LMFDB - Embedded newform 112.8.i.d.65.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,8,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.65"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,-27] 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:	$34.9871228542$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 $ x^10 - x^9 + 342x^8 + 2165x^7 + 113605x^6 + 319380x^5 + 1438128x^4 + 1705752x^3 + 7544016x^2 + 8573040x + 23619600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{4}\cdot 7^{5} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.5
Root			$10.0194 - 17.3541i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.8.i.d.81.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(42.0490 + 72.8309i) q^{3} +(58.1649 - 100.745i) q^{5} +(276.518 + 864.338i) q^{7} +(-2442.73 + 4230.93i) q^{9} +(-3407.25 - 5901.54i) q^{11} -11924.5 q^{13} +9783.10 q^{15} +(-3301.82 - 5718.92i) q^{17} +(-7387.57 + 12795.6i) q^{19} +(-51323.3 + 56483.6i) q^{21} +(-16134.9 + 27946.4i) q^{23} +(32296.2 + 55938.6i) q^{25} -226935. q^{27} -126792. q^{29} +(94753.8 + 164118. i) q^{31} +(286543. - 496307. i) q^{33} +(103161. + 22416.5i) q^{35} +(76077.0 - 131769. i) q^{37} +(-501412. - 868471. i) q^{39} -355426. q^{41} +650136. q^{43} +(284162. + 492184. i) q^{45} +(-50866.2 + 88102.8i) q^{47} +(-670618. + 478010. i) q^{49} +(277676. - 480950. i) q^{51} +(-1.02218e6 - 1.77047e6i) q^{53} -792730. q^{55} -1.24256e6 q^{57} +(-100356. - 173822. i) q^{59} +(-453981. + 786317. i) q^{61} +(-4.33242e6 - 941416. i) q^{63} +(-693586. + 1.20133e6i) q^{65} +(1.11616e6 + 1.93324e6i) q^{67} -2.71382e6 q^{69} -4.16441e6 q^{71} +(1.61704e6 + 2.80080e6i) q^{73} +(-2.71604e6 + 4.70432e6i) q^{75} +(4.15876e6 - 4.57690e6i) q^{77} +(-684840. + 1.18618e6i) q^{79} +(-4.20013e6 - 7.27485e6i) q^{81} +8.14674e6 q^{83} -768201. q^{85} +(-5.33148e6 - 9.23439e6i) q^{87} +(1.77793e6 - 3.07947e6i) q^{89} +(-3.29733e6 - 1.03068e7i) q^{91} +(-7.96860e6 + 1.38020e7i) q^{93} +(859394. + 1.48851e6i) q^{95} +1.48623e7 q^{97} +3.32920e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) $ 10 * q - 27 * q^3 + 249 * q^5 - 332 * q^7 - 5702 * q^9 - 6399 * q^11 - 26988 * q^13 - 19294 * q^15 + 3609 * q^17 + 12403 * q^19 + 16099 * q^21 + 13959 * q^23 - 162364 * q^25 - 161550 * q^27 + 26148 * q^29 + 20181 * q^31 + 264239 * q^33 - 791715 * q^35 - 54763 * q^37 - 4750 * q^39 - 1824468 * q^41 + 1938424 * q^43 + 1936430 * q^45 - 1217253 * q^47 + 1722058 * q^49 + 1353871 * q^51 - 2349159 * q^53 - 3123494 * q^55 - 9823194 * q^57 - 2188611 * q^59 + 6107445 * q^61 - 10275990 * q^63 - 3022014 * q^65 + 3171581 * q^67 - 11255938 * q^69 + 18938976 * q^71 + 8034853 * q^73 - 29173588 * q^75 + 24100191 * q^77 + 7589559 * q^79 - 9846485 * q^81 + 13452408 * q^83 - 48717670 * q^85 - 30033142 * q^87 + 29864757 * q^89 - 34267512 * q^91 - 20217029 * q^93 + 25812009 * q^95 + 285580 * q^97 + 119277812 * q^99

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	42.0490	+	72.8309i	0.899148	+	1.55737i	0.828586	+	0.559862i	$0.189146\pi$
$3$	42.0490	+	72.8309i	0.899148	+	1.55737i	0.0705614	+	0.997507i	$0.477521\pi$
$4$		0			0
$5$	58.1649	−	100.745i	0.208097	−	0.360435i	−0.743018	−	0.669271i	$-0.766606\pi$
$5$	58.1649	−	100.745i	0.208097	−	0.360435i	0.951115	+	0.308837i	$0.0999397\pi$
$6$		0			0
$7$	276.518	+	864.338i	0.304706	+	0.952447i
$7$	276.518	+	864.338i	0.304706	+	0.952447i
$8$		0			0
$9$	−2442.73	+	4230.93i	−1.11693	+	1.93458i
$10$		0			0
$11$	−3407.25	−	5901.54i	−0.771845	−	1.33687i	−0.936551	−	0.350532i	$-0.886001\pi$
$11$	−3407.25	−	5901.54i	−0.771845	−	1.33687i	0.164706	−	0.986343i	$-0.447332\pi$
$12$		0			0
$13$	−11924.5			−1.50535			−0.752675	−	0.658392i	$-0.771237\pi$
$13$	−11924.5			−1.50535			−0.752675	+	0.658392i	$0.771237\pi$
$14$		0			0
$15$	9783.10			0.748440
$16$		0			0
$17$	−3301.82	−	5718.92i	−0.162998	−	0.282321i	0.772944	−	0.634474i	$-0.218783\pi$
$17$	−3301.82	−	5718.92i	−0.162998	−	0.282321i	−0.935943	+	0.352153i	$0.885450\pi$
$18$		0			0
$19$	−7387.57	+	12795.6i	−0.247095	+	0.427981i	−0.962719	−	0.270505i	$-0.912809\pi$
$19$	−7387.57	+	12795.6i	−0.247095	+	0.427981i	0.715624	+	0.698486i	$0.246143\pi$
$20$		0			0
$21$	−51323.3	+	56483.6i	−1.20934	+	1.33093i
$22$		0			0
$23$	−16134.9	+	27946.4i	−0.276515	+	0.478938i	−0.970516	−	0.241036i	$-0.922513\pi$
$23$	−16134.9	+	27946.4i	−0.276515	+	0.478938i	0.694001	+	0.719974i	$0.255846\pi$
$24$		0			0
$25$	32296.2	+	55938.6i	0.413391	+	0.716015i
$26$		0			0
$27$	−226935.			−2.21885
$28$		0			0
$29$	−126792.			−0.965382			−0.482691	−	0.875791i	$-0.660341\pi$
$29$	−126792.			−0.965382			−0.482691	+	0.875791i	$0.660341\pi$
$30$		0			0
$31$	94753.8	+	164118.i	0.571256	+	0.989444i	0.996437	+	0.0843359i	$0.0268769\pi$
$31$	94753.8	+	164118.i	0.571256	+	0.989444i	−0.425182	+	0.905108i	$0.639790\pi$
$32$		0			0
$33$	286543.	−	496307.i	1.38800	−	2.40409i
$34$		0			0
$35$	103161.	+	22416.5i	0.406703	+	0.0883749i
$36$		0			0
$37$	76077.0	−	131769.i	0.246915	−	0.427669i	−0.715753	−	0.698353i	$-0.753917\pi$
$37$	76077.0	−	131769.i	0.246915	−	0.427669i	0.962668	+	0.270684i	$0.0872499\pi$
$38$		0			0
$39$	−501412.	−	868471.i	−1.35353	−	2.34439i
$40$		0			0
$41$	−355426.			−0.805389			−0.402694	−	0.915334i	$-0.631926\pi$
$41$	−355426.			−0.805389			−0.402694	+	0.915334i	$0.631926\pi$
$42$		0			0
$43$	650136.			1.24699			0.623497	−	0.781825i	$-0.285711\pi$
$43$	650136.			1.24699			0.623497	+	0.781825i	$0.285711\pi$
$44$		0			0
$45$	284162.	+	492184.i	0.464861	+	0.805162i
$46$		0			0
$47$	−50866.2	+	88102.8i	−0.0714638	+	0.123779i	−0.899543	−	0.436832i	$-0.856100\pi$
$47$	−50866.2	+	88102.8i	−0.0714638	+	0.123779i	0.828079	+	0.560611i	$0.189434\pi$
$48$		0			0
$49$	−670618.	+	478010.i	−0.814309	+	0.580432i
$50$		0			0
$51$	277676.	−	480950.i	0.293119	−	0.507696i
$52$		0			0
$53$	−1.02218e6	−	1.77047e6i	−0.943111	−	1.63352i	−0.759490	−	0.650519i	$-0.774552\pi$
$53$	−1.02218e6	−	1.77047e6i	−0.943111	−	1.63352i	−0.183620	−	0.982997i	$-0.558782\pi$
$54$		0			0
$55$	−792730.			−0.642475
$56$		0			0
$57$	−1.24256e6			−0.888699
$58$		0			0
$59$	−100356.	−	173822.i	−0.0636153	−	0.110185i	0.832464	−	0.554080i	$-0.186930\pi$
$59$	−100356.	−	173822.i	−0.0636153	−	0.110185i	−0.896079	+	0.443895i	$0.853596\pi$
$60$		0			0
$61$	−453981.	+	786317.i	−0.256084	+	0.443551i	−0.965189	−	0.261552i	$-0.915766\pi$
$61$	−453981.	+	786317.i	−0.256084	+	0.443551i	0.709105	+	0.705103i	$0.249099\pi$
$62$		0			0
$63$	−4.33242e6	−	941416.i	−2.18292	−	0.474340i
$64$		0			0
$65$	−693586.	+	1.20133e6i	−0.313259	+	0.542580i
$66$		0			0
$67$	1.11616e6	+	1.93324e6i	0.453382	+	0.785280i	0.998594	−	0.0530181i	$-0.0168841\pi$
$67$	1.11616e6	+	1.93324e6i	0.453382	+	0.785280i	−0.545212	+	0.838298i	$0.683551\pi$
$68$		0			0
$69$	−2.71382e6			−0.994511
$70$		0			0
