LMFDB - Embedded newform 8281.2.a.ch.1.3

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 = [6,4,0,4,-6,-4,0,12,4,-12,4,-2,0,0,20,8,4] 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:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.879640$ of defining polynomial
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)

$f(q)$	$=$	\(q+0.120360 q^{2} +0.582292 q^{3} -1.98551 q^{4} +1.68817 q^{5} +0.0700846 q^{6} -0.479696 q^{8} -2.66094 q^{9} +0.203187 q^{10} -0.364618 q^{11} -1.15615 q^{12} +0.983005 q^{15} +3.91329 q^{16} +3.18555 q^{17} -0.320270 q^{18} -1.44391 q^{19} -3.35188 q^{20} -0.0438854 q^{22} -5.08321 q^{23} -0.279323 q^{24} -2.15010 q^{25} -3.29632 q^{27} +8.19662 q^{29} +0.118314 q^{30} +4.69775 q^{31} +1.43040 q^{32} -0.212314 q^{33} +0.383412 q^{34} +5.28332 q^{36} +6.31584 q^{37} -0.173789 q^{38} -0.809806 q^{40} -5.82732 q^{41} -0.773122 q^{43} +0.723954 q^{44} -4.49210 q^{45} -0.611815 q^{46} -12.7905 q^{47} +2.27868 q^{48} -0.258786 q^{50} +1.85492 q^{51} +1.37110 q^{53} -0.396744 q^{54} -0.615536 q^{55} -0.840776 q^{57} +0.986544 q^{58} -9.36197 q^{59} -1.95177 q^{60} +9.02484 q^{61} +0.565421 q^{62} -7.65442 q^{64} -0.0255541 q^{66} +13.4759 q^{67} -6.32495 q^{68} -2.95991 q^{69} -7.08115 q^{71} +1.27644 q^{72} +2.16083 q^{73} +0.760173 q^{74} -1.25198 q^{75} +2.86690 q^{76} -6.88781 q^{79} +6.60628 q^{80} +6.06339 q^{81} -0.701376 q^{82} -0.567380 q^{83} +5.37773 q^{85} -0.0930528 q^{86} +4.77282 q^{87} +0.174906 q^{88} +1.13893 q^{89} -0.540669 q^{90} +10.0928 q^{92} +2.73547 q^{93} -1.53947 q^{94} -2.43755 q^{95} +0.832908 q^{96} -7.92785 q^{97} +0.970225 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 12 q^{8} + 4 q^{9} - 12 q^{10} + 4 q^{11} - 2 q^{12} + 20 q^{15} + 8 q^{16} + 4 q^{17} - 16 q^{18} - 2 q^{19} - 26 q^{20} - 6 q^{22} - 12 q^{23} - 2 q^{24}+ \cdots + 16 q^{99}+O(q^{100}) $ 6 * q + 4 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 12 * q^8 + 4 * q^9 - 12 * q^10 + 4 * q^11 - 2 * q^12 + 20 * q^15 + 8 * q^16 + 4 * q^17 - 16 * q^18 - 2 * q^19 - 26 * q^20 - 6 * q^22 - 12 * q^23 - 2 * q^24 + 10 * q^25 - 6 * q^27 - 8 * q^29 + 8 * q^30 + 14 * q^31 + 8 * q^32 - 16 * q^33 + 2 * q^34 - 10 * q^36 + 12 * q^37 + 2 * q^38 - 46 * q^40 - 28 * q^41 + 2 * q^43 - 20 * q^44 - 16 * q^45 + 20 * q^46 - 14 * q^47 - 2 * q^48 + 32 * q^50 - 26 * q^51 - 22 * q^53 - 14 * q^54 - 6 * q^55 + 4 * q^58 + 2 * q^59 + 14 * q^61 + 4 * q^62 + 26 * q^64 + 26 * q^66 + 24 * q^67 - 8 * q^68 - 4 * q^69 + 4 * q^71 + 8 * q^72 - 36 * q^73 - 6 * q^74 - 46 * q^75 + 26 * q^76 - 28 * q^79 - 36 * q^80 - 2 * q^81 - 14 * q^82 - 26 * q^83 - 20 * q^85 - 24 * q^86 - 2 * q^87 - 14 * q^88 - 42 * q^89 + 12 * q^90 + 12 * q^92 + 4 * q^94 - 22 * q^95 + 42 * q^96 - 24 * q^97 + 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 $

$2$	0.120360	0.0851073	0.0425536	−	0.999094i	$-0.486451\pi$
$2$	0.120360	0.0851073	0.0425536	+	0.999094i	$0.486451\pi$
$3$	0.582292	0.336186	0.168093	−	0.985771i	$-0.446239\pi$
$3$	0.582292	0.336186	0.168093	+	0.985771i	$0.446239\pi$
$4$	−1.98551	−0.992757
$5$	1.68817	0.754971	0.377485	−	0.926016i	$-0.376789\pi$
$5$	1.68817	0.754971	0.377485	+	0.926016i	$0.376789\pi$
$6$	0.0700846	0.0286119
$7$	0	0
$7$	0	0
$8$	−0.479696	−0.169598
$9$	−2.66094	−0.886979
$10$	0.203187	0.0642535
$11$	−0.364618	−0.109936	−0.0549682	−	0.998488i	$-0.517506\pi$
$11$	−0.364618	−0.109936	−0.0549682	+	0.998488i	$0.517506\pi$
$12$	−1.15615	−0.333751
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.983005	0.253811
$16$	3.91329	0.978323
$17$	3.18555	0.772609	0.386304	−	0.922371i	$-0.373751\pi$
$17$	3.18555	0.772609	0.386304	+	0.922371i	$0.373751\pi$
$18$	−0.320270	−0.0754884
$19$	−1.44391	−0.331255	−0.165628	−	0.986188i	$-0.552965\pi$
$19$	−1.44391	−0.331255	−0.165628	+	0.986188i	$0.552965\pi$
$20$	−3.35188	−0.749502
$21$	0	0
$22$	−0.0438854	−0.00935640
$23$	−5.08321	−1.05992	−0.529962	−	0.848022i	$-0.677794\pi$
$23$	−5.08321	−1.05992	−0.529962	+	0.848022i	$0.677794\pi$
$24$	−0.279323	−0.0570166
$25$	−2.15010	−0.430020
$26$	0	0
$27$	−3.29632	−0.634377
$28$	0	0
$29$	8.19662	1.52207	0.761037	−	0.648709i	$-0.224691\pi$
$29$	8.19662	1.52207	0.761037	+	0.648709i	$0.224691\pi$
$30$	0.118314	0.0216012
$31$	4.69775	0.843742	0.421871	−	0.906656i	$-0.361374\pi$
$31$	4.69775	0.843742	0.421871	+	0.906656i	$0.361374\pi$
$32$	1.43040	0.252861
$33$	−0.212314	−0.0369592
$34$	0.383412	0.0657546
$35$	0	0
$36$	5.28332	0.880554
$37$	6.31584	1.03832	0.519159	−	0.854678i	$-0.326245\pi$
$37$	6.31584	1.03832	0.519159	+	0.854678i	$0.326245\pi$
$38$	−0.173789	−0.0281922
$39$	0	0
$40$	−0.809806	−0.128042
$41$	−5.82732	−0.910074	−0.455037	−	0.890472i	$-0.650374\pi$
$41$	−5.82732	−0.910074	−0.455037	+	0.890472i	$0.650374\pi$
$42$	0	0
$43$	−0.773122	−0.117900	−0.0589500	−	0.998261i	$-0.518775\pi$
$43$	−0.773122	−0.117900	−0.0589500	+	0.998261i	$0.518775\pi$
$44$	0.723954	0.109140
$45$	−4.49210	−0.669643
$46$	−0.611815	−0.0902072
$47$	−12.7905	−1.86569	−0.932846	−	0.360275i	$-0.882683\pi$
$47$	−12.7905	−1.86569	−0.932846	+	0.360275i	$0.882683\pi$
$48$	2.27868	0.328899
$49$	0	0
$50$	−0.258786	−0.0365978
$51$	1.85492	0.259741
$52$	0	0
$53$	1.37110	0.188334	0.0941672	−	0.995556i	$-0.469981\pi$
$53$	1.37110	0.188334	0.0941672	+	0.995556i	$0.469981\pi$
$54$	−0.396744	−0.0539901
$55$	−0.615536	−0.0829988
$56$	0	0
$57$	−0.840776	−0.111363
$58$	0.986544	0.129540
$59$	−9.36197	−1.21882	−0.609412	−	0.792854i	$-0.708595\pi$
$59$	−9.36197	−1.21882	−0.609412	+	0.792854i	$0.708595\pi$
$60$	−1.95177	−0.251972
$61$	9.02484	1.15551	0.577756	−	0.816209i	$-0.303928\pi$
