LMFDB - Embedded newform 112.10.p.a.47.3

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,-162] 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			47.3
Character	$\chi$	$=$	112.47
Dual form			112.10.p.a.31.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-80.9515 - 140.212i) q^{3} +(-916.446 - 529.111i) q^{5} +(4644.00 + 4334.38i) q^{7} +(-3264.80 + 5654.80i) q^{9} +(6006.36 - 3467.77i) q^{11} -93825.2i q^{13} +171329. i q^{15} +(-119897. + 69222.8i) q^{17} +(415317. - 719350. i) q^{19} +(231794. - 1.00202e6i) q^{21} +(-2.26305e6 - 1.30657e6i) q^{23} +(-416647. - 721653. i) q^{25} -2.12958e6 q^{27} +6.68191e6 q^{29} +(-926240. - 1.60429e6i) q^{31} +(-972447. - 561443. i) q^{33} +(-1.96261e6 - 6.42942e6i) q^{35} +(-8.30683e6 + 1.43878e7i) q^{37} +(-1.31554e7 + 7.59529e6i) q^{39} -9.78601e6i q^{41} +1.04358e7i q^{43} +(5.98403e6 - 3.45488e6i) q^{45} +(1.27125e7 - 2.20187e7i) q^{47} +(2.77988e6 + 4.02577e7i) q^{49} +(1.94117e7 + 1.12074e7i) q^{51} +(8.09970e6 + 1.40291e7i) q^{53} -7.33934e6 q^{55} -1.34482e8 q^{57} +(2.26340e7 + 3.92032e7i) q^{59} +(8.25119e7 + 4.76383e7i) q^{61} +(-3.96718e7 + 1.21100e7i) q^{63} +(-4.96439e7 + 8.59857e7i) q^{65} +(-1.29397e7 + 7.47075e6i) q^{67} +4.23076e8i q^{69} +9.53158e7i q^{71} +(-1.78310e8 + 1.02947e8i) q^{73} +(-6.74564e7 + 1.16838e8i) q^{75} +(4.29242e7 + 9.92951e6i) q^{77} +(-4.33439e8 - 2.50246e8i) q^{79} +(2.36653e8 + 4.09896e8i) q^{81} -5.62286e8 q^{83} +1.46506e8 q^{85} +(-5.40911e8 - 9.36886e8i) q^{87} +(-6.08933e7 - 3.51568e7i) q^{89} +(4.06674e8 - 4.35724e8i) q^{91} +(-1.49961e8 + 2.59740e8i) q^{93} +(-7.61231e8 + 4.39497e8i) q^{95} +4.59671e8i q^{97} +4.52863e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) $ 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

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
$3$	−80.9515	−	140.212i	−0.577005	−	0.999402i	−0.995821	−	0.0913304i	$-0.970888\pi$
$3$	−80.9515	−	140.212i	−0.577005	−	0.999402i	0.418816	−	0.908071i	$-0.362445\pi$
$4$		0			0
$5$	−916.446	−	529.111i	−0.655756	−	0.378601i	0.134902	−	0.990859i	$-0.456928\pi$
$5$	−916.446	−	529.111i	−0.655756	−	0.378601i	−0.790658	+	0.612258i	$0.790261\pi$
$6$		0			0
$7$	4644.00	+	4334.38i	0.731057	+	0.682317i
$7$	4644.00	+	4334.38i	0.731057	+	0.682317i
$8$		0			0
$9$	−3264.80	+	5654.80i	−0.165869	+	0.287294i
$10$		0			0
$11$	6006.36	−	3467.77i	0.123693	−	0.0714140i	−0.436877	−	0.899521i	$-0.643916\pi$
$11$	6006.36	−	3467.77i	0.123693	−	0.0714140i	0.560570	+	0.828107i	$0.310582\pi$
$12$		0			0
$13$		−	93825.2i		−	0.911117i	−0.890206	−	0.455559i	$-0.849440\pi$
$13$		−	93825.2i		−	0.911117i	0.890206	−	0.455559i	$-0.150560\pi$
$14$		0			0
$15$			171329.i			0.873818i
$16$		0			0
$17$	−119897.	+	69222.8i	−0.348168	+	0.201015i	−0.663878	−	0.747841i	$-0.731091\pi$
$17$	−119897.	+	69222.8i	−0.348168	+	0.201015i	0.315710	+	0.948856i	$0.397757\pi$
$18$		0			0
$19$	415317.	−	719350.i	0.731119	−	1.26634i	−0.225286	−	0.974293i	$-0.572332\pi$
$19$	415317.	−	719350.i	0.731119	−	1.26634i	0.956405	−	0.292043i	$-0.0943350\pi$
$20$		0			0
$21$	231794.	−	1.00202e6i	0.260085	−	1.12432i
$22$		0			0
$23$	−2.26305e6	−	1.30657e6i	−1.68624	−	0.973550i	−0.957356	−	0.288911i	$-0.906707\pi$
$23$	−2.26305e6	−	1.30657e6i	−1.68624	−	0.973550i	−0.728882	−	0.684639i	$-0.759960\pi$
$24$		0			0
$25$	−416647.	−	721653.i	−0.213323	−	0.369486i
$26$		0			0
$27$	−2.12958e6			−0.771181
$28$		0			0
$29$	6.68191e6			1.75432			0.877162	−	0.480194i	$-0.159434\pi$
$29$	6.68191e6			1.75432			0.877162	+	0.480194i	$0.159434\pi$
$30$		0			0
$31$	−926240.	−	1.60429e6i	−0.180134	−	0.312001i	0.761792	−	0.647822i	$-0.224320\pi$
$31$	−926240.	−	1.60429e6i	−0.180134	−	0.312001i	−0.941926	+	0.335820i	$0.890986\pi$
$32$		0			0
$33$	−972447.	−	561443.i	−0.142743	−	0.0824124i
$34$		0			0
$35$	−1.96261e6	−	6.42942e6i	−0.221069	−	0.724212i
$36$		0			0
$37$	−8.30683e6	+	1.43878e7i	−0.728664	+	1.26208i	0.228784	+	0.973477i	$0.426525\pi$
$37$	−8.30683e6	+	1.43878e7i	−0.728664	+	1.26208i	−0.957448	+	0.288606i	$0.906808\pi$
$38$		0			0
$39$	−1.31554e7	+	7.59529e6i	−0.910572	+	0.525719i
$40$		0			0
$41$		−	9.78601e6i		−	0.540852i	−0.962741	−	0.270426i	$-0.912835\pi$
$41$		−	9.78601e6i		−	0.540852i	0.962741	−	0.270426i	$-0.0871645\pi$
$42$		0			0
$43$			1.04358e7i			0.465499i	0.972537	+	0.232750i	$0.0747723\pi$
$43$			1.04358e7i			0.465499i	−0.972537	+	0.232750i	$0.925228\pi$
$44$		0			0
$45$	5.98403e6	−	3.45488e6i	0.217539	−	0.125596i
$46$		0			0
$47$	1.27125e7	−	2.20187e7i	0.380006	−	0.658190i	−0.611057	−	0.791587i	$-0.709255\pi$
$47$	1.27125e7	−	2.20187e7i	0.380006	−	0.658190i	0.991063	+	0.133397i	$0.0425886\pi$
$48$		0			0
$49$	2.77988e6	+	4.02577e7i	0.0688880	+	0.997624i
$50$		0			0
$51$	1.94117e7	+	1.12074e7i	0.401790	+	0.231973i
$52$		0			0
$53$	8.09970e6	+	1.40291e7i	0.141003	+	0.244224i	0.927874	−	0.372893i	$-0.121634\pi$
$53$	8.09970e6	+	1.40291e7i	0.141003	+	0.244224i	−0.786872	+	0.617117i	$0.788301\pi$
$54$		0			0
$55$	−7.33934e6			−0.108150
$56$		0			0
$57$	−1.34482e8			−1.68744
$58$		0			0
$59$	2.26340e7	+	3.92032e7i	0.243179	+	0.421199i	0.961618	−	0.274391i	$-0.0884763\pi$
$59$	2.26340e7	+	3.92032e7i	0.243179	+	0.421199i	−0.718439	+	0.695590i	$0.755143\pi$
$60$		0			0
$61$	8.25119e7	+	4.76383e7i	0.763013	+	0.440526i	0.830377	−	0.557203i	$-0.188125\pi$
$61$	8.25119e7	+	4.76383e7i	0.763013	+	0.440526i	−0.0673632	+	0.997729i	$0.521459\pi$
$62$		0			0
$63$	−3.96718e7	+	1.21100e7i	−0.317285	+	0.0968527i
$64$		0			0
$65$	−4.96439e7	+	8.59857e7i	−0.344950	+	0.597470i
$66$		0			0
$67$	−1.29397e7	+	7.47075e6i	−0.0784492	+	0.0452927i	−0.538711	−	0.842490i	$-0.681089\pi$
$67$	−1.29397e7	+	7.47075e6i	−0.0784492	+	0.0452927i	0.460262	+	0.887783i	$0.347755\pi$
$68$		0			0
$69$			4.23076e8i			2.24697i
$70$		0			0
$71$			9.53158e7i			0.445146i	0.974916	+	0.222573i	$0.0714456\pi$
$71$			9.53158e7i			0.445146i	−0.974916	+	0.222573i	$0.928554\pi$
$72$		0			0
$73$	−1.78310e8	+	1.02947e8i	−0.734890	+	0.424289i	−0.820208	−	0.572065i	$-0.806143\pi$
$73$	−1.78310e8	+	1.02947e8i	−0.734890	+	0.424289i	0.0853185	+	0.996354i	$0.472809\pi$
$74$		0			0
$75$	−6.74564e7	+	1.16838e8i	−0.246177	+	0.426391i
$76$		0			0
$77$	4.29242e7	+	9.92951e6i	0.139153	+	0.0321899i
$78$		0			0
$79$	−4.33439e8	−	2.50246e8i	−1.25201	−	0.722846i	−0.280499	−	0.959854i	$-0.590500\pi$
$79$	−4.33439e8	−	2.50246e8i	−1.25201	−	0.722846i	−0.971508	+	0.237008i	$0.923833\pi$
$80$		0			0
$81$	2.36653e8	+	4.09896e8i	0.610844	+	1.05801i
$82$		0			0
$83$	−5.62286e8			−1.30049			−0.650244	−	0.759726i	$-0.725333\pi$
$83$	−5.62286e8			−1.30049			−0.650244	+	0.759726i	$0.725333\pi$
$84$		0			0
$85$	1.46506e8			0.304418
$86$		0			0
