LMFDB - Embedded newform 117.4.g.f.55.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(55,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.g (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:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 52 x^{14} + 1899 x^{12} + 33440 x^{10} + 424113 x^{8} + 2869882 x^{6} + 13705540 x^{4} + \cdots + 24920064 $ x^16 + 52x^14 + 1899x^12 + 33440x^10 + 424113x^8 + 2869882x^6 + 13705540x^4 + 21016320*x^2 + 24920064
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.4
Root			$-0.643348 - 1.11431i$ of defining polynomial
Character	$\chi$	$=$	117.55
Dual form			117.4.g.f.100.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.643348 - 1.11431i) q^{2} +(3.17221 - 5.49442i) q^{4} +12.4484 q^{5} +(4.50137 - 7.79659i) q^{7} -18.4569 q^{8} +O(q^{10})$	\(q+(-0.643348 - 1.11431i) q^{2} +(3.17221 - 5.49442i) q^{4} +12.4484 q^{5} +(4.50137 - 7.79659i) q^{7} -18.4569 q^{8} +(-8.00866 - 13.8714i) q^{10} +(25.5387 + 44.2343i) q^{11} +(2.33646 - 46.8139i) q^{13} -11.5838 q^{14} +(-13.5034 - 23.3886i) q^{16} +(3.43100 - 5.94266i) q^{17} +(41.5585 - 71.9815i) q^{19} +(39.4890 - 68.3969i) q^{20} +(32.8606 - 56.9162i) q^{22} +(-93.7893 - 162.448i) q^{23} +29.9631 q^{25} +(-53.6684 + 27.5141i) q^{26} +(-28.5585 - 49.4648i) q^{28} +(111.807 + 193.656i) q^{29} +57.3882 q^{31} +(-91.2024 + 157.967i) q^{32} -8.82930 q^{34} +(56.0349 - 97.0553i) q^{35} +(78.1293 + 135.324i) q^{37} -106.946 q^{38} -229.759 q^{40} +(-111.141 - 192.501i) q^{41} +(-173.987 + 301.354i) q^{43} +324.056 q^{44} +(-120.678 + 209.021i) q^{46} -45.0185 q^{47} +(130.975 + 226.856i) q^{49} +(-19.2767 - 33.3882i) q^{50} +(-249.804 - 161.341i) q^{52} +473.516 q^{53} +(317.917 + 550.648i) q^{55} +(-83.0813 + 143.901i) q^{56} +(143.862 - 249.176i) q^{58} +(-307.747 + 533.034i) q^{59} +(-97.6626 + 169.157i) q^{61} +(-36.9206 - 63.9483i) q^{62} +18.6446 q^{64} +(29.0852 - 582.759i) q^{65} +(177.548 + 307.523i) q^{67} +(-21.7677 - 37.7027i) q^{68} -144.200 q^{70} +(-381.710 + 661.141i) q^{71} +331.595 q^{73} +(100.529 - 174.121i) q^{74} +(-263.664 - 456.680i) q^{76} +459.836 q^{77} -207.777 q^{79} +(-168.096 - 291.152i) q^{80} +(-143.004 + 247.691i) q^{82} -251.185 q^{83} +(42.7105 - 73.9767i) q^{85} +447.737 q^{86} +(-471.365 - 816.429i) q^{88} +(-359.683 - 622.989i) q^{89} +(-354.472 - 228.943i) q^{91} -1190.08 q^{92} +(28.9626 + 50.1647i) q^{94} +(517.338 - 896.055i) q^{95} +(778.080 - 1347.67i) q^{97} +(168.526 - 291.895i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 40 q^{4} + 22 q^{7}+O(q^{10}) $ 16 * q - 40 * q^4 + 22 * q^7	$ 16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100}) $ 16 * q - 40 * q^4 + 22 * q^7 - 36 * q^10 + 36 * q^13 - 204 * q^16 - 244 * q^19 - 136 * q^22 + 708 * q^25 + 452 * q^28 + 484 * q^31 - 2584 * q^34 - 1018 * q^37 + 3400 * q^40 - 74 * q^43 + 896 * q^46 - 298 * q^49 - 1676 * q^52 - 1300 * q^55 - 812 * q^58 - 1148 * q^61 + 7272 * q^64 + 2198 * q^67 + 4400 * q^70 - 4352 * q^73 - 6936 * q^76 + 3724 * q^79 - 5436 * q^82 + 890 * q^85 - 3528 * q^88 - 4754 * q^91 + 3104 * q^94 + 4370 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	−0.643348	−	1.11431i	−0.227458	−	0.393969i	0.729596	−	0.683878i	$-0.239708\pi$
$2$	−0.643348	−	1.11431i	−0.227458	−	0.393969i	−0.957054	+	0.289910i	$0.906375\pi$
$3$		0			0
$3$		0			0
$4$	3.17221	−	5.49442i	0.396526	−	0.686803i
$5$	12.4484			1.11342			0.556710	−	0.830707i	$-0.312063\pi$
$5$	12.4484			1.11342			0.556710	+	0.830707i	$0.312063\pi$
$6$		0			0
$7$	4.50137	−	7.79659i	0.243051	−	0.420977i	−0.718531	−	0.695495i	$-0.755185\pi$
$7$	4.50137	−	7.79659i	0.243051	−	0.420977i	0.961582	+	0.274519i	$0.0885185\pi$
$8$	−18.4569			−0.815687
$9$		0			0
$10$	−8.00866	−	13.8714i	−0.253256	−	0.438653i
$11$	25.5387	+	44.2343i	0.700019	+	1.21247i	0.968459	+	0.249172i	$0.0801586\pi$
$11$	25.5387	+	44.2343i	0.700019	+	1.21247i	−0.268440	+	0.963296i	$0.586508\pi$
$12$		0			0
$13$	2.33646	−	46.8139i	0.0498475	−	0.998757i
$13$	2.33646	−	46.8139i	0.0498475	−	0.998757i
$14$	−11.5838			−0.221135
$15$		0			0
$16$	−13.5034	−	23.3886i	−0.210991	−	0.365448i
$17$	3.43100	−	5.94266i	0.0489493	−	0.0847827i	−0.840513	−	0.541792i	$-0.817746\pi$
$17$	3.43100	−	5.94266i	0.0489493	−	0.0847827i	0.889462	+	0.457009i	$0.151079\pi$
$18$		0			0
$19$	41.5585	−	71.9815i	0.501799	−	0.869141i	−0.498199	−	0.867063i	$-0.666005\pi$
$19$	41.5585	−	71.9815i	0.501799	−	0.869141i	0.999998	−	0.00207845i	$-0.000661591\pi$
$20$	39.4890	−	68.3969i	0.441500	−	0.764700i
$21$		0			0
$22$	32.8606	−	56.9162i	0.318450	−	0.551571i
$23$	−93.7893	−	162.448i	−0.850279	−	1.47273i	−0.880957	−	0.473197i	$-0.843100\pi$
$23$	−93.7893	−	162.448i	−0.850279	−	1.47273i	0.0306776	−	0.999529i	$-0.490233\pi$
$24$		0			0
$25$	29.9631			0.239705
$26$	−53.6684	+	27.5141i	−0.404817	+	0.207537i
$27$		0			0
$28$	−28.5585	−	49.4648i	−0.192752	−	0.333856i
$29$	111.807	+	193.656i	0.715933	+	1.24003i	0.962599	+	0.270932i	$0.0873318\pi$
$29$	111.807	+	193.656i	0.715933	+	1.24003i	−0.246665	+	0.969101i	$0.579335\pi$
$30$		0			0
$31$	57.3882			0.332491			0.166246	−	0.986084i	$-0.446836\pi$
$31$	57.3882			0.332491			0.166246	+	0.986084i	$0.446836\pi$
$32$	−91.2024	+	157.967i	−0.503827	+	0.872654i
$33$		0			0
$34$	−8.82930			−0.0445357
$35$	56.0349	−	97.0553i	0.270618	−	0.468724i
$36$		0			0
$37$	78.1293	+	135.324i	0.347146	+	0.601274i	0.985741	−	0.168269i	$-0.0538177\pi$
$37$	78.1293	+	135.324i	0.347146	+	0.601274i	−0.638596	+	0.769542i	$0.720484\pi$
$38$	−106.946			−0.456552
$39$		0			0
$40$	−229.759			−0.908203
$41$	−111.141	−	192.501i	−0.423348	−	0.733260i	0.572917	−	0.819613i	$-0.305812\pi$
$41$	−111.141	−	192.501i	−0.423348	−	0.733260i	−0.996265	+	0.0863539i	$0.972478\pi$
$42$		0			0
$43$	−173.987	+	301.354i	−0.617041	+	1.06875i	0.372982	+	0.927839i	$0.378335\pi$
$43$	−173.987	+	301.354i	−0.617041	+	1.06875i	−0.990023	+	0.140908i	$0.954998\pi$
$44$	324.056			1.11030
$45$		0			0
$46$	−120.678	+	209.021i	−0.386805	+	0.669966i
$47$	−45.0185			−0.139715			−0.0698577	−	0.997557i	$-0.522255\pi$
$47$	−45.0185			−0.139715			−0.0698577	+	0.997557i	$0.522255\pi$
$48$		0			0
$49$	130.975	+	226.856i	0.381853	+	0.661388i
$50$	−19.2767	−	33.3882i	−0.0545227	−	0.0944361i
$51$		0			0
$52$	−249.804	−	161.341i	−0.666183	−	0.430268i
$53$	473.516			1.22722			0.613608	−	0.789611i	$-0.289718\pi$
$53$	473.516			1.22722			0.613608	+	0.789611i	$0.289718\pi$
$54$		0			0
$55$	317.917	+	550.648i	0.779416	+	1.34999i
$56$	−83.0813	+	143.901i	−0.198254	+	0.343385i
$57$		0			0
$58$	143.862	−	249.176i	0.325689	−	0.564110i
$59$	−307.747	+	533.034i	−0.679072	+	1.17619i	0.296188	+	0.955130i	$0.404284\pi$
$59$	−307.747	+	533.034i	−0.679072	+	1.17619i	−0.975261	+	0.221058i	$0.929049\pi$
$60$		0			0
$61$	−97.6626	+	169.157i	−0.204990	+	0.355054i	−0.950130	−	0.311855i	$-0.899050\pi$
$61$	−97.6626	+	169.157i	−0.204990	+	0.355054i	0.745139	+	0.666909i	$0.232383\pi$
$62$	−36.9206	−	63.9483i	−0.0756277	−	0.130991i
$63$		0			0
$64$	18.6446			0.0364151
$65$	29.0852	−	582.759i	0.0555012	−	1.11204i
$66$		0			0
$67$	177.548	+	307.523i	0.323746	+	0.560745i	0.981258	−	0.192700i	$-0.0617243\pi$
$67$	177.548	+	307.523i	0.323746	+	0.560745i	−0.657512	+	0.753444i	$0.728391\pi$
