LMFDB - Embedded newform 5120.2.a.u.1.3

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			1.3
Root			$-2.56993$ 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)

$f(q)$	$=$	$q-0.296378 q^{3} -1.00000 q^{5} +1.73696 q^{7} -2.91216 q^{9} -0.714786 q^{11} +2.66933 q^{13} +0.296378 q^{15} -4.53524 q^{17} -4.55407 q^{19} -0.514799 q^{21} -8.85045 q^{23} +1.00000 q^{25} +1.75224 q^{27} -3.45151 q^{29} +5.70401 q^{31} +0.211847 q^{33} -1.73696 q^{35} +7.57552 q^{37} -0.791130 q^{39} +10.0343 q^{41} +2.97782 q^{43} +2.91216 q^{45} +4.32303 q^{47} -3.98295 q^{49} +1.34415 q^{51} -1.94396 q^{53} +0.714786 q^{55} +1.34973 q^{57} +9.39236 q^{59} +7.44171 q^{61} -5.05832 q^{63} -2.66933 q^{65} +14.9309 q^{67} +2.62308 q^{69} -14.0437 q^{71} -6.63830 q^{73} -0.296378 q^{75} -1.24156 q^{77} +4.27297 q^{79} +8.21715 q^{81} +12.9469 q^{83} +4.53524 q^{85} +1.02295 q^{87} -3.23826 q^{89} +4.63652 q^{91} -1.69055 q^{93} +4.55407 q^{95} +1.94129 q^{97} +2.08157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.296378	−0.171114	−0.0855571	−	0.996333i	$-0.527267\pi$
$3$	−0.296378	−0.171114	−0.0855571	+	0.996333i	$0.527267\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.73696	0.656511	0.328255	−	0.944589i	$-0.393539\pi$
$7$	1.73696	0.656511	0.328255	+	0.944589i	$0.393539\pi$
$8$	0	0
$9$	−2.91216	−0.970720
$10$	0	0
$11$	−0.714786	−0.215516	−0.107758	−	0.994177i	$-0.534367\pi$
$11$	−0.714786	−0.215516	−0.107758	+	0.994177i	$0.534367\pi$
$12$	0	0
$13$	2.66933	0.740338	0.370169	−	0.928964i	$-0.379300\pi$
$13$	2.66933	0.740338	0.370169	+	0.928964i	$0.379300\pi$
$14$	0	0
$15$	0.296378	0.0765246
$16$	0	0
$17$	−4.53524	−1.09996	−0.549979	−	0.835178i	$-0.685364\pi$
$17$	−4.53524	−1.09996	−0.549979	+	0.835178i	$0.685364\pi$
$18$	0	0
$19$	−4.55407	−1.04478	−0.522388	−	0.852708i	$-0.674959\pi$
$19$	−4.55407	−1.04478	−0.522388	+	0.852708i	$0.674959\pi$
$20$	0	0
$21$	−0.514799	−0.112338
$22$	0	0
$23$	−8.85045	−1.84545	−0.922723	−	0.385463i	$-0.874042\pi$
$23$	−8.85045	−1.84545	−0.922723	+	0.385463i	$0.874042\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.75224	0.337218
$28$	0	0
$29$	−3.45151	−0.640929	−0.320465	−	0.947260i	$-0.603839\pi$
$29$	−3.45151	−0.640929	−0.320465	+	0.947260i	$0.603839\pi$
$30$	0	0
$31$	5.70401	1.02447	0.512235	−	0.858845i	$-0.328818\pi$
$31$	5.70401	1.02447	0.512235	+	0.858845i	$0.328818\pi$
$32$	0	0
$33$	0.211847	0.0368779
$34$	0	0
$35$	−1.73696	−0.293601
$36$	0	0
$37$	7.57552	1.24541	0.622704	−	0.782458i	$-0.286034\pi$
$37$	7.57552	1.24541	0.622704	+	0.782458i	$0.286034\pi$
$38$	0	0
$39$	−0.791130	−0.126682
$40$	0	0
$41$	10.0343	1.56709	0.783545	−	0.621335i	$-0.213409\pi$
$41$	10.0343	1.56709	0.783545	+	0.621335i	$0.213409\pi$
$42$	0	0
$43$	2.97782	0.454113	0.227057	−	0.973882i	$-0.427090\pi$
$43$	2.97782	0.454113	0.227057	+	0.973882i	$0.427090\pi$
$44$	0	0
$45$	2.91216	0.434119
$46$	0	0
$47$	4.32303	0.630578	0.315289	−	0.948996i	$-0.397899\pi$
$47$	4.32303	0.630578	0.315289	+	0.948996i	$0.397899\pi$
$48$	0	0
$49$	−3.98295	−0.568993
$50$	0	0
$51$	1.34415	0.188218
$52$	0	0
$53$	−1.94396	−0.267023	−0.133511	−	0.991047i	$-0.542625\pi$
$53$	−1.94396	−0.267023	−0.133511	+	0.991047i	$0.542625\pi$
$54$	0	0
$55$	0.714786	0.0963817
$56$	0	0
$57$	1.34973	0.178776
$58$	0	0
$59$	9.39236	1.22278	0.611390	−	0.791329i	$-0.290611\pi$
$59$	9.39236	1.22278	0.611390	+	0.791329i	$0.290611\pi$
$60$	0	0
$61$	7.44171	0.952813	0.476407	−	0.879225i	$-0.341939\pi$
$61$	7.44171	0.952813	0.476407	+	0.879225i	$0.341939\pi$
$62$	0	0
$63$	−5.05832	−0.637288
$64$	0	0
$65$	−2.66933	−0.331089
$66$	0	0
$67$	14.9309	1.82411	0.912053	−	0.410073i	$-0.134497\pi$
$67$	14.9309	1.82411	0.912053	+	0.410073i	$0.134497\pi$
$68$	0	0
$69$	2.62308	0.315782
$70$	0	0
$71$	−14.0437	−1.66668	−0.833338	−	0.552764i	$-0.813573\pi$
$71$	−14.0437	−1.66668	−0.833338	+	0.552764i	$0.813573\pi$
$72$	0	0
$73$	−6.63830	−0.776954	−0.388477	−	0.921458i	$-0.626999\pi$
$73$	−6.63830	−0.776954	−0.388477	+	0.921458i	$0.626999\pi$
$74$	0	0
$75$	−0.296378	−0.0342228
$76$	0	0
$77$	−1.24156	−0.141489
$78$	0	0
$79$	4.27297	0.480746	0.240373	−	0.970681i	$-0.422730\pi$
$79$	4.27297	0.480746	0.240373	+	0.970681i	$0.422730\pi$
$80$	0	0
$81$	8.21715	0.913017
$82$	0	0
$83$	12.9469	1.42111	0.710553	−	0.703644i	$-0.248445\pi$
$83$	12.9469	1.42111	0.710553	+	0.703644i	$0.248445\pi$
$84$	0	0
$85$	4.53524	0.491916
$86$	0	0
$87$	1.02295	0.109672
$88$	0	0
$89$	−3.23826	−0.343255	−0.171627	−	0.985162i	$-0.554903\pi$
$89$	−3.23826	−0.343255	−0.171627	+	0.985162i	$0.554903\pi$
$90$	0	0
$91$	4.63652	0.486040
$92$	0	0
$93$	−1.69055	−0.175301
$94$	0	0
$95$	4.55407	0.467238
$96$	0	0
