LMFDB - Embedded newform 189.2.o.a.125.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(62,189)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([1, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("189.62");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	189.o (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.50917259820$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
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} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 $ x^12 - 7x^10 + 37x^8 - 78x^6 + 123x^4 - 36*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.4
Root			$-0.474636 - 0.274031i$ of defining polynomial
Character	$\chi$	$=$	189.125
Dual form			189.2.o.a.62.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.555632 - 0.320794i) q^{2} +(-0.794182 + 1.37556i) q^{4} +(1.10552 - 1.91482i) q^{5} +(2.60579 + 0.458109i) q^{7} +2.30225i q^{8} +O(q^{10})$	\(q+(0.555632 - 0.320794i) q^{2} +(-0.794182 + 1.37556i) q^{4} +(1.10552 - 1.91482i) q^{5} +(2.60579 + 0.458109i) q^{7} +2.30225i q^{8} -1.41858i q^{10} +(2.93818 - 1.69636i) q^{11} +(1.56060 + 0.901012i) q^{13} +(1.59482 - 0.581383i) q^{14} +(-0.849814 - 1.47192i) q^{16} -5.96901 q^{17} +1.64419i q^{19} +(1.75597 + 3.04144i) q^{20} +(1.08836 - 1.88510i) q^{22} +(-2.05563 - 1.18682i) q^{23} +(0.0556321 + 0.0963576i) q^{25} +1.15616 q^{26} +(-2.69963 + 3.22061i) q^{28} +(-2.44437 + 1.41126i) q^{29} +(-9.28558 - 5.36103i) q^{31} +(-4.93199 - 2.84748i) q^{32} +(-3.31657 + 1.91482i) q^{34} +(3.75796 - 4.48318i) q^{35} +1.69963 q^{37} +(0.527445 + 0.913562i) q^{38} +(4.40841 + 2.54520i) q^{40} +(-0.455074 + 0.788211i) q^{41} +(-1.96108 - 3.39669i) q^{43} +5.38887i q^{44} -1.52290 q^{46} +(0.123005 + 0.213051i) q^{47} +(6.58027 + 2.38747i) q^{49} +(0.0618219 + 0.0356929i) q^{50} +(-2.47880 + 1.43113i) q^{52} +7.87589i q^{53} -7.50146i q^{55} +(-1.05468 + 5.99919i) q^{56} +(-0.905446 + 1.56828i) q^{58} +(-5.39093 + 9.33736i) q^{59} +(-1.22853 + 0.709292i) q^{61} -6.87916 q^{62} -0.254572 q^{64} +(3.45056 - 1.99218i) q^{65} +(3.99381 - 6.91748i) q^{67} +(4.74048 - 8.21075i) q^{68} +(0.649865 - 3.69653i) q^{70} -12.1743i q^{71} -0.426103i q^{73} +(0.944368 - 0.545231i) q^{74} +(-2.26168 - 1.30578i) q^{76} +(8.43339 - 3.07435i) q^{77} +(2.49381 + 4.31941i) q^{79} -3.75796 q^{80} +0.583940i q^{82} +(-4.28541 - 7.42254i) q^{83} +(-6.59888 + 11.4296i) q^{85} +(-2.17928 - 1.25821i) q^{86} +(3.90545 + 6.76443i) q^{88} +10.5358 q^{89} +(3.65383 + 3.06277i) q^{91} +(3.26509 - 1.88510i) q^{92} +(0.136691 + 0.0789188i) q^{94} +(3.14833 + 1.81769i) q^{95} +(6.30108 - 3.63793i) q^{97} +(4.42210 - 0.784360i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} + 2 q^{4} - 2 q^{7}+O(q^{10}) $ 12 * q + 6 * q^2 + 2 * q^4 - 2 * q^7	$ 12 q + 6 q^{2} + 2 q^{4} - 2 q^{7} + 12 q^{14} + 2 q^{16} - 10 q^{22} - 24 q^{23} - 8 q^{28} - 30 q^{29} + 12 q^{32} - 4 q^{37} - 10 q^{43} - 40 q^{46} + 6 q^{49} + 36 q^{50} - 42 q^{56} + 2 q^{58} + 16 q^{64} + 78 q^{65} + 12 q^{67} + 18 q^{70} + 12 q^{74} + 24 q^{77} - 6 q^{79} - 6 q^{85} - 96 q^{86} + 34 q^{88} - 24 q^{91} - 30 q^{92} - 72 q^{95}+O(q^{100}) $ 12 * q + 6 * q^2 + 2 * q^4 - 2 * q^7 + 12 * q^14 + 2 * q^16 - 10 * q^22 - 24 * q^23 - 8 * q^28 - 30 * q^29 + 12 * q^32 - 4 * q^37 - 10 * q^43 - 40 * q^46 + 6 * q^49 + 36 * q^50 - 42 * q^56 + 2 * q^58 + 16 * q^64 + 78 * q^65 + 12 * q^67 + 18 * q^70 + 12 * q^74 + 24 * q^77 - 6 * q^79 - 6 * q^85 - 96 * q^86 + 34 * q^88 - 24 * q^91 - 30 * q^92 - 72 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$

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 $

$2$	0.555632	−	0.320794i	0.392891	−	0.226836i	−0.290521	−	0.956869i	$-0.593829\pi$
$2$	0.555632	−	0.320794i	0.392891	−	0.226836i	0.683412	+	0.730033i	$0.260495\pi$
$3$		0			0
$3$		0			0
$4$	−0.794182	+	1.37556i	−0.397091	+	0.687782i
$5$	1.10552	−	1.91482i	0.494405	−	0.856335i	−0.505574	−	0.862783i	$-0.668719\pi$
$5$	1.10552	−	1.91482i	0.494405	−	0.856335i	0.999979	+	0.00644798i	$0.00205247\pi$
$6$		0			0
$7$	2.60579	+	0.458109i	0.984896	+	0.173149i
$7$	2.60579	+	0.458109i	0.984896	+	0.173149i
$8$			2.30225i			0.813970i
$9$		0			0
$10$		−	1.41858i		−	0.448596i
$11$	2.93818	−	1.69636i	0.885894	−	0.511471i	0.0132968	−	0.999912i	$-0.495767\pi$
$11$	2.93818	−	1.69636i	0.885894	−	0.511471i	0.872597	+	0.488440i	$0.162434\pi$
$12$		0			0
$13$	1.56060	+	0.901012i	0.432832	+	0.249896i	0.700552	−	0.713601i	$-0.252937\pi$
$13$	1.56060	+	0.901012i	0.432832	+	0.249896i	−0.267720	+	0.963497i	$0.586270\pi$
$14$	1.59482	−	0.581383i	0.426233	−	0.155381i
$15$		0			0
$16$	−0.849814	−	1.47192i	−0.212454	−	0.367980i
$17$	−5.96901			−1.44770			−0.723849	−	0.689959i	$-0.757629\pi$
$17$	−5.96901			−1.44770			−0.723849	+	0.689959i	$0.757629\pi$
$18$		0			0
$19$			1.64419i			0.377202i	0.982054	+	0.188601i	$0.0603953\pi$
$19$			1.64419i			0.377202i	−0.982054	+	0.188601i	$0.939605\pi$
$20$	1.75597	+	3.04144i	0.392648	+	0.680086i
$21$		0			0
$22$	1.08836	−	1.88510i	0.232040	−	0.401905i
$23$	−2.05563	−	1.18682i	−0.428629	−	0.247469i	0.270133	−	0.962823i	$-0.412932\pi$
$23$	−2.05563	−	1.18682i	−0.428629	−	0.247469i	−0.698762	+	0.715354i	$0.746266\pi$
$24$		0			0
$25$	0.0556321	+	0.0963576i	0.0111264	+	0.0192715i
$26$	1.15616			0.226741
$27$		0			0
$28$	−2.69963	+	3.22061i	−0.510182	+	0.608638i
$29$	−2.44437	+	1.41126i	−0.453908	+	0.262064i	−0.709479	−	0.704726i	$-0.751070\pi$
$29$	−2.44437	+	1.41126i	−0.453908	+	0.262064i	0.255571	+	0.966790i	$0.417736\pi$
$30$		0			0
$31$	−9.28558	−	5.36103i	−1.66774	−	0.962870i	−0.968853	−	0.247638i	$-0.920346\pi$
$31$	−9.28558	−	5.36103i	−1.66774	−	0.962870i	−0.698887	−	0.715232i	$-0.746321\pi$
$32$	−4.93199	−	2.84748i	−0.871861	−	0.503369i
$33$		0			0
$34$	−3.31657	+	1.91482i	−0.568788	+	0.328390i
$35$	3.75796	−	4.48318i	0.635211	−	0.757795i
$36$		0			0
$37$	1.69963			0.279417			0.139709	−	0.990193i	$-0.455383\pi$
$37$	1.69963			0.279417			0.139709	+	0.990193i	$0.455383\pi$
$38$	0.527445	+	0.913562i	0.0855630	+	0.148199i
$39$		0			0
$40$	4.40841	+	2.54520i	0.697031	+	0.402431i
$41$	−0.455074	+	0.788211i	−0.0710706	+	0.123098i	−0.899371	−	0.437187i	$-0.855975\pi$
$41$	−0.455074	+	0.788211i	−0.0710706	+	0.123098i	0.828300	+	0.560285i	$0.189308\pi$
$42$		0			0
$43$	−1.96108	−	3.39669i	−0.299062	−	0.517990i	0.676860	−	0.736112i	$-0.263340\pi$
$43$	−1.96108	−	3.39669i	−0.299062	−	0.517990i	−0.975922	+	0.218122i	$0.930007\pi$
$44$			5.38887i			0.812402i
$45$		0			0
$46$	−1.52290			−0.224539
$47$	0.123005	+	0.213051i	0.0179422	+	0.0310767i	0.874857	−	0.484381i	$-0.160955\pi$
$47$	0.123005	+	0.213051i	0.0179422	+	0.0310767i	−0.856915	+	0.515458i	$0.827622\pi$
$48$		0			0
$49$	6.58027	+	2.38747i	0.940039	+	0.341067i
$50$	0.0618219	+	0.0356929i	0.00874294	+	0.00504774i
$51$		0			0
$52$	−2.47880	+	1.43113i	−0.343747	+	0.198463i
$53$			7.87589i			1.08184i	0.841075	+	0.540919i	$0.181923\pi$
$53$			7.87589i			1.08184i	−0.841075	+	0.540919i	$0.818077\pi$
$54$		0			0
$55$		−	7.50146i		−	1.01150i
$56$	−1.05468	+	5.99919i	−0.140938	+	0.801675i
$57$		0			0
$58$	−0.905446	+	1.56828i	−0.118891	+	0.205925i
$59$	−5.39093	+	9.33736i	−0.701839	+	1.21562i	0.265981	+	0.963978i	$0.414304\pi$
$59$	−5.39093	+	9.33736i	−0.701839	+	1.21562i	−0.967820	+	0.251643i	$0.919029\pi$
$60$		0			0
$61$	−1.22853	+	0.709292i	−0.157297	+	0.0908155i	−0.576582	−	0.817039i	$-0.695614\pi$
$61$	−1.22853	+	0.709292i	−0.157297	+	0.0908155i	0.419285	+	0.907855i	$0.362281\pi$
$62$	−6.87916			−0.873654
$63$		0			0
$64$	−0.254572			−0.0318214
$65$	3.45056	−	1.99218i	0.427989	−	0.247100i