$71$	−4.16441e6			−1.38086			−0.690430	−	0.723399i	$-0.742579\pi$
$71$	−4.16441e6			−1.38086			−0.690430	+	0.723399i	$0.742579\pi$
$72$		0			0
$73$	1.61704e6	+	2.80080e6i	0.486510	+	0.842660i	0.999880	−	0.0155074i	$-0.00493635\pi$
$73$	1.61704e6	+	2.80080e6i	0.486510	+	0.842660i	−0.513370	+	0.858168i	$0.671603\pi$
$74$		0			0
$75$	−2.71604e6	+	4.70432e6i	−0.743399	+	1.28761i
$76$		0			0
$77$	4.15876e6	−	4.57690e6i	1.03812	−	1.14249i
$78$		0			0
$79$	−684840.	+	1.18618e6i	−0.156277	+	0.270679i	−0.933523	−	0.358517i	$-0.883283\pi$
$79$	−684840.	+	1.18618e6i	−0.156277	+	0.270679i	0.777246	+	0.629196i	$0.216616\pi$
$80$		0			0
$81$	−4.20013e6	−	7.27485e6i	−0.878144	−	1.52099i
$82$		0			0
$83$	8.14674e6			1.56391			0.781953	−	0.623337i	$-0.214224\pi$
$83$	8.14674e6			1.56391			0.781953	+	0.623337i	$0.214224\pi$
$84$		0			0
$85$	−768201.			−0.135678
$86$		0			0
$87$	−5.33148e6	−	9.23439e6i	−0.868021	−	1.50346i
$88$		0			0
$89$	1.77793e6	−	3.07947e6i	0.267331	−	0.463031i	−0.700840	−	0.713318i	$-0.747192\pi$
$89$	1.77793e6	−	3.07947e6i	0.267331	−	0.463031i	0.968172	+	0.250287i	$0.0805249\pi$
$90$		0			0
$91$	−3.29733e6	−	1.03068e7i	−0.458689	−	1.43377i
$92$		0			0
$93$	−7.96860e6	+	1.38020e7i	−1.02729	+	1.77931i
$94$		0			0
$95$	859394.	+	1.48851e6i	0.102839	+	0.178123i
$96$		0			0
$97$	1.48623e7			1.65343			0.826715	−	0.562622i	$-0.190207\pi$
$97$	1.48623e7			1.65343			0.826715	+	0.562622i	$0.190207\pi$
$98$		0			0
$99$	3.32920e7			3.44839
$100$		0			0
$101$	−1.42329e6	−	2.46521e6i	−0.137458	−	0.238084i	0.789076	−	0.614296i	$-0.210560\pi$
$101$	−1.42329e6	−	2.46521e6i	−0.137458	−	0.238084i	−0.926534	+	0.376212i	$0.877226\pi$
$102$		0			0
$103$	−297848.	+	515888.i	−0.0268574	+	0.0465184i	−0.879142	−	0.476560i	$-0.841883\pi$
$103$	−297848.	+	515888.i	−0.0268574	+	0.0465184i	0.852284	+	0.523079i	$0.175217\pi$
$104$		0			0
$105$	2.70520e6	+	8.45591e6i	0.228054	+	0.712849i
$106$		0			0
$107$	−9.93320e6	+	1.72048e7i	−0.783873	+	1.35771i	0.145797	+	0.989315i	$0.453425\pi$
$107$	−9.93320e6	+	1.72048e7i	−0.783873	+	1.35771i	−0.929670	+	0.368393i	$0.879908\pi$
$108$		0			0
$109$	3.92925e6	+	6.80565e6i	0.290614	+	0.503358i	0.973955	−	0.226741i	$-0.0728071\pi$
$109$	3.92925e6	+	6.80565e6i	0.290614	+	0.503358i	−0.683341	+	0.730099i	$0.739474\pi$
$110$		0			0
$111$	1.27958e7			0.888051
$112$		0			0
$113$	−1.16317e7			−0.758347			−0.379174	−	0.925325i	$-0.623792\pi$
$113$	−1.16317e7			−0.758347			−0.379174	+	0.925325i	$0.623792\pi$
$114$		0			0
$115$	1.87697e6	+	3.25100e6i	0.115084	+	0.199331i
$116$		0			0
$117$	2.91283e7	−	5.04517e7i	1.68137	−	2.91223i
$118$		0			0
$119$	4.03007e6	−	4.43528e6i	0.219229	−	0.241272i
$120$		0			0
$121$	−1.34752e7	+	2.33397e7i	−0.691489	+	1.19769i
$122$		0			0
$123$	−1.49453e7	−	2.58860e7i	−0.724163	−	1.25429i
$124$		0			0
$125$	1.66023e7			0.760296
$126$		0			0
$127$	−1.02118e6			−0.0442372			−0.0221186	−	0.999755i	$-0.507041\pi$
$127$	−1.02118e6			−0.0442372			−0.0221186	+	0.999755i	$0.507041\pi$
$128$		0			0
$129$	2.73375e7	+	4.73500e7i	1.12123	+	1.94203i
$130$		0			0
$131$	−2.41713e7	+	4.18660e7i	−0.939401	+	1.62709i	−0.172810	+	0.984955i	$0.555285\pi$
$131$	−2.41713e7	+	4.18660e7i	−0.939401	+	1.62709i	−0.766591	+	0.642135i	$0.778049\pi$
$132$		0			0
$133$	−1.31026e7	−	2.84713e6i	−0.482920	−	0.104937i
$134$		0			0
$135$	−1.31997e7	+	2.28625e7i	−0.461737	+	0.799752i
$136$		0			0
$137$	−1.23915e7	−	2.14626e7i	−0.411719	−	0.713118i	0.583359	−	0.812214i	$-0.301738\pi$
$137$	−1.23915e7	−	2.14626e7i	−0.411719	−	0.713118i	−0.995078	+	0.0990967i	$0.968405\pi$
$138$		0			0
$139$	1.93687e7			0.611713			0.305857	−	0.952078i	$-0.401057\pi$
$139$	1.93687e7			0.611713			0.305857	+	0.952078i	$0.401057\pi$
$140$		0			0
$141$	−8.55548e6			−0.257026
$142$		0			0
$143$	4.06297e7	+	7.03727e7i	1.16190	+	2.01246i
$144$		0			0
$145$	−7.37485e6	+	1.27736e7i	−0.200893	+	0.347957i
$146$		0			0
$147$	−6.30128e7	−	2.87419e7i	−1.63613	−	0.746286i
$148$		0			0
$149$	−8.53357e6	+	1.47806e7i	−0.211339	+	0.366049i	−0.952134	−	0.305682i	$-0.901116\pi$
$149$	−8.53357e6	+	1.47806e7i	−0.211339	+	0.366049i	0.740795	+	0.671731i	$0.234449\pi$
$150$		0			0
$151$	1.31225e7	+	2.27288e7i	0.310167	+	0.537225i	0.978398	−	0.206728i	$-0.0662817\pi$
$151$	1.31225e7	+	2.27288e7i	0.310167	+	0.537225i	−0.668231	+	0.743954i	$0.732948\pi$
$152$		0			0
$153$	3.22619e7			0.728231
$154$		0			0
$155$	2.20454e7			0.475507
$156$		0			0
$157$	2.04968e7	+	3.55016e7i	0.422706	+	0.732148i	0.996203	−	0.0870598i	$-0.0277471\pi$
$157$	2.04968e7	+	3.55016e7i	0.422706	+	0.732148i	−0.573498	+	0.819207i	$0.694414\pi$
$158$		0			0
$159$	8.59634e7	−	1.48893e8i	1.69599	−	2.93754i
$160$		0			0
$161$	−2.86168e7	−	6.21830e6i	−0.540418	−	0.117431i
$162$		0			0
$163$	2.61227e7	−	4.52458e7i	0.472455	−	0.818317i	−0.527048	−	0.849836i	$-0.676701\pi$
$163$	2.61227e7	−	4.52458e7i	0.472455	−	0.818317i	0.999503	+	0.0315189i	$0.0100344\pi$
$164$		0			0
$165$	−3.33335e7	−	5.77353e7i	−0.577680	−	1.00057i
$166$		0			0
$167$	5.67268e7			0.942498			0.471249	−	0.882000i	$-0.343803\pi$
$167$	5.67268e7			0.942498			0.471249	+	0.882000i	$0.343803\pi$
$168$		0			0
$169$	7.94446e7			1.26608
$170$		0			0
$171$	−3.60917e7	−	6.25126e7i	−0.551977	−	0.956052i
$172$		0			0
$173$	774366.	−	1.34124e6i	0.0113706	−	0.0196945i	−0.860284	−	0.509815i	$-0.829714\pi$
$173$	774366.	−	1.34124e6i	0.0113706	−	0.0196945i	0.871655	+	0.490120i	$0.163047\pi$
$174$		0			0
$175$	−3.94194e7	+	4.33829e7i	−0.556003	+	0.611907i
$176$		0			0
$177$	8.43974e6	−	1.46181e7i	0.114399	−	0.198145i
$178$		0			0
$179$	1.87023e7	+	3.23933e7i	0.243730	+	0.422153i	0.961774	−	0.273845i	$-0.0882955\pi$
$179$	1.87023e7	+	3.23933e7i	0.243730	+	0.422153i	−0.718044	+	0.695998i	$0.754962\pi$
$180$		0			0
$181$	−1.08072e8			−1.35469			−0.677343	−	0.735667i	$-0.736869\pi$
$181$	−1.08072e8			−1.35469			−0.677343	+	0.735667i	$0.736869\pi$
$182$		0			0
$183$	−7.63576e7			−0.921030
$184$		0			0
$185$	−8.85002e6	−	1.53287e7i	−0.102765	−	0.177993i
$186$		0			0
$187$	−2.25003e7	+	3.89716e7i	−0.251618	+	0.435816i
$188$		0			0
$189$	−6.27517e7	−	1.96149e8i	−0.676097	−	2.11334i
$190$		0			0
$191$	−5.76760e7	+	9.98978e7i	−0.598933	+	1.03738i	0.394046	+	0.919091i	$0.371075\pi$
$191$	−5.76760e7	+	9.98978e7i	−0.598933	+	1.03738i	−0.992979	+	0.118292i	$0.962258\pi$
$192$		0			0
$193$	4.10645e7	+	7.11257e7i	0.411164	+	0.712158i	0.995017	−	0.0997017i	$-0.0317888\pi$
$193$	4.10645e7	+	7.11257e7i	0.411164	+	0.712158i	−0.583853	+	0.811859i	$0.698456\pi$
$194$		0			0
$195$	−1.16658e8			−1.12666
$196$		0			0
$197$	1.32327e8			1.23315			0.616576	−	0.787296i	$-0.288519\pi$
$197$	1.32327e8			1.23315			0.616576	+	0.787296i	$0.288519\pi$
$198$		0			0
$199$	3.70915e7	+	6.42444e7i	0.333648	+	0.577896i	0.983224	−	0.182401i	$-0.0583869\pi$
