LMFDB - Embedded newform 1386.2.k.v.991.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(793,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("1386.793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.k (of order $3$, degree $2$, 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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.21870000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 $ x^6 - 3x^5 + 24x^4 - 43x^3 + 138x^2 - 117*x + 73
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			991.2
Root			$0.500000 - 3.05087i$ of defining polynomial
Character	$\chi$	$=$	1386.991
Dual form			1386.2.k.v.793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.806615 - 1.39710i) q^{5} +(-0.806615 - 2.51980i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.806615 - 1.39710i) q^{5} +(-0.806615 - 2.51980i) q^{7} -1.00000 q^{8} +(0.806615 - 1.39710i) q^{10} +(-0.500000 + 0.866025i) q^{11} +(1.77890 - 1.95845i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.67103 + 6.35841i) q^{17} +(-0.585515 - 1.01414i) q^{19} +1.61323 q^{20} -1.00000 q^{22} +(1.77890 + 3.08115i) q^{23} +(1.19874 - 2.07629i) q^{25} +(2.58551 + 0.561349i) q^{28} -10.3975 q^{29} +(-3.17103 + 5.49238i) q^{31} +(0.500000 - 0.866025i) q^{32} -7.34206 q^{34} +(-2.86977 + 3.15943i) q^{35} +(-1.41449 - 2.44996i) q^{37} +(0.585515 - 1.01414i) q^{38} +(0.806615 + 1.39710i) q^{40} -4.94457 q^{41} -11.5131 q^{43} +(-0.500000 - 0.866025i) q^{44} +(-1.77890 + 3.08115i) q^{46} +(3.39213 + 5.87534i) q^{47} +(-5.69874 + 4.06501i) q^{49} +2.39749 q^{50} +(4.00000 - 6.92820i) q^{53} +1.61323 q^{55} +(0.806615 + 2.51980i) q^{56} +(-5.19874 - 9.00449i) q^{58} +(2.19874 - 3.80834i) q^{59} +(5.97764 + 10.3536i) q^{61} -6.34206 q^{62} +1.00000 q^{64} +(-1.47229 + 2.55007i) q^{67} +(-3.67103 - 6.35841i) q^{68} +(-4.17103 - 0.905585i) q^{70} -15.5131 q^{71} +(2.55780 - 4.43024i) q^{73} +(1.41449 - 2.44996i) q^{74} +1.17103 q^{76} +(2.58551 + 0.561349i) q^{77} +(-2.80661 - 4.86120i) q^{79} +(-0.806615 + 1.39710i) q^{80} +(-2.47229 - 4.28212i) q^{82} -5.05543 q^{83} +11.8444 q^{85} +(-5.75654 - 9.97063i) q^{86} +(0.500000 - 0.866025i) q^{88} +(-0.386770 - 0.669906i) q^{89} -3.55780 q^{92} +(-3.39213 + 5.87534i) q^{94} +(-0.944570 + 1.63604i) q^{95} +7.00000 q^{97} +(-6.36977 - 2.90275i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8}+O(q^{10}) $ 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8	$ 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8} - 3 q^{11} + 3 q^{14} - 3 q^{16} + 3 q^{17} + 9 q^{19} - 6 q^{22} + 3 q^{23} - 15 q^{25} + 3 q^{28} - 18 q^{29} + 6 q^{31} + 3 q^{32} + 6 q^{34} + 30 q^{35} - 21 q^{37} - 9 q^{38} - 24 q^{41} + 6 q^{43} - 3 q^{44} - 3 q^{46} + 3 q^{47} - 12 q^{49} - 30 q^{50} + 24 q^{53} - 9 q^{58} - 9 q^{59} + 6 q^{61} + 12 q^{62} + 6 q^{64} - 6 q^{67} + 3 q^{68} - 18 q^{71} + 21 q^{74} - 18 q^{76} + 3 q^{77} - 12 q^{79} - 12 q^{82} - 36 q^{83} + 3 q^{86} + 3 q^{88} - 12 q^{89} - 6 q^{92} - 3 q^{94} + 42 q^{97} + 9 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 - 3 * q^11 + 3 * q^14 - 3 * q^16 + 3 * q^17 + 9 * q^19 - 6 * q^22 + 3 * q^23 - 15 * q^25 + 3 * q^28 - 18 * q^29 + 6 * q^31 + 3 * q^32 + 6 * q^34 + 30 * q^35 - 21 * q^37 - 9 * q^38 - 24 * q^41 + 6 * q^43 - 3 * q^44 - 3 * q^46 + 3 * q^47 - 12 * q^49 - 30 * q^50 + 24 * q^53 - 9 * q^58 - 9 * q^59 + 6 * q^61 + 12 * q^62 + 6 * q^64 - 6 * q^67 + 3 * q^68 - 18 * q^71 + 21 * q^74 - 18 * q^76 + 3 * q^77 - 12 * q^79 - 12 * q^82 - 36 * q^83 + 3 * q^86 + 3 * q^88 - 12 * q^89 - 6 * q^92 - 3 * q^94 + 42 * q^97 + 9 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−0.806615	−	1.39710i	−0.360729	−	0.624801i	0.627352	−	0.778736i	$-0.284139\pi$
$5$	−0.806615	−	1.39710i	−0.360729	−	0.624801i	−0.988081	+	0.153935i	$0.950805\pi$
$6$		0			0
$7$	−0.806615	−	2.51980i	−0.304872	−	0.952393i
$7$	−0.806615	−	2.51980i	−0.304872	−	0.952393i
$8$	−1.00000			−0.353553
$9$		0			0
$10$	0.806615	−	1.39710i	0.255074	−	0.441801i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	1.77890	−	1.95845i	0.475431	−	0.523417i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−3.67103	+	6.35841i	−0.890356	+	1.54214i	−0.0509059	+	0.998703i	$0.516211\pi$
$17$	−3.67103	+	6.35841i	−0.890356	+	1.54214i	−0.839450	+	0.543438i	$0.817122\pi$
$18$		0			0
$19$	−0.585515	−	1.01414i	−0.134326	−	0.232660i	0.791014	−	0.611799i	$-0.209554\pi$
$19$	−0.585515	−	1.01414i	−0.134326	−	0.232660i	−0.925340	+	0.379139i	$0.876220\pi$
$20$	1.61323			0.360729
$21$		0			0
$22$	−1.00000			−0.213201
$23$	1.77890	+	3.08115i	0.370926	+	0.642463i	0.989708	−	0.143100i	$-0.0457069\pi$
$23$	1.77890	+	3.08115i	0.370926	+	0.642463i	−0.618782	+	0.785563i	$0.712374\pi$
$24$		0			0
$25$	1.19874	−	2.07629i	0.239749	−	0.415257i
$26$		0			0
$27$		0			0
$28$	2.58551	+	0.561349i	0.488616	+	0.106085i
$29$	−10.3975			−1.93077			−0.965383	−	0.260838i	$-0.916001\pi$
$29$	−10.3975			−1.93077			−0.965383	+	0.260838i	$0.916001\pi$
$30$		0			0
$31$	−3.17103	+	5.49238i	−0.569534	+	0.986461i	0.427078	+	0.904215i	$0.359543\pi$
$31$	−3.17103	+	5.49238i	−0.569534	+	0.986461i	−0.996612	+	0.0822467i	$0.973790\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$	−7.34206			−1.25915
$35$	−2.86977	+	3.15943i	−0.485080	+	0.534040i
$36$		0			0
$37$	−1.41449	−	2.44996i	−0.232540	−	0.402771i	0.726015	−	0.687679i	$-0.241370\pi$
$37$	−1.41449	−	2.44996i	−0.232540	−	0.402771i	−0.958555	+	0.284908i	$0.908037\pi$
$38$	0.585515	−	1.01414i	0.0949831	−	0.164515i
$39$		0			0
$40$	0.806615	+	1.39710i	0.127537	+	0.220901i
$41$	−4.94457			−0.772212			−0.386106	−	0.922454i	$-0.626180\pi$
$41$	−4.94457			−0.772212			−0.386106	+	0.922454i	$0.626180\pi$
$42$		0			0
$43$	−11.5131			−1.75573			−0.877865	−	0.478908i	$-0.841033\pi$
$43$	−11.5131			−1.75573			−0.877865	+	0.478908i	$0.841033\pi$
$44$	−0.500000	−	0.866025i	−0.0753778	−	0.130558i
$45$		0			0
$46$	−1.77890	+	3.08115i	−0.262284	+	0.454290i
$47$	3.39213	+	5.87534i	0.494793	+	0.857007i	0.999982	−	0.00600214i	$-0.00191055\pi$
$47$	3.39213	+	5.87534i	0.494793	+	0.857007i	−0.505189	+	0.863009i	$0.668577\pi$
$48$		0			0
$49$	−5.69874	+	4.06501i	−0.814106	+	0.580716i
$50$	2.39749			0.339056
$51$		0			0
$52$		0			0
$53$	4.00000	−	6.92820i	0.549442	−	0.951662i	−0.448871	−	0.893597i	$-0.648174\pi$
$53$	4.00000	−	6.92820i	0.549442	−	0.951662i	0.998313	−	0.0580651i	$-0.0184931\pi$
$54$		0			0
$55$	1.61323			0.217528
$56$	0.806615	+	2.51980i	0.107788	+	0.336722i
$57$		0			0
$58$	−5.19874	−	9.00449i	−0.682629	−	1.18235i
$59$	2.19874	−	3.80834i	0.286252	−	0.495803i	−0.686660	−	0.726979i	$-0.740924\pi$
$59$	2.19874	−	3.80834i	0.286252	−	0.495803i	0.972912	+	0.231175i	$0.0742572\pi$
$60$		0			0
$61$	5.97764	+	10.3536i	0.765359	+	1.32564i	0.940057	+	0.341019i	$0.110772\pi$
$61$	5.97764	+	10.3536i	0.765359	+	1.32564i	−0.174698	+	0.984622i	$0.555895\pi$
$62$	−6.34206			−0.805442
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−1.47229	+	2.55007i	−0.179868	+	0.311541i	−0.941835	−	0.336075i	$-0.890900\pi$
$67$	−1.47229	+	2.55007i	−0.179868	+	0.311541i	0.761967	+	0.647616i	$0.224234\pi$
