LMFDB - Embedded newform 416.2.bk.a.175.10

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			175.10
Character	$\chi$	$=$	416.175
Dual form			416.2.bk.a.271.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.928193 - 1.60768i) q^{3} +(1.51817 + 1.51817i) q^{5} +(2.97768 + 0.797866i) q^{7} +(-0.223086 - 0.386396i) q^{9} +(-0.865440 + 0.231894i) q^{11} +(-0.159516 + 3.60202i) q^{13} +(3.84988 - 1.03157i) q^{15} +(1.05666 - 0.610061i) q^{17} +(-6.68538 - 1.79134i) q^{19} +(4.04657 - 4.04657i) q^{21} +(-0.433955 + 0.751632i) q^{23} -0.390340i q^{25} +4.74089 q^{27} +(3.26808 + 1.88683i) q^{29} +(-5.06168 - 5.06168i) q^{31} +(-0.430485 + 1.60659i) q^{33} +(3.30932 + 5.73191i) q^{35} +(-2.52764 - 9.43328i) q^{37} +(5.64283 + 3.59982i) q^{39} +(-3.12278 - 11.6544i) q^{41} +(4.22972 - 2.44203i) q^{43} +(0.247932 - 0.925296i) q^{45} +(-4.24871 + 4.24871i) q^{47} +(2.16780 + 1.25158i) q^{49} -2.26502i q^{51} +2.16454i q^{53} +(-1.66594 - 0.961829i) q^{55} +(-9.08523 + 9.08523i) q^{57} +(-0.0382345 + 0.142693i) q^{59} +(-7.26272 + 4.19313i) q^{61} +(-0.355986 - 1.32856i) q^{63} +(-5.71064 + 5.22630i) q^{65} +(-0.422505 - 1.57681i) q^{67} +(0.805588 + 1.39532i) q^{69} +(-4.27941 + 15.9710i) q^{71} +(9.55184 + 9.55184i) q^{73} +(-0.627540 - 0.362311i) q^{75} -2.76202 q^{77} -6.37555i q^{79} +(5.06972 - 8.78102i) q^{81} +(-3.53261 + 3.53261i) q^{83} +(2.53035 + 0.678006i) q^{85} +(6.06683 - 3.50268i) q^{87} +(9.33624 - 2.50164i) q^{89} +(-3.34892 + 10.5984i) q^{91} +(-12.8358 + 3.43933i) q^{93} +(-7.42997 - 12.8691i) q^{95} +(-10.6893 - 2.86418i) q^{97} +(0.282671 + 0.282671i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.928193	−	1.60768i	0.535893	−	0.928193i	−0.463227	−	0.886240i	$-0.653308\pi$
$3$	0.928193	−	1.60768i	0.535893	−	0.928193i	0.999120	−	0.0419537i	$-0.0133582\pi$
$4$		0			0
$5$	1.51817	+	1.51817i	0.678945	+	0.678945i	0.959761	−	0.280817i	$-0.0906053\pi$
$5$	1.51817	+	1.51817i	0.678945	+	0.678945i	−0.280817	+	0.959761i	$0.590605\pi$
$6$		0			0
$7$	2.97768	+	0.797866i	1.12546	+	0.301565i	0.773090	−	0.634297i	$-0.218710\pi$
$7$	2.97768	+	0.797866i	1.12546	+	0.301565i	0.352367	+	0.935862i	$0.385377\pi$
$8$		0			0
$9$	−0.223086	−	0.386396i	−0.0743620	−	0.128799i
$10$		0			0
$11$	−0.865440	+	0.231894i	−0.260940	+	0.0699187i	−0.386917	−	0.922114i	$-0.626460\pi$
$11$	−0.865440	+	0.231894i	−0.260940	+	0.0699187i	0.125977	+	0.992033i	$0.459793\pi$
$12$		0			0
$13$	−0.159516	+	3.60202i	−0.0442419	+	0.999021i
$13$	−0.159516	+	3.60202i	−0.0442419	+	0.999021i
$14$		0			0
$15$	3.84988	−	1.03157i	0.994034	−	0.266351i
$16$		0			0
$17$	1.05666	−	0.610061i	0.256277	−	0.147961i	−0.366358	−	0.930474i	$-0.619396\pi$
$17$	1.05666	−	0.610061i	0.256277	−	0.147961i	0.622635	+	0.782512i	$0.286062\pi$
$18$		0			0
$19$	−6.68538	−	1.79134i	−1.53373	−	0.410962i	−0.609498	−	0.792788i	$-0.708629\pi$
$19$	−6.68538	−	1.79134i	−1.53373	−	0.410962i	−0.924235	+	0.381825i	$0.875296\pi$
$20$		0			0
$21$	4.04657	−	4.04657i	0.883035	−	0.883035i
$22$		0			0
$23$	−0.433955	+	0.751632i	−0.0904858	+	0.156726i	−0.907716	−	0.419586i	$-0.862175\pi$
$23$	−0.433955	+	0.751632i	−0.0904858	+	0.156726i	0.817230	+	0.576312i	$0.195509\pi$
$24$		0			0
$25$		−	0.390340i		−	0.0780679i
$26$		0			0
$27$	4.74089			0.912385
$28$		0			0
$29$	3.26808	+	1.88683i	0.606868	+	0.350375i	0.771739	−	0.635940i	$-0.219387\pi$
$29$	3.26808	+	1.88683i	0.606868	+	0.350375i	−0.164871	+	0.986315i	$0.552721\pi$
$30$		0			0
$31$	−5.06168	−	5.06168i	−0.909104	−	0.909104i	0.0870962	−	0.996200i	$-0.472241\pi$
$31$	−5.06168	−	5.06168i	−0.909104	−	0.909104i	−0.996200	+	0.0870962i	$0.972241\pi$
$32$		0			0
$33$	−0.430485	+	1.60659i	−0.0749378	+	0.279672i
$34$		0			0
$35$	3.30932	+	5.73191i	0.559377	+	0.968869i
$36$		0			0
$37$	−2.52764	−	9.43328i	−0.415542	−	1.55082i	−0.783749	−	0.621078i	$-0.786695\pi$
$37$	−2.52764	−	9.43328i	−0.415542	−	1.55082i	0.368207	−	0.929744i	$-0.379972\pi$
$38$		0			0
$39$	5.64283	+	3.59982i	0.903576	+	0.576433i
$40$		0			0
$41$	−3.12278	−	11.6544i	−0.487696	−	1.82011i	−0.567602	−	0.823303i	$-0.692129\pi$
$41$	−3.12278	−	11.6544i	−0.487696	−	1.82011i	0.0799065	−	0.996802i	$-0.474538\pi$
$42$		0			0
$43$	4.22972	−	2.44203i	0.645026	−	0.372406i	−0.141522	−	0.989935i	$-0.545200\pi$
$43$	4.22972	−	2.44203i	0.645026	−	0.372406i	0.786548	+	0.617529i	$0.211866\pi$
$44$		0			0
$45$	0.247932	−	0.925296i	0.0369596	−	0.137935i
$46$		0			0
$47$	−4.24871	+	4.24871i	−0.619738	+	0.619738i	−0.945464	−	0.325726i	$-0.894391\pi$
$47$	−4.24871	+	4.24871i	−0.619738	+	0.619738i	0.325726	+	0.945464i	$0.394391\pi$
$48$		0			0
$49$	2.16780	+	1.25158i	0.309685	+	0.178797i
$50$		0			0
$51$		−	2.26502i		−	0.317166i
$52$		0			0
$53$			2.16454i			0.297323i	0.988888	+	0.148662i	$0.0474965\pi$
$53$			2.16454i			0.297323i	−0.988888	+	0.148662i	$0.952504\pi$
$54$		0			0
$55$	−1.66594	−	0.961829i	−0.224635	−	0.129693i
$56$		0			0
$57$	−9.08523	+	9.08523i	−1.20337	+	1.20337i
$58$		0			0
$59$	−0.0382345	+	0.142693i	−0.00497771	+	0.0185771i	−0.968370	−	0.249519i	$-0.919728\pi$
$59$	−0.0382345	+	0.142693i	−0.00497771	+	0.0185771i	0.963392	+	0.268096i	$0.0863943\pi$
$60$		0			0
$61$	−7.26272	+	4.19313i	−0.929895	+	0.536875i	−0.886778	−	0.462195i	$-0.847062\pi$
$61$	−7.26272	+	4.19313i	−0.929895	+	0.536875i	−0.0431169	+	0.999070i	$0.513729\pi$
$62$		0			0
$63$	−0.355986	−	1.32856i	−0.0448500	−	0.167382i
$64$		0			0
$65$	−5.71064	+	5.22630i	−0.708318	+	0.648242i
$66$		0			0
$67$	−0.422505	−	1.57681i	−0.0516172	−	0.192638i	0.935303	−	0.353848i	$-0.115127\pi$
$67$	−0.422505	−	1.57681i	−0.0516172	−	0.192638i	−0.986920	+	0.161210i	$0.948460\pi$
$68$		0			0
$69$	0.805588	+	1.39532i	0.0969814	+	0.167977i
$70$		0			0
$71$	−4.27941	+	15.9710i	−0.507873	+	1.89541i	−0.0671978	+	0.997740i	$0.521406\pi$
$71$	−4.27941	+	15.9710i	−0.507873	+	1.89541i	−0.440675	+	0.897667i	$0.645261\pi$
$72$		0			0
$73$	9.55184	+	9.55184i	1.11796	+	1.11796i	0.992041	+	0.125918i	$0.0401876\pi$
$73$	9.55184	+	9.55184i	1.11796	+	1.11796i	0.125918	+	0.992041i	$0.459812\pi$
$74$		0			0
$75$	−0.627540	−	0.362311i	−0.0724621	−	0.0418360i
$76$		0			0
$77$	−2.76202			−0.314762
$78$		0			0
$79$		−	6.37555i		−	0.717305i	−0.933471	−	0.358653i	$-0.883236\pi$
$79$		−	6.37555i		−	0.717305i	0.933471	−	0.358653i	$-0.116764\pi$
$80$		0			0
$81$	5.06972	−	8.78102i	0.563303	−	0.975669i
$82$		0			0
$83$	−3.53261	+	3.53261i	−0.387754	+	0.387754i	−0.873886	−	0.486132i	$-0.838408\pi$
$83$	−3.53261	+	3.53261i	−0.387754	+	0.387754i	0.486132	+	0.873886i	$0.338408\pi$
$84$		0			0
$85$	2.53035	+	0.678006i	0.274455	+	0.0735401i
$86$		0			0
$87$	6.06683	−	3.50268i	0.650432	−	0.375527i