$97$	1.94129	0.197108	0.0985541	−	0.995132i	$-0.468578\pi$
$97$	1.94129	0.197108	0.0985541	+	0.995132i	$0.468578\pi$
$98$	0	0
$99$	2.08157	0.209206
$100$	0	0
$101$	−14.6223	−1.45497	−0.727485	−	0.686124i	$-0.759311\pi$
$101$	−14.6223	−1.45497	−0.727485	+	0.686124i	$0.759311\pi$
$102$	0	0
$103$	−4.96401	−0.489118	−0.244559	−	0.969634i	$-0.578643\pi$
$103$	−4.96401	−0.489118	−0.244559	+	0.969634i	$0.578643\pi$
$104$	0	0
$105$	0.514799	0.0502392
$106$	0	0
$107$	3.88387	0.375468	0.187734	−	0.982220i	$-0.439886\pi$
$107$	3.88387	0.375468	0.187734	+	0.982220i	$0.439886\pi$
$108$	0	0
$109$	−9.89891	−0.948144	−0.474072	−	0.880486i	$-0.657216\pi$
$109$	−9.89891	−0.948144	−0.474072	+	0.880486i	$0.657216\pi$
$110$	0	0
$111$	−2.24522	−0.213107
$112$	0	0
$113$	6.53194	0.614474	0.307237	−	0.951633i	$-0.400596\pi$
$113$	6.53194	0.614474	0.307237	+	0.951633i	$0.400596\pi$
$114$	0	0
$115$	8.85045	0.825309
$116$	0	0
$117$	−7.77350	−0.718661
$118$	0	0
$119$	−7.87756	−0.722134
$120$	0	0
$121$	−10.4891	−0.953553
$122$	0	0
$123$	−2.97394	−0.268151
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.50861	0.222603	0.111302	−	0.993787i	$-0.464498\pi$
$127$	2.50861	0.222603	0.111302	+	0.993787i	$0.464498\pi$
$128$	0	0
$129$	−0.882562	−0.0777053
$130$	0	0
$131$	−12.1026	−1.05741	−0.528704	−	0.848806i	$-0.677322\pi$
$131$	−12.1026	−1.05741	−0.528704	+	0.848806i	$0.677322\pi$
$132$	0	0
$133$	−7.91026	−0.685907
$134$	0	0
$135$	−1.75224	−0.150809
$136$	0	0
$137$	−6.47131	−0.552881	−0.276440	−	0.961031i	$-0.589155\pi$
$137$	−6.47131	−0.552881	−0.276440	+	0.961031i	$0.589155\pi$
$138$	0	0
$139$	23.2540	1.97238	0.986188	−	0.165629i	$-0.0529653\pi$
$139$	23.2540	1.97238	0.986188	+	0.165629i	$0.0529653\pi$
$140$	0	0
$141$	−1.28125	−0.107901
$142$	0	0
$143$	−1.90800	−0.159555
$144$	0	0
$145$	3.45151	0.286632
$146$	0	0
$147$	1.18046	0.0973628
$148$	0	0
$149$	−3.85801	−0.316061	−0.158030	−	0.987434i	$-0.550514\pi$
$149$	−3.85801	−0.316061	−0.158030	+	0.987434i	$0.550514\pi$
$150$	0	0
$151$	11.5196	0.937453	0.468726	−	0.883343i	$-0.344713\pi$
$151$	11.5196	0.937453	0.468726	+	0.883343i	$0.344713\pi$
$152$	0	0
$153$	13.2074	1.06775
$154$	0	0
$155$	−5.70401	−0.458157
$156$	0	0
$157$	4.63881	0.370217	0.185109	−	0.982718i	$-0.440736\pi$
$157$	4.63881	0.370217	0.185109	+	0.982718i	$0.440736\pi$
$158$	0	0
$159$	0.576147	0.0456914
$160$	0	0
$161$	−15.3729	−1.21156
$162$	0	0
$163$	13.1149	1.02724	0.513621	−	0.858017i	$-0.328304\pi$
$163$	13.1149	1.02724	0.513621	+	0.858017i	$0.328304\pi$
$164$	0	0
$165$	−0.211847	−0.0164923
$166$	0	0
$167$	7.08065	0.547917	0.273958	−	0.961742i	$-0.411667\pi$
$167$	7.08065	0.547917	0.273958	+	0.961742i	$0.411667\pi$
$168$	0	0
$169$	−5.87470	−0.451900
$170$	0	0
$171$	13.2622	1.01418
$172$	0	0
$173$	7.37471	0.560689	0.280345	−	0.959899i	$-0.409551\pi$
$173$	7.37471	0.560689	0.280345	+	0.959899i	$0.409551\pi$
$174$	0	0
$175$	1.73696	0.131302
$176$	0	0
$177$	−2.78369	−0.209235
$178$	0	0
$179$	8.94060	0.668252	0.334126	−	0.942528i	$-0.391559\pi$
$179$	8.94060	0.668252	0.334126	+	0.942528i	$0.391559\pi$
$180$	0	0
$181$	18.4831	1.37384	0.686918	−	0.726735i	$-0.258963\pi$
$181$	18.4831	1.37384	0.686918	+	0.726735i	$0.258963\pi$
$182$	0	0
$183$	−2.20556	−0.163040
$184$	0	0
$185$	−7.57552	−0.556963
$186$	0	0
$187$	3.24173	0.237059
$188$	0	0
$189$	3.04357	0.221387
$190$	0	0
$191$	22.1722	1.60433	0.802164	−	0.597104i	$-0.203682\pi$
$191$	22.1722	1.60433	0.802164	+	0.597104i	$0.203682\pi$
$192$	0	0
$193$	7.97695	0.574193	0.287097	−	0.957902i	$-0.407310\pi$
$193$	7.97695	0.574193	0.287097	+	0.957902i	$0.407310\pi$
$194$	0	0
$195$	0.791130	0.0566540
$196$	0	0
$197$	−8.15050	−0.580699	−0.290350	−	0.956921i	$-0.593772\pi$
$197$	−8.15050	−0.580699	−0.290350	+	0.956921i	$0.593772\pi$
$198$	0	0
$199$	−5.38869	−0.381994	−0.190997	−	0.981591i	$-0.561172\pi$
$199$	−5.38869	−0.381994	−0.190997	+	0.981591i	$0.561172\pi$
$200$	0	0
$201$	−4.42521	−0.312130
$202$	0	0
$203$	−5.99515	−0.420777
$204$	0	0
$205$	−10.0343	−0.700824
$206$	0	0
$207$	25.7739	1.79141
$208$	0	0
$209$	3.25519	0.225166
$210$	0	0
$211$	15.2094	1.04706	0.523530	−	0.852007i	$-0.324615\pi$
$211$	15.2094	1.04706	0.523530	+	0.852007i	$0.324615\pi$
$212$	0	0
$213$	4.16224	0.285192
$214$	0	0
$215$	−2.97782	−0.203086
$216$	0	0
$217$	9.90766	0.672576
$218$	0	0
$219$	1.96745	0.132948
$220$	0	0
$221$	−12.1060	−0.814340
$222$	0	0
$223$	3.98714	0.266998	0.133499	−	0.991049i	$-0.457379\pi$
$223$	3.98714	0.266998	0.133499	+	0.991049i	$0.457379\pi$
$224$	0	0
$225$	−2.91216	−0.194144
$226$	0	0
$227$	5.40375	0.358660	0.179330	−	0.983789i	$-0.442607\pi$
$227$	5.40375	0.358660	0.179330	+	0.983789i	$0.442607\pi$
$228$	0	0
