LMFDB - Embedded newform 112.10.p.b.47.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.7
Character	$\chi$	$=$	112.47
Dual form			112.10.p.b.31.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(29.1853 + 50.5504i) q^{3} +(2152.25 + 1242.60i) q^{5} +(-2216.90 - 5953.06i) q^{7} +(8137.94 - 14095.3i) q^{9} +(-71975.9 + 41555.3i) q^{11} -89784.7i q^{13} +145063. i q^{15} +(-329589. + 190288. i) q^{17} +(-517925. + 897073. i) q^{19} +(236229. - 285807. i) q^{21} +(-1.03488e6 - 597486. i) q^{23} +(2.11155e6 + 3.65731e6i) q^{25} +2.09894e6 q^{27} -3.28143e6 q^{29} +(215533. + 373313. i) q^{31} +(-4.20128e6 - 2.42561e6i) q^{33} +(2.62595e6 - 1.55672e7i) q^{35} +(1.55965e6 - 2.70140e6i) q^{37} +(4.53865e6 - 2.62039e6i) q^{39} +6.97186e6i q^{41} +1.88627e7i q^{43} +(3.50297e7 - 2.02244e7i) q^{45} +(-2.26013e7 + 3.91466e7i) q^{47} +(-3.05243e7 + 2.63947e7i) q^{49} +(-1.92383e7 - 1.11072e7i) q^{51} +(974600. + 1.68806e6i) q^{53} -2.06547e8 q^{55} -6.04632e7 q^{57} +(6.20310e7 + 1.07441e8i) q^{59} +(2.30614e7 + 1.33145e7i) q^{61} +(-1.01951e8 - 1.71977e7i) q^{63} +(1.11566e8 - 1.93239e8i) q^{65} +(7.12113e7 - 4.11139e7i) q^{67} -6.97512e7i q^{69} -7.73013e7i q^{71} +(-1.35016e8 + 7.79517e7i) q^{73} +(-1.23252e8 + 2.13479e8i) q^{75} +(4.06945e8 + 3.36353e8i) q^{77} +(-4.27239e8 - 2.46667e8i) q^{79} +(-9.89209e7 - 1.71336e8i) q^{81} -2.68553e8 q^{83} -9.45809e8 q^{85} +(-9.57693e7 - 1.65877e8i) q^{87} +(2.58286e8 + 1.49121e8i) q^{89} +(-5.34494e8 + 1.99044e8i) q^{91} +(-1.25808e7 + 2.17905e7i) q^{93} +(-2.22941e9 + 1.28715e9i) q^{95} +1.65606e9i q^{97} +1.35270e9i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57}+ \cdots - 1432348316 q^{93}+O(q^{100}) $ 24 * q + 1704 * q^5 - 80428 * q^9 - 1578672 * q^17 + 1219540 * q^21 + 8001240 * q^25 - 6709416 * q^29 - 29129772 * q^33 - 11130084 * q^37 + 57023292 * q^45 - 12671904 * q^49 - 93652164 * q^53 + 742621544 * q^57 - 593611308 * q^61 + 160281180 * q^65 - 1676922516 * q^73 + 1645751688 * q^77 - 528698056 * q^81 + 1370082456 * q^85 - 941603484 * q^89 - 1432348316 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$3$	29.1853	+	50.5504i	0.208026	+	0.360312i	0.951093	−	0.308906i	$-0.0999627\pi$
$3$	29.1853	+	50.5504i	0.208026	+	0.360312i	−0.743066	+	0.669218i	$0.766629\pi$
$4$		0			0
$5$	2152.25	+	1242.60i	1.54002	+	0.889132i	0.998836	+	0.0482296i	$0.0153579\pi$
$5$	2152.25	+	1242.60i	1.54002	+	0.889132i	0.541186	+	0.840903i	$0.317975\pi$
$6$		0			0
$7$	−2216.90	−	5953.06i	−0.348984	−	0.937129i
$7$	−2216.90	−	5953.06i	−0.348984	−	0.937129i
$8$		0			0
$9$	8137.94	−	14095.3i	0.413450	−	0.716117i
$10$		0			0
$11$	−71975.9	+	41555.3i	−1.48225	+	0.855775i	−0.999797	−	0.0201488i	$-0.993586\pi$
$11$	−71975.9	+	41555.3i	−1.48225	+	0.855775i	−0.482449	+	0.875924i	$0.660253\pi$
$12$		0			0
$13$		−	89784.7i		−	0.871881i	−0.899975	−	0.435941i	$-0.856416\pi$
$13$		−	89784.7i		−	0.871881i	0.899975	−	0.435941i	$-0.143584\pi$
$14$		0			0
$15$			145063.i			0.739851i
$16$		0			0
$17$	−329589.	+	190288.i	−0.957089	+	0.552576i	−0.895276	−	0.445512i	$-0.853022\pi$
$17$	−329589.	+	190288.i	−0.957089	+	0.552576i	−0.0618132	+	0.998088i	$0.519688\pi$
$18$		0			0
$19$	−517925.	+	897073.i	−0.911750	+	1.57920i	−0.100159	+	0.994971i	$0.531935\pi$
$19$	−517925.	+	897073.i	−0.911750	+	1.57920i	−0.811591	+	0.584226i	$0.801398\pi$
$20$		0			0
$21$	236229.	−	285807.i	0.265061	−	0.320691i
$22$		0			0
$23$	−1.03488e6	−	597486.i	−0.771105	−	0.445197i	0.0621640	−	0.998066i	$-0.480200\pi$
$23$	−1.03488e6	−	597486.i	−0.771105	−	0.445197i	−0.833268	+	0.552869i	$0.813533\pi$
$24$		0			0
$25$	2.11155e6	+	3.65731e6i	1.08111	+	1.87254i
$26$		0			0
$27$	2.09894e6			0.760086
$28$		0			0
$29$	−3.28143e6			−0.861532			−0.430766	−	0.902464i	$-0.641757\pi$
$29$	−3.28143e6			−0.861532			−0.430766	+	0.902464i	$0.641757\pi$
$30$		0			0
$31$	215533.	+	373313.i	0.0419165	+	0.0726015i	0.886223	−	0.463260i	$-0.153320\pi$
$31$	215533.	+	373313.i	0.0419165	+	0.0726015i	−0.844306	+	0.535861i	$0.819987\pi$
$32$		0			0
$33$	−4.20128e6	−	2.42561e6i	−0.616692	−	0.356047i
$34$		0			0
$35$	2.62595e6	−	1.55672e7i	0.295788	−	1.75349i
$36$		0			0
$37$	1.55965e6	−	2.70140e6i	0.136811	−	0.236963i	−0.789477	−	0.613780i	$-0.789648\pi$
$37$	1.55965e6	−	2.70140e6i	0.136811	−	0.236963i	0.926288	+	0.376817i	$0.122981\pi$
$38$		0			0
$39$	4.53865e6	−	2.62039e6i	0.314149	−	0.181374i
$40$		0			0
$41$			6.97186e6i			0.385320i	0.981266	+	0.192660i	$0.0617114\pi$
$41$			6.97186e6i			0.385320i	−0.981266	+	0.192660i	$0.938289\pi$
$42$		0			0
$43$			1.88627e7i			0.841389i	0.907202	+	0.420694i	$0.138213\pi$
$43$			1.88627e7i			0.841389i	−0.907202	+	0.420694i	$0.861787\pi$
$44$		0			0
$45$	3.50297e7	−	2.02244e7i	1.27345	−	0.735224i
$46$		0			0
$47$	−2.26013e7	+	3.91466e7i	−0.675606	+	1.17018i	0.300686	+	0.953723i	$0.402785\pi$
$47$	−2.26013e7	+	3.91466e7i	−0.675606	+	1.17018i	−0.976291	+	0.216460i	$0.930549\pi$
$48$		0			0
$49$	−3.05243e7	+	2.63947e7i	−0.756420	+	0.654086i
$50$		0			0
$51$	−1.92383e7	−	1.11072e7i	−0.398199	−	0.229900i
$52$		0			0
$53$	974600.	+	1.68806e6i	0.0169662	+	0.0293864i	0.874384	−	0.485235i	$-0.161266\pi$
$53$	974600.	+	1.68806e6i	0.0169662	+	0.0293864i	−0.857418	+	0.514621i	$0.827933\pi$
$54$		0			0
$55$	−2.06547e8			−3.04359
$56$		0			0
$57$	−6.04632e7			−0.758672
$58$		0			0
$59$	6.20310e7	+	1.07441e8i	0.666461	+	1.15435i	0.978887	+	0.204403i	$0.0655252\pi$
$59$	6.20310e7	+	1.07441e8i	0.666461	+	1.15435i	−0.312425	+	0.949942i	$0.601141\pi$
$60$		0			0
$61$	2.30614e7	+	1.33145e7i	0.213256	+	0.123123i	0.602824	−	0.797874i	$-0.294042\pi$
$61$	2.30614e7	+	1.33145e7i	0.213256	+	0.123123i	−0.389568	+	0.920998i	$0.627376\pi$
$62$		0			0
$63$	−1.01951e8	−	1.71977e7i	−0.815381	−	0.137542i
$64$		0			0
$65$	1.11566e8	−	1.93239e8i	0.775218	−	1.34272i
$66$		0			0
$67$	7.12113e7	−	4.11139e7i	0.431730	−	0.249259i	−0.268353	−	0.963321i	$-0.586479\pi$
$67$	7.12113e7	−	4.11139e7i	0.431730	−	0.249259i	0.700083	+	0.714061i	$0.253146\pi$
$68$		0			0
$69$		−	6.97512e7i		−	0.370451i
$70$		0			0
$71$		−	7.73013e7i		−	0.361014i	−0.983574	−	0.180507i	$-0.942226\pi$
$71$		−	7.73013e7i		−	0.361014i	0.983574	−	0.180507i	$-0.0577739\pi$
$72$		0			0
$73$	−1.35016e8	+	7.79517e7i	−0.556459	+	0.321272i	−0.751723	−	0.659479i	$-0.770777\pi$
$73$	−1.35016e8	+	7.79517e7i	−0.556459	+	0.321272i	0.195264	+	0.980751i	$0.437444\pi$
$74$		0			0
$75$	−1.23252e8	+	2.13479e8i	−0.449800	+	0.779076i
$76$		0			0
$77$	4.06945e8	+	3.36353e8i	1.31925	+	1.09040i
$78$		0			0
$79$	−4.27239e8	−	2.46667e8i	−1.23410	−	0.712506i	−0.266216	−	0.963913i	$-0.585773\pi$
$79$	−4.27239e8	−	2.46667e8i	−1.23410	−	0.712506i	−0.967881	+	0.251407i	$0.919107\pi$
$80$		0			0
$81$	−9.89209e7	−	1.71336e8i	−0.255332	−	0.442248i
$82$		0			0
