LMFDB - Embedded newform 1024.2.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1024,2,Mod(1,1024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1024, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1024.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,8,0,10,0,0,0,0,0,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.17668116698$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 512)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{3} +1.41421 q^{5} +4.00000 q^{7} +5.00000 q^{9} -2.82843 q^{11} +4.24264 q^{13} -4.00000 q^{15} -2.82843 q^{19} -11.3137 q^{21} +4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +4.24264 q^{29} -8.00000 q^{31} +8.00000 q^{33} +5.65685 q^{35} -1.41421 q^{37} -12.0000 q^{39} +8.00000 q^{41} +2.82843 q^{43} +7.07107 q^{45} +8.00000 q^{47} +9.00000 q^{49} +1.41421 q^{53} -4.00000 q^{55} +8.00000 q^{57} +8.48528 q^{59} -4.24264 q^{61} +20.0000 q^{63} +6.00000 q^{65} +2.82843 q^{67} -11.3137 q^{69} +12.0000 q^{71} -2.00000 q^{73} +8.48528 q^{75} -11.3137 q^{77} +1.00000 q^{81} +14.1421 q^{83} -12.0000 q^{87} +14.0000 q^{89} +16.9706 q^{91} +22.6274 q^{93} -4.00000 q^{95} -16.0000 q^{97} -14.1421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} + 10 q^{9} - 8 q^{15} + 8 q^{23} - 6 q^{25} - 16 q^{31} + 16 q^{33} - 24 q^{39} + 16 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{55} + 16 q^{57} + 40 q^{63} + 12 q^{65} + 24 q^{71} - 4 q^{73} + 2 q^{81}+ \cdots - 32 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 + 10 * q^9 - 8 * q^15 + 8 * q^23 - 6 * q^25 - 16 * q^31 + 16 * q^33 - 24 * q^39 + 16 * q^41 + 16 * q^47 + 18 * q^49 - 8 * q^55 + 16 * q^57 + 40 * q^63 + 12 * q^65 + 24 * q^71 - 4 * q^73 + 2 * q^81 - 24 * q^87 + 28 * q^89 - 8 * q^95 - 32 * q^97

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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	5.00000	1.66667
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	−4.00000	−1.03280
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−11.3137	−2.46885
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	5.65685	0.956183
$36$	0	0
$37$	−1.41421	−0.232495	−0.116248	−	0.993220i	$-0.537087\pi$
$37$	−1.41421	−0.232495	−0.116248	+	0.993220i	$0.537087\pi$
$38$	0	0
$39$	−12.0000	−1.92154
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	0	0
$45$	7.07107	1.05409
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.41421	0.194257	0.0971286	−	0.995272i	$-0.469034\pi$
$53$	1.41421	0.194257	0.0971286	+	0.995272i	$0.469034\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	−4.24264	−0.543214	−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264	−0.543214	−0.271607	+	0.962408i	$0.587555\pi$
$62$	0	0
$63$	20.0000	2.51976
$64$	0	0
$65$	6.00000	0.744208
$66$	0	0
$67$	2.82843	0.345547	0.172774	−	0.984962i	$-0.444727\pi$
$67$	2.82843	0.345547	0.172774	+	0.984962i	$0.444727\pi$
$68$	0	0
$69$	−11.3137	−1.36201
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	8.48528	0.979796
$76$	0	0
$77$	−11.3137	−1.28932
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.0000	−1.28654
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	16.9706	1.77900
$92$	0	0
$93$	22.6274	2.34635
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	−14.1421	−1.42134
$100$	0	0
$101$	−1.41421	−0.140720	−0.0703598	−	0.997522i	$-0.522415\pi$
$101$	−1.41421	−0.140720	−0.0703598	+	0.997522i	$0.522415\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−16.0000	−1.56144
$106$	0	0
$107$	−2.82843	−0.273434	−0.136717	−	0.990610i	$-0.543655\pi$
