LMFDB - Embedded newform 420.2.bv.b.317.2

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			317.2
Root			$-1.26217 + 1.18614i$ of defining polynomial
Character	$\chi$	$=$	420.317
Dual form			420.2.bv.b.53.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.68614 - 0.396143i) q^{3} +(0.133975 - 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(2.68614 - 1.33591i) q^{9} +O(q^{10})$	\(q+(1.68614 - 0.396143i) q^{3} +(0.133975 - 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(2.68614 - 1.33591i) q^{9} +(-2.00626 + 1.15831i) q^{11} +(-2.00000 + 2.00000i) q^{13} +(-0.658312 - 3.81662i) q^{15} +(-3.16457 - 0.847944i) q^{17} +(-0.274205 - 0.158312i) q^{19} +(2.24638 - 3.99422i) q^{21} +(1.58228 - 0.423972i) q^{23} +(-4.96410 - 0.598076i) q^{25} +(4.00000 - 3.31662i) q^{27} +7.31662 q^{29} +(3.00000 + 5.19615i) q^{31} +(-2.92397 + 2.74784i) q^{33} +(-4.10891 - 4.25639i) q^{35} +(5.46410 - 1.46410i) q^{37} +(-2.57999 + 4.16457i) q^{39} +9.63325i q^{41} +(1.84169 - 1.84169i) q^{43} +(-2.62194 - 6.17458i) q^{45} +(1.83013 + 6.83013i) q^{47} +(-0.591820 - 6.97494i) q^{49} +(-5.67181 - 0.176129i) q^{51} +(-2.67807 + 9.99470i) q^{53} +(2.31662 + 4.63325i) q^{55} +(-0.525063 - 0.158312i) q^{57} +(-6.15831 - 10.6665i) q^{59} +(-4.65831 + 8.06843i) q^{61} +(2.20542 - 7.62470i) q^{63} +(4.19615 + 4.73205i) q^{65} +(2.62012 - 9.77844i) q^{67} +(2.50000 - 1.34169i) q^{69} +2.31662i q^{71} +(9.06119 + 2.42794i) q^{73} +(-8.60710 + 0.958056i) q^{75} +(-1.33444 + 5.98218i) q^{77} +(-4.92195 - 2.84169i) q^{79} +(5.43070 - 7.17687i) q^{81} +(-3.84169 - 3.84169i) q^{83} +(-2.31662 + 6.94987i) q^{85} +(12.3369 - 2.89843i) q^{87} +(-8.29156 + 14.3614i) q^{89} +(0.316625 + 7.47661i) q^{91} +(7.11684 + 7.57301i) q^{93} +(-0.390098 + 0.590830i) q^{95} +(-9.31662 - 9.31662i) q^{97} +(-3.84169 + 5.79156i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} + 8 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10}) $ 8 * q + 2 * q^3 + 8 * q^5 + 6 * q^7 + 10 * q^9	$ 8 q + 2 q^{3} + 8 q^{5} + 6 q^{7} + 10 q^{9} - 16 q^{13} + 8 q^{15} + 4 q^{17} + 14 q^{21} - 2 q^{23} - 12 q^{25} + 32 q^{27} + 32 q^{29} + 24 q^{31} - 22 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} - 44 q^{57} - 36 q^{59} - 24 q^{61} + 22 q^{63} - 8 q^{65} - 22 q^{67} + 20 q^{69} + 6 q^{75} - 16 q^{77} - 14 q^{81} - 44 q^{83} + 8 q^{85} + 8 q^{87} - 24 q^{91} - 12 q^{93} - 12 q^{95} - 48 q^{97} - 44 q^{99}+O(q^{100}) $ 8 * q + 2 * q^3 + 8 * q^5 + 6 * q^7 + 10 * q^9 - 16 * q^13 + 8 * q^15 + 4 * q^17 + 14 * q^21 - 2 * q^23 - 12 * q^25 + 32 * q^27 + 32 * q^29 + 24 * q^31 - 22 * 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 - 44 * q^57 - 36 * q^59 - 24 * q^61 + 22 * q^63 - 8 * q^65 - 22 * q^67 + 20 * q^69 + 6 * q^75 - 16 * q^77 - 14 * q^81 - 44 * q^83 + 8 * q^85 + 8 * q^87 - 24 * q^91 - 12 * q^93 - 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 $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

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

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
$3$	1.68614	−	0.396143i	0.973494	−	0.228714i
$3$	1.68614	−	0.396143i	0.973494	−	0.228714i
$4$		0			0
$5$	0.133975	−	2.23205i	0.0599153	−	0.998203i
$5$	0.133975	−	2.23205i	0.0599153	−	0.998203i
$6$		0			0
$7$	1.79000	−	1.94831i	0.676555	−	0.736392i
$7$	1.79000	−	1.94831i	0.676555	−	0.736392i
$8$		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	−0.927794	−	0.373094i	$-0.878297\pi$
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	0.373094	+	0.927794i	$0.378297\pi$
$14$		0			0
$15$	−0.658312	−	3.81662i	−0.169976	−	0.985448i
$16$		0			0
$17$	−3.16457	−	0.847944i	−0.767521	−	0.205657i	−0.146245	−	0.989248i	$-0.546719\pi$
$17$	−3.16457	−	0.847944i	−0.767521	−	0.205657i	−0.621276	+	0.783592i	$0.713385\pi$
$18$		0			0
$19$	−0.274205	−	0.158312i	−0.0629070	−	0.0363194i	0.468217	−	0.883614i	$-0.344897\pi$
$19$	−0.274205	−	0.158312i	−0.0629070	−	0.0363194i	−0.531124	+	0.847294i	$0.678230\pi$
$20$		0			0
$21$	2.24638	−	3.99422i	0.490200	−	0.871610i
$22$		0			0
$23$	1.58228	−	0.423972i	0.329929	−	0.0884042i	−0.0900521	−	0.995937i	$-0.528703\pi$
$23$	1.58228	−	0.423972i	0.329929	−	0.0884042i	0.419981	+	0.907533i	$0.362037\pi$
$24$		0			0
$25$	−4.96410	−	0.598076i	−0.992820	−	0.119615i
$26$		0			0
$27$	4.00000	−	3.31662i	0.769800	−	0.638285i
$28$		0			0
$29$	7.31662			1.35866			0.679332	−	0.733831i	$-0.262270\pi$
$29$	7.31662			1.35866			0.679332	+	0.733831i	$0.262270\pi$
$30$		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
$33$	−2.92397	+	2.74784i	−0.508998	+	0.478338i
$34$		0			0
$35$	−4.10891	−	4.25639i	−0.694533	−	0.719461i
$36$		0			0
$37$	5.46410	−	1.46410i	0.898293	−	0.240697i	0.220010	−	0.975498i	$-0.429391\pi$
$37$	5.46410	−	1.46410i	0.898293	−	0.240697i	0.678283	+	0.734801i	$0.262724\pi$
$38$		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	−0.552595	−	0.833450i	$-0.686362\pi$
$43$	1.84169	−	1.84169i	0.280855	−	0.280855i	0.833450	+	0.552595i	$0.186362\pi$
$44$		0			0
$45$	−2.62194	−	6.17458i	−0.390855	−	0.920452i
$46$		0			0
$47$	1.83013	+	6.83013i	0.266951	+	0.996276i	0.961045	+	0.276392i	$0.0891387\pi$
$47$	1.83013	+	6.83013i	0.266951	+	0.996276i	−0.694094	+	0.719885i	$0.744195\pi$
$48$		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
$53$	−2.67807	+	9.99470i	−0.367861	+	1.37288i	0.495639	+	0.868529i	$0.334934\pi$
$53$	−2.67807	+	9.99470i	−0.367861	+	1.37288i	−0.863500	+	0.504348i	$0.831733\pi$
$54$		0			0
$55$	2.31662	+	4.63325i	0.312374	+	0.624747i
$56$		0			0
$57$	−0.525063	−	0.158312i	−0.0695463	−	0.0209690i
$58$		0			0
$59$	−6.15831	−	10.6665i	−0.801744	−	1.38866i	−0.918467	−	0.395497i	$-0.870572\pi$
$59$	−6.15831	−	10.6665i	−0.801744	−	1.38866i	0.116723	−	0.993164i	$-0.462761\pi$
