LMFDB - Embedded newform 144.4.k.b.109.1

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)

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")

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);

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:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			109.1
Character	$\chi$	$=$	144.109
Dual form			144.4.k.b.37.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.70307 - 0.832707i) q^{2} +(6.61320 + 4.50173i) q^{4} +(-11.7911 + 11.7911i) q^{5} +12.5754i q^{7} +(-14.1273 - 17.6754i) q^{8} +(41.6906 - 22.0536i) q^{10} +(17.0042 - 17.0042i) q^{11} +(-49.2384 - 49.2384i) q^{13} +(10.4716 - 33.9921i) q^{14} +(23.4688 + 59.5417i) q^{16} +51.8247 q^{17} +(-11.6655 - 11.6655i) q^{19} +(-131.057 + 24.8964i) q^{20} +(-60.1231 + 31.8041i) q^{22} -74.5524i q^{23} -153.058i q^{25} +(92.0939 + 174.096i) q^{26} +(-56.6109 + 83.1633i) q^{28} +(-211.183 - 211.183i) q^{29} -326.094 q^{31} +(-13.8571 - 180.488i) q^{32} +(-140.086 - 43.1547i) q^{34} +(-148.277 - 148.277i) q^{35} +(110.757 - 110.757i) q^{37} +(21.8187 + 41.2465i) q^{38} +(374.988 + 41.8350i) q^{40} +348.225i q^{41} +(205.838 - 205.838i) q^{43} +(189.000 - 35.9038i) q^{44} +(-62.0803 + 201.521i) q^{46} -254.983 q^{47} +184.861 q^{49} +(-127.453 + 413.728i) q^{50} +(-103.965 - 547.282i) q^{52} +(225.602 - 225.602i) q^{53} +400.995i q^{55} +(222.274 - 177.656i) q^{56} +(394.989 + 746.696i) q^{58} +(-285.442 + 285.442i) q^{59} +(286.952 + 286.952i) q^{61} +(881.456 + 271.541i) q^{62} +(-112.837 + 499.411i) q^{64} +1161.15 q^{65} +(-627.335 - 627.335i) q^{67} +(342.727 + 233.301i) q^{68} +(277.332 + 524.274i) q^{70} +274.784i q^{71} -298.190i q^{73} +(-391.612 + 207.156i) q^{74} +(-24.6312 - 129.661i) q^{76} +(213.834 + 213.834i) q^{77} -175.664 q^{79} +(-978.782 - 425.338i) q^{80} +(289.969 - 941.276i) q^{82} +(-125.254 - 125.254i) q^{83} +(-611.068 + 611.068i) q^{85} +(-727.796 + 384.991i) q^{86} +(-540.779 - 60.3313i) q^{88} -900.271i q^{89} +(619.190 - 619.190i) q^{91} +(335.615 - 493.030i) q^{92} +(689.237 + 212.326i) q^{94} +275.096 q^{95} +5.27858 q^{97} +(-499.691 - 153.935i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 20 q^{4} - 84 q^{8} + 72 q^{10} + 40 q^{11} + 348 q^{14} - 192 q^{16} + 24 q^{19} - 80 q^{20} + 704 q^{22} + 20 q^{26} - 344 q^{28} - 400 q^{29} - 744 q^{31} + 960 q^{32} - 704 q^{34} + 456 q^{35}+ \cdots - 6760 q^{98}+O(q^{100}) $ 24 * q - 20 * q^4 - 84 * q^8 + 72 * q^10 + 40 * q^11 + 348 * q^14 - 192 * q^16 + 24 * q^19 - 80 * q^20 + 704 * q^22 + 20 * q^26 - 344 * q^28 - 400 * q^29 - 744 * q^31 + 960 * q^32 - 704 * q^34 + 456 * q^35 + 16 * q^37 - 1256 * q^38 + 1744 * q^40 + 1240 * q^43 + 200 * q^44 - 1432 * q^46 - 1176 * q^49 - 708 * q^50 + 1008 * q^52 - 752 * q^53 - 1344 * q^56 + 1936 * q^58 + 1376 * q^59 - 912 * q^61 + 996 * q^62 - 56 * q^64 - 976 * q^65 - 2256 * q^67 + 1568 * q^68 - 1760 * q^70 + 2740 * q^74 - 1880 * q^76 - 1904 * q^77 + 5992 * q^79 - 712 * q^80 - 40 * q^82 - 2680 * q^83 - 240 * q^85 + 1712 * q^86 - 3936 * q^88 - 3496 * q^91 - 5296 * q^92 + 5272 * q^94 + 7728 * q^95 - 6760 * q^98

Character values

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

$n$	$37$	$65$	$127$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$	−2.70307	−	0.832707i	−0.955680	−	0.294406i
$2$	−2.70307	−	0.832707i	−0.955680	−	0.294406i
$3$		0			0
$3$		0			0
$4$	6.61320	+	4.50173i	0.826650	+	0.562717i
$5$	−11.7911	+	11.7911i	−1.05462	+	1.05462i	−0.0562056	+	0.998419i	$0.517900\pi$
$5$	−11.7911	+	11.7911i	−1.05462	+	1.05462i	−0.998419	+	0.0562056i	$0.982100\pi$
$6$		0			0
$7$			12.5754i			0.679005i	0.940605	+	0.339503i	$0.110259\pi$
$7$			12.5754i			0.679005i	−0.940605	+	0.339503i	$0.889741\pi$
$8$	−14.1273	−	17.6754i	−0.624346	−	0.781148i
$9$		0			0
$10$	41.6906	−	22.0536i	1.31837	−	0.697396i
$11$	17.0042	−	17.0042i	0.466087	−	0.466087i	−0.434557	−	0.900644i	$-0.643095\pi$
$11$	17.0042	−	17.0042i	0.466087	−	0.466087i	0.900644	+	0.434557i	$0.143095\pi$
$12$		0			0
$13$	−49.2384	−	49.2384i	−1.05048	−	1.05048i	−0.998656	−	0.0518270i	$-0.983496\pi$
$13$	−49.2384	−	49.2384i	−1.05048	−	1.05048i	−0.0518270	−	0.998656i	$-0.516504\pi$
$14$	10.4716	−	33.9921i	0.199903	−	0.648912i
$15$		0			0
$16$	23.4688	+	59.5417i	0.366700	+	0.930339i
$17$	51.8247			0.739372			0.369686	−	0.929157i	$-0.379465\pi$
$17$	51.8247			0.739372			0.369686	+	0.929157i	$0.379465\pi$
$18$		0			0
$19$	−11.6655	−	11.6655i	−0.140855	−	0.140855i	0.633163	−	0.774018i	$-0.281756\pi$
$19$	−11.6655	−	11.6655i	−0.140855	−	0.140855i	−0.774018	+	0.633163i	$0.781756\pi$
$20$	−131.057	+	24.8964i	−1.46526	+	0.278351i
$21$		0			0
$22$	−60.1231	+	31.8041i	−0.582649	+	0.308211i
$23$		−	74.5524i		−	0.675881i	−0.941168	−	0.337940i	$-0.890270\pi$
$23$		−	74.5524i		−	0.675881i	0.941168	−	0.337940i	$-0.109730\pi$
$24$		0			0
$25$		−	153.058i		−	1.22447i
$26$	92.0939	+	174.096i	0.694657	+	1.31319i
$27$		0			0
$28$	−56.6109	+	83.1633i	−0.382087	+	0.561300i
$29$	−211.183	−	211.183i	−1.35227	−	1.35227i	−0.883122	−	0.469143i	$-0.844563\pi$
$29$	−211.183	−	211.183i	−1.35227	−	1.35227i	−0.469143	−	0.883122i	$-0.655437\pi$
$30$		0			0
$31$	−326.094			−1.88930			−0.944648	−	0.328084i	$-0.893597\pi$
$31$	−326.094			−1.88930			−0.944648	+	0.328084i	$0.893597\pi$
$32$	−13.8571	−	180.488i	−0.0765506	−	0.997066i
$33$		0			0
$34$	−140.086	−	43.1547i	−0.706604	−	0.217676i
$35$	−148.277	−	148.277i	−0.716096	−	0.716096i
$36$		0			0
$37$	110.757	−	110.757i	0.492117	−	0.492117i	−0.416855	−	0.908973i	$-0.636868\pi$
$37$	110.757	−	110.757i	0.492117	−	0.492117i	0.908973	+	0.416855i	$0.136868\pi$
$38$	21.8187	+	41.2465i	0.0931436	+	0.176081i
$39$		0			0
$40$	374.988	+	41.8350i	1.48227	+	0.165367i
$41$			348.225i			1.32643i	0.748430	+	0.663214i	$0.230808\pi$
$41$			348.225i			1.32643i	−0.748430	+	0.663214i	$0.769192\pi$
$42$		0			0
$43$	205.838	−	205.838i	0.729998	−	0.729998i	−0.240621	−	0.970619i	$-0.577351\pi$
$43$	205.838	−	205.838i	0.729998	−	0.729998i	0.970619	+	0.240621i	$0.0773511\pi$
$44$	189.000	−	35.9038i	0.647566	−	0.123016i
$45$		0			0
$46$	−62.0803	+	201.521i	−0.198984	+	0.645926i
$47$	−254.983			−0.791342			−0.395671	−	0.918392i	$-0.629488\pi$
$47$	−254.983			−0.791342			−0.395671	+	0.918392i	$0.629488\pi$
$48$		0			0
$49$	184.861			0.538952
$50$	−127.453	+	413.728i	−0.360491	+	1.17020i
$51$		0			0
$52$	−103.965	−	547.282i	−0.277258	−	1.45951i
$53$	225.602	−	225.602i	0.584694	−	0.584694i	−0.351496	−	0.936189i	$-0.614327\pi$
$53$	225.602	−	225.602i	0.584694	−	0.584694i	0.936189	+	0.351496i	$0.114327\pi$
$54$		0			0
$55$			400.995i			0.983094i
$56$	222.274	−	177.656i	0.530404	−	0.423934i
$57$		0			0
$58$	394.989	+	746.696i	0.894218	+	1.69045i
$59$	−285.442	+	285.442i	−0.629854	+	0.629854i	−0.948031	−	0.318177i	$-0.896929\pi$
$59$	−285.442	+	285.442i	−0.629854	+	0.629854i	0.318177	+	0.948031i	$0.396929\pi$
$60$		0			0
$61$	286.952	+	286.952i	0.602301	+	0.602301i	0.940923	−	0.338621i	$-0.109961\pi$
$61$	286.952	+	286.952i	0.602301	+	0.602301i	−0.338621	+	0.940923i	$0.609961\pi$
$62$	881.456	+	271.541i	1.80556	+	0.556221i
$63$		0			0
$64$	−112.837	+	499.411i	−0.220385	+	0.975413i
$65$	1161.15			2.21573
$66$		0			0
$67$	−627.335	−	627.335i	−1.14390	−	1.14390i	−0.987731	−	0.156168i	$-0.950086\pi$
