.9293	−1.10052	−0.550262	−	0.834992i	$-0.685472\pi$
$691$	−28.9293	−1.10052	−0.550262	+	0.834992i	$0.685472\pi$
$692$	18.8478	0.716485
$693$	−16.3111	−0.619608
$694$	−69.4950	−2.63799
$695$	65.2031	2.47329
$696$	10.7130	0.406074
$697$	0.752827	0.0285153
$698$	21.8243	0.826063
$699$	−24.1695	−0.914173
$700$	−56.1347	−2.12169
$701$	7.45330	0.281507	0.140754	−	0.990045i	$-0.455047\pi$
$701$	7.45330	0.281507	0.140754	+	0.990045i	$0.455047\pi$
$702$	23.6717	0.893433
$703$	−45.9844	−1.73433
$704$	−42.7996	−1.61307
$705$	−30.7381	−1.15767
$706$	14.3622	0.540529
$707$	−41.9014	−1.57587
$708$	71.7801	2.69766
$709$	−42.5897	−1.59949	−0.799745	−	0.600339i	$-0.795032\pi$
$709$	−42.5897	−1.59949	−0.799745	+	0.600339i	$0.795032\pi$
$710$	41.8136	1.56924
$711$	−1.08252	−0.0405977
$712$	−14.1961	−0.532020
$713$	3.62355	0.135703
$714$	97.6836	3.65572
$715$	−33.3844	−1.24850
$716$	−8.57842	−0.320591
$717$	−41.4432	−1.54773
$718$	−36.6383	−1.36733
$719$	18.3830	0.685571	0.342785	−	0.939414i	$-0.388630\pi$
$719$	18.3830	0.685571	0.342785	+	0.939414i	$0.388630\pi$
$720$	−8.62132	−0.321298
$721$	−27.2250	−1.01391
$722$	−11.0454	−0.411066
$723$	8.97828	0.333906
$724$	25.4894	0.947305
$725$	20.8179	0.773159
$726$	7.68303	0.285144
$727$	−9.78506	−0.362908	−0.181454	−	0.983399i	$-0.558080\pi$
$727$	−9.78506	−0.362908	−0.181454	+	0.983399i	$0.558080\pi$
$728$	−15.7141	−0.582404
$729$	10.4234	0.386052
$730$	−83.0398	−3.07344
$731$	−2.27925	−0.0843013
$732$	37.1984	1.37489
$733$	−8.06558	−0.297909	−0.148954	−	0.988844i	$-0.547591\pi$
$733$	−8.06558	−0.297909	−0.148954	+	0.988844i	$0.547591\pi$
$734$	−17.3856	−0.641713
$735$	−59.1860	−2.18311
$736$	−23.9601	−0.883181
$737$	−8.92587	−0.328789
$738$	0.331377	0.0121981
$739$	−6.23030	−0.229185	−0.114593	−	0.993413i	$-0.536556\pi$
$739$	−6.23030	−0.229185	−0.114593	+	0.993413i	$0.536556\pi$
$740$	−79.0490	−2.90590
$741$	29.0664	1.06778
$742$	51.0993	1.87591
$743$	−0.00327381	−0.000120105 0	−6.00523e−5	−	1.00000i	$-0.500019\pi$
$743$	−0.00327381	−0.000120105 0	−6.00523e−5		1.00000i	$0.500019\pi$
$744$	−3.22046	−0.118068
$745$	−26.3771	−0.966382
$746$	11.5462	0.422737
$747$	−15.3153	−0.560358
$748$	−52.2722	−1.91126
$749$	6.36509	0.232575
$750$	4.69563	0.171460
$751$	13.8671	0.506017	0.253009	−	0.967464i	$-0.418580\pi$
$751$	13.8671	0.506017	0.253009	+	0.967464i	$0.418580\pi$
$752$	−11.0554	−0.403151
$753$	0.400074	0.0145795
$754$	24.4134	0.889083
$755$	41.4563	1.50875
