LMFDB - Embedded newform 4719.2.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4719,2,Mod(1,4719)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4719, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("4719.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4719 = 3 \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4719.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-1,5,3,-2,-1,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$37.6814047138$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.303952.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 6x^{3} + 2x^{2} + 5x + 1 $ x^5 - x^4 - 6x^3 + 2x^2 + 5*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.238066$ of defining polynomial
Character	$\chi$	$=$	4719.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.238066 q^{2} +1.00000 q^{3} -1.94332 q^{4} -2.57562 q^{5} +0.238066 q^{6} -3.96246 q^{7} -0.938770 q^{8} +1.00000 q^{9} -0.613168 q^{10} -1.94332 q^{12} +1.00000 q^{13} -0.943325 q^{14} -2.57562 q^{15} +3.66316 q^{16} +2.41792 q^{17} +0.238066 q^{18} +5.73860 q^{19} +5.00527 q^{20} -3.96246 q^{21} +5.60649 q^{23} -0.938770 q^{24} +1.63384 q^{25} +0.238066 q^{26} +1.00000 q^{27} +7.70034 q^{28} -1.34857 q^{29} -0.613168 q^{30} +2.22195 q^{31} +2.74961 q^{32} +0.575623 q^{34} +10.2058 q^{35} -1.94332 q^{36} -1.81067 q^{37} +1.36616 q^{38} +1.00000 q^{39} +2.41792 q^{40} -6.27194 q^{41} -0.943325 q^{42} -6.73333 q^{43} -2.57562 q^{45} +1.33471 q^{46} +9.82122 q^{47} +3.66316 q^{48} +8.70106 q^{49} +0.388960 q^{50} +2.41792 q^{51} -1.94332 q^{52} -5.46074 q^{53} +0.238066 q^{54} +3.71984 q^{56} +5.73860 q^{57} -0.321048 q^{58} -8.12697 q^{59} +5.00527 q^{60} +9.93172 q^{61} +0.528970 q^{62} -3.96246 q^{63} -6.67173 q^{64} -2.57562 q^{65} -2.57562 q^{67} -4.69880 q^{68} +5.60649 q^{69} +2.42965 q^{70} -11.5809 q^{71} -0.938770 q^{72} -11.2362 q^{73} -0.431059 q^{74} +1.63384 q^{75} -11.1520 q^{76} +0.238066 q^{78} -8.46529 q^{79} -9.43492 q^{80} +1.00000 q^{81} -1.49313 q^{82} +14.0165 q^{83} +7.70034 q^{84} -6.22765 q^{85} -1.60297 q^{86} -1.34857 q^{87} +1.50035 q^{89} -0.613168 q^{90} -3.96246 q^{91} -10.8952 q^{92} +2.22195 q^{93} +2.33809 q^{94} -14.7805 q^{95} +2.74961 q^{96} +12.5535 q^{97} +2.07142 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 5 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 6 q^{7} - 9 q^{8} + 5 q^{9} - 6 q^{10} + 3 q^{12} + 5 q^{13} + 8 q^{14} - 2 q^{15} + 11 q^{16} - 2 q^{17} - q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21}+ \cdots + q^{98}+O(q^{100}) $ 5 * q - q^2 + 5 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 6 * q^7 - 9 * q^8 + 5 * q^9 - 6 * q^10 + 3 * q^12 + 5 * q^13 + 8 * q^14 - 2 * q^15 + 11 * q^16 - 2 * q^17 - q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 + 8 * q^23 - 9 * q^24 + 7 * q^25 - q^26 + 5 * q^27 - 6 * q^29 - 6 * q^30 - 4 * q^31 - 19 * q^32 - 8 * q^34 + 2 * q^35 + 3 * q^36 - 10 * q^37 + 8 * q^38 + 5 * q^39 - 2 * q^40 - 6 * q^41 + 8 * q^42 - 26 * q^43 - 2 * q^45 - 26 * q^46 - 16 * q^47 + 11 * q^48 - 11 * q^49 + 39 * q^50 - 2 * q^51 + 3 * q^52 + 6 * q^53 - q^54 + 24 * q^56 - 12 * q^57 - 20 * q^58 + 6 * q^59 - 8 * q^60 - 14 * q^61 + 12 * q^62 - 6 * q^63 + 15 * q^64 - 2 * q^65 - 2 * q^67 + 12 * q^68 + 8 * q^69 - 10 * q^70 - 14 * q^71 - 9 * q^72 - 12 * q^73 - 36 * q^74 + 7 * q^75 - 18 * q^76 - q^78 + 8 * q^79 + 8 * q^80 + 5 * q^81 + 38 * q^82 + 22 * q^83 - 36 * q^85 - 16 * q^86 - 6 * q^87 - 8 * q^89 - 6 * q^90 - 6 * q^91 + 12 * q^92 - 4 * q^93 + 10 * q^94 - 32 * q^95 - 19 * q^96 + q^98

Coefficient data

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

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

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

$2$	0.238066	0.168338	0.0841689	−	0.996452i	$-0.473176\pi$
$2$	0.238066	0.168338	0.0841689	+	0.996452i	$0.473176\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.94332	−0.971662
$5$	−2.57562	−1.15185	−0.575927	−	0.817501i	$-0.695359\pi$
$5$	−2.57562	−1.15185	−0.575927	+	0.817501i	$0.695359\pi$
$6$	0.238066	0.0971899
$7$	−3.96246	−1.49767	−0.748834	−	0.662758i	$-0.769386\pi$
$7$	−3.96246	−1.49767	−0.748834	+	0.662758i	$0.769386\pi$
$8$	−0.938770	−0.331905
$9$	1.00000	0.333333
$10$	−0.613168	−0.193901
$11$	0	0
$11$	0	0
$12$	−1.94332	−0.560990
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.943325	−0.252114
$15$	−2.57562	−0.665023
$16$	3.66316	0.915790
$17$	2.41792	0.586431	0.293216	−	0.956046i	$-0.405275\pi$
$17$	2.41792	0.586431	0.293216	+	0.956046i	$0.405275\pi$
$18$	0.238066	0.0561126
$19$	5.73860	1.31653	0.658263	−	0.752788i	$-0.271292\pi$
$19$	5.73860	1.31653	0.658263	+	0.752788i	$0.271292\pi$
$20$	5.00527	1.11921
$21$	−3.96246	−0.864679
$22$	0	0
$23$	5.60649	1.16903	0.584516	−	0.811382i	$-0.301284\pi$
$23$	5.60649	1.16903	0.584516	+	0.811382i	$0.301284\pi$
$24$	−0.938770	−0.191626
$25$	1.63384	0.326767
$26$	0.238066	0.0466885
$27$	1.00000	0.192450
$28$	7.70034	1.45523
$29$	−1.34857	−0.250423	−0.125212	−	0.992130i	$-0.539961\pi$
$29$	−1.34857	−0.250423	−0.125212	+	0.992130i	$0.539961\pi$
$30$	−0.613168	−0.111949
$31$	2.22195	0.399074	0.199537	−	0.979890i	$-0.436056\pi$
$31$	2.22195	0.399074	0.199537	+	0.979890i	$0.436056\pi$
$32$	2.74961	0.486068
$33$	0	0
$34$	0.575623	0.0987186
$35$	10.2058	1.72509
$36$	−1.94332	−0.323887
$37$	−1.81067	−0.297673	−0.148836	−	0.988862i	$-0.547553\pi$
$37$	−1.81067	−0.297673	−0.148836	+	0.988862i	$0.547553\pi$
$38$	1.36616	0.221621
$39$	1.00000	0.160128
$40$	2.41792	0.382307
$41$	−6.27194	−0.979513	−0.489757	−	0.871859i	$-0.662914\pi$
$41$	−6.27194	−0.979513	−0.489757	+	0.871859i	$0.662914\pi$
$42$	−0.943325	−0.145558
$43$	−6.73333	−1.02682	−0.513411	−	0.858143i	$-0.671619\pi$
