LMFDB - Embedded newform 416.2.bk.a.175.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(15,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.15"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 6, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.bk (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$12$ over $\Q(\zeta_{12})$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			175.1
Character	$\chi$	$=$	416.175
Dual form			416.2.bk.a.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.11333 + 1.92835i) q^{3} +(-0.0693382 - 0.0693382i) q^{5} +(3.36551 + 0.901786i) q^{7} +(-0.979029 - 1.69573i) q^{9} +(0.305781 - 0.0819339i) q^{11} +(3.13155 + 1.78702i) q^{13} +(0.210905 - 0.0565119i) q^{15} +(-5.48889 + 3.16901i) q^{17} +(0.397336 + 0.106466i) q^{19} +(-5.48590 + 5.48590i) q^{21} +(-3.68562 + 6.38369i) q^{23} -4.99038i q^{25} -2.32006 q^{27} +(3.16915 + 1.82971i) q^{29} +(1.12080 + 1.12080i) q^{31} +(-0.182440 + 0.680874i) q^{33} +(-0.170830 - 0.295887i) q^{35} +(-0.804964 - 3.00417i) q^{37} +(-6.93245 + 4.04918i) q^{39} +(1.85187 + 6.91129i) q^{41} +(-4.88743 + 2.82176i) q^{43} +(-0.0496946 + 0.185463i) q^{45} +(6.15946 - 6.15946i) q^{47} +(4.45127 + 2.56994i) q^{49} -14.1127i q^{51} +3.89384i q^{53} +(-0.0268835 - 0.0155212i) q^{55} +(-0.647671 + 0.647671i) q^{57} +(2.51467 - 9.38488i) q^{59} +(11.9231 - 6.88378i) q^{61} +(-1.76575 - 6.58987i) q^{63} +(-0.0932273 - 0.341044i) q^{65} +(3.16917 + 11.8275i) q^{67} +(-8.20666 - 14.2144i) q^{69} +(1.81764 - 6.78351i) q^{71} +(1.49582 + 1.49582i) q^{73} +(9.62322 + 5.55597i) q^{75} +1.10300 q^{77} -10.3044i q^{79} +(5.52009 - 9.56108i) q^{81} +(-2.53771 + 2.53771i) q^{83} +(0.600323 + 0.160856i) q^{85} +(-7.05665 + 4.07416i) q^{87} +(0.110536 - 0.0296181i) q^{89} +(8.92775 + 8.83821i) q^{91} +(-3.40912 + 0.913470i) q^{93} +(-0.0201684 - 0.0349327i) q^{95} +(6.28973 + 1.68533i) q^{97} +(-0.438306 - 0.438306i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83}+ \cdots - 168 q^{99}+O(q^{100}) $ 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{12}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.11333	+	1.92835i	−0.642784	+	1.11333i	0.342024	+	0.939691i	$0.388887\pi$
$3$	−1.11333	+	1.92835i	−0.642784	+	1.11333i	−0.984809	+	0.173644i	$0.944446\pi$
$4$		0			0
$5$	−0.0693382	−	0.0693382i	−0.0310090	−	0.0310090i	0.691432	−	0.722441i	$-0.256980\pi$
$5$	−0.0693382	−	0.0693382i	−0.0310090	−	0.0310090i	−0.722441	+	0.691432i	$0.756980\pi$
$6$		0			0
$7$	3.36551	+	0.901786i	1.27204	+	0.340843i	0.830814	−	0.556551i	$-0.187876\pi$
$7$	3.36551	+	0.901786i	1.27204	+	0.340843i	0.441230	+	0.897394i	$0.354542\pi$
$8$		0			0
$9$	−0.979029	−	1.69573i	−0.326343	−	0.565243i
$10$		0			0
$11$	0.305781	−	0.0819339i	0.0921966	−	0.0247040i	−0.212426	−	0.977177i	$-0.568136\pi$
$11$	0.305781	−	0.0819339i	0.0921966	−	0.0247040i	0.304622	+	0.952473i	$0.401470\pi$
$12$		0			0
$13$	3.13155	+	1.78702i	0.868534	+	0.495629i
$13$	3.13155	+	1.78702i	0.868534	+	0.495629i
$14$		0			0
$15$	0.210905	−	0.0565119i	0.0544555	−	0.0145913i
$16$		0			0
$17$	−5.48889	+	3.16901i	−1.33125	+	0.768598i	−0.985491	−	0.169725i	$-0.945712\pi$
$17$	−5.48889	+	3.16901i	−1.33125	+	0.768598i	−0.345760	+	0.938323i	$0.612379\pi$
$18$		0			0
$19$	0.397336	+	0.106466i	0.0911550	+	0.0244249i	0.304108	−	0.952637i	$-0.401642\pi$
$19$	0.397336	+	0.106466i	0.0911550	+	0.0244249i	−0.212953	+	0.977062i	$0.568308\pi$
$20$		0			0
$21$	−5.48590	+	5.48590i	−1.19712	+	1.19712i
$22$		0			0
$23$	−3.68562	+	6.38369i	−0.768505	+	1.33109i	0.169868	+	0.985467i	$0.445666\pi$
$23$	−3.68562	+	6.38369i	−0.768505	+	1.33109i	−0.938373	+	0.345624i	$0.887667\pi$
$24$		0			0
$25$		−	4.99038i		−	0.998077i
$26$		0			0
$27$	−2.32006			−0.446496
$28$		0			0
$29$	3.16915	+	1.82971i	0.588496	+	0.339768i	0.764503	−	0.644621i	$-0.222985\pi$
$29$	3.16915	+	1.82971i	0.588496	+	0.339768i	−0.176006	+	0.984389i	$0.556318\pi$
$30$		0			0
$31$	1.12080	+	1.12080i	0.201301	+	0.201301i	0.800557	−	0.599256i	$-0.204537\pi$
$31$	1.12080	+	1.12080i	0.201301	+	0.201301i	−0.599256	+	0.800557i	$0.704537\pi$
$32$		0			0
$33$	−0.182440	+	0.680874i	−0.0317587	+	0.118525i
$34$		0			0
$35$	−0.170830	−	0.295887i	−0.0288756	−	0.0500140i
$36$		0			0
$37$	−0.804964	−	3.00417i	−0.132335	−	0.493882i	0.867659	−	0.497159i	$-0.165624\pi$
$37$	−0.804964	−	3.00417i	−0.132335	−	0.493882i	−0.999995	+	0.00327710i	$0.998957\pi$
$38$		0			0
$39$	−6.93245	+	4.04918i	−1.11008	+	0.648387i
$40$		0			0
$41$	1.85187	+	6.91129i	0.289214	+	1.07936i	0.945705	+	0.325027i	$0.105373\pi$
$41$	1.85187	+	6.91129i	0.289214	+	1.07936i	−0.656491	+	0.754334i	$0.727960\pi$
$42$		0			0
$43$	−4.88743	+	2.82176i	−0.745326	+	0.430314i	−0.824003	−	0.566586i	$-0.808264\pi$
$43$	−4.88743	+	2.82176i	−0.745326	+	0.430314i	0.0786766	+	0.996900i	$0.474931\pi$
$44$		0			0
$45$	−0.0496946	+	0.185463i	−0.00740804	+	0.0276472i
$46$		0			0
$47$	6.15946	−	6.15946i	0.898449	−	0.898449i	−0.0968500	−	0.995299i	$-0.530877\pi$
$47$	6.15946	−	6.15946i	0.898449	−	0.898449i	0.995299	+	0.0968500i	$0.0308767\pi$
$48$		0			0
$49$	4.45127	+	2.56994i	0.635896	+	0.367135i
$50$		0			0
$51$		−	14.1127i		−	1.97617i
$52$		0			0
$53$			3.89384i			0.534861i	0.963577	+	0.267430i	$0.0861745\pi$
$53$			3.89384i			0.534861i	−0.963577	+	0.267430i	$0.913825\pi$
$54$		0			0
$55$	−0.0268835	−	0.0155212i	−0.00362497	−	0.00209288i
$56$		0			0
$57$	−0.647671	+	0.647671i	−0.0857861	+	0.0857861i
$58$		0			0
$59$	2.51467	−	9.38488i	0.327382	−	1.22181i	−0.584513	−	0.811385i	$-0.698714\pi$
$59$	2.51467	−	9.38488i	0.327382	−	1.22181i	0.911895	−	0.410423i	$-0.134619\pi$
$60$		0			0
$61$	11.9231	−	6.88378i	1.52659	−	0.881378i	0.527089	−	0.849810i	$-0.323283\pi$
$61$	11.9231	−	6.88378i	1.52659	−	0.881378i	0.999502	−	0.0315679i	$-0.0100500\pi$
$62$		0			0
$63$	−1.76575	−	6.58987i	−0.222463	−	0.830245i
$64$		0			0
$65$	−0.0932273	−	0.341044i	−0.0115634	−	0.0423013i
$66$		0			0
$67$	3.16917	+	11.8275i	0.387176	+	1.44496i	0.834708	+	0.550693i	$0.185637\pi$
$67$	3.16917	+	11.8275i	0.387176	+	1.44496i	−0.447532	+	0.894268i	$0.647697\pi$
$68$		0			0
$69$	−8.20666	−	14.2144i	−0.987966	−	1.71121i
$70$		0			0
$71$	1.81764	−	6.78351i	0.215714	−	0.805055i	−0.770200	−	0.637802i	$-0.779844\pi$
$71$	1.81764	−	6.78351i	0.215714	−	0.805055i	0.985914	−	0.167253i	$-0.0534896\pi$
$72$		0			0
$73$	1.49582	+	1.49582i	0.175072	+	0.175072i	0.789204	−	0.614131i	$-0.210494\pi$
$73$	1.49582	+	1.49582i	0.175072	+	0.175072i	−0.614131	+	0.789204i	$0.710494\pi$
$74$		0			0
$75$	9.62322	+	5.55597i	1.11119	+	0.641548i
$76$		0			0
$77$	1.10300			0.125698
$78$		0			0
$79$		−	10.3044i		−	1.15933i	−0.814854	−	0.579667i	$-0.803183\pi$
$79$		−	10.3044i		−	1.15933i	0.814854	−	0.579667i	$-0.196817\pi$