$756$	−39.8297	−1.44859
$757$	11.8219	0.429674	0.214837	−	0.976650i	$-0.431078\pi$
$757$	11.8219	0.429674	0.214837	+	0.976650i	$0.431078\pi$
$758$	34.0648	1.23729
$759$	22.4424	0.814607
$760$	−21.3036	−0.772763
$761$	−30.9877	−1.12330	−0.561652	−	0.827374i	$-0.689834\pi$
$761$	−30.9877	−1.12330	−0.561652	+	0.827374i	$0.689834\pi$
$762$	−12.6743	−0.459140
$763$	30.2914	1.09662
$764$	38.2158	1.38260
$765$	−20.4169	−0.738173
$766$	20.6615	0.746532
$767$	39.0473	1.40992
$768$	−3.85537	−0.139119
$769$	1.44120	0.0519709	0.0259854	−	0.999662i	$-0.491728\pi$
$769$	1.44120	0.0519709	0.0259854	+	0.999662i	$0.491728\pi$
$770$	98.9354	3.56538
$771$	25.4414	0.916251
$772$	15.9550	0.574232
$773$	48.0864	1.72955	0.864775	−	0.502160i	$-0.167461\pi$
$773$	48.0864	1.72955	0.864775	+	0.502160i	$0.167461\pi$
$774$	−1.00327	−0.0360619
$775$	−6.25815	−0.224799
$776$	0	0
$777$	−76.2975	−2.73716
$778$	5.03582	0.180543
$779$	−0.663854	−0.0237850
$780$	49.9663	1.78908
$781$	−21.5963	−0.772778
$782$	−37.0101	−1.32348
$783$	14.7711	0.527877
$784$	−21.2872	−0.760256
$785$	−1.63106	−0.0582150
$786$	−68.1512	−2.43087
$787$	−21.3107	−0.759646	−0.379823	−	0.925059i	$-0.624015\pi$
$787$	−21.3107	−0.759646	−0.379823	+	0.925059i	$0.624015\pi$
$788$	−39.5883	−1.41028
$789$	1.33297	0.0474549
$790$	6.56605	0.233610
$791$	−11.5110	−0.409283
$792$	−5.49244	−0.195165
$793$	20.2354	0.718579
$794$	8.01059	0.284285
$795$	−38.7855	−1.37558
$796$	45.0458	1.59661
$797$	28.7917	1.01986	0.509928	−	0.860217i	$-0.329672\pi$
$797$	28.7917	1.01986	0.509928	+	0.860217i	$0.329672\pi$
$798$	−86.1389	−3.04928
$799$	−26.1813	−0.926228
$800$	41.3809	1.46304
$801$	11.9972	0.423902
$802$	−43.9656	−1.55248
$803$	42.8893	1.51353
$804$	13.3593	0.471147
$805$	39.7713	1.40175
$806$	−7.33900	−0.258505
$807$	59.2403	2.08536
$808$	−14.1095	−0.496369
$809$	34.6574	1.21849	0.609245	−	0.792982i	$-0.291472\pi$
$809$	34.6574	1.21849	0.609245	+	0.792982i	$0.291472\pi$
$810$	76.8962	2.70186
$811$	−23.6610	−0.830848	−0.415424	−	0.909628i	$-0.636367\pi$
$811$	−23.6610	−0.830848	−0.415424	+	0.909628i	$0.636367\pi$
$812$	−41.0776	−1.44154
$813$	54.4824	1.91078
$814$	71.9102	2.52045
$815$	23.2301	0.813715
$816$	−26.6671	−0.933534
$817$	2.00988	0.0703169
$818$	53.9080	1.88485
$819$	13.2802	0.464047
$820$	−1.14119	−0.0398521
$821$	52.2905	1.82495	0.912475	−	0.409132i	$-0.134168\pi$
$821$	52.2905	1.82495	0.912475	+	0.409132i	$0.134168\pi$
$822$	−51.0162	−1.77940
