LMFDB - Embedded newform 5120.2.a.u.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5120,2,Mod(1,5120)] 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("5120.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,4,0,-8,0,-4,0,8,0,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.8834058349$
Analytic rank:	$0$
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} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 $ x^8 - 12x^6 - 8x^5 + 21x^4 + 12x^3 - 10x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 80)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.187687$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58464 q^{3} -1.00000 q^{5} -4.50961 q^{7} +3.68037 q^{9} -2.32045 q^{11} +2.14759 q^{13} -2.58464 q^{15} -1.45616 q^{17} +3.78959 q^{19} -11.6557 q^{21} +2.37423 q^{23} +1.00000 q^{25} +1.75851 q^{27} -1.30810 q^{29} +7.20435 q^{31} -5.99752 q^{33} +4.50961 q^{35} -7.36979 q^{37} +5.55074 q^{39} +6.41166 q^{41} +10.8301 q^{43} -3.68037 q^{45} -2.51027 q^{47} +13.3366 q^{49} -3.76365 q^{51} -2.12574 q^{53} +2.32045 q^{55} +9.79472 q^{57} +7.52088 q^{59} +1.44489 q^{61} -16.5970 q^{63} -2.14759 q^{65} +7.39273 q^{67} +6.13653 q^{69} +1.92097 q^{71} +1.39412 q^{73} +2.58464 q^{75} +10.4643 q^{77} +5.06317 q^{79} -6.49599 q^{81} +3.46445 q^{83} +1.45616 q^{85} -3.38097 q^{87} +9.36007 q^{89} -9.68477 q^{91} +18.6207 q^{93} -3.78959 q^{95} +18.6313 q^{97} -8.54009 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 8 q^{5} - 4 q^{7} + 8 q^{9} + 8 q^{11} - 4 q^{15} + 16 q^{19} - 12 q^{23} + 8 q^{25} + 16 q^{27} + 4 q^{35} + 28 q^{43} - 8 q^{45} - 20 q^{47} + 8 q^{49} + 24 q^{51} - 8 q^{55} + 16 q^{59}+ \cdots + 40 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 8 * q^5 - 4 * q^7 + 8 * q^9 + 8 * q^11 - 4 * q^15 + 16 * q^19 - 12 * q^23 + 8 * q^25 + 16 * q^27 + 4 * q^35 + 28 * q^43 - 8 * q^45 - 20 * q^47 + 8 * q^49 + 24 * q^51 - 8 * q^55 + 16 * q^59 - 20 * q^63 + 36 * q^67 + 16 * q^69 + 8 * q^71 + 4 * q^75 + 8 * q^79 + 8 * q^81 + 20 * q^83 + 40 * q^91 - 16 * q^95 + 40 * 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
$3$	2.58464	1.49224	0.746122	−	0.665810i	$-0.231914\pi$
$3$	2.58464	1.49224	0.746122	+	0.665810i	$0.231914\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.50961	−1.70447	−0.852236	−	0.523158i	$-0.824754\pi$
$7$	−4.50961	−1.70447	−0.852236	+	0.523158i	$0.824754\pi$
$8$	0	0
$9$	3.68037	1.22679
$10$	0	0
$11$	−2.32045	−0.699641	−0.349820	−	0.936817i	$-0.613757\pi$
$11$	−2.32045	−0.699641	−0.349820	+	0.936817i	$0.613757\pi$
$12$	0	0
$13$	2.14759	0.595633	0.297817	−	0.954623i	$-0.403742\pi$
$13$	2.14759	0.595633	0.297817	+	0.954623i	$0.403742\pi$
$14$	0	0
$15$	−2.58464	−0.667351
$16$	0	0
$17$	−1.45616	−0.353170	−0.176585	−	0.984285i	$-0.556505\pi$
$17$	−1.45616	−0.353170	−0.176585	+	0.984285i	$0.556505\pi$
$18$	0	0
$19$	3.78959	0.869391	0.434695	−	0.900578i	$-0.356856\pi$
$19$	3.78959	0.869391	0.434695	+	0.900578i	$0.356856\pi$
$20$	0	0
$21$	−11.6557	−2.54349
$22$	0	0
$23$	2.37423	0.495061	0.247530	−	0.968880i	$-0.420381\pi$
$23$	2.37423	0.495061	0.247530	+	0.968880i	$0.420381\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.75851	0.338425
$28$	0	0
$29$	−1.30810	−0.242908	−0.121454	−	0.992597i	$-0.538756\pi$
$29$	−1.30810	−0.242908	−0.121454	+	0.992597i	$0.538756\pi$
$30$	0	0
$31$	7.20435	1.29394	0.646970	−	0.762515i	$-0.276036\pi$
$31$	7.20435	1.29394	0.646970	+	0.762515i	$0.276036\pi$
$32$	0	0
$33$	−5.99752	−1.04403
$34$	0	0
$35$	4.50961	0.762263
$36$	0	0
$37$	−7.36979	−1.21159	−0.605793	−	0.795622i	$-0.707144\pi$
$37$	−7.36979	−1.21159	−0.605793	+	0.795622i	$0.707144\pi$
$38$	0	0
$39$	5.55074	0.888830
$40$	0	0
$41$	6.41166	1.00133	0.500667	−	0.865640i	$-0.333088\pi$
$41$	6.41166	1.00133	0.500667	+	0.865640i	$0.333088\pi$
$42$	0	0
$43$	10.8301	1.65157	0.825784	−	0.563987i	$-0.190733\pi$
$43$	10.8301	1.65157	0.825784	+	0.563987i	$0.190733\pi$
$44$	0	0
$45$	−3.68037	−0.548637
$46$	0	0
$47$	−2.51027	−0.366161	−0.183081	−	0.983098i	$-0.558607\pi$
$47$	−2.51027	−0.366161	−0.183081	+	0.983098i	$0.558607\pi$
$48$	0	0
$49$	13.3366	1.90522
$50$	0	0
$51$	−3.76365	−0.527016
$52$	0	0
$53$	−2.12574	−0.291992	−0.145996	−	0.989285i	$-0.546639\pi$
$53$	−2.12574	−0.291992	−0.145996	+	0.989285i	$0.546639\pi$
$54$	0	0
$55$	2.32045	0.312889
$56$	0	0
$57$	9.79472	1.29734
$58$	0	0
$59$	7.52088	0.979135	0.489568	−	0.871965i	$-0.337155\pi$
$59$	7.52088	0.979135	0.489568	+	0.871965i	$0.337155\pi$
$60$	0	0
$61$	1.44489	0.185000	0.0924999	−	0.995713i	$-0.470514\pi$
$61$	1.44489	0.185000	0.0924999	+	0.995713i	$0.470514\pi$
$62$	0	0
$63$	−16.5970	−2.09103
$64$	0	0
$65$	−2.14759	−0.266375
$66$	0	0
$67$	7.39273	0.903166	0.451583	−	0.892229i	$-0.350859\pi$
$67$	7.39273	0.903166	0.451583	+	0.892229i	$0.350859\pi$
$68$	0	0
$69$	6.13653	0.738751
$70$	0	0
