LMFDB - Embedded newform 8281.2.a.cw.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] 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("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,1,0,23,13,14,0,0,26,-5,1,-5,0,0,-5,17,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8281.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.877203 q^{2} +0.755024 q^{3} -1.23052 q^{4} -0.265839 q^{5} -0.662309 q^{6} +2.83382 q^{8} -2.42994 q^{9} +0.233195 q^{10} -0.455666 q^{11} -0.929068 q^{12} -0.200715 q^{15} -0.0248023 q^{16} +3.65702 q^{17} +2.13155 q^{18} -3.55974 q^{19} +0.327119 q^{20} +0.399712 q^{22} -3.07952 q^{23} +2.13960 q^{24} -4.92933 q^{25} -4.09973 q^{27} -0.612999 q^{29} +0.176068 q^{30} -10.7090 q^{31} -5.64588 q^{32} -0.344039 q^{33} -3.20795 q^{34} +2.99008 q^{36} +2.18786 q^{37} +3.12262 q^{38} -0.753339 q^{40} +7.13669 q^{41} +6.76740 q^{43} +0.560704 q^{44} +0.645973 q^{45} +2.70137 q^{46} -7.47771 q^{47} -0.0187263 q^{48} +4.32402 q^{50} +2.76114 q^{51} -7.79148 q^{53} +3.59630 q^{54} +0.121134 q^{55} -2.68769 q^{57} +0.537725 q^{58} +11.6247 q^{59} +0.246983 q^{60} +1.59475 q^{61} +9.39396 q^{62} +5.00218 q^{64} +0.301792 q^{66} +9.39914 q^{67} -4.50002 q^{68} -2.32511 q^{69} +6.89894 q^{71} -6.88600 q^{72} +2.85100 q^{73} -1.91920 q^{74} -3.72176 q^{75} +4.38032 q^{76} +8.73448 q^{79} +0.00659341 q^{80} +4.19442 q^{81} -6.26033 q^{82} -7.51728 q^{83} -0.972179 q^{85} -5.93638 q^{86} -0.462829 q^{87} -1.29127 q^{88} +8.42769 q^{89} -0.566649 q^{90} +3.78940 q^{92} -8.08554 q^{93} +6.55947 q^{94} +0.946319 q^{95} -4.26277 q^{96} -1.61319 q^{97} +1.10724 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) $ 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * 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 $

$2$	−0.877203	−0.620276	−0.310138	−	0.950692i	$-0.600375\pi$
$2$	−0.877203	−0.620276	−0.310138	+	0.950692i	$0.600375\pi$
$3$	0.755024	0.435913	0.217957	−	0.975958i	$-0.430061\pi$
$3$	0.755024	0.435913	0.217957	+	0.975958i	$0.430061\pi$
$4$	−1.23052	−0.615258
$5$	−0.265839	−0.118887	−0.0594434	−	0.998232i	$-0.518933\pi$
$5$	−0.265839	−0.118887	−0.0594434	+	0.998232i	$0.518933\pi$
$6$	−0.662309	−0.270386
$7$	0	0
$7$	0	0
$8$	2.83382	1.00191
$9$	−2.42994	−0.809980
$10$	0.233195	0.0737427
$11$	−0.455666	−0.137388	−0.0686942	−	0.997638i	$-0.521883\pi$
$11$	−0.455666	−0.137388	−0.0686942	+	0.997638i	$0.521883\pi$
$12$	−0.929068	−0.268199
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.200715	−0.0518243
$16$	−0.0248023	−0.00620056
$17$	3.65702	0.886958	0.443479	−	0.896285i	$-0.353744\pi$
$17$	3.65702	0.886958	0.443479	+	0.896285i	$0.353744\pi$
$18$	2.13155	0.502411
$19$	−3.55974	−0.816661	−0.408331	−	0.912834i	$-0.633889\pi$
$19$	−3.55974	−0.816661	−0.408331	+	0.912834i	$0.633889\pi$
$20$	0.327119	0.0731460
$21$	0	0
$22$	0.399712	0.0852188
$23$	−3.07952	−0.642125	−0.321063	−	0.947058i	$-0.604040\pi$
$23$	−3.07952	−0.642125	−0.321063	+	0.947058i	$0.604040\pi$
$24$	2.13960	0.436744
$25$	−4.92933	−0.985866
$26$	0	0
$27$	−4.09973	−0.788994
$28$	0	0
$29$	−0.612999	−0.113831	−0.0569155	−	0.998379i	$-0.518127\pi$
$29$	−0.612999	−0.113831	−0.0569155	+	0.998379i	$0.518127\pi$
$30$	0.176068	0.0321454
$31$	−10.7090	−1.92339	−0.961696	−	0.274119i	$-0.911614\pi$
$31$	−10.7090	−1.92339	−0.961696	+	0.274119i	$0.911614\pi$
$32$	−5.64588	−0.998060
$33$	−0.344039	−0.0598894
$34$	−3.20795	−0.550159
$35$	0	0
$36$	2.99008	0.498346
$37$	2.18786	0.359682	0.179841	−	0.983696i	$-0.442442\pi$
$37$	2.18786	0.359682	0.179841	+	0.983696i	$0.442442\pi$
$38$	3.12262	0.506556
$39$	0	0
$40$	−0.753339	−0.119113
$41$	7.13669	1.11456	0.557282	−	0.830323i	$-0.311844\pi$
$41$	7.13669	1.11456	0.557282	+	0.830323i	$0.311844\pi$
$42$	0	0
$43$	6.76740	1.03202	0.516009	−	0.856583i	$-0.327417\pi$
$43$	6.76740	1.03202	0.516009	+	0.856583i	$0.327417\pi$
$44$	0.560704	0.0845293
$45$	0.645973	0.0962959
$46$	2.70137	0.398295
$47$	−7.47771	−1.09074	−0.545368	−	0.838197i	$-0.683610\pi$
$47$	−7.47771	−1.09074	−0.545368	+	0.838197i	$0.683610\pi$
$48$	−0.0187263	−0.00270291
$49$	0	0
$50$	4.32402	0.611509
$51$	2.76114	0.386637
$52$	0	0
$53$	−7.79148	−1.07024	−0.535121	−	0.844776i	$-0.679734\pi$
$53$	−7.79148	−1.07024	−0.535121	+	0.844776i	$0.679734\pi$
$54$	3.59630	0.489394
$55$	0.121134	0.0163337
$56$	0	0
$57$	−2.68769	−0.355993
$58$	0.537725	0.0706067
$59$	11.6247	1.51341	0.756703	−	0.653758i	$-0.226809\pi$
$59$	11.6247	1.51341	0.756703	+	0.653758i	$0.226809\pi$
$60$	0.246983	0.0318853
$61$	1.59475	0.204186	0.102093	−	0.994775i	$-0.467446\pi$
$61$	1.59475	0.204186	0.102093	+	0.994775i	$0.467446\pi$
$62$	9.39396	1.19303
$63$	0	0
$64$	5.00218	0.625273
$65$	0	0
$66$	0.301792	0.0371480
$67$	9.39914	1.14829	0.574144	−	0.818755i	$-0.305335\pi$
$67$	9.39914	1.14829	0.574144	+	0.818755i	$0.305335\pi$
$68$	−4.50002	−0.545708
$69$	−2.32511	−0.279911
$70$	0	0
$71$	6.89894	0.818754	0.409377	−	0.912365i	$-0.365746\pi$
$71$	6.89894	0.818754	0.409377	+	0.912365i	$0.365746\pi$
$72$	−6.88600	−0.811523
$73$	2.85100	0.333684	0.166842	−	0.985984i	$-0.446643\pi$
$73$	2.85100	0.333684	0.166842	+	0.985984i	$0.446643\pi$
$74$	−1.91920	−0.223102
$75$	−3.72176	−0.429752