$87$	−5.40911e8	−	9.36886e8i	−1.01225	−	1.75327i
$88$		0			0
$89$	−6.08933e7	−	3.51568e7i	−0.102876	−	0.0593956i	0.447679	−	0.894194i	$-0.352251\pi$
$89$	−6.08933e7	−	3.51568e7i	−0.102876	−	0.0593956i	−0.550555	+	0.834799i	$0.685584\pi$
$90$		0			0
$91$	4.06674e8	−	4.35724e8i	0.621670	−	0.666078i
$92$		0			0
$93$	−1.49961e8	+	2.59740e8i	−0.207876	+	0.360052i
$94$		0			0
$95$	−7.61231e8	+	4.39497e8i	−0.958871	+	0.553604i
$96$		0			0
$97$			4.59671e8i			0.527198i	0.964632	+	0.263599i	$0.0849096\pi$
$97$			4.59671e8i			0.527198i	−0.964632	+	0.263599i	$0.915090\pi$
$98$		0			0
$99$			4.52863e7i			0.0473815i
$100$		0			0
$101$	9.67458e6	−	5.58562e6i	0.00925094	−	0.00534103i	−0.495367	−	0.868684i	$-0.664967\pi$
$101$	9.67458e6	−	5.58562e6i	0.00925094	−	0.00534103i	0.504618	+	0.863343i	$0.331633\pi$
$102$		0			0
$103$	−3.63083e8	+	6.28877e8i	−0.317861	+	0.550552i	−0.980042	−	0.198792i	$-0.936298\pi$
$103$	−3.63083e8	+	6.28877e8i	−0.317861	+	0.550552i	0.662180	+	0.749345i	$0.269631\pi$
$104$		0			0
$105$	−7.42606e8	+	7.95653e8i	−0.596220	+	0.638810i
$106$		0			0
$107$	−1.86994e9	−	1.07961e9i	−1.37912	−	0.796233i	−0.387063	−	0.922053i	$-0.626510\pi$
$107$	−1.86994e9	−	1.07961e9i	−1.37912	−	0.796233i	−0.992053	+	0.125821i	$0.959844\pi$
$108$		0			0
$109$	8.58250e8	+	1.48653e9i	0.582364	+	1.00868i	0.995198	+	0.0978777i	$0.0312054\pi$
$109$	8.58250e8	+	1.48653e9i	0.582364	+	1.00868i	−0.412835	+	0.910806i	$0.635461\pi$
$110$		0			0
$111$	2.68980e9			1.68177
$112$		0			0
$113$	−2.22713e9			−1.28497			−0.642485	−	0.766298i	$-0.722097\pi$
$113$	−2.22713e9			−1.28497			−0.642485	+	0.766298i	$0.722097\pi$
$114$		0			0
$115$	1.38264e9	+	2.39481e9i	0.737173	+	1.27682i
$116$		0			0
$117$	5.30562e8	+	3.06320e8i	0.261758	+	0.151126i
$118$		0			0
$119$	−8.56841e8	−	1.98210e8i	−0.391687	−	0.0906077i
$120$		0			0
$121$	−1.15492e9	+	2.00039e9i	−0.489800	+	0.848359i
$122$		0			0
$123$	−1.37212e9	+	7.92192e8i	−0.540528	+	0.312074i
$124$		0			0
$125$			2.94865e9i			1.08026i
$126$		0			0
$127$			8.66110e8i			0.295431i	0.989030	+	0.147716i	$0.0471920\pi$
$127$			8.66110e8i			0.295431i	−0.989030	+	0.147716i	$0.952808\pi$
$128$		0			0
$129$	1.46323e9	−	8.44797e8i	0.465221	−	0.268595i
$130$		0			0
$131$	9.11507e8	−	1.57878e9i	0.270420	−	0.468382i	−0.698549	−	0.715562i	$-0.746171\pi$
$131$	9.11507e8	−	1.57878e9i	0.270420	−	0.468382i	0.968969	+	0.247180i	$0.0795040\pi$
$132$		0			0
$133$	5.04667e9	−	1.54052e9i	1.39853	−	0.426908i
$134$		0			0
$135$	1.95164e9	+	1.12678e9i	0.505706	+	0.291970i
$136$		0			0
$137$	−7.49973e8	−	1.29899e9i	−0.181887	−	0.315038i	0.760636	−	0.649179i	$-0.224887\pi$
$137$	−7.49973e8	−	1.29899e9i	−0.181887	−	0.315038i	−0.942523	+	0.334140i	$0.891554\pi$
$138$		0			0
$139$	−1.12735e9			−0.256149			−0.128075	−	0.991765i	$-0.540880\pi$
$139$	−1.12735e9			−0.256149			−0.128075	+	0.991765i	$0.540880\pi$
$140$		0			0
$141$	−4.11638e9			−0.877061
$142$		0			0
$143$	−3.25364e8	−	5.63547e8i	−0.0650665	−	0.112699i
$144$		0			0
$145$	−6.12362e9	−	3.53547e9i	−1.15041	−	0.664188i
$146$		0			0
$147$	5.41959e9	−	3.64870e9i	0.957279	−	0.644481i
$148$		0			0
$149$	2.72101e9	−	4.71293e9i	0.452264	−	0.783344i	−0.546263	−	0.837614i	$-0.683950\pi$
$149$	2.72101e9	−	4.71293e9i	0.452264	−	0.783344i	0.998526	+	0.0542703i	$0.0172833\pi$
$150$		0			0
$151$	−1.51712e9	+	8.75909e8i	−0.237478	+	0.137108i	−0.614017	−	0.789293i	$-0.710447\pi$
$151$	−1.51712e9	+	8.75909e8i	−0.237478	+	0.137108i	0.376539	+	0.926401i	$0.377114\pi$
$152$		0			0
$153$		−	9.03994e8i		−	0.133369i
$154$		0			0
$155$			1.96033e9i			0.272795i
$156$		0			0
$157$	1.19834e10	−	6.91859e9i	1.57409	−	0.908802i	0.578432	−	0.815731i	$-0.303665\pi$
$157$	1.19834e10	−	6.91859e9i	1.57409	−	0.908802i	0.995659	−	0.0930715i	$-0.0296685\pi$
$158$		0			0
$159$	1.31137e9	−	2.27135e9i	0.162718	−	0.281837i
$160$		0			0
$161$	−4.84642e9	−	1.58766e10i	−0.568466	−	1.86227i
$162$		0			0
$163$	−1.47245e9	−	8.50120e8i	−0.163379	−	0.0943270i	0.416081	−	0.909328i	$-0.363403\pi$
$163$	−1.47245e9	−	8.50120e8i	−0.163379	−	0.0943270i	−0.579460	+	0.815001i	$0.696737\pi$
$164$		0			0
$165$	5.94131e8	+	1.02906e9i	0.0624028	+	0.108085i
$166$		0			0
$167$	6.16125e9			0.612978			0.306489	−	0.951874i	$-0.400846\pi$
$167$	6.16125e9			0.612978			0.306489	+	0.951874i	$0.400846\pi$
$168$		0			0
$169$	1.80134e9			0.169866
$170$		0			0
$171$	2.71185e9	+	4.69706e9i	0.242540	+	0.420092i
$172$		0			0
$173$	3.11796e9	+	1.80016e9i	0.264645	+	0.152793i	0.626451	−	0.779460i	$-0.284507\pi$
$173$	3.11796e9	+	1.80016e9i	0.264645	+	0.152793i	−0.361807	+	0.932253i	$0.617840\pi$
$174$		0			0
$175$	1.19301e9	−	5.15726e9i	0.0961554	−	0.415669i
$176$		0			0
$177$	3.66451e9	−	6.34712e9i	0.280631	−	0.486068i
$178$		0			0
$179$	−7.51704e9	+	4.33996e9i	−0.547278	+	0.315971i	−0.748023	−	0.663672i	$-0.768997\pi$
$179$	−7.51704e9	+	4.33996e9i	−0.547278	+	0.315971i	0.200745	+	0.979643i	$0.435664\pi$
$180$		0			0
$181$		−	1.73505e10i		−	1.20160i	−0.799401	−	0.600798i	$-0.794849\pi$
$181$		−	1.73505e10i		−	1.20160i	0.799401	−	0.600798i	$-0.205151\pi$
$182$		0			0
$183$		−	1.54256e10i		−	1.01674i
$184$		0			0
$185$	1.52255e10	−	8.79046e9i	0.955651	−	0.551746i
$186$		0			0
$187$	−4.80098e8	+	8.31553e8i	−0.0287106	+	0.0497282i
$188$		0			0
$189$	−9.88975e9	−	9.23039e9i	−0.563777	−	0.526189i
$190$		0			0
$191$	−1.08502e10	−	6.26436e9i	−0.589912	−	0.340586i	0.175151	−	0.984542i	$-0.443959\pi$
$191$	−1.08502e10	−	6.26436e9i	−0.589912	−	0.340586i	−0.765063	+	0.643956i	$0.777292\pi$
$192$		0			0
$193$	1.36553e10	+	2.36516e10i	0.708422	+	1.22702i	0.965442	+	0.260616i	$0.0839257\pi$
$193$	1.36553e10	+	2.36516e10i	0.708422	+	1.22702i	−0.257021	+	0.966406i	$0.582741\pi$
$194$		0			0
$195$	1.60750e10			0.796150
$196$		0			0
$197$	−3.49380e10			−1.65272			−0.826362	−	0.563140i	$-0.809593\pi$
$197$	−3.49380e10			−1.65272			−0.826362	+	0.563140i	$0.809593\pi$
$198$		0			0
$199$	1.77894e10	+	3.08121e10i	0.804121	+	1.39278i	0.916883	+	0.399157i	$0.130697\pi$
$199$	1.77894e10	+	3.08121e10i	0.804121	+	1.39278i	−0.112761	+	0.993622i	$0.535970\pi$
$200$		0			0
$201$	2.09498e9	+	1.20954e9i	0.0905311	+	0.0522682i
$202$		0			0
$203$	3.10308e10	+	2.89620e10i	1.28251	+	1.19700i
$204$		0			0
$205$	−5.17788e9	+	8.96835e9i	−0.204767	+	0.354667i
$206$		0			0
$207$	1.47768e10	−	8.53139e9i	0.559389	−	0.322964i
$208$		0			0
$209$		−	5.76089e9i		−	0.208849i
$210$		0			0
$211$			4.67225e9i			0.162276i	0.996703	+	0.0811381i	$0.0258555\pi$
$211$			4.67225e9i			0.162276i	−0.996703	+	0.0811381i	$0.974145\pi$
$212$		0			0
$213$	1.33644e10	−	7.71596e9i	0.444880	−	0.256851i
$214$		0			0
$215$	5.52171e9	−	9.56388e9i	0.176238	−	0.305254i
$216$		0			0
