LMFDB - Embedded newform 5096.2.a.r.1.4

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 = [4,0,-1,0,-3,0,0,0,1,0,9] 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:	$4$
Coefficient field:	4.4.23252.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2 $ x^4 - x^3 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.88474$ 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)

$f(q)$	$=$	$q+1.88474 q^{3} -1.82358 q^{5} +0.552237 q^{9} -0.945893 q^{11} +1.00000 q^{13} -3.43697 q^{15} -2.37582 q^{17} +5.75869 q^{19} -3.70832 q^{23} -1.67455 q^{25} -4.61339 q^{27} +8.67720 q^{29} +1.33250 q^{31} -1.78276 q^{33} -9.81984 q^{37} +1.88474 q^{39} -4.61339 q^{41} -3.26056 q^{43} -1.00705 q^{45} -5.26761 q^{47} -4.47780 q^{51} +9.19566 q^{53} +1.72491 q^{55} +10.8536 q^{57} -3.45996 q^{59} -8.73570 q^{61} -1.82358 q^{65} -5.27839 q^{67} -6.98921 q^{69} +9.95809 q^{71} +6.24977 q^{73} -3.15608 q^{75} -13.1683 q^{79} -10.3517 q^{81} -4.40694 q^{83} +4.33250 q^{85} +16.3542 q^{87} -6.76243 q^{89} +2.51142 q^{93} -10.5014 q^{95} -12.6842 q^{97} -0.522357 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 3 q^{5} + q^{9} + 9 q^{11} + 4 q^{13} - 4 q^{15} - 4 q^{17} - 5 q^{19} - 2 q^{23} + 3 q^{25} - 13 q^{27} - 5 q^{29} - 2 q^{31} - 23 q^{33} - 7 q^{37} - q^{39} - 13 q^{41} + q^{43} + 13 q^{45}+ \cdots + 20 q^{99}+O(q^{100}) $ 4 * q - q^3 - 3 * q^5 + q^9 + 9 * q^11 + 4 * q^13 - 4 * q^15 - 4 * q^17 - 5 * q^19 - 2 * q^23 + 3 * q^25 - 13 * q^27 - 5 * q^29 - 2 * q^31 - 23 * q^33 - 7 * q^37 - q^39 - 13 * q^41 + q^43 + 13 * q^45 + 10 * q^47 + 12 * q^51 - q^53 - 18 * q^55 + 8 * q^57 + 10 * q^59 - 21 * q^61 - 3 * q^65 - q^67 - 17 * q^69 + 2 * q^71 - q^75 - 16 * q^79 + 8 * q^81 - 39 * q^83 + 10 * q^85 + 38 * q^87 - 27 * q^89 + 29 * q^93 + 17 * q^95 + 6 * q^97 + 20 * 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
$3$	1.88474	1.08815	0.544077	−	0.839035i	$-0.316880\pi$
$3$	1.88474	1.08815	0.544077	+	0.839035i	$0.316880\pi$
$4$	0	0
$5$	−1.82358	−0.815531	−0.407765	−	0.913087i	$-0.633692\pi$
$5$	−1.82358	−0.815531	−0.407765	+	0.913087i	$0.633692\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.552237	0.184079
$10$	0	0
$11$	−0.945893	−0.285198	−0.142599	−	0.989781i	$-0.545546\pi$
$11$	−0.945893	−0.285198	−0.142599	+	0.989781i	$0.545546\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.43697	−0.887423
$16$	0	0
$17$	−2.37582	−0.576221	−0.288110	−	0.957597i	$-0.593027\pi$
$17$	−2.37582	−0.576221	−0.288110	+	0.957597i	$0.593027\pi$
$18$	0	0
$19$	5.75869	1.32113	0.660567	−	0.750767i	$-0.270316\pi$
$19$	5.75869	1.32113	0.660567	+	0.750767i	$0.270316\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.70832	−0.773238	−0.386619	−	0.922239i	$-0.626357\pi$
$23$	−3.70832	−0.773238	−0.386619	+	0.922239i	$0.626357\pi$
$24$	0	0
$25$	−1.67455	−0.334910
$26$	0	0
$27$	−4.61339	−0.887848
$28$	0	0
$29$	8.67720	1.61132	0.805658	−	0.592382i	$-0.201812\pi$
$29$	8.67720	1.61132	0.805658	+	0.592382i	$0.201812\pi$
$30$	0	0
$31$	1.33250	0.239324	0.119662	−	0.992815i	$-0.461819\pi$
$31$	1.33250	0.239324	0.119662	+	0.992815i	$0.461819\pi$
$32$	0	0
$33$	−1.78276	−0.310339
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.81984	−1.61437	−0.807186	−	0.590297i	$-0.799011\pi$
$37$	−9.81984	−1.61437	−0.807186	+	0.590297i	$0.799011\pi$
$38$	0	0
$39$	1.88474	0.301800
$40$	0	0
$41$	−4.61339	−0.720491	−0.360245	−	0.932858i	$-0.617307\pi$
$41$	−4.61339	−0.720491	−0.360245	+	0.932858i	$0.617307\pi$
$42$	0	0
$43$	−3.26056	−0.497230	−0.248615	−	0.968602i	$-0.579975\pi$
$43$	−3.26056	−0.497230	−0.248615	+	0.968602i	$0.579975\pi$
$44$	0	0
$45$	−1.00705	−0.150122
$46$	0	0
$47$	−5.26761	−0.768359	−0.384180	−	0.923258i	$-0.625516\pi$
$47$	−5.26761	−0.768359	−0.384180	+	0.923258i	$0.625516\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.47780	−0.627017
$52$	0	0
$53$	9.19566	1.26312	0.631561	−	0.775327i	$-0.282415\pi$
$53$	9.19566	1.26312	0.631561	+	0.775327i	$0.282415\pi$
$54$	0	0
$55$	1.72491	0.232587
$56$	0	0
$57$	10.8536	1.43760
$58$	0	0
$59$	−3.45996	−0.450448	−0.225224	−	0.974307i	$-0.572311\pi$
$59$	−3.45996	−0.450448	−0.225224	+	0.974307i	$0.572311\pi$
$60$	0	0
$61$	−8.73570	−1.11849	−0.559246	−	0.829002i	$-0.688909\pi$
$61$	−8.73570	−1.11849	−0.559246	+	0.829002i	$0.688909\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.82358	−0.226188
$66$	0	0
$67$	−5.27839	−0.644859	−0.322429	−	0.946594i	$-0.604499\pi$
$67$	−5.27839	−0.644859	−0.322429	+	0.946594i	$0.604499\pi$
$68$	0	0
$69$	−6.98921	−0.841402
$70$	0	0
$71$	9.95809	1.18181	0.590904	−	0.806742i	$-0.298771\pi$
