LMFDB - Embedded newform 768.2.s.a.431.28

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,2,Mod(47,768)] 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("768.47"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 11, 8])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.s (of order $16$, degree $8$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$240$
Relative dimension:	$30$ over $\Q(\zeta_{16})$
Twist minimal:	no (minimal twist has level 192)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			431.28
Character	$\chi$	$=$	768.431
Dual form			768.2.s.a.335.28

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.67061 - 0.457221i) q^{3} +(-1.45822 + 2.18239i) q^{5} +(1.27779 + 3.08486i) q^{7} +(2.58190 - 1.52768i) q^{9} +(-2.31748 + 0.460975i) q^{11} +(-5.55652 + 3.71275i) q^{13} +(-1.43829 + 4.31265i) q^{15} +(-3.42067 + 3.42067i) q^{17} +(1.23054 - 0.822221i) q^{19} +(3.54516 + 4.56938i) q^{21} +(1.87568 - 4.52828i) q^{23} +(-0.722975 - 1.74542i) q^{25} +(3.61486 - 3.73266i) q^{27} +(-0.760797 + 3.82478i) q^{29} +0.0497279 q^{31} +(-3.66084 + 1.82971i) q^{33} +(-8.59567 - 1.70978i) q^{35} +(5.53510 - 8.28386i) q^{37} +(-7.58524 + 8.74312i) q^{39} +(2.28385 + 0.946002i) q^{41} +(0.522190 - 0.103870i) q^{43} +(-0.430996 + 7.86240i) q^{45} +(2.51930 + 2.51930i) q^{47} +(-2.93388 + 2.93388i) q^{49} +(-4.15061 + 7.27862i) q^{51} +(1.79038 + 9.00087i) q^{53} +(2.37338 - 5.72983i) q^{55} +(1.67982 - 1.93624i) q^{57} +(-2.56739 - 1.71547i) q^{59} +(-0.935842 + 4.70479i) q^{61} +(8.01181 + 6.01274i) q^{63} -17.5405i q^{65} +(3.65093 + 0.726215i) q^{67} +(1.06310 - 8.42261i) q^{69} +(12.6483 - 5.23910i) q^{71} +(13.6168 + 5.64025i) q^{73} +(-2.00585 - 2.58536i) q^{75} +(-4.38330 - 6.56007i) q^{77} +(1.69637 + 1.69637i) q^{79} +(4.33239 - 7.88863i) q^{81} +(5.63016 + 8.42612i) q^{83} +(-2.47712 - 12.4533i) q^{85} +(0.477776 + 6.73759i) q^{87} +(8.26075 - 3.42172i) q^{89} +(-18.5534 - 12.3970i) q^{91} +(0.0830760 - 0.0227366i) q^{93} +3.88450i q^{95} -3.93119i q^{97} +(-5.27927 + 4.73055i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 240 q + 8 q^{3} + 16 q^{7} - 8 q^{9} - 16 q^{13} + 8 q^{15} + 16 q^{19} - 8 q^{21} - 16 q^{25} + 8 q^{27} + 32 q^{31} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 8 q^{45} - 16 q^{49} + 8 q^{51} + 80 q^{55} - 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) $ 240 * q + 8 * q^3 + 16 * q^7 - 8 * q^9 - 16 * q^13 + 8 * q^15 + 16 * q^19 - 8 * q^21 - 16 * q^25 + 8 * q^27 + 32 * q^31 - 16 * q^37 + 8 * q^39 + 16 * q^43 - 8 * q^45 - 16 * q^49 + 8 * q^51 + 80 * q^55 - 8 * q^57 - 16 * q^61 + 144 * q^67 - 8 * q^69 - 16 * q^73 + 8 * q^75 + 48 * q^79 - 8 * q^81 - 16 * q^85 + 8 * q^87 + 16 * q^91 - 32 * q^93 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{16}\right)$

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
$3$	1.67061	−	0.457221i	0.964529	−	0.263977i
$3$	1.67061	−	0.457221i	0.964529	−	0.263977i
$4$		0			0
$5$	−1.45822	+	2.18239i	−0.652138	+	0.975993i	0.347133	+	0.937816i	$0.387155\pi$
$5$	−1.45822	+	2.18239i	−0.652138	+	0.975993i	−0.999271	+	0.0381770i	$0.987845\pi$
$6$		0			0
$7$	1.27779	+	3.08486i	0.482960	+	1.16597i	0.958197	+	0.286111i	$0.0923625\pi$
$7$	1.27779	+	3.08486i	0.482960	+	1.16597i	−0.475237	+	0.879858i	$0.657638\pi$
$8$		0			0
$9$	2.58190	−	1.52768i	0.860632	−	0.509227i
$10$		0			0
$11$	−2.31748	+	0.460975i	−0.698746	+	0.138989i	−0.531666	−	0.846954i	$-0.678434\pi$
$11$	−2.31748	+	0.460975i	−0.698746	+	0.138989i	−0.167080	+	0.985943i	$0.553434\pi$
$12$		0			0
$13$	−5.55652	+	3.71275i	−1.54110	+	1.02973i	−0.561811	+	0.827266i	$0.689895\pi$
$13$	−5.55652	+	3.71275i	−1.54110	+	1.02973i	−0.979289	+	0.202465i	$0.935105\pi$
$14$		0			0
$15$	−1.43829	+	4.31265i	−0.371366	+	1.11352i
$16$		0			0
$17$	−3.42067	+	3.42067i	−0.829634	+	0.829634i	−0.987466	−	0.157832i	$-0.949550\pi$
$17$	−3.42067	+	3.42067i	−0.829634	+	0.829634i	0.157832	+	0.987466i	$0.449550\pi$
$18$		0			0
$19$	1.23054	−	0.822221i	0.282305	−	0.188630i	−0.406353	−	0.913716i	$-0.633200\pi$
$19$	1.23054	−	0.822221i	0.282305	−	0.188630i	0.688659	+	0.725086i	$0.258200\pi$
$20$		0			0
$21$	3.54516	+	4.56938i	0.773618	+	0.997120i
$22$		0			0
$23$	1.87568	−	4.52828i	0.391105	−	0.944212i	−0.598594	−	0.801052i	$-0.704274\pi$
$23$	1.87568	−	4.52828i	0.391105	−	0.944212i	0.989700	−	0.143160i	$-0.0457263\pi$
$24$		0			0
$25$	−0.722975	−	1.74542i	−0.144595	−	0.349083i
$26$		0			0
$27$	3.61486	−	3.73266i	0.695681	−	0.718351i
$28$		0			0
$29$	−0.760797	+	3.82478i	−0.141276	+	0.710245i	0.843598	+	0.536975i	$0.180433\pi$
$29$	−0.760797	+	3.82478i	−0.141276	+	0.710245i	−0.984874	+	0.173269i	$0.944567\pi$
$30$		0			0
$31$	0.0497279			0.00893139			0.00446569	−	0.999990i	$-0.498579\pi$
$31$	0.0497279			0.00893139			0.00446569	+	0.999990i	$0.498579\pi$
$32$		0			0
$33$	−3.66084	+	1.82971i	−0.637271	+	0.318512i
$34$		0			0
$35$	−8.59567	−	1.70978i	−1.45293	−	0.289006i
$36$		0			0
$37$	5.53510	−	8.28386i	0.909965	−	1.36186i	−0.0221727	−	0.999754i	$-0.507058\pi$
$37$	5.53510	−	8.28386i	0.909965	−	1.36186i	0.932138	−	0.362104i	$-0.117942\pi$
$38$		0			0
$39$	−7.58524	+	8.74312i	−1.21461	+	1.40002i
$40$		0			0
$41$	2.28385	+	0.946002i	0.356678	+	0.147741i	0.553825	−	0.832633i	$-0.313168\pi$
$41$	2.28385	+	0.946002i	0.356678	+	0.147741i	−0.197147	+	0.980374i	$0.563168\pi$
$42$		0			0
$43$	0.522190	−	0.103870i	0.0796333	−	0.0158400i	−0.155113	−	0.987897i	$-0.549574\pi$
$43$	0.522190	−	0.103870i	0.0796333	−	0.0158400i	0.234746	+	0.972057i	$0.424574\pi$
$44$		0			0
$45$	−0.430996	+	7.86240i	−0.0642491	+	1.17206i
$46$		0			0
$47$	2.51930	+	2.51930i	0.367477	+	0.367477i	0.866556	−	0.499079i	$-0.166328\pi$
$47$	2.51930	+	2.51930i	0.367477	+	0.367477i	−0.499079	+	0.866556i	$0.666328\pi$
$48$		0			0
$49$	−2.93388	+	2.93388i	−0.419126	+	0.419126i
$50$		0			0
$51$	−4.15061	+	7.27862i	−0.581202	+	1.01921i
$52$		0			0
$53$	1.79038	+	9.00087i	0.245928	+	1.23636i	0.884404	+	0.466722i	$0.154565\pi$
$53$	1.79038	+	9.00087i	0.245928	+	1.23636i	−0.638476	+	0.769642i	$0.720435\pi$
$54$		0			0
$55$	2.37338	−	5.72983i	0.320026	−	0.772611i
$56$		0			0
$57$	1.67982	−	1.93624i	0.222498	−	0.256461i
$58$		0			0
$59$	−2.56739	−	1.71547i	−0.334245	−	0.223336i	0.377109	−	0.926169i	$-0.376918\pi$
$59$	−2.56739	−	1.71547i	−0.334245	−	0.223336i	−0.711355	+	0.702833i	$0.751918\pi$
$60$		0			0
$61$	−0.935842	+	4.70479i	−0.119822	+	0.602387i	0.873482	+	0.486857i	$0.161857\pi$
$61$	−0.935842	+	4.70479i	−0.119822	+	0.602387i	−0.993304	+	0.115530i	$0.963143\pi$
$62$		0			0
$63$	8.01181	+	6.01274i	1.00939	+	0.757534i
$64$		0			0
$65$		−	17.5405i		−	2.17563i
$66$		0			0
$67$	3.65093	+	0.726215i	0.446032	+	0.0887213i	0.412996	−	0.910733i	$-0.364482\pi$
$67$	3.65093	+	0.726215i	0.446032	+	0.0887213i	0.0330361	+	0.999454i	$0.489482\pi$
$68$		0			0
$69$	1.06310	−	8.42261i	0.127982	−	1.01396i
$70$		0			0
