LMFDB - Embedded newform 2142.2.p.f.1891.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2142,2,Mod(1135,2142)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2142, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2142.1135"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2142.p (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-12,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.1039561130$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 36x^{10} + 450x^{8} + 2400x^{6} + 5385x^{4} + 4428x^{2} + 1156 $ x^12 + 36x^10 + 450x^8 + 2400x^6 + 5385x^4 + 4428*x^2 + 1156
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 714)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1891.3
Root			$3.11211i$ of defining polynomial
Character	$\chi$	$=$	2142.1891
Dual form			2142.2.p.f.1135.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +(-0.240016 + 0.240016i) q^{5} +(0.707107 + 0.707107i) q^{7} +1.00000i q^{8} +(0.240016 + 0.240016i) q^{10} +(-1.93791 - 1.93791i) q^{11} +1.80644 q^{13} +(0.707107 - 0.707107i) q^{14} +1.00000 q^{16} +(-0.467090 - 4.09656i) q^{17} +1.82303i q^{19} +(0.240016 - 0.240016i) q^{20} +(-1.93791 + 1.93791i) q^{22} +(2.41421 + 2.41421i) q^{23} +4.88478i q^{25} -1.80644i q^{26} +(-0.707107 - 0.707107i) q^{28} +(5.19217 - 5.19217i) q^{29} +(2.39223 - 2.39223i) q^{31} -1.00000i q^{32} +(-4.09656 + 0.467090i) q^{34} -0.339434 q^{35} +(-3.69156 + 3.69156i) q^{37} +1.82303 q^{38} +(-0.240016 - 0.240016i) q^{40} +(-3.56009 - 3.56009i) q^{41} -5.52893i q^{43} +(1.93791 + 1.93791i) q^{44} +(2.41421 - 2.41421i) q^{46} +4.30442 q^{47} +1.00000i q^{49} +4.88478 q^{50} -1.80644 q^{52} -8.11455i q^{53} +0.930261 q^{55} +(-0.707107 + 0.707107i) q^{56} +(-5.19217 - 5.19217i) q^{58} +2.78067i q^{59} +(0.162466 + 0.162466i) q^{61} +(-2.39223 - 2.39223i) q^{62} -1.00000 q^{64} +(-0.433575 + 0.433575i) q^{65} +9.62930 q^{67} +(0.467090 + 4.09656i) q^{68} +0.339434i q^{70} +(10.8682 - 10.8682i) q^{71} +(5.33650 - 5.33650i) q^{73} +(3.69156 + 3.69156i) q^{74} -1.82303i q^{76} -2.74062i q^{77} +(-1.30165 - 1.30165i) q^{79} +(-0.240016 + 0.240016i) q^{80} +(-3.56009 + 3.56009i) q^{82} -3.31585i q^{83} +(1.09535 + 0.871133i) q^{85} -5.52893 q^{86} +(1.93791 - 1.93791i) q^{88} +5.45410 q^{89} +(1.27735 + 1.27735i) q^{91} +(-2.41421 - 2.41421i) q^{92} -4.30442i q^{94} +(-0.437557 - 0.437557i) q^{95} +(9.53533 - 9.53533i) q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} - 12 q^{13} + 12 q^{16} + 12 q^{23} - 12 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} - 12 q^{37} + 12 q^{46} + 24 q^{47} - 24 q^{50} + 12 q^{52} + 12 q^{55} + 12 q^{58} - 12 q^{61} - 12 q^{62}+ \cdots + 12 q^{98}+O(q^{100}) $ 12 * q - 12 * q^4 - 12 * q^13 + 12 * q^16 + 12 * q^23 - 12 * q^29 + 12 * q^31 + 12 * q^34 - 12 * q^35 - 12 * q^37 + 12 * q^46 + 24 * q^47 - 24 * q^50 + 12 * q^52 + 12 * q^55 + 12 * q^58 - 12 * q^61 - 12 * q^62 - 12 * q^64 - 36 * q^65 - 12 * q^67 + 36 * q^71 - 36 * q^73 + 12 * q^74 + 48 * q^79 - 48 * q^85 - 36 * q^86 - 12 * q^89 - 12 * q^92 + 12 * q^95 + 36 * q^97 + 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2142\mathbb{Z}\right)^\times$.

$n$	$1261$	$1667$	$1837$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−0.240016	+	0.240016i	−0.107339	+	0.107339i	−0.758736	−	0.651398i	$-0.774183\pi$
$5$	−0.240016	+	0.240016i	−0.107339	+	0.107339i	0.651398	+	0.758736i	$0.274183\pi$
$6$		0			0
$7$	0.707107	+	0.707107i	0.267261	+	0.267261i
$7$	0.707107	+	0.707107i	0.267261	+	0.267261i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	0.240016	+	0.240016i	0.0758999	+	0.0758999i
$11$	−1.93791	−	1.93791i	−0.584303	−	0.584303i	0.351780	−	0.936083i	$-0.385576\pi$
$11$	−1.93791	−	1.93791i	−0.584303	−	0.584303i	−0.936083	+	0.351780i	$0.885576\pi$
$12$		0			0
$13$	1.80644			0.501017			0.250508	−	0.968114i	$-0.419402\pi$
$13$	1.80644			0.501017			0.250508	+	0.968114i	$0.419402\pi$
$14$	0.707107	−	0.707107i	0.188982	−	0.188982i
$15$		0			0
$16$	1.00000			0.250000
$17$	−0.467090	−	4.09656i	−0.113286	−	0.993562i
$17$	−0.467090	−	4.09656i	−0.113286	−	0.993562i
$18$		0			0
$19$			1.82303i			0.418232i	0.977891	+	0.209116i	$0.0670586\pi$
$19$			1.82303i			0.418232i	−0.977891	+	0.209116i	$0.932941\pi$
$20$	0.240016	−	0.240016i	0.0536693	−	0.0536693i
$21$		0			0
$22$	−1.93791	+	1.93791i	−0.413164	+	0.413164i
$23$	2.41421	+	2.41421i	0.503398	+	0.503398i	0.912492	−	0.409094i	$-0.134155\pi$
$23$	2.41421	+	2.41421i	0.503398	+	0.503398i	−0.409094	+	0.912492i	$0.634155\pi$
$24$		0			0
$25$			4.88478i			0.976957i
$26$		−	1.80644i		−	0.354272i
$27$		0			0
$28$	−0.707107	−	0.707107i	−0.133631	−	0.133631i
$29$	5.19217	−	5.19217i	0.964161	−	0.964161i	−0.0352187	−	0.999380i	$-0.511213\pi$
$29$	5.19217	−	5.19217i	0.964161	−	0.964161i	0.999380	+	0.0352187i	$0.0112128\pi$
$30$		0			0
$31$	2.39223	−	2.39223i	0.429657	−	0.429657i	−0.458855	−	0.888511i	$-0.651740\pi$
$31$	2.39223	−	2.39223i	0.429657	−	0.429657i	0.888511	+	0.458855i	$0.151740\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−4.09656	+	0.467090i	−0.702555	+	0.0801053i
$35$	−0.339434			−0.0573749
$36$		0			0
$37$	−3.69156	+	3.69156i	−0.606889	+	0.606889i	−0.942132	−	0.335243i	$-0.891182\pi$
$37$	−3.69156	+	3.69156i	−0.606889	+	0.606889i	0.335243	+	0.942132i	$0.391182\pi$
$38$	1.82303			0.295735
$39$		0			0
$40$	−0.240016	−	0.240016i	−0.0379499	−	0.0379499i
$41$	−3.56009	−	3.56009i	−0.555992	−	0.555992i	0.372171	−	0.928164i	$-0.378613\pi$
$41$	−3.56009	−	3.56009i	−0.555992	−	0.555992i	−0.928164	+	0.372171i	$0.878613\pi$
$42$		0			0
$43$		−	5.52893i		−	0.843153i	−0.906793	−	0.421577i	$-0.861477\pi$
$43$		−	5.52893i		−	0.843153i	0.906793	−	0.421577i	$-0.138523\pi$
$44$	1.93791	+	1.93791i	0.292151	+	0.292151i
$45$		0			0
$46$	2.41421	−	2.41421i	0.355956	−	0.355956i
$47$	4.30442			0.627864			0.313932	−	0.949445i	$-0.398354\pi$
$47$	4.30442			0.627864			0.313932	+	0.949445i	$0.398354\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$	4.88478			0.690813
$51$		0			0
$52$	−1.80644			−0.250508
$53$		−	8.11455i		−	1.11462i	−0.830305	−	0.557309i	$-0.811834\pi$
$53$		−	8.11455i		−	1.11462i	0.830305	−	0.557309i	$-0.188166\pi$
$54$		0			0
$55$	0.930261			0.125436
$56$	−0.707107	+	0.707107i	−0.0944911	+	0.0944911i
$57$		0			0
$58$	−5.19217	−	5.19217i	−0.681765	−	0.681765i
$59$			2.78067i			0.362013i	0.983482	+	0.181007i	$0.0579355\pi$
$59$			2.78067i			0.362013i	−0.983482	+	0.181007i	$0.942064\pi$
$60$		0			0
$61$	0.162466	+	0.162466i	0.0208016	+	0.0208016i	0.717431	−	0.696629i	$-0.245318\pi$
$61$	0.162466	+	0.162466i	0.0208016	+	0.0208016i	−0.696629	+	0.717431i	$0.745318\pi$
$62$	−2.39223	−	2.39223i	−0.303813	−	0.303813i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−0.433575	+	0.433575i	−0.0537784	+	0.0537784i
$66$		0			0
$67$	9.62930			1.17641			0.588203	−	0.808713i	$-0.299835\pi$