$80$		0			0
$81$	5.52009	−	9.56108i	0.613344	−	1.06234i
$82$		0			0
$83$	−2.53771	+	2.53771i	−0.278550	+	0.278550i	−0.832530	−	0.553980i	$-0.813108\pi$
$83$	−2.53771	+	2.53771i	−0.278550	+	0.278550i	0.553980	+	0.832530i	$0.313108\pi$
$84$		0			0
$85$	0.600323	+	0.160856i	0.0651142	+	0.0174473i
$86$		0			0
$87$	−7.05665	+	4.07416i	−0.756552	+	0.436796i
$88$		0			0
$89$	0.110536	−	0.0296181i	0.0117168	−	0.00313951i	−0.252956	−	0.967478i	$-0.581403\pi$
$89$	0.110536	−	0.0296181i	0.0117168	−	0.00313951i	0.264673	+	0.964338i	$0.414736\pi$
$90$		0			0
$91$	8.92775	+	8.83821i	0.935882	+	0.926496i
$92$		0			0
$93$	−3.40912	+	0.913470i	−0.353509	+	0.0947224i
$94$		0			0
$95$	−0.0201684	−	0.0349327i	−0.00206923	−	0.00358402i
$96$		0			0
$97$	6.28973	+	1.68533i	0.638626	+	0.171119i	0.563581	−	0.826061i	$-0.309423\pi$
$97$	6.28973	+	1.68533i	0.638626	+	0.171119i	0.0750449	+	0.997180i	$0.476090\pi$
$98$		0			0
$99$	−0.438306	−	0.438306i	−0.0440515	−	0.0440515i
$100$		0			0
$101$	−3.22024	+	5.57761i	−0.320425	+	0.554993i	−0.980576	−	0.196140i	$-0.937159\pi$
$101$	−3.22024	+	5.57761i	−0.320425	+	0.554993i	0.660150	+	0.751133i	$0.270493\pi$
$102$		0			0
$103$	15.5401			1.53121			0.765604	−	0.643312i	$-0.222440\pi$
$103$	15.5401			1.53121			0.765604	+	0.643312i	$0.222440\pi$
$104$		0			0
$105$	0.760766			0.0742431
$106$		0			0
$107$	0.865163	−	1.49851i	0.0836385	−	0.144866i	−0.821172	−	0.570681i	$-0.806679\pi$
$107$	0.865163	−	1.49851i	0.0836385	−	0.144866i	0.904810	+	0.425815i	$0.140013\pi$
$108$		0			0
$109$	−13.2174	−	13.2174i	−1.26599	−	1.26599i	−0.948140	−	0.317853i	$-0.897038\pi$
$109$	−13.2174	−	13.2174i	−1.26599	−	1.26599i	−0.317853	−	0.948140i	$-0.602962\pi$
$110$		0			0
$111$	6.68929	+	1.79239i	0.634919	+	0.170126i
$112$		0			0
$113$	−6.88323	−	11.9221i	−0.647520	−	1.12154i	−0.983713	−	0.179744i	$-0.942473\pi$
$113$	−6.88323	−	11.9221i	−0.647520	−	1.12154i	0.336193	−	0.941793i	$-0.390860\pi$
$114$		0			0
$115$	0.698188	−	0.187079i	0.0651064	−	0.0174452i
$116$		0			0
$117$	−0.0355813	−	7.05979i	−0.00328949	−	0.652678i
$118$		0			0
$119$	−21.3307	+	5.71554i	−1.95538	+	0.523943i
$120$		0			0
$121$	−9.43949	+	5.44989i	−0.858135	+	0.495445i
$122$		0			0
$123$	−15.3891	−	4.12351i	−1.38759	−	0.371804i
$124$		0			0
$125$	−0.692716	+	0.692716i	−0.0619584	+	0.0619584i
$126$		0			0
$127$	4.23231	−	7.33057i	0.375557	−	0.650483i	−0.614853	−	0.788641i	$-0.710785\pi$
$127$	4.23231	−	7.33057i	0.375557	−	0.650483i	0.990410	+	0.138158i	$0.0441182\pi$
$128$		0			0
$129$		−	12.5662i		−	1.10640i
$130$		0			0
$131$	13.5858			1.18700			0.593499	−	0.804835i	$-0.297746\pi$
$131$	13.5858			1.18700			0.593499	+	0.804835i	$0.297746\pi$
$132$		0			0
$133$	1.24123	+	0.716623i	0.107628	+	0.0621391i
$134$		0			0
$135$	0.160869	+	0.160869i	0.0138454	+	0.0138454i
$136$		0			0
$137$	3.19234	−	11.9140i	0.272740	−	1.01788i	−0.684601	−	0.728918i	$-0.740024\pi$
$137$	3.19234	−	11.9140i	0.272740	−	1.01788i	0.957341	−	0.288961i	$-0.0933097\pi$
$138$		0			0
$139$	−5.36196	−	9.28718i	−0.454795	−	0.787729i	0.543881	−	0.839162i	$-0.316954\pi$
$139$	−5.36196	−	9.28718i	−0.454795	−	0.787729i	−0.998676	+	0.0514337i	$0.983621\pi$
$140$		0			0
$141$	5.02007	+	18.7351i	0.422766	+	1.57778i
$142$		0			0
$143$	1.10399	+	0.289857i	0.0923199	+	0.0242390i
$144$		0			0
$145$	−0.0928744	−	0.346612i	−0.00771280	−	0.0287846i
$146$		0			0
$147$	−9.91151	+	5.72241i	−0.817488	+	0.471977i
$148$		0			0
$149$	−2.96712	+	11.0734i	−0.243076	+	0.907172i	0.731265	+	0.682094i	$0.238930\pi$
$149$	−2.96712	+	11.0734i	−0.243076	+	0.907172i	−0.974341	+	0.225078i	$0.927736\pi$
$150$		0			0
$151$	−7.63837	+	7.63837i	−0.621602	+	0.621602i	−0.945941	−	0.324339i	$-0.894858\pi$
$151$	−7.63837	+	7.63837i	−0.621602	+	0.621602i	0.324339	+	0.945941i	$0.394858\pi$
$152$		0			0
$153$	10.7476	+	6.20511i	0.868889	+	0.501653i
$154$		0			0
$155$		−	0.155428i		−	0.0124843i
$156$		0			0
$157$		−	0.295631i		−	0.0235940i	−0.999930	−	0.0117970i	$-0.996245\pi$
$157$		−	0.295631i		−	0.0235940i	0.999930	−	0.0117970i	$-0.00375518\pi$
$158$		0			0
$159$	−7.50870	−	4.33515i	−0.595479	−	0.343800i
$160$		0			0
$161$	−18.1607	+	18.1607i	−1.43127	+	1.43127i
$162$		0			0
$163$	0.643966	−	2.40331i	0.0504393	−	0.188242i	−0.936110	−	0.351708i	$-0.885601\pi$
$163$	0.643966	−	2.40331i	0.0504393	−	0.188242i	0.986549	+	0.163466i	$0.0522675\pi$
$164$		0			0
$165$	0.0598607	−	0.0345606i	0.00466015	−	0.00269054i
$166$		0			0
$167$	2.40443	+	8.97345i	0.186060	+	0.694386i	0.994401	+	0.105672i	$0.0336992\pi$
$167$	2.40443	+	8.97345i	0.186060	+	0.694386i	−0.808341	+	0.588715i	$0.799634\pi$
$168$		0			0
$169$	6.61315	+	11.1922i	0.508704	+	0.860942i
$170$		0			0
$171$	−0.208466	−	0.778006i	−0.0159418	−	0.0594956i
$172$		0			0
$173$	−1.48266	−	2.56803i	−0.112724	−	0.195244i	0.804144	−	0.594435i	$-0.202624\pi$
$173$	−1.48266	−	2.56803i	−0.112724	−	0.195244i	−0.916868	+	0.399191i	$0.869291\pi$
$174$		0			0
$175$	4.50026	−	16.7952i	0.340188	−	1.26960i
$176$		0			0
$177$	15.2977	+	15.2977i	1.14984	+	1.14984i
$178$		0			0
$179$	−5.42364	−	3.13134i	−0.405382	−	0.234047i	0.283422	−	0.958995i	$-0.408530\pi$
$179$	−5.42364	−	3.13134i	−0.405382	−	0.234047i	−0.688803	+	0.724948i	$0.741864\pi$
$180$		0			0
$181$	19.6735			1.46232			0.731160	−	0.682206i	$-0.238979\pi$
$181$	19.6735			1.46232			0.731160	+	0.682206i	$0.238979\pi$
$182$		0			0
$183$			30.6558i			2.26614i
$184$		0			0
$185$	−0.152489	+	0.264119i	−0.0112112	+	0.0194184i
$186$		0			0
$187$	−1.41875	+	1.41875i	−0.103749	+	0.103749i
$188$		0			0
$189$	−7.80819	−	2.09220i	−0.567963	−	0.152185i
$190$		0			0
$191$	6.98944	−	4.03535i	0.505738	−	0.291988i	−0.225342	−	0.974280i	$-0.572350\pi$
$191$	6.98944	−	4.03535i	0.505738	−	0.291988i	0.731080	+	0.682292i	$0.239017\pi$
$192$		0			0
$193$	3.79172	−	1.01599i	0.272934	−	0.0731325i	−0.119756	−	0.992803i	$-0.538211\pi$
$193$	3.79172	−	1.01599i	0.272934	−	0.0731325i	0.392690	+	0.919671i	$0.371545\pi$
$194$		0			0
$195$	0.761447	+	0.199921i	0.0545283	+	0.0143167i
$196$		0			0
$197$	17.9395	−	4.80687i	1.27813	−	0.342475i	0.444993	−	0.895534i	$-0.353206\pi$
$197$	17.9395	−	4.80687i	1.27813	−	0.342475i	0.833142	+	0.553059i	$0.186540\pi$
$198$		0			0
$199$	0.747531	+	1.29476i	0.0529911	+	0.0917833i	0.891304	−	0.453406i	$-0.149791\pi$
$199$	0.747531	+	1.29476i	0.0529911	+	0.0917833i	−0.838313	+	0.545189i	$0.816458\pi$
$200$		0			0
$201$	−26.3360	−	7.05670i	−1.85760	−	0.497741i
$202$		0			0
$203$	9.01580	+	9.01580i	0.632785	+	0.632785i
$204$		0			0
$205$	0.350811	−	0.607622i	0.0245017	−	0.0424382i
$206$		0			0
$207$	14.4333			1.00319
$208$		0			0
$209$	0.130221			0.00900758
$210$		0			0
$211$	5.13613	−	8.89604i	0.353586	−	0.612429i	−0.633289	−	0.773915i	$-0.718296\pi$