$823$	−39.9011	−1.39086	−0.695432	−	0.718592i	$-0.744787\pi$
$823$	−39.9011	−1.39086	−0.695432	+	0.718592i	$0.744787\pi$
$824$	−9.16747	−0.319364
$825$	−38.7597	−1.34944
$826$	−115.718	−4.02633
$827$	10.2760	0.357331	0.178665	−	0.983910i	$-0.442822\pi$
$827$	10.2760	0.357331	0.178665	+	0.983910i	$0.442822\pi$
$828$	−9.24944	−0.321440
$829$	−42.2122	−1.46609	−0.733044	−	0.680181i	$-0.761901\pi$
$829$	−42.2122	−1.46609	−0.733044	+	0.680181i	$0.761901\pi$
$830$	92.8954	3.22444
$831$	20.9048	0.725179
$832$	34.8466	1.20809
$833$	−50.4119	−1.74667
$834$	88.7761	3.07407
$835$	−9.20063	−0.318401
$836$	46.0944	1.59421
$837$	−4.44040	−0.153483
$838$	59.8390	2.06710
$839$	0.789828	0.0272679	0.0136339	−	0.999907i	$-0.495660\pi$
$839$	0.789828	0.0272679	0.0136339	+	0.999907i	$0.495660\pi$
$840$	−35.3471	−1.21959
$841$	−13.7661	−0.474693
$842$	−15.6084	−0.537901
$843$	47.4417	1.63398
$844$	61.0776	2.10238
$845$	−14.6091	−0.502568
$846$	−11.5244	−0.396217
$847$	−7.03228	−0.241632
$848$	−13.9498	−0.479038
$849$	−6.48383	−0.222525
$850$	63.9192	2.19241
$851$	28.9074	0.990932
$852$	32.3232	1.10737
$853$	53.4439	1.82988	0.914942	−	0.403585i	$-0.132236\pi$
$853$	53.4439	1.82988	0.914942	+	0.403585i	$0.132236\pi$
$854$	−59.9680	−2.05206
$855$	18.0039	0.615721
$856$	2.14332	0.0732570
$857$	1.80725	0.0617346	0.0308673	−	0.999523i	$-0.490173\pi$
$857$	1.80725	0.0617346	0.0308673	+	0.999523i	$0.490173\pi$
$858$	−45.4539	−1.55177
$859$	39.3759	1.34349	0.671744	−	0.740784i	$-0.265546\pi$
$859$	39.3759	1.34349	0.671744	+	0.740784i	$0.265546\pi$
$860$	3.45507	0.117817
$861$	−1.10147	−0.0375380
$862$	−25.2950	−0.861551
$863$	17.2598	0.587531	0.293766	−	0.955877i	$-0.405091\pi$
$863$	17.2598	0.587531	0.293766	+	0.955877i	$0.405091\pi$
$864$	29.3614	0.998894
$865$	23.0627	0.784154
$866$	46.3877	1.57632
$867$	−28.5625	−0.970032
$868$	12.3485	0.419135
$869$	−3.39131	−0.115042
$870$	54.9150	1.86179
$871$	7.26726	0.246242
$872$	10.2000	0.345416
$873$	0	0
$874$	32.6360	1.10393
$875$	−4.29792	−0.145296
$876$	−64.1923	−2.16886
$877$	26.6773	0.900828	0.450414	−	0.892820i	$-0.351276\pi$
$877$	26.6773	0.900828	0.450414	+	0.892820i	$0.351276\pi$
$878$	−54.0035	−1.82253
$879$	22.4651	0.757727
$880$	−27.0088	−0.910466
$881$	23.2719	0.784051	0.392025	−	0.919954i	$-0.371775\pi$
$881$	23.2719	0.784051	0.392025	+	0.919954i	$0.371775\pi$
$882$	−22.1901	−0.747180
$883$	21.4634	0.722301	0.361150	−	0.932508i	$-0.382384\pi$
$883$	21.4634	0.722301	0.361150	+	0.932508i	$0.382384\pi$
