LMFDB - Embedded newform 351.2.i.b.242.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [351,2,Mod(161,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("351.161"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 351 = 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	351.i (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$2.80274911095$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 184x^{12} + 600x^{10} + 894x^{8} + 600x^{6} + 184x^{4} + 24x^{2} + 1 $ x^16 + 24x^14 + 184x^12 + 600x^10 + 894x^8 + 600x^6 + 184x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			242.3
Root			$-0.592624i$ of defining polynomial
Character	$\chi$	$=$	351.242
Dual form			351.2.i.b.161.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.14002 + 1.14002i) q^{2} -0.599280i q^{4} +(2.84492 - 2.84492i) q^{5} +(-2.82684 + 2.82684i) q^{7} +(-1.59685 - 1.59685i) q^{8} +6.48652i q^{10} +(2.08263 + 2.08263i) q^{11} +(3.59030 + 0.331331i) q^{13} -6.44529i q^{14} +4.83942 q^{16} +3.94914 q^{17} +(2.32235 + 2.32235i) q^{19} +(-1.70490 - 1.70490i) q^{20} -4.74846 q^{22} -0.755446 q^{23} -11.1872i q^{25} +(-4.47072 + 3.71528i) q^{26} +(1.69407 + 1.69407i) q^{28} +3.40981i q^{29} +(6.39174 + 6.39174i) q^{31} +(-2.32334 + 2.32334i) q^{32} +(-4.50208 + 4.50208i) q^{34} +16.0843i q^{35} +(-2.85824 + 2.85824i) q^{37} -5.29503 q^{38} -9.08580 q^{40} +(-2.08263 + 2.08263i) q^{41} -0.526914i q^{43} +(1.24808 - 1.24808i) q^{44} +(0.861221 - 0.861221i) q^{46} +(-3.18684 - 3.18684i) q^{47} -8.98203i q^{49} +(12.7536 + 12.7536i) q^{50} +(0.198560 - 2.15159i) q^{52} -10.3944i q^{53} +11.8498 q^{55} +9.02805 q^{56} +(-3.88724 - 3.88724i) q^{58} +(-7.20759 - 7.20759i) q^{59} +2.91264 q^{61} -14.5734 q^{62} +4.38156i q^{64} +(11.1567 - 9.27150i) q^{65} +(-5.13277 - 5.13277i) q^{67} -2.36664i q^{68} +(-18.3364 - 18.3364i) q^{70} +(2.08948 - 2.08948i) q^{71} +(-3.56787 + 3.56787i) q^{73} -6.51690i q^{74} +(1.39174 - 1.39174i) q^{76} -11.7745 q^{77} +5.79246 q^{79} +(13.7678 - 13.7678i) q^{80} -4.74846i q^{82} +(-7.18887 + 7.18887i) q^{83} +(11.2350 - 11.2350i) q^{85} +(0.600691 + 0.600691i) q^{86} -6.65127i q^{88} +(1.51774 + 1.51774i) q^{89} +(-11.0858 + 9.21257i) q^{91} +0.452723i q^{92} +7.26611 q^{94} +13.2138 q^{95} +(2.07237 + 2.07237i) q^{97} +(10.2397 + 10.2397i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{7} + 16 q^{13} - 8 q^{16} - 32 q^{19} + 8 q^{22} + 40 q^{28} + 16 q^{31} + 24 q^{34} - 32 q^{37} - 72 q^{40} + 48 q^{46} + 48 q^{52} + 32 q^{55} + 56 q^{58} - 64 q^{61} - 32 q^{67} - 40 q^{70}+ \cdots + 64 q^{97}+O(q^{100}) $ 16 * q - 8 * q^7 + 16 * q^13 - 8 * q^16 - 32 * q^19 + 8 * q^22 + 40 * q^28 + 16 * q^31 + 24 * q^34 - 32 * q^37 - 72 * q^40 + 48 * q^46 + 48 * q^52 + 32 * q^55 + 56 * q^58 - 64 * q^61 - 32 * q^67 - 40 * q^70 - 56 * q^73 - 64 * q^76 - 16 * q^79 - 104 * q^91 - 64 * q^94 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/351\mathbb{Z}\right)^\times$.

$n$	$28$	$326$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$

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 $

$2$	−1.14002	+	1.14002i	−0.806114	+	0.806114i	−0.984043	−	0.177929i	$-0.943060\pi$
$2$	−1.14002	+	1.14002i	−0.806114	+	0.806114i	0.177929	+	0.984043i	$0.443060\pi$
$3$		0			0
$3$		0			0
$4$		−	0.599280i		−	0.299640i
$5$	2.84492	−	2.84492i	1.27229	−	1.27229i	0.327403	−	0.944885i	$-0.393826\pi$
$5$	2.84492	−	2.84492i	1.27229	−	1.27229i	0.944885	−	0.327403i	$-0.106174\pi$
$6$		0			0
$7$	−2.82684	+	2.82684i	−1.06844	+	1.06844i	−0.0709657	+	0.997479i	$0.522608\pi$
$7$	−2.82684	+	2.82684i	−1.06844	+	1.06844i	−0.997479	+	0.0709657i	$0.977392\pi$
$8$	−1.59685	−	1.59685i	−0.564570	−	0.564570i
$9$		0			0
$10$			6.48652i			2.05122i
$11$	2.08263	+	2.08263i	0.627936	+	0.627936i	0.947548	−	0.319612i	$-0.103553\pi$
$11$	2.08263	+	2.08263i	0.627936	+	0.627936i	−0.319612	+	0.947548i	$0.603553\pi$
$12$		0			0
$13$	3.59030	+	0.331331i	0.995769	+	0.0918946i
$13$	3.59030	+	0.331331i	0.995769	+	0.0918946i
$14$		−	6.44529i		−	1.72258i
$15$		0			0
$16$	4.83942			1.20986
$17$	3.94914			0.957806			0.478903	−	0.877868i	$-0.341035\pi$
$17$	3.94914			0.957806			0.478903	+	0.877868i	$0.341035\pi$
$18$		0			0
$19$	2.32235	+	2.32235i	0.532783	+	0.532783i	0.921399	−	0.388617i	$-0.127047\pi$
$19$	2.32235	+	2.32235i	0.532783	+	0.532783i	−0.388617	+	0.921399i	$0.627047\pi$
$20$	−1.70490	−	1.70490i	−0.381228	−	0.381228i
$21$		0			0
$22$	−4.74846			−1.01238
$23$	−0.755446			−0.157521			−0.0787607	−	0.996894i	$-0.525096\pi$
$23$	−0.755446			−0.157521			−0.0787607	+	0.996894i	$0.525096\pi$
$24$		0			0
$25$		−	11.1872i		−	2.23743i
$26$	−4.47072	+	3.71528i	−0.876781	+	0.728626i
$27$		0			0
$28$	1.69407	+	1.69407i	0.320149	+	0.320149i
$29$			3.40981i			0.633186i	0.948562	+	0.316593i	$0.102539\pi$
$29$			3.40981i			0.633186i	−0.948562	+	0.316593i	$0.897461\pi$
$30$		0			0
$31$	6.39174	+	6.39174i	1.14799	+	1.14799i	0.986948	+	0.161042i	$0.0514855\pi$
$31$	6.39174	+	6.39174i	1.14799	+	1.14799i	0.161042	+	0.986948i	$0.448515\pi$
$32$	−2.32334	+	2.32334i	−0.410712	+	0.410712i
$33$		0			0
$34$	−4.50208	+	4.50208i	−0.772101	+	0.772101i
$35$			16.0843i			2.71874i
$36$		0			0
$37$	−2.85824	+	2.85824i	−0.469892	+	0.469892i	−0.901880	−	0.431987i	$-0.857813\pi$
$37$	−2.85824	+	2.85824i	−0.469892	+	0.469892i	0.431987	+	0.901880i	$0.357813\pi$
$38$	−5.29503			−0.858967
$39$		0			0
$40$	−9.08580			−1.43659
$41$	−2.08263	+	2.08263i	−0.325252	+	0.325252i	−0.850778	−	0.525526i	$-0.823869\pi$
$41$	−2.08263	+	2.08263i	−0.325252	+	0.325252i	0.525526	+	0.850778i	$0.323869\pi$
$42$		0			0
$43$		−	0.526914i		−	0.0803536i	−0.999193	−	0.0401768i	$-0.987208\pi$
$43$		−	0.526914i		−	0.0803536i	0.999193	−	0.0401768i	$-0.0127921\pi$
$44$	1.24808	−	1.24808i	0.188155	−	0.188155i
$45$		0			0
$46$	0.861221	−	0.861221i	0.126980	−	0.126980i
$47$	−3.18684	−	3.18684i	−0.464849	−	0.464849i	0.435392	−	0.900241i	$-0.356610\pi$
$47$	−3.18684	−	3.18684i	−0.464849	−	0.464849i	−0.900241	+	0.435392i	$0.856610\pi$
$48$		0			0
$49$		−	8.98203i		−	1.28315i
$50$	12.7536	+	12.7536i	1.80363	+	1.80363i
$51$		0			0
$52$	0.198560	−	2.15159i	0.0275353	−	0.298372i
$53$		−	10.3944i		−	1.42778i	−0.700256	−	0.713892i	$-0.746931\pi$
$53$		−	10.3944i		−	1.42778i	0.700256	−	0.713892i	$-0.253069\pi$
$54$		0			0
$55$	11.8498			1.59783
$56$	9.02805			1.20642
$57$		0			0
$58$	−3.88724	−	3.88724i	−0.510420	−	0.510420i
$59$	−7.20759	−	7.20759i	−0.938348	−	0.938348i	0.0598591	−	0.998207i	$-0.480935\pi$
$59$	−7.20759	−	7.20759i	−0.938348	−	0.938348i	−0.998207	+	0.0598591i	$0.980935\pi$
$60$		0			0
$61$	2.91264			0.372926			0.186463	−	0.982462i	$-0.440298\pi$
$61$	2.91264			0.372926			0.186463	+	0.982462i	$0.440298\pi$
$62$	−14.5734			−1.85082
$63$		0			0
$64$			4.38156i			0.547695i
$65$	11.1567	−	9.27150i	1.38382	−	1.14999i
