LMFDB - Embedded newform 416.2.bk.a.15.4

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			15.4
Character	$\chi$	$=$	416.15
Dual form			416.2.bk.a.111.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.793616 - 1.37458i) q^{3} +(0.203095 + 0.203095i) q^{5} +(0.207385 + 0.773970i) q^{7} +(0.240346 - 0.416291i) q^{9} +O(q^{10})$	\(q+(-0.793616 - 1.37458i) q^{3} +(0.203095 + 0.203095i) q^{5} +(0.207385 + 0.773970i) q^{7} +(0.240346 - 0.416291i) q^{9} +(0.587501 - 2.19258i) q^{11} +(3.31919 - 1.40818i) q^{13} +(0.117991 - 0.440350i) q^{15} +(-3.23700 - 1.86889i) q^{17} +(-0.792632 - 2.95814i) q^{19} +(0.899303 - 0.899303i) q^{21} +(-1.24741 - 2.16058i) q^{23} -4.91751i q^{25} -5.52467 q^{27} +(1.82695 - 1.05479i) q^{29} +(5.25282 + 5.25282i) q^{31} +(-3.48014 + 0.932501i) q^{33} +(-0.115071 + 0.199308i) q^{35} +(-7.30393 - 1.95708i) q^{37} +(-4.56983 - 3.44495i) q^{39} +(-2.91916 - 0.782185i) q^{41} +(3.07997 + 1.77822i) q^{43} +(0.133359 - 0.0357336i) q^{45} +(7.51561 - 7.51561i) q^{47} +(5.50616 - 3.17898i) q^{49} +5.93271i q^{51} +13.1159i q^{53} +(0.564621 - 0.325984i) q^{55} +(-3.43717 + 3.43717i) q^{57} +(6.44108 - 1.72588i) q^{59} +(-3.55817 - 2.05431i) q^{61} +(0.372041 + 0.0996881i) q^{63} +(0.960104 + 0.388116i) q^{65} +(10.5799 + 2.83488i) q^{67} +(-1.97993 + 3.42934i) q^{69} +(-7.61356 + 2.04005i) q^{71} +(10.3306 + 10.3306i) q^{73} +(-6.75952 + 3.90261i) q^{75} +1.81883 q^{77} +12.1132i q^{79} +(3.66343 + 6.34525i) q^{81} +(-2.07891 + 2.07891i) q^{83} +(-0.277858 - 1.03698i) q^{85} +(-2.89979 - 1.67420i) q^{87} +(0.497728 - 1.85755i) q^{89} +(1.77824 + 2.27692i) q^{91} +(3.05172 - 11.3892i) q^{93} +(0.439804 - 0.761763i) q^{95} +(1.50062 + 5.60039i) q^{97} +(-0.771550 - 0.771550i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

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

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	0.540671	−	0.841234i	$-0.318170\pi$
$3$	−0.793616	−	1.37458i	−0.458195	−	0.793616i	−0.998866	+	0.0476177i	$0.984837\pi$
$4$		0			0
$5$	0.203095	+	0.203095i	0.0908267	+	0.0908267i	0.751060	−	0.660234i	$-0.229543\pi$
$5$	0.203095	+	0.203095i	0.0908267	+	0.0908267i	−0.660234	+	0.751060i	$0.729543\pi$
$6$		0			0
$7$	0.207385	+	0.773970i	0.0783840	+	0.292533i	0.993979	−	0.109568i	$-0.0349467\pi$
$7$	0.207385	+	0.773970i	0.0783840	+	0.292533i	−0.915595	+	0.402101i	$0.868280\pi$
$8$		0			0
$9$	0.240346	−	0.416291i	0.0801153	−	0.138764i
$10$		0			0
$11$	0.587501	−	2.19258i	0.177138	−	0.661089i	−0.819039	−	0.573737i	$-0.805493\pi$
$11$	0.587501	−	2.19258i	0.177138	−	0.661089i	0.996178	−	0.0873516i	$-0.0278404\pi$
$12$		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	0.0531419	−	0.998587i	$-0.483076\pi$
$17$	−3.23700	−	1.86889i	−0.785089	−	0.453271i	−0.838231	+	0.545316i	$0.816410\pi$
$18$		0			0
$19$	−0.792632	−	2.95814i	−0.181842	−	0.678645i	−0.995284	−	0.0969990i	$-0.969076\pi$
$19$	−0.792632	−	2.95814i	−0.181842	−	0.678645i	0.813442	−	0.581646i	$-0.197591\pi$
$20$		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	$-0.250423\pi$
$23$	−1.24741	−	2.16058i	−0.260103	−	0.450511i	−0.966269	+	0.257535i	$0.917090\pi$
$24$		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	$-0.603914\pi$
$29$	1.82695	−	1.05479i	0.339256	−	0.195869i	0.659943	+	0.751316i	$0.270580\pi$
$30$		0			0
$31$	5.25282	+	5.25282i	0.943435	+	0.943435i	0.998484	−	0.0550487i	$-0.0175314\pi$
$31$	5.25282	+	5.25282i	0.943435	+	0.943435i	−0.0550487	+	0.998484i	$0.517531\pi$
$32$		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	$-0.630167\pi$
$37$	−7.30393	−	1.95708i	−1.20076	−	0.321742i	−0.803130	+	0.595804i	$0.796833\pi$
$38$		0			0
$39$	−4.56983	−	3.44495i	−0.731758	−	0.551634i
$40$		0			0
$41$	−2.91916	−	0.782185i	−0.455895	−	0.122157i	0.0235612	−	0.999722i	$-0.492500\pi$
