LMFDB - Embedded newform 416.2.bk.a.271.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			271.6
Character	$\chi$	$=$	416.271
Dual form			416.2.bk.a.175.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.0597334 - 0.103461i) q^{3} +(2.08890 - 2.08890i) q^{5} +(1.83497 - 0.491678i) q^{7} +(1.49286 - 2.58572i) q^{9} +(-5.53261 - 1.48246i) q^{11} +(-0.0282216 + 3.60544i) q^{13} +(-0.340898 - 0.0913433i) q^{15} +(-3.70190 - 2.13730i) q^{17} +(4.12503 - 1.10530i) q^{19} +(-0.160478 - 0.160478i) q^{21} +(1.56097 + 2.70367i) q^{23} -3.72704i q^{25} -0.715095 q^{27} +(3.41586 - 1.97215i) q^{29} +(5.91568 - 5.91568i) q^{31} +(0.177104 + 0.660963i) q^{33} +(2.80600 - 4.86014i) q^{35} +(0.0218323 - 0.0814791i) q^{37} +(0.374709 - 0.212445i) q^{39} +(-1.89625 + 7.07690i) q^{41} +(3.96129 + 2.28705i) q^{43} +(-2.28286 - 8.51976i) q^{45} +(-1.33607 - 1.33607i) q^{47} +(-2.93682 + 1.69557i) q^{49} +0.510672i q^{51} +7.65994i q^{53} +(-14.6538 + 8.46037i) q^{55} +(-0.360758 - 0.360758i) q^{57} +(-0.332032 - 1.23916i) q^{59} +(5.12445 + 2.95860i) q^{61} +(1.46802 - 5.47871i) q^{63} +(7.47247 + 7.59037i) q^{65} +(-0.943589 + 3.52152i) q^{67} +(0.186483 - 0.322999i) q^{69} +(1.87184 + 6.98581i) q^{71} +(-2.35363 + 2.35363i) q^{73} +(-0.385604 + 0.222629i) q^{75} -10.8810 q^{77} +4.48211i q^{79} +(-4.43588 - 7.68316i) q^{81} +(-0.871274 - 0.871274i) q^{83} +(-12.1975 + 3.26832i) q^{85} +(-0.408082 - 0.235606i) q^{87} +(-0.761451 - 0.204030i) q^{89} +(1.72093 + 6.62974i) q^{91} +(-0.965407 - 0.258680i) q^{93} +(6.30793 - 10.9257i) q^{95} +(11.8738 - 3.18157i) q^{97} +(-12.0926 + 12.0926i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83}+ \cdots - 168 q^{99}+O(q^{100}) $ 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Character values

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

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{12}\right)$

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
$3$	−0.0597334	−	0.103461i	−0.0344871	−	0.0597334i	0.848267	−	0.529569i	$-0.177646\pi$
$3$	−0.0597334	−	0.103461i	−0.0344871	−	0.0597334i	−0.882754	+	0.469836i	$0.844313\pi$
$4$		0			0
$5$	2.08890	−	2.08890i	0.934186	−	0.934186i	−0.0637779	−	0.997964i	$-0.520315\pi$
$5$	2.08890	−	2.08890i	0.934186	−	0.934186i	0.997964	+	0.0637779i	$0.0203149\pi$
$6$		0			0
$7$	1.83497	−	0.491678i	0.693552	−	0.185837i	0.105212	−	0.994450i	$-0.466448\pi$
$7$	1.83497	−	0.491678i	0.693552	−	0.185837i	0.588341	+	0.808613i	$0.299781\pi$
$8$		0			0
$9$	1.49286	−	2.58572i	0.497621	−	0.861905i
$10$		0			0
$11$	−5.53261	−	1.48246i	−1.66814	−	0.446978i	−0.703535	−	0.710661i	$-0.748396\pi$
$11$	−5.53261	−	1.48246i	−1.66814	−	0.446978i	−0.964609	+	0.263683i	$0.915063\pi$
$12$		0			0
$13$	−0.0282216	+	3.60544i	−0.00782726	+	0.999969i
$13$	−0.0282216	+	3.60544i	−0.00782726	+	0.999969i
$14$		0			0
$15$	−0.340898	−	0.0913433i	−0.0880194	−	0.0235847i
$16$		0			0
$17$	−3.70190	−	2.13730i	−0.897844	−	0.518370i	−0.0213437	−	0.999772i	$-0.506794\pi$
$17$	−3.70190	−	2.13730i	−0.897844	−	0.518370i	−0.876500	+	0.481402i	$0.840128\pi$
$18$		0			0
$19$	4.12503	−	1.10530i	0.946347	−	0.253573i	0.247535	−	0.968879i	$-0.420379\pi$
$19$	4.12503	−	1.10530i	0.946347	−	0.253573i	0.698811	+	0.715306i	$0.253713\pi$
$20$		0			0
$21$	−0.160478	−	0.160478i	−0.0350193	−	0.0350193i
$22$		0			0
$23$	1.56097	+	2.70367i	0.325484	+	0.563755i	0.981610	−	0.190897i	$-0.0611395\pi$
$23$	1.56097	+	2.70367i	0.325484	+	0.563755i	−0.656126	+	0.754651i	$0.727806\pi$
$24$		0			0
$25$		−	3.72704i		−	0.745408i
$26$		0			0
$27$	−0.715095			−0.137620
$28$		0			0
$29$	3.41586	−	1.97215i	0.634310	−	0.366219i	−0.148110	−	0.988971i	$-0.547319\pi$
$29$	3.41586	−	1.97215i	0.634310	−	0.366219i	0.782419	+	0.622752i	$0.213985\pi$
$30$		0			0
$31$	5.91568	−	5.91568i	1.06249	−	1.06249i	0.0645742	−	0.997913i	$-0.479431\pi$
$31$	5.91568	−	5.91568i	1.06249	−	1.06249i	0.997913	−	0.0645742i	$-0.0205689\pi$
$32$		0			0
$33$	0.177104	+	0.660963i	0.0308299	+	0.115059i
$34$		0			0
$35$	2.80600	−	4.86014i	0.474301	−	0.821513i
$36$		0			0
$37$	0.0218323	−	0.0814791i	0.00358920	−	0.0133951i	−0.964108	−	0.265510i	$-0.914460\pi$
$37$	0.0218323	−	0.0814791i	0.00358920	−	0.0133951i	0.967697	+	0.252115i	$0.0811262\pi$
$38$		0			0
$39$	0.374709	−	0.212445i	0.0600015	−	0.0340185i
$40$		0			0
$41$	−1.89625	+	7.07690i	−0.296144	+	1.10523i	0.644160	+	0.764891i	$0.277207\pi$
$41$	−1.89625	+	7.07690i	−0.296144	+	1.10523i	−0.940304	+	0.340335i	$0.889460\pi$
$42$		0			0
$43$	3.96129	+	2.28705i	0.604091	+	0.348772i	0.770649	−	0.637260i	$-0.219932\pi$
$43$	3.96129	+	2.28705i	0.604091	+	0.348772i	−0.166558	+	0.986032i	$0.553265\pi$
$44$		0			0
$45$	−2.28286	−	8.51976i	−0.340309	−	1.27005i
$46$		0			0
$47$	−1.33607	−	1.33607i	−0.194886	−	0.194886i	0.602917	−	0.797804i	$-0.294005\pi$
$47$	−1.33607	−	1.33607i	−0.194886	−	0.194886i	−0.797804	+	0.602917i	$0.794005\pi$
$48$		0			0
$49$	−2.93682	+	1.69557i	−0.419546	+	0.242225i
$50$		0			0
$51$			0.510672i			0.0715083i
$52$		0			0
$53$			7.65994i			1.05217i	0.850431	+	0.526087i	$0.176341\pi$
$53$			7.65994i			1.05217i	−0.850431	+	0.526087i	$0.823659\pi$
$54$		0			0
$55$	−14.6538	+	8.46037i	−1.97592	+	1.14080i
$56$		0			0
$57$	−0.360758	−	0.360758i	−0.0477835	−	0.0477835i
$58$		0			0
$59$	−0.332032	−	1.23916i	−0.0432269	−	0.161325i	0.940939	−	0.338577i	$-0.109946\pi$
$59$	−0.332032	−	1.23916i	−0.0432269	−	0.161325i	−0.984165	+	0.177253i	$0.943279\pi$
$60$		0			0
$61$	5.12445	+	2.95860i	0.656118	+	0.378810i	0.790796	−	0.612079i	$-0.209667\pi$
$61$	5.12445	+	2.95860i	0.656118	+	0.378810i	−0.134678	+	0.990889i	$0.543000\pi$
$62$		0			0
$63$	1.46802	−	5.47871i	0.184953	−	0.690253i
$64$		0			0
$65$	7.47247	+	7.59037i	0.926845	+	0.941470i
$66$		0			0
$67$	−0.943589	+	3.52152i	−0.115278	+	0.430222i	−0.999308	−	0.0372082i	$-0.988154\pi$
$67$	−0.943589	+	3.52152i	−0.115278	+	0.430222i	0.884030	+	0.467431i	$0.154820\pi$
$68$		0			0
$69$	0.186483	−	0.322999i	0.0224500	−	0.0388845i
$70$		0			0
$71$	1.87184	+	6.98581i	0.222147	+	0.829064i	0.983528	+	0.180758i	$0.0578552\pi$
$71$	1.87184	+	6.98581i	0.222147	+	0.829064i	−0.761381	+	0.648305i	$0.775478\pi$
$72$		0			0
$73$	−2.35363	+	2.35363i	−0.275472	+	0.275472i	−0.831298	−	0.555827i	$-0.812402\pi$
$73$	−2.35363	+	2.35363i	−0.275472	+	0.275472i	0.555827	+	0.831298i	$0.312402\pi$
$74$		0			0
$75$	−0.385604	+	0.222629i	−0.0445257	+	0.0257069i
$76$		0			0
$77$	−10.8810			−1.24001
$78$		0			0
$79$			4.48211i			0.504277i	0.967691	+	0.252139i	$0.0811339\pi$
$79$			4.48211i			0.504277i	−0.967691	+	0.252139i	$0.918866\pi$
$80$		0			0
$81$	−4.43588	−	7.68316i	−0.492875	−	0.853685i
$82$		0			0
$83$	−0.871274	−	0.871274i	−0.0956347	−	0.0956347i	0.657671	−	0.753305i	$-0.271542\pi$