$71$	12.6483	−	5.23910i	1.50108	−	0.621767i	0.527384	−	0.849627i	$-0.323173\pi$
$71$	12.6483	−	5.23910i	1.50108	−	0.621767i	0.973694	+	0.227860i	$0.0731728\pi$
$72$		0			0
$73$	13.6168	+	5.64025i	1.59372	+	0.660142i	0.990510	−	0.137439i	$-0.0438870\pi$
$73$	13.6168	+	5.64025i	1.59372	+	0.660142i	0.603213	+	0.797580i	$0.293887\pi$
$74$		0			0
$75$	−2.00585	−	2.58536i	−0.231616	−	0.298531i
$76$		0			0
$77$	−4.38330	−	6.56007i	−0.499523	−	0.747589i
$78$		0			0
$79$	1.69637	+	1.69637i	0.190856	+	0.190856i	0.796066	−	0.605210i	$-0.206911\pi$
$79$	1.69637	+	1.69637i	0.190856	+	0.190856i	−0.605210	+	0.796066i	$0.706911\pi$
$80$		0			0
$81$	4.33239	−	7.88863i	0.481376	−	0.876514i
$82$		0			0
$83$	5.63016	+	8.42612i	0.617990	+	0.924887i	1.00000		0.000728896i	$0.000232015\pi$
$83$	5.63016	+	8.42612i	0.617990	+	0.924887i	−0.382010	+	0.924158i	$0.624768\pi$
$84$		0			0
$85$	−2.47712	−	12.4533i	−0.268681	−	1.35075i
$86$		0			0
$87$	0.477776	+	6.73759i	0.0512229	+	0.722345i
$88$		0			0
$89$	8.26075	−	3.42172i	0.875638	−	0.362701i	0.100834	−	0.994903i	$-0.467849\pi$
$89$	8.26075	−	3.42172i	0.875638	−	0.362701i	0.774804	+	0.632202i	$0.217849\pi$
$90$		0			0
$91$	−18.5534	−	12.3970i	−1.94492	−	1.29956i
$92$		0			0
$93$	0.0830760	−	0.0227366i	0.00861458	−	0.00235768i
$94$		0			0
$95$			3.88450i			0.398541i
$96$		0			0
$97$		−	3.93119i		−	0.399152i	−0.979882	−	0.199576i	$-0.936044\pi$
$97$		−	3.93119i		−	0.399152i	0.979882	−	0.199576i	$-0.0639565\pi$
$98$		0			0
$99$	−5.27927	+	4.73055i	−0.530586	+	0.475439i
$100$		0			0
$101$	−1.14688	−	0.766317i	−0.114118	−	0.0762514i	0.497201	−	0.867635i	$-0.334361\pi$
$101$	−1.14688	−	0.766317i	−0.114118	−	0.0762514i	−0.611320	+	0.791384i	$0.709361\pi$
$102$		0			0
$103$	−3.42308	+	1.41789i	−0.337286	+	0.139708i	−0.544897	−	0.838503i	$-0.683431\pi$
$103$	−3.42308	+	1.41789i	−0.337286	+	0.139708i	0.207611	+	0.978212i	$0.433431\pi$
$104$		0			0
$105$	−15.1418	+	1.07373i	−1.47769	+	0.104786i
$106$		0			0
$107$	−0.353820	−	1.77877i	−0.0342051	−	0.171960i	0.959907	−	0.280319i	$-0.0904404\pi$
$107$	−0.353820	−	1.77877i	−0.0342051	−	0.171960i	−0.994112	+	0.108359i	$0.965440\pi$
$108$		0			0
$109$	−4.27438	−	6.39706i	−0.409411	−	0.612728i	0.568264	−	0.822846i	$-0.307615\pi$
$109$	−4.27438	−	6.39706i	−0.409411	−	0.612728i	−0.977676	+	0.210118i	$0.932615\pi$
$110$		0			0
$111$	5.45945	−	16.3699i	0.518188	−	1.55376i
$112$		0			0
$113$	7.68145	+	7.68145i	0.722610	+	0.722610i	0.969136	−	0.246526i	$-0.0792891\pi$
$113$	7.68145	+	7.68145i	0.722610	+	0.722610i	−0.246526	+	0.969136i	$0.579289\pi$
$114$		0			0
$115$	7.14731	+	10.6967i	0.666490	+	0.997472i
$116$		0			0
$117$	−8.67447	+	18.0745i	−0.801955	+	1.67099i
$118$		0			0
$119$	−14.9232	−	6.18139i	−1.36801	−	0.566647i
$120$		0			0
$121$	−5.00447	+	2.07292i	−0.454952	+	0.188447i
$122$		0			0
$123$	4.24796	+	0.536178i	0.383026	+	0.0483456i
$124$		0			0
$125$	−8.00807	−	1.59290i	−0.716263	−	0.142474i
$126$		0			0
$127$		−	2.11042i		−	0.187270i	−0.995607	−	0.0936348i	$-0.970151\pi$
$127$		−	2.11042i		−	0.187270i	0.995607	−	0.0936348i	$-0.0298486\pi$
$128$		0			0
$129$	0.824886	−	0.412283i	0.0726272	−	0.0362995i
$130$		0			0
$131$	−2.13696	+	10.7432i	−0.186707	+	0.938640i	0.767854	+	0.640625i	$0.221325\pi$
$131$	−2.13696	+	10.7432i	−0.186707	+	0.938640i	−0.954561	+	0.298015i	$0.903675\pi$
$132$		0			0
$133$	4.10881	+	2.74542i	0.356279	+	0.238058i
$134$		0			0
$135$	2.87483	+	13.3321i	0.247426	+	1.14744i
$136$		0			0
$137$	−1.35133	+	3.26240i	−0.115452	+	0.278726i	−0.971034	−	0.238942i	$-0.923200\pi$
$137$	−1.35133	+	3.26240i	−0.115452	+	0.278726i	0.855582	+	0.517667i	$0.173200\pi$
$138$		0			0
$139$	−2.25479	−	11.3356i	−0.191249	−	0.961475i	−0.950511	−	0.310690i	$-0.899440\pi$
$139$	−2.25479	−	11.3356i	−0.191249	−	0.961475i	0.759262	−	0.650785i	$-0.225560\pi$
$140$		0			0
$141$	5.36065	+	3.05689i	0.451448	+	0.257437i
$142$		0			0
$143$	11.1656	−	11.1656i	0.933716	−	0.933716i
$144$		0			0
$145$	−7.23774	−	7.23774i	−0.601062	−	0.601062i
$146$		0			0
$147$	−3.55995	+	6.24281i	−0.293619	+	0.514898i
$148$		0			0
$149$	−9.41669	+	1.87310i	−0.771445	+	0.153450i	−0.565092	−	0.825028i	$-0.691159\pi$
$149$	−9.41669	+	1.87310i	−0.771445	+	0.153450i	−0.206353	+	0.978478i	$0.566159\pi$
$150$		0			0
$151$	−15.4374	−	6.39437i	−1.25628	−	0.520367i	−0.347511	−	0.937676i	$-0.612973\pi$
$151$	−15.4374	−	6.39437i	−1.25628	−	0.520367i	−0.908765	+	0.417309i	$0.862973\pi$
$152$		0			0
$153$	−3.60613	+	14.0575i	−0.291538	+	1.13648i
$154$		0			0
$155$	−0.0725144	+	0.108525i	−0.00582449	+	0.00871697i
$156$		0			0
$157$	−1.10477	−	0.219752i	−0.0881699	−	0.0175381i	0.150808	−	0.988563i	$-0.451812\pi$
$157$	−1.10477	−	0.219752i	−0.0881699	−	0.0175381i	−0.238978	+	0.971025i	$0.576812\pi$
$158$		0			0
$159$	7.10643	+	14.2184i	0.563576	+	1.12759i
$160$		0			0
$161$	16.3659			1.28981
$162$		0			0
$163$	3.95820	−	19.8992i	0.310030	−	1.55863i	−0.440477	−	0.897764i	$-0.645191\pi$
$163$	3.95820	−	19.8992i	0.310030	−	1.55863i	0.750507	−	0.660862i	$-0.229809\pi$
$164$		0			0
$165$	1.34519	−	10.6575i	0.104723	−	0.829685i
$166$		0			0
$167$	0.109882	+	0.265279i	0.00850292	+	0.0205279i	0.928074	−	0.372396i	$-0.121464\pi$
$167$	0.109882	+	0.265279i	0.00850292	+	0.0205279i	−0.919571	+	0.392924i	$0.871464\pi$
$168$		0			0
$169$	12.1155	−	29.2494i	0.931963	−	2.24996i
$170$		0			0
$171$	1.92104	−	4.00276i	0.146905	−	0.306099i
$172$		0			0
$173$	−11.1530	+	7.45221i	−0.847948	+	0.566581i	−0.901891	−	0.431963i	$-0.857821\pi$
$173$	−11.1530	+	7.45221i	−0.847948	+	0.566581i	0.0539429	+	0.998544i	$0.482821\pi$
$174$		0			0
$175$	4.46056	−	4.46056i	0.337187	−	0.337187i
$176$		0			0
$177$	−5.07346	−	1.69203i	−0.381345	−	0.127181i
$178$		0			0
$179$	13.7986	−	9.21991i	1.03135	−	0.689128i	0.0798628	−	0.996806i	$-0.474552\pi$
$179$	13.7986	−	9.21991i	1.03135	−	0.689128i	0.951491	+	0.307677i	$0.0995518\pi$
$180$		0			0
$181$	−0.368628	+	0.0733247i	−0.0273999	+	0.00545018i	−0.208771	−	0.977965i	$-0.566946\pi$
$181$	−0.368628	+	0.0733247i	−0.0273999	+	0.00545018i	0.181371	+	0.983415i	$0.441946\pi$
$182$		0			0
$183$	0.587703	+	8.28778i	0.0434442	+	0.612650i
$184$		0			0
$185$	10.0072	+	24.1595i	0.735742	+	1.77624i
$186$		0			0
$187$	6.35048	−	9.50417i	0.464393	−	0.695014i
$188$		0			0
$189$	16.1338	+	6.38180i	1.17356	+	0.464207i
$190$		0			0
$191$	4.88024			0.353122			0.176561	−	0.984290i	$-0.443503\pi$
$191$	4.88024			0.353122			0.176561	+	0.984290i	$0.443503\pi$
$192$		0			0
$193$	−7.52929			−0.541970			−0.270985	−	0.962584i	$-0.587349\pi$
$193$	−7.52929			−0.541970			−0.270985	+	0.962584i	$0.587349\pi$
$194$		0			0
$195$	−8.01988	−	29.3034i	−0.574316	−	2.09846i
$196$		0			0
$197$	8.61841	−	12.8984i	0.614036	−	0.918970i	−0.385957	−	0.922517i	$-0.626129\pi$
$197$	8.61841	−	12.8984i	0.614036	−	0.918970i	0.999994	+	0.00354629i	$0.00112882\pi$
$198$		0			0
$199$	−4.43073	−	10.6967i	−0.314086	−	0.758271i	−0.999545	−	0.0301595i	$-0.990398\pi$
