LMFDB - Embedded newform 162.4.c.j.109.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,4,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.55"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,4,0,-8,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.55830942093$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	162.109
Dual form			162.4.c.j.55.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(0.401924 - 0.696152i) q^{5} +(1.19615 + 2.07180i) q^{7} -8.00000 q^{8} +1.60770 q^{10} +(29.7846 + 51.5885i) q^{11} +(-38.8731 + 67.3301i) q^{13} +(-2.39230 + 4.14359i) q^{14} +(-8.00000 - 13.8564i) q^{16} -2.84232 q^{17} -118.315 q^{19} +(1.60770 + 2.78461i) q^{20} +(-59.5692 + 103.177i) q^{22} +(55.3923 - 95.9423i) q^{23} +(62.1769 + 107.694i) q^{25} -155.492 q^{26} -9.56922 q^{28} +(62.8135 + 108.796i) q^{29} +(-31.5307 + 54.6128i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-2.84232 - 4.92305i) q^{34} +1.92305 q^{35} -227.392 q^{37} +(-118.315 - 204.928i) q^{38} +(-3.21539 + 5.56922i) q^{40} +(162.315 - 281.138i) q^{41} +(136.412 + 236.272i) q^{43} -238.277 q^{44} +221.569 q^{46} +(-2.46926 - 4.27688i) q^{47} +(168.638 - 292.090i) q^{49} +(-124.354 + 215.387i) q^{50} +(-155.492 - 269.321i) q^{52} +598.908 q^{53} +47.8846 q^{55} +(-9.56922 - 16.5744i) q^{56} +(-125.627 + 217.592i) q^{58} +(-335.138 + 580.477i) q^{59} +(-232.265 - 402.295i) q^{61} -126.123 q^{62} +64.0000 q^{64} +(31.2480 + 54.1232i) q^{65} +(384.535 - 666.033i) q^{67} +(5.68465 - 9.84610i) q^{68} +(1.92305 + 3.33082i) q^{70} +611.569 q^{71} +923.831 q^{73} +(-227.392 - 393.855i) q^{74} +(236.631 - 409.856i) q^{76} +(-71.2539 + 123.415i) q^{77} +(-19.9038 - 34.4744i) q^{79} -12.8616 q^{80} +649.261 q^{82} +(221.885 + 384.315i) q^{83} +(-1.14240 + 1.97869i) q^{85} +(-272.823 + 472.543i) q^{86} +(-238.277 - 412.708i) q^{88} +78.4576 q^{89} -185.992 q^{91} +(221.569 + 383.769i) q^{92} +(4.93851 - 8.55376i) q^{94} +(-47.5538 + 82.3655i) q^{95} +(45.5077 + 78.8217i) q^{97} +674.554 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{4} + 12 q^{5} - 16 q^{7} - 32 q^{8} + 48 q^{10} + 36 q^{11} - 10 q^{13} + 32 q^{14} - 32 q^{16} - 240 q^{17} - 16 q^{19} + 48 q^{20} - 72 q^{22} + 180 q^{23} + 124 q^{25} - 40 q^{26}+ \cdots + 1368 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^4 + 12 * q^5 - 16 * q^7 - 32 * q^8 + 48 * q^10 + 36 * q^11 - 10 * q^13 + 32 * q^14 - 32 * q^16 - 240 * q^17 - 16 * q^19 + 48 * q^20 - 72 * q^22 + 180 * q^23 + 124 * q^25 - 40 * q^26 + 128 * q^28 + 324 * q^29 + 248 * q^31 + 64 * q^32 - 240 * q^34 - 408 * q^35 - 868 * q^37 - 16 * q^38 - 96 * q^40 + 192 * q^41 + 608 * q^43 - 288 * q^44 + 720 * q^46 - 384 * q^47 + 342 * q^49 - 248 * q^50 - 40 * q^52 + 816 * q^53 - 432 * q^55 + 128 * q^56 - 648 * q^58 - 1008 * q^59 - 742 * q^61 + 992 * q^62 + 256 * q^64 - 696 * q^65 + 104 * q^67 + 480 * q^68 - 408 * q^70 + 2280 * q^71 + 1700 * q^73 - 868 * q^74 + 32 * q^76 - 576 * q^77 + 440 * q^79 - 384 * q^80 + 768 * q^82 + 264 * q^83 - 1314 * q^85 - 1216 * q^86 - 288 * q^88 - 1536 * q^89 - 2864 * q^91 + 720 * q^92 + 768 * q^94 + 1140 * q^95 + 764 * q^97 + 1368 * q^98

Character values

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

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	0.401924	−	0.696152i	0.0359492	−	0.0622658i	−0.847491	−	0.530810i	$-0.821888\pi$
$5$	0.401924	−	0.696152i	0.0359492	−	0.0622658i	0.883440	+	0.468544i	$0.155221\pi$
$6$		0			0
$7$	1.19615	+	2.07180i	0.0645862	+	0.111867i	0.896510	−	0.443023i	$-0.146094\pi$
$7$	1.19615	+	2.07180i	0.0645862	+	0.111867i	−0.831924	+	0.554889i	$0.812761\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	1.60770			0.0508398
$11$	29.7846	+	51.5885i	0.816400	+	1.41405i	0.908318	+	0.418279i	$0.137367\pi$
$11$	29.7846	+	51.5885i	0.816400	+	1.41405i	−0.0919186	+	0.995767i	$0.529300\pi$
$12$		0			0
$13$	−38.8731	+	67.3301i	−0.829342	+	1.43646i	0.0692128	+	0.997602i	$0.477951\pi$
$13$	−38.8731	+	67.3301i	−0.829342	+	1.43646i	−0.898555	+	0.438861i	$0.855382\pi$
$14$	−2.39230	+	4.14359i	−0.0456693	+	0.0791016i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	−2.84232			−0.0405509			−0.0202754	−	0.999794i	$-0.506454\pi$
$17$	−2.84232			−0.0405509			−0.0202754	+	0.999794i	$0.506454\pi$
$18$		0			0
$19$	−118.315			−1.42860			−0.714300	−	0.699840i	$-0.753255\pi$
$19$	−118.315			−1.42860			−0.714300	+	0.699840i	$0.753255\pi$
$20$	1.60770	+	2.78461i	0.0179746	+	0.0311329i
$21$		0			0
$22$	−59.5692	+	103.177i	−0.577282	+	0.999881i
$23$	55.3923	−	95.9423i	0.502178	−	0.869798i	−0.497819	−	0.867281i	$-0.665866\pi$
$23$	55.3923	−	95.9423i	0.502178	−	0.869798i	0.999997	−	0.00251677i	$-0.000801112\pi$
$24$		0			0
$25$	62.1769	+	107.694i	0.497415	+	0.861549i
$26$	−155.492			−1.17287
$27$		0			0
$28$	−9.56922			−0.0645862
$29$	62.8135	+	108.796i	0.402213	+	0.696653i	0.993993	−	0.109447i	$-0.0349079\pi$
$29$	62.8135	+	108.796i	0.402213	+	0.696653i	−0.591780	+	0.806100i	$0.701575\pi$
$30$		0			0
$31$	−31.5307	+	54.6128i	−0.182680	+	0.316412i	−0.942792	−	0.333380i	$-0.891811\pi$
$31$	−31.5307	+	54.6128i	−0.182680	+	0.316412i	0.760112	+	0.649792i	$0.225144\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	−2.84232	−	4.92305i	−0.0143369	−	0.0248322i
$35$	1.92305			0.00928727
$36$		0			0
$37$	−227.392			−1.01035			−0.505177	−	0.863016i	$-0.668573\pi$
$37$	−227.392			−1.01035			−0.505177	+	0.863016i	$0.668573\pi$
$38$	−118.315	−	204.928i	−0.505086	−	0.874835i
$39$		0			0
$40$	−3.21539	+	5.56922i	−0.0127099	+	0.0220143i
$41$	162.315	−	281.138i	0.618278	−	1.07089i	−0.371522	−	0.928424i	$-0.621164\pi$
$41$	162.315	−	281.138i	0.618278	−	1.07089i	0.989800	−	0.142465i	$-0.0455028\pi$
$42$		0			0
$43$	136.412	+	236.272i	0.483781	+	0.837933i	0.999826	−	0.0186284i	$-0.00592994\pi$
$43$	136.412	+	236.272i	0.483781	+	0.837933i	−0.516046	+	0.856561i	$0.672597\pi$
$44$	−238.277			−0.816400
$45$		0			0
$46$	221.569			0.710187
$47$	−2.46926	−	4.27688i	−0.00766336	−	0.0132733i	0.862168	−	0.506622i	$-0.169106\pi$
$47$	−2.46926	−	4.27688i	−0.00766336	−	0.0132733i	−0.869832	+	0.493349i	$0.835773\pi$
$48$		0			0
$49$	168.638	−	292.090i	0.491657	−	0.851575i
$50$	−124.354	+	215.387i	−0.351726	+	0.609207i
$51$		0			0
$52$	−155.492	−	269.321i	−0.414671	−	0.718231i
$53$	598.908			1.55219			0.776097	−	0.630614i	$-0.217197\pi$
$53$	598.908			1.55219			0.776097	+	0.630614i	$0.217197\pi$
$54$		0			0
$55$	47.8846			0.117396
$56$	−9.56922	−	16.5744i	−0.0228347	−	0.0395508i
$57$		0			0
$58$	−125.627	+	217.592i	−0.284407	+	0.492608i
$59$	−335.138	+	580.477i	−0.739514	+	1.28088i	0.213201	+	0.977008i	$0.431611\pi$
$59$	−335.138	+	580.477i	−0.739514	+	1.28088i	−0.952714	+	0.303867i	$0.901722\pi$
$60$		0			0
$61$	−232.265	−	402.295i	−0.487517	−	0.844404i	0.512380	−	0.858759i	$-0.328764\pi$
$61$	−232.265	−	402.295i	−0.487517	−	0.844404i	−0.999897	+	0.0143547i	$0.995431\pi$
$62$	−126.123			−0.258349
$63$		0			0
$64$	64.0000			0.125000
$65$	31.2480	+	54.1232i	0.0596283	+	0.103279i
$66$		0			0
$67$	384.535	−	666.033i	0.701170	−	1.21446i	−0.266886	−	0.963728i	$-0.585995\pi$
$67$	384.535	−	666.033i	0.701170	−	1.21446i	0.968056	−	0.250734i	$-0.0806718\pi$