$229$	12.4548	0.823036	0.411518	−	0.911402i	$-0.364999\pi$
$229$	12.4548	0.823036	0.411518	+	0.911402i	$0.364999\pi$
$230$	0	0
$231$	0.367971	0.0242107
$232$	0	0
$233$	16.6042	1.08778	0.543889	−	0.839157i	$-0.316951\pi$
$233$	16.6042	1.08778	0.543889	+	0.839157i	$0.316951\pi$
$234$	0	0
$235$	−4.32303	−0.282003
$236$	0	0
$237$	−1.26642	−0.0822625
$238$	0	0
$239$	3.81234	0.246600	0.123300	−	0.992369i	$-0.460652\pi$
$239$	3.81234	0.246600	0.123300	+	0.992369i	$0.460652\pi$
$240$	0	0
$241$	−9.54985	−0.615160	−0.307580	−	0.951522i	$-0.599519\pi$
$241$	−9.54985	−0.615160	−0.307580	+	0.951522i	$0.599519\pi$
$242$	0	0
$243$	−7.69210	−0.493448
$244$	0	0
$245$	3.98295	0.254462
$246$	0	0
$247$	−12.1563	−0.773487
$248$	0	0
$249$	−3.83718	−0.243171
$250$	0	0
$251$	16.9611	1.07057	0.535287	−	0.844670i	$-0.320203\pi$
$251$	16.9611	1.07057	0.535287	+	0.844670i	$0.320203\pi$
$252$	0	0
$253$	6.32618	0.397724
$254$	0	0
$255$	−1.34415	−0.0841738
$256$	0	0
$257$	18.8752	1.17740	0.588702	−	0.808350i	$-0.299639\pi$
$257$	18.8752	1.17740	0.588702	+	0.808350i	$0.299639\pi$
$258$	0	0
$259$	13.1584	0.817624
$260$	0	0
$261$	10.0513	0.622163
$262$	0	0
$263$	−23.1398	−1.42686	−0.713429	−	0.700727i	$-0.752859\pi$
$263$	−23.1398	−1.42686	−0.713429	+	0.700727i	$0.752859\pi$
$264$	0	0
$265$	1.94396	0.119416
$266$	0	0
$267$	0.959750	0.0587358
$268$	0	0
$269$	15.0428	0.917173	0.458586	−	0.888650i	$-0.348356\pi$
$269$	15.0428	0.917173	0.458586	+	0.888650i	$0.348356\pi$
$270$	0	0
$271$	19.9763	1.21348	0.606738	−	0.794902i	$-0.292478\pi$
$271$	19.9763	1.21348	0.606738	+	0.794902i	$0.292478\pi$
$272$	0	0
$273$	−1.37417	−0.0831683
$274$	0	0
$275$	−0.714786	−0.0431032
$276$	0	0
$277$	−22.8443	−1.37258	−0.686292	−	0.727326i	$-0.740763\pi$
$277$	−22.8443	−1.37258	−0.686292	+	0.727326i	$0.740763\pi$
$278$	0	0
$279$	−16.6110	−0.994474
$280$	0	0
$281$	9.43520	0.562857	0.281429	−	0.959582i	$-0.409192\pi$
$281$	9.43520	0.562857	0.281429	+	0.959582i	$0.409192\pi$
$282$	0	0
$283$	12.3219	0.732458	0.366229	−	0.930525i	$-0.380649\pi$
$283$	12.3219	0.732458	0.366229	+	0.930525i	$0.380649\pi$
$284$	0	0
$285$	−1.34973	−0.0799511
$286$	0	0
$287$	17.4292	1.02881
$288$	0	0
$289$	3.56843	0.209908
$290$	0	0
$291$	−0.575356	−0.0337280
$292$	0	0
$293$	15.7041	0.917441	0.458721	−	0.888581i	$-0.348308\pi$
$293$	15.7041	0.917441	0.458721	+	0.888581i	$0.348308\pi$
$294$	0	0
$295$	−9.39236	−0.546844
$296$	0	0
$297$	−1.25247	−0.0726759
$298$	0	0
$299$	−23.6247	−1.36625
$300$	0	0
$301$	5.17237	0.298130
$302$	0	0
$303$	4.33372	0.248966
$304$	0	0
$305$	−7.44171	−0.426111
$306$	0	0
$307$	−4.24057	−0.242022	−0.121011	−	0.992651i	$-0.538614\pi$
$307$	−4.24057	−0.242022	−0.121011	+	0.992651i	$0.538614\pi$
$308$	0	0
$309$	1.47123	0.0836951
$310$	0	0
$311$	−9.06099	−0.513802	−0.256901	−	0.966438i	$-0.582701\pi$
$311$	−9.06099	−0.513802	−0.256901	+	0.966438i	$0.582701\pi$
$312$	0	0
$313$	−19.5699	−1.10616	−0.553078	−	0.833129i	$-0.686547\pi$
$313$	−19.5699	−1.10616	−0.553078	+	0.833129i	$0.686547\pi$
$314$	0	0
$315$	5.05832	0.285004
$316$	0	0
$317$	−15.7005	−0.881827	−0.440914	−	0.897549i	$-0.645346\pi$
$317$	−15.7005	−0.881827	−0.440914	+	0.897549i	$0.645346\pi$
$318$	0	0
$319$	2.46709	0.138131
$320$	0	0
$321$	−1.15109	−0.0642478
$322$	0	0
$323$	20.6538	1.14921
$324$	0	0
$325$	2.66933	0.148068
$326$	0	0
$327$	2.93382	0.162241
$328$	0	0
$329$	7.50894	0.413981
$330$	0	0
$331$	11.5219	0.633299	0.316649	−	0.948543i	$-0.397442\pi$
$331$	11.5219	0.633299	0.316649	+	0.948543i	$0.397442\pi$
$332$	0	0
$333$	−22.0611	−1.20894
$334$	0	0
$335$	−14.9309	−0.815765
$336$	0	0
$337$	25.1380	1.36935	0.684677	−	0.728847i	$-0.259943\pi$
$337$	25.1380	1.36935	0.684677	+	0.728847i	$0.259943\pi$
$338$	0	0
$339$	−1.93593	−0.105145
$340$	0	0
$341$	−4.07715	−0.220790
$342$	0	0
$343$	−19.0770	−1.03006
$344$	0	0
$345$	−2.62308	−0.141222
$346$	0	0
$347$	10.4188	0.559309	0.279655	−	0.960101i	$-0.409780\pi$
$347$	10.4188	0.559309	0.279655	+	0.960101i	$0.409780\pi$
$348$	0	0
$349$	−4.61008	−0.246772	−0.123386	−	0.992359i	$-0.539375\pi$
$349$	−4.61008	−0.246772	−0.123386	+	0.992359i	$0.539375\pi$
$350$	0	0
$351$	4.67729	0.249655
$352$	0	0
$353$	0.502832	0.0267630	0.0133815	−	0.999910i	$-0.495740\pi$
$353$	0.502832	0.0267630	0.0133815	+	0.999910i	$0.495740\pi$
$354$	0	0
$355$	14.0437	0.745360
$356$	0	0
$357$	2.33474	0.123567
$358$	0	0
$359$	−5.95161	−0.314114	−0.157057	−	0.987590i	$-0.550201\pi$
$359$	−5.95161	−0.314114	−0.157057	+	0.987590i	$0.550201\pi$
$360$	0	0
$361$	1.73958	0.0915571
$362$	0	0
$363$	3.10874	0.163166
$364$	0	0
$365$	6.63830	0.347465
$366$	0	0
$367$	1.95365	0.101980	0.0509898	−	0.998699i	$-0.483762\pi$
$367$	1.95365	0.101980	0.0509898	+	0.998699i	$0.483762\pi$