$83$	−2.68553e8			−0.621125			−0.310562	−	0.950553i	$-0.600517\pi$
$83$	−2.68553e8			−0.621125			−0.310562	+	0.950553i	$0.600517\pi$
$84$		0			0
$85$	−9.45809e8			−1.96525
$86$		0			0
$87$	−9.57693e7	−	1.65877e8i	−0.179221	−	0.310420i
$88$		0			0
$89$	2.58286e8	+	1.49121e8i	0.436361	+	0.251933i	0.702053	−	0.712125i	$-0.252267\pi$
$89$	2.58286e8	+	1.49121e8i	0.436361	+	0.251933i	−0.265692	+	0.964058i	$0.585600\pi$
$90$		0			0
$91$	−5.34494e8	+	1.99044e8i	−0.817065	+	0.304273i
$92$		0			0
$93$	−1.25808e7	+	2.17905e7i	−0.0174395	+	0.0302061i
$94$		0			0
$95$	−2.22941e9	+	1.28715e9i	−2.80823	+	1.62133i
$96$		0			0
$97$			1.65606e9i			1.89935i	0.313243	+	0.949673i	$0.398584\pi$
$97$			1.65606e9i			1.89935i	−0.313243	+	0.949673i	$0.601416\pi$
$98$		0			0
$99$			1.35270e9i			1.41528i
$100$		0			0
$101$	1.48485e9	−	8.57279e8i	1.41983	−	0.819740i	0.423548	−	0.905874i	$-0.360785\pi$
$101$	1.48485e9	−	8.57279e8i	1.41983	−	0.819740i	0.996284	+	0.0861340i	$0.0274513\pi$
$102$		0			0
$103$	−2.42062e8	+	4.19263e8i	−0.211914	+	0.367045i	−0.952313	−	0.305122i	$-0.901303\pi$
$103$	−2.42062e8	+	4.19263e8i	−0.211914	+	0.367045i	0.740400	+	0.672167i	$0.234636\pi$
$104$		0			0
$105$	8.63566e8	−	3.21590e8i	0.693336	−	0.258197i
$106$		0			0
$107$	−8.43775e8	−	4.87154e8i	−0.622300	−	0.359285i	0.155464	−	0.987842i	$-0.450313\pi$
$107$	−8.43775e8	−	4.87154e8i	−0.622300	−	0.359285i	−0.777764	+	0.628556i	$0.783646\pi$
$108$		0			0
$109$	−8.66668e8	−	1.50111e9i	−0.588076	−	1.01858i	−0.994484	−	0.104886i	$-0.966552\pi$
$109$	−8.66668e8	−	1.50111e9i	−0.588076	−	1.01858i	0.406408	−	0.913692i	$-0.366781\pi$
$110$		0			0
$111$	1.82076e8			0.113841
$112$		0			0
$113$	−2.58942e9			−1.49400			−0.746998	−	0.664826i	$-0.768506\pi$
$113$	−2.58942e9			−1.49400			−0.746998	+	0.664826i	$0.768506\pi$
$114$		0			0
$115$	−1.48487e9	−	2.57188e9i	−0.791679	−	1.37123i
$116$		0			0
$117$	−1.26554e9	−	7.30662e8i	−0.624369	−	0.360479i
$118$		0			0
$119$	1.86346e9	+	1.54021e9i	0.851843	+	0.704075i
$120$		0			0
$121$	2.27472e9	−	3.93993e9i	0.964702	−	1.67091i
$122$		0			0
$123$	−3.52430e8	+	2.03476e8i	−0.138835	+	0.0801566i
$124$		0			0
$125$			5.64134e9i			2.06675i
$126$		0			0
$127$		−	4.99476e9i		−	1.70372i	−0.523770	−	0.851860i	$-0.675475\pi$
$127$		−	4.99476e9i		−	1.70372i	0.523770	−	0.851860i	$-0.324525\pi$
$128$		0			0
$129$	−9.53518e8	+	5.50514e8i	−0.303162	+	0.175031i
$130$		0			0
$131$	6.49262e8	−	1.12456e9i	0.192619	−	0.333626i	−0.753498	−	0.657450i	$-0.771635\pi$
$131$	6.49262e8	−	1.12456e9i	0.192619	−	0.333626i	0.946117	+	0.323824i	$0.104968\pi$
$132$		0			0
$133$	6.48852e9	+	1.09452e9i	1.79810	+	0.303312i
$134$		0			0
$135$	4.51744e9	+	2.60814e9i	1.17055	+	0.675817i
$136$		0			0
$137$	3.89707e8	+	6.74992e8i	0.0945138	+	0.163703i	0.909406	−	0.415910i	$-0.136537\pi$
$137$	3.89707e8	+	6.74992e8i	0.0945138	+	0.163703i	−0.814892	+	0.579613i	$0.803204\pi$
$138$		0			0
$139$	−2.33702e9			−0.531003			−0.265501	−	0.964111i	$-0.585537\pi$
$139$	−2.33702e9			−0.531003			−0.265501	+	0.964111i	$0.585537\pi$
$140$		0			0
$141$	−2.63850e9			−0.562175
$142$		0			0
$143$	3.73103e9	+	6.46234e9i	0.746134	+	1.29234i
$144$		0			0
$145$	−7.06244e9	−	4.07750e9i	−1.32678	−	0.766016i
$146$		0			0
$147$	−2.22512e9	−	7.72676e8i	−0.393030	−	0.136480i
$148$		0			0
$149$	3.14380e9	−	5.44522e9i	0.522537	−	0.905060i	−0.477120	−	0.878838i	$-0.658319\pi$
$149$	3.14380e9	−	5.44522e9i	0.522537	−	0.905060i	0.999656	−	0.0262215i	$-0.00834752\pi$
$150$		0			0
$151$	3.68614e9	−	2.12819e9i	0.577000	−	0.333131i	−0.182940	−	0.983124i	$-0.558562\pi$
$151$	3.68614e9	−	2.12819e9i	0.577000	−	0.333131i	0.759940	+	0.649993i	$0.225228\pi$
$152$		0			0
$153$			6.19422e9i			0.913850i
$154$		0			0
$155$			1.07128e9i			0.149077i
$156$		0			0
$157$	6.51421e8	−	3.76098e8i	0.0855684	−	0.0494030i	−0.456605	−	0.889669i	$-0.650935\pi$
$157$	6.51421e8	−	3.76098e8i	0.0855684	−	0.0494030i	0.542174	+	0.840266i	$0.317601\pi$
$158$		0			0
$159$	−5.68880e7	+	9.85328e7i	−0.00705884	+	0.0122263i
$160$		0			0
$161$	−1.26265e9	+	7.48525e9i	−0.148104	+	0.877991i
$162$		0			0
$163$	−6.95067e9	−	4.01297e9i	−0.771228	−	0.445269i	0.0620847	−	0.998071i	$-0.480225\pi$
$163$	−6.95067e9	−	4.01297e9i	−0.771228	−	0.445269i	−0.833312	+	0.552802i	$0.813558\pi$
$164$		0			0
$165$	−6.02812e9	−	1.04410e10i	−0.633147	−	1.09664i
$166$		0			0
$167$	1.00397e10			0.998842			0.499421	−	0.866359i	$-0.333546\pi$
$167$	1.00397e10			0.998842			0.499421	+	0.866359i	$0.333546\pi$
$168$		0			0
$169$	2.54321e9			0.239823
$170$		0			0
$171$	8.42969e9	+	1.46006e10i	0.753926	+	1.30584i
$172$		0			0
$173$	1.11373e10	+	6.43014e9i	0.945308	+	0.545774i	0.891620	−	0.452784i	$-0.149569\pi$
$173$	1.11373e10	+	6.43014e9i	0.945308	+	0.545774i	0.0536877	+	0.998558i	$0.482902\pi$
$174$		0			0
$175$	1.70911e10	−	2.06781e10i	1.37752	−	1.66663i
$176$		0			0
$177$	−3.62079e9	+	6.27139e9i	−0.277283	+	0.480268i
$178$		0			0
$179$	1.98066e10	−	1.14353e10i	1.44202	−	0.832549i	0.444032	−	0.896011i	$-0.353548\pi$
$179$	1.98066e10	−	1.14353e10i	1.44202	−	0.832549i	0.997984	+	0.0634620i	$0.0202142\pi$
$180$		0			0
$181$			8.35823e9i			0.578843i	0.957202	+	0.289421i	$0.0934629\pi$
$181$			8.35823e9i			0.578843i	−0.957202	+	0.289421i	$0.906537\pi$
$182$		0			0
$183$			1.55435e9i			0.102452i
$184$		0			0
$185$	6.71352e9	−	3.87605e9i	0.421383	−	0.243286i
$186$		0			0
$187$	1.58150e10	−	2.73923e10i	0.945761	−	1.63811i
$188$		0			0
$189$	−4.65315e9	−	1.24951e10i	−0.265258	−	0.712299i
$190$		0			0
$191$	5.61426e9	+	3.24140e9i	0.305241	+	0.176231i	0.644795	−	0.764356i	$-0.276943\pi$
$191$	5.61426e9	+	3.24140e9i	0.305241	+	0.176231i	−0.339554	+	0.940587i	$0.610276\pi$
$192$		0			0
$193$	7.60492e9	+	1.31721e10i	0.394536	+	0.683357i	0.993042	−	0.117762i	$-0.0375719\pi$
$193$	7.60492e9	+	1.31721e10i	0.394536	+	0.683357i	−0.598506	+	0.801119i	$0.704239\pi$
$194$		0			0
$195$	1.30244e10			0.645063
$196$		0			0
$197$	−2.52941e10			−1.19652			−0.598261	−	0.801302i	$-0.704141\pi$
$197$	−2.52941e10			−1.19652			−0.598261	+	0.801302i	$0.704141\pi$
$198$		0			0
$199$	−3.55837e8	−	6.16329e8i	−0.0160847	−	0.0278595i	0.857871	−	0.513865i	$-0.171787\pi$
$199$	−3.55837e8	−	6.16329e8i	−0.0160847	−	0.0278595i	−0.873956	+	0.486006i	$0.838453\pi$
$200$		0			0
$201$	4.15664e9	+	2.39984e9i	0.179622	+	0.103705i
$202$		0			0
$203$	7.27461e9	+	1.95345e10i	0.300661	+	0.807366i
$204$		0			0
$205$	−8.66323e9	+	1.50052e10i	−0.342600	+	0.593401i
$206$		0			0
$207$	−1.68435e10	+	9.72461e9i	−0.637627	+	0.368134i
$208$		0			0
$209$		−	8.60902e10i		−	3.12101i
$210$		0			0
$211$		−	2.31910e10i		−	0.805467i	−0.915317	−	0.402733i	$-0.868060\pi$
$211$		−	2.31910e10i		−	0.805467i	0.915317	−	0.402733i	$-0.131940\pi$
$212$		0			0
$213$	3.90761e9	−	2.25606e9i	0.130078	−	0.0751004i
