LMFDB - Embedded newform 108.4.i.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(13,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 8]))

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

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			49.2
Character	$\chi$	$=$	108.49
Dual form			108.4.i.a.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.14053 + 3.13943i) q^{3} +(-4.27250 + 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +(7.28791 - 25.9978i) q^{9} +O(q^{10})$	\(q+(-4.14053 + 3.13943i) q^{3} +(-4.27250 + 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +(7.28791 - 25.9978i) q^{9} +(50.9513 + 18.5448i) q^{11} +(42.8395 + 35.9466i) q^{13} +(12.8084 - 19.8520i) q^{15} +(49.3862 - 85.5394i) q^{17} +(-68.0936 - 117.942i) q^{19} +(103.047 + 95.5225i) q^{21} +(25.4482 - 144.324i) q^{23} +(-79.9195 + 67.0604i) q^{25} +(51.4427 + 130.525i) q^{27} +(-37.6146 + 31.5624i) q^{29} +(26.6026 - 150.871i) q^{31} +(-269.185 + 83.1733i) q^{33} +(61.4744 + 106.477i) q^{35} +(11.7745 - 20.3941i) q^{37} +(-290.230 - 14.3461i) q^{39} +(317.305 + 266.251i) q^{41} +(-269.878 - 98.2276i) q^{43} +(9.29066 + 122.409i) q^{45} +(-48.6186 - 275.730i) q^{47} +(-364.820 + 132.783i) q^{49} +(64.0605 + 509.223i) q^{51} +248.490 q^{53} -246.528 q^{55} +(652.213 + 274.565i) q^{57} +(243.543 - 88.6422i) q^{59} +(-57.5577 - 326.426i) q^{61} +(-726.556 - 72.0033i) q^{63} +(-238.931 - 86.9637i) q^{65} +(71.5526 + 60.0398i) q^{67} +(347.726 + 677.469i) q^{69} +(-333.619 + 577.845i) q^{71} +(-106.851 - 185.072i) q^{73} +(120.377 - 528.567i) q^{75} +(254.606 - 1443.94i) q^{77} +(-421.142 + 353.380i) q^{79} +(-622.773 - 378.939i) q^{81} +(310.120 - 260.222i) q^{83} +(-77.9834 + 442.266i) q^{85} +(56.6561 - 248.773i) q^{87} +(504.793 + 874.328i) q^{89} +(756.115 - 1309.63i) q^{91} +(363.501 + 708.203i) q^{93} +(474.337 + 398.016i) q^{95} +(-27.6645 - 10.0690i) q^{97} +(853.452 - 1189.47i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) $ 54 * q + 12 * q^5 - 48 * q^9	$ 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) $ 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{7}{9}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−4.14053	+	3.13943i	−0.796845	+	0.604184i
$3$	−4.14053	+	3.13943i	−0.796845	+	0.604184i
$4$		0			0
$5$	−4.27250	+	1.55506i	−0.382144	+	0.139089i	−0.525947	−	0.850517i	$-0.676289\pi$
$5$	−4.27250	+	1.55506i	−0.382144	+	0.139089i	0.143803	+	0.989606i	$0.454067\pi$
$6$		0			0
$7$	−4.69568	−	26.6305i	−0.253543	−	1.43791i	−0.799786	−	0.600285i	$-0.795054\pi$
$7$	−4.69568	−	26.6305i	−0.253543	−	1.43791i	0.546243	−	0.837627i	$-0.316057\pi$
$8$		0			0
$9$	7.28791	−	25.9978i	0.269922	−	0.962882i
$10$		0			0
