LMFDB - Embedded newform 112.7.s.c.17.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,7,Mod(17,112)] 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("112.17"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	112.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$25.7660573654$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 285x^{6} + 282x^{5} + 62091x^{4} + 29260x^{3} + 4838750x^{2} + 2401000x + 294122500 $ x^8 - 2x^7 + 285x^6 + 282x^5 + 62091x^4 + 29260x^3 + 4838750x^2 + 2401000*x + 294122500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.1
Root			$-4.86132 - 8.42006i$ of defining polynomial
Character	$\chi$	$=$	112.17
Dual form			112.7.s.c.33.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-27.6101 - 15.9407i) q^{3} +(111.836 - 64.5687i) q^{5} +(-298.743 + 168.528i) q^{7} +(143.713 + 248.917i) q^{9} +(1189.23 - 2059.81i) q^{11} -820.701i q^{13} -4117.09 q^{15} +(-2284.02 - 1318.68i) q^{17} +(4814.93 - 2779.90i) q^{19} +(10934.8 + 109.096i) q^{21} +(5696.37 + 9866.40i) q^{23} +(525.743 - 910.614i) q^{25} +14078.0i q^{27} -32822.6 q^{29} +(-40622.4 - 23453.4i) q^{31} +(-65669.6 + 37914.4i) q^{33} +(-22528.7 + 38137.0i) q^{35} +(-10862.8 - 18814.9i) q^{37} +(-13082.5 + 22659.6i) q^{39} +85846.0i q^{41} -113977. q^{43} +(32144.6 + 18558.7i) q^{45} +(-33367.3 + 19264.6i) q^{47} +(60845.6 - 100693. i) q^{49} +(42041.4 + 72817.9i) q^{51} +(-16620.4 + 28787.3i) q^{53} -307149. i q^{55} -177254. q^{57} +(289901. + 167374. i) q^{59} +(-145845. + 84203.9i) q^{61} +(-84882.6 - 50142.7i) q^{63} +(-52991.6 - 91784.1i) q^{65} +(-161261. + 279312. i) q^{67} -363217. i q^{69} -325510. q^{71} +(93404.3 + 53927.0i) q^{73} +(-29031.7 + 16761.4i) q^{75} +(-8138.93 + 815772. i) q^{77} +(-169525. - 293625. i) q^{79} +(329180. - 570157. i) q^{81} -224674. i q^{83} -340582. q^{85} +(906235. + 523215. i) q^{87} +(331888. - 191615. i) q^{89} +(138311. + 245178. i) q^{91} +(747726. + 1.29510e6i) q^{93} +(358989. - 621787. i) q^{95} -32664.1i q^{97} +683629. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9} + 1356 q^{11} - 27144 q^{15} - 17304 q^{17} + 32004 q^{19} + 9756 q^{21} + 4128 q^{23} + 4664 q^{25} - 30312 q^{29} + 3108 q^{31} + 3276 q^{33} - 98028 q^{35} - 6124 q^{37}+ \cdots + 4625928 q^{99}+O(q^{100}) $ 8 * q - 336 * q^5 - 652 * q^7 + 756 * q^9 + 1356 * q^11 - 27144 * q^15 - 17304 * q^17 + 32004 * q^19 + 9756 * q^21 + 4128 * q^23 + 4664 * q^25 - 30312 * q^29 + 3108 * q^31 + 3276 * q^33 - 98028 * q^35 - 6124 * q^37 - 100764 * q^39 + 297376 * q^43 - 172116 * q^45 - 313908 * q^47 + 32432 * q^49 - 253692 * q^51 + 278484 * q^53 - 81288 * q^57 + 835464 * q^59 - 995316 * q^61 - 1216188 * q^63 + 8316 * q^65 - 648808 * q^67 - 190128 * q^71 - 1617084 * q^73 + 2042208 * q^75 + 1456224 * q^77 - 70096 * q^79 + 1177920 * q^81 + 2190984 * q^85 + 2057076 * q^87 + 739116 * q^89 - 2233752 * q^91 - 23364 * q^93 - 725640 * q^95 + 4625928 * q^99

Character values

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

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

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
$3$	−27.6101	−	15.9407i	−1.02260	−	0.590397i	−0.107742	−	0.994179i	$-0.534362\pi$
$3$	−27.6101	−	15.9407i	−1.02260	−	0.590397i	−0.914855	+	0.403782i	$0.867695\pi$
$4$		0			0
$5$	111.836	−	64.5687i	0.894691	−	0.516550i	0.0192168	−	0.999815i	$-0.493883\pi$
$5$	111.836	−	64.5687i	0.894691	−	0.516550i	0.875474	+	0.483265i	$0.160549\pi$
$6$		0			0
$7$	−298.743	+	168.528i	−0.870971	+	0.491335i
$7$	−298.743	+	168.528i	−0.870971	+	0.491335i
$8$		0			0
$9$	143.713	+	248.917i	0.197136	+	0.341450i
$10$		0			0
$11$	1189.23	−	2059.81i	0.893487	−	1.54756i	0.0578204	−	0.998327i	$-0.481585\pi$
$11$	1189.23	−	2059.81i	0.893487	−	1.54756i	0.835666	−	0.549237i	$-0.185082\pi$
$12$		0			0
$13$		−	820.701i		−	0.373555i	−0.982402	−	0.186778i	$-0.940196\pi$
$13$		−	820.701i		−	0.373555i	0.982402	−	0.186778i	$-0.0598044\pi$
$14$		0			0
$15$	−4117.09			−1.21988
$16$		0			0
$17$	−2284.02	−	1318.68i	−0.464894	−	0.268406i	0.249206	−	0.968450i	$-0.419830\pi$
$17$	−2284.02	−	1318.68i	−0.464894	−	0.268406i	−0.714100	+	0.700044i	$0.753164\pi$
$18$		0			0
$19$	4814.93	−	2779.90i	0.701987	−	0.405292i	−0.106100	−	0.994355i	$-0.533836\pi$
$19$	4814.93	−	2779.90i	0.701987	−	0.405292i	0.808087	+	0.589063i	$0.200503\pi$
$20$		0			0
$21$	10934.8	+	109.096i	1.18073	+	0.0117801i
$22$		0			0
$23$	5696.37	+	9866.40i	0.468182	+	0.810914i	0.999339	−	0.0363589i	$-0.0115759\pi$
$23$	5696.37	+	9866.40i	0.468182	+	0.810914i	−0.531157	+	0.847273i	$0.678243\pi$
$24$		0			0
$25$	525.743	−	910.614i	0.0336476	−	0.0582793i
$26$		0			0
$27$			14078.0i			0.715238i
$28$		0			0
$29$	−32822.6			−1.34579			−0.672897	−	0.739736i	$-0.734950\pi$
$29$	−32822.6			−1.34579			−0.672897	+	0.739736i	$0.734950\pi$
$30$		0			0
$31$	−40622.4	−	23453.4i	−1.36358	−	0.787263i	−0.373481	−	0.927638i	$-0.621836\pi$
$31$	−40622.4	−	23453.4i	−1.36358	−	0.787263i	−0.990098	+	0.140375i	$0.955169\pi$
$32$		0			0
$33$	−65669.6	+	37914.4i	−1.82735	+	1.05502i
$34$		0			0
$35$	−22528.7	+	38137.0i	−0.525450	+	0.889493i
$36$		0			0
$37$	−10862.8	−	18814.9i	−0.214454	−	0.371446i	0.738649	−	0.674090i	$-0.235464\pi$
$37$	−10862.8	−	18814.9i	−0.214454	−	0.371446i	−0.953104	+	0.302644i	$0.902131\pi$
$38$		0			0
$39$	−13082.5	+	22659.6i	−0.220546	+	0.381996i
$40$		0			0
$41$			85846.0i			1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$41$			85846.0i			1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$42$		0			0
$43$	−113977.			−1.43355			−0.716774	−	0.697305i	$-0.754382\pi$
$43$	−113977.			−1.43355			−0.716774	+	0.697305i	$0.754382\pi$
$44$		0			0
$45$	32144.6	+	18558.7i	0.352752	+	0.203662i
$46$		0			0
$47$	−33367.3	+	19264.6i	−0.321387	+	0.185553i	−0.652011	−	0.758210i	$-0.726074\pi$
$47$	−33367.3	+	19264.6i	−0.321387	+	0.185553i	0.330624	+	0.943763i	$0.392741\pi$
$48$		0			0
$49$	60845.6	−	100693.i	0.517179	−	0.855877i
$50$		0			0
$51$	42041.4	+	72817.9i	0.316932	+	0.548943i
$52$		0			0
$53$	−16620.4	+	28787.3i	−0.111638	+	0.193363i	−0.916431	−	0.400193i	$-0.868943\pi$
$53$	−16620.4	+	28787.3i	−0.111638	+	0.193363i	0.804793	+	0.593556i	$0.202276\pi$
$54$		0			0
$55$		−	307149.i		−	1.84612i
$56$		0			0
$57$	−177254.			−0.957133
$58$		0			0
$59$	289901.	+	167374.i	1.41154	+	0.814953i	0.995534	−	0.0944082i	$-0.0300959\pi$
$59$	289901.	+	167374.i	1.41154	+	0.814953i	0.416007	+	0.909361i	$0.363429\pi$
$60$		0			0
$61$	−145845.	+	84203.9i	−0.642544	+	0.370973i	−0.785594	−	0.618742i	$-0.787642\pi$
$61$	−145845.	+	84203.9i	−0.642544	+	0.370973i	0.143050	+	0.989716i	$0.454309\pi$
$62$		0			0
$63$	−84882.6	−	50142.7i	−0.339467	−	0.200533i
$64$		0			0
$65$	−52991.6	−	91784.1i	−0.192960	−	0.334216i
$66$		0			0
$67$	−161261.	+	279312.i	−0.536173	+	0.928679i	0.462932	+	0.886394i	$0.346797\pi$
