LMFDB - Embedded newform 5096.2.a.bc.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5096,2,Mod(1,5096)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5096, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("5096.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5096 = 2^{3} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5096.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,-2,0,2,0,0,0,6,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.6917648700$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} + 20x^{5} + 56x^{4} - 60x^{3} - 74x^{2} + 60x - 2 $ x^8 - 2x^7 - 13x^6 + 20x^5 + 56x^4 - 60x^3 - 74x^2 + 60*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.75049$ of defining polynomial
Character	$\chi$	$=$	5096.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.75049 q^{3} +1.15117 q^{5} +0.0642006 q^{9} -2.15117 q^{11} +1.00000 q^{13} +2.01511 q^{15} +0.277589 q^{17} +3.01226 q^{19} -8.07284 q^{23} -3.67480 q^{25} -5.13908 q^{27} -4.51541 q^{29} -10.8566 q^{31} -3.76560 q^{33} -9.04636 q^{37} +1.75049 q^{39} -3.70349 q^{41} -5.16225 q^{43} +0.0739060 q^{45} +1.10775 q^{47} +0.485916 q^{51} +1.09278 q^{53} -2.47637 q^{55} +5.27291 q^{57} -4.26585 q^{59} +7.83823 q^{61} +1.15117 q^{65} -4.73127 q^{67} -14.1314 q^{69} -1.10022 q^{71} +3.23759 q^{73} -6.43269 q^{75} +16.6935 q^{79} -9.18848 q^{81} +10.8669 q^{83} +0.319553 q^{85} -7.90416 q^{87} +8.92564 q^{89} -19.0043 q^{93} +3.46763 q^{95} +4.03001 q^{97} -0.138107 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 2 q^{5} + 6 q^{9} - 10 q^{11} + 8 q^{13} - 4 q^{15} - 14 q^{19} - 8 q^{23} - 2 q^{25} - 14 q^{27} - 2 q^{29} - 20 q^{31} + 6 q^{33} + 2 q^{37} - 2 q^{39} + 2 q^{41} - 2 q^{43} - 18 q^{45}+ \cdots + 12 q^{99}+O(q^{100}) $ 8 * q - 2 * q^3 + 2 * q^5 + 6 * q^9 - 10 * q^11 + 8 * q^13 - 4 * q^15 - 14 * q^19 - 8 * q^23 - 2 * q^25 - 14 * q^27 - 2 * q^29 - 20 * q^31 + 6 * q^33 + 2 * q^37 - 2 * q^39 + 2 * q^41 - 2 * q^43 - 18 * q^45 - 24 * q^47 - 4 * q^51 + 6 * q^53 - 40 * q^55 + 4 * q^57 - 12 * q^59 - 10 * q^61 + 2 * q^65 + 14 * q^67 + 6 * q^69 - 24 * q^73 - 14 * q^75 + 8 * q^79 - 20 * q^81 - 18 * q^83 + 8 * q^85 - 56 * q^87 + 18 * q^89 + 10 * q^93 + 10 * q^95 - 16 * q^97 + 12 * 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	0
$2$	0	0
$3$	1.75049	1.01064	0.505322	−	0.862931i	$-0.331374\pi$
$3$	1.75049	1.01064	0.505322	+	0.862931i	$0.331374\pi$
$4$	0	0
$5$	1.15117	0.514820	0.257410	−	0.966302i	$-0.417131\pi$
$5$	1.15117	0.514820	0.257410	+	0.966302i	$0.417131\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.0642006	0.0214002
$10$	0	0
$11$	−2.15117	−0.648603	−0.324301	−	0.945954i	$-0.605129\pi$
$11$	−2.15117	−0.648603	−0.324301	+	0.945954i	$0.605129\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.01511	0.520299
$16$	0	0
$17$	0.277589	0.0673252	0.0336626	−	0.999433i	$-0.489283\pi$
$17$	0.277589	0.0673252	0.0336626	+	0.999433i	$0.489283\pi$
$18$	0	0
$19$	3.01226	0.691059	0.345530	−	0.938408i	$-0.387699\pi$
$19$	3.01226	0.691059	0.345530	+	0.938408i	$0.387699\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.07284	−1.68330	−0.841652	−	0.540020i	$-0.818417\pi$
$23$	−8.07284	−1.68330	−0.841652	+	0.540020i	$0.818417\pi$
$24$	0	0
$25$	−3.67480	−0.734961
$26$	0	0
$27$	−5.13908	−0.989015
$28$	0	0
$29$	−4.51541	−0.838490	−0.419245	−	0.907873i	$-0.637705\pi$
$29$	−4.51541	−0.838490	−0.419245	+	0.907873i	$0.637705\pi$
$30$	0	0
$31$	−10.8566	−1.94990	−0.974952	−	0.222417i	$-0.928605\pi$
$31$	−10.8566	−1.94990	−0.974952	+	0.222417i	$0.928605\pi$
$32$	0	0
$33$	−3.76560	−0.655506
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.04636	−1.48721	−0.743606	−	0.668618i	$-0.766886\pi$
$37$	−9.04636	−1.48721	−0.743606	+	0.668618i	$0.766886\pi$
$38$	0	0
$39$	1.75049	0.280302
$40$	0	0
$41$	−3.70349	−0.578388	−0.289194	−	0.957270i	$-0.593387\pi$
$41$	−3.70349	−0.578388	−0.289194	+	0.957270i	$0.593387\pi$
$42$	0	0
$43$	−5.16225	−0.787235	−0.393618	−	0.919274i	$-0.628777\pi$
$43$	−5.16225	−0.787235	−0.393618	+	0.919274i	$0.628777\pi$
$44$	0	0
$45$	0.0739060	0.0110173
$46$	0	0
$47$	1.10775	0.161581	0.0807907	−	0.996731i	$-0.474255\pi$
$47$	1.10775	0.161581	0.0807907	+	0.996731i	$0.474255\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.485916	0.0680418
$52$	0	0
$53$	1.09278	0.150105	0.0750523	−	0.997180i	$-0.476088\pi$
$53$	1.09278	0.150105	0.0750523	+	0.997180i	$0.476088\pi$
$54$	0	0
$55$	−2.47637	−0.333914
$56$	0	0
$57$	5.27291	0.698414
$58$	0	0
$59$	−4.26585	−0.555367	−0.277683	−	0.960673i	$-0.589567\pi$
$59$	−4.26585	−0.555367	−0.277683	+	0.960673i	$0.589567\pi$
$60$	0	0
$61$	7.83823	1.00358	0.501791	−	0.864989i	$-0.332675\pi$
