LMFDB - Embedded newform 144.4.k.c.37.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [144,4,Mod(37,144)] 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("144.37"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	144.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$8.49627504083$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.6
Character	$\chi$	$=$	144.37
Dual form			144.4.k.c.109.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.639782 - 2.75512i) q^{2} +(-7.18136 + 3.52535i) q^{4} +(5.16526 + 5.16526i) q^{5} -7.03833i q^{7} +(14.3073 + 17.5300i) q^{8} +(10.9263 - 17.5356i) q^{10} +(22.2198 + 22.2198i) q^{11} +(37.2029 - 37.2029i) q^{13} +(-19.3914 + 4.50300i) q^{14} +(39.1438 - 50.6336i) q^{16} +67.4608 q^{17} +(37.2366 - 37.2366i) q^{19} +(-55.3030 - 18.8842i) q^{20} +(47.0024 - 75.4340i) q^{22} +32.4348i q^{23} -71.6401i q^{25} +(-126.300 - 78.6966i) q^{26} +(24.8126 + 50.5448i) q^{28} +(94.7734 - 94.7734i) q^{29} -158.503 q^{31} +(-164.545 - 75.4513i) q^{32} +(-43.1602 - 185.863i) q^{34} +(36.3548 - 36.3548i) q^{35} +(192.501 + 192.501i) q^{37} +(-126.414 - 78.7678i) q^{38} +(-16.6465 + 164.448i) q^{40} +283.277i q^{41} +(-206.310 - 206.310i) q^{43} +(-237.901 - 81.2357i) q^{44} +(89.3617 - 20.7512i) q^{46} +584.883 q^{47} +293.462 q^{49} +(-197.377 + 45.8341i) q^{50} +(-136.014 + 398.320i) q^{52} +(82.5251 + 82.5251i) q^{53} +229.542i q^{55} +(123.382 - 100.699i) q^{56} +(-321.746 - 200.478i) q^{58} +(79.7567 + 79.7567i) q^{59} +(132.394 - 132.394i) q^{61} +(101.408 + 436.696i) q^{62} +(-102.604 + 501.614i) q^{64} +384.325 q^{65} +(-596.157 + 596.157i) q^{67} +(-484.460 + 237.823i) q^{68} +(-123.421 - 76.9027i) q^{70} -110.487i q^{71} +412.677i q^{73} +(407.204 - 653.522i) q^{74} +(-136.137 + 398.681i) q^{76} +(156.390 - 156.390i) q^{77} -1142.41 q^{79} +(463.724 - 59.3480i) q^{80} +(780.461 - 181.235i) q^{82} +(-656.133 + 656.133i) q^{83} +(348.453 + 348.453i) q^{85} +(-436.414 + 700.401i) q^{86} +(-71.6093 + 707.419i) q^{88} +474.926i q^{89} +(-261.846 - 261.846i) q^{91} +(-114.344 - 232.926i) q^{92} +(-374.198 - 1611.42i) q^{94} +384.673 q^{95} +1358.31 q^{97} +(-187.752 - 808.522i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 20 q^{4} + 48 q^{10} - 192 q^{16} + 24 q^{19} + 232 q^{22} + 416 q^{28} + 744 q^{31} + 296 q^{34} - 16 q^{37} + 104 q^{40} - 376 q^{43} - 32 q^{46} - 1176 q^{49} - 2088 q^{52} - 808 q^{58} - 912 q^{61}+ \cdots - 2896 q^{94}+O(q^{100}) $ 24 * q + 20 * q^4 + 48 * q^10 - 192 * q^16 + 24 * q^19 + 232 * q^22 + 416 * q^28 + 744 * q^31 + 296 * q^34 - 16 * q^37 + 104 * q^40 - 376 * q^43 - 32 * q^46 - 1176 * q^49 - 2088 * q^52 - 808 * q^58 - 912 * q^61 - 2968 * q^64 + 1440 * q^67 + 2408 * q^70 + 2408 * q^76 - 328 * q^79 + 5800 * q^82 - 240 * q^85 + 2160 * q^88 - 104 * q^91 - 2896 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$	−0.639782	−	2.75512i	−0.226197	−	0.974082i
$2$	−0.639782	−	2.75512i	−0.226197	−	0.974082i
$3$		0			0
$3$		0			0
$4$	−7.18136	+	3.52535i	−0.897670	+	0.440669i
$5$	5.16526	+	5.16526i	0.461995	+	0.461995i	0.899309	−	0.437314i	$-0.144070\pi$
$5$	5.16526	+	5.16526i	0.461995	+	0.461995i	−0.437314	+	0.899309i	$0.644070\pi$
$6$		0			0
$7$		−	7.03833i		−	0.380034i	−0.981781	−	0.190017i	$-0.939146\pi$
$7$		−	7.03833i		−	0.380034i	0.981781	−	0.190017i	$-0.0608543\pi$
$8$	14.3073	+	17.5300i	0.632298	+	0.774725i
$9$		0			0
$10$	10.9263	−	17.5356i	0.345519	−	0.554523i
$11$	22.2198	+	22.2198i	0.609047	+	0.609047i	0.942697	−	0.333650i	$-0.108280\pi$
$11$	22.2198	+	22.2198i	0.609047	+	0.609047i	−0.333650	+	0.942697i	$0.608280\pi$
$12$		0			0
$13$	37.2029	−	37.2029i	0.793710	−	0.793710i	−0.188386	−	0.982095i	$-0.560325\pi$
$13$	37.2029	−	37.2029i	0.793710	−	0.793710i	0.982095	+	0.188386i	$0.0603255\pi$
$14$	−19.3914	+	4.50300i	−0.370184	+	0.0859626i
$15$		0			0
$16$	39.1438	−	50.6336i	0.611622	−	0.791150i
$17$	67.4608			0.962450			0.481225	−	0.876597i	$-0.340192\pi$
$17$	67.4608			0.962450			0.481225	+	0.876597i	$0.340192\pi$
$18$		0			0
$19$	37.2366	−	37.2366i	0.449613	−	0.449613i	−0.445613	−	0.895226i	$-0.647014\pi$
$19$	37.2366	−	37.2366i	0.449613	−	0.449613i	0.895226	+	0.445613i	$0.147014\pi$
$20$	−55.3030	−	18.8842i	−0.618306	−	0.211132i
$21$		0			0
$22$	47.0024	−	75.4340i	0.455497	−	0.731027i
$23$			32.4348i			0.294049i	0.989133	+	0.147024i	$0.0469696\pi$
$23$			32.4348i			0.294049i	−0.989133	+	0.147024i	$0.953030\pi$
$24$		0			0
$25$		−	71.6401i		−	0.573121i
$26$	−126.300	−	78.6966i	−0.952673	−	0.593603i
$27$		0			0
$28$	24.8126	+	50.5448i	0.167469	+	0.341145i
$29$	94.7734	−	94.7734i	0.606861	−	0.606861i	−0.335263	−	0.942125i	$-0.608825\pi$
$29$	94.7734	−	94.7734i	0.606861	−	0.606861i	0.942125	+	0.335263i	$0.108825\pi$
$30$		0			0
$31$	−158.503			−0.918324			−0.459162	−	0.888353i	$-0.651850\pi$
$31$	−158.503			−0.918324			−0.459162	+	0.888353i	$0.651850\pi$
$32$	−164.545	−	75.4513i	−0.908992	−	0.416814i
$33$		0			0
$34$	−43.1602	−	185.863i	−0.217703	−	0.937505i
$35$	36.3548	−	36.3548i	0.175574	−	0.175574i
$36$		0			0
$37$	192.501	+	192.501i	0.855323	+	0.855323i	0.990783	−	0.135460i	$-0.0432511\pi$
$37$	192.501	+	192.501i	0.855323	+	0.855323i	−0.135460	+	0.990783i	$0.543251\pi$
$38$	−126.414	−	78.7678i	−0.539661	−	0.336259i
$39$		0			0
$40$	−16.6465	+	164.448i	−0.0658009	+	0.650038i
$41$			283.277i			1.07903i	0.841975	+	0.539517i	$0.181393\pi$
$41$			283.277i			1.07903i	−0.841975	+	0.539517i	$0.818607\pi$
$42$		0			0
$43$	−206.310	−	206.310i	−0.731672	−	0.731672i	0.239279	−	0.970951i	$-0.423089\pi$
$43$	−206.310	−	206.310i	−0.731672	−	0.731672i	−0.970951	+	0.239279i	$0.923089\pi$
$44$	−237.901	−	81.2357i	−0.815112	−	0.278335i
$45$		0			0
$46$	89.3617	−	20.7512i	0.286427	−	0.0665130i
$47$	584.883			1.81519			0.907596	−	0.419845i	$-0.137915\pi$
$47$	584.883			1.81519			0.907596	+	0.419845i	$0.137915\pi$
$48$		0			0
$49$	293.462			0.855574
$50$	−197.377	+	45.8341i	−0.558266	+	0.129638i
$51$		0			0
$52$	−136.014	+	398.320i	−0.362726	+	1.06225i
$53$	82.5251	+	82.5251i	0.213881	+	0.213881i	0.805914	−	0.592033i	$-0.201674\pi$
$53$	82.5251	+	82.5251i	0.213881	+	0.213881i	−0.592033	+	0.805914i	$0.701674\pi$
$54$		0			0
$55$			229.542i			0.562754i
$56$	123.382	−	100.699i	0.294422	−	0.240295i
$57$		0			0
$58$	−321.746	−	200.478i	−0.728403	−	0.453862i
$59$	79.7567	+	79.7567i	0.175990	+	0.175990i	0.789605	−	0.613615i	$-0.210285\pi$
$59$	79.7567	+	79.7567i	0.175990	+	0.175990i	−0.613615	+	0.789605i	$0.710285\pi$
$60$		0			0
$61$	132.394	−	132.394i	0.277889	−	0.277889i	−0.554377	−	0.832266i	$-0.687043\pi$
$61$	132.394	−	132.394i	0.277889	−	0.277889i	0.832266	+	0.554377i	$0.187043\pi$
$62$	101.408	+	436.696i	0.207722	+	0.894522i
$63$		0			0
$64$	−102.604	+	501.614i	−0.200399	+	0.979714i
$65$	384.325			0.733380
$66$		0			0
$67$	−596.157	+	596.157i	−1.08705	+	1.08705i	−0.0912161	+	0.995831i	$0.529075\pi$
$67$	−596.157	+	596.157i	−1.08705	+	1.08705i	−0.995831	+	0.0912161i	$0.970925\pi$
$68$	−484.460	+	237.823i	−0.863962	+	0.424122i
$69$		0			0
$70$	−123.421	−	76.9027i	−0.210738	−	0.131309i
$71$		−	110.487i		−	0.184682i	−0.995727	−	0.0923409i	$-0.970565\pi$
$71$		−	110.487i		−	0.184682i	0.995727	−	0.0923409i	$-0.0294350\pi$
