LMFDB - Embedded newform 416.2.bk.a.175.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			175.5
Character	$\chi$	$=$	416.175
Dual form			416.2.bk.a.271.5

$q$-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 $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.0597334	+	0.103461i	−0.0344871	+	0.0597334i	−0.882754	−	0.469836i	$-0.844313\pi$
$3$	−0.0597334	+	0.103461i	−0.0344871	+	0.0597334i	0.848267	+	0.529569i	$0.177646\pi$
$4$		0			0
$5$	−2.08890	−	2.08890i	−0.934186	−	0.934186i	0.0637779	−	0.997964i	$-0.479685\pi$
$5$	−2.08890	−	2.08890i	−0.934186	−	0.934186i	−0.997964	+	0.0637779i	$0.979685\pi$
$6$		0			0
$7$	−1.83497	−	0.491678i	−0.693552	−	0.185837i	−0.105212	−	0.994450i	$-0.533552\pi$
$7$	−1.83497	−	0.491678i	−0.693552	−	0.185837i	−0.588341	+	0.808613i	$0.700219\pi$
$8$		0			0
$9$	1.49286	+	2.58572i	0.497621	+	0.861905i
$10$		0			0
$11$	−5.53261	+	1.48246i	−1.66814	+	0.446978i	−0.964609	−	0.263683i	$-0.915063\pi$
$11$	−5.53261	+	1.48246i	−1.66814	+	0.446978i	−0.703535	+	0.710661i	$0.748396\pi$
$12$		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	−0.876500	−	0.481402i	$-0.840128\pi$
$17$	−3.70190	+	2.13730i	−0.897844	+	0.518370i	−0.0213437	+	0.999772i	$0.506794\pi$
$18$		0			0
$19$	4.12503	+	1.10530i	0.946347	+	0.253573i	0.698811	−	0.715306i	$-0.253713\pi$
$19$	4.12503	+	1.10530i	0.946347	+	0.253573i	0.247535	+	0.968879i	$0.420379\pi$
$20$		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	$-0.938861\pi$
$23$	−1.56097	+	2.70367i	−0.325484	+	0.563755i	0.656126	+	0.754651i	$0.272194\pi$
$24$		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	$-0.452681\pi$
$29$	−3.41586	−	1.97215i	−0.634310	−	0.366219i	−0.782419	+	0.622752i	$0.786015\pi$
$30$		0			0
$31$	−5.91568	−	5.91568i	−1.06249	−	1.06249i	−0.997913	−	0.0645742i	$-0.979431\pi$
$31$	−5.91568	−	5.91568i	−1.06249	−	1.06249i	−0.0645742	−	0.997913i	$-0.520569\pi$
$32$		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	$-0.0855405\pi$
$37$	−0.0218323	−	0.0814791i	−0.00358920	−	0.0133951i	−0.967697	+	0.252115i	$0.918874\pi$
$38$		0			0
$39$	−0.374709	−	0.212445i	−0.0600015	−	0.0340185i
$40$		0			0
$41$	−1.89625	−	7.07690i	−0.296144	−	1.10523i	−0.940304	−	0.340335i	$-0.889460\pi$
$41$	−1.89625	−	7.07690i	−0.296144	−	1.10523i	0.644160	−	0.764891i	$-0.277207\pi$
$42$		0			0
$43$	3.96129	−	2.28705i	0.604091	−	0.348772i	−0.166558	−	0.986032i	$-0.553265\pi$
$43$	3.96129	−	2.28705i	0.604091	−	0.348772i	0.770649	+	0.637260i	$0.219932\pi$
$44$		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	$-0.705995\pi$
$47$	1.33607	−	1.33607i	0.194886	−	0.194886i	0.797804	+	0.602917i	$0.205995\pi$
$48$		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	−0.984165	−	0.177253i	$-0.943279\pi$
$59$	−0.332032	+	1.23916i	−0.0432269	+	0.161325i	0.940939	+	0.338577i	$0.109946\pi$
$60$		0			0
$61$	−5.12445	+	2.95860i	−0.656118	+	0.378810i	−0.790796	−	0.612079i	$-0.790333\pi$
$61$	−5.12445	+	2.95860i	−0.656118	+	0.378810i	0.134678	+	0.990889i	$0.457000\pi$
$62$		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	0.884030	−	0.467431i	$-0.154820\pi$
$67$	−0.943589	−	3.52152i	−0.115278	−	0.430222i	−0.999308	+	0.0372082i	$0.988154\pi$
$68$		0			0
$69$	−0.186483	−	0.322999i	−0.0224500	−	0.0388845i
$70$		0			0
$71$	−1.87184	+	6.98581i	−0.222147	+	0.829064i	0.761381	+	0.648305i	$0.224522\pi$
$71$	−1.87184	+	6.98581i	−0.222147	+	0.829064i	−0.983528	+	0.180758i	$0.942145\pi$
$72$		0			0
$73$	−2.35363	−	2.35363i	−0.275472	−	0.275472i	0.555827	−	0.831298i	$-0.312402\pi$
$73$	−2.35363	−	2.35363i	−0.275472	−	0.275472i	−0.831298	+	0.555827i	$0.812402\pi$
