LMFDB - Embedded newform 420.2.bv.a.233.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(53,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 6, 9, 8]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("420.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.bv (of order $12$, degree $4$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			233.2
Root			$0.396143 + 1.68614i$ of defining polynomial
Character	$\chi$	$=$	420.233
Dual form			420.2.bv.a.137.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.396143 + 1.68614i) q^{3} +(-1.86603 + 1.23205i) q^{5} +(-1.94831 - 1.79000i) q^{7} +(-2.68614 + 1.33591i) q^{9} +O(q^{10})$	\(q+(0.396143 + 1.68614i) q^{3} +(-1.86603 + 1.23205i) q^{5} +(-1.94831 - 1.79000i) q^{7} +(-2.68614 + 1.33591i) q^{9} +(-2.00626 + 1.15831i) q^{11} +(-2.00000 - 2.00000i) q^{13} +(-2.81662 - 2.65831i) q^{15} +(-0.847944 + 3.16457i) q^{17} +(0.274205 + 0.158312i) q^{19} +(2.24638 - 3.99422i) q^{21} +(0.423972 + 1.58228i) q^{23} +(1.96410 - 4.59808i) q^{25} +(-3.31662 - 4.00000i) q^{27} -7.31662 q^{29} +(3.00000 + 5.19615i) q^{31} +(-2.74784 - 2.92397i) q^{33} +(5.84096 + 0.939764i) q^{35} +(-1.46410 - 5.46410i) q^{37} +(2.57999 - 4.16457i) q^{39} +9.63325i q^{41} +(1.84169 + 1.84169i) q^{43} +(3.36650 - 5.80230i) q^{45} +(6.83013 - 1.83013i) q^{47} +(0.591820 + 6.97494i) q^{49} +(-5.67181 - 0.176129i) q^{51} +(-9.99470 - 2.67807i) q^{53} +(2.31662 - 4.63325i) q^{55} +(-0.158312 + 0.525063i) q^{57} +(6.15831 + 10.6665i) q^{59} +(-4.65831 + 8.06843i) q^{61} +(7.62470 + 2.20542i) q^{63} +(6.19615 + 1.26795i) q^{65} +(-9.77844 - 2.62012i) q^{67} +(-2.50000 + 1.34169i) q^{69} +2.31662i q^{71} +(-2.42794 + 9.06119i) q^{73} +(8.53107 + 1.49025i) q^{75} +(5.98218 + 1.33444i) q^{77} +(4.92195 + 2.84169i) q^{79} +(5.43070 - 7.17687i) q^{81} +(3.84169 - 3.84169i) q^{83} +(-2.31662 - 6.94987i) q^{85} +(-2.89843 - 12.3369i) q^{87} +(8.29156 - 14.3614i) q^{89} +(0.316625 + 7.47661i) q^{91} +(-7.57301 + 7.11684i) q^{93} +(-0.706723 + 0.0424197i) q^{95} +(-9.31662 + 9.31662i) q^{97} +(3.84169 - 5.79156i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9	$ 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9} - 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{21} + 2 q^{23} - 12 q^{25} - 32 q^{29} + 24 q^{31} - 2 q^{33} + 16 q^{37} + 4 q^{39} + 28 q^{43} - 20 q^{45} + 20 q^{47} - 24 q^{51} - 16 q^{53} - 8 q^{55} + 12 q^{57} + 36 q^{59} - 24 q^{61} + 22 q^{63} + 8 q^{65} - 22 q^{67} - 20 q^{69} - 8 q^{75} + 16 q^{77} - 14 q^{81} + 44 q^{83} + 8 q^{85} - 22 q^{87} - 24 q^{91} + 12 q^{95} - 48 q^{97} + 44 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9 - 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^21 + 2 * q^23 - 12 * q^25 - 32 * q^29 + 24 * q^31 - 2 * q^33 + 16 * q^37 + 4 * q^39 + 28 * q^43 - 20 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 - 8 * q^55 + 12 * q^57 + 36 * q^59 - 24 * q^61 + 22 * q^63 + 8 * q^65 - 22 * q^67 - 20 * q^69 - 8 * q^75 + 16 * q^77 - 14 * q^81 + 44 * q^83 + 8 * q^85 - 22 * q^87 - 24 * q^91 + 12 * q^95 - 48 * q^97 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{3}{4}\right)$

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
$3$	0.396143	+	1.68614i	0.228714	+	0.973494i
$3$	0.396143	+	1.68614i	0.228714	+	0.973494i
$4$		0			0
$5$	−1.86603	+	1.23205i	−0.834512	+	0.550990i
$5$	−1.86603	+	1.23205i	−0.834512	+	0.550990i
$6$		0			0
$7$	−1.94831	−	1.79000i	−0.736392	−	0.676555i
$7$	−1.94831	−	1.79000i	−0.736392	−	0.676555i
$8$		0			0
$9$	−2.68614	+	1.33591i	−0.895380	+	0.445302i
$10$		0			0
$11$	−2.00626	+	1.15831i	−0.604909	+	0.349244i	−0.770970	−	0.636871i	$-0.780228\pi$
$11$	−2.00626	+	1.15831i	−0.604909	+	0.349244i	0.166061	+	0.986115i	$0.446895\pi$
$12$		0			0
$13$	−2.00000	−	2.00000i	−0.554700	−	0.554700i	0.373094	−	0.927794i	$-0.378297\pi$
$13$	−2.00000	−	2.00000i	−0.554700	−	0.554700i	−0.927794	+	0.373094i	$0.878297\pi$
$14$		0			0
$15$	−2.81662	−	2.65831i	−0.727249	−	0.686373i
$16$		0			0
$17$	−0.847944	+	3.16457i	−0.205657	+	0.767521i	0.783592	+	0.621276i	$0.213385\pi$
$17$	−0.847944	+	3.16457i	−0.205657	+	0.767521i	−0.989248	+	0.146245i	$0.953281\pi$
$18$		0			0
$19$	0.274205	+	0.158312i	0.0629070	+	0.0363194i	0.531124	−	0.847294i	$-0.321770\pi$
$19$	0.274205	+	0.158312i	0.0629070	+	0.0363194i	−0.468217	+	0.883614i	$0.655103\pi$
$20$		0			0
$21$	2.24638	−	3.99422i	0.490200	−	0.871610i
$22$		0			0
$23$	0.423972	+	1.58228i	0.0884042	+	0.329929i	0.995937	−	0.0900521i	$-0.0287034\pi$
$23$	0.423972	+	1.58228i	0.0884042	+	0.329929i	−0.907533	+	0.419981i	$0.862037\pi$
$24$		0			0
$25$	1.96410	−	4.59808i	0.392820	−	0.919615i
$26$		0			0
$27$	−3.31662	−	4.00000i	−0.638285	−	0.769800i
$28$		0			0
$29$	−7.31662			−1.35866			−0.679332	−	0.733831i	$-0.737730\pi$
$29$	−7.31662			−1.35866			−0.679332	+	0.733831i	$0.737730\pi$
$30$		0			0
$31$	3.00000	+	5.19615i	0.538816	+	0.933257i	0.998968	+	0.0454165i	$0.0144615\pi$
$31$	3.00000	+	5.19615i	0.538816	+	0.933257i	−0.460152	+	0.887840i	$0.652205\pi$
$32$		0			0
$33$	−2.74784	−	2.92397i	−0.478338	−	0.508998i
$34$		0			0
$35$	5.84096	+	0.939764i	0.987303	+	0.158849i
$36$		0			0
$37$	−1.46410	−	5.46410i	−0.240697	−	0.898293i	−0.975498	−	0.220010i	$-0.929391\pi$
$37$	−1.46410	−	5.46410i	−0.240697	−	0.898293i	0.734801	−	0.678283i	$-0.237276\pi$
$38$		0			0
$39$	2.57999	−	4.16457i	0.413130	−	0.666865i
$40$		0			0
$41$			9.63325i			1.50446i	0.658900	+	0.752230i	$0.271022\pi$
$41$			9.63325i			1.50446i	−0.658900	+	0.752230i	$0.728978\pi$
$42$		0			0
$43$	1.84169	+	1.84169i	0.280855	+	0.280855i	0.833450	−	0.552595i	$-0.186362\pi$
$43$	1.84169	+	1.84169i	0.280855	+	0.280855i	−0.552595	+	0.833450i	$0.686362\pi$
$44$		0			0
$45$	3.36650	−	5.80230i	0.501848	−	0.864956i
$46$		0			0
$47$	6.83013	−	1.83013i	0.996276	−	0.266951i	0.276392	−	0.961045i	$-0.410861\pi$
$47$	6.83013	−	1.83013i	0.996276	−	0.266951i	0.719885	+	0.694094i	$0.244195\pi$
$48$		0			0
$49$	0.591820	+	6.97494i	0.0845458	+	0.996420i
$50$		0			0
$51$	−5.67181	−	0.176129i	−0.794213	−	0.0246630i
$52$		0			0
$53$	−9.99470	−	2.67807i	−1.37288	−	0.367861i	−0.504348	−	0.863500i	$-0.668267\pi$
$53$	−9.99470	−	2.67807i	−1.37288	−	0.367861i	−0.868529	+	0.495639i	$0.834934\pi$
$54$		0			0
$55$	2.31662	−	4.63325i	0.312374	−	0.624747i
$56$		0			0
$57$	−0.158312	+	0.525063i	−0.0209690	+	0.0695463i
$58$		0			0
$59$	6.15831	+	10.6665i	0.801744	+	1.38866i	0.918467	+	0.395497i	$0.129428\pi$
$59$	6.15831	+	10.6665i	0.801744	+	1.38866i	−0.116723	+	0.993164i	$0.537239\pi$
$60$		0			0
$61$	−4.65831	+	8.06843i	−0.596436	+	1.03306i	0.396907	+	0.917859i	$0.370084\pi$
$61$	−4.65831	+	8.06843i	−0.596436	+	1.03306i	−0.993343	+	0.115198i	$0.963250\pi$
$62$		0			0
$63$	7.62470	+	2.20542i	0.960622	+	0.277857i
$64$		0			0
$65$	6.19615	+	1.26795i	0.768538	+	0.157270i
$66$		0			0
$67$	−9.77844	−	2.62012i	−1.19463	−	0.320099i	−0.393913	−	0.919148i	$-0.628879\pi$