$71$	9.95809	1.18181	0.590904	+	0.806742i	$0.298771\pi$
$72$	0	0
$73$	6.24977	0.731480	0.365740	−	0.930717i	$-0.380816\pi$
$73$	6.24977	0.731480	0.365740	+	0.930717i	$0.380816\pi$
$74$	0	0
$75$	−3.15608	−0.364433
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.1683	−1.48155	−0.740774	−	0.671755i	$-0.765541\pi$
$79$	−13.1683	−1.48155	−0.740774	+	0.671755i	$0.765541\pi$
$80$	0	0
$81$	−10.3517	−1.15019
$82$	0	0
$83$	−4.40694	−0.483725	−0.241862	−	0.970311i	$-0.577758\pi$
$83$	−4.40694	−0.483725	−0.241862	+	0.970311i	$0.577758\pi$
$84$	0	0
$85$	4.33250	0.469926
$86$	0	0
$87$	16.3542	1.75336
$88$	0	0
$89$	−6.76243	−0.716816	−0.358408	−	0.933565i	$-0.616680\pi$
$89$	−6.76243	−0.716816	−0.358408	+	0.933565i	$0.616680\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.51142	0.260422
$94$	0	0
$95$	−10.5014	−1.07742
$96$	0	0
$97$	−12.6842	−1.28789	−0.643945	−	0.765072i	$-0.722703\pi$
$97$	−12.6842	−1.28789	−0.643945	+	0.765072i	$0.722703\pi$
$98$	0	0
$99$	−0.522357	−0.0524989
$100$	0	0
$101$	1.76243	0.175368	0.0876840	−	0.996148i	$-0.472053\pi$
$101$	1.76243	0.175368	0.0876840	+	0.996148i	$0.472053\pi$
$102$	0	0
$103$	−8.52111	−0.839610	−0.419805	−	0.907614i	$-0.637902\pi$
$103$	−8.52111	−0.839610	−0.419805	+	0.907614i	$0.637902\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.45996	0.527834	0.263917	−	0.964545i	$-0.414985\pi$
$107$	5.45996	0.527834	0.263917	+	0.964545i	$0.414985\pi$
$108$	0	0
$109$	7.32876	0.701968	0.350984	−	0.936381i	$-0.385847\pi$
$109$	7.32876	0.701968	0.350984	+	0.936381i	$0.385847\pi$
$110$	0	0
$111$	−18.5078	−1.75669
$112$	0	0
$113$	−0.270102	−0.0254091	−0.0127045	−	0.999919i	$-0.504044\pi$
$113$	−0.270102	−0.0254091	−0.0127045	+	0.999919i	$0.504044\pi$
$114$	0	0
$115$	6.76243	0.630600
$116$	0	0
$117$	0.552237	0.0510543
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.1053	−0.918662
$122$	0	0
$123$	−8.69503	−0.784005
$124$	0	0
$125$	12.1716	1.08866
$126$	0	0
$127$	−10.4262	−0.925174	−0.462587	−	0.886574i	$-0.653079\pi$
$127$	−10.4262	−0.925174	−0.462587	+	0.886574i	$0.653079\pi$
$128$	0	0
$129$	−6.14529	−0.541063
$130$	0	0
$131$	−8.99626	−0.786007	−0.393003	−	0.919537i	$-0.628564\pi$
$131$	−8.99626	−0.786007	−0.393003	+	0.919537i	$0.628564\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.41290	0.724067
$136$	0	0
$137$	−3.19235	−0.272741	−0.136371	−	0.990658i	$-0.543544\pi$
$137$	−3.19235	−0.272741	−0.136371	+	0.990658i	$0.543544\pi$
$138$	0	0
$139$	−7.85471	−0.666227	−0.333114	−	0.942887i	$-0.608099\pi$
$139$	−7.85471	−0.666227	−0.333114	+	0.942887i	$0.608099\pi$
$140$	0	0
$141$	−9.92806	−0.836093
$142$	0	0
$143$	−0.945893	−0.0790996
$144$	0	0
$145$	−15.8236	−1.31408
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.88739	0.646160	0.323080	−	0.946372i	$-0.395282\pi$
$149$	7.88739	0.646160	0.323080	+	0.946372i	$0.395282\pi$
$150$	0	0
$151$	−4.33250	−0.352574	−0.176287	−	0.984339i	$-0.556409\pi$
$151$	−4.33250	−0.352574	−0.176287	+	0.984339i	$0.556409\pi$
$152$	0	0
$153$	−1.31201	−0.106070
$154$	0	0
$155$	−2.42993	−0.195176
$156$	0	0
$157$	−14.0530	−1.12155	−0.560776	−	0.827967i	$-0.689497\pi$
$157$	−14.0530	−1.12155	−0.560776	+	0.827967i	$0.689497\pi$
$158$	0	0
$159$	17.3314	1.37447
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.99626	0.391337	0.195669	−	0.980670i	$-0.437312\pi$
$163$	4.99626	0.391337	0.195669	+	0.980670i	$0.437312\pi$
$164$	0	0
$165$	3.25101	0.253091
$166$	0	0
$167$	1.51406	0.117162	0.0585809	−	0.998283i	$-0.481342\pi$
$167$	1.51406	0.117162	0.0585809	+	0.998283i	$0.481342\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.18016	0.243193
$172$	0	0
$173$	2.23193	0.169690	0.0848452	−	0.996394i	$-0.472960\pi$
$173$	2.23193	0.169690	0.0848452	+	0.996394i	$0.472960\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.52111	−0.490157
$178$	0	0
$179$	−8.43257	−0.630280	−0.315140	−	0.949045i	$-0.602052\pi$
$179$	−8.43257	−0.630280	−0.315140	+	0.949045i	$0.602052\pi$
$180$	0	0
$181$	10.7371	0.798083	0.399042	−	0.916933i	$-0.369343\pi$
$181$	10.7371	0.798083	0.399042	+	0.916933i	$0.369343\pi$
$182$	0	0
$183$	−16.4645	−1.21709
$184$	0	0
$185$	17.9073	1.31657
$186$	0	0
$187$	2.24727	0.164337
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.2257	−0.739905	−0.369953	−	0.929051i	$-0.620626\pi$
$191$	−10.2257	−0.739905	−0.369953	+	0.929051i	$0.620626\pi$
$192$	0	0
$193$	3.89179	0.280137	0.140068	−	0.990142i	$-0.455268\pi$
$193$	3.89179	0.280137	0.140068	+	0.990142i	$0.455268\pi$
$194$	0	0
$195$	−3.43697	−0.246127
$196$	0	0
$197$	12.5918	0.897130	0.448565	−	0.893750i	$-0.351935\pi$
