LMFDB - Embedded newform 1024.2.a.g.1.3

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 = [4,0,-4,0,0,0,0,0,4,0,-12,0,0,0,0,0,0] 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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 256)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.93185$ 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)

$f(q)$	$=$	$q+0.732051 q^{3} -2.44949 q^{5} +3.86370 q^{7} -2.46410 q^{9} -4.73205 q^{11} -0.378937 q^{13} -1.79315 q^{15} -3.46410 q^{17} -4.73205 q^{19} +2.82843 q^{21} +1.79315 q^{23} +1.00000 q^{25} -4.00000 q^{27} +2.44949 q^{29} -5.65685 q^{31} -3.46410 q^{33} -9.46410 q^{35} +5.27792 q^{37} -0.277401 q^{39} +6.92820 q^{41} +2.19615 q^{43} +6.03579 q^{45} -9.79796 q^{47} +7.92820 q^{49} -2.53590 q^{51} -10.9348 q^{53} +11.5911 q^{55} -3.46410 q^{57} -7.26795 q^{59} -5.27792 q^{61} -9.52056 q^{63} +0.928203 q^{65} -6.19615 q^{67} +1.31268 q^{69} -1.79315 q^{71} -2.53590 q^{73} +0.732051 q^{75} -18.2832 q^{77} +4.14110 q^{79} +4.46410 q^{81} +2.19615 q^{83} +8.48528 q^{85} +1.79315 q^{87} -2.53590 q^{89} -1.46410 q^{91} -4.14110 q^{93} +11.5911 q^{95} +3.46410 q^{97} +11.6603 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9} - 12 q^{11} - 12 q^{19} + 4 q^{25} - 16 q^{27} - 24 q^{35} - 12 q^{43} + 4 q^{49} - 24 q^{51} - 36 q^{59} - 24 q^{65} - 4 q^{67} - 24 q^{73} - 4 q^{75} + 4 q^{81} - 12 q^{83} - 24 q^{89}+ \cdots + 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 - 12 * q^11 - 12 * q^19 + 4 * q^25 - 16 * q^27 - 24 * q^35 - 12 * q^43 + 4 * q^49 - 24 * q^51 - 36 * q^59 - 24 * q^65 - 4 * q^67 - 24 * q^73 - 4 * q^75 + 4 * q^81 - 12 * q^83 - 24 * q^89 + 8 * q^91 + 12 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	−2.44949	−1.09545	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	3.86370	1.46034	0.730171	−	0.683264i	$-0.239440\pi$
$7$	3.86370	1.46034	0.730171	+	0.683264i	$0.239440\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	−0.378937	−0.105098	−0.0525492	−	0.998618i	$-0.516735\pi$
$13$	−0.378937	−0.105098	−0.0525492	+	0.998618i	$0.516735\pi$
$14$	0	0
$15$	−1.79315	−0.462990
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−4.73205	−1.08561	−0.542803	−	0.839860i	$-0.682637\pi$
$19$	−4.73205	−1.08561	−0.542803	+	0.839860i	$0.682637\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	1.79315	0.373898	0.186949	−	0.982370i	$-0.440140\pi$
$23$	1.79315	0.373898	0.186949	+	0.982370i	$0.440140\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	2.44949	0.454859	0.227429	−	0.973795i	$-0.426968\pi$
$29$	2.44949	0.454859	0.227429	+	0.973795i	$0.426968\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	−9.46410	−1.59973
$36$	0	0
$37$	5.27792	0.867684	0.433842	−	0.900989i	$-0.357158\pi$
$37$	5.27792	0.867684	0.433842	+	0.900989i	$0.357158\pi$
$38$	0	0
$39$	−0.277401	−0.0444198
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	2.19615	0.334910	0.167455	−	0.985880i	$-0.446445\pi$
$43$	2.19615	0.334910	0.167455	+	0.985880i	$0.446445\pi$
$44$	0	0
$45$	6.03579	0.899763
$46$	0	0
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	0	0
$49$	7.92820	1.13260
$50$	0	0
$51$	−2.53590	−0.355097
$52$	0	0
$53$	−10.9348	−1.50201	−0.751003	−	0.660299i	$-0.770430\pi$
$53$	−10.9348	−1.50201	−0.751003	+	0.660299i	$0.770430\pi$
$54$	0	0
$55$	11.5911	1.56294
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	−7.26795	−0.946206	−0.473103	−	0.881007i	$-0.656866\pi$
$59$	−7.26795	−0.946206	−0.473103	+	0.881007i	$0.656866\pi$
$60$	0	0
$61$	−5.27792	−0.675768	−0.337884	−	0.941188i	$-0.609711\pi$
$61$	−5.27792	−0.675768	−0.337884	+	0.941188i	$0.609711\pi$
$62$	0	0
$63$	−9.52056	−1.19948
$64$	0	0
$65$	0.928203	0.115129
$66$	0	0
$67$	−6.19615	−0.756980	−0.378490	−	0.925605i	$-0.623557\pi$
$67$	−6.19615	−0.756980	−0.378490	+	0.925605i	$0.623557\pi$
$68$	0	0
$69$	1.31268	0.158028
$70$	0	0
$71$	−1.79315	−0.212808	−0.106404	−	0.994323i	$-0.533934\pi$
$71$	−1.79315	−0.212808	−0.106404	+	0.994323i	$0.533934\pi$
$72$	0	0
$73$	−2.53590	−0.296804	−0.148402	−	0.988927i	$-0.547413\pi$
$73$	−2.53590	−0.296804	−0.148402	+	0.988927i	$0.547413\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	−18.2832	−2.08357
$78$	0	0
$79$	4.14110	0.465911	0.232955	−	0.972487i	$-0.425160\pi$
$79$	4.14110	0.465911	0.232955	+	0.972487i	$0.425160\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	2.19615	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$84$	0	0
$85$	8.48528	0.920358
$86$	0	0
$87$	1.79315	0.192246
$88$	0	0
$89$	−2.53590	−0.268805	−0.134402	−	0.990927i	$-0.542911\pi$
$89$	−2.53590	−0.268805	−0.134402	+	0.990927i	$0.542911\pi$
$90$	0	0