$66$		0			0
$67$	3.99381	−	6.91748i	0.487922	−	0.845105i	−0.511982	−	0.858996i	$-0.671089\pi$
$67$	3.99381	−	6.91748i	0.487922	−	0.845105i	0.999904	+	0.0138913i	$0.00442187\pi$
$68$	4.74048	−	8.21075i	0.574868	−	0.995700i
$69$		0			0
$70$	0.649865	−	3.69653i	0.0776738	−	0.441820i
$71$		−	12.1743i		−	1.44482i	−0.691463	−	0.722412i	$-0.743034\pi$
$71$		−	12.1743i		−	1.44482i	0.691463	−	0.722412i	$-0.256966\pi$
$72$		0			0
$73$		−	0.426103i		−	0.0498715i	−0.999689	−	0.0249358i	$-0.992062\pi$
$73$		−	0.426103i		−	0.0498715i	0.999689	−	0.0249358i	$-0.00793812\pi$
$74$	0.944368	−	0.545231i	0.109781	−	0.0633818i
$75$		0			0
$76$	−2.26168	−	1.30578i	−0.259433	−	0.149784i
$77$	8.43339	−	3.07435i	0.961074	−	0.350354i
$78$		0			0
$79$	2.49381	+	4.31941i	0.280576	+	0.485971i	0.971527	−	0.236930i	$-0.0761413\pi$
$79$	2.49381	+	4.31941i	0.280576	+	0.485971i	−0.690951	+	0.722902i	$0.742808\pi$
$80$	−3.75796			−0.420153
$81$		0			0
$82$			0.583940i			0.0644854i
$83$	−4.28541	−	7.42254i	−0.470384	−	0.814730i	0.529042	−	0.848596i	$-0.322551\pi$
$83$	−4.28541	−	7.42254i	−0.470384	−	0.814730i	−0.999426	+	0.0338660i	$0.989218\pi$
$84$		0			0
$85$	−6.59888	+	11.4296i	−0.715750	+	1.23971i
$86$	−2.17928	−	1.25821i	−0.234997	−	0.135676i
$87$		0			0
$88$	3.90545	+	6.76443i	0.416322	+	0.721091i
$89$	10.5358			1.11680			0.558399	−	0.829573i	$-0.311416\pi$
$89$	10.5358			1.11680			0.558399	+	0.829573i	$0.311416\pi$
$90$		0			0
$91$	3.65383	+	3.06277i	0.383025	+	0.321065i
$92$	3.26509	−	1.88510i	0.340409	−	0.196535i
$93$		0			0
$94$	0.136691	+	0.0789188i	0.0140986	+	0.00813985i
$95$	3.14833	+	1.81769i	0.323012	+	0.186491i
$96$		0			0
$97$	6.30108	−	3.63793i	0.639777	−	0.369376i	−0.144751	−	0.989468i	$-0.546238\pi$
$97$	6.30108	−	3.63793i	0.639777	−	0.369376i	0.784529	+	0.620092i	$0.212905\pi$
$98$	4.42210	−	0.784360i	0.446699	−	0.0792324i
$99$		0			0
$100$	−0.176728			−0.0176728
$101$	−2.33405	−	4.04270i	−0.232247	−	0.402264i	0.726222	−	0.687460i	$-0.241274\pi$
$101$	−2.33405	−	4.04270i	−0.232247	−	0.402264i	−0.958469	+	0.285197i	$0.907941\pi$
$102$		0			0
$103$	5.40462	+	3.12036i	0.532533	+	0.307458i	0.742047	−	0.670348i	$-0.233855\pi$
$103$	5.40462	+	3.12036i	0.532533	+	0.307458i	−0.209515	+	0.977806i	$0.567188\pi$
$104$	−2.07436	+	3.59289i	−0.203407	+	0.352312i
$105$		0			0
$106$	2.52654	+	4.37610i	0.245399	+	0.425044i
$107$			1.48939i			0.143985i	0.997405	+	0.0719925i	$0.0229358\pi$
$107$			1.48939i			0.143985i	−0.997405	+	0.0719925i	$0.977064\pi$
$108$		0			0
$109$	−4.38688			−0.420187			−0.210093	−	0.977681i	$-0.567377\pi$
$109$	−4.38688			−0.420187			−0.210093	+	0.977681i	$0.567377\pi$
$110$	−2.40643	−	4.16805i	−0.229444	−	0.397408i
$111$		0			0
$112$	−1.54014	−	4.22482i	−0.145529	−	0.399208i
$113$	14.8764	+	8.58887i	1.39945	+	0.807973i	0.994335	−	0.106293i	$-0.0338981\pi$
$113$	14.8764	+	8.58887i	1.39945	+	0.807973i	0.405115	+	0.914266i	$0.367231\pi$
$114$		0			0
$115$	−4.54510	+	2.62412i	−0.423833	+	0.244700i
$116$		−	4.48318i		−	0.416253i
$117$		0			0
$118$			6.91752i			0.636809i
$119$	−15.5540	−	2.73445i	−1.42583	−	0.250667i
$120$		0			0
$121$	0.255260	−	0.442124i	0.0232055	−	0.0401931i
$122$	−0.455074	+	0.788211i	−0.0412004	+	0.0713612i
$123$		0			0
$124$	14.7489	−	8.51527i	1.32449	−	0.764694i
$125$	11.3013			1.01081
$126$		0			0
$127$	6.32141			0.560935			0.280467	−	0.959864i	$-0.409511\pi$
$127$	6.32141			0.560935			0.280467	+	0.959864i	$0.409511\pi$
$128$	9.72253	−	5.61330i	0.859358	−	0.496151i
$129$		0			0
$130$	1.27816	−	2.21384i	0.112102	−	0.194167i
$131$	−8.51213	+	14.7434i	−0.743708	+	1.28814i	0.207088	+	0.978322i	$0.433601\pi$
$131$	−8.51213	+	14.7434i	−0.743708	+	1.28814i	−0.950796	+	0.309818i	$0.899732\pi$
$132$		0			0
$133$	−0.753215	+	4.28440i	−0.0653121	+	0.371505i
$134$		−	5.12477i		−	0.442712i
$135$		0			0
$136$		−	13.7422i		−	1.17838i
$137$	−5.42580	+	3.13259i	−0.463557	+	0.267635i	−0.713539	−	0.700616i	$-0.752909\pi$
$137$	−5.42580	+	3.13259i	−0.463557	+	0.267635i	0.249982	+	0.968251i	$0.419575\pi$
$138$		0			0
$139$	−6.65488	−	3.84220i	−0.564460	−	0.325891i	0.190474	−	0.981692i	$-0.438998\pi$
$139$	−6.65488	−	3.84220i	−0.564460	−	0.325891i	−0.754934	+	0.655801i	$0.772331\pi$
$140$	3.18239	+	8.72977i	0.268961	+	0.737800i
$141$		0			0
$142$	−3.90545	−	6.76443i	−0.327738	−	0.567658i
$143$	6.11375			0.511258
$144$		0			0
$145$			6.24071i			0.518263i
$146$	−0.136691	−	0.236756i	−0.0113127	−	0.0195941i
$147$		0			0
$148$	−1.34981	+	2.33795i	−0.110954	+	0.192178i
$149$	−13.3695	−	7.71887i	−1.09527	−	0.632355i	−0.160296	−	0.987069i	$-0.551245\pi$
$149$	−13.3695	−	7.71887i	−1.09527	−	0.632355i	−0.934975	+	0.354714i	$0.884578\pi$
$150$		0			0
$151$	−5.84362	−	10.1215i	−0.475547	−	0.823672i	0.524060	−	0.851681i	$-0.324417\pi$
$151$	−5.84362	−	10.1215i	−0.475547	−	0.823672i	−0.999608	+	0.0280089i	$0.991083\pi$
$152$	−3.78533			−0.307031
$153$		0			0
$154$	3.69963	−	4.41359i	0.298125	−	0.355657i
$155$	−20.5309	+	11.8535i	−1.64908	+	0.952096i
$156$		0			0
$157$	4.93586	+	2.84972i	0.393924	+	0.227432i	0.683859	−	0.729614i	$-0.260300\pi$
$157$	4.93586	+	2.84972i	0.393924	+	0.227432i	−0.289935	+	0.957046i	$0.593634\pi$
$158$	2.77128	+	1.60000i	0.220471	+	0.127289i
$159$		0			0
$160$	−10.9049	+	6.29593i	−0.862105	+	0.497737i
$161$	−4.81285	−	4.03430i	−0.379306	−	0.317948i
$162$		0			0
$163$	−10.2101			−0.799721			−0.399860	−	0.916576i	$-0.630941\pi$
$163$	−10.2101			−0.799721			−0.399860	+	0.916576i	$0.630941\pi$
$164$	−0.722823	−	1.25197i	−0.0564430	−	0.0977621i
$165$		0			0
$166$	−4.76222	−	2.74947i	−0.369620	−	0.213400i
$167$	1.80661	−	3.12914i	0.139800	−	0.242140i	−0.787621	−	0.616160i	$-0.788688\pi$
$167$	1.80661	−	3.12914i	0.139800	−	0.242140i	0.927421	+	0.374020i	$0.122021\pi$
$168$		0			0
$169$	−4.87636	−	8.44610i	−0.375104	−	0.649700i
$170$			8.46754i			0.649431i
$171$		0			0
$172$	6.22981			0.475019
$173$	9.03957	+	15.6570i	0.687266	+	1.19038i	0.972719	+	0.231987i	$0.0745226\pi$
$173$	9.03957	+	15.6570i	0.687266	+	1.19038i	−0.285453	+	0.958393i	$0.592144\pi$
$174$		0			0
$175$	0.100823	+	0.276573i	0.00762152	+	0.0209070i
$176$	−4.99381	−	2.88318i	−0.376423	−	0.217328i
$177$		0			0
$178$	5.85406	−	3.37984i	0.438780	−	0.253330i
$179$		−	5.03194i		−	0.376105i	−0.982159	−	0.188052i	$-0.939783\pi$
$179$		−	5.03194i		−	0.376105i	0.982159	−	0.188052i	$-0.0602175\pi$
$180$		0			0
$181$			13.5592i			1.00785i	0.863747	+	0.503925i	$0.168111\pi$
$181$			13.5592i			1.00785i	−0.863747	+	0.503925i	$0.831889\pi$
$182$	3.01270	+	0.529646i	0.223316	+	0.0392600i
$183$		0			0
$184$	2.73236	−	4.73259i	0.201432	−	0.348891i
$185$	1.87898	−	3.25449i	0.138145	−	0.239275i
$186$		0			0
$187$	−17.5380	+	10.1256i	−1.28251	+	0.740455i
$188$	−0.390754			−0.0284987
$189$		0			0
$190$	2.33242			0.169211
$191$	8.86948	−	5.12080i	0.641773	−	0.370528i	−0.143524	−	0.989647i	$-0.545844\pi$
$191$	8.86948	−	5.12080i	0.641773	−	0.370528i	0.785297	+	0.619119i	$0.212510\pi$
$192$		0			0
$193$	−8.06615	+	13.9710i	−0.580614	+	1.00565i	0.414792	+	0.909916i	$0.363854\pi$
$193$	−8.06615	+	13.9710i	−0.580614	+	1.00565i	−0.995407	+	0.0957374i	$0.969479\pi$
$194$	2.33405	−	4.04270i	0.167575	−	0.290249i
$195$		0			0
$196$	−8.51005	+	7.15550i	−0.607861	+	0.511107i
$197$		−	3.86303i		−	0.275230i	−0.990486	−	0.137615i	$-0.956056\pi$
$197$		−	3.86303i		−	0.275230i	0.990486	−	0.137615i	$-0.0439436\pi$
$198$		0			0
$199$			15.2034i			1.07774i	0.842388	+	0.538871i	$0.181149\pi$
$199$			15.2034i			1.07774i	−0.842388	+	0.538871i	$0.818851\pi$
$200$	−0.221840	+	0.128079i	−0.0156864	+	0.00905656i
$201$		0			0