$197$	12.5918	0.897130	0.448565	+	0.893750i	$0.351935\pi$
$198$	0	0
$199$	−8.15939	−0.578404	−0.289202	−	0.957268i	$-0.593390\pi$
$199$	−8.15939	−0.578404	−0.289202	+	0.957268i	$0.593390\pi$
$200$	0	0
$201$	−9.94839	−0.701705
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.41290	0.587582
$206$	0	0
$207$	−2.04787	−0.142337
$208$	0	0
$209$	−5.44710	−0.376784
$210$	0	0
$211$	−20.7205	−1.42646	−0.713230	−	0.700931i	$-0.752768\pi$
$211$	−20.7205	−1.42646	−0.713230	+	0.700931i	$0.752768\pi$
$212$	0	0
$213$	18.7684	1.28599
$214$	0	0
$215$	5.94589	0.405507
$216$	0	0
$217$	0	0
$218$	0	0
$219$	11.7792	0.795963
$220$	0	0
$221$	−2.37582	−0.159815
$222$	0	0
$223$	−11.7861	−0.789254	−0.394627	−	0.918841i	$-0.629126\pi$
$223$	−11.7861	−0.789254	−0.394627	+	0.918841i	$0.629126\pi$
$224$	0	0
$225$	−0.924747	−0.0616498
$226$	0	0
$227$	16.9096	1.12233	0.561166	−	0.827704i	$-0.310353\pi$
$227$	16.9096	1.12233	0.561166	+	0.827704i	$0.310353\pi$
$228$	0	0
$229$	−26.1250	−1.72639	−0.863193	−	0.504874i	$-0.831539\pi$
$229$	−26.1250	−1.72639	−0.863193	+	0.504874i	$0.831539\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.964324	0.0631750	0.0315875	−	0.999501i	$-0.489944\pi$
$233$	0.964324	0.0631750	0.0315875	+	0.999501i	$0.489944\pi$
$234$	0	0
$235$	9.60591	0.626621
$236$	0	0
$237$	−24.8188	−1.61215
$238$	0	0
$239$	19.5033	1.26156	0.630781	−	0.775961i	$-0.282735\pi$
$239$	19.5033	1.26156	0.630781	+	0.775961i	$0.282735\pi$
$240$	0	0
$241$	−29.3869	−1.89298	−0.946489	−	0.322736i	$-0.895397\pi$
$241$	−29.3869	−1.89298	−0.946489	+	0.322736i	$0.895397\pi$
$242$	0	0
$243$	−5.67015	−0.363740
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.75869	0.366416
$248$	0	0
$249$	−8.30593	−0.526367
$250$	0	0
$251$	−26.8047	−1.69189	−0.845947	−	0.533266i	$-0.820964\pi$
$251$	−26.8047	−1.69189	−0.845947	+	0.533266i	$0.820964\pi$
$252$	0	0
$253$	3.50768	0.220526
$254$	0	0
$255$	8.16563	0.511352
$256$	0	0
$257$	−7.50187	−0.467954	−0.233977	−	0.972242i	$-0.575174\pi$
$257$	−7.50187	−0.467954	−0.233977	+	0.972242i	$0.575174\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.79187	0.296609
$262$	0	0
$263$	22.9002	1.41209	0.706045	−	0.708167i	$-0.250478\pi$
$263$	22.9002	1.41209	0.706045	+	0.708167i	$0.250478\pi$
$264$	0	0
$265$	−16.7690	−1.03011
$266$	0	0
$267$	−12.7454	−0.780006
$268$	0	0
$269$	−6.72161	−0.409824	−0.204912	−	0.978780i	$-0.565691\pi$
$269$	−6.72161	−0.409824	−0.204912	+	0.978780i	$0.565691\pi$
$270$	0	0
$271$	3.75354	0.228011	0.114006	−	0.993480i	$-0.463632\pi$
$271$	3.75354	0.228011	0.114006	+	0.993480i	$0.463632\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.58394	0.0955154
$276$	0	0
$277$	17.7091	1.06404	0.532019	−	0.846732i	$-0.321433\pi$
$277$	17.7091	1.06404	0.532019	+	0.846732i	$0.321433\pi$
$278$	0	0
$279$	0.735856	0.0440545
$280$	0	0
$281$	1.97967	0.118097	0.0590485	−	0.998255i	$-0.481193\pi$
$281$	1.97967	0.118097	0.0590485	+	0.998255i	$0.481193\pi$
$282$	0	0
$283$	26.3882	1.56861	0.784307	−	0.620373i	$-0.213019\pi$
$283$	26.3882	1.56861	0.784307	+	0.620373i	$0.213019\pi$
$284$	0	0
$285$	−19.7925	−1.17240
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.3555	−0.667970
$290$	0	0
$291$	−23.9065	−1.40142
$292$	0	0
$293$	−11.8669	−0.693272	−0.346636	−	0.938000i	$-0.612676\pi$
$293$	−11.8669	−0.693272	−0.346636	+	0.938000i	$0.612676\pi$
$294$	0	0
$295$	6.30952	0.367354
$296$	0	0
$297$	4.36378	0.253212
$298$	0	0
$299$	−3.70832	−0.214458
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.32171	0.190827
$304$	0	0
$305$	15.9303	0.912165
$306$	0	0
$307$	−14.8198	−0.845813	−0.422907	−	0.906173i	$-0.638990\pi$
$307$	−14.8198	−0.845813	−0.422907	+	0.906173i	$0.638990\pi$
$308$	0	0
$309$	−16.0601	−0.913625
$310$	0	0
$311$	5.86880	0.332789	0.166395	−	0.986059i	$-0.446787\pi$
$311$	5.86880	0.332789	0.166395	+	0.986059i	$0.446787\pi$
$312$	0	0
$313$	−13.7274	−0.775919	−0.387959	−	0.921677i	$-0.626820\pi$
$313$	−13.7274	−0.775919	−0.387959	+	0.921677i	$0.626820\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.5230	0.984191	0.492095	−	0.870541i	$-0.336231\pi$
$317$	17.5230	0.984191	0.492095	+	0.870541i	$0.336231\pi$
$318$	0	0
$319$	−8.20770	−0.459543
$320$	0	0
$321$	10.2906	0.574365
$322$	0	0
$323$	−13.6816	−0.761264
$324$	0	0
$325$	−1.67455	−0.0928872
$326$	0	0
$327$	13.8128	0.763849
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0.689736	0.0379113	0.0189557	−	0.999820i	$-0.493966\pi$
$331$	0.689736	0.0379113	0.0189557	+	0.999820i	$0.493966\pi$
$332$	0	0
$333$	−5.42288	−0.297172
$334$	0	0