$68$	−3.67103	−	6.35841i	−0.445178	−	0.771070i
$69$		0			0
$70$	−4.17103	−	0.905585i	−0.498533	−	0.108238i
$71$	−15.5131			−1.84106			−0.920532	−	0.390666i	$-0.872245\pi$
$71$	−15.5131			−1.84106			−0.920532	+	0.390666i	$0.872245\pi$
$72$		0			0
$73$	2.55780	−	4.43024i	0.299368	−	0.518520i	−0.676624	−	0.736329i	$-0.736558\pi$
$73$	2.55780	−	4.43024i	0.299368	−	0.518520i	0.975991	+	0.217809i	$0.0698909\pi$
$74$	1.41449	−	2.44996i	0.164431	−	0.284802i
$75$		0			0
$76$	1.17103			0.134326
$77$	2.58551	+	0.561349i	0.294647	+	0.0639717i
$78$		0			0
$79$	−2.80661	−	4.86120i	−0.315769	−	0.546928i	0.663832	−	0.747882i	$-0.268929\pi$
$79$	−2.80661	−	4.86120i	−0.315769	−	0.546928i	−0.979601	+	0.200954i	$0.935596\pi$
$80$	−0.806615	+	1.39710i	−0.0901823	+	0.156200i
$81$		0			0
$82$	−2.47229	−	4.28212i	−0.273018	−	0.472881i
$83$	−5.05543			−0.554906			−0.277453	−	0.960739i	$-0.589490\pi$
$83$	−5.05543			−0.554906			−0.277453	+	0.960739i	$0.589490\pi$
$84$		0			0
$85$	11.8444			1.28471
$86$	−5.75654	−	9.97063i	−0.620744	−	1.07516i
$87$		0			0
$88$	0.500000	−	0.866025i	0.0533002	−	0.0923186i
$89$	−0.386770	−	0.669906i	−0.0409976	−	0.0710098i	0.844799	−	0.535085i	$-0.179720\pi$
$89$	−0.386770	−	0.669906i	−0.0409976	−	0.0710098i	−0.885796	+	0.464075i	$0.846387\pi$
$90$		0			0
$91$		0			0
$92$	−3.55780			−0.370926
$93$		0			0
$94$	−3.39213	+	5.87534i	−0.349871	+	0.605995i
$95$	−0.944570	+	1.63604i	−0.0969108	+	0.167855i
$96$		0			0
$97$	7.00000			0.710742			0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000			0.710742			0.355371	+	0.934725i	$0.384354\pi$
$98$	−6.36977	−	2.90275i	−0.643444	−	0.293222i
$99$		0			0
$100$	1.19874	+	2.07629i	0.119874	+	0.207629i
$101$	6.81197	−	11.7987i	0.677817	−	1.17401i	−0.297820	−	0.954622i	$-0.596260\pi$
$101$	6.81197	−	11.7987i	0.677817	−	1.17401i	0.975637	−	0.219391i	$-0.0704071\pi$
$102$		0			0
$103$	6.17103	+	10.6885i	0.608050	+	1.05317i	0.991562	+	0.129637i	$0.0413812\pi$
$103$	6.17103	+	10.6885i	0.608050	+	1.05317i	−0.383512	+	0.923536i	$0.625285\pi$
$104$		0			0
$105$		0			0
$106$	8.00000			0.777029
$107$	−7.86977	−	13.6308i	−0.760800	−	1.31774i	−0.942439	−	0.334379i	$-0.891474\pi$
$107$	−7.86977	−	13.6308i	−0.760800	−	1.31774i	0.181639	−	0.983365i	$-0.441860\pi$
$108$		0			0
$109$	−1.97764	+	3.42538i	−0.189424	+	0.328092i	−0.945058	−	0.326902i	$-0.893995\pi$
$109$	−1.97764	+	3.42538i	−0.189424	+	0.328092i	0.755634	+	0.654994i	$0.227329\pi$
$110$	0.806615	+	1.39710i	0.0769077	+	0.133208i
$111$		0			0
$112$	−1.77890	+	1.95845i	−0.168090	+	0.185056i
$113$	4.34206			0.408467			0.204233	−	0.978922i	$-0.434530\pi$
$113$	4.34206			0.408467			0.204233	+	0.978922i	$0.434530\pi$
$114$		0			0
$115$	2.86977	−	4.97060i	0.267608	−	0.463510i
$116$	5.19874	−	9.00449i	0.482691	−	0.836046i
$117$		0			0
$118$	4.39749			0.404822
$119$	18.9830	+	4.12146i	1.74017	+	0.377813i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$	−5.97764	+	10.3536i	−0.541191	+	0.937369i
$123$		0			0
$124$	−3.17103	−	5.49238i	−0.284767	−	0.493231i
$125$	−11.9339			−1.06740
$126$		0			0
$127$	−0.442200			−0.0392389			−0.0196195	−	0.999808i	$-0.506245\pi$
$127$	−0.442200			−0.0392389			−0.0196195	+	0.999808i	$0.506245\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$		0			0
$131$	−1.17103	−	2.02828i	−0.102313	−	0.177212i	0.810324	−	0.585982i	$-0.199291\pi$
$131$	−1.17103	−	2.02828i	−0.102313	−	0.177212i	−0.912637	+	0.408770i	$0.865958\pi$
$132$		0			0
$133$	−2.08314	+	2.29340i	−0.180632	+	0.198863i
$134$	−2.94457			−0.254372
$135$		0			0
$136$	3.67103	−	6.35841i	0.314788	−	0.545229i
$137$	1.61323	−	2.79420i	0.137828	−	0.238724i	−0.788847	−	0.614590i	$-0.789321\pi$
$137$	1.61323	−	2.79420i	0.137828	−	0.238724i	0.926674	+	0.375866i	$0.122655\pi$
$138$		0			0
$139$	18.3975			1.56045			0.780227	−	0.625496i	$-0.215103\pi$
$139$	18.3975			1.56045			0.780227	+	0.625496i	$0.215103\pi$
$140$	−1.30126	−	4.06501i	−0.109976	−	0.343556i
$141$		0			0
$142$	−7.75654	−	13.4347i	−0.650915	−	1.12742i
$143$		0			0
$144$		0			0
$145$	8.38677	+	14.5263i	0.696483	+	1.20634i
$146$	5.11560			0.423370
$147$		0			0
$148$	2.82897			0.232540
$149$	9.81197	+	16.9948i	0.803828	+	1.39227i	0.917079	+	0.398705i	$0.130540\pi$
$149$	9.81197	+	16.9948i	0.803828	+	1.39227i	−0.113251	+	0.993566i	$0.536126\pi$
$150$		0			0
$151$	0.165670	−	0.286949i	0.0134820	−	0.0233516i	−0.859206	−	0.511630i	$-0.829042\pi$
$151$	0.165670	−	0.286949i	0.0134820	−	0.0233516i	0.872688	+	0.488279i	$0.162375\pi$
$152$	0.585515	+	1.01414i	0.0474915	+	0.0822577i
$153$		0			0
$154$	0.806615	+	2.51980i	0.0649989	+	0.203051i
$155$	10.2312			0.821790
$156$		0			0
$157$	−9.98300	+	17.2911i	−0.796730	+	1.37998i	0.125004	+	0.992156i	$0.460106\pi$
$157$	−9.98300	+	17.2911i	−0.796730	+	1.37998i	−0.921735	+	0.387821i	$0.873228\pi$
$158$	2.80661	−	4.86120i	0.223282	−	0.386736i
$159$		0			0
$160$	−1.61323			−0.127537
$161$	6.32897	−	6.96776i	0.498793	−	0.549137i
$162$		0			0
$163$	−10.6433	−	18.4348i	−0.833649	−	1.44392i	−0.895126	−	0.445814i	$-0.852914\pi$
$163$	−10.6433	−	18.4348i	−0.833649	−	1.44392i	0.0614771	−	0.998108i	$-0.480419\pi$
$164$	2.47229	−	4.28212i	0.193053	−	0.334378i
$165$		0			0
$166$	−2.52771	−	4.37813i	−0.196189	−	0.339809i
$167$	−22.0214			−1.70407			−0.852035	−	0.523485i	$-0.824632\pi$
$167$	−22.0214			−1.70407			−0.852035	+	0.523485i	$0.824632\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$	5.92221	+	10.2576i	0.454213	+	0.786720i
$171$		0			0
$172$	5.75654	−	9.97063i	0.438932	−	0.760253i
$173$	0.828970	+	1.43582i	0.0630254	+	0.109163i	0.895816	−	0.444424i	$-0.146592\pi$
$173$	0.828970	+	1.43582i	0.0630254	+	0.109163i	−0.832791	+	0.553588i	$0.813258\pi$
$174$		0			0
$175$	−6.19874	−	1.34583i	−0.468581	−	0.101735i
$176$	1.00000			0.0753778
$177$		0			0
$178$	0.386770	−	0.669906i	0.0289896	−	0.0502115i
$179$	7.53008	−	13.0425i	0.562825	−	0.974841i	−0.434423	−	0.900709i	$-0.643048\pi$
$179$	7.53008	−	13.0425i	0.562825	−	0.974841i	0.997248	−	0.0741327i	$-0.0236188\pi$
$180$		0			0
$181$	15.1156			1.12353			0.561767	−	0.827296i	$-0.310122\pi$
$181$	15.1156			1.12353			0.561767	+	0.827296i	$0.310122\pi$
$182$		0			0
$183$		0			0
$184$	−1.77890	−	3.08115i	−0.131142	−	0.227145i
$185$	−2.28189	+	3.95235i	−0.167768	+	0.290582i
$186$		0			0
$187$	−3.67103	−	6.35841i	−0.268452	−	0.464973i
$188$	−6.78426			−0.494793
$189$		0			0
$190$	−1.88914			−0.137053
$191$	−2.38677	−	4.13401i	−0.172701	−	0.299126i	0.766663	−	0.642050i	$-0.221916\pi$
$191$	−2.38677	−	4.13401i	−0.172701	−	0.299126i	−0.939363	+	0.342924i	$0.888583\pi$
$192$		0			0
$193$	−1.55780	+	2.69819i	−0.112133	+	0.194220i	−0.916630	−	0.399737i	$-0.869102\pi$
$193$	−1.55780	+	2.69819i	−0.112133	+	0.194220i	0.804497	+	0.593957i	$0.202435\pi$
$194$	3.50000	+	6.06218i	0.251285	+	0.435239i
$195$		0			0
$196$	−0.671030	−	6.96776i	−0.0479307	−	0.497697i
$197$	4.82897			0.344050			0.172025	−	0.985093i	$-0.444969\pi$
$197$	4.82897			0.344050			0.172025	+	0.985093i	$0.444969\pi$
$198$		0			0
$199$	−2.94457	+	5.10015i	−0.208735	+	0.361540i	−0.951316	−	0.308216i	$-0.900268\pi$
