LMFDB - Embedded newform 420.2.bv.a.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.a.53.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.26217 + 1.18614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(0.186141 + 2.99422i) q^{9} +O(q^{10})$	\(q+(1.26217 + 1.18614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(0.186141 + 2.99422i) q^{9} +(2.00626 - 1.15831i) q^{11} +(-2.00000 + 2.00000i) q^{13} +(-2.81662 + 2.65831i) q^{15} +(3.16457 + 0.847944i) q^{17} +(-0.274205 - 0.158312i) q^{19} +(4.57025 - 0.335907i) q^{21} +(-1.58228 + 0.423972i) q^{23} +(-4.96410 - 0.598076i) q^{25} +(-3.31662 + 4.00000i) q^{27} -7.31662 q^{29} +(3.00000 + 5.19615i) q^{31} +(3.90616 + 0.917716i) q^{33} +(4.10891 + 4.25639i) q^{35} +(5.46410 - 1.46410i) q^{37} +(-4.89662 + 0.152056i) q^{39} -9.63325i q^{41} +(1.84169 - 1.84169i) q^{43} +(-6.70819 + 0.0143260i) q^{45} +(-1.83013 - 6.83013i) q^{47} +(-0.591820 - 6.97494i) q^{49} +(2.98844 + 4.82387i) q^{51} +(2.67807 - 9.99470i) q^{53} +(2.31662 + 4.63325i) q^{55} +(-0.158312 - 0.525063i) q^{57} +(6.15831 + 10.6665i) q^{59} +(-4.65831 + 8.06843i) q^{61} +(6.16686 + 4.99699i) q^{63} +(-4.19615 - 4.73205i) q^{65} +(2.62012 - 9.77844i) q^{67} +(-2.50000 - 1.34169i) q^{69} -2.31662i q^{71} +(9.06119 + 2.42794i) q^{73} +(-5.55613 - 6.64300i) q^{75} +(1.33444 - 5.98218i) q^{77} +(-4.92195 - 2.84169i) q^{79} +(-8.93070 + 1.11469i) q^{81} +(3.84169 + 3.84169i) q^{83} +(-2.31662 + 6.94987i) q^{85} +(-9.23482 - 8.67855i) q^{87} +(8.29156 - 14.3614i) q^{89} +(0.316625 + 7.47661i) q^{91} +(-2.37686 + 10.1168i) q^{93} +(0.390098 - 0.590830i) q^{95} +(-9.31662 - 9.31662i) q^{97} +(3.84169 + 5.79156i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 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 $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

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

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
$3$	1.26217	+	1.18614i	0.728714	+	0.684819i
$3$	1.26217	+	1.18614i	0.728714	+	0.684819i
$4$		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
$9$	0.186141	+	2.99422i	0.0620469	+	0.998073i
$10$		0			0
$11$	2.00626	−	1.15831i	0.604909	−	0.349244i	−0.166061	−	0.986115i	$-0.553105\pi$
$11$	2.00626	−	1.15831i	0.604909	−	0.349244i	0.770970	+	0.636871i	$0.219772\pi$
$12$		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
$15$	−2.81662	+	2.65831i	−0.727249	+	0.686373i
$16$		0			0
$17$	3.16457	+	0.847944i	0.767521	+	0.205657i	0.621276	−	0.783592i	$-0.286615\pi$
$17$	3.16457	+	0.847944i	0.767521	+	0.205657i	0.146245	+	0.989248i	$0.453281\pi$
$18$		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
$21$	4.57025	−	0.335907i	0.997310	−	0.0733010i
$22$		0			0
$23$	−1.58228	+	0.423972i	−0.329929	+	0.0884042i	−0.419981	−	0.907533i	$-0.637963\pi$
$23$	−1.58228	+	0.423972i	−0.329929	+	0.0884042i	0.0900521	+	0.995937i	$0.471297\pi$
$24$		0			0
$25$	−4.96410	−	0.598076i	−0.992820	−	0.119615i
$26$		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
$33$	3.90616	+	0.917716i	0.679974	+	0.159754i
$34$		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
$39$	−4.89662	+	0.152056i	−0.784087	+	0.0243485i
$40$		0			0
$41$		−	9.63325i		−	1.50446i	−0.658900	−	0.752230i	$-0.728978\pi$
$41$		−	9.63325i		−	1.50446i	0.658900	−	0.752230i	$-0.271022\pi$
$42$		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
$45$	−6.70819	+	0.0143260i	−0.999998	+	0.00213560i
$46$		0			0
$47$	−1.83013	−	6.83013i	−0.266951	−	0.996276i	−0.961045	−	0.276392i	$-0.910861\pi$
$47$	−1.83013	−	6.83013i	−0.266951	−	0.996276i	0.694094	−	0.719885i	$-0.255805\pi$
$48$		0			0
$49$	−0.591820	−	6.97494i	−0.0845458	−	0.996420i
$50$		0			0
$51$	2.98844	+	4.82387i	0.418465	+	0.675477i
$52$		0			0
$53$	2.67807	−	9.99470i	0.367861	−	1.37288i	−0.495639	−	0.868529i	$-0.665066\pi$