$43$	−6.73333	−1.02682	−0.513411	+	0.858143i	$0.671619\pi$
$44$	0	0
$45$	−2.57562	−0.383951
$46$	1.33471	0.196793
$47$	9.82122	1.43257	0.716286	−	0.697807i	$-0.245841\pi$
$47$	9.82122	1.43257	0.716286	+	0.697807i	$0.245841\pi$
$48$	3.66316	0.528732
$49$	8.70106	1.24301
$50$	0.388960	0.0550073
$51$	2.41792	0.338576
$52$	−1.94332	−0.269491
$53$	−5.46074	−0.750090	−0.375045	−	0.927007i	$-0.622373\pi$
$53$	−5.46074	−0.750090	−0.375045	+	0.927007i	$0.622373\pi$
$54$	0.238066	0.0323966
$55$	0	0
$56$	3.71984	0.497084
$57$	5.73860	0.760096
$58$	−0.321048	−0.0421557
$59$	−8.12697	−1.05804	−0.529021	−	0.848609i	$-0.677441\pi$
$59$	−8.12697	−1.05804	−0.529021	+	0.848609i	$0.677441\pi$
$60$	5.00527	0.646178
$61$	9.93172	1.27163	0.635813	−	0.771843i	$-0.280665\pi$
$61$	9.93172	1.27163	0.635813	+	0.771843i	$0.280665\pi$
$62$	0.528970	0.0671793
$63$	−3.96246	−0.499223
$64$	−6.67173	−0.833967
$65$	−2.57562	−0.319467
$66$	0	0
$67$	−2.57562	−0.314662	−0.157331	−	0.987546i	$-0.550289\pi$
$67$	−2.57562	−0.314662	−0.157331	+	0.987546i	$0.550289\pi$
$68$	−4.69880	−0.569813
$69$	5.60649	0.674942
$70$	2.42965	0.290399
$71$	−11.5809	−1.37440	−0.687200	−	0.726469i	$-0.741160\pi$
$71$	−11.5809	−1.37440	−0.687200	+	0.726469i	$0.741160\pi$
$72$	−0.938770	−0.110635
$73$	−11.2362	−1.31509	−0.657547	−	0.753414i	$-0.728406\pi$
$73$	−11.2362	−1.31509	−0.657547	+	0.753414i	$0.728406\pi$
$74$	−0.431059	−0.0501096
$75$	1.63384	0.188659
$76$	−11.1520	−1.27922
$77$	0	0
$78$	0.238066	0.0269556
$79$	−8.46529	−0.952420	−0.476210	−	0.879332i	$-0.657990\pi$
$79$	−8.46529	−0.952420	−0.476210	+	0.879332i	$0.657990\pi$
$80$	−9.43492	−1.05486
$81$	1.00000	0.111111
$82$	−1.49313	−0.164889
$83$	14.0165	1.53851	0.769254	−	0.638944i	$-0.220628\pi$
$83$	14.0165	1.53851	0.769254	+	0.638944i	$0.220628\pi$
$84$	7.70034	0.840176
$85$	−6.22765	−0.675483
$86$	−1.60297	−0.172853
$87$	−1.34857	−0.144582
$88$	0	0
$89$	1.50035	0.159037	0.0795186	−	0.996833i	$-0.474662\pi$
$89$	1.50035	0.159037	0.0795186	+	0.996833i	$0.474662\pi$
$90$	−0.613168	−0.0646335
$91$	−3.96246	−0.415378
$92$	−10.8952	−1.13591
$93$	2.22195	0.230406
$94$	2.33809	0.241156
$95$	−14.7805	−1.51644
$96$	2.74961	0.280631
$97$	12.5535	1.27462	0.637310	−	0.770608i	$-0.280047\pi$
$97$	12.5535	1.27462	0.637310	+	0.770608i	$0.280047\pi$
$98$	2.07142	0.209245
$99$	0	0
$100$	−3.17507	−0.317507
$101$	9.21549	0.916976	0.458488	−	0.888701i	$-0.348391\pi$
$101$	9.21549	0.916976	0.458488	+	0.888701i	$0.348391\pi$
$102$	0.575623	0.0569952
$103$	12.4365	1.22540	0.612700	−	0.790315i	$-0.290083\pi$
$103$	12.4365	1.22540	0.612700	+	0.790315i	$0.290083\pi$
$104$	−0.938770	−0.0920540
$105$	10.2058	0.995984
$106$	−1.30001	−0.126268
$107$	−19.2400	−1.86000	−0.929998	−	0.367564i	$-0.880192\pi$
$107$	−19.2400	−1.86000	−0.929998	+	0.367564i	$0.880192\pi$
$108$	−1.94332	−0.186997
$109$	−11.0193	−1.05546	−0.527729	−	0.849413i	$-0.676956\pi$
$109$	−11.0193	−1.05546	−0.527729	+	0.849413i	$0.676956\pi$
$110$	0	0
$111$	−1.81067	−0.171861
$112$	−14.5151	−1.37155
$113$	−7.81441	−0.735118	−0.367559	−	0.930000i	$-0.619806\pi$
$113$	−7.81441	−0.735118	−0.367559	+	0.930000i	$0.619806\pi$
$114$	1.36616	0.127953
$115$	−14.4402	−1.34656
$116$	2.62071	0.243327
$117$	1.00000	0.0924500
$118$	−1.93475	−0.178108
$119$	−9.58090	−0.878279
$120$	2.41792	0.220725
$121$	0	0
$122$	2.36440	0.214063
$123$	−6.27194	−0.565522
$124$	−4.31797	−0.387765
$125$	8.66997	0.775466
$126$	−0.943325	−0.0840380
$127$	−12.2654	−1.08838	−0.544191	−	0.838961i	$-0.683163\pi$
$127$	−12.2654	−1.08838	−0.544191	+	0.838961i	$0.683163\pi$
$128$	−7.08754	−0.626456
$129$	−6.73333	−0.592836
$130$	−0.613168	−0.0537783
$131$	13.4389	1.17417	0.587083	−	0.809527i	$-0.300276\pi$
$131$	13.4389	1.17417	0.587083	+	0.809527i	$0.300276\pi$
$132$	0	0
$133$	−22.7390	−1.97172
$134$	−0.613168	−0.0529696
$135$	−2.57562	−0.221674
$136$	−2.26987	−0.194640
$137$	−12.2077	−1.04297	−0.521487	−	0.853259i	$-0.674622\pi$
$137$	−12.2077	−1.04297	−0.521487	+	0.853259i	$0.674622\pi$
$138$	1.33471	0.113618
$139$	−15.4683	−1.31201	−0.656003	−	0.754758i	$-0.727754\pi$
$139$	−15.4683	−1.31201	−0.656003	+	0.754758i	$0.727754\pi$
$140$	−19.8332	−1.67621
$141$	9.82122	0.827096
$142$	−2.75701	−0.231363
$143$	0	0
$144$	3.66316	0.305263
$145$	3.47341	0.288451
$146$	−2.67494	−0.221380
$147$	8.70106	0.717651
$148$	3.51872	0.289237
$149$	−8.16595	−0.668981	−0.334490	−	0.942399i	$-0.608564\pi$
$149$	−8.16595	−0.668981	−0.334490	+	0.942399i	$0.608564\pi$
$150$	0.388960	0.0317585
$151$	−1.33305	−0.108482	−0.0542409	−	0.998528i	$-0.517274\pi$
$151$	−1.33305	−0.108482	−0.0542409	+	0.998528i	$0.517274\pi$
$152$	−5.38723	−0.436962
$153$	2.41792	0.195477
$154$	0	0
$155$	−5.72291	−0.459675
$156$	−1.94332	−0.155591
$157$	17.0633	1.36180	0.680900	−	0.732377i	$-0.261589\pi$
$157$	17.0633	1.36180	0.680900	+	0.732377i	$0.261589\pi$
$158$	−2.01529	−0.160328
$159$	−5.46074	−0.433064
$160$	−7.08197	−0.559879
$161$	−22.2155	−1.75082
$162$	0.238066	0.0187042
$163$	−14.7786	−1.15755	−0.578774	−	0.815488i	$-0.696468\pi$
$163$	−14.7786	−1.15755	−0.578774	+	0.815488i	$0.696468\pi$
$164$	12.1884	0.951756
$165$	0	0
$166$	3.33684	0.258989
$167$	−19.1853	−1.48460	−0.742300	−	0.670068i	$-0.766265\pi$
$167$	−19.1853	−1.48460	−0.742300	+	0.670068i	$0.766265\pi$
$168$	3.71984	0.286992
$169$	1.00000	0.0769231
$170$	−1.48259	−0.113709
$171$	5.73860	0.438842
$172$	13.0850	0.997725
$173$	10.8190	0.822550	0.411275	−	0.911511i	$-0.365084\pi$
$173$	10.8190	0.822550	0.411275	+	0.911511i	$0.365084\pi$
$174$	−0.321048	−0.0243386
$175$	−6.47400	−0.489389
$176$	0	0
$177$	−8.12697	−0.610861
$178$	0.357183	0.0267720
$179$	−4.60020	−0.343835	−0.171917	−	0.985111i	$-0.554996\pi$
$179$	−4.60020	−0.343835	−0.171917	+	0.985111i	$0.554996\pi$