$107$	−2.82843	−0.273434	−0.136717	+	0.990610i	$0.543655\pi$
$108$	0	0
$109$	18.3848	1.76094	0.880471	−	0.474100i	$-0.157226\pi$
$109$	18.3848	1.76094	0.880471	+	0.474100i	$0.157226\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	0	0
$117$	21.2132	1.96116
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	−22.6274	−2.04025
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	14.1421	1.23560	0.617802	−	0.786334i	$-0.288023\pi$
$131$	14.1421	1.23560	0.617802	+	0.786334i	$0.288023\pi$
$132$	0	0
$133$	−11.3137	−0.981023
$134$	0	0
$135$	−8.00000	−0.688530
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−8.48528	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	−8.48528	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	−22.6274	−1.90557
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	−25.4558	−2.09956
$148$	0	0
$149$	−1.41421	−0.115857	−0.0579284	−	0.998321i	$-0.518450\pi$
$149$	−1.41421	−0.115857	−0.0579284	+	0.998321i	$0.518450\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−11.3137	−0.908739
$156$	0	0
$157$	−4.24264	−0.338600	−0.169300	−	0.985565i	$-0.554151\pi$
$157$	−4.24264	−0.338600	−0.169300	+	0.985565i	$0.554151\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	8.48528	0.664619	0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528	0.664619	0.332309	+	0.943170i	$0.392172\pi$
$164$	0	0
$165$	11.3137	0.880771
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	−14.1421	−1.08148
$172$	0	0
$173$	−18.3848	−1.39777	−0.698884	−	0.715235i	$-0.746320\pi$
$173$	−18.3848	−1.39777	−0.698884	+	0.715235i	$0.746320\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	−24.0000	−1.80395
$178$	0	0
$179$	−19.7990	−1.47985	−0.739923	−	0.672692i	$-0.765138\pi$
$179$	−19.7990	−1.47985	−0.739923	+	0.672692i	$0.765138\pi$
$180$	0	0
$181$	21.2132	1.57676	0.788382	−	0.615185i	$-0.210919\pi$
$181$	21.2132	1.57676	0.788382	+	0.615185i	$0.210919\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−22.6274	−1.64590
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	−16.9706	−1.21529
$196$	0	0
$197$	24.0416	1.71290	0.856448	−	0.516234i	$-0.172666\pi$
$197$	24.0416	1.71290	0.856448	+	0.516234i	$0.172666\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	16.9706	1.19110
$204$	0	0
$205$	11.3137	0.790184
$206$	0	0
$207$	20.0000	1.39010
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	14.1421	0.973585	0.486792	−	0.873518i	$-0.338167\pi$
$211$	14.1421	0.973585	0.486792	+	0.873518i	$0.338167\pi$
$212$	0	0
$213$	−33.9411	−2.32561
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−32.0000	−2.17230
$218$	0	0
$219$	5.65685	0.382255
$220$	0	0
$221$	0	0
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	−25.4558	−1.68956	−0.844782	−	0.535111i	$-0.820270\pi$
$227$	−25.4558	−1.68956	−0.844782	+	0.535111i	$0.820270\pi$
$228$	0	0
$229$	1.41421	0.0934539	0.0467269	−	0.998908i	$-0.485121\pi$
$229$	1.41421	0.0934539	0.0467269	+	0.998908i	$0.485121\pi$
$230$	0	0
$231$	32.0000	2.10545
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	11.3137	0.738025
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	0	0
$245$	12.7279	0.813157
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	−40.0000	−2.53490
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−5.65685	−0.351500
$260$	0	0
$261$	21.2132	1.31306
$262$	0	0
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−39.5980	−2.42336
$268$	0	0
$269$	−4.24264	−0.258678	−0.129339	−	0.991600i	$-0.541286\pi$