$91$	−1.46410	−0.153480
$92$	0	0
$93$	−4.14110	−0.429413
$94$	0	0
$95$	11.5911	1.18922
$96$	0	0
$97$	3.46410	0.351726	0.175863	−	0.984415i	$-0.443728\pi$
$97$	3.46410	0.351726	0.175863	+	0.984415i	$0.443728\pi$
$98$	0	0
$99$	11.6603	1.17190
$100$	0	0
$101$	1.13681	0.113117	0.0565585	−	0.998399i	$-0.481987\pi$
$101$	1.13681	0.113117	0.0565585	+	0.998399i	$0.481987\pi$
$102$	0	0
$103$	9.52056	0.938088	0.469044	−	0.883175i	$-0.344598\pi$
$103$	9.52056	0.938088	0.469044	+	0.883175i	$0.344598\pi$
$104$	0	0
$105$	−6.92820	−0.676123
$106$	0	0
$107$	−18.5885	−1.79701	−0.898507	−	0.438959i	$-0.855347\pi$
$107$	−18.5885	−1.79701	−0.898507	+	0.438959i	$0.855347\pi$
$108$	0	0
$109$	18.6622	1.78751	0.893756	−	0.448553i	$-0.148060\pi$
$109$	18.6622	1.78751	0.893756	+	0.448553i	$0.148060\pi$
$110$	0	0
$111$	3.86370	0.366726
$112$	0	0
$113$	−12.9282	−1.21618	−0.608092	−	0.793867i	$-0.708065\pi$
$113$	−12.9282	−1.21618	−0.608092	+	0.793867i	$0.708065\pi$
$114$	0	0
$115$	−4.39230	−0.409585
$116$	0	0
$117$	0.933740	0.0863243
$118$	0	0
$119$	−13.3843	−1.22693
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	5.07180	0.457309
$124$	0	0
$125$	9.79796	0.876356
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	1.60770	0.141550
$130$	0	0
$131$	−0.339746	−0.0296837	−0.0148419	−	0.999890i	$-0.504724\pi$
$131$	−0.339746	−0.0296837	−0.0148419	+	0.999890i	$0.504724\pi$
$132$	0	0
$133$	−18.2832	−1.58536
$134$	0	0
$135$	9.79796	0.843274
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	10.1962	0.864826	0.432413	−	0.901676i	$-0.357662\pi$
$139$	10.1962	0.864826	0.432413	+	0.901676i	$0.357662\pi$
$140$	0	0
$141$	−7.17260	−0.604042
$142$	0	0
$143$	1.79315	0.149951
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	5.80385	0.478693
$148$	0	0
$149$	19.4201	1.59095	0.795476	−	0.605985i	$-0.207221\pi$
$149$	19.4201	1.59095	0.795476	+	0.605985i	$0.207221\pi$
$150$	0	0
$151$	−9.52056	−0.774772	−0.387386	−	0.921918i	$-0.626622\pi$
$151$	−9.52056	−0.774772	−0.387386	+	0.921918i	$0.626622\pi$
$152$	0	0
$153$	8.53590	0.690086
$154$	0	0
$155$	13.8564	1.11297
$156$	0	0
$157$	10.1769	0.812205	0.406102	−	0.913828i	$-0.366888\pi$
$157$	10.1769	0.812205	0.406102	+	0.913828i	$0.366888\pi$
$158$	0	0
$159$	−8.00481	−0.634823
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	16.7321	1.31056	0.655278	−	0.755388i	$-0.272552\pi$
$163$	16.7321	1.31056	0.655278	+	0.755388i	$0.272552\pi$
$164$	0	0
$165$	8.48528	0.660578
$166$	0	0
$167$	−18.7637	−1.45198	−0.725990	−	0.687705i	$-0.758618\pi$
$167$	−18.7637	−1.45198	−0.725990	+	0.687705i	$0.758618\pi$
$168$	0	0
$169$	−12.8564	−0.988954
$170$	0	0
$171$	11.6603	0.891682
$172$	0	0
$173$	10.9348	0.831355	0.415678	−	0.909512i	$-0.363544\pi$
$173$	10.9348	0.831355	0.415678	+	0.909512i	$0.363544\pi$
$174$	0	0
$175$	3.86370	0.292069
$176$	0	0
$177$	−5.32051	−0.399914
$178$	0	0
$179$	2.19615	0.164148	0.0820741	−	0.996626i	$-0.473846\pi$
$179$	2.19615	0.164148	0.0820741	+	0.996626i	$0.473846\pi$
$180$	0	0
$181$	−22.8033	−1.69495	−0.847477	−	0.530832i	$-0.821880\pi$
$181$	−22.8033	−1.69495	−0.847477	+	0.530832i	$0.821880\pi$
$182$	0	0
$183$	−3.86370	−0.285613
$184$	0	0
$185$	−12.9282	−0.950500
$186$	0	0
$187$	16.3923	1.19872
$188$	0	0
$189$	−15.4548	−1.12417
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	−18.3923	−1.32391	−0.661954	−	0.749545i	$-0.730272\pi$
$193$	−18.3923	−1.32391	−0.661954	+	0.749545i	$0.730272\pi$
$194$	0	0
$195$	0.679492	0.0486594
$196$	0	0
$197$	−2.44949	−0.174519	−0.0872595	−	0.996186i	$-0.527811\pi$
$197$	−2.44949	−0.174519	−0.0872595	+	0.996186i	$0.527811\pi$
$198$	0	0
$199$	−15.7322	−1.11523	−0.557614	−	0.830101i	$-0.688283\pi$
$199$	−15.7322	−1.11523	−0.557614	+	0.830101i	$0.688283\pi$
$200$	0	0
$201$	−4.53590	−0.319938
$202$	0	0
$203$	9.46410	0.664250
$204$	0	0
$205$	−16.9706	−1.18528
$206$	0	0
$207$	−4.41851	−0.307107
$208$	0	0
$209$	22.3923	1.54891
$210$	0	0
$211$	10.1962	0.701932	0.350966	−	0.936388i	$-0.385853\pi$
$211$	10.1962	0.701932	0.350966	+	0.936388i	$0.385853\pi$
$212$	0	0
$213$	−1.31268	−0.0899432
$214$	0	0
$215$	−5.37945	−0.366876
$216$	0	0
$217$	−21.8564	−1.48371
$218$	0	0
$219$	−1.85641	−0.125444
$220$	0	0
$221$	1.31268	0.0883003
$222$	0	0
$223$	−5.65685	−0.378811	−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685	−0.378811	−0.189405	+	0.981899i	$0.560656\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	−21.1244	−1.40207	−0.701036	−	0.713126i	$-0.747279\pi$
$227$	−21.1244	−1.40207	−0.701036	+	0.713126i	$0.747279\pi$
$228$	0	0
$229$	13.0053	0.859416	0.429708	−	0.902968i	$-0.358617\pi$
$229$	13.0053	0.859416	0.429708	+	0.902968i	$0.358617\pi$
$230$	0	0