$180$	5.00527	0.373071
$181$	11.6717	0.867553	0.433777	−	0.901020i	$-0.357181\pi$
$181$	11.6717	0.867553	0.433777	+	0.901020i	$0.357181\pi$
$182$	−0.943325	−0.0699239
$183$	9.93172	0.734174
$184$	−5.26320	−0.388008
$185$	4.66361	0.342875
$186$	0.528970	0.0387860
$187$	0	0
$188$	−19.0858	−1.39198
$189$	−3.96246	−0.288226
$190$	−3.51872	−0.255275
$191$	9.10706	0.658964	0.329482	−	0.944162i	$-0.393126\pi$
$191$	9.10706	0.658964	0.329482	+	0.944162i	$0.393126\pi$
$192$	−6.67173	−0.481491
$193$	−2.30178	−0.165685	−0.0828427	−	0.996563i	$-0.526400\pi$
$193$	−2.30178	−0.165685	−0.0828427	+	0.996563i	$0.526400\pi$
$194$	2.98857	0.214567
$195$	−2.57562	−0.184444
$196$	−16.9090	−1.20778
$197$	18.4273	1.31289	0.656447	−	0.754372i	$-0.272059\pi$
$197$	18.4273	1.31289	0.656447	+	0.754372i	$0.272059\pi$
$198$	0	0
$199$	23.3299	1.65381	0.826905	−	0.562341i	$-0.190099\pi$
$199$	23.3299	1.65381	0.826905	+	0.562341i	$0.190099\pi$
$200$	−1.53380	−0.108456
$201$	−2.57562	−0.181670
$202$	2.19389	0.154362
$203$	5.34365	0.375051
$204$	−4.69880	−0.328982
$205$	16.1542	1.12826
$206$	2.96069	0.206281
$207$	5.60649	0.389678
$208$	3.66316	0.253994
$209$	0	0
$210$	2.42965	0.167662
$211$	−11.5692	−0.796454	−0.398227	−	0.917287i	$-0.630374\pi$
$211$	−11.5692	−0.796454	−0.398227	+	0.917287i	$0.630374\pi$
$212$	10.6120	0.728834
$213$	−11.5809	−0.793510
$214$	−4.58037	−0.313108
$215$	17.3425	1.18275
$216$	−0.938770	−0.0638752
$217$	−8.80439	−0.597681
$218$	−2.62332	−0.177673
$219$	−11.2362	−0.759269
$220$	0	0
$221$	2.41792	0.162647
$222$	−0.431059	−0.0289308
$223$	10.9915	0.736043	0.368022	−	0.929817i	$-0.380035\pi$
$223$	10.9915	0.736043	0.368022	+	0.929817i	$0.380035\pi$
$224$	−10.8952	−0.727968
$225$	1.63384	0.108922
$226$	−1.86034	−0.123748
$227$	−26.6099	−1.76616	−0.883079	−	0.469224i	$-0.844534\pi$
$227$	−26.6099	−1.76616	−0.883079	+	0.469224i	$0.844534\pi$
$228$	−11.1520	−0.738557
$229$	−3.98619	−0.263415	−0.131708	−	0.991289i	$-0.542046\pi$
$229$	−3.98619	−0.263415	−0.131708	+	0.991289i	$0.542046\pi$
$230$	−3.43771	−0.226676
$231$	0	0
$232$	1.26600	0.0831168
$233$	−2.07106	−0.135679	−0.0678397	−	0.997696i	$-0.521611\pi$
$233$	−2.07106	−0.135679	−0.0678397	+	0.997696i	$0.521611\pi$
$234$	0.238066	0.0155628
$235$	−25.2958	−1.65011
$236$	15.7933	1.02806
$237$	−8.46529	−0.549880
$238$	−2.28088	−0.147848
$239$	−10.4860	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$239$	−10.4860	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$240$	−9.43492	−0.609022
$241$	1.27230	0.0819558	0.0409779	−	0.999160i	$-0.486953\pi$
$241$	1.27230	0.0819558	0.0409779	+	0.999160i	$0.486953\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−19.3006	−1.23559
$245$	−22.4106	−1.43176
$246$	−1.49313	−0.0951988
$247$	5.73860	0.365138
$248$	−2.08590	−0.132455
$249$	14.0165	0.888257
$250$	2.06402	0.130540
$251$	−7.14073	−0.450719	−0.225359	−	0.974276i	$-0.572356\pi$
$251$	−7.14073	−0.450719	−0.225359	+	0.974276i	$0.572356\pi$
$252$	7.70034	0.485076
$253$	0	0
$254$	−2.91998	−0.183216
$255$	−6.22765	−0.389990
$256$	11.6562	0.728510
$257$	6.96873	0.434697	0.217349	−	0.976094i	$-0.430259\pi$
$257$	6.96873	0.434697	0.217349	+	0.976094i	$0.430259\pi$
$258$	−1.60297	−0.0997968
$259$	7.17471	0.445815
$260$	5.00527	0.310414
$261$	−1.34857	−0.0834744
$262$	3.19935	0.197657
$263$	6.59552	0.406697	0.203348	−	0.979106i	$-0.434818\pi$
$263$	6.59552	0.406697	0.203348	+	0.979106i	$0.434818\pi$
$264$	0	0
$265$	14.0648	0.863994
$266$	−5.41336	−0.331915
$267$	1.50035	0.0918202
$268$	5.00527	0.305746
$269$	1.50492	0.0917565	0.0458783	−	0.998947i	$-0.485391\pi$
$269$	1.50492	0.0917565	0.0458783	+	0.998947i	$0.485391\pi$
$270$	−0.613168	−0.0373162
$271$	−18.6849	−1.13503	−0.567515	−	0.823363i	$-0.692095\pi$
$271$	−18.6849	−1.13503	−0.567515	+	0.823363i	$0.692095\pi$
$272$	8.85722	0.537048
$273$	−3.96246	−0.239819
$274$	−2.90623	−0.175572
$275$	0	0
$276$	−10.8952	−0.655815
$277$	−32.5192	−1.95389	−0.976946	−	0.213486i	$-0.931518\pi$
$277$	−32.5192	−1.95389	−0.976946	+	0.213486i	$0.931518\pi$
$278$	−3.68248	−0.220860
$279$	2.22195	0.133025
$280$	−9.58090	−0.572568
$281$	−19.7127	−1.17596	−0.587979	−	0.808876i	$-0.700076\pi$
$281$	−19.7127	−1.17596	−0.587979	+	0.808876i	$0.700076\pi$
$282$	2.33809	0.139232
$283$	−31.0570	−1.84615	−0.923073	−	0.384625i	$-0.874331\pi$
$283$	−31.0570	−1.84615	−0.923073	+	0.384625i	$0.874331\pi$
$284$	22.5054	1.33545
$285$	−14.7805	−0.875520
$286$	0	0
$287$	24.8523	1.46699
$288$	2.74961	0.162023
$289$	−11.1537	−0.656098
$290$	0.826899	0.0485572
$291$	12.5535	0.735902
$292$	21.8355	1.27783
$293$	1.33899	0.0782249	0.0391124	−	0.999235i	$-0.487547\pi$
$293$	1.33899	0.0782249	0.0391124	+	0.999235i	$0.487547\pi$
$294$	2.07142	0.120808
$295$	20.9320	1.21871
$296$	1.69981	0.0987992
$297$	0	0
$298$	−1.94403	−0.112615
$299$	5.60649	0.324231
$300$	−3.17507	−0.183313
$301$	26.6805	1.53784
$302$	−0.317353	−0.0182616
$303$	9.21549	0.529416
$304$	21.0214	1.20566
$305$	−25.5804	−1.46473
$306$	0.575623	0.0329062
$307$	−26.8143	−1.53037	−0.765187	−	0.643808i	$-0.777353\pi$
$307$	−26.8143	−1.53037	−0.765187	+	0.643808i	$0.777353\pi$
$308$	0	0
$309$	12.4365	0.707485
$310$	−1.36243	−0.0773807
$311$	27.8859	1.58126	0.790631	−	0.612293i	$-0.209753\pi$
$311$	27.8859	1.58126	0.790631	+	0.612293i	$0.209753\pi$
$312$	−0.938770	−0.0531474
$313$	1.71284	0.0968155	0.0484077	−	0.998828i	$-0.484585\pi$
$313$	1.71284	0.0968155	0.0484077	+	0.998828i	$0.484585\pi$
$314$	4.06219	0.229242
$315$	10.2058	0.575031
$316$	16.4508	0.925430
$317$	32.9427	1.85025	0.925124	−	0.379665i	$-0.123961\pi$
$317$	32.9427	1.85025	0.925124	+	0.379665i	$0.123961\pi$
$318$	−1.30001	−0.0729011
$319$	0	0
$320$	17.1839	0.960608
$321$	−19.2400	−1.07387