$214$		0			0
$215$	−2.34388e10	+	4.05973e10i	−0.748106	+	1.29576i
$216$		0			0
$217$	1.74454e9	−	2.11068e9i	0.0534088	−	0.0646180i
$218$		0			0
$219$	−7.88098e9	−	4.55008e9i	−0.231516	−	0.133666i
$220$		0			0
$221$	1.70850e10	+	2.95920e10i	0.481780	+	0.834468i
$222$		0			0
$223$	−2.99761e10			−0.811714			−0.405857	−	0.913937i	$-0.633027\pi$
$223$	−2.99761e10			−0.811714			−0.405857	+	0.913937i	$0.633027\pi$
$224$		0			0
$225$	6.87346e10			1.78795
$226$		0			0
$227$	−2.89596e9	−	5.01595e9i	−0.0723896	−	0.125382i	0.827559	−	0.561379i	$-0.189729\pi$
$227$	−2.89596e9	−	5.01595e9i	−0.0723896	−	0.125382i	−0.899948	+	0.435997i	$0.856396\pi$
$228$		0			0
$229$	−3.07465e9	−	1.77515e9i	−0.0738816	−	0.0426556i	0.462604	−	0.886565i	$-0.346915\pi$
$229$	−3.07465e9	−	1.77515e9i	−0.0738816	−	0.0426556i	−0.536486	+	0.843909i	$0.680248\pi$
$230$		0			0
$231$	−5.12596e9	+	3.03878e10i	−0.118446	+	0.702175i
$232$		0			0
$233$	−2.81224e10	+	4.87094e10i	−0.625101	+	1.08271i	0.363420	+	0.931625i	$0.381609\pi$
$233$	−2.81224e10	+	4.87094e10i	−0.625101	+	1.08271i	−0.988521	+	0.151082i	$0.951724\pi$
$234$		0			0
$235$	−9.72872e10	+	5.61688e10i	−2.08090	+	1.20141i
$236$		0			0
$237$		−	2.87962e10i		−	0.592880i
$238$		0			0
$239$		−	4.15634e10i		−	0.823988i	−0.911187	−	0.411994i	$-0.864833\pi$
$239$		−	4.15634e10i		−	0.823988i	0.911187	−	0.411994i	$-0.135167\pi$
$240$		0			0
$241$	−5.60364e10	+	3.23526e10i	−1.07002	+	0.617779i	−0.928188	−	0.372112i	$-0.878634\pi$
$241$	−5.60364e10	+	3.23526e10i	−1.07002	+	0.617779i	−0.141836	+	0.989890i	$0.545301\pi$
$242$		0			0
$243$	2.64308e10	−	4.57795e10i	0.486275	−	0.842253i
$244$		0			0
$245$	−9.84939e10	+	1.88785e10i	−1.74647	+	0.334750i
$246$		0			0
$247$	8.05434e10	+	4.65018e10i	1.37687	+	0.794938i
$248$		0			0
$249$	−7.83780e9	−	1.35755e10i	−0.129210	−	0.223799i
$250$		0			0
$251$	−1.85774e10			−0.295429			−0.147714	−	0.989030i	$-0.547192\pi$
$251$	−1.85774e10			−0.295429			−0.147714	+	0.989030i	$0.547192\pi$
$252$		0			0
$253$	9.93150e10			1.52396
$254$		0			0
$255$	−2.76037e10	−	4.78110e10i	−0.408824	−	0.708104i
$256$		0			0
$257$	2.25283e10	+	1.30067e10i	0.322129	+	0.185981i	0.652341	−	0.757925i	$-0.273787\pi$
$257$	2.25283e10	+	1.30067e10i	0.322129	+	0.185981i	−0.330212	+	0.943907i	$0.607120\pi$
$258$		0			0
$259$	−1.95392e10	−	3.29597e9i	−0.269810	−	0.0455129i
$260$		0			0
$261$	−2.67040e10	+	4.62528e10i	−0.356201	+	0.616958i
$262$		0			0
$263$	−8.27723e9	+	4.77886e9i	−0.106680	+	0.0615919i	−0.552391	−	0.833585i	$-0.686284\pi$
$263$	−8.27723e9	+	4.77886e9i	−0.106680	+	0.0615919i	0.445711	+	0.895177i	$0.352951\pi$
$264$		0			0
$265$			4.84415e9i			0.0603409i
$266$		0			0
$267$			1.74086e10i			0.209635i
$268$		0			0
$269$	8.12919e10	−	4.69339e10i	0.946591	−	0.546514i	0.0545705	−	0.998510i	$-0.482621\pi$
$269$	8.12919e10	−	4.69339e10i	0.946591	−	0.546514i	0.892020	+	0.451996i	$0.149288\pi$
$270$		0			0
$271$	−3.83247e10	+	6.63803e10i	−0.431635	+	0.747614i	−0.997014	−	0.0772173i	$-0.975396\pi$
$271$	−3.83247e10	+	6.63803e10i	−0.431635	+	0.747614i	0.565379	+	0.824831i	$0.308730\pi$
$272$		0			0
$273$	−2.56611e10	−	2.12097e10i	−0.279604	−	0.231101i
$274$		0			0
$275$	−3.03961e11	−	1.75492e11i	−3.20495	−	1.85038i
$276$		0			0
$277$	3.62619e9	+	6.28075e9i	0.0370077	+	0.0640992i	0.883936	−	0.467608i	$-0.154884\pi$
$277$	3.62619e9	+	6.28075e9i	0.0370077	+	0.0640992i	−0.846928	+	0.531707i	$0.821551\pi$
$278$		0			0
$279$	7.01596e9			0.0693216
$280$		0			0
$281$	9.54552e10			0.913316			0.456658	−	0.889642i	$-0.349046\pi$
$281$	9.54552e10			0.913316			0.456658	+	0.889642i	$0.349046\pi$
$282$		0			0
$283$	2.85912e10	+	4.95214e10i	0.264968	+	0.458938i	0.967555	−	0.252660i	$-0.0813053\pi$
$283$	2.85912e10	+	4.95214e10i	0.264968	+	0.458938i	−0.702587	+	0.711598i	$0.747972\pi$
$284$		0			0
$285$	−1.30132e11	−	7.51315e10i	−1.16837	−	0.674560i
$286$		0			0
$287$	4.15039e10	−	1.54559e10i	0.361094	−	0.134471i
$288$		0			0
$289$	1.31253e10	−	2.27336e10i	0.110680	−	0.191703i
$290$		0			0
$291$	−8.37146e10	+	4.83327e10i	−0.684357	+	0.395114i
$292$		0			0
$293$		−	4.51828e10i		−	0.358153i	−0.983835	−	0.179077i	$-0.942689\pi$
$293$		−	4.51828e10i		−	0.358153i	0.983835	−	0.179077i	$-0.0573110\pi$
$294$		0			0
$295$			3.08319e11i			2.37029i
$296$		0			0
$297$	−1.51073e11	+	8.72221e10i	−1.12664	+	0.650463i
$298$		0			0
$299$	−5.36451e10	+	9.29161e10i	−0.388159	+	0.672311i
$300$		0			0
$301$	1.12291e11	−	4.18169e10i	0.788489	−	0.293631i
$302$		0			0
$303$	8.66716e10	+	5.00399e10i	0.590724	+	0.341055i
$304$		0			0
$305$	3.30892e10	+	5.73121e10i	0.218946	+	0.379225i
$306$		0			0
$307$	−2.64255e11			−1.69785			−0.848926	−	0.528511i	$-0.822750\pi$
$307$	−2.64255e11			−1.69785			−0.848926	+	0.528511i	$0.822750\pi$
$308$		0			0
$309$	−2.82586e10			−0.176334
$310$		0			0
$311$	1.06956e11	+	1.85253e11i	0.648310	+	1.12291i	0.983526	+	0.180764i	$0.0578572\pi$
$311$	1.06956e11	+	1.85253e11i	0.648310	+	1.12291i	−0.335217	+	0.942141i	$0.608810\pi$
$312$		0			0
$313$	−7.19103e10	−	4.15174e10i	−0.423488	−	0.244501i	0.273080	−	0.961991i	$-0.411957\pi$
$313$	−7.19103e10	−	4.15174e10i	−0.423488	−	0.244501i	−0.696569	+	0.717490i	$0.745291\pi$
$314$		0			0
$315$	−1.98055e11	−	1.63698e11i	−1.13341	−	0.936800i
$316$		0			0
$317$	−1.17673e11	+	2.03815e11i	−0.654498	+	1.13362i	0.327521	+	0.944844i	$0.393787\pi$
$317$	−1.17673e11	+	2.03815e11i	−0.654498	+	1.13362i	−0.982019	+	0.188780i	$0.939547\pi$
$318$		0			0
$319$	2.36184e11	−	1.36361e11i	1.27700	−	0.737278i
$320$		0			0
$321$		−	5.68709e10i		−	0.298963i
$322$		0			0
$323$		−	3.94220e11i		−	2.01524i
$324$		0			0
$325$	3.28370e11	−	1.89585e11i	1.63263	−	0.942602i
$326$		0			0
$327$	5.05879e10	−	8.76208e10i	0.244670	−	0.423782i
$328$		0			0
$329$	2.83147e11	+	4.77627e10i	1.33239	+	0.224754i
$330$		0			0
$331$	1.87403e11	+	1.08197e11i	0.858125	+	0.495439i	0.863384	−	0.504547i	$-0.168341\pi$
$331$	1.87403e11	+	1.08197e11i	0.858125	+	0.495439i	−0.00525888	+	0.999986i	$0.501674\pi$
$332$		0			0
$333$	−2.53847e10	−	4.39676e10i	−0.113129	−	0.195945i
$334$		0			0
$335$	2.04352e11			0.886499
$336$		0			0
$337$	−3.41693e11			−1.44312			−0.721559	−	0.692352i	$-0.756574\pi$
$337$	−3.41693e11			−1.44312			−0.721559	+	0.692352i	$0.756574\pi$
$338$		0			0
$339$	−7.55730e10	−	1.30896e11i	−0.310791	−	0.538305i
$340$		0			0
$341$	−3.10263e10	−	1.79131e10i	−0.124261	−	0.0717422i
$342$		0			0
$343$	2.24799e11	+	1.23198e11i	0.876942	+	0.480597i
$344$		0			0
$345$	8.66729e10	−	1.50122e11i	0.329380	−	0.570503i
$346$		0			0
$347$	4.04359e11	−	2.33457e11i	1.49722	−	0.864418i	0.497220	−	0.867624i	$-0.334354\pi$
$347$	4.04359e11	−	2.33457e11i	1.49722	−	0.864418i	0.999995	+	0.00320653i	$0.00102067\pi$
$348$		0			0
$349$			4.37317e11i			1.57791i	0.614451	+	0.788955i	$0.289378\pi$