$41$	−2.91916	−	0.782185i	−0.455895	−	0.122157i	−0.479457	+	0.877566i	$0.659166\pi$
$42$		0			0
$43$	3.07997	+	1.77822i	0.469691	+	0.271176i	0.716110	−	0.697987i	$-0.245921\pi$
$43$	3.07997	+	1.77822i	0.469691	+	0.271176i	−0.246419	+	0.969163i	$0.579254\pi$
$44$		0			0
$45$	0.133359	−	0.0357336i	0.0198801	−	0.00532684i
$46$		0			0
$47$	7.51561	−	7.51561i	1.09626	−	1.09626i	0.101420	−	0.994844i	$-0.467661\pi$
$47$	7.51561	−	7.51561i	1.09626	−	1.09626i	0.994844	−	0.101420i	$-0.0323387\pi$
$48$		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	0.186114	−	0.982528i	$-0.440411\pi$
$59$	6.44108	−	1.72588i	0.838558	−	0.224691i	0.652444	+	0.757837i	$0.273744\pi$
$60$		0			0
$61$	−3.55817	−	2.05431i	−0.455577	−	0.263028i	0.254606	−	0.967045i	$-0.418054\pi$
$61$	−3.55817	−	2.05431i	−0.455577	−	0.263028i	−0.710183	+	0.704017i	$0.751388\pi$
$62$		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	0.838626	−	0.544708i	$-0.183360\pi$
$67$	10.5799	+	2.83488i	1.29254	+	0.346336i	0.453917	+	0.891044i	$0.350026\pi$
$68$		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	$-0.738257\pi$
$71$	−7.61356	+	2.04005i	−0.903563	+	0.242109i	−0.223017	+	0.974814i	$0.571591\pi$
$72$		0			0
$73$	10.3306	+	10.3306i	1.20910	+	1.20910i	0.971319	+	0.237781i	$0.0764201\pi$
$73$	10.3306	+	10.3306i	1.20910	+	1.20910i	0.237781	+	0.971319i	$0.423580\pi$
$74$		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	−0.811936	−	0.583746i	$-0.801586\pi$
$83$	−2.07891	+	2.07891i	−0.228190	+	0.228190i	0.583746	+	0.811936i	$0.301586\pi$
$84$		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	−0.934516	−	0.355921i	$-0.884167\pi$
$89$	0.497728	−	1.85755i	0.0527590	−	0.196899i	0.987275	+	0.159021i	$0.0508339\pi$
$90$		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	0.999317	+	0.0369639i	$0.0117687\pi$
$97$	1.50062	+	5.60039i	0.152365	+	0.568634i	−0.846952	+	0.531670i	$0.821565\pi$
$98$		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	$-0.299309\pi$
$101$	−4.06770	−	7.04546i	−0.404751	−	0.701050i	−0.994293	+	0.106688i	$0.965975\pi$
$102$		0			0
$103$	−17.4099			−1.71545			−0.857723	−	0.514113i	$-0.828121\pi$
$103$	−17.4099			−1.71545			−0.857723	+	0.514113i	$0.828121\pi$
$104$		0			0
$105$	0.365287			0.0356484
$106$		0			0
$107$	0.736948	+	1.27643i	0.0712434	+	0.123397i	0.899447	−	0.437031i	$-0.143970\pi$
$107$	0.736948	+	1.27643i	0.0712434	+	0.123397i	−0.828203	+	0.560428i	$0.810637\pi$
$108$		0			0
$109$	8.43246	+	8.43246i	0.807683	+	0.807683i	0.984283	−	0.176600i	$-0.0565098\pi$
$109$	8.43246	+	8.43246i	0.807683	+	0.807683i	−0.176600	+	0.984283i	$0.556510\pi$
$110$		0			0
$111$	3.10634	+	11.5930i	0.294841	+	1.10036i
$112$		0			0
$113$	−2.67940	+	4.64087i	−0.252057	+	0.436576i	−0.964092	−	0.265568i	$-0.914440\pi$
$113$	−2.67940	+	4.64087i	−0.252057	+	0.436576i	0.712035	+	0.702144i	$0.247774\pi$
$114$		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	$-0.105528\pi$
$127$	2.15128	+	3.72613i	0.190895	+	0.330640i	−0.754652	+	0.656125i	$0.772194\pi$
$128$		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	−0.845206	−	0.534440i	$-0.820522\pi$
$137$	−15.3326	+	4.10837i	−1.30996	+	0.351002i	−0.464750	+	0.885442i	$0.653856\pi$
$138$		0			0
$139$	4.87379	−	8.44165i	0.413390	−	0.716012i	−0.581868	−	0.813283i	$-0.697678\pi$
$139$	4.87379	−	8.44165i	0.413390	−	0.716012i	0.995258	+	0.0972713i	$0.0310115\pi$
$140$		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	$-0.473569\pi$
$149$	7.97144	−	2.13594i	0.653046	−	0.174983i	0.570106	+	0.821571i	$0.306902\pi$
$150$		0			0
$151$	10.3500	−	10.3500i	0.842268	−	0.842268i	−0.146886	−	0.989153i	$-0.546925\pi$
$151$	10.3500	−	10.3500i	0.842268	−	0.842268i	0.989153	+	0.146886i	$0.0469249\pi$
$152$		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	−0.101469	−	0.994839i	$-0.532354\pi$
$163$	3.93325	−	1.05391i	0.308076	−	0.0825487i	0.409545	+	0.912290i	$0.365688\pi$