$68$	5.68465	−	9.84610i	0.0101377	−	0.0175590i
$69$		0			0
$70$	1.92305	+	3.33082i	0.00328355	+	0.00568727i
$71$	611.569			1.02225			0.511126	−	0.859506i	$-0.329228\pi$
$71$	611.569			1.02225			0.511126	+	0.859506i	$0.329228\pi$
$72$		0			0
$73$	923.831			1.48118			0.740590	−	0.671957i	$-0.234546\pi$
$73$	923.831			1.48118			0.740590	+	0.671957i	$0.234546\pi$
$74$	−227.392	−	393.855i	−0.357214	−	0.618712i
$75$		0			0
$76$	236.631	−	409.856i	0.357150	−	0.618602i
$77$	−71.2539	+	123.415i	−0.105456	+	0.182656i
$78$		0			0
$79$	−19.9038	−	34.4744i	−0.0283462	−	0.0490971i	0.851504	−	0.524348i	$-0.175691\pi$
$79$	−19.9038	−	34.4744i	−0.0283462	−	0.0490971i	−0.879851	+	0.475251i	$0.842357\pi$
$80$	−12.8616			−0.0179746
$81$		0			0
$82$	649.261			0.874377
$83$	221.885	+	384.315i	0.293434	+	0.508242i	0.974619	−	0.223869i	$-0.0718687\pi$
$83$	221.885	+	384.315i	0.293434	+	0.508242i	−0.681186	+	0.732111i	$0.738535\pi$
$84$		0			0
$85$	−1.14240	+	1.97869i	−0.00145777	+	0.00252493i
$86$	−272.823	+	472.543i	−0.342085	+	0.592508i
$87$		0			0
$88$	−238.277	−	412.708i	−0.288641	−	0.499941i
$89$	78.4576			0.0934437			0.0467218	−	0.998908i	$-0.485123\pi$
$89$	78.4576			0.0934437			0.0467218	+	0.998908i	$0.485123\pi$
$90$		0			0
$91$	−185.992			−0.214256
$92$	221.569	+	383.769i	0.251089	+	0.434899i
$93$		0			0
$94$	4.93851	−	8.55376i	0.00541882	−	0.00938566i
$95$	−47.5538	+	82.3655i	−0.0513570	+	0.0889529i
$96$		0			0
$97$	45.5077	+	78.8217i	0.0476352	+	0.0825065i	0.888860	−	0.458179i	$-0.151498\pi$
$97$	45.5077	+	78.8217i	0.0476352	+	0.0825065i	−0.841225	+	0.540686i	$0.818165\pi$
$98$	674.554			0.695308
$99$		0			0
$100$	−497.415			−0.497415
$101$	−811.261	−	1405.15i	−0.799243	−	1.38433i	−0.920110	−	0.391661i	$-0.871901\pi$
$101$	−811.261	−	1405.15i	−0.799243	−	1.38433i	0.120867	−	0.992669i	$-0.461433\pi$
$102$		0			0
$103$	374.115	−	647.987i	0.357890	−	0.619884i	−0.629718	−	0.776824i	$-0.716830\pi$
$103$	374.115	−	647.987i	0.357890	−	0.619884i	0.987608	+	0.156940i	$0.0501629\pi$
$104$	310.985	−	538.641i	0.293217	−	0.507866i
$105$		0			0
$106$	598.908	+	1037.34i	0.548783	+	0.950521i
$107$	−1435.00			−1.29651			−0.648255	−	0.761423i	$-0.724501\pi$
$107$	−1435.00			−1.29651			−0.648255	+	0.761423i	$0.724501\pi$
$108$		0			0
$109$	−83.1305			−0.0730501			−0.0365250	−	0.999333i	$-0.511629\pi$
$109$	−83.1305			−0.0730501			−0.0365250	+	0.999333i	$0.511629\pi$
$110$	47.8846	+	82.9385i	0.0415056	+	0.0718898i
$111$		0			0
$112$	19.1384	−	33.1487i	0.0161465	−	0.0279666i
$113$	486.098	−	841.946i	0.404675	−	0.700917i	−0.589609	−	0.807689i	$-0.700718\pi$
$113$	486.098	−	841.946i	0.404675	−	0.700917i	0.994284	+	0.106772i	$0.0340514\pi$
$114$		0			0
$115$	−44.5270	−	77.1230i	−0.0361058	−	0.0625370i
$116$	−502.508			−0.402213
$117$		0			0
$118$	−1340.55			−1.04583
$119$	−3.39985	−	5.88872i	−0.00261902	−	0.00453628i
$120$		0			0
$121$	−1108.75	+	1920.40i	−0.833017	+	1.44283i
$122$	464.531	−	804.591i	0.344727	−	0.597084i
$123$		0			0
$124$	−126.123	−	218.451i	−0.0913401	−	0.158206i
$125$	200.442			0.143425
$126$		0			0
$127$	−1316.13			−0.919588			−0.459794	−	0.888026i	$-0.652077\pi$
$127$	−1316.13			−0.919588			−0.459794	+	0.888026i	$0.652077\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	−62.4960	+	108.246i	−0.0421636	+	0.0730295i
$131$	−557.638	+	965.858i	−0.371917	+	0.644179i	−0.989860	−	0.142044i	$-0.954633\pi$
$131$	−557.638	+	965.858i	−0.371917	+	0.644179i	0.617944	+	0.786222i	$0.287966\pi$
$132$		0			0
$133$	−141.523	−	245.125i	−0.0922678	−	0.159813i
$134$	1538.14			0.991604
$135$		0			0
$136$	22.7386			0.0143369
$137$	308.571	+	534.461i	0.192431	+	0.333300i	0.946055	−	0.324005i	$-0.105030\pi$
$137$	308.571	+	534.461i	0.192431	+	0.333300i	−0.753624	+	0.657305i	$0.771696\pi$
$138$		0			0
$139$	−1078.28	+	1867.64i	−0.657978	+	1.13965i	0.323161	+	0.946344i	$0.395255\pi$
$139$	−1078.28	+	1867.64i	−0.657978	+	1.13965i	−0.981138	+	0.193307i	$0.938079\pi$
$140$	−3.84610	+	6.66164i	−0.00232182	+	0.00402151i
$141$		0			0
$142$	611.569	+	1059.27i	0.361421	+	0.625999i
$143$	−4631.28			−2.70830
$144$		0			0
$145$	100.985			0.0578368
$146$	923.831	+	1600.12i	0.523676	+	0.907034i
$147$		0			0
$148$	454.785	−	787.710i	0.252588	−	0.437496i
$149$	1785.84	−	3093.17i	0.981893	−	1.70069i	0.326889	−	0.945063i	$-0.394000\pi$
$149$	1785.84	−	3093.17i	0.981893	−	1.70069i	0.655004	−	0.755626i	$-0.272667\pi$
$150$		0			0
$151$	1130.72	+	1958.46i	0.609379	+	1.05548i	0.991343	+	0.131299i	$0.0419147\pi$
$151$	1130.72	+	1958.46i	0.609379	+	1.05548i	−0.381963	+	0.924177i	$0.624752\pi$
$152$	946.523			0.505086
$153$		0			0
$154$	−285.015			−0.149138
$155$	25.3459	+	43.9004i	0.0131344	+	0.0227495i
$156$		0			0
$157$	−914.627	+	1584.18i	−0.464937	+	0.805295i	−0.999199	−	0.0400243i	$-0.987256\pi$
$157$	−914.627	+	1584.18i	−0.464937	+	0.805295i	0.534261	+	0.845319i	$0.320590\pi$
$158$	39.8076	−	68.9488i	0.0200438	−	0.0347169i
$159$		0			0
$160$	−12.8616	−	22.2769i	−0.00635497	−	0.0110071i
$161$	265.031			0.129735
$162$		0			0
$163$	268.554			0.129048			0.0645238	−	0.997916i	$-0.479447\pi$
$163$	268.554			0.129048			0.0645238	+	0.997916i	$0.479447\pi$
$164$	649.261	+	1124.55i	0.309139	+	0.535444i
$165$		0			0
$166$	−443.769	+	768.631i	−0.207489	+	0.359381i
$167$	−2027.26	+	3511.32i	−0.939366	+	1.62703i	−0.172710	+	0.984973i	$0.555252\pi$
$167$	−2027.26	+	3511.32i	−0.939366	+	1.62703i	−0.766657	+	0.642057i	$0.778081\pi$
$168$		0			0
$169$	−1923.73	−	3332.00i	−0.875617	−	1.51661i
$170$	−4.56959			−0.00206160
$171$		0			0
$172$	−1091.29			−0.483781
$173$	873.675	+	1513.25i	0.383955	+	0.665030i	0.991624	−	0.129161i	$-0.0412283\pi$
$173$	873.675	+	1513.25i	0.383955	+	0.665030i	−0.607668	+	0.794191i	$0.707895\pi$
$174$		0			0
$175$	−148.746	+	257.636i	−0.0642523	+	0.111288i
$176$	476.554	−	825.415i	0.204100	−	0.353511i
$177$		0			0
$178$	78.4576	+	135.892i	0.0330373	+	0.0572223i
$179$	2037.71			0.850868			0.425434	−	0.904989i	$-0.360121\pi$
$179$	2037.71			0.850868			0.425434	+	0.904989i	$0.360121\pi$
$180$		0			0
$181$	2820.68			1.15834			0.579169	−	0.815207i	$-0.303377\pi$
$181$	2820.68			1.15834			0.579169	+	0.815207i	$0.303377\pi$
$182$	−185.992	−	322.148i	−0.0757510	−	0.131205i
$183$		0			0
$184$	−443.138	+	767.538i	−0.177547	+	0.307520i
$185$	−91.3944	+	158.300i	−0.0363213	+	0.0629104i
$186$		0			0
$187$	−84.6575	−	146.631i	−0.0331057	−	0.0573408i
$188$	19.7541			0.00766336
$189$		0			0
$190$	−190.215			−0.0726297
$191$	219.862	+	380.811i	0.0832912	+	0.144265i	0.904662	−	0.426130i	$-0.140123\pi$
$191$	219.862	+	380.811i	0.0832912	+	0.144265i	−0.821371	+	0.570395i	$0.806790\pi$
$192$		0			0
$193$	231.338	−	400.689i	0.0862802	−	0.149442i	−0.819656	−	0.572856i	$-0.805835\pi$
$193$	231.338	−	400.689i	0.0862802	−	0.149442i	0.905936	+	0.423415i	$0.139169\pi$
$194$	−91.0155	+	157.643i	−0.0336831	+	0.0583409i
$195$		0			0
$196$	674.554	+	1168.36i	0.245829	+	0.425788i
$197$	−1036.60			−0.374896			−0.187448	−	0.982275i	$-0.560022\pi$
$197$	−1036.60			−0.374896			−0.187448	+	0.982275i	$0.560022\pi$
$198$		0			0
$199$	−1190.22			−0.423980			−0.211990	−	0.977272i	$-0.567994\pi$