$66$		0			0
$67$	−5.13277	−	5.13277i	−0.627068	−	0.627068i	0.320261	−	0.947329i	$-0.396229\pi$
$67$	−5.13277	−	5.13277i	−0.627068	−	0.627068i	−0.947329	+	0.320261i	$0.896229\pi$
$68$		−	2.36664i		−	0.286997i
$69$		0			0
$70$	−18.3364	−	18.3364i	−2.19161	−	2.19161i
$71$	2.08948	−	2.08948i	0.247975	−	0.247975i	−0.572164	−	0.820139i	$-0.693896\pi$
$71$	2.08948	−	2.08948i	0.247975	−	0.247975i	0.820139	+	0.572164i	$0.193896\pi$
$72$		0			0
$73$	−3.56787	+	3.56787i	−0.417588	+	0.417588i	−0.884372	−	0.466784i	$-0.845413\pi$
$73$	−3.56787	+	3.56787i	−0.417588	+	0.417588i	0.466784	+	0.884372i	$0.345413\pi$
$74$		−	6.51690i		−	0.757574i
$75$		0			0
$76$	1.39174	−	1.39174i	0.159643	−	0.159643i
$77$	−11.7745			−1.34183
$78$		0			0
$79$	5.79246			0.651702			0.325851	−	0.945421i	$-0.394349\pi$
$79$	5.79246			0.651702			0.325851	+	0.945421i	$0.394349\pi$
$80$	13.7678	−	13.7678i	1.53929	−	1.53929i
$81$		0			0
$82$		−	4.74846i		−	0.524380i
$83$	−7.18887	+	7.18887i	−0.789081	+	0.789081i	−0.981344	−	0.192262i	$-0.938418\pi$
$83$	−7.18887	+	7.18887i	−0.789081	+	0.789081i	0.192262	+	0.981344i	$0.438418\pi$
$84$		0			0
$85$	11.2350	−	11.2350i	1.21861	−	1.21861i
$86$	0.600691	+	0.600691i	0.0647742	+	0.0647742i
$87$		0			0
$88$		−	6.65127i		−	0.709028i
$89$	1.51774	+	1.51774i	0.160880	+	0.160880i	0.782957	−	0.622076i	$-0.213711\pi$
$89$	1.51774	+	1.51774i	0.160880	+	0.160880i	−0.622076	+	0.782957i	$0.713711\pi$
$90$		0			0
$91$	−11.0858	+	9.21257i	−1.16211	+	0.965739i
$92$			0.452723i			0.0471997i
$93$		0			0
$94$	7.26611			0.749442
$95$	13.2138			1.35571
$96$		0			0
$97$	2.07237	+	2.07237i	0.210417	+	0.210417i	0.804445	−	0.594028i	$-0.202463\pi$
$97$	2.07237	+	2.07237i	0.210417	+	0.210417i	−0.594028	+	0.804445i	$0.702463\pi$
$98$	10.2397	+	10.2397i	1.03436	+	1.03436i
$99$		0			0
$100$	−6.70425			−0.670425
$101$	9.92671			0.987744			0.493872	−	0.869535i	$-0.335581\pi$
$101$	9.92671			0.987744			0.493872	+	0.869535i	$0.335581\pi$
$102$		0			0
$103$		−	8.93719i		−	0.880607i	−0.897849	−	0.440304i	$-0.854871\pi$
$103$		−	8.93719i		−	0.880607i	0.897849	−	0.440304i	$-0.145129\pi$
$104$	−5.20406	−	6.26223i	−0.510300	−	0.614062i
$105$		0			0
$106$	11.8498	+	11.8498i	1.15096	+	1.15096i
$107$		−	12.3513i		−	1.19404i	−0.802226	−	0.597020i	$-0.796351\pi$
$107$		−	12.3513i		−	1.19404i	0.802226	−	0.597020i	$-0.203649\pi$
$108$		0			0
$109$	−8.62828	−	8.62828i	−0.826439	−	0.826439i	0.160583	−	0.987022i	$-0.448663\pi$
$109$	−8.62828	−	8.62828i	−0.826439	−	0.826439i	−0.987022	+	0.160583i	$0.948663\pi$
$110$	−13.5090	+	13.5090i	−1.28803	+	1.28803i
$111$		0			0
$112$	−13.6803	+	13.6803i	−1.29266	+	1.29266i
$113$			10.9338i			1.02856i	0.857622	+	0.514281i	$0.171941\pi$
$113$			10.9338i			1.02856i	−0.857622	+	0.514281i	$0.828059\pi$
$114$		0			0
$115$	−2.14918	+	2.14918i	−0.200412	+	0.200412i
$116$	2.04343			0.189728
$117$		0			0
$118$	16.4335			1.51283
$119$	−11.1636	+	11.1636i	−1.02336	+	1.02336i
$120$		0			0
$121$		−	2.32532i		−	0.211393i
$122$	−3.32046	+	3.32046i	−0.300621	+	0.300621i
$123$		0			0
$124$	3.83044	−	3.83044i	0.343984	−	0.343984i
$125$	−17.6020	−	17.6020i	−1.57437	−	1.57437i
$126$		0			0
$127$		−	6.58429i		−	0.584261i	−0.956378	−	0.292131i	$-0.905636\pi$
$127$		−	6.58429i		−	0.584261i	0.956378	−	0.292131i	$-0.0943642\pi$
$128$	−9.64173	−	9.64173i	−0.852216	−	0.852216i
$129$		0			0
$130$	−2.14918	+	23.2885i	−0.188496	+	2.04254i
$131$		−	1.84780i		−	0.161443i	−0.996737	−	0.0807214i	$-0.974278\pi$
$131$		−	1.84780i		−	0.161443i	0.996737	−	0.0807214i	$-0.0257224\pi$
$132$		0			0
$133$	−13.1298			−1.13850
$134$	11.7029			1.01098
$135$		0			0
$136$	−6.30616	−	6.30616i	−0.540749	−	0.540749i
$137$	−0.932368	−	0.932368i	−0.0796575	−	0.0796575i	0.666155	−	0.745813i	$-0.267939\pi$
$137$	−0.932368	−	0.932368i	−0.0796575	−	0.0796575i	−0.745813	+	0.666155i	$0.767939\pi$
$138$		0			0
$139$	−1.59985			−0.135697			−0.0678487	−	0.997696i	$-0.521614\pi$
$139$	−1.59985			−0.135697			−0.0678487	+	0.997696i	$0.521614\pi$
$140$	9.63898			0.814643
$141$		0			0
$142$			4.76408i			0.399793i
$143$	6.78721	+	8.16729i	0.567575	+	0.682983i
$144$		0			0
$145$	9.70064	+	9.70064i	0.805595	+	0.805595i
$146$		−	8.13488i		−	0.673247i
$147$		0			0
$148$	1.71289	+	1.71289i	0.140799	+	0.140799i
$149$	−10.6174	+	10.6174i	−0.869811	+	0.869811i	−0.992451	−	0.122640i	$-0.960864\pi$
$149$	−10.6174	+	10.6174i	−0.869811	+	0.869811i	0.122640	+	0.992451i	$0.460864\pi$
$150$		0			0
$151$	−4.67908	+	4.67908i	−0.380778	+	0.380778i	−0.871382	−	0.490605i	$-0.836776\pi$
$151$	−4.67908	+	4.67908i	−0.380778	+	0.380778i	0.490605	+	0.871382i	$0.336776\pi$
$152$		−	7.41686i		−	0.601586i
$153$		0			0
$154$	13.4231	−	13.4231i	1.08167	−	1.08167i
$155$	36.3680			2.92115
$156$		0			0
$157$	−4.46410			−0.356274			−0.178137	−	0.984006i	$-0.557007\pi$
$157$	−4.46410			−0.356274			−0.178137	+	0.984006i	$0.557007\pi$
$158$	−6.60350	+	6.60350i	−0.525346	+	0.525346i
$159$		0			0
$160$			13.2194i			1.04509i
$161$	2.13552	−	2.13552i	0.168303	−	0.168303i
$162$		0			0
$163$	5.14776	−	5.14776i	0.403204	−	0.403204i	−0.476157	−	0.879361i	$-0.657970\pi$
$163$	5.14776	−	5.14776i	0.403204	−	0.403204i	0.879361	+	0.476157i	$0.157970\pi$
$164$	1.24808	+	1.24808i	0.0974584	+	0.0974584i
$165$		0			0
$166$		−	16.3909i		−	1.27218i
$167$	−6.61536	−	6.61536i	−0.511912	−	0.511912i	0.403200	−	0.915112i	$-0.367898\pi$
$167$	−6.61536	−	6.61536i	−0.511912	−	0.511912i	−0.915112	+	0.403200i	$0.867898\pi$
$168$		0			0
$169$	12.7804	+	2.37915i	0.983111	+	0.183012i
$170$			25.6162i			1.96467i
$171$		0			0
$172$	−0.315769			−0.0240771
$173$	−5.15052			−0.391587			−0.195793	−	0.980645i	$-0.562728\pi$
$173$	−5.15052			−0.391587			−0.195793	+	0.980645i	$0.562728\pi$
$174$		0			0
$175$	31.6243	+	31.6243i	2.39057	+	2.39057i
$176$	10.0787	+	10.0787i	0.759712	+	0.759712i
$177$		0			0
$178$	−3.46050			−0.259376
$179$	−24.9678			−1.86618			−0.933091	−	0.359641i	$-0.882899\pi$
$179$	−24.9678			−1.86618			−0.933091	+	0.359641i	$0.882899\pi$
$180$		0			0
$181$		−	12.9012i		−	0.958942i	−0.877558	−	0.479471i	$-0.840829\pi$
$181$		−	12.9012i		−	0.958942i	0.877558	−	0.479471i	$-0.159171\pi$
$182$	2.13552	−	23.1405i	0.158295	−	1.71529i
$183$		0			0
$184$	1.20633	+	1.20633i	0.0889318	+	0.0889318i
$185$			16.2630i			1.19568i
$186$		0			0
$187$	8.22458	+	8.22458i	0.601441	+	0.601441i
$188$	−1.90981	+	1.90981i	−0.139287	+	0.139287i
$189$		0			0
$190$	−15.0640	+	15.0640i	−1.09285	+	1.09285i
$191$		−	4.73877i		−	0.342885i	−0.985194	−	0.171443i	$-0.945157\pi$
$191$		−	4.73877i		−	0.342885i	0.985194	−	0.171443i	$-0.0548428\pi$
$192$		0			0
$193$	−13.2715	+	13.2715i	−0.955307	+	0.955307i	−0.999043	−	0.0437361i	$-0.986074\pi$
$193$	−13.2715	+	13.2715i	−0.955307	+	0.955307i	0.0437361	+	0.999043i	$0.486074\pi$
$194$	−4.72507			−0.339240