$67$	9.62930			1.17641			0.588203	+	0.808713i	$0.299835\pi$
$68$	0.467090	+	4.09656i	0.0566430	+	0.496781i
$69$		0			0
$70$			0.339434i			0.0405702i
$71$	10.8682	−	10.8682i	1.28982	−	1.28982i	0.354921	−	0.934896i	$-0.384508\pi$
$71$	10.8682	−	10.8682i	1.28982	−	1.28982i	0.934896	−	0.354921i	$-0.115492\pi$
$72$		0			0
$73$	5.33650	−	5.33650i	0.624590	−	0.624590i	−0.322112	−	0.946702i	$-0.604393\pi$
$73$	5.33650	−	5.33650i	0.624590	−	0.624590i	0.946702	+	0.322112i	$0.104393\pi$
$74$	3.69156	+	3.69156i	0.429135	+	0.429135i
$75$		0			0
$76$		−	1.82303i		−	0.209116i
$77$		−	2.74062i		−	0.312323i
$78$		0			0
$79$	−1.30165	−	1.30165i	−0.146447	−	0.146447i	0.630082	−	0.776529i	$-0.283021\pi$
$79$	−1.30165	−	1.30165i	−0.146447	−	0.146447i	−0.776529	+	0.630082i	$0.783021\pi$
$80$	−0.240016	+	0.240016i	−0.0268346	+	0.0268346i
$81$		0			0
$82$	−3.56009	+	3.56009i	−0.393146	+	0.393146i
$83$		−	3.31585i		−	0.363962i	−0.983302	−	0.181981i	$-0.941749\pi$
$83$		−	3.31585i		−	0.363962i	0.983302	−	0.181981i	$-0.0582510\pi$
$84$		0			0
$85$	1.09535	+	0.871133i	0.118808	+	0.0944876i
$86$	−5.52893			−0.596200
$87$		0			0
$88$	1.93791	−	1.93791i	0.206582	−	0.206582i
$89$	5.45410			0.578133			0.289067	−	0.957309i	$-0.406655\pi$
$89$	5.45410			0.578133			0.289067	+	0.957309i	$0.406655\pi$
$90$		0			0
$91$	1.27735	+	1.27735i	0.133902	+	0.133902i
$92$	−2.41421	−	2.41421i	−0.251699	−	0.251699i
$93$		0			0
$94$		−	4.30442i		−	0.443967i
$95$	−0.437557	−	0.437557i	−0.0448924	−	0.0448924i
$96$		0			0
$97$	9.53533	−	9.53533i	0.968166	−	0.968166i	−0.0313425	−	0.999509i	$-0.509978\pi$
$97$	9.53533	−	9.53533i	0.968166	−	0.968166i	0.999509	+	0.0313425i	$0.00997826\pi$
$98$	1.00000			0.101015
$99$		0			0
$100$		−	4.88478i		−	0.488478i
$101$	−4.60749			−0.458462			−0.229231	−	0.973372i	$-0.573621\pi$
$101$	−4.60749			−0.458462			−0.229231	+	0.973372i	$0.573621\pi$
$102$		0			0
$103$	8.84243			0.871270			0.435635	−	0.900123i	$-0.356524\pi$
$103$	8.84243			0.871270			0.435635	+	0.900123i	$0.356524\pi$
$104$			1.80644i			0.177136i
$105$		0			0
$106$	−8.11455			−0.788154
$107$	2.12122	−	2.12122i	0.205066	−	0.205066i	−0.597101	−	0.802166i	$-0.703681\pi$
$107$	2.12122	−	2.12122i	0.205066	−	0.205066i	0.802166	+	0.597101i	$0.203681\pi$
$108$		0			0
$109$	1.74173	+	1.74173i	0.166827	+	0.166827i	0.785583	−	0.618756i	$-0.212363\pi$
$109$	1.74173	+	1.74173i	0.166827	+	0.166827i	−0.618756	+	0.785583i	$0.712363\pi$
$110$		−	0.930261i		−	0.0886969i
$111$		0			0
$112$	0.707107	+	0.707107i	0.0668153	+	0.0668153i
$113$	−5.65951	−	5.65951i	−0.532402	−	0.532402i	0.388885	−	0.921286i	$-0.372860\pi$
$113$	−5.65951	−	5.65951i	−0.532402	−	0.532402i	−0.921286	+	0.388885i	$0.872860\pi$
$114$		0			0
$115$	−1.15890			−0.108068
$116$	−5.19217	+	5.19217i	−0.482080	+	0.482080i
$117$		0			0
$118$	2.78067			0.255982
$119$	2.56642	−	3.22699i	0.235264	−	0.295818i
$120$		0			0
$121$		−	3.48899i		−	0.317181i
$122$	0.162466	−	0.162466i	0.0147090	−	0.0147090i
$123$		0			0
$124$	−2.39223	+	2.39223i	−0.214828	+	0.214828i
$125$	−2.37251	−	2.37251i	−0.212204	−	0.212204i
$126$		0			0
$127$		−	11.4578i		−	1.01671i	−0.861147	−	0.508357i	$-0.830253\pi$
$127$		−	11.4578i		−	1.01671i	0.861147	−	0.508357i	$-0.169747\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	0.433575	+	0.433575i	0.0380271	+	0.0380271i
$131$	−4.06938	+	4.06938i	−0.355544	+	0.355544i	−0.862167	−	0.506624i	$-0.830893\pi$
$131$	−4.06938	+	4.06938i	−0.355544	+	0.355544i	0.506624	+	0.862167i	$0.330893\pi$
$132$		0			0
$133$	−1.28908	+	1.28908i	−0.111777	+	0.111777i
$134$		−	9.62930i		−	0.831845i
$135$		0			0
$136$	4.09656	−	0.467090i	0.351277	−	0.0400527i
$137$	−21.7675			−1.85972			−0.929859	−	0.367915i	$-0.880072\pi$
$137$	−21.7675			−1.85972			−0.929859	+	0.367915i	$0.880072\pi$
$138$		0			0
$139$	2.21924	−	2.21924i	0.188234	−	0.188234i	−0.606698	−	0.794932i	$-0.707506\pi$
$139$	2.21924	−	2.21924i	0.188234	−	0.188234i	0.794932	+	0.606698i	$0.207506\pi$
$140$	0.339434			0.0286874
$141$		0			0
$142$	−10.8682	−	10.8682i	−0.912039	−	0.912039i
$143$	−3.50072	−	3.50072i	−0.292745	−	0.292745i
$144$		0			0
$145$			2.49241i			0.206983i
$146$	−5.33650	−	5.33650i	−0.441651	−	0.441651i
$147$		0			0
$148$	3.69156	−	3.69156i	0.303444	−	0.303444i
$149$	0.0163825			0.00134210			0.000671052	−	1.00000i	$-0.499786\pi$
$149$	0.0163825			0.00134210			0.000671052		1.00000i	$0.499786\pi$
$150$		0			0
$151$			7.33169i			0.596644i	0.954465	+	0.298322i	$0.0964269\pi$
$151$			7.33169i			0.596644i	−0.954465	+	0.298322i	$0.903573\pi$
$152$	−1.82303			−0.147867
$153$		0			0
$154$	−2.74062			−0.220846
$155$			1.14835i			0.0922375i
$156$		0			0
$157$	10.0288			0.800388			0.400194	−	0.916431i	$-0.368943\pi$
$157$	10.0288			0.800388			0.400194	+	0.916431i	$0.368943\pi$
$158$	−1.30165	+	1.30165i	−0.103554	+	0.103554i
$159$		0			0
$160$	0.240016	+	0.240016i	0.0189750	+	0.0189750i
$161$			3.41421i			0.269078i
$162$		0			0
$163$	−13.4512	−	13.4512i	−1.05358	−	1.05358i	−0.998481	−	0.0551008i	$-0.982452\pi$
$163$	−13.4512	−	13.4512i	−1.05358	−	1.05358i	−0.0551008	−	0.998481i	$-0.517548\pi$
$164$	3.56009	+	3.56009i	0.277996	+	0.277996i
$165$		0			0
$166$	−3.31585			−0.257360
$167$	−3.55746	+	3.55746i	−0.275285	+	0.275285i	−0.831223	−	0.555939i	$-0.812359\pi$
$167$	−3.55746	+	3.55746i	−0.275285	+	0.275285i	0.555939	+	0.831223i	$0.312359\pi$
$168$		0			0
$169$	−9.73677			−0.748982
$170$	0.871133	−	1.09535i	0.0668128	−	0.0840096i
$171$		0			0
$172$			5.52893i			0.421577i
$173$	−4.77128	+	4.77128i	−0.362754	+	0.362754i	−0.864826	−	0.502072i	$-0.832571\pi$
$173$	−4.77128	+	4.77128i	−0.362754	+	0.362754i	0.502072	+	0.864826i	$0.332571\pi$
$174$		0			0
$175$	−3.45406	+	3.45406i	−0.261103	+	0.261103i
$176$	−1.93791	−	1.93791i	−0.146076	−	0.146076i
$177$		0			0
$178$		−	5.45410i		−	0.408802i
$179$			3.58407i			0.267886i	0.990989	+	0.133943i	$0.0427639\pi$
$179$			3.58407i			0.267886i	−0.990989	+	0.133943i	$0.957236\pi$
$180$		0			0
$181$	17.9339	+	17.9339i	1.33301	+	1.33301i	0.902661	+	0.430353i	$0.141611\pi$
$181$	17.9339	+	17.9339i	1.33301	+	1.33301i	0.430353	+	0.902661i	$0.358389\pi$
$182$	1.27735	−	1.27735i	0.0946832	−	0.0946832i
$183$		0			0
$184$	−2.41421	+	2.41421i	−0.177978	+	0.177978i
$185$		−	1.77207i		−	0.130285i
$186$		0			0
$187$	−7.03360	+	8.84396i	−0.514348	+	0.646734i
$188$	−4.30442			−0.313932
$189$		0			0
$190$	−0.437557	+	0.437557i	−0.0317438	+	0.0317438i
$191$	4.50725			0.326133			0.163066	−	0.986615i	$-0.447861\pi$
$191$	4.50725			0.326133			0.163066	+	0.986615i	$0.447861\pi$
$192$		0			0
$193$	2.98733	+	2.98733i	0.215033	+	0.215033i	0.806401	−	0.591369i	$-0.201412\pi$
$193$	2.98733	+	2.98733i	0.215033	+	0.215033i	−0.591369	+	0.806401i	$0.701412\pi$
$194$	−9.53533	−	9.53533i	−0.684597	−	0.684597i
$195$		0			0
$196$		−	1.00000i		−	0.0714286i