$67$	−627.335	−	627.335i	−1.14390	−	1.14390i	−0.156168	−	0.987731i	$-0.549914\pi$
$68$	342.727	+	233.301i	0.611202	+	0.416057i
$69$		0			0
$70$	277.332	+	524.274i	0.473536	+	0.895182i
$71$			274.784i			0.459308i	0.973272	+	0.229654i	$0.0737595\pi$
$71$			274.784i			0.459308i	−0.973272	+	0.229654i	$0.926241\pi$
$72$		0			0
$73$		−	298.190i		−	0.478089i	−0.971009	−	0.239044i	$-0.923166\pi$
$73$		−	298.190i		−	0.478089i	0.971009	−	0.239044i	$-0.0768341\pi$
$74$	−391.612	+	207.156i	−0.615189	+	0.325424i
$75$		0			0
$76$	−24.6312	−	129.661i	−0.0371763	−	0.195699i
$77$	213.834	+	213.834i	0.316475	+	0.316475i
$78$		0			0
$79$	−175.664			−0.250174			−0.125087	−	0.992146i	$-0.539921\pi$
$79$	−175.664			−0.250174			−0.125087	+	0.992146i	$0.539921\pi$
$80$	−978.782	−	425.338i	−1.36789	−	0.594428i
$81$		0			0
$82$	289.969	−	941.276i	0.390509	−	1.26764i
$83$	−125.254	−	125.254i	−0.165644	−	0.165644i	0.619418	−	0.785062i	$-0.287369\pi$
$83$	−125.254	−	125.254i	−0.165644	−	0.165644i	−0.785062	+	0.619418i	$0.787369\pi$
$84$		0			0
$85$	−611.068	+	611.068i	−0.779760	+	0.779760i
$86$	−727.796	+	384.991i	−0.912561	+	0.482729i
$87$		0			0
$88$	−540.779	−	60.3313i	−0.655082	−	0.0730834i
$89$		−	900.271i		−	1.07223i	−0.844145	−	0.536116i	$-0.819891\pi$
$89$		−	900.271i		−	1.07223i	0.844145	−	0.536116i	$-0.180109\pi$
$90$		0			0
$91$	619.190	−	619.190i	0.713283	−	0.713283i
$92$	335.615	−	493.030i	0.380329	−	0.558717i
$93$		0			0
$94$	689.237	+	212.326i	0.756270	+	0.232976i
$95$	275.096			0.297098
$96$		0			0
$97$	5.27858			0.00552534			0.00276267	−	0.999996i	$-0.499121\pi$
$97$	5.27858			0.00552534			0.00276267	+	0.999996i	$0.499121\pi$
$98$	−499.691	−	153.935i	−0.515066	−	0.158671i
$99$		0			0
$100$	689.028	−	1012.21i	0.689028	−	1.01221i
$101$	459.109	−	459.109i	0.452307	−	0.452307i	−0.443812	−	0.896120i	$-0.646374\pi$
$101$	459.109	−	459.109i	0.452307	−	0.452307i	0.896120	+	0.443812i	$0.146374\pi$
$102$		0			0
$103$			791.173i			0.756860i	0.925630	+	0.378430i	$0.123536\pi$
$103$			791.173i			0.756860i	−0.925630	+	0.378430i	$0.876464\pi$
$104$	−174.699	+	1565.91i	−0.164718	+	1.47645i
$105$		0			0
$106$	−797.678	+	421.958i	−0.730918	+	0.386643i
$107$	−1525.81	+	1525.81i	−1.37855	+	1.37855i	−0.531487	+	0.847067i	$0.678366\pi$
$107$	−1525.81	+	1525.81i	−1.37855	+	1.37855i	−0.847067	+	0.531487i	$0.821634\pi$
$108$		0			0
$109$	−413.307	−	413.307i	−0.363189	−	0.363189i	0.501796	−	0.864986i	$-0.332673\pi$
$109$	−413.307	−	413.307i	−0.363189	−	0.363189i	−0.864986	+	0.501796i	$0.832673\pi$
$110$	333.911	−	1083.92i	0.289429	−	0.939523i
$111$		0			0
$112$	−748.758	+	295.129i	−0.631705	+	0.248991i
$113$	−210.248			−0.175031			−0.0875154	−	0.996163i	$-0.527893\pi$
$113$	−210.248			−0.175031			−0.0875154	+	0.996163i	$0.527893\pi$
$114$		0			0
$115$	879.053	+	879.053i	0.712801	+	0.712801i
$116$	−445.906	−	2347.28i	−0.356908	−	1.87879i
$117$		0			0
$118$	1009.26	−	533.881i	0.787372	−	0.416506i
$119$			651.714i			0.502038i
$120$		0			0
$121$			752.715i			0.565526i
$122$	−536.704	−	1014.60i	−0.398286	−	0.752929i
$123$		0			0
$124$	−2156.52	−	1467.99i	−1.56179	−	1.06314i
$125$	330.838	+	330.838i	0.236729	+	0.236729i
$126$		0			0
$127$	177.367			0.123927			0.0619637	−	0.998078i	$-0.480264\pi$
$127$	177.367			0.123927			0.0619637	+	0.998078i	$0.480264\pi$
$128$	720.869	−	1255.99i	0.497785	−	0.867301i
$129$		0			0
$130$	−3138.66	−	966.895i	−2.11753	−	0.652325i
$131$	−938.828	−	938.828i	−0.626151	−	0.626151i	0.320946	−	0.947097i	$-0.395999\pi$
$131$	−938.828	−	938.828i	−0.626151	−	0.626151i	−0.947097	+	0.320946i	$0.895999\pi$
$132$		0			0
$133$	146.697	−	146.697i	0.0956411	−	0.0956411i
$134$	1173.35	+	2218.12i	0.756430	+	1.42997i
$135$		0			0
$136$	−732.145	−	916.020i	−0.461624	−	0.577559i
$137$		−	307.429i		−	0.191718i	−0.995395	−	0.0958592i	$-0.969440\pi$
$137$		−	307.429i		−	0.191718i	0.995395	−	0.0958592i	$-0.0305598\pi$
$138$		0			0
$139$	−1253.43	+	1253.43i	−0.764852	+	0.764852i	−0.977195	−	0.212343i	$-0.931891\pi$
$139$	−1253.43	+	1253.43i	−0.764852	+	0.764852i	0.212343	+	0.977195i	$0.431891\pi$
$140$	−313.082	−	1648.09i	−0.189002	−	0.994919i
$141$		0			0
$142$	228.815	−	742.761i	0.135223	−	0.438952i
$143$	−1674.52			−0.979233
$144$		0			0
$145$	4980.14			2.85226
$146$	−248.305	+	806.028i	−0.140752	+	0.456900i
$147$		0			0
$148$	1231.06	−	233.860i	0.683731	−	0.129886i
$149$	−1686.80	+	1686.80i	−0.927436	+	0.927436i	−0.997540	−	0.0701033i	$-0.977667\pi$
$149$	−1686.80	+	1686.80i	−0.927436	+	0.927436i	0.0701033	+	0.997540i	$0.477667\pi$
$150$		0			0
$151$			1682.23i			0.906611i	0.891355	+	0.453306i	$0.149755\pi$
$151$			1682.23i			0.906611i	−0.891355	+	0.453306i	$0.850245\pi$
$152$	−41.3894	+	370.993i	−0.0220863	+	0.197971i
$153$		0			0
$154$	−399.947	−	756.069i	−0.209277	−	0.395622i
$155$	3844.99	−	3844.99i	1.99250	−	1.99250i
$156$		0			0
$157$	−1066.81	−	1066.81i	−0.542300	−	0.542300i	0.381903	−	0.924202i	$-0.375269\pi$
$157$	−1066.81	−	1066.81i	−0.542300	−	0.542300i	−0.924202	+	0.381903i	$0.875269\pi$
$158$	474.833	+	146.277i	0.239087	+	0.0736529i
$159$		0			0
$160$	2291.54	+	1964.76i	1.13226	+	0.970798i
$161$	937.523			0.458927
$162$		0			0
$163$	−2379.54	−	2379.54i	−1.14343	−	1.14343i	−0.987817	−	0.155618i	$-0.950263\pi$
$163$	−2379.54	−	2379.54i	−1.14343	−	1.14343i	−0.155618	−	0.987817i	$-0.549737\pi$
$164$	−1567.61	+	2302.88i	−0.746403	+	1.09649i
$165$		0			0
$166$	234.271	+	442.871i	0.109536	+	0.207069i
$167$			839.991i			0.389224i	0.980880	+	0.194612i	$0.0623448\pi$
$167$			839.991i			0.389224i	−0.980880	+	0.194612i	$0.937655\pi$
$168$		0			0
$169$			2651.84i			1.20703i
$170$	2160.60	−	1142.92i	0.974768	−	0.515635i
$171$		0			0
$172$	2287.87	−	434.619i	1.01424	−	0.192671i
$173$	450.278	+	450.278i	0.197885	+	0.197885i	0.799093	−	0.601208i	$-0.205314\pi$
$173$	450.278	+	450.278i	0.197885	+	0.197885i	−0.601208	+	0.799093i	$0.705314\pi$
$174$		0			0
$175$	1924.76			0.831419
$176$	1411.53	+	613.390i	0.604533	+	0.262705i
$177$		0			0
$178$	−749.662	+	2433.50i	−0.315672	+	1.02471i
$179$	647.759	+	647.759i	0.270479	+	0.270479i	0.829293	−	0.558814i	$-0.188743\pi$
$179$	647.759	+	647.759i	0.270479	+	0.270479i	−0.558814	+	0.829293i	$0.688743\pi$
$180$		0			0
$181$	−755.750	+	755.750i	−0.310356	+	0.310356i	−0.845047	−	0.534691i	$-0.820428\pi$
$181$	−755.750	+	755.750i	−0.310356	+	0.310356i	0.534691	+	0.845047i	$0.320428\pi$
$182$	−2189.32	+	1158.11i	−0.891666	+	0.471676i
$183$		0			0
$184$	−1317.74	+	1053.23i	−0.527963	+	0.421983i
$185$			2611.89i			1.03800i
$186$		0			0
$187$	881.237	−	881.237i	0.344612	−	0.344612i
$188$	−1686.25	−	1147.86i	−0.654163	−	0.445301i
$189$		0			0
$190$	−743.605	−	229.075i	−0.283931	−	0.0874675i
$191$	−760.485			−0.288098			−0.144049	−	0.989571i	$-0.546012\pi$
$191$	−760.485			−0.288098			−0.144049	+	0.989571i	$0.546012\pi$
$192$		0			0
$193$	2184.99			0.814915			0.407458	−	0.913224i	$-0.366415\pi$
$193$	2184.99			0.814915			0.407458	+	0.913224i	$0.366415\pi$
$194$	−14.2684	−	4.39551i	−0.00528046	−	0.00162670i
$195$		0			0
$196$	1222.52	+	832.193i	0.445525	+	0.303277i
$197$	−83.4121	+	83.4121i	−0.0301668	+	0.0301668i	−0.722029	−	0.691862i	$-0.756790\pi$
$197$	−83.4121	+	83.4121i	−0.0301668	+	0.0301668i	0.691862	+	0.722029i	$0.256790\pi$
$198$		0			0
$199$			1734.18i			0.617754i	0.951102	+	0.308877i	$0.0999531\pi$