$322$	−5.28874	−0.294730
$323$	13.8755	0.772052
$324$	−1.94332	−0.107962
$325$	1.63384	0.0906289
$326$	−3.51827	−0.194859
$327$	−11.0193	−0.609369
$328$	5.88792	0.325106
$329$	−38.9161	−2.14552
$330$	0	0
$331$	−7.18387	−0.394861	−0.197431	−	0.980317i	$-0.563260\pi$
$331$	−7.18387	−0.394861	−0.197431	+	0.980317i	$0.563260\pi$
$332$	−27.2385	−1.49491
$333$	−1.81067	−0.0992242
$334$	−4.56735	−0.249914
$335$	6.63384	0.362445
$336$	−14.5151	−0.791864
$337$	−8.79298	−0.478984	−0.239492	−	0.970898i	$-0.576981\pi$
$337$	−8.79298	−0.478984	−0.239492	+	0.970898i	$0.576981\pi$
$338$	0.238066	0.0129491
$339$	−7.81441	−0.424420
$340$	12.1023	0.656342
$341$	0	0
$342$	1.36616	0.0738737
$343$	−6.74036	−0.363945
$344$	6.32105	0.340808
$345$	−14.4402	−0.777434
$346$	2.57562	0.138466
$347$	19.1713	1.02917	0.514586	−	0.857439i	$-0.327946\pi$
$347$	19.1713	1.02917	0.514586	+	0.857439i	$0.327946\pi$
$348$	2.62071	0.140485
$349$	17.1498	0.918010	0.459005	−	0.888434i	$-0.348206\pi$
$349$	17.1498	0.918010	0.459005	+	0.888434i	$0.348206\pi$
$350$	−1.54124	−0.0823826
$351$	1.00000	0.0533761
$352$	0	0
$353$	−24.3489	−1.29596	−0.647981	−	0.761656i	$-0.724386\pi$
$353$	−24.3489	−1.29596	−0.647981	+	0.761656i	$0.724386\pi$
$354$	−1.93475	−0.102831
$355$	29.8280	1.58311
$356$	−2.91568	−0.154530
$357$	−9.58090	−0.507075
$358$	−1.09515	−0.0578804
$359$	−14.2222	−0.750618	−0.375309	−	0.926900i	$-0.622463\pi$
$359$	−14.2222	−0.750618	−0.375309	+	0.926900i	$0.622463\pi$
$360$	2.41792	0.127436
$361$	13.9315	0.733239
$362$	2.77864	0.146042
$363$	0	0
$364$	7.70034	0.403607
$365$	28.9401	1.51480
$366$	2.36440	0.123589
$367$	−31.1312	−1.62504	−0.812518	−	0.582935i	$-0.801904\pi$
$367$	−31.1312	−1.62504	−0.812518	+	0.582935i	$0.801904\pi$
$368$	20.5375	1.07059
$369$	−6.27194	−0.326504
$370$	1.11025	0.0577189
$371$	21.6379	1.12339
$372$	−4.31797	−0.223876
$373$	21.0326	1.08903	0.544513	−	0.838753i	$-0.316715\pi$
$373$	21.0326	1.08903	0.544513	+	0.838753i	$0.316715\pi$
$374$	0	0
$375$	8.66997	0.447715
$376$	−9.21987	−0.475478
$377$	−1.34857	−0.0694549
$378$	−0.943325	−0.0485194
$379$	0.403809	0.0207423	0.0103711	−	0.999946i	$-0.496699\pi$
$379$	0.403809	0.0207423	0.0103711	+	0.999946i	$0.496699\pi$
$380$	28.7233	1.47347
$381$	−12.2654	−0.628378
$382$	2.16808	0.110929
$383$	8.54282	0.436517	0.218259	−	0.975891i	$-0.429962\pi$
$383$	8.54282	0.436517	0.218259	+	0.975891i	$0.429962\pi$
$384$	−7.08754	−0.361684
$385$	0	0
$386$	−0.547974	−0.0278911
$387$	−6.73333	−0.342274
$388$	−24.3956	−1.23850
$389$	−6.29897	−0.319370	−0.159685	−	0.987168i	$-0.551048\pi$
$389$	−6.29897	−0.319370	−0.159685	+	0.987168i	$0.551048\pi$
$390$	−0.613168	−0.0310489
$391$	13.5560	0.685558
$392$	−8.16829	−0.412561
$393$	13.4389	0.677905
$394$	4.38692	0.221010
$395$	21.8034	1.09705
$396$	0	0
$397$	−15.5293	−0.779392	−0.389696	−	0.920944i	$-0.627420\pi$
$397$	−15.5293	−0.779392	−0.389696	+	0.920944i	$0.627420\pi$
$398$	5.55404	0.278399
$399$	−22.7390	−1.13837
$400$	5.98500	0.299250
$401$	3.19768	0.159684	0.0798421	−	0.996808i	$-0.474558\pi$
$401$	3.19768	0.159684	0.0798421	+	0.996808i	$0.474558\pi$
$402$	−0.613168	−0.0305820
$403$	2.22195	0.110683
$404$	−17.9087	−0.890991
$405$	−2.57562	−0.127984
$406$	1.27214	0.0631352
$407$	0	0
$408$	−2.26987	−0.112375
$409$	−1.79677	−0.0888448	−0.0444224	−	0.999013i	$-0.514145\pi$
$409$	−1.79677	−0.0888448	−0.0444224	+	0.999013i	$0.514145\pi$
$410$	3.84575	0.189928
$411$	−12.2077	−0.602161
$412$	−24.1681	−1.19068
$413$	32.2028	1.58459
$414$	1.33471	0.0655975
$415$	−36.1011	−1.77214
$416$	2.74961	0.134811
$417$	−15.4683	−0.757487
$418$	0	0
$419$	−26.4076	−1.29010	−0.645048	−	0.764142i	$-0.723163\pi$
$419$	−26.4076	−1.29010	−0.645048	+	0.764142i	$0.723163\pi$
$420$	−19.8332	−0.967760
$421$	−31.1102	−1.51622	−0.758108	−	0.652129i	$-0.773876\pi$
$421$	−31.1102	−1.51622	−0.758108	+	0.652129i	$0.773876\pi$
$422$	−2.75422	−0.134073
$423$	9.82122	0.477524
$424$	5.12638	0.248959
$425$	3.95048	0.191627
$426$	−2.75701	−0.133578
$427$	−39.3540	−1.90447
$428$	37.3895	1.80729
$429$	0	0
$430$	4.12866	0.199102
$431$	19.5564	0.942001	0.471000	−	0.882133i	$-0.343893\pi$
$431$	19.5564	0.942001	0.471000	+	0.882133i	$0.343893\pi$
$432$	3.66316	0.176244
$433$	−11.7186	−0.563160	−0.281580	−	0.959538i	$-0.590858\pi$
$433$	−11.7186	−0.563160	−0.281580	+	0.959538i	$0.590858\pi$
$434$	−2.09602	−0.100612
$435$	3.47341	0.166537
$436$	21.4141	1.02555
$437$	32.1734	1.53906
$438$	−2.67494	−0.127814
$439$	16.0319	0.765162	0.382581	−	0.923922i	$-0.375035\pi$
$439$	16.0319	0.765162	0.382581	+	0.923922i	$0.375035\pi$
$440$	0	0
$441$	8.70106	0.414336
$442$	0.575623	0.0273796
$443$	31.9047	1.51584	0.757918	−	0.652350i	$-0.226217\pi$
$443$	31.9047	1.51584	0.757918	+	0.652350i	$0.226217\pi$
$444$	3.51872	0.166991
$445$	−3.86435	−0.183188
$446$	2.61669	0.123904
$447$	−8.16595	−0.386236
$448$	26.4364	1.24900
$449$	−11.1360	−0.525538	−0.262769	−	0.964859i	$-0.584636\pi$
$449$	−11.1360	−0.525538	−0.262769	+	0.964859i	$0.584636\pi$
$450$	0.388960	0.0183358
$451$	0	0
$452$	15.1859	0.714286
$453$	−1.33305	−0.0626320
$454$	−6.33489	−0.297311
$455$	10.2058	0.478455
$456$	−5.38723	−0.252280
$457$	−24.5856	−1.15007	−0.575033	−	0.818130i	$-0.695011\pi$
$457$	−24.5856	−1.15007	−0.575033	+	0.818130i	$0.695011\pi$
$458$	−0.948976	−0.0443427
$459$	2.41792	0.112859
$460$	28.0620	1.30840
$461$	−28.4979	−1.32728	−0.663640	−	0.748052i	$-0.730989\pi$
$461$	−28.4979	−1.32728	−0.663640	+	0.748052i	$0.730989\pi$
$462$	0	0
$463$	39.7372	1.84675	0.923373	−	0.383904i	$-0.125421\pi$
$463$	39.7372	1.84675	0.923373	+	0.383904i	$0.125421\pi$
$464$	−4.94003	−0.229335
$465$	−5.72291	−0.265394
$466$	−0.493048	−0.0228400
$467$	29.9707	1.38688	0.693439	−	0.720515i	$-0.256095\pi$