$199$	3.70915e7	+	6.42444e7i	0.333648	+	0.577896i	−0.649576	+	0.760297i	$0.725054\pi$
$200$		0			0
$201$	−9.38666e7	+	1.62582e8i	−0.815314	+	1.41217i
$202$		0			0
$203$	−3.50603e7	−	1.09591e8i	−0.294157	−	0.919475i
$204$		0			0
$205$	−2.06733e7	+	3.58072e7i	−0.167599	+	0.290290i
$206$		0			0
$207$	−7.88264e7	−	1.36531e8i	−0.617697	−	1.06988i
$208$		0			0
$209$	1.00685e8			0.762876
$210$		0			0
$211$	−5.84260e7			−0.428171			−0.214086	−	0.976815i	$-0.568677\pi$
$211$	−5.84260e7			−0.428171			−0.214086	+	0.976815i	$0.568677\pi$
$212$		0			0
$213$	−1.75109e8	−	3.03298e8i	−1.24160	−	2.15051i
$214$		0			0
$215$	3.78151e7	−	6.54977e7i	0.259496	−	0.449460i
$216$		0			0
$217$	−1.15653e8	+	1.27281e8i	−0.768328	+	0.845580i
$218$		0			0
$219$	−1.35990e8	+	2.35542e8i	−0.874889	+	1.51535i
$220$		0			0
$221$	3.93725e7	+	6.81952e7i	0.245369	+	0.424992i
$222$		0			0
$223$	−2.61801e8			−1.58090			−0.790449	−	0.612528i	$-0.790153\pi$
$223$	−2.61801e8			−1.58090			−0.790449	+	0.612528i	$0.790153\pi$
$224$		0			0
$225$	−3.15564e8			−1.84692
$226$		0			0
$227$	−2.89678e7	−	5.01737e7i	−0.164371	−	0.284699i	0.772061	−	0.635549i	$-0.219226\pi$
$227$	−2.89678e7	−	5.01737e7i	−0.164371	−	0.284699i	−0.936432	+	0.350850i	$0.885893\pi$
$228$		0			0
$229$	8.34771e7	−	1.44587e8i	0.459349	−	0.795616i	−0.539577	−	0.841936i	$-0.681416\pi$
$229$	8.34771e7	−	1.44587e8i	0.459349	−	0.795616i	0.998927	+	0.0463198i	$0.0147493\pi$
$230$		0			0
$231$	5.08211e8	+	1.10432e8i	2.71270	+	0.589459i
$232$		0			0
$233$	−1.08941e8	+	1.88691e8i	−0.564217	+	0.977252i	0.432906	+	0.901439i	$0.357488\pi$
$233$	−1.08941e8	+	1.88691e8i	−0.564217	+	0.977252i	−0.997122	+	0.0758125i	$0.975845\pi$
$234$		0			0
$235$	5.91725e6	+	1.02490e7i	0.0297428	+	0.0515161i
$236$		0			0
$237$	−1.15187e8			−0.562064
$238$		0			0
$239$	−2.23970e8			−1.06120			−0.530600	−	0.847622i	$-0.678033\pi$
$239$	−2.23970e8			−1.06120			−0.530600	+	0.847622i	$0.678033\pi$
$240$		0			0
$241$	−4.93506e7	−	8.54778e7i	−0.227108	−	0.393363i	0.729842	−	0.683616i	$-0.239594\pi$
$241$	−4.93506e7	−	8.54778e7i	−0.227108	−	0.393363i	−0.956950	+	0.290253i	$0.906260\pi$
$242$		0			0
$243$	1.05069e8	−	1.81985e8i	0.469735	−	0.813605i
$244$		0			0
$245$	9.15049e6	+	9.53646e7i	0.0397524	+	0.414291i
$246$		0			0
$247$	8.80929e7	−	1.52581e8i	0.371964	−	0.644261i
$248$		0			0
$249$	3.42562e8	+	5.93335e8i	1.40618	+	2.43558i
$250$		0			0
$251$	2.65751e8			1.06076			0.530379	−	0.847760i	$-0.322050\pi$
$251$	2.65751e8			1.06076			0.530379	+	0.847760i	$0.322050\pi$
$252$		0			0
$253$	2.19903e8			0.853706
$254$		0			0
$255$	−3.23020e7	−	5.59488e7i	−0.121994	−	0.211300i
$256$		0			0
$257$	−1.52841e7	+	2.64728e7i	−0.0561660	+	0.0972824i	−0.892741	−	0.450570i	$-0.851221\pi$
$257$	−1.52841e7	+	2.64728e7i	−0.0561660	+	0.0972824i	0.836575	+	0.547852i	$0.184554\pi$
$258$		0			0
$259$	1.34930e8	+	2.93197e7i	0.482568	+	0.104860i
$260$		0			0
$261$	3.09719e8	−	5.36449e8i	1.07827	−	1.86761i
$262$		0			0
$263$	−3.61770e6	−	6.26604e6i	−0.0122627	−	0.0212397i	0.859829	−	0.510582i	$-0.170570\pi$
$263$	−3.61770e6	−	6.26604e6i	−0.0122627	−	0.0212397i	−0.872092	+	0.489343i	$0.837237\pi$
$264$		0			0
$265$	−2.37820e8			−0.785034
$266$		0			0
$267$	2.99041e8			0.961481
$268$		0			0
$269$	−3.42076e7	−	5.92494e7i	−0.107150	−	0.185588i	0.807465	−	0.589916i	$-0.200839\pi$
$269$	−3.42076e7	−	5.92494e7i	−0.107150	−	0.185588i	−0.914614	+	0.404327i	$0.867506\pi$
$270$		0			0
$271$	−4.88220e7	+	8.45622e7i	−0.149013	+	0.258097i	−0.930863	−	0.365369i	$-0.880943\pi$
$271$	−4.88220e7	+	8.45622e7i	−0.149013	+	0.258097i	0.781850	+	0.623466i	$0.214276\pi$
$272$		0			0
$273$	6.12003e8	−	6.73537e8i	1.82047	−	2.00351i
$274$		0			0
$275$	2.20083e8	−	3.81194e8i	0.638148	−	1.10530i
$276$		0			0
$277$	−1.93980e8	−	3.35983e8i	−0.548375	−	0.949814i	−0.998386	−	0.0567907i	$-0.981913\pi$
$277$	−1.93980e8	−	3.35983e8i	−0.548375	−	0.949814i	0.450011	−	0.893023i	$-0.351420\pi$
$278$		0			0
$279$	−9.25832e8			−2.55222
$280$		0			0
$281$	4.44469e8			1.19500			0.597502	−	0.801868i	$-0.296160\pi$
$281$	4.44469e8			1.19500			0.597502	+	0.801868i	$0.296160\pi$
$282$		0			0
$283$	3.11056e8	+	5.38765e8i	0.815805	+	1.41302i	0.908748	+	0.417345i	$0.137039\pi$
$283$	3.11056e8	+	5.38765e8i	0.815805	+	1.41302i	−0.0929429	+	0.995671i	$0.529627\pi$
$284$		0			0
$285$	−7.22733e7	+	1.25181e8i	−0.184936	+	0.320318i
$286$		0			0
$287$	−9.82817e7	−	3.07208e8i	−0.245407	−	0.767090i
$288$		0			0
$289$	1.83365e8	−	3.17598e8i	0.446863	−	0.773990i
$290$		0			0
$291$	6.24945e8	+	1.08244e9i	1.48668	+	2.57500i
$292$		0			0
$293$	−3.26197e8			−0.757606			−0.378803	−	0.925477i	$-0.623664\pi$
$293$	−3.26197e8			−0.757606			−0.378803	+	0.925477i	$0.623664\pi$
$294$		0			0
$295$	−2.33488e7			−0.0529526
$296$		0			0
$297$	7.73225e8	+	1.33927e9i	1.71261	+	2.96633i
$298$		0			0
$299$	1.92400e8	−	3.33247e8i	0.416252	−	0.720969i
$300$		0			0
$301$	1.79774e8	+	5.61937e8i	0.379966	+	1.18770i
$302$		0			0
$303$	1.19696e8	−	2.07319e8i	0.247190	−	0.428145i
$304$		0			0
$305$	5.28115e7	+	9.14721e7i	0.106581	+	0.184603i
$306$		0			0
$307$	−2.17327e8			−0.428675			−0.214338	−	0.976760i	$-0.568759\pi$
$307$	−2.17327e8			−0.428675			−0.214338	+	0.976760i	$0.568759\pi$
$308$		0			0
$309$	−5.00968e7			−0.0965952
$310$		0			0
$311$	−1.26348e8	−	2.18842e8i	−0.238182	−	0.412543i	0.722011	−	0.691882i	$-0.243218\pi$
$311$	−1.26348e8	−	2.18842e8i	−0.238182	−	0.412543i	−0.960193	+	0.279339i	$0.909885\pi$
$312$		0			0
$313$	−1.71496e8	+	2.97039e8i	−0.316117	+	0.547531i	−0.979674	−	0.200595i	$-0.935713\pi$
$313$	−1.71496e8	+	2.97039e8i	−0.316117	+	0.547531i	0.663557	+	0.748125i	$0.269046\pi$
$314$		0			0
$315$	−3.46837e8	+	3.81710e8i	−0.625229	+	0.688093i
$316$		0			0
$317$	−1.70616e8	+	2.95516e8i	−0.300824	+	0.521042i	−0.976323	−	0.216319i	$-0.930595\pi$
$317$	−1.70616e8	+	2.95516e8i	−0.300824	+	0.521042i	0.675499	+	0.737361i	$0.263928\pi$
$318$		0			0
$319$	4.32013e8	+	7.48268e8i	0.745125	+	1.29059i
$320$		0			0
$321$	−1.67072e9			−2.81927
$322$		0			0
$323$	9.75697e7			0.161104
$324$		0			0
$325$	−3.85115e8	−	6.67039e8i	−0.622299	−	1.07785i
$326$		0			0
$327$	−3.30441e8	+	5.72341e8i	−0.522610	+	0.905186i
$328$		0			0
$329$	−9.02160e7	−	1.96036e7i	−0.139668	−	0.0303493i
$330$		0			0
$331$	2.70854e8	−	4.69133e8i	0.410523	−	0.711046i	−0.584424	−	0.811448i	$-0.698680\pi$
$331$	2.70854e8	−	4.69133e8i	0.410523	−	0.711046i	0.994947	+	0.100402i	$0.0320130\pi$
$332$		0			0
$333$	3.71671e8	+	6.43753e8i	0.551574	+	0.955355i
$334$		0			0
$335$	2.59685e8			0.377390
$336$		0			0
$337$	5.08120e7			0.0723205			0.0361602	−	0.999346i	$-0.488487\pi$
$337$	5.08120e7			0.0723205			0.0361602	+	0.999346i	$0.488487\pi$
$338$		0			0
$339$	−4.89101e8	−	8.47147e8i	−0.681866	−	1.18103i
$340$		0			0
$341$	6.45700e8	−	1.11839e9i	0.881842	−	1.52739i
$342$		0			0