$211$	5.13613	−	8.89604i	0.353586	−	0.612429i	0.986875	+	0.161487i	$0.0516289\pi$
$212$		0			0
$213$	11.0574	+	11.0574i	0.757638	+	0.757638i
$214$		0			0
$215$	0.534542	+	0.143230i	0.0364554	+	0.00976820i
$216$		0			0
$217$	2.76134	+	4.78278i	0.187452	+	0.324676i
$218$		0			0
$219$	−4.54982	+	1.21912i	−0.307448	+	0.0823805i
$220$		0			0
$221$	−22.8518	+	0.115173i	−1.53718	+	0.00774737i
$222$		0			0
$223$	−4.08977	+	1.09585i	−0.273871	+	0.0733836i	−0.393141	−	0.919478i	$-0.628611\pi$
$223$	−4.08977	+	1.09585i	−0.273871	+	0.0733836i	0.119269	+	0.992862i	$0.461945\pi$
$224$		0			0
$225$	−8.46233	+	4.88573i	−0.564155	+	0.325715i
$226$		0			0
$227$	−20.5241	−	5.49942i	−1.36223	−	0.365009i	−0.497598	−	0.867408i	$-0.665784\pi$
$227$	−20.5241	−	5.49942i	−1.36223	−	0.365009i	−0.864636	+	0.502399i	$0.832451\pi$
$228$		0			0
$229$	2.65946	−	2.65946i	0.175742	−	0.175742i	−0.613755	−	0.789497i	$-0.710342\pi$
$229$	2.65946	−	2.65946i	0.175742	−	0.175742i	0.789497	+	0.613755i	$0.210342\pi$
$230$		0			0
$231$	−1.22801	+	2.12697i	−0.0807969	+	0.139944i
$232$		0			0
$233$		−	12.9659i		−	0.849423i	−0.905329	−	0.424711i	$-0.860376\pi$
$233$		−	12.9659i		−	0.849423i	0.905329	−	0.424711i	$-0.139624\pi$
$234$		0			0
$235$	−0.854172			−0.0557200
$236$		0			0
$237$	19.8705	+	11.4722i	1.29073	+	0.745201i
$238$		0			0
$239$	−2.00711	−	2.00711i	−0.129829	−	0.129829i	0.639206	−	0.769035i	$-0.279263\pi$
$239$	−2.00711	−	2.00711i	−0.129829	−	0.129829i	−0.769035	+	0.639206i	$0.779263\pi$
$240$		0			0
$241$	−2.01768	+	7.53010i	−0.129970	+	0.485056i	−0.999968	−	0.00799977i	$-0.997454\pi$
$241$	−2.01768	+	7.53010i	−0.129970	+	0.485056i	0.869998	+	0.493056i	$0.164120\pi$
$242$		0			0
$243$	8.81133	+	15.2617i	0.565247	+	0.979036i
$244$		0			0
$245$	−0.130448	−	0.486839i	−0.00833402	−	0.0311030i
$246$		0			0
$247$	1.05402	+	1.04345i	0.0670656	+	0.0663929i
$248$		0			0
$249$	−2.06828	−	7.71891i	−0.131072	−	0.489166i
$250$		0			0
$251$	8.49344	−	4.90369i	0.536101	−	0.309518i	−0.207396	−	0.978257i	$-0.566499\pi$
$251$	8.49344	−	4.90369i	0.536101	−	0.309518i	0.743497	+	0.668739i	$0.233166\pi$
$252$		0			0
$253$	−0.603955	+	2.25399i	−0.0379703	+	0.141707i
$254$		0			0
$255$	−0.978548	+	0.978548i	−0.0612791	+	0.0612791i
$256$		0			0
$257$	8.22290	+	4.74749i	0.512930	+	0.296140i	0.734037	−	0.679109i	$-0.237634\pi$
$257$	8.22290	+	4.74749i	0.512930	+	0.296140i	−0.221107	+	0.975250i	$0.570967\pi$
$258$		0			0
$259$		−	10.8365i		−	0.673345i
$260$		0			0
$261$		−	7.16535i		−	0.443524i
$262$		0			0
$263$	0.383169	+	0.221223i	0.0236272	+	0.0136412i	0.511767	−	0.859124i	$-0.328991\pi$
$263$	0.383169	+	0.221223i	0.0236272	+	0.0136412i	−0.488140	+	0.872765i	$0.662324\pi$
$264$		0			0
$265$	0.269992	−	0.269992i	0.0165855	−	0.0165855i
$266$		0			0
$267$	−0.0659497	+	0.246128i	−0.00403606	+	0.0150628i
$268$		0			0
$269$	−12.2972	+	7.09981i	−0.749776	+	0.432883i	−0.825613	−	0.564237i	$-0.809170\pi$
$269$	−12.2972	+	7.09981i	−0.749776	+	0.432883i	0.0758371	+	0.997120i	$0.475837\pi$
$270$		0			0
$271$	1.28851	+	4.80880i	0.0782717	+	0.292114i	0.993955	−	0.109786i	$-0.0350166\pi$
$271$	1.28851	+	4.80880i	0.0782717	+	0.292114i	−0.915684	+	0.401900i	$0.868350\pi$
$272$		0			0
$273$	−26.9827	+	7.37596i	−1.63307	+	0.446413i
$274$		0			0
$275$	−0.408882	−	1.52597i	−0.0246565	−	0.0920193i
$276$		0			0
$277$	5.99110	+	10.3769i	0.359970	+	0.623487i	0.987955	−	0.154739i	$-0.0494535\pi$
$277$	5.99110	+	10.3769i	0.359970	+	0.623487i	−0.627985	+	0.778225i	$0.716120\pi$
$278$		0			0
$279$	0.803275	−	2.99786i	0.0480908	−	0.179477i
$280$		0			0
$281$	−5.12419	−	5.12419i	−0.305684	−	0.305684i	0.537549	−	0.843233i	$-0.319350\pi$
$281$	−5.12419	−	5.12419i	−0.305684	−	0.305684i	−0.843233	+	0.537549i	$0.819350\pi$
$282$		0			0
$283$	10.7351	+	6.19789i	0.638133	+	0.368426i	0.783895	−	0.620893i	$-0.213230\pi$
$283$	10.7351	+	6.19789i	0.638133	+	0.368426i	−0.145762	+	0.989320i	$0.546563\pi$
$284$		0			0
$285$	0.0898167			0.00532028
$286$		0			0
$287$			24.9300i			1.47157i
$288$		0			0
$289$	11.5853	−	20.0663i	0.681486	−	1.18037i
$290$		0			0
$291$	−10.2525	+	10.2525i	−0.601012	+	0.601012i
$292$		0			0
$293$	−2.75972	−	0.739465i	−0.161225	−	0.0432000i	0.177304	−	0.984156i	$-0.443262\pi$
$293$	−2.75972	−	0.739465i	−0.161225	−	0.0432000i	−0.338528	+	0.940956i	$0.609929\pi$
$294$		0			0
$295$	−0.825094	+	0.476368i	−0.0480388	+	0.0277352i
$296$		0			0
$297$	−0.709432	+	0.190092i	−0.0411654	+	0.0110302i
$298$		0			0
$299$	−22.9494	+	13.4045i	−1.32720	+	0.775204i
$300$		0			0
$301$	−18.9933	+	5.08925i	−1.09476	+	0.293339i
$302$		0			0
$303$	−7.17040	−	12.4195i	−0.411929	−	0.713481i
$304$		0			0
$305$	−1.30403	−	0.349415i	−0.0746687	−	0.0200074i
$306$		0			0
$307$	−1.31454	−	1.31454i	−0.0750248	−	0.0750248i	0.668599	−	0.743623i	$-0.266894\pi$
$307$	−1.31454	−	1.31454i	−0.0750248	−	0.0750248i	−0.743623	+	0.668599i	$0.766894\pi$
$308$		0			0
$309$	−17.3013	+	29.9667i	−0.984236	+	1.70475i
$310$		0			0
$311$	−26.3976			−1.49687			−0.748436	−	0.663207i	$-0.769195\pi$
$311$	−26.3976			−1.49687			−0.748436	+	0.663207i	$0.769195\pi$
$312$		0			0
$313$	23.0930			1.30529			0.652647	−	0.757662i	$-0.273658\pi$
$313$	23.0930			1.30529			0.652647	+	0.757662i	$0.273658\pi$
$314$		0			0
$315$	−0.334496	+	0.579364i	−0.0188467	+	0.0326434i
$316$		0			0
$317$	−13.9691	−	13.9691i	−0.784583	−	0.784583i	0.196018	−	0.980600i	$-0.437199\pi$
$317$	−13.9691	−	13.9691i	−0.784583	−	0.784583i	−0.980600	+	0.196018i	$0.937199\pi$
$318$		0			0
$319$	1.11898	+	0.299830i	0.0626510	+	0.0167873i
$320$		0			0
$321$	1.92643	+	3.33668i	0.107523	+	0.186235i
$322$		0			0
$323$	−2.51832	+	0.674782i	−0.140123	+	0.0375459i
$324$		0			0
$325$	8.91789	−	15.6276i	0.494676	−	0.866864i
$326$		0			0
$327$	40.2031	−	10.7724i	2.22323	−	0.595714i
$328$		0			0
$329$	26.2842	−	15.1752i	1.44910	−	0.836636i
$330$		0			0
$331$	−8.82798	−	2.36545i	−0.485229	−	0.130017i	0.00790714	−	0.999969i	$-0.497483\pi$
$331$	−8.82798	−	2.36545i	−0.485229	−	0.130017i	−0.493137	+	0.869952i	$0.664150\pi$
$332$		0			0
$333$	−4.30617	+	4.30617i	−0.235976	+	0.235976i
$334$		0			0
$335$	0.600354	−	1.03984i	0.0328009	−	0.0568127i
$336$		0			0
$337$			8.38992i			0.457028i	0.973541	+	0.228514i	$0.0733867\pi$
$337$			8.38992i			0.457028i	−0.973541	+	0.228514i	$0.926613\pi$
$338$		0			0
$339$	30.6534			1.66486
$340$		0			0
$341$	0.434551	+	0.250888i	0.0235322	+	0.0135863i
$342$		0			0
$343$	−4.58281	−	4.58281i	−0.247448	−	0.247448i
$344$		0			0
$345$	−0.416563	+	1.55463i	−0.0224270	+	0.0836987i
$346$		0			0
$347$	−1.54479	−	2.67566i	−0.0829287	−	0.143637i	0.821578	−	0.570096i	$-0.193094\pi$
$347$	−1.54479	−	2.67566i	−0.0829287	−	0.143637i	−0.904507	+	0.426459i	$0.859761\pi$
