LMFDB - Embedded newform 112.10.p.a.31.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [24,0,-162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

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

Embedding invariants

Embedding label			31.8
Character	$\chi$	$=$	112.31
Dual form			112.10.p.a.47.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(22.7708 - 39.4402i) q^{3} +(-1500.67 + 866.412i) q^{5} +(-6349.75 - 185.284i) q^{7} +(8804.48 + 15249.8i) q^{9} +(-71855.0 - 41485.5i) q^{11} -151879. i q^{13} +78915.7i q^{15} +(-325208. - 187759. i) q^{17} +(167329. + 289822. i) q^{19} +(-151897. + 246216. i) q^{21} +(-1.23177e6 + 711164. i) q^{23} +(524777. - 908940. i) q^{25} +1.69834e6 q^{27} +5.11171e6 q^{29} +(-443123. + 767512. i) q^{31} +(-3.27240e6 + 1.88932e6i) q^{33} +(9.68940e6 - 5.22345e6i) q^{35} +(7.46397e6 + 1.29280e7i) q^{37} +(-5.99014e6 - 3.45841e6i) q^{39} +1.60174e7i q^{41} +2.69275e7i q^{43} +(-2.64252e7 - 1.52566e7i) q^{45} +(-2.88694e7 - 5.00033e7i) q^{47} +(4.02849e7 + 2.35301e6i) q^{49} +(-1.48105e7 + 8.55084e6i) q^{51} +(4.55716e7 - 7.89324e7i) q^{53} +1.43774e8 q^{55} +1.52408e7 q^{57} +(-1.65678e6 + 2.86964e6i) q^{59} +(-7.99730e7 + 4.61724e7i) q^{61} +(-5.30807e7 - 9.84637e7i) q^{63} +(1.31590e8 + 2.27920e8i) q^{65} +(-4.61774e7 - 2.66605e7i) q^{67} +6.47752e7i q^{69} -1.85803e8i q^{71} +(2.17356e8 + 1.25491e8i) q^{73} +(-2.38992e7 - 4.13946e7i) q^{75} +(4.48575e8 + 2.76736e8i) q^{77} +(3.29407e8 - 1.90183e8i) q^{79} +(-1.34626e8 + 2.33179e8i) q^{81} +6.02933e8 q^{83} +6.50705e8 q^{85} +(1.16398e8 - 2.01607e8i) q^{87} +(3.63585e8 - 2.09916e8i) q^{89} +(-2.81407e7 + 9.64393e8i) q^{91} +(2.01806e7 + 3.49538e7i) q^{93} +(-5.02210e8 - 2.89951e8i) q^{95} -2.18214e8i q^{97} -1.46103e9i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) $ 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

Character values

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

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

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
$3$	22.7708	−	39.4402i	0.162306	−	0.281121i	−0.773390	−	0.633931i	$-0.781440\pi$
$3$	22.7708	−	39.4402i	0.162306	−	0.281121i	0.935695	+	0.352810i	$0.114774\pi$
$4$		0			0
$5$	−1500.67	+	866.412i	−1.07379	+	0.619954i	−0.929215	−	0.369540i	$-0.879515\pi$
$5$	−1500.67	+	866.412i	−1.07379	+	0.619954i	−0.144577	+	0.989494i	$0.546182\pi$
$6$		0			0
$7$	−6349.75	−	185.284i	−0.999575	−	0.0291673i
$7$	−6349.75	−	185.284i	−0.999575	−	0.0291673i
$8$		0			0
$9$	8804.48	+	15249.8i	0.447314	+	0.774770i
$10$		0			0
$11$	−71855.0	−	41485.5i	−1.47976	−	0.854338i	−0.480019	−	0.877258i	$-0.659370\pi$
$11$	−71855.0	−	41485.5i	−1.47976	−	0.854338i	−0.999737	+	0.0229207i	$0.992703\pi$
$12$		0			0
$13$		−	151879.i		−	1.47487i	−0.675420	−	0.737433i	$-0.736038\pi$
$13$		−	151879.i		−	1.47487i	0.675420	−	0.737433i	$-0.263962\pi$
$14$		0			0
$15$			78915.7i			0.402488i
$16$		0			0
$17$	−325208.	−	187759.i	−0.944366	−	0.545230i	−0.0530399	−	0.998592i	$-0.516891\pi$
$17$	−325208.	−	187759.i	−0.944366	−	0.545230i	−0.891326	+	0.453362i	$0.850224\pi$
$18$		0			0
$19$	167329.	+	289822.i	0.294564	+	0.510199i	0.974883	−	0.222716i	$-0.0714924\pi$
$19$	167329.	+	289822.i	0.294564	+	0.510199i	−0.680320	+	0.732915i	$0.738159\pi$
$20$		0			0
$21$	−151897.	+	246216.i	−0.170436	+	0.276268i
$22$		0			0
$23$	−1.23177e6	+	711164.i	−0.917815	+	0.529901i	−0.882937	−	0.469491i	$-0.844438\pi$
$23$	−1.23177e6	+	711164.i	−0.917815	+	0.529901i	−0.0348780	+	0.999392i	$0.511104\pi$
$24$		0			0
$25$	524777.	−	908940.i	0.268686	−	0.465377i
$26$		0			0
$27$	1.69834e6			0.615017
$28$		0			0
$29$	5.11171e6			1.34207			0.671036	−	0.741425i	$-0.265850\pi$
$29$	5.11171e6			1.34207			0.671036	+	0.741425i	$0.265850\pi$
$30$		0			0
$31$	−443123.	+	767512.i	−0.0861781	+	0.149265i	−0.905893	−	0.423507i	$-0.860799\pi$
$31$	−443123.	+	767512.i	−0.0861781	+	0.149265i	0.819715	+	0.572772i	$0.194132\pi$
$32$		0			0
$33$	−3.27240e6	+	1.88932e6i	−0.480345	+	0.277327i
$34$		0			0
$35$	9.68940e6	−	5.22345e6i	1.09142	−	0.588371i
$36$		0			0
$37$	7.46397e6	+	1.29280e7i	0.654729	+	1.13402i	0.981962	+	0.189081i	$0.0605508\pi$
$37$	7.46397e6	+	1.29280e7i	0.654729	+	1.13402i	−0.327232	+	0.944944i	$0.606116\pi$
$38$		0			0
$39$	−5.99014e6	−	3.45841e6i	−0.414617	−	0.239379i
$40$		0			0
$41$			1.60174e7i			0.885246i	0.896708	+	0.442623i	$0.145952\pi$
$41$			1.60174e7i			0.885246i	−0.896708	+	0.442623i	$0.854048\pi$
$42$		0			0
$43$			2.69275e7i			1.20112i	0.799579	+	0.600561i	$0.205056\pi$
$43$			2.69275e7i			1.20112i	−0.799579	+	0.600561i	$0.794944\pi$
$44$		0			0
$45$	−2.64252e7	−	1.52566e7i	−0.960644	−	0.554628i
$46$		0			0
$47$	−2.88694e7	−	5.00033e7i	−0.862973	−	1.49471i	−0.869045	−	0.494732i	$-0.835266\pi$
$47$	−2.88694e7	−	5.00033e7i	−0.862973	−	1.49471i	0.00607192	−	0.999982i	$-0.498067\pi$
$48$		0			0
$49$	4.02849e7	+	2.35301e6i	0.998299	+	0.0583098i
$50$		0			0
$51$	−1.48105e7	+	8.55084e6i	−0.306552	+	0.176988i
$52$		0			0
$53$	4.55716e7	−	7.89324e7i	0.793329	−	1.37409i	−0.130567	−	0.991440i	$-0.541680\pi$
$53$	4.55716e7	−	7.89324e7i	0.793329	−	1.37409i	0.923895	−	0.382646i	$-0.124987\pi$
$54$		0			0
$55$	1.43774e8			2.11860
$56$		0			0
$57$	1.52408e7			0.191237
$58$		0			0
$59$	−1.65678e6	+	2.86964e6i	−0.0178005	+	0.0308314i	−0.874788	−	0.484505i	$-0.839000\pi$
$59$	−1.65678e6	+	2.86964e6i	−0.0178005	+	0.0308314i	0.856988	+	0.515336i	$0.172333\pi$
$60$		0			0
$61$	−7.99730e7	+	4.61724e7i	−0.739536	+	0.426971i	−0.821900	−	0.569631i	$-0.807086\pi$
$61$	−7.99730e7	+	4.61724e7i	−0.739536	+	0.426971i	0.0823648	+	0.996602i	$0.473753\pi$
$62$		0			0
$63$	−5.30807e7	−	9.84637e7i	−0.424526	−	0.787488i
$64$		0			0
$65$	1.31590e8	+	2.27920e8i	0.914349	+	1.58370i
$66$		0			0
$67$	−4.61774e7	−	2.66605e7i	−0.279958	−	0.161634i	0.353447	−	0.935455i	$-0.385010\pi$
$67$	−4.61774e7	−	2.66605e7i	−0.279958	−	0.161634i	−0.633404	+	0.773821i	$0.718343\pi$
$68$		0			0
$69$			6.47752e7i			0.344023i
$70$		0			0
$71$		−	1.85803e8i		−	0.867743i	−0.900975	−	0.433871i	$-0.857147\pi$
$71$		−	1.85803e8i		−	0.867743i	0.900975	−	0.433871i	$-0.142853\pi$
$72$		0			0
$73$	2.17356e8	+	1.25491e8i	0.895817	+	0.517200i	0.875841	−	0.482600i	$-0.160308\pi$
$73$	2.17356e8	+	1.25491e8i	0.895817	+	0.517200i	0.0199765	+	0.999800i	$0.493641\pi$
$74$		0			0
$75$	−2.38992e7	−	4.13946e7i	−0.0872183	−	0.151067i
$76$		0			0
$77$	4.48575e8	+	2.76736e8i	1.45421	+	0.897135i
$78$		0			0
$79$	3.29407e8	−	1.90183e8i	0.951505	−	0.549352i	0.0579567	−	0.998319i	$-0.481541\pi$
$79$	3.29407e8	−	1.90183e8i	0.951505	−	0.549352i	0.893548	+	0.448968i	$0.148208\pi$
$80$		0			0
$81$	−1.34626e8	+	2.33179e8i	−0.347493	+	0.601876i
$82$		0			0