$199$	−2.94457	+	5.10015i	−0.208735	+	0.361540i	0.742581	+	0.669756i	$0.233601\pi$
$200$	−1.19874	+	2.07629i	−0.0847641	+	0.146816i
$201$		0			0
$202$	13.6239			0.958578
$203$	8.38677	+	26.1996i	0.588636	+	1.83885i
$204$		0			0
$205$	3.98836	+	6.90805i	0.278559	+	0.482479i
$206$	−6.17103	+	10.6885i	−0.429956	+	0.744706i
$207$		0			0
$208$		0			0
$209$	1.17103			0.0810018
$210$		0			0
$211$	−13.6794			−0.941727			−0.470864	−	0.882206i	$-0.656058\pi$
$211$	−13.6794			−0.941727			−0.470864	+	0.882206i	$0.656058\pi$
$212$	4.00000	+	6.92820i	0.274721	+	0.475831i
$213$		0			0
$214$	7.86977	−	13.6308i	0.537967	−	0.931786i
$215$	9.28663	+	16.0849i	0.633343	+	1.09698i
$216$		0			0
$217$	16.3975	+	3.56011i	1.11313	+	0.241676i
$218$	−3.95529			−0.267886
$219$		0			0
$220$	−0.806615	+	1.39710i	−0.0543820	+	0.0941923i
$221$		0			0
$222$		0			0
$223$	−5.11560			−0.342566			−0.171283	−	0.985222i	$-0.554791\pi$
$223$	−5.11560			−0.342566			−0.171283	+	0.985222i	$0.554791\pi$
$224$	−2.58551	−	0.561349i	−0.172752	−	0.0375067i
$225$		0			0
$226$	2.17103	+	3.76033i	0.144415	+	0.250134i
$227$	−2.64331	+	4.57836i	−0.175443	+	0.303876i	−0.940314	−	0.340307i	$-0.889469\pi$
$227$	−2.64331	+	4.57836i	−0.175443	+	0.303876i	0.764872	+	0.644183i	$0.222802\pi$
$228$		0			0
$229$	2.44220	+	4.23001i	0.161385	+	0.279527i	0.935366	−	0.353682i	$-0.115071\pi$
$229$	2.44220	+	4.23001i	0.161385	+	0.279527i	−0.773981	+	0.633209i	$0.781737\pi$
$230$	5.73955			0.378455
$231$		0			0
$232$	10.3975			0.682629
$233$	4.89749	+	8.48270i	0.320845	+	0.555720i	0.980663	−	0.195706i	$-0.0626997\pi$
$233$	4.89749	+	8.48270i	0.320845	+	0.555720i	−0.659817	+	0.751426i	$0.729366\pi$
$234$		0			0
$235$	5.47229	−	9.47828i	0.356973	−	0.618295i
$236$	2.19874	+	3.80834i	0.143126	+	0.247902i
$237$		0			0
$238$	5.92221	+	18.5005i	0.383880	+	1.19921i
$239$	23.9106			1.54665			0.773323	−	0.634012i	$-0.218593\pi$
$239$	23.9106			1.54665			0.773323	+	0.634012i	$0.218593\pi$
$240$		0			0
$241$	−0.944570	+	1.63604i	−0.0608451	+	0.105387i	−0.894843	−	0.446380i	$-0.852713\pi$
$241$	−0.944570	+	1.63604i	−0.0608451	+	0.105387i	0.833998	+	0.551767i	$0.186046\pi$
$242$	0.500000	−	0.866025i	0.0321412	−	0.0556702i
$243$		0			0
$244$	−11.9553			−0.765359
$245$	10.2759	+	4.68281i	0.656504	+	0.299174i
$246$		0			0
$247$		0			0
$248$	3.17103	−	5.49238i	0.201361	−	0.348767i
$249$		0			0
$250$	−5.96693	−	10.3350i	−0.377381	−	0.653644i
$251$	−0.939830			−0.0593215			−0.0296608	−	0.999560i	$-0.509443\pi$
$251$	−0.939830			−0.0593215			−0.0296608	+	0.999560i	$0.509443\pi$
$252$		0			0
$253$	−3.55780			−0.223677
$254$	−0.221100	−	0.382956i	−0.0138730	−	0.0240288i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−5.61323	−	9.72240i	−0.350144	−	0.606467i	0.636131	−	0.771581i	$-0.280534\pi$
$257$	−5.61323	−	9.72240i	−0.350144	−	0.606467i	−0.986274	+	0.165115i	$0.947201\pi$
$258$		0			0
$259$	−5.03246	+	5.54039i	−0.312702	+	0.344263i
$260$		0			0
$261$		0			0
$262$	1.17103	−	2.02828i	0.0723465	−	0.125308i
$263$	−0.613230	+	1.06215i	−0.0378134	+	0.0654947i	−0.884313	−	0.466895i	$-0.845373\pi$
$263$	−0.613230	+	1.06215i	−0.0378134	+	0.0654947i	0.846499	+	0.532390i	$0.178706\pi$
$264$		0			0
$265$	−12.9058			−0.792799
$266$	−3.02771	−	0.657356i	−0.185641	−	0.0403051i
$267$		0			0
$268$	−1.47229	−	2.55007i	−0.0899341	−	0.155770i
$269$	4.75119	−	8.22929i	0.289685	−	0.501749i	−0.684050	−	0.729435i	$-0.739783\pi$
$269$	4.75119	−	8.22929i	0.289685	−	0.501749i	0.973734	+	0.227687i	$0.0731162\pi$
$270$		0			0
$271$	−1.61323	−	2.79420i	−0.0979967	−	0.169735i	0.812859	−	0.582461i	$-0.197910\pi$
$271$	−1.61323	−	2.79420i	−0.0979967	−	0.169735i	−0.910855	+	0.412726i	$0.864577\pi$
$272$	7.34206			0.445178
$273$		0			0
$274$	3.22646			0.194918
$275$	1.19874	+	2.07629i	0.0722870	+	0.125205i
$276$		0			0
$277$	8.00000	−	13.8564i	0.480673	−	0.832551i	−0.519081	−	0.854725i	$-0.673726\pi$
$277$	8.00000	−	13.8564i	0.480673	−	0.832551i	0.999754	+	0.0221745i	$0.00705893\pi$
$278$	9.19874	+	15.9327i	0.551704	+	0.955579i
$279$		0			0
$280$	2.86977	−	3.15943i	0.171502	−	0.188812i
$281$	11.2312			0.669997			0.334999	−	0.942219i	$-0.391264\pi$
$281$	11.2312			0.669997			0.334999	+	0.942219i	$0.391264\pi$
$282$		0			0
$283$	12.7288	−	22.0470i	0.756650	−	1.31056i	−0.187899	−	0.982188i	$-0.560168\pi$
$283$	12.7288	−	22.0470i	0.756650	−	1.31056i	0.944550	−	0.328369i	$-0.106499\pi$
$284$	7.75654	−	13.4347i	0.460266	−	0.797205i
$285$		0			0
$286$		0			0
$287$	3.98836	+	12.4593i	0.235426	+	0.735450i
$288$		0			0
$289$	−18.4529	−	31.9614i	−1.08547	−	1.88008i
$290$	−8.38677	+	14.5263i	−0.492488	+	0.853014i
$291$		0			0
$292$	2.55780	+	4.43024i	0.149684	+	0.259260i
$293$	−17.9446			−1.04833			−0.524166	−	0.851616i	$-0.675623\pi$
$293$	−17.9446			−1.04833			−0.524166	+	0.851616i	$0.675623\pi$
$294$		0			0
$295$	−7.09416			−0.413038
$296$	1.41449	+	2.44996i	0.0822153	+	0.142401i
$297$		0			0
$298$	−9.81197	+	16.9948i	−0.568392	+	0.984485i
$299$		0			0
$300$		0			0
$301$	9.28663	+	29.0106i	0.535272	+	1.67215i
$302$	0.331340			0.0190665
$303$		0			0
$304$	−0.585515	+	1.01414i	−0.0335816	+	0.0581650i
$305$	9.64331	−	16.7027i	0.552175	−	0.956394i
$306$		0			0
$307$	−33.0262			−1.88490			−0.942452	−	0.334342i	$-0.891486\pi$
$307$	−33.0262			−1.88490			−0.942452	+	0.334342i	$0.891486\pi$
$308$	−1.77890	+	1.95845i	−0.101362	+	0.111593i
$309$		0			0
$310$	5.11560	+	8.86048i	0.290547	+	0.503241i
$311$	−0.165670	+	0.286949i	−0.00939429	+	0.0162714i	−0.870684	−	0.491842i	$-0.836324\pi$
$311$	−0.165670	+	0.286949i	−0.00939429	+	0.0162714i	0.861290	+	0.508114i	$0.169657\pi$
$312$		0			0
$313$	7.19874	+	12.4686i	0.406897	+	0.704766i	0.994540	−	0.104354i	$-0.0332774\pi$
$313$	7.19874	+	12.4686i	0.406897	+	0.704766i	−0.587643	+	0.809120i	$0.699944\pi$
$314$	−19.9660			−1.12675
$315$		0			0
$316$	5.61323			0.315769
$317$	8.76190	+	15.1761i	0.492118	+	0.852373i	0.999959	−	0.00907805i	$-0.00288967\pi$
$317$	8.76190	+	15.1761i	0.492118	+	0.852373i	−0.507841	+	0.861451i	$0.669556\pi$
$318$		0			0
$319$	5.19874	−	9.00449i	0.291074	−	0.504155i
$320$	−0.806615	−	1.39710i	−0.0450911	−	0.0781002i
$321$		0			0
$322$	9.19874	+	1.99717i	0.512626	+	0.111298i
$323$	8.59777			0.478393
$324$		0			0
$325$		0			0
$326$	10.6433	−	18.4348i	0.589479	−	1.02101i
$327$		0			0
$328$	4.94457			0.273018
$329$	12.0685	−	13.2866i	0.665359	−	0.732515i
$330$		0			0
$331$	4.47229	+	7.74622i	0.245819	+	0.425771i	0.962362	−	0.271772i	$-0.0876098\pi$
$331$	4.47229	+	7.74622i	0.245819	+	0.425771i	−0.716543	+	0.697543i	$0.754276\pi$
$332$	2.52771	−	4.37813i	0.138726	−	0.240281i
$333$		0			0
$334$	−11.0107	−	19.0711i	−0.602480	−	1.04353i
$335$	4.75027			0.259535
$336$		0			0
$337$	−3.33732			−0.181795			−0.0908977	−	0.995860i	$-0.528974\pi$
$337$	−3.33732			−0.181795			−0.0908977	+	0.995860i	$0.528974\pi$
$338$	−6.50000	−	11.2583i	−0.353553	−	0.612372i
$339$		0			0
$340$	−5.92221	+	10.2576i	−0.321177	+	0.556295i
$341$	−3.17103	−	5.49238i	−0.171721	−	0.297429i
$342$		0			0
$343$	14.8397	+	11.0808i	0.801268	+	0.598306i
$344$	11.5131			0.620744