$83$	−0.871274	−	0.871274i	−0.0956347	−	0.0956347i	−0.753305	+	0.657671i	$0.771542\pi$
$84$		0			0
$85$	−12.1975	+	3.26832i	−1.32301	+	0.354499i
$86$		0			0
$87$	−0.408082	−	0.235606i	−0.0437510	−	0.0252596i
$88$		0			0
$89$	−0.761451	−	0.204030i	−0.0807137	−	0.0216272i	0.218236	−	0.975896i	$-0.429970\pi$
$89$	−0.761451	−	0.204030i	−0.0807137	−	0.0216272i	−0.298950	+	0.954269i	$0.596636\pi$
$90$		0			0
$91$	1.72093	+	6.62974i	0.180403	+	0.694986i
$92$		0			0
$93$	−0.965407	−	0.258680i	−0.100108	−	0.0268239i
$94$		0			0
$95$	6.30793	−	10.9257i	0.647180	−	1.12095i
$96$		0			0
$97$	11.8738	−	3.18157i	1.20560	−	0.323039i	0.400566	−	0.916268i	$-0.368814\pi$
$97$	11.8738	−	3.18157i	1.20560	−	0.323039i	0.805034	+	0.593229i	$0.202147\pi$
$98$		0			0
$99$	−12.0926	+	12.0926i	−1.21536	+	1.21536i
$100$		0			0
$101$	−2.64360	−	4.57885i	−0.263048	−	0.455613i	0.704002	−	0.710198i	$-0.251394\pi$
$101$	−2.64360	−	4.57885i	−0.263048	−	0.455613i	−0.967050	+	0.254585i	$0.918061\pi$
$102$		0			0
$103$	−15.1762			−1.49535			−0.747676	−	0.664064i	$-0.768830\pi$
$103$	−15.1762			−1.49535			−0.747676	+	0.664064i	$0.768830\pi$
$104$		0			0
$105$	−0.670448			−0.0654290
$106$		0			0
$107$	3.13866	+	5.43632i	0.303426	+	0.525549i	0.976910	−	0.213653i	$-0.0685361\pi$
$107$	3.13866	+	5.43632i	0.303426	+	0.525549i	−0.673484	+	0.739202i	$0.735203\pi$
$108$		0			0
$109$	−2.77694	+	2.77694i	−0.265983	+	0.265983i	−0.827479	−	0.561497i	$-0.810226\pi$
$109$	−2.77694	+	2.77694i	−0.265983	+	0.265983i	0.561497	+	0.827479i	$0.310226\pi$
$110$		0			0
$111$	−0.00973405	+	0.00260823i	−0.000923915	+	0.000247562i
$112$		0			0
$113$	−4.95084	+	8.57511i	−0.465736	+	0.806678i	−0.999234	−	0.0391228i	$-0.987544\pi$
$113$	−4.95084	+	8.57511i	−0.465736	+	0.806678i	0.533499	+	0.845801i	$0.320877\pi$
$114$		0			0
$115$	8.90842	+	2.38700i	0.830714	+	0.222589i
$116$		0			0
$117$	9.28052	+	5.45541i	0.857984	+	0.504352i
$118$		0			0
$119$	−7.84373	−	2.10172i	−0.719034	−	0.192665i
$120$		0			0
$121$	18.8858	+	10.9037i	1.71689	+	0.991247i
$122$		0			0
$123$	0.845454	−	0.226539i	0.0762320	−	0.0204263i
$124$		0			0
$125$	2.65909	+	2.65909i	0.237837	+	0.237837i
$126$		0			0
$127$	4.55719	+	7.89328i	0.404385	+	0.700416i	0.994250	−	0.107087i	$-0.0341522\pi$
$127$	4.55719	+	7.89328i	0.404385	+	0.700416i	−0.589865	+	0.807502i	$0.700819\pi$
$128$		0			0
$129$		−	0.546453i		−	0.0481125i
$130$		0			0
$131$	−9.19603			−0.803461			−0.401730	−	0.915758i	$-0.631591\pi$
$131$	−9.19603			−0.803461			−0.401730	+	0.915758i	$0.631591\pi$
$132$		0			0
$133$	7.02584	−	4.05637i	0.609218	−	0.351732i
$134$		0			0
$135$	−1.49377	+	1.49377i	−0.128563	+	0.128563i
$136$		0			0
$137$	−2.93822	−	10.9656i	−0.251029	−	0.936852i	−0.970257	−	0.242078i	$-0.922171\pi$
$137$	−2.93822	−	10.9656i	−0.251029	−	0.936852i	0.719228	−	0.694774i	$-0.244496\pi$
$138$		0			0
$139$	−1.53429	+	2.65747i	−0.130137	+	0.225404i	−0.923729	−	0.383046i	$-0.874875\pi$
$139$	−1.53429	+	2.65747i	−0.130137	+	0.225404i	0.793592	+	0.608450i	$0.208208\pi$
$140$		0			0
$141$	−0.0584236	+	0.218040i	−0.00492015	+	0.0183623i
$142$		0			0
$143$	5.50105	−	19.9057i	0.460021	−	1.66459i
$144$		0			0
$145$	3.01578	−	11.2550i	0.250447	−	0.934680i
$146$		0			0
$147$	0.350852	+	0.202565i	0.0289378	+	0.0167073i
$148$		0			0
$149$	−4.78484	−	17.8573i	−0.391990	−	1.46292i	−0.826848	−	0.562426i	$-0.809868\pi$
$149$	−4.78484	−	17.8573i	−0.391990	−	1.46292i	0.434858	−	0.900499i	$-0.356799\pi$
$150$		0			0
$151$	16.0837	+	16.0837i	1.30887	+	1.30887i	0.922229	+	0.386643i	$0.126365\pi$
$151$	16.0837	+	16.0837i	1.30887	+	1.30887i	0.386643	+	0.922229i	$0.373635\pi$
$152$		0			0
$153$	−11.0529	+	6.38138i	−0.893572	+	0.515904i
$154$		0			0
$155$		−	24.7146i		−	1.98512i
$156$		0			0
$157$			7.07494i			0.564641i	0.959320	+	0.282321i	$0.0911042\pi$
$157$			7.07494i			0.564641i	−0.959320	+	0.282321i	$0.908896\pi$
$158$		0			0
$159$	0.792507	−	0.457554i	0.0628499	−	0.0362864i
$160$		0			0
$161$	4.19366	+	4.19366i	0.330506	+	0.330506i
$162$		0			0
$163$	−1.63996	−	6.12043i	−0.128452	−	0.479389i	0.871487	−	0.490418i	$-0.163156\pi$
$163$	−1.63996	−	6.12043i	−0.128452	−	0.479389i	−0.999939	+	0.0110292i	$0.996489\pi$
$164$		0			0
$165$	1.75064	+	1.01073i	0.136287	+	0.0786855i
$166$		0			0
$167$	−4.89526	+	18.2694i	−0.378806	+	1.41372i	0.468896	+	0.883253i	$0.344652\pi$
$167$	−4.89526	+	18.2694i	−0.378806	+	1.41372i	−0.847703	+	0.530472i	$0.822015\pi$
$168$		0			0
$169$	−12.9984	−	0.203502i	−0.999877	−	0.0156540i
$170$		0			0
$171$	3.30012	−	12.3162i	0.252366	−	0.941845i
$172$		0			0
$173$	2.93213	−	5.07860i	0.222926	−	0.386119i	−0.732769	−	0.680477i	$-0.761773\pi$
$173$	2.93213	−	5.07860i	0.222926	−	0.386119i	0.955695	+	0.294358i	$0.0951059\pi$
$174$		0			0
$175$	−1.83250	−	6.83899i	−0.138524	−	0.516979i
$176$		0			0
$177$	−0.108372	+	0.108372i	−0.00814571	+	0.00814571i
$178$		0			0
$179$	−7.92052	+	4.57292i	−0.592008	+	0.341796i	−0.765891	−	0.642970i	$-0.777702\pi$
$179$	−7.92052	+	4.57292i	−0.592008	+	0.341796i	0.173883	+	0.984766i	$0.444368\pi$
$180$		0			0
$181$	−11.5513			−0.858599			−0.429300	−	0.903162i	$-0.641240\pi$
$181$	−11.5513			−0.858599			−0.429300	+	0.903162i	$0.641240\pi$
$182$		0			0
$183$		−	0.706909i		−	0.0522562i
$184$		0			0
$185$	−0.124597	−	0.215808i	−0.00916052	−	0.0158665i
$186$		0			0
$187$	17.3127	+	17.3127i	1.26603	+	1.26603i
$188$		0			0
$189$	−1.31218	+	0.351597i	−0.0954468	+	0.0255749i
$190$		0			0
$191$	−18.0500	−	10.4212i	−1.30605	−	0.754048i	−0.324615	−	0.945846i	$-0.605235\pi$
$191$	−18.0500	−	10.4212i	−1.30605	−	0.754048i	−0.981434	+	0.191798i	$0.938568\pi$
$192$		0			0
$193$	−2.91871	−	0.782065i	−0.210093	−	0.0562943i	0.152237	−	0.988344i	$-0.451352\pi$
$193$	−2.91871	−	0.782065i	−0.210093	−	0.0562943i	−0.362331	+	0.932050i	$0.618019\pi$
$194$		0			0
$195$	0.338954	−	1.22651i	0.0242730	−	0.0878321i
$196$		0			0
$197$	20.3251	+	5.44609i	1.44810	+	0.388018i	0.895363	−	0.445337i	$-0.146916\pi$
$197$	20.3251	+	5.44609i	1.44810	+	0.388018i	0.552738	+	0.833355i	$0.313583\pi$
$198$		0			0
$199$	2.68040	−	4.64259i	0.190009	−	0.329105i	−0.755244	−	0.655443i	$-0.772482\pi$
$199$	2.68040	−	4.64259i	0.190009	−	0.329105i	0.945253	+	0.326339i	$0.105815\pi$
$200$		0			0
$201$	0.420705	−	0.112727i	0.0296742	−	0.00795118i
$202$		0			0
$203$	5.29833	−	5.29833i	0.371870	−	0.371870i
$204$		0			0
$205$	10.8219	+	18.7440i	0.755833	+	1.30914i
$206$		0			0
$207$	9.32124			0.647871
$208$		0			0
$209$	−24.4607			−1.69198
$210$		0			0
$211$	−12.8300	−	22.2223i	−0.883256	−	1.52984i	−0.847699	−	0.530477i	$-0.822013\pi$
$211$	−12.8300	−	22.2223i	−0.883256	−	1.52984i	−0.0355568	−	0.999368i	$-0.511320\pi$
$212$		0			0