$88$		0			0
$89$	9.33624	−	2.50164i	0.989639	−	0.265173i	0.272540	−	0.962144i	$-0.412136\pi$
$89$	9.33624	−	2.50164i	0.989639	−	0.265173i	0.717099	+	0.696971i	$0.245470\pi$
$90$		0			0
$91$	−3.34892	+	10.5984i	−0.351062	+	1.11101i
$92$		0			0
$93$	−12.8358	+	3.43933i	−1.33101	+	0.356642i
$94$		0			0
$95$	−7.42997	−	12.8691i	−0.762299	−	1.32034i
$96$		0			0
$97$	−10.6893	−	2.86418i	−1.08533	−	0.290813i	−0.328552	−	0.944486i	$-0.606561\pi$
$97$	−10.6893	−	2.86418i	−1.08533	−	0.290813i	−0.756777	+	0.653673i	$0.773227\pi$
$98$		0			0
$99$	0.282671	+	0.282671i	0.0284095	+	0.0284095i
$100$		0			0
$101$	−2.21864	+	3.84279i	−0.220763	+	0.382372i	−0.955040	−	0.296478i	$-0.904188\pi$
$101$	−2.21864	+	3.84279i	−0.220763	+	0.382372i	0.734277	+	0.678850i	$0.237521\pi$
$102$		0			0
$103$	−6.24017			−0.614862			−0.307431	−	0.951570i	$-0.599469\pi$
$103$	−6.24017			−0.614862			−0.307431	+	0.951570i	$0.599469\pi$
$104$		0			0
$105$	12.2867			1.19906
$106$		0			0
$107$	−0.706753	+	1.22413i	−0.0683244	+	0.118341i	−0.898164	−	0.439661i	$-0.855099\pi$
$107$	−0.706753	+	1.22413i	−0.0683244	+	0.118341i	0.829839	+	0.558002i	$0.188432\pi$
$108$		0			0
$109$	−0.960216	−	0.960216i	−0.0919720	−	0.0919720i	0.659624	−	0.751596i	$-0.270716\pi$
$109$	−0.960216	−	0.960216i	−0.0919720	−	0.0919720i	−0.751596	+	0.659624i	$0.770716\pi$
$110$		0			0
$111$	−17.5118	−	4.69228i	−1.66215	−	0.445371i
$112$		0			0
$113$	1.53001	+	2.65005i	0.143931	+	0.249296i	0.928974	−	0.370146i	$-0.120692\pi$
$113$	1.53001	+	2.65005i	0.143931	+	0.249296i	−0.785043	+	0.619442i	$0.787359\pi$
$114$		0			0
$115$	−1.79992	+	0.482287i	−0.167843	+	0.0449735i
$116$		0			0
$117$	1.42739	−	0.741924i	0.131963	−	0.0685909i
$118$		0			0
$119$	3.63313	−	0.973494i	0.333048	−	0.0892400i
$120$		0			0
$121$	−8.83107	+	5.09862i	−0.802824	+	0.463511i
$122$		0			0
$123$	−21.6350	−	5.79708i	−1.95076	−	0.522705i
$124$		0			0
$125$	8.18343	−	8.18343i	0.731949	−	0.731949i
$126$		0			0
$127$	3.69229	−	6.39523i	0.327637	−	0.567485i	−0.654405	−	0.756144i	$-0.727081\pi$
$127$	3.69229	−	6.39523i	0.327637	−	0.567485i	0.982043	+	0.188659i	$0.0604142\pi$
$128$		0			0
$129$		−	9.06670i		−	0.798278i
$130$		0			0
$131$	15.9912			1.39716			0.698580	−	0.715532i	$-0.253816\pi$
$131$	15.9912			1.39716			0.698580	+	0.715532i	$0.253816\pi$
$132$		0			0
$133$	−18.4777	−	10.6681i	−1.60222	−	0.925040i
$134$		0			0
$135$	7.19747	+	7.19747i	0.619459	+	0.619459i
$136$		0			0
$137$	−2.60171	+	9.70972i	−0.222279	+	0.829558i	0.761197	+	0.648521i	$0.224612\pi$
$137$	−2.60171	+	9.70972i	−0.222279	+	0.829558i	−0.983476	+	0.181037i	$0.942055\pi$
$138$		0			0
$139$	1.25261	+	2.16959i	0.106245	+	0.184022i	0.914246	−	0.405159i	$-0.132784\pi$
$139$	1.25261	+	2.16959i	0.106245	+	0.184022i	−0.808001	+	0.589181i	$0.799450\pi$
$140$		0			0
$141$	2.88694	+	10.7742i	0.243124	+	0.907350i
$142$		0			0
$143$	−0.697235	−	3.15433i	−0.0583057	−	0.263778i
$144$		0			0
$145$	2.09697	+	7.82602i	0.174144	+	0.649915i
$146$		0			0
$147$	4.02427	−	2.32341i	0.331916	−	0.191632i
$148$		0			0
$149$	−1.27183	+	4.74654i	−0.104192	+	0.388851i	−0.998252	−	0.0590960i	$-0.981178\pi$
$149$	−1.27183	+	4.74654i	−0.104192	+	0.388851i	0.894060	+	0.447947i	$0.147845\pi$
$150$		0			0
$151$	6.92678	−	6.92678i	0.563693	−	0.563693i	−0.366661	−	0.930355i	$-0.619499\pi$
$151$	6.92678	−	6.92678i	0.563693	−	0.563693i	0.930355	+	0.366661i	$0.119499\pi$
$152$		0			0
$153$	−0.471450	−	0.272192i	−0.0381145	−	0.0220054i
$154$		0			0
$155$		−	15.3689i		−	1.23446i
$156$		0			0
$157$		−	18.0396i		−	1.43972i	−0.694121	−	0.719858i	$-0.744207\pi$
$157$		−	18.0396i		−	1.43972i	0.694121	−	0.719858i	$-0.255793\pi$
$158$		0			0
$159$	3.47989	+	2.00912i	0.275973	+	0.159333i
$160$		0			0
$161$	−1.89188	+	1.89188i	−0.149101	+	0.149101i
$162$		0			0
$163$	1.71282	−	6.39232i	0.134158	−	0.500685i	−0.865842	−	0.500318i	$-0.833216\pi$
$163$	1.71282	−	6.39232i	0.134158	−	0.500685i	1.00000		0.000367214i	$-0.000116888\pi$
$164$		0			0
$165$	−3.09262	+	1.78553i	−0.240760	+	0.139003i
$166$		0			0
$167$	3.00756	+	11.2244i	0.232732	+	0.868569i	0.979158	+	0.203100i	$0.0651017\pi$
$167$	3.00756	+	11.2244i	0.232732	+	0.868569i	−0.746426	+	0.665469i	$0.768232\pi$
$168$		0			0
$169$	−12.9491	−	1.14916i	−0.996085	−	0.0883971i
$170$		0			0
$171$	0.799247	+	2.98283i	0.0611200	+	0.228103i
$172$		0			0
$173$	6.46840	+	11.2036i	0.491784	+	0.851794i	0.999955	−	0.00946175i	$-0.00301181\pi$
$173$	6.46840	+	11.2036i	0.491784	+	0.851794i	−0.508172	+	0.861256i	$0.669678\pi$
$174$		0			0
$175$	0.311439	−	1.16231i	0.0235426	−	0.0878620i
$176$		0			0
$177$	0.193916	+	0.193916i	0.0145756	+	0.0145756i
$178$		0			0
$179$	4.56241	+	2.63411i	0.341011	+	0.196883i	0.660719	−	0.750633i	$-0.270252\pi$
$179$	4.56241	+	2.63411i	0.341011	+	0.196883i	−0.319708	+	0.947516i	$0.603585\pi$
$180$		0			0
$181$	12.7764			0.949664			0.474832	−	0.880077i	$-0.342509\pi$
$181$	12.7764			0.949664			0.474832	+	0.880077i	$0.342509\pi$
$182$		0			0
$183$			15.5681i			1.15083i
$184$		0			0
$185$	10.4839	−	18.1587i	0.770793	−	1.33505i
$186$		0			0
$187$	−0.773003	+	0.773003i	−0.0565276	+	0.0565276i
$188$		0			0
$189$	14.1169	+	3.78260i	1.02685	+	0.275144i
$190$		0			0
$191$	4.76820	−	2.75292i	0.345015	−	0.199194i	−0.317473	−	0.948267i	$-0.602834\pi$
$191$	4.76820	−	2.75292i	0.345015	−	0.199194i	0.662487	+	0.749073i	$0.269501\pi$
$192$		0			0
$193$	−3.03099	+	0.812152i	−0.218176	+	0.0584600i	−0.366251	−	0.930516i	$-0.619359\pi$
$193$	−3.03099	+	0.812152i	−0.218176	+	0.0584600i	0.148075	+	0.988976i	$0.452692\pi$
$194$		0			0
$195$	3.10162	+	14.0319i	0.222112	+	1.00484i
$196$		0			0
$197$	6.20134	−	1.66164i	0.441827	−	0.118387i	−0.0310457	−	0.999518i	$-0.509884\pi$
$197$	6.20134	−	1.66164i	0.441827	−	0.118387i	0.472873	+	0.881131i	$0.343217\pi$
$198$		0			0
$199$	8.24489	+	14.2806i	0.584465	+	1.01232i	0.994942	+	0.100452i	$0.0320287\pi$
$199$	8.24489	+	14.2806i	0.584465	+	1.01232i	−0.410477	+	0.911871i	$0.634638\pi$
$200$		0			0
$201$	−2.92717	−	0.784333i	−0.206467	−	0.0553226i
$202$		0			0
$203$	8.22586	+	8.22586i	0.577342	+	0.577342i
$204$		0			0
$205$	12.9524	−	22.4342i	0.904633	−	1.56687i
$206$		0			0
$207$	0.387237			0.0269148
$208$		0			0
$209$	6.20120			0.428946
$210$		0			0
$211$	−1.39689	+	2.41949i	−0.0961660	+	0.166564i	−0.910095	−	0.414400i	$-0.863991\pi$
$211$	−1.39689	+	2.41949i	−0.0961660	+	0.166564i	0.813929	+	0.580965i	$0.197325\pi$
$212$		0			0
$213$	21.7041	+	21.7041i	1.48714	+	1.48714i
$214$		0			0
$215$	10.1288	+	2.71401i	0.690780	+	0.185094i
$216$		0			0
$217$	−11.0335	−	19.1106i	−0.749003	−	1.29731i
$218$		0			0
$219$	24.2222	−	6.49033i	1.63679	−	0.438576i