$74$		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	−0.753305	−	0.657671i	$-0.771542\pi$
$83$	−0.871274	+	0.871274i	−0.0956347	+	0.0956347i	0.657671	+	0.753305i	$0.271542\pi$
$84$		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	−0.298950	−	0.954269i	$-0.596636\pi$
$89$	−0.761451	+	0.204030i	−0.0807137	+	0.0216272i	0.218236	+	0.975896i	$0.429970\pi$
$90$		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	0.805034	−	0.593229i	$-0.202147\pi$
$97$	11.8738	+	3.18157i	1.20560	+	0.323039i	0.400566	+	0.916268i	$0.368814\pi$
$98$		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	$-0.748606\pi$
$101$	2.64360	−	4.57885i	0.263048	−	0.455613i	0.967050	+	0.254585i	$0.0819389\pi$
$102$		0			0
$103$	15.1762			1.49535			0.747676	−	0.664064i	$-0.231170\pi$
$103$	15.1762			1.49535			0.747676	+	0.664064i	$0.231170\pi$
$104$		0			0
$105$	−0.670448			−0.0654290
$106$		0			0
$107$	3.13866	−	5.43632i	0.303426	−	0.525549i	−0.673484	−	0.739202i	$-0.735203\pi$
$107$	3.13866	−	5.43632i	0.303426	−	0.525549i	0.976910	+	0.213653i	$0.0685361\pi$
$108$		0			0
$109$	2.77694	+	2.77694i	0.265983	+	0.265983i	0.827479	−	0.561497i	$-0.189774\pi$
$109$	2.77694	+	2.77694i	0.265983	+	0.265983i	−0.561497	+	0.827479i	$0.689774\pi$
$110$		0			0
$111$	0.00973405	+	0.00260823i	0.000923915	+	0.000247562i
$112$		0			0
$113$	−4.95084	−	8.57511i	−0.465736	−	0.806678i	0.533499	−	0.845801i	$-0.320877\pi$
$113$	−4.95084	−	8.57511i	−0.465736	−	0.806678i	−0.999234	+	0.0391228i	$0.987544\pi$
$114$		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	$-0.965848\pi$
$127$	−4.55719	+	7.89328i	−0.404385	+	0.700416i	0.589865	+	0.807502i	$0.299181\pi$
$128$		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	0.719228	+	0.694774i	$0.244496\pi$
$137$	−2.93822	+	10.9656i	−0.251029	+	0.936852i	−0.970257	+	0.242078i	$0.922171\pi$
$138$		0			0
$139$	−1.53429	−	2.65747i	−0.130137	−	0.225404i	0.793592	−	0.608450i	$-0.208208\pi$
$139$	−1.53429	−	2.65747i	−0.130137	−	0.225404i	−0.923729	+	0.383046i	$0.874875\pi$
$140$		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	−0.434858	−	0.900499i	$-0.643201\pi$
$149$	4.78484	−	17.8573i	0.391990	−	1.46292i	0.826848	−	0.562426i	$-0.190132\pi$
$150$		0			0
$151$	−16.0837	+	16.0837i	−1.30887	+	1.30887i	−0.386643	+	0.922229i	$0.626365\pi$
$151$	−16.0837	+	16.0837i	−1.30887	+	1.30887i	−0.922229	+	0.386643i	$0.873635\pi$
$152$		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	−0.999939	−	0.0110292i	$-0.996489\pi$
$163$	−1.63996	+	6.12043i	−0.128452	+	0.479389i	0.871487	+	0.490418i	$0.163156\pi$
$164$		0			0
$165$	−1.75064	+	1.01073i	−0.136287	+	0.0786855i
$166$		0			0
$167$	4.89526	+	18.2694i	0.378806	+	1.41372i	0.847703	+	0.530472i	$0.177985\pi$
$167$	4.89526	+	18.2694i	0.378806	+	1.41372i	−0.468896	+	0.883253i	$0.655348\pi$
$168$		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	$-0.238227\pi$
$173$	−2.93213	−	5.07860i	−0.222926	−	0.386119i	−0.955695	+	0.294358i	$0.904894\pi$
$174$		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	0.173883	−	0.984766i	$-0.444368\pi$
$179$	−7.92052	−	4.57292i	−0.592008	−	0.341796i	−0.765891	+	0.642970i	$0.777702\pi$
$180$		0			0
$181$	11.5513			0.858599			0.429300	−	0.903162i	$-0.358760\pi$
$181$	11.5513			0.858599			0.429300	+	0.903162i	$0.358760\pi$
$182$		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	$-0.394765\pi$
$191$	18.0500	−	10.4212i	1.30605	−	0.754048i	0.981434	+	0.191798i	$0.0614318\pi$
$192$		0			0
$193$	−2.91871	+	0.782065i	−0.210093	+	0.0562943i	−0.362331	−	0.932050i	$-0.618019\pi$
$193$	−2.91871	+	0.782065i	−0.210093	+	0.0562943i	0.152237	+	0.988344i	$0.451352\pi$
$194$		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	$-0.853084\pi$
