LMFDB - Embedded newform 416.2.bk.a.111.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(15,416)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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			111.3
Character	$\chi$	$=$	416.111
Dual form			416.2.bk.a.15.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.793616 + 1.37458i) q^{3} +(-0.203095 + 0.203095i) q^{5} +(-0.207385 + 0.773970i) q^{7} +(0.240346 + 0.416291i) q^{9} +O(q^{10})$	\(q+(-0.793616 + 1.37458i) q^{3} +(-0.203095 + 0.203095i) q^{5} +(-0.207385 + 0.773970i) q^{7} +(0.240346 + 0.416291i) q^{9} +(0.587501 + 2.19258i) q^{11} +(-3.31919 - 1.40818i) q^{13} +(-0.117991 - 0.440350i) q^{15} +(-3.23700 + 1.86889i) q^{17} +(-0.792632 + 2.95814i) q^{19} +(-0.899303 - 0.899303i) q^{21} +(1.24741 - 2.16058i) q^{23} +4.91751i q^{25} -5.52467 q^{27} +(-1.82695 - 1.05479i) q^{29} +(-5.25282 + 5.25282i) q^{31} +(-3.48014 - 0.932501i) q^{33} +(-0.115071 - 0.199308i) q^{35} +(7.30393 - 1.95708i) q^{37} +(4.56983 - 3.44495i) q^{39} +(-2.91916 + 0.782185i) q^{41} +(3.07997 - 1.77822i) q^{43} +(-0.133359 - 0.0357336i) q^{45} +(-7.51561 - 7.51561i) q^{47} +(5.50616 + 3.17898i) q^{49} -5.93271i q^{51} +13.1159i q^{53} +(-0.564621 - 0.325984i) q^{55} +(-3.43717 - 3.43717i) q^{57} +(6.44108 + 1.72588i) q^{59} +(3.55817 - 2.05431i) q^{61} +(-0.372041 + 0.0996881i) q^{63} +(0.960104 - 0.388116i) q^{65} +(10.5799 - 2.83488i) q^{67} +(1.97993 + 3.42934i) q^{69} +(7.61356 + 2.04005i) q^{71} +(10.3306 - 10.3306i) q^{73} +(-6.75952 - 3.90261i) q^{75} -1.81883 q^{77} +12.1132i q^{79} +(3.66343 - 6.34525i) q^{81} +(-2.07891 - 2.07891i) q^{83} +(0.277858 - 1.03698i) q^{85} +(2.89979 - 1.67420i) q^{87} +(0.497728 + 1.85755i) q^{89} +(1.77824 - 2.27692i) q^{91} +(-3.05172 - 11.3892i) q^{93} +(-0.439804 - 0.761763i) q^{95} +(1.50062 - 5.60039i) q^{97} +(-0.771550 + 0.771550i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{3} - 20 q^{9}+O(q^{10}) $ 48 * q + 4 * q^3 - 20 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{11}{12}\right)$

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
$3$	−0.793616	+	1.37458i	−0.458195	+	0.793616i	−0.998866	−	0.0476177i	$-0.984837\pi$
$3$	−0.793616	+	1.37458i	−0.458195	+	0.793616i	0.540671	+	0.841234i	$0.318170\pi$
$4$		0			0
$5$	−0.203095	+	0.203095i	−0.0908267	+	0.0908267i	−0.751060	−	0.660234i	$-0.770457\pi$
$5$	−0.203095	+	0.203095i	−0.0908267	+	0.0908267i	0.660234	+	0.751060i	$0.270457\pi$
$6$		0			0
$7$	−0.207385	+	0.773970i	−0.0783840	+	0.292533i	−0.993979	−	0.109568i	$-0.965053\pi$
$7$	−0.207385	+	0.773970i	−0.0783840	+	0.292533i	0.915595	+	0.402101i	$0.131720\pi$
$8$		0			0
$9$	0.240346	+	0.416291i	0.0801153	+	0.138764i
$10$		0			0
$11$	0.587501	+	2.19258i	0.177138	+	0.661089i	0.996178	+	0.0873516i	$0.0278404\pi$
$11$	0.587501	+	2.19258i	0.177138	+	0.661089i	−0.819039	+	0.573737i	$0.805493\pi$
$12$		0			0
$13$	−3.31919	−	1.40818i	−0.920578	−	0.390559i
$13$	−3.31919	−	1.40818i	−0.920578	−	0.390559i
$14$		0			0
$15$	−0.117991	−	0.440350i	−0.0304653	−	0.113698i
$16$		0			0
$17$	−3.23700	+	1.86889i	−0.785089	+	0.453271i	−0.838231	−	0.545316i	$-0.816410\pi$
$17$	−3.23700	+	1.86889i	−0.785089	+	0.453271i	0.0531419	+	0.998587i	$0.483076\pi$
$18$		0			0
$19$	−0.792632	+	2.95814i	−0.181842	+	0.678645i	0.813442	+	0.581646i	$0.197591\pi$
$19$	−0.792632	+	2.95814i	−0.181842	+	0.678645i	−0.995284	+	0.0969990i	$0.969076\pi$
$20$		0			0
$21$	−0.899303	−	0.899303i	−0.196244	−	0.196244i
$22$		0			0
$23$	1.24741	−	2.16058i	0.260103	−	0.450511i	−0.706166	−	0.708046i	$-0.749577\pi$
$23$	1.24741	−	2.16058i	0.260103	−	0.450511i	0.966269	+	0.257535i	$0.0829102\pi$
$24$		0			0
$25$			4.91751i			0.983501i
$26$		0			0
$27$	−5.52467			−1.06322
$28$		0			0
$29$	−1.82695	−	1.05479i	−0.339256	−	0.195869i	0.320687	−	0.947185i	$-0.396086\pi$
$29$	−1.82695	−	1.05479i	−0.339256	−	0.195869i	−0.659943	+	0.751316i	$0.729420\pi$
$30$		0			0
$31$	−5.25282	+	5.25282i	−0.943435	+	0.943435i	−0.998484	−	0.0550487i	$-0.982469\pi$
$31$	−5.25282	+	5.25282i	−0.943435	+	0.943435i	0.0550487	+	0.998484i	$0.482469\pi$
$32$		0			0
$33$	−3.48014	−	0.932501i	−0.605815	−	0.162328i
$34$		0			0
$35$	−0.115071	−	0.199308i	−0.0194505	−	0.0336892i
$36$		0			0
$37$	7.30393	−	1.95708i	1.20076	−	0.321742i	0.397629	−	0.917546i	$-0.369833\pi$
$37$	7.30393	−	1.95708i	1.20076	−	0.321742i	0.803130	+	0.595804i	$0.203167\pi$
$38$		0			0
$39$	4.56983	−	3.44495i	0.731758	−	0.551634i
$40$		0			0
$41$	−2.91916	+	0.782185i	−0.455895	+	0.122157i	−0.479457	−	0.877566i	$-0.659166\pi$
$41$	−2.91916	+	0.782185i	−0.455895	+	0.122157i	0.0235612	+	0.999722i	$0.492500\pi$
$42$		0			0
$43$	3.07997	−	1.77822i	0.469691	−	0.271176i	−0.246419	−	0.969163i	$-0.579254\pi$
$43$	3.07997	−	1.77822i	0.469691	−	0.271176i	0.716110	+	0.697987i	$0.245921\pi$
$44$		0			0
$45$	−0.133359	−	0.0357336i	−0.0198801	−	0.00532684i
$46$		0			0
$47$	−7.51561	−	7.51561i	−1.09626	−	1.09626i	−0.994844	−	0.101420i	$-0.967661\pi$
$47$	−7.51561	−	7.51561i	−1.09626	−	1.09626i	−0.101420	−	0.994844i	$-0.532339\pi$
$48$		0			0
$49$	5.50616	+	3.17898i	0.786594	+	0.454140i
$50$		0			0
$51$		−	5.93271i		−	0.830746i
$52$		0			0
$53$			13.1159i			1.80160i	0.434230	+	0.900802i	$0.357020\pi$
$53$			13.1159i			1.80160i	−0.434230	+	0.900802i	$0.642980\pi$
$54$		0			0
$55$	−0.564621	−	0.325984i	−0.0761334	−	0.0439556i
$56$		0			0
$57$	−3.43717	−	3.43717i	−0.455265	−	0.455265i
$58$		0			0
$59$	6.44108	+	1.72588i	0.838558	+	0.224691i	0.652444	−	0.757837i	$-0.273744\pi$
$59$	6.44108	+	1.72588i	0.838558	+	0.224691i	0.186114	+	0.982528i	$0.440411\pi$
$60$		0			0
$61$	3.55817	−	2.05431i	0.455577	−	0.263028i	−0.254606	−	0.967045i	$-0.581946\pi$
$61$	3.55817	−	2.05431i	0.455577	−	0.263028i	0.710183	+	0.704017i	$0.248612\pi$
$62$		0			0
$63$	−0.372041	+	0.0996881i	−0.0468728	+	0.0125595i
$64$		0			0
$65$	0.960104	−	0.388116i	0.119086	−	0.0481399i
$66$		0			0
$67$	10.5799	−	2.83488i	1.29254	−	0.346336i	0.453917	−	0.891044i	$-0.350026\pi$
$67$	10.5799	−	2.83488i	1.29254	−	0.346336i	0.838626	+	0.544708i	$0.183360\pi$
$68$		0			0
$69$	1.97993	+	3.42934i	0.238355	+	0.412844i
$70$		0			0
$71$	7.61356	+	2.04005i	0.903563	+	0.242109i	0.680546	−	0.732705i	$-0.261743\pi$
$71$	7.61356	+	2.04005i	0.903563	+	0.242109i	0.223017	+	0.974814i	$0.428409\pi$
$72$		0			0
$73$	10.3306	−	10.3306i	1.20910	−	1.20910i	0.237781	−	0.971319i	$-0.423580\pi$
$73$	10.3306	−	10.3306i	1.20910	−	1.20910i	0.971319	−	0.237781i	$-0.0764201\pi$
$74$		0			0
$75$	−6.75952	−	3.90261i	−0.780523	−	0.450635i
$76$		0			0
$77$	−1.81883			−0.207275
$78$		0			0
$79$			12.1132i			1.36284i	0.731892	+	0.681421i	$0.238637\pi$
$79$			12.1132i			1.36284i	−0.731892	+	0.681421i	$0.761363\pi$