$269$	−4.24264	−0.258678	−0.129339	+	0.991600i	$0.541286\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	−48.0000	−2.90509
$274$	0	0
$275$	8.48528	0.511682
$276$	0	0
$277$	−21.2132	−1.27458	−0.637289	−	0.770625i	$-0.719944\pi$
$277$	−21.2132	−1.27458	−0.637289	+	0.770625i	$0.719944\pi$
$278$	0	0
$279$	−40.0000	−2.39474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−19.7990	−1.17693	−0.588464	−	0.808523i	$-0.700267\pi$
$283$	−19.7990	−1.17693	−0.588464	+	0.808523i	$0.700267\pi$
$284$	0	0
$285$	11.3137	0.670166
$286$	0	0
$287$	32.0000	1.88890
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	45.2548	2.65289
$292$	0	0
$293$	24.0416	1.40453	0.702264	−	0.711917i	$-0.252173\pi$
$293$	24.0416	1.40453	0.702264	+	0.711917i	$0.252173\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	16.0000	0.928414
$298$	0	0
$299$	16.9706	0.981433
$300$	0	0
$301$	11.3137	0.652111
$302$	0	0
$303$	4.00000	0.229794
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	8.48528	0.484281	0.242140	−	0.970241i	$-0.422151\pi$
$307$	8.48528	0.484281	0.242140	+	0.970241i	$0.422151\pi$
$308$	0	0
$309$	11.3137	0.643614
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	28.2843	1.59364
$316$	0	0
$317$	−18.3848	−1.03259	−0.516296	−	0.856410i	$-0.672690\pi$
$317$	−18.3848	−1.03259	−0.516296	+	0.856410i	$0.672690\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−12.7279	−0.706018
$326$	0	0
$327$	−52.0000	−2.87561
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	31.1127	1.71011	0.855054	−	0.518538i	$-0.173524\pi$
$331$	31.1127	1.71011	0.855054	+	0.518538i	$0.173524\pi$
$332$	0	0
$333$	−7.07107	−0.387492
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	5.65685	0.307238
$340$	0	0
$341$	22.6274	1.22534
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	−2.82843	−0.151838	−0.0759190	−	0.997114i	$-0.524189\pi$
$347$	−2.82843	−0.151838	−0.0759190	+	0.997114i	$0.524189\pi$
$348$	0	0
$349$	−18.3848	−0.984115	−0.492057	−	0.870563i	$-0.663755\pi$
$349$	−18.3848	−0.984115	−0.492057	+	0.870563i	$0.663755\pi$
$350$	0	0
$351$	−24.0000	−1.28103
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	16.9706	0.900704
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	8.48528	0.445362
$364$	0	0
$365$	−2.82843	−0.148047
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	40.0000	2.08232
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	−21.2132	−1.09838	−0.549189	−	0.835698i	$-0.685063\pi$
$373$	−21.2132	−1.09838	−0.549189	+	0.835698i	$0.685063\pi$
$374$	0	0
$375$	32.0000	1.65247
$376$	0	0
$377$	18.0000	0.927047
$378$	0	0
$379$	−25.4558	−1.30758	−0.653789	−	0.756677i	$-0.726822\pi$
$379$	−25.4558	−1.30758	−0.653789	+	0.756677i	$0.726822\pi$
$380$	0	0
$381$	22.6274	1.15924
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	14.1421	0.718885
$388$	0	0
$389$	−1.41421	−0.0717035	−0.0358517	−	0.999357i	$-0.511414\pi$
$389$	−1.41421	−0.0717035	−0.0358517	+	0.999357i	$0.511414\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−40.0000	−2.01773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.24264	−0.212932	−0.106466	−	0.994316i	$-0.533954\pi$
$397$	−4.24264	−0.212932	−0.106466	+	0.994316i	$0.533954\pi$
$398$	0	0
$399$	32.0000	1.60200
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−33.9411	−1.69073
$404$	0	0
$405$	1.41421	0.0702728
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	8.00000	0.395575	0.197787	−	0.980245i	$-0.436624\pi$
$409$	8.00000	0.395575	0.197787	+	0.980245i	$0.436624\pi$
$410$	0	0
$411$	22.6274	1.11613
$412$	0	0
$413$	33.9411	1.67013
$414$	0	0
$415$	20.0000	0.981761