$60$		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
$63$	2.20542	−	7.62470i	0.277857	−	0.960622i
$64$		0			0
$65$	4.19615	+	4.73205i	0.520469	+	0.586939i
$66$		0			0
$67$	2.62012	−	9.77844i	0.320099	−	1.19463i	−0.599048	−	0.800713i	$-0.704454\pi$
$67$	2.62012	−	9.77844i	0.320099	−	1.19463i	0.919148	−	0.393913i	$-0.128879\pi$
$68$		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
$73$	9.06119	+	2.42794i	1.06053	+	0.284169i	0.746597	−	0.665276i	$-0.231686\pi$
$73$	9.06119	+	2.42794i	1.06053	+	0.284169i	0.313934	+	0.949445i	$0.398353\pi$
$74$		0			0
$75$	−8.60710	+	0.958056i	−0.993862	+	0.110627i
$76$		0			0
$77$	−1.33444	+	5.98218i	−0.152074	+	0.681733i
$78$		0			0
$79$	−4.92195	−	2.84169i	−0.553762	−	0.319715i	0.196876	−	0.980428i	$-0.436920\pi$
$79$	−4.92195	−	2.84169i	−0.553762	−	0.319715i	−0.750638	+	0.660714i	$0.770254\pi$
$80$		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	$-0.346377\pi$
$83$	−3.84169	−	3.84169i	−0.421680	−	0.421680i	−0.885782	+	0.464102i	$0.846377\pi$
$84$		0			0
$85$	−2.31662	+	6.94987i	−0.251273	+	0.753820i
$86$		0			0
$87$	12.3369	−	2.89843i	1.32265	−	0.310745i
$88$		0			0
$89$	−8.29156	+	14.3614i	−0.878904	+	1.52231i	−0.0263586	+	0.999653i	$0.508391\pi$
$89$	−8.29156	+	14.3614i	−0.878904	+	1.52231i	−0.852545	+	0.522654i	$0.824942\pi$
$90$		0			0
$91$	0.316625	+	7.47661i	0.0331913	+	0.783762i
$92$		0			0
$93$	7.11684	+	7.57301i	0.737982	+	0.785285i
$94$		0			0
$95$	−0.390098	+	0.590830i	−0.0400232	+	0.0606179i
$96$		0			0
$97$	−9.31662	−	9.31662i	−0.945960	−	0.945960i	0.0526529	−	0.998613i	$-0.483232\pi$
$97$	−9.31662	−	9.31662i	−0.945960	−	0.945960i	−0.998613	+	0.0526529i	$0.983232\pi$
$98$		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
$103$	−3.69985	−	13.8080i	−0.364557	−	1.36055i	−0.868020	−	0.496530i	$-0.834607\pi$
$103$	−3.69985	−	13.8080i	−0.364557	−	1.36055i	0.503462	−	0.864017i	$-0.332059\pi$
$104$		0			0
$105$	−8.61434	−	5.54915i	−0.840674	−	0.541542i
$106$		0			0
$107$	1.40616	+	5.24784i	0.135938	+	0.507328i	0.999992	+	0.00392920i	$0.00125070\pi$
$107$	1.40616	+	5.24784i	0.135938	+	0.507328i	−0.864054	+	0.503399i	$0.832083\pi$
$108$		0			0
$109$	−16.0935	+	9.29156i	−1.54147	+	0.889970i	−0.542728	+	0.839909i	$0.682608\pi$
$109$	−16.0935	+	9.29156i	−1.54147	+	0.889970i	−0.998746	+	0.0500614i	$0.984058\pi$
$110$		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	$-0.0286052\pi$
$113$	9.63325	+	9.63325i	0.906220	+	0.906220i	−0.0897449	+	0.995965i	$0.528605\pi$
$114$		0			0
$115$	−0.734341	−	3.58854i	−0.0684776	−	0.334633i
$116$		0			0
$117$	−2.70047	+	8.04410i	−0.249658	+	0.743677i
$118$		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
$123$	3.81615	+	16.2430i	0.344091	+	1.46458i
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$	5.63325	+	5.63325i	0.499870	+	0.499870i	0.911397	−	0.411527i	$-0.135005\pi$
$127$	5.63325	+	5.63325i	0.499870	+	0.499870i	−0.411527	+	0.911397i	$0.635005\pi$
$128$		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
$133$	−0.799268	+	0.250858i	−0.0693053	+	0.0217521i
$134$		0			0
$135$	−6.86698	−	9.37255i	−0.591015	−	0.806660i
$136$		0			0
$137$	9.49370	+	2.54383i	0.811102	+	0.217334i	0.640453	−	0.767998i	$-0.278747\pi$
$137$	9.49370	+	2.54383i	0.811102	+	0.217334i	0.170649	+	0.985332i	$0.445413\pi$
$138$		0			0
$139$			5.68338i			0.482058i	0.970518	+	0.241029i	$0.0774848\pi$
$139$			5.68338i			0.482058i	−0.970518	+	0.241029i	$0.922515\pi$
$140$		0			0
$141$	5.79156	+	10.7916i	0.487738	+	0.908813i
$142$		0			0
$143$	1.69589	−	6.32914i	0.141817	−	0.529269i
$144$		0			0
$145$	0.980242	−	16.3311i	0.0814047	−	1.35622i
$146$		0			0
$147$	−3.76097	−	11.5263i	−0.310199	−	0.950672i
$148$		0			0
$149$	4.81662	−	8.34264i	0.394593	−	0.683456i	−0.598456	−	0.801156i	$-0.704219\pi$
$149$	4.81662	−	8.34264i	0.394593	−	0.683456i	0.993049	+	0.117700i	$0.0375522\pi$
$150$		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
$153$	−9.63325	+	1.94987i	−0.778802	+	0.157638i
$154$		0			0
$155$	12.0000	−	6.00000i	0.963863	−	0.481932i
$156$		0			0
$157$	1.94602	−	7.26264i	0.155309	−	0.579622i	−0.843769	−	0.536706i	$-0.819668\pi$
$157$	1.94602	−	7.26264i	0.155309	−	0.579622i	0.999079	−	0.0429162i	$-0.0136649\pi$
$158$		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
$163$	−0.366025	−	1.36603i	−0.0286693	−	0.106995i	0.950109	−	0.311919i	$-0.100972\pi$
$163$	−0.366025	−	1.36603i	−0.0286693	−	0.106995i	−0.978778	+	0.204924i	$0.934305\pi$
$164$		0			0
$165$	5.74159	+	6.89459i	0.446982	+	0.536743i
$166$		0			0
$167$	13.1082	−	13.1082i	1.01434	−	1.01434i	0.0144463	−	0.999896i	$-0.495401\pi$
$167$	13.1082	−	13.1082i	1.01434	−	1.01434i	0.999896	−	0.0144463i	$-0.00459856\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	−0.948044	−	0.0589367i	−0.0724988	−	0.00450701i
$172$		0			0
$173$	−9.49370	+	2.54383i	−0.721793	+	0.193404i	−0.600972	−	0.799270i	$-0.705220\pi$
$173$	−9.49370	+	2.54383i	−0.721793	+	0.193404i	−0.120821	+	0.992674i	$0.538553\pi$
$174$		0			0
$175$	−10.0510	+	8.60105i	−0.759782	+	0.650178i
$176$		0			0
$177$	−14.6092	−	15.5457i	−1.09810	−	1.16848i
$178$		0			0
$179$	−9.63325	−	16.6853i	−0.720023	−	1.24712i	−0.960990	−	0.276583i	$-0.910798\pi$
$179$	−9.63325	−	16.6853i	−0.720023	−	1.24712i	0.240967	−	0.970533i	$-0.422535\pi$
$180$		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
$183$	−4.65831	+	15.4499i	−0.344352	+	1.14209i
$184$		0			0
$185$	−2.53590	−	12.3923i	−0.186443	−	0.911100i
$186$		0			0
$187$	7.33112	−	1.96437i	0.536104	−	0.143649i
$188$		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
$193$	19.0559	+	5.10601i	1.37167	+	0.367539i	0.868089	−	0.496408i	$-0.165348\pi$