$67$	−161261.	+	279312.i	−0.536173	+	0.928679i	−0.999106	+	0.0422855i	$0.986536\pi$
$68$		0			0
$69$		−	363217.i		−	1.10565i
$70$		0			0
$71$	−325510.			−0.909473			−0.454737	−	0.890626i	$-0.650267\pi$
$71$	−325510.			−0.909473			−0.454737	+	0.890626i	$0.650267\pi$
$72$		0			0
$73$	93404.3	+	53927.0i	0.240103	+	0.138624i	0.615224	−	0.788352i	$-0.289065\pi$
$73$	93404.3	+	53927.0i	0.240103	+	0.138624i	−0.375121	+	0.926976i	$0.622399\pi$
$74$		0			0
$75$	−29031.7	+	16761.4i	−0.0688158	+	0.0397308i
$76$		0			0
$77$	−8138.93	+	815772.i	−0.0178277	+	1.78688i
$78$		0			0
$79$	−169525.	−	293625.i	−0.343836	−	0.595542i	0.641305	−	0.767286i	$-0.278393\pi$
$79$	−169525.	−	293625.i	−0.343836	−	0.595542i	−0.985142	+	0.171744i	$0.945060\pi$
$80$		0			0
$81$	329180.	−	570157.i	0.619411	−	1.07285i
$82$		0			0
$83$		−	224674.i		−	0.392934i	−0.980510	−	0.196467i	$-0.937053\pi$
$83$		−	224674.i		−	0.392934i	0.980510	−	0.196467i	$-0.0629468\pi$
$84$		0			0
$85$	−340582.			−0.554581
$86$		0			0
$87$	906235.	+	523215.i	1.37620	+	0.794552i
$88$		0			0
$89$	331888.	−	191615.i	0.470783	−	0.271807i	−0.245784	−	0.969325i	$-0.579045\pi$
$89$	331888.	−	191615.i	0.470783	−	0.271807i	0.716568	+	0.697518i	$0.245712\pi$
$90$		0			0
$91$	138311.	+	245178.i	0.183541	+	0.325355i
$92$		0			0
$93$	747726.	+	1.29510e6i	0.929595	+	1.61011i
$94$		0			0
$95$	358989.	−	621787.i	0.418707	−	0.725222i
$96$		0			0
$97$		−	32664.1i		−	0.0357895i	−0.999840	−	0.0178948i	$-0.994304\pi$
$97$		−	32664.1i		−	0.0357895i	0.999840	−	0.0178948i	$-0.00569638\pi$
$98$		0			0
$99$	683629.			0.704555
$100$		0			0
$101$	−433003.	−	249994.i	−0.420268	−	0.242642i	0.274924	−	0.961466i	$-0.411347\pi$
$101$	−433003.	−	249994.i	−0.420268	−	0.242642i	−0.695192	+	0.718824i	$0.744681\pi$
$102$		0			0
$103$	−386249.	+	223001.i	−0.353472	+	0.204077i	−0.666214	−	0.745761i	$-0.732086\pi$
$103$	−386249.	+	223001.i	−0.353472	+	0.204077i	0.312741	+	0.949838i	$0.398753\pi$
$104$		0			0
$105$	1.22995e6	−	693844.i	1.06248	−	0.599369i
$106$		0			0
$107$	−944735.	−	1.63633e6i	−0.771186	−	1.33573i	−0.936914	−	0.349561i	$-0.886331\pi$
$107$	−944735.	−	1.63633e6i	−0.771186	−	1.33573i	0.165728	−	0.986171i	$-0.447003\pi$
$108$		0			0
$109$	960930.	−	1.66438e6i	0.742014	−	1.28521i	−0.209563	−	0.977795i	$-0.567204\pi$
$109$	960930.	−	1.66438e6i	0.742014	−	1.28521i	0.951577	−	0.307411i	$-0.0994626\pi$
$110$		0			0
$111$			692640.i			0.506453i
$112$		0			0
$113$	−472481.			−0.327453			−0.163726	−	0.986506i	$-0.552351\pi$
$113$	−472481.			−0.327453			−0.163726	+	0.986506i	$0.552351\pi$
$114$		0			0
$115$	1.27412e6	+	735614.i	0.837756	+	0.483678i
$116$		0			0
$117$	204287.	−	117945.i	0.127551	−	0.0736413i
$118$		0			0
$119$	904570.	+	9024.86i	0.536786	+	0.00535550i
$120$		0			0
$121$	−1.94276e6	−	3.36496e6i	−1.09664	−	1.89943i
$122$		0			0
$123$	1.36845e6	−	2.37022e6i	0.735381	−	1.27372i
$124$		0			0
$125$			1.88199e6i			0.963577i
$126$		0			0
$127$	632211.			0.308639			0.154319	−	0.988021i	$-0.450681\pi$
$127$	632211.			0.308639			0.154319	+	0.988021i	$0.450681\pi$
$128$		0			0
$129$	3.14692e6	+	1.81688e6i	1.46594	+	0.846362i
$130$		0			0
$131$	−111928.	+	64621.5i	−0.0497879	+	0.0287451i	−0.524687	−	0.851295i	$-0.675818\pi$
$131$	−111928.	+	64621.5i	−0.0497879	+	0.0287451i	0.474899	+	0.880040i	$0.342484\pi$
$132$		0			0
$133$	−969934.	+	1.64193e6i	−0.412275	+	0.697908i
$134$		0			0
$135$	909001.	+	1.57444e6i	0.369456	+	0.639917i
$136$		0			0
$137$	798004.	−	1.38218e6i	0.310344	−	0.537532i	−0.668093	−	0.744078i	$-0.732889\pi$
$137$	798004.	−	1.38218e6i	0.310344	−	0.537532i	0.978437	+	0.206546i	$0.0662224\pi$
$138$		0			0
$139$		−	688428.i		−	0.256339i	−0.991752	−	0.128169i	$-0.959090\pi$
$139$		−	688428.i		−	0.256339i	0.991752	−	0.128169i	$-0.0409101\pi$
$140$		0			0
$141$	1.22837e6			0.438199
$142$		0			0
$143$	−1.69049e6	−	976002.i	−0.578101	−	0.333767i
$144$		0			0
$145$	−3.67075e6	+	2.11931e6i	−1.20407	+	0.695170i
$146$		0			0
$147$	−3.28507e6	+	1.81023e6i	−1.03417	+	0.569876i
$148$		0			0
$149$	−316745.	−	548619.i	−0.0957527	−	0.165849i	0.814170	−	0.580627i	$-0.197192\pi$
$149$	−316745.	−	548619.i	−0.0957527	−	0.165849i	−0.909923	+	0.414778i	$0.863859\pi$
$150$		0			0
$151$	−1.24696e6	+	2.15980e6i	−0.362178	+	0.627311i	−0.988319	−	0.152399i	$-0.951300\pi$
$151$	−1.24696e6	+	2.15980e6i	−0.362178	+	0.627311i	0.626141	+	0.779710i	$0.284633\pi$
$152$		0			0
$153$		−	758044.i		−	0.211651i
$154$		0			0
$155$	−6.05741e6			−1.62664
$156$		0			0
$157$	2.23038e6	+	1.28771e6i	0.576342	+	0.332751i	0.759678	−	0.650299i	$-0.225356\pi$
$157$	2.23038e6	+	1.28771e6i	0.576342	+	0.332751i	−0.183336	+	0.983050i	$0.558690\pi$
$158$		0			0
$159$	917781.	−	529881.i	0.228322	−	0.131822i
$160$		0			0
$161$	−3.36451e6	−	1.98752e6i	−0.806203	−	0.476248i
$162$		0			0
$163$	−437858.	−	758391.i	−0.101104	−	0.175118i	0.811036	−	0.584997i	$-0.198904\pi$
$163$	−437858.	−	758391.i	−0.101104	−	0.175118i	−0.912140	+	0.409879i	$0.865571\pi$
$164$		0			0
$165$	−4.89617e6	+	8.48041e6i	−1.08994	+	1.88784i
$166$		0			0
$167$			4.20699e6i			0.903280i	0.892200	+	0.451640i	$0.149161\pi$
$167$			4.20699e6i			0.903280i	−0.892200	+	0.451640i	$0.850839\pi$
$168$		0			0
$169$	4.15326e6			0.860457
$170$		0			0
$171$	1.38393e6	+	799013.i	0.276774	+	0.159796i
$172$		0			0
$173$	−3.28647e6	+	1.89744e6i	−0.634732	+	0.366463i	−0.782583	−	0.622547i	$-0.786098\pi$
$173$	−3.28647e6	+	1.89744e6i	−0.634732	+	0.366463i	0.147850	+	0.989010i	$0.452765\pi$
$174$		0			0
$175$	−3598.11	+	360642.i	−0.000671368	+	0.0672918i
$176$		0			0
$177$	−5.33613e6	−	9.24245e6i	−0.962291	−	1.66674i
$178$		0			0
$179$	662601.	−	1.14766e6i	0.115530	−	0.200103i	−0.802462	−	0.596704i	$-0.796477\pi$
$179$	662601.	−	1.14766e6i	0.115530	−	0.200103i	0.917991	+	0.396600i	$0.129810\pi$
$180$		0			0
$181$			4.90685e6i			0.827499i	0.910391	+	0.413749i	$0.135781\pi$
$181$			4.90685e6i			0.827499i	−0.910391	+	0.413749i	$0.864219\pi$
$182$		0			0
$183$	5.36908e6			0.876085
$184$		0			0
$185$	−2.42970e6	−	1.40279e6i	−0.383741	−	0.221553i
$186$		0			0
$187$	−5.43246e6	+	3.13643e6i	−0.830752	+	0.479635i
$188$		0			0
$189$	−2.37254e6	−	4.20571e6i	−0.351422	−	0.622952i
$190$		0			0
$191$	−3.15391e6	−	5.46273e6i	−0.452636	−	0.783989i	0.545913	−	0.837842i	$-0.316183\pi$
$191$	−3.15391e6	−	5.46273e6i	−0.452636	−	0.783989i	−0.998549	+	0.0538534i	$0.982850\pi$
$192$		0			0
$193$	−437455.	+	757694.i	−0.0608501	+	0.105395i	−0.894846	−	0.446376i	$-0.852714\pi$
$193$	−437455.	+	757694.i	−0.0608501	+	0.105395i	0.833996	+	0.551771i	$0.186048\pi$
$194$		0			0
$195$			3.37889e6i			0.455691i
$196$		0			0
$197$	−6.85421e6			−0.896518			−0.448259	−	0.893904i	$-0.647956\pi$
$197$	−6.85421e6			−0.896518			−0.448259	+	0.893904i	$0.647956\pi$
$198$		0			0
$199$	−8.24491e6	−	4.76020e6i	−1.04623	−	0.604040i	−0.124637	−	0.992202i	$-0.539777\pi$