$61$	7.83823	1.00358	0.501791	+	0.864989i	$0.332675\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.15117	0.142785
$66$	0	0
$67$	−4.73127	−0.578016	−0.289008	−	0.957327i	$-0.593325\pi$
$67$	−4.73127	−0.578016	−0.289008	+	0.957327i	$0.593325\pi$
$68$	0	0
$69$	−14.1314	−1.70122
$70$	0	0
$71$	−1.10022	−0.130573	−0.0652863	−	0.997867i	$-0.520796\pi$
$71$	−1.10022	−0.130573	−0.0652863	+	0.997867i	$0.520796\pi$
$72$	0	0
$73$	3.23759	0.378931	0.189466	−	0.981887i	$-0.439324\pi$
$73$	3.23759	0.378931	0.189466	+	0.981887i	$0.439324\pi$
$74$	0	0
$75$	−6.43269	−0.742783
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.6935	1.87817	0.939083	−	0.343690i	$-0.111677\pi$
$79$	16.6935	1.87817	0.939083	+	0.343690i	$0.111677\pi$
$80$	0	0
$81$	−9.18848	−1.02094
$82$	0	0
$83$	10.8669	1.19280	0.596399	−	0.802688i	$-0.296597\pi$
$83$	10.8669	1.19280	0.596399	+	0.802688i	$0.296597\pi$
$84$	0	0
$85$	0.319553	0.0346604
$86$	0	0
$87$	−7.90416	−0.847414
$88$	0	0
$89$	8.92564	0.946116	0.473058	−	0.881031i	$-0.343150\pi$
$89$	8.92564	0.946116	0.473058	+	0.881031i	$0.343150\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−19.0043	−1.97066
$94$	0	0
$95$	3.46763	0.355771
$96$	0	0
$97$	4.03001	0.409185	0.204593	−	0.978847i	$-0.434413\pi$
$97$	4.03001	0.409185	0.204593	+	0.978847i	$0.434413\pi$
$98$	0	0
$99$	−0.138107	−0.0138802
$100$	0	0
$101$	8.38900	0.834737	0.417369	−	0.908737i	$-0.362952\pi$
$101$	8.38900	0.834737	0.417369	+	0.908737i	$0.362952\pi$
$102$	0	0
$103$	4.77695	0.470687	0.235343	−	0.971912i	$-0.424379\pi$
$103$	4.77695	0.470687	0.235343	+	0.971912i	$0.424379\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.82255	0.272866	0.136433	−	0.990649i	$-0.456436\pi$
$107$	2.82255	0.272866	0.136433	+	0.990649i	$0.456436\pi$
$108$	0	0
$109$	7.75463	0.742759	0.371380	−	0.928481i	$-0.378885\pi$
$109$	7.75463	0.742759	0.371380	+	0.928481i	$0.378885\pi$
$110$	0	0
$111$	−15.8355	−1.50304
$112$	0	0
$113$	19.2213	1.80819	0.904093	−	0.427337i	$-0.140548\pi$
$113$	19.2213	1.80819	0.904093	+	0.427337i	$0.140548\pi$
$114$	0	0
$115$	−9.29323	−0.866598
$116$	0	0
$117$	0.0642006	0.00593535
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.37246	−0.579314
$122$	0	0
$123$	−6.48291	−0.584544
$124$	0	0
$125$	−9.98619	−0.893192
$126$	0	0
$127$	3.13730	0.278390	0.139195	−	0.990265i	$-0.455548\pi$
$127$	3.13730	0.278390	0.139195	+	0.990265i	$0.455548\pi$
$128$	0	0
$129$	−9.03644	−0.795614
$130$	0	0
$131$	1.23406	0.107820	0.0539099	−	0.998546i	$-0.482832\pi$
$131$	1.23406	0.107820	0.0539099	+	0.998546i	$0.482832\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.91596	−0.509165
$136$	0	0
$137$	−16.0509	−1.37132	−0.685659	−	0.727923i	$-0.740486\pi$
$137$	−16.0509	−1.37132	−0.685659	+	0.727923i	$0.740486\pi$
$138$	0	0
$139$	−18.3234	−1.55417	−0.777085	−	0.629396i	$-0.783302\pi$
$139$	−18.3234	−1.55417	−0.777085	+	0.629396i	$0.783302\pi$
$140$	0	0
$141$	1.93909	0.163301
$142$	0	0
$143$	−2.15117	−0.179890
$144$	0	0
$145$	−5.19801	−0.431671
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.8510	−1.05280	−0.526398	−	0.850238i	$-0.676458\pi$
$149$	−12.8510	−1.05280	−0.526398	+	0.850238i	$0.676458\pi$
$150$	0	0
$151$	−18.8615	−1.53493	−0.767465	−	0.641091i	$-0.778482\pi$
$151$	−18.8615	−1.53493	−0.767465	+	0.641091i	$0.778482\pi$
$152$	0	0
$153$	0.0178214	0.00144077
$154$	0	0
$155$	−12.4978	−1.00385
$156$	0	0
$157$	−18.4088	−1.46918	−0.734590	−	0.678511i	$-0.762625\pi$
$157$	−18.4088	−1.46918	−0.734590	+	0.678511i	$0.762625\pi$
$158$	0	0
$159$	1.91289	0.151702
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.95863	0.388390	0.194195	−	0.980963i	$-0.437791\pi$
$163$	4.95863	0.388390	0.194195	+	0.980963i	$0.437791\pi$
$164$	0	0
$165$	−4.33485	−0.337468
$166$	0	0
$167$	12.9778	1.00425	0.502126	−	0.864794i	$-0.332551\pi$
$167$	12.9778	1.00425	0.502126	+	0.864794i	$0.332551\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.193389	0.0147888
$172$	0	0
$173$	2.53742	0.192917	0.0964584	−	0.995337i	$-0.469249\pi$
$173$	2.53742	0.192917	0.0964584	+	0.995337i	$0.469249\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.46731	−0.561278
$178$	0	0
$179$	−0.702711	−0.0525231	−0.0262616	−	0.999655i	$-0.508360\pi$
$179$	−0.702711	−0.0525231	−0.0262616	+	0.999655i	$0.508360\pi$
$180$	0	0
$181$	1.53149	0.113835	0.0569175	−	0.998379i	$-0.481873\pi$
$181$	1.53149	0.113835	0.0569175	+	0.998379i	$0.481873\pi$
$182$	0	0
$183$	13.7207	1.01426
$184$	0	0
$185$	−10.4139	−0.765646
$186$	0	0
$187$	−0.597142	−0.0436673
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.59657	0.404954	0.202477	−	0.979287i	$-0.435101\pi$