$61$	9.02484	1.15551	0.577756	+	0.816209i	$0.303928\pi$
$62$	0.565421	0.0718086
$63$	0	0
$64$	−7.65442	−0.956802
$65$	0	0
$66$	−0.0255541	−0.00314549
$67$	13.4759	1.64635	0.823174	−	0.567789i	$-0.192201\pi$
$67$	13.4759	1.64635	0.823174	+	0.567789i	$0.192201\pi$
$68$	−6.32495	−0.767013
$69$	−2.95991	−0.356332
$70$	0	0
$71$	−7.08115	−0.840378	−0.420189	−	0.907437i	$-0.638036\pi$
$71$	−7.08115	−0.840378	−0.420189	+	0.907437i	$0.638036\pi$
$72$	1.27644	0.150430
$73$	2.16083	0.252906	0.126453	−	0.991973i	$-0.459641\pi$
$73$	2.16083	0.252906	0.126453	+	0.991973i	$0.459641\pi$
$74$	0.760173	0.0883684
$75$	−1.25198	−0.144567
$76$	2.86690	0.328856
$77$	0	0
$78$	0	0
$79$	−6.88781	−0.774940	−0.387470	−	0.921882i	$-0.626651\pi$
$79$	−6.88781	−0.774940	−0.387470	+	0.921882i	$0.626651\pi$
$80$	6.60628	0.738605
$81$	6.06339	0.673710
$82$	−0.701376	−0.0774540
$83$	−0.567380	−0.0622780	−0.0311390	−	0.999515i	$-0.509913\pi$
$83$	−0.567380	−0.0622780	−0.0311390	+	0.999515i	$0.509913\pi$
$84$	0	0
$85$	5.37773	0.583297
$86$	−0.0930528	−0.0100341
$87$	4.77282	0.511700
$88$	0.174906	0.0186450
$89$	1.13893	0.120727	0.0603634	−	0.998176i	$-0.480774\pi$
$89$	1.13893	0.120727	0.0603634	+	0.998176i	$0.480774\pi$
$90$	−0.540669	−0.0569915
$91$	0	0
$92$	10.0928	1.05225
$93$	2.73547	0.283655
$94$	−1.53947	−0.158784
$95$	−2.43755	−0.250088
$96$	0.832908	0.0850083
$97$	−7.92785	−0.804952	−0.402476	−	0.915431i	$-0.631850\pi$
$97$	−7.92785	−0.804952	−0.402476	+	0.915431i	$0.631850\pi$
$98$	0	0
$99$	0.970225	0.0975113
$100$	4.26905	0.426905
$101$	15.5464	1.54693	0.773465	−	0.633839i	$-0.218522\pi$
$101$	15.5464	1.54693	0.773465	+	0.633839i	$0.218522\pi$
$102$	0.223258	0.0221058
$103$	−10.2982	−1.01471	−0.507354	−	0.861738i	$-0.669376\pi$
$103$	−10.2982	−1.01471	−0.507354	+	0.861738i	$0.669376\pi$
$104$	0	0
$105$	0	0
$106$	0.165025	0.0160286
$107$	−13.1244	−1.26878	−0.634391	−	0.773012i	$-0.718749\pi$
$107$	−13.1244	−1.26878	−0.634391	+	0.773012i	$0.718749\pi$
$108$	6.54488	0.629782
$109$	10.4459	1.00054	0.500268	−	0.865871i	$-0.333235\pi$
$109$	10.4459	1.00054	0.500268	+	0.865871i	$0.333235\pi$
$110$	−0.0740858	−0.00706380
$111$	3.67766	0.349068
$112$	0	0
$113$	4.95262	0.465903	0.232952	−	0.972488i	$-0.425162\pi$
$113$	4.95262	0.465903	0.232952	+	0.972488i	$0.425162\pi$
$114$	−0.101196	−0.00947784
$115$	−8.58130	−0.800211
$116$	−16.2745	−1.51105
$117$	0	0
$118$	−1.12681	−0.103731
$119$	0	0
$120$	−0.471544	−0.0430458
$121$	−10.8671	−0.987914
$122$	1.08623	0.0983425
$123$	−3.39320	−0.305955
$124$	−9.32746	−0.837630
$125$	−12.0705	−1.07962
$126$	0	0
$127$	8.06731	0.715858	0.357929	−	0.933749i	$-0.383483\pi$
$127$	8.06731	0.715858	0.357929	+	0.933749i	$0.383483\pi$
$128$	−3.78208	−0.334291
$129$	−0.450183	−0.0396364
$130$	0	0
$131$	−18.9039	−1.65164	−0.825820	−	0.563934i	$-0.809287\pi$
$131$	−18.9039	−1.65164	−0.825820	+	0.563934i	$0.809287\pi$
$132$	0.421553	0.0366915
$133$	0	0
$134$	1.62196	0.140116
$135$	−5.56473	−0.478936
$136$	−1.52809	−0.131033
$137$	−18.2255	−1.55711	−0.778554	−	0.627577i	$-0.784047\pi$
$137$	−18.2255	−1.55711	−0.778554	+	0.627577i	$0.784047\pi$
$138$	−0.356255	−0.0303264
$139$	−5.25085	−0.445371	−0.222686	−	0.974890i	$-0.571482\pi$
$139$	−5.25085	−0.445371	−0.222686	+	0.974890i	$0.571482\pi$
$140$	0	0
$141$	−7.44783	−0.627220
$142$	−0.852287	−0.0715223
$143$	0	0
$144$	−10.4130	−0.867751
$145$	13.8372	1.14912
$146$	0.260077	0.0215241
$147$	0	0
$148$	−12.5402	−1.03080
$149$	9.27309	0.759681	0.379841	−	0.925052i	$-0.375979\pi$
$149$	9.27309	0.759681	0.379841	+	0.925052i	$0.375979\pi$
$150$	−0.150689	−0.0123037
$151$	−14.0132	−1.14038	−0.570189	−	0.821513i	$-0.693130\pi$
$151$	−14.0132	−1.14038	−0.570189	+	0.821513i	$0.693130\pi$
$152$	0.692636	0.0561802
$153$	−8.47654	−0.685287
$154$	0	0
$155$	7.93059	0.637000
$156$	0	0
$157$	17.1825	1.37131	0.685656	−	0.727925i	$-0.259515\pi$
$157$	17.1825	1.37131	0.685656	+	0.727925i	$0.259515\pi$
$158$	−0.829017	−0.0659530
$159$	0.798378	0.0633155
$160$	2.41474	0.190902
$161$	0	0
$162$	0.729789	0.0573376
$163$	−11.7927	−0.923679	−0.461840	−	0.886963i	$-0.652810\pi$
$163$	−11.7927	−0.923679	−0.461840	+	0.886963i	$0.652810\pi$
$164$	11.5702	0.903482
$165$	−0.358422	−0.0279031
$166$	−0.0682898	−0.00530031
$167$	−4.31687	−0.334049	−0.167025	−	0.985953i	$-0.553416\pi$
$167$	−4.31687	−0.334049	−0.167025	+	0.985953i	$0.553416\pi$
$168$	0	0
$169$	0	0
$170$	0.647263	0.0496428
$171$	3.84214	0.293816
$172$	1.53504	0.117046
$173$	−12.5197	−0.951855	−0.475928	−	0.879484i	$-0.657888\pi$
$173$	−12.5197	−0.951855	−0.475928	+	0.879484i	$0.657888\pi$
$174$	0.574457	0.0435494
$175$	0	0
$176$	−1.42686	−0.107553
$177$	−5.45140	−0.409752
$178$	0.137082	0.0102747
$179$	−6.59534	−0.492959	−0.246479	−	0.969148i	$-0.579274\pi$
$179$	−6.59534	−0.492959	−0.246479	+	0.969148i	$0.579274\pi$
$180$	8.91913	0.664792
$181$	−11.0157	−0.818791	−0.409395	−	0.912357i	$-0.634260\pi$
$181$	−11.0157	−0.818791	−0.409395	+	0.912357i	$0.634260\pi$
$182$	0	0
$183$	5.25509	0.388468
$184$	2.43840	0.179761
$185$	10.6622	0.783899
$186$	0.329240	0.0241411
$187$	−1.16151	−0.0849379
$188$	25.3958	1.85218
$189$	0	0
$190$	−0.293384	−0.0212843
$191$	5.93213	0.429234	0.214617	−	0.976698i	$-0.431150\pi$
$191$	5.93213	0.429234	0.214617	+	0.976698i	$0.431150\pi$
$192$	−4.45711	−0.321664
$193$	4.19595	0.302031	0.151016	−	0.988531i	$-0.451746\pi$
$193$	4.19595	0.302031	0.151016	+	0.988531i	$0.451746\pi$
$194$	−0.954196	−0.0685073
$195$	0	0
$196$	0	0
$197$	−5.78494	−0.412160	−0.206080	−	0.978535i	$-0.566071\pi$
$197$	−5.78494	−0.412160	−0.206080	+	0.978535i	$0.566071\pi$