$199$	−4.43073	−	10.6967i	−0.314086	−	0.758271i	0.685459	−	0.728111i	$-0.259602\pi$
$200$		0			0
$201$	6.43133	−	0.456058i	0.453631	−	0.0321679i
$202$		0			0
$203$	−12.7711	+	2.54032i	−0.896354	+	0.178296i
$204$		0			0
$205$	−5.39491	+	3.60476i	−0.376797	+	0.251768i
$206$		0			0
$207$	−2.07496	−	14.5570i	−0.144220	−	1.01178i
$208$		0			0
$209$	−2.47273	+	2.47273i	−0.171042	+	0.171042i
$210$		0			0
$211$	−17.8899	+	11.9536i	−1.23159	+	0.822923i	−0.989102	−	0.147232i	$-0.952964\pi$
$211$	−17.8899	+	11.9536i	−1.23159	+	0.822923i	−0.242489	+	0.970154i	$0.577964\pi$
$212$		0			0
$213$	18.7350	−	14.5356i	1.28370	−	0.995962i
$214$		0			0
$215$	−0.534786	+	1.29109i	−0.0364721	+	0.0880514i
$216$		0			0
$217$	0.0635419	+	0.153404i	0.00431350	+	0.0104137i
$218$		0			0
$219$	25.3272	+	3.19680i	1.71145	+	0.216020i
$220$		0			0
$221$	6.30693	−	31.7071i	0.424250	−	2.13285i
$222$		0			0
$223$	1.45744			0.0975977			0.0487989	−	0.998809i	$-0.484461\pi$
$223$	1.45744			0.0975977			0.0487989	+	0.998809i	$0.484461\pi$
$224$		0			0
$225$	−4.53309	−	3.40201i	−0.302206	−	0.226801i
$226$		0			0
$227$	−19.3684	−	3.85261i	−1.28552	−	0.255707i	−0.495406	−	0.868662i	$-0.664981\pi$
$227$	−19.3684	−	3.85261i	−1.28552	−	0.255707i	−0.790118	+	0.612955i	$0.789981\pi$
$228$		0			0
$229$	0.936069	−	1.40093i	0.0618572	−	0.0925758i	−0.799252	−	0.600997i	$-0.794771\pi$
$229$	0.936069	−	1.40093i	0.0618572	−	0.0925758i	0.861109	+	0.508421i	$0.169771\pi$
$230$		0			0
$231$	−10.3222	−	8.95520i	−0.679151	−	0.589209i
$232$		0			0
$233$	−14.4154	−	5.97104i	−0.944382	−	0.391176i	−0.143266	−	0.989684i	$-0.545760\pi$
$233$	−14.4154	−	5.97104i	−0.944382	−	0.391176i	−0.801117	+	0.598508i	$0.795760\pi$
$234$		0			0
$235$	−9.17178	+	1.82438i	−0.598301	+	0.119009i
$236$		0			0
$237$	3.60959	+	2.05836i	0.234468	+	0.133705i
$238$		0			0
$239$	−0.690266	−	0.690266i	−0.0446496	−	0.0446496i	0.684429	−	0.729079i	$-0.260051\pi$
$239$	−0.690266	−	0.690266i	−0.0446496	−	0.0446496i	−0.729079	+	0.684429i	$0.760051\pi$
$240$		0			0
$241$	−10.6103	+	10.6103i	−0.683472	+	0.683472i	−0.960781	−	0.277309i	$-0.910558\pi$
$241$	−10.6103	+	10.6103i	−0.683472	+	0.683472i	0.277309	+	0.960781i	$0.410558\pi$
$242$		0			0
$243$	3.63090	−	15.1597i	0.232922	−	0.972495i
$244$		0			0
$245$	−2.12461	−	10.6811i	−0.135736	−	0.682391i
$246$		0			0
$247$	−3.78482	+	9.13737i	−0.240822	+	0.581397i
$248$		0			0
$249$	13.2584	+	11.5026i	0.840218	+	0.728945i
$250$		0			0
$251$	22.7440	+	15.1971i	1.43559	+	0.959232i	0.998201	+	0.0599600i	$0.0190973\pi$
$251$	22.7440	+	15.1971i	1.43559	+	0.959232i	0.437391	+	0.899272i	$0.355903\pi$
$252$		0			0
$253$	−2.25941	+	11.3588i	−0.142048	+	0.714124i
$254$		0			0
$255$	−9.83224	−	19.6721i	−0.615719	−	1.23191i
$256$		0			0
$257$		−	4.51818i		−	0.281836i	−0.990021	−	0.140918i	$-0.954995\pi$
$257$		−	4.51818i		−	0.281836i	0.990021	−	0.140918i	$-0.0450054\pi$
$258$		0			0
$259$	32.6273	+	6.48997i	2.02736	+	0.403267i
$260$		0			0
$261$	3.87875	+	11.0375i	0.240088	+	0.683201i
$262$		0			0
$263$	2.04943	−	0.848904i	0.126374	−	0.0523456i	−0.318601	−	0.947889i	$-0.603213\pi$
$263$	2.04943	−	0.848904i	0.126374	−	0.0523456i	0.444974	+	0.895543i	$0.353213\pi$
$264$		0			0
$265$	−22.2541	−	9.21797i	−1.36706	−	0.566255i
$266$		0			0
$267$	12.2360	−	9.49335i	0.748833	−	0.580984i
$268$		0			0
$269$	12.8156	+	19.1799i	0.781379	+	1.16942i	0.981839	+	0.189715i	$0.0607564\pi$
$269$	12.8156	+	19.1799i	0.781379	+	1.16942i	−0.200460	+	0.979702i	$0.564244\pi$
$270$		0			0
$271$	4.68156	+	4.68156i	0.284384	+	0.284384i	0.834855	−	0.550470i	$-0.185552\pi$
$271$	4.68156	+	4.68156i	0.284384	+	0.284384i	−0.550470	+	0.834855i	$0.685552\pi$
$272$		0			0
$273$	−36.6637	−	12.2275i	−2.21899	−	0.740045i
$274$		0			0
$275$	2.48007	+	3.71169i	0.149554	+	0.223823i
$276$		0			0
$277$	0.351539	+	1.76730i	0.0211219	+	0.106187i	0.989908	−	0.141711i	$-0.0452603\pi$
$277$	0.351539	+	1.76730i	0.0211219	+	0.106187i	−0.968786	+	0.247898i	$0.920260\pi$
$278$		0			0
$279$	0.128392	−	0.0759683i	0.00768664	−	0.00454810i
$280$		0			0
$281$	6.33662	−	2.62472i	0.378011	−	0.156577i	−0.185584	−	0.982628i	$-0.559418\pi$
$281$	6.33662	−	2.62472i	0.378011	−	0.156577i	0.563595	+	0.826051i	$0.309418\pi$
$282$		0			0
$283$	19.0458	+	12.7260i	1.13215	+	0.756481i	0.973007	−	0.230774i	$-0.0741259\pi$
$283$	19.0458	+	12.7260i	1.13215	+	0.756481i	0.159146	+	0.987255i	$0.449126\pi$
$284$		0			0
$285$	1.77607	+	6.48949i	0.105206	+	0.384404i
$286$		0			0
$287$			8.25416i			0.487228i
$288$		0			0
$289$		−	6.40197i		−	0.376586i
$290$		0			0
$291$	−1.79742	−	6.56750i	−0.105367	−	0.384994i
$292$		0			0
$293$	−10.4198	−	6.96228i	−0.608731	−	0.406741i	0.212644	−	0.977130i	$-0.431793\pi$
$293$	−10.4198	−	6.96228i	−0.608731	−	0.406741i	−0.821375	+	0.570389i	$0.806793\pi$
$294$		0			0
$295$	7.48765	−	3.10149i	0.435948	−	0.180575i
$296$		0			0
$297$	−6.65670	+	10.3167i	−0.386261	+	0.598637i
$298$		0			0
$299$	6.39014	+	32.1254i	0.369551	+	1.85786i
$300$		0			0
$301$	0.987675	+	1.47816i	0.0569287	+	0.0851998i
$302$		0			0
$303$	−2.26636	−	0.755844i	−0.130199	−	0.0434221i
$304$		0			0
$305$	−8.90301	−	8.90301i	−0.509785	−	0.509785i
$306$		0			0
$307$	7.76821	+	11.6259i	0.443355	+	0.663528i	0.984091	−	0.177665i	$-0.0568542\pi$
$307$	7.76821	+	11.6259i	0.443355	+	0.663528i	−0.540736	+	0.841192i	$0.681854\pi$
$308$		0			0
$309$	−5.07036	+	3.93385i	−0.288442	+	0.223789i
$310$		0			0
$311$	−30.5106	−	12.6379i	−1.73010	−	0.716630i	−0.999425	−	0.0339024i	$-0.989206\pi$
$311$	−30.5106	−	12.6379i	−1.73010	−	0.716630i	−0.730673	−	0.682728i	$-0.760794\pi$
$312$		0			0
$313$	−0.773283	+	0.320305i	−0.0437086	+	0.0181047i	−0.404431	−	0.914569i	$-0.632530\pi$
$313$	−0.773283	+	0.320305i	−0.0437086	+	0.0181047i	0.360722	+	0.932673i	$0.382530\pi$
$314$		0			0
$315$	−24.8051	+	8.71694i	−1.39761	+	0.491144i
$316$		0			0
$317$	13.4190	+	2.66920i	0.753685	+	0.149917i	0.556958	−	0.830541i	$-0.311969\pi$
$317$	13.4190	+	2.66920i	0.753685	+	0.149917i	0.196728	+	0.980458i	$0.436969\pi$
$318$		0			0
$319$		−	9.21456i		−	0.515916i
$320$		0			0
$321$	−1.40439	−	2.80987i	−0.0783853	−	0.156831i
$322$		0			0
$323$	−1.39673	+	7.02182i	−0.0777159	+	0.390704i
$324$		0			0
$325$	10.4975	+	7.01422i	0.582297	+	0.389079i
$326$		0			0
$327$	−10.0657	−	8.73268i	−0.556635	−	0.482918i
$328$		0			0
$329$	−4.55255	+	10.9908i	−0.250990	+	0.605944i
$330$		0			0
$331$	−2.41877	−	12.1600i	−0.132947	−	0.668372i	−0.988569	−	0.150770i	$-0.951825\pi$
$331$	−2.41877	−	12.1600i	−0.132947	−	0.668372i	0.855621	−	0.517602i	$-0.173175\pi$
$332$		0			0
$333$	1.63597	−	29.8439i	0.0896505	−	1.63544i
$334$		0			0
$335$	−6.90875	+	6.90875i	−0.377466	+	0.377466i
$336$		0			0
$337$	22.7974	+	22.7974i	1.24185	+	1.24185i	0.959231	+	0.282622i	$0.0912041\pi$
$337$	22.7974	+	22.7974i	1.24185	+	1.24185i	0.282622	+	0.959231i	$0.408796\pi$
$338$		0			0