$467$	29.9707	1.38688	0.693439	+	0.720515i	$0.256095\pi$
$468$	−1.94332	−0.0898302
$469$	10.2058	0.471260
$470$	−6.02205	−0.277777
$471$	17.0633	0.786235
$472$	7.62936	0.351170
$473$	0	0
$474$	−2.01529	−0.0925656
$475$	9.37593	0.430197
$476$	18.6188	0.853391
$477$	−5.46074	−0.250030
$478$	−2.49635	−0.114180
$479$	5.77861	0.264032	0.132016	−	0.991248i	$-0.457855\pi$
$479$	5.77861	0.264032	0.132016	+	0.991248i	$0.457855\pi$
$480$	−7.08197	−0.323246
$481$	−1.81067	−0.0825595
$482$	0.302890	0.0137963
$483$	−22.2155	−1.01084
$484$	0	0
$485$	−32.3332	−1.46818
$486$	0.238066	0.0107989
$487$	−27.0163	−1.22422	−0.612112	−	0.790771i	$-0.709680\pi$
$487$	−27.0163	−1.22422	−0.612112	+	0.790771i	$0.709680\pi$
$488$	−9.32361	−0.422060
$489$	−14.7786	−0.668310
$490$	−5.33521	−0.241020
$491$	18.7819	0.847616	0.423808	−	0.905752i	$-0.360693\pi$
$491$	18.7819	0.847616	0.423808	+	0.905752i	$0.360693\pi$
$492$	12.1884	0.549497
$493$	−3.26073	−0.146856
$494$	1.36616	0.0614666
$495$	0	0
$496$	8.13937	0.365468
$497$	45.8888	2.05839
$498$	3.33684	0.149527
$499$	−9.05708	−0.405451	−0.202725	−	0.979236i	$-0.564980\pi$
$499$	−9.05708	−0.405451	−0.202725	+	0.979236i	$0.564980\pi$
$500$	−16.8486	−0.753491
$501$	−19.1853	−0.857134
$502$	−1.69996	−0.0758730
$503$	−5.67441	−0.253009	−0.126505	−	0.991966i	$-0.540376\pi$
$503$	−5.67441	−0.253009	−0.126505	+	0.991966i	$0.540376\pi$
$504$	3.71984	0.165695
$505$	−23.7356	−1.05622
$506$	0	0
$507$	1.00000	0.0444116
$508$	23.8357	1.05754
$509$	−30.6372	−1.35797	−0.678984	−	0.734153i	$-0.737580\pi$
$509$	−30.6372	−1.35797	−0.678984	+	0.734153i	$0.737580\pi$
$510$	−1.48259	−0.0656502
$511$	44.5228	1.96957
$512$	16.9500	0.749092
$513$	5.73860	0.253365
$514$	1.65902	0.0731760
$515$	−32.0316	−1.41148
$516$	13.0850	0.576037
$517$	0	0
$518$	1.70805	0.0750475
$519$	10.8190	0.474900
$520$	2.41792	0.106033
$521$	3.07201	0.134587	0.0672936	−	0.997733i	$-0.478564\pi$
$521$	3.07201	0.134587	0.0672936	+	0.997733i	$0.478564\pi$
$522$	−0.321048	−0.0140519
$523$	−6.96310	−0.304475	−0.152238	−	0.988344i	$-0.548648\pi$
$523$	−6.96310	−0.304475	−0.152238	+	0.988344i	$0.548648\pi$
$524$	−26.1162	−1.14089
$525$	−6.47400	−0.282549
$526$	1.57017	0.0684625
$527$	5.37250	0.234030
$528$	0	0
$529$	8.43268	0.366638
$530$	3.34835	0.145443
$531$	−8.12697	−0.352680
$532$	44.1892	1.91584
$533$	−6.27194	−0.271668
$534$	0.357183	0.0154568
$535$	49.5549	2.14244
$536$	2.41792	0.104438
$537$	−4.60020	−0.198513
$538$	0.358269	0.0154461
$539$	0	0
$540$	5.00527	0.215393
$541$	−0.375581	−0.0161475	−0.00807375	−	0.999967i	$-0.502570\pi$
$541$	−0.375581	−0.0161475	−0.00807375	+	0.999967i	$0.502570\pi$
$542$	−4.44824	−0.191068
$543$	11.6717	0.500882
$544$	6.64834	0.285045
$545$	28.3816	1.21573
$546$	−0.943325	−0.0403706
$547$	−22.3507	−0.955648	−0.477824	−	0.878456i	$-0.658574\pi$
$547$	−22.3507	−0.955648	−0.477824	+	0.878456i	$0.658574\pi$
$548$	23.7235	1.01342
$549$	9.93172	0.423876
$550$	0	0
$551$	−7.73891	−0.329688
$552$	−5.26320	−0.224017
$553$	33.5433	1.42641
$554$	−7.74172	−0.328914
$555$	4.66361	0.197959
$556$	30.0600	1.27483
$557$	−28.5757	−1.21079	−0.605395	−	0.795925i	$-0.706985\pi$
$557$	−28.5757	−1.21079	−0.605395	+	0.795925i	$0.706985\pi$
$558$	0.528970	0.0223931
$559$	−6.73333	−0.284789
$560$	37.3855	1.57982
$561$	0	0
$562$	−4.69291	−0.197958
$563$	44.9158	1.89297	0.946487	−	0.322741i	$-0.104604\pi$
$563$	44.9158	1.89297	0.946487	+	0.322741i	$0.104604\pi$
$564$	−19.0858	−0.803658
$565$	20.1270	0.846748
$566$	−7.39360	−0.310776
$567$	−3.96246	−0.166408
$568$	10.8718	0.456171
$569$	33.8306	1.41825	0.709126	−	0.705082i	$-0.249090\pi$
$569$	33.8306	1.41825	0.709126	+	0.705082i	$0.249090\pi$
$570$	−3.51872	−0.147383
$571$	−8.16423	−0.341663	−0.170831	−	0.985300i	$-0.554645\pi$
$571$	−8.16423	−0.341663	−0.170831	+	0.985300i	$0.554645\pi$
$572$	0	0
$573$	9.10706	0.380453
$574$	5.91648	0.246949
$575$	9.16008	0.382002
$576$	−6.67173	−0.277989
$577$	−44.1822	−1.83933	−0.919664	−	0.392706i	$-0.871539\pi$
$577$	−44.1822	−1.83933	−0.919664	+	0.392706i	$0.871539\pi$
$578$	−2.65531	−0.110446
$579$	−2.30178	−0.0956585
$580$	−6.74996	−0.280277
$581$	−55.5396	−2.30417
$582$	2.98857	0.123880
$583$	0	0
$584$	10.5482	0.436487
$585$	−2.57562	−0.106489
$586$	0.318769	0.0131682
$587$	43.6118	1.80005	0.900027	−	0.435835i	$-0.143547\pi$
$587$	43.6118	1.80005	0.900027	+	0.435835i	$0.143547\pi$
$588$	−16.9090	−0.697315
$589$	12.7509	0.525391
$590$	4.98319	0.205155
$591$	18.4273	0.757999
$592$	−6.63278	−0.272606
$593$	−33.2765	−1.36650	−0.683252	−	0.730183i	$-0.739435\pi$
$593$	−33.2765	−1.36650	−0.683252	+	0.730183i	$0.739435\pi$
$594$	0	0
$595$	24.6768	1.01165
$596$	15.8691	0.650023
$597$	23.3299	0.954828
$598$	1.33471	0.0545804
$599$	−35.8898	−1.46642	−0.733208	−	0.680005i	$-0.761978\pi$
$599$	−35.8898	−1.46642	−0.733208	+	0.680005i	$0.761978\pi$
$600$	−1.53380	−0.0626170
$601$	−2.29897	−0.0937769	−0.0468885	−	0.998900i	$-0.514931\pi$
$601$	−2.29897	−0.0937769	−0.0468885	+	0.998900i	$0.514931\pi$
$602$	6.35172	0.258877
$603$	−2.57562	−0.104887
$604$	2.59054	0.105408
$605$	0	0
$606$	2.19389	0.0891208
$607$	−32.6616	−1.32569	−0.662847	−	0.748755i	$-0.730652\pi$
$607$	−32.6616	−1.32569	−0.662847	+	0.748755i	$0.730652\pi$
$608$	15.7789	0.639920
$609$	5.34365	0.216536
$610$	−6.08981	−0.246569
$611$	9.82122	0.397324
$612$	−4.69880	−0.189938
$613$	−4.91723	−0.198605	−0.0993025	−	0.995057i	$-0.531661\pi$
$613$	−4.91723	−0.198605	−0.0993025	+	0.995057i	$0.531661\pi$
$614$	−6.38357	−0.257620
$615$	16.1542	0.651399
$616$	0	0
$617$	−26.6795	−1.07408	−0.537038	−	0.843558i	$-0.680457\pi$
$617$	−26.6795	−1.07408	−0.537038	+	0.843558i	$0.680457\pi$
$618$	2.96069	0.119097
$619$	−14.5859	−0.586257	−0.293129	−	0.956073i	$-0.594696\pi$
