LMFDB - Embedded newform 112.10.p.b.31.2

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			31.2
Character	$\chi$	$=$	112.31
Dual form			112.10.p.b.47.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-104.726 + 181.391i) q^{3} +(1126.36 - 650.304i) q^{5} +(-5633.01 - 2936.47i) q^{7} +(-12093.7 - 20946.9i) q^{9} +(-40016.3 - 23103.4i) q^{11} -78895.6i q^{13} +272416. i q^{15} +(305883. + 176602. i) q^{17} +(61180.6 + 105968. i) q^{19} +(1.12257e6 - 714253. i) q^{21} +(-1.09732e6 + 633539. i) q^{23} +(-130773. + 226505. i) q^{25} +943463. q^{27} +6.92650e6 q^{29} +(3.06287e6 - 5.30505e6i) q^{31} +(8.38153e6 - 4.83908e6i) q^{33} +(-8.25438e6 + 355647. i) q^{35} +(-555787. - 962651. i) q^{37} +(1.43110e7 + 8.26244e6i) q^{39} -9.79928e6i q^{41} -2.93595e6i q^{43} +(-2.72437e7 - 1.57292e7i) q^{45} +(2.03658e7 + 3.52745e7i) q^{47} +(2.31079e7 + 3.30823e7i) q^{49} +(-6.40680e7 + 3.69897e7i) q^{51} +(-4.25860e7 + 7.37611e7i) q^{53} -6.00970e7 q^{55} -2.56289e7 q^{57} +(-8.04025e6 + 1.39261e7i) q^{59} +(-1.68625e8 + 9.73559e7i) q^{61} +(6.61397e6 + 1.53507e8i) q^{63} +(-5.13061e7 - 8.88647e7i) q^{65} +(1.61154e8 + 9.30424e7i) q^{67} -2.65393e8i q^{69} +2.86511e8i q^{71} +(8.20997e7 + 4.74003e7i) q^{73} +(-2.73907e7 - 4.74422e7i) q^{75} +(1.57570e8 + 2.47649e8i) q^{77} +(5.77449e7 - 3.33390e7i) q^{79} +(1.39235e8 - 2.41162e8i) q^{81} -4.63680e7 q^{83} +4.59379e8 q^{85} +(-7.25387e8 + 1.25641e9i) q^{87} +(1.60330e8 - 9.25666e7i) q^{89} +(-2.31674e8 + 4.44419e8i) q^{91} +(6.41527e8 + 1.11116e9i) q^{93} +(1.37823e8 + 7.95719e7i) q^{95} -1.37549e9i q^{97} +1.11763e9i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−104.726	+	181.391i	−0.746466	+	1.29292i	0.203040	+	0.979170i	$0.434918\pi$
$3$	−104.726	+	181.391i	−0.746466	+	1.29292i	−0.949507	+	0.313747i	$0.898416\pi$
$4$		0			0
$5$	1126.36	−	650.304i	0.805957	−	0.465319i	−0.0395930	−	0.999216i	$-0.512606\pi$
$5$	1126.36	−	650.304i	0.805957	−	0.465319i	0.845550	+	0.533897i	$0.179273\pi$
$6$		0			0
$7$	−5633.01	−	2936.47i	−0.886746	−	0.462258i
$7$	−5633.01	−	2936.47i	−0.886746	−	0.462258i
$8$		0			0
$9$	−12093.7	−	20946.9i	−0.614424	−	1.06421i
$10$		0			0
$11$	−40016.3	−	23103.4i	−0.824081	−	0.475784i	0.0277404	−	0.999615i	$-0.491169\pi$
$11$	−40016.3	−	23103.4i	−0.824081	−	0.475784i	−0.851822	+	0.523831i	$0.824502\pi$
$12$		0			0
$13$		−	78895.6i		−	0.766139i	−0.923720	−	0.383069i	$-0.874867\pi$
$13$		−	78895.6i		−	0.766139i	0.923720	−	0.383069i	$-0.125133\pi$
$14$		0			0
$15$			272416.i			1.38938i
$16$		0			0
$17$	305883.	+	176602.i	0.888250	+	0.512831i	0.873370	−	0.487058i	$-0.161930\pi$
$17$	305883.	+	176602.i	0.888250	+	0.512831i	0.0148803	+	0.999889i	$0.495263\pi$
$18$		0			0
$19$	61180.6	+	105968.i	0.107702	+	0.186545i	0.914839	−	0.403819i	$-0.132318\pi$
$19$	61180.6	+	105968.i	0.107702	+	0.186545i	−0.807137	+	0.590364i	$0.798984\pi$
$20$		0			0
$21$	1.12257e6	−	714253.i	1.25959	−	0.801429i
$22$		0			0
$23$	−1.09732e6	+	633539.i	−0.817634	+	0.472061i	−0.849600	−	0.527428i	$-0.823157\pi$
$23$	−1.09732e6	+	633539.i	−0.817634	+	0.472061i	0.0319660	+	0.999489i	$0.489823\pi$
$24$		0			0
$25$	−130773.	+	226505.i	−0.0669557	+	0.115971i
$26$		0			0
$27$	943463.			0.341655
$28$		0			0
$29$	6.92650e6			1.81854			0.909270	−	0.416206i	$-0.136641\pi$
$29$	6.92650e6			1.81854			0.909270	+	0.416206i	$0.136641\pi$
$30$		0			0
$31$	3.06287e6	−	5.30505e6i	0.595664	−	1.03172i	−0.397789	−	0.917477i	$-0.630222\pi$
$31$	3.06287e6	−	5.30505e6i	0.595664	−	1.03172i	0.993453	−	0.114243i	$-0.0364442\pi$
$32$		0			0
$33$	8.38153e6	−	4.83908e6i	1.23030	−	0.710313i
$34$		0			0
$35$	−8.25438e6	+	355647.i	−0.929776	+	0.0400602i
$36$		0			0
$37$	−555787.	−	962651.i	−0.0487529	−	0.0844425i	0.840619	−	0.541627i	$-0.182191\pi$
$37$	−555787.	−	962651.i	−0.0487529	−	0.0844425i	−0.889372	+	0.457184i	$0.848858\pi$
$38$		0			0
$39$	1.43110e7	+	8.26244e6i	0.990555	+	0.571897i
$40$		0			0
$41$		−	9.79928e6i		−	0.541585i	−0.962638	−	0.270793i	$-0.912714\pi$
$41$		−	9.79928e6i		−	0.541585i	0.962638	−	0.270793i	$-0.0872858\pi$
$42$		0			0
$43$		−	2.93595e6i		−	0.130961i	−0.997854	−	0.0654803i	$-0.979142\pi$
$43$		−	2.93595e6i		−	0.130961i	0.997854	−	0.0654803i	$-0.0208580\pi$
$44$		0			0
$45$	−2.72437e7	−	1.57292e7i	−0.990399	−	0.571807i
$46$		0			0
$47$	2.03658e7	+	3.52745e7i	0.608780	+	1.05444i	0.991442	+	0.130550i	$0.0416742\pi$
$47$	2.03658e7	+	3.52745e7i	0.608780	+	1.05444i	−0.382662	+	0.923889i	$0.624992\pi$
$48$		0			0
$49$	2.31079e7	+	3.30823e7i	0.572636	+	0.819810i
$50$		0			0
$51$	−6.40680e7	+	3.69897e7i	−1.32610	+	0.765623i
$52$		0			0
$53$	−4.25860e7	+	7.37611e7i	−0.741353	+	1.28406i	0.210526	+	0.977588i	$0.432482\pi$
$53$	−4.25860e7	+	7.37611e7i	−0.741353	+	1.28406i	−0.951879	+	0.306474i	$0.900851\pi$
$54$		0			0
$55$	−6.00970e7			−0.885565
$56$		0			0
$57$	−2.56289e7			−0.321583
$58$		0			0
$59$	−8.04025e6	+	1.39261e7i	−0.0863845	+	0.149622i	−0.905980	−	0.423320i	$-0.860865\pi$
$59$	−8.04025e6	+	1.39261e7i	−0.0863845	+	0.149622i	0.819596	+	0.572942i	$0.194198\pi$
$60$		0			0
$61$	−1.68625e8	+	9.73559e7i	−1.55933	+	0.900281i	−0.562011	+	0.827130i	$0.689972\pi$
$61$	−1.68625e8	+	9.73559e7i	−1.55933	+	0.900281i	−0.997321	+	0.0731509i	$0.976695\pi$
$62$		0			0
$63$	6.61397e6	+	1.53507e8i	0.0528968	+	1.22771i
$64$		0			0
$65$	−5.13061e7	−	8.88647e7i	−0.356499	−	0.617475i
$66$		0			0
$67$	1.61154e8	+	9.30424e7i	0.977024	+	0.564085i	0.901370	−	0.433049i	$-0.142562\pi$
$67$	1.61154e8	+	9.30424e7i	0.977024	+	0.564085i	0.0756535	+	0.997134i	$0.475896\pi$
$68$		0			0
$69$		−	2.65393e8i		−	1.40951i
$70$		0			0
$71$			2.86511e8i			1.33807i	0.743231	+	0.669035i	$0.233292\pi$
$71$			2.86511e8i			1.33807i	−0.743231	+	0.669035i	$0.766708\pi$
$72$		0			0
$73$	8.20997e7	+	4.74003e7i	0.338367	+	0.195357i	0.659550	−	0.751661i	$-0.270747\pi$
$73$	8.20997e7	+	4.74003e7i	0.338367	+	0.195357i	−0.321182	+	0.947017i	$0.604080\pi$
$74$		0			0
$75$	−2.73907e7	−	4.74422e7i	−0.0999604	−	0.173137i
$76$		0			0
$77$	1.57570e8	+	2.47649e8i	0.510816	+	0.802837i
$78$		0			0
$79$	5.77449e7	−	3.33390e7i	0.166798	−	0.0963011i	−0.414277	−	0.910151i	$-0.635966\pi$
$79$	5.77449e7	−	3.33390e7i	0.166798	−	0.0963011i	0.581075	+	0.813850i	$0.302632\pi$
$80$		0			0
$81$	1.39235e8	−	2.41162e8i	0.359390	−	0.622482i
$82$		0			0
$83$	−4.63680e7			−0.107242			−0.0536212	−	0.998561i	$-0.517076\pi$
$83$	−4.63680e7			−0.107242			−0.0536212	+	0.998561i	$0.517076\pi$
$84$		0			0
$85$	4.59379e8			0.954522
$86$		0			0
$87$	−7.25387e8	+	1.25641e9i	−1.35748	+	2.35122i
$88$		0			0
$89$	1.60330e8	−	9.25666e7i	0.270869	−	0.156387i	−0.358413	−	0.933563i	$-0.616682\pi$