$345$		0			0
$346$	−0.828970	+	1.43582i	−0.0445657	+	0.0771901i
$347$	−15.2118	+	26.3477i	−0.816614	+	1.41442i	0.0915491	+	0.995801i	$0.470818\pi$
$347$	−15.2118	+	26.3477i	−0.816614	+	1.41442i	−0.908163	+	0.418616i	$0.862515\pi$
$348$		0			0
$349$	17.5238			0.938028			0.469014	−	0.883191i	$-0.344609\pi$
$349$	17.5238			0.938028			0.469014	+	0.883191i	$0.344609\pi$
$350$	−1.93385	−	6.04118i	−0.103369	−	0.322915i
$351$		0			0
$352$	0.500000	+	0.866025i	0.0266501	+	0.0461593i
$353$	2.55780	−	4.43024i	0.136138	−	0.235798i	−0.789894	−	0.613244i	$-0.789864\pi$
$353$	2.55780	−	4.43024i	0.136138	−	0.235798i	0.926032	+	0.377446i	$0.123198\pi$
$354$		0			0
$355$	12.5131	+	21.6733i	0.664126	+	1.15030i
$356$	0.773540			0.0409976
$357$		0			0
$358$	15.0602			0.795955
$359$	−15.3421	−	26.5732i	−0.809723	−	1.40248i	−0.913056	−	0.407834i	$-0.866284\pi$
$359$	−15.3421	−	26.5732i	−0.809723	−	1.40248i	0.103333	−	0.994647i	$-0.467049\pi$
$360$		0			0
$361$	8.81434	−	15.2669i	0.463913	−	0.803521i
$362$	7.55780	+	13.0905i	0.397229	+	0.688021i
$363$		0			0
$364$		0			0
$365$	−8.25264			−0.431963
$366$		0			0
$367$	−0.728830	+	1.26237i	−0.0380446	+	0.0658952i	−0.884421	−	0.466690i	$-0.845446\pi$
$367$	−0.728830	+	1.26237i	−0.0380446	+	0.0658952i	0.846376	+	0.532586i	$0.178780\pi$
$368$	1.77890	−	3.08115i	0.0927316	−	0.160616i
$369$		0			0
$370$	−4.56378			−0.237260
$371$	−20.6841	−	4.49079i	−1.07387	−	0.233150i
$372$		0			0
$373$	17.0932	+	29.6064i	0.885055	+	1.53296i	0.845651	+	0.533736i	$0.179212\pi$
$373$	17.0932	+	29.6064i	0.885055	+	1.53296i	0.0394037	+	0.999223i	$0.487454\pi$
$374$	3.67103	−	6.35841i	0.189824	−	0.328786i
$375$		0			0
$376$	−3.39213	−	5.87534i	−0.174936	−	0.302998i
$377$		0			0
$378$		0			0
$379$	−9.28663			−0.477022			−0.238511	−	0.971140i	$-0.576659\pi$
$379$	−9.28663			−0.477022			−0.238511	+	0.971140i	$0.576659\pi$
$380$	−0.944570	−	1.63604i	−0.0484554	−	0.0839273i
$381$		0			0
$382$	2.38677	−	4.13401i	0.122118	−	0.211514i
$383$	−1.81197	−	3.13843i	−0.0925876	−	0.160366i	0.816012	−	0.578035i	$-0.196180\pi$
$383$	−1.81197	−	3.13843i	−0.0925876	−	0.160366i	−0.908599	+	0.417669i	$0.862847\pi$
$384$		0			0
$385$	−1.30126	−	4.06501i	−0.0663181	−	0.207172i
$386$	−3.11560			−0.158580
$387$		0			0
$388$	−3.50000	+	6.06218i	−0.177686	+	0.307760i
$389$	3.58016	−	6.20101i	0.181521	−	0.314404i	−0.760878	−	0.648895i	$-0.775231\pi$
$389$	3.58016	−	6.20101i	0.181521	−	0.314404i	0.942399	+	0.334492i	$0.108565\pi$
$390$		0			0
$391$	−26.1216			−1.32103
$392$	5.69874	−	4.06501i	0.287830	−	0.205314i
$393$		0			0
$394$	2.41449	+	4.18201i	0.121640	+	0.210687i
$395$	−4.52771	+	7.84223i	−0.227814	+	0.394586i
$396$		0			0
$397$	15.1987	+	26.3250i	0.762803	+	1.32121i	0.941400	+	0.337291i	$0.109511\pi$
$397$	15.1987	+	26.3250i	0.762803	+	1.32121i	−0.178597	+	0.983922i	$0.557156\pi$
$398$	−5.88914			−0.295196
$399$		0			0
$400$	−2.39749			−0.119874
$401$	−6.01072	−	10.4109i	−0.300161	−	0.519894i	0.676011	−	0.736891i	$-0.263707\pi$
$401$	−6.01072	−	10.4109i	−0.300161	−	0.519894i	−0.976172	+	0.216997i	$0.930374\pi$
$402$		0			0
$403$		0			0
$404$	6.81197	+	11.7987i	0.338908	+	0.587007i
$405$		0			0
$406$	−18.4961	+	20.3629i	−0.917946	+	1.01060i
$407$	2.82897			0.140227
$408$		0			0
$409$	−8.78426	+	15.2148i	−0.434354	+	0.752323i	−0.997243	−	0.0742099i	$-0.976357\pi$
$409$	−8.78426	+	15.2148i	−0.434354	+	0.752323i	0.562889	+	0.826533i	$0.309690\pi$
$410$	−3.98836	+	6.90805i	−0.196971	+	0.341164i
$411$		0			0
$412$	−12.3421			−0.608050
$413$	−11.3698	−	2.46853i	−0.559470	−	0.121468i
$414$		0			0
$415$	4.07779	+	7.06293i	0.200171	+	0.346706i
$416$		0			0
$417$		0			0
$418$	0.585515	+	1.01414i	0.0286385	+	0.0496033i
$419$	27.1710			1.32739			0.663696	−	0.748003i	$-0.268987\pi$
$419$	27.1710			1.32739			0.663696	+	0.748003i	$0.268987\pi$
$420$		0			0
$421$	23.1710			1.12929			0.564643	−	0.825335i	$-0.309014\pi$
$421$	23.1710			1.12929			0.564643	+	0.825335i	$0.309014\pi$
$422$	−6.83969	−	11.8467i	−0.332951	−	0.576688i
$423$		0			0
$424$	−4.00000	+	6.92820i	−0.194257	+	0.336463i
$425$	8.80126	+	15.2442i	0.426924	+	0.739453i
$426$		0			0
$427$	21.2673	−	23.4138i	1.02920	−	1.13307i
$428$	15.7395			0.760800
$429$		0			0
$430$	−9.28663	+	16.0849i	−0.447841	+	0.775683i
$431$	−8.38677	+	14.5263i	−0.403977	+	0.699708i	−0.994202	−	0.107531i	$-0.965706\pi$
$431$	−8.38677	+	14.5263i	−0.403977	+	0.699708i	0.590225	+	0.807239i	$0.299039\pi$
$432$		0			0
$433$	17.3421			0.833406			0.416703	−	0.909043i	$-0.363185\pi$
$433$	17.3421			0.833406			0.416703	+	0.909043i	$0.363185\pi$
$434$	5.11560	+	15.9807i	0.245557	+	0.767098i
$435$		0			0
$436$	−1.97764	−	3.42538i	−0.0947120	−	0.164046i
$437$	2.08314	−	3.60811i	0.0996503	−	0.172599i
$438$		0			0
$439$	−16.5739	−	28.7068i	−0.791028	−	1.37010i	−0.925331	−	0.379160i	$-0.876213\pi$
$439$	−16.5739	−	28.7068i	−0.791028	−	1.37010i	0.134303	−	0.990940i	$-0.457121\pi$
$440$	−1.61323			−0.0769077
$441$		0			0
$442$		0			0
$443$	−6.02771	−	10.4403i	−0.286385	−	0.496034i	0.686559	−	0.727074i	$-0.259120\pi$
$443$	−6.02771	−	10.4403i	−0.286385	−	0.496034i	−0.972944	+	0.231040i	$0.925787\pi$
$444$		0			0
$445$	−0.623949	+	1.08071i	−0.0295780	+	0.0512306i
$446$	−2.55780	−	4.43024i	−0.121115	−	0.209778i
$447$		0			0
$448$	−0.806615	−	2.51980i	−0.0381090	−	0.119049i
$449$	36.5947			1.72701			0.863505	−	0.504340i	$-0.168264\pi$
$449$	36.5947			1.72701			0.863505	+	0.504340i	$0.168264\pi$
$450$		0			0
$451$	2.47229	−	4.28212i	0.116415	−	0.201637i
$452$	−2.17103	+	3.76033i	−0.102117	+	0.176871i
$453$		0			0
$454$	−5.28663			−0.248114
$455$		0			0
$456$		0			0
$457$	−18.9106	−	32.7541i	−0.884600	−	1.53217i	−0.846172	−	0.532910i	$-0.821098\pi$
$457$	−18.9106	−	32.7541i	−0.884600	−	1.53217i	−0.0384276	−	0.999261i	$-0.512235\pi$
$458$	−2.44220	+	4.23001i	−0.114117	+	0.197656i
$459$		0			0
$460$	2.86977	+	4.97060i	0.133804	+	0.231755i
$461$	−34.9707			−1.62875			−0.814375	−	0.580339i	$-0.802920\pi$
$461$	−34.9707			−1.62875			−0.814375	+	0.580339i	$0.802920\pi$
$462$		0			0
$463$	7.56852			0.351739			0.175869	−	0.984413i	$-0.443726\pi$
$463$	7.56852			0.351739			0.175869	+	0.984413i	$0.443726\pi$
$464$	5.19874	+	9.00449i	0.241346	+	0.418023i
$465$		0			0
$466$	−4.89749	+	8.48270i	−0.226872	+	0.392954i
$467$	−7.31434	−	12.6688i	−0.338468	−	0.586243i	0.645677	−	0.763610i	$-0.276575\pi$
$467$	−7.31434	−	12.6688i	−0.338468	−	0.586243i	−0.984145	+	0.177368i	$0.943242\pi$
$468$		0			0
$469$	7.61323	+	1.65293i	0.351546	+	0.0763253i
$470$	10.9446			0.504835
$471$		0			0
$472$	−2.19874	+	3.80834i	−0.101205	+	0.175293i
$473$	5.75654	−	9.97063i	0.264686	−	0.458450i
$474$		0			0
$475$	−2.80753			−0.128818
$476$	−13.0608	+	14.3790i	−0.598640	+	0.659062i
$477$		0			0
$478$	11.9553	+	20.7072i	0.546822	+	0.947124i
$479$	−0.613230	+	1.06215i	−0.0280192	+	0.0485307i	−0.879695	−	0.475538i	$-0.842253\pi$
$479$	−0.613230	+	1.06215i	−0.0280192	+	0.0485307i	0.851676	+	0.524069i	$0.175587\pi$
$480$		0			0
$481$		0			0
$482$	−1.88914			−0.0860480
$483$		0			0
$484$	1.00000			0.0454545