$197$	−1.09775	−	1.09775i	−0.0782117	−	0.0782117i	0.666919	−	0.745130i	$-0.267613\pi$
$197$	−1.09775	−	1.09775i	−0.0782117	−	0.0782117i	−0.745130	+	0.666919i	$0.767613\pi$
$198$		0			0
$199$	14.0544	−	14.0544i	0.996287	−	0.996287i	−0.00370606	−	0.999993i	$-0.501180\pi$
$199$	14.0544	−	14.0544i	0.996287	−	0.996287i	0.999993	+	0.00370606i	$0.00117968\pi$
$200$	−4.88478			−0.345406
$201$		0			0
$202$			4.60749i			0.324182i
$203$	7.34283			0.515366
$204$		0			0
$205$	1.70896			0.119359
$206$		−	8.84243i		−	0.616081i
$207$		0			0
$208$	1.80644			0.125254
$209$	3.53287	−	3.53287i	0.244374	−	0.244374i
$210$		0			0
$211$	−0.183324	−	0.183324i	−0.0126206	−	0.0126206i	0.700768	−	0.713389i	$-0.252841\pi$
$211$	−0.183324	−	0.183324i	−0.0126206	−	0.0126206i	−0.713389	+	0.700768i	$0.752841\pi$
$212$			8.11455i			0.557309i
$213$		0			0
$214$	−2.12122	−	2.12122i	−0.145003	−	0.145003i
$215$	1.32703	+	1.32703i	0.0905029	+	0.0905029i
$216$		0			0
$217$	3.38312			0.229661
$218$	1.74173	−	1.74173i	0.117965	−	0.117965i
$219$		0			0
$220$	−0.930261			−0.0627182
$221$	−0.843771	−	7.40020i	−0.0567582	−	0.497791i
$222$		0			0
$223$		−	24.8385i		−	1.66331i	−0.555292	−	0.831655i	$-0.687394\pi$
$223$		−	24.8385i		−	1.66331i	0.555292	−	0.831655i	$-0.312606\pi$
$224$	0.707107	−	0.707107i	0.0472456	−	0.0472456i
$225$		0			0
$226$	−5.65951	+	5.65951i	−0.376465	+	0.376465i
$227$	10.7733	+	10.7733i	0.715048	+	0.715048i	0.967587	−	0.252539i	$-0.0812656\pi$
$227$	10.7733	+	10.7733i	0.715048	+	0.715048i	−0.252539	+	0.967587i	$0.581266\pi$
$228$		0			0
$229$		−	22.9959i		−	1.51961i	−0.650149	−	0.759807i	$-0.725293\pi$
$229$		−	22.9959i		−	1.51961i	0.650149	−	0.759807i	$-0.274707\pi$
$230$			1.15890i			0.0764157i
$231$		0			0
$232$	5.19217	+	5.19217i	0.340882	+	0.340882i
$233$	12.1448	−	12.1448i	0.795632	−	0.795632i	−0.186772	−	0.982403i	$-0.559802\pi$
$233$	12.1448	−	12.1448i	0.795632	−	0.795632i	0.982403	+	0.186772i	$0.0598025\pi$
$234$		0			0
$235$	−1.03313	+	1.03313i	−0.0673941	+	0.0673941i
$236$		−	2.78067i		−	0.181007i
$237$		0			0
$238$	−3.22699	−	2.56642i	−0.209175	−	0.166357i
$239$	14.0808			0.910813			0.455407	−	0.890284i	$-0.349494\pi$
$239$	14.0808			0.910813			0.455407	+	0.890284i	$0.349494\pi$
$240$		0			0
$241$	−13.7186	+	13.7186i	−0.883695	+	0.883695i	−0.993908	−	0.110213i	$-0.964847\pi$
$241$	−13.7186	+	13.7186i	−0.883695	+	0.883695i	0.110213	+	0.993908i	$0.464847\pi$
$242$	−3.48899			−0.224281
$243$		0			0
$244$	−0.162466	−	0.162466i	−0.0104008	−	0.0104008i
$245$	−0.240016	−	0.240016i	−0.0153341	−	0.0153341i
$246$		0			0
$247$			3.29320i			0.209541i
$248$	2.39223	+	2.39223i	0.151907	+	0.151907i
$249$		0			0
$250$	−2.37251	+	2.37251i	−0.150051	+	0.150051i
$251$	−27.7237			−1.74990			−0.874952	−	0.484210i	$-0.839107\pi$
$251$	−27.7237			−1.74990			−0.874952	+	0.484210i	$0.839107\pi$
$252$		0			0
$253$		−	9.35707i		−	0.588274i
$254$	−11.4578			−0.718925
$255$		0			0
$256$	1.00000			0.0625000
$257$			27.8870i			1.73955i	0.493452	+	0.869773i	$0.335735\pi$
$257$			27.8870i			1.73955i	−0.493452	+	0.869773i	$0.664265\pi$
$258$		0			0
$259$	−5.22065			−0.324396
$260$	0.433575	−	0.433575i	0.0268892	−	0.0268892i
$261$		0			0
$262$	4.06938	+	4.06938i	0.251407	+	0.251407i
$263$			30.0521i			1.85309i	0.376184	+	0.926545i	$0.377236\pi$
$263$			30.0521i			1.85309i	−0.376184	+	0.926545i	$0.622764\pi$
$264$		0			0
$265$	1.94762	+	1.94762i	0.119642	+	0.119642i
$266$	1.28908	+	1.28908i	0.0790384	+	0.0790384i
$267$		0			0
$268$	−9.62930			−0.588203
$269$	1.34283	−	1.34283i	0.0818738	−	0.0818738i	−0.664984	−	0.746858i	$-0.731562\pi$
$269$	1.34283	−	1.34283i	0.0818738	−	0.0818738i	0.746858	+	0.664984i	$0.231562\pi$
$270$		0			0
$271$	−2.70813			−0.164507			−0.0822536	−	0.996611i	$-0.526212\pi$
$271$	−2.70813			−0.164507			−0.0822536	+	0.996611i	$0.526212\pi$
$272$	−0.467090	−	4.09656i	−0.0283215	−	0.248391i
$273$		0			0
$274$			21.7675i			1.31502i
$275$	9.46628	−	9.46628i	0.570838	−	0.570838i
$276$		0			0
$277$	−20.4512	+	20.4512i	−1.22880	+	1.22880i	−0.264375	+	0.964420i	$0.585166\pi$
$277$	−20.4512	+	20.4512i	−1.22880	+	1.22880i	−0.964420	+	0.264375i	$0.914834\pi$
$278$	−2.21924	−	2.21924i	−0.133101	−	0.133101i
$279$		0			0
$280$		−	0.339434i		−	0.0202851i
$281$			26.7565i			1.59616i	0.602551	+	0.798080i	$0.294151\pi$
$281$			26.7565i			1.59616i	−0.602551	+	0.798080i	$0.705849\pi$
$282$		0			0
$283$	10.0171	+	10.0171i	0.595456	+	0.595456i	0.939100	−	0.343644i	$-0.111661\pi$
$283$	10.0171	+	10.0171i	0.595456	+	0.595456i	−0.343644	+	0.939100i	$0.611661\pi$
$284$	−10.8682	+	10.8682i	−0.644909	+	0.644909i
$285$		0			0
$286$	−3.50072	+	3.50072i	−0.207002	+	0.207002i
$287$		−	5.03473i		−	0.297190i
$288$		0			0
$289$	−16.5637	+	3.82693i	−0.974333	+	0.225114i
$290$	2.49241			0.146359
$291$		0			0
$292$	−5.33650	+	5.33650i	−0.312295	+	0.312295i
$293$	4.48392			0.261954			0.130977	−	0.991385i	$-0.458189\pi$
$293$	4.48392			0.261954			0.130977	+	0.991385i	$0.458189\pi$
$294$		0			0
$295$	−0.667408	−	0.667408i	−0.0388580	−	0.0388580i
$296$	−3.69156	−	3.69156i	−0.214568	−	0.214568i
$297$		0			0
$298$		−	0.0163825i		−	0.000949011i
$299$	4.36113	+	4.36113i	0.252211	+	0.252211i
$300$		0			0
$301$	3.90954	−	3.90954i	0.225342	−	0.225342i
$302$	7.33169			0.421891
$303$		0			0
$304$			1.82303i			0.104558i
$305$	−0.0779890			−0.00446563
$306$		0			0
$307$	16.2491			0.927384			0.463692	−	0.885996i	$-0.346524\pi$
$307$	16.2491			0.927384			0.463692	+	0.885996i	$0.346524\pi$
$308$			2.74062i			0.156161i
$309$		0			0
$310$	1.14835			0.0652218
$311$	17.7089	−	17.7089i	1.00418	−	1.00418i	0.00419081	−	0.999991i	$-0.498666\pi$
$311$	17.7089	−	17.7089i	1.00418	−	1.00418i	0.999991	−	0.00419081i	$-0.00133398\pi$
$312$		0			0
$313$	−8.00508	−	8.00508i	−0.452474	−	0.452474i	0.443701	−	0.896175i	$-0.353665\pi$
$313$	−8.00508	−	8.00508i	−0.452474	−	0.452474i	−0.896175	+	0.443701i	$0.853665\pi$
$314$		−	10.0288i		−	0.565960i
$315$		0			0
$316$	1.30165	+	1.30165i	0.0732236	+	0.0732236i
$317$	−8.93918	−	8.93918i	−0.502074	−	0.502074i	0.410008	−	0.912082i	$-0.365526\pi$
$317$	−8.93918	−	8.93918i	−0.502074	−	0.502074i	−0.912082	+	0.410008i	$0.865526\pi$
$318$		0			0
$319$	−20.1239			−1.12672
$320$	0.240016	−	0.240016i	0.0134173	−	0.0134173i
$321$		0			0
$322$	3.41421			0.190267
$323$	7.46816	−	0.851520i	0.415540	−	0.0473799i
$324$		0			0
$325$			8.82407i			0.489472i
$326$	−13.4512	+	13.4512i	−0.744995	+	0.744995i
$327$		0			0
$328$	3.56009	−	3.56009i	0.196573	−	0.196573i
$329$	3.04369	+	3.04369i	0.167804	+	0.167804i
$330$		0			0
$331$			6.15048i			0.338061i	0.985611	+	0.169031i	$0.0540637\pi$
$331$			6.15048i			0.338061i	−0.985611	+	0.169031i	$0.945936\pi$
$332$			3.31585i			0.181981i
$333$		0			0
$334$	3.55746	+	3.55746i	0.194656	+	0.194656i
$335$	−2.31119	+	2.31119i	−0.126274	+	0.126274i
$336$		0			0
$337$	5.26410	−	5.26410i	0.286754	−	0.286754i	−0.549041	−	0.835795i	$-0.685007\pi$