$195$		0			0
$196$	−5.38275			−0.384482
$197$	−11.3728	+	11.3728i	−0.810281	+	0.810281i	−0.984676	−	0.174395i	$-0.944203\pi$
$197$	−11.3728	+	11.3728i	−0.810281	+	0.810281i	0.174395	+	0.984676i	$0.444203\pi$
$198$		0			0
$199$			3.44854i			0.244460i	0.992502	+	0.122230i	$0.0390046\pi$
$199$			3.44854i			0.244460i	−0.992502	+	0.122230i	$0.960995\pi$
$200$	−17.8642	+	17.8642i	−1.26319	+	1.26319i
$201$		0			0
$202$	−11.3166	+	11.3166i	−0.796235	+	0.796235i
$203$	−9.63898	−	9.63898i	−0.676524	−	0.676524i
$204$		0			0
$205$			11.8498i			0.827628i
$206$	10.1886	+	10.1886i	0.709870	+	0.709870i
$207$		0			0
$208$	17.3750	+	1.60345i	1.20474	+	0.111179i
$209$			9.67316i			0.669107i
$210$		0			0
$211$	11.4485			0.788150			0.394075	−	0.919078i	$-0.371065\pi$
$211$	11.4485			0.788150			0.394075	+	0.919078i	$0.371065\pi$
$212$	−6.22917			−0.427821
$213$		0			0
$214$	14.0806	+	14.0806i	0.962533	+	0.962533i
$215$	−1.49903	−	1.49903i	−0.102233	−	0.102233i
$216$		0			0
$217$	−36.1368			−2.45313
$218$	19.6728			1.33241
$219$		0			0
$220$		−	7.10136i		−	0.478774i
$221$	14.1786	+	1.30847i	0.953754	+	0.0880172i
$222$		0			0
$223$	4.29695	+	4.29695i	0.287745	+	0.287745i	0.836188	−	0.548443i	$-0.184779\pi$
$223$	4.29695	+	4.29695i	0.287745	+	0.287745i	−0.548443	+	0.836188i	$0.684779\pi$
$224$		−	13.1354i		−	0.877645i
$225$		0			0
$226$	−12.4647	−	12.4647i	−0.829138	−	0.829138i
$227$	0.709400	−	0.709400i	0.0470845	−	0.0470845i	−0.683173	−	0.730257i	$-0.739400\pi$
$227$	0.709400	−	0.709400i	0.0470845	−	0.0470845i	0.730257	+	0.683173i	$0.239400\pi$
$228$		0			0
$229$	13.7455	−	13.7455i	0.908327	−	0.908327i	−0.0878099	−	0.996137i	$-0.527987\pi$
$229$	13.7455	−	13.7455i	0.908327	−	0.908327i	0.996137	+	0.0878099i	$0.0279868\pi$
$230$		−	4.90022i		−	0.323111i
$231$		0			0
$232$	5.44494	−	5.44494i	0.357478	−	0.357478i
$233$	23.6593			1.54997			0.774987	−	0.631977i	$-0.217756\pi$
$233$	23.6593			1.54997			0.774987	+	0.631977i	$0.217756\pi$
$234$		0			0
$235$	−18.1326			−1.18284
$236$	−4.31936	+	4.31936i	−0.281166	+	0.281166i
$237$		0			0
$238$		−	25.4533i		−	1.64989i
$239$	10.6634	−	10.6634i	0.689761	−	0.689761i	−0.272418	−	0.962179i	$-0.587823\pi$
$239$	10.6634	−	10.6634i	0.689761	−	0.689761i	0.962179	+	0.272418i	$0.0878234\pi$
$240$		0			0
$241$	−13.1118	+	13.1118i	−0.844607	+	0.844607i	−0.989454	−	0.144847i	$-0.953731\pi$
$241$	−13.1118	+	13.1118i	−0.844607	+	0.844607i	0.144847	+	0.989454i	$0.453731\pi$
$242$	2.65091	+	2.65091i	0.170407	+	0.170407i
$243$		0			0
$244$		−	1.74549i		−	0.111743i
$245$	−25.5532	−	25.5532i	−1.63253	−	1.63253i
$246$		0			0
$247$	7.56844	+	9.10737i	0.481569	+	0.579488i
$248$		−	20.4132i		−	1.29624i
$249$		0			0
$250$	40.1332			2.53825
$251$	−19.9605			−1.25990			−0.629948	−	0.776637i	$-0.716924\pi$
$251$	−19.9605			−1.25990			−0.629948	+	0.776637i	$0.716924\pi$
$252$		0			0
$253$	−1.57331	−	1.57331i	−0.0989133	−	0.0989133i
$254$	7.50620	+	7.50620i	0.470981	+	0.470981i
$255$		0			0
$256$	13.2204			0.826272
$257$	3.84205			0.239660			0.119830	−	0.992794i	$-0.461765\pi$
$257$	3.84205			0.239660			0.119830	+	0.992794i	$0.461765\pi$
$258$		0			0
$259$		−	16.1596i		−	1.00411i
$260$	−5.55622	−	6.68600i	−0.344582	−	0.414648i
$261$		0			0
$262$	2.10652	+	2.10652i	0.130141	+	0.130141i
$263$		−	16.9250i		−	1.04364i	−0.853055	−	0.521821i	$-0.825253\pi$
$263$		−	16.9250i		−	1.04364i	0.853055	−	0.521821i	$-0.174747\pi$
$264$		0			0
$265$	−29.5713	−	29.5713i	−1.81655	−	1.81655i
$266$	14.9682	−	14.9682i	0.917759	−	0.917759i
$267$		0			0
$268$	−3.07597	+	3.07597i	−0.187895	+	0.187895i
$269$			13.1409i			0.801214i	0.916250	+	0.400607i	$0.131201\pi$
$269$			13.1409i			0.801214i	−0.916250	+	0.400607i	$0.868799\pi$
$270$		0			0
$271$	3.45455	−	3.45455i	0.209849	−	0.209849i	−0.594354	−	0.804203i	$-0.702592\pi$
$271$	3.45455	−	3.45455i	0.209849	−	0.209849i	0.804203	+	0.594354i	$0.202592\pi$
$272$	19.1115			1.15881
$273$		0			0
$274$	2.12583			0.128426
$275$	23.2987	−	23.2987i	1.40496	−	1.40496i
$276$		0			0
$277$		−	2.38332i		−	0.143200i	−0.997433	−	0.0715999i	$-0.977190\pi$
$277$		−	2.38332i		−	0.143200i	0.997433	−	0.0715999i	$-0.0228105\pi$
$278$	1.82386	−	1.82386i	0.109388	−	0.109388i
$279$		0			0
$280$	25.6841	−	25.6841i	1.53492	−	1.53492i
$281$	20.8144	+	20.8144i	1.24168	+	1.24168i	0.959302	+	0.282382i	$0.0911245\pi$
$281$	20.8144	+	20.8144i	1.24168	+	1.24168i	0.282382	+	0.959302i	$0.408875\pi$
$282$		0			0
$283$			29.2051i			1.73606i	0.496511	+	0.868030i	$0.334614\pi$
$283$			29.2051i			1.73606i	−0.496511	+	0.868030i	$0.665386\pi$
$284$	−1.25218	−	1.25218i	−0.0743033	−	0.0743033i
$285$		0			0
$286$	−17.0484	−	1.57331i	−1.00809	−	0.0930319i
$287$		−	11.7745i		−	0.695027i
$288$		0			0
$289$	−1.40432			−0.0826071
$290$	−22.1178			−1.29880
$291$		0			0
$292$	2.13816	+	2.13816i	0.125126	+	0.125126i
$293$	−23.6406	−	23.6406i	−1.38110	−	1.38110i	−0.842670	−	0.538430i	$-0.819018\pi$
$293$	−23.6406	−	23.6406i	−1.38110	−	1.38110i	−0.538430	−	0.842670i	$-0.680982\pi$
$294$		0			0
$295$	−41.0100			−2.38770
$296$	9.12835			0.530575
$297$		0			0
$298$		−	24.2080i		−	1.40233i
$299$	−2.71227	−	0.250302i	−0.156855	−	0.0144754i
$300$		0			0
$301$	1.48950	+	1.48950i	0.0858533	+	0.0858533i
$302$		−	10.6685i		−	0.613901i
$303$		0			0
$304$	11.2388	+	11.2388i	0.644590	+	0.644590i
$305$	8.28624	−	8.28624i	0.474469	−	0.474469i
$306$		0			0
$307$	−10.5089	+	10.5089i	−0.599777	+	0.599777i	−0.940253	−	0.340476i	$-0.889412\pi$
$307$	−10.5089	+	10.5089i	−0.599777	+	0.599777i	0.340476	+	0.940253i	$0.389412\pi$
$308$			7.05622i			0.402066i
$309$		0			0
$310$	−41.4601	+	41.4601i	−2.35478	+	2.35478i
$311$	−4.95488			−0.280966			−0.140483	−	0.990083i	$-0.544865\pi$
$311$	−4.95488			−0.280966			−0.140483	+	0.990083i	$0.544865\pi$
$312$		0			0
$313$	4.30078			0.243094			0.121547	−	0.992586i	$-0.461214\pi$
$313$	4.30078			0.243094			0.121547	+	0.992586i	$0.461214\pi$
$314$	5.08915	−	5.08915i	0.287198	−	0.287198i
$315$		0			0
$316$		−	3.47130i		−	0.195276i
$317$	−18.9565	+	18.9565i	−1.06470	+	1.06470i	−0.0669483	+	0.997756i	$0.521326\pi$
$317$	−18.9565	+	18.9565i	−1.06470	+	1.06470i	−0.997756	+	0.0669483i	$0.978674\pi$
$318$		0			0
$319$	−7.10136	+	7.10136i	−0.397600	+	0.397600i
$320$	12.4652	+	12.4652i	0.696825	+	0.696825i
$321$		0			0
$322$			4.86907i			0.271343i
$323$	9.17126	+	9.17126i	0.510303	+	0.510303i
$324$		0			0
$325$	3.70665	−	40.1652i	0.205608	−	2.22797i
$326$			11.7371i			0.650057i
$327$		0			0
$328$	6.65127			0.367255
$329$	18.0174			0.993330
$330$		0			0
$331$	−21.1471	−	21.1471i	−1.16235	−	1.16235i	−0.983960	−	0.178387i	$-0.942912\pi$
$331$	−21.1471	−	21.1471i	−1.16235	−	1.16235i	−0.178387	−	0.983960i	$-0.557088\pi$
$332$	4.30815	+	4.30815i	0.236440	+	0.236440i
$333$		0			0
$334$	15.0833			0.825319
$335$	−29.2047			−1.59562
