LMFDB - Embedded newform 162.5.d.d.107.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,5,Mod(53,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("162.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	162.d (of order $6$, degree $2$, not 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:	$16.7459340196$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	162.107
Dual form			162.5.d.d.53.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.44949 + 1.41421i) q^{2} +(4.00000 + 6.92820i) q^{4} +(-7.97262 + 4.60300i) q^{5} +(-27.2750 + 47.2417i) q^{7} +22.6274i q^{8} +O(q^{10})$	\(q+(2.44949 + 1.41421i) q^{2} +(4.00000 + 6.92820i) q^{4} +(-7.97262 + 4.60300i) q^{5} +(-27.2750 + 47.2417i) q^{7} +22.6274i q^{8} -26.0385 q^{10} +(-34.3164 - 19.8126i) q^{11} +(-107.873 - 186.842i) q^{13} +(-133.620 + 77.1453i) q^{14} +(-32.0000 + 55.4256i) q^{16} -28.2272i q^{17} -254.435 q^{19} +(-63.7810 - 36.8240i) q^{20} +(-56.0385 - 97.0615i) q^{22} +(-801.737 + 462.883i) q^{23} +(-270.125 + 467.870i) q^{25} -610.222i q^{26} -436.400 q^{28} +(1164.65 + 672.411i) q^{29} +(-588.296 - 1018.96i) q^{31} +(-156.767 + 90.5097i) q^{32} +(39.9193 - 69.1422i) q^{34} -502.187i q^{35} -2344.80 q^{37} +(-623.235 - 359.825i) q^{38} +(-104.154 - 180.400i) q^{40} +(1257.92 - 726.261i) q^{41} +(-837.359 + 1450.35i) q^{43} -317.001i q^{44} -2618.46 q^{46} +(2878.47 + 1661.88i) q^{47} +(-287.350 - 497.705i) q^{49} +(-1323.34 + 764.028i) q^{50} +(862.985 - 1494.73i) q^{52} -1743.30i q^{53} +364.789 q^{55} +(-1068.96 - 617.163i) q^{56} +(1901.86 + 3294.13i) q^{58} +(709.264 - 409.494i) q^{59} +(-393.925 + 682.298i) q^{61} -3327.91i q^{62} -512.000 q^{64} +(1720.06 + 993.079i) q^{65} +(-1428.92 - 2474.97i) q^{67} +(195.564 - 112.909i) q^{68} +(710.199 - 1230.10i) q^{70} +7906.95i q^{71} +7112.40 q^{73} +(-5743.57 - 3316.05i) q^{74} +(-1017.74 - 1762.77i) q^{76} +(1871.96 - 1080.78i) q^{77} +(1737.80 - 3009.96i) q^{79} -589.183i q^{80} +4108.35 q^{82} +(996.961 + 575.596i) q^{83} +(129.930 + 225.045i) q^{85} +(-4102.21 + 2368.41i) q^{86} +(448.308 - 776.492i) q^{88} +10793.5i q^{89} +11768.9 q^{91} +(-6413.89 - 3703.06i) q^{92} +(4700.51 + 8141.53i) q^{94} +(2028.51 - 1171.16i) q^{95} +(-4982.60 + 8630.11i) q^{97} -1625.50i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{4} + 52 q^{7}+O(q^{10}) $ 8 * q + 32 * q^4 + 52 * q^7	$ 8 q + 32 q^{4} + 52 q^{7} - 624 q^{10} - 572 q^{13} - 256 q^{16} - 248 q^{19} - 864 q^{22} + 1892 q^{25} + 832 q^{28} - 3584 q^{31} + 1608 q^{34} - 18800 q^{37} - 2496 q^{40} - 3020 q^{43} - 4320 q^{46} - 9324 q^{49} + 4576 q^{52} + 22248 q^{55} + 2952 q^{58} + 4144 q^{61} - 4096 q^{64} + 12076 q^{67} - 18096 q^{70} - 3584 q^{73} - 992 q^{76} + 15004 q^{79} + 10752 q^{82} + 24048 q^{85} + 6912 q^{88} + 24440 q^{91} + 12912 q^{94} - 46304 q^{97}+O(q^{100}) $ 8 * q + 32 * q^4 + 52 * q^7 - 624 * q^10 - 572 * q^13 - 256 * q^16 - 248 * q^19 - 864 * q^22 + 1892 * q^25 + 832 * q^28 - 3584 * q^31 + 1608 * q^34 - 18800 * q^37 - 2496 * q^40 - 3020 * q^43 - 4320 * q^46 - 9324 * q^49 + 4576 * q^52 + 22248 * q^55 + 2952 * q^58 + 4144 * q^61 - 4096 * q^64 + 12076 * q^67 - 18096 * q^70 - 3584 * q^73 - 992 * q^76 + 15004 * q^79 + 10752 * q^82 + 24048 * q^85 + 6912 * q^88 + 24440 * q^91 + 12912 * q^94 - 46304 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{6}\right)$

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 $

$2$	2.44949	+	1.41421i	0.612372	+	0.353553i
$2$	2.44949	+	1.41421i	0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	4.00000	+	6.92820i	0.250000	+	0.433013i
$5$	−7.97262	+	4.60300i	−0.318905	+	0.184120i	−0.650904	−	0.759160i	$-0.725610\pi$
$5$	−7.97262	+	4.60300i	−0.318905	+	0.184120i	0.331999	+	0.943280i	$0.392277\pi$
$6$		0			0
$7$	−27.2750	+	47.2417i	−0.556632	+	0.964116i	0.441142	+	0.897437i	$0.354573\pi$
$7$	−27.2750	+	47.2417i	−0.556632	+	0.964116i	−0.997775	+	0.0666784i	$0.978760\pi$
$8$			22.6274i			0.353553i
$9$		0			0
$10$	−26.0385			−0.260385
$11$	−34.3164	−	19.8126i	−0.283607	−	0.163740i	0.351448	−	0.936207i	$-0.385689\pi$
$11$	−34.3164	−	19.8126i	−0.283607	−	0.163740i	−0.635055	+	0.772467i	$0.719023\pi$
$12$		0			0
$13$	−107.873	−	186.842i	−0.638302	−	1.10557i	−0.985805	−	0.167893i	$-0.946304\pi$
$13$	−107.873	−	186.842i	−0.638302	−	1.10557i	0.347503	−	0.937679i	$-0.387030\pi$
$14$	−133.620	+	77.1453i	−0.681733	+	0.393599i
$15$		0			0
$16$	−32.0000	+	55.4256i	−0.125000	+	0.216506i
$17$		−	28.2272i		−	0.0976719i	−0.998807	−	0.0488360i	$-0.984449\pi$
$17$		−	28.2272i		−	0.0976719i	0.998807	−	0.0488360i	$-0.0155512\pi$
$18$		0			0
$19$	−254.435			−0.704805			−0.352402	−	0.935849i	$-0.614635\pi$
$19$	−254.435			−0.704805			−0.352402	+	0.935849i	$0.614635\pi$
$20$	−63.7810	−	36.8240i	−0.159452	−	0.0920599i
$21$		0			0
$22$	−56.0385	−	97.0615i	−0.115782	−	0.200540i
$23$	−801.737	+	462.883i	−1.51557	+	0.875015i	−0.515737	+	0.856747i	$0.672482\pi$
$23$	−801.737	+	462.883i	−1.51557	+	0.875015i	−0.999833	+	0.0182680i	$0.994185\pi$
$24$		0			0
$25$	−270.125	+	467.870i	−0.432200	+	0.748592i
$26$		−	610.222i		−	0.902696i
$27$		0			0
$28$	−436.400			−0.556632
$29$	1164.65	+	672.411i	1.38484	+	0.799537i	0.992728	−	0.120381i	$-0.0384115\pi$
$29$	1164.65	+	672.411i	1.38484	+	0.799537i	0.392111	+	0.919918i	$0.371745\pi$
$30$		0			0
$31$	−588.296	−	1018.96i	−0.612171	−	1.06031i	−0.990874	−	0.134793i	$-0.956963\pi$
$31$	−588.296	−	1018.96i	−0.612171	−	1.06031i	0.378703	−	0.925518i	$-0.376370\pi$
$32$	−156.767	+	90.5097i	−0.153093	+	0.0883883i
$33$		0			0
$34$	39.9193	−	69.1422i	0.0345322	−	0.0598116i
$35$		−	502.187i		−	0.409948i
$36$		0			0
$37$	−2344.80			−1.71279			−0.856393	−	0.516325i	$-0.827300\pi$
$37$	−2344.80			−1.71279			−0.856393	+	0.516325i	$0.827300\pi$
$38$	−623.235	−	359.825i	−0.431603	−	0.249186i
$39$		0			0
$40$	−104.154	−	180.400i	−0.0650962	−	0.112750i
$41$	1257.92	−	726.261i	0.748317	−	0.432041i	−0.0767684	−	0.997049i	$-0.524460\pi$
$41$	1257.92	−	726.261i	0.748317	−	0.432041i	0.825086	+	0.565008i	$0.191127\pi$
$42$		0			0
$43$	−837.359	+	1450.35i	−0.452872	+	0.784397i	−0.998563	−	0.0535896i	$-0.982934\pi$
$43$	−837.359	+	1450.35i	−0.452872	+	0.784397i	0.545691	+	0.837986i	$0.316267\pi$
$44$		−	317.001i		−	0.163740i
$45$		0			0
$46$	−2618.46			−1.23746
$47$	2878.47	+	1661.88i	1.30306	+	0.752324i	0.980928	−	0.194370i	$-0.0622663\pi$
$47$	2878.47	+	1661.88i	1.30306	+	0.752324i	0.322134	+	0.946694i	$0.395600\pi$
$48$		0			0
$49$	−287.350	−	497.705i	−0.119679	−	0.207291i
$50$	−1323.34	+	764.028i	−0.529334	+	0.305611i
$51$		0			0
$52$	862.985	−	1494.73i	0.319151	−	0.552786i
$53$		−	1743.30i		−	0.620611i	−0.950637	−	0.310305i	$-0.899569\pi$
$53$		−	1743.30i		−	0.620611i	0.950637	−	0.310305i	$-0.100431\pi$
$54$		0			0
$55$	364.789			0.120591
$56$	−1068.96	−	617.163i	−0.340866	−	0.196799i
$57$		0			0
$58$	1901.86	+	3294.13i	0.565358	+	0.979229i
$59$	709.264	−	409.494i	0.203753	−	0.117637i	−0.394652	−	0.918831i	$-0.629135\pi$
$59$	709.264	−	409.494i	0.203753	−	0.117637i	0.598405	+	0.801194i	$0.295801\pi$
$60$		0			0
$61$	−393.925	+	682.298i	−0.105865	+	0.183364i	−0.914091	−	0.405508i	$-0.867095\pi$
$61$	−393.925	+	682.298i	−0.105865	+	0.183364i	0.808226	+	0.588872i	$0.200428\pi$
$62$		−	3327.91i		−	0.865740i