$199$	−8.24491e6	−	4.76020e6i	−1.04623	−	0.604040i	−0.921591	+	0.388162i	$0.873110\pi$
$200$		0			0
$201$	8.90487e6	−	5.14123e6i	1.09658	−	0.633110i
$202$		0			0
$203$	9.80551e6	−	5.53152e6i	1.17215	−	0.661236i
$204$		0			0
$205$	5.54297e6	+	9.60071e6i	0.643400	+	1.11440i
$206$		0			0
$207$	−1.63728e6	+	2.83585e6i	−0.184591	+	0.319722i
$208$		0			0
$209$		−	1.32238e7i		−	1.44849i
$210$		0			0
$211$	1.02482e7			1.09094			0.545469	−	0.838131i	$-0.316351\pi$
$211$	1.02482e7			1.09094			0.545469	+	0.838131i	$0.316351\pi$
$212$		0			0
$213$	8.98738e6	+	5.18887e6i	0.930025	+	0.536950i
$214$		0			0
$215$	−1.27468e7	+	7.35936e6i	−1.28258	+	0.740499i
$216$		0			0
$217$	1.60882e7	+	160511.i	1.57445	+	0.0157082i
$218$		0			0
$219$	−1.71927e6	−	2.97786e6i	−0.163686	−	0.283512i
$220$		0			0
$221$	−1.08224e6	+	1.87450e6i	−0.100265	+	0.173663i
$222$		0			0
$223$			1.75966e6i			0.158677i	0.996848	+	0.0793385i	$0.0252808\pi$
$223$			1.75966e6i			0.158677i	−0.996848	+	0.0793385i	$0.974719\pi$
$224$		0			0
$225$	302224.			0.0265327
$226$		0			0
$227$	8.44566e6	+	4.87610e6i	0.722031	+	0.416865i	0.815500	−	0.578757i	$-0.196462\pi$
$227$	8.44566e6	+	4.87610e6i	0.722031	+	0.416865i	−0.0934685	+	0.995622i	$0.529795\pi$
$228$		0			0
$229$	−1.75825e7	+	1.01513e7i	−1.46411	+	0.845306i	−0.999198	−	0.0400494i	$-0.987248\pi$
$229$	−1.75825e7	+	1.01513e7i	−1.46411	+	0.845306i	−0.464915	+	0.885355i	$0.653915\pi$
$230$		0			0
$231$	1.32287e7	−	2.23938e7i	1.07320	−	1.81674i
$232$		0			0
$233$	2.71453e6	+	4.70171e6i	0.214599	+	0.371696i	0.953148	−	0.302503i	$-0.0978223\pi$
$233$	2.71453e6	+	4.70171e6i	0.214599	+	0.371696i	−0.738550	+	0.674199i	$0.764489\pi$
$234$		0			0
$235$	−2.48779e6	+	4.30897e6i	−0.191694	+	0.332025i
$236$		0			0
$237$			1.08094e7i			0.811999i
$238$		0			0
$239$	920423.			0.0674208			0.0337104	−	0.999432i	$-0.489268\pi$
$239$	920423.			0.0674208			0.0337104	+	0.999432i	$0.489268\pi$
$240$		0			0
$241$	−2.15894e7	−	1.24646e7i	−1.54237	−	0.890488i	−0.998689	−	0.0511961i	$-0.983697\pi$
$241$	−2.15894e7	−	1.24646e7i	−1.54237	−	0.890488i	−0.543681	−	0.839292i	$-0.682970\pi$
$242$		0			0
$243$	−9.28949e6	+	5.36329e6i	−0.647401	+	0.373777i
$244$		0			0
$245$	303128.	−	1.51899e7i	0.0206124	−	1.03289i
$246$		0			0
$247$	−2.28146e6	−	3.95161e6i	−0.151399	−	0.262231i
$248$		0			0
$249$	−3.58147e6	+	6.20329e6i	−0.231987	+	0.401813i
$250$		0			0
$251$			9.00977e6i			0.569761i	0.958563	+	0.284880i	$0.0919539\pi$
$251$			9.00977e6i			0.569761i	−0.958563	+	0.284880i	$0.908046\pi$
$252$		0			0
$253$	2.70972e7			1.67326
$254$		0			0
$255$	9.40351e6	+	5.42912e6i	0.567113	+	0.327423i
$256$		0			0
$257$	1.52035e7	−	8.77772e6i	0.895659	−	0.517109i	0.0198699	−	0.999803i	$-0.493675\pi$
$257$	1.52035e7	−	8.77772e6i	0.895659	−	0.517109i	0.875789	+	0.482693i	$0.160341\pi$
$258$		0			0
$259$	6.41600e6	+	3.79012e6i	0.369288	+	0.218149i
$260$		0			0
$261$	−4.71701e6	−	8.17011e6i	−0.265305	−	0.459522i
$262$		0			0
$263$	−4.02078e6	+	6.96420e6i	−0.221026	+	0.382828i	−0.955120	−	0.296220i	$-0.904274\pi$
$263$	−4.02078e6	+	6.96420e6i	−0.221026	+	0.382828i	0.734094	+	0.679048i	$0.237607\pi$
$264$		0			0
$265$			4.29263e6i			0.230667i
$266$		0			0
$267$	−1.22179e7			−0.641895
$268$		0			0
$269$	1.41178e7	+	8.15090e6i	0.725286	+	0.418744i	0.816695	−	0.577069i	$-0.195804\pi$
$269$	1.41178e7	+	8.15090e6i	0.725286	+	0.418744i	−0.0914090	+	0.995813i	$0.529137\pi$
$270$		0			0
$271$	2.67932e7	−	1.54691e7i	1.34622	−	0.777243i	0.358511	−	0.933525i	$-0.383284\pi$
$271$	2.67932e7	−	1.54691e7i	1.34622	−	0.777243i	0.987712	+	0.156283i	$0.0499511\pi$
$272$		0			0
$273$	89535.1	−	8.97418e6i	0.00440053	−	0.441069i
$274$		0			0
$275$	−1.25046e6	−	2.16586e6i	−0.0601273	−	0.104144i
$276$		0			0
$277$	6.70800e6	−	1.16186e7i	0.315612	−	0.546656i	−0.663955	−	0.747772i	$-0.731123\pi$
$277$	6.70800e6	−	1.16186e7i	0.315612	−	0.546656i	0.979567	+	0.201116i	$0.0644568\pi$
$278$		0			0
$279$		−	1.34822e7i		−	0.620793i
$280$		0			0
$281$	−1.17351e6			−0.0528895			−0.0264447	−	0.999650i	$-0.508419\pi$
$281$	−1.17351e6			−0.0528895			−0.0264447	+	0.999650i	$0.508419\pi$
$282$		0			0
$283$	2.44869e7	+	1.41375e7i	1.08038	+	0.623756i	0.930998	−	0.365024i	$-0.118939\pi$
$283$	2.44869e7	+	1.41375e7i	1.08038	+	0.623756i	0.149379	+	0.988780i	$0.452273\pi$
$284$		0			0
$285$	−1.98235e7	+	1.14451e7i	−0.856338	+	0.494407i
$286$		0			0
$287$	−1.44675e7	−	2.56459e7i	−0.611993	−	1.08486i
$288$		0			0
$289$	−8.59095e6	−	1.48800e7i	−0.355916	−	0.616465i
$290$		0			0
$291$	−520689.	+	901860.i	−0.0211300	+	0.0365982i
$292$		0			0
$293$		−	5.58159e6i		−	0.221899i	−0.993826	−	0.110949i	$-0.964611\pi$
$293$		−	5.58159e6i		−	0.221899i	0.993826	−	0.110949i	$-0.0353892\pi$
$294$		0			0
$295$	4.32286e7			1.68386
$296$		0			0
$297$	2.89981e7	+	1.67420e7i	1.10688	+	0.639056i
$298$		0			0
$299$	8.09736e6	−	4.67501e6i	0.302921	−	0.174892i
$300$		0			0
$301$	3.40499e7	−	1.92083e7i	1.24858	−	0.704353i
$302$		0			0
$303$	7.97017e6	+	1.38047e7i	0.286510	+	0.496250i
$304$		0			0
$305$	−1.08739e7	+	1.88341e7i	−0.383252	+	0.663812i
$306$		0			0
$307$		−	4.40807e7i		−	1.52347i	−0.647891	−	0.761733i	$-0.724349\pi$
$307$		−	4.40807e7i		−	1.52347i	0.647891	−	0.761733i	$-0.275651\pi$
$308$		0			0
$309$	1.42192e7			0.481946
$310$		0			0
$311$	−2.44238e7	−	1.41011e7i	−0.811956	−	0.468783i	0.0356786	−	0.999363i	$-0.488641\pi$
$311$	−2.44238e7	−	1.41011e7i	−0.811956	−	0.468783i	−0.847635	+	0.530580i	$0.821974\pi$
$312$		0			0
$313$	−5.13461e6	+	2.96447e6i	−0.167446	+	0.0966750i	−0.581381	−	0.813631i	$-0.697487\pi$
$313$	−5.13461e6	+	2.96447e6i	−0.167446	+	0.0966750i	0.413935	+	0.910306i	$0.364154\pi$
$314$		0			0
$315$	−1.27306e7	−	127013.i	−0.407303	−	0.00406365i
$316$		0			0
$317$	−2.26491e7	−	3.92294e7i	−0.711007	−	1.23150i	−0.964480	−	0.264157i	$-0.914906\pi$
$317$	−2.26491e7	−	3.92294e7i	−0.711007	−	1.23150i	0.253473	−	0.967342i	$-0.418427\pi$
$318$		0			0
$319$	−3.90336e7	+	6.76082e7i	−1.20245	+	2.08270i
$320$		0			0
$321$			6.02390e7i			1.82122i
$322$		0			0
$323$	−1.46632e7			−0.435132
$324$		0			0
$325$	−747342.	−	431478.i	−0.0217705	−	0.0125692i
$326$		0			0
$327$	−5.30628e7	+	3.06358e7i	−1.51756	+	0.876165i
$328$		0			0
$329$	6.72162e6	−	1.13785e7i	0.188750	−	0.319520i
$330$		0			0
$331$	−3.34507e7	−	5.79383e7i	−0.922404	−	1.59765i	−0.795685	−	0.605711i	$-0.792889\pi$
$331$	−3.34507e7	−	5.79383e7i	−0.922404	−	1.59765i	−0.126719	−	0.991939i	$-0.540445\pi$
$332$		0			0
$333$	3.12223e6	−	5.40786e6i	0.0845536	−	0.146451i
$334$		0			0
$335$			4.16497e7i			1.10784i
$336$		0			0
$337$	−3.67215e7			−0.959469			−0.479735	−	0.877414i	$-0.659267\pi$
$337$	−3.67215e7			−0.959469			−0.479735	+	0.877414i	$0.659267\pi$
$338$		0			0
$339$	1.30452e7	+	7.53168e6i	0.334852	+	0.193327i
$340$		0			0
$341$	−9.66188e7	+	5.57829e7i	−2.43668	+	1.40682i
$342$		0			0