$193$	19.0559	+	5.10601i	1.37167	+	0.367539i	0.503583	+	0.863947i	$0.332015\pi$
$194$		0			0
$195$	8.94987	+	6.31662i	0.640914	+	0.452343i
$196$		0			0
$197$	15.0000	−	15.0000i	1.06871	−	1.06871i	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	15.0000	−	15.0000i	1.06871	−	1.06871i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	2.82887	−	1.63325i	0.200533	−	0.115778i	−0.396371	−	0.918090i	$-0.629731\pi$
$199$	2.82887	−	1.63325i	0.200533	−	0.115778i	0.596904	+	0.802312i	$0.296397\pi$
$200$		0			0
$201$	0.544234	−	17.5258i	0.0383873	−	1.23617i
$202$		0			0
$203$	13.0967	−	14.2551i	0.919211	−	1.00051i
$204$		0			0
$205$	21.5019	+	1.29061i	1.50176	+	0.0901402i
$206$		0			0
$207$	3.68385	−	3.25263i	0.256045	−	0.226074i
$208$		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
$213$	0.917716	+	3.90616i	0.0628809	+	0.267645i
$214$		0			0
$215$	−3.86400	−	4.35748i	−0.263523	−	0.297178i
$216$		0			0
$217$	15.4937	+	3.45617i	1.05178	+	0.234620i
$218$		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	−0.218108	−	0.975925i	$-0.569988\pi$
$223$	11.3166	−	11.3166i	0.757817	−	0.757817i	0.975925	+	0.218108i	$0.0699883\pi$
$224$		0			0
$225$	−14.1332	+	5.02506i	−0.942217	+	0.335004i
$226$		0			0
$227$	−26.8195	−	7.18627i	−1.78007	−	0.476969i	−0.789477	−	0.613780i	$-0.789648\pi$
$227$	−26.8195	−	7.18627i	−1.78007	−	0.476969i	−0.990597	+	0.136811i	$0.956315\pi$
$228$		0			0
$229$	−6.92820	−	4.00000i	−0.457829	−	0.264327i	0.253302	−	0.967387i	$-0.418483\pi$
$229$	−6.92820	−	4.00000i	−0.457829	−	0.264327i	−0.711131	+	0.703060i	$0.751817\pi$
$230$		0			0
$231$	0.119747	+	10.6154i	0.00787880	+	0.698444i
$232$		0			0
$233$	−16.8248	+	4.50820i	−1.10223	+	0.295342i	−0.763673	−	0.645604i	$-0.776606\pi$
$233$	−16.8248	+	4.50820i	−1.10223	+	0.295342i	−0.338558	+	0.940945i	$0.609939\pi$
$234$		0			0
$235$	15.4904	−	3.16987i	1.01048	−	0.206780i
$236$		0			0
$237$	−9.42481	−	2.84169i	−0.612207	−	0.184587i
$238$		0			0
$239$	28.5330			1.84565			0.922823	−	0.385224i	$-0.125876\pi$
$239$	28.5330			1.84565			0.922823	+	0.385224i	$0.125876\pi$
$240$		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
$243$	6.31386	−	14.2525i	0.405034	−	0.914302i
$244$		0			0
$245$	−15.6477	+	0.386509i	−0.999695	+	0.0246931i
$246$		0			0
$247$	0.865035	−	0.231785i	0.0550409	−	0.0147482i
$248$		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
$255$	−1.15301	+	12.6362i	−0.0722041	+	0.791308i
$256$		0			0
$257$	−5.35614	−	19.9894i	−0.334107	−	1.24690i	−0.904834	−	0.425764i	$-0.860005\pi$
$257$	−5.35614	−	19.9894i	−0.334107	−	1.24690i	0.570727	−	0.821140i	$-0.306661\pi$
$258$		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
$263$	−1.40616	+	5.24784i	−0.0867072	+	0.323596i	−0.995632	−	0.0933643i	$-0.970238\pi$
$263$	−1.40616	+	5.24784i	−0.0867072	+	0.323596i	0.908925	+	0.416960i	$0.136905\pi$
$264$		0			0
$265$	21.9499	+	7.31662i	1.34837	+	0.449457i
$266$		0			0
$267$	−8.29156	+	27.5000i	−0.507435	+	1.68297i
$268$		0			0
$269$	2.13325	+	3.69490i	0.130067	+	0.225282i	0.923702	−	0.383112i	$-0.125148\pi$
$269$	2.13325	+	3.69490i	0.130067	+	0.225282i	−0.793635	+	0.608394i	$0.791814\pi$
$270$		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
$273$	3.49569	+	12.4812i	0.211569	+	0.755396i
$274$		0			0
$275$	10.6520	−	4.55009i	0.642341	−	0.274381i
$276$		0			0
$277$	−2.31205	+	8.62867i	−0.138917	+	0.518447i	0.861034	+	0.508548i	$0.169818\pi$
$277$	−2.31205	+	8.62867i	−0.138917	+	0.518447i	−0.999951	+	0.00989859i	$0.996849\pi$
$278$		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
$283$	−24.0875	−	6.45422i	−1.43185	−	0.383663i	−0.542179	−	0.840263i	$-0.682401\pi$
$283$	−24.0875	−	6.45422i	−1.43185	−	0.383663i	−0.889672	+	0.456600i	$0.849067\pi$
$284$		0			0
$285$	−0.423706	+	1.15076i	−0.0250982	+	0.0681650i
$286$		0			0
$287$	18.7686	+	17.2435i	1.10787	+	1.01785i
$288$		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	$-0.321160\pi$
$293$	−5.36675	−	5.36675i	−0.313529	−	0.313529i	−0.846275	+	0.532746i	$0.821160\pi$
$294$		0			0
$295$	−24.6332	+	12.3166i	−1.43420	+	0.717102i
$296$		0			0
$297$	−4.18334	+	11.2872i	−0.242742	+	0.654953i
$298$		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
$303$	−3.38687	+	3.18286i	−0.194571	+	0.182851i
$304$		0			0
$305$	17.3851	+	11.4786i	0.995466	+	0.657260i
$306$		0			0
$307$	9.47494	+	9.47494i	0.540763	+	0.540763i	0.923753	−	0.382989i	$-0.125105\pi$
$307$	9.47494	+	9.47494i	0.540763	+	0.540763i	−0.382989	+	0.923753i	$0.625105\pi$
$308$		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
$313$	−3.27588	−	12.2258i	−0.185164	−	0.691041i	−0.994595	−	0.103826i	$-0.966891\pi$
$313$	−3.27588	−	12.2258i	−0.185164	−	0.691041i	0.809432	−	0.587214i	$-0.199775\pi$
$314$		0			0
$315$	−16.7233	−	5.94413i	−0.942249	−	0.334914i
$316$		0			0
$317$	2.54383	+	9.49370i	0.142876	+	0.533220i	0.999841	+	0.0178450i	$0.00568054\pi$
$317$	2.54383	+	9.49370i	0.142876	+	0.533220i	−0.856965	+	0.515375i	$0.827653\pi$
$318$		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
$325$	11.1244	−	8.73205i	0.617068	−	0.484367i
$326$		0			0
$327$	−23.4550	+	22.0422i	−1.29707	+	1.21894i
$328$		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
$333$	12.7214	−	11.2323i	0.697131	−	0.615527i
$334$		0			0
$335$	−21.4749	−	7.15831i	−1.17330	−	0.391100i
$336$		0			0
$337$	−4.36675	−	4.36675i	−0.237872	−	0.237872i	0.578096	−	0.815968i	$-0.303796\pi$
$337$	−4.36675	−	4.36675i	−0.237872	−	0.237872i	−0.815968	+	0.578096i	$0.803796\pi$
$338$		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
$343$	−14.6487	−	11.3321i	−0.790955	−	0.611874i
$344$		0			0
$345$	−2.65978	−	5.75988i	−0.143198	−	0.310101i
$346$		0			0
$347$	−15.7435	−	4.21847i	−0.845157	−	0.226459i	−0.189842	−	0.981815i	$-0.560797\pi$