$336$		0			0
$337$		−	26.3468i		−	1.43520i	−0.696456	−	0.717600i	$-0.745241\pi$
$337$		−	26.3468i		−	1.43520i	0.696456	−	0.717600i	$-0.254759\pi$
$338$	−17.2822	+	11.8577i	−0.940028	+	0.644971i
$339$		0			0
$340$	−6.73290	−	6.73290i	−0.365143	−	0.365143i
$341$			26.6232i			1.44173i
$342$		0			0
$343$	5.60288	+	5.60288i	0.302527	+	0.302527i
$344$	−0.841400	+	0.841400i	−0.0453652	+	0.0453652i
$345$		0			0
$346$	5.87168	−	5.87168i	0.315664	−	0.315664i
$347$		−	0.164998i		−	0.00885753i	−0.999990	−	0.00442877i	$-0.998590\pi$
$347$		−	0.164998i		−	0.00885753i	0.999990	−	0.00442877i	$-0.00140972\pi$
$348$		0			0
$349$	25.4287	−	25.4287i	1.36117	−	1.36117i	0.488739	−	0.872430i	$-0.337457\pi$
$349$	25.4287	−	25.4287i	1.36117	−	1.36117i	0.872430	−	0.488739i	$-0.162543\pi$
$350$	−72.1045			−3.85415
$351$		0			0
$352$	−9.67729			−0.515801
$353$	−8.52107	+	8.52107i	−0.453531	+	0.453531i	−0.896525	−	0.442994i	$-0.853916\pi$
$353$	−8.52107	+	8.52107i	−0.453531	+	0.453531i	0.442994	+	0.896525i	$0.353916\pi$
$354$		0			0
$355$		−	11.8888i		−	0.630992i
$356$	0.909551	−	0.909551i	0.0482061	−	0.0482061i
$357$		0			0
$358$	28.4637	−	28.4637i	1.50436	−	1.50436i
$359$	20.2377	+	20.2377i	1.06810	+	1.06810i	0.997505	+	0.0705983i	$0.0224908\pi$
$359$	20.2377	+	20.2377i	1.06810	+	1.06810i	0.0705983	+	0.997505i	$0.477509\pi$
$360$		0			0
$361$		−	8.21342i		−	0.432285i
$362$	14.7077	+	14.7077i	0.773017	+	0.773017i
$363$		0			0
$364$	5.52091	+	6.64350i	0.289374	+	0.348214i
$365$			20.3006i			1.06258i
$366$		0			0
$367$	−6.88506			−0.359397			−0.179699	−	0.983722i	$-0.557512\pi$
$367$	−6.88506			−0.359397			−0.179699	+	0.983722i	$0.557512\pi$
$368$	−3.65592			−0.190578
$369$		0			0
$370$	−18.5401	−	18.5401i	−0.963852	−	0.963852i
$371$	29.3834	+	29.3834i	1.52551	+	1.52551i
$372$		0			0
$373$	32.1812			1.66628			0.833140	−	0.553062i	$-0.186541\pi$
$373$	32.1812			1.66628			0.833140	+	0.553062i	$0.186541\pi$
$374$	−18.7523			−0.969660
$375$		0			0
$376$			10.1778i			0.524879i
$377$	−1.12977	+	12.2422i	−0.0581864	+	0.630507i
$378$		0			0
$379$	14.4969	+	14.4969i	0.744657	+	0.744657i	0.973470	−	0.228813i	$-0.0734845\pi$
$379$	14.4969	+	14.4969i	0.744657	+	0.744657i	−0.228813	+	0.973470i	$0.573484\pi$
$380$		−	7.91876i		−	0.406224i
$381$		0			0
$382$	5.40228	+	5.40228i	0.276404	+	0.276404i
$383$	19.6199	−	19.6199i	1.00253	−	1.00253i	0.00253237	−	0.999997i	$-0.499194\pi$
$383$	19.6199	−	19.6199i	1.00253	−	1.00253i	0.999997	−	0.00253237i	$-0.000806078\pi$
$384$		0			0
$385$	−33.4976	+	33.4976i	−1.70719	+	1.70719i
$386$		−	30.2596i		−	1.54017i
$387$		0			0
$388$	1.24193	−	1.24193i	0.0630493	−	0.0630493i
$389$	−5.17424			−0.262345			−0.131172	−	0.991360i	$-0.541874\pi$
$389$	−5.17424			−0.262345			−0.131172	+	0.991360i	$0.541874\pi$
$390$		0			0
$391$	−2.98336			−0.150875
$392$	−14.3429	+	14.3429i	−0.724427	+	0.724427i
$393$		0			0
$394$		−	25.9305i		−	1.30636i
$395$	16.4791	−	16.4791i	0.829153	−	0.829153i
$396$		0			0
$397$	−26.9826	+	26.9826i	−1.35422	+	1.35422i	−0.473336	+	0.880882i	$0.656950\pi$
$397$	−26.9826	+	26.9826i	−1.35422	+	1.35422i	−0.880882	+	0.473336i	$0.843050\pi$
$398$	−3.93140	−	3.93140i	−0.197063	−	0.197063i
$399$		0			0
$400$		−	54.1394i		−	2.70697i
$401$	11.4444	+	11.4444i	0.571508	+	0.571508i	0.932550	−	0.361041i	$-0.117579\pi$
$401$	11.4444	+	11.4444i	0.571508	+	0.571508i	−0.361041	+	0.932550i	$0.617579\pi$
$402$		0			0
$403$	20.8304	+	25.0660i	1.03764	+	1.24863i
$404$		−	5.94888i		−	0.295968i
$405$		0			0
$406$	21.9772			1.09071
$407$	−11.9053			−0.590125
$408$		0			0
$409$	−16.7065	−	16.7065i	−0.826084	−	0.826084i	0.160888	−	0.986973i	$-0.448564\pi$
$409$	−16.7065	−	16.7065i	−0.826084	−	0.826084i	−0.986973	+	0.160888i	$0.948564\pi$
$410$	−13.5090	−	13.5090i	−0.667163	−	0.667163i
$411$		0			0
$412$	−5.35588			−0.263865
$413$	40.7494			2.00514
$414$		0			0
$415$			40.9036i			2.00788i
$416$	−9.11126	+	7.57167i	−0.446716	+	0.371232i
$417$		0			0
$418$	−11.0276	−	11.0276i	−0.539376	−	0.539376i
$419$			39.8507i			1.94683i	0.229040	+	0.973417i	$0.426441\pi$
$419$			39.8507i			1.94683i	−0.229040	+	0.973417i	$0.573559\pi$
$420$		0			0
$421$	−1.73565	−	1.73565i	−0.0845905	−	0.0845905i	0.663545	−	0.748136i	$-0.269051\pi$
$421$	−1.73565	−	1.73565i	−0.0845905	−	0.0845905i	−0.748136	+	0.663545i	$0.769051\pi$
$422$	−13.0515	+	13.0515i	−0.635339	+	0.635339i
$423$		0			0
$424$	−16.5983	+	16.5983i	−0.806084	+	0.806084i
$425$		−	44.1797i		−	2.14303i
$426$		0			0
$427$	−8.23357	+	8.23357i	−0.398450	+	0.398450i
$428$	−7.40186			−0.357782
$429$		0			0
$430$	3.41784			0.164823
$431$	5.07206	−	5.07206i	0.244313	−	0.244313i	−0.574319	−	0.818632i	$-0.694733\pi$
$431$	5.07206	−	5.07206i	0.244313	−	0.244313i	0.818632	+	0.574319i	$0.194733\pi$
$432$		0			0
$433$			10.4803i			0.503650i	0.967773	+	0.251825i	$0.0810308\pi$
$433$			10.4803i			0.503650i	−0.967773	+	0.251825i	$0.918969\pi$
$434$	41.1966	−	41.1966i	1.97750	−	1.97750i
$435$		0			0
$436$	−5.17075	+	5.17075i	−0.247634	+	0.247634i
$437$	−1.75441	−	1.75441i	−0.0839246	−	0.0839246i
$438$		0			0
$439$			2.71763i			0.129705i	0.997895	+	0.0648526i	$0.0206577\pi$
$439$			2.71763i			0.129705i	−0.997895	+	0.0648526i	$0.979342\pi$
$440$	−18.9223	−	18.9223i	−0.902087	−	0.902087i
$441$		0			0
$442$	−17.6555	+	14.6721i	−0.839786	+	0.697882i
$443$		−	4.55328i		−	0.216333i	−0.994133	−	0.108166i	$-0.965502\pi$
$443$		−	4.55328i		−	0.216333i	0.994133	−	0.108166i	$-0.0344980\pi$
$444$		0			0
$445$	8.63571			0.409372
$446$	−9.79719			−0.463910
$447$		0			0
$448$	−12.3860	−	12.3860i	−0.585181	−	0.585181i
$449$	−28.2655	−	28.2655i	−1.33393	−	1.33393i	−0.901822	−	0.432107i	$-0.857770\pi$
$449$	−28.2655	−	28.2655i	−1.33393	−	1.33393i	−0.432107	−	0.901822i	$-0.642230\pi$
$450$		0			0
$451$	−8.67468			−0.408475
$452$	6.55238			0.308198
$453$		0			0
$454$			1.61746i			0.0759110i
$455$	−5.32921	+	57.7473i	−0.249837	+	2.70723i
$456$		0			0
$457$	6.20457	+	6.20457i	0.290237	+	0.290237i	0.837174	−	0.546937i	$-0.184206\pi$
$457$	6.20457	+	6.20457i	0.290237	+	0.290237i	−0.546937	+	0.837174i	$0.684206\pi$
$458$			31.3402i			1.46443i
$459$		0			0
$460$	1.28796	+	1.28796i	0.0600516	+	0.0600516i
$461$	24.9610	−	24.9610i	1.16255	−	1.16255i	0.178632	−	0.983916i	$-0.442833\pi$
$461$	24.9610	−	24.9610i	1.16255	−	1.16255i	0.983916	−	0.178632i	$-0.0571671\pi$
$462$		0			0
$463$	−0.184419	+	0.184419i	−0.00857068	+	0.00857068i	−0.711379	−	0.702808i	$-0.751929\pi$
$463$	−0.184419	+	0.184419i	−0.00857068	+	0.00857068i	0.702808	+	0.711379i	$0.251929\pi$
$464$			16.5015i			0.766064i
$465$		0			0
$466$	−26.9721	+	26.9721i	−1.24946	+	1.24946i
$467$	3.26530			0.151100			0.0755500	−	0.997142i	$-0.475929\pi$
$467$	3.26530			0.151100			0.0755500	+	0.997142i	$0.475929\pi$
$468$		0			0
$469$	29.0190			1.33997