$83$	6.02933e8			1.39450			0.697249	−	0.716829i	$-0.254407\pi$
$83$	6.02933e8			1.39450			0.697249	+	0.716829i	$0.254407\pi$
$84$		0			0
$85$	6.50705e8			1.35207
$86$		0			0
$87$	1.16398e8	−	2.01607e8i	0.217826	−	0.377285i
$88$		0			0
$89$	3.63585e8	−	2.09916e8i	0.614258	−	0.354642i	−0.160372	−	0.987057i	$-0.551269\pi$
$89$	3.63585e8	−	2.09916e8i	0.614258	−	0.354642i	0.774630	+	0.632415i	$0.217936\pi$
$90$		0			0
$91$	−2.81407e7	+	9.64393e8i	−0.0430179	+	1.47424i
$92$		0			0
$93$	2.01806e7	+	3.49538e7i	0.0279743	+	0.0484530i
$94$		0			0
$95$	−5.02210e8	−	2.89951e8i	−0.632600	−	0.365232i
$96$		0			0
$97$		−	2.18214e8i		−	0.250271i	−0.992140	−	0.125135i	$-0.960064\pi$
$97$		−	2.18214e8i		−	0.250271i	0.992140	−	0.125135i	$-0.0399365\pi$
$98$		0			0
$99$		−	1.46103e9i		−	1.52863i
$100$		0			0
$101$	1.44885e9	+	8.36492e8i	1.38540	+	0.799863i	0.992793	−	0.119841i	$-0.0382386\pi$
$101$	1.44885e9	+	8.36492e8i	1.38540	+	0.799863i	0.392611	+	0.919705i	$0.371572\pi$
$102$		0			0
$103$	−1.20613e8	−	2.08908e8i	−0.105591	−	0.182889i	0.808389	−	0.588649i	$-0.200340\pi$
$103$	−1.20613e8	−	2.08908e8i	−0.105591	−	0.182889i	−0.913979	+	0.405761i	$0.867007\pi$
$104$		0			0
$105$	1.46218e7	−	5.01095e8i	0.0117395	−	0.402316i
$106$		0			0
$107$	−1.00885e9	+	5.82462e8i	−0.744048	+	0.429577i	−0.823539	−	0.567259i	$-0.808004\pi$
$107$	−1.00885e9	+	5.82462e8i	−0.744048	+	0.429577i	0.0794911	+	0.996836i	$0.474670\pi$
$108$		0			0
$109$	−1.25277e9	+	2.16987e9i	−0.850067	+	1.47236i	0.0310799	+	0.999517i	$0.490105\pi$
$109$	−1.25277e9	+	2.16987e9i	−0.850067	+	1.47236i	−0.881147	+	0.472842i	$0.843228\pi$
$110$		0			0
$111$	6.79843e8			0.425065
$112$		0			0
$113$	−9.63673e8			−0.556002			−0.278001	−	0.960581i	$-0.589672\pi$
$113$	−9.63673e8			−0.556002			−0.278001	+	0.960581i	$0.589672\pi$
$114$		0			0
$115$	1.23232e9	−	2.13445e9i	0.657028	−	1.13801i
$116$		0			0
$117$	2.31613e9	−	1.33722e9i	1.14268	−	0.659728i
$118$		0			0
$119$	2.03020e9	+	1.25248e9i	0.928062	+	0.572543i
$120$		0			0
$121$	2.26312e9	+	3.91984e9i	0.959785	+	1.66240i
$122$		0			0
$123$	6.31729e8	+	3.64729e8i	0.248862	+	0.143680i
$124$		0			0
$125$		−	1.56573e9i		−	0.573617i
$126$		0			0
$127$		−	4.36434e9i		−	1.48868i	−0.667801	−	0.744340i	$-0.732764\pi$
$127$		−	4.36434e9i		−	1.48868i	0.667801	−	0.744340i	$-0.267236\pi$
$128$		0			0
$129$	1.06203e9	+	6.13160e8i	0.337661	+	0.194949i
$130$		0			0
$131$	−3.32537e8	−	5.75972e8i	−0.0986551	−	0.170876i	0.812473	−	0.582999i	$-0.198121\pi$
$131$	−3.32537e8	−	5.75972e8i	−0.0986551	−	0.170876i	−0.911128	+	0.412123i	$0.864787\pi$
$132$		0			0
$133$	−1.00879e9	−	1.87130e9i	−0.279557	−	0.518574i
$134$		0			0
$135$	−2.54864e9	+	1.47146e9i	−0.660400	+	0.381282i
$136$		0			0
$137$	6.32515e8	−	1.09555e9i	0.153401	−	0.265698i	−0.779075	−	0.626931i	$-0.784311\pi$
$137$	6.32515e8	−	1.09555e9i	0.153401	−	0.265698i	0.932476	+	0.361233i	$0.117644\pi$
$138$		0			0
$139$	2.88275e9			0.655000			0.327500	−	0.944851i	$-0.393794\pi$
$139$	2.88275e9			0.655000			0.327500	+	0.944851i	$0.393794\pi$
$140$		0			0
$141$	−2.62952e9			−0.560261
$142$		0			0
$143$	−6.30078e9	+	1.09133e10i	−1.26003	+	2.18244i
$144$		0			0
$145$	−7.67099e9	+	4.42885e9i	−1.44110	+	0.832022i
$146$		0			0
$147$	1.01012e9	−	1.53527e9i	0.178421	−	0.271179i
$148$		0			0
$149$	−2.35256e9	−	4.07475e9i	−0.391023	−	0.677271i	0.601562	−	0.798826i	$-0.294545\pi$
$149$	−2.35256e9	−	4.07475e9i	−0.391023	−	0.677271i	−0.992585	+	0.121555i	$0.961212\pi$
$150$		0			0
$151$	1.90746e9	+	1.10127e9i	0.298579	+	0.172385i	0.641804	−	0.766868i	$-0.278186\pi$
$151$	1.90746e9	+	1.10127e9i	0.298579	+	0.172385i	−0.343225	+	0.939253i	$0.611519\pi$
$152$		0			0
$153$		−	6.61247e9i		−	0.975556i
$154$		0			0
$155$		−	1.53571e9i		−	0.213706i
$156$		0			0
$157$	−1.03562e10	−	5.97918e9i	−1.36036	−	0.785404i	−0.370687	−	0.928758i	$-0.620878\pi$
$157$	−1.03562e10	−	5.97918e9i	−1.36036	−	0.785404i	−0.989671	+	0.143354i	$0.954211\pi$
$158$		0			0
$159$	−2.07541e9	−	3.59471e9i	−0.257523	−	0.446043i
$160$		0			0
$161$	7.95321e9	−	4.28749e9i	0.932881	−	0.502905i
$162$		0			0
$163$	−1.19142e10	+	6.87866e9i	−1.32197	+	0.763237i	−0.984042	−	0.177937i	$-0.943058\pi$
$163$	−1.19142e10	+	6.87866e9i	−1.32197	+	0.763237i	−0.337923	+	0.941174i	$0.609724\pi$
$164$		0			0
$165$	3.27386e9	−	5.67049e9i	0.343860	−	0.595584i
$166$		0			0
$167$	1.23344e9			0.122714			0.0613570	−	0.998116i	$-0.480457\pi$
$167$	1.23344e9			0.122714			0.0613570	+	0.998116i	$0.480457\pi$
$168$		0			0
$169$	−1.24627e10			−1.17523
$170$		0			0
$171$	−2.94648e9	+	5.10346e9i	−0.263525	+	0.456438i
$172$		0			0
$173$	2.47079e9	−	1.42651e9i	0.209714	−	0.121079i	−0.391464	−	0.920193i	$-0.628031\pi$
$173$	2.47079e9	−	1.42651e9i	0.209714	−	0.121079i	0.601179	+	0.799115i	$0.294698\pi$
$174$		0			0
$175$	−3.50061e9	+	5.67430e9i	−0.282145	+	0.457342i
$176$		0			0
$177$	7.54527e7	+	1.30688e8i	0.00577824	+	0.0100082i
$178$		0			0
$179$	−6.20782e9	−	3.58409e9i	−0.451960	−	0.260939i	0.256697	−	0.966492i	$-0.417366\pi$
$179$	−6.20782e9	−	3.58409e9i	−0.451960	−	0.260939i	−0.708658	+	0.705552i	$0.750699\pi$
$180$		0			0
$181$			1.94335e10i			1.34585i	0.739710	+	0.672926i	$0.234963\pi$
$181$			1.94335e10i			1.34585i	−0.739710	+	0.672926i	$0.765037\pi$
$182$		0			0
$183$			4.20554e9i			0.277199i
$184$		0			0
$185$	−2.24019e10	−	1.29337e10i	−1.40609	−	0.811804i
$186$		0			0
$187$	1.55785e10	+	2.69828e10i	0.931621	+	1.61362i
$188$		0			0
$189$	−1.07840e10	−	3.14674e8i	−0.614755	−	0.0179384i
$190$		0			0
$191$	1.94679e10	−	1.12398e10i	1.05845	−	0.611095i	0.133445	−	0.991056i	$-0.457396\pi$
$191$	1.94679e10	−	1.12398e10i	1.05845	−	0.611095i	0.925002	+	0.379962i	$0.124063\pi$
$192$		0			0
$193$	−2.04218e9	+	3.53717e9i	−0.105947	+	0.183505i	−0.914125	−	0.405433i	$-0.867121\pi$
$193$	−2.04218e9	+	3.53717e9i	−0.105947	+	0.183505i	0.808178	+	0.588938i	$0.200454\pi$
$194$		0			0
$195$	1.19856e10			0.593616
$196$		0			0
$197$	1.76297e10			0.833962			0.416981	−	0.908915i	$-0.363088\pi$
$197$	1.76297e10			0.833962			0.416981	+	0.908915i	$0.363088\pi$
$198$		0			0
$199$	8.57463e9	−	1.48517e10i	0.387594	−	0.671332i	−0.604532	−	0.796581i	$-0.706640\pi$
$199$	8.57463e9	−	1.48517e10i	0.387594	−	0.671332i	0.992125	+	0.125249i	$0.0399730\pi$
$200$		0			0
$201$	−2.10299e9	+	1.21416e9i	−0.0908774	+	0.0524681i
$202$		0			0
$203$	−3.24581e10	−	9.47118e8i	−1.34150	−	0.0391446i
$204$		0			0
$205$	−1.38776e10	−	2.40368e10i	−0.548812	−	0.950570i
$206$		0			0
$207$	−2.16902e10	−	1.25229e10i	−0.821103	−	0.474064i
$208$		0			0
$209$		−	2.77669e10i		−	1.00663i
$210$		0			0
$211$			3.97909e10i			1.38201i	0.722848	+	0.691007i	$0.242833\pi$
$211$			3.97909e10i			1.38201i	−0.722848	+	0.691007i	$0.757167\pi$