$63$		0			0
$64$	−512.000			−0.125000
$65$	1720.06	+	993.079i	0.407115	+	0.235048i
$66$		0			0
$67$	−1428.92	−	2474.97i	−0.318317	−	0.551341i	0.661820	−	0.749663i	$-0.269784\pi$
$67$	−1428.92	−	2474.97i	−0.318317	−	0.551341i	−0.980137	+	0.198322i	$0.936451\pi$
$68$	195.564	−	112.909i	0.0422932	−	0.0244180i
$69$		0			0
$70$	710.199	−	1230.10i	0.144939	−	0.251041i
$71$			7906.95i			1.56853i	0.620427	+	0.784264i	$0.286959\pi$
$71$			7906.95i			1.56853i	−0.620427	+	0.784264i	$0.713041\pi$
$72$		0			0
$73$	7112.40			1.33466			0.667330	−	0.744762i	$-0.267437\pi$
$73$	7112.40			1.33466			0.667330	+	0.744762i	$0.267437\pi$
$74$	−5743.57	−	3316.05i	−1.04886	−	0.605561i
$75$		0			0
$76$	−1017.74	−	1762.77i	−0.176201	−	0.305189i
$77$	1871.96	−	1080.78i	0.315729	−	0.182286i
$78$		0			0
$79$	1737.80	−	3009.96i	0.278449	−	0.482288i	−0.692550	−	0.721370i	$-0.743513\pi$
$79$	1737.80	−	3009.96i	0.278449	−	0.482288i	0.971000	+	0.239081i	$0.0768463\pi$
$80$		−	589.183i		−	0.0920599i
$81$		0			0
$82$	4108.35			0.610998
$83$	996.961	+	575.596i	0.144718	+	0.0835529i	0.570611	−	0.821221i	$-0.306707\pi$
$83$	996.961	+	575.596i	0.144718	+	0.0835529i	−0.425893	+	0.904774i	$0.640040\pi$
$84$		0			0
$85$	129.930	+	225.045i	0.0179833	+	0.0311481i
$86$	−4102.21	+	2368.41i	−0.554652	+	0.320229i
$87$		0			0
$88$	448.308	−	776.492i	0.0578910	−	0.100270i
$89$			10793.5i			1.36265i	0.731983	+	0.681323i	$0.238595\pi$
$89$			10793.5i			1.36265i	−0.731983	+	0.681323i	$0.761405\pi$
$90$		0			0
$91$	11768.9			1.42120
$92$	−6413.89	−	3703.06i	−0.757785	−	0.437507i
$93$		0			0
$94$	4700.51	+	8141.53i	0.531973	+	0.921404i
$95$	2028.51	−	1171.16i	0.224766	−	0.129769i
$96$		0			0
$97$	−4982.60	+	8630.11i	−0.529556	+	0.917219i	0.469849	+	0.882747i	$0.344308\pi$
$97$	−4982.60	+	8630.11i	−0.529556	+	0.917219i	−0.999406	+	0.0344720i	$0.989025\pi$
$98$		−	1625.50i		−	0.169252i
$99$		0			0
$100$	−4322.00			−0.432200
$101$	−8556.31	−	4939.99i	−0.838772	−	0.484265i	0.0180748	−	0.999837i	$-0.494246\pi$
$101$	−8556.31	−	4939.99i	−0.838772	−	0.484265i	−0.856847	+	0.515572i	$0.827580\pi$
$102$		0			0
$103$	6752.51	+	11695.7i	0.636489	+	1.10243i	0.986198	+	0.165572i	$0.0529471\pi$
$103$	6752.51	+	11695.7i	0.636489	+	1.10243i	−0.349709	+	0.936858i	$0.613720\pi$
$104$	4227.74	−	2440.89i	0.390879	−	0.225674i
$105$		0			0
$106$	2465.39	−	4270.18i	0.219419	−	0.380045i
$107$		−	9.72286i		−	0.000849233i	−1.00000		0.000424616i	$-0.999865\pi$
$107$		−	9.72286i		−	0.000849233i	1.00000		0.000424616i	$-0.000135160\pi$
$108$		0			0
$109$	18929.2			1.59324			0.796618	−	0.604484i	$-0.206620\pi$
$109$	18929.2			1.59324			0.796618	+	0.604484i	$0.206620\pi$
$110$	893.547	+	515.890i	0.0738469	+	0.0426355i
$111$		0			0
$112$	−1745.60	−	3023.47i	−0.139158	−	0.241029i
$113$	5106.30	−	2948.13i	0.399898	−	0.230881i	−0.286542	−	0.958068i	$-0.592506\pi$
$113$	5106.30	−	2948.13i	0.399898	−	0.230881i	0.686440	+	0.727186i	$0.259172\pi$
$114$		0			0
$115$	4261.30	−	7380.78i	0.322215	−	0.558093i
$116$			10758.6i			0.799537i
$117$		0			0
$118$	2316.45			0.166363
$119$	1333.50	+	769.896i	0.0941670	+	0.0543674i
$120$		0			0
$121$	−6535.42	−	11319.7i	−0.446378	−	0.773150i
$122$	−1929.83	+	1114.19i	−0.129658	+	0.0748581i
$123$		0			0
$124$	4706.37	−	8151.67i	0.306085	−	0.530155i
$125$		−	10727.3i		−	0.686546i
$126$		0			0
$127$	−20216.9			−1.25345			−0.626724	−	0.779241i	$-0.715605\pi$
$127$	−20216.9			−1.25345			−0.626724	+	0.779241i	$0.715605\pi$
$128$	−1254.14	−	724.077i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	2808.85	+	4865.07i	0.166204	+	0.287874i
$131$	12179.1	−	7031.58i	0.709694	−	0.409742i	−0.101254	−	0.994861i	$-0.532285\pi$
$131$	12179.1	−	7031.58i	0.709694	−	0.409742i	0.810948	+	0.585119i	$0.198952\pi$
$132$		0			0
$133$	6939.70	−	12019.9i	0.392317	−	0.679513i
$134$		−	8083.22i		−	0.450168i
$135$		0			0
$136$	638.708			0.0345322
$137$	12773.7	+	7374.92i	0.680576	+	0.392931i	0.800072	−	0.599904i	$-0.204795\pi$
$137$	12773.7	+	7374.92i	0.680576	+	0.392931i	−0.119496	+	0.992835i	$0.538128\pi$
$138$		0			0
$139$	7069.12	+	12244.1i	0.365878	+	0.633719i	0.988917	−	0.148472i	$-0.0474355\pi$
$139$	7069.12	+	12244.1i	0.365878	+	0.633719i	−0.623039	+	0.782191i	$0.714102\pi$
$140$	3479.25	−	2008.75i	0.177513	−	0.102487i
$141$		0			0
$142$	−11182.1	+	19368.0i	−0.554559	+	0.960524i
$143$			8548.98i			0.418064i
$144$		0			0
$145$	−12380.4			−0.588843
$146$	17421.8	+	10058.5i	0.817309	+	0.471873i
$147$		0			0
$148$	−9379.22	−	16245.3i	−0.428196	−	0.741658i
$149$	−9451.21	+	5456.66i	−0.425711	+	0.245784i	−0.697518	−	0.716567i	$-0.745712\pi$
$149$	−9451.21	+	5456.66i	−0.425711	+	0.245784i	0.271807	+	0.962352i	$0.412379\pi$
$150$		0			0
$151$	−1394.36	+	2415.10i	−0.0611534	+	0.105921i	−0.894981	−	0.446104i	$-0.852811\pi$
$151$	−1394.36	+	2415.10i	−0.0611534	+	0.105921i	0.833828	+	0.552025i	$0.186145\pi$
$152$		−	5757.20i		−	0.249186i
$153$		0			0
$154$	6113.80			0.257792
$155$	9380.53	+	5415.85i	0.390449	+	0.225426i
$156$		0			0
$157$	1238.42	+	2145.01i	0.0502423	+	0.0870222i	0.890053	−	0.455857i	$-0.150667\pi$
$157$	1238.42	+	2145.01i	0.0502423	+	0.0870222i	−0.839811	+	0.542880i	$0.817334\pi$
$158$	8513.46	−	4915.25i	0.341029	−	0.196893i
$159$		0			0
$160$	833.231	−	1443.20i	0.0325481	−	0.0563750i
$161$		−	50500.5i		−	1.94825i
$162$		0			0
$163$	−35547.0			−1.33791			−0.668957	−	0.743301i	$-0.733259\pi$
$163$	−35547.0			−1.33791			−0.668957	+	0.743301i	$0.733259\pi$
$164$	10063.4	+	5810.09i	0.374159	+	0.216021i
$165$		0			0
$166$	1628.03	+	2819.83i	0.0590808	+	0.102331i
$167$	−45892.8	+	26496.2i	−1.64555	+	0.950060i	−0.666743	+	0.745287i	$0.732312\pi$
$167$	−45892.8	+	26496.2i	−1.64555	+	0.950060i	−0.978810	+	0.204773i	$0.934354\pi$
$168$		0			0
$169$	−8992.70	+	15575.8i	−0.314859	+	0.545352i
$170$			734.993i			0.0254323i
$171$		0			0
$172$	−13397.8			−0.452872
$173$	−46242.8	−	26698.3i	−1.54508	−	0.892054i	−0.998506	−	0.0546443i	$-0.982598\pi$
$173$	−46242.8	−	26698.3i	−1.54508	−	0.892054i	−0.546576	−	0.837409i	$-0.684069\pi$
$174$		0			0
$175$	−14735.3	−	25522.3i	−0.481153	−	0.833381i
$176$	2196.25	−	1268.01i	0.0709017	−	0.0409351i
$177$		0			0
$178$	−15264.3	+	26438.6i	−0.481768	+	0.834447i
$179$			32448.3i			1.01271i	0.862324	+	0.506356i	$0.169008\pi$
$179$			32448.3i			1.01271i	−0.862324	+	0.506356i	$0.830992\pi$
$180$		0			0
$181$	−50381.3			−1.53784			−0.768921	−	0.639344i	$-0.779206\pi$
$181$	−50381.3			−1.53784			−0.768921	+	0.639344i	$0.779206\pi$
$182$	28827.9	+	16643.8i	0.870303	+	0.502470i
$183$		0			0
$184$	−10473.8	−	18141.2i	−0.309364	−	0.535835i
$185$	18694.2	−	10793.1i	0.546216	−	0.315358i
$186$		0			0
$187$	−559.254	+	968.656i	−0.0159928	+	0.0277004i
$188$			26590.1i			0.752324i
$189$		0			0
$190$	6625.09			0.183520
$191$	1953.74	+	1127.99i	0.0535551	+	0.0309201i	0.526538	−	0.850151i	$-0.323490\pi$
$191$	1953.74	+	1127.99i	0.0535551	+	0.0309201i	−0.472983	+	0.881071i	$0.656823\pi$
$192$		0			0
$193$	−12016.0	−	20812.3i	−0.322586	−	0.558736i	0.658435	−	0.752638i	$-0.271219\pi$
$193$	−12016.0	−	20812.3i	−0.322586	−	0.558736i	−0.981021	+	0.193902i	$0.937886\pi$
$194$	−24409.6	+	14092.9i	−0.648572	+	0.374453i
$195$		0			0
$196$	2298.80	−	3981.64i	0.0598397	−	0.103645i
$197$		−	66824.6i		−	1.72189i	−0.508702	−	0.860943i	$-0.669875\pi$