$67$	−9.77844	−	2.62012i	−1.19463	−	0.320099i	−0.800713	+	0.599048i	$0.795546\pi$
$68$		0			0
$69$	−2.50000	+	1.34169i	−0.300965	+	0.161520i
$70$		0			0
$71$			2.31662i			0.274933i	0.990506	+	0.137466i	$0.0438959\pi$
$71$			2.31662i			0.274933i	−0.990506	+	0.137466i	$0.956104\pi$
$72$		0			0
$73$	−2.42794	+	9.06119i	−0.284169	+	1.06053i	0.665276	+	0.746597i	$0.268314\pi$
$73$	−2.42794	+	9.06119i	−0.284169	+	1.06053i	−0.949445	+	0.313934i	$0.898353\pi$
$74$		0			0
$75$	8.53107	+	1.49025i	0.985083	+	0.172080i
$76$		0			0
$77$	5.98218	+	1.33444i	0.681733	+	0.152074i
$78$		0			0
$79$	4.92195	+	2.84169i	0.553762	+	0.319715i	0.750638	−	0.660714i	$-0.229746\pi$
$79$	4.92195	+	2.84169i	0.553762	+	0.319715i	−0.196876	+	0.980428i	$0.563080\pi$
$80$		0			0
$81$	5.43070	−	7.17687i	0.603411	−	0.797430i
$82$		0			0
$83$	3.84169	−	3.84169i	0.421680	−	0.421680i	−0.464102	−	0.885782i	$-0.653623\pi$
$83$	3.84169	−	3.84169i	0.421680	−	0.421680i	0.885782	+	0.464102i	$0.153623\pi$
$84$		0			0
$85$	−2.31662	−	6.94987i	−0.251273	−	0.753820i
$86$		0			0
$87$	−2.89843	−	12.3369i	−0.310745	−	1.32265i
$88$		0			0
$89$	8.29156	−	14.3614i	0.878904	−	1.52231i	0.0263586	−	0.999653i	$-0.491609\pi$
$89$	8.29156	−	14.3614i	0.878904	−	1.52231i	0.852545	−	0.522654i	$-0.175058\pi$
$90$		0			0
$91$	0.316625	+	7.47661i	0.0331913	+	0.783762i
$92$		0			0
$93$	−7.57301	+	7.11684i	−0.785285	+	0.737982i
$94$		0			0
$95$	−0.706723	+	0.0424197i	−0.0725082	+	0.00435217i
$96$		0			0
$97$	−9.31662	+	9.31662i	−0.945960	+	0.945960i	−0.998613	−	0.0526529i	$-0.983232\pi$
$97$	−9.31662	+	9.31662i	−0.945960	+	0.945960i	0.0526529	+	0.998613i	$0.483232\pi$
$98$		0			0
$99$	3.84169	−	5.79156i	0.386104	−	0.582074i
$100$		0			0
$101$	−2.32387	+	1.34169i	−0.231234	+	0.133503i	−0.611141	−	0.791522i	$-0.709289\pi$
$101$	−2.32387	+	1.34169i	−0.231234	+	0.133503i	0.379907	+	0.925025i	$0.375956\pi$
$102$		0			0
$103$	13.8080	−	3.69985i	1.36055	−	0.364557i	0.496530	−	0.868020i	$-0.334607\pi$
$103$	13.8080	−	3.69985i	1.36055	−	0.364557i	0.864017	+	0.503462i	$0.167941\pi$
$104$		0			0
$105$	0.729285	+	10.2210i	0.0711710	+	0.997464i
$106$		0			0
$107$	5.24784	−	1.40616i	0.507328	−	0.135938i	0.00392920	−	0.999992i	$-0.498749\pi$
$107$	5.24784	−	1.40616i	0.507328	−	0.135938i	0.503399	+	0.864054i	$0.332083\pi$
$108$		0			0
$109$	16.0935	−	9.29156i	1.54147	−	0.889970i	0.542728	−	0.839909i	$-0.317392\pi$
$109$	16.0935	−	9.29156i	1.54147	−	0.889970i	0.998746	−	0.0500614i	$-0.0159417\pi$
$110$		0			0
$111$	8.63325	−	4.63325i	0.819432	−	0.439769i
$112$		0			0
$113$	−9.63325	+	9.63325i	−0.906220	+	0.906220i	−0.995965	−	0.0897449i	$-0.971395\pi$
$113$	−9.63325	+	9.63325i	−0.906220	+	0.906220i	0.0897449	+	0.995965i	$0.471395\pi$
$114$		0			0
$115$	−2.74060	−	2.43023i	−0.255562	−	0.226620i
$116$		0			0
$117$	8.04410	+	2.70047i	0.743677	+	0.249658i
$118$		0			0
$119$	7.31662	−	4.64774i	0.670714	−	0.426058i
$120$		0			0
$121$	−2.81662	+	4.87854i	−0.256057	+	0.443503i
$122$		0			0
$123$	−16.2430	+	3.81615i	−1.46458	+	0.344091i
$124$		0			0
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$	5.63325	−	5.63325i	0.499870	−	0.499870i	−0.411527	−	0.911397i	$-0.635005\pi$
$127$	5.63325	−	5.63325i	0.499870	−	0.499870i	0.911397	+	0.411527i	$0.135005\pi$
$128$		0			0
$129$	−2.37577	+	3.83492i	−0.209175	+	0.337646i
$130$		0			0
$131$	−8.66025	−	5.00000i	−0.756650	−	0.436852i	0.0714417	−	0.997445i	$-0.477240\pi$
$131$	−8.66025	−	5.00000i	−0.756650	−	0.436852i	−0.828092	+	0.560593i	$0.810573\pi$
$132$		0			0
$133$	−0.250858	−	0.799268i	−0.0217521	−	0.0693053i
$134$		0			0
$135$	11.1171	+	3.37785i	0.956808	+	0.290719i
$136$		0			0
$137$	2.54383	−	9.49370i	0.217334	−	0.811102i	−0.767998	−	0.640453i	$-0.778747\pi$
$137$	2.54383	−	9.49370i	0.217334	−	0.811102i	0.985332	−	0.170649i	$-0.0545865\pi$
$138$		0			0
$139$		−	5.68338i		−	0.482058i	−0.970518	−	0.241029i	$-0.922515\pi$
$139$		−	5.68338i		−	0.482058i	0.970518	−	0.241029i	$-0.0774848\pi$
$140$		0			0
$141$	5.79156	+	10.7916i	0.487738	+	0.908813i
$142$		0			0
$143$	6.32914	+	1.69589i	0.529269	+	0.141817i
$144$		0			0
$145$	13.6530	−	9.01445i	1.13382	−	0.748610i
$146$		0			0
$147$	−11.5263	+	3.76097i	−0.950672	+	0.310199i
$148$		0			0
$149$	−4.81662	+	8.34264i	−0.394593	+	0.683456i	−0.993049	−	0.117700i	$-0.962448\pi$
$149$	−4.81662	+	8.34264i	−0.394593	+	0.683456i	0.598456	+	0.801156i	$0.295781\pi$
$150$		0			0
$151$	1.68338	+	2.91569i	0.136991	+	0.237276i	0.926356	−	0.376648i	$-0.122923\pi$
$151$	1.68338	+	2.91569i	0.136991	+	0.237276i	−0.789365	+	0.613924i	$0.789590\pi$
$152$		0			0
$153$	−1.94987	−	9.63325i	−0.157638	−	0.778802i
$154$		0			0
$155$	−12.0000	−	6.00000i	−0.963863	−	0.481932i
$156$		0			0
$157$	−7.26264	−	1.94602i	−0.579622	−	0.155309i	−0.0429162	−	0.999079i	$-0.513665\pi$
$157$	−7.26264	−	1.94602i	−0.579622	−	0.155309i	−0.536706	+	0.843769i	$0.680332\pi$
$158$		0			0
$159$	0.556270	−	17.9134i	0.0441151	−	1.42062i
$160$		0			0
$161$	2.00626	−	3.84169i	0.158115	−	0.302767i
$162$		0			0
$163$	1.36603	−	0.366025i	0.106995	−	0.0286693i	−0.204924	−	0.978778i	$-0.565695\pi$
$163$	1.36603	−	0.366025i	0.106995	−	0.0286693i	0.311919	+	0.950109i	$0.399028\pi$
$164$		0			0
$165$	8.73003	+	2.07072i	0.679632	+	0.161206i
$166$		0			0
$167$	−13.1082	−	13.1082i	−1.01434	−	1.01434i	−0.999896	−	0.0144463i	$-0.995401\pi$
$167$	−13.1082	−	13.1082i	−1.01434	−	1.01434i	−0.0144463	−	0.999896i	$-0.504599\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$	−0.948044	−	0.0589367i	−0.0724988	−	0.00450701i
$172$		0			0
$173$	−2.54383	−	9.49370i	−0.193404	−	0.721793i	−0.992674	−	0.120821i	$-0.961447\pi$
$173$	−2.54383	−	9.49370i	−0.193404	−	0.721793i	0.799270	−	0.600972i	$-0.205220\pi$
$174$		0			0
$175$	−12.0572	+	5.44274i	−0.911440	+	0.411432i
$176$		0			0
$177$	−15.5457	+	14.6092i	−1.16848	+	1.09810i
$178$		0			0
$179$	9.63325	+	16.6853i	0.720023	+	1.24712i	0.960990	+	0.276583i	$0.0892021\pi$
$179$	9.63325	+	16.6853i	0.720023	+	1.24712i	−0.240967	+	0.970533i	$0.577465\pi$
$180$		0			0
$181$	−10.2665			−0.763103			−0.381551	−	0.924348i	$-0.624610\pi$
$181$	−10.2665			−0.763103			−0.381551	+	0.924348i	$0.624610\pi$
$182$		0			0
$183$	−15.4499	−	4.65831i	−1.14209	−	0.344352i
$184$		0			0
$185$	9.46410	+	8.39230i	0.695815	+	0.617015i
$186$		0			0
$187$	−1.96437	−	7.33112i	−0.143649	−	0.536104i
$188$		0			0
$189$	−0.698177	+	13.7300i	−0.0507849	+	0.998710i
$190$		0			0
$191$	−2.00626	−	1.15831i	−0.145168	−	0.0838125i	0.425657	−	0.904885i	$-0.360043\pi$
$191$	−2.00626	−	1.15831i	−0.145168	−	0.0838125i	−0.570825	+	0.821072i	$0.693376\pi$
$192$		0			0
$193$	−5.10601	+	19.0559i	−0.367539	+	1.37167i	0.496408	+	0.868089i	$0.334652\pi$
$193$	−5.10601	+	19.0559i	−0.367539	+	1.37167i	−0.863947	+	0.503583i	$0.832015\pi$
$194$		0			0
$195$	0.316625	+	10.9499i	0.0226740	+	0.784137i
$196$		0			0
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.0712470	−	0.997459i	$-0.522698\pi$
$198$		0			0