$335$	9.62559	0.525902
$336$	0	0
$337$	15.5570	0.847441	0.423721	−	0.905793i	$-0.360724\pi$
$337$	15.5570	0.847441	0.423721	+	0.905793i	$0.360724\pi$
$338$	0	0
$339$	−0.509072	−0.0276490
$340$	0	0
$341$	−1.26040	−0.0682547
$342$	0	0
$343$	0	0
$344$	0	0
$345$	12.7454	0.686189
$346$	0	0
$347$	−14.2549	−0.765244	−0.382622	−	0.923905i	$-0.624979\pi$
$347$	−14.2549	−0.765244	−0.382622	+	0.923905i	$0.624979\pi$
$348$	0	0
$349$	0.431169	0.0230800	0.0115400	−	0.999933i	$-0.496327\pi$
$349$	0.431169	0.0230800	0.0115400	+	0.999933i	$0.496327\pi$
$350$	0	0
$351$	−4.61339	−0.246245
$352$	0	0
$353$	7.26695	0.386780	0.193390	−	0.981122i	$-0.438052\pi$
$353$	7.26695	0.386780	0.193390	+	0.981122i	$0.438052\pi$
$354$	0	0
$355$	−18.1594	−0.963800
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.4489	1.87092	0.935460	−	0.353431i	$-0.114985\pi$
$359$	35.4489	1.87092	0.935460	+	0.353431i	$0.114985\pi$
$360$	0	0
$361$	14.1625	0.745393
$362$	0	0
$363$	−19.0458	−0.999646
$364$	0	0
$365$	−11.3970	−0.596544
$366$	0	0
$367$	−6.96448	−0.363543	−0.181771	−	0.983341i	$-0.558183\pi$
$367$	−6.96448	−0.363543	−0.181771	+	0.983341i	$0.558183\pi$
$368$	0	0
$369$	−2.54768	−0.132627
$370$	0	0
$371$	0	0
$372$	0	0
$373$	24.2257	1.25436	0.627179	−	0.778875i	$-0.284209\pi$
$373$	24.2257	1.25436	0.627179	+	0.778875i	$0.284209\pi$
$374$	0	0
$375$	22.9403	1.18463
$376$	0	0
$377$	8.67720	0.446898
$378$	0	0
$379$	26.9900	1.38638	0.693192	−	0.720753i	$-0.256204\pi$
$379$	26.9900	1.38638	0.693192	+	0.720753i	$0.256204\pi$
$380$	0	0
$381$	−19.6506	−1.00673
$382$	0	0
$383$	35.1805	1.79764	0.898819	−	0.438320i	$-0.144426\pi$
$383$	35.1805	1.79764	0.898819	+	0.438320i	$0.144426\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.80060	−0.0915296
$388$	0	0
$389$	13.2510	0.671853	0.335926	−	0.941888i	$-0.390951\pi$
$389$	13.2510	0.671853	0.335926	+	0.941888i	$0.390951\pi$
$390$	0	0
$391$	8.81030	0.445556
$392$	0	0
$393$	−16.9556	−0.855296
$394$	0	0
$395$	24.0134	1.20825
$396$	0	0
$397$	−8.18787	−0.410937	−0.205469	−	0.978664i	$-0.565872\pi$
$397$	−8.18787	−0.410937	−0.205469	+	0.978664i	$0.565872\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.7072	−1.58338	−0.791692	−	0.610921i	$-0.790799\pi$
$401$	−31.7072	−1.58338	−0.791692	+	0.610921i	$0.790799\pi$
$402$	0	0
$403$	1.33250	0.0663766
$404$	0	0
$405$	18.8773	0.938018
$406$	0	0
$407$	9.28852	0.460415
$408$	0	0
$409$	10.5704	0.522672	0.261336	−	0.965248i	$-0.415837\pi$
$409$	10.5704	0.522672	0.261336	+	0.965248i	$0.415837\pi$
$410$	0	0
$411$	−6.01675	−0.296784
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.03642	0.394492
$416$	0	0
$417$	−14.8041	−0.724958
$418$	0	0
$419$	−38.6098	−1.88621	−0.943106	−	0.332492i	$-0.892111\pi$
$419$	−38.6098	−1.88621	−0.943106	+	0.332492i	$0.892111\pi$
$420$	0	0
$421$	−39.6738	−1.93358	−0.966791	−	0.255570i	$-0.917737\pi$
$421$	−39.6738	−1.93358	−0.966791	+	0.255570i	$0.917737\pi$
$422$	0	0
$423$	−2.90896	−0.141439
$424$	0	0
$425$	3.97842	0.192982
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.78276	−0.0860725
$430$	0	0
$431$	14.9340	0.719346	0.359673	−	0.933078i	$-0.382888\pi$
$431$	14.9340	0.719346	0.359673	+	0.933078i	$0.382888\pi$
$432$	0	0
$433$	33.0347	1.58755	0.793774	−	0.608212i	$-0.208113\pi$
$433$	33.0347	1.58755	0.793774	+	0.608212i	$0.208113\pi$
$434$	0	0
$435$	−29.8233	−1.42992
$436$	0	0
$437$	−21.3551	−1.02155
$438$	0	0
$439$	−19.9162	−0.950547	−0.475274	−	0.879838i	$-0.657651\pi$
$439$	−19.9162	−0.950547	−0.475274	+	0.879838i	$0.657651\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.9667	1.32874	0.664369	−	0.747405i	$-0.268700\pi$
$443$	27.9667	1.32874	0.664369	+	0.747405i	$0.268700\pi$
$444$	0	0
$445$	12.3318	0.584585
$446$	0	0
$447$	14.8657	0.703121
$448$	0	0
$449$	4.33374	0.204522	0.102261	−	0.994758i	$-0.467392\pi$
$449$	4.33374	0.204522	0.102261	+	0.994758i	$0.467392\pi$
$450$	0	0
$451$	4.36378	0.205482
$452$	0	0
$453$	−8.16563	−0.383655
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.77571	0.176620	0.0883102	−	0.996093i	$-0.471853\pi$
$457$	3.77571	0.176620	0.0883102	+	0.996093i	$0.471853\pi$
$458$	0	0
$459$	10.9606	0.511596
$460$	0	0
$461$	−31.2462	−1.45528	−0.727640	−	0.685959i	$-0.759383\pi$
$461$	−31.2462	−1.45528	−0.727640	+	0.685959i	$0.759383\pi$
$462$	0	0
$463$	19.4066	0.901901	0.450951	−	0.892549i	$-0.351085\pi$
$463$	19.4066	0.901901	0.450951	+	0.892549i	$0.351085\pi$
$464$	0	0
$465$	−4.57977	−0.212382
$466$	0	0
$467$	−16.1735	−0.748420	−0.374210	−	0.927344i	$-0.622086\pi$
$467$	−16.1735	−0.748420	−0.374210	+	0.927344i	$0.622086\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−26.4863	−1.22042