$347$	−15.7435	−	4.21847i	−0.845157	−	0.226459i	−0.655315	+	0.755356i	$0.727464\pi$
$348$		0			0
$349$		−	11.0000i		−	0.588817i	−0.955680	−	0.294408i	$-0.904877\pi$
$349$		−	11.0000i		−	0.588817i	0.955680	−	0.294408i	$-0.0951225\pi$
$350$		0			0
$351$	−1.36675	+	14.6332i	−0.0729517	+	0.781065i
$352$		0			0
$353$	3.52601	−	13.1593i	0.187671	−	0.700397i	−0.806372	−	0.591408i	$-0.798572\pi$
$353$	3.52601	−	13.1593i	0.187671	−	0.700397i	0.994043	−	0.108989i	$-0.0347612\pi$
$354$		0			0
$355$	5.17082	+	0.310369i	0.274439	+	0.0164727i
$356$		0			0
$357$	−10.4957	+	10.7352i	−0.555491	+	0.568166i
$358$		0			0
$359$	−14.6332	+	25.3455i	−0.772313	+	1.33769i	0.163979	+	0.986464i	$0.447567\pi$
$359$	−14.6332	+	25.3455i	−0.772313	+	1.33769i	−0.936292	+	0.351222i	$0.885766\pi$
$360$		0			0
$361$	−9.44987	−	16.3677i	−0.497362	−	0.861456i
$362$		0			0
$363$	−2.81662	+	9.34169i	−0.147834	+	0.490311i
$364$		0			0
$365$	6.63325	−	19.8997i	0.347200	−	1.04160i
$366$		0			0
$367$	−7.10997	+	26.5348i	−0.371138	+	1.38510i	0.487769	+	0.872973i	$0.337811\pi$
$367$	−7.10997	+	26.5348i	−0.371138	+	1.38510i	−0.858907	+	0.512132i	$0.828856\pi$
$368$		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
$373$	−1.57999	−	5.89662i	−0.0818090	−	0.305315i	0.912882	−	0.408224i	$-0.133852\pi$
$373$	−1.57999	−	5.89662i	−0.0818090	−	0.305315i	−0.994691	+	0.102909i	$0.967185\pi$
$374$		0			0
$375$	0.985297	+	19.3398i	0.0508805	+	0.998705i
$376$		0			0
$377$	−14.6332	+	14.6332i	−0.753651	+	0.753651i
$378$		0			0
$379$		−	38.2164i		−	1.96304i	−0.191351	−	0.981522i	$-0.561287\pi$
$379$		−	38.2164i		−	1.96304i	0.191351	−	0.981522i	$-0.438713\pi$
$380$		0			0
$381$	11.7300	+	7.26688i	0.600947	+	0.372293i
$382$		0			0
$383$	31.0654	−	8.32394i	1.58737	−	0.425334i	0.646170	−	0.763193i	$-0.276370\pi$
$383$	31.0654	−	8.32394i	1.58737	−	0.425334i	0.941197	+	0.337860i	$0.109703\pi$
$384$		0			0
$385$	13.1738	+	3.78000i	0.671397	+	0.192647i
$386$		0			0
$387$	2.48671	−	7.40736i	0.126406	−	0.376537i
$388$		0			0
$389$	−2.31662	−	4.01251i	−0.117458	−	0.203442i	0.801302	−	0.598260i	$-0.204141\pi$
$389$	−2.31662	−	4.01251i	−0.117458	−	0.203442i	−0.918759	+	0.394818i	$0.870808\pi$
$390$		0			0
$391$	−5.36675			−0.271408
$392$		0			0
$393$	−16.5831	−	5.00000i	−0.836508	−	0.252217i
$394$		0			0
$395$	−7.00221	+	10.6053i	−0.352319	+	0.533612i
$396$		0			0
$397$	−8.69714	+	2.33039i	−0.436497	+	0.116959i	−0.470374	−	0.882467i	$-0.655881\pi$
$397$	−8.69714	+	2.33039i	−0.436497	+	0.116959i	0.0338773	+	0.999426i	$0.489214\pi$
$398$		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
$403$	−16.3923	−	4.39230i	−0.816559	−	0.218796i
$404$		0			0
$405$	−15.2916	−	13.0831i	−0.759844	−	0.650106i
$406$		0			0
$407$	−9.26650	+	9.26650i	−0.459323	+	0.459323i
$408$		0			0
$409$	5.23956	−	3.02506i	0.259080	−	0.149580i	−0.364835	−	0.931072i	$-0.618875\pi$
$409$	5.23956	−	3.02506i	0.259080	−	0.149580i	0.623915	+	0.781492i	$0.285541\pi$
$410$		0			0
$411$	17.0154	+	0.528387i	0.839310	+	0.0260634i
$412$		0			0
$413$	−31.8050	−	7.09472i	−1.56502	−	0.349109i
$414$		0			0
$415$	−9.08953	+	8.06015i	−0.446188	+	0.395657i
$416$		0			0
$417$	2.25143	+	9.58297i	0.110253	+	0.469280i
$418$		0			0
$419$	−0.733501			−0.0358339			−0.0179169	−	0.999839i	$-0.505703\pi$
$419$	−0.733501			−0.0358339			−0.0179169	+	0.999839i	$0.505703\pi$
$420$		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
$423$	14.0404	+	15.9018i	0.682667	+	0.773172i
$424$		0			0
$425$	15.2021	+	6.10193i	0.737410	+	0.295987i
$426$		0			0
$427$	7.38144	+	23.5183i	0.357213	+	1.13813i
$428$		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	−0.957805	−	0.287417i	$-0.907203\pi$
$433$	−13.9499	+	13.9499i	−0.670388	+	0.670388i	0.287417	+	0.957805i	$0.407203\pi$
$434$		0			0
$435$	−4.81662	−	27.9248i	−0.230939	−	1.33889i
$436$		0			0
$437$	−0.500990	−	0.134240i	−0.0239656	−	0.00642157i
$438$		0			0
$439$	33.9190	+	19.5831i	1.61886	+	0.934652i	0.987215	+	0.159397i	$0.0509551\pi$
$439$	33.9190	+	19.5831i	1.61886	+	0.934652i	0.631649	+	0.775254i	$0.282378\pi$
$440$		0			0
$441$	−10.9076	−	17.9450i	−0.519409	−	0.854526i
$442$		0			0
$443$	−4.74685	+	1.27192i	−0.225530	+	0.0604305i	−0.369814	−	0.929106i	$-0.620579\pi$
$443$	−4.74685	+	1.27192i	−0.225530	+	0.0604305i	0.144284	+	0.989536i	$0.453912\pi$
$444$		0			0
$445$	30.9445	+	20.4313i	1.46691	+	0.968534i
$446$		0			0
$447$	4.81662	−	15.9749i	0.227819	−	0.755589i
$448$		0			0
$449$	−8.89975			−0.420005			−0.210003	−	0.977701i	$-0.567347\pi$
$449$	−8.89975			−0.420005			−0.210003	+	0.977701i	$0.567347\pi$
$450$		0			0
$451$	−11.1583	−	19.3268i	−0.525424	−	0.910062i
$452$		0			0
$453$	3.99344	+	4.24941i	0.187628	+	0.199655i
$454$		0			0
$455$	16.7306	+	0.294954i	0.784343	+	0.0138276i
$456$		0			0
$457$	−37.6792	+	10.0961i	−1.76256	+	0.472277i	−0.987233	−	0.159282i	$-0.949082\pi$
$457$	−37.6792	+	10.0961i	−1.76256	+	0.472277i	−0.775328	+	0.631559i	$0.782415\pi$
$458$		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	−0.461511	−	0.887134i	$-0.652693\pi$
$463$	9.15831	−	9.15831i	0.425623	−	0.425623i	0.887134	+	0.461511i	$0.152693\pi$
$464$		0			0
$465$	17.8568	−	14.8706i	0.828091	−	0.689606i
$466$		0			0
$467$	−0.558212	−	2.08327i	−0.0258310	−	0.0964024i	0.951807	−	0.306698i	$-0.0992241\pi$
$467$	−0.558212	−	2.08327i	−0.0258310	−	0.0964024i	−0.977638	+	0.210295i	$0.932557\pi$
$468$		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
$473$	−1.56165	+	5.82815i	−0.0718046	+	0.267978i
$474$		0			0
$475$	1.26650	+	0.949874i	0.0581110	+	0.0435832i
$476$		0			0
$477$	6.15831	+	30.4248i	0.281970	+	1.39306i
$478$		0			0
$479$	4.63325	+	8.02502i	0.211699	+	0.366673i	0.952246	−	0.305331i	$-0.0987672\pi$