$89$	1.60330e8	−	9.25666e7i	0.270869	−	0.156387i	0.629283	+	0.777177i	$0.283349\pi$
$90$		0			0
$91$	−2.31674e8	+	4.44419e8i	−0.354154	+	0.679370i
$92$		0			0
$93$	6.41527e8	+	1.11116e9i	0.889286	+	1.54029i
$94$		0			0
$95$	1.37823e8	+	7.95719e7i	0.173606	+	0.100231i
$96$		0			0
$97$		−	1.37549e9i		−	1.57755i	−0.614679	−	0.788777i	$-0.710715\pi$
$97$		−	1.37549e9i		−	1.57755i	0.614679	−	0.788777i	$-0.289285\pi$
$98$		0			0
$99$			1.11763e9i			1.16933i
$100$		0			0
$101$	6.33073e7	+	3.65505e7i	0.0605352	+	0.0349500i	0.529962	−	0.848021i	$-0.322206\pi$
$101$	6.33073e7	+	3.65505e7i	0.0605352	+	0.0349500i	−0.469427	+	0.882971i	$0.655539\pi$
$102$		0			0
$103$	8.38437e8	+	1.45222e9i	0.734012	+	1.27135i	0.955156	+	0.296105i	$0.0956877\pi$
$103$	8.38437e8	+	1.45222e9i	0.734012	+	1.27135i	−0.221144	+	0.975241i	$0.570979\pi$
$104$		0			0
$105$	7.99940e8	−	1.53452e9i	0.642252	−	1.23203i
$106$		0			0
$107$	−8.34324e8	+	4.81697e8i	−0.615330	+	0.355261i	−0.775048	−	0.631902i	$-0.782275\pi$
$107$	−8.34324e8	+	4.81697e8i	−0.615330	+	0.355261i	0.159719	+	0.987163i	$0.448941\pi$
$108$		0			0
$109$	−7.03053e8	+	1.21772e9i	−0.477055	+	0.826284i	−0.999654	−	0.0262947i	$-0.991629\pi$
$109$	−7.03053e8	+	1.21772e9i	−0.477055	+	0.826284i	0.522599	+	0.852579i	$0.324962\pi$
$110$		0			0
$111$	2.32822e8			0.145570
$112$		0			0
$113$	−3.09479e9			−1.78557			−0.892787	−	0.450480i	$-0.851253\pi$
$113$	−3.09479e9			−1.78557			−0.892787	+	0.450480i	$0.851253\pi$
$114$		0			0
$115$	−8.23986e8	+	1.42719e9i	−0.439318	+	0.760922i
$116$		0			0
$117$	−1.65262e9	+	9.54140e8i	−0.815336	+	0.470734i
$118$		0			0
$119$	−1.20446e9	−	1.89301e9i	−0.550592	−	0.865351i
$120$		0			0
$121$	−1.11436e8	−	1.93013e8i	−0.0472598	−	0.0818564i
$122$		0			0
$123$	1.77751e9	+	1.02624e9i	0.700225	+	0.404275i
$124$		0			0
$125$			2.88042e9i			1.05526i
$126$		0			0
$127$		−	4.23682e9i		−	1.44518i	−0.691275	−	0.722592i	$-0.742951\pi$
$127$		−	4.23682e9i		−	1.44518i	0.691275	−	0.722592i	$-0.257049\pi$
$128$		0			0
$129$	5.32556e8	+	3.07472e8i	0.169321	+	0.0977578i
$130$		0			0
$131$	3.61344e7	+	6.25866e7i	0.0107201	+	0.0185678i	0.871336	−	0.490687i	$-0.163254\pi$
$131$	3.61344e7	+	6.25866e7i	0.0107201	+	0.0185678i	−0.860616	+	0.509255i	$0.829921\pi$
$132$		0			0
$133$	−3.34593e7	−	7.76573e8i	−0.00927223	−	0.215204i
$134$		0			0
$135$	1.06268e9	−	6.13537e8i	0.275359	−	0.158979i
$136$		0			0
$137$	−1.10483e9	+	1.91362e9i	−0.267949	+	0.464101i	−0.968332	−	0.249666i	$-0.919679\pi$
$137$	−1.10483e9	+	1.91362e9i	−0.267949	+	0.464101i	0.700383	+	0.713767i	$0.253013\pi$
$138$		0			0
$139$	5.11827e9			1.16294			0.581469	−	0.813569i	$-0.302478\pi$
$139$	5.11827e9			1.16294			0.581469	+	0.813569i	$0.302478\pi$
$140$		0			0
$141$	−8.53133e9			−1.81774
$142$		0			0
$143$	−1.82276e9	+	3.15711e9i	−0.364516	+	0.631361i
$144$		0			0
$145$	7.80173e9	−	4.50433e9i	1.46567	−	0.846202i
$146$		0			0
$147$	−8.42085e9	+	7.26988e8i	−1.48740	+	0.128410i
$148$		0			0
$149$	2.31338e9	+	4.00689e9i	0.384511	+	0.665992i	0.991701	−	0.128564i	$-0.0410369\pi$
$149$	2.31338e9	+	4.00689e9i	0.384511	+	0.665992i	−0.607191	+	0.794556i	$0.707704\pi$
$150$		0			0
$151$	6.07438e9	+	3.50705e9i	0.950837	+	0.548966i	0.893341	−	0.449380i	$-0.148355\pi$
$151$	6.07438e9	+	3.50705e9i	0.950837	+	0.548966i	0.0574960	+	0.998346i	$0.481688\pi$
$152$		0			0
$153$		−	8.54308e9i		−	1.26038i
$154$		0			0
$155$		−	7.96718e9i		−	1.10870i
$156$		0			0
$157$	6.68521e9	+	3.85971e9i	0.878146	+	0.506998i	0.870047	−	0.492970i	$-0.164089\pi$
$157$	6.68521e9	+	3.85971e9i	0.878146	+	0.506998i	0.00809907	+	0.999967i	$0.497422\pi$
$158$		0			0
$159$	−8.91975e9	−	1.54495e10i	−1.10679	−	1.91702i
$160$		0			0
$161$	8.04159e9	−	3.46478e8i	0.943247	−	0.0406406i
$162$		0			0
$163$	1.31778e10	−	7.60819e9i	1.46217	−	0.844184i	0.463057	−	0.886328i	$-0.346752\pi$
$163$	1.31778e10	−	7.60819e9i	1.46217	−	0.844184i	0.999112	+	0.0421446i	$0.0134190\pi$
$164$		0			0
$165$	6.29374e9	−	1.09011e10i	0.661045	−	1.14496i
$166$		0			0
$167$	−1.18996e10			−1.18388			−0.591942	−	0.805981i	$-0.701638\pi$
$167$	−1.18996e10			−1.18388			−0.591942	+	0.805981i	$0.701638\pi$
$168$		0			0
$169$	4.37999e9			0.413031
$170$		0			0
$171$	1.47980e9	−	2.56309e9i	0.132349	−	0.229235i
$172$		0			0
$173$	−4.83114e9	+	2.78926e9i	−0.410055	+	0.236746i	−0.690814	−	0.723033i	$-0.742747\pi$
$173$	−4.83114e9	+	2.78926e9i	−0.410055	+	0.236746i	0.280758	+	0.959779i	$0.409414\pi$
$174$		0			0
$175$	1.40177e9	−	8.91896e8i	0.112981	−	0.0718857i
$176$		0			0
$177$	−1.68405e9	−	2.91686e9i	−0.128966	−	0.223376i
$178$		0			0
$179$	1.71294e10	+	9.88964e9i	1.24710	+	0.720015i	0.970531	−	0.240978i	$-0.0774681\pi$
$179$	1.71294e10	+	9.88964e9i	1.24710	+	0.720015i	0.276572	+	0.960993i	$0.410801\pi$
$180$		0			0
$181$			5.32364e9i			0.368685i	0.982862	+	0.184342i	$0.0590155\pi$
$181$			5.32364e9i			0.368685i	−0.982862	+	0.184342i	$0.940984\pi$
$182$		0			0
$183$		−	4.07829e10i		−	2.68812i
$184$		0			0
$185$	−1.25203e9	−	7.22860e8i	−0.0785855	−	0.0453713i
$186$		0			0
$187$	−8.16021e9	−	1.41339e10i	−0.487994	−	0.845230i
$188$		0			0
$189$	−5.31453e9	−	2.77045e9i	−0.302961	−	0.157933i
$190$		0			0
$191$	2.29678e10	−	1.32605e10i	1.24873	−	0.720956i	0.277876	−	0.960617i	$-0.410369\pi$
$191$	2.29678e10	−	1.32605e10i	1.24873	−	0.720956i	0.970857	+	0.239660i	$0.0770361\pi$
$192$		0			0
$193$	−1.55255e10	+	2.68910e10i	−0.805449	+	1.39508i	0.110539	+	0.993872i	$0.464742\pi$
$193$	−1.55255e10	+	2.68910e10i	−0.805449	+	1.39508i	−0.915988	+	0.401206i	$0.868591\pi$
$194$		0			0
$195$	2.14924e10			1.06446
$196$		0			0
$197$	3.13231e10			1.48172			0.740861	−	0.671658i	$-0.234418\pi$
$197$	3.13231e10			1.48172			0.740861	+	0.671658i	$0.234418\pi$
$198$		0			0
$199$	−1.95419e10	+	3.38476e10i	−0.883340	+	1.52999i	−0.0357352	+	0.999361i	$0.511377\pi$
$199$	−1.95419e10	+	3.38476e10i	−0.883340	+	1.52999i	−0.847605	+	0.530628i	$0.821956\pi$
$200$		0			0
$201$	−3.37542e10	+	1.94880e10i	−1.45863	+	0.842141i
$202$		0			0
$203$	−3.90170e10	−	2.03395e10i	−1.61258	−	0.840634i
$204$		0			0
$205$	−6.37251e9	−	1.10375e10i	−0.252010	−	0.436495i
$206$		0			0
$207$	2.65414e10	+	1.53237e10i	1.00475	+	0.580092i
$208$		0			0
$209$		−	5.65393e9i		−	0.204971i
$210$		0			0
$211$		−	2.15299e10i		−	0.747776i	−0.927474	−	0.373888i	$-0.878024\pi$
$211$		−	2.15299e10i		−	0.747776i	0.927474	−	0.373888i	$-0.121976\pi$
$212$		0			0
$213$	−5.19706e10	−	3.00052e10i	−1.73001	−	0.998824i
$214$		0			0
$215$	−1.90926e9	−	3.30694e9i	−0.0609385	−	0.105549i
$216$		0			0
$217$	−3.28313e10	+	2.08893e10i	−1.00512	+	0.639523i
$218$		0			0
$219$	−1.71960e10	+	9.92811e9i	−0.505160	+	0.291654i
$220$		0			0
$221$	1.39331e10	−	2.41328e10i	0.392900	−	0.680523i