$197$	−20.3251	+	5.44609i	−1.44810	+	0.388018i	−0.552738	+	0.833355i	$0.686417\pi$
$198$		0			0
$199$	−2.68040	−	4.64259i	−0.190009	−	0.329105i	0.755244	−	0.655443i	$-0.227518\pi$
$199$	−2.68040	−	4.64259i	−0.190009	−	0.329105i	−0.945253	+	0.326339i	$0.894185\pi$
$200$		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	−0.0355568	+	0.999368i	$0.511320\pi$
$211$	−12.8300	+	22.2223i	−0.883256	+	1.52984i	−0.847699	+	0.530477i	$0.822013\pi$
$212$		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	$-0.727732\pi$
$223$	−12.6427	+	3.38761i	−0.846621	+	0.226851i	−0.190669	+	0.981654i	$0.561066\pi$
$224$		0			0
$225$	−9.63706	+	5.56396i	−0.642471	+	0.370931i
$226$		0			0
$227$	26.0325	+	6.97540i	1.72784	+	0.462973i	0.979684	−	0.200549i	$-0.0642725\pi$
$227$	26.0325	+	6.97540i	1.72784	+	0.462973i	0.748157	+	0.663522i	$0.230939\pi$
$228$		0			0
$229$	0.157142	−	0.157142i	0.0103842	−	0.0103842i	−0.701896	−	0.712280i	$-0.747663\pi$
$229$	0.157142	−	0.157142i	0.0103842	−	0.0103842i	0.712280	+	0.701896i	$0.247663\pi$
$230$		0			0
$231$	−0.649962	+	1.12577i	−0.0427643	+	0.0740700i
$232$		0			0
$233$			24.0418i			1.57503i	0.616297	+	0.787514i	$0.288632\pi$
$233$			24.0418i			1.57503i	−0.616297	+	0.787514i	$0.711368\pi$
$234$		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	−0.951474	−	0.307729i	$-0.900431\pi$
$239$	−19.4668	−	19.4668i	−1.25920	−	1.25920i	−0.307729	−	0.951474i	$-0.599569\pi$
$240$		0			0
$241$	3.69069	−	13.7738i	0.237738	−	0.887251i	−0.739157	−	0.673533i	$-0.764776\pi$
$241$	3.69069	−	13.7738i	0.237738	−	0.887251i	0.976895	−	0.213718i	$-0.0685573\pi$
$242$		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	−0.543569	−	0.839365i	$-0.682927\pi$
$251$	−1.40119	+	0.808976i	−0.0884422	+	0.0510621i	0.455127	+	0.890427i	$0.349594\pi$
$252$		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	0.770905	−	0.636950i	$-0.219804\pi$
$257$	9.69476	+	5.59727i	0.604743	+	0.349148i	−0.166162	+	0.986098i	$0.553138\pi$
$258$		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	$-0.252822\pi$
$263$	7.02915	+	4.05828i	0.433436	+	0.250245i	−0.267373	+	0.963593i	$0.586156\pi$
$264$		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	$-0.581967\pi$
$269$	7.47014	−	4.31289i	0.455462	−	0.262961i	0.710134	+	0.704066i	$0.248634\pi$
$270$		0			0
$271$	0.638081	+	2.38135i	0.0387607	+	0.144657i	0.982594	−	0.185764i	$-0.0594760\pi$
$271$	0.638081	+	2.38135i	0.0387607	+	0.144657i	−0.943834	+	0.330421i	$0.892809\pi$
$272$		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	$-0.0219647\pi$
$277$	7.30805	+	12.6579i	0.439098	+	0.760540i	−0.558522	+	0.829490i	$0.688631\pi$
$278$		0			0
$279$	6.46496	−	24.1276i	0.387047	−	1.44448i
$280$		0			0
$281$	−3.44904	−	3.44904i	−0.205753	−	0.205753i	0.596707	−	0.802459i	$-0.296476\pi$
$281$	−3.44904	−	3.44904i	−0.205753	−	0.205753i	−0.802459	+	0.596707i	$0.796476\pi$
$282$		0			0
$283$	−16.0402	−	9.26082i	−0.953492	−	0.550499i	−0.0593279	−	0.998239i	$-0.518896\pi$
$283$	−16.0402	−	9.26082i	−0.953492	−	0.550499i	−0.894164	+	0.447740i	$0.852229\pi$
$284$		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	$-0.581508\pi$
$293$	−16.3695	−	4.38618i	−0.956314	−	0.256244i	−0.703039	+	0.711151i	$0.748174\pi$
$294$		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	0.988213	−	0.153087i	$-0.0489216\pi$
$307$	14.6326	+	14.6326i	0.835125	+	0.835125i	−0.153087	+	0.988213i	$0.548922\pi$
$308$		0			0
$309$	−0.906524	+	1.57014i	−0.0515703	+	0.0893224i
$310$		0			0
$311$	19.2142			1.08954			0.544768	−	0.838587i	$-0.316618\pi$
$311$	19.2142			1.08954			0.544768	+	0.838587i	$0.316618\pi$