$68$	−21.7677	−	37.7027i	−0.0388194	−	0.0672371i
$69$		0			0
$70$	−144.200			−0.246217
$71$	−381.710	+	661.141i	−0.638038	+	1.10511i	0.347825	+	0.937559i	$0.386920\pi$
$71$	−381.710	+	661.141i	−0.638038	+	1.10511i	−0.985863	+	0.167554i	$0.946413\pi$
$72$		0			0
$73$	331.595			0.531647			0.265824	−	0.964022i	$-0.414356\pi$
$73$	331.595			0.531647			0.265824	+	0.964022i	$0.414356\pi$
$74$	100.529	−	174.121i	0.157922	−	0.273529i
$75$		0			0
$76$	−263.664	−	456.680i	−0.397952	−	0.689274i
$77$	459.836			0.680561
$78$		0			0
$79$	−207.777			−0.295908			−0.147954	−	0.988994i	$-0.547269\pi$
$79$	−207.777			−0.295908			−0.147954	+	0.988994i	$0.547269\pi$
$80$	−168.096	−	291.152i	−0.234922	−	0.406897i
$81$		0			0
$82$	−143.004	+	247.691i	−0.192587	+	0.333571i
$83$	−251.185			−0.332183			−0.166091	−	0.986110i	$-0.553115\pi$
$83$	−251.185			−0.332183			−0.166091	+	0.986110i	$0.553115\pi$
$84$		0			0
$85$	42.7105	−	73.9767i	0.0545012	−	0.0943988i
$86$	447.737			0.561403
$87$		0			0
$88$	−471.365	−	816.429i	−0.570997	−	0.988996i
$89$	−359.683	−	622.989i	−0.428385	−	0.741985i	0.568345	−	0.822791i	$-0.307584\pi$
$89$	−359.683	−	622.989i	−0.428385	−	0.741985i	−0.996730	+	0.0808054i	$0.974251\pi$
$90$		0			0
$91$	−354.472	−	228.943i	−0.408338	−	0.263733i
$92$	−1190.08			−1.34863
$93$		0			0
$94$	28.9626	+	50.1647i	0.0317794	+	0.0550435i
$95$	517.338	−	896.055i	0.558713	−	0.967719i
$96$		0			0
$97$	778.080	−	1347.67i	0.814454	−	1.41068i	−0.0952647	−	0.995452i	$-0.530370\pi$
$97$	778.080	−	1347.67i	0.814454	−	1.41068i	0.909719	−	0.415224i	$-0.136297\pi$
$98$	168.526	−	291.895i	0.173711	−	0.300876i
$99$		0			0
$100$	95.0491	−	164.630i	0.0950491	−	0.164630i
$101$	391.693	+	678.431i	0.385890	+	0.668381i	0.991892	−	0.127082i	$-0.0405612\pi$
$101$	391.693	+	678.431i	0.385890	+	0.668381i	−0.606002	+	0.795463i	$0.707228\pi$
$102$		0			0
$103$	1033.54			0.988720			0.494360	−	0.869257i	$-0.335402\pi$
$103$	1033.54			0.988720			0.494360	+	0.869257i	$0.335402\pi$
$104$	−43.1238	+	864.039i	−0.0406599	+	0.814673i
$105$		0			0
$106$	−304.635	−	527.644i	−0.279140	−	0.483484i
$107$	120.679	+	209.022i	0.109033	+	0.188850i	0.915379	−	0.402594i	$-0.131891\pi$
$107$	120.679	+	209.022i	0.109033	+	0.188850i	−0.806346	+	0.591444i	$0.798558\pi$
$108$		0			0
$109$	1763.93			1.55003			0.775016	−	0.631941i	$-0.217742\pi$
$109$	1763.93			1.55003			0.775016	+	0.631941i	$0.217742\pi$
$110$	409.062	−	708.516i	0.354568	−	0.614131i
$111$		0			0
$112$	−243.136			−0.205126
$113$	−933.917	+	1617.59i	−0.777482	+	1.34664i	0.155907	+	0.987772i	$0.450170\pi$
$113$	−933.917	+	1617.59i	−0.777482	+	1.34664i	−0.933389	+	0.358867i	$0.883163\pi$
$114$		0			0
$115$	−1167.53	−	2022.22i	−0.946718	−	1.63976i
$116$	1418.70			1.13554
$117$		0			0
$118$	791.954			0.617841
$119$	−30.8883	−	53.5002i	−0.0237944	−	0.0412130i
$120$		0			0
$121$	−638.952	+	1106.70i	−0.480054	+	0.831478i
$122$	251.324			0.186507
$123$		0			0
$124$	182.047	−	315.315i	0.131841	−	0.228356i
$125$	−1183.06			−0.846528
$126$		0			0
$127$	−119.442	−	206.880i	−0.0834551	−	0.144548i	0.821277	−	0.570530i	$-0.193262\pi$
$127$	−119.442	−	206.880i	−0.0834551	−	0.144548i	−0.904732	+	0.425982i	$0.859929\pi$
$128$	717.624	+	1242.96i	0.495544	+	0.858308i
$129$		0			0
$130$	−668.087	+	342.507i	−0.450732	+	0.231076i
$131$	−1158.72			−0.772807			−0.386403	−	0.922330i	$-0.626283\pi$
$131$	−1158.72			−0.772807			−0.386403	+	0.922330i	$0.626283\pi$
$132$		0			0
$133$	−374.140	−	648.030i	−0.243925	−	0.422491i
$134$	228.451	−	395.688i	0.147277	−	0.255092i
$135$		0			0
$136$	−63.3256	+	109.683i	−0.0399274	+	0.0691562i
$137$	456.260	−	790.265i	0.284532	−	0.492824i	−0.687963	−	0.725745i	$-0.741495\pi$
$137$	456.260	−	790.265i	0.284532	−	0.492824i	0.972496	+	0.232921i	$0.0748284\pi$
$138$		0			0
$139$	−1157.52	+	2004.88i	−0.706326	+	1.22339i	0.259884	+	0.965640i	$0.416316\pi$
$139$	−1157.52	+	2004.88i	−0.706326	+	1.22339i	−0.966211	+	0.257753i	$0.917018\pi$
$140$	−355.508	−	615.759i	−0.214614	−	0.371722i
$141$		0			0
$142$	982.290			0.580507
$143$	2130.45	−	1092.21i	1.24586	−	0.638711i
$144$		0			0
$145$	1391.82	+	2410.71i	0.797134	+	1.38068i
$146$	−213.331	−	369.500i	−0.120927	−	0.209452i
$147$		0			0
$148$	991.370			0.550609
$149$	462.704	−	801.427i	0.254404	−	0.440641i	−0.710329	−	0.703869i	$-0.751454\pi$
$149$	462.704	−	801.427i	0.254404	−	0.440641i	0.964733	+	0.263229i	$0.0847874\pi$
$150$		0			0
$151$	−3523.78			−1.89908			−0.949541	−	0.313644i	$-0.898450\pi$
$151$	−3523.78			−1.89908			−0.949541	+	0.313644i	$0.898450\pi$
$152$	−767.042	+	1328.55i	−0.409311	+	0.708948i
$153$		0			0
$154$	−295.835	−	512.401i	−0.154799	−	0.268120i
$155$	714.392			0.370202
$156$		0			0
$157$	1615.83			0.821384			0.410692	−	0.911774i	$-0.365287\pi$
$157$	1615.83			0.821384			0.410692	+	0.911774i	$0.365287\pi$
$158$	133.673	+	231.528i	0.0673067	+	0.116579i
$159$		0			0
$160$	−1135.33	+	1966.44i	−0.560971	+	0.971631i
$161$	−1688.72			−0.826644
$162$		0			0
$163$	−504.003	+	872.959i	−0.242188	+	0.419481i	−0.961337	−	0.275374i	$-0.911198\pi$
$163$	−504.003	+	872.959i	−0.242188	+	0.419481i	0.719150	+	0.694855i	$0.244532\pi$
$164$	−1410.24			−0.671473
$165$		0			0
$166$	161.600	+	279.899i	0.0755576	+	0.130870i
$167$	−1177.02	−	2038.66i	−0.545392	−	0.944647i	−0.998582	−	0.0532328i	$-0.983047\pi$
$167$	−1177.02	−	2038.66i	−0.545392	−	0.944647i	0.453190	−	0.891414i	$-0.350286\pi$
$168$		0			0
$169$	−2186.08	−	218.757i	−0.995030	−	0.0995710i
$170$	−109.911			−0.0495869
$171$		0			0
$172$	1103.84	+	1911.92i	0.489345	+	0.847571i
$173$	−1566.16	+	2712.67i	−0.688284	+	1.19214i	0.284109	+	0.958792i	$0.408302\pi$
$173$	−1566.16	+	2712.67i	−0.688284	+	1.19214i	−0.972393	+	0.233350i	$0.925031\pi$
$174$		0			0
$175$	134.875	−	233.610i	0.0582604	−	0.100910i
$176$	689.721	−	1194.63i	0.295396	−	0.511641i
$177$		0			0
$178$	−462.802	+	801.597i	−0.194879	+	0.337541i
$179$	1590.41	+	2754.67i	0.664093	+	1.15024i	0.979530	+	0.201297i	$0.0645156\pi$
$179$	1590.41	+	2754.67i	0.664093	+	1.15024i	−0.315437	+	0.948947i	$0.602151\pi$
$180$		0			0
$181$	4230.52			1.73731			0.868653	−	0.495421i	$-0.164986\pi$
$181$	4230.52			1.73731			0.868653	+	0.495421i	$0.164986\pi$
$182$	−27.0650	+	542.282i	−0.0110230	+	0.220860i
$183$		0			0
$184$	1731.06	+	2998.28i	0.693562	+	1.20128i
$185$	972.587	+	1684.57i	0.386519	+	0.669470i
$186$		0			0
$187$	350.493			0.137062
$188$	−142.808	+	247.351i	−0.0554008	+	0.0959570i
$189$		0			0
$190$	−1331.31			−0.508335
$191$	1976.40	−	3423.22i	0.748729	−	1.29684i	−0.199703	−	0.979856i	$-0.563998\pi$
$191$	1976.40	−	3423.22i	0.748729	−	1.29684i	0.948432	−	0.316980i	$-0.102669\pi$
$192$		0			0
$193$	−2521.22	−	4366.89i	−0.940319	−	1.62868i	−0.764863	−	0.644193i	$-0.777193\pi$
$193$	−2521.22	−	4366.89i	−0.940319	−	1.62868i	−0.175456	−	0.984487i	$-0.556140\pi$
$194$	−2002.31			−0.741016
$195$		0			0
$196$	1661.92			0.605658
$197$	−1786.06	−	3093.55i	−0.645947	−	1.11881i	−0.984082	−	0.177715i	$-0.943129\pi$
$197$	−1786.06	−	3093.55i	−0.645947	−	1.11881i	0.338135	−	0.941098i	$-0.390204\pi$
$198$		0			0
$199$	−2503.42	+	4336.05i	−0.891773	+	1.54460i	−0.0540255	+	0.998540i	$0.517205\pi$