$220$		0			0
$221$	2.02890	+	3.90341i	0.136478	+	0.262572i
$222$		0			0
$223$	−17.8074	+	4.77148i	−1.19247	+	0.319522i	−0.799863	−	0.600183i	$-0.795094\pi$
$223$	−17.8074	+	4.77148i	−1.19247	+	0.319522i	−0.392610	+	0.919705i	$0.628428\pi$
$224$		0			0
$225$	−0.150826	+	0.0870793i	−0.0100551	+	0.00580529i
$226$		0			0
$227$	−11.3561	−	3.04286i	−0.753731	−	0.201962i	−0.138558	−	0.990354i	$-0.544247\pi$
$227$	−11.3561	−	3.04286i	−0.753731	−	0.201962i	−0.615172	+	0.788393i	$0.710914\pi$
$228$		0			0
$229$	−12.3457	+	12.3457i	−0.815827	+	0.815827i	−0.985500	−	0.169673i	$-0.945729\pi$
$229$	−12.3457	+	12.3457i	−0.815827	+	0.815827i	0.169673	+	0.985500i	$0.445729\pi$
$230$		0			0
$231$	−2.56369	+	4.44044i	−0.168679	+	0.292160i
$232$		0			0
$233$		−	12.3861i		−	0.811440i	−0.913997	−	0.405720i	$-0.867021\pi$
$233$		−	12.3861i		−	0.811440i	0.913997	−	0.405720i	$-0.132979\pi$
$234$		0			0
$235$	−12.9005			−0.841536
$236$		0			0
$237$	−10.2498	−	5.91774i	−0.665798	−	0.384399i
$238$		0			0
$239$	11.8203	+	11.8203i	0.764589	+	0.764589i	0.977148	−	0.212559i	$-0.0681798\pi$
$239$	11.8203	+	11.8203i	0.764589	+	0.764589i	−0.212559	+	0.977148i	$0.568180\pi$
$240$		0			0
$241$	4.17425	−	15.5785i	0.268887	−	1.00350i	−0.690941	−	0.722911i	$-0.742804\pi$
$241$	4.17425	−	15.5785i	0.268887	−	1.00350i	0.959828	−	0.280589i	$-0.0905296\pi$
$242$		0			0
$243$	−2.30003	−	3.98377i	−0.147547	−	0.255559i
$244$		0			0
$245$	1.39097	+	5.19118i	0.0888660	+	0.331652i
$246$		0			0
$247$	7.51889	−	23.7951i	0.478415	−	1.51405i
$248$		0			0
$249$	2.40035	+	8.95823i	0.152116	+	0.567705i
$250$		0			0
$251$	−15.0162	+	8.66963i	−0.947817	+	0.547222i	−0.892402	−	0.451241i	$-0.850981\pi$
$251$	−15.0162	+	8.66963i	−0.947817	+	0.547222i	−0.0554146	+	0.998463i	$0.517648\pi$
$252$		0			0
$253$	0.201263	−	0.751124i	0.0126533	−	0.0472228i
$254$		0			0
$255$	3.43867	−	3.43867i	0.215338	−	0.215338i
$256$		0			0
$257$	15.3868	+	8.88359i	0.959804	+	0.554143i	0.896113	−	0.443826i	$-0.146379\pi$
$257$	15.3868	+	8.88359i	0.959804	+	0.554143i	0.0636915	+	0.997970i	$0.479713\pi$
$258$		0			0
$259$		−	30.1060i		−	1.87070i
$260$		0			0
$261$		−	1.68370i		−	0.104218i
$262$		0			0
$263$	13.6169	+	7.86171i	0.839653	+	0.484774i	0.857146	−	0.515073i	$-0.172235\pi$
$263$	13.6169	+	7.86171i	0.839653	+	0.484774i	−0.0174932	+	0.999847i	$0.505569\pi$
$264$		0			0
$265$	−3.28614	+	3.28614i	−0.201866	+	0.201866i
$266$		0			0
$267$	4.64401	−	17.3317i	0.284209	−	1.06068i
$268$		0			0
$269$	20.3903	−	11.7723i	1.24322	−	0.717771i	0.273469	−	0.961881i	$-0.411829\pi$
$269$	20.3903	−	11.7723i	1.24322	−	0.717771i	0.969748	+	0.244110i	$0.0784957\pi$
$270$		0			0
$271$	−3.15771	−	11.7847i	−0.191817	−	0.715872i	−0.993068	−	0.117543i	$-0.962498\pi$
$271$	−3.15771	−	11.7847i	−0.191817	−	0.715872i	0.801251	−	0.598329i	$-0.204168\pi$
$272$		0			0
$273$	13.9303	+	15.2213i	0.843103	+	0.921237i
$274$		0			0
$275$	0.0905174	+	0.337816i	0.00545841	+	0.0203711i
$276$		0			0
$277$	0.828359	+	1.43476i	0.0497713	+	0.0862063i	0.889838	−	0.456277i	$-0.150817\pi$
$277$	0.828359	+	1.43476i	0.0497713	+	0.0862063i	−0.840066	+	0.542483i	$0.817484\pi$
$278$		0			0
$279$	−0.826624	+	3.08500i	−0.0494887	+	0.184694i
$280$		0			0
$281$	9.02164	+	9.02164i	0.538186	+	0.538186i	0.922996	−	0.384810i	$-0.125733\pi$
$281$	9.02164	+	9.02164i	0.538186	+	0.538186i	−0.384810	+	0.922996i	$0.625733\pi$
$282$		0			0
$283$	18.6957	+	10.7940i	1.11134	+	0.641634i	0.939177	−	0.343434i	$-0.111590\pi$
$283$	18.6957	+	10.7940i	1.11134	+	0.641634i	0.172166	+	0.985068i	$0.444924\pi$
$284$		0			0
$285$	−27.5858			−1.63404
$286$		0			0
$287$		−	37.1945i		−	2.19552i
$288$		0			0
$289$	−7.75565	+	13.4332i	−0.456215	+	0.790187i
$290$		0			0
$291$	−14.5264	+	14.5264i	−0.851551	+	0.851551i
$292$		0			0
$293$	−22.9272	−	6.14333i	−1.33942	−	0.358897i	−0.483202	−	0.875509i	$-0.660526\pi$
$293$	−22.9272	−	6.14333i	−1.33942	−	0.358897i	−0.856220	+	0.516612i	$0.827193\pi$
$294$		0			0
$295$	−0.274678	+	0.158586i	−0.0159924	+	0.00923321i
$296$		0			0
$297$	−4.10296	+	1.09938i	−0.238078	+	0.0637928i
$298$		0			0
$299$	−2.63817	−	1.68301i	−0.152569	−	0.0973311i
$300$		0			0
$301$	14.5432	−	3.89683i	0.838253	−	0.224609i
$302$		0			0
$303$	4.11865	+	7.13371i	0.236610	+	0.409821i
$304$		0			0
$305$	−17.3919	−	4.66014i	−0.995856	−	0.266839i
$306$		0			0
$307$	3.16487	+	3.16487i	0.180629	+	0.180629i	0.791630	−	0.611001i	$-0.209233\pi$
$307$	3.16487	+	3.16487i	0.180629	+	0.180629i	−0.611001	+	0.791630i	$0.709233\pi$
$308$		0			0
$309$	−5.79208	+	10.0322i	−0.329500	+	0.570711i
$310$		0			0
$311$	−20.2792			−1.14993			−0.574965	−	0.818178i	$-0.694984\pi$
$311$	−20.2792			−1.14993			−0.574965	+	0.818178i	$0.694984\pi$
$312$		0			0
$313$	−2.16882			−0.122589			−0.0612945	−	0.998120i	$-0.519523\pi$
$313$	−2.16882			−0.122589			−0.0612945	+	0.998120i	$0.519523\pi$
$314$		0			0
$315$	1.47652	−	2.55742i	0.0831927	−	0.144094i
$316$		0			0
$317$	−12.9527	−	12.9527i	−0.727494	−	0.727494i	0.242626	−	0.970120i	$-0.421991\pi$
$317$	−12.9527	−	12.9527i	−0.727494	−	0.727494i	−0.970120	+	0.242626i	$0.921991\pi$
$318$		0			0
$319$	−3.26588	−	0.875089i	−0.182854	−	0.0489956i
$320$		0			0
$321$	1.31201	+	2.27246i	0.0732291	+	0.126837i
$322$		0			0
$323$	−8.15698	+	2.18566i	−0.453866	+	0.121613i
$324$		0			0
$325$	1.40601	+	0.0622656i	0.0779915	+	0.00345387i
$326$		0			0
$327$	−2.43498	+	0.652452i	−0.134655	+	0.0360807i
$328$		0			0
$329$	−16.0412	+	9.26139i	−0.884380	+	0.510597i
$330$		0			0
$331$	1.73058	+	0.463707i	0.0951211	+	0.0254876i	0.306066	−	0.952010i	$-0.400987\pi$
$331$	1.73058	+	0.463707i	0.0951211	+	0.0254876i	−0.210945	+	0.977498i	$0.567654\pi$
$332$		0			0
$333$	−3.08110	+	3.08110i	−0.168843	+	0.168843i
$334$		0			0
$335$	1.75243	−	3.03529i	0.0957454	−	0.165836i
$336$		0			0
$337$		−	26.4251i		−	1.43947i	−0.694250	−	0.719734i	$-0.744264\pi$
$337$		−	26.4251i		−	1.43947i	0.694250	−	0.719734i	$-0.255736\pi$
$338$		0			0
$339$	5.68057			0.308526
$340$		0			0
$341$	5.55435	+	3.20681i	0.300785	+	0.173658i
$342$		0			0
$343$	−9.80227	−	9.80227i	−0.529273	−	0.529273i
$344$		0			0
$345$	−0.895310	+	3.34134i	−0.0482019	+	0.179892i
$346$		0			0
$347$	−13.6071	−	23.5682i	−0.730469	−	1.26521i	−0.956683	−	0.291131i	$-0.905968\pi$
$347$	−13.6071	−	23.5682i	−0.730469	−	1.26521i	0.226214	−	0.974078i	$-0.427365\pi$
$348$		0			0
$349$	1.42060	+	5.30175i	0.0760429	+	0.283796i	0.993468	−	0.114113i	$-0.0364026\pi$
$349$	1.42060	+	5.30175i	0.0760429	+	0.283796i	−0.917425	+	0.397909i	$0.869736\pi$
$350$		0			0
$351$	−0.756250	+	17.0768i	−0.0403656	+	0.911492i
$352$		0			0