$212$		0			0
$213$	−7.32813e9	−	4.23090e9i	−0.243941	−	0.140839i
$214$		0			0
$215$	−2.33303e10	−	4.04092e10i	−0.744640	−	1.28975i
$216$		0			0
$217$	2.95593e9	−	4.79140e9i	0.0904950	−	0.146688i
$218$		0			0
$219$	9.89877e9	−	5.71506e9i	0.290792	−	0.167889i
$220$		0			0
$221$	−2.85166e10	+	4.93922e10i	−0.804142	+	1.39281i
$222$		0			0
$223$	−6.75244e10			−1.82848			−0.914238	−	0.405178i	$-0.867209\pi$
$223$	−6.75244e10			−1.82848			−0.914238	+	0.405178i	$0.867209\pi$
$224$		0			0
$225$	1.84815e10			0.480747
$226$		0			0
$227$	7.84953e9	−	1.35958e10i	0.196213	−	0.339850i	−0.751085	−	0.660206i	$-0.770469\pi$
$227$	7.84953e9	−	1.35958e10i	0.196213	−	0.339850i	0.947297	+	0.320356i	$0.103802\pi$
$228$		0			0
$229$	2.45655e10	−	1.41829e10i	0.590292	−	0.340805i	−0.174921	−	0.984582i	$-0.555967\pi$
$229$	2.45655e10	−	1.41829e10i	0.590292	−	0.340805i	0.765213	+	0.643777i	$0.222634\pi$
$230$		0			0
$231$	2.11290e10	−	1.13904e10i	0.488230	−	0.263199i
$232$		0			0
$233$	1.00851e10	+	1.74679e10i	0.224170	+	0.388274i	0.956070	−	0.293138i	$-0.0946995\pi$
$233$	1.00851e10	+	1.74679e10i	0.224170	+	0.388274i	−0.731900	+	0.681412i	$0.761366\pi$
$234$		0			0
$235$	8.66469e10	+	5.00256e10i	1.85331	+	1.07001i
$236$		0			0
$237$		−	1.73225e10i		−	0.356651i
$238$		0			0
$239$		−	1.43373e10i		−	0.284234i	−0.989850	−	0.142117i	$-0.954609\pi$
$239$		−	1.43373e10i		−	0.284234i	0.989850	−	0.142117i	$-0.0453909\pi$
$240$		0			0
$241$	5.39213e10	+	3.11315e10i	1.02964	+	0.594461i	0.916880	−	0.399162i	$-0.130699\pi$
$241$	5.39213e10	+	3.11315e10i	1.02964	+	0.594461i	0.112756	+	0.993623i	$0.464032\pi$
$242$		0			0
$243$	2.28453e10	+	3.95692e10i	0.420309	+	0.727996i
$244$		0			0
$245$	−6.24931e10	+	3.13723e10i	−1.10811	+	0.556287i
$246$		0			0
$247$	4.40178e10	−	2.54137e10i	0.752475	−	0.434442i
$248$		0			0
$249$	1.37293e10	−	2.37798e10i	0.226335	−	0.392023i
$250$		0			0
$251$	1.58215e10			0.251604			0.125802	−	0.992055i	$-0.459850\pi$
$251$	1.58215e10			0.251604			0.125802	+	0.992055i	$0.459850\pi$
$252$		0			0
$253$	1.18012e11			1.81086
$254$		0			0
$255$	1.48171e10	−	2.56640e10i	0.219448	−	0.380096i
$256$		0			0
$257$	4.52041e10	−	2.60986e10i	0.646367	−	0.373180i	−0.140696	−	0.990053i	$-0.544934\pi$
$257$	4.52041e10	−	2.60986e10i	0.646367	−	0.373180i	0.787063	+	0.616873i	$0.211601\pi$
$258$		0			0
$259$	−4.49989e10	−	8.34723e10i	−0.621374	−	1.15264i
$260$		0			0
$261$	4.50060e10	+	7.79526e10i	0.600327	+	1.03980i
$262$		0			0
$263$	6.84368e10	+	3.95120e10i	0.882041	+	0.509247i	0.871331	−	0.490696i	$-0.163257\pi$
$263$	6.84368e10	+	3.95120e10i	0.882041	+	0.509247i	0.0107103	+	0.999943i	$0.496591\pi$
$264$		0			0
$265$			1.57935e11i			1.96731i
$266$		0			0
$267$		−	1.91198e10i		−	0.230241i
$268$		0			0
$269$	7.01776e10	+	4.05170e10i	0.817171	+	0.471794i	0.849440	−	0.527685i	$-0.176940\pi$
$269$	7.01776e10	+	4.05170e10i	0.817171	+	0.471794i	−0.0322689	+	0.999479i	$0.510273\pi$
$270$		0			0
$271$	−4.69082e10	−	8.12474e10i	−0.528308	−	0.915056i	−0.999455	−	0.0330013i	$-0.989493\pi$
$271$	−4.69082e10	−	8.12474e10i	−0.528308	−	0.915056i	0.471148	−	0.882054i	$-0.343840\pi$
$272$		0			0
$273$	3.73951e10	+	2.30699e10i	0.407458	+	0.251370i
$274$		0			0
$275$	−7.54157e10	+	4.35413e10i	−0.795178	+	0.459096i
$276$		0			0
$277$	1.65232e10	−	2.86190e10i	0.168630	−	0.292076i	−0.769308	−	0.638878i	$-0.779399\pi$
$277$	1.65232e10	−	2.86190e10i	0.168630	−	0.292076i	0.937938	+	0.346802i	$0.112732\pi$
$278$		0			0
$279$	−1.56059e10			−0.154195
$280$		0			0
$281$	−2.97809e10			−0.284944			−0.142472	−	0.989799i	$-0.545505\pi$
$281$	−2.97809e10			−0.284944			−0.142472	+	0.989799i	$0.545505\pi$
$282$		0			0
$283$	2.89160e10	−	5.00840e10i	0.267978	−	0.464152i	−0.700361	−	0.713789i	$-0.746978\pi$
$283$	2.89160e10	−	5.00840e10i	0.267978	−	0.464152i	0.968340	+	0.249636i	$0.0803111\pi$
$284$		0			0
$285$	−2.28715e10	+	1.32049e10i	−0.205349	+	0.118558i
$286$		0			0
$287$	2.96776e9	−	1.01706e11i	0.0258202	−	0.884869i
$288$		0			0
$289$	1.12127e10	+	1.94210e10i	0.0945518	+	0.163768i
$290$		0			0
$291$	−8.60641e9	−	4.96891e9i	−0.0703564	−	0.0406203i
$292$		0			0
$293$			2.52839e10i			0.200420i	0.994966	+	0.100210i	$0.0319514\pi$
$293$			2.52839e10i			0.200420i	−0.994966	+	0.100210i	$0.968049\pi$
$294$		0			0
$295$		−	5.74183e9i		−	0.0441419i
$296$		0			0
$297$	−1.22034e11	−	7.04564e10i	−0.910075	−	0.525432i
$298$		0			0
$299$	1.08011e11	+	1.87080e11i	0.781533	+	1.35366i
$300$		0			0
$301$	4.98922e9	−	1.70982e11i	0.0350335	−	1.20061i
$302$		0			0
$303$	6.59829e10	−	3.80953e10i	0.449717	−	0.259644i
$304$		0			0
$305$	8.00087e10	−	1.38579e11i	0.529405	−	0.916956i
$306$		0			0
$307$	4.28902e10			0.275572			0.137786	−	0.990462i	$-0.456001\pi$
$307$	4.28902e10			0.275572			0.137786	+	0.990462i	$0.456001\pi$
$308$		0			0
$309$	−1.09858e10			−0.0685519
$310$		0			0
$311$	−2.90092e10	+	5.02454e10i	−0.175838	+	0.304561i	−0.940451	−	0.339929i	$-0.889597\pi$
$311$	−2.90092e10	+	5.02454e10i	−0.175838	+	0.304561i	0.764613	+	0.644490i	$0.222930\pi$
$312$		0			0
$313$	4.14519e10	−	2.39323e10i	0.244116	−	0.140940i	−0.372951	−	0.927851i	$-0.621654\pi$
$313$	4.14519e10	−	2.39323e10i	0.244116	−	0.140940i	0.617067	+	0.786911i	$0.288321\pi$
$314$		0			0
$315$	1.64967e11	+	1.01772e11i	0.944058	+	0.582411i
$316$		0			0
$317$	2.01989e10	+	3.49855e10i	0.112347	+	0.194590i	0.916716	−	0.399539i	$-0.130830\pi$
$317$	2.01989e10	+	3.49855e10i	0.112347	+	0.194590i	−0.804369	+	0.594130i	$0.797497\pi$
$318$		0			0
$319$	−3.67302e11	−	2.12062e11i	−1.98594	−	1.14658i
$320$		0			0
$321$			5.30526e10i			0.278891i
$322$		0			0
$323$		−	1.25670e11i		−	0.642420i
$324$		0			0
$325$	−1.38049e11	−	7.97026e10i	−0.686369	−	0.396275i
$326$		0			0
$327$	5.70534e10	+	9.88193e10i	0.275941	+	0.477944i
$328$		0			0
$329$	1.74049e11	+	3.22857e11i	0.819010	+	1.51925i
$330$		0			0
$331$	−6.73267e10	+	3.88711e10i	−0.308291	+	0.177992i	−0.646162	−	0.763201i	$-0.723627\pi$
$331$	−6.73267e10	+	3.88711e10i	−0.308291	+	0.177992i	0.337870	+	0.941193i	$0.390293\pi$
$332$		0			0
$333$	−1.31433e11	+	2.27648e11i	−0.585739	+	1.01453i
$334$		0			0
$335$	9.23959e10			0.400822
$336$		0			0
$337$	−2.55640e11			−1.07968			−0.539840	−	0.841768i	$-0.681515\pi$
$337$	−2.55640e11			−1.07968			−0.539840	+	0.841768i	$0.681515\pi$
$338$		0			0
$339$	−2.19436e10	+	3.80075e10i	−0.0902422	+	0.156304i
$340$		0			0
$341$	6.36812e10	−	3.67664e10i	0.255045	−	0.147250i
$342$		0			0
$343$	−2.55363e11	−	2.24052e10i	−0.996173	−	0.0874026i
$344$		0			0
$345$	−5.61220e10	−	9.72062e10i	−0.213279	−	0.369409i
$346$		0			0
$347$	2.13731e11	+	1.23398e11i	0.791379	+	0.456903i	0.840448	−	0.541893i	$-0.182292\pi$
$347$	2.13731e11	+	1.23398e11i	0.791379	+	0.456903i	−0.0490688	+	0.998795i	$0.515625\pi$