$80$		0			0
$81$	3.66343	−	6.34525i	0.407048	−	0.705027i
$82$		0			0
$83$	−2.07891	−	2.07891i	−0.228190	−	0.228190i	0.583746	−	0.811936i	$-0.301586\pi$
$83$	−2.07891	−	2.07891i	−0.228190	−	0.228190i	−0.811936	+	0.583746i	$0.801586\pi$
$84$		0			0
$85$	0.277858	−	1.03698i	0.0301379	−	0.112476i
$86$		0			0
$87$	2.89979	−	1.67420i	0.310890	−	0.179493i
$88$		0			0
$89$	0.497728	+	1.85755i	0.0527590	+	0.196899i	0.987275	−	0.159021i	$-0.0508339\pi$
$89$	0.497728	+	1.85755i	0.0527590	+	0.196899i	−0.934516	+	0.355921i	$0.884167\pi$
$90$		0			0
$91$	1.77824	−	2.27692i	0.186410	−	0.238686i
$92$		0			0
$93$	−3.05172	−	11.3892i	−0.316449	−	1.18100i
$94$		0			0
$95$	−0.439804	−	0.761763i	−0.0451229	−	0.0781552i
$96$		0			0
$97$	1.50062	−	5.60039i	0.152365	−	0.568634i	−0.846952	−	0.531670i	$-0.821565\pi$
$97$	1.50062	−	5.60039i	0.152365	−	0.568634i	0.999317	−	0.0369639i	$-0.0117687\pi$
$98$		0			0
$99$	−0.771550	+	0.771550i	−0.0775437	+	0.0775437i
$100$		0			0
$101$	4.06770	−	7.04546i	0.404751	−	0.701050i	−0.589541	−	0.807738i	$-0.700691\pi$
$101$	4.06770	−	7.04546i	0.404751	−	0.701050i	0.994293	+	0.106688i	$0.0340247\pi$
$102$		0			0
$103$	17.4099			1.71545			0.857723	−	0.514113i	$-0.171879\pi$
$103$	17.4099			1.71545			0.857723	+	0.514113i	$0.171879\pi$
$104$		0			0
$105$	0.365287			0.0356484
$106$		0			0
$107$	0.736948	−	1.27643i	0.0712434	−	0.123397i	−0.828203	−	0.560428i	$-0.810637\pi$
$107$	0.736948	−	1.27643i	0.0712434	−	0.123397i	0.899447	+	0.437031i	$0.143970\pi$
$108$		0			0
$109$	−8.43246	+	8.43246i	−0.807683	+	0.807683i	−0.984283	−	0.176600i	$-0.943490\pi$
$109$	−8.43246	+	8.43246i	−0.807683	+	0.807683i	0.176600	+	0.984283i	$0.443490\pi$
$110$		0			0
$111$	−3.10634	+	11.5930i	−0.294841	+	1.10036i
$112$		0			0
$113$	−2.67940	−	4.64087i	−0.252057	−	0.436576i	0.712035	−	0.702144i	$-0.247774\pi$
$113$	−2.67940	−	4.64087i	−0.252057	−	0.436576i	−0.964092	+	0.265568i	$0.914440\pi$
$114$		0			0
$115$	0.185459	+	0.692143i	0.0172942	+	0.0645427i
$116$		0			0
$117$	−0.211541	−	1.72020i	−0.0195570	−	0.159033i
$118$		0			0
$119$	−0.775156	−	2.89292i	−0.0710585	−	0.265194i
$120$		0			0
$121$	5.06401	−	2.92371i	0.460365	−	0.265792i
$122$		0			0
$123$	1.24151	−	4.63338i	0.111943	−	0.417778i
$124$		0			0
$125$	−2.01419	−	2.01419i	−0.180155	−	0.180155i
$126$		0			0
$127$	−2.15128	+	3.72613i	−0.190895	+	0.330640i	−0.945547	−	0.325485i	$-0.894472\pi$
$127$	−2.15128	+	3.72613i	−0.190895	+	0.330640i	0.754652	+	0.656125i	$0.227806\pi$
$128$		0			0
$129$			5.64490i			0.497006i
$130$		0			0
$131$	−14.6076			−1.27627			−0.638136	−	0.769924i	$-0.720294\pi$
$131$	−14.6076			−1.27627			−0.638136	+	0.769924i	$0.720294\pi$
$132$		0			0
$133$	−2.12514	−	1.22695i	−0.184273	−	0.106390i
$134$		0			0
$135$	1.12203	−	1.12203i	0.0965690	−	0.0965690i
$136$		0			0
$137$	−15.3326	−	4.10837i	−1.30996	−	0.351002i	−0.464750	−	0.885442i	$-0.653856\pi$
$137$	−15.3326	−	4.10837i	−1.30996	−	0.351002i	−0.845206	+	0.534440i	$0.820522\pi$
$138$		0			0
$139$	4.87379	+	8.44165i	0.413390	+	0.716012i	0.995258	−	0.0972713i	$-0.0310115\pi$
$139$	4.87379	+	8.44165i	0.413390	+	0.716012i	−0.581868	+	0.813283i	$0.697678\pi$
$140$		0			0
$141$	16.2953	−	4.36632i	1.37232	−	0.367711i
$142$		0			0
$143$	1.13752	−	8.10491i	0.0951244	−	0.677767i
$144$		0			0
$145$	0.585265	−	0.156821i	0.0486036	−	0.0130233i
$146$		0			0
$147$	−8.73955	+	5.04578i	−0.720826	+	0.416169i
$148$		0			0
$149$	−7.97144	−	2.13594i	−0.653046	−	0.174983i	−0.0829402	−	0.996555i	$-0.526431\pi$
$149$	−7.97144	−	2.13594i	−0.653046	−	0.174983i	−0.570106	+	0.821571i	$0.693098\pi$
$150$		0			0
$151$	−10.3500	−	10.3500i	−0.842268	−	0.842268i	0.146886	−	0.989153i	$-0.453075\pi$
$151$	−10.3500	−	10.3500i	−0.842268	−	0.842268i	−0.989153	+	0.146886i	$0.953075\pi$
$152$		0			0
$153$	−1.55600	−	0.898357i	−0.125795	−	0.0726279i
$154$		0			0
$155$		−	2.13364i		−	0.171378i
$156$		0			0
$157$			18.5981i			1.48429i	0.670239	+	0.742146i	$0.266192\pi$
$157$			18.5981i			1.48429i	−0.670239	+	0.742146i	$0.733808\pi$
$158$		0			0
$159$	−18.0289	−	10.4090i	−1.42978	−	0.825485i
$160$		0			0
$161$	1.41353	+	1.41353i	0.111402	+	0.111402i
$162$		0			0
$163$	3.93325	+	1.05391i	0.308076	+	0.0825487i	0.409545	−	0.912290i	$-0.365688\pi$
$163$	3.93325	+	1.05391i	0.308076	+	0.0825487i	−0.101469	+	0.994839i	$0.532354\pi$
$164$		0			0
$165$	0.896184	−	0.517412i	0.0697679	−	0.0402805i
$166$		0			0
$167$	21.6215	−	5.79347i	1.67312	−	0.448312i	0.707173	−	0.707040i	$-0.249970\pi$
$167$	21.6215	−	5.79347i	1.67312	−	0.448312i	0.965950	+	0.258728i	$0.0833033\pi$
$168$		0			0
$169$	9.03406	+	9.34803i	0.694928	+	0.719079i
$170$		0			0
$171$	−1.42196	+	0.381012i	−0.108740	+	0.0291367i
$172$		0			0
$173$	7.52545	+	13.0345i	0.572149	+	0.990992i	0.996345	+	0.0854210i	$0.0272235\pi$
$173$	7.52545	+	13.0345i	0.572149	+	0.990992i	−0.424196	+	0.905571i	$0.639443\pi$
$174$		0			0
$175$	−3.80600	−	1.01982i	−0.287707	−	0.0770908i
$176$		0			0
$177$	−7.48412	+	7.48412i	−0.562541	+	0.562541i
$178$		0			0
$179$	14.1062	+	8.14424i	1.05435	+	0.608729i	0.923864	−	0.382721i	$-0.125013\pi$
$179$	14.1062	+	8.14424i	1.05435	+	0.608729i	0.130486	+	0.991450i	$0.458346\pi$
$180$		0			0
$181$	−3.59813			−0.267447			−0.133724	−	0.991019i	$-0.542693\pi$
$181$	−3.59813			−0.267447			−0.133724	+	0.991019i	$0.542693\pi$
$182$		0			0
$183$			6.52134i			0.482071i
$184$		0			0
$185$	−1.08592	+	1.88086i	−0.0798381	+	0.138284i
$186$		0			0
$187$	−5.99943	−	5.99943i	−0.438722	−	0.438722i
$188$		0			0
$189$	1.14573	−	4.27593i	0.0833397	−	0.311028i
$190$		0			0
$191$	−23.1278	+	13.3529i	−1.67347	+	0.966180i	−0.707801	+	0.706411i	$0.750313\pi$
$191$	−23.1278	+	13.3529i	−1.67347	+	0.966180i	−0.965671	+	0.259768i	$0.916354\pi$
$192$		0			0
$193$	2.87004	+	10.7111i	0.206590	+	0.771003i	0.988959	+	0.148189i	$0.0473443\pi$
$193$	2.87004	+	10.7111i	0.206590	+	0.771003i	−0.782369	+	0.622815i	$0.785989\pi$
$194$		0			0
$195$	−0.228455	+	1.62776i	−0.0163600	+	0.116566i
$196$		0			0
$197$	−4.30867	−	16.0802i	−0.306980	−	1.14567i	−0.931227	−	0.364439i	$-0.881261\pi$
$197$	−4.30867	−	16.0802i	−0.306980	−	1.14567i	0.624247	−	0.781227i	$-0.285406\pi$
$198$		0			0
$199$	3.05770	+	5.29609i	0.216755	+	0.375430i	0.953814	−	0.300398i	$-0.0971195\pi$
$199$	3.05770	+	5.29609i	0.216755	+	0.375430i	−0.737059	+	0.675828i	$0.763786\pi$
$200$		0			0
$201$	−4.49961	+	16.7928i	−0.317378	+	1.18447i
$202$		0			0
$203$	1.19526	−	1.19526i	0.0838905	−	0.0838905i
$204$		0			0
$205$	0.434007	−	0.751723i	0.0303124	−	0.0525026i
$206$		0			0
$207$	1.19924			0.0833528
$208$		0			0
$209$	−6.95165			−0.480856
$210$		0			0
$211$	6.47658	−	11.2178i	0.445866	−	0.772263i	−0.552246	−	0.833681i	$-0.686229\pi$