$53$	2.67807	−	9.99470i	0.367861	−	1.37288i	0.863500	−	0.504348i	$-0.168267\pi$
$54$		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
$63$	6.16686	+	4.99699i	0.776951	+	0.629561i
$64$		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	$-0.956104\pi$
$71$		−	2.31662i		−	0.274933i	0.990506	−	0.137466i	$-0.0438959\pi$
$72$		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
$75$	−5.55613	−	6.64300i	−0.641567	−	0.767067i
$76$		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
$81$	−8.93070	+	1.11469i	−0.992300	+	0.123855i
$82$		0			0
$83$	3.84169	+	3.84169i	0.421680	+	0.421680i	0.885782	−	0.464102i	$-0.153623\pi$
$83$	3.84169	+	3.84169i	0.421680	+	0.421680i	−0.464102	+	0.885782i	$0.653623\pi$
$84$		0			0
$85$	−2.31662	+	6.94987i	−0.251273	+	0.753820i
$86$		0			0
$87$	−9.23482	−	8.67855i	−0.990076	−	0.930438i
$88$		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
$93$	−2.37686	+	10.1168i	−0.246469	+	1.04907i
$94$		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	−0.379907	−	0.925025i	$-0.624044\pi$
$101$	2.32387	−	1.34169i	0.231234	−	0.133503i	0.611141	+	0.791522i	$0.290711\pi$
$102$		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
$105$	0.137465	+	10.2460i	0.0134152	+	0.999910i
$106$		0			0
$107$	−1.40616	−	5.24784i	−0.135938	−	0.507328i	−0.999992	−	0.00392920i	$-0.998749\pi$
$107$	−1.40616	−	5.24784i	−0.135938	−	0.507328i	0.864054	−	0.503399i	$-0.167917\pi$
$108$		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	0.0897449	−	0.995965i	$-0.471395\pi$
$113$	−9.63325	−	9.63325i	−0.906220	−	0.906220i	−0.995965	+	0.0897449i	$0.971395\pi$
$114$		0			0
$115$	−0.734341	−	3.58854i	−0.0684776	−	0.334633i
$116$		0			0
$117$	−6.36072	−	5.61616i	−0.588049	−	0.519214i
$118$		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
$123$	11.4264	−	12.1588i	1.03028	−	1.09632i
$124$		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
$129$	4.50902	−	0.140020i	0.396997	−	0.0123281i
$130$		0			0
$131$	8.66025	+	5.00000i	0.756650	+	0.436852i	0.828092	−	0.560593i	$-0.189427\pi$
$131$	8.66025	+	5.00000i	0.756650	+	0.436852i	−0.0714417	+	0.997445i	$0.522760\pi$
$132$		0			0
$133$	−0.799268	+	0.250858i	−0.0693053	+	0.0217521i
$134$		0			0
$135$	−8.48386	−	7.93877i	−0.730174	−	0.683261i
$136$		0			0
$137$	−9.49370	−	2.54383i	−0.811102	−	0.217334i	−0.170649	−	0.985332i	$-0.554587\pi$
$137$	−9.49370	−	2.54383i	−0.811102	−	0.217334i	−0.640453	+	0.767998i	$0.721253\pi$
$138$		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
$147$	7.52628	−	9.50553i	0.620757	−	0.784003i
$148$		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
$157$	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
$159$	15.2353	−	9.43843i	1.20824	−	0.748516i
$160$		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
$165$	−2.57171	+	8.59579i	−0.200208	+	0.669181i
$166$		0			0
$167$	−13.1082	+	13.1082i	−1.01434	+	1.01434i	−0.0144463	+	0.999896i	$0.504599\pi$
$167$	−13.1082	+	13.1082i	−1.01434	+	1.01434i	−0.999896	+	0.0144463i	$0.995401\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	0.422981	−	0.850499i	0.0323462	−	0.0650393i
$172$		0			0
$173$	9.49370	−	2.54383i	0.721793	−	0.193404i	0.120821	−	0.992674i	$-0.461447\pi$
$173$	9.49370	−	2.54383i	0.721793	−	0.193404i	0.600972	+	0.799270i	$0.294780\pi$
$174$		0			0
$175$	−10.0510	+	8.60105i	−0.759782	+	0.650178i
$176$		0			0
$177$	−4.87915	+	20.7676i	−0.366739	+	1.56099i
$178$		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
$185$	2.53590	+	12.3923i	0.186443	+	0.911100i
$186$		0			0
$187$	7.33112	−	1.96437i	0.536104	−	0.143649i
$188$		0			0
$189$	1.85649	+	13.6218i	0.135040	+	0.990840i
$190$		0			0
$191$	2.00626	+	1.15831i	0.145168	+	0.0838125i	0.570825	−	0.821072i	$-0.306624\pi$
$191$	2.00626	+	1.15831i	0.145168	+	0.0838125i	−0.425657	+	0.904885i	$0.639957\pi$