$348$		0			0
$349$		−	3.44751e11i		−	1.24392i	−0.783050	−	0.621959i	$-0.786337\pi$
$349$		−	3.44751e11i		−	1.24392i	0.783050	−	0.621959i	$-0.213663\pi$
$350$		0			0
$351$		−	2.57942e11i		−	0.907068i
$352$		0			0
$353$	−8.14773e10	−	4.70410e10i	−0.279287	−	0.161246i	0.353814	−	0.935316i	$-0.384885\pi$
$353$	−8.14773e10	−	4.70410e10i	−0.279287	−	0.161246i	−0.633101	+	0.774070i	$0.718218\pi$
$354$		0			0
$355$	1.60982e11	+	2.78829e11i	0.537961	+	0.931775i
$356$		0			0
$357$	9.56272e10	−	5.15515e10i	0.311584	−	0.167971i
$358$		0			0
$359$	−4.88810e11	+	2.82215e11i	−1.55316	+	0.896715i	−0.555275	+	0.831667i	$0.687387\pi$
$359$	−4.88810e11	+	2.82215e11i	−1.55316	+	0.896715i	−0.997882	+	0.0650484i	$0.979280\pi$
$360$		0			0
$361$	1.05346e11	−	1.82465e11i	0.326465	−	0.565453i
$362$		0			0
$363$	2.06133e11			0.623114
$364$		0			0
$365$	−4.34907e11			−1.28256
$366$		0			0
$367$	2.22369e11	−	3.85154e11i	0.639848	−	1.10825i	−0.345618	−	0.938375i	$-0.612331\pi$
$367$	2.22369e11	−	3.85154e11i	0.639848	−	1.10825i	0.985466	−	0.169874i	$-0.0543361\pi$
$368$		0			0
$369$	−2.44262e11	+	1.41025e11i	−0.685862	+	0.395983i
$370$		0			0
$371$	−3.03993e11	+	4.92757e11i	−0.833069	+	1.35036i
$372$		0			0
$373$	1.15395e11	+	1.99870e11i	0.308672	+	0.534636i	0.978072	−	0.208266i	$-0.0667821\pi$
$373$	1.15395e11	+	1.99870e11i	0.308672	+	0.534636i	−0.669400	+	0.742902i	$0.733449\pi$
$374$		0			0
$375$	−6.17528e10	−	3.56530e10i	−0.161256	−	0.0931012i
$376$		0			0
$377$		−	7.76362e11i		−	1.97938i
$378$		0			0
$379$			7.54247e11i			1.87775i	0.344263	+	0.938873i	$0.388129\pi$
$379$			7.54247e11i			1.87775i	−0.344263	+	0.938873i	$0.611871\pi$
$380$		0			0
$381$	−1.72130e11	−	9.93796e10i	−0.418500	−	0.241621i
$382$		0			0
$383$	−1.98093e11	−	3.43108e11i	−0.470409	−	0.814772i	0.529019	−	0.848610i	$-0.322560\pi$
$383$	−1.98093e11	−	3.43108e11i	−0.470409	−	0.814772i	−0.999427	+	0.0338384i	$0.989227\pi$
$384$		0			0
$385$	−9.12930e11	−	2.66390e10i	−2.11770	−	0.0617938i
$386$		0			0
$387$	−4.10638e11	+	2.37082e11i	−0.930594	+	0.537279i
$388$		0			0
$389$	−5.07159e10	+	8.78425e10i	−0.112298	+	0.194505i	−0.916696	−	0.399585i	$-0.869154\pi$
$389$	−5.07159e10	+	8.78425e10i	−0.112298	+	0.194505i	0.804399	+	0.594090i	$0.202488\pi$
$390$		0			0
$391$	5.34109e11			1.15567
$392$		0			0
$393$	−3.02886e10			−0.0640491
$394$		0			0
$395$	−3.29554e11	+	5.70804e11i	−0.681145	+	1.17978i
$396$		0			0
$397$	4.39290e11	−	2.53624e11i	0.887552	−	0.512428i	0.0144108	−	0.999896i	$-0.495413\pi$
$397$	4.39290e11	−	2.53624e11i	0.887552	−	0.512428i	0.873141	+	0.487468i	$0.162079\pi$
$398$		0			0
$399$	−9.67755e10	−	2.82388e9i	−0.191156	−	0.00557787i
$400$		0			0
$401$	−2.59688e11	−	4.49793e11i	−0.501537	−	0.868687i	−0.999998	−	0.00177532i	$-0.999435\pi$
$401$	−2.59688e11	−	4.49793e11i	−0.501537	−	0.868687i	0.498462	−	0.866912i	$-0.333898\pi$
$402$		0			0
$403$	1.16569e11	+	6.73011e10i	0.220146	+	0.127101i
$404$		0			0
$405$		−	4.66566e11i		−	0.861719i
$406$		0			0
$407$		−	1.23859e12i		−	2.23744i
$408$		0			0
$409$	7.03601e11	+	4.06224e11i	1.24329	+	0.717813i	0.969762	−	0.244052i	$-0.0784768\pi$
$409$	7.03601e11	+	4.06224e11i	1.24329	+	0.717813i	0.273526	+	0.961865i	$0.411810\pi$
$410$		0			0
$411$	−2.88058e10	−	4.98931e10i	−0.0497956	−	0.0862486i
$412$		0			0
$413$	1.10519e10	−	1.79145e10i	0.0186922	−	0.0302991i
$414$		0			0
$415$	−9.04803e11	+	5.22388e11i	−1.49740	+	0.864524i
$416$		0			0
$417$	6.56427e10	−	1.13697e11i	0.106310	−	0.184134i
$418$		0			0
$419$	2.53960e11			0.402534			0.201267	−	0.979536i	$-0.435494\pi$
$419$	2.53960e11			0.402534			0.201267	+	0.979536i	$0.435494\pi$
$420$		0			0
$421$	−3.17135e11			−0.492011			−0.246006	−	0.969268i	$-0.579118\pi$
$421$	−3.17135e11			−0.492011			−0.246006	+	0.969268i	$0.579118\pi$
$422$		0			0
$423$	5.08360e11	−	8.80505e11i	0.772040	−	1.33721i
$424$		0			0
$425$	−3.41323e11	+	1.97063e11i	−0.507475	+	0.292991i
$426$		0			0
$427$	5.16363e11	−	2.78366e11i	0.751675	−	0.405219i
$428$		0			0
$429$	2.86948e11	+	4.97009e11i	0.409021	+	0.708445i
$430$		0			0
$431$	3.58037e11	+	2.06713e11i	0.499781	+	0.288549i	0.728623	−	0.684915i	$-0.240161\pi$
$431$	3.58037e11	+	2.06713e11i	0.499781	+	0.288549i	−0.228842	+	0.973464i	$0.573494\pi$
$432$		0			0
$433$			7.97087e11i			1.08971i	0.838531	+	0.544854i	$0.183415\pi$
$433$			7.97087e11i			1.08971i	−0.838531	+	0.544854i	$0.816585\pi$
$434$		0			0
$435$			4.03394e11i			0.540167i
$436$		0			0
$437$	−4.12222e11	−	2.37996e11i	−0.540710	−	0.312179i
$438$		0			0
$439$	3.21289e11	+	5.56488e11i	0.412862	+	0.715098i	0.995201	−	0.0978478i	$-0.0311959\pi$
$439$	3.21289e11	+	5.56488e11i	0.412862	+	0.715098i	−0.582339	+	0.812946i	$0.697863\pi$
$440$		0			0
$441$	3.18805e11	+	6.35055e11i	0.401376	+	0.799535i
$442$		0			0
$443$	−6.85528e10	+	3.95790e10i	−0.0845684	+	0.0488256i	−0.541688	−	0.840580i	$-0.682215\pi$
$443$	−6.85528e10	+	3.95790e10i	−0.0845684	+	0.0488256i	0.457119	+	0.889405i	$0.348881\pi$
$444$		0			0
$445$	−3.63747e11	+	6.30029e11i	−0.439724	+	0.761624i
$446$		0			0
$447$	−2.14279e11			−0.253861
$448$		0			0
$449$	2.80684e11			0.325919			0.162959	−	0.986633i	$-0.447896\pi$
$449$	2.80684e11			0.325919			0.162959	+	0.986633i	$0.447896\pi$
$450$		0			0
$451$	6.64489e11	−	1.15093e12i	0.756299	−	1.30995i
$452$		0			0
$453$	8.68690e10	−	5.01538e10i	0.0969221	−	0.0559580i
$454$		0			0
$455$	−7.93332e11	−	1.47162e12i	−0.867768	−	1.60969i
$456$		0			0
$457$	7.41362e11	+	1.28408e12i	0.795074	+	1.37711i	0.922792	+	0.385297i	$0.125901\pi$
$457$	7.41362e11	+	1.28408e12i	0.795074	+	1.37711i	−0.127719	+	0.991810i	$0.540766\pi$
$458$		0			0
$459$	−5.52312e11	−	3.18878e11i	−0.580801	−	0.335326i
$460$		0			0
$461$		−	2.09388e11i		−	0.215923i	−0.994155	−	0.107961i	$-0.965568\pi$
$461$		−	2.09388e11i		−	0.215923i	0.994155	−	0.107961i	$-0.0344323\pi$
$462$		0			0
$463$			5.00270e11i			0.505929i	0.967476	+	0.252965i	$0.0814056\pi$
$463$			5.00270e11i			0.505929i	−0.967476	+	0.252965i	$0.918594\pi$
$464$		0			0
$465$	−6.05687e10	−	3.49694e10i	−0.0600772	−	0.0346856i
$466$		0			0
$467$	9.06637e11	+	1.57034e12i	0.882079	+	1.52781i	0.849025	+	0.528352i	$0.177190\pi$
$467$	9.06637e11	+	1.57034e12i	0.882079	+	1.52781i	0.0330541	+	0.999454i	$0.489477\pi$
$468$		0			0
$469$	2.88275e11	+	1.77843e11i	0.275124	+	0.169731i
$470$		0			0
$471$	−4.71640e11	+	2.72302e11i	−0.441587	+	0.254951i
$472$		0			0
$473$	1.11710e12	−	1.93487e12i	1.02616	−	1.77737i
$474$		0			0
$475$	3.51241e11			0.316580
$476$		0			0
$477$	1.60494e12			1.41947
$478$		0			0
$479$	4.22930e11	−	7.32537e11i	0.367079	−	0.635799i	−0.622029	−	0.782994i	$-0.713691\pi$
$479$	4.22930e11	−	7.32537e11i	0.367079	−	0.635799i	0.989107	+	0.147196i	$0.0470247\pi$