$339$	16.3449	+	9.32061i	0.887731	+	0.506226i
$340$		0			0
$341$	−0.115243	+	0.0229233i	−0.00624077	+	0.00124137i
$342$		0			0
$343$	8.79454	+	3.64282i	0.474860	+	0.196694i
$344$		0			0
$345$	16.8311	+	14.6021i	0.906158	+	0.786153i
$346$		0			0
$347$	15.3279	−	22.9398i	0.822843	−	1.23147i	−0.147338	−	0.989086i	$-0.547070\pi$
$347$	15.3279	−	22.9398i	0.822843	−	1.23147i	0.970180	−	0.242385i	$-0.0779296\pi$
$348$		0			0
$349$	8.15254	+	1.62164i	0.436395	+	0.0868044i	0.408399	−	0.912804i	$-0.366087\pi$
$349$	8.15254	+	1.62164i	0.436395	+	0.0868044i	0.0279965	+	0.999608i	$0.491087\pi$
$350$		0			0
$351$	−6.22763	+	34.1617i	−0.332406	+	1.82341i
$352$		0			0
$353$	12.2210			0.650460			0.325230	−	0.945635i	$-0.394558\pi$
$353$	12.2210			0.650460			0.325230	+	0.945635i	$0.394558\pi$
$354$		0			0
$355$	−7.01032	+	35.2433i	−0.372069	+	1.87052i
$356$		0			0
$357$	−27.7572	−	3.50351i	−1.46907	−	0.185425i
$358$		0			0
$359$	3.41180	+	8.23682i	0.180068	+	0.434723i	0.987980	−	0.154580i	$-0.0494025\pi$
$359$	3.41180	+	8.23682i	0.180068	+	0.434723i	−0.807912	+	0.589303i	$0.799402\pi$
$360$		0			0
$361$	−6.43280	+	15.5302i	−0.338569	+	0.817377i
$362$		0			0
$363$	−7.41275	+	5.75120i	−0.389069	+	0.301860i
$364$		0			0
$365$	−32.1655	+	21.4923i	−1.68362	+	1.12496i
$366$		0			0
$367$	−11.6703	+	11.6703i	−0.609183	+	0.609183i	−0.942733	−	0.333550i	$-0.891754\pi$
$367$	−11.6703	+	11.6703i	−0.609183	+	0.609183i	0.333550	+	0.942733i	$0.391754\pi$
$368$		0			0
$369$	7.34186	−	1.04651i	0.382202	−	0.0544793i
$370$		0			0
$371$	−25.4787	+	17.0243i	−1.32279	+	0.883859i
$372$		0			0
$373$	30.2561	−	6.01831i	1.56660	−	0.311616i	0.665894	−	0.746046i	$-0.268050\pi$
$373$	30.2561	−	6.01831i	1.56660	−	0.311616i	0.900706	+	0.434430i	$0.143050\pi$
$374$		0			0
$375$	−14.1067	+	1.00033i	−0.728466	+	0.0516570i
$376$		0			0
$377$	−9.97307	−	24.0771i	−0.513639	−	1.24004i
$378$		0			0
$379$	−12.8100	+	19.1715i	−0.658005	+	0.984774i	0.340995	+	0.940065i	$0.389236\pi$
$379$	−12.8100	+	19.1715i	−0.658005	+	0.984774i	−0.999000	+	0.0447089i	$0.985764\pi$
$380$		0			0
$381$	−0.964929	−	3.52570i	−0.0494348	−	0.180627i
$382$		0			0
$383$	−11.5359			−0.589459			−0.294729	−	0.955581i	$-0.595230\pi$
$383$	−11.5359			−0.589459			−0.294729	+	0.955581i	$0.595230\pi$
$384$		0			0
$385$	20.7084			1.05540
$386$		0			0
$387$	1.18956	−	1.06592i	0.0604688	−	0.0541838i
$388$		0			0
$389$	12.8942	−	19.2976i	0.653764	−	0.978427i	−0.345436	−	0.938442i	$-0.612269\pi$
$389$	12.8942	−	19.2976i	0.653764	−	0.978427i	0.999200	−	0.0399850i	$-0.0127310\pi$
$390$		0			0
$391$	9.07369	+	21.9058i	0.458876	+	1.10783i
$392$		0			0
$393$	1.34200	+	18.9248i	0.0676948	+	0.954632i
$394$		0			0
$395$	−6.17581	+	1.22844i	−0.310739	+	0.0618097i
$396$		0			0
$397$	2.60492	−	1.74055i	0.130737	−	0.0873557i	−0.488482	−	0.872574i	$-0.662449\pi$
$397$	2.60492	−	1.74055i	0.130737	−	0.0873557i	0.619219	+	0.785218i	$0.287449\pi$
$398$		0			0
$399$	8.11950	+	2.70790i	0.406483	+	0.135565i
$400$		0			0
$401$	−24.3210	+	24.3210i	−1.21453	+	1.21453i	−0.245012	+	0.969520i	$0.578792\pi$
$401$	−24.3210	+	24.3210i	−1.21453	+	1.21453i	−0.969520	+	0.245012i	$0.921208\pi$
$402$		0			0
$403$	−0.276314	+	0.184627i	−0.0137642	+	0.00919692i
$404$		0			0
$405$	10.8984	+	20.9583i	0.541548	+	1.04143i
$406$		0			0
$407$	−9.00882	+	21.7492i	−0.446550	+	1.07807i
$408$		0			0
$409$	3.24175	+	7.82629i	0.160294	+	0.386985i	0.983538	−	0.180704i	$-0.0578376\pi$
$409$	3.24175	+	7.82629i	0.160294	+	0.386985i	−0.823243	+	0.567689i	$0.807838\pi$
$410$		0			0
$411$	−0.765911	+	6.06807i	−0.0377796	+	0.299316i
$412$		0			0
$413$	2.01141	−	10.1121i	0.0989751	−	0.497582i
$414$		0			0
$415$	−26.5991			−1.30570
$416$		0			0
$417$	−8.94978	−	17.9065i	−0.438272	−	0.876885i
$418$		0			0
$419$	30.5931	+	6.08535i	1.49457	+	0.297289i	0.873639	−	0.486575i	$-0.161754\pi$
$419$	30.5931	+	6.08535i	1.49457	+	0.297289i	0.620933	+	0.783864i	$0.286754\pi$
$420$		0			0
$421$	−3.52363	+	5.27349i	−0.171731	+	0.257014i	−0.907345	−	0.420388i	$-0.861894\pi$
$421$	−3.52363	+	5.27349i	−0.171731	+	0.257014i	0.735613	+	0.677402i	$0.236894\pi$
$422$		0			0
$423$	10.3532	+	2.65589i	0.503392	+	0.129134i
$424$		0			0
$425$	8.44356	+	3.49744i	0.409573	+	0.169651i
$426$		0			0
$427$	−15.7095	+	3.12481i	−0.760234	+	0.151220i
$428$		0			0
$429$	13.5483	−	23.7586i	0.654117	−	1.14708i
$430$		0			0
$431$	−25.0198	−	25.0198i	−1.20516	−	1.20516i	−0.972576	−	0.232586i	$-0.925281\pi$
$431$	−25.0198	−	25.0198i	−1.20516	−	1.20516i	−0.232586	−	0.972576i	$-0.574719\pi$
$432$		0			0
$433$	−0.182313	+	0.182313i	−0.00876139	+	0.00876139i	−0.711474	−	0.702713i	$-0.751972\pi$
$433$	−0.182313	+	0.182313i	−0.00876139	+	0.00876139i	0.702713	+	0.711474i	$0.251972\pi$
$434$		0			0
$435$	−15.4007	−	8.78222i	−0.738408	−	0.421075i
$436$		0			0
$437$	−1.41515	−	7.11445i	−0.0676959	−	0.340330i
$438$		0			0
$439$	15.4119	−	37.2076i	0.735570	−	1.77582i	0.112508	−	0.993651i	$-0.464112\pi$
$439$	15.4119	−	37.2076i	0.735570	−	1.77582i	0.623062	−	0.782172i	$-0.285888\pi$
$440$		0			0
$441$	−3.09295	+	12.0570i	−0.147283	+	0.574143i
$442$		0			0
$443$	5.50223	+	3.67647i	0.261419	+	0.174674i	0.679372	−	0.733794i	$-0.262252\pi$
$443$	5.50223	+	3.67647i	0.261419	+	0.174674i	−0.417953	+	0.908469i	$0.637252\pi$
$444$		0			0
$445$	−4.57852	+	23.0178i	−0.217043	+	1.09115i
$446$		0			0
$447$	−14.8752	+	7.43473i	−0.703574	+	0.351651i
$448$		0			0
$449$			13.3291i			0.629041i	0.949251	+	0.314520i	$0.101844\pi$
$449$			13.3291i			0.629041i	−0.949251	+	0.314520i	$0.898156\pi$
$450$		0			0
$451$	−5.72886	−	1.13954i	−0.269761	−	0.0536589i
$452$		0			0
$453$	−28.7135	−	3.62422i	−1.34908	−	0.170281i
$454$		0			0
$455$	54.1100	−	22.4131i	2.53672	−	1.05074i
$456$		0			0
$457$	−18.5018	−	7.66368i	−0.865476	−	0.358492i	−0.0946295	−	0.995513i	$-0.530167\pi$
$457$	−18.5018	−	7.66368i	−0.865476	−	0.358492i	−0.770847	+	0.637020i	$0.780167\pi$
$458$		0			0
$459$	0.402945	+	25.1335i	0.0188079	+	1.17313i
$460$		0			0
$461$	−19.4038	−	29.0398i	−0.903724	−	1.35252i	−0.935613	−	0.353027i	$-0.885152\pi$
$461$	−19.4038	−	29.0398i	−0.903724	−	1.35252i	0.0318892	−	0.999491i	$-0.489848\pi$
$462$		0			0
$463$	26.4304	+	26.4304i	1.22833	+	1.22833i	0.964597	+	0.263728i	$0.0849522\pi$
$463$	26.4304	+	26.4304i	1.22833	+	1.22833i	0.263728	+	0.964597i	$0.415048\pi$
$464$		0			0
$465$	−0.0715233	+	0.214459i	−0.00331681	+	0.00994530i
$466$		0			0
$467$	−6.62269	−	9.91156i	−0.306462	−	0.458652i	0.645988	−	0.763347i	$-0.276445\pi$
$467$	−6.62269	−	9.91156i	−0.306462	−	0.458652i	−0.952450	+	0.304695i	$0.901445\pi$
$468$		0			0
$469$	2.42485	+	12.1906i	0.111969	+	0.562908i
$470$		0			0
$471$	−1.94611	+	0.138003i	−0.0896721	+	0.00635882i
$472$		0			0
$473$	−1.16228	+	0.481433i	−0.0534418	+	0.0221363i
$474$		0			0
$475$	−2.32477	−	1.55336i	−0.106668	−	0.0712731i
$476$		0			0
$477$	18.3730	+	20.5042i	0.841243	+	0.938822i
$478$		0			0