$192$		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
$195$	0.316625	−	10.9499i	0.0226740	−	0.784137i
$196$		0			0
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.0712470	+	0.997459i	$0.522698\pi$
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		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
$201$	14.9056	−	9.23420i	1.05136	−	0.651330i
$202$		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
$207$	−1.56399	−	4.65879i	−0.108705	−	0.323808i
$208$		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
$213$	2.74784	−	2.92397i	0.188279	−	0.200347i
$214$		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
$219$	8.55687	+	13.8123i	0.578220	+	0.933349i
$220$		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
$225$	0.866750	−	14.9749i	0.0577834	−	0.998329i
$226$		0			0
$227$	26.8195	+	7.18627i	1.78007	+	0.476969i	0.990597	−	0.136811i	$-0.0436853\pi$
$227$	26.8195	+	7.18627i	1.78007	+	0.476969i	0.789477	+	0.613780i	$0.210352\pi$
$228$		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
$231$	8.78000	−	5.96769i	0.577682	−	0.392645i
$232$		0			0
$233$	16.8248	−	4.50820i	1.10223	−	0.295342i	0.338558	−	0.940945i	$-0.390061\pi$
$233$	16.8248	−	4.50820i	1.10223	−	0.295342i	0.763673	+	0.645604i	$0.223394\pi$
$234$		0			0
$235$	15.4904	−	3.16987i	1.01048	−	0.206780i
$236$		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
$243$	−12.5942	−	9.18614i	−0.807921	−	0.589291i
$244$		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
$249$	0.292077	+	9.40564i	0.0185096	+	0.596058i
$250$		0			0
$251$			9.26650i			0.584896i	0.956281	+	0.292448i	$0.0944699\pi$
$251$			9.26650i			0.584896i	−0.956281	+	0.292448i	$0.905530\pi$
$252$		0			0
$253$	−2.68338	+	2.68338i	−0.168702	+	0.168702i
$254$		0			0
$255$	−11.1675	+	6.02407i	−0.699336	+	0.377242i
$256$		0			0
$257$	5.35614	+	19.9894i	0.334107	+	1.24690i	0.904834	+	0.425764i	$0.139995\pi$
$257$	5.35614	+	19.9894i	0.334107	+	1.24690i	−0.570727	+	0.821140i	$0.693339\pi$
$258$		0			0
$259$	6.92820	−	13.2665i	0.430498	−	0.824340i
$260$		0			0
$261$	−1.36192	−	21.9076i	−0.0843008	−	1.35605i
$262$		0			0
$263$	1.40616	−	5.24784i	0.0867072	−	0.323596i	−0.908925	−	0.416960i	$-0.863095\pi$
$263$	1.40616	−	5.24784i	0.0867072	−	0.323596i	0.995632	+	0.0933643i	$0.0297621\pi$
$264$		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
$273$	−8.46868	+	9.81231i	−0.512548	+	0.593868i
$274$		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	$0.262700\pi$
$281$			24.6332i			1.46950i	−0.678340	+	0.734748i	$0.737300\pi$
$282$		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
$285$	1.19318	−	0.283016i	0.0706777	−	0.0167644i
$286$		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
$291$	−0.708327	−	22.8100i	−0.0415228	−	1.33714i
$292$		0			0
$293$	5.36675	+	5.36675i	0.313529	+	0.313529i	0.846275	−	0.532746i	$-0.178840\pi$
$293$	5.36675	+	5.36675i	0.313529	+	0.313529i	−0.532746	+	0.846275i	$0.678840\pi$
$294$		0			0
$295$	−24.6332	+	12.3166i	−1.43420	+	0.717102i
$296$		0			0
$297$	−2.02075	+	11.8667i	−0.117256	+	0.688576i
$298$		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
$303$	4.52455	+	1.06300i	0.259928	+	0.0610678i
$304$		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	−0.555802	−	0.831314i	$-0.687589\pi$
$311$	−2.00626	+	1.15831i	−0.113764	+	0.0656819i	0.442038	+	0.896996i	$0.354256\pi$
$312$		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
$315$	−11.9797	+	13.0953i	−0.674981	+	0.737835i
$316$		0			0
$317$	−2.54383	−	9.49370i	−0.142876	−	0.533220i	−0.999841	−	0.0178450i	$-0.994319\pi$
$317$	−2.54383	−	9.49370i	−0.142876	−	0.533220i	0.856965	−	0.515375i	$-0.172347\pi$
$318$		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
$327$	−31.3338	−	7.36158i	−1.73276	−	0.407097i
$328$		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