$222$		0			0
$223$	6.78259e9			0.183664			0.0918319	−	0.995775i	$-0.470728\pi$
$223$	6.78259e9			0.183664			0.0918319	+	0.995775i	$0.470728\pi$
$224$		0			0
$225$	6.32612e9			0.164557
$226$		0			0
$227$	−2.87680e10	+	4.98277e10i	−0.719108	+	1.24553i	0.242246	+	0.970215i	$0.422116\pi$
$227$	−2.87680e10	+	4.98277e10i	−0.719108	+	1.24553i	−0.961354	+	0.275316i	$0.911217\pi$
$228$		0			0
$229$	1.01654e10	−	5.86901e9i	0.244267	−	0.141028i	−0.372869	−	0.927884i	$-0.621626\pi$
$229$	1.01654e10	−	5.86901e9i	0.244267	−	0.141028i	0.617137	+	0.786856i	$0.288293\pi$
$230$		0			0
$231$	−6.14230e10	+	2.64646e9i	−1.41931	+	0.0611521i
$232$		0			0
$233$	4.22178e9	+	7.31234e9i	0.0938414	+	0.162538i	0.909124	−	0.416525i	$-0.136752\pi$
$233$	4.22178e9	+	7.31234e9i	0.0938414	+	0.162538i	−0.815283	+	0.579063i	$0.803419\pi$
$234$		0			0
$235$	4.58783e10	+	2.64879e10i	0.981301	+	0.566554i
$236$		0			0
$237$			1.39659e10i			0.287542i
$238$		0			0
$239$		−	4.69299e9i		−	0.0930377i	−0.998917	−	0.0465188i	$-0.985187\pi$
$239$		−	4.69299e9i		−	0.0930377i	0.998917	−	0.0465188i	$-0.0148127\pi$
$240$		0			0
$241$	7.58488e10	+	4.37913e10i	1.44834	+	0.836202i	0.998383	−	0.0568454i	$-0.0181042\pi$
$241$	7.58488e10	+	4.37913e10i	1.44834	+	0.836202i	0.449962	+	0.893048i	$0.351438\pi$
$242$		0			0
$243$	3.84483e10	+	6.65943e10i	0.707373	+	1.22521i
$244$		0			0
$245$	4.75413e10	+	2.22354e10i	0.842993	+	0.394273i
$246$		0			0
$247$	8.36040e9	−	4.82688e9i	0.142919	−	0.0825145i
$248$		0			0
$249$	4.85595e9	−	8.41074e9i	0.0800528	−	0.138656i
$250$		0			0
$251$	8.04549e10			1.27944			0.639721	−	0.768607i	$-0.279050\pi$
$251$	8.04549e10			1.27944			0.639721	+	0.768607i	$0.279050\pi$
$252$		0			0
$253$	5.85477e10			0.898396
$254$		0			0
$255$	−4.81091e10	+	8.33273e10i	−0.712518	+	1.23412i
$256$		0			0
$257$	6.48181e10	−	3.74228e10i	0.926825	−	0.535102i	0.0410187	−	0.999158i	$-0.486940\pi$
$257$	6.48181e10	−	3.74228e10i	0.926825	−	0.535102i	0.885806	+	0.464056i	$0.153606\pi$
$258$		0			0
$259$	3.03956e8	+	7.05467e9i	0.00419722	+	0.0974154i
$260$		0			0
$261$	−8.37671e10	−	1.45089e11i	−1.11736	−	1.93532i
$262$		0			0
$263$	−1.22792e10	−	7.08940e9i	−0.158259	−	0.0913711i	0.418779	−	0.908088i	$-0.362458\pi$
$263$	−1.22792e10	−	7.08940e9i	−0.158259	−	0.0913711i	−0.577038	+	0.816717i	$0.695792\pi$
$264$		0			0
$265$			1.10775e11i			1.37986i
$266$		0			0
$267$			3.87766e10i			0.466949i
$268$		0			0
$269$	2.56061e10	+	1.47837e10i	0.298166	+	0.172146i	0.641619	−	0.767024i	$-0.278263\pi$
$269$	2.56061e10	+	1.47837e10i	0.298166	+	0.172146i	−0.343453	+	0.939170i	$0.611596\pi$
$270$		0			0
$271$	−3.77000e10	−	6.52983e10i	−0.424600	−	0.735428i	0.571783	−	0.820405i	$-0.306252\pi$
$271$	−3.77000e10	−	6.52983e10i	−0.424600	−	0.735428i	−0.996383	+	0.0849765i	$0.972918\pi$
$272$		0			0
$273$	−5.63514e10	−	8.85661e10i	−0.614006	−	0.965019i
$274$		0			0
$275$	1.04661e10	−	6.04261e9i	0.110354	−	0.0637129i
$276$		0			0
$277$	5.67411e10	−	9.82785e10i	0.579080	−	1.00300i	−0.416505	−	0.909133i	$-0.636745\pi$
$277$	5.67411e10	−	9.82785e10i	0.579080	−	1.00300i	0.995585	−	0.0938629i	$-0.0299215\pi$
$278$		0			0
$279$	−1.48166e11			−1.46396
$280$		0			0
$281$	3.48271e9			0.0333226			0.0166613	−	0.999861i	$-0.494696\pi$
$281$	3.48271e9			0.0333226			0.0166613	+	0.999861i	$0.494696\pi$
$282$		0			0
$283$	−1.01330e11	+	1.75509e11i	−0.939071	+	1.62652i	−0.171863	+	0.985121i	$0.554979\pi$
$283$	−1.01330e11	+	1.75509e11i	−0.939071	+	1.62652i	−0.767208	+	0.641398i	$0.778355\pi$
$284$		0			0
$285$	−2.88673e10	+	1.66666e10i	−0.259182	+	0.149639i
$286$		0			0
$287$	−2.87753e10	+	5.51994e10i	−0.250352	+	0.480249i
$288$		0			0
$289$	3.08235e9	+	5.33878e9i	0.0259921	+	0.0450196i
$290$		0			0
$291$	2.49502e11	+	1.44050e11i	2.03965	+	1.17759i
$292$		0			0
$293$			1.48160e11i			1.17443i	0.809431	+	0.587216i	$0.199776\pi$
$293$			1.48160e11i			1.17443i	−0.809431	+	0.587216i	$0.800224\pi$
$294$		0			0
$295$			2.09144e10i			0.160785i
$296$		0			0
$297$	−3.77539e10	−	2.17972e10i	−0.281552	−	0.162554i
$298$		0			0
$299$	4.99834e10	+	8.65739e10i	0.361664	+	0.626421i
$300$		0			0
$301$	−8.62133e9	+	1.65382e10i	−0.0605376	+	0.116129i
$302$		0			0
$303$	−1.32599e10	+	7.65560e9i	−0.0903750	+	0.0521780i
$304$		0			0
$305$	−1.26622e11	+	2.19315e11i	−0.837836	+	1.45117i
$306$		0			0
$307$	−1.00403e11			−0.645095			−0.322547	−	0.946553i	$-0.604539\pi$
$307$	−1.00403e11			−0.645095			−0.322547	+	0.946553i	$0.604539\pi$
$308$		0			0
$309$	−3.51226e11			−2.19166
$310$		0			0
$311$	7.70723e10	−	1.33493e11i	0.467172	−	0.809165i	−0.532125	−	0.846666i	$-0.678606\pi$
$311$	7.70723e10	−	1.33493e11i	0.467172	−	0.809165i	0.999297	+	0.0375006i	$0.0119396\pi$
$312$		0			0
$313$	−6.29271e10	+	3.63310e10i	−0.370585	+	0.213957i	−0.673714	−	0.738992i	$-0.735302\pi$
$313$	−6.29271e10	+	3.63310e10i	−0.370585	+	0.213957i	0.303129	+	0.952950i	$0.401969\pi$
$314$		0			0
$315$	1.07276e11	+	1.68603e11i	0.613909	+	0.964867i
$316$		0			0
$317$	1.26222e11	+	2.18623e11i	0.702050	+	1.21599i	0.967746	+	0.251929i	$0.0810650\pi$
$317$	1.26222e11	+	2.18623e11i	0.702050	+	1.21599i	−0.265696	+	0.964057i	$0.585602\pi$
$318$		0			0
$319$	−2.77173e11	−	1.60026e11i	−1.49863	−	0.865232i
$320$		0			0
$321$		−	2.01786e11i		−	1.06076i
$322$		0			0
$323$			4.32184e10i			0.220931i
$324$		0			0
$325$	1.78703e10	+	1.03174e10i	0.0888497	+	0.0512974i
$326$		0			0
$327$	−1.47256e11	−	2.55055e11i	−0.712211	−	1.23359i
$328$		0			0
$329$	−1.11379e10	−	2.58505e11i	−0.0524109	−	1.21643i
$330$		0			0
$331$	−3.41634e10	+	1.97243e10i	−0.156436	+	0.0903181i	−0.576174	−	0.817327i	$-0.695455\pi$
$331$	−3.41634e10	+	1.97243e10i	−0.156436	+	0.0903181i	0.419739	+	0.907645i	$0.362122\pi$
$332$		0			0
$333$	−1.34430e10	+	2.32840e10i	−0.0599099	+	0.103767i
$334$		0			0
$335$	2.42023e11			1.04992
$336$		0			0
$337$	3.81413e11			1.61087			0.805437	−	0.592682i	$-0.201931\pi$
$337$	3.81413e11			1.61087			0.805437	+	0.592682i	$0.201931\pi$
$338$		0			0
$339$	3.24106e11	−	5.61367e11i	1.33287	−	2.30860i
$340$		0			0
$341$	−2.45130e11	+	1.41526e11i	−0.981751	+	0.566814i
$342$		0			0
$343$	−3.30219e10	−	2.54208e11i	−0.128819	−	0.991668i
$344$		0			0
$345$	−1.72586e11	−	2.98928e11i	−0.655873	−	1.13601i
$346$		0			0
$347$	−1.31049e11	−	7.56611e10i	−0.485233	−	0.280149i	0.237362	−	0.971421i	$-0.423717\pi$
$347$	−1.31049e11	−	7.56611e10i	−0.485233	−	0.280149i	−0.722595	+	0.691272i	$0.757051\pi$
$348$		0			0
$349$			3.88593e10i			0.140211i	0.997540	+	0.0701053i	$0.0223335\pi$
$349$			3.88593e10i			0.140211i	−0.997540	+	0.0701053i	$0.977666\pi$
$350$		0			0
$351$		−	7.44351e10i		−	0.261755i
$352$		0			0