$164$		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	$-0.750030\pi$
$167$	−21.6215	−	5.79347i	−1.67312	−	0.448312i	−0.965950	+	0.258728i	$0.916697\pi$
$168$		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	0.424196	+	0.905571i	$0.360557\pi$
$173$	−7.52545	+	13.0345i	−0.572149	+	0.990992i	−0.996345	+	0.0854210i	$0.972776\pi$
$174$		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	0.130486	−	0.991450i	$-0.458346\pi$
$179$	14.1062	−	8.14424i	1.05435	−	0.608729i	0.923864	+	0.382721i	$0.125013\pi$
$180$		0			0
$181$	3.59813			0.267447			0.133724	−	0.991019i	$-0.457307\pi$
$181$	3.59813			0.267447			0.133724	+	0.991019i	$0.457307\pi$
$182$		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	0.965671	+	0.259768i	$0.0836462\pi$
$191$	23.1278	+	13.3529i	1.67347	+	0.966180i	0.707801	+	0.706411i	$0.249687\pi$
$192$		0			0
$193$	2.87004	−	10.7111i	0.206590	−	0.771003i	−0.782369	−	0.622815i	$-0.785989\pi$
$193$	2.87004	−	10.7111i	0.206590	−	0.771003i	0.988959	−	0.148189i	$-0.0473443\pi$
$194$		0			0
$195$	−0.228455	−	1.62776i	−0.0163600	−	0.116566i
$196$		0			0
$197$	4.30867	−	16.0802i	0.306980	−	1.14567i	−0.624247	−	0.781227i	$-0.714594\pi$
$197$	4.30867	−	16.0802i	0.306980	−	1.14567i	0.931227	−	0.364439i	$-0.118739\pi$
$198$		0			0
$199$	−3.05770	+	5.29609i	−0.216755	+	0.375430i	−0.953814	−	0.300398i	$-0.902880\pi$
$199$	−3.05770	+	5.29609i	−0.216755	+	0.375430i	0.737059	+	0.675828i	$0.236214\pi$
$200$		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	0.998112	−	0.0614182i	$-0.0195623\pi$
$211$	6.47658	+	11.2178i	0.445866	+	0.772263i	−0.552246	+	0.833681i	$0.686229\pi$
$212$		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	$-0.862321\pi$
$223$	1.31334	−	4.90147i	0.0879480	−	0.328226i	0.995856	+	0.0909428i	$0.0289880\pi$
$224$		0			0
$225$	−2.04711	−	1.18190i	−0.136474	−	0.0787935i
$226$		0			0
$227$	0.177177	+	0.661235i	0.0117597	+	0.0438877i	0.971556	−	0.236808i	$-0.0761014\pi$
$227$	0.177177	+	0.661235i	0.0117597	+	0.0438877i	−0.959797	+	0.280696i	$0.909435\pi$
$228$		0			0
$229$	3.26943	−	3.26943i	0.216050	−	0.216050i	−0.590782	−	0.806832i	$-0.701180\pi$
$229$	3.26943	−	3.26943i	0.216050	−	0.216050i	0.806832	+	0.590782i	$0.201180\pi$
$230$		0			0
$231$	−1.44346	−	2.50014i	−0.0949725	−	0.164497i
$232$		0			0
$233$		−	15.3725i		−	1.00708i	−0.863971	−	0.503542i	$-0.832030\pi$
$233$		−	15.3725i		−	1.00708i	0.863971	−	0.503542i	$-0.167970\pi$
$234$		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	$-0.167514\pi$
$239$	5.60237	+	5.60237i	0.362387	+	0.362387i	−0.502304	+	0.864691i	$0.667514\pi$
$240$		0			0
$241$	15.3608	−	4.11592i	0.989476	−	0.265129i	0.272446	−	0.962171i	$-0.412167\pi$
$241$	15.3608	−	4.11592i	0.989476	−	0.265129i	0.717031	+	0.697042i	$0.245501\pi$
$242$		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	−0.0786312	−	0.996904i	$-0.525055\pi$
$251$	−15.5466	−	8.97582i	−0.981291	−	0.566549i	−0.902660	+	0.430355i	$0.858388\pi$
$252$		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	−0.964774	−	0.263078i	$-0.915262\pi$
$257$	−19.5473	+	11.2857i	−1.21933	+	0.703980i	−0.254555	+	0.967058i	$0.581929\pi$
$258$		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	$-0.632683\pi$
$263$	2.99321	−	1.72813i	0.184569	−	0.106561i	0.589438	+	0.807814i	$0.299349\pi$
$264$		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	$-0.204141\pi$
$269$	11.2159	+	6.47548i	0.683843	+	0.394817i	−0.117458	+	0.993078i	$0.537475\pi$
$270$		0			0
$271$	−9.57438	−	2.56545i	−0.581602	−	0.155840i	−0.0439898	−	0.999032i	$-0.514007\pi$
$271$	−9.57438	−	2.56545i	−0.581602	−	0.155840i	−0.537612	+	0.843192i	$0.680674\pi$
$272$		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	$-0.925026\pi$
$277$	−4.72824	+	8.18955i	−0.284092	+	0.492062i	0.688296	+	0.725430i	$0.258359\pi$