$217$	2.65217e9	−	1.14650e10i	0.0811955	−	0.350999i
$218$		0			0
$219$	2.88689e10	+	1.66675e10i	0.848070	+	0.489633i
$220$		0			0
$221$	6.49484e9	+	1.12494e10i	0.183148	+	0.317222i
$222$		0			0
$223$	−6.06636e10			−1.64269			−0.821347	−	0.570429i	$-0.806777\pi$
$223$	−6.06636e10			−1.64269			−0.821347	+	0.570429i	$0.806777\pi$
$224$		0			0
$225$	5.44107e9			0.141535
$226$		0			0
$227$	7.06266e9	+	1.22329e10i	0.176544	+	0.305782i	0.940694	−	0.339255i	$-0.110175\pi$
$227$	7.06266e9	+	1.22329e10i	0.176544	+	0.305782i	−0.764151	+	0.645038i	$0.776842\pi$
$228$		0			0
$229$	−6.90916e10	−	3.98900e10i	−1.66022	−	0.958528i	−0.972609	−	0.232449i	$-0.925326\pi$
$229$	−6.90916e10	−	3.98900e10i	−1.66022	−	0.958528i	−0.687611	−	0.726080i	$-0.741340\pi$
$230$		0			0
$231$	−2.08254e9	−	6.82230e9i	−0.0481215	−	0.157644i
$232$		0			0
$233$	−2.72352e10	+	4.71728e10i	−0.605381	+	1.04855i	0.386610	+	0.922243i	$0.373646\pi$
$233$	−2.72352e10	+	4.71728e10i	−0.605381	+	1.04855i	−0.991991	+	0.126308i	$0.959687\pi$
$234$		0			0
$235$	−2.33006e10	+	1.34526e10i	−0.498382	+	0.287741i
$236$		0			0
$237$			8.10313e10i			1.66834i
$238$		0			0
$239$		−	7.91902e10i		−	1.56993i	−0.619539	−	0.784966i	$-0.712680\pi$
$239$		−	7.91902e10i		−	1.56993i	0.619539	−	0.784966i	$-0.287320\pi$
$240$		0			0
$241$	5.15708e10	−	2.97744e10i	0.984753	−	0.568547i	0.0810510	−	0.996710i	$-0.474172\pi$
$241$	5.15708e10	−	2.97744e10i	0.984753	−	0.568547i	0.903702	+	0.428163i	$0.140839\pi$
$242$		0			0
$243$	1.73567e10	−	3.00627e10i	0.319329	−	0.553095i
$244$		0			0
$245$	1.87532e10	−	3.83649e10i	0.332528	−	0.680279i
$246$		0			0
$247$	−6.74931e10	−	3.89672e10i	−1.15378	−	0.666135i
$248$		0			0
$249$	4.55179e10	+	7.88394e10i	0.750388	+	1.29971i
$250$		0			0
$251$	−5.73362e10			−0.911795			−0.455898	−	0.890032i	$-0.650682\pi$
$251$	−5.73362e10			−0.911795			−0.455898	+	0.890032i	$0.650682\pi$
$252$		0			0
$253$	−1.81236e10			−0.278100
$254$		0			0
$255$	−1.18599e10	−	2.05419e10i	−0.175651	−	0.304236i
$256$		0			0
$257$	9.47085e10	+	5.46800e10i	1.35422	+	0.781860i	0.988838	−	0.148996i	$-0.0476043\pi$
$257$	9.47085e10	+	5.46800e10i	1.35422	+	0.781860i	0.365384	+	0.930857i	$0.380938\pi$
$258$		0			0
$259$	−1.00939e11	+	3.08122e10i	−1.39384	+	0.425475i
$260$		0			0
$261$	−2.18151e10	+	3.77849e10i	−0.290988	+	0.504006i
$262$		0			0
$263$	−9.79765e10	+	5.65668e10i	−1.26276	+	0.729056i	−0.973608	−	0.228227i	$-0.926707\pi$
$263$	−9.79765e10	+	5.65668e10i	−1.26276	+	0.729056i	−0.289153	+	0.957283i	$0.593374\pi$
$264$		0			0
$265$		−	1.71425e10i		−	0.213535i
$266$		0			0
$267$			1.13840e10i			0.137086i
$268$		0			0
$269$	−9.18922e10	+	5.30540e10i	−1.07002	+	0.617778i	−0.928188	−	0.372112i	$-0.878634\pi$
$269$	−9.18922e10	+	5.30540e10i	−1.07002	+	0.617778i	−0.141836	+	0.989890i	$0.545300\pi$
$270$		0			0
$271$	4.25260e10	−	7.36571e10i	0.478952	−	0.829570i	−0.520756	−	0.853705i	$-0.674350\pi$
$271$	4.25260e10	−	7.36571e10i	0.478952	−	0.829570i	0.999709	+	0.0241354i	$0.00768330\pi$
$272$		0			0
$273$	−9.40147e10	−	2.17481e10i	−1.02439	−	0.236968i
$274$		0			0
$275$	−5.00506e9	−	2.88967e9i	−0.0527730	−	0.0304685i
$276$		0			0
$277$	5.75883e10	+	9.97459e10i	0.587727	+	1.01797i	0.994529	+	0.104457i	$0.0333104\pi$
$277$	5.75883e10	+	9.97459e10i	0.587727	+	1.01797i	−0.406803	+	0.913516i	$0.633356\pi$
$278$		0			0
$279$	1.20959e10			0.119515
$280$		0			0
$281$	−8.91206e10			−0.852707			−0.426354	−	0.904557i	$-0.640202\pi$
$281$	−8.91206e10			−0.852707			−0.426354	+	0.904557i	$0.640202\pi$
$282$		0			0
$283$	4.63405e10	+	8.02642e10i	0.429459	+	0.743845i	0.996825	−	0.0796203i	$-0.0253708\pi$
$283$	4.63405e10	+	8.02642e10i	0.429459	+	0.743845i	−0.567366	+	0.823466i	$0.692037\pi$
$284$		0			0
$285$	1.23246e11	+	7.11559e10i	1.10655	+	0.638865i
$286$		0			0
$287$	4.24163e10	−	4.54462e10i	0.369032	−	0.395393i
$288$		0			0
$289$	−4.97104e10	+	8.61009e10i	−0.419186	+	0.726051i
$290$		0			0
$291$	6.44514e10	−	3.72111e10i	0.526883	−	0.304196i
$292$		0			0
$293$		−	6.64592e10i		−	0.526806i	−0.964686	−	0.263403i	$-0.915155\pi$
$293$		−	6.64592e10i		−	0.526806i	0.964686	−	0.263403i	$-0.0848449\pi$
$294$		0			0
$295$		−	4.79035e10i		−	0.368271i
$296$		0			0
$297$	−1.27910e10	+	7.38488e9i	−0.0953894	+	0.0550731i
$298$		0			0
$299$	−1.22589e11	+	2.12331e11i	−0.887018	+	1.53636i
$300$		0			0
$301$	−4.52329e10	+	4.84640e10i	−0.317618	+	0.340306i
$302$		0			0
$303$	−1.56634e9	−	9.04329e8i	−0.0106757	−	0.00616360i
$304$		0			0
$305$	−5.04118e10	−	8.73158e10i	−0.333567	−	0.577755i
$306$		0			0
$307$	1.08333e11			0.696046			0.348023	−	0.937486i	$-0.386853\pi$
$307$	1.08333e11			0.696046			0.348023	+	0.937486i	$0.386853\pi$
$308$		0			0
$309$	1.17568e11			0.733630
$310$		0			0
$311$	−3.18414e10	−	5.51510e10i	−0.193006	−	0.334296i	0.753239	−	0.657747i	$-0.228490\pi$
$311$	−3.18414e10	−	5.51510e10i	−0.193006	−	0.334296i	−0.946245	+	0.323451i	$0.895157\pi$
$312$		0			0
$313$	−1.71378e11	−	9.89449e10i	−1.00926	−	0.582698i	−0.0982873	−	0.995158i	$-0.531336\pi$
$313$	−1.71378e11	−	9.89449e10i	−1.00926	−	0.582698i	−0.910976	+	0.412460i	$0.864670\pi$
$314$		0			0
$315$	4.27646e10	+	9.89259e9i	0.244730	+	0.0566125i
$316$		0			0
$317$	7.16976e10	−	1.24184e11i	0.398784	−	0.690714i	−0.594792	−	0.803879i	$-0.702766\pi$
$317$	7.16976e10	−	1.24184e11i	0.398784	−	0.690714i	0.993576	+	0.113165i	$0.0360990\pi$
$318$		0			0
$319$	4.01340e10	−	2.31714e10i	0.216997	−	0.125283i
$320$		0			0
$321$			3.49584e11i			1.83772i
$322$		0			0
$323$			1.14997e11i			0.587864i
$324$		0			0
$325$	−6.77092e10	+	3.90919e10i	−0.336645	+	0.194362i
$326$		0			0
$327$	1.38953e11	−	2.40674e11i	0.672053	−	1.16403i
$328$		0			0
$329$	1.54474e11	−	4.71540e10i	0.726900	−	0.221890i
$330$		0			0
$331$	−7.37062e9	−	4.25543e9i	−0.0337503	−	0.0194858i	0.483030	−	0.875604i	$-0.339536\pi$
$331$	−7.37062e9	−	4.25543e9i	−0.0337503	−	0.0194858i	−0.516780	+	0.856118i	$0.672870\pi$
$332$		0			0
$333$	−5.42402e10	−	9.39469e10i	−0.241726	−	0.418681i
$334$		0			0
$335$	1.58114e10			0.0685913
$336$		0			0
$337$	−9.73131e9			−0.0410995			−0.0205498	−	0.999789i	$-0.506542\pi$
$337$	−9.73131e9			−0.0410995			−0.0205498	+	0.999789i	$0.506542\pi$
$338$		0			0
$339$	1.80290e11	+	3.12271e11i	0.741434	+	1.28420i
$340$		0			0
$341$	−1.11267e10	−	6.42398e9i	−0.0445625	−	0.0257282i
$342$		0			0
$343$	−1.61583e11	+	1.99006e11i	−0.630335	+	0.776324i
$344$		0			0
$345$	2.23854e11	−	3.87727e11i	0.850705	−	1.47346i
$346$		0			0
$347$	1.03687e11	−	5.98639e10i	0.383922	−	0.221658i	−0.295601	−	0.955311i	$-0.595520\pi$
$347$	1.03687e11	−	5.98639e10i	0.383922	−	0.221658i	0.679523	+	0.733654i	$0.262187\pi$
$348$		0			0
$349$		−	1.47764e11i		−	0.533156i	−0.963813	−	0.266578i	$-0.914107\pi$