$348$		0			0
$349$	−7.75487	−	28.9416i	−0.415109	−	1.54921i	−0.784618	−	0.619980i	$-0.787141\pi$
$349$	−7.75487	−	28.9416i	−0.415109	−	1.54921i	0.369509	−	0.929227i	$-0.379526\pi$
$350$		0			0
$351$	−7.26538	−	4.14599i	−0.387797	−	0.221296i
$352$		0			0
$353$	−6.52845	−	24.3645i	−0.347474	−	1.29679i	−0.889695	−	0.456556i	$-0.849083\pi$
$353$	−6.52845	−	24.3645i	−0.347474	−	1.29679i	0.542220	−	0.840236i	$-0.317584\pi$
$354$		0			0
$355$	−0.596389	+	0.344325i	−0.0316530	+	0.0182749i
$356$		0			0
$357$	12.7266	−	47.4964i	0.673564	−	2.51378i
$358$		0			0
$359$	−9.78353	+	9.78353i	−0.516355	+	0.516355i	−0.916467	−	0.400111i	$-0.868971\pi$
$359$	−9.78353	+	9.78353i	−0.516355	+	0.516355i	0.400111	+	0.916467i	$0.368971\pi$
$360$		0			0
$361$	−16.3079	−	9.41539i	−0.858313	−	0.495547i
$362$		0			0
$363$		−	24.2702i		−	1.27386i
$364$		0			0
$365$		−	0.207435i		−	0.0108576i
$366$		0			0
$367$	7.32432	+	4.22870i	0.382327	+	0.220736i	0.678830	−	0.734295i	$-0.262487\pi$
$367$	7.32432	+	4.22870i	0.382327	+	0.220736i	−0.296504	+	0.955032i	$0.595821\pi$
$368$		0			0
$369$	9.90662	−	9.90662i	0.515718	−	0.515718i
$370$		0			0
$371$	−3.51141	+	13.1048i	−0.182304	+	0.680366i
$372$		0			0
$373$	−8.98736	+	5.18886i	−0.465348	+	0.268669i	−0.714290	−	0.699850i	$-0.753250\pi$
$373$	−8.98736	+	5.18886i	−0.465348	+	0.268669i	0.248942	+	0.968518i	$0.419917\pi$
$374$		0			0
$375$	−0.564575	−	2.10702i	−0.0291546	−	0.108806i
$376$		0			0
$377$	6.65461	+	11.3931i	0.342730	+	0.586776i
$378$		0			0
$379$	−1.21827	−	4.54665i	−0.0625784	−	0.233546i	0.927552	−	0.373694i	$-0.121909\pi$
$379$	−1.21827	−	4.54665i	−0.0625784	−	0.233546i	−0.990131	+	0.140148i	$0.955242\pi$
$380$		0			0
$381$	9.42395	+	16.3228i	0.482804	+	0.836241i
$382$		0			0
$383$	5.80376	−	21.6599i	0.296558	−	1.10677i	−0.643414	−	0.765519i	$-0.722482\pi$
$383$	5.80376	−	21.6599i	0.296558	−	1.10677i	0.939972	−	0.341252i	$-0.110851\pi$
$384$		0			0
$385$	−0.0764799	−	0.0764799i	−0.00389778	−	0.00389778i
$386$		0			0
$387$	9.56987	+	5.52517i	0.486464	+	0.280860i
$388$		0			0
$389$	−0.255974			−0.0129784			−0.00648920	−	0.999979i	$-0.502066\pi$
$389$	−0.255974			−0.0129784			−0.00648920	+	0.999979i	$0.502066\pi$
$390$		0			0
$391$		−	46.7191i		−	2.36269i
$392$		0			0
$393$	−15.1255	+	26.1982i	−0.762983	+	1.32153i
$394$		0			0
$395$	−0.714488	+	0.714488i	−0.0359498	+	0.0359498i
$396$		0			0
$397$	29.9538	+	8.02610i	1.50334	+	0.402818i	0.914217	−	0.405226i	$-0.132807\pi$
$397$	29.9538	+	8.02610i	1.50334	+	0.402818i	0.589122	+	0.808044i	$0.299474\pi$
$398$		0			0
$399$	−2.76380	+	1.59568i	−0.138363	+	0.0798841i
$400$		0			0
$401$	20.6707	−	5.53870i	1.03225	−	0.276590i	0.297350	−	0.954769i	$-0.403897\pi$
$401$	20.6707	−	5.53870i	1.03225	−	0.276590i	0.734897	+	0.678179i	$0.237231\pi$
$402$		0			0
$403$	1.50695	+	5.51271i	0.0750663	+	0.274608i
$404$		0			0
$405$	−1.04570	+	0.280195i	−0.0519613	+	0.0139230i
$406$		0			0
$407$	−0.492286	−	0.852665i	−0.0244017	−	0.0422650i
$408$		0			0
$409$	−31.2652	−	8.37750i	−1.54597	−	0.414241i	−0.617779	−	0.786352i	$-0.711967\pi$
$409$	−31.2652	−	8.37750i	−1.54597	−	0.414241i	−0.928188	+	0.372112i	$0.878634\pi$
$410$		0			0
$411$	19.4202	+	19.4202i	0.957927	+	0.957927i
$412$		0			0
$413$	16.9263	−	29.3172i	0.832889	−	1.44261i
$414$		0			0
$415$	0.351920			0.0172751
$416$		0			0
$417$	23.8786			1.16934
$418$		0			0
$419$	10.4525	−	18.1043i	0.510640	−	0.884454i	−0.489284	−	0.872125i	$-0.662742\pi$
$419$	10.4525	−	18.1043i	0.510640	−	0.884454i	0.999924	−	0.0123300i	$-0.00392485\pi$
$420$		0			0
$421$	−8.42426	−	8.42426i	−0.410573	−	0.410573i	0.471365	−	0.881938i	$-0.343761\pi$
$421$	−8.42426	−	8.42426i	−0.410573	−	0.410573i	−0.881938	+	0.471365i	$0.843761\pi$
$422$		0			0
$423$	−16.4750	−	4.41447i	−0.801044	−	0.214639i
$424$		0			0
$425$	15.8146	+	27.3917i	0.767120	+	1.32869i
$426$		0			0
$427$	46.3349	−	12.4154i	2.24230	−	0.600823i
$428$		0			0
$429$	−1.78805	+	1.80617i	−0.0863279	+	0.0872025i
$430$		0			0
$431$	21.5916	−	5.78544i	1.04003	−	0.278675i	0.301904	−	0.953338i	$-0.402378\pi$
$431$	21.5916	−	5.78544i	1.04003	−	0.278675i	0.738125	+	0.674663i	$0.235711\pi$
$432$		0			0
$433$	−12.3592	+	7.13560i	−0.593946	+	0.342915i	−0.766656	−	0.642058i	$-0.778081\pi$
$433$	−12.3592	+	7.13560i	−0.593946	+	0.342915i	0.172710	+	0.984973i	$0.444748\pi$
$434$		0			0
$435$	0.771790	+	0.206801i	0.0370045	+	0.00991533i
$436$		0			0
$437$	−2.14407	+	2.14407i	−0.102565	+	0.102565i
$438$		0			0
$439$	1.78605	−	3.09353i	0.0852434	−	0.147646i	−0.820251	−	0.572003i	$-0.806167\pi$
$439$	1.78605	−	3.09353i	0.0852434	−	0.147646i	0.905495	+	0.424357i	$0.139500\pi$
$440$		0			0
$441$		−	10.0642i		−	0.479247i
$442$		0			0
$443$	−8.73925			−0.415214			−0.207607	−	0.978212i	$-0.566568\pi$
$443$	−8.73925			−0.415214			−0.207607	+	0.978212i	$0.566568\pi$
$444$		0			0
$445$	−0.00971805	−	0.00561072i	−0.000460680	−	0.000265974i
$446$		0			0
$447$	−18.0501	−	18.0501i	−0.853740	−	0.853740i
$448$		0			0
$449$	−6.78232	+	25.3120i	−0.320077	+	1.19455i	0.599091	+	0.800681i	$0.295529\pi$
$449$	−6.78232	+	25.3120i	−0.320077	+	1.19455i	−0.919169	+	0.393864i	$0.871138\pi$
$450$		0			0
$451$	1.13254	+	1.96161i	0.0533291	+	0.0923687i
$452$		0			0
$453$	−6.22541	−	23.2335i	−0.292495	−	1.09161i
$454$		0			0
$455$	−0.00620857	−	1.23186i	−0.000291062	−	0.0577505i
$456$		0			0
$457$	8.37320	+	31.2492i	0.391682	+	1.46178i	0.827359	+	0.561674i	$0.189842\pi$
$457$	8.37320	+	31.2492i	0.391682	+	1.46178i	−0.435677	+	0.900103i	$0.643491\pi$
$458$		0			0
$459$	12.7346	−	7.35230i	0.594398	−	0.343176i
$460$		0			0
$461$	4.21669	−	15.7369i	0.196391	−	0.732941i	−0.795511	−	0.605939i	$-0.792798\pi$
$461$	4.21669	−	15.7369i	0.196391	−	0.732941i	0.991902	−	0.127003i	$-0.0405356\pi$
$462$		0			0
$463$	−24.8178	+	24.8178i	−1.15338	+	1.15338i	−0.167509	+	0.985871i	$0.553572\pi$
$463$	−24.8178	+	24.8178i	−1.15338	+	1.15338i	−0.985871	+	0.167509i	$0.946428\pi$
$464$		0			0
$465$	0.299721	+	0.173044i	0.0138992	+	0.00802471i
$466$		0			0
$467$			10.5453i			0.487976i	0.969778	+	0.243988i	$0.0784558\pi$
$467$			10.5453i			0.487976i	−0.969778	+	0.243988i	$0.921544\pi$
$468$		0			0
$469$			42.6636i			1.97002i
$470$		0			0
$471$	0.570082	+	0.329137i	0.0262680	+	0.0151658i
$472$		0			0
$473$	−1.26329	+	1.26329i	−0.0580860	+	0.0580860i
$474$		0			0
$475$	0.531305	−	1.98286i	0.0243779	−	0.0909797i
$476$		0			0
$477$	6.60290	−	3.81219i	0.302326	−	0.174548i
$478$		0			0
$479$	2.68034	+	10.0032i	0.122468	+	0.457057i	0.999737	−	0.0229425i	$-0.00730348\pi$
$479$	2.68034	+	10.0032i	0.122468	+	0.457057i	−0.877269	+	0.480000i	$0.840637\pi$
$480$		0			0
$481$	2.84771	−	10.8462i	0.129845	−	0.494543i
$482$		0			0
