LMFDB - Embedded newform 112.10.p.b.31.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			31.9
Character	$\chi$	$=$	112.31
Dual form			112.10.p.b.47.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(48.7996 - 84.5234i) q^{3} +(-62.4476 + 36.0541i) q^{5} +(-6259.38 - 1083.41i) q^{7} +(5078.69 + 8796.55i) q^{9} +(44516.5 + 25701.6i) q^{11} -152010. i q^{13} +7037.71i q^{15} +(-93083.9 - 53742.0i) q^{17} +(-208545. - 361210. i) q^{19} +(-397029. + 476194. i) q^{21} +(-685413. + 395724. i) q^{23} +(-973963. + 1.68695e6i) q^{25} +2.91240e6 q^{27} -4.61586e6 q^{29} +(-1.95313e6 + 3.38291e6i) q^{31} +(4.34478e6 - 2.50846e6i) q^{33} +(429944. - 158020. i) q^{35} +(-8.08419e6 - 1.40022e7i) q^{37} +(-1.28484e7 - 7.41805e6i) q^{39} +1.48715e7i q^{41} +7.60496e6i q^{43} +(-634304. - 366216. i) q^{45} +(1.72883e7 + 2.99443e7i) q^{47} +(3.80061e7 + 1.35629e7i) q^{49} +(-9.08492e6 + 5.24518e6i) q^{51} +(-9.73612e6 + 1.68635e7i) q^{53} -3.70660e6 q^{55} -4.07076e7 q^{57} +(-9.12057e7 + 1.57973e8i) q^{59} +(1.50849e8 - 8.70927e7i) q^{61} +(-2.22592e7 - 6.05633e7i) q^{63} +(5.48060e6 + 9.49268e6i) q^{65} +(-5.72063e7 - 3.30281e7i) q^{67} +7.72447e7i q^{69} +3.98123e8i q^{71} +(5.72087e7 + 3.30295e7i) q^{73} +(9.50580e7 + 1.64645e8i) q^{75} +(-2.50800e8 - 2.09106e8i) q^{77} +(-3.94490e8 + 2.27759e8i) q^{79} +(4.21601e7 - 7.30234e7i) q^{81} -2.37486e8 q^{83} +7.75049e6 q^{85} +(-2.25252e8 + 3.90148e8i) q^{87} +(-8.45694e8 + 4.88262e8i) q^{89} +(-1.64689e8 + 9.51491e8i) q^{91} +(1.90624e8 + 3.30170e8i) q^{93} +(2.60462e7 + 1.50378e7i) q^{95} -1.18899e8i q^{97} +5.22122e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57}+ \cdots - 1432348316 q^{93}+O(q^{100}) $ 24 * q + 1704 * q^5 - 80428 * q^9 - 1578672 * q^17 + 1219540 * q^21 + 8001240 * q^25 - 6709416 * q^29 - 29129772 * q^33 - 11130084 * q^37 + 57023292 * q^45 - 12671904 * q^49 - 93652164 * q^53 + 742621544 * q^57 - 593611308 * q^61 + 160281180 * q^65 - 1676922516 * q^73 + 1645751688 * q^77 - 528698056 * q^81 + 1370082456 * q^85 - 941603484 * q^89 - 1432348316 * q^93

Character values

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

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

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
$3$	48.7996	−	84.5234i	0.347833	−	0.602465i	−0.638031	−	0.770011i	$-0.720251\pi$
$3$	48.7996	−	84.5234i	0.347833	−	0.602465i	0.985864	+	0.167546i	$0.0535843\pi$
$4$		0			0
$5$	−62.4476	+	36.0541i	−0.0446838	+	0.0257982i	−0.522176	−	0.852838i	$-0.674879\pi$
$5$	−62.4476	+	36.0541i	−0.0446838	+	0.0257982i	0.477492	+	0.878636i	$0.341546\pi$
$6$		0			0
$7$	−6259.38	−	1083.41i	−0.985349	−	0.170550i
$7$	−6259.38	−	1083.41i	−0.985349	−	0.170550i
$8$		0			0
$9$	5078.69	+	8796.55i	0.258024	+	0.446911i
$10$		0			0
$11$	44516.5	+	25701.6i	0.916756	+	0.529290i	0.882599	−	0.470127i	$-0.155792\pi$
$11$	44516.5	+	25701.6i	0.916756	+	0.529290i	0.0341576	+	0.999416i	$0.489125\pi$
$12$		0			0
$13$		−	152010.i		−	1.47614i	−0.674723	−	0.738071i	$-0.735737\pi$
$13$		−	152010.i		−	1.47614i	0.674723	−	0.738071i	$-0.264263\pi$
$14$		0			0
$15$			7037.71i			0.0358939i
$16$		0			0
$17$	−93083.9	−	53742.0i	−0.270305	−	0.156061i	0.358721	−	0.933445i	$-0.383213\pi$
$17$	−93083.9	−	53742.0i	−0.270305	−	0.156061i	−0.629026	+	0.777384i	$0.716546\pi$
$18$		0			0
$19$	−208545.	−	361210.i	−0.367120	−	0.635870i	0.621994	−	0.783022i	$-0.286323\pi$
$19$	−208545.	−	361210.i	−0.367120	−	0.635870i	−0.989114	+	0.147152i	$0.952989\pi$
$20$		0			0
$21$	−397029.	+	476194.i	−0.445487	+	0.534315i
$22$		0			0
$23$	−685413.	+	395724.i	−0.510713	+	0.294861i	−0.733127	−	0.680092i	$-0.761940\pi$
$23$	−685413.	+	395724.i	−0.510713	+	0.294861i	0.222413	+	0.974952i	$0.428607\pi$
$24$		0			0
$25$	−973963.	+	1.68695e6i	−0.498669	+	0.863720i
$26$		0			0
$27$	2.91240e6			1.05466
$28$		0			0
$29$	−4.61586e6			−1.21189			−0.605943	−	0.795508i	$-0.707204\pi$
$29$	−4.61586e6			−1.21189			−0.605943	+	0.795508i	$0.707204\pi$
$30$		0			0
$31$	−1.95313e6	+	3.38291e6i	−0.379842	+	0.657905i	−0.991039	−	0.133573i	$-0.957355\pi$
$31$	−1.95313e6	+	3.38291e6i	−0.379842	+	0.657905i	0.611197	+	0.791478i	$0.290688\pi$
$32$		0			0
$33$	4.34478e6	−	2.50846e6i	0.637756	−	0.368209i
$34$		0			0
$35$	429944.	−	158020.i	0.0484291	−	0.0177995i
$36$		0			0
$37$	−8.08419e6	−	1.40022e7i	−0.709134	−	1.22826i	−0.965179	−	0.261592i	$-0.915753\pi$
$37$	−8.08419e6	−	1.40022e7i	−0.709134	−	1.22826i	0.256044	−	0.966665i	$-0.417581\pi$
$38$		0			0
$39$	−1.28484e7	−	7.41805e6i	−0.889324	−	0.513451i
$40$		0			0
$41$			1.48715e7i			0.821914i	0.911655	+	0.410957i	$0.134805\pi$
$41$			1.48715e7i			0.821914i	−0.911655	+	0.410957i	$0.865195\pi$
$42$		0			0
$43$			7.60496e6i			0.339226i	0.985511	+	0.169613i	$0.0542518\pi$
$43$			7.60496e6i			0.339226i	−0.985511	+	0.169613i	$0.945748\pi$
$44$		0			0
$45$	−634304.	−	366216.i	−0.0230590	−	0.0133131i
$46$		0			0
$47$	1.72883e7	+	2.99443e7i	0.516788	+	0.895104i	0.999810	+	0.0194954i	$0.00620596\pi$
$47$	1.72883e7	+	2.99443e7i	0.516788	+	0.895104i	−0.483021	+	0.875608i	$0.660461\pi$
$48$		0			0
$49$	3.80061e7	+	1.35629e7i	0.941826	+	0.336102i
$50$		0			0
$51$	−9.08492e6	+	5.24518e6i	−0.188042	+	0.108566i
$52$		0			0
$53$	−9.73612e6	+	1.68635e7i	−0.169490	+	0.293566i	−0.938241	−	0.345983i	$-0.887545\pi$
$53$	−9.73612e6	+	1.68635e7i	−0.169490	+	0.293566i	0.768751	+	0.639549i	$0.220879\pi$
$54$		0			0
$55$	−3.70660e6			−0.0546189
$56$		0			0
$57$	−4.07076e7			−0.510786
$58$		0			0
$59$	−9.12057e7	+	1.57973e8i	−0.979914	+	1.69726i	−0.317253	+	0.948341i	$0.602760\pi$
$59$	−9.12057e7	+	1.57973e8i	−0.979914	+	1.69726i	−0.662661	+	0.748920i	$0.730573\pi$
$60$		0			0
$61$	1.50849e8	−	8.70927e7i	1.39495	−	0.805374i	0.401091	−	0.916038i	$-0.368631\pi$
$61$	1.50849e8	−	8.70927e7i	1.39495	−	0.805374i	0.993858	+	0.110664i	$0.0352977\pi$
$62$		0			0
$63$	−2.22592e7	−	6.05633e7i	−0.178023	−	0.484370i
$64$		0			0
$65$	5.48060e6	+	9.49268e6i	0.0380819	+	0.0659597i
$66$		0			0
$67$	−5.72063e7	−	3.30281e7i	−0.346822	−	0.200238i	0.316463	−	0.948605i	$-0.397505\pi$
$67$	−5.72063e7	−	3.30281e7i	−0.346822	−	0.200238i	−0.663285	+	0.748367i	$0.730838\pi$
$68$		0			0
$69$			7.72447e7i			0.410249i
$70$		0			0
$71$			3.98123e8i			1.85932i	0.368413	+	0.929662i	$0.379901\pi$
$71$			3.98123e8i			1.85932i	−0.368413	+	0.929662i	$0.620099\pi$
$72$		0			0
$73$	5.72087e7	+	3.30295e7i	0.235781	+	0.136128i	0.613236	−	0.789900i	$-0.289867\pi$
$73$	5.72087e7	+	3.30295e7i	0.235781	+	0.136128i	−0.377455	+	0.926028i	$0.623201\pi$
$74$		0			0
$75$	9.50580e7	+	1.64645e8i	0.346907	+	0.600861i
$76$		0			0
$77$	−2.50800e8	−	2.09106e8i	−0.813055	−	0.677887i
$78$		0			0
$79$	−3.94490e8	+	2.27759e8i	−1.13950	+	0.657890i	−0.946307	−	0.323271i	$-0.895218\pi$
$79$	−3.94490e8	+	2.27759e8i	−1.13950	+	0.657890i	−0.193193	+	0.981161i	$0.561884\pi$