$312$		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	$-0.251637\pi$
$317$	−0.129483	−	0.129483i	−0.00727249	−	0.00727249i	−0.710734	+	0.703461i	$0.751637\pi$
$318$		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	−0.0999013	−	0.994997i	$-0.531853\pi$
$331$	−12.4428	−	3.33403i	−0.683917	−	0.183255i	−0.584016	+	0.811742i	$0.698519\pi$
$332$		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	$0.119766\pi$
$337$			13.4906i			0.734881i	−0.930047	+	0.367441i	$0.880234\pi$
$338$		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	−0.972788	−	0.231696i	$-0.925572\pi$
$347$	−12.7983	−	22.1673i	−0.687049	−	1.19000i	0.285739	−	0.958307i	$-0.407761\pi$
$348$		0			0
$349$	−2.09309	−	7.81152i	−0.112041	−	0.418141i	0.887008	−	0.461754i	$-0.152780\pi$
$349$	−2.09309	−	7.81152i	−0.112041	−	0.418141i	−0.999048	+	0.0436131i	$0.986113\pi$
$350$		0			0
$351$	−0.0201811	−	2.57823i	−0.00107719	−	0.137616i
$352$		0			0
$353$	−0.526505	−	1.96494i	−0.0280230	−	0.104583i	0.950498	−	0.310731i	$-0.100574\pi$
$353$	−0.526505	−	1.96494i	−0.0280230	−	0.104583i	−0.978521	+	0.206148i	$0.933907\pi$
$354$		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	$-0.870781\pi$
$359$	−9.92523	+	9.92523i	−0.523833	+	0.523833i	0.394894	+	0.918727i	$0.370781\pi$
$360$		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	$-0.461298\pi$
$367$	−12.9829	−	7.49570i	−0.677703	−	0.391272i	−0.798989	+	0.601345i	$0.794632\pi$
$368$		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	$-0.535576\pi$
$373$	13.3903	−	7.73087i	0.693322	−	0.400289i	0.804855	+	0.593471i	$0.202243\pi$
$374$		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	0.979067	−	0.203537i	$-0.0652436\pi$
$379$	0.572393	+	2.13620i	0.0294018	+	0.109729i	−0.949665	+	0.313266i	$0.898577\pi$
$380$		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	$-0.908702\pi$
$383$	0.253448	−	0.945882i	0.0129506	−	0.0483323i	0.972099	+	0.234572i	$0.0753688\pi$
$384$		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	$-0.528400\pi$
$389$	−3.51480			−0.178208			−0.0891038	+	0.996022i	$0.528400\pi$
$390$		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	$-0.155435\pi$
$397$	28.1611	+	7.54575i	1.41337	+	0.378710i	0.530241	+	0.847847i	$0.322102\pi$
$398$		0			0
$399$	0.839355	−	0.484602i	0.0420203	−	0.0242604i
$400$		0			0
$401$	−8.07154	+	2.16276i	−0.403073	+	0.108003i	−0.454659	−	0.890665i	$-0.650239\pi$
$401$	−8.07154	+	2.16276i	−0.403073	+	0.108003i	0.0515861	+	0.998669i	$0.483572\pi$
$402$		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	0.181949	−	0.983308i	$-0.441759\pi$
$409$	−3.07670	−	0.824398i	−0.152133	−	0.0407639i	−0.334082	+	0.942544i	$0.608426\pi$
$410$		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	−0.483789	−	0.875185i	$-0.660740\pi$
$419$	10.5630	−	18.2957i	0.516038	−	0.893804i	0.999827	−	0.0186190i	$-0.00592695\pi$
$420$		0			0
$421$	−22.8080	−	22.8080i	−1.11159	−	1.11159i	−0.992935	−	0.118658i	$-0.962141\pi$
$421$	−22.8080	−	22.8080i	−1.11159	−	1.11159i	−0.118658	−	0.992935i	$-0.537859\pi$
$422$		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	$-0.422312\pi$
$431$	19.4340	−	5.20732i	0.936101	−	0.250828i	0.694454	+	0.719537i	$0.255646\pi$
$432$		0			0
$433$	19.6101	−	11.3219i	0.942402	−	0.544096i	0.0516897	−	0.998663i	$-0.483539\pi$
$433$	19.6101	−	11.3219i	0.942402	−	0.544096i	0.890713	+	0.454567i	$0.150206\pi$
$434$		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	−0.353788	−	0.935326i	$-0.615106\pi$
$439$	13.2654	−	22.9763i	0.633122	−	1.09660i	0.986910	−	0.161274i	$-0.0515603\pi$
$440$		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	0.524370	+	0.851490i	$0.324301\pi$