$349$			4.37317e11i			1.57791i	−0.614451	+	0.788955i	$0.710622\pi$
$350$		0			0
$351$		−	1.88453e11i		−	0.662705i
$352$		0			0
$353$	1.10179e11	−	6.36117e10i	0.377669	−	0.218048i	−0.299134	−	0.954211i	$-0.596698\pi$
$353$	1.10179e11	−	6.36117e10i	0.377669	−	0.218048i	0.676804	+	0.736163i	$0.263365\pi$
$354$		0			0
$355$	9.60546e10	−	1.66371e11i	0.320989	−	0.555970i
$356$		0			0
$357$	−2.34726e10	+	1.39150e11i	−0.0764810	+	0.453396i
$358$		0			0
$359$	2.61664e11	+	1.51072e11i	0.831418	+	0.480019i	0.854338	−	0.519718i	$-0.173963\pi$
$359$	2.61664e11	+	1.51072e11i	0.831418	+	0.480019i	−0.0229198	+	0.999737i	$0.507296\pi$
$360$		0			0
$361$	−3.75149e11	−	6.49777e11i	−1.16258	−	2.01364i
$362$		0			0
$363$	2.65553e11			0.802734
$364$		0			0
$365$	−3.87451e11			−1.14261
$366$		0			0
$367$	3.18846e10	+	5.52258e10i	0.0917453	+	0.158908i	0.908246	−	0.418437i	$-0.137422\pi$
$367$	3.18846e10	+	5.52258e10i	0.0917453	+	0.158908i	−0.816500	+	0.577345i	$0.804089\pi$
$368$		0			0
$369$	9.82706e10	+	5.67366e10i	0.275934	+	0.159310i
$370$		0			0
$371$	7.88851e9	−	9.54412e9i	0.0216178	−	0.0261549i
$372$		0			0
$373$	6.70986e10	−	1.16218e11i	0.179483	−	0.310874i	−0.762220	−	0.647317i	$-0.775891\pi$
$373$	6.70986e10	−	1.16218e11i	0.179483	−	0.310874i	0.941704	+	0.336444i	$0.109224\pi$
$374$		0			0
$375$	−2.85172e11	+	1.64644e11i	−0.744673	+	0.429937i
$376$		0			0
$377$			2.94622e11i			0.751154i
$378$		0			0
$379$			5.13591e11i			1.27862i	0.768950	+	0.639309i	$0.220780\pi$
$379$			5.13591e11i			1.27862i	−0.768950	+	0.639309i	$0.779220\pi$
$380$		0			0
$381$	2.52487e11	−	1.45774e11i	0.613871	−	0.354418i
$382$		0			0
$383$	2.86881e10	−	4.96892e10i	0.0681251	−	0.117996i	−0.829951	−	0.557836i	$-0.811632\pi$
$383$	2.86881e10	−	4.96892e10i	0.0681251	−	0.117996i	0.898076	+	0.439840i	$0.144965\pi$
$384$		0			0
$385$	4.57894e11	+	1.22959e12i	1.06217	+	2.85223i
$386$		0			0
$387$	2.65876e11	+	1.53504e11i	0.602532	+	0.347872i
$388$		0			0
$389$	1.94868e11	+	3.37521e11i	0.431486	+	0.747355i	0.997001	−	0.0773823i	$-0.0246562\pi$
$389$	1.94868e11	+	3.37521e11i	0.431486	+	0.747355i	−0.565516	+	0.824737i	$0.691323\pi$
$390$		0			0
$391$	4.54778e11			0.984021
$392$		0			0
$393$	7.57956e10			0.160279
$394$		0			0
$395$	−6.13016e11	−	1.06178e12i	−1.26703	−	2.19455i
$396$		0			0
$397$	−4.64609e11	−	2.68242e11i	−0.938708	−	0.541964i	−0.0491532	−	0.998791i	$-0.515652\pi$
$397$	−4.64609e11	−	2.68242e11i	−0.938708	−	0.541964i	−0.889555	+	0.456828i	$0.848986\pi$
$398$		0			0
$399$	1.34041e11	+	3.59941e11i	0.264765	+	0.710973i
$400$		0			0
$401$	3.41886e10	−	5.92164e10i	0.0660285	−	0.114365i	−0.831121	−	0.556091i	$-0.812301\pi$
$401$	3.41886e10	−	5.92164e10i	0.0660285	−	0.114365i	0.897150	+	0.441727i	$0.145634\pi$
$402$		0			0
$403$	3.35178e10	−	1.93515e10i	0.0632999	−	0.0365462i
$404$		0			0
$405$		−	4.91677e11i		−	0.908097i
$406$		0			0
$407$			2.59248e11i			0.468317i
$408$		0			0
$409$	5.31461e10	−	3.06839e10i	0.0939110	−	0.0542196i	−0.452309	−	0.891861i	$-0.649400\pi$
$409$	5.31461e10	−	3.06839e10i	0.0939110	−	0.0542196i	0.546220	+	0.837642i	$0.316066\pi$
$410$		0			0
$411$	−2.27474e10	+	3.93996e10i	−0.0393227	+	0.0681089i
$412$		0			0
$413$	5.02085e11	−	6.07461e11i	0.849185	−	1.02741i
$414$		0			0
$415$	−5.77993e11	−	3.33704e11i	−0.956546	−	0.552262i
$416$		0			0
$417$	−6.82067e10	−	1.18137e11i	−0.110462	−	0.191327i
$418$		0			0
$419$	6.07126e11			0.962312			0.481156	−	0.876635i	$-0.340217\pi$
$419$	6.07126e11			0.962312			0.481156	+	0.876635i	$0.340217\pi$
$420$		0			0
$421$	−5.49372e11			−0.852308			−0.426154	−	0.904651i	$-0.640132\pi$
$421$	−5.49372e11			−0.852308			−0.426154	+	0.904651i	$0.640132\pi$
$422$		0			0
$423$	3.67856e11	+	6.37146e11i	0.558659	+	0.967625i
$424$		0			0
$425$	−1.39189e12	−	8.03606e11i	−2.06944	−	1.19479i
$426$		0			0
$427$	2.81371e10	−	1.66803e11i	0.0409594	−	0.242816i
$428$		0			0
$429$	−2.17782e11	+	3.77210e11i	−0.310431	+	0.537682i
$430$		0			0
$431$	3.78947e11	−	2.18785e11i	0.528969	−	0.305401i	−0.211627	−	0.977350i	$-0.567876\pi$
$431$	3.78947e11	−	2.18785e11i	0.528969	−	0.305401i	0.740597	+	0.671950i	$0.234543\pi$
$432$		0			0
$433$		−	4.02080e11i		−	0.549689i	−0.961489	−	0.274844i	$-0.911374\pi$
$433$		−	4.02080e11i		−	0.549689i	0.961489	−	0.274844i	$-0.0886263\pi$
$434$		0			0
$435$		−	4.76012e11i		−	0.637406i
$436$		0			0
$437$	1.07198e12	−	6.18906e11i	1.40611	−	0.811818i
$438$		0			0
$439$	4.96649e11	−	8.60222e11i	0.638204	−	1.10540i	−0.347623	−	0.937635i	$-0.613011\pi$
$439$	4.96649e11	−	8.60222e11i	0.638204	−	1.10540i	0.985827	−	0.167767i	$-0.0536557\pi$
$440$		0			0
$441$	1.23638e11	+	6.45048e11i	0.155660	+	0.812117i
$442$		0			0
$443$	2.92029e11	+	1.68603e11i	0.360254	+	0.207993i	0.669192	−	0.743089i	$-0.266640\pi$
$443$	2.92029e11	+	1.68603e11i	0.360254	+	0.207993i	−0.308938	+	0.951082i	$0.599974\pi$
$444$		0			0
$445$	3.70597e11	+	6.41892e11i	0.448004	+	0.775965i
$446$		0			0
$447$	3.67011e11			0.434805
$448$		0			0
$449$	1.37315e12			1.59444			0.797220	−	0.603688i	$-0.206303\pi$
$449$	1.37315e12			1.59444			0.797220	+	0.603688i	$0.206303\pi$
$450$		0			0
$451$	−2.89718e11	−	5.01806e11i	−0.329747	−	0.571139i
$452$		0			0
$453$	2.15162e11	+	1.24224e11i	0.240062	+	0.138600i
$454$		0			0
$455$	−1.39769e12	−	2.35770e11i	−1.52884	−	0.257892i
$456$		0			0
$457$	−3.27135e11	+	5.66614e11i	−0.350836	+	0.607665i	−0.986396	−	0.164386i	$-0.947436\pi$
$457$	−3.27135e11	+	5.66614e11i	−0.350836	+	0.607665i	0.635560	+	0.772051i	$0.280769\pi$
$458$		0			0
$459$	−6.91787e11	+	3.99403e11i	−0.727470	+	0.420005i
$460$		0			0
$461$			2.56491e11i			0.264495i	0.991217	+	0.132248i	$0.0422194\pi$
$461$			2.56491e11i			0.264495i	−0.991217	+	0.132248i	$0.957781\pi$
$462$		0			0
$463$			8.11617e11i			0.820799i	0.911906	+	0.410399i	$0.134611\pi$
$463$			8.11617e11i			0.820799i	−0.911906	+	0.410399i	$0.865389\pi$
$464$		0			0
$465$	−5.41538e10	+	3.12657e10i	−0.0537144	+	0.0310120i
$466$		0			0
$467$	−7.21242e11	+	1.24923e12i	−0.701706	+	1.21539i	0.266161	+	0.963929i	$0.414245\pi$
$467$	−7.21242e11	+	1.24923e12i	−0.701706	+	1.21539i	−0.967867	+	0.251462i	$0.919089\pi$
$468$		0			0
$469$	−4.02622e11	−	3.32780e11i	−0.384255	−	0.317599i
$470$		0			0
$471$	3.80238e10	+	2.19531e10i	0.0356010	+	0.0205542i
$472$		0			0
$473$	−7.83847e11	−	1.35766e12i	−0.720039	−	1.24714i
$474$		0			0
$475$	−4.37450e12			−3.94282
$476$		0			0
$477$	3.17250e10			0.0280587
$478$		0			0
$479$	7.50986e11	+	1.30075e12i	0.651811	+	1.12897i	0.982683	+	0.185294i	$0.0593239\pi$
$479$	7.50986e11	+	1.30075e12i	0.651811	+	1.12897i	−0.330872	+	0.943676i	$0.607343\pi$
$480$		0			0
$481$	−2.42544e11	−	1.40033e11i	−0.206604	−	0.119283i
$482$		0			0
$483$	−4.15233e11	+	1.54632e11i	−0.347160	+	0.129282i
$484$		0			0