$343$	−1.20760e6	+	4.03355e7i	−0.0299254	+	0.999552i
$344$		0			0
$345$	−2.34524e7	−	4.06208e7i	−0.571124	−	0.989216i
$346$		0			0
$347$	3.31773e7	−	5.74648e7i	0.794059	−	1.37535i	−0.129375	−	0.991596i	$-0.541297\pi$
$347$	3.31773e7	−	5.74648e7i	0.794059	−	1.37535i	0.923435	−	0.383756i	$-0.125370\pi$
$348$		0			0
$349$			9.44163e6i			0.222111i	0.993814	+	0.111056i	$0.0354232\pi$
$349$			9.44163e6i			0.222111i	−0.993814	+	0.111056i	$0.964577\pi$
$350$		0			0
$351$	1.15539e7			0.267181
$352$		0			0
$353$	9.53150e6	+	5.50301e6i	0.216689	+	0.125105i	0.604416	−	0.796669i	$-0.293406\pi$
$353$	9.53150e6	+	5.50301e6i	0.216689	+	0.125105i	−0.387727	+	0.921774i	$0.626740\pi$
$354$		0			0
$355$	−3.64039e7	+	2.10178e7i	−0.813697	+	0.469788i
$356$		0			0
$357$	−2.48314e7	−	1.46687e7i	−0.545754	−	0.322393i
$358$		0			0
$359$	−2.73638e7	−	4.73954e7i	−0.591415	−	1.02436i	−0.994042	−	0.108997i	$-0.965236\pi$
$359$	−2.73638e7	−	4.73954e7i	−0.591415	−	1.02436i	0.402627	−	0.915364i	$-0.368097\pi$
$360$		0			0
$361$	−8.06726e6	+	1.39729e7i	−0.171476	+	0.297006i
$362$		0			0
$363$			1.23876e8i			2.58980i
$364$		0			0
$365$	1.39280e7			0.286424
$366$		0			0
$367$	8.26059e7	+	4.76926e7i	1.67114	+	0.964834i	0.967001	+	0.254773i	$0.0820007\pi$
$367$	8.26059e7	+	4.76926e7i	1.67114	+	0.964834i	0.704140	+	0.710061i	$0.251333\pi$
$368$		0			0
$369$	−2.13686e7	+	1.23371e7i	−0.425301	+	0.245548i
$370$		0			0
$371$	113747.	−	1.14010e7i	0.00222751	−	0.223265i
$372$		0			0
$373$	2.11254e7	+	3.65902e7i	0.407078	+	0.705081i	0.994561	−	0.104156i	$-0.0332142\pi$
$373$	2.11254e7	+	3.65902e7i	0.407078	+	0.705081i	−0.587483	+	0.809237i	$0.699881\pi$
$374$		0			0
$375$	3.00002e7	−	5.19619e7i	0.568893	−	0.985351i
$376$		0			0
$377$			2.69375e7i			0.502728i
$378$		0			0
$379$	−2.92437e7			−0.537173			−0.268587	−	0.963256i	$-0.586557\pi$
$379$	−2.92437e7			−0.537173			−0.268587	+	0.963256i	$0.586557\pi$
$380$		0			0
$381$	−1.74554e7	−	1.00779e7i	−0.315613	−	0.182219i
$382$		0			0
$383$	−2.53968e7	+	1.46629e7i	−0.452047	+	0.260989i	−0.708694	−	0.705516i	$-0.750715\pi$
$383$	−2.53968e7	+	1.46629e7i	−0.452047	+	0.260989i	0.256647	+	0.966505i	$0.417382\pi$
$384$		0			0
$385$	5.17631e7	+	9.17584e7i	0.907065	+	1.60792i
$386$		0			0
$387$	−1.63799e7	−	2.83709e7i	−0.282605	−	0.489486i
$388$		0			0
$389$	5.20331e7	−	9.01241e7i	0.883957	−	1.53106i	0.0370533	−	0.999313i	$-0.488203\pi$
$389$	5.20331e7	−	9.01241e7i	0.883957	−	1.53106i	0.846904	−	0.531746i	$-0.178464\pi$
$390$		0			0
$391$		−	3.00468e7i		−	0.502652i
$392$		0			0
$393$	4.12045e6			0.0678840
$394$		0			0
$395$	−3.79180e7	−	2.18920e7i	−0.615254	−	0.355217i
$396$		0			0
$397$	6.06717e7	−	3.50288e7i	0.969649	−	0.559827i	0.0705197	−	0.997510i	$-0.477534\pi$
$397$	6.06717e7	−	3.50288e7i	0.969649	−	0.559827i	0.899129	+	0.437683i	$0.144201\pi$
$398$		0			0
$399$	5.29535e7	−	2.98723e7i	0.833634	−	0.470273i
$400$		0			0
$401$	−3.84740e7	−	6.66389e7i	−0.596670	−	1.03346i	−0.993309	−	0.115488i	$-0.963157\pi$
$401$	−3.84740e7	−	6.66389e7i	−0.596670	−	1.03346i	0.396639	−	0.917975i	$-0.370177\pi$
$402$		0			0
$403$	−1.92482e7	+	3.33388e7i	−0.294086	+	0.509372i
$404$		0			0
$405$		−	8.50190e7i		−	1.27983i
$406$		0			0
$407$	−5.16733e7			−0.766449
$408$		0			0
$409$	3.50627e7	+	2.02435e7i	0.512479	+	0.295880i	0.733852	−	0.679309i	$-0.237721\pi$
$409$	3.50627e7	+	2.02435e7i	0.512479	+	0.295880i	−0.221373	+	0.975189i	$0.571054\pi$
$410$		0			0
$411$	−4.40660e7	+	2.54415e7i	−0.634714	+	0.366452i
$412$		0			0
$413$	−1.14813e8	−	1.14549e6i	−1.62983	−	0.0162607i
$414$		0			0
$415$	−1.45069e7	−	2.51268e7i	−0.202970	−	0.351554i
$416$		0			0
$417$	−1.09740e7	+	1.90076e7i	−0.151342	+	0.262131i
$418$		0			0
$419$		−	5.02960e7i		−	0.683741i	−0.939747	−	0.341870i	$-0.888940\pi$
$419$		−	5.02960e7i		−	0.683741i	0.939747	−	0.341870i	$-0.111060\pi$
$420$		0			0
$421$	1.61043e6			0.0215822			0.0107911	−	0.999942i	$-0.496565\pi$
$421$	1.61043e6			0.0215822			0.0107911	+	0.999942i	$0.496565\pi$
$422$		0			0
$423$	−9.59061e6	−	5.53714e6i	−0.126714	−	0.0731584i
$424$		0			0
$425$	−2.40162e6	+	1.38658e6i	−0.0312851	+	0.0180624i
$426$		0			0
$427$	2.93796e7	−	4.97343e7i	0.377365	−	0.638811i
$428$		0			0
$429$	3.11163e7	+	5.38951e7i	0.394109	+	0.682617i
$430$		0			0
$431$	3.45151e7	−	5.97819e7i	0.431099	−	0.746686i	−0.565869	−	0.824495i	$-0.691459\pi$
$431$	3.45151e7	−	5.97819e7i	0.431099	−	0.746686i	0.996968	+	0.0778092i	$0.0247925\pi$
$432$		0			0
$433$			7.91462e7i			0.974915i	0.873147	+	0.487457i	$0.162076\pi$
$433$			7.91462e7i			0.974915i	−0.873147	+	0.487457i	$0.837924\pi$
$434$		0			0
$435$	1.35133e8			1.64170
$436$		0			0
$437$	5.48552e7	+	3.16707e7i	0.657315	+	0.379501i
$438$		0			0
$439$	−2.09035e7	+	1.20687e7i	−0.247073	+	0.142648i	−0.618423	−	0.785845i	$-0.712228\pi$
$439$	−2.09035e7	+	1.20687e7i	−0.247073	+	0.142648i	0.371350	+	0.928493i	$0.378895\pi$
$440$		0			0
$441$	3.38085e7	+	674680.i	0.394195	+	0.00786651i
$442$		0			0
$443$	2.35504e7	+	4.07904e7i	0.270886	+	0.469188i	0.969089	−	0.246712i	$-0.0793500\pi$
$443$	2.35504e7	+	4.07904e7i	0.270886	+	0.469188i	−0.698203	+	0.715900i	$0.746017\pi$
$444$		0			0
$445$	2.47447e7	−	4.28591e7i	0.280804	−	0.486366i
$446$		0			0
$447$			2.01966e7i			0.226128i
$448$		0			0
$449$	−4.69720e6			−0.0518920			−0.0259460	−	0.999663i	$-0.508260\pi$
$449$	−4.69720e6			−0.0518920			−0.0259460	+	0.999663i	$0.508260\pi$
$450$		0			0
$451$	1.76826e8	+	1.02091e8i	1.92760	+	1.11290i
$452$		0			0
$453$	6.88575e7	−	3.97549e7i	0.740724	−	0.427657i
$454$		0			0
$455$	3.12991e7	+	1.84893e7i	0.332275	+	0.196285i
$456$		0			0
$457$	−5.91484e7	−	1.02448e8i	−0.619718	−	1.07338i	−0.989537	−	0.144280i	$-0.953914\pi$
$457$	−5.91484e7	−	1.02448e8i	−0.619718	−	1.07338i	0.369819	−	0.929104i	$-0.379420\pi$
$458$		0			0
$459$	1.85644e7	−	3.21545e7i	0.191975	−	0.332510i
$460$		0			0
$461$		−	1.71817e8i		−	1.75373i	−0.480733	−	0.876867i	$-0.659629\pi$
$461$		−	1.71817e8i		−	1.75373i	0.480733	−	0.876867i	$-0.340371\pi$
$462$		0			0
$463$	−4.58717e7			−0.462170			−0.231085	−	0.972934i	$-0.574228\pi$
$463$	−4.58717e7			−0.462170			−0.231085	+	0.972934i	$0.574228\pi$
$464$		0			0
$465$	1.67246e8	+	9.65595e7i	1.66340	+	0.960364i
$466$		0			0
$467$	−1.12372e7	+	6.48779e6i	−0.110333	+	0.0637010i	−0.554151	−	0.832416i	$-0.686957\pi$
$467$	−1.12372e7	+	6.48779e6i	−0.110333	+	0.0637010i	0.443818	+	0.896117i	$0.353624\pi$
$468$		0			0
$469$	1.10365e6	−	1.10620e8i	0.0106982	−	1.07229i
$470$		0			0
$471$	−4.10541e7	−	7.11077e7i	−0.392910	−	0.680541i
$472$		0			0
$473$	−1.35545e8	+	2.34771e8i	−1.28086	+	2.21851i
$474$		0			0
$475$		−	5.84605e6i		−	0.0545484i
$476$		0			0
$477$	−9.55422e6			−0.0880319
$478$		0			0
$479$	2.00019e7	+	1.15481e7i	0.181997	+	0.105076i	0.588231	−	0.808693i	$-0.299825\pi$
$479$	2.00019e7	+	1.15481e7i	0.181997	+	0.105076i	−0.406233	+	0.913769i	$0.633158\pi$