$449$	−7.53276	+	28.1126i	−0.355493	+	1.32672i	−0.879863	+	0.475227i	$0.842366\pi$
$450$		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	−0.988242	−	0.152896i	$-0.951140\pi$
$457$	−4.46465	−	16.6623i	−0.208848	−	0.779430i	0.779395	−	0.626533i	$-0.215527\pi$
$458$		0			0
$459$	2.64722	−	1.52837i	0.123561	−	0.0713382i
$460$		0			0
$461$	−7.40872	+	27.6497i	−0.345058	+	1.28778i	0.547485	+	0.836815i	$0.315585\pi$
$461$	−7.40872	+	27.6497i	−0.345058	+	1.28778i	−0.892544	+	0.450961i	$0.851081\pi$
$462$		0			0
$463$	8.53804	−	8.53804i	0.396796	−	0.396796i	−0.480305	−	0.877101i	$-0.659474\pi$
$463$	8.53804	−	8.53804i	0.396796	−	0.396796i	0.877101	+	0.480305i	$0.159474\pi$
$464$		0			0
$465$	−2.55700	−	1.47628i	−0.118578	−	0.0684610i
$466$		0			0
$467$			1.47727i			0.0683599i	0.999416	+	0.0341799i	$0.0108819\pi$
$467$			1.47727i			0.0683599i	−0.999416	+	0.0341799i	$0.989118\pi$
$468$		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	$-0.0826084\pi$
$479$	0.0258024	+	0.0962960i	0.00117894	+	0.00439988i	−0.965334	+	0.261018i	$0.915942\pi$
$480$		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	−0.681966	−	0.731384i	$-0.738875\pi$
$487$	6.05385	−	22.5933i	0.274326	−	1.02380i	0.956292	−	0.292414i	$-0.0944586\pi$
$488$		0			0
$489$	−0.535266	−	0.535266i	−0.0242056	−	0.0242056i
$490$		0			0
$491$	12.4224	+	7.17210i	0.560617	+	0.323672i	0.753393	−	0.657570i	$-0.228416\pi$
$491$	12.4224	+	7.17210i	0.560617	+	0.323672i	−0.192776	+	0.981243i	$0.561749\pi$
$492$		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	−0.445932	+	0.895067i	$0.647128\pi$
$499$	−29.9556	+	29.9556i	−1.34100	+	1.34100i	−0.895067	+	0.445932i	$0.852872\pi$
$500$		0			0
$501$	−2.18258	−	0.584821i	−0.0975105	−	0.0261279i
$502$		0			0
$503$	−36.4031	+	21.0173i	−1.62313	+	0.937116i	−0.637057	+	0.770817i	$0.719848\pi$
$503$	−36.4031	+	21.0173i	−1.62313	+	0.937116i	−0.986075	+	0.166299i	$0.946818\pi$
$504$		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	$-0.346986\pi$
$509$	29.4691	−	7.89622i	1.30619	−	0.349994i	0.843789	+	0.536675i	$0.180320\pi$
$510$		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	−0.736019	−	0.676960i	$-0.763297\pi$
$523$	4.99132	−	8.64522i	0.218255	−	0.378029i	0.954275	+	0.298931i	$0.0966301\pi$
$524$		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	$-0.839365\pi$
$541$	−9.11401	+	9.11401i	−0.391842	+	0.391842i	0.483502	+	0.875343i	$0.339365\pi$
$542$		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	$-0.163723\pi$
$557$	−3.05265	−	11.3926i	−0.129345	−	0.482721i	−0.999957	+	0.00924808i	$0.997056\pi$
$558$		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	−0.992766	−	0.120062i	$-0.961691\pi$
$563$	−32.8669	+	18.9757i	−1.38517	+	0.799730i	−0.392407	+	0.919792i	$0.628357\pi$
$564$		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	0.501026	−	0.865432i	$-0.332956\pi$
$569$	0.0489692	+	0.0282724i	0.00205290	+	0.00118524i	−0.498973	+	0.866617i	$0.666289\pi$
$570$		0			0
$571$			7.98125i			0.334005i	0.985956	+	0.167002i	$0.0534088\pi$
$571$			7.98125i			0.334005i	−0.985956	+	0.167002i	$0.946591\pi$
$572$		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	−0.399564	+	0.916705i	$0.630839\pi$
$577$	−31.6179	+	31.6179i	−1.31627	+	1.31627i	−0.916705	+	0.399564i	$0.869161\pi$
$578$		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	0.975108	+	0.221730i	$0.0711704\pi$
$587$	5.85120	+	21.8370i	0.241505	+	0.901308i	−0.733603	+	0.679578i	$0.762163\pi$
$588$		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	0.997737	−	0.0672444i	$-0.0214207\pi$
$593$	22.6590	+	22.6590i	0.930492	+	0.930492i	−0.0672444	+	0.997737i	$0.521421\pi$