$202$	−2.59375	−	1.49750i	−0.182496	−	0.105364i
$203$	−7.01602	+	2.55765i	−0.492428	+	0.179512i
$204$		0			0
$205$	1.00619	+	1.74277i	0.0702753	+	0.121720i
$206$	4.00397			0.278970
$207$		0			0
$208$		−	3.06277i		−	0.212365i
$209$	2.78913	+	4.83091i	0.192928	+	0.334161i
$210$		0			0
$211$	11.9523	−	20.7021i	0.822833	−	1.42519i	−0.0807311	−	0.996736i	$-0.525726\pi$
$211$	11.9523	−	20.7021i	0.822833	−	1.42519i	0.903564	−	0.428453i	$-0.140941\pi$
$212$	−10.8338	−	6.25489i	−0.744068	−	0.429588i
$213$		0			0
$214$	0.477789	+	0.827554i	0.0326610	+	0.0565704i
$215$	−8.67208			−0.591431
$216$		0			0
$217$	−21.7403	−	18.2235i	−1.47583	−	1.23709i
$218$	−2.43749	+	1.40729i	−0.165088	+	0.0953134i
$219$		0			0
$220$	10.3187	+	5.95752i	0.695689	+	0.401656i
$221$	−9.31522	−	5.37815i	−0.626610	−	0.361773i
$222$		0			0
$223$	16.6198	−	9.59545i	1.11294	−	0.642559i	0.173354	−	0.984860i	$-0.444539\pi$
$223$	16.6198	−	9.59545i	1.11294	−	0.642559i	0.939591	+	0.342300i	$0.111206\pi$
$224$	−11.5473	−	9.67933i	−0.771534	−	0.646727i
$225$		0			0
$226$	11.0210			0.733109
$227$	4.33604	+	7.51024i	0.287793	+	0.498472i	0.973283	−	0.229610i	$-0.0737451\pi$
$227$	4.33604	+	7.51024i	0.287793	+	0.498472i	−0.685490	+	0.728082i	$0.740412\pi$
$228$		0			0
$229$	−12.4437	−	7.18439i	−0.822304	−	0.474758i	0.0289060	−	0.999582i	$-0.490798\pi$
$229$	−12.4437	−	7.18439i	−0.822304	−	0.474758i	−0.851211	+	0.524824i	$0.824131\pi$
$230$	−1.68360	+	2.91609i	−0.111014	+	0.192281i
$231$		0			0
$232$	−3.24907	−	5.62755i	−0.213312	−	0.369467i
$233$			29.7160i			1.94676i	0.229194	+	0.973381i	$0.426391\pi$
$233$			29.7160i			1.94676i	−0.229194	+	0.973381i	$0.573609\pi$
$234$		0			0
$235$	0.543941			0.0354828
$236$	−8.56276	−	14.8311i	−0.557388	−	0.965425i
$237$		0			0
$238$	−9.51949	+	3.47028i	−0.617057	+	0.224945i
$239$	13.7101	+	7.91556i	0.886836	+	0.512015i	0.872906	−	0.487888i	$-0.162233\pi$
$239$	13.7101	+	7.91556i	0.886836	+	0.512015i	0.0139296	+	0.999903i	$0.495566\pi$
$240$		0			0
$241$	4.34973	−	2.51132i	0.280190	−	0.161768i	−0.353319	−	0.935503i	$-0.614947\pi$
$241$	4.34973	−	2.51132i	0.280190	−	0.161768i	0.633510	+	0.773735i	$0.281614\pi$
$242$		−	0.327544i		−	0.0210553i
$243$		0			0
$244$		−	2.25323i		−	0.144248i
$245$	11.8462	−	9.96066i	0.756828	−	0.636363i
$246$		0			0
$247$	−1.48143	+	2.56591i	−0.0942612	+	0.163265i
$248$	12.3425	−	21.3778i	0.783747	−	1.35749i
$249$		0			0
$250$	6.27934	−	3.62538i	0.397140	−	0.229289i
$251$	−7.29728			−0.460600			−0.230300	−	0.973120i	$-0.573971\pi$
$251$	−7.29728			−0.460600			−0.230300	+	0.973120i	$0.573971\pi$
$252$		0			0
$253$	−8.05308			−0.506293
$254$	3.51238	−	2.02787i	0.220386	−	0.127240i
$255$		0			0
$256$	3.85600	−	6.67879i	0.241000	−	0.417425i
$257$	−4.00397	+	6.93508i	−0.249761	+	0.432598i	−0.963459	−	0.267855i	$-0.913685\pi$
$257$	−4.00397	+	6.93508i	−0.249761	+	0.432598i	0.713699	+	0.700453i	$0.247019\pi$
$258$		0			0
$259$	4.42887	+	0.778614i	0.275197	+	0.0483807i
$260$			6.32862i			0.392484i
$261$		0			0
$262$			10.9226i			0.674798i
$263$	13.6051	−	7.85489i	0.838925	−	0.484353i	−0.0179738	−	0.999838i	$-0.505722\pi$
$263$	13.6051	−	7.85489i	0.838925	−	0.484353i	0.856899	+	0.515485i	$0.172388\pi$
$264$		0			0
$265$	15.0810	+	8.70699i	0.926416	+	0.534866i
$266$	0.955901	+	2.62218i	0.0586100	+	0.160776i
$267$		0			0
$268$	6.34362	+	10.9875i	0.387499	+	0.671167i
$269$	10.4924			0.639731			0.319866	−	0.947463i	$-0.396362\pi$
$269$	10.4924			0.639731			0.319866	+	0.947463i	$0.396362\pi$
$270$		0			0
$271$		−	22.2537i		−	1.35181i	−0.736987	−	0.675907i	$-0.763752\pi$
$271$		−	22.2537i		−	1.35181i	0.736987	−	0.675907i	$-0.236248\pi$
$272$	5.07255	+	8.78591i	0.307568	+	0.532724i
$273$		0			0
$274$	−2.00983	+	3.48113i	−0.121418	+	0.210303i
$275$	0.326914	+	0.188744i	0.0197137	+	0.0113817i
$276$		0			0
$277$	11.4251	+	19.7889i	0.686468	+	1.18900i	0.972973	+	0.230919i	$0.0741733\pi$
$277$	11.4251	+	19.7889i	0.686468	+	1.18900i	−0.286505	+	0.958079i	$0.592493\pi$
$278$	−4.93022			−0.295695
$279$		0			0
$280$	10.3214	+	8.65178i	0.616822	+	0.517043i
$281$	−0.796041	+	0.459595i	−0.0474878	+	0.0274171i	−0.523556	−	0.851991i	$-0.675395\pi$
$281$	−0.796041	+	0.459595i	−0.0474878	+	0.0274171i	0.476068	+	0.879408i	$0.342062\pi$
$282$		0			0
$283$	19.1573	+	11.0605i	1.13878	+	0.657477i	0.946129	−	0.323790i	$-0.104957\pi$
$283$	19.1573	+	11.0605i	1.13878	+	0.657477i	0.192654	+	0.981267i	$0.438290\pi$
$284$	16.7465	+	9.66861i	0.993723	+	0.573726i
$285$		0			0
$286$	3.39700	−	1.96126i	0.200869	−	0.115972i
$287$	−1.54691	+	1.84544i	−0.0913113	+	0.108933i
$288$		0			0
$289$	18.6291			1.09583
$290$	2.00199	+	3.46754i	0.117561	+	0.203621i
$291$		0			0
$292$	0.586131	+	0.338403i	0.0343007	+	0.0198035i
$293$	14.6259	−	25.3328i	0.854453	−	1.47996i	−0.0226986	−	0.999742i	$-0.507226\pi$
$293$	14.6259	−	25.3328i	0.854453	−	1.47996i	0.877152	−	0.480214i	$-0.159441\pi$
$294$		0			0
$295$	11.9196	+	20.6454i	0.693986	+	1.20202i
$296$			3.91298i			0.227437i
$297$		0			0
$298$	−9.90468			−0.573763
$299$	−2.13868	−	3.70430i	−0.123683	−	0.214225i
$300$		0			0
$301$	−3.55410	−	9.74944i	−0.204855	−	0.561948i
$302$	−6.49381	−	3.74920i	−0.373677	−	0.215742i
$303$		0			0
$304$	2.42011	−	1.39725i	0.138803	−	0.0801379i
$305$			3.13656i			0.179599i
$306$		0			0
$307$		−	14.8451i		−	0.847254i	−0.905837	−	0.423627i	$-0.860757\pi$
$307$		−	14.8451i		−	0.847254i	0.905837	−	0.423627i	$-0.139243\pi$
$308$	−2.46869	+	14.0423i	−0.140666	+	0.800132i
$309$		0			0
$310$	−7.60507	+	13.1724i	−0.431939	+	0.748141i
$311$	9.69002	−	16.7836i	0.549471	−	0.951711i	−0.448840	−	0.893612i	$-0.648163\pi$
$311$	9.69002	−	16.7836i	0.549471	−	0.951711i	0.998311	−	0.0580991i	$-0.0185040\pi$
$312$		0			0
$313$	−12.6608	+	7.30974i	−0.715633	+	0.413171i	−0.813143	−	0.582064i	$-0.802245\pi$
$313$	−12.6608	+	7.30974i	−0.715633	+	0.413171i	0.0975102	+	0.995235i	$0.468912\pi$
$314$	3.65669			0.206359
$315$		0			0
$316$	−7.92216			−0.445656
$317$	14.7046	−	8.48973i	0.825895	−	0.476831i	−0.0265499	−	0.999647i	$-0.508452\pi$
$317$	14.7046	−	8.48973i	0.825895	−	0.476831i	0.852445	+	0.522817i	$0.175119\pi$
$318$		0			0
$319$	−4.78799	+	8.29305i	−0.268076	+	0.464321i
$320$	−0.281435	+	0.487460i	−0.0157327	+	0.0272498i
$321$		0			0
$322$	−3.96836	−	0.697654i	−0.221148	−	0.0388787i
$323$		−	9.81416i		−	0.546074i
$324$		0			0
$325$			0.200501i			0.0111218i
$326$	−5.67309	+	3.27536i	−0.314203	+	0.181405i
$327$		0			0
$328$	−1.81466	−	1.04769i	−0.100198	−	0.0578493i
$329$	0.222925	+	0.611517i	0.0122903	+	0.0337140i
$330$		0			0
$331$	−9.94801	−	17.2305i	−0.546792	−	0.947072i	−0.998492	−	0.0549016i	$-0.982515\pi$
$331$	−9.94801	−	17.2305i	−0.546792	−	0.947072i	0.451700	−	0.892170i	$-0.350818\pi$
$332$	13.6136			0.747142
$333$		0			0
$334$		−	2.31820i		−	0.126846i
$335$	−8.83051	−	15.2949i	−0.482462	−	0.835649i
$336$		0			0
$337$	0.490168	−	0.848996i	0.0267012	−	0.0462478i	−0.852366	−	0.522946i	$-0.824833\pi$
$337$	0.490168	−	0.848996i	0.0267012	−	0.0462478i	0.879067	+	0.476698i	$0.158166\pi$
$338$	−5.41892	−	3.12861i	−0.294750	−	0.170174i
$339$		0			0
$340$	−10.4814	−	18.1544i	−0.568435	−	0.984559i
$341$	−36.3769			−1.96992
$342$		0			0
$343$	16.0531	+	9.23572i	0.866785	+	0.498682i
$344$	7.82004	−	4.51490i	0.421628	−	0.243427i
$345$		0			0
$346$	10.0454	+	5.79969i	0.540041	+	0.311793i
$347$	−18.3702	−	10.6060i	−0.986162	−	0.569361i	−0.0820373	−	0.996629i	$-0.526143\pi$
$347$	−18.3702	−	10.6060i	−0.986162	−	0.569361i	−0.904125	+	0.427268i	$0.859476\pi$
$348$		0			0
$349$	−8.69945	+	5.02263i	−0.465671	+	0.268855i	−0.714426	−	0.699711i	$-0.753312\pi$