$211$	6.47658	−	11.2178i	0.445866	−	0.772263i	0.998112	+	0.0614182i	$0.0195623\pi$
$212$		0			0
$213$	−8.84646	+	8.84646i	−0.606150	+	0.606150i
$214$		0			0
$215$	−0.264378	+	0.986673i	−0.0180304	+	0.0672905i
$216$		0			0
$217$	−2.97617	−	5.15488i	−0.202036	−	0.349936i
$218$		0			0
$219$	6.00172	+	22.3987i	0.405558	+	1.51356i
$220$		0			0
$221$	13.3760	−	1.64491i	0.899764	−	0.110648i
$222$		0			0
$223$	−1.31334	−	4.90147i	−0.0879480	−	0.328226i	0.907908	−	0.419169i	$-0.137679\pi$
$223$	−1.31334	−	4.90147i	−0.0879480	−	0.328226i	−0.995856	+	0.0909428i	$0.971012\pi$
$224$		0			0
$225$	−2.04711	+	1.18190i	−0.136474	+	0.0787935i
$226$		0			0
$227$	0.177177	−	0.661235i	0.0117597	−	0.0438877i	−0.959797	−	0.280696i	$-0.909435\pi$
$227$	0.177177	−	0.661235i	0.0117597	−	0.0438877i	0.971556	+	0.236808i	$0.0761014\pi$
$228$		0			0
$229$	−3.26943	−	3.26943i	−0.216050	−	0.216050i	0.590782	−	0.806832i	$-0.298820\pi$
$229$	−3.26943	−	3.26943i	−0.216050	−	0.216050i	−0.806832	+	0.590782i	$0.798820\pi$
$230$		0			0
$231$	1.44346	−	2.50014i	0.0949725	−	0.164497i
$232$		0			0
$233$			15.3725i			1.00708i	0.863971	+	0.503542i	$0.167970\pi$
$233$			15.3725i			1.00708i	−0.863971	+	0.503542i	$0.832030\pi$
$234$		0			0
$235$	3.05276			0.199140
$236$		0			0
$237$	−16.6506	−	9.61323i	−1.08157	−	0.624447i
$238$		0			0
$239$	−5.60237	+	5.60237i	−0.362387	+	0.362387i	−0.864691	−	0.502304i	$-0.832486\pi$
$239$	−5.60237	+	5.60237i	−0.362387	+	0.362387i	0.502304	+	0.864691i	$0.332486\pi$
$240$		0			0
$241$	15.3608	+	4.11592i	0.989476	+	0.265129i	0.717031	−	0.697042i	$-0.245501\pi$
$241$	15.3608	+	4.11592i	0.989476	+	0.265129i	0.272446	+	0.962171i	$0.412167\pi$
$242$		0			0
$243$	−2.47229	−	4.28212i	−0.158597	−	0.274698i
$244$		0			0
$245$	−1.76391	+	0.472637i	−0.112692	+	0.0301957i
$246$		0			0
$247$	6.79650	−	8.70248i	0.432451	−	0.553726i
$248$		0			0
$249$	4.50750	−	1.20778i	0.285651	−	0.0765400i
$250$		0			0
$251$	−15.5466	+	8.97582i	−0.981291	+	0.566549i	−0.902660	−	0.430355i	$-0.858388\pi$
$251$	−15.5466	+	8.97582i	−0.981291	+	0.566549i	−0.0786312	+	0.996904i	$0.525055\pi$
$252$		0			0
$253$	5.47010	+	1.46571i	0.343902	+	0.0921483i
$254$		0			0
$255$	1.20490	+	1.20490i	0.0754539	+	0.0754539i
$256$		0			0
$257$	−19.5473	−	11.2857i	−1.21933	−	0.703980i	−0.254555	−	0.967058i	$-0.581929\pi$
$257$	−19.5473	−	11.2857i	−1.21933	−	0.703980i	−0.964774	+	0.263078i	$0.915262\pi$
$258$		0			0
$259$			6.05889i			0.376481i
$260$		0			0
$261$		−	1.01406i		−	0.0627685i
$262$		0			0
$263$	−2.99321	−	1.72813i	−0.184569	−	0.106561i	0.404868	−	0.914375i	$-0.367317\pi$
$263$	−2.99321	−	1.72813i	−0.184569	−	0.106561i	−0.589438	+	0.807814i	$0.700651\pi$
$264$		0			0
$265$	−2.66376	−	2.66376i	−0.163634	−	0.163634i
$266$		0			0
$267$	−2.94836	−	0.790010i	−0.180437	−	0.0483478i
$268$		0			0
$269$	−11.2159	+	6.47548i	−0.683843	+	0.394817i	−0.801301	−	0.598261i	$-0.795859\pi$
$269$	−11.2159	+	6.47548i	−0.683843	+	0.394817i	0.117458	+	0.993078i	$0.462525\pi$
$270$		0			0
$271$	9.57438	−	2.56545i	0.581602	−	0.155840i	0.0439898	−	0.999032i	$-0.485993\pi$
$271$	9.57438	−	2.56545i	0.581602	−	0.155840i	0.537612	+	0.843192i	$0.319326\pi$
$272$		0			0
$273$	1.71858	+	4.25134i	0.104013	+	0.257303i
$274$		0			0
$275$	−10.7820	+	2.88904i	−0.650182	+	0.174216i
$276$		0			0
$277$	4.72824	+	8.18955i	0.284092	+	0.492062i	0.972389	−	0.233367i	$-0.0749744\pi$
$277$	4.72824	+	8.18955i	0.284092	+	0.492062i	−0.688296	+	0.725430i	$0.741641\pi$
$278$		0			0
$279$	−3.44920	−	0.924210i	−0.206498	−	0.0553310i
$280$		0			0
$281$	5.83221	−	5.83221i	0.347921	−	0.347921i	−0.511414	−	0.859334i	$-0.670878\pi$
$281$	5.83221	−	5.83221i	0.347921	−	0.347921i	0.859334	+	0.511414i	$0.170878\pi$
$282$		0			0
$283$	0.217451	+	0.125545i	0.0129261	+	0.00746290i	0.506449	−	0.862270i	$-0.330958\pi$
$283$	0.217451	+	0.125545i	0.0129261	+	0.00746290i	−0.493523	+	0.869733i	$0.664291\pi$
$284$		0			0
$285$	1.39614			0.0827004
$286$		0			0
$287$		−	2.42155i		−	0.142940i
$288$		0			0
$289$	−1.51454	+	2.62325i	−0.0890904	+	0.154309i
$290$		0			0
$291$	6.50730	+	6.50730i	0.381464	+	0.381464i
$292$		0			0
$293$	5.41972	−	20.2267i	0.316624	−	1.18166i	−0.605845	−	0.795583i	$-0.707165\pi$
$293$	5.41972	−	20.2267i	0.316624	−	1.18166i	0.922468	−	0.386072i	$-0.126169\pi$
$294$		0			0
$295$	−1.65867	+	0.957632i	−0.0965714	+	0.0557555i
$296$		0			0
$297$	−3.24575	−	12.1133i	−0.188337	−	0.702885i
$298$		0			0
$299$	−7.18287	+	5.41479i	−0.415396	+	0.313145i
$300$		0			0
$301$	0.737552	+	2.75258i	0.0425118	+	0.158656i
$302$		0			0
$303$	6.45639	+	11.1828i	0.370910	+	0.642435i
$304$		0			0
$305$	−0.305426	+	1.13987i	−0.0174886	+	0.0652685i
$306$		0			0
$307$	−10.5407	+	10.5407i	−0.601588	+	0.601588i	−0.940734	−	0.339146i	$-0.889862\pi$
$307$	−10.5407	+	10.5407i	−0.601588	+	0.601588i	0.339146	+	0.940734i	$0.389862\pi$
$308$		0			0
$309$	−13.8168	+	23.9313i	−0.786008	+	1.36141i
$310$		0			0
$311$	1.71529			0.0972653			0.0486327	−	0.998817i	$-0.484514\pi$
$311$	1.71529			0.0972653			0.0486327	+	0.998817i	$0.484514\pi$
$312$		0			0
$313$	−8.25608			−0.466661			−0.233331	−	0.972397i	$-0.574962\pi$
$313$	−8.25608			−0.466661			−0.233331	+	0.972397i	$0.574962\pi$
$314$		0			0
$315$	0.0553134	−	0.0958057i	0.00311656	−	0.00539804i
$316$		0			0
$317$	9.53707	−	9.53707i	0.535655	−	0.535655i	−0.386595	−	0.922250i	$-0.626349\pi$
$317$	9.53707	−	9.53707i	0.535655	−	0.535655i	0.922250	+	0.386595i	$0.126349\pi$
$318$		0			0
$319$	1.23938	−	4.62543i	0.0693919	−	0.258974i
$320$		0			0
$321$	1.16971	+	2.02599i	0.0652867	+	0.113080i
$322$		0			0
$323$	−2.96268	−	11.0569i	−0.164848	−	0.615220i
$324$		0			0
$325$	6.92473	−	16.3221i	0.384115	−	0.905389i
$326$		0			0
$327$	−4.89899	−	18.2833i	−0.270915	−	1.01107i
$328$		0			0
$329$	7.37548	−	4.25823i	0.406623	−	0.234764i
$330$		0			0
$331$	5.08215	−	18.9669i	0.279340	−	1.04251i	−0.673536	−	0.739154i	$-0.735225\pi$
$331$	5.08215	−	18.9669i	0.279340	−	1.04251i	0.952877	−	0.303358i	$-0.0981079\pi$
$332$		0			0
$333$	2.57018	+	2.57018i	0.140845	+	0.140845i
$334$		0			0
$335$	−1.57298	+	2.72447i	−0.0859408	+	0.148854i
$336$		0			0
$337$		−	34.2794i		−	1.86732i	−0.358165	−	0.933658i	$-0.616598\pi$
$337$		−	34.2794i		−	1.86732i	0.358165	−	0.933658i	$-0.383402\pi$
$338$		0			0
$339$	8.50568			0.461965
$340$		0			0
$341$	−14.6033	−	8.43122i	−0.790813	−	0.456576i
$342$		0			0
$343$	−7.56843	+	7.56843i	−0.408657	+	0.408657i
$344$		0			0
$345$	−1.09859	−	0.294367i	−0.0591462	−	0.0158482i
$346$		0			0
$347$	−10.3503	−	17.9272i	−0.555633	−	0.962384i	−0.997854	−	0.0654782i	$-0.979143\pi$
$347$	−10.3503	−	17.9272i	−0.555633	−	0.962384i	0.442221	−	0.896906i	$-0.354191\pi$
$348$		0			0