$199$			1734.18i			0.617754i	−0.951102	+	0.308877i	$0.900047\pi$
$200$	−2705.36	+	2162.31i	−0.956490	+	0.764491i
$201$		0			0
$202$	−1623.31	+	858.701i	−0.565423	+	0.299099i
$203$	2655.70	−	2655.70i	0.918195	−	0.918195i
$204$		0			0
$205$	−4105.94	−	4105.94i	−1.39888	−	1.39888i
$206$	658.815	−	2138.60i	0.222824	−	0.723316i
$207$		0			0
$208$	1776.17	−	4087.31i	0.592093	−	1.36252i
$209$	−396.723			−0.131301
$210$		0			0
$211$	−365.900	−	365.900i	−0.119382	−	0.119382i	0.644892	−	0.764274i	$-0.276902\pi$
$211$	−365.900	−	365.900i	−0.119382	−	0.119382i	−0.764274	+	0.644892i	$0.776902\pi$
$212$	2507.55	−	476.350i	0.812354	−	0.154320i
$213$		0			0
$214$	5394.91	−	2853.81i	1.72331	−	0.911602i
$215$			4854.09i			1.53975i
$216$		0			0
$217$		−	4100.75i		−	1.28284i
$218$	773.035	+	1461.36i	0.240168	+	0.454018i
$219$		0			0
$220$	−1805.17	+	2651.86i	−0.553203	+	0.812674i
$221$	−2551.77	−	2551.77i	−0.776698	−	0.776698i
$222$		0			0
$223$	−4037.17			−1.21233			−0.606164	−	0.795340i	$-0.707292\pi$
$223$	−4037.17			−1.21233			−0.606164	+	0.795340i	$0.707292\pi$
$224$	2269.70	−	174.258i	0.677013	−	0.0519782i
$225$		0			0
$226$	568.316	+	175.075i	0.167274	+	0.0515302i
$227$	1502.81	+	1502.81i	0.439407	+	0.439407i	0.891812	−	0.452406i	$-0.149434\pi$
$227$	1502.81	+	1502.81i	0.439407	+	0.439407i	−0.452406	+	0.891812i	$0.649434\pi$
$228$		0			0
$229$	−3443.65	+	3443.65i	−0.993723	+	0.993723i	−0.999980	−	0.00625772i	$-0.998008\pi$
$229$	−3443.65	+	3443.65i	−0.993723	+	0.993723i	0.00625772	+	0.999980i	$0.498008\pi$
$230$	−1644.15	−	3108.14i	−0.471357	−	0.891063i
$231$		0			0
$232$	−749.283	+	6716.19i	−0.212038	+	1.90060i
$233$		−	1139.00i		−	0.320250i	−0.987097	−	0.160125i	$-0.948810\pi$
$233$		−	1139.00i		−	0.320250i	0.987097	−	0.160125i	$-0.0511898\pi$
$234$		0			0
$235$	3006.52	−	3006.52i	0.834569	−	0.834569i
$236$	−3172.67	+	602.702i	−0.875098	+	0.166240i
$237$		0			0
$238$	542.686	−	1761.63i	0.147803	−	0.479788i
$239$	4603.80			1.24600			0.623002	−	0.782220i	$-0.285913\pi$
$239$	4603.80			1.24600			0.623002	+	0.782220i	$0.285913\pi$
$240$		0			0
$241$	−4681.58			−1.25132			−0.625658	−	0.780097i	$-0.715170\pi$
$241$	−4681.58			−1.25132			−0.625658	+	0.780097i	$0.715170\pi$
$242$	626.791	−	2034.64i	0.166494	−	0.540462i
$243$		0			0
$244$	605.889	+	3189.45i	0.158967	+	0.836817i
$245$	−2179.70	+	2179.70i	−0.568392	+	0.568392i
$246$		0			0
$247$			1148.78i			0.295931i
$248$	4606.84	+	5763.83i	1.17957	+	1.47582i
$249$		0			0
$250$	−618.788	−	1169.77i	−0.156542	−	0.295931i
$251$	476.440	−	476.440i	0.119811	−	0.119811i	−0.644659	−	0.764470i	$-0.723001\pi$
$251$	476.440	−	476.440i	0.119811	−	0.119811i	0.764470	+	0.644659i	$0.223001\pi$
$252$		0			0
$253$	−1267.70	−	1267.70i	−0.315019	−	0.315019i
$254$	−479.436	−	147.695i	−0.118435	−	0.0364850i
$255$		0			0
$256$	−2994.43	+	2794.75i	−0.731062	+	0.682311i
$257$	−925.326			−0.224592			−0.112296	−	0.993675i	$-0.535821\pi$
$257$	−925.326			−0.224592			−0.112296	+	0.993675i	$0.535821\pi$
$258$		0			0
$259$	1392.81	+	1392.81i	0.334150	+	0.334150i
$260$	7678.89	+	5227.17i	1.83163	+	1.24683i
$261$		0			0
$262$	1755.95	+	3319.49i	0.414058	+	0.782743i
$263$		−	3477.65i		−	0.815365i	−0.913124	−	0.407683i	$-0.866337\pi$
$263$		−	3477.65i		−	0.815365i	0.913124	−	0.407683i	$-0.133663\pi$
$264$		0			0
$265$			5320.17i			1.23327i
$266$	−518.689	+	274.378i	−0.119560	+	0.0632450i
$267$		0			0
$268$	−1324.60	−	6972.79i	−0.301913	−	1.58929i
$269$	−3773.37	−	3773.37i	−0.855266	−	0.855266i	0.135510	−	0.990776i	$-0.456733\pi$
$269$	−3773.37	−	3773.37i	−0.855266	−	0.855266i	−0.990776	+	0.135510i	$0.956733\pi$
$270$		0			0
$271$	8007.09			1.79482			0.897410	−	0.441198i	$-0.145446\pi$
$271$	8007.09			1.79482			0.897410	+	0.441198i	$0.145446\pi$
$272$	1216.26	+	3085.73i	0.271128	+	0.687867i
$273$		0			0
$274$	−255.998	+	831.002i	−0.0564431	+	0.183221i
$275$	−2602.63	−	2602.63i	−0.570708	−	0.570708i
$276$		0			0
$277$	4550.73	−	4550.73i	0.987101	−	0.987101i	−0.0128173	−	0.999918i	$-0.504080\pi$
$277$	4550.73	−	4550.73i	0.987101	−	0.987101i	0.999918	+	0.0128173i	$0.00407998\pi$
$278$	4431.85	−	2344.37i	0.956131	−	0.505777i
$279$		0			0
$280$	−526.090	+	4715.60i	−0.112285	+	1.00647i
$281$		−	3333.05i		−	0.707592i	−0.935323	−	0.353796i	$-0.884891\pi$
$281$		−	3333.05i		−	0.707592i	0.935323	−	0.353796i	$-0.115109\pi$
$282$		0			0
$283$	6242.23	−	6242.23i	1.31117	−	1.31117i	0.390621	−	0.920552i	$-0.372260\pi$
$283$	6242.23	−	6242.23i	1.31117	−	1.31117i	0.920552	−	0.390621i	$-0.127740\pi$
$284$	−1237.00	+	1817.20i	−0.258460	+	0.379687i
$285$		0			0
$286$	4526.35	+	1394.38i	0.935834	+	0.288292i
$287$	−4379.05			−0.900651
$288$		0			0
$289$	−2227.20			−0.453329
$290$	−13461.7	−	4147.00i	−2.72585	−	0.839725i
$291$		0			0
$292$	1342.37	−	1971.99i	0.269028	−	0.395212i
$293$	2458.24	−	2458.24i	0.490143	−	0.490143i	−0.418208	−	0.908351i	$-0.637342\pi$
$293$	2458.24	−	2458.24i	0.490143	−	0.490143i	0.908351	+	0.418208i	$0.137342\pi$
$294$		0			0
$295$		−	6731.33i		−	1.32852i
$296$	−3522.37	−	392.969i	−0.691668	−	0.0771651i
$297$		0			0
$298$	5964.15	−	3154.93i	1.15938	−	0.613290i
$299$	−3670.84	+	3670.84i	−0.710001	+	0.710001i
$300$		0			0
$301$	2588.48	+	2588.48i	0.495673	+	0.495673i
$302$	1400.81	−	4547.20i	0.266912	−	0.866431i
$303$		0			0
$304$	420.807	−	968.356i	0.0793912	−	0.182694i
$305$	−6766.93			−1.27040
$306$		0			0
$307$	−3008.49	−	3008.49i	−0.559295	−	0.559295i	0.369812	−	0.929107i	$-0.379422\pi$
$307$	−3008.49	−	3008.49i	−0.559295	−	0.559295i	−0.929107	+	0.369812i	$0.879422\pi$
$308$	451.503	+	2376.75i	0.0835284	+	0.439700i
$309$		0			0
$310$	−13595.1	+	7191.55i	−2.49080	+	1.31759i
$311$		−	4507.09i		−	0.821779i	−0.911685	−	0.410890i	$-0.865218\pi$
$311$		−	4507.09i		−	0.821779i	0.911685	−	0.410890i	$-0.134782\pi$
$312$		0			0
$313$		−	5237.61i		−	0.945837i	−0.881106	−	0.472919i	$-0.843200\pi$
$313$		−	5237.61i		−	0.945837i	0.881106	−	0.472919i	$-0.156800\pi$
$314$	1995.33	+	3772.02i	0.358609	+	0.677922i
$315$		0			0
$316$	−1161.70	−	790.794i	−0.206807	−	0.140777i
$317$	989.530	+	989.530i	0.175323	+	0.175323i	0.789314	−	0.613990i	$-0.210437\pi$
$317$	989.530	+	989.530i	0.175323	+	0.175323i	−0.613990	+	0.789314i	$0.710437\pi$
$318$		0			0
$319$	−7181.99			−1.26055
$320$	−4558.13	−	7219.06i	−0.796272	−	1.26112i
$321$		0			0
$322$	−2534.19	−	780.682i	−0.438587	−	0.135111i
$323$	−604.559	−	604.559i	−0.104144	−	0.104144i
$324$		0			0
$325$	−7536.35	+	7536.35i	−1.28628	+	1.28628i
$326$	4450.61	+	8413.52i	0.756124	+	1.42939i
$327$		0			0
$328$	6154.99	−	4919.48i	1.03614	−	0.828150i
$329$		−	3206.50i		−	0.537325i
$330$		0			0
$331$	−5652.55	+	5652.55i	−0.938647	+	0.938647i	−0.998224	−	0.0595765i	$-0.981025\pi$
$331$	−5652.55	+	5652.55i	−0.938647	+	0.938647i	0.0595765	+	0.998224i	$0.481025\pi$
$332$	−264.470	−	1392.19i	−0.0437189	−	0.230140i
$333$		0			0
$334$	699.466	−	2270.56i	0.114590	−	0.371974i
$335$	14793.9			2.41277
$336$		0			0
$337$	291.315			0.0470888			0.0235444	−	0.999723i	$-0.492505\pi$
$337$	291.315			0.0470888			0.0235444	+	0.999723i	$0.492505\pi$
$338$	2208.21	−	7168.13i	0.355357	−	1.15353i
$339$		0			0
$340$	−6791.98	+	1290.25i	−1.08337	+	0.205805i
$341$	−5544.96	+	5544.96i	−0.880577	+	0.880577i
$342$		0			0
$343$			6638.03i			1.04496i