$76$	4.38032	0.502457
$77$	0	0
$78$	0	0
$79$	8.73448	0.982706	0.491353	−	0.870960i	$-0.336502\pi$
$79$	8.73448	0.982706	0.491353	+	0.870960i	$0.336502\pi$
$80$	0.00659341	0.000737165 0
$81$	4.19442	0.466047
$82$	−6.26033	−0.691337
$83$	−7.51728	−0.825129	−0.412564	−	0.910928i	$-0.635367\pi$
$83$	−7.51728	−0.825129	−0.412564	+	0.910928i	$0.635367\pi$
$84$	0	0
$85$	−0.972179	−0.105448
$86$	−5.93638	−0.640136
$87$	−0.462829	−0.0496205
$88$	−1.29127	−0.137650
$89$	8.42769	0.893333	0.446666	−	0.894701i	$-0.352611\pi$
$89$	8.42769	0.893333	0.446666	+	0.894701i	$0.352611\pi$
$90$	−0.566649	−0.0597301
$91$	0	0
$92$	3.78940	0.395072
$93$	−8.08554	−0.838432
$94$	6.55947	0.676557
$95$	0.946319	0.0970903
$96$	−4.26277	−0.435067
$97$	−1.61319	−0.163795	−0.0818975	−	0.996641i	$-0.526098\pi$
$97$	−1.61319	−0.163795	−0.0818975	+	0.996641i	$0.526098\pi$
$98$	0	0
$99$	1.10724	0.111282
$100$	6.06561	0.606561
$101$	−7.29769	−0.726147	−0.363074	−	0.931760i	$-0.618273\pi$
$101$	−7.29769	−0.726147	−0.363074	+	0.931760i	$0.618273\pi$
$102$	−2.42208	−0.239822
$103$	8.16853	0.804869	0.402434	−	0.915449i	$-0.368164\pi$
$103$	8.16853	0.804869	0.402434	+	0.915449i	$0.368164\pi$
$104$	0	0
$105$	0	0
$106$	6.83471	0.663845
$107$	−16.4079	−1.58621	−0.793104	−	0.609087i	$-0.791536\pi$
$107$	−16.4079	−1.58621	−0.793104	+	0.609087i	$0.791536\pi$
$108$	5.04478	0.485434
$109$	5.96790	0.571621	0.285811	−	0.958286i	$-0.407737\pi$
$109$	5.96790	0.571621	0.285811	+	0.958286i	$0.407737\pi$
$110$	−0.106259	−0.0101314
$111$	1.65189	0.156790
$112$	0	0
$113$	−16.9748	−1.59685	−0.798426	−	0.602093i	$-0.794334\pi$
$113$	−16.9748	−1.59685	−0.798426	+	0.602093i	$0.794334\pi$
$114$	2.35765	0.220814
$115$	0.818658	0.0763402
$116$	0.754305	0.0700354
$117$	0	0
$118$	−10.1972	−0.938730
$119$	0	0
$120$	−0.568789	−0.0519231
$121$	−10.7924	−0.981124
$122$	−1.39892	−0.126652
$123$	5.38837	0.485853
$124$	13.1776	1.18338
$125$	2.63960	0.236093
$126$	0	0
$127$	13.3247	1.18238	0.591188	−	0.806534i	$-0.298659\pi$
$127$	13.3247	1.18238	0.591188	+	0.806534i	$0.298659\pi$
$128$	6.90382	0.610218
$129$	5.10955	0.449870
$130$	0	0
$131$	−15.2631	−1.33355	−0.666773	−	0.745260i	$-0.732325\pi$
$131$	−15.2631	−1.33355	−0.666773	+	0.745260i	$0.732325\pi$
$132$	0.423345	0.0368474
$133$	0	0
$134$	−8.24495	−0.712255
$135$	1.08987	0.0938010
$136$	10.3633	0.888648
$137$	13.6118	1.16293	0.581466	−	0.813571i	$-0.302479\pi$
$137$	13.6118	1.16293	0.581466	+	0.813571i	$0.302479\pi$
$138$	2.03960	0.173622
$139$	−14.3332	−1.21573	−0.607863	−	0.794042i	$-0.707973\pi$
$139$	−14.3332	−1.21573	−0.607863	+	0.794042i	$0.707973\pi$
$140$	0	0
$141$	−5.64585	−0.475466
$142$	−6.05177	−0.507854
$143$	0	0
$144$	0.0602680	0.00502233
$145$	0.162959	0.0135330
$146$	−2.50090	−0.206976
$147$	0	0
$148$	−2.69220	−0.221297
$149$	−19.6559	−1.61027	−0.805136	−	0.593090i	$-0.797908\pi$
$149$	−19.6559	−1.61027	−0.805136	+	0.593090i	$0.797908\pi$
$150$	3.26474	0.266565
$151$	−1.93740	−0.157664	−0.0788319	−	0.996888i	$-0.525119\pi$
$151$	−1.93740	−0.157664	−0.0788319	+	0.996888i	$0.525119\pi$
$152$	−10.0877	−0.818218
$153$	−8.88634	−0.718418
$154$	0	0
$155$	2.84687	0.228666
$156$	0	0
$157$	−24.7415	−1.97458	−0.987291	−	0.158920i	$-0.949199\pi$
$157$	−24.7415	−1.97458	−0.987291	+	0.158920i	$0.949199\pi$
$158$	−7.66191	−0.609549
$159$	−5.88275	−0.466532
$160$	1.50089	0.118656
$161$	0	0
$162$	−3.67936	−0.289078
$163$	−6.74233	−0.528100	−0.264050	−	0.964509i	$-0.585058\pi$
$163$	−6.74233	−0.528100	−0.264050	+	0.964509i	$0.585058\pi$
$164$	−8.78181	−0.685744
$165$	0.0914589	0.00712006
$166$	6.59418	0.511808
$167$	2.06354	0.159681	0.0798407	−	0.996808i	$-0.474559\pi$
$167$	2.06354	0.159681	0.0798407	+	0.996808i	$0.474559\pi$
$168$	0	0
$169$	0	0
$170$	0.852798	0.0654066
$171$	8.64996	0.661479
$172$	−8.32738	−0.634957
$173$	13.5227	1.02811	0.514057	−	0.857756i	$-0.328142\pi$
$173$	13.5227	1.02811	0.514057	+	0.857756i	$0.328142\pi$
$174$	0.405995	0.0307784
$175$	0	0
$176$	0.0113015	0.000851886 0
$177$	8.77692	0.659714
$178$	−7.39279	−0.554113
$179$	−8.62973	−0.645017	−0.322508	−	0.946567i	$-0.604526\pi$
$179$	−8.62973	−0.645017	−0.322508	+	0.946567i	$0.604526\pi$
$180$	−0.794879	−0.0592468
$181$	23.4579	1.74361	0.871806	−	0.489852i	$-0.162949\pi$
$181$	23.4579	1.74361	0.871806	+	0.489852i	$0.162949\pi$
$182$	0	0
$183$	1.20407	0.0890075
$184$	−8.72681	−0.643349
$185$	−0.581619	−0.0427615
$186$	7.09266	0.520059
$187$	−1.66638	−0.121858
$188$	9.20143	0.671084
$189$	0	0
$190$	−0.830114	−0.0602228
$191$	17.1685	1.24227	0.621134	−	0.783705i	$-0.286672\pi$
$191$	17.1685	1.24227	0.621134	+	0.783705i	$0.286672\pi$
$192$	3.77677	0.272565
$193$	16.1895	1.16534	0.582672	−	0.812707i	$-0.302007\pi$
$193$	16.1895	1.16534	0.582672	+	0.812707i	$0.302007\pi$
$194$	1.41510	0.101598
$195$	0	0
$196$	0	0
$197$	14.1272	1.00652	0.503262	−	0.864134i	$-0.332133\pi$
$197$	14.1272	1.00652	0.503262	+	0.864134i	$0.332133\pi$
$198$	−0.971275	−0.0690255
$199$	−13.7329	−0.973499	−0.486749	−	0.873542i	$-0.661817\pi$
$199$	−13.7329	−0.973499	−0.486749	+	0.873542i	$0.661817\pi$
$200$	−13.9688	−0.987745
$201$	7.09657	0.500554
$202$	6.40155	0.450412
$203$	0	0
$204$	−3.39762	−0.237881
$205$	−1.89721	−0.132507