$480$		0			0
$481$	−1.54414e7	+	8.91507e6i	−0.138756	+	0.0801105i
$482$		0			0
$483$	6.12121e7	+	1.08508e8i	0.543246	+	0.962990i
$484$		0			0
$485$	−2.10908e6	−	3.65304e6i	−0.0184871	−	0.0320205i
$486$		0			0
$487$	−1.67086e7	+	2.89401e7i	−0.144661	+	0.250561i	−0.929246	−	0.369460i	$-0.879543\pi$
$487$	−1.67086e7	+	2.89401e7i	−0.144661	+	0.250561i	0.784585	+	0.620021i	$0.212876\pi$
$488$		0			0
$489$			2.79190e7i			0.238767i
$490$		0			0
$491$	−1.96184e8			−1.65737			−0.828686	−	0.559714i	$-0.810911\pi$
$491$	−1.96184e8			−1.65737			−0.828686	+	0.559714i	$0.810911\pi$
$492$		0			0
$493$	7.49674e7	+	4.32825e7i	0.625651	+	0.361220i
$494$		0			0
$495$	7.64546e7	−	4.41411e7i	0.630359	−	0.363938i
$496$		0			0
$497$	9.72439e7	−	5.48576e7i	0.792124	−	0.446856i
$498$		0			0
$499$	4.63697e7	+	8.03147e7i	0.373192	+	0.646388i	0.990055	−	0.140683i	$-0.0449298\pi$
$499$	4.63697e7	+	8.03147e7i	0.373192	+	0.646388i	−0.616862	+	0.787071i	$0.711597\pi$
$500$		0			0
$501$	6.70625e7	−	1.16156e8i	0.533294	−	0.923692i
$502$		0			0
$503$		−	1.26998e8i		−	0.997914i	−0.866627	−	0.498957i	$-0.833717\pi$
$503$		−	1.26998e8i		−	0.997914i	0.866627	−	0.498957i	$-0.166283\pi$
$504$		0			0
$505$	−6.45673e7			−0.501347
$506$		0			0
$507$	−1.14672e8	−	6.62059e7i	−0.879900	−	0.508011i
$508$		0			0
$509$	−1.24437e8	+	7.18438e7i	−0.943619	+	0.544799i	−0.891093	−	0.453821i	$-0.850061\pi$
$509$	−1.24437e8	+	7.18438e7i	−0.943619	+	0.544799i	−0.0525259	+	0.998620i	$0.516727\pi$
$510$		0			0
$511$	−3.69921e7	−	369069.i	−0.277234	−	0.00276595i
$512$		0			0
$513$	3.91355e7	+	6.77847e7i	0.289881	+	0.502088i
$514$		0			0
$515$	−2.87978e7	+	4.98792e7i	−0.210832	+	0.365172i
$516$		0			0
$517$			9.16404e7i			0.663156i
$518$		0			0
$519$	1.20986e8			0.865434
$520$		0			0
$521$	−1.12128e8	−	6.47370e7i	−0.792866	−	0.457761i	0.0481045	−	0.998842i	$-0.484682\pi$
$521$	−1.12128e8	−	6.47370e7i	−0.792866	−	0.457761i	−0.840971	+	0.541081i	$0.818015\pi$
$522$		0			0
$523$	2.11656e8	−	1.22200e8i	1.47954	−	0.854210i	0.479804	−	0.877376i	$-0.340708\pi$
$523$	2.11656e8	−	1.22200e8i	1.47954	−	0.854210i	0.999732	+	0.0231657i	$0.00737454\pi$
$524$		0			0
$525$	5.84823e6	−	9.90001e6i	0.0404154	−	0.0684160i
$526$		0			0
$527$	6.18550e7	+	1.07136e8i	0.422613	+	0.731987i
$528$		0			0
$529$	9.12076e6	−	1.57976e7i	0.0616118	−	0.106715i
$530$		0			0
$531$			9.62151e7i			0.642628i
$532$		0			0
$533$	7.04539e7			0.465290
$534$		0			0
$535$	−2.11311e8	−	1.22001e8i	−1.37995	−	0.796712i
$536$		0			0
$537$	−3.65890e7	+	2.11247e7i	−0.236280	+	0.136417i
$538$		0			0
$539$	−1.35049e8	−	2.45078e8i	−0.862432	−	1.56508i
$540$		0			0
$541$	2.21709e6	+	3.84011e6i	0.0140020	+	0.0242522i	0.872942	−	0.487825i	$-0.162210\pi$
$541$	2.21709e6	+	3.84011e6i	0.0140020	+	0.0242522i	−0.858939	+	0.512077i	$0.828876\pi$
$542$		0			0
$543$	7.82187e7	−	1.35479e8i	0.488553	−	0.846198i
$544$		0			0
$545$		−	2.48184e8i		−	1.53315i
$546$		0			0
$547$	7.96292e7			0.486531			0.243265	−	0.969960i	$-0.421781\pi$
$547$	7.96292e7			0.486531			0.243265	+	0.969960i	$0.421781\pi$
$548$		0			0
$549$	−4.19196e7	−	2.42023e7i	−0.253338	−	0.146265i
$550$		0			0
$551$	−1.58038e8	+	9.12434e7i	−0.944729	+	0.545440i
$552$		0			0
$553$	1.00128e8	+	5.91488e7i	0.592082	+	0.349761i
$554$		0			0
$555$	4.47229e7	+	7.74624e7i	0.261608	+	0.453119i
$556$		0			0
$557$	−1.46346e7	+	2.53478e7i	−0.0846865	+	0.146681i	−0.905258	−	0.424863i	$-0.860322\pi$
$557$	−1.46346e7	+	2.53478e7i	−0.0846865	+	0.146681i	0.820571	+	0.571544i	$0.193656\pi$
$558$		0			0
$559$			9.35411e7i			0.535509i
$560$		0			0
$561$	1.99988e8			1.13270
$562$		0			0
$563$	−1.62041e8	−	9.35545e7i	−0.908030	−	0.524251i	−0.0282330	−	0.999601i	$-0.508988\pi$
$563$	−1.62041e8	−	9.35545e7i	−0.908030	−	0.524251i	−0.879797	+	0.475350i	$0.842321\pi$
$564$		0			0
$565$	−5.28405e7	+	3.05075e7i	−0.292969	+	0.169146i
$566$		0			0
$567$	−2.25286e6	+	2.25806e8i	−0.0123591	+	1.23876i
$568$		0			0
$569$	−3.18400e7	−	5.51485e7i	−0.172837	−	0.299362i	0.766574	−	0.642156i	$-0.221960\pi$
$569$	−3.18400e7	−	5.51485e7i	−0.172837	−	0.299362i	−0.939410	+	0.342794i	$0.888627\pi$
$570$		0			0
$571$	−7.74210e7	+	1.34097e8i	−0.415863	+	0.720296i	−0.995519	−	0.0945655i	$-0.969854\pi$
$571$	−7.74210e7	+	1.34097e8i	−0.415863	+	0.720296i	0.579655	+	0.814862i	$0.303187\pi$
$572$		0			0
$573$			2.01102e8i			1.06894i
$574$		0			0
$575$	1.19793e7			0.0630127
$576$		0			0
$577$	−2.73237e8	−	1.57754e8i	−1.42237	−	0.821205i	−0.425868	−	0.904785i	$-0.640031\pi$
$577$	−2.73237e8	−	1.57754e8i	−1.42237	−	0.821205i	−0.996501	+	0.0835798i	$0.973365\pi$
$578$		0			0
$579$	2.41564e7	−	1.39467e7i	0.124450	−	0.0718514i
$580$		0			0
$581$	3.78639e7	+	6.71199e7i	0.193062	+	0.342234i
$582$		0			0
$583$	3.95309e7	+	6.84696e7i	0.199495	+	0.345535i
$584$		0			0
$585$	1.52311e7	−	2.63811e7i	0.0760789	−	0.131772i
$586$		0			0
$587$		−	5.57952e7i		−	0.275856i	−0.990442	−	0.137928i	$-0.955956\pi$
$587$		−	5.57952e7i		−	0.275856i	0.990442	−	0.137928i	$-0.0440443\pi$
$588$		0			0
$589$	−2.60792e8			−1.27629
$590$		0			0
$591$	1.89246e8	+	1.09261e8i	0.916776	+	0.529301i
$592$		0			0
$593$	2.39874e8	−	1.38491e8i	1.15032	−	0.664137i	0.201354	−	0.979518i	$-0.435466\pi$
$593$	2.39874e8	−	1.38491e8i	1.15032	−	0.664137i	0.948965	+	0.315381i	$0.102132\pi$
$594$		0			0
$595$	1.01747e8	−	5.73976e7i	0.483024	−	0.272485i
$596$		0			0
$597$	1.51762e8	+	2.62859e8i	0.713247	+	1.23538i
$598$		0			0
$599$	−9.32594e7	+	1.61530e8i	−0.433923	+	0.751576i	−0.997207	−	0.0746871i	$-0.976204\pi$
$599$	−9.32594e7	+	1.61530e8i	−0.433923	+	0.751576i	0.563284	+	0.826263i	$0.309538\pi$
$600$		0			0
$601$			2.43621e8i			1.12226i	0.827729	+	0.561128i	$0.189632\pi$
$601$			2.43621e8i			1.12226i	−0.827729	+	0.561128i	$0.810368\pi$
$602$		0			0
$603$	−9.27009e7			−0.422797
$604$		0			0
$605$	−4.34542e8	−	2.50883e8i	−1.96230	−	1.13294i
$606$		0			0
$607$	−9.01650e7	+	5.20568e7i	−0.403155	+	0.232762i	−0.687844	−	0.725858i	$-0.741443\pi$
$607$	−9.01650e7	+	5.20568e7i	−0.403155	+	0.232762i	0.284689	+	0.958620i	$0.408110\pi$
$608$		0			0
$609$	−3.58908e8	−	3.58081e6i	−1.58903	−	0.0158537i
$610$		0			0
$611$	1.58105e7	+	2.73846e7i	0.0693142	+	0.120056i
$612$		0			0
$613$	−1.42147e8	+	2.46207e8i	−0.617103	+	1.06885i	0.372909	+	0.927868i	$0.378361\pi$
$613$	−1.42147e8	+	2.46207e8i	−0.617103	+	1.06885i	−0.990012	+	0.140986i	$0.954973\pi$
$614$		0			0
$615$		−	3.53436e8i		−	1.51944i
$616$		0			0
$617$	−1.78930e8			−0.761776			−0.380888	−	0.924621i	$-0.624382\pi$
$617$	−1.78930e8			−0.761776			−0.380888	+	0.924621i	$0.624382\pi$
$618$		0			0
$619$	−2.04695e8	−	1.18181e8i	−0.863048	−	0.498281i	0.00198390	−	0.999998i	$-0.499369\pi$
$619$	−2.04695e8	−	1.18181e8i	−0.863048	−	0.498281i	−0.865032	+	0.501717i	$0.832702\pi$
$620$		0			0
$621$	−1.38900e8	+	8.01937e7i	−0.579997	+	0.334862i
$622$		0			0