$199$	−1190.22			−0.423980			−0.211990	+	0.977272i	$0.567994\pi$
$200$	−497.415	−	861.549i	−0.175863	−	0.304603i
$201$		0			0
$202$	1622.52	−	2810.29i	0.565150	−	0.978869i
$203$	−150.269	+	260.273i	−0.0519547	+	0.0899883i
$204$		0			0
$205$	−130.477	−	225.992i	−0.0444531	−	0.0769951i
$206$	1496.46			0.506133
$207$		0			0
$208$	1243.94			0.414671
$209$	−3523.98	−	6103.71i	−1.16631	−	2.02011i
$210$		0			0
$211$	2464.00	−	4267.78i	0.803929	−	1.39245i	−0.113083	−	0.993586i	$-0.536073\pi$
$211$	2464.00	−	4267.78i	0.803929	−	1.39245i	0.917012	−	0.398860i	$-0.130594\pi$
$212$	−1197.82	+	2074.68i	−0.388049	+	0.672120i
$213$		0			0
$214$	−1435.00	−	2485.49i	−0.458386	−	0.793947i
$215$	219.308			0.0695660
$216$		0			0
$217$	−150.862			−0.0471945
$218$	−83.1305	−	143.986i	−0.0258271	−	0.0447339i
$219$		0			0
$220$	−95.7691	+	165.877i	−0.0293489	+	0.0508338i
$221$	110.490	−	191.374i	0.0336305	−	0.0582498i
$222$		0			0
$223$	735.235	+	1273.46i	0.220785	+	0.382410i	0.955046	−	0.296456i	$-0.0958049\pi$
$223$	735.235	+	1273.46i	0.220785	+	0.382410i	−0.734262	+	0.678866i	$0.762472\pi$
$224$	76.5538			0.0228347
$225$		0			0
$226$	1944.39			0.572296
$227$	1051.13	+	1820.61i	0.307339	+	0.532327i	0.977779	−	0.209637i	$-0.0672281\pi$
$227$	1051.13	+	1820.61i	0.307339	+	0.532327i	−0.670440	+	0.741964i	$0.733895\pi$
$228$		0			0
$229$	500.519	−	866.924i	0.144433	−	0.250166i	−0.784728	−	0.619840i	$-0.787197\pi$
$229$	500.519	−	866.924i	0.144433	−	0.250166i	0.929161	+	0.369674i	$0.120531\pi$
$230$	89.0539	−	154.246i	0.0255306	−	0.0442203i
$231$		0			0
$232$	−502.508	−	870.369i	−0.142204	−	0.246304i
$233$	6280.78			1.76596			0.882978	−	0.469415i	$-0.155535\pi$
$233$	6280.78			1.76596			0.882978	+	0.469415i	$0.155535\pi$
$234$		0			0
$235$	−3.96981			−0.00110197
$236$	−1340.55	−	2321.91i	−0.369757	−	0.640438i
$237$		0			0
$238$	6.79970	−	11.7774i	0.00185193	−	0.00320764i
$239$	−508.685	+	881.068i	−0.137674	+	0.238458i	−0.926616	−	0.376010i	$-0.877296\pi$
$239$	−508.685	+	881.068i	−0.137674	+	0.238458i	0.788942	+	0.614468i	$0.210629\pi$
$240$		0			0
$241$	460.646	+	797.862i	0.123124	+	0.213257i	0.920998	−	0.389567i	$-0.127375\pi$
$241$	460.646	+	797.862i	0.123124	+	0.213257i	−0.797874	+	0.602824i	$0.794042\pi$
$242$	−4434.98			−1.17806
$243$		0			0
$244$	1858.12			0.487517
$245$	−135.560	−	234.796i	−0.0353493	−	0.0612268i
$246$		0			0
$247$	4599.28	−	7966.19i	1.18480	−	2.05213i
$248$	252.246	−	436.903i	0.0645872	−	0.111868i
$249$		0			0
$250$	200.442	+	347.177i	0.0507084	+	0.0878295i
$251$	−2262.40			−0.568930			−0.284465	−	0.958686i	$-0.591816\pi$
$251$	−2262.40			−0.568930			−0.284465	+	0.958686i	$0.591816\pi$
$252$		0			0
$253$	6599.35			1.63991
$254$	−1316.13	−	2279.60i	−0.325124	−	0.563130i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	2393.32	−	4145.36i	0.580901	−	1.00615i	−0.414472	−	0.910062i	$-0.636034\pi$
$257$	2393.32	−	4145.36i	0.580901	−	1.00615i	0.995373	−	0.0960877i	$-0.0306329\pi$
$258$		0			0
$259$	−271.996	−	471.111i	−0.0652548	−	0.113025i
$260$	−249.984			−0.0596283
$261$		0			0
$262$	−2230.55			−0.525970
$263$	−1503.96	−	2604.94i	−0.352617	−	0.610750i	0.634090	−	0.773259i	$-0.281375\pi$
$263$	−1503.96	−	2604.94i	−0.352617	−	0.610750i	−0.986707	+	0.162509i	$0.948041\pi$
$264$		0			0
$265$	240.715	−	416.931i	0.0558001	−	0.0966485i
$266$	283.046	−	490.251i	0.0652432	−	0.113005i
$267$		0			0
$268$	1538.14	+	2664.13i	0.350585	+	0.607231i
$269$	6559.73			1.48682			0.743409	−	0.668837i	$-0.233208\pi$
$269$	6559.73			1.48682			0.743409	+	0.668837i	$0.233208\pi$
$270$		0			0
$271$	−5147.64			−1.15386			−0.576931	−	0.816793i	$-0.695750\pi$
$271$	−5147.64			−1.15386			−0.576931	+	0.816793i	$0.695750\pi$
$272$	22.7386	+	39.3844i	0.00506886	+	0.00877952i
$273$		0			0
$274$	−617.143	+	1068.92i	−0.136069	+	0.235679i
$275$	−3703.83	+	6415.22i	−0.812180	+	1.40674i
$276$		0			0
$277$	533.831	+	924.622i	0.115793	+	0.200560i	0.918097	−	0.396357i	$-0.129726\pi$
$277$	533.831	+	924.622i	0.115793	+	0.200560i	−0.802303	+	0.596917i	$0.796392\pi$
$278$	−4313.14			−0.930521
$279$		0			0
$280$	−15.3844			−0.00328355
$281$	2093.24	+	3625.59i	0.444384	+	0.769696i	0.998009	−	0.0630699i	$-0.0200891\pi$
$281$	2093.24	+	3625.59i	0.444384	+	0.769696i	−0.553625	+	0.832766i	$0.686756\pi$
$282$		0			0
$283$	−2678.48	+	4639.26i	−0.562611	+	0.974471i	0.434657	+	0.900596i	$0.356870\pi$
$283$	−2678.48	+	4639.26i	−0.562611	+	0.974471i	−0.997268	+	0.0738746i	$0.976464\pi$
$284$	−1223.14	+	2118.54i	−0.255563	+	0.442648i
$285$		0			0
$286$	−4631.28	−	8021.61i	−0.957528	−	1.65849i
$287$	776.616			0.159729
$288$		0			0
$289$	−4904.92			−0.998356
$290$	100.985	+	174.911i	0.0204484	+	0.0354177i
$291$		0			0
$292$	−1847.66	+	3200.24i	−0.370295	+	0.641370i
$293$	1447.89	−	2507.83i	0.288692	−	0.500030i	−0.684805	−	0.728726i	$-0.740113\pi$
$293$	1447.89	−	2507.83i	0.288692	−	0.500030i	0.973498	+	0.228696i	$0.0734462\pi$
$294$		0			0
$295$	269.400	+	466.615i	0.0531698	+	0.0920928i
$296$	1819.14			0.357214
$297$		0			0
$298$	7143.38			1.38861
$299$	4306.54	+	7459.14i	0.832955	+	1.44272i
$300$		0			0
$301$	−326.338	+	565.234i	−0.0624911	+	0.108238i
$302$	−2261.43	+	3916.91i	−0.430896	+	0.746334i
$303$		0			0
$304$	946.523	+	1639.43i	0.178575	+	0.309301i
$305$	−373.412			−0.0701033
$306$		0			0
$307$	1855.49			0.344947			0.172473	−	0.985014i	$-0.444824\pi$
$307$	1855.49			0.344947			0.172473	+	0.985014i	$0.444824\pi$
$308$	−285.015	−	493.661i	−0.0527281	−	0.0913278i
$309$		0			0
$310$	−50.6918	+	87.8008i	−0.00928743	+	0.0160863i
$311$	3585.85	−	6210.87i	0.653809	−	1.13243i	−0.328382	−	0.944545i	$-0.606503\pi$
$311$	3585.85	−	6210.87i	0.653809	−	1.13243i	0.982191	−	0.187886i	$-0.0601635\pi$
$312$		0			0
$313$	655.415	+	1135.21i	0.118359	+	0.205003i	0.919117	−	0.393984i	$-0.128903\pi$
$313$	655.415	+	1135.21i	0.118359	+	0.205003i	−0.800759	+	0.598987i	$0.795570\pi$
$314$	−3658.51			−0.657521
$315$		0			0
$316$	159.230			0.0283462
$317$	1844.63	+	3195.00i	0.326829	+	0.566085i	0.981881	−	0.189499i	$-0.0606865\pi$
$317$	1844.63	+	3195.00i	0.326829	+	0.566085i	−0.655052	+	0.755584i	$0.727353\pi$
$318$		0			0
$319$	−3741.75	+	6480.90i	−0.656733	+	1.13749i
$320$	25.7231	−	44.5538i	0.00449364	−	0.00778322i
$321$		0			0
$322$	265.031	+	459.046i	0.0458682	+	0.0794461i
$323$	336.290			0.0579310
$324$		0			0
$325$	−9668.03			−1.65011
$326$	268.554	+	465.149i	0.0456252	+	0.0790252i
$327$		0			0
$328$	−1298.52	+	2249.11i	−0.218594	+	0.378616i
$329$	5.90721	−	10.2316i	0.000989894	−	0.00171455i
$330$		0			0
$331$	1622.48	+	2810.22i	0.269425	+	0.466657i	0.968713	−	0.248182i	$-0.0798331\pi$
$331$	1622.48	+	2810.22i	0.269425	+	0.466657i	−0.699289	+	0.714839i	$0.746500\pi$
$332$	−1775.08			−0.293434
$333$		0			0
$334$	−8109.04			−1.32846
$335$	−309.107	−	535.389i	−0.0504129	−	0.0873177i
$336$		0			0
$337$	−2150.83	+	3725.35i	−0.347665	+	0.602174i	−0.985834	−	0.167722i	$-0.946359\pi$
$337$	−2150.83	+	3725.35i	−0.347665	+	0.602174i	0.638169	+	0.769896i	$0.279692\pi$
$338$	3847.46	−	6664.00i	0.619155	−	1.07241i
$339$		0			0
$340$	−4.56959	−	7.91476i	−0.000728885	−	0.00126247i
$341$	−3756.52			−0.596561
$342$		0			0