$343$	−5.98601e8	−	4.47463e8i	−0.800955	−	0.598725i
$344$		0			0
$345$	−1.57849e8	+	2.73403e8i	−0.206955	+	0.358456i
$346$		0			0
$347$	−3.39930e8	−	5.88776e8i	−0.436753	−	0.756479i	0.560684	−	0.828030i	$-0.310538\pi$
$347$	−3.39930e8	−	5.88776e8i	−0.436753	−	0.756479i	−0.997437	+	0.0715511i	$0.977205\pi$
$348$		0			0
$349$	−4.75862e8			−0.599228			−0.299614	−	0.954060i	$-0.596858\pi$
$349$	−4.75862e8			−0.599228			−0.299614	+	0.954060i	$0.596858\pi$
$350$		0			0
$351$	2.70608e9			3.34015
$352$		0			0
$353$	−2.08969e8	−	3.61945e8i	−0.252855	−	0.437957i	0.711456	−	0.702731i	$-0.248036\pi$
$353$	−2.08969e8	−	3.61945e8i	−0.252855	−	0.437957i	−0.964311	+	0.264774i	$0.914703\pi$
$354$		0			0
$355$	−2.42223e8	+	4.19542e8i	−0.287353	+	0.497710i
$356$		0			0
$357$	4.92486e8	+	1.07015e8i	0.572868	+	0.124482i
$358$		0			0
$359$	1.89840e8	−	3.28812e8i	0.216549	−	0.375074i	−0.737201	−	0.675673i	$-0.763853\pi$
$359$	1.89840e8	−	3.28812e8i	0.216549	−	0.375074i	0.953751	+	0.300599i	$0.0971865\pi$
$360$		0			0
$361$	3.37784e8	+	5.85058e8i	0.377888	+	0.654521i
$362$		0			0
$363$	−2.26647e9			−2.48700
$364$		0			0
$365$	3.76221e8			0.404965
$366$		0			0
$367$	−2.24517e8	−	3.88875e8i	−0.237093	−	0.410656i	0.722786	−	0.691072i	$-0.242861\pi$
$367$	−2.24517e8	−	3.88875e8i	−0.237093	−	0.410656i	−0.959879	+	0.280415i	$0.909528\pi$
$368$		0			0
$369$	8.68210e8	−	1.50378e9i	0.899565	−	1.55809i
$370$		0			0
$371$	1.24763e9	−	1.37308e9i	1.26847	−	1.39600i
$372$		0			0
$373$	−1.92208e8	+	3.32913e8i	−0.191774	+	0.332162i	−0.945838	−	0.324638i	$-0.894757\pi$
$373$	−1.92208e8	+	3.32913e8i	−0.191774	+	0.332162i	0.754064	+	0.656801i	$0.228091\pi$
$374$		0			0
$375$	6.98109e8	+	1.20916e9i	0.683618	+	1.18406i
$376$		0			0
$377$	1.51193e9			1.45324
$378$		0			0
$379$	−1.06012e9			−1.00027			−0.500137	−	0.865946i	$-0.666717\pi$
$379$	−1.06012e9			−1.00027			−0.500137	+	0.865946i	$0.666717\pi$
$380$		0			0
$381$	−4.29394e7	−	7.43732e7i	−0.0397758	−	0.0688936i
$382$		0			0
$383$	−4.71522e8	+	8.16701e8i	−0.428851	+	0.742792i	−0.996771	−	0.0802915i	$-0.974415\pi$
$383$	−4.71522e8	+	8.16701e8i	−0.428851	+	0.742792i	0.567920	+	0.823084i	$0.307748\pi$
$384$		0			0
$385$	−2.19204e8	−	6.85187e8i	−0.195766	−	0.611923i
$386$		0			0
$387$	−1.58811e9	+	2.75068e9i	−1.39281	+	2.41242i
$388$		0			0
$389$	−2.74414e8	−	4.75300e8i	−0.236365	−	0.409396i	0.723303	−	0.690530i	$-0.242623\pi$
$389$	−2.74414e8	−	4.75300e8i	−0.236365	−	0.409396i	−0.959669	+	0.281134i	$0.909289\pi$
$390$		0			0
$391$	2.13098e8			0.180286
$392$		0			0
$393$	−4.06552e9			−3.37864
$394$		0			0
$395$	7.96673e7	+	1.37988e8i	0.0650415	+	0.112655i
$396$		0			0
$397$	7.09112e8	−	1.22822e9i	0.568785	−	0.985164i	−0.427902	−	0.903825i	$-0.640747\pi$
$397$	7.09112e8	−	1.22822e9i	0.568785	−	0.985164i	0.996687	−	0.0813387i	$-0.0259196\pi$
$398$		0			0
$399$	−3.43590e8	−	1.07399e9i	−0.270792	−	0.846439i
$400$		0			0
$401$	5.03119e8	−	8.71428e8i	0.389642	−	0.674880i	−0.602759	−	0.797923i	$-0.705932\pi$
$401$	5.03119e8	−	8.71428e8i	0.389642	−	0.674880i	0.992401	+	0.123043i	$0.0392654\pi$
$402$		0			0
$403$	−1.12989e9	−	1.95703e9i	−0.859940	−	1.48946i
$404$		0			0
$405$	−9.77202e8			−0.730957
$406$		0			0
$407$	−1.03685e9			−0.762320
$408$		0			0
$409$	4.83763e8	+	8.37902e8i	0.349624	+	0.605566i	0.986183	−	0.165662i	$-0.0529761\pi$
$409$	4.83763e8	+	8.37902e8i	0.349624	+	0.605566i	−0.636559	+	0.771228i	$0.719643\pi$
$410$		0			0
$411$	1.04210e9	−	1.80496e9i	0.740392	−	1.28240i
$412$		0			0
$413$	1.22491e8	−	1.34806e8i	0.0855613	−	0.0941641i
$414$		0			0
$415$	4.73854e8	−	8.20740e8i	0.325444	−	0.563686i
$416$		0			0
$417$	8.14433e8	+	1.41064e9i	0.550021	+	0.952664i
$418$		0			0
$419$	1.33931e9			0.889473			0.444736	−	0.895662i	$-0.353297\pi$
$419$	1.33931e9			0.889473			0.444736	+	0.895662i	$0.353297\pi$
$420$		0			0
$421$	4.59395e8			0.300054			0.150027	−	0.988682i	$-0.452064\pi$
$421$	4.59395e8			0.300054			0.150027	+	0.988682i	$0.452064\pi$
$422$		0			0
$423$	−2.48505e8	−	4.30423e8i	−0.159641	−	0.276506i
$424$		0			0
$425$	2.13273e8	−	3.69399e8i	0.134764	−	0.233418i
$426$		0			0
$427$	−8.05178e8	−	1.74962e8i	−0.500489	−	0.108754i
$428$		0			0
$429$	−3.41687e9	+	5.91820e9i	−2.08943	+	3.61900i
$430$		0			0
$431$	1.25854e9	+	2.17985e9i	0.757173	+	1.31146i	0.944287	+	0.329123i	$0.106753\pi$
$431$	1.25854e9	+	2.17985e9i	0.757173	+	1.31146i	−0.187114	+	0.982338i	$0.559913\pi$
$432$		0			0
$433$	−1.94780e9			−1.15302			−0.576511	−	0.817090i	$-0.695586\pi$
$433$	−1.94780e9			−1.15302			−0.576511	+	0.817090i	$0.695586\pi$
$434$		0			0
$435$	−1.24042e9			−0.722530
$436$		0			0
$437$	−2.38395e8	−	4.12913e8i	−0.136651	−	0.236686i
$438$		0			0
$439$	6.98148e8	−	1.20923e9i	0.393842	−	0.682154i	−0.599111	−	0.800666i	$-0.704479\pi$
$439$	6.98148e8	−	1.20923e9i	0.393842	−	0.682154i	0.992953	+	0.118512i	$0.0378124\pi$
$440$		0			0
$441$	−3.84290e8	−	4.00499e9i	−0.213365	−	2.22365i
$442$		0			0
$443$	−9.85364e8	+	1.70670e9i	−0.538498	+	0.932705i	0.460488	+	0.887666i	$0.347675\pi$
$443$	−9.85364e8	+	1.70670e9i	−0.538498	+	0.932705i	−0.998985	+	0.0450392i	$0.985659\pi$
$444$		0			0
$445$	−2.06826e8	−	3.58234e8i	−0.111262	−	0.192711i
$446$		0			0
$447$	−1.43531e9			−0.760098
$448$		0			0
$449$	8.24242e8			0.429727			0.214864	−	0.976644i	$-0.431069\pi$
$449$	8.24242e8			0.429727			0.214864	+	0.976644i	$0.431069\pi$
$450$		0			0
$451$	1.21103e9	+	2.09756e9i	0.621635	+	1.07670i
$452$		0			0
$453$	−1.10357e9	+	1.91144e9i	−0.557772	+	0.966089i
$454$		0			0
$455$	−1.23014e9	−	2.67304e8i	−0.612231	−	0.133035i
$456$		0			0
$457$	3.05260e8	−	5.28726e8i	0.149611	−	0.259134i	−0.781473	−	0.623939i	$-0.785531\pi$
$457$	3.05260e8	−	5.28726e8i	0.149611	−	0.259134i	0.931084	+	0.364806i	$0.118865\pi$
$458$		0			0
$459$	7.49299e8	+	1.29782e9i	0.361669	+	0.626429i
$460$		0			0
$461$	−1.46019e9			−0.694155			−0.347078	−	0.937836i	$-0.612826\pi$
$461$	−1.46019e9			−0.694155			−0.347078	+	0.937836i	$0.612826\pi$
$462$		0			0
$463$	6.99630e7			0.0327593			0.0163797	−	0.999866i	$-0.494786\pi$
$463$	6.99630e7			0.0327593			0.0163797	+	0.999866i	$0.494786\pi$
$464$		0			0
$465$	9.26985e8	+	1.60559e9i	0.427551	+	0.740539i
$466$		0			0
$467$	1.79060e9	−	3.10141e9i	0.813559	−	1.40913i	−0.0967986	−	0.995304i	$-0.530860\pi$
$467$	1.79060e9	−	3.10141e9i	0.813559	−	1.40913i	0.910358	−	0.413822i	$-0.135806\pi$
$468$		0			0
$469$	−1.36234e9	+	1.49932e9i	−0.609790	+	0.671101i
$470$		0			0
$471$	−1.72374e9	+	2.98561e9i	−0.760149	+	1.31662i
$472$		0			0
$473$	−2.21518e9	−	3.83680e9i	−0.962486	−	1.66708i
$474$		0			0
$475$	−9.54361e8			−0.408588
$476$		0			0
$477$	9.98766e9			4.21356
$478$		0			0
$479$	9.65832e7	+	1.67287e8i	0.0401539	+	0.0695485i	0.885404	−	0.464822i	$-0.153882\pi$
$479$	9.65832e7	+	1.67287e8i	0.0401539	+	0.0695485i	−0.845250	+	0.534371i	$0.820549\pi$
$480$		0			0