$416$	0	0
$417$	24.0000	1.17529
$418$	0	0
$419$	−8.48528	−0.414533	−0.207267	−	0.978285i	$-0.566457\pi$
$419$	−8.48528	−0.414533	−0.207267	+	0.978285i	$0.566457\pi$
$420$	0	0
$421$	−21.2132	−1.03387	−0.516934	−	0.856025i	$-0.672927\pi$
$421$	−21.2132	−1.03387	−0.516934	+	0.856025i	$0.672927\pi$
$422$	0	0
$423$	40.0000	1.94487
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.9706	−0.821263
$428$	0	0
$429$	33.9411	1.63869
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−32.0000	−1.53782	−0.768911	−	0.639356i	$-0.779201\pi$
$433$	−32.0000	−1.53782	−0.768911	+	0.639356i	$0.779201\pi$
$434$	0	0
$435$	−16.9706	−0.813676
$436$	0	0
$437$	−11.3137	−0.541208
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	45.0000	2.14286
$442$	0	0
$443$	−8.48528	−0.403148	−0.201574	−	0.979473i	$-0.564606\pi$
$443$	−8.48528	−0.403148	−0.201574	+	0.979473i	$0.564606\pi$
$444$	0	0
$445$	19.7990	0.938562
$446$	0	0
$447$	4.00000	0.189194
$448$	0	0
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	−22.6274	−1.06548
$452$	0	0
$453$	−56.5685	−2.65782
$454$	0	0
$455$	24.0000	1.12514
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−26.8701	−1.25146	−0.625732	−	0.780038i	$-0.715200\pi$
$461$	−26.8701	−1.25146	−0.625732	+	0.780038i	$0.715200\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	0	0
$465$	32.0000	1.48396
$466$	0	0
$467$	−2.82843	−0.130884	−0.0654420	−	0.997856i	$-0.520846\pi$
$467$	−2.82843	−0.130884	−0.0654420	+	0.997856i	$0.520846\pi$
$468$	0	0
$469$	11.3137	0.522419
$470$	0	0
$471$	12.0000	0.552931
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	7.07107	0.323762
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	−45.2548	−2.05917
$484$	0	0
$485$	−22.6274	−1.02746
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	−19.7990	−0.893516	−0.446758	−	0.894655i	$-0.647421\pi$
$491$	−19.7990	−0.893516	−0.446758	+	0.894655i	$0.647421\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−20.0000	−0.898933
$496$	0	0
$497$	48.0000	2.15309
$498$	0	0
$499$	2.82843	0.126618	0.0633089	−	0.997994i	$-0.479835\pi$
$499$	2.82843	0.126618	0.0633089	+	0.997994i	$0.479835\pi$
$500$	0	0
$501$	−11.3137	−0.505459
$502$	0	0
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	−14.1421	−0.628074
$508$	0	0
$509$	−26.8701	−1.19099	−0.595497	−	0.803357i	$-0.703045\pi$
$509$	−26.8701	−1.19099	−0.595497	+	0.803357i	$0.703045\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	16.0000	0.706417
$514$	0	0
$515$	−5.65685	−0.249271
$516$	0	0
$517$	−22.6274	−0.995153
$518$	0	0
$519$	52.0000	2.28255
$520$	0	0
$521$	24.0000	1.05146	0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000	1.05146	0.525730	+	0.850652i	$0.323792\pi$
$522$	0	0
$523$	−25.4558	−1.11311	−0.556553	−	0.830812i	$-0.687876\pi$
$523$	−25.4558	−1.11311	−0.556553	+	0.830812i	$0.687876\pi$
$524$	0	0
$525$	33.9411	1.48131
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	42.4264	1.84115
$532$	0	0
$533$	33.9411	1.47015
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	56.0000	2.41658
$538$	0	0
$539$	−25.4558	−1.09646
$540$	0	0
$541$	−4.24264	−0.182405	−0.0912027	−	0.995832i	$-0.529071\pi$
$541$	−4.24264	−0.182405	−0.0912027	+	0.995832i	$0.529071\pi$
$542$	0	0
$543$	−60.0000	−2.57485
$544$	0	0
$545$	26.0000	1.11372
$546$	0	0
$547$	−42.4264	−1.81402	−0.907011	−	0.421107i	$-0.861642\pi$
$547$	−42.4264	−1.81402	−0.907011	+	0.421107i	$0.861642\pi$
$548$	0	0
$549$	−21.2132	−0.905357
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.65685	0.240120
$556$	0	0