$11$	50.9513	+	18.5448i	1.39658	+	0.508315i	0.927162	−	0.374660i	$-0.122241\pi$
$11$	50.9513	+	18.5448i	1.39658	+	0.508315i	0.469420	+	0.882975i	$0.344463\pi$
$12$		0			0
$13$	42.8395	+	35.9466i	0.913964	+	0.766906i	0.972869	−	0.231357i	$-0.0743166\pi$
$13$	42.8395	+	35.9466i	0.913964	+	0.766906i	−0.0589054	+	0.998264i	$0.518761\pi$
$14$		0			0
$15$	12.8084	−	19.8520i	0.220474	−	0.341718i
$16$		0			0
$17$	49.3862	−	85.5394i	0.704583	−	1.22037i	−0.262259	−	0.964998i	$-0.584467\pi$
$17$	49.3862	−	85.5394i	0.704583	−	1.22037i	0.966842	−	0.255376i	$-0.0821993\pi$
$18$		0			0
$19$	−68.0936	−	117.942i	−0.822197	−	1.42409i	−0.904043	−	0.427442i	$-0.859415\pi$
$19$	−68.0936	−	117.942i	−0.822197	−	1.42409i	0.0818453	−	0.996645i	$-0.473919\pi$
$20$		0			0
$21$	103.047	+	95.5225i	1.07080	+	0.992606i
$22$		0			0
$23$	25.4482	−	144.324i	0.230709	−	1.30842i	−0.620756	−	0.784004i	$-0.713174\pi$
$23$	25.4482	−	144.324i	0.230709	−	1.30842i	0.851465	−	0.524412i	$-0.175715\pi$
$24$		0			0
$25$	−79.9195	+	67.0604i	−0.639356	+	0.536483i
$26$		0			0
$27$	51.4427	+	130.525i	0.366672	+	0.930350i
$28$		0			0
$29$	−37.6146	+	31.5624i	−0.240857	+	0.202103i	−0.755223	−	0.655468i	$-0.772472\pi$
$29$	−37.6146	+	31.5624i	−0.240857	+	0.202103i	0.514366	+	0.857570i	$0.328027\pi$
$30$		0			0
$31$	26.6026	−	150.871i	0.154128	−	0.874105i	−0.805450	−	0.592663i	$-0.798076\pi$
$31$	26.6026	−	150.871i	0.154128	−	0.874105i	0.959578	−	0.281441i	$-0.0908125\pi$
$32$		0			0
$33$	−269.185	+	83.1733i	−1.41997	+	0.438746i
$34$		0			0
$35$	61.4744	+	106.477i	0.296888	+	0.514225i
$36$		0			0
$37$	11.7745	−	20.3941i	0.0523168	−	0.0906154i	−0.838681	−	0.544623i	$-0.816673\pi$
$37$	11.7745	−	20.3941i	0.0523168	−	0.0906154i	0.890998	+	0.454008i	$0.150006\pi$
$38$		0			0
$39$	−290.230	−	14.3461i	−1.19164	−	0.0589027i
$40$		0			0
$41$	317.305	+	266.251i	1.20865	+	1.01418i	0.999340	+	0.0363307i	$0.0115670\pi$
$41$	317.305	+	266.251i	1.20865	+	1.01418i	0.209312	+	0.977849i	$0.432877\pi$
$42$		0			0
$43$	−269.878	−	98.2276i	−0.957117	−	0.348362i	−0.184214	−	0.982886i	$-0.558974\pi$
$43$	−269.878	−	98.2276i	−0.957117	−	0.348362i	−0.772903	+	0.634524i	$0.781196\pi$
$44$		0			0
$45$	9.29066	+	122.409i	0.0307771	+	0.405503i
$46$		0			0
$47$	−48.6186	−	275.730i	−0.150888	−	0.855730i	−0.962449	−	0.271462i	$-0.912493\pi$
$47$	−48.6186	−	275.730i	−0.150888	−	0.855730i	0.811561	−	0.584268i	$-0.198618\pi$
$48$		0			0
$49$	−364.820	+	132.783i	−1.06361	+	0.387124i
$50$		0			0
$51$	64.0605	+	509.223i	0.175888	+	1.39815i
$52$		0			0
$53$	248.490			0.644015			0.322007	−	0.946737i	$-0.395642\pi$
$53$	248.490			0.644015			0.322007	+	0.946737i	$0.395642\pi$
$54$		0			0
$55$	−246.528			−0.604397
$56$		0			0