$619$	−14.5859	−0.586257	−0.293129	+	0.956073i	$0.594696\pi$
$620$	11.1215	0.446649
$621$	5.60649	0.224981
$622$	6.63867	0.266186
$623$	−5.94509	−0.238185
$624$	3.66316	0.146644
$625$	−30.4998	−1.21999
$626$	0.407769	0.0162977
$627$	0	0
$628$	−33.1595	−1.32321
$629$	−4.37806	−0.174565
$630$	2.42965	0.0967995
$631$	−10.1596	−0.404447	−0.202224	−	0.979339i	$-0.564817\pi$
$631$	−10.1596	−0.404447	−0.202224	+	0.979339i	$0.564817\pi$
$632$	7.94696	0.316113
$633$	−11.5692	−0.459833
$634$	7.84253	0.311467
$635$	31.5912	1.25366
$636$	10.6120	0.420792
$637$	8.70106	0.344748
$638$	0	0
$639$	−11.5809	−0.458133
$640$	18.2548	0.721585
$641$	−28.7390	−1.13512	−0.567561	−	0.823332i	$-0.692113\pi$
$641$	−28.7390	−1.13512	−0.567561	+	0.823332i	$0.692113\pi$
$642$	−4.58037	−0.180773
$643$	20.2176	0.797305	0.398652	−	0.917102i	$-0.369478\pi$
$643$	20.2176	0.797305	0.398652	+	0.917102i	$0.369478\pi$
$644$	43.1718	1.70121
$645$	17.3425	0.682861
$646$	3.30327	0.129966
$647$	46.7968	1.83977	0.919886	−	0.392187i	$-0.128281\pi$
$647$	46.7968	1.83977	0.919886	+	0.392187i	$0.128281\pi$
$648$	−0.938770	−0.0368784
$649$	0	0
$650$	0.388960	0.0152563
$651$	−8.80439	−0.345071
$652$	28.7196	1.12474
$653$	15.7046	0.614568	0.307284	−	0.951618i	$-0.400580\pi$
$653$	15.7046	0.614568	0.307284	+	0.951618i	$0.400580\pi$
$654$	−2.62332	−0.102580
$655$	−34.6136	−1.35247
$656$	−22.9751	−0.897028
$657$	−11.2362	−0.438364
$658$	−9.26460	−0.361172
$659$	−23.0124	−0.896436	−0.448218	−	0.893924i	$-0.647941\pi$
$659$	−23.0124	−0.896436	−0.448218	+	0.893924i	$0.647941\pi$
$660$	0	0
$661$	−35.2126	−1.36961	−0.684806	−	0.728726i	$-0.740113\pi$
$661$	−35.2126	−1.36961	−0.684806	+	0.728726i	$0.740113\pi$
$662$	−1.71023	−0.0664701
$663$	2.41792	0.0939042
$664$	−13.1582	−0.510639
$665$	58.5670	2.27113
$666$	−0.431059	−0.0167032
$667$	−7.56074	−0.292753
$668$	37.2832	1.44253
$669$	10.9915	0.424955
$670$	1.57929	0.0610132
$671$	0	0
$672$	−10.8952	−0.420292
$673$	26.9620	1.03931	0.519655	−	0.854376i	$-0.326060\pi$
$673$	26.9620	1.03931	0.519655	+	0.854376i	$0.326060\pi$
$674$	−2.09331	−0.0806311
$675$	1.63384	0.0628864
$676$	−1.94332	−0.0747433
$677$	−25.0237	−0.961738	−0.480869	−	0.876792i	$-0.659679\pi$
$677$	−25.0237	−0.961738	−0.480869	+	0.876792i	$0.659679\pi$
$678$	−1.86034	−0.0714460
$679$	−49.7429	−1.90896
$680$	5.84633	0.224197
$681$	−26.6099	−1.01969
$682$	0	0
$683$	−41.9106	−1.60367	−0.801833	−	0.597548i	$-0.796142\pi$
$683$	−41.9106	−1.60367	−0.801833	+	0.597548i	$0.796142\pi$
$684$	−11.1520	−0.426406
$685$	31.4424	1.20135
$686$	−1.60465	−0.0612658
$687$	−3.98619	−0.152083
$688$	−24.6653	−0.940354
$689$	−5.46074	−0.208037
$690$	−3.43771	−0.130872
$691$	25.3836	0.965636	0.482818	−	0.875721i	$-0.339613\pi$
$691$	25.3836	0.965636	0.482818	+	0.875721i	$0.339613\pi$
$692$	−21.0248	−0.799241
$693$	0	0
$694$	4.56404	0.173248
$695$	39.8406	1.51124
$696$	1.26600	0.0479875
$697$	−15.1651	−0.574417
$698$	4.08279	0.154536
$699$	−2.07106	−0.0783346
$700$	12.5811	0.475521
$701$	46.2825	1.74807	0.874034	−	0.485865i	$-0.161495\pi$
$701$	46.2825	1.74807	0.874034	+	0.485865i	$0.161495\pi$
$702$	0.238066	0.00898521
$703$	−10.3907	−0.391894
$704$	0	0
$705$	−25.2958	−0.952693
$706$	−5.79664	−0.218160
$707$	−36.5160	−1.37333
$708$	15.7933	0.593550
$709$	−44.1487	−1.65804	−0.829020	−	0.559220i	$-0.811101\pi$
$709$	−44.1487	−1.65804	−0.829020	+	0.559220i	$0.811101\pi$
$710$	7.10103	0.266497
$711$	−8.46529	−0.317473
$712$	−1.40849	−0.0527853
$713$	12.4573	0.466531
$714$	−2.28088	−0.0853599
$715$	0	0
$716$	8.93968	0.334091
$717$	−10.4860	−0.391605
$718$	−3.38581	−0.126357
$719$	−9.25157	−0.345026	−0.172513	−	0.985007i	$-0.555189\pi$
$719$	−9.25157	−0.345026	−0.172513	+	0.985007i	$0.555189\pi$
$720$	−9.43492	−0.351619
$721$	−49.2789	−1.83524
$722$	3.31662	0.123432
$723$	1.27230	0.0473172
$724$	−22.6820	−0.842969
$725$	−2.20334	−0.0818301
$726$	0	0
$727$	−33.2301	−1.23244	−0.616218	−	0.787575i	$-0.711336\pi$
$727$	−33.2301	−1.23244	−0.616218	+	0.787575i	$0.711336\pi$
$728$	3.71984	0.137866
$729$	1.00000	0.0370370
$730$	6.88965	0.254997
$731$	−16.2806	−0.602161
$732$	−19.3006	−0.713369
$733$	−26.2204	−0.968474	−0.484237	−	0.874937i	$-0.660903\pi$
$733$	−26.2204	−0.968474	−0.484237	+	0.874937i	$0.660903\pi$
$734$	−7.41128	−0.273555
$735$	−22.4106	−0.826629
$736$	15.4157	0.568229
$737$	0	0
$738$	−1.49313	−0.0549630
$739$	14.1542	0.520672	0.260336	−	0.965518i	$-0.416167\pi$
$739$	14.1542	0.520672	0.260336	+	0.965518i	$0.416167\pi$
$740$	−9.06291	−0.333159
$741$	5.73860	0.210813
$742$	5.15125	0.189108
$743$	6.63959	0.243583	0.121792	−	0.992556i	$-0.461136\pi$
$743$	6.63959	0.243583	0.121792	+	0.992556i	$0.461136\pi$
$744$	−2.08590	−0.0764729
$745$	21.0324	0.770568
$746$	5.00713	0.183324
$747$	14.0165	0.512836
$748$	0	0
$749$	76.2375	2.78566
$750$	2.06402	0.0753674
$751$	−26.8488	−0.979726	−0.489863	−	0.871799i	$-0.662953\pi$
$751$	−26.8488	−0.979726	−0.489863	+	0.871799i	$0.662953\pi$
$752$	35.9767	1.31194
$753$	−7.14073	−0.260223
$754$	−0.321048	−0.0116919
$755$	3.43343	0.124955
$756$	7.70034	0.280059
$757$	11.9856	0.435624	0.217812	−	0.975991i	$-0.430108\pi$
$757$	11.9856	0.435624	0.217812	+	0.975991i	$0.430108\pi$
$758$	0.0961330	0.00349171
$759$	0	0
$760$	13.8755	0.503316
$761$	13.6453	0.494641	0.247320	−	0.968934i	$-0.420450\pi$
$761$	13.6453	0.494641	0.247320	+	0.968934i	$0.420450\pi$
$762$	−2.91998	−0.105780
$763$	43.6635	1.58072
$764$	−17.6980	−0.640291
$765$	−6.22765	−0.225161
$766$	2.03375	0.0734824
$767$	−8.12697	−0.293448
$768$	11.6562	0.420606
$769$	−9.26305	−0.334034	−0.167017	−	0.985954i	$-0.553414\pi$
$769$	−9.26305	−0.334034	−0.167017	+	0.985954i	$0.553414\pi$
$770$	0	0
$771$	6.96873	0.250973
$772$	4.47310	0.160990