$333$	5.40093	+	16.0882i	0.295969	+	0.881627i
$334$		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
$339$	−0.732399	−	23.5852i	−0.0397785	−	1.28097i
$340$		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
$345$	3.32965	−	5.40037i	0.179262	−	0.290746i
$346$		0			0
$347$	15.7435	+	4.21847i	0.845157	+	0.226459i	0.655315	−	0.755356i	$-0.272536\pi$
$347$	15.7435	+	4.21847i	0.845157	+	0.226459i	0.189842	+	0.981815i	$0.439203\pi$
$348$		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	$0.201428\pi$
$353$	−3.52601	+	13.1593i	−0.187671	+	0.700397i	−0.994043	+	0.108989i	$0.965239\pi$
$354$		0			0
$355$	5.17082	+	0.310369i	0.274439	+	0.0164727i
$356$		0			0
$357$	14.7477	+	2.81231i	0.780531	+	0.148843i
$358$		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
$367$	−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
$369$	28.8441	−	1.79314i	1.50156	−	0.0933471i
$370$		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
$375$	15.5719	−	11.5116i	0.804129	−	0.594455i
$376$		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
$381$	0.428286	+	13.7919i	0.0219418	+	0.706582i
$382$		0			0
$383$	−31.0654	+	8.32394i	−1.58737	+	0.425334i	−0.941197	−	0.337860i	$-0.890297\pi$
$383$	−31.0654	+	8.32394i	−1.58737	+	0.425334i	−0.646170	+	0.763193i	$0.723630\pi$
$384$		0			0
$385$	13.1738	+	3.78000i	0.671397	+	0.192647i
$386$		0			0
$387$	5.85723	+	5.17160i	0.297740	+	0.262887i
$388$		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
$395$	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
$399$	−1.30636	−	0.631419i	−0.0654000	−	0.0316105i
$400$		0			0
$401$	−28.3046	−	16.3417i	−1.41347	−	0.816065i	−0.417753	−	0.908561i	$-0.637182\pi$
$401$	−28.3046	−	16.3417i	−1.41347	−	0.816065i	−0.995713	+	0.0924958i	$0.970516\pi$
$402$		0			0
$403$	−16.3923	−	4.39230i	−0.816559	−	0.218796i
$404$		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	−0.364835	−	0.931072i	$-0.618875\pi$
$409$	5.23956	−	3.02506i	0.259080	−	0.149580i	0.623915	+	0.781492i	$0.285541\pi$
$410$		0			0
$411$	−8.96532	−	14.4716i	−0.442227	−	0.713832i
$412$		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
$417$	−6.74128	+	7.17338i	−0.330122	+	0.351282i
$418$		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
$423$	20.1102	−	6.75117i	0.977793	−	0.328253i
$424$		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
$429$	−9.64774	+	5.97688i	−0.465797	+	0.288566i
$430$		0			0
$431$	−10.6665	+	6.15831i	−0.513788	+	0.296635i	−0.734389	−	0.678729i	$-0.762531\pi$
$431$	−10.6665	+	6.15831i	−0.513788	+	0.296635i	0.220602	+	0.975364i	$0.429198\pi$
$432$		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
$435$	20.6082	−	19.4499i	0.988087	−	0.932550i
$436$		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
$441$	20.7743	−	3.07036i	0.989254	−	0.146208i
$442$		0			0
$443$	4.74685	−	1.27192i	0.225530	−	0.0604305i	−0.144284	−	0.989536i	$-0.546088\pi$
$443$	4.74685	−	1.27192i	0.225530	−	0.0604305i	0.369814	+	0.929106i	$0.379421\pi$
$444$		0			0
$445$	30.9445	+	20.4313i	1.46691	+	0.968534i
$446$		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
$453$	−1.33372	+	5.67681i	−0.0626635	+	0.266720i
$454$		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
$459$	−13.8875	+	9.84596i	−0.648211	+	0.459570i
$460$		0			0
$461$		−	5.36675i		−	0.249954i	−0.992160	−	0.124977i	$-0.960114\pi$
$461$		−	5.36675i		−	0.249954i	0.992160	−	0.124977i	$-0.0398858\pi$
$462$		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
$465$	−22.2629	−	6.66067i	−1.03242	−	0.308881i
$466$		0			0
$467$	0.558212	+	2.08327i	0.0258310	+	0.0964024i	0.977638	−	0.210295i	$-0.0674426\pi$
$467$	0.558212	+	2.08327i	0.0258310	+	0.0964024i	−0.951807	+	0.306698i	$0.900776\pi$
$468$		0			0
$469$	−14.3614	−	22.6082i	−0.663148	−	1.04395i
$470$		0			0
$471$	11.0707	−	6.85843i	0.510112	−	0.316020i