$197$		−	66824.6i		−	1.72189i	0.508702	−	0.860943i	$-0.330125\pi$
$198$		0			0
$199$	−8425.29			−0.212754			−0.106377	−	0.994326i	$-0.533925\pi$
$199$	−8425.29			−0.212754			−0.106377	+	0.994326i	$0.533925\pi$
$200$	−10586.7	−	6112.23i	−0.264667	−	0.152806i
$201$		0			0
$202$	−13972.4	−	24200.9i	−0.342427	−	0.593101i
$203$	−63531.6	+	36680.0i	−1.54169	+	0.890097i
$204$		0			0
$205$	−6685.95	+	11580.4i	−0.159095	+	0.275560i
$206$			38197.9i			0.900131i
$207$		0			0
$208$	13807.8			0.319151
$209$	8731.28	+	5041.01i	0.199887	+	0.115405i
$210$		0			0
$211$	16850.0	+	29185.0i	0.378473	+	0.655534i	0.990840	−	0.135039i	$-0.0431160\pi$
$211$	16850.0	+	29185.0i	0.378473	+	0.655534i	−0.612367	+	0.790573i	$0.709783\pi$
$212$	12077.9	−	6973.18i	0.268732	−	0.155153i
$213$		0			0
$214$	13.7502	−	23.8161i	0.000300249	−	0.000520047i
$215$		−	15417.4i		−	0.333531i
$216$		0			0
$217$	64183.1			1.36302
$218$	46367.0	+	26770.0i	0.975653	+	0.563294i
$219$		0			0
$220$	1459.16	+	2527.33i	0.0301479	+	0.0522176i
$221$	−5274.01	+	3044.95i	−0.107983	+	0.0623442i
$222$		0			0
$223$	10265.6	−	17780.6i	0.206432	−	0.357550i	−0.744156	−	0.668006i	$-0.767148\pi$
$223$	10265.6	−	17780.6i	0.206432	−	0.357550i	0.950588	+	0.310455i	$0.100482\pi$
$224$		−	9874.60i		−	0.196799i
$225$		0			0
$226$	16677.1			0.326516
$227$	−50720.2	−	29283.3i	−0.984305	−	0.568289i	−0.0807378	−	0.996735i	$-0.525728\pi$
$227$	−50720.2	−	29283.3i	−0.984305	−	0.568289i	−0.903567	+	0.428447i	$0.859061\pi$
$228$		0			0
$229$	3644.28	+	6312.07i	0.0694929	+	0.120365i	0.898678	−	0.438609i	$-0.144529\pi$
$229$	3644.28	+	6312.07i	0.0694929	+	0.120365i	−0.829185	+	0.558974i	$0.811195\pi$
$230$	20876.0	−	12052.8i	0.394631	−	0.227841i
$231$		0			0
$232$	−15214.9	+	26353.0i	−0.282679	+	0.489615i
$233$			13584.9i			0.250232i	0.992142	+	0.125116i	$0.0399304\pi$
$233$			13584.9i			0.250232i	−0.992142	+	0.125116i	$0.960070\pi$
$234$		0			0
$235$	−30598.6			−0.554071
$236$	5674.11	+	3275.95i	0.101876	+	0.0588184i
$237$		0			0
$238$	2177.60	+	3771.71i	0.0384435	+	0.0665862i
$239$	22418.8	−	12943.5i	0.392479	−	0.226598i	−0.290755	−	0.956798i	$-0.593906\pi$
$239$	22418.8	−	12943.5i	0.392479	−	0.226598i	0.683234	+	0.730200i	$0.260573\pi$
$240$		0			0
$241$	4680.48	−	8106.84i	0.0805855	−	0.139578i	−0.822916	−	0.568163i	$-0.807654\pi$
$241$	4680.48	−	8106.84i	0.0805855	−	0.139578i	0.903502	+	0.428585i	$0.140988\pi$
$242$		−	36969.9i		−	0.631274i
$243$		0			0
$244$	−6302.80			−0.105865
$245$	4581.87	+	2645.34i	0.0763327	+	0.0440707i
$246$		0			0
$247$	27446.6	+	47539.0i	0.449878	+	0.779212i
$248$	23056.4	−	13311.6i	0.374877	−	0.216435i
$249$		0			0
$250$	15170.7	−	26276.4i	0.242731	−	0.420422i
$251$			109546.i			1.73879i	0.494117	+	0.869396i	$0.335492\pi$
$251$			109546.i			1.73879i	−0.494117	+	0.869396i	$0.664508\pi$
$252$		0			0
$253$	36683.6			0.573101
$254$	−49521.0	−	28591.0i	−0.767577	−	0.443161i
$255$		0			0
$256$	−2048.00	−	3547.24i	−0.0312500	−	0.0541266i
$257$	83266.7	−	48074.1i	1.26068	−	0.727855i	0.287475	−	0.957788i	$-0.407184\pi$
$257$	83266.7	−	48074.1i	1.26068	−	0.727855i	0.973206	+	0.229933i	$0.0738508\pi$
$258$		0			0
$259$	63954.5	−	110772.i	0.953392	−	1.65132i
$260$			15889.3i			0.235048i
$261$		0			0
$262$	39776.6			0.579463
$263$	11913.5	+	6878.28i	0.172238	+	0.0994417i	0.583641	−	0.812012i	$-0.301628\pi$
$263$	11913.5	+	6878.28i	0.172238	+	0.0994417i	−0.411403	+	0.911454i	$0.634961\pi$
$264$		0			0
$265$	8024.38	+	13898.6i	0.114267	+	0.197916i
$266$	33997.4	−	19628.4i	0.480489	−	0.277410i
$267$		0			0
$268$	11431.4	−	19799.8i	0.159158	−	0.275670i
$269$		−	65136.7i		−	0.900163i	−0.892988	−	0.450081i	$-0.851395\pi$
$269$		−	65136.7i		−	0.900163i	0.892988	−	0.450081i	$-0.148605\pi$
$270$		0			0
$271$	9569.63			0.130304			0.0651518	−	0.997875i	$-0.479247\pi$
$271$	9569.63			0.130304			0.0651518	+	0.997875i	$0.479247\pi$
$272$	1564.51	+	903.270i	0.0211466	+	0.0122090i
$273$		0			0
$274$	20859.4	+	36129.6i	0.277844	+	0.481240i
$275$	18539.4	−	10703.7i	0.245150	−	0.141537i
$276$		0			0
$277$	−22168.3	+	38396.6i	−0.288916	+	0.500418i	−0.973551	−	0.228468i	$-0.926628\pi$
$277$	−22168.3	+	38396.6i	−0.288916	+	0.500418i	0.684635	+	0.728886i	$0.259962\pi$
$278$			39989.0i			0.517429i
$279$		0			0
$280$	11363.2			0.144939
$281$	−105192.	−	60732.4i	−1.33220	−	0.769145i	−0.346561	−	0.938027i	$-0.612651\pi$
$281$	−105192.	−	60732.4i	−1.33220	−	0.769145i	−0.985636	+	0.168883i	$0.945984\pi$
$282$		0			0
$283$	58191.1	+	100790.i	0.726581	+	1.25848i	0.958320	+	0.285697i	$0.0922251\pi$
$283$	58191.1	+	100790.i	0.726581	+	1.25848i	−0.231739	+	0.972778i	$0.574442\pi$
$284$	−54781.0	+	31627.8i	−0.679193	+	0.392132i
$285$		0			0
$286$	−12090.1	+	20940.6i	−0.147808	+	0.256011i
$287$			79235.1i			0.961952i
$288$		0			0
$289$	82724.2			0.990460
$290$	−30325.7	−	17508.6i	−0.360591	−	0.208187i
$291$		0			0
$292$	28449.6	+	49276.2i	0.333665	+	0.577925i
$293$	137541.	−	79409.4i	1.60213	−	0.924990i	0.611068	−	0.791578i	$-0.290740\pi$
$293$	137541.	−	79409.4i	1.60213	−	0.924990i	0.991061	−	0.133412i	$-0.0425933\pi$
$294$		0			0
$295$	−3769.79	+	6529.47i	−0.0433185	+	0.0750299i
$296$		−	53056.9i		−	0.605561i
$297$		0			0
$298$	−30867.5			−0.347592
$299$	172972.	+	99865.2i	1.93478	+	1.11705i
$300$		0			0
$301$	−45677.9	−	79116.5i	−0.504166	−	0.873241i
$302$	−6830.94	+	3943.84i	−0.0748973	+	0.0432420i
$303$		0			0
$304$	8141.91	−	14102.2i	0.0881006	−	0.152595i
$305$		−	7252.94i		−	0.0779676i
$306$		0			0
$307$	100162.			1.06274			0.531370	−	0.847140i	$-0.321677\pi$
$307$	100162.			1.06274			0.531370	+	0.847140i	$0.321677\pi$
$308$	14975.7	+	8646.21i	0.157865	+	0.0911432i
$309$		0			0
$310$	15318.3	+	26532.1i	0.159400	+	0.276089i
$311$	−46738.2	+	26984.3i	−0.483227	+	0.278991i	−0.721760	−	0.692143i	$-0.756667\pi$
$311$	−46738.2	+	26984.3i	−0.483227	+	0.278991i	0.238534	+	0.971134i	$0.423333\pi$
$312$		0			0
$313$	−37756.9	+	65396.9i	−0.385396	+	0.667526i	−0.991824	−	0.127613i	$-0.959269\pi$
$313$	−37756.9	+	65396.9i	−0.385396	+	0.667526i	0.606428	+	0.795139i	$0.292602\pi$
$314$			7005.58i			0.0710534i
$315$		0			0
$316$	27804.8			0.278449
$317$	−57648.6	−	33283.4i	−0.573680	−	0.331215i	0.184938	−	0.982750i	$-0.440792\pi$
$317$	−57648.6	−	33283.4i	−0.573680	−	0.331215i	−0.758618	+	0.651536i	$0.774125\pi$
$318$		0			0
$319$	−26644.4	−	46149.5i	−0.261833	−	0.453508i
$320$	4081.98	−	2356.73i	0.0398631	−	0.0230150i
$321$		0			0
$322$	71418.5	−	123700.i	0.688809	−	1.19305i
$323$			7181.97i			0.0688397i
$324$		0			0
$325$	116557.			1.10350
$326$	−87072.1	−	50271.1i	−0.819302	−	0.473024i
$327$		0			0
$328$	16433.4	+	28463.5i	0.152750	+	0.264570i
$329$	−157020.	+	90655.7i	−1.45065	+	0.837535i
$330$		0			0
$331$	34710.1	−	60119.7i	0.316811	−	0.548732i	−0.663010	−	0.748611i	$-0.730721\pi$
$331$	34710.1	−	60119.7i	0.316811	−	0.548732i	0.979821	+	0.199878i	$0.0640546\pi$
$332$			9209.53i			0.0835529i
$333$		0			0
$334$	−149885.			−1.34359
$335$	22784.5	+	13154.7i	0.203026	+	0.117217i
$336$		0			0
$337$	−44393.0	−	76890.9i	−0.390890	−	0.677041i	0.601677	−	0.798739i	$-0.294499\pi$
$337$	−44393.0	−	76890.9i	−0.390890	−	0.677041i	−0.992567	+	0.121698i	$0.961166\pi$
$338$	−44055.0	+	25435.2i	−0.385622	+	0.222639i
$339$		0			0