$349$	14.7302	−	3.94695i	0.788490	−	0.211275i	0.157966	−	0.987445i	$-0.449506\pi$
$349$	14.7302	−	3.94695i	0.788490	−	0.211275i	0.630525	+	0.776169i	$0.282840\pi$
$350$		0			0
$351$	18.3374	+	7.77972i	0.978780	+	0.415251i
$352$		0			0
$353$	26.0283	−	6.97426i	1.38535	−	0.371202i	0.512286	−	0.858815i	$-0.328799\pi$
$353$	26.0283	−	6.97426i	1.38535	−	0.371202i	0.873060	+	0.487612i	$0.162132\pi$
$354$		0			0
$355$	−1.96060	+	1.13195i	−0.104058	+	0.0600777i
$356$		0			0
$357$	4.59174	+	1.23035i	0.243021	+	0.0651172i
$358$		0			0
$359$	17.2783	+	17.2783i	0.911913	+	0.911913i	0.996423	−	0.0845098i	$-0.0269324\pi$
$359$	17.2783	+	17.2783i	0.911913	+	0.911913i	−0.0845098	+	0.996423i	$0.526932\pi$
$360$		0			0
$361$	8.33213	+	4.81056i	0.438533	+	0.253187i
$362$		0			0
$363$			9.28121i			0.487137i
$364$		0			0
$365$			4.19616i			0.219637i
$366$		0			0
$367$	0.905546	+	0.522817i	0.0472691	+	0.0272908i	0.523448	−	0.852057i	$-0.324645\pi$
$367$	0.905546	+	0.522817i	0.0472691	+	0.0272908i	−0.476179	+	0.879348i	$0.657979\pi$
$368$		0			0
$369$	−1.02722	−	1.02722i	−0.0534751	−	0.0534751i
$370$		0			0
$371$	−10.1513	−	2.72003i	−0.527029	−	0.141217i
$372$		0			0
$373$	−24.2536	+	14.0028i	−1.25580	+	0.725039i	−0.972256	−	0.233919i	$-0.924845\pi$
$373$	−24.2536	+	14.0028i	−1.25580	+	0.725039i	−0.283548	+	0.958958i	$0.591512\pi$
$374$		0			0
$375$	4.36717	−	1.17018i	0.225520	−	0.0604279i
$376$		0			0
$377$	4.57866	+	6.07371i	0.235813	+	0.312812i
$378$		0			0
$379$	24.2227	−	6.49045i	1.24424	−	0.333392i	0.424129	−	0.905602i	$-0.360580\pi$
$379$	24.2227	−	6.49045i	1.24424	−	0.333392i	0.820108	+	0.572209i	$0.193914\pi$
$380$		0			0
$381$	−3.41458	−	5.91423i	−0.174934	−	0.302995i
$382$		0			0
$383$	−18.0059	−	4.82468i	−0.920061	−	0.246530i	−0.232449	−	0.972608i	$-0.574674\pi$
$383$	−18.0059	−	4.82468i	−0.920061	−	0.246530i	−0.687611	+	0.726079i	$0.741341\pi$
$384$		0			0
$385$	0.369395	−	0.369395i	0.0188261	−	0.0188261i
$386$		0			0
$387$	1.48052	+	0.854776i	0.0752588	+	0.0434507i
$388$		0			0
$389$	27.5165			1.39514			0.697572	−	0.716515i	$-0.254264\pi$
$389$	27.5165			1.39514			0.697572	+	0.716515i	$0.254264\pi$
$390$		0			0
$391$			9.32505i			0.471588i
$392$		0			0
$393$	11.5928	−	20.0794i	0.584781	−	1.01287i
$394$		0			0
$395$	−2.46013	−	2.46013i	−0.123782	−	0.123782i
$396$		0			0
$397$	−3.63406	+	13.5625i	−0.182388	+	0.680681i	0.812787	+	0.582562i	$0.197950\pi$
$397$	−3.63406	+	13.5625i	−0.182388	+	0.680681i	−0.995175	+	0.0981198i	$0.968717\pi$
$398$		0			0
$399$	3.37309	−	1.94745i	0.168865	−	0.0974945i
$400$		0			0
$401$	2.77041	+	10.3393i	0.138348	+	0.516321i	0.999962	+	0.00875651i	$0.00278732\pi$
$401$	2.77041	+	10.3393i	0.138348	+	0.516321i	−0.861614	+	0.507564i	$0.830546\pi$
$402$		0			0
$403$	24.8320	−	10.0382i	1.23697	−	0.500039i
$404$		0			0
$405$	0.544663	+	2.03271i	0.0270645	+	0.101006i
$406$		0			0
$407$	8.58213	+	14.8647i	0.425401	+	0.736815i
$408$		0			0
$409$	1.81350	−	6.76808i	0.0896718	−	0.334660i	−0.906486	−	0.422236i	$-0.861245\pi$
$409$	1.81350	−	6.76808i	0.0896718	−	0.334660i	0.996158	+	0.0875760i	$0.0279121\pi$
$410$		0			0
$411$	17.8155	−	17.8155i	0.878775	−	0.878775i
$412$		0			0
$413$	−2.67156	+	4.62729i	−0.131459	+	0.227694i
$414$		0			0
$415$	0.844432			0.0414516
$416$		0			0
$417$	−15.4717			−0.757652
$418$		0			0
$419$	11.8842	−	20.5840i	0.580580	−	1.00559i	−0.414830	−	0.909899i	$-0.636159\pi$
$419$	11.8842	−	20.5840i	0.580580	−	1.00559i	0.995411	−	0.0956958i	$-0.0305076\pi$
$420$		0			0
$421$	10.0457	−	10.0457i	0.489596	−	0.489596i	−0.418583	−	0.908179i	$-0.637473\pi$
$421$	10.0457	−	10.0457i	0.489596	−	0.489596i	0.908179	+	0.418583i	$0.137473\pi$
$422$		0			0
$423$	1.32234	−	4.93503i	0.0642942	−	0.239949i
$424$		0			0
$425$	−9.19025	−	15.9180i	−0.445793	−	0.772136i
$426$		0			0
$427$	0.852066	+	3.17995i	0.0412343	+	0.153889i
$428$		0			0
$429$	10.2381	+	7.99581i	0.494302	+	0.386041i
$430$		0			0
$431$	−1.94441	−	7.25665i	−0.0936591	−	0.349540i	0.903154	−	0.429317i	$-0.141246\pi$
$431$	−1.94441	−	7.25665i	−0.0936591	−	0.349540i	−0.996813	+	0.0797770i	$0.974579\pi$
$432$		0			0
$433$	−11.3356	+	6.54458i	−0.544752	+	0.314513i	−0.747003	−	0.664821i	$-0.768508\pi$
$433$	−11.3356	+	6.54458i	−0.544752	+	0.314513i	0.202251	+	0.979334i	$0.435174\pi$
$434$		0			0
$435$	−0.248912	+	0.928952i	−0.0119344	+	0.0445399i
$436$		0			0
$437$	5.40256	+	5.40256i	0.258439	+	0.258439i
$438$		0			0
$439$	0.725737	−	1.25701i	0.0346376	−	0.0599940i	−0.848187	−	0.529697i	$-0.822306\pi$
$439$	0.725737	−	1.25701i	0.0346376	−	0.0599940i	0.882825	+	0.469703i	$0.155639\pi$
$440$		0			0
$441$			3.05622i			0.145534i
$442$		0			0
$443$	18.0569			0.857909			0.428954	−	0.903326i	$-0.358882\pi$
$443$	18.0569			0.857909			0.428954	+	0.903326i	$0.358882\pi$
$444$		0			0
$445$	−0.478343	−	0.276172i	−0.0226757	−	0.0130918i
$446$		0			0
$447$	9.26230	−	9.26230i	0.438092	−	0.438092i
$448$		0			0
$449$	33.1966	+	8.89499i	1.56664	+	0.419780i	0.934759	−	0.355283i	$-0.115616\pi$
$449$	33.1966	+	8.89499i	1.56664	+	0.419780i	0.631883	+	0.775064i	$0.282282\pi$
$450$		0			0
$451$	−3.43001	−	5.94096i	−0.161513	−	0.279749i
$452$		0			0
$453$	22.4408	−	6.01299i	1.05436	−	0.282515i
$454$		0			0
$455$	0.101280	+	0.823581i	0.00474807	+	0.0386101i
$456$		0			0
$457$	−21.1085	+	5.65600i	−0.987414	+	0.264577i	−0.716164	−	0.697932i	$-0.754104\pi$
$457$	−21.1085	+	5.65600i	−0.987414	+	0.264577i	−0.271250	+	0.962509i	$0.587437\pi$
$458$		0			0
$459$	17.8834	−	10.3250i	0.834724	−	0.481928i
$460$		0			0
$461$	−15.3365	−	4.10940i	−0.714292	−	0.191394i	−0.116669	−	0.993171i	$-0.537222\pi$
$461$	−15.3365	−	4.10940i	−0.714292	−	0.191394i	−0.597623	+	0.801777i	$0.703888\pi$
$462$		0			0
$463$	3.31898	+	3.31898i	0.154246	+	0.154246i	0.780011	−	0.625765i	$-0.215213\pi$
$463$	3.31898	+	3.31898i	0.154246	+	0.154246i	−0.625765	+	0.780011i	$0.715213\pi$
$464$		0			0
$465$	2.93287	+	1.69329i	0.136009	+	0.0785246i
$466$		0			0
$467$			4.31326i			0.199594i	0.995008	+	0.0997968i	$0.0318193\pi$
$467$			4.31326i			0.199594i	−0.995008	+	0.0997968i	$0.968181\pi$
$468$		0			0
$469$			8.77645i			0.405259i
$470$		0			0
$471$	−25.5647	−	14.7598i	−1.17796	−	0.680094i
$472$		0			0
$473$	5.70839	+	5.70839i	0.262472	+	0.262472i
$474$		0			0
$475$	−14.5467	−	3.89777i	−0.667448	−	0.178842i
$476$		0			0
$477$	−5.46002	+	3.15235i	−0.249997	+	0.144336i
$478$		0			0
$479$	14.5501	−	3.89868i	0.664810	−	0.178135i	0.0893943	−	0.995996i	$-0.471507\pi$
$479$	14.5501	−	3.89868i	0.664810	−	0.178135i	0.575416	+	0.817861i	$0.304840\pi$
$480$		0			0
$481$	−26.9991	−	3.78931i	−1.23105	−	0.172778i
$482$		0			0
$483$	−3.06481	+	0.821214i	−0.139454	+	0.0373665i
$484$		0			0
$485$	0.832642	+	1.44218i	0.0378083	+	0.0654860i