$368$	0	0
$369$	−29.2214	−1.52121
$370$	0	0
$371$	−3.37658	−0.175303
$372$	0	0
$373$	−26.3764	−1.36572	−0.682859	−	0.730551i	$-0.739264\pi$
$373$	−26.3764	−1.36572	−0.682859	+	0.730551i	$0.739264\pi$
$374$	0	0
$375$	0.296378	0.0153049
$376$	0	0
$377$	−9.21320	−0.474504
$378$	0	0
$379$	−5.44674	−0.279780	−0.139890	−	0.990167i	$-0.544675\pi$
$379$	−5.44674	−0.279780	−0.139890	+	0.990167i	$0.544675\pi$
$380$	0	0
$381$	−0.743498	−0.0380906
$382$	0	0
$383$	−2.29258	−0.117145	−0.0585726	−	0.998283i	$-0.518655\pi$
$383$	−2.29258	−0.117145	−0.0585726	+	0.998283i	$0.518655\pi$
$384$	0	0
$385$	1.24156	0.0632757
$386$	0	0
$387$	−8.67189	−0.440817
$388$	0	0
$389$	−6.93671	−0.351705	−0.175853	−	0.984417i	$-0.556268\pi$
$389$	−6.93671	−0.351705	−0.175853	+	0.984417i	$0.556268\pi$
$390$	0	0
$391$	40.1389	2.02991
$392$	0	0
$393$	3.58695	0.180938
$394$	0	0
$395$	−4.27297	−0.214996
$396$	0	0
$397$	15.3606	0.770925	0.385462	−	0.922724i	$-0.374042\pi$
$397$	15.3606	0.770925	0.385462	+	0.922724i	$0.374042\pi$
$398$	0	0
$399$	2.34443	0.117368
$400$	0	0
$401$	7.10783	0.354948	0.177474	−	0.984125i	$-0.443207\pi$
$401$	7.10783	0.354948	0.177474	+	0.984125i	$0.443207\pi$
$402$	0	0
$403$	15.2259	0.758454
$404$	0	0
$405$	−8.21715	−0.408314
$406$	0	0
$407$	−5.41487	−0.268405
$408$	0	0
$409$	29.1697	1.44235	0.721176	−	0.692752i	$-0.243602\pi$
$409$	29.1697	1.44235	0.721176	+	0.692752i	$0.243602\pi$
$410$	0	0
$411$	1.91796	0.0946058
$412$	0	0
$413$	16.3142	0.802769
$414$	0	0
$415$	−12.9469	−0.635538
$416$	0	0
$417$	−6.89198	−0.337502
$418$	0	0
$419$	−4.33621	−0.211837	−0.105919	−	0.994375i	$-0.533778\pi$
$419$	−4.33621	−0.211837	−0.105919	+	0.994375i	$0.533778\pi$
$420$	0	0
$421$	0.752703	0.0366845	0.0183423	−	0.999832i	$-0.494161\pi$
$421$	0.752703	0.0366845	0.0183423	+	0.999832i	$0.494161\pi$
$422$	0	0
$423$	−12.5893	−0.612115
$424$	0	0
$425$	−4.53524	−0.219992
$426$	0	0
$427$	12.9260	0.625532
$428$	0	0
$429$	0.565489	0.0273021
$430$	0	0
$431$	16.7237	0.805555	0.402777	−	0.915298i	$-0.368045\pi$
$431$	16.7237	0.805555	0.402777	+	0.915298i	$0.368045\pi$
$432$	0	0
$433$	−28.3675	−1.36326	−0.681628	−	0.731699i	$-0.738728\pi$
$433$	−28.3675	−1.36326	−0.681628	+	0.731699i	$0.738728\pi$
$434$	0	0
$435$	−1.02295	−0.0490469
$436$	0	0
$437$	40.3056	1.92808
$438$	0	0
$439$	13.5018	0.644405	0.322203	−	0.946671i	$-0.395577\pi$
$439$	13.5018	0.644405	0.322203	+	0.946671i	$0.395577\pi$
$440$	0	0
$441$	11.5990	0.552333
$442$	0	0
$443$	−13.5092	−0.641842	−0.320921	−	0.947106i	$-0.603992\pi$
$443$	−13.5092	−0.641842	−0.320921	+	0.947106i	$0.603992\pi$
$444$	0	0
$445$	3.23826	0.153508
$446$	0	0
$447$	1.14343	0.0540824
$448$	0	0
$449$	−9.35573	−0.441524	−0.220762	−	0.975328i	$-0.570854\pi$
$449$	−9.35573	−0.441524	−0.220762	+	0.975328i	$0.570854\pi$
$450$	0	0
$451$	−7.17236	−0.337733
$452$	0	0
$453$	−3.41417	−0.160411
$454$	0	0
$455$	−4.63652	−0.217364
$456$	0	0
$457$	−6.84779	−0.320326	−0.160163	−	0.987091i	$-0.551202\pi$
$457$	−6.84779	−0.320326	−0.160163	+	0.987091i	$0.551202\pi$
$458$	0	0
$459$	−7.94682	−0.370926
$460$	0	0
$461$	16.6033	0.773293	0.386646	−	0.922228i	$-0.373633\pi$
$461$	16.6033	0.773293	0.386646	+	0.922228i	$0.373633\pi$
$462$	0	0
$463$	−26.6096	−1.23665	−0.618326	−	0.785922i	$-0.712189\pi$
$463$	−26.6096	−1.23665	−0.618326	+	0.785922i	$0.712189\pi$
$464$	0	0
$465$	1.69055	0.0783972
$466$	0	0
$467$	2.08714	0.0965813	0.0482907	−	0.998833i	$-0.484623\pi$
$467$	2.08714	0.0965813	0.0482907	+	0.998833i	$0.484623\pi$
$468$	0	0
$469$	25.9345	1.19755
$470$	0	0
$471$	−1.37484	−0.0633494
$472$	0	0
$473$	−2.12851	−0.0978688
$474$	0	0
$475$	−4.55407	−0.208955
$476$	0	0
$477$	5.66111	0.259204
$478$	0	0
$479$	2.78600	0.127296	0.0636479	−	0.997972i	$-0.479727\pi$
$479$	2.78600	0.127296	0.0636479	+	0.997972i	$0.479727\pi$
$480$	0	0
$481$	20.2215	0.922022
$482$	0	0
$483$	4.55620	0.207314
$484$	0	0
$485$	−1.94129	−0.0881494
$486$	0	0
$487$	−16.9499	−0.768073	−0.384036	−	0.923318i	$-0.625466\pi$
$487$	−16.9499	−0.768073	−0.384036	+	0.923318i	$0.625466\pi$
$488$	0	0
$489$	−3.88699	−0.175776
$490$	0	0
$491$	32.2992	1.45764	0.728822	−	0.684703i	$-0.240068\pi$
$491$	32.2992	1.45764	0.728822	+	0.684703i	$0.240068\pi$
$492$	0	0
$493$	15.6534	0.704995
$494$	0	0
$495$	−2.08157	−0.0935597
$496$	0	0
$497$	−24.3933	−1.09419
$498$	0	0
$499$	3.30793	0.148083	0.0740417	−	0.997255i	$-0.476410\pi$
$499$	3.30793	0.148083	0.0740417	+	0.997255i	$0.476410\pi$
$500$	0	0
$501$	−2.09855	−0.0937564
$502$	0	0
$503$	−1.58801	−0.0708057	−0.0354029	−	0.999373i	$-0.511271\pi$
$503$	−1.58801	−0.0708057	−0.0354029	+	0.999373i	$0.511271\pi$
$504$	0	0
$505$	14.6223	0.650682
$506$	0	0