$344$	−6546.19	−	730.317i	−1.02601	−	0.114465i
$345$		0			0
$346$	−842.185	−	1592.08i	−0.130856	−	0.247373i
$347$	−6530.87	+	6530.87i	−1.01036	+	1.01036i	−0.0104158	+	0.999946i	$0.503316\pi$
$347$	−6530.87	+	6530.87i	−1.01036	+	1.01036i	−0.999946	+	0.0104158i	$0.996684\pi$
$348$		0			0
$349$	7267.79	+	7267.79i	1.11472	+	1.11472i	0.992504	+	0.122213i	$0.0389990\pi$
$349$	7267.79	+	7267.79i	1.11472	+	1.11472i	0.122213	+	0.992504i	$0.461001\pi$
$350$	−5202.77	−	1602.76i	−0.794571	−	0.244775i
$351$		0			0
$352$	−3304.68	−	2833.43i	−0.500399	−	0.429040i
$353$	8253.63			1.24447			0.622233	−	0.782832i	$-0.286226\pi$
$353$	8253.63			1.24447			0.622233	+	0.782832i	$0.286226\pi$
$354$		0			0
$355$	−3240.00	−	3240.00i	−0.484398	−	0.484398i
$356$	4052.78	−	5953.67i	0.603362	−	0.886360i
$357$		0			0
$358$	−1211.55	−	2290.33i	−0.178861	−	0.338122i
$359$		−	4827.70i		−	0.709738i	−0.934916	−	0.354869i	$-0.884525\pi$
$359$		−	4827.70i		−	0.709738i	0.934916	−	0.354869i	$-0.115475\pi$
$360$		0			0
$361$		−	6586.83i		−	0.960320i
$362$	2672.16	−	1413.53i	0.387972	−	0.205230i
$363$		0			0
$364$	6882.26	−	1307.40i	0.991012	−	0.188259i
$365$	3515.97	+	3515.97i	0.504204	+	0.504204i
$366$		0			0
$367$	−2556.37			−0.363600			−0.181800	−	0.983336i	$-0.558192\pi$
$367$	−2556.37			−0.363600			−0.181800	+	0.983336i	$0.558192\pi$
$368$	4438.98	−	1749.66i	0.628798	−	0.247846i
$369$		0			0
$370$	2174.93	−	7060.12i	0.305593	−	0.991995i
$371$	2837.02	+	2837.02i	0.397010	+	0.397010i
$372$		0			0
$373$	1141.57	−	1141.57i	0.158468	−	0.158468i	−0.623420	−	0.781887i	$-0.714257\pi$
$373$	1141.57	−	1141.57i	0.158468	−	0.158468i	0.781887	+	0.623420i	$0.214257\pi$
$374$	−3115.86	+	1648.23i	−0.430795	+	0.227883i
$375$		0			0
$376$	3602.23	+	4506.92i	0.494071	+	0.618155i
$377$			20796.6i			2.84106i
$378$		0			0
$379$	−2602.11	+	2602.11i	−0.352668	+	0.352668i	−0.861101	−	0.508433i	$-0.830225\pi$
$379$	−2602.11	+	2602.11i	−0.352668	+	0.352668i	0.508433	+	0.861101i	$0.330225\pi$
$380$	1819.27	+	1238.41i	0.245596	+	0.167182i
$381$		0			0
$382$	2055.65	+	633.261i	0.275330	+	0.0848179i
$383$	370.858			0.0494777			0.0247388	−	0.999694i	$-0.492125\pi$
$383$	370.858			0.0494777			0.0247388	+	0.999694i	$0.492125\pi$
$384$		0			0
$385$	−5042.65			−0.667526
$386$	−5906.17	−	1819.45i	−0.778799	−	0.239916i
$387$		0			0
$388$	34.9083	+	23.7627i	0.00456752	+	0.00310920i
$389$	6457.55	−	6457.55i	0.841673	−	0.841673i	−0.147403	−	0.989077i	$-0.547091\pi$
$389$	6457.55	−	6457.55i	0.841673	−	0.841673i	0.989077	+	0.147403i	$0.0470914\pi$
$390$		0			0
$391$		−	3863.66i		−	0.499728i
$392$	−2611.59	−	3267.48i	−0.336492	−	0.421001i
$393$		0			0
$394$	294.927	−	156.011i	0.0377112	−	0.0199486i
$395$	2071.27	−	2071.27i	0.263840	−	0.263840i
$396$		0			0
$397$	−232.353	−	232.353i	−0.0293739	−	0.0293739i	0.692267	−	0.721641i	$-0.256612\pi$
$397$	−232.353	−	232.353i	−0.0293739	−	0.0293739i	−0.721641	+	0.692267i	$0.756612\pi$
$398$	1444.07	−	4687.62i	0.181871	−	0.590375i
$399$		0			0
$400$	9113.36	−	3592.10i	1.13917	−	0.449012i
$401$	119.615			0.0148960			0.00744801	−	0.999972i	$-0.497629\pi$
$401$	119.615			0.0148960			0.00744801	+	0.999972i	$0.497629\pi$
$402$		0			0
$403$	16056.4	+	16056.4i	1.98467	+	1.98467i
$404$	5102.96	−	969.393i	0.628421	−	0.119379i
$405$		0			0
$406$	−9389.97	+	4967.13i	−1.14782	+	0.607179i
$407$		−	3766.67i		−	0.458739i
$408$		0			0
$409$		−	626.952i		−	0.0757965i	−0.999282	−	0.0378983i	$-0.987934\pi$
$409$		−	626.952i		−	0.0757965i	0.999282	−	0.0378983i	$-0.0120663\pi$
$410$	7679.60	+	14517.7i	0.925045	+	1.74873i
$411$		0			0
$412$	−3561.65	+	5232.18i	−0.425897	+	0.625658i
$413$	−3589.54	−	3589.54i	−0.427674	−	0.427674i
$414$		0			0
$415$	2953.76			0.349384
$416$	−8204.65	+	9569.26i	−0.966986	+	1.12782i
$417$		0			0
$418$	1072.37	+	330.354i	0.125482	+	0.0386559i
$419$	4981.58	+	4981.58i	0.580826	+	0.580826i	0.935130	−	0.354304i	$-0.115282\pi$
$419$	4981.58	+	4981.58i	0.580826	+	0.580826i	−0.354304	+	0.935130i	$0.615282\pi$
$420$		0			0
$421$	10770.4	−	10770.4i	1.24683	−	1.24683i	0.289723	−	0.957110i	$-0.406437\pi$
$421$	10770.4	−	10770.4i	1.24683	−	1.24683i	0.957110	−	0.289723i	$-0.0935632\pi$
$422$	684.366	+	1293.74i	0.0789442	+	0.149238i
$423$		0			0
$424$	−7174.74	−	800.441i	−0.821783	−	0.0916813i
$425$		−	7932.20i		−	0.905337i
$426$		0			0
$427$	−3608.52	+	3608.52i	−0.408966	+	0.408966i
$428$	−16959.2	+	3221.69i	−1.91532	+	0.363846i
$429$		0			0
$430$	4042.03	−	13120.9i	0.453312	−	1.47151i
$431$	−7546.94			−0.843442			−0.421721	−	0.906726i	$-0.638574\pi$
$431$	−7546.94			−0.843442			−0.421721	+	0.906726i	$0.638574\pi$
$432$		0			0
$433$	5614.17			0.623094			0.311547	−	0.950231i	$-0.399153\pi$
$433$	5614.17			0.623094			0.311547	+	0.950231i	$0.399153\pi$
$434$	−3414.72	+	11084.6i	−0.377677	+	1.22599i
$435$		0			0
$436$	−872.684	−	4593.88i	−0.0958578	−	0.504603i
$437$	−869.689	+	869.689i	−0.0952010	+	0.0952010i
$438$		0			0
$439$			7198.92i			0.782656i	0.920251	+	0.391328i	$0.127984\pi$
$439$			7198.92i			0.782656i	−0.920251	+	0.391328i	$0.872016\pi$
$440$	7087.73	−	5664.99i	0.767942	−	0.613790i
$441$		0			0
$442$	4772.73	+	9022.48i	0.513610	+	0.970940i
$443$	5582.95	−	5582.95i	0.598767	−	0.598767i	−0.341217	−	0.939984i	$-0.610839\pi$
$443$	5582.95	−	5582.95i	0.598767	−	0.598767i	0.939984	+	0.341217i	$0.110839\pi$
$444$		0			0
$445$	10615.2	+	10615.2i	1.13080	+	1.13080i
$446$	10912.8	+	3361.78i	1.15860	+	0.356917i
$447$		0			0
$448$	−6280.28	−	1418.96i	−0.662311	−	0.149642i
$449$	−1224.53			−0.128706			−0.0643530	−	0.997927i	$-0.520498\pi$
$449$	−1224.53			−0.128706			−0.0643530	+	0.997927i	$0.520498\pi$
$450$		0			0
$451$	5921.28	+	5921.28i	0.618231	+	0.618231i
$452$	−1390.41	−	946.481i	−0.144689	−	0.0984927i
$453$		0			0
$454$	−2810.81	−	5313.62i	−0.290568	−	0.549297i
$455$			14601.8i			1.50449i
$456$		0			0
$457$			11182.3i			1.14461i	0.820040	+	0.572307i	$0.193951\pi$
$457$			11182.3i			1.14461i	−0.820040	+	0.572307i	$0.806049\pi$
$458$	12176.0	−	6440.88i	1.24224	−	0.657123i
$459$		0			0
$460$	1856.09	+	9770.61i	0.188132	+	0.990341i
$461$	11334.9	+	11334.9i	1.14516	+	1.14516i	0.987492	+	0.157670i	$0.0503982\pi$
$461$	11334.9	+	11334.9i	1.14516	+	1.14516i	0.157670	+	0.987492i	$0.449602\pi$
$462$		0			0
$463$	3013.37			0.302469			0.151234	−	0.988498i	$-0.451675\pi$
$463$	3013.37			0.302469			0.151234	+	0.988498i	$0.451675\pi$
$464$	7617.98	−	17530.4i	0.762189	−	1.75394i
$465$		0			0
$466$	−948.453	+	3078.80i	−0.0942837	+	0.306057i
$467$	843.917	+	843.917i	0.0836228	+	0.0836228i	0.747681	−	0.664058i	$-0.231167\pi$
$467$	843.917	+	843.917i	0.0836228	+	0.0836228i	−0.664058	+	0.747681i	$0.731167\pi$
$468$		0			0
$469$	7888.96	−	7888.96i	0.776713	−	0.776713i
$470$	−10630.4	+	5623.29i	−1.04328	+	0.551879i
$471$		0			0
$472$	9077.83	+	1012.76i	0.885256	+	0.0987625i
$473$		−	7000.20i		−	0.680485i
$474$		0			0
$475$	−1785.50	+	1785.50i	−0.172472	+	0.172472i
$476$	−2933.84	+	4309.91i	−0.282505	+	0.415009i
$477$		0			0
$478$	−12444.4	−	3833.62i	−1.19078	−	0.366832i
$479$	−10569.4			−1.00821			−0.504103	−	0.863644i	$-0.668177\pi$
$479$	−10569.4			−1.00821			−0.504103	+	0.863644i	$0.668177\pi$
$480$		0			0
$481$	−10907.0			−1.03392
$482$	12654.7	+	3898.39i	1.19586	+	0.368395i
$483$		0			0
$484$	−3388.52	+	4977.85i	−0.318231	+	0.467492i