$191$	5.59657	0.404954	0.202477	+	0.979287i	$0.435101\pi$
$192$	0	0
$193$	19.1089	1.37549	0.687743	−	0.725954i	$-0.258602\pi$
$193$	19.1089	1.37549	0.687743	+	0.725954i	$0.258602\pi$
$194$	0	0
$195$	2.01511	0.144305
$196$	0	0
$197$	6.04551	0.430724	0.215362	−	0.976534i	$-0.430907\pi$
$197$	6.04551	0.430724	0.215362	+	0.976534i	$0.430907\pi$
$198$	0	0
$199$	−22.1575	−1.57071	−0.785353	−	0.619048i	$-0.787519\pi$
$199$	−22.1575	−1.57071	−0.785353	+	0.619048i	$0.787519\pi$
$200$	0	0
$201$	−8.28202	−0.584168
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.26336	−0.297766
$206$	0	0
$207$	−0.518282	−0.0360231
$208$	0	0
$209$	−6.47988	−0.448223
$210$	0	0
$211$	18.5581	1.27759	0.638795	−	0.769377i	$-0.279433\pi$
$211$	18.5581	1.27759	0.638795	+	0.769377i	$0.279433\pi$
$212$	0	0
$213$	−1.92593	−0.131962
$214$	0	0
$215$	−5.94263	−0.405284
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.66736	0.382964
$220$	0	0
$221$	0.277589	0.0186727
$222$	0	0
$223$	−16.8866	−1.13081	−0.565406	−	0.824813i	$-0.691280\pi$
$223$	−16.8866	−1.13081	−0.565406	+	0.824813i	$0.691280\pi$
$224$	0	0
$225$	−0.235925	−0.0157283
$226$	0	0
$227$	−19.4371	−1.29009	−0.645043	−	0.764146i	$-0.723161\pi$
$227$	−19.4371	−1.29009	−0.645043	+	0.764146i	$0.723161\pi$
$228$	0	0
$229$	−6.55450	−0.433134	−0.216567	−	0.976268i	$-0.569486\pi$
$229$	−6.55450	−0.433134	−0.216567	+	0.976268i	$0.569486\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.24416	0.474581	0.237290	−	0.971439i	$-0.423741\pi$
$233$	7.24416	0.474581	0.237290	+	0.971439i	$0.423741\pi$
$234$	0	0
$235$	1.27521	0.0831853
$236$	0	0
$237$	29.2217	1.89816
$238$	0	0
$239$	19.2566	1.24561	0.622804	−	0.782378i	$-0.285993\pi$
$239$	19.2566	1.24561	0.622804	+	0.782378i	$0.285993\pi$
$240$	0	0
$241$	16.6327	1.07141	0.535705	−	0.844405i	$-0.320046\pi$
$241$	16.6327	1.07141	0.535705	+	0.844405i	$0.320046\pi$
$242$	0	0
$243$	−0.667079	−0.0427931
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.01226	0.191665
$248$	0	0
$249$	19.0224	1.20549
$250$	0	0
$251$	−4.26389	−0.269134	−0.134567	−	0.990904i	$-0.542964\pi$
$251$	−4.26389	−0.269134	−0.134567	+	0.990904i	$0.542964\pi$
$252$	0	0
$253$	17.3661	1.09180
$254$	0	0
$255$	0.559373	0.0350293
$256$	0	0
$257$	0.690745	0.0430875	0.0215437	−	0.999768i	$-0.493142\pi$
$257$	0.690745	0.0430875	0.0215437	+	0.999768i	$0.493142\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.289892	−0.0179439
$262$	0	0
$263$	−12.1588	−0.749743	−0.374871	−	0.927077i	$-0.622313\pi$
$263$	−12.1588	−0.749743	−0.374871	+	0.927077i	$0.622313\pi$
$264$	0	0
$265$	1.25798	0.0772768
$266$	0	0
$267$	15.6242	0.956186
$268$	0	0
$269$	6.58964	0.401777	0.200889	−	0.979614i	$-0.435617\pi$
$269$	6.58964	0.401777	0.200889	+	0.979614i	$0.435617\pi$
$270$	0	0
$271$	−12.1686	−0.739189	−0.369594	−	0.929193i	$-0.620503\pi$
$271$	−12.1686	−0.739189	−0.369594	+	0.929193i	$0.620503\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.90513	0.476697
$276$	0	0
$277$	−22.5460	−1.35466	−0.677328	−	0.735681i	$-0.736862\pi$
$277$	−22.5460	−1.35466	−0.677328	+	0.735681i	$0.736862\pi$
$278$	0	0
$279$	−0.697001	−0.0417283
$280$	0	0
$281$	10.2755	0.612983	0.306491	−	0.951873i	$-0.400845\pi$
$281$	10.2755	0.612983	0.306491	+	0.951873i	$0.400845\pi$
$282$	0	0
$283$	−11.4001	−0.677668	−0.338834	−	0.940846i	$-0.610032\pi$
$283$	−11.4001	−0.677668	−0.338834	+	0.940846i	$0.610032\pi$
$284$	0	0
$285$	6.07003	0.359558
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9229	−0.995467
$290$	0	0
$291$	7.05447	0.413540
$292$	0	0
$293$	26.4525	1.54537	0.772686	−	0.634788i	$-0.218913\pi$
$293$	26.4525	1.54537	0.772686	+	0.634788i	$0.218913\pi$
$294$	0	0
$295$	−4.91073	−0.285914
$296$	0	0
$297$	11.0550	0.641478
$298$	0	0
$299$	−8.07284	−0.466865
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.6848	0.843622
$304$	0	0
$305$	9.02315	0.516664
$306$	0	0
$307$	−6.91144	−0.394457	−0.197228	−	0.980358i	$-0.563194\pi$
$307$	−6.91144	−0.394457	−0.197228	+	0.980358i	$0.563194\pi$
$308$	0	0
$309$	8.36198	0.475697
$310$	0	0
$311$	−21.5264	−1.22065	−0.610324	−	0.792152i	$-0.708961\pi$
$311$	−21.5264	−1.22065	−0.610324	+	0.792152i	$0.708961\pi$
$312$	0	0
$313$	−25.6908	−1.45213	−0.726066	−	0.687625i	$-0.758653\pi$
$313$	−25.6908	−1.45213	−0.726066	+	0.687625i	$0.758653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.2807	−0.745916	−0.372958	−	0.927848i	$-0.621656\pi$
$317$	−13.2807	−0.745916	−0.372958	+	0.927848i	$0.621656\pi$
$318$	0	0
$319$	9.71342	0.543847
$320$	0	0
$321$	4.94083	0.275770
$322$	0	0
$323$	0.836170	0.0465257
$324$	0	0
$325$	−3.67480	−0.203841
$326$	0	0
$327$	13.5744	0.750665