$80$		0			0
$81$	4.21601e7	−	7.30234e7i	0.108823	−	0.188486i
$82$		0			0
$83$	−2.37486e8			−0.549272			−0.274636	−	0.961548i	$-0.588557\pi$
$83$	−2.37486e8			−0.549272			−0.274636	+	0.961548i	$0.588557\pi$
$84$		0			0
$85$	7.75049e6			0.0161044
$86$		0			0
$87$	−2.25252e8	+	3.90148e8i	−0.421534	+	0.730118i
$88$		0			0
$89$	−8.45694e8	+	4.88262e8i	−1.42876	+	0.824893i	−0.997023	−	0.0771078i	$-0.975431\pi$
$89$	−8.45694e8	+	4.88262e8i	−1.42876	+	0.824893i	−0.431734	+	0.902001i	$0.642098\pi$
$90$		0			0
$91$	−1.64689e8	+	9.51491e8i	−0.251756	+	1.45452i
$92$		0			0
$93$	1.90624e8	+	3.30170e8i	0.264243	+	0.457682i
$94$		0			0
$95$	2.60462e7	+	1.50378e7i	0.0328087	+	0.0189421i
$96$		0			0
$97$		−	1.18899e8i		−	0.136366i	−0.997673	−	0.0681828i	$-0.978280\pi$
$97$		−	1.18899e8i		−	0.136366i	0.997673	−	0.0681828i	$-0.0217201\pi$
$98$		0			0
$99$			5.22122e8i			0.546278i
$100$		0			0
$101$	−1.24465e8	−	7.18597e7i	−0.119014	−	0.0687130i	0.439311	−	0.898335i	$-0.355223\pi$
$101$	−1.24465e8	−	7.18597e7i	−0.119014	−	0.0687130i	−0.558325	+	0.829622i	$0.688556\pi$
$102$		0			0
$103$	3.48873e8	+	6.04265e8i	0.305421	+	0.529005i	0.977355	−	0.211606i	$-0.0678693\pi$
$103$	3.48873e8	+	6.04265e8i	0.305421	+	0.529005i	−0.671934	+	0.740611i	$0.734536\pi$
$104$		0			0
$105$	7.62471e6	−	4.40517e7i	0.00612169	−	0.0353680i
$106$		0			0
$107$	−1.29804e9	+	7.49424e8i	−0.957329	+	0.552714i	−0.895350	−	0.445363i	$-0.853075\pi$
$107$	−1.29804e9	+	7.49424e8i	−0.957329	+	0.552714i	−0.0619793	+	0.998077i	$0.519741\pi$
$108$		0			0
$109$	6.36594e8	−	1.10261e9i	0.431960	−	0.748176i	−0.565082	−	0.825035i	$-0.691156\pi$
$109$	6.36594e8	−	1.10261e9i	0.431960	−	0.748176i	0.997042	+	0.0768584i	$0.0244889\pi$
$110$		0			0
$111$	−1.57802e9			−0.986641
$112$		0			0
$113$	2.39657e8			0.138273			0.0691365	−	0.997607i	$-0.477976\pi$
$113$	2.39657e8			0.138273			0.0691365	+	0.997607i	$0.477976\pi$
$114$		0			0
$115$	2.85349e7	−	4.94240e7i	0.0152138	−	0.0263510i
$116$		0			0
$117$	1.33717e9	−	7.72014e8i	0.659705	−	0.380881i
$118$		0			0
$119$	5.24423e8	+	4.37240e8i	0.239729	+	0.199875i
$120$		0			0
$121$	1.42172e8	+	2.46249e8i	0.0602948	+	0.104434i
$122$		0			0
$123$	1.25699e9	+	7.25722e8i	0.495174	+	0.285889i
$124$		0			0
$125$		−	2.81298e8i		−	0.103056i
$126$		0			0
$127$			1.78644e9i			0.609356i	0.952455	+	0.304678i	$0.0985488\pi$
$127$			1.78644e9i			0.609356i	−0.952455	+	0.304678i	$0.901451\pi$
$128$		0			0
$129$	6.42798e8	+	3.71119e8i	0.204372	+	0.117994i
$130$		0			0
$131$	−5.81656e8	−	1.00746e9i	−0.172562	−	0.298887i	0.766753	−	0.641943i	$-0.221871\pi$
$131$	−5.81656e8	−	1.00746e9i	−0.172562	−	0.298887i	−0.939315	+	0.343056i	$0.888538\pi$
$132$		0			0
$133$	9.14023e8	+	2.48689e9i	0.253294	+	0.689166i
$134$		0			0
$135$	−1.81872e8	+	1.05004e8i	−0.0471264	+	0.0272085i
$136$		0			0
$137$	−2.65942e9	+	4.60625e9i	−0.644977	+	1.11713i	0.339330	+	0.940667i	$0.389800\pi$
$137$	−2.65942e9	+	4.60625e9i	−0.644977	+	1.11713i	−0.984307	+	0.176465i	$0.943534\pi$
$138$		0			0
$139$	2.59205e8			0.0588948			0.0294474	−	0.999566i	$-0.490625\pi$
$139$	2.59205e8			0.0588948			0.0294474	+	0.999566i	$0.490625\pi$
$140$		0			0
$141$	3.37466e9			0.719024
$142$		0			0
$143$	3.90691e9	−	6.76697e9i	0.781307	−	1.35326i
$144$		0			0
$145$	2.88249e8	−	1.66421e8i	0.0541517	−	0.0312645i
$146$		0			0
$147$	3.00107e9	−	2.55054e9i	0.530088	−	0.450509i
$148$		0			0
$149$	−4.97838e9	−	8.62281e9i	−0.827465	−	1.43321i	−0.900020	−	0.435848i	$-0.856449\pi$
$149$	−4.97838e9	−	8.62281e9i	−0.827465	−	1.43321i	0.0725551	−	0.997364i	$-0.476885\pi$
$150$		0			0
$151$	−8.76599e9	−	5.06105e9i	−1.37216	−	0.792217i	−0.380960	−	0.924591i	$-0.624407\pi$
$151$	−8.76599e9	−	5.06105e9i	−1.37216	−	0.792217i	−0.991200	+	0.132374i	$0.957740\pi$
$152$		0			0
$153$		−	1.09176e9i		−	0.161070i
$154$		0			0
$155$		−	2.81673e8i		−	0.0391970i
$156$		0			0
$157$	−3.82395e9	−	2.20776e9i	−0.502301	−	0.290004i	0.227362	−	0.973810i	$-0.426990\pi$
$157$	−3.82395e9	−	2.20776e9i	−0.502301	−	0.290004i	−0.729663	+	0.683807i	$0.760323\pi$
$158$		0			0
$159$	9.50238e8	+	1.64586e9i	0.117909	+	0.204224i
$160$		0			0
$161$	4.71899e9	−	1.73440e9i	0.553519	−	0.203439i
$162$		0			0
$163$	1.13789e10	−	6.56962e9i	1.26257	−	0.728947i	0.289001	−	0.957329i	$-0.406677\pi$
$163$	1.13789e10	−	6.56962e9i	1.26257	−	0.728947i	0.973572	+	0.228382i	$0.0733436\pi$
$164$		0			0
$165$	−1.80881e8	+	3.13294e8i	−0.0189983	+	0.0329060i
$166$		0			0
$167$	1.16636e10			1.16041			0.580203	−	0.814472i	$-0.302973\pi$
$167$	1.16636e10			1.16041			0.580203	+	0.814472i	$0.302973\pi$
$168$		0			0
$169$	−1.25027e10			−1.17900
$170$		0			0
$171$	2.11827e9	−	3.66895e9i	0.189452	−	0.328140i
$172$		0			0
$173$	−2.83830e9	+	1.63869e9i	−0.240908	+	0.139088i	−0.615594	−	0.788064i	$-0.711084\pi$
$173$	−2.83830e9	+	1.63869e9i	−0.240908	+	0.139088i	0.374686	+	0.927152i	$0.377750\pi$
$174$		0			0
$175$	7.92406e9	−	9.50408e9i	0.638670	−	0.766018i
$176$		0			0
$177$	8.90161e9	+	1.54180e10i	0.681693	+	1.18073i
$178$		0			0
$179$	−1.08411e10	−	6.25909e9i	−0.789284	−	0.455694i	0.0504262	−	0.998728i	$-0.483942\pi$
$179$	−1.08411e10	−	6.25909e9i	−0.789284	−	0.455694i	−0.839711	+	0.543034i	$0.817275\pi$
$180$		0			0
$181$			9.48698e9i			0.657014i	0.944502	+	0.328507i	$0.106545\pi$
$181$			9.48698e9i			0.657014i	−0.944502	+	0.328507i	$0.893455\pi$
$182$		0			0
$183$		−	1.70004e10i		−	1.12054i
$184$		0			0
$185$	1.00968e9	+	5.82936e8i	0.0633737	+	0.0365888i
$186$		0			0
$187$	−2.76251e9	−	4.78482e9i	−0.165203	−	0.286140i
$188$		0			0
$189$	−1.82298e10	−	3.15532e9i	−1.03921	−	0.179873i
$190$		0			0
$191$	−1.17903e10	+	6.80715e9i	−0.641026	+	0.370097i	−0.785010	−	0.619483i	$-0.787342\pi$
$191$	−1.17903e10	+	6.80715e9i	−0.641026	+	0.370097i	0.143983	+	0.989580i	$0.454009\pi$
$192$		0			0
$193$	1.18533e10	−	2.05306e10i	0.614940	−	1.06511i	−0.375455	−	0.926841i	$-0.622513\pi$
$193$	1.18533e10	−	2.05306e10i	0.614940	−	1.06511i	0.990395	−	0.138267i	$-0.0441532\pi$
$194$		0			0
$195$	1.06981e9			0.0529845
$196$		0			0
$197$	2.43994e10			1.15420			0.577100	−	0.816673i	$-0.304184\pi$
$197$	2.43994e10			1.15420			0.577100	+	0.816673i	$0.304184\pi$
$198$		0			0
$199$	1.94957e10	−	3.37675e10i	0.881250	−	1.52637i	0.0312977	−	0.999510i	$-0.490036\pi$
$199$	1.94957e10	−	3.37675e10i	0.881250	−	1.52637i	0.849952	−	0.526860i	$-0.176631\pi$
$200$		0			0
$201$	−5.58329e9	+	3.22351e9i	−0.241272	+	0.139299i
$202$		0			0
$203$	2.88924e10	+	5.00086e9i	1.19413	+	0.206687i
$204$		0			0
$205$	−5.36178e8	−	9.28687e8i	−0.0212039	−	0.0367263i
$206$		0			0
$207$	−6.96201e9	−	4.01952e9i	−0.263553	−	0.152162i
$208$		0			0
$209$		−	2.14397e10i		−	0.777251i
$210$		0			0
$211$			4.21725e10i			1.46473i	0.680911	+	0.732367i	$0.261584\pi$
$211$			4.21725e10i			1.46473i	−0.680911	+	0.732367i	$0.738416\pi$