$72$		0			0
$73$			412.677i			0.661646i	0.943693	+	0.330823i	$0.107326\pi$
$73$			412.677i			0.661646i	−0.943693	+	0.330823i	$0.892674\pi$
$74$	407.204	−	653.522i	0.639683	−	1.02663i
$75$		0			0
$76$	−136.137	+	398.681i	−0.205474	+	0.601735i
$77$	156.390	−	156.390i	0.231459	−	0.231459i
$78$		0			0
$79$	−1142.41			−1.62698			−0.813491	−	0.581577i	$-0.802436\pi$
$79$	−1142.41			−1.62698			−0.813491	+	0.581577i	$0.802436\pi$
$80$	463.724	−	59.3480i	0.648074	−	0.0829413i
$81$		0			0
$82$	780.461	−	181.235i	1.05107	−	0.244074i
$83$	−656.133	+	656.133i	−0.867711	+	0.867711i	−0.992219	−	0.124508i	$-0.960265\pi$
$83$	−656.133	+	656.133i	−0.867711	+	0.867711i	0.124508	+	0.992219i	$0.460265\pi$
$84$		0			0
$85$	348.453	+	348.453i	0.444647	+	0.444647i
$86$	−436.414	+	700.401i	−0.547206	+	0.878211i
$87$		0			0
$88$	−71.6093	+	707.419i	−0.0867452	+	0.856944i
$89$			474.926i			0.565641i	0.959173	+	0.282820i	$0.0912700\pi$
$89$			474.926i			0.565641i	−0.959173	+	0.282820i	$0.908730\pi$
$90$		0			0
$91$	−261.846	−	261.846i	−0.301637	−	0.301637i
$92$	−114.344	−	232.926i	−0.129578	−	0.263959i
$93$		0			0
$94$	−374.198	−	1611.42i	−0.410591	−	1.76814i
$95$	384.673			0.415438
$96$		0			0
$97$	1358.31			1.42181			0.710906	−	0.703287i	$-0.248285\pi$
$97$	1358.31			1.42181			0.710906	+	0.703287i	$0.248285\pi$
$98$	−187.752	−	808.522i	−0.193528	−	0.833399i
$99$		0			0
$100$	252.557	+	514.473i	0.252557	+	0.514473i
$101$	−1345.17	−	1345.17i	−1.32524	−	1.32524i	−0.909470	−	0.415771i	$-0.863512\pi$
$101$	−1345.17	−	1345.17i	−1.32524	−	1.32524i	−0.415771	−	0.909470i	$-0.636488\pi$
$102$		0			0
$103$		−	1077.73i		−	1.03099i	−0.856894	−	0.515493i	$-0.827609\pi$
$103$		−	1077.73i		−	1.03099i	0.856894	−	0.515493i	$-0.172391\pi$
$104$	1184.44	+	119.896i	1.11677	+	0.113046i
$105$		0			0
$106$	174.568	−	280.165i	0.159958	−	0.256717i
$107$	−1505.13	−	1505.13i	−1.35988	−	1.35988i	−0.874061	−	0.485816i	$-0.838523\pi$
$107$	−1505.13	−	1505.13i	−1.35988	−	1.35988i	−0.485816	−	0.874061i	$-0.661477\pi$
$108$		0			0
$109$	885.884	−	885.884i	0.778462	−	0.778462i	−0.201107	−	0.979569i	$-0.564454\pi$
$109$	885.884	−	885.884i	0.778462	−	0.778462i	0.979569	+	0.201107i	$0.0644540\pi$
$110$	632.416	−	146.857i	0.548168	−	0.127293i
$111$		0			0
$112$	−356.376	−	275.507i	−0.300664	−	0.232437i
$113$	−1002.89			−0.834902			−0.417451	−	0.908699i	$-0.637076\pi$
$113$	−1002.89			−0.834902			−0.417451	+	0.908699i	$0.637076\pi$
$114$		0			0
$115$	−167.534	+	167.534i	−0.135849	+	0.135849i
$116$	−346.492	+	1014.71i	−0.277336	+	0.812186i
$117$		0			0
$118$	168.712	−	270.766i	0.131621	−	0.211238i
$119$		−	474.812i		−	0.365764i
$120$		0			0
$121$		−	343.561i		−	0.258122i
$122$	−449.463	−	280.057i	−0.333545	−	0.207829i
$123$		0			0
$124$	1138.27	−	558.780i	0.824352	−	0.404677i
$125$	1015.70	−	1015.70i	0.726774	−	0.726774i
$126$		0			0
$127$	657.435			0.459354			0.229677	−	0.973267i	$-0.426233\pi$
$127$	657.435			0.459354			0.229677	+	0.973267i	$0.426233\pi$
$128$	1447.65	−	38.2366i	0.999651	−	0.0264037i
$129$		0			0
$130$	−245.885	−	1058.86i	−0.165888	−	0.714372i
$131$	−1708.93	+	1708.93i	−1.13977	+	1.13977i	−0.151279	+	0.988491i	$0.548339\pi$
$131$	−1708.93	+	1708.93i	−1.13977	+	1.13977i	−0.988491	+	0.151279i	$0.951661\pi$
$132$		0			0
$133$	−262.083	−	262.083i	−0.170868	−	0.170868i
$134$	2023.89	+	1261.07i	1.30476	+	0.812986i
$135$		0			0
$136$	965.180	+	1182.59i	0.608555	+	0.745635i
$137$			2990.96i			1.86522i	0.360891	+	0.932608i	$0.382473\pi$
$137$			2990.96i			1.86522i	−0.360891	+	0.932608i	$0.617527\pi$
$138$		0			0
$139$	−898.789	−	898.789i	−0.548448	−	0.548448i	0.377543	−	0.925992i	$-0.376769\pi$
$139$	−898.789	−	898.789i	−0.548448	−	0.548448i	−0.925992	+	0.377543i	$0.876769\pi$
$140$	−132.914	+	389.241i	−0.0802374	+	0.234977i
$141$		0			0
$142$	−304.405	+	70.6877i	−0.179895	+	0.0417745i
$143$	1653.28			0.966814
$144$		0			0
$145$	979.059			0.560734
$146$	1136.97	−	264.023i	0.644497	−	0.149662i
$147$		0			0
$148$	−2061.05	−	703.785i	−1.14471	−	0.390883i
$149$	−1949.10	−	1949.10i	−1.07165	−	1.07165i	−0.997227	−	0.0744264i	$-0.976287\pi$
$149$	−1949.10	−	1949.10i	−1.07165	−	1.07165i	−0.0744264	−	0.997227i	$-0.523713\pi$
$150$		0			0
$151$			389.789i			0.210070i	0.994469	+	0.105035i	$0.0334955\pi$
$151$			389.789i			0.210070i	−0.994469	+	0.105035i	$0.966505\pi$
$152$	1185.51	+	120.005i	0.632616	+	0.0640373i
$153$		0			0
$154$	−530.930	−	330.818i	−0.277815	−	0.173104i
$155$	−818.712	−	818.712i	−0.424261	−	0.424261i
$156$		0			0
$157$	−573.333	+	573.333i	−0.291446	+	0.291446i	−0.837651	−	0.546206i	$-0.816072\pi$
$157$	−573.333	+	573.333i	−0.291446	+	0.291446i	0.546206	+	0.837651i	$0.316072\pi$
$158$	730.896	+	3147.49i	0.368019	+	1.58481i
$159$		0			0
$160$	−460.193	−	1239.64i	−0.227384	−	0.612516i
$161$	228.287			0.111749
$162$		0			0
$163$	94.7841	−	94.7841i	0.0455464	−	0.0455464i	−0.683967	−	0.729513i	$-0.739747\pi$
$163$	94.7841	−	94.7841i	0.0455464	−	0.0455464i	0.729513	+	0.683967i	$0.239747\pi$
$164$	−998.650	−	2034.31i	−0.475497	−	0.968616i
$165$		0			0
$166$	2227.51	+	1387.94i	1.04149	+	0.648947i
$167$		−	396.734i		−	0.183834i	−0.995767	−	0.0919169i	$-0.970701\pi$
$167$		−	396.734i		−	0.183834i	0.995767	−	0.0919169i	$-0.0292994\pi$
$168$		0			0
$169$		−	571.109i		−	0.259950i
$170$	737.095	−	1182.96i	0.332545	−	0.533701i
$171$		0			0
$172$	2208.90	+	754.269i	0.979225	+	0.334375i
$173$	−193.034	+	193.034i	−0.0848329	+	0.0848329i	−0.748250	−	0.663417i	$-0.769106\pi$
$173$	−193.034	+	193.034i	−0.0848329	+	0.0848329i	0.663417	+	0.748250i	$0.269106\pi$
$174$		0			0
$175$	−504.227			−0.217805
$176$	1994.84	−	255.302i	0.854355	−	0.109341i
$177$		0			0
$178$	1308.48	−	303.849i	0.550980	−	0.127946i
$179$	−114.931	+	114.931i	−0.0479909	+	0.0479909i	−0.730695	−	0.682704i	$-0.760804\pi$
$179$	−114.931	+	114.931i	−0.0479909	+	0.0479909i	0.682704	+	0.730695i	$0.260804\pi$
$180$		0			0
$181$	764.431	+	764.431i	0.313921	+	0.313921i	0.846427	−	0.532505i	$-0.178749\pi$
$181$	764.431	+	764.431i	0.313921	+	0.313921i	−0.532505	+	0.846427i	$0.678749\pi$
$182$	−553.893	+	888.942i	−0.225589	+	0.362048i
$183$		0			0
$184$	−568.583	+	464.053i	−0.227807	+	0.185926i
$185$			1988.64i			0.790310i
$186$		0			0
$187$	1498.97	+	1498.97i	0.586178	+	0.586178i
$188$	−4200.26	+	2061.92i	−1.62944	+	0.799899i
$189$		0			0
$190$	−246.107	−	1059.82i	−0.0939710	−	0.404671i
$191$	2413.08			0.914159			0.457080	−	0.889426i	$-0.348895\pi$
$191$	2413.08			0.914159			0.457080	+	0.889426i	$0.348895\pi$
$192$		0			0
$193$	−4996.88			−1.86364			−0.931822	−	0.362915i	$-0.881781\pi$
$193$	−4996.88			−1.86364			−0.931822	+	0.362915i	$0.881781\pi$
$194$	−869.025	−	3742.32i	−0.321610	−	1.38496i
$195$		0			0
$196$	−2107.45	+	1034.56i	−0.768023	+	0.377025i
$197$	−1208.70	−	1208.70i	−0.437138	−	0.437138i	0.453910	−	0.891048i	$-0.350029\pi$
$197$	−1208.70	−	1208.70i	−0.437138	−	0.437138i	−0.891048	+	0.453910i	$0.850029\pi$
$198$		0			0
$199$			3020.23i			1.07587i	0.842986	+	0.537935i	$0.180795\pi$
$199$			3020.23i			1.07587i	−0.842986	+	0.537935i	$0.819205\pi$
$200$	1255.85	−	1024.97i	0.444011	−	0.362383i
$201$		0			0
$202$	−2845.48	+	4566.71i	−0.991126	+	1.59066i