$884$	42.5589	1.43141
$885$	87.8322	2.95245
$886$	−64.1469	−2.15506
$887$	−31.3387	−1.05225	−0.526126	−	0.850407i	$-0.676356\pi$
$887$	−31.3387	−1.05225	−0.526126	+	0.850407i	$0.676356\pi$
$888$	−25.6917	−0.862156
$889$	11.6008	0.389077
$890$	−72.7695	−2.43924
$891$	−39.7163	−1.33054
$892$	−6.38736	−0.213865
$893$	23.0871	0.772580
$894$	−35.9133	−1.20112
$895$	−10.4968	−0.350870
$896$	−41.1075	−1.37330
$897$	−18.2721	−0.610089
$898$	−33.1117	−1.10495
$899$	−4.57952	−0.152736
$900$	15.9745	0.532483
$901$	−33.0357	−1.10058
$902$	1.03813	0.0345660
$903$	3.33481	0.110975
$904$	−3.87609	−0.128917
$905$	31.1895	1.03677
$906$	56.4442	1.87523
$907$	3.21745	0.106834	0.0534168	−	0.998572i	$-0.482989\pi$
$907$	3.21745	0.106834	0.0534168	+	0.998572i	$0.482989\pi$
$908$	−25.2399	−0.837617
$909$	11.9241	0.395496
$910$	−80.5512	−2.67025
$911$	15.0217	0.497690	0.248845	−	0.968543i	$-0.419949\pi$
$911$	15.0217	0.497690	0.248845	+	0.968543i	$0.419949\pi$
$912$	23.5154	0.778674
$913$	−47.9797	−1.58789
$914$	89.4555	2.95893
$915$	45.5171	1.50475
$916$	71.2326	2.35359
$917$	62.3789	2.05993
$918$	45.3531	1.49688
$919$	27.9727	0.922733	0.461366	−	0.887210i	$-0.347359\pi$
$919$	27.9727	0.922733	0.461366	+	0.887210i	$0.347359\pi$
$920$	13.3922	0.441527
$921$	12.1803	0.401355
$922$	−24.6402	−0.811482
$923$	17.5833	0.578762
$924$	76.4801	2.51601
$925$	−49.9252	−1.64153
$926$	−30.8805	−1.01480
$927$	7.74753	0.254462
$928$	30.2813	0.994031
$929$	−1.74910	−0.0573863	−0.0286931	−	0.999588i	$-0.509135\pi$
$929$	−1.74910	−0.0573863	−0.0286931	+	0.999588i	$0.509135\pi$
$930$	−16.5082	−0.541325
$931$	44.4540	1.45692
$932$	31.2064	1.02220
$933$	−8.06025	−0.263881
$934$	−46.9758	−1.53709
$935$	−63.9617	−2.09177
$936$	4.47183	0.146166
$937$	50.2385	1.64122	0.820610	−	0.571488i	$-0.193634\pi$
$937$	50.2385	1.64122	0.820610	+	0.571488i	$0.193634\pi$
$938$	−21.5367	−0.703198
$939$	51.5501	1.68227
$940$	39.6876	1.29447
$941$	−7.80880	−0.254560	−0.127280	−	0.991867i	$-0.540625\pi$
$941$	−7.80880	−0.254560	−0.127280	+	0.991867i	$0.540625\pi$
$942$	−2.22074	−0.0723557
$943$	0.417322	0.0135899
$944$	31.5902	1.02817
$945$	−48.7368	−1.58541
$946$	−3.14305	−0.102189
$947$	24.4392	0.794168	0.397084	−	0.917782i	$-0.370022\pi$
$947$	24.4392	0.794168	0.397084	+	0.917782i	$0.370022\pi$
$948$	5.07576	0.164853
$949$	−34.9196	−1.13354
$950$	−56.3649	−1.82872
$951$	24.7391	0.802222
$952$	−30.1070	−0.975773