$773$	−30.8916	−1.11109	−0.555546	−	0.831486i	$-0.687491\pi$
$773$	−30.8916	−1.11109	−0.555546	+	0.831486i	$0.687491\pi$
$774$	−1.60297	−0.0576177
$775$	3.63030	0.130404
$776$	−11.7849	−0.423053
$777$	7.17471	0.257391
$778$	−1.49957	−0.0537621
$779$	−35.9922	−1.28955
$780$	5.00527	0.179218
$781$	0	0
$782$	3.22722	0.115405
$783$	−1.34857	−0.0481940
$784$	31.8734	1.13833
$785$	−43.9486	−1.56859
$786$	3.19935	0.114117
$787$	−44.7373	−1.59471	−0.797356	−	0.603510i	$-0.793769\pi$
$787$	−44.7373	−1.59471	−0.797356	+	0.603510i	$0.793769\pi$
$788$	−35.8103	−1.27569
$789$	6.59552	0.234807
$790$	5.19064	0.184675
$791$	30.9642	1.10096
$792$	0	0
$793$	9.93172	0.352686
$794$	−3.69699	−0.131201
$795$	14.0648	0.498827
$796$	−45.3375	−1.60695
$797$	14.6912	0.520390	0.260195	−	0.965556i	$-0.416213\pi$
$797$	14.6912	0.520390	0.260195	+	0.965556i	$0.416213\pi$
$798$	−5.41336	−0.191631
$799$	23.7469	0.840105
$800$	4.49242	0.158831
$801$	1.50035	0.0530124
$802$	0.761257	0.0268809
$803$	0	0
$804$	5.00527	0.176522
$805$	57.2186	2.01669
$806$	0.528970	0.0186322
$807$	1.50492	0.0529756
$808$	−8.65123	−0.304349
$809$	−30.2159	−1.06233	−0.531167	−	0.847267i	$-0.678246\pi$
$809$	−30.2159	−1.06233	−0.531167	+	0.847267i	$0.678246\pi$
$810$	−0.613168	−0.0215445
$811$	−20.5495	−0.721589	−0.360794	−	0.932645i	$-0.617494\pi$
$811$	−20.5495	−0.721589	−0.360794	+	0.932645i	$0.617494\pi$
$812$	−10.3844	−0.364423
$813$	−18.6849	−0.655310
$814$	0	0
$815$	38.0640	1.33333
$816$	8.85722	0.310065
$817$	−38.6399	−1.35184
$818$	−0.427750	−0.0149559
$819$	−3.96246	−0.138459
$820$	−31.3928	−1.09628
$821$	30.1178	1.05112	0.525560	−	0.850756i	$-0.323856\pi$
$821$	30.1178	1.05112	0.525560	+	0.850756i	$0.323856\pi$
$822$	−2.90623	−0.101367
$823$	4.58345	0.159769	0.0798845	−	0.996804i	$-0.474545\pi$
$823$	4.58345	0.159769	0.0798845	+	0.996804i	$0.474545\pi$
$824$	−11.6750	−0.406717
$825$	0	0
$826$	7.66637	0.266747
$827$	10.0800	0.350514	0.175257	−	0.984523i	$-0.443924\pi$
$827$	10.0800	0.350514	0.175257	+	0.984523i	$0.443924\pi$
$828$	−10.8952	−0.378635
$829$	−3.55302	−0.123402	−0.0617008	−	0.998095i	$-0.519652\pi$
$829$	−3.55302	−0.123402	−0.0617008	+	0.998095i	$0.519652\pi$
$830$	−8.59444	−0.298317
$831$	−32.5192	−1.12808
$832$	−6.67173	−0.231301
$833$	21.0384	0.728939
$834$	−3.68248	−0.127514
$835$	49.4140	1.71004
$836$	0	0
$837$	2.22195	0.0768019
$838$	−6.28675	−0.217172
$839$	14.6898	0.507149	0.253575	−	0.967316i	$-0.418394\pi$
$839$	14.6898	0.507149	0.253575	+	0.967316i	$0.418394\pi$
$840$	−9.58090	−0.330572
$841$	−27.1814	−0.937288
$842$	−7.40626	−0.255237
$843$	−19.7127	−0.678940
$844$	22.4826	0.773885
$845$	−2.57562	−0.0886041
$846$	2.33809	0.0803854
$847$	0	0
$848$	−20.0036	−0.686925
$849$	−31.0570	−1.06587
$850$	0.940474	0.0322580
$851$	−10.1515	−0.347989
$852$	22.5054	0.771024
$853$	17.0215	0.582804	0.291402	−	0.956601i	$-0.405878\pi$
$853$	17.0215	0.582804	0.291402	+	0.956601i	$0.405878\pi$
$854$	−9.36884	−0.320595
$855$	−14.7805	−0.505482
$856$	18.0619	0.617343
$857$	27.3037	0.932676	0.466338	−	0.884607i	$-0.345573\pi$
$857$	27.3037	0.932676	0.466338	+	0.884607i	$0.345573\pi$
$858$	0	0
$859$	30.8050	1.05105	0.525527	−	0.850777i	$-0.323868\pi$
$859$	30.8050	1.05105	0.525527	+	0.850777i	$0.323868\pi$
$860$	−33.7021	−1.14923
$861$	24.8523	0.846964
$862$	4.65572	0.158574
$863$	−14.6190	−0.497636	−0.248818	−	0.968550i	$-0.580042\pi$
$863$	−14.6190	−0.497636	−0.248818	+	0.968550i	$0.580042\pi$
$864$	2.74961	0.0935437
$865$	−27.8656	−0.947458
$866$	−2.78980	−0.0948011
$867$	−11.1537	−0.378798
$868$	17.1098	0.580744
$869$	0	0
$870$	0.826899	0.0280345
$871$	−2.57562	−0.0872717
$872$	10.3446	0.350312
$873$	12.5535	0.424873
$874$	7.65938	0.259082
$875$	−34.3544	−1.16139
$876$	21.8355	0.737754
$877$	−0.259385	−0.00875879	−0.00437940	−	0.999990i	$-0.501394\pi$
$877$	−0.259385	−0.00875879	−0.00437940	+	0.999990i	$0.501394\pi$
$878$	3.81665	0.128806
$879$	1.33899	0.0451632
$880$	0	0
$881$	−7.65546	−0.257919	−0.128959	−	0.991650i	$-0.541164\pi$
$881$	−7.65546	−0.257919	−0.128959	+	0.991650i	$0.541164\pi$
$882$	2.07142	0.0697484
$883$	7.81748	0.263079	0.131540	−	0.991311i	$-0.458008\pi$
$883$	7.81748	0.263079	0.131540	+	0.991311i	$0.458008\pi$
$884$	−4.69880	−0.158038
$885$	20.9320	0.703622
$886$	7.59541	0.255173
$887$	−47.2910	−1.58788	−0.793938	−	0.607999i	$-0.791972\pi$
$887$	−47.2910	−1.58788	−0.793938	+	0.607999i	$0.791972\pi$
$888$	1.69981	0.0570417
$889$	48.6013	1.63003
$890$	−0.919968	−0.0308374
$891$	0	0
$892$	−21.3600	−0.715185
$893$	56.3600	1.88602
$894$	−1.94403	−0.0650182
$895$	11.8484	0.396048
$896$	28.0841	0.938222
$897$	5.60649	0.187195
$898$	−2.65109	−0.0884679
$899$	−2.99646	−0.0999375
$900$	−3.17507	−0.105836
$901$	−13.2036	−0.439876
$902$	0	0
$903$	26.6805	0.887872
$904$	7.33593	0.243990
$905$	−30.0620	−0.999294
$906$	−0.317353	−0.0105433
$907$	29.6393	0.984158	0.492079	−	0.870551i	$-0.336237\pi$
$907$	29.6393	0.984158	0.492079	+	0.870551i	$0.336237\pi$
$908$	51.7116	1.71611
$909$	9.21549	0.305659
$910$	2.42965	0.0805421
$911$	−1.48572	−0.0492241	−0.0246120	−	0.999697i	$-0.507835\pi$
$911$	−1.48572	−0.0492241	−0.0246120	+	0.999697i	$0.507835\pi$
$912$	21.0214	0.696089
$913$	0	0
$914$	−5.85298	−0.193600
$915$	−25.5804	−0.845661
$916$	7.74647	0.255951
$917$	−53.2512	−1.75851
$918$	0.575623	0.0189984
$919$	−22.3108	−0.735966	−0.367983	−	0.929833i	$-0.619952\pi$
$919$	−22.3108	−0.735966	−0.367983	+	0.929833i	$0.619952\pi$
$920$	13.5560	0.446929
$921$	−26.8143	−0.883562
$922$	−6.78437	−0.223431
$923$	−11.5809	−0.381190
$924$	0	0
$925$	−2.95834	−0.0972697
$926$	9.46007	0.310877
$927$	12.4365	0.408467
$928$	−3.70805	−0.121723
$929$	−31.0951	−1.02020	−0.510098	−	0.860116i	$-0.670391\pi$
$929$	−31.0951	−1.02020	−0.510098	+	0.860116i	$0.670391\pi$