$353$	2.30377	+	8.59780i	0.122618	+	0.457615i	0.999744	−	0.0226454i	$-0.00720886\pi$
$353$	2.30377	+	8.59780i	0.122618	+	0.457615i	−0.877126	+	0.480260i	$0.840542\pi$
$354$		0			0
$355$	−30.7435	+	17.7497i	−1.63169	+	0.942059i
$356$		0			0
$357$	1.80718	−	6.74449i	0.0956461	−	0.356956i
$358$		0			0
$359$	−2.49366	+	2.49366i	−0.131611	+	0.131611i	−0.769843	−	0.638233i	$-0.779666\pi$
$359$	−2.49366	+	2.49366i	−0.131611	+	0.131611i	0.638233	+	0.769843i	$0.279666\pi$
$360$		0			0
$361$	25.0310	+	14.4516i	1.31742	+	0.760613i
$362$		0			0
$363$			18.9300i			0.993568i
$364$		0			0
$365$			29.0026i			1.51806i
$366$		0			0
$367$	−3.94804	−	2.27940i	−0.206086	−	0.118984i	0.393405	−	0.919365i	$-0.371297\pi$
$367$	−3.94804	−	2.27940i	−0.206086	−	0.118984i	−0.599491	+	0.800381i	$0.704630\pi$
$368$		0			0
$369$	−3.80655	+	3.80655i	−0.198161	+	0.198161i
$370$		0			0
$371$	−1.72702	+	6.44532i	−0.0896623	+	0.334624i
$372$		0			0
$373$	−6.61761	+	3.82068i	−0.342647	+	0.197827i	−0.661442	−	0.749997i	$-0.730055\pi$
$373$	−6.61761	+	3.82068i	−0.342647	+	0.197827i	0.318795	+	0.947824i	$0.396722\pi$
$374$		0			0
$375$	−5.56052	−	20.7521i	−0.287144	−	1.07164i
$376$		0			0
$377$	−7.31771	+	11.4707i	−0.376881	+	0.590772i
$378$		0			0
$379$	6.19755	+	23.1296i	0.318347	+	1.18809i	0.920833	+	0.389957i	$0.127510\pi$
$379$	6.19755	+	23.1296i	0.318347	+	1.18809i	−0.602486	+	0.798129i	$0.705823\pi$
$380$		0			0
$381$	−6.85431	−	11.8720i	−0.351157	−	0.608222i
$382$		0			0
$383$	−1.99580	+	7.44844i	−0.101981	+	0.380598i	−0.997985	−	0.0634476i	$-0.979790\pi$
$383$	−1.99580	+	7.44844i	−0.101981	+	0.380598i	0.896004	+	0.444045i	$0.146457\pi$
$384$		0			0
$385$	−4.19321	−	4.19321i	−0.213706	−	0.213706i
$386$		0			0
$387$	−1.88718	−	1.08956i	−0.0959308	−	0.0553857i
$388$		0			0
$389$	−31.2509			−1.58448			−0.792242	−	0.610207i	$-0.791086\pi$
$389$	−31.2509			−1.58448			−0.792242	+	0.610207i	$0.791086\pi$
$390$		0			0
$391$			1.05895i			0.0535536i
$392$		0			0
$393$	14.8429	−	25.7087i	0.748727	−	1.29683i
$394$		0			0
$395$	9.67914	−	9.67914i	0.487011	−	0.487011i
$396$		0			0
$397$	−9.70876	−	2.60145i	−0.487269	−	0.130563i	0.00681640	−	0.999977i	$-0.497830\pi$
$397$	−9.70876	−	2.60145i	−0.487269	−	0.130563i	−0.494085	+	0.869413i	$0.664497\pi$
$398$		0			0
$399$	−34.3017	+	19.8041i	−1.71723	+	0.991445i
$400$		0			0
$401$	−0.321158	+	0.0860540i	−0.0160379	+	0.00429733i	−0.266829	−	0.963744i	$-0.585976\pi$
$401$	−0.321158	+	0.0860540i	−0.0160379	+	0.00429733i	0.250791	+	0.968041i	$0.419309\pi$
$402$		0			0
$403$	19.0397	−	17.4248i	0.948434	−	0.867993i
$404$		0			0
$405$	21.0277	−	5.63436i	1.04488	−	0.279974i
$406$		0			0
$407$	4.37504	+	7.57780i	0.216863	+	0.375618i
$408$		0			0
$409$	12.2976	+	3.29512i	0.608076	+	0.162933i	0.549701	−	0.835361i	$-0.314742\pi$
$409$	12.2976	+	3.29512i	0.608076	+	0.162933i	0.0583746	+	0.998295i	$0.481408\pi$
$410$		0			0
$411$	13.1952	+	13.1952i	0.650872	+	0.650872i
$412$		0			0
$413$	−0.227700	+	0.394388i	−0.0112044	+	0.0194066i
$414$		0			0
$415$	−10.7262			−0.526527
$416$		0			0
$417$	4.65067			0.227744
$418$		0			0
$419$	11.7449	−	20.3428i	0.573777	−	0.993812i	−0.422396	−	0.906411i	$-0.638811\pi$
$419$	11.7449	−	20.3428i	0.573777	−	0.993812i	0.996173	−	0.0874001i	$-0.0278559\pi$
$420$		0			0
$421$	−7.20576	−	7.20576i	−0.351187	−	0.351187i	0.509364	−	0.860551i	$-0.329881\pi$
$421$	−7.20576	−	7.20576i	−0.351187	−	0.351187i	−0.860551	+	0.509364i	$0.829881\pi$
$422$		0			0
$423$	2.58951	+	0.693858i	0.125907	+	0.0337366i
$424$		0			0
$425$	−0.238131	−	0.412455i	−0.0115510	−	0.0200070i
$426$		0			0
$427$	−24.9716	+	6.69112i	−1.20846	+	0.323806i
$428$		0			0
$429$	−5.71831	−	1.80689i	−0.276083	−	0.0872377i
$430$		0			0
$431$	33.7738	−	9.04966i	1.62683	−	0.435907i	0.673829	−	0.738887i	$-0.264648\pi$
$431$	33.7738	−	9.04966i	1.62683	−	0.435907i	0.952997	+	0.302981i	$0.0979817\pi$
$432$		0			0
$433$	−15.6775	+	9.05142i	−0.753414	+	0.434984i	−0.826926	−	0.562311i	$-0.809913\pi$
$433$	−15.6775	+	9.05142i	−0.753414	+	0.434984i	0.0735123	+	0.997294i	$0.476579\pi$
$434$		0			0
$435$	14.5281	+	3.89280i	0.696570	+	0.186645i
$436$		0			0
$437$	4.24758	−	4.24758i	0.203190	−	0.203190i
$438$		0			0
$439$	−8.05255	+	13.9474i	−0.384327	+	0.665674i	−0.991676	−	0.128761i	$-0.958900\pi$
$439$	−8.05255	+	13.9474i	−0.384327	+	0.665674i	0.607348	+	0.794436i	$0.292233\pi$
$440$		0			0
$441$		−	1.11684i		−	0.0531828i
$442$		0			0
$443$	17.9120			0.851025			0.425512	−	0.904953i	$-0.360094\pi$
$443$	17.9120			0.851025			0.425512	+	0.904953i	$0.360094\pi$
$444$		0			0
$445$	17.9719	+	10.3761i	0.851948	+	0.491872i
$446$		0			0
$447$	6.45040	+	6.45040i	0.305093	+	0.305093i
$448$		0			0
$449$	6.95151	−	25.9434i	0.328062	−	1.22434i	−0.583136	−	0.812375i	$-0.698174\pi$
$449$	6.95151	−	25.9434i	0.328062	−	1.22434i	0.911198	−	0.411969i	$-0.135159\pi$
$450$		0			0
$451$	5.40515	+	9.36200i	0.254519	+	0.440839i
$452$		0			0
$453$	−4.70664	−	17.5654i	−0.221137	−	0.825296i
$454$		0			0
$455$	−21.1743	+	11.0059i	−0.992668	+	0.515964i
$456$		0			0
$457$	4.15957	+	15.5237i	0.194577	+	0.726170i	0.992376	+	0.123247i	$0.0393307\pi$
$457$	4.15957	+	15.5237i	0.194577	+	0.726170i	−0.797799	+	0.602923i	$0.794003\pi$
$458$		0			0
$459$	5.00949	−	2.89223i	0.233823	−	0.134998i
$460$		0			0
$461$	−3.69416	+	13.7868i	−0.172054	+	0.642115i	0.824981	+	0.565161i	$0.191186\pi$
$461$	−3.69416	+	13.7868i	−0.172054	+	0.642115i	−0.997035	+	0.0769536i	$0.975481\pi$
$462$		0			0
$463$	−4.18298	+	4.18298i	−0.194400	+	0.194400i	−0.797594	−	0.603194i	$-0.793894\pi$
$463$	−4.18298	+	4.18298i	−0.194400	+	0.194400i	0.603194	+	0.797594i	$0.293894\pi$
$464$		0			0
$465$	−24.7083	−	14.2653i	−1.14582	−	0.661539i
$466$		0			0
$467$		−	3.76421i		−	0.174187i	−0.996200	−	0.0870935i	$-0.972242\pi$
$467$		−	3.76421i		−	0.174187i	0.996200	−	0.0870935i	$-0.0277579\pi$
$468$		0			0
$469$		−	5.03234i		−	0.232372i
$470$		0			0
$471$	−29.0019	−	16.7442i	−1.33634	−	0.771534i
$472$		0			0
$473$	−3.09428	+	3.09428i	−0.142275	+	0.142275i
$474$		0			0
$475$	−0.699232	+	2.60957i	−0.0320830	+	0.119735i
$476$		0			0
$477$	0.836372	−	0.482880i	0.0382948	−	0.0221095i
$478$		0			0
$479$	2.57796	+	9.62109i	0.117790	+	0.439599i	0.999480	−	0.0322298i	$-0.0102608\pi$
$479$	2.57796	+	9.62109i	0.117790	+	0.439599i	−0.881690	+	0.471828i	$0.843594\pi$
$480$		0			0
$481$	34.3821	−	7.59985i	1.56769	−	0.346523i
$482$		0			0
$483$	1.28550	+	4.79756i	0.0584924	+	0.218297i
$484$		0			0
$485$	−11.8798	−	20.5764i	−0.539433	−	0.934325i
$486$		0			0
$487$	7.18195	−	26.8034i	0.325445	−	1.21458i	−0.588418	−	0.808557i	$-0.700249\pi$
$487$	7.18195	−	26.8034i	0.325445	−	1.21458i	0.913864	−	0.406021i	$-0.133084\pi$