$486$		0			0
$487$	−17.2756	−	4.62898i	−0.782831	−	0.209759i	−0.154798	−	0.987946i	$-0.549473\pi$
$487$	−17.2756	−	4.62898i	−0.782831	−	0.209759i	−0.628032	+	0.778187i	$0.716139\pi$
$488$		0			0
$489$	−4.57018	+	4.57018i	−0.206671	+	0.206671i
$490$		0			0
$491$	−15.4867	−	8.94124i	−0.698904	−	0.403512i	0.108035	−	0.994147i	$-0.465544\pi$
$491$	−15.4867	−	8.94124i	−0.698904	−	0.403512i	−0.806939	+	0.590635i	$0.798877\pi$
$492$		0			0
$493$	7.88512			0.355128
$494$		0			0
$495$		−	0.313395i		−	0.0140861i
$496$		0			0
$497$	−3.15787	+	5.46959i	−0.141650	+	0.245345i
$498$		0			0
$499$	2.29278	+	2.29278i	0.102639	+	0.102639i	0.756561	−	0.653923i	$-0.226878\pi$
$499$	2.29278	+	2.29278i	0.102639	+	0.102639i	−0.653923	+	0.756561i	$0.726878\pi$
$500$		0			0
$501$	−9.19559	+	34.3184i	−0.410828	+	1.53323i
$502$		0			0
$503$	−17.2984	+	9.98723i	−0.771297	+	0.445309i	−0.833337	−	0.552765i	$-0.813573\pi$
$503$	−17.2984	+	9.98723i	−0.771297	+	0.445309i	0.0620400	+	0.998074i	$0.480239\pi$
$504$		0			0
$505$	0.604768	+	2.25702i	0.0269118	+	0.100436i
$506$		0			0
$507$	−20.0192	+	4.99933i	−0.889086	+	0.222028i
$508$		0			0
$509$	−0.745854	−	2.78356i	−0.0330594	−	0.123379i	0.947424	−	0.319980i	$-0.103676\pi$
$509$	−0.745854	−	2.78356i	−0.0330594	−	0.123379i	−0.980484	+	0.196601i	$0.937010\pi$
$510$		0			0
$511$	5.85314	+	10.1379i	0.258928	+	0.448476i
$512$		0			0
$513$	4.37903	−	16.3428i	0.193339	−	0.721551i
$514$		0			0
$515$	−3.53585	+	3.53585i	−0.155808	+	0.155808i
$516$		0			0
$517$	12.0632	−	20.8940i	0.530538	−	0.918918i
$518$		0			0
$519$	−23.8893			−1.04862
$520$		0			0
$521$	24.0994			1.05581			0.527906	−	0.849303i	$-0.322977\pi$
$521$	24.0994			1.05581			0.527906	+	0.849303i	$0.322977\pi$
$522$		0			0
$523$	−17.5798	+	30.4491i	−0.768712	+	1.33145i	0.169550	+	0.985522i	$0.445769\pi$
$523$	−17.5798	+	30.4491i	−0.768712	+	1.33145i	−0.938262	+	0.345926i	$0.887565\pi$
$524$		0			0
$525$	4.42233	−	4.42233i	0.193006	−	0.193006i
$526$		0			0
$527$	7.18649	−	26.8203i	0.313048	−	1.16831i
$528$		0			0
$529$	8.38794	+	14.5283i	0.364693	+	0.631667i
$530$		0			0
$531$	0.829618	+	3.09618i	0.0360023	+	0.134363i
$532$		0			0
$533$	10.7907	+	1.51447i	0.467397	+	0.0655990i
$534$		0			0
$535$	0.109566	+	0.408906i	0.00473696	+	0.0176786i
$536$		0			0
$537$	−22.3899	+	12.9268i	−0.966195	+	0.557833i
$538$		0			0
$539$	−3.73531	+	13.9404i	−0.160891	+	0.600454i
$540$		0			0
$541$	−11.1945	−	11.1945i	−0.481288	−	0.481288i	0.424255	−	0.905543i	$-0.360536\pi$
$541$	−11.1945	−	11.1945i	−0.481288	−	0.481288i	−0.905543	+	0.424255i	$0.860536\pi$
$542$		0			0
$543$	2.85554	−	4.94594i	0.122543	−	0.212250i
$544$		0			0
$545$		−	3.42518i		−	0.146718i
$546$		0			0
$547$	−21.0720			−0.900973			−0.450486	−	0.892783i	$-0.648749\pi$
$547$	−21.0720			−0.900973			−0.450486	+	0.892783i	$0.648749\pi$
$548$		0			0
$549$	1.71038	+	0.987490i	0.0729974	+	0.0421451i
$550$		0			0
$551$	4.56832	−	4.56832i	0.194617	−	0.194617i
$552$		0			0
$553$	−9.37525	−	2.51209i	−0.398676	−	0.106825i
$554$		0			0
$555$	−1.72360	−	2.98537i	−0.0731628	−	0.126722i
$556$		0			0
$557$	−35.3694	+	9.47720i	−1.49865	+	0.401562i	−0.912647	−	0.408748i	$-0.865965\pi$
$557$	−35.3694	+	9.47720i	−1.49865	+	0.401562i	−0.586002	+	0.810310i	$0.699299\pi$
$558$		0			0
$559$	−12.7271	+	1.56511i	−0.538297	+	0.0661970i
$560$		0			0
$561$	13.0080	−	3.48548i	0.549197	−	0.147157i
$562$		0			0
$563$	7.36457	−	4.25194i	0.310380	−	0.179198i	−0.336717	−	0.941606i	$-0.609316\pi$
$563$	7.36457	−	4.25194i	0.310380	−	0.179198i	0.647096	+	0.762408i	$0.275983\pi$
$564$		0			0
$565$	1.48671	+	0.398362i	0.0625463	+	0.0167592i
$566$		0			0
$567$	4.15129	+	4.15129i	0.174338	+	0.174338i
$568$		0			0
$569$	−2.15664	−	1.24514i	−0.0904110	−	0.0521988i	0.454113	−	0.890944i	$-0.349956\pi$
$569$	−2.15664	−	1.24514i	−0.0904110	−	0.0521988i	−0.544524	+	0.838745i	$0.683290\pi$
$570$		0			0
$571$			32.2266i			1.34864i	0.738439	+	0.674320i	$0.235563\pi$
$571$			32.2266i			1.34864i	−0.738439	+	0.674320i	$0.764437\pi$
$572$		0			0
$573$		−	42.3882i		−	1.77079i
$574$		0			0
$575$	10.6246	+	6.13414i	0.443078	+	0.255811i
$576$		0			0
$577$	−3.92020	−	3.92020i	−0.163200	−	0.163200i	0.620783	−	0.783983i	$-0.286815\pi$
$577$	−3.92020	−	3.92020i	−0.163200	−	0.163200i	−0.783983	+	0.620783i	$0.786815\pi$
$578$		0			0
$579$	−17.0010	−	4.55542i	−0.706539	−	0.189317i
$580$		0			0
$581$	2.04015	−	1.17788i	0.0846397	−	0.0488668i
$582$		0			0
$583$	−28.7577	+	7.70559i	−1.19102	+	0.319133i
$584$		0			0
$585$	0.392326	+	0.306401i	0.0162207	+	0.0126681i
$586$		0			0
$587$	−20.1329	+	5.39459i	−0.830972	+	0.222658i	−0.649138	−	0.760671i	$-0.724870\pi$
$587$	−20.1329	+	5.39459i	−0.830972	+	0.222658i	−0.181835	+	0.983329i	$0.558204\pi$
$588$		0			0
$589$	−11.3751	−	19.7022i	−0.468701	−	0.811814i
$590$		0			0
$591$	25.5230	+	6.83887i	1.04988	+	0.281313i
$592$		0			0
$593$	−26.9784	+	26.9784i	−1.10787	+	1.10787i	−0.114441	+	0.993430i	$0.536508\pi$
$593$	−26.9784	+	26.9784i	−1.10787	+	1.10787i	−0.993430	+	0.114441i	$0.963492\pi$
$594$		0			0
$595$	0.744967	+	0.430107i	0.0305407	+	0.0176327i
$596$		0			0
$597$	−9.70656			−0.397263
$598$		0			0
$599$			8.59618i			0.351230i	0.984459	+	0.175615i	$0.0561914\pi$
$599$			8.59618i			0.351230i	−0.984459	+	0.175615i	$0.943809\pi$
$600$		0			0
$601$	−3.37794	+	5.85076i	−0.137789	+	0.238658i	−0.926659	−	0.375902i	$-0.877333\pi$
$601$	−3.37794	+	5.85076i	−0.137789	+	0.238658i	0.788870	+	0.614560i	$0.210666\pi$
$602$		0			0
$603$	3.72297	+	3.72297i	0.151611	+	0.151611i
$604$		0			0
$605$	−0.434684	+	1.62226i	−0.0176724	+	0.0659544i
$606$		0			0
$607$	38.1905	−	22.0493i	1.55010	−	0.894953i	0.551971	−	0.833863i	$-0.313876\pi$
$607$	38.1905	−	22.0493i	1.55010	−	0.894953i	0.998132	−	0.0610892i	$-0.0194574\pi$
$608$		0			0
$609$	0.694405	+	2.59155i	0.0281387	+	0.105015i
$610$		0			0
$611$	14.3624	+	35.5291i	0.581041	+	1.43735i
$612$		0			0
$613$	2.58306	+	9.64013i	0.104329	+	0.389361i	0.998268	−	0.0588272i	$-0.0187361\pi$
$613$	2.58306	+	9.64013i	0.104329	+	0.389361i	−0.893939	+	0.448188i	$0.852069\pi$
$614$		0			0
$615$	0.688871	+	1.19316i	0.0277779	+	0.0481128i
$616$		0			0
$617$	3.14881	−	11.7515i	0.126766	−	0.473099i	−0.873130	−	0.487487i	$-0.837913\pi$
$617$	3.14881	−	11.7515i	0.126766	−	0.473099i	0.999896	+	0.0143885i	$0.00458015\pi$
$618$		0			0
$619$	11.9483	−	11.9483i	0.480243	−	0.480243i	−0.424966	−	0.905209i	$-0.639714\pi$
$619$	11.9483	−	11.9483i	0.480243	−	0.480243i	0.905209	+	0.424966i	$0.139714\pi$
$620$		0			0
$621$	−6.89152	+	11.9365i	−0.276547	+	0.478994i
$622$		0			0
$623$	−1.54091			−0.0617351
$624$		0			0
$625$	−23.7694			−0.950775
$626$		0			0
$627$	5.51695	−	9.55563i	0.220326	−	0.381615i