$71$	1.92097	0.227978	0.113989	−	0.993482i	$-0.463637\pi$
$71$	1.92097	0.227978	0.113989	+	0.993482i	$0.463637\pi$
$72$	0	0
$73$	1.39412	0.163169	0.0815847	−	0.996666i	$-0.474002\pi$
$73$	1.39412	0.163169	0.0815847	+	0.996666i	$0.474002\pi$
$74$	0	0
$75$	2.58464	0.298449
$76$	0	0
$77$	10.4643	1.19252
$78$	0	0
$79$	5.06317	0.569651	0.284825	−	0.958579i	$-0.408064\pi$
$79$	5.06317	0.569651	0.284825	+	0.958579i	$0.408064\pi$
$80$	0	0
$81$	−6.49599	−0.721777
$82$	0	0
$83$	3.46445	0.380273	0.190137	−	0.981758i	$-0.439107\pi$
$83$	3.46445	0.380273	0.190137	+	0.981758i	$0.439107\pi$
$84$	0	0
$85$	1.45616	0.157943
$86$	0	0
$87$	−3.38097	−0.362478
$88$	0	0
$89$	9.36007	0.992165	0.496083	−	0.868275i	$-0.334771\pi$
$89$	9.36007	0.992165	0.496083	+	0.868275i	$0.334771\pi$
$90$	0	0
$91$	−9.68477	−1.01524
$92$	0	0
$93$	18.6207	1.93087
$94$	0	0
$95$	−3.78959	−0.388803
$96$	0	0
$97$	18.6313	1.89172	0.945859	−	0.324579i	$-0.105223\pi$
$97$	18.6313	1.89172	0.945859	+	0.324579i	$0.105223\pi$
$98$	0	0
$99$	−8.54009	−0.858312
$100$	0	0
$101$	−6.84632	−0.681234	−0.340617	−	0.940202i	$-0.610636\pi$
$101$	−6.84632	−0.681234	−0.340617	+	0.940202i	$0.610636\pi$
$102$	0	0
$103$	9.12540	0.899153	0.449576	−	0.893242i	$-0.351575\pi$
$103$	9.12540	0.899153	0.449576	+	0.893242i	$0.351575\pi$
$104$	0	0
$105$	11.6557	1.13748
$106$	0	0
$107$	−14.3550	−1.38775	−0.693877	−	0.720094i	$-0.744099\pi$
$107$	−14.3550	−1.38775	−0.693877	+	0.720094i	$0.744099\pi$
$108$	0	0
$109$	1.91611	0.183530	0.0917650	−	0.995781i	$-0.470749\pi$
$109$	1.91611	0.183530	0.0917650	+	0.995781i	$0.470749\pi$
$110$	0	0
$111$	−19.0483	−1.80798
$112$	0	0
$113$	2.56039	0.240861	0.120431	−	0.992722i	$-0.461572\pi$
$113$	2.56039	0.240861	0.120431	+	0.992722i	$0.461572\pi$
$114$	0	0
$115$	−2.37423	−0.221398
$116$	0	0
$117$	7.90391	0.730717
$118$	0	0
$119$	6.56670	0.601969
$120$	0	0
$121$	−5.61553	−0.510503
$122$	0	0
$123$	16.5718	1.49423
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.7354	−1.21882	−0.609409	−	0.792856i	$-0.708593\pi$
$127$	−13.7354	−1.21882	−0.609409	+	0.792856i	$0.708593\pi$
$128$	0	0
$129$	27.9918	2.46454
$130$	0	0
$131$	7.36417	0.643411	0.321705	−	0.946840i	$-0.395744\pi$
$131$	7.36417	0.643411	0.321705	+	0.946840i	$0.395744\pi$
$132$	0	0
$133$	−17.0895	−1.48185
$134$	0	0
$135$	−1.75851	−0.151348
$136$	0	0
$137$	−22.7563	−1.94420	−0.972102	−	0.234559i	$-0.924635\pi$
$137$	−22.7563	−1.94420	−0.972102	+	0.234559i	$0.924635\pi$
$138$	0	0
$139$	8.88246	0.753400	0.376700	−	0.926335i	$-0.377059\pi$
$139$	8.88246	0.753400	0.376700	+	0.926335i	$0.377059\pi$
$140$	0	0
$141$	−6.48816	−0.546402
$142$	0	0
$143$	−4.98336	−0.416729
$144$	0	0
$145$	1.30810	0.108632
$146$	0	0
$147$	34.4702	2.84305
$148$	0	0
$149$	18.3245	1.50121	0.750603	−	0.660754i	$-0.229763\pi$
$149$	18.3245	1.50121	0.750603	+	0.660754i	$0.229763\pi$
$150$	0	0
$151$	14.3417	1.16711	0.583555	−	0.812073i	$-0.301661\pi$
$151$	14.3417	1.16711	0.583555	+	0.812073i	$0.301661\pi$
$152$	0	0
$153$	−5.35920	−0.433266
$154$	0	0
$155$	−7.20435	−0.578668
$156$	0	0
$157$	2.97783	0.237656	0.118828	−	0.992915i	$-0.462086\pi$
$157$	2.97783	0.237656	0.118828	+	0.992915i	$0.462086\pi$
$158$	0	0
$159$	−5.49426	−0.435723
$160$	0	0
$161$	−10.7068	−0.843817
$162$	0	0
$163$	−7.55196	−0.591515	−0.295758	−	0.955263i	$-0.595572\pi$
$163$	−7.55196	−0.591515	−0.295758	+	0.955263i	$0.595572\pi$
$164$	0	0
$165$	5.99752	0.466906
$166$	0	0
$167$	16.0686	1.24343	0.621714	−	0.783245i	$-0.286437\pi$
$167$	16.0686	1.24343	0.621714	+	0.783245i	$0.286437\pi$
$168$	0	0
$169$	−8.38787	−0.645221
$170$	0	0
$171$	13.9471	1.06656
$172$	0	0
$173$	24.2019	1.84004	0.920018	−	0.391876i	$-0.128174\pi$
$173$	24.2019	1.84004	0.920018	+	0.391876i	$0.128174\pi$
$174$	0	0
$175$	−4.50961	−0.340894
$176$	0	0
$177$	19.4388	1.46111
$178$	0	0
$179$	−1.47760	−0.110441	−0.0552203	−	0.998474i	$-0.517586\pi$
$179$	−1.47760	−0.110441	−0.0552203	+	0.998474i	$0.517586\pi$
$180$	0	0
$181$	−16.9544	−1.26021	−0.630106	−	0.776509i	$-0.716988\pi$
$181$	−16.9544	−1.26021	−0.630106	+	0.776509i	$0.716988\pi$
$182$	0	0
$183$	3.73453	0.276065
$184$	0	0
$185$	7.36979	0.541838
$186$	0	0
$187$	3.37894	0.247092
$188$	0	0
$189$	−7.93018	−0.576835
$190$	0	0
$191$	−0.0667471	−0.00482965	−0.00241483	−	0.999997i	$-0.500769\pi$
$191$	−0.0667471	−0.00482965	−0.00241483	+	0.999997i	$0.500769\pi$
$192$	0	0
$193$	−1.09895	−0.0791039	−0.0395520	−	0.999218i	$-0.512593\pi$
$193$	−1.09895	−0.0791039	−0.0395520	+	0.999218i	$0.512593\pi$
$194$	0	0
$195$	−5.55074	−0.397497
$196$	0	0
$197$	16.8700	1.20194	0.600969	−	0.799272i	$-0.294782\pi$
$197$	16.8700	1.20194	0.600969	+	0.799272i	$0.294782\pi$
$198$	0	0
$199$	11.0397	0.782584	0.391292	−	0.920267i	$-0.372028\pi$
$199$	11.0397	0.782584	0.391292	+	0.920267i	$0.372028\pi$
$200$	0	0
$201$	19.1076	1.34774
$202$	0	0