$472$	0	0
$473$	3.08414	0.141809
$474$	0	0
$475$	−9.64320	−0.442460
$476$	0	0
$477$	5.07818	0.232514
$478$	0	0
$479$	28.9334	1.32200	0.660999	−	0.750386i	$-0.270133\pi$
$479$	28.9334	1.32200	0.660999	+	0.750386i	$0.270133\pi$
$480$	0	0
$481$	−9.81984	−0.447746
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.1308	1.05031
$486$	0	0
$487$	13.5249	0.612870	0.306435	−	0.951892i	$-0.400864\pi$
$487$	13.5249	0.612870	0.306435	+	0.951892i	$0.400864\pi$
$488$	0	0
$489$	9.41664	0.425835
$490$	0	0
$491$	24.0768	1.08657	0.543286	−	0.839548i	$-0.317180\pi$
$491$	24.0768	1.08657	0.543286	+	0.839548i	$0.317180\pi$
$492$	0	0
$493$	−20.6154	−0.928473
$494$	0	0
$495$	0.952561	0.0428144
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.7613	−0.571276	−0.285638	−	0.958338i	$-0.592205\pi$
$499$	−12.7613	−0.571276	−0.285638	+	0.958338i	$0.592205\pi$
$500$	0	0
$501$	2.85361	0.127490
$502$	0	0
$503$	3.53365	0.157558	0.0787789	−	0.996892i	$-0.474898\pi$
$503$	3.53365	0.157558	0.0787789	+	0.996892i	$0.474898\pi$
$504$	0	0
$505$	−3.21393	−0.143018
$506$	0	0
$507$	1.88474	0.0837041
$508$	0	0
$509$	−12.3473	−0.547286	−0.273643	−	0.961831i	$-0.588229\pi$
$509$	−12.3473	−0.547286	−0.273643	+	0.961831i	$0.588229\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−26.5671	−1.17297
$514$	0	0
$515$	15.5390	0.684728
$516$	0	0
$517$	4.98259	0.219134
$518$	0	0
$519$	4.20660	0.184649
$520$	0	0
$521$	−38.1558	−1.67164	−0.835818	−	0.549006i	$-0.815006\pi$
$521$	−38.1558	−1.67164	−0.835818	+	0.549006i	$0.815006\pi$
$522$	0	0
$523$	45.1605	1.97473	0.987366	−	0.158457i	$-0.0506520\pi$
$523$	45.1605	1.97473	0.987366	+	0.158457i	$0.0506520\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.16578	−0.137904
$528$	0	0
$529$	−9.24836	−0.402103
$530$	0	0
$531$	−1.91072	−0.0829180
$532$	0	0
$533$	−4.61339	−0.199828
$534$	0	0
$535$	−9.95668	−0.430465
$536$	0	0
$537$	−15.8932	−0.685842
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19.8640	0.854019	0.427009	−	0.904247i	$-0.359567\pi$
$541$	19.8640	0.854019	0.427009	+	0.904247i	$0.359567\pi$
$542$	0	0
$543$	20.2366	0.868437
$544$	0	0
$545$	−13.3646	−0.572477
$546$	0	0
$547$	35.5247	1.51893	0.759464	−	0.650549i	$-0.225461\pi$
$547$	35.5247	1.51893	0.759464	+	0.650549i	$0.225461\pi$
$548$	0	0
$549$	−4.82418	−0.205891
$550$	0	0
$551$	49.9693	2.12876
$552$	0	0
$553$	0	0
$554$	0	0
$555$	33.7505	1.43263
$556$	0	0
$557$	3.29249	0.139507	0.0697537	−	0.997564i	$-0.477779\pi$
$557$	3.29249	0.139507	0.0697537	+	0.997564i	$0.477779\pi$
$558$	0	0
$559$	−3.26056	−0.137907
$560$	0	0
$561$	4.23552	0.178824
$562$	0	0
$563$	9.89494	0.417022	0.208511	−	0.978020i	$-0.433138\pi$
$563$	9.89494	0.417022	0.208511	+	0.978020i	$0.433138\pi$
$564$	0	0
$565$	0.492553	0.0207219
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.8080	−1.29154	−0.645768	−	0.763533i	$-0.723463\pi$
$569$	−30.8080	−1.29154	−0.645768	+	0.763533i	$0.723463\pi$
$570$	0	0
$571$	16.8976	0.707142	0.353571	−	0.935408i	$-0.384967\pi$
$571$	16.8976	0.707142	0.353571	+	0.935408i	$0.384967\pi$
$572$	0	0
$573$	−19.2728	−0.805131
$574$	0	0
$575$	6.20976	0.258965
$576$	0	0
$577$	27.2462	1.13427	0.567137	−	0.823624i	$-0.308051\pi$
$577$	27.2462	1.13427	0.567137	+	0.823624i	$0.308051\pi$
$578$	0	0
$579$	7.33500	0.304832
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−8.69811	−0.360239
$584$	0	0
$585$	−1.00705	−0.0416363
$586$	0	0
$587$	17.6033	0.726568	0.363284	−	0.931678i	$-0.381655\pi$
$587$	17.6033	0.726568	0.363284	+	0.931678i	$0.381655\pi$
$588$	0	0
$589$	7.67346	0.316179
$590$	0	0
$591$	23.7323	0.976215
$592$	0	0
$593$	−13.6072	−0.558779	−0.279389	−	0.960178i	$-0.590132\pi$
$593$	−13.6072	−0.558779	−0.279389	+	0.960178i	$0.590132\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.3783	−0.629393
$598$	0	0
$599$	−8.03693	−0.328380	−0.164190	−	0.986429i	$-0.552501\pi$
$599$	−8.03693	−0.328380	−0.164190	+	0.986429i	$0.552501\pi$
$600$	0	0
$601$	7.71603	0.314743	0.157372	−	0.987539i	$-0.449698\pi$
$601$	7.71603	0.314743	0.157372	+	0.987539i	$0.449698\pi$
$602$	0	0
$603$	−2.91492	−0.118705
$604$	0	0
$605$	18.4278	0.749197
$606$	0	0
$607$	2.17187	0.0881533	0.0440766	−	0.999028i	$-0.485965\pi$
$607$	2.17187	0.0881533	0.0440766	+	0.999028i	$0.485965\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.26761	−0.213105
$612$	0	0
$613$	12.4964	0.504724	0.252362	−	0.967633i	$-0.418793\pi$
$613$	12.4964	0.504724	0.252362	+	0.967633i	$0.418793\pi$
$614$	0	0
$615$	15.8561	0.639380
$616$	0	0