$594$		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	0.136813	+	0.990597i	$0.456314\pi$
$601$	−19.3542	+	33.5225i	−0.789476	+	1.36741i	−0.926288	+	0.376815i	$0.877019\pi$
$602$		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	$-0.861456\pi$
$607$	−24.5142	+	14.1533i	−0.994999	+	0.574463i	−0.0882341	+	0.996100i	$0.528122\pi$
$608$		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	$-0.691317\pi$
$613$	−15.9165	+	4.26482i	−0.642863	+	0.172255i	−0.0773631	+	0.997003i	$0.524650\pi$
$614$		0			0
$615$	−1.29285	−	2.23929i	−0.0521329	−	0.0902969i
$616$		0			0
$617$	24.3342	+	6.52032i	0.979656	+	0.262498i	0.712900	−	0.701266i	$-0.247381\pi$
$617$	24.3342	+	6.52032i	0.979656	+	0.262498i	0.266756	+	0.963764i	$0.414048\pi$
$618$		0			0
$619$	−8.60779	−	8.60779i	−0.345976	−	0.345976i	0.512632	−	0.858608i	$-0.328671\pi$
$619$	−8.60779	−	8.60779i	−0.345976	−	0.345976i	−0.858608	+	0.512632i	$0.828671\pi$
$620$		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	−0.999950	−	0.0100000i	$-0.996817\pi$
$631$	−46.9973	−	12.5929i	−1.87093	−	0.501315i	−0.870982	−	0.491315i	$-0.836516\pi$
$632$		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	−0.452453	−	0.891788i	$-0.649451\pi$
$641$	2.37058	−	1.36865i	0.0936322	−	0.0540586i	0.546085	+	0.837730i	$0.316118\pi$
$642$		0			0
$643$	−4.87893	−	1.30730i	−0.192406	−	0.0515550i	0.161329	−	0.986901i	$-0.448422\pi$
$643$	−4.87893	−	1.30730i	−0.192406	−	0.0515550i	−0.353735	+	0.935346i	$0.615089\pi$
$644$		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	$-0.949782\pi$
$647$	−9.09928	+	15.7604i	−0.357730	+	0.619606i	0.629851	+	0.776716i	$0.283116\pi$
$648$		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	$-0.158515\pi$
$653$	23.1046	+	13.3394i	0.904151	+	0.522012i	0.0256065	+	0.999672i	$0.491848\pi$
$654$		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	0.957299	−	0.289100i	$-0.0933559\pi$
$659$	5.86022	+	10.1502i	0.228282	+	0.395396i	−0.729017	+	0.684495i	$0.760023\pi$
$660$		0			0
$661$	4.42914	+	16.5298i	0.172273	+	0.642933i	0.997000	+	0.0774016i	$0.0246623\pi$
$661$	4.42914	+	16.5298i	0.172273	+	0.642933i	−0.824727	+	0.565532i	$0.808671\pi$
$662$		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	−0.00167601	−	0.999999i	$-0.500533\pi$
$673$	−22.5318	−	13.0088i	−0.868538	−	0.501451i	−0.866862	+	0.498548i	$0.833867\pi$
$674$		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	−0.961710	−	0.274069i	$-0.911630\pi$
$683$	0.214029	−	0.798768i	0.00818960	−	0.0305640i	0.969900	+	0.243505i	$0.0782971\pi$
$684$		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	−0.940692	−	0.339261i	$-0.889823\pi$
$691$	−7.77196	−	29.0054i	−0.295659	−	1.10342i	0.645033	−	0.764155i	$-0.276844\pi$
$692$		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	$-0.682138\pi$
$701$	−28.6732			−1.08297			−0.541486	+	0.840710i	$0.682138\pi$
$702$		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	$-0.382602\pi$
$709$	5.49446	+	1.47224i	0.206349	+	0.0552910i	−0.154164	+	0.988045i	$0.549268\pi$
$710$		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	0.883576	+	0.468287i	$0.155129\pi$
$719$	22.7206	+	39.3533i	0.847337	+	1.46763i	−0.0362397	+	0.999343i	$0.511538\pi$
$720$		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	$-0.372258\pi$
$727$	21.0650			0.781256			0.390628	+	0.920549i	$0.372258\pi$
$728$		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	$-0.444863\pi$
$733$	−22.0025	−	22.0025i	−0.812682	−	0.812682i	−0.985035	+	0.172353i	$0.944863\pi$
$734$		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	−0.991183	−	0.132501i	$-0.957699\pi$