$343$	1627.43			0.256189
$344$	−1091.29	−	1890.17i	−0.171042	−	0.296254i
$345$		0			0
$346$	−1747.35	+	3026.50i	−0.271497	+	0.470247i
$347$	−1652.10	+	2861.52i	−0.255589	+	0.442693i	−0.965055	−	0.262046i	$-0.915603\pi$
$347$	−1652.10	+	2861.52i	−0.255589	+	0.442693i	0.709466	+	0.704739i	$0.248936\pi$
$348$		0			0
$349$	−2254.35	−	3904.65i	−0.345767	−	0.598886i	0.639726	−	0.768603i	$-0.279048\pi$
$349$	−2254.35	−	3904.65i	−0.345767	−	0.598886i	−0.985493	+	0.169717i	$0.945715\pi$
$350$	−594.985			−0.0908665
$351$		0			0
$352$	1906.22			0.288641
$353$	3443.37	+	5964.09i	0.519184	+	0.899254i	0.999751	+	0.0222956i	$0.00709751\pi$
$353$	3443.37	+	5964.09i	0.519184	+	0.899254i	−0.480567	+	0.876958i	$0.659569\pi$
$354$		0			0
$355$	245.804	−	425.745i	0.0367491	−	0.0636513i
$356$	−156.915	+	271.785i	−0.0233609	+	0.0404623i
$357$		0			0
$358$	2037.71	+	3529.41i	0.300827	+	0.521048i
$359$	1529.31			0.224829			0.112415	−	0.993661i	$-0.464141\pi$
$359$	1529.31			0.224829			0.112415	+	0.993661i	$0.464141\pi$
$360$		0			0
$361$	7139.52			1.04090
$362$	2820.68	+	4885.56i	0.409534	+	0.709335i
$363$		0			0
$364$	371.985	−	644.297i	0.0535640	−	0.0927756i
$365$	371.310	−	643.127i	0.0532472	−	0.0922269i
$366$		0			0
$367$	5115.68	+	8860.62i	0.727620	+	1.26027i	0.957886	+	0.287147i	$0.0927069\pi$
$367$	5115.68	+	8860.62i	0.727620	+	1.26027i	−0.230267	+	0.973128i	$0.573960\pi$
$368$	−1772.55			−0.251089
$369$		0			0
$370$	−365.578			−0.0513661
$371$	716.385	+	1240.81i	0.100250	+	0.173639i
$372$		0			0
$373$	−473.215	+	819.632i	−0.0656894	+	0.113777i	−0.897000	−	0.442031i	$-0.854258\pi$
$373$	−473.215	+	819.632i	−0.0656894	+	0.113777i	0.831310	+	0.555809i	$0.187591\pi$
$374$	169.315	−	293.262i	0.0234093	−	0.0405461i
$375$		0			0
$376$	19.7541	+	34.2150i	0.00270941	+	0.00469283i
$377$	−9767.01			−1.33429
$378$		0			0
$379$	−8717.46			−1.18149			−0.590746	−	0.806857i	$-0.701167\pi$
$379$	−8717.46			−1.18149			−0.590746	+	0.806857i	$0.701167\pi$
$380$	−190.215	−	329.462i	−0.0256785	−	0.0444764i
$381$		0			0
$382$	−439.723	+	761.623i	−0.0588958	+	0.102011i
$383$	6130.77	−	10618.8i	0.817931	−	1.41670i	−0.0892728	−	0.996007i	$-0.528454\pi$
$383$	6130.77	−	10618.8i	0.817931	−	1.41670i	0.907204	−	0.420691i	$-0.138212\pi$
$384$		0			0
$385$	57.2772	+	99.2071i	0.00758213	+	0.0131326i
$386$	925.353			0.122019
$387$		0			0
$388$	−364.062			−0.0476352
$389$	−6605.53	−	11441.1i	−0.860961	−	1.49123i	−0.871003	−	0.491278i	$-0.836530\pi$
$389$	−6605.53	−	11441.1i	−0.860961	−	1.49123i	0.0100423	−	0.999950i	$-0.496803\pi$
$390$		0			0
$391$	−157.443	+	272.699i	−0.0203638	+	0.0352711i
$392$	−1349.11	+	2336.72i	−0.173827	+	0.301077i
$393$		0			0
$394$	−1036.60	−	1795.44i	−0.132546	−	0.229576i
$395$	−31.9993			−0.00407609
$396$		0			0
$397$	−5211.93			−0.658890			−0.329445	−	0.944175i	$-0.606862\pi$
$397$	−5211.93			−0.658890			−0.329445	+	0.944175i	$0.606862\pi$
$398$	−1190.22	−	2061.51i	−0.149900	−	0.259634i
$399$		0			0
$400$	994.831	−	1723.10i	0.124354	−	0.215387i
$401$	492.817	−	853.584i	0.0613719	−	0.106299i	−0.833707	−	0.552207i	$-0.813786\pi$
$401$	492.817	−	853.584i	0.0613719	−	0.106299i	0.895079	+	0.445908i	$0.147119\pi$
$402$		0			0
$403$	−2451.39	−	4245.94i	−0.303009	−	0.524827i
$404$	6490.09			0.799243
$405$		0			0
$406$	−601.076			−0.0734751
$407$	−6772.79	−	11730.8i	−0.824852	−	1.42869i
$408$		0			0
$409$	337.323	−	584.261i	0.0407813	−	0.0706354i	−0.844914	−	0.534902i	$-0.820349\pi$
$409$	337.323	−	584.261i	0.0407813	−	0.0706354i	0.885696	+	0.464266i	$0.153682\pi$
$410$	260.954	−	451.985i	0.0314331	−	0.0544438i
$411$		0			0
$412$	1496.46	+	2591.95i	0.178945	+	0.309942i
$413$	−1603.51			−0.191049
$414$		0			0
$415$	356.723			0.0421948
$416$	1243.94	+	2154.56i	0.146608	+	0.253933i
$417$		0			0
$418$	7047.95	−	12207.4i	0.824705	−	1.42843i
$419$	−3752.82	+	6500.07i	−0.437559	+	0.757874i	−0.997501	−	0.0706581i	$-0.977490\pi$
$419$	−3752.82	+	6500.07i	−0.437559	+	0.757874i	0.559942	+	0.828532i	$0.310823\pi$
$420$		0			0
$421$	676.312	+	1171.41i	0.0782932	+	0.135608i	0.902514	−	0.430661i	$-0.141720\pi$
$421$	676.312	+	1171.41i	0.0782932	+	0.135608i	−0.824220	+	0.566269i	$0.808386\pi$
$422$	9856.01			1.13693
$423$		0			0
$424$	−4791.26			−0.548783
$425$	−176.727	−	306.100i	−0.0201706	−	0.0349365i
$426$		0			0
$427$	555.650	−	962.413i	0.0629737	−	0.109074i
$428$	2870.00	−	4970.98i	0.324128	−	0.561406i
$429$		0			0
$430$	219.308	+	379.853i	0.0245953	+	0.0426003i
$431$	−4003.21			−0.447397			−0.223698	−	0.974658i	$-0.571813\pi$
$431$	−4003.21			−0.447397			−0.223698	+	0.974658i	$0.571813\pi$
$432$		0			0
$433$	9975.38			1.10713			0.553564	−	0.832807i	$-0.313267\pi$
$433$	9975.38			1.10713			0.553564	+	0.832807i	$0.313267\pi$
$434$	−150.862	−	261.301i	−0.0166858	−	0.0289006i
$435$		0			0
$436$	166.261	−	287.973i	0.0182625	−	0.0316316i
$437$	−6553.76	+	11351.4i	−0.717412	+	1.24259i
$438$		0			0
$439$	−8680.46	−	15035.0i	−0.943726	−	1.63458i	−0.758282	−	0.651927i	$-0.773961\pi$
$439$	−8680.46	−	15035.0i	−0.943726	−	1.63458i	−0.185444	−	0.982655i	$-0.559372\pi$
$440$	−383.077			−0.0415056
$441$		0			0
$442$	441.959			0.0475608
$443$	4287.71	+	7426.54i	0.459854	+	0.796491i	0.998953	−	0.0457517i	$-0.0145683\pi$
$443$	4287.71	+	7426.54i	0.459854	+	0.796491i	−0.539099	+	0.842243i	$0.681235\pi$
$444$		0			0
$445$	31.5340	−	54.6184i	0.00335922	−	0.00581834i
$446$	−1470.47	+	2546.93i	−0.156118	+	0.270405i
$447$		0			0
$448$	76.5538	+	132.595i	0.00807327	+	0.0139833i
$449$	4703.77			0.494398			0.247199	−	0.968965i	$-0.420490\pi$
$449$	4703.77			0.494398			0.247199	+	0.968965i	$0.420490\pi$
$450$		0			0
$451$	19338.0			2.01905
$452$	1944.39	+	3367.78i	0.202337	+	0.350459i
$453$		0			0
$454$	−2102.26	+	3641.22i	−0.217322	+	0.376412i
$455$	−74.7548	+	129.479i	−0.00770233	+	0.0133408i
$456$		0			0
$457$	3523.40	+	6102.71i	0.360651	+	0.624667i	0.988068	−	0.154017i	$-0.0492211\pi$
$457$	3523.40	+	6102.71i	0.360651	+	0.624667i	−0.627417	+	0.778684i	$0.715888\pi$
$458$	2002.08			0.204259
$459$		0			0
$460$	356.216			0.0361058
$461$	−2885.13	−	4997.19i	−0.291484	−	0.504864i	0.682677	−	0.730720i	$-0.260816\pi$
$461$	−2885.13	−	4997.19i	−0.291484	−	0.504864i	−0.974161	+	0.225856i	$0.927482\pi$
$462$		0			0
$463$	6799.57	−	11777.2i	0.682511	−	1.18214i	−0.291701	−	0.956510i	$-0.594221\pi$
$463$	6799.57	−	11777.2i	0.682511	−	1.18214i	0.974212	−	0.225634i	$-0.0724455\pi$
$464$	1005.02	−	1740.74i	0.100553	−	0.174163i
$465$		0			0
$466$	6280.78	+	10878.6i	0.624359	+	1.08142i
$467$	5730.52			0.567831			0.283915	−	0.958849i	$-0.408367\pi$
$467$	5730.52			0.567831			0.283915	+	0.958849i	$0.408367\pi$
$468$		0			0
$469$	1839.85			0.181143
$470$	−3.96981	−	6.87592i	−0.000389604	−	0.000674813i
$471$		0			0
$472$	2681.11	−	4643.81i	0.261458	−	0.452858i
$473$	−8125.93	+	14074.5i	−0.789917	+	1.36818i
$474$		0			0
$475$	−7356.48	−	12741.8i	−0.710608	−	1.23081i
$476$	27.1988			0.00261902
$477$		0			0
$478$	−2034.74			−0.194701
$479$	−7284.14	−	12616.5i	−0.694824	−	1.20347i	−0.970240	−	0.242145i	$-0.922149\pi$
$479$	−7284.14	−	12616.5i	−0.694824	−	1.20347i	0.275416	−	0.961325i	$-0.411184\pi$
$480$		0			0