$485$	−5.64630	−	9.77969i	−0.256385	−	0.444073i
$486$		0			0
$487$	−15.3528	+	26.5918i	−0.695701	+	1.20499i	0.274243	+	0.961660i	$0.411573\pi$
$487$	−15.3528	+	26.5918i	−0.695701	+	1.20499i	−0.969944	+	0.243329i	$0.921761\pi$
$488$	−5.97764	−	10.3536i	−0.270595	−	0.468685i
$489$		0			0
$490$	1.08253	+	11.2406i	0.0489035	+	0.507799i
$491$	−30.9446			−1.39651			−0.698254	−	0.715850i	$-0.746040\pi$
$491$	−30.9446			−1.39651			−0.698254	+	0.715850i	$0.746040\pi$
$492$		0			0
$493$	38.1695	−	66.1115i	1.71907	−	2.97751i
$494$		0			0
$495$		0			0
$496$	6.34206			0.284767
$497$	12.5131	+	39.0898i	0.561289	+	1.75342i
$498$		0			0
$499$	5.17103	+	8.95649i	0.231487	+	0.400947i	0.958246	−	0.285945i	$-0.0923076\pi$
$499$	5.17103	+	8.95649i	0.231487	+	0.400947i	−0.726759	+	0.686893i	$0.758974\pi$
$500$	5.96693	−	10.3350i	0.266849	−	0.462196i
$501$		0			0
$502$	−0.469915	−	0.813917i	−0.0209733	−	0.0363269i
$503$	−4.66268			−0.207899			−0.103949	−	0.994583i	$-0.533148\pi$
$503$	−4.66268			−0.207899			−0.103949	+	0.994583i	$0.533148\pi$
$504$		0			0
$505$	−21.9786			−0.978033
$506$	−1.77890	−	3.08115i	−0.0790817	−	0.136974i
$507$		0			0
$508$	0.221100	−	0.382956i	0.00980973	−	0.0169909i
$509$	12.7950	+	22.1616i	0.567127	+	0.982294i	0.996848	+	0.0793318i	$0.0252787\pi$
$509$	12.7950	+	22.1616i	0.567127	+	0.982294i	−0.429721	+	0.902962i	$0.641388\pi$
$510$		0			0
$511$	−13.2265	−	2.87164i	−0.585104	−	0.127034i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	5.61323	−	9.72240i	0.247589	−	0.428837i
$515$	9.95529	−	17.2431i	0.438682	−	0.759820i
$516$		0			0
$517$	−6.78426			−0.298371
$518$	−7.31434	−	1.58804i	−0.321374	−	0.0697745i
$519$		0			0
$520$		0			0
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	−0.993011	−	0.118020i	$-0.962345\pi$
$521$	−9.00000	+	15.5885i	−0.394297	+	0.682943i	0.598714	+	0.800963i	$0.295679\pi$
$522$		0			0
$523$	−15.1817	−	26.2956i	−0.663852	−	1.14982i	−0.979595	−	0.200980i	$-0.935587\pi$
$523$	−15.1817	−	26.2956i	−0.663852	−	1.14982i	0.315744	−	0.948844i	$-0.397746\pi$
$524$	2.34206			0.102313
$525$		0			0
$526$	−1.22646			−0.0534762
$527$	−23.2819	−	40.3254i	−1.01418	−	1.75660i
$528$		0			0
$529$	5.17103	−	8.95649i	0.224827	−	0.389412i
$530$	−6.45292	−	11.1768i	−0.280297	−	0.485488i
$531$		0			0
$532$	−0.944570	−	2.95076i	−0.0409523	−	0.127932i
$533$		0			0
$534$		0			0
$535$	−12.6958	+	21.9897i	−0.548885	+	0.950697i
$536$	1.47229	−	2.55007i	0.0635930	−	0.110146i
$537$		0			0
$538$	9.50237			0.409676
$539$	−0.671030	−	6.96776i	−0.0289033	−	0.300123i
$540$		0			0
$541$	−0.806615	−	1.39710i	−0.0346791	−	0.0600659i	0.848165	−	0.529732i	$-0.177708\pi$
$541$	−0.806615	−	1.39710i	−0.0346791	−	0.0600659i	−0.882844	+	0.469666i	$0.844374\pi$
$542$	1.61323	−	2.79420i	0.0692942	−	0.120021i
$543$		0			0
$544$	3.67103	+	6.35841i	0.157394	+	0.272615i
$545$	6.38079			0.273323
$546$		0			0
$547$	15.7348			0.672772			0.336386	−	0.941724i	$-0.390795\pi$
$547$	15.7348			0.672772			0.336386	+	0.941724i	$0.390795\pi$
$548$	1.61323	+	2.79420i	0.0689138	+	0.119362i
$549$		0			0
$550$	−1.19874	+	2.07629i	−0.0511146	+	0.0885332i
$551$	6.08788	+	10.5445i	0.259353	+	0.449212i
$552$		0			0
$553$	−9.98537	+	10.9932i	−0.424621	+	0.467479i
$554$	16.0000			0.679775
$555$		0			0
$556$	−9.19874	+	15.9327i	−0.390114	+	0.675697i
$557$	5.64094	−	9.77040i	0.239015	−	0.413985i	−0.721417	−	0.692501i	$-0.756509\pi$
$557$	5.64094	−	9.77040i	0.239015	−	0.413985i	0.960432	+	0.278515i	$0.0898423\pi$
$558$		0			0
$559$		0			0
$560$	4.17103	+	0.905585i	0.176258	+	0.0382680i
$561$		0			0
$562$	5.61560	+	9.72650i	0.236880	+	0.410288i
$563$	10.4529	−	18.1050i	0.440538	−	0.763034i	−0.557191	−	0.830384i	$-0.688121\pi$
$563$	10.4529	−	18.1050i	0.440538	−	0.763034i	0.997729	+	0.0673499i	$0.0214544\pi$
$564$		0			0
$565$	−3.50237	−	6.06628i	−0.147346	−	0.255210i
$566$	25.4577			1.07007
$567$		0			0
$568$	15.5131			0.650915
$569$	−3.91686	−	6.78419i	−0.164203	−	0.284408i	0.772169	−	0.635417i	$-0.219172\pi$
$569$	−3.91686	−	6.78419i	−0.164203	−	0.284408i	−0.936372	+	0.351009i	$0.885839\pi$
$570$		0			0
$571$	−1.19874	+	2.07629i	−0.0501659	+	0.0868899i	−0.890018	−	0.455926i	$-0.849308\pi$
$571$	−1.19874	+	2.07629i	−0.0501659	+	0.0868899i	0.839852	+	0.542815i	$0.182642\pi$
$572$		0			0
$573$		0			0
$574$	−8.79590	+	9.68368i	−0.367134	+	0.404189i
$575$	8.52979			0.355717
$576$		0			0
$577$	−12.8143	+	22.1951i	−0.533468	+	0.923994i	0.465768	+	0.884907i	$0.345778\pi$
$577$	−12.8143	+	22.1951i	−0.533468	+	0.923994i	−0.999236	+	0.0390869i	$0.987555\pi$
$578$	18.4529	−	31.9614i	0.767540	−	1.32942i
$579$		0			0
$580$	−16.7735			−0.696483
$581$	4.07779	+	12.7387i	0.169175	+	0.528488i
$582$		0			0
$583$	4.00000	+	6.92820i	0.165663	+	0.286937i
$584$	−2.55780	+	4.43024i	−0.105843	+	0.183325i
$585$		0			0
$586$	−8.97229	−	15.5405i	−0.370642	−	0.641970i
$587$	−5.79972			−0.239380			−0.119690	−	0.992811i	$-0.538190\pi$
$587$	−5.79972			−0.239380			−0.119690	+	0.992811i	$0.538190\pi$
$588$		0			0
$589$	7.42674			0.306014
$590$	−3.54708	−	6.14372i	−0.146031	−	0.252933i
$591$		0			0
$592$	−1.41449	+	2.44996i	−0.0581350	+	0.100693i
$593$	15.5962	+	27.0135i	0.640461	+	1.10931i	0.985330	+	0.170660i	$0.0545898\pi$
$593$	15.5962	+	27.0135i	0.640461	+	1.10931i	−0.344869	+	0.938651i	$0.612077\pi$
$594$		0			0
$595$	−9.55389	−	29.8455i	−0.391671	−	1.22355i
$596$	−19.6239			−0.803828
$597$		0			0
$598$		0			0
$599$	−15.6463	+	27.1002i	−0.639291	+	1.10728i	0.346298	+	0.938125i	$0.387439\pi$
$599$	−15.6463	+	27.1002i	−0.639291	+	1.10728i	−0.985589	+	0.169159i	$0.945895\pi$
$600$		0			0
$601$	−22.1203			−0.902307			−0.451154	−	0.892446i	$-0.648987\pi$
$601$	−22.1203			−0.902307			−0.451154	+	0.892446i	$0.648987\pi$
$602$	−20.4806	+	22.5478i	−0.834728	+	0.918979i
$603$		0			0
$604$	0.165670	+	0.286949i	0.00674102	+	0.0116758i
$605$	−0.806615	+	1.39710i	−0.0327936	+	0.0568001i
$606$		0			0
$607$	10.3090	+	17.8557i	0.418429	+	0.724740i	0.995782	−	0.0917548i	$-0.0292476\pi$
$607$	10.3090	+	17.8557i	0.418429	+	0.724740i	−0.577353	+	0.816495i	$0.695914\pi$
$608$	−1.17103			−0.0474915
$609$		0			0
$610$	19.2866			0.780893
$611$		0			0
$612$		0			0
$613$	7.87750	−	13.6442i	0.318169	−	0.551086i	−0.661937	−	0.749560i	$-0.730265\pi$
$613$	7.87750	−	13.6442i	0.318169	−	0.551086i	0.980106	+	0.198474i	$0.0635986\pi$
$614$	−16.5131	−	28.6015i	−0.666414	−	1.15426i
$615$		0			0
$616$	−2.58551	−	0.561349i	−0.104173	−	0.0226174i
$617$	−34.2526			−1.37896			−0.689480	−	0.724305i	$-0.742161\pi$
$617$	−34.2526			−1.37896			−0.689480	+	0.724305i	$0.742161\pi$
$618$		0			0
$619$	10.9854	−	19.0272i	0.441539	−	0.764769i	−0.556264	−	0.831005i	$-0.687766\pi$
$619$	10.9854	−	19.0272i	0.441539	−	0.764769i	0.997804	+	0.0662365i	$0.0210992\pi$
$620$	−5.11560	+	8.86048i	−0.205447	+	0.355845i
$621$		0			0
$622$	−0.331340			−0.0132855
$623$	−1.37605	+	1.51494i	−0.0551303	+	0.0606947i
$624$		0			0
$625$	3.63230	+	6.29133i	0.145292	+	0.251653i
$626$	−7.19874	+	12.4686i	−0.287720	+	0.498345i
$627$		0			0
$628$	−9.98300	−	17.2911i	−0.398365	−	0.689989i
$629$	20.7705			0.828173
$630$		0			0