$198$	0.116776	0.00829893
$199$	−11.9598	−0.847805	−0.423903	−	0.905708i	$-0.639340\pi$
$199$	−11.9598	−0.847805	−0.423903	+	0.905708i	$0.639340\pi$
$200$	1.03139	0.0729305
$201$	7.84693	0.553480
$202$	1.87117	0.131655
$203$	0	0
$204$	−3.68297	−0.257859
$205$	−9.83748	−0.687079
$206$	−1.23949	−0.0863590
$207$	13.5261	0.940129
$208$	0	0
$209$	0.526475	0.0364170
$210$	0	0
$211$	−8.23591	−0.566983	−0.283492	−	0.958975i	$-0.591493\pi$
$211$	−8.23591	−0.566983	−0.283492	+	0.958975i	$0.591493\pi$
$212$	−2.72233	−0.186970
$213$	−4.12330	−0.282524
$214$	−1.57965	−0.107983
$215$	−1.30516	−0.0890110
$216$	1.58123	0.107589
$217$	0	0
$218$	1.25727	0.0851528
$219$	1.25823	0.0850234
$220$	1.22215	0.0823976
$221$	0	0
$222$	0.442643	0.0297082
$223$	15.3015	1.02466	0.512331	−	0.858788i	$-0.328782\pi$
$223$	15.3015	1.02466	0.512331	+	0.858788i	$0.328782\pi$
$224$	0	0
$225$	5.72127	0.381418
$226$	0.596097	0.0396518
$227$	6.95467	0.461598	0.230799	−	0.973002i	$-0.425866\pi$
$227$	6.95467	0.461598	0.230799	+	0.973002i	$0.425866\pi$
$228$	1.66937	0.110557
$229$	−27.4219	−1.81209	−0.906045	−	0.423180i	$-0.860914\pi$
$229$	−27.4219	−1.81209	−0.906045	+	0.423180i	$0.860914\pi$
$230$	−1.03284	−0.0681038
$231$	0	0
$232$	−3.93188	−0.258141
$233$	−6.85333	−0.448976	−0.224488	−	0.974477i	$-0.572071\pi$
$233$	−6.85333	−0.448976	−0.224488	+	0.974477i	$0.572071\pi$
$234$	0	0
$235$	−21.5926	−1.40854
$236$	18.5883	1.21000
$237$	−4.01072	−0.260524
$238$	0	0
$239$	22.0754	1.42794	0.713970	−	0.700177i	$-0.246895\pi$
$239$	22.0754	1.42794	0.713970	+	0.700177i	$0.246895\pi$
$240$	3.84679	0.248309
$241$	15.7971	1.01758	0.508790	−	0.860890i	$-0.330093\pi$
$241$	15.7971	1.01758	0.508790	+	0.860890i	$0.330093\pi$
$242$	−1.30796	−0.0840787
$243$	13.4196	0.860869
$244$	−17.9189	−1.14714
$245$	0	0
$246$	−0.408405	−0.0260390
$247$	0	0
$248$	−2.25349	−0.143097
$249$	−0.330381	−0.0209370
$250$	−1.45281	−0.0918838
$251$	−22.5567	−1.42376	−0.711882	−	0.702299i	$-0.752157\pi$
$251$	−22.5567	−1.42376	−0.711882	+	0.702299i	$0.752157\pi$
$252$	0	0
$253$	1.85343	0.116524
$254$	0.970981	0.0609247
$255$	3.13141	0.196097
$256$	14.8536	0.928352
$257$	−20.4129	−1.27332	−0.636660	−	0.771145i	$-0.719685\pi$
$257$	−20.4129	−1.27332	−0.636660	+	0.771145i	$0.719685\pi$
$258$	−0.0541839	−0.00337334
$259$	0	0
$260$	0	0
$261$	−21.8107	−1.35005
$262$	−2.27527	−0.140567
$263$	−29.5402	−1.82153	−0.910764	−	0.412927i	$-0.864506\pi$
$263$	−29.5402	−1.82153	−0.910764	+	0.412927i	$0.864506\pi$
$264$	0.101846	0.00626820
$265$	2.31464	0.142187
$266$	0	0
$267$	0.663193	0.0405867
$268$	−26.7567	−1.63442
$269$	−27.9163	−1.70209	−0.851043	−	0.525096i	$-0.824029\pi$
$269$	−27.9163	−1.70209	−0.851043	+	0.525096i	$0.824029\pi$
$270$	−0.669770	−0.0407609
$271$	29.4491	1.78890	0.894451	−	0.447165i	$-0.147566\pi$
$271$	29.4491	1.78890	0.894451	+	0.447165i	$0.147566\pi$
$272$	12.4660	0.755861
$273$	0	0
$274$	−2.19362	−0.132521
$275$	0.783965	0.0472748
$276$	5.87695	0.353751
$277$	6.85854	0.412090	0.206045	−	0.978543i	$-0.433941\pi$
$277$	6.85854	0.412090	0.206045	+	0.978543i	$0.433941\pi$
$278$	−0.631992	−0.0379043
$279$	−12.5004	−0.748381
$280$	0	0
$281$	29.0940	1.73561	0.867803	−	0.496909i	$-0.165532\pi$
$281$	29.0940	1.73561	0.867803	+	0.496909i	$0.165532\pi$
$282$	−0.896420	−0.0533810
$283$	−11.6102	−0.690156	−0.345078	−	0.938574i	$-0.612147\pi$
$283$	−11.6102	−0.690156	−0.345078	+	0.938574i	$0.612147\pi$
$284$	14.0597	0.834291
$285$	−1.41937	−0.0840761
$286$	0	0
$287$	0	0
$288$	−3.80619	−0.224282
$289$	−6.85229	−0.403076
$290$	1.66545	0.0977985
$291$	−4.61633	−0.270614
$292$	−4.29035	−0.251074
$293$	−17.7886	−1.03922	−0.519610	−	0.854403i	$-0.673923\pi$
$293$	−17.7886	−1.03922	−0.519610	+	0.854403i	$0.673923\pi$
$294$	0	0
$295$	−15.8046	−0.920177
$296$	−3.02968	−0.176097
$297$	1.20190	0.0697412
$298$	1.11611	0.0646544
$299$	0	0
$300$	2.48583	0.143520
$301$	0	0
$302$	−1.68663	−0.0970546
$303$	9.05257	0.520057
$304$	−5.65043	−0.324074
$305$	15.2354	0.872378
$306$	−1.02024	−0.0583230
$307$	9.07966	0.518204	0.259102	−	0.965850i	$-0.416573\pi$
$307$	9.07966	0.518204	0.259102	+	0.965850i	$0.416573\pi$
$308$	0	0
$309$	−5.99654	−0.341131
$310$	0.954525	0.0542134
$311$	−1.57073	−0.0890677	−0.0445338	−	0.999008i	$-0.514180\pi$
$311$	−1.57073	−0.0890677	−0.0445338	+	0.999008i	$0.514180\pi$
$312$	0	0
$313$	−20.6232	−1.16569	−0.582846	−	0.812582i	$-0.698061\pi$
$313$	−20.6232	−1.16569	−0.582846	+	0.812582i	$0.698061\pi$
$314$	2.06808	0.116709
$315$	0	0
$316$	13.6758	0.769327
$317$	30.5435	1.71549	0.857747	−	0.514072i	$-0.171863\pi$
$317$	30.5435	1.71549	0.857747	+	0.514072i	$0.171863\pi$
$318$	0.0960927	0.00538861
$319$	−2.98863	−0.167331
$320$	−12.9219	−0.722358
$321$	−7.64223	−0.426548
$322$	0	0
$323$	−4.59964	−0.255931
$324$	−12.0389	−0.668830
$325$	0	0
$326$	−1.41937	−0.0786118
$327$	6.08256	0.336366
$328$	2.79534	0.154347
$329$	0	0
$330$	−0.0431396	−0.00237476
$331$	−25.8531	−1.42101	−0.710507	−	0.703690i	$-0.751534\pi$
$331$	−25.8531	−1.42101	−0.710507	+	0.703690i	$0.751534\pi$
$332$	1.12654	0.0618269
$333$	−16.8060	−0.920965
$334$	−0.519578	−0.0284300
$335$	22.7496	1.24294
$336$	0	0
$337$	21.3954	1.16548	0.582742	−	0.812657i	$-0.301980\pi$
$337$	21.3954	1.16548	0.582742	+	0.812657i	$0.301980\pi$
$338$	0	0
$339$	2.88387	0.156630
$340$	−10.6776	−0.579072
$341$	−1.71289	−0.0927580
$342$	0.462440	0.0250059
$343$	0	0
$344$	0.370863	0.0199956
$345$	−4.99682	−0.269020
$346$	−1.50687	−0.0810098
$347$	2.20883	0.118576	0.0592882	−	0.998241i	$-0.481117\pi$
$347$	2.20883	0.118576	0.0592882	+	0.998241i	$0.481117\pi$