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.01061	−0.0555481	−0.0277741	−	0.999614i	$-0.508842\pi$
$331$	−1.01061	−0.0555481	−0.0277741	+	0.999614i	$0.508842\pi$
$332$	0	0
$333$	−0.580782	−0.0318267
$334$	0	0
$335$	−5.44650	−0.297574
$336$	0	0
$337$	14.0001	0.762633	0.381316	−	0.924445i	$-0.375471\pi$
$337$	14.0001	0.762633	0.381316	+	0.924445i	$0.375471\pi$
$338$	0	0
$339$	33.6466	1.82743
$340$	0	0
$341$	23.3544	1.26471
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−16.2677	−0.875822
$346$	0	0
$347$	8.65254	0.464493	0.232246	−	0.972657i	$-0.425392\pi$
$347$	8.65254	0.464493	0.232246	+	0.972657i	$0.425392\pi$
$348$	0	0
$349$	−23.8062	−1.27432	−0.637158	−	0.770734i	$-0.719890\pi$
$349$	−23.8062	−1.27432	−0.637158	+	0.770734i	$0.719890\pi$
$350$	0	0
$351$	−5.13908	−0.274304
$352$	0	0
$353$	−25.1275	−1.33740	−0.668702	−	0.743531i	$-0.733150\pi$
$353$	−25.1275	−1.33740	−0.668702	+	0.743531i	$0.733150\pi$
$354$	0	0
$355$	−1.26655	−0.0672214
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.1431	1.48534	0.742668	−	0.669660i	$-0.233560\pi$
$359$	28.1431	1.48534	0.742668	+	0.669660i	$0.233560\pi$
$360$	0	0
$361$	−9.92631	−0.522437
$362$	0	0
$363$	−11.1549	−0.585480
$364$	0	0
$365$	3.72702	0.195081
$366$	0	0
$367$	7.25779	0.378854	0.189427	−	0.981895i	$-0.439337\pi$
$367$	7.25779	0.378854	0.189427	+	0.981895i	$0.439337\pi$
$368$	0	0
$369$	−0.237767	−0.0123776
$370$	0	0
$371$	0	0
$372$	0	0
$373$	8.57217	0.443850	0.221925	−	0.975064i	$-0.428766\pi$
$373$	8.57217	0.443850	0.221925	+	0.975064i	$0.428766\pi$
$374$	0	0
$375$	−17.4807	−0.902699
$376$	0	0
$377$	−4.51541	−0.232555
$378$	0	0
$379$	−7.23298	−0.371533	−0.185767	−	0.982594i	$-0.559477\pi$
$379$	−7.23298	−0.371533	−0.185767	+	0.982594i	$0.559477\pi$
$380$	0	0
$381$	5.49180	0.281353
$382$	0	0
$383$	−3.30880	−0.169072	−0.0845358	−	0.996420i	$-0.526941\pi$
$383$	−3.30880	−0.169072	−0.0845358	+	0.996420i	$0.526941\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.331420	−0.0168470
$388$	0	0
$389$	−22.0416	−1.11756	−0.558778	−	0.829318i	$-0.688729\pi$
$389$	−22.0416	−1.11756	−0.558778	+	0.829318i	$0.688729\pi$
$390$	0	0
$391$	−2.24093	−0.113329
$392$	0	0
$393$	2.16020	0.108967
$394$	0	0
$395$	19.2171	0.966917
$396$	0	0
$397$	13.7354	0.689360	0.344680	−	0.938720i	$-0.387987\pi$
$397$	13.7354	0.689360	0.344680	+	0.938720i	$0.387987\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.1233	−0.954971	−0.477486	−	0.878640i	$-0.658452\pi$
$401$	−19.1233	−0.954971	−0.477486	+	0.878640i	$0.658452\pi$
$402$	0	0
$403$	−10.8566	−0.540806
$404$	0	0
$405$	−10.5775	−0.525601
$406$	0	0
$407$	19.4603	0.964610
$408$	0	0
$409$	15.4547	0.764184	0.382092	−	0.924124i	$-0.375204\pi$
$409$	15.4547	0.764184	0.382092	+	0.924124i	$0.375204\pi$
$410$	0	0
$411$	−28.0968	−1.38591
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.5097	0.614077
$416$	0	0
$417$	−32.0748	−1.57071
$418$	0	0
$419$	12.9026	0.630335	0.315167	−	0.949036i	$-0.397939\pi$
$419$	12.9026	0.630335	0.315167	+	0.949036i	$0.397939\pi$
$420$	0	0
$421$	16.7332	0.815524	0.407762	−	0.913088i	$-0.366309\pi$
$421$	16.7332	0.815524	0.407762	+	0.913088i	$0.366309\pi$
$422$	0	0
$423$	0.0711180	0.00345787
$424$	0	0
$425$	−1.02009	−0.0494814
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.76560	−0.181805
$430$	0	0
$431$	0.339922	0.0163735	0.00818673	−	0.999966i	$-0.497394\pi$
$431$	0.339922	0.0163735	0.00818673	+	0.999966i	$0.497394\pi$
$432$	0	0
$433$	20.8557	1.00226	0.501130	−	0.865372i	$-0.332918\pi$
$433$	20.8557	1.00226	0.501130	+	0.865372i	$0.332918\pi$
$434$	0	0
$435$	−9.09904	−0.436266
$436$	0	0
$437$	−24.3175	−1.16326
$438$	0	0
$439$	−39.8047	−1.89977	−0.949887	−	0.312592i	$-0.898803\pi$
$439$	−39.8047	−1.89977	−0.949887	+	0.312592i	$0.898803\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.88642	0.137138	0.0685690	−	0.997646i	$-0.478157\pi$
$443$	2.88642	0.137138	0.0685690	+	0.997646i	$0.478157\pi$
$444$	0	0
$445$	10.2749	0.487079
$446$	0	0
$447$	−22.4955	−1.06400
$448$	0	0
$449$	28.4309	1.34174	0.670869	−	0.741576i	$-0.265921\pi$
$449$	28.4309	1.34174	0.670869	+	0.741576i	$0.265921\pi$
$450$	0	0
$451$	7.96685	0.375144
$452$	0	0
$453$	−33.0168	−1.55127
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.9619	1.58867	0.794334	−	0.607481i	$-0.207820\pi$
$457$	33.9619	1.58867	0.794334	+	0.607481i	$0.207820\pi$
$458$	0	0
$459$	−1.42655	−0.0665857
$460$	0	0
$461$	23.2828	1.08439	0.542193	−	0.840254i	$-0.317594\pi$
$461$	23.2828	1.08439	0.542193	+	0.840254i	$0.317594\pi$
$462$	0	0
$463$	−7.65725	−0.355863	−0.177931	−	0.984043i	$-0.556941\pi$
$463$	−7.65725	−0.355863	−0.177931	+	0.984043i	$0.556941\pi$
$464$	0	0