$953$	−21.7005	−0.702949	−0.351475	−	0.936197i	$-0.614320\pi$
$953$	−21.7005	−0.702949	−0.351475	+	0.936197i	$0.614320\pi$
$954$	−14.5415	−0.470799
$955$	46.7619	1.51318
$956$	53.5095	1.73062
$957$	−28.3631	−0.916850
$958$	−8.33456	−0.269277
$959$	46.6952	1.50787
$960$	78.3832	2.52981
$961$	−29.6233	−0.955591
$962$	−58.5478	−1.88766
$963$	−1.81134	−0.0583696
$964$	−11.5923	−0.373364
$965$	19.5230	0.628466
$966$	54.1499	1.74225
$967$	35.4982	1.14154	0.570772	−	0.821108i	$-0.306644\pi$
$967$	35.4982	1.14154	0.570772	+	0.821108i	$0.306644\pi$
$968$	−2.36798	−0.0761098
$969$	55.6888	1.78898
$970$	0	0
$971$	14.4422	0.463472	0.231736	−	0.972779i	$-0.425559\pi$
$971$	14.4422	0.463472	0.231736	+	0.972779i	$0.425559\pi$
$972$	29.6163	0.949942
$973$	−81.2568	−2.60497
$974$	21.2550	0.681055
$975$	31.5574	1.01064
$976$	16.3709	0.524020
$977$	7.75433	0.248083	0.124042	−	0.992277i	$-0.460414\pi$
$977$	7.75433	0.248083	0.124042	+	0.992277i	$0.460414\pi$
$978$	31.6285	1.01137
$979$	37.5848	1.20122
$980$	76.4181	2.44109
$981$	−8.62015	−0.275220
$982$	14.9418	0.476811
$983$	−8.01225	−0.255551	−0.127776	−	0.991803i	$-0.540784\pi$
$983$	−8.01225	−0.255551	−0.127776	+	0.991803i	$0.540784\pi$
$984$	−0.370898	−0.0118238
$985$	−48.4414	−1.54347
$986$	46.7741	1.48959
$987$	38.3062	1.21930
$988$	−37.5291	−1.19396
$989$	−1.26348	−0.0401764
$990$	−28.1544	−0.894807
$991$	−58.7603	−1.86658	−0.933291	−	0.359121i	$-0.883077\pi$
$991$	−58.7603	−1.86658	−0.933291	+	0.359121i	$0.883077\pi$
$992$	−9.10296	−0.289019
$993$	−44.9476	−1.42637
$994$	−52.1086	−1.65278
$995$	55.1194	1.74740
$996$	71.8110	2.27542
$997$	58.6776	1.85834	0.929170	−	0.369653i	$-0.120524\pi$
$997$	58.6776	1.85834	0.929170	+	0.369653i	$0.120524\pi$
$998$	41.4308	1.31147
$999$	−35.4239	−1.12076

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9409.2.a.n.1.6		56
97.42	odd	32		97.2.h.a.18.1	✓	56
97.67	odd	32		97.2.h.a.27.1	yes	56
97.96	even	2	inner	9409.2.a.n.1.5		56
291.164	even	32		873.2.bh.b.415.7		56
291.236	even	32		873.2.bh.b.406.7		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
97.2.h.a.18.1	✓	56	97.42	odd	32
97.2.h.a.27.1	yes	56	97.67	odd	32
873.2.bh.b.406.7		56	291.236	even	32
873.2.bh.b.415.7		56	291.164	even	32
9409.2.a.n.1.5		56	97.96	even	2	inner