$206$	−7.16546	−0.499241
$207$	7.48306	0.520109
$208$	0	0
$209$	1.62205	0.112200
$210$	0	0
$211$	18.7380	1.28998	0.644988	−	0.764193i	$-0.276863\pi$
$211$	18.7380	1.28998	0.644988	+	0.764193i	$0.276863\pi$
$212$	9.58753	0.658474
$213$	5.20887	0.356906
$214$	14.3930	0.983886
$215$	−1.79904	−0.122693
$216$	−11.6179	−0.790497
$217$	0	0
$218$	−5.23506	−0.354563
$219$	2.15257	0.145457
$220$	−0.149057	−0.0100494
$221$	0	0
$222$	−1.44904	−0.0972533
$223$	−3.78482	−0.253450	−0.126725	−	0.991938i	$-0.540447\pi$
$223$	−3.78482	−0.253450	−0.126725	+	0.991938i	$0.540447\pi$
$224$	0	0
$225$	11.9780	0.798531
$226$	14.8903	0.990489
$227$	9.00380	0.597603	0.298802	−	0.954315i	$-0.403413\pi$
$227$	9.00380	0.597603	0.298802	+	0.954315i	$0.403413\pi$
$228$	3.30724	0.219028
$229$	1.08397	0.0716306	0.0358153	−	0.999358i	$-0.488597\pi$
$229$	1.08397	0.0716306	0.0358153	+	0.999358i	$0.488597\pi$
$230$	−0.718129	−0.0473520
$231$	0	0
$232$	−1.73713	−0.114048
$233$	−14.1071	−0.924186	−0.462093	−	0.886831i	$-0.652901\pi$
$233$	−14.1071	−0.924186	−0.462093	+	0.886831i	$0.652901\pi$
$234$	0	0
$235$	1.98787	0.129674
$236$	−14.3044	−0.931135
$237$	6.59474	0.428375
$238$	0	0
$239$	18.9679	1.22693	0.613467	−	0.789721i	$-0.289775\pi$
$239$	18.9679	1.22693	0.613467	+	0.789721i	$0.289775\pi$
$240$	0.00497818	0.000321340 0
$241$	25.1138	1.61772	0.808862	−	0.587999i	$-0.200084\pi$
$241$	25.1138	1.61772	0.808862	+	0.587999i	$0.200084\pi$
$242$	9.46710	0.608568
$243$	15.4661	0.992150
$244$	−1.96236	−0.125627
$245$	0	0
$246$	−4.72670	−0.301363
$247$	0	0
$248$	−30.3473	−1.92706
$249$	−5.67572	−0.359684
$250$	−2.31547	−0.146443
$251$	21.2412	1.34073	0.670367	−	0.742030i	$-0.266137\pi$
$251$	21.2412	1.34073	0.670367	+	0.742030i	$0.266137\pi$
$252$	0	0
$253$	1.40323	0.0882206
$254$	−11.6885	−0.733399
$255$	−0.734018	−0.0459660
$256$	−16.0604	−1.00378
$257$	−25.3217	−1.57952	−0.789762	−	0.613413i	$-0.789796\pi$
$257$	−25.3217	−1.57952	−0.789762	+	0.613413i	$0.789796\pi$
$258$	−4.48211	−0.279044
$259$	0	0
$260$	0	0
$261$	1.48955	0.0922009
$262$	13.3889	0.827167
$263$	10.4189	0.642458	0.321229	−	0.947002i	$-0.395904\pi$
$263$	10.4189	0.642458	0.321229	+	0.947002i	$0.395904\pi$
$264$	−0.974943	−0.0600036
$265$	2.07128	0.127238
$266$	0	0
$267$	6.36310	0.389416
$268$	−11.5658	−0.706492
$269$	−29.5305	−1.80051	−0.900254	−	0.435365i	$-0.856619\pi$
$269$	−29.5305	−1.80051	−0.900254	+	0.435365i	$0.856619\pi$
$270$	−0.956036	−0.0581825
$271$	25.4321	1.54489	0.772446	−	0.635081i	$-0.219033\pi$
$271$	25.4321	1.54489	0.772446	+	0.635081i	$0.219033\pi$
$272$	−0.0907024	−0.00549964
$273$	0	0
$274$	−11.9403	−0.721339
$275$	2.24613	0.135447
$276$	2.86109	0.172217
$277$	11.4076	0.685416	0.342708	−	0.939442i	$-0.388656\pi$
$277$	11.4076	0.685416	0.342708	+	0.939442i	$0.388656\pi$
$278$	12.5731	0.754086
$279$	26.0222	1.55791
$280$	0	0
$281$	−11.9773	−0.714507	−0.357253	−	0.934008i	$-0.616287\pi$
$281$	−11.9773	−0.714507	−0.357253	+	0.934008i	$0.616287\pi$
$282$	4.95255	0.294920
$283$	−2.32905	−0.138448	−0.0692239	−	0.997601i	$-0.522052\pi$
$283$	−2.32905	−0.138448	−0.0692239	+	0.997601i	$0.522052\pi$
$284$	−8.48925	−0.503745
$285$	0.714493	0.0423229
$286$	0	0
$287$	0	0
$288$	13.7191	0.808408
$289$	−3.62619	−0.213305
$290$	−0.142948	−0.00839421
$291$	−1.21800	−0.0714004
$292$	−3.50820	−0.205302
$293$	29.8421	1.74339	0.871696	−	0.490047i	$-0.163021\pi$
$293$	29.8421	1.74339	0.871696	+	0.490047i	$0.163021\pi$
$294$	0	0
$295$	−3.09030	−0.179924
$296$	6.20000	0.360368
$297$	1.86811	0.108399
$298$	17.2422	0.998813
$299$	0	0
$300$	4.57968	0.264408
$301$	0	0
$302$	1.69950	0.0977950
$303$	−5.50993	−0.316537
$304$	0.0882897	0.00506376
$305$	−0.423946	−0.0242750
$306$	7.79512	0.445618
$307$	13.3825	0.763780	0.381890	−	0.924208i	$-0.375273\pi$
$307$	13.3825	0.763780	0.381890	+	0.924208i	$0.375273\pi$
$308$	0	0
$309$	6.16743	0.350853
$310$	−2.49728	−0.141836
$311$	20.2744	1.14966	0.574828	−	0.818275i	$-0.305069\pi$
$311$	20.2744	1.14966	0.574828	+	0.818275i	$0.305069\pi$
$312$	0	0
$313$	−1.18573	−0.0670215	−0.0335107	−	0.999438i	$-0.510669\pi$
$313$	−1.18573	−0.0670215	−0.0335107	+	0.999438i	$0.510669\pi$
$314$	21.7033	1.22479
$315$	0	0
$316$	−10.7479	−0.604617
$317$	−3.81328	−0.214175	−0.107087	−	0.994250i	$-0.534152\pi$
$317$	−3.81328	−0.214175	−0.107087	+	0.994250i	$0.534152\pi$
$318$	5.16036	0.289379
$319$	0.279323	0.0156391
$320$	−1.32978	−0.0743367
$321$	−12.3883	−0.691448
$322$	0	0
$323$	−13.0181	−0.724344
$324$	−5.16130	−0.286739
$325$	0	0
$326$	5.91439	0.327568
$327$	4.50591	0.249177
$328$	20.2241	1.11669
$329$	0	0
$330$	−0.0802280	−0.00441641
$331$	−9.78226	−0.537682	−0.268841	−	0.963185i	$-0.586641\pi$
$331$	−9.78226	−0.537682	−0.268841	+	0.963185i	$0.586641\pi$
$332$	9.25013	0.507667
$333$	−5.31637	−0.291335
$334$	−1.81014	−0.0990466
$335$	−2.49866	−0.136516
$336$	0	0
$337$	−7.21841	−0.393212	−0.196606	−	0.980483i	$-0.562992\pi$
$337$	−7.21841	−0.393212	−0.196606	+	0.980483i	$0.562992\pi$
$338$	0	0
$339$	−12.8164	−0.696089
$340$	1.19628	0.0648775
$341$	4.87972	0.264252
$342$	−7.58777	−0.410300
$343$	0	0
$344$	19.1776	1.03399
$345$	0.618106	0.0332777
$346$	−11.8622	−0.637714
$347$	14.8788	0.798736	0.399368	−	0.916791i	$-0.369230\pi$