$203$	−667.047	−	667.047i	−0.230628	−	0.230628i
$204$		0			0
$205$	−1463.20	+	1463.20i	−0.498508	+	0.498508i
$206$	−2969.27	+	689.510i	−1.00426	+	0.233206i
$207$		0			0
$208$	−427.455	−	3339.98i	−0.142494	−	1.11339i
$209$	1654.78			0.547672
$210$		0			0
$211$	1874.16	−	1874.16i	0.611482	−	0.611482i	−0.331850	−	0.943332i	$-0.607673\pi$
$211$	1874.16	−	1874.16i	0.611482	−	0.611482i	0.943332	+	0.331850i	$0.107673\pi$
$212$	−883.572	−	301.712i	−0.286245	−	0.0977438i
$213$		0			0
$214$	−3183.87	+	5109.78i	−1.01703	+	1.63223i
$215$		−	2131.29i		−	0.676058i
$216$		0			0
$217$			1115.60i			0.348994i
$218$	−3007.49	−	1873.94i	−0.934371	−	0.582199i
$219$		0			0
$220$	−809.217	−	1648.43i	−0.247988	−	0.505167i
$221$	2509.74	−	2509.74i	0.763906	−	0.763906i
$222$		0			0
$223$	974.233			0.292554			0.146277	−	0.989244i	$-0.453271\pi$
$223$	974.233			0.292554			0.146277	+	0.989244i	$0.453271\pi$
$224$	−531.051	+	1158.12i	−0.158403	+	0.345448i
$225$		0			0
$226$	641.631	+	2763.08i	0.188852	+	0.813263i
$227$	−3640.99	+	3640.99i	−1.06459	+	1.06459i	−0.0668201	+	0.997765i	$0.521285\pi$
$227$	−3640.99	+	3640.99i	−1.06459	+	1.06459i	−0.997765	+	0.0668201i	$0.978715\pi$
$228$		0			0
$229$	−1480.42	−	1480.42i	−0.427202	−	0.427202i	0.460472	−	0.887674i	$-0.347680\pi$
$229$	−1480.42	−	1480.42i	−0.427202	−	0.427202i	−0.887674	+	0.460472i	$0.847680\pi$
$230$	568.762	+	354.391i	0.163057	+	0.101599i
$231$		0			0
$232$	3017.33	+	305.433i	0.853868	+	0.0864338i
$233$			1623.75i			0.456545i	0.973597	+	0.228273i	$0.0733078\pi$
$233$			1623.75i			0.456545i	−0.973597	+	0.228273i	$0.926692\pi$
$234$		0			0
$235$	3021.08	+	3021.08i	0.838610	+	0.838610i
$236$	−853.932	−	291.591i	−0.235535	−	0.0804278i
$237$		0			0
$238$	−1308.16	+	303.776i	−0.356284	+	0.0827348i
$239$	6868.93			1.85905			0.929527	−	0.368753i	$-0.120215\pi$
$239$	6868.93			1.85905			0.929527	+	0.368753i	$0.120215\pi$
$240$		0			0
$241$	−2767.43			−0.739693			−0.369847	−	0.929093i	$-0.620590\pi$
$241$	−2767.43			−0.739693			−0.369847	+	0.929093i	$0.620590\pi$
$242$	−946.551	+	219.804i	−0.251432	+	0.0583865i
$243$		0			0
$244$	−484.032	+	1417.50i	−0.126996	+	0.371910i
$245$	1515.81	+	1515.81i	0.395271	+	0.395271i
$246$		0			0
$247$		−	2770.61i		−	0.713725i
$248$	−2267.75	−	2778.57i	−0.580654	−	0.711449i
$249$		0			0
$250$	−3448.19	−	2148.54i	−0.872332	−	0.543543i
$251$	35.4718	+	35.4718i	0.00892016	+	0.00892016i	0.711553	−	0.702633i	$-0.247992\pi$
$251$	35.4718	+	35.4718i	0.00892016	+	0.00892016i	−0.702633	+	0.711553i	$0.747992\pi$
$252$		0			0
$253$	−720.694	+	720.694i	−0.179090	+	0.179090i
$254$	−420.615	−	1811.31i	−0.103905	−	0.447448i
$255$		0			0
$256$	−1031.53	−	3963.98i	−0.251838	−	0.967770i
$257$	3415.73			0.829056			0.414528	−	0.910037i	$-0.363947\pi$
$257$	3415.73			0.829056			0.414528	+	0.910037i	$0.363947\pi$
$258$		0			0
$259$	1354.89	−	1354.89i	0.325052	−	0.325052i
$260$	−2759.98	+	1354.88i	−0.658333	+	0.323178i
$261$		0			0
$262$	5801.65	+	3614.96i	1.36804	+	0.852416i
$263$			6780.71i			1.58980i	0.606742	+	0.794899i	$0.292476\pi$
$263$			6780.71i			1.58980i	−0.606742	+	0.794899i	$0.707524\pi$
$264$		0			0
$265$			852.528i			0.197624i
$266$	−554.394	+	889.747i	−0.127790	+	0.205090i
$267$		0			0
$268$	2179.55	−	6382.88i	0.496781	−	1.45484i
$269$	3834.98	−	3834.98i	0.869229	−	0.869229i	−0.123158	−	0.992387i	$-0.539302\pi$
$269$	3834.98	−	3834.98i	0.869229	−	0.869229i	0.992387	+	0.123158i	$0.0393022\pi$
$270$		0			0
$271$	7031.32			1.57610			0.788048	−	0.615614i	$-0.211092\pi$
$271$	7031.32			1.57610			0.788048	+	0.615614i	$0.211092\pi$
$272$	2640.67	−	3415.79i	0.588655	−	0.761443i
$273$		0			0
$274$	8240.44	−	1913.56i	1.81687	−	0.421907i
$275$	1591.83	−	1591.83i	0.349058	−	0.349058i
$276$		0			0
$277$	−3030.30	−	3030.30i	−0.657304	−	0.657304i	0.297437	−	0.954741i	$-0.403868\pi$
$277$	−3030.30	−	3030.30i	−0.657304	−	0.657304i	−0.954741	+	0.297437i	$0.903868\pi$
$278$	−1901.24	+	3051.30i	−0.410176	+	0.658291i
$279$		0			0
$280$	1157.44	+	117.163i	0.247037	+	0.0250066i
$281$			4719.06i			1.00183i	0.865495	+	0.500917i	$0.167004\pi$
$281$			4719.06i			1.00183i	−0.865495	+	0.500917i	$0.832996\pi$
$282$		0			0
$283$	−2696.52	−	2696.52i	−0.566400	−	0.566400i	0.364718	−	0.931118i	$-0.381165\pi$
$283$	−2696.52	−	2696.52i	−0.566400	−	0.566400i	−0.931118	+	0.364718i	$0.881165\pi$
$284$	389.506	+	793.448i	0.0813836	+	0.165783i
$285$		0			0
$286$	−1057.74	−	4554.99i	−0.218690	−	0.941755i
$287$	1993.79			0.410070
$288$		0			0
$289$	−362.038			−0.0736898
$290$	−626.385	−	2697.42i	−0.126836	−	0.546201i
$291$		0			0
$292$	−1454.83	−	2963.58i	−0.291567	−	0.593939i
$293$	865.453	+	865.453i	0.172561	+	0.172561i	0.788103	−	0.615543i	$-0.211063\pi$
$293$	865.453	+	865.453i	0.172561	+	0.172561i	−0.615543	+	0.788103i	$0.711063\pi$
$294$		0			0
$295$			823.929i			0.162614i
$296$	−620.386	+	6128.71i	−0.121822	+	1.20346i
$297$		0			0
$298$	−4123.00	+	6616.99i	−0.801473	+	1.28628i
$299$	1206.67	+	1206.67i	0.233389	+	0.233389i
$300$		0			0
$301$	−1452.08	+	1452.08i	−0.278061	+	0.278061i
$302$	1073.92	−	249.380i	0.204626	−	0.0475173i
$303$		0			0
$304$	−427.842	−	3343.00i	−0.0807184	−	0.630705i
$305$	1367.70			0.256767
$306$		0			0
$307$	703.076	−	703.076i	0.130706	−	0.130706i	−0.638727	−	0.769433i	$-0.720539\pi$
$307$	703.076	−	703.076i	0.130706	−	0.130706i	0.769433	+	0.638727i	$0.220539\pi$
$308$	−571.764	+	1674.43i	−0.105777	+	0.309770i
$309$		0			0
$310$	−1731.85	+	2779.44i	−0.317298	+	0.509232i
$311$		−	6484.26i		−	1.18228i	−0.806570	−	0.591139i	$-0.798678\pi$
$311$		−	6484.26i		−	1.18228i	0.806570	−	0.591139i	$-0.201322\pi$
$312$		0			0
$313$		−	7115.56i		−	1.28497i	−0.766299	−	0.642484i	$-0.777904\pi$
$313$		−	7115.56i		−	1.28497i	0.766299	−	0.642484i	$-0.222096\pi$
$314$	1946.41	+	1212.79i	0.349816	+	0.217968i
$315$		0			0
$316$	8204.09	−	4027.41i	1.46049	−	0.716961i
$317$	−5573.78	+	5573.78i	−0.987554	+	0.987554i	−0.999924	−	0.0123690i	$-0.996063\pi$
$317$	−5573.78	+	5573.78i	−0.987554	+	0.987554i	0.0123690	+	0.999924i	$0.496063\pi$
$318$		0			0
$319$	4211.69			0.739215
$320$	−3120.95	+	2060.99i	−0.545207	+	0.360040i
$321$		0			0
$322$	−146.054	−	628.957i	−0.0252772	−	0.108852i
$323$	2512.01	−	2512.01i	0.432730	−	0.432730i
$324$		0			0
$325$	−2665.22	−	2665.22i	−0.454891	−	0.454891i
$326$	−321.783	−	200.500i	−0.0546684	−	0.0340635i
$327$		0			0
$328$	−4965.85	+	4052.91i	−0.835955	+	0.682270i
$329$		−	4116.60i		−	0.689835i
$330$		0			0
$331$	5774.02	+	5774.02i	0.958819	+	0.958819i	0.999185	−	0.0403663i	$-0.0128525\pi$
$331$	5774.02	+	5774.02i	0.958819	+	0.958819i	−0.0403663	+	0.999185i	$0.512852\pi$
$332$	2398.83	−	7025.03i	0.396545	−	1.16129i
$333$		0			0
$334$	−1093.05	+	253.824i	−0.179069	+	0.0415827i
$335$	−6158.62			−1.00442
$336$		0			0
$337$	2111.30			0.341275			0.170638	−	0.985334i	$-0.445417\pi$
$337$	2111.30			0.341275			0.170638	+	0.985334i	$0.445417\pi$
$338$	−1573.47	+	365.385i	−0.253212	+	0.0587999i
$339$		0			0
$340$	−3730.78	−	1273.95i	−0.595089	−	0.203204i
$341$	−3521.91	−	3521.91i	−0.559303	−	0.559303i
$342$		0			0
$343$		−	4479.63i		−	0.705182i
$344$	664.888	−	6568.34i	0.104210	−	1.02948i
$345$		0			0
$346$	655.331	+	408.332i	0.101823	+	0.0634452i