$485$	−62.2400	+	62.2400i	−0.00582716	+	0.00582716i
$486$		0			0
$487$		−	1579.10i		−	0.146932i	−0.997298	−	0.0734661i	$-0.976594\pi$
$487$		−	1579.10i		−	0.146932i	0.997298	−	0.0734661i	$-0.0234061\pi$
$488$	1018.11	−	9125.83i	0.0944422	−	0.846531i
$489$		0			0
$490$	7706.95	−	4076.84i	0.710539	−	0.375863i
$491$	−8650.50	+	8650.50i	−0.795095	+	0.795095i	−0.982318	−	0.187222i	$-0.940052\pi$
$491$	−8650.50	+	8650.50i	−0.795095	+	0.795095i	0.187222	+	0.982318i	$0.440052\pi$
$492$		0			0
$493$	−10944.5	−	10944.5i	−0.999828	−	0.999828i
$494$	956.595	−	3105.23i	0.0871240	−	0.282816i
$495$		0			0
$496$	−7653.04	−	19416.2i	−0.692806	−	1.75769i
$497$	−3455.51			−0.311873
$498$		0			0
$499$	−11942.3	−	11942.3i	−1.07137	−	1.07137i	−0.997250	−	0.0741168i	$-0.976386\pi$
$499$	−11942.3	−	11942.3i	−1.07137	−	1.07137i	−0.0741168	−	0.997250i	$-0.523614\pi$
$500$	698.554	+	3677.24i	0.0624806	+	0.328903i
$501$		0			0
$502$	−1684.59	+	891.117i	−0.149775	+	0.0792281i
$503$		−	3290.09i		−	0.291646i	−0.989311	−	0.145823i	$-0.953417\pi$
$503$		−	3290.09i		−	0.291646i	0.989311	−	0.145823i	$-0.0465830\pi$
$504$		0			0
$505$			10826.8i			0.954029i
$506$	2371.07	+	4482.32i	0.208314	+	0.393801i
$507$		0			0
$508$	1172.96	+	798.458i	0.102445	+	0.0697360i
$509$	−1392.67	−	1392.67i	−0.121275	−	0.121275i	0.643864	−	0.765140i	$-0.277330\pi$
$509$	−1392.67	−	1392.67i	−0.121275	−	0.121275i	−0.765140	+	0.643864i	$0.777330\pi$
$510$		0			0
$511$	3749.84			0.324625
$512$	10421.4	−	5060.92i	0.899538	−	0.436842i
$513$		0			0
$514$	2501.22	+	770.525i	0.214639	+	0.0661214i
$515$	−9328.77	−	9328.77i	−0.798203	−	0.798203i
$516$		0			0
$517$	−4335.78	+	4335.78i	−0.368834	+	0.368834i
$518$	−2605.06	−	4924.66i	−0.220965	−	0.417717i
$519$		0			0
$520$	−16403.9	−	20523.7i	−1.38338	−	1.73081i
$521$			1175.72i			0.0988661i	0.998777	+	0.0494331i	$0.0157414\pi$
$521$			1175.72i			0.0988661i	−0.998777	+	0.0494331i	$0.984259\pi$
$522$		0			0
$523$	−9451.05	+	9451.05i	−0.790183	+	0.790183i	−0.981524	−	0.191341i	$-0.938716\pi$
$523$	−9451.05	+	9451.05i	−0.790183	+	0.790183i	0.191341	+	0.981524i	$0.438716\pi$
$524$	−1982.30	−	10435.0i	−0.165262	−	0.869953i
$525$		0			0
$526$	−2895.86	+	9400.34i	−0.240049	+	0.779228i
$527$	−16899.7			−1.39689
$528$		0			0
$529$	6608.93			0.543185
$530$	4430.14	−	14380.8i	0.363081	−	1.17861i
$531$		0			0
$532$	1630.53	−	309.747i	0.132881	−	0.0252429i
$533$	17146.0	−	17146.0i	1.39339	−	1.39339i
$534$		0			0
$535$		−	35981.7i		−	2.90771i
$536$	−2225.80	+	19951.0i	−0.179366	+	1.60774i
$537$		0			0
$538$	7057.58	+	13341.8i	0.565565	+	1.06916i
$539$	3143.40	−	3143.40i	0.251198	−	0.251198i
$540$		0			0
$541$	2893.97	+	2893.97i	0.229984	+	0.229984i	0.812686	−	0.582702i	$-0.198004\pi$
$541$	2893.97	+	2893.97i	0.229984	+	0.229984i	−0.582702	+	0.812686i	$0.698004\pi$
$542$	−21643.7	−	6667.56i	−1.71527	−	0.528406i
$543$		0			0
$544$	−718.141	−	9353.74i	−0.0565994	−	0.737203i
$545$	9746.66			0.766057
$546$		0			0
$547$	−956.857	−	956.857i	−0.0747939	−	0.0747939i	0.668720	−	0.743514i	$-0.266842\pi$
$547$	−956.857	−	956.857i	−0.0747939	−	0.0747939i	−0.743514	+	0.668720i	$0.766842\pi$
$548$	1383.96	−	2033.09i	0.107883	−	0.158484i
$549$		0			0
$550$	4867.88	+	9202.34i	0.377394	+	0.713435i
$551$			4927.09i			0.380946i
$552$		0			0
$553$		−	2209.04i		−	0.169870i
$554$	−16090.4	+	8511.53i	−1.23396	+	0.652744i
$555$		0			0
$556$	−13931.8	+	2646.57i	−1.06266	+	0.201870i
$557$	8482.07	+	8482.07i	0.645236	+	0.645236i	0.951838	−	0.306602i	$-0.0991919\pi$
$557$	8482.07	+	8482.07i	0.645236	+	0.645236i	−0.306602	+	0.951838i	$0.599192\pi$
$558$		0			0
$559$	−20270.2			−1.53370
$560$	5348.77	−	12308.5i	0.403619	−	0.928804i
$561$		0			0
$562$	−2775.46	+	9009.48i	−0.208319	+	0.676232i
$563$	−9461.78	−	9461.78i	−0.708289	−	0.708289i	0.257886	−	0.966175i	$-0.416974\pi$
$563$	−9461.78	−	9461.78i	−0.708289	−	0.708289i	−0.966175	+	0.257886i	$0.916974\pi$
$564$		0			0
$565$	2479.05	−	2479.05i	0.184592	−	0.184592i
$566$	−22071.1	+	11675.2i	−1.63908	+	0.867044i
$567$		0			0
$568$	4856.91	−	3881.97i	0.358788	−	0.286767i
$569$		−	7197.61i		−	0.530298i	−0.964207	−	0.265149i	$-0.914579\pi$
$569$		−	7197.61i		−	0.530298i	0.964207	−	0.265149i	$-0.0854211\pi$
$570$		0			0
$571$	990.936	−	990.936i	0.0726259	−	0.0726259i	−0.669861	−	0.742487i	$-0.733646\pi$
$571$	990.936	−	990.936i	0.0726259	−	0.0726259i	0.742487	+	0.669861i	$0.233646\pi$
$572$	−11073.9	−	7538.24i	−0.809483	−	0.551031i
$573$		0			0
$574$	11836.9	+	3646.46i	0.860735	+	0.265157i
$575$	−11410.9			−0.827594
$576$		0			0
$577$	−14836.1			−1.07043			−0.535214	−	0.844717i	$-0.679769\pi$
$577$	−14836.1			−1.07043			−0.535214	+	0.844717i	$0.679769\pi$
$578$	6020.29	+	1854.61i	0.433237	+	0.133463i
$579$		0			0
$580$	32934.7	+	22419.3i	2.35782	+	1.60502i
$581$	1575.11	−	1575.11i	0.112473	−	0.112473i
$582$		0			0
$583$		−	7672.35i		−	0.545036i
$584$	−5270.61	+	4212.63i	−0.373458	+	0.298493i
$585$		0			0
$586$	−8691.80	+	4597.81i	−0.612721	+	0.324119i
$587$	−4038.29	+	4038.29i	−0.283949	+	0.283949i	−0.834682	−	0.550733i	$-0.814348\pi$
$587$	−4038.29	+	4038.29i	−0.283949	+	0.283949i	0.550733	+	0.834682i	$0.314348\pi$
$588$		0			0
$589$	3804.04	+	3804.04i	0.266116	+	0.266116i
$590$	−5605.23	+	18195.3i	−0.391125	+	1.26964i
$591$		0			0
$592$	9194.00	+	3995.33i	0.638296	+	0.277376i
$593$	−11081.1			−0.767361			−0.383681	−	0.923466i	$-0.625344\pi$
$593$	−11081.1			−0.767361			−0.383681	+	0.923466i	$0.625344\pi$
$594$		0			0
$595$	−7684.40	−	7684.40i	−0.529461	−	0.529461i
$596$	−18748.7	+	3561.62i	−1.28855	+	0.244781i
$597$		0			0
$598$	12979.3	−	6865.82i	0.887563	−	0.469506i
$599$		−	7036.50i		−	0.479973i	−0.970776	−	0.239986i	$-0.922857\pi$
$599$		−	7036.50i		−	0.479973i	0.970776	−	0.239986i	$-0.0771430\pi$
$600$		0			0
$601$		−	24290.7i		−	1.64865i	−0.566117	−	0.824325i	$-0.691555\pi$
$601$		−	24290.7i		−	1.64865i	0.566117	−	0.824325i	$-0.308445\pi$
$602$	−4841.40	−	9152.29i	−0.327775	−	0.619634i
$603$		0			0
$604$	−7572.97	+	11125.0i	−0.510165	+	0.749450i
$605$	−8875.31	−	8875.31i	−0.596418	−	0.596418i
$606$		0			0
$607$	10931.5			0.730967			0.365484	−	0.930818i	$-0.380904\pi$
$607$	10931.5			0.730967			0.365484	+	0.930818i	$0.380904\pi$
$608$	−1943.83	+	2267.13i	−0.129659	+	0.151224i
$609$		0			0
$610$	18291.5	+	5634.87i	1.21410	+	0.374015i
$611$	12555.0	+	12555.0i	0.831292	+	0.831292i
$612$		0			0
$613$	−2217.95	+	2217.95i	−0.146137	+	0.146137i	−0.776390	−	0.630253i	$-0.782951\pi$
$613$	−2217.95	+	2217.95i	−0.146137	+	0.146137i	0.630253	+	0.776390i	$0.282951\pi$
$614$	5626.97	+	10637.4i	0.369847	+	0.699167i
$615$		0			0
$616$	758.688	−	6800.49i	0.0496240	−	0.444804i
$617$		−	26271.5i		−	1.71418i	−0.515164	−	0.857092i	$-0.672269\pi$
$617$		−	26271.5i		−	1.71418i	0.515164	−	0.857092i	$-0.327731\pi$
$618$		0			0
$619$	−11171.8	+	11171.8i	−0.725417	+	0.725417i	−0.969703	−	0.244287i	$-0.921446\pi$
$619$	−11171.8	+	11171.8i	−0.725417	+	0.725417i	0.244287	+	0.969703i	$0.421446\pi$
$620$	42736.9	−	8118.58i	2.76831	−	0.525887i
$621$		0			0
$622$	−3753.08	+	12183.0i	−0.241937	+	0.785358i
$623$	11321.2			0.728050
$624$		0			0
$625$	11330.4			0.725148
$626$	−4361.39	+	14157.6i	−0.278460	+	0.903918i
$627$		0			0
$628$	−2252.54	−	11857.6i	−0.143131	−	0.753453i
$629$	5739.95	−	5739.95i	0.363858	−	0.363858i
$630$		0			0