$631$	−27.5685			−1.09749			−0.548743	−	0.835991i	$-0.684893\pi$
$631$	−27.5685			−1.09749			−0.548743	+	0.835991i	$0.684893\pi$
$632$	2.80661	+	4.86120i	0.111641	+	0.193368i
$633$		0			0
$634$	−8.76190	+	15.1761i	−0.347980	+	0.602718i
$635$	0.356685	+	0.617797i	0.0141546	+	0.0245165i
$636$		0			0
$637$		0			0
$638$	10.3975			0.411641
$639$		0			0
$640$	0.806615	−	1.39710i	0.0318843	−	0.0552251i
$641$	2.82897	−	4.89992i	0.111738	−	0.193535i	−0.804733	−	0.593636i	$-0.797692\pi$
$641$	2.82897	−	4.89992i	0.111738	−	0.193535i	0.916471	+	0.400101i	$0.131025\pi$
$642$		0			0
$643$	17.4791			0.689308			0.344654	−	0.938730i	$-0.387996\pi$
$643$	17.4791			0.689308			0.344654	+	0.938730i	$0.387996\pi$
$644$	2.86977	+	8.96493i	0.113085	+	0.353268i
$645$		0			0
$646$	4.29889	+	7.44589i	0.169137	+	0.292955i
$647$	−4.41984	+	7.65540i	−0.173762	+	0.300965i	−0.939732	−	0.341912i	$-0.888926\pi$
$647$	−4.41984	+	7.65540i	−0.173762	+	0.300965i	0.765970	+	0.642876i	$0.222259\pi$
$648$		0			0
$649$	2.19874	+	3.80834i	0.0863083	+	0.149490i
$650$		0			0
$651$		0			0
$652$	21.2866			0.833649
$653$	−12.3751	−	21.4344i	−0.484276	−	0.838791i	0.515560	−	0.856853i	$-0.327584\pi$
$653$	−12.3751	−	21.4344i	−0.484276	−	0.838791i	−0.999837	+	0.0180618i	$0.994250\pi$
$654$		0			0
$655$	−1.88914	+	3.27209i	−0.0738148	+	0.127851i
$656$	2.47229	+	4.28212i	0.0965265	+	0.167189i
$657$		0			0
$658$	17.5408	+	3.80834i	0.683812	+	0.148464i
$659$	−5.62869			−0.219263			−0.109631	−	0.993972i	$-0.534967\pi$
$659$	−5.62869			−0.219263			−0.109631	+	0.993972i	$0.534967\pi$
$660$		0			0
$661$	6.70111	−	11.6067i	0.260643	−	0.451447i	−0.705770	−	0.708441i	$-0.749399\pi$
$661$	6.70111	−	11.6067i	0.260643	−	0.451447i	0.966413	+	0.256994i	$0.0827321\pi$
$662$	−4.47229	+	7.74622i	−0.173820	+	0.301066i
$663$		0			0
$664$	5.05543			0.196189
$665$	4.88440	+	1.06047i	0.189409	+	0.0411231i
$666$		0			0
$667$	−18.4961	−	32.0362i	−0.716172	−	1.24045i
$668$	11.0107	−	19.0711i	0.426018	−	0.737884i
$669$		0			0
$670$	2.37513	+	4.11385i	0.0917594	+	0.158932i
$671$	−11.9553			−0.461529
$672$		0			0
$673$	−29.0262			−1.11888			−0.559438	−	0.828872i	$-0.688983\pi$
$673$	−29.0262			−1.11888			−0.559438	+	0.828872i	$0.688983\pi$
$674$	−1.66866	−	2.89020i	−0.0642744	−	0.111326i
$675$		0			0
$676$	6.50000	−	11.2583i	0.250000	−	0.433013i
$677$	−10.1540	−	17.5873i	−0.390251	−	0.675935i	0.602231	−	0.798322i	$-0.294279\pi$
$677$	−10.1540	−	17.5873i	−0.390251	−	0.675935i	−0.992482	+	0.122387i	$0.960945\pi$
$678$		0			0
$679$	−5.64630	−	17.6386i	−0.216685	−	0.676906i
$680$	−11.8444			−0.454213
$681$		0			0
$682$	3.17103	−	5.49238i	0.121425	−	0.210314i
$683$	−5.36977	+	9.30072i	−0.205469	+	0.355882i	−0.950282	−	0.311391i	$-0.899205\pi$
$683$	−5.36977	+	9.30072i	−0.205469	+	0.355882i	0.744813	+	0.667273i	$0.232539\pi$
$684$		0			0
$685$	−5.20502			−0.198874
$686$	−2.17639	+	18.3919i	−0.0830949	+	0.702207i
$687$		0			0
$688$	5.75654	+	9.97063i	0.219466	+	0.380127i
$689$		0			0
$690$		0			0
$691$	−2.98537	−	5.17082i	−0.113569	−	0.196707i	0.803638	−	0.595119i	$-0.202895\pi$
$691$	−2.98537	−	5.17082i	−0.113569	−	0.196707i	−0.917207	+	0.398411i	$0.869562\pi$
$692$	−1.65794			−0.0630254
$693$		0			0
$694$	−30.4237			−1.15487
$695$	−14.8397	−	25.7031i	−0.562902	−	0.974974i
$696$		0			0
$697$	18.1517	−	31.4396i	0.687543	−	1.19086i
$698$	8.76190	+	15.1761i	0.331643	+	0.574422i
$699$		0			0
$700$	4.26489	−	4.69536i	0.161198	−	0.177468i
$701$	5.28189			0.199494			0.0997471	−	0.995013i	$-0.468197\pi$
$701$	5.28189			0.199494			0.0997471	+	0.995013i	$0.468197\pi$
$702$		0			0
$703$	−1.65640	+	2.86898i	−0.0624725	+	0.108205i
$704$	−0.500000	+	0.866025i	−0.0188445	+	0.0326396i
$705$		0			0
$706$	5.11560			0.192528
$707$	−35.2249	−	7.64779i	−1.32477	−	0.287625i
$708$		0			0
$709$	−19.6564	−	34.0459i	−0.738212	−	1.27862i	−0.953300	−	0.302026i	$-0.902337\pi$
$709$	−19.6564	−	34.0459i	−0.738212	−	1.27862i	0.215088	−	0.976595i	$-0.430996\pi$
$710$	−12.5131	+	21.6733i	−0.469608	+	0.813385i
$711$		0			0
$712$	0.386770	+	0.669906i	0.0144948	+	0.0251058i
$713$	−22.5638			−0.845020
$714$		0			0
$715$		0			0
$716$	7.53008	+	13.0425i	0.281412	+	0.487421i
$717$		0			0
$718$	15.3421	−	26.5732i	0.572561	−	0.991704i
$719$	−17.6788	−	30.6205i	−0.659306	−	1.14195i	−0.980795	−	0.195039i	$-0.937517\pi$
$719$	−17.6788	−	30.6205i	−0.659306	−	1.14195i	0.321489	−	0.946913i	$-0.395817\pi$
$720$		0			0
$721$	21.9553	−	24.1713i	0.817658	−	0.900185i
$722$	17.6287			0.656072
$723$		0			0
$724$	−7.55780	+	13.0905i	−0.280883	+	0.486504i
$725$	−12.4639	+	21.5882i	−0.462899	+	0.801765i
$726$		0			0
$727$	−40.2312			−1.49209			−0.746046	−	0.665894i	$-0.768050\pi$
$727$	−40.2312			−1.49209			−0.746046	+	0.665894i	$0.768050\pi$
$728$		0			0
$729$		0			0
$730$	−4.12632	−	7.14699i	−0.152722	−	0.264522i
$731$	42.2649	−	73.2049i	1.56322	−	2.70758i
$732$		0			0
$733$	−14.3644	−	24.8799i	−0.530562	−	0.918960i	−0.999364	−	0.0356568i	$-0.988648\pi$
$733$	−14.3644	−	24.8799i	−0.530562	−	0.918960i	0.468802	−	0.883303i	$-0.344686\pi$
$734$	−1.45766			−0.0538032
$735$		0			0
$736$	3.55780			0.131142
$737$	−1.47229	−	2.55007i	−0.0542323	−	0.0939331i
$738$		0			0
$739$	8.17103	−	14.1526i	0.300576	−	0.520613i	−0.675690	−	0.737186i	$-0.736154\pi$
$739$	8.17103	−	14.1526i	0.300576	−	0.520613i	0.976267	+	0.216572i	$0.0694877\pi$
$740$	−2.28189	−	3.95235i	−0.0838839	−	0.145291i
$741$		0			0
$742$	−6.45292	−	20.1584i	−0.236894	−	0.740037i
$743$	−45.3897			−1.66519			−0.832593	−	0.553885i	$-0.813145\pi$
$743$	−45.3897			−1.66519			−0.832593	+	0.553885i	$0.813145\pi$
$744$		0			0
$745$	15.8290	−	27.4166i	0.579929	−	1.00447i
$746$	−17.0932	+	29.6064i	−0.625828	+	1.08397i
$747$		0			0
$748$	7.34206			0.268452
$749$	−27.9991	+	30.8251i	−1.02306	+	1.12632i
$750$		0			0
$751$	13.8999	+	24.0753i	0.507213	+	0.878519i	0.999965	+	0.00834907i	$0.00265762\pi$
$751$	13.8999	+	24.0753i	0.507213	+	0.878519i	−0.492752	+	0.870170i	$0.664009\pi$
$752$	3.39213	−	5.87534i	0.123698	−	0.214252i
$753$		0			0
$754$		0			0
$755$	−0.534528			−0.0194535
$756$		0			0
$757$	19.0816			0.693533			0.346766	−	0.937952i	$-0.387280\pi$
$757$	19.0816			0.693533			0.346766	+	0.937952i	$0.387280\pi$
$758$	−4.64331	−	8.04246i	−0.168653	−	0.292115i
$759$		0			0
$760$	0.944570	−	1.63604i	0.0342632	−	0.0593455i
$761$	23.6093	+	40.8925i	0.855837	+	1.48235i	0.875866	+	0.482554i	$0.160291\pi$
$761$	23.6093	+	40.8925i	0.855837	+	1.48235i	−0.0200287	+	0.999799i	$0.506376\pi$
$762$		0			0
$763$	10.2265	+	2.22030i	0.370223	+	0.0803802i
$764$	4.77354			0.172701
$765$		0			0
$766$	1.81197	−	3.13843i	0.0654693	−	0.113396i
$767$		0			0
$768$		0			0
$769$	−9.56852			−0.345050			−0.172525	−	0.985005i	$-0.555192\pi$
$769$	−9.56852			−0.345050			−0.172525	+	0.985005i	$0.555192\pi$
$770$	2.86977	−	3.15943i	0.103419	−	0.113858i
$771$		0			0
$772$	−1.55780	−	2.69819i	−0.0560664	−	0.0971099i
$773$	7.97764	−	13.8177i	0.286936	−	0.496988i	−0.686141	−	0.727469i	$-0.740697\pi$
$773$	7.97764	−	13.8177i	0.286936	−	0.496988i	0.973077	+	0.230481i	$0.0740300\pi$
$774$		0			0