$481$	−9.07178e8	+	1.57128e9i	−0.371693	+	0.643792i
$482$		0			0
$483$	−7.50421e8	−	2.34566e9i	−0.303033	−	0.947218i
$484$		0			0
$485$	8.64465e8	−	1.49730e9i	0.344074	−	0.595953i
$486$		0			0
$487$	−1.79836e8	−	3.11486e8i	−0.0705547	−	0.122204i	0.828590	−	0.559856i	$-0.189144\pi$
$487$	−1.79836e8	−	3.11486e8i	−0.0705547	−	0.122204i	−0.899145	+	0.437652i	$0.855810\pi$
$488$		0			0
$489$	4.39372e9			1.69923
$490$		0			0
$491$	2.79812e9			1.06680			0.533398	−	0.845865i	$-0.320915\pi$
$491$	2.79812e9			1.06680			0.533398	+	0.845865i	$0.320915\pi$
$492$		0			0
$493$	4.18645e8	+	7.25114e8i	0.157355	+	0.272548i
$494$		0			0
$495$	1.93643e9	−	3.35399e9i	0.717601	−	1.24292i
$496$		0			0
$497$	−1.15154e9	−	3.59946e9i	−0.420756	−	1.31520i
$498$		0			0
$499$	1.02578e8	−	1.77670e8i	0.0369573	−	0.0640120i	−0.846955	−	0.531664i	$-0.821567\pi$
$499$	1.02578e8	−	1.77670e8i	0.0369573	−	0.0640120i	0.883912	+	0.467652i	$0.154900\pi$
$500$		0			0
$501$	2.38530e9	+	4.13146e9i	0.847444	+	1.46782i
$502$		0			0
$503$	2.67473e9			0.937114			0.468557	−	0.883433i	$-0.344774\pi$
$503$	2.67473e9			0.937114			0.468557	+	0.883433i	$0.344774\pi$
$504$		0			0
$505$	−3.31143e8			−0.114418
$506$		0			0
$507$	3.34056e9	+	5.78602e9i	1.13839	+	1.97175i
$508$		0			0
$509$	2.49115e9	−	4.31481e9i	0.837314	−	1.45027i	−0.0548181	−	0.998496i	$-0.517458\pi$
$509$	2.49115e9	−	4.31481e9i	0.837314	−	1.45027i	0.892132	−	0.451774i	$-0.149209\pi$
$510$		0			0
$511$	−1.97370e9	+	2.17215e9i	−0.654346	+	0.720138i
$512$		0			0
$513$	1.67650e9	−	2.90378e9i	0.548267	−	0.949627i
$514$		0			0
$515$	3.46486e7	+	6.00131e7i	0.0111779	+	0.0193607i
$516$		0			0
$517$	6.93255e8			0.220636
$518$		0			0
$519$	1.30245e8			0.0408955
$520$		0			0
$521$	9.19617e8	+	1.59282e9i	0.284888	+	0.493441i	0.972582	−	0.232560i	$-0.0747101\pi$
$521$	9.19617e8	+	1.59282e9i	0.284888	+	0.493441i	−0.687694	+	0.726001i	$0.741377\pi$
$522$		0			0
$523$	−2.73502e8	+	4.73720e8i	−0.0835997	+	0.144799i	−0.904794	−	0.425850i	$-0.859975\pi$
$523$	−2.73502e8	+	4.73720e8i	−0.0835997	+	0.144799i	0.821194	+	0.570649i	$0.193308\pi$
$524$		0			0
$525$	−4.81716e9	−	1.04675e9i	−1.45289	−	0.315708i
$526$		0			0
$527$	6.25720e8	−	1.08378e9i	0.186227	−	0.322555i
$528$		0			0
$529$	1.18174e9	+	2.04684e9i	0.347079	+	0.601158i
$530$		0			0
$531$	9.80571e8			0.284216
$532$		0			0
$533$	4.23827e9			1.21239
$534$		0			0
$535$	1.15553e9	+	2.00143e9i	0.326243	+	0.565070i
$536$		0			0
$537$	−1.57283e9	+	2.72421e9i	−0.438299	+	0.759156i
$538$		0			0
$539$	5.10596e9	+	2.32898e9i	1.40448	+	0.640626i
$540$		0			0
$541$	−3.54288e9	+	6.13644e9i	−0.961979	+	1.66620i	−0.244457	+	0.969660i	$0.578610\pi$
$541$	−3.54288e9	+	6.13644e9i	−0.961979	+	1.66620i	−0.717522	+	0.696536i	$0.754724\pi$
$542$		0			0
$543$	−4.54432e9	−	7.87100e9i	−1.21806	−	2.10975i
$544$		0			0
$545$	9.14177e8			0.241904
$546$		0			0
$547$	6.74238e9			1.76140			0.880700	−	0.473675i	$-0.157073\pi$
$547$	6.74238e9			1.76140			0.880700	+	0.473675i	$0.157073\pi$
$548$		0			0
$549$	−2.21790e9	−	3.84152e9i	−0.572058	−	0.990833i
$550$		0			0
$551$	9.36685e8	−	1.62239e9i	0.238541	−	0.413165i
$552$		0			0
$553$	−1.21463e9	−	2.63934e8i	−0.305426	−	0.0663678i
$554$		0			0
$555$	7.44268e8	−	1.28911e9i	0.184801	−	0.320085i
$556$		0			0
$557$	3.68771e9	+	6.38730e9i	0.904198	+	1.56612i	0.821990	+	0.569502i	$0.192864\pi$
$557$	3.68771e9	+	6.38730e9i	0.904198	+	1.56612i	0.0822077	+	0.996615i	$0.473803\pi$
$558$		0			0
$559$	−7.75253e9			−1.87716
$560$		0			0
$561$	−3.78446e9			−0.904968
$562$		0			0
$563$	−3.19279e9	−	5.53008e9i	−0.754034	−	1.30603i	−0.945853	−	0.324596i	$-0.894772\pi$
$563$	−3.19279e9	−	5.53008e9i	−0.754034	−	1.30603i	0.191818	−	0.981430i	$-0.438562\pi$
$564$		0			0
$565$	−6.76556e8	+	1.17183e9i	−0.157810	+	0.273335i
$566$		0			0
$567$	5.12652e9	−	5.64196e9i	1.18109	−	1.29984i
$568$		0			0
$569$	3.00015e9	−	5.19642e9i	0.682733	−	1.18253i	−0.291411	−	0.956598i	$-0.594125\pi$
$569$	3.00015e9	−	5.19642e9i	0.682733	−	1.18253i	0.974144	−	0.225929i	$-0.0725419\pi$
$570$		0			0
$571$	1.47621e9	+	2.55687e9i	0.331835	+	0.574755i	0.982872	−	0.184291i	$-0.0589990\pi$
$571$	1.47621e9	+	2.55687e9i	0.331835	+	0.574755i	−0.651037	+	0.759046i	$0.725666\pi$
$572$		0			0
$573$	−9.70086e9			−2.15412
$574$		0			0
$575$	−2.08438e9			−0.457235
$576$		0			0
$577$	−4.88420e8	−	8.45969e8i	−0.105847	−	0.183332i	0.808237	−	0.588857i	$-0.200422\pi$
$577$	−4.88420e8	−	8.45969e8i	−0.105847	−	0.183332i	−0.914084	+	0.405525i	$0.867089\pi$
$578$		0			0
$579$	−3.45344e9	+	5.98153e9i	−0.739395	+	1.28067i
$580$		0			0
$581$	2.25272e9	+	7.04154e9i	0.476531	+	1.48954i
$582$		0			0
$583$	−6.96567e9	+	1.20649e10i	−1.45587	+	2.52164i
$584$		0			0
$585$	−3.38849e9	−	5.86903e9i	−0.699778	−	1.21205i
$586$		0			0
$587$	−1.30141e9			−0.265572			−0.132786	−	0.991145i	$-0.542392\pi$
$587$	−1.30141e9			−0.265572			−0.132786	+	0.991145i	$0.542392\pi$
$588$		0			0
$589$	−2.80000e9			−0.564618
$590$		0			0
$591$	5.56421e9	+	9.63749e9i	1.10878	+	1.92047i
$592$		0			0
$593$	−1.46216e8	+	2.53254e8i	−0.0287942	+	0.0498730i	−0.880063	−	0.474856i	$-0.842500\pi$
$593$	−1.46216e8	+	2.53254e8i	−0.0287942	+	0.0498730i	0.851269	+	0.524729i	$0.175833\pi$
$594$		0			0
$595$	−2.12421e8	−	6.63985e8i	−0.0413417	−	0.129226i
$596$		0			0
$597$	−3.11932e9	+	5.40282e9i	−0.599998	+	1.03923i
$598$		0			0
$599$	6.08228e8	+	1.05348e9i	0.115631	+	0.200278i	0.918032	−	0.396507i	$-0.129778\pi$
$599$	6.08228e8	+	1.05348e9i	0.115631	+	0.200278i	−0.802401	+	0.596785i	$0.796444\pi$
$600$		0			0
$601$	8.70073e9			1.63491			0.817457	−	0.575990i	$-0.195383\pi$
$601$	8.70073e9			1.63491			0.817457	+	0.575990i	$0.195383\pi$
$602$		0			0
$603$	−1.09059e10			−2.02559
$604$		0			0
$605$	1.56756e9	+	2.71510e9i	0.287794	+	0.498473i
$606$		0			0
$607$	1.73761e9	−	3.00963e9i	0.315350	−	0.546202i	−0.664162	−	0.747589i	$-0.731212\pi$
$607$	1.73761e9	−	3.00963e9i	0.315350	−	0.546202i	0.979512	+	0.201387i	$0.0645448\pi$
$608$		0			0
$609$	6.50739e9	−	7.16168e9i	1.16747	−	1.28485i
$610$		0			0
$611$	6.06552e8	−	1.05058e9i	0.107578	−	0.186331i
$612$		0			0
$613$	−9.58454e8	−	1.66009e9i	−0.168058	−	0.291085i	0.769679	−	0.638431i	$-0.220416\pi$
$613$	−9.58454e8	−	1.66009e9i	−0.168058	−	0.291085i	−0.937737	+	0.347346i	$0.887083\pi$
$614$		0			0
$615$	−3.47717e9			−0.602785
$616$		0			0
$617$	−9.94935e9			−1.70528			−0.852642	−	0.522496i	$-0.825001\pi$
$617$	−9.94935e9			−1.70528			−0.852642	+	0.522496i	$0.825001\pi$
$618$		0			0
$619$	−5.68330e9	−	9.84377e9i	−0.963127	−	1.66818i	−0.714565	−	0.699569i	$-0.753375\pi$
$619$	−5.68330e9	−	9.84377e9i	−0.963127	−	1.66818i	−0.248562	−	0.968616i	$-0.579958\pi$
$620$		0			0
$621$	3.66157e9	−	6.34203e9i	0.613546	−	1.06269i
$622$		0			0
$623$	3.15333e9	+	6.85206e8i	0.522470	+	0.113531i
$624$		0			0