$617$	1.01784	0.0409766	0.0204883	−	0.999790i	$-0.493478\pi$
$617$	1.01784	0.0409766	0.0204883	+	0.999790i	$0.493478\pi$
$618$	0	0
$619$	−49.3203	−1.98235	−0.991176	−	0.132553i	$-0.957683\pi$
$619$	−49.3203	−1.98235	−0.991176	+	0.132553i	$0.957683\pi$
$620$	0	0
$621$	17.1079	0.686518
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.8232	−0.552926
$626$	0	0
$627$	−10.2664	−0.409999
$628$	0	0
$629$	23.3302	0.930235
$630$	0	0
$631$	30.6028	1.21828	0.609138	−	0.793064i	$-0.291515\pi$
$631$	30.6028	1.21828	0.609138	+	0.793064i	$0.291515\pi$
$632$	0	0
$633$	−39.0527	−1.55221
$634$	0	0
$635$	19.0130	0.754508
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.49922	0.217546
$640$	0	0
$641$	−29.5189	−1.16593	−0.582964	−	0.812498i	$-0.698107\pi$
$641$	−29.5189	−1.16593	−0.582964	+	0.812498i	$0.698107\pi$
$642$	0	0
$643$	23.0066	0.907290	0.453645	−	0.891183i	$-0.350123\pi$
$643$	23.0066	0.907290	0.453645	+	0.891183i	$0.350123\pi$
$644$	0	0
$645$	11.2065	0.441254
$646$	0	0
$647$	32.3352	1.27123	0.635613	−	0.772008i	$-0.280747\pi$
$647$	32.3352	1.27123	0.635613	+	0.772008i	$0.280747\pi$
$648$	0	0
$649$	3.27275	0.128467
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.6575	−0.964923	−0.482462	−	0.875917i	$-0.660257\pi$
$653$	−24.6575	−0.964923	−0.482462	+	0.875917i	$0.660257\pi$
$654$	0	0
$655$	16.4054	0.641013
$656$	0	0
$657$	3.45135	0.134650
$658$	0	0
$659$	40.1322	1.56333	0.781664	−	0.623700i	$-0.214371\pi$
$659$	40.1322	1.56333	0.781664	+	0.623700i	$0.214371\pi$
$660$	0	0
$661$	41.7733	1.62479	0.812396	−	0.583107i	$-0.198163\pi$
$661$	41.7733	1.62479	0.812396	+	0.583107i	$0.198163\pi$
$662$	0	0
$663$	−4.47780	−0.173903
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−32.1778	−1.24593
$668$	0	0
$669$	−22.2137	−0.858829
$670$	0	0
$671$	8.26304	0.318991
$672$	0	0
$673$	−22.8676	−0.881480	−0.440740	−	0.897635i	$-0.645284\pi$
$673$	−22.8676	−0.881480	−0.440740	+	0.897635i	$0.645284\pi$
$674$	0	0
$675$	7.72535	0.297349
$676$	0	0
$677$	−38.4443	−1.47754	−0.738768	−	0.673960i	$-0.764592\pi$
$677$	−38.4443	−1.47754	−0.738768	+	0.673960i	$0.764592\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	31.8702	1.22127
$682$	0	0
$683$	13.3791	0.511938	0.255969	−	0.966685i	$-0.417605\pi$
$683$	13.3791	0.511938	0.255969	+	0.966685i	$0.417605\pi$
$684$	0	0
$685$	5.82152	0.222429
$686$	0	0
$687$	−49.2387	−1.87857
$688$	0	0
$689$	9.19566	0.350327
$690$	0	0
$691$	−5.55473	−0.211312	−0.105656	−	0.994403i	$-0.533694\pi$
$691$	−5.55473	−0.211312	−0.105656	+	0.994403i	$0.533694\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.3237	0.543329
$696$	0	0
$697$	10.9606	0.415162
$698$	0	0
$699$	1.81750	0.0687441
$700$	0	0
$701$	−46.8737	−1.77039	−0.885197	−	0.465216i	$-0.845977\pi$
$701$	−46.8737	−1.77039	−0.885197	+	0.465216i	$0.845977\pi$
$702$	0	0
$703$	−56.5494	−2.13280
$704$	0	0
$705$	18.1046	0.681860
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14.1320	0.530739	0.265369	−	0.964147i	$-0.414506\pi$
$709$	14.1320	0.530739	0.265369	+	0.964147i	$0.414506\pi$
$710$	0	0
$711$	−7.27201	−0.272722
$712$	0	0
$713$	−4.94134	−0.185055
$714$	0	0
$715$	1.72491	0.0645081
$716$	0	0
$717$	36.7586	1.37277
$718$	0	0
$719$	−20.4337	−0.762048	−0.381024	−	0.924565i	$-0.624429\pi$
$719$	−20.4337	−0.762048	−0.381024	+	0.924565i	$0.624429\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−55.3866	−2.05985
$724$	0	0
$725$	−14.5304	−0.539645
$726$	0	0
$727$	28.3669	1.05207	0.526036	−	0.850463i	$-0.323678\pi$
$727$	28.3669	1.05207	0.526036	+	0.850463i	$0.323678\pi$
$728$	0	0
$729$	20.3685	0.754389
$730$	0	0
$731$	7.74649	0.286514
$732$	0	0
$733$	5.83768	0.215620	0.107810	−	0.994172i	$-0.465616\pi$
$733$	5.83768	0.215620	0.107810	+	0.994172i	$0.465616\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.99280	0.183912
$738$	0	0
$739$	−15.6794	−0.576777	−0.288388	−	0.957514i	$-0.593119\pi$
$739$	−15.6794	−0.576777	−0.288388	+	0.957514i	$0.593119\pi$
$740$	0	0
$741$	10.8536	0.398717
$742$	0	0
$743$	20.8905	0.766399	0.383200	−	0.923666i	$-0.374822\pi$
$743$	20.8905	0.766399	0.383200	+	0.923666i	$0.374822\pi$
$744$	0	0
$745$	−14.3833	−0.526963
$746$	0	0
$747$	−2.43367	−0.0890435
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29.8014	−1.08747	−0.543734	−	0.839257i	$-0.682990\pi$
$751$	−29.8014	−1.08747	−0.543734	+	0.839257i	$0.682990\pi$
$752$	0	0
$753$	−50.5197	−1.84104
$754$	0	0
$755$	7.90067	0.287535
$756$	0	0
$757$	−37.3422	−1.35723	−0.678613	−	0.734496i	$-0.737418\pi$
$757$	−37.3422	−1.35723	−0.678613	+	0.734496i	$0.737418\pi$
$758$	0	0
$759$	6.61105	0.239966
$760$	0	0