$353$	1.59811e11	+	9.22669e10i	0.547798	+	0.316271i	0.748233	−	0.663436i	$-0.230902\pi$
$353$	1.59811e11	+	9.22669e10i	0.547798	+	0.316271i	−0.200436	+	0.979707i	$0.564236\pi$
$354$		0			0
$355$	1.86319e11	+	3.22714e11i	0.622630	+	1.07843i
$356$		0			0
$357$	4.69515e11	−	2.02294e10i	1.52983	−	0.0659138i
$358$		0			0
$359$	−7.43325e10	+	4.29159e10i	−0.236186	+	0.136362i	−0.613422	−	0.789755i	$-0.710208\pi$
$359$	−7.43325e10	+	4.29159e10i	−0.236186	+	0.136362i	0.377237	+	0.926117i	$0.376874\pi$
$360$		0			0
$361$	1.53858e11	−	2.66489e11i	0.476801	−	0.825843i
$362$		0			0
$363$	4.66812e10			0.141111
$364$		0			0
$365$	1.23298e11			0.363613
$366$		0			0
$367$	2.10555e11	−	3.64693e11i	0.605855	−	1.04937i	−0.386060	−	0.922474i	$-0.626164\pi$
$367$	2.10555e11	−	3.64693e11i	0.605855	−	1.04937i	0.991916	−	0.126899i	$-0.0405024\pi$
$368$		0			0
$369$	−2.05265e11	+	1.18510e11i	−0.576363	+	0.332763i
$370$		0			0
$371$	4.56484e11	−	2.90444e11i	1.25096	−	0.795940i
$372$		0			0
$373$	−3.57354e10	−	6.18955e10i	−0.0955892	−	0.165565i	0.814265	−	0.580493i	$-0.197140\pi$
$373$	−3.57354e10	−	6.18955e10i	−0.0955892	−	0.165565i	−0.909854	+	0.414928i	$0.863807\pi$
$374$		0			0
$375$	−5.22483e11	−	3.01656e11i	−1.36437	−	0.787718i
$376$		0			0
$377$		−	5.46470e11i		−	1.39325i
$378$		0			0
$379$			1.77866e10i			0.0442809i	0.999755	+	0.0221404i	$0.00704810\pi$
$379$			1.77866e10i			0.0442809i	−0.999755	+	0.0221404i	$0.992952\pi$
$380$		0			0
$381$	7.68522e11	+	4.43706e11i	1.86850	+	1.07878i
$382$		0			0
$383$	6.17954e9	+	1.07033e10i	0.0146744	+	0.0254169i	0.873269	−	0.487238i	$-0.161995\pi$
$383$	6.17954e9	+	1.07033e10i	0.0146744	+	0.0254169i	−0.858595	+	0.512655i	$0.828662\pi$
$384$		0			0
$385$	3.38527e11	+	1.76473e11i	0.785271	+	0.409359i
$386$		0			0
$387$	−6.14992e10	+	3.55066e10i	−0.139370	+	0.0804654i
$388$		0			0
$389$	−1.18565e11	+	2.05361e11i	−0.262533	+	0.454721i	−0.966914	−	0.255101i	$-0.917891\pi$
$389$	−1.18565e11	+	2.05361e11i	−0.262533	+	0.454721i	0.704381	+	0.709822i	$0.251225\pi$
$390$		0			0
$391$	−4.47536e11			−0.968351
$392$		0			0
$393$	−1.51369e10			−0.0320088
$394$		0			0
$395$	4.33610e10	−	7.51034e10i	0.0896215	−	0.155229i
$396$		0			0
$397$	−4.05153e11	+	2.33915e11i	−0.818581	+	0.472608i	−0.849927	−	0.526901i	$-0.823354\pi$
$397$	−4.05153e11	+	2.33915e11i	−0.818581	+	0.472608i	0.0313461	+	0.999509i	$0.490021\pi$
$398$		0			0
$399$	1.44368e11	+	7.52584e10i	0.285162	+	0.148654i
$400$		0			0
$401$	−7.28646e10	−	1.26205e11i	−0.140724	−	0.243740i	0.787046	−	0.616895i	$-0.211610\pi$
$401$	−7.28646e10	−	1.26205e11i	−0.140724	−	0.243740i	−0.927769	+	0.373154i	$0.878276\pi$
$402$		0			0
$403$	−4.18545e11	−	2.41647e11i	−0.790441	−	0.456361i
$404$		0			0
$405$		−	3.62180e11i		−	0.668925i
$406$		0			0
$407$			5.13623e10i			0.0927833i
$408$		0			0
$409$	−5.26483e11	−	3.03965e11i	−0.930313	−	0.537117i	−0.0434027	−	0.999058i	$-0.513820\pi$
$409$	−5.26483e11	−	3.03965e11i	−0.930313	−	0.537117i	−0.886911	+	0.461941i	$0.847153\pi$
$410$		0			0
$411$	−2.31409e11	−	4.00813e11i	−0.400030	−	0.692872i
$412$		0			0
$413$	8.61844e10	−	5.48360e10i	0.145765	−	0.0927450i
$414$		0			0
$415$	−5.22270e10	+	3.01532e10i	−0.0864327	+	0.0499020i
$416$		0			0
$417$	−5.36017e11	+	9.28410e11i	−0.868094	+	1.50358i
$418$		0			0
$419$	6.24074e11			0.989175			0.494588	−	0.869128i	$-0.335319\pi$
$419$	6.24074e11			0.989175			0.494588	+	0.869128i	$0.335319\pi$
$420$		0			0
$421$	−7.61834e11			−1.18193			−0.590964	−	0.806698i	$-0.701252\pi$
$421$	−7.61834e11			−1.18193			−0.590964	+	0.806698i	$0.701252\pi$
$422$		0			0
$423$	4.92595e11	−	8.53200e11i	0.748098	−	1.29574i
$424$		0			0
$425$	−8.00024e10	+	4.61894e10i	−0.118947	+	0.0686740i
$426$		0			0
$427$	1.23575e12	−	5.32433e10i	1.79889	−	0.0775067i
$428$		0			0
$429$	−3.81782e11	−	6.61265e11i	−0.544199	−	0.942579i
$430$		0			0
$431$	−1.10975e12	−	6.40714e11i	−1.54909	−	0.894368i	−0.998211	−	0.0597835i	$-0.980959\pi$
$431$	−1.10975e12	−	6.40714e11i	−1.54909	−	0.894368i	−0.550880	−	0.834585i	$-0.685708\pi$
$432$		0			0
$433$			1.20788e12i			1.65131i	0.564176	+	0.825655i	$0.309194\pi$
$433$			1.20788e12i			1.65131i	−0.564176	+	0.825655i	$0.690806\pi$
$434$		0			0
$435$			1.88689e12i			2.52665i
$436$		0			0
$437$	−1.34270e11	−	7.75206e10i	−0.176121	−	0.101684i
$438$		0			0
$439$	4.58240e11	+	7.93695e11i	0.588847	+	1.01991i	0.994384	+	0.105834i	$0.0337513\pi$
$439$	4.58240e11	+	7.93695e11i	0.588847	+	1.01991i	−0.405537	+	0.914079i	$0.632915\pi$
$440$		0			0
$441$	4.13512e11	−	8.84127e11i	0.520612	−	1.11312i
$442$		0			0
$443$	5.81248e11	−	3.35584e11i	0.717042	−	0.413985i	−0.0966208	−	0.995321i	$-0.530803\pi$
$443$	5.81248e11	−	3.35584e11i	0.717042	−	0.413985i	0.813663	+	0.581337i	$0.197470\pi$
$444$		0			0
$445$	1.20393e11	−	2.08526e11i	0.145539	−	0.252082i
$446$		0			0
$447$	−9.69086e11			−1.14810
$448$		0			0
$449$	2.98106e11			0.346149			0.173074	−	0.984909i	$-0.444630\pi$
$449$	2.98106e11			0.346149			0.173074	+	0.984909i	$0.444630\pi$
$450$		0			0
$451$	−2.26397e11	+	3.92131e11i	−0.257678	+	0.446311i
$452$		0			0
$453$	−1.27230e12	+	7.34560e11i	−1.41954	+	0.819569i
$454$		0			0
$455$	2.80590e10	+	6.51234e11i	0.0306916	+	0.712338i
$456$		0			0
$457$	−4.11645e11	−	7.12990e11i	−0.441469	−	0.764647i	0.556330	−	0.830962i	$-0.312209\pi$
$457$	−4.11645e11	−	7.12990e11i	−0.441469	−	0.764647i	−0.997799	+	0.0663150i	$0.978876\pi$
$458$		0			0
$459$	2.88589e11	+	1.66617e11i	0.303475	+	0.175211i
$460$		0			0
$461$			5.38478e11i			0.555282i	0.960685	+	0.277641i	$0.0895525\pi$
$461$			5.38478e11i			0.555282i	−0.960685	+	0.277641i	$0.910447\pi$
$462$		0			0
$463$		−	1.65299e11i		−	0.167169i	−0.996501	−	0.0835844i	$-0.973363\pi$
$463$		−	1.65299e11i		−	0.167169i	0.996501	−	0.0835844i	$-0.0266368\pi$
$464$		0			0
$465$	1.44518e12	+	8.34374e11i	1.43345	+	0.827604i
$466$		0			0
$467$	−2.41102e11	−	4.17601e11i	−0.234571	−	0.406289i	0.724577	−	0.689194i	$-0.242035\pi$
$467$	−2.41102e11	−	4.17601e11i	−0.234571	−	0.406289i	−0.959148	+	0.282905i	$0.908702\pi$
$468$		0			0
$469$	−6.34567e11	−	9.97333e11i	−0.605619	−	0.951837i
$470$		0			0
$471$	−1.40023e12	+	8.08426e11i	−1.31101	+	0.756913i
$472$		0			0
$473$	−6.78306e10	+	1.17486e11i	−0.0623090	+	0.107922i
$474$		0			0
$475$	−3.20031e10			−0.0288450
$476$		0			0
$477$	2.06009e12			1.82202
$478$		0			0
$479$	2.00198e11	−	3.46752e11i	0.173760	−	0.300961i	−0.765972	−	0.642874i	$-0.777742\pi$
$479$	2.00198e11	−	3.46752e11i	0.173760	−	0.300961i	0.939731	+	0.341914i	$0.111075\pi$
$480$		0			0
$481$	−7.59489e10	+	4.38491e10i	−0.0646947	+	0.0373515i
$482$		0			0
$483$	−7.79318e11	+	1.49496e12i	−0.651557	+	1.24988i
$484$		0			0
$485$	−8.94485e11	−	1.54929e12i	−0.734066	−	1.27144i