$347$	2375.98	+	2375.98i	0.367577	+	0.367577i	0.866593	−	0.499016i	$-0.166305\pi$
$347$	2375.98	+	2375.98i	0.367577	+	0.367577i	−0.499016	+	0.866593i	$0.666305\pi$
$348$		0			0
$349$	−212.228	+	212.228i	−0.0325511	+	0.0325511i	−0.723195	−	0.690644i	$-0.757327\pi$
$349$	−212.228	+	212.228i	−0.0325511	+	0.0325511i	0.690644	+	0.723195i	$0.257327\pi$
$350$	322.595	+	1389.20i	0.0492670	+	0.212160i
$351$		0			0
$352$	−1979.65	−	5332.67i	−0.299760	−	0.807479i
$353$	−8492.56			−1.28049			−0.640246	−	0.768170i	$-0.721167\pi$
$353$	−8492.56			−1.28049			−0.640246	+	0.768170i	$0.721167\pi$
$354$		0			0
$355$	570.695	−	570.695i	0.0853221	−	0.0853221i
$356$	−1674.28	−	3410.61i	−0.249260	−	0.507758i
$357$		0			0
$358$	390.180	+	243.118i	0.0576024	+	0.0358916i
$359$		−	2300.75i		−	0.338242i	−0.985595	−	0.169121i	$-0.945907\pi$
$359$		−	2300.75i		−	0.338242i	0.985595	−	0.169121i	$-0.0540929\pi$
$360$		0			0
$361$			4085.88i			0.595696i
$362$	1617.03	−	2595.17i	0.234777	−	0.376793i
$363$		0			0
$364$	2803.51	+	957.311i	0.403692	+	0.137848i
$365$	−2131.58	+	2131.58i	−0.305677	+	0.305677i
$366$		0			0
$367$	−4035.86			−0.574033			−0.287016	−	0.957926i	$-0.592663\pi$
$367$	−4035.86			−0.574033			−0.287016	+	0.957926i	$0.592663\pi$
$368$	1642.29	+	1269.62i	0.232637	+	0.179847i
$369$		0			0
$370$	5478.93	−	1272.29i	0.769827	−	0.178766i
$371$	580.839	−	580.839i	0.0812821	−	0.0812821i
$372$		0			0
$373$	1491.28	+	1491.28i	0.207013	+	0.207013i	0.802997	−	0.595984i	$-0.203238\pi$
$373$	1491.28	+	1491.28i	0.207013	+	0.207013i	−0.595984	+	0.802997i	$0.703238\pi$
$374$	3170.82	−	5088.84i	0.438393	−	0.703577i
$375$		0			0
$376$	8368.08	+	10253.0i	1.14774	+	1.40628i
$377$		−	7051.69i		−	0.963343i
$378$		0			0
$379$	−4343.13	−	4343.13i	−0.588632	−	0.588632i	0.348629	−	0.937261i	$-0.386647\pi$
$379$	−4343.13	−	4343.13i	−0.588632	−	0.588632i	−0.937261	+	0.348629i	$0.886647\pi$
$380$	−2762.48	+	1356.11i	−0.372926	+	0.183071i
$381$		0			0
$382$	−1543.85	−	6648.33i	−0.206780	−	0.890466i
$383$	−13342.2			−1.78004			−0.890022	−	0.455917i	$-0.849311\pi$
$383$	−13342.2			−1.78004			−0.890022	+	0.455917i	$0.849311\pi$
$384$		0			0
$385$	1615.59			0.213866
$386$	3196.92	+	13767.0i	0.421551	+	1.81534i
$387$		0			0
$388$	−9754.54	+	4788.53i	−1.27632	+	0.626549i
$389$	3336.92	+	3336.92i	0.434932	+	0.434932i	0.890302	−	0.455370i	$-0.150493\pi$
$389$	3336.92	+	3336.92i	0.434932	+	0.434932i	−0.455370	+	0.890302i	$0.650493\pi$
$390$		0			0
$391$			2188.08i			0.283007i
$392$	4198.64	+	5144.40i	0.540978	+	0.662835i
$393$		0			0
$394$	−2556.80	+	4103.41i	−0.326928	+	0.524687i
$395$	−5900.87	−	5900.87i	−0.751658	−	0.751658i
$396$		0			0
$397$	3228.02	−	3228.02i	0.408085	−	0.408085i	−0.472986	−	0.881070i	$-0.656824\pi$
$397$	3228.02	−	3228.02i	0.408085	−	0.408085i	0.881070	+	0.472986i	$0.156824\pi$
$398$	8321.08	−	1932.29i	1.04799	−	0.243359i
$399$		0			0
$400$	−3627.40	−	2804.27i	−0.453425	−	0.350533i
$401$	52.2560			0.00650759			0.00325379	−	0.999995i	$-0.498964\pi$
$401$	52.2560			0.00650759			0.00325379	+	0.999995i	$0.498964\pi$
$402$		0			0
$403$	−5896.78	+	5896.78i	−0.728882	+	0.728882i
$404$	14402.3	+	4917.94i	1.77362	+	0.605636i
$405$		0			0
$406$	−1411.03	+	2264.56i	−0.172483	+	0.276818i
$407$			8554.66i			1.04186i
$408$		0			0
$409$			14516.7i			1.75502i	0.479558	+	0.877510i	$0.340797\pi$
$409$			14516.7i			1.75502i	−0.479558	+	0.877510i	$0.659203\pi$
$410$	4967.41	+	3095.16i	0.598349	+	0.372827i
$411$		0			0
$412$	3799.37	+	7739.54i	0.454324	+	0.925485i
$413$	561.354	−	561.354i	0.0668824	−	0.0668824i
$414$		0			0
$415$	−6778.20			−0.801757
$416$	−8928.56	+	3314.55i	−1.05230	+	0.390647i
$417$		0			0
$418$	−1058.70	−	4559.11i	−0.123882	−	0.533477i
$419$	−409.663	+	409.663i	−0.0477646	+	0.0477646i	−0.730586	−	0.682821i	$-0.760753\pi$
$419$	−409.663	+	409.663i	−0.0477646	+	0.0477646i	0.682821	+	0.730586i	$0.260753\pi$
$420$		0			0
$421$	4465.76	+	4465.76i	0.516978	+	0.516978i	0.916656	−	0.399678i	$-0.130878\pi$
$421$	4465.76	+	4465.76i	0.516978	+	0.516978i	−0.399678	+	0.916656i	$0.630878\pi$
$422$	−6362.60	−	3964.49i	−0.733949	−	0.457318i
$423$		0			0
$424$	−265.959	+	2627.38i	−0.0304626	+	0.300936i
$425$		−	4832.90i		−	0.551600i
$426$		0			0
$427$	−931.830	−	931.830i	−0.105607	−	0.105607i
$428$	16115.0	+	5502.78i	1.81998	+	0.621465i
$429$		0			0
$430$	−5871.95	+	1363.56i	−0.658536	+	0.152922i
$431$	−2298.21			−0.256846			−0.128423	−	0.991719i	$-0.540992\pi$
$431$	−2298.21			−0.256846			−0.128423	+	0.991719i	$0.540992\pi$
$432$		0			0
$433$	−4951.41			−0.549538			−0.274769	−	0.961510i	$-0.588601\pi$
$433$	−4951.41			−0.549538			−0.274769	+	0.961510i	$0.588601\pi$
$434$	3073.61	−	713.740i	0.339949	−	0.0789416i
$435$		0			0
$436$	−3238.80	+	9484.91i	−0.355758	+	1.04185i
$437$	1207.76	+	1207.76i	0.132208	+	0.132208i
$438$		0			0
$439$			2937.90i			0.319404i	0.987165	+	0.159702i	$0.0510533\pi$
$439$			2937.90i			0.319404i	−0.987165	+	0.159702i	$0.948947\pi$
$440$	−4023.88	+	3284.12i	−0.435980	+	0.355828i
$441$		0			0
$442$	−8520.31	−	5308.94i	−0.916900	−	0.571313i
$443$	4175.48	+	4175.48i	0.447818	+	0.447818i	0.894628	−	0.446811i	$-0.147440\pi$
$443$	4175.48	+	4175.48i	0.447818	+	0.447818i	−0.446811	+	0.894628i	$0.647440\pi$
$444$		0			0
$445$	−2453.12	+	2453.12i	−0.261323	+	0.261323i
$446$	−623.297	−	2684.13i	−0.0661748	−	0.284971i
$447$		0			0
$448$	3530.52	+	722.163i	0.372325	+	0.0761585i
$449$	−6835.79			−0.718487			−0.359244	−	0.933244i	$-0.616965\pi$
$449$	−6835.79			−0.718487			−0.359244	+	0.933244i	$0.616965\pi$
$450$		0			0
$451$	−6294.35	+	6294.35i	−0.657183	+	0.657183i
$452$	7202.11	−	3535.54i	0.749466	−	0.367915i
$453$		0			0
$454$	12360.8	+	7701.91i	1.27780	+	0.796187i
$455$		−	2705.01i		−	0.278709i
$456$		0			0
$457$		−	11521.0i		−	1.17928i	−0.807668	−	0.589638i	$-0.799271\pi$
$457$		−	11521.0i		−	1.17928i	0.807668	−	0.589638i	$-0.200729\pi$
$458$	−3131.59	+	5025.89i	−0.319497	+	0.512761i
$459$		0			0
$460$	612.506	−	1793.74i	0.0620831	−	0.181812i
$461$	−4875.43	+	4875.43i	−0.492563	+	0.492563i	−0.909113	−	0.416550i	$-0.863239\pi$
$461$	−4875.43	+	4875.43i	−0.492563	+	0.492563i	0.416550	+	0.909113i	$0.363239\pi$
$462$		0			0
$463$	−13485.6			−1.35362			−0.676812	−	0.736156i	$-0.736639\pi$
$463$	−13485.6			−1.35362			−0.676812	+	0.736156i	$0.736639\pi$
$464$	−1088.93	−	8508.51i	−0.108949	−	0.851288i
$465$		0			0
$466$	4473.61	−	1038.84i	0.444713	−	0.103269i
$467$	6043.84	−	6043.84i	0.598877	−	0.598877i	−0.341137	−	0.940014i	$-0.610812\pi$
$467$	6043.84	−	6043.84i	0.598877	−	0.598877i	0.940014	+	0.341137i	$0.110812\pi$
$468$		0			0
$469$	4195.95	+	4195.95i	0.413115	+	0.413115i
$470$	6390.59	−	10256.3i	0.627183	−	1.00657i
$471$		0			0
$472$	−257.037	+	2539.24i	−0.0250659	+	0.247623i
$473$		−	9168.32i		−	0.891246i
$474$		0			0
$475$	−2667.63	−	2667.63i	−0.257683	−	0.257683i
$476$	1673.88	+	3409.79i	0.161181	+	0.328335i
$477$		0			0
$478$	−4394.62	−	18924.7i	−0.420513	−	1.81087i
$479$	−12502.9			−1.19263			−0.596315	−	0.802750i	$-0.703369\pi$
$479$	−12502.9			−1.19263			−0.596315	+	0.802750i	$0.703369\pi$
$480$		0			0
$481$	14323.2			1.35776
$482$	1770.56	+	7624.61i	0.167317	+	0.720522i
$483$		0			0
$484$	1211.17	+	2467.23i	0.113747	+	0.231709i