$485$	−2.05782e12	+	3.56426e12i	−1.68877	+	2.92504i
$486$		0			0
$487$	1.02043e12	−	5.89145e11i	0.822058	−	0.474615i	−0.0290676	−	0.999577i	$-0.509254\pi$
$487$	1.02043e12	−	5.89145e11i	0.822058	−	0.474615i	0.851126	+	0.524962i	$0.175920\pi$
$488$		0			0
$489$		−	4.68479e11i		−	0.370510i
$490$		0			0
$491$		−	8.44909e11i		−	0.656059i	−0.944667	−	0.328030i	$-0.893615\pi$
$491$		−	8.44909e11i		−	0.656059i	0.944667	−	0.328030i	$-0.106385\pi$
$492$		0			0
$493$	1.08152e12	−	6.24417e11i	0.824563	−	0.476062i
$494$		0			0
$495$	−1.68086e12	+	2.91134e12i	−1.25837	+	2.17957i
$496$		0			0
$497$	−4.60179e11	+	1.71370e11i	−0.338317	+	0.125988i
$498$		0			0
$499$	−1.38358e12	−	7.98810e11i	−0.998968	−	0.576755i	−0.0910253	−	0.995849i	$-0.529014\pi$
$499$	−1.38358e12	−	7.98810e11i	−0.998968	−	0.576755i	−0.907943	+	0.419094i	$0.862348\pi$
$500$		0			0
$501$	2.93011e11	+	5.07511e11i	0.207785	+	0.359895i
$502$		0			0
$503$	−9.48827e11			−0.660893			−0.330447	−	0.943825i	$-0.607199\pi$
$503$	−9.48827e11			−0.660893			−0.330447	+	0.943825i	$0.607199\pi$
$504$		0			0
$505$	4.26102e12			2.91543
$506$		0			0
$507$	7.42242e10	+	1.28560e11i	0.0498896	+	0.0864113i
$508$		0			0
$509$	−2.33878e12	−	1.35029e12i	−1.54440	−	0.891658i	−0.998553	−	0.0537715i	$-0.982876\pi$
$509$	−2.33878e12	−	1.35029e12i	−1.54440	−	0.891658i	−0.545844	−	0.837887i	$-0.683791\pi$
$510$		0			0
$511$	7.63370e11	+	6.30949e11i	0.495269	+	0.409355i
$512$		0			0
$513$	−1.08709e12	+	1.88290e12i	−0.693009	+	1.20033i
$514$		0			0
$515$	−1.04195e12	+	6.01572e11i	−0.652703	+	0.376839i
$516$		0			0
$517$		−	3.75682e12i		−	2.31267i
$518$		0			0
$519$			7.50661e11i			0.454141i
$520$		0			0
$521$	−9.17021e11	+	5.29443e11i	−0.545268	+	0.314810i	−0.747211	−	0.664587i	$-0.768608\pi$
$521$	−9.17021e11	+	5.29443e11i	−0.545268	+	0.314810i	0.201943	+	0.979397i	$0.435274\pi$
$522$		0			0
$523$	7.07915e11	−	1.22615e12i	0.413736	−	0.716613i	−0.581558	−	0.813505i	$-0.697557\pi$
$523$	7.07915e11	−	1.22615e12i	0.413736	−	0.716613i	0.995295	+	0.0968921i	$0.0308902\pi$
$524$		0			0
$525$	1.54409e12	+	2.60465e11i	0.887067	+	0.149635i
$526$		0			0
$527$	−1.42074e11	−	8.20266e10i	−0.0802357	−	0.0463241i
$528$		0			0
$529$	−1.86597e11	−	3.23195e11i	−0.103599	−	0.179438i
$530$		0			0
$531$	2.01922e12			1.10219
$532$		0			0
$533$	6.25966e11			0.335953
$534$		0			0
$535$	−1.21068e12	−	2.09695e12i	−0.638904	−	1.10661i
$536$		0			0
$537$	1.15612e12	+	6.67486e11i	0.599955	+	0.346384i
$538$		0			0
$539$	1.10017e12	−	3.16823e12i	0.561450	−	1.61684i
$540$		0			0
$541$	−1.22382e12	+	2.11972e12i	−0.614228	+	1.06387i	0.376291	+	0.926502i	$0.377199\pi$
$541$	−1.22382e12	+	2.11972e12i	−0.614228	+	1.06387i	−0.990519	+	0.137373i	$0.956134\pi$
$542$		0			0
$543$	−4.22512e11	+	2.43937e11i	−0.208564	+	0.120414i
$544$		0			0
$545$		−	4.30769e12i		−	2.09151i
$546$		0			0
$547$			2.83833e12i			1.35556i	0.735264	+	0.677781i	$0.237058\pi$
$547$			2.83833e12i			1.35556i	−0.735264	+	0.677781i	$0.762942\pi$
$548$		0			0
$549$	3.75344e11	−	2.16705e11i	0.176341	−	0.101811i
$550$		0			0
$551$	1.69953e12	−	2.94368e12i	0.785502	−	1.36053i
$552$		0			0
$553$	−5.21273e11	+	3.09022e12i	−0.237030	+	1.40516i
$554$		0			0
$555$	3.91872e11	+	2.26247e11i	0.175318	+	0.101220i
$556$		0			0
$557$	1.33865e12	+	2.31861e12i	0.589277	+	1.02066i	0.994327	+	0.106363i	$0.0339205\pi$
$557$	1.33865e12	+	2.31861e12i	0.589277	+	1.02066i	−0.405051	+	0.914294i	$0.632746\pi$
$558$		0			0
$559$	1.69358e12			0.733591
$560$		0			0
$561$	1.84626e12			0.786972
$562$		0			0
$563$	7.47974e11	+	1.29553e12i	0.313761	+	0.543450i	0.979173	−	0.203026i	$-0.0650777\pi$
$563$	7.47974e11	+	1.29553e12i	0.313761	+	0.543450i	−0.665413	+	0.746476i	$0.731744\pi$
$564$		0			0
$565$	−5.57307e12	−	3.21762e12i	−2.30079	−	1.32836i
$566$		0			0
$567$	−8.00676e11	+	9.68718e11i	−0.325337	+	0.393617i
$568$		0			0
$569$	−7.93951e11	+	1.37516e12i	−0.317533	+	0.549983i	−0.979973	−	0.199132i	$-0.936188\pi$
$569$	−7.93951e11	+	1.37516e12i	−0.317533	+	0.549983i	0.662440	+	0.749115i	$0.269521\pi$
$570$		0			0
$571$	−1.50607e12	+	8.69530e11i	−0.592902	+	0.342312i	−0.766244	−	0.642550i	$-0.777877\pi$
$571$	−1.50607e12	+	8.69530e11i	−0.592902	+	0.342312i	0.173342	+	0.984862i	$0.444543\pi$
$572$		0			0
$573$			3.78404e11i			0.146643i
$574$		0			0
$575$		−	5.04648e12i		−	1.92523i
$576$		0			0
$577$	1.47729e12	−	8.52914e11i	0.554849	−	0.320342i	−0.196227	−	0.980559i	$-0.562869\pi$
$577$	1.47729e12	−	8.52914e11i	0.554849	−	0.320342i	0.751075	+	0.660216i	$0.229535\pi$
$578$		0			0
$579$	−4.43903e11	+	7.68863e11i	−0.164148	+	0.284312i
$580$		0			0
$581$	5.95357e11	+	1.59871e12i	0.216763	+	0.582074i
$582$		0			0
$583$	−1.40296e11	−	8.09997e10i	−0.0502962	−	0.0290385i
$584$		0			0
$585$	−1.81584e12	−	3.14513e12i	−0.641028	−	1.11029i
$586$		0			0
$587$	4.02356e12			1.39875			0.699374	−	0.714756i	$-0.253462\pi$
$587$	4.02356e12			1.39875			0.699374	+	0.714756i	$0.253462\pi$
$588$		0			0
$589$	−4.46519e11			−0.152870
$590$		0			0
$591$	−7.38214e11	−	1.27862e12i	−0.248908	−	0.431121i
$592$		0			0
$593$	6.89387e11	+	3.98018e11i	0.228938	+	0.132177i	0.610082	−	0.792338i	$-0.291137\pi$
$593$	6.89387e11	+	3.98018e11i	0.228938	+	0.132177i	−0.381144	+	0.924516i	$0.624470\pi$
$594$		0			0
$595$	2.09677e12	+	5.63046e12i	0.685842	+	1.84169i
$596$		0			0
$597$	2.07704e10	−	3.59754e10i	0.00669208	−	0.0115910i
$598$		0			0
$599$	−2.11011e12	+	1.21827e12i	−0.669705	+	0.386654i	−0.795965	−	0.605343i	$-0.793036\pi$
$599$	−2.11011e12	+	1.21827e12i	−0.669705	+	0.386654i	0.126260	+	0.991997i	$0.459703\pi$
$600$		0			0
$601$		−	7.25690e11i		−	0.226890i	−0.993544	−	0.113445i	$-0.963811\pi$
$601$		−	7.25690e11i		−	0.226890i	0.993544	−	0.113445i	$-0.0361886\pi$
$602$		0			0
$603$		−	1.33833e12i		−	0.412225i
$604$		0			0
$605$	9.79151e12	−	5.65313e12i	2.97133	−	1.71550i
$606$		0			0
$607$	−7.63980e11	+	1.32325e12i	−0.228419	+	0.395634i	−0.957340	−	0.288964i	$-0.906689\pi$
$607$	−7.63980e11	+	1.32325e12i	−0.228419	+	0.395634i	0.728920	+	0.684598i	$0.240022\pi$
$608$		0			0
$609$	−7.75166e11	+	9.37855e11i	−0.228358	+	0.276285i
$610$		0			0
$611$	3.51477e12	+	2.02925e12i	1.02026	+	0.589048i
$612$		0			0
$613$	1.73311e12	+	3.00183e12i	0.495739	+	0.858645i	0.999988	−	0.00491362i	$-0.00156406\pi$
$613$	1.73311e12	+	3.00183e12i	0.495739	+	0.858645i	−0.504249	+	0.863558i	$0.668231\pi$
$614$		0			0
$615$	−1.01136e12			−0.285079
$616$		0			0
$617$	−3.35261e12			−0.931321			−0.465661	−	0.884963i	$-0.654183\pi$
$617$	−3.35261e12			−0.931321			−0.465661	+	0.884963i	$0.654183\pi$
$618$		0			0
$619$	3.28716e12	+	5.69353e12i	0.899939	+	1.55874i	0.827571	+	0.561361i	$0.189722\pi$
$619$	3.28716e12	+	5.69353e12i	0.899939	+	1.55874i	0.0723679	+	0.997378i	$0.476944\pi$
$620$		0			0