$480$		0			0
$481$	1.96349e12	−	1.13362e12i	1.67254	−	0.965639i
$482$		0			0
$483$	1.20018e10	−	4.11306e11i	0.0100342	−	0.343877i
$484$		0			0
$485$	1.89063e11	+	3.27467e11i	0.155156	+	0.268738i
$486$		0			0
$487$	1.58566e11	+	9.15479e10i	0.127741	+	0.0737511i	0.562509	−	0.826791i	$-0.309836\pi$
$487$	1.58566e11	+	9.15479e10i	0.127741	+	0.0737511i	−0.434768	+	0.900542i	$0.643170\pi$
$488$		0			0
$489$			6.26531e11i			0.495510i
$490$		0			0
$491$		−	1.77766e12i		−	1.38033i	−0.723654	−	0.690163i	$-0.757539\pi$
$491$		−	1.77766e12i		−	1.38033i	0.723654	−	0.690163i	$-0.242461\pi$
$492$		0			0
$493$	−1.66237e12	−	9.59768e11i	−1.26741	−	0.731738i
$494$		0			0
$495$	1.26586e12	+	2.19253e12i	0.947679	+	1.64143i
$496$		0			0
$497$	−3.44264e10	+	1.17980e12i	−0.0253097	+	0.867374i
$498$		0			0
$499$	1.19124e12	−	6.87763e11i	0.860096	−	0.496577i	−0.00394819	−	0.999992i	$-0.501257\pi$
$499$	1.19124e12	−	6.87763e11i	0.860096	−	0.496577i	0.864045	+	0.503415i	$0.167923\pi$
$500$		0			0
$501$	2.80865e10	−	4.86472e10i	0.0199171	−	0.0344975i
$502$		0			0
$503$	−1.43021e12			−0.996194			−0.498097	−	0.867121i	$-0.665968\pi$
$503$	−1.43021e12			−0.996194			−0.498097	+	0.867121i	$0.665968\pi$
$504$		0			0
$505$	−2.89899e12			−1.98351
$506$		0			0
$507$	−2.83787e11	+	4.91533e11i	−0.190746	+	0.330383i
$508$		0			0
$509$	1.22938e12	−	7.09784e11i	0.811814	−	0.468701i	−0.0357713	−	0.999360i	$-0.511389\pi$
$509$	1.22938e12	−	7.09784e11i	0.811814	−	0.468701i	0.847586	+	0.530659i	$0.178055\pi$
$510$		0			0
$511$	−1.35691e12	−	8.37107e11i	−0.880351	−	0.543109i
$512$		0			0
$513$	2.84180e11	+	4.92215e11i	0.181162	+	0.313781i
$514$		0			0
$515$	3.62000e11	+	2.09001e11i	0.226765	+	0.130923i
$516$		0			0
$517$			4.79065e12i			2.94908i
$518$		0			0
$519$		−	1.29931e11i		−	0.0786069i
$520$		0			0
$521$	1.94785e12	+	1.12459e12i	1.15820	+	0.668689i	0.950873	−	0.309582i	$-0.100189\pi$
$521$	1.94785e12	+	1.12459e12i	1.15820	+	0.668689i	0.207330	+	0.978271i	$0.433522\pi$
$522$		0			0
$523$	8.86935e9	+	1.53622e10i	0.00518363	+	0.00897831i	0.868606	−	0.495504i	$-0.165017\pi$
$523$	8.86935e9	+	1.53622e10i	0.00518363	+	0.00897831i	−0.863422	+	0.504482i	$0.831683\pi$
$524$		0			0
$525$	1.44084e11	+	2.67274e11i	0.0827750	+	0.153546i
$526$		0			0
$527$	2.88214e11	−	1.66400e11i	0.162767	−	0.0939738i
$528$		0			0
$529$	1.10933e11	−	1.92142e11i	0.0615901	−	0.106677i
$530$		0			0
$531$	−5.83485e10			−0.0318496
$532$		0			0
$533$	2.43270e12			1.30562
$534$		0			0
$535$	1.00930e12	−	1.74817e12i	0.532635	−	0.922551i
$536$		0			0
$537$	−2.82714e11	+	1.63225e11i	−0.146711	+	0.0847038i
$538$		0			0
$539$	−2.79706e12	−	1.84032e12i	−1.42742	−	0.939168i
$540$		0			0
$541$	1.17269e12	+	2.03116e12i	0.588566	+	1.01943i	0.994420	+	0.105489i	$0.0336407\pi$
$541$	1.17269e12	+	2.03116e12i	0.588566	+	1.01943i	−0.405854	+	0.913938i	$0.633026\pi$
$542$		0			0
$543$	7.66462e11	+	4.42517e11i	0.378348	+	0.218439i
$544$		0			0
$545$		−	4.34167e12i		−	2.10801i
$546$		0			0
$547$			8.78562e11i			0.419594i	0.977745	+	0.209797i	$0.0672803\pi$
$547$			8.78562e11i			0.419594i	−0.977745	+	0.209797i	$0.932720\pi$
$548$		0			0
$549$	−1.40824e12	−	8.13048e11i	−0.661609	−	0.381980i
$550$		0			0
$551$	8.55336e11	+	1.48149e12i	0.395325	+	0.684723i
$552$		0			0
$553$	−2.12689e12	+	1.14658e12i	−0.967123	+	0.521365i
$554$		0			0
$555$	−1.02022e12	+	5.89024e11i	−0.456431	+	0.263521i
$556$		0			0
$557$	2.50713e11	−	4.34247e11i	0.110364	−	0.191156i	−0.805553	−	0.592524i	$-0.798132\pi$
$557$	2.50713e11	−	4.34247e11i	0.110364	−	0.191156i	0.915917	+	0.401368i	$0.131465\pi$
$558$		0			0
$559$	4.08971e12			1.77149
$560$		0			0
$561$	1.41894e12			0.604829
$562$		0			0
$563$	1.45457e12	−	2.51939e12i	0.610164	−	1.05684i	−0.381048	−	0.924555i	$-0.624437\pi$
$563$	1.45457e12	−	2.51939e12i	0.610164	−	1.05684i	0.991212	−	0.132280i	$-0.0422298\pi$
$564$		0			0
$565$	1.44615e12	−	8.34938e11i	0.597031	−	0.344696i
$566$		0			0
$567$	8.98045e11	−	1.45568e12i	0.364900	−	0.591484i
$568$		0			0
$569$	−1.13834e12	−	1.97167e12i	−0.455269	−	0.788549i	0.543435	−	0.839451i	$-0.317124\pi$
$569$	−1.13834e12	−	1.97167e12i	−0.455269	−	0.788549i	−0.998704	+	0.0509025i	$0.983790\pi$
$570$		0			0
$571$	2.77798e12	+	1.60387e12i	1.09362	+	0.631402i	0.934538	−	0.355863i	$-0.115813\pi$
$571$	2.77798e12	+	1.60387e12i	1.09362	+	0.631402i	0.159083	+	0.987265i	$0.449146\pi$
$572$		0			0
$573$		−	1.02376e12i		−	0.396736i
$574$		0			0
$575$			1.49281e12i			0.569507i
$576$		0			0
$577$	−1.10796e12	−	6.39682e11i	−0.416134	−	0.240255i	0.277288	−	0.960787i	$-0.410564\pi$
$577$	−1.10796e12	−	6.39682e11i	−0.416134	−	0.240255i	−0.693422	+	0.720532i	$0.743898\pi$
$578$		0			0
$579$	9.30045e10	+	1.61088e11i	0.0343914	+	0.0595677i
$580$		0			0
$581$	−3.82847e12	−	1.11714e11i	−1.39390	−	0.0406737i
$582$		0			0
$583$	−6.54910e12	+	3.78113e12i	−2.34787	+	1.35554i
$584$		0			0
$585$	−2.31716e12	+	4.01344e12i	−0.818002	+	1.41682i
$586$		0			0
$587$	−2.01269e12			−0.699690			−0.349845	−	0.936808i	$-0.613766\pi$
$587$	−2.01269e12			−0.699690			−0.349845	+	0.936808i	$0.613766\pi$
$588$		0			0
$589$	−2.96589e11			−0.101540
$590$		0			0
$591$	4.01442e11	−	6.95318e11i	0.135357	−	0.234444i
$592$		0			0
$593$	1.10703e12	−	6.39147e11i	0.367633	−	0.212253i	−0.304791	−	0.952419i	$-0.598586\pi$
$593$	1.10703e12	−	6.39147e11i	0.367633	−	0.212253i	0.672424	+	0.740166i	$0.265253\pi$
$594$		0			0
$595$	−4.13181e12	−	1.20565e11i	−1.35149	−	0.0394362i
$596$		0			0
$597$	−3.90503e11	−	6.76371e11i	−0.125817	−	0.217922i
$598$		0			0
$599$	1.08604e12	+	6.27024e11i	0.344687	+	0.199005i	0.662343	−	0.749201i	$-0.269562\pi$
$599$	1.08604e12	+	6.27024e11i	0.344687	+	0.199005i	−0.317656	+	0.948206i	$0.602896\pi$
$600$		0			0
$601$		−	5.17545e12i		−	1.61813i	−0.587720	−	0.809065i	$-0.699974\pi$
$601$		−	5.17545e12i		−	1.61813i	0.587720	−	0.809065i	$-0.300026\pi$
$602$		0			0
$603$		−	9.38928e11i		−	0.289204i
$604$		0			0
$605$	−6.79240e12	−	3.92159e12i	−2.06122	−	1.19005i
$606$		0			0
$607$	8.83052e10	+	1.52949e11i	0.0264020	+	0.0457296i	0.878925	−	0.476961i	$-0.158262\pi$
$607$	8.83052e10	+	1.52949e11i	0.0264020	+	0.0457296i	−0.852523	+	0.522691i	$0.824928\pi$
$608$		0			0
$609$	−7.76452e11	+	1.25859e12i	−0.228737	+	0.370771i
$610$		0			0
$611$	−7.59445e12	+	4.38466e12i	−2.20450	+	1.27277i
$612$		0			0
$613$	4.87539e11	−	8.44443e11i	0.139456	−	0.241545i	−0.787835	−	0.615887i	$-0.788798\pi$
$613$	4.87539e11	−	8.44443e11i	0.139456	−	0.241545i	0.927291	+	0.374342i	$0.122131\pi$
$614$		0			0
$615$	−1.26402e12			−0.356301
$616$		0			0
$617$	−2.89535e12			−0.804300			−0.402150	−	0.915574i	$-0.631737\pi$
$617$	−2.89535e12			−0.804300			−0.402150	+	0.915574i	$0.631737\pi$