$199$	−2503.42	+	4336.05i	−0.891773	+	1.54460i	−0.837748	+	0.546057i	$0.816128\pi$
$200$	−553.026			−0.195524
$201$		0			0
$202$	503.989	−	872.935i	0.175547	−	0.304057i
$203$	2013.14			0.696033
$204$		0			0
$205$	−1383.52	−	2396.34i	−0.471364	−	0.816426i
$206$	−664.929	−	1151.69i	−0.224892	−	0.389525i
$207$		0			0
$208$	−1126.46	+	577.502i	−0.375511	+	0.192512i
$209$	4245.40			1.40508
$210$		0			0
$211$	−879.615	−	1523.54i	−0.286992	−	0.497084i	0.686099	−	0.727509i	$-0.259322\pi$
$211$	−879.615	−	1523.54i	−0.286992	−	0.497084i	−0.973090	+	0.230425i	$0.925989\pi$
$212$	1502.09	−	2601.70i	0.486622	−	0.842855i
$213$		0			0
$214$	155.277	−	268.948i	0.0496007	−	0.0859109i
$215$	−2165.86	+	3751.38i	−0.687026	+	1.18996i
$216$		0			0
$217$	258.325	−	447.433i	0.0808123	−	0.139971i
$218$	−1134.82	−	1965.56i	−0.352567	−	0.610664i
$219$		0			0
$220$	4033.99			1.23623
$221$	−270.183	−	174.503i	−0.0822373	−	0.0531147i
$222$		0			0
$223$	−856.260	−	1483.09i	−0.257127	−	0.445358i	0.708344	−	0.705868i	$-0.249443\pi$
$223$	−856.260	−	1483.09i	−0.257127	−	0.445358i	−0.965471	+	0.260510i	$0.916109\pi$
$224$	821.071	+	1422.14i	0.244911	+	0.424199i
$225$		0			0
$226$	2403.33			0.707378
$227$	−1288.33	+	2231.45i	−0.376693	+	0.652451i	−0.990579	−	0.136944i	$-0.956272\pi$
$227$	−1288.33	+	2231.45i	−0.376693	+	0.652451i	0.613886	+	0.789394i	$0.289605\pi$
$228$		0			0
$229$	−1963.10			−0.566486			−0.283243	−	0.959048i	$-0.591410\pi$
$229$	−1963.10			−0.566486			−0.283243	+	0.959048i	$0.591410\pi$
$230$	−1502.25	+	2601.98i	−0.430677	+	0.745954i
$231$		0			0
$232$	−2063.61	−	3574.28i	−0.583978	−	1.01148i
$233$	−1078.11			−0.303131			−0.151565	−	0.988447i	$-0.548431\pi$
$233$	−1078.11			−0.303131			−0.151565	+	0.988447i	$0.548431\pi$
$234$		0			0
$235$	−560.409			−0.155562
$236$	1952.48	+	3381.79i	0.538539	+	0.932778i
$237$		0			0
$238$	−39.7439	+	68.8385i	−0.0108244	+	0.0187485i
$239$	3998.43			1.08216			0.541081	−	0.840970i	$-0.318015\pi$
$239$	3998.43			1.08216			0.541081	+	0.840970i	$0.318015\pi$
$240$		0			0
$241$	1787.36	−	3095.80i	0.477735	−	0.827461i	−0.521939	−	0.852983i	$-0.674791\pi$
$241$	1787.36	−	3095.80i	0.477735	−	0.827461i	0.999674	+	0.0255213i	$0.00812457\pi$
$242$	1644.27			0.436768
$243$		0			0
$244$	619.612	+	1073.20i	0.162568	+	0.281576i
$245$	1630.44	+	2824.00i	0.425162	+	0.736403i
$246$		0			0
$247$	−3272.63	−	2113.70i	−0.843047	−	0.544500i
$248$	−1059.21			−0.271209
$249$		0			0
$250$	761.119	+	1318.30i	0.192550	+	0.333506i
$251$	−1075.35	+	1862.55i	−0.270419	+	0.468380i	−0.968969	−	0.247181i	$-0.920496\pi$
$251$	−1075.35	+	1862.55i	−0.270419	+	0.468380i	0.698550	+	0.715561i	$0.253829\pi$
$252$		0			0
$253$	4790.52	−	8297.42i	1.19042	−	2.06187i
$254$	−153.686	+	266.192i	−0.0379650	+	0.0657574i
$255$		0			0
$256$	997.943	−	1728.49i	0.243638	−	0.421994i
$257$	−2137.40	−	3702.09i	−0.518784	−	0.898560i	−0.999762	−	0.0218271i	$-0.993052\pi$
$257$	−2137.40	−	3702.09i	−0.518784	−	0.898560i	0.480978	−	0.876733i	$-0.340282\pi$
$258$		0			0
$259$	1406.75			0.337496
$260$	−3109.66	−	2008.44i	−0.741742	−	0.479069i
$261$		0			0
$262$	745.459	+	1291.17i	0.175781	+	0.304462i
$263$	−4185.82	−	7250.06i	−0.981403	−	1.69984i	−0.656944	−	0.753940i	$-0.728151\pi$
$263$	−4185.82	−	7250.06i	−0.981403	−	1.69984i	−0.324459	−	0.945900i	$-0.605182\pi$
$264$		0			0
$265$	5894.52			1.36641
$266$	−481.405	+	833.818i	−0.110965	+	0.192198i
$267$		0			0
$268$	2252.88			0.513495
$269$	2539.82	−	4399.10i	0.575672	−	0.997092i	−0.420297	−	0.907387i	$-0.638074\pi$
$269$	2539.82	−	4399.10i	0.575672	−	0.997092i	0.995968	−	0.0897057i	$-0.0285926\pi$
$270$		0			0
$271$	−2465.97	−	4271.18i	−0.552756	−	0.957401i	−0.998074	−	0.0620290i	$-0.980243\pi$
$271$	−2465.97	−	4271.18i	−0.552756	−	0.957401i	0.445318	−	0.895372i	$-0.353090\pi$
$272$	−185.321			−0.0413115
$273$		0			0
$274$	−1174.14			−0.258876
$275$	765.219	+	1325.40i	0.167798	+	0.290635i
$276$		0			0
$277$	−2586.26	+	4479.54i	−0.560987	+	0.971658i	0.436424	+	0.899741i	$0.356245\pi$
$277$	−2586.26	+	4479.54i	−0.560987	+	0.971658i	−0.997411	+	0.0719165i	$0.977088\pi$
$278$	2978.75			0.642638
$279$		0			0
$280$	−1034.23	+	1791.34i	−0.220740	+	0.382332i
$281$	−217.515			−0.0461773			−0.0230887	−	0.999733i	$-0.507350\pi$
$281$	−217.515			−0.0461773			−0.0230887	+	0.999733i	$0.507350\pi$
$282$		0			0
$283$	−2461.49	−	4263.42i	−0.517033	−	0.895527i	−0.999804	−	0.0197808i	$-0.993703\pi$
$283$	−2461.49	−	4263.42i	−0.517033	−	0.895527i	0.482771	−	0.875746i	$-0.339630\pi$
$284$	2421.73	+	4194.55i	0.505997	+	0.876412i
$285$		0			0
$286$	−2587.69	−	1671.31i	−0.535012	−	0.345548i
$287$	−2001.14			−0.411580
$288$		0			0
$289$	2432.96	+	4214.00i	0.495208	+	0.857725i
$290$	1790.85	−	3101.85i	0.362629	−	0.628092i
$291$		0			0
$292$	1051.89	−	1821.92i	0.210812	−	0.365137i
$293$	1321.13	−	2288.26i	0.263416	−	0.456250i	−0.703731	−	0.710466i	$-0.748484\pi$
$293$	1321.13	−	2288.26i	0.263416	−	0.456250i	0.967147	+	0.254216i	$0.0818174\pi$
$294$		0			0
$295$	−3830.97	+	6635.43i	−0.756093	+	1.30959i
$296$	−1442.03	−	2497.66i	−0.283162	−	0.490451i
$297$		0			0
$298$	−1190.72			−0.231465
$299$	−7823.95	+	4011.09i	−1.51328	+	0.775810i
$300$		0			0
$301$	1566.36	+	2713.01i	0.299945	+	0.519519i
$302$	2267.02	+	3926.59i	0.431961	+	0.748178i
$303$		0			0
$304$	−2244.73			−0.423501
$305$	−1215.74	+	2105.73i	−0.228240	+	0.395324i
$306$		0			0
$307$	−3386.22			−0.629517			−0.314759	−	0.949172i	$-0.601924\pi$
$307$	−3386.22			−0.629517			−0.314759	+	0.949172i	$0.601924\pi$
$308$	1458.70	−	2526.54i	0.269860	−	0.467411i
$309$		0			0
$310$	−459.603	−	796.056i	−0.0842055	−	0.145848i
$311$	−3.29813			−0.000601349			−0.000300675	−	1.00000i	$-0.500096\pi$
$311$	−3.29813			−0.000601349			−0.000300675		1.00000i	$0.500096\pi$
$312$		0			0
$313$	2882.62			0.520560			0.260280	−	0.965533i	$-0.416185\pi$
$313$	2882.62			0.520560			0.260280	+	0.965533i	$0.416185\pi$
$314$	−1039.54	−	1800.54i	−0.186830	−	0.323599i
$315$		0			0
$316$	−659.112	+	1141.62i	−0.117335	+	0.203231i
$317$	−1442.70			−0.255616			−0.127808	−	0.991799i	$-0.540794\pi$
$317$	−1442.70			−0.255616			−0.127808	+	0.991799i	$0.540794\pi$
$318$		0			0
$319$	−5710.82	+	9891.43i	−1.00233	+	1.73609i
$320$	232.095			0.0405454
$321$		0			0
$322$	1086.43	+	1881.76i	0.188027	+	0.325672i
$323$	−285.174	−	493.936i	−0.0491255	−	0.0850878i
$324$		0			0
$325$	70.0075	−	1402.69i	0.0119487	−	0.239407i
$326$	1297.00			0.220350
$327$		0			0
$328$	2051.31	+	3552.98i	0.345319	+	0.598111i
$329$	−202.645	+	350.991i	−0.0339580	+	0.0588169i
$330$		0			0
$331$	1912.49	−	3312.53i	0.317583	−	0.550070i	−0.662400	−	0.749150i	$-0.730462\pi$
$331$	1912.49	−	3312.53i	0.317583	−	0.550070i	0.979983	+	0.199080i	$0.0637954\pi$
$332$	−796.812	+	1380.12i	−0.131719	+	0.228144i
$333$		0			0
$334$	−1514.47	+	2623.13i	−0.248107	+	0.429735i
$335$	2210.20	+	3828.17i	0.360465	+	0.624344i
$336$		0			0
$337$	2290.22			0.370197			0.185098	−	0.982720i	$-0.440740\pi$
$337$	2290.22			0.370197			0.185098	+	0.982720i	$0.440740\pi$
$338$	1162.65	+	2576.71i	0.187100	+	0.414659i
$339$		0			0
$340$	−270.973	−	469.339i	−0.0432223	−	0.0748631i
$341$	1465.62	+	2538.53i	0.232750	+	0.403135i
$342$		0			0
$343$	5446.21			0.857340