$485$	7016.05	+	7016.05i	0.656871	+	0.656871i
$486$		0			0
$487$			10204.1i			0.949469i	0.880129	+	0.474734i	$0.157456\pi$
$487$			10204.1i			0.949469i	−0.880129	+	0.474734i	$0.842544\pi$
$488$	4215.05	+	426.674i	0.390997	+	0.0395791i
$489$		0			0
$490$	3206.44	−	5146.02i	0.295617	−	0.474435i
$491$	3102.01	+	3102.01i	0.285116	+	0.285116i	0.835145	−	0.550030i	$-0.185384\pi$
$491$	3102.01	+	3102.01i	0.285116	+	0.285116i	−0.550030	+	0.835145i	$0.685384\pi$
$492$		0			0
$493$	6393.49	−	6393.49i	0.584074	−	0.584074i
$494$	−7633.37	+	1772.59i	−0.695226	+	0.161442i
$495$		0			0
$496$	−6204.42	+	8025.60i	−0.561667	+	0.726532i
$497$	−777.645			−0.0701854
$498$		0			0
$499$	13549.5	−	13549.5i	1.21555	−	1.21555i	0.246379	−	0.969174i	$-0.420759\pi$
$499$	13549.5	−	13549.5i	1.21555	−	1.21555i	0.969174	−	0.246379i	$-0.0792407\pi$
$500$	−3713.40	+	10874.8i	−0.332136	+	0.972670i
$501$		0			0
$502$	75.0348	−	120.423i	0.00667125	−	0.0107067i
$503$			3300.94i			0.292607i	0.989240	+	0.146304i	$0.0467377\pi$
$503$			3300.94i			0.292607i	−0.989240	+	0.146304i	$0.953262\pi$
$504$		0			0
$505$		−	13896.3i		−	1.22451i
$506$	2446.69	+	1524.51i	0.214957	+	0.133938i
$507$		0			0
$508$	−4721.28	+	2317.69i	−0.412348	+	0.202423i
$509$	−10081.2	+	10081.2i	−0.877880	+	0.877880i	−0.993315	−	0.115435i	$-0.963174\pi$
$509$	−10081.2	+	10081.2i	−0.877880	+	0.877880i	0.115435	+	0.993315i	$0.463174\pi$
$510$		0			0
$511$	2904.55			0.251448
$512$	−10261.3	+	5378.07i	−0.885721	+	0.464217i
$513$		0			0
$514$	−2185.32	−	9410.74i	−0.187530	−	0.807568i
$515$	5566.74	−	5566.74i	0.476311	−	0.476311i
$516$		0			0
$517$	12996.0	+	12996.0i	1.10554	+	1.10554i
$518$	−4599.70	−	2866.04i	−0.390153	−	0.243101i
$519$		0			0
$520$	5498.65	+	6737.24i	0.463715	+	0.568168i
$521$		−	22396.4i		−	1.88331i	−0.336579	−	0.941655i	$-0.609270\pi$
$521$		−	22396.4i		−	1.88331i	0.336579	−	0.941655i	$-0.390730\pi$
$522$		0			0
$523$	−10114.9	−	10114.9i	−0.845688	−	0.845688i	0.143904	−	0.989592i	$-0.454034\pi$
$523$	−10114.9	−	10114.9i	−0.845688	−	0.845688i	−0.989592	+	0.143904i	$0.954034\pi$
$524$	6247.86	−	18297.0i	0.520876	−	1.52540i
$525$		0			0
$526$	18681.7	−	4338.18i	1.54859	−	0.359608i
$527$	−10692.8			−0.883841
$528$		0			0
$529$	11115.0			0.913535
$530$	2348.82	−	545.432i	0.192502	−	0.0447020i
$531$		0			0
$532$	2806.05	+	958.178i	0.228680	+	0.0780870i
$533$	10538.7	+	10538.7i	0.856439	+	0.856439i
$534$		0			0
$535$		−	15548.8i		−	1.25651i
$536$	−18980.0	−	1921.28i	−1.52950	−	0.154826i
$537$		0			0
$538$	−13019.4	−	8112.27i	−1.04332	−	0.650083i
$539$	6520.66	+	6520.66i	0.521085	+	0.521085i
$540$		0			0
$541$	10004.1	−	10004.1i	0.795024	−	0.795024i	−0.187282	−	0.982306i	$-0.559968\pi$
$541$	10004.1	−	10004.1i	0.795024	−	0.795024i	0.982306	+	0.187282i	$0.0599679\pi$
$542$	−4498.51	−	19372.1i	−0.356508	−	1.53525i
$543$		0			0
$544$	−11100.3	−	5090.01i	−0.874859	−	0.401162i
$545$	9151.65			0.719291
$546$		0			0
$547$	−11671.6	+	11671.6i	−0.912325	+	0.912325i	−0.996455	−	0.0841298i	$-0.973189\pi$
$547$	−11671.6	+	11671.6i	−0.912325	+	0.912325i	0.0841298	+	0.996455i	$0.473189\pi$
$548$	−10544.2	−	21479.1i	−0.821943	−	1.67435i
$549$		0			0
$550$	−5404.10	−	3367.25i	−0.418967	−	0.261055i
$551$		−	7058.07i		−	0.545706i
$552$		0			0
$553$			8040.69i			0.618309i
$554$	−6410.11	+	10287.6i	−0.491588	+	0.788949i
$555$		0			0
$556$	9623.08	+	3285.98i	0.734010	+	0.250641i
$557$	117.575	−	117.575i	0.00894401	−	0.00894401i	−0.702621	−	0.711565i	$-0.747987\pi$
$557$	117.575	−	117.575i	0.00894401	−	0.00894401i	0.711565	+	0.702621i	$0.247987\pi$
$558$		0			0
$559$	−15350.6			−1.16147
$560$	−417.711	−	3263.84i	−0.0315205	−	0.246290i
$561$		0			0
$562$	13001.6	−	3019.17i	0.975869	−	0.226612i
$563$	6510.39	−	6510.39i	0.487354	−	0.487354i	−0.420116	−	0.907470i	$-0.638011\pi$
$563$	6510.39	−	6510.39i	0.487354	−	0.487354i	0.907470	+	0.420116i	$0.138011\pi$
$564$		0			0
$565$	−5180.19	−	5180.19i	−0.385721	−	0.385721i
$566$	−5704.04	+	9154.41i	−0.423602	+	0.679838i
$567$		0			0
$568$	1936.84	−	1580.77i	0.143078	−	0.116774i
$569$		−	1080.96i		−	0.0796421i	−0.999207	−	0.0398210i	$-0.987321\pi$
$569$		−	1080.96i		−	0.0796421i	0.999207	−	0.0398210i	$-0.0126788\pi$
$570$		0			0
$571$	−6064.11	−	6064.11i	−0.444440	−	0.444440i	0.449061	−	0.893501i	$-0.351758\pi$
$571$	−6064.11	−	6064.11i	−0.444440	−	0.444440i	−0.893501	+	0.449061i	$0.851758\pi$
$572$	−11872.8	+	5828.40i	−0.867879	+	0.426045i
$573$		0			0
$574$	−1275.59	−	5493.14i	−0.0927566	−	0.399441i
$575$	2323.63			0.168525
$576$		0			0
$577$	75.0142			0.00541227			0.00270614	−	0.999996i	$-0.499139\pi$
$577$	75.0142			0.00541227			0.00270614	+	0.999996i	$0.499139\pi$
$578$	231.625	+	997.458i	0.0166684	+	0.0717799i
$579$		0			0
$580$	−7030.97	+	3451.53i	−0.503354	+	0.247098i
$581$	4618.08	+	4618.08i	0.329760	+	0.329760i
$582$		0			0
$583$			3667.38i			0.260527i
$584$	−7234.24	+	5904.27i	−0.512594	+	0.418357i
$585$		0			0
$586$	1830.72	−	2938.13i	0.129055	−	0.207121i
$587$	15487.5	+	15487.5i	1.08899	+	1.08899i	0.995633	+	0.0933554i	$0.0297593\pi$
$587$	15487.5	+	15487.5i	1.08899	+	1.08899i	0.0933554	+	0.995633i	$0.470241\pi$
$588$		0			0
$589$	−5902.12	+	5902.12i	−0.412891	+	0.412891i
$590$	2270.02	−	527.135i	0.158399	−	0.0367827i
$591$		0			0
$592$	17282.2	−	2211.80i	1.19982	−	0.153555i
$593$	−6675.53			−0.462278			−0.231139	−	0.972921i	$-0.574245\pi$
$593$	−6675.53			−0.462278			−0.231139	+	0.972921i	$0.574245\pi$
$594$		0			0
$595$	2452.53	−	2452.53i	0.168981	−	0.168981i
$596$	20868.4	+	7125.91i	1.43423	+	0.489746i
$597$		0			0
$598$	2552.51	−	4096.52i	0.174548	−	0.280132i
$599$			4338.77i			0.295955i	0.988991	+	0.147978i	$0.0472763\pi$
$599$			4338.77i			0.295955i	−0.988991	+	0.147978i	$0.952724\pi$
$600$		0			0
$601$			4339.37i			0.294520i	0.989098	+	0.147260i	$0.0470455\pi$
$601$			4339.37i			0.294520i	−0.989098	+	0.147260i	$0.952955\pi$
$602$	4929.65	+	3071.63i	0.333750	+	0.207957i
$603$		0			0
$604$	−1374.14	−	2799.22i	−0.0925714	−	0.188574i
$605$	1774.58	−	1774.58i	0.119251	−	0.119251i
$606$		0			0
$607$	19248.1			1.28708			0.643539	−	0.765414i	$-0.277466\pi$
$607$	19248.1			1.28708			0.643539	+	0.765414i	$0.277466\pi$
$608$	−8936.64	+	3317.55i	−0.596100	+	0.221290i
$609$		0			0
$610$	−875.027	−	3768.16i	−0.0580800	−	0.250112i
$611$	21759.4	−	21759.4i	1.44074	−	1.44074i
$612$		0			0
$613$	−9056.59	−	9056.59i	−0.596724	−	0.596724i	0.342715	−	0.939439i	$-0.388653\pi$
$613$	−9056.59	−	9056.59i	−0.596724	−	0.596724i	−0.939439	+	0.342715i	$0.888653\pi$
$614$	−2386.87	−	1487.24i	−0.156883	−	0.0977528i
$615$		0			0
$616$	4979.05	+	504.010i	0.325668	+	0.0329661i
$617$			12587.6i			0.821325i	0.911787	+	0.410662i	$0.134703\pi$
$617$			12587.6i			0.821325i	−0.911787	+	0.410662i	$0.865297\pi$
$618$		0			0
$619$	11044.3	+	11044.3i	0.717136	+	0.717136i	0.968018	−	0.250881i	$-0.0807204\pi$
$619$	11044.3	+	11044.3i	0.717136	+	0.717136i	−0.250881	+	0.968018i	$0.580720\pi$
$620$	8765.71	+	2993.21i	0.567805	+	0.193888i
$621$		0			0
$622$	−17864.9	+	4148.51i	−1.15164	+	0.267428i
$623$	3342.68			0.214963
$624$		0			0
$625$	1537.68			0.0984117
$626$	−19604.2	+	4552.41i	−1.25166	+	0.290656i
$627$		0			0