$212$		0			0
$213$	3.36508e10	+	1.94283e10i	1.12018	+	0.646735i
$214$		0			0
$215$	−2.74190e8	−	4.74912e8i	−0.00875143	−	0.0151579i
$216$		0			0
$217$	1.58904e10	−	1.90589e10i	0.486482	−	0.583484i
$218$		0			0
$219$	5.58353e9	−	3.22365e9i	0.164025	−	0.0946999i
$220$		0			0
$221$	−8.16935e9	+	1.41497e10i	−0.230368	+	0.399009i
$222$		0			0
$223$	−3.52016e10			−0.953214			−0.476607	−	0.879116i	$-0.658134\pi$
$223$	−3.52016e10			−0.953214			−0.476607	+	0.879116i	$0.658134\pi$
$224$		0			0
$225$	−1.97858e10			−0.514675
$226$		0			0
$227$	6.10064e9	−	1.05666e10i	0.152496	−	0.264131i	−0.779648	−	0.626218i	$-0.784602\pi$
$227$	6.10064e9	−	1.05666e10i	0.152496	−	0.264131i	0.932145	+	0.362086i	$0.117935\pi$
$228$		0			0
$229$	−2.23605e9	+	1.29099e9i	−0.0537307	+	0.0310214i	−0.526625	−	0.850098i	$-0.676543\pi$
$229$	−2.23605e9	+	1.29099e9i	−0.0537307	+	0.0310214i	0.472894	+	0.881119i	$0.343209\pi$
$230$		0			0
$231$	−2.99133e10	+	1.09942e10i	−0.691210	+	0.254045i
$232$		0			0
$233$	−6.09373e9	−	1.05546e10i	−0.135451	−	0.234608i	0.790319	−	0.612696i	$-0.209915\pi$
$233$	−6.09373e9	−	1.05546e10i	−0.135451	−	0.234608i	−0.925770	+	0.378088i	$0.876582\pi$
$234$		0			0
$235$	−2.15923e9	−	1.24663e9i	−0.0461842	−	0.0266645i
$236$		0			0
$237$			4.44582e10i			0.915344i
$238$		0			0
$239$			2.44231e10i			0.484185i	0.970253	+	0.242092i	$0.0778337\pi$
$239$			2.44231e10i			0.484185i	−0.970253	+	0.242092i	$0.922166\pi$
$240$		0			0
$241$	−2.20902e10	−	1.27538e10i	−0.421816	−	0.243536i	0.274038	−	0.961719i	$-0.411641\pi$
$241$	−2.20902e10	−	1.27538e10i	−0.421816	−	0.243536i	−0.695854	+	0.718183i	$0.744974\pi$
$242$		0			0
$243$	2.45476e10	+	4.25177e10i	0.451628	+	0.782242i
$244$		0			0
$245$	−2.86239e9	+	5.23304e8i	−0.0507552	+	0.00927911i
$246$		0			0
$247$	−5.49077e10	+	3.17010e10i	−0.938635	+	0.541921i
$248$		0			0
$249$	−1.15892e10	+	2.00732e10i	−0.191055	+	0.330917i
$250$		0			0
$251$	−3.10608e10			−0.493947			−0.246974	−	0.969022i	$-0.579436\pi$
$251$	−3.10608e10			−0.493947			−0.246974	+	0.969022i	$0.579436\pi$
$252$		0			0
$253$	−4.06829e10			−0.624266
$254$		0			0
$255$	3.78221e8	−	6.55098e8i	0.00560163	−	0.00970232i
$256$		0			0
$257$	7.91694e9	−	4.57085e9i	0.113203	−	0.0653578i	−0.442329	−	0.896853i	$-0.645848\pi$
$257$	7.91694e9	−	4.57085e9i	0.113203	−	0.0653578i	0.555532	+	0.831495i	$0.312514\pi$
$258$		0			0
$259$	3.54319e10	+	9.64037e10i	0.489266	+	1.33120i
$260$		0			0
$261$	−2.34425e10	−	4.06037e10i	−0.312696	−	0.541605i
$262$		0			0
$263$	−1.21844e10	−	7.03465e9i	−0.157037	−	0.0906654i	0.419422	−	0.907791i	$-0.362233\pi$
$263$	−1.21844e10	−	7.03465e9i	−0.157037	−	0.0906654i	−0.576459	+	0.817126i	$0.695566\pi$
$264$		0			0
$265$		−	1.40411e9i		−	0.0174902i
$266$		0			0
$267$			9.53080e10i			1.14770i
$268$		0			0
$269$	−6.87971e10	−	3.97200e10i	−0.801097	−	0.462513i	0.0427579	−	0.999085i	$-0.486386\pi$
$269$	−6.87971e10	−	3.97200e10i	−0.801097	−	0.462513i	−0.843854	+	0.536572i	$0.819719\pi$
$270$		0			0
$271$	−5.55245e9	−	9.61712e9i	−0.0625349	−	0.108314i	0.833063	−	0.553178i	$-0.186585\pi$
$271$	−5.55245e9	−	9.61712e9i	−0.0625349	−	0.108314i	−0.895598	+	0.444865i	$0.853252\pi$
$272$		0			0
$273$	7.23865e10	+	6.03525e10i	0.788725	+	0.657603i
$274$		0			0
$275$	−8.67148e10	+	5.00648e10i	−0.914316	+	0.527880i
$276$		0			0
$277$	−2.01780e10	+	3.49493e10i	−0.205930	+	0.356681i	−0.950429	−	0.310943i	$-0.899355\pi$
$277$	−2.01780e10	+	3.49493e10i	−0.205930	+	0.356681i	0.744499	+	0.667624i	$0.232688\pi$
$278$		0			0
$279$	−3.96773e10			−0.392034
$280$		0			0
$281$	8.75112e10			0.837308			0.418654	−	0.908146i	$-0.362502\pi$
$281$	8.75112e10			0.837308			0.418654	+	0.908146i	$0.362502\pi$
$282$		0			0
$283$	−7.08902e10	+	1.22785e11i	−0.656973	+	1.13791i	0.324423	+	0.945912i	$0.394830\pi$
$283$	−7.08902e10	+	1.22785e11i	−0.656973	+	1.13791i	−0.981395	+	0.191998i	$0.938503\pi$
$284$		0			0
$285$	2.54209e9	−	1.46768e9i	0.0228239	−	0.0131774i
$286$		0			0
$287$	1.61119e10	−	9.30862e10i	0.140177	−	0.809872i
$288$		0			0
$289$	−5.35175e10	−	9.26951e10i	−0.451290	−	0.781657i
$290$		0			0
$291$	−1.00497e10	−	5.80222e9i	−0.0821555	−	0.0474325i
$292$		0			0
$293$		−	8.36333e10i		−	0.662941i	−0.943466	−	0.331471i	$-0.892455\pi$
$293$		−	8.36333e10i		−	0.662941i	0.943466	−	0.331471i	$-0.107545\pi$
$294$		0			0
$295$		−	1.31534e10i		−	0.101120i
$296$		0			0
$297$	1.29650e11	+	7.48534e10i	0.966870	+	0.558222i
$298$		0			0
$299$	6.01541e10	+	1.04190e11i	0.435256	+	0.753886i
$300$		0			0
$301$	8.23928e9	−	4.76024e10i	0.0578549	−	0.334256i
$302$		0			0
$303$	−1.21477e10	+	7.01345e9i	−0.0827943	+	0.0478013i
$304$		0			0
$305$	−6.28011e9	+	1.08775e10i	−0.0415545	+	0.0719744i
$306$		0			0
$307$	1.47456e11			0.947414			0.473707	−	0.880683i	$-0.342916\pi$
$307$	1.47456e11			0.947414			0.473707	+	0.880683i	$0.342916\pi$
$308$		0			0
$309$	6.80994e10			0.424943
$310$		0			0
$311$	−1.59072e10	+	2.75520e10i	−0.0964208	+	0.167006i	−0.910201	−	0.414167i	$-0.864073\pi$
$311$	−1.59072e10	+	2.75520e10i	−0.0964208	+	0.167006i	0.813780	+	0.581173i	$0.197406\pi$
$312$		0			0
$313$	−1.66358e11	+	9.60466e10i	−0.979700	+	0.565630i	−0.902180	−	0.431361i	$-0.858034\pi$
$313$	−1.66358e11	+	9.60466e10i	−0.979700	+	0.565630i	−0.0775206	+	0.996991i	$0.524700\pi$
$314$		0			0
$315$	3.57359e9	+	2.97949e9i	0.0204507	+	0.0170508i
$316$		0			0
$317$	4.47297e10	+	7.74742e10i	0.248788	+	0.430914i	0.963190	−	0.268822i	$-0.0866344\pi$
$317$	4.47297e10	+	7.74742e10i	0.248788	+	0.430914i	−0.714402	+	0.699736i	$0.753301\pi$
$318$		0			0
$319$	−2.05482e11	−	1.18635e11i	−1.11100	−	0.641438i
$320$		0			0
$321$			1.46286e11i			0.769009i
$322$		0			0
$323$			4.48305e10i			0.229172i
$324$		0			0
$325$	2.56434e11	+	1.48053e11i	1.27497	+	0.736106i
$326$		0			0
$327$	−6.21311e10	−	1.07614e11i	−0.300500	−	0.520481i
$328$		0			0
$329$	−7.57724e10	−	2.06163e11i	−0.356557	−	0.970128i
$330$		0			0
$331$	2.56920e11	−	1.48333e11i	1.17645	−	0.679221i	0.221256	−	0.975216i	$-0.428984\pi$
$331$	2.56920e11	−	1.48333e11i	1.17645	−	0.679221i	0.955190	+	0.295994i	$0.0956510\pi$
$332$		0			0
$333$	8.21142e10	−	1.42226e11i	0.365948	−	0.633840i
$334$		0			0
$335$	4.76319e9			0.0206631
$336$		0			0
$337$	2.11598e10			0.0893668			0.0446834	−	0.999001i	$-0.485772\pi$
$337$	2.11598e10			0.0893668			0.0446834	+	0.999001i	$0.485772\pi$
$338$		0			0
$339$	1.16952e10	−	2.02567e10i	0.0480959	−	0.0833046i
$340$		0			0
$341$	−1.73893e11	+	1.00397e11i	−0.696445	+	0.402092i
$342$		0			0
$343$	−2.23200e11	−	1.26072e11i	−0.870705	−	0.491806i
$344$		0			0
$345$	−2.78499e9	−	4.82374e9i	−0.0105837	−	0.0183315i
$346$		0			0
$347$	4.16624e11	+	2.40538e11i	1.54263	+	0.890638i	0.998672	+	0.0515258i	$0.0164085\pi$
$347$	4.16624e11	+	2.40538e11i	1.54263	+	0.890638i	0.543959	+	0.839112i	$0.316925\pi$