$618$		0			0
$619$	2.76348e12	−	4.78649e12i	0.756570	−	1.31042i	−0.188021	−	0.982165i	$-0.560207\pi$
$619$	2.76348e12	−	4.78649e12i	0.756570	−	1.31042i	0.944590	−	0.328252i	$-0.106459\pi$
$620$		0			0
$621$	−2.09197e12	+	1.20780e12i	−0.564472	+	0.325898i
$622$		0			0
$623$	−2.34757e12	+	1.26555e12i	−0.624341	+	0.336575i
$624$		0			0
$625$	2.38152e12	+	4.12492e12i	0.624302	+	1.08132i
$626$		0			0
$627$	−1.09513e12	−	6.32274e11i	−0.282984	−	0.163381i
$628$		0			0
$629$		−	5.60570e12i		−	1.42791i
$630$		0			0
$631$			4.28386e11i			0.107573i	0.998552	+	0.0537864i	$0.0171290\pi$
$631$			4.28386e11i			0.107573i	−0.998552	+	0.0537864i	$0.982871\pi$
$632$		0			0
$633$	1.56936e12	+	9.06071e11i	0.388514	+	0.224308i
$634$		0			0
$635$	3.78131e12	+	6.54943e12i	0.922913	+	1.59853i
$636$		0			0
$637$	3.57373e11	−	6.11844e12i	0.0859991	−	1.47236i
$638$		0			0
$639$	2.83347e12	−	1.63590e12i	0.672301	−	0.388153i
$640$		0			0
$641$	4.05915e11	−	7.03066e11i	0.0949674	−	0.164488i	−0.814628	−	0.579984i	$-0.803059\pi$
$641$	4.05915e11	−	7.03066e11i	0.0949674	−	0.164488i	0.909595	+	0.415496i	$0.136392\pi$
$642$		0			0
$643$	7.07574e12			1.63238			0.816192	−	0.577780i	$-0.196081\pi$
$643$	7.07574e12			1.63238			0.816192	+	0.577780i	$0.196081\pi$
$644$		0			0
$645$	−2.12500e12			−0.483437
$646$		0			0
$647$	−2.31746e12	+	4.01396e12i	−0.519928	+	0.900542i	0.479803	+	0.877376i	$0.340708\pi$
$647$	−2.31746e12	+	4.01396e12i	−0.519928	+	0.900542i	−0.999732	+	0.0231661i	$0.992625\pi$
$648$		0			0
$649$	2.38097e11	−	1.37465e11i	0.0526808	−	0.0304153i
$650$		0			0
$651$	−1.21665e11	−	2.25687e11i	−0.0265492	−	0.0492483i
$652$		0			0
$653$	1.68840e12	+	2.92439e12i	0.363383	+	0.629399i	0.988515	−	0.151121i	$-0.0482882\pi$
$653$	1.68840e12	+	2.92439e12i	0.363383	+	0.629399i	−0.625132	+	0.780519i	$0.714955\pi$
$654$		0			0
$655$	9.98058e11	+	5.76229e11i	0.211870	+	0.122323i
$656$		0			0
$657$			4.41952e12i			0.925404i
$658$		0			0
$659$		−	4.33660e12i		−	0.895705i	−0.894107	−	0.447853i	$-0.852189\pi$
$659$		−	4.33660e12i		−	0.895705i	0.894107	−	0.447853i	$-0.147811\pi$
$660$		0			0
$661$	6.25703e11	+	3.61250e11i	0.127486	+	0.0736039i	0.562387	−	0.826874i	$-0.309883\pi$
$661$	6.25703e11	+	3.61250e11i	0.127486	+	0.0736039i	−0.434901	+	0.900478i	$0.643217\pi$
$662$		0			0
$663$	1.29869e12	+	2.24940e12i	0.261033	+	0.452123i
$664$		0			0
$665$	3.13518e12	+	1.93417e12i	0.621678	+	0.383527i
$666$		0			0
$667$	−6.29647e12	+	3.63527e12i	−1.23177	+	0.711165i
$668$		0			0
$669$	−1.53759e12	+	2.66318e12i	−0.296772	+	0.514024i
$670$		0			0
$671$	7.66195e12			1.45911
$672$		0			0
$673$	−5.83276e12			−1.09599			−0.547995	−	0.836482i	$-0.684609\pi$
$673$	−5.83276e12			−1.09599			−0.547995	+	0.836482i	$0.684609\pi$
$674$		0			0
$675$	8.91248e11	−	1.54369e12i	0.165246	−	0.286215i
$676$		0			0
$677$	7.68759e12	−	4.43843e12i	1.40650	−	0.812046i	0.411455	−	0.911430i	$-0.365021\pi$
$677$	7.68759e12	−	4.43843e12i	1.40650	−	0.812046i	0.995049	+	0.0993845i	$0.0316874\pi$
$678$		0			0
$679$	−4.04315e10	+	1.38560e12i	−0.00729972	+	0.250164i
$680$		0			0
$681$	−3.57480e11	−	6.19174e11i	−0.0636928	−	0.110319i
$682$		0			0
$683$	−8.48026e12	−	4.89608e12i	−1.49113	−	0.860906i	−0.491184	−	0.871056i	$-0.663436\pi$
$683$	−8.48026e12	−	4.89608e12i	−1.49113	−	0.860906i	−0.999948	+	0.0101504i	$0.996769\pi$
$684$		0			0
$685$			2.19207e12i			0.380406i
$686$		0			0
$687$		−	1.29183e12i		−	0.221258i
$688$		0			0
$689$	−1.19882e13	−	6.92137e12i	−2.02659	−	1.17005i
$690$		0			0
$691$	1.95856e12	+	3.39232e12i	0.326802	+	0.566038i	0.981875	−	0.189528i	$-0.0606956\pi$
$691$	1.95856e12	+	3.39232e12i	0.326802	+	0.566038i	−0.655073	+	0.755565i	$0.727362\pi$
$692$		0			0
$693$	−2.70706e11	+	9.27719e12i	−0.0445860	+	1.52798i
$694$		0			0
$695$	−4.32606e12	+	2.49765e12i	−0.703333	+	0.406070i
$696$		0			0
$697$	3.00740e12	−	5.20897e12i	0.482663	−	0.835997i
$698$		0			0
$699$	9.18583e11			0.145536
$700$		0			0
$701$	−1.21201e12			−0.189572			−0.0947860	−	0.995498i	$-0.530217\pi$
$701$	−1.21201e12			−0.189572			−0.0947860	+	0.995498i	$0.530217\pi$
$702$		0			0
$703$	−2.49787e12	+	4.32644e12i	−0.385719	+	0.668085i
$704$		0			0
$705$	3.94604e12	−	2.27825e12i	0.601604	−	0.347336i
$706$		0			0
$707$	−9.04482e12	−	5.57996e12i	−1.36148	−	0.839932i
$708$		0			0
$709$	3.50414e12	+	6.06935e12i	0.520803	+	0.902058i	0.999707	+	0.0241906i	$0.00770085\pi$
$709$	3.50414e12	+	6.06935e12i	0.520803	+	0.902058i	−0.478904	+	0.877867i	$0.658966\pi$
$710$		0			0
$711$	5.80051e12	+	3.34893e12i	0.851243	+	0.491465i
$712$		0			0
$713$		−	1.26053e12i		−	0.182663i
$714$		0			0
$715$		−	2.18363e13i		−	3.12465i
$716$		0			0
$717$	−5.65465e11	−	3.26472e11i	−0.0799042	−	0.0461327i
$718$		0			0
$719$	4.05381e12	+	7.02140e12i	0.565696	+	0.979814i	0.996985	+	0.0776001i	$0.0247257\pi$
$719$	4.05381e12	+	7.02140e12i	0.565696	+	0.979814i	−0.431289	+	0.902214i	$0.641941\pi$
$720$		0			0
$721$	7.27154e11	+	1.34886e12i	0.100212	+	0.185891i
$722$		0			0
$723$	2.45567e12	−	1.41778e12i	0.334231	−	0.192968i
$724$		0			0
$725$	2.68251e12	−	4.64624e12i	0.360595	−	0.624569i
$726$		0			0
$727$	3.54300e12			0.470399			0.235200	−	0.971947i	$-0.424426\pi$
$727$	3.54300e12			0.470399			0.235200	+	0.971947i	$0.424426\pi$
$728$		0			0
$729$	−3.21886e12			−0.422113
$730$		0			0
$731$	5.05586e12	−	8.75701e12i	0.654888	−	1.13430i
$732$		0			0
$733$	−1.19145e12	+	6.87884e11i	−0.152443	+	0.0880131i	−0.574281	−	0.818658i	$-0.694718\pi$
$733$	−1.19145e12	+	6.87884e11i	−0.152443	+	0.0880131i	0.421838	+	0.906671i	$0.361385\pi$
$734$		0			0
$735$	−1.85689e11	+	3.17911e12i	−0.0234690	+	0.401803i
$736$		0			0
$737$	2.21205e12	+	3.83138e12i	0.276179	+	0.478357i
$738$		0			0
$739$	−1.24391e12	−	7.18171e11i	−0.153422	−	0.0885784i	0.421323	−	0.906910i	$-0.361566\pi$
$739$	−1.24391e12	−	7.18171e11i	−0.153422	−	0.0885784i	−0.574746	+	0.818332i	$0.694899\pi$
$740$		0			0
$741$		−	2.31476e12i		−	0.282049i
$742$		0			0
$743$		−	3.46371e12i		−	0.416958i	−0.978027	−	0.208479i	$-0.933149\pi$
$743$		−	3.46371e12i		−	0.416958i	0.978027	−	0.208479i	$-0.0668513\pi$
$744$		0			0
$745$	7.06082e12	+	4.07657e12i	0.839754	+	0.484832i
$746$		0			0
$747$	5.30851e12	+	9.19461e12i	0.623778	+	1.08042i
$748$		0			0
$749$	6.51389e12	−	3.51156e12i	0.756261	−	0.407692i
$750$		0			0
$751$	−1.03546e13	+	5.97822e12i	−1.18782	+	0.685791i	−0.957812	−	0.287396i	$-0.907210\pi$
$751$	−1.03546e13	+	5.97822e12i	−1.18782	+	0.685791i	−0.230013	+	0.973187i	$0.573877\pi$
$752$		0			0
$753$	3.60269e11	−	6.24005e11i	0.0408366	−	0.0707311i
$754$		0			0
$755$	−3.81662e12			−0.427482
$756$		0			0
$757$	1.36803e12			0.151413			0.0757066	−	0.997130i	$-0.475879\pi$