$623$	−6.68565e7	+	1.13176e8i	−0.276490	+	0.468048i
$624$		0			0
$625$	1.29732e8	+	2.24703e8i	0.531383	+	0.920383i
$626$		0			0
$627$	−2.10796e8	+	3.65110e8i	−0.855185	+	1.48122i
$628$		0			0
$629$			5.72980e7i			0.230244i
$630$		0			0
$631$	−1.51341e8			−0.602378			−0.301189	−	0.953564i	$-0.597384\pi$
$631$	−1.51341e8			−0.602378			−0.301189	+	0.953564i	$0.597384\pi$
$632$		0			0
$633$	−2.82954e8	−	1.63364e8i	−1.11559	−	0.644087i
$634$		0			0
$635$	7.07041e7	−	4.08211e7i	0.276136	−	0.159427i
$636$		0			0
$637$	−8.26389e7	−	4.99360e7i	−0.319717	−	0.193195i
$638$		0			0
$639$	−4.67799e7	−	8.10252e7i	−0.179290	−	0.310540i
$640$		0			0
$641$	1.15332e8	−	1.99760e8i	0.437899	−	0.758463i	−0.559628	−	0.828744i	$-0.689056\pi$
$641$	1.15332e8	−	1.99760e8i	0.437899	−	0.758463i	0.997527	+	0.0702803i	$0.0223894\pi$
$642$		0			0
$643$			1.68539e8i			0.633967i	0.948431	+	0.316983i	$0.102670\pi$
$643$			1.68539e8i			0.633967i	−0.948431	+	0.316983i	$0.897330\pi$
$644$		0			0
$645$	4.69254e8			1.74875
$646$		0			0
$647$	1.43881e8	+	8.30696e7i	0.531239	+	0.306711i	0.741521	−	0.670930i	$-0.234105\pi$
$647$	1.43881e8	+	8.30696e7i	0.531239	+	0.306711i	−0.210282	+	0.977641i	$0.567438\pi$
$648$		0			0
$649$	6.89518e8	−	3.98093e8i	2.52239	−	1.45630i
$650$		0			0
$651$	−4.41638e8	−	2.60889e8i	−1.60075	−	0.945612i
$652$		0			0
$653$	1.18804e8	+	2.05775e8i	0.426670	+	0.739014i	0.996575	−	0.0826970i	$-0.0263534\pi$
$653$	1.18804e8	+	2.05775e8i	0.426670	+	0.739014i	−0.569905	+	0.821711i	$0.693020\pi$
$654$		0			0
$655$	−8.34506e6	+	1.44541e7i	−0.0296965	+	0.0514359i
$656$		0			0
$657$			3.09999e7i			0.109311i
$658$		0			0
$659$	3.19568e8			1.11662			0.558312	−	0.829631i	$-0.311449\pi$
$659$	3.19568e8			1.11662			0.558312	+	0.829631i	$0.311449\pi$
$660$		0			0
$661$	3.66423e8	+	2.11554e8i	1.26876	+	0.732517i	0.974753	−	0.223287i	$-0.0716787\pi$
$661$	3.66423e8	+	2.11554e8i	1.26876	+	0.732517i	0.294004	+	0.955804i	$0.405012\pi$
$662$		0			0
$663$	5.97617e7	−	3.45034e7i	0.205061	−	0.118392i
$664$		0			0
$665$	−2.45687e6	+	2.46254e8i	−0.00835444	+	0.837373i
$666$		0			0
$667$	−1.86969e8	−	3.23840e8i	−0.630076	−	1.09132i
$668$		0			0
$669$	2.80502e7	−	4.85844e7i	0.0936824	−	0.162263i
$670$		0			0
$671$			4.00551e8i			1.32584i
$672$		0			0
$673$	−2.60395e8			−0.854256			−0.427128	−	0.904191i	$-0.640475\pi$
$673$	−2.60395e8			−0.854256			−0.427128	+	0.904191i	$0.640475\pi$
$674$		0			0
$675$	1.28197e7	+	7.40144e6i	0.0416836	+	0.0240660i
$676$		0			0
$677$	−2.43309e8	+	1.40475e8i	−0.784139	+	0.452723i	−0.837895	−	0.545831i	$-0.816214\pi$
$677$	−2.43309e8	+	1.40475e8i	−0.784139	+	0.452723i	0.0537564	+	0.998554i	$0.482881\pi$
$678$		0			0
$679$	5.50482e6	+	9.75817e6i	0.0175846	+	0.0311716i
$680$		0			0
$681$	−1.55457e8	−	2.69260e8i	−0.492231	−	0.852570i
$682$		0			0
$683$	1.08581e8	−	1.88068e8i	0.340794	−	0.590272i	−0.643787	−	0.765205i	$-0.722638\pi$
$683$	1.08581e8	−	1.88068e8i	0.340794	−	0.590272i	0.984580	+	0.174933i	$0.0559709\pi$
$684$		0			0
$685$		−	2.06105e8i		−	0.641233i
$686$		0			0
$687$	6.47274e8			1.99626
$688$		0			0
$689$	2.36258e7	+	1.36403e7i	0.0722318	+	0.0417031i
$690$		0			0
$691$	4.52106e8	−	2.61024e8i	1.37027	−	0.791126i	0.379309	−	0.925270i	$-0.376162\pi$
$691$	4.52106e8	−	2.61024e8i	1.37027	−	0.791126i	0.990962	+	0.134144i	$0.0428285\pi$
$692$		0			0
$693$	−2.04229e8	+	1.15211e8i	−0.613647	+	0.346173i
$694$		0			0
$695$	−4.44509e7	−	7.69913e7i	−0.132412	−	0.229344i
$696$		0			0
$697$	1.13204e8	−	1.96074e8i	0.334319	−	0.579058i
$698$		0			0
$699$		−	1.73086e8i		−	0.506794i
$700$		0			0
$701$	3.81338e8			1.10702			0.553511	−	0.832842i	$-0.313288\pi$
$701$	3.81338e8			1.10702			0.553511	+	0.832842i	$0.313288\pi$
$702$		0			0
$703$	−1.04607e8	−	6.03948e7i	−0.301088	−	0.173833i
$704$		0			0
$705$	1.37376e8	−	7.93142e7i	0.392052	−	0.226352i
$706$		0			0
$707$	1.71488e8	+	1.71093e6i	0.485260	+	0.00484142i
$708$		0			0
$709$	−1.19766e8	−	2.07441e8i	−0.336043	−	0.582043i	0.647642	−	0.761945i	$-0.275755\pi$
$709$	−1.19766e8	−	2.07441e8i	−0.336043	−	0.582043i	−0.983685	+	0.179902i	$0.942422\pi$
$710$		0			0
$711$	4.87256e7	−	8.43953e7i	0.135565	−	0.234806i
$712$		0			0
$713$		−	5.34396e8i		−	1.47433i
$714$		0			0
$715$	−2.52077e8			−0.689628
$716$		0			0
$717$	−2.54130e7	−	1.46722e7i	−0.0689443	−	0.0398050i
$718$		0			0
$719$	4.18017e8	−	2.41342e8i	1.12462	−	0.649302i	0.182047	−	0.983290i	$-0.441728\pi$
$719$	4.18017e8	−	2.41342e8i	1.12462	−	0.649302i	0.942577	+	0.333988i	$0.108394\pi$
$720$		0			0
$721$	7.78072e7	−	1.31714e8i	0.207594	−	0.351419i
$722$		0			0
$723$	3.97390e8	+	6.88299e8i	1.05148	+	1.82122i
$724$		0			0
$725$	−1.72562e7	+	2.98887e7i	−0.0452827	+	0.0784319i
$726$		0			0
$727$		−	5.49464e8i		−	1.43000i	−0.699124	−	0.715000i	$-0.746426\pi$
$727$		−	5.49464e8i		−	1.43000i	0.699124	−	0.715000i	$-0.253574\pi$
$728$		0			0
$729$	−1.37966e8			−0.356115
$730$		0			0
$731$	2.60326e8	+	1.50299e8i	0.666447	+	0.384774i
$732$		0			0
$733$	5.09314e8	−	2.94053e8i	1.29322	−	0.746643i	0.313999	−	0.949423i	$-0.398331\pi$
$733$	5.09314e8	−	2.94053e8i	1.29322	−	0.746643i	0.979224	+	0.202780i	$0.0649977\pi$
$734$		0			0
$735$	−2.50507e8	+	4.14562e8i	−0.630895	+	1.04407i
$736$		0			0
$737$	3.83553e8	+	6.64334e8i	0.958127	+	1.65952i
$738$		0			0
$739$	3.92818e8	−	6.80381e8i	0.973325	−	1.68585i	0.287972	−	0.957639i	$-0.407019\pi$
$739$	3.92818e8	−	6.80381e8i	0.973325	−	1.68585i	0.685354	−	0.728210i	$-0.259648\pi$
$740$		0			0
$741$			1.45473e8i			0.357542i
$742$		0			0
$743$	−5.34716e8			−1.30364			−0.651819	−	0.758375i	$-0.725994\pi$
$743$	−5.34716e8			−1.30364			−0.651819	+	0.758375i	$0.725994\pi$
$744$		0			0
$745$	−7.08472e7	−	4.09037e7i	−0.171338	−	0.0989221i
$746$		0			0
$747$	5.59253e7	−	3.22885e7i	0.134167	−	0.0774616i
$748$		0			0
$749$	5.58000e8	+	3.29628e8i	1.32797	+	0.784473i
$750$		0			0
$751$	3.36530e8	+	5.82887e8i	0.794518	+	1.37615i	0.923145	+	0.384453i	$0.125610\pi$
$751$	3.36530e8	+	5.82887e8i	0.794518	+	1.37615i	−0.128626	+	0.991693i	$0.541057\pi$
$752$		0			0
$753$	1.43622e8	−	2.48761e8i	0.336385	−	0.582636i
$754$		0			0
$755$			3.22059e8i			0.748332i
$756$		0			0
$757$	−4.17395e8			−0.962188			−0.481094	−	0.876669i	$-0.659760\pi$
$757$	−4.17395e8			−0.962188			−0.481094	+	0.876669i	$0.659760\pi$
$758$		0			0
$759$	−7.48156e8	−	4.31948e8i	−1.71107	−	0.987885i
$760$		0			0
$761$	−3.96918e8	+	2.29161e8i	−0.900631	+	0.519980i	−0.877405	−	0.479751i	$-0.840727\pi$
$761$	−3.96918e8	+	2.29161e8i	−0.900631	+	0.519980i	−0.0232261	+	0.999730i	$0.507394\pi$
$762$		0			0
$763$	−6.57647e6	+	6.59165e8i	−0.0148054	+	1.48395i
$764$		0			0
$765$	−4.89459e7	−	8.47768e7i	−0.109328	−	0.189362i
$766$		0			0
$767$	1.37364e8	−	2.37922e8i	0.304430	−	0.527288i
$768$		0			0
$769$		−	4.67013e8i		−	1.02695i	−0.858104	−	0.513475i	$-0.828358\pi$
$769$		−	4.67013e8i		−	1.02695i	0.858104	−	0.513475i	$-0.171642\pi$