$231$	−13.3843	−0.880620
$232$	0	0
$233$	−2.53590	−0.166132	−0.0830661	−	0.996544i	$-0.526471\pi$
$233$	−2.53590	−0.166132	−0.0830661	+	0.996544i	$0.526471\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	3.03150	0.196917
$238$	0	0
$239$	26.7685	1.73151	0.865756	−	0.500467i	$-0.166838\pi$
$239$	26.7685	1.73151	0.865756	+	0.500467i	$0.166838\pi$
$240$	0	0
$241$	2.39230	0.154102	0.0770510	−	0.997027i	$-0.475450\pi$
$241$	2.39230	0.154102	0.0770510	+	0.997027i	$0.475450\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	−19.4201	−1.24070
$246$	0	0
$247$	1.79315	0.114095
$248$	0	0
$249$	1.60770	0.101884
$250$	0	0
$251$	−21.1244	−1.33336	−0.666679	−	0.745345i	$-0.732285\pi$
$251$	−21.1244	−1.33336	−0.666679	+	0.745345i	$0.732285\pi$
$252$	0	0
$253$	−8.48528	−0.533465
$254$	0	0
$255$	6.21166	0.388989
$256$	0	0
$257$	12.9282	0.806439	0.403220	−	0.915103i	$-0.367891\pi$
$257$	12.9282	0.806439	0.403220	+	0.915103i	$0.367891\pi$
$258$	0	0
$259$	20.3923	1.26712
$260$	0	0
$261$	−6.03579	−0.373606
$262$	0	0
$263$	−21.3891	−1.31891	−0.659453	−	0.751746i	$-0.729212\pi$
$263$	−21.3891	−1.31891	−0.659453	+	0.751746i	$0.729212\pi$
$264$	0	0
$265$	26.7846	1.64537
$266$	0	0
$267$	−1.85641	−0.113610
$268$	0	0
$269$	7.34847	0.448044	0.224022	−	0.974584i	$-0.428081\pi$
$269$	7.34847	0.448044	0.224022	+	0.974584i	$0.428081\pi$
$270$	0	0
$271$	1.51575	0.0920752	0.0460376	−	0.998940i	$-0.485341\pi$
$271$	1.51575	0.0920752	0.0460376	+	0.998940i	$0.485341\pi$
$272$	0	0
$273$	−1.07180	−0.0648681
$274$	0	0
$275$	−4.73205	−0.285353
$276$	0	0
$277$	4.52004	0.271583	0.135792	−	0.990737i	$-0.456642\pi$
$277$	4.52004	0.271583	0.135792	+	0.990737i	$0.456642\pi$
$278$	0	0
$279$	13.9391	0.834510
$280$	0	0
$281$	16.3923	0.977883	0.488941	−	0.872317i	$-0.337383\pi$
$281$	16.3923	0.977883	0.488941	+	0.872317i	$0.337383\pi$
$282$	0	0
$283$	1.80385	0.107228	0.0536138	−	0.998562i	$-0.482926\pi$
$283$	1.80385	0.107228	0.0536138	+	0.998562i	$0.482926\pi$
$284$	0	0
$285$	8.48528	0.502625
$286$	0	0
$287$	26.7685	1.58010
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	2.53590	0.148657
$292$	0	0
$293$	17.1464	1.00171	0.500853	−	0.865533i	$-0.333020\pi$
$293$	17.1464	1.00171	0.500853	+	0.865533i	$0.333020\pi$
$294$	0	0
$295$	17.8028	1.03652
$296$	0	0
$297$	18.9282	1.09833
$298$	0	0
$299$	−0.679492	−0.0392960
$300$	0	0
$301$	8.48528	0.489083
$302$	0	0
$303$	0.832204	0.0478089
$304$	0	0
$305$	12.9282	0.740267
$306$	0	0
$307$	30.9808	1.76817	0.884083	−	0.467330i	$-0.154784\pi$
$307$	30.9808	1.76817	0.884083	+	0.467330i	$0.154784\pi$
$308$	0	0
$309$	6.96953	0.396483
$310$	0	0
$311$	−15.1774	−0.860632	−0.430316	−	0.902678i	$-0.641598\pi$
$311$	−15.1774	−0.860632	−0.430316	+	0.902678i	$0.641598\pi$
$312$	0	0
$313$	12.7846	0.722629	0.361314	−	0.932444i	$-0.382328\pi$
$313$	12.7846	0.722629	0.361314	+	0.932444i	$0.382328\pi$
$314$	0	0
$315$	23.3205	1.31396
$316$	0	0
$317$	9.62209	0.540431	0.270215	−	0.962800i	$-0.412905\pi$
$317$	9.62209	0.540431	0.270215	+	0.962800i	$0.412905\pi$
$318$	0	0
$319$	−11.5911	−0.648978
$320$	0	0
$321$	−13.6077	−0.759507
$322$	0	0
$323$	16.3923	0.912092
$324$	0	0
$325$	−0.378937	−0.0210197
$326$	0	0
$327$	13.6617	0.755492
$328$	0	0
$329$	−37.8564	−2.08709
$330$	0	0
$331$	−26.5885	−1.46143	−0.730717	−	0.682681i	$-0.760814\pi$
$331$	−26.5885	−1.46143	−0.730717	+	0.682681i	$0.760814\pi$
$332$	0	0
$333$	−13.0053	−0.712687
$334$	0	0
$335$	15.1774	0.829231
$336$	0	0
$337$	−21.7128	−1.18277	−0.591386	−	0.806389i	$-0.701419\pi$
$337$	−21.7128	−1.18277	−0.591386	+	0.806389i	$0.701419\pi$
$338$	0	0
$339$	−9.46410	−0.514019
$340$	0	0
$341$	26.7685	1.44960
$342$	0	0
$343$	3.58630	0.193642
$344$	0	0
$345$	−3.21539	−0.173111
$346$	0	0
$347$	−7.26795	−0.390164	−0.195082	−	0.980787i	$-0.562497\pi$
$347$	−7.26795	−0.390164	−0.195082	+	0.980787i	$0.562497\pi$
$348$	0	0
$349$	−5.83272	−0.312218	−0.156109	−	0.987740i	$-0.549895\pi$
$349$	−5.83272	−0.312218	−0.156109	+	0.987740i	$0.549895\pi$
$350$	0	0
$351$	1.51575	0.0809047
$352$	0	0
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	4.39230	0.233119
$356$	0	0
$357$	−9.79796	−0.518563
$358$	0	0
$359$	17.8028	0.939594	0.469797	−	0.882774i	$-0.344327\pi$
$359$	17.8028	0.939594	0.469797	+	0.882774i	$0.344327\pi$
$360$	0	0
$361$	3.39230	0.178542
$362$	0	0
$363$	8.33975	0.437723
$364$	0	0
$365$	6.21166	0.325133
$366$	0	0
$367$	−32.4254	−1.69259	−0.846295	−	0.532714i	$-0.821172\pi$
$367$	−32.4254	−1.69259	−0.846295	+	0.532714i	$0.821172\pi$
$368$	0	0
$369$	−17.0718	−0.888722
$370$	0	0
$371$	−42.2487	−2.19344
$372$	0	0
$373$	19.9749	1.03426	0.517129	−	0.855907i	$-0.327001\pi$