$479$		−	37.9382i		−	1.73344i	−0.498795	−	0.866720i	$-0.666224\pi$
$479$		−	37.9382i		−	1.73344i	0.498795	−	0.866720i	$-0.333776\pi$
$480$		0			0
$481$			66.5798i			3.03578i
$482$		0			0
$483$	27.3410	−	7.48282i	1.24406	−	0.340480i
$484$		0			0
$485$	8.57938	+	5.73256i	0.389569	+	0.260302i
$486$		0			0
$487$	20.1298	−	8.33804i	0.912168	−	0.377833i	0.123282	−	0.992372i	$-0.460658\pi$
$487$	20.1298	−	8.33804i	0.912168	−	0.377833i	0.788886	+	0.614539i	$0.210658\pi$
$488$		0			0
$489$	−2.48572	−	35.0536i	−0.112408	−	1.58518i
$490$		0			0
$491$	−3.99742	−	20.0964i	−0.180401	−	0.906936i	−0.959859	−	0.280482i	$-0.909506\pi$
$491$	−3.99742	−	20.0964i	−0.180401	−	0.906936i	0.779458	−	0.626454i	$-0.215494\pi$
$492$		0			0
$493$	−10.4809	−	15.6858i	−0.472036	−	0.706451i
$494$		0			0
$495$	−2.62554	−	18.4196i	−0.118009	−	0.827900i
$496$		0			0
$497$	32.3238	+	32.3238i	1.44992	+	1.44992i
$498$		0			0
$499$	−10.8139	−	16.1841i	−0.484095	−	0.724500i	0.506362	−	0.862321i	$-0.330990\pi$
$499$	−10.8139	−	16.1841i	−0.484095	−	0.724500i	−0.990457	+	0.137821i	$0.955990\pi$
$500$		0			0
$501$	0.304861	+	0.392937i	0.0136202	+	0.0175551i
$502$		0			0
$503$	13.0766	+	5.41650i	0.583056	+	0.241510i	0.654660	−	0.755923i	$-0.272812\pi$
$503$	13.0766	+	5.41650i	0.583056	+	0.241510i	−0.0716041	+	0.997433i	$0.522812\pi$
$504$		0			0
$505$	3.34480	−	1.38546i	0.148842	−	0.0616522i
$506$		0			0
$507$	6.86687	−	54.4040i	0.304969	−	2.41617i
$508$		0			0
$509$	21.9603	+	4.36817i	0.973371	+	0.193616i	0.656062	−	0.754707i	$-0.272221\pi$
$509$	21.9603	+	4.36817i	0.973371	+	0.193616i	0.317309	+	0.948322i	$0.397221\pi$
$510$		0			0
$511$			49.2130i			2.17705i
$512$		0			0
$513$	1.37916	−	7.56540i	0.0608916	−	0.334021i
$514$		0			0
$515$	1.89724	−	9.53808i	0.0836025	−	0.420298i
$516$		0			0
$517$	−6.99975	−	4.67708i	−0.307848	−	0.205698i
$518$		0			0
$519$	−15.2251	+	17.5492i	−0.668307	+	0.770323i
$520$		0			0
$521$	−0.213242	+	0.514813i	−0.00934232	+	0.0225544i	−0.928482	−	0.371378i	$-0.878885\pi$
$521$	−0.213242	+	0.514813i	−0.00934232	+	0.0225544i	0.919140	+	0.393932i	$0.128885\pi$
$522$		0			0
$523$	0.636430	+	3.19955i	0.0278291	+	0.139907i	0.992202	−	0.124640i	$-0.0397777\pi$
$523$	0.636430	+	3.19955i	0.0278291	+	0.139907i	−0.964373	+	0.264547i	$0.914778\pi$
$524$		0			0
$525$	5.41241	−	9.49134i	0.236217	−	0.414236i
$526$		0			0
$527$	−0.170103	+	0.170103i	−0.00740979	+	0.00740979i
$528$		0			0
$529$	−0.723726	−	0.723726i	−0.0314664	−	0.0314664i
$530$		0			0
$531$	−9.24942	−	0.507029i	−0.401391	−	0.0220032i
$532$		0			0
$533$	−16.2025	+	3.22288i	−0.701809	+	0.139599i
$534$		0			0
$535$	4.39792	+	1.82168i	0.190139	+	0.0787580i
$536$		0			0
$537$	18.8365	−	21.7119i	0.812856	−	0.936938i
$538$		0			0
$539$	5.44675	−	8.15164i	0.234608	−	0.351116i
$540$		0			0
$541$	−18.6138	−	3.70251i	−0.800270	−	0.159184i	−0.222019	−	0.975042i	$-0.571265\pi$
$541$	−18.6138	−	3.70251i	−0.800270	−	0.159184i	−0.578251	+	0.815859i	$0.696265\pi$
$542$		0			0
$543$	−0.582309	+	0.291042i	−0.0249893	+	0.0124898i
$544$		0			0
$545$	20.1939			0.865010
$546$		0			0
$547$	−0.321617	+	1.61688i	−0.0137513	+	0.0691327i	−0.987052	−	0.160403i	$-0.948721\pi$
$547$	−0.321617	+	1.61688i	−0.0137513	+	0.0691327i	0.973300	+	0.229535i	$0.0737207\pi$
$548$		0			0
$549$	4.77117	+	13.5770i	0.203629	+	0.579451i
$550$		0			0
$551$	2.20863	+	5.33209i	0.0940906	+	0.227155i
$552$		0			0
$553$	−3.06545	+	7.40066i	−0.130356	+	0.314708i
$554$		0			0
$555$	27.7643	+	35.7856i	1.17853	+	1.51901i
$556$		0			0
$557$	−7.81765	+	5.22359i	−0.331245	+	0.221330i	−0.710059	−	0.704142i	$-0.751332\pi$
$557$	−7.81765	+	5.22359i	−0.331245	+	0.221330i	0.378815	+	0.925473i	$0.376332\pi$
$558$		0			0
$559$	−2.51592	+	2.51592i	−0.106412	+	0.106412i
$560$		0			0
$561$	6.26369	−	18.7814i	0.264453	−	0.792950i
$562$		0			0
$563$	25.4030	−	16.9737i	1.07061	−	0.715357i	0.110186	−	0.993911i	$-0.464855\pi$
$563$	25.4030	−	16.9737i	1.07061	−	0.715357i	0.960420	+	0.278554i	$0.0898552\pi$
$564$		0			0
$565$	−27.9652	+	5.56262i	−1.17650	+	0.234021i
$566$		0			0
$567$	29.8712	+	3.28480i	1.25447	+	0.137949i
$568$		0			0
$569$	−11.5727	−	27.9391i	−0.485154	−	1.17127i	−0.957131	−	0.289655i	$-0.906459\pi$
$569$	−11.5727	−	27.9391i	−0.485154	−	1.17127i	0.471977	−	0.881611i	$-0.343541\pi$
$570$		0			0
$571$	−7.03644	+	10.5308i	−0.294466	+	0.440700i	−0.948973	−	0.315356i	$-0.897876\pi$
$571$	−7.03644	+	10.5308i	−0.294466	+	0.440700i	0.654507	+	0.756056i	$0.272876\pi$
$572$		0			0
$573$	8.15299	−	2.23135i	0.340596	−	0.0932160i
$574$		0			0
$575$	−9.25981			−0.386161
$576$		0			0
$577$	9.10864			0.379198			0.189599	−	0.981862i	$-0.439281\pi$
$577$	9.10864			0.379198			0.189599	+	0.981862i	$0.439281\pi$
$578$		0			0
$579$	−12.5785	+	3.44255i	−0.522746	+	0.143068i
$580$		0			0
$581$	−18.7993	+	28.1351i	−0.779925	+	1.16724i
$582$		0			0
$583$	−8.29835	−	20.0340i	−0.343682	−	0.829723i
$584$		0			0
$585$	−26.7962	−	45.2877i	−1.10789	−	1.87242i
$586$		0			0
$587$	−43.4391	+	8.64058i	−1.79292	+	0.356635i	−0.975602	−	0.219546i	$-0.929543\pi$
$587$	−43.4391	+	8.64058i	−1.79292	+	0.356635i	−0.817322	+	0.576180i	$0.804543\pi$
$588$		0			0
$589$	0.0611921	−	0.0408873i	0.00252138	−	0.00168473i
$590$		0			0
$591$	8.50062	−	25.4887i	0.349669	−	1.04846i
$592$		0			0
$593$	−12.4091	+	12.4091i	−0.509580	+	0.509580i	−0.914398	−	0.404817i	$-0.867335\pi$
$593$	−12.4091	+	12.4091i	−0.509580	+	0.509580i	0.404817	+	0.914398i	$0.367335\pi$
$594$		0			0
$595$	35.2516	−	23.5543i	1.44517	−	0.965634i
$596$		0			0
$597$	−12.2928	−	15.8443i	−0.503111	−	0.648463i
$598$		0			0
$599$	−5.32882	+	12.8649i	−0.217730	+	0.525646i	−0.994572	−	0.104049i	$-0.966820\pi$
$599$	−5.32882	+	12.8649i	−0.217730	+	0.525646i	0.776843	+	0.629695i	$0.216820\pi$
$600$		0			0
$601$	−9.61422	−	23.2108i	−0.392172	−	0.946788i	−0.989466	−	0.144765i	$-0.953757\pi$
$601$	−9.61422	−	23.2108i	−0.392172	−	0.946788i	0.597294	−	0.802023i	$-0.296243\pi$
$602$		0			0
$603$	10.5357	−	3.70244i	0.429049	−	0.150775i
$604$		0			0
$605$	2.77373	−	13.9445i	0.112768	−	0.566923i
$606$		0			0
$607$	−7.83446			−0.317991			−0.158995	−	0.987279i	$-0.550825\pi$
$607$	−7.83446			−0.317991			−0.158995	+	0.987279i	$0.550825\pi$
$608$		0			0
$609$	−20.1740	+	10.0831i	−0.817493	+	0.408588i
$610$		0			0
$611$	−23.3520	−	4.64501i	−0.944722	−	0.187917i
$612$		0			0
$613$	16.6418	−	24.9062i	0.672154	−	1.00595i	−0.326009	−	0.945367i	$-0.605704\pi$
$613$	16.6418	−	24.9062i	0.672154	−	1.00595i	0.998163	−	0.0605835i	$-0.0192961\pi$
$614$		0			0
$615$	−7.36463	+	8.48883i	−0.296971	+	0.342303i
$616$		0			0
$617$	34.0519	+	14.1048i	1.37088	+	0.567836i	0.942027	−	0.335538i	$-0.108918\pi$
$617$	34.0519	+	14.1048i	1.37088	+	0.567836i	0.428852	+	0.903375i	$0.358918\pi$
$618$		0			0
$619$	−29.7566	+	5.91895i	−1.19602	+	0.237903i	−0.752640	−	0.658433i	$-0.771220\pi$
$619$	−29.7566	+	5.91895i	−1.19602	+	0.237903i	−0.443377	+	0.896335i	$0.646220\pi$
