LMFDB - Embedded newform 351.2.e.b.118.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [351,2,Mod(118,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("351.118"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 351 = 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	351.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$2.80274911095$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	10.0.487558322307.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 13x^{8} + 43x^{6} + 48x^{4} + 21x^{2} + 3 $ x^10 + 13x^8 + 43x^6 + 48x^4 + 21x^2 + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			118.4
Root			$-0.775434i$ of defining polynomial
Character	$\chi$	$=$	351.118
Dual form			351.2.e.b.235.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.754070 + 1.30609i) q^{2} +(-0.137244 + 0.237713i) q^{4} +(-1.32157 + 2.28903i) q^{5} +(-0.171546 - 0.297126i) q^{7} +2.60231 q^{8} -3.98624 q^{10} +(2.94380 + 5.09881i) q^{11} +(-0.500000 + 0.866025i) q^{13} +(0.258715 - 0.448108i) q^{14} +(2.23682 + 3.87428i) q^{16} -7.72803 q^{17} +1.60678 q^{19} +(-0.362756 - 0.628312i) q^{20} +(-4.43966 + 7.68972i) q^{22} +(2.61683 - 4.53248i) q^{23} +(-0.993121 - 1.72014i) q^{25} -1.50814 q^{26} +0.0941744 q^{28} +(-0.204749 - 0.354635i) q^{29} +(1.31419 - 2.27624i) q^{31} +(-0.771118 + 1.33562i) q^{32} +(-5.82748 - 10.0935i) q^{34} +0.906842 q^{35} +5.94321 q^{37} +(1.21163 + 2.09860i) q^{38} +(-3.43915 + 5.95679i) q^{40} +(3.30656 - 5.72712i) q^{41} +(-0.655427 - 1.13523i) q^{43} -1.61607 q^{44} +7.89308 q^{46} +(-1.80043 - 3.11843i) q^{47} +(3.44114 - 5.96024i) q^{49} +(1.49777 - 2.59421i) q^{50} +(-0.137244 - 0.237713i) q^{52} +3.87831 q^{53} -15.5618 q^{55} +(-0.446416 - 0.773215i) q^{56} +(0.308790 - 0.534839i) q^{58} +(0.226265 - 0.391903i) q^{59} +(-1.87014 - 3.23918i) q^{61} +3.96396 q^{62} +6.62136 q^{64} +(-1.32157 - 2.28903i) q^{65} +(1.40485 - 2.43328i) q^{67} +(1.06063 - 1.83706i) q^{68} +(0.683823 + 1.18442i) q^{70} -16.5889 q^{71} -5.15974 q^{73} +(4.48160 + 7.76235i) q^{74} +(-0.220521 + 0.381954i) q^{76} +(1.00999 - 1.74936i) q^{77} +(4.64906 + 8.05240i) q^{79} -11.8245 q^{80} +9.97350 q^{82} +(-0.569519 - 0.986435i) q^{83} +(10.2132 - 17.6897i) q^{85} +(0.988476 - 1.71209i) q^{86} +(7.66069 + 13.2687i) q^{88} +9.01137 q^{89} +0.343091 q^{91} +(0.718287 + 1.24411i) q^{92} +(2.71530 - 4.70304i) q^{94} +(-2.12349 + 3.67798i) q^{95} +(-2.16133 - 3.74354i) q^{97} +10.3795 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} - 4 q^{4} + q^{5} + 2 q^{7} - 24 q^{8} - 4 q^{10} + 11 q^{11} - 5 q^{13} - 5 q^{14} - 10 q^{16} - 14 q^{17} + 6 q^{19} - q^{20} + 7 q^{22} + 18 q^{23} + 8 q^{25} - 4 q^{26} - 38 q^{28}+ \cdots + 78 q^{98}+O(q^{100}) $ 10 * q + 2 * q^2 - 4 * q^4 + q^5 + 2 * q^7 - 24 * q^8 - 4 * q^10 + 11 * q^11 - 5 * q^13 - 5 * q^14 - 10 * q^16 - 14 * q^17 + 6 * q^19 - q^20 + 7 * q^22 + 18 * q^23 + 8 * q^25 - 4 * q^26 - 38 * q^28 + 4 * q^29 + 16 * q^31 + 31 * q^32 - 13 * q^34 - 22 * q^35 + 10 * q^37 + 24 * q^38 - 18 * q^40 + 12 * q^41 - 2 * q^44 + 12 * q^46 + 15 * q^47 - 3 * q^49 - 5 * q^50 - 4 * q^52 + 6 * q^53 - 34 * q^55 - 24 * q^56 + 2 * q^58 + 3 * q^59 + q^61 + 8 * q^62 + 124 * q^64 + q^65 + 4 * q^67 - 16 * q^68 + 25 * q^70 - 30 * q^71 - 10 * q^73 + 11 * q^74 - 9 * q^76 + q^77 - 13 * q^79 - 64 * q^80 - 42 * q^82 + 26 * q^83 + 7 * q^85 - 6 * q^86 - 3 * q^88 - 58 * q^89 - 4 * q^91 - 6 * q^92 - 12 * q^94 + 12 * q^95 + 16 * q^97 + 78 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/351\mathbb{Z}\right)^\times$.

$n$	$28$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	0.754070	+	1.30609i	0.533208	+	0.923544i	0.999248	+	0.0387798i	$0.0123471\pi$
$2$	0.754070	+	1.30609i	0.533208	+	0.923544i	−0.466040	+	0.884764i	$0.654320\pi$
$3$		0			0
$3$		0			0
$4$	−0.137244	+	0.237713i	−0.0686219	+	0.118857i
$5$	−1.32157	+	2.28903i	−0.591026	+	1.02369i	0.403068	+	0.915170i	$0.367944\pi$
$5$	−1.32157	+	2.28903i	−0.591026	+	1.02369i	−0.994095	+	0.108518i	$0.965390\pi$
$6$		0			0
$7$	−0.171546	−	0.297126i	−0.0648382	−	0.112303i	0.831784	−	0.555099i	$-0.187320\pi$
$7$	−0.171546	−	0.297126i	−0.0648382	−	0.112303i	−0.896622	+	0.442796i	$0.853986\pi$
$8$	2.60231			0.920057
$9$		0			0
$10$	−3.98624			−1.26056
$11$	2.94380	+	5.09881i	0.887589	+	1.53735i	0.842717	+	0.538356i	$0.180954\pi$
$11$	2.94380	+	5.09881i	0.887589	+	1.53735i	0.0448714	+	0.998993i	$0.485712\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i
$13$	−0.500000	+	0.866025i	−0.138675	+	0.240192i
$14$	0.258715	−	0.448108i	0.0691445	−	0.119762i
$15$		0			0
$16$	2.23682	+	3.87428i	0.559204	+	0.968570i
$17$	−7.72803			−1.87432			−0.937162	−	0.348895i	$-0.886557\pi$
$17$	−7.72803			−1.87432			−0.937162	+	0.348895i	$0.886557\pi$
$18$		0			0
$19$	1.60678			0.368621			0.184311	−	0.982868i	$-0.440995\pi$
$19$	1.60678			0.368621			0.184311	+	0.982868i	$0.440995\pi$
$20$	−0.362756	−	0.628312i	−0.0811147	−	0.140495i
$21$		0			0
$22$	−4.43966	+	7.68972i	−0.946539	+	1.63945i
$23$	2.61683	−	4.53248i	0.545646	−	0.945087i	−0.452920	−	0.891551i	$-0.649618\pi$
$23$	2.61683	−	4.53248i	0.545646	−	0.945087i	0.998566	−	0.0535354i	$-0.0170490\pi$
$24$		0			0
$25$	−0.993121	−	1.72014i	−0.198624	−	0.344027i
$26$	−1.50814			−0.295771
$27$		0			0
$28$	0.0941744			0.0177973
$29$	−0.204749	−	0.354635i	−0.0380209	−	0.0658541i	0.846389	−	0.532566i	$-0.178772\pi$
$29$	−0.204749	−	0.354635i	−0.0380209	−	0.0658541i	−0.884410	+	0.466711i	$0.845439\pi$
$30$		0			0
$31$	1.31419	−	2.27624i	0.236035	−	0.408825i	−0.723538	−	0.690285i	$-0.757485\pi$
$31$	1.31419	−	2.27624i	0.236035	−	0.408825i	0.959573	+	0.281460i	$0.0908187\pi$
$32$	−0.771118	+	1.33562i	−0.136316	+	0.236106i
$33$		0			0
$34$	−5.82748	−	10.0935i	−0.999405	−	1.73102i
$35$	0.906842			0.153284
$36$		0			0
$37$	5.94321			0.977057			0.488529	−	0.872548i	$-0.337534\pi$
$37$	5.94321			0.977057			0.488529	+	0.872548i	$0.337534\pi$
$38$	1.21163	+	2.09860i	0.196552	+	0.340438i
$39$		0			0
$40$	−3.43915	+	5.95679i	−0.543778	+	0.941851i
$41$	3.30656	−	5.72712i	0.516397	−	0.894426i	−0.483422	−	0.875388i	$-0.660606\pi$
$41$	3.30656	−	5.72712i	0.516397	−	0.894426i	0.999819	−	0.0190384i	$-0.00606049\pi$
$42$		0			0
$43$	−0.655427	−	1.13523i	−0.0999517	−	0.173121i	0.811713	−	0.584057i	$-0.198535\pi$
$43$	−0.655427	−	1.13523i	−0.0999517	−	0.173121i	−0.911664	+	0.410935i	$0.865202\pi$
$44$	−1.61607			−0.243632
$45$		0			0
$46$	7.89308			1.16377
$47$	−1.80043	−	3.11843i	−0.262620	−	0.454870i	0.704318	−	0.709885i	$-0.251253\pi$
$47$	−1.80043	−	3.11843i	−0.262620	−	0.454870i	−0.966937	+	0.255015i	$0.917920\pi$
$48$		0			0
$49$	3.44114	−	5.96024i	0.491592	−	0.851462i
$50$	1.49777	−	2.59421i	0.211816	−	0.366876i
$51$		0			0
$52$	−0.137244	−	0.237713i	−0.0190323	−	0.0329649i
$53$	3.87831			0.532727			0.266363	−	0.963873i	$-0.414178\pi$
$53$	3.87831			0.532727			0.266363	+	0.963873i	$0.414178\pi$
$54$		0			0
$55$	−15.5618			−2.09835
$56$	−0.446416	−	0.773215i	−0.0596549	−	0.103325i
$57$		0			0
$58$	0.308790	−	0.534839i	0.0405461	−	0.0702278i
$59$	0.226265	−	0.391903i	0.0294572	−	0.0510214i	−0.850921	−	0.525294i	$-0.823955\pi$
$59$	0.226265	−	0.391903i	0.0294572	−	0.0510214i	0.880378	+	0.474272i	$0.157289\pi$