$337$	5.26410	−	5.26410i	0.286754	−	0.286754i	0.835795	+	0.549041i	$0.185007\pi$
$338$			9.73677i			0.529611i
$339$		0			0
$340$	−1.09535	−	0.871133i	−0.0594038	−	0.0472438i
$341$	−9.27185			−0.502099
$342$		0			0
$343$	−0.707107	+	0.707107i	−0.0381802	+	0.0381802i
$344$	5.52893			0.298100
$345$		0			0
$346$	4.77128	+	4.77128i	0.256506	+	0.256506i
$347$	8.84658	+	8.84658i	0.474909	+	0.474909i	0.903499	−	0.428590i	$-0.140989\pi$
$347$	8.84658	+	8.84658i	0.474909	+	0.474909i	−0.428590	+	0.903499i	$0.640989\pi$
$348$		0			0
$349$			2.30228i			0.123238i	0.998100	+	0.0616190i	$0.0196264\pi$
$349$			2.30228i			0.123238i	−0.998100	+	0.0616190i	$0.980374\pi$
$350$	3.45406	+	3.45406i	0.184627	+	0.184627i
$351$		0			0
$352$	−1.93791	+	1.93791i	−0.103291	+	0.103291i
$353$	7.77879			0.414023			0.207012	−	0.978338i	$-0.433626\pi$
$353$	7.77879			0.414023			0.207012	+	0.978338i	$0.433626\pi$
$354$		0			0
$355$			5.21709i			0.276894i
$356$	−5.45410			−0.289067
$357$		0			0
$358$	3.58407			0.189424
$359$			14.7356i			0.777713i	0.921298	+	0.388856i	$0.127130\pi$
$359$			14.7356i			0.777713i	−0.921298	+	0.388856i	$0.872870\pi$
$360$		0			0
$361$	15.6766			0.825082
$362$	17.9339	−	17.9339i	0.942583	−	0.942583i
$363$		0			0
$364$	−1.27735	−	1.27735i	−0.0669512	−	0.0669512i
$365$			2.56169i			0.134085i
$366$		0			0
$367$	−2.81236	−	2.81236i	−0.146804	−	0.146804i	0.629885	−	0.776689i	$-0.283102\pi$
$367$	−2.81236	−	2.81236i	−0.146804	−	0.146804i	−0.776689	+	0.629885i	$0.783102\pi$
$368$	2.41421	+	2.41421i	0.125850	+	0.125850i
$369$		0			0
$370$	−1.77207			−0.0921255
$371$	5.73785	−	5.73785i	0.297894	−	0.297894i
$372$		0			0
$373$	−15.8364			−0.819979			−0.409990	−	0.912090i	$-0.634468\pi$
$373$	−15.8364			−0.819979			−0.409990	+	0.912090i	$0.634468\pi$
$374$	8.84396	+	7.03360i	0.457310	+	0.363699i
$375$		0			0
$376$			4.30442i			0.221984i
$377$	9.37934	−	9.37934i	0.483061	−	0.483061i
$378$		0			0
$379$	−5.66927	+	5.66927i	−0.291211	+	0.291211i	−0.837559	−	0.546348i	$-0.816018\pi$
$379$	−5.66927	+	5.66927i	−0.291211	+	0.291211i	0.546348	+	0.837559i	$0.316018\pi$
$380$	0.437557	+	0.437557i	0.0224462	+	0.0224462i
$381$		0			0
$382$		−	4.50725i		−	0.230611i
$383$			30.7579i			1.57165i	0.618447	+	0.785827i	$0.287762\pi$
$383$			30.7579i			1.57165i	−0.618447	+	0.785827i	$0.712238\pi$
$384$		0			0
$385$	0.657794	+	0.657794i	0.0335243	+	0.0335243i
$386$	2.98733	−	2.98733i	0.152051	−	0.152051i
$387$		0			0
$388$	−9.53533	+	9.53533i	−0.484083	+	0.484083i
$389$		−	8.15477i		−	0.413463i	−0.978398	−	0.206732i	$-0.933717\pi$
$389$		−	8.15477i		−	0.413463i	0.978398	−	0.206732i	$-0.0662827\pi$
$390$		0			0
$391$	8.76232	−	11.0176i	0.443130	−	0.557186i
$392$	−1.00000			−0.0505076
$393$		0			0
$394$	−1.09775	+	1.09775i	−0.0553040	+	0.0553040i
$395$	0.624835			0.0314389
$396$		0			0
$397$	13.1845	+	13.1845i	0.661709	+	0.661709i	0.955783	−	0.294074i	$-0.0950112\pi$
$397$	13.1845	+	13.1845i	0.661709	+	0.661709i	−0.294074	+	0.955783i	$0.595011\pi$
$398$	−14.0544	−	14.0544i	−0.704481	−	0.704481i
$399$		0			0
$400$			4.88478i			0.244239i
$401$	3.96168	+	3.96168i	0.197837	+	0.197837i	0.799072	−	0.601235i	$-0.205325\pi$
$401$	3.96168	+	3.96168i	0.197837	+	0.197837i	−0.601235	+	0.799072i	$0.705325\pi$
$402$		0			0
$403$	4.32142	−	4.32142i	0.215265	−	0.215265i
$404$	4.60749			0.229231
$405$		0			0
$406$		−	7.34283i		−	0.364419i
$407$	14.3078			0.709213
$408$		0			0
$409$	10.4073			0.514610			0.257305	−	0.966330i	$-0.417165\pi$
$409$	10.4073			0.514610			0.257305	+	0.966330i	$0.417165\pi$
$410$		−	1.70896i		−	0.0843995i
$411$		0			0
$412$	−8.84243			−0.435635
$413$	−1.96623	+	1.96623i	−0.0967520	+	0.0967520i
$414$		0			0
$415$	0.795859	+	0.795859i	0.0390672	+	0.0390672i
$416$		−	1.80644i		−	0.0885681i
$417$		0			0
$418$	−3.53287	−	3.53287i	−0.172799	−	0.172799i
$419$	1.30036	+	1.30036i	0.0635265	+	0.0635265i	0.738156	−	0.674630i	$-0.235697\pi$
$419$	1.30036	+	1.30036i	0.0635265	+	0.0635265i	−0.674630	+	0.738156i	$0.735697\pi$
$420$		0			0
$421$	−37.4523			−1.82532			−0.912658	−	0.408724i	$-0.865974\pi$
$421$	−37.4523			−1.82532			−0.912658	+	0.408724i	$0.865974\pi$
$422$	−0.183324	+	0.183324i	−0.00892409	+	0.00892409i
$423$		0			0
$424$	8.11455			0.394077
$425$	20.0108	−	2.28164i	0.970668	−	0.110676i
$426$		0			0
$427$			0.229761i			0.0111189i
$428$	−2.12122	+	2.12122i	−0.102533	+	0.102533i
$429$		0			0
$430$	1.32703	−	1.32703i	0.0639952	−	0.0639952i
$431$	6.92272	+	6.92272i	0.333456	+	0.333456i	0.853897	−	0.520442i	$-0.174233\pi$
$431$	6.92272	+	6.92272i	0.333456	+	0.333456i	−0.520442	+	0.853897i	$0.674233\pi$
$432$		0			0
$433$			3.12042i			0.149958i	0.997185	+	0.0749788i	$0.0238889\pi$
$433$			3.12042i			0.149958i	−0.997185	+	0.0749788i	$0.976111\pi$
$434$		−	3.38312i		−	0.162395i
$435$		0			0
$436$	−1.74173	−	1.74173i	−0.0834137	−	0.0834137i
$437$	−4.40119	+	4.40119i	−0.210537	+	0.210537i
$438$		0			0
$439$	−12.7659	+	12.7659i	−0.609283	+	0.609283i	−0.942759	−	0.333476i	$-0.891778\pi$
$439$	−12.7659	+	12.7659i	−0.609283	+	0.609283i	0.333476	+	0.942759i	$0.391778\pi$
$440$			0.930261i			0.0443485i
$441$		0			0
$442$	−7.40020	+	0.843771i	−0.351992	+	0.0401341i
$443$	−36.6892			−1.74316			−0.871578	−	0.490258i	$-0.836903\pi$
$443$	−36.6892			−1.74316			−0.871578	+	0.490258i	$0.836903\pi$
$444$		0			0
$445$	−1.30907	+	1.30907i	−0.0620560	+	0.0620560i
$446$	−24.8385			−1.17614
$447$		0			0
$448$	−0.707107	−	0.707107i	−0.0334077	−	0.0334077i
$449$	−6.08539	−	6.08539i	−0.287187	−	0.287187i	0.548780	−	0.835967i	$-0.315093\pi$
$449$	−6.08539	−	6.08539i	−0.287187	−	0.287187i	−0.835967	+	0.548780i	$0.815093\pi$
$450$		0			0
$451$			13.7983i			0.649736i
$452$	5.65951	+	5.65951i	0.266201	+	0.266201i
$453$		0			0
$454$	10.7733	−	10.7733i	0.505615	−	0.505615i
$455$	−0.613168			−0.0287458
$456$		0			0
$457$			1.66033i			0.0776669i	0.999246	+	0.0388334i	$0.0123642\pi$
$457$			1.66033i			0.0776669i	−0.999246	+	0.0388334i	$0.987636\pi$
$458$	−22.9959			−1.07453
$459$		0			0
$460$	1.15890			0.0540341
$461$		−	18.1548i		−	0.845551i	−0.906234	−	0.422776i	$-0.861056\pi$
$461$		−	18.1548i		−	0.845551i	0.906234	−	0.422776i	$-0.138944\pi$
$462$		0			0
$463$	−6.89976			−0.320659			−0.160330	−	0.987064i	$-0.551256\pi$
$463$	−6.89976			−0.320659			−0.160330	+	0.987064i	$0.551256\pi$
$464$	5.19217	−	5.19217i	0.241040	−	0.241040i
$465$		0			0
$466$	−12.1448	−	12.1448i	−0.562596	−	0.562596i
$467$			11.5008i			0.532192i	0.963947	+	0.266096i	$0.0857338\pi$
$467$			11.5008i			0.532192i	−0.963947	+	0.266096i	$0.914266\pi$
$468$		0			0
$469$	6.80895	+	6.80895i	0.314408	+	0.314408i
$470$	1.03313	+	1.03313i	0.0476548	+	0.0476548i
$471$		0			0
$472$	−2.78067			−0.127991
$473$	−10.7146	+	10.7146i	−0.492657	+	0.492657i
$474$		0			0
$475$	−8.90512			−0.408595
$476$	−2.56642	+	3.22699i	−0.117632	+	0.147909i