$344$	3211.26	−	5562.06i	0.503313	−	0.871763i
$345$		0			0
$346$	4030.35			0.626222
$347$	167.496	−	290.112i	0.0259126	−	0.0448820i	−0.852778	−	0.522273i	$-0.825084\pi$
$347$	167.496	−	290.112i	0.0259126	−	0.0448820i	0.878691	+	0.477391i	$0.158418\pi$
$348$		0			0
$349$	4487.64	+	7772.81i	0.688303	+	1.19218i	0.972387	+	0.233376i	$0.0749773\pi$
$349$	4487.64	+	7772.81i	0.688303	+	1.19218i	−0.284084	+	0.958799i	$0.591689\pi$
$350$	−347.086			−0.0530072
$351$		0			0
$352$	−9316.77			−1.41075
$353$	4962.94	+	8596.06i	0.748302	+	1.29610i	0.948636	+	0.316369i	$0.102464\pi$
$353$	4962.94	+	8596.06i	0.748302	+	1.29610i	−0.200334	+	0.979728i	$0.564203\pi$
$354$		0			0
$355$	−4751.69	+	8230.16i	−0.710404	+	1.23046i
$356$	−4563.95			−0.679463
$357$		0			0
$358$	2046.37	−	3544.42i	0.302107	−	0.523264i
$359$	215.661			0.0317052			0.0158526	−	0.999874i	$-0.494954\pi$
$359$	215.661			0.0317052			0.0158526	+	0.999874i	$0.494954\pi$
$360$		0			0
$361$	−24.7218	−	42.8195i	−0.00360429	−	0.00624282i
$362$	−2721.70	−	4714.12i	−0.395164	−	0.684444i
$363$		0			0
$364$	−2382.37	+	1221.36i	−0.343049	+	0.175870i
$365$	4127.83			0.591947
$366$		0			0
$367$	1430.76	+	2478.15i	0.203502	+	0.352475i	0.949654	−	0.313300i	$-0.101434\pi$
$367$	1430.76	+	2478.15i	0.203502	+	0.352475i	−0.746153	+	0.665775i	$0.768101\pi$
$368$	−2532.96	+	4387.21i	−0.358803	+	0.621465i
$369$		0			0
$370$	1251.42	−	2167.53i	0.175834	−	0.304553i
$371$	2131.47	−	3691.81i	0.298276	−	0.516629i
$372$		0			0
$373$	8.50723	−	14.7350i	0.00118093	−	0.00204544i	−0.865434	−	0.501022i	$-0.832957\pi$
$373$	8.50723	−	14.7350i	0.00118093	−	0.00204544i	0.866615	+	0.498977i	$0.166291\pi$
$374$	−225.489	−	390.558i	−0.0311758	−	0.0539981i
$375$		0			0
$376$	830.902			0.113964
$377$	9327.01	−	4781.66i	1.27418	−	0.653231i
$378$		0			0
$379$	950.983	+	1647.15i	0.128888	+	0.223241i	0.923246	−	0.384209i	$-0.125526\pi$
$379$	950.983	+	1647.15i	0.128888	+	0.223241i	−0.794358	+	0.607450i	$0.792192\pi$
$380$	−3282.20	−	5684.95i	−0.443088	−	0.767451i
$381$		0			0
$382$	−5086.05			−0.681217
$383$	3864.89	−	6694.18i	0.515631	−	0.893099i	−0.484205	−	0.874955i	$-0.660891\pi$
$383$	3864.89	−	6694.18i	0.515631	−	0.893099i	0.999835	−	0.0181440i	$-0.00577574\pi$
$384$		0			0
$385$	5724.23			0.757751
$386$	−3244.05	+	5618.85i	−0.427766	+	0.740912i
$387$		0			0
$388$	−4936.46	−	8550.20i	−0.645904	−	1.11874i
$389$	−14445.5			−1.88282			−0.941408	−	0.337271i	$-0.890496\pi$
$389$	−14445.5			−1.88282			−0.941408	+	0.337271i	$0.890496\pi$
$390$		0			0
$391$	−1287.16			−0.166482
$392$	−2417.40	−	4187.06i	−0.311472	−	0.539486i
$393$		0			0
$394$	−2298.12	+	3980.46i	−0.293851	+	0.508966i
$395$	−2586.50			−0.329470
$396$		0			0
$397$	−516.534	+	894.664i	−0.0653000	+	0.113103i	−0.896827	−	0.442381i	$-0.854134\pi$
$397$	−516.534	+	894.664i	−0.0653000	+	0.113103i	0.831527	+	0.555484i	$0.187467\pi$
$398$	6442.29			0.811364
$399$		0			0
$400$	−404.605	−	700.796i	−0.0505756	−	0.0875995i
$401$	2305.86	+	3993.87i	0.287155	+	0.497368i	0.973130	−	0.230258i	$-0.0739571\pi$
$401$	2305.86	+	3993.87i	0.287155	+	0.497368i	−0.685974	+	0.727626i	$0.740624\pi$
$402$		0			0
$403$	134.085	−	2686.57i	0.0165738	−	0.332078i
$404$	4970.12			0.612061
$405$		0			0
$406$	−1295.15	−	2243.26i	−0.158318	−	0.274215i
$407$	−3990.65	+	6912.00i	−0.486017	+	0.841806i
$408$		0			0
$409$	−1334.28	+	2311.04i	−0.161311	+	0.279398i	−0.935339	−	0.353753i	$-0.884905\pi$
$409$	−1334.28	+	2311.04i	−0.161311	+	0.279398i	0.774028	+	0.633151i	$0.218239\pi$
$410$	−1780.18	+	3083.36i	−0.214431	+	0.371405i
$411$		0			0
$412$	3278.62	−	5678.73i	0.392053	−	0.679056i
$413$	2770.57	+	4798.76i	0.330098	+	0.571747i
$414$		0			0
$415$	−3126.86			−0.369859
$416$	7181.97	+	4638.62i	0.846455	+	0.546700i
$417$		0			0
$418$	−2731.27	−	4730.70i	−0.319596	−	0.553556i
$419$	−3094.91	−	5360.55i	−0.360851	−	0.625012i	0.627250	−	0.778818i	$-0.284180\pi$
$419$	−3094.91	−	5360.55i	−0.360851	−	0.625012i	−0.988101	+	0.153806i	$0.950847\pi$
$420$		0			0
$421$	8991.13			1.04086			0.520429	−	0.853905i	$-0.325772\pi$
$421$	8991.13			1.04086			0.520429	+	0.853905i	$0.325772\pi$
$422$	−1131.80	+	1960.33i	−0.130557	+	0.226131i
$423$		0			0
$424$	−8739.63			−1.00102
$425$	102.803	−	178.060i	0.0117334	−	0.0203228i
$426$		0			0
$427$	879.230	+	1522.87i	0.0996462	+	0.172592i
$428$	1531.28			0.172937
$429$		0			0
$430$	5573.61			0.625078
$431$	−1034.51	−	1791.82i	−0.115616	−	0.200253i	0.802410	−	0.596773i	$-0.203551\pi$
$431$	−1034.51	−	1791.82i	−0.115616	−	0.200253i	−0.918026	+	0.396521i	$0.870217\pi$
$432$		0			0
$433$	7467.33	−	12933.8i	0.828769	−	1.43547i	−0.0702363	−	0.997530i	$-0.522375\pi$
$433$	7467.33	−	12933.8i	0.828769	−	1.43547i	0.899005	−	0.437939i	$-0.144291\pi$
$434$	−664.772			−0.0735256
$435$		0			0
$436$	5595.54	−	9691.76i	0.614628	−	1.06457i
$437$	−15591.0			−1.70668
$438$		0			0
$439$	3000.12	+	5196.36i	0.326168	+	0.564940i	0.981748	−	0.190186i	$-0.0609091\pi$
$439$	3000.12	+	5196.36i	0.326168	+	0.564940i	−0.655580	+	0.755126i	$0.727576\pi$
$440$	−5867.75	−	10163.2i	−0.635760	−	1.10117i
$441$		0			0
$442$	−20.6293	+	413.334i	−0.00221999	+	0.0444803i
$443$	−5392.60			−0.578352			−0.289176	−	0.957276i	$-0.593381\pi$
$443$	−5392.60			−0.578352			−0.289176	+	0.957276i	$0.593381\pi$
$444$		0			0
$445$	−4477.48	−	7755.23i	−0.476973	−	0.826141i
$446$	−1101.75	+	1908.28i	−0.116971	+	0.202600i
$447$		0			0
$448$	83.9259	−	145.364i	0.00885073	−	0.0153299i
$449$	3023.30	−	5236.51i	0.317769	−	0.550392i	−0.662253	−	0.749280i	$-0.730400\pi$
$449$	3023.30	−	5236.51i	0.317769	−	0.550392i	0.980022	+	0.198888i	$0.0637329\pi$
$450$		0			0
$451$	5676.78	−	9832.47i	0.592703	−	1.02659i
$452$	5925.15	+	10262.7i	0.616584	+	1.06795i
$453$		0			0
$454$	3315.37			0.342727
$455$	−4412.61	−	2849.98i	−0.454651	−	0.293646i
$456$		0			0
$457$	−507.286	−	878.646i	−0.0519253	−	0.0899372i	0.838894	−	0.544294i	$-0.183202\pi$
$457$	−507.286	−	878.646i	−0.0519253	−	0.0899372i	−0.890820	+	0.454357i	$0.849869\pi$
$458$	1262.96	+	2187.51i	0.128852	+	0.223178i
$459$		0			0
$460$	−14814.6			−1.50159
$461$	7989.45	−	13838.1i	0.807171	−	1.39806i	−0.107644	−	0.994189i	$-0.534331\pi$
$461$	7989.45	−	13838.1i	0.807171	−	1.39806i	0.914816	−	0.403872i	$-0.132336\pi$
$462$		0			0
$463$	−128.805			−0.0129289			−0.00646445	−	0.999979i	$-0.502058\pi$
$463$	−128.805			−0.0129289			−0.00646445	+	0.999979i	$0.502058\pi$
$464$	3019.56	−	5230.03i	0.302111	−	0.523272i
$465$		0			0
$466$	693.601	+	1201.35i	0.0689494	+	0.119424i
$467$	18832.4			1.86608			0.933039	−	0.359776i	$-0.117147\pi$
$467$	18832.4			1.86608			0.933039	+	0.359776i	$0.117147\pi$
$468$		0			0
$469$	3196.84			0.314747
$470$	360.538	+	624.471i	0.0353838	+	0.0612865i
$471$		0			0
$472$	5680.06	−	9838.15i	0.553911	−	0.959402i
$473$	−17773.6			−1.72776
$474$		0			0
$475$	1245.22	−	2156.79i	0.120284	−	0.208337i
$476$	−391.937			−0.0377403
$477$		0			0
$478$	−2572.38	−	4455.49i	−0.246146	−	0.426338i
$479$	4615.79	+	7994.79i	0.440294	+	0.762612i	0.997711	−	0.0676205i	$-0.0215407\pi$
$479$	4615.79	+	7994.79i	0.440294	+	0.762612i	−0.557417	+	0.830233i	$0.688207\pi$
$480$		0			0
$481$	6517.59	−	3341.36i	0.617831	−	0.316742i
$482$	−4599.58			−0.434658
$483$		0			0