$340$	−1039.44	+	1800.36i	−0.00899167	+	0.0155740i
$341$			46622.7i			0.400948i
$342$		0			0
$343$	−99624.6			−0.846795
$344$	−32817.7	−	18947.3i	−0.277326	−	0.160114i
$345$		0			0
$346$	−75514.1	−	130794.i	−0.630777	−	1.09254i
$347$	95033.8	−	54867.8i	0.789259	−	0.455679i	−0.0504428	−	0.998727i	$-0.516063\pi$
$347$	95033.8	−	54867.8i	0.789259	−	0.455679i	0.839701	+	0.543048i	$0.182730\pi$
$348$		0			0
$349$	−69389.6	+	120186.i	−0.569696	+	0.986743i	0.426899	+	0.904299i	$0.359606\pi$
$349$	−69389.6	+	120186.i	−0.569696	+	0.986743i	−0.996596	+	0.0824438i	$0.973728\pi$
$350$		−	83355.5i		−	0.680453i
$351$		0			0
$352$	7172.92			0.0578910
$353$	67397.6	+	38912.0i	0.540873	+	0.312273i	0.745433	−	0.666581i	$-0.232243\pi$
$353$	67397.6	+	38912.0i	0.540873	+	0.312273i	−0.204560	+	0.978854i	$0.565576\pi$
$354$		0			0
$355$	−36395.7	−	63039.2i	−0.288797	−	0.500212i
$356$	−74779.7	+	43174.1i	−0.590043	+	0.340662i
$357$		0			0
$358$	−45888.9	+	79481.8i	−0.358048	+	0.620157i
$359$			195283.i			1.51522i	0.652708	+	0.757610i	$0.273633\pi$
$359$			195283.i			1.51522i	−0.652708	+	0.757610i	$0.726367\pi$
$360$		0			0
$361$	−65584.1			−0.503250
$362$	−123408.	−	71249.8i	−0.941732	−	0.543709i
$363$		0			0
$364$	47075.8	+	81537.7i	0.355300	+	0.615397i
$365$	−56704.5	+	32738.4i	−0.425630	+	0.245737i
$366$		0			0
$367$	−42902.7	+	74309.6i	−0.318531	+	0.551713i	−0.980182	−	0.198100i	$-0.936523\pi$
$367$	−42902.7	+	74309.6i	−0.318531	+	0.551713i	0.661650	+	0.749812i	$0.269856\pi$
$368$		−	59249.0i		−	0.437507i
$369$		0			0
$370$	61055.1			0.445983
$371$	82356.2	+	47548.4i	0.598340	+	0.345452i
$372$		0			0
$373$	−6159.02	−	10667.7i	−0.0442684	−	0.0766751i	0.843042	−	0.537847i	$-0.180762\pi$
$373$	−6159.02	−	10667.7i	−0.0442684	−	0.0766751i	−0.887311	+	0.461172i	$0.847429\pi$
$374$	−2739.77	+	1581.81i	−0.0195872	+	0.0113086i
$375$		0			0
$376$	−37604.1	+	65132.2i	−0.265987	+	0.460702i
$377$		−	290140.i		−	2.04139i
$378$		0			0
$379$	−216174.			−1.50496			−0.752479	−	0.658617i	$-0.771142\pi$
$379$	−216174.			−1.50496			−0.752479	+	0.658617i	$0.771142\pi$
$380$	16228.1	+	9369.29i	0.112383	+	0.0648843i
$381$		0			0
$382$	3190.45	+	5526.02i	0.0218638	+	0.0378692i
$383$	−205867.	+	118857.i	−1.40342	+	0.810266i	−0.994742	−	0.102411i	$-0.967344\pi$
$383$	−205867.	+	118857.i	−1.40342	+	0.810266i	−0.408680	+	0.912678i	$0.634011\pi$
$384$		0			0
$385$	−9949.62	+	17233.2i	−0.0671251	+	0.116264i
$386$		−	67972.8i		−	0.456206i
$387$		0			0
$388$	−79721.5			−0.529556
$389$	161345.	+	93152.6i	1.06624	+	0.615596i	0.927153	−	0.374684i	$-0.122249\pi$
$389$	161345.	+	93152.6i	1.06624	+	0.615596i	0.139091	+	0.990280i	$0.455582\pi$
$390$		0			0
$391$	13065.9	+	22630.8i	0.0854644	+	0.148029i
$392$	11261.8	−	6501.99i	0.0732884	−	0.0423131i
$393$		0			0
$394$	94504.3	−	163686.i	0.608778	−	1.05444i
$395$			31996.4i			0.205072i
$396$		0			0
$397$	5004.70			0.0317539			0.0158769	−	0.999874i	$-0.494946\pi$
$397$	5004.70			0.0317539			0.0158769	+	0.999874i	$0.494946\pi$
$398$	−20637.7	−	11915.2i	−0.130285	−	0.0752200i
$399$		0			0
$400$	−17288.0	−	29943.7i	−0.108050	−	0.187148i
$401$	152835.	−	88239.4i	0.950461	−	0.548749i	0.0572371	−	0.998361i	$-0.481771\pi$
$401$	152835.	−	88239.4i	0.950461	−	0.548749i	0.893224	+	0.449612i	$0.148438\pi$
$402$		0			0
$403$	−126923.	+	219836.i	−0.781500	+	1.35360i
$404$		−	79039.8i		−	0.484265i
$405$		0			0
$406$	−207493.			−1.25879
$407$	80465.3	+	46456.6i	0.485758	+	0.280452i
$408$		0			0
$409$	−94356.8	−	163431.i	−0.564062	−	0.976984i	−0.997136	−	0.0756254i	$-0.975905\pi$
$409$	−94356.8	−	163431.i	−0.564062	−	0.976984i	0.433075	−	0.901358i	$-0.357429\pi$
$410$	−32754.3	+	18910.7i	−0.194850	+	0.112497i
$411$		0			0
$412$	−54020.1	+	93565.5i	−0.318244	+	0.551215i
$413$			44675.7i			0.261922i
$414$		0			0
$415$	−10597.9			−0.0615349
$416$	33821.9	+	19527.1i	0.195439	+	0.112837i
$417$		0			0
$418$	14258.1	+	24695.8i	0.0816037	+	0.141342i
$419$	−10397.0	+	6002.70i	−0.0592215	+	0.0341915i	−0.529318	−	0.848423i	$-0.677552\pi$
$419$	−10397.0	+	6002.70i	−0.0592215	+	0.0341915i	0.470097	+	0.882615i	$0.344219\pi$
$420$		0			0
$421$	27797.6	−	48146.8i	0.156835	−	0.271646i	−0.776891	−	0.629635i	$-0.783204\pi$
$421$	27797.6	−	48146.8i	0.156835	−	0.271646i	0.933726	+	0.357989i	$0.116538\pi$
$422$			95317.9i			0.535241i
$423$		0			0
$424$	39446.3			0.219419
$425$	13206.7	+	7624.87i	0.0731164	+	0.0422138i
$426$		0			0
$427$	−21488.6	−	37219.3i	−0.117856	−	0.204133i
$428$	67.3620	−	38.8915i	0.000367728	−	0.000212308i
$429$		0			0
$430$	21803.6	−	37764.9i	0.117921	−	0.204245i
$431$		−	52071.6i		−	0.280315i	−0.990129	−	0.140158i	$-0.955239\pi$
$431$		−	52071.6i		−	0.280315i	0.990129	−	0.140158i	$-0.0447609\pi$
$432$		0			0
$433$	−205361.			−1.09532			−0.547662	−	0.836700i	$-0.684482\pi$
$433$	−205361.			−1.09532			−0.547662	+	0.836700i	$0.684482\pi$
$434$	157216.	+	90768.6i	0.834674	+	0.481899i
$435$		0			0
$436$	75716.9	+	131146.i	0.398309	+	0.689891i
$437$	203990.	−	117773.i	1.06818	−	0.616715i
$438$		0			0
$439$	162282.	−	281081.i	0.842058	−	1.45849i	−0.0460931	−	0.998937i	$-0.514677\pi$
$439$	162282.	−	281081.i	0.842058	−	1.45849i	0.888152	−	0.459551i	$-0.151990\pi$
$440$			8254.24i			0.0426355i
$441$		0			0
$442$	−17224.9			−0.0881680
$443$	−148975.	−	86010.6i	−0.759111	−	0.438273i	0.0698657	−	0.997556i	$-0.477743\pi$
$443$	−148975.	−	86010.6i	−0.759111	−	0.438273i	−0.828976	+	0.559284i	$0.811076\pi$
$444$		0			0
$445$	−49682.5	−	86052.7i	−0.250890	−	0.434555i
$446$	50291.2	−	29035.6i	0.252826	−	0.145969i
$447$		0			0
$448$	13964.8	−	24187.7i	0.0695791	−	0.120514i
$449$		−	121635.i		−	0.603345i	−0.953412	−	0.301672i	$-0.902455\pi$
$449$		−	121635.i		−	0.603345i	0.953412	−	0.301672i	$-0.0975449\pi$
$450$		0			0
$451$	−57556.5			−0.282970
$452$	40850.4	+	23585.0i	0.199949	+	0.115441i
$453$		0			0
$454$	−82825.8	−	143459.i	−0.401841	−	0.696009i
$455$	−93829.4	+	54172.4i	−0.453227	+	0.261671i
$456$		0			0
$457$	−23274.4	+	40312.4i	−0.111441	+	0.193022i	−0.916352	−	0.400375i	$-0.868880\pi$
$457$	−23274.4	+	40312.4i	−0.111441	+	0.193022i	0.804910	+	0.593396i	$0.202213\pi$
$458$			20615.1i			0.0982778i
$459$		0			0
$460$	68180.7			0.322215
$461$	176904.	+	102136.i	0.832409	+	0.480591i	0.854677	−	0.519161i	$-0.173755\pi$
$461$	176904.	+	102136.i	0.832409	+	0.480591i	−0.0222679	+	0.999752i	$0.507089\pi$
$462$		0			0
$463$	8635.53	+	14957.2i	0.0402835	+	0.0697731i	0.885464	−	0.464708i	$-0.153841\pi$
$463$	8635.53	+	14957.2i	0.0402835	+	0.0697731i	−0.845181	+	0.534481i	$0.820507\pi$
$464$	−74537.6	+	43034.3i	−0.346210	+	0.199884i
$465$		0			0
$466$	−19211.9	+	33276.0i	−0.0884705	+	0.153235i
$467$		−	219664.i		−	1.00722i	−0.863930	−	0.503611i	$-0.832004\pi$
$467$		−	219664.i		−	1.00722i	0.863930	−	0.503611i	$-0.167996\pi$
$468$		0			0
$469$	155896.			0.708742
$470$	−74950.9	−	43272.9i	−0.339298	−	0.195894i
$471$		0			0
$472$	9265.78	+	16048.8i	0.0415909	+	0.0720375i
$473$	57470.4	−	33180.5i	0.256875	−	0.148307i
$474$		0			0
$475$	68729.1	−	119042.i	0.304617	−	0.527611i
$476$			12318.3i			0.0543674i
$477$		0			0
$478$	73219.4			0.320457
$479$	−78854.6	−	45526.7i	−0.343682	−	0.198425i	0.318217	−	0.948018i	$-0.396916\pi$
$479$	−78854.6	−	45526.7i	−0.343682	−	0.198425i	−0.661899	+	0.749593i	$0.730249\pi$