$465$	−21.8773	−1.01453
$466$	0	0
$467$	−31.8356	−1.47318	−0.736588	−	0.676342i	$-0.763564\pi$
$467$	−31.8356	−1.47318	−0.736588	+	0.676342i	$0.763564\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−32.2243	−1.48482
$472$	0	0
$473$	11.1049	0.510603
$474$	0	0
$475$	−11.0695	−0.507901
$476$	0	0
$477$	0.0701570	0.00321227
$478$	0	0
$479$	−20.6657	−0.944242	−0.472121	−	0.881534i	$-0.656511\pi$
$479$	−20.6657	−0.944242	−0.472121	+	0.881534i	$0.656511\pi$
$480$	0	0
$481$	−9.04636	−0.412478
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.63923	0.210657
$486$	0	0
$487$	14.0233	0.635456	0.317728	−	0.948182i	$-0.397080\pi$
$487$	14.0233	0.635456	0.317728	+	0.948182i	$0.397080\pi$
$488$	0	0
$489$	8.68001	0.392524
$490$	0	0
$491$	−19.0073	−0.857790	−0.428895	−	0.903354i	$-0.641097\pi$
$491$	−19.0073	−0.857790	−0.428895	+	0.903354i	$0.641097\pi$
$492$	0	0
$493$	−1.25343	−0.0564515
$494$	0	0
$495$	−0.158984	−0.00714582
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.07654	0.182491	0.0912456	−	0.995828i	$-0.470915\pi$
$499$	4.07654	0.182491	0.0912456	+	0.995828i	$0.470915\pi$
$500$	0	0
$501$	22.7175	1.01494
$502$	0	0
$503$	−22.4568	−1.00130	−0.500649	−	0.865651i	$-0.666905\pi$
$503$	−22.4568	−1.00130	−0.500649	+	0.865651i	$0.666905\pi$
$504$	0	0
$505$	9.65719	0.429739
$506$	0	0
$507$	1.75049	0.0777418
$508$	0	0
$509$	11.4832	0.508982	0.254491	−	0.967075i	$-0.418092\pi$
$509$	11.4832	0.508982	0.254491	+	0.967075i	$0.418092\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−15.4802	−0.683468
$514$	0	0
$515$	5.49909	0.242319
$516$	0	0
$517$	−2.38295	−0.104802
$518$	0	0
$519$	4.44172	0.194970
$520$	0	0
$521$	6.51060	0.285235	0.142617	−	0.989778i	$-0.454448\pi$
$521$	6.51060	0.285235	0.142617	+	0.989778i	$0.454448\pi$
$522$	0	0
$523$	18.1105	0.791917	0.395958	−	0.918268i	$-0.370412\pi$
$523$	18.1105	0.791917	0.395958	+	0.918268i	$0.370412\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.01367	−0.131278
$528$	0	0
$529$	42.1708	1.83351
$530$	0	0
$531$	−0.273870	−0.0118850
$532$	0	0
$533$	−3.70349	−0.160416
$534$	0	0
$535$	3.24924	0.140477
$536$	0	0
$537$	−1.23009	−0.0530821
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.3984	0.748017	0.374009	−	0.927425i	$-0.377983\pi$
$541$	17.3984	0.748017	0.374009	+	0.927425i	$0.377983\pi$
$542$	0	0
$543$	2.68086	0.115047
$544$	0	0
$545$	8.92692	0.382387
$546$	0	0
$547$	−35.7778	−1.52975	−0.764874	−	0.644180i	$-0.777199\pi$
$547$	−35.7778	−1.52975	−0.764874	+	0.644180i	$0.777199\pi$
$548$	0	0
$549$	0.503219	0.0214769
$550$	0	0
$551$	−13.6016	−0.579446
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−18.2294	−0.773795
$556$	0	0
$557$	−17.1063	−0.724816	−0.362408	−	0.932019i	$-0.618045\pi$
$557$	−17.1063	−0.724816	−0.362408	+	0.932019i	$0.618045\pi$
$558$	0	0
$559$	−5.16225	−0.218340
$560$	0	0
$561$	−1.04529	−0.0441321
$562$	0	0
$563$	28.9259	1.21908	0.609542	−	0.792754i	$-0.291354\pi$
$563$	28.9259	1.21908	0.609542	+	0.792754i	$0.291354\pi$
$564$	0	0
$565$	22.1270	0.930890
$566$	0	0
$567$	0	0
$568$	0	0
$569$	29.5554	1.23903	0.619514	−	0.784986i	$-0.287330\pi$
$569$	29.5554	1.23903	0.619514	+	0.784986i	$0.287330\pi$
$570$	0	0
$571$	−22.6210	−0.946661	−0.473330	−	0.880885i	$-0.656948\pi$
$571$	−22.6210	−0.946661	−0.473330	+	0.880885i	$0.656948\pi$
$572$	0	0
$573$	9.79672	0.409264
$574$	0	0
$575$	29.6661	1.23716
$576$	0	0
$577$	35.4983	1.47781	0.738907	−	0.673808i	$-0.235342\pi$
$577$	35.4983	1.47781	0.738907	+	0.673808i	$0.235342\pi$
$578$	0	0
$579$	33.4498	1.39013
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2.35075	−0.0973582
$584$	0	0
$585$	0.0739060	0.00305564
$586$	0	0
$587$	−4.30442	−0.177662	−0.0888312	−	0.996047i	$-0.528313\pi$
$587$	−4.30442	−0.177662	−0.0888312	+	0.996047i	$0.528313\pi$
$588$	0	0
$589$	−32.7029	−1.34750
$590$	0	0
$591$	10.5826	0.435309
$592$	0	0
$593$	−25.4430	−1.04482	−0.522409	−	0.852695i	$-0.674967\pi$
$593$	−25.4430	−1.04482	−0.522409	+	0.852695i	$0.674967\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−38.7865	−1.58742
$598$	0	0
$599$	−30.4434	−1.24388	−0.621942	−	0.783063i	$-0.713656\pi$
$599$	−30.4434	−1.24388	−0.621942	+	0.783063i	$0.713656\pi$
$600$	0	0
$601$	26.6195	1.08583	0.542916	−	0.839787i	$-0.317320\pi$
$601$	26.6195	1.08583	0.542916	+	0.839787i	$0.317320\pi$
$602$	0	0
$603$	−0.303750	−0.0123697
$604$	0	0
$605$	−7.33580	−0.298243
$606$	0	0
$607$	−3.25515	−0.132123	−0.0660613	−	0.997816i	$-0.521043\pi$
$607$	−3.25515	−0.132123	−0.0660613	+	0.997816i	$0.521043\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.10775	0.0448146
$612$	0	0
$613$	27.0816	1.09381	0.546907	−	0.837193i	$-0.315805\pi$