$203$	5.89901	0.414030
$204$	0	0
$205$	−6.41166	−0.447810
$206$	0	0
$207$	8.73803	0.607335
$208$	0	0
$209$	−8.79353	−0.608261
$210$	0	0
$211$	12.1585	0.837027	0.418514	−	0.908211i	$-0.362551\pi$
$211$	12.1585	0.837027	0.418514	+	0.908211i	$0.362551\pi$
$212$	0	0
$213$	4.96503	0.340198
$214$	0	0
$215$	−10.8301	−0.738603
$216$	0	0
$217$	−32.4888	−2.20548
$218$	0	0
$219$	3.60330	0.243488
$220$	0	0
$221$	−3.12723	−0.210360
$222$	0	0
$223$	21.4238	1.43465	0.717323	−	0.696741i	$-0.245367\pi$
$223$	21.4238	1.43465	0.717323	+	0.696741i	$0.245367\pi$
$224$	0	0
$225$	3.68037	0.245358
$226$	0	0
$227$	−11.4032	−0.756860	−0.378430	−	0.925630i	$-0.623536\pi$
$227$	−11.4032	−0.756860	−0.378430	+	0.925630i	$0.623536\pi$
$228$	0	0
$229$	6.55020	0.432849	0.216425	−	0.976299i	$-0.430560\pi$
$229$	6.55020	0.432849	0.216425	+	0.976299i	$0.430560\pi$
$230$	0	0
$231$	27.0465	1.77953
$232$	0	0
$233$	−26.0672	−1.70772	−0.853860	−	0.520502i	$-0.825745\pi$
$233$	−26.0672	−1.70772	−0.853860	+	0.520502i	$0.825745\pi$
$234$	0	0
$235$	2.51027	0.163752
$236$	0	0
$237$	13.0865	0.850058
$238$	0	0
$239$	5.12209	0.331320	0.165660	−	0.986183i	$-0.447025\pi$
$239$	5.12209	0.331320	0.165660	+	0.986183i	$0.447025\pi$
$240$	0	0
$241$	−11.4987	−0.740695	−0.370347	−	0.928893i	$-0.620761\pi$
$241$	−11.4987	−0.740695	−0.370347	+	0.928893i	$0.620761\pi$
$242$	0	0
$243$	−22.0653	−1.41549
$244$	0	0
$245$	−13.3366	−0.852041
$246$	0	0
$247$	8.13847	0.517838
$248$	0	0
$249$	8.95437	0.567460
$250$	0	0
$251$	28.0396	1.76985	0.884923	−	0.465738i	$-0.154211\pi$
$251$	28.0396	1.76985	0.884923	+	0.465738i	$0.154211\pi$
$252$	0	0
$253$	−5.50927	−0.346365
$254$	0	0
$255$	3.76365	0.235689
$256$	0	0
$257$	−24.2494	−1.51264	−0.756319	−	0.654203i	$-0.773004\pi$
$257$	−24.2494	−1.51264	−0.756319	+	0.654203i	$0.773004\pi$
$258$	0	0
$259$	33.2348	2.06511
$260$	0	0
$261$	−4.81429	−0.297997
$262$	0	0
$263$	−22.5680	−1.39160	−0.695802	−	0.718234i	$-0.744951\pi$
$263$	−22.5680	−1.39160	−0.695802	+	0.718234i	$0.744951\pi$
$264$	0	0
$265$	2.12574	0.130583
$266$	0	0
$267$	24.1924	1.48055
$268$	0	0
$269$	−7.22039	−0.440235	−0.220117	−	0.975473i	$-0.570644\pi$
$269$	−7.22039	−0.440235	−0.220117	+	0.975473i	$0.570644\pi$
$270$	0	0
$271$	6.67920	0.405733	0.202866	−	0.979206i	$-0.434974\pi$
$271$	6.67920	0.405733	0.202866	+	0.979206i	$0.434974\pi$
$272$	0	0
$273$	−25.0317	−1.51498
$274$	0	0
$275$	−2.32045	−0.139928
$276$	0	0
$277$	16.7618	1.00712	0.503560	−	0.863960i	$-0.332023\pi$
$277$	16.7618	1.00712	0.503560	+	0.863960i	$0.332023\pi$
$278$	0	0
$279$	26.5147	1.58739
$280$	0	0
$281$	0.477460	0.0284829	0.0142414	−	0.999899i	$-0.495467\pi$
$281$	0.477460	0.0284829	0.0142414	+	0.999899i	$0.495467\pi$
$282$	0	0
$283$	0.682944	0.0405968	0.0202984	−	0.999794i	$-0.493538\pi$
$283$	0.682944	0.0405968	0.0202984	+	0.999794i	$0.493538\pi$
$284$	0	0
$285$	−9.79472	−0.580189
$286$	0	0
$287$	−28.9141	−1.70674
$288$	0	0
$289$	−14.8796	−0.875271
$290$	0	0
$291$	48.1551	2.82290
$292$	0	0
$293$	10.5591	0.616866	0.308433	−	0.951246i	$-0.400195\pi$
$293$	10.5591	0.616866	0.308433	+	0.951246i	$0.400195\pi$
$294$	0	0
$295$	−7.52088	−0.437883
$296$	0	0
$297$	−4.08052	−0.236776
$298$	0	0
$299$	5.09886	0.294875
$300$	0	0
$301$	−48.8393	−2.81505
$302$	0	0
$303$	−17.6953	−1.01657
$304$	0	0
$305$	−1.44489	−0.0827344
$306$	0	0
$307$	−3.38491	−0.193187	−0.0965935	−	0.995324i	$-0.530795\pi$
$307$	−3.38491	−0.193187	−0.0965935	+	0.995324i	$0.530795\pi$
$308$	0	0
$309$	23.5859	1.34175
$310$	0	0
$311$	20.4404	1.15907	0.579534	−	0.814948i	$-0.303235\pi$
$311$	20.4404	1.15907	0.579534	+	0.814948i	$0.303235\pi$
$312$	0	0
$313$	−2.46975	−0.139598	−0.0697992	−	0.997561i	$-0.522236\pi$
$313$	−2.46975	−0.139598	−0.0697992	+	0.997561i	$0.522236\pi$
$314$	0	0
$315$	16.5970	0.935136
$316$	0	0
$317$	22.9443	1.28868	0.644340	−	0.764739i	$-0.277132\pi$
$317$	22.9443	1.28868	0.644340	+	0.764739i	$0.277132\pi$
$318$	0	0
$319$	3.03537	0.169948
$320$	0	0
$321$	−37.1026	−2.07086
$322$	0	0
$323$	−5.51824	−0.307043
$324$	0	0
$325$	2.14759	0.119127
$326$	0	0
$327$	4.95246	0.273871
$328$	0	0
$329$	11.3204	0.624111
$330$	0	0
$331$	4.84143	0.266109	0.133054	−	0.991109i	$-0.457521\pi$
$331$	4.84143	0.266109	0.133054	+	0.991109i	$0.457521\pi$
$332$	0	0
$333$	−27.1235	−1.48636
$334$	0	0
$335$	−7.39273	−0.403908
$336$	0	0
$337$	5.40017	0.294166	0.147083	−	0.989124i	$-0.453012\pi$
$337$	5.40017	0.294166	0.147083	+	0.989124i	$0.453012\pi$
$338$	0	0
$339$	6.61769	0.359423
$340$	0	0
$341$	−16.7173	−0.905293
$342$	0	0
$343$	−28.5754	−1.54292
$344$	0	0
$345$	−6.13653	−0.330379
$346$	0	0
$347$	5.76336	0.309393	0.154697	−	0.987962i	$-0.450560\pi$
$347$	5.76336	0.309393	0.154697	+	0.987962i	$0.450560\pi$
$348$	0	0
$349$	−2.20166	−0.117852	−0.0589260	−	0.998262i	$-0.518768\pi$
$349$	−2.20166	−0.117852	−0.0589260	+	0.998262i	$0.518768\pi$