$213$	0.610950	−	0.610950i	0.0418616	−	0.0418616i
$214$		0			0
$215$	13.0522	−	3.49732i	0.890151	−	0.238515i
$216$		0			0
$217$	7.94647	−	13.7637i	0.539441	−	0.934340i
$218$		0			0
$219$	0.384100	+	0.102919i	0.0259551	+	0.00695464i
$220$		0			0
$221$	7.81037	−	13.2867i	0.525382	−	0.893759i
$222$		0			0
$223$	12.6427	+	3.38761i	0.846621	+	0.226851i	0.655952	−	0.754803i	$-0.272268\pi$
$223$	12.6427	+	3.38761i	0.846621	+	0.226851i	0.190669	+	0.981654i	$0.438934\pi$
$224$		0			0
$225$	−9.63706	−	5.56396i	−0.642471	−	0.370931i
$226$		0			0
$227$	26.0325	−	6.97540i	1.72784	−	0.462973i	0.748157	−	0.663522i	$-0.230939\pi$
$227$	26.0325	−	6.97540i	1.72784	−	0.462973i	0.979684	+	0.200549i	$0.0642725\pi$
$228$		0			0
$229$	−0.157142	−	0.157142i	−0.0103842	−	0.0103842i	0.701896	−	0.712280i	$-0.252337\pi$
$229$	−0.157142	−	0.157142i	−0.0103842	−	0.0103842i	−0.712280	+	0.701896i	$0.752337\pi$
$230$		0			0
$231$	0.649962	+	1.12577i	0.0427643	+	0.0740700i
$232$		0			0
$233$		−	24.0418i		−	1.57503i	−0.616297	−	0.787514i	$-0.711368\pi$
$233$		−	24.0418i		−	1.57503i	0.616297	−	0.787514i	$-0.288632\pi$
$234$		0			0
$235$	−5.58185			−0.364120
$236$		0			0
$237$	0.463725	−	0.267732i	0.0301222	−	0.0173910i
$238$		0			0
$239$	19.4668	−	19.4668i	1.25920	−	1.25920i	0.307729	−	0.951474i	$-0.400431\pi$
$239$	19.4668	−	19.4668i	1.25920	−	1.25920i	0.951474	−	0.307729i	$-0.0995689\pi$
$240$		0			0
$241$	3.69069	+	13.7738i	0.237738	+	0.887251i	0.976895	+	0.213718i	$0.0685573\pi$
$241$	3.69069	+	13.7738i	0.237738	+	0.887251i	−0.739157	+	0.673533i	$0.764776\pi$
$242$		0			0
$243$	−1.60258	+	2.77576i	−0.102806	+	0.178065i
$244$		0			0
$245$	−2.59284	+	9.67663i	−0.165651	+	0.618217i
$246$		0			0
$247$	3.86867	+	14.9037i	0.246158	+	0.948303i
$248$		0			0
$249$	−0.0380989	+	0.142187i	−0.00241442	+	0.00901075i
$250$		0			0
$251$	−1.40119	−	0.808976i	−0.0884422	−	0.0510621i	0.455127	−	0.890427i	$-0.349594\pi$
$251$	−1.40119	−	0.808976i	−0.0884422	−	0.0510621i	−0.543569	+	0.839365i	$0.682927\pi$
$252$		0			0
$253$	−4.62813	−	17.2724i	−0.290968	−	1.08591i
$254$		0			0
$255$	1.06674	+	1.06674i	0.0668021	+	0.0668021i
$256$		0			0
$257$	9.69476	−	5.59727i	0.604743	−	0.349148i	−0.166162	−	0.986098i	$-0.553138\pi$
$257$	9.69476	−	5.59727i	0.604743	−	0.349148i	0.770905	+	0.636950i	$0.219804\pi$
$258$		0			0
$259$		−	0.160246i		−	0.00995720i
$260$		0			0
$261$		−	11.7766i		−	0.728953i
$262$		0			0
$263$	−7.02915	+	4.05828i	−0.433436	+	0.250245i	−0.700809	−	0.713349i	$-0.747178\pi$
$263$	−7.02915	+	4.05828i	−0.433436	+	0.250245i	0.267373	+	0.963593i	$0.413844\pi$
$264$		0			0
$265$	16.0009	+	16.0009i	0.982927	+	0.982927i
$266$		0			0
$267$	0.0243748	+	0.0909681i	0.00149172	+	0.00556716i
$268$		0			0
$269$	−7.47014	−	4.31289i	−0.455462	−	0.262961i	0.254672	−	0.967028i	$-0.418033\pi$
$269$	−7.47014	−	4.31289i	−0.455462	−	0.262961i	−0.710134	+	0.704066i	$0.751366\pi$
$270$		0			0
$271$	−0.638081	+	2.38135i	−0.0387607	+	0.144657i	−0.982594	−	0.185764i	$-0.940524\pi$
$271$	−0.638081	+	2.38135i	−0.0387607	+	0.144657i	0.943834	+	0.330421i	$0.107191\pi$
$272$		0			0
$273$	0.583124	−	0.574066i	0.0352923	−	0.0347441i
$274$		0			0
$275$	−5.52518	+	20.6202i	−0.333181	+	1.24345i
$276$		0			0
$277$	−7.30805	+	12.6579i	−0.439098	+	0.760540i	−0.997620	−	0.0689494i	$-0.978035\pi$
$277$	−7.30805	+	12.6579i	−0.439098	+	0.760540i	0.558522	+	0.829490i	$0.311369\pi$
$278$		0			0
$279$	−6.46496	−	24.1276i	−0.387047	−	1.44448i
$280$		0			0
$281$	−3.44904	+	3.44904i	−0.205753	+	0.205753i	−0.802459	−	0.596707i	$-0.796476\pi$
$281$	−3.44904	+	3.44904i	−0.205753	+	0.205753i	0.596707	+	0.802459i	$0.296476\pi$
$282$		0			0
$283$	−16.0402	+	9.26082i	−0.953492	+	0.550499i	−0.894164	−	0.447740i	$-0.852229\pi$
$283$	−16.0402	+	9.26082i	−0.953492	+	0.550499i	−0.0593279	+	0.998239i	$0.518896\pi$
$284$		0			0
$285$	−1.50718			−0.0892774
$286$		0			0
$287$			13.9182i			0.821566i
$288$		0			0
$289$	0.636064	+	1.10170i	0.0374155	+	0.0648056i
$290$		0			0
$291$	−1.03843	−	1.03843i	−0.0608738	−	0.0608738i
$292$		0			0
$293$	16.3695	−	4.38618i	0.956314	−	0.256244i	0.253275	−	0.967394i	$-0.418492\pi$
$293$	16.3695	−	4.38618i	0.956314	−	0.256244i	0.703039	+	0.711151i	$0.251826\pi$
$294$		0			0
$295$	−3.28207	−	1.89490i	−0.191089	−	0.110326i
$296$		0			0
$297$	3.95634	+	1.06010i	0.229570	+	0.0615132i
$298$		0			0
$299$	−9.79198	+	5.55167i	−0.566285	+	0.321061i
$300$		0			0
$301$	8.39333	+	2.24899i	0.483783	+	0.129629i
$302$		0			0
$303$	−0.315822	+	0.547020i	−0.0181435	+	0.0314255i
$304$		0			0
$305$	16.8847	−	4.52424i	0.966815	−	0.259057i
$306$		0			0
$307$	14.6326	−	14.6326i	0.835125	−	0.835125i	−0.153087	−	0.988213i	$-0.548922\pi$
$307$	14.6326	−	14.6326i	0.835125	−	0.835125i	0.988213	+	0.153087i	$0.0489216\pi$
$308$		0			0
$309$	0.906524	+	1.57014i	0.0515703	+	0.0893224i
$310$		0			0
$311$	−19.2142			−1.08954			−0.544768	−	0.838587i	$-0.683382\pi$
$311$	−19.2142			−1.08954			−0.544768	+	0.838587i	$0.683382\pi$
$312$		0			0
$313$	−1.56387			−0.0883950			−0.0441975	−	0.999023i	$-0.514073\pi$
$313$	−1.56387			−0.0883950			−0.0441975	+	0.999023i	$0.514073\pi$
$314$		0			0
$315$	−8.37796	−	14.5110i	−0.472044	−	0.817605i
$316$		0			0
$317$	0.129483	−	0.129483i	0.00727249	−	0.00727249i	−0.703461	−	0.710734i	$-0.748363\pi$
$317$	0.129483	−	0.129483i	0.00727249	−	0.00727249i	0.710734	+	0.703461i	$0.248363\pi$
$318$		0			0
$319$	−21.8223	+	5.84726i	−1.22181	+	0.327383i
$320$		0			0
$321$	0.374966	−	0.649460i	0.0209286	−	0.0362493i
$322$		0			0
$323$	−17.6328	−	4.72470i	−0.981116	−	0.262889i
$324$		0			0
$325$	13.4376	+	0.105183i	0.745385	+	0.00583450i
$326$		0			0
$327$	0.453182	+	0.121430i	0.0250610	+	0.00671507i
$328$		0			0
$329$	−3.10857	−	1.79473i	−0.171381	−	0.0989468i
$330$		0			0
$331$	−12.4428	+	3.33403i	−0.683917	+	0.183255i	−0.584016	−	0.811742i	$-0.698519\pi$
$331$	−12.4428	+	3.33403i	−0.683917	+	0.183255i	−0.0999013	+	0.994997i	$0.531853\pi$
$332$		0			0
$333$	−0.178089	−	0.178089i	−0.00975923	−	0.00975923i
$334$		0			0
$335$	5.38505	+	9.32719i	0.294217	+	0.509599i
$336$		0			0
$337$		−	13.4906i		−	0.734881i	−0.930047	−	0.367441i	$-0.880234\pi$
$337$		−	13.4906i		−	0.734881i	0.930047	−	0.367441i	$-0.119766\pi$
$338$		0			0
$339$	1.18292			0.0642475
$340$		0			0
$341$	−41.4989	+	23.9594i	−2.24729	+	1.29747i
$342$		0			0
$343$	−13.9583	+	13.9583i	−0.753678	+	0.753678i
$344$		0			0
$345$	−0.285168	−	1.06426i	−0.0153529	−	0.0572978i
$346$		0			0
$347$	−12.7983	+	22.1673i	−0.687049	+	1.19000i	0.285739	+	0.958307i	$0.407761\pi$
$347$	−12.7983	+	22.1673i	−0.687049	+	1.19000i	−0.972788	+	0.231696i	$0.925572\pi$
$348$		0			0
$349$	2.09309	−	7.81152i	0.112041	−	0.418141i	−0.887008	−	0.461754i	$-0.847220\pi$