$349$		−	1.47764e11i		−	0.533156i	0.963813	−	0.266578i	$-0.0858930\pi$
$350$		0			0
$351$			1.99808e11i			0.702636i
$352$		0			0
$353$	−3.37242e10	+	1.94707e10i	−0.115599	+	0.0667413i	−0.556685	−	0.830724i	$-0.687927\pi$
$353$	−3.37242e10	+	1.94707e10i	−0.115599	+	0.0667413i	0.441085	+	0.897465i	$0.354594\pi$
$354$		0			0
$355$	5.04326e10	−	8.73518e10i	0.168533	−	0.291907i
$356$		0			0
$357$	4.15711e10	+	1.36185e11i	0.135452	+	0.443734i
$358$		0			0
$359$	−4.85695e11	−	2.80416e11i	−1.54326	−	0.891000i	−0.998630	−	0.0523219i	$-0.983338\pi$
$359$	−4.85695e11	−	2.80416e11i	−1.54326	−	0.891000i	−0.544627	−	0.838678i	$-0.683329\pi$
$360$		0			0
$361$	−1.83632e11	−	3.18060e11i	−0.569070	−	0.985659i
$362$		0			0
$363$	3.73971e11			1.13047
$364$		0			0
$365$	2.17882e11			0.642544
$366$		0			0
$367$	−3.40438e11	−	5.89657e11i	−0.979583	−	1.69669i	−0.663896	−	0.747825i	$-0.731098\pi$
$367$	−3.40438e11	−	5.89657e11i	−0.979583	−	1.69669i	−0.315687	−	0.948863i	$-0.602235\pi$
$368$		0			0
$369$	5.53379e10	+	3.19494e10i	0.155383	+	0.0897105i
$370$		0			0
$371$	−2.31924e10	+	1.00258e11i	−0.0635570	+	0.274750i
$372$		0			0
$373$	2.71398e11	−	4.70074e11i	0.725966	−	1.25741i	−0.232609	−	0.972570i	$-0.574726\pi$
$373$	2.71398e11	−	4.70074e11i	0.725966	−	1.25741i	0.958575	−	0.284840i	$-0.0919404\pi$
$374$		0			0
$375$	4.13436e11	−	2.38697e11i	1.07961	−	0.623314i
$376$		0			0
$377$		−	6.26932e11i		−	1.59840i
$378$		0			0
$379$			5.07966e11i			1.26462i	0.774717	+	0.632308i	$0.217892\pi$
$379$			5.07966e11i			1.26462i	−0.774717	+	0.632308i	$0.782108\pi$
$380$		0			0
$381$	1.21439e11	−	7.01129e10i	0.295254	−	0.170465i
$382$		0			0
$383$	2.24294e11	−	3.88489e11i	0.532627	−	0.922538i	−0.466647	−	0.884444i	$-0.654538\pi$
$383$	2.24294e11	−	3.88489e11i	0.532627	−	0.922538i	0.999274	−	0.0380941i	$-0.0121286\pi$
$384$		0			0
$385$	−3.40839e10	−	3.18115e10i	−0.0790635	−	0.0737922i
$386$		0			0
$387$	−5.90125e10	−	3.40709e10i	−0.133735	−	0.0772119i
$388$		0			0
$389$	−3.91569e11	−	6.78218e11i	−0.867033	−	1.50174i	−0.865014	−	0.501747i	$-0.832691\pi$
$389$	−3.91569e11	−	6.78218e11i	−0.867033	−	1.50174i	−0.00201837	−	0.999998i	$-0.500642\pi$
$390$		0			0
$391$	3.61778e11			0.782793
$392$		0			0
$393$	−2.95152e11			−0.624135
$394$		0			0
$395$	2.64816e11	+	4.58675e11i	0.547340	+	0.948021i
$396$		0			0
$397$	6.54080e11	+	3.77633e11i	1.32152	+	0.762980i	0.983971	−	0.178328i	$-0.0570687\pi$
$397$	6.54080e11	+	3.77633e11i	1.32152	+	0.762980i	0.337549	+	0.941308i	$0.390402\pi$
$398$		0			0
$399$	−6.24535e11	−	5.82897e11i	−1.23361	−	1.15137i
$400$		0			0
$401$	9.77646e10	−	1.69333e11i	0.188813	−	0.327034i	−0.756042	−	0.654523i	$-0.772869\pi$
$401$	9.77646e10	−	1.69333e11i	0.188813	−	0.327034i	0.944855	+	0.327490i	$0.106203\pi$
$402$		0			0
$403$	−1.50523e11	+	8.69046e10i	−0.284270	+	0.164123i
$404$		0			0
$405$		−	5.00863e11i		−	0.925064i
$406$		0			0
$407$			1.15225e11i			0.208147i
$408$		0			0
$409$	−5.40799e11	+	3.12231e11i	−0.955611	+	0.551722i	−0.894820	−	0.446428i	$-0.852696\pi$
$409$	−5.40799e11	+	3.12231e11i	−0.955611	+	0.551722i	−0.0607918	+	0.998150i	$0.519363\pi$
$410$		0			0
$411$	−1.21423e11	+	2.10311e11i	−0.209900	+	0.363557i
$412$		0			0
$413$	−6.48094e10	+	2.80164e11i	−0.109613	+	0.473846i
$414$		0			0
$415$	5.15305e11	+	2.97512e11i	0.852802	+	0.492365i
$416$		0			0
$417$	9.12610e10	+	1.58069e11i	0.147799	+	0.255996i
$418$		0			0
$419$	8.55564e11			1.35609			0.678047	−	0.735019i	$-0.262827\pi$
$419$	8.55564e11			1.35609			0.678047	+	0.735019i	$0.262827\pi$
$420$		0			0
$421$	6.80832e11			1.05626			0.528129	−	0.849164i	$-0.322894\pi$
$421$	6.80832e11			1.05626			0.528129	+	0.849164i	$0.322894\pi$
$422$		0			0
$423$	8.30075e10	+	1.43773e11i	0.126062	+	0.218347i
$424$		0			0
$425$	9.99096e10	+	5.76829e10i	0.148545	+	0.0857623i
$426$		0			0
$427$	1.76703e11	+	5.78870e11i	0.257228	+	0.842666i
$428$		0			0
$429$	−5.26775e10	+	9.12400e10i	−0.0750874	+	0.130055i
$430$		0			0
$431$	1.54207e11	−	8.90314e10i	0.215257	−	0.124278i	−0.388495	−	0.921451i	$-0.627005\pi$
$431$	1.54207e11	−	8.90314e10i	0.215257	−	0.124278i	0.603752	+	0.797172i	$0.293672\pi$
$432$		0			0
$433$			1.14363e12i			1.56348i	0.623607	+	0.781738i	$0.285667\pi$
$433$			1.14363e12i			1.56348i	−0.623607	+	0.781738i	$0.714333\pi$
$434$		0			0
$435$			1.14481e12i			1.53296i
$436$		0			0
$437$	−1.87976e12	+	1.08528e12i	−2.46568	+	1.42356i
$438$		0			0
$439$	2.19362e11	−	3.79946e11i	0.281884	−	0.488238i	−0.689964	−	0.723843i	$-0.742374\pi$
$439$	2.19362e11	−	3.79946e11i	0.281884	−	0.488238i	0.971849	+	0.235605i	$0.0757072\pi$
$440$		0			0
$441$	−2.36725e11	−	1.15714e11i	−0.298037	−	0.145684i
$442$		0			0
$443$	−3.33909e11	−	1.92782e11i	−0.411919	−	0.237821i	0.279695	−	0.960089i	$-0.409767\pi$
$443$	−3.33909e11	−	1.92782e11i	−0.411919	−	0.237821i	−0.691614	+	0.722268i	$0.743100\pi$
$444$		0			0
$445$	3.72036e10	+	6.44386e10i	0.0449744	+	0.0778979i
$446$		0			0
$447$	−8.81079e11			−1.04383
$448$		0			0
$449$	1.27314e11			0.147832			0.0739160	−	0.997264i	$-0.476450\pi$
$449$	1.27314e11			0.147832			0.0739160	+	0.997264i	$0.476450\pi$
$450$		0			0
$451$	−3.39356e10	−	5.87783e10i	−0.0386244	−	0.0668994i
$452$		0			0
$453$	2.45626e11	+	1.41812e11i	0.274052	+	0.158224i
$454$		0			0
$455$	−6.03241e11	+	1.84142e11i	−0.659842	+	0.201420i
$456$		0			0
$457$	4.63705e11	−	8.03161e11i	0.497301	−	0.861350i	−0.502694	−	0.864464i	$-0.667658\pi$
$457$	4.63705e11	−	8.03161e11i	0.497301	−	0.861350i	0.999995	+	0.00311394i	$0.000991201\pi$
$458$		0			0
$459$	2.55331e11	−	1.47415e11i	0.268501	−	0.155019i
$460$		0			0
$461$			1.36015e12i			1.40260i	0.712867	+	0.701300i	$0.247397\pi$
$461$			1.36015e12i			1.40260i	−0.712867	+	0.701300i	$0.752603\pi$
$462$		0			0
$463$		−	2.57213e11i		−	0.260123i	−0.991506	−	0.130061i	$-0.958483\pi$
$463$		−	2.57213e11i		−	0.260123i	0.991506	−	0.130061i	$-0.0415175\pi$
$464$		0			0
$465$	2.74863e11	−	1.58692e11i	0.272632	−	0.157404i
$466$		0			0
$467$	9.27100e11	−	1.60578e12i	0.901988	−	1.56229i	0.0770775	−	0.997025i	$-0.475441\pi$
$467$	9.27100e11	−	1.60578e12i	0.901988	−	1.56229i	0.824910	−	0.565264i	$-0.191226\pi$
$468$		0			0
$469$	−9.24732e10	−	2.13915e10i	−0.0882547	−	0.0204157i
$470$		0			0
$471$	−1.94014e12	−	1.12014e12i	−1.81652	−	1.04877i
$472$		0			0
$473$	3.61891e10	+	6.26813e10i	0.0332432	+	0.0575789i
$474$		0			0
$475$	−6.92161e11			−0.623858
$476$		0			0
$477$	−1.05776e11			−0.0935519
$478$		0			0
$479$	5.28772e11	+	9.15859e11i	0.458942	+	0.794912i	0.998905	−	0.0467771i	$-0.0148951\pi$
$479$	5.28772e11	+	9.15859e11i	0.458942	+	0.794912i	−0.539963	+	0.841689i	$0.681562\pi$
$480$		0			0
$481$	1.34994e12	+	7.79389e11i	1.14991	+	0.663898i