$350$	0	0
$351$	3.77655	0.201577
$352$	0	0
$353$	1.34919	0.0718103	0.0359052	−	0.999355i	$-0.488569\pi$
$353$	1.34919	0.0718103	0.0359052	+	0.999355i	$0.488569\pi$
$354$	0	0
$355$	−1.92097	−0.101955
$356$	0	0
$357$	16.9726	0.898284
$358$	0	0
$359$	−23.2192	−1.22546	−0.612732	−	0.790291i	$-0.709929\pi$
$359$	−23.2192	−1.22546	−0.612732	+	0.790291i	$0.709929\pi$
$360$	0	0
$361$	−4.63903	−0.244159
$362$	0	0
$363$	−14.5141	−0.761794
$364$	0	0
$365$	−1.39412	−0.0729716
$366$	0	0
$367$	−5.16452	−0.269586	−0.134793	−	0.990874i	$-0.543037\pi$
$367$	−5.16452	−0.269586	−0.134793	+	0.990874i	$0.543037\pi$
$368$	0	0
$369$	23.5973	1.22843
$370$	0	0
$371$	9.58623	0.497692
$372$	0	0
$373$	−26.1709	−1.35508	−0.677539	−	0.735487i	$-0.736954\pi$
$373$	−26.1709	−1.35508	−0.677539	+	0.735487i	$0.736954\pi$
$374$	0	0
$375$	−2.58464	−0.133470
$376$	0	0
$377$	−2.80926	−0.144684
$378$	0	0
$379$	19.1278	0.982530	0.491265	−	0.871010i	$-0.336535\pi$
$379$	19.1278	0.982530	0.491265	+	0.871010i	$0.336535\pi$
$380$	0	0
$381$	−35.5011	−1.81877
$382$	0	0
$383$	−21.9051	−1.11930	−0.559650	−	0.828729i	$-0.689064\pi$
$383$	−21.9051	−1.11930	−0.559650	+	0.828729i	$0.689064\pi$
$384$	0	0
$385$	−10.4643	−0.533310
$386$	0	0
$387$	39.8586	2.02613
$388$	0	0
$389$	6.34761	0.321837	0.160918	−	0.986968i	$-0.448554\pi$
$389$	6.34761	0.321837	0.160918	+	0.986968i	$0.448554\pi$
$390$	0	0
$391$	−3.45725	−0.174841
$392$	0	0
$393$	19.0337	0.960126
$394$	0	0
$395$	−5.06317	−0.254756
$396$	0	0
$397$	16.6724	0.836764	0.418382	−	0.908271i	$-0.362597\pi$
$397$	16.6724	0.836764	0.418382	+	0.908271i	$0.362597\pi$
$398$	0	0
$399$	−44.1703	−2.21128
$400$	0	0
$401$	−24.9259	−1.24474	−0.622371	−	0.782722i	$-0.713830\pi$
$401$	−24.9259	−1.24474	−0.622371	+	0.782722i	$0.713830\pi$
$402$	0	0
$403$	15.4720	0.770714
$404$	0	0
$405$	6.49599	0.322789
$406$	0	0
$407$	17.1012	0.847675
$408$	0	0
$409$	−21.5355	−1.06486	−0.532430	−	0.846474i	$-0.678721\pi$
$409$	−21.5355	−1.06486	−0.532430	+	0.846474i	$0.678721\pi$
$410$	0	0
$411$	−58.8169	−2.90122
$412$	0	0
$413$	−33.9162	−1.66891
$414$	0	0
$415$	−3.46445	−0.170063
$416$	0	0
$417$	22.9580	1.12426
$418$	0	0
$419$	−24.4630	−1.19509	−0.597547	−	0.801834i	$-0.703858\pi$
$419$	−24.4630	−1.19509	−0.597547	+	0.801834i	$0.703858\pi$
$420$	0	0
$421$	−27.4824	−1.33941	−0.669704	−	0.742628i	$-0.733579\pi$
$421$	−27.4824	−1.33941	−0.669704	+	0.742628i	$0.733579\pi$
$422$	0	0
$423$	−9.23874	−0.449203
$424$	0	0
$425$	−1.45616	−0.0706341
$426$	0	0
$427$	−6.51591	−0.315327
$428$	0	0
$429$	−12.8802	−0.621862
$430$	0	0
$431$	28.3769	1.36687	0.683433	−	0.730013i	$-0.260486\pi$
$431$	28.3769	1.36687	0.683433	+	0.730013i	$0.260486\pi$
$432$	0	0
$433$	−9.04007	−0.434438	−0.217219	−	0.976123i	$-0.569699\pi$
$433$	−9.04007	−0.434438	−0.217219	+	0.976123i	$0.569699\pi$
$434$	0	0
$435$	3.38097	0.162105
$436$	0	0
$437$	8.99734	0.430401
$438$	0	0
$439$	−28.2949	−1.35044	−0.675221	−	0.737615i	$-0.735952\pi$
$439$	−28.2949	−1.35044	−0.675221	+	0.737615i	$0.735952\pi$
$440$	0	0
$441$	49.0834	2.33731
$442$	0	0
$443$	18.5591	0.881768	0.440884	−	0.897564i	$-0.354665\pi$
$443$	18.5591	0.881768	0.440884	+	0.897564i	$0.354665\pi$
$444$	0	0
$445$	−9.36007	−0.443710
$446$	0	0
$447$	47.3624	2.24016
$448$	0	0
$449$	−14.3902	−0.679116	−0.339558	−	0.940585i	$-0.610277\pi$
$449$	−14.3902	−0.679116	−0.339558	+	0.940585i	$0.610277\pi$
$450$	0	0
$451$	−14.8779	−0.700574
$452$	0	0
$453$	37.0681	1.74161
$454$	0	0
$455$	9.68477	0.454029
$456$	0	0
$457$	−4.54538	−0.212624	−0.106312	−	0.994333i	$-0.533904\pi$
$457$	−4.54538	−0.212624	−0.106312	+	0.994333i	$0.533904\pi$
$458$	0	0
$459$	−2.56067	−0.119522
$460$	0	0
$461$	28.0080	1.30446	0.652230	−	0.758021i	$-0.273833\pi$
$461$	28.0080	1.30446	0.652230	+	0.758021i	$0.273833\pi$
$462$	0	0
$463$	−14.5997	−0.678506	−0.339253	−	0.940695i	$-0.610174\pi$
$463$	−14.5997	−0.678506	−0.339253	+	0.940695i	$0.610174\pi$
$464$	0	0
$465$	−18.6207	−0.863513
$466$	0	0
$467$	−28.0163	−1.29644	−0.648221	−	0.761453i	$-0.724487\pi$
$467$	−28.0163	−1.29644	−0.648221	+	0.761453i	$0.724487\pi$
$468$	0	0
$469$	−33.3383	−1.53942
$470$	0	0
$471$	7.69661	0.354641
$472$	0	0
$473$	−25.1306	−1.15550
$474$	0	0
$475$	3.78959	0.173878
$476$	0	0
$477$	−7.82349	−0.358213
$478$	0	0
$479$	−21.0378	−0.961243	−0.480621	−	0.876928i	$-0.659589\pi$
$479$	−21.0378	−0.961243	−0.480621	+	0.876928i	$0.659589\pi$
$480$	0	0
$481$	−15.8273	−0.721661
$482$	0	0
$483$	−27.6733	−1.25918
$484$	0	0
$485$	−18.6313	−0.846002
$486$	0	0
$487$	10.2724	0.465485	0.232743	−	0.972538i	$-0.425230\pi$
$487$	10.2724	0.465485	0.232743	+	0.972538i	$0.425230\pi$
$488$	0	0
$489$	−19.5191	−0.882685
$490$	0	0
$491$	−8.42420	−0.380179	−0.190089	−	0.981767i	$-0.560878\pi$
$491$	−8.42420	−0.380179	−0.190089	+	0.981767i	$0.560878\pi$
$492$	0	0
$493$	1.90480	0.0857879