$278$		0			0
$279$	3.44920	−	0.924210i	0.206498	−	0.0553310i
$280$		0			0
$281$	5.83221	+	5.83221i	0.347921	+	0.347921i	0.859334	−	0.511414i	$-0.170878\pi$
$281$	5.83221	+	5.83221i	0.347921	+	0.347921i	−0.511414	+	0.859334i	$0.670878\pi$
$282$		0			0
$283$	0.217451	−	0.125545i	0.0129261	−	0.00746290i	−0.493523	−	0.869733i	$-0.664291\pi$
$283$	0.217451	−	0.125545i	0.0129261	−	0.00746290i	0.506449	+	0.862270i	$0.330958\pi$
$284$		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	−0.922468	−	0.386072i	$-0.873831\pi$
$293$	−5.41972	−	20.2267i	−0.316624	−	1.18166i	0.605845	−	0.795583i	$-0.292835\pi$
$294$		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	0.339146	−	0.940734i	$-0.389862\pi$
$307$	−10.5407	−	10.5407i	−0.601588	−	0.601588i	−0.940734	+	0.339146i	$0.889862\pi$
$308$		0			0
$309$	13.8168	+	23.9313i	0.786008	+	1.36141i
$310$		0			0
$311$	−1.71529			−0.0972653			−0.0486327	−	0.998817i	$-0.515486\pi$
$311$	−1.71529			−0.0972653			−0.0486327	+	0.998817i	$0.515486\pi$
$312$		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	$-0.373651\pi$
$317$	−9.53707	−	9.53707i	−0.535655	−	0.535655i	−0.922250	+	0.386595i	$0.873651\pi$
$318$		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	0.952877	+	0.303358i	$0.0981079\pi$
$331$	5.08215	+	18.9669i	0.279340	+	1.04251i	−0.673536	+	0.739154i	$0.735225\pi$
$332$		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	$0.383402\pi$
$337$			34.2794i			1.86732i	−0.358165	+	0.933658i	$0.616598\pi$
$338$		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	0.442221	+	0.896906i	$0.354191\pi$
$347$	−10.3503	+	17.9272i	−0.555633	+	0.962384i	−0.997854	+	0.0654782i	$0.979143\pi$
$348$		0			0
$349$	−14.7302	−	3.94695i	−0.788490	−	0.211275i	−0.157966	−	0.987445i	$-0.550494\pi$
$349$	−14.7302	−	3.94695i	−0.788490	−	0.211275i	−0.630525	+	0.776169i	$0.717160\pi$
$350$		0			0
$351$	−18.3374	+	7.77972i	−0.978780	+	0.415251i
$352$		0			0
$353$	26.0283	+	6.97426i	1.38535	+	0.371202i	0.873060	−	0.487612i	$-0.162132\pi$
$353$	26.0283	+	6.97426i	1.38535	+	0.371202i	0.512286	+	0.858815i	$0.328799\pi$
$354$		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	$-0.973068\pi$
$359$	−17.2783	+	17.2783i	−0.911913	+	0.911913i	0.0845098	+	0.996423i	$0.473068\pi$
$360$		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	$-0.675355\pi$
$367$	−0.905546	+	0.522817i	−0.0472691	+	0.0272908i	0.476179	+	0.879348i	$0.342021\pi$
$368$		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	$-0.0751551\pi$
$373$	24.2536	+	14.0028i	1.25580	+	0.725039i	0.283548	+	0.958958i	$0.408488\pi$
$374$		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	0.820108	−	0.572209i	$-0.193914\pi$
$379$	24.2227	+	6.49045i	1.24424	+	0.333392i	0.424129	+	0.905602i	$0.360580\pi$
$380$		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	$-0.425326\pi$
$383$	18.0059	−	4.82468i	0.920061	−	0.246530i	0.687611	+	0.726079i	$0.258659\pi$
$384$		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	$-0.745736\pi$
$389$	−27.5165			−1.39514			−0.697572	+	0.716515i	$0.745736\pi$
$390$		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	0.995175	+	0.0981198i	$0.0312828\pi$
$397$	3.63406	+	13.5625i	0.182388	+	0.680681i	−0.812787	+	0.582562i	$0.802050\pi$
$398$		0			0
$399$	−3.37309	−	1.94745i	−0.168865	−	0.0974945i
$400$		0			0
$401$	2.77041	−	10.3393i	0.138348	−	0.516321i	−0.861614	−	0.507564i	$-0.830546\pi$
$401$	2.77041	−	10.3393i	0.138348	−	0.516321i	0.999962	−	0.00875651i	$-0.00278732\pi$
$402$		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	0.996158	−	0.0875760i	$-0.0279121\pi$
$409$	1.81350	+	6.76808i	0.0896718	+	0.334660i	−0.906486	+	0.422236i	$0.861245\pi$
$410$		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	0.995411	+	0.0956958i	$0.0305076\pi$
$419$	11.8842	+	20.5840i	0.580580	+	1.00559i	−0.414830	+	0.909899i	$0.636159\pi$