$348$		0			0
$349$		−	1.98773e11i		−	0.717204i	−0.933490	−	0.358602i	$-0.883253\pi$
$349$		−	1.98773e11i		−	0.717204i	0.933490	−	0.358602i	$-0.116747\pi$
$350$		0			0
$351$		−	4.42715e11i		−	1.55683i
$352$		0			0
$353$	−3.20642e11	−	1.85123e11i	−1.09909	−	0.634562i	−0.163111	−	0.986608i	$-0.552153\pi$
$353$	−3.20642e11	−	1.85123e11i	−1.09909	−	0.634562i	−0.935983	+	0.352046i	$0.885486\pi$
$354$		0			0
$355$	−1.43540e10	−	2.48618e10i	−0.0479673	−	0.0830818i
$356$		0			0
$357$	6.25487e10	−	2.29889e10i	0.203803	−	0.0749051i
$358$		0			0
$359$	2.40132e11	−	1.38640e11i	0.763000	−	0.440518i	−0.0673720	−	0.997728i	$-0.521461\pi$
$359$	2.40132e11	−	1.38640e11i	0.763000	−	0.440518i	0.830372	+	0.557210i	$0.188128\pi$
$360$		0			0
$361$	7.43621e10	−	1.28799e11i	0.230446	−	0.399144i
$362$		0			0
$363$	2.77518e10			0.0838901
$364$		0			0
$365$	−4.76339e9			−0.0140475
$366$		0			0
$367$	1.79004e11	−	3.10044e11i	0.515069	−	0.892125i	−0.484778	−	0.874637i	$-0.661100\pi$
$367$	1.79004e11	−	3.10044e11i	0.515069	−	0.892125i	0.999847	−	0.0174880i	$-0.00556689\pi$
$368$		0			0
$369$	−1.30818e11	+	7.55276e10i	−0.367323	+	0.212074i
$370$		0			0
$371$	7.92121e10	−	9.50066e10i	0.217075	−	0.260358i
$372$		0			0
$373$	−1.58340e11	−	2.74253e11i	−0.423546	−	0.733604i	0.572737	−	0.819739i	$-0.305882\pi$
$373$	−1.58340e11	−	2.74253e11i	−0.423546	−	0.733604i	−0.996283	+	0.0861352i	$0.972548\pi$
$374$		0			0
$375$	−2.37763e10	−	1.37272e10i	−0.0620873	−	0.0358461i
$376$		0			0
$377$			7.01659e11i			1.78892i
$378$		0			0
$379$		−	1.66427e11i		−	0.414331i	−0.978306	−	0.207165i	$-0.933576\pi$
$379$		−	1.66427e11i		−	0.414331i	0.978306	−	0.207165i	$-0.0664238\pi$
$380$		0			0
$381$	1.50996e11	+	8.71775e10i	0.367115	+	0.211954i
$382$		0			0
$383$	−3.48979e11	−	6.04450e11i	−0.828715	−	1.43538i	−0.899047	−	0.437852i	$-0.855739\pi$
$383$	−3.48979e11	−	6.04450e11i	−0.828715	−	1.43538i	0.0703324	−	0.997524i	$-0.477594\pi$
$384$		0			0
$385$	2.32010e10	+	4.01576e9i	0.0538187	+	0.00931524i
$386$		0			0
$387$	−6.68975e10	+	3.86233e10i	−0.151604	+	0.0875285i
$388$		0			0
$389$	−3.53548e11	+	6.12363e11i	−0.782844	+	1.35592i	0.147435	+	0.989072i	$0.452898\pi$
$389$	−3.53548e11	+	6.12363e11i	−0.782844	+	1.35592i	−0.930279	+	0.366853i	$0.880435\pi$
$390$		0			0
$391$	8.50680e10			0.184065
$392$		0			0
$393$	−1.13538e11			−0.240091
$394$		0			0
$395$	1.64233e10	−	2.84460e10i	0.0339448	−	0.0587941i
$396$		0			0
$397$	−4.53041e10	+	2.61563e10i	−0.0915335	+	0.0528469i	−0.545068	−	0.838392i	$-0.683496\pi$
$397$	−4.53041e10	+	2.61563e10i	−0.0915335	+	0.0528469i	0.453535	+	0.891239i	$0.350163\pi$
$398$		0			0
$399$	2.54804e11	+	4.41029e10i	0.503302	+	0.0871143i
$400$		0			0
$401$	−3.34552e11	−	5.79461e11i	−0.646121	−	1.11911i	−0.984041	−	0.177940i	$-0.943057\pi$
$401$	−3.34552e11	−	5.79461e11i	−0.646121	−	1.11911i	0.337920	−	0.941175i	$-0.390277\pi$
$402$		0			0
$403$	5.14238e11	+	2.96896e11i	0.971162	+	0.560701i
$404$		0			0
$405$			6.08018e9i			0.0112297i
$406$		0			0
$407$		−	8.31106e11i		−	1.50135i
$408$		0			0
$409$	8.65580e11	+	4.99743e11i	1.52951	+	0.883063i	0.999382	+	0.0351448i	$0.0111893\pi$
$409$	8.65580e11	+	4.99743e11i	1.52951	+	0.883063i	0.530127	+	0.847918i	$0.322144\pi$
$410$		0			0
$411$	2.59557e11	+	4.49566e11i	0.448688	+	0.777151i
$412$		0			0
$413$	7.42040e11	−	8.89999e11i	1.25502	−	1.50527i
$414$		0			0
$415$	1.48304e10	−	8.56236e9i	0.0245436	−	0.0141702i
$416$		0			0
$417$	1.26491e10	−	2.19089e10i	0.0204855	−	0.0354820i
$418$		0			0
$419$	3.34939e11			0.530887			0.265444	−	0.964126i	$-0.414482\pi$
$419$	3.34939e11			0.530887			0.265444	+	0.964126i	$0.414482\pi$
$420$		0			0
$421$	−1.25766e11			−0.195117			−0.0975584	−	0.995230i	$-0.531103\pi$
$421$	−1.25766e11			−0.195117			−0.0975584	+	0.995230i	$0.531103\pi$
$422$		0			0
$423$	−1.75604e11	+	3.04156e11i	−0.266688	+	0.461917i
$424$		0			0
$425$	1.81321e11	−	1.04685e11i	0.269586	−	0.155645i
$426$		0			0
$427$	−1.03858e12	+	3.81715e11i	−1.51187	+	0.555667i
$428$		0			0
$429$	−3.81312e11	−	6.60452e11i	−0.543529	−	0.941419i
$430$		0			0
$431$	3.26607e11	+	1.88567e11i	0.455909	+	0.263219i	0.710322	−	0.703876i	$-0.248549\pi$
$431$	3.26607e11	+	1.88567e11i	0.455909	+	0.263219i	−0.254414	+	0.967095i	$0.581882\pi$
$432$		0			0
$433$		−	9.74859e10i		−	0.133274i	−0.997777	−	0.0666372i	$-0.978773\pi$
$433$		−	9.74859e10i		−	0.133274i	0.997777	−	0.0666372i	$-0.0212270\pi$
$434$		0			0
$435$		−	3.24851e10i		−	0.0434993i
$436$		0			0
$437$	2.85879e11	+	1.65052e11i	0.374986	+	0.216498i
$438$		0			0
$439$	−4.74842e11	−	8.22451e11i	−0.610181	−	1.05686i	−0.991210	−	0.132301i	$-0.957763\pi$
$439$	−4.74842e11	−	8.22451e11i	−0.610181	−	1.05686i	0.381028	−	0.924563i	$-0.375570\pi$
$440$		0			0
$441$	7.37141e10	+	4.03204e11i	0.0928062	+	0.507635i
$442$		0			0
$443$	4.32032e11	−	2.49434e11i	0.532965	−	0.307708i	−0.209258	−	0.977861i	$-0.567105\pi$
$443$	4.32032e11	−	2.49434e11i	0.532965	−	0.307708i	0.742223	+	0.670153i	$0.233771\pi$
$444$		0			0
$445$	3.52077e10	−	6.09815e10i	0.0425616	−	0.0737188i
$446$		0			0
$447$	−9.71772e11			−1.15128
$448$		0			0
$449$	1.29243e12			1.50071			0.750355	−	0.661035i	$-0.229882\pi$
$449$	1.29243e12			1.50071			0.750355	+	0.661035i	$0.229882\pi$
$450$		0			0
$451$	−3.82221e11	+	6.62026e11i	−0.435031	+	0.753495i
$452$		0			0
$453$	−8.55554e11	+	4.93955e11i	−0.954565	+	0.551119i
$454$		0			0
$455$	−2.40207e10	−	6.53560e10i	−0.0262745	−	0.0714882i
$456$		0			0
$457$	−3.45494e11	−	5.98413e11i	−0.370525	−	0.641768i	0.619121	−	0.785295i	$-0.287489\pi$
$457$	−3.45494e11	−	5.98413e11i	−0.370525	−	0.641768i	−0.989646	+	0.143527i	$0.954156\pi$
$458$		0			0
$459$	−2.71098e11	−	1.56518e11i	−0.285081	−	0.164592i
$460$		0			0
$461$			1.01991e12i			1.05174i	0.850565	+	0.525870i	$0.176260\pi$
$461$			1.01991e12i			1.05174i	−0.850565	+	0.525870i	$0.823740\pi$
$462$		0			0
$463$			1.55917e12i			1.57681i	0.615159	+	0.788403i	$0.289092\pi$
$463$			1.55917e12i			1.57681i	−0.615159	+	0.788403i	$0.710908\pi$
$464$		0			0
$465$	−2.38080e10	−	1.37455e10i	−0.0236148	−	0.0136340i
$466$		0			0
$467$	9.37989e11	+	1.62464e12i	0.912581	+	1.58064i	0.810404	+	0.585871i	$0.199247\pi$
$467$	9.37989e11	+	1.62464e12i	0.912581	+	1.58064i	0.102177	+	0.994766i	$0.467419\pi$
$468$		0			0
$469$	3.22293e11	+	2.68713e11i	0.307591	+	0.256455i
$470$		0			0
$471$	−3.73215e11	+	2.15476e11i	−0.349434	+	0.201746i
$472$		0			0
$473$	−1.95460e11	+	3.38546e11i	−0.179549	+	0.310988i
$474$		0			0
$475$	8.12459e11			0.732285
$476$		0			0
$477$	−1.97787e11			−0.174930
$478$		0			0
$479$	−7.39485e11	+	1.28083e12i	−0.641829	+	1.11168i	0.343195	+	0.939264i	$0.388491\pi$
$479$	−7.39485e11	+	1.28083e12i	−0.641829	+	1.11168i	−0.985024	+	0.172416i	$0.944843\pi$
$480$		0			0
$481$	−2.12848e12	+	1.22888e12i	−1.81308	+	1.04678i
$482$		0			0