$613$	27.0816	1.09381	0.546907	+	0.837193i	$0.315805\pi$
$614$	0	0
$615$	−7.46295	−0.300935
$616$	0	0
$617$	33.6782	1.35583	0.677916	−	0.735140i	$-0.262883\pi$
$617$	33.6782	1.35583	0.677916	+	0.735140i	$0.262883\pi$
$618$	0	0
$619$	1.19738	0.0481268	0.0240634	−	0.999710i	$-0.492340\pi$
$619$	1.19738	0.0481268	0.0240634	+	0.999710i	$0.492340\pi$
$620$	0	0
$621$	41.4869	1.66481
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.87819	0.275128
$626$	0	0
$627$	−11.3429	−0.452994
$628$	0	0
$629$	−2.51117	−0.100127
$630$	0	0
$631$	5.33704	0.212464	0.106232	−	0.994341i	$-0.466121\pi$
$631$	5.33704	0.212464	0.106232	+	0.994341i	$0.466121\pi$
$632$	0	0
$633$	32.4856	1.29119
$634$	0	0
$635$	3.61157	0.143321
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.0706351	−0.00279428
$640$	0	0
$641$	9.33636	0.368764	0.184382	−	0.982855i	$-0.440972\pi$
$641$	9.33636	0.368764	0.184382	+	0.982855i	$0.440972\pi$
$642$	0	0
$643$	1.65709	0.0653491	0.0326746	−	0.999466i	$-0.489598\pi$
$643$	1.65709	0.0653491	0.0326746	+	0.999466i	$0.489598\pi$
$644$	0	0
$645$	−10.4025	−0.409598
$646$	0	0
$647$	−28.2282	−1.10977	−0.554883	−	0.831929i	$-0.687237\pi$
$647$	−28.2282	−1.10977	−0.554883	+	0.831929i	$0.687237\pi$
$648$	0	0
$649$	9.17658	0.360212
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.6210	−1.70702	−0.853511	−	0.521075i	$-0.825531\pi$
$653$	−43.6210	−1.70702	−0.853511	+	0.521075i	$0.825531\pi$
$654$	0	0
$655$	1.42061	0.0555078
$656$	0	0
$657$	0.207855	0.00810921
$658$	0	0
$659$	−12.5006	−0.486956	−0.243478	−	0.969906i	$-0.578288\pi$
$659$	−12.5006	−0.486956	−0.243478	+	0.969906i	$0.578288\pi$
$660$	0	0
$661$	−40.2077	−1.56390	−0.781949	−	0.623343i	$-0.785774\pi$
$661$	−40.2077	−1.56390	−0.781949	+	0.623343i	$0.785774\pi$
$662$	0	0
$663$	0.485916	0.0188714
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.4522	1.41143
$668$	0	0
$669$	−29.5598	−1.14285
$670$	0	0
$671$	−16.8614	−0.650927
$672$	0	0
$673$	−28.8405	−1.11172	−0.555859	−	0.831277i	$-0.687611\pi$
$673$	−28.8405	−1.11172	−0.555859	+	0.831277i	$0.687611\pi$
$674$	0	0
$675$	18.8851	0.726887
$676$	0	0
$677$	44.2757	1.70166	0.850828	−	0.525445i	$-0.176101\pi$
$677$	44.2757	1.70166	0.850828	+	0.525445i	$0.176101\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−34.0244	−1.30382
$682$	0	0
$683$	−20.2385	−0.774403	−0.387201	−	0.921995i	$-0.626558\pi$
$683$	−20.2385	−0.774403	−0.387201	+	0.921995i	$0.626558\pi$
$684$	0	0
$685$	−18.4773	−0.705982
$686$	0	0
$687$	−11.4736	−0.437744
$688$	0	0
$689$	1.09278	0.0416315
$690$	0	0
$691$	2.45852	0.0935266	0.0467633	−	0.998906i	$-0.485109\pi$
$691$	2.45852	0.0935266	0.0467633	+	0.998906i	$0.485109\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.0934	−0.800117
$696$	0	0
$697$	−1.02805	−0.0389401
$698$	0	0
$699$	12.6808	0.479632
$700$	0	0
$701$	−44.6070	−1.68478	−0.842392	−	0.538866i	$-0.818853\pi$
$701$	−44.6070	−1.68478	−0.842392	+	0.538866i	$0.818853\pi$
$702$	0	0
$703$	−27.2500	−1.02775
$704$	0	0
$705$	2.23223	0.0840706
$706$	0	0
$707$	0	0
$708$	0	0
$709$	34.4310	1.29308	0.646542	−	0.762878i	$-0.276215\pi$
$709$	34.4310	1.29308	0.646542	+	0.762878i	$0.276215\pi$
$710$	0	0
$711$	1.07173	0.0401932
$712$	0	0
$713$	87.6436	3.28228
$714$	0	0
$715$	−2.47637	−0.0926110
$716$	0	0
$717$	33.7085	1.25887
$718$	0	0
$719$	−12.4355	−0.463768	−0.231884	−	0.972743i	$-0.574489\pi$
$719$	−12.4355	−0.463768	−0.231884	+	0.972743i	$0.574489\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	29.1154	1.08281
$724$	0	0
$725$	16.5932	0.616257
$726$	0	0
$727$	−20.6343	−0.765286	−0.382643	−	0.923896i	$-0.624986\pi$
$727$	−20.6343	−0.765286	−0.382643	+	0.923896i	$0.624986\pi$
$728$	0	0
$729$	26.3977	0.977694
$730$	0	0
$731$	−1.43298	−0.0530008
$732$	0	0
$733$	33.3515	1.23187	0.615933	−	0.787798i	$-0.288779\pi$
$733$	33.3515	1.23187	0.615933	+	0.787798i	$0.288779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.1778	0.374903
$738$	0	0
$739$	16.9868	0.624871	0.312436	−	0.949939i	$-0.398855\pi$
$739$	16.9868	0.624871	0.312436	+	0.949939i	$0.398855\pi$
$740$	0	0
$741$	5.27291	0.193705
$742$	0	0
$743$	−22.6209	−0.829880	−0.414940	−	0.909849i	$-0.636197\pi$
$743$	−22.6209	−0.829880	−0.414940	+	0.909849i	$0.636197\pi$
$744$	0	0
$745$	−14.7937	−0.542000
$746$	0	0
$747$	0.697663	0.0255262
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.7578	0.392558	0.196279	−	0.980548i	$-0.437114\pi$
$751$	10.7578	0.392558	0.196279	+	0.980548i	$0.437114\pi$
$752$	0	0
$753$	−7.46388	−0.271999
$754$	0	0
$755$	−21.7129	−0.790212
$756$	0	0
$757$	4.32885	0.157335	0.0786673	−	0.996901i	$-0.474934\pi$
$757$	4.32885	0.157335	0.0786673	+	0.996901i	$0.474934\pi$