$349$	−8.69945	+	5.02263i	−0.465671	+	0.268855i	0.248755	+	0.968566i	$0.419979\pi$
$350$	0.144744	+	0.121329i	0.00773688	+	0.00648533i
$351$		0			0
$352$	−19.3214			−1.02983
$353$	1.37327	+	2.37858i	0.0730920	+	0.126599i	0.900255	−	0.435363i	$-0.143380\pi$
$353$	1.37327	+	2.37858i	0.0730920	+	0.126599i	−0.827163	+	0.561962i	$0.810047\pi$
$354$		0			0
$355$	−23.3116	−	13.4590i	−1.23725	−	0.714329i
$356$	−8.36738	+	14.4927i	−0.443470	+	0.768113i
$357$		0			0
$358$	−1.61422	−	2.79591i	−0.0853140	−	0.147768i
$359$		−	10.0013i		−	0.527849i	−0.964543	−	0.263925i	$-0.914983\pi$
$359$		−	10.0013i		−	0.527849i	0.964543	−	0.263925i	$-0.0850170\pi$
$360$		0			0
$361$	16.2967			0.857719
$362$	4.34973	+	7.53395i	0.228616	+	0.395975i
$363$		0			0
$364$	−7.11484	+	2.59368i	−0.372919	+	0.135946i
$365$	−0.815912	−	0.471067i	−0.0427068	−	0.0246568i
$366$		0			0
$367$	5.03560	−	2.90731i	0.262856	−	0.151760i	−0.362781	−	0.931875i	$-0.618173\pi$
$367$	5.03560	−	2.90731i	0.262856	−	0.151760i	0.625637	+	0.780114i	$0.284839\pi$
$368$			4.03430i			0.210303i
$369$		0			0
$370$		−	2.41106i		−	0.125345i
$371$	−3.60801	+	20.5229i	−0.187319	+	1.06550i
$372$		0			0
$373$	7.75959	−	13.4400i	0.401776	−	0.695897i	−0.592164	−	0.805817i	$-0.701726\pi$
$373$	7.75959	−	13.4400i	0.401776	−	0.695897i	0.993940	+	0.109920i	$0.0350596\pi$
$374$	−6.49645	+	11.2522i	−0.335924	+	0.581837i
$375$		0			0
$376$	−0.490498	+	0.283189i	−0.0252955	+	0.0146044i
$377$	−5.08623			−0.261954
$378$		0			0
$379$	2.79714			0.143679			0.0718396	−	0.997416i	$-0.477113\pi$
$379$	2.79714			0.143679			0.0718396	+	0.997416i	$0.477113\pi$
$380$	−5.00069	+	2.88715i	−0.256530	+	0.148108i
$381$		0			0
$382$	3.28544	−	5.69056i	0.168098	−	0.291154i
$383$	−1.74229	+	3.01773i	−0.0890268	+	0.154199i	−0.907100	−	0.420915i	$-0.861709\pi$
$383$	−1.74229	+	3.01773i	−0.0890268	+	0.154199i	0.818073	+	0.575114i	$0.195042\pi$
$384$		0			0
$385$	3.43648	−	19.5472i	0.175139	−	0.996219i
$386$			10.3503i			0.526817i
$387$		0			0
$388$			11.5567i			0.586703i
$389$	6.37017	−	3.67782i	0.322980	−	0.186473i	−0.329740	−	0.944072i	$-0.606961\pi$
$389$	6.37017	−	3.67782i	0.322980	−	0.186473i	0.652720	+	0.757599i	$0.273628\pi$
$390$		0			0
$391$	12.2701	+	7.08414i	0.620525	+	0.358260i
$392$	−5.49656	+	15.1495i	−0.277618	+	0.765163i
$393$		0			0
$394$	−1.23924	−	2.14642i	−0.0624319	−	0.108135i
$395$	11.0279			0.554872
$396$		0			0
$397$			19.2838i			0.967825i	0.875116	+	0.483912i	$0.160785\pi$
$397$			19.2838i			0.967825i	−0.875116	+	0.483912i	$0.839215\pi$
$398$	4.87717	+	8.44751i	0.244470	+	0.423435i
$399$		0			0
$400$	0.0945538	−	0.163772i	0.00472769	−	0.00818860i
$401$	−9.60576	−	5.54589i	−0.479689	−	0.276949i	0.240598	−	0.970625i	$-0.422656\pi$
$401$	−9.60576	−	5.54589i	−0.479689	−	0.276949i	−0.720287	+	0.693676i	$0.755990\pi$
$402$		0			0
$403$	−9.66071	−	16.7328i	−0.481234	−	0.833522i
$404$	7.41465			0.368893
$405$		0			0
$406$	−3.07784	+	3.67181i	−0.152751	+	0.182229i
$407$	4.99381	−	2.88318i	0.247534	−	0.142914i
$408$		0			0
$409$	−17.5597	−	10.1381i	−0.868274	−	0.501298i	−0.00149954	−	0.999999i	$-0.500477\pi$
$409$	−17.5597	−	10.1381i	−0.868274	−	0.501298i	−0.866774	+	0.498701i	$0.833811\pi$
$410$	1.11814	+	0.645560i	0.0552211	+	0.0318819i
$411$		0			0
$412$	−8.58450	+	4.95626i	−0.422928	+	0.244178i
$413$	−18.3252	+	21.8616i	−0.901722	+	1.07574i
$414$		0			0
$415$	−18.9505			−0.930243
$416$	−5.13123	−	8.88756i	−0.251579	−	0.435748i
$417$		0			0
$418$	3.09946	+	1.78947i	0.151599	+	0.0875260i
$419$	−5.54936	+	9.61177i	−0.271104	+	0.469566i	−0.969145	−	0.246492i	$-0.920722\pi$
$419$	−5.54936	+	9.61177i	−0.271104	+	0.469566i	0.698041	+	0.716058i	$0.254055\pi$
$420$		0			0
$421$	4.59269	+	7.95478i	0.223834	+	0.387692i	0.955969	−	0.293467i	$-0.0948092\pi$
$421$	4.59269	+	7.95478i	0.223834	+	0.387692i	−0.732135	+	0.681160i	$0.761476\pi$
$422$		−	15.3370i		−	0.746592i
$423$		0			0
$424$	−18.1323			−0.880583
$425$	−0.332068	−	0.575159i	−0.0161077	−	0.0278993i
$426$		0			0
$427$	−3.52622	+	1.28547i	−0.170646	+	0.0622080i
$428$	−2.04875	−	1.18285i	−0.0990302	−	0.0571751i
$429$		0			0
$430$	−4.81849	+	2.78195i	−0.232368	+	0.134158i
$431$			15.1102i			0.727833i	0.931432	+	0.363916i	$0.118561\pi$
$431$			15.1102i			0.727833i	−0.931432	+	0.363916i	$0.881439\pi$
$432$		0			0
$433$		−	3.33578i		−	0.160307i	−0.996783	−	0.0801537i	$-0.974459\pi$
$433$		−	3.33578i		−	0.160307i	0.996783	−	0.0801537i	$-0.0255411\pi$
$434$	−17.9256	−	3.15140i	−0.860458	−	0.151272i
$435$		0			0
$436$	3.48398	−	6.03443i	0.166852	−	0.288997i
$437$	1.95135	−	3.37984i	0.0933458	−	0.161680i
$438$		0			0
$439$	−5.91032	+	3.41233i	−0.282084	+	0.162861i	−0.634367	−	0.773032i	$-0.718739\pi$
$439$	−5.91032	+	3.41233i	−0.282084	+	0.162861i	0.352282	+	0.935894i	$0.385406\pi$
$440$	17.2703			0.823327
$441$		0			0
$442$	−6.90112			−0.328253
$443$	−9.77747	+	5.64503i	−0.464542	+	0.268203i	−0.713952	−	0.700195i	$-0.753097\pi$
$443$	−9.77747	+	5.64503i	−0.464542	+	0.268203i	0.249410	+	0.968398i	$0.419763\pi$
$444$		0			0
$445$	11.6476	−	20.1743i	0.552151	−	0.956354i
$446$	6.15633	−	10.6631i	0.291511	−	0.504912i
$447$		0			0
$448$	−0.663360	−	0.116621i	−0.0313408	−	0.00550984i
$449$		−	24.8554i		−	1.17300i	−0.809950	−	0.586498i	$-0.800506\pi$
$449$		−	24.8554i		−	1.17300i	0.809950	−	0.586498i	$-0.199494\pi$
$450$		0			0
$451$			3.08787i			0.145402i
$452$	−23.6291	+	13.6422i	−1.11142	+	0.641677i
$453$		0			0
$454$	4.81849	+	2.78195i	0.226143	+	0.130564i
$455$	9.90406	−	3.61047i	0.464310	−	0.169262i
$456$		0			0
$457$	6.30470	+	10.9201i	0.294922	+	0.510819i	0.974967	−	0.222351i	$-0.0713732\pi$
$457$	6.30470	+	10.9201i	0.294922	+	0.510819i	−0.680045	+	0.733170i	$0.738040\pi$
$458$	−9.21884			−0.430768
$459$		0			0
$460$		−	8.33610i		−	0.388673i
$461$	14.4031	+	24.9470i	0.670821	+	1.16190i	0.977672	+	0.210138i	$0.0673913\pi$
$461$	14.4031	+	24.9470i	0.670821	+	1.16190i	−0.306851	+	0.951758i	$0.599275\pi$
$462$		0			0
$463$	−12.5858	+	21.7993i	−0.584912	+	1.01310i	0.409974	+	0.912097i	$0.365538\pi$
$463$	−12.5858	+	21.7993i	−0.584912	+	1.01310i	−0.994886	+	0.101001i	$0.967796\pi$
$464$	4.15452	+	2.39861i	0.192869	+	0.111353i
$465$		0			0
$466$	9.53273	+	16.5112i	0.441595	+	0.764865i
$467$	25.5951			1.18440			0.592199	−	0.805792i	$-0.298260\pi$
$467$	25.5951			1.18440			0.592199	+	0.805792i	$0.298260\pi$
$468$		0			0
$469$	13.5760	−	16.1959i	0.626881	−	0.747857i
$470$	0.302231	−	0.174493i	0.0139409	−	0.00804877i
$471$		0			0
$472$	−21.4970	−	12.4113i	−0.989479	−	0.571276i
$473$	−11.5240	−	6.65338i	−0.529874	−	0.305923i
$474$		0			0
$475$	−0.158430	+	0.0914695i	−0.00726926	+	0.00419691i
$476$	16.1141	−	19.2238i	0.738589	−	0.881123i
$477$		0			0
$478$	10.1571			0.464573
$479$	−0.267749	−	0.463755i	−0.0122338	−	0.0211895i	0.859844	−	0.510557i	$-0.170561\pi$
$479$	−0.267749	−	0.463755i	−0.0122338	−	0.0211895i	−0.872077	+	0.489368i	$0.837228\pi$
$480$		0			0
$481$	2.65244	+	1.53138i	0.120941	+	0.0698251i
$482$	1.61123	−	2.79073i	0.0733896	−	0.127114i
$483$		0			0
$484$	0.405446	+	0.702253i	0.0184294	+	0.0319206i
$485$		−	16.0873i		−	0.730485i
$486$		0			0
$487$	34.1323			1.54668			0.773341	−	0.633990i	$-0.218584\pi$
$487$	34.1323			1.54668			0.773341	+	0.633990i	$0.218584\pi$
$488$	−1.63297	−	2.82839i	−0.0739211	−	0.128035i
$489$		0			0
$490$	3.38682	−	9.33467i	0.153001	−	0.421697i
$491$	−5.86948	−	3.38874i	−0.264886	−	0.152932i	0.361675	−	0.932304i	$-0.382205\pi$
$491$	−5.86948	−	3.38874i	−0.264886	−	0.152932i	−0.626561	+	0.779372i	$0.715538\pi$
$492$		0			0
$493$	14.5905	−	8.42380i	0.657121	−	0.379389i