$483$	8.36875e10	−	4.83504e11i	0.0699678	−	0.404238i
$484$		0			0
$485$	4.28679e9	+	7.42495e9i	0.00351799	+	0.00609334i
$486$		0			0
$487$	−5.36952e11	−	3.10009e11i	−0.432569	−	0.249744i	0.267872	−	0.963455i	$-0.413680\pi$
$487$	−5.36952e11	−	3.10009e11i	−0.432569	−	0.249744i	−0.700440	+	0.713711i	$0.747013\pi$
$488$		0			0
$489$		−	1.28238e12i		−	1.01421i
$490$		0			0
$491$			5.50450e11i			0.427417i	0.976898	+	0.213708i	$0.0685542\pi$
$491$			5.50450e11i			0.427417i	−0.976898	+	0.213708i	$0.931446\pi$
$492$		0			0
$493$	4.29662e11	+	2.48066e11i	0.327579	+	0.189128i
$494$		0			0
$495$	−1.88247e10	−	3.26053e10i	−0.0140930	−	0.0244098i
$496$		0			0
$497$	4.31330e11	−	2.49201e12i	0.317107	−	1.83208i
$498$		0			0
$499$	−1.62133e12	+	9.36077e11i	−1.17063	+	0.675864i	−0.953829	−	0.300352i	$-0.902896\pi$
$499$	−1.62133e12	+	9.36077e11i	−1.17063	+	0.675864i	−0.216802	+	0.976216i	$0.569563\pi$
$500$		0			0
$501$	5.69182e11	−	9.85851e11i	0.403628	−	0.699104i
$502$		0			0
$503$	8.54975e10			0.0595522			0.0297761	−	0.999557i	$-0.490521\pi$
$503$	8.54975e10			0.0595522			0.0297761	+	0.999557i	$0.490521\pi$
$504$		0			0
$505$	1.03634e10			0.00709070
$506$		0			0
$507$	−6.10126e11	+	1.05677e12i	−0.410094	+	0.710304i
$508$		0			0
$509$	−2.14505e12	+	1.23845e12i	−1.41647	+	0.817801i	−0.995987	−	0.0894989i	$-0.971473\pi$
$509$	−2.14505e12	+	1.23845e12i	−1.41647	+	0.817801i	−0.420485	+	0.907299i	$0.638140\pi$
$510$		0			0
$511$	−3.22307e11	−	2.68724e11i	−0.209110	−	0.174346i
$512$		0			0
$513$	−6.07365e11	−	1.05199e12i	−0.387188	−	0.670629i
$514$		0			0
$515$	−4.35725e10	−	2.51566e10i	−0.0272948	−	0.0157587i
$516$		0			0
$517$			1.77735e12i			1.09412i
$518$		0			0
$519$			3.19870e11i			0.193518i
$520$		0			0
$521$	2.08692e12	+	1.20488e12i	1.24090	+	0.716431i	0.969277	−	0.245973i	$-0.0791076\pi$
$521$	2.08692e12	+	1.20488e12i	1.24090	+	0.716431i	0.271619	+	0.962405i	$0.412441\pi$
$522$		0			0
$523$	−2.19179e11	−	3.79628e11i	−0.128097	−	0.221871i	0.794842	−	0.606816i	$-0.207554\pi$
$523$	−2.19179e11	−	3.79628e11i	−0.128097	−	0.221871i	−0.922939	+	0.384945i	$0.874220\pi$
$524$		0			0
$525$	−4.16626e11	−	1.13356e12i	−0.239348	−	0.651222i
$526$		0			0
$527$	3.63609e11	−	2.09930e11i	0.205346	−	0.118557i
$528$		0			0
$529$	−5.87382e11	+	1.01738e12i	−0.326114	+	0.564847i
$530$		0			0
$531$	−1.85282e12			−1.01137
$532$		0			0
$533$	2.26062e12			1.21326
$534$		0			0
$535$	5.40397e10	−	9.35995e10i	0.0285181	−	0.0493948i
$536$		0			0
$537$	−1.05808e12	+	6.10883e11i	−0.549078	+	0.317011i
$538$		0			0
$539$	1.34331e12	+	1.58059e12i	0.685529	+	0.806622i
$540$		0			0
$541$	−1.46060e12	−	2.52983e12i	−0.733066	−	1.26971i	−0.955567	−	0.294775i	$-0.904755\pi$
$541$	−1.46060e12	−	2.52983e12i	−0.733066	−	1.26971i	0.222501	−	0.974932i	$-0.428578\pi$
$542$		0			0
$543$	8.01872e11	+	4.62961e11i	0.395827	+	0.228531i
$544$		0			0
$545$			9.18073e10i			0.0445752i
$546$		0			0
$547$			6.64645e10i			0.0317429i	0.999874	+	0.0158715i	$0.00505225\pi$
$547$			6.64645e10i			0.0317429i	−0.999874	+	0.0158715i	$0.994948\pi$
$548$		0			0
$549$	1.53223e12	+	8.84635e11i	0.719862	+	0.415612i
$550$		0			0
$551$	9.62613e11	+	1.66729e12i	0.444907	+	0.770602i
$552$		0			0
$553$	2.71602e12	−	9.98236e11i	1.23501	−	0.453910i
$554$		0			0
$555$	9.85436e10	−	5.68942e10i	0.0440869	−	0.0254536i
$556$		0			0
$557$	−1.13156e12	+	1.95992e12i	−0.498115	+	0.862761i	−0.999998	−	0.00217510i	$-0.999308\pi$
$557$	−1.13156e12	+	1.95992e12i	−0.498115	+	0.862761i	0.501883	+	0.864936i	$0.332641\pi$
$558$		0			0
$559$	1.15603e12			0.500746
$560$		0			0
$561$	−5.39239e11			−0.229852
$562$		0			0
$563$	1.97456e12	−	3.42004e12i	0.828290	−	1.43464i	−0.0710888	−	0.997470i	$-0.522647\pi$
$563$	1.97456e12	−	3.42004e12i	0.828290	−	1.43464i	0.899379	−	0.437170i	$-0.144019\pi$
$564$		0			0
$565$	−1.49660e10	+	8.64063e9i	−0.00617857	+	0.00356720i
$566$		0			0
$567$	−3.43010e11	+	4.11405e11i	−0.139374	+	0.167165i
$568$		0			0
$569$	−1.26169e12	−	2.18531e12i	−0.504599	−	0.873991i	−0.999986	−	0.00531851i	$-0.998307\pi$
$569$	−1.26169e12	−	2.18531e12i	−0.504599	−	0.873991i	0.495387	−	0.868672i	$-0.335026\pi$
$570$		0			0
$571$	−2.86036e12	−	1.65143e12i	−1.12605	−	0.650126i	−0.183112	−	0.983092i	$-0.558617\pi$
$571$	−2.86036e12	−	1.65143e12i	−1.12605	−	0.650126i	−0.942939	+	0.332967i	$0.891950\pi$
$572$		0			0
$573$			1.32875e12i			0.514928i
$574$		0			0
$575$		−	1.54168e12i		−	0.588151i
$576$		0			0
$577$	2.14328e12	+	1.23742e12i	0.804983	+	0.464757i	0.845211	−	0.534433i	$-0.179475\pi$
$577$	2.14328e12	+	1.23742e12i	0.804983	+	0.464757i	−0.0402275	+	0.999191i	$0.512808\pi$
$578$		0			0
$579$	−1.15688e12	−	2.00377e12i	−0.427793	−	0.740959i
$580$		0			0
$581$	1.48652e12	+	2.57294e11i	0.541224	+	0.0936781i
$582$		0			0
$583$	−8.66836e11	+	5.00468e11i	−0.310762	+	0.179419i
$584$		0			0
$585$	−5.56686e10	+	9.64209e10i	−0.0196521	+	0.0340384i
$586$		0			0
$587$	7.58274e11			0.263606			0.131803	−	0.991276i	$-0.457923\pi$
$587$	7.58274e11			0.263606			0.131803	+	0.991276i	$0.457923\pi$
$588$		0			0
$589$	1.62926e12			0.557790
$590$		0			0
$591$	1.19068e12	−	2.06232e12i	0.401469	−	0.695365i
$592$		0			0
$593$	−8.64858e10	+	4.99326e10i	−0.0287209	+	0.0165820i	−0.514292	−	0.857615i	$-0.671945\pi$
$593$	−8.64858e10	+	4.99326e10i	−0.0287209	+	0.0165820i	0.485571	+	0.874197i	$0.338612\pi$
$594$		0			0
$595$	−4.85133e10	−	8.39694e9i	−0.0158684	−	0.00274660i
$596$		0			0
$597$	−1.90276e12	−	3.29568e12i	−0.613056	−	1.06184i
$598$		0			0
$599$	1.88135e12	+	1.08620e12i	0.597102	+	0.344737i	0.767901	−	0.640569i	$-0.221302\pi$
$599$	1.88135e12	+	1.08620e12i	0.597102	+	0.344737i	−0.170799	+	0.985306i	$0.554635\pi$
$600$		0			0
$601$			4.36599e12i			1.36505i	0.730863	+	0.682524i	$0.239118\pi$
$601$			4.36599e12i			1.36505i	−0.730863	+	0.682524i	$0.760882\pi$
$602$		0			0
$603$		−	6.70957e11i		−	0.206665i
$604$		0			0
$605$	−1.77566e10	−	1.02518e10i	−0.00538841	−	0.00311100i
$606$		0			0
$607$	−1.04817e12	−	1.81549e12i	−0.313389	−	0.542806i	0.665705	−	0.746215i	$-0.268131\pi$
$607$	−1.04817e12	−	1.81549e12i	−0.313389	−	0.542806i	−0.979094	+	0.203410i	$0.934798\pi$
$608$		0			0
$609$	1.83263e12	−	2.19805e12i	0.539879	−	0.647529i
$610$		0			0
$611$	4.55184e12	−	2.62801e12i	1.32130	−	0.762854i
$612$		0			0
$613$	1.37092e12	−	2.37450e12i	0.392138	−	0.679204i	−0.600593	−	0.799555i	$-0.705069\pi$
$613$	1.37092e12	−	2.37450e12i	0.392138	−	0.679204i	0.992731	+	0.120351i	$0.0384021\pi$
$614$		0			0
$615$	−1.04661e11			−0.0295017
$616$		0			0
$617$	−3.83016e12			−1.06398			−0.531990	−	0.846751i	$-0.678556\pi$
$617$	−3.83016e12			−1.06398			−0.531990	+	0.846751i	$0.678556\pi$
$618$		0			0
$619$	3.22983e12	−	5.59423e12i	0.884244	−	1.53156i	0.0376658	−	0.999290i	$-0.488008\pi$
$619$	3.22983e12	−	5.59423e12i	0.884244	−	1.53156i	0.846578	−	0.532265i	$-0.178659\pi$