$483$	−14.8013	−	55.2392i	−0.673483	−	2.51347i
$484$		0			0
$485$	−0.319261	−	0.552977i	−0.0144969	−	0.0251094i
$486$		0			0
$487$	−6.29726	+	23.5017i	−0.285356	+	1.06496i	0.663222	+	0.748422i	$0.269188\pi$
$487$	−6.29726	+	23.5017i	−0.285356	+	1.06496i	−0.948579	+	0.316541i	$0.897478\pi$
$488$		0			0
$489$	3.91748	+	3.91748i	0.177155	+	0.177155i
$490$		0			0
$491$	−28.1909	−	16.2760i	−1.27224	−	0.734525i	−0.296827	−	0.954931i	$-0.595928\pi$
$491$	−28.1909	−	16.2760i	−1.27224	−	0.734525i	−0.975408	+	0.220406i	$0.929262\pi$
$492$		0			0
$493$	−23.1935			−1.04458
$494$		0			0
$495$			0.0607828i			0.00273198i
$496$		0			0
$497$	12.2346	−	21.1909i	0.548795	−	0.950541i
$498$		0			0
$499$	−20.4541	+	20.4541i	−0.915653	+	0.915653i	−0.996710	−	0.0810563i	$-0.974171\pi$
$499$	−20.4541	+	20.4541i	−0.915653	+	0.915653i	0.0810563	+	0.996710i	$0.474171\pi$
$500$		0			0
$501$	−19.9809	−	5.35387i	−0.892681	−	0.239193i
$502$		0			0
$503$	22.7444	−	13.1315i	1.01412	−	0.585504i	0.101726	−	0.994812i	$-0.467563\pi$
$503$	22.7444	−	13.1315i	1.01412	−	0.585504i	0.912396	+	0.409309i	$0.134230\pi$
$504$		0			0
$505$	0.610027	−	0.163456i	0.0271459	−	0.00727371i
$506$		0			0
$507$	−28.9452	+	0.291775i	−1.28550	+	0.0129582i
$508$		0			0
$509$	13.3661	−	3.58144i	0.592443	−	0.158745i	0.0498735	−	0.998756i	$-0.484118\pi$
$509$	13.3661	−	3.58144i	0.592443	−	0.158745i	0.542569	+	0.840011i	$0.317452\pi$
$510$		0			0
$511$	3.68529	+	6.38311i	0.163028	+	0.282372i
$512$		0			0
$513$	−0.921843	−	0.247007i	−0.0407004	−	0.0109056i
$514$		0			0
$515$	−1.07752	−	1.07752i	−0.0474812	−	0.0474812i
$516$		0			0
$517$	1.37878	−	2.38812i	0.0606386	−	0.105029i
$518$		0			0
$519$	6.60277			0.289829
$520$		0			0
$521$	0.605870			0.0265437			0.0132718	−	0.999912i	$-0.495775\pi$
$521$	0.605870			0.0265437			0.0132718	+	0.999912i	$0.495775\pi$
$522$		0			0
$523$	−6.97171	+	12.0753i	−0.304851	+	0.528018i	−0.977228	−	0.212191i	$-0.931940\pi$
$523$	−6.97171	+	12.0753i	−0.304851	+	0.528018i	0.672377	+	0.740209i	$0.265273\pi$
$524$		0			0
$525$	27.3768	+	27.3768i	1.19482	+	1.19482i
$526$		0			0
$527$	−9.70376	−	2.60011i	−0.422702	−	0.113263i
$528$		0			0
$529$	−15.6676	−	27.1371i	−0.681201	−	1.17987i
$530$		0			0
$531$	−18.3761	+	4.92387i	−0.797457	+	0.213678i
$532$		0			0
$533$	−6.55135	+	24.9523i	−0.283770	+	1.08081i
$534$		0			0
$535$	−0.163893	+	0.0439149i	−0.00708570	+	0.00189861i
$536$		0			0
$537$	12.0766	−	6.97245i	0.521146	−	0.300884i
$538$		0			0
$539$	1.57168	+	0.421131i	0.0676971	+	0.0181394i
$540$		0			0
$541$	8.59605	−	8.59605i	0.369573	−	0.369573i	−0.497748	−	0.867321i	$-0.665840\pi$
$541$	8.59605	−	8.59605i	0.369573	−	0.369573i	0.867321	+	0.497748i	$0.165840\pi$
$542$		0			0
$543$	−21.9032	+	37.9374i	−0.939956	+	1.62805i
$544$		0			0
$545$			1.83294i			0.0785144i
$546$		0			0
$547$	−8.02828			−0.343264			−0.171632	−	0.985161i	$-0.554904\pi$
$547$	−8.02828			−0.343264			−0.171632	+	0.985161i	$0.554904\pi$
$548$		0			0
$549$	−23.3460	−	13.4788i	−0.996384	−	0.575263i
$550$		0			0
$551$	1.06441	+	1.06441i	0.0453456	+	0.0453456i
$552$		0			0
$553$	9.29235	−	34.6795i	0.395151	−	1.47472i
$554$		0			0
$555$	−0.339542	−	0.588105i	−0.0144128	−	0.0249637i
$556$		0			0
$557$	9.27198	+	34.6035i	0.392866	+	1.46620i	0.825384	+	0.564572i	$0.190959\pi$
$557$	9.27198	+	34.6035i	0.392866	+	1.46620i	−0.432517	+	0.901626i	$0.642375\pi$
$558$		0			0
$559$	−20.3477	+	0.102553i	−0.860617	+	0.00433751i
$560$		0			0
$561$	−1.15631	−	4.31540i	−0.0488193	−	0.182196i
$562$		0			0
$563$	5.28880	−	3.05349i	0.222897	−	0.128689i	−0.384394	−	0.923169i	$-0.625590\pi$
$563$	5.28880	−	3.05349i	0.222897	−	0.128689i	0.607291	+	0.794480i	$0.292256\pi$
$564$		0			0
$565$	−0.349387	+	1.30393i	−0.0146988	+	0.0548567i
$566$		0			0
$567$	27.2000	−	27.2000i	1.14229	−	1.14229i
$568$		0			0
$569$	−16.0840	−	9.28610i	−0.674276	−	0.389293i	0.123419	−	0.992355i	$-0.460614\pi$
$569$	−16.0840	−	9.28610i	−0.674276	−	0.389293i	−0.797695	+	0.603061i	$0.793947\pi$
$570$		0			0
$571$			30.2053i			1.26405i	0.774947	+	0.632026i	$0.217777\pi$
$571$			30.2053i			1.26405i	−0.774947	+	0.632026i	$0.782223\pi$
$572$		0			0
$573$			17.9708i			0.750741i
$574$		0			0
$575$	31.8570	+	18.3927i	1.32853	+	0.767027i
$576$		0			0
$577$	−26.5827	+	26.5827i	−1.10665	+	1.10665i	−0.113063	+	0.993588i	$0.536066\pi$
$577$	−26.5827	+	26.5827i	−1.10665	+	1.10665i	−0.993588	+	0.113063i	$0.963934\pi$
$578$		0			0
$579$	−2.26227	+	8.44291i	−0.0940168	+	0.350875i
$580$		0			0
$581$	−10.8292	+	6.25222i	−0.449269	+	0.259386i
$582$		0			0
$583$	0.319038	+	1.19067i	0.0132132	+	0.0493123i
$584$		0			0
$585$	−0.487046	+	0.491980i	−0.0201369	+	0.0203409i
$586$		0			0
$587$	9.64475	+	35.9947i	0.398081	+	1.48566i	0.816468	+	0.577391i	$0.195929\pi$
$587$	9.64475	+	35.9947i	0.398081	+	1.48566i	−0.418387	+	0.908269i	$0.637404\pi$
$588$		0			0
$589$	0.326006	+	0.564659i	0.0134329	+	0.0232664i
$590$		0			0
$591$	−10.7033	+	39.9453i	−0.440275	+	1.64313i
$592$		0			0
$593$	11.5292	+	11.5292i	0.473447	+	0.473447i	0.903028	−	0.429581i	$-0.141339\pi$
$593$	11.5292	+	11.5292i	0.473447	+	0.473447i	−0.429581	+	0.903028i	$0.641339\pi$
$594$		0			0
$595$	1.87534	+	1.08273i	0.0768814	+	0.0443875i
$596$		0			0
$597$	−3.32901			−0.136247
$598$		0			0
$599$			16.7701i			0.685208i	0.939480	+	0.342604i	$0.111309\pi$
$599$			16.7701i			0.685208i	−0.939480	+	0.342604i	$0.888691\pi$
$600$		0			0
$601$	−8.91511	+	15.4414i	−0.363655	+	0.629869i	−0.988559	−	0.150832i	$-0.951805\pi$
$601$	−8.91511	+	15.4414i	−0.363655	+	0.629869i	0.624904	+	0.780701i	$0.285138\pi$
$602$		0			0
$603$	16.9535	−	16.9535i	0.690401	−	0.690401i
$604$		0			0
$605$	1.03240	+	0.276632i	0.0419732	+	0.0112467i
$606$		0			0
$607$	−7.27664	+	4.20117i	−0.295350	+	0.170520i	−0.640352	−	0.768082i	$-0.721211\pi$
$607$	−7.27664	+	4.20117i	−0.295350	+	0.170520i	0.345002	+	0.938602i	$0.387878\pi$
$608$		0			0
$609$	−27.4232	+	7.34804i	−1.11125	+	0.297757i
$610$		0			0
$611$	30.2957	−	8.28157i	1.22563	−	0.335037i
$612$		0			0
$613$	−28.9567	+	7.75893i	−1.16955	+	0.313380i	−0.790774	−	0.612108i	$-0.790322\pi$
$613$	−28.9567	+	7.75893i	−1.16955	+	0.313380i	−0.378776	+	0.925488i	$0.623655\pi$
$614$		0			0
$615$	0.781140	+	1.35297i	0.0314986	+	0.0545572i
$616$		0			0
$617$	−29.7623	−	7.97479i	−1.19819	−	0.321053i	−0.396069	−	0.918221i	$-0.629626\pi$
$617$	−29.7623	−	7.97479i	−1.19819	−	0.321053i	−0.802116	+	0.597168i	$0.796293\pi$
$618$		0			0
$619$	−16.2444	−	16.2444i	−0.652916	−	0.652916i	0.300778	−	0.953694i	$-0.402754\pi$
$619$	−16.2444	−	16.2444i	−0.652916	−	0.652916i	−0.953694	+	0.300778i	$0.902754\pi$
$620$		0			0
$621$	8.55087	−	14.8105i	0.343135	−	0.594327i
$622$		0			0
$623$	0.398720			0.0159744