$557$	−18.3848	−0.778988	−0.389494	−	0.921029i	$-0.627350\pi$
$557$	−18.3848	−0.778988	−0.389494	+	0.921029i	$0.627350\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.82843	−0.119204	−0.0596020	−	0.998222i	$-0.518983\pi$
$563$	−2.82843	−0.119204	−0.0596020	+	0.998222i	$0.518983\pi$
$564$	0	0
$565$	−2.82843	−0.118993
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−8.48528	−0.355098	−0.177549	−	0.984112i	$-0.556817\pi$
$571$	−8.48528	−0.355098	−0.177549	+	0.984112i	$0.556817\pi$
$572$	0	0
$573$	67.8823	2.83582
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	45.2548	1.88073
$580$	0	0
$581$	56.5685	2.34686
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	30.0000	1.24035
$586$	0	0
$587$	−14.1421	−0.583708	−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421	−0.583708	−0.291854	+	0.956463i	$0.594272\pi$
$588$	0	0
$589$	22.6274	0.932346
$590$	0	0
$591$	−68.0000	−2.79715
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.3137	−0.463039
$598$	0	0
$599$	28.0000	1.14405	0.572024	−	0.820237i	$-0.306158\pi$
$599$	28.0000	1.14405	0.572024	+	0.820237i	$0.306158\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	14.1421	0.575912
$604$	0	0
$605$	−4.24264	−0.172488
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	−48.0000	−1.94506
$610$	0	0
$611$	33.9411	1.37311
$612$	0	0
$613$	24.0416	0.971032	0.485516	−	0.874228i	$-0.338632\pi$
$613$	24.0416	0.971032	0.485516	+	0.874228i	$0.338632\pi$
$614$	0	0
$615$	−32.0000	−1.29036
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	−25.4558	−1.02316	−0.511578	−	0.859237i	$-0.670939\pi$
$619$	−25.4558	−1.02316	−0.511578	+	0.859237i	$0.670939\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	0	0
$623$	56.0000	2.24359
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	−22.6274	−0.903652
$628$	0	0
$629$	0	0
$630$	0	0
$631$	4.00000	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$632$	0	0
$633$	−40.0000	−1.58986
$634$	0	0
$635$	−11.3137	−0.448971
$636$	0	0
$637$	38.1838	1.51290
$638$	0	0
$639$	60.0000	2.37356
$640$	0	0
$641$	16.0000	0.631962	0.315981	−	0.948766i	$-0.397666\pi$
$641$	16.0000	0.631962	0.315981	+	0.948766i	$0.397666\pi$
$642$	0	0
$643$	25.4558	1.00388	0.501940	−	0.864902i	$-0.332620\pi$
$643$	25.4558	1.00388	0.501940	+	0.864902i	$0.332620\pi$
$644$	0	0
$645$	−11.3137	−0.445477
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	90.5097	3.54735
$652$	0	0
$653$	41.0122	1.60493	0.802466	−	0.596698i	$-0.203521\pi$
$653$	41.0122	1.60493	0.802466	+	0.596698i	$0.203521\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	−31.1127	−1.21198	−0.605989	−	0.795473i	$-0.707223\pi$
$659$	−31.1127	−1.21198	−0.605989	+	0.795473i	$0.707223\pi$
$660$	0	0
$661$	−24.0416	−0.935111	−0.467556	−	0.883964i	$-0.654865\pi$
$661$	−24.0416	−0.935111	−0.467556	+	0.883964i	$0.654865\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	16.9706	0.657103
$668$	0	0
$669$	−67.8823	−2.62448
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	0	0
$675$	16.9706	0.653197
$676$	0	0
$677$	−46.6690	−1.79364	−0.896819	−	0.442398i	$-0.854128\pi$
$677$	−46.6690	−1.79364	−0.896819	+	0.442398i	$0.854128\pi$
$678$	0	0
$679$	−64.0000	−2.45609
$680$	0	0
$681$	72.0000	2.75905
$682$	0	0
$683$	−8.48528	−0.324680	−0.162340	−	0.986735i	$-0.551904\pi$
$683$	−8.48528	−0.324680	−0.162340	+	0.986735i	$0.551904\pi$
$684$	0	0
$685$	−11.3137	−0.432275
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	−8.48528	−0.322795	−0.161398	−	0.986889i	$-0.551600\pi$
$691$	−8.48528	−0.322795	−0.161398	+	0.986889i	$0.551600\pi$
$692$	0	0