$625$	−1.55747e9	+	2.69762e9i	−0.255176	+	0.441977i
$626$		0			0
$627$	4.23371e9	+	7.33300e9i	0.685938	+	1.18808i
$628$		0			0
$629$	−1.00477e9			−0.160987
$630$		0			0
$631$	−7.79638e9			−1.23535			−0.617676	−	0.786433i	$-0.711925\pi$
$631$	−7.79638e9			−1.23535			−0.617676	+	0.786433i	$0.711925\pi$
$632$		0			0
$633$	−2.45675e9	−	4.25522e9i	−0.384989	−	0.666821i
$634$		0			0
$635$	−5.93966e7	+	1.02878e8i	−0.00920563	+	0.0159446i
$636$		0			0
$637$	7.99677e9	−	5.70002e9i	1.22582	−	0.873753i
$638$		0			0
$639$	1.01725e10	−	1.76194e10i	1.54233	−	2.67139i
$640$		0			0
$641$	5.82020e9	+	1.00809e10i	0.872841	+	1.51181i	0.859045	+	0.511900i	$0.171058\pi$
$641$	5.82020e9	+	1.00809e10i	0.872841	+	1.51181i	0.0137961	+	0.999905i	$0.495608\pi$
$642$		0			0
$643$	9.21864e9			1.36750			0.683752	−	0.729714i	$-0.260347\pi$
$643$	9.21864e9			1.36750			0.683752	+	0.729714i	$0.260347\pi$
$644$		0			0
$645$	6.36034e9			0.933301
$646$		0			0
$647$	−5.05598e9	−	8.75721e9i	−0.733905	−	1.27116i	−0.955202	−	0.295955i	$-0.904362\pi$
$647$	−5.05598e9	−	8.75721e9i	−0.733905	−	1.27116i	0.221297	−	0.975206i	$-0.428971\pi$
$648$		0			0
$649$	−6.83877e8	+	1.18451e9i	−0.0982023	+	0.170091i
$650$		0			0
$651$	−1.41331e10	−	3.07106e9i	−2.00772	−	0.436269i
$652$		0			0
$653$	6.45809e9	−	1.11857e10i	0.907629	−	1.57206i	0.0902794	−	0.995916i	$-0.471224\pi$
$653$	6.45809e9	−	1.11857e10i	0.907629	−	1.57206i	0.817349	−	0.576142i	$-0.195443\pi$
$654$		0			0
$655$	2.81185e9	+	4.87026e9i	0.390973	+	0.677186i
$656$		0			0
$657$	−1.58000e10			−2.17360
$658$		0			0
$659$	1.30898e9			0.178170			0.0890848	−	0.996024i	$-0.471606\pi$
$659$	1.30898e9			0.178170			0.0890848	+	0.996024i	$0.471606\pi$
$660$		0			0
$661$	−5.09754e9	−	8.82919e9i	−0.686523	−	1.18909i	−0.972956	−	0.230993i	$-0.925803\pi$
$661$	−5.09754e9	−	8.82919e9i	−0.686523	−	1.18909i	0.286432	−	0.958100i	$-0.407531\pi$
$662$		0			0
$663$	−3.31115e9	+	5.73507e9i	−0.441246	+	0.764261i
$664$		0			0
$665$	−1.04894e9	+	1.15441e9i	−0.138317	+	0.152224i
$666$		0			0
$667$	2.04578e9	−	3.54339e9i	0.266943	−	0.462358i
$668$		0			0
$669$	−1.10085e10	−	1.90672e10i	−1.42146	−	2.46204i
$670$		0			0
$671$	6.18731e9			0.790629
$672$		0			0
$673$	−1.13920e10			−1.44061			−0.720305	−	0.693658i	$-0.755998\pi$
$673$	−1.13920e10			−1.44061			−0.720305	+	0.693658i	$0.755998\pi$
$674$		0			0
$675$	−7.32914e9	−	1.26944e10i	−0.917254	−	1.58873i
$676$		0			0
$677$	−4.89269e9	+	8.47439e9i	−0.606021	+	1.04966i	0.385868	+	0.922554i	$0.373902\pi$
$677$	−4.89269e9	+	8.47439e9i	−0.606021	+	1.04966i	−0.991889	+	0.127105i	$0.959431\pi$
$678$		0			0
$679$	4.10970e9	+	1.28461e10i	0.503809	+	1.57480i
$680$		0			0
$681$	2.43613e9	−	4.21950e9i	0.295587	−	0.511972i
$682$		0			0
$683$	3.21469e9	+	5.56801e9i	0.386071	+	0.668694i	0.991917	−	0.126888i	$-0.0404988\pi$
$683$	3.21469e9	+	5.56801e9i	0.386071	+	0.668694i	−0.605847	+	0.795582i	$0.707165\pi$
$684$		0			0
$685$	−2.88299e9			−0.342710
$686$		0			0
$687$	1.40405e10			1.65209
$688$		0			0
$689$	1.21890e10	+	2.11119e10i	1.41971	+	2.45901i
$690$		0			0
$691$	−8.28687e7	+	1.43533e8i	−0.00955471	+	0.0165492i	−0.870763	−	0.491703i	$-0.836375\pi$
$691$	−8.28687e7	+	1.43533e8i	−0.00955471	+	0.0165492i	0.861209	+	0.508252i	$0.169708\pi$
$692$		0			0
$693$	9.20585e9	+	2.87756e10i	1.05075	+	3.28441i
$694$		0			0
$695$	1.12658e9	−	1.95129e9i	0.127296	−	0.220483i
$696$		0			0
$697$	1.17355e9	+	2.03265e9i	0.131277	+	0.227378i
$698$		0			0
$699$	−1.83234e10			−2.02926
$700$		0			0
$701$	2.81761e9			0.308935			0.154468	−	0.987998i	$-0.450634\pi$
$701$	2.81761e9			0.308935			0.154468	+	0.987998i	$0.450634\pi$
$702$		0			0
$703$	1.12405e9	+	1.94691e9i	0.122023	+	0.211350i
$704$		0			0
$705$	−4.97628e8	+	8.61918e8i	−0.0534864	+	0.0926411i
$706$		0			0
$707$	1.73721e9	−	1.91188e9i	0.184878	−	0.203467i
$708$		0			0
$709$	−3.67090e9	+	6.35819e9i	−0.386822	+	0.669995i	−0.992020	−	0.126080i	$-0.959761\pi$
$709$	−3.67090e9	+	6.35819e9i	−0.386822	+	0.669995i	0.605198	+	0.796075i	$0.293094\pi$
$710$		0			0
$711$	−3.34576e9	−	5.79503e9i	−0.349101	−	0.604661i
$712$		0			0
$713$	−6.11537e9			−0.631843
$714$		0			0
$715$	9.45289e9			0.967149
$716$		0			0
$717$	−9.41771e9	−	1.63119e10i	−0.954175	−	1.65268i
$718$		0			0
$719$	−9.73646e8	+	1.68640e9i	−0.0976899	+	0.169204i	−0.910728	−	0.413006i	$-0.864479\pi$
$719$	−9.73646e8	+	1.68640e9i	−0.0976899	+	0.169204i	0.813038	+	0.582210i	$0.197812\pi$
$720$		0			0
$721$	−5.28262e8	−	1.14789e8i	−0.0524899	−	0.0114058i
$722$		0			0
$723$	4.15029e9	−	7.18851e9i	0.408408	−	0.707383i
$724$		0			0
$725$	−4.09490e9	−	7.09258e9i	−0.399080	−	0.691228i
$726$		0			0
$727$	−8.55516e9			−0.825768			−0.412884	−	0.910784i	$-0.635478\pi$
$727$	−8.55516e9			−0.825768			−0.412884	+	0.910784i	$0.635478\pi$
$728$		0			0
$729$	−6.99205e8			−0.0668434
$730$		0			0
$731$	−2.14663e9	−	3.71808e9i	−0.203258	−	0.352053i
$732$		0			0
$733$	5.23062e9	−	9.05970e9i	0.490557	−	0.849669i	−0.509384	−	0.860539i	$-0.670127\pi$
$733$	5.23062e9	−	9.05970e9i	0.490557	−	0.849669i	0.999941	+	0.0108699i	$0.00346006\pi$
$734$		0			0
$735$	−6.56073e9	+	4.67642e9i	−0.609461	+	0.434418i
$736$		0			0
$737$	7.60607e9	−	1.31741e10i	0.699881	−	1.21223i
$738$		0			0
$739$	−6.64787e8	−	1.15144e9i	−0.0605936	−	0.104951i	0.834137	−	0.551557i	$-0.185966\pi$
$739$	−6.64787e8	−	1.15144e9i	−0.0605936	−	0.104951i	−0.894731	+	0.446606i	$0.852633\pi$
$740$		0			0
$741$	1.48169e10			1.33780
$742$		0			0
$743$	1.68408e9			0.150627			0.0753133	−	0.997160i	$-0.476004\pi$
$743$	1.68408e9			0.150627			0.0753133	+	0.997160i	$0.476004\pi$
$744$		0			0
$745$	9.92708e8	+	1.71942e9i	0.0879579	+	0.152348i
$746$		0			0
$747$	−1.99003e10	+	3.44683e10i	−1.74678	+	3.02551i
$748$		0			0
$749$	−1.76175e10	−	3.82820e9i	−1.53199	−	0.332896i
$750$		0			0
$751$	2.22383e9	−	3.85178e9i	0.191585	−	0.331835i	−0.754191	−	0.656655i	$-0.771971\pi$
$751$	2.22383e9	−	3.85178e9i	0.191585	−	0.331835i	0.945776	+	0.324821i	$0.105304\pi$
$752$		0			0
$753$	1.11745e10	+	1.93549e10i	0.953778	+	1.65199i
$754$		0			0
$755$	3.05307e9			0.258179
$756$		0			0
$757$	−7.99647e9			−0.669981			−0.334991	−	0.942221i	$-0.608733\pi$
$757$	−7.99647e9			−0.669981			−0.334991	+	0.942221i	$0.608733\pi$
$758$		0			0
$759$	9.24668e9	+	1.60157e10i	0.767608	+	1.32954i
$760$		0			0
$761$	4.22630e9	−	7.32016e9i	0.347627	−	0.602108i	−0.638200	−	0.769870i	$-0.720321\pi$
$761$	4.22630e9	−	7.32016e9i	0.347627	−	0.602108i	0.985827	+	0.167762i	$0.0536541\pi$
$762$		0			0
$763$	−4.79588e9	+	5.27808e9i	−0.390870	+	0.430170i
$764$		0			0
$765$	1.87651e9	−	3.25021e9i	0.151543	−	0.262480i
$766$		0			0
$767$	1.19669e9	+	2.07273e9i	0.0957633	+	0.165867i
$768$		0			0
$769$	4.93932e9			0.391674			0.195837	−	0.980636i	$-0.437258\pi$
$769$	4.93932e9			0.391674			0.195837	+	0.980636i	$0.437258\pi$
$770$		0			0
$771$	−2.57072e9			−0.202006