$470$	20.6715	−	20.6715i	0.953506	−	0.953506i
$471$		0			0
$472$			23.0188i			1.05953i
$473$	1.09737	−	1.09737i	0.0504569	−	0.0504569i
$474$		0			0
$475$	25.9805	−	25.9805i	1.19207	−	1.19207i
$476$	6.69010	+	6.69010i	0.306640	+	0.306640i
$477$		0			0
$478$			24.3130i			1.11205i
$479$	−15.2846	−	15.2846i	−0.698369	−	0.698369i	0.265690	−	0.964059i	$-0.414400\pi$
$479$	−15.2846	−	15.2846i	−0.698369	−	0.698369i	−0.964059	+	0.265690i	$0.914400\pi$
$480$		0			0
$481$	−11.2090	+	9.31492i	−0.511085	+	0.424724i
$482$		−	29.8954i		−	1.36170i
$483$		0			0
$484$	−1.39352			−0.0633418
$485$	11.7914			0.535422
$486$		0			0
$487$	11.7321	+	11.7321i	0.531630	+	0.531630i	0.921057	−	0.389427i	$-0.127327\pi$
$487$	11.7321	+	11.7321i	0.531630	+	0.531630i	−0.389427	+	0.921057i	$0.627327\pi$
$488$	−4.65104	−	4.65104i	−0.210543	−	0.210543i
$489$		0			0
$490$	58.2621			2.63202
$491$	−25.7233			−1.16087			−0.580437	−	0.814305i	$-0.697118\pi$
$491$	−25.7233			−1.16087			−0.580437	+	0.814305i	$0.697118\pi$
$492$		0			0
$493$			13.4658i			0.606469i
$494$	−19.0107	−	1.75441i	−0.855333	−	0.0789345i
$495$		0			0
$496$	30.9323	+	30.9323i	1.38890	+	1.38890i
$497$			11.8132i			0.529896i
$498$		0			0
$499$	16.0744	+	16.0744i	0.719589	+	0.719589i	0.968521	−	0.248932i	$-0.0800794\pi$
$499$	16.0744	+	16.0744i	0.719589	+	0.719589i	−0.248932	+	0.968521i	$0.580079\pi$
$500$	−10.5485	+	10.5485i	−0.471745	+	0.471745i
$501$		0			0
$502$	22.7553	−	22.7553i	1.01562	−	1.01562i
$503$			18.0242i			0.803658i	0.915715	+	0.401829i	$0.131625\pi$
$503$			18.0242i			0.803658i	−0.915715	+	0.401829i	$0.868375\pi$
$504$		0			0
$505$	28.2407	−	28.2407i	1.25670	−	1.25670i
$506$	3.58721			0.159471
$507$		0			0
$508$	−3.94583			−0.175068
$509$	22.7730	−	22.7730i	1.00940	−	1.00940i	0.00944076	−	0.999955i	$-0.496995\pi$
$509$	22.7730	−	22.7730i	1.00940	−	1.00940i	0.999955	−	0.00944076i	$-0.00300513\pi$
$510$		0			0
$511$		−	20.1716i		−	0.892339i
$512$	4.21201	−	4.21201i	0.186146	−	0.186146i
$513$		0			0
$514$	−4.38000	+	4.38000i	−0.193194	+	0.193194i
$515$	−25.4256	−	25.4256i	−1.12039	−	1.12039i
$516$		0			0
$517$		−	13.2740i		−	0.583790i
$518$	18.4222	+	18.4222i	0.809426	+	0.809426i
$519$		0			0
$520$	−32.6207	−	3.01041i	−1.43051	−	0.132015i
$521$		−	34.8024i		−	1.52472i	−0.647152	−	0.762361i	$-0.724040\pi$
$521$		−	34.8024i		−	1.52472i	0.647152	−	0.762361i	$-0.275960\pi$
$522$		0			0
$523$	−0.671646			−0.0293690			−0.0146845	−	0.999892i	$-0.504674\pi$
$523$	−0.671646			−0.0293690			−0.0146845	+	0.999892i	$0.504674\pi$
$524$	−1.10735			−0.0483747
$525$		0			0
$526$	19.2948	+	19.2948i	0.841294	+	0.841294i
$527$	25.2418	+	25.2418i	1.09955	+	1.09955i
$528$		0			0
$529$	−22.4293			−0.975187
$530$	67.4237			2.92870
$531$		0			0
$532$			7.86842i			0.341139i
$533$	−8.16729	+	6.78721i	−0.353764	+	0.293987i
$534$		0			0
$535$	−35.1384	−	35.1384i	−1.51916	−	1.51916i
$536$			16.3925i			0.708047i
$537$		0			0
$538$	−14.9808	−	14.9808i	−0.645870	−	0.645870i
$539$	18.7062	−	18.7062i	0.805734	−	0.805734i
$540$		0			0
$541$	23.7311	−	23.7311i	1.02028	−	1.02028i	0.0204881	−	0.999790i	$-0.493478\pi$
$541$	23.7311	−	23.7311i	1.02028	−	1.02028i	0.999790	−	0.0204881i	$-0.00652201\pi$
$542$			7.87649i			0.338324i
$543$		0			0
$544$	−9.17517	+	9.17517i	−0.393382	+	0.393382i
$545$	−49.0936			−2.10294
$546$		0			0
$547$	−12.5878			−0.538217			−0.269108	−	0.963110i	$-0.586729\pi$
$547$	−12.5878			−0.538217			−0.269108	+	0.963110i	$0.586729\pi$
$548$	−0.558749	+	0.558749i	−0.0238686	+	0.0238686i
$549$		0			0
$550$			53.1219i			2.26512i
$551$	−7.91876	+	7.91876i	−0.337350	+	0.337350i
$552$		0			0
$553$	−16.3743	+	16.3743i	−0.696308	+	0.696308i
$554$	2.71703	+	2.71703i	0.115435	+	0.115435i
$555$		0			0
$556$			0.958758i			0.0406604i
$557$	−9.28336	−	9.28336i	−0.393349	−	0.393349i	0.482530	−	0.875879i	$-0.339718\pi$
$557$	−9.28336	−	9.28336i	−0.393349	−	0.393349i	−0.875879	+	0.482530i	$0.839718\pi$
$558$		0			0
$559$	0.174583	−	1.89178i	0.00738406	−	0.0800136i
$560$			77.8386i			3.28928i
$561$		0			0
$562$	−47.4576			−2.00188
$563$	−24.7006			−1.04100			−0.520502	−	0.853860i	$-0.674255\pi$
$563$	−24.7006			−1.04100			−0.520502	+	0.853860i	$0.674255\pi$
$564$		0			0
$565$	31.1057	+	31.1057i	1.30863	+	1.30863i
$566$	−33.2943	−	33.2943i	−1.39946	−	1.39946i
$567$		0			0
$568$	−6.67314			−0.279999
$569$	−11.6163			−0.486980			−0.243490	−	0.969903i	$-0.578292\pi$
$569$	−11.6163			−0.486980			−0.243490	+	0.969903i	$0.578292\pi$
$570$		0			0
$571$			4.79069i			0.200484i	0.994963	+	0.100242i	$0.0319617\pi$
$571$			4.79069i			0.200484i	−0.994963	+	0.100242i	$0.968038\pi$
$572$	4.89449	−	4.06744i	0.204649	−	0.170068i
$573$		0			0
$574$	13.4231	+	13.4231i	0.560271	+	0.560271i
$575$			8.45130i			0.352443i
$576$		0			0
$577$	−10.9082	−	10.9082i	−0.454116	−	0.454116i	0.442602	−	0.896718i	$-0.354055\pi$
$577$	−10.9082	−	10.9082i	−0.454116	−	0.454116i	−0.896718	+	0.442602i	$0.854055\pi$
$578$	1.60095	−	1.60095i	0.0665907	−	0.0665907i
$579$		0			0
$580$	5.81340	−	5.81340i	0.241388	−	0.241388i
$581$		−	40.6436i		−	1.68618i
$582$		0			0
$583$	21.6477	−	21.6477i	0.896557	−	0.896557i
$584$	11.3947			0.471515
$585$		0			0
$586$	53.9015			2.22665
$587$	21.9578	−	21.9578i	0.906296	−	0.906296i	−0.0896746	−	0.995971i	$-0.528583\pi$
$587$	21.9578	−	21.9578i	0.906296	−	0.906296i	0.995971	+	0.0896746i	$0.0285827\pi$
$588$		0			0
$589$			29.6876i			1.22326i
$590$	46.7522	−	46.7522i	1.92476	−	1.92476i
$591$		0			0
$592$	−13.8323	+	13.8323i	−0.568502	+	0.568502i
$593$	24.5051	+	24.5051i	1.00630	+	1.00630i	0.999980	+	0.00632392i	$0.00201298\pi$
$593$	24.5051	+	24.5051i	1.00630	+	1.00630i	0.00632392	+	0.999980i	$0.497987\pi$
$594$		0			0
$595$			63.5190i			2.60402i
$596$	6.36279	+	6.36279i	0.260630	+	0.260630i
$597$		0			0
$598$	3.37739	−	2.80669i	0.138112	−	0.114774i
$599$			23.0963i			0.943689i	0.881682	+	0.471844i	$0.156412\pi$
$599$			23.0963i			0.943689i	−0.881682	+	0.471844i	$0.843588\pi$
$600$		0			0
$601$	18.0484			0.736209			0.368105	−	0.929784i	$-0.380007\pi$
$601$	18.0484			0.736209			0.368105	+	0.929784i	$0.380007\pi$
$602$	−3.39611			−0.138415
$603$		0			0
$604$	2.80408	+	2.80408i	0.114096	+	0.114096i
$605$	−6.61536	−	6.61536i	−0.268953	−	0.268953i
$606$		0			0
$607$	−5.55379			−0.225421			−0.112711	−	0.993628i	$-0.535953\pi$
$607$	−5.55379			−0.225421			−0.112711	+	0.993628i	$0.535953\pi$
$608$	−10.7912			−0.437640
$609$		0			0
$610$			18.8929i			0.764952i
$611$	−10.3858	−	12.4976i	−0.420165	−	0.505599i
$612$		0			0
$613$	−15.8678	−	15.8678i	−0.640894	−	0.640894i	0.309881	−	0.950775i	$-0.399711\pi$
$613$	−15.8678	−	15.8678i	−0.640894	−	0.640894i	−0.950775	+	0.309881i	$0.899711\pi$
$614$		−	23.9608i		−	0.966978i
$615$		0			0
$616$	18.8021	+	18.8021i	0.757557	+	0.757557i