$775$	7.60251	+	13.1679i	0.273090	+	0.473006i
$776$	−7.00000			−0.251285
$777$		0			0
$778$	7.16031			0.256710
$779$	2.89512	+	5.01449i	0.103728	+	0.179663i
$780$		0			0
$781$	7.75654	−	13.4347i	0.277551	−	0.480732i
$782$	−13.0608	−	22.6220i	−0.467053	−	0.808959i
$783$		0			0
$784$	6.36977	+	2.90275i	0.227492	+	0.103670i
$785$	32.2098			1.14962
$786$		0			0
$787$	−24.9276	+	43.1758i	−0.888572	+	1.53905i	−0.0470081	+	0.998895i	$0.514969\pi$
$787$	−24.9276	+	43.1758i	−0.888572	+	1.53905i	−0.841564	+	0.540157i	$0.818365\pi$
$788$	−2.41449	+	4.18201i	−0.0860125	+	0.148978i
$789$		0			0
$790$	−9.05543			−0.322178
$791$	−3.50237	−	10.9411i	−0.124530	−	0.389021i
$792$		0			0
$793$		0			0
$794$	−15.1987	+	26.3250i	−0.539383	+	0.934239i
$795$		0			0
$796$	−2.94457	−	5.10015i	−0.104368	−	0.180770i
$797$	−36.4177			−1.28998			−0.644990	−	0.764191i	$-0.723139\pi$
$797$	−36.4177			−1.28998			−0.644990	+	0.764191i	$0.723139\pi$
$798$		0			0
$799$	−49.8104			−1.76217
$800$	−1.19874	−	2.07629i	−0.0423820	−	0.0734078i
$801$		0			0
$802$	6.01072	−	10.4109i	0.212246	−	0.367621i
$803$	2.55780	+	4.43024i	0.0902628	+	0.156340i
$804$		0			0
$805$	−14.8397	−	3.22189i	−0.523030	−	0.113557i
$806$		0			0
$807$		0			0
$808$	−6.81197	+	11.7987i	−0.239644	+	0.415076i
$809$	24.2118	−	41.9361i	0.851243	−	1.47440i	−0.0288446	−	0.999584i	$-0.509183\pi$
$809$	24.2118	−	41.9361i	0.851243	−	1.47440i	0.880087	−	0.474812i	$-0.157484\pi$
$810$		0			0
$811$	−41.0262			−1.44062			−0.720312	−	0.693650i	$-0.756001\pi$
$811$	−41.0262			−1.44062			−0.720312	+	0.693650i	$0.756001\pi$
$812$	−26.8829	−	5.83662i	−0.943404	−	0.204825i
$813$		0			0
$814$	1.41449	+	2.44996i	0.0495777	+	0.0858711i
$815$	−17.1701	+	29.7395i	−0.601443	+	1.04173i
$816$		0			0
$817$	6.74109	+	11.6759i	0.235841	+	0.408488i
$818$	−17.5685			−0.614269
$819$		0			0
$820$	−7.97673			−0.278559
$821$	19.9660	+	34.5821i	0.696819	+	1.20693i	0.969564	+	0.244839i	$0.0787350\pi$
$821$	19.9660	+	34.5821i	0.696819	+	1.20693i	−0.272745	+	0.962086i	$0.587932\pi$
$822$		0			0
$823$	7.89986	−	13.6830i	0.275372	−	0.476958i	−0.694857	−	0.719148i	$-0.744532\pi$
$823$	7.89986	−	13.6830i	0.275372	−	0.476958i	0.970229	+	0.242190i	$0.0778657\pi$
$824$	−6.17103	−	10.6885i	−0.214978	−	0.372353i
$825$		0			0
$826$	−3.54708	−	11.0808i	−0.123419	−	0.385550i
$827$	26.8766			0.934591			0.467295	−	0.884101i	$-0.345229\pi$
$827$	26.8766			0.934591			0.467295	+	0.884101i	$0.345229\pi$
$828$		0			0
$829$	4.15403	−	7.19500i	0.144276	−	0.249893i	−0.784827	−	0.619715i	$-0.787248\pi$
$829$	4.15403	−	7.19500i	0.144276	−	0.249893i	0.929102	+	0.369822i	$0.120581\pi$
$830$	−4.07779	+	7.06293i	−0.141542	+	0.245158i
$831$		0			0
$832$		0			0
$833$	−4.92674	−	51.1577i	−0.170701	−	1.77251i
$834$		0			0
$835$	17.7628	+	30.7661i	0.614708	+	1.06471i
$836$	−0.585515	+	1.01414i	−0.0202505	+	0.0350748i
$837$		0			0
$838$	13.5855	+	23.5308i	0.469304	+	0.812858i
$839$	4.28787			0.148034			0.0740168	−	0.997257i	$-0.476418\pi$
$839$	4.28787			0.148034			0.0740168	+	0.997257i	$0.476418\pi$
$840$		0			0
$841$	79.1078			2.72785
$842$	11.5855	+	20.0667i	0.399263	+	0.691544i
$843$		0			0
$844$	6.83969	−	11.8467i	0.235432	−	0.407780i
$845$	10.4860	+	18.1623i	0.360729	+	0.624801i
$846$		0			0
$847$	−1.77890	+	1.95845i	−0.0611237	+	0.0672930i
$848$	−8.00000			−0.274721
$849$		0			0
$850$	−8.80126	+	15.2442i	−0.301881	+	0.522872i
$851$	5.03246	−	8.71647i	0.172510	−	0.298797i
$852$		0			0
$853$	−46.2973			−1.58519			−0.792596	−	0.609748i	$-0.791271\pi$
$853$	−46.2973			−1.58519			−0.792596	+	0.609748i	$0.791271\pi$
$854$	30.9106	+	6.71109i	1.05774	+	0.229649i
$855$		0			0
$856$	7.86977	+	13.6308i	0.268983	+	0.465893i
$857$	−13.7312	+	23.7831i	−0.469049	+	0.812416i	−0.999374	−	0.0353779i	$-0.988737\pi$
$857$	−13.7312	+	23.7831i	−0.469049	+	0.812416i	0.530325	+	0.847794i	$0.322070\pi$
$858$		0			0
$859$	6.09623	+	10.5590i	0.208001	+	0.360268i	0.951085	−	0.308930i	$-0.0999710\pi$
$859$	6.09623	+	10.5590i	0.208001	+	0.360268i	−0.743084	+	0.669198i	$0.766638\pi$
$860$	−18.5733			−0.633343
$861$		0			0
$862$	−16.7735			−0.571309
$863$	0.419845	+	0.727192i	0.0142917	+	0.0247539i	0.873083	−	0.487572i	$-0.162117\pi$
$863$	0.419845	+	0.727192i	0.0142917	+	0.0247539i	−0.858791	+	0.512326i	$0.828784\pi$
$864$		0			0
$865$	1.33732	−	2.31631i	0.0454702	−	0.0787567i
$866$	8.67103	+	15.0187i	0.294654	+	0.510355i
$867$		0			0
$868$	−11.2819	+	12.4206i	−0.382932	+	0.421582i
$869$	5.61323			0.190416
$870$		0			0
$871$		0			0
$872$	1.97764	−	3.42538i	0.0669715	−	0.115998i
$873$		0			0
$874$	4.16629			0.140927
$875$	9.62602	+	30.0709i	0.325419	+	1.01658i
$876$		0			0
$877$	10.2643	+	17.7782i	0.346600	+	0.600329i	0.985643	−	0.168842i	$-0.0540028\pi$
$877$	10.2643	+	17.7782i	0.346600	+	0.600329i	−0.639043	+	0.769171i	$0.720669\pi$
$878$	16.5739	−	28.7068i	0.559341	−	0.968808i
$879$		0			0
$880$	−0.806615	−	1.39710i	−0.0271910	−	0.0470962i
$881$	−1.88914			−0.0636468			−0.0318234	−	0.999494i	$-0.510131\pi$
$881$	−1.88914			−0.0636468			−0.0318234	+	0.999494i	$0.510131\pi$
$882$		0			0
$883$	26.5345			0.892958			0.446479	−	0.894794i	$-0.352678\pi$
$883$	26.5345			0.892958			0.446479	+	0.894794i	$0.352678\pi$
$884$		0			0
$885$		0			0
$886$	6.02771	−	10.4403i	0.202505	−	0.350749i
$887$	8.55780	+	14.8225i	0.287343	+	0.497692i	0.973175	−	0.230068i	$-0.0738948\pi$
$887$	8.55780	+	14.8225i	0.287343	+	0.497692i	−0.685832	+	0.727760i	$0.740561\pi$
$888$		0			0
$889$	0.356685	+	1.11425i	0.0119628	+	0.0373709i
$890$	−1.24790			−0.0418296
$891$		0			0
$892$	2.55780	−	4.43024i	0.0856415	−	0.148335i
$893$	3.97229	−	6.88020i	0.132927	−	0.230237i
$894$		0			0
$895$	−24.2955			−0.812110
$896$	1.77890	−	1.95845i	0.0594289	−	0.0654271i
$897$		0			0
$898$	18.2973	+	31.6919i	0.610590	+	1.05757i
$899$	32.9707	−	57.1070i	1.09964	−	1.90463i
$900$		0			0
$901$	29.3682	+	50.8673i	0.978398	+	1.69463i
$902$	4.94457			0.164636
$903$		0			0
$904$	−4.34206			−0.144415
$905$	−12.1925	−	21.1180i	−0.405291	−	0.701985i
$906$		0			0
$907$	−1.64331	+	2.84630i	−0.0545654	+	0.0945100i	−0.892018	−	0.452000i	$-0.850711\pi$
$907$	−1.64331	+	2.84630i	−0.0545654	+	0.0945100i	0.837453	+	0.546510i	$0.184044\pi$
$908$	−2.64331	−	4.57836i	−0.0877215	−	0.151938i
$909$		0			0
$910$		0			0
$911$	−14.0107			−0.464196			−0.232098	−	0.972692i	$-0.574559\pi$
$911$	−14.0107			−0.464196			−0.232098	+	0.972692i	$0.574559\pi$
$912$		0			0
$913$	2.52771	−	4.37813i	0.0836552	−	0.144895i
$914$	18.9106	−	32.7541i	0.625506	−	1.08341i
$915$		0			0
$916$	−4.88440			−0.161385
$917$	−4.16629	+	4.58680i	−0.137583	+	0.151469i
$918$		0			0
$919$	23.2580	+	40.2840i	0.767211	+	1.32885i	0.939070	+	0.343726i	$0.111689\pi$
$919$	23.2580	+	40.2840i	0.767211	+	1.32885i	−0.171859	+	0.985121i	$0.554977\pi$
$920$	−2.86977	+	4.97060i	−0.0946137	+	0.163876i
$921$		0			0
$922$	−17.4854	−	30.2856i	−0.575850	−	0.997402i
$923$		0			0
$924$		0			0
$925$	−6.78243			−0.223005
$926$	3.78426	+	6.55453i	0.124358	+	0.215395i
$927$		0			0
$928$	−5.19874	+	9.00449i	−0.170657	+	0.295587i