$693$	−56.5685	−2.14886
$694$	0	0
$695$	−12.0000	−0.455186
$696$	0	0
$697$	0	0
$698$	0	0
$699$	5.65685	0.213962
$700$	0	0
$701$	−26.8701	−1.01487	−0.507434	−	0.861691i	$-0.669406\pi$
$701$	−26.8701	−1.01487	−0.507434	+	0.861691i	$0.669406\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	−32.0000	−1.20519
$706$	0	0
$707$	−5.65685	−0.212748
$708$	0	0
$709$	21.2132	0.796679	0.398339	−	0.917238i	$-0.369587\pi$
$709$	21.2132	0.796679	0.398339	+	0.917238i	$0.369587\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−16.9706	−0.634663
$716$	0	0
$717$	45.2548	1.69007
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.7279	−0.472703
$726$	0	0
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	0	0
$729$	−43.0000	−1.59259
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.8701	0.992468	0.496234	−	0.868189i	$-0.334716\pi$
$733$	26.8701	0.992468	0.496234	+	0.868189i	$0.334716\pi$
$734$	0	0
$735$	−36.0000	−1.32788
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	8.48528	0.312136	0.156068	−	0.987746i	$-0.450118\pi$
$739$	8.48528	0.312136	0.156068	+	0.987746i	$0.450118\pi$
$740$	0	0
$741$	33.9411	1.24686
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	70.7107	2.58717
$748$	0	0
$749$	−11.3137	−0.413394
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−56.0000	−2.04075
$754$	0	0
$755$	28.2843	1.02937
$756$	0	0
$757$	−1.41421	−0.0514005	−0.0257002	−	0.999670i	$-0.508182\pi$
$757$	−1.41421	−0.0514005	−0.0257002	+	0.999670i	$0.508182\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	0	0
$763$	73.5391	2.66229
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.0000	1.29988
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	−50.9117	−1.83354
$772$	0	0
$773$	21.2132	0.762986	0.381493	−	0.924372i	$-0.375410\pi$
$773$	21.2132	0.762986	0.381493	+	0.924372i	$0.375410\pi$
$774$	0	0
$775$	24.0000	0.862105
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	−22.6274	−0.810711
$780$	0	0
$781$	−33.9411	−1.21451
$782$	0	0
$783$	−24.0000	−0.857690
$784$	0	0
$785$	−6.00000	−0.214149
$786$	0	0
$787$	25.4558	0.907403	0.453701	−	0.891154i	$-0.350103\pi$
$787$	25.4558	0.907403	0.453701	+	0.891154i	$0.350103\pi$
$788$	0	0
$789$	56.5685	2.01389
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	−18.0000	−0.639199
$794$	0	0
$795$	−5.65685	−0.200628
$796$	0	0
$797$	−26.8701	−0.951786	−0.475893	−	0.879503i	$-0.657875\pi$
$797$	−26.8701	−0.951786	−0.475893	+	0.879503i	$0.657875\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	70.0000	2.47333
$802$	0	0
$803$	5.65685	0.199626
$804$	0	0
$805$	22.6274	0.797512
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	40.0000	1.40633	0.703163	−	0.711029i	$-0.251771\pi$
$809$	40.0000	1.40633	0.703163	+	0.711029i	$0.251771\pi$
$810$	0	0
$811$	31.1127	1.09251	0.546257	−	0.837617i	$-0.316052\pi$
$811$	31.1127	1.09251	0.546257	+	0.837617i	$0.316052\pi$
$812$	0	0
$813$	−22.6274	−0.793578
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	84.8528	2.96500
$820$	0	0
$821$	−21.2132	−0.740346	−0.370173	−	0.928963i	$-0.620702\pi$
$821$	−21.2132	−0.740346	−0.370173	+	0.928963i	$0.620702\pi$
$822$	0	0
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	−19.7990	−0.688478	−0.344239	−	0.938882i	$-0.611863\pi$
$827$	−19.7990	−0.688478	−0.344239	+	0.938882i	$0.611863\pi$
$828$	0	0
$829$	18.3848	0.638530	0.319265	−	0.947666i	$-0.396564\pi$
$829$	18.3848	0.638530	0.319265	+	0.947666i	$0.396564\pi$
$830$	0	0
$831$	60.0000	2.08138
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.65685	0.195764