$199$	−2.82887	+	1.63325i	−0.200533	+	0.115778i	−0.596904	−	0.802312i	$-0.703603\pi$
$199$	−2.82887	+	1.63325i	−0.200533	+	0.115778i	0.396371	+	0.918090i	$0.370269\pi$
$200$		0			0
$201$	0.544234	−	17.5258i	0.0383873	−	1.23617i
$202$		0			0
$203$	14.2551	+	13.0967i	1.00051	+	0.919211i
$204$		0			0
$205$	−11.8687	−	17.9759i	−0.828943	−	1.25549i
$206$		0			0
$207$	−3.25263	−	3.68385i	−0.226074	−	0.256045i
$208$		0			0
$209$	−0.733501			−0.0507373
$210$		0			0
$211$	−4.94987			−0.340763			−0.170382	−	0.985378i	$-0.554500\pi$
$211$	−4.94987			−0.340763			−0.170382	+	0.985378i	$0.554500\pi$
$212$		0			0
$213$	−3.90616	+	0.917716i	−0.267645	+	0.0628809i
$214$		0			0
$215$	−5.70569	−	1.16758i	−0.389125	−	0.0796285i
$216$		0			0
$217$	3.45617	−	15.4937i	0.234620	−	1.05178i
$218$		0			0
$219$	−16.2402	−	0.504314i	−1.09741	−	0.0340784i
$220$		0			0
$221$	8.02502	−	4.63325i	0.539822	−	0.311666i
$222$		0			0
$223$	11.3166	+	11.3166i	0.757817	+	0.757817i	0.975925	−	0.218108i	$-0.0699883\pi$
$223$	11.3166	+	11.3166i	0.757817	+	0.757817i	−0.218108	+	0.975925i	$0.569988\pi$
$224$		0			0
$225$	0.866750	+	14.9749i	0.0577834	+	0.998329i
$226$		0			0
$227$	−7.18627	+	26.8195i	−0.476969	+	1.78007i	0.136811	+	0.990597i	$0.456315\pi$
$227$	−7.18627	+	26.8195i	−0.476969	+	1.78007i	−0.613780	+	0.789477i	$0.710352\pi$
$228$		0			0
$229$	6.92820	+	4.00000i	0.457829	+	0.264327i	0.711131	−	0.703060i	$-0.248183\pi$
$229$	6.92820	+	4.00000i	0.457829	+	0.264327i	−0.253302	+	0.967387i	$0.581517\pi$
$230$		0			0
$231$	0.119747	+	10.6154i	0.00787880	+	0.698444i
$232$		0			0
$233$	−4.50820	−	16.8248i	−0.295342	−	1.10223i	−0.940945	−	0.338558i	$-0.890061\pi$
$233$	−4.50820	−	16.8248i	−0.295342	−	1.10223i	0.645604	−	0.763673i	$-0.276606\pi$
$234$		0			0
$235$	−10.4904	+	11.8301i	−0.684317	+	0.771712i
$236$		0			0
$237$	−2.84169	+	9.42481i	−0.184587	+	0.612207i
$238$		0			0
$239$	−28.5330			−1.84565			−0.922823	−	0.385224i	$-0.874124\pi$
$239$	−28.5330			−1.84565			−0.922823	+	0.385224i	$0.874124\pi$
$240$		0			0
$241$	1.31662	+	2.28046i	0.0848113	+	0.146897i	0.905311	−	0.424750i	$-0.139638\pi$
$241$	1.31662	+	2.28046i	0.0848113	+	0.146897i	−0.820500	+	0.571647i	$0.806305\pi$
$242$		0			0
$243$	14.2525	+	6.31386i	0.914302	+	0.405034i
$244$		0			0
$245$	−9.69783	−	12.2863i	−0.619572	−	0.784940i
$246$		0			0
$247$	−0.231785	−	0.865035i	−0.0147482	−	0.0550409i
$248$		0			0
$249$	7.99949	+	4.95577i	0.506947	+	0.314059i
$250$		0			0
$251$		−	9.26650i		−	0.584896i	−0.956281	−	0.292448i	$-0.905530\pi$
$251$		−	9.26650i		−	0.584896i	0.956281	−	0.292448i	$-0.0944699\pi$
$252$		0			0
$253$	−2.68338	−	2.68338i	−0.168702	−	0.168702i
$254$		0			0
$255$	10.8007	−	6.65930i	0.676369	−	0.417022i
$256$		0			0
$257$	−19.9894	+	5.35614i	−1.24690	+	0.334107i	−0.821140	−	0.570727i	$-0.806661\pi$
$257$	−19.9894	+	5.35614i	−1.24690	+	0.334107i	−0.425764	+	0.904834i	$0.639995\pi$
$258$		0			0
$259$	−6.92820	+	13.2665i	−0.430498	+	0.824340i
$260$		0			0
$261$	19.6535	−	9.77433i	1.21652	−	0.605016i
$262$		0			0
$263$	−5.24784	−	1.40616i	−0.323596	−	0.0867072i	0.0933643	−	0.995632i	$-0.470238\pi$
$263$	−5.24784	−	1.40616i	−0.323596	−	0.0867072i	−0.416960	+	0.908925i	$0.636905\pi$
$264$		0			0
$265$	21.9499	−	7.31662i	1.34837	−	0.449457i
$266$		0			0
$267$	27.5000	+	8.29156i	1.68297	+	0.507435i
$268$		0			0
$269$	−2.13325	−	3.69490i	−0.130067	−	0.225282i	0.793635	−	0.608394i	$-0.208186\pi$
$269$	−2.13325	−	3.69490i	−0.130067	−	0.225282i	−0.923702	+	0.383112i	$0.874852\pi$
$270$		0			0
$271$	−14.1082	+	24.4361i	−0.857011	+	1.48439i	0.0177551	+	0.999842i	$0.494348\pi$
$271$	−14.1082	+	24.4361i	−0.857011	+	1.48439i	−0.874766	+	0.484545i	$0.838985\pi$
$272$		0			0
$273$	−12.4812	+	3.49569i	−0.755396	+	0.211569i
$274$		0			0
$275$	1.38552	+	11.5000i	0.0835499	+	0.693474i
$276$		0			0
$277$	8.62867	+	2.31205i	0.518447	+	0.138917i	0.508548	−	0.861034i	$-0.330182\pi$
$277$	8.62867	+	2.31205i	0.518447	+	0.138917i	0.00989859	+	0.999951i	$0.496849\pi$
$278$		0			0
$279$	−15.0000	−	9.94987i	−0.898027	−	0.595683i
$280$		0			0
$281$		−	24.6332i		−	1.46950i	−0.678340	−	0.734748i	$-0.737300\pi$
$281$		−	24.6332i		−	1.46950i	0.678340	−	0.734748i	$-0.262700\pi$
$282$		0			0
$283$	6.45422	−	24.0875i	0.383663	−	1.43185i	−0.456600	−	0.889672i	$-0.650933\pi$
$283$	6.45422	−	24.0875i	0.383663	−	1.43185i	0.840263	−	0.542179i	$-0.182401\pi$
$284$		0			0
$285$	−0.351489	−	1.17483i	−0.0208204	−	0.0695909i
$286$		0			0
$287$	17.2435	−	18.7686i	1.01785	−	1.10787i
$288$		0			0
$289$	5.42695	+	3.13325i	0.319232	+	0.184309i
$290$		0			0
$291$	−19.3999	−	12.0184i	−1.13724	−	0.704532i
$292$		0			0
$293$	5.36675	−	5.36675i	0.313529	−	0.313529i	−0.532746	−	0.846275i	$-0.678840\pi$
$293$	5.36675	−	5.36675i	0.313529	−	0.313529i	0.846275	+	0.532746i	$0.178840\pi$
$294$		0			0
$295$	−24.6332	−	12.3166i	−1.43420	−	0.717102i
$296$		0			0
$297$	11.2872	+	4.18334i	0.654953	+	0.242742i
$298$		0			0
$299$	2.31662	−	4.01251i	0.133974	−	0.232050i
$300$		0			0
$301$	−0.291562	−	6.88479i	−0.0168054	−	0.396833i
$302$		0			0
$303$	−3.18286	−	3.38687i	−0.182851	−	0.194571i
$304$		0			0
$305$	−1.24819	−	20.7952i	−0.0714712	−	1.19073i
$306$		0			0
$307$	9.47494	−	9.47494i	0.540763	−	0.540763i	−0.382989	−	0.923753i	$-0.625105\pi$
$307$	9.47494	−	9.47494i	0.540763	−	0.540763i	0.923753	+	0.382989i	$0.125105\pi$
$308$		0			0
$309$	11.7084	+	21.8166i	0.666070	+	1.24110i
$310$		0			0
$311$	2.00626	−	1.15831i	0.113764	−	0.0656819i	−0.442038	−	0.896996i	$-0.645744\pi$
$311$	2.00626	−	1.15831i	0.113764	−	0.0656819i	0.555802	+	0.831314i	$0.312411\pi$
$312$		0			0
$313$	12.2258	−	3.27588i	0.691041	−	0.185164i	0.103826	−	0.994595i	$-0.466891\pi$
$313$	12.2258	−	3.27588i	0.691041	−	0.185164i	0.587214	+	0.809432i	$0.300225\pi$
$314$		0			0
$315$	−16.9451	+	5.27865i	−0.954747	+	0.297418i
$316$		0			0
$317$	9.49370	−	2.54383i	0.533220	−	0.142876i	0.0178450	−	0.999841i	$-0.494319\pi$
$317$	9.49370	−	2.54383i	0.533220	−	0.142876i	0.515375	+	0.856965i	$0.327653\pi$
$318$		0			0
$319$	14.6790	−	8.47494i	0.821867	−	0.474505i
$320$		0			0
$321$	4.44987	+	8.29156i	0.248368	+	0.462790i
$322$		0			0
$323$	−0.733501	+	0.733501i	−0.0408131	+	0.0408131i
$324$		0			0
$325$	−13.1244	+	5.26795i	−0.728008	+	0.292213i
$326$		0			0
$327$	22.0422	+	23.4550i	1.21894	+	1.29707i
$328$		0			0
$329$	−16.5831	−	8.66025i	−0.914257	−	0.477455i
$330$		0			0
$331$	−4.31662	+	7.47661i	−0.237263	+	0.410952i	−0.959928	−	0.280247i	$-0.909584\pi$
$331$	−4.31662	+	7.47661i	−0.237263	+	0.410952i	0.722665	+	0.691199i	$0.242917\pi$
$332$		0			0
$333$	11.2323	+	12.7214i	0.615527	+	0.697131i
$334$		0			0
$335$	21.4749	−	7.15831i	1.17330	−	0.391100i
$336$		0			0
$337$	−4.36675	+	4.36675i	−0.237872	+	0.237872i	−0.815968	−	0.578096i	$-0.803796\pi$
$337$	−4.36675	+	4.36675i	−0.237872	+	0.237872i	0.578096	+	0.815968i	$0.303796\pi$
$338$		0			0
$339$	−20.0592	−	12.4269i	−1.08946	−	0.674935i
$340$		0			0
$341$	−12.0375	−	6.94987i	−0.651869	−	0.376357i