$929$	−12.4422	−	21.5505i	−0.408215	−	0.707050i	0.586475	−	0.809968i	$-0.300515\pi$
$929$	−12.4422	−	21.5505i	−0.408215	−	0.707050i	−0.994690	+	0.102918i	$0.967182\pi$
$930$		0			0
$931$	7.45920	+	3.39921i	0.244465	+	0.111405i
$932$	−9.79498			−0.320845
$933$		0			0
$934$	7.31434	−	12.6688i	0.239333	−	0.414536i
$935$	−5.92221	+	10.2576i	−0.193677	+	0.335459i
$936$		0			0
$937$	−46.7950			−1.52873			−0.764363	−	0.644787i	$-0.776946\pi$
$937$	−46.7950			−1.52873			−0.764363	+	0.644787i	$0.776946\pi$
$938$	2.37513	+	7.41972i	0.0775509	+	0.242262i
$939$		0			0
$940$	5.47229	+	9.47828i	0.178486	+	0.309147i
$941$	8.74583	−	15.1482i	0.285106	−	0.493818i	−0.687529	−	0.726157i	$-0.741305\pi$
$941$	8.74583	−	15.1482i	0.285106	−	0.493818i	0.972635	+	0.232339i	$0.0746379\pi$
$942$		0			0
$943$	−8.79590	−	15.2349i	−0.286434	−	0.496118i
$944$	−4.39749			−0.143126
$945$		0			0
$946$	11.5131			0.374323
$947$	7.30960	+	12.6606i	0.237530	+	0.411414i	0.960005	−	0.279983i	$-0.0903288\pi$
$947$	7.30960	+	12.6606i	0.237530	+	0.411414i	−0.722475	+	0.691397i	$0.756995\pi$
$948$		0			0
$949$		0			0
$950$	−1.40377	−	2.43139i	−0.0455442	−	0.0788848i
$951$		0			0
$952$	−18.9830	−	4.12146i	−0.615243	−	0.133577i
$953$	20.0816			0.650507			0.325254	−	0.945627i	$-0.394550\pi$
$953$	20.0816			0.650507			0.325254	+	0.945627i	$0.394550\pi$
$954$		0			0
$955$	−3.85041	+	6.66910i	−0.124596	+	0.215807i
$956$	−11.9553	+	20.7072i	−0.386662	+	0.669718i
$957$		0			0
$958$	−1.22646			−0.0396251
$959$	−8.34206	−	1.81117i	−0.269379	−	0.0584857i
$960$		0			0
$961$	−4.61086	−	7.98624i	−0.148737	−	0.257621i
$962$		0			0
$963$		0			0
$964$	−0.944570	−	1.63604i	−0.0304226	−	0.0526934i
$965$	5.02618			0.161798
$966$		0			0
$967$	−60.2634			−1.93794			−0.968969	−	0.247180i	$-0.920496\pi$
$967$	−60.2634			−1.93794			−0.968969	+	0.247180i	$0.920496\pi$
$968$	0.500000	+	0.866025i	0.0160706	+	0.0278351i
$969$		0			0
$970$	5.64630	−	9.77969i	0.181292	−	0.314007i
$971$	−26.4237	−	45.7671i	−0.847976	−	1.46874i	−0.883012	−	0.469351i	$-0.844488\pi$
$971$	−26.4237	−	45.7671i	−0.847976	−	1.46874i	0.0350357	−	0.999386i	$-0.488846\pi$
$972$		0			0
$973$	−14.8397	−	46.3579i	−0.475739	−	1.48617i
$974$	−30.7056			−0.983870
$975$		0			0
$976$	5.97764	−	10.3536i	0.191340	−	0.331410i
$977$	12.4529	−	21.5691i	0.398404	−	0.690056i	−0.595125	−	0.803633i	$-0.702897\pi$
$977$	12.4529	−	21.5691i	0.398404	−	0.690056i	0.993529	+	0.113577i	$0.0362308\pi$
$978$		0			0
$979$	0.773540			0.0247225
$980$	−9.19339	+	6.55780i	−0.293672	+	0.209481i
$981$		0			0
$982$	−15.4723	−	26.7988i	−0.493740	−	0.855184i
$983$	−12.5632	+	21.7600i	−0.400703	+	0.694037i	−0.993811	−	0.111086i	$-0.964567\pi$
$983$	−12.5632	+	21.7600i	−0.400703	+	0.694037i	0.593108	+	0.805123i	$0.297901\pi$
$984$		0			0
$985$	−3.89512	−	6.74654i	−0.124109	−	0.214963i
$986$	76.3390			2.43113
$987$		0			0
$988$		0			0
$989$	−20.4806	−	35.4735i	−0.651246	−	1.12799i
$990$		0			0
$991$	−28.9600	+	50.1602i	−0.919946	+	1.59339i	−0.120451	+	0.992719i	$0.538434\pi$
$991$	−28.9600	+	50.1602i	−0.919946	+	1.59339i	−0.799495	+	0.600673i	$0.794899\pi$
$992$	3.17103	+	5.49238i	0.100680	+	0.174383i
$993$		0			0
$994$	−27.5962	+	30.3816i	−0.875299	+	0.963645i
$995$	9.50054			0.301187
$996$		0			0
$997$	−8.68412	+	15.0413i	−0.275029	+	0.476364i	−0.970142	−	0.242536i	$-0.922021\pi$
$997$	−8.68412	+	15.0413i	−0.275029	+	0.476364i	0.695114	+	0.718900i	$0.255354\pi$
$998$	−5.17103	+	8.95649i	−0.163686	+	0.283513i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.k.v.991.2		6
3.2	odd	2		462.2.i.g.67.2	✓	6
7.2	even	3	inner	1386.2.k.v.793.2		6
7.3	odd	6		9702.2.a.dw.1.2		3
7.4	even	3		9702.2.a.dv.1.2		3
21.2	odd	6		462.2.i.g.331.2	yes	6
21.11	odd	6		3234.2.a.bf.1.2		3
21.17	even	6		3234.2.a.bh.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.i.g.67.2	✓	6	3.2	odd	2
462.2.i.g.331.2	yes	6	21.2	odd	6
1386.2.k.v.793.2		6	7.2	even	3	inner
1386.2.k.v.991.2		6	1.1	even	1	trivial
3234.2.a.bf.1.2		3	21.11	odd	6
3234.2.a.bh.1.2		3	21.17	even	6
9702.2.a.dv.1.2		3	7.4	even	3
9702.2.a.dw.1.2		3	7.3	odd	6

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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.806615 - 1.39710i) q^{5} +(-0.806615 - 2.51980i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.806615 - 1.39710i) q^{5} +(-0.806615 - 2.51980i) q^{7} -1.00000 q^{8} +(0.806615 - 1.39710i) q^{10} +(-0.500000 + 0.866025i) q^{11} +(1.77890 - 1.95845i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.67103 + 6.35841i) q^{17} +(-0.585515 - 1.01414i) q^{19} +1.61323 q^{20} -1.00000 q^{22} +(1.77890 + 3.08115i) q^{23} +(1.19874 - 2.07629i) q^{25} +(2.58551 + 0.561349i) q^{28} -10.3975 q^{29} +(-3.17103 + 5.49238i) q^{31} +(0.500000 - 0.866025i) q^{32} -7.34206 q^{34} +(-2.86977 + 3.15943i) q^{35} +(-1.41449 - 2.44996i) q^{37} +(0.585515 - 1.01414i) q^{38} +(0.806615 + 1.39710i) q^{40} -4.94457 q^{41} -11.5131 q^{43} +(-0.500000 - 0.866025i) q^{44} +(-1.77890 + 3.08115i) q^{46} +(3.39213 + 5.87534i) q^{47} +(-5.69874 + 4.06501i) q^{49} +2.39749 q^{50} +(4.00000 - 6.92820i) q^{53} +1.61323 q^{55} +(0.806615 + 2.51980i) q^{56} +(-5.19874 - 9.00449i) q^{58} +(2.19874 - 3.80834i) q^{59} +(5.97764 + 10.3536i) q^{61} -6.34206 q^{62} +1.00000 q^{64} +(-1.47229 + 2.55007i) q^{67} +(-3.67103 - 6.35841i) q^{68} +(-4.17103 - 0.905585i) q^{70} -15.5131 q^{71} +(2.55780 - 4.43024i) q^{73} +(1.41449 - 2.44996i) q^{74} +1.17103 q^{76} +(2.58551 + 0.561349i) q^{77} +(-2.80661 - 4.86120i) q^{79} +(-0.806615 + 1.39710i) q^{80} +(-2.47229 - 4.28212i) q^{82} -5.05543 q^{83} +11.8444 q^{85} +(-5.75654 - 9.97063i) q^{86} +(0.500000 - 0.866025i) q^{88} +(-0.386770 - 0.669906i) q^{89} -3.55780 q^{92} +(-3.39213 + 5.87534i) q^{94} +(-0.944570 + 1.63604i) q^{95} +7.00000 q^{97} +(-6.36977 - 2.90275i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8}+O(q^{10}) \) 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8	\( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8} - 3 q^{11} + 3 q^{14} - 3 q^{16} + 3 q^{17} + 9 q^{19} - 6 q^{22} + 3 q^{23} - 15 q^{25} + 3 q^{28} - 18 q^{29} + 6 q^{31} + 3 q^{32} + 6 q^{34} + 30 q^{35} - 21 q^{37} - 9 q^{38} - 24 q^{41} + 6 q^{43} - 3 q^{44} - 3 q^{46} + 3 q^{47} - 12 q^{49} - 30 q^{50} + 24 q^{53} - 9 q^{58} - 9 q^{59} + 6 q^{61} + 12 q^{62} + 6 q^{64} - 6 q^{67} + 3 q^{68} - 18 q^{71} + 21 q^{74} - 18 q^{76} + 3 q^{77} - 12 q^{79} - 12 q^{82} - 36 q^{83} + 3 q^{86} + 3 q^{88} - 12 q^{89} - 6 q^{92} - 3 q^{94} + 42 q^{97} + 9 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 - 3 * q^11 + 3 * q^14 - 3 * q^16 + 3 * q^17 + 9 * q^19 - 6 * q^22 + 3 * q^23 - 15 * q^25 + 3 * q^28 - 18 * q^29 + 6 * q^31 + 3 * q^32 + 6 * q^34 + 30 * q^35 - 21 * q^37 - 9 * q^38 - 24 * q^41 + 6 * q^43 - 3 * q^44 - 3 * q^46 + 3 * q^47 - 12 * q^49 - 30 * q^50 + 24 * q^53 - 9 * q^58 - 9 * q^59 + 6 * q^61 + 12 * q^62 + 6 * q^64 - 6 * q^67 + 3 * q^68 - 18 * q^71 + 21 * q^74 - 18 * q^76 + 3 * q^77 - 12 * q^79 - 12 * q^82 - 36 * q^83 + 3 * q^86 + 3 * q^88 - 12 * q^89 - 6 * q^92 - 3 * q^94 + 42 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more