$481$	8839.44	−	15310.4i	0.837928	−	1.45133i
$482$	−921.292	+	1595.72i	−0.0870616	+	0.150795i
$483$		0			0
$484$	−4434.98	−	7681.62i	−0.416509	−	0.721414i
$485$	73.1626			0.00684977
$486$		0			0
$487$	7164.51			0.666642			0.333321	−	0.942813i	$-0.391831\pi$
$487$	7164.51			0.666642			0.333321	+	0.942813i	$0.391831\pi$
$488$	1858.12	+	3218.36i	0.172363	+	0.298542i
$489$		0			0
$490$	271.119	−	469.592i	0.0249957	−	0.0432939i
$491$	4755.03	−	8235.96i	0.437050	−	0.756993i	−0.560410	−	0.828215i	$-0.689357\pi$
$491$	4755.03	−	8235.96i	0.437050	−	0.756993i	0.997460	+	0.0712222i	$0.0226899\pi$
$492$		0			0
$493$	−178.536	−	309.234i	−0.0163101	−	0.0282499i
$494$	18397.1			1.67556
$495$		0			0
$496$	1008.98			0.0913401
$497$	731.530	+	1267.05i	0.0660234	+	0.114356i
$498$		0			0
$499$	−10742.8	+	18607.1i	−0.963756	+	1.66927i	−0.250835	+	0.968030i	$0.580705\pi$
$499$	−10742.8	+	18607.1i	−0.963756	+	1.66927i	−0.712921	+	0.701245i	$0.752628\pi$
$500$	−400.885	+	694.353i	−0.0358562	+	0.0621048i
$501$		0			0
$502$	−2262.40	−	3918.59i	−0.201147	−	0.348397i
$503$	−11603.2			−1.02855			−0.514274	−	0.857626i	$-0.671939\pi$
$503$	−11603.2			−1.02855			−0.514274	+	0.857626i	$0.671939\pi$
$504$		0			0
$505$	−1304.26			−0.114928
$506$	6599.35	+	11430.4i	0.579796	+	1.00424i
$507$		0			0
$508$	2632.26	−	4559.21i	0.229897	−	0.398193i
$509$	−4469.87	+	7742.04i	−0.389241	+	0.674184i	−0.992348	−	0.123476i	$-0.960596\pi$
$509$	−4469.87	+	7742.04i	−0.389241	+	0.674184i	0.603107	+	0.797660i	$0.293929\pi$
$510$		0			0
$511$	1105.04	+	1913.99i	0.0956638	+	0.165695i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	9573.30			0.821518
$515$	−300.732	−	520.883i	−0.0257317	−	0.0445686i
$516$		0			0
$517$	147.092	−	254.770i	0.0125127	−	0.0216727i
$518$	543.992	−	942.221i	0.0461421	−	0.0799205i
$519$		0			0
$520$	−249.984	−	432.985i	−0.0210818	−	0.0365147i
$521$	−7410.10			−0.623114			−0.311557	−	0.950227i	$-0.600851\pi$
$521$	−7410.10			−0.623114			−0.311557	+	0.950227i	$0.600851\pi$
$522$		0			0
$523$	18970.8			1.58611			0.793054	−	0.609151i	$-0.208490\pi$
$523$	18970.8			1.58611			0.793054	+	0.609151i	$0.208490\pi$
$524$	−2230.55	−	3863.43i	−0.185958	−	0.322089i
$525$		0			0
$526$	3007.92	−	5209.88i	0.249338	−	0.431866i
$527$	89.6206	−	155.227i	0.00740784	−	0.0128308i
$528$		0			0
$529$	−53.1149	−	91.9977i	−0.00436549	−	0.00756124i
$530$	962.861			0.0789132
$531$		0			0
$532$	1132.19			0.0922678
$533$	12619.4	+	21857.4i	1.02553	+	1.77627i
$534$		0			0
$535$	−576.760	+	998.978i	−0.0466085	+	0.0807282i
$536$	−3076.28	+	5328.27i	−0.247901	+	0.429377i
$537$		0			0
$538$	6559.73	+	11361.8i	0.525669	+	0.910486i
$539$	20091.3			1.60556
$540$		0			0
$541$	−11440.4			−0.909169			−0.454584	−	0.890704i	$-0.650212\pi$
$541$	−11440.4			−0.909169			−0.454584	+	0.890704i	$0.650212\pi$
$542$	−5147.64	−	8915.97i	−0.407952	−	0.706594i
$543$		0			0
$544$	−45.4772	+	78.7688i	−0.00358422	+	0.00620806i
$545$	−33.4121	+	57.8715i	−0.00262609	+	0.00454852i
$546$		0			0
$547$	−5126.48	−	8879.32i	−0.400717	−	0.694063i	0.593095	−	0.805132i	$-0.297906\pi$
$547$	−5126.48	−	8879.32i	−0.400717	−	0.694063i	−0.993813	+	0.111070i	$0.964572\pi$
$548$	−2468.57			−0.192431
$549$		0			0
$550$	−14815.3			−1.14860
$551$	−7431.80	−	12872.3i	−0.574601	−	0.995238i
$552$		0			0
$553$	47.6160	−	82.4733i	0.00366155	−	0.00634199i
$554$	−1067.66	+	1849.24i	−0.0818783	+	0.141817i
$555$		0			0
$556$	−4313.14	−	7470.57i	−0.328989	−	0.569825i
$557$	−15336.5			−1.16666			−0.583328	−	0.812236i	$-0.698250\pi$
$557$	−15336.5			−1.16666			−0.583328	+	0.812236i	$0.698250\pi$
$558$		0			0
$559$	−21210.9			−1.60488
$560$	−15.3844	−	26.6465i	−0.00116091	−	0.00201075i
$561$		0			0
$562$	−4186.47	+	7251.18i	−0.314227	+	0.544257i
$563$	3850.69	−	6669.58i	0.288254	−	0.499271i	−0.685139	−	0.728412i	$-0.740259\pi$
$563$	3850.69	−	6669.58i	0.288254	−	0.499271i	0.973393	+	0.229142i	$0.0735919\pi$
$564$		0			0
$565$	−390.749	−	676.796i	−0.0290954	−	0.0503948i
$566$	−10713.9			−0.795652
$567$		0			0
$568$	−4892.55			−0.361421
$569$	−8869.63	−	15362.6i	−0.653487	−	1.13187i	−0.982271	−	0.187467i	$-0.939972\pi$
$569$	−8869.63	−	15362.6i	−0.653487	−	1.13187i	0.328784	−	0.944405i	$-0.393361\pi$
$570$		0			0
$571$	−6911.55	+	11971.2i	−0.506549	+	0.877369i	0.493422	+	0.869790i	$0.335746\pi$
$571$	−6911.55	+	11971.2i	−0.506549	+	0.877369i	−0.999971	+	0.00757865i	$0.997588\pi$
$572$	9262.55	−	16043.2i	0.677075	−	1.17273i
$573$		0			0
$574$	776.616	+	1345.14i	0.0564727	+	0.0978135i
$575$	13776.5			0.999164
$576$		0			0
$577$	−15339.1			−1.10672			−0.553359	−	0.832943i	$-0.686654\pi$
$577$	−15339.1			−1.10672			−0.553359	+	0.832943i	$0.686654\pi$
$578$	−4904.92	−	8495.57i	−0.352972	−	0.611365i
$579$		0			0
$580$	−201.970	+	349.822i	−0.0144592	+	0.0250441i
$581$	−530.816	+	919.399i	−0.0379035	+	0.0656508i
$582$		0			0
$583$	17838.2	+	30896.7i	1.26721	+	2.19487i
$584$	−7390.65			−0.523676
$585$		0			0
$586$	5791.58			0.408273
$587$	7383.26	+	12788.2i	0.519148	+	0.899191i	0.999752	+	0.0222531i	$0.00708396\pi$
$587$	7383.26	+	12788.2i	0.519148	+	0.899191i	−0.480604	+	0.876937i	$0.659583\pi$
$588$		0			0
$589$	3730.57	−	6461.54i	0.260977	−	0.452026i
$590$	−538.800	+	933.230i	−0.0375967	+	0.0651194i
$591$		0			0
$592$	1819.14	+	3150.84i	0.126294	+	0.218748i
$593$	−17603.6			−1.21904			−0.609522	−	0.792769i	$-0.708639\pi$
$593$	−17603.6			−1.21904			−0.609522	+	0.792769i	$0.708639\pi$
$594$		0			0
$595$	−5.46593			−0.000376607
$596$	7143.38	+	12372.7i	0.490946	+	0.850344i
$597$		0			0
$598$	−8613.08	+	14918.3i	−0.588988	+	1.02016i
$599$	11400.6	−	19746.3i	0.777653	−	1.34693i	−0.155638	−	0.987814i	$-0.549743\pi$
$599$	11400.6	−	19746.3i	0.777653	−	1.34693i	0.933291	−	0.359120i	$-0.116923\pi$
$600$		0			0
$601$	−952.624	−	1649.99i	−0.0646562	−	0.111988i	0.831885	−	0.554948i	$-0.187262\pi$
$601$	−952.624	−	1649.99i	−0.0646562	−	0.111988i	−0.896541	+	0.442960i	$0.853928\pi$
$602$	−1305.35			−0.0883757
$603$		0			0
$604$	−9045.72			−0.609379
$605$	891.263	+	1543.71i	0.0598925	+	0.103737i
$606$		0			0
$607$	1294.61	−	2242.33i	0.0865678	−	0.149940i	−0.819490	−	0.573093i	$-0.805743\pi$
$607$	1294.61	−	2242.33i	0.0865678	−	0.149940i	0.906058	+	0.423153i	$0.139077\pi$
$608$	−1893.05	+	3278.85i	−0.126272	+	0.218709i
$609$		0			0
$610$	−373.412	−	646.768i	−0.0247853	−	0.0429293i
$611$	383.950			0.0254222
$612$		0			0
$613$	16930.4			1.11552			0.557759	−	0.830003i	$-0.311661\pi$
$613$	16930.4			1.11552			0.557759	+	0.830003i	$0.311661\pi$
$614$	1855.49	+	3213.81i	0.121957	+	0.211236i
$615$		0			0
$616$	570.031	−	987.323i	0.0372844	−	0.0645785i
$617$	6406.72	−	11096.8i	0.418030	−	0.724050i	−0.577711	−	0.816241i	$-0.696054\pi$
$617$	6406.72	−	11096.8i	0.418030	−	0.724050i	0.995741	+	0.0921917i	$0.0293872\pi$
$618$		0			0
$619$	−5133.79	−	8891.99i	−0.333351	−	0.577382i	0.649815	−	0.760092i	$-0.274846\pi$
$619$	−5133.79	−	8891.99i	−0.333351	−	0.577382i	−0.983167	+	0.182711i	$0.941513\pi$
$620$	−202.767			−0.0131344
$621$		0			0
$622$	14343.4			0.924626
$623$	93.8472	+	162.548i	0.00603517	+	0.0104532i
$624$		0			0
$625$	−7691.55	+	13322.2i	−0.492259	+	0.852618i