$342$		0			0
$343$	11.3321	−	14.6487i	0.611874	−	0.790955i
$344$		0			0
$345$	3.01204	−	5.58375i	0.162163	−	0.300619i
$346$		0			0
$347$	−4.21847	+	15.7435i	−0.226459	+	0.845157i	0.755356	+	0.655315i	$0.227464\pi$
$347$	−4.21847	+	15.7435i	−0.226459	+	0.845157i	−0.981815	+	0.189842i	$0.939203\pi$
$348$		0			0
$349$			11.0000i			0.588817i	0.955680	+	0.294408i	$0.0951225\pi$
$349$			11.0000i			0.588817i	−0.955680	+	0.294408i	$0.904877\pi$
$350$		0			0
$351$	−1.36675	+	14.6332i	−0.0729517	+	0.781065i
$352$		0			0
$353$	13.1593	+	3.52601i	0.700397	+	0.187671i	0.591408	−	0.806372i	$-0.298572\pi$
$353$	13.1593	+	3.52601i	0.700397	+	0.187671i	0.108989	+	0.994043i	$0.465239\pi$
$354$		0			0
$355$	−2.85420	−	4.32288i	−0.151485	−	0.229435i
$356$		0			0
$357$	10.7352	+	10.4957i	0.568166	+	0.555491i
$358$		0			0
$359$	14.6332	−	25.3455i	0.772313	−	1.33769i	−0.163979	−	0.986464i	$-0.552433\pi$
$359$	14.6332	−	25.3455i	0.772313	−	1.33769i	0.936292	−	0.351222i	$-0.114234\pi$
$360$		0			0
$361$	−9.44987	−	16.3677i	−0.497362	−	0.861456i
$362$		0			0
$363$	−9.34169	−	2.81662i	−0.490311	−	0.147834i
$364$		0			0
$365$	−6.63325	−	19.8997i	−0.347200	−	1.04160i
$366$		0			0
$367$	26.5348	+	7.10997i	1.38510	+	0.371138i	0.872973	−	0.487769i	$-0.162189\pi$
$367$	26.5348	+	7.10997i	1.38510	+	0.371138i	0.512132	+	0.858907i	$0.328856\pi$
$368$		0			0
$369$	−12.8691	−	25.8763i	−0.669940	−	1.34706i
$370$		0			0
$371$	14.6790	+	23.1082i	0.762097	+	1.19972i
$372$		0			0
$373$	5.89662	−	1.57999i	0.305315	−	0.0818090i	−0.102909	−	0.994691i	$-0.532815\pi$
$373$	5.89662	−	1.57999i	0.305315	−	0.0818090i	0.408224	+	0.912882i	$0.366148\pi$
$374$		0			0
$375$	−17.7553	+	7.72986i	−0.916878	+	0.399168i
$376$		0			0
$377$	14.6332	+	14.6332i	0.753651	+	0.753651i
$378$		0			0
$379$			38.2164i			1.96304i	0.191351	+	0.981522i	$0.438713\pi$
$379$			38.2164i			1.96304i	−0.191351	+	0.981522i	$0.561287\pi$
$380$		0			0
$381$	11.7300	+	7.26688i	0.600947	+	0.372293i
$382$		0			0
$383$	8.32394	+	31.0654i	0.425334	+	1.58737i	0.763193	+	0.646170i	$0.223630\pi$
$383$	8.32394	+	31.0654i	0.425334	+	1.58737i	−0.337860	+	0.941197i	$0.609703\pi$
$384$		0			0
$385$	−12.8070	+	4.88025i	−0.652705	+	0.248721i
$386$		0			0
$387$	−7.40736	−	2.48671i	−0.376537	−	0.126406i
$388$		0			0
$389$	2.31662	+	4.01251i	0.117458	+	0.203442i	0.918759	−	0.394818i	$-0.129192\pi$
$389$	2.31662	+	4.01251i	0.117458	+	0.203442i	−0.801302	+	0.598260i	$0.795859\pi$
$390$		0			0
$391$	−5.36675			−0.271408
$392$		0			0
$393$	5.00000	−	16.5831i	0.252217	−	0.836508i
$394$		0			0
$395$	−12.6856	+	0.761428i	−0.638281	+	0.0383116i
$396$		0			0
$397$	2.33039	+	8.69714i	0.116959	+	0.436497i	0.999426	−	0.0338773i	$-0.0107855\pi$
$397$	2.33039	+	8.69714i	0.116959	+	0.436497i	−0.882467	+	0.470374i	$0.844119\pi$
$398$		0			0
$399$	1.24830	−	0.739606i	0.0624933	−	0.0370266i
$400$		0			0
$401$	28.3046	+	16.3417i	1.41347	+	0.816065i	0.995713	−	0.0924958i	$-0.0294845\pi$
$401$	28.3046	+	16.3417i	1.41347	+	0.816065i	0.417753	+	0.908561i	$0.362818\pi$
$402$		0			0
$403$	4.39230	−	16.3923i	0.218796	−	0.816559i
$404$		0			0
$405$	−1.29156	+	20.0831i	−0.0641782	+	0.997938i
$406$		0			0
$407$	9.26650	+	9.26650i	0.459323	+	0.459323i
$408$		0			0
$409$	−5.23956	+	3.02506i	−0.259080	+	0.149580i	−0.623915	−	0.781492i	$-0.714459\pi$
$409$	−5.23956	+	3.02506i	−0.259080	+	0.149580i	0.364835	+	0.931072i	$0.381125\pi$
$410$		0			0
$411$	17.0154	+	0.528387i	0.839310	+	0.0260634i
$412$		0			0
$413$	7.09472	−	31.8050i	0.349109	−	1.56502i
$414$		0			0
$415$	−2.43553	+	11.9018i	−0.119556	+	0.584238i
$416$		0			0
$417$	9.58297	−	2.25143i	0.469280	−	0.110253i
$418$		0			0
$419$	0.733501			0.0358339			0.0179169	−	0.999839i	$-0.494297\pi$
$419$	0.733501			0.0358339			0.0179169	+	0.999839i	$0.494297\pi$
$420$		0			0
$421$	21.3166			1.03891			0.519454	−	0.854498i	$-0.326135\pi$
$421$	21.3166			1.03891			0.519454	+	0.854498i	$0.326135\pi$
$422$		0			0
$423$	−15.9018	+	14.0404i	−0.773172	+	0.682667i
$424$		0			0
$425$	12.8855	+	10.1144i	0.625038	+	0.490623i
$426$		0			0
$427$	23.5183	−	7.38144i	1.13813	−	0.357213i
$428$		0			0
$429$	−0.352258	+	11.3436i	−0.0170072	+	0.547676i
$430$		0			0
$431$	10.6665	−	6.15831i	0.513788	−	0.296635i	−0.220602	−	0.975364i	$-0.570802\pi$
$431$	10.6665	−	6.15831i	0.513788	−	0.296635i	0.734389	+	0.678729i	$0.237469\pi$
$432$		0			0
$433$	−13.9499	−	13.9499i	−0.670388	−	0.670388i	0.287417	−	0.957805i	$-0.407203\pi$
$433$	−13.9499	−	13.9499i	−0.670388	−	0.670388i	−0.957805	+	0.287417i	$0.907203\pi$
$434$		0			0
$435$	20.6082	+	19.4499i	0.988087	+	0.932550i
$436$		0			0
$437$	−0.134240	+	0.500990i	−0.00642157	+	0.0239656i
$438$		0			0
$439$	−33.9190	−	19.5831i	−1.61886	−	0.934652i	−0.987215	−	0.159397i	$-0.949045\pi$
$439$	−33.9190	−	19.5831i	−1.61886	−	0.934652i	−0.631649	−	0.775254i	$-0.717622\pi$
$440$		0			0
$441$	−10.9076	−	17.9450i	−0.519409	−	0.854526i
$442$		0			0
$443$	−1.27192	−	4.74685i	−0.0604305	−	0.225530i	0.929106	−	0.369814i	$-0.120579\pi$
$443$	−1.27192	−	4.74685i	−0.0604305	−	0.225530i	−0.989536	+	0.144284i	$0.953912\pi$
$444$		0			0
$445$	2.22172	+	37.0144i	0.105320	+	1.75465i
$446$		0			0
$447$	−15.9749	−	4.81662i	−0.755589	−	0.227819i
$448$		0			0
$449$	8.89975			0.420005			0.210003	−	0.977701i	$-0.432653\pi$
$449$	8.89975			0.420005			0.210003	+	0.977701i	$0.432653\pi$
$450$		0			0
$451$	−11.1583	−	19.3268i	−0.525424	−	0.910062i
$452$		0			0
$453$	−4.24941	+	3.99344i	−0.199655	+	0.187628i
$454$		0			0
$455$	−9.80240	−	13.5615i	−0.459543	−	0.635771i
$456$		0			0
$457$	10.0961	+	37.6792i	0.472277	+	1.76256i	0.631559	+	0.775328i	$0.282415\pi$
$457$	10.0961	+	37.6792i	0.472277	+	1.76256i	−0.159282	+	0.987233i	$0.550918\pi$
$458$		0			0
$459$	15.4706	−	7.10391i	0.722105	−	0.331582i
$460$		0			0
$461$			5.36675i			0.249954i	0.992160	+	0.124977i	$0.0398858\pi$
$461$			5.36675i			0.249954i	−0.992160	+	0.124977i	$0.960114\pi$
$462$		0			0
$463$	9.15831	+	9.15831i	0.425623	+	0.425623i	0.887134	−	0.461511i	$-0.152693\pi$
$463$	9.15831	+	9.15831i	0.425623	+	0.425623i	−0.461511	+	0.887134i	$0.652693\pi$
$464$		0			0
$465$	5.36312	−	22.6105i	0.248709	−	1.04854i
$466$		0			0
$467$	−2.08327	+	0.558212i	−0.0964024	+	0.0258310i	−0.306698	−	0.951807i	$-0.599224\pi$
$467$	−2.08327	+	0.558212i	−0.0964024	+	0.0258310i	0.210295	+	0.977638i	$0.432557\pi$
$468$		0			0
$469$	14.3614	+	22.6082i	0.663148	+	1.04395i
$470$		0			0
$471$	0.404214	−	13.0167i	0.0186252	−	0.599780i
$472$		0			0
$473$	−5.82815	−	1.56165i	−0.267978	−	0.0718046i
$474$		0			0
$475$	1.26650	−	0.949874i	0.0581110	−	0.0435832i
$476$		0			0
$477$	30.4248	−	6.15831i	1.39306	−	0.281970i
$478$		0			0
$479$	−4.63325	−	8.02502i	−0.211699	−	0.366673i	0.740548	−	0.672004i	$-0.234566\pi$
$479$	−4.63325	−	8.02502i	−0.211699	−	0.366673i	−0.952246	+	0.305331i	$0.901233\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$		0			0
$483$	7.27239	+	1.86097i	0.330905	+	0.0846771i