$620$		0			0
$621$	−1.99620e12	+	1.15251e12i	−0.538631	+	0.310979i
$622$		0			0
$623$	5.82251e12	−	2.13998e12i	1.54851	−	0.569134i
$624$		0			0
$625$	−1.89213e12	−	3.27726e12i	−0.496010	−	0.859115i
$626$		0			0
$627$	−1.81216e12	−	1.04625e12i	−0.468266	−	0.270354i
$628$		0			0
$629$			1.73784e12i			0.442672i
$630$		0			0
$631$			4.58234e12i			1.15068i	0.817913	+	0.575341i	$0.195131\pi$
$631$			4.58234e12i			1.15068i	−0.817913	+	0.575341i	$0.804869\pi$
$632$		0			0
$633$	3.56457e12	+	2.05800e12i	0.882450	+	0.509483i
$634$		0			0
$635$	−6.44084e10	−	1.11559e11i	−0.0157203	−	0.0272284i
$636$		0			0
$637$	2.06171e12	−	5.77732e12i	0.496134	−	1.39027i
$638$		0			0
$639$	−3.50211e12	+	2.02195e12i	−0.830953	+	0.479751i
$640$		0			0
$641$	−1.97507e12	+	3.42093e12i	−0.462085	+	0.800355i	−0.999065	−	0.0432402i	$-0.986232\pi$
$641$	−1.97507e12	+	3.42093e12i	−0.462085	+	0.800355i	0.536979	+	0.843595i	$0.319565\pi$
$642$		0			0
$643$	−5.41792e12			−1.24992			−0.624961	−	0.780656i	$-0.714885\pi$
$643$	−5.41792e12			−1.24992			−0.624961	+	0.780656i	$0.714885\pi$
$644$		0			0
$645$	−5.35215e10			−0.0121761
$646$		0			0
$647$	2.41856e12	−	4.18907e12i	0.542610	−	0.939828i	−0.456143	−	0.889906i	$-0.650769\pi$
$647$	2.41856e12	−	4.18907e12i	0.542610	−	0.939828i	0.998753	−	0.0499218i	$-0.0158972\pi$
$648$		0			0
$649$	−8.12032e12	+	4.68827e12i	−1.79668	+	1.03732i
$650$		0			0
$651$	−8.35477e11	−	2.27318e12i	−0.182314	−	0.496043i
$652$		0			0
$653$	−3.08329e12	−	5.34042e12i	−0.663598	−	1.14939i	−0.979663	−	0.200648i	$-0.935695\pi$
$653$	−3.08329e12	−	5.34042e12i	−0.663598	−	1.14939i	0.316065	−	0.948737i	$-0.397638\pi$
$654$		0			0
$655$	7.26461e10	+	4.19422e10i	0.0154215	+	0.00890360i
$656$		0			0
$657$			6.70986e11i			0.140498i
$658$		0			0
$659$		−	5.47527e12i		−	1.13089i	−0.824785	−	0.565446i	$-0.808704\pi$
$659$		−	5.47527e12i		−	1.13089i	0.824785	−	0.565446i	$-0.191296\pi$
$660$		0			0
$661$	4.49391e12	+	2.59456e12i	0.915625	+	0.528636i	0.882237	−	0.470806i	$-0.156037\pi$
$661$	4.49391e12	+	2.59456e12i	0.915625	+	0.528636i	0.0333884	+	0.999442i	$0.489370\pi$
$662$		0			0
$663$	7.97323e11	+	1.38100e12i	0.160259	+	0.277577i
$664$		0			0
$665$	−1.46741e11	−	1.22346e11i	−0.0290974	−	0.0242601i
$666$		0			0
$667$	3.16377e12	−	1.82660e12i	0.618926	−	0.357337i
$668$		0			0
$669$	−1.71782e12	+	2.97536e12i	−0.331559	+	0.574278i
$670$		0			0
$671$	8.95370e12			1.70510
$672$		0			0
$673$	6.77917e12			1.27382			0.636911	−	0.770937i	$-0.280212\pi$
$673$	6.77917e12			1.27382			0.636911	+	0.770937i	$0.280212\pi$
$674$		0			0
$675$	−2.83657e12	+	4.91308e12i	−0.525928	+	0.910934i
$676$		0			0
$677$	−8.88766e12	+	5.13129e12i	−1.62607	+	0.938810i	−0.640816	+	0.767695i	$0.721404\pi$
$677$	−8.88766e12	+	5.13129e12i	−1.62607	+	0.938810i	−0.985251	+	0.171115i	$0.945263\pi$
$678$		0			0
$679$	−1.28816e11	+	7.44233e11i	−0.0232571	+	0.134368i
$680$		0			0
$681$	−5.95418e11	−	1.03129e12i	−0.106086	−	0.183747i
$682$		0			0
$683$	−3.57354e12	−	2.06318e12i	−0.628356	−	0.362781i	0.151759	−	0.988417i	$-0.451506\pi$
$683$	−3.57354e12	−	2.06318e12i	−0.628356	−	0.362781i	−0.780115	+	0.625636i	$0.784839\pi$
$684$		0			0
$685$		−	3.83532e11i		−	0.0665570i
$686$		0			0
$687$			2.51999e11i			0.0431611i
$688$		0			0
$689$	2.56342e12	+	1.47999e12i	0.433345	+	0.250192i
$690$		0			0
$691$	−4.85220e11	−	8.40426e11i	−0.0809632	−	0.140232i	0.822701	−	0.568475i	$-0.192466\pi$
$691$	−4.85220e11	−	8.40426e11i	−0.0809632	−	0.140232i	−0.903664	+	0.428242i	$0.859133\pi$
$692$		0			0
$693$	5.65672e11	−	3.26816e12i	0.0931676	−	0.538275i
$694$		0			0
$695$	−1.61867e10	+	9.34541e9i	−0.00263164	+	0.00151938i
$696$		0			0
$697$	7.99223e11	−	1.38429e12i	0.128269	−	0.222168i
$698$		0			0
$699$	−1.18949e12			−0.188457
$700$		0			0
$701$	7.79861e12			1.21979			0.609897	−	0.792481i	$-0.291211\pi$
$701$	7.79861e12			1.21979			0.609897	+	0.792481i	$0.291211\pi$
$702$		0			0
$703$	−3.37183e12	+	5.84018e12i	−0.520675	+	0.901835i
$704$		0			0
$705$	−2.10739e11	+	1.21670e11i	−0.0321288	+	0.0185496i
$706$		0			0
$707$	7.01218e11	+	5.84643e11i	0.105552	+	0.0880042i
$708$		0			0
$709$	−8.36199e11	−	1.44834e12i	−0.124280	−	0.215260i	0.797171	−	0.603753i	$-0.206329\pi$
$709$	−8.36199e11	−	1.44834e12i	−0.124280	−	0.215260i	−0.921451	+	0.388494i	$0.872995\pi$
$710$		0			0
$711$	−4.00699e12	−	2.31343e12i	−0.588037	−	0.339503i
$712$		0			0
$713$		−	3.09159e12i		−	0.448001i
$714$		0			0
$715$			5.63441e11i			0.0806253i
$716$		0			0
$717$	2.06433e12	+	1.19184e12i	0.291704	+	0.168415i
$718$		0			0
$719$	5.38709e12	+	9.33071e12i	0.751751	+	1.30207i	0.946973	+	0.321312i	$0.104124\pi$
$719$	5.38709e12	+	9.33071e12i	0.751751	+	1.30207i	−0.195222	+	0.980759i	$0.562543\pi$
$720$		0			0
$721$	−1.52906e12	−	4.16030e12i	−0.210725	−	0.573345i
$722$		0			0
$723$	−2.15599e12	+	1.24476e12i	−0.293443	+	0.169420i
$724$		0			0
$725$	4.49568e12	−	7.78674e12i	0.604330	−	1.04673i
$726$		0			0
$727$	5.40436e11			0.0717529			0.0358764	−	0.999356i	$-0.488578\pi$
$727$	5.40436e11			0.0717529			0.0358764	+	0.999356i	$0.488578\pi$
$728$		0			0
$729$	6.45133e12			0.846009
$730$		0			0
$731$	4.08706e11	−	7.07900e11i	0.0529399	−	0.0916946i
$732$		0			0
$733$	−1.29761e12	+	7.49173e11i	−0.166026	+	0.0958549i	−0.580710	−	0.814110i	$-0.697225\pi$
$733$	−1.29761e12	+	7.49173e11i	−0.166026	+	0.0958549i	0.414685	+	0.909965i	$0.363892\pi$
$734$		0			0
$735$	−9.54519e10	+	2.67476e11i	−0.0120640	+	0.0338058i
$736$		0			0
$737$	−1.69775e12	−	2.94059e12i	−0.211968	−	0.367139i
$738$		0			0
$739$	−9.39840e12	−	5.42617e12i	−1.15919	−	0.669258i	−0.208079	−	0.978112i	$-0.566721\pi$
$739$	−9.39840e12	−	5.42617e12i	−1.15919	−	0.669258i	−0.951109	+	0.308855i	$0.900054\pi$
$740$		0			0
$741$			6.18798e12i			0.753993i
$742$		0			0
$743$			1.57187e13i			1.89220i	0.323872	+	0.946101i	$0.395015\pi$
$743$			1.57187e13i			1.89220i	−0.323872	+	0.946101i	$0.604985\pi$
$744$		0			0
$745$	6.21776e11	+	3.58982e11i	0.0739487	+	0.0426943i
$746$		0			0
$747$	−1.20612e12	−	2.08906e12i	−0.141725	−	0.245476i
$748$		0			0
$749$	8.93686e12	−	3.28462e12i	1.03757	−	0.381344i
$750$		0			0
$751$	−1.11534e13	+	6.43940e12i	−1.27946	+	0.738696i	−0.976749	−	0.214387i	$-0.931225\pi$
$751$	−1.11534e13	+	6.43940e12i	−1.27946	+	0.738696i	−0.302710	+	0.953083i	$0.597891\pi$
$752$		0			0
$753$	−1.51575e12	+	2.62536e12i	−0.171811	+	0.297586i
$754$		0			0
$755$	7.29887e11			0.0817512
$756$		0			0
$757$	−1.05114e13			−1.16340			−0.581702	−	0.813402i	$-0.697613\pi$
$757$	−1.05114e13			−1.16340			−0.581702	+	0.813402i	$0.697613\pi$
$758$		0			0
$759$	−1.98531e12	+	3.43866e12i	−0.217141	+	0.376098i
$760$		0			0
$761$	−9.98791e12	+	5.76652e12i	−1.07955	+	0.623280i	−0.930776	−	0.365591i	$-0.880867\pi$
$761$	−9.98791e12	+	5.76652e12i	−1.07955	+	0.623280i	−0.148777	+	0.988871i	$0.547534\pi$