$628$		0			0
$629$	−19.9853	+	19.9853i	−0.796866	+	0.796866i
$630$		0			0
$631$	4.96268	−	18.5210i	0.197561	−	0.737309i	−0.794028	−	0.607882i	$-0.792020\pi$
$631$	4.96268	−	18.5210i	0.197561	−	0.737309i	0.991589	−	0.129427i	$-0.0413138\pi$
$632$		0			0
$633$	10.2798	+	17.8052i	0.408587	+	0.707694i
$634$		0			0
$635$	−0.319843	−	1.19367i	−0.0126926	−	0.0473693i
$636$		0			0
$637$	−13.7994	−	18.3053i	−0.546753	−	0.725282i
$638$		0			0
$639$	0.980634	+	3.65977i	0.0387933	+	0.144778i
$640$		0			0
$641$	−3.97192	+	2.29319i	−0.156881	+	0.0905755i	−0.576386	−	0.817178i	$-0.695537\pi$
$641$	−3.97192	+	2.29319i	−0.156881	+	0.0905755i	0.419504	+	0.907753i	$0.362204\pi$
$642$		0			0
$643$	−10.7606	+	40.1589i	−0.424355	+	1.58371i	0.340973	+	0.940073i	$0.389244\pi$
$643$	−10.7606	+	40.1589i	−0.424355	+	1.58371i	−0.765328	+	0.643641i	$0.777423\pi$
$644$		0			0
$645$	−1.14645	−	1.14645i	−0.0451414	−	0.0451414i
$646$		0			0
$647$	−20.7662	+	35.9681i	−0.816402	+	1.41405i	0.0919140	+	0.995767i	$0.470702\pi$
$647$	−20.7662	+	35.9681i	−0.816402	+	1.41405i	−0.908316	+	0.418284i	$0.862632\pi$
$648$		0			0
$649$			15.1366i			0.594163i
$650$		0			0
$651$	9.44776			0.370287
$652$		0			0
$653$	4.70452	+	2.71615i	0.184102	+	0.106291i	0.589218	−	0.807974i	$-0.299436\pi$
$653$	4.70452	+	2.71615i	0.184102	+	0.106291i	−0.405117	+	0.914265i	$0.632769\pi$
$654$		0			0
$655$	2.96672	−	2.96672i	0.115920	−	0.115920i
$656$		0			0
$657$	6.78342	+	1.81761i	0.264647	+	0.0709118i
$658$		0			0
$659$	7.46965	+	12.9378i	0.290976	+	0.503986i	0.974041	−	0.226372i	$-0.0726866\pi$
$659$	7.46965	+	12.9378i	0.290976	+	0.503986i	−0.683065	+	0.730358i	$0.739353\pi$
$660$		0			0
$661$	−14.3738	+	3.85145i	−0.559076	+	0.149804i	−0.527282	−	0.849690i	$-0.676789\pi$
$661$	−14.3738	+	3.85145i	−0.559076	+	0.149804i	−0.0317943	+	0.999494i	$0.510122\pi$
$662$		0			0
$663$	−8.35432	+	19.6918i	−0.324455	+	0.764766i
$664$		0			0
$665$	0.680790	−	0.182417i	0.0263999	−	0.00707384i
$666$		0			0
$667$	−4.55790	+	2.63151i	−0.176483	+	0.101892i
$668$		0			0
$669$	7.77977	+	2.08458i	0.300783	+	0.0805946i
$670$		0			0
$671$	6.59468	+	6.59468i	0.254585	+	0.254585i
$672$		0			0
$673$	19.6041	+	11.3184i	0.755682	+	0.436293i	0.827743	−	0.561107i	$-0.189624\pi$
$673$	19.6041	+	11.3184i	0.755682	+	0.436293i	−0.0720614	+	0.997400i	$0.522958\pi$
$674$		0			0
$675$		−	27.1676i		−	1.04568i
$676$		0			0
$677$		−	28.1241i		−	1.08090i	−0.841377	−	0.540448i	$-0.818255\pi$
$677$		−	28.1241i		−	1.08090i	0.841377	−	0.540448i	$-0.181745\pi$
$678$		0			0
$679$	4.02333	+	2.32287i	0.154401	+	0.0891437i
$680$		0			0
$681$	0.768312	+	0.768312i	0.0294418	+	0.0294418i
$682$		0			0
$683$	43.1843	+	11.5712i	1.65240	+	0.442760i	0.960284	−	0.279025i	$-0.0900112\pi$
$683$	43.1843	+	11.5712i	1.65240	+	0.442760i	0.692118	+	0.721785i	$0.256678\pi$
$684$		0			0
$685$	3.94837	−	2.27959i	0.150859	−	0.0870986i
$686$		0			0
$687$	7.08878	−	1.89943i	0.270454	−	0.0724679i
$688$		0			0
$689$	18.4695	−	43.5341i	0.703632	−	1.65852i
$690$		0			0
$691$	−18.4100	+	4.93295i	−0.700350	+	0.187658i	−0.591387	−	0.806388i	$-0.701420\pi$
$691$	−18.4100	+	4.93295i	−0.700350	+	0.187658i	−0.108963	+	0.994046i	$0.534753\pi$
$692$		0			0
$693$	−0.437149	−	0.757164i	−0.0166059	−	0.0287623i
$694$		0			0
$695$	−2.70430	−	0.724614i	−0.102580	−	0.0274862i
$696$		0			0
$697$	7.98750	−	7.98750i	0.302548	−	0.302548i
$698$		0			0
$699$	−21.1307	−	12.1998i	−0.799238	−	0.461440i
$700$		0			0
$701$	8.11837			0.306627			0.153313	−	0.988178i	$-0.451006\pi$
$701$	8.11837			0.306627			0.153313	+	0.988178i	$0.451006\pi$
$702$		0			0
$703$			23.1573i			0.873395i
$704$		0			0
$705$	−2.42272	+	4.19627i	−0.0912449	+	0.158041i
$706$		0			0
$707$	4.60940	+	4.60940i	0.173354	+	0.173354i
$708$		0			0
$709$	0.488166	−	1.82186i	0.0183335	−	0.0684215i	−0.956153	−	0.292867i	$-0.905391\pi$
$709$	0.488166	−	1.82186i	0.0183335	−	0.0684215i	0.974487	+	0.224446i	$0.0720572\pi$
$710$		0			0
$711$	−5.04262	+	2.91136i	−0.189113	+	0.109184i
$712$		0			0
$713$	4.79670	+	17.9015i	0.179638	+	0.670418i
$714$		0			0
$715$	1.41504	+	1.87709i	0.0529195	+	0.0701992i
$716$		0			0
$717$	−3.25480	−	12.1471i	−0.121553	−	0.453640i
$718$		0			0
$719$	−17.1820	−	29.7601i	−0.640780	−	1.10986i	−0.985259	−	0.171069i	$-0.945278\pi$
$719$	−17.1820	−	29.7601i	−0.640780	−	1.10986i	0.344479	−	0.938794i	$-0.388056\pi$
$720$		0			0
$721$	−3.61054	+	13.4747i	−0.134464	+	0.501825i
$722$		0			0
$723$	−17.8483	+	17.8483i	−0.663784	+	0.663784i
$724$		0			0
$725$	5.18693	−	8.98402i	0.192638	−	0.333658i
$726$		0			0
$727$	−20.4072			−0.756861			−0.378430	−	0.925630i	$-0.623536\pi$
$727$	−20.4072			−0.756861			−0.378430	+	0.925630i	$0.623536\pi$
$728$		0			0
$729$	29.8288			1.10477
$730$		0			0
$731$	−6.64658	+	11.5122i	−0.245833	+	0.425795i
$732$		0			0
$733$	−16.3864	+	16.3864i	−0.605247	+	0.605247i	−0.941700	−	0.336453i	$-0.890773\pi$
$733$	−16.3864	+	16.3864i	−0.605247	+	0.605247i	0.336453	+	0.941700i	$0.390773\pi$
$734$		0			0
$735$	0.750185	−	2.79973i	0.0276710	−	0.103270i
$736$		0			0
$737$	12.4314	+	21.5319i	0.457917	+	0.793136i
$738$		0			0
$739$	−10.5673	−	39.4378i	−0.388725	−	1.45074i	−0.832211	−	0.554460i	$-0.812925\pi$
$739$	−10.5673	−	39.4378i	−0.388725	−	1.45074i	0.443486	−	0.896282i	$-0.353742\pi$
$740$		0			0
$741$	6.56848	+	16.2488i	0.241299	+	0.596914i
$742$		0			0
$743$	1.61384	+	6.02293i	0.0592060	+	0.220960i	0.989190	−	0.146641i	$-0.0468461\pi$
$743$	1.61384	+	6.02293i	0.0592060	+	0.220960i	−0.929984	+	0.367600i	$0.880179\pi$
$744$		0			0
$745$	2.05276	−	1.18516i	0.0752071	−	0.0434209i
$746$		0			0
$747$	0.365775	−	1.36509i	0.0133830	−	0.0499461i
$748$		0			0
$749$	0.835088	+	0.835088i	0.0305134	+	0.0305134i
$750$		0			0
$751$	15.2301	−	26.3794i	0.555755	−	0.962597i	−0.442089	−	0.896971i	$-0.645762\pi$
$751$	15.2301	−	26.3794i	0.555755	−	0.962597i	0.997844	−	0.0656254i	$-0.0209042\pi$
$752$		0			0
$753$		−	28.4934i		−	1.03836i
$754$		0			0
$755$	4.20404			0.153001
$756$		0			0
$757$	29.4578	+	17.0074i	1.07066	+	0.618146i	0.928362	−	0.371678i	$-0.121217\pi$
$757$	29.4578	+	17.0074i	1.07066	+	0.618146i	0.142299	+	0.989824i	$0.454551\pi$
$758$		0			0
$759$	−6.35590	+	6.35590i	−0.230705	+	0.230705i
$760$		0			0
$761$	−29.3342	−	7.86008i	−1.06336	−	0.284928i	−0.315600	−	0.948892i	$-0.602206\pi$
$761$	−29.3342	−	7.86008i	−1.06336	−	0.284928i	−0.747764	+	0.663965i	$0.768872\pi$
$762$		0			0
$763$	−4.77771	−	8.27524i	−0.172965	−	0.299584i
$764$		0			0
$765$	0.498467	−	0.133564i	0.0180221	−	0.00482901i
$766$		0			0
$767$	−18.9488	−	14.7987i	−0.684203	−	0.534351i
$768$		0			0
$769$	−14.4158	+	3.86269i	−0.519846	+	0.139292i	−0.509195	−	0.860651i	$-0.670057\pi$