$761$	45.3569	1.64419	0.822093	−	0.569353i	$-0.192806\pi$
$761$	45.3569	1.64419	0.822093	+	0.569353i	$0.192806\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.39257	0.0865034
$766$	0	0
$767$	−3.45996	−0.124932
$768$	0	0
$769$	17.8893	0.645104	0.322552	−	0.946552i	$-0.395459\pi$
$769$	17.8893	0.645104	0.322552	+	0.946552i	$0.395459\pi$
$770$	0	0
$771$	−14.1391	−0.509206
$772$	0	0
$773$	−28.3291	−1.01893	−0.509463	−	0.860493i	$-0.670156\pi$
$773$	−28.3291	−1.01893	−0.509463	+	0.860493i	$0.670156\pi$
$774$	0	0
$775$	−2.23134	−0.0801520
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−26.5671	−0.951864
$780$	0	0
$781$	−9.41929	−0.337049
$782$	0	0
$783$	−40.0313	−1.43060
$784$	0	0
$785$	25.6268	0.914661
$786$	0	0
$787$	3.25432	0.116004	0.0580020	−	0.998316i	$-0.481527\pi$
$787$	3.25432	0.116004	0.0580020	+	0.998316i	$0.481527\pi$
$788$	0	0
$789$	43.1610	1.53657
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.73570	−0.310214
$794$	0	0
$795$	−31.6053	−1.12092
$796$	0	0
$797$	6.59430	0.233582	0.116791	−	0.993157i	$-0.462739\pi$
$797$	6.59430	0.233582	0.116791	+	0.993157i	$0.462739\pi$
$798$	0	0
$799$	12.5149	0.442745
$800$	0	0
$801$	−3.73446	−0.131951
$802$	0	0
$803$	−5.91161	−0.208616
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.6685	−0.445951
$808$	0	0
$809$	−3.68094	−0.129415	−0.0647074	−	0.997904i	$-0.520611\pi$
$809$	−3.68094	−0.129415	−0.0647074	+	0.997904i	$0.520611\pi$
$810$	0	0
$811$	−31.6997	−1.11313	−0.556564	−	0.830804i	$-0.687881\pi$
$811$	−31.6997	−1.11313	−0.556564	+	0.830804i	$0.687881\pi$
$812$	0	0
$813$	7.07444	0.248111
$814$	0	0
$815$	−9.11109	−0.319148
$816$	0	0
$817$	−18.7765	−0.656907
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.3469	1.30342	0.651708	−	0.758470i	$-0.274053\pi$
$821$	37.3469	1.30342	0.651708	+	0.758470i	$0.274053\pi$
$822$	0	0
$823$	−23.5982	−0.822582	−0.411291	−	0.911504i	$-0.634922\pi$
$823$	−23.5982	−0.822582	−0.411291	+	0.911504i	$0.634922\pi$
$824$	0	0
$825$	2.98532	0.103935
$826$	0	0
$827$	−20.0028	−0.695566	−0.347783	−	0.937575i	$-0.613065\pi$
$827$	−20.0028	−0.695566	−0.347783	+	0.937575i	$0.613065\pi$
$828$	0	0
$829$	−43.7174	−1.51837	−0.759185	−	0.650875i	$-0.774402\pi$
$829$	−43.7174	−1.51837	−0.759185	+	0.650875i	$0.774402\pi$
$830$	0	0
$831$	33.3771	1.15784
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.76102	−0.0955491
$836$	0	0
$837$	−6.14735	−0.212484
$838$	0	0
$839$	−0.957428	−0.0330541	−0.0165270	−	0.999863i	$-0.505261\pi$
$839$	−0.957428	−0.0330541	−0.0165270	+	0.999863i	$0.505261\pi$
$840$	0	0
$841$	46.2937	1.59634
$842$	0	0
$843$	3.73115	0.128508
$844$	0	0
$845$	−1.82358	−0.0627331
$846$	0	0
$847$	0	0
$848$	0	0
$849$	49.7348	1.70689
$850$	0	0
$851$	36.4151	1.24829
$852$	0	0
$853$	18.7955	0.643544	0.321772	−	0.946817i	$-0.395722\pi$
$853$	18.7955	0.643544	0.321772	+	0.946817i	$0.395722\pi$
$854$	0	0
$855$	−5.79928	−0.198331
$856$	0	0
$857$	−48.9000	−1.67039	−0.835195	−	0.549953i	$-0.814646\pi$
$857$	−48.9000	−1.67039	−0.835195	+	0.549953i	$0.814646\pi$
$858$	0	0
$859$	−1.47705	−0.0503963	−0.0251981	−	0.999682i	$-0.508022\pi$
$859$	−1.47705	−0.0503963	−0.0251981	+	0.999682i	$0.508022\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.3225	−0.930070	−0.465035	−	0.885292i	$-0.653958\pi$
$863$	−27.3225	−0.930070	−0.465035	+	0.885292i	$0.653958\pi$
$864$	0	0
$865$	−4.07011	−0.138388
$866$	0	0
$867$	−21.4021	−0.726854
$868$	0	0
$869$	12.4558	0.422534
$870$	0	0
$871$	−5.27839	−0.178852
$872$	0	0
$873$	−7.00471	−0.237073
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.6747	−0.630600	−0.315300	−	0.948992i	$-0.602105\pi$
$877$	−18.6747	−0.630600	−0.315300	+	0.948992i	$0.602105\pi$
$878$	0	0
$879$	−22.3660	−0.754386
$880$	0	0
$881$	15.3669	0.517723	0.258861	−	0.965914i	$-0.416653\pi$
$881$	15.3669	0.517723	0.258861	+	0.965914i	$0.416653\pi$
$882$	0	0
$883$	−54.4184	−1.83132	−0.915662	−	0.401950i	$-0.868333\pi$
$883$	−54.4184	−1.83132	−0.915662	+	0.401950i	$0.868333\pi$
$884$	0	0
$885$	11.8918	0.399738
$886$	0	0
$887$	45.5340	1.52888	0.764440	−	0.644694i	$-0.223015\pi$
$887$	45.5340	1.52888	0.764440	+	0.644694i	$0.223015\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.79165	0.328033
$892$	0	0
$893$	−30.3345	−1.01511
$894$	0	0
$895$	15.3775	0.514013
$896$	0	0
$897$	−6.98921	−0.233363
$898$	0	0
$899$	11.5624	0.385627
$900$	0	0
$901$	−21.8472	−0.727837
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.5800	−0.650861
$906$	0	0
$907$	9.04678	0.300393	0.150197	−	0.988656i	$-0.452009\pi$
$907$	9.04678	0.300393	0.150197	+	0.988656i	$0.452009\pi$