$484$		0			0
$485$	5.90650	−	28.8636i	0.268201	−	1.31063i
$486$		0			0
$487$	−34.0137	−	9.11394i	−1.54131	−	0.412992i	−0.614619	−	0.788824i	$-0.710690\pi$
$487$	−34.0137	−	9.11394i	−1.54131	−	0.412992i	−0.926688	+	0.375832i	$0.877357\pi$
$488$		0			0
$489$	1.15831	+	2.15831i	0.0523807	+	0.0976023i
$490$		0			0
$491$			3.89975i			0.175993i	0.996121	+	0.0879966i	$0.0280465\pi$
$491$			3.89975i			0.175993i	−0.996121	+	0.0879966i	$0.971954\pi$
$492$		0			0
$493$	6.20408	−	23.1540i	0.279418	−	1.04280i
$494$		0			0
$495$	−0.0331881	+	15.5404i	−0.00149169	+	0.698487i
$496$		0			0
$497$	4.14675	−	4.51350i	0.186007	−	0.202458i
$498$		0			0
$499$	−13.0338	−	7.52506i	−0.583473	−	0.336868i	0.179040	−	0.983842i	$-0.442701\pi$
$499$	−13.0338	−	7.52506i	−0.583473	−	0.336868i	−0.762512	+	0.646974i	$0.776034\pi$
$500$		0			0
$501$	16.9095	−	27.2950i	0.755462	−	1.21945i
$502$		0			0
$503$	−4.20844	+	4.20844i	−0.187645	+	0.187645i	−0.794677	−	0.607032i	$-0.792360\pi$
$503$	−4.20844	+	4.20844i	−0.187645	+	0.187645i	0.607032	+	0.794677i	$0.292360\pi$
$504$		0			0
$505$	2.68338	−	5.36675i	0.119409	−	0.238817i
$506$		0			0
$507$	8.43070	−	1.98072i	0.374421	−	0.0879668i
$508$		0			0
$509$	−20.6082	+	35.6944i	−0.913442	+	1.58213i	−0.104275	+	0.994548i	$0.533252\pi$
$509$	−20.6082	+	35.6944i	−0.913442	+	1.58213i	−0.809167	+	0.587579i	$0.800081\pi$
$510$		0			0
$511$	20.9499	−	13.3080i	0.926768	−	0.588711i
$512$		0			0
$513$	−0.276186	−	1.62188i	−0.0121939	−	0.0716079i
$514$		0			0
$515$	−21.2077	+	23.9162i	−0.934525	+	1.05387i
$516$		0			0
$517$	−11.5831	+	11.5831i	−0.509425	+	0.509425i
$518$		0			0
$519$	15.0000	−	8.05013i	0.658427	−	0.353361i
$520$		0			0
$521$	−16.6853	+	9.63325i	−0.730995	+	0.422040i	−0.818786	−	0.574099i	$-0.805353\pi$
$521$	−16.6853	+	9.63325i	−0.730995	+	0.422040i	0.0877909	+	0.996139i	$0.472019\pi$
$522$		0			0
$523$	−10.4272	+	2.79396i	−0.455950	+	0.122171i	−0.479482	−	0.877552i	$-0.659175\pi$
$523$	−10.4272	+	2.79396i	−0.455950	+	0.122171i	0.0235320	+	0.999723i	$0.492509\pi$
$524$		0			0
$525$	−13.9536	−	18.1741i	−0.608986	−	0.793181i
$526$		0			0
$527$	−18.9874	+	5.08766i	−0.827105	+	0.221622i
$528$		0			0
$529$	17.5947	−	10.1583i	0.764988	−	0.441666i
$530$		0			0
$531$	−30.7916	−	20.4248i	−1.33624	−	0.886361i
$532$		0			0
$533$	19.2665	−	19.2665i	0.834525	−	0.834525i
$534$		0			0
$535$	−8.06015	+	9.08953i	−0.348471	+	0.392975i
$536$		0			0
$537$	−24.3176	+	22.8528i	−1.04938	+	0.986170i
$538$		0			0
$539$	−9.26650	−	13.3080i	−0.399136	−	0.573216i
$540$		0			0
$541$	−9.13325	+	15.8193i	−0.392669	+	0.680123i	−0.992801	−	0.119779i	$-0.961782\pi$
$541$	−9.13325	+	15.8193i	−0.392669	+	0.680123i	0.600132	+	0.799901i	$0.295115\pi$
$542$		0			0
$543$	−4.06701	−	17.3108i	−0.174532	−	0.742876i
$544$		0			0
$545$	−18.5831	+	37.1662i	−0.796014	+	1.59203i
$546$		0			0
$547$	16.1082	−	16.1082i	0.688736	−	0.688736i	−0.273216	−	0.961953i	$-0.588087\pi$
$547$	16.1082	−	16.1082i	0.688736	−	0.688736i	0.961953	+	0.273216i	$0.0880875\pi$
$548$		0			0
$549$	1.73420	−	27.8960i	0.0740140	−	1.19057i
$550$		0			0
$551$	−2.00626	−	1.15831i	−0.0854694	−	0.0493458i
$552$		0			0
$553$	−4.50286	−	14.3468i	−0.191481	−	0.610086i
$554$		0			0
$555$	−10.4015	+	19.2824i	−0.441518	+	0.818491i
$556$		0			0
$557$	−2.67807	+	9.99470i	−0.113473	+	0.423489i	−0.999168	−	0.0407793i	$-0.987016\pi$
$557$	−2.67807	+	9.99470i	−0.113473	+	0.423489i	0.885695	+	0.464268i	$0.153683\pi$
$558$		0			0
$559$		−	7.36675i		−	0.311580i
$560$		0			0
$561$	11.5831	−	6.21637i	0.489040	−	0.262455i
$562$		0			0
$563$	−10.5750	−	2.83356i	−0.445683	−	0.119420i	0.0289957	−	0.999580i	$-0.490769\pi$
$563$	−10.5750	−	2.83356i	−0.445683	−	0.119420i	−0.474679	+	0.880159i	$0.657436\pi$
$564$		0			0
$565$	6.10724	−	29.8445i	0.256933	−	1.25557i
$566$		0			0
$567$	−23.4273	+	4.26182i	−0.983853	+	0.178980i
$568$		0			0
$569$	12.3166	−	21.3330i	0.516340	−	0.894327i	−0.483480	−	0.875355i	$-0.660628\pi$
$569$	12.3166	−	21.3330i	0.516340	−	0.894327i	0.999820	−	0.0189715i	$-0.00603916\pi$
$570$		0			0
$571$	−10.4749	−	18.1431i	−0.438362	−	0.759266i	0.559201	−	0.829032i	$-0.311108\pi$
$571$	−10.4749	−	18.1431i	−0.438362	−	0.759266i	−0.997563	+	0.0697661i	$0.977775\pi$
$572$		0			0
$573$	1.15831	−	3.84169i	0.0483892	−	0.160489i
$574$		0			0
$575$	8.10819	+	1.15831i	0.338135	+	0.0483050i
$576$		0			0
$577$	25.9545	+	6.95448i	1.08050	+	0.289519i	0.754800	−	0.655955i	$-0.227734\pi$
$577$	25.9545	+	6.95448i	1.08050	+	0.289519i	0.325699	+	0.945474i	$0.394400\pi$
$578$		0			0
$579$	−34.1536	−	1.06058i	−1.41938	−	0.0440764i
$580$		0			0
$581$	−14.3614	+	0.608187i	−0.595812	+	0.0252318i
$582$		0			0
$583$	23.1540	−	6.20408i	0.958939	−	0.256947i
$584$		0			0
$585$	−18.3376	+	4.87160i	−0.758166	+	0.201416i
$586$		0			0
$587$	−0.366750	−	0.366750i	−0.0151374	−	0.0151374i	0.699498	−	0.714635i	$-0.253407\pi$
$587$	−0.366750	−	0.366750i	−0.0151374	−	0.0151374i	−0.714635	+	0.699498i	$0.753407\pi$
$588$		0			0
$589$			1.89975i			0.0782778i
$590$		0			0
$591$	19.3500	−	31.2343i	0.795951	−	1.28481i
$592$		0			0
$593$	9.73010	+	36.3132i	0.399567	+	1.49121i	0.813860	+	0.581061i	$0.197362\pi$
$593$	9.73010	+	36.3132i	0.399567	+	1.49121i	−0.414292	+	0.910144i	$0.635971\pi$
$594$		0			0
$595$	−7.92675	+	17.6873i	−0.324965	+	0.725107i
$596$		0			0
$597$	−3.87453	−	4.12287i	−0.158574	−	0.168738i
$598$		0			0
$599$	14.2665	+	24.7103i	0.582913	+	1.00964i	0.995132	+	0.0985506i	$0.0314206\pi$
$599$	14.2665	+	24.7103i	0.582913	+	1.00964i	−0.412219	+	0.911085i	$0.635246\pi$
$600$		0			0
$601$	−4.53300			−0.184905			−0.0924524	−	0.995717i	$-0.529471\pi$
$601$	−4.53300			−0.184905			−0.0924524	+	0.995717i	$0.529471\pi$
$602$		0			0
$603$	29.7665	−	6.02506i	1.21219	−	0.245360i
$604$		0			0
$605$	−0.754712	−	12.5737i	−0.0306834	−	0.511194i
$606$		0			0
$607$	−2.87026	−	10.7119i	−0.116500	−	0.434784i	0.882895	−	0.469571i	$-0.155591\pi$
$607$	−2.87026	−	10.7119i	−0.116500	−	0.434784i	−0.999395	+	0.0347868i	$0.988925\pi$
$608$		0			0
$609$	−16.4359	+	29.2242i	−0.666016	+	1.18422i
$610$		0			0
$611$	−17.3205	−	10.0000i	−0.700713	−	0.404557i
$612$		0			0
$613$	−5.58793	+	20.8544i	−0.225694	+	0.842302i	0.756431	+	0.654074i	$0.226941\pi$
$613$	−5.58793	+	20.8544i	−0.225694	+	0.842302i	−0.982125	+	0.188229i	$0.939725\pi$
$614$		0			0
$615$	25.6082	−	27.1332i	1.03262	−	1.09412i
$616$		0			0
$617$	1.58312	+	1.58312i	0.0637342	+	0.0637342i	0.738255	−	0.674521i	$-0.235650\pi$
$617$	1.58312	+	1.58312i	0.0637342	+	0.0637342i	−0.674521	+	0.738255i	$0.735650\pi$
$618$		0			0
$619$	−29.6322	+	17.1082i	−1.19102	+	0.687636i	−0.958538	−	0.284964i	$-0.908018\pi$
$619$	−29.6322	+	17.1082i	−1.19102	+	0.687636i	−0.232483	+	0.972601i	$0.574685\pi$
$620$		0			0
$621$	4.92298	−	6.94373i	0.197552	−	0.278642i
$622$		0			0
$623$	−41.8614	+	13.1386i	−1.67714	+	0.526387i
$624$		0			0
$625$	−17.2846	−	18.0622i	−0.691384	−	0.722487i
$626$		0			0