$486$		0			0
$487$	8.99483e11	+	5.19317e11i	0.724624	+	0.418362i	0.816452	−	0.577413i	$-0.195938\pi$
$487$	8.99483e11	+	5.19317e11i	0.724624	+	0.418362i	−0.0918281	+	0.995775i	$0.529271\pi$
$488$		0			0
$489$			3.18711e12i			2.52062i
$490$		0			0
$491$			1.55476e12i			1.20725i	0.797268	+	0.603625i	$0.206278\pi$
$491$			1.55476e12i			1.20725i	−0.797268	+	0.603625i	$0.793722\pi$
$492$		0			0
$493$	2.11870e12	+	1.22323e12i	1.61532	+	0.932605i
$494$		0			0
$495$	7.26796e11	+	1.25885e12i	0.544113	+	0.942431i
$496$		0			0
$497$	8.41331e11	−	1.61392e12i	0.618533	−	1.18653i
$498$		0			0
$499$	−2.37534e12	+	1.37140e12i	−1.71504	+	0.990176i	−0.787605	+	0.616180i	$0.788679\pi$
$499$	−2.37534e12	+	1.37140e12i	−1.71504	+	0.990176i	−0.927430	+	0.373996i	$0.877987\pi$
$500$		0			0
$501$	1.24620e12	−	2.15849e12i	0.883729	−	1.53066i
$502$		0			0
$503$	−3.77979e11			−0.263276			−0.131638	−	0.991298i	$-0.542024\pi$
$503$	−3.77979e11			−0.263276			−0.131638	+	0.991298i	$0.542024\pi$
$504$		0			0
$505$	9.50757e10			0.0650517
$506$		0			0
$507$	−4.58700e11	+	7.94492e11i	−0.308314	+	0.534015i
$508$		0			0
$509$	1.04354e12	−	6.02486e11i	0.689092	−	0.397848i	−0.114180	−	0.993460i	$-0.536424\pi$
$509$	1.04354e12	−	6.02486e11i	0.689092	−	0.397848i	0.803272	+	0.595612i	$0.203091\pi$
$510$		0			0
$511$	−3.23279e11	−	5.08089e11i	−0.209741	−	0.329645i
$512$		0			0
$513$	5.77216e10	+	9.99768e10i	0.0367968	+	0.0637340i
$514$		0			0
$515$	1.88876e12	+	1.09048e12i	1.18316	+	0.683100i
$516$		0			0
$517$		−	1.88208e12i		−	1.15859i
$518$		0			0
$519$		−	1.16844e12i		−	0.706891i
$520$		0			0
$521$	−1.50028e12	−	8.66186e11i	−0.892077	−	0.515041i	−0.0174556	−	0.999848i	$-0.505557\pi$
$521$	−1.50028e12	−	8.66186e11i	−0.892077	−	0.515041i	−0.874621	+	0.484807i	$0.838890\pi$
$522$		0			0
$523$	5.85295e11	+	1.01376e12i	0.342071	+	0.592485i	0.984817	−	0.173594i	$-0.0555382\pi$
$523$	5.85295e11	+	1.01376e12i	0.342071	+	0.592485i	−0.642746	+	0.766080i	$0.722205\pi$
$524$		0			0
$525$	1.49798e10	+	3.47674e11i	0.00860577	+	0.199736i
$526$		0			0
$527$	1.87376e12	−	1.08182e12i	1.05820	−	0.610950i
$528$		0			0
$529$	−9.78324e10	+	1.69451e11i	−0.0543166	+	0.0940790i
$530$		0			0
$531$	3.88946e11			0.212307
$532$		0			0
$533$	−7.73120e11			−0.414930
$534$		0			0
$535$	−6.26499e11	+	1.08513e12i	−0.330619	+	0.572650i
$536$		0			0
$537$	−3.58779e12	+	2.07141e12i	−1.86184	+	1.07493i
$538$		0			0
$539$	−1.60379e11	−	1.85770e12i	−0.0818462	−	0.948041i
$540$		0			0
$541$	−1.73390e12	−	3.00320e12i	−0.870233	−	1.50729i	−0.861755	−	0.507324i	$-0.830635\pi$
$541$	−1.73390e12	−	3.00320e12i	−0.870233	−	1.50729i	−0.00847833	−	0.999964i	$-0.502699\pi$
$542$		0			0
$543$	−9.65663e11	−	5.57526e11i	−0.476679	−	0.275211i
$544$		0			0
$545$			1.82879e12i			0.887932i
$546$		0			0
$547$		−	1.60618e12i		−	0.767096i	−0.923521	−	0.383548i	$-0.874702\pi$
$547$		−	1.60618e12i		−	0.767096i	0.923521	−	0.383548i	$-0.125298\pi$
$548$		0			0
$549$	4.07861e12	+	2.35479e12i	1.91618	+	1.10631i
$550$		0			0
$551$	4.23768e11	+	7.33987e11i	0.195860	+	0.339239i
$552$		0			0
$553$	−4.23176e11	+	1.82329e10i	−0.192424	+	0.00829073i
$554$		0			0
$555$	2.62241e11	−	1.51405e11i	0.117323	−	0.0677364i
$556$		0			0
$557$	−9.79821e11	+	1.69710e12i	−0.431319	+	0.747066i	−0.996987	−	0.0775667i	$-0.975285\pi$
$557$	−9.79821e11	+	1.69710e12i	−0.431319	+	0.747066i	0.565668	+	0.824633i	$0.308618\pi$
$558$		0			0
$559$	−2.31634e11			−0.100334
$560$		0			0
$561$	3.41836e12			1.45708
$562$		0			0
$563$	5.33850e11	−	9.24656e11i	0.223940	−	0.387876i	−0.732061	−	0.681239i	$-0.761441\pi$
$563$	5.33850e11	−	9.24656e11i	0.223940	−	0.387876i	0.956001	+	0.293364i	$0.0947747\pi$
$564$		0			0
$565$	−3.48584e12	+	2.01255e12i	−1.43910	+	0.830862i
$566$		0			0
$567$	−1.49248e12	+	9.49609e11i	−0.606435	+	0.385852i
$568$		0			0
$569$	−1.99429e12	−	3.45421e12i	−0.797595	−	1.38148i	−0.921178	−	0.389142i	$-0.872772\pi$
$569$	−1.99429e12	−	3.45421e12i	−0.797595	−	1.38148i	0.123582	−	0.992334i	$-0.460562\pi$
$570$		0			0
$571$	3.93494e10	+	2.27184e10i	0.0154909	+	0.00894366i	0.507725	−	0.861519i	$-0.330486\pi$
$571$	3.93494e10	+	2.27184e10i	0.0154909	+	0.00894366i	−0.492235	+	0.870463i	$0.663820\pi$
$572$		0			0
$573$			5.55489e12i			2.15268i
$574$		0			0
$575$		−	3.31399e11i		−	0.126429i
$576$		0			0
$577$	−1.15892e12	−	6.69103e11i	−0.435274	−	0.251305i	0.266317	−	0.963885i	$-0.414193\pi$
$577$	−1.15892e12	−	6.69103e11i	−0.435274	−	0.251305i	−0.701591	+	0.712580i	$0.747527\pi$
$578$		0			0
$579$	−3.25186e12	−	5.63238e12i	−1.20248	−	2.08276i
$580$		0			0
$581$	2.61191e11	+	1.36158e11i	0.0950967	+	0.0495736i
$582$		0			0
$583$	3.40827e12	−	1.96777e12i	1.22187	−	0.705448i
$584$		0			0
$585$	−1.24096e12	+	2.14941e12i	−0.438084	+	0.758783i
$586$		0			0
$587$	3.26586e12			1.13534			0.567670	−	0.823256i	$-0.307845\pi$
$587$	3.26586e12			1.13534			0.567670	+	0.823256i	$0.307845\pi$
$588$		0			0
$589$	7.49553e11			0.256616
$590$		0			0
$591$	−3.28035e12	+	5.68174e12i	−1.10606	+	1.91575i
$592$		0			0
$593$	−3.99754e12	+	2.30798e12i	−1.32754	+	0.766454i	−0.984918	−	0.173020i	$-0.944647\pi$
$593$	−3.99754e12	+	2.30798e12i	−1.32754	+	0.766454i	−0.342619	+	0.939474i	$0.611314\pi$
$594$		0			0
$595$	−2.58768e12	−	1.34895e12i	−0.846418	−	0.441235i
$596$		0			0
$597$	−4.09310e12	−	7.08946e12i	−1.31877	−	2.28417i
$598$		0			0
$599$	3.08168e12	+	1.77921e12i	0.978062	+	0.564684i	0.901684	−	0.432395i	$-0.142331\pi$
$599$	3.08168e12	+	1.77921e12i	0.978062	+	0.564684i	0.0763774	+	0.997079i	$0.475665\pi$
$600$		0			0
$601$		−	4.89515e12i		−	1.53049i	−0.643738	−	0.765246i	$-0.722617\pi$
$601$		−	4.89515e12i		−	1.53049i	0.643738	−	0.765246i	$-0.277383\pi$
$602$		0			0
$603$		−	4.50091e12i		−	1.38635i
$604$		0			0
$605$	−2.51034e11	−	1.44935e11i	−0.0761787	−	0.0439818i
$606$		0			0
$607$	−6.56410e11	−	1.13694e12i	−0.196258	−	0.339928i	0.751055	−	0.660240i	$-0.229546\pi$
$607$	−6.56410e11	−	1.13694e12i	−0.196258	−	0.339928i	−0.947312	+	0.320312i	$0.896212\pi$
$608$		0			0
$609$	7.77551e12	−	4.94727e12i	2.29061	−	1.45743i
$610$		0			0
$611$	2.78301e12	−	1.60677e12i	0.807846	−	0.466410i
$612$		0			0
$613$	−1.95369e12	+	3.38390e12i	−0.558836	+	0.967932i	0.438758	+	0.898605i	$0.355419\pi$
$613$	−1.95369e12	+	3.38390e12i	−0.558836	+	0.967932i	−0.997594	+	0.0693271i	$0.977915\pi$
$614$		0			0
$615$	2.66948e12			0.752469
$616$		0			0
$617$	3.61913e12			1.00536			0.502680	−	0.864473i	$-0.332347\pi$
$617$	3.61913e12			1.00536			0.502680	+	0.864473i	$0.332347\pi$
$618$		0			0
$619$	−2.21807e12	+	3.84182e12i	−0.607251	+	1.05179i	0.384440	+	0.923150i	$0.374394\pi$
$619$	−2.21807e12	+	3.84182e12i	−0.607251	+	1.05179i	−0.991691	+	0.128640i	$0.958939\pi$