$348$	−9.47651	−0.507994
$349$	11.2912	0.604402	0.302201	−	0.953244i	$-0.402279\pi$
$349$	11.2912	0.604402	0.302201	+	0.953244i	$0.402279\pi$
$350$	0	0
$351$	0	0
$352$	−0.521548	−0.0277986
$353$	−35.6433	−1.89710	−0.948552	−	0.316623i	$-0.897451\pi$
$353$	−35.6433	−1.89710	−0.948552	+	0.316623i	$0.897451\pi$
$354$	−0.656130	−0.0348729
$355$	−11.9542	−0.634461
$356$	−2.26137	−0.119852
$357$	0	0
$358$	−0.793815	−0.0419544
$359$	−19.3218	−1.01976	−0.509882	−	0.860244i	$-0.670311\pi$
$359$	−19.3218	−1.01976	−0.509882	+	0.860244i	$0.670311\pi$
$360$	2.15484	0.113570
$361$	−16.9151	−0.890270
$362$	−1.32585	−0.0696851
$363$	−6.32780	−0.332123
$364$	0	0
$365$	3.64783	0.190936
$366$	0.632502	0.0330614
$367$	3.72065	0.194216	0.0971082	−	0.995274i	$-0.469041\pi$
$367$	3.72065	0.194216	0.0971082	+	0.995274i	$0.469041\pi$
$368$	−19.8921	−1.03695
$369$	15.5061	0.807217
$370$	1.28330	0.0667155
$371$	0	0
$372$	−5.43130	−0.281600
$373$	−3.51276	−0.181884	−0.0909420	−	0.995856i	$-0.528988\pi$
$373$	−3.51276	−0.181884	−0.0909420	+	0.995856i	$0.528988\pi$
$374$	−0.139799	−0.00722884
$375$	−7.02858	−0.362954
$376$	6.13557	0.316418
$377$	0	0
$378$	0	0
$379$	−25.0163	−1.28500	−0.642500	−	0.766285i	$-0.722103\pi$
$379$	−25.0163	−1.28500	−0.642500	+	0.766285i	$0.722103\pi$
$380$	4.83980	0.248276
$381$	4.69753	0.240662
$382$	0.713990	0.0365309
$383$	22.4654	1.14793	0.573964	−	0.818881i	$-0.305405\pi$
$383$	22.4654	1.14793	0.573964	+	0.818881i	$0.305405\pi$
$384$	−2.20227	−0.112384
$385$	0	0
$386$	0.505024	0.0257051
$387$	2.05723	0.104575
$388$	15.7409	0.799121
$389$	−13.3364	−0.676184	−0.338092	−	0.941113i	$-0.609781\pi$
$389$	−13.3364	−0.676184	−0.338092	+	0.941113i	$0.609781\pi$
$390$	0	0
$391$	−16.1928	−0.818906
$392$	0	0
$393$	−11.0076	−0.555259
$394$	−0.696274	−0.0350778
$395$	−11.6278	−0.585057
$396$	−1.92640	−0.0968050
$397$	−25.8333	−1.29654	−0.648268	−	0.761412i	$-0.724506\pi$
$397$	−25.8333	−1.29654	−0.648268	+	0.761412i	$0.724506\pi$
$398$	−1.43948	−0.0721544
$399$	0	0
$400$	−8.41396	−0.420698
$401$	17.5605	0.876930	0.438465	−	0.898748i	$-0.355522\pi$
$401$	17.5605	0.876930	0.438465	+	0.898748i	$0.355522\pi$
$402$	0.944456	0.0471052
$403$	0	0
$404$	−30.8677	−1.53572
$405$	10.2360	0.508631
$406$	0	0
$407$	−2.30287	−0.114149
$408$	−0.889797	−0.0440515
$409$	−14.5282	−0.718373	−0.359186	−	0.933266i	$-0.616946\pi$
$409$	−14.5282	−0.718373	−0.359186	+	0.933266i	$0.616946\pi$
$410$	−1.18404	−0.0584755
$411$	−10.6126	−0.523479
$412$	20.4471	1.00736
$413$	0	0
$414$	1.62800	0.0800119
$415$	−0.957831	−0.0470181
$416$	0	0
$417$	−3.05753	−0.149728
$418$	0.0633664	0.00309935
$419$	4.60192	0.224819	0.112409	−	0.993662i	$-0.464143\pi$
$419$	4.60192	0.224819	0.112409	+	0.993662i	$0.464143\pi$
$420$	0	0
$421$	−19.2645	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$421$	−19.2645	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$422$	−0.991273	−0.0482544
$423$	34.0348	1.65483
$424$	−0.657709	−0.0319412
$425$	−6.84924	−0.332237
$426$	−0.496280	−0.0240448
$427$	0	0
$428$	26.0587	1.25959
$429$	0	0
$430$	−0.157089	−0.00757548
$431$	28.3651	1.36630	0.683149	−	0.730279i	$-0.260610\pi$
$431$	28.3651	1.36630	0.683149	+	0.730279i	$0.260610\pi$
$432$	−12.8995	−0.620625
$433$	12.5203	0.601686	0.300843	−	0.953674i	$-0.402732\pi$
$433$	12.5203	0.601686	0.300843	+	0.953674i	$0.402732\pi$
$434$	0	0
$435$	8.05732	0.386319
$436$	−20.7405	−0.993288
$437$	7.33969	0.351105
$438$	0.151441	0.00723611
$439$	−31.7273	−1.51426	−0.757132	−	0.653262i	$-0.773400\pi$
$439$	−31.7273	−1.51426	−0.757132	+	0.653262i	$0.773400\pi$
$440$	0.295270	0.0140764
$441$	0	0
$442$	0	0
$443$	1.73048	0.0822177	0.0411088	−	0.999155i	$-0.486911\pi$
$443$	1.73048	0.0822177	0.0411088	+	0.999155i	$0.486911\pi$
$444$	−7.30205	−0.346540
$445$	1.92271	0.0911452
$446$	1.84168	0.0872062
$447$	5.39965	0.255394
$448$	0	0
$449$	10.5564	0.498186	0.249093	−	0.968480i	$-0.419868\pi$
$449$	10.5564	0.498186	0.249093	+	0.968480i	$0.419868\pi$
$450$	0.688612	0.0324615
$451$	2.12475	0.100050
$452$	−9.83349	−0.462528
$453$	−8.15978	−0.383380
$454$	0.837063	0.0392853
$455$	0	0
$456$	0.403317	0.0188870
$457$	7.94894	0.371836	0.185918	−	0.982565i	$-0.440474\pi$
$457$	7.94894	0.371836	0.185918	+	0.982565i	$0.440474\pi$
$458$	−3.30050	−0.154222
$459$	−10.5006	−0.490125
$460$	17.0383	0.794415
$461$	−10.8918	−0.507284	−0.253642	−	0.967298i	$-0.581629\pi$
$461$	−10.8918	−0.507284	−0.253642	+	0.967298i	$0.581629\pi$
$462$	0	0
$463$	35.8227	1.66482	0.832411	−	0.554158i	$-0.186960\pi$
$463$	35.8227	1.66482	0.832411	+	0.554158i	$0.186960\pi$
$464$	32.0757	1.48908
$465$	4.61792	0.214151
$466$	−0.824866	−0.0382112
$467$	−19.8983	−0.920785	−0.460393	−	0.887715i	$-0.652291\pi$
$467$	−19.8983	−0.920785	−0.460393	+	0.887715i	$0.652291\pi$
$468$	0	0
$469$	0	0
$470$	−2.59888	−0.119877
$471$	10.0052	0.461017
$472$	4.49090	0.206710
$473$	0.281894	0.0129615
$474$	−0.482730	−0.0221725
$475$	3.10454	0.142446
$476$	0	0
$477$	−3.64840	−0.167049
$478$	2.65699	0.121528
$479$	−26.2902	−1.20123	−0.600615	−	0.799538i	$-0.705078\pi$
$479$	−26.2902	−1.20123	−0.600615	+	0.799538i	$0.705078\pi$
$480$	1.40609	0.0641788
$481$	0	0
$482$	1.90134	0.0866035
$483$	0	0
$484$	21.5767	0.980758
$485$	−13.3835	−0.607715
$486$	1.61518	0.0732662
$487$	6.37962	0.289088	0.144544	−	0.989498i	$-0.453828\pi$
$487$	6.37962	0.289088	0.144544	+	0.989498i	$0.453828\pi$
$488$	−4.32918	−0.195973
$489$	−6.86682	−0.310528
$490$	0	0
$491$	−2.96768	−0.133929	−0.0669647	−	0.997755i	$-0.521331\pi$
$491$	−2.96768	−0.133929	−0.0669647	+	0.997755i	$0.521331\pi$