$420$		0			0
$421$	−10.0457	−	10.0457i	−0.489596	−	0.489596i	0.418583	−	0.908179i	$-0.362527\pi$
$421$	−10.0457	−	10.0457i	−0.489596	−	0.489596i	−0.908179	+	0.418583i	$0.862527\pi$
$422$		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	$-0.858754\pi$
$431$	1.94441	−	7.25665i	0.0936591	−	0.349540i	0.996813	+	0.0797770i	$0.0254208\pi$
$432$		0			0
$433$	−11.3356	−	6.54458i	−0.544752	−	0.314513i	0.202251	−	0.979334i	$-0.435174\pi$
$433$	−11.3356	−	6.54458i	−0.544752	−	0.314513i	−0.747003	+	0.664821i	$0.768508\pi$
$434$		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	$-0.177694\pi$
$439$	−0.725737	−	1.25701i	−0.0346376	−	0.0599940i	−0.882825	+	0.469703i	$0.844361\pi$
$440$		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	0.631883	−	0.775064i	$-0.282282\pi$
$449$	33.1966	−	8.89499i	1.56664	−	0.419780i	0.934759	+	0.355283i	$0.115616\pi$
$450$		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	−0.271250	−	0.962509i	$-0.587437\pi$
$457$	−21.1085	−	5.65600i	−0.987414	−	0.264577i	−0.716164	+	0.697932i	$0.754104\pi$
$458$		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	$-0.462778\pi$
$461$	15.3365	−	4.10940i	0.714292	−	0.191394i	0.597623	+	0.801777i	$0.296112\pi$
$462$		0			0
$463$	−3.31898	+	3.31898i	−0.154246	+	0.154246i	−0.780011	−	0.625765i	$-0.784787\pi$
$463$	−3.31898	+	3.31898i	−0.154246	+	0.154246i	0.625765	+	0.780011i	$0.284787\pi$
$464$		0			0
$465$	2.93287	−	1.69329i	0.136009	−	0.0785246i
$466$		0			0
$467$		−	4.31326i		−	0.199594i	−0.995008	−	0.0997968i	$-0.968181\pi$
$467$		−	4.31326i		−	0.199594i	0.995008	−	0.0997968i	$-0.0318193\pi$
$468$		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	$-0.528493\pi$
$479$	−14.5501	−	3.89868i	−0.664810	−	0.178135i	−0.575416	+	0.817861i	$0.695160\pi$
$480$		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	$-0.450527\pi$
$487$	17.2756	−	4.62898i	0.782831	−	0.209759i	0.628032	+	0.778187i	$0.283861\pi$
$488$		0			0
$489$	−4.57018	−	4.57018i	−0.206671	−	0.206671i
$490$		0			0
$491$	−15.4867	+	8.94124i	−0.698904	+	0.403512i	−0.806939	−	0.590635i	$-0.798877\pi$
$491$	−15.4867	+	8.94124i	−0.698904	+	0.403512i	0.108035	+	0.994147i	$0.465544\pi$
$492$		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	−0.653923	−	0.756561i	$-0.726878\pi$
$499$	2.29278	−	2.29278i	0.102639	−	0.102639i	0.756561	+	0.653923i	$0.226878\pi$
$500$		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	$-0.186427\pi$
$503$	17.2984	+	9.98723i	0.771297	+	0.445309i	−0.0620400	+	0.998074i	$0.519761\pi$
$504$		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	$-0.896324\pi$
$509$	0.745854	−	2.78356i	0.0330594	−	0.123379i	0.980484	+	0.196601i	$0.0629904\pi$
$510$		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	−0.938262	−	0.345926i	$-0.887565\pi$
$523$	−17.5798	−	30.4491i	−0.768712	−	1.33145i	0.169550	−	0.985522i	$-0.445769\pi$
$524$		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	$-0.639464\pi$
$541$	11.1945	−	11.1945i	0.481288	−	0.481288i	0.905543	+	0.424255i	$0.139464\pi$
$542$		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	$-0.134035\pi$
$557$	35.3694	+	9.47720i	1.49865	+	0.401562i	0.586002	+	0.810310i	$0.300701\pi$
$558$		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	0.647096	−	0.762408i	$-0.275983\pi$
$563$	7.36457	+	4.25194i	0.310380	+	0.179198i	−0.336717	+	0.941606i	$0.609316\pi$
$564$		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	−0.544524	−	0.838745i	$-0.683290\pi$
$569$	−2.15664	+	1.24514i	−0.0904110	+	0.0521988i	0.454113	+	0.890944i	$0.349956\pi$
$570$		0			0
$571$		−	32.2266i		−	1.34864i	−0.738439	−	0.674320i	$-0.764437\pi$
$571$		−	32.2266i		−	1.34864i	0.738439	−	0.674320i	$-0.235563\pi$
$572$		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	−0.783983	−	0.620783i	$-0.786815\pi$