$617$	15.6265	−	15.6265i	0.629098	−	0.629098i	−0.318743	−	0.947841i	$-0.603261\pi$
$617$	15.6265	−	15.6265i	0.629098	−	0.629098i	0.947841	+	0.318743i	$0.103261\pi$
$618$		0			0
$619$	30.6482	−	30.6482i	1.23185	−	1.23185i	0.268602	−	0.963251i	$-0.413438\pi$
$619$	30.6482	−	30.6482i	1.23185	−	1.23185i	0.963251	−	0.268602i	$-0.0865617\pi$
$620$		−	21.7946i		−	0.875292i
$621$		0			0
$622$	5.64866	−	5.64866i	0.226490	−	0.226490i
$623$	−8.58081			−0.343783
$624$		0			0
$625$	−44.2169			−1.76868
$626$	−4.90296	+	4.90296i	−0.195962	+	0.195962i
$627$		0			0
$628$			2.67525i			0.106754i
$629$	−11.2876	+	11.2876i	−0.450066	+	0.450066i
$630$		0			0
$631$	−11.2885	+	11.2885i	−0.449389	+	0.449389i	−0.895151	−	0.445762i	$-0.852933\pi$
$631$	−11.2885	+	11.2885i	−0.449389	+	0.449389i	0.445762	+	0.895151i	$0.352933\pi$
$632$	−9.24966	−	9.24966i	−0.367932	−	0.367932i
$633$		0			0
$634$		−	43.2216i		−	1.71655i
$635$	−18.7318	−	18.7318i	−0.743348	−	0.743348i
$636$		0			0
$637$	2.97602	−	32.2481i	0.117914	−	1.27772i
$638$		−	16.1914i		−	0.641022i
$639$		0			0
$640$	−54.8599			−2.16853
$641$	−20.8879			−0.825021			−0.412510	−	0.910953i	$-0.635348\pi$
$641$	−20.8879			−0.825021			−0.412510	+	0.910953i	$0.635348\pi$
$642$		0			0
$643$	32.4927	+	32.4927i	1.28139	+	1.28139i	0.939879	+	0.341507i	$0.110937\pi$
$643$	32.4927	+	32.4927i	1.28139	+	1.28139i	0.341507	+	0.939879i	$0.389063\pi$
$644$	−1.27978	−	1.27978i	−0.0504302	−	0.0504302i
$645$		0			0
$646$	−20.9108			−0.822724
$647$	−11.9714			−0.470646			−0.235323	−	0.971917i	$-0.575615\pi$
$647$	−11.9714			−0.470646			−0.235323	+	0.971917i	$0.575615\pi$
$648$		0			0
$649$		−	30.0214i		−	1.17844i
$650$	41.5634	+	50.0147i	1.63025	+	1.96174i
$651$		0			0
$652$	−3.08495	−	3.08495i	−0.120816	−	0.120816i
$653$		−	9.71059i		−	0.380005i	−0.981784	−	0.190002i	$-0.939150\pi$
$653$		−	9.71059i		−	0.380005i	0.981784	−	0.190002i	$-0.0608496\pi$
$654$		0			0
$655$	−5.25684	−	5.25684i	−0.205402	−	0.205402i
$656$	−10.0787	+	10.0787i	−0.393508	+	0.393508i
$657$		0			0
$658$	−20.5401	+	20.5401i	−0.800737	+	0.800737i
$659$			29.0022i			1.12977i	0.825171	+	0.564884i	$0.191079\pi$
$659$			29.0022i			1.12977i	−0.825171	+	0.564884i	$0.808921\pi$
$660$		0			0
$661$	22.9631	−	22.9631i	0.893160	−	0.893160i	−0.101659	−	0.994819i	$-0.532415\pi$
$661$	22.9631	−	22.9631i	0.893160	−	0.893160i	0.994819	+	0.101659i	$0.0324151\pi$
$662$	48.2160			1.87397
$663$		0			0
$664$	22.9590			0.890984
$665$	−37.3532	+	37.3532i	−1.44850	+	1.44850i
$666$		0			0
$667$		−	2.57593i		−	0.0997403i
$668$	−3.96445	+	3.96445i	−0.153389	+	0.153389i
$669$		0			0
$670$	33.2938	−	33.2938i	1.28625	−	1.28625i
$671$	6.06595	+	6.06595i	0.234173	+	0.234173i
$672$		0			0
$673$			12.5669i			0.484420i	0.970224	+	0.242210i	$0.0778723\pi$
$673$			12.5669i			0.484420i	−0.970224	+	0.242210i	$0.922128\pi$
$674$	30.0358	+	30.0358i	1.15693	+	1.15693i
$675$		0			0
$676$	1.42578	−	7.65906i	0.0548376	−	0.294579i
$677$			21.1632i			0.813367i	0.913569	+	0.406684i	$0.133315\pi$
$677$			21.1632i			0.813367i	−0.913569	+	0.406684i	$0.866685\pi$
$678$		0			0
$679$	−11.7165			−0.449638
$680$	−35.8811			−1.37598
$681$		0			0
$682$	−30.3509	−	30.3509i	−1.16220	−	1.16220i
$683$	−6.59843	−	6.59843i	−0.252482	−	0.252482i	0.569506	−	0.821987i	$-0.307135\pi$
$683$	−6.59843	−	6.59843i	−0.252482	−	0.252482i	−0.821987	+	0.569506i	$0.807135\pi$
$684$		0			0
$685$	−5.30503			−0.202695
$686$	−12.7748			−0.487743
$687$		0			0
$688$		−	2.54996i		−	0.0972163i
$689$	3.44399	−	37.3191i	0.131206	−	1.42174i
$690$		0			0
$691$	−34.7891	−	34.7891i	−1.32344	−	1.32344i	−0.910974	−	0.412465i	$-0.864668\pi$
$691$	−34.7891	−	34.7891i	−1.32344	−	1.32344i	−0.412465	−	0.910974i	$-0.635332\pi$
$692$			3.08660i			0.117335i
$693$		0			0
$694$	0.188100	+	0.188100i	0.00714018	+	0.00714018i
$695$	−4.55145	+	4.55145i	−0.172646	+	0.172646i
$696$		0			0
$697$	−8.22458	+	8.22458i	−0.311528	+	0.311528i
$698$			57.9784i			2.19451i
$699$		0			0
$700$	18.9518	−	18.9518i	0.716311	−	0.716311i
$701$	−31.6224			−1.19436			−0.597181	−	0.802106i	$-0.703713\pi$
$701$	−31.6224			−1.19436			−0.597181	+	0.802106i	$0.703713\pi$
$702$		0			0
$703$	−13.2757			−0.500701
$704$	−9.12515	+	9.12515i	−0.343917	+	0.343917i
$705$		0			0
$706$		−	19.4283i		−	0.731195i
$707$	−28.0612	+	28.0612i	−1.05535	+	1.05535i
$708$		0			0
$709$	−7.91321	+	7.91321i	−0.297187	+	0.297187i	−0.839911	−	0.542724i	$-0.817393\pi$
$709$	−7.91321	+	7.91321i	−0.297187	+	0.297187i	0.542724	+	0.839911i	$0.317393\pi$
$710$	13.5534	+	13.5534i	0.508652	+	0.508652i
$711$		0			0
$712$		−	4.84719i		−	0.181656i
$713$	−4.82861	−	4.82861i	−0.180833	−	0.180833i
$714$		0			0
$715$	42.5444	+	3.92621i	1.59107	+	0.146832i
$716$			14.9627i			0.559183i
$717$		0			0
$718$	−46.1426			−1.72203
$719$	−7.86279			−0.293233			−0.146616	−	0.989193i	$-0.546838\pi$
$719$	−7.86279			−0.293233			−0.146616	+	0.989193i	$0.546838\pi$
$720$		0			0
$721$	25.2640	+	25.2640i	0.940880	+	0.940880i
$722$	9.36344	+	9.36344i	0.348471	+	0.348471i
$723$		0			0
$724$	−7.73146			−0.287337
$725$	38.1461			1.41671
$726$		0			0
$727$		−	8.92509i		−	0.331013i	−0.986209	−	0.165507i	$-0.947074\pi$
$727$		−	8.92509i		−	0.331013i	0.986209	−	0.165507i	$-0.0529259\pi$
$728$	32.4134	+	2.99127i	1.20132	+	0.110864i
$729$		0			0
$730$	−23.1431	−	23.1431i	−0.856564	−	0.856564i
$731$		−	2.08085i		−	0.0769632i
$732$		0			0
$733$	13.8333	+	13.8333i	0.510944	+	0.510944i	0.914816	−	0.403872i	$-0.132336\pi$
$733$	13.8333	+	13.8333i	0.510944	+	0.510944i	−0.403872	+	0.914816i	$0.632336\pi$
$734$	7.84909	−	7.84909i	0.289715	−	0.289715i
$735$		0			0
$736$	1.75515	−	1.75515i	0.0646959	−	0.0646959i
$737$		−	21.3793i		−	0.787517i
$738$		0			0
$739$	10.3041	−	10.3041i	0.379042	−	0.379042i	−0.491714	−	0.870757i	$-0.663630\pi$
$739$	10.3041	−	10.3041i	0.379042	−	0.379042i	0.870757	+	0.491714i	$0.163630\pi$
$740$	9.74607			0.358273
$741$		0			0
$742$	−66.9951			−2.45947
$743$	1.22694	−	1.22694i	0.0450122	−	0.0450122i	−0.684242	−	0.729255i	$-0.739867\pi$
$743$	1.22694	−	1.22694i	0.0450122	−	0.0450122i	0.729255	+	0.684242i	$0.239867\pi$
$744$		0			0
$745$			60.4113i			2.21330i
$746$	−36.6871	+	36.6871i	−1.34321	+	1.34321i
$747$		0			0
$748$	4.92883	−	4.92883i	0.180216	−	0.180216i
$749$	34.9150	+	34.9150i	1.27577	+	1.27577i
$750$		0			0
$751$		−	50.9443i		−	1.85898i	−0.368842	−	0.929492i	$-0.620246\pi$
$751$		−	50.9443i		−	1.85898i	0.368842	−	0.929492i	$-0.379754\pi$
$752$	−15.4225	−	15.4225i	−0.562400	−	0.562400i
$753$		0			0
$754$	−12.6684	−	15.2443i	−0.461355	−	0.555165i
$755$			26.6232i			0.968918i
$756$		0			0
$757$	13.7057			0.498143			0.249071	−	0.968485i	$-0.419875\pi$
$757$	13.7057			0.498143			0.249071	+	0.968485i	$0.419875\pi$
$758$	−33.0535			−1.20056
$759$		0			0
$760$	−21.1004	−	21.1004i	−0.765391	−	0.765391i