$930$	−1.36243	−0.0446758
$931$	49.9319	1.63645
$932$	4.02474	0.131835
$933$	27.8859	0.912942
$934$	7.13500	0.233464
$935$	0	0
$936$	−0.938770	−0.0306847
$937$	−1.99119	−0.0650493	−0.0325247	−	0.999471i	$-0.510355\pi$
$937$	−1.99119	−0.0650493	−0.0325247	+	0.999471i	$0.510355\pi$
$938$	2.42965	0.0793309
$939$	1.71284	0.0558964
$940$	49.1579	1.60335
$941$	−25.1503	−0.819876	−0.409938	−	0.912113i	$-0.634450\pi$
$941$	−25.1503	−0.819876	−0.409938	+	0.912113i	$0.634450\pi$
$942$	4.06219	0.132353
$943$	−35.1636	−1.14508
$944$	−29.7704	−0.968944
$945$	10.2058	0.331995
$946$	0	0
$947$	39.5801	1.28618	0.643090	−	0.765791i	$-0.277652\pi$
$947$	39.5801	1.28618	0.643090	+	0.765791i	$0.277652\pi$
$948$	16.4508	0.534297
$949$	−11.2362	−0.364741
$950$	2.23209	0.0724185
$951$	32.9427	1.06824
$952$	8.99426	0.291506
$953$	12.0479	0.390268	0.195134	−	0.980777i	$-0.437486\pi$
$953$	12.0479	0.390268	0.195134	+	0.980777i	$0.437486\pi$
$954$	−1.30001	−0.0420895
$955$	−23.4564	−0.759030
$956$	20.3776	0.659059
$957$	0	0
$958$	1.37569	0.0444465
$959$	48.3725	1.56203
$960$	17.1839	0.554607
$961$	−26.0629	−0.840740
$962$	−0.431059	−0.0138979
$963$	−19.2400	−0.619999
$964$	−2.47249	−0.0796334
$965$	5.92851	0.190845
$966$	−5.28874	−0.170162
$967$	27.7373	0.891972	0.445986	−	0.895040i	$-0.352853\pi$
$967$	27.7373	0.891972	0.445986	+	0.895040i	$0.352853\pi$
$968$	0	0
$969$	13.8755	0.445744
$970$	−7.69743	−0.247149
$971$	46.9084	1.50536	0.752681	−	0.658385i	$-0.228760\pi$
$971$	46.9084	1.50536	0.752681	+	0.658385i	$0.228760\pi$
$972$	−1.94332	−0.0623322
$973$	61.2925	1.96495
$974$	−6.43165	−0.206083
$975$	1.63384	0.0523246
$976$	36.3815	1.16454
$977$	11.7035	0.374428	0.187214	−	0.982319i	$-0.440054\pi$
$977$	11.7035	0.374428	0.187214	+	0.982319i	$0.440054\pi$
$978$	−3.51827	−0.112502
$979$	0	0
$980$	43.5512	1.39119
$981$	−11.0193	−0.351819
$982$	4.47133	0.142686
$983$	10.1992	0.325305	0.162653	−	0.986683i	$-0.447995\pi$
$983$	10.1992	0.325305	0.162653	+	0.986683i	$0.447995\pi$
$984$	5.88792	0.187700
$985$	−47.4619	−1.51226
$986$	−0.776269	−0.0247214
$987$	−38.9161	−1.23871
$988$	−11.1520	−0.354791
$989$	−37.7503	−1.20039
$990$	0	0
$991$	−55.4946	−1.76284	−0.881422	−	0.472330i	$-0.843413\pi$
$991$	−55.4946	−1.76284	−0.881422	+	0.472330i	$0.843413\pi$
$992$	6.10951	0.193977
$993$	−7.18387	−0.227973
$994$	10.9245	0.346506
$995$	−60.0890	−1.90495
$996$	−27.2385	−0.863086
$997$	31.8411	1.00842	0.504208	−	0.863582i	$-0.331784\pi$
$997$	31.8411	1.00842	0.504208	+	0.863582i	$0.331784\pi$
$998$	−2.15618	−0.0682527
$999$	−1.81067	−0.0572871

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4719.2.a.bc.1.3	✓	5
11.10	odd	2		4719.2.a.be.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4719.2.a.bc.1.3	✓	5	1.1	even	1	trivial
4719.2.a.be.1.3	yes	5	11.10	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4719.a (trivial)

\(f(q)\)	\(=\)	\(q+0.238066 q^{2} +1.00000 q^{3} -1.94332 q^{4} -2.57562 q^{5} +0.238066 q^{6} -3.96246 q^{7} -0.938770 q^{8} +1.00000 q^{9} -0.613168 q^{10} -1.94332 q^{12} +1.00000 q^{13} -0.943325 q^{14} -2.57562 q^{15} +3.66316 q^{16} +2.41792 q^{17} +0.238066 q^{18} +5.73860 q^{19} +5.00527 q^{20} -3.96246 q^{21} +5.60649 q^{23} -0.938770 q^{24} +1.63384 q^{25} +0.238066 q^{26} +1.00000 q^{27} +7.70034 q^{28} -1.34857 q^{29} -0.613168 q^{30} +2.22195 q^{31} +2.74961 q^{32} +0.575623 q^{34} +10.2058 q^{35} -1.94332 q^{36} -1.81067 q^{37} +1.36616 q^{38} +1.00000 q^{39} +2.41792 q^{40} -6.27194 q^{41} -0.943325 q^{42} -6.73333 q^{43} -2.57562 q^{45} +1.33471 q^{46} +9.82122 q^{47} +3.66316 q^{48} +8.70106 q^{49} +0.388960 q^{50} +2.41792 q^{51} -1.94332 q^{52} -5.46074 q^{53} +0.238066 q^{54} +3.71984 q^{56} +5.73860 q^{57} -0.321048 q^{58} -8.12697 q^{59} +5.00527 q^{60} +9.93172 q^{61} +0.528970 q^{62} -3.96246 q^{63} -6.67173 q^{64} -2.57562 q^{65} -2.57562 q^{67} -4.69880 q^{68} +5.60649 q^{69} +2.42965 q^{70} -11.5809 q^{71} -0.938770 q^{72} -11.2362 q^{73} -0.431059 q^{74} +1.63384 q^{75} -11.1520 q^{76} +0.238066 q^{78} -8.46529 q^{79} -9.43492 q^{80} +1.00000 q^{81} -1.49313 q^{82} +14.0165 q^{83} +7.70034 q^{84} -6.22765 q^{85} -1.60297 q^{86} -1.34857 q^{87} +1.50035 q^{89} -0.613168 q^{90} -3.96246 q^{91} -10.8952 q^{92} +2.22195 q^{93} +2.33809 q^{94} -14.7805 q^{95} +2.74961 q^{96} +12.5535 q^{97} +2.07142 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 5 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 6 q^{7} - 9 q^{8} + 5 q^{9} - 6 q^{10} + 3 q^{12} + 5 q^{13} + 8 q^{14} - 2 q^{15} + 11 q^{16} - 2 q^{17} - q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21}+ \cdots + q^{98}+O(q^{100}) \) 5 * q - q^2 + 5 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 6 * q^7 - 9 * q^8 + 5 * q^9 - 6 * q^10 + 3 * q^12 + 5 * q^13 + 8 * q^14 - 2 * q^15 + 11 * q^16 - 2 * q^17 - q^18 - 12 * q^19 - 8 * q^20 - 6 * q^21 + 8 * q^23 - 9 * q^24 + 7 * q^25 - q^26 + 5 * q^27 - 6 * q^29 - 6 * q^30 - 4 * q^31 - 19 * q^32 - 8 * q^34 + 2 * q^35 + 3 * q^36 - 10 * q^37 + 8 * q^38 + 5 * q^39 - 2 * q^40 - 6 * q^41 + 8 * q^42 - 26 * q^43 - 2 * q^45 - 26 * q^46 - 16 * q^47 + 11 * q^48 - 11 * q^49 + 39 * q^50 - 2 * q^51 + 3 * q^52 + 6 * q^53 - q^54 + 24 * q^56 - 12 * q^57 - 20 * q^58 + 6 * q^59 - 8 * q^60 - 14 * q^61 + 12 * q^62 - 6 * q^63 + 15 * q^64 - 2 * q^65 - 2 * q^67 + 12 * q^68 + 8 * q^69 - 10 * q^70 - 14 * q^71 - 9 * q^72 - 12 * q^73 - 36 * q^74 + 7 * q^75 - 18 * q^76 - q^78 + 8 * q^79 + 8 * q^80 + 5 * q^81 + 38 * q^82 + 22 * q^83 - 36 * q^85 - 16 * q^86 - 6 * q^87 - 8 * q^89 - 6 * q^90 - 6 * q^91 + 12 * q^92 - 4 * q^93 + 10 * q^94 - 32 * q^95 - 19 * q^96 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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