$492$	6.73725	0.303739
$493$	26.1107	1.17597
$494$	0	0
$495$	1.63790	0.0736182
$496$	18.3837	0.825452
$497$	0	0
$498$	−0.0397646	−0.00178189
$499$	−28.1331	−1.25941	−0.629704	−	0.776835i	$-0.716824\pi$
$499$	−28.1331	−1.25941	−0.629704	+	0.776835i	$0.716824\pi$
$500$	23.9662	1.07180
$501$	−2.51368	−0.112303
$502$	−2.71492	−0.121173
$503$	31.5376	1.40619	0.703097	−	0.711094i	$-0.251800\pi$
$503$	31.5376	1.40619	0.703097	+	0.711094i	$0.251800\pi$
$504$	0	0
$505$	26.2450	1.16789
$506$	0.223079	0.00991706
$507$	0	0
$508$	−16.0178	−0.710673
$509$	13.5944	0.602560	0.301280	−	0.953536i	$-0.402586\pi$
$509$	13.5944	0.602560	0.301280	+	0.953536i	$0.402586\pi$
$510$	0.376896	0.0166892
$511$	0	0
$512$	9.35193	0.413301
$513$	4.75958	0.210140
$514$	−2.45689	−0.108369
$515$	−17.3850	−0.766074
$516$	0.893844	0.0393493
$517$	4.66366	0.205108
$518$	0	0
$519$	−7.29012	−0.320001
$520$	0	0
$521$	−8.78344	−0.384810	−0.192405	−	0.981316i	$-0.561629\pi$
$521$	−8.78344	−0.384810	−0.192405	+	0.981316i	$0.561629\pi$
$522$	−2.62513	−0.114899
$523$	−32.5698	−1.42418	−0.712088	−	0.702090i	$-0.752251\pi$
$523$	−32.5698	−1.42418	−0.712088	+	0.702090i	$0.752251\pi$
$524$	37.5339	1.63968
$525$	0	0
$526$	−3.55546	−0.155025
$527$	14.9649	0.651882
$528$	−0.830847	−0.0361580
$529$	2.83905	0.123437
$530$	0.278589	0.0121011
$531$	24.9116	1.08107
$532$	0	0
$533$	0	0
$534$	0.0798218	0.00345423
$535$	−22.1561	−0.957894
$536$	−6.46435	−0.279218
$537$	−3.84041	−0.165726
$538$	−3.36000	−0.144860
$539$	0	0
$540$	11.0488	0.475467
$541$	6.94870	0.298748	0.149374	−	0.988781i	$-0.452274\pi$
$541$	6.94870	0.298748	0.149374	+	0.988781i	$0.452274\pi$
$542$	3.54448	0.152249
$543$	−6.41436	−0.275266
$544$	4.55659	0.195362
$545$	17.6344	0.755375
$546$	0	0
$547$	10.9095	0.466457	0.233229	−	0.972422i	$-0.425071\pi$
$547$	10.9095	0.466457	0.233229	+	0.972422i	$0.425071\pi$
$548$	36.1870	1.54583
$549$	−24.0145	−1.02491
$550$	0.0943579	0.00402343
$551$	−11.8352	−0.504194
$552$	1.41986	0.0604332
$553$	0	0
$554$	0.825493	0.0350718
$555$	6.20850	0.263536
$556$	10.4256	0.442145
$557$	34.6295	1.46730	0.733650	−	0.679527i	$-0.237815\pi$
$557$	34.6295	1.46730	0.733650	+	0.679527i	$0.237815\pi$
$558$	−1.50455	−0.0636927
$559$	0	0
$560$	0	0
$561$	−0.676337	−0.0285550
$562$	3.50176	0.147713
$563$	−9.13679	−0.385070	−0.192535	−	0.981290i	$-0.561671\pi$
$563$	−9.13679	−0.385070	−0.192535	+	0.981290i	$0.561671\pi$
$564$	14.7878	0.622677
$565$	8.36084	0.351743
$566$	−1.39740	−0.0587373
$567$	0	0
$568$	3.39680	0.142527
$569$	18.3000	0.767176	0.383588	−	0.923504i	$-0.374688\pi$
$569$	18.3000	0.767176	0.383588	+	0.923504i	$0.374688\pi$
$570$	−0.170835	−0.00715549
$571$	10.1791	0.425981	0.212990	−	0.977054i	$-0.431680\pi$
$571$	10.1791	0.425981	0.212990	+	0.977054i	$0.431680\pi$
$572$	0	0
$573$	3.45423	0.144303
$574$	0	0
$575$	10.9294	0.455788
$576$	20.3679	0.848663
$577$	−19.5165	−0.812482	−0.406241	−	0.913766i	$-0.633161\pi$
$577$	−19.5165	−0.812482	−0.406241	+	0.913766i	$0.633161\pi$
$578$	−0.824740	−0.0343047
$579$	2.44327	0.101539
$580$	−27.4740	−1.14080
$581$	0	0
$582$	−0.555621	−0.0230312
$583$	−0.499926	−0.0207048
$584$	−1.03654	−0.0428923
$585$	0	0
$586$	−2.14103	−0.0884453
$587$	−35.3900	−1.46070	−0.730351	−	0.683072i	$-0.760643\pi$
$587$	−35.3900	−1.46070	−0.730351	+	0.683072i	$0.760643\pi$
$588$	0	0
$589$	−6.78312	−0.279494
$590$	−1.90223	−0.0783137
$591$	−3.36852	−0.138562
$592$	24.7157	1.01581
$593$	−18.0881	−0.742790	−0.371395	−	0.928475i	$-0.621120\pi$
$593$	−18.0881	−0.742790	−0.371395	+	0.928475i	$0.621120\pi$
$594$	0.144660	0.00593548
$595$	0	0
$596$	−18.4118	−0.754179
$597$	−6.96407	−0.285021
$598$	0	0
$599$	9.05992	0.370178	0.185089	−	0.982722i	$-0.440743\pi$
$599$	9.05992	0.370178	0.185089	+	0.982722i	$0.440743\pi$
$600$	0.600572	0.0245182
$601$	−29.2881	−1.19469	−0.597343	−	0.801986i	$-0.703777\pi$
$601$	−29.2881	−1.19469	−0.597343	+	0.801986i	$0.703777\pi$
$602$	0	0
$603$	−35.8586	−1.46028
$604$	27.8234	1.13212
$605$	−18.3454	−0.745846
$606$	1.08957	0.0442606
$607$	39.3650	1.59777	0.798887	−	0.601481i	$-0.205422\pi$
$607$	39.3650	1.59777	0.798887	+	0.601481i	$0.205422\pi$
$608$	−2.06536	−0.0837613
$609$	0	0
$610$	1.83373	0.0742457
$611$	0	0
$612$	16.8303	0.680324
$613$	−5.53316	−0.223482	−0.111741	−	0.993737i	$-0.535643\pi$
$613$	−5.53316	−0.223482	−0.111741	+	0.993737i	$0.535643\pi$
$614$	1.09283	0.0441029
$615$	−5.72829	−0.230987
$616$	0	0
$617$	12.5815	0.506514	0.253257	−	0.967399i	$-0.418498\pi$
$617$	12.5815	0.506514	0.253257	+	0.967399i	$0.418498\pi$
$618$	−0.721742	−0.0290327
$619$	22.3955	0.900149	0.450075	−	0.892991i	$-0.351397\pi$
$619$	22.3955	0.900149	0.450075	+	0.892991i	$0.351397\pi$
$620$	−15.7463	−0.632386
$621$	16.7559	0.672390
$622$	−0.189052	−0.00758031
$623$	0	0
$624$	0	0
$625$	−9.62659	−0.385064
$626$	−2.48221	−0.0992090
$627$	0.306562	0.0122429
$628$	−34.1161	−1.36138
$629$	20.1194	0.802213
$630$	0	0
$631$	−1.94888	−0.0775836	−0.0387918	−	0.999247i	$-0.512351\pi$
$631$	−1.94888	−0.0775836	−0.0387918	+	0.999247i	$0.512351\pi$
$632$	3.30406	0.131428
$633$	−4.79570	−0.190612
$634$	3.67621	0.146001
$635$	13.6190	0.540452
$636$	−1.58519	−0.0628569
$637$	0	0
$638$	−0.359712	−0.0142411
$639$	18.8425	0.745397
$640$	−6.38477	−0.252380
$641$	10.4210	0.411605	0.205803	−	0.978594i	$-0.434019\pi$
$641$	10.4210	0.411605	0.205803	+	0.978594i	$0.434019\pi$
$642$	−0.919818	−0.0363023
$643$	−15.2706	−0.602214	−0.301107	−	0.953590i	$-0.597356\pi$
$643$	−15.2706	−0.602214	−0.301107	+	0.953590i	$0.597356\pi$
$644$	0	0
$645$	−0.759983	−0.0299243
$646$	−0.553612	−0.0217816