$507$	1.74114	0.0773265
$508$	0	0
$509$	−5.11398	−0.226673	−0.113337	−	0.993557i	$-0.536154\pi$
$509$	−5.11398	−0.226673	−0.113337	+	0.993557i	$0.536154\pi$
$510$	0	0
$511$	−11.5305	−0.510079
$512$	0	0
$513$	−7.97981	−0.352317
$514$	0	0
$515$	4.96401	0.218740
$516$	0	0
$517$	−3.09004	−0.135900
$518$	0	0
$519$	−2.18571	−0.0959419
$520$	0	0
$521$	−8.93031	−0.391244	−0.195622	−	0.980679i	$-0.562673\pi$
$521$	−8.93031	−0.391244	−0.195622	+	0.980679i	$0.562673\pi$
$522$	0	0
$523$	−21.2633	−0.929781	−0.464891	−	0.885368i	$-0.653906\pi$
$523$	−21.2633	−0.929781	−0.464891	+	0.885368i	$0.653906\pi$
$524$	0	0
$525$	−0.514799	−0.0224677
$526$	0	0
$527$	−25.8691	−1.12687
$528$	0	0
$529$	55.3305	2.40567
$530$	0	0
$531$	−27.3520	−1.18698
$532$	0	0
$533$	26.7847	1.16018
$534$	0	0
$535$	−3.88387	−0.167914
$536$	0	0
$537$	−2.64980	−0.114347
$538$	0	0
$539$	2.84696	0.122627
$540$	0	0
$541$	7.88552	0.339025	0.169513	−	0.985528i	$-0.445781\pi$
$541$	7.88552	0.339025	0.169513	+	0.985528i	$0.445781\pi$
$542$	0	0
$543$	−5.47798	−0.235083
$544$	0	0
$545$	9.89891	0.424023
$546$	0	0
$547$	−46.4380	−1.98555	−0.992773	−	0.120010i	$-0.961707\pi$
$547$	−46.4380	−1.98555	−0.992773	+	0.120010i	$0.961707\pi$
$548$	0	0
$549$	−21.6714	−0.924915
$550$	0	0
$551$	15.7184	0.669628
$552$	0	0
$553$	7.42199	0.315615
$554$	0	0
$555$	2.24522	0.0953043
$556$	0	0
$557$	−34.2349	−1.45058	−0.725289	−	0.688444i	$-0.758294\pi$
$557$	−34.2349	−1.45058	−0.725289	+	0.688444i	$0.758294\pi$
$558$	0	0
$559$	7.94877	0.336197
$560$	0	0
$561$	−0.960778	−0.0405641
$562$	0	0
$563$	31.5945	1.33155	0.665774	−	0.746153i	$-0.268101\pi$
$563$	31.5945	1.33155	0.665774	+	0.746153i	$0.268101\pi$
$564$	0	0
$565$	−6.53194	−0.274801
$566$	0	0
$567$	14.2729	0.599406
$568$	0	0
$569$	−29.3339	−1.22974	−0.614870	−	0.788629i	$-0.710791\pi$
$569$	−29.3339	−1.22974	−0.614870	+	0.788629i	$0.710791\pi$
$570$	0	0
$571$	33.9318	1.42000	0.710000	−	0.704201i	$-0.248695\pi$
$571$	33.9318	1.42000	0.710000	+	0.704201i	$0.248695\pi$
$572$	0	0
$573$	−6.57138	−0.274523
$574$	0	0
$575$	−8.85045	−0.369089
$576$	0	0
$577$	−31.9232	−1.32898	−0.664490	−	0.747297i	$-0.731351\pi$
$577$	−31.9232	−1.32898	−0.664490	+	0.747297i	$0.731351\pi$
$578$	0	0
$579$	−2.36420	−0.0982526
$580$	0	0
$581$	22.4883	0.932972
$582$	0	0
$583$	1.38951	0.0575477
$584$	0	0
$585$	7.77350	0.321395
$586$	0	0
$587$	37.1721	1.53426	0.767129	−	0.641493i	$-0.221685\pi$
$587$	37.1721	1.53426	0.767129	+	0.641493i	$0.221685\pi$
$588$	0	0
$589$	−25.9765	−1.07034
$590$	0	0
$591$	2.41563	0.0993658
$592$	0	0
$593$	38.2085	1.56904	0.784518	−	0.620106i	$-0.212910\pi$
$593$	38.2085	1.56904	0.784518	+	0.620106i	$0.212910\pi$
$594$	0	0
$595$	7.87756	0.322948
$596$	0	0
$597$	1.59709	0.0653646
$598$	0	0
$599$	25.1150	1.02617	0.513086	−	0.858337i	$-0.328502\pi$
$599$	25.1150	1.02617	0.513086	+	0.858337i	$0.328502\pi$
$600$	0	0
$601$	22.2022	0.905647	0.452823	−	0.891600i	$-0.350417\pi$
$601$	22.2022	0.905647	0.452823	+	0.891600i	$0.350417\pi$
$602$	0	0
$603$	−43.4813	−1.77070
$604$	0	0
$605$	10.4891	0.426442
$606$	0	0
$607$	−12.9648	−0.526226	−0.263113	−	0.964765i	$-0.584749\pi$
$607$	−12.9648	−0.526226	−0.263113	+	0.964765i	$0.584749\pi$
$608$	0	0
$609$	1.77683	0.0720009
$610$	0	0
$611$	11.5396	0.466841
$612$	0	0
$613$	−10.5048	−0.424287	−0.212143	−	0.977239i	$-0.568044\pi$
$613$	−10.5048	−0.424287	−0.212143	+	0.977239i	$0.568044\pi$
$614$	0	0
$615$	2.97394	0.119921
$616$	0	0
$617$	23.2743	0.936989	0.468494	−	0.883467i	$-0.344797\pi$
$617$	23.2743	0.936989	0.468494	+	0.883467i	$0.344797\pi$
$618$	0	0
$619$	44.7193	1.79742	0.898710	−	0.438544i	$-0.144506\pi$
$619$	44.7193	1.79742	0.898710	+	0.438544i	$0.144506\pi$
$620$	0	0
$621$	−15.5081	−0.622318
$622$	0	0
$623$	−5.62474	−0.225351
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.964768	−0.0385291
$628$	0	0
$629$	−34.3568	−1.36990
$630$	0	0
$631$	29.9258	1.19133	0.595663	−	0.803234i	$-0.296889\pi$
$631$	29.9258	1.19133	0.595663	+	0.803234i	$0.296889\pi$
$632$	0	0
$633$	−4.50775	−0.179167
$634$	0	0
$635$	−2.50861	−0.0995512
$636$	0	0
$637$	−10.6318	−0.421247
$638$	0	0
$639$	40.8974	1.61788
$640$	0	0
$641$	10.2240	0.403825	0.201912	−	0.979404i	$-0.435284\pi$
$641$	10.2240	0.403825	0.201912	+	0.979404i	$0.435284\pi$
$642$	0	0
$643$	19.4034	0.765194	0.382597	−	0.923915i	$-0.375030\pi$
$643$	19.4034	0.765194	0.382597	+	0.923915i	$0.375030\pi$
$644$	0	0
$645$	0.882562	0.0347508
$646$	0	0
$647$	−18.6767	−0.734255	−0.367128	−	0.930171i	$-0.619659\pi$
$647$	−18.6767	−0.734255	−0.367128	+	0.930171i	$0.619659\pi$
$648$	0	0
$649$	−6.71353	−0.263529
$650$	0	0
$651$	−2.93642	−0.115087
$652$	0	0
$653$	−18.0927	−0.708022	−0.354011	−	0.935241i	$-0.615183\pi$