$480$		0			0
$481$	252941.	+	438107.i	1.09327	+	1.89361i
$482$	22929.6	−	13238.4i	0.0986966	−	0.0569825i
$483$		0			0
$484$	52283.4	−	90557.5i	0.223189	−	0.386575i
$485$		−	91739.5i		−	0.390007i
$486$		0			0
$487$	146600.			0.618126			0.309063	−	0.951042i	$-0.399985\pi$
$487$	146600.			0.618126			0.309063	+	0.951042i	$0.399985\pi$
$488$	−15438.6	−	8913.50i	−0.0648290	−	0.0374290i
$489$		0			0
$490$	7482.16	+	12959.5i	0.0311627	+	0.0539754i
$491$	41966.3	−	24229.2i	0.174075	−	0.100502i	−0.410431	−	0.911892i	$-0.634622\pi$
$491$	41966.3	−	24229.2i	0.174075	−	0.100502i	0.584506	+	0.811389i	$0.301288\pi$
$492$		0			0
$493$	18980.3	−	32874.8i	0.0780924	−	0.135260i
$494$			155262.i			0.636224i
$495$		0			0
$496$	75301.9			0.306085
$497$	−373538.	−	215662.i	−1.51224	−	0.873094i
$498$		0			0
$499$	124007.	+	214786.i	0.498016	+	0.862590i	0.999997	−	0.00228889i	$-0.000728578\pi$
$499$	124007.	+	214786.i	0.498016	+	0.862590i	−0.501981	+	0.864879i	$0.667395\pi$
$500$	74320.8	−	42909.1i	0.297283	−	0.171636i
$501$		0			0
$502$	−154921.	+	268331.i	−0.614755	+	1.06479i
$503$			486322.i			1.92215i	0.276286	+	0.961076i	$0.410896\pi$
$503$			486322.i			1.92215i	−0.276286	+	0.961076i	$0.589104\pi$
$504$		0			0
$505$	90955.0			0.356651
$506$	89856.2	+	51878.5i	0.350951	+	0.202622i
$507$		0			0
$508$	−80867.4	−	140066.i	−0.313362	−	0.542759i
$509$	−246292.	+	142196.i	−0.950635	+	0.548850i	−0.893278	−	0.449504i	$-0.851601\pi$
$509$	−246292.	+	142196.i	−0.950635	+	0.548850i	−0.0573571	+	0.998354i	$0.518267\pi$
$510$		0			0
$511$	−193991.	+	336002.i	−0.742915	+	1.28677i
$512$		−	11585.2i		−	0.0441942i
$513$		0			0
$514$	271948.			1.02934
$515$	−107670.	−	62163.5i	−0.405959	−	0.234380i
$516$		0			0
$517$	−65852.4	−	114060.i	−0.246372	−	0.426728i
$518$	313312.	−	180891.i	1.16766	−	0.674150i
$519$		0			0
$520$	−22470.8	+	38920.6i	−0.0831021	+	0.143937i
$521$			248390.i			0.915079i	0.889189	+	0.457539i	$0.151269\pi$
$521$			248390.i			0.915079i	−0.889189	+	0.457539i	$0.848731\pi$
$522$		0			0
$523$	113565.			0.415186			0.207593	−	0.978215i	$-0.433437\pi$
$523$	113565.			0.415186			0.207593	+	0.978215i	$0.433437\pi$
$524$	97432.5	+	56252.7i	0.354847	+	0.204871i
$525$		0			0
$526$	19454.7	+	33696.6i	0.0703159	+	0.121791i
$527$	−28762.3	+	16605.9i	−0.103563	+	0.0597919i
$528$		0			0
$529$	288601.	−	499871.i	1.03130	−	1.78627i
$530$			45392.8i			0.161598i
$531$		0			0
$532$	111035.			0.392317
$533$	−271392.	−	156688.i	−0.955305	−	0.551546i
$534$		0			0
$535$	44.7543	+	77.5167i	0.000156361	+	0.000270824i
$536$	56002.2	−	32332.9i	0.194928	−	0.112542i
$537$		0			0
$538$	92117.2	−	159552.i	0.318256	−	0.551235i
$539$			22772.6i			0.0783854i
$540$		0			0
$541$	68803.4			0.235080			0.117540	−	0.993068i	$-0.462499\pi$
$541$	68803.4			0.235080			0.117540	+	0.993068i	$0.462499\pi$
$542$	23440.7	+	13533.5i	0.0797943	+	0.0460693i
$543$		0			0
$544$	2554.83	+	4425.10i	0.00863306	+	0.0149529i
$545$	−150916.	+	87131.2i	−0.508091	+	0.293346i
$546$		0			0
$547$	70749.2	−	122541.i	0.236454	−	0.409551i	−0.723240	−	0.690597i	$-0.757348\pi$
$547$	70749.2	−	122541.i	0.236454	−	0.409551i	0.959694	+	0.281046i	$0.0906814\pi$
$548$			117999.i			0.392931i
$549$		0			0
$550$	60549.5			0.200164
$551$	−296327.	−	171085.i	−0.976041	−	0.563518i
$552$		0			0
$553$	94797.1	+	164193.i	0.309988	+	0.536915i
$554$	−108602.	+	62701.3i	−0.353849	+	0.204295i
$555$		0			0
$556$	−56553.0	+	97952.7i	−0.182939	+	0.316859i
$557$			59163.4i			0.190697i	0.995444	+	0.0953483i	$0.0303965\pi$
$557$			59163.4i			0.190697i	−0.995444	+	0.0953483i	$0.969604\pi$
$558$		0			0
$559$	361314.			1.15628
$560$	27834.0	+	16070.0i	0.0887564	+	0.0512435i
$561$		0			0
$562$	−171777.	−	297527.i	−0.543867	−	0.942006i
$563$	359369.	−	207482.i	1.13377	−	0.654580i	0.188886	−	0.981999i	$-0.439512\pi$
$563$	359369.	−	207482.i	1.13377	−	0.654580i	0.944879	+	0.327419i	$0.106179\pi$
$564$		0			0
$565$	−27140.4	+	47008.6i	−0.0850197	+	0.147258i
$566$			329179.i			1.02754i
$567$		0			0
$568$	−178914.			−0.554559
$569$	−229601.	−	132560.i	−0.709169	−	0.409439i	0.101584	−	0.994827i	$-0.467609\pi$
$569$	−229601.	−	132560.i	−0.709169	−	0.409439i	−0.810753	+	0.585388i	$0.800942\pi$
$570$		0			0
$571$	16370.1	+	28353.8i	0.0502086	+	0.0869639i	0.890037	−	0.455887i	$-0.150678\pi$
$571$	16370.1	+	28353.8i	0.0502086	+	0.0869639i	−0.839829	+	0.542851i	$0.817345\pi$
$572$	−59229.1	+	34195.9i	−0.181027	+	0.104516i
$573$		0			0
$574$	−112055.	+	194085.i	−0.340102	+	0.589073i
$575$		−	500145.i		−	1.51272i
$576$		0			0
$577$	136483.			0.409946			0.204973	−	0.978768i	$-0.434289\pi$
$577$	136483.			0.409946			0.204973	+	0.978768i	$0.434289\pi$
$578$	202632.	+	116990.i	0.606531	+	0.350181i
$579$		0			0
$580$	−49521.7	−	85774.0i	−0.147211	−	0.254976i
$581$	−54384.2	+	31398.7i	−0.161109	+	0.0930165i
$582$		0			0
$583$	−34539.2	+	59823.7i	−0.101619	+	0.176009i
$584$			160935.i			0.471873i
$585$		0			0
$586$	449208.			1.30813
$587$	249463.	+	144027.i	0.723984	+	0.417992i	0.816217	−	0.577745i	$-0.196067\pi$
$587$	249463.	+	144027.i	0.723984	+	0.417992i	−0.0922331	+	0.995737i	$0.529401\pi$
$588$		0			0
$589$	149683.	+	259258.i	0.431461	+	0.747312i
$590$	−18468.1	+	10662.6i	−0.0530541	+	0.0306308i
$591$		0			0
$592$	75033.7	−	129962.i	0.214098	−	0.370829i
$593$			272778.i			0.775711i	0.921720	+	0.387856i	$0.126784\pi$
$593$			272778.i			0.775711i	−0.921720	+	0.387856i	$0.873216\pi$
$594$		0			0
$595$	−14175.3			−0.0400404
$596$	−75609.7	−	43653.3i	−0.212856	−	0.122892i
$597$		0			0
$598$	282461.	+	489238.i	0.789872	+	1.36810i
$599$	−68974.2	+	39822.3i	−0.192235	+	0.110987i	−0.593029	−	0.805181i	$-0.702068\pi$
$599$	−68974.2	+	39822.3i	−0.192235	+	0.110987i	0.400793	+	0.916169i	$0.368735\pi$
$600$		0			0
$601$	−59878.8	+	103713.i	−0.165777	+	0.287134i	−0.936931	−	0.349515i	$-0.886347\pi$
$601$	−59878.8	+	103713.i	−0.165777	+	0.287134i	0.771154	+	0.636649i	$0.219680\pi$
$602$		−	258393.i		−	0.712998i
$603$		0			0
$604$	−22309.7			−0.0611534
$605$	104209.	+	60165.0i	0.284704	+	0.164374i
$606$		0			0
$607$	−57220.1	−	99108.1i	−0.155300	−	0.268987i	0.777868	−	0.628427i	$-0.216301\pi$
$607$	−57220.1	−	99108.1i	−0.155300	−	0.268987i	−0.933168	+	0.359440i	$0.882968\pi$
$608$	39887.0	−	23028.8i	0.107901	−	0.0622965i
$609$		0			0
$610$	10257.2	−	17766.0i	0.0275657	−	0.0477452i
$611$		−	717090.i		−	1.92084i
$612$		0			0
$613$	−598912.			−1.59383			−0.796915	−	0.604091i	$-0.793536\pi$
$613$	−598912.			−1.59383			−0.796915	+	0.604091i	$0.793536\pi$
$614$	245346.	+	141651.i	0.650793	+	0.375735i
$615$		0			0
$616$	24455.2	+	42357.6i	0.0644480	+	0.111627i
$617$	−150704.	+	87008.9i	−0.395871	+	0.228556i	−0.684701	−	0.728824i	$-0.740067\pi$
$617$	−150704.	+	87008.9i	−0.395871	+	0.228556i	0.288830	+	0.957381i	$0.406734\pi$
$618$		0			0
$619$	368424.	−	638130.i	0.961540	−	1.66544i	0.242902	−	0.970051i	$-0.421901\pi$
$619$	368424.	−	638130.i	0.961540	−	1.66544i	0.718638	−	0.695384i	$-0.244766\pi$
$620$			86653.6i			0.225426i
$621$		0			0
$622$	−152646.			−0.394553
$623$	−509904.	−	294393.i	−1.31375	−	0.758493i
$624$		0			0
$625$	−119450.	−	206894.i	−0.305793	−	0.529649i
$626$	−184970.	+	106793.i	−0.472012	+	0.272516i
$627$		0			0
$628$	−9907.38	+	17160.1i	−0.0251212	+	0.0435111i