$770$		0			0
$771$	−5.59692e8			−1.22120
$772$		0			0
$773$	−4.73930e7	−	2.73624e7i	−0.102607	−	0.0592400i	0.447819	−	0.894124i	$-0.352201\pi$
$773$	−4.73930e7	−	2.73624e7i	−0.102607	−	0.0592400i	−0.550425	+	0.834884i	$0.685534\pi$
$774$		0			0
$775$	−4.27139e7	+	2.46609e7i	−0.0917623	+	0.0529790i
$776$		0			0
$777$	−1.16729e8	−	2.06921e8i	−0.248838	−	0.441105i
$778$		0			0
$779$	2.38643e8	+	4.13342e8i	0.504820	+	0.874375i
$780$		0			0
$781$	−3.87107e8	+	6.70489e8i	−0.812602	+	1.40747i
$782$		0			0
$783$		−	4.62077e8i		−	0.962563i
$784$		0			0
$785$	3.32584e8			0.687530
$786$		0			0
$787$	−3.05483e8	−	1.76371e8i	−0.626704	−	0.361828i	0.152770	−	0.988262i	$-0.451181\pi$
$787$	−3.05483e8	−	1.76371e8i	−0.626704	−	0.361828i	−0.779475	+	0.626434i	$0.784514\pi$
$788$		0			0
$789$	2.22029e8	−	1.28188e8i	0.452041	−	0.260986i
$790$		0			0
$791$	1.41150e8	−	7.96262e7i	0.285202	−	0.160889i
$792$		0			0
$793$	6.91062e7	+	1.19695e8i	0.138579	+	0.240026i
$794$		0			0
$795$	6.84275e7	−	1.18520e8i	0.136185	−	0.235879i
$796$		0			0
$797$			6.57525e8i			1.29879i	0.760453	+	0.649393i	$0.224977\pi$
$797$			6.57525e8i			1.29879i	−0.760453	+	0.649393i	$0.775023\pi$
$798$		0			0
$799$	1.01616e8			0.199214
$800$		0			0
$801$	9.53928e7	+	5.50751e7i	0.185617	+	0.107166i
$802$		0			0
$803$	2.22159e8	−	1.28263e8i	0.429058	−	0.247717i
$804$		0			0
$805$	−5.04606e8	−	5.03444e6i	−0.967309	−	0.00965081i
$806$		0			0
$807$	−2.59862e8	−	4.50095e8i	−0.494450	−	0.856413i
$808$		0			0
$809$	3.44264e8	−	5.96282e8i	0.650198	−	1.12618i	−0.332876	−	0.942971i	$-0.608019\pi$
$809$	3.44264e8	−	5.96282e8i	0.650198	−	1.12618i	0.983075	−	0.183206i	$-0.0586475\pi$
$810$		0			0
$811$			4.12718e8i			0.773732i	0.922136	+	0.386866i	$0.126442\pi$
$811$			4.12718e8i			0.773732i	−0.922136	+	0.386866i	$0.873558\pi$
$812$		0			0
$813$	−9.86353e8			−1.83553
$814$		0			0
$815$	−9.79368e7	−	5.65438e7i	−0.180914	−	0.104451i
$816$		0			0
$817$	−5.48792e8	+	3.16845e8i	−1.00633	+	0.581006i
$818$		0			0
$819$	−4.11522e7	+	6.96632e7i	−0.0749102	+	0.126810i
$820$		0			0
$821$	1.98839e8	+	3.44399e8i	0.359312	+	0.622347i	0.987846	−	0.155435i	$-0.0496780\pi$
$821$	1.98839e8	+	3.44399e8i	0.359312	+	0.622347i	−0.628534	+	0.777782i	$0.716345\pi$
$822$		0			0
$823$	−1.08211e8	+	1.87427e8i	−0.194121	+	0.336228i	−0.946612	−	0.322375i	$-0.895519\pi$
$823$	−1.08211e8	+	1.87427e8i	−0.194121	+	0.336228i	0.752491	+	0.658603i	$0.228852\pi$
$824$		0			0
$825$			7.97329e7i			0.141996i
$826$		0			0
$827$	−4.58785e8			−0.811134			−0.405567	−	0.914065i	$-0.632926\pi$
$827$	−4.58785e8			−0.811134			−0.405567	+	0.914065i	$0.632926\pi$
$828$		0			0
$829$	6.95641e8	+	4.01628e8i	1.22102	+	0.704954i	0.965135	−	0.261754i	$-0.0843010\pi$
$829$	6.95641e8	+	4.01628e8i	1.22102	+	0.704954i	0.255881	+	0.966708i	$0.417634\pi$
$830$		0			0
$831$	−3.70417e8	+	2.13861e8i	−0.645488	+	0.372673i
$832$		0			0
$833$	−2.71755e8	+	1.49749e8i	−0.470156	+	0.259077i
$834$		0			0
$835$	2.71640e8	+	4.70495e8i	0.466589	+	0.808156i
$836$		0			0
$837$	3.30177e8	−	5.71884e8i	0.563081	−	0.975284i
$838$		0			0
$839$		−	1.09867e9i		−	1.86030i	−0.367185	−	0.930148i	$-0.619678\pi$
$839$		−	1.09867e9i		−	1.86030i	0.367185	−	0.930148i	$-0.380322\pi$
$840$		0			0
$841$	4.82497e8			0.811160
$842$		0			0
$843$	3.24009e7	+	1.87066e7i	0.0540846	+	0.0312258i
$844$		0			0
$845$	4.64485e8	−	2.68171e8i	0.769842	−	0.444469i
$846$		0			0
$847$	1.14748e9	+	6.77848e8i	1.88840	+	1.11553i
$848$		0			0
$849$	−4.50725e8	−	7.80679e8i	−0.736527	−	1.27570i
$850$		0			0
$851$	1.23757e8	−	2.14353e8i	0.200807	−	0.347808i
$852$		0			0
$853$		−	8.99416e8i		−	1.44915i	−0.689196	−	0.724575i	$-0.742036\pi$
$853$		−	8.99416e8i		−	1.44915i	0.689196	−	0.724575i	$-0.257964\pi$
$854$		0			0
$855$	2.06365e8			0.330170
$856$		0			0
$857$	8.09008e8	+	4.67081e8i	1.28532	+	0.742078i	0.977815	−	0.209470i	$-0.0671738\pi$
$857$	8.09008e8	+	4.67081e8i	1.28532	+	0.742078i	0.307501	+	0.951548i	$0.400507\pi$
$858$		0			0
$859$	−3.89837e8	+	2.25073e8i	−0.615041	+	0.355094i	−0.774936	−	0.632040i	$-0.782218\pi$
$859$	−3.89837e8	+	2.25073e8i	−0.615041	+	0.355094i	0.159895	+	0.987134i	$0.448884\pi$
$860$		0			0
$861$	−9.36546e6	+	9.38708e8i	−0.0146730	+	1.47069i
$862$		0			0
$863$	2.32956e8	+	4.03492e8i	0.362445	+	0.627773i	0.988363	−	0.152116i	$-0.0486087\pi$
$863$	2.32956e8	+	4.03492e8i	0.362445	+	0.627773i	−0.625918	+	0.779889i	$0.715275\pi$
$864$		0			0
$865$	−2.45031e8	+	4.24406e8i	−0.378593	+	0.655742i
$866$		0			0
$867$			5.47783e8i			0.840527i
$868$		0			0
$869$	−8.06416e8			−1.22885
$870$		0			0
$871$	2.29232e8	+	1.32347e8i	0.346913	+	0.200290i
$872$		0			0
$873$	8.13067e6	−	4.69424e6i	0.0122203	−	0.00705542i
$874$		0			0
$875$	−3.17167e8	−	5.62230e8i	−0.473439	−	0.839247i
$876$		0			0
$877$	−1.87763e8	−	3.25216e8i	−0.278363	−	0.482140i	0.692615	−	0.721308i	$-0.256459\pi$
$877$	−1.87763e8	−	3.25216e8i	−0.278363	−	0.482140i	−0.970978	+	0.239168i	$0.923125\pi$
$878$		0			0
$879$	−8.89745e7	+	1.54108e8i	−0.131008	+	0.226913i
$880$		0			0
$881$		−	6.74362e8i		−	0.986201i	−0.869972	−	0.493101i	$-0.835863\pi$
$881$		−	6.74362e8i		−	0.986201i	0.869972	−	0.493101i	$-0.164137\pi$
$882$		0			0
$883$	−1.23657e9			−1.79612			−0.898061	−	0.439871i	$-0.855024\pi$
$883$	−1.23657e9			−1.79612			−0.898061	+	0.439871i	$0.855024\pi$
$884$		0			0
$885$	−1.19355e9	−	6.89094e8i	−1.72191	−	0.994143i
$886$		0			0
$887$	−4.15490e8	+	2.39883e8i	−0.595374	+	0.343739i	−0.767219	−	0.641385i	$-0.778360\pi$
$887$	−4.15490e8	+	2.39883e8i	−0.595374	+	0.343739i	0.171846	+	0.985124i	$0.445027\pi$
$888$		0			0
$889$	−1.88868e8	+	1.06545e8i	−0.268815	+	0.151645i
$890$		0			0
$891$	−7.82943e8	−	1.35610e9i	−1.10687	−	1.91716i
$892$		0			0
$893$	−1.07108e8	+	1.85516e8i	−0.150406	+	0.260511i
$894$		0			0
$895$		−	1.71133e8i		−	0.238707i
$896$		0			0
$897$	−2.98092e8			−0.413022
$898$		0			0
$899$	1.33333e9	+	7.69799e8i	1.83510	+	1.05949i
$900$		0			0
$901$	7.59226e7	−	4.38339e7i	0.103800	−	0.0599289i
$902$		0			0
$903$	−1.24632e9	−	1.24344e7i	−1.69264	−	0.0168874i
$904$		0			0
$905$	3.16829e8	+	5.48765e8i	0.427444	+	0.740355i
$906$		0			0
$907$	−2.16162e6	+	3.74403e6i	−0.00289705	+	0.00501784i	−0.867470	−	0.497489i	$-0.834255\pi$
$907$	−2.16162e6	+	3.74403e6i	−0.00289705	+	0.00501784i	0.864573	+	0.502507i	$0.167589\pi$
$908$		0			0
$909$		−	1.43709e8i		−	0.191334i
$910$		0			0
$911$	4.54738e8			0.601459			0.300730	−	0.953709i	$-0.402770\pi$
$911$	4.54738e8			0.601459			0.300730	+	0.953709i	$0.402770\pi$
$912$		0			0
$913$	−4.62786e8	−	2.67190e8i	−0.608090	−	0.351081i
$914$		0			0
$915$	6.00458e8	−	3.46675e8i	0.783825	−	0.452542i
$916$		0			0
$917$	2.25471e7	−	3.81682e7i	0.0292403	−	0.0494987i
$918$		0			0
$919$	−3.08907e8	−	5.35042e8i	−0.397998	−	0.689353i	0.595481	−	0.803369i	$-0.296962\pi$