$60$		0			0
$61$	−1.87014	−	3.23918i	−0.239447	−	0.414735i	0.721108	−	0.692822i	$-0.243633\pi$
$61$	−1.87014	−	3.23918i	−0.239447	−	0.414735i	−0.960556	+	0.278087i	$0.910300\pi$
$62$	3.96396			0.503423
$63$		0			0
$64$	6.62136			0.827669
$65$	−1.32157	−	2.28903i	−0.163921	−	0.283920i
$66$		0			0
$67$	1.40485	−	2.43328i	0.171630	−	0.297272i	−0.767360	−	0.641217i	$-0.778430\pi$
$67$	1.40485	−	2.43328i	0.171630	−	0.297272i	0.938990	+	0.343945i	$0.111763\pi$
$68$	1.06063	−	1.83706i	0.128620	−	0.222776i
$69$		0			0
$70$	0.683823	+	1.18442i	0.0817325	+	0.141565i
$71$	−16.5889			−1.96874			−0.984369	−	0.176119i	$-0.943646\pi$
$71$	−16.5889			−1.96874			−0.984369	+	0.176119i	$0.943646\pi$
$72$		0			0
$73$	−5.15974			−0.603902			−0.301951	−	0.953323i	$-0.597638\pi$
$73$	−5.15974			−0.603902			−0.301951	+	0.953323i	$0.597638\pi$
$74$	4.48160	+	7.76235i	0.520975	+	0.902355i
$75$		0			0
$76$	−0.220521	+	0.381954i	−0.0252955	+	0.0438131i
$77$	1.00999	−	1.74936i	0.115099	−	0.199358i
$78$		0			0
$79$	4.64906	+	8.05240i	0.523060	+	0.905966i	0.999640	+	0.0268349i	$0.00854284\pi$
$79$	4.64906	+	8.05240i	0.523060	+	0.905966i	−0.476580	+	0.879131i	$0.658124\pi$
$80$	−11.8245			−1.32202
$81$		0			0
$82$	9.97350			1.10139
$83$	−0.569519	−	0.986435i	−0.0625128	−	0.108275i	0.833075	−	0.553160i	$-0.186578\pi$
$83$	−0.569519	−	0.986435i	−0.0625128	−	0.108275i	−0.895588	+	0.444884i	$0.853245\pi$
$84$		0			0
$85$	10.2132	−	17.6897i	1.10777	−	1.91872i
$86$	0.988476	−	1.71209i	0.106590	−	0.184619i
$87$		0			0
$88$	7.66069	+	13.2687i	0.816633	+	1.41445i
$89$	9.01137			0.955203			0.477602	−	0.878576i	$-0.341506\pi$
$89$	9.01137			0.955203			0.477602	+	0.878576i	$0.341506\pi$
$90$		0			0
$91$	0.343091			0.0359658
$92$	0.718287	+	1.24411i	0.0748866	+	0.129707i
$93$		0			0
$94$	2.71530	−	4.70304i	0.280062	−	0.485081i
$95$	−2.12349	+	3.67798i	−0.217865	+	0.377353i
$96$		0			0
$97$	−2.16133	−	3.74354i	−0.219450	−	0.380099i	0.735190	−	0.677861i	$-0.237093\pi$
$97$	−2.16133	−	3.74354i	−0.219450	−	0.380099i	−0.954640	+	0.297762i	$0.903760\pi$
$98$	10.3795			1.04848
$99$		0			0
$100$	0.545199			0.0545199
$101$	3.31579	+	5.74312i	0.329934	+	0.571462i	0.982498	−	0.186271i	$-0.0596402\pi$
$101$	3.31579	+	5.74312i	0.329934	+	0.571462i	−0.652565	+	0.757733i	$0.726307\pi$
$102$		0			0
$103$	7.11512	−	12.3238i	0.701074	−	1.21430i	−0.267016	−	0.963692i	$-0.586038\pi$
$103$	7.11512	−	12.3238i	0.701074	−	1.21430i	0.968090	−	0.250604i	$-0.0806291\pi$
$104$	−1.30116	+	2.25367i	−0.127589	+	0.220991i
$105$		0			0
$106$	2.92452	+	5.06541i	0.284054	+	0.491996i
$107$	0.957659			0.0925804			0.0462902	−	0.998928i	$-0.485260\pi$
$107$	0.957659			0.0925804			0.0462902	+	0.998928i	$0.485260\pi$
$108$		0			0
$109$	5.77436			0.553083			0.276541	−	0.961002i	$-0.410812\pi$
$109$	5.77436			0.553083			0.276541	+	0.961002i	$0.410812\pi$
$110$	−11.7347	−	20.3251i	−1.11886	−	1.93792i
$111$		0			0
$112$	0.767433	−	1.32923i	0.0725156	−	0.125601i
$113$	5.93856	−	10.2859i	0.558653	−	0.967615i	−0.438956	−	0.898508i	$-0.644652\pi$
$113$	5.93856	−	10.2859i	0.558653	−	0.967615i	0.997609	−	0.0691070i	$-0.0220150\pi$
$114$		0			0
$115$	6.91666	+	11.9800i	0.644982	+	1.11714i
$116$	0.112402			0.0104363
$117$		0			0
$118$	0.682479			0.0628273
$119$	1.32571	+	2.29620i	0.121528	+	0.210492i
$120$		0			0
$121$	−11.8319	+	20.4935i	−1.07563	+	1.86304i
$122$	2.82044	−	4.88514i	0.255351	−	0.442280i
$123$		0			0
$124$	0.360728	+	0.624800i	0.0323944	+	0.0561087i
$125$	−7.96582			−0.712484
$126$		0			0
$127$	−11.9783			−1.06290			−0.531451	−	0.847089i	$-0.678353\pi$
$127$	−11.9783			−1.06290			−0.531451	+	0.847089i	$0.678353\pi$
$128$	6.53520	+	11.3193i	0.577636	+	1.00049i
$129$		0			0
$130$	1.99312	−	3.45219i	0.174808	−	0.302777i
$131$	−6.01906	+	10.4253i	−0.525888	+	0.910864i	0.473657	+	0.880709i	$0.342934\pi$
$131$	−6.01906	+	10.4253i	−0.525888	+	0.910864i	−0.999545	+	0.0301553i	$0.990400\pi$
$132$		0			0
$133$	−0.275637	−	0.477417i	−0.0239008	−	0.0413973i
$134$	4.23743			0.366058
$135$		0			0
$136$	−20.1108			−1.72449
$137$	−0.0153660	−	0.0266147i	−0.00131280	−	0.00227384i	0.865368	−	0.501136i	$-0.167085\pi$
$137$	−0.0153660	−	0.0266147i	−0.00131280	−	0.00227384i	−0.866681	+	0.498863i	$0.833751\pi$
$138$		0			0
$139$	−5.64943	+	9.78509i	−0.479178	+	0.829961i	−0.999715	−	0.0238785i	$-0.992399\pi$
$139$	−5.64943	+	9.78509i	−0.479178	+	0.829961i	0.520537	+	0.853839i	$0.325732\pi$
$140$	−0.124459	+	0.215569i	−0.0105187	+	0.0182189i
$141$		0			0
$142$	−12.5092	−	21.6665i	−1.04975	−	1.81822i
$143$	−5.88760			−0.492346
$144$		0			0
$145$	1.08236			0.0898853
$146$	−3.89081	−	6.73907i	−0.322005	−	0.557730i
$147$		0			0
$148$	−0.815669	+	1.41278i	−0.0670476	+	0.116130i
$149$	−0.759394	+	1.31531i	−0.0622120	+	0.107754i	−0.895454	−	0.445154i	$-0.853149\pi$
$149$	−0.759394	+	1.31531i	−0.0622120	+	0.107754i	0.833242	+	0.552909i	$0.186482\pi$
$150$		0			0
$151$	4.57599	+	7.92585i	0.372389	+	0.644996i	0.989933	−	0.141540i	$-0.0452054\pi$
$151$	4.57599	+	7.92585i	0.372389	+	0.644996i	−0.617544	+	0.786537i	$0.711872\pi$
$152$	4.18136			0.339153
$153$		0			0
$154$	3.04642			0.245488
$155$	3.47359	+	6.01644i	0.279006	+	0.483252i
$156$		0			0
$157$	−1.01015	+	1.74964i	−0.0806191	+	0.139636i	−0.903516	−	0.428554i	$-0.859023\pi$
$157$	−1.01015	+	1.74964i	−0.0806191	+	0.139636i	0.822897	+	0.568191i	$0.192356\pi$
$158$	−7.01143	+	12.1442i	−0.557799	+	0.966137i
$159$		0			0
$160$	−2.03818	−	3.53023i	−0.161132	−	0.279089i
$161$	−1.79562			−0.141515
$162$		0			0
$163$	−13.4360			−1.05239			−0.526194	−	0.850365i	$-0.676381\pi$
$163$	−13.4360			−1.05239			−0.526194	+	0.850365i	$0.676381\pi$
$164$	0.907609	+	1.57202i	0.0708723	+	0.122755i
$165$		0			0
$166$	0.858914	−	1.48768i	0.0666647	−	0.115467i
$167$	5.91837	−	10.2509i	0.457977	−	0.793239i	−0.540877	−	0.841102i	$-0.681908\pi$
$167$	5.91837	−	10.2509i	0.457977	−	0.793239i	0.998854	+	0.0478624i	$0.0152409\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.0384615	−	0.0666173i
$170$	30.8058			2.36270
$171$		0			0
$172$	0.359813			0.0274355
$173$	−9.25849	−	16.0362i	−0.703910	−	1.21921i	−0.967083	−	0.254460i	$-0.918102\pi$
$173$	−9.25849	−	16.0362i	−0.703910	−	1.21921i	0.263173	−	0.964749i	$-0.415231\pi$
$174$		0			0
$175$	−0.340731	+	0.590164i	−0.0257569	+	0.0446122i
$176$	−13.1695	+	22.8102i	−0.992686	+	1.71938i
$177$		0			0
$178$	6.79521	+	11.7696i	0.509322	+	0.882172i
$179$	−18.6972			−1.39749			−0.698745	−	0.715370i	$-0.746258\pi$
$179$	−18.6972			−1.39749			−0.698745	+	0.715370i	$0.746258\pi$
$180$		0			0
$181$	5.93713			0.441304			0.220652	−	0.975353i	$-0.429182\pi$
$181$	5.93713			0.441304			0.220652	+	0.975353i	$0.429182\pi$
$182$	0.258715	+	0.448108i	0.0191772	+	0.0332160i
$183$		0			0
$184$	6.80981	−	11.7949i	0.502026	−	0.869534i
$185$	−7.85440	+	13.6042i	−0.577467	+	1.00020i
$186$		0			0
$187$	−22.7498	−	39.4038i	−1.66363	−	2.88149i
$188$	0.988391			0.0720858
$189$		0			0
$190$	−6.40503			−0.464670
$191$	−2.58212	−	4.47236i	−0.186835	−	0.323609i	0.757358	−	0.653000i	$-0.226490\pi$
$191$	−2.58212	−	4.47236i	−0.186835	−	0.323609i	−0.944193	+	0.329391i	$0.893156\pi$
$192$		0			0
$193$	−11.4596	+	19.8487i	−0.824883	+	1.42874i	0.0771259	+	0.997021i	$0.475426\pi$