$349$	2.09309	−	7.81152i	0.112041	−	0.418141i	0.999048	+	0.0436131i	$0.0138869\pi$
$350$		0			0
$351$	0.0201811	−	2.57823i	0.00107719	−	0.137616i
$352$		0			0
$353$	−0.526505	+	1.96494i	−0.0280230	+	0.104583i	−0.978521	−	0.206148i	$-0.933907\pi$
$353$	−0.526505	+	1.96494i	−0.0280230	+	0.104583i	0.950498	+	0.310731i	$0.100574\pi$
$354$		0			0
$355$	18.5028	+	10.6826i	0.982026	+	0.566973i
$356$		0			0
$357$	0.251086	+	0.937065i	0.0132889	+	0.0495948i
$358$		0			0
$359$	9.92523	+	9.92523i	0.523833	+	0.523833i	0.918727	−	0.394894i	$-0.129219\pi$
$359$	9.92523	+	9.92523i	0.523833	+	0.523833i	−0.394894	+	0.918727i	$0.629219\pi$
$360$		0			0
$361$	−0.660296	+	0.381222i	−0.0347524	+	0.0200643i
$362$		0			0
$363$		−	2.60526i		−	0.136741i
$364$		0			0
$365$			9.83302i			0.514684i
$366$		0			0
$367$	12.9829	−	7.49570i	0.677703	−	0.391272i	−0.121286	−	0.992618i	$-0.538702\pi$
$367$	12.9829	−	7.49570i	0.677703	−	0.391272i	0.798989	+	0.601345i	$0.205368\pi$
$368$		0			0
$369$	15.4680	+	15.4680i	0.805232	+	0.805232i
$370$		0			0
$371$	3.76623	+	14.0557i	0.195533	+	0.729738i
$372$		0			0
$373$	−13.3903	−	7.73087i	−0.693322	−	0.400289i	0.111534	−	0.993761i	$-0.464424\pi$
$373$	−13.3903	−	7.73087i	−0.693322	−	0.400289i	−0.804855	+	0.593471i	$0.797757\pi$
$374$		0			0
$375$	0.116277	−	0.433950i	0.00600449	−	0.0224091i
$376$		0			0
$377$	7.01406	+	12.3713i	0.361243	+	0.637157i
$378$		0			0
$379$	0.572393	−	2.13620i	0.0294018	−	0.109729i	−0.949665	−	0.313266i	$-0.898577\pi$
$379$	0.572393	−	2.13620i	0.0294018	−	0.109729i	0.979067	+	0.203537i	$0.0652436\pi$
$380$		0			0
$381$	0.544433	−	0.942985i	0.0278921	−	0.0483106i
$382$		0			0
$383$	−0.253448	−	0.945882i	−0.0129506	−	0.0483323i	0.959148	−	0.282904i	$-0.0912978\pi$
$383$	−0.253448	−	0.945882i	−0.0129506	−	0.0483323i	−0.972099	+	0.234572i	$0.924631\pi$
$384$		0			0
$385$	−22.7295	+	22.7295i	−1.15840	+	1.15840i
$386$		0			0
$387$	11.8273	−	6.82851i	0.601217	−	0.347113i
$388$		0			0
$389$	3.51480			0.178208			0.0891038	−	0.996022i	$-0.471600\pi$
$389$	3.51480			0.178208			0.0891038	+	0.996022i	$0.471600\pi$
$390$		0			0
$391$		−	13.3450i		−	0.674885i
$392$		0			0
$393$	0.549310	+	0.951432i	0.0277090	+	0.0479934i
$394$		0			0
$395$	9.36270	+	9.36270i	0.471089	+	0.471089i
$396$		0			0
$397$	−28.1611	+	7.54575i	−1.41337	+	0.378710i	−0.883126	−	0.469137i	$-0.844565\pi$
$397$	−28.1611	+	7.54575i	−1.41337	+	0.378710i	−0.530241	+	0.847847i	$0.677898\pi$
$398$		0			0
$399$	−0.839355	−	0.484602i	−0.0420203	−	0.0242604i
$400$		0			0
$401$	−8.07154	−	2.16276i	−0.403073	−	0.108003i	0.0515861	−	0.998669i	$-0.483572\pi$
$401$	−8.07154	−	2.16276i	−0.403073	−	0.108003i	−0.454659	+	0.890665i	$0.650239\pi$
$402$		0			0
$403$	21.1617	+	21.4956i	1.05414	+	1.07077i
$404$		0			0
$405$	−25.3155	−	6.78327i	−1.25794	−	0.337063i
$406$		0			0
$407$	−0.241579	+	0.418427i	−0.0119746	+	0.0207406i
$408$		0			0
$409$	−3.07670	+	0.824398i	−0.152133	+	0.0407639i	−0.334082	−	0.942544i	$-0.608426\pi$
$409$	−3.07670	+	0.824398i	−0.152133	+	0.0407639i	0.181949	+	0.983308i	$0.441759\pi$
$410$		0			0
$411$	−0.959002	+	0.959002i	−0.0473041	+	0.0473041i
$412$		0			0
$413$	−1.21854	−	2.11056i	−0.0599602	−	0.103854i
$414$		0			0
$415$	−3.64001			−0.178681
$416$		0			0
$417$	0.366594			0.0179522
$418$		0			0
$419$	10.5630	+	18.2957i	0.516038	+	0.893804i	0.999827	+	0.0186190i	$0.00592695\pi$
$419$	10.5630	+	18.2957i	0.516038	+	0.893804i	−0.483789	+	0.875185i	$0.660740\pi$
$420$		0			0
$421$	22.8080	−	22.8080i	1.11159	−	1.11159i	0.118658	−	0.992935i	$-0.462141\pi$
$421$	22.8080	−	22.8080i	1.11159	−	1.11159i	0.992935	−	0.118658i	$-0.0378591\pi$
$422$		0			0
$423$	−5.44928	+	1.46013i	−0.264953	+	0.0709939i
$424$		0			0
$425$	−7.96578	+	13.7971i	−0.386397	+	0.669260i
$426$		0			0
$427$	10.8579	+	2.90936i	0.525449	+	0.140794i
$428$		0			0
$429$	−2.38806	+	0.619886i	−0.115297	+	0.0299284i
$430$		0			0
$431$	−19.4340	−	5.20732i	−0.936101	−	0.250828i	−0.241647	−	0.970364i	$-0.577688\pi$
$431$	−19.4340	−	5.20732i	−0.936101	−	0.250828i	−0.694454	+	0.719537i	$0.744354\pi$
$432$		0			0
$433$	19.6101	+	11.3219i	0.942402	+	0.544096i	0.890713	−	0.454567i	$-0.150206\pi$
$433$	19.6101	+	11.3219i	0.942402	+	0.544096i	0.0516897	+	0.998663i	$0.483539\pi$
$434$		0			0
$435$	−1.34460	+	0.360285i	−0.0644688	+	0.0172744i
$436$		0			0
$437$	9.42739	+	9.42739i	0.450973	+	0.450973i
$438$		0			0
$439$	−13.2654	−	22.9763i	−0.633122	−	1.09660i	−0.986910	−	0.161274i	$-0.948440\pi$
$439$	−13.2654	−	22.9763i	−0.633122	−	1.09660i	0.353788	−	0.935326i	$-0.384894\pi$
$440$		0			0
$441$			10.1250i			0.482145i
$442$		0			0
$443$	−6.75908			−0.321134			−0.160567	−	0.987025i	$-0.551332\pi$
$443$	−6.75908			−0.321134			−0.160567	+	0.987025i	$0.551332\pi$
$444$		0			0
$445$	−2.01680	+	1.16440i	−0.0956054	+	0.0551978i
$446$		0			0
$447$	−1.56172	+	1.56172i	−0.0738669	+	0.0738669i
$448$		0			0
$449$	−7.53276	−	28.1126i	−0.355493	−	1.32672i	−0.879863	−	0.475227i	$-0.842366\pi$
$449$	−7.53276	−	28.1126i	−0.355493	−	1.32672i	0.524370	−	0.851490i	$-0.324301\pi$
$450$		0			0
$451$	20.9824	−	36.3426i	0.988023	−	1.71131i
$452$		0			0
$453$	0.703306	−	2.62477i	0.0330442	−	0.123323i
$454$		0			0
$455$	17.4438	+	10.2540i	0.817776	+	0.480717i
$456$		0			0
$457$	−4.46465	+	16.6623i	−0.208848	+	0.779430i	0.779395	+	0.626533i	$0.215527\pi$
$457$	−4.46465	+	16.6623i	−0.208848	+	0.779430i	−0.988242	+	0.152896i	$0.951140\pi$
$458$		0			0
$459$	2.64722	+	1.52837i	0.123561	+	0.0713382i
$460$		0			0
$461$	7.40872	+	27.6497i	0.345058	+	1.28778i	0.892544	+	0.450961i	$0.148919\pi$
$461$	7.40872	+	27.6497i	0.345058	+	1.28778i	−0.547485	+	0.836815i	$0.684415\pi$
$462$		0			0
$463$	−8.53804	−	8.53804i	−0.396796	−	0.396796i	0.480305	−	0.877101i	$-0.340526\pi$
$463$	−8.53804	−	8.53804i	−0.396796	−	0.396796i	−0.877101	+	0.480305i	$0.840526\pi$
$464$		0			0
$465$	−2.55700	+	1.47628i	−0.118578	+	0.0684610i
$466$		0			0
$467$		−	1.47727i		−	0.0683599i	−0.999416	−	0.0341799i	$-0.989118\pi$
$467$		−	1.47727i		−	0.0683599i	0.999416	−	0.0341799i	$-0.0108819\pi$
$468$		0			0
$469$			6.92582i			0.319805i
$470$		0			0
$471$	0.731982	−	0.422610i	0.0337279	−	0.0194728i
$472$		0			0
$473$	−18.5258	−	18.5258i	−0.851817	−	0.851817i
$474$		0			0
$475$	−4.11949	−	15.3741i	−0.189015	−	0.705414i
$476$		0			0
$477$	19.8064	+	11.4353i	0.906874	+	0.523584i
$478$		0			0
$479$	−0.0258024	+	0.0962960i	−0.00117894	+	0.00439988i	−0.966513	−	0.256618i	$-0.917392\pi$
$479$	−0.0258024	+	0.0962960i	−0.00117894	+	0.00439988i	0.965334	+	0.261018i	$0.0840583\pi$
$480$		0			0
$481$	0.293152	+	0.0810144i	0.0133666	+	0.00369394i
$482$		0			0
$483$	0.183380	−	0.684382i	0.00834406	−	0.0311405i
$484$		0			0
$485$	18.1572	−	31.4492i	0.824475	−	1.42803i