$57$	652.213	+	274.565i	1.51557	+	0.638018i
$58$		0			0
$59$	243.543	−	88.6422i	0.537399	−	0.195597i	−0.0590401	−	0.998256i	$-0.518804\pi$
$59$	243.543	−	88.6422i	0.537399	−	0.195597i	0.596439	+	0.802658i	$0.296582\pi$
$60$		0			0
$61$	−57.5577	−	326.426i	−0.120812	−	0.685156i	−0.983708	−	0.179775i	$-0.942463\pi$
$61$	−57.5577	−	326.426i	−0.120812	−	0.685156i	0.862896	−	0.505381i	$-0.168648\pi$
$62$		0			0
$63$	−726.556	−	72.0033i	−1.45298	−	0.143993i
$64$		0			0
$65$	−238.931	−	86.9637i	−0.455934	−	0.165946i
$66$		0			0
$67$	71.5526	+	60.0398i	0.130471	+	0.109478i	0.705688	−	0.708523i	$-0.250638\pi$
$67$	71.5526	+	60.0398i	0.130471	+	0.109478i	−0.575217	+	0.818001i	$0.695082\pi$
$68$		0			0
$69$	347.726	+	677.469i	0.606685	+	1.18199i
$70$		0			0
$71$	−333.619	+	577.845i	−0.557651	+	0.965881i	0.440040	+	0.897978i	$0.354964\pi$
$71$	−333.619	+	577.845i	−0.557651	+	0.965881i	−0.997692	+	0.0679028i	$0.978369\pi$
$72$		0			0
$73$	−106.851	−	185.072i	−0.171315	−	0.296726i	0.767565	−	0.640971i	$-0.221468\pi$
$73$	−106.851	−	185.072i	−0.171315	−	0.296726i	−0.938880	+	0.344245i	$0.888135\pi$
$74$		0			0
$75$	120.377	−	528.567i	0.185332	−	0.813783i
$76$		0			0
$77$	254.606	−	1443.94i	0.376818	−	2.13704i
$78$		0			0
$79$	−421.142	+	353.380i	−0.599775	+	0.503271i	−0.891373	−	0.453270i	$-0.850257\pi$
$79$	−421.142	+	353.380i	−0.599775	+	0.503271i	0.291598	+	0.956541i	$0.405813\pi$
$80$		0			0
$81$	−622.773	−	378.939i	−0.854284	−	0.519807i
$82$		0			0
$83$	310.120	−	260.222i	0.410122	−	0.344133i	−0.414268	−	0.910155i	$-0.635962\pi$
$83$	310.120	−	260.222i	0.410122	−	0.344133i	0.824391	+	0.566021i	$0.191518\pi$
$84$		0			0
$85$	−77.9834	+	442.266i	−0.0995116	+	0.564358i
$86$		0			0
$87$	56.6561	−	248.773i	0.0698180	−	0.306566i
$88$		0			0
$89$	504.793	+	874.328i	0.601213	+	1.04133i	0.992638	+	0.121122i	$0.0386492\pi$
$89$	504.793	+	874.328i	0.601213	+	1.04133i	−0.391424	+	0.920210i	$0.628017\pi$
$90$		0			0
$91$	756.115	−	1309.63i	0.871015	−	1.50864i
$92$		0			0
$93$	363.501	+	708.203i	0.405304	+	0.789647i
$94$		0			0
$95$	474.337	+	398.016i	0.512273	+	0.429848i
$96$		0			0
$97$	−27.6645	−	10.0690i	−0.0289578	−	0.0105398i	0.327501	−	0.944851i	$-0.393794\pi$
$97$	−27.6645	−	10.0690i	−0.0289578	−	0.0105398i	−0.356458	+	0.934311i	$0.616016\pi$
$98$		0			0
$99$	853.452	−	1189.47i	0.866416	−	1.20754i
$100$		0			0
$101$	−16.0416	−	90.9764i	−0.0158039	−	0.0896286i	0.975886	−	0.218283i	$-0.0700455\pi$
$101$	−16.0416	−	90.9764i	−0.0158039	−	0.0896286i	−0.991690	+	0.128654i	$0.958934\pi$
$102$		0			0
$103$	1917.18	−	697.795i	1.83403	−	0.667532i	0.842330	−	0.538963i	$-0.181184\pi$
$103$	1917.18	−	697.795i	1.83403	−	0.667532i	0.991701	−	0.128569i	$-0.0410385\pi$