$193$	−11.4596	+	19.8487i	−0.824883	+	1.42874i	−0.902009	+	0.431718i	$0.857908\pi$
$194$	3.25959	−	5.64578i	0.234025	−	0.405344i
$195$		0			0
$196$	0.944552	+	1.63601i	0.0674680	+	0.116858i
$197$	14.5085			1.03369			0.516843	−	0.856080i	$-0.327107\pi$
$197$	14.5085			1.03369			0.516843	+	0.856080i	$0.327107\pi$
$198$		0			0
$199$	−9.40273			−0.666542			−0.333271	−	0.942831i	$-0.608152\pi$
$199$	−9.40273			−0.666542			−0.333271	+	0.942831i	$0.608152\pi$
$200$	−2.58441	−	4.47633i	−0.182746	−	0.316525i
$201$		0			0
$202$	−5.00068	+	8.66144i	−0.351847	+	0.609417i
$203$	−0.0702475	+	0.121672i	−0.00493041	+	0.00853972i
$204$		0			0
$205$	8.73972	+	15.1376i	0.610409	+	1.05726i
$206$	21.4612			1.49527
$207$		0			0
$208$	−4.47363			−0.310191
$209$	4.73005	+	8.19268i	0.327184	+	0.566700i
$210$		0			0
$211$	−7.78297	+	13.4805i	−0.535801	+	0.928035i	0.463323	+	0.886190i	$0.346657\pi$
$211$	−7.78297	+	13.4805i	−0.535801	+	0.928035i	−0.999124	+	0.0418457i	$0.986676\pi$
$212$	−0.532274	+	0.921926i	−0.0365567	+	0.0633181i
$213$		0			0
$214$	0.722142	+	1.25079i	0.0493646	+	0.0855021i
$215$	3.46478			0.236296
$216$		0			0
$217$	−0.901773			−0.0612163
$218$	4.35427	+	7.54182i	0.294908	+	0.510796i
$219$		0			0
$220$	2.13576	−	3.69925i	0.143993	−	0.249403i
$221$	3.86402	−	6.69267i	0.259922	−	0.450198i
$222$		0			0
$223$	1.92148	+	3.32810i	0.128672	+	0.222866i	0.923162	−	0.384411i	$-0.125595\pi$
$223$	1.92148	+	3.32810i	0.128672	+	0.222866i	−0.794490	+	0.607277i	$0.792262\pi$
$224$	0.529128			0.0353538
$225$		0			0
$226$	17.9124			1.19151
$227$	12.0292	+	20.8352i	0.798406	+	1.38288i	0.920654	+	0.390379i	$0.127656\pi$
$227$	12.0292	+	20.8352i	0.798406	+	1.38288i	−0.122249	+	0.992500i	$0.539011\pi$
$228$		0			0
$229$	10.0730	−	17.4469i	0.665641	−	1.15292i	−0.313471	−	0.949598i	$-0.601492\pi$
$229$	10.0730	−	17.4469i	0.665641	−	1.15292i	0.979111	−	0.203325i	$-0.0651749\pi$
$230$	−10.4313	+	18.0675i	−0.687820	+	1.19134i
$231$		0			0
$232$	−0.532820	−	0.922872i	−0.0349814	−	0.0605895i
$233$	−5.86085			−0.383957			−0.191978	−	0.981399i	$-0.561490\pi$
$233$	−5.86085			−0.383957			−0.191978	+	0.981399i	$0.561490\pi$
$234$		0			0
$235$	9.51761			0.620860
$236$	0.0621070	+	0.107572i	0.00404282	+	0.00700237i
$237$		0			0
$238$	−1.99936	+	3.46299i	−0.129599	+	0.224472i
$239$	4.58472	−	7.94097i	0.296561	−	0.513659i	−0.678786	−	0.734336i	$-0.737494\pi$
$239$	4.58472	−	7.94097i	0.296561	−	0.513659i	0.975347	+	0.220678i	$0.0708268\pi$
$240$		0			0
$241$	12.1866	+	21.1079i	0.785010	+	1.35968i	0.928993	+	0.370097i	$0.120676\pi$
$241$	12.1866	+	21.1079i	0.785010	+	1.35968i	−0.143983	+	0.989580i	$0.545991\pi$
$242$	−35.6884			−2.29413
$243$		0			0
$244$	1.02666			0.0657254
$245$	9.09546	+	15.7538i	0.581088	+	1.00647i
$246$		0			0
$247$	−0.803392	+	1.39152i	−0.0511186	+	0.0885400i
$248$	3.41993	−	5.92349i	0.217166	−	0.376142i
$249$		0			0
$250$	−6.00678	−	10.4041i	−0.379902	−	0.658010i
$251$	11.2925			0.712775			0.356387	−	0.934338i	$-0.384008\pi$
$251$	11.2925			0.712775			0.356387	+	0.934338i	$0.384008\pi$
$252$		0			0
$253$	30.8136			1.93724
$254$	−9.03247	−	15.6447i	−0.566748	−	0.981636i
$255$		0			0
$256$	−3.23465	+	5.60258i	−0.202166	+	0.350161i
$257$	−12.3174	+	21.3343i	−0.768337	+	1.33080i	0.170127	+	0.985422i	$0.445582\pi$
$257$	−12.3174	+	21.3343i	−0.768337	+	1.33080i	−0.938464	+	0.345377i	$0.887751\pi$
$258$		0			0
$259$	−1.01953	−	1.76588i	−0.0633506	−	0.109727i
$260$	0.725512			0.0449944
$261$		0			0
$262$	−18.1552			−1.12163
$263$	−11.1876	−	19.3775i	−0.689856	−	1.19487i	−0.971884	−	0.235459i	$-0.924341\pi$
$263$	−11.1876	−	19.3775i	−0.689856	−	1.19487i	0.282028	−	0.959406i	$-0.408993\pi$
$264$		0			0
$265$	−5.12548	+	8.87758i	−0.314855	+	0.545346i
$266$	0.415699	−	0.720012i	0.0254882	−	0.0441468i
$267$		0			0
$268$	0.385615	+	0.667904i	0.0235552	+	0.0407988i
$269$	24.7265			1.50760			0.753799	−	0.657105i	$-0.228219\pi$
$269$	24.7265			1.50760			0.753799	+	0.657105i	$0.228219\pi$
$270$		0			0
$271$	10.1047			0.613817			0.306909	−	0.951739i	$-0.400705\pi$
$271$	10.1047			0.613817			0.306909	+	0.951739i	$0.400705\pi$
$272$	−17.2862	−	29.9406i	−1.04813	−	1.81541i
$273$		0			0
$274$	0.0231741	−	0.0401386i	0.00140000	−	0.00242486i
$275$	5.84710	−	10.1275i	0.352593	−	0.610709i
$276$		0			0
$277$	−5.29702	−	9.17471i	−0.318267	−	0.551255i	0.661859	−	0.749628i	$-0.269768\pi$
$277$	−5.29702	−	9.17471i	−0.318267	−	0.551255i	−0.980127	+	0.198373i	$0.936434\pi$
$278$	−17.0403			−1.02201
$279$		0			0
$280$	2.35989			0.141030
$281$	−10.6053	−	18.3690i	−0.632662	−	1.09580i	−0.987005	−	0.160687i	$-0.948629\pi$
$281$	−10.6053	−	18.3690i	−0.632662	−	1.09580i	0.354344	−	0.935115i	$-0.384704\pi$
$282$		0			0
$283$	−0.105435	+	0.182618i	−0.00626745	+	0.0108555i	−0.869142	−	0.494562i	$-0.835328\pi$
$283$	−0.105435	+	0.182618i	−0.00626745	+	0.0108555i	0.862875	+	0.505418i	$0.168662\pi$
$284$	2.27672	−	3.94340i	0.135099	−	0.233998i
$285$		0			0
$286$	−4.43966	−	7.68972i	−0.262523	−	0.454703i
$287$	−2.26890			−0.133929
$288$		0			0
$289$	42.7225			2.51309
$290$	0.816177	+	1.41366i	0.0479276	+	0.0830130i
$291$		0			0
$292$	0.708143	−	1.22654i	0.0414409	−	0.0717778i
$293$	13.2463	−	22.9433i	0.773858	−	1.34036i	−0.161576	−	0.986860i	$-0.551658\pi$
$293$	13.2463	−	22.9433i	0.773858	−	1.34036i	0.935434	−	0.353502i	$-0.115009\pi$
$294$		0			0
$295$	0.598053	+	1.03586i	0.0348200	+	0.0603100i
$296$	15.4661			0.898949
$297$		0			0
$298$	−2.29054			−0.132688
$299$	2.61683	+	4.53248i	0.151335	+	0.262120i
$300$		0			0
$301$	−0.224871	+	0.389489i	−0.0129614	+	0.0224498i
$302$	−6.90124	+	11.9533i	−0.397122	+	0.687835i
$303$		0			0
$304$	3.59408	+	6.22513i	0.206135	+	0.357036i
$305$	9.88614			0.566079
$306$		0			0
$307$	−6.00218			−0.342563			−0.171281	−	0.985222i	$-0.554791\pi$
$307$	−6.00218			−0.342563			−0.171281	+	0.985222i	$0.554791\pi$
$308$	0.277231	+	0.480177i	0.0157967	+	0.0273606i
$309$		0			0
$310$	−5.23867	+	9.07364i	−0.297536	+	0.515348i
$311$	−14.5076	+	25.1279i	−0.822652	+	1.42487i	0.0810487	+	0.996710i	$0.474173\pi$
$311$	−14.5076	+	25.1279i	−0.822652	+	1.42487i	−0.903701	+	0.428165i	$0.859160\pi$
$312$		0			0
$313$	3.40992	+	5.90615i	0.192740	+	0.333835i	0.946157	−	0.323707i	$-0.104929\pi$
$313$	3.40992	+	5.90615i	0.192740	+	0.333835i	−0.753417	+	0.657543i	$0.771596\pi$
$314$	−3.04691			−0.171947
$315$		0			0
$316$	−2.55222			−0.143573
$317$	8.63590	+	14.9578i	0.485040	+	0.840115i	0.999852	−	0.0171885i	$-0.00547153\pi$
$317$	8.63590	+	14.9578i	0.485040	+	0.840115i	−0.514812	+	0.857303i	$0.672138\pi$
$318$		0			0
$319$	1.20548	−	2.08795i	0.0674938	−	0.116903i
$320$	−8.75062	+	15.1565i	−0.489174	+	0.847275i
$321$		0			0
$322$	−1.35402	−	2.34524i	−0.0754569	−	0.130695i
$323$	−12.4173			−0.690916
$324$		0			0
$325$	1.98624			0.110177
$326$	−10.1317	−	17.5486i	−0.561142	−	0.971926i
$327$		0			0
$328$	8.60470	−	14.9038i	0.475115	−	0.822923i
$329$	−0.617712	+	1.06991i	−0.0340556	+	0.0589859i
$330$		0			0
$331$	−15.1048	−	26.1623i	−0.830237	−	1.43801i	−0.897850	−	0.440301i	$-0.854872\pi$
$331$	−15.1048	−	26.1623i	−0.830237	−	1.43801i	0.0676133	−	0.997712i	$-0.478462\pi$
$332$	0.312652			0.0171590
$333$		0			0