$620$		0			0
$621$	−1.03528e12	+	5.97721e11i	−0.279349	+	0.161282i
$622$		0			0
$623$	−1.17496e12	+	5.06241e10i	−0.312483	+	0.0134636i
$624$		0			0
$625$	1.61773e12	+	2.80199e12i	0.424078	+	0.734525i
$626$		0			0
$627$	1.02557e12	+	5.92115e11i	0.265010	+	0.153004i
$628$		0			0
$629$		−	3.92611e11i		−	0.100008i
$630$		0			0
$631$		−	5.28844e12i		−	1.32799i	−0.747736	−	0.663997i	$-0.768859\pi$
$631$		−	5.28844e12i		−	1.32799i	0.747736	−	0.663997i	$-0.231141\pi$
$632$		0			0
$633$	3.90534e12	+	2.25475e12i	0.966813	+	0.558190i
$634$		0			0
$635$	−2.75522e12	−	4.77218e12i	−0.672472	−	1.16476i
$636$		0			0
$637$	2.61005e12	−	1.82311e12i	0.628088	−	0.438718i
$638$		0			0
$639$	6.00152e12	−	3.46498e12i	1.42399	−	0.822142i
$640$		0			0
$641$	1.72914e12	−	2.99497e12i	0.404548	−	0.700698i	−0.589721	−	0.807607i	$-0.700762\pi$
$641$	1.72914e12	−	2.99497e12i	0.404548	−	0.700698i	0.994269	+	0.106909i	$0.0340955\pi$
$642$		0			0
$643$	−7.09662e12			−1.63720			−0.818601	−	0.574362i	$-0.805250\pi$
$643$	−7.09662e12			−1.63720			−0.818601	+	0.574362i	$0.805250\pi$
$644$		0			0
$645$	7.99799e11			0.181954
$646$		0			0
$647$	2.56813e12	−	4.44814e12i	0.576167	−	0.997950i	−0.419747	−	0.907641i	$-0.637881\pi$
$647$	2.56813e12	−	4.44814e12i	0.576167	−	0.997950i	0.995914	−	0.0903091i	$-0.0287855\pi$
$648$		0			0
$649$	6.43483e11	−	3.71515e11i	0.142376	−	0.0822006i
$650$		0			0
$651$	−3.50847e11	−	8.14297e12i	−0.0765602	−	1.77692i
$652$		0			0
$653$	1.91637e12	+	3.31924e12i	0.412448	+	0.714380i	0.995157	−	0.0983004i	$-0.0313406\pi$
$653$	1.91637e12	+	3.31924e12i	0.412448	+	0.714380i	−0.582709	+	0.812681i	$0.698007\pi$
$654$		0			0
$655$	8.14005e10	+	4.69966e10i	0.0172799	+	0.00997656i
$656$		0			0
$657$		−	2.29298e12i		−	0.480127i
$658$		0			0
$659$		−	8.38730e12i		−	1.73236i	−0.499734	−	0.866179i	$-0.666569\pi$
$659$		−	8.38730e12i		−	1.73236i	0.499734	−	0.866179i	$-0.333431\pi$
$660$		0			0
$661$	7.22103e12	+	4.16906e12i	1.47127	+	0.849439i	0.999479	−	0.0322713i	$-0.0102740\pi$
$661$	7.22103e12	+	4.16906e12i	1.47127	+	0.849439i	0.471792	+	0.881710i	$0.343607\pi$
$662$		0			0
$663$	2.91832e12	+	5.05468e12i	0.586573	+	1.01598i
$664$		0			0
$665$	−5.42695e11	−	8.52941e11i	−0.107611	−	0.169130i
$666$		0			0
$667$	−7.60060e12	+	4.38821e12i	−1.48690	+	0.858462i
$668$		0			0
$669$	−7.10316e11	+	1.23030e12i	−0.137099	+	0.237462i
$670$		0			0
$671$	8.99702e12			1.71336
$672$		0			0
$673$	−2.47455e12			−0.464973			−0.232487	−	0.972600i	$-0.574686\pi$
$673$	−2.47455e12			−0.464973			−0.232487	+	0.972600i	$0.574686\pi$
$674$		0			0
$675$	−1.23379e11	+	2.13699e11i	−0.0228758	+	0.0396220i
$676$		0			0
$677$	−4.89750e12	+	2.82757e12i	−0.896036	+	0.517327i	−0.875912	−	0.482471i	$-0.839739\pi$
$677$	−4.89750e12	+	2.82757e12i	−0.896036	+	0.517327i	−0.0201240	+	0.999797i	$0.506406\pi$
$678$		0			0
$679$	−4.03908e12	+	7.74814e12i	−0.729237	+	1.39889i
$680$		0			0
$681$	−6.02554e12	−	1.04365e13i	−1.07358	−	1.85949i
$682$		0			0
$683$	9.39444e12	+	5.42388e12i	1.65188	+	0.953712i	0.976299	+	0.216424i	$0.0694393\pi$
$683$	9.39444e12	+	5.42388e12i	1.65188	+	0.953712i	0.675578	+	0.737288i	$0.263894\pi$
$684$		0			0
$685$			2.87390e12i			0.498728i
$686$		0			0
$687$			2.45856e12i			0.421090i
$688$		0			0
$689$	5.81942e12	+	3.35985e12i	0.983770	+	0.567980i
$690$		0			0
$691$	−2.35649e12	−	4.08155e12i	−0.393200	−	0.681043i	0.599669	−	0.800248i	$-0.295299\pi$
$691$	−2.35649e12	−	4.08155e12i	−0.393200	−	0.681043i	−0.992870	+	0.119205i	$0.961965\pi$
$692$		0			0
$693$	3.28187e12	−	6.29559e12i	0.540533	−	1.03690i
$694$		0			0
$695$	5.76501e12	−	3.32843e12i	0.937278	−	0.541137i
$696$		0			0
$697$	1.73057e12	−	2.99743e12i	0.277742	−	0.481063i
$698$		0			0
$699$	−1.76853e12			−0.280198
$700$		0			0
$701$	−5.00006e12			−0.782068			−0.391034	−	0.920376i	$-0.627883\pi$
$701$	−5.00006e12			−0.782068			−0.391034	+	0.920376i	$0.627883\pi$
$702$		0			0
$703$	6.80067e10	−	1.17791e11i	0.0105015	−	0.0181892i
$704$		0			0
$705$	−9.60934e12	+	5.54795e12i	−1.46502	+	0.845828i
$706$		0			0
$707$	−2.49281e11	−	3.91789e11i	−0.0375234	−	0.0589746i
$708$		0			0
$709$	1.24345e12	+	2.15372e12i	0.184808	+	0.320096i	0.943512	−	0.331339i	$-0.107500\pi$
$709$	1.24345e12	+	2.15372e12i	0.184808	+	0.320096i	−0.758704	+	0.651435i	$0.774167\pi$
$710$		0			0
$711$	−1.39670e12	−	8.06385e11i	−0.204970	−	0.118339i
$712$		0			0
$713$			7.76180e12i			1.12476i
$714$		0			0
$715$			4.74139e12i			0.678466i
$716$		0			0
$717$	8.51267e11	+	4.91479e11i	0.120290	+	0.0694495i
$718$		0			0
$719$	4.45965e12	+	7.72435e12i	0.622331	+	1.07791i	0.989051	+	0.147577i	$0.0471474\pi$
$719$	4.45965e12	+	7.72435e12i	0.622331	+	1.07791i	−0.366720	+	0.930331i	$0.619519\pi$
$720$		0			0
$721$	−4.58536e11	−	1.06424e13i	−0.0631924	−	1.46666i
$722$		0			0
$723$	−1.58867e13	+	9.17221e12i	−2.16228	+	1.24839i
$724$		0			0
$725$	−9.05799e11	+	1.56889e12i	−0.121762	+	0.210898i
$726$		0			0
$727$	−1.36348e13			−1.81028			−0.905139	−	0.425115i	$-0.860234\pi$
$727$	−1.36348e13			−1.81028			−0.905139	+	0.425115i	$0.860234\pi$
$728$		0			0
$729$	−1.06250e13			−1.39334
$730$		0			0
$731$	5.18494e11	−	8.98058e11i	0.0671608	−	0.116326i
$732$		0			0
$733$	−1.23396e13	+	7.12425e12i	−1.57882	+	0.911530i	−0.583792	+	0.811904i	$0.698432\pi$
$733$	−1.23396e13	+	7.12425e12i	−1.57882	+	0.911530i	−0.995025	+	0.0996266i	$0.968235\pi$
$734$		0			0
$735$	−9.01213e12	+	6.29496e12i	−1.13903	+	0.795609i
$736$		0			0
$737$	−4.29920e12	−	7.44643e12i	−0.536765	−	0.929704i
$738$		0			0
$739$	2.40417e12	+	1.38805e12i	0.296527	+	0.171200i	0.640882	−	0.767640i	$-0.278569\pi$
$739$	2.40417e12	+	1.38805e12i	0.296527	+	0.171200i	−0.344354	+	0.938840i	$0.611902\pi$
$740$		0			0
$741$			2.02201e12i			0.246377i
$742$		0			0
$743$			3.05533e12i			0.367797i	0.982945	+	0.183899i	$0.0588718\pi$
$743$			3.05533e12i			0.367797i	−0.982945	+	0.183899i	$0.941128\pi$
$744$		0			0
$745$	5.21139e12	+	3.00880e12i	0.619798	+	0.357840i
$746$		0			0
$747$	5.60761e11	+	9.71266e11i	0.0658923	+	0.114129i
$748$		0			0
$749$	6.11424e12	−	2.63437e11i	0.709863	−	0.0305850i
$750$		0			0
$751$	7.61480e12	−	4.39641e12i	0.873532	−	0.504334i	0.00501163	−	0.999987i	$-0.498405\pi$
$751$	7.61480e12	−	4.39641e12i	0.873532	−	0.504334i	0.868520	+	0.495654i	$0.165071\pi$
$752$		0			0
$753$	−8.42575e12	+	1.45938e13i	−0.955061	+	1.65421i
$754$		0			0
$755$	9.12258e12			1.02178
$756$		0			0
$757$	−2.68080e11			−0.0296711			−0.0148355	−	0.999890i	$-0.504722\pi$
$757$	−2.68080e11			−0.0296711			−0.0148355	+	0.999890i	$0.504722\pi$
$758$		0			0
$759$	−6.13149e12	+	1.06201e13i	−0.670622	+	1.16155i
$760$		0			0
