LMFDB - Embedded newform 5120.2.a.v.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	$-0.606461\pi$
$7$	−1.73696	−0.656511	−0.328255	+	0.944589i	$0.606461\pi$
$8$	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	$-0.620700\pi$
$13$	−2.66933	−0.740338	−0.370169	+	0.928964i	$0.620700\pi$
$14$	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	$-0.125958\pi$
$23$	8.85045	1.84545	0.922723	+	0.385463i	$0.125958\pi$
$24$	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	$-0.396161\pi$
$29$	3.45151	0.640929	0.320465	+	0.947260i	$0.396161\pi$
$30$	0	0
$31$	−5.70401	−1.02447	−0.512235	−	0.858845i	$-0.671182\pi$
$31$	−5.70401	−1.02447	−0.512235	+	0.858845i	$0.671182\pi$
$32$	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	$-0.713966\pi$
$37$	−7.57552	−1.24541	−0.622704	+	0.782458i	$0.713966\pi$
$38$	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	$-0.602101\pi$
$47$	−4.32303	−0.630578	−0.315289	+	0.948996i	$0.602101\pi$
$48$	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	$-0.457375\pi$
$53$	1.94396	0.267023	0.133511	+	0.991047i	$0.457375\pi$
$54$	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	$-0.658061\pi$
$61$	−7.44171	−0.952813	−0.476407	+	0.879225i	$0.658061\pi$
$62$	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	$-0.186427\pi$
$71$	14.0437	1.66668	0.833338	+	0.552764i	$0.186427\pi$
$72$	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	$-0.577270\pi$
$79$	−4.27297	−0.480746	−0.240373	+	0.970681i	$0.577270\pi$
$80$	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	$-0.240689\pi$
$101$	14.6223	1.45497	0.727485	+	0.686124i	$0.240689\pi$
$102$	0	0
$103$	4.96401	0.489118	0.244559	−	0.969634i	$-0.421357\pi$
$103$	4.96401	0.489118	0.244559	+	0.969634i	$0.421357\pi$
$104$	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	$-0.342784\pi$
$109$	9.89891	0.948144	0.474072	+	0.880486i	$0.342784\pi$
$110$	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	$-0.535502\pi$
$127$	−2.50861	−0.222603	−0.111302	+	0.993787i	$0.535502\pi$
$128$	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	$-0.449486\pi$
$149$	3.85801	0.316061	0.158030	+	0.987434i	$0.449486\pi$
$150$	0	0
$151$	−11.5196	−0.937453	−0.468726	−	0.883343i	$-0.655287\pi$
$151$	−11.5196	−0.937453	−0.468726	+	0.883343i	$0.655287\pi$
$152$	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	$-0.559264\pi$
$157$	−4.63881	−0.370217	−0.185109	+	0.982718i	$0.559264\pi$
$158$	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	$-0.588333\pi$
$167$	−7.08065	−0.547917	−0.273958	+	0.961742i	$0.588333\pi$
$168$	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	$-0.590449\pi$
$173$	−7.37471	−0.560689	−0.280345	+	0.959899i	$0.590449\pi$
$174$	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	$-0.741037\pi$
$181$	−18.4831	−1.37384	−0.686918	+	0.726735i	$0.741037\pi$
$182$	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	$-0.796318\pi$
$191$	−22.1722	−1.60433	−0.802164	+	0.597104i	$0.796318\pi$
$192$	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	$-0.406228\pi$
$197$	8.15050	0.580699	0.290350	+	0.956921i	$0.406228\pi$
$198$	0	0
$199$	5.38869	0.381994	0.190997	−	0.981591i	$-0.438828\pi$
$199$	5.38869	0.381994	0.190997	+	0.981591i	$0.438828\pi$
$200$	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	$-0.542621\pi$
$223$	−3.98714	−0.266998	−0.133499	+	0.991049i	$0.542621\pi$