$347$	14.8788	0.798736	0.399368	+	0.916791i	$0.369230\pi$
$348$	0.569518	0.0305294
$349$	6.61098	0.353878	0.176939	−	0.984222i	$-0.443380\pi$
$349$	6.61098	0.353878	0.176939	+	0.984222i	$0.443380\pi$
$350$	0	0
$351$	0	0
$352$	2.57263	0.137122
$353$	35.3388	1.88089	0.940447	−	0.339939i	$-0.110407\pi$
$353$	35.3388	1.88089	0.940447	+	0.339939i	$0.110407\pi$
$354$	−7.69914	−0.409205
$355$	−1.83401	−0.0973391
$356$	−10.3704	−0.549630
$357$	0	0
$358$	7.57003	0.400088
$359$	14.0258	0.740252	0.370126	−	0.928981i	$-0.379314\pi$
$359$	14.0258	0.740252	0.370126	+	0.928981i	$0.379314\pi$
$360$	1.83057	0.0964794
$361$	−6.32822	−0.333064
$362$	−20.5773	−1.08152
$363$	−8.14849	−0.427685
$364$	0	0
$365$	−0.757907	−0.0396707
$366$	−1.05621	−0.0552092
$367$	25.3700	1.32430	0.662152	−	0.749369i	$-0.269643\pi$
$367$	25.3700	1.32430	0.662152	+	0.749369i	$0.269643\pi$
$368$	0.0763792	0.00398154
$369$	−17.3417	−0.902774
$370$	0.510198	0.0265239
$371$	0	0
$372$	9.94938	0.515851
$373$	18.2160	0.943188	0.471594	−	0.881816i	$-0.343679\pi$
$373$	18.2160	0.943188	0.471594	+	0.881816i	$0.343679\pi$
$374$	1.46175	0.0755855
$375$	1.99296	0.102916
$376$	−21.1905	−1.09281
$377$	0	0
$378$	0	0
$379$	−7.24143	−0.371968	−0.185984	−	0.982553i	$-0.559547\pi$
$379$	−7.24143	−0.371968	−0.185984	+	0.982553i	$0.559547\pi$
$380$	−1.16446	−0.0597355
$381$	10.0605	0.515413
$382$	−15.0602	−0.770549
$383$	−0.367176	−0.0187618	−0.00938091	−	0.999956i	$-0.502986\pi$
$383$	−0.367176	−0.0187618	−0.00938091	+	0.999956i	$0.502986\pi$
$384$	5.21255	0.266002
$385$	0	0
$386$	−14.2015	−0.722835
$387$	−16.4444	−0.835914
$388$	1.98506	0.100776
$389$	11.1560	0.565631	0.282815	−	0.959174i	$-0.408732\pi$
$389$	11.1560	0.565631	0.282815	+	0.959174i	$0.408732\pi$
$390$	0	0
$391$	−11.2619	−0.569538
$392$	0	0
$393$	−11.5240	−0.581311
$394$	−12.3924	−0.624322
$395$	−2.32197	−0.116831
$396$	−1.36248	−0.0684670
$397$	3.17876	0.159537	0.0797686	−	0.996813i	$-0.474582\pi$
$397$	3.17876	0.159537	0.0797686	+	0.996813i	$0.474582\pi$
$398$	12.0465	0.603838
$399$	0	0
$400$	0.122258	0.00611292
$401$	−8.09236	−0.404113	−0.202057	−	0.979374i	$-0.564763\pi$
$401$	−8.09236	−0.404113	−0.202057	+	0.979374i	$0.564763\pi$
$402$	−6.22513	−0.310481
$403$	0	0
$404$	8.97992	0.446767
$405$	−1.11504	−0.0554068
$406$	0	0
$407$	−0.996934	−0.0494162
$408$	7.82456	0.387374
$409$	3.13028	0.154783	0.0773913	−	0.997001i	$-0.475341\pi$
$409$	3.13028	0.154783	0.0773913	+	0.997001i	$0.475341\pi$
$410$	1.66424	0.0821909
$411$	10.2772	0.506937
$412$	−10.0515	−0.495202
$413$	0	0
$414$	−6.56416	−0.322611
$415$	1.99839	0.0980969
$416$	0	0
$417$	−10.8219	−0.529951
$418$	−1.42287	−0.0695949
$419$	−11.4655	−0.560128	−0.280064	−	0.959981i	$-0.590356\pi$
$419$	−11.4655	−0.560128	−0.280064	+	0.959981i	$0.590356\pi$
$420$	0	0
$421$	−22.1642	−1.08022	−0.540108	−	0.841596i	$-0.681617\pi$
$421$	−22.1642	−1.08022	−0.540108	+	0.841596i	$0.681617\pi$
$422$	−16.4370	−0.800141
$423$	18.1704	0.883474
$424$	−22.0796	−1.07228
$425$	−18.0267	−0.874422
$426$	−4.56923	−0.221380
$427$	0	0
$428$	20.1901	0.975926
$429$	0	0
$430$	1.57812	0.0761038
$431$	21.3135	1.02664	0.513319	−	0.858198i	$-0.328416\pi$
$431$	21.3135	1.02664	0.513319	+	0.858198i	$0.328416\pi$
$432$	0.101683	0.00489221
$433$	−0.0447167	−0.00214895	−0.00107447	−	0.999999i	$-0.500342\pi$
$433$	−0.0447167	−0.00214895	−0.00107447	+	0.999999i	$0.500342\pi$
$434$	0	0
$435$	0.123038	0.00589922
$436$	−7.34359	−0.351694
$437$	10.9623	0.524399
$438$	−1.88824	−0.0902237
$439$	3.32925	0.158896	0.0794481	−	0.996839i	$-0.474684\pi$
$439$	3.32925	0.158896	0.0794481	+	0.996839i	$0.474684\pi$
$440$	0.343271	0.0163648
$441$	0	0
$442$	0	0
$443$	30.4552	1.44697	0.723485	−	0.690340i	$-0.242539\pi$
$443$	30.4552	1.44697	0.723485	+	0.690340i	$0.242539\pi$
$444$	−2.03267	−0.0964664
$445$	−2.24041	−0.106206
$446$	3.32006	0.157209
$447$	−14.8407	−0.701939
$448$	0	0
$449$	6.77363	0.319667	0.159834	−	0.987144i	$-0.448904\pi$
$449$	6.77363	0.319667	0.159834	+	0.987144i	$0.448904\pi$
$450$	−10.5071	−0.495310
$451$	−3.25195	−0.153128
$452$	20.8877	0.982475
$453$	−1.46279	−0.0687277
$454$	−7.89816	−0.370679
$455$	0	0
$456$	−7.61643	−0.356672
$457$	−32.1702	−1.50486	−0.752430	−	0.658672i	$-0.771118\pi$
$457$	−32.1702	−1.50486	−0.752430	+	0.658672i	$0.771118\pi$
$458$	−0.950859	−0.0444307
$459$	−14.9928	−0.699805
$460$	−1.00737	−0.0469689
$461$	−24.4018	−1.13651	−0.568253	−	0.822854i	$-0.692380\pi$
$461$	−24.4018	−1.13651	−0.568253	+	0.822854i	$0.692380\pi$
$462$	0	0
$463$	23.4370	1.08921	0.544604	−	0.838693i	$-0.316680\pi$
$463$	23.4370	1.08921	0.544604	+	0.838693i	$0.316680\pi$
$464$	0.0152038	0.000705817 0
$465$	2.14945	0.0996785
$466$	12.3748	0.573251
$467$	27.8751	1.28990	0.644952	−	0.764223i	$-0.276877\pi$
$467$	27.8751	1.28990	0.644952	+	0.764223i	$0.276877\pi$
$468$	0	0
$469$	0	0
$470$	−1.74376	−0.0804338
$471$	−18.6804	−0.860747
$472$	32.9423	1.51629
$473$	−3.08367	−0.141787
$474$	−5.78493	−0.265710
$475$	17.5472	0.805119
$476$	0	0
$477$	18.9328	0.866874
$478$	−16.6387	−0.761037
$479$	2.07202	0.0946729	0.0473365	−	0.998879i	$-0.484927\pi$
$479$	2.07202	0.0946729	0.0473365	+	0.998879i	$0.484927\pi$
$480$	1.13321	0.0517238