$626$	−1310.83	+	2270.42i	−0.0836922	+	0.144959i
$627$		0			0
$628$	−3658.51	−	6336.72i	−0.232469	−	0.402648i
$629$	646.322			0.0409707
$630$		0			0
$631$	−21887.5			−1.38087			−0.690433	−	0.723397i	$-0.742580\pi$
$631$	−21887.5			−1.38087			−0.690433	+	0.723397i	$0.742580\pi$
$632$	159.230	+	275.795i	0.0100219	+	0.0173585i
$633$		0			0
$634$	−3689.26	+	6389.99i	−0.231103	+	0.400282i
$635$	−528.984	+	916.227i	−0.0330584	+	0.0572589i
$636$		0			0
$637$	13111.0	+	22708.9i	0.815504	+	1.41249i
$638$	−14967.0			−0.928760
$639$		0			0
$640$	102.892			0.00635497
$641$	8601.84	+	14898.8i	0.530035	+	0.918047i	0.999386	+	0.0350356i	$0.0111545\pi$
$641$	8601.84	+	14898.8i	0.530035	+	0.918047i	−0.469351	+	0.883012i	$0.655512\pi$
$642$		0			0
$643$	−6282.79	+	10882.1i	−0.385333	+	0.667416i	−0.991815	−	0.127681i	$-0.959247\pi$
$643$	−6282.79	+	10882.1i	−0.385333	+	0.667416i	0.606482	+	0.795097i	$0.292580\pi$
$644$	−530.061	+	918.093i	−0.0324337	+	0.0561769i
$645$		0			0
$646$	336.290	+	582.472i	0.0204817	+	0.0354753i
$647$	−3552.55			−0.215866			−0.107933	−	0.994158i	$-0.534423\pi$
$647$	−3552.55			−0.215866			−0.107933	+	0.994158i	$0.534423\pi$
$648$		0			0
$649$	−39927.9			−2.41496
$650$	−9668.03	−	16745.5i	−0.583402	−	1.01048i
$651$		0			0
$652$	−537.108	+	930.297i	−0.0322619	+	0.0558792i
$653$	2885.50	−	4997.83i	0.172922	−	0.299511i	−0.766518	−	0.642223i	$-0.778012\pi$
$653$	2885.50	−	4997.83i	0.172922	−	0.299511i	0.939440	+	0.342712i	$0.111346\pi$
$654$		0			0
$655$	448.256	+	776.402i	0.0267402	+	0.0463153i
$656$	−5194.09			−0.309139
$657$		0			0
$658$	23.6289			0.00139992
$659$	8239.20	+	14270.7i	0.487032	+	0.843564i	0.999889	−	0.0149104i	$-0.00474631\pi$
$659$	8239.20	+	14270.7i	0.487032	+	0.843564i	−0.512857	+	0.858474i	$0.671413\pi$
$660$		0			0
$661$	2281.62	−	3951.88i	0.134258	−	0.232542i	−0.791056	−	0.611744i	$-0.790468\pi$
$661$	2281.62	−	3951.88i	0.134258	−	0.232542i	0.925314	+	0.379202i	$0.123801\pi$
$662$	−3244.96	+	5620.44i	−0.190512	+	0.329977i
$663$		0			0
$664$	−1775.08	−	3074.52i	−0.103744	−	0.179691i
$665$	−227.526			−0.0132678
$666$		0			0
$667$	13917.5			0.807929
$668$	−8109.04	−	14045.3i	−0.469683	−	0.813515i
$669$		0			0
$670$	618.214	−	1070.78i	0.0356473	−	0.0617430i
$671$	13835.9	−	23964.4i	0.796017	−	1.37874i
$672$		0			0
$673$	6576.14	+	11390.2i	0.376659	+	0.652393i	0.990574	−	0.136980i	$-0.0437395\pi$
$673$	6576.14	+	11390.2i	0.376659	+	0.652393i	−0.613915	+	0.789372i	$0.710406\pi$
$674$	−8603.32			−0.491673
$675$		0			0
$676$	15389.8			0.875617
$677$	7438.07	+	12883.1i	0.422257	+	0.731371i	0.996160	−	0.0875524i	$-0.0279045\pi$
$677$	7438.07	+	12883.1i	0.422257	+	0.731371i	−0.573903	+	0.818924i	$0.694571\pi$
$678$		0			0
$679$	−108.868	+	188.566i	−0.00615314	+	0.0106576i
$680$	9.13918	−	15.8295i	0.000515399	−	0.000892698i
$681$		0			0
$682$	−3756.52	−	6506.49i	−0.210916	−	0.365317i
$683$	24082.4			1.34918			0.674588	−	0.738195i	$-0.264321\pi$
$683$	24082.4			1.34918			0.674588	+	0.738195i	$0.264321\pi$
$684$		0			0
$685$	496.089			0.0276709
$686$	1627.43	+	2818.79i	0.0905766	+	0.156883i
$687$		0			0
$688$	2182.58	−	3780.35i	0.120945	−	0.209483i
$689$	−23281.4	+	40324.5i	−1.28730	+	2.22967i
$690$		0			0
$691$	−6445.10	−	11163.2i	−0.354824	−	0.614573i	0.632264	−	0.774753i	$-0.282126\pi$
$691$	−6445.10	−	11163.2i	−0.354824	−	0.614573i	−0.987088	+	0.160180i	$0.948792\pi$
$692$	−6989.40			−0.383955
$693$		0			0
$694$	−6608.40			−0.361457
$695$	866.776	+	1501.30i	0.0473075	+	0.0819390i
$696$		0			0
$697$	−461.353	+	799.086i	−0.0250717	+	0.0434255i
$698$	4508.71	−	7809.31i	0.244494	−	0.423477i
$699$		0			0
$700$	−594.985	−	1030.54i	−0.0321261	−	0.0556441i
$701$	11887.0			0.640467			0.320233	−	0.947339i	$-0.396239\pi$
$701$	11887.0			0.640467			0.320233	+	0.947339i	$0.396239\pi$
$702$		0			0
$703$	26904.0			1.44339
$704$	1906.22	+	3301.66i	0.102050	+	0.176756i
$705$		0			0
$706$	−6886.74	+	11928.2i	−0.367119	+	0.635868i
$707$	1940.78	−	3361.54i	0.103240	−	0.178817i
$708$		0			0
$709$	−2907.22	−	5035.45i	−0.153996	−	0.266728i	0.778697	−	0.627400i	$-0.215881\pi$
$709$	−2907.22	−	5035.45i	−0.153996	−	0.266728i	−0.932693	+	0.360672i	$0.882547\pi$
$710$	983.217			0.0519711
$711$		0			0
$712$	−627.661			−0.0330373
$713$	3493.12	+	6050.26i	0.183476	+	0.317790i
$714$		0			0
$715$	−1861.42	+	3224.07i	−0.0973611	+	0.168634i
$716$	−4075.42	+	7058.83i	−0.212717	+	0.368437i
$717$		0			0
$718$	1529.31	+	2648.84i	0.0794892	+	0.137679i
$719$	−4488.23			−0.232800			−0.116400	−	0.993202i	$-0.537135\pi$
$719$	−4488.23			−0.232800			−0.116400	+	0.993202i	$0.537135\pi$
$720$		0			0
$721$	1790.00			0.0924590
$722$	7139.52	+	12366.0i	0.368013	+	0.637418i
$723$		0			0
$724$	−5641.35	+	9771.11i	−0.289585	+	0.501575i
$725$	−7811.10	+	13529.2i	−0.400134	+	0.693052i
$726$		0			0
$727$	−11592.0	−	20077.9i	−0.591366	−	1.02428i	−0.994049	−	0.108937i	$-0.965255\pi$
$727$	−11592.0	−	20077.9i	−0.591366	−	1.02428i	0.402682	−	0.915340i	$-0.368078\pi$
$728$	1487.94			0.0757510
$729$		0			0
$730$	1485.24			0.0753029
$731$	−387.726	−	671.561i	−0.0196177	−	0.0339789i
$732$		0			0
$733$	10338.7	−	17907.2i	0.520967	−	0.902341i	−0.478736	−	0.877959i	$-0.658905\pi$
$733$	10338.7	−	17907.2i	0.520967	−	0.902341i	0.999703	−	0.0243822i	$-0.00776185\pi$
$734$	−10231.4	+	17721.2i	−0.514505	+	0.891149i
$735$		0			0
$736$	−1772.55	−	3070.15i	−0.0887734	−	0.153760i
$737$	45812.8			2.28974
$738$		0			0
$739$	−13001.7			−0.647191			−0.323595	−	0.946196i	$-0.604892\pi$
$739$	−13001.7			−0.647191			−0.323595	+	0.946196i	$0.604892\pi$
$740$	−365.578	−	633.199i	−0.0181607	−	0.0314552i
$741$		0			0
$742$	−1432.77	+	2481.63i	−0.0708876	+	0.122781i
$743$	4633.55	−	8025.54i	0.228786	−	0.396270i	−0.728662	−	0.684873i	$-0.759858\pi$
$743$	4633.55	−	8025.54i	0.228786	−	0.396270i	0.957449	+	0.288603i	$0.0931909\pi$
$744$		0			0
$745$	−1435.55	−	2486.44i	−0.0705964	−	0.122277i
$746$	−1892.86			−0.0928988
$747$		0			0
$748$	677.260			0.0331057
$749$	−1716.48	−	2973.03i	−0.0837367	−	0.145036i
$750$		0			0
$751$	15361.1	−	26606.3i	0.746385	−	1.29278i	−0.203159	−	0.979146i	$-0.565121\pi$
$751$	15361.1	−	26606.3i	0.746385	−	1.29278i	0.949545	−	0.313632i	$-0.101546\pi$
$752$	−39.5081	+	68.4300i	−0.00191584	+	0.00331833i
$753$		0			0
$754$	−9767.01	−	16917.0i	−0.471742	−	0.817081i
$755$	1817.85			0.0876267
$756$		0			0
$757$	5356.86			0.257197			0.128599	−	0.991697i	$-0.458952\pi$
$757$	5356.86			0.257197			0.128599	+	0.991697i	$0.458952\pi$
$758$	−8717.46	−	15099.1i	−0.417721	−	0.723514i
$759$		0			0
$760$	380.430	−	658.924i	0.0181574	−	0.0314496i
$761$	5009.51	−	8676.72i	0.238626	−	0.413313i	−0.721694	−	0.692212i	$-0.756636\pi$
$761$	5009.51	−	8676.72i	0.238626	−	0.413313i	0.960320	+	0.278899i	$0.0899696\pi$
$762$		0			0
$763$	−99.4368	−	172.230i	−0.00471802	−	0.00817186i
$764$	−1758.89			−0.0832912
$765$		0			0
$766$	24523.1			1.15673
$767$	−26055.7	−	45129.8i	−1.22662	−	2.12457i
$768$		0			0
$769$	8775.55	−	15199.7i	0.411514	−	0.712764i	−0.583541	−	0.812084i	$-0.698333\pi$
$769$	8775.55	−	15199.7i	0.411514	−	0.712764i	0.995056	+	0.0993198i	$0.0316667\pi$
$770$	−114.554	+	198.414i	−0.00536137	+	0.00928617i
$771$		0			0