$919$	−3.08907e8	−	5.35042e8i	−0.397998	−	0.689353i	−0.993479	+	0.114017i	$0.963628\pi$
$920$		0			0
$921$	−7.02677e8	+	1.21707e9i	−0.899450	+	1.55789i
$922$		0			0
$923$			2.67147e8i			0.339738i
$924$		0			0
$925$	−2.28441e7			−0.0288635
$926$		0			0
$927$	−1.11018e8	−	6.40960e7i	−0.139365	−	0.0804622i
$928$		0			0
$929$	−6.08765e8	+	3.51471e8i	−0.759282	+	0.438371i	−0.829038	−	0.559193i	$-0.811111\pi$
$929$	−6.08765e8	+	3.51471e8i	−0.759282	+	0.438371i	0.0697562	+	0.997564i	$0.477778\pi$
$930$		0			0
$931$	1.30507e7	−	6.53975e8i	0.0161727	−	0.810423i
$932$		0			0
$933$	4.49563e8	+	7.78666e8i	0.553536	+	0.958752i
$934$		0			0
$935$	−4.05031e8	+	7.01534e8i	−0.495511	+	0.858250i
$936$		0			0
$937$			2.35920e8i			0.286779i	0.989666	+	0.143389i	$0.0458001\pi$
$937$			2.35920e8i			0.286779i	−0.989666	+	0.143389i	$0.954200\pi$
$938$		0			0
$939$	1.89023e8			0.228306
$940$		0			0
$941$	−3.18894e8	−	1.84114e8i	−0.382717	−	0.220962i	0.296283	−	0.955100i	$-0.404253\pi$
$941$	−3.18894e8	−	1.84114e8i	−0.382717	−	0.220962i	−0.679000	+	0.734139i	$0.737586\pi$
$942$		0			0
$943$	−8.46991e8	+	4.89010e8i	−1.01005	+	0.583154i
$944$		0			0
$945$	−5.36894e8	−	3.17159e8i	−0.636199	−	0.375822i
$946$		0			0
$947$	3.01831e8	+	5.22786e8i	0.355397	+	0.615566i	0.987186	−	0.159575i	$-0.0510123\pi$
$947$	3.01831e8	+	5.22786e8i	0.355397	+	0.615566i	−0.631789	+	0.775141i	$0.717679\pi$
$948$		0			0
$949$	4.42579e7	−	7.66569e7i	0.0517836	−	0.0896918i
$950$		0			0
$951$			1.44417e9i			1.67910i
$952$		0			0
$953$	2.87119e8			0.331729			0.165864	−	0.986149i	$-0.446959\pi$
$953$	2.87119e8			0.331729			0.165864	+	0.986149i	$0.446959\pi$
$954$		0			0
$955$	−7.05443e8	−	4.07288e8i	−0.809938	−	0.467618i
$956$		0			0
$957$	2.15544e9	−	1.24445e9i	2.45924	−	1.41984i
$958$		0			0
$959$	−5.46143e6	+	5.47404e8i	−0.00619228	+	0.620657i
$960$		0			0
$961$	6.56367e8	+	1.13686e9i	0.739566	+	1.28097i
$962$		0			0
$963$	2.71541e8	−	4.70322e8i	0.304058	−	0.526643i
$964$		0			0
$965$			1.12984e8i			0.125728i
$966$		0			0
$967$	7.68357e8			0.849735			0.424868	−	0.905256i	$-0.360321\pi$
$967$	7.68357e8			0.849735			0.424868	+	0.905256i	$0.360321\pi$
$968$		0			0
$969$	4.04853e8	+	2.33742e8i	0.444965	+	0.256901i
$970$		0			0
$971$	7.15458e8	−	4.13070e8i	0.781496	−	0.451197i	−0.0554643	−	0.998461i	$-0.517664\pi$
$971$	7.15458e8	−	4.13070e8i	0.781496	−	0.451197i	0.836960	+	0.547264i	$0.184331\pi$
$972$		0			0
$973$	1.16019e8	+	2.05663e8i	0.125948	+	0.223264i
$974$		0			0
$975$	1.37561e7	+	2.38263e7i	0.0148417	+	0.0257065i
$976$		0			0
$977$	4.17857e8	−	7.23750e8i	0.448068	−	0.776077i	−0.550192	−	0.835038i	$-0.685445\pi$
$977$	4.17857e8	−	7.23750e8i	0.448068	−	0.776077i	0.998260	+	0.0589610i	$0.0187788\pi$
$978$		0			0
$979$		−	9.11500e8i		−	0.971423i
$980$		0			0
$981$	5.52390e8			0.585112
$982$		0			0
$983$	−1.91250e8	−	1.10418e8i	−0.201345	−	0.116247i	0.395938	−	0.918277i	$-0.370420\pi$
$983$	−1.91250e8	−	1.10418e8i	−0.201345	−	0.116247i	−0.597283	+	0.802031i	$0.703753\pi$
$984$		0			0
$985$	−7.66550e8	+	4.42568e8i	−0.802106	+	0.463096i
$986$		0			0
$987$	−3.66966e8	+	2.07014e8i	−0.381658	+	0.215303i
$988$		0			0
$989$	−6.49256e8	−	1.12454e9i	−0.671161	−	1.16249i
$990$		0			0
$991$	−3.64147e8	+	6.30721e8i	−0.374158	+	0.648061i	−0.990201	−	0.139652i	$-0.955402\pi$
$991$	−3.64147e8	+	6.30721e8i	−0.374158	+	0.648061i	0.616042	+	0.787713i	$0.288735\pi$
$992$		0			0
$993$			2.13291e9i			2.17834i
$994$		0			0
$995$	−1.22944e9			−1.24807
$996$		0			0
$997$	−1.47059e8	−	8.49047e7i	−0.148391	−	0.0856735i	0.423966	−	0.905678i	$-0.360638\pi$
$997$	−1.47059e8	−	8.49047e7i	−0.148391	−	0.0856735i	−0.572357	+	0.820005i	$0.693971\pi$
$998$		0			0
$999$	2.64876e8	−	1.52926e8i	0.265672	−	0.153386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.7.s.c.17.1		8
4.3	odd	2		14.7.d.a.3.2	✓	8
7.5	odd	6	inner	112.7.s.c.33.1		8
12.11	even	2		126.7.n.c.73.3		8
28.3	even	6		98.7.b.c.97.8		8
28.11	odd	6		98.7.b.c.97.5		8
28.19	even	6		14.7.d.a.5.2	yes	8
28.23	odd	6		98.7.d.c.19.1		8
28.27	even	2		98.7.d.c.31.1		8
84.47	odd	6		126.7.n.c.19.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.7.d.a.3.2	✓	8	4.3	odd	2
14.7.d.a.5.2	yes	8	28.19	even	6
98.7.b.c.97.5		8	28.11	odd	6
98.7.b.c.97.8		8	28.3	even	6
98.7.d.c.19.1		8	28.23	odd	6
98.7.d.c.31.1		8	28.27	even	2
112.7.s.c.17.1		8	1.1	even	1	trivial
112.7.s.c.33.1		8	7.5	odd	6	inner
126.7.n.c.19.3		8	84.47	odd	6
126.7.n.c.73.3		8	12.11	even	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	112.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-27.6101 - 15.9407i) q^{3} +(111.836 - 64.5687i) q^{5} +(-298.743 + 168.528i) q^{7} +(143.713 + 248.917i) q^{9} +(1189.23 - 2059.81i) q^{11} -820.701i q^{13} -4117.09 q^{15} +(-2284.02 - 1318.68i) q^{17} +(4814.93 - 2779.90i) q^{19} +(10934.8 + 109.096i) q^{21} +(5696.37 + 9866.40i) q^{23} +(525.743 - 910.614i) q^{25} +14078.0i q^{27} -32822.6 q^{29} +(-40622.4 - 23453.4i) q^{31} +(-65669.6 + 37914.4i) q^{33} +(-22528.7 + 38137.0i) q^{35} +(-10862.8 - 18814.9i) q^{37} +(-13082.5 + 22659.6i) q^{39} +85846.0i q^{41} -113977. q^{43} +(32144.6 + 18558.7i) q^{45} +(-33367.3 + 19264.6i) q^{47} +(60845.6 - 100693. i) q^{49} +(42041.4 + 72817.9i) q^{51} +(-16620.4 + 28787.3i) q^{53} -307149. i q^{55} -177254. q^{57} +(289901. + 167374. i) q^{59} +(-145845. + 84203.9i) q^{61} +(-84882.6 - 50142.7i) q^{63} +(-52991.6 - 91784.1i) q^{65} +(-161261. + 279312. i) q^{67} -363217. i q^{69} -325510. q^{71} +(93404.3 + 53927.0i) q^{73} +(-29031.7 + 16761.4i) q^{75} +(-8138.93 + 815772. i) q^{77} +(-169525. - 293625. i) q^{79} +(329180. - 570157. i) q^{81} -224674. i q^{83} -340582. q^{85} +(906235. + 523215. i) q^{87} +(331888. - 191615. i) q^{89} +(138311. + 245178. i) q^{91} +(747726. + 1.29510e6i) q^{93} +(358989. - 621787. i) q^{95} -32664.1i q^{97} +683629. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 336 q^{5} - 652 q^{7} + 756 q^{9} + 1356 q^{11} - 27144 q^{15} - 17304 q^{17} + 32004 q^{19} + 9756 q^{21} + 4128 q^{23} + 4664 q^{25} - 30312 q^{29} + 3108 q^{31} + 3276 q^{33} - 98028 q^{35} - 6124 q^{37}+ \cdots + 4625928 q^{99}+O(q^{100}) \) 8 * q - 336 * q^5 - 652 * q^7 + 756 * q^9 + 1356 * q^11 - 27144 * q^15 - 17304 * q^17 + 32004 * q^19 + 9756 * q^21 + 4128 * q^23 + 4664 * q^25 - 30312 * q^29 + 3108 * q^31 + 3276 * q^33 - 98028 * q^35 - 6124 * q^37 - 100764 * q^39 + 297376 * q^43 - 172116 * q^45 - 313908 * q^47 + 32432 * q^49 - 253692 * q^51 + 278484 * q^53 - 81288 * q^57 + 835464 * q^59 - 995316 * q^61 - 1216188 * q^63 + 8316 * q^65 - 648808 * q^67 - 190128 * q^71 - 1617084 * q^73 + 2042208 * q^75 + 1456224 * q^77 - 70096 * q^79 + 1177920 * q^81 + 2190984 * q^85 + 2057076 * q^87 + 739116 * q^89 - 2233752 * q^91 - 23364 * q^93 - 725640 * q^95 + 4625928 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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