$577$	−3.92020	+	3.92020i	−0.163200	+	0.163200i	0.620783	+	0.783983i	$0.286815\pi$
$578$		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	−0.181835	−	0.983329i	$-0.558204\pi$
$587$	−20.1329	−	5.39459i	−0.830972	−	0.222658i	−0.649138	+	0.760671i	$0.724870\pi$
$588$		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	−0.993430	−	0.114441i	$-0.963492\pi$
$593$	−26.9784	−	26.9784i	−1.10787	−	1.10787i	−0.114441	−	0.993430i	$-0.536508\pi$
$594$		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	0.788870	−	0.614560i	$-0.210666\pi$
$601$	−3.37794	−	5.85076i	−0.137789	−	0.238658i	−0.926659	+	0.375902i	$0.877333\pi$
$602$		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	−0.998132	−	0.0610892i	$-0.980543\pi$
$607$	−38.1905	−	22.0493i	−1.55010	−	0.894953i	−0.551971	−	0.833863i	$-0.686124\pi$
$608$		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	$-0.981264\pi$
$613$	−2.58306	+	9.64013i	−0.104329	+	0.389361i	0.893939	+	0.448188i	$0.147931\pi$
$614$		0			0
$615$	−0.688871	+	1.19316i	−0.0277779	+	0.0481128i
$616$		0			0
$617$	3.14881	+	11.7515i	0.126766	+	0.473099i	0.999896	−	0.0143885i	$-0.00458015\pi$
$617$	3.14881	+	11.7515i	0.126766	+	0.473099i	−0.873130	+	0.487487i	$0.837913\pi$
$618$		0			0
$619$	11.9483	+	11.9483i	0.480243	+	0.480243i	0.905209	−	0.424966i	$-0.139714\pi$
$619$	11.9483	+	11.9483i	0.480243	+	0.480243i	−0.424966	+	0.905209i	$0.639714\pi$
$620$		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	−0.991589	−	0.129427i	$-0.958686\pi$
$631$	−4.96268	−	18.5210i	−0.197561	−	0.737309i	0.794028	−	0.607882i	$-0.207980\pi$
$632$		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	0.419504	−	0.907753i	$-0.362204\pi$
$641$	−3.97192	−	2.29319i	−0.156881	−	0.0905755i	−0.576386	+	0.817178i	$0.695537\pi$
$642$		0			0
$643$	−10.7606	−	40.1589i	−0.424355	−	1.58371i	−0.765328	−	0.643641i	$-0.777423\pi$
$643$	−10.7606	−	40.1589i	−0.424355	−	1.58371i	0.340973	−	0.940073i	$-0.389244\pi$
$644$		0			0
$645$	1.14645	−	1.14645i	0.0451414	−	0.0451414i
$646$		0			0
$647$	20.7662	+	35.9681i	0.816402	+	1.41405i	0.908316	+	0.418284i	$0.137368\pi$
$647$	20.7662	+	35.9681i	0.816402	+	1.41405i	−0.0919140	+	0.995767i	$0.529298\pi$
$648$		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	$-0.700564\pi$
$653$	−4.70452	+	2.71615i	−0.184102	+	0.106291i	0.405117	+	0.914265i	$0.367231\pi$
$654$		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	−0.683065	−	0.730358i	$-0.739353\pi$
$659$	7.46965	−	12.9378i	0.290976	−	0.503986i	0.974041	+	0.226372i	$0.0726866\pi$
$660$		0			0
$661$	14.3738	+	3.85145i	0.559076	+	0.149804i	0.527282	−	0.849690i	$-0.323211\pi$
$661$	14.3738	+	3.85145i	0.559076	+	0.149804i	0.0317943	+	0.999494i	$0.489878\pi$
$662$		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	−0.0720614	−	0.997400i	$-0.522958\pi$
$673$	19.6041	−	11.3184i	0.755682	−	0.436293i	0.827743	+	0.561107i	$0.189624\pi$
$674$		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	0.692118	−	0.721785i	$-0.256678\pi$
$683$	43.1843	−	11.5712i	1.65240	−	0.442760i	0.960284	+	0.279025i	$0.0900112\pi$
$684$		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	−0.108963	−	0.994046i	$-0.534753\pi$
$691$	−18.4100	−	4.93295i	−0.700350	−	0.187658i	−0.591387	+	0.806388i	$0.701420\pi$
$692$		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	$-0.548994\pi$
$701$	−8.11837			−0.306627			−0.153313	+	0.988178i	$0.548994\pi$
$702$		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	$-0.0946094\pi$
$709$	−0.488166	−	1.82186i	−0.0183335	−	0.0684215i	−0.974487	+	0.224446i	$0.927943\pi$
$710$		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	−0.344479	−	0.938794i	$-0.611944\pi$
$719$	17.1820	−	29.7601i	0.640780	−	1.10986i	0.985259	−	0.171069i	$-0.0547222\pi$
$720$		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	$-0.376464\pi$
$727$	20.4072			0.756861			0.378430	+	0.925630i	$0.376464\pi$