$488$		0			0
$489$	−8.68697	−	8.68697i	−0.392838	−	0.392838i
$490$		0			0
$491$	4.70703	+	2.71760i	0.212425	+	0.122644i	0.602438	−	0.798166i	$-0.294196\pi$
$491$	4.70703	+	2.71760i	0.212425	+	0.122644i	−0.390013	+	0.920809i	$0.627529\pi$
$492$		0			0
$493$	4.60432			0.207368
$494$		0			0
$495$			0.858282i			0.0385769i
$496$		0			0
$497$	−25.4854	+	44.1420i	−1.14318	+	1.98004i
$498$		0			0
$499$	−14.7930	+	14.7930i	−0.662226	+	0.662226i	−0.955904	−	0.293678i	$-0.905121\pi$
$499$	−14.7930	+	14.7930i	−0.662226	+	0.662226i	0.293678	+	0.955904i	$0.405121\pi$
$500$		0			0
$501$	20.8368	+	5.58320i	0.930920	+	0.249439i
$502$		0			0
$503$	1.65389	−	0.954873i	0.0737432	−	0.0425757i	−0.462675	−	0.886528i	$-0.653110\pi$
$503$	1.65389	−	0.954873i	0.0737432	−	0.0425757i	0.536418	+	0.843952i	$0.319777\pi$
$504$		0			0
$505$	−9.20226	+	2.46574i	−0.409495	+	0.109724i
$506$		0			0
$507$	−13.8668	+	19.7514i	−0.615845	+	0.877188i
$508$		0			0
$509$	−25.0168	+	6.70324i	−1.10885	+	0.297116i	−0.766364	−	0.642406i	$-0.777936\pi$
$509$	−25.0168	+	6.70324i	−1.10885	+	0.297116i	−0.342488	+	0.939522i	$0.611270\pi$
$510$		0			0
$511$	20.8212	+	36.0634i	0.921076	+	1.59535i
$512$		0			0
$513$	−31.6947	−	8.49257i	−1.39935	−	0.374956i
$514$		0			0
$515$	−9.47362	−	9.47362i	−0.417457	−	0.417457i
$516$		0			0
$517$	2.69176	−	4.66226i	0.118383	−	0.205046i
$518$		0			0
$519$	24.0157			1.05417
$520$		0			0
$521$	−34.4885			−1.51097			−0.755485	−	0.655166i	$-0.772599\pi$
$521$	−34.4885			−1.51097			−0.755485	+	0.655166i	$0.772599\pi$
$522$		0			0
$523$	4.27922	−	7.41182i	0.187117	−	0.324096i	−0.757171	−	0.653217i	$-0.773419\pi$
$523$	4.27922	−	7.41182i	0.187117	−	0.324096i	0.944288	+	0.329121i	$0.106752\pi$
$524$		0			0
$525$	−1.57954	−	1.57954i	−0.0689367	−	0.0689367i
$526$		0			0
$527$	−8.43638	−	2.26052i	−0.367494	−	0.0984698i
$528$		0			0
$529$	11.1234	+	19.2662i	0.483625	+	0.837662i
$530$		0			0
$531$	0.0636657	−	0.0170592i	0.00276285	−	0.000740305i
$532$		0			0
$533$	42.4774	−	9.38924i	1.83990	−	0.406693i
$534$		0			0
$535$	−2.93141	+	0.785468i	−0.126736	+	0.0339587i
$536$		0			0
$537$	8.46961	−	4.88993i	0.365490	−	0.211016i
$538$		0			0
$539$	−2.16633	−	0.580467i	−0.0933105	−	0.0250025i
$540$		0			0
$541$	−4.38710	+	4.38710i	−0.188616	+	0.188616i	−0.795098	−	0.606481i	$-0.792580\pi$
$541$	−4.38710	+	4.38710i	−0.188616	+	0.188616i	0.606481	+	0.795098i	$0.292580\pi$
$542$		0			0
$543$	11.8590	−	20.5404i	0.508918	−	0.881472i
$544$		0			0
$545$		−	2.91553i		−	0.124888i
$546$		0			0
$547$	7.06581			0.302112			0.151056	−	0.988525i	$-0.451733\pi$
$547$	7.06581			0.302112			0.151056	+	0.988525i	$0.451733\pi$
$548$		0			0
$549$	3.24042	+	1.87086i	0.138298	+	0.0798462i
$550$		0			0
$551$	−18.4684	−	18.4684i	−0.786782	−	0.786782i
$552$		0			0
$553$	5.08683	−	18.9843i	0.216314	−	0.807296i
$554$		0			0
$555$	−19.4622	−	33.7095i	−0.826124	−	1.43089i
$556$		0			0
$557$	−5.75717	−	21.4861i	−0.243939	−	0.910394i	−0.973914	−	0.226919i	$-0.927135\pi$
$557$	−5.75717	−	21.4861i	−0.243939	−	0.910394i	0.729974	−	0.683475i	$-0.239532\pi$
$558$		0			0
$559$	8.12153	+	15.6251i	0.343504	+	0.660870i
$560$		0			0
$561$	0.525244	+	1.96024i	0.0221758	+	0.0827613i
$562$		0			0
$563$	−7.46259	+	4.30853i	−0.314511	+	0.181583i	−0.648943	−	0.760837i	$-0.724789\pi$
$563$	−7.46259	+	4.30853i	−0.314511	+	0.181583i	0.334432	+	0.942420i	$0.391455\pi$
$564$		0			0
$565$	−1.70041	+	6.34603i	−0.0715369	+	0.266979i
$566$		0			0
$567$	22.1021	−	22.1021i	0.928200	−	0.928200i
$568$		0			0
$569$	18.5329	+	10.7000i	0.776940	+	0.448566i	0.835345	−	0.549727i	$-0.185268\pi$
$569$	18.5329	+	10.7000i	0.776940	+	0.448566i	−0.0584048	+	0.998293i	$0.518601\pi$
$570$		0			0
$571$		−	32.6657i		−	1.36702i	−0.729943	−	0.683508i	$-0.760454\pi$
$571$		−	32.6657i		−	1.36702i	0.729943	−	0.683508i	$-0.239546\pi$
$572$		0			0
$573$		−	10.2210i		−	0.426987i
$574$		0			0
$575$	0.293392	+	0.169390i	0.0122353	+	0.00706404i
$576$		0			0
$577$	14.9528	−	14.9528i	0.622495	−	0.622495i	−0.323674	−	0.946169i	$-0.604918\pi$
$577$	14.9528	−	14.9528i	0.622495	−	0.622495i	0.946169	+	0.323674i	$0.104918\pi$
$578$		0			0
$579$	−1.50767	+	5.62670i	−0.0626566	+	0.233838i
$580$		0			0
$581$	−13.3375	+	7.70041i	−0.553333	+	0.319467i
$582$		0			0
$583$	−0.501945	−	1.87328i	−0.0207884	−	0.0775835i
$584$		0			0
$585$	3.29338	+	1.04066i	0.136165	+	0.0430259i
$586$		0			0
$587$	−2.77481	−	10.3557i	−0.114529	−	0.427427i	0.884722	−	0.466118i	$-0.154348\pi$
$587$	−2.77481	−	10.3557i	−0.114529	−	0.427427i	−0.999251	+	0.0386910i	$0.987681\pi$
$588$		0			0
$589$	24.7720	+	42.9064i	1.02071	+	1.76793i
$590$		0			0
$591$	3.08465	−	11.5121i	0.126886	−	0.473544i
$592$		0			0
$593$	−1.75970	−	1.75970i	−0.0722623	−	0.0722623i	0.670052	−	0.742314i	$-0.266272\pi$
$593$	−1.75970	−	1.75970i	−0.0722623	−	0.0722623i	−0.742314	+	0.670052i	$0.766272\pi$
$594$		0			0
$595$	6.99362	+	4.03777i	0.286710	+	0.165532i
$596$		0			0
$597$	30.6114			1.25284
$598$		0			0
$599$			27.9976i			1.14395i	0.820271	+	0.571975i	$0.193823\pi$
$599$			27.9976i			1.14395i	−0.820271	+	0.571975i	$0.806177\pi$
$600$		0			0
$601$	0.221136	−	0.383018i	0.00902032	−	0.0156236i	−0.861480	−	0.507791i	$-0.830462\pi$
$601$	0.221136	−	0.383018i	0.00902032	−	0.0156236i	0.870500	+	0.492168i	$0.163795\pi$
$602$		0			0
$603$	−0.515019	+	0.515019i	−0.0209732	+	0.0209732i
$604$		0			0
$605$	−21.1476	−	5.66648i	−0.859772	−	0.230375i
$606$		0			0
$607$	5.63658	−	3.25428i	0.228782	−	0.132087i	−0.381228	−	0.924481i	$-0.624499\pi$
$607$	5.63658	−	3.25428i	0.228782	−	0.132087i	0.610010	+	0.792394i	$0.291165\pi$
$608$		0			0
$609$	20.8597	−	5.58935i	0.845279	−	0.226492i
$610$		0			0
$611$	−14.6262	−	15.9817i	−0.591713	−	0.646550i
$612$		0			0
$613$	−0.889978	+	0.238469i	−0.0359459	+	0.00963167i	−0.276747	−	0.960943i	$-0.589256\pi$
$613$	−0.889978	+	0.238469i	−0.0359459	+	0.00963167i	0.240801	+	0.970574i	$0.422590\pi$
$614$		0			0
$615$	−24.0446	−	41.6465i	−0.969572	−	1.67935i
$616$		0			0
$617$	31.2554	+	8.37486i	1.25830	+	0.337159i	0.825536	−	0.564350i	$-0.190873\pi$
$617$	31.2554	+	8.37486i	1.25830	+	0.337159i	0.432760	+	0.901509i	$0.357540\pi$
$618$		0			0
$619$	5.65589	+	5.65589i	0.227329	+	0.227329i	0.811576	−	0.584247i	$-0.198610\pi$
$619$	5.65589	+	5.65589i	0.227329	+	0.227329i	−0.584247	+	0.811576i	$0.698610\pi$
$620$		0			0
$621$	−2.05733	+	3.56341i	−0.0825579	+	0.142995i
$622$		0			0
$623$	29.7963			1.19376
$624$		0			0
$625$	22.8959			0.915837
$626$		0			0
$627$	5.75592	−	9.96954i	0.229869	−	0.398145i
$628$		0			0
$629$	−8.42572	−	8.42572i	−0.335955	−	0.335955i
$630$		0			0