$620$		0			0
$621$	−10.1222	−	23.3704i	−0.406191	−	0.937821i
$622$		0			0
$623$	21.1110	+	21.1110i	0.845796	+	0.845796i
$624$		0			0
$625$	21.8333	−	21.8333i	0.873332	−	0.873332i
$626$		0			0
$627$	−3.00038	+	5.26155i	−0.119824	+	0.210126i
$628$		0			0
$629$	9.40261	+	47.2701i	0.374907	+	1.88478i
$630$		0			0
$631$	5.45257	−	13.1637i	0.217063	−	0.524037i	−0.777414	−	0.628989i	$-0.783469\pi$
$631$	5.45257	−	13.1637i	0.217063	−	0.524037i	0.994477	+	0.104952i	$0.0334689\pi$
$632$		0			0
$633$	−24.4216	+	28.1496i	−0.970672	+	1.11884i
$634$		0			0
$635$	4.60575	+	3.07747i	0.182774	+	0.122126i
$636$		0			0
$637$	5.40940	−	27.1949i	0.214328	−	1.07750i
$638$		0			0
$639$	24.6530	−	32.8494i	0.975256	−	1.29950i
$640$		0			0
$641$		−	7.05243i		−	0.278554i	−0.990253	−	0.139277i	$-0.955522\pi$
$641$		−	7.05243i		−	0.278554i	0.990253	−	0.139277i	$-0.0444779\pi$
$642$		0			0
$643$	−29.0442	−	5.77725i	−1.14539	−	0.227832i	−0.414318	−	0.910132i	$-0.635980\pi$
$643$	−29.0442	−	5.77725i	−1.14539	−	0.227832i	−0.731072	+	0.682300i	$0.760980\pi$
$644$		0			0
$645$	−0.303107	+	2.40142i	−0.0119348	+	0.0945559i
$646$		0			0
$647$	−43.7051	+	18.1032i	−1.71822	+	0.711711i	−0.718352	+	0.695680i	$0.755103\pi$
$647$	−43.7051	+	18.1032i	−1.71822	+	0.711711i	−0.999871	+	0.0160315i	$0.994897\pi$
$648$		0			0
$649$	6.74065	+	2.79207i	0.264594	+	0.109598i
$650$		0			0
$651$	0.176293	+	0.227225i	0.00690948	+	0.00890567i
$652$		0			0
$653$	14.1402	+	21.1623i	0.553348	+	0.828143i	0.997705	−	0.0677063i	$-0.0215681\pi$
$653$	14.1402	+	21.1623i	0.553348	+	0.828143i	−0.444358	+	0.895849i	$0.646568\pi$
$654$		0			0
$655$	−20.3297	−	20.3297i	−0.794347	−	0.794347i
$656$		0			0
$657$	43.7736	−	6.23952i	1.70777	−	0.243427i
$658$		0			0
$659$	18.1263	+	27.1279i	0.706100	+	1.05675i	0.995047	+	0.0994083i	$0.0316950\pi$
$659$	18.1263	+	27.1279i	0.706100	+	1.05675i	−0.288947	+	0.957345i	$0.593305\pi$
$660$		0			0
$661$	5.91065	+	29.7149i	0.229898	+	1.15577i	0.907405	+	0.420258i	$0.138060\pi$
$661$	5.91065	+	29.7149i	0.229898	+	1.15577i	−0.677507	+	0.735516i	$0.736940\pi$
$662$		0			0
$663$	−3.96071	−	55.8540i	−0.153821	−	2.16919i
$664$		0			0
$665$	−11.9831	+	4.96358i	−0.464686	+	0.192479i
$666$		0			0
$667$	15.8927	+	10.6192i	0.615368	+	0.411176i
$668$		0			0
$669$	2.43483	−	0.666375i	0.0941358	−	0.0257635i
$670$		0			0
$671$		−	11.3347i		−	0.437569i
$672$		0			0
$673$			11.0359i			0.425402i	0.977117	+	0.212701i	$0.0682259\pi$
$673$			11.0359i			0.425402i	−0.977117	+	0.212701i	$0.931774\pi$
$674$		0			0
$675$	−9.12851	−	3.61082i	−0.351356	−	0.138981i
$676$		0			0
$677$	16.1303	+	10.7779i	0.619939	+	0.414230i	0.825490	−	0.564416i	$-0.190899\pi$
$677$	16.1303	+	10.7779i	0.619939	+	0.414230i	−0.205551	+	0.978646i	$0.565899\pi$
$678$		0			0
$679$	12.1272	−	5.02324i	0.465399	−	0.192774i
$680$		0			0
$681$	−34.1185	+	2.41941i	−1.30743	+	0.0927121i
$682$		0			0
$683$	−9.44062	−	47.4612i	−0.361235	−	1.81605i	−0.551331	−	0.834287i	$-0.685880\pi$
$683$	−9.44062	−	47.4612i	−0.361235	−	1.81605i	0.190095	−	0.981766i	$-0.439120\pi$
$684$		0			0
$685$	−5.14927	−	7.70643i	−0.196744	−	0.294448i
$686$		0			0
$687$	0.923276	−	2.76840i	0.0352252	−	0.105621i
$688$		0			0
$689$	−43.3662	−	43.3662i	−1.65212	−	1.65212i
$690$		0			0
$691$	2.62103	+	3.92264i	0.0997085	+	0.149224i	0.877969	−	0.478717i	$-0.158898\pi$
$691$	2.62103	+	3.92264i	0.0997085	+	0.149224i	−0.778261	+	0.627941i	$0.783898\pi$
$692$		0			0
$693$	−21.3389	−	10.2411i	−0.810598	−	0.389029i
$694$		0			0
$695$	28.0267	+	11.6090i	1.06311	+	0.440356i
$696$		0			0
$697$	−11.0483	+	4.57634i	−0.418483	+	0.173341i
$698$		0			0
$699$	−26.8126	−	3.38429i	−1.01415	−	0.128005i
$700$		0			0
$701$	0.420755	+	0.0836934i	0.0158917	+	0.00316106i	0.203030	−	0.979173i	$-0.434921\pi$
$701$	0.420755	+	0.0836934i	0.0158917	+	0.00316106i	−0.187138	+	0.982334i	$0.559921\pi$
$702$		0			0
$703$		−	14.7447i		−	0.556107i
$704$		0			0
$705$	−14.4883	+	7.24137i	−0.545663	+	0.272726i
$706$		0			0
$707$	0.898516	−	4.51715i	0.0337922	−	0.169885i
$708$		0			0
$709$	25.3814	+	16.9593i	0.953218	+	0.636920i	0.931847	−	0.362852i	$-0.118197\pi$
$709$	25.3814	+	16.9593i	0.953218	+	0.636920i	0.0213708	+	0.999772i	$0.493197\pi$
$710$		0			0
$711$	6.97135	+	1.78834i	0.261446	+	0.0670679i
$712$		0			0
$713$	0.0932734	−	0.225182i	0.00349312	−	0.00843313i
$714$		0			0
$715$	8.08572	+	40.6497i	0.302389	+	1.52021i
$716$		0			0
$717$	−1.46877	−	0.837563i	−0.0548523	−	0.0312794i
$718$		0			0
$719$	−4.47853	+	4.47853i	−0.167021	+	0.167021i	−0.785669	−	0.618648i	$-0.787681\pi$
$719$	−4.47853	+	4.47853i	−0.167021	+	0.167021i	0.618648	+	0.785669i	$0.287681\pi$
$720$		0			0
$721$	−8.74797	−	8.74797i	−0.325791	−	0.325791i
$722$		0			0
$723$	−12.8745	+	22.5770i	−0.478808	+	0.839649i
$724$		0			0
$725$	7.22588	−	1.43732i	0.268363	−	0.0533806i
$726$		0			0
$727$	3.41297	+	1.41370i	0.126580	+	0.0524311i	0.445074	−	0.895494i	$-0.353177\pi$
$727$	3.41297	+	1.41370i	0.126580	+	0.0524311i	−0.318494	+	0.947925i	$0.603177\pi$
$728$		0			0
$729$	−0.865518	−	26.9861i	−0.0320562	−	0.999486i
$730$		0			0
$731$	−1.43094	+	2.14155i	−0.0529250	+	0.0792079i
$732$		0			0
$733$	−1.76647	−	0.351374i	−0.0652462	−	0.0129783i	0.162359	−	0.986732i	$-0.448090\pi$
$733$	−1.76647	−	0.351374i	−0.0652462	−	0.0129783i	−0.227605	+	0.973753i	$0.573090\pi$
$734$		0			0
$735$	−8.43303	−	16.8726i	−0.311057	−	0.622355i
$736$		0			0
$737$	−8.79571			−0.323994
$738$		0			0
$739$	7.50862	−	37.7484i	0.276209	−	1.38860i	−0.554636	−	0.832093i	$-0.687143\pi$
$739$	7.50862	−	37.7484i	0.276209	−	1.38860i	0.830845	−	0.556504i	$-0.187857\pi$
$740$		0			0
$741$	−2.14517	+	16.9955i	−0.0788049	+	0.624345i
$742$		0			0
$743$	−13.3142	−	32.1432i	−0.488449	−	1.17922i	−0.955500	−	0.294990i	$-0.904684\pi$
$743$	−13.3142	−	32.1432i	−0.488449	−	1.17922i	0.467051	−	0.884230i	$-0.345316\pi$
$744$		0			0
$745$	9.64382	−	23.2822i	0.353322	−	0.852995i
$746$		0			0
$747$	27.4089	+	13.1543i	1.00284	+	0.481291i
$748$		0			0
$749$	5.03516	−	3.36439i	0.183981	−	0.122932i
$750$		0			0
$751$	−30.0300	+	30.0300i	−1.09581	+	1.09581i	−0.100914	+	0.994895i	$0.532177\pi$
$751$	−30.0300	+	30.0300i	−1.09581	+	1.09581i	−0.994895	+	0.100914i	$0.967823\pi$
$752$		0			0
$753$	44.9449	+	14.9894i	1.63788	+	0.546244i
$754$		0			0
$755$	36.4661	−	24.3659i	1.32714	−	0.886766i
$756$		0			0
$757$	37.4839	−	7.45600i	1.36237	−	0.270993i	0.540827	−	0.841134i	$-0.318111\pi$
$757$	37.4839	−	7.45600i	1.36237	−	0.270993i	0.821547	+	0.570141i	$0.193111\pi$
$758$		0			0
$759$	1.41890	+	20.0093i	0.0515027	+	0.726290i
$760$		0			0
$761$	−5.61157	−	13.5475i	−0.203419	−	0.491097i	0.788941	−	0.614468i	$-0.210629\pi$
$761$	−5.61157	−	13.5475i	−0.203419	−	0.491097i	−0.992361	+	0.123371i	$0.960629\pi$
$762$		0			0
$763$	14.2723	−	21.3600i	0.516692	−	0.773284i
$764$		0			0
$765$	−25.4204	−	28.3690i	−0.919075	−	1.02568i
$766$		0			0