$739$	−48.4788	+	12.9899i	−1.78332	+	0.477840i	−0.792139	+	0.610341i	$0.791033\pi$
$740$		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	$-0.493800\pi$
$743$	14.6170	−	3.91663i	0.536247	−	0.143687i	0.516772	+	0.856123i	$0.327134\pi$
$744$		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	$-0.928032\pi$
$751$	−8.03320	+	13.9139i	−0.293136	+	0.507726i	0.681414	+	0.731898i	$0.261365\pi$
$752$		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	$-0.500336\pi$
$757$	−23.8711	−	13.7820i	−0.867609	−	0.500914i	−0.866553	+	0.499085i	$0.833669\pi$
$758$		0			0
$759$	1.51057	+	1.51057i	0.0548303	+	0.0548303i
$760$		0			0
$761$	8.71969	−	32.5423i	0.316089	−	1.17966i	−0.606883	−	0.794791i	$-0.707580\pi$
$761$	8.71969	−	32.5423i	0.316089	−	1.17966i	0.922972	−	0.384868i	$-0.125753\pi$
$762$		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	−0.908521	−	0.417839i	$-0.862788\pi$
$769$	−9.16888	−	34.2187i	−0.330638	−	1.23396i	0.577883	−	0.816120i	$-0.303879\pi$
$770$		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	$-0.898939\pi$
$773$	0.801104	−	2.98976i	0.0288137	−	0.107534i	0.978835	+	0.204651i	$0.0656058\pi$
$774$		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	0.739942	+	0.672671i	$0.234853\pi$
$787$	−6.65435	+	24.8344i	−0.237202	+	0.885250i	−0.977144	+	0.212579i	$0.931814\pi$
$788$		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	0.985176	+	0.171547i	$0.0548766\pi$
$797$	18.1005	+	31.3510i	0.641152	+	1.11051i	−0.344024	+	0.938961i	$0.611790\pi$
$798$		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	−0.703661	−	0.710536i	$-0.748453\pi$
$809$	7.49504	−	12.9818i	0.263512	−	0.456415i	0.967173	+	0.254120i	$0.0817859\pi$
$810$		0			0
$811$	22.2096	−	22.2096i	0.779883	−	0.779883i	−0.199928	−	0.979811i	$-0.564071\pi$
$811$	22.2096	−	22.2096i	0.779883	−	0.779883i	0.979811	+	0.199928i	$0.0640707\pi$
$812$		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	$-0.479937\pi$
$821$	17.6660	−	4.73358i	0.616547	−	0.165203i	0.553557	+	0.832811i	$0.313270\pi$
$822$		0			0
$823$	−12.2951	−	21.2957i	−0.428580	−	0.742323i	0.568167	−	0.822913i	$-0.307653\pi$
$823$	−12.2951	−	21.2957i	−0.428580	−	0.742323i	−0.996747	+	0.0805905i	$0.974319\pi$
$824$		0			0
$825$	2.46343	+	0.660075i	0.0857657	+	0.0229809i
$826$		0			0
$827$	35.8846	+	35.8846i	1.24783	+	1.24783i	0.956677	+	0.291152i	$0.0940386\pi$
$827$	35.8846	+	35.8846i	1.24783	+	1.24783i	0.291152	+	0.956677i	$0.405961\pi$
$828$		0			0
$829$	6.84382	−	11.8539i	0.237696	−	0.411701i	−0.722357	−	0.691520i	$-0.756941\pi$
$829$	6.84382	−	11.8539i	0.237696	−	0.411701i	0.960053	+	0.279819i	$0.0902745\pi$
$830$		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	$-0.331821\pi$
$839$	14.7393	+	3.94939i	0.508858	+	0.136348i	0.00474988	+	0.999989i	$0.498488\pi$
$840$		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	$-0.559727\pi$
$853$	23.2454	−	23.2454i	0.795909	−	0.795909i	0.982448	+	0.186539i	$0.0597271\pi$
$854$		0			0
$855$	18.8338	−	32.6210i	0.644101	−	1.11562i
$856$		0			0
$857$		−	46.4375i		−	1.58627i	−0.609043	−	0.793137i	$-0.708446\pi$
$857$		−	46.4375i		−	1.58627i	0.609043	−	0.793137i	$-0.291554\pi$
$858$		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	$-0.113455\pi$
$863$	17.2801	+	17.2801i	0.588220	+	0.588220i	−0.348929	+	0.937149i	$0.613455\pi$
$864$		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	0.474845	+	0.880069i	$0.342504\pi$
$877$	−11.1473	+	41.6023i	−0.376417	+	1.40481i	−0.851263	+	0.524740i	$0.824163\pi$
$878$		0			0
$879$	1.43160	−	1.43160i	0.0482868	−	0.0482868i
$880$		0			0
$881$	−7.04095	−	4.06510i	−0.237216	−	0.136957i	0.376681	−	0.926343i	$-0.377065\pi$