$479$	4.63325	+	8.02502i	0.211699	+	0.366673i	−0.740548	+	0.672004i	$0.765434\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$		0			0
$483$	1.86097	−	7.27239i	0.0846771	−	0.330905i
$484$		0			0
$485$	−22.0434	+	19.5470i	−1.00094	+	0.887583i
$486$		0			0
$487$	9.11394	−	34.0137i	0.412992	−	1.54131i	−0.375832	−	0.926688i	$-0.622643\pi$
$487$	9.11394	−	34.0137i	0.412992	−	1.54131i	0.788824	−	0.614619i	$-0.210690\pi$
$488$		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
$493$	−23.1540	−	6.20408i	−1.04280	−	0.279418i
$494$		0			0
$495$	12.4124	+	9.35076i	0.557895	+	0.420286i
$496$		0			0
$497$	4.51350	+	4.14675i	0.202458	+	0.186007i
$498$		0			0
$499$	13.0338	+	7.52506i	0.583473	+	0.336868i	0.762512	−	0.646974i	$-0.223966\pi$
$499$	13.0338	+	7.52506i	0.583473	+	0.336868i	−0.179040	+	0.983842i	$0.557299\pi$
$500$		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	$-0.207640\pi$
$503$	4.20844	+	4.20844i	0.187645	+	0.187645i	−0.607032	+	0.794677i	$0.707640\pi$
$504$		0			0
$505$	2.68338	+	5.36675i	0.119409	+	0.238817i
$506$		0			0
$507$	1.98072	+	8.43070i	0.0879668	+	0.374421i
$508$		0			0
$509$	20.6082	−	35.6944i	0.913442	−	1.58213i	0.104275	−	0.994548i	$-0.466748\pi$
$509$	20.6082	−	35.6944i	0.913442	−	1.58213i	0.809167	−	0.587579i	$-0.199919\pi$
$510$		0			0
$511$	20.9499	−	13.3080i	0.926768	−	0.588711i
$512$		0			0
$513$	−1.62188	+	0.276186i	−0.0716079	+	0.0121939i
$514$		0			0
$515$	−31.3159	+	6.40833i	−1.37994	+	0.282385i
$516$		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
$523$	2.79396	+	10.4272i	0.122171	+	0.455950i	0.999723	−	0.0235320i	$-0.00749117\pi$
$523$	2.79396	+	10.4272i	0.122171	+	0.455950i	−0.877552	+	0.479482i	$0.840825\pi$
$524$		0			0
$525$	−13.5401	+	18.4842i	−0.590938	+	0.806717i
$526$		0			0
$527$	−5.08766	−	18.9874i	−0.221622	−	0.827105i
$528$		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
$535$	11.9018	−	2.43553i	0.514561	−	0.105297i
$536$		0			0
$537$	−22.8528	−	24.3176i	−0.986170	−	1.04938i
$538$		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
$543$	−17.3108	+	4.06701i	−0.742876	+	0.174532i
$544$		0			0
$545$	18.5831	+	37.1662i	0.796014	+	1.59203i
$546$		0			0
$547$	16.1082	+	16.1082i	0.688736	+	0.688736i	0.961953	−	0.273216i	$-0.0880875\pi$
$547$	16.1082	+	16.1082i	0.688736	+	0.688736i	−0.273216	+	0.961953i	$0.588087\pi$
$548$		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
$553$	−14.3468	+	4.50286i	−0.610086	+	0.191481i
$554$		0			0
$555$	−9.18501	−	19.8906i	−0.389882	−	0.844309i
$556$		0			0
$557$	−9.99470	−	2.67807i	−0.423489	−	0.113473i	0.0407793	−	0.999168i	$-0.487016\pi$
$557$	−9.99470	−	2.67807i	−0.423489	−	0.113473i	−0.464268	+	0.885695i	$0.653683\pi$
$558$		0			0
$559$			7.36675i			0.311580i
$560$		0			0
$561$	11.5831	−	6.21637i	0.489040	−	0.262455i
$562$		0			0
$563$	−2.83356	+	10.5750i	−0.119420	+	0.445683i	−0.999580	−	0.0289957i	$-0.990769\pi$
$563$	−2.83356	+	10.5750i	−0.119420	+	0.445683i	0.880159	+	0.474679i	$0.157436\pi$
$564$		0			0
$565$	22.7925	−	20.2113i	0.958888	−	0.850295i
$566$		0			0
$567$	−4.26182	−	23.4273i	−0.178980	−	0.983853i
$568$		0			0
$569$	−12.3166	+	21.3330i	−0.516340	+	0.894327i	0.483480	+	0.875355i	$0.339372\pi$
$569$	−12.3166	+	21.3330i	−0.516340	+	0.894327i	−0.999820	+	0.0189715i	$0.993961\pi$
$570$		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
$573$	−3.84169	−	1.15831i	−0.160489	−	0.0483892i
$574$		0			0
$575$	−8.10819	+	1.15831i	−0.338135	+	0.0483050i
$576$		0			0
$577$	−6.95448	+	25.9545i	−0.289519	+	1.08050i	0.655955	+	0.754800i	$0.272266\pi$
$577$	−6.95448	+	25.9545i	−0.289519	+	1.08050i	−0.945474	+	0.325699i	$0.894400\pi$
$578$		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
$583$	−6.20408	−	23.1540i	−0.256947	−	0.958939i
$584$		0			0
$585$	17.5930	+	7.10528i	0.727383	+	0.293767i
$586$		0			0
$587$	0.366750	−	0.366750i	0.0151374	−	0.0151374i	−0.699498	−	0.714635i	$-0.746593\pi$
$587$	0.366750	−	0.366750i	0.0151374	−	0.0151374i	0.714635	+	0.699498i	$0.246593\pi$
$588$		0			0
$589$		−	1.89975i		−	0.0782778i
$590$		0			0
$591$	19.3500	−	31.2343i	0.795951	−	1.28481i
$592$		0			0
$593$	36.3132	−	9.73010i	1.49121	−	0.399567i	0.581061	−	0.813860i	$-0.302638\pi$
$593$	36.3132	−	9.73010i	1.49121	−	0.399567i	0.910144	+	0.414292i	$0.135971\pi$
$594$		0			0
$595$	9.39375	+	16.9538i	0.385106	+	0.695036i
$596$		0			0
$597$	4.12287	−	3.87453i	0.168738	−	0.158574i
$598$		0			0
$599$	−14.2665	−	24.7103i	−0.582913	−	1.00964i	−0.995132	−	0.0985506i	$-0.968579\pi$
$599$	−14.2665	−	24.7103i	−0.582913	−	1.00964i	0.412219	−	0.911085i	$-0.364754\pi$
$600$		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
$603$	−6.02506	−	29.7665i	−0.245360	−	1.21219i
$604$		0			0
$605$	10.5118	+	6.94045i	0.427365	+	0.282169i
$606$		0			0
$607$	10.7119	−	2.87026i	0.434784	−	0.116500i	−0.0347868	−	0.999395i	$-0.511075\pi$
$607$	10.7119	−	2.87026i	0.434784	−	0.116500i	0.469571	+	0.882895i	$0.344409\pi$
$608$		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
$613$	20.8544	+	5.58793i	0.842302	+	0.225694i	0.654074	−	0.756431i	$-0.273059\pi$
$613$	20.8544	+	5.58793i	0.842302	+	0.225694i	0.188229	+	0.982125i	$0.439725\pi$
$614$		0			0
$615$	36.7665	−	6.34169i	1.48257	−	0.255722i
$616$		0			0
$617$	−1.58312	+	1.58312i	−0.0637342	+	0.0637342i	−0.738255	−	0.674521i	$-0.764350\pi$
$617$	−1.58312	+	1.58312i	−0.0637342	+	0.0637342i	0.674521	+	0.738255i	$0.264350\pi$
$618$		0			0
$619$	29.6322	−	17.1082i	1.19102	−	0.687636i	0.232483	−	0.972601i	$-0.425315\pi$
$619$	29.6322	−	17.1082i	1.19102	−	0.687636i	0.958538	+	0.284964i	$0.0919818\pi$
$620$		0			0
$621$	4.92298	−	6.94373i	0.197552	−	0.278642i
$622$		0			0
$623$	13.1386	+	41.8614i	0.526387	+	1.67714i
$624$		0			0