$472$		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
$477$	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
$483$	−7.08902	+	2.46916i	−0.322561	+	0.112351i
$484$		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	$-0.971954\pi$
$491$		−	3.89975i		−	0.175993i	0.996121	−	0.0879966i	$-0.0280465\pi$
$492$		0			0
$493$	−23.1540	−	6.20408i	−1.04280	−	0.279418i
$494$		0			0
$495$	−13.4417	+	7.79892i	−0.604162	+	0.350535i
$496$		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
$501$	−32.0929	+	0.996592i	−1.43380	+	0.0445245i
$502$		0			0
$503$	−4.20844	−	4.20844i	−0.187645	−	0.187645i	0.607032	−	0.794677i	$-0.292360\pi$
$503$	−4.20844	−	4.20844i	−0.187645	−	0.187645i	−0.794677	+	0.607032i	$0.792360\pi$
$504$		0			0
$505$	2.68338	+	5.36675i	0.119409	+	0.238817i
$506$		0			0
$507$	−5.93070	+	6.31084i	−0.263392	+	0.280274i
$508$		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
$513$	1.54269	−	0.571758i	0.0681112	−	0.0252437i
$514$		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	−0.0877909	−	0.996139i	$-0.527981\pi$
$521$	16.6853	−	9.63325i	0.730995	−	0.422040i	0.818786	+	0.574099i	$0.194647\pi$
$522$		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
$525$	−22.8881	−	1.06588i	−0.998917	−	0.0465188i
$526$		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
$537$	−7.63230	+	32.4860i	−0.329358	+	1.40188i
$538$		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
$543$	−12.9581	−	12.1775i	−0.556083	−	0.522587i
$544$		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
$549$	−25.0258	−	12.4461i	−1.06807	−	0.531189i
$550$		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
$555$	−11.4983	+	18.6491i	−0.488075	+	0.791611i
$556$		0			0
$557$	9.99470	+	2.67807i	0.423489	+	0.113473i	0.464268	−	0.885695i	$-0.346317\pi$
$557$	9.99470	+	2.67807i	0.423489	+	0.113473i	−0.0407793	+	0.999168i	$0.512984\pi$
$558$		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	−0.880159	−	0.474679i	$-0.842564\pi$
$563$	2.83356	−	10.5750i	0.119420	−	0.445683i	0.999580	+	0.0289957i	$0.00923091\pi$
$564$		0			0
$565$	22.7925	−	20.2113i	0.958888	−	0.850295i
$566$		0			0
$567$	−13.8142	+	19.3951i	−0.580141	+	0.814516i
$568$		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
$577$	−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
$579$	17.9953	+	29.0476i	0.747859	+	1.20718i
$580$		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
$585$	13.3877	−	13.4450i	0.553514	−	0.555884i
$586$		0			0
$587$	−0.366750	+	0.366750i	−0.0151374	+	0.0151374i	−0.714635	−	0.699498i	$-0.753407\pi$
$587$	−0.366750	+	0.366750i	−0.0151374	+	0.0151374i	0.699498	+	0.714635i	$0.253407\pi$
$588$		0			0
$589$		−	1.89975i		−	0.0782778i
$590$		0			0
$591$	−36.7246	+	1.14042i	−1.51065	+	0.0469108i
$592$		0			0
$593$	−36.3132	+	9.73010i	−1.49121	+	0.399567i	−0.910144	−	0.414292i	$-0.864029\pi$
$593$	−36.3132	+	9.73010i	−1.49121	+	0.399567i	−0.581061	+	0.813860i	$0.697362\pi$
$594$		0			0
$595$	9.39375	+	16.9538i	0.385106	+	0.695036i
$596$		0			0
$597$	5.50778	+	1.29400i	0.225418	+	0.0529600i
$598$		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
$605$	−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
$609$	−33.4388	+	2.45771i	−1.35501	+	0.0995913i
$610$		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
$615$	25.6082	+	27.1332i	1.03262	+	1.09412i
$616$		0			0
$617$	1.58312	−	1.58312i	0.0637342	−	0.0637342i	−0.674521	−	0.738255i	$-0.735650\pi$
$617$	1.58312	−	1.58312i	0.0637342	−	0.0637342i	0.738255	+	0.674521i	$0.235650\pi$
$618$		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
$621$	3.55196	−	7.73529i	0.142535	−	0.310407i
$622$		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
$627$	−0.925802	−	0.870035i	−0.0369730	−	0.0347459i
$628$		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