$334$	17.8515			0.976788
$335$	3.71324	+	6.43151i	0.202876	+	0.351391i
$336$		0			0
$337$	−4.70433	+	8.14814i	−0.256261	+	0.443857i	−0.965237	−	0.261375i	$-0.915824\pi$
$337$	−4.70433	+	8.14814i	−0.256261	+	0.443857i	0.708976	+	0.705232i	$0.249157\pi$
$338$	0.754070	−	1.30609i	0.0410160	−	0.0710418i
$339$		0			0
$340$	2.80339	+	4.85562i	0.152035	+	0.263333i
$341$	15.4748			0.838008
$342$		0			0
$343$	−4.76289			−0.257172
$344$	−1.70563	−	2.95423i	−0.0919613	−	0.159282i
$345$		0			0
$346$	13.9631	−	24.1848i	0.750661	−	1.30018i
$347$	5.84166	−	10.1181i	0.313597	−	0.543165i	−0.665541	−	0.746361i	$-0.731799\pi$
$347$	5.84166	−	10.1181i	0.313597	−	0.543165i	0.979138	+	0.203195i	$0.0651327\pi$
$348$		0			0
$349$	2.37484	+	4.11334i	0.127122	+	0.220182i	0.922560	−	0.385853i	$-0.126093\pi$
$349$	2.37484	+	4.11334i	0.127122	+	0.220182i	−0.795438	+	0.606035i	$0.792759\pi$
$350$	−1.02774			−0.0549351
$351$		0			0
$352$	−9.08006			−0.483969
$353$	0.136823	+	0.236985i	0.00728238	+	0.0126134i	0.869644	−	0.493680i	$-0.164349\pi$
$353$	0.136823	+	0.236985i	0.00728238	+	0.0126134i	−0.862361	+	0.506293i	$0.831015\pi$
$354$		0			0
$355$	21.9234	−	37.9725i	1.16358	−	2.01537i
$356$	−1.23676	+	2.14212i	−0.0655479	+	0.113532i
$357$		0			0
$358$	−14.0990	−	24.4201i	−0.745154	−	1.29064i
$359$	11.3893			0.601105			0.300553	−	0.953765i	$-0.402829\pi$
$359$	11.3893			0.601105			0.300553	+	0.953765i	$0.402829\pi$
$360$		0			0
$361$	−16.4182			−0.864118
$362$	4.47702	+	7.75442i	0.235307	+	0.407563i
$363$		0			0
$364$	−0.0470872	+	0.0815574i	−0.00246804	+	0.00427477i
$365$	6.81898	−	11.8108i	0.356922	−	0.618207i
$366$		0			0
$367$	3.80454	+	6.58966i	0.198595	+	0.343977i	0.948073	−	0.318052i	$-0.103029\pi$
$367$	3.80454	+	6.58966i	0.198595	+	0.343977i	−0.749478	+	0.662029i	$0.769695\pi$
$368$	23.4134			1.22051
$369$		0			0
$370$	−23.6911			−1.23164
$371$	−0.665307	−	1.15235i	−0.0345410	−	0.0598268i
$372$		0			0
$373$	14.0594	−	24.3517i	0.727970	−	1.26088i	−0.229769	−	0.973245i	$-0.573797\pi$
$373$	14.0594	−	24.3517i	0.727970	−	1.26088i	0.957740	−	0.287637i	$-0.0928695\pi$
$374$	34.3099	−	59.4264i	1.77412	−	3.07287i
$375$		0			0
$376$	−4.68528	−	8.11515i	−0.241625	−	0.418507i
$377$	0.409497			0.0210902
$378$		0			0
$379$	−16.8213			−0.864052			−0.432026	−	0.901861i	$-0.642201\pi$
$379$	−16.8213			−0.864052			−0.432026	+	0.901861i	$0.642201\pi$
$380$	−0.582871	−	1.00956i	−0.0299006	−	0.0517894i
$381$		0			0
$382$	3.89420	−	6.74495i	0.199244	−	0.345101i
$383$	2.42358	−	4.19776i	0.123839	−	0.214495i	−0.797440	−	0.603399i	$-0.793813\pi$
$383$	2.42358	−	4.19776i	0.123839	−	0.214495i	0.921278	+	0.388903i	$0.127146\pi$
$384$		0			0
$385$	2.66956	+	4.62382i	0.136053	+	0.235651i
$386$	−34.5655			−1.75934
$387$		0			0
$388$	1.18652			0.0602364
$389$	−18.1834	−	31.4946i	−0.921937	−	1.59684i	−0.796415	−	0.604751i	$-0.793273\pi$
$389$	−18.1834	−	31.4946i	−0.921937	−	1.59684i	−0.125522	−	0.992091i	$-0.540061\pi$
$390$		0			0
$391$	−20.2229	+	35.0271i	−1.02272	+	1.77140i
$392$	8.95494	−	15.5104i	0.452293	−	0.783394i
$393$		0			0
$394$	10.9404	+	18.9493i	0.551169	+	0.954654i
$395$	−24.5763			−1.23657
$396$		0			0
$397$	−29.0866			−1.45981			−0.729907	−	0.683547i	$-0.760437\pi$
$397$	−29.0866			−1.45981			−0.729907	+	0.683547i	$0.760437\pi$
$398$	−7.09032	−	12.2808i	−0.355406	−	0.615581i
$399$		0			0
$400$	4.44286	−	7.69525i	0.222143	−	0.384763i
$401$	2.19451	−	3.80101i	0.109589	−	0.189813i	−0.806015	−	0.591895i	$-0.798380\pi$
$401$	2.19451	−	3.80101i	0.109589	−	0.189813i	0.915604	+	0.402082i	$0.131713\pi$
$402$		0			0
$403$	1.31419	+	2.27624i	0.0654643	+	0.113388i
$404$	−1.82029			−0.0905628
$405$		0			0
$406$	−0.211886			−0.0105157
$407$	17.4956	+	30.3033i	0.867225	+	1.50208i
$408$		0			0
$409$	−0.278736	+	0.482785i	−0.0137826	+	0.0238722i	−0.872834	−	0.488016i	$-0.837721\pi$
$409$	−0.278736	+	0.482785i	−0.0137826	+	0.0238722i	0.859052	+	0.511889i	$0.171054\pi$
$410$	−13.1807	+	22.8297i	−0.650950	+	1.12748i
$411$		0			0
$412$	1.95301	+	3.38272i	0.0962181	+	0.166655i
$413$	−0.155259			−0.00763981
$414$		0			0
$415$	3.01065			0.147787
$416$	−0.771118	−	1.33562i	−0.0378072	−	0.0654839i
$417$		0			0
$418$	−7.13358	+	12.3557i	−0.348915	+	0.604338i
$419$	−3.99151	+	6.91350i	−0.194998	+	0.337747i	−0.946900	−	0.321528i	$-0.895803\pi$
$419$	−3.99151	+	6.91350i	−0.194998	+	0.337747i	0.751902	+	0.659275i	$0.229137\pi$
$420$		0			0
$421$	6.51423	+	11.2830i	0.317484	+	0.549899i	0.979962	−	0.199182i	$-0.0638286\pi$
$421$	6.51423	+	11.2830i	0.317484	+	0.549899i	−0.662478	+	0.749081i	$0.730495\pi$
$422$	−23.4756			−1.14277
$423$		0			0
$424$	10.0926			0.490139
$425$	7.67487	+	13.2933i	0.372286	+	0.644818i
$426$		0			0
$427$	−0.641630	+	1.11134i	−0.0310507	+	0.0537813i
$428$	−0.131433	+	0.227648i	−0.00635305	+	0.0110038i
$429$		0			0
$430$	2.61269	+	4.52531i	0.125995	+	0.218230i
$431$	−13.8846			−0.668796			−0.334398	−	0.942432i	$-0.608533\pi$
$431$	−13.8846			−0.668796			−0.334398	+	0.942432i	$0.608533\pi$
$432$		0			0
$433$	−8.20792			−0.394447			−0.197224	−	0.980359i	$-0.563193\pi$
$433$	−8.20792			−0.394447			−0.197224	+	0.980359i	$0.563193\pi$
$434$	−0.680000	−	1.17779i	−0.0326411	−	0.0565360i
$435$		0			0
$436$	−0.792495	+	1.37264i	−0.0379536	+	0.0657376i
$437$	4.20467	−	7.28271i	0.201137	−	0.348379i
$438$		0			0
$439$	5.34748	+	9.26210i	0.255221	+	0.442056i	0.964956	−	0.262413i	$-0.0845184\pi$
$439$	5.34748	+	9.26210i	0.255221	+	0.442056i	−0.709734	+	0.704469i	$0.751185\pi$
$440$	−40.4967			−1.93061
$441$		0			0
$442$	11.6550			0.554370
$443$	9.54960	+	16.5404i	0.453715	+	0.785857i	0.998613	−	0.0526447i	$-0.0167651\pi$
$443$	9.54960	+	16.5404i	0.453715	+	0.785857i	−0.544898	+	0.838502i	$0.683432\pi$
$444$		0			0
$445$	−11.9092	+	20.6273i	−0.564550	+	0.977830i
$446$	−2.89786	+	5.01924i	−0.137218	+	0.237668i
$447$		0			0
$448$	−1.13587	−	1.96738i	−0.0536646	−	0.0929498i
$449$	28.5873			1.34912			0.674558	−	0.738221i	$-0.264334\pi$
$449$	28.5873			1.34912			0.674558	+	0.738221i	$0.264334\pi$
$450$		0			0
$451$	38.9353			1.83339
$452$	1.63006	+	2.82335i	0.0766717	+	0.132799i
$453$		0			0
$454$	−18.1417	+	31.4224i	−0.851433	+	1.47472i
$455$	−0.453421	+	0.785348i	−0.0212567	+	0.0368177i
$456$		0			0
$457$	−3.54691	−	6.14343i	−0.165918	−	0.287378i	0.771063	−	0.636759i	$-0.219725\pi$
$457$	−3.54691	−	6.14343i	−0.165918	−	0.287378i	−0.936981	+	0.349381i	$0.886392\pi$
$458$	30.3829			1.41970
$459$		0			0
$460$	−3.79708			−0.177040
$461$	−10.4066	−	18.0248i	−0.484685	−	0.839500i	0.515160	−	0.857094i	$-0.327733\pi$
$461$	−10.4066	−	18.0248i	−0.484685	−	0.839500i	−0.999845	+	0.0175945i	$0.994399\pi$
$462$		0			0
$463$	9.78714	−	16.9518i	0.454847	−	0.787818i	−0.543832	−	0.839194i	$-0.683027\pi$
$463$	9.78714	−	16.9518i	0.454847	−	0.787818i	0.998679	+	0.0513757i	$0.0163606\pi$
$464$	0.915970	−	1.58651i	0.0425228	−	0.0736517i
$465$		0			0
$466$	−4.41949	−	7.65478i	−0.204729	−	0.354601i
$467$	−32.2324			−1.49154			−0.745768	−	0.666206i	$-0.767917\pi$
$467$	−32.2324			−1.49154			−0.745768	+	0.666206i	$0.767917\pi$
$468$		0			0
$469$	−0.963986			−0.0445127
$470$	7.17694	+	12.4308i	0.331048	+	0.573391i
$471$		0			0
$472$	0.588813	−	1.01985i	0.0271023	−	0.0469426i