$625$	24.2846	+	5.93782i	0.971384	+	0.237513i
$626$		0			0
$627$	1.23679	−	0.290572i	0.0493925	−	0.0116043i
$628$		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
$633$	−8.34618	+	1.96086i	−0.331731	+	0.0779372i
$634$		0			0
$635$	13.3284	−	11.8190i	0.528922	−	0.469022i
$636$		0			0
$637$	15.1335	+	12.7662i	0.599612	+	0.505817i
$638$		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	−0.290434	−	0.956895i	$-0.593800\pi$
$643$	16.8997	−	16.8997i	0.666461	−	0.666461i	0.956895	+	0.290434i	$0.0937997\pi$
$644$		0			0
$645$	−8.24144	−	5.81662i	−0.324506	−	0.229029i
$646$		0			0
$647$	−18.4071	−	4.93217i	−0.723658	−	0.193904i	−0.121855	−	0.992548i	$-0.538884\pi$
$647$	−18.4071	−	4.93217i	−0.723658	−	0.193904i	−0.601803	+	0.798644i	$0.705551\pi$
$648$		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
$653$	46.3079	−	12.4082i	1.81217	−	0.485569i	0.816402	−	0.577484i	$-0.195965\pi$
$653$	46.3079	−	12.4082i	1.81217	−	0.485569i	0.995767	+	0.0919148i	$0.0292988\pi$
$654$		0			0
$655$	−12.3205	+	18.6603i	−0.481402	+	0.729116i
$656$		0			0
$657$	27.5831	−	5.58312i	1.07612	−	0.217818i
$658$		0			0
$659$	3.05013			0.118816			0.0594080	−	0.998234i	$-0.481079\pi$
$659$	3.05013			0.118816			0.0594080	+	0.998234i	$0.481079\pi$
$660$		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
$663$	11.6959	−	10.9914i	0.454231	−	0.426869i
$664$		0			0
$665$	0.452846	+	1.81762i	0.0175606	+	0.0704841i
$666$		0			0
$667$	11.5770	−	3.10204i	0.448262	−	0.120112i
$668$		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	−0.553047	−	0.833150i	$-0.686535\pi$
$673$	7.26650	−	7.26650i	0.280103	−	0.280103i	0.833150	+	0.553047i	$0.186535\pi$
$674$		0			0
$675$	−21.8400	+	14.0718i	−0.840622	+	0.541622i
$676$		0			0
$677$	−10.4438	−	38.9768i	−0.401388	−	1.49800i	−0.810622	−	0.585570i	$-0.800871\pi$
$677$	−10.4438	−	38.9768i	−0.401388	−	1.49800i	0.409234	−	0.912430i	$-0.365796\pi$
$678$		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
$683$	5.20065	−	19.4091i	0.198997	−	0.742668i	−0.792198	−	0.610264i	$-0.791064\pi$
$683$	5.20065	−	19.4091i	0.198997	−	0.742668i	0.991196	−	0.132405i	$-0.0422698\pi$
$684$		0			0
$685$	6.94987	−	20.8496i	0.265541	−	0.796623i
$686$		0			0
$687$	−13.2665	−	4.00000i	−0.506149	−	0.152610i
$688$		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
$693$	4.40715	+	17.8517i	0.167414	+	0.678129i
$694$		0			0
$695$	12.6856	+	0.761428i	0.481192	+	0.0288826i
$696$		0			0
$697$	8.16845	−	30.4851i	0.309402	−	1.15470i
$698$		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
$703$	−1.73007	−	0.463571i	−0.0652508	−	0.0174839i
$704$		0			0
$705$	24.8632	−	11.4813i	0.936404	−	0.432409i
$706$		0			0
$707$	−1.54570	+	6.92924i	−0.0581320	+	0.260601i
$708$		0			0
$709$	8.25582	+	4.76650i	0.310054	+	0.179010i	0.646951	−	0.762532i	$-0.276044\pi$
$709$	8.25582	+	4.76650i	0.310054	+	0.179010i	−0.336897	+	0.941542i	$0.609377\pi$
$710$		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
$717$	48.1106	−	11.3032i	1.79673	−	0.422124i
$718$		0			0
$719$	−5.79156	+	10.0313i	−0.215989	+	0.374104i	−0.953578	−	0.301146i	$-0.902631\pi$
$719$	−5.79156	+	10.0313i	−0.215989	+	0.374104i	0.737589	+	0.675250i	$0.235964\pi$
$720$		0			0
$721$	−33.5251	−	17.5079i	−1.24854	−	0.652028i
$722$		0			0
$723$	3.12340	+	3.32361i	0.116161	+	0.123606i
$724$		0			0
$725$	−36.3205	−	4.37590i	−1.34891	−	0.162517i
$726$		0			0
$727$	−8.20844	−	8.20844i	−0.304434	−	0.304434i	0.538312	−	0.842746i	$-0.319062\pi$
$727$	−8.20844	−	8.20844i	−0.304434	−	0.304434i	−0.842746	+	0.538312i	$0.819062\pi$
$728$		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
$733$	4.48985	+	16.7563i	0.165836	+	0.618910i	0.997932	+	0.0642777i	$0.0204744\pi$
$733$	4.48985	+	16.7563i	0.165836	+	0.618910i	−0.832096	+	0.554632i	$0.812859\pi$
$734$		0			0
$735$	−26.2311	+	6.85044i	−0.967549	+	0.252682i
$736$		0			0
$737$	6.06984	+	22.6530i	0.223586	+	0.834433i
$738$		0			0
$739$	−16.4111	+	9.47494i	−0.603691	+	0.348541i	−0.770492	−	0.637449i	$-0.779990\pi$
$739$	−16.4111	+	9.47494i	−0.603691	+	0.348541i	0.166801	+	0.985991i	$0.446656\pi$
$740$		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	$-0.301070\pi$
$743$	−6.15831	−	6.15831i	−0.225927	−	0.225927i	−0.810989	+	0.585062i	$0.801070\pi$
$744$		0			0
$745$	−17.9759	−	11.8687i	−0.658586	−	0.434834i
$746$		0			0
$747$	−15.4515	−	5.18717i	−0.565339	−	0.189789i
$748$		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
$753$	−3.67086	−	15.6246i	−0.133774	−	0.569393i
$754$		0			0
$755$	6.73350	−	3.36675i	0.245057	−	0.122529i
$756$		0			0
$757$	−14.3166	−	14.3166i	−0.520347	−	0.520347i	0.397329	−	0.917676i	$-0.369937\pi$
$757$	−14.3166	−	14.3166i	−0.520347	−	0.520347i	−0.917676	+	0.397329i	$0.869937\pi$
$758$		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
$763$	−10.7044	+	47.9869i	−0.387525	+	1.73724i
$764$		0			0
$765$	3.06161	+	21.7631i	0.110693	+	0.786848i
$766$		0			0
$767$	33.6496	+	9.01640i	1.21502	+	0.325563i
$768$		0			0
$769$			46.5330i			1.67802i	0.544114	+	0.839011i	$0.316866\pi$
$769$			46.5330i			1.67802i	−0.544114	+	0.839011i	$0.683134\pi$
$770$		0			0
$771$	−16.9499	−	31.5831i	−0.610435	−	1.13744i
$772$		0			0
$773$	−9.86434	+	36.8142i	−0.354796	+	1.32412i	0.525946	+	0.850518i	$0.323711\pi$
$773$	−9.86434	+	36.8142i	−0.354796	+	1.32412i	−0.880742	+	0.473597i	$0.842955\pi$
$774$		0			0
$775$	−11.7846	−	27.5885i	−0.423316	−	0.991007i
$776$		0			0
$777$	6.42649	−	25.1137i	0.230549	−	0.900951i
$778$		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
$783$	29.2665	−	24.2665i	1.04590	−	0.867214i
$784$		0			0
$785$	−15.9499	−	5.31662i	−0.569275	−	0.189758i
$786$		0			0
$787$	−5.66422	+	21.1392i	−0.201908	+	0.753530i	0.788462	+	0.615083i	$0.210878\pi$