$757$	1.36803e12			0.151413			0.0757066	+	0.997130i	$0.475879\pi$
$758$		0			0
$759$	2.68723e12	−	4.65443e12i	0.293912	−	0.509071i
$760$		0			0
$761$	3.93155e12	−	2.26988e12i	0.424945	−	0.245342i	−0.272246	−	0.962228i	$-0.587766\pi$
$761$	3.93155e12	−	2.26988e12i	0.424945	−	0.245342i	0.697191	+	0.716886i	$0.254433\pi$
$762$		0			0
$763$	8.35683e12	−	1.35460e13i	0.892650	−	1.44694i
$764$		0			0
$765$	5.72912e12	+	9.92313e12i	0.604800	+	1.04754i
$766$		0			0
$767$	4.35837e11	+	2.51631e11i	0.0454721	+	0.0262534i
$768$		0			0
$769$			8.44287e12i			0.870606i	0.900284	+	0.435303i	$0.143359\pi$
$769$			8.44287e12i			0.870606i	−0.900284	+	0.435303i	$0.856641\pi$
$770$		0			0
$771$		−	2.37715e12i		−	0.242277i
$772$		0			0
$773$	5.59408e12	+	3.22975e12i	0.563536	+	0.325357i	0.754563	−	0.656227i	$-0.227849\pi$
$773$	5.59408e12	+	3.22975e12i	0.563536	+	0.325357i	−0.191028	+	0.981585i	$0.561182\pi$
$774$		0			0
$775$	4.65081e11	+	8.05544e11i	0.0463096	+	0.0802106i
$776$		0			0
$777$	−4.31683e12	−	1.25964e11i	−0.424884	−	0.0123980i
$778$		0			0
$779$	−4.64218e12	+	2.68017e12i	−0.451652	+	0.260761i
$780$		0			0
$781$	−7.70815e12	+	1.33509e13i	−0.741345	+	1.28405i
$782$		0			0
$783$	8.68142e12			0.825397
$784$		0			0
$785$	2.07217e13			1.94766
$786$		0			0
$787$	4.92098e12	−	8.52338e12i	0.457262	−	0.792001i	−0.541553	−	0.840666i	$-0.682164\pi$
$787$	4.92098e12	−	8.52338e12i	0.457262	−	0.792001i	0.998815	+	0.0486657i	$0.0154969\pi$
$788$		0			0
$789$	3.11673e12	−	1.79944e12i	0.286320	−	0.165307i
$790$		0			0
$791$	6.11908e12	+	1.78553e11i	0.555766	+	0.0162171i
$792$		0			0
$793$	7.01262e12	+	1.21462e13i	0.629725	+	1.09072i
$794$		0			0
$795$	6.22900e12	+	3.59632e12i	0.553052	+	0.319305i
$796$		0			0
$797$		−	7.77271e12i		−	0.682355i	−0.939999	−	0.341177i	$-0.889174\pi$
$797$		−	7.77271e12i		−	0.682355i	0.939999	−	0.341177i	$-0.110826\pi$
$798$		0			0
$799$			2.16819e13i			1.88208i
$800$		0			0
$801$	6.40235e12	+	3.69640e12i	0.549532	+	0.317273i
$802$		0			0
$803$	−1.04121e13	−	1.80343e13i	−0.883727	−	1.53066i
$804$		0			0
$805$	−8.22041e12	+	1.33249e13i	−0.689941	+	1.11836i
$806$		0			0
$807$	3.19600e12	−	1.84521e12i	0.265263	−	0.153149i
$808$		0			0
$809$	4.59800e12	−	7.96397e12i	0.377399	−	0.653674i	−0.613284	−	0.789863i	$-0.710152\pi$
$809$	4.59800e12	−	7.96397e12i	0.377399	−	0.653674i	0.990683	+	0.136188i	$0.0434852\pi$
$810$		0			0
$811$	6.84596e12			0.555700			0.277850	−	0.960624i	$-0.410378\pi$
$811$	6.84596e12			0.555700			0.277850	+	0.960624i	$0.410378\pi$
$812$		0			0
$813$	−4.27255e12			−0.342989
$814$		0			0
$815$	1.19195e13	−	2.06452e13i	0.946343	−	1.63911i
$816$		0			0
$817$	−7.80416e12	+	4.50573e12i	−0.612811	+	0.353807i
$818$		0			0
$819$	−1.49546e13	+	8.06184e12i	−1.16144	+	0.626119i
$820$		0			0
$821$	−7.57450e12	−	1.31194e13i	−0.581848	−	1.00779i	−0.995260	−	0.0972467i	$-0.968996\pi$
$821$	−7.57450e12	−	1.31194e13i	−0.581848	−	1.00779i	0.413412	−	0.910544i	$-0.364337\pi$
$822$		0			0
$823$	1.68348e13	+	9.71957e12i	1.27911	+	0.738496i	0.976685	−	0.214677i	$-0.0688699\pi$
$823$	1.68348e13	+	9.71957e12i	1.27911	+	0.738496i	0.302427	+	0.953173i	$0.402203\pi$
$824$		0			0
$825$			3.96588e12i			0.298056i
$826$		0			0
$827$			1.88160e13i			1.39879i	0.714734	+	0.699396i	$0.246548\pi$
$827$			1.88160e13i			1.39879i	−0.714734	+	0.699396i	$0.753452\pi$
$828$		0			0
$829$	1.46989e13	+	8.48642e12i	1.08091	+	0.624064i	0.931142	−	0.364657i	$-0.118814\pi$
$829$	1.46989e13	+	8.48642e12i	1.08091	+	0.624064i	0.149769	+	0.988721i	$0.452147\pi$
$830$		0			0
$831$	−7.52493e11	−	1.30336e12i	−0.0547392	−	0.0948110i
$832$		0			0
$833$	−1.26592e13	−	8.32906e12i	−0.910967	−	0.599368i
$834$		0			0
$835$	−1.85099e12	+	1.06867e12i	−0.131769	+	0.0760770i
$836$		0			0
$837$	−7.52573e11	+	1.30349e12i	−0.0530010	+	0.0918004i
$838$		0			0
$839$	−2.77217e13			−1.93148			−0.965740	−	0.259511i	$-0.916439\pi$
$839$	−2.77217e13			−1.93148			−0.965740	+	0.259511i	$0.916439\pi$
$840$		0			0
$841$	1.16225e13			0.801155
$842$		0			0
$843$	−6.78136e11	+	1.17457e12i	−0.0462479	+	0.0801038i
$844$		0			0
$845$	1.87025e13	−	1.07979e13i	1.26195	−	0.728589i
$846$		0			0
$847$	−1.36440e13	−	2.53093e13i	−0.910889	−	1.68968i
$848$		0			0
$849$	−1.31688e12	−	2.28091e12i	−0.0869887	−	0.150669i
$850$		0			0
$851$	−1.83878e13	−	1.06162e13i	−1.20184	−	0.693884i
$852$		0			0
$853$		−	1.50000e13i		−	0.970108i	−0.874484	−	0.485054i	$-0.838800\pi$
$853$		−	1.50000e13i		−	0.970108i	0.874484	−	0.485054i	$-0.161200\pi$
$854$		0			0
$855$		−	1.02115e13i		−	0.653493i
$856$		0			0
$857$	−1.66520e13	−	9.61406e12i	−1.05452	−	0.608826i	−0.130607	−	0.991434i	$-0.541692\pi$
$857$	−1.66520e13	−	9.61406e12i	−1.05452	−	0.608826i	−0.923911	+	0.382609i	$0.875026\pi$
$858$		0			0
$859$	9.45709e12	+	1.63802e13i	0.592636	+	1.02648i	0.993876	+	0.110503i	$0.0352462\pi$
$859$	9.45709e12	+	1.63802e13i	0.592636	+	1.02648i	−0.401240	+	0.915973i	$0.631421\pi$
$860$		0			0
$861$	−3.94374e12	−	2.43299e12i	−0.244565	−	0.150878i
$862$		0			0
$863$	−8.63920e12	+	4.98784e12i	−0.530182	+	0.306101i	−0.741091	−	0.671405i	$-0.765691\pi$
$863$	−8.63920e12	+	4.98784e12i	−0.530182	+	0.306101i	0.210909	+	0.977506i	$0.432358\pi$
$864$		0			0
$865$	−2.47189e12	+	4.28144e12i	−0.150126	+	0.260026i
$866$		0			0
$867$	1.02129e12			0.0613851
$868$		0			0
$869$	−3.15594e13			−1.87733
$870$		0			0
$871$	−4.04917e12	+	7.01337e12i	−0.238388	+	0.412900i
$872$		0			0
$873$	3.32772e12	−	1.92126e12i	0.193902	−	0.111950i
$874$		0			0
$875$	−2.90104e11	+	9.94199e12i	−0.0167309	+	0.573373i
$876$		0			0
$877$	−7.30706e12	−	1.26562e13i	−0.417104	−	0.722446i	0.578543	−	0.815652i	$-0.303622\pi$
$877$	−7.30706e12	−	1.26562e13i	−0.417104	−	0.722446i	−0.995647	+	0.0932065i	$0.970288\pi$
$878$		0			0
$879$	9.97204e11	+	5.75736e11i	0.0563422	+	0.0325292i
$880$		0			0
$881$		−	1.00111e13i		−	0.559873i	−0.960018	−	0.279937i	$-0.909687\pi$
$881$		−	1.00111e13i		−	0.559873i	0.960018	−	0.279937i	$-0.0903134\pi$
$882$		0			0
$883$			1.93746e13i			1.07253i	0.844050	+	0.536264i	$0.180165\pi$
$883$			1.93746e13i			1.07253i	−0.844050	+	0.536264i	$0.819835\pi$
$884$		0			0
$885$	−2.26459e11	−	1.30746e11i	−0.0124092	−	0.00716448i
$886$		0			0
$887$	1.02114e13	+	1.76866e13i	0.553895	+	0.959375i	0.997989	+	0.0633944i	$0.0201926\pi$
$887$	1.02114e13	+	1.76866e13i	0.553895	+	0.959375i	−0.444093	+	0.895981i	$0.646474\pi$
$888$		0			0
$889$	−8.08641e11	+	2.77124e13i	−0.0434208	+	1.48805i
$890$		0			0
$891$	1.93471e13	−	1.11701e13i	1.02841	−	0.593753i
$892$		0			0
$893$	9.66135e12	−	1.67340e13i	0.508401	−	0.880576i
$894$		0			0
$895$	1.24212e13			0.647082
$896$		0			0
$897$	9.83800e12			0.507389
$898$		0			0
$899$	−2.26512e12	+	3.92330e12i	−0.115657	+	0.200324i