$761$	20.9931	+	20.9931i	0.761000	+	0.761000i	0.976503	−	0.215503i	$-0.0691392\pi$
$761$	20.9931	+	20.9931i	0.761000	+	0.761000i	−0.215503	+	0.976503i	$0.569139\pi$
$762$		0			0
$763$	48.7815			1.76601
$764$	−2.83985			−0.102742
$765$		0			0
$766$			44.7340i			1.61631i
$767$	−23.4893	−	28.2655i	−0.848148	−	1.02061i
$768$		0			0
$769$	−5.06281	−	5.06281i	−0.182570	−	0.182570i	0.609905	−	0.792475i	$-0.291208\pi$
$769$	−5.06281	−	5.06281i	−0.182570	−	0.182570i	−0.792475	+	0.609905i	$0.791208\pi$
$770$		−	76.3756i		−	2.75239i
$771$		0			0
$772$	7.95337	+	7.95337i	0.286248	+	0.286248i
$773$	−11.2488	+	11.2488i	−0.404592	+	0.404592i	−0.879848	−	0.475256i	$-0.842355\pi$
$773$	−11.2488	+	11.2488i	−0.404592	+	0.404592i	0.475256	+	0.879848i	$0.342355\pi$
$774$		0			0
$775$	71.5054	−	71.5054i	2.56855	−	2.56855i
$776$		−	6.61850i		−	0.237590i
$777$		0			0
$778$	5.89873	−	5.89873i	0.211480	−	0.211480i
$779$	−9.67316			−0.346577
$780$		0			0
$781$	8.70320			0.311425
$782$	3.40108	−	3.40108i	0.121622	−	0.121622i
$783$		0			0
$784$		−	43.4678i		−	1.55242i
$785$	−12.7000	+	12.7000i	−0.453283	+	0.453283i
$786$		0			0
$787$	−16.4021	+	16.4021i	−0.584673	+	0.584673i	−0.936184	−	0.351511i	$-0.885668\pi$
$787$	−16.4021	+	16.4021i	−0.584673	+	0.584673i	0.351511	+	0.936184i	$0.385668\pi$
$788$	6.81552	+	6.81552i	0.242793	+	0.242793i
$789$		0			0
$790$			37.5729i			1.33678i
$791$	−30.9080	−	30.9080i	−1.09896	−	1.09896i
$792$		0			0
$793$	10.4572	+	0.965048i	0.371348	+	0.0342698i
$794$		−	61.5213i		−	2.18331i
$795$		0			0
$796$	2.06664			0.0732501
$797$	51.6968			1.83119			0.915597	−	0.402096i	$-0.131718\pi$
$797$	51.6968			1.83119			0.915597	+	0.402096i	$0.131718\pi$
$798$		0			0
$799$	−12.5853	−	12.5853i	−0.445235	−	0.445235i
$800$	25.9916	+	25.9916i	0.918940	+	0.918940i
$801$		0			0
$802$	−26.0937			−0.921402
$803$	−14.8611			−0.524437
$804$		0			0
$805$		−	12.1508i		−	0.428259i
$806$	−52.3227	−	4.82861i	−1.84299	−	0.170080i
$807$		0			0
$808$	−15.8514	−	15.8514i	−0.557651	−	0.557651i
$809$			33.3494i			1.17250i	0.810129	+	0.586252i	$0.199397\pi$
$809$			33.3494i			1.17250i	−0.810129	+	0.586252i	$0.800603\pi$
$810$		0			0
$811$	−14.8364	−	14.8364i	−0.520978	−	0.520978i	0.396889	−	0.917867i	$-0.370090\pi$
$811$	−14.8364	−	14.8364i	−0.520978	−	0.520978i	−0.917867	+	0.396889i	$0.870090\pi$
$812$	−5.77645	+	5.77645i	−0.202714	+	0.202714i
$813$		0			0
$814$	13.5723	−	13.5723i	0.475708	−	0.475708i
$815$		−	29.2900i		−	1.02598i
$816$		0			0
$817$	1.22368	−	1.22368i	0.0428110	−	0.0428110i
$818$	38.0914			1.33184
$819$		0			0
$820$	7.10136			0.247990
$821$	19.3172	−	19.3172i	0.674174	−	0.674174i	−0.284502	−	0.958675i	$-0.591828\pi$
$821$	19.3172	−	19.3172i	0.674174	−	0.674174i	0.958675	+	0.284502i	$0.0918282\pi$
$822$		0			0
$823$		−	43.3067i		−	1.50958i	−0.655968	−	0.754788i	$-0.727739\pi$
$823$		−	43.3067i		−	1.50958i	0.655968	−	0.754788i	$-0.272261\pi$
$824$	−14.2713	+	14.2713i	−0.497165	+	0.497165i
$825$		0			0
$826$	−46.4550	+	46.4550i	−1.61638	+	1.61638i
$827$	−0.987036	−	0.987036i	−0.0343226	−	0.0343226i	0.689737	−	0.724060i	$-0.257726\pi$
$827$	−0.987036	−	0.987036i	−0.0343226	−	0.0343226i	−0.724060	+	0.689737i	$0.757726\pi$
$828$		0			0
$829$			32.5598i			1.13085i	0.824800	+	0.565424i	$0.191288\pi$
$829$			32.5598i			1.13085i	−0.824800	+	0.565424i	$0.808712\pi$
$830$	−46.6308	−	46.6308i	−1.61858	−	1.61858i
$831$		0			0
$832$	−1.45174	+	15.7311i	−0.0503302	+	0.545377i
$833$		−	35.4713i		−	1.22901i
$834$		0			0
$835$	−37.6404			−1.30260
$836$	5.79693			0.200491
$837$		0			0
$838$	−45.4305	−	45.4305i	−1.56937	−	1.56937i
$839$	22.3969	+	22.3969i	0.773227	+	0.773227i	0.978669	−	0.205442i	$-0.0658632\pi$
$839$	22.3969	+	22.3969i	0.773227	+	0.773227i	−0.205442	+	0.978669i	$0.565863\pi$
$840$		0			0
$841$	17.3732			0.599076
$842$	3.95735			0.136379
$843$		0			0
$844$		−	6.86088i		−	0.236161i
$845$	43.1279	−	29.5909i	1.48364	−	1.01796i
$846$		0			0
$847$	6.57331	+	6.57331i	0.225862	+	0.225862i
$848$		−	50.3030i		−	1.72741i
$849$		0			0
$850$	50.3656	+	50.3656i	1.72753	+	1.72753i
$851$	2.15925	−	2.15925i	0.0740181	−	0.0740181i
$852$		0			0
$853$	8.37348	−	8.37348i	0.286703	−	0.286703i	−0.549072	−	0.835775i	$-0.685019\pi$
$853$	8.37348	−	8.37348i	0.286703	−	0.286703i	0.835775	+	0.549072i	$0.185019\pi$
$854$		−	18.7728i		−	0.642393i
$855$		0			0
$856$	−19.7230	+	19.7230i	−0.674120	+	0.674120i
$857$	41.8824			1.43067			0.715337	−	0.698779i	$-0.246273\pi$
$857$	41.8824			1.43067			0.715337	+	0.698779i	$0.246273\pi$
$858$		0			0
$859$	17.4849			0.596578			0.298289	−	0.954476i	$-0.403584\pi$
$859$	17.4849			0.596578			0.298289	+	0.954476i	$0.403584\pi$
$860$	−0.898338	+	0.898338i	−0.0306331	+	0.0306331i
$861$		0			0
$862$			11.5645i			0.393888i
$863$	25.9343	−	25.9343i	0.882814	−	0.882814i	−0.111006	−	0.993820i	$-0.535407\pi$
$863$	25.9343	−	25.9343i	0.882814	−	0.882814i	0.993820	+	0.111006i	$0.0354073\pi$
$864$		0			0
$865$	−14.6528	+	14.6528i	−0.498211	+	0.498211i
$866$	−11.9477	−	11.9477i	−0.406000	−	0.406000i
$867$		0			0
$868$			21.6561i			0.735055i
$869$	12.0635	+	12.0635i	0.409227	+	0.409227i
$870$		0			0
$871$	−16.7275	−	20.1288i	−0.566790	−	0.682039i
$872$			27.5561i			0.933166i
$873$		0			0
$874$	4.00011			0.135306
$875$	99.5161			3.36426
$876$		0			0
$877$	−19.4440	−	19.4440i	−0.656576	−	0.656576i	0.297993	−	0.954568i	$-0.403683\pi$
$877$	−19.4440	−	19.4440i	−0.656576	−	0.656576i	−0.954568	+	0.297993i	$0.903683\pi$
$878$	−3.09814	−	3.09814i	−0.104557	−	0.104557i
$879$		0			0
$880$	57.3463			1.93314
$881$	−38.6717			−1.30288			−0.651442	−	0.758698i	$-0.725836\pi$
$881$	−38.6717			−1.30288			−0.651442	+	0.758698i	$0.725836\pi$
$882$		0			0
$883$		−	32.1536i		−	1.08206i	−0.841005	−	0.541028i	$-0.818035\pi$
$883$		−	32.1536i		−	1.08206i	0.841005	−	0.541028i	$-0.181965\pi$
$884$	0.784140	−	8.49693i	0.0263735	−	0.285783i
$885$		0			0
$886$	5.19082	+	5.19082i	0.174389	+	0.174389i
$887$		−	15.4017i		−	0.517140i	−0.965993	−	0.258570i	$-0.916749\pi$
$887$		−	15.4017i		−	0.517140i	0.965993	−	0.258570i	$-0.0832512\pi$
$888$		0			0
$889$	18.6127	+	18.6127i	0.624250	+	0.624250i
$890$	−9.84486	+	9.84486i	−0.330000	+	0.330000i
$891$		0			0
$892$	2.57507	−	2.57507i	0.0862199	−	0.0862199i
$893$		−	14.8019i		−	0.495327i
$894$		0			0
$895$	−71.0315	+	71.0315i	−2.37432	+	2.37432i
$896$	54.5112			1.82109
$897$		0			0
$898$	64.4462			2.15060
$899$	−21.7946	+	21.7946i	−0.726891	+	0.726891i
$900$		0			0
$901$		−	41.0490i		−	1.36754i
$902$	9.88928	−	9.88928i	0.329277	−	0.329277i
$903$		0			0
$904$	17.4595	−	17.4595i	0.580695	−	0.580695i
$905$	−36.7031	−	36.7031i	−1.22005	−	1.22005i
$906$		0			0
$907$		−	41.1690i		−	1.36699i	−0.729953	−	0.683497i	$-0.760458\pi$
$907$		−	41.1690i		−	1.36699i	0.729953	−	0.683497i	$-0.239542\pi$
$908$	−0.425129	−	0.425129i	−0.0141084	−	0.0141084i