$631$		−	17415.1i		−	1.09871i	−0.835590	−	0.549354i	$-0.814874\pi$
$631$		−	17415.1i		−	1.09871i	0.835590	−	0.549354i	$-0.185126\pi$
$632$	2481.67	+	3104.93i	0.156195	+	0.195423i
$633$		0			0
$634$	−1850.78	−	3498.76i	−0.115937	−	0.219169i
$635$	−2091.34	+	2091.34i	−0.130697	+	0.130697i
$636$		0			0
$637$	−9102.24	−	9102.24i	−0.566160	−	0.566160i
$638$	19413.4	+	5980.49i	1.20468	+	0.371113i
$639$		0			0
$640$	6309.59	+	23309.2i	0.389700	+	1.43965i
$641$	2727.28			0.168051			0.0840257	−	0.996464i	$-0.473222\pi$
$641$	2727.28			0.168051			0.0840257	+	0.996464i	$0.473222\pi$
$642$		0			0
$643$	11681.7	+	11681.7i	0.716458	+	0.716458i	0.967878	−	0.251420i	$-0.0808975\pi$
$643$	11681.7	+	11681.7i	0.716458	+	0.716458i	−0.251420	+	0.967878i	$0.580897\pi$
$644$	6200.03	+	4220.48i	0.379372	+	0.258246i
$645$		0			0
$646$	1130.75	+	2137.59i	0.0688678	+	0.130189i
$647$		−	4399.67i		−	0.267340i	−0.991026	−	0.133670i	$-0.957324\pi$
$647$		−	4399.67i		−	0.267340i	0.991026	−	0.133670i	$-0.0426762\pi$
$648$		0			0
$649$			9707.43i			0.587134i
$650$	26646.9	−	14095.7i	1.60796	−	0.850585i
$651$		0			0
$652$	−5024.32	−	26448.4i	−0.301791	−	1.58865i
$653$	22924.8	+	22924.8i	1.37384	+	1.37384i	0.854673	+	0.519167i	$0.173758\pi$
$653$	22924.8	+	22924.8i	1.37384	+	1.37384i	0.519167	+	0.854673i	$0.326242\pi$
$654$		0			0
$655$	22139.6			1.32071
$656$	−20733.9	+	8172.42i	−1.23403	+	0.486401i
$657$		0			0
$658$	−2670.07	+	8667.40i	−0.158192	+	0.513511i
$659$	11508.4	+	11508.4i	0.680276	+	0.680276i	0.960062	−	0.279786i	$-0.0902636\pi$
$659$	11508.4	+	11508.4i	0.680276	+	0.680276i	−0.279786	+	0.960062i	$0.590264\pi$
$660$		0			0
$661$	10207.0	−	10207.0i	0.600615	−	0.600615i	−0.339861	−	0.940476i	$-0.610380\pi$
$661$	10207.0	−	10207.0i	0.600615	−	0.600615i	0.940476	+	0.339861i	$0.110380\pi$
$662$	19986.2	−	10572.3i	1.17339	−	0.620703i
$663$		0			0
$664$	−444.405	+	3983.42i	−0.0259733	+	0.232811i
$665$			3459.43i			0.201731i
$666$		0			0
$667$	−15744.2	+	15744.2i	−0.913970	+	0.913970i
$668$	−3781.42	+	5555.03i	−0.219023	+	0.321752i
$669$		0			0
$670$	−39989.0	−	12319.0i	−2.30583	−	0.710334i
$671$	9758.76			0.561450
$672$		0			0
$673$	−23908.4			−1.36939			−0.684695	−	0.728830i	$-0.740065\pi$
$673$	−23908.4			−1.36939			−0.684695	+	0.728830i	$0.740065\pi$
$674$	−787.445	−	242.580i	−0.0450019	−	0.0138632i
$675$		0			0
$676$	−11937.9	+	17537.2i	−0.679215	+	0.997791i
$677$	−10411.1	+	10411.1i	−0.591035	+	0.591035i	−0.937911	−	0.346876i	$-0.887242\pi$
$677$	−10411.1	+	10411.1i	−0.591035	+	0.591035i	0.346876	+	0.937911i	$0.387242\pi$
$678$		0			0
$679$			66.3800i			0.00375174i
$680$	19433.6	+	2168.09i	1.09595	+	0.122268i
$681$		0			0
$682$	19605.8	−	10371.1i	1.10080	−	0.582303i
$683$	−11126.8	+	11126.8i	−0.623358	+	0.623358i	−0.946389	−	0.323030i	$-0.895298\pi$
$683$	−11126.8	+	11126.8i	−0.623358	+	0.623358i	0.323030	+	0.946389i	$0.395298\pi$
$684$		0			0
$685$	3624.91	+	3624.91i	0.202191	+	0.202191i
$686$	5527.53	−	17943.1i	0.307642	−	0.998644i
$687$		0			0
$688$	17086.7	+	7425.15i	0.946836	+	0.411455i
$689$	−22216.5			−1.22842
$690$		0			0
$691$	2722.85	+	2722.85i	0.149901	+	0.149901i	0.778074	−	0.628173i	$-0.216197\pi$
$691$	2722.85	+	2722.85i	0.149901	+	0.149901i	−0.628173	+	0.778074i	$0.716197\pi$
$692$	950.748	+	5004.81i	0.0522283	+	0.274934i
$693$		0			0
$694$	23091.7	−	12215.1i	1.26304	−	0.668126i
$695$		−	29558.5i		−	1.61326i
$696$		0			0
$697$			18046.6i			0.980724i
$698$	−13593.4	−	25697.3i	−0.737133	−	1.39349i
$699$		0			0
$700$	12728.8	+	8664.77i	0.687293	+	0.467853i
$701$	−2736.38	−	2736.38i	−0.147435	−	0.147435i	0.629536	−	0.776971i	$-0.283245\pi$
$701$	−2736.38	−	2736.38i	−0.147435	−	0.147435i	−0.776971	+	0.629536i	$0.783245\pi$
$702$		0			0
$703$	−2584.06			−0.138634
$704$	6573.39	+	10410.8i	0.351909	+	0.557346i
$705$		0			0
$706$	−22310.1	−	6872.85i	−1.18931	−	0.366378i
$707$	5773.45	+	5773.45i	0.307119	+	0.307119i
$708$		0			0
$709$	−783.090	+	783.090i	−0.0414803	+	0.0414803i	−0.727543	−	0.686062i	$-0.759338\pi$
$709$	−783.090	+	783.090i	−0.0414803	+	0.0414803i	0.686062	+	0.727543i	$0.259338\pi$
$710$	6059.98	+	11455.9i	0.320320	+	0.605539i
$711$		0			0
$712$	−15912.6	+	12718.4i	−0.837571	+	0.669443i
$713$			24311.1i			1.27694i
$714$		0			0
$715$	19744.4	−	19744.4i	1.03272	−	1.03272i
$716$	1367.72	+	7199.79i	0.0713885	+	0.375795i
$717$		0			0
$718$	−4020.06	+	13049.6i	−0.208951	+	0.678283i
$719$	11021.5			0.571671			0.285835	−	0.958279i	$-0.407729\pi$
$719$	11021.5			0.571671			0.285835	+	0.958279i	$0.407729\pi$
$720$		0			0
$721$	−9949.27			−0.513912
$722$	−5484.90	+	17804.7i	−0.282724	+	0.917759i
$723$		0			0
$724$	−8400.11	+	1595.74i	−0.431198	+	0.0819133i
$725$	−32323.3	+	32323.3i	−1.65580	+	1.65580i
$726$		0			0
$727$			21740.4i			1.10909i	0.832154	+	0.554544i	$0.187107\pi$
$727$			21740.4i			1.10909i	−0.832154	+	0.554544i	$0.812893\pi$
$728$	−19691.9	−	2196.91i	−1.00252	−	0.111844i
$729$		0			0
$730$	−6576.16	−	12431.7i	−0.333417	−	0.630299i
$731$	10667.5	−	10667.5i	0.539741	−	0.539741i
$732$		0			0
$733$	13194.8	+	13194.8i	0.664886	+	0.664886i	0.956528	−	0.291642i	$-0.0942015\pi$
$733$	13194.8	+	13194.8i	0.664886	+	0.664886i	−0.291642	+	0.956528i	$0.594202\pi$
$734$	6910.04	+	2128.70i	0.347485	+	0.107046i
$735$		0			0
$736$	−13455.8	+	1033.08i	−0.673898	+	0.0517391i
$737$	−21334.7			−1.06631
$738$		0			0
$739$	21786.6	+	21786.6i	1.08448	+	1.08448i	0.996085	+	0.0883973i	$0.0281745\pi$
$739$	21786.6	+	21786.6i	1.08448	+	1.08448i	0.0883973	+	0.996085i	$0.471825\pi$
$740$	−11758.0	+	17272.9i	−0.584099	+	0.858061i
$741$		0			0
$742$	−5306.26	−	10031.1i	−0.262532	−	0.496297i
$743$			7418.24i			0.366284i	0.983086	+	0.183142i	$0.0586268\pi$
$743$			7418.24i			0.366284i	−0.983086	+	0.183142i	$0.941373\pi$
$744$		0			0
$745$		−	39778.3i		−	1.95619i
$746$	−4036.35	+	2135.16i	−0.198098	+	0.104790i
$747$		0			0
$748$	9794.89	−	1860.70i	0.478792	−	0.0909546i
$749$	−19187.5	−	19187.5i	−0.936045	−	0.936045i
$750$		0			0
$751$	−9320.68			−0.452885			−0.226442	−	0.974025i	$-0.572709\pi$
$751$	−9320.68			−0.452885			−0.226442	+	0.974025i	$0.572709\pi$
$752$	−5984.15	−	15182.1i	−0.290185	−	0.736217i
$753$		0			0
$754$	17317.5	−	56214.8i	0.836427	−	2.71515i
$755$	−19835.3	−	19835.3i	−0.956135	−	0.956135i
$756$		0			0
$757$	2105.29	−	2105.29i	0.101081	−	0.101081i	−0.654758	−	0.755839i	$-0.727229\pi$
$757$	2105.29	−	2105.29i	0.101081	−	0.101081i	0.755839	+	0.654758i	$0.227229\pi$
$758$	9200.47	−	4866.89i	0.440866	−	0.233210i
$759$		0			0
$760$	−3886.38	−	4862.43i	−0.185492	−	0.232077i
$761$			13011.0i			0.619776i	0.950773	+	0.309888i	$0.100292\pi$
$761$			13011.0i			0.619776i	−0.950773	+	0.309888i	$0.899708\pi$
$762$		0			0
$763$	5197.48	−	5197.48i	0.246607	−	0.246607i
$764$	−5029.24	−	3423.50i	−0.238156	−	0.162118i
$765$		0			0
$766$	−1002.46	−	308.816i	−0.0472849	−	0.0145665i
$767$	28109.4			1.32330
$768$		0			0
$769$	26438.9			1.23981			0.619904	−	0.784677i	$-0.287171\pi$
$769$	26438.9			1.23981			0.619904	+	0.784677i	$0.287171\pi$
$770$	13630.7	+	4199.05i	0.637941	+	0.196524i
$771$		0			0
$772$	14449.7	+	9836.22i	0.673650	+	0.458566i
$773$	21614.3	−	21614.3i	1.00571	−	1.00571i	0.00572318	−	0.999984i	$-0.498178\pi$
$773$	21614.3	−	21614.3i	1.00571	−	1.00571i	0.999984	−	0.00572318i	$-0.00182175\pi$
$774$		0			0