$373$	19.9749	1.03426	0.517129	+	0.855907i	$0.327001\pi$
$374$	0	0
$375$	7.17260	0.370392
$376$	0	0
$377$	−0.928203	−0.0478049
$378$	0	0
$379$	2.87564	0.147712	0.0738560	−	0.997269i	$-0.476469\pi$
$379$	2.87564	0.147712	0.0738560	+	0.997269i	$0.476469\pi$
$380$	0	0
$381$	4.14110	0.212155
$382$	0	0
$383$	−19.5959	−1.00130	−0.500652	−	0.865648i	$-0.666906\pi$
$383$	−19.5959	−1.00130	−0.500652	+	0.865648i	$0.666906\pi$
$384$	0	0
$385$	44.7846	2.28244
$386$	0	0
$387$	−5.41154	−0.275084
$388$	0	0
$389$	−6.03579	−0.306027	−0.153013	−	0.988224i	$-0.548898\pi$
$389$	−6.03579	−0.306027	−0.153013	+	0.988224i	$0.548898\pi$
$390$	0	0
$391$	−6.21166	−0.314137
$392$	0	0
$393$	−0.248711	−0.0125458
$394$	0	0
$395$	−10.1436	−0.510380
$396$	0	0
$397$	−27.7023	−1.39034	−0.695168	−	0.718847i	$-0.744670\pi$
$397$	−27.7023	−1.39034	−0.695168	+	0.718847i	$0.744670\pi$
$398$	0	0
$399$	−13.3843	−0.670051
$400$	0	0
$401$	−13.6077	−0.679536	−0.339768	−	0.940509i	$-0.610349\pi$
$401$	−13.6077	−0.679536	−0.339768	+	0.940509i	$0.610349\pi$
$402$	0	0
$403$	2.14359	0.106780
$404$	0	0
$405$	−10.9348	−0.543353
$406$	0	0
$407$	−24.9754	−1.23798
$408$	0	0
$409$	17.0718	0.844146	0.422073	−	0.906562i	$-0.361303\pi$
$409$	17.0718	0.844146	0.422073	+	0.906562i	$0.361303\pi$
$410$	0	0
$411$	8.78461	0.433313
$412$	0	0
$413$	−28.0812	−1.38179
$414$	0	0
$415$	−5.37945	−0.264067
$416$	0	0
$417$	7.46410	0.365519
$418$	0	0
$419$	−14.1962	−0.693527	−0.346764	−	0.937953i	$-0.612719\pi$
$419$	−14.1962	−0.693527	−0.346764	+	0.937953i	$0.612719\pi$
$420$	0	0
$421$	−26.1865	−1.27625	−0.638126	−	0.769932i	$-0.720290\pi$
$421$	−26.1865	−1.27625	−0.638126	+	0.769932i	$0.720290\pi$
$422$	0	0
$423$	24.1432	1.17388
$424$	0	0
$425$	−3.46410	−0.168034
$426$	0	0
$427$	−20.3923	−0.986853
$428$	0	0
$429$	1.31268	0.0633767
$430$	0	0
$431$	26.7685	1.28939	0.644697	−	0.764438i	$-0.276983\pi$
$431$	26.7685	1.28939	0.644697	+	0.764438i	$0.276983\pi$
$432$	0	0
$433$	34.3923	1.65279	0.826394	−	0.563092i	$-0.190388\pi$
$433$	34.3923	1.65279	0.826394	+	0.563092i	$0.190388\pi$
$434$	0	0
$435$	−4.39230	−0.210595
$436$	0	0
$437$	−8.48528	−0.405906
$438$	0	0
$439$	−12.1459	−0.579693	−0.289846	−	0.957073i	$-0.593604\pi$
$439$	−12.1459	−0.579693	−0.289846	+	0.957073i	$0.593604\pi$
$440$	0	0
$441$	−19.5359	−0.930281
$442$	0	0
$443$	−14.1962	−0.674480	−0.337240	−	0.941419i	$-0.609493\pi$
$443$	−14.1962	−0.674480	−0.337240	+	0.941419i	$0.609493\pi$
$444$	0	0
$445$	6.21166	0.294461
$446$	0	0
$447$	14.2165	0.672416
$448$	0	0
$449$	−24.2487	−1.14437	−0.572184	−	0.820125i	$-0.693904\pi$
$449$	−24.2487	−1.14437	−0.572184	+	0.820125i	$0.693904\pi$
$450$	0	0
$451$	−32.7846	−1.54377
$452$	0	0
$453$	−6.96953	−0.327457
$454$	0	0
$455$	3.58630	0.168128
$456$	0	0
$457$	−20.7846	−0.972263	−0.486132	−	0.873886i	$-0.661592\pi$
$457$	−20.7846	−0.972263	−0.486132	+	0.873886i	$0.661592\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	34.1170	1.58899	0.794493	−	0.607273i	$-0.207737\pi$
$461$	34.1170	1.58899	0.794493	+	0.607273i	$0.207737\pi$
$462$	0	0
$463$	−38.0822	−1.76983	−0.884916	−	0.465751i	$-0.845784\pi$
$463$	−38.0822	−1.76983	−0.884916	+	0.465751i	$0.845784\pi$
$464$	0	0
$465$	10.1436	0.470398
$466$	0	0
$467$	−28.7321	−1.32956	−0.664780	−	0.747039i	$-0.731475\pi$
$467$	−28.7321	−1.32956	−0.664780	+	0.747039i	$0.731475\pi$
$468$	0	0
$469$	−23.9401	−1.10545
$470$	0	0
$471$	7.45001	0.343278
$472$	0	0
$473$	−10.3923	−0.477839
$474$	0	0
$475$	−4.73205	−0.217121
$476$	0	0
$477$	26.9444	1.23370
$478$	0	0
$479$	−36.5665	−1.67077	−0.835383	−	0.549669i	$-0.814754\pi$
$479$	−36.5665	−1.67077	−0.835383	+	0.549669i	$0.814754\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	5.07180	0.230775
$484$	0	0
$485$	−8.48528	−0.385297
$486$	0	0
$487$	−27.0459	−1.22557	−0.612784	−	0.790251i	$-0.709950\pi$
$487$	−27.0459	−1.22557	−0.612784	+	0.790251i	$0.709950\pi$
$488$	0	0
$489$	12.2487	0.553906
$490$	0	0
$491$	−21.8038	−0.983994	−0.491997	−	0.870597i	$-0.663733\pi$
$491$	−21.8038	−0.983994	−0.491997	+	0.870597i	$0.663733\pi$
$492$	0	0
$493$	−8.48528	−0.382158
$494$	0	0
$495$	−28.5617	−1.28375
$496$	0	0
$497$	−6.92820	−0.310772
$498$	0	0
$499$	−5.80385	−0.259816	−0.129908	−	0.991526i	$-0.541468\pi$
$499$	−5.80385	−0.259816	−0.129908	+	0.991526i	$0.541468\pi$
$500$	0	0
$501$	−13.7360	−0.613679
$502$	0	0
$503$	40.9850	1.82743	0.913715	−	0.406355i	$-0.133200\pi$
$503$	40.9850	1.82743	0.913715	+	0.406355i	$0.133200\pi$
$504$	0	0
$505$	−2.78461	−0.123914
$506$	0	0
$507$	−9.41154	−0.417981
$508$	0	0
$509$	−13.5601	−0.601042	−0.300521	−	0.953775i	$-0.597161\pi$
$509$	−13.5601	−0.601042	−0.300521	+	0.953775i	$0.597161\pi$