$769$	−14.4158	+	3.86269i	−0.519846	+	0.139292i	−0.0106506	+	0.999943i	$0.503390\pi$
$770$		0			0
$771$	31.0262	−	17.9130i	1.11738	−	0.645120i
$772$		0			0
$773$	8.50020	+	2.27762i	0.305731	+	0.0819204i	0.408423	−	0.912793i	$-0.366079\pi$
$773$	8.50020	+	2.27762i	0.305731	+	0.0819204i	−0.102692	+	0.994713i	$0.532746\pi$
$774$		0			0
$775$	−25.8308	−	25.8308i	−0.927869	−	0.927869i
$776$		0			0
$777$	−8.32846	−	4.80844i	−0.298782	−	0.172502i
$778$		0			0
$779$		−	9.25527i		−	0.331604i
$780$		0			0
$781$			17.8919i			0.640223i
$782$		0			0
$783$	10.0933	+	5.82736i	0.360704	+	0.208253i
$784$		0			0
$785$	−3.77718	−	3.77718i	−0.134813	−	0.134813i
$786$		0			0
$787$	−22.2999	−	5.97524i	−0.794906	−	0.212994i	−0.161560	−	0.986863i	$-0.551652\pi$
$787$	−22.2999	−	5.97524i	−0.794906	−	0.212994i	−0.633346	+	0.773869i	$0.718319\pi$
$788$		0			0
$789$	4.75093	−	2.74295i	0.169138	−	0.0976516i
$790$		0			0
$791$	4.14756	−	1.11134i	0.147470	−	0.0395145i
$792$		0			0
$793$	−14.7031	+	1.80811i	−0.522122	+	0.0642079i
$794$		0			0
$795$	5.77558	−	1.54756i	0.204839	−	0.0548863i
$796$		0			0
$797$	17.2646	+	29.9032i	0.611544	+	1.05923i	0.990980	+	0.134007i	$0.0427846\pi$
$797$	17.2646	+	29.9032i	0.611544	+	1.05923i	−0.379436	+	0.925218i	$0.623882\pi$
$798$		0			0
$799$	38.3739	+	10.2822i	1.35757	+	0.363760i
$800$		0			0
$801$	−0.653653	+	0.653653i	−0.0230957	+	0.0230957i
$802$		0			0
$803$	28.7198	+	16.5814i	1.01350	+	0.585145i
$804$		0			0
$805$	−0.574160			−0.0202365
$806$		0			0
$807$		−	20.5562i		−	0.723612i
$808$		0			0
$809$	−6.45115	+	11.1737i	−0.226810	+	0.392847i	−0.956861	−	0.290546i	$-0.906163\pi$
$809$	−6.45115	+	11.1737i	−0.226810	+	0.392847i	0.730051	+	0.683393i	$0.239496\pi$
$810$		0			0
$811$	−18.8656	−	18.8656i	−0.662463	−	0.662463i	0.293497	−	0.955960i	$-0.405181\pi$
$811$	−18.8656	−	18.8656i	−0.662463	−	0.662463i	−0.955960	+	0.293497i	$0.905181\pi$
$812$		0			0
$813$	−4.07196	+	15.1968i	−0.142810	+	0.532974i
$814$		0			0
$815$	−1.01287	+	0.584778i	−0.0354791	+	0.0204839i
$816$		0			0
$817$	2.81895	+	10.5205i	0.0986226	+	0.368065i
$818$		0			0
$819$	1.37525	+	0.193016i	0.0480553	+	0.00674454i
$820$		0			0
$821$	12.7023	+	47.4056i	0.443313	+	1.65447i	0.720353	+	0.693608i	$0.243980\pi$
$821$	12.7023	+	47.4056i	0.443313	+	1.65447i	−0.277040	+	0.960858i	$0.589353\pi$
$822$		0			0
$823$	−14.9212	−	25.8443i	−0.520121	−	0.900876i	−0.999726	−	0.0233914i	$-0.992554\pi$
$823$	−14.9212	−	25.8443i	−0.520121	−	0.900876i	0.479606	−	0.877484i	$-0.340780\pi$
$824$		0			0
$825$	4.58558	−	17.1136i	0.159649	−	0.595820i
$826$		0			0
$827$	39.0150	−	39.0150i	1.35668	−	1.35668i	0.478714	−	0.877971i	$-0.341103\pi$
$827$	39.0150	−	39.0150i	1.35668	−	1.35668i	0.877971	−	0.478714i	$-0.158897\pi$
$828$		0			0
$829$	17.4500	−	30.2242i	0.606063	−	1.04973i	−0.385820	−	0.922574i	$-0.626081\pi$
$829$	17.4500	−	30.2242i	0.606063	−	1.04973i	0.991883	−	0.127157i	$-0.0405853\pi$
$830$		0			0
$831$	−15.0096			−0.520678
$832$		0			0
$833$	−23.7646			−0.823395
$834$		0			0
$835$	−3.21459	+	5.56784i	−0.111246	+	0.192683i
$836$		0			0
$837$	29.0201	−	29.0201i	1.00308	−	1.00308i
$838$		0			0
$839$	−7.72729	+	28.8387i	−0.266776	+	0.995621i	0.694379	+	0.719610i	$0.255679\pi$
$839$	−7.72729	+	28.8387i	−0.266776	+	0.995621i	−0.961154	+	0.276011i	$0.910987\pi$
$840$		0			0
$841$	−12.2748	−	21.2606i	−0.423270	−	0.733126i
$842$		0			0
$843$	3.38833	+	12.6454i	0.116700	+	0.435531i
$844$		0			0
$845$	−3.73331	−	0.0637651i	−0.128430	−	0.00219359i
$846$		0			0
$847$	1.21266	+	4.52573i	0.0416677	+	0.155506i
$848$		0			0
$849$	−0.345145	+	0.199270i	−0.0118454	+	0.00683892i
$850$		0			0
$851$	4.88256	−	18.2220i	0.167372	−	0.624641i
$852$		0			0
$853$	−23.9776	−	23.9776i	−0.820976	−	0.820976i	0.165272	−	0.986248i	$-0.447150\pi$
$853$	−23.9776	−	23.9776i	−0.820976	−	0.820976i	−0.986248	+	0.165272i	$0.947150\pi$
$854$		0			0
$855$	0.211410	−	0.366173i	0.00723007	−	0.0125229i
$856$		0			0
$857$			14.2812i			0.487836i	0.969796	+	0.243918i	$0.0784328\pi$
$857$			14.2812i			0.487836i	−0.969796	+	0.243918i	$0.921567\pi$
$858$		0			0
$859$	7.95874			0.271549			0.135774	−	0.990740i	$-0.456648\pi$
$859$	7.95874			0.271549			0.135774	+	0.990740i	$0.456648\pi$
$860$		0			0
$861$	3.32863	+	1.92178i	0.113439	+	0.0654942i
$862$		0			0
$863$	21.5136	−	21.5136i	0.732331	−	0.732331i	−0.238750	−	0.971081i	$-0.576738\pi$
$863$	21.5136	−	21.5136i	0.732331	−	0.732331i	0.971081	+	0.238750i	$0.0767377\pi$
$864$		0			0
$865$	−4.17561	−	1.11885i	−0.141975	−	0.0380421i
$866$		0			0
$867$	−2.40392	−	4.16372i	−0.0816415	−	0.141407i
$868$		0			0
$869$	−26.5592	+	7.11652i	−0.900959	+	0.241411i
$870$		0			0
$871$	−39.1088	−	5.48891i	−1.32515	−	0.185984i
$872$		0			0
$873$	2.69206	−	0.721336i	0.0911125	−	0.0244135i
$874$		0			0
$875$	1.97664	−	1.14121i	0.0668225	−	0.0385800i
$876$		0			0
$877$	15.4279	+	4.13388i	0.520962	+	0.139591i	0.509711	−	0.860345i	$-0.329752\pi$
$877$	15.4279	+	4.13388i	0.520962	+	0.139591i	0.0112504	+	0.999937i	$0.496419\pi$
$878$		0			0
$879$	23.5021	+	23.5021i	0.792706	+	0.792706i
$880$		0			0
$881$	12.4608	+	7.19424i	0.419814	+	0.242380i	0.694998	−	0.719012i	$-0.255405\pi$
$881$	12.4608	+	7.19424i	0.419814	+	0.242380i	−0.275183	+	0.961392i	$0.588739\pi$
$882$		0			0
$883$		−	3.55013i		−	0.119472i	−0.998214	−	0.0597358i	$-0.980974\pi$
$883$		−	3.55013i		−	0.119472i	0.998214	−	0.0597358i	$-0.0190258\pi$
$884$		0			0
$885$		−	3.03997i		−	0.102187i
$886$		0			0
$887$	47.6836	+	27.5301i	1.60106	+	0.924371i	0.991276	+	0.131802i	$0.0420764\pi$
$887$	47.6836	+	27.5301i	1.60106	+	0.924371i	0.609782	+	0.792569i	$0.291257\pi$
$888$		0			0
$889$	−2.43777	−	2.43777i	−0.0817601	−	0.0817601i
$890$		0			0
$891$	16.0648	+	4.30454i	0.538190	+	0.144207i
$892$		0			0
$893$	28.1894	−	16.2751i	0.943321	−	0.544627i
$894$		0			0
$895$	−4.51895	+	1.21085i	−0.151052	+	0.0404742i
$896$		0			0
$897$	−1.74264	−	14.1707i	−0.0581851	−	0.473146i
$898$		0			0
$899$	15.1373	−	4.05601i	0.504856	−	0.135276i
$900$		0			0
$901$	−24.5121	−	42.4561i	−0.816615	−	1.41442i
$902$		0			0
$903$	−4.36899	−	1.17067i	−0.145391	−	0.0389573i
$904$		0			0
$905$	0.730762	−	0.730762i	0.0242913	−	0.0242913i
$906$		0			0
$907$	−38.6201	−	22.2973i	−1.28236	−	0.740370i	−0.305079	−	0.952327i	$-0.598683\pi$
$907$	−38.6201	−	22.2973i	−1.28236	−	0.740370i	−0.977279	+	0.211957i	$0.932016\pi$
$908$		0			0
$909$	3.91062			0.129707
$910$		0			0
$911$		−	22.8758i		−	0.757910i	−0.925415	−	0.378955i	$-0.876283\pi$
$911$		−	22.8758i		−	0.757910i	0.925415	−	0.378955i	$-0.123717\pi$
$912$		0			0
$913$	3.33683	−	5.77956i	0.110433	−	0.191275i
$914$		0			0
$915$	−1.32445	−	1.32445i	−0.0437850	−	0.0437850i
$916$		0			0
$917$	3.02939	−	11.3058i	0.100039	−	0.373352i