$104$		0			0
$105$	−588.813	−	247.875i	−0.547260	−	0.230382i
$106$		0			0
$107$	−351.086			−0.317204			−0.158602	−	0.987343i	$-0.550699\pi$
$107$	−351.086			−0.317204			−0.158602	+	0.987343i	$0.550699\pi$
$108$		0			0
$109$	656.929			0.577270			0.288635	−	0.957439i	$-0.406799\pi$
$109$	656.929			0.577270			0.288635	+	0.957439i	$0.406799\pi$
$110$		0			0
$111$	15.2732	+	121.408i	0.0130600	+	0.103815i
$112$		0			0
$113$	−119.983	+	43.6704i	−0.0998857	+	0.0363554i	−0.391480	−	0.920187i	$-0.628037\pi$
$113$	−119.983	+	43.6704i	−0.0998857	+	0.0363554i	0.291594	+	0.956542i	$0.405814\pi$
$114$		0			0
$115$	115.705	+	656.196i	0.0938223	+	0.532092i
$116$		0			0
$117$	1246.74	−	851.757i	0.985140	−	0.673034i
$118$		0			0
$119$	−2509.86	−	913.514i	−1.93343	−	0.703711i
$120$		0			0
$121$	1232.53	+	1034.21i	0.926015	+	0.777019i
$122$		0			0
$123$	−2149.69	−	106.259i	−1.57586	−	0.0778947i
$124$		0			0
$125$	521.342	−	902.991i	0.373042	−	0.646127i
$126$		0			0
$127$	−649.026	−	1124.15i	−0.453478	−	0.785447i	0.545121	−	0.838357i	$-0.316484\pi$
$127$	−649.026	−	1124.15i	−0.453478	−	0.785447i	−0.998599	+	0.0529102i	$0.983150\pi$
$128$		0			0
$129$	1425.82	−	440.551i	0.973148	−	0.300685i
$130$		0			0
$131$	−196.126	+	1112.28i	−0.130806	+	0.741837i	0.846883	+	0.531779i	$0.178476\pi$
$131$	−196.126	+	1112.28i	−0.130806	+	0.741837i	−0.977689	+	0.210058i	$0.932635\pi$
$132$		0			0
$133$	−2821.10	+	2367.18i	−1.83925	+	1.54331i
$134$		0			0
$135$	−422.763	−	477.670i	−0.269523	−	0.304528i
$136$		0			0
$137$	−1978.63	+	1660.27i	−1.23391	+	1.03538i	−0.235937	+	0.971768i	$0.575816\pi$
$137$	−1978.63	+	1660.27i	−1.23391	+	1.03538i	−0.997975	+	0.0636072i	$0.979740\pi$
$138$		0			0
$139$	−487.444	+	2764.43i	−0.297442	+	1.68688i	0.359666	+	0.933081i	$0.382891\pi$
$139$	−487.444	+	2764.43i	−0.297442	+	1.68688i	−0.657108	+	0.753797i	$0.728220\pi$
$140$		0			0
$141$	1066.94	+	989.031i	0.637253	+	0.590720i
$142$		0			0
$143$	1516.11	+	2625.97i	0.886596	+	1.53563i
$144$		0			0
$145$	111.627	−	193.343i	0.0639317	−	0.110733i
$146$		0			0
$147$	1093.68	−	1695.12i	0.613641	−	0.951096i
$148$		0			0
$149$	−2123.02	−	1781.43i	−1.16728	−	0.979464i	−0.167300	−	0.985906i	$-0.553505\pi$
$149$	−2123.02	−	1781.43i	−1.16728	−	0.979464i	−0.999979	+	0.00644242i	$0.997949\pi$
$150$		0			0
$151$	−711.587	−	258.996i	−0.383497	−	0.139582i	0.143075	−	0.989712i	$-0.454301\pi$
$151$	−711.587	−	258.996i	−0.383497	−	0.139582i	−0.526573	+	0.850130i	$0.676523\pi$
$152$		0			0
$153$	−1863.92	−	1907.34i	−0.984893	−	1.00784i
$154$		0			0
$155$	120.954	+	685.966i	0.0626792	+	0.355472i
$156$		0			0
$157$	1105.36	−	402.318i	0.561894	−	0.204513i	−0.0454293	−	0.998968i	$-0.514466\pi$