$481$	0	0
$482$	−22.0299	−1.00343
$483$	0	0
$484$	13.2802	0.603644
$485$	0.428850	0.0194731
$486$	−13.5669	−0.615407
$487$	−6.01962	−0.272775	−0.136388	−	0.990656i	$-0.543549\pi$
$487$	−6.01962	−0.272775	−0.136388	+	0.990656i	$0.543549\pi$
$488$	4.51922	0.204575
$489$	−5.09062	−0.230206
$490$	0	0
$491$	2.01551	0.0909588	0.0454794	−	0.998965i	$-0.485518\pi$
$491$	2.01551	0.0909588	0.0454794	+	0.998965i	$0.485518\pi$
$492$	−6.63047	−0.298925
$493$	−2.24175	−0.100963
$494$	0	0
$495$	−0.294348	−0.0132299
$496$	0.265607	0.0119261
$497$	0	0
$498$	4.97876	0.223104
$499$	7.51174	0.336272	0.168136	−	0.985764i	$-0.446225\pi$
$499$	7.51174	0.336272	0.168136	+	0.985764i	$0.446225\pi$
$500$	−3.24807	−0.145258
$501$	1.55802	0.0696072
$502$	−18.6329	−0.831625
$503$	15.1154	0.673962	0.336981	−	0.941511i	$-0.390594\pi$
$503$	15.1154	0.673962	0.336981	+	0.941511i	$0.390594\pi$
$504$	0	0
$505$	1.94001	0.0863293
$506$	−1.23092	−0.0547211
$507$	0	0
$508$	−16.3962	−0.727466
$509$	36.6699	1.62537	0.812683	−	0.582706i	$-0.198006\pi$
$509$	36.6699	1.62537	0.812683	+	0.582706i	$0.198006\pi$
$510$	0.643883	0.0285116
$511$	0	0
$512$	0.280601	0.0124009
$513$	14.5940	0.644341
$514$	22.2123	0.979742
$515$	−2.17151	−0.0956883
$516$	−6.28737	−0.276786
$517$	3.40734	0.149855
$518$	0	0
$519$	10.2100	0.448168
$520$	0	0
$521$	31.8083	1.39355	0.696774	−	0.717291i	$-0.254618\pi$
$521$	31.8083	1.39355	0.696774	+	0.717291i	$0.254618\pi$
$522$	−1.30664	−0.0571900
$523$	39.5596	1.72982	0.864909	−	0.501928i	$-0.167376\pi$
$523$	39.5596	1.72982	0.864909	+	0.501928i	$0.167376\pi$
$524$	18.7815	0.820475
$525$	0	0
$526$	−9.13951	−0.398501
$527$	−39.1630	−1.70597
$528$	0.00853293	0.000371348 0
$529$	−13.5165	−0.587675
$530$	−1.81693	−0.0789224
$531$	−28.2473	−1.22583
$532$	0	0
$533$	0	0
$534$	−5.58173	−0.241545
$535$	4.36185	0.188579
$536$	26.6354	1.15048
$537$	−6.51565	−0.281171
$538$	25.9043	1.11681
$539$	0	0
$540$	−1.34110	−0.0577118
$541$	−22.2284	−0.955673	−0.477836	−	0.878449i	$-0.658579\pi$
$541$	−22.2284	−0.955673	−0.477836	+	0.878449i	$0.658579\pi$
$542$	−22.3091	−0.958259
$543$	17.7113	0.760063
$544$	−20.6471	−0.885237
$545$	−1.58650	−0.0679582
$546$	0	0
$547$	−14.1989	−0.607100	−0.303550	−	0.952816i	$-0.598172\pi$
$547$	−14.1989	−0.607100	−0.303550	+	0.952816i	$0.598172\pi$
$548$	−16.7495	−0.715503
$549$	−3.87513	−0.165387
$550$	−1.97031	−0.0840143
$551$	2.18212	0.0929615
$552$	−6.58895	−0.280444
$553$	0	0
$554$	−10.0068	−0.425147
$555$	−0.439136	−0.0186403
$556$	17.6372	0.747985
$557$	−26.6179	−1.12784	−0.563918	−	0.825831i	$-0.690707\pi$
$557$	−26.6179	−1.12784	−0.563918	+	0.825831i	$0.690707\pi$
$558$	−22.8267	−0.966333
$559$	0	0
$560$	0	0
$561$	−1.25816	−0.0531194
$562$	10.5065	0.443191
$563$	20.9514	0.882996	0.441498	−	0.897262i	$-0.354447\pi$
$563$	20.9514	0.882996	0.441498	+	0.897262i	$0.354447\pi$
$564$	6.94730	0.292534
$565$	4.51256	0.189845
$566$	2.04305	0.0858759
$567$	0	0
$568$	19.5503	0.820314
$569$	−1.12515	−0.0471686	−0.0235843	−	0.999722i	$-0.507508\pi$
$569$	−1.12515	−0.0471686	−0.0235843	+	0.999722i	$0.507508\pi$
$570$	−0.626755	−0.0262519
$571$	−41.6288	−1.74211	−0.871055	−	0.491185i	$-0.836564\pi$
$571$	−41.6288	−1.74211	−0.871055	+	0.491185i	$0.836564\pi$
$572$	0	0
$573$	12.9626	0.541521
$574$	0	0
$575$	15.1800	0.633049
$576$	−12.1550	−0.506459
$577$	−29.4246	−1.22496	−0.612481	−	0.790485i	$-0.709829\pi$
$577$	−29.4246	−1.22496	−0.612481	+	0.790485i	$0.709829\pi$
$578$	3.18090	0.132308
$579$	12.2234	0.507989
$580$	−0.200524	−0.00832629
$581$	0	0
$582$	1.06843	0.0442879
$583$	3.55031	0.147039
$584$	8.07921	0.334320
$585$	0	0
$586$	−26.1775	−1.08138
$587$	36.0961	1.48985	0.744923	−	0.667150i	$-0.232486\pi$
$587$	36.0961	1.48985	0.744923	+	0.667150i	$0.232486\pi$
$588$	0	0
$589$	38.1213	1.57076
$590$	2.71082	0.111603
$591$	10.6664	0.438757
$592$	−0.0542639	−0.00223023
$593$	4.07766	0.167449	0.0837246	−	0.996489i	$-0.473318\pi$
$593$	4.07766	0.167449	0.0837246	+	0.996489i	$0.473318\pi$
$594$	−1.63871	−0.0672371
$595$	0	0
$596$	24.1869	0.990732
$597$	−10.3687	−0.424361
$598$	0	0
$599$	−25.3993	−1.03779	−0.518894	−	0.854839i	$-0.673656\pi$
$599$	−25.3993	−1.03779	−0.518894	+	0.854839i	$0.673656\pi$
$600$	−10.5468	−0.430571
$601$	43.4761	1.77343	0.886713	−	0.462321i	$-0.152983\pi$
$601$	43.4761	1.77343	0.886713	+	0.462321i	$0.152983\pi$
$602$	0	0
$603$	−22.8393	−0.930089
$604$	2.38401	0.0970038
$605$	2.86903	0.116643
$606$	4.83332	0.196340
$607$	18.2340	0.740094	0.370047	−	0.929013i	$-0.379342\pi$
$607$	18.2340	0.740094	0.370047	+	0.929013i	$0.379342\pi$
$608$	20.0979	0.815077
$609$	0	0
$610$	0.371886	0.0150572
$611$	0	0
$612$	10.9348	0.442012
$613$	32.2272	1.30165	0.650823	−	0.759230i	$-0.274424\pi$
$613$	32.2272	1.30165	0.650823	+	0.759230i	$0.274424\pi$
$614$	−11.7392	−0.473755
$615$	−1.43244	−0.0577615
$616$	0	0
$617$	37.0753	1.49260	0.746298	−	0.665612i	$-0.231830\pi$
$617$	37.0753	1.49260	0.746298	+	0.665612i	$0.231830\pi$
$618$	−5.41009	−0.217626
$619$	33.8527	1.36066	0.680328	−	0.732908i	$-0.261837\pi$
$619$	33.8527	1.36066	0.680328	+	0.732908i	$0.261837\pi$
$620$	−3.50311	−0.140688
$621$	12.6252	0.506633
$622$	−17.7848	−0.713104
$623$	0	0
$624$	0	0
$625$	23.9449	0.957798