$653$	−18.0927	−0.708022	−0.354011	+	0.935241i	$0.615183\pi$
$654$	0	0
$655$	12.1026	0.472888
$656$	0	0
$657$	19.3318	0.754205
$658$	0	0
$659$	17.4222	0.678671	0.339336	−	0.940665i	$-0.389798\pi$
$659$	17.4222	0.678671	0.339336	+	0.940665i	$0.389798\pi$
$660$	0	0
$661$	−33.9910	−1.32210	−0.661048	−	0.750344i	$-0.729888\pi$
$661$	−33.9910	−1.32210	−0.661048	+	0.750344i	$0.729888\pi$
$662$	0	0
$663$	3.58797	0.139345
$664$	0	0
$665$	7.91026	0.306747
$666$	0	0
$667$	30.5474	1.18280
$668$	0	0
$669$	−1.18170	−0.0456872
$670$	0	0
$671$	−5.31923	−0.205347
$672$	0	0
$673$	−21.5360	−0.830150	−0.415075	−	0.909787i	$-0.636245\pi$
$673$	−21.5360	−0.830150	−0.415075	+	0.909787i	$0.636245\pi$
$674$	0	0
$675$	1.75224	0.0674436
$676$	0	0
$677$	−18.6467	−0.716652	−0.358326	−	0.933596i	$-0.616652\pi$
$677$	−18.6467	−0.716652	−0.358326	+	0.933596i	$0.616652\pi$
$678$	0	0
$679$	3.37195	0.129404
$680$	0	0
$681$	−1.60156	−0.0613717
$682$	0	0
$683$	−43.2765	−1.65593	−0.827964	−	0.560781i	$-0.810501\pi$
$683$	−43.2765	−1.65593	−0.827964	+	0.560781i	$0.810501\pi$
$684$	0	0
$685$	6.47131	0.247256
$686$	0	0
$687$	−3.69133	−0.140833
$688$	0	0
$689$	−5.18905	−0.197687
$690$	0	0
$691$	−35.7336	−1.35937	−0.679685	−	0.733504i	$-0.737884\pi$
$691$	−35.7336	−1.35937	−0.679685	+	0.733504i	$0.737884\pi$
$692$	0	0
$693$	3.61562	0.137346
$694$	0	0
$695$	−23.2540	−0.882074
$696$	0	0
$697$	−45.5079	−1.72373
$698$	0	0
$699$	−4.92113	−0.186134
$700$	0	0
$701$	26.2454	0.991275	0.495637	−	0.868530i	$-0.334935\pi$
$701$	26.2454	0.991275	0.495637	+	0.868530i	$0.334935\pi$
$702$	0	0
$703$	−34.4995	−1.30117
$704$	0	0
$705$	1.28125	0.0482547
$706$	0	0
$707$	−25.3983	−0.955203
$708$	0	0
$709$	−6.19558	−0.232680	−0.116340	−	0.993209i	$-0.537116\pi$
$709$	−6.19558	−0.232680	−0.116340	+	0.993209i	$0.537116\pi$
$710$	0	0
$711$	−12.4436	−0.466670
$712$	0	0
$713$	−50.4831	−1.89061
$714$	0	0
$715$	1.90800	0.0713550
$716$	0	0
$717$	−1.12990	−0.0421967
$718$	0	0
$719$	1.61691	0.0603007	0.0301503	−	0.999545i	$-0.490401\pi$
$719$	1.61691	0.0603007	0.0301503	+	0.999545i	$0.490401\pi$
$720$	0	0
$721$	−8.62231	−0.321112
$722$	0	0
$723$	2.83037	0.105263
$724$	0	0
$725$	−3.45151	−0.128186
$726$	0	0
$727$	39.3600	1.45978	0.729891	−	0.683563i	$-0.239571\pi$
$727$	39.3600	1.45978	0.729891	+	0.683563i	$0.239571\pi$
$728$	0	0
$729$	−22.3717	−0.828581
$730$	0	0
$731$	−13.5051	−0.499506
$732$	0	0
$733$	−48.1946	−1.78011	−0.890055	−	0.455854i	$-0.849334\pi$
$733$	−48.1946	−1.78011	−0.890055	+	0.455854i	$0.849334\pi$
$734$	0	0
$735$	−1.18046	−0.0435420
$736$	0	0
$737$	−10.6724	−0.393124
$738$	0	0
$739$	−21.8182	−0.802595	−0.401298	−	0.915948i	$-0.631441\pi$
$739$	−21.8182	−0.802595	−0.401298	+	0.915948i	$0.631441\pi$
$740$	0	0
$741$	3.60287	0.132355
$742$	0	0
$743$	−23.5004	−0.862147	−0.431074	−	0.902317i	$-0.641865\pi$
$743$	−23.5004	−0.862147	−0.431074	+	0.902317i	$0.641865\pi$
$744$	0	0
$745$	3.85801	0.141347
$746$	0	0
$747$	−37.7034	−1.37950
$748$	0	0
$749$	6.74614	0.246499
$750$	0	0
$751$	10.8586	0.396236	0.198118	−	0.980178i	$-0.436517\pi$
$751$	10.8586	0.396236	0.198118	+	0.980178i	$0.436517\pi$
$752$	0	0
$753$	−5.02690	−0.183190
$754$	0	0
$755$	−11.5196	−0.419242
$756$	0	0
$757$	26.6486	0.968558	0.484279	−	0.874914i	$-0.339082\pi$
$757$	26.6486	0.968558	0.484279	+	0.874914i	$0.339082\pi$
$758$	0	0
$759$	−1.87494	−0.0680561
$760$	0	0
$761$	−22.2837	−0.807783	−0.403891	−	0.914807i	$-0.632343\pi$
$761$	−22.2837	−0.807783	−0.403891	+	0.914807i	$0.632343\pi$
$762$	0	0
$763$	−17.1941	−0.622467
$764$	0	0
$765$	−13.2074	−0.477513
$766$	0	0
$767$	25.0713	0.905271
$768$	0	0
$769$	10.5399	0.380077	0.190039	−	0.981777i	$-0.439139\pi$
$769$	10.5399	0.380077	0.190039	+	0.981777i	$0.439139\pi$
$770$	0	0
$771$	−5.59421	−0.201471
$772$	0	0
$773$	−5.76671	−0.207414	−0.103707	−	0.994608i	$-0.533070\pi$
$773$	−5.76671	−0.207414	−0.103707	+	0.994608i	$0.533070\pi$
$774$	0	0
$775$	5.70401	0.204894
$776$	0	0
$777$	−3.89987	−0.139907
$778$	0	0
$779$	−45.6968	−1.63726
$780$	0	0
$781$	10.0382	0.359196
$782$	0	0
$783$	−6.04786	−0.216133
$784$	0	0
$785$	−4.63881	−0.165566
$786$	0	0
$787$	−11.5494	−0.411693	−0.205847	−	0.978584i	$-0.565995\pi$
$787$	−11.5494	−0.411693	−0.205847	+	0.978584i	$0.565995\pi$
$788$	0	0
$789$	6.85812	0.244156
$790$	0	0
$791$	11.3458	0.403409
$792$	0	0
$793$	19.8643	0.705403
$794$	0	0
$795$	−0.576147	−0.0204338
$796$	0	0
$797$	25.4518	0.901548	0.450774	−	0.892638i	$-0.351148\pi$
$797$	25.4518	0.901548	0.450774	+	0.892638i	$0.351148\pi$
$798$	0	0
$799$	−19.6060	−0.693609
$800$	0	0
$801$	9.43033	0.333204
$802$	0	0
$803$	4.74496	0.167446
$804$	0	0
$805$	15.3729	0.541824
$806$	0	0