$494$	0	0
$495$	8.54009	0.383849
$496$	0	0
$497$	−8.66284	−0.388581
$498$	0	0
$499$	3.98054	0.178193	0.0890966	−	0.996023i	$-0.471602\pi$
$499$	3.98054	0.178193	0.0890966	+	0.996023i	$0.471602\pi$
$500$	0	0
$501$	41.5316	1.85550
$502$	0	0
$503$	5.49759	0.245125	0.122563	−	0.992461i	$-0.460889\pi$
$503$	5.49759	0.245125	0.122563	+	0.992461i	$0.460889\pi$
$504$	0	0
$505$	6.84632	0.304657
$506$	0	0
$507$	−21.6796	−0.962826
$508$	0	0
$509$	6.18829	0.274291	0.137146	−	0.990551i	$-0.456207\pi$
$509$	6.18829	0.274291	0.137146	+	0.990551i	$0.456207\pi$
$510$	0	0
$511$	−6.28693	−0.278118
$512$	0	0
$513$	6.66402	0.294224
$514$	0	0
$515$	−9.12540	−0.402113
$516$	0	0
$517$	5.82496	0.256181
$518$	0	0
$519$	62.5532	2.74578
$520$	0	0
$521$	33.8729	1.48400	0.741999	−	0.670401i	$-0.233878\pi$
$521$	33.8729	1.48400	0.741999	+	0.670401i	$0.233878\pi$
$522$	0	0
$523$	39.3236	1.71950	0.859749	−	0.510716i	$-0.170620\pi$
$523$	39.3236	1.71950	0.859749	+	0.510716i	$0.170620\pi$
$524$	0	0
$525$	−11.6557	−0.508697
$526$	0	0
$527$	−10.4907	−0.456981
$528$	0	0
$529$	−17.3630	−0.754915
$530$	0	0
$531$	27.6796	1.20119
$532$	0	0
$533$	13.7696	0.596428
$534$	0	0
$535$	14.3550	0.620622
$536$	0	0
$537$	−3.81905	−0.164804
$538$	0	0
$539$	−30.9468	−1.33297
$540$	0	0
$541$	−4.28600	−0.184270	−0.0921349	−	0.995747i	$-0.529369\pi$
$541$	−4.28600	−0.184270	−0.0921349	+	0.995747i	$0.529369\pi$
$542$	0	0
$543$	−43.8211	−1.88054
$544$	0	0
$545$	−1.91611	−0.0820771
$546$	0	0
$547$	−26.1322	−1.11733	−0.558667	−	0.829392i	$-0.688687\pi$
$547$	−26.1322	−1.11733	−0.558667	+	0.829392i	$0.688687\pi$
$548$	0	0
$549$	5.31774	0.226956
$550$	0	0
$551$	−4.95716	−0.211182
$552$	0	0
$553$	−22.8329	−0.970953
$554$	0	0
$555$	19.0483	0.808553
$556$	0	0
$557$	−42.7177	−1.81001	−0.905003	−	0.425405i	$-0.860132\pi$
$557$	−42.7177	−1.81001	−0.905003	+	0.425405i	$0.860132\pi$
$558$	0	0
$559$	23.2585	0.983729
$560$	0	0
$561$	8.73334	0.368722
$562$	0	0
$563$	−4.05522	−0.170907	−0.0854536	−	0.996342i	$-0.527234\pi$
$563$	−4.05522	−0.170907	−0.0854536	+	0.996342i	$0.527234\pi$
$564$	0	0
$565$	−2.56039	−0.107716
$566$	0	0
$567$	29.2944	1.23025
$568$	0	0
$569$	−35.8628	−1.50345	−0.751723	−	0.659479i	$-0.770777\pi$
$569$	−35.8628	−1.50345	−0.751723	+	0.659479i	$0.770777\pi$
$570$	0	0
$571$	24.9621	1.04463	0.522317	−	0.852752i	$-0.325068\pi$
$571$	24.9621	1.04463	0.522317	+	0.852752i	$0.325068\pi$
$572$	0	0
$573$	−0.172517	−0.00720702
$574$	0	0
$575$	2.37423	0.0990122
$576$	0	0
$577$	36.1387	1.50448	0.752238	−	0.658892i	$-0.228975\pi$
$577$	36.1387	1.50448	0.752238	+	0.658892i	$0.228975\pi$
$578$	0	0
$579$	−2.84038	−0.118042
$580$	0	0
$581$	−15.6233	−0.648165
$582$	0	0
$583$	4.93265	0.204290
$584$	0	0
$585$	−7.90391	−0.326786
$586$	0	0
$587$	−16.1228	−0.665459	−0.332729	−	0.943022i	$-0.607970\pi$
$587$	−16.1228	−0.665459	−0.332729	+	0.943022i	$0.607970\pi$
$588$	0	0
$589$	27.3015	1.12494
$590$	0	0
$591$	43.6029	1.79358
$592$	0	0
$593$	35.0454	1.43914	0.719572	−	0.694418i	$-0.244338\pi$
$593$	35.0454	1.43914	0.719572	+	0.694418i	$0.244338\pi$
$594$	0	0
$595$	−6.56670	−0.269209
$596$	0	0
$597$	28.5337	1.16781
$598$	0	0
$599$	−18.2753	−0.746707	−0.373354	−	0.927689i	$-0.621792\pi$
$599$	−18.2753	−0.746707	−0.373354	+	0.927689i	$0.621792\pi$
$600$	0	0
$601$	0.480142	0.0195854	0.00979269	−	0.999952i	$-0.496883\pi$
$601$	0.480142	0.0195854	0.00979269	+	0.999952i	$0.496883\pi$
$602$	0	0
$603$	27.2080	1.10799
$604$	0	0
$605$	5.61553	0.228304
$606$	0	0
$607$	38.6107	1.56716	0.783581	−	0.621290i	$-0.213391\pi$
$607$	38.6107	1.56716	0.783581	+	0.621290i	$0.213391\pi$
$608$	0	0
$609$	15.2468	0.617833
$610$	0	0
$611$	−5.39103	−0.218098
$612$	0	0
$613$	−7.82897	−0.316209	−0.158105	−	0.987422i	$-0.550538\pi$
$613$	−7.82897	−0.316209	−0.158105	+	0.987422i	$0.550538\pi$
$614$	0	0
$615$	−16.5718	−0.668241
$616$	0	0
$617$	−31.8836	−1.28358	−0.641792	−	0.766879i	$-0.721809\pi$
$617$	−31.8836	−1.28358	−0.641792	+	0.766879i	$0.721809\pi$
$618$	0	0
$619$	−41.5855	−1.67146	−0.835730	−	0.549140i	$-0.814955\pi$
$619$	−41.5855	−1.67146	−0.835730	+	0.549140i	$0.814955\pi$
$620$	0	0
$621$	4.17510	0.167541
$622$	0	0
$623$	−42.2102	−1.69112
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−22.7281	−0.907674
$628$	0	0
$629$	10.7316	0.427896
$630$	0	0
$631$	−30.7381	−1.22367	−0.611833	−	0.790987i	$-0.709568\pi$
$631$	−30.7381	−1.22367	−0.611833	+	0.790987i	$0.709568\pi$
$632$	0	0
$633$	31.4254	1.24905
$634$	0	0
$635$	13.7354	0.545072
$636$	0	0
$637$	28.6414	1.13481
$638$	0	0
$639$	7.06989	0.279681
$640$	0	0
$641$	13.6348	0.538540	0.269270	−	0.963065i	$-0.413218\pi$
$641$	13.6348	0.538540	0.269270	+	0.963065i	$0.413218\pi$
$642$	0	0
$643$	21.1034	0.832237	0.416118	−	0.909310i	$-0.363390\pi$
$643$	21.1034	0.832237	0.416118	+	0.909310i	$0.363390\pi$
$644$	0	0
$645$	−27.9918	−1.10218