$772$		0			0
$773$	2.76862e9	+	4.79539e9i	0.215593	+	0.373418i	0.953456	−	0.301532i	$-0.0974982\pi$
$773$	2.76862e9	+	4.79539e9i	0.215593	+	0.373418i	−0.737863	+	0.674951i	$0.764165\pi$
$774$		0			0
$775$	−6.12037e9	+	1.06008e10i	−0.472304	+	0.818055i
$776$		0			0
$777$	3.53828e9	+	1.10599e10i	0.270594	+	0.845822i
$778$		0			0
$779$	2.62573e9	−	4.54790e9i	0.199008	−	0.344691i
$780$		0			0
$781$	1.41892e10	+	2.45764e10i	1.06581	+	1.84604i
$782$		0			0
$783$	2.87736e10			2.14204
$784$		0			0
$785$	4.76879e9			0.351855
$786$		0			0
$787$	2.39855e9	+	4.15440e9i	0.175403	+	0.303806i	0.940301	−	0.340345i	$-0.110544\pi$
$787$	2.39855e9	+	4.15440e9i	0.175403	+	0.303806i	−0.764898	+	0.644152i	$0.777211\pi$
$788$		0			0
$789$	3.04241e8	−	5.26961e8i	0.0220520	−	0.0381952i
$790$		0			0
$791$	−3.21637e9	−	1.00537e10i	−0.231073	−	0.722285i
$792$		0			0
$793$	5.41348e9	−	9.37642e9i	0.385496	−	0.667699i
$794$		0			0
$795$	−1.00001e10	−	1.73207e10i	−0.705862	−	1.22259i
$796$		0			0
$797$	1.15558e9			0.0808526			0.0404263	−	0.999183i	$-0.487128\pi$
$797$	1.15558e9			0.0808526			0.0404263	+	0.999183i	$0.487128\pi$
$798$		0			0
$799$	6.71804e8			0.0465939
$800$		0			0
$801$	8.68601e9	+	1.50446e10i	0.597182	+	1.03435i
$802$		0			0
$803$	1.10194e10	−	1.90861e10i	0.751021	−	1.30081i
$804$		0			0
$805$	−2.29095e9	+	2.52130e9i	−0.154786	+	0.170349i
$806$		0			0
$807$	2.87679e9	−	4.98275e9i	0.192686	−	0.333743i
$808$		0			0
$809$	−1.19483e10	−	2.06951e10i	−0.793390	−	1.37419i	−0.923857	−	0.382739i	$-0.874981\pi$
$809$	−1.19483e10	−	2.06951e10i	−0.793390	−	1.37419i	0.130467	−	0.991453i	$-0.458352\pi$
$810$		0			0
$811$	1.09421e10			0.720326			0.360163	−	0.932889i	$-0.382721\pi$
$811$	1.09421e10			0.720326			0.360163	+	0.932889i	$0.382721\pi$
$812$		0			0
$813$	−8.21166e9			−0.535937
$814$		0			0
$815$	−3.03884e9	−	5.26343e9i	−0.196633	−	0.340579i
$816$		0			0
$817$	−4.80292e9	+	8.31891e9i	−0.308126	+	0.533690i
$818$		0			0
$819$	5.16618e10	+	1.12259e10i	3.28606	+	0.714048i
$820$		0			0
$821$	1.11318e10	−	1.92808e10i	0.702043	−	1.21597i	−0.265705	−	0.964054i	$-0.585605\pi$
$821$	1.11318e10	−	1.92808e10i	0.702043	−	1.21597i	0.967748	−	0.251920i	$-0.0810620\pi$
$822$		0			0
$823$	−1.83529e9	−	3.17881e9i	−0.114764	−	0.198777i	0.802921	−	0.596085i	$-0.203278\pi$
$823$	−1.83529e9	−	3.17881e9i	−0.114764	−	0.198777i	−0.917685	+	0.397308i	$0.869944\pi$
$824$		0			0
$825$	3.70170e10			2.29516
$826$		0			0
$827$	−8.88318e9			−0.546134			−0.273067	−	0.961995i	$-0.588038\pi$
$827$	−8.88318e9			−0.546134			−0.273067	+	0.961995i	$0.588038\pi$
$828$		0			0
$829$	−6.36202e9	−	1.10193e10i	−0.387842	−	0.671761i	0.604317	−	0.796744i	$-0.293446\pi$
$829$	−6.36202e9	−	1.10193e10i	−0.387842	−	0.671761i	−0.992159	+	0.124982i	$0.960113\pi$
$830$		0			0
$831$	1.63133e10	−	2.82555e10i	0.986140	−	1.70805i
$832$		0			0
$833$	4.94797e9	+	2.25691e9i	0.296599	+	0.135287i
$834$		0			0
$835$	3.29951e9	−	5.71491e9i	0.196131	−	0.339709i
$836$		0			0
$837$	−2.15030e10	−	3.72442e10i	−1.26753	−	2.19543i
$838$		0			0
$839$	−2.40186e9			−0.140404			−0.0702022	−	0.997533i	$-0.522364\pi$
$839$	−2.40186e9			−0.140404			−0.0702022	+	0.997533i	$0.522364\pi$
$840$		0			0
$841$	−1.17364e9			−0.0680375
$842$		0			0
$843$	1.86895e10	+	3.23711e10i	1.07448	+	1.86106i
$844$		0			0
$845$	4.62089e9	−	8.00361e9i	0.263467	−	0.456339i
$846$		0			0
$847$	−2.38995e10	−	5.19326e9i	−1.35144	−	0.293662i
$848$		0			0
$849$	−2.61592e10	+	4.53091e10i	−1.46706	+	2.54102i
$850$		0			0
$851$	2.45499e9	+	4.25216e9i	0.136551	+	0.236514i
$852$		0			0
$853$	2.38086e10			1.31344			0.656722	−	0.754133i	$-0.271942\pi$
$853$	2.38086e10			1.31344			0.656722	+	0.754133i	$0.271942\pi$
$854$		0			0
$855$	−8.39708e9			−0.459459
$856$		0			0
$857$	9.45036e8	+	1.63685e9i	0.0512880	+	0.0888334i	0.890530	−	0.454925i	$-0.150334\pi$
$857$	9.45036e8	+	1.63685e9i	0.0512880	+	0.0888334i	−0.839242	+	0.543759i	$0.817001\pi$
$858$		0			0
$859$	1.62617e10	−	2.81661e10i	0.875368	−	1.51618i	0.0189969	−	0.999820i	$-0.493953\pi$
$859$	1.62617e10	−	2.81661e10i	0.875368	−	1.51618i	0.856371	−	0.516362i	$-0.172714\pi$
$860$		0			0
$861$	1.82416e10	−	2.00757e10i	0.973986	−	1.07192i
$862$		0			0
$863$	−1.57375e10	+	2.72581e10i	−0.833483	+	1.44364i	0.0617761	+	0.998090i	$0.480324\pi$
$863$	−1.57375e10	+	2.72581e10i	−0.833483	+	1.44364i	−0.895259	+	0.445545i	$0.853010\pi$
$864$		0			0
$865$	−9.00818e7	−	1.56026e8i	−0.00473239	−	0.00819675i
$866$		0			0
$867$	3.08413e10			1.60718
$868$		0			0
$869$	9.33370e9			0.482486
$870$		0			0
$871$	−1.33096e10	−	2.30529e10i	−0.682498	−	1.18212i
$872$		0			0
$873$	−3.63047e10	+	6.28815e10i	−1.84677	+	3.19870i
$874$		0			0
$875$	4.59083e9	+	1.43500e10i	0.231666	+	0.724142i
$876$		0			0
$877$	−4.66652e9	+	8.08264e9i	−0.233612	+	0.404627i	−0.958868	−	0.283851i	$-0.908388\pi$
$877$	−4.66652e9	+	8.08264e9i	−0.233612	+	0.404627i	0.725257	+	0.688479i	$0.241721\pi$
$878$		0			0
$879$	−1.37163e10	−	2.37572e10i	−0.681200	−	1.17987i
$880$		0			0
$881$	−2.08746e10			−1.02850			−0.514248	−	0.857641i	$-0.671929\pi$
$881$	−2.08746e10			−1.02850			−0.514248	+	0.857641i	$0.671929\pi$
$882$		0			0
$883$	−2.12927e10			−1.04080			−0.520402	−	0.853922i	$-0.674218\pi$
$883$	−2.12927e10			−1.04080			−0.520402	+	0.853922i	$0.674218\pi$
$884$		0			0
$885$	−9.81793e8	−	1.70051e9i	−0.0476122	−	0.0824668i
$886$		0			0
$887$	−1.74469e10	+	3.02188e10i	−0.839430	+	1.45394i	0.0509420	+	0.998702i	$0.483778\pi$
$887$	−1.74469e10	+	3.02188e10i	−0.839430	+	1.45394i	−0.890372	+	0.455234i	$0.849556\pi$
$888$		0			0
$889$	−2.82374e8	−	8.82642e8i	−0.0134793	−	0.0421336i
$890$		0			0
$891$	−2.86218e10	+	4.95745e10i	−1.35558	+	2.34794i
$892$		0			0
$893$	−7.51554e8	−	1.30173e9i	−0.0353167	−	0.0611703i
$894$		0			0
$895$	4.35127e9			0.202878
$896$		0			0
$897$	3.23609e10			1.49709
$898$		0			0
$899$	−1.20140e10	−	2.08089e10i	−0.551480	−	0.955191i
$900$		0			0
$901$	−6.75013e9	+	1.16916e10i	−0.307450	+	0.532520i
$902$		0			0
$903$	−3.33671e10	+	3.67220e10i	−1.50804	+	1.65966i
$904$		0			0
$905$	−6.28601e9	+	1.08877e10i	−0.281906	+	0.488276i
$906$		0			0
$907$	−1.66490e9	−	2.88370e9i	−0.0740907	−	0.128329i	0.826600	−	0.562790i	$-0.190272\pi$
$907$	−1.66490e9	−	2.88370e9i	−0.0740907	−	0.128329i	−0.900691	+	0.434461i	$0.856939\pi$
$908$		0			0
$909$	1.39069e10			0.614124
$910$		0			0
$911$	−4.05647e10			−1.77760			−0.888799	−	0.458297i	$-0.848460\pi$
$911$	−4.05647e10			−1.77760			−0.888799	+	0.458297i	$0.848460\pi$
$912$		0			0
$913$	−2.77580e10	−	4.80783e10i	−1.20709	−	2.09075i
$914$		0			0
$915$	−4.44134e9	+	7.69262e9i	−0.191664	+	0.331971i
$916$		0			0
$917$	−4.28702e10	−	9.31551e9i	−1.83596	−	0.398946i
$918$		0			0
$919$	1.14032e10	−	1.97509e10i	0.484644	−	0.839428i	−0.515200	−	0.857070i	$-0.672283\pi$