$900$		0			0
$901$	−2.96405e13	+	1.71129e13i	−1.49839	+	0.865093i
$902$		0			0
$903$	−6.62998e12	−	4.09019e12i	−0.331831	−	0.204714i
$904$		0			0
$905$	−1.68374e13	−	2.91633e13i	−0.834366	−	1.44516i
$906$		0			0
$907$	2.90893e13	+	1.67947e13i	1.42725	+	0.824023i	0.996903	−	0.0786409i	$-0.0250580\pi$
$907$	2.90893e13	+	1.67947e13i	1.42725	+	0.824023i	0.430347	+	0.902664i	$0.358391\pi$
$908$		0			0
$909$			2.94595e13i			1.43116i
$910$		0			0
$911$		−	2.84450e12i		−	0.136828i	−0.997657	−	0.0684138i	$-0.978206\pi$
$911$		−	2.84450e12i		−	0.136828i	0.997657	−	0.0684138i	$-0.0217938\pi$
$912$		0			0
$913$	−4.33238e13	−	2.50130e13i	−2.06352	−	1.19137i
$914$		0			0
$915$	−3.64373e12	−	6.31112e12i	−0.171851	−	0.297654i
$916$		0			0
$917$	2.00481e12	+	3.71889e12i	0.0936292	+	0.173681i
$918$		0			0
$919$	4.76309e12	−	2.74997e12i	0.220277	−	0.127177i	−0.385802	−	0.922582i	$-0.626075\pi$
$919$	4.76309e12	−	2.74997e12i	0.220277	−	0.127177i	0.606078	+	0.795405i	$0.292742\pi$
$920$		0			0
$921$	9.76645e11	−	1.69160e12i	0.0447269	−	0.0774692i
$922$		0			0
$923$	−2.82196e13			−1.27980
$924$		0			0
$925$	1.56677e13			0.703666
$926$		0			0
$927$	2.12387e12	−	3.67865e12i	0.0944645	−	0.163617i
$928$		0			0
$929$	9.90845e12	−	5.72064e12i	0.436450	−	0.251985i	−0.265640	−	0.964072i	$-0.585583\pi$
$929$	9.90845e12	−	5.72064e12i	0.436450	−	0.251985i	0.702091	+	0.712087i	$0.252250\pi$
$930$		0			0
$931$	6.05887e12	+	1.20692e13i	0.264313	+	0.526507i
$932$		0			0
$933$	1.32113e12	+	2.28826e12i	0.0570791	+	0.0988639i
$934$		0			0
$935$	−4.67565e13	−	2.69949e13i	−2.00073	−	1.15512i
$936$		0			0
$937$			6.35103e12i			0.269163i	0.990902	+	0.134582i	$0.0429691\pi$
$937$			6.35103e12i			0.269163i	−0.990902	+	0.134582i	$0.957031\pi$
$938$		0			0
$939$		−	2.17983e12i		−	0.0915015i
$940$		0			0
$941$	6.71682e12	+	3.87796e12i	0.279261	+	0.161231i	0.633089	−	0.774079i	$-0.281787\pi$
$941$	6.71682e12	+	3.87796e12i	0.279261	+	0.161231i	−0.353828	+	0.935311i	$0.615120\pi$
$942$		0			0
$943$	−1.13910e13	−	1.97298e13i	−0.469093	−	0.812493i
$944$		0			0
$945$	1.64559e13	−	8.87118e12i	0.671240	−	0.361858i
$946$		0			0
$947$	2.18625e13	−	1.26223e13i	0.883334	−	0.509993i	0.0115777	−	0.999933i	$-0.496315\pi$
$947$	2.18625e13	−	1.26223e13i	0.883334	−	0.509993i	0.871756	+	0.489940i	$0.162981\pi$
$948$		0			0
$949$	1.90594e13	−	3.30119e13i	0.762802	−	1.32121i
$950$		0			0
$951$	1.83978e12			0.0729380
$952$		0			0
$953$	−2.68632e13			−1.05497			−0.527484	−	0.849565i	$-0.676865\pi$
$953$	−2.68632e13			−1.05497			−0.527484	+	0.849565i	$0.676865\pi$
$954$		0			0
$955$	−1.94766e13	+	3.37345e13i	−0.757701	+	1.31238i
$956$		0			0
$957$	−1.67276e13	+	9.65766e12i	−0.644657	+	0.372193i
$958$		0			0
$959$	−4.21930e12	+	6.83926e12i	−0.161085	+	0.261111i
$960$		0			0
$961$	1.28271e13	+	2.22172e13i	0.485147	+	0.840299i
$962$		0			0
$963$	−1.77649e13	−	1.02565e13i	−0.665646	−	0.384311i
$964$		0			0
$965$		−	7.07749e12i		−	0.262728i
$966$		0			0
$967$			2.19533e12i			0.0807385i	0.999185	+	0.0403692i	$0.0128534\pi$
$967$			2.19533e12i			0.0807385i	−0.999185	+	0.0403692i	$0.987147\pi$
$968$		0			0
$969$	−4.95644e12	−	2.86160e12i	−0.180598	−	0.104268i
$970$		0			0
$971$	1.41273e13	+	2.44693e13i	0.510005	+	0.883354i	0.999933	+	0.0115912i	$0.00368967\pi$
$971$	1.41273e13	+	2.44693e13i	0.510005	+	0.883354i	−0.489928	+	0.871763i	$0.662977\pi$
$972$		0			0
$973$	−1.83048e13	−	5.34128e11i	−0.654721	−	0.0191046i
$974$		0			0
$975$	−6.28698e12	+	3.62979e12i	−0.222803	+	0.128635i
$976$		0			0
$977$	−1.27634e13	+	2.21068e13i	−0.448166	+	0.776247i	−0.998267	−	0.0588517i	$-0.981256\pi$
$977$	−1.27634e13	+	2.21068e13i	−0.448166	+	0.776247i	0.550100	+	0.835098i	$0.314589\pi$
$978$		0			0
$979$	−3.48339e13			−1.21194
$980$		0			0
$981$	−4.41201e13			−1.52099
$982$		0			0
$983$	−8.44092e12	+	1.46201e13i	−0.288336	+	0.499413i	−0.973413	−	0.229058i	$-0.926435\pi$
$983$	−8.44092e12	+	1.46201e13i	−0.288336	+	0.499413i	0.685077	+	0.728471i	$0.259769\pi$
$984$		0			0
$985$	−2.64563e13	+	1.52746e13i	−0.895501	+	0.517018i
$986$		0			0
$987$	1.66968e13	+	4.87208e11i	0.560023	+	0.0163413i
$988$		0			0
$989$	−1.91498e13	−	3.31685e13i	−0.636476	−	1.10241i
$990$		0			0
$991$	1.80360e13	+	1.04131e13i	0.594032	+	0.342965i	0.766690	−	0.642017i	$-0.221902\pi$
$991$	1.80360e13	+	1.04131e13i	0.594032	+	0.342965i	−0.172658	+	0.984982i	$0.555236\pi$
$992$		0			0
$993$			3.54051e12i			0.115556i
$994$		0			0
$995$			2.97167e13i			0.961161i
$996$		0			0
$997$	6.71761e12	+	3.87841e12i	0.215321	+	0.124316i	0.603782	−	0.797150i	$-0.293660\pi$
$997$	6.71761e12	+	3.87841e12i	0.215321	+	0.124316i	−0.388461	+	0.921465i	$0.626993\pi$
$998$		0			0
$999$	1.26763e13	+	2.19561e13i	0.402670	+	0.697444i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.a.31.8	✓	24
4.3	odd	2		112.10.p.c.31.5	yes	24
7.5	odd	6		112.10.p.c.47.5	yes	24
28.19	even	6	inner	112.10.p.a.47.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.a.31.8	✓	24	1.1	even	1	trivial
112.10.p.a.47.8	yes	24	28.19	even	6	inner
112.10.p.c.31.5	yes	24	4.3	odd	2
112.10.p.c.47.5	yes	24	7.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(22.7708 - 39.4402i) q^{3} +(-1500.67 + 866.412i) q^{5} +(-6349.75 - 185.284i) q^{7} +(8804.48 + 15249.8i) q^{9} +(-71855.0 - 41485.5i) q^{11} -151879. i q^{13} +78915.7i q^{15} +(-325208. - 187759. i) q^{17} +(167329. + 289822. i) q^{19} +(-151897. + 246216. i) q^{21} +(-1.23177e6 + 711164. i) q^{23} +(524777. - 908940. i) q^{25} +1.69834e6 q^{27} +5.11171e6 q^{29} +(-443123. + 767512. i) q^{31} +(-3.27240e6 + 1.88932e6i) q^{33} +(9.68940e6 - 5.22345e6i) q^{35} +(7.46397e6 + 1.29280e7i) q^{37} +(-5.99014e6 - 3.45841e6i) q^{39} +1.60174e7i q^{41} +2.69275e7i q^{43} +(-2.64252e7 - 1.52566e7i) q^{45} +(-2.88694e7 - 5.00033e7i) q^{47} +(4.02849e7 + 2.35301e6i) q^{49} +(-1.48105e7 + 8.55084e6i) q^{51} +(4.55716e7 - 7.89324e7i) q^{53} +1.43774e8 q^{55} +1.52408e7 q^{57} +(-1.65678e6 + 2.86964e6i) q^{59} +(-7.99730e7 + 4.61724e7i) q^{61} +(-5.30807e7 - 9.84637e7i) q^{63} +(1.31590e8 + 2.27920e8i) q^{65} +(-4.61774e7 - 2.66605e7i) q^{67} +6.47752e7i q^{69} -1.85803e8i q^{71} +(2.17356e8 + 1.25491e8i) q^{73} +(-2.38992e7 - 4.13946e7i) q^{75} +(4.48575e8 + 2.76736e8i) q^{77} +(3.29407e8 - 1.90183e8i) q^{79} +(-1.34626e8 + 2.33179e8i) q^{81} +6.02933e8 q^{83} +6.50705e8 q^{85} +(1.16398e8 - 2.01607e8i) q^{87} +(3.63585e8 - 2.09916e8i) q^{89} +(-2.81407e7 + 9.64393e8i) q^{91} +(2.01806e7 + 3.49538e7i) q^{93} +(-5.02210e8 - 2.89951e8i) q^{95} -2.18214e8i q^{97} -1.46103e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) \) 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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