$621$	−2.17214e12	−	1.25409e12i	−0.586106	−	0.338388i
$622$		0			0
$623$	3.15134e11	−	1.86818e12i	0.0838106	−	0.496847i
$624$		0			0
$625$	−2.88581e12	+	4.99837e12i	−0.756497	+	1.31029i
$626$		0			0
$627$	4.35189e12	−	2.51257e12i	1.12454	−	0.649253i
$628$		0			0
$629$			1.18713e12i			0.302393i
$630$		0			0
$631$			2.96004e11i			0.0743303i	0.999309	+	0.0371652i	$0.0118328\pi$
$631$			2.96004e11i			0.0743303i	−0.999309	+	0.0371652i	$0.988167\pi$
$632$		0			0
$633$	1.17231e12	−	6.76835e11i	0.290219	−	0.167558i
$634$		0			0
$635$	6.20649e12	−	1.07500e13i	1.51483	−	2.62377i
$636$		0			0
$637$	2.36984e12	+	2.74061e12i	0.570285	+	0.659508i
$638$		0			0
$639$	−1.08959e12	−	6.29073e11i	−0.258528	−	0.149261i
$640$		0			0
$641$	−2.43613e12	−	4.21950e12i	−0.569954	−	0.987188i	−0.996570	−	0.0827557i	$-0.973628\pi$
$641$	−2.43613e12	−	4.21950e12i	−0.569954	−	0.987188i	0.426616	−	0.904433i	$-0.359705\pi$
$642$		0			0
$643$	2.28925e12			0.528133			0.264066	−	0.964505i	$-0.414936\pi$
$643$	2.28925e12			0.528133			0.264066	+	0.964505i	$0.414936\pi$
$644$		0			0
$645$	−2.73628e12			−0.622503
$646$		0			0
$647$	−3.35031e12	−	5.80291e12i	−0.751650	−	1.30190i	−0.947022	−	0.321167i	$-0.895925\pi$
$647$	−3.35031e12	−	5.80291e12i	−0.751650	−	1.30190i	0.195372	−	0.980729i	$-0.437409\pi$
$648$		0			0
$649$	−8.92949e12	−	5.15544e12i	−1.97572	−	1.14068i
$650$		0			0
$651$	1.57611e11	+	2.65865e10i	0.0343931	+	0.00580159i
$652$		0			0
$653$	−1.72814e12	+	2.99323e12i	−0.371938	+	0.644215i	−0.989864	−	0.142021i	$-0.954640\pi$
$653$	−1.72814e12	+	2.99323e12i	−0.371938	+	0.644215i	0.617926	+	0.786236i	$0.287973\pi$
$654$		0			0
$655$	2.79475e12	−	1.61355e12i	0.593276	−	0.342528i
$656$		0			0
$657$			2.53747e12i			0.531320i
$658$		0			0
$659$		−	5.65948e12i		−	1.16894i	−0.811415	−	0.584470i	$-0.801302\pi$
$659$		−	5.65948e12i		−	1.16894i	0.811415	−	0.584470i	$-0.198698\pi$
$660$		0			0
$661$	3.74441e11	−	2.16184e11i	0.0762916	−	0.0440470i	−0.461369	−	0.887208i	$-0.652642\pi$
$661$	3.74441e11	−	2.16184e11i	0.0762916	−	0.0440470i	0.537661	+	0.843161i	$0.319308\pi$
$662$		0			0
$663$	−9.97259e11	+	1.72730e12i	−0.200446	+	0.347182i
$664$		0			0
$665$	1.26049e13	+	1.04183e13i	2.49943	+	2.06585i
$666$		0			0
$667$	3.39587e12	+	1.96061e12i	0.664331	+	0.383552i
$668$		0			0
$669$	−8.74861e11	−	1.51530e12i	−0.168858	−	0.292470i
$670$		0			0
$671$	−2.21315e12			−0.421463
$672$		0			0
$673$	3.60551e12			0.677483			0.338742	−	0.940879i	$-0.389999\pi$
$673$	3.60551e12			0.677483			0.338742	+	0.940879i	$0.389999\pi$
$674$		0			0
$675$	4.43201e12	+	7.67647e12i	0.821739	+	1.42329i
$676$		0			0
$677$	−2.14791e11	−	1.24010e11i	−0.0392978	−	0.0226886i	0.480222	−	0.877147i	$-0.340556\pi$
$677$	−2.14791e11	−	1.24010e11i	−0.0392978	−	0.0226886i	−0.519520	+	0.854458i	$0.673889\pi$
$678$		0			0
$679$	9.85864e12	−	3.67133e12i	1.77993	−	0.662842i
$680$		0			0
$681$	1.69039e11	−	2.92784e11i	0.0301179	−	0.0521657i
$682$		0			0
$683$	4.72312e12	−	2.72690e12i	0.830493	−	0.479485i	−0.0235284	−	0.999723i	$-0.507490\pi$
$683$	4.72312e12	−	2.72690e12i	0.830493	−	0.479485i	0.854021	+	0.520238i	$0.174157\pi$
$684$		0			0
$685$			1.93700e12i			0.336141i
$686$		0			0
$687$		−	2.07233e11i		−	0.0354939i
$688$		0			0
$689$	1.51562e11	−	8.75042e10i	0.0256214	−	0.0147925i
$690$		0			0
$691$	−5.16449e12	+	8.94516e12i	−0.861740	+	1.49258i	0.00850730	+	0.999964i	$0.497292\pi$
$691$	−5.16449e12	+	8.94516e12i	−0.861740	+	1.49258i	−0.870248	+	0.492614i	$0.836041\pi$
$692$		0			0
$693$	8.05270e12	−	2.99881e12i	1.32630	−	0.493911i
$694$		0			0
$695$	−5.02985e12	−	2.90399e12i	−0.817756	−	0.472132i
$696$		0			0
$697$	−1.32666e12	−	2.29785e12i	−0.212918	−	0.368785i
$698$		0			0
$699$	−3.28304e12			−0.520150
$700$		0			0
$701$	−7.88262e12			−1.23293			−0.616467	−	0.787381i	$-0.711437\pi$
$701$	−7.88262e12			−1.23293			−0.616467	+	0.787381i	$0.711437\pi$
$702$		0			0
$703$	1.61557e12	+	2.79825e12i	0.249475	+	0.432103i
$704$		0			0
$705$	−5.67871e12	−	3.27860e12i	−0.865762	−	0.499848i
$706$		0			0
$707$	−8.39521e12	−	6.93890e12i	−1.26370	−	1.04449i
$708$		0			0
$709$	2.05623e12	−	3.56150e12i	0.305608	−	0.529328i	−0.671789	−	0.740743i	$-0.734474\pi$
$709$	2.05623e12	−	3.56150e12i	0.305608	−	0.529328i	0.977396	+	0.211415i	$0.0678070\pi$
$710$		0			0
$711$	−6.95370e12	+	4.01472e12i	−1.02048	+	0.589172i
$712$		0			0
$713$		−	5.15111e11i		−	0.0746445i
$714$		0			0
$715$			1.85447e13i			2.65365i
$716$		0			0
$717$	2.10105e12	−	1.21304e12i	0.296893	−	0.171411i
$718$		0			0
$719$	−1.88257e12	+	3.26071e12i	−0.262707	+	0.455022i	−0.966960	−	0.254927i	$-0.917949\pi$
$719$	−1.88257e12	+	3.26071e12i	−0.262707	+	0.455022i	0.704253	+	0.709949i	$0.251282\pi$
$720$		0			0
$721$	3.03253e12	+	5.11542e11i	0.417923	+	0.0704973i
$722$		0			0
$723$	−3.27087e12	−	1.88844e12i	−0.445186	−	0.257028i
$724$		0			0
$725$	−6.92889e12	−	1.20012e13i	−0.931414	−	1.61326i
$726$		0			0
$727$	−1.10216e13			−1.46332			−0.731661	−	0.681668i	$-0.761255\pi$
$727$	−1.10216e13			−1.46332			−0.731661	+	0.681668i	$0.761255\pi$
$728$		0			0
$729$	−8.08563e11			−0.106033
$730$		0			0
$731$	−3.58936e12	−	6.21695e12i	−0.464931	−	0.805284i
$732$		0			0
$733$	−5.56265e12	−	3.21160e12i	−0.711728	−	0.410916i	0.0999726	−	0.994990i	$-0.468124\pi$
$733$	−5.56265e12	−	3.21160e12i	−0.711728	−	0.410916i	−0.811700	+	0.584074i	$0.801458\pi$
$734$		0			0
$735$	−3.82889e12	−	4.42793e12i	−0.483927	−	0.559638i
$736$		0			0
$737$	−3.41700e12	+	5.91842e12i	−0.426620	+	0.738928i
$738$		0			0
$739$	−1.24518e13	+	7.18904e12i	−1.53579	+	0.886689i	−0.536711	+	0.843766i	$0.680334\pi$
$739$	−1.24518e13	+	7.18904e12i	−1.53579	+	0.886689i	−0.999078	+	0.0429225i	$0.986333\pi$
$740$		0			0
$741$			5.42867e12i			0.661472i
$742$		0			0
$743$		−	7.40249e12i		−	0.891103i	−0.895256	−	0.445552i	$-0.853008\pi$
$743$		−	7.40249e12i		−	0.891103i	0.895256	−	0.445552i	$-0.146992\pi$
$744$		0			0
$745$	1.35325e13	−	7.81297e12i	1.60944	−	0.929208i
$746$		0			0
$747$	−2.18547e12	+	3.78534e12i	−0.256804	+	0.444798i
$748$		0			0
$749$	−1.02949e12	+	6.10302e12i	−0.119523	+	0.708560i
$750$		0			0
$751$	−3.01374e12	−	1.73998e12i	−0.345721	−	0.199602i	0.317078	−	0.948400i	$-0.397298\pi$
$751$	−3.01374e12	−	1.73998e12i	−0.345721	−	0.199602i	−0.662799	+	0.748797i	$0.730632\pi$
$752$		0			0
$753$	−5.42186e11	−	9.39093e11i	−0.0614569	−	0.106446i
$754$		0			0
$755$	1.05780e13			1.18479
$756$		0			0
$757$	1.57468e13			1.74285			0.871427	−	0.490526i	$-0.163195\pi$
$757$	1.57468e13			1.74285			0.871427	+	0.490526i	$0.163195\pi$
$758$		0			0
$759$	2.89853e12	+	5.02041e12i	0.317023	+	0.549100i
$760$		0			0
$761$	−5.70866e12	−	3.29590e12i	−0.617026	−	0.356240i	0.158684	−	0.987329i	$-0.449275\pi$
$761$	−5.70866e12	−	3.29590e12i	−0.617026	−	0.356240i	−0.775710	+	0.631089i	$0.782608\pi$