$473$	3.85889	−	6.68379i	0.177432	−	0.307321i
$474$		0			0
$475$	−1.59573	−	2.76389i	−0.0732171	−	0.126816i
$476$	−0.727783			−0.0333579
$477$		0			0
$478$	13.8288			0.632515
$479$	−0.525387	−	0.909996i	−0.0240055	−	0.0415788i	0.853773	−	0.520645i	$-0.174309\pi$
$479$	−0.525387	−	0.909996i	−0.0240055	−	0.0415788i	−0.877779	+	0.479067i	$0.840975\pi$
$480$		0			0
$481$	−2.97160	+	5.14697i	−0.135493	+	0.234682i
$482$	−18.3792	+	31.8336i	−0.837148	+	1.44998i
$483$		0			0
$484$	−3.24771	−	5.62520i	−0.147623	−	0.255691i
$485$	11.4255			0.518803
$486$		0			0
$487$	18.2019			0.824808			0.412404	−	0.911001i	$-0.364689\pi$
$487$	18.2019			0.824808			0.412404	+	0.911001i	$0.364689\pi$
$488$	−4.86670	−	8.42938i	−0.220305	−	0.381580i
$489$		0			0
$490$	−13.7172	+	23.7589i	−0.619681	+	1.07332i
$491$	0.687377	−	1.19057i	0.0310209	−	0.0537297i	−0.850098	−	0.526624i	$-0.823457\pi$
$491$	0.687377	−	1.19057i	0.0310209	−	0.0537297i	0.881119	+	0.472895i	$0.156791\pi$
$492$		0			0
$493$	1.58230	+	2.74063i	0.0712634	+	0.123432i
$494$	−2.42326			−0.109027
$495$		0			0
$496$	11.7584			0.527967
$497$	2.84575	+	4.92899i	0.127649	+	0.221095i
$498$		0			0
$499$	−11.8467	+	20.5191i	−0.530330	+	0.918559i	0.469044	+	0.883175i	$0.344599\pi$
$499$	−11.8467	+	20.5191i	−0.530330	+	0.918559i	−0.999374	+	0.0353839i	$0.988735\pi$
$500$	1.09326	−	1.89358i	0.0488920	−	0.0846835i
$501$		0			0
$502$	8.51532	+	14.7490i	0.380057	+	0.658279i
$503$	8.31698			0.370836			0.185418	−	0.982660i	$-0.440636\pi$
$503$	8.31698			0.370836			0.185418	+	0.982660i	$0.440636\pi$
$504$		0			0
$505$	−17.5283			−0.779998
$506$	23.2357	+	40.2453i	1.03295	+	1.78912i
$507$		0			0
$508$	1.64395	−	2.84740i	0.0729384	−	0.126333i
$509$	4.83988	−	8.38292i	0.214524	−	0.371567i	−0.738601	−	0.674143i	$-0.764513\pi$
$509$	4.83988	−	8.38292i	0.214524	−	0.371567i	0.953125	+	0.302576i	$0.0978466\pi$
$510$		0			0
$511$	0.885131	+	1.53309i	0.0391559	+	0.0678200i
$512$	16.3842			0.724086
$513$		0			0
$514$	−37.1527			−1.63873
$515$	18.8063	+	32.5735i	0.828706	+	1.43536i
$516$		0			0
$517$	10.6002	−	18.3601i	0.466196	−	0.807476i
$518$	1.53760	−	2.66320i	0.0675582	−	0.117014i
$519$		0			0
$520$	−3.43915	−	5.95679i	−0.150817	−	0.261223i
$521$	13.7581			0.602753			0.301377	−	0.953505i	$-0.402554\pi$
$521$	13.7581			0.602753			0.301377	+	0.953505i	$0.402554\pi$
$522$		0			0
$523$	−11.2006			−0.489769			−0.244885	−	0.969552i	$-0.578750\pi$
$523$	−11.2006			−0.489769			−0.244885	+	0.969552i	$0.578750\pi$
$524$	−1.65216	−	2.86162i	−0.0721749	−	0.125011i
$525$		0			0
$526$	16.8724	−	29.2239i	0.735674	−	1.27422i
$527$	−10.1561	+	17.5909i	−0.442406	+	0.766270i
$528$		0			0
$529$	−2.19556	−	3.80282i	−0.0954592	−	0.165340i
$530$	−15.4599			−0.671534
$531$		0			0
$532$	0.151318			0.00656046
$533$	3.30656	+	5.72712i	0.143223	+	0.248069i
$534$		0			0
$535$	−1.26562	+	2.19212i	−0.0547175	+	0.0947734i
$536$	3.65587	−	6.33215i	0.157909	−	0.273507i
$537$		0			0
$538$	18.6455	+	32.2949i	0.803864	+	1.39233i
$539$	40.5201			1.74533
$540$		0			0
$541$	−24.0677			−1.03475			−0.517375	−	0.855759i	$-0.673091\pi$
$541$	−24.0677			−1.03475			−0.517375	+	0.855759i	$0.673091\pi$
$542$	7.61966	+	13.1976i	0.327292	+	0.566887i
$543$		0			0
$544$	5.95923	−	10.3217i	0.255500	−	0.442538i
$545$	−7.63124	+	13.2177i	−0.326887	+	0.566184i
$546$		0			0
$547$	−10.8189	−	18.7389i	−0.462582	−	0.801216i	0.536507	−	0.843896i	$-0.319744\pi$
$547$	−10.8189	−	18.7389i	−0.462582	−	0.801216i	−0.999089	+	0.0426803i	$0.986410\pi$
$548$	0.00843555			0.000360349
$549$		0			0
$550$	17.6365			0.752022
$551$	−0.328987	−	0.569822i	−0.0140153	−	0.0242752i
$552$		0			0
$553$	1.59505	−	2.76271i	0.0678285	−	0.117482i
$554$	7.98865	−	13.8368i	0.339405	−	0.587867i
$555$		0			0
$556$	−1.55070	−	2.68589i	−0.0657643	−	0.113907i
$557$	−23.1251			−0.979840			−0.489920	−	0.871767i	$-0.662974\pi$
$557$	−23.1251			−0.979840			−0.489920	+	0.871767i	$0.662974\pi$
$558$		0			0
$559$	1.31085			0.0554432
$560$	2.02844	+	3.51336i	0.0857172	+	0.148467i
$561$		0			0
$562$	15.9944	−	27.7030i	0.674681	−	1.16858i
$563$	−12.6167	+	21.8528i	−0.531732	+	0.920987i	0.467582	+	0.883950i	$0.345125\pi$
$563$	−12.6167	+	21.8528i	−0.531732	+	0.920987i	−0.999314	+	0.0370372i	$0.988208\pi$
$564$		0			0
$565$	15.6965	+	27.1872i	0.660357	+	1.14377i
$566$	−0.318021			−0.0133674
$567$		0			0
$568$	−43.1695			−1.81135
$569$	16.7024	+	28.9294i	0.700200	+	1.21278i	0.968396	+	0.249418i	$0.0802394\pi$
$569$	16.7024	+	28.9294i	0.700200	+	1.21278i	−0.268195	+	0.963365i	$0.586427\pi$
$570$		0			0
$571$	−19.2445	+	33.3324i	−0.805355	+	1.39492i	0.110695	+	0.993854i	$0.464692\pi$
$571$	−19.2445	+	33.3324i	−0.805355	+	1.39492i	−0.916051	+	0.401062i	$0.868641\pi$
$572$	0.808037	−	1.39956i	0.0337857	−	0.0585186i
$573$		0			0
$574$	−1.71091	−	2.96339i	−0.0714121	−	0.123689i
$575$	−10.3953			−0.433514
$576$		0			0
$577$	−40.5935			−1.68993			−0.844965	−	0.534821i	$-0.820379\pi$
$577$	−40.5935			−1.68993			−0.844965	+	0.534821i	$0.820379\pi$
$578$	32.2158	+	55.7994i	1.34000	+	2.32095i
$579$		0			0
$580$	−0.148548	+	0.257292i	−0.00616810	+	0.0106835i
$581$	−0.195397	+	0.338438i	−0.00810644	+	0.0140408i
$582$		0			0
$583$	11.4170	+	19.7748i	0.472842	+	0.818987i
$584$	−13.4273			−0.555624
$585$		0			0
$586$	39.9546			1.65051
$587$	−4.86211	−	8.42142i	−0.200681	−	0.347589i	0.748067	−	0.663623i	$-0.230982\pi$
$587$	−4.86211	−	8.42142i	−0.200681	−	0.347589i	−0.948748	+	0.316034i	$0.897649\pi$
$588$		0			0
$589$	2.11161	−	3.65742i	0.0870076	−	0.150702i
$590$	−0.901947	+	1.56222i	−0.0371326	+	0.0643155i
$591$		0			0
$592$	13.2939	+	23.0256i	0.546374	+	0.946348i
$593$	−12.0667			−0.495518			−0.247759	−	0.968822i	$-0.579694\pi$
$593$	−12.0667			−0.495518			−0.247759	+	0.968822i	$0.579694\pi$
$594$		0			0
$595$	−7.00811			−0.287304
$596$	−0.208444	−	0.361036i	−0.00853821	−	0.0147886i
$597$		0			0
$598$	−3.94654	+	6.83561i	−0.161386	+	0.279529i
$599$	−2.33962	+	4.05234i	−0.0955943	+	0.165574i	−0.909856	−	0.414923i	$-0.863808\pi$
$599$	−2.33962	+	4.05234i	−0.0955943	+	0.165574i	0.814262	+	0.580497i	$0.197142\pi$
$600$		0			0
$601$	−9.74305	−	16.8755i	−0.397427	−	0.688365i	0.595980	−	0.802999i	$-0.296764\pi$
$601$	−9.74305	−	16.8755i	−0.397427	−	0.688365i	−0.993408	+	0.114634i	$0.963430\pi$
$602$	−0.678275			−0.0276444
$603$		0			0
$604$	−2.51211			−0.102216
$605$	−31.2735	−	54.1673i	−1.27145	−	2.20221i
$606$		0			0
$607$	5.80961	−	10.0625i	0.235805	−	0.408426i	−0.723701	−	0.690113i	$-0.757561\pi$
$607$	5.80961	−	10.0625i	0.235805	−	0.408426i	0.959506	+	0.281687i	$0.0908941\pi$
$608$	−1.23902	+	2.14605i	−0.0502489	+	0.0870336i
$609$		0			0
$610$	7.45484	+	12.9122i	0.301838	+	0.522798i
$611$	3.60086			0.145675
$612$		0			0
$613$	24.0957			0.973218			0.486609	−	0.873620i	$-0.338234\pi$
$613$	24.0957			0.973218			0.486609	+	0.873620i	$0.338234\pi$
$614$	−4.52607	−	7.83938i	−0.182657	−	0.316372i
$615$		0			0
$616$	2.62832	−	4.55238i	0.105898	−	0.183421i
$617$	3.35045	−	5.80316i	0.134884	−	0.233626i	−0.790669	−	0.612244i	$-0.790267\pi$
$617$	3.35045	−	5.80316i	0.134884	−	0.233626i	0.925553	+	0.378618i	$0.123600\pi$
$618$		0			0
$619$	−5.29020	−	9.16290i	−0.212631	−	0.368288i	0.739906	−	0.672710i	$-0.234870\pi$