$482$		0			0
$483$	−1.83377e12	+	1.96477e12i	−1.53315	+	1.64266i
$484$		0			0
$485$	2.43217e11	−	4.21264e11i	0.199598	−	0.345713i
$486$		0			0
$487$	4.08557e11	−	2.35880e11i	0.329133	−	0.190025i	−0.326323	−	0.945258i	$-0.605810\pi$
$487$	4.08557e11	−	2.35880e11i	0.329133	−	0.190025i	0.655456	+	0.755233i	$0.272476\pi$
$488$		0			0
$489$			2.75274e11i			0.217709i
$490$		0			0
$491$		−	1.93898e12i		−	1.50559i	−0.658255	−	0.752795i	$-0.728705\pi$
$491$		−	1.93898e12i		−	1.50559i	0.658255	−	0.752795i	$-0.271295\pi$
$492$		0			0
$493$	−8.01144e11	+	4.62541e11i	−0.610801	+	0.352646i
$494$		0			0
$495$	2.39615e10	−	4.15025e10i	0.0179387	−	0.0310707i
$496$		0			0
$497$	−4.13135e11	+	4.42647e11i	−0.303731	+	0.325427i
$498$		0			0
$499$	−1.68826e12	−	9.74717e11i	−1.21895	−	0.703762i	−0.254258	−	0.967136i	$-0.581831\pi$
$499$	−1.68826e12	−	9.74717e11i	−1.21895	−	0.703762i	−0.964694	+	0.263374i	$0.915165\pi$
$500$		0			0
$501$	−4.98762e11	−	8.63882e11i	−0.353691	−	0.612611i
$502$		0			0
$503$	8.84756e11			0.616265			0.308133	−	0.951343i	$-0.400296\pi$
$503$	8.84756e11			0.616265			0.308133	+	0.951343i	$0.400296\pi$
$504$		0			0
$505$	−1.18216e10			−0.00808848
$506$		0			0
$507$	−1.45821e11	−	2.52570e11i	−0.0980133	−	0.169764i
$508$		0			0
$509$	−6.27221e11	−	3.62126e11i	−0.414181	−	0.239128i	0.278404	−	0.960464i	$-0.410195\pi$
$509$	−6.27221e11	−	3.62126e11i	−0.414181	−	0.239128i	−0.692585	+	0.721337i	$0.743528\pi$
$510$		0			0
$511$	−1.27428e12	−	2.94776e11i	−0.826746	−	0.191248i
$512$		0			0
$513$	−8.84448e11	+	1.53191e12i	−0.563825	+	0.976574i
$514$		0			0
$515$	6.65491e11	−	3.84222e11i	0.416879	−	0.240685i
$516$		0			0
$517$		−	1.76336e11i		−	0.108551i
$518$		0			0
$519$		−	5.82901e11i		−	0.352648i
$520$		0			0
$521$	4.23157e11	−	2.44310e11i	0.251612	−	0.145268i	−0.368890	−	0.929473i	$-0.620262\pi$
$521$	4.23157e11	−	2.44310e11i	0.251612	−	0.145268i	0.620502	+	0.784205i	$0.286929\pi$
$522$		0			0
$523$	−9.35351e11	+	1.62007e12i	−0.546659	+	0.946842i	0.451841	+	0.892098i	$0.350767\pi$
$523$	−9.35351e11	+	1.62007e12i	−0.546659	+	0.946842i	−0.998500	+	0.0547434i	$0.982566\pi$
$524$		0			0
$525$	−8.19687e11	+	2.50213e11i	−0.470903	+	0.143745i
$526$		0			0
$527$	2.22107e11	+	1.28234e11i	0.125434	+	0.0724194i
$528$		0			0
$529$	2.51369e12	+	4.35383e12i	1.39560	+	2.41725i
$530$		0			0
$531$	−2.95581e11			−0.161344
$532$		0			0
$533$	−9.18174e11			−0.492779
$534$		0			0
$535$	1.14247e12	+	1.97881e12i	0.602909	+	1.04427i
$536$		0			0
$537$	1.21703e12	+	7.02653e11i	0.631564	+	0.364634i
$538$		0			0
$539$	1.56302e11	+	2.32162e11i	0.0797653	+	0.118479i
$540$		0			0
$541$	−1.98694e11	+	3.44148e11i	−0.0997234	+	0.172726i	−0.911570	−	0.411145i	$-0.865129\pi$
$541$	−1.98694e11	+	3.44148e11i	−0.0997234	+	0.172726i	0.811847	+	0.583871i	$0.198462\pi$
$542$		0			0
$543$	−2.43275e12	+	1.40455e12i	−1.20088	+	0.693327i
$544$		0			0
$545$		−	1.81644e12i		−	0.881933i
$546$		0			0
$547$			1.73293e12i			0.827631i	0.910361	+	0.413816i	$0.135804\pi$
$547$			1.73293e12i			0.827631i	−0.910361	+	0.413816i	$0.864196\pi$
$548$		0			0
$549$	−5.38769e11	+	3.11059e11i	−0.253121	+	0.146139i
$550$		0			0
$551$	2.77511e12	−	4.80663e12i	1.28262	−	2.22156i
$552$		0			0
$553$	−9.28230e11	−	3.04084e12i	−0.422078	−	1.38271i
$554$		0			0
$555$	−2.46506e12	−	1.42320e12i	−1.10283	−	0.636720i
$556$		0			0
$557$	3.12630e11	+	5.41491e11i	0.137620	+	0.238365i	0.926595	−	0.376060i	$-0.122721\pi$
$557$	3.12630e11	+	5.41491e11i	0.137620	+	0.238365i	−0.788975	+	0.614425i	$0.789388\pi$
$558$		0			0
$559$	9.79144e11			0.424124
$560$		0			0
$561$	1.55459e11			0.0662646
$562$		0			0
$563$	1.34886e12	+	2.33629e12i	0.565820	+	0.980029i	0.996973	+	0.0777503i	$0.0247737\pi$
$563$	1.34886e12	+	2.33629e12i	0.565820	+	0.980029i	−0.431153	+	0.902279i	$0.641893\pi$
$564$		0			0
$565$	2.04105e12	+	1.17840e12i	0.842627	+	0.486491i
$566$		0			0
$567$	−6.77626e11	+	2.92930e12i	−0.275338	+	1.19026i
$568$		0			0
$569$	5.38286e11	−	9.32339e11i	0.215282	−	0.372880i	−0.738078	−	0.674716i	$-0.764266\pi$
$569$	5.38286e11	−	9.32339e11i	0.215282	−	0.372880i	0.953360	+	0.301836i	$0.0975996\pi$
$570$		0			0
$571$	3.87705e12	−	2.23842e12i	1.52630	−	0.881208i	0.526784	−	0.849999i	$-0.323398\pi$
$571$	3.87705e12	−	2.23842e12i	1.52630	−	0.881208i	0.999513	−	0.0312091i	$-0.00993579\pi$
$572$		0			0
$573$			2.02844e12i			0.786079i
$574$		0			0
$575$			2.17752e12i			0.830723i
$576$		0			0
$577$	−6.02473e11	+	3.47838e11i	−0.226280	+	0.130643i	−0.608855	−	0.793282i	$-0.708371\pi$
$577$	−6.02473e11	+	3.47838e11i	−0.226280	+	0.130643i	0.382575	+	0.923925i	$0.375037\pi$
$578$		0			0
$579$	2.21083e12	−	3.82926e12i	0.817525	−	1.41600i
$580$		0			0
$581$	−2.61126e12	−	2.43716e12i	−0.950730	−	0.887344i
$582$		0			0
$583$	9.72993e10	+	5.61758e10i	0.0348820	+	0.0201391i
$584$		0			0
$585$	−3.24155e11	−	5.61452e11i	−0.114433	−	0.198204i
$586$		0			0
$587$	9.96340e11			0.346367			0.173183	−	0.984890i	$-0.444595\pi$
$587$	9.96340e11			0.346367			0.173183	+	0.984890i	$0.444595\pi$
$588$		0			0
$589$	−1.53873e12			−0.526798
$590$		0			0
$591$	2.82829e12	+	4.89873e12i	0.953629	+	1.65173i
$592$		0			0
$593$	−2.93493e12	−	1.69448e12i	−0.974656	−	0.562718i	−0.0740032	−	0.997258i	$-0.523578\pi$
$593$	−2.93493e12	−	1.69448e12i	−0.974656	−	0.562718i	−0.900652	+	0.434540i	$0.856911\pi$
$594$		0			0
$595$	6.80374e11	+	6.35013e11i	0.222547	+	0.207709i
$596$		0			0
$597$	2.88015e12	−	4.98857e12i	0.927964	−	1.60728i
$598$		0			0
$599$	−2.56167e12	+	1.47898e12i	−0.813023	+	0.469399i	−0.848005	−	0.529989i	$-0.822196\pi$
$599$	−2.56167e12	+	1.47898e12i	−0.813023	+	0.469399i	0.0349815	+	0.999388i	$0.488863\pi$
$600$		0			0
$601$			1.96519e12i			0.614426i	0.951641	+	0.307213i	$0.0993963\pi$
$601$			1.96519e12i			0.614426i	−0.951641	+	0.307213i	$0.900604\pi$
$602$		0			0
$603$		−	9.75620e10i		−	0.0300506i
$604$		0			0
$605$	2.11685e12	−	1.22216e12i	0.642378	−	0.370877i
$606$		0			0
$607$	2.27593e12	−	3.94202e12i	0.680470	−	1.17861i	−0.294367	−	0.955692i	$-0.595109\pi$
$607$	2.27593e12	−	3.94202e12i	0.680470	−	1.17861i	0.974837	−	0.222917i	$-0.0715579\pi$
$608$		0			0
$609$	1.54883e12	−	6.69541e12i	0.456274	−	1.97242i
$610$		0			0
$611$	−2.06591e12	−	1.19275e12i	−0.599688	−	0.346230i
$612$		0			0
$613$	−7.98134e11	−	1.38241e12i	−0.228299	−	0.395425i	0.729005	−	0.684508i	$-0.239983\pi$
$613$	−7.98134e11	−	1.38241e12i	−0.228299	−	0.395425i	−0.957304	+	0.289083i	$0.906650\pi$
$614$		0			0
$615$	1.67663e12			0.472606
$616$		0			0
$617$	−5.06588e12			−1.40725			−0.703626	−	0.710571i	$-0.748437\pi$
$617$	−5.06588e12			−1.40725			−0.703626	+	0.710571i	$0.748437\pi$
$618$		0			0