$628$	2096.11	−	6138.51i	0.133191	−	0.390053i
$629$	12986.3	+	12986.3i	0.823206	+	0.823206i
$630$		0			0
$631$		−	7554.44i		−	0.476605i	−0.971191	−	0.238302i	$-0.923409\pi$
$631$		−	7554.44i		−	0.476605i	0.971191	−	0.238302i	$-0.0765909\pi$
$632$	−16344.8	−	20026.6i	−1.02874	−	1.26047i
$633$		0			0
$634$	18922.4	+	11790.4i	1.18534	+	0.738577i
$635$	3395.83	+	3395.83i	0.212219	+	0.212219i
$636$		0			0
$637$	10917.6	−	10917.6i	0.679077	−	0.679077i
$638$	−2694.57	−	11603.7i	−0.167208	−	0.720055i
$639$		0			0
$640$	7674.99	+	7279.99i	0.474033	+	0.449636i
$641$	18511.6			1.14066			0.570330	−	0.821416i	$-0.306815\pi$
$641$	18511.6			1.14066			0.570330	+	0.821416i	$0.306815\pi$
$642$		0			0
$643$	−7895.63	+	7895.63i	−0.484251	+	0.484251i	−0.906486	−	0.422236i	$-0.861245\pi$
$643$	−7895.63	+	7895.63i	−0.484251	+	0.484251i	0.422236	+	0.906486i	$0.361245\pi$
$644$	−1639.41	+	804.791i	−0.100313	+	0.0492441i
$645$		0			0
$646$	−8528.02	−	5313.74i	−0.519397	−	0.323632i
$647$			25087.9i			1.52443i	0.647323	+	0.762216i	$0.275888\pi$
$647$			25087.9i			1.52443i	−0.647323	+	0.762216i	$0.724112\pi$
$648$		0			0
$649$			3544.36i			0.214373i
$650$	−5637.83	+	9048.15i	−0.340206	+	0.545997i
$651$		0			0
$652$	−346.531	+	1014.83i	−0.0208148	+	0.0609565i
$653$	−3966.20	+	3966.20i	−0.237687	+	0.237687i	−0.815892	−	0.578205i	$-0.803753\pi$
$653$	−3966.20	+	3966.20i	−0.237687	+	0.237687i	0.578205	+	0.815892i	$0.303753\pi$
$654$		0			0
$655$	−17654.1			−1.05314
$656$	14343.3	+	11088.5i	0.853678	+	0.659960i
$657$		0			0
$658$	−11341.7	+	2633.73i	−0.671955	+	0.156039i
$659$	−8121.04	+	8121.04i	−0.480047	+	0.480047i	−0.905147	−	0.425100i	$-0.860239\pi$
$659$	−8121.04	+	8121.04i	−0.480047	+	0.480047i	0.425100	+	0.905147i	$0.360239\pi$
$660$		0			0
$661$	6417.92	+	6417.92i	0.377652	+	0.377652i	0.870255	−	0.492602i	$-0.163954\pi$
$661$	6417.92	+	6417.92i	0.377652	+	0.377652i	−0.492602	+	0.870255i	$0.663954\pi$
$662$	12214.0	−	19602.2i	0.717086	−	1.15085i
$663$		0			0
$664$	−20889.5	−	2114.57i	−1.22089	−	0.123586i
$665$		−	2707.46i		−	0.157881i
$666$		0			0
$667$	3073.95	+	3073.95i	0.178447	+	0.178447i
$668$	1398.63	+	2849.09i	0.0810098	+	0.165022i
$669$		0			0
$670$	3940.17	+	16967.7i	0.227197	+	0.978388i
$671$	5883.52			0.338496
$672$		0			0
$673$	−4207.12			−0.240969			−0.120485	−	0.992715i	$-0.538445\pi$
$673$	−4207.12			−0.240969			−0.120485	+	0.992715i	$0.538445\pi$
$674$	−1350.77	−	5816.88i	−0.0771955	−	0.332430i
$675$		0			0
$676$	2013.36	+	4101.34i	0.114552	+	0.233349i
$677$	−2686.58	−	2686.58i	−0.152517	−	0.152517i	0.626724	−	0.779241i	$-0.284395\pi$
$677$	−2686.58	−	2686.58i	−0.152517	−	0.152517i	−0.779241	+	0.626724i	$0.784395\pi$
$678$		0			0
$679$		−	9560.26i		−	0.540338i
$680$	−1122.98	+	11093.8i	−0.0633301	+	0.625629i
$681$		0			0
$682$	−7450.03	+	11956.5i	−0.418294	+	0.671319i
$683$	−1806.99	−	1806.99i	−0.101234	−	0.101234i	0.654676	−	0.755910i	$-0.272805\pi$
$683$	−1806.99	−	1806.99i	−0.101234	−	0.101234i	−0.755910	+	0.654676i	$0.772805\pi$
$684$		0			0
$685$	−15449.1	+	15449.1i	−0.861721	+	0.861721i
$686$	−12341.9	+	2865.99i	−0.686904	+	0.159510i
$687$		0			0
$688$	−18521.9	+	2370.46i	−1.02637	+	0.131356i
$689$	6140.34			0.339519
$690$		0			0
$691$	3964.38	−	3964.38i	0.218252	−	0.218252i	−0.589509	−	0.807761i	$-0.700679\pi$
$691$	3964.38	−	3964.38i	0.218252	−	0.218252i	0.807761	+	0.589509i	$0.200679\pi$
$692$	705.733	−	2066.76i	0.0387687	−	0.113535i
$693$		0			0
$694$	5025.99	−	8066.20i	0.274905	−	0.441194i
$695$		−	9284.97i		−	0.506761i
$696$		0			0
$697$			19110.1i			1.03852i
$698$	720.494	+	448.934i	0.0390704	+	0.0243444i
$699$		0			0
$700$	3621.03	−	1777.58i	0.195517	−	0.0959801i
$701$	20882.0	−	20882.0i	1.12511	−	1.12511i	0.134149	−	0.990961i	$-0.457170\pi$
$701$	20882.0	−	20882.0i	1.12511	−	1.12511i	0.990961	−	0.134149i	$-0.0428300\pi$
$702$		0			0
$703$	14336.1			0.769129
$704$	−13425.6	+	8865.91i	−0.718745	+	0.474640i
$705$		0			0
$706$	5433.39	+	23398.0i	0.289644	+	1.24730i
$707$	−9467.74	+	9467.74i	−0.503637	+	0.503637i
$708$		0			0
$709$	−8482.93	−	8482.93i	−0.449342	−	0.449342i	0.445794	−	0.895136i	$-0.352921\pi$
$709$	−8482.93	−	8482.93i	−0.449342	−	0.449342i	−0.895136	+	0.445794i	$0.852921\pi$
$710$	−1937.45	−	1207.21i	−0.102410	−	0.0638111i
$711$		0			0
$712$	−8325.46	+	6794.89i	−0.438216	+	0.357653i
$713$		−	5141.02i		−	0.270032i
$714$		0			0
$715$	8539.63	+	8539.63i	0.446663	+	0.446663i
$716$	420.190	−	1230.54i	0.0219319	−	0.0642281i
$717$		0			0
$718$	−6338.84	+	1471.98i	−0.329476	+	0.0765094i
$719$	−23765.0			−1.23266			−0.616332	−	0.787486i	$-0.711382\pi$
$719$	−23765.0			−1.23266			−0.616332	+	0.787486i	$0.711382\pi$
$720$		0			0
$721$	−7585.40			−0.391810
$722$	11257.1	−	2614.07i	0.580256	−	0.134745i
$723$		0			0
$724$	−8184.54	−	2794.77i	−0.420133	−	0.143462i
$725$	−6789.58	−	6789.58i	−0.347805	−	0.347805i
$726$		0			0
$727$			27979.3i			1.42737i	0.700468	+	0.713684i	$0.252975\pi$
$727$			27979.3i			1.42737i	−0.700468	+	0.713684i	$0.747025\pi$
$728$	843.870	−	8336.48i	0.0429614	−	0.424410i
$729$		0			0
$730$	7236.51	+	4509.02i	0.366898	+	0.228611i
$731$	−13917.8	−	13917.8i	−0.704198	−	0.704198i
$732$		0			0
$733$	−17513.6	+	17513.6i	−0.882509	+	0.882509i	−0.993789	−	0.111280i	$-0.964505\pi$
$733$	−17513.6	+	17513.6i	−0.882509	+	0.882509i	0.111280	+	0.993789i	$0.464505\pi$
$734$	2582.07	+	11119.3i	0.129845	+	0.559155i
$735$		0			0
$736$	2447.25	−	5336.99i	0.122563	−	0.267288i
$737$	−26493.0			−1.32413
$738$		0			0
$739$	−9683.07	+	9683.07i	−0.481999	+	0.481999i	−0.905770	−	0.423770i	$-0.860706\pi$
$739$	−9683.07	+	9683.07i	−0.481999	+	0.481999i	0.423770	+	0.905770i	$0.360706\pi$
$740$	−7010.64	−	14281.1i	−0.348265	−	0.709438i
$741$		0			0
$742$	−1971.89	−	1228.67i	−0.0975612	−	0.0607896i
$743$		−	19107.6i		−	0.943460i	−0.881743	−	0.471730i	$-0.843630\pi$
$743$		−	19107.6i		−	0.943460i	0.881743	−	0.471730i	$-0.156370\pi$
$744$		0			0
$745$		−	20135.2i		−	0.990197i
$746$	3154.57	−	5062.76i	0.154822	−	0.248473i
$747$		0			0
$748$	−16049.0	−	5480.23i	−0.784504	−	0.267884i
$749$	−10593.6	+	10593.6i	−0.516800	+	0.516800i
$750$		0			0
$751$	10823.0			0.525883			0.262942	−	0.964812i	$-0.415307\pi$
$751$	10823.0			0.525883			0.262942	+	0.964812i	$0.415307\pi$
$752$	22894.6	−	29614.8i	1.11021	−	1.43609i
$753$		0			0
$754$	−19428.2	+	4511.54i	−0.938375	+	0.217906i
$755$	−2013.36	+	2013.36i	−0.0970514	+	0.0970514i
$756$		0			0
$757$	−12578.0	−	12578.0i	−0.603904	−	0.603904i	0.337442	−	0.941346i	$-0.390438\pi$
$757$	−12578.0	−	12578.0i	−0.603904	−	0.603904i	−0.941346	+	0.337442i	$0.890438\pi$
$758$	−9187.17	+	14744.5i	−0.440228	+	0.706522i
$759$		0			0
$760$	5503.62	+	6743.34i	0.262681	+	0.321851i
$761$		−	30860.2i		−	1.47002i	−0.678058	−	0.735009i	$-0.737178\pi$
$761$		−	30860.2i		−	1.47002i	0.678058	−	0.735009i	$-0.262822\pi$
$762$		0			0
$763$	−6235.15	−	6235.15i	−0.295842	−	0.295842i
$764$	−17329.2	+	8506.96i	−0.820613	+	0.402842i
$765$		0			0
$766$	8536.13	+	36759.5i	0.402641	+	1.73391i
$767$	5934.36			0.279371
$768$		0			0
$769$	−19357.6			−0.907741			−0.453870	−	0.891068i	$-0.649957\pi$
$769$	−19357.6			−0.907741			−0.453870	+	0.891068i	$0.649957\pi$
$770$	−1033.63	−	4451.15i	−0.0483758	−	0.208323i