$619$	−5.29020	−	9.16290i	−0.212631	−	0.368288i	−0.952537	+	0.304422i	$0.901537\pi$
$620$	−1.90692			−0.0765837
$621$		0			0
$622$	−43.7591			−1.75458
$623$	−1.54586	−	2.67751i	−0.0619337	−	0.107272i
$624$		0			0
$625$	15.4930	−	26.8347i	0.619721	−	1.07339i
$626$	−5.14263	+	8.90730i	−0.205541	+	0.356007i
$627$		0			0
$628$	−0.277275	−	0.480254i	−0.0110645	−	0.0191642i
$629$	−45.9293			−1.83132
$630$		0			0
$631$	31.1707			1.24089			0.620443	−	0.784252i	$-0.286953\pi$
$631$	31.1707			1.24089			0.620443	+	0.784252i	$0.286953\pi$
$632$	12.0983	+	20.9549i	0.481245	+	0.833541i
$633$		0			0
$634$	−13.0242	+	22.5585i	−0.517255	+	0.895912i
$635$	15.8302	−	27.4187i	0.628203	−	1.08808i
$636$		0			0
$637$	3.44114	+	5.96024i	0.136343	+	0.236153i
$638$	3.63606			0.143953
$639$		0			0
$640$	−34.5470			−1.36559
$641$	−7.95094	−	13.7714i	−0.314043	−	0.543939i	0.665191	−	0.746674i	$-0.268350\pi$
$641$	−7.95094	−	13.7714i	−0.314043	−	0.543939i	−0.979234	+	0.202735i	$0.935017\pi$
$642$		0			0
$643$	11.9550	−	20.7067i	0.471460	−	0.816593i	−0.528007	−	0.849240i	$-0.677061\pi$
$643$	11.9550	−	20.7067i	0.471460	−	0.816593i	0.999467	+	0.0326474i	$0.0103938\pi$
$644$	0.246438	−	0.426843i	0.00971102	−	0.0168200i
$645$		0			0
$646$	−9.36350	−	16.2181i	−0.368402	−	0.638091i
$647$	−41.8996			−1.64725			−0.823623	−	0.567138i	$-0.808050\pi$
$647$	−41.8996			−1.64725			−0.823623	+	0.567138i	$0.808050\pi$
$648$		0			0
$649$	2.66432			0.104584
$650$	1.49777	+	2.59421i	0.0587472	+	0.101753i
$651$		0			0
$652$	1.84401	−	3.19391i	0.0722169	−	0.125083i
$653$	−1.59273	+	2.75868i	−0.0623282	+	0.107956i	−0.895506	−	0.445050i	$-0.853186\pi$
$653$	−1.59273	+	2.75868i	−0.0623282	+	0.107956i	0.833177	+	0.553006i	$0.186519\pi$
$654$		0			0
$655$	−15.9093	−	27.5557i	−0.621627	−	1.07669i
$656$	29.5846			1.15509
$657$		0			0
$658$	−1.86319			−0.0726348
$659$	−7.21287	−	12.4931i	−0.280973	−	0.486660i	0.690651	−	0.723188i	$-0.257324\pi$
$659$	−7.21287	−	12.4931i	−0.280973	−	0.486660i	−0.971625	+	0.236528i	$0.923991\pi$
$660$		0			0
$661$	−6.13531	+	10.6267i	−0.238636	+	0.413330i	−0.960323	−	0.278890i	$-0.910034\pi$
$661$	−6.13531	+	10.6267i	−0.238636	+	0.413330i	0.721687	+	0.692219i	$0.243367\pi$
$662$	22.7802	−	39.4565i	0.885378	−	1.53352i
$663$		0			0
$664$	−1.48207	−	2.56702i	−0.0575154	−	0.0996195i
$665$	1.45710			0.0565039
$666$		0			0
$667$	−2.14317			−0.0829837
$668$	1.62452	+	2.81375i	0.0628545	+	0.108867i
$669$		0			0
$670$	−5.60008	+	9.69962i	−0.216350	+	0.374729i
$671$	11.0107	−	19.0710i	0.425062	−	0.736228i
$672$		0			0
$673$	−21.5015	−	37.2417i	−0.828821	−	1.43556i	−0.898964	−	0.438023i	$-0.855679\pi$
$673$	−21.5015	−	37.2417i	−0.828821	−	1.43556i	0.0701426	−	0.997537i	$-0.477655\pi$
$674$	−14.1896			−0.546562
$675$		0			0
$676$	0.274488			0.0105572
$677$	−19.5460	−	33.8547i	−0.751213	−	1.30114i	−0.947235	−	0.320540i	$-0.896136\pi$
$677$	−19.5460	−	33.8547i	−0.751213	−	1.30114i	0.196021	−	0.980600i	$-0.437198\pi$
$678$		0			0
$679$	−0.741535	+	1.28438i	−0.0284575	+	0.0492898i
$680$	26.5779	−	46.0343i	1.01922	−	1.76533i
$681$		0			0
$682$	11.6691	+	20.2115i	0.446833	+	0.773937i
$683$	−12.2513			−0.468783			−0.234391	−	0.972142i	$-0.575310\pi$
$683$	−12.2513			−0.468783			−0.234391	+	0.972142i	$0.575310\pi$
$684$		0			0
$685$	0.0812292			0.00310361
$686$	−3.59156	−	6.22076i	−0.137126	−	0.237510i
$687$		0			0
$688$	2.93214	−	5.07861i	0.111787	−	0.193620i
$689$	−1.93915	+	3.35871i	−0.0738759	+	0.127957i
$690$		0			0
$691$	−11.5612	−	20.0247i	−0.439810	−	0.761774i	0.557864	−	0.829932i	$-0.311621\pi$
$691$	−11.5612	−	20.0247i	−0.439810	−	0.761774i	−0.997675	+	0.0681586i	$0.978288\pi$
$692$	5.08269			0.193215
$693$		0			0
$694$	17.6201			0.668849
$695$	−14.9323	−	25.8635i	−0.566414	−	0.981057i
$696$		0			0
$697$	−25.5532	+	44.2594i	−0.967895	+	1.67644i
$698$	−3.58159	+	6.20349i	−0.135565	+	0.234806i
$699$		0			0
$700$	−0.0935266	−	0.161993i	−0.00353497	−	0.00612275i
$701$	16.6318			0.628173			0.314086	−	0.949394i	$-0.398302\pi$
$701$	16.6318			0.628173			0.314086	+	0.949394i	$0.398302\pi$
$702$		0			0
$703$	9.54945			0.360164
$704$	19.4919	+	33.7610i	0.734630	+	1.27242i
$705$		0			0
$706$	−0.206349	+	0.357407i	−0.00776605	+	0.0134512i
$707$	1.13762	−	1.97042i	0.0427846	−	0.0741052i
$708$		0			0
$709$	13.2968	+	23.0307i	0.499371	+	0.864936i	1.00000		0.000725798i	$-0.000231029\pi$
$709$	13.2968	+	23.0307i	0.499371	+	0.864936i	−0.500628	+	0.865662i	$0.666898\pi$
$710$	66.1273			2.48171
$711$		0			0
$712$	23.4504			0.878842
$713$	−6.87800	−	11.9130i	−0.257583	−	0.446147i
$714$		0			0
$715$	7.78090	−	13.4769i	0.290989	−	0.504008i
$716$	2.56607	−	4.44456i	0.0958985	−	0.166101i
$717$		0			0
$718$	8.58835	+	14.8755i	0.320514	+	0.555147i
$719$	−27.2922			−1.01783			−0.508914	−	0.860817i	$-0.669953\pi$
$719$	−27.2922			−1.01783			−0.508914	+	0.860817i	$0.669953\pi$
$720$		0			0
$721$	−4.88228			−0.181826
$722$	−12.3805	−	21.4437i	−0.460755	−	0.798051i
$723$		0			0
$724$	−0.814835	+	1.41134i	−0.0302831	+	0.0524519i
$725$	−0.406680	+	0.704391i	−0.0151037	+	0.0261604i
$726$		0			0
$727$	−3.33939	−	5.78399i	−0.123851	−	0.214516i	0.797432	−	0.603409i	$-0.206191\pi$
$727$	−3.33939	−	5.78399i	−0.123851	−	0.214516i	−0.921283	+	0.388892i	$0.872858\pi$
$728$	0.892832			0.0330906
$729$		0			0
$730$	20.5680			0.761255
$731$	5.06516	+	8.77312i	0.187342	+	0.324485i
$732$		0			0
$733$	9.14541	−	15.8403i	0.337793	−	0.585075i	−0.646224	−	0.763148i	$-0.723653\pi$
$733$	9.14541	−	15.8403i	0.337793	−	0.585075i	0.984017	+	0.178073i	$0.0569862\pi$
$734$	−5.73778	+	9.93813i	−0.211785	+	0.366823i
$735$		0			0
$736$	4.03576	+	6.99015i	0.148760	+	0.257660i
$737$	16.5424			0.609348
$738$		0			0
$739$	1.08400			0.0398757			0.0199378	−	0.999801i	$-0.493653\pi$
$739$	1.08400			0.0398757			0.0199378	+	0.999801i	$0.493653\pi$
$740$	−2.15594	−	3.73419i	−0.0792538	−	0.137272i
$741$		0			0
$742$	1.00338	−	1.73790i	0.0368351	−	0.0638003i
$743$	6.43088	−	11.1386i	0.235926	−	0.408636i	−0.723615	−	0.690204i	$-0.757521\pi$
$743$	6.43088	−	11.1386i	0.235926	−	0.408636i	0.959541	+	0.281567i	$0.0908543\pi$
$744$		0			0
$745$	−2.00719	−	3.47656i	−0.0735378	−	0.127371i
$746$	42.4072			1.55264
$747$		0			0
$748$	12.4891			0.456646
$749$	−0.164282	−	0.284545i	−0.00600275	−	0.0103971i
$750$		0			0
$751$	4.57485	−	7.92388i	0.166939	−	0.289146i	−0.770403	−	0.637557i	$-0.779945\pi$
$751$	4.57485	−	7.92388i	0.166939	−	0.289146i	0.937342	+	0.348410i	$0.113278\pi$
$752$	8.05446	−	13.9507i	0.293716	−	0.508731i
$753$		0			0
$754$	0.308790	+	0.534839i	0.0112455	+	0.0194777i
$755$	−24.1901			−0.880367
$756$		0			0
$757$	31.3956			1.14109			0.570546	−	0.821266i	$-0.306732\pi$
$757$	31.3956			1.14109			0.570546	+	0.821266i	$0.306732\pi$
$758$	−12.6844	−	21.9701i	−0.460720	−	0.797990i
$759$		0			0
$760$	−5.52598	+	9.57127i	−0.200448	+	0.347187i
$761$	−21.3439	+	36.9687i	−0.773714	+	1.34011i	0.161800	+	0.986824i	$0.448270\pi$
$761$	−21.3439	+	36.9687i	−0.773714	+	1.34011i	−0.935514	+	0.353289i	$0.885063\pi$
$762$		0			0
$763$	−0.990566	−	1.71571i	−0.0358609	−	0.0621129i
$764$	1.41752			0.0512841
$765$		0			0
$766$	7.31019			0.264128
$767$	0.226265	+	0.391903i	0.00816996	+	0.0141508i