$631$	37.9511	+	10.1690i	1.51081	+	0.404820i	0.916704	−	0.399567i	$-0.130839\pi$
$631$	37.9511	+	10.1690i	1.51081	+	0.404820i	0.594106	+	0.804387i	$0.297506\pi$
$632$		0			0
$633$	2.59317	+	4.49151i	0.103069	+	0.178521i
$634$		0			0
$635$	15.3145	−	4.10351i	0.607738	−	0.162843i
$636$		0			0
$637$	−4.85401	+	7.60880i	−0.192323	+	0.301472i
$638$		0			0
$639$	7.12580	−	1.90935i	0.281892	−	0.0755328i
$640$		0			0
$641$	30.9704	−	17.8808i	1.22326	−	0.706248i	0.257647	−	0.966239i	$-0.417053\pi$
$641$	30.9704	−	17.8808i	1.22326	−	0.706248i	0.965611	+	0.259991i	$0.0837196\pi$
$642$		0			0
$643$	38.3465	+	10.2749i	1.51224	+	0.405203i	0.917178	−	0.398477i	$-0.130461\pi$
$643$	38.3465	+	10.2749i	1.51224	+	0.405203i	0.595061	+	0.803680i	$0.297128\pi$
$644$		0			0
$645$	13.7648	−	13.7648i	0.541987	−	0.541987i
$646$		0			0
$647$	10.7499	−	18.6195i	0.422624	−	0.732006i	−0.573571	−	0.819156i	$-0.694442\pi$
$647$	10.7499	−	18.6195i	0.422624	−	0.732006i	0.996195	+	0.0871493i	$0.0277757\pi$
$648$		0			0
$649$		−	0.132359i		−	0.00519554i
$650$		0			0
$651$	−40.9649			−1.60554
$652$		0			0
$653$	35.0221	+	20.2200i	1.37052	+	0.791270i	0.990993	−	0.133911i	$-0.0427538\pi$
$653$	35.0221	+	20.2200i	1.37052	+	0.791270i	0.379526	+	0.925181i	$0.376087\pi$
$654$		0			0
$655$	24.2773	+	24.2773i	0.948594	+	0.948594i
$656$		0			0
$657$	1.55991	−	5.82168i	0.0608580	−	0.227125i
$658$		0			0
$659$	19.3441	+	33.5049i	0.753539	+	1.30517i	0.946098	+	0.323882i	$0.104988\pi$
$659$	19.3441	+	33.5049i	0.753539	+	1.30517i	−0.192559	+	0.981285i	$0.561679\pi$
$660$		0			0
$661$	6.62710	+	24.7327i	0.257764	+	0.961990i	0.966532	+	0.256548i	$0.0825851\pi$
$661$	6.62710	+	24.7327i	0.257764	+	0.961990i	−0.708767	+	0.705443i	$0.750748\pi$
$662$		0			0
$663$	8.15864	+	0.361307i	0.316855	+	0.0140320i
$664$		0			0
$665$	−11.8562	−	44.2481i	−0.459766	−	1.71587i
$666$		0			0
$667$	−2.83640	+	1.63760i	−0.109826	+	0.0634080i
$668$		0			0
$669$	−8.85771	+	33.0574i	−0.342459	+	1.27807i
$670$		0			0
$671$	5.31309	−	5.31309i	0.205109	−	0.205109i
$672$		0			0
$673$	−14.8421	−	8.56910i	−0.572121	−	0.330314i	0.185875	−	0.982573i	$-0.440488\pi$
$673$	−14.8421	−	8.56910i	−0.572121	−	0.330314i	−0.757996	+	0.652259i	$0.773821\pi$
$674$		0			0
$675$		−	1.85056i		−	0.0712280i
$676$		0			0
$677$		−	3.78614i		−	0.145513i	−0.997350	−	0.0727565i	$-0.976820\pi$
$677$		−	3.78614i		−	0.145513i	0.997350	−	0.0727565i	$-0.0231796\pi$
$678$		0			0
$679$	−29.5439	−	17.0572i	−1.13379	−	0.654595i
$680$		0			0
$681$	−15.4326	+	15.4326i	−0.591378	+	0.591378i
$682$		0			0
$683$	−6.18965	+	23.1001i	−0.236840	+	0.883900i	0.740470	+	0.672089i	$0.234603\pi$
$683$	−6.18965	+	23.1001i	−0.236840	+	0.883900i	−0.977311	+	0.211811i	$0.932064\pi$
$684$		0			0
$685$	−18.6908	+	10.7911i	−0.714139	+	0.412309i
$686$		0			0
$687$	8.38872	+	31.3071i	0.320050	+	1.19444i
$688$		0			0
$689$	−7.79674	−	0.345280i	−0.297032	−	0.0131541i
$690$		0			0
$691$	−8.71060	−	32.5084i	−0.331367	−	1.23668i	−0.907755	−	0.419502i	$-0.862205\pi$
$691$	−8.71060	−	32.5084i	−0.331367	−	1.23668i	0.576388	−	0.817176i	$-0.304462\pi$
$692$		0			0
$693$	0.616169	+	1.06724i	0.0234063	+	0.0405409i
$694$		0			0
$695$	−1.39212	+	5.19547i	−0.0528062	+	0.197076i
$696$		0			0
$697$	−10.4096	−	10.4096i	−0.394290	−	0.394290i
$698$		0			0
$699$	−19.9129	−	11.4967i	−0.753173	−	0.434845i
$700$		0			0
$701$	42.4122			1.60189			0.800943	−	0.598741i	$-0.204332\pi$
$701$	42.4122			1.60189			0.800943	+	0.598741i	$0.204332\pi$
$702$		0			0
$703$			67.5930i			2.54932i
$704$		0			0
$705$	−11.9742	+	20.7399i	−0.450973	+	0.781109i
$706$		0			0
$707$	−9.67242	+	9.67242i	−0.363769	+	0.363769i
$708$		0			0
$709$	−16.5841	−	4.44369i	−0.622828	−	0.166886i	−0.0664152	−	0.997792i	$-0.521156\pi$
$709$	−16.5841	−	4.44369i	−0.622828	−	0.166886i	−0.556413	+	0.830906i	$0.687823\pi$
$710$		0			0
$711$	−2.46349	+	1.42230i	−0.0923880	+	0.0533402i
$712$		0			0
$713$	6.00105	−	1.60798i	0.224741	−	0.0602192i
$714$		0			0
$715$	3.73027	−	5.84731i	0.139504	−	0.218677i
$716$		0			0
$717$	29.9747	−	8.03169i	1.11942	−	0.299949i
$718$		0			0
$719$	−12.9144	−	22.3684i	−0.481625	−	0.834199i	0.518152	−	0.855288i	$-0.326620\pi$
$719$	−12.9144	−	22.3684i	−0.481625	−	0.834199i	−0.999778	+	0.0210891i	$0.993287\pi$
$720$		0			0
$721$	−18.5812	−	4.97882i	−0.692000	−	0.185421i
$722$		0			0
$723$	−21.1707	−	21.1707i	−0.787347	−	0.787347i
$724$		0			0
$725$	0.736504	−	1.27566i	0.0273531	−	0.0473769i
$726$		0			0
$727$	−20.4783			−0.759499			−0.379749	−	0.925089i	$-0.623990\pi$
$727$	−20.4783			−0.759499			−0.379749	+	0.925089i	$0.623990\pi$
$728$		0			0
$729$	21.8789			0.810328
$730$		0			0
$731$	2.97957	−	5.16077i	0.110203	−	0.190878i
$732$		0			0
$733$	3.94018	+	3.94018i	0.145534	+	0.145534i	0.776120	−	0.630586i	$-0.217185\pi$
$733$	3.94018	+	3.94018i	0.145534	+	0.145534i	−0.630586	+	0.776120i	$0.717185\pi$
$734$		0			0
$735$	9.63684	+	2.58218i	0.355460	+	0.0952452i
$736$		0			0
$737$	0.731306	+	1.26666i	0.0269380	+	0.0466580i
$738$		0			0
$739$	7.07345	−	1.89532i	0.260201	−	0.0697206i	−0.126360	−	0.991984i	$-0.540329\pi$
$739$	7.07345	−	1.89532i	0.260201	−	0.0697206i	0.386561	+	0.922264i	$0.373663\pi$
$740$		0			0
$741$	−31.2760	−	34.1744i	−1.14895	−	1.25543i
$742$		0			0
$743$	−18.7680	+	5.02887i	−0.688530	+	0.184491i	−0.586088	−	0.810248i	$-0.699333\pi$
$743$	−18.7680	+	5.02887i	−0.688530	+	0.184491i	−0.102443	+	0.994739i	$0.532666\pi$
$744$		0			0
$745$	−9.13688	+	5.27518i	−0.334750	+	0.193268i
$746$		0			0
$747$	2.15306	+	0.576911i	0.0787764	+	0.0211081i
$748$		0			0
$749$	−3.08118	+	3.08118i	−0.112584	+	0.112584i
$750$		0			0
$751$	−1.09140	+	1.89037i	−0.0398259	+	0.0689805i	−0.885251	−	0.465113i	$-0.846014\pi$
$751$	−1.09140	+	1.89037i	−0.0398259	+	0.0689805i	0.845425	+	0.534094i	$0.179347\pi$
$752$		0			0
$753$			32.1884i			1.17301i
$754$		0			0
$755$	21.0320			0.765433
$756$		0			0
$757$	30.6174	+	17.6769i	1.11281	+	0.642479i	0.939555	−	0.342398i	$-0.111239\pi$
$757$	30.6174	+	17.6769i	1.11281	+	0.642479i	0.173252	+	0.984878i	$0.444572\pi$
$758$		0			0
$759$	−1.02075	−	1.02075i	−0.0370510	−	0.0370510i
$760$		0			0
$761$	−12.7004	+	47.3984i	−0.460388	+	1.71819i	0.211356	+	0.977409i	$0.432212\pi$
$761$	−12.7004	+	47.3984i	−0.460388	+	1.71819i	−0.671744	+	0.740783i	$0.734455\pi$
$762$		0			0
$763$	−2.09309	−	3.62534i	−0.0757749	−	0.131246i
$764$		0			0
$765$	−0.302507	−	1.12897i	−0.0109372	−	0.0408181i
$766$		0			0
$767$	−0.507885	−	0.160483i	−0.0183386	−	0.00579472i
$768$		0			0
$769$	9.20326	+	34.3470i	0.331878	+	1.23859i	0.907214	+	0.420669i	$0.138205\pi$
$769$	9.20326	+	34.3470i	0.331878	+	1.23859i	−0.575336	+	0.817917i	$0.695129\pi$