$633$	−6.24758	−	5.87125i	−0.248319	−	0.233361i
$634$		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
$639$	6.93648	−	0.431218i	0.274403	−	0.0170587i
$640$		0			0
$641$	0.952846	−	0.550126i	0.0376351	−	0.0217287i	−0.481064	−	0.876685i	$-0.659750\pi$
$641$	0.952846	−	0.550126i	0.0376351	−	0.0217287i	0.518700	+	0.854957i	$0.326416\pi$
$642$		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
$645$	−0.291562	+	10.0831i	−0.0114802	+	0.397023i
$646$		0			0
$647$	18.4071	+	4.93217i	0.723658	+	0.193904i	0.601803	−	0.798644i	$-0.294449\pi$
$647$	18.4071	+	4.93217i	0.723658	+	0.193904i	0.121855	+	0.992548i	$0.461116\pi$
$648$		0			0
$649$	24.7103	+	14.2665i	0.969964	+	0.560009i
$650$		0			0
$651$	15.4562	+	22.7400i	0.605775	+	0.891250i
$652$		0			0
$653$	−46.3079	+	12.4082i	−1.81217	+	0.485569i	−0.995767	−	0.0919148i	$-0.970701\pi$
$653$	−46.3079	+	12.4082i	−1.81217	+	0.485569i	−0.816402	+	0.577484i	$0.804035\pi$
$654$		0			0
$655$	−12.3205	+	18.6603i	−0.481402	+	0.729116i
$656$		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
$663$	−15.6246	−	3.67086i	−0.606810	−	0.142565i
$664$		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
$669$	27.7066	−	0.860383i	1.07120	−	0.0332643i
$670$		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
$675$	18.8564	−	17.8728i	0.725782	−	0.687925i
$676$		0			0
$677$	10.4438	+	38.9768i	0.401388	+	1.49800i	0.810622	+	0.585570i	$0.199129\pi$
$677$	10.4438	+	38.9768i	0.401388	+	1.49800i	−0.409234	+	0.912430i	$0.634204\pi$
$678$		0			0
$679$	−34.8284	+	1.47494i	−1.33659	+	0.0566029i
$680$		0			0
$681$	25.3268	+	40.8820i	0.970526	+	1.56660i
$682$		0			0
$683$	−5.20065	+	19.4091i	−0.198997	+	0.742668i	0.792198	+	0.610264i	$0.208936\pi$
$683$	−5.20065	+	19.4091i	−0.198997	+	0.742668i	−0.991196	+	0.132405i	$0.957730\pi$
$684$		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
$693$	18.1604	+	2.88208i	0.689855	+	0.109481i
$694$		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	$-0.879094\pi$
$701$		−	19.6332i		−	0.741538i	0.928725	−	0.370769i	$-0.120906\pi$
$702$		0			0
$703$	−1.73007	−	0.463571i	−0.0652508	−	0.0174839i
$704$		0			0
$705$	23.3114	+	14.3729i	0.877958	+	0.541313i
$706$		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
$711$	7.59246	−	15.2663i	0.284740	−	0.572533i
$712$		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
$717$	−36.0135	−	33.8441i	−1.34495	−	1.26393i
$718$		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
$723$	−1.04314	+	4.44003i	−0.0387950	+	0.165126i
$724$		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
$735$	20.2085	+	18.0725i	0.745402	+	0.666616i
$736$		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	0.810989	−	0.585062i	$-0.198930\pi$
$743$	6.15831	+	6.15831i	0.225927	+	0.225927i	−0.585062	+	0.810989i	$0.698930\pi$
$744$		0			0
$745$	−17.9759	−	11.8687i	−0.658586	−	0.434834i
$746$		0			0
$747$	−10.7878	+	12.2180i	−0.394704	+	0.447031i
$748$		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
$753$	−10.9914	+	11.6959i	−0.400548	+	0.426222i
$754$		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
$759$	−6.56973	+	0.204012i	−0.238466	+	0.00740518i
$760$		0			0
$761$	−17.3205	−	10.0000i	−0.627868	−	0.362500i	0.152058	−	0.988372i	$-0.451410\pi$
$761$	−17.3205	−	10.0000i	−0.627868	−	0.362500i	−0.779926	+	0.625872i	$0.784743\pi$
$762$		0			0
$763$	−10.7044	+	47.9869i	−0.387525	+	1.73724i
$764$		0			0
$765$	−21.2407	−	5.64283i	−0.767958	−	0.204017i
$766$		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	$-0.676289\pi$
$773$	9.86434	−	36.8142i	0.354796	−	1.32412i	0.880742	−	0.473597i	$-0.157045\pi$
$774$		0			0
$775$	−11.7846	−	27.5885i	−0.423316	−	0.991007i
$776$		0			0
$777$	24.4805	−	8.52674i	0.878233	−	0.305895i
$778$		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
$787$	−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