$224$	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	$-0.635001\pi$
$229$	−12.4548	−0.823036	−0.411518	+	0.911402i	$0.635001\pi$
$230$	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	$-0.539348\pi$
$239$	−3.81234	−0.246600	−0.123300	+	0.992369i	$0.539348\pi$
$240$	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	$-0.247141\pi$
$263$	23.1398	1.42686	0.713429	+	0.700727i	$0.247141\pi$
$264$	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	$-0.651644\pi$
$269$	−15.0428	−0.917173	−0.458586	+	0.888650i	$0.651644\pi$
$270$	0	0
$271$	−19.9763	−1.21348	−0.606738	−	0.794902i	$-0.707522\pi$
$271$	−19.9763	−1.21348	−0.606738	+	0.794902i	$0.707522\pi$
$272$	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	$-0.259237\pi$
$277$	22.8443	1.37258	0.686292	+	0.727326i	$0.259237\pi$
$278$	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	$-0.651692\pi$
$293$	−15.7041	−0.917441	−0.458721	+	0.888581i	$0.651692\pi$
$294$	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	$-0.417299\pi$
$311$	9.06099	0.513802	0.256901	+	0.966438i	$0.417299\pi$
$312$	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	$-0.354654\pi$
$317$	15.7005	0.881827	0.440914	+	0.897549i	$0.354654\pi$
$318$	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	$-0.460625\pi$
$349$	4.61008	0.246772	0.123386	+	0.992359i	$0.460625\pi$
$350$	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	$-0.449799\pi$
$359$	5.95161	0.314114	0.157057	+	0.987590i	$0.449799\pi$
$360$	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	$-0.516238\pi$
$367$	−1.95365	−0.101980	−0.0509898	+	0.998699i	$0.516238\pi$
$368$	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	$-0.260736\pi$
$373$	26.3764	1.36572	0.682859	+	0.730551i	$0.260736\pi$
$374$	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	$-0.481345\pi$
$383$	2.29258	0.117145	0.0585726	+	0.998283i	$0.481345\pi$
$384$	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	$-0.443732\pi$
$389$	6.93671	0.351705	0.175853	+	0.984417i	$0.443732\pi$
$390$	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	$-0.625958\pi$
$397$	−15.3606	−0.770925	−0.385462	+	0.922724i	$0.625958\pi$
$398$	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	$-0.505839\pi$
$421$	−0.752703	−0.0366845	−0.0183423	+	0.999832i	$0.505839\pi$
$422$	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	$-0.631955\pi$
$431$	−16.7237	−0.805555	−0.402777	+	0.915298i	$0.631955\pi$
$432$	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	$-0.604423\pi$
$439$	−13.5018	−0.644405	−0.322203	+	0.946671i	$0.604423\pi$
$440$	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	$-0.626367\pi$
$461$	−16.6033	−0.773293	−0.386646	+	0.922228i	$0.626367\pi$
$462$	0	0
$463$	26.6096	1.23665	0.618326	−	0.785922i	$-0.287811\pi$
$463$	26.6096	1.23665	0.618326	+	0.785922i	$0.287811\pi$
$464$	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	$-0.520273\pi$
$479$	−2.78600	−0.127296	−0.0636479	+	0.997972i	$0.520273\pi$
$480$	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	$-0.374534\pi$
$487$	16.9499	0.768073	0.384036	+	0.923318i	$0.374534\pi$
$488$	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	$-0.488729\pi$
$503$	1.58801	0.0708057	0.0354029	+	0.999373i	$0.488729\pi$
$504$	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	$-0.463846\pi$
$509$	5.11398	0.226673	0.113337	+	0.993557i	$0.463846\pi$