$758$	0	0
$759$	30.3991	1.10342
$760$	0	0
$761$	11.7791	0.426992	0.213496	−	0.976944i	$-0.431515\pi$
$761$	11.7791	0.426992	0.213496	+	0.976944i	$0.431515\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.0205155	0.000741739 0
$766$	0	0
$767$	−4.26585	−0.154031
$768$	0	0
$769$	−18.0232	−0.649935	−0.324967	−	0.945725i	$-0.605353\pi$
$769$	−18.0232	−0.649935	−0.324967	+	0.945725i	$0.605353\pi$
$770$	0	0
$771$	1.20914	0.0435461
$772$	0	0
$773$	8.19242	0.294661	0.147330	−	0.989087i	$-0.452932\pi$
$773$	8.19242	0.294661	0.147330	+	0.989087i	$0.452932\pi$
$774$	0	0
$775$	39.8959	1.43310
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.1559	−0.399701
$780$	0	0
$781$	2.36677	0.0846898
$782$	0	0
$783$	23.2050	0.829280
$784$	0	0
$785$	−21.1917	−0.756363
$786$	0	0
$787$	30.0481	1.07110	0.535550	−	0.844504i	$-0.320104\pi$
$787$	30.0481	1.07110	0.535550	+	0.844504i	$0.320104\pi$
$788$	0	0
$789$	−21.2838	−0.757723
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.83823	0.278344
$794$	0	0
$795$	2.20207	0.0780993
$796$	0	0
$797$	−35.3707	−1.25289	−0.626447	−	0.779464i	$-0.715492\pi$
$797$	−35.3707	−1.25289	−0.626447	+	0.779464i	$0.715492\pi$
$798$	0	0
$799$	0.307498	0.0108785
$800$	0	0
$801$	0.573032	0.0202471
$802$	0	0
$803$	−6.96461	−0.245776
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.5351	0.406054
$808$	0	0
$809$	29.8850	1.05070	0.525350	−	0.850886i	$-0.323934\pi$
$809$	29.8850	1.05070	0.525350	+	0.850886i	$0.323934\pi$
$810$	0	0
$811$	27.3665	0.960966	0.480483	−	0.877004i	$-0.340461\pi$
$811$	27.3665	0.960966	0.480483	+	0.877004i	$0.340461\pi$
$812$	0	0
$813$	−21.3009	−0.747056
$814$	0	0
$815$	5.70824	0.199951
$816$	0	0
$817$	−15.5500	−0.544026
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.7956	−1.17948	−0.589738	−	0.807595i	$-0.700769\pi$
$821$	−33.7956	−1.17948	−0.589738	+	0.807595i	$0.700769\pi$
$822$	0	0
$823$	−40.1006	−1.39782	−0.698910	−	0.715210i	$-0.746331\pi$
$823$	−40.1006	−1.39782	−0.698910	+	0.715210i	$0.746331\pi$
$824$	0	0
$825$	13.8378	0.481771
$826$	0	0
$827$	−30.8171	−1.07162	−0.535808	−	0.844340i	$-0.679993\pi$
$827$	−30.8171	−1.07162	−0.535808	+	0.844340i	$0.679993\pi$
$828$	0	0
$829$	−34.3633	−1.19349	−0.596744	−	0.802432i	$-0.703539\pi$
$829$	−34.3633	−1.19349	−0.596744	+	0.802432i	$0.703539\pi$
$830$	0	0
$831$	−39.4664	−1.36907
$832$	0	0
$833$	0	0
$834$	0	0
$835$	14.9397	0.517009
$836$	0	0
$837$	55.7929	1.92848
$838$	0	0
$839$	31.6835	1.09384	0.546918	−	0.837186i	$-0.315801\pi$
$839$	31.6835	1.09384	0.546918	+	0.837186i	$0.315801\pi$
$840$	0	0
$841$	−8.61110	−0.296935
$842$	0	0
$843$	17.9870	0.619507
$844$	0	0
$845$	1.15117	0.0396015
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−19.9558	−0.684880
$850$	0	0
$851$	73.0298	2.50343
$852$	0	0
$853$	−30.7057	−1.05134	−0.525672	−	0.850688i	$-0.676186\pi$
$853$	−30.7057	−1.05134	−0.525672	+	0.850688i	$0.676186\pi$
$854$	0	0
$855$	0.222624	0.00761357
$856$	0	0
$857$	7.39148	0.252488	0.126244	−	0.991999i	$-0.459708\pi$
$857$	7.39148	0.252488	0.126244	+	0.991999i	$0.459708\pi$
$858$	0	0
$859$	−52.6089	−1.79499	−0.897496	−	0.441023i	$-0.854616\pi$
$859$	−52.6089	−1.79499	−0.897496	+	0.441023i	$0.854616\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.5546	−1.27837	−0.639187	−	0.769051i	$-0.720729\pi$
$863$	−37.5546	−1.27837	−0.639187	+	0.769051i	$0.720729\pi$
$864$	0	0
$865$	2.92101	0.0993173
$866$	0	0
$867$	−29.6234	−1.00606
$868$	0	0
$869$	−35.9106	−1.21818
$870$	0	0
$871$	−4.73127	−0.160313
$872$	0	0
$873$	0.258729	0.00875665
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.0254	1.21649	0.608245	−	0.793749i	$-0.291874\pi$
$877$	36.0254	1.21649	0.608245	+	0.793749i	$0.291874\pi$
$878$	0	0
$879$	46.3048	1.56182
$880$	0	0
$881$	41.9078	1.41191	0.705956	−	0.708256i	$-0.250518\pi$
$881$	41.9078	1.41191	0.705956	+	0.708256i	$0.250518\pi$
$882$	0	0
$883$	−52.1294	−1.75429	−0.877147	−	0.480222i	$-0.840556\pi$
$883$	−52.1294	−1.75429	−0.877147	+	0.480222i	$0.840556\pi$
$884$	0	0
$885$	−8.59616	−0.288957
$886$	0	0
$887$	8.50230	0.285479	0.142740	−	0.989760i	$-0.454409\pi$
$887$	8.50230	0.285479	0.142740	+	0.989760i	$0.454409\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	19.7660	0.662186
$892$	0	0
$893$	3.33681	0.111662
$894$	0	0
$895$	−0.808942	−0.0270399
$896$	0	0
$897$	−14.1314	−0.471834
$898$	0	0
$899$	49.0220	1.63497
$900$	0	0
$901$	0.303343	0.0101058
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.76301	0.0586045
$906$	0	0
$907$	−14.9906	−0.497755	−0.248877	−	0.968535i	$-0.580062\pi$
$907$	−14.9906	−0.497755	−0.248877	+	0.968535i	$0.580062\pi$
$908$	0	0
$909$	0.538579	0.0178636
$910$	0	0