$619$	1.49967e12	+	2.59750e12i	0.410570	+	0.711129i	0.994952	−	0.100350i	$-0.0319963\pi$
$619$	1.49967e12	+	2.59750e12i	0.410570	+	0.711129i	−0.584382	+	0.811479i	$0.698663\pi$
$620$		0			0
$621$	4.81934e12	+	2.78245e12i	1.30039	+	0.750783i
$622$		0			0
$623$	−1.30406e11	−	4.27203e11i	−0.0346817	−	0.113616i
$624$		0			0
$625$	7.46397e11	−	1.29280e12i	0.195664	−	0.338899i
$626$		0			0
$627$	−8.07747e11	+	4.66353e11i	−0.208724	+	0.120507i
$628$		0			0
$629$		−	2.30009e12i		−	0.585890i
$630$		0			0
$631$		−	7.34129e10i		−	0.0184349i	−0.999958	−	0.00921744i	$-0.997066\pi$
$631$		−	7.34129e10i		−	0.0184349i	0.999958	−	0.00921744i	$-0.00293404\pi$
$632$		0			0
$633$	6.55106e11	−	3.78226e11i	0.162179	−	0.0936341i
$634$		0			0
$635$	4.58268e11	−	7.93743e11i	0.111850	−	0.193731i
$636$		0			0
$637$	3.77719e12	−	2.60823e11i	0.908953	−	0.0627650i
$638$		0			0
$639$	−5.38992e11	−	3.11187e11i	−0.127888	−	0.0738359i
$640$		0			0
$641$	−5.83190e11	−	1.01011e12i	−0.136442	−	0.236325i	0.789705	−	0.613486i	$-0.210233\pi$
$641$	−5.83190e11	−	1.01011e12i	−0.136442	−	0.236325i	−0.926147	+	0.377162i	$0.876900\pi$
$642$		0			0
$643$	3.28207e12			0.757179			0.378590	−	0.925565i	$-0.376409\pi$
$643$	3.28207e12			0.757179			0.378590	+	0.925565i	$0.376409\pi$
$644$		0			0
$645$	−1.78796e12			−0.406761
$646$		0			0
$647$	−6.30415e10	−	1.09191e11i	−0.0141435	−	0.0244973i	0.858867	−	0.512199i	$-0.171169\pi$
$647$	−6.30415e10	−	1.09191e11i	−0.0141435	−	0.0244973i	−0.873011	+	0.487701i	$0.837836\pi$
$648$		0			0
$649$	2.71895e11	+	1.56979e11i	0.0601590	+	0.0347328i
$650$		0			0
$651$	−1.82223e12	+	5.56245e11i	−0.397639	+	0.121381i
$652$		0			0
$653$	2.98767e12	−	5.17480e12i	0.643019	−	1.11374i	−0.341736	−	0.939796i	$-0.611015\pi$
$653$	2.98767e12	−	5.17480e12i	0.643019	−	1.11374i	0.984755	−	0.173946i	$-0.0556517\pi$
$654$		0			0
$655$	−1.67069e12	+	9.64576e11i	−0.354659	+	0.204763i
$656$		0			0
$657$		−	1.34441e12i		−	0.281505i
$658$		0			0
$659$		−	7.31369e12i		−	1.51061i	−0.655374	−	0.755305i	$-0.727489\pi$
$659$		−	7.31369e12i		−	1.51061i	0.655374	−	0.755305i	$-0.272511\pi$
$660$		0			0
$661$	6.92901e12	−	4.00046e12i	1.41177	−	0.815087i	0.416216	−	0.909266i	$-0.363356\pi$
$661$	6.92901e12	−	4.00046e12i	1.41177	−	0.815087i	0.995555	+	0.0941789i	$0.0300226\pi$
$662$		0			0
$663$	1.05153e12	−	1.82131e12i	0.211355	−	0.366078i
$664$		0			0
$665$	−5.44010e12	−	1.25844e12i	−1.07872	−	0.249537i
$666$		0			0
$667$	−1.51215e13	−	8.73041e12i	−2.95821	−	1.70792i
$668$		0			0
$669$	4.91081e12	+	8.50578e12i	0.947842	+	1.64171i
$670$		0			0
$671$	6.60794e11			0.125839
$672$		0			0
$673$	−4.31192e12			−0.810219			−0.405109	−	0.914268i	$-0.632767\pi$
$673$	−4.31192e12			−0.810219			−0.405109	+	0.914268i	$0.632767\pi$
$674$		0			0
$675$	8.87280e11	+	1.53681e12i	0.164511	+	0.284941i
$676$		0			0
$677$	−2.67983e12	−	1.54720e12i	−0.490296	−	0.283073i	0.234401	−	0.972140i	$-0.424687\pi$
$677$	−2.67983e12	−	1.54720e12i	−0.490296	−	0.283073i	−0.724697	+	0.689067i	$0.758020\pi$
$678$		0			0
$679$	−1.99239e12	+	2.13471e12i	−0.359716	+	0.385412i
$680$		0			0
$681$	1.14347e12	−	1.98054e12i	0.203733	−	0.352876i
$682$		0			0
$683$	3.41583e12	−	1.97213e12i	0.600625	−	0.346771i	−0.168662	−	0.985674i	$-0.553945\pi$
$683$	3.41583e12	−	1.97213e12i	0.600625	−	0.346771i	0.769288	+	0.638903i	$0.220611\pi$
$684$		0			0
$685$			1.58727e12i			0.275451i
$686$		0			0
$687$			1.29166e13i			2.21230i
$688$		0			0
$689$	1.31628e12	−	7.59955e11i	0.222517	−	0.128470i
$690$		0			0
$691$	5.27311e11	−	9.13329e11i	0.0879864	−	0.152397i	−0.818673	−	0.574259i	$-0.805290\pi$
$691$	5.27311e11	−	9.13329e11i	0.0879864	−	0.152397i	0.906660	+	0.421862i	$0.138623\pi$
$692$		0			0
$693$	−1.96288e11	+	2.10310e11i	−0.0323292	+	0.0346385i
$694$		0			0
$695$	1.03316e12	+	5.96495e11i	0.167971	+	0.0969784i
$696$		0			0
$697$	6.77415e11	+	1.17332e12i	0.108719	+	0.188308i
$698$		0			0
$699$	8.81893e12			1.39723
$700$		0			0
$701$	−1.51493e12			−0.236952			−0.118476	−	0.992957i	$-0.537801\pi$
$701$	−1.51493e12			−0.236952			−0.118476	+	0.992957i	$0.537801\pi$
$702$		0			0
$703$	6.89993e12	+	1.19510e13i	1.06548	+	1.84547i
$704$		0			0
$705$	3.77245e12	+	2.17802e12i	0.575138	+	0.332056i
$706$		0			0
$707$	6.91390e10	+	1.59937e10i	0.0104072	+	0.00240747i
$708$		0			0
$709$	−3.38441e12	+	5.86198e12i	−0.503008	+	0.871236i	0.496985	+	0.867759i	$0.334440\pi$
$709$	−3.38441e12	+	5.86198e12i	−0.503008	+	0.871236i	−0.999994	+	0.00347738i	$0.998893\pi$
$710$		0			0
$711$	2.83019e12	−	1.63401e12i	0.415338	−	0.239796i
$712$		0			0
$713$			4.84080e12i			0.701478i
$714$		0			0
$715$			6.88615e11i			0.0985369i
$716$		0			0
$717$	−1.11034e13	+	6.41057e12i	−1.56899	+	0.905858i
$718$		0			0
$719$	4.13383e12	−	7.16001e12i	0.576864	−	0.999157i	−0.418973	−	0.907999i	$-0.637610\pi$
$719$	4.13383e12	−	7.16001e12i	0.576864	−	0.999157i	0.995836	−	0.0911582i	$-0.0290569\pi$
$720$		0			0
$721$	−4.41195e12	+	1.34677e12i	−0.608026	+	0.185603i
$722$		0			0
$723$	−8.34947e12	−	4.82057e12i	−1.13641	−	0.656109i
$724$		0			0
$725$	−2.78400e12	−	4.82202e12i	−0.374238	−	0.648199i
$726$		0			0
$727$	6.66096e12			0.884366			0.442183	−	0.896925i	$-0.354204\pi$
$727$	6.66096e12			0.884366			0.442183	+	0.896925i	$0.354204\pi$
$728$		0			0
$729$	3.69590e12			0.484670
$730$		0			0
$731$	−7.22397e11	−	1.25123e12i	−0.0935724	−	0.162072i
$732$		0			0
$733$	−3.47424e12	−	2.00585e12i	−0.444521	−	0.256644i	0.260993	−	0.965341i	$-0.415950\pi$
$733$	−3.47424e12	−	2.00585e12i	−0.444521	−	0.256644i	−0.705513	+	0.708697i	$0.749283\pi$
$734$		0			0
$735$	−6.89733e12	+	4.76275e11i	−0.871742	+	0.0601955i
$736$		0			0
$737$	−5.18137e10	+	8.97440e10i	−0.00646906	+	0.0112047i
$738$		0			0
$739$	−6.43111e12	+	3.71300e12i	−0.793205	+	0.457957i	−0.841090	−	0.540896i	$-0.818085\pi$
$739$	−6.43111e12	+	3.71300e12i	−0.793205	+	0.457957i	0.0478844	+	0.998853i	$0.484752\pi$
$740$		0			0
$741$			1.26178e13i			1.53745i
$742$		0			0
$743$			1.37727e13i			1.65795i	0.559288	+	0.828973i	$0.311074\pi$
$743$			1.37727e13i			1.65795i	−0.559288	+	0.828973i	$0.688926\pi$
$744$		0			0
$745$	−4.98732e12	+	2.87943e12i	−0.593149	+	0.342455i
$746$		0			0
$747$	1.83575e12	−	3.17962e12i	0.215711	−	0.373622i
$748$		0			0
$749$	−4.00456e12	−	1.31187e13i	−0.464929	−	1.52308i
$750$		0			0
$751$	−4.74753e12	−	2.74099e12i	−0.544613	−	0.314433i	0.202333	−	0.979317i	$-0.435148\pi$
$751$	−4.74753e12	−	2.74099e12i	−0.544613	−	0.314433i	−0.746947	+	0.664884i	$0.768481\pi$
$752$		0			0
$753$	4.64146e12	+	8.03924e12i	0.526110	+	0.911250i
$754$		0			0
$755$	1.85381e12			0.207637
$756$		0			0
$757$	1.21160e13			1.34100			0.670498	−	0.741911i	$-0.266080\pi$
$757$	1.21160e13			1.34100			0.670498	+	0.741911i	$0.266080\pi$