$629$			66187.2i			0.167291i
$630$		0			0
$631$	−26188.6			−0.0657739			−0.0328869	−	0.999459i	$-0.510470\pi$
$631$	−26188.6			−0.0657739			−0.0328869	+	0.999459i	$0.510470\pi$
$632$	68107.6	+	39322.0i	0.170515	+	0.0984467i
$633$		0			0
$634$	−94139.7	−	163055.i	−0.234204	−	0.405653i
$635$	161181.	−	93058.1i	0.399731	−	0.230785i
$636$		0			0
$637$	−61994.7	+	107378.i	−0.152783	+	0.264628i
$638$		−	150724.i		−	0.370288i
$639$		0			0
$640$	13331.7			0.0325481
$641$	343687.	+	198428.i	0.836462	+	0.482932i	0.856060	−	0.516876i	$-0.172905\pi$
$641$	343687.	+	198428.i	0.836462	+	0.482932i	−0.0195978	+	0.999808i	$0.506239\pi$
$642$		0			0
$643$	32175.0	+	55728.7i	0.0778209	+	0.134790i	0.902309	−	0.431089i	$-0.141870\pi$
$643$	32175.0	+	55728.7i	0.0778209	+	0.134790i	−0.824489	+	0.565879i	$0.808537\pi$
$644$	349878.	−	202002.i	0.843616	−	0.487062i
$645$		0			0
$646$	−10156.8	+	17592.2i	−0.0243385	+	0.0421555i
$647$			16209.2i			0.0387215i	0.999813	+	0.0193607i	$0.00616310\pi$
$647$			16209.2i			0.0387215i	−0.999813	+	0.0193607i	$0.993837\pi$
$648$		0			0
$649$	−32452.5			−0.0770476
$650$	285505.	+	164836.i	0.675751	+	0.390145i
$651$		0			0
$652$	−142188.	−	246277.i	−0.334478	−	0.579334i
$653$	96132.7	−	55502.2i	0.225447	−	0.130162i	−0.383023	−	0.923739i	$-0.625117\pi$
$653$	96132.7	−	55502.2i	0.225447	−	0.130162i	0.608470	+	0.793577i	$0.291784\pi$
$654$		0			0
$655$	−64732.7	+	112120.i	−0.150883	+	0.261337i
$656$			92961.4i			0.216021i
$657$		0			0
$658$	−512826.			−1.18445
$659$	625726.	+	361263.i	1.44083	+	0.831865i	0.997905	−	0.0646920i	$-0.0206065\pi$
$659$	625726.	+	361263.i	1.44083	+	0.831865i	0.442928	+	0.896557i	$0.353940\pi$
$660$		0			0
$661$	−1621.86	−	2809.15i	−0.00371203	−	0.00642942i	0.864163	−	0.503211i	$-0.167848\pi$
$661$	−1621.86	−	2809.15i	−0.00371203	−	0.00642942i	−0.867875	+	0.496782i	$0.834515\pi$
$662$	170044.	−	98175.0i	0.388012	−	0.224019i
$663$		0			0
$664$	−13024.2	+	22558.6i	−0.0295404	+	0.0511655i
$665$			127774.i			0.288934i
$666$		0			0
$667$	−1.24499e6			−2.79843
$668$	−367143.	−	211970.i	−0.822776	−	0.475030i
$669$		0			0
$670$	37207.0	+	64444.4i	0.0828848	+	0.143561i
$671$	27036.2	−	15609.3i	0.0600482	−	0.0346689i
$672$		0			0
$673$	−340188.	+	589223.i	−0.751085	+	1.30092i	0.196212	+	0.980561i	$0.437136\pi$
$673$	−340188.	+	589223.i	−0.751085	+	1.30092i	−0.947297	+	0.320356i	$0.896198\pi$
$674$		−	251125.i		−	0.552802i
$675$		0			0
$676$	−143883.			−0.314859
$677$	71954.5	+	41542.9i	0.156993	+	0.0906400i	0.576438	−	0.817141i	$-0.304442\pi$
$677$	71954.5	+	41542.9i	0.156993	+	0.0906400i	−0.419445	+	0.907781i	$0.637775\pi$
$678$		0			0
$679$	−271801.	−	470772.i	−0.589537	−	1.02111i
$680$	−5092.18	+	2939.97i	−0.0110125	+	0.00635807i
$681$		0			0
$682$	−65934.4	+	114202.i	−0.141757	+	0.245530i
$683$		−	6386.51i		−	0.0136906i	−0.999977	−	0.00684529i	$-0.997821\pi$
$683$		−	6386.51i		−	0.0136906i	0.999977	−	0.00684529i	$-0.00217894\pi$
$684$		0			0
$685$	−135787.			−0.289385
$686$	−244029.	−	140890.i	−0.518554	−	0.299387i
$687$		0			0
$688$	−53591.0	−	92822.3i	−0.113218	−	0.196099i
$689$	−325720.	+	188055.i	−0.686130	+	0.396137i
$690$		0			0
$691$	−224531.	+	388900.i	−0.470242	+	0.814483i	−0.999421	−	0.0340275i	$-0.989167\pi$
$691$	−224531.	+	388900.i	−0.470242	+	0.814483i	0.529179	+	0.848510i	$0.322500\pi$
$692$		−	427172.i		−	0.892054i
$693$		0			0
$694$	310379.			0.644427
$695$	−112719.	−	65078.3i	−0.233360	−	0.134731i
$696$		0			0
$697$	−20500.3	−	35507.6i	−0.0421983	−	0.0730896i
$698$	−339938.	+	196263.i	−0.697733	+	0.402836i
$699$		0			0
$700$	117882.	−	204178.i	0.240576	−	0.416691i
$701$			408122.i			0.830527i	0.909701	+	0.415264i	$0.136311\pi$
$701$			408122.i			0.830527i	−0.909701	+	0.415264i	$0.863689\pi$
$702$		0			0
$703$	596599.			1.20718
$704$	17570.0	+	10144.0i	0.0354508	+	0.0204676i
$705$		0			0
$706$	110060.	+	190629.i	0.220810	+	0.382455i
$707$	466747.	−	269476.i	0.933775	−	0.539115i
$708$		0			0
$709$	425844.	−	737584.i	0.847146	−	1.46730i	−0.0365985	−	0.999330i	$-0.511652\pi$
$709$	425844.	−	737584.i	0.847146	−	1.46730i	0.883744	−	0.467970i	$-0.155014\pi$
$710$		−	205885.i		−	0.408421i
$711$		0			0
$712$	−244230.			−0.481768
$713$	943317.	+	544624.i	1.85558	+	1.07132i
$714$		0			0
$715$	−39350.9	−	68157.8i	−0.0769738	−	0.133323i
$716$	−224809.	+	129793.i	−0.438517	+	0.253178i
$717$		0			0
$718$	−276172.	+	478344.i	−0.535711	+	0.927879i
$719$		−	951431.i		−	1.84043i	−0.391412	−	0.920215i	$-0.628013\pi$
$719$		−	951431.i		−	1.84043i	0.391412	−	0.920215i	$-0.371987\pi$
$720$		0			0
$721$	−736698.			−1.41716
$722$	−160647.	−	92749.9i	−0.308176	−	0.177926i
$723$		0			0
$724$	−201525.	−	349052.i	−0.384461	−	0.665905i
$725$	−629202.	+	363270.i	−1.19705	+	0.691120i
$726$		0			0
$727$	−410863.	+	711636.i	−0.777371	+	1.34645i	0.156081	+	0.987744i	$0.450114\pi$
$727$	−410863.	+	711636.i	−0.777371	+	1.34645i	−0.933452	+	0.358702i	$0.883219\pi$
$728$			266301.i			0.502470i
$729$		0			0
$730$	−185196.			−0.347525
$731$	40939.3	+	23636.3i	0.0766135	+	0.0442328i
$732$		0			0
$733$	224965.	+	389651.i	0.418704	+	0.725217i	0.995809	−	0.0914530i	$-0.0291511\pi$
$733$	224965.	+	389651.i	0.418704	+	0.725217i	−0.577105	+	0.816670i	$0.695818\pi$
$734$	−210179.	+	121347.i	−0.390120	+	0.225236i
$735$		0			0
$736$	83790.8	−	145130.i	0.154682	−	0.267917i
$737$			113243.i			0.208485i
$738$		0			0
$739$	533367.			0.976647			0.488323	−	0.872663i	$-0.337609\pi$
$739$	533367.			0.976647			0.488323	+	0.872663i	$0.337609\pi$
$740$	149554.	+	86345.0i	0.273108	+	0.157679i
$741$		0			0
$742$	134487.	+	232938.i	0.244271	+	0.423091i
$743$	−505677.	+	291952.i	−0.915999	+	0.528852i	−0.882357	−	0.470581i	$-0.844044\pi$
$743$	−505677.	+	291952.i	−0.915999	+	0.528852i	−0.0336428	+	0.999434i	$0.510711\pi$
$744$		0			0
$745$	50234.0	−	87007.8i	0.0905076	−	0.156764i
$746$		−	34840.7i		−	0.0626050i
$747$		0			0
$748$	−8948.06			−0.0159928
$749$	459.324	+	265.191i	0.000818758	+	0.000472710i
$750$		0			0
$751$	−296313.	−	513229.i	−0.525377	−	0.909979i	−0.999563	−	0.0295548i	$-0.990591\pi$
$751$	−296313.	−	513229.i	−0.525377	−	0.909979i	0.474186	−	0.880424i	$-0.342742\pi$
$752$	−184222.	+	106361.i	−0.325766	+	0.188081i
$753$		0			0
$754$	410320.	−	710695.i	0.721739	−	1.25009i
$755$		−	25672.9i		−	0.0450382i
$756$		0			0
$757$	−30185.2			−0.0526747			−0.0263373	−	0.999653i	$-0.508384\pi$
$757$	−30185.2			−0.0526747			−0.0263373	+	0.999653i	$0.508384\pi$
$758$	−529515.	−	305716.i	−0.921594	−	0.532083i
$759$		0			0
$760$	26500.4	+	45900.0i	0.0458801	+	0.0794667i
$761$	265213.	−	153121.i	0.457957	−	0.264402i	−0.253228	−	0.967407i	$-0.581492\pi$
$761$	265213.	−	153121.i	0.457957	−	0.264402i	0.711185	+	0.703005i	$0.248159\pi$
$762$		0			0
$763$	−516295.	+	894248.i	−0.886846	+	1.53606i
$764$			18047.9i			0.0309201i
$765$		0			0
$766$	−672358.			−1.14589
$767$	−153021.	−	88346.6i	−0.260112	−	0.150176i
$768$		0			0
$769$	−444182.	−	769346.i	−0.751118	−	1.30097i	−0.947281	−	0.320403i	$-0.896182\pi$
$769$	−444182.	−	769346.i	−0.751118	−	1.30097i	0.196163	−	0.980571i	$-0.437152\pi$
$770$	−48743.0	+	28141.8i	−0.0822111	+	0.0474646i
$771$		0			0
$772$	96128.1	−	166499.i	0.161293	−	0.279368i
$773$			456476.i			0.763939i	0.924175	+	0.381970i	$0.124754\pi$