$624$		0			0
$625$	−24.8559			−0.994234
$626$		0			0
$627$	−0.144980	+	0.251112i	−0.00578993	+	0.0100284i
$628$		0			0
$629$	13.9386	+	13.9386i	0.555768	+	0.555768i
$630$		0			0
$631$	18.9019	+	5.06474i	0.752472	+	0.201624i	0.614614	−	0.788828i	$-0.289312\pi$
$631$	18.9019	+	5.06474i	0.752472	+	0.201624i	0.137858	+	0.990452i	$0.455978\pi$
$632$		0			0
$633$	11.4365	+	19.8085i	0.454559	+	0.787319i
$634$		0			0
$635$	−0.801750	+	0.214828i	−0.0318165	+	0.00852520i
$636$		0			0
$637$	9.34683	+	16.0024i	0.370335	+	0.634038i
$638$		0			0
$639$	−13.2825	+	3.55904i	−0.525448	+	0.140793i
$640$		0			0
$641$	15.3793	−	8.87927i	0.607448	−	0.350710i	−0.164518	−	0.986374i	$-0.552607\pi$
$641$	15.3793	−	8.87927i	0.607448	−	0.350710i	0.771966	+	0.635664i	$0.219274\pi$
$642$		0			0
$643$	32.3334	+	8.66372i	1.27511	+	0.341664i	0.831985	−	0.554798i	$-0.187204\pi$
$643$	32.3334	+	8.66372i	1.27511	+	0.341664i	0.443121	+	0.896462i	$0.353871\pi$
$644$		0			0
$645$	−0.871322	+	0.871322i	−0.0343083	+	0.0343083i
$646$		0			0
$647$	13.7124	−	23.7505i	0.539089	−	0.933730i	−0.459864	−	0.887989i	$-0.652102\pi$
$647$	13.7124	−	23.7505i	0.539089	−	0.933730i	0.998953	−	0.0457408i	$-0.0145648\pi$
$648$		0			0
$649$		−	3.07576i		−	0.120734i
$650$		0			0
$651$	−12.2972			−0.481964
$652$		0			0
$653$	−15.1849	−	8.76701i	−0.594232	−	0.343080i	0.172537	−	0.985003i	$-0.444803\pi$
$653$	−15.1849	−	8.76701i	−0.594232	−	0.343080i	−0.766769	+	0.641923i	$0.778137\pi$
$654$		0			0
$655$	−0.942016	−	0.942016i	−0.0368076	−	0.0368076i
$656$		0			0
$657$	1.07205	−	4.00095i	0.0418247	−	0.156092i
$658$		0			0
$659$	1.44190	+	2.49744i	0.0561684	+	0.0972865i	0.892742	−	0.450567i	$-0.148778\pi$
$659$	1.44190	+	2.49744i	0.0561684	+	0.0972865i	−0.836574	+	0.547854i	$0.815445\pi$
$660$		0			0
$661$	10.3684	+	38.6953i	0.403283	+	1.50507i	0.807200	+	0.590278i	$0.200982\pi$
$661$	10.3684	+	38.6953i	0.403283	+	1.50507i	−0.403917	+	0.914796i	$0.632351\pi$
$662$		0			0
$663$	25.2196	−	44.1945i	0.979447	−	1.71637i
$664$		0			0
$665$	−0.0363752	−	0.135754i	−0.00141057	−	0.00526431i
$666$		0			0
$667$	−23.3606	+	13.4872i	−0.904525	+	0.522228i
$668$		0			0
$669$	2.44010	−	9.10656i	0.0943396	−	0.352080i
$670$		0			0
$671$	3.08184	−	3.08184i	0.118973	−	0.118973i
$672$		0			0
$673$	6.34609	+	3.66391i	0.244624	+	0.141234i	0.617300	−	0.786728i	$-0.288226\pi$
$673$	6.34609	+	3.66391i	0.244624	+	0.141234i	−0.372676	+	0.927961i	$0.621560\pi$
$674$		0			0
$675$			11.5780i			0.445637i
$676$		0			0
$677$		−	45.6313i		−	1.75376i	−0.480714	−	0.876878i	$-0.659622\pi$
$677$		−	45.6313i		−	1.75376i	0.480714	−	0.876878i	$-0.340378\pi$
$678$		0			0
$679$	19.6484	+	11.3440i	0.754035	+	0.435342i
$680$		0			0
$681$	33.4550	−	33.4550i	1.28200	−	1.28200i
$682$		0			0
$683$	−5.80580	+	21.6676i	−0.222153	+	0.829086i	0.761372	+	0.648315i	$0.224526\pi$
$683$	−5.80580	+	21.6676i	−0.222153	+	0.829086i	−0.983525	+	0.180771i	$0.942141\pi$
$684$		0			0
$685$	−1.04744	+	0.604742i	−0.0400208	+	0.0231060i
$686$		0			0
$687$	2.16751	+	8.08925i	0.0826956	+	0.308624i
$688$		0			0
$689$	−6.95836	+	12.1937i	−0.265092	+	0.464545i
$690$		0			0
$691$	1.10893	+	4.13859i	0.0421857	+	0.157439i	0.983806	−	0.179238i	$-0.0573634\pi$
$691$	1.10893	+	4.13859i	0.0421857	+	0.157439i	−0.941620	+	0.336678i	$0.890697\pi$
$692$		0			0
$693$	−1.07987	−	1.87038i	−0.0410207	−	0.0710500i
$694$		0			0
$695$	−0.272168	+	1.01575i	−0.0103239	+	0.0385294i
$696$		0			0
$697$	−32.0667	−	32.0667i	−1.21461	−	1.21461i
$698$		0			0
$699$	25.0028	+	14.4354i	0.945692	+	0.545995i
$700$		0			0
$701$	36.4870			1.37810			0.689048	−	0.724716i	$-0.258029\pi$
$701$	36.4870			1.37810			0.689048	+	0.724716i	$0.258029\pi$
$702$		0			0
$703$		−	1.27936i		−	0.0482521i
$704$		0			0
$705$	0.950979	−	1.64714i	0.0358159	−	0.0620350i
$706$		0			0
$707$	−15.8676	+	15.8676i	−0.596761	+	0.596761i
$708$		0			0
$709$	14.8664	+	3.98345i	0.558321	+	0.149602i	0.526935	−	0.849906i	$-0.323341\pi$
$709$	14.8664	+	3.98345i	0.558321	+	0.149602i	0.0313862	+	0.999507i	$0.490008\pi$
$710$		0			0
$711$	−17.4734	+	10.0883i	−0.655305	+	0.378340i
$712$		0			0
$713$	−11.2857	+	3.02398i	−0.422651	+	0.113249i
$714$		0			0
$715$	−0.0564503	−	0.0966466i	−0.00211112	−	0.00361438i
$716$		0			0
$717$	6.10501	−	1.63583i	0.227996	−	0.0610913i
$718$		0			0
$719$	11.9512	+	20.7000i	0.445703	+	0.771980i	0.998101	−	0.0616005i	$-0.0196205\pi$
$719$	11.9512	+	20.7000i	0.445703	+	0.771980i	−0.552398	+	0.833580i	$0.686287\pi$
$720$		0			0
$721$	52.3002	+	14.0138i	1.94776	+	0.521902i
$722$		0			0
$723$	−12.2743	−	12.2743i	−0.456487	−	0.456487i
$724$		0			0
$725$	9.13095	−	15.8153i	0.339115	−	0.587364i
$726$		0			0
$727$	21.0545			0.780868			0.390434	−	0.920631i	$-0.372325\pi$
$727$	21.0545			0.780868			0.390434	+	0.920631i	$0.372325\pi$
$728$		0			0
$729$	−6.11928			−0.226640
$730$		0			0
$731$	17.8844	−	30.9766i	0.661477	−	1.14571i
$732$		0			0
$733$	−2.18046	−	2.18046i	−0.0805370	−	0.0805370i	0.665691	−	0.746228i	$-0.268137\pi$
$733$	−2.18046	−	2.18046i	−0.0805370	−	0.0805370i	−0.746228	+	0.665691i	$0.768137\pi$
$734$		0			0
$735$	1.08403	+	0.290465i	0.0399850	+	0.0107140i
$736$		0			0
$737$	1.93815	+	3.35697i	0.0713926	+	0.123656i
$738$		0			0
$739$	−45.3273	+	12.1454i	−1.66739	+	0.446776i	−0.964405	−	0.264429i	$-0.914817\pi$
$739$	−45.3273	+	12.1454i	−1.66739	+	0.446776i	−0.702985	+	0.711204i	$0.748150\pi$
$740$		0			0
$741$	−3.18561	+	0.870813i	−0.117026	+	0.0319901i
$742$		0			0
$743$	−41.4704	+	11.1120i	−1.52140	+	0.407658i	−0.920203	−	0.391442i	$-0.871976\pi$
$743$	−41.4704	+	11.1120i	−1.52140	+	0.407658i	−0.601198	+	0.799100i	$0.705310\pi$
$744$		0			0
$745$	0.973548	−	0.562078i	0.0356680	−	0.0205929i
$746$		0			0
$747$	6.78775	+	1.81877i	0.248351	+	0.0665454i
$748$		0			0
$749$	4.26305	−	4.26305i	0.155768	−	0.155768i
$750$		0			0
$751$	−3.78589	+	6.55735i	−0.138149	+	0.239281i	−0.926796	−	0.375565i	$-0.877449\pi$
$751$	−3.78589	+	6.55735i	−0.138149	+	0.239281i	0.788647	+	0.614846i	$0.210782\pi$
$752$		0			0
$753$			21.8378i			0.795814i
$754$		0			0
$755$	1.05926			0.0385505
$756$		0			0
$757$	−18.5122	−	10.6880i	−0.672836	−	0.388462i	0.124314	−	0.992243i	$-0.460327\pi$
$757$	−18.5122	−	10.6880i	−0.672836	−	0.388462i	−0.797151	+	0.603781i	$0.793660\pi$
$758$		0			0
$759$	−3.67408	−	3.67408i	−0.133361	−	0.133361i
$760$		0			0
$761$	0.991139	−	3.69898i	0.0359288	−	0.134088i	−0.945632	−	0.325239i	$-0.894555\pi$
$761$	0.991139	−	3.69898i	0.0359288	−	0.134088i	0.981561	+	0.191151i	$0.0612220\pi$
$762$		0			0
$763$	−32.5639	−	56.4024i	−1.17889	−	2.04190i
$764$		0			0
$765$	−0.314966	−	1.17547i	−0.0113876	−	0.0424991i
$766$		0			0
$767$	24.6457	−	24.8954i	0.889906	−	0.898922i
$768$		0			0