$918$		0			0
$919$	24.1895	−	13.9658i	0.797939	−	0.460690i	−0.0448109	−	0.998995i	$-0.514269\pi$
$919$	24.1895	−	13.9658i	0.797939	−	0.460690i	0.842750	+	0.538305i	$0.180935\pi$
$920$		0			0
$921$	−6.12379	−	22.8543i	−0.201786	−	0.753074i
$922$		0			0
$923$	−22.3981	−	17.4926i	−0.737243	−	0.575775i
$924$		0			0
$925$	9.62396	+	35.9171i	0.316434	+	1.18095i
$926$		0			0
$927$	4.18439	+	7.24757i	0.137433	+	0.238042i
$928$		0			0
$929$	7.54479	−	28.1575i	0.247536	−	0.923818i	−0.724555	−	0.689217i	$-0.757955\pi$
$929$	7.54479	−	28.1575i	0.247536	−	0.923818i	0.972092	−	0.234602i	$-0.0753785\pi$
$930$		0			0
$931$	−13.7682	+	13.7682i	−0.451236	+	0.451236i
$932$		0			0
$933$	−1.36128	+	2.35781i	−0.0445665	+	0.0771914i
$934$		0			0
$935$	2.43691			0.0796953
$936$		0			0
$937$	7.52598			0.245863			0.122931	−	0.992415i	$-0.460770\pi$
$937$	7.52598			0.245863			0.122931	+	0.992415i	$0.460770\pi$
$938$		0			0
$939$	6.55216	−	11.3487i	0.213822	−	0.370350i
$940$		0			0
$941$	−1.73699	+	1.73699i	−0.0566241	+	0.0566241i	−0.734852	−	0.678228i	$-0.762748\pi$
$941$	−1.73699	+	1.73699i	−0.0566241	+	0.0566241i	0.678228	+	0.734852i	$0.262748\pi$
$942$		0			0
$943$	−1.95141	+	7.28276i	−0.0635466	+	0.237159i
$944$		0			0
$945$	0.635726	+	1.10111i	0.0206802	+	0.0358191i
$946$		0			0
$947$	−14.1576	−	52.8369i	−0.460060	−	1.71697i	−0.672769	−	0.739852i	$-0.734895\pi$
$947$	−14.1576	−	52.8369i	−0.460060	−	1.71697i	0.212709	−	0.977116i	$-0.431771\pi$
$948$		0			0
$949$	−48.8364	+	19.7418i	−1.58530	+	0.640847i
$950$		0			0
$951$	5.54073	+	20.6783i	0.179670	+	0.670539i
$952$		0			0
$953$	49.3129	−	28.4708i	1.59740	−	0.922261i	0.605418	−	0.795908i	$-0.293006\pi$
$953$	49.3129	−	28.4708i	1.59740	−	0.922261i	0.991985	−	0.126353i	$-0.0403273\pi$
$954$		0			0
$955$	1.98525	−	7.40904i	0.0642411	−	0.239751i
$956$		0			0
$957$	5.37445	+	5.37445i	0.173731	+	0.173731i
$958$		0			0
$959$	6.35951	−	11.0150i	0.205359	−	0.355693i
$960$		0			0
$961$		−	24.1843i		−	0.780139i
$962$		0			0
$963$	0.708489			0.0228307
$964$		0			0
$965$	−2.75826	−	1.59248i	−0.0887915	−	0.0512638i
$966$		0			0
$967$	−28.7702	+	28.7702i	−0.925189	+	0.925189i	−0.997390	−	0.0722014i	$-0.976998\pi$
$967$	−28.7702	+	28.7702i	−0.925189	+	0.925189i	0.0722014	+	0.997390i	$0.476998\pi$
$968$		0			0
$969$	17.5498	+	4.70246i	0.563781	+	0.151065i
$970$		0			0
$971$	27.3131	+	47.3076i	0.876518	+	1.51817i	0.855137	+	0.518402i	$0.173473\pi$
$971$	27.3131	+	47.3076i	0.876518	+	1.51817i	0.0213803	+	0.999771i	$0.493194\pi$
$972$		0			0
$973$	−7.54434	+	2.02150i	−0.241860	+	0.0648063i
$974$		0			0
$975$	16.9406	+	22.4721i	0.542533	+	0.719684i
$976$		0			0
$977$	−3.60921	+	0.967085i	−0.115469	+	0.0309398i	−0.316091	−	0.948729i	$-0.602370\pi$
$977$	−3.60921	+	0.967085i	−0.115469	+	0.0309398i	0.200622	+	0.979669i	$0.435704\pi$
$978$		0			0
$979$	−3.78041	+	2.18262i	−0.120822	+	0.0697569i
$980$		0			0
$981$	−5.53706	−	1.48365i	−0.176785	−	0.0473694i
$982$		0			0
$983$	20.7574	+	20.7574i	0.662057	+	0.662057i	0.955865	−	0.293808i	$-0.0949225\pi$
$983$	20.7574	+	20.7574i	0.662057	+	0.662057i	−0.293808	+	0.955865i	$0.594922\pi$
$984$		0			0
$985$	4.14087	+	2.39073i	0.131939	+	0.0761751i
$986$		0			0
$987$			13.5176i			0.430271i
$988$		0			0
$989$		−	8.87268i		−	0.282135i
$990$		0			0
$991$	48.8698	+	28.2150i	1.55240	+	0.896279i	0.997946	+	0.0640655i	$0.0204067\pi$
$991$	48.8698	+	28.2150i	1.55240	+	0.896279i	0.554455	+	0.832214i	$0.312927\pi$
$992$		0			0
$993$	22.0383	+	22.0383i	0.699363	+	0.699363i
$994$		0			0
$995$	−1.69661	−	0.454605i	−0.0537862	−	0.0144120i
$996$		0			0
$997$	52.4375	−	30.2748i	1.66071	−	0.958812i	0.688336	−	0.725392i	$-0.258342\pi$
$997$	52.4375	−	30.2748i	1.66071	−	0.958812i	0.972376	−	0.233420i	$-0.0749918\pi$
$998$		0			0
$999$	−40.3518	+	10.8122i	−1.27667	+	0.342084i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bk.a.111.3		48
4.3	odd	2		104.2.u.a.59.10	yes	48
8.3	odd	2	inner	416.2.bk.a.111.4		48
8.5	even	2		104.2.u.a.59.8	✓	48
12.11	even	2		936.2.ed.d.163.3		48
13.2	odd	12	inner	416.2.bk.a.15.4		48
24.5	odd	2		936.2.ed.d.163.5		48
52.15	even	12		104.2.u.a.67.8	yes	48
104.67	even	12	inner	416.2.bk.a.15.3		48
104.93	odd	12		104.2.u.a.67.10	yes	48
156.119	odd	12		936.2.ed.d.379.5		48
312.197	even	12		936.2.ed.d.379.3		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.59.8	✓	48	8.5	even	2
104.2.u.a.59.10	yes	48	4.3	odd	2
104.2.u.a.67.8	yes	48	52.15	even	12
104.2.u.a.67.10	yes	48	104.93	odd	12
416.2.bk.a.15.3		48	104.67	even	12	inner
416.2.bk.a.15.4		48	13.2	odd	12	inner
416.2.bk.a.111.3		48	1.1	even	1	trivial
416.2.bk.a.111.4		48	8.3	odd	2	inner
936.2.ed.d.163.3		48	12.11	even	2
936.2.ed.d.163.5		48	24.5	odd	2
936.2.ed.d.379.3		48	312.197	even	12
936.2.ed.d.379.5		48	156.119	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-0.793616 + 1.37458i) q^{3} +(-0.203095 + 0.203095i) q^{5} +(-0.207385 + 0.773970i) q^{7} +(0.240346 + 0.416291i) q^{9} +O(q^{10})\)	\(q+(-0.793616 + 1.37458i) q^{3} +(-0.203095 + 0.203095i) q^{5} +(-0.207385 + 0.773970i) q^{7} +(0.240346 + 0.416291i) q^{9} +(0.587501 + 2.19258i) q^{11} +(-3.31919 - 1.40818i) q^{13} +(-0.117991 - 0.440350i) q^{15} +(-3.23700 + 1.86889i) q^{17} +(-0.792632 + 2.95814i) q^{19} +(-0.899303 - 0.899303i) q^{21} +(1.24741 - 2.16058i) q^{23} +4.91751i q^{25} -5.52467 q^{27} +(-1.82695 - 1.05479i) q^{29} +(-5.25282 + 5.25282i) q^{31} +(-3.48014 - 0.932501i) q^{33} +(-0.115071 - 0.199308i) q^{35} +(7.30393 - 1.95708i) q^{37} +(4.56983 - 3.44495i) q^{39} +(-2.91916 + 0.782185i) q^{41} +(3.07997 - 1.77822i) q^{43} +(-0.133359 - 0.0357336i) q^{45} +(-7.51561 - 7.51561i) q^{47} +(5.50616 + 3.17898i) q^{49} -5.93271i q^{51} +13.1159i q^{53} +(-0.564621 - 0.325984i) q^{55} +(-3.43717 - 3.43717i) q^{57} +(6.44108 + 1.72588i) q^{59} +(3.55817 - 2.05431i) q^{61} +(-0.372041 + 0.0996881i) q^{63} +(0.960104 - 0.388116i) q^{65} +(10.5799 - 2.83488i) q^{67} +(1.97993 + 3.42934i) q^{69} +(7.61356 + 2.04005i) q^{71} +(10.3306 - 10.3306i) q^{73} +(-6.75952 - 3.90261i) q^{75} -1.81883 q^{77} +12.1132i q^{79} +(3.66343 - 6.34525i) q^{81} +(-2.07891 - 2.07891i) q^{83} +(0.277858 - 1.03698i) q^{85} +(2.89979 - 1.67420i) q^{87} +(0.497728 + 1.85755i) q^{89} +(1.77824 - 2.27692i) q^{91} +(-3.05172 - 11.3892i) q^{93} +(-0.439804 - 0.761763i) q^{95} +(1.50062 - 5.60039i) q^{97} +(-0.771550 + 0.771550i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3} - 20 q^{9}+O(q^{10}) \) 48 * q + 4 * q^3 - 20 * q^9	\( 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83} + 12 q^{89} - 104 q^{91} + 4 q^{97} - 168 q^{99}+O(q^{100}) \) 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

Related objects

Downloads

Learn more