$646$	0	0
$647$	4.87972	0.191841	0.0959207	−	0.995389i	$-0.469420\pi$
$647$	4.87972	0.191841	0.0959207	+	0.995389i	$0.469420\pi$
$648$	0	0
$649$	−17.4518	−0.685043
$650$	0	0
$651$	−83.9719	−3.29112
$652$	0	0
$653$	−14.5541	−0.569548	−0.284774	−	0.958595i	$-0.591919\pi$
$653$	−14.5541	−0.569548	−0.284774	+	0.958595i	$0.591919\pi$
$654$	0	0
$655$	−7.36417	−0.287742
$656$	0	0
$657$	5.13088	0.200174
$658$	0	0
$659$	30.9748	1.20661	0.603304	−	0.797511i	$-0.293851\pi$
$659$	30.9748	1.20661	0.603304	+	0.797511i	$0.293851\pi$
$660$	0	0
$661$	−7.64517	−0.297363	−0.148681	−	0.988885i	$-0.547503\pi$
$661$	−7.64517	−0.297363	−0.148681	+	0.988885i	$0.547503\pi$
$662$	0	0
$663$	−8.08276	−0.313908
$664$	0	0
$665$	17.0895	0.662704
$666$	0	0
$667$	−3.10573	−0.120254
$668$	0	0
$669$	55.3729	2.14084
$670$	0	0
$671$	−3.35280	−0.129433
$672$	0	0
$673$	35.3820	1.36388	0.681938	−	0.731410i	$-0.261138\pi$
$673$	35.3820	1.36388	0.681938	+	0.731410i	$0.261138\pi$
$674$	0	0
$675$	1.75851	0.0676850
$676$	0	0
$677$	7.31235	0.281036	0.140518	−	0.990078i	$-0.455123\pi$
$677$	7.31235	0.281036	0.140518	+	0.990078i	$0.455123\pi$
$678$	0	0
$679$	−84.0196	−3.22438
$680$	0	0
$681$	−29.4733	−1.12942
$682$	0	0
$683$	37.6166	1.43936	0.719679	−	0.694307i	$-0.244289\pi$
$683$	37.6166	1.43936	0.719679	+	0.694307i	$0.244289\pi$
$684$	0	0
$685$	22.7563	0.869474
$686$	0	0
$687$	16.9299	0.645916
$688$	0	0
$689$	−4.56520	−0.173920
$690$	0	0
$691$	30.8283	1.17276	0.586382	−	0.810034i	$-0.300552\pi$
$691$	30.8283	1.17276	0.586382	+	0.810034i	$0.300552\pi$
$692$	0	0
$693$	38.5125	1.46297
$694$	0	0
$695$	−8.88246	−0.336931
$696$	0	0
$697$	−9.33640	−0.353641
$698$	0	0
$699$	−67.3744	−2.54833
$700$	0	0
$701$	21.5208	0.812828	0.406414	−	0.913689i	$-0.366779\pi$
$701$	21.5208	0.812828	0.406414	+	0.913689i	$0.366779\pi$
$702$	0	0
$703$	−27.9285	−1.05334
$704$	0	0
$705$	6.48816	0.244358
$706$	0	0
$707$	30.8742	1.16114
$708$	0	0
$709$	6.89217	0.258841	0.129421	−	0.991590i	$-0.458688\pi$
$709$	6.89217	0.258841	0.129421	+	0.991590i	$0.458688\pi$
$710$	0	0
$711$	18.6343	0.698842
$712$	0	0
$713$	17.1048	0.640579
$714$	0	0
$715$	4.98336	0.186367
$716$	0	0
$717$	13.2388	0.494410
$718$	0	0
$719$	9.27351	0.345843	0.172922	−	0.984936i	$-0.444679\pi$
$719$	9.27351	0.345843	0.172922	+	0.984936i	$0.444679\pi$
$720$	0	0
$721$	−41.1520	−1.53258
$722$	0	0
$723$	−29.7199	−1.10530
$724$	0	0
$725$	−1.30810	−0.0485816
$726$	0	0
$727$	−10.6056	−0.393341	−0.196670	−	0.980470i	$-0.563013\pi$
$727$	−10.6056	−0.393341	−0.196670	+	0.980470i	$0.563013\pi$
$728$	0	0
$729$	−37.5430	−1.39048
$730$	0	0
$731$	−15.7703	−0.583285
$732$	0	0
$733$	41.9357	1.54893	0.774465	−	0.632617i	$-0.218019\pi$
$733$	41.9357	1.54893	0.774465	+	0.632617i	$0.218019\pi$
$734$	0	0
$735$	−34.4702	−1.27145
$736$	0	0
$737$	−17.1544	−0.631892
$738$	0	0
$739$	43.6641	1.60621	0.803104	−	0.595839i	$-0.203180\pi$
$739$	43.6641	1.60621	0.803104	+	0.595839i	$0.203180\pi$
$740$	0	0
$741$	21.0350	0.772741
$742$	0	0
$743$	22.3956	0.821617	0.410808	−	0.911722i	$-0.365247\pi$
$743$	22.3956	0.821617	0.410808	+	0.911722i	$0.365247\pi$
$744$	0	0
$745$	−18.3245	−0.671360
$746$	0	0
$747$	12.7505	0.466515
$748$	0	0
$749$	64.7355	2.36539
$750$	0	0
$751$	20.6448	0.753341	0.376670	−	0.926347i	$-0.377069\pi$
$751$	20.6448	0.753341	0.376670	+	0.926347i	$0.377069\pi$
$752$	0	0
$753$	72.4724	2.64104
$754$	0	0
$755$	−14.3417	−0.521948
$756$	0	0
$757$	−34.1282	−1.24041	−0.620206	−	0.784439i	$-0.712951\pi$
$757$	−34.1282	−1.24041	−0.620206	+	0.784439i	$0.712951\pi$
$758$	0	0
$759$	−14.2395	−0.516860
$760$	0	0
$761$	50.1874	1.81929	0.909647	−	0.415383i	$-0.136352\pi$
$761$	50.1874	1.81929	0.909647	+	0.415383i	$0.136352\pi$
$762$	0	0
$763$	−8.64090	−0.312822
$764$	0	0
$765$	5.35920	0.193762
$766$	0	0
$767$	16.1517	0.583206
$768$	0	0
$769$	−28.6887	−1.03454	−0.517270	−	0.855822i	$-0.673052\pi$
$769$	−28.6887	−1.03454	−0.517270	+	0.855822i	$0.673052\pi$
$770$	0	0
$771$	−62.6761	−2.25722
$772$	0	0
$773$	−53.1683	−1.91233	−0.956166	−	0.292826i	$-0.905404\pi$
$773$	−53.1683	−1.91233	−0.956166	+	0.292826i	$0.905404\pi$
$774$	0	0
$775$	7.20435	0.258788
$776$	0	0
$777$	85.9001	3.08165
$778$	0	0
$779$	24.2976	0.870550
$780$	0	0
$781$	−4.45752	−0.159502
$782$	0	0
$783$	−2.30030	−0.0822061
$784$	0	0
$785$	−2.97783	−0.106283
$786$	0	0
$787$	4.43052	0.157931	0.0789655	−	0.996877i	$-0.474838\pi$
$787$	4.43052	0.157931	0.0789655	+	0.996877i	$0.474838\pi$
$788$	0	0
$789$	−58.3302	−2.07661
$790$	0	0
$791$	−11.5463	−0.410541
$792$	0	0
$793$	3.10304	0.110192
$794$	0	0
$795$	5.49426	0.194861
$796$	0	0
$797$	−0.0795064	−0.00281626	−0.00140813	−	0.999999i	$-0.500448\pi$
$797$	−0.0795064	−0.00281626	−0.00140813	+	0.999999i	$0.500448\pi$
$798$	0	0
$799$	3.65536	0.129317
$800$	0	0
$801$	34.4485	1.21718
$802$	0	0
$803$	−3.23498	−0.114160
$804$	0	0