$484$	4053.77	+	7021.34i	0.380708	+	0.659405i
$485$	9685.87	−	16776.4i	0.906830	−	1.57068i
$486$		0			0
$487$	−3616.50	+	6263.97i	−0.336508	+	0.582849i	−0.983773	−	0.179416i	$-0.942579\pi$
$487$	−3616.50	+	6263.97i	−0.336508	+	0.582849i	0.647265	+	0.762265i	$0.275913\pi$
$488$	1802.55	−	3122.11i	0.167208	−	0.289613i
$489$		0			0
$490$	2097.88	−	3633.63i	0.193413	−	0.335001i
$491$	327.565	+	567.360i	0.0301076	+	0.0521479i	0.880687	−	0.473699i	$-0.157082\pi$
$491$	327.565	+	567.360i	0.0301076	+	0.0521479i	−0.850579	+	0.525847i	$0.823748\pi$
$492$		0			0
$493$	1534.44			0.140178
$494$	−249.876	+	5006.58i	−0.0227580	+	0.455985i
$495$		0			0
$496$	−774.938	−	1342.23i	−0.0701527	−	0.121508i
$497$	3436.43	+	5952.08i	0.310151	+	0.537198i
$498$		0			0
$499$	−4560.07			−0.409092			−0.204546	−	0.978857i	$-0.565572\pi$
$499$	−4560.07			−0.409092			−0.204546	+	0.978857i	$0.565572\pi$
$500$	−3752.91	+	6500.23i	−0.335670	+	0.581398i
$501$		0			0
$502$	2767.29			0.246036
$503$	−4534.34	+	7853.70i	−0.401941	+	0.696181i	−0.993960	−	0.109742i	$-0.964997\pi$
$503$	−4534.34	+	7853.70i	−0.401941	+	0.696181i	0.592020	+	0.805924i	$0.298331\pi$
$504$		0			0
$505$	4875.95	+	8445.40i	0.429657	+	0.744189i
$506$	−12327.9			−1.08308
$507$		0			0
$508$	−1515.58			−0.132368
$509$	7940.44	+	13753.2i	0.691461	+	1.19765i	0.971359	+	0.237616i	$0.0763661\pi$
$509$	7940.44	+	13753.2i	0.691461	+	1.19765i	−0.279898	+	0.960030i	$0.590301\pi$
$510$		0			0
$511$	1492.63	−	2585.31i	0.129217	−	0.223811i
$512$	8913.89			0.769418
$513$		0			0
$514$	−2750.19	+	4763.46i	−0.236003	+	0.408769i
$515$	12866.0			1.10086
$516$		0			0
$517$	−1149.72	−	1991.36i	−0.0978035	−	0.169401i
$518$	−905.033	−	1567.56i	−0.0767662	−	0.132963i
$519$		0			0
$520$	−536.823	+	10755.9i	−0.0452716	+	0.907074i
$521$	−10998.0			−0.924821			−0.462410	−	0.886666i	$-0.653015\pi$
$521$	−10998.0			−0.924821			−0.462410	+	0.886666i	$0.653015\pi$
$522$		0			0
$523$	−9315.88	−	16135.6i	−0.778882	−	1.34906i	−0.932587	−	0.360946i	$-0.882454\pi$
$523$	−9315.88	−	16135.6i	−0.778882	−	1.34906i	0.153705	−	0.988117i	$-0.450879\pi$
$524$	−3675.69	+	6366.49i	−0.306438	+	0.530766i
$525$		0			0
$526$	−5385.88	+	9328.62i	−0.446456	+	0.773284i
$527$	196.899	−	341.039i	0.0162752	−	0.0281895i
$528$		0			0
$529$	−11509.4	+	19934.8i	−0.945949	+	1.63843i
$530$	−3792.23	−	6568.33i	−0.310800	−	0.538321i
$531$		0			0
$532$	−4747.40			−0.386891
$533$	−9271.41	+	4753.16i	−0.753451	+	0.386270i
$534$		0			0
$535$	1502.26	+	2602.00i	0.121399	+	0.210270i
$536$	−3276.99	−	5675.92i	−0.264076	−	0.457392i
$537$		0			0
$538$	−6535.96			−0.523764
$539$	−6689.89	+	11587.2i	−0.534608	+	0.925969i
$540$		0			0
$541$	−14270.1			−1.13405			−0.567023	−	0.823702i	$-0.691905\pi$
$541$	−14270.1			−1.13405			−0.567023	+	0.823702i	$0.691905\pi$
$542$	−3172.95	+	5495.71i	−0.251457	+	0.435537i
$543$		0			0
$544$	625.830	+	1083.97i	0.0493240	+	0.0854317i
$545$	21958.1			1.72584
$546$		0			0
$547$	14379.7			1.12400			0.562002	−	0.827136i	$-0.310031\pi$
$547$	14379.7			1.12400			0.562002	+	0.827136i	$0.310031\pi$
$548$	−2894.70	−	5013.77i	−0.225649	−	0.390835i
$549$		0			0
$550$	984.604	−	1705.38i	0.0763339	−	0.132214i
$551$	18586.2			1.43702
$552$		0			0
$553$	−935.281	+	1619.95i	−0.0719208	+	0.124570i
$554$	6655.46			0.510404
$555$		0			0
$556$	7343.77	+	12719.8i	0.560153	+	0.970214i
$557$	4517.55	+	7824.62i	0.343653	+	0.595224i	0.985108	−	0.171936i	$-0.0550023\pi$
$557$	4517.55	+	7824.62i	0.343653	+	0.595224i	−0.641455	+	0.767160i	$0.721669\pi$
$558$		0			0
$559$	13701.1	+	8849.11i	1.03666	+	0.669548i
$560$	−3026.65			−0.228392
$561$		0			0
$562$	139.938	+	242.379i	0.0105034	+	0.0181924i
$563$	3646.73	−	6316.33i	0.272987	−	0.472827i	−0.696638	−	0.717422i	$-0.745322\pi$
$563$	3646.73	−	6316.33i	0.272987	−	0.472827i	0.969625	+	0.244595i	$0.0786551\pi$
$564$		0			0
$565$	−11625.8	+	20136.4i	−0.865664	+	1.49937i
$566$	−3167.19	+	5485.73i	−0.235206	+	0.407389i
$567$		0			0
$568$	7045.19	−	12202.6i	0.520439	−	0.901427i
$569$	−13207.9	−	22876.7i	−0.973117	−	1.68549i	−0.686013	−	0.727589i	$-0.740641\pi$
$569$	−13207.9	−	22876.7i	−0.973117	−	1.68549i	−0.287104	−	0.957899i	$-0.592693\pi$
$570$		0			0
$571$	−23940.9			−1.75463			−0.877317	−	0.479912i	$-0.840669\pi$
$571$	−23940.9			−1.75463			−0.877317	+	0.479912i	$0.840669\pi$
$572$	757.144	−	15170.3i	0.0553458	−	1.10892i
$573$		0			0
$574$	1287.43	+	2229.89i	0.0936171	+	0.162150i
$575$	−2810.22	−	4867.44i	−0.203816	−	0.353019i
$576$		0			0
$577$	10182.8			0.734691			0.367346	−	0.930085i	$-0.380267\pi$
$577$	10182.8			0.734691			0.367346	+	0.930085i	$0.380267\pi$
$578$	3130.48	−	5422.14i	0.225278	−	0.390193i
$579$		0			0
$580$	17660.6			1.26434
$581$	−1130.68	+	1958.39i	−0.0807373	+	0.139841i
$582$		0			0
$583$	12093.0	+	20945.7i	0.859074	+	1.48796i
$584$	−6120.22			−0.433658
$585$		0			0
$586$	−3399.77			−0.239664
$587$	−9553.74	−	16547.6i	−0.671763	−	1.16353i	−0.977404	−	0.211381i	$-0.932204\pi$
$587$	−9553.74	−	16547.6i	−0.671763	−	1.16353i	0.305641	−	0.952147i	$-0.401129\pi$
$588$		0			0
$589$	2384.97	−	4130.89i	0.166844	−	0.288982i
$590$	9858.58			0.687917
$591$		0			0
$592$	2110.03	−	3654.68i	0.146489	−	0.253727i
$593$	−13597.5			−0.941621			−0.470810	−	0.882234i	$-0.656038\pi$
$593$	−13597.5			−0.941621			−0.470810	+	0.882234i	$0.656038\pi$
$594$		0			0
$595$	−384.511	−	665.992i	−0.0264931	−	0.0458874i
$596$	−2935.59	−	5084.58i	−0.201755	−	0.349451i
$597$		0			0
$598$	9503.13	+	6137.79i	0.649852	+	0.419721i
$599$	4150.37			0.283104			0.141552	−	0.989931i	$-0.454791\pi$
$599$	4150.37			0.283104			0.141552	+	0.989931i	$0.454791\pi$
$600$		0			0
$601$	−9588.73	−	16608.2i	−0.650803	−	1.12722i	−0.982928	−	0.183988i	$-0.941099\pi$
$601$	−9588.73	−	16608.2i	−0.650803	−	1.12722i	0.332126	−	0.943235i	$-0.392234\pi$
$602$	2015.43	−	3490.82i	0.136450	−	0.236338i
$603$		0			0
$604$	−11178.2	+	19361.1i	−0.753035	+	1.30429i
$605$	−7953.94	+	13776.6i	−0.534502	+	0.925784i
$606$		0			0
$607$	−3997.18	+	6923.32i	−0.267282	+	0.462947i	−0.968159	−	0.250336i	$-0.919459\pi$
$607$	−3997.18	+	6923.32i	−0.267282	+	0.462947i	0.700877	+	0.713282i	$0.252792\pi$
$608$	7580.48	+	13129.8i	0.505640	+	0.875794i
$609$		0			0
$610$	3128.59			0.207660
$611$	−105.184	+	2107.49i	−0.00696446	+	0.139542i
$612$		0			0
$613$	−7806.69	−	13521.6i	−0.514371	−	0.890916i	−0.999861	−	0.0166742i	$-0.994692\pi$
$613$	−7806.69	−	13521.6i	−0.514371	−	0.890916i	0.485490	−	0.874242i	$-0.338641\pi$
$614$	2178.52	+	3773.30i	0.143189	+	0.248010i
$615$		0			0
$616$	−8487.15			−0.555125
$617$	−10818.1	+	18737.5i	−0.705868	+	1.22260i	0.260509	+	0.965471i	$0.416110\pi$
$617$	−10818.1	+	18737.5i	−0.705868	+	1.22260i	−0.966377	+	0.257128i	$0.917224\pi$
$618$		0			0
$619$	10394.0			0.674911			0.337456	−	0.941341i	$-0.390434\pi$
$619$	10394.0			0.674911			0.337456	+	0.941341i	$0.390434\pi$
$620$	2266.20	−	3925.17i	0.146795	−	0.254256i
$621$		0			0
$622$	2.12184	+	3.67514i	0.000136782	+	0.000236913i
$623$	−6476.25			−0.416478
$624$		0			0
$625$	−18472.6			−1.18225
$626$	−1854.53	−	3212.14i	−0.118405	−	0.205084i
$627$		0			0
$628$	5125.74	−	8878.05i	0.325700	−	0.564129i
$629$	1072.25			0.0679702
$630$		0			0