$494$			1.90094i			0.0855272i
$495$		0			0
$496$			18.2235i			0.818260i
$497$	5.57715	−	31.7237i	0.250169	−	1.42300i
$498$		0			0
$499$	−4.30037	+	7.44846i	−0.192511	+	0.333439i	−0.946082	−	0.323928i	$-0.894996\pi$
$499$	−4.30037	+	7.44846i	−0.192511	+	0.333439i	0.753571	+	0.657367i	$0.228330\pi$
$500$	−8.97525	+	15.5456i	−0.401385	+	0.695220i
$501$		0			0
$502$	−4.05460	+	2.34093i	−0.180966	+	0.104481i
$503$	−2.96518			−0.132211			−0.0661055	−	0.997813i	$-0.521057\pi$
$503$	−2.96518			−0.132211			−0.0661055	+	0.997813i	$0.521057\pi$
$504$		0			0
$505$	−10.3214			−0.459297
$506$	−4.47455	+	2.58338i	−0.198918	+	0.114845i
$507$		0			0
$508$	−5.02035	+	8.69551i	−0.222742	+	0.385801i
$509$	−3.04882	+	5.28072i	−0.135137	+	0.234064i	−0.925650	−	0.378382i	$-0.876481\pi$
$509$	−3.04882	+	5.28072i	−0.135137	+	0.234064i	0.790513	+	0.612445i	$0.209814\pi$
$510$		0			0
$511$	0.195201	−	1.11033i	0.00863519	−	0.0491183i
$512$			17.5053i			0.773631i
$513$		0			0
$514$			5.13780i			0.226619i
$515$	11.9499	−	6.89926i	0.526574	−	0.304018i
$516$		0			0
$517$	0.722823	+	0.417322i	0.0317897	+	0.0183538i
$518$	2.71060	−	0.988134i	0.119097	−	0.0434161i
$519$		0			0
$520$	4.58650	+	7.94406i	0.201132	+	0.348370i
$521$	−32.6929			−1.43230			−0.716150	−	0.697946i	$-0.754097\pi$
$521$	−32.6929			−1.43230			−0.716150	+	0.697946i	$0.754097\pi$
$522$		0			0
$523$			2.00252i			0.0875643i	0.999041	+	0.0437821i	$0.0139407\pi$
$523$			2.00252i			0.0875643i	−0.999041	+	0.0437821i	$0.986059\pi$
$524$	−13.5204	−	23.4179i	−0.590639	−	1.02302i
$525$		0			0
$526$	5.03961	−	8.72886i	0.219737	−	0.380596i
$527$	55.4257	+	32.0001i	2.41438	+	1.39394i
$528$		0			0
$529$	−8.68292	−	15.0393i	−0.377518	−	0.653881i
$530$	11.1726			0.485307
$531$		0			0
$532$	−5.29528	−	4.43869i	−0.229579	−	0.192442i
$533$	−1.42037	+	0.820053i	−0.0615232	+	0.0355204i
$534$		0			0
$535$	2.85192	+	1.64656i	0.123299	+	0.0711870i
$536$	15.9258	+	9.19476i	0.687890	+	0.397153i
$537$		0			0
$538$	5.82990	−	3.36589i	0.251345	−	0.145114i
$539$	23.3840	−	4.14769i	1.00722	−	0.178654i
$540$		0			0
$541$	−11.4451			−0.492061			−0.246031	−	0.969262i	$-0.579126\pi$
$541$	−11.4451			−0.492061			−0.246031	+	0.969262i	$0.579126\pi$
$542$	−7.13885	−	12.3649i	−0.306640	−	0.531116i
$543$		0			0
$544$	29.4391	+	16.9967i	1.26219	+	0.728726i
$545$	−4.84980	+	8.40010i	−0.207743	+	0.359821i
$546$		0			0
$547$	−3.91961	−	6.78896i	−0.167590	−	0.290275i	0.769982	−	0.638066i	$-0.220265\pi$
$547$	−3.91961	−	6.78896i	−0.167590	−	0.290275i	−0.937572	+	0.347791i	$0.886932\pi$
$548$		−	9.95137i		−	0.425102i
$549$		0			0
$550$	0.242192			0.0103271
$551$	−2.32037	−	4.01899i	−0.0988510	−	0.171215i
$552$		0			0
$553$	4.51959	+	12.3979i	0.192192	+	0.527212i
$554$	12.6963	+	7.33022i	0.539415	+	0.311431i
$555$		0			0
$556$	10.5704	−	6.10281i	0.448284	−	0.258817i
$557$		−	0.0134996i		−	0.000571997i	−1.00000		0.000285998i	$-0.999909\pi$
$557$		−	0.0134996i		−	0.000571997i	1.00000		0.000285998i	$-9.10361e-5\pi$
$558$		0			0
$559$		−	7.06782i		−	0.298937i
$560$	−9.79245	−	1.72155i	−0.413807	−	0.0727489i
$561$		0			0
$562$	−0.294871	+	0.510731i	−0.0124384	+	0.0215439i
$563$	−9.54528	+	16.5329i	−0.402286	+	0.696779i	−0.994001	−	0.109368i	$-0.965117\pi$
$563$	−9.54528	+	16.5329i	−0.402286	+	0.696779i	0.591716	+	0.806147i	$0.298451\pi$
$564$		0			0
$565$	32.8923	−	18.9904i	1.38379	−	0.798932i
$566$	14.1925			0.596557
$567$		0			0
$568$	28.0283			1.17604
$569$	−32.3406	+	18.6719i	−1.35579	+	0.782765i	−0.989053	−	0.147561i	$-0.952858\pi$
$569$	−32.3406	+	18.6719i	−1.35579	+	0.782765i	−0.366735	+	0.930325i	$0.619525\pi$
$570$		0			0
$571$	−22.6421	+	39.2173i	−0.947544	+	1.64119i	−0.196968	+	0.980410i	$0.563110\pi$
$571$	−22.6421	+	39.2173i	−0.947544	+	1.64119i	−0.750576	+	0.660784i	$0.770224\pi$
$572$	−4.85543	+	8.40986i	−0.203016	+	0.351634i
$573$		0			0
$574$	−0.267508	+	1.52162i	−0.0111656	+	0.0635114i
$575$		−	0.264101i		−	0.0110138i
$576$		0			0
$577$		−	37.0988i		−	1.54444i	−0.635354	−	0.772221i	$-0.719146\pi$
$577$		−	37.0988i		−	1.54444i	0.635354	−	0.772221i	$-0.280854\pi$
$578$	10.3509	−	5.97610i	0.430541	−	0.248573i
$579$		0			0
$580$	−8.58450	−	4.95626i	−0.356452	−	0.205798i
$581$	−7.76653	−	21.3048i	−0.322210	−	0.883870i
$582$		0			0
$583$	13.3603	+	23.1408i	0.553329	+	0.958393i
$584$	0.980996			0.0405939
$585$		0			0
$586$		−	18.7676i		−	0.775282i
$587$	−17.0612	−	29.5509i	−0.704191	−	1.21969i	−0.966983	−	0.254842i	$-0.917977\pi$
$587$	−17.0612	−	29.5509i	−0.704191	−	1.21969i	0.262792	−	0.964853i	$-0.415357\pi$
$588$		0			0
$589$	8.81453	−	15.2672i	0.363197	−	0.629075i
$590$	13.2458	+	7.64749i	0.545322	+	0.314842i
$591$		0			0
$592$	−1.44437	−	2.50172i	−0.0593632	−	0.102820i
$593$	19.6999			0.808980			0.404490	−	0.914542i	$-0.367449\pi$
$593$	19.6999			0.808980			0.404490	+	0.914542i	$0.367449\pi$
$594$		0			0
$595$	−22.4313	+	26.7601i	−0.919594	+	1.09706i
$596$	21.2356	−	12.2604i	0.869844	−	0.502205i
$597$		0			0
$598$	−2.37663	−	1.37215i	−0.0971878	−	0.0561114i
$599$	9.74033	+	5.62358i	0.397979	+	0.229773i	0.685612	−	0.727967i	$-0.259535\pi$
$599$	9.74033	+	5.62358i	0.397979	+	0.229773i	−0.287632	+	0.957741i	$0.592868\pi$
$600$		0			0
$601$	−29.7646	+	17.1846i	−1.21412	+	0.700975i	−0.963655	−	0.267150i	$-0.913918\pi$
$601$	−29.7646	+	17.1846i	−1.21412	+	0.700975i	−0.250469	+	0.968125i	$0.580585\pi$
$602$	−5.10234	−	4.27696i	−0.207956	−	0.174316i
$603$		0			0
$604$	18.5636			0.755342
$605$	−0.564393	−	0.977557i	−0.0229458	−	0.0397433i
$606$		0			0
$607$	33.7888	+	19.5080i	1.37145	+	0.791804i	0.991110	−	0.133044i	$-0.0424753\pi$
$607$	33.7888	+	19.5080i	1.37145	+	0.791804i	0.380335	+	0.924849i	$0.375809\pi$
$608$	4.68179	−	8.10910i	0.189872	−	0.328868i
$609$		0			0
$610$	1.00619	+	1.74277i	0.0407394	+	0.0705628i
$611$			0.443317i			0.0179347i
$612$		0			0
$613$	−16.1099			−0.650672			−0.325336	−	0.945598i	$-0.605477\pi$
$613$	−16.1099			−0.650672			−0.325336	+	0.945598i	$0.605477\pi$
$614$	−4.76222	−	8.24840i	−0.192187	−	0.332878i
$615$		0			0
$616$	7.07793	+	19.4158i	0.285178	+	0.782285i
$617$	−7.03569	−	4.06205i	−0.283246	−	0.163532i	0.351646	−	0.936133i	$-0.385622\pi$
$617$	−7.03569	−	4.06205i	−0.283246	−	0.163532i	−0.634892	+	0.772601i	$0.718955\pi$
$618$		0			0
$619$	−32.4018	+	18.7072i	−1.30234	+	0.751906i	−0.980805	−	0.194991i	$-0.937532\pi$
$619$	−32.4018	+	18.7072i	−1.30234	+	0.751906i	−0.321535	+	0.946898i	$0.604199\pi$
$620$		−	37.6554i		−	1.51228i
$621$		0			0
$622$		−	12.4340i		−	0.498559i
$623$	27.4542	+	4.82656i	1.09993	+	0.193372i
$624$		0			0
$625$	12.2156	−	21.1581i	0.488626	−	0.846325i
$626$	−4.68985	+	8.12305i	−0.187444	+	0.324662i
$627$		0			0
$628$	−7.83994	+	4.52639i	−0.312848	+	0.180623i
$629$	−10.1451			−0.404511
$630$		0			0
$631$	−19.8268			−0.789294			−0.394647	−	0.918833i	$-0.629133\pi$
$631$	−19.8268			−0.789294			−0.394647	+	0.918833i	$0.629133\pi$
$632$	−9.94437	+	5.74138i	−0.395566	+	0.228380i
$633$		0			0
$634$	5.44692	−	9.43434i	0.216325	−	0.374685i
$635$	6.98848	−	12.1044i	0.277329	−	0.480348i
$636$		0			0
$637$	8.11802	+	9.65478i	0.321648	+	0.382536i
$638$			6.14384i			0.243237i
$639$		0			0
$640$		−	24.8226i		−	0.981199i
$641$	8.01849	−	4.62948i	0.316711	−	0.182853i	−0.333214	−	0.942851i	$-0.608133\pi$
$641$	8.01849	−	4.62948i	0.316711	−	0.182853i	0.649926	+	0.759998i	$0.274800\pi$
$642$		0			0
$643$	−36.3456	−	20.9841i	−1.43333	−	0.827534i	−0.435958	−	0.899967i	$-0.643590\pi$
$643$	−36.3456	−	20.9841i	−1.43333	−	0.827534i	−0.997373	+	0.0724332i	$0.976924\pi$
$644$	9.37172	−	3.41641i	0.369298	−	0.134625i
$645$		0			0