$626$	1.04013	0.0415718
$627$	1.22469	0.0489094
$628$	30.4447	1.21488
$629$	8.00106	0.319023
$630$	0	0
$631$	33.2239	1.32262	0.661311	−	0.750111i	$-0.270000\pi$
$631$	33.2239	1.32262	0.661311	+	0.750111i	$0.270000\pi$
$632$	24.7519	0.984579
$633$	14.1476	0.562317
$634$	3.34502	0.132848
$635$	−3.54222	−0.140569
$636$	7.23881	0.287038
$637$	0	0
$638$	−0.245023	−0.00970055
$639$	−16.7640	−0.663174
$640$	−1.83531	−0.0725468
$641$	−11.8377	−0.467561	−0.233780	−	0.972289i	$-0.575110\pi$
$641$	−11.8377	−0.467561	−0.233780	+	0.972289i	$0.575110\pi$
$642$	10.8671	0.428889
$643$	40.1562	1.58360	0.791802	−	0.610778i	$-0.209143\pi$
$643$	40.1562	1.58360	0.791802	+	0.610778i	$0.209143\pi$
$644$	0	0
$645$	−1.35832	−0.0534837
$646$	11.4195	0.449294
$647$	−3.17699	−0.124900	−0.0624501	−	0.998048i	$-0.519891\pi$
$647$	−3.17699	−0.124900	−0.0624501	+	0.998048i	$0.519891\pi$
$648$	11.8862	0.466935
$649$	−5.29698	−0.207925
$650$	0	0
$651$	0	0
$652$	8.29654	0.324918
$653$	30.6776	1.20051	0.600253	−	0.799810i	$-0.295067\pi$
$653$	30.6776	1.20051	0.600253	+	0.799810i	$0.295067\pi$
$654$	−3.95259	−0.154559
$655$	4.05754	0.158541
$656$	−0.177006	−0.00691093
$657$	−6.92775	−0.270277
$658$	0	0
$659$	−8.19638	−0.319286	−0.159643	−	0.987175i	$-0.551034\pi$
$659$	−8.19638	−0.319286	−0.159643	+	0.987175i	$0.551034\pi$
$660$	−0.112542	−0.00438067
$661$	30.8005	1.19800	0.599000	−	0.800749i	$-0.295565\pi$
$661$	30.8005	1.19800	0.599000	+	0.800749i	$0.295565\pi$
$662$	8.58102	0.333511
$663$	0	0
$664$	−21.3026	−0.826701
$665$	0	0
$666$	4.66354	0.180708
$667$	1.88775	0.0730938
$668$	−2.53922	−0.0982452
$669$	−2.85763	−0.110482
$670$	2.19183	0.0846778
$671$	−0.726671	−0.0280528
$672$	0	0
$673$	−29.7292	−1.14598	−0.572988	−	0.819564i	$-0.694216\pi$
$673$	−29.7292	−1.14598	−0.572988	+	0.819564i	$0.694216\pi$
$674$	6.33201	0.243900
$675$	20.2089	0.777842
$676$	0	0
$677$	27.8067	1.06870	0.534349	−	0.845264i	$-0.320557\pi$
$677$	27.8067	1.06870	0.534349	+	0.845264i	$0.320557\pi$
$678$	11.2425	0.431767
$679$	0	0
$680$	−2.75498	−0.105649
$681$	6.79808	0.260503
$682$	−4.28051	−0.163909
$683$	−13.8232	−0.528929	−0.264465	−	0.964395i	$-0.585195\pi$
$683$	−13.8232	−0.528929	−0.264465	+	0.964395i	$0.585195\pi$
$684$	−10.6439	−0.406980
$685$	−3.61854	−0.138257
$686$	0	0
$687$	0.818421	0.0312247
$688$	−0.167847	−0.00639910
$689$	0	0
$690$	−0.542204	−0.0206414
$691$	−2.04187	−0.0776762	−0.0388381	−	0.999246i	$-0.512366\pi$
$691$	−2.04187	−0.0776762	−0.0388381	+	0.999246i	$0.512366\pi$
$692$	−16.6399	−0.632555
$693$	0	0
$694$	−13.0517	−0.495437
$695$	3.81032	0.144534
$696$	−1.31157	−0.0497150
$697$	26.0990	0.988572
$698$	−5.79917	−0.219502
$699$	−10.6512	−0.402865
$700$	0	0
$701$	−27.0161	−1.02039	−0.510193	−	0.860060i	$-0.670426\pi$
$701$	−27.0161	−1.02039	−0.510193	+	0.860060i	$0.670426\pi$
$702$	0	0
$703$	−7.78823	−0.293739
$704$	−2.27933	−0.0859053
$705$	1.50089	0.0565267
$706$	−30.9993	−1.16667
$707$	0	0
$708$	−10.8001	−0.405894
$709$	2.83830	0.106595	0.0532973	−	0.998579i	$-0.483027\pi$
$709$	2.83830	0.106595	0.0532973	+	0.998579i	$0.483027\pi$
$710$	1.60880	0.0603771
$711$	−21.2243	−0.795972
$712$	23.8825	0.895035
$713$	32.9786	1.23506
$714$	0	0
$715$	0	0
$716$	10.6190	0.396851
$717$	14.3212	0.534836
$718$	−12.3035	−0.459161
$719$	−39.2389	−1.46336	−0.731682	−	0.681646i	$-0.761264\pi$
$719$	−39.2389	−1.46336	−0.731682	+	0.681646i	$0.761264\pi$
$720$	−0.0160216	−0.000597089 0
$721$	0	0
$722$	5.55113	0.206592
$723$	18.9615	0.705187
$724$	−28.8653	−1.07277
$725$	3.02168	0.112222
$726$	7.14788	0.265283
$727$	−12.9172	−0.479074	−0.239537	−	0.970887i	$-0.576996\pi$
$727$	−12.9172	−0.479074	−0.239537	+	0.970887i	$0.576996\pi$
$728$	0	0
$729$	−0.906005	−0.0335558
$730$	0.664838	0.0246068
$731$	24.7485	0.915357
$732$	−1.48163	−0.0547625
$733$	−7.21963	−0.266663	−0.133332	−	0.991071i	$-0.542568\pi$
$733$	−7.21963	−0.266663	−0.133332	+	0.991071i	$0.542568\pi$
$734$	−22.2547	−0.821434
$735$	0	0
$736$	17.3866	0.640879
$737$	−4.28287	−0.157761
$738$	15.2122	0.559969
$739$	−24.4029	−0.897677	−0.448838	−	0.893613i	$-0.648162\pi$
$739$	−24.4029	−0.897677	−0.448838	+	0.893613i	$0.648162\pi$
$740$	0.715691	0.0263093
$741$	0	0
$742$	0	0
$743$	−43.6054	−1.59973	−0.799864	−	0.600182i	$-0.795095\pi$
$743$	−43.6054	−1.59973	−0.799864	+	0.600182i	$0.795095\pi$
$744$	−22.9129	−0.840029
$745$	5.22530	0.191440
$746$	−15.9791	−0.585037
$747$	18.2665	0.668338
$748$	2.05051	0.0749739
$749$	0	0
$750$	−1.74823	−0.0638364
$751$	0.527048	0.0192322	0.00961612	−	0.999954i	$-0.496939\pi$
$751$	0.527048	0.0192322	0.00961612	+	0.999954i	$0.496939\pi$
$752$	0.185464	0.00676318
$753$	16.0376	0.584444
$754$	0	0
$755$	0.515038	0.0187441
$756$	0	0
$757$	−17.4591	−0.634563	−0.317282	−	0.948331i	$-0.602770\pi$
$757$	−17.4591	−0.634563	−0.317282	+	0.948331i	$0.602770\pi$
$758$	6.35221	0.230723
$759$	1.05948	0.0384565
$760$	2.68169	0.0972753
$761$	−36.3311	−1.31700	−0.658501	−	0.752580i	$-0.728809\pi$
$761$	−36.3311	−1.31700	−0.658501	+	0.752580i	$0.728809\pi$
$762$	−8.82507	−0.319698
$763$	0	0
$764$	−21.1261	−0.764314
$765$	2.36234	0.0854104
$766$	0.322088	0.0116375
$767$	0	0
$768$	−12.1260	−0.437559
$769$	−30.7750	−1.10977	−0.554887	−	0.831926i	$-0.687239\pi$