$911$	−52.4345	−1.73723	−0.868616	−	0.495486i	$-0.834990\pi$
$911$	−52.4345	−1.73723	−0.868616	+	0.495486i	$0.834990\pi$
$912$	0	0
$913$	−23.3766	−0.773653
$914$	0	0
$915$	15.7949	0.522163
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−12.5300	−0.413327	−0.206663	−	0.978412i	$-0.566260\pi$
$919$	−12.5300	−0.413327	−0.206663	+	0.978412i	$0.566260\pi$
$920$	0	0
$921$	−12.0984	−0.398655
$922$	0	0
$923$	−1.10022	−0.0362143
$924$	0	0
$925$	33.2436	1.09304
$926$	0	0
$927$	0.306683	0.0100728
$928$	0	0
$929$	56.6403	1.85831	0.929154	−	0.369694i	$-0.120537\pi$
$929$	56.6403	1.85831	0.929154	+	0.369694i	$0.120537\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−37.6816	−1.23364
$934$	0	0
$935$	−0.687413	−0.0224808
$936$	0	0
$937$	−34.2713	−1.11959	−0.559797	−	0.828630i	$-0.689121\pi$
$937$	−34.2713	−1.11959	−0.559797	+	0.828630i	$0.689121\pi$
$938$	0	0
$939$	−44.9714	−1.46759
$940$	0	0
$941$	−27.2877	−0.889554	−0.444777	−	0.895641i	$-0.646717\pi$
$941$	−27.2877	−0.889554	−0.444777	+	0.895641i	$0.646717\pi$
$942$	0	0
$943$	29.8977	0.973603
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.25569	−0.268274	−0.134137	−	0.990963i	$-0.542826\pi$
$947$	−8.25569	−0.268274	−0.134137	+	0.990963i	$0.542826\pi$
$948$	0	0
$949$	3.23759	0.105097
$950$	0	0
$951$	−23.2476	−0.753855
$952$	0	0
$953$	19.6589	0.636813	0.318406	−	0.947954i	$-0.396852\pi$
$953$	19.6589	0.636813	0.318406	+	0.947954i	$0.396852\pi$
$954$	0	0
$955$	6.44262	0.208478
$956$	0	0
$957$	17.0032	0.549635
$958$	0	0
$959$	0	0
$960$	0	0
$961$	86.8658	2.80212
$962$	0	0
$963$	0.181210	0.00583939
$964$	0	0
$965$	21.9976	0.708127
$966$	0	0
$967$	33.3260	1.07169	0.535846	−	0.844316i	$-0.319993\pi$
$967$	33.3260	1.07169	0.535846	+	0.844316i	$0.319993\pi$
$968$	0	0
$969$	1.46370	0.0470209
$970$	0	0
$971$	50.0974	1.60770	0.803851	−	0.594831i	$-0.202781\pi$
$971$	50.0974	1.60770	0.803851	+	0.594831i	$0.202781\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6.43269	−0.206011
$976$	0	0
$977$	31.7318	1.01519	0.507595	−	0.861596i	$-0.330535\pi$
$977$	31.7318	1.01519	0.507595	+	0.861596i	$0.330535\pi$
$978$	0	0
$979$	−19.2006	−0.613653
$980$	0	0
$981$	0.497852	0.0158952
$982$	0	0
$983$	−21.5721	−0.688042	−0.344021	−	0.938962i	$-0.611789\pi$
$983$	−21.5721	−0.688042	−0.344021	+	0.938962i	$0.611789\pi$
$984$	0	0
$985$	6.95942	0.221745
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41.6740	1.32516
$990$	0	0
$991$	−62.6841	−1.99123	−0.995613	−	0.0935694i	$-0.970172\pi$
$991$	−62.6841	−1.99123	−0.995613	+	0.0935694i	$0.970172\pi$
$992$	0	0
$993$	−1.76906	−0.0561393
$994$	0	0
$995$	−25.5071	−0.808631
$996$	0	0
$997$	10.9614	0.347150	0.173575	−	0.984821i	$-0.444468\pi$
$997$	10.9614	0.347150	0.173575	+	0.984821i	$0.444468\pi$
$998$	0	0
$999$	46.4899	1.47088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5096.2.a.bc.1.6	✓	8
7.6	odd	2		5096.2.a.bd.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5096.2.a.bc.1.6	✓	8	1.1	even	1	trivial
5096.2.a.bd.1.3	yes	8	7.6	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:	\( N \)	\(=\)	\( 5096 = 2^{3} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5096.a (trivial)

\(f(q)\)	\(=\)	\(q+1.75049 q^{3} +1.15117 q^{5} +0.0642006 q^{9} -2.15117 q^{11} +1.00000 q^{13} +2.01511 q^{15} +0.277589 q^{17} +3.01226 q^{19} -8.07284 q^{23} -3.67480 q^{25} -5.13908 q^{27} -4.51541 q^{29} -10.8566 q^{31} -3.76560 q^{33} -9.04636 q^{37} +1.75049 q^{39} -3.70349 q^{41} -5.16225 q^{43} +0.0739060 q^{45} +1.10775 q^{47} +0.485916 q^{51} +1.09278 q^{53} -2.47637 q^{55} +5.27291 q^{57} -4.26585 q^{59} +7.83823 q^{61} +1.15117 q^{65} -4.73127 q^{67} -14.1314 q^{69} -1.10022 q^{71} +3.23759 q^{73} -6.43269 q^{75} +16.6935 q^{79} -9.18848 q^{81} +10.8669 q^{83} +0.319553 q^{85} -7.90416 q^{87} +8.92564 q^{89} -19.0043 q^{93} +3.46763 q^{95} +4.03001 q^{97} -0.138107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 2 q^{5} + 6 q^{9} - 10 q^{11} + 8 q^{13} - 4 q^{15} - 14 q^{19} - 8 q^{23} - 2 q^{25} - 14 q^{27} - 2 q^{29} - 20 q^{31} + 6 q^{33} + 2 q^{37} - 2 q^{39} + 2 q^{41} - 2 q^{43} - 18 q^{45}+ \cdots + 12 q^{99}+O(q^{100}) \) 8 * q - 2 * q^3 + 2 * q^5 + 6 * q^9 - 10 * q^11 + 8 * q^13 - 4 * q^15 - 14 * q^19 - 8 * q^23 - 2 * q^25 - 14 * q^27 - 2 * q^29 - 20 * q^31 + 6 * q^33 + 2 * q^37 - 2 * q^39 + 2 * q^41 - 2 * q^43 - 18 * q^45 - 24 * q^47 - 4 * q^51 + 6 * q^53 - 40 * q^55 + 4 * q^57 - 12 * q^59 - 10 * q^61 + 2 * q^65 + 14 * q^67 + 6 * q^69 - 24 * q^73 - 14 * q^75 + 8 * q^79 - 20 * q^81 - 18 * q^83 + 8 * q^85 - 56 * q^87 + 18 * q^89 + 10 * q^93 + 10 * q^95 - 16 * q^97 + 12 * q^99

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