$646$	−3.14833	−	5.45306i	−0.123869	−	0.214548i
$647$	6.28587			0.247123			0.123561	−	0.992337i	$-0.460568\pi$
$647$	6.28587			0.247123			0.123561	+	0.992337i	$0.460568\pi$
$648$		0			0
$649$			36.5798i			1.43588i
$650$	0.0643195	+	0.111405i	0.00252282	+	0.00436965i
$651$		0			0
$652$	8.10872	−	14.0447i	0.317562	−	0.550033i
$653$	−20.1668	−	11.6433i	−0.789189	−	0.455638i	0.0504882	−	0.998725i	$-0.483922\pi$
$653$	−20.1668	−	11.6433i	−0.789189	−	0.455638i	−0.839677	+	0.543086i	$0.817256\pi$
$654$		0			0
$655$	18.8207	+	32.5985i	0.735387	+	1.27373i
$656$	1.54691			0.0603968
$657$		0			0
$658$	0.320035	+	0.268265i	0.0124763	+	0.0104581i
$659$	−25.8880	+	14.9464i	−1.00845	+	0.582230i	−0.910738	−	0.412984i	$-0.864486\pi$
$659$	−25.8880	+	14.9464i	−1.00845	+	0.582230i	−0.0977141	+	0.995215i	$0.531153\pi$
$660$		0			0
$661$	−17.6184	−	10.1720i	−0.685278	−	0.395645i	0.116563	−	0.993183i	$-0.462812\pi$
$661$	−17.6184	−	10.1720i	−0.685278	−	0.395645i	−0.801841	+	0.597538i	$0.796146\pi$
$662$	−11.0549	−	6.38253i	−0.429660	−	0.248064i
$663$		0			0
$664$	17.0886	−	9.86609i	0.663165	−	0.382879i
$665$	7.37118	+	6.17878i	0.285842	+	0.239603i
$666$		0			0
$667$	6.69963			0.259411
$668$	2.86955	+	4.97021i	0.111026	+	0.192303i
$669$		0			0
$670$	−9.81303	−	5.66555i	−0.379110	−	0.218879i
$671$	−2.40643	+	4.16805i	−0.0928990	+	0.160906i
$672$		0			0
$673$	−8.55996	−	14.8263i	−0.329962	−	0.571511i	0.652542	−	0.757753i	$-0.273703\pi$
$673$	−8.55996	−	14.8263i	−0.329962	−	0.571511i	−0.982504	+	0.186241i	$0.940369\pi$
$674$		−	0.628973i		−	0.0242271i
$675$		0			0
$676$	15.4909			0.595802
$677$	14.2078	+	24.6085i	0.546048	+	0.945783i	0.998540	+	0.0540148i	$0.0172018\pi$
$677$	14.2078	+	24.6085i	0.546048	+	0.945783i	−0.452492	+	0.891769i	$0.649465\pi$
$678$		0			0
$679$	18.0858	−	6.59310i	0.694071	−	0.253020i
$680$	−26.3138	−	15.1923i	−1.00909	−	0.582598i
$681$		0			0
$682$	−20.2122	+	11.6695i	−0.773965	+	0.446849i
$683$			20.9274i			0.800764i	0.916348	+	0.400382i	$0.131123\pi$
$683$			20.9274i			0.800764i	−0.916348	+	0.400382i	$0.868877\pi$
$684$		0			0
$685$			13.8526i			0.529281i
$686$	11.8824	−	0.0180771i	0.453671	−	0.000690186i
$687$		0			0
$688$	−3.33310	+	5.77311i	−0.127073	+	0.220098i
$689$	−7.09627	+	12.2911i	−0.270346	+	0.468254i
$690$		0			0
$691$	20.7918	−	12.0041i	0.790957	−	0.456659i	−0.0493424	−	0.998782i	$-0.515713\pi$
$691$	20.7918	−	12.0041i	0.790957	−	0.456659i	0.840299	+	0.542123i	$0.182379\pi$
$692$	−28.7163			−1.09163
$693$		0			0
$694$	−13.6094			−0.516606
$695$	−14.7143	+	8.49529i	−0.558144	+	0.322245i
$696$		0			0
$697$	2.71634	−	4.70484i	0.102889	−	0.178208i
$698$	−3.22246	+	5.58147i	−0.121972	+	0.211262i
$699$		0			0
$700$	−0.460516	−	0.0809606i	−0.0174059	−	0.00306002i
$701$			42.0117i			1.58676i	0.608728	+	0.793379i	$0.291680\pi$
$701$			42.0117i			1.58676i	−0.608728	+	0.793379i	$0.708320\pi$
$702$		0			0
$703$			2.79450i			0.105397i
$704$	−0.747976	+	0.431844i	−0.0281904	+	0.0162757i
$705$		0			0
$706$	1.52607	+	0.881077i	0.0574344	+	0.0331598i
$707$	−4.23006	−	11.6037i	−0.159088	−	0.436401i
$708$		0			0
$709$	−18.6094	−	32.2324i	−0.698891	−	1.21051i	−0.968851	−	0.247643i	$-0.920344\pi$
$709$	−18.6094	−	32.2324i	−0.698891	−	1.21051i	0.269960	−	0.962871i	$-0.412989\pi$
$710$	−17.2703			−0.648141
$711$		0			0
$712$			24.2562i			0.909039i
$713$	12.7252	+	22.0406i	0.476561	+	0.825428i
$714$		0			0
$715$	6.75890	−	11.7068i	0.252769	−	0.437808i
$716$	6.92175	+	3.99627i	0.258678	+	0.149348i
$717$		0			0
$718$	−3.20837	−	5.55705i	−0.119735	−	0.207387i
$719$	18.2978			0.682392			0.341196	−	0.939992i	$-0.389168\pi$
$719$	18.2978			0.682392			0.341196	+	0.939992i	$0.389168\pi$
$720$		0			0
$721$	12.6538	+	10.6069i	0.471253	+	0.395021i
$722$	9.05494	−	5.22787i	0.336990	−	0.194561i
$723$		0			0
$724$	−18.6516	−	10.7685i	−0.693181	−	0.400208i
$725$	−0.271971	−	0.157022i	−0.0101007	−	0.00583166i
$726$		0			0
$727$	28.3214	−	16.3514i	1.05038	−	0.606439i	0.127626	−	0.991822i	$-0.459264\pi$
$727$	28.3214	−	16.3514i	1.05038	−	0.606439i	0.922756	+	0.385384i	$0.125931\pi$
$728$	−7.05127	+	8.41204i	−0.261338	+	0.311771i
$729$		0			0
$730$	−0.604462			−0.0223722
$731$	11.7057	+	20.2749i	0.432951	+	0.749893i
$732$		0			0
$733$	−0.431812	−	0.249307i	−0.0159494	−	0.00920836i	0.492004	−	0.870593i	$-0.336264\pi$
$733$	−0.431812	−	0.249307i	−0.0159494	−	0.00920836i	−0.507953	+	0.861385i	$0.669598\pi$
$734$	1.86529	−	3.23078i	0.0688493	−	0.119250i
$735$		0			0
$736$	6.75890	+	11.7068i	0.249136	+	0.431517i
$737$		−	27.0997i		−	0.998231i
$738$		0			0
$739$	−47.7046			−1.75484			−0.877421	−	0.479722i	$-0.840737\pi$
$739$	−47.7046			−1.75484			−0.877421	+	0.479722i	$0.840737\pi$
$740$	2.98450	+	5.16931i	0.109713	+	0.190028i
$741$		0			0
$742$	4.57891	+	12.5606i	0.168097	+	0.461115i
$743$	−9.20534	−	5.31470i	−0.337711	−	0.194978i	0.321548	−	0.946893i	$-0.395797\pi$
$743$	−9.20534	−	5.31470i	−0.337711	−	0.194978i	−0.659259	+	0.751916i	$0.729130\pi$
$744$		0			0
$745$	−29.5606	+	17.0668i	−1.08302	+	0.625279i
$746$		−	9.95693i		−	0.364549i
$747$		0			0
$748$		−	32.1662i		−	1.17611i
$749$	−0.682303	+	3.88104i	−0.0249308	+	0.141810i
$750$		0			0
$751$	−9.55927	+	16.5571i	−0.348823	+	0.604179i	−0.986041	−	0.166505i	$-0.946752\pi$
$751$	−9.55927	+	16.5571i	−0.348823	+	0.604179i	0.637218	+	0.770684i	$0.280085\pi$
$752$	0.209063	−	0.362108i	0.00762375	−	0.0132047i
$753$		0			0
$754$	−2.82607	+	1.63164i	−0.102920	+	0.0594206i
$755$	−25.8411			−0.940453
$756$		0			0
$757$	28.5388			1.03726			0.518631	−	0.854998i	$-0.326442\pi$
$757$	28.5388			1.03726			0.518631	+	0.854998i	$0.326442\pi$
$758$	1.55418	−	0.897305i	0.0564503	−	0.0325916i
$759$		0			0
$760$	−4.18478	+	7.24825i	−0.151798	+	0.262922i
$761$	21.6650	−	37.5249i	0.785355	−	1.36028i	−0.143431	−	0.989660i	$-0.545814\pi$
$761$	21.6650	−	37.5249i	0.785355	−	1.36028i	0.928787	−	0.370615i	$-0.120853\pi$
$762$		0			0
$763$	−11.4313	−	2.00967i	−0.413840	−	0.0727548i
$764$			16.2674i			0.588533i
$765$		0			0
$766$			2.23567i			0.0807779i
$767$	−16.8261	+	9.71458i	−0.607557	+	0.350773i
$768$		0			0
$769$	−5.75189	−	3.32086i	−0.207419	−	0.119753i	0.392693	−	0.919670i	$-0.371544\pi$
$769$	−5.75189	−	3.32086i	−0.207419	−	0.119753i	−0.600111	+	0.799917i	$0.704877\pi$
$770$	−4.36122	−	11.9635i	−0.157167	−	0.431133i
$771$		0			0
$772$	−12.8120	−	22.1910i	−0.461113	−	0.798672i
$773$	−44.4831			−1.59995			−0.799973	−	0.600036i	$-0.795153\pi$
$773$	−44.4831			−1.59995			−0.799973	+	0.600036i	$0.795153\pi$
$774$		0			0
$775$		−	1.19298i		−	0.0428532i
$776$	8.37543	+	14.5067i	0.300661	+	0.520759i
$777$		0			0
$778$	2.35965	−	4.08703i	0.0845974	−	0.146527i
$779$	−1.29596	−	0.748226i	−0.0464328	−	0.0268080i
$780$		0			0
$781$	−20.6520	−	35.7703i	−0.738986	−	1.27996i
$782$	9.09020			0.325065
$783$		0			0
$784$	−2.07784	−	11.7145i	−0.0742087	−	0.418377i
$785$	10.9134	−	6.30087i	0.389517	−	0.224888i
$786$		0			0
$787$	19.0399	+	10.9927i	0.678700	+	0.391848i	0.799365	−	0.600846i	$-0.205169\pi$
$787$	19.0399	+	10.9927i	0.678700	+	0.391848i	−0.120665	+	0.992693i	$0.538503\pi$
$788$	5.31385	+	3.06795i	0.189298	+	0.109291i
$789$		0			0
$790$	6.12744	−	3.53768i	0.218004	−	0.125865i
$791$	34.8300	+	29.1958i	1.23841	+	1.03808i
$792$		0			0
$793$	−2.55632			−0.0907776
$794$	6.18612	+	10.7147i	0.219537	+	0.380250i
$795$		0			0
$796$	−20.9133	−	12.0743i	−0.741251	−	0.427962i
$797$	−9.71892	+	16.8337i	−0.344262	+	0.596279i	−0.985219	−	0.171297i	$-0.945204\pi$
$797$	−9.71892	+	16.8337i	−0.344262	+	0.596279i	0.640958	+	0.767576i	$0.278537\pi$
$798$		0			0