$881$	−7.04095	−	4.06510i	−0.237216	−	0.136957i	−0.613896	+	0.789387i	$0.710399\pi$
$882$		0			0
$883$		−	38.1442i		−	1.28365i	−0.766850	−	0.641827i	$-0.778177\pi$
$883$		−	38.1442i		−	1.28365i	0.766850	−	0.641827i	$-0.221823\pi$
$884$		0			0
$885$			0.452756i			0.0152192i
$886$		0			0
$887$	−11.1078	−	6.41307i	−0.372962	−	0.215330i	0.301790	−	0.953375i	$-0.402416\pi$
$887$	−11.1078	−	6.41307i	−0.372962	−	0.215330i	−0.674752	+	0.738045i	$0.735749\pi$
$888$		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	0.872018	−	0.489473i	$-0.162811\pi$
$907$	26.6269	+	15.3730i	0.884132	+	0.510454i	0.0121133	+	0.999927i	$0.496144\pi$
$908$		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	$-0.420043\pi$
$919$	36.7323	−	21.2074i	1.21169	−	0.699567i	0.963126	+	0.269049i	$0.0867095\pi$
$920$		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	−0.458549	−	0.888669i	$-0.651631\pi$
$929$	−39.6233	−	10.6170i	−1.30000	−	0.348334i	−0.841450	+	0.540335i	$0.818297\pi$
$930$		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	$-0.00866119\pi$
$941$	29.8298	+	29.8298i	0.972423	+	0.972423i	−0.0272066	+	0.999630i	$0.508661\pi$
$942$		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	−0.0680969	−	0.997679i	$-0.521693\pi$
$947$	11.4406	−	3.06549i	0.371769	−	0.0996152i	0.439866	+	0.898064i	$0.355026\pi$
$948$		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	−0.635387	−	0.772194i	$-0.719159\pi$
$953$	−8.77779	+	5.06786i	−0.284341	+	0.164164i	0.351046	+	0.936358i	$0.385826\pi$
$954$		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	$-0.218466\pi$
$967$	4.34955	+	4.34955i	0.139872	+	0.139872i	−0.633704	+	0.773576i	$0.718466\pi$
$968$		0			0
$969$	0.564444	−	2.10654i	0.0181326	−	0.0676717i
$970$		0			0
$971$	−27.9858	−	48.4728i	−0.898107	−	1.55557i	−0.829912	−	0.557894i	$-0.811609\pi$
$971$	−27.9858	−	48.4728i	−0.898107	−	1.55557i	−0.0681947	−	0.997672i	$-0.521724\pi$
$972$		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	−0.973846	−	0.227208i	$-0.927040\pi$
$977$	−7.62905	−	28.4720i	−0.244075	−	0.910899i	0.729771	−	0.683691i	$-0.239626\pi$
$978$		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	$-0.946838\pi$
$983$	−25.7045	+	25.7045i	−0.819846	+	0.819846i	0.166240	+	0.986085i	$0.446838\pi$
$984$		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	$-0.176568\pi$
$991$	25.7808	+	14.8846i	0.818955	+	0.472824i	−0.0311012	+	0.999516i	$0.509901\pi$
$992$		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	$-0.641298\pi$
$997$	4.35409	−	2.51384i	0.137895	−	0.0796140i	0.567361	+	0.823469i	$0.307965\pi$
$998$		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.175.5		48
4.3	odd	2		104.2.u.a.19.8	yes	48
8.3	odd	2	inner	416.2.bk.a.175.6		48
8.5	even	2		104.2.u.a.19.3	yes	48
12.11	even	2		936.2.ed.d.19.5		48
13.11	odd	12	inner	416.2.bk.a.271.6		48
24.5	odd	2		936.2.ed.d.19.10		48
52.11	even	12		104.2.u.a.11.3	✓	48
104.11	even	12	inner	416.2.bk.a.271.5		48
104.37	odd	12		104.2.u.a.11.8	yes	48
156.11	odd	12		936.2.ed.d.739.10		48
312.245	even	12		936.2.ed.d.739.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.11.3	✓	48	52.11	even	12
104.2.u.a.11.8	yes	48	104.37	odd	12
104.2.u.a.19.3	yes	48	8.5	even	2
104.2.u.a.19.8	yes	48	4.3	odd	2
416.2.bk.a.175.5		48	1.1	even	1	trivial
416.2.bk.a.175.6		48	8.3	odd	2	inner
416.2.bk.a.271.5		48	104.11	even	12	inner
416.2.bk.a.271.6		48	13.11	odd	12	inner
936.2.ed.d.19.5		48	12.11	even	2
936.2.ed.d.19.10		48	24.5	odd	2
936.2.ed.d.739.5		48	312.245	even	12
936.2.ed.d.739.10		48	156.11	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

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