$773$			456476.i			0.763939i	−0.924175	+	0.381970i	$0.875246\pi$
$774$		0			0
$775$	635654.			1.05832
$776$	−195277.	−	112743.i	−0.324286	−	0.187226i
$777$		0			0
$778$	263475.	+	456353.i	0.435292	+	0.753948i
$779$	−320059.	+	184786.i	−0.527418	+	0.304505i
$780$		0			0
$781$	156657.	−	271338.i	0.256832	−	0.444845i
$782$			73911.8i			0.120865i
$783$		0			0
$784$	36780.8			0.0598397
$785$	−19747.0	−	11400.9i	−0.0320450	−	0.0185012i
$786$		0			0
$787$	−269736.	−	467196.i	−0.435501	−	0.754310i	0.561835	−	0.827249i	$-0.310095\pi$
$787$	−269736.	−	467196.i	−0.435501	−	0.754310i	−0.997336	+	0.0729393i	$0.976762\pi$
$788$	462975.	−	267299.i	0.745598	−	0.430471i
$789$		0			0
$790$	−45249.7	+	78374.8i	−0.0725039	+	0.125581i
$791$			321640.i			0.514064i
$792$		0			0
$793$	169975.			0.270296
$794$	12258.9	+	7077.71i	0.0194452	+	0.0112267i
$795$		0			0
$796$	−33701.1	−	58372.1i	−0.0531886	−	0.0921254i
$797$	435416.	−	251387.i	0.685468	−	0.395755i	−0.116444	−	0.993197i	$-0.537150\pi$
$797$	435416.	−	251387.i	0.685468	−	0.395755i	0.801912	+	0.597442i	$0.203816\pi$
$798$		0			0
$799$	46910.3	−	81251.0i	0.0734809	−	0.127273i
$800$		−	97795.6i		−	0.152806i
$801$		0			0
$802$	499157.			0.776048
$803$	−244072.	−	140915.i	−0.378519	−	0.218538i
$804$		0			0
$805$	232454.	+	402621.i	0.358711	+	0.621305i
$806$	−621791.	+	358991.i	−0.957138	+	0.552604i
$807$		0			0
$808$	111779.	−	193607.i	0.171214	−	0.296551i
$809$			699315.i			1.06850i	0.845326	+	0.534251i	$0.179406\pi$
$809$			699315.i			1.06850i	−0.845326	+	0.534251i	$0.820594\pi$
$810$		0			0
$811$	741168.			1.12687			0.563437	−	0.826159i	$-0.309479\pi$
$811$	741168.			1.12687			0.563437	+	0.826159i	$0.309479\pi$
$812$	−508253.	−	293440.i	−0.770846	−	0.445048i
$813$		0			0
$814$	131399.	+	227590.i	0.198310	+	0.343483i
$815$	283403.	−	163623.i	0.426667	−	0.246336i
$816$		0			0
$817$	213053.	−	369019.i	0.319186	−	0.552846i
$818$		−	533763.i		−	0.797704i
$819$		0			0
$820$	−106975.			−0.159095
$821$	58936.9	+	34027.3i	0.0874382	+	0.0504825i	0.543082	−	0.839680i	$-0.317257\pi$
$821$	58936.9	+	34027.3i	0.0874382	+	0.0504825i	−0.455643	+	0.890162i	$0.650591\pi$
$822$		0			0
$823$	−325915.	−	564502.i	−0.481177	−	0.833423i	0.518590	−	0.855023i	$-0.326457\pi$
$823$	−325915.	−	564502.i	−0.481177	−	0.833423i	−0.999767	+	0.0216000i	$0.993124\pi$
$824$	−264643.	+	152792.i	−0.389768	+	0.225033i
$825$		0			0
$826$	−63181.0	+	109433.i	−0.0926033	+	0.160394i
$827$		−	36453.4i		−	0.0532999i	−0.999645	−	0.0266500i	$-0.991516\pi$
$827$		−	36453.4i		−	0.0532999i	0.999645	−	0.0266500i	$-0.00848395\pi$
$828$		0			0
$829$	78480.8			0.114197			0.0570984	−	0.998369i	$-0.481815\pi$
$829$	78480.8			0.114197			0.0570984	+	0.998369i	$0.481815\pi$
$830$	−25959.3	−	14987.6i	−0.0376823	−	0.0217559i
$831$		0			0
$832$	55231.0	+	95662.9i	0.0797878	+	0.138196i
$833$	−14048.8	+	8111.09i	−0.0202465	+	0.0116893i
$834$		0			0
$835$	243924.	−	422489.i	0.349850	−	0.605958i
$836$			80656.1i			0.115405i
$837$		0			0
$838$	−33956.4			−0.0483541
$839$	59633.8	+	34429.6i	0.0847166	+	0.0489111i	0.541760	−	0.840533i	$-0.317758\pi$
$839$	59633.8	+	34429.6i	0.0847166	+	0.0489111i	−0.457043	+	0.889444i	$0.651092\pi$
$840$		0			0
$841$	550632.	+	953723.i	0.778520	+	1.34844i
$842$	136180.	−	78623.4i	0.192083	−	0.110899i
$843$		0			0
$844$	−134800.	+	233480.i	−0.189236	+	0.327767i
$845$		−	165573.i		−	0.231887i
$846$		0			0
$847$	713014.			0.993874
$848$	96623.2	+	55785.5i	0.134366	+	0.0775763i
$849$		0			0
$850$	21566.4	+	37354.1i	0.0298497	+	0.0517011i
$851$	1.87992e6	−	1.08537e6i	2.59585	−	1.49871i
$852$		0			0
$853$	−412386.	+	714273.i	−0.566768	+	0.981671i	0.430115	+	0.902774i	$0.358473\pi$
$853$	−412386.	+	714273.i	−0.566768	+	0.981671i	−0.996883	+	0.0788970i	$0.974860\pi$
$854$		−	121558.i		−	0.166674i
$855$		0			0
$856$	220.003			0.000300249
$857$	561077.	+	323938.i	0.763942	+	0.441062i	0.830709	−	0.556706i	$-0.187935\pi$
$857$	561077.	+	323938.i	0.763942	+	0.441062i	−0.0667670	+	0.997769i	$0.521268\pi$
$858$		0			0
$859$	657566.	+	1.13894e6i	0.891154	+	1.54352i	0.838493	+	0.544912i	$0.183437\pi$
$859$	657566.	+	1.13894e6i	0.891154	+	1.54352i	0.0526607	+	0.998612i	$0.483230\pi$
$860$	106815.	−	61669.8i	0.144423	−	0.0833826i
$861$		0			0
$862$	73640.4	−	127549.i	0.0991063	−	0.171657i
$863$			1.27319e6i			1.70951i	0.519028	+	0.854757i	$0.326294\pi$
$863$			1.27319e6i			1.70951i	−0.519028	+	0.854757i	$0.673706\pi$
$864$		0			0
$865$	491568.			0.656979
$866$	−503030.	−	290424.i	−0.670746	−	0.387255i
$867$		0			0
$868$	256732.	+	444673.i	0.340754	+	0.590203i
$869$	−119270.	+	68860.7i	−0.157940	+	0.0911868i
$870$		0			0
$871$	−308285.	+	533965.i	−0.406365	+	0.703844i
$872$			428320.i			0.563294i
$873$		0			0
$874$	666227.			0.872166
$875$	506775.	+	292586.i	0.661910	+	0.382154i
$876$		0			0
$877$	352514.	+	610572.i	0.458329	+	0.793848i	0.998873	−	0.0474673i	$-0.0151150\pi$
$877$	352514.	+	610572.i	0.458329	+	0.793848i	−0.540544	+	0.841316i	$0.681782\pi$
$878$	795018.	−	459004.i	1.03131	−	0.595425i
$879$		0			0
$880$	−11673.3	+	20218.7i	−0.0150739	+	0.0261088i
$881$			750647.i			0.967128i	0.875309	+	0.483564i	$0.160658\pi$
$881$			750647.i			0.967128i	−0.875309	+	0.483564i	$0.839342\pi$
$882$		0			0
$883$	−638249.			−0.818594			−0.409297	−	0.912401i	$-0.634226\pi$
$883$	−638249.			−0.818594			−0.409297	+	0.912401i	$0.634226\pi$
$884$	−42192.1	−	24359.6i	−0.0539917	−	0.0311721i
$885$		0			0
$886$	−243275.	−	421364.i	−0.309906	−	0.536772i
$887$	−985132.	+	568766.i	−1.25212	+	0.722914i	−0.971531	−	0.236914i	$-0.923864\pi$
$887$	−985132.	+	568766.i	−1.25212	+	0.722914i	−0.280592	+	0.959827i	$0.590531\pi$
$888$		0			0
$889$	551415.	−	955078.i	0.697710	−	1.20847i
$890$		−	281047.i		−	0.354812i
$891$		0			0
$892$	164250.			0.206432
$893$	−732381.	−	422840.i	−0.918405	−	0.530241i
$894$		0			0
$895$	−149359.	−	258698.i	−0.186460	−	0.322959i
$896$	68413.2	−	39498.4i	0.0852166	−	0.0491998i
$897$		0			0
$898$	172018.	−	297944.i	0.213315	−	0.369472i
$899$		−	1.58231e6i		−	1.95781i
$900$		0			0
$901$	−49208.3			−0.0606162
$902$	−140984.	−	81397.1i	−0.173283	−	0.100045i
$903$		0			0
$904$	66708.5	+	115542.i	0.0816289	+	0.141385i
$905$	401671.	−	231905.i	0.490425	−	0.283147i
$906$		0			0
$907$	652106.	−	1.12948e6i	0.792690	−	1.37298i	−0.131606	−	0.991302i	$-0.542013\pi$
$907$	652106.	−	1.12948e6i	0.792690	−	1.37298i	0.924296	−	0.381677i	$-0.124653\pi$
$908$		−	468534.i		−	0.568289i
$909$		0			0
$910$	−306445.			−0.370059
$911$	−936156.	−	540490.i	−1.12801	−	0.651254i	−0.184573	−	0.982819i	$-0.559090\pi$
$911$	−936156.	−	540490.i	−1.12801	−	0.651254i	−0.943433	+	0.331564i	$0.892424\pi$
$912$		0			0
$913$	−22808.1	−	39504.8i	−0.0273620	−	0.0473923i
$914$	−114021.	+	65829.8i	−0.136487	+	0.0788008i
$915$		0			0
$916$	−29154.2	+	50496.6i	−0.0347465	+	0.0601826i
$917$			767145.i			0.912303i
$918$		0			0
$919$	558574.			0.661378			0.330689	−	0.943740i	$-0.392719\pi$
$919$	558574.			0.661378			0.330689	+	0.943740i	$0.392719\pi$
$920$	167008.	+	96422.1i	0.197316	+	0.113920i
$921$		0			0
$922$	288884.	+	500361.i	0.339829	+	0.588602i
$923$	1.47735e6	−	852947.i	1.73412	−	1.00120i
$924$		0			0
$925$	633390.	−	1.09706e6i	0.740266	−	1.28218i
$926$			48849.9i			0.0569695i
$927$		0			0