$761$	8.68094e12	−	5.01194e12i	0.938287	−	0.541720i	0.0488643	−	0.998805i	$-0.484440\pi$
$761$	8.68094e12	−	5.01194e12i	0.938287	−	0.541720i	0.889423	+	0.457085i	$0.151106\pi$
$762$		0			0
$763$	7.53611e12	−	4.79495e12i	0.804983	−	0.512181i
$764$		0			0
$765$	−5.55559e12	−	9.62257e12i	−0.586481	−	1.01582i
$766$		0			0
$767$	1.09871e12	+	6.34340e11i	0.114631	+	0.0661825i
$768$		0			0
$769$			7.07038e12i			0.729078i	0.931188	+	0.364539i	$0.118773\pi$
$769$			7.07038e12i			0.729078i	−0.931188	+	0.364539i	$0.881227\pi$
$770$		0			0
$771$			1.56766e13i			1.59774i
$772$		0			0
$773$	7.35243e12	+	4.24493e12i	0.740667	+	0.427624i	0.822312	−	0.569037i	$-0.192684\pi$
$773$	7.35243e12	+	4.24493e12i	0.740667	+	0.427624i	−0.0816448	+	0.996661i	$0.526017\pi$
$774$		0			0
$775$	8.01081e11	+	1.38751e12i	0.0797662	+	0.138159i
$776$		0			0
$777$	−1.31149e12	−	6.83675e11i	−0.129083	−	0.0672907i
$778$		0			0
$779$	1.03841e12	−	5.99526e11i	0.101030	−	0.0583297i
$780$		0			0
$781$	6.61939e12	−	1.14651e13i	0.636632	−	1.10268i
$782$		0			0
$783$	6.53490e12			0.621314
$784$		0			0
$785$	1.00399e13			0.943663
$786$		0			0
$787$	−3.05806e12	+	5.29672e12i	−0.284158	+	0.492176i	−0.972405	−	0.233301i	$-0.925047\pi$
$787$	−3.05806e12	+	5.29672e12i	−0.284158	+	0.492176i	0.688247	+	0.725477i	$0.258381\pi$
$788$		0			0
$789$	2.57191e12	−	1.48489e12i	0.236271	−	0.136411i
$790$		0			0
$791$	1.74330e13	+	9.08774e12i	1.58335	+	0.825395i
$792$		0			0
$793$	7.68095e12	+	1.33038e13i	0.689740	+	1.19466i
$794$		0			0
$795$	−2.00937e13	−	1.16011e13i	−1.78405	−	1.03002i
$796$		0			0
$797$		−	9.13834e12i		−	0.802241i	−0.916025	−	0.401120i	$-0.868621\pi$
$797$		−	9.13834e12i		−	0.802241i	0.916025	−	0.401120i	$-0.131379\pi$
$798$		0			0
$799$			1.43865e13i			1.24881i
$800$		0			0
$801$	−3.87797e12	−	2.23895e12i	−0.332857	−	0.192175i
$802$		0			0
$803$	−2.19022e12	−	3.79357e12i	−0.185895	−	0.321979i
$804$		0			0
$805$	8.83240e12	−	5.61973e12i	0.741306	−	0.471666i
$806$		0			0
$807$	−5.36327e12	+	3.09649e12i	−0.445142	+	0.257003i
$808$		0			0
$809$	−2.00177e12	+	3.46717e12i	−0.164303	+	0.284581i	−0.936408	−	0.350914i	$-0.885871\pi$
$809$	−2.00177e12	+	3.46717e12i	−0.164303	+	0.284581i	0.772105	+	0.635496i	$0.219204\pi$
$810$		0			0
$811$	−5.87382e12			−0.476790			−0.238395	−	0.971168i	$-0.576621\pi$
$811$	−5.87382e12			−0.476790			−0.238395	+	0.971168i	$0.576621\pi$
$812$		0			0
$813$	1.57927e13			1.26780
$814$		0			0
$815$	9.89526e12	−	1.71391e13i	0.785630	−	1.36075i
$816$		0			0
$817$	3.11117e11	−	1.79623e11i	0.0244300	−	0.0141047i
$818$		0			0
$819$	1.21110e13	−	5.21813e11i	0.940596	−	0.0405263i
$820$		0			0
$821$	−6.64814e12	−	1.15149e13i	−0.510688	−	0.884538i	−0.999923	−	0.0123862i	$-0.996057\pi$
$821$	−6.64814e12	−	1.15149e13i	−0.510688	−	0.884538i	0.489235	−	0.872152i	$-0.337276\pi$
$822$		0			0
$823$	−1.31671e13	−	7.60201e12i	−1.00044	−	0.577603i	−0.0920599	−	0.995753i	$-0.529345\pi$
$823$	−1.31671e13	−	7.60201e12i	−1.00044	−	0.577603i	−0.908378	+	0.418151i	$0.862678\pi$
$824$		0			0
$825$			2.53128e12i			0.190238i
$826$		0			0
$827$			5.92688e11i			0.0440607i	0.999757	+	0.0220303i	$0.00701304\pi$
$827$			5.92688e11i			0.0440607i	−0.999757	+	0.0220303i	$0.992987\pi$
$828$		0			0
$829$	−1.85597e13	−	1.07155e13i	−1.36482	−	0.787980i	−0.374560	−	0.927203i	$-0.622206\pi$
$829$	−1.85597e13	−	1.07155e13i	−1.36482	−	0.787980i	−0.990261	+	0.139223i	$0.955540\pi$
$830$		0			0
$831$	1.18846e13	+	2.05847e13i	0.864528	+	1.49741i
$832$		0			0
$833$	1.22593e12	+	1.42002e13i	0.0882192	+	1.02186i
$834$		0			0
$835$	−1.34032e13	+	7.73837e12i	−0.954159	+	0.550884i
$836$		0			0
$837$	2.88971e12	−	5.00512e12i	0.203512	−	0.352492i
$838$		0			0
$839$	−1.79319e13			−1.24939			−0.624693	−	0.780870i	$-0.714776\pi$
$839$	−1.79319e13			−1.24939			−0.624693	+	0.780870i	$0.714776\pi$
$840$		0			0
$841$	3.34693e13			2.30709
$842$		0			0
$843$	−3.64732e11	+	6.31734e11i	−0.0248742	+	0.0430834i
$844$		0			0
$845$	4.93344e12	−	2.84832e12i	0.332885	−	0.192191i
$846$		0			0
$847$	6.09437e10	+	1.41447e12i	0.00406868	+	0.0944320i
$848$		0			0
$849$	−2.12238e13	−	3.67607e13i	−1.40197	−	2.42828i
$850$		0			0
$851$	1.21975e12	+	7.04225e11i	0.0797240	+	0.0460287i
$852$		0			0
$853$			1.67452e13i			1.08298i	0.840707	+	0.541490i	$0.182140\pi$
$853$			1.67452e13i			1.08298i	−0.840707	+	0.541490i	$0.817860\pi$
$854$		0			0
$855$		−	3.84928e12i		−	0.246338i
$856$		0			0
$857$	2.44348e13	+	1.41075e13i	1.54738	+	0.893378i	0.998341	+	0.0575784i	$0.0183379\pi$
$857$	2.44348e13	+	1.41075e13i	1.54738	+	0.893378i	0.549035	+	0.835799i	$0.314995\pi$
$858$		0			0
$859$	−5.23811e12	−	9.07268e12i	−0.328251	−	0.568547i	0.653914	−	0.756569i	$-0.273126\pi$
$859$	−5.23811e12	−	9.07268e12i	−0.328251	−	0.568547i	−0.982165	+	0.188022i	$0.939792\pi$
$860$		0			0
$861$	−6.99917e12	−	1.10004e13i	−0.434042	−	0.682174i
$862$		0			0
$863$	1.27285e13	−	7.34879e12i	0.781139	−	0.450991i	−0.0556951	−	0.998448i	$-0.517737\pi$
$863$	1.27285e13	−	7.34879e12i	0.781139	−	0.450991i	0.836834	+	0.547457i	$0.184404\pi$
$864$		0			0
$865$	−3.62774e12	+	6.28342e12i	−0.220325	+	0.381614i
$866$		0			0
$867$	−1.29121e12			−0.0776089
$868$		0			0
$869$	−3.08099e12			−0.183274
$870$		0			0
$871$	7.34064e12	−	1.27144e13i	0.432167	−	0.748536i
$872$		0			0
$873$	−2.88123e13	+	1.66348e13i	−1.67885	+	0.969287i
$874$		0			0
$875$	8.45825e12	−	1.62254e13i	0.487803	−	0.935749i
$876$		0			0
$877$	3.01054e12	+	5.21440e12i	0.171848	+	0.297650i	0.939066	−	0.343737i	$-0.111693\pi$
$877$	3.01054e12	+	5.21440e12i	0.171848	+	0.297650i	−0.767218	+	0.641387i	$0.778359\pi$
$878$		0			0
$879$	−2.68750e13	−	1.55163e13i	−1.51844	−	0.876673i
$880$		0			0
$881$		−	2.00824e12i		−	0.112311i	−0.998422	−	0.0561557i	$-0.982116\pi$
$881$		−	2.00824e12i		−	0.112311i	0.998422	−	0.0561557i	$-0.0178843\pi$
$882$		0			0
$883$			2.79790e13i			1.54885i	0.632666	+	0.774425i	$0.281961\pi$
$883$			2.79790e13i			1.54885i	−0.632666	+	0.774425i	$0.718039\pi$
$884$		0			0
$885$	−3.79370e12	−	2.19029e12i	−0.207882	−	0.120021i
$886$		0			0
$887$	1.38133e13	+	2.39254e13i	0.749276	+	1.29778i	0.948170	+	0.317763i	$0.102932\pi$
$887$	1.38133e13	+	2.39254e13i	0.749276	+	1.29778i	−0.198894	+	0.980021i	$0.563735\pi$
$888$		0			0
$889$	−1.24413e13	+	2.38660e13i	−0.668047	+	1.28151i
$890$		0			0
$891$	−1.11434e13	+	6.43362e12i	−0.592333	+	0.341984i
$892$		0			0
$893$	−2.49198e12	+	4.31624e12i	−0.131133	+	0.227130i
$894$		0			0
$895$	2.57251e13			1.34015
$896$		0			0
$897$	−2.09383e13			−1.07988
$898$		0			0
$899$	2.12150e13	−	3.67454e13i	1.08324	−	1.87622i
$900$		0			0
$901$	−2.60527e13	+	1.50415e13i	−1.31701	+	0.760379i
$902$		0			0
$903$	−2.09701e12	−	3.29582e12i	−0.104956	−	0.164956i