$510$	0	0
$511$	−9.79796	−0.433436
$512$	0	0
$513$	18.9282	0.835701
$514$	0	0
$515$	−23.3205	−1.02762
$516$	0	0
$517$	46.3644	2.03911
$518$	0	0
$519$	8.00481	0.351372
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	19.2679	0.842529	0.421264	−	0.906938i	$-0.361586\pi$
$523$	19.2679	0.842529	0.421264	+	0.906938i	$0.361586\pi$
$524$	0	0
$525$	2.82843	0.123443
$526$	0	0
$527$	19.5959	0.853612
$528$	0	0
$529$	−19.7846	−0.860200
$530$	0	0
$531$	17.9090	0.777183
$532$	0	0
$533$	−2.62536	−0.113717
$534$	0	0
$535$	45.5322	1.96853
$536$	0	0
$537$	1.60770	0.0693772
$538$	0	0
$539$	−37.5167	−1.61596
$540$	0	0
$541$	14.3180	0.615579	0.307789	−	0.951454i	$-0.400411\pi$
$541$	14.3180	0.615579	0.307789	+	0.951454i	$0.400411\pi$
$542$	0	0
$543$	−16.6932	−0.716372
$544$	0	0
$545$	−45.7128	−1.95812
$546$	0	0
$547$	−0.339746	−0.0145265	−0.00726324	−	0.999974i	$-0.502312\pi$
$547$	−0.339746	−0.0145265	−0.00726324	+	0.999974i	$0.502312\pi$
$548$	0	0
$549$	13.0053	0.555054
$550$	0	0
$551$	−11.5911	−0.493798
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	−9.46410	−0.401729
$556$	0	0
$557$	−15.8338	−0.670898	−0.335449	−	0.942058i	$-0.608888\pi$
$557$	−15.8338	−0.670898	−0.335449	+	0.942058i	$0.608888\pi$
$558$	0	0
$559$	−0.832204	−0.0351985
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−14.8756	−0.626934	−0.313467	−	0.949599i	$-0.601490\pi$
$563$	−14.8756	−0.626934	−0.313467	+	0.949599i	$0.601490\pi$
$564$	0	0
$565$	31.6675	1.33226
$566$	0	0
$567$	17.2480	0.724346
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−30.1962	−1.26367	−0.631835	−	0.775103i	$-0.717698\pi$
$571$	−30.1962	−1.26367	−0.631835	+	0.775103i	$0.717698\pi$
$572$	0	0
$573$	12.4233	0.518991
$574$	0	0
$575$	1.79315	0.0747796
$576$	0	0
$577$	−43.8564	−1.82577	−0.912883	−	0.408221i	$-0.866149\pi$
$577$	−43.8564	−1.82577	−0.912883	+	0.408221i	$0.866149\pi$
$578$	0	0
$579$	−13.4641	−0.559549
$580$	0	0
$581$	8.48528	0.352029
$582$	0	0
$583$	51.7439	2.14301
$584$	0	0
$585$	−2.28719	−0.0945635
$586$	0	0
$587$	9.12436	0.376602	0.188301	−	0.982111i	$-0.439702\pi$
$587$	9.12436	0.376602	0.188301	+	0.982111i	$0.439702\pi$
$588$	0	0
$589$	26.7685	1.10298
$590$	0	0
$591$	−1.79315	−0.0737604
$592$	0	0
$593$	19.8564	0.815405	0.407702	−	0.913115i	$-0.366330\pi$
$593$	19.8564	0.815405	0.407702	+	0.913115i	$0.366330\pi$
$594$	0	0
$595$	32.7846	1.34404
$596$	0	0
$597$	−11.5168	−0.471350
$598$	0	0
$599$	4.41851	0.180535	0.0902676	−	0.995918i	$-0.471228\pi$
$599$	4.41851	0.180535	0.0902676	+	0.995918i	$0.471228\pi$
$600$	0	0
$601$	−40.3923	−1.64764	−0.823818	−	0.566854i	$-0.808160\pi$
$601$	−40.3923	−1.64764	−0.823818	+	0.566854i	$0.808160\pi$
$602$	0	0
$603$	15.2679	0.621759
$604$	0	0
$605$	−27.9053	−1.13451
$606$	0	0
$607$	8.28221	0.336165	0.168082	−	0.985773i	$-0.446243\pi$
$607$	8.28221	0.336165	0.168082	+	0.985773i	$0.446243\pi$
$608$	0	0
$609$	6.92820	0.280745
$610$	0	0
$611$	3.71281	0.150204
$612$	0	0
$613$	21.2875	0.859795	0.429898	−	0.902878i	$-0.358550\pi$
$613$	21.2875	0.859795	0.429898	+	0.902878i	$0.358550\pi$
$614$	0	0
$615$	−12.4233	−0.500956
$616$	0	0
$617$	−11.3205	−0.455746	−0.227873	−	0.973691i	$-0.573177\pi$
$617$	−11.3205	−0.455746	−0.227873	+	0.973691i	$0.573177\pi$
$618$	0	0
$619$	−26.5885	−1.06868	−0.534340	−	0.845270i	$-0.679440\pi$
$619$	−26.5885	−1.06868	−0.534340	+	0.845270i	$0.679440\pi$
$620$	0	0
$621$	−7.17260	−0.287827
$622$	0	0
$623$	−9.79796	−0.392547
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	16.3923	0.654646
$628$	0	0
$629$	−18.2832	−0.729001
$630$	0	0
$631$	−1.23835	−0.0492979	−0.0246489	−	0.999696i	$-0.507847\pi$
$631$	−1.23835	−0.0492979	−0.0246489	+	0.999696i	$0.507847\pi$
$632$	0	0
$633$	7.46410	0.296671
$634$	0	0
$635$	−13.8564	−0.549875
$636$	0	0
$637$	−3.00429	−0.119034
$638$	0	0
$639$	4.41851	0.174793
$640$	0	0
$641$	13.6077	0.537472	0.268736	−	0.963214i	$-0.413394\pi$
$641$	13.6077	0.537472	0.268736	+	0.963214i	$0.413394\pi$
$642$	0	0
$643$	9.80385	0.386626	0.193313	−	0.981137i	$-0.438077\pi$
$643$	9.80385	0.386626	0.193313	+	0.981137i	$0.438077\pi$
$644$	0	0
$645$	−3.93803	−0.155060
$646$	0	0
$647$	−1.79315	−0.0704960	−0.0352480	−	0.999379i	$-0.511222\pi$
$647$	−1.79315	−0.0704960	−0.0352480	+	0.999379i	$0.511222\pi$
$648$	0	0
$649$	34.3923	1.35002
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	21.6937	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$653$	21.6937	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$654$	0	0
$655$	0.832204	0.0325169
$656$	0	0
$657$	6.24871	0.243785
$658$	0	0
$659$	−25.5167	−0.993988	−0.496994	−	0.867754i	$-0.665563\pi$
$659$	−25.5167	−0.993988	−0.496994	+	0.867754i	$0.665563\pi$