$789$	7.99949	−	4.95577i	0.284789	−	0.176430i
$790$		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
$795$	19.0259	+	35.2705i	0.674779	+	1.25091i
$796$		0			0
$797$	15.3668	−	15.3668i	0.544318	−	0.544318i	−0.380474	−	0.924792i	$-0.624239\pi$
$797$	15.3668	−	15.3668i	0.544318	−	0.544318i	0.924792	+	0.380474i	$0.124239\pi$
$798$		0			0
$799$		−	23.1662i		−	0.819563i
$800$		0			0
$801$	44.5446	+	22.1535i	1.57391	+	0.782756i
$802$		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
$807$	1.69015	−	7.19392i	0.0594960	−	0.253238i
$808$		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
$815$	3.09808	−	0.633975i	0.108521	−	0.0222072i
$816$		0			0
$817$	−0.796562	+	0.213438i	−0.0278682	+	0.00746726i
$818$		0			0
$819$	−22.3277	+	2.33975i	−0.780193	+	0.0817573i
$820$		0			0
$821$	8.02502	+	4.63325i	0.280075	+	0.161702i	0.633457	−	0.773777i	$-0.281635\pi$
$821$	8.02502	+	4.63325i	0.280075	+	0.161702i	−0.353382	+	0.935479i	$0.614968\pi$
$822$		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
$825$	−18.8417	−	6.89181i	−0.655983	−	0.239942i
$826$		0			0
$827$	6.52506	−	6.52506i	0.226899	−	0.226899i	−0.584497	−	0.811396i	$-0.698708\pi$
$827$	6.52506	−	6.52506i	0.226899	−	0.226899i	0.811396	+	0.584497i	$0.198708\pi$
$828$		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
$831$	−13.1530	+	8.14843i	−0.456273	+	0.282666i
$832$		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
$837$	−30.7345	−	5.23369i	−1.06234	−	0.180903i
$838$		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
$843$	−29.2185	+	31.0913i	−1.00634	+	1.07084i
$844$		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
$849$	−22.7468	−	36.7174i	−0.780670	−	1.26014i
$850$		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
$855$	1.84169	+	1.05806i	0.0629844	+	0.0361849i
$856$		0			0
$857$	−23.6549	−	6.33832i	−0.808038	−	0.216513i	−0.168928	−	0.985628i	$-0.554030\pi$
$857$	−23.6549	−	6.33832i	−0.808038	−	0.216513i	−0.639110	+	0.769115i	$0.720697\pi$
$858$		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
$861$	−3.23588	−	44.0263i	−0.110278	−	1.50041i
$862$		0			0
$863$	51.5558	−	13.8143i	1.75498	−	0.470245i	0.769301	−	0.638886i	$-0.220605\pi$
$863$	51.5558	−	13.8143i	1.75498	−	0.470245i	0.985677	+	0.168641i	$0.0539380\pi$
$864$		0			0
$865$	4.40604	+	21.5312i	0.149810	+	0.732084i
$866$		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
$873$	26.1618	−	29.6302i	0.885443	−	1.00283i
$874$		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
$879$	0.408025	+	13.1395i	0.0137623	+	0.443183i
$880$		0			0
$881$		−	33.5330i		−	1.12976i	−0.825175	−	0.564878i	$-0.808923\pi$
$881$		−	33.5330i		−	1.12976i	0.825175	−	0.564878i	$-0.191077\pi$
$882$		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
$885$	−45.7006	−	13.6728i	−1.53621	−	0.459607i
$886$		0			0
$887$	1.27192	+	4.74685i	0.0427067	+	0.159384i	0.983986	−	0.178245i	$-0.0570419\pi$
$887$	1.27192	+	4.74685i	0.0427067	+	0.159384i	−0.941279	+	0.337629i	$0.890375\pi$
$888$		0			0
$889$	21.0588	−	0.891813i	0.706290	−	0.0299105i
$890$		0			0
$891$	−16.6261	+	12.5809i	−0.556996	+	0.421476i
$892$		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
$897$	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
$903$	7.79833	−	9.03561i	0.259512	−	0.300686i
$904$		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	$-0.852690\pi$
$911$		−	26.9499i		−	0.892889i	0.894811	−	0.446445i	$-0.147310\pi$
$912$		0			0
$913$	12.1573	+	3.25753i	0.402347	+	0.107809i
$914$		0			0
$915$	−8.32770	−	35.1090i	−0.275305	−	1.16067i
$916$		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
$921$	0.720363	+	23.1976i	0.0237368	+	0.764386i
$922$		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
$927$	40.6556	−	13.6484i	1.33531	−	0.448273i
$928$		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