$787$	−5.66422	+	21.1392i	−0.201908	+	0.753530i	−0.990370	+	0.138446i	$0.955789\pi$
$788$		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
$793$	−6.82024	−	25.4535i	−0.242194	−	0.903880i
$794$		0			0
$795$	39.9090	+	3.64156i	1.41543	+	0.129153i
$796$		0			0
$797$	−15.3668	+	15.3668i	−0.544318	+	0.544318i	−0.924792	−	0.380474i	$-0.875761\pi$
$797$	−15.3668	+	15.3668i	−0.544318	+	0.544318i	0.380474	+	0.924792i	$0.375761\pi$
$798$		0			0
$799$		−	23.1662i		−	0.819563i
$800$		0			0
$801$	−3.08679	+	49.6535i	−0.109066	+	1.75442i
$802$		0			0
$803$	−20.9914	+	5.62462i	−0.740769	+	0.198489i
$804$		0			0
$805$	−8.30605	−	4.99275i	−0.292750	−	0.175971i
$806$		0			0
$807$	5.06067	+	5.38504i	0.178144	+	0.189562i
$808$		0			0
$809$	−0.608187	−	1.05341i	−0.0213827	−	0.0370359i	0.855136	−	0.518404i	$-0.173474\pi$
$809$	−0.608187	−	1.05341i	−0.0213827	−	0.0370359i	−0.876519	+	0.481368i	$0.840140\pi$
$810$		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
$813$	−14.1082	+	46.7916i	−0.494796	+	1.64105i
$814$		0			0
$815$	−3.09808	+	0.633975i	−0.108521	+	0.0222072i
$816$		0			0
$817$	−0.796562	+	0.213438i	−0.0278682	+	0.00746726i
$818$		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
$823$	37.5999	+	10.0749i	1.31065	+	0.351188i	0.845468	−	0.534026i	$-0.179322\pi$
$823$	37.5999	+	10.0749i	1.31065	+	0.351188i	0.465183	+	0.885214i	$0.345988\pi$
$824$		0			0
$825$	16.1583	−	11.8918i	0.562560	−	0.414020i
$826$		0			0
$827$	−6.52506	+	6.52506i	−0.226899	+	0.226899i	−0.811396	−	0.584497i	$-0.801292\pi$
$827$	−6.52506	+	6.52506i	−0.226899	+	0.226899i	0.584497	+	0.811396i	$0.301292\pi$
$828$		0			0
$829$	−25.8071	+	14.8997i	−0.896318	+	0.517490i	−0.876004	−	0.482304i	$-0.839800\pi$
$829$	−25.8071	+	14.8997i	−0.896318	+	0.517490i	−0.0203145	+	0.999794i	$0.506467\pi$
$830$		0			0
$831$	−0.480242	+	15.4651i	−0.0166594	+	0.536477i
$832$		0			0
$833$	−4.04150	+	22.5745i	−0.140030	+	0.782160i
$834$		0			0
$835$	−27.5020	−	31.0143i	−0.951745	−	1.07329i
$836$		0			0
$837$	29.2337	+	10.8347i	1.01046	+	0.374503i
$838$		0			0
$839$	−46.9499			−1.62089			−0.810445	−	0.585815i	$-0.800775\pi$
$839$	−46.9499			−1.62089			−0.810445	+	0.585815i	$0.800775\pi$
$840$		0			0
$841$	24.5330			0.845965
$842$		0			0
$843$	−9.75830	−	41.5351i	−0.336094	−	1.43055i
$844$		0			0
$845$	11.1603	+	0.669873i	0.383924	+	0.0230443i
$846$		0			0
$847$	4.46315	+	14.2202i	0.153356	+	0.488613i
$848$		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	−0.564248	−	0.825605i	$-0.690834\pi$
$853$	7.63325	−	7.63325i	0.261357	−	0.261357i	0.825605	+	0.564248i	$0.190834\pi$
$854$		0			0
$855$	−0.258564	+	2.10819i	−0.00884269	+	0.0720985i
$856$		0			0
$857$	23.6549	+	6.33832i	0.808038	+	0.216513i	0.639110	−	0.769115i	$-0.279303\pi$
$857$	23.6549	+	6.33832i	0.808038	+	0.216513i	0.168928	+	0.985628i	$0.445970\pi$
$858$		0			0
$859$	−38.1051	−	22.0000i	−1.30013	−	0.750630i	−0.319704	−	0.947518i	$-0.603583\pi$
$859$	−38.1051	−	22.0000i	−1.30013	−	0.750630i	−0.980426	+	0.196887i	$0.936917\pi$
$860$		0			0
$861$	38.4773	+	21.6399i	1.31130	+	0.737486i
$862$		0			0
$863$	−51.5558	+	13.8143i	−1.75498	+	0.470245i	−0.985677	−	0.168641i	$-0.946062\pi$
$863$	−51.5558	+	13.8143i	−1.75498	+	0.470245i	−0.769301	+	0.638886i	$0.779395\pi$
$864$		0			0
$865$	4.40604	+	21.5312i	0.149810	+	0.732084i
$866$		0			0
$867$	−10.3918	−	3.13325i	−0.352924	−	0.106411i
$868$		0			0
$869$	13.1662			0.446634
$870$		0			0
$871$	14.3166	+	24.7971i	0.485100	+	0.840218i
$872$		0			0
$873$	−37.4719	−	12.5796i	−1.26823	−	0.425756i
$874$		0			0
$875$	17.8514	+	23.5866i	0.603488	+	0.797372i
$876$		0			0
$877$	−8.26463	+	2.21450i	−0.279077	+	0.0747783i	−0.395642	−	0.918405i	$-0.629478\pi$
$877$	−8.26463	+	2.21450i	−0.279077	+	0.0747783i	0.116566	+	0.993183i	$0.462811\pi$
$878$		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	−0.804643	−	0.593759i	$-0.797643\pi$
$883$	−6.26650	+	6.26650i	−0.210884	+	0.210884i	0.593759	+	0.804643i	$0.297643\pi$
$884$		0			0
$885$	−36.6560	+	30.5259i	−1.23218	+	1.02612i
$886$		0			0
$887$	−1.27192	−	4.74685i	−0.0427067	−	0.159384i	0.941279	−	0.337629i	$-0.109625\pi$
$887$	−1.27192	−	4.74685i	−0.0427067	−	0.159384i	−0.983986	+	0.178245i	$0.942958\pi$
$888$		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
$893$	0.579464	−	2.16259i	0.0193910	−	0.0723682i
$894$		0			0
$895$	−38.5330	+	19.2665i	−1.28802	+	0.644008i
$896$		0			0
$897$	−2.31662	+	7.68338i	−0.0773499	+	0.256540i
$898$		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
$903$	−3.21898	−	11.4932i	−0.107121	−	0.382471i
$904$		0			0
$905$	−1.37545	+	22.9153i	−0.0457215	+	0.761732i
$906$		0			0
$907$	5.02971	−	18.7712i	0.167009	−	0.623286i	−0.830766	−	0.556621i	$-0.812098\pi$
$907$	5.02971	−	18.7712i	0.167009	−	0.623286i	0.997775	−	0.0666648i	$-0.0212358\pi$
$908$		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
$913$	12.1573	+	3.25753i	0.402347	+	0.107809i
$914$		0			0
$915$	33.8608	+	12.4675i	1.11940	+	0.412162i
$916$		0			0
$917$	−25.2434	+	7.92287i	−0.833610	+	0.261636i
$918$		0			0
$919$	33.0964	+	19.1082i	1.09175	+	0.630321i	0.934041	−	0.357165i	$-0.116257\pi$
$919$	33.0964	+	19.1082i	1.09175	+	0.630321i	0.157707	+	0.987486i	$0.449590\pi$
$920$		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
$927$	−28.3846	−	32.1477i	−0.932272	−	1.05587i
$928$		0			0
$929$	0.550126	−	0.952846i	0.0180490	−	0.0312618i	−0.856860	−	0.515549i	$-0.827588\pi$
$929$	0.550126	−	0.952846i	0.0180490	−	0.0312618i	0.874909	+	0.484288i	$0.160921\pi$
$930$		0			0
$931$	−0.941939	+	2.00626i	−0.0308708	+	0.0657524i
$932$		0			0
$933$	2.92397	−	2.74784i	0.0957265	−	0.0899603i
$934$		0			0
$935$	−3.40238	−	16.6266i	−0.111270	−	0.543748i
$936$		0			0