$928$	−243439.			−0.282679
$929$	764452.	+	441357.i	0.885765	+	0.511397i	0.872555	−	0.488516i	$-0.162462\pi$
$929$	764452.	+	441357.i	0.885765	+	0.511397i	0.0132104	+	0.999913i	$0.495795\pi$
$930$		0			0
$931$	73111.8	+	126633.i	0.0843506	+	0.146100i
$932$	−94118.7	+	54339.5i	−0.108354	+	0.0625581i
$933$		0			0
$934$	310652.	−	538065.i	0.356107	−	0.616795i
$935$		−	10297.0i		−	0.0117784i
$936$		0			0
$937$	893683.			1.01790			0.508949	−	0.860797i	$-0.330034\pi$
$937$	893683.			1.01790			0.508949	+	0.860797i	$0.330034\pi$
$938$	381865.	+	220470.i	0.434014	+	0.250578i
$939$		0			0
$940$	−122394.	−	211993.i	−0.138518	−	0.239920i
$941$	−408726.	+	235978.i	−0.461587	+	0.266497i	−0.712711	−	0.701458i	$-0.752533\pi$
$941$	−408726.	+	235978.i	−0.461587	+	0.266497i	0.251125	+	0.967955i	$0.419200\pi$
$942$		0			0
$943$	−672348.	+	1.16454e6i	−0.756085	+	1.30958i
$944$			52415.2i			0.0588184i
$945$		0			0
$946$	187697.			0.209737
$947$	−629058.	−	363187.i	−0.701441	−	0.404977i	0.106443	−	0.994319i	$-0.466054\pi$
$947$	−629058.	−	363187.i	−0.701441	−	0.404977i	−0.807884	+	0.589342i	$0.799387\pi$
$948$		0			0
$949$	−767237.	−	1.32889e6i	−0.851916	−	1.47556i
$950$	336702.	−	194395.i	0.373078	−	0.215396i
$951$		0			0
$952$	−17420.8	+	30173.7i	−0.0192218	+	0.0332931i
$953$			41574.8i			0.0457767i	0.999738	+	0.0228884i	$0.00728623\pi$
$953$			41574.8i			0.0457767i	−0.999738	+	0.0228884i	$0.992714\pi$
$954$		0			0
$955$	−20768.6			−0.0227720
$956$	179350.	+	103548.i	0.196239	+	0.113299i
$957$		0			0
$958$	−128769.	−	223035.i	−0.140307	−	0.243020i
$959$	−696807.	+	402302.i	−0.757662	+	0.437436i
$960$		0			0
$961$	−230424.	+	399106.i	−0.249506	+	0.432157i
$962$			1.43085e6i			1.54612i
$963$		0			0
$964$	74887.7			0.0805855
$965$	191598.	+	110619.i	0.205749	+	0.118789i
$966$		0			0
$967$	600282.	+	1.03972e6i	0.641952	+	1.11189i	0.984996	+	0.172575i	$0.0552086\pi$
$967$	600282.	+	1.03972e6i	0.641952	+	1.11189i	−0.343044	+	0.939319i	$0.611458\pi$
$968$	256135.	−	147880.i	0.273350	−	0.157819i
$969$		0			0
$970$	129739.	−	224715.i	0.137888	−	0.238830i
$971$		−	1.38244e6i		−	1.46625i	−0.680092	−	0.733127i	$-0.738060\pi$
$971$		−	1.38244e6i		−	1.46625i	0.680092	−	0.733127i	$-0.261940\pi$
$972$		0			0
$973$	−771241.			−0.814638
$974$	359096.	+	207324.i	0.378523	+	0.218541i
$975$		0			0
$976$	−25211.2	−	43667.1i	−0.0264663	−	0.0458410i
$977$	−124404.	+	71824.7i	−0.130330	+	0.0752463i	−0.563748	−	0.825947i	$-0.690641\pi$
$977$	−124404.	+	71824.7i	−0.130330	+	0.0752463i	0.433417	+	0.901193i	$0.357308\pi$
$978$		0			0
$979$	213848.	−	370395.i	0.223120	−	0.386456i
$980$			42325.5i			0.0440707i
$981$		0			0
$982$	137061.			0.142132
$983$	−492098.	−	284113.i	−0.509266	−	0.294025i	0.223266	−	0.974758i	$-0.428328\pi$
$983$	−492098.	−	284113.i	−0.509266	−	0.294025i	−0.732532	+	0.680733i	$0.761661\pi$
$984$		0			0
$985$	307594.	+	532768.i	0.317033	+	0.549118i
$986$	92983.9	−	53684.3i	0.0956432	−	0.0552196i
$987$		0			0
$988$	−219573.	+	380312.i	−0.224939	+	0.389606i
$989$		−	1.55040e6i		−	1.58508i
$990$		0			0
$991$	−920749.			−0.937549			−0.468774	−	0.883318i	$-0.655304\pi$
$991$	−920749.			−0.937549			−0.468774	+	0.883318i	$0.655304\pi$
$992$	184451.	+	106493.i	0.187438	+	0.108218i
$993$		0			0
$994$	−609985.	−	1.05652e6i	−0.617371	−	1.06932i
$995$	67171.6	−	38781.6i	0.0678484	−	0.0391723i
$996$		0			0
$997$	−500709.	+	867254.i	−0.503727	+	0.872481i	0.496264	+	0.868172i	$0.334705\pi$
$997$	−500709.	+	867254.i	−0.503727	+	0.872481i	−0.999991	+	0.00430895i	$0.998628\pi$
$998$			701487.i			0.704302i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.5.d.d.107.4		8
3.2	odd	2	inner	162.5.d.d.107.1		8
9.2	odd	6		162.5.b.a.161.3	yes	4
9.4	even	3	inner	162.5.d.d.53.1		8
9.5	odd	6	inner	162.5.d.d.53.4		8
9.7	even	3		162.5.b.a.161.2	✓	4
36.7	odd	6		1296.5.e.b.161.2		4
36.11	even	6		1296.5.e.b.161.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.5.b.a.161.2	✓	4	9.7	even	3
162.5.b.a.161.3	yes	4	9.2	odd	6
162.5.d.d.53.1		8	9.4	even	3	inner
162.5.d.d.53.4		8	9.5	odd	6	inner
162.5.d.d.107.1		8	3.2	odd	2	inner
162.5.d.d.107.4		8	1.1	even	1	trivial
1296.5.e.b.161.2		4	36.7	odd	6
1296.5.e.b.161.3		4	36.11	even	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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.44949 + 1.41421i) q^{2} +(4.00000 + 6.92820i) q^{4} +(-7.97262 + 4.60300i) q^{5} +(-27.2750 + 47.2417i) q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q+(2.44949 + 1.41421i) q^{2} +(4.00000 + 6.92820i) q^{4} +(-7.97262 + 4.60300i) q^{5} +(-27.2750 + 47.2417i) q^{7} +22.6274i q^{8} -26.0385 q^{10} +(-34.3164 - 19.8126i) q^{11} +(-107.873 - 186.842i) q^{13} +(-133.620 + 77.1453i) q^{14} +(-32.0000 + 55.4256i) q^{16} -28.2272i q^{17} -254.435 q^{19} +(-63.7810 - 36.8240i) q^{20} +(-56.0385 - 97.0615i) q^{22} +(-801.737 + 462.883i) q^{23} +(-270.125 + 467.870i) q^{25} -610.222i q^{26} -436.400 q^{28} +(1164.65 + 672.411i) q^{29} +(-588.296 - 1018.96i) q^{31} +(-156.767 + 90.5097i) q^{32} +(39.9193 - 69.1422i) q^{34} -502.187i q^{35} -2344.80 q^{37} +(-623.235 - 359.825i) q^{38} +(-104.154 - 180.400i) q^{40} +(1257.92 - 726.261i) q^{41} +(-837.359 + 1450.35i) q^{43} -317.001i q^{44} -2618.46 q^{46} +(2878.47 + 1661.88i) q^{47} +(-287.350 - 497.705i) q^{49} +(-1323.34 + 764.028i) q^{50} +(862.985 - 1494.73i) q^{52} -1743.30i q^{53} +364.789 q^{55} +(-1068.96 - 617.163i) q^{56} +(1901.86 + 3294.13i) q^{58} +(709.264 - 409.494i) q^{59} +(-393.925 + 682.298i) q^{61} -3327.91i q^{62} -512.000 q^{64} +(1720.06 + 993.079i) q^{65} +(-1428.92 - 2474.97i) q^{67} +(195.564 - 112.909i) q^{68} +(710.199 - 1230.10i) q^{70} +7906.95i q^{71} +7112.40 q^{73} +(-5743.57 - 3316.05i) q^{74} +(-1017.74 - 1762.77i) q^{76} +(1871.96 - 1080.78i) q^{77} +(1737.80 - 3009.96i) q^{79} -589.183i q^{80} +4108.35 q^{82} +(996.961 + 575.596i) q^{83} +(129.930 + 225.045i) q^{85} +(-4102.21 + 2368.41i) q^{86} +(448.308 - 776.492i) q^{88} +10793.5i q^{89} +11768.9 q^{91} +(-6413.89 - 3703.06i) q^{92} +(4700.51 + 8141.53i) q^{94} +(2028.51 - 1171.16i) q^{95} +(-4982.60 + 8630.11i) q^{97} -1625.50i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{4} + 52 q^{7}+O(q^{10}) \) 8 * q + 32 * q^4 + 52 * q^7	\( 8 q + 32 q^{4} + 52 q^{7} - 624 q^{10} - 572 q^{13} - 256 q^{16} - 248 q^{19} - 864 q^{22} + 1892 q^{25} + 832 q^{28} - 3584 q^{31} + 1608 q^{34} - 18800 q^{37} - 2496 q^{40} - 3020 q^{43} - 4320 q^{46} - 9324 q^{49} + 4576 q^{52} + 22248 q^{55} + 2952 q^{58} + 4144 q^{61} - 4096 q^{64} + 12076 q^{67} - 18096 q^{70} - 3584 q^{73} - 992 q^{76} + 15004 q^{79} + 10752 q^{82} + 24048 q^{85} + 6912 q^{88} + 24440 q^{91} + 12912 q^{94} - 46304 q^{97}+O(q^{100}) \) 8 * q + 32 * q^4 + 52 * q^7 - 624 * q^10 - 572 * q^13 - 256 * q^16 - 248 * q^19 - 864 * q^22 + 1892 * q^25 + 832 * q^28 - 3584 * q^31 + 1608 * q^34 - 18800 * q^37 - 2496 * q^40 - 3020 * q^43 - 4320 * q^46 - 9324 * q^49 + 4576 * q^52 + 22248 * q^55 + 2952 * q^58 + 4144 * q^61 - 4096 * q^64 + 12076 * q^67 - 18096 * q^70 - 3584 * q^73 - 992 * q^76 + 15004 * q^79 + 10752 * q^82 + 24048 * q^85 + 6912 * q^88 + 24440 * q^91 + 12912 * q^94 - 46304 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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