$799$	−0.734219	−	1.27171i	−0.0259748	−	0.0449897i
$800$		−	0.633646i		−	0.0224028i
$801$		0			0
$802$	−7.11636			−0.251287
$803$	−0.722823	−	1.25197i	−0.0255079	−	0.0441809i
$804$		0			0
$805$	−13.0457	+	4.75574i	−0.459801	+	0.167618i
$806$	−10.7356	−	6.19820i	−0.378145	−	0.218322i
$807$		0			0
$808$	9.30732	−	5.37358i	0.327430	−	0.189042i
$809$		−	21.0058i		−	0.738526i	−0.929325	−	0.369263i	$-0.879610\pi$
$809$		−	21.0058i		−	0.738526i	0.929325	−	0.369263i	$-0.120390\pi$
$810$		0			0
$811$			37.3291i			1.31080i	0.755281	+	0.655401i	$0.227500\pi$
$811$			37.3291i			1.31080i	−0.755281	+	0.655401i	$0.772500\pi$
$812$	2.05378	−	11.6822i	0.0720736	−	0.409965i
$813$		0			0
$814$	1.84981	−	3.20397i	0.0648359	−	0.112299i
$815$	−11.2876	+	19.5506i	−0.395386	+	0.684829i
$816$		0			0
$817$	5.58478	−	3.22438i	0.195387	−	0.112807i
$818$	−13.0090			−0.454849
$819$		0			0
$820$	−3.19639			−0.111623
$821$	10.9017	−	6.29412i	0.380473	−	0.219666i	−0.297551	−	0.954706i	$-0.596170\pi$
$821$	10.9017	−	6.29412i	0.380473	−	0.219666i	0.678024	+	0.735040i	$0.262837\pi$
$822$		0			0
$823$	22.4189	−	38.8307i	0.781474	−	1.35355i	−0.149608	−	0.988745i	$-0.547801\pi$
$823$	22.4189	−	38.8307i	0.781474	−	1.35355i	0.931083	−	0.364808i	$-0.118865\pi$
$824$	−7.18385	+	12.4428i	−0.250261	+	0.433465i
$825$		0			0
$826$	−3.16897	+	18.0256i	−0.110263	+	0.627191i
$827$			25.7293i			0.894695i	0.894360	+	0.447347i	$0.147631\pi$
$827$			25.7293i			0.894695i	−0.894360	+	0.447347i	$0.852369\pi$
$828$		0			0
$829$		−	16.9628i		−	0.589142i	−0.955630	−	0.294571i	$-0.904823\pi$
$829$		−	16.9628i		−	0.589142i	0.955630	−	0.294571i	$-0.0951767\pi$
$830$	−10.5295	+	6.07921i	−0.365484	+	0.211012i
$831$		0			0
$832$	−0.397284	−	0.229372i	−0.0137733	−	0.00795204i
$833$	−39.2777	−	14.2508i	−1.36089	−	0.493762i
$834$		0			0
$835$	−3.99450	−	6.91867i	−0.138235	−	0.239431i
$836$	−8.86030			−0.306440
$837$		0			0
$838$			7.12081i			0.245984i
$839$	−13.3539	−	23.1296i	−0.461027	−	0.798522i	0.537986	−	0.842954i	$-0.319185\pi$
$839$	−13.3539	−	23.1296i	−0.461027	−	0.798522i	−0.999012	+	0.0444321i	$0.985852\pi$
$840$		0			0
$841$	−10.5167	+	18.2155i	−0.362645	+	0.628120i
$842$	5.10370	+	2.94662i	0.175885	+	0.101547i
$843$		0			0
$844$	18.9847	+	32.8824i	0.653479	+	1.13186i
$845$	−21.5637			−0.741815
$846$		0			0
$847$	0.867695	−	1.03514i	0.0298144	−	0.0355680i
$848$	11.5927	−	6.69305i	0.398095	−	0.229840i
$849$		0			0
$850$	−0.369016	−	0.213051i	−0.0126571	−	0.00730760i
$851$	−3.49381	−	2.01715i	−0.119766	−	0.0691471i
$852$		0			0
$853$	−37.6287	+	21.7249i	−1.28838	+	0.743848i	−0.978366	−	0.206883i	$-0.933668\pi$
$853$	−37.6287	+	21.7249i	−1.28838	+	0.743848i	−0.310017	+	0.950731i	$0.600335\pi$
$854$	−1.54691	+	1.84544i	−0.0529342	+	0.0631496i
$855$		0			0
$856$	−3.42896			−0.117199
$857$	−7.83430	−	13.5694i	−0.267615	−	0.463522i	0.700631	−	0.713524i	$-0.252902\pi$
$857$	−7.83430	−	13.5694i	−0.267615	−	0.463522i	−0.968245	+	0.250002i	$0.919569\pi$
$858$		0			0
$859$	17.3578	+	10.0216i	0.592242	+	0.341931i	0.765984	−	0.642860i	$-0.222252\pi$
$859$	17.3578	+	10.0216i	0.592242	+	0.341931i	−0.173742	+	0.984791i	$0.555586\pi$
$860$	6.88721	−	11.9290i	0.234852	−	0.406775i
$861$		0			0
$862$	4.84727	+	8.39571i	0.165099	+	0.285959i
$863$		−	40.0219i		−	1.36236i	−0.732115	−	0.681181i	$-0.761467\pi$
$863$		−	40.0219i		−	1.36236i	0.732115	−	0.681181i	$-0.238533\pi$
$864$		0			0
$865$	39.9739			1.35915
$866$	−1.07010	−	1.85347i	−0.0363635	−	0.0629834i
$867$		0			0
$868$	42.3334	−	15.4324i	1.43689	−	0.523810i
$869$	14.6545	+	8.46079i	0.497120	+	0.287013i
$870$		0			0
$871$	12.4655	−	7.19694i	0.422376	−	0.243859i
$872$		−	10.0997i		−	0.342019i
$873$		0			0
$874$		−	2.50393i		−	0.0846967i
$875$	29.4487	+	5.17720i	0.995547	+	0.175021i
$876$		0			0
$877$	−22.6353	+	39.2054i	−0.764338	+	1.32387i	0.176257	+	0.984344i	$0.443601\pi$
$877$	−22.6353	+	39.2054i	−0.764338	+	1.32387i	−0.940596	+	0.339529i	$0.889732\pi$
$878$	−2.18931	+	3.79200i	−0.0738856	+	0.127974i
$879$		0			0
$880$	−11.0416	+	6.37485i	−0.372211	+	0.214896i
$881$	45.3385			1.52749			0.763746	−	0.645517i	$-0.223358\pi$
$881$	45.3385			1.52749			0.763746	+	0.645517i	$0.223358\pi$
$882$		0			0
$883$	12.5650			0.422845			0.211423	−	0.977395i	$-0.432190\pi$
$883$	12.5650			0.422845			0.211423	+	0.977395i	$0.432190\pi$
$884$	14.7960	−	8.54245i	0.497642	−	0.287314i
$885$		0			0
$886$	−3.62178	+	6.27311i	−0.121676	+	0.210749i
$887$	17.8620	−	30.9379i	0.599748	−	1.03879i	−0.393110	−	0.919492i	$-0.628601\pi$
$887$	17.8620	−	30.9379i	0.599748	−	1.03879i	0.992858	−	0.119303i	$-0.0380659\pi$
$888$		0			0
$889$	16.4723	+	2.89589i	0.552462	+	0.0971251i
$890$		−	14.9460i		−	0.500991i
$891$		0			0
$892$			30.4821i			1.02062i
$893$	−0.350296	+	0.202243i	−0.0117222	+	0.00676782i
$894$		0			0
$895$	−9.63528	−	5.56293i	−0.322072	−	0.185948i
$896$	27.9064	−	10.1731i	0.932286	−	0.339860i
$897$		0			0
$898$	−7.97346	−	13.8104i	−0.266078	−	0.460860i
$899$	30.2632			1.00933
$900$		0			0
$901$		−	47.0113i		−	1.56617i
$902$	0.990571	+	1.71572i	0.0329824	+	0.0571272i
$903$		0			0
$904$	−19.7738	+	34.2491i	−0.657665	+	1.13911i
$905$	25.9635	+	14.9901i	0.863058	+	0.498287i
$906$		0			0
$907$	4.52104	+	7.83067i	0.150119	+	0.260013i	0.931271	−	0.364327i	$-0.118701\pi$
$907$	4.52104	+	7.83067i	0.150119	+	0.260013i	−0.781152	+	0.624341i	$0.785368\pi$
$908$	−13.7744			−0.457120
$909$		0			0
$910$	4.34479	−	5.18326i	0.144029	−	0.171823i
$911$	35.5171	−	20.5058i	1.17673	−	0.679388i	0.221478	−	0.975165i	$-0.428912\pi$
$911$	35.5171	−	20.5058i	1.17673	−	0.679388i	0.955257	+	0.295777i	$0.0955787\pi$
$912$		0			0
$913$	−25.1826	−	14.5392i	−0.833421	−	0.481176i
$914$	7.00619	+	4.04503i	0.231744	+	0.133798i
$915$		0			0
$916$	19.7652	−	11.4114i	0.653059	−	0.377044i
$917$	−28.9349	+	34.5188i	−0.955515	+	1.13991i
$918$		0			0
$919$	10.2326			0.337541			0.168771	−	0.985655i	$-0.446020\pi$
$919$	10.2326			0.337541			0.168771	+	0.985655i	$0.446020\pi$
$920$	−6.04138	−	10.4640i	−0.199178	−	0.344987i
$921$		0			0
$922$	16.0057	+	9.24088i	0.527119	+	0.304332i
$923$	10.9692	−	18.9992i	0.361055	−	0.625366i
$924$		0			0
$925$	0.0945538	+	0.163772i	0.00310891	+	0.00538479i
$926$			16.1498i			0.530716i
$927$		0			0
$928$	16.0741			0.527659
$929$	−12.8330	−	22.2273i	−0.421036	−	0.729255i	0.575005	−	0.818150i	$-0.305000\pi$
$929$	−12.8330	−	22.2273i	−0.421036	−	0.729255i	−0.996041	+	0.0888945i	$0.971667\pi$
$930$		0			0
$931$	−3.92544	+	10.8192i	−0.128651	+	0.354585i
$932$	−40.8763	−	23.5999i	−1.33895	−	0.773041i
$933$		0			0
$934$	14.2214	−	8.21075i	0.465340	−	0.268664i
$935$			44.7763i			1.46434i
$936$		0			0
$937$			15.9276i			0.520333i	0.965564	+	0.260167i	$0.0837775\pi$
$937$			15.9276i			0.520333i	−0.965564	+	0.260167i	$0.916223\pi$
$938$	2.34770	−	13.3541i	0.0766551	−	0.436026i
$939$		0			0
$940$	−0.431988	+	0.748226i	−0.0140899	+	0.0244044i
$941$	19.6767	−	34.0810i	0.641442	−	1.11101i	−0.343669	−	0.939091i	$-0.611670\pi$
$941$	19.6767	−	34.0810i	0.641442	−	1.11101i	0.985111	−	0.171919i	$-0.0549967\pi$
$942$		0			0
$943$	1.87093	−	1.08018i	0.0609258	−	0.0351755i
$944$	18.3252			0.596433
$945$		0			0
$946$	−8.53747			−0.277577
$947$	−28.9086	+	16.6904i	−0.939403	+	0.542365i	−0.889773	−	0.456403i	$-0.849138\pi$
$947$	−28.9086	+	16.6904i	−0.939403	+	0.542365i	−0.0496302	+	0.998768i	$0.515804\pi$
$948$		0			0
$949$	0.383923	−	0.664975i	0.0124627	−	0.0215860i
$950$	−0.0586858	+	0.101647i	−0.00190402	+	0.00329786i
$951$		0			0
$952$	6.29541	−	35.8092i	0.204035	−	1.16058i
$953$		−	44.4622i		−	1.44027i	−0.693832	−	0.720137i	$-0.744079\pi$