9409.2.a.n.1.6		56	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9409 = 97^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9409.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15107 q^{2} -2.03471 q^{3} +2.62711 q^{4} +3.21461 q^{5} +4.37680 q^{6} -4.00609 q^{7} -1.34897 q^{8} +1.14003 q^{9} -6.91487 q^{10} +3.57147 q^{11} -5.34541 q^{12} -2.90782 q^{13} +8.61739 q^{14} -6.54079 q^{15} -2.35250 q^{16} -5.57114 q^{17} -2.45229 q^{18} +4.91272 q^{19} +8.44516 q^{20} +8.15121 q^{21} -7.68249 q^{22} -3.08831 q^{23} +2.74476 q^{24} +5.33374 q^{25} +6.25493 q^{26} +3.78449 q^{27} -10.5245 q^{28} +3.90306 q^{29} +14.0697 q^{30} -1.17331 q^{31} +7.75833 q^{32} -7.26689 q^{33} +11.9839 q^{34} -12.8780 q^{35} +2.99499 q^{36} -9.36027 q^{37} -10.5676 q^{38} +5.91656 q^{39} -4.33642 q^{40} -0.135130 q^{41} -17.5339 q^{42} +0.409118 q^{43} +9.38266 q^{44} +3.66475 q^{45} +6.64317 q^{46} +4.69945 q^{47} +4.78664 q^{48} +9.04875 q^{49} -11.4733 q^{50} +11.3356 q^{51} -7.63917 q^{52} +5.92979 q^{53} -8.14072 q^{54} +11.4809 q^{55} +5.40409 q^{56} -9.99594 q^{57} -8.39578 q^{58} -13.4284 q^{59} -17.1834 q^{60} -6.95895 q^{61} +2.52388 q^{62} -4.56706 q^{63} -11.9837 q^{64} -9.34751 q^{65} +15.6316 q^{66} -2.49921 q^{67} -14.6360 q^{68} +6.28380 q^{69} +27.7016 q^{70} -6.04691 q^{71} -1.53786 q^{72} +12.0089 q^{73} +20.1346 q^{74} -10.8526 q^{75} +12.9063 q^{76} -14.3076 q^{77} -12.7269 q^{78} -0.949556 q^{79} -7.56237 q^{80} -11.1204 q^{81} +0.290674 q^{82} -13.4341 q^{83} +21.4142 q^{84} -17.9091 q^{85} -0.880043 q^{86} -7.94159 q^{87} -4.81781 q^{88} +10.5236 q^{89} -7.88315 q^{90} +11.6490 q^{91} -8.11334 q^{92} +2.38735 q^{93} -10.1089 q^{94} +15.7925 q^{95} -15.7859 q^{96} -19.4645 q^{98} +4.07158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{2} + 16 q^{3} + 40 q^{4} + 16 q^{6} + 24 q^{8} + 40 q^{9} + 32 q^{11} + 48 q^{12} + 8 q^{16} + 24 q^{18} + 16 q^{22} + 8 q^{25} + 64 q^{27} + 48 q^{31} + 56 q^{32} + 80 q^{33} + 48 q^{35} + 8 q^{36}+ \cdots - 16 q^{99}+O(q^{100}) \) 56 * q + 8 * q^2 + 16 * q^3 + 40 * q^4 + 16 * q^6 + 24 * q^8 + 40 * q^9 + 32 * q^11 + 48 * q^12 + 8 * q^16 + 24 * q^18 + 16 * q^22 + 8 * q^25 + 64 * q^27 + 48 * q^31 + 56 * q^32 + 80 * q^33 + 48 * q^35 + 8 * q^36 + 32 * q^43 + 80 * q^44 + 16 * q^47 + 112 * q^48 + 8 * q^49 + 24 * q^50 + 48 * q^53 + 96 * q^54 + 32 * q^61 + 80 * q^62 - 56 * q^64 + 128 * q^65 + 192 * q^66 + 80 * q^70 - 88 * q^72 + 32 * q^73 + 80 * q^75 + 48 * q^79 + 72 * q^81 + 144 * q^86 + 176 * q^88 + 96 * q^89 + 64 * q^91 + 64 * q^94 + 176 * q^95 + 32 * q^96 - 136 * q^98 - 16 * q^99

Introduction

L-functions

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