$477$		0			0
$478$		−	14.0808i		−	0.644042i
$479$	26.7468	−	26.7468i	1.22209	−	1.22209i	0.255203	−	0.966887i	$-0.417858\pi$
$479$	26.7468	−	26.7468i	1.22209	−	1.22209i	0.966887	−	0.255203i	$-0.0821424\pi$
$480$		0			0
$481$	−6.66859	+	6.66859i	−0.304061	+	0.304061i
$482$	13.7186	+	13.7186i	0.624867	+	0.624867i
$483$		0			0
$484$			3.48899i			0.158591i
$485$			4.57727i			0.207843i
$486$		0			0
$487$	17.7810	+	17.7810i	0.805735	+	0.805735i	0.983985	−	0.178251i	$-0.0570437\pi$
$487$	17.7810	+	17.7810i	0.805735	+	0.805735i	−0.178251	+	0.983985i	$0.557044\pi$
$488$	−0.162466	+	0.162466i	−0.00735449	+	0.00735449i
$489$		0			0
$490$	−0.240016	+	0.240016i	−0.0108428	+	0.0108428i
$491$		−	25.2378i		−	1.13897i	−0.822003	−	0.569483i	$-0.807143\pi$
$491$		−	25.2378i		−	1.13897i	0.822003	−	0.569483i	$-0.192857\pi$
$492$		0			0
$493$	−23.6952	−	18.8448i	−1.06718	−	0.848728i
$494$	3.29320			0.148168
$495$		0			0
$496$	2.39223	−	2.39223i	0.107414	−	0.107414i
$497$	15.3699			0.689436
$498$		0			0
$499$	28.0346	+	28.0346i	1.25500	+	1.25500i	0.953449	+	0.301553i	$0.0975051\pi$
$499$	28.0346	+	28.0346i	1.25500	+	1.25500i	0.301553	+	0.953449i	$0.402495\pi$
$500$	2.37251	+	2.37251i	0.106102	+	0.106102i
$501$		0			0
$502$			27.7237i			1.23737i
$503$	24.8897	+	24.8897i	1.10978	+	1.10978i	0.993179	+	0.116600i	$0.0371995\pi$
$503$	24.8897	+	24.8897i	1.10978	+	1.10978i	0.116600	+	0.993179i	$0.462800\pi$
$504$		0			0
$505$	1.10587	−	1.10587i	0.0492107	−	0.0492107i
$506$	−9.35707			−0.415972
$507$		0			0
$508$			11.4578i			0.508357i
$509$	−6.32811			−0.280489			−0.140244	−	0.990117i	$-0.544789\pi$
$509$	−6.32811			−0.280489			−0.140244	+	0.990117i	$0.544789\pi$
$510$		0			0
$511$	7.54694			0.333857
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	27.8870			1.23004
$515$	−2.12233	+	2.12233i	−0.0935209	+	0.0935209i
$516$		0			0
$517$	−8.34159	−	8.34159i	−0.366863	−	0.366863i
$518$			5.22065i			0.229382i
$519$		0			0
$520$	−0.433575	−	0.433575i	−0.0190135	−	0.0190135i
$521$	−16.5607	−	16.5607i	−0.725536	−	0.725536i	0.244191	−	0.969727i	$-0.421477\pi$
$521$	−16.5607	−	16.5607i	−0.725536	−	0.725536i	−0.969727	+	0.244191i	$0.921477\pi$
$522$		0			0
$523$	21.4372			0.937384			0.468692	−	0.883362i	$-0.344725\pi$
$523$	21.4372			0.937384			0.468692	+	0.883362i	$0.344725\pi$
$524$	4.06938	−	4.06938i	0.177772	−	0.177772i
$525$		0			0
$526$	30.0521			1.31033
$527$	−10.9173	−	8.68252i	−0.475565	−	0.378217i
$528$		0			0
$529$		−	11.3431i		−	0.493180i
$530$	1.94762	−	1.94762i	0.0845994	−	0.0845994i
$531$		0			0
$532$	1.28908	−	1.28908i	0.0558886	−	0.0558886i
$533$	−6.43109	−	6.43109i	−0.278561	−	0.278561i
$534$		0			0
$535$			1.01825i			0.0440229i
$536$			9.62930i			0.415923i
$537$		0			0
$538$	−1.34283	−	1.34283i	−0.0578936	−	0.0578936i
$539$	1.93791	−	1.93791i	0.0834718	−	0.0834718i
$540$		0			0
$541$	3.27906	−	3.27906i	0.140978	−	0.140978i	−0.633096	−	0.774074i	$-0.718216\pi$
$541$	3.27906	−	3.27906i	0.140978	−	0.140978i	0.774074	+	0.633096i	$0.218216\pi$
$542$			2.70813i			0.116324i
$543$		0			0
$544$	−4.09656	+	0.467090i	−0.175639	+	0.0200263i
$545$	−0.836087			−0.0358140
$546$		0			0
$547$	23.2835	−	23.2835i	0.995529	−	0.995529i	−0.00446133	−	0.999990i	$-0.501420\pi$
$547$	23.2835	−	23.2835i	0.995529	−	0.995529i	0.999990	+	0.00446133i	$0.00142009\pi$
$548$	21.7675			0.929859
$549$		0			0
$550$	−9.46628	−	9.46628i	−0.403644	−	0.403644i
$551$	9.46548	+	9.46548i	0.403243	+	0.403243i
$552$		0			0
$553$		−	1.84081i		−	0.0782793i
$554$	20.4512	+	20.4512i	0.868889	+	0.868889i
$555$		0			0
$556$	−2.21924	+	2.21924i	−0.0941168	+	0.0941168i
$557$	−39.9988			−1.69481			−0.847403	−	0.530951i	$-0.821835\pi$
$557$	−39.9988			−1.69481			−0.847403	+	0.530951i	$0.821835\pi$
$558$		0			0
$559$		−	9.98768i		−	0.422434i
$560$	−0.339434			−0.0143437
$561$		0			0
$562$	26.7565			1.12866
$563$			5.95374i			0.250920i	0.992099	+	0.125460i	$0.0400407\pi$
$563$			5.95374i			0.250920i	−0.992099	+	0.125460i	$0.959959\pi$
$564$		0			0
$565$	2.71675			0.114294
$566$	10.0171	−	10.0171i	0.421051	−	0.421051i
$567$		0			0
$568$	10.8682	+	10.8682i	0.456019	+	0.456019i
$569$			15.5842i			0.653322i	0.945141	+	0.326661i	$0.105924\pi$
$569$			15.5842i			0.653322i	−0.945141	+	0.326661i	$0.894076\pi$
$570$		0			0
$571$	10.5646	+	10.5646i	0.442113	+	0.442113i	0.892722	−	0.450608i	$-0.148793\pi$
$571$	10.5646	+	10.5646i	0.442113	+	0.442113i	−0.450608	+	0.892722i	$0.648793\pi$
$572$	3.50072	+	3.50072i	0.146373	+	0.146373i
$573$		0			0
$574$	−5.03473			−0.210145
$575$	−11.7929	+	11.7929i	−0.491798	+	0.491798i
$576$		0			0
$577$	25.4088			1.05778			0.528891	−	0.848690i	$-0.322608\pi$
$577$	25.4088			1.05778			0.528891	+	0.848690i	$0.322608\pi$
$578$	3.82693	+	16.5637i	0.159179	+	0.688957i
$579$		0			0
$580$		−	2.49241i		−	0.103492i
$581$	2.34466	−	2.34466i	0.0972730	−	0.0972730i
$582$		0			0
$583$	−15.7253	+	15.7253i	−0.651274	+	0.651274i
$584$	5.33650	+	5.33650i	0.220826	+	0.220826i
$585$		0			0
$586$		−	4.48392i		−	0.185229i
$587$			7.57429i			0.312624i	0.987708	+	0.156312i	$0.0499606\pi$
$587$			7.57429i			0.312624i	−0.987708	+	0.156312i	$0.950039\pi$
$588$		0			0
$589$	4.36111	+	4.36111i	0.179696	+	0.179696i
$590$	−0.667408	+	0.667408i	−0.0274767	+	0.0274767i
$591$		0			0
$592$	−3.69156	+	3.69156i	−0.151722	+	0.151722i
$593$			17.6670i			0.725497i	0.931887	+	0.362748i	$0.118162\pi$
$593$			17.6670i			0.725497i	−0.931887	+	0.362748i	$0.881838\pi$
$594$		0			0
$595$	0.158547	+	1.39051i	0.00649978	+	0.0570055i
$596$	−0.0163825			−0.000671052
$597$		0			0
$598$	4.36113	−	4.36113i	0.178340	−	0.178340i
$599$	−44.4534			−1.81632			−0.908159	−	0.418625i	$-0.862512\pi$
$599$	−44.4534			−1.81632			−0.908159	+	0.418625i	$0.862512\pi$
$600$		0			0
$601$	25.6822	+	25.6822i	1.04760	+	1.04760i	0.998809	+	0.0487900i	$0.0155365\pi$
$601$	25.6822	+	25.6822i	1.04760	+	1.04760i	0.0487900	+	0.998809i	$0.484463\pi$
$602$	−3.90954	−	3.90954i	−0.159341	−	0.159341i
$603$		0			0
$604$		−	7.33169i		−	0.298322i
$605$	0.837415	+	0.837415i	0.0340458	+	0.0340458i
$606$		0			0
$607$	10.6018	−	10.6018i	0.430315	−	0.430315i	−0.458421	−	0.888735i	$-0.651585\pi$
$607$	10.6018	−	10.6018i	0.430315	−	0.430315i	0.888735	+	0.458421i	$0.151585\pi$
$608$	1.82303			0.0739337
$609$		0			0
$610$			0.0779890i			0.00315768i
$611$	7.77568			0.314571
$612$		0			0
$613$	−2.21309			−0.0893859			−0.0446930	−	0.999001i	$-0.514231\pi$
$613$	−2.21309			−0.0893859			−0.0446930	+	0.999001i	$0.514231\pi$
$614$		−	16.2491i		−	0.655759i
$615$		0			0
$616$	2.74062			0.110423
$617$	−7.40060	+	7.40060i	−0.297937	+	0.297937i	−0.840205	−	0.542268i	$-0.817566\pi$
$617$	−7.40060	+	7.40060i	−0.297937	+	0.297937i	0.542268	+	0.840205i	$0.317566\pi$
$618$		0			0
$619$	14.5663	+	14.5663i	0.585468	+	0.585468i	0.936401	−	0.350933i	$-0.114135\pi$