$953$		−	44.4622i		−	1.44027i	0.693832	−	0.720137i	$-0.255921\pi$
$954$		0			0
$955$		−	22.6447i		−	0.732764i
$956$	−21.7767	+	12.5728i	−0.704309	+	0.406633i
$957$		0			0
$958$	−0.297540	−	0.171785i	−0.00961307	−	0.00555011i
$959$	−15.5736	+	5.67725i	−0.502896	+	0.183328i
$960$		0			0
$961$	41.9814	+	72.7138i	1.35424	+	2.34561i
$962$	1.96504			0.0633554
$963$		0			0
$964$			7.97776i			0.256947i
$965$	17.8347	+	30.8905i	0.574118	+	0.994401i
$966$		0			0
$967$	−20.0556	+	34.7372i	−0.644943	+	1.11707i	0.339371	+	0.940652i	$0.389786\pi$
$967$	−20.0556	+	34.7372i	−0.644943	+	1.11707i	−0.984315	+	0.176422i	$0.943548\pi$
$968$	1.01788	+	0.587674i	0.0327159	+	0.0188886i
$969$		0			0
$970$	−5.16071	−	8.93861i	−0.165700	−	0.287001i
$971$	−46.0026			−1.47629			−0.738147	−	0.674640i	$-0.764299\pi$
$971$	−46.0026			−1.47629			−0.738147	+	0.674640i	$0.764299\pi$
$972$		0			0
$973$	−15.5811	−	13.0606i	−0.499506	−	0.418704i
$974$	18.9650	−	10.9494i	0.607678	−	0.350843i
$975$		0			0
$976$	2.08804	+	1.20553i	0.0668366	+	0.0385882i
$977$	46.8323	+	27.0386i	1.49830	+	0.865042i	0.999998	−	0.00196335i	$-0.000624955\pi$
$977$	46.8323	+	27.0386i	1.49830	+	0.865042i	0.498299	+	0.867005i	$0.333958\pi$
$978$		0			0
$979$	30.9562	−	17.8726i	0.989365	−	0.571210i
$980$	4.29346	+	24.2058i	0.137149	+	0.773227i
$981$		0			0
$982$	−4.34836			−0.138762
$983$	−6.97890	−	12.0878i	−0.222592	−	0.385541i	0.733002	−	0.680226i	$-0.238119\pi$
$983$	−6.97890	−	12.0878i	−0.222592	−	0.385541i	−0.955594	+	0.294685i	$0.904785\pi$
$984$		0			0
$985$	−7.39703	−	4.27068i	−0.235689	−	0.136075i
$986$	5.40462	−	9.36107i	0.172118	−	0.298117i
$987$		0			0
$988$	−2.35305	−	4.07560i	−0.0748605	−	0.129662i
$989$			9.30979i			0.296034i
$990$		0			0
$991$	−37.0297			−1.17629			−0.588144	−	0.808756i	$-0.700141\pi$
$991$	−37.0297			−1.17629			−0.588144	+	0.808756i	$0.700141\pi$
$992$	30.5309	+	52.8811i	0.969358	+	1.67898i
$993$		0			0
$994$	−7.07793	−	19.4158i	−0.224498	−	0.615832i
$995$	29.1119	+	16.8077i	0.922909	+	0.532841i
$996$		0			0
$997$	43.4282	−	25.0733i	1.37538	−	0.794079i	0.383785	−	0.923422i	$-0.374620\pi$
$997$	43.4282	−	25.0733i	1.37538	−	0.794079i	0.991600	+	0.129344i	$0.0412871\pi$
$998$			5.51814i			0.174674i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.o.a.125.4		12
3.2	odd	2		63.2.o.a.41.3	yes	12
4.3	odd	2		3024.2.cc.a.881.5		12
7.2	even	3		1323.2.s.c.962.3		12
7.3	odd	6		1323.2.i.c.1097.3		12
7.4	even	3		1323.2.i.c.1097.4		12
7.5	odd	6		1323.2.s.c.962.4		12
7.6	odd	2	inner	189.2.o.a.125.3		12
9.2	odd	6	inner	189.2.o.a.62.3		12
9.4	even	3		567.2.c.c.566.7		12
9.5	odd	6		567.2.c.c.566.6		12
9.7	even	3		63.2.o.a.20.4	yes	12
12.11	even	2		1008.2.cc.a.545.6		12
21.2	odd	6		441.2.s.c.374.4		12
21.5	even	6		441.2.s.c.374.3		12
21.11	odd	6		441.2.i.c.68.4		12
21.17	even	6		441.2.i.c.68.3		12
21.20	even	2		63.2.o.a.41.4	yes	12
28.27	even	2		3024.2.cc.a.881.2		12
36.7	odd	6		1008.2.cc.a.209.1		12
36.11	even	6		3024.2.cc.a.2897.2		12
63.2	odd	6		1323.2.i.c.521.3		12
63.11	odd	6		1323.2.s.c.656.4		12
63.13	odd	6		567.2.c.c.566.8		12
63.16	even	3		441.2.i.c.227.3		12
63.20	even	6	inner	189.2.o.a.62.4		12
63.25	even	3		441.2.s.c.362.3		12
63.34	odd	6		63.2.o.a.20.3	✓	12
63.38	even	6		1323.2.s.c.656.3		12
63.41	even	6		567.2.c.c.566.5		12
63.47	even	6		1323.2.i.c.521.4		12
63.52	odd	6		441.2.s.c.362.4		12
63.61	odd	6		441.2.i.c.227.4		12
84.83	odd	2		1008.2.cc.a.545.1		12
252.83	odd	6		3024.2.cc.a.2897.5		12
252.223	even	6		1008.2.cc.a.209.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.o.a.20.3	✓	12	63.34	odd	6
63.2.o.a.20.4	yes	12	9.7	even	3
63.2.o.a.41.3	yes	12	3.2	odd	2
63.2.o.a.41.4	yes	12	21.20	even	2
189.2.o.a.62.3		12	9.2	odd	6	inner
189.2.o.a.62.4		12	63.20	even	6	inner
189.2.o.a.125.3		12	7.6	odd	2	inner
189.2.o.a.125.4		12	1.1	even	1	trivial
441.2.i.c.68.3		12	21.17	even	6
441.2.i.c.68.4		12	21.11	odd	6
441.2.i.c.227.3		12	63.16	even	3
441.2.i.c.227.4		12	63.61	odd	6
441.2.s.c.362.3		12	63.25	even	3
441.2.s.c.362.4		12	63.52	odd	6
441.2.s.c.374.3		12	21.5	even	6
441.2.s.c.374.4		12	21.2	odd	6
567.2.c.c.566.5		12	63.41	even	6
567.2.c.c.566.6		12	9.5	odd	6
567.2.c.c.566.7		12	9.4	even	3
567.2.c.c.566.8		12	63.13	odd	6
1008.2.cc.a.209.1		12	36.7	odd	6
1008.2.cc.a.209.6		12	252.223	even	6
1008.2.cc.a.545.1		12	84.83	odd	2
1008.2.cc.a.545.6		12	12.11	even	2
1323.2.i.c.521.3		12	63.2	odd	6
1323.2.i.c.521.4		12	63.47	even	6
1323.2.i.c.1097.3		12	7.3	odd	6
1323.2.i.c.1097.4		12	7.4	even	3
1323.2.s.c.656.3		12	63.38	even	6
1323.2.s.c.656.4		12	63.11	odd	6
1323.2.s.c.962.3		12	7.2	even	3
1323.2.s.c.962.4		12	7.5	odd	6
3024.2.cc.a.881.2		12	28.27	even	2
3024.2.cc.a.881.5		12	4.3	odd	2
3024.2.cc.a.2897.2		12	36.11	even	6
3024.2.cc.a.2897.5		12	252.83	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.555632 - 0.320794i) q^{2} +(-0.794182 + 1.37556i) q^{4} +(1.10552 - 1.91482i) q^{5} +(2.60579 + 0.458109i) q^{7} +2.30225i q^{8} +O(q^{10})\)	\(q+(0.555632 - 0.320794i) q^{2} +(-0.794182 + 1.37556i) q^{4} +(1.10552 - 1.91482i) q^{5} +(2.60579 + 0.458109i) q^{7} +2.30225i q^{8} -1.41858i q^{10} +(2.93818 - 1.69636i) q^{11} +(1.56060 + 0.901012i) q^{13} +(1.59482 - 0.581383i) q^{14} +(-0.849814 - 1.47192i) q^{16} -5.96901 q^{17} +1.64419i q^{19} +(1.75597 + 3.04144i) q^{20} +(1.08836 - 1.88510i) q^{22} +(-2.05563 - 1.18682i) q^{23} +(0.0556321 + 0.0963576i) q^{25} +1.15616 q^{26} +(-2.69963 + 3.22061i) q^{28} +(-2.44437 + 1.41126i) q^{29} +(-9.28558 - 5.36103i) q^{31} +(-4.93199 - 2.84748i) q^{32} +(-3.31657 + 1.91482i) q^{34} +(3.75796 - 4.48318i) q^{35} +1.69963 q^{37} +(0.527445 + 0.913562i) q^{38} +(4.40841 + 2.54520i) q^{40} +(-0.455074 + 0.788211i) q^{41} +(-1.96108 - 3.39669i) q^{43} +5.38887i q^{44} -1.52290 q^{46} +(0.123005 + 0.213051i) q^{47} +(6.58027 + 2.38747i) q^{49} +(0.0618219 + 0.0356929i) q^{50} +(-2.47880 + 1.43113i) q^{52} +7.87589i q^{53} -7.50146i q^{55} +(-1.05468 + 5.99919i) q^{56} +(-0.905446 + 1.56828i) q^{58} +(-5.39093 + 9.33736i) q^{59} +(-1.22853 + 0.709292i) q^{61} -6.87916 q^{62} -0.254572 q^{64} +(3.45056 - 1.99218i) q^{65} +(3.99381 - 6.91748i) q^{67} +(4.74048 - 8.21075i) q^{68} +(0.649865 - 3.69653i) q^{70} -12.1743i q^{71} -0.426103i q^{73} +(0.944368 - 0.545231i) q^{74} +(-2.26168 - 1.30578i) q^{76} +(8.43339 - 3.07435i) q^{77} +(2.49381 + 4.31941i) q^{79} -3.75796 q^{80} +0.583940i q^{82} +(-4.28541 - 7.42254i) q^{83} +(-6.59888 + 11.4296i) q^{85} +(-2.17928 - 1.25821i) q^{86} +(3.90545 + 6.76443i) q^{88} +10.5358 q^{89} +(3.65383 + 3.06277i) q^{91} +(3.26509 - 1.88510i) q^{92} +(0.136691 + 0.0789188i) q^{94} +(3.14833 + 1.81769i) q^{95} +(6.30108 - 3.63793i) q^{97} +(4.42210 - 0.784360i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 2 q^{4} - 2 q^{7}+O(q^{10}) \) 12 * q + 6 * q^2 + 2 * q^4 - 2 * q^7	\( 12 q + 6 q^{2} + 2 q^{4} - 2 q^{7} + 12 q^{14} + 2 q^{16} - 10 q^{22} - 24 q^{23} - 8 q^{28} - 30 q^{29} + 12 q^{32} - 4 q^{37} - 10 q^{43} - 40 q^{46} + 6 q^{49} + 36 q^{50} - 42 q^{56} + 2 q^{58} + 16 q^{64} + 78 q^{65} + 12 q^{67} + 18 q^{70} + 12 q^{74} + 24 q^{77} - 6 q^{79} - 6 q^{85} - 96 q^{86} + 34 q^{88} - 24 q^{91} - 30 q^{92} - 72 q^{95}+O(q^{100}) \) 12 * q + 6 * q^2 + 2 * q^4 - 2 * q^7 + 12 * q^14 + 2 * q^16 - 10 * q^22 - 24 * q^23 - 8 * q^28 - 30 * q^29 + 12 * q^32 - 4 * q^37 - 10 * q^43 - 40 * q^46 + 6 * q^49 + 36 * q^50 - 42 * q^56 + 2 * q^58 + 16 * q^64 + 78 * q^65 + 12 * q^67 + 18 * q^70 + 12 * q^74 + 24 * q^77 - 6 * q^79 - 6 * q^85 - 96 * q^86 + 34 * q^88 - 24 * q^91 - 30 * q^92 - 72 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more