$775$			49911.4i			2.31338i
$776$	−74.5722	−	93.3007i	−0.00344972	−	0.00431611i
$777$		0			0
$778$	−22832.5	+	12078.0i	−1.05216	+	0.556577i
$779$	4062.20	−	4062.20i	0.186834	−	0.186834i
$780$		0			0
$781$	4672.48	+	4672.48i	0.214077	+	0.214077i
$782$	−3217.29	+	10443.7i	−0.147123	+	0.477580i
$783$		0			0
$784$	4338.46	+	11006.9i	0.197634	+	0.501408i
$785$	25157.8			1.14385
$786$		0			0
$787$	−20949.4	−	20949.4i	−0.948874	−	0.948874i	0.0498808	−	0.998755i	$-0.484116\pi$
$787$	−20949.4	−	20949.4i	−0.948874	−	0.948874i	−0.998755	+	0.0498808i	$0.984116\pi$
$788$	−927.120	+	176.122i	−0.0419128	+	0.00796204i
$789$		0			0
$790$	−7323.55	+	3874.03i	−0.329823	+	0.174471i
$791$		−	2643.94i		−	0.118847i
$792$		0			0
$793$		−	28258.1i		−	1.26541i
$794$	434.584	+	821.548i	0.0194242	+	0.0367199i
$795$		0			0
$796$	−7806.83	+	11468.5i	−0.347620	+	0.510666i
$797$	−1576.38	−	1576.38i	−0.0700603	−	0.0700603i	0.671208	−	0.741269i	$-0.265776\pi$
$797$	−1576.38	−	1576.38i	−0.0700603	−	0.0700603i	−0.741269	+	0.671208i	$0.765776\pi$
$798$		0			0
$799$	−13214.4			−0.585097
$800$	−27625.2	+	2120.95i	−1.22087	+	0.0937336i
$801$		0			0
$802$	−323.329	−	99.6044i	−0.0142358	−	0.00438548i
$803$	−5070.48	−	5070.48i	−0.222831	−	0.222831i
$804$		0			0
$805$	−11054.4	+	11054.4i	−0.483995	+	0.483995i
$806$	−30031.2	−	56771.7i	−1.31241	−	2.48101i
$807$		0			0
$808$	−14600.9	−	1628.93i	−0.635715	−	0.0709228i
$809$			8102.89i			0.352142i	0.984378	+	0.176071i	$0.0563388\pi$
$809$			8102.89i			0.352142i	−0.984378	+	0.176071i	$0.943661\pi$
$810$		0			0
$811$	20649.6	−	20649.6i	0.894090	−	0.894090i	−0.100816	−	0.994905i	$-0.532145\pi$
$811$	20649.6	−	20649.6i	0.894090	−	0.894090i	0.994905	+	0.100816i	$0.0321452\pi$
$812$	29517.9	−	5607.42i	1.27571	−	0.242342i
$813$		0			0
$814$	−3136.53	+	10181.6i	−0.135056	+	0.438408i
$815$	56114.6			2.41179
$816$		0			0
$817$	−4802.38			−0.205647
$818$	−522.067	+	1694.70i	−0.0223150	+	0.0724373i
$819$		0			0
$820$	−8669.55	−	45637.2i	−0.369212	−	1.94356i
$821$	−13405.1	+	13405.1i	−0.569844	+	0.569844i	−0.932085	−	0.362240i	$-0.882012\pi$
$821$	−13405.1	+	13405.1i	−0.569844	+	0.569844i	0.362240	+	0.932085i	$0.382012\pi$
$822$		0			0
$823$		−	17203.4i		−	0.728643i	−0.931273	−	0.364321i	$-0.881301\pi$
$823$		−	17203.4i		−	0.728643i	0.931273	−	0.364321i	$-0.118699\pi$
$824$	13984.3	−	11177.2i	0.591219	−	0.472542i
$825$		0			0
$826$	6713.74	+	12691.8i	0.282810	+	0.534630i
$827$	11127.2	−	11127.2i	0.467872	−	0.467872i	−0.433353	−	0.901224i	$-0.642670\pi$
$827$	11127.2	−	11127.2i	0.467872	−	0.467872i	0.901224	+	0.433353i	$0.142670\pi$
$828$		0			0
$829$	−5752.09	−	5752.09i	−0.240987	−	0.240987i	0.576271	−	0.817259i	$-0.304507\pi$
$829$	−5752.09	−	5752.09i	−0.240987	−	0.240987i	−0.817259	+	0.576271i	$0.804507\pi$
$830$	−7984.23	−	2459.61i	−0.333899	−	0.102861i
$831$		0			0
$832$	30146.1	−	19034.3i	1.25617	−	0.793145i
$833$	9580.34			0.398486
$834$		0			0
$835$	−9904.39	−	9904.39i	−0.410486	−	0.410486i
$836$	−2623.61	−	1785.94i	−0.108540	−	0.0738853i
$837$		0			0
$838$	−9317.38	−	17613.8i	−0.384086	−	0.726083i
$839$			45875.9i			1.88774i	0.330318	+	0.943870i	$0.392844\pi$
$839$			45875.9i			1.88774i	−0.330318	+	0.943870i	$0.607156\pi$
$840$		0			0
$841$			64807.5i			2.65724i
$842$	−38081.7	+	20144.6i	−1.55865	+	0.824499i
$843$		0			0
$844$	−772.585	−	4066.95i	−0.0315089	−	0.165865i
$845$	−31268.1	−	31268.1i	−1.27296	−	1.27296i
$846$		0			0
$847$	−9465.66			−0.383995
$848$	18727.3	+	8138.10i	0.758371	+	0.329556i
$849$		0			0
$850$	−6605.20	+	21441.3i	−0.266537	+	0.865213i
$851$	−8257.21	−	8257.21i	−0.332613	−	0.332613i
$852$		0			0
$853$	−27266.2	+	27266.2i	−1.09446	+	1.09446i	−0.0994157	+	0.995046i	$0.531697\pi$
$853$	−27266.2	+	27266.2i	−1.09446	+	1.09446i	−0.995046	+	0.0994157i	$0.968303\pi$
$854$	12758.9	−	6749.24i	0.511243	−	0.270438i
$855$		0			0
$856$	48524.7	+	5413.60i	1.93755	+	0.216160i
$857$			28960.0i			1.15432i	0.816629	+	0.577162i	$0.195840\pi$
$857$			28960.0i			1.15432i	−0.816629	+	0.577162i	$0.804160\pi$
$858$		0			0
$859$	17255.5	−	17255.5i	0.685388	−	0.685388i	−0.275821	−	0.961209i	$-0.588950\pi$
$859$	17255.5	−	17255.5i	0.685388	−	0.685388i	0.961209	+	0.275821i	$0.0889496\pi$
$860$	−21851.8	+	32101.0i	−0.866442	+	1.27283i
$861$		0			0
$862$	20399.9	+	6284.39i	0.806061	+	0.248314i
$863$	−39885.9			−1.57327			−0.786635	−	0.617418i	$-0.788179\pi$
$863$	−39885.9			−1.57327			−0.786635	+	0.617418i	$0.788179\pi$
$864$		0			0
$865$	−10618.5			−0.417388
$866$	−15175.5	−	4674.95i	−0.595478	−	0.183443i
$867$		0			0
$868$	18460.5	−	27119.1i	0.721877	−	1.06046i
$869$	−2987.03	+	2987.03i	−0.116603	+	0.116603i
$870$		0			0
$871$			61778.0i			2.40329i
$872$	−1466.43	+	13144.3i	−0.0569489	+	0.510460i
$873$		0			0
$874$	3075.03	−	1626.64i	0.119010	−	0.0629540i
$875$	−4160.41	+	4160.41i	−0.160740	+	0.160740i
$876$		0			0
$877$	8855.51	+	8855.51i	0.340968	+	0.340968i	0.856731	−	0.515763i	$-0.172491\pi$
$877$	8855.51	+	8855.51i	0.340968	+	0.340968i	−0.515763	+	0.856731i	$0.672491\pi$
$878$	5994.59	−	19459.2i	0.230419	−	0.747969i
$879$		0			0
$880$	−23875.9	+	9410.88i	−0.914611	+	0.360501i
$881$	43343.8			1.65753			0.828767	−	0.559593i	$-0.189043\pi$
$881$	43343.8			1.65753			0.828767	+	0.559593i	$0.189043\pi$
$882$		0			0
$883$	11229.3	+	11229.3i	0.427968	+	0.427968i	0.887936	−	0.459968i	$-0.152139\pi$
$883$	11229.3	+	11229.3i	0.427968	+	0.427968i	−0.459968	+	0.887936i	$0.652139\pi$
$884$	−5387.97	−	28362.7i	−0.204997	−	1.07912i
$885$		0			0
$886$	−19740.1	+	10442.2i	−0.748511	+	0.395949i
$887$		−	36912.9i		−	1.39731i	−0.715459	−	0.698655i	$-0.753782\pi$
$887$		−	36912.9i		−	1.39731i	0.715459	−	0.698655i	$-0.246218\pi$
$888$		0			0
$889$			2230.45i			0.0841473i
$890$	−19854.2	−	37532.8i	−0.747770	−	1.41360i
$891$		0			0
$892$	−26698.6	−	18174.3i	−1.00217	−	0.682197i
$893$	2974.49	+	2974.49i	0.111464	+	0.111464i
$894$		0			0
$895$	−15275.5			−0.570508
$896$	15794.5	+	9065.19i	0.588902	+	0.337998i
$897$		0			0
$898$	3309.98	+	1019.67i	0.123002	+	0.0378918i
$899$	68865.5	+	68865.5i	2.55483	+	2.55483i
$900$		0			0
$901$	11691.7	−	11691.7i	0.432306	−	0.432306i
$902$	−11075.0	−	20936.3i	−0.408820	−	0.772842i
$903$		0			0
$904$	2970.25	+	3716.21i	0.109280	+	0.136725i
$905$		−	17822.2i		−	0.654618i
$906$		0			0
$907$	14588.0	−	14588.0i	0.534053	−	0.534053i	−0.387723	−	0.921776i	$-0.626738\pi$
$907$	14588.0	−	14588.0i	0.534053	−	0.534053i	0.921776	+	0.387723i	$0.126738\pi$
$908$	3173.14	+	16703.7i	0.115974	+	0.610497i
$909$		0			0
$910$	12159.0	−	39469.8i	0.442932	−	1.43781i
$911$	−50418.8			−1.83365			−0.916823	−	0.399295i	$-0.869255\pi$
$911$	−50418.8			−1.83365			−0.916823	+	0.399295i	$0.869255\pi$
$912$		0			0
$913$	−4259.69			−0.154409
$914$	9311.61	−	30226.7i	0.336981	−	1.09388i
$915$		0			0
$916$	−38275.9	+	7271.14i	−1.38064	+	0.262277i
$917$	11806.1	−	11806.1i	0.425160	−	0.425160i
$918$		0			0
$919$		−	53290.4i		−	1.91283i	−0.292018	−	0.956413i	$-0.594327\pi$
$919$		−	53290.4i		−	1.91283i	0.292018	−	0.956413i	$-0.405673\pi$
$920$	3118.90	−	27956.2i	0.111769	−	1.00184i
$921$		0			0
$922$	−21200.4	−	40077.8i	−0.757266	−	1.43155i
$923$	13529.9	−	13529.9i	0.482495	−	0.482495i
$924$		0			0
$925$	−16952.3	−	16952.3i	−0.602581	−	0.602581i
$926$	−8145.35	−	2509.25i	−0.289064	−	0.0890488i
$927$		0			0