$486$		0			0
$487$	−6.05385	−	22.5933i	−0.274326	−	1.02380i	−0.956292	−	0.292414i	$-0.905541\pi$
$487$	−6.05385	−	22.5933i	−0.274326	−	1.02380i	0.681966	−	0.731384i	$-0.261125\pi$
$488$		0			0
$489$	−0.535266	+	0.535266i	−0.0242056	+	0.0242056i
$490$		0			0
$491$	12.4224	−	7.17210i	0.560617	−	0.323672i	−0.192776	−	0.981243i	$-0.561749\pi$
$491$	12.4224	−	7.17210i	0.560617	−	0.323672i	0.753393	+	0.657570i	$0.228416\pi$
$492$		0			0
$493$	−16.8603			−0.759348
$494$		0			0
$495$			50.5208i			2.27074i
$496$		0			0
$497$	6.86954	+	11.8984i	0.308141	+	0.533716i
$498$		0			0
$499$	−29.9556	−	29.9556i	−1.34100	−	1.34100i	−0.895067	−	0.445932i	$-0.852872\pi$
$499$	−29.9556	−	29.9556i	−1.34100	−	1.34100i	−0.445932	−	0.895067i	$-0.647128\pi$
$500$		0			0
$501$	2.18258	−	0.584821i	0.0975105	−	0.0261279i
$502$		0			0
$503$	36.4031	+	21.0173i	1.62313	+	0.937116i	0.986075	+	0.166299i	$0.0531818\pi$
$503$	36.4031	+	21.0173i	1.62313	+	0.937116i	0.637057	+	0.770817i	$0.280152\pi$
$504$		0			0
$505$	−15.0870	−	4.04255i	−0.671363	−	0.179891i
$506$		0			0
$507$	0.755384	+	1.35699i	0.0335478	+	0.0602659i
$508$		0			0
$509$	−29.4691	−	7.89622i	−1.30619	−	0.349994i	−0.462406	−	0.886668i	$-0.653014\pi$
$509$	−29.4691	−	7.89622i	−1.30619	−	0.349994i	−0.843789	+	0.536675i	$0.819680\pi$
$510$		0			0
$511$	−3.16161	+	5.47607i	−0.139861	+	0.242247i
$512$		0			0
$513$	−2.94979	+	0.790394i	−0.130236	+	0.0348967i
$514$		0			0
$515$	−31.7015	+	31.7015i	−1.39694	+	1.39694i
$516$		0			0
$517$	5.41129	+	9.37264i	0.237988	+	0.412208i
$518$		0			0
$519$	−0.700584			−0.0307522
$520$		0			0
$521$	7.97376			0.349337			0.174668	−	0.984627i	$-0.444115\pi$
$521$	7.97376			0.349337			0.174668	+	0.984627i	$0.444115\pi$
$522$		0			0
$523$	4.99132	+	8.64522i	0.218255	+	0.378029i	0.954275	−	0.298931i	$-0.0966301\pi$
$523$	4.99132	+	8.64522i	0.218255	+	0.378029i	−0.736019	+	0.676960i	$0.763297\pi$
$524$		0			0
$525$	−0.598109	+	0.598109i	−0.0261036	+	0.0261036i
$526$		0			0
$527$	−34.5428	+	9.25572i	−1.50471	+	0.403186i
$528$		0			0
$529$	6.62677	−	11.4779i	0.288121	−	0.499039i
$530$		0			0
$531$	−3.69979	−	0.991357i	−0.160557	−	0.0430212i
$532$		0			0
$533$	−25.4618	−	7.03653i	−1.10287	−	0.304786i
$534$		0			0
$535$	17.9123	+	4.79959i	0.774417	+	0.207504i
$536$		0			0
$537$	0.946239	+	0.546312i	0.0408332	+	0.0235751i
$538$		0			0
$539$	18.7619	−	5.02723i	0.808132	−	0.216538i
$540$		0			0
$541$	9.11401	+	9.11401i	0.391842	+	0.391842i	0.875343	−	0.483502i	$-0.160635\pi$
$541$	9.11401	+	9.11401i	0.391842	+	0.391842i	−0.483502	+	0.875343i	$0.660635\pi$
$542$		0			0
$543$	0.689996	+	1.19511i	0.0296106	+	0.0512870i
$544$		0			0
$545$			11.6015i			0.496954i
$546$		0			0
$547$	−4.62739			−0.197853			−0.0989264	−	0.995095i	$-0.531541\pi$
$547$	−4.62739			−0.197853			−0.0989264	+	0.995095i	$0.531541\pi$
$548$		0			0
$549$	15.3002	−	8.83357i	0.652997	−	0.377008i
$550$		0			0
$551$	11.9107	−	11.9107i	0.507414	−	0.507414i
$552$		0			0
$553$	2.20376	+	8.22453i	0.0937133	+	0.349743i
$554$		0			0
$555$	−0.0148851	+	0.0257818i	−0.000631839	+	0.00109438i
$556$		0			0
$557$	3.05265	−	11.3926i	0.129345	−	0.482721i	−0.870612	−	0.491970i	$-0.836277\pi$
$557$	3.05265	−	11.3926i	0.129345	−	0.482721i	0.999957	+	0.00924808i	$0.00294380\pi$
$558$		0			0
$559$	−8.35762	+	14.2176i	−0.353490	+	0.601343i
$560$		0			0
$561$	0.757049	−	2.82535i	0.0319626	−	0.119286i
$562$		0			0
$563$	−32.8669	−	18.9757i	−1.38517	−	0.799730i	−0.392407	−	0.919792i	$-0.628357\pi$
$563$	−32.8669	−	18.9757i	−1.38517	−	0.799730i	−0.992766	+	0.120062i	$0.961691\pi$
$564$		0			0
$565$	7.57074	+	28.2544i	0.318504	+	1.18867i
$566$		0			0
$567$	−11.9173	−	11.9173i	−0.500481	−	0.500481i
$568$		0			0
$569$	0.0489692	−	0.0282724i	0.00205290	−	0.00118524i	−0.498973	−	0.866617i	$-0.666289\pi$
$569$	0.0489692	−	0.0282724i	0.00205290	−	0.00118524i	0.501026	+	0.865432i	$0.332956\pi$
$570$		0			0
$571$		−	7.98125i		−	0.334005i	−0.985956	−	0.167002i	$-0.946591\pi$
$571$		−	7.98125i		−	0.334005i	0.985956	−	0.167002i	$-0.0534088\pi$
$572$		0			0
$573$			2.48996i			0.104020i
$574$		0			0
$575$	10.0767	−	5.81778i	0.420227	−	0.242618i
$576$		0			0
$577$	−31.6179	−	31.6179i	−1.31627	−	1.31627i	−0.916705	−	0.399564i	$-0.869161\pi$
$577$	−31.6179	−	31.6179i	−1.31627	−	1.31627i	−0.399564	−	0.916705i	$-0.630839\pi$
$578$		0			0
$579$	0.0934308	+	0.348688i	0.00388285	+	0.0144910i
$580$		0			0
$581$	−2.02714	−	1.17037i	−0.0841001	−	0.0485552i
$582$		0			0
$583$	11.3555	−	42.3795i	0.470299	−	1.75518i
$584$		0			0
$585$	30.7819	−	7.99029i	1.27268	−	0.330358i
$586$		0			0
$587$	5.85120	−	21.8370i	0.241505	−	0.901308i	−0.733603	−	0.679578i	$-0.762163\pi$
$587$	5.85120	−	21.8370i	0.241505	−	0.901308i	0.975108	−	0.221730i	$-0.0711704\pi$
$588$		0			0
$589$	17.8638	−	30.9409i	0.736063	−	1.27490i
$590$		0			0
$591$	−0.650626	−	2.42817i	−0.0267632	−	0.0998816i
$592$		0			0
$593$	22.6590	−	22.6590i	0.930492	−	0.930492i	−0.0672444	−	0.997737i	$-0.521421\pi$
$593$	22.6590	−	22.6590i	0.930492	−	0.930492i	0.997737	+	0.0672444i	$0.0214207\pi$
$594$		0			0
$595$	−20.7751	+	11.9945i	−0.851696	+	0.491727i
$596$		0			0
$597$	−0.640438			−0.0262114
$598$		0			0
$599$		−	35.5478i		−	1.45244i	−0.687461	−	0.726221i	$-0.741275\pi$
$599$		−	35.5478i		−	1.45244i	0.687461	−	0.726221i	$-0.258725\pi$
$600$		0			0
$601$	−19.3542	−	33.5225i	−0.789476	−	1.36741i	−0.926288	−	0.376815i	$-0.877019\pi$
$601$	−19.3542	−	33.5225i	−0.789476	−	1.36741i	0.136813	−	0.990597i	$-0.456314\pi$
$602$		0			0
$603$	7.69701	+	7.69701i	0.313446	+	0.313446i
$604$		0			0
$605$	62.2274	−	16.6738i	2.52990	−	0.677886i
$606$		0			0
$607$	24.5142	+	14.1533i	0.994999	+	0.574463i	0.906765	−	0.421637i	$-0.138544\pi$
$607$	24.5142	+	14.1533i	0.994999	+	0.574463i	0.0882341	+	0.996100i	$0.471878\pi$
$608$		0			0
$609$	−0.864659	−	0.231685i	−0.0350378	−	0.00938834i
$610$		0			0
$611$	4.85484	−	4.77942i	0.196406	−	0.193355i
$612$		0			0
$613$	15.9165	+	4.26482i	0.642863	+	0.172255i	0.565500	−	0.824748i	$-0.308683\pi$
$613$	15.9165	+	4.26482i	0.642863	+	0.172255i	0.0773631	+	0.997003i	$0.475350\pi$
$614$		0			0
$615$	1.29285	−	2.23929i	0.0521329	−	0.0902969i
$616$		0			0
$617$	24.3342	−	6.52032i	0.979656	−	0.262498i	0.266756	−	0.963764i	$-0.414048\pi$
$617$	24.3342	−	6.52032i	0.979656	−	0.262498i	0.712900	+	0.701266i	$0.247381\pi$
$618$		0			0
$619$	−8.60779	+	8.60779i	−0.345976	+	0.345976i	−0.858608	−	0.512632i	$-0.828671\pi$
$619$	−8.60779	+	8.60779i	−0.345976	+	0.345976i	0.512632	+	0.858608i	$0.328671\pi$
$620$		0			0
$621$	−1.11624	−	1.93338i	−0.0447931	−	0.0775840i
$622$		0			0
$623$	−1.49756			−0.0599983
$624$		0			0
$625$	29.7444			1.18978
$626$		0			0
$627$	1.46112	+	2.53074i	0.0583516	+	0.101068i