$510$	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	$-0.554219\pi$
$541$	−7.88552	−0.339025	−0.169513	+	0.985528i	$0.554219\pi$
$542$	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	$-0.241706\pi$
$557$	34.2349	1.45058	0.725289	+	0.688444i	$0.241706\pi$
$558$	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	$-0.671498\pi$
$599$	−25.1150	−1.02617	−0.513086	+	0.858337i	$0.671498\pi$
$600$	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	$-0.415251\pi$
$607$	12.9648	0.526226	0.263113	+	0.964765i	$0.415251\pi$
$608$	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	$-0.431956\pi$
$613$	10.5048	0.424287	0.212143	+	0.977239i	$0.431956\pi$
$614$	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	$-0.703111\pi$
$631$	−29.9258	−1.19133	−0.595663	+	0.803234i	$0.703111\pi$
$632$	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	$-0.380341\pi$
$647$	18.6767	0.734255	0.367128	+	0.930171i	$0.380341\pi$
$648$	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	$-0.384817\pi$
$653$	18.0927	0.708022	0.354011	+	0.935241i	$0.384817\pi$
$654$	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	$-0.270112\pi$
$661$	33.9910	1.32210	0.661048	+	0.750344i	$0.270112\pi$
$662$	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	$-0.383348\pi$
$677$	18.6467	0.716652	0.358326	+	0.933596i	$0.383348\pi$
$678$	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	$-0.665065\pi$
$701$	−26.2454	−0.991275	−0.495637	+	0.868530i	$0.665065\pi$
$702$	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	$-0.462884\pi$
$709$	6.19558	0.232680	0.116340	+	0.993209i	$0.462884\pi$
$710$	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	$-0.509599\pi$
$719$	−1.61691	−0.0603007	−0.0301503	+	0.999545i	$0.509599\pi$
$720$	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	$-0.760429\pi$
$727$	−39.3600	−1.45978	−0.729891	+	0.683563i	$0.760429\pi$
$728$	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	$-0.150666\pi$
$733$	48.1946	1.78011	0.890055	+	0.455854i	$0.150666\pi$
$734$	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	$-0.358135\pi$
$743$	23.5004	0.862147	0.431074	+	0.902317i	$0.358135\pi$
$744$	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	$-0.563483\pi$
$751$	−10.8586	−0.396236	−0.198118	+	0.980178i	$0.563483\pi$
$752$	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	$-0.660918\pi$
$757$	−26.6486	−0.968558	−0.484279	+	0.874914i	$0.660918\pi$
$758$	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	$-0.466930\pi$
$773$	5.76671	0.207414	0.103707	+	0.994608i	$0.466930\pi$
$774$	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	$-0.648852\pi$
$797$	−25.4518	−0.901548	−0.450774	+	0.892638i	$0.648852\pi$
$798$	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	$-0.322854\pi$
$821$	30.2712	1.05647	0.528236	+	0.849098i	$0.322854\pi$
$822$	0	0
$823$	43.7323	1.52441	0.762206	−	0.647334i	$-0.224116\pi$
$823$	43.7323	1.52441	0.762206	+	0.647334i	$0.224116\pi$
$824$	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	$-0.780156\pi$
$829$	−44.3878	−1.54165	−0.770826	+	0.637046i	$0.780156\pi$
$830$	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	$-0.890445\pi$
$839$	−54.5335	−1.88271	−0.941353	+	0.337423i	$0.890445\pi$
$840$	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	$-0.675109\pi$
$853$	−30.5373	−1.04558	−0.522789	+	0.852462i	$0.675109\pi$