$772$	925.353	+	1602.76i	0.0431401	+	0.0747209i
$773$	−17144.5			−0.797731			−0.398866	−	0.917009i	$-0.630596\pi$
$773$	−17144.5			−0.797731			−0.398866	+	0.917009i	$0.630596\pi$
$774$		0			0
$775$	−7841.94			−0.363472
$776$	−364.062	−	630.574i	−0.0168416	−	0.0291705i
$777$		0			0
$778$	13211.1	−	22882.2i	0.608791	−	1.05446i
$779$	−19204.4	+	33263.0i	−0.883272	+	1.52987i
$780$		0			0
$781$	18215.4	+	31549.9i	0.834567	+	1.44551i
$782$	−629.771			−0.0287987
$783$		0			0
$784$	−5396.43			−0.245829
$785$	735.220	+	1273.44i	0.0334282	+	0.0578994i
$786$		0			0
$787$	4297.73	−	7443.89i	0.194660	−	0.337161i	−0.752129	−	0.659016i	$-0.770973\pi$
$787$	4297.73	−	7443.89i	0.194660	−	0.337161i	0.946789	+	0.321855i	$0.104306\pi$
$788$	2073.19	−	3590.88i	0.0937239	−	0.162335i
$789$		0			0
$790$	−31.9993	−	55.4243i	−0.00144112	−	0.00249609i
$791$	2325.79			0.104546
$792$		0			0
$793$	36115.5			1.61727
$794$	−5211.93	−	9027.33i	−0.232953	−	0.403486i
$795$		0			0
$796$	2380.43	−	4123.03i	0.105995	−	0.183589i
$797$	10418.9	−	18046.1i	0.463057	−	0.802039i	−0.536054	−	0.844184i	$-0.680086\pi$
$797$	10418.9	−	18046.1i	0.463057	−	0.802039i	0.999112	+	0.0421447i	$0.0134190\pi$
$798$		0			0
$799$	7.01843	+	12.1563i	0.000310756	+	0.000538245i
$800$	3979.32			0.175863
$801$		0			0
$802$	1971.27			0.0867929
$803$	27515.9	+	47659.0i	1.20924	+	2.09446i
$804$		0			0
$805$	106.522	−	184.502i	0.00466386	−	0.00807805i
$806$	4902.79	−	8491.88i	0.214260	−	0.371109i
$807$		0			0
$808$	6490.09	+	11241.2i	0.282575	+	0.489434i
$809$	−6529.43			−0.283761			−0.141880	−	0.989884i	$-0.545315\pi$
$809$	−6529.43			−0.283761			−0.141880	+	0.989884i	$0.545315\pi$
$810$		0			0
$811$	29764.0			1.28873			0.644363	−	0.764720i	$-0.277123\pi$
$811$	29764.0			1.28873			0.644363	+	0.764720i	$0.277123\pi$
$812$	−601.076	−	1041.09i	−0.0259774	−	0.0449941i
$813$		0			0
$814$	13545.6	−	23461.6i	0.583258	−	1.01023i
$815$	107.938	−	186.954i	0.00463915	−	0.00803525i
$816$		0			0
$817$	−16139.6	−	27954.6i	−0.691129	−	1.19707i
$818$	1349.29			0.0576735
$819$		0			0
$820$	1043.81			0.0444531
$821$	−14785.6	−	25609.4i	−0.628527	−	1.08864i	−0.987847	−	0.155426i	$-0.950325\pi$
$821$	−14785.6	−	25609.4i	−0.628527	−	1.08864i	0.359321	−	0.933214i	$-0.383008\pi$
$822$		0			0
$823$	−3056.22	+	5293.52i	−0.129445	+	0.224205i	−0.923462	−	0.383691i	$-0.874653\pi$
$823$	−3056.22	+	5293.52i	−0.129445	+	0.224205i	0.794017	+	0.607896i	$0.207986\pi$
$824$	−2992.92	+	5183.90i	−0.126533	+	0.219162i
$825$		0			0
$826$	−1603.51	−	2777.35i	−0.0675462	−	0.116993i
$827$	3265.87			0.137322			0.0686612	−	0.997640i	$-0.478127\pi$
$827$	3265.87			0.137322			0.0686612	+	0.997640i	$0.478127\pi$
$828$		0			0
$829$	−19327.8			−0.809747			−0.404874	−	0.914373i	$-0.632685\pi$
$829$	−19327.8			−0.809747			−0.404874	+	0.914373i	$0.632685\pi$
$830$	356.723	+	617.862i	0.0149181	+	0.0258389i
$831$		0			0
$832$	−2487.88	+	4309.13i	−0.103668	+	0.179558i
$833$	−479.325	+	830.215i	−0.0199371	+	0.0345321i
$834$		0			0
$835$	1629.61	+	2822.57i	0.0675389	+	0.116981i
$836$	28191.8			1.16631
$837$		0			0
$838$	−15011.3			−0.618801
$839$	−18819.1	−	32595.6i	−0.774382	−	1.34127i	−0.935141	−	0.354275i	$-0.884728\pi$
$839$	−18819.1	−	32595.6i	−0.774382	−	1.34127i	0.160760	−	0.986994i	$-0.448606\pi$
$840$		0			0
$841$	4303.44	−	7453.77i	0.176450	−	0.305620i
$842$	−1352.62	+	2342.81i	−0.0553617	+	0.0958892i
$843$		0			0
$844$	9856.01	+	17071.1i	0.401964	+	0.696223i
$845$	−3092.77			−0.125911
$846$		0			0
$847$	−5304.92			−0.215206
$848$	−4791.26	−	8298.71i	−0.194024	−	0.336060i
$849$		0			0
$850$	353.454	−	612.200i	0.0142628	−	0.0247039i
$851$	−12595.8	+	21816.5i	−0.507377	+	0.878803i
$852$		0			0
$853$	−20934.8	−	36260.2i	−0.840322	−	1.45548i	−0.889623	−	0.456696i	$-0.849033\pi$
$853$	−20934.8	−	36260.2i	−0.840322	−	1.45548i	0.0493012	−	0.998784i	$-0.484301\pi$
$854$	2222.60			0.0890583
$855$		0			0
$856$	11480.0			0.458386
$857$	−16536.5	−	28642.0i	−0.659131	−	1.14165i	−0.980841	−	0.194810i	$-0.937591\pi$
$857$	−16536.5	−	28642.0i	−0.659131	−	1.14165i	0.321711	−	0.946838i	$-0.395742\pi$
$858$		0			0
$859$	7304.84	−	12652.4i	0.290149	−	0.502553i	−0.683696	−	0.729767i	$-0.739628\pi$
$859$	7304.84	−	12652.4i	0.290149	−	0.502553i	0.973845	+	0.227214i	$0.0729618\pi$
$860$	−438.616	+	759.706i	−0.0173915	+	0.0301230i
$861$		0			0
$862$	−4003.21	−	6933.77i	−0.158179	−	0.273973i
$863$	118.786			0.00468543			0.00234271	−	0.999997i	$-0.499254\pi$
$863$	118.786			0.00468543			0.00234271	+	0.999997i	$0.499254\pi$
$864$		0			0
$865$	1404.60			0.0552115
$866$	9975.38	+	17277.9i	0.391429	+	0.677974i
$867$		0			0
$868$	301.725	−	522.602i	0.0117986	−	0.0204358i
$869$	1185.65	−	2053.61i	0.0462837	−	0.0801658i
$870$		0			0
$871$	29896.1	+	51781.5i	1.16302	+	2.01441i
$872$	665.044			0.0258271
$873$		0			0
$874$	−26215.0			−1.01457
$875$	239.760	+	415.276i	0.00926327	+	0.0160444i
$876$		0			0
$877$	3180.09	−	5508.08i	0.122445	−	0.212080i	−0.798287	−	0.602278i	$-0.794260\pi$
$877$	3180.09	−	5508.08i	0.122445	−	0.212080i	0.920731	+	0.390197i	$0.127593\pi$
$878$	17360.9	−	30070.0i	0.667315	−	1.15582i
$879$		0			0
$880$	−383.077	−	663.508i	−0.0146744	−	0.0254169i
$881$	−32601.5			−1.24673			−0.623367	−	0.781930i	$-0.714236\pi$
$881$	−32601.5			−1.24673			−0.623367	+	0.781930i	$0.714236\pi$
$882$		0			0
$883$	1478.26			0.0563392			0.0281696	−	0.999603i	$-0.491032\pi$
$883$	1478.26			0.0563392			0.0281696	+	0.999603i	$0.491032\pi$
$884$	441.959	+	765.496i	0.0168153	+	0.0291249i
$885$		0			0
$886$	−8575.43	+	14853.1i	−0.325166	+	0.563204i
$887$	22633.6	−	39202.5i	0.856776	−	1.48398i	−0.0182111	−	0.999834i	$-0.505797\pi$
$887$	22633.6	−	39202.5i	0.856776	−	1.48398i	0.874987	−	0.484146i	$-0.160870\pi$
$888$		0			0
$889$	−1574.29	−	2726.75i	−0.0593927	−	0.102871i
$890$	126.136			0.00475066
$891$		0			0
$892$	−5881.88			−0.220785
$893$	292.151	+	506.020i	0.0109479	+	0.0189623i
$894$		0			0
$895$	819.003	−	1418.56i	0.0305880	−	0.0529800i
$896$	−153.108	+	265.190i	−0.00570866	+	0.00988770i
$897$		0			0
$898$	4703.77	+	8147.17i	0.174796	+	0.302756i
$899$	−7922.22			−0.293905
$900$		0			0
$901$	−1702.29			−0.0629428
$902$	19338.0	+	33494.4i	0.713841	+	1.23641i
$903$		0			0
$904$	−3888.78	+	6735.57i	−0.143074	+	0.247812i
$905$	1133.70	−	1963.62i	0.0416413	−	0.0721248i
$906$		0			0
$907$	−3630.76	−	6288.66i	−0.132919	−	0.230222i	0.791882	−	0.610675i	$-0.209102\pi$
$907$	−3630.76	−	6288.66i	−0.132919	−	0.230222i	−0.924801	+	0.380452i	$0.875768\pi$
$908$	−8409.04			−0.307339
$909$		0			0
$910$	−299.019			−0.0108927
$911$	−9717.20	−	16830.7i	−0.353398	−	0.612103i	0.633445	−	0.773788i	$-0.281640\pi$
$911$	−9717.20	−	16830.7i	−0.353398	−	0.612103i	−0.986842	+	0.161685i	$0.948307\pi$
$912$		0			0
$913$	−13217.5	+	22893.4i	−0.479118	+	0.829857i
$914$	−7046.80	+	12205.4i	−0.255019	+	0.441706i
$915$		0			0
$916$	2002.08	+	3467.70i	0.0722166	+	0.125083i
$917$	−2668.08			−0.0960827
$918$		0			0
$919$	−24010.2			−0.861830			−0.430915	−	0.902392i	$-0.641809\pi$
$919$	−24010.2			−0.861830			−0.430915	+	0.902392i	$0.641809\pi$
$920$	356.216	+	616.984i	0.0127653	+	0.0221102i
$921$		0			0