$758$		0			0
$759$	1.46713e12	+	2.54115e12i	0.160465	+	0.277934i
$760$		0			0
$761$	6.86780e12	+	3.96513e12i	0.742313	+	0.428574i	0.822910	−	0.568172i	$-0.192349\pi$
$761$	6.86780e12	+	3.96513e12i	0.742313	+	0.428574i	−0.0805970	+	0.996747i	$0.525683\pi$
$762$		0			0
$763$	−2.45748e12	+	1.06234e13i	−0.262501	+	1.13476i
$764$		0			0
$765$	−4.78313e11	+	8.28462e11i	−0.0504935	+	0.0874573i
$766$		0			0
$767$	3.67825e12	−	2.12364e12i	0.383762	−	0.221565i
$768$		0			0
$769$			1.21650e13i			1.25443i	0.778848	+	0.627213i	$0.215805\pi$
$769$			1.21650e13i			1.25443i	−0.778848	+	0.627213i	$0.784195\pi$
$770$		0			0
$771$		−	1.77057e13i		−	1.80455i
$772$		0			0
$773$	−1.47792e13	+	8.53280e12i	−1.48883	+	0.859575i	−0.999919	−	0.0127600i	$-0.995938\pi$
$773$	−1.47792e13	+	8.53280e12i	−1.48883	+	0.859575i	−0.488909	+	0.872335i	$0.662605\pi$
$774$		0			0
$775$	−7.71829e11	+	1.33685e12i	−0.0768535	+	0.133114i
$776$		0			0
$777$	1.24914e13	+	1.16586e13i	1.22947	+	1.14750i
$778$		0			0
$779$	−7.03956e12	−	4.06429e12i	−0.684900	−	0.395427i
$780$		0			0
$781$	3.30534e11	+	5.72501e11i	0.0317897	+	0.0550613i
$782$		0			0
$783$	−1.42296e13			−1.35290
$784$		0			0
$785$	−1.46428e13			−1.37629
$786$		0			0
$787$	−9.03253e12	−	1.56448e13i	−0.839311	−	1.45373i	−0.890471	−	0.455039i	$-0.849625\pi$
$787$	−9.03253e12	−	1.56448e13i	−0.839311	−	1.45373i	0.0511599	−	0.998690i	$-0.483708\pi$
$788$		0			0
$789$	1.58627e13	+	9.15833e12i	1.45724	+	0.841337i
$790$		0			0
$791$	−1.03428e13	−	9.65324e12i	−0.939386	−	0.876757i
$792$		0			0
$793$	4.46967e12	−	7.74169e12i	0.401371	−	0.695195i
$794$		0			0
$795$	−2.40359e12	+	1.38771e12i	−0.213407	+	0.123211i
$796$		0			0
$797$		−	1.21166e13i		−	1.06370i	−0.846839	−	0.531850i	$-0.821497\pi$
$797$		−	1.21166e13i		−	1.06370i	0.846839	−	0.531850i	$-0.178503\pi$
$798$		0			0
$799$			3.51998e12i			0.305548i
$800$		0			0
$801$	3.97609e11	−	2.29560e11i	0.0341279	−	0.0197038i
$802$		0			0
$803$	−7.13995e11	+	1.23668e12i	−0.0606003	+	0.104963i
$804$		0			0
$805$	−3.95902e12	+	1.71144e13i	−0.332281	+	1.43641i
$806$		0			0
$807$	1.48776e13	+	8.58960e12i	1.23482	+	0.712922i
$808$		0			0
$809$	−8.79845e12	−	1.52394e13i	−0.722167	−	1.25083i	−0.960129	−	0.279556i	$-0.909813\pi$
$809$	−8.79845e12	−	1.52394e13i	−0.722167	−	1.25083i	0.237962	−	0.971274i	$-0.423520\pi$
$810$		0			0
$811$	−1.80406e13			−1.46439			−0.732195	−	0.681096i	$-0.761504\pi$
$811$	−1.80406e13			−1.46439			−0.732195	+	0.681096i	$0.761504\pi$
$812$		0			0
$813$	−1.37702e13			−1.10543
$814$		0			0
$815$	8.99615e11	+	1.55818e12i	0.0714246	+	0.123711i
$816$		0			0
$817$	7.50701e12	+	4.33418e12i	0.589478	+	0.340335i
$818$		0			0
$819$	1.13622e12	+	3.72221e12i	0.0882441	+	0.289084i
$820$		0			0
$821$	−8.22389e11	+	1.42442e12i	−0.0631732	+	0.109419i	−0.895882	−	0.444292i	$-0.853455\pi$
$821$	−8.22389e11	+	1.42442e12i	−0.0631732	+	0.109419i	0.832709	+	0.553711i	$0.186789\pi$
$822$		0			0
$823$	−1.14442e13	+	6.60733e12i	−0.869536	+	0.502027i	−0.867194	−	0.497970i	$-0.834079\pi$
$823$	−1.14442e13	+	6.60733e12i	−0.869536	+	0.502027i	−0.00234191	+	0.999997i	$0.500745\pi$
$824$		0			0
$825$			9.35693e11i			0.0703219i
$826$		0			0
$827$			1.40536e12i			0.104475i	0.998635	+	0.0522375i	$0.0166353\pi$
$827$			1.40536e12i			0.104475i	−0.998635	+	0.0522375i	$0.983365\pi$
$828$		0			0
$829$	4.30907e12	−	2.48784e12i	0.316875	−	0.182948i	−0.333124	−	0.942883i	$-0.608103\pi$
$829$	4.30907e12	−	2.48784e12i	0.316875	−	0.182948i	0.649999	+	0.759935i	$0.274769\pi$
$830$		0			0
$831$	9.32373e12	−	1.61492e13i	0.678242	−	1.17475i
$832$		0			0
$833$	−3.12005e12	−	4.63437e12i	−0.224522	−	0.333494i
$834$		0			0
$835$	−5.64645e12	−	3.25998e12i	−0.401963	−	0.232074i
$836$		0			0
$837$	1.97250e12	+	3.41647e12i	0.138916	+	0.240609i
$838$		0			0
$839$	5.65499e12			0.394006			0.197003	−	0.980403i	$-0.436879\pi$
$839$	5.65499e12			0.394006			0.197003	+	0.980403i	$0.436879\pi$
$840$		0			0
$841$	3.01408e13			2.07765
$842$		0			0
$843$	7.21445e12	+	1.24958e13i	0.492016	+	0.852197i
$844$		0			0
$845$	−1.65083e12	−	9.53108e11i	−0.111390	−	0.0643112i
$846$		0			0
$847$	−1.40339e13	+	4.28391e12i	−0.936921	+	0.286000i
$848$		0			0
$849$	7.50267e12	−	1.29950e13i	0.495600	−	0.858405i
$850$		0			0
$851$	3.75975e13	−	2.17069e13i	2.45740	−	1.41878i
$852$		0			0
$853$		−	1.43204e13i		−	0.926156i	−0.886318	−	0.463078i	$-0.846745\pi$
$853$		−	1.43204e13i		−	0.926156i	0.886318	−	0.463078i	$-0.153255\pi$
$854$		0			0
$855$		−	5.73948e12i		−	0.367303i
$856$		0			0
$857$	−2.23150e13	+	1.28836e13i	−1.41314	+	0.815874i	−0.995682	−	0.0928245i	$-0.970410\pi$
$857$	−2.23150e13	+	1.28836e13i	−1.41314	+	0.815874i	−0.417453	+	0.908699i	$0.637077\pi$
$858$		0			0
$859$	−4.71934e12	+	8.17414e12i	−0.295741	+	0.512239i	−0.975157	−	0.221515i	$-0.928900\pi$
$859$	−4.71934e12	+	8.17414e12i	−0.295741	+	0.512239i	0.679416	+	0.733754i	$0.262233\pi$
$860$		0			0
$861$	−9.80578e12	−	2.26834e12i	−0.608090	−	0.140667i
$862$		0			0
$863$	−1.21515e11	−	7.01565e10i	−0.00745728	−	0.00430546i	0.496267	−	0.868170i	$-0.334704\pi$
$863$	−1.21515e11	−	7.01565e10i	−0.00745728	−	0.00430546i	−0.503724	+	0.863865i	$0.668037\pi$
$864$		0			0
$865$	−1.90496e12	−	3.29949e12i	−0.115695	−	0.200389i
$866$		0			0
$867$	1.60965e13			0.967489
$868$		0			0
$869$	−3.47119e12			−0.206485
$870$		0			0
$871$	7.00945e11	+	1.21407e12i	0.0412669	+	0.0714764i
$872$		0			0
$873$	−2.59935e12	−	1.50073e12i	−0.151461	−	0.0874459i
$874$		0			0
$875$	−1.27806e13	+	1.36935e13i	−0.737078	+	0.789730i
$876$		0			0
$877$	6.84426e12	−	1.18546e13i	0.390686	−	0.676689i	−0.601854	−	0.798606i	$-0.705571\pi$
$877$	6.84426e12	−	1.18546e13i	0.390686	−	0.676689i	0.992540	+	0.121918i	$0.0389044\pi$
$878$		0			0
$879$	−9.31839e12	+	5.37997e12i	−0.526491	+	0.303970i
$880$		0			0
$881$		−	1.95110e13i		−	1.09116i	−0.838060	−	0.545579i	$-0.816310\pi$
$881$		−	1.95110e13i		−	1.09116i	0.838060	−	0.545579i	$-0.183690\pi$
$882$		0			0
$883$			2.17674e13i			1.20499i	0.798123	+	0.602494i	$0.205826\pi$
$883$			2.17674e13i			1.20499i	−0.798123	+	0.602494i	$0.794174\pi$
$884$		0			0
$885$	−6.71665e12	+	3.87786e12i	−0.368051	+	0.212494i
$886$		0			0
$887$	1.08701e13	−	1.88275e13i	0.589625	−	1.02126i	−0.404656	−	0.914469i	$-0.632609\pi$
$887$	1.08701e13	−	1.88275e13i	0.589625	−	1.02126i	0.994281	−	0.106792i	$-0.0340578\pi$
$888$		0			0
$889$	−3.75405e12	+	4.02221e12i	−0.201578	+	0.215977i
$890$		0			0
$891$	2.84285e12	+	1.64132e12i	0.151114	+	0.0872456i
$892$		0			0
$893$	−1.05594e13	−	1.82895e13i	−0.555659	−	0.962430i
$894$		0			0
$895$	9.18528e12			0.478508
$896$		0			0
$897$	3.96952e13			2.04725
$898$		0			0
$899$	−6.18906e12	−	1.07198e13i	−0.316014	−	0.547352i