$631$	−2322.07	+	4021.95i	−0.146498	+	0.253742i	−0.929931	−	0.367735i	$-0.880134\pi$
$631$	−2322.07	+	4021.95i	−0.146498	+	0.253742i	0.783433	+	0.621476i	$0.213467\pi$
$632$	3834.92			0.241369
$633$		0			0
$634$	928.160	+	1607.62i	0.0581419	+	0.100705i
$635$	−1486.87	−	2575.33i	−0.0929206	−	0.160943i
$636$		0			0
$637$	10926.0	−	5601.43i	0.679600	−	0.348409i
$638$	14696.2			0.911955
$639$		0			0
$640$	8933.29	+	15472.9i	0.551749	+	0.955657i
$641$	−76.9612	+	133.301i	−0.00474225	+	0.00821383i	−0.868387	−	0.495887i	$-0.834843\pi$
$641$	−76.9612	+	133.301i	−0.00474225	+	0.00821383i	0.863645	+	0.504101i	$0.168176\pi$
$642$		0			0
$643$	4219.02	−	7307.56i	0.258759	−	0.448184i	−0.707151	−	0.707063i	$-0.750020\pi$
$643$	4219.02	−	7307.56i	0.258759	−	0.448184i	0.965910	+	0.258879i	$0.0833531\pi$
$644$	−5356.97	+	9278.54i	−0.327786	+	0.567742i
$645$		0			0
$646$	−366.933	+	635.546i	−0.0223479	+	0.0387078i
$647$	−332.618	−	576.111i	−0.0202111	−	0.0350066i	0.855743	−	0.517401i	$-0.173100\pi$
$647$	−332.618	−	576.111i	−0.0202111	−	0.0350066i	−0.875954	+	0.482395i	$0.839767\pi$
$648$		0			0
$649$	−31437.9			−1.90145
$650$	−1608.07	+	824.407i	−0.0970366	+	0.0497475i
$651$		0			0
$652$	3197.60	+	5538.41i	0.192067	+	0.332670i
$653$	11207.4	+	19411.7i	0.671636	+	1.16331i	0.977440	+	0.211213i	$0.0677412\pi$
$653$	11207.4	+	19411.7i	0.671636	+	1.16331i	−0.305805	+	0.952094i	$0.598925\pi$
$654$		0			0
$655$	−14424.2			−0.860458
$656$	−3001.56	+	5198.86i	−0.178645	+	0.309423i
$657$		0			0
$658$	521.485			0.0308960
$659$	12792.8	−	22157.8i	0.756202	−	1.30978i	−0.188572	−	0.982059i	$-0.560386\pi$
$659$	12792.8	−	22157.8i	0.756202	−	1.30978i	0.944775	−	0.327721i	$-0.106281\pi$
$660$		0			0
$661$	6658.75	+	11533.3i	0.391824	+	0.678658i	0.992690	−	0.120691i	$-0.0385110\pi$
$661$	6658.75	+	11533.3i	0.391824	+	0.678658i	−0.600866	+	0.799349i	$0.705178\pi$
$662$	−4921.59			−0.288947
$663$		0			0
$664$	4636.10			0.270957
$665$	−4657.45	−	8066.95i	−0.271591	−	0.470410i
$666$		0			0
$667$	20972.6	−	36325.6i	1.21749	−	2.10875i
$668$	−14935.0			−0.865048
$669$		0			0
$670$	2843.85	−	4925.69i	0.163981	−	0.284024i
$671$	−9976.71			−0.573989
$672$		0			0
$673$	−4889.14	−	8468.24i	−0.280034	−	0.485032i	0.691359	−	0.722511i	$-0.257012\pi$
$673$	−4889.14	−	8468.24i	−0.280034	−	0.485032i	−0.971393	+	0.237479i	$0.923679\pi$
$674$	−1473.41	−	2552.02i	−0.0842042	−	0.145846i
$675$		0			0
$676$	−8136.65	+	11317.3i	−0.462941	+	0.643907i
$677$	12392.8			0.703537			0.351769	−	0.936087i	$-0.385580\pi$
$677$	12392.8			0.703537			0.351769	+	0.936087i	$0.385580\pi$
$678$		0			0
$679$	−7004.85	−	12132.8i	−0.395908	−	0.685732i
$680$	−788.303	+	1365.38i	−0.0444559	+	0.0769999i
$681$		0			0
$682$	1885.81	−	3266.32i	0.105882	−	0.183393i
$683$	9470.84	−	16404.0i	0.530588	−	0.919006i	−0.468775	−	0.883318i	$-0.655304\pi$
$683$	9470.84	−	16404.0i	0.530588	−	0.919006i	0.999363	−	0.0356881i	$-0.0113623\pi$
$684$		0			0
$685$	5679.71	−	9837.55i	0.316804	−	0.548721i
$686$	−3503.81	−	6068.77i	−0.195009	−	0.337765i
$687$		0			0
$688$	9397.69			0.520761
$689$	1106.35	−	22167.1i	0.0611735	−	1.22569i
$690$		0			0
$691$	−16188.8	−	28039.8i	−0.891247	−	1.54368i	−0.838382	−	0.545083i	$-0.816498\pi$
$691$	−16188.8	−	28039.8i	−0.891247	−	1.54368i	−0.0528645	−	0.998602i	$-0.516835\pi$
$692$	9936.38	+	17210.3i	0.545844	+	0.945430i
$693$		0			0
$694$	−431.034			−0.0235761
$695$	−14409.3	+	24957.6i	−0.786438	+	1.36215i
$696$		0			0
$697$	−1525.29			−0.0828903
$698$	5774.22	−	10001.2i	0.313120	−	0.542339i
$699$		0			0
$700$	−855.702	−	1482.12i	−0.0462035	−	0.0800269i
$701$	6072.34			0.327174			0.163587	−	0.986529i	$-0.447694\pi$
$701$	6072.34			0.327174			0.163587	+	0.986529i	$0.447694\pi$
$702$		0			0
$703$	12987.8			0.696789
$704$	476.158	+	824.730i	0.0254913	+	0.0441522i
$705$		0			0
$706$	6385.79	−	11060.5i	0.340414	−	0.589615i
$707$	7052.61			0.375163
$708$		0			0
$709$	1699.26	−	2943.21i	0.0900100	−	0.155902i	−0.817505	−	0.575921i	$-0.804643\pi$
$709$	1699.26	−	2943.21i	0.0900100	−	0.155902i	0.907515	+	0.420019i	$0.137977\pi$
$710$	12228.0			0.646348
$711$		0			0
$712$	6638.63	+	11498.4i	0.349429	+	0.605228i
$713$	−5382.40	−	9322.59i	−0.282710	−	0.489669i
$714$		0			0
$715$	26520.8	−	13596.3i	1.38716	−	0.711153i
$716$	20180.4			1.05332
$717$		0			0
$718$	−138.745	−	240.314i	−0.00721160	−	0.0124909i
$719$	−14416.8	+	24970.6i	−0.747782	+	1.29520i	0.201102	+	0.979570i	$0.435548\pi$
$719$	−14416.8	+	24970.6i	−0.747782	+	1.29520i	−0.948884	+	0.315626i	$0.897786\pi$
$720$		0			0
$721$	4652.36	−	8058.13i	0.240309	−	0.416228i
$722$	−31.8095	+	55.0957i	−0.00163965	+	0.00283996i
$723$		0			0
$724$	13420.1	−	23244.3i	0.688886	−	1.19319i
$725$	3350.09	+	5802.52i	0.171613	+	0.297242i
$726$		0			0
$727$	−12894.7			−0.657822			−0.328911	−	0.944361i	$-0.606682\pi$
$727$	−12894.7			−0.657822			−0.328911	+	0.944361i	$0.606682\pi$
$728$	6542.45	+	4225.58i	0.333076	+	0.215124i
$729$		0			0
$730$	−2655.63	−	4599.69i	−0.134643	−	0.233209i
$731$	1193.90	+	2067.89i	0.0604075	+	0.104629i
$732$		0			0
$733$	−23754.7			−1.19700			−0.598499	−	0.801124i	$-0.704236\pi$
$733$	−23754.7			−1.19700			−0.598499	+	0.801124i	$0.704236\pi$
$734$	1840.95	−	3188.63i	0.0925761	−	0.160347i
$735$		0			0
$736$	34215.2			1.71357
$737$	−9068.71	+	15707.5i	−0.453257	+	0.785064i
$738$		0			0
$739$	13023.6	+	22557.5i	0.648282	+	1.12286i	0.983533	+	0.180728i	$0.0578455\pi$
$739$	13023.6	+	22557.5i	0.648282	+	1.12286i	−0.335251	+	0.942129i	$0.608821\pi$
$740$	12341.0			0.613059
$741$		0			0
$742$	−5485.10			−0.271381
$743$	−7959.65	−	13786.5i	−0.393016	−	0.680725i	0.599829	−	0.800128i	$-0.295235\pi$
$743$	−7959.65	−	13786.5i	−0.393016	−	0.680725i	−0.992846	+	0.119403i	$0.961902\pi$
$744$		0			0
$745$	5759.93	−	9976.50i	0.283259	−	0.490618i
$746$	−21.8924			−0.00107445
$747$		0			0
$748$	1111.84	−	1925.76i	0.0543486	−	0.0941345i
$749$	2172.88			0.106002
$750$		0			0
$751$	−1118.79	−	1937.81i	−0.0543613	−	0.0941566i	0.837564	−	0.546339i	$-0.183979\pi$
$751$	−1118.79	−	1937.81i	−0.0543613	−	0.0941566i	−0.891926	+	0.452182i	$0.850646\pi$
$752$	607.905	+	1052.92i	0.0294787	+	0.0510587i
$753$		0			0
$754$	−11328.8	−	7316.92i	−0.547174	−	0.353404i
$755$	−43865.5			−2.11448
$756$		0			0
$757$	−1631.95	−	2826.63i	−0.0783545	−	0.135714i	0.824186	−	0.566320i	$-0.191633\pi$
$757$	−1631.95	−	2826.63i	−0.0783545	−	0.135714i	−0.902540	+	0.430606i	$0.858300\pi$
$758$	1223.63	−	2119.38i	0.0586334	−	0.101556i
$759$		0			0
$760$	−9548.45	+	16538.4i	−0.455735	+	0.789357i
$761$	952.456	−	1649.70i	0.0453699	−	0.0785830i	−0.842449	−	0.538777i	$-0.818887\pi$
$761$	952.456	−	1649.70i	0.0453699	−	0.0785830i	0.887819	+	0.460194i	$0.152220\pi$
$762$		0			0
$763$	7940.08	−	13752.6i	0.376737	−	0.652527i
$764$	−12539.1	−	21718.3i	−0.593781	−	1.02846i
$765$		0			0
$766$	−9945.88			−0.469137
$767$	24234.3	+	15652.3i	1.14088	+	0.736858i
$768$		0			0
$769$	−1714.94	−	2970.36i	−0.0804190	−	0.139290i	0.823011	−	0.568025i	$-0.192292\pi$
$769$	−1714.94	−	2970.36i	−0.0804190	−	0.139290i	−0.903430	+	0.428736i	$0.858959\pi$
$770$	−3682.68	−	6378.58i	−0.172356	−	0.298530i
$771$		0			0
$772$	−31991.4			−1.49144
$773$	14121.8	−	24459.6i	0.657082	−	1.13810i	−0.324285	−	0.945959i	$-0.605124\pi$