$647$	−17.5066	−0.688254	−0.344127	−	0.938923i	$-0.611825\pi$
$647$	−17.5066	−0.688254	−0.344127	+	0.938923i	$0.611825\pi$
$648$	−2.90858	−0.114260
$649$	3.41354	0.133993
$650$	0	0
$651$	0	0
$652$	23.4147	0.916989
$653$	−10.1834	−0.398506	−0.199253	−	0.979948i	$-0.563852\pi$
$653$	−10.1834	−0.398506	−0.199253	+	0.979948i	$0.563852\pi$
$654$	0.732096	0.0286272
$655$	−31.9129	−1.24694
$656$	−22.8040	−0.890346
$657$	−5.74982	−0.224322
$658$	0	0
$659$	−43.8587	−1.70849	−0.854247	−	0.519867i	$-0.825981\pi$
$659$	−43.8587	−1.70849	−0.854247	+	0.519867i	$0.825981\pi$
$660$	0.711651	0.0277010
$661$	32.9270	1.28071	0.640356	−	0.768078i	$-0.278787\pi$
$661$	32.9270	1.28071	0.640356	+	0.768078i	$0.278787\pi$
$662$	−3.11167	−0.120939
$663$	0	0
$664$	0.272170	0.0105622
$665$	0	0
$666$	−2.02277	−0.0783808
$667$	−41.6651	−1.61328
$668$	8.57120	0.331630
$669$	8.90992	0.344478
$670$	2.73814	0.105784
$671$	−3.29062	−0.127033
$672$	0	0
$673$	26.6845	1.02861	0.514307	−	0.857606i	$-0.328049\pi$
$673$	26.6845	1.02861	0.514307	+	0.857606i	$0.328049\pi$
$674$	2.57515	0.0991912
$675$	7.08740	0.272794
$676$	0	0
$677$	−29.5328	−1.13504	−0.567519	−	0.823361i	$-0.692096\pi$
$677$	−29.5328	−1.13504	−0.567519	+	0.823361i	$0.692096\pi$
$678$	0.347102	0.0133304
$679$	0	0
$680$	−2.57968	−0.0989261
$681$	4.04965	0.155183
$682$	−0.206163	−0.00789438
$683$	18.2880	0.699771	0.349885	−	0.936793i	$-0.386221\pi$
$683$	18.2880	0.699771	0.349885	+	0.936793i	$0.386221\pi$
$684$	−7.62863	−0.291688
$685$	−30.7676	−1.17557
$686$	0	0
$687$	−15.9676	−0.609200
$688$	−3.02545	−0.115344
$689$	0	0
$690$	−0.601417	−0.0228956
$691$	−10.3406	−0.393376	−0.196688	−	0.980466i	$-0.563019\pi$
$691$	−10.3406	−0.393376	−0.196688	+	0.980466i	$0.563019\pi$
$692$	24.8580	0.944961
$693$	0	0
$694$	0.265855	0.0100917
$695$	−8.86430	−0.336242
$696$	−2.28950	−0.0867834
$697$	−18.5632	−0.703131
$698$	1.35900	0.0514390
$699$	−3.99064	−0.150940
$700$	0	0
$701$	−41.6959	−1.57483	−0.787415	−	0.616423i	$-0.788581\pi$
$701$	−41.6959	−1.57483	−0.787415	+	0.616423i	$0.788581\pi$
$702$	0	0
$703$	−9.11948	−0.343948
$704$	2.79094	0.105188
$705$	−12.5732	−0.473533
$706$	−4.29003	−0.161457
$707$	0	0
$708$	10.8238	0.406784
$709$	−0.0109463	−0.000411095 0	−0.000205548	−	1.00000i	$-0.500065\pi$
$709$	−0.0109463	−0.000411095 0	−0.000205548		1.00000i	$0.500065\pi$
$710$	−1.43880	−0.0539972
$711$	18.3280	0.687355
$712$	−0.546342	−0.0204750
$713$	−23.8797	−0.894301
$714$	0	0
$715$	0	0
$716$	13.0951	0.489388
$717$	12.8543	0.480054
$718$	−2.32557	−0.0867894
$719$	25.4660	0.949722	0.474861	−	0.880061i	$-0.342498\pi$
$719$	25.4660	0.949722	0.474861	+	0.880061i	$0.342498\pi$
$720$	−17.5789	−0.655127
$721$	0	0
$722$	−2.03590	−0.0757685
$723$	9.19853	0.342097
$724$	21.8718	0.812860
$725$	−17.6235	−0.654521
$726$	−0.761613	−0.0282661
$727$	−23.5565	−0.873663	−0.436831	−	0.899543i	$-0.643899\pi$
$727$	−23.5565	−0.873663	−0.436831	+	0.899543i	$0.643899\pi$
$728$	0	0
$729$	−10.3760	−0.384297
$730$	0.439053	0.0162501
$731$	−2.46282	−0.0910905
$732$	−10.4341	−0.385654
$733$	−6.23249	−0.230202	−0.115101	−	0.993354i	$-0.536719\pi$
$733$	−6.23249	−0.230202	−0.115101	+	0.993354i	$0.536719\pi$
$734$	0.447817	0.0165292
$735$	0	0
$736$	−7.27100	−0.268013
$737$	−4.91357	−0.180994
$738$	1.86632	0.0687000
$739$	1.29718	0.0477174	0.0238587	−	0.999715i	$-0.492405\pi$
$739$	1.29718	0.0477174	0.0238587	+	0.999715i	$0.492405\pi$
$740$	−21.1699	−0.778221
$741$	0	0
$742$	0	0
$743$	−6.06942	−0.222665	−0.111333	−	0.993783i	$-0.535512\pi$
$743$	−6.06942	−0.222665	−0.111333	+	0.993783i	$0.535512\pi$
$744$	−1.31219	−0.0481073
$745$	15.6545	0.573537
$746$	−0.422796	−0.0154797
$747$	1.50976	0.0552393
$748$	2.30619	0.0843227
$749$	0	0
$750$	−0.845960	−0.0308901
$751$	−36.6046	−1.33572	−0.667860	−	0.744287i	$-0.732790\pi$
$751$	−36.6046	−1.33572	−0.667860	+	0.744287i	$0.732790\pi$
$752$	−50.0531	−1.82525
$753$	−13.1346	−0.478650
$754$	0	0
$755$	−23.6566	−0.860952
$756$	0	0
$757$	11.6798	0.424510	0.212255	−	0.977214i	$-0.431919\pi$
$757$	11.6798	0.424510	0.212255	+	0.977214i	$0.431919\pi$
$758$	−3.01096	−0.109363
$759$	1.07924	0.0391739
$760$	1.16928	0.0424144
$761$	−39.7688	−1.44162	−0.720810	−	0.693133i	$-0.756230\pi$
$761$	−39.7688	−1.44162	−0.720810	+	0.693133i	$0.756230\pi$
$762$	0.565394	0.0204821
$763$	0	0
$764$	−11.7783	−0.426125
$765$	−14.3098	−0.517372
$766$	2.70393	0.0976970
$767$	0	0
$768$	8.64915	0.312099
$769$	−9.95937	−0.359144	−0.179572	−	0.983745i	$-0.557471\pi$
$769$	−9.95937	−0.359144	−0.179572	+	0.983745i	$0.557471\pi$
$770$	0	0
$771$	−11.8863	−0.428073
$772$	−8.33112	−0.299843
$773$	−12.7518	−0.458649	−0.229324	−	0.973350i	$-0.573652\pi$
$773$	−12.7518	−0.458649	−0.229324	+	0.973350i	$0.573652\pi$
$774$	0.247608	0.00890007
$775$	−10.1006	−0.362825
$776$	3.80296	0.136518
$777$	0	0
$778$	−1.60517	−0.0575482
$779$	8.41411	0.301467
$780$	0	0
$781$	2.58192	0.0923882
$782$	−1.94897	−0.0696949
$783$	−27.0186	−0.965568
$784$	0	0
$785$	29.0069	1.03530
$786$	−1.32487	−0.0472566
$787$	8.68773	0.309684	0.154842	−	0.987939i	$-0.450513\pi$
$787$	8.68773	0.309684	0.154842	+	0.987939i	$0.450513\pi$
$788$	11.4861	0.409174
$789$	−17.2010	−0.612373
$790$	−1.39952	−0.0497926
$791$	0	0
$792$	−0.465413	−0.0165377
$793$	0	0
$794$	−3.10929	−0.110345
$795$	1.34779	0.0478013
$796$	23.7463	0.841664
$797$	−38.7438	−1.37237	−0.686187	−	0.727425i	$-0.740717\pi$
$797$	−38.7438	−1.37237	−0.686187	+	0.727425i	$0.740717\pi$
$798$	0	0
$799$	−40.7449	−1.44145
$800$	−3.07549	−0.108735
$801$	−3.03063	−0.107082
$802$	2.11358	0.0746331
$803$	−0.787876	−0.0278036