$628$		0			0
$629$	−0.254966	+	0.254966i	−0.0101662	+	0.0101662i
$630$		0			0
$631$	46.9973	−	12.5929i	1.87093	−	0.501315i	0.870982	−	0.491315i	$-0.163484\pi$
$631$	46.9973	−	12.5929i	1.87093	−	0.501315i	0.999950	−	0.0100000i	$-0.00318316\pi$
$632$		0			0
$633$	−1.53276	+	2.65482i	−0.0609219	+	0.105520i
$634$		0			0
$635$	26.0078	+	6.96878i	1.03209	+	0.276548i
$636$		0			0
$637$	−6.03041	−	10.6364i	−0.238934	−	0.421429i
$638$		0			0
$639$	20.8577	+	5.58881i	0.825119	+	0.221090i
$640$		0			0
$641$	2.37058	+	1.36865i	0.0936322	+	0.0540586i	0.546085	−	0.837730i	$-0.316118\pi$
$641$	2.37058	+	1.36865i	0.0936322	+	0.0540586i	−0.452453	+	0.891788i	$0.649451\pi$
$642$		0			0
$643$	−4.87893	+	1.30730i	−0.192406	+	0.0515550i	−0.353735	−	0.935346i	$-0.615089\pi$
$643$	−4.87893	+	1.30730i	−0.192406	+	0.0515550i	0.161329	+	0.986901i	$0.448422\pi$
$644$		0			0
$645$	−1.14149	−	1.14149i	−0.0449461	−	0.0449461i
$646$		0			0
$647$	9.09928	+	15.7604i	0.357730	+	0.619606i	0.987581	−	0.157109i	$-0.0502176\pi$
$647$	9.09928	+	15.7604i	0.357730	+	0.619606i	−0.629851	+	0.776716i	$0.716884\pi$
$648$		0			0
$649$			7.34801i			0.288435i
$650$		0			0
$651$	−1.89868			−0.0744150
$652$		0			0
$653$	−23.1046	+	13.3394i	−0.904151	+	0.522012i	−0.878545	−	0.477660i	$-0.841485\pi$
$653$	−23.1046	+	13.3394i	−0.904151	+	0.522012i	−0.0256065	+	0.999672i	$0.508152\pi$
$654$		0			0
$655$	−19.2096	+	19.2096i	−0.750582	+	0.750582i
$656$		0			0
$657$	2.57217	+	9.59948i	0.100350	+	0.374511i
$658$		0			0
$659$	5.86022	−	10.1502i	0.228282	−	0.395396i	−0.729017	−	0.684495i	$-0.760023\pi$
$659$	5.86022	−	10.1502i	0.228282	−	0.395396i	0.957299	+	0.289100i	$0.0933559\pi$
$660$		0			0
$661$	−4.42914	+	16.5298i	−0.172273	+	0.642933i	0.824727	+	0.565532i	$0.191329\pi$
$661$	−4.42914	+	16.5298i	−0.172273	+	0.642933i	−0.997000	+	0.0774016i	$0.975338\pi$
$662$		0			0
$663$	−1.84120	−	0.0144120i	−0.0715061	−	0.000559714i
$664$		0			0
$665$	6.20294	−	23.1497i	0.240540	−	0.897706i
$666$		0			0
$667$	10.6641	+	6.15691i	0.412915	+	0.238397i
$668$		0			0
$669$	−0.404707	−	1.51039i	−0.0156469	−	0.0583950i
$670$		0			0
$671$	−23.9656	−	23.9656i	−0.925180	−	0.925180i
$672$		0			0
$673$	−22.5318	+	13.0088i	−0.868538	+	0.501451i	−0.866862	−	0.498548i	$-0.833867\pi$
$673$	−22.5318	+	13.0088i	−0.868538	+	0.501451i	−0.00167601	+	0.999999i	$0.500533\pi$
$674$		0			0
$675$			2.66519i			0.102583i
$676$		0			0
$677$			15.2979i			0.587945i	0.955814	+	0.293972i	$0.0949774\pi$
$677$			15.2979i			0.587945i	−0.955814	+	0.293972i	$0.905023\pi$
$678$		0			0
$679$	20.2237	−	11.6762i	0.776114	−	0.448090i
$680$		0			0
$681$	−2.27669	−	2.27669i	−0.0872431	−	0.0872431i
$682$		0			0
$683$	0.214029	+	0.798768i	0.00818960	+	0.0305640i	0.969900	−	0.243505i	$-0.0782971\pi$
$683$	0.214029	+	0.798768i	0.00818960	+	0.0305640i	−0.961710	+	0.274069i	$0.911630\pi$
$684$		0			0
$685$	−29.0437	−	16.7684i	−1.10970	−	0.640686i
$686$		0			0
$687$	−0.00687147	+	0.0256447i	−0.000262163	+	0.000978406i
$688$		0			0
$689$	−27.6175	−	0.216176i	−1.05214	−	0.00823564i
$690$		0			0
$691$	−7.77196	+	29.0054i	−0.295659	+	1.10342i	0.645033	+	0.764155i	$0.276844\pi$
$691$	−7.77196	+	29.0054i	−0.295659	+	1.10342i	−0.940692	+	0.339261i	$0.889823\pi$
$692$		0			0
$693$	−16.2439	+	28.1353i	−0.617056	+	1.06877i
$694$		0			0
$695$	2.34621	+	8.75619i	0.0889970	+	0.332141i
$696$		0			0
$697$	22.1452	−	22.1452i	0.838807	−	0.838807i
$698$		0			0
$699$	−2.48739	+	1.43610i	−0.0940817	+	0.0543181i
$700$		0			0
$701$	28.6732			1.08297			0.541486	−	0.840710i	$-0.317862\pi$
$701$	28.6732			1.08297			0.541486	+	0.840710i	$0.317862\pi$
$702$		0			0
$703$		−	0.360235i		−	0.0135865i
$704$		0			0
$705$	0.333423	+	0.577505i	0.0125574	+	0.0217501i
$706$		0			0
$707$	−7.10224	−	7.10224i	−0.267107	−	0.267107i
$708$		0			0
$709$	−5.49446	+	1.47224i	−0.206349	+	0.0552910i	−0.360513	−	0.932754i	$-0.617398\pi$
$709$	−5.49446	+	1.47224i	−0.206349	+	0.0552910i	0.154164	+	0.988045i	$0.450732\pi$
$710$		0			0
$711$	11.5895	+	6.69119i	0.434639	+	0.250939i
$712$		0			0
$713$	25.2282	+	6.75988i	0.944804	+	0.253160i
$714$		0			0
$715$	−30.0898	−	53.0722i	−1.12530	−	1.98479i
$716$		0			0
$717$	−3.17688	−	0.851242i	−0.118643	−	0.0317902i
$718$		0			0
$719$	−22.7206	+	39.3533i	−0.847337	+	1.46763i	0.0362397	+	0.999343i	$0.488462\pi$
$719$	−22.7206	+	39.3533i	−0.847337	+	1.46763i	−0.883576	+	0.468287i	$0.844871\pi$
$720$		0			0
$721$	−27.8478	+	7.46179i	−1.03710	+	0.277891i
$722$		0			0
$723$	1.20460	−	1.20460i	0.0447996	−	0.0447996i
$724$		0			0
$725$	−7.35027	−	12.7310i	−0.272982	−	0.472819i
$726$		0			0
$727$	−21.0650			−0.781256			−0.390628	−	0.920549i	$-0.627742\pi$
$727$	−21.0650			−0.781256			−0.390628	+	0.920549i	$0.627742\pi$
$728$		0			0
$729$	−26.2323			−0.971568
$730$		0			0
$731$	−9.77621	−	16.9329i	−0.361586	−	0.626286i
$732$		0			0
$733$	22.0025	−	22.0025i	0.812682	−	0.812682i	−0.172353	−	0.985035i	$-0.555137\pi$
$733$	22.0025	−	22.0025i	0.812682	−	0.812682i	0.985035	+	0.172353i	$0.0551370\pi$
$734$		0			0
$735$	1.15604	−	0.309759i	0.0426410	−	0.0114256i
$736$		0			0
$737$	10.4410	−	18.0844i	0.384600	−	0.666146i
$738$		0			0
$739$	−48.4788	−	12.9899i	−1.78332	−	0.477840i	−0.792139	−	0.610341i	$-0.791033\pi$
$739$	−48.4788	−	12.9899i	−1.78332	−	0.477840i	−0.991183	+	0.132501i	$0.957699\pi$
$740$		0			0
$741$	1.31087	−	1.29051i	0.0481560	−	0.0474080i
$742$		0			0
$743$	−14.6170	−	3.91663i	−0.536247	−	0.143687i	−0.0194758	−	0.999810i	$-0.506200\pi$
$743$	−14.6170	−	3.91663i	−0.536247	−	0.143687i	−0.516772	+	0.856123i	$0.672866\pi$
$744$		0			0
$745$	−47.2972	−	27.3071i	−1.73284	−	1.00045i
$746$		0			0
$747$	−3.55356	+	0.952173i	−0.130018	+	0.0348382i
$748$		0			0
$749$	8.43226	+	8.43226i	0.308108	+	0.308108i
$750$		0			0
$751$	8.03320	+	13.9139i	0.293136	+	0.507726i	0.974549	−	0.224173i	$-0.0719679\pi$
$751$	8.03320	+	13.9139i	0.293136	+	0.507726i	−0.681414	+	0.731898i	$0.738635\pi$
$752$		0			0
$753$			0.193292i			0.00704393i
$754$		0			0
$755$	67.1946			2.44546
$756$		0			0
$757$	23.8711	−	13.7820i	0.867609	−	0.500914i	0.00105597	−	0.999999i	$-0.499664\pi$
$757$	23.8711	−	13.7820i	0.867609	−	0.500914i	0.866553	+	0.499085i	$0.166331\pi$
$758$		0			0
$759$	−1.51057	+	1.51057i	−0.0548303	+	0.0548303i
$760$		0			0
$761$	8.71969	+	32.5423i	0.316089	+	1.17966i	0.922972	+	0.384868i	$0.125753\pi$
$761$	8.71969	+	32.5423i	0.316089	+	1.17966i	−0.606883	+	0.794791i	$0.707580\pi$
$762$		0			0
$763$	−3.73023	+	6.46095i	−0.135043	+	0.233902i
$764$		0			0
$765$	−9.75831	+	36.4185i	−0.352812	+	1.31671i
$766$		0			0
$767$	4.47709	−	1.16215i	0.161658	−	0.0419628i
$768$		0			0
$769$	−9.16888	+	34.2187i	−0.330638	+	1.23396i	0.577883	+	0.816120i	$0.303879\pi$