$627$	−0.290572	−	1.23679i	−0.0116043	−	0.0493925i
$628$		0			0
$629$	18.5330			0.738959
$630$		0			0
$631$	4.31662			0.171842			0.0859211	−	0.996302i	$-0.472617\pi$
$631$	4.31662			0.171842			0.0859211	+	0.996302i	$0.472617\pi$
$632$		0			0
$633$	−1.96086	−	8.34618i	−0.0779372	−	0.331731i
$634$		0			0
$635$	−3.57134	+	17.4522i	−0.141724	+	0.692571i
$636$		0			0
$637$	12.7662	−	15.1335i	0.505817	−	0.599612i
$638$		0			0
$639$	−3.09480	−	6.22278i	−0.122428	−	0.246169i
$640$		0			0
$641$	−0.952846	+	0.550126i	−0.0376351	+	0.0217287i	−0.518700	−	0.854957i	$-0.673584\pi$
$641$	−0.952846	+	0.550126i	−0.0376351	+	0.0217287i	0.481064	+	0.876685i	$0.340250\pi$
$642$		0			0
$643$	16.8997	+	16.8997i	0.666461	+	0.666461i	0.956895	−	0.290434i	$-0.0937997\pi$
$643$	16.8997	+	16.8997i	0.666461	+	0.666461i	−0.290434	+	0.956895i	$0.593800\pi$
$644$		0			0
$645$	−0.291562	−	10.0831i	−0.0114802	−	0.397023i
$646$		0			0
$647$	−4.93217	+	18.4071i	−0.193904	+	0.723658i	0.798644	+	0.601803i	$0.205551\pi$
$647$	−4.93217	+	18.4071i	−0.193904	+	0.723658i	−0.992548	+	0.121855i	$0.961116\pi$
$648$		0			0
$649$	−24.7103	−	14.2665i	−0.969964	−	0.560009i
$650$		0			0
$651$	27.4937	−	0.310143i	1.07756	−	0.0121554i
$652$		0			0
$653$	12.4082	+	46.3079i	0.485569	+	1.81217i	0.577484	+	0.816402i	$0.304035\pi$
$653$	12.4082	+	46.3079i	0.485569	+	1.81217i	−0.0919148	+	0.995767i	$0.529299\pi$
$654$		0			0
$655$	22.3205	−	1.33975i	0.872134	−	0.0523482i
$656$		0			0
$657$	−5.58312	−	27.5831i	−0.217818	−	1.07612i
$658$		0			0
$659$	−3.05013			−0.118816			−0.0594080	−	0.998234i	$-0.518921\pi$
$659$	−3.05013			−0.118816			−0.0594080	+	0.998234i	$0.518921\pi$
$660$		0			0
$661$	−0.658312	−	1.14023i	−0.0256054	−	0.0443498i	0.852939	−	0.522011i	$-0.174818\pi$
$661$	−0.658312	−	1.14023i	−0.0256054	−	0.0443498i	−0.878544	+	0.477661i	$0.841485\pi$
$662$		0			0
$663$	10.9914	+	11.6959i	0.426869	+	0.454231i
$664$		0			0
$665$	1.45285	+	1.18238i	0.0563389	+	0.0458509i
$666$		0			0
$667$	−3.10204	−	11.5770i	−0.120112	−	0.448262i
$668$		0			0
$669$	−14.5984	+	23.5644i	−0.564407	+	0.911053i
$670$		0			0
$671$		−	21.5831i		−	0.833207i
$672$		0			0
$673$	7.26650	+	7.26650i	0.280103	+	0.280103i	0.833150	−	0.553047i	$-0.186535\pi$
$673$	7.26650	+	7.26650i	0.280103	+	0.280103i	−0.553047	+	0.833150i	$0.686535\pi$
$674$		0			0
$675$	−24.9065	+	7.39369i	−0.958651	+	0.284583i
$676$		0			0
$677$	−38.9768	+	10.4438i	−1.49800	+	0.401388i	−0.912430	−	0.409234i	$-0.865796\pi$
$677$	−38.9768	+	10.4438i	−1.49800	+	0.401388i	−0.585570	+	0.810622i	$0.699129\pi$
$678$		0			0
$679$	34.8284	−	1.47494i	1.33659	−	0.0566029i
$680$		0			0
$681$	−48.0683	−	1.49268i	−1.84198	−	0.0571997i
$682$		0			0
$683$	19.4091	+	5.20065i	0.742668	+	0.198997i	0.610264	−	0.792198i	$-0.291064\pi$
$683$	19.4091	+	5.20065i	0.742668	+	0.198997i	0.132405	+	0.991196i	$0.457730\pi$
$684$		0			0
$685$	6.94987	+	20.8496i	0.265541	+	0.796623i
$686$		0			0
$687$	−4.00000	+	13.2665i	−0.152610	+	0.506149i
$688$		0			0
$689$	14.6332	+	25.3455i	0.557482	+	0.965588i
$690$		0			0
$691$	−22.4248	+	38.8409i	−0.853080	+	1.47758i	0.0253348	+	0.999679i	$0.491935\pi$
$691$	−22.4248	+	38.8409i	−0.853080	+	1.47758i	−0.878415	+	0.477899i	$0.841399\pi$
$692$		0			0
$693$	−17.8517	+	4.40715i	−0.678129	+	0.167414i
$694$		0			0
$695$	7.00221	+	10.6053i	0.265609	+	0.402283i
$696$		0			0
$697$	−30.4851	−	8.16845i	−1.15470	−	0.309402i
$698$		0			0
$699$	26.5831	−	14.2665i	1.00547	−	0.539609i
$700$		0			0
$701$			19.6332i			0.741538i	0.928725	+	0.370769i	$0.120906\pi$
$701$			19.6332i			0.741538i	−0.928725	+	0.370769i	$0.879094\pi$
$702$		0			0
$703$	0.463571	−	1.73007i	0.0174839	−	0.0652508i
$704$		0			0
$705$	−24.1030	−	13.0018i	−0.907770	−	0.489677i
$706$		0			0
$707$	6.92924	+	1.54570i	0.260601	+	0.0581320i
$708$		0			0
$709$	−8.25582	−	4.76650i	−0.310054	−	0.179010i	0.336897	−	0.941542i	$-0.390623\pi$
$709$	−8.25582	−	4.76650i	−0.310054	−	0.179010i	−0.646951	+	0.762532i	$0.723956\pi$
$710$		0			0
$711$	−17.0173	−	1.05791i	−0.638198	−	0.0396746i
$712$		0			0
$713$	−6.94987	+	6.94987i	−0.260275	+	0.260275i
$714$		0			0
$715$	−13.8997	+	4.63325i	−0.519821	+	0.173274i
$716$		0			0
$717$	−11.3032	−	48.1106i	−0.422124	−	1.79673i
$718$		0			0
$719$	5.79156	−	10.0313i	0.215989	−	0.374104i	−0.737589	−	0.675250i	$-0.764036\pi$
$719$	5.79156	−	10.0313i	0.215989	−	0.374104i	0.953578	+	0.301146i	$0.0973692\pi$
$720$		0			0
$721$	−33.5251	−	17.5079i	−1.24854	−	0.652028i
$722$		0			0
$723$	−3.32361	+	3.12340i	−0.123606	+	0.116161i
$724$		0			0
$725$	−14.3706	+	33.6424i	−0.533710	+	1.24945i
$726$		0			0
$727$	−8.20844	+	8.20844i	−0.304434	+	0.304434i	−0.842746	−	0.538312i	$-0.819062\pi$
$727$	−8.20844	+	8.20844i	−0.304434	+	0.304434i	0.538312	+	0.842746i	$0.319062\pi$
$728$		0			0
$729$	−5.00000	+	26.5330i	−0.185185	+	0.982704i
$730$		0			0
$731$	−7.38979	+	4.26650i	−0.273321	+	0.157802i
$732$		0			0
$733$	−16.7563	+	4.48985i	−0.618910	+	0.165836i	−0.554632	−	0.832096i	$-0.687141\pi$
$733$	−16.7563	+	4.48985i	−0.618910	+	0.165836i	−0.0642777	+	0.997932i	$0.520474\pi$
$734$		0			0
$735$	16.8746	−	21.2190i	0.622430	−	0.782675i
$736$		0			0
$737$	22.6530	−	6.06984i	0.834433	−	0.223586i
$738$		0			0
$739$	16.4111	−	9.47494i	0.603691	−	0.348541i	−0.166801	−	0.985991i	$-0.553344\pi$
$739$	16.4111	−	9.47494i	0.603691	−	0.348541i	0.770492	+	0.637449i	$0.220010\pi$
$740$		0			0
$741$	1.36675	−	0.733501i	0.0502088	−	0.0269458i
$742$		0			0
$743$	6.15831	−	6.15831i	0.225927	−	0.225927i	−0.585062	−	0.810989i	$-0.698930\pi$
$743$	6.15831	−	6.15831i	0.225927	−	0.225927i	0.810989	+	0.585062i	$0.198930\pi$
$744$		0			0
$745$	−1.29061	−	21.5019i	−0.0472843	−	0.787769i
$746$		0			0
$747$	−5.18717	+	15.4515i	−0.189789	+	0.565339i
$748$		0			0
$749$	−12.7414	−	6.65400i	−0.465562	−	0.243132i
$750$		0			0
$751$	19.9499	−	34.5542i	0.727981	−	1.26090i	−0.229754	−	0.973249i	$-0.573792\pi$
$751$	19.9499	−	34.5542i	0.727981	−	1.26090i	0.957735	−	0.287652i	$-0.0928746\pi$
$752$		0			0
$753$	15.6246	−	3.67086i	0.569393	−	0.133774i
$754$		0			0
$755$	−6.73350	−	3.36675i	−0.245057	−	0.122529i
$756$		0			0
$757$	−14.3166	+	14.3166i	−0.520347	+	0.520347i	−0.917676	−	0.397329i	$-0.869937\pi$
$757$	−14.3166	+	14.3166i	−0.520347	+	0.520347i	0.397329	+	0.917676i	$0.369937\pi$
$758$		0			0
$759$	3.46155	−	5.58755i	0.125646	−	0.202815i
$760$		0			0
$761$	17.3205	+	10.0000i	0.627868	+	0.362500i	0.779926	−	0.625872i	$-0.215257\pi$
$761$	17.3205	+	10.0000i	0.627868	+	0.362500i	−0.152058	+	0.988372i	$0.548590\pi$
$762$		0			0
$763$	−47.9869	−	10.7044i	−1.73724	−	0.387525i
$764$		0			0
$765$	15.5072	+	15.5735i	0.560663	+	0.563063i
$766$		0			0
$767$	9.01640	−	33.6496i	0.325563	−	1.21502i
$768$		0			0
$769$		−	46.5330i		−	1.67802i	−0.544114	−	0.839011i	$-0.683134\pi$
$769$		−	46.5330i		−	1.67802i	0.544114	−	0.839011i	$-0.316866\pi$
$770$		0			0
$771$	−16.9499	−	31.5831i	−0.610435	−	1.13744i
$772$		0			0