$767$	20.6348			0.745081
$768$		0			0
$769$	17.9581			0.647587			0.323793	−	0.946128i	$-0.395042\pi$
$769$	17.9581			0.647587			0.323793	+	0.946128i	$0.395042\pi$
$770$		0			0
$771$	−2.06581	−	7.54813i	−0.0743982	−	0.271839i
$772$		0			0
$773$	9.82391	−	14.7025i	0.353341	−	0.528813i	−0.611638	−	0.791138i	$-0.709489\pi$
$773$	9.82391	−	14.7025i	0.353341	−	0.528813i	0.964980	+	0.262325i	$0.0844892\pi$
$774$		0			0
$775$	−0.0359520	−	0.0867959i	−0.00129144	−	0.00311780i
$776$		0			0
$777$	57.4749	−	4.07566i	2.06190	−	0.146214i
$778$		0			0
$779$	3.58819	−	0.713736i	0.128560	−	0.0255722i
$780$		0			0
$781$	−26.8971	+	17.9721i	−0.962453	+	0.643091i
$782$		0			0
$783$	11.5264	+	16.6659i	0.411922	+	0.595590i
$784$		0			0
$785$	2.09058	−	2.09058i	0.0746160	−	0.0746160i
$786$		0			0
$787$	30.2625	−	20.2207i	1.07874	−	0.720791i	0.116554	−	0.993184i	$-0.462815\pi$
$787$	30.2625	−	20.2207i	1.07874	−	0.720791i	0.962186	+	0.272393i	$0.0878151\pi$
$788$		0			0
$789$	3.03568	−	2.35523i	0.108073	−	0.0838486i
$790$		0			0
$791$	−13.8809	+	33.5115i	−0.493549	+	1.19153i
$792$		0			0
$793$	−12.2677	−	29.6168i	−0.435638	−	1.05172i
$794$		0			0
$795$	−41.3927	−	5.22459i	−1.46805	−	0.185297i
$796$		0			0
$797$	−4.54990	+	22.8739i	−0.161166	+	0.810234i	0.812625	+	0.582787i	$0.198038\pi$
$797$	−4.54990	+	22.8739i	−0.161166	+	0.810234i	−0.973790	+	0.227447i	$0.926962\pi$
$798$		0			0
$799$	−17.2354			−0.609744
$800$		0			0
$801$	16.1011	−	21.4543i	0.568905	−	0.758051i
$802$		0			0
$803$	−34.1566	−	6.79417i	−1.20536	−	0.239761i
$804$		0			0
$805$	−23.8651	+	35.7166i	−0.841134	+	1.25885i
$806$		0			0
$807$	30.1793	+	26.1826i	1.06236	+	0.921671i
$808$		0			0
$809$	−18.3664	−	7.60762i	−0.645729	−	0.267470i	0.0356905	−	0.999363i	$-0.488637\pi$
$809$	−18.3664	−	7.60762i	−0.645729	−	0.267470i	−0.681419	+	0.731893i	$0.738637\pi$
$810$		0			0
$811$	46.0682	−	9.16353i	1.61767	−	0.321775i	0.698494	−	0.715616i	$-0.253854\pi$
$811$	46.0682	−	9.16353i	1.61767	−	0.321775i	0.919179	+	0.393840i	$0.128854\pi$
$812$		0			0
$813$	9.96158	+	5.68056i	0.349368	+	0.199226i
$814$		0			0
$815$	37.6558	+	37.6558i	1.31903	+	1.31903i
$816$		0			0
$817$	0.557172	−	0.557172i	0.0194930	−	0.0194930i
$818$		0			0
$819$	−66.8416	−	3.66408i	−2.33563	−	0.128033i
$820$		0			0
$821$	−7.82812	−	39.3546i	−0.273203	−	1.37349i	−0.836833	−	0.547458i	$-0.815596\pi$
$821$	−7.82812	−	39.3546i	−0.273203	−	1.37349i	0.563630	−	0.826028i	$-0.309404\pi$
$822$		0			0
$823$	−10.4572	+	25.2460i	−0.364516	+	0.880019i	0.630112	+	0.776504i	$0.283009\pi$
$823$	−10.4572	+	25.2460i	−0.364516	+	0.880019i	−0.994628	+	0.103515i	$0.966991\pi$
$824$		0			0
$825$	5.84031	+	5.06686i	0.203333	+	0.176405i
$826$		0			0
$827$	13.3380	+	8.91218i	0.463809	+	0.309907i	0.765428	−	0.643521i	$-0.222527\pi$
$827$	13.3380	+	8.91218i	0.463809	+	0.309907i	−0.301620	+	0.953428i	$0.597527\pi$
$828$		0			0
$829$	4.63371	−	23.2952i	0.160935	−	0.809077i	−0.813002	−	0.582261i	$-0.802168\pi$
$829$	4.63371	−	23.2952i	0.160935	−	0.809077i	0.973938	−	0.226816i	$-0.0728317\pi$
$830$		0			0
$831$	1.39533	+	2.79175i	0.0484036	+	0.0968448i
$832$		0			0
$833$		−	20.0717i		−	0.695442i
$834$		0			0
$835$	−0.739173	−	0.147031i	−0.0255801	−	0.00508820i
$836$		0			0
$837$	0.179759	−	0.185617i	0.00621340	−	0.00641587i
$838$		0			0
$839$	27.5815	−	11.4246i	0.952220	−	0.394423i	0.148155	−	0.988964i	$-0.452666\pi$
$839$	27.5815	−	11.4246i	0.952220	−	0.394423i	0.804065	+	0.594542i	$0.202666\pi$
$840$		0			0
$841$	12.7423	+	5.27805i	0.439391	+	0.182002i
$842$		0			0
$843$	9.38597	−	7.28212i	0.323270	−	0.250810i
$844$		0			0
$845$	46.1664	+	69.0930i	1.58817	+	2.37687i
$846$		0			0
$847$	−12.7894	−	12.7894i	−0.439447	−	0.439447i
$848$		0			0
$849$	37.6367	+	12.5521i	1.29169	+	0.430785i
$850$		0			0
$851$	−27.1296	−	40.6023i	−0.929991	−	1.39183i
$852$		0			0
$853$	0.300445	+	1.51044i	0.0102870	+	0.0517164i	0.985588	−	0.169164i	$-0.0541068\pi$
$853$	0.300445	+	1.51044i	0.0102870	+	0.0517164i	−0.975301	+	0.220881i	$0.929107\pi$
$854$		0			0
$855$	5.93427	+	10.0294i	0.202948	+	0.342997i
$856$		0			0
$857$	6.92900	−	2.87008i	0.236690	−	0.0980402i	−0.261186	−	0.965289i	$-0.584114\pi$
$857$	6.92900	−	2.87008i	0.236690	−	0.0980402i	0.497876	+	0.867248i	$0.334114\pi$
$858$		0			0
$859$	−12.3876	−	8.27714i	−0.422660	−	0.282412i	0.325996	−	0.945371i	$-0.394300\pi$
$859$	−12.3876	−	8.27714i	−0.422660	−	0.282412i	−0.748656	+	0.662959i	$0.769300\pi$
$860$		0			0
$861$	3.77398	+	13.7895i	0.128617	+	0.469945i
$862$		0			0
$863$			31.0444i			1.05676i	0.849007	+	0.528382i	$0.177201\pi$
$863$			31.0444i			1.05676i	−0.849007	+	0.528382i	$0.822799\pi$
$864$		0			0
$865$		−	35.2072i		−	1.19708i
$866$		0			0
$867$	−2.92712	−	10.6952i	−0.0994101	−	0.363228i
$868$		0			0
$869$	−4.71327	−	3.14931i	−0.159887	−	0.106833i
$870$		0			0
$871$	−22.9827	+	9.51975i	−0.778739	+	0.322564i
$872$		0			0
$873$	−6.00560	−	10.1499i	−0.203259	−	0.343523i
$874$		0			0
$875$	−5.31875	−	26.7392i	−0.179807	−	0.903949i
$876$		0			0
$877$	13.8663	+	20.7524i	0.468232	+	0.700759i	0.988156	−	0.153454i	$-0.0490397\pi$
$877$	13.8663	+	20.7524i	0.468232	+	0.700759i	−0.519924	+	0.854213i	$0.674040\pi$
$878$		0			0
$879$	−20.5907	−	6.86713i	−0.694509	−	0.231623i
$880$		0			0
$881$	−22.9018	−	22.9018i	−0.771582	−	0.771582i	0.206801	−	0.978383i	$-0.433695\pi$
$881$	−22.9018	−	22.9018i	−0.771582	−	0.771582i	−0.978383	+	0.206801i	$0.933695\pi$
$882$		0			0
$883$	−10.2732	−	15.3750i	−0.345722	−	0.517409i	0.617338	−	0.786698i	$-0.288211\pi$
$883$	−10.2732	−	15.3750i	−0.345722	−	0.517409i	−0.963059	+	0.269289i	$0.913211\pi$
$884$		0			0
$885$	11.0909	−	8.60490i	0.372817	−	0.289250i
$886$		0			0
$887$	9.52810	+	3.94667i	0.319922	+	0.132516i	0.536865	−	0.843668i	$-0.319608\pi$
$887$	9.52810	+	3.94667i	0.319922	+	0.132516i	−0.216943	+	0.976184i	$0.569608\pi$
$888$		0			0
$889$	6.51036	−	2.69668i	0.218350	−	0.0904437i
$890$		0			0
$891$	−6.40375	+	20.2788i	−0.214534	+	0.679366i
$892$		0			0
$893$	5.17151	+	1.02868i	0.173058	+	0.0344234i
$894$		0			0
$895$			43.5585i			1.45600i
$896$		0			0
$897$	25.3639	+	50.7474i	0.846875	+	1.69441i
$898$		0			0
$899$	−0.0378328	+	0.190198i	−0.00126179	+	0.00634347i
$900$		0			0
$901$	−36.9133	−	24.6647i	−1.22976	−	0.821700i
$902$		0			0
$903$	2.32587	+	2.01785i	0.0774001	+	0.0671498i
$904$		0			0
$905$	0.377519	−	0.911413i	0.0125492	−	0.0302964i
$906$		0			0
$907$	−6.38322	−	32.0906i	−0.211951	−	1.06555i	−0.929438	−	0.368979i	$-0.879708\pi$
$907$	−6.38322	−	32.0906i	−0.211951	−	1.06555i	0.717487	−	0.696572i	$-0.245292\pi$
$908$		0			0
$909$	−4.13180	−	0.226495i	−0.137043	−	0.00751235i
$910$		0			0
$911$	−20.1753	+	20.1753i	−0.668438	+	0.668438i	−0.957354	−	0.288916i	$-0.906705\pi$
$911$	−20.1753	+	20.1753i	−0.668438	+	0.668438i	0.288916	+	0.957354i	$0.406705\pi$
$912$		0			0
$913$	−16.9320	−	16.9320i	−0.560367	−	0.560367i
$914$		0			0
$915$	−18.9441	−	10.8028i	−0.626274	−	0.357131i