$660$	0	0
$661$	−25.8348	−1.00486	−0.502428	−	0.864619i	$-0.667560\pi$
$661$	−25.8348	−1.00486	−0.502428	+	0.864619i	$0.667560\pi$
$662$	0	0
$663$	0.960947	0.0373201
$664$	0	0
$665$	44.7846	1.73667
$666$	0	0
$667$	4.39230	0.170071
$668$	0	0
$669$	−4.14110	−0.160104
$670$	0	0
$671$	24.9754	0.964163
$672$	0	0
$673$	41.3205	1.59279	0.796394	−	0.604778i	$-0.206738\pi$
$673$	41.3205	1.59279	0.796394	+	0.604778i	$0.206738\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−39.9769	−1.53644	−0.768219	−	0.640187i	$-0.778857\pi$
$677$	−39.9769	−1.53644	−0.768219	+	0.640187i	$0.778857\pi$
$678$	0	0
$679$	13.3843	0.513641
$680$	0	0
$681$	−15.4641	−0.592586
$682$	0	0
$683$	12.3397	0.472167	0.236084	−	0.971733i	$-0.424136\pi$
$683$	12.3397	0.472167	0.236084	+	0.971733i	$0.424136\pi$
$684$	0	0
$685$	−29.3939	−1.12308
$686$	0	0
$687$	9.52056	0.363232
$688$	0	0
$689$	4.14359	0.157858
$690$	0	0
$691$	−39.3731	−1.49782	−0.748911	−	0.662671i	$-0.769423\pi$
$691$	−39.3731	−1.49782	−0.748911	+	0.662671i	$0.769423\pi$
$692$	0	0
$693$	45.0518	1.71137
$694$	0	0
$695$	−24.9754	−0.947370
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	−1.85641	−0.0702157
$700$	0	0
$701$	13.2084	0.498874	0.249437	−	0.968391i	$-0.419754\pi$
$701$	13.2084	0.498874	0.249437	+	0.968391i	$0.419754\pi$
$702$	0	0
$703$	−24.9754	−0.941964
$704$	0	0
$705$	17.5692	0.661695
$706$	0	0
$707$	4.39230	0.165190
$708$	0	0
$709$	−41.0865	−1.54304	−0.771518	−	0.636207i	$-0.780502\pi$
$709$	−41.0865	−1.54304	−0.771518	+	0.636207i	$0.780502\pi$
$710$	0	0
$711$	−10.2041	−0.382684
$712$	0	0
$713$	−10.1436	−0.379881
$714$	0	0
$715$	−4.39230	−0.164263
$716$	0	0
$717$	19.5959	0.731823
$718$	0	0
$719$	12.4233	0.463311	0.231656	−	0.972798i	$-0.425586\pi$
$719$	12.4233	0.463311	0.231656	+	0.972798i	$0.425586\pi$
$720$	0	0
$721$	36.7846	1.36993
$722$	0	0
$723$	1.75129	0.0651311
$724$	0	0
$725$	2.44949	0.0909718
$726$	0	0
$727$	−23.4596	−0.870069	−0.435035	−	0.900414i	$-0.643264\pi$
$727$	−23.4596	−0.870069	−0.435035	+	0.900414i	$0.643264\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−7.60770	−0.281381
$732$	0	0
$733$	20.9358	0.773281	0.386641	−	0.922230i	$-0.373635\pi$
$733$	20.9358	0.773281	0.386641	+	0.922230i	$0.373635\pi$
$734$	0	0
$735$	−14.2165	−0.524382
$736$	0	0
$737$	29.3205	1.08003
$738$	0	0
$739$	−2.98076	−0.109649	−0.0548246	−	0.998496i	$-0.517460\pi$
$739$	−2.98076	−0.109649	−0.0548246	+	0.998496i	$0.517460\pi$
$740$	0	0
$741$	1.31268	0.0482224
$742$	0	0
$743$	15.1774	0.556805	0.278403	−	0.960464i	$-0.410195\pi$
$743$	15.1774	0.556805	0.278403	+	0.960464i	$0.410195\pi$
$744$	0	0
$745$	−47.5692	−1.74280
$746$	0	0
$747$	−5.41154	−0.197998
$748$	0	0
$749$	−71.8203	−2.62426
$750$	0	0
$751$	−4.14110	−0.151111	−0.0755555	−	0.997142i	$-0.524073\pi$
$751$	−4.14110	−0.151111	−0.0755555	+	0.997142i	$0.524073\pi$
$752$	0	0
$753$	−15.4641	−0.563543
$754$	0	0
$755$	23.3205	0.848720
$756$	0	0
$757$	30.7338	1.11704	0.558519	−	0.829492i	$-0.311370\pi$
$757$	30.7338	1.11704	0.558519	+	0.829492i	$0.311370\pi$
$758$	0	0
$759$	−6.21166	−0.225469
$760$	0	0
$761$	34.6410	1.25574	0.627868	−	0.778320i	$-0.283928\pi$
$761$	34.6410	1.25574	0.627868	+	0.778320i	$0.283928\pi$
$762$	0	0
$763$	72.1051	2.61038
$764$	0	0
$765$	−20.9086	−0.755952
$766$	0	0
$767$	2.75410	0.0994447
$768$	0	0
$769$	−2.39230	−0.0862687	−0.0431344	−	0.999069i	$-0.513734\pi$
$769$	−2.39230	−0.0862687	−0.0431344	+	0.999069i	$0.513734\pi$
$770$	0	0
$771$	9.46410	0.340841
$772$	0	0
$773$	20.7327	0.745704	0.372852	−	0.927891i	$-0.378380\pi$
$773$	20.7327	0.745704	0.372852	+	0.927891i	$0.378380\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0	0
$777$	14.9282	0.535546
$778$	0	0
$779$	−32.7846	−1.17463
$780$	0	0
$781$	8.48528	0.303627
$782$	0	0
$783$	−9.79796	−0.350150
$784$	0	0
$785$	−24.9282	−0.889726
$786$	0	0
$787$	−35.6603	−1.27115	−0.635575	−	0.772039i	$-0.719237\pi$
$787$	−35.6603	−1.27115	−0.635575	+	0.772039i	$0.719237\pi$
$788$	0	0
$789$	−15.6579	−0.557435
$790$	0	0
$791$	−49.9507	−1.77604
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	19.6077	0.695413
$796$	0	0
$797$	−47.5013	−1.68258	−0.841290	−	0.540584i	$-0.818203\pi$
$797$	−47.5013	−1.68258	−0.841290	+	0.540584i	$0.818203\pi$
$798$	0	0
$799$	33.9411	1.20075
$800$	0	0
$801$	6.24871	0.220787
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−16.9706	−0.598134
$806$	0	0
$807$	5.37945	0.189366
$808$	0	0
$809$	17.0718	0.600212	0.300106	−	0.953906i	$-0.402978\pi$
$809$	17.0718	0.600212	0.300106	+	0.953906i	$0.402978\pi$
$810$	0	0
$811$	46.9808	1.64972	0.824859	−	0.565339i	$-0.191255\pi$