$619$	14.5663	+	14.5663i	0.585468	+	0.585468i	−0.350933	+	0.936401i	$0.614135\pi$
$620$		−	1.14835i		−	0.0461188i
$621$		0			0
$622$	−17.7089	−	17.7089i	−0.710064	−	0.710064i
$623$	3.85663	+	3.85663i	0.154513	+	0.154513i
$624$		0			0
$625$	−23.2850			−0.931402
$626$	−8.00508	+	8.00508i	−0.319947	+	0.319947i
$627$		0			0
$628$	−10.0288			−0.400194
$629$	16.8470	+	13.3984i	0.671734	+	0.534230i
$630$		0			0
$631$		−	29.1584i		−	1.16078i	−0.814339	−	0.580390i	$-0.802900\pi$
$631$		−	29.1584i		−	1.16078i	0.814339	−	0.580390i	$-0.197100\pi$
$632$	1.30165	−	1.30165i	0.0517769	−	0.0517769i
$633$		0			0
$634$	−8.93918	+	8.93918i	−0.355020	+	0.355020i
$635$	2.75006	+	2.75006i	0.109133	+	0.109133i
$636$		0			0
$637$			1.80644i			0.0715738i
$638$			20.1239i			0.796714i
$639$		0			0
$640$	−0.240016	−	0.240016i	−0.00948748	−	0.00948748i
$641$	−26.2647	+	26.2647i	−1.03739	+	1.03739i	−0.0381213	+	0.999273i	$0.512137\pi$
$641$	−26.2647	+	26.2647i	−1.03739	+	1.03739i	−0.999273	+	0.0381213i	$0.987863\pi$
$642$		0			0
$643$	−0.665070	+	0.665070i	−0.0262278	+	0.0262278i	−0.720099	−	0.693871i	$-0.755904\pi$
$643$	−0.665070	+	0.665070i	−0.0262278	+	0.0262278i	0.693871	+	0.720099i	$0.255904\pi$
$644$		−	3.41421i		−	0.134539i
$645$		0			0
$646$	−0.851520	−	7.46816i	−0.0335026	−	0.293831i
$647$	28.8151			1.13284			0.566420	−	0.824117i	$-0.308328\pi$
$647$	28.8151			1.13284			0.566420	+	0.824117i	$0.308328\pi$
$648$		0			0
$649$	5.38870	−	5.38870i	0.211525	−	0.211525i
$650$	8.82407			0.346109
$651$		0			0
$652$	13.4512	+	13.4512i	0.526791	+	0.526791i
$653$	−0.282235	−	0.282235i	−0.0110447	−	0.0110447i	0.701563	−	0.712608i	$-0.252486\pi$
$653$	−0.282235	−	0.282235i	−0.0110447	−	0.0110447i	−0.712608	+	0.701563i	$0.752486\pi$
$654$		0			0
$655$		−	1.95344i		−	0.0763271i
$656$	−3.56009	−	3.56009i	−0.138998	−	0.138998i
$657$		0			0
$658$	3.04369	−	3.04369i	0.118655	−	0.118655i
$659$	−9.98709			−0.389042			−0.194521	−	0.980898i	$-0.562315\pi$
$659$	−9.98709			−0.389042			−0.194521	+	0.980898i	$0.562315\pi$
$660$		0			0
$661$			21.5683i			0.838908i	0.907777	+	0.419454i	$0.137778\pi$
$661$			21.5683i			0.838908i	−0.907777	+	0.419454i	$0.862222\pi$
$662$	6.15048			0.239045
$663$		0			0
$664$	3.31585			0.128680
$665$		−	0.618800i		−	0.0239960i
$666$		0			0
$667$	25.0700			0.970714
$668$	3.55746	−	3.55746i	0.137642	−	0.137642i
$669$		0			0
$670$	2.31119	+	2.31119i	0.0892891	+	0.0892891i
$671$		−	0.629689i		−	0.0243089i
$672$		0			0
$673$	−24.4532	−	24.4532i	−0.942600	−	0.942600i	0.0558396	−	0.998440i	$-0.482216\pi$
$673$	−24.4532	−	24.4532i	−0.942600	−	0.942600i	−0.998440	+	0.0558396i	$0.982216\pi$
$674$	−5.26410	−	5.26410i	−0.202766	−	0.202766i
$675$		0			0
$676$	9.73677			0.374491
$677$	−30.7462	+	30.7462i	−1.18167	+	1.18167i	−0.202360	+	0.979311i	$0.564861\pi$
$677$	−30.7462	+	30.7462i	−1.18167	+	1.18167i	−0.979311	+	0.202360i	$0.935139\pi$
$678$		0			0
$679$	13.4850			0.517507
$680$	−0.871133	+	1.09535i	−0.0334064	+	0.0420048i
$681$		0			0
$682$			9.27185i			0.355038i
$683$	−1.16832	+	1.16832i	−0.0447045	+	0.0447045i	−0.729106	−	0.684401i	$-0.760064\pi$
$683$	−1.16832	+	1.16832i	−0.0447045	+	0.0447045i	0.684401	+	0.729106i	$0.260064\pi$
$684$		0			0
$685$	5.22455	−	5.22455i	0.199620	−	0.199620i
$686$	0.707107	+	0.707107i	0.0269975	+	0.0269975i
$687$		0			0
$688$		−	5.52893i		−	0.210788i
$689$		−	14.6584i		−	0.558442i
$690$		0			0
$691$	−3.85442	−	3.85442i	−0.146629	−	0.146629i	0.629981	−	0.776610i	$-0.283063\pi$
$691$	−3.85442	−	3.85442i	−0.146629	−	0.146629i	−0.776610	+	0.629981i	$0.783063\pi$
$692$	4.77128	−	4.77128i	0.181377	−	0.181377i
$693$		0			0
$694$	8.84658	−	8.84658i	0.335812	−	0.335812i
$695$			1.06531i			0.0404095i
$696$		0			0
$697$	−12.9212	+	16.2470i	−0.489427	+	0.615399i
$698$	2.30228			0.0871424
$699$		0			0
$700$	3.45406	−	3.45406i	0.130551	−	0.130551i
$701$	−11.4410			−0.432121			−0.216061	−	0.976380i	$-0.569321\pi$
$701$	−11.4410			−0.432121			−0.216061	+	0.976380i	$0.569321\pi$
$702$		0			0
$703$	−6.72983	−	6.72983i	−0.253820	−	0.253820i
$704$	1.93791	+	1.93791i	0.0730378	+	0.0730378i
$705$		0			0
$706$		−	7.77879i		−	0.292759i
$707$	−3.25798	−	3.25798i	−0.122529	−	0.122529i
$708$		0			0
$709$	−3.58597	+	3.58597i	−0.134674	+	0.134674i	−0.771230	−	0.636556i	$-0.780358\pi$
$709$	−3.58597	+	3.58597i	−0.134674	+	0.134674i	0.636556	+	0.771230i	$0.280358\pi$
$710$	5.21709			0.195794
$711$		0			0
$712$			5.45410i			0.204401i
$713$	11.5507			0.432577
$714$		0			0
$715$	1.68046			0.0628457
$716$		−	3.58407i		−	0.133943i
$717$		0			0
$718$	14.7356			0.549926
$719$	8.16781	−	8.16781i	0.304608	−	0.304608i	−0.538206	−	0.842814i	$-0.680898\pi$
$719$	8.16781	−	8.16781i	0.304608	−	0.304608i	0.842814	+	0.538206i	$0.180898\pi$
$720$		0			0
$721$	6.25254	+	6.25254i	0.232857	+	0.232857i
$722$		−	15.6766i		−	0.583421i
$723$		0			0
$724$	−17.9339	−	17.9339i	−0.666507	−	0.666507i
$725$	25.3626	+	25.3626i	0.941944	+	0.941944i
$726$		0			0
$727$	−5.63439			−0.208968			−0.104484	−	0.994527i	$-0.533319\pi$
$727$	−5.63439			−0.208968			−0.104484	+	0.994527i	$0.533319\pi$
$728$	−1.27735	+	1.27735i	−0.0473416	+	0.0473416i
$729$		0			0
$730$	2.56169			0.0948125
$731$	−22.6496	+	2.58251i	−0.837726	+	0.0955175i
$732$		0			0
$733$			11.0582i			0.408444i	0.978925	+	0.204222i	$0.0654665\pi$
$733$			11.0582i			0.408444i	−0.978925	+	0.204222i	$0.934534\pi$
$734$	−2.81236	+	2.81236i	−0.103806	+	0.103806i
$735$		0			0
$736$	2.41421	−	2.41421i	0.0889891	−	0.0889891i
$737$	−18.6607	−	18.6607i	−0.687377	−	0.687377i
$738$		0			0
$739$			6.14152i			0.225920i	0.993600	+	0.112960i	$0.0360331\pi$
$739$			6.14152i			0.225920i	−0.993600	+	0.112960i	$0.963967\pi$
$740$			1.77207i			0.0651426i
$741$		0			0
$742$	−5.73785	−	5.73785i	−0.210643	−	0.210643i
$743$	8.12217	−	8.12217i	0.297973	−	0.297973i	−0.542246	−	0.840220i	$-0.682426\pi$
$743$	8.12217	−	8.12217i	0.297973	−	0.297973i	0.840220	+	0.542246i	$0.182426\pi$
$744$		0			0
$745$	−0.00393206	+	0.00393206i	−0.000144060	+	0.000144060i
$746$			15.8364i			0.579813i
$747$		0			0
$748$	7.03360	−	8.84396i	0.257174	−	0.323367i
$749$	2.99985			0.109612
$750$		0			0
$751$	1.80430	−	1.80430i	0.0658397	−	0.0658397i	−0.673420	−	0.739260i	$-0.735176\pi$
$751$	1.80430	−	1.80430i	0.0658397	−	0.0658397i	0.739260	+	0.673420i	$0.235176\pi$
$752$	4.30442			0.156966
$753$		0			0
$754$	−9.37934	−	9.37934i	−0.341575	−	0.341575i
$755$	−1.75972	−	1.75972i	−0.0640429	−	0.0640429i
$756$		0			0
$757$		−	18.4814i		−	0.671719i	−0.941912	−	0.335859i	$-0.890973\pi$
$757$		−	18.4814i		−	0.671719i	0.941912	−	0.335859i	$-0.109027\pi$
$758$	5.66927	+	5.66927i	0.205917	+	0.205917i
$759$		0			0
$760$	0.437557	−	0.437557i	0.0158719	−	0.0158719i
$761$	−27.9865			−1.01451			−0.507255	−	0.861796i	$-0.669340\pi$
$761$	−27.9865			−1.01451			−0.507255	+	0.861796i	$0.669340\pi$
$762$		0			0
$763$			2.46318i			0.0891730i