$768$		0			0
$769$	4.50243	−	7.79845i	0.162362	−	0.281219i	−0.773353	−	0.633975i	$-0.781422\pi$
$769$	4.50243	−	7.79845i	0.162362	−	0.281219i	0.935715	+	0.352756i	$0.114755\pi$
$770$	−4.02607	+	6.97336i	−0.145090	+	0.251303i
$771$		0			0
$772$	−3.14553	−	5.44822i	−0.113210	−	0.196086i
$773$	−18.7179			−0.673235			−0.336617	−	0.941641i	$-0.609283\pi$
$773$	−18.7179			−0.673235			−0.336617	+	0.941641i	$0.609283\pi$
$774$		0			0
$775$	−5.22059			−0.187529
$776$	−5.62447	−	9.74187i	−0.201907	−	0.349713i
$777$		0			0
$778$	27.4232	−	47.4983i	0.983169	−	1.70290i
$779$	5.31292	−	9.20225i	0.190355	−	0.329705i
$780$		0			0
$781$	−48.8343	−	84.5835i	−1.74743	−	3.02664i
$782$	−60.9980			−2.18128
$783$		0			0
$784$	30.7888			1.09960
$785$	−2.66999	−	4.62456i	−0.0952960	−	0.165058i
$786$		0			0
$787$	−15.2163	+	26.3553i	−0.542401	+	0.939466i	0.456365	+	0.889793i	$0.349151\pi$
$787$	−15.2163	+	26.3553i	−0.542401	+	0.939466i	−0.998766	+	0.0496732i	$0.984182\pi$
$788$	−1.99120	+	3.44886i	−0.0709335	+	0.122860i
$789$		0			0
$790$	−18.5323	−	32.0988i	−0.659348	−	1.14202i
$791$	−4.07494			−0.144888
$792$		0			0
$793$	3.74029			0.132822
$794$	−21.9333	−	37.9896i	−0.778384	−	1.34820i
$795$		0			0
$796$	1.29047	−	2.23515i	0.0457394	−	0.0792230i
$797$	−13.8933	+	24.0639i	−0.492126	+	0.852387i	−0.999959	−	0.00906828i	$-0.997113\pi$
$797$	−13.8933	+	24.0639i	−0.492126	+	0.852387i	0.507833	+	0.861456i	$0.330447\pi$
$798$		0			0
$799$	13.9138	+	24.0994i	0.492234	+	0.852574i
$800$	3.06325			0.108302
$801$		0			0
$802$	6.61926			0.233734
$803$	−15.1892	−	26.3085i	−0.536016	−	0.928408i
$804$		0			0
$805$	2.37305	−	4.11024i	0.0836390	−	0.144867i
$806$	−1.98198	+	3.43289i	−0.0698122	+	0.120918i
$807$		0			0
$808$	8.62874	+	14.9454i	0.303558	+	0.525778i
$809$	25.1521			0.884299			0.442150	−	0.896941i	$-0.354216\pi$
$809$	25.1521			0.884299			0.442150	+	0.896941i	$0.354216\pi$
$810$		0			0
$811$	46.6061			1.63656			0.818281	−	0.574819i	$-0.194928\pi$
$811$	46.6061			1.63656			0.818281	+	0.574819i	$0.194928\pi$
$812$	−0.0192821	−	0.0333975i	−0.000676668	−	0.00117202i
$813$		0			0
$814$	−26.3858	+	45.7016i	−0.924823	+	1.60184i
$815$	17.7567	−	30.7554i	0.621989	−	1.07732i
$816$		0			0
$817$	−1.05313	−	1.82407i	−0.0368443	−	0.0638163i
$818$	−0.840747			−0.0293960
$819$		0			0
$820$	−4.79789			−0.167550
$821$	5.19777	+	9.00280i	0.181403	+	0.314200i	0.942359	−	0.334605i	$-0.108603\pi$
$821$	5.19777	+	9.00280i	0.181403	+	0.314200i	−0.760955	+	0.648804i	$0.775269\pi$
$822$		0			0
$823$	−3.66966	+	6.35603i	−0.127916	+	0.221557i	−0.922869	−	0.385114i	$-0.874162\pi$
$823$	−3.66966	+	6.35603i	−0.127916	+	0.221557i	0.794953	+	0.606671i	$0.207496\pi$
$824$	18.5158	−	32.0703i	0.645028	−	1.11722i
$825$		0			0
$826$	−0.117076	−	0.202782i	−0.00407361	−	0.00705570i
$827$	11.6262			0.404281			0.202141	−	0.979357i	$-0.435210\pi$
$827$	11.6262			0.404281			0.202141	+	0.979357i	$0.435210\pi$
$828$		0			0
$829$	5.35678			0.186049			0.0930243	−	0.995664i	$-0.470347\pi$
$829$	5.35678			0.186049			0.0930243	+	0.995664i	$0.470347\pi$
$830$	2.27024	+	3.93217i	0.0788012	+	0.136488i
$831$		0			0
$832$	−3.31068	+	5.73426i	−0.114777	+	0.198800i
$833$	−26.5933	+	46.0609i	−0.921403	+	1.59592i
$834$		0			0
$835$	15.6431	+	27.0947i	0.541353	+	0.937651i
$836$	−2.59668			−0.0898081
$837$		0			0
$838$	−12.0395			−0.415899
$839$	24.0328	+	41.6261i	0.829705	+	1.43709i	0.898270	+	0.439444i	$0.144825\pi$
$839$	24.0328	+	41.6261i	0.829705	+	1.43709i	−0.0685648	+	0.997647i	$0.521842\pi$
$840$		0			0
$841$	14.4162	−	24.9695i	0.497109	−	0.861018i
$842$	−9.82437	+	17.0163i	−0.338570	+	0.586421i
$843$		0			0
$844$	−2.13633	−	3.70023i	−0.0735355	−	0.127367i
$845$	2.64315			0.0909271
$846$		0			0
$847$	8.11885			0.278967
$848$	8.67506	+	15.0256i	0.297903	+	0.515983i
$849$		0			0
$850$	−11.5748	+	20.0481i	−0.397012	+	0.687645i
$851$	15.5523	−	26.9375i	0.533127	−	0.923404i
$852$		0			0
$853$	12.3049	+	21.3126i	0.421310	+	0.729731i	0.996068	−	0.0885930i	$-0.0282371\pi$
$853$	12.3049	+	21.3126i	0.421310	+	0.729731i	−0.574758	+	0.818324i	$0.694904\pi$
$854$	−1.93534			−0.0662259
$855$		0			0
$856$	2.49213			0.0851793
$857$	16.6402	+	28.8216i	0.568417	+	0.984528i	0.996723	+	0.0808938i	$0.0257775\pi$
$857$	16.6402	+	28.8216i	0.568417	+	0.984528i	−0.428305	+	0.903634i	$0.640889\pi$
$858$		0			0
$859$	−6.91335	+	11.9743i	−0.235881	+	0.408557i	−0.959528	−	0.281613i	$-0.909131\pi$
$859$	−6.91335	+	11.9743i	−0.235881	+	0.408557i	0.723648	+	0.690170i	$0.242464\pi$
$860$	−0.475520	+	0.823625i	−0.0162151	+	0.0280854i
$861$		0			0
$862$	−10.4699	−	18.1345i	−0.356607	−	0.617662i
$863$	23.1145			0.786827			0.393413	−	0.919362i	$-0.371294\pi$
$863$	23.1145			0.786827			0.393413	+	0.919362i	$0.371294\pi$
$864$		0			0
$865$	48.9432			1.66412
$866$	−6.18935	−	10.7203i	−0.210323	−	0.364289i
$867$		0			0
$868$	0.123763	−	0.214363i	0.00420078	−	0.00727597i
$869$	−27.3718	+	47.4093i	−0.928524	+	1.60825i
$870$		0			0
$871$	1.40485	+	2.43328i	0.0476016	+	0.0824484i
$872$	15.0267			0.508868
$873$		0			0
$874$	12.6825			0.428991
$875$	1.36650	+	2.36685i	0.0461962	+	0.0800142i
$876$		0			0
$877$	−23.4704	+	40.6520i	−0.792540	+	1.37272i	0.131849	+	0.991270i	$0.457909\pi$
$877$	−23.4704	+	40.6520i	−0.792540	+	1.37272i	−0.924389	+	0.381450i	$0.875425\pi$
$878$	−8.06475	+	13.9686i	−0.272172	+	0.471416i
$879$		0			0
$880$	−34.8089	−	60.2908i	−1.17341	−	2.03240i
$881$	10.1247			0.341109			0.170555	−	0.985348i	$-0.445444\pi$
$881$	10.1247			0.341109			0.170555	+	0.985348i	$0.445444\pi$
$882$		0			0
$883$	−34.3946			−1.15747			−0.578736	−	0.815515i	$-0.696454\pi$
$883$	−34.3946			−1.15747			−0.578736	+	0.815515i	$0.696454\pi$
$884$	1.06063	+	1.83706i	0.0356727	+	0.0617869i
$885$		0			0
$886$	−14.4021	+	24.9452i	−0.483849	+	0.838051i
$887$	−14.4606	+	25.0465i	−0.485539	+	0.840978i	−0.999862	−	0.0166184i	$-0.994710\pi$
$887$	−14.4606	+	25.0465i	−0.485539	+	0.840978i	0.514323	+	0.857597i	$0.328043\pi$
$888$		0			0
$889$	2.05483	+	3.55906i	0.0689166	+	0.119367i
$890$	−35.9215			−1.20409
$891$		0			0
$892$	−1.05485			−0.0353188
$893$	−2.89290	−	5.01065i	−0.0968072	−	0.167675i
$894$		0			0
$895$	24.7097	−	42.7984i	0.825954	−	1.43059i
$896$	2.24217	−	3.88356i	0.0749057	−	0.129741i
$897$		0			0
$898$	21.5568	+	37.3375i	0.719360	+	1.24597i
$899$	−1.07631			−0.0358970
$900$		0			0
$901$	−29.9717			−0.998502
$902$	29.3600	+	50.8530i	0.977580	+	1.69322i
$903$		0			0
$904$	15.4540	−	26.7671i	0.513993	−	0.890262i
$905$	−7.84637	+	13.5903i	−0.260822	+	0.451757i
$906$		0			0
$907$	20.0348	+	34.7013i	0.665244	+	1.15224i	0.979219	+	0.202806i	$0.0650059\pi$
$907$	20.0348	+	34.7013i	0.665244	+	1.15224i	−0.313975	+	0.949431i	$0.601661\pi$
$908$	−6.60373			−0.219153
$909$		0			0
$910$	−1.36765			−0.0453370
$911$	−28.3462	−	49.0970i	−0.939151	−	1.62666i	−0.767060	−	0.641576i	$-0.778281\pi$
$911$	−28.3462	−	49.0970i	−0.939151	−	1.62666i	−0.172091	−	0.985081i	$-0.555052\pi$
$912$		0			0
$913$	3.35310	−	5.80774i	0.110971	−	0.192208i
$914$	5.34924	−	9.26516i	0.176937	−	0.306464i
$915$		0			0
$916$	2.76491	+	4.78896i	0.0913551	+	0.158232i
$917$	4.13018			0.136390
$918$		0			0