$807$	−4.45835	−0.156941
$808$	0	0
$809$	42.0296	1.47768	0.738841	−	0.673879i	$-0.235373\pi$
$809$	42.0296	1.47768	0.738841	+	0.673879i	$0.235373\pi$
$810$	0	0
$811$	−26.5308	−0.931623	−0.465812	−	0.884884i	$-0.654238\pi$
$811$	−26.5308	−0.931623	−0.465812	+	0.884884i	$0.654238\pi$
$812$	0	0
$813$	−5.92056	−0.207643
$814$	0	0
$815$	−13.1149	−0.459397
$816$	0	0
$817$	−13.5612	−0.474447
$818$	0	0
$819$	−13.5023	−0.471809
$820$	0	0
$821$	−30.2712	−1.05647	−0.528236	−	0.849098i	$-0.677146\pi$
$821$	−30.2712	−1.05647	−0.528236	+	0.849098i	$0.677146\pi$
$822$	0	0
$823$	−43.7323	−1.52441	−0.762206	−	0.647334i	$-0.775884\pi$
$823$	−43.7323	−1.52441	−0.762206	+	0.647334i	$0.775884\pi$
$824$	0	0
$825$	0.211847	0.00737557
$826$	0	0
$827$	−28.2306	−0.981675	−0.490838	−	0.871251i	$-0.663309\pi$
$827$	−28.2306	−0.981675	−0.490838	+	0.871251i	$0.663309\pi$
$828$	0	0
$829$	44.3878	1.54165	0.770826	−	0.637046i	$-0.219844\pi$
$829$	44.3878	1.54165	0.770826	+	0.637046i	$0.219844\pi$
$830$	0	0
$831$	6.77057	0.234868
$832$	0	0
$833$	18.0637	0.625869
$834$	0	0
$835$	−7.08065	−0.245036
$836$	0	0
$837$	9.99477	0.345470
$838$	0	0
$839$	54.5335	1.88271	0.941353	−	0.337423i	$-0.109555\pi$
$839$	54.5335	1.88271	0.941353	+	0.337423i	$0.109555\pi$
$840$	0	0
$841$	−17.0871	−0.589210
$842$	0	0
$843$	−2.79639	−0.0963128
$844$	0	0
$845$	5.87470	0.202096
$846$	0	0
$847$	−18.2192	−0.626018
$848$	0	0
$849$	−3.65193	−0.125334
$850$	0	0
$851$	−67.0467	−2.29833
$852$	0	0
$853$	30.5373	1.04558	0.522789	−	0.852462i	$-0.324891\pi$
$853$	30.5373	1.04558	0.522789	+	0.852462i	$0.324891\pi$
$854$	0	0
$855$	−13.2622	−0.453557
$856$	0	0
$857$	41.3609	1.41286	0.706431	−	0.707782i	$-0.250304\pi$
$857$	41.3609	1.41286	0.706431	+	0.707782i	$0.250304\pi$
$858$	0	0
$859$	−0.990792	−0.0338054	−0.0169027	−	0.999857i	$-0.505381\pi$
$859$	−0.990792	−0.0338054	−0.0169027	+	0.999857i	$0.505381\pi$
$860$	0	0
$861$	−5.16563	−0.176044
$862$	0	0
$863$	−55.0780	−1.87488	−0.937439	−	0.348150i	$-0.886810\pi$
$863$	−55.0780	−1.87488	−0.937439	+	0.348150i	$0.886810\pi$
$864$	0	0
$865$	−7.37471	−0.250748
$866$	0	0
$867$	−1.05761	−0.0359182
$868$	0	0
$869$	−3.05426	−0.103609
$870$	0	0
$871$	39.8556	1.35045
$872$	0	0
$873$	−5.65335	−0.191337
$874$	0	0
$875$	−1.73696	−0.0587201
$876$	0	0
$877$	51.6329	1.74352	0.871760	−	0.489933i	$-0.162979\pi$
$877$	51.6329	1.74352	0.871760	+	0.489933i	$0.162979\pi$
$878$	0	0
$879$	−4.65435	−0.156987
$880$	0	0
$881$	−54.3503	−1.83111	−0.915554	−	0.402196i	$-0.868247\pi$
$881$	−54.3503	−1.83111	−0.915554	+	0.402196i	$0.868247\pi$
$882$	0	0
$883$	50.2720	1.69179	0.845893	−	0.533353i	$-0.179068\pi$
$883$	50.2720	1.69179	0.845893	+	0.533353i	$0.179068\pi$
$884$	0	0
$885$	2.78369	0.0935728
$886$	0	0
$887$	0.817003	0.0274323	0.0137161	−	0.999906i	$-0.495634\pi$
$887$	0.817003	0.0274323	0.0137161	+	0.999906i	$0.495634\pi$
$888$	0	0
$889$	4.35737	0.146141
$890$	0	0
$891$	−5.87351	−0.196770
$892$	0	0
$893$	−19.6874	−0.658813
$894$	0	0
$895$	−8.94060	−0.298851
$896$	0	0
$897$	7.00186	0.233785
$898$	0	0
$899$	−19.6874	−0.656613
$900$	0	0
$901$	8.81631	0.293714
$902$	0	0
$903$	−1.53298	−0.0510144
$904$	0	0
$905$	−18.4831	−0.614398
$906$	0	0
$907$	4.75400	0.157854	0.0789271	−	0.996880i	$-0.474851\pi$
$907$	4.75400	0.157854	0.0789271	+	0.996880i	$0.474851\pi$
$908$	0	0
$909$	42.5824	1.41237
$910$	0	0
$911$	34.6568	1.14823	0.574116	−	0.818774i	$-0.305346\pi$
$911$	34.6568	1.14823	0.574116	+	0.818774i	$0.305346\pi$
$912$	0	0
$913$	−9.25426	−0.306271
$914$	0	0
$915$	2.20556	0.0729136
$916$	0	0
$917$	−21.0218	−0.694200
$918$	0	0
$919$	24.3452	0.803074	0.401537	−	0.915843i	$-0.368476\pi$
$919$	24.3452	0.803074	0.401537	+	0.915843i	$0.368476\pi$
$920$	0	0
$921$	1.25681	0.0414135
$922$	0	0
$923$	−37.4871	−1.23390
$924$	0	0
$925$	7.57552	0.249081
$926$	0	0
$927$	14.4560	0.474797
$928$	0	0
$929$	3.16600	0.103873	0.0519366	−	0.998650i	$-0.483461\pi$
$929$	3.16600	0.103873	0.0519366	+	0.998650i	$0.483461\pi$
$930$	0	0
$931$	18.1387	0.594471
$932$	0	0
$933$	2.68548	0.0879188
$934$	0	0
$935$	−3.24173	−0.106016
$936$	0	0
$937$	23.4847	0.767211	0.383606	−	0.923497i	$-0.374682\pi$
$937$	23.4847	0.767211	0.383606	+	0.923497i	$0.374682\pi$
$938$	0	0
$939$	5.80010	0.189279
$940$	0	0
$941$	−39.2562	−1.27972	−0.639858	−	0.768493i	$-0.721007\pi$
$941$	−39.2562	−1.27972	−0.639858	+	0.768493i	$0.721007\pi$
$942$	0	0
$943$	−88.8078	−2.89198
$944$	0	0
$945$	−3.04357	−0.0990074
$946$	0	0
$947$	−38.5961	−1.25421	−0.627103	−	0.778937i	$-0.715759\pi$
$947$	−38.5961	−1.25421	−0.627103	+	0.778937i	$0.715759\pi$
$948$	0	0
$949$	−17.7198	−0.575209
$950$	0	0
$951$	4.65329	0.150893
$952$	0	0