$728$		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	$-0.109227\pi$
$733$	16.3864	+	16.3864i	0.605247	+	0.605247i	−0.336453	+	0.941700i	$0.609227\pi$
$734$		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	0.443486	+	0.896282i	$0.353742\pi$
$739$	−10.5673	+	39.4378i	−0.388725	+	1.45074i	−0.832211	+	0.554460i	$0.812925\pi$
$740$		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	$-0.953154\pi$
$743$	−1.61384	+	6.02293i	−0.0592060	+	0.220960i	0.929984	+	0.367600i	$0.119821\pi$
$744$		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	−0.997844	−	0.0656254i	$-0.979096\pi$
$751$	−15.2301	−	26.3794i	−0.555755	−	0.962597i	0.442089	−	0.896971i	$-0.354238\pi$
$752$		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	$-0.878783\pi$
$757$	−29.4578	+	17.0074i	−1.07066	+	0.618146i	−0.142299	+	0.989824i	$0.545449\pi$
$758$		0			0
$759$	6.35590	+	6.35590i	0.230705	+	0.230705i
$760$		0			0
$761$	−29.3342	+	7.86008i	−1.06336	+	0.284928i	−0.747764	−	0.663965i	$-0.768872\pi$
$761$	−29.3342	+	7.86008i	−1.06336	+	0.284928i	−0.315600	+	0.948892i	$0.602206\pi$
$762$		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	−0.0106506	−	0.999943i	$-0.503390\pi$
$769$	−14.4158	−	3.86269i	−0.519846	−	0.139292i	−0.509195	+	0.860651i	$0.670057\pi$
$770$		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	$-0.633921\pi$
$773$	−8.50020	+	2.27762i	−0.305731	+	0.0819204i	0.102692	+	0.994713i	$0.467254\pi$
$774$		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	−0.633346	−	0.773869i	$-0.718319\pi$
$787$	−22.2999	+	5.97524i	−0.794906	+	0.212994i	−0.161560	+	0.986863i	$0.551652\pi$
$788$		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	0.379436	+	0.925218i	$0.376118\pi$
$797$	−17.2646	+	29.9032i	−0.611544	+	1.05923i	−0.990980	+	0.134007i	$0.957215\pi$
$798$		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	0.730051	−	0.683393i	$-0.239496\pi$
$809$	−6.45115	−	11.1737i	−0.226810	−	0.392847i	−0.956861	+	0.290546i	$0.906163\pi$
$810$		0			0
$811$	−18.8656	+	18.8656i	−0.662463	+	0.662463i	−0.955960	−	0.293497i	$-0.905181\pi$
$811$	−18.8656	+	18.8656i	−0.662463	+	0.662463i	0.293497	+	0.955960i	$0.405181\pi$
$812$		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	0.277040	+	0.960858i	$0.410647\pi$
$821$	−12.7023	+	47.4056i	−0.443313	+	1.65447i	−0.720353	+	0.693608i	$0.756020\pi$
$822$		0			0
$823$	14.9212	−	25.8443i	0.520121	−	0.900876i	−0.479606	−	0.877484i	$-0.659220\pi$
$823$	14.9212	−	25.8443i	0.520121	−	0.900876i	0.999726	−	0.0233914i	$-0.00744639\pi$
$824$		0			0
$825$	4.58558	+	17.1136i	0.159649	+	0.595820i
$826$		0			0
$827$	39.0150	+	39.0150i	1.35668	+	1.35668i	0.877971	+	0.478714i	$0.158897\pi$
$827$	39.0150	+	39.0150i	1.35668	+	1.35668i	0.478714	+	0.877971i	$0.341103\pi$
$828$		0			0
$829$	−17.4500	−	30.2242i	−0.606063	−	1.04973i	−0.991883	−	0.127157i	$-0.959415\pi$
$829$	−17.4500	−	30.2242i	−0.606063	−	1.04973i	0.385820	−	0.922574i	$-0.373919\pi$
$830$		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	0.961154	+	0.276011i	$0.0890127\pi$
$839$	7.72729	+	28.8387i	0.266776	+	0.995621i	−0.694379	+	0.719610i	$0.744321\pi$
$840$		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	$-0.552850\pi$
$853$	23.9776	−	23.9776i	0.820976	−	0.820976i	0.986248	+	0.165272i	$0.0528502\pi$
$854$		0			0
$855$	−0.211410	−	0.366173i	−0.00723007	−	0.0125229i
$856$		0			0
$857$		−	14.2812i		−	0.487836i	−0.969796	−	0.243918i	$-0.921567\pi$
$857$		−	14.2812i		−	0.487836i	0.969796	−	0.243918i	$-0.0784328\pi$
$858$		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	$-0.423262\pi$
$863$	−21.5136	−	21.5136i	−0.732331	−	0.732331i	−0.971081	+	0.238750i	$0.923262\pi$
$864$		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	$-0.670248\pi$
$877$	−15.4279	+	4.13388i	−0.520962	+	0.139591i	−0.0112504	+	0.999937i	$0.503581\pi$