$919$	−26.3183			−0.868162			−0.434081	−	0.900874i	$-0.642927\pi$
$919$	−26.3183			−0.868162			−0.434081	+	0.900874i	$0.642927\pi$
$920$	17.9993	+	31.1758i	0.593421	+	1.02783i
$921$		0			0
$922$	15.6947	−	27.1840i	0.516876	−	0.895256i
$923$	8.29444	−	14.3664i	0.273015	−	0.472876i
$924$		0			0
$925$	−5.90232	−	10.2231i	−0.194067	−	0.336134i
$926$	29.5208			0.970113
$927$		0			0
$928$	0.631541			0.0207314
$929$	−3.12768	−	5.41730i	−0.102616	−	0.177736i	0.810146	−	0.586228i	$-0.199388\pi$
$929$	−3.12768	−	5.41730i	−0.102616	−	0.177736i	−0.912762	+	0.408493i	$0.866055\pi$
$930$		0			0
$931$	5.52917	−	9.57681i	0.181211	−	0.313867i
$932$	0.804365	−	1.39320i	0.0263479	−	0.0456358i
$933$		0			0
$934$	−24.3055	−	42.0983i	−0.795299	−	1.37750i
$935$	120.262			3.93299
$936$		0			0
$937$	−29.4349			−0.961598			−0.480799	−	0.876831i	$-0.659653\pi$
$937$	−29.4349			−0.961598			−0.480799	+	0.876831i	$0.659653\pi$
$938$	−0.726913	−	1.25905i	−0.0237345	−	0.0411094i
$939$		0			0
$940$	−1.30623	+	2.26246i	−0.0426046	+	0.0737934i
$941$	−3.38544	+	5.86375i	−0.110362	+	0.191153i	−0.915916	−	0.401369i	$-0.868534\pi$
$941$	−3.38544	+	5.86375i	−0.110362	+	0.191153i	0.805554	+	0.592522i	$0.201868\pi$
$942$		0			0
$943$	−17.3054	−	29.9738i	−0.563540	−	0.976080i
$944$	2.02445			0.0658904
$945$		0			0
$946$	11.6395			0.378433
$947$	−21.8045	−	37.7666i	−0.708552	−	1.22725i	−0.965394	−	0.260795i	$-0.916015\pi$
$947$	−21.8045	−	37.7666i	−0.708552	−	1.22725i	0.256842	−	0.966454i	$-0.417318\pi$
$948$		0			0
$949$	2.57987	−	4.46847i	0.0837461	−	0.145053i
$950$	2.40659	−	4.16833i	0.0780799	−	0.135238i
$951$		0			0
$952$	3.44992	+	5.97543i	0.111813	+	0.193665i
$953$	−11.8773			−0.384744			−0.192372	−	0.981322i	$-0.561618\pi$
$953$	−11.8773			−0.384744			−0.192372	+	0.981322i	$0.561618\pi$
$954$		0			0
$955$	13.6498			0.441699
$956$	1.25845	+	2.17970i	0.0407012	+	0.0704965i
$957$		0			0
$958$	0.792357	−	1.37240i	0.0255999	−	0.0443403i
$959$	−0.00527194	+	0.00913126i	−0.000170240	+	0.000294864i
$960$		0			0
$961$	12.0458	+	20.8640i	0.388575	+	0.673032i
$962$	−8.96319			−0.288985
$963$		0			0
$964$	−6.69016			−0.215476
$965$	−30.2895	−	52.4630i	−0.975055	−	1.68884i
$966$		0			0
$967$	−10.2599	+	17.7706i	−0.329935	+	0.571464i	−0.982499	−	0.186270i	$-0.940360\pi$
$967$	−10.2599	+	17.7706i	−0.329935	+	0.571464i	0.652564	+	0.757734i	$0.273694\pi$
$968$	−30.7903	+	53.3304i	−0.989639	+	1.71411i
$969$		0			0
$970$	8.61559	+	14.9226i	0.276630	+	0.479137i
$971$	17.5599			0.563526			0.281763	−	0.959484i	$-0.409081\pi$
$971$	17.5599			0.563526			0.281763	+	0.959484i	$0.409081\pi$
$972$		0			0
$973$	3.87654			0.124276
$974$	13.7255	+	23.7733i	0.439794	+	0.761746i
$975$		0			0
$976$	8.36633	−	14.4909i	0.267800	−	0.463843i
$977$	−7.19723	+	12.4660i	−0.230260	+	0.398822i	−0.957884	−	0.287154i	$-0.907291\pi$
$977$	−7.19723	+	12.4660i	−0.230260	+	0.398822i	0.727625	+	0.685975i	$0.240624\pi$
$978$		0			0
$979$	26.5277	+	45.9473i	0.847828	+	1.46848i
$980$	−4.99318			−0.159501
$981$		0			0
$982$	2.07332			0.0661623
$983$	10.8945	+	18.8699i	0.347482	+	0.601857i	0.985801	−	0.167915i	$-0.0537034\pi$
$983$	10.8945	+	18.8699i	0.347482	+	0.601857i	−0.638319	+	0.769772i	$0.720370\pi$
$984$		0			0
$985$	−19.1740	+	33.2104i	−0.610935	+	1.05817i
$986$	−2.38634	+	4.13326i	−0.0759964	+	0.131630i
$987$		0			0
$988$	−0.220521	−	0.381954i	−0.00701572	−	0.0121516i
$989$	−6.86055			−0.218153
$990$		0			0
$991$	−57.2494			−1.81859			−0.909294	−	0.416154i	$-0.863378\pi$
$991$	−57.2494			−1.81859			−0.909294	+	0.416154i	$0.863378\pi$
$992$	2.02679	+	3.51050i	0.0643505	+	0.111458i
$993$		0			0
$994$	−4.29179	+	7.43360i	−0.136127	+	0.235780i
$995$	12.4264	−	21.5232i	0.393944	−	0.682331i
$996$		0			0
$997$	11.7445	+	20.3420i	0.371951	+	0.644239i	0.989866	−	0.142007i	$-0.0453555\pi$
$997$	11.7445	+	20.3420i	0.371951	+	0.644239i	−0.617914	+	0.786245i	$0.712022\pi$
$998$	−35.7329			−1.13111
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	351.2.e.b.118.4		10
3.2	odd	2		117.2.e.b.40.2	✓	10
9.2	odd	6		117.2.e.b.79.2	yes	10
9.4	even	3		1053.2.a.j.1.2		5
9.5	odd	6		1053.2.a.k.1.4		5
9.7	even	3	inner	351.2.e.b.235.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.e.b.40.2	✓	10	3.2	odd	2
117.2.e.b.79.2	yes	10	9.2	odd	6
351.2.e.b.118.4		10	1.1	even	1	trivial
351.2.e.b.235.4		10	9.7	even	3	inner
1053.2.a.j.1.2		5	9.4	even	3
1053.2.a.k.1.4		5	9.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.754070 + 1.30609i) q^{2} +(-0.137244 + 0.237713i) q^{4} +(-1.32157 + 2.28903i) q^{5} +(-0.171546 - 0.297126i) q^{7} +2.60231 q^{8} -3.98624 q^{10} +(2.94380 + 5.09881i) q^{11} +(-0.500000 + 0.866025i) q^{13} +(0.258715 - 0.448108i) q^{14} +(2.23682 + 3.87428i) q^{16} -7.72803 q^{17} +1.60678 q^{19} +(-0.362756 - 0.628312i) q^{20} +(-4.43966 + 7.68972i) q^{22} +(2.61683 - 4.53248i) q^{23} +(-0.993121 - 1.72014i) q^{25} -1.50814 q^{26} +0.0941744 q^{28} +(-0.204749 - 0.354635i) q^{29} +(1.31419 - 2.27624i) q^{31} +(-0.771118 + 1.33562i) q^{32} +(-5.82748 - 10.0935i) q^{34} +0.906842 q^{35} +5.94321 q^{37} +(1.21163 + 2.09860i) q^{38} +(-3.43915 + 5.95679i) q^{40} +(3.30656 - 5.72712i) q^{41} +(-0.655427 - 1.13523i) q^{43} -1.61607 q^{44} +7.89308 q^{46} +(-1.80043 - 3.11843i) q^{47} +(3.44114 - 5.96024i) q^{49} +(1.49777 - 2.59421i) q^{50} +(-0.137244 - 0.237713i) q^{52} +3.87831 q^{53} -15.5618 q^{55} +(-0.446416 - 0.773215i) q^{56} +(0.308790 - 0.534839i) q^{58} +(0.226265 - 0.391903i) q^{59} +(-1.87014 - 3.23918i) q^{61} +3.96396 q^{62} +6.62136 q^{64} +(-1.32157 - 2.28903i) q^{65} +(1.40485 - 2.43328i) q^{67} +(1.06063 - 1.83706i) q^{68} +(0.683823 + 1.18442i) q^{70} -16.5889 q^{71} -5.15974 q^{73} +(4.48160 + 7.76235i) q^{74} +(-0.220521 + 0.381954i) q^{76} +(1.00999 - 1.74936i) q^{77} +(4.64906 + 8.05240i) q^{79} -11.8245 q^{80} +9.97350 q^{82} +(-0.569519 - 0.986435i) q^{83} +(10.2132 - 17.6897i) q^{85} +(0.988476 - 1.71209i) q^{86} +(7.66069 + 13.2687i) q^{88} +9.01137 q^{89} +0.343091 q^{91} +(0.718287 + 1.24411i) q^{92} +(2.71530 - 4.70304i) q^{94} +(-2.12349 + 3.67798i) q^{95} +(-2.16133 - 3.74354i) q^{97} +10.3795 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} - 4 q^{4} + q^{5} + 2 q^{7} - 24 q^{8} - 4 q^{10} + 11 q^{11} - 5 q^{13} - 5 q^{14} - 10 q^{16} - 14 q^{17} + 6 q^{19} - q^{20} + 7 q^{22} + 18 q^{23} + 8 q^{25} - 4 q^{26} - 38 q^{28}+ \cdots + 78 q^{98}+O(q^{100}) \) 10 * q + 2 * q^2 - 4 * q^4 + q^5 + 2 * q^7 - 24 * q^8 - 4 * q^10 + 11 * q^11 - 5 * q^13 - 5 * q^14 - 10 * q^16 - 14 * q^17 + 6 * q^19 - q^20 + 7 * q^22 + 18 * q^23 + 8 * q^25 - 4 * q^26 - 38 * q^28 + 4 * q^29 + 16 * q^31 + 31 * q^32 - 13 * q^34 - 22 * q^35 + 10 * q^37 + 24 * q^38 - 18 * q^40 + 12 * q^41 - 2 * q^44 + 12 * q^46 + 15 * q^47 - 3 * q^49 - 5 * q^50 - 4 * q^52 + 6 * q^53 - 34 * q^55 - 24 * q^56 + 2 * q^58 + 3 * q^59 + q^61 + 8 * q^62 + 124 * q^64 + q^65 + 4 * q^67 - 16 * q^68 + 25 * q^70 - 30 * q^71 - 10 * q^73 + 11 * q^74 - 9 * q^76 + q^77 - 13 * q^79 - 64 * q^80 - 42 * q^82 + 26 * q^83 + 7 * q^85 - 6 * q^86 - 3 * q^88 - 58 * q^89 - 4 * q^91 - 6 * q^92 - 12 * q^94 + 12 * q^95 + 16 * q^97 + 78 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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