$909$		0			0
$910$	−59.7575	−	71.9083i	−1.98094	−	2.38374i
$911$		−	18.6022i		−	0.616319i	−0.951335	−	0.308159i	$-0.900287\pi$
$911$		−	18.6022i		−	0.616319i	0.951335	−	0.308159i	$-0.0997130\pi$
$912$		0			0
$913$	−29.9435			−0.990985
$914$	−14.1466			−0.467929
$915$		0			0
$916$	−8.23739	−	8.23739i	−0.272171	−	0.272171i
$917$	5.22342	+	5.22342i	0.172493	+	0.172493i
$918$		0			0
$919$	−3.42602			−0.113014			−0.0565071	−	0.998402i	$-0.517996\pi$
$919$	−3.42602			−0.113014			−0.0565071	+	0.998402i	$0.517996\pi$
$920$	6.86383			0.226294
$921$		0			0
$922$			56.9119i			1.87429i
$923$	8.19415	−	6.80953i	0.269714	−	0.224138i
$924$		0			0
$925$	31.9757	+	31.9757i	1.05135	+	1.05135i
$926$		−	0.420482i		−	0.0138179i
$927$		0			0
$928$	−7.92214	−	7.92214i	−0.260057	−	0.260057i
$929$	−12.4259	+	12.4259i	−0.407682	+	0.407682i	−0.880929	−	0.473248i	$-0.843082\pi$
$929$	−12.4259	+	12.4259i	−0.407682	+	0.407682i	0.473248	+	0.880929i	$0.343082\pi$
$930$		0			0
$931$	20.8594	−	20.8594i	0.683639	−	0.683639i
$932$		−	14.1786i		−	0.464434i
$933$		0			0
$934$	−3.72250	+	3.72250i	−0.121804	+	0.121804i
$935$	46.7966			1.53041
$936$		0			0
$937$	13.8756			0.453298			0.226649	−	0.973977i	$-0.427223\pi$
$937$	13.8756			0.453298			0.226649	+	0.973977i	$0.427223\pi$
$938$	−33.0822	+	33.0822i	−1.08017	+	1.08017i
$939$		0			0
$940$			10.8665i			0.354427i
$941$	−24.9933	+	24.9933i	−0.814759	+	0.814759i	−0.985343	−	0.170585i	$-0.945434\pi$
$941$	−24.9933	+	24.9933i	−0.814759	+	0.814759i	0.170585	+	0.985343i	$0.445434\pi$
$942$		0			0
$943$	1.57331	−	1.57331i	0.0512341	−	0.0512341i
$944$	−34.8806	−	34.8806i	−1.13527	−	1.13527i
$945$		0			0
$946$			2.50203i			0.0813480i
$947$	42.4253	+	42.4253i	1.37864	+	1.37864i	0.846925	+	0.531712i	$0.178451\pi$
$947$	42.4253	+	42.4253i	1.37864	+	1.37864i	0.531712	+	0.846925i	$0.321549\pi$
$948$		0			0
$949$	−13.9919	+	11.6276i	−0.454195	+	0.377447i
$950$			59.2364i			1.92188i
$951$		0			0
$952$	35.6530			1.15552
$953$	−49.0629			−1.58930			−0.794652	−	0.607065i	$-0.792347\pi$
$953$	−49.0629			−1.58930			−0.794652	+	0.607065i	$0.792347\pi$
$954$		0			0
$955$	−13.4814	−	13.4814i	−0.436249	−	0.436249i
$956$	−6.39039	−	6.39039i	−0.206680	−	0.206680i
$957$		0			0
$958$	34.8493			1.12593
$959$	5.27130			0.170219
$960$		0			0
$961$			50.7086i			1.63576i
$962$	2.15925	−	23.3976i	0.0696170	−	0.754368i
$963$		0			0
$964$	7.85765	+	7.85765i	0.253078	+	0.253078i
$965$			75.5131i			2.43085i
$966$		0			0
$967$	23.7884	+	23.7884i	0.764985	+	0.764985i	0.977219	−	0.212234i	$-0.0680740\pi$
$967$	23.7884	+	23.7884i	0.764985	+	0.764985i	−0.212234	+	0.977219i	$0.568074\pi$
$968$	−3.71318	+	3.71318i	−0.119346	+	0.119346i
$969$		0			0
$970$	−13.4425	+	13.4425i	−0.431611	+	0.431611i
$971$		−	14.8419i		−	0.476300i	−0.971228	−	0.238150i	$-0.923459\pi$
$971$		−	14.8419i		−	0.476300i	0.971228	−	0.238150i	$-0.0765410\pi$
$972$		0			0
$973$	4.52252	−	4.52252i	0.144985	−	0.144985i
$974$	−26.7495			−0.857109
$975$		0			0
$976$	14.0955			0.451186
$977$	−29.6113	+	29.6113i	−0.947351	+	0.947351i	−0.998682	−	0.0513308i	$-0.983654\pi$
$977$	−29.6113	+	29.6113i	−0.947351	+	0.947351i	0.0513308	+	0.998682i	$0.483654\pi$
$978$		0			0
$979$			6.32178i			0.202045i
$980$	−15.3135	+	15.3135i	−0.489172	+	0.489172i
$981$		0			0
$982$	29.3250	−	29.3250i	0.935797	−	0.935797i
$983$	−23.4450	−	23.4450i	−0.747779	−	0.747779i	0.226283	−	0.974062i	$-0.427343\pi$
$983$	−23.4450	−	23.4450i	−0.747779	−	0.747779i	−0.974062	+	0.226283i	$0.927343\pi$
$984$		0			0
$985$			64.7097i			2.06182i
$986$	−15.3513	−	15.3513i	−0.488884	−	0.488884i
$987$		0			0
$988$	5.45787	−	4.53562i	0.173638	−	0.144297i
$989$			0.398055i			0.0126574i
$990$		0			0
$991$	18.6854			0.593560			0.296780	−	0.954946i	$-0.404087\pi$
$991$	18.6854			0.593560			0.296780	+	0.954946i	$0.404087\pi$
$992$	−29.7003			−0.942986
$993$		0			0
$994$	−13.4673	−	13.4673i	−0.427156	−	0.427156i
$995$	9.81083	+	9.81083i	0.311024	+	0.311024i
$996$		0			0
$997$	45.8968			1.45357			0.726783	−	0.686867i	$-0.241014\pi$
$997$	45.8968			1.45357			0.726783	+	0.686867i	$0.241014\pi$
$998$	−36.6502			−1.16014
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	351.2.i.b.242.3	yes	16
3.2	odd	2	inner	351.2.i.b.242.6	yes	16
13.5	odd	4	inner	351.2.i.b.161.6	yes	16
39.5	even	4	inner	351.2.i.b.161.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
351.2.i.b.161.3	✓	16	39.5	even	4	inner
351.2.i.b.161.6	yes	16	13.5	odd	4	inner
351.2.i.b.242.3	yes	16	1.1	even	1	trivial
351.2.i.b.242.6	yes	16	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.14002 + 1.14002i) q^{2} -0.599280i q^{4} +(2.84492 - 2.84492i) q^{5} +(-2.82684 + 2.82684i) q^{7} +(-1.59685 - 1.59685i) q^{8} +6.48652i q^{10} +(2.08263 + 2.08263i) q^{11} +(3.59030 + 0.331331i) q^{13} -6.44529i q^{14} +4.83942 q^{16} +3.94914 q^{17} +(2.32235 + 2.32235i) q^{19} +(-1.70490 - 1.70490i) q^{20} -4.74846 q^{22} -0.755446 q^{23} -11.1872i q^{25} +(-4.47072 + 3.71528i) q^{26} +(1.69407 + 1.69407i) q^{28} +3.40981i q^{29} +(6.39174 + 6.39174i) q^{31} +(-2.32334 + 2.32334i) q^{32} +(-4.50208 + 4.50208i) q^{34} +16.0843i q^{35} +(-2.85824 + 2.85824i) q^{37} -5.29503 q^{38} -9.08580 q^{40} +(-2.08263 + 2.08263i) q^{41} -0.526914i q^{43} +(1.24808 - 1.24808i) q^{44} +(0.861221 - 0.861221i) q^{46} +(-3.18684 - 3.18684i) q^{47} -8.98203i q^{49} +(12.7536 + 12.7536i) q^{50} +(0.198560 - 2.15159i) q^{52} -10.3944i q^{53} +11.8498 q^{55} +9.02805 q^{56} +(-3.88724 - 3.88724i) q^{58} +(-7.20759 - 7.20759i) q^{59} +2.91264 q^{61} -14.5734 q^{62} +4.38156i q^{64} +(11.1567 - 9.27150i) q^{65} +(-5.13277 - 5.13277i) q^{67} -2.36664i q^{68} +(-18.3364 - 18.3364i) q^{70} +(2.08948 - 2.08948i) q^{71} +(-3.56787 + 3.56787i) q^{73} -6.51690i q^{74} +(1.39174 - 1.39174i) q^{76} -11.7745 q^{77} +5.79246 q^{79} +(13.7678 - 13.7678i) q^{80} -4.74846i q^{82} +(-7.18887 + 7.18887i) q^{83} +(11.2350 - 11.2350i) q^{85} +(0.600691 + 0.600691i) q^{86} -6.65127i q^{88} +(1.51774 + 1.51774i) q^{89} +(-11.0858 + 9.21257i) q^{91} +0.452723i q^{92} +7.26611 q^{94} +13.2138 q^{95} +(2.07237 + 2.07237i) q^{97} +(10.2397 + 10.2397i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{7} + 16 q^{13} - 8 q^{16} - 32 q^{19} + 8 q^{22} + 40 q^{28} + 16 q^{31} + 24 q^{34} - 32 q^{37} - 72 q^{40} + 48 q^{46} + 48 q^{52} + 32 q^{55} + 56 q^{58} - 64 q^{61} - 32 q^{67} - 40 q^{70}+ \cdots + 64 q^{97}+O(q^{100}) \) 16 * q - 8 * q^7 + 16 * q^13 - 8 * q^16 - 32 * q^19 + 8 * q^22 + 40 * q^28 + 16 * q^31 + 24 * q^34 - 32 * q^37 - 72 * q^40 + 48 * q^46 + 48 * q^52 + 32 * q^55 + 56 * q^58 - 64 * q^61 - 32 * q^67 - 40 * q^70 - 56 * q^73 - 64 * q^76 - 16 * q^79 - 104 * q^91 - 64 * q^94 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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