$762$		0			0
$763$	−5.17926e12	+	6.21198e12i	−0.553232	+	0.663544i
$764$		0			0
$765$	3.93624e10	+	6.81776e10i	0.00415532	+	0.00719723i
$766$		0			0
$767$	2.40135e13	+	1.38642e13i	2.50540	+	1.44649i
$768$		0			0
$769$			6.73610e12i			0.694609i	0.937752	+	0.347304i	$0.112903\pi$
$769$			6.73610e12i			0.694609i	−0.937752	+	0.347304i	$0.887097\pi$
$770$		0			0
$771$		−	8.92223e11i		−	0.0909345i
$772$		0			0
$773$	1.27287e13	+	7.34892e12i	1.28226	+	0.740313i	0.977261	−	0.212039i	$-0.0680104\pi$
$773$	1.27287e13	+	7.34892e12i	1.28226	+	0.740313i	0.304999	+	0.952353i	$0.401344\pi$
$774$		0			0
$775$	−3.80454e12	−	6.58966e12i	−0.378830	−	0.656154i
$776$		0			0
$777$	9.87743e12	+	1.70964e12i	0.972186	+	0.168271i
$778$		0			0
$779$	5.37172e12	−	3.10137e12i	0.522631	−	0.301741i
$780$		0			0
$781$	−1.02324e13	+	1.77231e13i	−0.984121	+	1.70455i
$782$		0			0
$783$	−1.34432e13			−1.27813
$784$		0			0
$785$	3.18395e11			0.0299263
$786$		0			0
$787$	−7.36414e12	+	1.27551e13i	−0.684283	+	1.18521i	0.289379	+	0.957215i	$0.406551\pi$
$787$	−7.36414e12	+	1.27551e13i	−0.684283	+	1.18521i	−0.973662	+	0.227998i	$0.926782\pi$
$788$		0			0
$789$	−1.18919e12	+	6.86576e11i	−0.109245	+	0.0630728i
$790$		0			0
$791$	−1.50011e12	−	2.59647e11i	−0.136247	−	0.0235824i
$792$		0			0
$793$	−1.32390e13	−	2.29306e13i	−1.18885	−	2.05914i
$794$		0			0
$795$	−1.18680e11	−	6.85200e10i	−0.0105372	−	0.00608367i
$796$		0			0
$797$		−	1.71463e12i		−	0.150525i	−0.997164	−	0.0752625i	$-0.976021\pi$
$797$		−	1.71463e12i		−	0.150525i	0.997164	−	0.0752625i	$-0.0239795\pi$
$798$		0			0
$799$		−	3.71644e12i		−	0.322602i
$800$		0			0
$801$	−8.59004e12	−	4.95946e12i	−0.737308	−	0.425685i
$802$		0			0
$803$	1.69782e12	+	2.94071e12i	0.144103	+	0.249593i
$804$		0			0
$805$	−2.32157e11	+	2.78448e11i	−0.0194850	+	0.0233702i
$806$		0			0
$807$	−6.71454e12	+	3.87664e12i	−0.557296	+	0.321755i
$808$		0			0
$809$	5.95793e12	−	1.03194e13i	0.489020	−	0.847008i	−0.510900	−	0.859640i	$-0.670688\pi$
$809$	5.95793e12	−	1.03194e13i	0.489020	−	0.847008i	0.999920	+	0.0126325i	$0.00402114\pi$
$810$		0			0
$811$	−7.25756e12			−0.589110			−0.294555	−	0.955634i	$-0.595171\pi$
$811$	−7.25756e12			−0.589110			−0.294555	+	0.955634i	$0.595171\pi$
$812$		0			0
$813$	−1.08383e12			−0.0870068
$814$		0			0
$815$	−4.73723e11	+	8.20513e11i	−0.0376111	+	0.0651443i
$816$		0			0
$817$	2.74699e12	−	1.58597e12i	0.215704	−	0.124537i
$818$		0			0
$819$	−9.20625e12	+	3.38363e12i	−0.714999	+	0.262788i
$820$		0			0
$821$	−9.54978e12	−	1.65407e13i	−0.733583	−	1.27060i	−0.955342	−	0.295501i	$-0.904513\pi$
$821$	−9.54978e12	−	1.65407e13i	−0.733583	−	1.27060i	0.221760	−	0.975101i	$-0.428820\pi$
$822$		0			0
$823$	4.68988e12	+	2.70770e12i	0.356338	+	0.205732i	0.667473	−	0.744634i	$-0.267376\pi$
$823$	4.68988e12	+	2.70770e12i	0.356338	+	0.205732i	−0.311135	+	0.950366i	$0.600709\pi$
$824$		0			0
$825$			9.77258e12i			0.734457i
$826$		0			0
$827$		−	6.28870e12i		−	0.467505i	−0.972296	−	0.233752i	$-0.924899\pi$
$827$		−	6.28870e12i		−	0.467505i	0.972296	−	0.233752i	$-0.0751005\pi$
$828$		0			0
$829$	9.34566e11	+	5.39572e11i	0.0687250	+	0.0396784i	0.533969	−	0.845504i	$-0.320700\pi$
$829$	9.34566e11	+	5.39572e11i	0.0687250	+	0.0396784i	−0.465244	+	0.885183i	$0.654033\pi$
$830$		0			0
$831$	1.96936e12	+	3.41103e12i	0.143258	+	0.248131i
$832$		0			0
$833$	−2.80886e12	−	3.30501e12i	−0.202128	−	0.237832i
$834$		0			0
$835$	−7.28366e11	+	4.20523e11i	−0.0518514	+	0.0299364i
$836$		0			0
$837$	−5.68828e12	+	9.85239e12i	−0.400605	+	0.693869i
$838$		0			0
$839$	1.32667e13			0.924345			0.462173	−	0.886790i	$-0.347070\pi$
$839$	1.32667e13			0.924345			0.462173	+	0.886790i	$0.347070\pi$
$840$		0			0
$841$	6.79901e12			0.468667
$842$		0			0
$843$	4.27051e12	−	7.39675e12i	0.291243	−	0.504448i
$844$		0			0
$845$	7.80762e11	−	4.50773e11i	0.0526821	−	0.0304160i
$846$		0			0
$847$	−6.23121e11	−	1.69540e12i	−0.0416003	−	0.113187i
$848$		0			0
$849$	6.91883e12	+	1.19838e13i	0.457034	+	0.791606i
$850$		0			0
$851$	1.10820e13	+	6.39821e12i	0.724329	+	0.418191i
$852$		0			0
$853$		−	1.26415e13i		−	0.817577i	−0.912629	−	0.408789i	$-0.865951\pi$
$853$		−	1.26415e13i		−	0.817577i	0.912629	−	0.408789i	$-0.134049\pi$
$854$		0			0
$855$			3.05489e11i			0.0195501i
$856$		0			0
$857$	1.62106e13	+	9.35920e12i	1.02656	+	0.592687i	0.915998	−	0.401182i	$-0.131401\pi$
$857$	1.62106e13	+	9.35920e12i	1.02656	+	0.592687i	0.110565	+	0.993869i	$0.464734\pi$
$858$		0			0
$859$	3.41840e12	+	5.92084e12i	0.214217	+	0.371034i	0.953030	−	0.302876i	$-0.0979468\pi$
$859$	3.41840e12	+	5.92084e12i	0.214217	+	0.371034i	−0.738813	+	0.673910i	$0.764613\pi$
$860$		0			0
$861$	−7.08171e12	−	5.90440e12i	−0.439161	−	0.366152i
$862$		0			0
$863$	2.33827e13	−	1.35000e13i	1.43498	−	0.828485i	0.437483	−	0.899227i	$-0.355870\pi$
$863$	2.33827e13	−	1.35000e13i	1.43498	−	0.828485i	0.997495	+	0.0707418i	$0.0225366\pi$
$864$		0			0
$865$	1.18163e11	−	2.04665e11i	0.00717646	−	0.0124300i
$866$		0			0
$867$	−1.04465e13			−0.627894
$868$		0			0
$869$	−2.34151e13			−1.39286
$870$		0			0
$871$	−5.02061e12	+	8.69595e12i	−0.295580	+	0.511959i
$872$		0			0
$873$	1.04590e12	−	6.03851e11i	0.0609433	−	0.0351857i
$874$		0			0
$875$	−3.04760e11	+	1.76075e12i	−0.0175761	+	0.101546i
$876$		0			0
$877$	1.38831e13	+	2.40463e13i	0.792481	+	1.37262i	0.924426	+	0.381361i	$0.124544\pi$
$877$	1.38831e13	+	2.40463e13i	0.792481	+	1.37262i	−0.131945	+	0.991257i	$0.542122\pi$
$878$		0			0
$879$	−7.06897e12	−	4.08127e12i	−0.399399	−	0.230593i
$880$		0			0
$881$			1.84371e13i			1.03110i	0.856859	+	0.515550i	$0.172412\pi$
$881$			1.84371e13i			1.03110i	−0.856859	+	0.515550i	$0.827588\pi$
$882$		0			0
$883$		−	2.63903e13i		−	1.46090i	−0.682966	−	0.730450i	$-0.739310\pi$
$883$		−	2.63903e13i		−	1.46090i	0.682966	−	0.730450i	$-0.260690\pi$
$884$		0			0
$885$	−1.11177e12	−	6.41879e11i	−0.0609213	−	0.0351729i
$886$		0			0
$887$	−1.70404e13	−	2.95148e13i	−0.924321	−	1.60097i	−0.792649	−	0.609678i	$-0.791299\pi$
$887$	−1.70404e13	−	2.95148e13i	−0.924321	−	1.60097i	−0.131672	−	0.991293i	$-0.542035\pi$
$888$		0			0
$889$	1.93544e12	−	1.11820e13i	0.103925	−	0.600428i
$890$		0			0
$891$	3.75364e12	−	2.16716e12i	0.199528	−	0.115197i
$892$		0			0
$893$	7.21078e12	−	1.24894e13i	0.379447	−	0.657221i
$894$		0			0
$895$	9.02665e11			0.0470244
$896$		0			0
$897$	1.17420e13			0.605586
$898$		0			0
$899$	9.01535e12	−	1.56151e13i	0.460325	−	0.797306i
$900$		0			0
$901$	1.81255e12	−	1.04648e12i	0.0916282	−	0.0529016i
$902$		0			0
$903$	−3.62144e12	−	3.01939e12i	−0.181254	−	0.151121i
$904$		0			0
$905$	−3.42045e11	−	5.92439e11i	−0.0169498	−	0.0293579i
$906$		0			0
$907$	3.76434e12	+	2.17334e12i	0.184695	+	0.106634i	0.589497	−	0.807771i	$-0.299326\pi$