$811$	46.9808	1.64972	0.824859	+	0.565339i	$0.191255\pi$
$812$	0	0
$813$	1.10961	0.0389156
$814$	0	0
$815$	−40.9850	−1.43564
$816$	0	0
$817$	−10.3923	−0.363581
$818$	0	0
$819$	3.60770	0.126063
$820$	0	0
$821$	28.2571	0.986178	0.493089	−	0.869979i	$-0.335868\pi$
$821$	28.2571	0.986178	0.493089	+	0.869979i	$0.335868\pi$
$822$	0	0
$823$	15.7322	0.548391	0.274195	−	0.961674i	$-0.411589\pi$
$823$	15.7322	0.548391	0.274195	+	0.961674i	$0.411589\pi$
$824$	0	0
$825$	−3.46410	−0.120605
$826$	0	0
$827$	23.6603	0.822748	0.411374	−	0.911467i	$-0.365049\pi$
$827$	23.6603	0.822748	0.411374	+	0.911467i	$0.365049\pi$
$828$	0	0
$829$	3.00429	0.104343	0.0521717	−	0.998638i	$-0.483386\pi$
$829$	3.00429	0.104343	0.0521717	+	0.998638i	$0.483386\pi$
$830$	0	0
$831$	3.30890	0.114784
$832$	0	0
$833$	−27.4641	−0.951575
$834$	0	0
$835$	45.9615	1.59056
$836$	0	0
$837$	22.6274	0.782118
$838$	0	0
$839$	−38.3596	−1.32432	−0.662161	−	0.749362i	$-0.730360\pi$
$839$	−38.3596	−1.32432	−0.662161	+	0.749362i	$0.730360\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	0	0
$843$	12.0000	0.413302
$844$	0	0
$845$	31.4916	1.08335
$846$	0	0
$847$	44.0165	1.51242
$848$	0	0
$849$	1.32051	0.0453197
$850$	0	0
$851$	9.46410	0.324425
$852$	0	0
$853$	−13.9663	−0.478196	−0.239098	−	0.970995i	$-0.576852\pi$
$853$	−13.9663	−0.478196	−0.239098	+	0.970995i	$0.576852\pi$
$854$	0	0
$855$	−28.5617	−0.976789
$856$	0	0
$857$	−25.8564	−0.883238	−0.441619	−	0.897203i	$-0.645596\pi$
$857$	−25.8564	−0.883238	−0.441619	+	0.897203i	$0.645596\pi$
$858$	0	0
$859$	20.4449	0.697570	0.348785	−	0.937203i	$-0.386594\pi$
$859$	20.4449	0.697570	0.348785	+	0.937203i	$0.386594\pi$
$860$	0	0
$861$	19.5959	0.667827
$862$	0	0
$863$	2.62536	0.0893681	0.0446841	−	0.999001i	$-0.485772\pi$
$863$	2.62536	0.0893681	0.0446841	+	0.999001i	$0.485772\pi$
$864$	0	0
$865$	−26.7846	−0.910704
$866$	0	0
$867$	−3.66025	−0.124309
$868$	0	0
$869$	−19.5959	−0.664746
$870$	0	0
$871$	2.34795	0.0795574
$872$	0	0
$873$	−8.53590	−0.288896
$874$	0	0
$875$	37.8564	1.27978
$876$	0	0
$877$	22.8033	0.770012	0.385006	−	0.922914i	$-0.374199\pi$
$877$	22.8033	0.770012	0.385006	+	0.922914i	$0.374199\pi$
$878$	0	0
$879$	12.5521	0.423370
$880$	0	0
$881$	−31.8564	−1.07327	−0.536635	−	0.843815i	$-0.680305\pi$
$881$	−31.8564	−1.07327	−0.536635	+	0.843815i	$0.680305\pi$
$882$	0	0
$883$	57.1244	1.92239	0.961194	−	0.275874i	$-0.0889672\pi$
$883$	57.1244	1.92239	0.961194	+	0.275874i	$0.0889672\pi$
$884$	0	0
$885$	13.0325	0.438084
$886$	0	0
$887$	−15.1774	−0.509608	−0.254804	−	0.966993i	$-0.582011\pi$
$887$	−15.1774	−0.509608	−0.254804	+	0.966993i	$0.582011\pi$
$888$	0	0
$889$	21.8564	0.733040
$890$	0	0
$891$	−21.1244	−0.707693
$892$	0	0
$893$	46.3644	1.55153
$894$	0	0
$895$	−5.37945	−0.179815
$896$	0	0
$897$	−0.497423	−0.0166085
$898$	0	0
$899$	−13.8564	−0.462137
$900$	0	0
$901$	37.8792	1.26194
$902$	0	0
$903$	6.21166	0.206711
$904$	0	0
$905$	55.8564	1.85673
$906$	0	0
$907$	29.9090	0.993111	0.496555	−	0.868005i	$-0.334598\pi$
$907$	29.9090	0.993111	0.496555	+	0.868005i	$0.334598\pi$
$908$	0	0
$909$	−2.80122	−0.0929106
$910$	0	0
$911$	24.1432	0.799899	0.399949	−	0.916537i	$-0.369028\pi$
$911$	24.1432	0.799899	0.399949	+	0.916537i	$0.369028\pi$
$912$	0	0
$913$	−10.3923	−0.343935
$914$	0	0
$915$	9.46410	0.312874
$916$	0	0
$917$	−1.31268	−0.0433484
$918$	0	0
$919$	18.3576	0.605560	0.302780	−	0.953060i	$-0.402085\pi$
$919$	18.3576	0.605560	0.302780	+	0.953060i	$0.402085\pi$
$920$	0	0
$921$	22.6795	0.747315
$922$	0	0
$923$	0.679492	0.0223657
$924$	0	0
$925$	5.27792	0.173537
$926$	0	0
$927$	−23.4596	−0.770515
$928$	0	0
$929$	3.46410	0.113653	0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410	0.113653	0.0568267	+	0.998384i	$0.481902\pi$
$930$	0	0
$931$	−37.5167	−1.22956
$932$	0	0
$933$	−11.1106	−0.363746
$934$	0	0
$935$	−40.1528	−1.31314
$936$	0	0
$937$	9.17691	0.299797	0.149898	−	0.988701i	$-0.452105\pi$
$937$	9.17691	0.299797	0.149898	+	0.988701i	$0.452105\pi$
$938$	0	0
$939$	9.35898	0.305419
$940$	0	0
$941$	−26.5927	−0.866896	−0.433448	−	0.901179i	$-0.642703\pi$
$941$	−26.5927	−0.866896	−0.433448	+	0.901179i	$0.642703\pi$
$942$	0	0
$943$	12.4233	0.404559
$944$	0	0
$945$	37.8564	1.23147
$946$	0	0
$947$	19.2679	0.626124	0.313062	−	0.949733i	$-0.398645\pi$
$947$	19.2679	0.626124	0.313062	+	0.949733i	$0.398645\pi$
$948$	0	0
$949$	0.960947	0.0311936
$950$	0	0
$951$	7.04386	0.228413
$952$	0	0
$953$	−53.5692	−1.73528	−0.867639	−	0.497195i	$-0.834363\pi$
$953$	−53.5692	−1.73528	−0.867639	+	0.497195i	$0.834363\pi$
$954$	0	0
$955$	−41.5692	−1.34515
$956$	0	0
$957$	−8.48528	−0.274290
$958$	0	0
$959$	46.3644	1.49719