$769$	4.33512	+	16.1789i	0.156329	+	0.583426i	0.998988	+	0.0449792i	$0.0143221\pi$
$769$	4.33512	+	16.1789i	0.156329	+	0.583426i	−0.842659	+	0.538447i	$0.819011\pi$
$770$		0			0
$771$	−18.3097	+	10.5711i	−0.659407	+	0.380709i
$772$		0			0
$773$	−6.29416	+	23.4901i	−0.226385	+	0.844882i	0.755459	+	0.655195i	$0.227414\pi$
$773$	−6.29416	+	23.4901i	−0.226385	+	0.844882i	−0.981845	+	0.189686i	$0.939253\pi$
$774$		0			0
$775$	5.59321	−	5.59321i	0.200914	−	0.200914i
$776$		0			0
$777$	20.8965	+	12.0646i	0.749659	+	0.432816i
$778$		0			0
$779$			2.94326i			0.105453i
$780$		0			0
$781$		−	2.22320i		−	0.0795523i
$782$		0			0
$783$	−7.35262	−	4.24504i	−0.262761	−	0.151705i
$784$		0			0
$785$	−0.0204986	+	0.0204986i	−0.000731625	+	0.000731625i
$786$		0			0
$787$	9.30815	−	34.7385i	0.331800	−	1.23829i	−0.575498	−	0.817803i	$-0.695192\pi$
$787$	9.30815	−	34.7385i	0.331800	−	1.23829i	0.907297	−	0.420490i	$-0.138142\pi$
$788$		0			0
$789$	−0.853191	+	0.492590i	−0.0303744	+	0.0175367i
$790$		0			0
$791$	−12.4144	−	46.3312i	−0.441405	−	1.64735i
$792$		0			0
$793$	49.6390	−	0.250181i	1.76273	−	0.00888417i
$794$		0			0
$795$	0.220048	+	0.821232i	0.00780431	+	0.0291261i
$796$		0			0
$797$	−5.36008	−	9.28393i	−0.189864	−	0.328854i	0.755341	−	0.655332i	$-0.227471\pi$
$797$	−5.36008	−	9.28393i	−0.189864	−	0.328854i	−0.945205	+	0.326478i	$0.894138\pi$
$798$		0			0
$799$	−14.2892	+	53.3280i	−0.505515	+	1.88661i
$800$		0			0
$801$	−0.158442	−	0.158442i	−0.00559829	−	0.00559829i
$802$		0			0
$803$	0.579952	+	0.334836i	0.0204661	+	0.0118161i
$804$		0			0
$805$	2.51847			0.0887642
$806$		0			0
$807$		−	31.6179i		−	1.11300i
$808$		0			0
$809$	−19.6641	+	34.0593i	−0.691355	+	1.19746i	0.280040	+	0.959988i	$0.409652\pi$
$809$	−19.6641	+	34.0593i	−0.691355	+	1.19746i	−0.971394	+	0.237473i	$0.923681\pi$
$810$		0			0
$811$	38.2641	−	38.2641i	1.34364	−	1.34364i	0.451226	−	0.892410i	$-0.350987\pi$
$811$	38.2641	−	38.2641i	1.34364	−	1.34364i	0.892410	−	0.451226i	$-0.149013\pi$
$812$		0			0
$813$	−10.7076	−	2.86910i	−0.375532	−	0.100624i
$814$		0			0
$815$	−0.211293	+	0.121990i	−0.00740127	+	0.00427312i
$816$		0			0
$817$	−2.24237	+	0.600841i	−0.0784506	+	0.0210208i
$818$		0			0
$819$	6.24667	−	23.7919i	0.218276	−	0.831356i
$820$		0			0
$821$	19.9781	−	5.35310i	0.697239	−	0.186825i	0.107245	−	0.994233i	$-0.465797\pi$
$821$	19.9781	−	5.35310i	0.697239	−	0.186825i	0.589993	+	0.807408i	$0.299130\pi$
$822$		0			0
$823$	−11.8141	−	20.4626i	−0.411813	−	0.713281i	0.583275	−	0.812275i	$-0.301771\pi$
$823$	−11.8141	−	20.4626i	−0.411813	−	0.713281i	−0.995088	+	0.0989937i	$0.968438\pi$
$824$		0			0
$825$	3.39782	+	0.910444i	0.118297	+	0.0316976i
$826$		0			0
$827$	17.7598	+	17.7598i	0.617568	+	0.617568i	0.944907	−	0.327339i	$-0.106152\pi$
$827$	17.7598	+	17.7598i	0.617568	+	0.617568i	−0.327339	+	0.944907i	$0.606152\pi$
$828$		0			0
$829$	−16.4735	+	28.5329i	−0.572147	+	0.990988i	0.424198	+	0.905569i	$0.360556\pi$
$829$	−16.4735	+	28.5329i	−0.572147	+	0.990988i	−0.996345	+	0.0854184i	$0.972777\pi$
$830$		0			0
$831$	−26.6804			−0.925533
$832$		0			0
$833$	−32.5767			−1.12872
$834$		0			0
$835$	0.455484	−	0.788922i	0.0157627	−	0.0273018i
$836$		0			0
$837$	−2.60032	−	2.60032i	−0.0898802	−	0.0898802i
$838$		0			0
$839$	−42.5633	−	11.4048i	−1.46945	−	0.393737i	−0.566706	−	0.823920i	$-0.691782\pi$
$839$	−42.5633	−	11.4048i	−1.46945	−	0.393737i	−0.902742	+	0.430183i	$0.858449\pi$
$840$		0			0
$841$	−7.80433	−	13.5175i	−0.269115	−	0.466121i
$842$		0			0
$843$	15.5862	−	4.17630i	0.536817	−	0.143840i
$844$		0			0
$845$	0.317506	−	1.23459i	0.0109225	−	0.0424713i
$846$		0			0
$847$	−36.6834	+	9.82927i	−1.26045	+	0.337738i
$848$		0			0
$849$	−23.9034	+	13.8007i	−0.820364	+	0.473637i
$850$		0			0
$851$	22.1445	+	5.93359i	0.759102	+	0.203401i
$852$		0			0
$853$	−0.463907	+	0.463907i	−0.0158839	+	0.0158839i	−0.715004	−	0.699120i	$-0.753575\pi$
$853$	−0.463907	+	0.463907i	−0.0158839	+	0.0158839i	0.699120	+	0.715004i	$0.253575\pi$
$854$		0			0
$855$	−0.0394909	+	0.0684002i	−0.00135056	+	0.00233924i
$856$		0			0
$857$			33.1382i			1.13198i	0.824412	+	0.565990i	$0.191506\pi$
$857$			33.1382i			1.13198i	−0.824412	+	0.565990i	$0.808494\pi$
$858$		0			0
$859$	−21.3046			−0.726903			−0.363451	−	0.931613i	$-0.618402\pi$
$859$	−21.3046			−0.726903			−0.363451	+	0.931613i	$0.618402\pi$
$860$		0			0
$861$	−48.0738	−	27.7554i	−1.63835	−	0.945903i
$862$		0			0
$863$	27.2032	+	27.2032i	0.926008	+	0.926008i	0.997445	−	0.0714372i	$-0.0227585\pi$
$863$	27.2032	+	27.2032i	0.926008	+	0.926008i	−0.0714372	+	0.997445i	$0.522759\pi$
$864$		0			0
$865$	−0.0752583	+	0.280868i	−0.00255886	+	0.00954979i
$866$		0			0
$867$	25.7966	+	44.6809i	0.876097	+	1.51744i
$868$		0			0
$869$	−0.844278	−	3.15089i	−0.0286402	−	0.106887i
$870$		0			0
$871$	−11.2115	+	42.7018i	−0.379889	+	1.44689i
$872$		0			0
$873$	−3.29997	−	12.3157i	−0.111687	−	0.416822i
$874$		0			0
$875$	−2.95602	+	1.70666i	−0.0999318	+	0.0576957i
$876$		0			0
$877$	−4.86299	+	18.1489i	−0.164212	+	0.612846i	0.833928	+	0.551874i	$0.186087\pi$
$877$	−4.86299	+	18.1489i	−0.164212	+	0.612846i	−0.998139	+	0.0609726i	$0.980580\pi$
$878$		0			0
$879$	4.49844	−	4.49844i	0.151729	−	0.151729i
$880$		0			0
$881$	−10.8962	−	6.29091i	−0.367101	−	0.211946i	0.305090	−	0.952324i	$-0.401313\pi$
$881$	−10.8962	−	6.29091i	−0.367101	−	0.211946i	−0.672191	+	0.740377i	$0.734647\pi$
$882$		0			0
$883$		−	44.2186i		−	1.48807i	−0.668139	−	0.744036i	$-0.732909\pi$
$883$		−	44.2186i		−	1.48807i	0.668139	−	0.744036i	$-0.267091\pi$
$884$		0			0
$885$		−	2.12143i		−	0.0713111i
$886$		0			0
$887$	−5.54684	−	3.20247i	−0.186245	−	0.107528i	0.403979	−	0.914768i	$-0.367627\pi$
$887$	−5.54684	−	3.20247i	−0.186245	−	0.107528i	−0.590223	+	0.807240i	$0.700960\pi$
$888$		0			0
$889$	20.8545	−	20.8545i	0.699437	−	0.699437i
$890$		0			0
$891$	0.904565	−	3.37588i	0.0303041	−	0.113096i
$892$		0			0
$893$	3.10314	−	1.79160i	0.103843	−	0.0599536i
$894$		0			0
$895$	0.158944	+	0.593187i	0.00531291	+	0.0198280i
$896$		0			0
$897$	−0.298259	−	59.1783i	−0.00995857	−	1.97591i
$898$		0			0
$899$	1.50124	+	5.60271i	0.0500692	+	0.186861i
$900$		0			0
$901$	−12.3396	−	21.3729i	−0.411093	−	0.712034i
$902$		0			0
$903$	11.3321	−	42.2919i	0.377108	−	1.40738i
$904$		0			0
$905$	−1.36413	−	1.36413i	−0.0453451	−	0.0453451i
$906$		0			0
$907$	−11.0865	−	6.40082i	−0.368123	−	0.212536i	0.304515	−	0.952507i	$-0.401506\pi$
$907$	−11.0865	−	6.40082i	−0.368123	−	0.212536i	−0.672638	+	0.739972i	$0.734839\pi$
$908$		0			0
$909$	12.6108			0.418274
$910$		0			0
$911$			4.52483i			0.149914i	0.997187	+	0.0749572i	$0.0238820\pi$
$911$			4.52483i			0.149914i	−0.997187	+	0.0749572i	$0.976118\pi$
$912$		0			0