$764$	−4.50725			−0.163066
$765$		0			0
$766$	30.7579			1.11133
$767$			5.02312i			0.181375i
$768$		0			0
$769$	−24.3818			−0.879230			−0.439615	−	0.898186i	$-0.644885\pi$
$769$	−24.3818			−0.879230			−0.439615	+	0.898186i	$0.644885\pi$
$770$	0.657794	−	0.657794i	0.0237053	−	0.0237053i
$771$		0			0
$772$	−2.98733	−	2.98733i	−0.107516	−	0.107516i
$773$			4.72562i			0.169969i	0.996382	+	0.0849843i	$0.0270840\pi$
$773$			4.72562i			0.169969i	−0.996382	+	0.0849843i	$0.972916\pi$
$774$		0			0
$775$	11.6855	+	11.6855i	0.419756	+	0.419756i
$776$	9.53533	+	9.53533i	0.342298	+	0.342298i
$777$		0			0
$778$	−8.15477			−0.292363
$779$	6.49015	−	6.49015i	0.232534	−	0.232534i
$780$		0			0
$781$	−42.1232			−1.50729
$782$	−11.0176	−	8.76232i	−0.393990	−	0.313340i
$783$		0			0
$784$			1.00000i			0.0357143i
$785$	−2.40708	+	2.40708i	−0.0859125	+	0.0859125i
$786$		0			0
$787$	−0.262150	+	0.262150i	−0.00934463	+	0.00934463i	−0.711764	−	0.702419i	$-0.752103\pi$
$787$	−0.262150	+	0.262150i	−0.00934463	+	0.00934463i	0.702419	+	0.711764i	$0.252103\pi$
$788$	1.09775	+	1.09775i	0.0391059	+	0.0391059i
$789$		0			0
$790$		−	0.624835i		−	0.0222306i
$791$		−	8.00375i		−	0.284581i
$792$		0			0
$793$	0.293485	+	0.293485i	0.0104220	+	0.0104220i
$794$	13.1845	−	13.1845i	0.467899	−	0.467899i
$795$		0			0
$796$	−14.0544	+	14.0544i	−0.498144	+	0.498144i
$797$		−	42.2758i		−	1.49749i	−0.662860	−	0.748743i	$-0.730658\pi$
$797$		−	42.2758i		−	1.49749i	0.662860	−	0.748743i	$-0.269342\pi$
$798$		0			0
$799$	−2.01055	−	17.6333i	−0.0711283	−	0.623823i
$800$	4.88478			0.172703
$801$		0			0
$802$	3.96168	−	3.96168i	0.139892	−	0.139892i
$803$	−20.6833			−0.729898
$804$		0			0
$805$	−0.819467	−	0.819467i	−0.0288824	−	0.0288824i
$806$	−4.32142	−	4.32142i	−0.152215	−	0.152215i
$807$		0			0
$808$		−	4.60749i		−	0.162091i
$809$	11.4011	+	11.4011i	0.400840	+	0.400840i	0.878529	−	0.477689i	$-0.158525\pi$
$809$	11.4011	+	11.4011i	0.400840	+	0.400840i	−0.477689	+	0.878529i	$0.658525\pi$
$810$		0			0
$811$	18.0124	−	18.0124i	0.632502	−	0.632502i	−0.316193	−	0.948695i	$-0.602405\pi$
$811$	18.0124	−	18.0124i	0.632502	−	0.632502i	0.948695	+	0.316193i	$0.102405\pi$
$812$	−7.34283			−0.257683
$813$		0			0
$814$		−	14.3078i		−	0.501489i
$815$	6.45703			0.226180
$816$		0			0
$817$	10.0794			0.352634
$818$		−	10.4073i		−	0.363884i
$819$		0			0
$820$	−1.70896			−0.0596795
$821$	−26.5179	+	26.5179i	−0.925481	+	0.925481i	−0.997410	−	0.0719292i	$-0.977084\pi$
$821$	−26.5179	+	26.5179i	−0.925481	+	0.925481i	0.0719292	+	0.997410i	$0.477084\pi$
$822$		0			0
$823$	−34.5400	−	34.5400i	−1.20399	−	1.20399i	−0.972943	−	0.231047i	$-0.925785\pi$
$823$	−34.5400	−	34.5400i	−1.20399	−	1.20399i	−0.231047	−	0.972943i	$-0.574215\pi$
$824$			8.84243i			0.308041i
$825$		0			0
$826$	1.96623	+	1.96623i	0.0684140	+	0.0684140i
$827$	2.89831	+	2.89831i	0.100784	+	0.100784i	0.755701	−	0.654917i	$-0.227296\pi$
$827$	2.89831	+	2.89831i	0.100784	+	0.100784i	−0.654917	+	0.755701i	$0.727296\pi$
$828$		0			0
$829$	−23.0796			−0.801589			−0.400795	−	0.916168i	$-0.631266\pi$
$829$	−23.0796			−0.801589			−0.400795	+	0.916168i	$0.631266\pi$
$830$	0.795859	−	0.795859i	0.0276247	−	0.0276247i
$831$		0			0
$832$	−1.80644			−0.0626271
$833$	4.09656	−	0.467090i	0.141937	−	0.0161837i
$834$		0			0
$835$		−	1.70770i		−	0.0590974i
$836$	−3.53287	+	3.53287i	−0.122187	+	0.122187i
$837$		0			0
$838$	1.30036	−	1.30036i	0.0449200	−	0.0449200i
$839$	30.6274	+	30.6274i	1.05737	+	1.05737i	0.998251	+	0.0591232i	$0.0188305\pi$
$839$	30.6274	+	30.6274i	1.05737	+	1.05737i	0.0591232	+	0.998251i	$0.481170\pi$
$840$		0			0
$841$		−	24.9172i		−	0.859213i
$842$			37.4523i			1.29069i
$843$		0			0
$844$	0.183324	+	0.183324i	0.00631028	+	0.00631028i
$845$	2.33698	−	2.33698i	0.0803947	−	0.0803947i
$846$		0			0
$847$	2.46709	−	2.46709i	0.0847702	−	0.0847702i
$848$		−	8.11455i		−	0.278655i
$849$		0			0
$850$	−2.28164	−	20.0108i	−0.0782595	−	0.686366i
$851$	−17.8244			−0.611014
$852$		0			0
$853$	−33.2922	+	33.2922i	−1.13990	+	1.13990i	−0.151434	+	0.988467i	$0.548389\pi$
$853$	−33.2922	+	33.2922i	−1.13990	+	1.13990i	−0.988467	+	0.151434i	$0.951611\pi$
$854$	0.229761			0.00786228
$855$		0			0
$856$	2.12122	+	2.12122i	0.0725017	+	0.0725017i
$857$	−14.3154	−	14.3154i	−0.489005	−	0.489005i	0.418987	−	0.907992i	$-0.362385\pi$
$857$	−14.3154	−	14.3154i	−0.489005	−	0.489005i	−0.907992	+	0.418987i	$0.862385\pi$
$858$		0			0
$859$			10.0218i			0.341940i	0.985276	+	0.170970i	$0.0546901\pi$
$859$			10.0218i			0.341940i	−0.985276	+	0.170970i	$0.945310\pi$
$860$	−1.32703	−	1.32703i	−0.0452515	−	0.0452515i
$861$		0			0
$862$	6.92272	−	6.92272i	0.235789	−	0.235789i
$863$	−29.6685			−1.00993			−0.504963	−	0.863141i	$-0.668494\pi$
$863$	−29.6685			−1.00993			−0.504963	+	0.863141i	$0.668494\pi$
$864$		0			0
$865$		−	2.29037i		−	0.0778750i
$866$	3.12042			0.106036
$867$		0			0
$868$	−3.38312			−0.114831
$869$			5.04497i			0.171139i
$870$		0			0
$871$	17.3948			0.589399
$872$	−1.74173	+	1.74173i	−0.0589824	+	0.0589824i
$873$		0			0
$874$	4.40119	+	4.40119i	0.148872	+	0.148872i
$875$		−	3.35524i		−	0.113428i
$876$		0			0
$877$	−22.2798	−	22.2798i	−0.752337	−	0.752337i	0.222578	−	0.974915i	$-0.428553\pi$
$877$	−22.2798	−	22.2798i	−0.752337	−	0.752337i	−0.974915	+	0.222578i	$0.928553\pi$
$878$	12.7659	+	12.7659i	0.430828	+	0.430828i
$879$		0			0
$880$	0.930261			0.0313591
$881$	38.1670	−	38.1670i	1.28588	−	1.28588i	0.348610	−	0.937268i	$-0.386654\pi$
$881$	38.1670	−	38.1670i	1.28588	−	1.28588i	0.937268	−	0.348610i	$-0.113346\pi$
$882$		0			0
$883$	−8.84706			−0.297727			−0.148864	−	0.988858i	$-0.547562\pi$
$883$	−8.84706			−0.297727			−0.148864	+	0.988858i	$0.547562\pi$
$884$	0.843771	+	7.40020i	0.0283791	+	0.248896i
$885$		0			0
$886$			36.6892i			1.23260i
$887$	16.0113	−	16.0113i	0.537608	−	0.537608i	−0.385217	−	0.922826i	$-0.625874\pi$
$887$	16.0113	−	16.0113i	0.537608	−	0.537608i	0.922826	+	0.385217i	$0.125874\pi$
$888$		0			0
$889$	8.10187	−	8.10187i	0.271728	−	0.271728i
$890$	1.30907	+	1.30907i	0.0438802	+	0.0438802i
$891$		0			0
$892$			24.8385i			0.831655i
$893$			7.84710i			0.262593i
$894$		0			0
$895$	−0.860236	−	0.860236i	−0.0287545	−	0.0287545i
$896$	−0.707107	+	0.707107i	−0.0236228	+	0.0236228i
$897$		0			0
$898$	−6.08539	+	6.08539i	−0.203072	+	0.203072i
$899$		−	24.8417i		−	0.828516i
$900$		0			0
$901$	−33.2417	+	3.79023i	−1.10744	+	0.126271i
$902$	13.7983			0.459432
$903$		0			0
$904$	5.65951	−	5.65951i	0.188232	−	0.188232i
$905$	−8.60884			−0.286168
$906$		0			0
$907$	26.4720	+	26.4720i	0.878987	+	0.878987i	0.993430	−	0.114443i	$-0.0365083\pi$
$907$	26.4720	+	26.4720i	0.878987	+	0.878987i	−0.114443	+	0.993430i	$0.536508\pi$
$908$	−10.7733	−	10.7733i	−0.357524	−	0.357524i
$909$		0			0
$910$			0.613168i			0.0203263i
$911$	−16.1451	−	16.1451i	−0.534910	−	0.534910i	0.387120	−	0.922029i	$-0.373470\pi$