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	45.8038	1.47601
$964$	0	0
$965$	45.0518	1.45027
$966$	0	0
$967$	−27.0459	−0.869738	−0.434869	−	0.900494i	$-0.643205\pi$
$967$	−27.0459	−0.869738	−0.434869	+	0.900494i	$0.643205\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−39.3731	−1.26354	−0.631771	−	0.775155i	$-0.717672\pi$
$971$	−39.3731	−1.26354	−0.631771	+	0.775155i	$0.717672\pi$
$972$	0	0
$973$	39.3949	1.26294
$974$	0	0
$975$	−0.277401	−0.00888396
$976$	0	0
$977$	44.5359	1.42483	0.712415	−	0.701759i	$-0.247601\pi$
$977$	44.5359	1.42483	0.712415	+	0.701759i	$0.247601\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−45.9855	−1.46820
$982$	0	0
$983$	52.7048	1.68102	0.840512	−	0.541793i	$-0.182255\pi$
$983$	52.7048	1.68102	0.840512	+	0.541793i	$0.182255\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	−27.7128	−0.882109
$988$	0	0
$989$	3.93803	0.125222
$990$	0	0
$991$	61.8193	1.96375	0.981877	−	0.189521i	$-0.0606936\pi$
$991$	61.8193	1.96375	0.981877	+	0.189521i	$0.0606936\pi$
$992$	0	0
$993$	−19.4641	−0.617675
$994$	0	0
$995$	38.5359	1.22167
$996$	0	0
$997$	−21.4906	−0.680614	−0.340307	−	0.940314i	$-0.610531\pi$
$997$	−21.4906	−0.680614	−0.340307	+	0.940314i	$0.610531\pi$
$998$	0	0
$999$	−21.1117	−0.667944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.a.g.1.3		4
3.2	odd	2		9216.2.a.bk.1.4		4
4.3	odd	2		1024.2.a.j.1.1		4
8.3	odd	2	inner	1024.2.a.g.1.4		4
8.5	even	2		1024.2.a.j.1.2		4
12.11	even	2		9216.2.a.bb.1.3		4
16.3	odd	4		1024.2.b.h.513.4		8
16.5	even	4		1024.2.b.h.513.3		8
16.11	odd	4		1024.2.b.h.513.5		8
16.13	even	4		1024.2.b.h.513.6		8
24.5	odd	2		9216.2.a.bb.1.2		4
24.11	even	2		9216.2.a.bk.1.1		4
32.3	odd	8		256.2.e.b.65.2	yes	8
32.5	even	8		256.2.e.a.193.2	yes	8
32.11	odd	8		256.2.e.b.193.2	yes	8
32.13	even	8		256.2.e.a.65.2	✓	8
32.19	odd	8		256.2.e.a.65.3	yes	8
32.21	even	8		256.2.e.b.193.3	yes	8
32.27	odd	8		256.2.e.a.193.3	yes	8
32.29	even	8		256.2.e.b.65.3	yes	8
96.5	odd	8		2304.2.k.f.1729.4		8
96.11	even	8		2304.2.k.k.1729.1		8
96.29	odd	8		2304.2.k.k.577.1		8
96.35	even	8		2304.2.k.k.577.2		8
96.53	odd	8		2304.2.k.k.1729.2		8
96.59	even	8		2304.2.k.f.1729.3		8
96.77	odd	8		2304.2.k.f.577.3		8
96.83	even	8		2304.2.k.f.577.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.e.a.65.2	✓	8	32.13	even	8
256.2.e.a.65.3	yes	8	32.19	odd	8
256.2.e.a.193.2	yes	8	32.5	even	8
256.2.e.a.193.3	yes	8	32.27	odd	8
256.2.e.b.65.2	yes	8	32.3	odd	8
256.2.e.b.65.3	yes	8	32.29	even	8
256.2.e.b.193.2	yes	8	32.11	odd	8
256.2.e.b.193.3	yes	8	32.21	even	8
1024.2.a.g.1.3		4	1.1	even	1	trivial
1024.2.a.g.1.4		4	8.3	odd	2	inner
1024.2.a.j.1.1		4	4.3	odd	2
1024.2.a.j.1.2		4	8.5	even	2
1024.2.b.h.513.3		8	16.5	even	4
1024.2.b.h.513.4		8	16.3	odd	4
1024.2.b.h.513.5		8	16.11	odd	4
1024.2.b.h.513.6		8	16.13	even	4
2304.2.k.f.577.3		8	96.77	odd	8
2304.2.k.f.577.4		8	96.83	even	8
2304.2.k.f.1729.3		8	96.59	even	8
2304.2.k.f.1729.4		8	96.5	odd	8
2304.2.k.k.577.1		8	96.29	odd	8
2304.2.k.k.577.2		8	96.35	even	8
2304.2.k.k.1729.1		8	96.11	even	8
2304.2.k.k.1729.2		8	96.53	odd	8
9216.2.a.bb.1.2		4	24.5	odd	2
9216.2.a.bb.1.3		4	12.11	even	2
9216.2.a.bk.1.1		4	24.11	even	2
9216.2.a.bk.1.4		4	3.2	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)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -2.44949 q^{5} +3.86370 q^{7} -2.46410 q^{9} -4.73205 q^{11} -0.378937 q^{13} -1.79315 q^{15} -3.46410 q^{17} -4.73205 q^{19} +2.82843 q^{21} +1.79315 q^{23} +1.00000 q^{25} -4.00000 q^{27} +2.44949 q^{29} -5.65685 q^{31} -3.46410 q^{33} -9.46410 q^{35} +5.27792 q^{37} -0.277401 q^{39} +6.92820 q^{41} +2.19615 q^{43} +6.03579 q^{45} -9.79796 q^{47} +7.92820 q^{49} -2.53590 q^{51} -10.9348 q^{53} +11.5911 q^{55} -3.46410 q^{57} -7.26795 q^{59} -5.27792 q^{61} -9.52056 q^{63} +0.928203 q^{65} -6.19615 q^{67} +1.31268 q^{69} -1.79315 q^{71} -2.53590 q^{73} +0.732051 q^{75} -18.2832 q^{77} +4.14110 q^{79} +4.46410 q^{81} +2.19615 q^{83} +8.48528 q^{85} +1.79315 q^{87} -2.53590 q^{89} -1.46410 q^{91} -4.14110 q^{93} +11.5911 q^{95} +3.46410 q^{97} +11.6603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9} - 12 q^{11} - 12 q^{19} + 4 q^{25} - 16 q^{27} - 24 q^{35} - 12 q^{43} + 4 q^{49} - 24 q^{51} - 36 q^{59} - 24 q^{65} - 4 q^{67} - 24 q^{73} - 4 q^{75} + 4 q^{81} - 12 q^{83} - 24 q^{89}+ \cdots + 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 - 12 * q^11 - 12 * q^19 + 4 * q^25 - 16 * q^27 - 24 * q^35 - 12 * q^43 + 4 * q^49 - 24 * q^51 - 36 * q^59 - 24 * q^65 - 4 * q^67 - 24 * q^73 - 4 * q^75 + 4 * q^81 - 12 * q^83 - 24 * q^89 + 8 * q^91 + 12 * q^99

Introduction

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