$804$	−15.5802	−0.549471
$805$	0	0
$806$	0	0
$807$	−16.2554	−0.572218
$808$	−7.45757	−0.262356
$809$	28.8550	1.01449	0.507244	−	0.861802i	$-0.330664\pi$
$809$	28.8550	1.01449	0.507244	+	0.861802i	$0.330664\pi$
$810$	1.23200	0.0432882
$811$	12.3917	0.435131	0.217566	−	0.976046i	$-0.430188\pi$
$811$	12.3917	0.435131	0.217566	+	0.976046i	$0.430188\pi$
$812$	0	0
$813$	17.1479	0.601405
$814$	−0.277173	−0.00971491
$815$	−19.9081	−0.697351
$816$	7.25884	0.254110
$817$	1.11632	0.0390549
$818$	−1.74861	−0.0611388
$819$	0	0
$820$	19.5324	0.682103
$821$	−41.0238	−1.43174	−0.715870	−	0.698233i	$-0.753970\pi$
$821$	−41.0238	−1.43174	−0.715870	+	0.698233i	$0.753970\pi$
$822$	−1.27733	−0.0445519
$823$	−2.13613	−0.0744608	−0.0372304	−	0.999307i	$-0.511854\pi$
$823$	−2.13613	−0.0744608	−0.0372304	+	0.999307i	$0.511854\pi$
$824$	4.93999	0.172093
$825$	0.456496	0.0158932
$826$	0	0
$827$	−8.54938	−0.297291	−0.148645	−	0.988891i	$-0.547491\pi$
$827$	−8.54938	−0.297291	−0.148645	+	0.988891i	$0.547491\pi$
$828$	−26.8563	−0.933320
$829$	14.7569	0.512528	0.256264	−	0.966607i	$-0.417508\pi$
$829$	14.7569	0.512528	0.256264	+	0.966607i	$0.417508\pi$
$830$	−0.115284	−0.00400158
$831$	3.99367	0.138539
$832$	0	0
$833$	0	0
$834$	−0.368004	−0.0127429
$835$	−7.28758	−0.252197
$836$	−1.04532	−0.0361532
$837$	−15.4853	−0.535250
$838$	0.553887	0.0191337
$839$	26.9716	0.931164	0.465582	−	0.885005i	$-0.345845\pi$
$839$	26.9716	0.931164	0.465582	+	0.885005i	$0.345845\pi$
$840$	0	0
$841$	38.1845	1.31671
$842$	−2.31867	−0.0799068
$843$	16.9412	0.583487
$844$	16.3525	0.562877
$845$	0	0
$846$	4.09643	0.140838
$847$	0	0
$848$	5.36549	0.184252
$849$	−6.76053	−0.232021
$850$	−0.824374	−0.0282758
$851$	−32.1047	−1.10054
$852$	8.18686	0.280477
$853$	25.6332	0.877665	0.438832	−	0.898569i	$-0.355392\pi$
$853$	25.6332	0.877665	0.438832	+	0.898569i	$0.355392\pi$
$854$	0	0
$855$	6.48618	0.221823
$856$	6.29572	0.215183
$857$	−11.7653	−0.401894	−0.200947	−	0.979602i	$-0.564402\pi$
$857$	−11.7653	−0.401894	−0.200947	+	0.979602i	$0.564402\pi$
$858$	0	0
$859$	21.7761	0.742992	0.371496	−	0.928435i	$-0.378845\pi$
$859$	21.7761	0.742992	0.371496	+	0.928435i	$0.378845\pi$
$860$	2.59141	0.0883663
$861$	0	0
$862$	3.41402	0.116282
$863$	41.0575	1.39761	0.698807	−	0.715310i	$-0.253715\pi$
$863$	41.0575	1.39761	0.698807	+	0.715310i	$0.253715\pi$
$864$	−4.71504	−0.160409
$865$	−21.1353	−0.718623
$866$	1.50694	0.0512079
$867$	−3.99003	−0.135509
$868$	0	0
$869$	2.51142	0.0851942
$870$	0.969778	0.0328785
$871$	0	0
$872$	−5.01085	−0.169689
$873$	21.0955	0.713975
$874$	0.883404	0.0298816
$875$	0	0
$876$	−2.49824	−0.0844076
$877$	6.89112	0.232696	0.116348	−	0.993208i	$-0.462881\pi$
$877$	6.89112	0.232696	0.116348	+	0.993208i	$0.462881\pi$
$878$	−3.81870	−0.128875
$879$	−10.3582	−0.349372
$880$	−2.40877	−0.0811996
$881$	10.6458	0.358665	0.179332	−	0.983789i	$-0.442606\pi$
$881$	10.6458	0.358665	0.179332	+	0.983789i	$0.442606\pi$
$882$	0	0
$883$	−21.3844	−0.719641	−0.359821	−	0.933022i	$-0.617162\pi$
$883$	−21.3844	−0.719641	−0.359821	+	0.933022i	$0.617162\pi$
$884$	0	0
$885$	−9.20286	−0.309351
$886$	0.208281	0.00699732
$887$	34.1150	1.14547	0.572735	−	0.819740i	$-0.305882\pi$
$887$	34.1150	1.14547	0.572735	+	0.819740i	$0.305882\pi$
$888$	−1.76416	−0.0592013
$889$	0	0
$890$	0.231417	0.00775712
$891$	−2.21082	−0.0740653
$892$	−30.3813	−1.01724
$893$	18.4684	0.618020
$894$	0.649901	0.0217359
$895$	−11.1340	−0.372169
$896$	0	0
$897$	0	0
$898$	1.27056	0.0423992
$899$	38.5057	1.28424
$900$	−11.3597	−0.378655
$901$	4.36769	0.145509
$902$	0.255734	0.00851502
$903$	0	0
$904$	−2.37575	−0.0790163
$905$	−18.5963	−0.618163
$906$	−0.982110	−0.0326284
$907$	42.1515	1.39962	0.699810	−	0.714329i	$-0.253268\pi$
$907$	42.1515	1.39962	0.699810	+	0.714329i	$0.253268\pi$
$908$	−13.8086	−0.458254
$909$	−41.3681	−1.37209
$910$	0	0
$911$	20.9947	0.695584	0.347792	−	0.937572i	$-0.386932\pi$
$911$	20.9947	0.695584	0.347792	+	0.937572i	$0.386932\pi$
$912$	−3.29020	−0.108949
$913$	0.206877	0.00684663
$914$	0.956734	0.0316459
$915$	8.87147	0.293282
$916$	54.4466	1.79897
$917$	0	0
$918$	−1.26385	−0.0417132
$919$	−14.2940	−0.471515	−0.235757	−	0.971812i	$-0.575757\pi$
$919$	−14.2940	−0.471515	−0.235757	+	0.971812i	$0.575757\pi$
$920$	4.11642	0.135714
$921$	5.28701	0.174213
$922$	−1.31094	−0.0431736
$923$	0	0
$924$	0	0
$925$	−13.5797	−0.446497
$926$	4.31162	0.141689
$927$	27.4027	0.900024
$928$	11.7244	0.384872
$929$	−6.80723	−0.223338	−0.111669	−	0.993745i	$-0.535620\pi$
$929$	−6.80723	−0.223338	−0.111669	+	0.993745i	$0.535620\pi$
$930$	0.555812	0.0182258
$931$	0	0
$932$	13.6074	0.445724
$933$	−0.914621	−0.0299433
$934$	−2.39496	−0.0783655
$935$	−1.96082	−0.0641256
$936$	0	0
$937$	5.22890	0.170821	0.0854104	−	0.996346i	$-0.472780\pi$
$937$	5.22890	0.170821	0.0854104	+	0.996346i	$0.472780\pi$
$938$	0	0
$939$	−12.0087	−0.391890
$940$	42.8723	1.39834
$941$	56.4403	1.83990	0.919951	−	0.392033i	$-0.128228\pi$
$941$	56.4403	1.83990	0.919951	+	0.392033i	$0.128228\pi$
$942$	1.20423	0.0392359
$943$	29.6215	0.964609
$944$	−36.6361	−1.19240
$945$	0	0
$946$	0.0339287	0.00110312
$947$	−5.85027	−0.190108	−0.0950541	−	0.995472i	$-0.530302\pi$
$947$	−5.85027	−0.190108	−0.0950541	+	0.995472i	$0.530302\pi$
$948$	7.96334	0.258637
$949$	0	0
$950$	0.373662	0.0121232
$951$	17.7852	0.576726
$952$	0	0
$953$	−21.7484	−0.704499	−0.352249	−	0.935906i	$-0.614583\pi$
$953$	−21.7484	−0.704499	−0.352249	+	0.935906i	$0.614583\pi$
$954$	−0.439121	−0.0142171
$955$	10.0144	0.324059
$956$	−43.8310	−1.41760
$957$	−1.74026	−0.0562546
$958$	−3.16429	−0.102233
$959$	0	0