$773$	14121.8	−	24459.6i	0.657082	−	1.13810i	0.981367	−	0.192141i	$-0.0615430\pi$
$774$		0			0
$775$	1719.53			0.0796997
$776$	−14360.9	+	24873.9i	−0.664340	+	1.15067i
$777$		0			0
$778$	9293.48	+	16096.8i	0.428261	+	0.741770i
$779$	−18475.4			−0.849741
$780$		0			0
$781$	−38993.5			−1.78655
$782$	828.094	+	1434.30i	0.0378677	+	0.0655888i
$783$		0			0
$784$	3537.24	−	6126.68i	0.161135	−	0.279094i
$785$	20114.5			0.914545
$786$		0			0
$787$	−3313.36	+	5738.91i	−0.150074	+	0.259937i	−0.931255	−	0.364369i	$-0.881285\pi$
$787$	−3313.36	+	5738.91i	−0.150074	+	0.259937i	0.781180	+	0.624306i	$0.214618\pi$
$788$	−22663.0			−1.02454
$789$		0			0
$790$	1664.02	+	2882.16i	0.0749406	+	0.129801i
$791$	8407.80	+	14562.7i	0.377936	+	0.654603i
$792$		0			0
$793$	7690.69	+	4967.19i	0.344394	+	0.222434i
$794$	1329.25			0.0594120
$795$		0			0
$796$	15882.7	+	27509.7i	0.707222	+	1.22495i
$797$	10375.4	−	17970.8i	0.461125	−	0.798691i	−0.537893	−	0.843013i	$-0.680779\pi$
$797$	10375.4	−	17970.8i	0.461125	−	0.798691i	0.999017	+	0.0443221i	$0.0141128\pi$
$798$		0			0
$799$	−154.458	+	267.530i	−0.00683898	+	0.0118455i
$800$	−2732.71	+	4733.19i	−0.120770	+	0.209179i
$801$		0			0
$802$	2966.94	−	5138.90i	0.130631	−	0.226260i
$803$	8468.51	+	14667.9i	0.372163	+	0.644606i
$804$		0			0
$805$	−21021.9			−0.920403
$806$	−3079.93	+	1578.98i	−0.134598	+	0.0690041i
$807$		0			0
$808$	−7229.43	−	12521.7i	−0.314765	−	0.545190i
$809$	−5002.66	−	8664.86i	−0.217409	−	0.376564i	0.736606	−	0.676322i	$-0.236427\pi$
$809$	−5002.66	−	8664.86i	−0.217409	−	0.376564i	−0.954015	+	0.299758i	$0.903094\pi$
$810$		0			0
$811$	32773.5			1.41903			0.709515	−	0.704690i	$-0.248914\pi$
$811$	32773.5			1.41903			0.709515	+	0.704690i	$0.248914\pi$
$812$	6386.09	−	11061.0i	0.275995	−	0.478037i
$813$		0			0
$814$	10269.5			0.442194
$815$	−6274.04	+	10867.0i	−0.269657	+	0.467059i
$816$		0			0
$817$	14461.3	+	25047.7i	0.619261	+	1.07259i
$818$	3433.63			0.146765
$819$		0			0
$820$	−17555.3			−0.747632
$821$	1515.87	+	2625.57i	0.0644388	+	0.111611i	0.896445	−	0.443155i	$-0.146141\pi$
$821$	1515.87	+	2625.57i	0.0644388	+	0.111611i	−0.832006	+	0.554766i	$0.812808\pi$
$822$		0			0
$823$	−15528.5	+	26896.2i	−0.657704	+	1.13918i	0.323505	+	0.946226i	$0.395139\pi$
$823$	−15528.5	+	26896.2i	−0.657704	+	1.13918i	−0.981209	+	0.192950i	$0.938195\pi$
$824$	−19076.0			−0.806486
$825$		0			0
$826$	3564.88	−	6174.55i	0.150167	−	0.260097i
$827$	−8770.13			−0.368764			−0.184382	−	0.982855i	$-0.559028\pi$
$827$	−8770.13			−0.368764			−0.184382	+	0.982855i	$0.559028\pi$
$828$		0			0
$829$	19349.4	+	33514.2i	0.810656	+	1.40410i	0.912406	+	0.409287i	$0.134223\pi$
$829$	19349.4	+	33514.2i	0.810656	+	1.40410i	−0.101750	+	0.994810i	$0.532444\pi$
$830$	2011.66	+	3484.30i	0.0841274	+	0.145713i
$831$		0			0
$832$	43.5622	−	872.824i	0.00181520	−	0.0363699i
$833$	1797.50			0.0747657
$834$		0			0
$835$	−14652.0	−	25378.1i	−0.607251	−	1.05179i
$836$	13467.3	−	23326.0i	0.557149	−	0.965010i
$837$		0			0
$838$	−3982.21	+	6897.40i	−0.164157	+	0.284328i
$839$	11271.9	−	19523.4i	0.463824	−	0.803366i	−0.535324	−	0.844647i	$-0.679810\pi$
$839$	11271.9	−	19523.4i	0.463824	−	0.803366i	0.999148	+	0.0412806i	$0.0131438\pi$
$840$		0			0
$841$	−12807.2	+	22182.7i	−0.525121	+	0.909535i
$842$	−5784.43	−	10018.9i	−0.236751	−	0.410065i
$843$		0			0
$844$	−11161.3			−0.455198
$845$	−27213.3	−	2723.18i	−1.10789	−	0.110864i
$846$		0			0
$847$	5752.31	+	9963.29i	0.233355	+	0.404183i
$848$	−6394.09	−	11074.9i	−0.258932	−	0.448483i
$849$		0			0
$850$	−264.553			−0.0106754
$851$	14655.4	−	25383.9i	0.590341	−	1.02250i
$852$		0			0
$853$	−7751.80			−0.311156			−0.155578	−	0.987824i	$-0.549724\pi$
$853$	−7751.80			−0.311156			−0.155578	+	0.987824i	$0.549724\pi$
$854$	1131.30	−	1959.47i	0.0453306	−	0.0785150i
$855$		0			0
$856$	−2227.36	−	3857.91i	−0.0889366	−	0.154043i
$857$	30235.6			1.20517			0.602583	−	0.798056i	$-0.294138\pi$
$857$	30235.6			1.20517			0.602583	+	0.798056i	$0.294138\pi$
$858$		0			0
$859$	3568.09			0.141725			0.0708624	−	0.997486i	$-0.477425\pi$
$859$	3568.09			0.141725			0.0708624	+	0.997486i	$0.477425\pi$
$860$	13741.1	+	23800.3i	0.544847	+	0.943703i
$861$		0			0
$862$	−1331.10	+	2305.53i	−0.0525955	+	0.0910980i
$863$	−406.509			−0.0160345			−0.00801723	−	0.999968i	$-0.502552\pi$
$863$	−406.509			−0.0160345			−0.00801723	+	0.999968i	$0.502552\pi$
$864$		0			0
$865$	−19496.2	+	33768.5i	−0.766349	+	1.32736i
$866$	−19216.4			−0.754040
$867$		0			0
$868$	−1638.92	−	2838.70i	−0.0640883	−	0.111004i
$869$	−5306.36	−	9190.89i	−0.207141	−	0.358780i
$870$		0			0
$871$	14811.2	−	7593.22i	0.576185	−	0.295392i
$872$	−32556.6			−1.26434
$873$		0			0
$874$	10030.4	+	17373.2i	0.388197	+	0.672377i
$875$	−5325.38	+	9223.83i	−0.205749	+	0.356368i
$876$		0			0
$877$	−9480.97	+	16421.5i	−0.365051	+	0.632287i	−0.988784	−	0.149350i	$-0.952282\pi$
$877$	−9480.97	+	16421.5i	−0.365051	+	0.632287i	0.623733	+	0.781637i	$0.285615\pi$
$878$	3860.24	−	6686.13i	0.148379	−	0.257000i
$879$		0			0
$880$	8585.94	−	14871.3i	0.328900	−	0.569671i
$881$	15514.5	+	26871.8i	0.593298	+	1.02762i	0.993785	+	0.111320i	$0.0355078\pi$
$881$	15514.5	+	26871.8i	0.593298	+	1.02762i	−0.400487	+	0.916303i	$0.631159\pi$
$882$		0			0
$883$	33896.2			1.29184			0.645922	−	0.763404i	$-0.276473\pi$
$883$	33896.2			1.29184			0.645922	+	0.763404i	$0.276473\pi$
$884$	−1815.87	+	930.938i	−0.0690886	+	0.0354195i
$885$		0			0
$886$	3469.32	+	6009.03i	0.131551	+	0.227853i
$887$	886.675	+	1535.77i	0.0335644	+	0.0581353i	0.882320	−	0.470651i	$-0.155981\pi$
$887$	886.675	+	1535.77i	0.0335644	+	0.0581353i	−0.848755	+	0.528786i	$0.822647\pi$
$888$		0			0
$889$	−2150.62			−0.0811354
$890$	−5761.16	+	9978.62i	−0.216983	+	0.375825i
$891$		0			0
$892$	−10864.9			−0.407831
$893$	−1870.90	+	3240.50i	−0.0701091	+	0.121432i
$894$		0			0
$895$	19798.1	+	34291.3i	0.739415	+	1.28070i
$896$	12921.2			0.481770
$897$		0			0
$898$	−7780.14			−0.289116
$899$	6416.41	+	11113.5i	0.238041	+	0.412300i
$900$		0			0
$901$	1624.63	−	2813.94i	0.0600714	−	0.104047i
$902$	−14608.6			−0.539260
$903$		0			0
$904$	17237.2	−	29855.7i	0.634182	−	1.09844i
$905$	52663.3			1.93435
$906$		0			0
$907$	−7200.82	−	12472.2i	−0.263615	−	0.456595i	0.703584	−	0.710612i	$-0.251582\pi$
$907$	−7200.82	−	12472.2i	−0.263615	−	0.456595i	−0.967200	+	0.254016i	$0.918248\pi$
$908$	8173.67	+	14157.2i	0.298737	+	0.517427i
$909$		0			0
$910$	−336.917	+	6750.55i	−0.0122733	+	0.245911i
$911$	−19324.5			−0.702799			−0.351400	−	0.936226i	$-0.614294\pi$
$911$	−19324.5			−0.702799			−0.351400	+	0.936226i	$0.614294\pi$
$912$		0			0
$913$	−6414.95	−	11111.0i	−0.232534	−	0.402761i
$914$	−652.723	+	1130.55i	−0.0236216	+	0.0409139i
$915$		0			0
$916$	−6227.36	+	10786.1i	−0.224626	+	0.389064i
$917$	−5215.81	+	9034.05i	−0.187831	+	0.325333i
$918$		0			0
$919$	−9324.57	+	16150.6i	−0.334700	+	0.579717i	−0.983427	−	0.181304i	$-0.941968\pi$
$919$	−9324.57	+	16150.6i	−0.334700	+	0.579717i	0.648727	+	0.761021i	$0.275302\pi$
$920$	21548.9	+	37323.9i	0.772226	+	1.33753i
$921$		0			0
$922$	−20560.0			−0.734390
$923$	30058.8	+	19414.1i	1.07194	+	0.692331i
$924$		0			0
$925$	2341.00	+	4054.72i	0.0832124	+	0.144128i