$953$	12.1516	0.393630	0.196815	−	0.980441i	$-0.436940\pi$
$953$	12.1516	0.393630	0.196815	+	0.980441i	$0.436940\pi$
$954$	0	0
$955$	−22.1722	−0.717477
$956$	0	0
$957$	−0.731193	−0.0236361
$958$	0	0
$959$	−11.2404	−0.362972
$960$	0	0
$961$	1.53571	0.0495392
$962$	0	0
$963$	−11.3104	−0.364474
$964$	0	0
$965$	−7.97695	−0.256787
$966$	0	0
$967$	48.2694	1.55224	0.776120	−	0.630585i	$-0.217185\pi$
$967$	48.2694	1.55224	0.776120	+	0.630585i	$0.217185\pi$
$968$	0	0
$969$	−6.12135	−0.196646
$970$	0	0
$971$	−8.37281	−0.268696	−0.134348	−	0.990934i	$-0.542894\pi$
$971$	−8.37281	−0.268696	−0.134348	+	0.990934i	$0.542894\pi$
$972$	0	0
$973$	40.3913	1.29489
$974$	0	0
$975$	−0.791130	−0.0253365
$976$	0	0
$977$	27.7522	0.887872	0.443936	−	0.896059i	$-0.353582\pi$
$977$	27.7522	0.887872	0.443936	+	0.896059i	$0.353582\pi$
$978$	0	0
$979$	2.31466	0.0739769
$980$	0	0
$981$	28.8272	0.920382
$982$	0	0
$983$	−28.3604	−0.904556	−0.452278	−	0.891877i	$-0.649389\pi$
$983$	−28.3604	−0.904556	−0.452278	+	0.891877i	$0.649389\pi$
$984$	0	0
$985$	8.15050	0.259697
$986$	0	0
$987$	−2.22549	−0.0708381
$988$	0	0
$989$	−26.3551	−0.838042
$990$	0	0
$991$	43.7506	1.38979	0.694893	−	0.719114i	$-0.255452\pi$
$991$	43.7506	1.38979	0.694893	+	0.719114i	$0.255452\pi$
$992$	0	0
$993$	−3.41483	−0.108366
$994$	0	0
$995$	5.38869	0.170833
$996$	0	0
$997$	−14.9301	−0.472840	−0.236420	−	0.971651i	$-0.575974\pi$
$997$	−14.9301	−0.472840	−0.236420	+	0.971651i	$0.575974\pi$
$998$	0	0
$999$	13.2741	0.419974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.u.1.3		8
4.3	odd	2		5120.2.a.s.1.6		8
8.3	odd	2		5120.2.a.v.1.3		8
8.5	even	2		5120.2.a.t.1.6		8
32.3	odd	8		80.2.l.a.61.6	yes	16
32.5	even	8		640.2.l.a.481.5		16
32.11	odd	8		80.2.l.a.21.6	✓	16
32.13	even	8		640.2.l.a.161.5		16
32.19	odd	8		640.2.l.b.161.4		16
32.21	even	8		320.2.l.a.241.4		16
32.27	odd	8		640.2.l.b.481.4		16
32.29	even	8		320.2.l.a.81.4		16
96.11	even	8		720.2.t.c.181.3		16
96.29	odd	8		2880.2.t.c.721.1		16
96.35	even	8		720.2.t.c.541.3		16
96.53	odd	8		2880.2.t.c.2161.4		16
160.3	even	8		400.2.q.g.349.7		16
160.29	even	8		1600.2.l.i.401.5		16
160.43	even	8		400.2.q.h.149.2		16
160.53	odd	8		1600.2.q.g.49.4		16
160.67	even	8		400.2.q.h.349.2		16
160.93	odd	8		1600.2.q.h.849.5		16
160.99	odd	8		400.2.l.h.301.3		16
160.107	even	8		400.2.q.g.149.7		16
160.117	odd	8		1600.2.q.h.49.5		16
160.139	odd	8		400.2.l.h.101.3		16
160.149	even	8		1600.2.l.i.1201.5		16
160.157	odd	8		1600.2.q.g.849.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.6	✓	16	32.11	odd	8
80.2.l.a.61.6	yes	16	32.3	odd	8
320.2.l.a.81.4		16	32.29	even	8
320.2.l.a.241.4		16	32.21	even	8
400.2.l.h.101.3		16	160.139	odd	8
400.2.l.h.301.3		16	160.99	odd	8
400.2.q.g.149.7		16	160.107	even	8
400.2.q.g.349.7		16	160.3	even	8
400.2.q.h.149.2		16	160.43	even	8
400.2.q.h.349.2		16	160.67	even	8
640.2.l.a.161.5		16	32.13	even	8
640.2.l.a.481.5		16	32.5	even	8
640.2.l.b.161.4		16	32.19	odd	8
640.2.l.b.481.4		16	32.27	odd	8
720.2.t.c.181.3		16	96.11	even	8
720.2.t.c.541.3		16	96.35	even	8
1600.2.l.i.401.5		16	160.29	even	8
1600.2.l.i.1201.5		16	160.149	even	8
1600.2.q.g.49.4		16	160.53	odd	8
1600.2.q.g.849.4		16	160.157	odd	8
1600.2.q.h.49.5		16	160.117	odd	8
1600.2.q.h.849.5		16	160.93	odd	8
2880.2.t.c.721.1		16	96.29	odd	8
2880.2.t.c.2161.4		16	96.53	odd	8
5120.2.a.s.1.6		8	4.3	odd	2
5120.2.a.t.1.6		8	8.5	even	2
5120.2.a.u.1.3		8	1.1	even	1	trivial
5120.2.a.v.1.3		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)

\(f(q)\)	\(=\)	\(q-0.296378 q^{3} -1.00000 q^{5} +1.73696 q^{7} -2.91216 q^{9} -0.714786 q^{11} +2.66933 q^{13} +0.296378 q^{15} -4.53524 q^{17} -4.55407 q^{19} -0.514799 q^{21} -8.85045 q^{23} +1.00000 q^{25} +1.75224 q^{27} -3.45151 q^{29} +5.70401 q^{31} +0.211847 q^{33} -1.73696 q^{35} +7.57552 q^{37} -0.791130 q^{39} +10.0343 q^{41} +2.97782 q^{43} +2.91216 q^{45} +4.32303 q^{47} -3.98295 q^{49} +1.34415 q^{51} -1.94396 q^{53} +0.714786 q^{55} +1.34973 q^{57} +9.39236 q^{59} +7.44171 q^{61} -5.05832 q^{63} -2.66933 q^{65} +14.9309 q^{67} +2.62308 q^{69} -14.0437 q^{71} -6.63830 q^{73} -0.296378 q^{75} -1.24156 q^{77} +4.27297 q^{79} +8.21715 q^{81} +12.9469 q^{83} +4.53524 q^{85} +1.02295 q^{87} -3.23826 q^{89} +4.63652 q^{91} -1.69055 q^{93} +4.55407 q^{95} +1.94129 q^{97} +2.08157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 8 q^{5} - 4 q^{7} + 8 q^{9} + 8 q^{11} - 4 q^{15} + 16 q^{19} - 12 q^{23} + 8 q^{25} + 16 q^{27} + 4 q^{35} + 28 q^{43} - 8 q^{45} - 20 q^{47} + 8 q^{49} + 24 q^{51} - 8 q^{55} + 16 q^{59}+ \cdots + 40 q^{99}+O(q^{100}) \) 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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