$773$	−36.8142	−	9.86434i	−1.32412	−	0.354796i	−0.473597	−	0.880742i	$-0.657045\pi$
$773$	−36.8142	−	9.86434i	−1.32412	−	0.354796i	−0.850518	+	0.525946i	$0.823711\pi$
$774$		0			0
$775$	29.7846	−	3.58846i	1.06989	−	0.128901i
$776$		0			0
$777$	−25.1137	−	6.42649i	−0.900951	−	0.230549i
$778$		0			0
$779$	−1.52506	+	2.64149i	−0.0546410	+	0.0946411i
$780$		0			0
$781$	−2.68338	−	4.64774i	−0.0960187	−	0.166309i
$782$		0			0
$783$	24.2665	+	29.2665i	0.867214	+	1.04590i
$784$		0			0
$785$	15.9499	−	5.31662i	0.569275	−	0.189758i
$786$		0			0
$787$	21.1392	+	5.66422i	0.753530	+	0.201908i	0.615083	−	0.788462i	$-0.289122\pi$
$787$	21.1392	+	5.66422i	0.753530	+	0.201908i	0.138446	+	0.990370i	$0.455789\pi$
$788$		0			0
$789$	0.292077	−	9.40564i	0.0103982	−	0.334850i
$790$		0			0
$791$	36.0120	−	1.52506i	1.28044	−	0.0542250i
$792$		0			0
$793$	25.4535	−	6.82024i	0.903880	−	0.242194i
$794$		0			0
$795$	21.0322	+	34.1121i	0.745934	+	1.20983i
$796$		0			0
$797$	15.3668	+	15.3668i	0.544318	+	0.544318i	0.924792	−	0.380474i	$-0.124239\pi$
$797$	15.3668	+	15.3668i	0.544318	+	0.544318i	−0.380474	+	0.924792i	$0.624239\pi$
$798$		0			0
$799$			23.1662i			0.819563i
$800$		0			0
$801$	−3.08679	+	49.6535i	−0.109066	+	1.75442i
$802$		0			0
$803$	−5.62462	−	20.9914i	−0.198489	−	0.740769i
$804$		0			0
$805$	0.989430	+	9.64050i	0.0348728	+	0.339783i
$806$		0			0
$807$	5.38504	−	5.06067i	0.189562	−	0.178144i
$808$		0			0
$809$	0.608187	+	1.05341i	0.0213827	+	0.0370359i	0.876519	−	0.481368i	$-0.159860\pi$
$809$	0.608187	+	1.05341i	0.0213827	+	0.0370359i	−0.855136	+	0.518404i	$0.826526\pi$
$810$		0			0
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$	−46.7916	−	14.1082i	−1.64105	−	0.494796i
$814$		0			0
$815$	−2.09808	+	2.36603i	−0.0734924	+	0.0828783i
$816$		0			0
$817$	0.213438	+	0.796562i	0.00746726	+	0.0278682i
$818$		0			0
$819$	−10.8386	−	19.6603i	−0.378730	−	0.686985i
$820$		0			0
$821$	−8.02502	−	4.63325i	−0.280075	−	0.161702i	0.353382	−	0.935479i	$-0.385032\pi$
$821$	−8.02502	−	4.63325i	−0.280075	−	0.161702i	−0.633457	+	0.773777i	$0.718365\pi$
$822$		0			0
$823$	−10.0749	+	37.5999i	−0.351188	+	1.31065i	0.534026	+	0.845468i	$0.320678\pi$
$823$	−10.0749	+	37.5999i	−0.351188	+	1.31065i	−0.885214	+	0.465183i	$0.845988\pi$
$824$		0			0
$825$	−18.8417	+	6.89181i	−0.655983	+	0.239942i
$826$		0			0
$827$	6.52506	+	6.52506i	0.226899	+	0.226899i	0.811396	−	0.584497i	$-0.198708\pi$
$827$	6.52506	+	6.52506i	0.226899	+	0.226899i	−0.584497	+	0.811396i	$0.698708\pi$
$828$		0			0
$829$	25.8071	−	14.8997i	0.896318	−	0.517490i	0.0203145	−	0.999794i	$-0.493533\pi$
$829$	25.8071	−	14.8997i	0.896318	−	0.517490i	0.876004	+	0.482304i	$0.160200\pi$
$830$		0			0
$831$	−0.480242	+	15.4651i	−0.0166594	+	0.536477i
$832$		0			0
$833$	−22.5745	−	4.04150i	−0.782160	−	0.140030i
$834$		0			0
$835$	40.6102	+	8.31026i	1.40537	+	0.287588i
$836$		0			0
$837$	10.8347	−	29.2337i	0.374503	−	1.01046i
$838$		0			0
$839$	46.9499			1.62089			0.810445	−	0.585815i	$-0.199225\pi$
$839$	46.9499			1.62089			0.810445	+	0.585815i	$0.199225\pi$
$840$		0			0
$841$	24.5330			0.845965
$842$		0			0
$843$	41.5351	−	9.75830i	1.43055	−	0.336094i
$844$		0			0
$845$	6.16025	+	9.33013i	0.211919	+	0.320966i
$846$		0			0
$847$	14.2202	−	4.46315i	0.488613	−	0.153356i
$848$		0			0
$849$	43.1717	+	1.34062i	1.48165	+	0.0460101i
$850$		0			0
$851$	8.02502	−	4.63325i	0.275094	−	0.158826i
$852$		0			0
$853$	7.63325	+	7.63325i	0.261357	+	0.261357i	0.825605	−	0.564248i	$-0.190834\pi$
$853$	7.63325	+	7.63325i	0.261357	+	0.261357i	−0.564248	+	0.825605i	$0.690834\pi$
$854$		0			0
$855$	1.84169	−	1.05806i	0.0629844	−	0.0361849i
$856$		0			0
$857$	6.33832	−	23.6549i	0.216513	−	0.808038i	−0.769115	−	0.639110i	$-0.779303\pi$
$857$	6.33832	−	23.6549i	0.216513	−	0.808038i	0.985628	−	0.168928i	$-0.0540305\pi$
$858$		0			0
$859$	38.1051	+	22.0000i	1.30013	+	0.750630i	0.980426	−	0.196887i	$-0.0630833\pi$
$859$	38.1051	+	22.0000i	1.30013	+	0.750630i	0.319704	+	0.947518i	$0.396417\pi$
$860$		0			0
$861$	38.4773	+	21.6399i	1.31130	+	0.737486i
$862$		0			0
$863$	−13.8143	−	51.5558i	−0.470245	−	1.75498i	−0.638886	−	0.769301i	$-0.720605\pi$
$863$	−13.8143	−	51.5558i	−0.470245	−	1.75498i	0.168641	−	0.985677i	$-0.446062\pi$
$864$		0			0
$865$	16.4436	+	14.5814i	0.559098	+	0.495781i
$866$		0			0
$867$	−3.13325	+	10.3918i	−0.106411	+	0.352924i
$868$		0			0
$869$	−13.1662			−0.446634
$870$		0			0
$871$	14.3166	+	24.7971i	0.485100	+	0.840218i
$872$		0			0
$873$	12.5796	−	37.4719i	0.425756	−	1.26823i
$874$		0			0
$875$	15.7933	−	25.0114i	0.533913	−	0.845540i
$876$		0			0
$877$	2.21450	+	8.26463i	0.0747783	+	0.279077i	0.993183	−	0.116566i	$-0.0371886\pi$
$877$	2.21450	+	8.26463i	0.0747783	+	0.279077i	−0.918405	+	0.395642i	$0.870522\pi$
$878$		0			0
$879$	11.1751	+	6.92309i	0.376927	+	0.233510i
$880$		0			0
$881$			33.5330i			1.12976i	0.825175	+	0.564878i	$0.191077\pi$
$881$			33.5330i			1.12976i	−0.825175	+	0.564878i	$0.808923\pi$
$882$		0			0
$883$	−6.26650	−	6.26650i	−0.210884	−	0.210884i	0.593759	−	0.804643i	$-0.297643\pi$
$883$	−6.26650	−	6.26650i	−0.210884	−	0.210884i	−0.804643	+	0.593759i	$0.797643\pi$
$884$		0			0
$885$	11.0093	−	46.4143i	0.370072	−	1.56020i
$886$		0			0
$887$	−4.74685	+	1.27192i	−0.159384	+	0.0427067i	−0.337629	−	0.941279i	$-0.609625\pi$
$887$	−4.74685	+	1.27192i	−0.159384	+	0.0427067i	0.178245	+	0.983986i	$0.442958\pi$
$888$		0			0
$889$	−21.0588	+	0.891813i	−0.706290	+	0.0299105i
$890$		0			0
$891$	−2.58232	+	20.6891i	−0.0865111	+	0.693111i
$892$		0			0
$893$	2.16259	+	0.579464i	0.0723682	+	0.0193910i
$894$		0			0
$895$	−38.5330	−	19.2665i	−1.28802	−	0.644008i
$896$		0			0
$897$	7.68338	+	2.31662i	0.256540	+	0.0773499i
$898$		0			0
$899$	−21.9499	−	38.0183i	−0.732069	−	1.26798i
$900$		0			0
$901$	16.9499	−	29.3580i	0.564682	−	0.978058i
$902$		0			0
$903$	11.4932	−	3.21898i	0.382471	−	0.107121i
$904$		0			0
$905$	19.1575	−	12.6488i	0.636818	−	0.420462i
$906$		0			0
$907$	−18.7712	−	5.02971i	−0.623286	−	0.167009i	−0.0666648	−	0.997775i	$-0.521236\pi$
$907$	−18.7712	−	5.02971i	−0.623286	−	0.167009i	−0.556621	+	0.830766i	$0.687902\pi$
$908$		0			0
$909$	4.44987	−	6.70844i	0.147593	−	0.222505i
$910$		0			0
$911$			26.9499i			0.892889i	0.894811	+	0.446445i	$0.147310\pi$
$911$			26.9499i			0.892889i	−0.894811	+	0.446445i	$0.852690\pi$
$912$		0			0
$913$	−3.25753	+	12.1573i	−0.107809	+	0.402347i
$914$		0			0
$915$	34.5691	−	10.3425i	1.14282	−	0.341913i
$916$		0			0
$917$	7.92287	+	25.2434i	0.261636	+	0.833610i
$918$		0			0
$919$	−33.0964	−	19.1082i	−1.09175	−	0.630321i	−0.157707	−	0.987486i	$-0.550410\pi$
$919$	−33.0964	−	19.1082i	−1.09175	−	0.630321i	−0.934041	+	0.357165i	$0.883743\pi$
$920$		0			0
$921$	19.7295	+	12.2226i	0.650110	+	0.402750i
$922$		0			0
$923$	4.63325	−	4.63325i	0.152505	−	0.152505i
$924$		0			0
$925$	−28.0000	−	4.00000i	−0.920634	−	0.131519i
$926$		0			0
$927$	−32.1477	+	28.3846i	−1.05587	+	0.932272i
$928$		0			0