$913$	−0.568060	+	0.983909i	−0.0188000	+	0.0325626i
$914$		0			0
$915$	2.12562	−	2.12562i	0.0702708	−	0.0702708i
$916$		0			0
$917$	45.7232	+	12.2515i	1.50991	+	0.404580i
$918$		0			0
$919$	44.0101	−	25.4093i	1.45176	−	0.838174i	0.453179	−	0.891420i	$-0.350290\pi$
$919$	44.0101	−	25.4093i	1.45176	−	0.838174i	0.998581	+	0.0532452i	$0.0169565\pi$
$920$		0			0
$921$	3.99842	−	1.07137i	0.131753	−	0.0353030i
$922$		0			0
$923$	17.8143	−	17.9947i	0.586363	−	0.592304i
$924$		0			0
$925$	−14.9920	+	4.01708i	−0.492932	+	0.132081i
$926$		0			0
$927$	−15.2142	−	26.3517i	−0.499699	−	0.865504i
$928$		0			0
$929$	58.1272	+	15.5751i	1.90709	+	0.511003i	0.994851	+	0.101347i	$0.0323152\pi$
$929$	58.1272	+	15.5751i	1.90709	+	0.511003i	0.912240	+	0.409657i	$0.134351\pi$
$930$		0			0
$931$	1.49504	+	1.49504i	0.0489979	+	0.0489979i
$932$		0			0
$933$	29.3894	−	50.9040i	0.962166	−	1.66652i
$934$		0			0
$935$	0.196747			0.00643433
$936$		0			0
$937$	−31.3424			−1.02391			−0.511956	−	0.859012i	$-0.671079\pi$
$937$	−31.3424			−1.02391			−0.511956	+	0.859012i	$0.671079\pi$
$938$		0			0
$939$	−25.7103	+	44.5315i	−0.839023	+	1.45323i
$940$		0			0
$941$	−22.8653	−	22.8653i	−0.745387	−	0.745387i	0.228222	−	0.973609i	$-0.426709\pi$
$941$	−22.8653	−	22.8653i	−0.745387	−	0.745387i	−0.973609	+	0.228222i	$0.926709\pi$
$942$		0			0
$943$	−50.9448	−	13.6506i	−1.65899	−	0.444525i
$944$		0			0
$945$	0.396337	+	0.686476i	0.0128928	+	0.0223311i
$946$		0			0
$947$	20.0408	−	5.36992i	0.651239	−	0.174499i	0.0819499	−	0.996636i	$-0.473885\pi$
$947$	20.0408	−	5.36992i	0.651239	−	0.174499i	0.569289	+	0.822138i	$0.307219\pi$
$948$		0			0
$949$	2.01117	+	7.35728i	0.0652855	+	0.238827i
$950$		0			0
$951$	42.4896	−	11.3851i	1.37782	−	0.369186i
$952$		0			0
$953$	2.83553	−	1.63710i	0.0918520	−	0.0530308i	−0.453370	−	0.891322i	$-0.649779\pi$
$953$	2.83553	−	1.63710i	0.0918520	−	0.0530308i	0.545222	+	0.838291i	$0.316445\pi$
$954$		0			0
$955$	−0.764439	−	0.204831i	−0.0247367	−	0.00662817i
$956$		0			0
$957$	−1.82398	+	1.82398i	−0.0589609	+	0.0589609i
$958$		0			0
$959$	21.4877	−	37.2178i	0.693874	−	1.20182i
$960$		0			0
$961$		−	28.4876i		−	0.918956i
$962$		0			0
$963$	−3.38808			−0.109179
$964$		0			0
$965$	−0.333358	−	0.192464i	−0.0107312	−	0.00619565i
$966$		0			0
$967$	15.9826	+	15.9826i	0.513966	+	0.513966i	0.915739	−	0.401773i	$-0.131606\pi$
$967$	15.9826	+	15.9826i	0.513966	+	0.513966i	−0.401773	+	0.915739i	$0.631606\pi$
$968$		0			0
$969$	1.50252	−	5.60747i	0.0482678	−	0.180138i
$970$		0			0
$971$	−12.5870	−	21.8014i	−0.403938	−	0.699640i	0.590260	−	0.807213i	$-0.299025\pi$
$971$	−12.5870	−	21.8014i	−0.403938	−	0.699640i	−0.994197	+	0.107573i	$0.965692\pi$
$972$		0			0
$973$	−9.67068	−	36.0915i	−0.310028	−	1.15704i
$974$		0			0
$975$	20.2069	+	34.5956i	0.647140	+	1.10795i
$976$		0			0
$977$	−5.42229	−	20.2363i	−0.173475	−	0.647416i	−0.996806	−	0.0798556i	$-0.974554\pi$
$977$	−5.42229	−	20.2363i	−0.173475	−	0.647416i	0.823332	−	0.567560i	$-0.192113\pi$
$978$		0			0
$979$	0.0313732	−	0.0181133i	0.00100269	−	0.000578905i
$980$		0			0
$981$	−9.47286	+	35.3532i	−0.302445	+	1.12874i
$982$		0			0
$983$	14.2586	−	14.2586i	0.454777	−	0.454777i	−0.442159	−	0.896937i	$-0.645787\pi$
$983$	14.2586	−	14.2586i	0.454777	−	0.454777i	0.896937	+	0.442159i	$0.145787\pi$
$984$		0			0
$985$	−1.57719	−	0.910592i	−0.0502535	−	0.0290139i
$986$		0			0
$987$			67.5804i			2.15111i
$988$		0			0
$989$		−	41.5997i		−	1.32280i
$990$		0			0
$991$	−23.1803	−	13.3832i	−0.736348	−	0.425131i	0.0843920	−	0.996433i	$-0.473105\pi$
$991$	−23.1803	−	13.3832i	−0.736348	−	0.425131i	−0.820740	+	0.571302i	$0.806439\pi$
$992$		0			0
$993$	14.3899	−	14.3899i	0.456650	−	0.456650i
$994$		0			0
$995$	0.0379440	−	0.141609i	0.00120291	−	0.00448931i
$996$		0			0
$997$	31.1545	−	17.9870i	0.986672	−	0.569655i	0.0823942	−	0.996600i	$-0.473743\pi$
$997$	31.1545	−	17.9870i	0.986672	−	0.569655i	0.904278	+	0.426944i	$0.140410\pi$
$998$		0			0
$999$	1.86757	+	6.96985i	0.0590872	+	0.220516i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bk.a.175.1		48
4.3	odd	2		104.2.u.a.19.11	yes	48
8.3	odd	2	inner	416.2.bk.a.175.2		48
8.5	even	2		104.2.u.a.19.5	yes	48
12.11	even	2		936.2.ed.d.19.2		48
13.11	odd	12	inner	416.2.bk.a.271.2		48
24.5	odd	2		936.2.ed.d.19.8		48
52.11	even	12		104.2.u.a.11.5	✓	48
104.11	even	12	inner	416.2.bk.a.271.1		48
104.37	odd	12		104.2.u.a.11.11	yes	48
156.11	odd	12		936.2.ed.d.739.8		48
312.245	even	12		936.2.ed.d.739.2		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.11.5	✓	48	52.11	even	12
104.2.u.a.11.11	yes	48	104.37	odd	12
104.2.u.a.19.5	yes	48	8.5	even	2
104.2.u.a.19.11	yes	48	4.3	odd	2
416.2.bk.a.175.1		48	1.1	even	1	trivial
416.2.bk.a.175.2		48	8.3	odd	2	inner
416.2.bk.a.271.1		48	104.11	even	12	inner
416.2.bk.a.271.2		48	13.11	odd	12	inner
936.2.ed.d.19.2		48	12.11	even	2
936.2.ed.d.19.8		48	24.5	odd	2
936.2.ed.d.739.2		48	312.245	even	12
936.2.ed.d.739.8		48	156.11	odd	12

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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bk (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.11333 + 1.92835i) q^{3} +(-0.0693382 - 0.0693382i) q^{5} +(3.36551 + 0.901786i) q^{7} +(-0.979029 - 1.69573i) q^{9} +(0.305781 - 0.0819339i) q^{11} +(3.13155 + 1.78702i) q^{13} +(0.210905 - 0.0565119i) q^{15} +(-5.48889 + 3.16901i) q^{17} +(0.397336 + 0.106466i) q^{19} +(-5.48590 + 5.48590i) q^{21} +(-3.68562 + 6.38369i) q^{23} -4.99038i q^{25} -2.32006 q^{27} +(3.16915 + 1.82971i) q^{29} +(1.12080 + 1.12080i) q^{31} +(-0.182440 + 0.680874i) q^{33} +(-0.170830 - 0.295887i) q^{35} +(-0.804964 - 3.00417i) q^{37} +(-6.93245 + 4.04918i) q^{39} +(1.85187 + 6.91129i) q^{41} +(-4.88743 + 2.82176i) q^{43} +(-0.0496946 + 0.185463i) q^{45} +(6.15946 - 6.15946i) q^{47} +(4.45127 + 2.56994i) q^{49} -14.1127i q^{51} +3.89384i q^{53} +(-0.0268835 - 0.0155212i) q^{55} +(-0.647671 + 0.647671i) q^{57} +(2.51467 - 9.38488i) q^{59} +(11.9231 - 6.88378i) q^{61} +(-1.76575 - 6.58987i) q^{63} +(-0.0932273 - 0.341044i) q^{65} +(3.16917 + 11.8275i) q^{67} +(-8.20666 - 14.2144i) q^{69} +(1.81764 - 6.78351i) q^{71} +(1.49582 + 1.49582i) q^{73} +(9.62322 + 5.55597i) q^{75} +1.10300 q^{77} -10.3044i q^{79} +(5.52009 - 9.56108i) q^{81} +(-2.53771 + 2.53771i) q^{83} +(0.600323 + 0.160856i) q^{85} +(-7.05665 + 4.07416i) q^{87} +(0.110536 - 0.0296181i) q^{89} +(8.92775 + 8.83821i) q^{91} +(-3.40912 + 0.913470i) q^{93} +(-0.0201684 - 0.0349327i) q^{95} +(6.28973 + 1.68533i) q^{97} +(-0.438306 - 0.438306i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83}+ \cdots - 168 q^{99}+O(q^{100}) \) 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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