$922$	5770.26	−	9994.38i	0.206110	−	0.356993i
$923$	−23773.6	+	41177.0i	−0.847797	+	1.46843i
$924$		0			0
$925$	−14138.6	−	24488.7i	−0.502565	−	0.870468i
$926$	27198.3			0.965217
$927$		0			0
$928$	4020.06			0.142204
$929$	20777.3	+	35987.4i	0.733780	+	1.27094i	0.955257	+	0.295778i	$0.0955790\pi$
$929$	20777.3	+	35987.4i	0.733780	+	1.27094i	−0.221477	+	0.975166i	$0.571088\pi$
$930$		0			0
$931$	−19952.5	+	34558.8i	−0.702382	+	1.21656i
$932$	−12561.6	+	21757.3i	−0.441489	+	0.764681i
$933$		0			0
$934$	5730.52	+	9925.55i	0.200758	+	0.347724i
$935$	−136.103			−0.00476049
$936$		0			0
$937$	16811.0			0.586116			0.293058	−	0.956095i	$-0.405327\pi$
$937$	16811.0			0.586116			0.293058	+	0.956095i	$0.405327\pi$
$938$	1839.85	+	3186.71i	0.0640439	+	0.110927i
$939$		0			0
$940$	7.93962	−	13.7518i	0.000275491	−	0.000477165i
$941$	−5366.43	+	9294.93i	−0.185909	+	0.322004i	−0.943883	−	0.330281i	$-0.892856\pi$
$941$	−5366.43	+	9294.93i	−0.185909	+	0.322004i	0.757973	+	0.652286i	$0.226190\pi$
$942$		0			0
$943$	−17982.0	−	31145.8i	−0.620971	−	1.07555i
$944$	10724.4			0.369757
$945$		0			0
$946$	−32503.7			−1.11711
$947$	15774.7	+	27322.5i	0.541297	+	0.937553i	0.998830	+	0.0483606i	$0.0153997\pi$
$947$	15774.7	+	27322.5i	0.541297	+	0.937553i	−0.457533	+	0.889192i	$0.651267\pi$
$948$		0			0
$949$	−35912.1	+	62201.6i	−1.22841	+	2.12766i
$950$	14713.0	−	25483.6i	0.502475	−	0.870313i
$951$		0			0
$952$	27.1988	+	47.1097i	0.000925965	+	0.00160382i
$953$	−31955.2			−1.08618			−0.543090	−	0.839675i	$-0.682746\pi$
$953$	−31955.2			−1.08618			−0.543090	+	0.839675i	$0.682746\pi$
$954$		0			0
$955$	353.470			0.0119770
$956$	−2034.74	−	3524.27i	−0.0688370	−	0.119229i
$957$		0			0
$958$	14568.3	−	25233.0i	0.491315	−	0.850982i
$959$	−738.197	+	1278.59i	−0.0248567	+	0.0430531i
$960$		0			0
$961$	12907.1	+	22355.8i	0.433256	+	0.750421i
$962$	35357.7			1.18501
$963$		0			0
$964$	−3685.17			−0.123124
$965$	−185.961	−	322.093i	−0.00620340	−	0.0107446i
$966$		0			0
$967$	−19186.1	+	33231.3i	−0.638038	+	1.10511i	0.347825	+	0.937560i	$0.386920\pi$
$967$	−19186.1	+	33231.3i	−0.638038	+	1.10511i	−0.985863	+	0.167555i	$0.946413\pi$
$968$	8869.97	−	15363.2i	0.294516	−	0.510117i
$969$		0			0
$970$	73.1626	+	126.721i	0.00242176	+	0.00419461i
$971$	−46445.9			−1.53503			−0.767517	−	0.641028i	$-0.778508\pi$
$971$	−46445.9			−1.53503			−0.767517	+	0.641028i	$0.778508\pi$
$972$		0			0
$973$	−5159.17			−0.169985
$974$	7164.51	+	12409.3i	0.235694	+	0.408233i
$975$		0			0
$976$	−3716.25	+	6436.73i	−0.121879	+	0.211101i
$977$	23954.1	−	41489.8i	0.784402	−	1.35862i	−0.144954	−	0.989438i	$-0.546303\pi$
$977$	23954.1	−	41489.8i	0.784402	−	1.35862i	0.929356	−	0.369185i	$-0.120363\pi$
$978$		0			0
$979$	2336.83	+	4047.50i	0.0762874	+	0.132134i
$980$	1084.48			0.0353493
$981$		0			0
$982$	19020.1			0.618082
$983$	20273.8	+	35115.3i	0.657818	+	1.13937i	0.981179	+	0.193099i	$0.0618538\pi$
$983$	20273.8	+	35115.3i	0.657818	+	1.13937i	−0.323361	+	0.946276i	$0.604813\pi$
$984$		0			0
$985$	−416.633	+	721.629i	−0.0134772	+	0.0233432i
$986$	357.072	−	618.467i	0.0115330	−	0.0199757i
$987$		0			0
$988$	18397.1	+	31864.8i	0.592399	+	1.02607i
$989$	30224.6			0.971776
$990$		0			0
$991$	−10651.5			−0.341428			−0.170714	−	0.985321i	$-0.554607\pi$
$991$	−10651.5			−0.341428			−0.170714	+	0.985321i	$0.554607\pi$
$992$	1008.98	+	1747.61i	0.0322936	+	0.0559342i
$993$		0			0
$994$	−1463.06	+	2534.09i	−0.0466856	+	0.0808618i
$995$	−478.376	+	828.571i	−0.0152417	+	0.0263995i
$996$		0			0
$997$	9379.18	+	16245.2i	0.297935	+	0.516039i	0.975663	−	0.219273i	$-0.0703687\pi$
$997$	9379.18	+	16245.2i	0.297935	+	0.516039i	−0.677728	+	0.735313i	$0.737035\pi$
$998$	−42971.2			−1.36296
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.c.j.109.1		4
3.2	odd	2		162.4.c.i.109.2		4
9.2	odd	6		162.4.c.i.55.2		4
9.4	even	3		162.4.a.e.1.2	✓	2
9.5	odd	6		162.4.a.h.1.1	yes	2
9.7	even	3	inner	162.4.c.j.55.1		4
36.23	even	6		1296.4.a.s.1.1		2
36.31	odd	6		1296.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.a.e.1.2	✓	2	9.4	even	3
162.4.a.h.1.1	yes	2	9.5	odd	6
162.4.c.i.55.2		4	9.2	odd	6
162.4.c.i.109.2		4	3.2	odd	2
162.4.c.j.55.1		4	9.7	even	3	inner
162.4.c.j.109.1		4	1.1	even	1	trivial
1296.4.a.j.1.2		2	36.31	odd	6
1296.4.a.s.1.1		2	36.23	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(0.401924 - 0.696152i) q^{5} +(1.19615 + 2.07180i) q^{7} -8.00000 q^{8} +1.60770 q^{10} +(29.7846 + 51.5885i) q^{11} +(-38.8731 + 67.3301i) q^{13} +(-2.39230 + 4.14359i) q^{14} +(-8.00000 - 13.8564i) q^{16} -2.84232 q^{17} -118.315 q^{19} +(1.60770 + 2.78461i) q^{20} +(-59.5692 + 103.177i) q^{22} +(55.3923 - 95.9423i) q^{23} +(62.1769 + 107.694i) q^{25} -155.492 q^{26} -9.56922 q^{28} +(62.8135 + 108.796i) q^{29} +(-31.5307 + 54.6128i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-2.84232 - 4.92305i) q^{34} +1.92305 q^{35} -227.392 q^{37} +(-118.315 - 204.928i) q^{38} +(-3.21539 + 5.56922i) q^{40} +(162.315 - 281.138i) q^{41} +(136.412 + 236.272i) q^{43} -238.277 q^{44} +221.569 q^{46} +(-2.46926 - 4.27688i) q^{47} +(168.638 - 292.090i) q^{49} +(-124.354 + 215.387i) q^{50} +(-155.492 - 269.321i) q^{52} +598.908 q^{53} +47.8846 q^{55} +(-9.56922 - 16.5744i) q^{56} +(-125.627 + 217.592i) q^{58} +(-335.138 + 580.477i) q^{59} +(-232.265 - 402.295i) q^{61} -126.123 q^{62} +64.0000 q^{64} +(31.2480 + 54.1232i) q^{65} +(384.535 - 666.033i) q^{67} +(5.68465 - 9.84610i) q^{68} +(1.92305 + 3.33082i) q^{70} +611.569 q^{71} +923.831 q^{73} +(-227.392 - 393.855i) q^{74} +(236.631 - 409.856i) q^{76} +(-71.2539 + 123.415i) q^{77} +(-19.9038 - 34.4744i) q^{79} -12.8616 q^{80} +649.261 q^{82} +(221.885 + 384.315i) q^{83} +(-1.14240 + 1.97869i) q^{85} +(-272.823 + 472.543i) q^{86} +(-238.277 - 412.708i) q^{88} +78.4576 q^{89} -185.992 q^{91} +(221.569 + 383.769i) q^{92} +(4.93851 - 8.55376i) q^{94} +(-47.5538 + 82.3655i) q^{95} +(45.5077 + 78.8217i) q^{97} +674.554 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{4} + 12 q^{5} - 16 q^{7} - 32 q^{8} + 48 q^{10} + 36 q^{11} - 10 q^{13} + 32 q^{14} - 32 q^{16} - 240 q^{17} - 16 q^{19} + 48 q^{20} - 72 q^{22} + 180 q^{23} + 124 q^{25} - 40 q^{26}+ \cdots + 1368 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^4 + 12 * q^5 - 16 * q^7 - 32 * q^8 + 48 * q^10 + 36 * q^11 - 10 * q^13 + 32 * q^14 - 32 * q^16 - 240 * q^17 - 16 * q^19 + 48 * q^20 - 72 * q^22 + 180 * q^23 + 124 * q^25 - 40 * q^26 + 128 * q^28 + 324 * q^29 + 248 * q^31 + 64 * q^32 - 240 * q^34 - 408 * q^35 - 868 * q^37 - 16 * q^38 - 96 * q^40 + 192 * q^41 + 608 * q^43 - 288 * q^44 + 720 * q^46 - 384 * q^47 + 342 * q^49 - 248 * q^50 - 40 * q^52 + 816 * q^53 - 432 * q^55 + 128 * q^56 - 648 * q^58 - 1008 * q^59 - 742 * q^61 + 992 * q^62 + 256 * q^64 - 696 * q^65 + 104 * q^67 + 480 * q^68 - 408 * q^70 + 2280 * q^71 + 1700 * q^73 - 868 * q^74 + 32 * q^76 - 576 * q^77 + 440 * q^79 - 384 * q^80 + 768 * q^82 + 264 * q^83 - 1314 * q^85 - 1216 * q^86 - 288 * q^88 - 1536 * q^89 - 2864 * q^91 + 720 * q^92 + 768 * q^94 + 1140 * q^95 + 764 * q^97 + 1368 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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