$928$	−35189.6	+	41042.4i	−1.24478	+	1.45181i
$929$	−44859.2			−1.58427			−0.792133	−	0.610349i	$-0.791029\pi$
$929$	−44859.2			−1.58427			−0.792133	+	0.610349i	$0.791029\pi$
$930$		0			0
$931$	−2156.48	−	2156.48i	−0.0759140	−	0.0759140i
$932$	5127.47	−	7532.43i	0.180210	−	0.264735i
$933$		0			0
$934$	−1578.43	−	2983.91i	−0.0552976	−	0.104536i
$935$			20781.4i			0.726872i
$936$		0			0
$937$			22463.9i			0.783204i	0.920135	+	0.391602i	$0.128079\pi$
$937$			22463.9i			0.783204i	−0.920135	+	0.391602i	$0.871921\pi$
$938$	−27893.6	+	14755.2i	−0.970958	+	0.513620i
$939$		0			0
$940$	33417.3	−	6348.17i	1.15952	−	0.220271i
$941$	−20653.7	−	20653.7i	−0.715507	−	0.715507i	0.252175	−	0.967682i	$-0.418854\pi$
$941$	−20653.7	−	20653.7i	−0.715507	−	0.715507i	−0.967682	+	0.252175i	$0.918854\pi$
$942$		0			0
$943$	25961.0			0.896507
$944$	−23694.7	−	10296.7i	−0.816946	−	0.355010i
$945$		0			0
$946$	−5829.11	+	18922.0i	−0.200339	+	0.650326i
$947$	−33123.2	−	33123.2i	−1.13660	−	1.13660i	−0.989055	−	0.147544i	$-0.952863\pi$
$947$	−33123.2	−	33123.2i	−1.13660	−	1.13660i	−0.147544	−	0.989055i	$-0.547137\pi$
$948$		0			0
$949$	−14682.4	+	14682.4i	−0.502224	+	0.502224i
$950$	6313.12	−	3339.53i	0.215605	−	0.114051i
$951$		0			0
$952$	11519.3	−	9206.97i	0.392166	−	0.313445i
$953$			10871.9i			0.369544i	0.982781	+	0.184772i	$0.0591546\pi$
$953$			10871.9i			0.369544i	−0.982781	+	0.184772i	$0.940845\pi$
$954$		0			0
$955$	8966.93	−	8966.93i	0.303836	−	0.303836i
$956$	30445.9	+	20725.1i	1.03001	+	0.701147i
$957$		0			0
$958$	28570.0	+	8801.25i	0.963522	+	0.296822i
$959$	3866.02			0.130178
$960$		0			0
$961$	76546.3			2.56944
$962$	29482.4	+	9082.33i	0.988099	+	0.304393i
$963$		0			0
$964$	−30960.2	−	21075.2i	−1.03440	−	0.704137i
$965$	−25763.3	+	25763.3i	−0.859430	+	0.859430i
$966$		0			0
$967$		−	6501.58i		−	0.216212i	−0.994139	−	0.108106i	$-0.965521\pi$
$967$		−	6501.58i		−	0.216212i	0.994139	−	0.108106i	$-0.0344786\pi$
$968$	13304.5	−	10633.9i	0.441759	−	0.353084i
$969$		0			0
$970$	220.067	−	116.412i	0.00728446	−	0.00385335i
$971$	16761.7	−	16761.7i	0.553975	−	0.553975i	−0.373611	−	0.927586i	$-0.621880\pi$
$971$	16761.7	−	16761.7i	0.553975	−	0.553975i	0.927586	+	0.373611i	$0.121880\pi$
$972$		0			0
$973$	−15762.3	−	15762.3i	−0.519338	−	0.519338i
$974$	−1314.93	+	4268.43i	−0.0432578	+	0.140420i
$975$		0			0
$976$	−10351.2	+	23820.0i	−0.339480	+	0.781208i
$977$	7407.34			0.242561			0.121280	−	0.992618i	$-0.461300\pi$
$977$	7407.34			0.242561			0.121280	+	0.992618i	$0.461300\pi$
$978$		0			0
$979$	−15308.4	−	15308.4i	−0.499753	−	0.499753i
$980$	−24227.2	+	4602.37i	−0.789705	+	0.150018i
$981$		0			0
$982$	30586.3	−	16179.6i	0.993938	−	0.525776i
$983$			35694.8i			1.15818i	0.815265	+	0.579088i	$0.196591\pi$
$983$			35694.8i			1.15818i	−0.815265	+	0.579088i	$0.803409\pi$
$984$		0			0
$985$		−	1967.04i		−	0.0636294i
$986$	20470.2	+	38697.3i	0.661160	+	1.24987i
$987$		0			0
$988$	−5171.49	+	7597.10i	−0.166525	+	0.244631i
$989$	−15345.7	−	15345.7i	−0.493392	−	0.493392i
$990$		0			0
$991$	46662.2			1.49573			0.747867	−	0.663849i	$-0.231078\pi$
$991$	46662.2			1.49573			0.747867	+	0.663849i	$0.231078\pi$
$992$	4518.73	+	58856.1i	0.144627	+	1.88375i
$993$		0			0
$994$	9340.48	+	2877.42i	0.298050	+	0.0918172i
$995$	−20447.9	−	20447.9i	−0.651498	−	0.651498i
$996$		0			0
$997$	−22390.7	+	22390.7i	−0.711255	+	0.711255i	−0.966798	−	0.255543i	$-0.917746\pi$
$997$	−22390.7	+	22390.7i	−0.711255	+	0.711255i	0.255543	+	0.966798i	$0.417746\pi$
$998$	22336.5	+	42225.4i	0.708467	+	1.33930i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.k.b.109.1		24
3.2	odd	2		48.4.j.a.13.12	✓	24
4.3	odd	2		576.4.k.b.145.2		24
12.11	even	2		192.4.j.a.145.12		24
16.5	even	4	inner	144.4.k.b.37.1		24
16.11	odd	4		576.4.k.b.433.2		24
24.5	odd	2		384.4.j.b.289.7		24
24.11	even	2		384.4.j.a.289.6		24
48.5	odd	4		48.4.j.a.37.12	yes	24
48.11	even	4		192.4.j.a.49.12		24
48.29	odd	4		384.4.j.b.97.7		24
48.35	even	4		384.4.j.a.97.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.12	✓	24	3.2	odd	2
48.4.j.a.37.12	yes	24	48.5	odd	4
144.4.k.b.37.1		24	16.5	even	4	inner
144.4.k.b.109.1		24	1.1	even	1	trivial
192.4.j.a.49.12		24	48.11	even	4
192.4.j.a.145.12		24	12.11	even	2
384.4.j.a.97.6		24	48.35	even	4
384.4.j.a.289.6		24	24.11	even	2
384.4.j.b.97.7		24	48.29	odd	4
384.4.j.b.289.7		24	24.5	odd	2
576.4.k.b.145.2		24	4.3	odd	2
576.4.k.b.433.2		24	16.11	odd	4

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)

\(f(q)\)	\(=\)	\(q+(-2.70307 - 0.832707i) q^{2} +(6.61320 + 4.50173i) q^{4} +(-11.7911 + 11.7911i) q^{5} +12.5754i q^{7} +(-14.1273 - 17.6754i) q^{8} +(41.6906 - 22.0536i) q^{10} +(17.0042 - 17.0042i) q^{11} +(-49.2384 - 49.2384i) q^{13} +(10.4716 - 33.9921i) q^{14} +(23.4688 + 59.5417i) q^{16} +51.8247 q^{17} +(-11.6655 - 11.6655i) q^{19} +(-131.057 + 24.8964i) q^{20} +(-60.1231 + 31.8041i) q^{22} -74.5524i q^{23} -153.058i q^{25} +(92.0939 + 174.096i) q^{26} +(-56.6109 + 83.1633i) q^{28} +(-211.183 - 211.183i) q^{29} -326.094 q^{31} +(-13.8571 - 180.488i) q^{32} +(-140.086 - 43.1547i) q^{34} +(-148.277 - 148.277i) q^{35} +(110.757 - 110.757i) q^{37} +(21.8187 + 41.2465i) q^{38} +(374.988 + 41.8350i) q^{40} +348.225i q^{41} +(205.838 - 205.838i) q^{43} +(189.000 - 35.9038i) q^{44} +(-62.0803 + 201.521i) q^{46} -254.983 q^{47} +184.861 q^{49} +(-127.453 + 413.728i) q^{50} +(-103.965 - 547.282i) q^{52} +(225.602 - 225.602i) q^{53} +400.995i q^{55} +(222.274 - 177.656i) q^{56} +(394.989 + 746.696i) q^{58} +(-285.442 + 285.442i) q^{59} +(286.952 + 286.952i) q^{61} +(881.456 + 271.541i) q^{62} +(-112.837 + 499.411i) q^{64} +1161.15 q^{65} +(-627.335 - 627.335i) q^{67} +(342.727 + 233.301i) q^{68} +(277.332 + 524.274i) q^{70} +274.784i q^{71} -298.190i q^{73} +(-391.612 + 207.156i) q^{74} +(-24.6312 - 129.661i) q^{76} +(213.834 + 213.834i) q^{77} -175.664 q^{79} +(-978.782 - 425.338i) q^{80} +(289.969 - 941.276i) q^{82} +(-125.254 - 125.254i) q^{83} +(-611.068 + 611.068i) q^{85} +(-727.796 + 384.991i) q^{86} +(-540.779 - 60.3313i) q^{88} -900.271i q^{89} +(619.190 - 619.190i) q^{91} +(335.615 - 493.030i) q^{92} +(689.237 + 212.326i) q^{94} +275.096 q^{95} +5.27858 q^{97} +(-499.691 - 153.935i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{4} - 84 q^{8} + 72 q^{10} + 40 q^{11} + 348 q^{14} - 192 q^{16} + 24 q^{19} - 80 q^{20} + 704 q^{22} + 20 q^{26} - 344 q^{28} - 400 q^{29} - 744 q^{31} + 960 q^{32} - 704 q^{34} + 456 q^{35}+ \cdots - 6760 q^{98}+O(q^{100}) \) 24 * q - 20 * q^4 - 84 * q^8 + 72 * q^10 + 40 * q^11 + 348 * q^14 - 192 * q^16 + 24 * q^19 - 80 * q^20 + 704 * q^22 + 20 * q^26 - 344 * q^28 - 400 * q^29 - 744 * q^31 + 960 * q^32 - 704 * q^34 + 456 * q^35 + 16 * q^37 - 1256 * q^38 + 1744 * q^40 + 1240 * q^43 + 200 * q^44 - 1432 * q^46 - 1176 * q^49 - 708 * q^50 + 1008 * q^52 - 752 * q^53 - 1344 * q^56 + 1936 * q^58 + 1376 * q^59 - 912 * q^61 + 996 * q^62 - 56 * q^64 - 976 * q^65 - 2256 * q^67 + 1568 * q^68 - 1760 * q^70 + 2740 * q^74 - 1880 * q^76 - 1904 * q^77 + 5992 * q^79 - 712 * q^80 - 40 * q^82 - 2680 * q^83 - 240 * q^85 + 1712 * q^86 - 3936 * q^88 - 3496 * q^91 - 5296 * q^92 + 5272 * q^94 + 7728 * q^95 - 6760 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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