$805$	10.7068	0.377366
$806$	0	0
$807$	−18.6621	−0.656937
$808$	0	0
$809$	3.59856	0.126518	0.0632592	−	0.997997i	$-0.479851\pi$
$809$	3.59856	0.126518	0.0632592	+	0.997997i	$0.479851\pi$
$810$	0	0
$811$	−10.4125	−0.365632	−0.182816	−	0.983147i	$-0.558521\pi$
$811$	−10.4125	−0.365632	−0.182816	+	0.983147i	$0.558521\pi$
$812$	0	0
$813$	17.2633	0.605452
$814$	0	0
$815$	7.55196	0.264534
$816$	0	0
$817$	41.0414	1.43586
$818$	0	0
$819$	−35.6435	−1.24549
$820$	0	0
$821$	−20.9020	−0.729485	−0.364742	−	0.931108i	$-0.618843\pi$
$821$	−20.9020	−0.729485	−0.364742	+	0.931108i	$0.618843\pi$
$822$	0	0
$823$	−52.7544	−1.83890	−0.919452	−	0.393203i	$-0.871367\pi$
$823$	−52.7544	−1.83890	−0.919452	+	0.393203i	$0.871367\pi$
$824$	0	0
$825$	−5.99752	−0.208807
$826$	0	0
$827$	−23.8837	−0.830518	−0.415259	−	0.909703i	$-0.636309\pi$
$827$	−23.8837	−0.830518	−0.415259	+	0.909703i	$0.636309\pi$
$828$	0	0
$829$	12.1053	0.420434	0.210217	−	0.977655i	$-0.432583\pi$
$829$	12.1053	0.420434	0.210217	+	0.977655i	$0.432583\pi$
$830$	0	0
$831$	43.3232	1.50287
$832$	0	0
$833$	−19.4201	−0.672868
$834$	0	0
$835$	−16.0686	−0.556078
$836$	0	0
$837$	12.6689	0.437902
$838$	0	0
$839$	17.5407	0.605572	0.302786	−	0.953059i	$-0.402083\pi$
$839$	17.5407	0.605572	0.302786	+	0.953059i	$0.402083\pi$
$840$	0	0
$841$	−27.2889	−0.940996
$842$	0	0
$843$	1.23406	0.0425034
$844$	0	0
$845$	8.38787	0.288552
$846$	0	0
$847$	25.3238	0.870137
$848$	0	0
$849$	1.76516	0.0605803
$850$	0	0
$851$	−17.4976	−0.599808
$852$	0	0
$853$	21.7191	0.743648	0.371824	−	0.928303i	$-0.378732\pi$
$853$	21.7191	0.743648	0.371824	+	0.928303i	$0.378732\pi$
$854$	0	0
$855$	−13.9471	−0.476980
$856$	0	0
$857$	−17.9553	−0.613341	−0.306671	−	0.951816i	$-0.599215\pi$
$857$	−17.9553	−0.613341	−0.306671	+	0.951816i	$0.599215\pi$
$858$	0	0
$859$	47.1001	1.60703	0.803517	−	0.595282i	$-0.202960\pi$
$859$	47.1001	1.60703	0.803517	+	0.595282i	$0.202960\pi$
$860$	0	0
$861$	−74.7325	−2.54688
$862$	0	0
$863$	32.3557	1.10140	0.550701	−	0.834703i	$-0.314361\pi$
$863$	32.3557	1.10140	0.550701	+	0.834703i	$0.314361\pi$
$864$	0	0
$865$	−24.2019	−0.822889
$866$	0	0
$867$	−38.4584	−1.30612
$868$	0	0
$869$	−11.7488	−0.398551
$870$	0	0
$871$	15.8765	0.537956
$872$	0	0
$873$	68.5699	2.32074
$874$	0	0
$875$	4.50961	0.152453
$876$	0	0
$877$	37.0944	1.25259	0.626295	−	0.779586i	$-0.284571\pi$
$877$	37.0944	1.25259	0.626295	+	0.779586i	$0.284571\pi$
$878$	0	0
$879$	27.2914	0.920515
$880$	0	0
$881$	47.3359	1.59479	0.797394	−	0.603459i	$-0.206211\pi$
$881$	47.3359	1.59479	0.797394	+	0.603459i	$0.206211\pi$
$882$	0	0
$883$	−11.4321	−0.384721	−0.192360	−	0.981324i	$-0.561614\pi$
$883$	−11.4321	−0.384721	−0.192360	+	0.981324i	$0.561614\pi$
$884$	0	0
$885$	−19.4388	−0.653427
$886$	0	0
$887$	−12.9255	−0.433994	−0.216997	−	0.976172i	$-0.569626\pi$
$887$	−12.9255	−0.433994	−0.216997	+	0.976172i	$0.569626\pi$
$888$	0	0
$889$	61.9412	2.07744
$890$	0	0
$891$	15.0736	0.504985
$892$	0	0
$893$	−9.51291	−0.318337
$894$	0	0
$895$	1.47760	0.0493906
$896$	0	0
$897$	13.1787	0.440025
$898$	0	0
$899$	−9.42401	−0.314308
$900$	0	0
$901$	3.09541	0.103123
$902$	0	0
$903$	−126.232	−4.20074
$904$	0	0
$905$	16.9544	0.563584
$906$	0	0
$907$	23.3332	0.774768	0.387384	−	0.921918i	$-0.373379\pi$
$907$	23.3332	0.774768	0.387384	+	0.921918i	$0.373379\pi$
$908$	0	0
$909$	−25.1970	−0.835730
$910$	0	0
$911$	−26.6745	−0.883765	−0.441883	−	0.897073i	$-0.645689\pi$
$911$	−26.6745	−0.883765	−0.441883	+	0.897073i	$0.645689\pi$
$912$	0	0
$913$	−8.03908	−0.266055
$914$	0	0
$915$	−3.73453	−0.123460
$916$	0	0
$917$	−33.2095	−1.09668
$918$	0	0
$919$	57.7425	1.90475	0.952374	−	0.304932i	$-0.0986336\pi$
$919$	57.7425	1.90475	0.952374	+	0.304932i	$0.0986336\pi$
$920$	0	0
$921$	−8.74878	−0.288282
$922$	0	0
$923$	4.12546	0.135791
$924$	0	0
$925$	−7.36979	−0.242317
$926$	0	0
$927$	33.5848	1.10307
$928$	0	0
$929$	42.5386	1.39565	0.697823	−	0.716270i	$-0.254152\pi$
$929$	42.5386	1.39565	0.697823	+	0.716270i	$0.254152\pi$
$930$	0	0
$931$	50.5400	1.65638
$932$	0	0
$933$	52.8310	1.72961
$934$	0	0
$935$	−3.37894	−0.110503
$936$	0	0
$937$	16.6795	0.544894	0.272447	−	0.962171i	$-0.412167\pi$
$937$	16.6795	0.544894	0.272447	+	0.962171i	$0.412167\pi$
$938$	0	0
$939$	−6.38342	−0.208315
$940$	0	0
$941$	13.6210	0.444033	0.222016	−	0.975043i	$-0.428736\pi$
$941$	13.6210	0.444033	0.222016	+	0.975043i	$0.428736\pi$
$942$	0	0
$943$	15.2227	0.495721
$944$	0	0
$945$	7.93018	0.257969
$946$	0	0
$947$	4.86538	0.158104	0.0790518	−	0.996871i	$-0.474811\pi$
$947$	4.86538	0.158104	0.0790518	+	0.996871i	$0.474811\pi$
$948$	0	0
$949$	2.99399	0.0971891
$950$	0	0
$951$	59.3028	1.92302
$952$	0	0
$953$	17.6965	0.573247	0.286623	−	0.958043i	$-0.407467\pi$
$953$	17.6965	0.573247	0.286623	+	0.958043i	$0.407467\pi$
$954$	0	0
$955$	0.0667471	0.00215989
$956$	0	0
$957$	7.84535	0.253604