$770$		0			0
$771$	28.5639	−	16.4914i	1.02870	−	0.593923i
$772$		0			0
$773$	−2.31295	+	8.63204i	−0.0831910	+	0.310473i	−0.994965	−	0.100218i	$-0.968046\pi$
$773$	−2.31295	+	8.63204i	−0.0831910	+	0.310473i	0.911775	+	0.410691i	$0.134713\pi$
$774$		0			0
$775$	−1.97577	+	1.97577i	−0.0709718	+	0.0709718i
$776$		0			0
$777$	−48.4007	−	27.9442i	−1.73637	−	1.00249i
$778$		0			0
$779$			83.5078i			2.99198i
$780$		0			0
$781$		−	14.8143i		−	0.530097i
$782$		0			0
$783$	15.4936	+	8.94525i	0.553697	+	0.319677i
$784$		0			0
$785$	27.3871	−	27.3871i	0.977488	−	0.977488i
$786$		0			0
$787$	−0.134667	+	0.502585i	−0.00480037	+	0.0179152i	−0.968284	−	0.249850i	$-0.919619\pi$
$787$	−0.134667	+	0.502585i	−0.00480037	+	0.0179152i	0.963484	+	0.267766i	$0.0862853\pi$
$788$		0			0
$789$	25.2782	−	14.5944i	0.899928	−	0.519574i
$790$		0			0
$791$	2.44148	+	9.11174i	0.0868092	+	0.323976i
$792$		0			0
$793$	−13.9452	−	26.8293i	−0.495209	−	0.952737i
$794$		0			0
$795$	2.23288	+	8.33323i	0.0791922	+	0.295549i
$796$		0			0
$797$	−26.5438	−	45.9751i	−0.940228	−	1.62852i	−0.765035	−	0.643989i	$-0.777278\pi$
$797$	−26.5438	−	45.9751i	−0.940228	−	1.62852i	−0.175193	−	0.984534i	$-0.556055\pi$
$798$		0			0
$799$	−1.89746	+	7.08140i	−0.0671271	+	0.250522i
$800$		0			0
$801$	−3.04941	−	3.04941i	−0.107745	−	0.107745i
$802$		0			0
$803$	−10.4816	−	6.05153i	−0.369886	−	0.213554i
$804$		0			0
$805$	−5.74438			−0.202463
$806$		0			0
$807$		−	43.7080i		−	1.53859i
$808$		0			0
$809$	−0.520277	+	0.901146i	−0.0182920	+	0.0316826i	−0.875026	−	0.484075i	$-0.839156\pi$
$809$	−0.520277	+	0.901146i	−0.0182920	+	0.0316826i	0.856735	+	0.515758i	$0.172490\pi$
$810$		0			0
$811$	37.2714	−	37.2714i	1.30878	−	1.30878i	0.386479	−	0.922298i	$-0.373691\pi$
$811$	37.2714	−	37.2714i	1.30878	−	1.30878i	0.922298	−	0.386479i	$-0.126309\pi$
$812$		0			0
$813$	−21.8770	−	5.86193i	−0.767261	−	0.205587i
$814$		0			0
$815$	12.3050	−	7.10427i	0.431024	−	0.248852i
$816$		0			0
$817$	−32.6518	+	8.74903i	−1.14234	+	0.306090i
$818$		0			0
$819$	4.84227	−	1.07034i	0.169203	−	0.0374007i
$820$		0			0
$821$	−35.4311	+	9.49372i	−1.23655	+	0.331333i	−0.817128	−	0.576457i	$-0.804435\pi$
$821$	−35.4311	+	9.49372i	−1.23655	+	0.331333i	−0.419425	+	0.907790i	$0.637768\pi$
$822$		0			0
$823$	−5.46329	−	9.46269i	−0.190438	−	0.329849i	0.754957	−	0.655774i	$-0.227658\pi$
$823$	−5.46329	−	9.46269i	−0.190438	−	0.329849i	−0.945396	+	0.325925i	$0.894324\pi$
$824$		0			0
$825$	0.627117	+	0.168035i	0.0218334	+	0.00585024i
$826$		0			0
$827$	7.38922	+	7.38922i	0.256948	+	0.256948i	0.823812	−	0.566864i	$-0.191843\pi$
$827$	7.38922	+	7.38922i	0.256948	+	0.256948i	−0.566864	+	0.823812i	$0.691843\pi$
$828$		0			0
$829$	−9.96688	+	17.2631i	−0.346164	+	0.599574i	−0.985565	−	0.169300i	$-0.945849\pi$
$829$	−9.96688	+	17.2631i	−0.346164	+	0.599574i	0.639401	+	0.768874i	$0.279183\pi$
$830$		0			0
$831$	3.07551			0.106688
$832$		0			0
$833$	3.05415			0.105820
$834$		0			0
$835$	−12.4745	+	21.6065i	−0.431698	+	0.747723i
$836$		0			0
$837$	−23.9969	−	23.9969i	−0.829453	−	0.829453i
$838$		0			0
$839$	−46.7341	−	12.5224i	−1.61344	−	0.432320i	−0.664375	−	0.747399i	$-0.731302\pi$
$839$	−46.7341	−	12.5224i	−1.61344	−	0.432320i	−0.949065	+	0.315079i	$0.897969\pi$
$840$		0			0
$841$	−7.37975	−	12.7821i	−0.254474	−	0.440762i
$842$		0			0
$843$	22.8777	−	6.13007i	0.787951	−	0.211131i
$844$		0			0
$845$	−17.9143	−	21.4035i	−0.616270	−	0.736304i
$846$		0			0
$847$	−30.3641	+	8.13603i	−1.04332	+	0.279557i
$848$		0			0
$849$	34.7064	−	20.0378i	1.19112	−	0.687694i
$850$		0			0
$851$	8.18724	+	2.19376i	0.280655	+	0.0752012i
$852$		0			0
$853$	−0.897526	+	0.897526i	−0.0307307	+	0.0307307i	−0.722305	−	0.691574i	$-0.756917\pi$
$853$	−0.897526	+	0.897526i	−0.0307307	+	0.0307307i	0.691574	+	0.722305i	$0.256917\pi$
$854$		0			0
$855$	−3.31504	+	5.74183i	−0.113372	+	0.196366i
$856$		0			0
$857$			11.9475i			0.408120i	0.978958	+	0.204060i	$0.0654137\pi$
$857$			11.9475i			0.408120i	−0.978958	+	0.204060i	$0.934586\pi$
$858$		0			0
$859$	−46.0184			−1.57013			−0.785065	−	0.619414i	$-0.787370\pi$
$859$	−46.0184			−1.57013			−0.785065	+	0.619414i	$0.787370\pi$
$860$		0			0
$861$	−59.7968	−	34.5237i	−2.03787	−	1.17656i
$862$		0			0
$863$	−15.2947	−	15.2947i	−0.520639	−	0.520639i	0.397125	−	0.917764i	$-0.370008\pi$
$863$	−15.2947	−	15.2947i	−0.520639	−	0.520639i	−0.917764	+	0.397125i	$0.870008\pi$
$864$		0			0
$865$	−7.18882	+	26.8290i	−0.244427	+	0.912215i
$866$		0			0
$867$	14.3975	+	24.9372i	0.488964	+	0.846911i
$868$		0			0
$869$	1.47845	+	5.51766i	0.0501530	+	0.187174i
$870$		0			0
$871$	5.74710	−	1.27034i	0.194733	−	0.0430440i
$872$		0			0
$873$	1.27792	+	4.76925i	0.0432509	+	0.161415i
$874$		0			0
$875$	30.8969	−	17.8383i	1.04451	−	0.603046i
$876$		0			0
$877$	4.34149	−	16.2027i	0.146602	−	0.547125i	−0.853077	−	0.521785i	$-0.825266\pi$
$877$	4.34149	−	16.2027i	0.146602	−	0.547125i	0.999679	−	0.0253405i	$-0.00806700\pi$
$878$		0			0
$879$	−31.1574	+	31.1574i	−1.05091	+	1.05091i
$880$		0			0
$881$	−11.9739	−	6.91316i	−0.403412	−	0.232910i	0.284543	−	0.958663i	$-0.408158\pi$
$881$	−11.9739	−	6.91316i	−0.403412	−	0.232910i	−0.687955	+	0.725753i	$0.741491\pi$
$882$		0			0
$883$			9.15740i			0.308171i	0.988058	+	0.154086i	$0.0492431\pi$
$883$			9.15740i			0.308171i	−0.988058	+	0.154086i	$0.950757\pi$
$884$		0			0
$885$			0.588792i			0.0197920i
$886$		0			0
$887$	−23.0479	−	13.3067i	−0.773873	−	0.446796i	0.0603813	−	0.998175i	$-0.480768\pi$
$887$	−23.0479	−	13.3067i	−0.773873	−	0.446796i	−0.834255	+	0.551379i	$0.814102\pi$
$888$		0			0
$889$	16.0970	−	16.0970i	0.539875	−	0.539875i
$890$		0			0
$891$	−2.35128	+	8.77509i	−0.0787708	+	0.293976i
$892$		0			0
$893$	36.0152	−	20.7934i	1.20520	−	0.695824i
$894$		0			0
$895$	2.92749	+	10.9255i	0.0978551	+	0.365200i
$896$		0			0
$897$	−5.15447	+	2.67917i	−0.172103	+	0.0894548i
$898$		0			0
$899$	−6.99146	−	26.0925i	−0.233178	−	0.870233i
$900$		0			0
$901$	1.32050	+	2.28718i	0.0439923	+	0.0761970i
$902$		0			0
$903$	7.23402	−	26.9977i	0.240733	−	0.898428i
$904$		0			0
$905$	19.3967	+	19.3967i	0.644769	+	0.644769i
$906$		0			0
$907$	24.0477	+	13.8839i	0.798490	+	0.461008i	0.842943	−	0.538003i	$-0.180821\pi$
$907$	24.0477	+	13.8839i	0.798490	+	0.461008i	−0.0444530	+	0.999011i	$0.514154\pi$
$908$		0			0
$909$	1.97979			0.0656654
$910$		0			0
$911$		−	49.1603i		−	1.62875i	−0.580336	−	0.814377i	$-0.697079\pi$
$911$		−	49.1603i		−	1.62875i	0.580336	−	0.814377i	$-0.302921\pi$
$912$		0			0
$913$	2.23807	−	3.87645i	0.0740693	−	0.128292i
$914$		0			0
$915$	−23.6350	+	23.6350i	−0.781350	+	0.781350i
$916$		0			0