$762$		0			0
$763$	−7.01490e12	+	8.48715e12i	−0.749309	+	0.906570i
$764$		0			0
$765$	−7.69693e12	+	1.33315e13i	−0.812534	+	1.40735i
$766$		0			0
$767$	9.64655e12	−	5.56944e12i	1.00645	−	0.581075i
$768$		0			0
$769$			8.20496e12i			0.846073i	0.906113	+	0.423036i	$0.139036\pi$
$769$			8.20496e12i			0.846073i	−0.906113	+	0.423036i	$0.860964\pi$
$770$		0			0
$771$			1.51842e12i			0.154756i
$772$		0			0
$773$	3.52090e11	−	2.03279e11i	0.0354687	−	0.0204779i	−0.482161	−	0.876083i	$-0.660148\pi$
$773$	3.52090e11	−	2.03279e11i	0.0354687	−	0.0204779i	0.517630	+	0.855605i	$0.326815\pi$
$774$		0			0
$775$	−9.10215e11	+	1.57654e12i	−0.0906330	+	0.156981i
$776$		0			0
$777$	−4.03644e11	−	1.08391e12i	−0.0397287	−	0.106684i
$778$		0			0
$779$	−6.25426e12	−	3.61090e12i	−0.608496	−	0.351315i
$780$		0			0
$781$	3.21228e12	+	5.56383e12i	0.308947	+	0.535112i
$782$		0			0
$783$	−6.88751e12			−0.654839
$784$		0			0
$785$	1.86936e12			0.175703
$786$		0			0
$787$	−3.98054e12	−	6.89450e12i	−0.369876	−	0.640644i	0.619670	−	0.784863i	$-0.287266\pi$
$787$	−3.98054e12	−	6.89450e12i	−0.369876	−	0.640644i	−0.989546	+	0.144219i	$0.953933\pi$
$788$		0			0
$789$	−4.83146e11	−	2.78945e11i	−0.0443846	−	0.0256255i
$790$		0			0
$791$	5.74050e12	+	1.54150e13i	0.521381	+	1.40007i
$792$		0			0
$793$	1.19544e12	−	2.07056e12i	0.107349	−	0.185934i
$794$		0			0
$795$	−2.44874e11	+	1.41378e11i	−0.0217415	+	0.0125525i
$796$		0			0
$797$		−	1.89899e13i		−	1.66709i	−0.552450	−	0.833546i	$-0.686307\pi$
$797$		−	1.89899e13i		−	1.66709i	0.552450	−	0.833546i	$-0.313693\pi$
$798$		0			0
$799$		−	1.72031e13i		−	1.49329i
$800$		0			0
$801$	4.20383e12	−	2.42708e12i	0.360827	−	0.208323i
$802$		0			0
$803$	6.47862e12	−	1.12213e13i	0.549873	−	0.952408i
$804$		0			0
$805$	−1.20187e13	+	1.45411e13i	−1.00873	+	1.22044i
$806$		0			0
$807$	4.74505e12	+	2.73956e12i	0.393831	+	0.227379i
$808$		0			0
$809$	9.20201e12	+	1.59383e13i	0.755291	+	1.30820i	0.945229	+	0.326407i	$0.105838\pi$
$809$	9.20201e12	+	1.59383e13i	0.755291	+	1.30820i	−0.189938	+	0.981796i	$0.560829\pi$
$810$		0			0
$811$	−6.96575e12			−0.565424			−0.282712	−	0.959205i	$-0.591234\pi$
$811$	−6.96575e12			−0.565424			−0.282712	+	0.959205i	$0.591234\pi$
$812$		0			0
$813$	−4.47406e12			−0.359166
$814$		0			0
$815$	−9.97304e12	−	1.72738e13i	−0.791805	−	1.37145i
$816$		0			0
$817$	−1.69212e13	−	9.76949e12i	−1.32872	−	0.767136i
$818$		0			0
$819$	−1.54409e12	+	9.15367e12i	−0.119921	+	0.710915i
$820$		0			0
$821$	−8.32207e12	+	1.44143e13i	−0.639274	+	1.10726i	0.346318	+	0.938117i	$0.387432\pi$
$821$	−8.32207e12	+	1.44143e13i	−0.639274	+	1.10726i	−0.985592	+	0.169138i	$0.945902\pi$
$822$		0			0
$823$	−1.29668e13	+	7.48636e12i	−0.985218	+	0.568816i	−0.903841	−	0.427868i	$-0.859265\pi$
$823$	−1.29668e13	+	7.48636e12i	−0.985218	+	0.568816i	−0.0813765	+	0.996683i	$0.525932\pi$
$824$		0			0
$825$		−	2.04872e13i		−	1.53971i
$826$		0			0
$827$			2.84386e12i			0.211414i	0.994397	+	0.105707i	$0.0337105\pi$
$827$			2.84386e12i			0.211414i	−0.994397	+	0.105707i	$0.966290\pi$
$828$		0			0
$829$	6.59291e12	−	3.80642e12i	0.484821	−	0.279912i	−0.237602	−	0.971363i	$-0.576361\pi$
$829$	6.59291e12	−	3.80642e12i	0.484821	−	0.279912i	0.722424	+	0.691451i	$0.243028\pi$
$830$		0			0
$831$	−2.11663e11	+	3.66611e11i	−0.0153971	+	0.0266686i
$832$		0			0
$833$	5.03785e12	−	1.45078e13i	0.362529	−	1.04400i
$834$		0			0
$835$	2.16079e13	+	1.24753e13i	1.53824	+	0.888103i
$836$		0			0
$837$	4.52390e11	+	7.83562e11i	0.0318602	+	0.0551834i
$838$		0			0
$839$	8.65465e11			0.0603004			0.0301502	−	0.999545i	$-0.490401\pi$
$839$	8.65465e11			0.0603004			0.0301502	+	0.999545i	$0.490401\pi$
$840$		0			0
$841$	−3.73939e12			−0.257762
$842$		0			0
$843$	2.78589e12	+	4.82530e12i	0.189994	+	0.329079i
$844$		0			0
$845$	5.47361e12	+	3.16019e12i	0.369333	+	0.213235i
$846$		0			0
$847$	−2.84975e13	−	4.80709e12i	−1.90253	−	0.320928i
$848$		0			0
$849$	−1.66888e12	+	2.89059e12i	−0.110241	+	0.190942i
$850$		0			0
$851$	−3.22810e12	+	1.86374e12i	−0.210991	+	0.121816i
$852$		0			0
$853$		−	1.59456e13i		−	1.03127i	−0.856810	−	0.515633i	$-0.827557\pi$
$853$		−	1.59456e13i		−	1.03127i	0.856810	−	0.515633i	$-0.172443\pi$
$854$		0			0
$855$			4.18989e13i			2.68136i
$856$		0			0
$857$	−1.67500e13	+	9.67061e12i	−1.06072	+	0.612407i	−0.925631	−	0.378427i	$-0.876465\pi$
$857$	−1.67500e13	+	9.67061e12i	−1.06072	+	0.612407i	−0.135088	+	0.990834i	$0.543132\pi$
$858$		0			0
$859$	5.69926e11	−	9.87141e11i	0.0357149	−	0.0618600i	−0.847615	−	0.530611i	$-0.821963\pi$
$859$	5.69926e11	−	9.87141e11i	0.0357149	−	0.0618600i	0.883330	+	0.468751i	$0.155296\pi$
$860$		0			0
$861$	1.99261e12	+	1.64695e12i	0.123568	+	0.102133i
$862$		0			0
$863$	−2.10087e13	−	1.21294e13i	−1.28929	−	0.744371i	−0.310761	−	0.950488i	$-0.600584\pi$
$863$	−2.10087e13	−	1.21294e13i	−1.28929	−	0.744371i	−0.978527	+	0.206117i	$0.933917\pi$
$864$		0			0
$865$	1.59802e13	+	2.76785e13i	0.970530	+	1.68101i
$866$		0			0
$867$	1.53226e12			0.0920971
$868$		0			0
$869$	4.10013e13			2.43898
$870$		0			0
$871$	−3.69140e12	−	6.39369e12i	−0.217325	−	0.376417i
$872$		0			0
$873$	2.33427e13	+	1.34769e13i	1.36015	+	0.785285i
$874$		0			0
$875$	3.35832e13	−	1.25063e13i	1.93681	−	0.721262i
$876$		0			0
$877$	−1.59793e12	+	2.76769e12i	−0.0912134	+	0.157986i	−0.908022	−	0.418923i	$-0.862408\pi$
$877$	−1.59793e12	+	2.76769e12i	−0.0912134	+	0.157986i	0.816809	+	0.576909i	$0.195741\pi$
$878$		0			0
$879$	2.28401e12	−	1.31867e12i	0.129047	−	0.0745052i
$880$		0			0
$881$		−	9.42039e12i		−	0.526838i	−0.964681	−	0.263419i	$-0.915150\pi$
$881$		−	9.42039e12i		−	0.526838i	0.964681	−	0.263419i	$-0.0848502\pi$
$882$		0			0
$883$		−	1.06804e13i		−	0.591239i	−0.955306	−	0.295619i	$-0.904474\pi$
$883$		−	1.06804e13i		−	0.591239i	0.955306	−	0.295619i	$-0.0955260\pi$
$884$		0			0
$885$	−1.55857e13	+	8.99838e12i	−0.854044	+	0.493082i
$886$		0			0
$887$	−3.80802e12	+	6.59568e12i	−0.206558	+	0.357770i	−0.950628	−	0.310332i	$-0.899560\pi$
$887$	−3.80802e12	+	6.59568e12i	−0.206558	+	0.357770i	0.744070	+	0.668102i	$0.232893\pi$
$888$		0			0
$889$	−2.97341e13	+	1.10729e13i	−1.59660	+	0.594571i
$890$		0			0
$891$	1.42399e13	+	8.22139e12i	0.756930	+	0.437014i
$892$		0			0
$893$	−2.34116e13	−	4.05501e13i	−1.23197	−	2.13383i
$894$		0			0
$895$	5.68381e13			2.96098
$896$		0			0
$897$	−6.26259e12			−0.322989
$898$		0			0
$899$	−7.07254e11	−	1.22500e12i	−0.0361124	−	0.0625486i
$900$		0			0
$901$	−6.42435e11	−	3.70910e11i	−0.0324764	−	0.0187502i
$902$		0			0
$903$	5.39110e12	+	4.45592e12i	0.269825	+	0.223019i
$904$		0			0
$905$	−1.03859e13	+	1.79890e13i	−0.514668	+	0.891431i
$906$		0			0
$907$	2.36956e13	−	1.36806e13i	1.16261	−	0.671234i	0.210683	−	0.977555i	$-0.432431\pi$