$769$	−30.7750	−1.10977	−0.554887	+	0.831926i	$0.687239\pi$
$770$	0	0
$771$	−19.1185	−0.688536
$772$	−19.9214	−0.716987
$773$	29.6721	1.06723	0.533616	−	0.845727i	$-0.320833\pi$
$773$	29.6721	1.06723	0.533616	+	0.845727i	$0.320833\pi$
$774$	14.4250	0.518497
$775$	52.7881	1.89621
$776$	−4.57149	−0.164107
$777$	0	0
$778$	−9.78606	−0.350847
$779$	−25.4048	−0.910221
$780$	0	0
$781$	−3.14361	−0.112487
$782$	9.87896	0.353271
$783$	2.51313	0.0898120
$784$	0	0
$785$	6.57724	0.234752
$786$	10.1089	0.360573
$787$	−27.5755	−0.982959	−0.491479	−	0.870889i	$-0.663544\pi$
$787$	−27.5755	−0.982959	−0.491479	+	0.870889i	$0.663544\pi$
$788$	−17.3838	−0.619271
$789$	7.86653	0.280056
$790$	2.03684	0.0724674
$791$	0	0
$792$	3.13772	0.111494
$793$	0	0
$794$	−2.78842	−0.0989572
$795$	1.56386	0.0554645
$796$	16.8985	0.598952
$797$	15.4319	0.546627	0.273314	−	0.961925i	$-0.411880\pi$
$797$	15.4319	0.546627	0.273314	+	0.961925i	$0.411880\pi$
$798$	0	0
$799$	−27.3461	−0.967437
$800$	27.8304	0.983953
$801$	−20.4788	−0.723582
$802$	7.09864	0.250662
$803$	−1.29910	−0.0458444
$804$	−8.73244	−0.307969
$805$	0	0
$806$	0	0
$807$	−22.2962	−0.784865
$808$	−20.6803	−0.727531
$809$	6.45513	0.226950	0.113475	−	0.993541i	$-0.463802\pi$
$809$	6.45513	0.226950	0.113475	+	0.993541i	$0.463802\pi$
$810$	0.978117	0.0343675
$811$	−21.6726	−0.761027	−0.380513	−	0.924775i	$-0.624253\pi$
$811$	−21.6726	−0.761027	−0.380513	+	0.924775i	$0.624253\pi$
$812$	0	0
$813$	19.2019	0.673438
$814$	0.874514	0.0306517
$815$	1.79238	0.0627842
$816$	−0.0684825	−0.00239737
$817$	−24.0902	−0.842810
$818$	−2.74589	−0.0960079
$819$	0	0
$820$	2.33455	0.0815259
$821$	24.7869	0.865068	0.432534	−	0.901618i	$-0.357620\pi$
$821$	24.7869	0.865068	0.432534	+	0.901618i	$0.357620\pi$
$822$	−9.01520	−0.314441
$823$	−12.1129	−0.422230	−0.211115	−	0.977461i	$-0.567709\pi$
$823$	−12.1129	−0.422230	−0.211115	+	0.977461i	$0.567709\pi$
$824$	23.1481	0.806403
$825$	1.69588	0.0590430
$826$	0	0
$827$	−44.8946	−1.56114	−0.780569	−	0.625069i	$-0.785071\pi$
$827$	−44.8946	−1.56114	−0.780569	+	0.625069i	$0.785071\pi$
$828$	−9.20802	−0.320001
$829$	1.74161	0.0604885	0.0302443	−	0.999543i	$-0.490371\pi$
$829$	1.74161	0.0604885	0.0302443	+	0.999543i	$0.490371\pi$
$830$	−1.75299	−0.0608472
$831$	8.61300	0.298782
$832$	0	0
$833$	0	0
$834$	9.49301	0.328716
$835$	−0.548569	−0.0189840
$836$	−1.99596	−0.0690318
$837$	43.9040	1.51754
$838$	10.0576	0.347434
$839$	−6.96538	−0.240472	−0.120236	−	0.992745i	$-0.538365\pi$
$839$	−6.96538	−0.240472	−0.120236	+	0.992745i	$0.538365\pi$
$840$	0	0
$841$	−28.6242	−0.987042
$842$	19.4425	0.670033
$843$	−9.04316	−0.311463
$844$	−23.0574	−0.793667
$845$	0	0
$846$	−15.9391	−0.547998
$847$	0	0
$848$	0.193246	0.00663610
$849$	−1.75849	−0.0603512
$850$	15.8130	0.542383
$851$	−6.73758	−0.230961
$852$	−6.40959	−0.219589
$853$	10.5953	0.362774	0.181387	−	0.983412i	$-0.441941\pi$
$853$	10.5953	0.362774	0.181387	+	0.983412i	$0.441941\pi$
$854$	0	0
$855$	−2.29950	−0.0786412
$856$	−46.4969	−1.58923
$857$	18.4590	0.630548	0.315274	−	0.949001i	$-0.397904\pi$
$857$	18.4590	0.630548	0.315274	+	0.949001i	$0.397904\pi$
$858$	0	0
$859$	38.9766	1.32986	0.664932	−	0.746904i	$-0.268460\pi$
$859$	38.9766	1.32986	0.664932	+	0.746904i	$0.268460\pi$
$860$	2.21374	0.0754880
$861$	0	0
$862$	−18.6963	−0.636799
$863$	−4.16284	−0.141705	−0.0708524	−	0.997487i	$-0.522572\pi$
$863$	−4.16284	−0.141705	−0.0708524	+	0.997487i	$0.522572\pi$
$864$	23.1466	0.787463
$865$	−3.59487	−0.122229
$866$	0.0392256	0.00133294
$867$	−2.73786	−0.0929825
$868$	0	0
$869$	−3.98001	−0.135012
$870$	−0.107929	−0.00365914
$871$	0	0
$872$	16.9119	0.572711
$873$	3.91996	0.132671
$874$	−9.61618	−0.325272
$875$	0	0
$876$	−2.64877	−0.0894937
$877$	−3.57332	−0.120662	−0.0603312	−	0.998178i	$-0.519216\pi$
$877$	−3.57332	−0.120662	−0.0603312	+	0.998178i	$0.519216\pi$
$878$	−2.92042	−0.0985595
$879$	22.5315	0.759967
$880$	−0.00300439	−0.000101278 0
$881$	21.4938	0.724144	0.362072	−	0.932150i	$-0.382069\pi$
$881$	21.4938	0.724144	0.362072	+	0.932150i	$0.382069\pi$
$882$	0	0
$883$	7.48321	0.251830	0.125915	−	0.992041i	$-0.459813\pi$
$883$	7.48321	0.251830	0.125915	+	0.992041i	$0.459813\pi$
$884$	0	0
$885$	−2.33325	−0.0784313
$886$	−26.7154	−0.897521
$887$	−29.4870	−0.990076	−0.495038	−	0.868871i	$-0.664846\pi$
$887$	−29.4870	−0.990076	−0.495038	+	0.868871i	$0.664846\pi$
$888$	4.68115	0.157089
$889$	0	0
$890$	1.96529	0.0658767
$891$	−1.91126	−0.0640295
$892$	4.65728	0.155937
$893$	26.6187	0.890762
$894$	13.0183	0.435396
$895$	2.29412	0.0766840
$896$	0	0
$897$	0	0
$898$	−5.94185	−0.198282
$899$	6.56460	0.218942
$900$	−14.7391	−0.491303
$901$	−28.4936	−0.949259
$902$	2.85262	0.0949818
$903$	0	0
$904$	−48.1034	−1.59990
$905$	−6.23602	−0.207292
$906$	1.28316	0.0426301
$907$	−24.7641	−0.822280	−0.411140	−	0.911572i	$-0.634869\pi$
$907$	−24.7641	−0.822280	−0.411140	+	0.911572i	$0.634869\pi$
$908$	−11.0793	−0.367680
$909$	17.7329	0.588164
$910$	0	0
$911$	−5.42252	−0.179656	−0.0898281	−	0.995957i	$-0.528632\pi$
$911$	−5.42252	−0.179656	−0.0898281	+	0.995957i	$0.528632\pi$
$912$	0.0666608	0.00220736
$913$	3.42537	0.113363
$914$	28.2198	0.933429
$915$	−0.320089	−0.0105818