$904$		0			0
$905$	3.46198e12	+	5.99633e12i	0.171556	+	0.297144i
$906$		0			0
$907$	7.65416e12	+	4.41913e12i	0.375547	+	0.216822i	0.675879	−	0.737012i	$-0.263764\pi$
$907$	7.65416e12	+	4.41913e12i	0.375547	+	0.216822i	−0.300332	+	0.953835i	$0.597097\pi$
$908$		0			0
$909$		−	1.76813e12i		−	0.0858965i
$910$		0			0
$911$		−	2.52442e13i		−	1.21431i	−0.794584	−	0.607154i	$-0.792311\pi$
$911$		−	2.52442e13i		−	1.21431i	0.794584	−	0.607154i	$-0.207689\pi$
$912$		0			0
$913$	1.85548e12	+	1.07126e12i	0.0883765	+	0.0510242i
$914$		0			0
$915$	−2.65213e13	−	4.59362e13i	−1.25083	−	2.16651i
$916$		0			0
$917$	−1.97616e10	−	4.58658e11i	−0.000922914	−	0.0214204i
$918$		0			0
$919$	−8.87350e12	+	5.12311e12i	−0.410369	+	0.236927i	−0.690948	−	0.722904i	$-0.742807\pi$
$919$	−8.87350e12	+	5.12311e12i	−0.410369	+	0.236927i	0.280579	+	0.959831i	$0.409474\pi$
$920$		0			0
$921$	1.05148e13	−	1.82122e13i	0.481542	−	0.834055i
$922$		0			0
$923$	2.26045e13			1.02515
$924$		0			0
$925$	2.90727e11			0.0130571
$926$		0			0
$927$	2.02796e13	−	3.51254e13i	0.901989	−	1.56229i
$928$		0			0
$929$	−9.77079e11	+	5.64117e11i	−0.0430387	+	0.0248484i	−0.521365	−	0.853334i	$-0.674577\pi$
$929$	−9.77079e11	+	5.64117e11i	−0.0430387	+	0.0248484i	0.478326	+	0.878182i	$0.341244\pi$
$930$		0			0
$931$	−2.09191e12	+	4.47269e12i	−0.0912575	+	0.195117i
$932$		0			0
$933$	1.61430e13	+	2.79605e13i	0.697456	+	1.20803i
$934$		0			0
$935$	−1.83827e13	−	1.06132e13i	−0.786604	−	0.454146i
$936$		0			0
$937$		−	2.11381e13i		−	0.895854i	−0.894070	−	0.447927i	$-0.852162\pi$
$937$		−	2.11381e13i		−	0.895854i	0.894070	−	0.447927i	$-0.147838\pi$
$938$		0			0
$939$		−	1.52192e13i		−	0.638848i
$940$		0			0
$941$	−3.12346e13	−	1.80333e13i	−1.29862	−	0.749759i	−0.318455	−	0.947938i	$-0.603164\pi$
$941$	−3.12346e13	−	1.80333e13i	−1.29862	−	0.749759i	−0.980166	+	0.198179i	$0.936497\pi$
$942$		0			0
$943$	6.20823e12	+	1.07530e13i	0.255661	+	0.442819i
$944$		0			0
$945$	−7.78770e12	+	3.35540e11i	−0.317663	+	0.0136868i
$946$		0			0
$947$	1.53616e13	−	8.86903e12i	0.620672	−	0.358345i	−0.156459	−	0.987684i	$-0.550008\pi$
$947$	1.53616e13	−	8.86903e12i	0.620672	−	0.358345i	0.777131	+	0.629339i	$0.216674\pi$
$948$		0			0
$949$	3.73967e12	−	6.47730e12i	0.149670	−	0.259237i
$950$		0			0
$951$	−5.28750e13			−2.09623
$952$		0			0
$953$	−1.83252e13			−0.719665			−0.359832	−	0.933017i	$-0.617166\pi$
$953$	−1.83252e13			−0.719665			−0.359832	+	0.933017i	$0.617166\pi$
$954$		0			0
$955$	1.72467e13	−	2.98721e13i	0.670950	−	1.16212i
$956$		0			0
$957$	5.80547e13	−	3.35179e13i	2.23735	−	1.29173i
$958$		0			0
$959$	1.18428e13	−	7.53513e12i	0.452137	−	0.287678i
$960$		0			0
$961$	−5.54255e12	−	9.59998e12i	−0.209630	−	0.363091i
$962$		0			0
$963$	2.01802e13	+	1.16510e13i	0.756147	+	0.436562i
$964$		0			0
$965$			4.03852e13i			1.49916i
$966$		0			0
$967$		−	1.32710e13i		−	0.488073i	−0.969766	−	0.244037i	$-0.921528\pi$
$967$		−	1.32710e13i		−	0.488073i	0.969766	−	0.244037i	$-0.0784717\pi$
$968$		0			0
$969$	−7.83944e12	−	4.52610e12i	−0.285646	−	0.164918i
$970$		0			0
$971$	−3.15601e12	−	5.46637e12i	−0.113934	−	0.197339i	0.803419	−	0.595414i	$-0.203012\pi$
$971$	−3.15601e12	−	5.46637e12i	−0.113934	−	0.197339i	−0.917353	+	0.398075i	$0.869678\pi$
$972$		0			0
$973$	−2.88312e13	−	1.50296e13i	−1.03123	−	0.537577i
$974$		0			0
$975$	−3.74298e12	+	2.16101e12i	−0.132647	+	0.0765836i
$976$		0			0
$977$	8.05901e12	−	1.39586e13i	0.282980	−	0.490136i	−0.689137	−	0.724631i	$-0.742010\pi$
$977$	8.05901e12	−	1.39586e13i	0.282980	−	0.490136i	0.972117	+	0.234495i	$0.0753436\pi$
$978$		0			0
$979$	−8.55443e12			−0.297625
$980$		0			0
$981$	3.40101e13			1.17246
$982$		0			0
$983$	−3.42506e12	+	5.93238e12i	−0.116998	+	0.202646i	−0.918577	−	0.395243i	$-0.870660\pi$
$983$	−3.42506e12	+	5.93238e12i	−0.116998	+	0.202646i	0.801579	+	0.597889i	$0.203994\pi$
$984$		0			0
$985$	3.52811e13	−	2.03695e13i	1.19420	−	0.689474i
$986$		0			0
$987$	4.80570e13	+	2.50520e13i	1.61187	+	0.840262i
$988$		0			0
$989$	1.86004e12	+	3.22169e12i	0.0618215	+	0.107078i
$990$		0			0
$991$	−2.92356e13	−	1.68792e13i	−0.962900	−	0.555930i	−0.0658355	−	0.997830i	$-0.520971\pi$
$991$	−2.92356e13	−	1.68792e13i	−0.962900	−	0.555930i	−0.897064	+	0.441900i	$0.854305\pi$
$992$		0			0
$993$		−	8.26260e12i		−	0.269678i
$994$		0			0
$995$			5.08327e13i			1.64414i
$996$		0			0
$997$	3.41953e13	+	1.97427e13i	1.09607	+	0.632816i	0.935186	−	0.354158i	$-0.115232\pi$
$997$	3.41953e13	+	1.97427e13i	1.09607	+	0.632816i	0.160883	+	0.986973i	$0.448566\pi$
$998$		0			0
$999$	−5.24364e11	−	9.08225e11i	−0.0166567	−	0.0288502i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.b.31.2	✓	24
4.3	odd	2	inner	112.10.p.b.31.11	yes	24
7.5	odd	6	inner	112.10.p.b.47.11	yes	24
28.19	even	6	inner	112.10.p.b.47.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.b.31.2	✓	24	1.1	even	1	trivial
112.10.p.b.31.11	yes	24	4.3	odd	2	inner
112.10.p.b.47.2	yes	24	28.19	even	6	inner
112.10.p.b.47.11	yes	24	7.5	odd	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)

\(f(q)\)	\(=\)	\(q+(-104.726 + 181.391i) q^{3} +(1126.36 - 650.304i) q^{5} +(-5633.01 - 2936.47i) q^{7} +(-12093.7 - 20946.9i) q^{9} +(-40016.3 - 23103.4i) q^{11} -78895.6i q^{13} +272416. i q^{15} +(305883. + 176602. i) q^{17} +(61180.6 + 105968. i) q^{19} +(1.12257e6 - 714253. i) q^{21} +(-1.09732e6 + 633539. i) q^{23} +(-130773. + 226505. i) q^{25} +943463. q^{27} +6.92650e6 q^{29} +(3.06287e6 - 5.30505e6i) q^{31} +(8.38153e6 - 4.83908e6i) q^{33} +(-8.25438e6 + 355647. i) q^{35} +(-555787. - 962651. i) q^{37} +(1.43110e7 + 8.26244e6i) q^{39} -9.79928e6i q^{41} -2.93595e6i q^{43} +(-2.72437e7 - 1.57292e7i) q^{45} +(2.03658e7 + 3.52745e7i) q^{47} +(2.31079e7 + 3.30823e7i) q^{49} +(-6.40680e7 + 3.69897e7i) q^{51} +(-4.25860e7 + 7.37611e7i) q^{53} -6.00970e7 q^{55} -2.56289e7 q^{57} +(-8.04025e6 + 1.39261e7i) q^{59} +(-1.68625e8 + 9.73559e7i) q^{61} +(6.61397e6 + 1.53507e8i) q^{63} +(-5.13061e7 - 8.88647e7i) q^{65} +(1.61154e8 + 9.30424e7i) q^{67} -2.65393e8i q^{69} +2.86511e8i q^{71} +(8.20997e7 + 4.74003e7i) q^{73} +(-2.73907e7 - 4.74422e7i) q^{75} +(1.57570e8 + 2.47649e8i) q^{77} +(5.77449e7 - 3.33390e7i) q^{79} +(1.39235e8 - 2.41162e8i) q^{81} -4.63680e7 q^{83} +4.59379e8 q^{85} +(-7.25387e8 + 1.25641e9i) q^{87} +(1.60330e8 - 9.25666e7i) q^{89} +(-2.31674e8 + 4.44419e8i) q^{91} +(6.41527e8 + 1.11116e9i) q^{93} +(1.37823e8 + 7.95719e7i) q^{95} -1.37549e9i q^{97} +1.11763e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57}+ \cdots - 1432348316 q^{93}+O(q^{100}) \) 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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