$917$	47.6167	+	12.7589i	1.57244	+	0.421334i
$918$		0			0
$919$	1.24049	−	0.716196i	0.0409199	−	0.0236251i	−0.479400	−	0.877596i	$-0.659146\pi$
$919$	1.24049	−	0.716196i	0.0409199	−	0.0236251i	0.520320	+	0.853971i	$0.325813\pi$
$920$		0			0
$921$	8.02570	−	2.15048i	0.264456	−	0.0708607i
$922$		0			0
$923$	−56.8452	−	17.9622i	−1.87108	−	0.591232i
$924$		0			0
$925$	−3.68218	+	0.986638i	−0.121069	+	0.0324405i
$926$		0			0
$927$	1.39209	+	2.41118i	0.0457224	+	0.0791935i
$928$		0			0
$929$	−21.5871	−	5.78426i	−0.708251	−	0.189775i	−0.113328	−	0.993558i	$-0.536151\pi$
$929$	−21.5871	−	5.78426i	−0.708251	−	0.189775i	−0.594924	+	0.803782i	$0.702818\pi$
$930$		0			0
$931$	−12.2505	−	12.2505i	−0.401495	−	0.401495i
$932$		0			0
$933$	−18.8231	+	32.6025i	−0.616239	+	1.06736i
$934$		0			0
$935$	−2.34710			−0.0767582
$936$		0			0
$937$	13.0538			0.426448			0.213224	−	0.977003i	$-0.431604\pi$
$937$	13.0538			0.426448			0.213224	+	0.977003i	$0.431604\pi$
$938$		0			0
$939$	−2.01308	+	3.48676i	−0.0656945	+	0.113786i
$940$		0			0
$941$	16.6807	+	16.6807i	0.543777	+	0.543777i	0.924634	−	0.380857i	$-0.124371\pi$
$941$	16.6807	+	16.6807i	0.543777	+	0.543777i	−0.380857	+	0.924634i	$0.624371\pi$
$942$		0			0
$943$	10.1149	+	2.71029i	0.329387	+	0.0882591i
$944$		0			0
$945$	15.6891	+	27.1743i	0.510367	+	0.883982i
$946$		0			0
$947$	−53.5470	+	14.3479i	−1.74004	+	0.466243i	−0.982456	−	0.186493i	$-0.940288\pi$
$947$	−53.5470	+	14.3479i	−1.74004	+	0.466243i	−0.757586	+	0.652735i	$0.773621\pi$
$948$		0			0
$949$	−35.9296	+	32.8823i	−1.16632	+	1.06740i
$950$		0			0
$951$	−32.8463	+	8.80114i	−1.06511	+	0.285396i
$952$		0			0
$953$	−0.170196	+	0.0982626i	−0.00551318	+	0.00318304i	−0.502754	−	0.864429i	$-0.667680\pi$
$953$	−0.170196	+	0.0982626i	−0.00551318	+	0.00318304i	0.497241	+	0.867613i	$0.334347\pi$
$954$		0			0
$955$	11.4183	+	3.05953i	0.369488	+	0.0990040i
$956$		0			0
$957$	−4.43823	+	4.43823i	−0.143467	+	0.143467i
$958$		0			0
$959$	−15.4941	+	26.8366i	−0.500331	+	0.866599i
$960$		0			0
$961$			20.2411i			0.652939i
$962$		0			0
$963$	0.630667			0.0203230
$964$		0			0
$965$	−5.83454	−	3.36857i	−0.187820	−	0.108438i
$966$		0			0
$967$	8.39214	+	8.39214i	0.269873	+	0.269873i	0.829049	−	0.559176i	$-0.188882\pi$
$967$	8.39214	+	8.39214i	0.269873	+	0.269873i	−0.559176	+	0.829049i	$0.688882\pi$
$968$		0			0
$969$	−4.05742	+	15.1425i	−0.130343	+	0.486447i
$970$		0			0
$971$	4.01139	+	6.94792i	0.128731	+	0.222969i	0.923185	−	0.384355i	$-0.125576\pi$
$971$	4.01139	+	6.94792i	0.128731	+	0.222969i	−0.794454	+	0.607324i	$0.792243\pi$
$972$		0			0
$973$	1.99884	+	7.45976i	0.0640797	+	0.239149i
$974$		0			0
$975$	1.40515	−	2.20262i	0.0450009	−	0.0705403i
$976$		0			0
$977$	−10.4415	−	38.9682i	−0.334053	−	1.24670i	−0.904892	−	0.425641i	$-0.860049\pi$
$977$	−10.4415	−	38.9682i	−0.334053	−	1.24670i	0.570839	−	0.821062i	$-0.306618\pi$
$978$		0			0
$979$	−7.49984	+	4.33004i	−0.239696	+	0.138389i
$980$		0			0
$981$	−0.156813	+	0.585234i	−0.00500666	+	0.0186851i
$982$		0			0
$983$	−19.1993	+	19.1993i	−0.612363	+	0.612363i	−0.943561	−	0.331198i	$-0.892547\pi$
$983$	−19.1993	+	19.1993i	−0.612363	+	0.612363i	0.331198	+	0.943561i	$0.392547\pi$
$984$		0			0
$985$	11.9373	+	6.89201i	0.380354	+	0.219598i
$986$		0			0
$987$			34.3855i			1.09450i
$988$		0			0
$989$			4.23892i			0.134790i
$990$		0			0
$991$	22.0555	+	12.7338i	0.700616	+	0.404501i	0.807577	−	0.589762i	$-0.200779\pi$
$991$	22.0555	+	12.7338i	0.700616	+	0.404501i	−0.106961	+	0.994263i	$0.534112\pi$
$992$		0			0
$993$	2.35180	−	2.35180i	0.0746321	−	0.0746321i
$994$		0			0
$995$	−9.16316	+	34.1974i	−0.290492	+	1.08413i
$996$		0			0
$997$	−10.7663	+	6.21593i	−0.340972	+	0.196860i	−0.660702	−	0.750648i	$-0.729741\pi$
$997$	−10.7663	+	6.21593i	−0.340972	+	0.196860i	0.319730	+	0.947509i	$0.396408\pi$
$998$		0			0
$999$	−11.9833	−	44.7222i	−0.379134	−	1.41495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.bk.a.175.10		48
4.3	odd	2		104.2.u.a.19.6	yes	48
8.3	odd	2	inner	416.2.bk.a.175.9		48
8.5	even	2		104.2.u.a.19.9	yes	48
12.11	even	2		936.2.ed.d.19.7		48
13.11	odd	12	inner	416.2.bk.a.271.9		48
24.5	odd	2		936.2.ed.d.19.4		48
52.11	even	12		104.2.u.a.11.9	yes	48
104.11	even	12	inner	416.2.bk.a.271.10		48
104.37	odd	12		104.2.u.a.11.6	✓	48
156.11	odd	12		936.2.ed.d.739.4		48
312.245	even	12		936.2.ed.d.739.7		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.u.a.11.6	✓	48	104.37	odd	12
104.2.u.a.11.9	yes	48	52.11	even	12
104.2.u.a.19.6	yes	48	4.3	odd	2
104.2.u.a.19.9	yes	48	8.5	even	2
416.2.bk.a.175.9		48	8.3	odd	2	inner
416.2.bk.a.175.10		48	1.1	even	1	trivial
416.2.bk.a.271.9		48	13.11	odd	12	inner
416.2.bk.a.271.10		48	104.11	even	12	inner
936.2.ed.d.19.4		48	24.5	odd	2
936.2.ed.d.19.7		48	12.11	even	2
936.2.ed.d.739.4		48	156.11	odd	12
936.2.ed.d.739.7		48	312.245	even	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)

\(f(q)\)	\(=\)	\(q+(0.928193 - 1.60768i) q^{3} +(1.51817 + 1.51817i) q^{5} +(2.97768 + 0.797866i) q^{7} +(-0.223086 - 0.386396i) q^{9} +(-0.865440 + 0.231894i) q^{11} +(-0.159516 + 3.60202i) q^{13} +(3.84988 - 1.03157i) q^{15} +(1.05666 - 0.610061i) q^{17} +(-6.68538 - 1.79134i) q^{19} +(4.04657 - 4.04657i) q^{21} +(-0.433955 + 0.751632i) q^{23} -0.390340i q^{25} +4.74089 q^{27} +(3.26808 + 1.88683i) q^{29} +(-5.06168 - 5.06168i) q^{31} +(-0.430485 + 1.60659i) q^{33} +(3.30932 + 5.73191i) q^{35} +(-2.52764 - 9.43328i) q^{37} +(5.64283 + 3.59982i) q^{39} +(-3.12278 - 11.6544i) q^{41} +(4.22972 - 2.44203i) q^{43} +(0.247932 - 0.925296i) q^{45} +(-4.24871 + 4.24871i) q^{47} +(2.16780 + 1.25158i) q^{49} -2.26502i q^{51} +2.16454i q^{53} +(-1.66594 - 0.961829i) q^{55} +(-9.08523 + 9.08523i) q^{57} +(-0.0382345 + 0.142693i) q^{59} +(-7.26272 + 4.19313i) q^{61} +(-0.355986 - 1.32856i) q^{63} +(-5.71064 + 5.22630i) q^{65} +(-0.422505 - 1.57681i) q^{67} +(0.805588 + 1.39532i) q^{69} +(-4.27941 + 15.9710i) q^{71} +(9.55184 + 9.55184i) q^{73} +(-0.627540 - 0.362311i) q^{75} -2.76202 q^{77} -6.37555i q^{79} +(5.06972 - 8.78102i) q^{81} +(-3.53261 + 3.53261i) q^{83} +(2.53035 + 0.678006i) q^{85} +(6.06683 - 3.50268i) q^{87} +(9.33624 - 2.50164i) q^{89} +(-3.34892 + 10.5984i) q^{91} +(-12.8358 + 3.43933i) q^{93} +(-7.42997 - 12.8691i) q^{95} +(-10.6893 - 2.86418i) q^{97} +(0.282671 + 0.282671i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{3} - 20 q^{9} + 8 q^{11} - 12 q^{17} + 8 q^{19} - 8 q^{27} + 4 q^{33} + 4 q^{35} + 12 q^{43} - 60 q^{49} + 36 q^{57} + 64 q^{59} - 16 q^{65} + 8 q^{67} - 12 q^{73} - 24 q^{75} - 8 q^{81} + 48 q^{83}+ \cdots - 168 q^{99}+O(q^{100}) \) 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