$836$	0	0
$837$	45.2548	1.56424
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	−50.9117	−1.75349
$844$	0	0
$845$	7.07107	0.243252
$846$	0	0
$847$	−12.0000	−0.412325
$848$	0	0
$849$	56.0000	1.92192
$850$	0	0
$851$	−5.65685	−0.193914
$852$	0	0
$853$	1.41421	0.0484218	0.0242109	−	0.999707i	$-0.492293\pi$
$853$	1.41421	0.0484218	0.0242109	+	0.999707i	$0.492293\pi$
$854$	0	0
$855$	−20.0000	−0.683986
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	−25.4558	−0.868542	−0.434271	−	0.900782i	$-0.642994\pi$
$859$	−25.4558	−0.868542	−0.434271	+	0.900782i	$0.642994\pi$
$860$	0	0
$861$	−90.5097	−3.08456
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	−26.0000	−0.884027
$866$	0	0
$867$	48.0833	1.63299
$868$	0	0
$869$	0	0
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	−80.0000	−2.70759
$874$	0	0
$875$	−45.2548	−1.52989
$876$	0	0
$877$	−26.8701	−0.907337	−0.453669	−	0.891170i	$-0.649885\pi$
$877$	−26.8701	−0.907337	−0.453669	+	0.891170i	$0.649885\pi$
$878$	0	0
$879$	−68.0000	−2.29358
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−48.0833	−1.61813	−0.809065	−	0.587719i	$-0.800026\pi$
$883$	−48.0833	−1.61813	−0.809065	+	0.587719i	$0.800026\pi$
$884$	0	0
$885$	−33.9411	−1.14092
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	−2.82843	−0.0947559
$892$	0	0
$893$	−22.6274	−0.757198
$894$	0	0
$895$	−28.0000	−0.935937
$896$	0	0
$897$	−48.0000	−1.60267
$898$	0	0
$899$	−33.9411	−1.13200
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−32.0000	−1.06489
$904$	0	0
$905$	30.0000	0.997234
$906$	0	0
$907$	14.1421	0.469582	0.234791	−	0.972046i	$-0.424559\pi$
$907$	14.1421	0.469582	0.234791	+	0.972046i	$0.424559\pi$
$908$	0	0
$909$	−7.07107	−0.234533
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	−40.0000	−1.32381
$914$	0	0
$915$	16.9706	0.561029
$916$	0	0
$917$	56.5685	1.86806
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	50.9117	1.67578
$924$	0	0
$925$	4.24264	0.139497
$926$	0	0
$927$	−20.0000	−0.656886
$928$	0	0
$929$	−48.0000	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$929$	−48.0000	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$930$	0	0
$931$	−25.4558	−0.834282
$932$	0	0
$933$	11.3137	0.370394
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	−22.6274	−0.738418
$940$	0	0
$941$	18.3848	0.599327	0.299663	−	0.954045i	$-0.403126\pi$
$941$	18.3848	0.599327	0.299663	+	0.954045i	$0.403126\pi$
$942$	0	0
$943$	32.0000	1.04206
$944$	0	0
$945$	−32.0000	−1.04096
$946$	0	0
$947$	−14.1421	−0.459558	−0.229779	−	0.973243i	$-0.573800\pi$
$947$	−14.1421	−0.459558	−0.229779	+	0.973243i	$0.573800\pi$
$948$	0	0
$949$	−8.48528	−0.275444
$950$	0	0
$951$	52.0000	1.68622
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	−33.9411	−1.09831
$956$	0	0
$957$	33.9411	1.09716
$958$	0	0
$959$	−32.0000	−1.03333
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−14.1421	−0.455724
$964$	0	0
$965$	−22.6274	−0.728402
$966$	0	0
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.7990	−0.635380	−0.317690	−	0.948195i	$-0.602907\pi$
$971$	−19.7990	−0.635380	−0.317690	+	0.948195i	$0.602907\pi$
$972$	0	0
$973$	−33.9411	−1.08810
$974$	0	0
$975$	36.0000	1.15292
$976$	0	0
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	0	0
$979$	−39.5980	−1.26556
$980$	0	0
$981$	91.9239	2.93490
$982$	0	0
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	0	0
$985$	34.0000	1.08333
$986$	0	0
$987$	−90.5097	−2.88095
$988$	0	0
$989$	11.3137	0.359755
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−88.0000	−2.79260
$994$	0	0