$771$		0			0
$772$	35884.4	−	17615.8i	1.67294	−	0.821250i
$773$	−2959.96	−	2959.96i	−0.137726	−	0.137726i	0.634882	−	0.772609i	$-0.281049\pi$
$773$	−2959.96	−	2959.96i	−0.137726	−	0.137726i	−0.772609	+	0.634882i	$0.781049\pi$
$774$		0			0
$775$			11355.2i			0.526311i
$776$	19433.8	+	23811.3i	0.899009	+	1.10151i
$777$		0			0
$778$	7058.71	−	11328.5i	0.325279	−	0.522040i
$779$	10548.2	+	10548.2i	0.485148	+	0.485148i
$780$		0			0
$781$	2455.00	−	2455.00i	0.112480	−	0.112480i
$782$	6028.41	−	1399.89i	0.275672	−	0.0640154i
$783$		0			0
$784$	11487.2	−	14859.0i	0.523288	−	0.676888i
$785$	−5922.83			−0.269293
$786$		0			0
$787$	21346.3	−	21346.3i	0.966853	−	0.966853i	−0.0326147	−	0.999468i	$-0.510383\pi$
$787$	21346.3	−	21346.3i	0.966853	−	0.966853i	0.999468	+	0.0326147i	$0.0103834\pi$
$788$	12941.2	+	4419.00i	0.585038	+	0.199772i
$789$		0			0
$790$	−12482.3	+	20032.9i	−0.562154	+	0.902199i
$791$			7058.67i			0.317291i
$792$		0			0
$793$		−	9850.84i		−	0.441127i
$794$	−10958.8	−	6828.34i	−0.489815	−	0.305200i
$795$		0			0
$796$	−10647.4	−	21689.3i	−0.474103	−	0.965776i
$797$	−22434.9	+	22434.9i	−0.997097	+	0.997097i	−0.999996	−	0.00289926i	$-0.999077\pi$
$797$	−22434.9	+	22434.9i	−0.997097	+	0.997097i	0.00289926	+	0.999996i	$0.499077\pi$
$798$		0			0
$799$	39456.7			1.74703
$800$	−5405.34	+	11788.0i	−0.238885	+	0.520962i
$801$		0			0
$802$	−33.4325	−	143.972i	−0.00147200	−	0.00633892i
$803$	−9169.59	+	9169.59i	−0.402974	+	0.402974i
$804$		0			0
$805$	1179.16	+	1179.16i	0.0516273	+	0.0516273i
$806$	20019.0	+	12473.7i	0.874862	+	0.545120i
$807$		0			0
$808$	4335.17	−	42826.5i	0.188751	−	1.86464i
$809$			36085.5i			1.56823i	0.620616	+	0.784115i	$0.286883\pi$
$809$			36085.5i			1.56823i	−0.620616	+	0.784115i	$0.713117\pi$
$810$		0			0
$811$	23874.2	+	23874.2i	1.03371	+	1.03371i	0.999412	+	0.0342973i	$0.0109193\pi$
$811$	23874.2	+	23874.2i	1.03371	+	1.03371i	0.0342973	+	0.999412i	$0.489081\pi$
$812$	7141.87	+	2438.73i	0.308658	+	0.105397i
$813$		0			0
$814$	23569.1	−	5473.12i	1.01486	−	0.235667i
$815$	979.170			0.0420845
$816$		0			0
$817$	−15364.5			−0.657939
$818$	39995.2	−	9287.51i	1.70953	−	0.396981i
$819$		0			0
$820$	5349.46	−	15666.0i	0.227819	−	0.667173i
$821$	−17774.8	−	17774.8i	−0.755596	−	0.755596i	0.219922	−	0.975518i	$-0.429420\pi$
$821$	−17774.8	−	17774.8i	−0.755596	−	0.755596i	−0.975518	+	0.219922i	$0.929420\pi$
$822$		0			0
$823$		−	44331.9i		−	1.87766i	−0.344384	−	0.938829i	$-0.611912\pi$
$823$		−	44331.9i		−	1.87766i	0.344384	−	0.938829i	$-0.388088\pi$
$824$	18892.6	−	15419.3i	0.798731	−	0.651890i
$825$		0			0
$826$	−1905.74	−	1187.45i	−0.0802775	−	0.0500203i
$827$	−29773.5	−	29773.5i	−1.25191	−	1.25191i	−0.954864	−	0.297044i	$-0.903999\pi$
$827$	−29773.5	−	29773.5i	−1.25191	−	1.25191i	−0.297044	−	0.954864i	$-0.596001\pi$
$828$		0			0
$829$	−9794.56	+	9794.56i	−0.410349	+	0.410349i	−0.881860	−	0.471511i	$-0.843709\pi$
$829$	−9794.56	+	9794.56i	−0.410349	+	0.410349i	0.471511	+	0.881860i	$0.343709\pi$
$830$	4336.57	+	18674.8i	0.181355	+	0.780976i
$831$		0			0
$832$	14844.3	+	22478.7i	0.618550	+	0.936667i
$833$	19797.2			0.823447
$834$		0			0
$835$	2049.24	−	2049.24i	0.0849303	−	0.0849303i
$836$	−11883.6	+	5833.67i	−0.491628	+	0.241342i
$837$		0			0
$838$	1390.77	+	866.575i	0.0573308	+	0.0357224i
$839$		−	20248.1i		−	0.833183i	−0.909094	−	0.416591i	$-0.863225\pi$
$839$		−	20248.1i		−	0.833183i	0.909094	−	0.416591i	$-0.136775\pi$
$840$		0			0
$841$			6425.00i			0.263439i
$842$	9446.59	−	15160.8i	0.386640	−	0.620518i
$843$		0			0
$844$	−6851.95	+	20066.1i	−0.279448	+	0.818370i
$845$	2949.93	−	2949.93i	0.120095	−	0.120095i
$846$		0			0
$847$	−2418.10			−0.0980953
$848$	7408.89	−	948.199i	0.300026	−	0.0383977i
$849$		0			0
$850$	−13315.2	+	3092.00i	−0.537304	+	0.124770i
$851$	−6243.72	+	6243.72i	−0.251507	+	0.251507i
$852$		0			0
$853$	−10983.4	−	10983.4i	−0.440873	−	0.440873i	0.451433	−	0.892305i	$-0.350913\pi$
$853$	−10983.4	−	10983.4i	−0.440873	−	0.440873i	−0.892305	+	0.451433i	$0.850913\pi$
$854$	−1971.13	+	3163.47i	−0.0789822	+	0.126758i
$855$		0			0
$856$	4850.70	−	47919.4i	0.193684	−	1.91338i
$857$			26782.3i			1.06752i	0.845635	+	0.533761i	$0.179222\pi$
$857$			26782.3i			1.06752i	−0.845635	+	0.533761i	$0.820778\pi$
$858$		0			0
$859$	−26073.4	−	26073.4i	−1.03564	−	1.03564i	−0.999341	−	0.0362968i	$-0.988444\pi$
$859$	−26073.4	−	26073.4i	−1.03564	−	1.03564i	−0.0362968	−	0.999341i	$-0.511556\pi$
$860$	7513.53	+	15305.5i	0.297918	+	0.606877i
$861$		0			0
$862$	1470.35	+	6331.83i	0.0580979	+	0.250189i
$863$	30417.7			1.19980			0.599902	−	0.800073i	$-0.295206\pi$
$863$	30417.7			1.19980			0.599902	+	0.800073i	$0.295206\pi$
$864$		0			0
$865$	−1994.14			−0.0783848
$866$	3167.83	+	13641.7i	0.124304	+	0.535294i
$867$		0			0
$868$	−3932.88	−	8011.52i	−0.153791	−	0.313282i
$869$	−25384.2	−	25384.2i	−0.990910	−	0.990910i
$870$		0			0
$871$			44357.5i			1.72560i
$872$	28204.2	+	2855.00i	1.09531	+	0.110874i
$873$		0			0
$874$	2554.82	−	4100.22i	0.0988764	−	0.158687i
$875$	−7148.82	−	7148.82i	−0.276199	−	0.276199i
$876$		0			0
$877$	11483.1	−	11483.1i	0.442142	−	0.442142i	−0.450590	−	0.892731i	$-0.648786\pi$
$877$	11483.1	−	11483.1i	0.442142	−	0.442142i	0.892731	+	0.450590i	$0.148786\pi$
$878$	8094.26	−	1879.61i	0.311125	−	0.0722482i
$879$		0			0
$880$	11622.6	+	8985.15i	0.445223	+	0.344193i
$881$	−34502.7			−1.31944			−0.659720	−	0.751511i	$-0.729325\pi$
$881$	−34502.7			−1.31944			−0.659720	+	0.751511i	$0.729325\pi$
$882$		0			0
$883$	35667.9	−	35667.9i	1.35937	−	1.35937i	0.484667	−	0.874699i	$-0.338941\pi$
$883$	35667.9	−	35667.9i	1.35937	−	1.35937i	0.874699	−	0.484667i	$-0.161059\pi$
$884$	−9175.61	+	26871.0i	−0.349106	+	1.02236i
$885$		0			0
$886$	8832.55	−	14175.4i	0.334916	−	0.537506i
$887$			4004.04i			0.151570i	0.997124	+	0.0757850i	$0.0241463\pi$
$887$			4004.04i			0.151570i	−0.997124	+	0.0757850i	$0.975854\pi$
$888$		0			0
$889$		−	4627.25i		−	0.174570i
$890$	8328.08	+	5189.17i	0.313661	+	0.195440i
$891$		0			0
$892$	−6996.32	+	3434.51i	−0.262617	+	0.128919i
$893$	21779.0	−	21779.0i	0.816134	−	0.816134i
$894$		0			0
$895$	−1187.30			−0.0443431
$896$	−269.122	−	10189.0i	−0.0100343	−	0.379902i
$897$		0			0
$898$	4373.41	+	18833.4i	0.162520	+	0.699865i
$899$	−15021.9	+	15021.9i	−0.557295	+	0.557295i
$900$		0			0
$901$	5567.21	+	5567.21i	0.205850	+	0.205850i
$902$	21368.7	+	13314.7i	0.788802	+	0.491497i
$903$		0			0
$904$	−14348.6	−	17580.7i	−0.527907	−	0.646820i
$905$			7896.98i			0.290060i
$906$		0			0
$907$	26847.6	+	26847.6i	0.982865	+	0.982865i	0.999856	−	0.0169907i	$-0.00540855\pi$
$907$	26847.6	+	26847.6i	0.982865	+	0.982865i	−0.0169907	+	0.999856i	$0.505409\pi$
$908$	13311.5	−	38983.0i	0.486516	−	1.42478i
$909$		0			0
$910$	−7452.62	+	1730.62i	−0.271486	+	0.0630433i
$911$	15303.2			0.556552			0.278276	−	0.960501i	$-0.410237\pi$
$911$	15303.2			0.556552			0.278276	+	0.960501i	$0.410237\pi$
$912$		0			0
$913$	−29158.3			−1.05695
$914$	−31741.7	+	7370.92i	−1.14871	+	0.266749i
$915$		0			0
$916$	15850.5	+	5412.44i	0.571740	+	0.195231i
$917$	12028.0	+	12028.0i	0.433152	+	0.433152i
$918$		0			0
$919$		−	43252.1i		−	1.55251i	−0.630421	−	0.776254i	$-0.717118\pi$