$907$	2.36956e13	−	1.36806e13i	1.16261	−	0.671234i	0.951928	+	0.306321i	$0.0990980\pi$
$908$		0			0
$909$		−	2.79059e13i		−	1.35569i
$910$		0			0
$911$		−	2.65027e13i		−	1.27484i	−0.770515	−	0.637422i	$-0.780001\pi$
$911$		−	2.65027e13i		−	1.27484i	0.770515	−	0.637422i	$-0.219999\pi$
$912$		0			0
$913$	1.93294e13	−	1.11598e13i	0.920659	−	0.531543i
$914$		0			0
$915$	−1.93143e12	+	3.34534e12i	−0.0910930	+	0.157778i
$916$		0			0
$917$	−8.13390e12	−	1.37207e12i	−0.379872	−	0.0640786i
$918$		0			0
$919$	3.27652e12	+	1.89170e12i	0.151528	+	0.0874848i	0.573847	−	0.818963i	$-0.305450\pi$
$919$	3.27652e12	+	1.89170e12i	0.151528	+	0.0874848i	−0.422319	+	0.906447i	$0.638784\pi$
$920$		0			0
$921$	−7.71235e12	−	1.33582e13i	−0.353198	−	0.611757i
$922$		0			0
$923$	−6.94047e12			−0.314761
$924$		0			0
$925$	1.31731e13			0.591632
$926$		0			0
$927$	3.93977e12	+	6.82388e12i	0.175231	+	0.303510i
$928$		0			0
$929$	1.44452e13	+	8.33993e12i	0.636286	+	0.367360i	0.783182	−	0.621792i	$-0.213595\pi$
$929$	1.44452e13	+	8.33993e12i	0.636286	+	0.367360i	−0.146897	+	0.989152i	$0.546928\pi$
$930$		0			0
$931$	−7.86871e12	−	4.10530e13i	−0.343266	−	1.79090i
$932$		0			0
$933$	−6.24307e12	+	1.08133e13i	−0.269731	+	0.467188i
$934$		0			0
$935$	6.80755e13	−	3.93034e13i	2.91299	−	1.68181i
$936$		0			0
$937$			7.13660e12i			0.302456i	0.988499	+	0.151228i	$0.0483228\pi$
$937$			7.13660e12i			0.302456i	−0.988499	+	0.151228i	$0.951677\pi$
$938$		0			0
$939$		−	4.84679e12i		−	0.203451i
$940$		0			0
$941$	−9.29370e12	+	5.36572e12i	−0.386398	+	0.223087i	−0.680598	−	0.732657i	$-0.738280\pi$
$941$	−9.29370e12	+	5.36572e12i	−0.386398	+	0.223087i	0.294200	+	0.955744i	$0.404947\pi$
$942$		0			0
$943$	4.16559e12	−	7.21501e12i	0.171543	−	0.297122i
$944$		0			0
$945$	5.51171e12	−	3.26746e13i	0.224824	−	1.33281i
$946$		0			0
$947$	1.13857e13	+	6.57351e12i	0.460027	+	0.265597i	0.712055	−	0.702123i	$-0.247764\pi$
$947$	1.13857e13	+	6.57351e12i	0.460027	+	0.265597i	−0.252029	+	0.967720i	$0.581098\pi$
$948$		0			0
$949$	6.99887e12	+	1.21224e13i	0.280111	+	0.485166i
$950$		0			0
$951$	−1.37372e13			−0.544611
$952$		0			0
$953$	1.09968e13			0.431866			0.215933	−	0.976408i	$-0.430721\pi$
$953$	1.09968e13			0.431866			0.215933	+	0.976408i	$0.430721\pi$
$954$		0			0
$955$	8.05552e12	+	1.39526e13i	0.313385	+	0.542799i
$956$		0			0
$957$	1.37862e13	+	7.95945e12i	0.531300	+	0.306746i
$958$		0			0
$959$	3.15433e12	−	3.81634e12i	0.120427	−	0.145701i
$960$		0			0
$961$	1.31269e13	−	2.27365e13i	0.496486	−	0.859939i
$962$		0			0
$963$	−1.37332e13	+	7.92886e12i	−0.514580	+	0.297093i
$964$		0			0
$965$			3.77995e13i			1.40318i
$966$		0			0
$967$			2.61514e13i			0.961779i	0.876781	+	0.480890i	$0.159686\pi$
$967$			2.61514e13i			0.961779i	−0.876781	+	0.480890i	$0.840314\pi$
$968$		0			0
$969$	1.99280e13	−	1.15054e13i	0.726116	−	0.419224i
$970$		0			0
$971$	7.99792e12	−	1.38528e13i	0.288729	−	0.500093i	−0.684778	−	0.728752i	$-0.740101\pi$
$971$	7.99792e12	−	1.38528e13i	0.288729	−	0.500093i	0.973507	+	0.228659i	$0.0734340\pi$
$972$		0			0
$973$	5.18096e12	+	1.39124e13i	0.185312	+	0.497618i
$974$		0			0
$975$	1.91672e13	+	1.10662e13i	0.679262	+	0.392172i
$976$		0			0
$977$	7.74111e12	+	1.34080e13i	0.271818	+	0.470802i	0.969327	−	0.245773i	$-0.0790419\pi$
$977$	7.74111e12	+	1.34080e13i	0.271818	+	0.470802i	−0.697510	+	0.716575i	$0.745709\pi$
$978$		0			0
$979$	−2.47872e13			−0.862392
$980$		0			0
$981$	−2.82116e13			−0.972560
$982$		0			0
$983$	1.47957e13	+	2.56270e13i	0.505413	+	0.875400i	0.999980	+	0.00626135i	$0.00199306\pi$
$983$	1.47957e13	+	2.56270e13i	0.505413	+	0.875400i	−0.494568	+	0.869139i	$0.664674\pi$
$984$		0			0
$985$	−5.44391e13	−	3.14304e13i	−1.84267	−	1.06387i
$986$		0			0
$987$	5.84931e12	+	1.57072e13i	0.196190	+	0.526830i
$988$		0			0
$989$	1.12702e13	−	1.95206e13i	0.374584	−	0.648798i
$990$		0			0
$991$	−1.08807e13	+	6.28199e12i	−0.358366	+	0.206903i	−0.668364	−	0.743835i	$-0.733005\pi$
$991$	−1.08807e13	+	6.28199e12i	−0.358366	+	0.206903i	0.309998	+	0.950737i	$0.399672\pi$
$992$		0			0
$993$			1.26311e13i			0.412257i
$994$		0			0
$995$		−	1.76865e12i		−	0.0572057i
$996$		0			0
$997$	−3.97938e13	+	2.29750e13i	−1.27552	+	0.736422i	−0.976021	−	0.217674i	$-0.930153\pi$
$997$	−3.97938e13	+	2.29750e13i	−1.27552	+	0.736422i	−0.299499	+	0.954097i	$0.596820\pi$
$998$		0			0
$999$	3.27362e12	−	5.67007e12i	0.103988	−	0.180113i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.b.47.7	yes	24
4.3	odd	2	inner	112.10.p.b.47.6	yes	24
7.3	odd	6	inner	112.10.p.b.31.6	✓	24
28.3	even	6	inner	112.10.p.b.31.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.b.31.6	✓	24	7.3	odd	6	inner
112.10.p.b.31.7	yes	24	28.3	even	6	inner
112.10.p.b.47.6	yes	24	4.3	odd	2	inner
112.10.p.b.47.7	yes	24	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(29.1853 + 50.5504i) q^{3} +(2152.25 + 1242.60i) q^{5} +(-2216.90 - 5953.06i) q^{7} +(8137.94 - 14095.3i) q^{9} +(-71975.9 + 41555.3i) q^{11} -89784.7i q^{13} +145063. i q^{15} +(-329589. + 190288. i) q^{17} +(-517925. + 897073. i) q^{19} +(236229. - 285807. i) q^{21} +(-1.03488e6 - 597486. i) q^{23} +(2.11155e6 + 3.65731e6i) q^{25} +2.09894e6 q^{27} -3.28143e6 q^{29} +(215533. + 373313. i) q^{31} +(-4.20128e6 - 2.42561e6i) q^{33} +(2.62595e6 - 1.55672e7i) q^{35} +(1.55965e6 - 2.70140e6i) q^{37} +(4.53865e6 - 2.62039e6i) q^{39} +6.97186e6i q^{41} +1.88627e7i q^{43} +(3.50297e7 - 2.02244e7i) q^{45} +(-2.26013e7 + 3.91466e7i) q^{47} +(-3.05243e7 + 2.63947e7i) q^{49} +(-1.92383e7 - 1.11072e7i) q^{51} +(974600. + 1.68806e6i) q^{53} -2.06547e8 q^{55} -6.04632e7 q^{57} +(6.20310e7 + 1.07441e8i) q^{59} +(2.30614e7 + 1.33145e7i) q^{61} +(-1.01951e8 - 1.71977e7i) q^{63} +(1.11566e8 - 1.93239e8i) q^{65} +(7.12113e7 - 4.11139e7i) q^{67} -6.97512e7i q^{69} -7.73013e7i q^{71} +(-1.35016e8 + 7.79517e7i) q^{73} +(-1.23252e8 + 2.13479e8i) q^{75} +(4.06945e8 + 3.36353e8i) q^{77} +(-4.27239e8 - 2.46667e8i) q^{79} +(-9.89209e7 - 1.71336e8i) q^{81} -2.68553e8 q^{83} -9.45809e8 q^{85} +(-9.57693e7 - 1.65877e8i) q^{87} +(2.58286e8 + 1.49121e8i) q^{89} +(-5.34494e8 + 1.99044e8i) q^{91} +(-1.25808e7 + 2.17905e7i) q^{93} +(-2.22941e9 + 1.28715e9i) q^{95} +1.65606e9i q^{97} +1.35270e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57}+ \cdots - 1432348316 q^{93}+O(q^{100}) \) 24 * q + 1704 * q^5 - 80428 * q^9 - 1578672 * q^17 + 1219540 * q^21 + 8001240 * q^25 - 6709416 * q^29 - 29129772 * q^33 - 11130084 * q^37 + 57023292 * q^45 - 12671904 * q^49 - 93652164 * q^53 + 742621544 * q^57 - 593611308 * q^61 + 160281180 * q^65 - 1676922516 * q^73 + 1645751688 * q^77 - 528698056 * q^81 + 1370082456 * q^85 - 941603484 * q^89 - 1432348316 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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