$916$		0			0
$917$	−35.8720	+	7.13538i	−1.18460	+	0.235631i
$918$		0			0
$919$	44.0591	+	18.2499i	1.45338	+	0.602008i	0.963000	−	0.269502i	$-0.0868591\pi$
$919$	44.0591	+	18.2499i	1.45338	+	0.602008i	0.490377	+	0.871510i	$0.336859\pi$
$920$		0			0
$921$	18.2933	+	15.8707i	0.602785	+	0.522956i
$922$		0			0
$923$	−50.8291	+	76.0711i	−1.67306	+	2.50391i
$924$		0			0
$925$	−18.4605	−	3.67203i	−0.606979	−	0.120736i
$926$		0			0
$927$	−6.67197	+	8.89021i	−0.219136	+	0.291993i
$928$		0			0
$929$	−28.1038			−0.922057			−0.461028	−	0.887385i	$-0.652519\pi$
$929$	−28.1038			−0.922057			−0.461028	+	0.887385i	$0.652519\pi$
$930$		0			0
$931$	−1.19796	+	6.02255i	−0.0392616	+	0.197381i
$932$		0			0
$933$	−56.7498	−	7.16296i	−1.85790	−	0.234505i
$934$		0			0
$935$	11.4813	+	27.7184i	0.375480	+	0.906489i
$936$		0			0
$937$	−7.86396	+	18.9853i	−0.256904	+	0.620222i	−0.998731	−	0.0503700i	$-0.983960\pi$
$937$	−7.86396	+	18.9853i	−0.256904	+	0.620222i	0.741826	+	0.670592i	$0.233960\pi$
$938$		0			0
$939$	−1.14541	+	0.888667i	−0.0373790	+	0.0290005i
$940$		0			0
$941$	43.2971	−	28.9302i	1.41145	−	0.943098i	0.411955	−	0.911204i	$-0.364846\pi$
$941$	43.2971	−	28.9302i	1.41145	−	0.943098i	0.999491	−	0.0318944i	$-0.0101540\pi$
$942$		0			0
$943$	8.56753	−	8.56753i	0.278997	−	0.278997i
$944$		0			0
$945$	−37.4542	+	25.9041i	−1.21839	+	0.842660i
$946$		0			0
$947$	13.0677	−	8.73155i	0.424643	−	0.283737i	−0.324830	−	0.945772i	$-0.605307\pi$
$947$	13.0677	−	8.73155i	0.424643	−	0.283737i	0.749473	+	0.662035i	$0.230307\pi$
$948$		0			0
$949$	−96.6027	+	19.2155i	−3.13586	+	0.623760i
$950$		0			0
$951$	23.6384	−	1.67624i	0.766526	−	0.0543559i
$952$		0			0
$953$	−18.0842	−	43.6591i	−0.585804	−	1.41426i	−0.887480	−	0.460846i	$-0.847546\pi$
$953$	−18.0842	−	43.6591i	−0.585804	−	1.41426i	0.301676	−	0.953411i	$-0.402454\pi$
$954$		0			0
$955$	−7.11648	+	10.6506i	−0.230284	+	0.344644i
$956$		0			0
$957$	−4.21309	−	15.3940i	−0.136190	−	0.497616i
$958$		0			0
$959$	−11.7908			−0.380744
$960$		0			0
$961$	−30.9975			−0.999920
$962$		0			0
$963$	−3.63092	−	4.05208i	−0.117005	−	0.130577i
$964$		0			0
$965$	10.9794	−	16.4318i	0.353439	−	0.528959i
$966$		0			0
$967$	−7.60618	−	18.3629i	−0.244598	−	0.590512i	0.753131	−	0.657871i	$-0.228543\pi$
$967$	−7.60618	−	18.3629i	−0.244598	−	0.590512i	−0.997729	+	0.0673588i	$0.978543\pi$
$968$		0			0
$969$	0.877135	+	12.3694i	0.0281776	+	0.397361i
$970$		0			0
$971$	23.2362	−	4.62196i	0.745685	−	0.148326i	0.192398	−	0.981317i	$-0.438373\pi$
$971$	23.2362	−	4.62196i	0.745685	−	0.148326i	0.553286	+	0.832991i	$0.313373\pi$
$972$		0			0
$973$	32.0877	−	21.4403i	1.02868	−	0.687344i
$974$		0			0
$975$	20.7443	+	6.91835i	0.664351	+	0.221565i
$976$		0			0
$977$	18.8219	−	18.8219i	0.602164	−	0.602164i	−0.338722	−	0.940886i	$-0.609995\pi$
$977$	18.8219	−	18.8219i	0.602164	−	0.602164i	0.940886	+	0.338722i	$0.109995\pi$
$978$		0			0
$979$	−17.5668	+	11.7377i	−0.561437	+	0.375140i
$980$		0			0
$981$	−20.8087	−	9.98667i	−0.664370	−	0.318850i
$982$		0			0
$983$	13.8501	−	33.4371i	0.441750	−	1.06648i	−0.533585	−	0.845746i	$-0.679156\pi$
$983$	13.8501	−	33.4371i	0.441750	−	1.06648i	0.975335	−	0.220731i	$-0.0708444\pi$
$984$		0			0
$985$	15.5816	+	37.6174i	0.496472	+	1.19859i
$986$		0			0
$987$	−2.58031	+	20.4429i	−0.0821321	+	0.650706i
$988$		0			0
$989$	0.509107	−	2.55945i	0.0161886	−	0.0813858i
$990$		0			0
$991$	−3.20062			−0.101671			−0.0508356	−	0.998707i	$-0.516188\pi$
$991$	−3.20062			−0.101671			−0.0508356	+	0.998707i	$0.516188\pi$
$992$		0			0
$993$	−9.60062	−	19.2087i	−0.304667	−	0.609569i
$994$		0			0
$995$	29.8054	+	5.92866i	0.944894	+	0.187951i
$996$		0			0
$997$	25.8757	−	38.7257i	0.819492	−	1.22646i	−0.151764	−	0.988417i	$-0.548495\pi$
$997$	25.8757	−	38.7257i	0.819492	−	1.22646i	0.971256	−	0.238039i	$-0.0765047\pi$
$998$		0			0
$999$	−10.9122	−	50.6057i	−0.345247	−	1.60109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.s.a.431.28		240
3.2	odd	2	inner	768.2.s.a.431.24		240
4.3	odd	2		192.2.s.a.131.18	yes	240
12.11	even	2		192.2.s.a.131.13	yes	240
64.21	even	16		192.2.s.a.107.13	✓	240
64.43	odd	16	inner	768.2.s.a.335.24		240
192.107	even	16	inner	768.2.s.a.335.28		240
192.149	odd	16		192.2.s.a.107.18	yes	240

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.s.a.107.13	✓	240	64.21	even	16
192.2.s.a.107.18	yes	240	192.149	odd	16
192.2.s.a.131.13	yes	240	12.11	even	2
192.2.s.a.131.18	yes	240	4.3	odd	2
768.2.s.a.335.24		240	64.43	odd	16	inner
768.2.s.a.335.28		240	192.107	even	16	inner
768.2.s.a.431.24		240	3.2	odd	2	inner
768.2.s.a.431.28		240	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.s (of order \(16\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.67061 - 0.457221i) q^{3} +(-1.45822 + 2.18239i) q^{5} +(1.27779 + 3.08486i) q^{7} +(2.58190 - 1.52768i) q^{9} +(-2.31748 + 0.460975i) q^{11} +(-5.55652 + 3.71275i) q^{13} +(-1.43829 + 4.31265i) q^{15} +(-3.42067 + 3.42067i) q^{17} +(1.23054 - 0.822221i) q^{19} +(3.54516 + 4.56938i) q^{21} +(1.87568 - 4.52828i) q^{23} +(-0.722975 - 1.74542i) q^{25} +(3.61486 - 3.73266i) q^{27} +(-0.760797 + 3.82478i) q^{29} +0.0497279 q^{31} +(-3.66084 + 1.82971i) q^{33} +(-8.59567 - 1.70978i) q^{35} +(5.53510 - 8.28386i) q^{37} +(-7.58524 + 8.74312i) q^{39} +(2.28385 + 0.946002i) q^{41} +(0.522190 - 0.103870i) q^{43} +(-0.430996 + 7.86240i) q^{45} +(2.51930 + 2.51930i) q^{47} +(-2.93388 + 2.93388i) q^{49} +(-4.15061 + 7.27862i) q^{51} +(1.79038 + 9.00087i) q^{53} +(2.37338 - 5.72983i) q^{55} +(1.67982 - 1.93624i) q^{57} +(-2.56739 - 1.71547i) q^{59} +(-0.935842 + 4.70479i) q^{61} +(8.01181 + 6.01274i) q^{63} -17.5405i q^{65} +(3.65093 + 0.726215i) q^{67} +(1.06310 - 8.42261i) q^{69} +(12.6483 - 5.23910i) q^{71} +(13.6168 + 5.64025i) q^{73} +(-2.00585 - 2.58536i) q^{75} +(-4.38330 - 6.56007i) q^{77} +(1.69637 + 1.69637i) q^{79} +(4.33239 - 7.88863i) q^{81} +(5.63016 + 8.42612i) q^{83} +(-2.47712 - 12.4533i) q^{85} +(0.477776 + 6.73759i) q^{87} +(8.26075 - 3.42172i) q^{89} +(-18.5534 - 12.3970i) q^{91} +(0.0830760 - 0.0227366i) q^{93} +3.88450i q^{95} -3.93119i q^{97} +(-5.27927 + 4.73055i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q + 8 q^{3} + 16 q^{7} - 8 q^{9} - 16 q^{13} + 8 q^{15} + 16 q^{19} - 8 q^{21} - 16 q^{25} + 8 q^{27} + 32 q^{31} - 16 q^{37} + 8 q^{39} + 16 q^{43} - 8 q^{45} - 16 q^{49} + 8 q^{51} + 80 q^{55} - 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) \) 240 * q + 8 * q^3 + 16 * q^7 - 8 * q^9 - 16 * q^13 + 8 * q^15 + 16 * q^19 - 8 * q^21 - 16 * q^25 + 8 * q^27 + 32 * q^31 - 16 * q^37 + 8 * q^39 + 16 * q^43 - 8 * q^45 - 16 * q^49 + 8 * q^51 + 80 * q^55 - 8 * q^57 - 16 * q^61 + 144 * q^67 - 8 * q^69 - 16 * q^73 + 8 * q^75 + 48 * q^79 - 8 * q^81 - 16 * q^85 + 8 * q^87 + 16 * q^91 - 32 * q^93 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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