$919$	1.14032e10	−	1.97509e10i	0.484644	−	0.839428i	0.999844	+	0.0176418i	$0.00561585\pi$
$920$		0			0
$921$	−9.13836e9	−	1.58281e10i	−0.385442	−	0.667606i
$922$		0			0
$923$	4.96585e10			2.07868
$924$		0			0
$925$	9.82798e9			0.408290
$926$		0			0
$927$	−1.45513e9	−	2.52035e9i	−0.0599959	−	0.103916i
$928$		0			0
$929$	1.60242e10	−	2.77547e10i	0.655724	−	1.13575i	−0.325987	−	0.945374i	$-0.605697\pi$
$929$	1.60242e10	−	2.77547e10i	0.655724	−	1.13575i	0.981712	−	0.190374i	$-0.0609700\pi$
$930$		0			0
$931$	−1.16221e9	−	1.21123e10i	−0.0472021	−	0.491931i
$932$		0			0
$933$	1.06256e10	−	1.84041e10i	0.428321	−	0.741873i
$934$		0			0
$935$	2.61745e9	+	4.53356e9i	0.104722	+	0.181384i
$936$		0			0
$937$	2.73774e10			1.08718			0.543592	−	0.839350i	$-0.317064\pi$
$937$	2.73774e10			1.08718			0.543592	+	0.839350i	$0.317064\pi$
$938$		0			0
$939$	−2.88448e10			−1.13694
$940$		0			0
$941$	4.29951e9	+	7.44697e9i	0.168211	+	0.291350i	0.937791	−	0.347200i	$-0.112868\pi$
$941$	4.29951e9	+	7.44697e9i	0.168211	+	0.291350i	−0.769580	+	0.638551i	$0.779534\pi$
$942$		0			0
$943$	5.73476e9	−	9.93289e9i	0.222702	−	0.385731i
$944$		0			0
$945$	−2.34109e10	−	5.08708e9i	−0.902415	−	0.196091i
$946$		0			0
$947$	−1.99474e10	+	3.45500e10i	−0.763242	+	1.32197i	0.177930	+	0.984043i	$0.443060\pi$
$947$	−1.99474e10	+	3.45500e10i	−0.763242	+	1.32197i	−0.941171	+	0.337930i	$0.890273\pi$
$948$		0			0
$949$	−1.92824e10	−	3.33981e10i	−0.732368	−	1.26850i
$950$		0			0
$951$	−2.86969e10			−1.08194
$952$		0			0
$953$	−7.66469e9			−0.286860			−0.143430	−	0.989660i	$-0.545813\pi$
$953$	−7.66469e9			−0.286860			−0.143430	+	0.989660i	$0.545813\pi$
$954$		0			0
$955$	6.70944e9	+	1.16211e10i	0.249272	+	0.431753i
$956$		0			0
$957$	−3.63314e10	+	6.29278e10i	−1.33995	+	2.32087i
$958$		0			0
$959$	1.51245e10	−	1.66452e10i	0.553753	−	0.609431i
$960$		0			0
$961$	−4.20025e9	+	7.27504e9i	−0.152666	+	0.264426i
$962$		0			0
$963$	−4.85283e10	−	8.40534e10i	−1.75107	−	3.03294i
$964$		0			0
$965$	9.55404e9			0.342249
$966$		0			0
$967$	5.31576e10			1.89048			0.945241	−	0.326373i	$-0.105826\pi$
$967$	5.31576e10			1.89048			0.945241	+	0.326373i	$0.105826\pi$
$968$		0			0
$969$	4.10271e9	+	7.10610e9i	0.144856	+	0.250898i
$970$		0			0
$971$	4.20926e9	−	7.29066e9i	0.147550	−	0.255564i	−0.782772	−	0.622309i	$-0.786195\pi$
$971$	4.20926e9	−	7.29066e9i	0.147550	−	0.255564i	0.930321	+	0.366745i	$0.119528\pi$
$972$		0			0
$973$	5.35579e9	+	1.67411e10i	0.186393	+	0.582624i
$974$		0			0
$975$	3.23874e10	−	5.60966e10i	1.11908	−	1.93830i
$976$		0			0
$977$	−7.60872e9	−	1.31787e10i	−0.261024	−	0.452107i	0.705490	−	0.708720i	$-0.250727\pi$
$977$	−7.60872e9	−	1.31787e10i	−0.261024	−	0.452107i	−0.966514	+	0.256613i	$0.917393\pi$
$978$		0			0
$979$	−2.42314e10			−0.825353
$980$		0			0
$981$	−3.83924e10			−1.29838
$982$		0			0
$983$	1.96526e10	+	3.40393e10i	0.659907	+	1.14299i	0.980639	+	0.195822i	$0.0627375\pi$
$983$	1.96526e10	+	3.40393e10i	0.659907	+	1.14299i	−0.320733	+	0.947170i	$0.603929\pi$
$984$		0			0
$985$	7.69678e9	−	1.33312e10i	0.256615	−	0.444470i
$986$		0			0
$987$	−2.36574e9	−	7.39483e9i	−0.0783173	−	0.244804i
$988$		0			0
$989$	−1.04899e10	+	1.81690e10i	−0.344813	+	0.597233i
$990$		0			0
$991$	−4.22919e9	−	7.32516e9i	−0.138038	−	0.239089i	0.788716	−	0.614758i	$-0.210746\pi$
$991$	−4.22919e9	−	7.32516e9i	−0.138038	−	0.239089i	−0.926754	+	0.375669i	$0.877413\pi$
$992$		0			0
$993$	4.55565e10			1.47648
$994$		0			0
$995$	8.62970e9			0.277725
$996$		0			0
$997$	−4.10207e8	−	7.10500e8i	−0.0131090	−	0.0227055i	0.859396	−	0.511310i	$-0.170840\pi$
$997$	−4.10207e8	−	7.10500e8i	−0.0131090	−	0.0227055i	−0.872505	+	0.488604i	$0.837506\pi$
$998$		0			0
$999$	−1.72645e10	+	2.99031e10i	−0.547868	+	0.948935i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.8.i.d.65.5		10
4.3	odd	2		28.8.e.a.9.1	✓	10
7.4	even	3	inner	112.8.i.d.81.5		10
12.11	even	2		252.8.k.c.37.3		10
28.3	even	6		196.8.e.f.165.5		10
28.11	odd	6		28.8.e.a.25.1	yes	10
28.19	even	6		196.8.a.e.1.1		5
28.23	odd	6		196.8.a.d.1.5		5
28.27	even	2		196.8.e.f.177.5		10
84.11	even	6		252.8.k.c.109.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.e.a.9.1	✓	10	4.3	odd	2
28.8.e.a.25.1	yes	10	28.11	odd	6
112.8.i.d.65.5		10	1.1	even	1	trivial
112.8.i.d.81.5		10	7.4	even	3	inner
196.8.a.d.1.5		5	28.23	odd	6
196.8.a.e.1.1		5	28.19	even	6
196.8.e.f.165.5		10	28.3	even	6
196.8.e.f.177.5		10	28.27	even	2
252.8.k.c.37.3		10	12.11	even	2
252.8.k.c.109.3		10	84.11	even	6

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(42.0490 + 72.8309i) q^{3} +(58.1649 - 100.745i) q^{5} +(276.518 + 864.338i) q^{7} +(-2442.73 + 4230.93i) q^{9} +(-3407.25 - 5901.54i) q^{11} -11924.5 q^{13} +9783.10 q^{15} +(-3301.82 - 5718.92i) q^{17} +(-7387.57 + 12795.6i) q^{19} +(-51323.3 + 56483.6i) q^{21} +(-16134.9 + 27946.4i) q^{23} +(32296.2 + 55938.6i) q^{25} -226935. q^{27} -126792. q^{29} +(94753.8 + 164118. i) q^{31} +(286543. - 496307. i) q^{33} +(103161. + 22416.5i) q^{35} +(76077.0 - 131769. i) q^{37} +(-501412. - 868471. i) q^{39} -355426. q^{41} +650136. q^{43} +(284162. + 492184. i) q^{45} +(-50866.2 + 88102.8i) q^{47} +(-670618. + 478010. i) q^{49} +(277676. - 480950. i) q^{51} +(-1.02218e6 - 1.77047e6i) q^{53} -792730. q^{55} -1.24256e6 q^{57} +(-100356. - 173822. i) q^{59} +(-453981. + 786317. i) q^{61} +(-4.33242e6 - 941416. i) q^{63} +(-693586. + 1.20133e6i) q^{65} +(1.11616e6 + 1.93324e6i) q^{67} -2.71382e6 q^{69} -4.16441e6 q^{71} +(1.61704e6 + 2.80080e6i) q^{73} +(-2.71604e6 + 4.70432e6i) q^{75} +(4.15876e6 - 4.57690e6i) q^{77} +(-684840. + 1.18618e6i) q^{79} +(-4.20013e6 - 7.27485e6i) q^{81} +8.14674e6 q^{83} -768201. q^{85} +(-5.33148e6 - 9.23439e6i) q^{87} +(1.77793e6 - 3.07947e6i) q^{89} +(-3.29733e6 - 1.03068e7i) q^{91} +(-7.96860e6 + 1.38020e7i) q^{93} +(859394. + 1.48851e6i) q^{95} +1.48623e7 q^{97} +3.32920e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) \) 10 * q - 27 * q^3 + 249 * q^5 - 332 * q^7 - 5702 * q^9 - 6399 * q^11 - 26988 * q^13 - 19294 * q^15 + 3609 * q^17 + 12403 * q^19 + 16099 * q^21 + 13959 * q^23 - 162364 * q^25 - 161550 * q^27 + 26148 * q^29 + 20181 * q^31 + 264239 * q^33 - 791715 * q^35 - 54763 * q^37 - 4750 * q^39 - 1824468 * q^41 + 1938424 * q^43 + 1936430 * q^45 - 1217253 * q^47 + 1722058 * q^49 + 1353871 * q^51 - 2349159 * q^53 - 3123494 * q^55 - 9823194 * q^57 - 2188611 * q^59 + 6107445 * q^61 - 10275990 * q^63 - 3022014 * q^65 + 3171581 * q^67 - 11255938 * q^69 + 18938976 * q^71 + 8034853 * q^73 - 29173588 * q^75 + 24100191 * q^77 + 7589559 * q^79 - 9846485 * q^81 + 13452408 * q^83 - 48717670 * q^85 - 30033142 * q^87 + 29864757 * q^89 - 34267512 * q^91 - 20217029 * q^93 + 25812009 * q^95 + 285580 * q^97 + 119277812 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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