$907$	3.76434e12	+	2.17334e12i	0.184695	+	0.106634i	−0.404802	+	0.914405i	$0.632659\pi$
$908$		0			0
$909$		−	1.45981e12i		−	0.0709186i
$910$		0			0
$911$		−	2.13135e13i		−	1.02523i	−0.858617	−	0.512617i	$-0.828676\pi$
$911$		−	2.13135e13i		−	1.02523i	0.858617	−	0.512617i	$-0.171324\pi$
$912$		0			0
$913$	−1.05721e13	−	6.10378e12i	−0.503548	−	0.290724i
$914$		0			0
$915$	6.12934e11	+	1.06163e12i	0.0289080	+	0.0500702i
$916$		0			0
$917$	2.54932e12	+	6.93624e12i	0.119059	+	0.323938i
$918$		0			0
$919$	−7.30324e12	+	4.21653e12i	−0.337750	+	0.195000i	−0.659277	−	0.751900i	$-0.729137\pi$
$919$	−7.30324e12	+	4.21653e12i	−0.337750	+	0.195000i	0.321526	+	0.946901i	$0.395804\pi$
$920$		0			0
$921$	7.19580e12	−	1.24635e13i	0.329542	−	0.570783i
$922$		0			0
$923$	6.05189e13			2.74463
$924$		0			0
$925$	3.14948e13			1.41449
$926$		0			0
$927$	−3.54363e12	+	6.13776e12i	−0.157612	+	0.272993i
$928$		0			0
$929$	−7.18120e12	+	4.14607e12i	−0.316320	+	0.182627i	−0.649751	−	0.760147i	$-0.725127\pi$
$929$	−7.18120e12	+	4.14607e12i	−0.316320	+	0.182627i	0.333431	+	0.942774i	$0.391794\pi$
$930$		0			0
$931$	−3.02690e12	−	1.65566e13i	−0.132046	−	0.722269i
$932$		0			0
$933$	1.55253e12	+	2.68906e12i	0.0670767	+	0.116180i
$934$		0			0
$935$	3.45025e11	+	1.99200e11i	0.0147638	+	0.00852388i
$936$		0			0
$937$		−	2.25567e13i		−	0.955978i	−0.878366	−	0.477989i	$-0.841366\pi$
$937$		−	2.25567e13i		−	0.955978i	0.878366	−	0.477989i	$-0.158634\pi$
$938$		0			0
$939$			1.87482e13i			0.786979i
$940$		0			0
$941$	−1.94352e12	−	1.12209e12i	−0.0808047	−	0.0466526i	0.459053	−	0.888409i	$-0.348189\pi$
$941$	−1.94352e12	−	1.12209e12i	−0.0808047	−	0.0466526i	−0.539858	+	0.841756i	$0.681522\pi$
$942$		0			0
$943$	−5.88499e12	−	1.01931e13i	−0.242350	−	0.419763i
$944$		0			0
$945$	1.25217e12	−	4.60218e11i	0.0510764	−	0.0187724i
$946$		0			0
$947$	1.29414e13	−	7.47171e12i	0.522885	−	0.301888i	−0.215229	−	0.976564i	$-0.569050\pi$
$947$	1.29414e13	−	7.47171e12i	0.522885	−	0.301888i	0.738114	+	0.674676i	$0.235717\pi$
$948$		0			0
$949$	5.02082e12	−	8.69632e12i	0.200945	−	0.348047i
$950$		0			0
$951$	8.73118e12			0.346147
$952$		0			0
$953$	−1.94651e13			−0.764433			−0.382216	−	0.924073i	$-0.624839\pi$
$953$	−1.94651e13			−0.764433			−0.382216	+	0.924073i	$0.624839\pi$
$954$		0			0
$955$	4.90852e11	−	8.50180e11i	0.0190957	−	0.0330747i
$956$		0			0
$957$	−2.00549e13	+	1.15787e13i	−0.772888	+	0.446227i
$958$		0			0
$959$	2.16368e13	−	2.59510e13i	0.826054	−	0.990765i
$960$		0			0
$961$	5.59041e12	+	9.68287e12i	0.211441	+	0.366226i
$962$		0			0
$963$	−1.31847e13	−	7.61219e12i	−0.494029	−	0.285227i
$964$		0			0
$965$			1.70945e12i			0.0634575i
$966$		0			0
$967$		−	2.68758e13i		−	0.988421i	−0.869342	−	0.494210i	$-0.835457\pi$
$967$		−	2.68758e13i		−	0.988421i	0.869342	−	0.494210i	$-0.164543\pi$
$968$		0			0
$969$	3.78923e12	+	2.18771e12i	0.138068	+	0.0797137i
$970$		0			0
$971$	1.53534e13	+	2.65929e13i	0.554266	+	0.960017i	0.997960	+	0.0638387i	$0.0203343\pi$
$971$	1.53534e13	+	2.65929e13i	0.554266	+	0.960017i	−0.443694	+	0.896178i	$0.646332\pi$
$972$		0			0
$973$	−1.62246e12	−	2.80825e11i	−0.0580319	−	0.0100445i
$974$		0			0
$975$	2.50278e13	−	1.44498e13i	0.886956	−	0.512084i
$976$		0			0
$977$	−1.20379e13	+	2.08503e13i	−0.422695	+	0.732129i	−0.996202	−	0.0870716i	$-0.972249\pi$
$977$	−1.20379e13	+	2.08503e13i	−0.422695	+	0.732129i	0.573507	+	0.819201i	$0.305582\pi$
$978$		0			0
$979$	−5.01965e13			−1.74643
$980$		0			0
$981$	1.29323e13			0.445824
$982$		0			0
$983$	−7.76577e12	+	1.34507e13i	−0.265273	+	0.459467i	−0.967635	−	0.252353i	$-0.918796\pi$
$983$	−7.76577e12	+	1.34507e13i	−0.265273	+	0.459467i	0.702362	+	0.711820i	$0.252129\pi$
$984$		0			0
$985$	−1.52368e12	+	8.79700e11i	−0.0515741	+	0.0297763i
$986$		0			0
$987$	−2.11233e13	−	3.65613e12i	−0.708490	−	0.122629i
$988$		0			0
$989$	−3.00946e12	−	5.21254e12i	−0.100024	−	0.173247i
$990$		0			0
$991$	−4.71861e11	−	2.72429e11i	−0.0155411	−	0.00897268i	0.492209	−	0.870477i	$-0.336189\pi$
$991$	−4.71861e11	−	2.72429e11i	−0.0155411	−	0.00897268i	−0.507750	+	0.861504i	$0.669523\pi$
$992$		0			0
$993$		−	2.89543e13i		−	0.945022i
$994$		0			0
$995$			2.81160e12i			0.0909388i
$996$		0			0
$997$	−1.84745e13	−	1.06663e13i	−0.592168	−	0.341888i	0.173786	−	0.984783i	$-0.444400\pi$
$997$	−1.84745e13	−	1.06663e13i	−0.592168	−	0.341888i	−0.765954	+	0.642895i	$0.777733\pi$
$998$		0			0
$999$	−2.35444e13	−	4.07801e13i	−0.747898	−	1.29540i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.b.31.9	yes	24
4.3	odd	2	inner	112.10.p.b.31.4	✓	24
7.5	odd	6	inner	112.10.p.b.47.4	yes	24
28.19	even	6	inner	112.10.p.b.47.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.b.31.4	✓	24	4.3	odd	2	inner
112.10.p.b.31.9	yes	24	1.1	even	1	trivial
112.10.p.b.47.4	yes	24	7.5	odd	6	inner
112.10.p.b.47.9	yes	24	28.19	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(48.7996 - 84.5234i) q^{3} +(-62.4476 + 36.0541i) q^{5} +(-6259.38 - 1083.41i) q^{7} +(5078.69 + 8796.55i) q^{9} +(44516.5 + 25701.6i) q^{11} -152010. i q^{13} +7037.71i q^{15} +(-93083.9 - 53742.0i) q^{17} +(-208545. - 361210. i) q^{19} +(-397029. + 476194. i) q^{21} +(-685413. + 395724. i) q^{23} +(-973963. + 1.68695e6i) q^{25} +2.91240e6 q^{27} -4.61586e6 q^{29} +(-1.95313e6 + 3.38291e6i) q^{31} +(4.34478e6 - 2.50846e6i) q^{33} +(429944. - 158020. i) q^{35} +(-8.08419e6 - 1.40022e7i) q^{37} +(-1.28484e7 - 7.41805e6i) q^{39} +1.48715e7i q^{41} +7.60496e6i q^{43} +(-634304. - 366216. i) q^{45} +(1.72883e7 + 2.99443e7i) q^{47} +(3.80061e7 + 1.35629e7i) q^{49} +(-9.08492e6 + 5.24518e6i) q^{51} +(-9.73612e6 + 1.68635e7i) q^{53} -3.70660e6 q^{55} -4.07076e7 q^{57} +(-9.12057e7 + 1.57973e8i) q^{59} +(1.50849e8 - 8.70927e7i) q^{61} +(-2.22592e7 - 6.05633e7i) q^{63} +(5.48060e6 + 9.49268e6i) q^{65} +(-5.72063e7 - 3.30281e7i) q^{67} +7.72447e7i q^{69} +3.98123e8i q^{71} +(5.72087e7 + 3.30295e7i) q^{73} +(9.50580e7 + 1.64645e8i) q^{75} +(-2.50800e8 - 2.09106e8i) q^{77} +(-3.94490e8 + 2.27759e8i) q^{79} +(4.21601e7 - 7.30234e7i) q^{81} -2.37486e8 q^{83} +7.75049e6 q^{85} +(-2.25252e8 + 3.90148e8i) q^{87} +(-8.45694e8 + 4.88262e8i) q^{89} +(-1.64689e8 + 9.51491e8i) q^{91} +(1.90624e8 + 3.30170e8i) q^{93} +(2.60462e7 + 1.50378e7i) q^{95} -1.18899e8i q^{97} +5.22122e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57}+ \cdots - 1432348316 q^{93}+O(q^{100}) \) 24 * q + 1704 * q^5 - 80428 * q^9 - 1578672 * q^17 + 1219540 * q^21 + 8001240 * q^25 - 6709416 * q^29 - 29129772 * q^33 - 11130084 * q^37 + 57023292 * q^45 - 12671904 * q^49 - 93652164 * q^53 + 742621544 * q^57 - 593611308 * q^61 + 160281180 * q^65 - 1676922516 * q^73 + 1645751688 * q^77 - 528698056 * q^81 + 1370082456 * q^85 - 941603484 * q^89 - 1432348316 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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