$769$	−9.16888	+	34.2187i	−0.330638	+	1.23396i	−0.908521	+	0.417839i	$0.862788\pi$
$770$		0			0
$771$	−1.15820	−	0.668688i	−0.0417116	−	0.0240822i
$772$		0			0
$773$	−0.801104	−	2.98976i	−0.0288137	−	0.107534i	0.950021	−	0.312185i	$-0.101061\pi$
$773$	−0.801104	−	2.98976i	−0.0288137	−	0.107534i	−0.978835	+	0.204651i	$0.934394\pi$
$774$		0			0
$775$	−22.0480	−	22.0480i	−0.791986	−	0.791986i
$776$		0			0
$777$	−0.0165793	+	0.00957204i	−0.000594777	+	0.000343395i
$778$		0			0
$779$			31.2883i			1.12102i
$780$		0			0
$781$		−	41.4247i		−	1.48229i
$782$		0			0
$783$	−2.44267	+	1.41027i	−0.0872938	+	0.0503991i
$784$		0			0
$785$	14.7789	+	14.7789i	0.527480	+	0.527480i
$786$		0			0
$787$	−6.65435	−	24.8344i	−0.237202	−	0.885250i	−0.977144	−	0.212579i	$-0.931814\pi$
$787$	−6.65435	−	24.8344i	−0.237202	−	0.885250i	0.739942	−	0.672671i	$-0.234853\pi$
$788$		0			0
$789$	0.839750	+	0.484830i	0.0298959	+	0.0172604i
$790$		0			0
$791$	−4.86844	+	18.1693i	−0.173102	+	0.646025i
$792$		0			0
$793$	−10.8117	+	18.3924i	−0.383934	+	0.653133i
$794$		0			0
$795$	0.699685	−	2.61126i	0.0248153	−	0.0926118i
$796$		0			0
$797$	−18.1005	+	31.3510i	−0.641152	+	1.11051i	0.344024	+	0.938961i	$0.388210\pi$
$797$	−18.1005	+	31.3510i	−0.641152	+	1.11051i	−0.985176	+	0.171547i	$0.945123\pi$
$798$		0			0
$799$	2.09043	+	7.80159i	0.0739541	+	0.276001i
$800$		0			0
$801$	−1.66431	+	1.66431i	−0.0588054	+	0.0588054i
$802$		0			0
$803$	16.5109	−	9.53256i	0.582656	−	0.336397i
$804$		0			0
$805$	17.5203			0.617509
$806$		0			0
$807$			1.03049i			0.0362751i
$808$		0			0
$809$	7.49504	+	12.9818i	0.263512	+	0.456415i	0.967173	−	0.254120i	$-0.0817859\pi$
$809$	7.49504	+	12.9818i	0.263512	+	0.456415i	−0.703661	+	0.710536i	$0.748453\pi$
$810$		0			0
$811$	22.2096	+	22.2096i	0.779883	+	0.779883i	0.979811	−	0.199928i	$-0.0640707\pi$
$811$	22.2096	+	22.2096i	0.779883	+	0.779883i	−0.199928	+	0.979811i	$0.564071\pi$
$812$		0			0
$813$	0.284492	−	0.0762295i	0.00997758	−	0.00267348i
$814$		0			0
$815$	−16.2107	−	9.35926i	−0.567836	−	0.327840i
$816$		0			0
$817$	18.8683	+	5.05575i	0.660119	+	0.176878i
$818$		0			0
$819$	19.7117	+	5.44746i	0.688784	+	0.190350i
$820$		0			0
$821$	−17.6660	−	4.73358i	−0.616547	−	0.165203i	−0.0629893	−	0.998014i	$-0.520063\pi$
$821$	−17.6660	−	4.73358i	−0.616547	−	0.165203i	−0.553557	+	0.832811i	$0.686730\pi$
$822$		0			0
$823$	12.2951	−	21.2957i	0.428580	−	0.742323i	−0.568167	−	0.822913i	$-0.692347\pi$
$823$	12.2951	−	21.2957i	0.428580	−	0.742323i	0.996747	+	0.0805905i	$0.0256806\pi$
$824$		0			0
$825$	2.46343	−	0.660075i	0.0857657	−	0.0229809i
$826$		0			0
$827$	35.8846	−	35.8846i	1.24783	−	1.24783i	0.291152	−	0.956677i	$-0.405961\pi$
$827$	35.8846	−	35.8846i	1.24783	−	1.24783i	0.956677	−	0.291152i	$-0.0940386\pi$
$828$		0			0
$829$	−6.84382	−	11.8539i	−0.237696	−	0.411701i	0.722357	−	0.691520i	$-0.243059\pi$
$829$	−6.84382	−	11.8539i	−0.237696	−	0.411701i	−0.960053	+	0.279819i	$0.909725\pi$
$830$		0			0
$831$	1.74614			0.0605728
$832$		0			0
$833$	14.4958			0.502249
$834$		0			0
$835$	27.9372	+	48.3886i	0.966807	+	1.67456i
$836$		0			0
$837$	−4.23027	+	4.23027i	−0.146220	+	0.146220i
$838$		0			0
$839$	−14.7393	+	3.94939i	−0.508858	+	0.136348i	−0.504108	−	0.863641i	$-0.668179\pi$
$839$	−14.7393	+	3.94939i	−0.508858	+	0.136348i	−0.00474988	+	0.999989i	$0.501512\pi$
$840$		0			0
$841$	−6.72126	+	11.6416i	−0.231768	+	0.401433i
$842$		0			0
$843$	0.562865	+	0.150819i	0.0193861	+	0.00519449i
$844$		0			0
$845$	−27.5775	+	26.7273i	−0.948695	+	0.919448i
$846$		0			0
$847$	40.0159	+	10.7222i	1.37496	+	0.368420i
$848$		0			0
$849$	1.91627	+	1.10636i	0.0657663	+	0.0379702i
$850$		0			0
$851$	0.254372	−	0.0681588i	0.00871977	−	0.00233646i
$852$		0			0
$853$	−23.2454	−	23.2454i	−0.795909	−	0.795909i	0.186539	−	0.982448i	$-0.440273\pi$
$853$	−23.2454	−	23.2454i	−0.795909	−	0.795909i	−0.982448	+	0.186539i	$0.940273\pi$
$854$		0			0
$855$	−18.8338	−	32.6210i	−0.644101	−	1.11562i
$856$		0			0
$857$			46.4375i			1.58627i	0.609043	+	0.793137i	$0.291554\pi$
$857$			46.4375i			1.58627i	−0.609043	+	0.793137i	$0.708446\pi$
$858$		0			0
$859$	−37.0596			−1.26446			−0.632228	−	0.774782i	$-0.717859\pi$
$859$	−37.0596			−1.26446			−0.632228	+	0.774782i	$0.717859\pi$
$860$		0			0
$861$	1.44000	−	0.831382i	0.0490749	−	0.0283334i
$862$		0			0
$863$	−17.2801	+	17.2801i	−0.588220	+	0.588220i	−0.937149	−	0.348929i	$-0.886545\pi$
$863$	−17.2801	+	17.2801i	−0.588220	+	0.588220i	0.348929	+	0.937149i	$0.386545\pi$
$864$		0			0
$865$	−4.48377	−	16.7336i	−0.152453	−	0.568961i
$866$		0			0
$867$	0.0759885	−	0.131616i	0.00258071	−	0.00446991i
$868$		0			0
$869$	6.64454	−	24.7978i	0.225401	−	0.841207i
$870$		0			0
$871$	−12.6700	−	3.50144i	−0.429307	−	0.118642i
$872$		0			0
$873$	9.49930	−	35.4519i	0.321503	−	1.19986i
$874$		0			0
$875$	6.18677	+	3.57193i	0.209151	+	0.120753i
$876$		0			0
$877$	11.1473	+	41.6023i	0.376417	+	1.40481i	0.851263	+	0.524740i	$0.175837\pi$
$877$	11.1473	+	41.6023i	0.376417	+	1.40481i	−0.474845	+	0.880069i	$0.657496\pi$
$878$		0			0
$879$	−1.43160	−	1.43160i	−0.0482868	−	0.0482868i
$880$		0			0
$881$	−7.04095	+	4.06510i	−0.237216	+	0.136957i	−0.613896	−	0.789387i	$-0.710399\pi$
$881$	−7.04095	+	4.06510i	−0.237216	+	0.136957i	0.376681	+	0.926343i	$0.377065\pi$
$882$		0			0
$883$			38.1442i			1.28365i	0.766850	+	0.641827i	$0.221823\pi$
$883$			38.1442i			1.28365i	−0.766850	+	0.641827i	$0.778177\pi$
$884$		0			0
$885$			0.452756i			0.0152192i
$886$		0			0
$887$	11.1078	−	6.41307i	0.372962	−	0.215330i	−0.301790	−	0.953375i	$-0.597584\pi$
$887$	11.1078	−	6.41307i	0.372962	−	0.215330i	0.674752	+	0.738045i	$0.264251\pi$
$888$		0			0
$889$	12.2432	+	12.2432i	0.410625	+	0.410625i
$890$		0			0
$891$	13.1520	+	49.0839i	0.440609	+	1.64437i
$892$		0			0
$893$	−6.98810	−	4.03458i	−0.233848	−	0.135012i
$894$		0			0
$895$	−6.99283	+	26.0976i	−0.233745	+	0.872347i
$896$		0			0
$897$	1.15929	+	0.681471i	0.0387076	+	0.0227536i
$898$		0			0
$899$	8.54054	−	31.8737i	0.284843	−	1.06305i
$900$		0			0
$901$	16.3716	−	28.3564i	0.545416	−	0.944688i
$902$		0			0
$903$	−0.268679	−	1.00272i	−0.00894108	−	0.0333686i
$904$		0			0
$905$	−24.1295	+	24.1295i	−0.802091	+	0.802091i
$906$		0			0
$907$	26.6269	−	15.3730i	0.884132	−	0.510454i	0.0121133	−	0.999927i	$-0.496144\pi$
$907$	26.6269	−	15.3730i	0.884132	−	0.510454i	0.872018	+	0.489473i	$0.162811\pi$
$908$		0			0
$909$	−15.7861			−0.523593
$910$		0			0
$911$		−	41.5292i		−	1.37593i	−0.725746	−	0.687963i	$-0.758505\pi$
$911$		−	41.5292i		−	1.37593i	0.725746	−	0.687963i	$-0.241495\pi$
$912$		0			0
$913$	3.52879	+	6.11204i	0.116786	+	0.202279i
$914$		0			0
$915$	−1.47666	−	1.47666i	−0.0488170	−	0.0488170i
$916$		0			0