$911$	−16.1451	−	16.1451i	−0.534910	−	0.534910i	−0.922029	+	0.387120i	$0.873470\pi$
$912$		0			0
$913$	−6.42583	+	6.42583i	−0.212664	+	0.212664i
$914$	1.66033			0.0549188
$915$		0			0
$916$			22.9959i			0.759807i
$917$	−5.75498			−0.190046
$918$		0			0
$919$	45.9692			1.51638			0.758192	−	0.652032i	$-0.226083\pi$
$919$	45.9692			1.51638			0.758192	+	0.652032i	$0.226083\pi$
$920$		−	1.15890i		−	0.0382079i
$921$		0			0
$922$	−18.1548			−0.597895
$923$	19.6328	−	19.6328i	0.646220	−	0.646220i
$924$		0			0
$925$	−18.0325	−	18.0325i	−0.592904	−	0.592904i
$926$			6.89976i			0.226740i
$927$		0			0
$928$	−5.19217	−	5.19217i	−0.170441	−	0.170441i
$929$	0.379222	+	0.379222i	0.0124419	+	0.0124419i	0.713300	−	0.700858i	$-0.247200\pi$
$929$	0.379222	+	0.379222i	0.0124419	+	0.0124419i	−0.700858	+	0.713300i	$0.747200\pi$
$930$		0			0
$931$	−1.82303			−0.0597474
$932$	−12.1448	+	12.1448i	−0.397816	+	0.397816i
$933$		0			0
$934$	11.5008			0.376316
$935$	−0.434516	−	3.81087i	−0.0142102	−	0.124629i
$936$		0			0
$937$		−	2.04680i		−	0.0668660i	−0.999441	−	0.0334330i	$-0.989356\pi$
$937$		−	2.04680i		−	0.0668660i	0.999441	−	0.0334330i	$-0.0106440\pi$
$938$	6.80895	−	6.80895i	0.222320	−	0.222320i
$939$		0			0
$940$	1.03313	−	1.03313i	0.0336970	−	0.0336970i
$941$	−17.4256	−	17.4256i	−0.568059	−	0.568059i	0.363526	−	0.931584i	$-0.381573\pi$
$941$	−17.4256	−	17.4256i	−0.568059	−	0.568059i	−0.931584	+	0.363526i	$0.881573\pi$
$942$		0			0
$943$		−	17.1896i		−	0.559771i
$944$			2.78067i			0.0905033i
$945$		0			0
$946$	10.7146	+	10.7146i	0.348361	+	0.348361i
$947$	19.7960	−	19.7960i	0.643285	−	0.643285i	−0.308077	−	0.951361i	$-0.599685\pi$
$947$	19.7960	−	19.7960i	0.643285	−	0.643285i	0.951361	+	0.308077i	$0.0996854\pi$
$948$		0			0
$949$	9.64006	−	9.64006i	0.312930	−	0.312930i
$950$			8.90512i			0.288920i
$951$		0			0
$952$	3.22699	+	2.56642i	0.104587	+	0.0831783i
$953$	−14.4240			−0.467239			−0.233619	−	0.972328i	$-0.575057\pi$
$953$	−14.4240			−0.467239			−0.233619	+	0.972328i	$0.575057\pi$
$954$		0			0
$955$	−1.08181	+	1.08181i	−0.0350067	+	0.0350067i
$956$	−14.0808			−0.455407
$957$		0			0
$958$	−26.7468	−	26.7468i	−0.864149	−	0.864149i
$959$	−15.3919	−	15.3919i	−0.497031	−	0.497031i
$960$		0			0
$961$			19.5545i			0.630790i
$962$	6.66859	+	6.66859i	0.215004	+	0.215004i
$963$		0			0
$964$	13.7186	−	13.7186i	0.441847	−	0.441847i
$965$	−1.43402			−0.0461626
$966$		0			0
$967$			52.7678i			1.69690i	0.529277	+	0.848449i	$0.322463\pi$
$967$			52.7678i			1.69690i	−0.529277	+	0.848449i	$0.677537\pi$
$968$	3.48899			0.112140
$969$		0			0
$970$	4.57727			0.146967
$971$		−	44.5103i		−	1.42840i	−0.699940	−	0.714201i	$-0.746790\pi$
$971$		−	44.5103i		−	1.42840i	0.699940	−	0.714201i	$-0.253210\pi$
$972$		0			0
$973$	3.13848			0.100615
$974$	17.7810	−	17.7810i	0.569740	−	0.569740i
$975$		0			0
$976$	0.162466	+	0.162466i	0.00520041	+	0.00520041i
$977$		−	8.02101i		−	0.256615i	−0.991734	−	0.128307i	$-0.959046\pi$
$977$		−	8.02101i		−	0.256615i	0.991734	−	0.128307i	$-0.0409544\pi$
$978$		0			0
$979$	−10.5696	−	10.5696i	−0.337805	−	0.337805i
$980$	0.240016	+	0.240016i	0.00766704	+	0.00766704i
$981$		0			0
$982$	−25.2378			−0.805371
$983$	4.63151	−	4.63151i	0.147722	−	0.147722i	−0.629377	−	0.777100i	$-0.716690\pi$
$983$	4.63151	−	4.63151i	0.147722	−	0.147722i	0.777100	+	0.629377i	$0.216690\pi$
$984$		0			0
$985$	0.526958			0.0167903
$986$	−18.8448	+	23.6952i	−0.600141	+	0.754610i
$987$		0			0
$988$		−	3.29320i		−	0.104771i
$989$	13.3480	−	13.3480i	0.424442	−	0.424442i
$990$		0			0
$991$	−15.8288	+	15.8288i	−0.502818	+	0.502818i	−0.912313	−	0.409495i	$-0.865705\pi$
$991$	−15.8288	+	15.8288i	−0.502818	+	0.502818i	0.409495	+	0.912313i	$0.365705\pi$
$992$	−2.39223	−	2.39223i	−0.0759533	−	0.0759533i
$993$		0			0
$994$		−	15.3699i		−	0.487505i
$995$			6.74655i			0.213880i
$996$		0			0
$997$	−32.5563	−	32.5563i	−1.03107	−	1.03107i	−0.999502	−	0.0315660i	$-0.989951\pi$
$997$	−32.5563	−	32.5563i	−1.03107	−	1.03107i	−0.0315660	−	0.999502i	$-0.510049\pi$
$998$	28.0346	−	28.0346i	0.887421	−	0.887421i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2142.2.p.f.1891.3		12
3.2	odd	2		714.2.m.e.463.2	yes	12
17.13	even	4	inner	2142.2.p.f.1135.3		12
51.47	odd	4		714.2.m.e.421.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
714.2.m.e.421.2	✓	12	51.47	odd	4
714.2.m.e.463.2	yes	12	3.2	odd	2
2142.2.p.f.1135.3		12	17.13	even	4	inner
2142.2.p.f.1891.3		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2142.p (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-0.240016 + 0.240016i) q^{5} +(0.707107 + 0.707107i) q^{7} +1.00000i q^{8} +(0.240016 + 0.240016i) q^{10} +(-1.93791 - 1.93791i) q^{11} +1.80644 q^{13} +(0.707107 - 0.707107i) q^{14} +1.00000 q^{16} +(-0.467090 - 4.09656i) q^{17} +1.82303i q^{19} +(0.240016 - 0.240016i) q^{20} +(-1.93791 + 1.93791i) q^{22} +(2.41421 + 2.41421i) q^{23} +4.88478i q^{25} -1.80644i q^{26} +(-0.707107 - 0.707107i) q^{28} +(5.19217 - 5.19217i) q^{29} +(2.39223 - 2.39223i) q^{31} -1.00000i q^{32} +(-4.09656 + 0.467090i) q^{34} -0.339434 q^{35} +(-3.69156 + 3.69156i) q^{37} +1.82303 q^{38} +(-0.240016 - 0.240016i) q^{40} +(-3.56009 - 3.56009i) q^{41} -5.52893i q^{43} +(1.93791 + 1.93791i) q^{44} +(2.41421 - 2.41421i) q^{46} +4.30442 q^{47} +1.00000i q^{49} +4.88478 q^{50} -1.80644 q^{52} -8.11455i q^{53} +0.930261 q^{55} +(-0.707107 + 0.707107i) q^{56} +(-5.19217 - 5.19217i) q^{58} +2.78067i q^{59} +(0.162466 + 0.162466i) q^{61} +(-2.39223 - 2.39223i) q^{62} -1.00000 q^{64} +(-0.433575 + 0.433575i) q^{65} +9.62930 q^{67} +(0.467090 + 4.09656i) q^{68} +0.339434i q^{70} +(10.8682 - 10.8682i) q^{71} +(5.33650 - 5.33650i) q^{73} +(3.69156 + 3.69156i) q^{74} -1.82303i q^{76} -2.74062i q^{77} +(-1.30165 - 1.30165i) q^{79} +(-0.240016 + 0.240016i) q^{80} +(-3.56009 + 3.56009i) q^{82} -3.31585i q^{83} +(1.09535 + 0.871133i) q^{85} -5.52893 q^{86} +(1.93791 - 1.93791i) q^{88} +5.45410 q^{89} +(1.27735 + 1.27735i) q^{91} +(-2.41421 - 2.41421i) q^{92} -4.30442i q^{94} +(-0.437557 - 0.437557i) q^{95} +(9.53533 - 9.53533i) q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} - 12 q^{13} + 12 q^{16} + 12 q^{23} - 12 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} - 12 q^{37} + 12 q^{46} + 24 q^{47} - 24 q^{50} + 12 q^{52} + 12 q^{55} + 12 q^{58} - 12 q^{61} - 12 q^{62}+ \cdots + 12 q^{98}+O(q^{100}) \) 12 * q - 12 * q^4 - 12 * q^13 + 12 * q^16 + 12 * q^23 - 12 * q^29 + 12 * q^31 + 12 * q^34 - 12 * q^35 - 12 * q^37 + 12 * q^46 + 24 * q^47 - 24 * q^50 + 12 * q^52 + 12 * q^55 + 12 * q^58 - 12 * q^61 - 12 * q^62 - 12 * q^64 - 36 * q^65 - 12 * q^67 + 36 * q^71 - 36 * q^73 + 12 * q^74 + 48 * q^79 - 48 * q^85 - 36 * q^86 - 12 * q^89 - 12 * q^92 + 12 * q^95 + 36 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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