$908$	0	0
$909$	0.973277	0.0322815
$910$	0	0
$911$	−2.14728	−0.0711427	−0.0355713	−	0.999367i	$-0.511325\pi$
$911$	−2.14728	−0.0711427	−0.0355713	+	0.999367i	$0.511325\pi$
$912$	0	0
$913$	4.16850	0.137957
$914$	0	0
$915$	30.0244	0.992576
$916$	0	0
$917$	0	0
$918$	0	0
$919$	41.4007	1.36568	0.682842	−	0.730566i	$-0.260744\pi$
$919$	41.4007	1.36568	0.682842	+	0.730566i	$0.260744\pi$
$920$	0	0
$921$	−27.9315	−0.920375
$922$	0	0
$923$	9.95809	0.327774
$924$	0	0
$925$	16.4438	0.540669
$926$	0	0
$927$	−4.70567	−0.154555
$928$	0	0
$929$	43.7952	1.43687	0.718437	−	0.695593i	$-0.244858\pi$
$929$	43.7952	1.43687	0.718437	+	0.695593i	$0.244858\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	11.0612	0.362126
$934$	0	0
$935$	−4.09808	−0.134022
$936$	0	0
$937$	27.3543	0.893626	0.446813	−	0.894627i	$-0.352559\pi$
$937$	27.3543	0.893626	0.446813	+	0.894627i	$0.352559\pi$
$938$	0	0
$939$	−25.8726	−0.844319
$940$	0	0
$941$	12.9615	0.422534	0.211267	−	0.977428i	$-0.432241\pi$
$941$	12.9615	0.422534	0.211267	+	0.977428i	$0.432241\pi$
$942$	0	0
$943$	17.1079	0.557111
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.6116	−0.344829	−0.172415	−	0.985024i	$-0.555157\pi$
$947$	−10.6116	−0.344829	−0.172415	+	0.985024i	$0.555157\pi$
$948$	0	0
$949$	6.24977	0.202876
$950$	0	0
$951$	33.0263	1.07095
$952$	0	0
$953$	4.45167	0.144204	0.0721018	−	0.997397i	$-0.477029\pi$
$953$	4.45167	0.144204	0.0721018	+	0.997397i	$0.477029\pi$
$954$	0	0
$955$	18.6474	0.603415
$956$	0	0
$957$	−15.4694	−0.500054
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.2244	−0.942724
$962$	0	0
$963$	3.01519	0.0971631
$964$	0	0
$965$	−7.09699	−0.228460
$966$	0	0
$967$	48.3585	1.55511	0.777553	−	0.628818i	$-0.216461\pi$
$967$	48.3585	1.55511	0.777553	+	0.628818i	$0.216461\pi$
$968$	0	0
$969$	−25.7862	−0.828373
$970$	0	0
$971$	−51.7617	−1.66111	−0.830555	−	0.556936i	$-0.811977\pi$
$971$	−51.7617	−1.66111	−0.830555	+	0.556936i	$0.811977\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−3.15608	−0.101076
$976$	0	0
$977$	−41.9365	−1.34167	−0.670834	−	0.741608i	$-0.734064\pi$
$977$	−41.9365	−1.34167	−0.670834	+	0.741608i	$0.734064\pi$
$978$	0	0
$979$	6.39653	0.204434
$980$	0	0
$981$	4.04721	0.129217
$982$	0	0
$983$	−48.8892	−1.55932	−0.779661	−	0.626202i	$-0.784609\pi$
$983$	−48.8892	−1.55932	−0.779661	+	0.626202i	$0.784609\pi$
$984$	0	0
$985$	−22.9622	−0.731637
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.0912	0.384477
$990$	0	0
$991$	7.41715	0.235613	0.117807	−	0.993037i	$-0.462414\pi$
$991$	7.41715	0.235613	0.117807	+	0.993037i	$0.462414\pi$
$992$	0	0
$993$	1.29997	0.0412534
$994$	0	0
$995$	14.8793	0.471706
$996$	0	0
$997$	−23.1539	−0.733291	−0.366646	−	0.930361i	$-0.619494\pi$
$997$	−23.1539	−0.733291	−0.366646	+	0.930361i	$0.619494\pi$
$998$	0	0
$999$	45.3028	1.43332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5096.2.a.r.1.4	✓	4
7.6	odd	2		5096.2.a.w.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5096.2.a.r.1.4	✓	4	1.1	even	1	trivial
5096.2.a.w.1.1	yes	4	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)

\(f(q)\)	\(=\)	\(q+1.88474 q^{3} -1.82358 q^{5} +0.552237 q^{9} -0.945893 q^{11} +1.00000 q^{13} -3.43697 q^{15} -2.37582 q^{17} +5.75869 q^{19} -3.70832 q^{23} -1.67455 q^{25} -4.61339 q^{27} +8.67720 q^{29} +1.33250 q^{31} -1.78276 q^{33} -9.81984 q^{37} +1.88474 q^{39} -4.61339 q^{41} -3.26056 q^{43} -1.00705 q^{45} -5.26761 q^{47} -4.47780 q^{51} +9.19566 q^{53} +1.72491 q^{55} +10.8536 q^{57} -3.45996 q^{59} -8.73570 q^{61} -1.82358 q^{65} -5.27839 q^{67} -6.98921 q^{69} +9.95809 q^{71} +6.24977 q^{73} -3.15608 q^{75} -13.1683 q^{79} -10.3517 q^{81} -4.40694 q^{83} +4.33250 q^{85} +16.3542 q^{87} -6.76243 q^{89} +2.51142 q^{93} -10.5014 q^{95} -12.6842 q^{97} -0.522357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 3 q^{5} + q^{9} + 9 q^{11} + 4 q^{13} - 4 q^{15} - 4 q^{17} - 5 q^{19} - 2 q^{23} + 3 q^{25} - 13 q^{27} - 5 q^{29} - 2 q^{31} - 23 q^{33} - 7 q^{37} - q^{39} - 13 q^{41} + q^{43} + 13 q^{45}+ \cdots + 20 q^{99}+O(q^{100}) \) 4 * q - q^3 - 3 * q^5 + q^9 + 9 * q^11 + 4 * q^13 - 4 * q^15 - 4 * q^17 - 5 * q^19 - 2 * q^23 + 3 * q^25 - 13 * q^27 - 5 * q^29 - 2 * q^31 - 23 * q^33 - 7 * q^37 - q^39 - 13 * q^41 + q^43 + 13 * q^45 + 10 * q^47 + 12 * q^51 - q^53 - 18 * q^55 + 8 * q^57 + 10 * q^59 - 21 * q^61 - 3 * q^65 - q^67 - 17 * q^69 + 2 * q^71 - q^75 - 16 * q^79 + 8 * q^81 - 39 * q^83 + 10 * q^85 + 38 * q^87 - 27 * q^89 + 29 * q^93 + 17 * q^95 + 6 * q^97 + 20 * q^99

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