$916$	−1.33384	−0.0440713
$917$	0	0
$918$	13.1517	0.434072
$919$	13.8605	0.457215	0.228607	−	0.973519i	$-0.426583\pi$
$919$	13.8605	0.457215	0.228607	+	0.973519i	$0.426583\pi$
$920$	2.31993	0.0764857
$921$	10.1041	0.332942
$922$	21.4053	0.704947
$923$	0	0
$924$	0	0
$925$	−10.7847	−0.354599
$926$	−20.5590	−0.675610
$927$	−19.8490	−0.651927
$928$	3.46092	0.113610
$929$	13.5503	0.444572	0.222286	−	0.974982i	$-0.428648\pi$
$929$	13.5503	0.444572	0.222286	+	0.974982i	$0.428648\pi$
$930$	−1.88551	−0.0618282
$931$	0	0
$932$	17.3590	0.568613
$933$	15.3076	0.501150
$934$	−24.4521	−0.800097
$935$	0.442989	0.0144873
$936$	0	0
$937$	9.85431	0.321926	0.160963	−	0.986960i	$-0.448540\pi$
$937$	9.85431	0.321926	0.160963	+	0.986960i	$0.448540\pi$
$938$	0	0
$939$	−0.895255	−0.0292155
$940$	−2.44610	−0.0797830
$941$	38.6166	1.25887	0.629433	−	0.777055i	$-0.283287\pi$
$941$	38.6166	1.25887	0.629433	+	0.777055i	$0.283287\pi$
$942$	16.3865	0.533901
$943$	−21.9776	−0.715690
$944$	−0.288319	−0.00938398
$945$	0	0
$946$	2.70501	0.0879473
$947$	34.7023	1.12767	0.563837	−	0.825886i	$-0.309325\pi$
$947$	34.7023	1.12767	0.563837	+	0.825886i	$0.309325\pi$
$948$	−8.11493	−0.263561
$949$	0	0
$950$	−15.3924	−0.499396
$951$	−2.87911	−0.0933617
$952$	0	0
$953$	−8.61339	−0.279015	−0.139508	−	0.990221i	$-0.544552\pi$
$953$	−8.61339	−0.279015	−0.139508	+	0.990221i	$0.544552\pi$
$954$	−16.6079	−0.537701
$955$	−4.56405	−0.147689
$956$	−23.3403	−0.754880
$957$	0.210895	0.00681728
$958$	−1.81758	−0.0587234
$959$	0	0
$960$	−1.00401	−0.0324044
$961$	83.6825	2.69943
$962$	0	0
$963$	39.8701	1.28480
$964$	−30.9029	−0.995316
$965$	−4.30379	−0.138544
$966$	0	0
$967$	12.9432	0.416225	0.208113	−	0.978105i	$-0.433268\pi$
$967$	12.9432	0.416225	0.208113	+	0.978105i	$0.433268\pi$
$968$	−30.5836	−0.982994
$969$	−9.82895	−0.315751
$970$	−0.376188	−0.0120787
$971$	−31.7833	−1.01997	−0.509987	−	0.860182i	$-0.670350\pi$
$971$	−31.7833	−1.01997	−0.509987	+	0.860182i	$0.670350\pi$
$972$	−19.0313	−0.610428
$973$	0	0
$974$	5.28043	0.169196
$975$	0	0
$976$	−0.0395533	−0.00126607
$977$	−18.7110	−0.598618	−0.299309	−	0.954156i	$-0.596756\pi$
$977$	−18.7110	−0.598618	−0.299309	+	0.954156i	$0.596756\pi$
$978$	4.46551	0.142791
$979$	−3.84021	−0.122734
$980$	0	0
$981$	−14.5016	−0.463002
$982$	−1.76801	−0.0564196
$983$	6.32468	0.201726	0.100863	−	0.994900i	$-0.467840\pi$
$983$	6.32468	0.201726	0.100863	+	0.994900i	$0.467840\pi$
$984$	15.2697	0.486779
$985$	−3.75557	−0.119662
$986$	1.96647	0.0626252
$987$	0	0
$988$	0	0
$989$	−20.8404	−0.662685
$990$	0.258203	0.00820622
$991$	15.5948	0.495387	0.247693	−	0.968838i	$-0.420327\pi$
$991$	15.5948	0.495387	0.247693	+	0.968838i	$0.420327\pi$
$992$	60.4617	1.91966
$993$	−7.38584	−0.234382
$994$	0	0
$995$	3.65074	0.115736
$996$	6.98407	0.221299
$997$	−31.1034	−0.985056	−0.492528	−	0.870297i	$-0.663927\pi$
$997$	−31.1034	−0.985056	−0.492528	+	0.870297i	$0.663927\pi$
$998$	−6.58932	−0.208581
$999$	−8.96965	−0.283787

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cw.1.10		24
7.3	odd	6		1183.2.e.k.170.15	✓	48
7.5	odd	6		1183.2.e.k.508.15	yes	48
7.6	odd	2		8281.2.a.cv.1.10		24
13.12	even	2		8281.2.a.ct.1.15		24
91.12	odd	6		1183.2.e.l.508.10	yes	48
91.38	odd	6		1183.2.e.l.170.10	yes	48
91.90	odd	2		8281.2.a.cu.1.15		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.15	✓	48	7.3	odd	6
1183.2.e.k.508.15	yes	48	7.5	odd	6
1183.2.e.l.170.10	yes	48	91.38	odd	6
1183.2.e.l.508.10	yes	48	91.12	odd	6
8281.2.a.ct.1.15		24	13.12	even	2
8281.2.a.cu.1.15		24	91.90	odd	2
8281.2.a.cv.1.10		24	7.6	odd	2
8281.2.a.cw.1.10		24	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-0.877203 q^{2} +0.755024 q^{3} -1.23052 q^{4} -0.265839 q^{5} -0.662309 q^{6} +2.83382 q^{8} -2.42994 q^{9} +0.233195 q^{10} -0.455666 q^{11} -0.929068 q^{12} -0.200715 q^{15} -0.0248023 q^{16} +3.65702 q^{17} +2.13155 q^{18} -3.55974 q^{19} +0.327119 q^{20} +0.399712 q^{22} -3.07952 q^{23} +2.13960 q^{24} -4.92933 q^{25} -4.09973 q^{27} -0.612999 q^{29} +0.176068 q^{30} -10.7090 q^{31} -5.64588 q^{32} -0.344039 q^{33} -3.20795 q^{34} +2.99008 q^{36} +2.18786 q^{37} +3.12262 q^{38} -0.753339 q^{40} +7.13669 q^{41} +6.76740 q^{43} +0.560704 q^{44} +0.645973 q^{45} +2.70137 q^{46} -7.47771 q^{47} -0.0187263 q^{48} +4.32402 q^{50} +2.76114 q^{51} -7.79148 q^{53} +3.59630 q^{54} +0.121134 q^{55} -2.68769 q^{57} +0.537725 q^{58} +11.6247 q^{59} +0.246983 q^{60} +1.59475 q^{61} +9.39396 q^{62} +5.00218 q^{64} +0.301792 q^{66} +9.39914 q^{67} -4.50002 q^{68} -2.32511 q^{69} +6.89894 q^{71} -6.88600 q^{72} +2.85100 q^{73} -1.91920 q^{74} -3.72176 q^{75} +4.38032 q^{76} +8.73448 q^{79} +0.00659341 q^{80} +4.19442 q^{81} -6.26033 q^{82} -7.51728 q^{83} -0.972179 q^{85} -5.93638 q^{86} -0.462829 q^{87} -1.29127 q^{88} +8.42769 q^{89} -0.566649 q^{90} +3.78940 q^{92} -8.08554 q^{93} +6.55947 q^{94} +0.946319 q^{95} -4.26277 q^{96} -1.61319 q^{97} +1.10724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) \) 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

Introduction

L-functions

Modular forms

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Database

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