$958$	0	0
$959$	102.622	3.31384
$960$	0	0
$961$	20.9027	0.674281
$962$	0	0
$963$	−52.8318	−1.70248
$964$	0	0
$965$	1.09895	0.0353763
$966$	0	0
$967$	15.0023	0.482442	0.241221	−	0.970470i	$-0.422452\pi$
$967$	15.0023	0.482442	0.241221	+	0.970470i	$0.422452\pi$
$968$	0	0
$969$	−14.2627	−0.458183
$970$	0	0
$971$	−20.2423	−0.649606	−0.324803	−	0.945782i	$-0.605298\pi$
$971$	−20.2423	−0.649606	−0.324803	+	0.945782i	$0.605298\pi$
$972$	0	0
$973$	−40.0564	−1.28415
$974$	0	0
$975$	5.55074	0.177766
$976$	0	0
$977$	−48.1433	−1.54024	−0.770120	−	0.637900i	$-0.779804\pi$
$977$	−48.1433	−1.54024	−0.770120	+	0.637900i	$0.779804\pi$
$978$	0	0
$979$	−21.7195	−0.694159
$980$	0	0
$981$	7.05199	0.225153
$982$	0	0
$983$	0.791292	0.0252383	0.0126191	−	0.999920i	$-0.495983\pi$
$983$	0.791292	0.0252383	0.0126191	+	0.999920i	$0.495983\pi$
$984$	0	0
$985$	−16.8700	−0.537523
$986$	0	0
$987$	29.2590	0.931326
$988$	0	0
$989$	25.7130	0.817626
$990$	0	0
$991$	60.2424	1.91366	0.956832	−	0.290643i	$-0.0938690\pi$
$991$	60.2424	1.91366	0.956832	+	0.290643i	$0.0938690\pi$
$992$	0	0
$993$	12.5133	0.397099
$994$	0	0
$995$	−11.0397	−0.349982
$996$	0	0
$997$	1.63727	0.0518529	0.0259264	−	0.999664i	$-0.491746\pi$
$997$	1.63727	0.0518529	0.0259264	+	0.999664i	$0.491746\pi$
$998$	0	0
$999$	−12.9598	−0.410031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.u.1.7		8
4.3	odd	2		5120.2.a.s.1.2		8
8.3	odd	2		5120.2.a.v.1.7		8
8.5	even	2		5120.2.a.t.1.2		8
32.3	odd	8		640.2.l.b.161.1		16
32.5	even	8		320.2.l.a.241.1		16
32.11	odd	8		640.2.l.b.481.1		16
32.13	even	8		320.2.l.a.81.1		16
32.19	odd	8		80.2.l.a.61.3	yes	16
32.21	even	8		640.2.l.a.481.8		16
32.27	odd	8		80.2.l.a.21.3	✓	16
32.29	even	8		640.2.l.a.161.8		16
96.5	odd	8		2880.2.t.c.2161.5		16
96.59	even	8		720.2.t.c.181.6		16
96.77	odd	8		2880.2.t.c.721.8		16
96.83	even	8		720.2.t.c.541.6		16
160.13	odd	8		1600.2.q.h.849.8		16
160.19	odd	8		400.2.l.h.301.6		16
160.27	even	8		400.2.q.g.149.8		16
160.37	odd	8		1600.2.q.h.49.8		16
160.59	odd	8		400.2.l.h.101.6		16
160.69	even	8		1600.2.l.i.1201.8		16
160.77	odd	8		1600.2.q.g.849.1		16
160.83	even	8		400.2.q.g.349.8		16
160.109	even	8		1600.2.l.i.401.8		16
160.123	even	8		400.2.q.h.149.1		16
160.133	odd	8		1600.2.q.g.49.1		16
160.147	even	8		400.2.q.h.349.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.3	✓	16	32.27	odd	8
80.2.l.a.61.3	yes	16	32.19	odd	8
320.2.l.a.81.1		16	32.13	even	8
320.2.l.a.241.1		16	32.5	even	8
400.2.l.h.101.6		16	160.59	odd	8
400.2.l.h.301.6		16	160.19	odd	8
400.2.q.g.149.8		16	160.27	even	8
400.2.q.g.349.8		16	160.83	even	8
400.2.q.h.149.1		16	160.123	even	8
400.2.q.h.349.1		16	160.147	even	8
640.2.l.a.161.8		16	32.29	even	8
640.2.l.a.481.8		16	32.21	even	8
640.2.l.b.161.1		16	32.3	odd	8
640.2.l.b.481.1		16	32.11	odd	8
720.2.t.c.181.6		16	96.59	even	8
720.2.t.c.541.6		16	96.83	even	8
1600.2.l.i.401.8		16	160.109	even	8
1600.2.l.i.1201.8		16	160.69	even	8
1600.2.q.g.49.1		16	160.133	odd	8
1600.2.q.g.849.1		16	160.77	odd	8
1600.2.q.h.49.8		16	160.37	odd	8
1600.2.q.h.849.8		16	160.13	odd	8
2880.2.t.c.721.8		16	96.77	odd	8
2880.2.t.c.2161.5		16	96.5	odd	8
5120.2.a.s.1.2		8	4.3	odd	2
5120.2.a.t.1.2		8	8.5	even	2
5120.2.a.u.1.7		8	1.1	even	1	trivial
5120.2.a.v.1.7		8	8.3	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:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58464 q^{3} -1.00000 q^{5} -4.50961 q^{7} +3.68037 q^{9} -2.32045 q^{11} +2.14759 q^{13} -2.58464 q^{15} -1.45616 q^{17} +3.78959 q^{19} -11.6557 q^{21} +2.37423 q^{23} +1.00000 q^{25} +1.75851 q^{27} -1.30810 q^{29} +7.20435 q^{31} -5.99752 q^{33} +4.50961 q^{35} -7.36979 q^{37} +5.55074 q^{39} +6.41166 q^{41} +10.8301 q^{43} -3.68037 q^{45} -2.51027 q^{47} +13.3366 q^{49} -3.76365 q^{51} -2.12574 q^{53} +2.32045 q^{55} +9.79472 q^{57} +7.52088 q^{59} +1.44489 q^{61} -16.5970 q^{63} -2.14759 q^{65} +7.39273 q^{67} +6.13653 q^{69} +1.92097 q^{71} +1.39412 q^{73} +2.58464 q^{75} +10.4643 q^{77} +5.06317 q^{79} -6.49599 q^{81} +3.46445 q^{83} +1.45616 q^{85} -3.38097 q^{87} +9.36007 q^{89} -9.68477 q^{91} +18.6207 q^{93} -3.78959 q^{95} +18.6313 q^{97} -8.54009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{5} - 4 q^{7} + 8 q^{9} + 8 q^{11} - 4 q^{15} + 16 q^{19} - 12 q^{23} + 8 q^{25} + 16 q^{27} + 4 q^{35} + 28 q^{43} - 8 q^{45} - 20 q^{47} + 8 q^{49} + 24 q^{51} - 8 q^{55} + 16 q^{59}+ \cdots + 40 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 8 * q^5 - 4 * q^7 + 8 * q^9 + 8 * q^11 - 4 * q^15 + 16 * q^19 - 12 * q^23 + 8 * q^25 + 16 * q^27 + 4 * q^35 + 28 * q^43 - 8 * q^45 - 20 * q^47 + 8 * q^49 + 24 * q^51 - 8 * q^55 + 16 * q^59 - 20 * q^63 + 36 * q^67 + 16 * q^69 + 8 * q^71 + 4 * q^75 + 8 * q^79 + 8 * q^81 + 20 * q^83 + 40 * q^91 - 16 * q^95 + 40 * q^99

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