$929$	−0.550126	+	0.952846i	−0.0180490	+	0.0312618i	−0.874909	−	0.484288i	$-0.839079\pi$
$929$	−0.550126	+	0.952846i	−0.0180490	+	0.0312618i	0.856860	+	0.515549i	$0.172412\pi$
$930$		0			0
$931$	−0.941939	+	2.00626i	−0.0308708	+	0.0657524i
$932$		0			0
$933$	2.74784	+	2.92397i	0.0899603	+	0.0957265i
$934$		0			0
$935$	12.6979	+	11.2599i	0.415265	+	0.368237i
$936$		0			0
$937$	−14.0000	+	14.0000i	−0.457360	+	0.457360i	−0.897788	−	0.440428i	$-0.854827\pi$
$937$	−14.0000	+	14.0000i	−0.457360	+	0.457360i	0.440428	+	0.897788i	$0.354827\pi$
$938$		0			0
$939$	10.3668	+	19.3166i	0.338306	+	0.630374i
$940$		0			0
$941$	−16.6853	+	9.63325i	−0.543925	+	0.314035i	−0.746668	−	0.665197i	$-0.768348\pi$
$941$	−16.6853	+	9.63325i	−0.543925	+	0.314035i	0.202743	+	0.979232i	$0.435014\pi$
$942$		0			0
$943$	−15.2425	+	4.08423i	−0.496365	+	0.133001i
$944$		0			0
$945$	−15.6132	−	26.4807i	−0.507898	−	0.861417i
$946$		0			0
$947$	43.0640	−	11.5390i	1.39939	−	0.374966i	0.521266	−	0.853394i	$-0.325460\pi$
$947$	43.0640	−	11.5390i	1.39939	−	0.374966i	0.878127	+	0.478428i	$0.158793\pi$
$948$		0			0
$949$	22.9783	−	13.2665i	0.745906	−	0.430649i
$950$		0			0
$951$	8.05013	+	15.0000i	0.261043	+	0.486408i
$952$		0			0
$953$	20.0000	−	20.0000i	0.647864	−	0.647864i	−0.304613	−	0.952476i	$-0.598527\pi$
$953$	20.0000	−	20.0000i	0.647864	−	0.647864i	0.952476	+	0.304613i	$0.0985270\pi$
$954$		0			0
$955$	5.17082	−	0.310369i	0.167324	−	0.0100433i
$956$		0			0
$957$	20.1049	+	21.3936i	0.649900	+	0.691557i
$958$		0			0
$959$	−21.9499	+	13.9432i	−0.708798	+	0.450250i
$960$		0			0
$961$	−2.50000	+	4.33013i	−0.0806452	+	0.139682i
$962$		0			0
$963$	−12.2180	+	10.7878i	−0.393718	+	0.347631i
$964$		0			0
$965$	−13.9499	−	41.8496i	−0.449062	−	1.34719i
$966$		0			0
$967$	−27.1082	+	27.1082i	−0.871741	+	0.871741i	−0.992662	−	0.120922i	$-0.961415\pi$
$967$	−27.1082	+	27.1082i	−0.871741	+	0.871741i	0.120922	+	0.992662i	$0.461415\pi$
$968$		0			0
$969$	−1.52736	−	0.946214i	−0.0490658	−	0.0303968i
$970$		0			0
$971$	−12.0375	−	6.94987i	−0.386303	−	0.223032i	0.294254	−	0.955727i	$-0.404929\pi$
$971$	−12.0375	−	6.94987i	−0.386303	−	0.223032i	−0.680557	+	0.732695i	$0.738262\pi$
$972$		0			0
$973$	−10.1732	+	11.0730i	−0.326139	+	0.354983i
$974$		0			0
$975$	−14.0816	−	20.0426i	−0.450973	−	0.641878i
$976$		0			0
$977$	1.83013	−	6.83013i	0.0585510	−	0.218515i	−0.930451	−	0.366416i	$-0.880585\pi$
$977$	1.83013	−	6.83013i	0.0585510	−	0.218515i	0.989002	+	0.147900i	$0.0472515\pi$
$978$		0			0
$979$			38.4169i			1.22781i
$980$		0			0
$981$	−30.8166	+	46.4578i	−0.983899	+	1.48328i
$982$		0			0
$983$	14.7415	+	3.94999i	0.470182	+	0.125985i	0.486128	−	0.873888i	$-0.338409\pi$
$983$	14.7415	+	3.94999i	0.470182	+	0.125985i	−0.0159452	+	0.999873i	$0.505076\pi$
$984$		0			0
$985$	46.4711	+	9.50962i	1.48069	+	0.303002i
$986$		0			0
$987$	8.03311	−	31.3922i	0.255697	−	0.999224i
$988$		0			0
$989$	−2.13325	+	3.69490i	−0.0678334	+	0.117491i
$990$		0			0
$991$	27.2665	+	47.2270i	0.866149	+	1.50021i	0.865902	+	0.500214i	$0.166745\pi$
$991$	27.2665	+	47.2270i	0.866149	+	1.50021i	0.000247028		1.00000i	$0.499921\pi$
$992$		0			0
$993$	−14.3166	−	4.31662i	−0.454324	−	0.136984i
$994$		0			0
$995$	3.26650	−	6.53300i	0.103555	−	0.207110i
$996$		0			0
$997$	5.96509	+	1.59834i	0.188916	+	0.0506200i	0.352037	−	0.935986i	$-0.385489\pi$
$997$	5.96509	+	1.59834i	0.188916	+	0.0506200i	−0.163120	+	0.986606i	$0.552156\pi$
$998$		0			0
$999$	−17.0005	+	23.9788i	−0.537873	+	0.758655i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bv.a.233.2	yes	8
3.2	odd	2		420.2.bv.b.233.1	yes	8
5.2	odd	4		420.2.bv.b.317.2	yes	8
7.4	even	3	inner	420.2.bv.a.53.2	✓	8
15.2	even	4	inner	420.2.bv.a.317.2	yes	8
21.11	odd	6		420.2.bv.b.53.2	yes	8
35.32	odd	12		420.2.bv.b.137.1	yes	8
105.32	even	12	inner	420.2.bv.a.137.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.a.53.2	✓	8	7.4	even	3	inner
420.2.bv.a.137.2	yes	8	105.32	even	12	inner
420.2.bv.a.233.2	yes	8	1.1	even	1	trivial
420.2.bv.a.317.2	yes	8	15.2	even	4	inner
420.2.bv.b.53.2	yes	8	21.11	odd	6
420.2.bv.b.137.1	yes	8	35.32	odd	12
420.2.bv.b.233.1	yes	8	3.2	odd	2
420.2.bv.b.317.2	yes	8	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.bv (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.396143 + 1.68614i) q^{3} +(-1.86603 + 1.23205i) q^{5} +(-1.94831 - 1.79000i) q^{7} +(-2.68614 + 1.33591i) q^{9} +O(q^{10})\)	\(q+(0.396143 + 1.68614i) q^{3} +(-1.86603 + 1.23205i) q^{5} +(-1.94831 - 1.79000i) q^{7} +(-2.68614 + 1.33591i) q^{9} +(-2.00626 + 1.15831i) q^{11} +(-2.00000 - 2.00000i) q^{13} +(-2.81662 - 2.65831i) q^{15} +(-0.847944 + 3.16457i) q^{17} +(0.274205 + 0.158312i) q^{19} +(2.24638 - 3.99422i) q^{21} +(0.423972 + 1.58228i) q^{23} +(1.96410 - 4.59808i) q^{25} +(-3.31662 - 4.00000i) q^{27} -7.31662 q^{29} +(3.00000 + 5.19615i) q^{31} +(-2.74784 - 2.92397i) q^{33} +(5.84096 + 0.939764i) q^{35} +(-1.46410 - 5.46410i) q^{37} +(2.57999 - 4.16457i) q^{39} +9.63325i q^{41} +(1.84169 + 1.84169i) q^{43} +(3.36650 - 5.80230i) q^{45} +(6.83013 - 1.83013i) q^{47} +(0.591820 + 6.97494i) q^{49} +(-5.67181 - 0.176129i) q^{51} +(-9.99470 - 2.67807i) q^{53} +(2.31662 - 4.63325i) q^{55} +(-0.158312 + 0.525063i) q^{57} +(6.15831 + 10.6665i) q^{59} +(-4.65831 + 8.06843i) q^{61} +(7.62470 + 2.20542i) q^{63} +(6.19615 + 1.26795i) q^{65} +(-9.77844 - 2.62012i) q^{67} +(-2.50000 + 1.34169i) q^{69} +2.31662i q^{71} +(-2.42794 + 9.06119i) q^{73} +(8.53107 + 1.49025i) q^{75} +(5.98218 + 1.33444i) q^{77} +(4.92195 + 2.84169i) q^{79} +(5.43070 - 7.17687i) q^{81} +(3.84169 - 3.84169i) q^{83} +(-2.31662 - 6.94987i) q^{85} +(-2.89843 - 12.3369i) q^{87} +(8.29156 - 14.3614i) q^{89} +(0.316625 + 7.47661i) q^{91} +(-7.57301 + 7.11684i) q^{93} +(-0.706723 + 0.0424197i) q^{95} +(-9.31662 + 9.31662i) q^{97} +(3.84169 - 5.79156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9	\( 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9} - 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{21} + 2 q^{23} - 12 q^{25} - 32 q^{29} + 24 q^{31} - 2 q^{33} + 16 q^{37} + 4 q^{39} + 28 q^{43} - 20 q^{45} + 20 q^{47} - 24 q^{51} - 16 q^{53} - 8 q^{55} + 12 q^{57} + 36 q^{59} - 24 q^{61} + 22 q^{63} + 8 q^{65} - 22 q^{67} - 20 q^{69} - 8 q^{75} + 16 q^{77} - 14 q^{81} + 44 q^{83} + 8 q^{85} - 22 q^{87} - 24 q^{91} + 12 q^{95} - 48 q^{97} + 44 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9 - 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^21 + 2 * q^23 - 12 * q^25 - 32 * q^29 + 24 * q^31 - 2 * q^33 + 16 * q^37 + 4 * q^39 + 28 * q^43 - 20 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 - 8 * q^55 + 12 * q^57 + 36 * q^59 - 24 * q^61 + 22 * q^63 + 8 * q^65 - 22 * q^67 - 20 * q^69 - 8 * q^75 + 16 * q^77 - 14 * q^81 + 44 * q^83 + 8 * q^85 - 22 * q^87 - 24 * q^91 + 12 * q^95 - 48 * q^97 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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