$854$	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	$-0.113190\pi$
$863$	55.0780	1.87488	0.937439	+	0.348150i	$0.113190\pi$
$864$	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	$-0.837021\pi$
$877$	−51.6329	−1.74352	−0.871760	+	0.489933i	$0.837021\pi$
$878$	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	$-0.504366\pi$
$887$	−0.817003	−0.0274323	−0.0137161	+	0.999906i	$0.504366\pi$
$888$	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	$-0.694654\pi$
$911$	−34.6568	−1.14823	−0.574116	+	0.818774i	$0.694654\pi$
$912$	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	$-0.631524\pi$
$919$	−24.3452	−0.803074	−0.401537	+	0.915843i	$0.631524\pi$
$920$	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	$-0.278993\pi$
$941$	39.2562	1.27972	0.639858	+	0.768493i	$0.278993\pi$
$942$	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	$-0.782815\pi$
$967$	−48.2694	−1.55224	−0.776120	+	0.630585i	$0.782815\pi$
$968$	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	$-0.350611\pi$
$983$	28.3604	0.904556	0.452278	+	0.891877i	$0.350611\pi$
$984$	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	$-0.744548\pi$
$991$	−43.7506	−1.38979	−0.694893	+	0.719114i	$0.744548\pi$
$992$	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	$-0.424026\pi$
$997$	14.9301	0.472840	0.236420	+	0.971651i	$0.424026\pi$
$998$	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.v.1.3		8
4.3	odd	2		5120.2.a.t.1.6		8
8.3	odd	2		5120.2.a.u.1.3		8
8.5	even	2		5120.2.a.s.1.6		8
32.3	odd	8		640.2.l.a.161.5		16
32.5	even	8		80.2.l.a.21.6	✓	16
32.11	odd	8		640.2.l.a.481.5		16
32.13	even	8		80.2.l.a.61.6	yes	16
32.19	odd	8		320.2.l.a.81.4		16
32.21	even	8		640.2.l.b.481.4		16
32.27	odd	8		320.2.l.a.241.4		16
32.29	even	8		640.2.l.b.161.4		16
96.5	odd	8		720.2.t.c.181.3		16
96.59	even	8		2880.2.t.c.2161.4		16
96.77	odd	8		720.2.t.c.541.3		16
96.83	even	8		2880.2.t.c.721.1		16
160.13	odd	8		400.2.q.g.349.7		16
160.19	odd	8		1600.2.l.i.401.5		16
160.27	even	8		1600.2.q.h.49.5		16
160.37	odd	8		400.2.q.g.149.7		16
160.59	odd	8		1600.2.l.i.1201.5		16
160.69	even	8		400.2.l.h.101.3		16
160.77	odd	8		400.2.q.h.349.2		16
160.83	even	8		1600.2.q.h.849.5		16
160.109	even	8		400.2.l.h.301.3		16
160.123	even	8		1600.2.q.g.49.4		16
160.133	odd	8		400.2.q.h.149.2		16
160.147	even	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	32.5	even	8
80.2.l.a.61.6	yes	16	32.13	even	8
320.2.l.a.81.4		16	32.19	odd	8
320.2.l.a.241.4		16	32.27	odd	8
400.2.l.h.101.3		16	160.69	even	8
400.2.l.h.301.3		16	160.109	even	8
400.2.q.g.149.7		16	160.37	odd	8
400.2.q.g.349.7		16	160.13	odd	8
400.2.q.h.149.2		16	160.133	odd	8
400.2.q.h.349.2		16	160.77	odd	8
640.2.l.a.161.5		16	32.3	odd	8
640.2.l.a.481.5		16	32.11	odd	8
640.2.l.b.161.4		16	32.29	even	8
640.2.l.b.481.4		16	32.21	even	8
720.2.t.c.181.3		16	96.5	odd	8
720.2.t.c.541.3		16	96.77	odd	8
1600.2.l.i.401.5		16	160.19	odd	8
1600.2.l.i.1201.5		16	160.59	odd	8
1600.2.q.g.49.4		16	160.123	even	8
1600.2.q.g.849.4		16	160.147	even	8
1600.2.q.h.49.5		16	160.27	even	8
1600.2.q.h.849.5		16	160.83	even	8
2880.2.t.c.721.1		16	96.83	even	8
2880.2.t.c.2161.4		16	96.59	even	8
5120.2.a.s.1.6		8	8.5	even	2
5120.2.a.t.1.6		8	4.3	odd	2
5120.2.a.u.1.3		8	8.3	odd	2
5120.2.a.v.1.3		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
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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