$878$		0			0
$879$	−23.5021	+	23.5021i	−0.792706	+	0.792706i
$880$		0			0
$881$	12.4608	−	7.19424i	0.419814	−	0.242380i	−0.275183	−	0.961392i	$-0.588739\pi$
$881$	12.4608	−	7.19424i	0.419814	−	0.242380i	0.694998	+	0.719012i	$0.255405\pi$
$882$		0			0
$883$			3.55013i			0.119472i	0.998214	+	0.0597358i	$0.0190258\pi$
$883$			3.55013i			0.119472i	−0.998214	+	0.0597358i	$0.980974\pi$
$884$		0			0
$885$		−	3.03997i		−	0.102187i
$886$		0			0
$887$	−47.6836	+	27.5301i	−1.60106	+	0.924371i	−0.609782	+	0.792569i	$0.708743\pi$
$887$	−47.6836	+	27.5301i	−1.60106	+	0.924371i	−0.991276	+	0.131802i	$0.957924\pi$
$888$		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	−0.977279	−	0.211957i	$-0.932016\pi$
$907$	−38.6201	+	22.2973i	−1.28236	+	0.740370i	−0.305079	+	0.952327i	$0.598683\pi$
$908$		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	$-0.485731\pi$
$919$	−24.1895	−	13.9658i	−0.797939	−	0.460690i	−0.842750	+	0.538305i	$0.819065\pi$
$920$		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	0.972092	+	0.234602i	$0.0753785\pi$
$929$	7.54479	+	28.1575i	0.247536	+	0.923818i	−0.724555	+	0.689217i	$0.757955\pi$
$930$		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	$-0.237252\pi$
$941$	1.73699	+	1.73699i	0.0566241	+	0.0566241i	−0.678228	+	0.734852i	$0.737252\pi$
$942$		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	0.212709	+	0.977116i	$0.431771\pi$
$947$	−14.1576	+	52.8369i	−0.460060	+	1.71697i	−0.672769	+	0.739852i	$0.734895\pi$
$948$		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	0.991985	+	0.126353i	$0.0403273\pi$
$953$	49.3129	+	28.4708i	1.59740	+	0.922261i	0.605418	+	0.795908i	$0.293006\pi$
$954$		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	$-0.0230024\pi$
$967$	28.7702	+	28.7702i	0.925189	+	0.925189i	−0.0722014	+	0.997390i	$0.523002\pi$
$968$		0			0
$969$	17.5498	−	4.70246i	0.563781	−	0.151065i
$970$		0			0
$971$	27.3131	−	47.3076i	0.876518	−	1.51817i	0.0213803	−	0.999771i	$-0.493194\pi$
$971$	27.3131	−	47.3076i	0.876518	−	1.51817i	0.855137	−	0.518402i	$-0.173473\pi$
$972$		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	0.200622	−	0.979669i	$-0.435704\pi$
$977$	−3.60921	−	0.967085i	−0.115469	−	0.0309398i	−0.316091	+	0.948729i	$0.602370\pi$
$978$		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	$-0.905078\pi$
$983$	−20.7574	+	20.7574i	−0.662057	+	0.662057i	0.293808	+	0.955865i	$0.405078\pi$
$984$		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	−0.554455	+	0.832214i	$0.687073\pi$
$991$	−48.8698	+	28.2150i	−1.55240	+	0.896279i	−0.997946	+	0.0640655i	$0.979593\pi$
$992$		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	−0.972376	−	0.233420i	$-0.925008\pi$
$997$	−52.4375	−	30.2748i	−1.66071	−	0.958812i	−0.688336	−	0.725392i	$-0.741658\pi$
$998$		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.15.4		48
4.3	odd	2		104.2.u.a.67.8	yes	48
8.3	odd	2	inner	416.2.bk.a.15.3		48
8.5	even	2		104.2.u.a.67.10	yes	48
12.11	even	2		936.2.ed.d.379.5		48
13.7	odd	12	inner	416.2.bk.a.111.3		48
24.5	odd	2		936.2.ed.d.379.3		48
52.7	even	12		104.2.u.a.59.10	yes	48
104.59	even	12	inner	416.2.bk.a.111.4		48
104.85	odd	12		104.2.u.a.59.8	✓	48
156.59	odd	12		936.2.ed.d.163.3		48
312.293	even	12		936.2.ed.d.163.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.59.8	✓	48	104.85	odd	12
104.2.u.a.59.10	yes	48	52.7	even	12
104.2.u.a.67.8	yes	48	4.3	odd	2
104.2.u.a.67.10	yes	48	8.5	even	2
416.2.bk.a.15.3		48	8.3	odd	2	inner
416.2.bk.a.15.4		48	1.1	even	1	trivial
416.2.bk.a.111.3		48	13.7	odd	12	inner
416.2.bk.a.111.4		48	104.59	even	12	inner
936.2.ed.d.163.3		48	156.59	odd	12
936.2.ed.d.163.5		48	312.293	even	12
936.2.ed.d.379.3		48	24.5	odd	2
936.2.ed.d.379.5		48	12.11	even	2

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

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