$960$	−7.52433	−0.242847
$961$	−8.93110	−0.288100
$962$	0	0
$963$	34.9232	1.12538
$964$	−31.3654	−1.01021
$965$	7.08346	0.228025
$966$	0	0
$967$	13.3251	0.428507	0.214253	−	0.976778i	$-0.431268\pi$
$967$	13.3251	0.428507	0.214253	+	0.976778i	$0.431268\pi$
$968$	5.21288	0.167548
$969$	−2.67833	−0.0860404
$970$	−1.61084	−0.0517210
$971$	−7.46185	−0.239462	−0.119731	−	0.992806i	$-0.538203\pi$
$971$	−7.46185	−0.239462	−0.119731	+	0.992806i	$0.538203\pi$
$972$	−26.6448	−0.854633
$973$	0	0
$974$	0.767850	0.0246035
$975$	0	0
$976$	35.3168	1.13046
$977$	10.9605	0.350656	0.175328	−	0.984510i	$-0.443901\pi$
$977$	10.9605	0.350656	0.175328	+	0.984510i	$0.443901\pi$
$978$	−0.826490	−0.0264282
$979$	−0.415276	−0.0132723
$980$	0	0
$981$	−27.7959	−0.887453
$982$	−0.357189	−0.0113984
$983$	16.1441	0.514918	0.257459	−	0.966289i	$-0.417115\pi$
$983$	16.1441	0.514918	0.257459	+	0.966289i	$0.417115\pi$
$984$	1.62771	0.0518893
$985$	−9.76593	−0.311168
$986$	3.14268	0.100083
$987$	0	0
$988$	0	0
$989$	3.92994	0.124965
$990$	0.197138	0.00626544
$991$	6.71496	0.213308	0.106654	−	0.994296i	$-0.465986\pi$
$991$	6.71496	0.213308	0.106654	+	0.994296i	$0.465986\pi$
$992$	6.71965	0.213349
$993$	−15.0540	−0.477725
$994$	0	0
$995$	−20.1901	−0.640068
$996$	0.655975	0.0207854
$997$	18.4411	0.584037	0.292018	−	0.956413i	$-0.405673\pi$
$997$	18.4411	0.584037	0.292018	+	0.956413i	$0.405673\pi$
$998$	−3.38609	−0.107185
$999$	−20.8190	−0.658684

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ch.1.3		6
7.6	odd	2		1183.2.a.p.1.3		6
13.6	odd	12		637.2.q.h.491.3		12
13.11	odd	12		637.2.q.h.589.3		12
13.12	even	2		8281.2.a.by.1.4		6
91.6	even	12		91.2.q.a.36.3	✓	12
91.11	odd	12		637.2.u.i.30.4		12
91.19	even	12		637.2.u.h.361.4		12
91.24	even	12		637.2.u.h.30.4		12
91.32	odd	12		637.2.k.g.569.3		12
91.34	even	4		1183.2.c.i.337.7		12
91.37	odd	12		637.2.k.g.459.4		12
91.45	even	12		637.2.k.h.569.3		12
91.58	odd	12		637.2.u.i.361.4		12
91.76	even	12		91.2.q.a.43.3	yes	12
91.83	even	4		1183.2.c.i.337.6		12
91.89	even	12		637.2.k.h.459.4		12
91.90	odd	2		1183.2.a.m.1.4		6
273.167	odd	12		819.2.ct.a.316.4		12
273.188	odd	12		819.2.ct.a.127.4		12
364.167	odd	12		1456.2.cc.c.225.3		12
364.279	odd	12		1456.2.cc.c.673.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.q.a.36.3	✓	12	91.6	even	12
91.2.q.a.43.3	yes	12	91.76	even	12
637.2.k.g.459.4		12	91.37	odd	12
637.2.k.g.569.3		12	91.32	odd	12
637.2.k.h.459.4		12	91.89	even	12
637.2.k.h.569.3		12	91.45	even	12
637.2.q.h.491.3		12	13.6	odd	12
637.2.q.h.589.3		12	13.11	odd	12
637.2.u.h.30.4		12	91.24	even	12
637.2.u.h.361.4		12	91.19	even	12
637.2.u.i.30.4		12	91.11	odd	12
637.2.u.i.361.4		12	91.58	odd	12
819.2.ct.a.127.4		12	273.188	odd	12
819.2.ct.a.316.4		12	273.167	odd	12
1183.2.a.m.1.4		6	91.90	odd	2
1183.2.a.p.1.3		6	7.6	odd	2
1183.2.c.i.337.6		12	91.83	even	4
1183.2.c.i.337.7		12	91.34	even	4
1456.2.cc.c.225.3		12	364.167	odd	12
1456.2.cc.c.673.3		12	364.279	odd	12
8281.2.a.by.1.4		6	13.12	even	2
8281.2.a.ch.1.3		6	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)

\(f(q)\)	\(=\)	\(q+0.120360 q^{2} +0.582292 q^{3} -1.98551 q^{4} +1.68817 q^{5} +0.0700846 q^{6} -0.479696 q^{8} -2.66094 q^{9} +0.203187 q^{10} -0.364618 q^{11} -1.15615 q^{12} +0.983005 q^{15} +3.91329 q^{16} +3.18555 q^{17} -0.320270 q^{18} -1.44391 q^{19} -3.35188 q^{20} -0.0438854 q^{22} -5.08321 q^{23} -0.279323 q^{24} -2.15010 q^{25} -3.29632 q^{27} +8.19662 q^{29} +0.118314 q^{30} +4.69775 q^{31} +1.43040 q^{32} -0.212314 q^{33} +0.383412 q^{34} +5.28332 q^{36} +6.31584 q^{37} -0.173789 q^{38} -0.809806 q^{40} -5.82732 q^{41} -0.773122 q^{43} +0.723954 q^{44} -4.49210 q^{45} -0.611815 q^{46} -12.7905 q^{47} +2.27868 q^{48} -0.258786 q^{50} +1.85492 q^{51} +1.37110 q^{53} -0.396744 q^{54} -0.615536 q^{55} -0.840776 q^{57} +0.986544 q^{58} -9.36197 q^{59} -1.95177 q^{60} +9.02484 q^{61} +0.565421 q^{62} -7.65442 q^{64} -0.0255541 q^{66} +13.4759 q^{67} -6.32495 q^{68} -2.95991 q^{69} -7.08115 q^{71} +1.27644 q^{72} +2.16083 q^{73} +0.760173 q^{74} -1.25198 q^{75} +2.86690 q^{76} -6.88781 q^{79} +6.60628 q^{80} +6.06339 q^{81} -0.701376 q^{82} -0.567380 q^{83} +5.37773 q^{85} -0.0930528 q^{86} +4.77282 q^{87} +0.174906 q^{88} +1.13893 q^{89} -0.540669 q^{90} +10.0928 q^{92} +2.73547 q^{93} -1.53947 q^{94} -2.43755 q^{95} +0.832908 q^{96} -7.92785 q^{97} +0.970225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 12 q^{8} + 4 q^{9} - 12 q^{10} + 4 q^{11} - 2 q^{12} + 20 q^{15} + 8 q^{16} + 4 q^{17} - 16 q^{18} - 2 q^{19} - 26 q^{20} - 6 q^{22} - 12 q^{23} - 2 q^{24}+ \cdots + 16 q^{99}+O(q^{100}) \) 6 * q + 4 * q^2 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 12 * q^8 + 4 * q^9 - 12 * q^10 + 4 * q^11 - 2 * q^12 + 20 * q^15 + 8 * q^16 + 4 * q^17 - 16 * q^18 - 2 * q^19 - 26 * q^20 - 6 * q^22 - 12 * q^23 - 2 * q^24 + 10 * q^25 - 6 * q^27 - 8 * q^29 + 8 * q^30 + 14 * q^31 + 8 * q^32 - 16 * q^33 + 2 * q^34 - 10 * q^36 + 12 * q^37 + 2 * q^38 - 46 * q^40 - 28 * q^41 + 2 * q^43 - 20 * q^44 - 16 * q^45 + 20 * q^46 - 14 * q^47 - 2 * q^48 + 32 * q^50 - 26 * q^51 - 22 * q^53 - 14 * q^54 - 6 * q^55 + 4 * q^58 + 2 * q^59 + 14 * q^61 + 4 * q^62 + 26 * q^64 + 26 * q^66 + 24 * q^67 - 8 * q^68 - 4 * q^69 + 4 * q^71 + 8 * q^72 - 36 * q^73 - 6 * q^74 - 46 * q^75 + 26 * q^76 - 28 * q^79 - 36 * q^80 - 2 * q^81 - 14 * q^82 - 26 * q^83 - 20 * q^85 - 24 * q^86 - 2 * q^87 - 14 * q^88 - 42 * q^89 + 12 * q^90 + 12 * q^92 + 4 * q^94 - 22 * q^95 + 42 * q^96 - 24 * q^97 + 16 * q^99

Introduction

L-functions

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