$937$	−14.0000	−	14.0000i	−0.457360	−	0.457360i	0.440428	−	0.897788i	$-0.354827\pi$
$937$	−14.0000	−	14.0000i	−0.457360	−	0.457360i	−0.897788	+	0.440428i	$0.854827\pi$
$938$		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
$943$	4.08423	+	15.2425i	0.133001	+	0.496365i
$944$		0			0
$945$	−30.5525	−	3.39784i	−0.993873	−	0.110532i
$946$		0			0
$947$	11.5390	+	43.0640i	0.374966	+	1.39939i	0.853394	+	0.521266i	$0.174540\pi$
$947$	11.5390	+	43.0640i	0.374966	+	1.39939i	−0.478428	+	0.878127i	$0.658793\pi$
$948$		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	$-0.401473\pi$
$953$	−20.0000	−	20.0000i	−0.647864	−	0.647864i	−0.952476	+	0.304613i	$0.901473\pi$
$954$		0			0
$955$	−2.85420	+	4.32288i	−0.0923597	+	0.139885i
$956$		0			0
$957$	−21.3936	+	20.1049i	−0.691557	+	0.649900i
$958$		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
$963$	10.7878	+	12.2180i	0.347631	+	0.393718i
$964$		0			0
$965$	13.9499	−	41.8496i	0.449062	−	1.34719i
$966$		0			0
$967$	−27.1082	−	27.1082i	−0.871741	−	0.871741i	0.120922	−	0.992662i	$-0.461415\pi$
$967$	−27.1082	−	27.1082i	−0.871741	−	0.871741i	−0.992662	+	0.120922i	$0.961415\pi$
$968$		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
$973$	11.0730	+	10.1732i	0.354983	+	0.326139i
$974$		0			0
$975$	15.2981	−	19.1303i	0.489931	−	0.612660i
$976$		0			0
$977$	6.83013	+	1.83013i	0.218515	+	0.0585510i	0.366416	−	0.930451i	$-0.380585\pi$
$977$	6.83013	+	1.83013i	0.218515	+	0.0585510i	−0.147900	+	0.989002i	$0.547252\pi$
$978$		0			0
$979$		−	38.4169i		−	1.22781i
$980$		0			0
$981$	−30.8166	+	46.4578i	−0.983899	+	1.48328i
$982$		0			0
$983$	3.94999	−	14.7415i	0.125985	−	0.470182i	−0.873888	−	0.486128i	$-0.838409\pi$
$983$	3.94999	−	14.7415i	0.125985	−	0.470182i	0.999873	+	0.0159452i	$0.00507572\pi$
$984$		0			0
$985$	−31.4711	−	35.4904i	−1.00275	−	1.13082i
$986$		0			0
$987$	31.3922	+	8.03311i	0.999224	+	0.255697i
$988$		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
$993$	−4.31662	+	14.3166i	−0.136984	+	0.454324i
$994$		0			0
$995$	−3.26650	−	6.53300i	−0.103555	−	0.207110i
$996$		0			0
$997$	−1.59834	+	5.96509i	−0.0506200	+	0.188916i	−0.986606	−	0.163120i	$-0.947844\pi$
$997$	−1.59834	+	5.96509i	−0.0506200	+	0.188916i	0.935986	+	0.352037i	$0.114511\pi$
$998$		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.b.317.2	yes	8
3.2	odd	2		420.2.bv.a.317.2	yes	8
5.3	odd	4		420.2.bv.a.233.2	yes	8
7.4	even	3	inner	420.2.bv.b.137.1	yes	8
15.8	even	4	inner	420.2.bv.b.233.1	yes	8
21.11	odd	6		420.2.bv.a.137.2	yes	8
35.18	odd	12		420.2.bv.a.53.2	✓	8
105.53	even	12	inner	420.2.bv.b.53.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.a.53.2	✓	8	35.18	odd	12
420.2.bv.a.137.2	yes	8	21.11	odd	6
420.2.bv.a.233.2	yes	8	5.3	odd	4
420.2.bv.a.317.2	yes	8	3.2	odd	2
420.2.bv.b.53.2	yes	8	105.53	even	12	inner
420.2.bv.b.137.1	yes	8	7.4	even	3	inner
420.2.bv.b.233.1	yes	8	15.8	even	4	inner
420.2.bv.b.317.2	yes	8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(1.68614 - 0.396143i) q^{3} +(0.133975 - 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(2.68614 - 1.33591i) q^{9} +O(q^{10})\)	\(q+(1.68614 - 0.396143i) q^{3} +(0.133975 - 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(2.68614 - 1.33591i) q^{9} +(-2.00626 + 1.15831i) q^{11} +(-2.00000 + 2.00000i) q^{13} +(-0.658312 - 3.81662i) q^{15} +(-3.16457 - 0.847944i) q^{17} +(-0.274205 - 0.158312i) q^{19} +(2.24638 - 3.99422i) q^{21} +(1.58228 - 0.423972i) q^{23} +(-4.96410 - 0.598076i) q^{25} +(4.00000 - 3.31662i) q^{27} +7.31662 q^{29} +(3.00000 + 5.19615i) q^{31} +(-2.92397 + 2.74784i) q^{33} +(-4.10891 - 4.25639i) q^{35} +(5.46410 - 1.46410i) q^{37} +(-2.57999 + 4.16457i) q^{39} +9.63325i q^{41} +(1.84169 - 1.84169i) q^{43} +(-2.62194 - 6.17458i) q^{45} +(1.83013 + 6.83013i) q^{47} +(-0.591820 - 6.97494i) q^{49} +(-5.67181 - 0.176129i) q^{51} +(-2.67807 + 9.99470i) q^{53} +(2.31662 + 4.63325i) q^{55} +(-0.525063 - 0.158312i) q^{57} +(-6.15831 - 10.6665i) q^{59} +(-4.65831 + 8.06843i) q^{61} +(2.20542 - 7.62470i) q^{63} +(4.19615 + 4.73205i) q^{65} +(2.62012 - 9.77844i) q^{67} +(2.50000 - 1.34169i) q^{69} +2.31662i q^{71} +(9.06119 + 2.42794i) q^{73} +(-8.60710 + 0.958056i) q^{75} +(-1.33444 + 5.98218i) q^{77} +(-4.92195 - 2.84169i) q^{79} +(5.43070 - 7.17687i) q^{81} +(-3.84169 - 3.84169i) q^{83} +(-2.31662 + 6.94987i) q^{85} +(12.3369 - 2.89843i) q^{87} +(-8.29156 + 14.3614i) q^{89} +(0.316625 + 7.47661i) q^{91} +(7.11684 + 7.57301i) q^{93} +(-0.390098 + 0.590830i) q^{95} +(-9.31662 - 9.31662i) q^{97} +(-3.84169 + 5.79156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} + 8 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) 8 * q + 2 * q^3 + 8 * q^5 + 6 * q^7 + 10 * q^9	\( 8 q + 2 q^{3} + 8 q^{5} + 6 q^{7} + 10 q^{9} - 16 q^{13} + 8 q^{15} + 4 q^{17} + 14 q^{21} - 2 q^{23} - 12 q^{25} + 32 q^{27} + 32 q^{29} + 24 q^{31} - 22 q^{33} + 16 q^{37} - 4 q^{39} + 28 q^{43} - 20 q^{45} - 20 q^{47} - 24 q^{51} + 16 q^{53} - 8 q^{55} - 44 q^{57} - 36 q^{59} - 24 q^{61} + 22 q^{63} - 8 q^{65} - 22 q^{67} + 20 q^{69} + 6 q^{75} - 16 q^{77} - 14 q^{81} - 44 q^{83} + 8 q^{85} + 8 q^{87} - 24 q^{91} - 12 q^{93} - 12 q^{95} - 48 q^{97} - 44 q^{99}+O(q^{100}) \) 8 * q + 2 * q^3 + 8 * q^5 + 6 * q^7 + 10 * q^9 - 16 * q^13 + 8 * q^15 + 4 * q^17 + 14 * q^21 - 2 * q^23 - 12 * q^25 + 32 * q^27 + 32 * q^29 + 24 * q^31 - 22 * 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 - 44 * q^57 - 36 * q^59 - 24 * q^61 + 22 * q^63 - 8 * q^65 - 22 * q^67 + 20 * q^69 + 6 * q^75 - 16 * q^77 - 14 * q^81 - 44 * q^83 + 8 * q^85 + 8 * q^87 - 24 * q^91 - 12 * q^93 - 12 * q^95 - 48 * q^97 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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