$900$		0			0
$901$	−1.94226e12	−	1.12137e12i	−0.0981854	−	0.0566874i
$902$		0			0
$903$	1.04569e13	+	2.41896e12i	0.523370	+	0.121069i
$904$		0			0
$905$	−9.18034e12	+	1.59008e13i	−0.454925	+	0.787954i
$906$		0			0
$907$	2.75517e13	−	1.59070e13i	1.35181	−	0.780469i	0.363308	−	0.931669i	$-0.381647\pi$
$907$	2.75517e13	−	1.59070e13i	1.35181	−	0.780469i	0.988503	+	0.151200i	$0.0483139\pi$
$908$		0			0
$909$			7.29437e10i			0.00354365i
$910$		0			0
$911$		−	1.13495e12i		−	0.0545941i	−0.999627	−	0.0272971i	$-0.991310\pi$
$911$		−	1.13495e12i		−	0.0545941i	0.999627	−	0.0272971i	$-0.00869000\pi$
$912$		0			0
$913$	−3.37729e12	+	1.94988e12i	−0.160861	+	0.0928730i
$914$		0			0
$915$	−8.16182e12	+	1.41367e13i	−0.384939	+	0.666735i
$916$		0			0
$917$	1.10761e13	−	3.38102e12i	0.517277	−	0.157901i
$918$		0			0
$919$	2.08232e13	+	1.20223e13i	0.963004	+	0.555991i	0.897096	−	0.441835i	$-0.145672\pi$
$919$	2.08232e13	+	1.20223e13i	0.963004	+	0.555991i	0.0659079	+	0.997826i	$0.479006\pi$
$920$		0			0
$921$	−8.76972e12	−	1.51896e13i	−0.401622	−	0.695630i
$922$		0			0
$923$	8.94302e12			0.405580
$924$		0			0
$925$	1.38440e13			0.621763
$926$		0			0
$927$	−2.37078e12	−	4.10632e12i	−0.105447	−	0.182639i
$928$		0			0
$929$	−2.53437e13	−	1.46322e13i	−1.11635	−	0.644522i	−0.175880	−	0.984412i	$-0.556277\pi$
$929$	−2.53437e13	−	1.46322e13i	−1.11635	−	0.644522i	−0.940465	+	0.339889i	$0.889610\pi$
$930$		0			0
$931$	3.01139e13	+	1.47200e13i	1.31369	+	0.642147i
$932$		0			0
$933$	−5.15522e12	+	8.92911e12i	−0.222731	+	0.385781i
$934$		0			0
$935$	8.79967e11	−	5.08049e11i	0.0376543	−	0.0217397i
$936$		0			0
$937$			9.07724e11i			0.0384703i	0.999815	+	0.0192351i	$0.00612312\pi$
$937$			9.07724e11i			0.0384703i	−0.999815	+	0.0192351i	$0.993877\pi$
$938$		0			0
$939$			3.20390e13i			1.34488i
$940$		0			0
$941$	2.11094e13	−	1.21875e13i	0.877653	−	0.506713i	0.00776889	−	0.999970i	$-0.497527\pi$
$941$	2.11094e13	−	1.21875e13i	0.877653	−	0.506713i	0.869884	+	0.493257i	$0.164194\pi$
$942$		0			0
$943$	−1.27861e13	+	2.21462e13i	−0.526546	+	0.912005i
$944$		0			0
$945$	4.17953e12	+	1.36919e13i	0.170484	+	0.558498i
$946$		0			0
$947$	3.86617e12	+	2.23213e12i	0.156209	+	0.0901872i	0.576067	−	0.817403i	$-0.304587\pi$
$947$	3.86617e12	+	2.23213e12i	0.156209	+	0.0901872i	−0.419858	+	0.907590i	$0.637920\pi$
$948$		0			0
$949$	9.65904e12	+	1.67299e13i	0.386577	+	0.669571i
$950$		0			0
$951$	−2.32161e13			−0.920401
$952$		0			0
$953$	−8.56121e12			−0.336215			−0.168107	−	0.985769i	$-0.553766\pi$
$953$	−8.56121e12			−0.336215			−0.168107	+	0.985769i	$0.553766\pi$
$954$		0			0
$955$	6.62908e12	+	1.14819e13i	0.257892	+	0.446682i
$956$		0			0
$957$	−6.49781e12	−	3.75151e12i	−0.250417	−	0.144578i
$958$		0			0
$959$	2.14745e12	−	9.28318e12i	0.0819858	−	0.354416i
$960$		0			0
$961$	1.15040e13	−	1.99255e13i	0.435103	−	0.753621i
$962$		0			0
$963$	1.22100e13	−	7.04942e12i	0.457505	−	0.264141i
$964$		0			0
$965$		−	2.89005e13i		−	1.07284i
$966$		0			0
$967$		−	4.62732e13i		−	1.70181i	−0.525322	−	0.850903i	$-0.676055\pi$
$967$		−	4.62732e13i		−	1.70181i	0.525322	−	0.850903i	$-0.323945\pi$
$968$		0			0
$969$	1.61240e13	−	9.30922e12i	0.587512	−	0.339200i
$970$		0			0
$971$	−8.58114e12	+	1.48630e13i	−0.309784	+	0.536561i	−0.978315	−	0.207123i	$-0.933590\pi$
$971$	−8.58114e12	+	1.48630e13i	−0.309784	+	0.536561i	0.668531	+	0.743684i	$0.266923\pi$
$972$		0			0
$973$	−5.23543e12	−	4.88638e12i	−0.187260	−	0.174775i
$974$		0			0
$975$	1.09623e13	+	6.32910e12i	0.388492	+	0.224296i
$976$		0			0
$977$	−2.75592e13	−	4.77340e13i	−0.967702	−	1.67611i	−0.702173	−	0.712006i	$-0.747787\pi$
$977$	−2.75592e13	−	4.77340e13i	−0.967702	−	1.67611i	−0.265529	−	0.964103i	$-0.585546\pi$
$978$		0			0
$979$	−4.87663e11			−0.0169667
$980$		0			0
$981$	−1.12081e13			−0.386384
$982$		0			0
$983$	−1.40548e13	−	2.43436e13i	−0.480102	−	0.831562i	0.519637	−	0.854387i	$-0.326067\pi$
$983$	−1.40548e13	−	2.43436e13i	−0.480102	−	0.831562i	−0.999739	+	0.0228254i	$0.992734\pi$
$984$		0			0
$985$	3.20188e13	+	1.84861e13i	1.08378	+	0.625722i
$986$		0			0
$987$	−1.91165e13	−	1.78420e13i	−0.641182	−	0.598434i
$988$		0			0
$989$	1.36352e13	−	2.36168e13i	0.453187	−	0.784943i
$990$		0			0
$991$	−1.33850e13	+	7.72782e12i	−0.440845	+	0.254522i	−0.703956	−	0.710244i	$-0.748585\pi$
$991$	−1.33850e13	+	7.72782e12i	−0.440845	+	0.254522i	0.263111	+	0.964766i	$0.415251\pi$
$992$		0			0
$993$			1.37793e12i			0.0449735i
$994$		0			0
$995$		−	3.76502e13i		−	1.21776i
$996$		0			0
$997$	2.00838e12	−	1.15954e12i	0.0643752	−	0.0371670i	−0.467467	−	0.884011i	$-0.654833\pi$
$997$	2.00838e12	−	1.15954e12i	0.0643752	−	0.0371670i	0.531842	+	0.846844i	$0.321500\pi$
$998$		0			0
$999$	1.76900e13	−	3.06400e13i	0.561932	−	0.973294i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.a.47.3	yes	24
4.3	odd	2		112.10.p.c.47.10	yes	24
7.3	odd	6		112.10.p.c.31.10	yes	24
28.3	even	6	inner	112.10.p.a.31.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.a.31.3	✓	24	28.3	even	6	inner
112.10.p.a.47.3	yes	24	1.1	even	1	trivial
112.10.p.c.31.10	yes	24	7.3	odd	6
112.10.p.c.47.10	yes	24	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-80.9515 - 140.212i) q^{3} +(-916.446 - 529.111i) q^{5} +(4644.00 + 4334.38i) q^{7} +(-3264.80 + 5654.80i) q^{9} +(6006.36 - 3467.77i) q^{11} -93825.2i q^{13} +171329. i q^{15} +(-119897. + 69222.8i) q^{17} +(415317. - 719350. i) q^{19} +(231794. - 1.00202e6i) q^{21} +(-2.26305e6 - 1.30657e6i) q^{23} +(-416647. - 721653. i) q^{25} -2.12958e6 q^{27} +6.68191e6 q^{29} +(-926240. - 1.60429e6i) q^{31} +(-972447. - 561443. i) q^{33} +(-1.96261e6 - 6.42942e6i) q^{35} +(-8.30683e6 + 1.43878e7i) q^{37} +(-1.31554e7 + 7.59529e6i) q^{39} -9.78601e6i q^{41} +1.04358e7i q^{43} +(5.98403e6 - 3.45488e6i) q^{45} +(1.27125e7 - 2.20187e7i) q^{47} +(2.77988e6 + 4.02577e7i) q^{49} +(1.94117e7 + 1.12074e7i) q^{51} +(8.09970e6 + 1.40291e7i) q^{53} -7.33934e6 q^{55} -1.34482e8 q^{57} +(2.26340e7 + 3.92032e7i) q^{59} +(8.25119e7 + 4.76383e7i) q^{61} +(-3.96718e7 + 1.21100e7i) q^{63} +(-4.96439e7 + 8.59857e7i) q^{65} +(-1.29397e7 + 7.47075e6i) q^{67} +4.23076e8i q^{69} +9.53158e7i q^{71} +(-1.78310e8 + 1.02947e8i) q^{73} +(-6.74564e7 + 1.16838e8i) q^{75} +(4.29242e7 + 9.92951e6i) q^{77} +(-4.33439e8 - 2.50246e8i) q^{79} +(2.36653e8 + 4.09896e8i) q^{81} -5.62286e8 q^{83} +1.46506e8 q^{85} +(-5.40911e8 - 9.36886e8i) q^{87} +(-6.08933e7 - 3.51568e7i) q^{89} +(4.06674e8 - 4.35724e8i) q^{91} +(-1.49961e8 + 2.59740e8i) q^{93} +(-7.61231e8 + 4.39497e8i) q^{95} +4.59671e8i q^{97} +4.52863e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) \) 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more