$933$	−3.90616	−	0.917716i	−0.127882	−	0.0300447i
$934$		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	−0.202743	−	0.979232i	$-0.564986\pi$
$941$	16.6853	−	9.63325i	0.543925	−	0.314035i	0.746668	+	0.665197i	$0.231652\pi$
$942$		0			0
$943$	4.08423	+	15.2425i	0.133001	+	0.496365i
$944$		0			0
$945$	−30.6533	+	2.31880i	−0.997151	+	0.0754307i
$946$		0			0
$947$	−11.5390	−	43.0640i	−0.374966	−	1.39939i	−0.853394	−	0.521266i	$-0.825460\pi$
$947$	−11.5390	−	43.0640i	−0.374966	−	1.39939i	0.478428	−	0.878127i	$-0.341207\pi$
$948$		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	0.952476	−	0.304613i	$-0.0985270\pi$
$953$	20.0000	+	20.0000i	0.647864	+	0.647864i	−0.304613	+	0.952476i	$0.598527\pi$
$954$		0			0
$955$	−2.85420	+	4.32288i	−0.0923597	+	0.139885i
$956$		0			0
$957$	−28.5799	−	6.71458i	−0.923856	−	0.217052i
$958$		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
$963$	15.4515	−	5.18717i	0.497916	−	0.167154i
$964$		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
$969$	−0.0557668	−	1.79584i	−0.00179149	−	0.0576906i
$970$		0			0
$971$	12.0375	+	6.94987i	0.386303	+	0.223032i	0.680557	−	0.732695i	$-0.261738\pi$
$971$	12.0375	+	6.94987i	0.386303	+	0.223032i	−0.294254	+	0.955727i	$0.595071\pi$
$972$		0			0
$973$	11.0730	+	10.1732i	0.354983	+	0.326139i
$974$		0			0
$975$	24.3983	+	2.17373i	0.781370	+	0.0696150i
$976$		0			0
$977$	−6.83013	−	1.83013i	−0.218515	−	0.0585510i	0.147900	−	0.989002i	$-0.452748\pi$
$977$	−6.83013	−	1.83013i	−0.218515	−	0.0585510i	−0.366416	+	0.930451i	$0.619415\pi$
$978$		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	−0.999873	−	0.0159452i	$-0.994924\pi$
$983$	−3.94999	+	14.7415i	−0.125985	+	0.470182i	0.873888	+	0.486128i	$0.161591\pi$
$984$		0			0
$985$	−31.4711	−	35.4904i	−1.00275	−	1.13082i
$986$		0			0
$987$	−10.6584	−	30.6006i	−0.339261	−	0.974028i
$988$		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
$997$	−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
$999$	−12.2660	+	26.7123i	−0.388078	+	0.845139i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+(1.26217 + 1.18614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(0.186141 + 2.99422i) q^{9} +O(q^{10})\)	\(q+(1.26217 + 1.18614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(1.79000 - 1.94831i) q^{7} +(0.186141 + 2.99422i) q^{9} +(2.00626 - 1.15831i) q^{11} +(-2.00000 + 2.00000i) q^{13} +(-2.81662 + 2.65831i) q^{15} +(3.16457 + 0.847944i) q^{17} +(-0.274205 - 0.158312i) q^{19} +(4.57025 - 0.335907i) q^{21} +(-1.58228 + 0.423972i) q^{23} +(-4.96410 - 0.598076i) q^{25} +(-3.31662 + 4.00000i) q^{27} -7.31662 q^{29} +(3.00000 + 5.19615i) q^{31} +(3.90616 + 0.917716i) q^{33} +(4.10891 + 4.25639i) q^{35} +(5.46410 - 1.46410i) q^{37} +(-4.89662 + 0.152056i) q^{39} -9.63325i q^{41} +(1.84169 - 1.84169i) q^{43} +(-6.70819 + 0.0143260i) q^{45} +(-1.83013 - 6.83013i) q^{47} +(-0.591820 - 6.97494i) q^{49} +(2.98844 + 4.82387i) q^{51} +(2.67807 - 9.99470i) q^{53} +(2.31662 + 4.63325i) q^{55} +(-0.158312 - 0.525063i) q^{57} +(6.15831 + 10.6665i) q^{59} +(-4.65831 + 8.06843i) q^{61} +(6.16686 + 4.99699i) q^{63} +(-4.19615 - 4.73205i) q^{65} +(2.62012 - 9.77844i) q^{67} +(-2.50000 - 1.34169i) q^{69} -2.31662i q^{71} +(9.06119 + 2.42794i) q^{73} +(-5.55613 - 6.64300i) q^{75} +(1.33444 - 5.98218i) q^{77} +(-4.92195 - 2.84169i) q^{79} +(-8.93070 + 1.11469i) q^{81} +(3.84169 + 3.84169i) q^{83} +(-2.31662 + 6.94987i) q^{85} +(-9.23482 - 8.67855i) q^{87} +(8.29156 - 14.3614i) q^{89} +(0.316625 + 7.47661i) q^{91} +(-2.37686 + 10.1168i) q^{93} +(0.390098 - 0.590830i) q^{95} +(-9.31662 - 9.31662i) q^{97} +(3.84169 + 5.79156i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9}+O(q^{10}) \) 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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