$926$	82.8666	+	143.529i	0.00294078	+	0.00509358i
$927$		0			0
$928$	−40788.3			−1.44283
$929$	−13663.2	+	23665.4i	−0.482535	+	0.835775i	−0.999799	−	0.0200506i	$-0.993617\pi$
$929$	−13663.2	+	23665.4i	−0.482535	+	0.835775i	0.517264	+	0.855826i	$0.326951\pi$
$930$		0			0
$931$	21772.6			0.766453
$932$	−3419.99	+	5923.60i	−0.120199	+	0.208191i
$933$		0			0
$934$	−12115.8	−	20985.1i	−0.424454	−	0.735176i
$935$	4363.08			0.152608
$936$		0			0
$937$	28699.1			1.00060			0.500298	−	0.865853i	$-0.333224\pi$
$937$	28699.1			1.00060			0.500298	+	0.865853i	$0.333224\pi$
$938$	−2056.68	−	3562.28i	−0.0715917	−	0.124000i
$939$		0			0
$940$	−1777.73	+	3079.13i	−0.0616844	+	0.106840i
$941$	−55401.1			−1.91926			−0.959630	−	0.281264i	$-0.909246\pi$
$941$	−55401.1			−1.91926			−0.959630	+	0.281264i	$0.909246\pi$
$942$		0			0
$943$	−20847.6	+	36109.1i	−0.719927	+	1.24695i
$944$	16622.6			0.573113
$945$		0			0
$946$	11434.6	+	19805.3i	0.392993	+	0.680684i
$947$	−16629.2	−	28802.6i	−0.570619	−	0.988341i	−0.996502	−	0.0835631i	$-0.973370\pi$
$947$	−16629.2	−	28802.6i	−0.570619	−	0.988341i	0.425883	−	0.904778i	$-0.359963\pi$
$948$		0			0
$949$	774.758	−	15523.3i	0.0265013	−	0.530986i
$950$	−3204.44			−0.109438
$951$		0			0
$952$	570.103	+	987.447i	0.0194088	+	0.0336170i
$953$	−2205.95	+	3820.81i	−0.0749817	+	0.129872i	−0.901078	−	0.433656i	$-0.857223\pi$
$953$	−2205.95	+	3820.81i	−0.0749817	+	0.129872i	0.826097	+	0.563529i	$0.190557\pi$
$954$		0			0
$955$	24603.0	−	42613.7i	0.833650	−	1.44392i
$956$	12683.8	−	21969.1i	0.429105	−	0.743232i
$957$		0			0
$958$	5939.13	−	10286.9i	0.200297	−	0.346924i
$959$	−4107.59	−	7114.55i	−0.138312	−	0.239563i
$960$		0			0
$961$	−26497.6			−0.889450
$962$	−7916.39	−	5112.97i	−0.265317	−	0.171360i
$963$		0			0
$964$	−11339.8	−	19641.1i	−0.378869	−	0.656220i
$965$	−31385.2	−	54360.8i	−1.04697	−	1.81341i
$966$		0			0
$967$	26014.3			0.865114			0.432557	−	0.901607i	$-0.357611\pi$
$967$	26014.3			0.865114			0.432557	+	0.901607i	$0.357611\pi$
$968$	11793.1	−	20426.2i	0.391574	−	0.678226i
$969$		0			0
$970$	−24925.5			−0.825063
$971$	18102.5	−	31354.4i	0.598286	−	1.03626i	−0.394788	−	0.918772i	$-0.629182\pi$
$971$	18102.5	−	31354.4i	0.598286	−	1.03626i	0.993074	−	0.117490i	$-0.0374847\pi$
$972$		0			0
$973$	10420.8	+	18049.4i	0.343347	+	0.594694i
$974$	9306.69			0.306166
$975$		0			0
$976$	5275.12			0.173005
$977$	6958.19	+	12051.9i	0.227853	+	0.394653i	0.957172	−	0.289522i	$-0.0934962\pi$
$977$	6958.19	+	12051.9i	0.227853	+	0.394653i	−0.729319	+	0.684174i	$0.760163\pi$
$978$		0			0
$979$	18371.7	−	31820.7i	0.599756	−	1.03881i
$980$	20688.3			0.674351
$981$		0			0
$982$	421.477	−	730.020i	0.0136964	−	0.0237229i
$983$	−13585.3			−0.440796			−0.220398	−	0.975410i	$-0.570736\pi$
$983$	−13585.3			−0.440796			−0.220398	+	0.975410i	$0.570736\pi$
$984$		0			0
$985$	−22233.6	−	38509.8i	−0.719210	−	1.24571i
$986$	−987.178	−	1709.84i	−0.0318846	−	0.0552257i
$987$		0			0
$988$	−21995.0	+	11276.1i	−0.708254	+	0.363099i
$989$	65272.4			2.09863
$990$		0			0
$991$	512.246	+	887.237i	0.0164198	+	0.0284400i	0.874119	−	0.485713i	$-0.161440\pi$
$991$	512.246	+	887.237i	0.0164198	+	0.0284400i	−0.857699	+	0.514153i	$0.828107\pi$
$992$	−5233.94	+	9065.46i	−0.167518	+	0.290150i
$993$		0			0
$994$	4421.65	−	7658.52i	0.141093	−	0.244380i
$995$	−31163.6	+	53977.0i	−0.992919	+	1.71979i
$996$		0			0
$997$	−9257.43	+	16034.3i	−0.294068	+	0.509341i	−0.974768	−	0.223222i	$-0.928343\pi$
$997$	−9257.43	+	16034.3i	−0.294068	+	0.509341i	0.680700	+	0.732563i	$0.261676\pi$
$998$	2933.71	+	5081.34i	0.0930512	+	0.161169i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.g.f.55.4	✓	16
3.2	odd	2	inner	117.4.g.f.55.5	yes	16
13.3	even	3		1521.4.a.bc.1.5		8
13.9	even	3	inner	117.4.g.f.100.4	yes	16
13.10	even	6		1521.4.a.bd.1.4		8
39.23	odd	6		1521.4.a.bd.1.5		8
39.29	odd	6		1521.4.a.bc.1.4		8
39.35	odd	6	inner	117.4.g.f.100.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.g.f.55.4	✓	16	1.1	even	1	trivial
117.4.g.f.55.5	yes	16	3.2	odd	2	inner
117.4.g.f.100.4	yes	16	13.9	even	3	inner
117.4.g.f.100.5	yes	16	39.35	odd	6	inner
1521.4.a.bc.1.4		8	39.29	odd	6
1521.4.a.bc.1.5		8	13.3	even	3
1521.4.a.bd.1.4		8	13.10	even	6
1521.4.a.bd.1.5		8	39.23	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.643348 - 1.11431i) q^{2} +(3.17221 - 5.49442i) q^{4} +12.4484 q^{5} +(4.50137 - 7.79659i) q^{7} -18.4569 q^{8} +O(q^{10})\)	\(q+(-0.643348 - 1.11431i) q^{2} +(3.17221 - 5.49442i) q^{4} +12.4484 q^{5} +(4.50137 - 7.79659i) q^{7} -18.4569 q^{8} +(-8.00866 - 13.8714i) q^{10} +(25.5387 + 44.2343i) q^{11} +(2.33646 - 46.8139i) q^{13} -11.5838 q^{14} +(-13.5034 - 23.3886i) q^{16} +(3.43100 - 5.94266i) q^{17} +(41.5585 - 71.9815i) q^{19} +(39.4890 - 68.3969i) q^{20} +(32.8606 - 56.9162i) q^{22} +(-93.7893 - 162.448i) q^{23} +29.9631 q^{25} +(-53.6684 + 27.5141i) q^{26} +(-28.5585 - 49.4648i) q^{28} +(111.807 + 193.656i) q^{29} +57.3882 q^{31} +(-91.2024 + 157.967i) q^{32} -8.82930 q^{34} +(56.0349 - 97.0553i) q^{35} +(78.1293 + 135.324i) q^{37} -106.946 q^{38} -229.759 q^{40} +(-111.141 - 192.501i) q^{41} +(-173.987 + 301.354i) q^{43} +324.056 q^{44} +(-120.678 + 209.021i) q^{46} -45.0185 q^{47} +(130.975 + 226.856i) q^{49} +(-19.2767 - 33.3882i) q^{50} +(-249.804 - 161.341i) q^{52} +473.516 q^{53} +(317.917 + 550.648i) q^{55} +(-83.0813 + 143.901i) q^{56} +(143.862 - 249.176i) q^{58} +(-307.747 + 533.034i) q^{59} +(-97.6626 + 169.157i) q^{61} +(-36.9206 - 63.9483i) q^{62} +18.6446 q^{64} +(29.0852 - 582.759i) q^{65} +(177.548 + 307.523i) q^{67} +(-21.7677 - 37.7027i) q^{68} -144.200 q^{70} +(-381.710 + 661.141i) q^{71} +331.595 q^{73} +(100.529 - 174.121i) q^{74} +(-263.664 - 456.680i) q^{76} +459.836 q^{77} -207.777 q^{79} +(-168.096 - 291.152i) q^{80} +(-143.004 + 247.691i) q^{82} -251.185 q^{83} +(42.7105 - 73.9767i) q^{85} +447.737 q^{86} +(-471.365 - 816.429i) q^{88} +(-359.683 - 622.989i) q^{89} +(-354.472 - 228.943i) q^{91} -1190.08 q^{92} +(28.9626 + 50.1647i) q^{94} +(517.338 - 896.055i) q^{95} +(778.080 - 1347.67i) q^{97} +(168.526 - 291.895i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 40 q^{4} + 22 q^{7}+O(q^{10}) \) 16 * q - 40 * q^4 + 22 * q^7	\( 16 q - 40 q^{4} + 22 q^{7} - 36 q^{10} + 36 q^{13} - 204 q^{16} - 244 q^{19} - 136 q^{22} + 708 q^{25} + 452 q^{28} + 484 q^{31} - 2584 q^{34} - 1018 q^{37} + 3400 q^{40} - 74 q^{43} + 896 q^{46} - 298 q^{49} - 1676 q^{52} - 1300 q^{55} - 812 q^{58} - 1148 q^{61} + 7272 q^{64} + 2198 q^{67} + 4400 q^{70} - 4352 q^{73} - 6936 q^{76} + 3724 q^{79} - 5436 q^{82} + 890 q^{85} - 3528 q^{88} - 4754 q^{91} + 3104 q^{94} + 4370 q^{97}+O(q^{100}) \) 16 * q - 40 * q^4 + 22 * q^7 - 36 * q^10 + 36 * q^13 - 204 * q^16 - 244 * q^19 - 136 * q^22 + 708 * q^25 + 452 * q^28 + 484 * q^31 - 2584 * q^34 - 1018 * q^37 + 3400 * q^40 - 74 * q^43 + 896 * q^46 - 298 * q^49 - 1676 * q^52 - 1300 * q^55 - 812 * q^58 - 1148 * q^61 + 7272 * q^64 + 2198 * q^67 + 4400 * q^70 - 4352 * q^73 - 6936 * q^76 + 3724 * q^79 - 5436 * q^82 + 890 * q^85 - 3528 * q^88 - 4754 * q^91 + 3104 * q^94 + 4370 * q^97

Introduction

L-functions

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