$917$	−16.8744	+	4.52148i	−0.557242	+	0.149313i
$918$		0			0
$919$	−36.7323	−	21.2074i	−1.21169	−	0.699567i	−0.248560	−	0.968617i	$-0.579957\pi$
$919$	−36.7323	−	21.2074i	−1.21169	−	0.699567i	−0.963126	+	0.269049i	$0.913291\pi$
$920$		0			0
$921$	−2.38796	−	0.639851i	−0.0786859	−	0.0210838i
$922$		0			0
$923$	−25.2398	+	6.55167i	−0.830777	+	0.215651i
$924$		0			0
$925$	−0.303676	−	0.0813697i	−0.00998480	−	0.00267542i
$926$		0			0
$927$	−22.6559	+	39.2413i	−0.744119	+	1.28885i
$928$		0			0
$929$	−39.6233	+	10.6170i	−1.30000	+	0.348334i	−0.841450	−	0.540335i	$-0.818297\pi$
$929$	−39.6233	+	10.6170i	−1.30000	+	0.348334i	−0.458549	+	0.888669i	$0.651631\pi$
$930$		0			0
$931$	−10.2404	+	10.2404i	−0.335614	+	0.335614i
$932$		0			0
$933$	1.14773	+	1.98792i	0.0375749	+	0.0650816i
$934$		0			0
$935$	72.3293			2.36542
$936$		0			0
$937$	14.4265			0.471292			0.235646	−	0.971839i	$-0.424279\pi$
$937$	14.4265			0.471292			0.235646	+	0.971839i	$0.424279\pi$
$938$		0			0
$939$	0.0934150	+	0.161800i	0.00304848	+	0.00528013i
$940$		0			0
$941$	−29.8298	+	29.8298i	−0.972423	+	0.972423i	−0.999630	−	0.0272066i	$-0.991339\pi$
$941$	−29.8298	+	29.8298i	−0.972423	+	0.972423i	0.0272066	+	0.999630i	$0.491339\pi$
$942$		0			0
$943$	−22.0936	+	5.91996i	−0.719466	+	0.192780i
$944$		0			0
$945$	−2.00656	+	3.47546i	−0.0652734	+	0.113057i
$946$		0			0
$947$	11.4406	+	3.06549i	0.371769	+	0.0996152i	0.439866	−	0.898064i	$-0.355026\pi$
$947$	11.4406	+	3.06549i	0.371769	+	0.0996152i	−0.0680969	+	0.997679i	$0.521693\pi$
$948$		0			0
$949$	−8.41946	−	8.55230i	−0.273307	−	0.277620i
$950$		0			0
$951$	−0.0211309	−	0.00566201i	−0.000685217	−	0.000183603i
$952$		0			0
$953$	−8.77779	−	5.06786i	−0.284341	−	0.164164i	0.351046	−	0.936358i	$-0.385826\pi$
$953$	−8.77779	−	5.06786i	−0.284341	−	0.164164i	−0.635387	+	0.772194i	$0.719159\pi$
$954$		0			0
$955$	−59.4734	+	15.9359i	−1.92451	+	0.515672i
$956$		0			0
$957$	1.90848	+	1.90848i	0.0616924	+	0.0616924i
$958$		0			0
$959$	−10.7831	−	18.6768i	−0.348203	−	0.603105i
$960$		0			0
$961$		−	38.9905i		−	1.25776i
$962$		0			0
$963$	18.7424			0.603965
$964$		0			0
$965$	−7.73056	+	4.46324i	−0.248855	+	0.143677i
$966$		0			0
$967$	−4.34955	+	4.34955i	−0.139872	+	0.139872i	−0.773576	−	0.633704i	$-0.781534\pi$
$967$	−4.34955	+	4.34955i	−0.139872	+	0.139872i	0.633704	+	0.773576i	$0.281534\pi$
$968$		0			0
$969$	0.564444	+	2.10654i	0.0181326	+	0.0676717i
$970$		0			0
$971$	−27.9858	+	48.4728i	−0.898107	+	1.55557i	−0.0681947	+	0.997672i	$0.521724\pi$
$971$	−27.9858	+	48.4728i	−0.898107	+	1.55557i	−0.829912	+	0.557894i	$0.811609\pi$
$972$		0			0
$973$	−1.50876	+	5.63075i	−0.0483685	+	0.180514i
$974$		0			0
$975$	−0.791792	−	1.39656i	−0.0253576	−	0.0447256i
$976$		0			0
$977$	−7.62905	+	28.4720i	−0.244075	+	0.910899i	0.729771	+	0.683691i	$0.239626\pi$
$977$	−7.62905	+	28.4720i	−0.244075	+	0.910899i	−0.973846	+	0.227208i	$0.927040\pi$
$978$		0			0
$979$	3.91034	+	2.25764i	0.124975	+	0.0721544i
$980$		0			0
$981$	3.03478	+	11.3260i	0.0968932	+	0.361610i
$982$		0			0
$983$	25.7045	+	25.7045i	0.819846	+	0.819846i	0.986085	−	0.166240i	$-0.0531625\pi$
$983$	25.7045	+	25.7045i	0.819846	+	0.819846i	−0.166240	+	0.986085i	$0.553162\pi$
$984$		0			0
$985$	53.8335	−	31.0808i	1.71528	−	0.990316i
$986$		0			0
$987$			0.428821i			0.0136495i
$988$		0			0
$989$			14.2800i			0.454079i
$990$		0			0
$991$	−25.7808	+	14.8846i	−0.818955	+	0.472824i	−0.850056	−	0.526693i	$-0.823432\pi$
$991$	−25.7808	+	14.8846i	−0.818955	+	0.472824i	0.0311012	+	0.999516i	$0.490099\pi$
$992$		0			0
$993$	1.08819	+	1.08819i	0.0345327	+	0.0345327i
$994$		0			0
$995$	−4.09883	−	15.2970i	−0.129942	−	0.484948i
$996$		0			0
$997$	−4.35409	−	2.51384i	−0.137895	−	0.0796140i	0.429465	−	0.903083i	$-0.358702\pi$
$997$	−4.35409	−	2.51384i	−0.137895	−	0.0796140i	−0.567361	+	0.823469i	$0.692035\pi$
$998$		0			0
$999$	−0.0156122	+	0.0582654i	−0.000493947	+	0.00184343i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bk.a.271.6		48
4.3	odd	2		104.2.u.a.11.3	✓	48
8.3	odd	2	inner	416.2.bk.a.271.5		48
8.5	even	2		104.2.u.a.11.8	yes	48
12.11	even	2		936.2.ed.d.739.10		48
13.6	odd	12	inner	416.2.bk.a.175.5		48
24.5	odd	2		936.2.ed.d.739.5		48
52.19	even	12		104.2.u.a.19.8	yes	48
104.19	even	12	inner	416.2.bk.a.175.6		48
104.45	odd	12		104.2.u.a.19.3	yes	48
156.71	odd	12		936.2.ed.d.19.5		48
312.149	even	12		936.2.ed.d.19.10		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.11.3	✓	48	4.3	odd	2
104.2.u.a.11.8	yes	48	8.5	even	2
104.2.u.a.19.3	yes	48	104.45	odd	12
104.2.u.a.19.8	yes	48	52.19	even	12
416.2.bk.a.175.5		48	13.6	odd	12	inner
416.2.bk.a.175.6		48	104.19	even	12	inner
416.2.bk.a.271.5		48	8.3	odd	2	inner
416.2.bk.a.271.6		48	1.1	even	1	trivial
936.2.ed.d.19.5		48	156.71	odd	12
936.2.ed.d.19.10		48	312.149	even	12
936.2.ed.d.739.5		48	24.5	odd	2
936.2.ed.d.739.10		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-0.0597334 - 0.103461i) q^{3} +(2.08890 - 2.08890i) q^{5} +(1.83497 - 0.491678i) q^{7} +(1.49286 - 2.58572i) q^{9} +(-5.53261 - 1.48246i) q^{11} +(-0.0282216 + 3.60544i) q^{13} +(-0.340898 - 0.0913433i) q^{15} +(-3.70190 - 2.13730i) q^{17} +(4.12503 - 1.10530i) q^{19} +(-0.160478 - 0.160478i) q^{21} +(1.56097 + 2.70367i) q^{23} -3.72704i q^{25} -0.715095 q^{27} +(3.41586 - 1.97215i) q^{29} +(5.91568 - 5.91568i) q^{31} +(0.177104 + 0.660963i) q^{33} +(2.80600 - 4.86014i) q^{35} +(0.0218323 - 0.0814791i) q^{37} +(0.374709 - 0.212445i) q^{39} +(-1.89625 + 7.07690i) q^{41} +(3.96129 + 2.28705i) q^{43} +(-2.28286 - 8.51976i) q^{45} +(-1.33607 - 1.33607i) q^{47} +(-2.93682 + 1.69557i) q^{49} +0.510672i q^{51} +7.65994i q^{53} +(-14.6538 + 8.46037i) q^{55} +(-0.360758 - 0.360758i) q^{57} +(-0.332032 - 1.23916i) q^{59} +(5.12445 + 2.95860i) q^{61} +(1.46802 - 5.47871i) q^{63} +(7.47247 + 7.59037i) q^{65} +(-0.943589 + 3.52152i) q^{67} +(0.186483 - 0.322999i) q^{69} +(1.87184 + 6.98581i) q^{71} +(-2.35363 + 2.35363i) q^{73} +(-0.385604 + 0.222629i) q^{75} -10.8810 q^{77} +4.48211i q^{79} +(-4.43588 - 7.68316i) q^{81} +(-0.871274 - 0.871274i) q^{83} +(-12.1975 + 3.26832i) q^{85} +(-0.408082 - 0.235606i) q^{87} +(-0.761451 - 0.204030i) q^{89} +(1.72093 + 6.62974i) q^{91} +(-0.965407 - 0.258680i) q^{93} +(6.30793 - 10.9257i) q^{95} +(11.8738 - 3.18157i) q^{97} +(-12.0926 + 12.0926i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83}+ \cdots - 168 q^{99}+O(q^{100}) \) 48 * q + 4 * q^3 - 20 * q^9 + 8 * q^11 - 12 * q^17 + 8 * q^19 - 8 * q^27 + 4 * q^33 + 4 * q^35 + 12 * q^43 - 60 * q^49 + 36 * q^57 + 64 * q^59 - 16 * q^65 + 8 * q^67 - 12 * q^73 - 24 * q^75 - 8 * q^81 + 48 * q^83 + 12 * q^89 - 104 * q^91 + 4 * q^97 - 168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more