$919$		−	43252.1i		−	1.55251i	0.630421	−	0.776254i	$-0.282882\pi$
$920$	−5333.84	−	539.924i	−0.191143	−	0.0193487i
$921$		0			0
$922$	16551.6	+	10313.2i	0.591213	+	0.368380i
$923$	−4110.44	−	4110.44i	−0.146584	−	0.146584i
$924$		0			0
$925$	13790.8	−	13790.8i	0.490203	−	0.490203i
$926$	8627.83	+	37154.4i	0.306186	+	1.31854i
$927$		0			0
$928$	−22745.3	+	8443.72i	−0.804580	+	0.298684i
$929$	−13328.9			−0.470729			−0.235365	−	0.971907i	$-0.575628\pi$
$929$	−13328.9			−0.470729			−0.235365	+	0.971907i	$0.575628\pi$
$930$		0			0
$931$	10927.5	−	10927.5i	0.384677	−	0.384677i
$932$	−5724.27	−	11660.7i	−0.201185	−	0.409827i
$933$		0			0
$934$	−20518.2	−	12784.8i	−0.718819	−	0.447891i
$935$			15485.1i			0.541623i
$936$		0			0
$937$		−	15781.5i		−	0.550225i	−0.961412	−	0.275112i	$-0.911285\pi$
$937$		−	15781.5i		−	0.550225i	0.961412	−	0.275112i	$-0.0887151\pi$
$938$	8875.85	−	14244.8i	0.308962	−	0.495853i
$939$		0			0
$940$	−32345.8	−	11045.1i	−1.12234	−	0.383245i
$941$	24664.1	−	24664.1i	0.854440	−	0.854440i	−0.136236	−	0.990676i	$-0.543501\pi$
$941$	24664.1	−	24664.1i	0.854440	−	0.854440i	0.990676	+	0.136236i	$0.0435006\pi$
$942$		0			0
$943$	−9188.01			−0.317288
$944$	7160.35	−	916.391i	0.246875	−	0.0315953i
$945$		0			0
$946$	−25259.8	+	5865.72i	−0.868147	+	0.201597i
$947$	−16466.0	+	16466.0i	−0.565017	+	0.565017i	−0.930728	−	0.365711i	$-0.880826\pi$
$947$	−16466.0	+	16466.0i	−0.565017	+	0.565017i	0.365711	+	0.930728i	$0.380826\pi$
$948$		0			0
$949$	15352.8	+	15352.8i	0.525155	+	0.525155i
$950$	−5642.94	+	9056.34i	−0.192717	+	0.309291i
$951$		0			0
$952$	8323.46	−	6793.26i	0.283367	−	0.231272i
$953$			42180.3i			1.43374i	0.697207	+	0.716870i	$0.254426\pi$
$953$			42180.3i			1.43374i	−0.697207	+	0.716870i	$0.745574\pi$
$954$		0			0
$955$	12464.2	+	12464.2i	0.422337	+	0.422337i
$956$	−49328.2	+	24215.4i	−1.66882	+	0.819228i
$957$		0			0
$958$	7999.10	+	34446.9i	0.269770	+	1.16172i
$959$	21051.3			0.708846
$960$		0			0
$961$	−4667.69			−0.156681
$962$	−9163.71	−	39462.1i	−0.307121	−	1.32257i
$963$		0			0
$964$	19873.9	−	9756.18i	0.664000	−	0.325960i
$965$	−25810.2	−	25810.2i	−0.860995	−	0.860995i
$966$		0			0
$967$			29742.4i			0.989091i	0.869152	+	0.494546i	$0.164666\pi$
$967$			29742.4i			0.989091i	−0.869152	+	0.494546i	$0.835334\pi$
$968$	6022.63	−	4915.42i	0.199974	−	0.163210i
$969$		0			0
$970$	14841.3	−	23818.8i	0.491263	−	0.788428i
$971$	−30803.6	−	30803.6i	−1.01806	−	1.01806i	−0.999834	−	0.0182247i	$-0.994199\pi$
$971$	−30803.6	−	30803.6i	−1.01806	−	1.01806i	−0.0182247	−	0.999834i	$-0.505801\pi$
$972$		0			0
$973$	−6325.98	+	6325.98i	−0.208429	+	0.208429i
$974$	28113.5	−	6528.39i	0.924860	−	0.214767i
$975$		0			0
$976$	−1521.18	−	11886.0i	−0.0498891	−	0.389816i
$977$	37576.0			1.23046			0.615232	−	0.788346i	$-0.289062\pi$
$977$	37576.0			1.23046			0.615232	+	0.788346i	$0.289062\pi$
$978$		0			0
$979$	−10552.8	+	10552.8i	−0.344502	+	0.344502i
$980$	−16229.3	−	5541.80i	−0.529007	−	0.180639i
$981$		0			0
$982$	6561.80	−	10531.0i	0.213234	−	0.342218i
$983$		−	3002.36i		−	0.0974164i	−0.998813	−	0.0487082i	$-0.984490\pi$
$983$		−	3002.36i		−	0.0974164i	0.998813	−	0.0487082i	$-0.0155104\pi$
$984$		0			0
$985$		−	12486.5i		−	0.403911i
$986$	−21705.3	−	13524.4i	−0.701051	−	0.436820i
$987$		0			0
$988$	9767.39	+	19896.8i	0.314516	+	0.640689i
$989$	6691.61	−	6691.61i	0.215147	−	0.215147i
$990$		0			0
$991$	4854.04			0.155594			0.0777971	−	0.996969i	$-0.475211\pi$
$991$	4854.04			0.155594			0.0777971	+	0.996969i	$0.475211\pi$
$992$	26081.0	+	11959.3i	0.834749	+	0.382770i
$993$		0			0
$994$	497.524	+	2142.50i	0.0158757	+	0.0683663i
$995$	−15600.3	+	15600.3i	−0.497047	+	0.497047i
$996$		0			0
$997$	−24273.6	−	24273.6i	−0.771067	−	0.771067i	0.207226	−	0.978293i	$-0.433557\pi$
$997$	−24273.6	−	24273.6i	−0.771067	−	0.771067i	−0.978293	+	0.207226i	$0.933557\pi$
$998$	−45999.3	−	28661.8i	−1.45900	−	0.909093i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.k.c.37.6	✓	24
3.2	odd	2	inner	144.4.k.c.37.7	yes	24
4.3	odd	2		576.4.k.c.433.8		24
12.11	even	2		576.4.k.c.433.5		24
16.3	odd	4		576.4.k.c.145.8		24
16.13	even	4	inner	144.4.k.c.109.6	yes	24
48.29	odd	4	inner	144.4.k.c.109.7	yes	24
48.35	even	4		576.4.k.c.145.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.k.c.37.6	✓	24	1.1	even	1	trivial
144.4.k.c.37.7	yes	24	3.2	odd	2	inner
144.4.k.c.109.6	yes	24	16.13	even	4	inner
144.4.k.c.109.7	yes	24	48.29	odd	4	inner
576.4.k.c.145.5		24	48.35	even	4
576.4.k.c.145.8		24	16.3	odd	4
576.4.k.c.433.5		24	12.11	even	2
576.4.k.c.433.8		24	4.3	odd	2

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:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	144.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.639782 - 2.75512i) q^{2} +(-7.18136 + 3.52535i) q^{4} +(5.16526 + 5.16526i) q^{5} -7.03833i q^{7} +(14.3073 + 17.5300i) q^{8} +(10.9263 - 17.5356i) q^{10} +(22.2198 + 22.2198i) q^{11} +(37.2029 - 37.2029i) q^{13} +(-19.3914 + 4.50300i) q^{14} +(39.1438 - 50.6336i) q^{16} +67.4608 q^{17} +(37.2366 - 37.2366i) q^{19} +(-55.3030 - 18.8842i) q^{20} +(47.0024 - 75.4340i) q^{22} +32.4348i q^{23} -71.6401i q^{25} +(-126.300 - 78.6966i) q^{26} +(24.8126 + 50.5448i) q^{28} +(94.7734 - 94.7734i) q^{29} -158.503 q^{31} +(-164.545 - 75.4513i) q^{32} +(-43.1602 - 185.863i) q^{34} +(36.3548 - 36.3548i) q^{35} +(192.501 + 192.501i) q^{37} +(-126.414 - 78.7678i) q^{38} +(-16.6465 + 164.448i) q^{40} +283.277i q^{41} +(-206.310 - 206.310i) q^{43} +(-237.901 - 81.2357i) q^{44} +(89.3617 - 20.7512i) q^{46} +584.883 q^{47} +293.462 q^{49} +(-197.377 + 45.8341i) q^{50} +(-136.014 + 398.320i) q^{52} +(82.5251 + 82.5251i) q^{53} +229.542i q^{55} +(123.382 - 100.699i) q^{56} +(-321.746 - 200.478i) q^{58} +(79.7567 + 79.7567i) q^{59} +(132.394 - 132.394i) q^{61} +(101.408 + 436.696i) q^{62} +(-102.604 + 501.614i) q^{64} +384.325 q^{65} +(-596.157 + 596.157i) q^{67} +(-484.460 + 237.823i) q^{68} +(-123.421 - 76.9027i) q^{70} -110.487i q^{71} +412.677i q^{73} +(407.204 - 653.522i) q^{74} +(-136.137 + 398.681i) q^{76} +(156.390 - 156.390i) q^{77} -1142.41 q^{79} +(463.724 - 59.3480i) q^{80} +(780.461 - 181.235i) q^{82} +(-656.133 + 656.133i) q^{83} +(348.453 + 348.453i) q^{85} +(-436.414 + 700.401i) q^{86} +(-71.6093 + 707.419i) q^{88} +474.926i q^{89} +(-261.846 - 261.846i) q^{91} +(-114.344 - 232.926i) q^{92} +(-374.198 - 1611.42i) q^{94} +384.673 q^{95} +1358.31 q^{97} +(-187.752 - 808.522i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{4} + 48 q^{10} - 192 q^{16} + 24 q^{19} + 232 q^{22} + 416 q^{28} + 744 q^{31} + 296 q^{34} - 16 q^{37} + 104 q^{40} - 376 q^{43} - 32 q^{46} - 1176 q^{49} - 2088 q^{52} - 808 q^{58} - 912 q^{61}+ \cdots - 2896 q^{94}+O(q^{100}) \) 24 * q + 20 * q^4 + 48 * q^10 - 192 * q^16 + 24 * q^19 + 232 * q^22 + 416 * q^28 + 744 * q^31 + 296 * q^34 - 16 * q^37 + 104 * q^40 - 376 * q^43 - 32 * q^46 - 1176 * q^49 - 2088 * q^52 - 808 * q^58 - 912 * q^61 - 2968 * q^64 + 1440 * q^67 + 2408 * q^70 + 2408 * q^76 - 328 * q^79 + 5800 * q^82 - 240 * q^85 + 2160 * q^88 - 104 * q^91 - 2896 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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