$995$	5.65685	0.179334
$996$	0	0
$997$	21.2132	0.671829	0.335914	−	0.941893i	$-0.390955\pi$
$997$	21.2132	0.671829	0.335914	+	0.941893i	$0.390955\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.a.f.1.1		2
3.2	odd	2		9216.2.a.u.1.1		2
4.3	odd	2		1024.2.a.a.1.2		2
8.3	odd	2		1024.2.a.a.1.1		2
8.5	even	2	inner	1024.2.a.f.1.2		2
12.11	even	2		9216.2.a.b.1.1		2
16.3	odd	4		1024.2.b.f.513.2		2
16.5	even	4		1024.2.b.a.513.2		2
16.11	odd	4		1024.2.b.f.513.1		2
16.13	even	4		1024.2.b.a.513.1		2
24.5	odd	2		9216.2.a.u.1.2		2
24.11	even	2		9216.2.a.b.1.2		2
32.3	odd	8		512.2.e.g.129.1	yes	2
32.5	even	8		512.2.e.h.385.1	yes	2
32.11	odd	8		512.2.e.g.385.1	yes	2
32.13	even	8		512.2.e.h.129.1	yes	2
32.19	odd	8		512.2.e.b.129.1	yes	2
32.21	even	8		512.2.e.a.385.1	yes	2
32.27	odd	8		512.2.e.b.385.1	yes	2
32.29	even	8		512.2.e.a.129.1	✓	2
96.5	odd	8		4608.2.k.f.3457.1		2
96.11	even	8		4608.2.k.o.3457.1		2
96.29	odd	8		4608.2.k.s.1153.1		2
96.35	even	8		4608.2.k.o.1153.1		2
96.53	odd	8		4608.2.k.s.3457.1		2
96.59	even	8		4608.2.k.j.3457.1		2
96.77	odd	8		4608.2.k.f.1153.1		2
96.83	even	8		4608.2.k.j.1153.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.e.a.129.1	✓	2	32.29	even	8
512.2.e.a.385.1	yes	2	32.21	even	8
512.2.e.b.129.1	yes	2	32.19	odd	8
512.2.e.b.385.1	yes	2	32.27	odd	8
512.2.e.g.129.1	yes	2	32.3	odd	8
512.2.e.g.385.1	yes	2	32.11	odd	8
512.2.e.h.129.1	yes	2	32.13	even	8
512.2.e.h.385.1	yes	2	32.5	even	8
1024.2.a.a.1.1		2	8.3	odd	2
1024.2.a.a.1.2		2	4.3	odd	2
1024.2.a.f.1.1		2	1.1	even	1	trivial
1024.2.a.f.1.2		2	8.5	even	2	inner
1024.2.b.a.513.1		2	16.13	even	4
1024.2.b.a.513.2		2	16.5	even	4
1024.2.b.f.513.1		2	16.11	odd	4
1024.2.b.f.513.2		2	16.3	odd	4
4608.2.k.f.1153.1		2	96.77	odd	8
4608.2.k.f.3457.1		2	96.5	odd	8
4608.2.k.j.1153.1		2	96.83	even	8
4608.2.k.j.3457.1		2	96.59	even	8
4608.2.k.o.1153.1		2	96.35	even	8
4608.2.k.o.3457.1		2	96.11	even	8
4608.2.k.s.1153.1		2	96.29	odd	8
4608.2.k.s.3457.1		2	96.53	odd	8
9216.2.a.b.1.1		2	12.11	even	2
9216.2.a.b.1.2		2	24.11	even	2
9216.2.a.u.1.1		2	3.2	odd	2
9216.2.a.u.1.2		2	24.5	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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{3} +1.41421 q^{5} +4.00000 q^{7} +5.00000 q^{9} -2.82843 q^{11} +4.24264 q^{13} -4.00000 q^{15} -2.82843 q^{19} -11.3137 q^{21} +4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +4.24264 q^{29} -8.00000 q^{31} +8.00000 q^{33} +5.65685 q^{35} -1.41421 q^{37} -12.0000 q^{39} +8.00000 q^{41} +2.82843 q^{43} +7.07107 q^{45} +8.00000 q^{47} +9.00000 q^{49} +1.41421 q^{53} -4.00000 q^{55} +8.00000 q^{57} +8.48528 q^{59} -4.24264 q^{61} +20.0000 q^{63} +6.00000 q^{65} +2.82843 q^{67} -11.3137 q^{69} +12.0000 q^{71} -2.00000 q^{73} +8.48528 q^{75} -11.3137 q^{77} +1.00000 q^{81} +14.1421 q^{83} -12.0000 q^{87} +14.0000 q^{89} +16.9706 q^{91} +22.6274 q^{93} -4.00000 q^{95} -16.0000 q^{97} -14.1421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} + 10 q^{9} - 8 q^{15} + 8 q^{23} - 6 q^{25} - 16 q^{31} + 16 q^{33} - 24 q^{39} + 16 q^{41} + 16 q^{47} + 18 q^{49} - 8 q^{55} + 16 q^{57} + 40 q^{63} + 12 q^{65} + 24 q^{71} - 4 q^{73} + 2 q^{81}+ \cdots - 32 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 + 10 * q^9 - 8 * q^15 + 8 * q^23 - 6 * q^25 - 16 * q^31 + 16 * q^33 - 24 * q^39 + 16 * q^41 + 16 * q^47 + 18 * q^49 - 8 * q^55 + 16 * q^57 + 40 * q^63 + 12 * q^65 + 24 * q^71 - 4 * q^73 + 2 * q^81 - 24 * q^87 + 28 * q^89 - 8 * q^95 - 32 * q^97

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