$157$	1105.36	−	402.318i	0.561894	−	0.204513i	0.607323	+	0.794455i	$0.292243\pi$
$158$		0			0
$159$	−1028.88	+	780.119i	−0.513179	+	0.389104i
$160$		0			0
$161$	−3962.91			−1.93988
$162$		0			0
$163$	2323.45			1.11648			0.558240	−	0.829679i	$-0.311477\pi$
$163$	2323.45			1.11648			0.558240	+	0.829679i	$0.311477\pi$
$164$		0			0
$165$	1020.76	−	773.958i	0.481610	−	0.365167i
$166$		0			0
$167$	3377.16	−	1229.19i	1.56487	−	0.569565i	0.593022	−	0.805186i	$-0.297935\pi$
$167$	3377.16	−	1229.19i	1.56487	−	0.569565i	0.971845	+	0.235621i	$0.0757126\pi$
$168$		0			0
$169$	161.558	+	916.240i	0.0735356	+	0.417041i
$170$		0			0
$171$	−3562.48	+	910.738i	−1.59316	+	0.407286i
$172$		0			0
$173$	−315.163	−	114.710i	−0.138505	−	0.0504118i	0.271837	−	0.962343i	$-0.412369\pi$
$173$	−315.163	−	114.710i	−0.138505	−	0.0504118i	−0.410343	+	0.911931i	$0.634591\pi$
$174$		0			0
$175$	2161.13	+	1813.40i	0.933520	+	0.783316i
$176$		0			0
$177$	−730.108	+	1131.61i	−0.310047	+	0.480549i
$178$		0			0
$179$	−1838.52	+	3184.41i	−0.767695	+	1.32969i	0.171115	+	0.985251i	$0.445263\pi$
$179$	−1838.52	+	3184.41i	−0.767695	+	1.32969i	−0.938810	+	0.344436i	$0.888070\pi$
$180$		0			0
$181$	−208.366	−	360.900i	−0.0855675	−	0.148207i	0.820065	−	0.572270i	$-0.193937\pi$
$181$	−208.366	−	360.900i	−0.0855675	−	0.148207i	−0.905633	+	0.424063i	$0.860604\pi$
$182$		0			0
$183$	1263.11	+	1170.88i	0.510229	+	0.472971i
$184$		0			0
$185$	−18.5926	+	105.444i	−0.00738896	+	0.0419049i
$186$		0			0
$187$	4102.60	−	3442.49i	1.60434	−	1.34620i
$188$		0			0
$189$	3234.38	−	1982.84i	1.24479	−	0.763126i
$190$		0			0
$191$	−1409.52	+	1182.73i	−0.533977	+	0.448060i	−0.869472	−	0.493982i	$-0.835541\pi$
$191$	−1409.52	+	1182.73i	−0.533977	+	0.448060i	0.335495	+	0.942042i	$0.391096\pi$
$192$		0			0
$193$	54.6854	−	310.136i	0.0203956	−	0.115669i	−0.972910	−	0.231183i	$-0.925740\pi$
$193$	54.6854	−	310.136i	0.0203956	−	0.115669i	0.993306	+	0.115514i	$0.0368515\pi$
$194$		0			0
$195$	1262.32	−	390.032i	0.463571	−	0.143235i
$196$		0			0
$197$	839.214	+	1453.56i	0.303510	+	0.525695i	0.976929	−	0.213566i	$-0.0685079\pi$
$197$	839.214	+	1453.56i	0.303510	+	0.525695i	−0.673418	+	0.739262i	$0.735175\pi$
$198$		0			0
$199$	−255.716	+	442.913i	−0.0910917	+	0.157775i	−0.907971	−	0.419034i	$-0.862369\pi$
$199$	−255.716	+	442.913i	−0.0910917	+	0.157775i	0.816879	+	0.576809i	$0.195702\pi$
$200$		0			0
$201$	−484.756	−	23.9615i	−0.170110	−	0.00840853i
$202$		0			0
$203$	1017.15	+	853.488i	0.351674	+	0.295089i
$204$		0			0
$205$	−1769.72	−	644.127i	−0.602941	−	0.219452i
$206$		0			0
$207$	−3566.64	−	1713.41i	−1.19758	−	0.575316i
$208$		0			0
$209$	−1282.26	−	7272.07i	−0.424382	−	2.40679i
$210$		0			0
$211$	−2331.59	+	848.631i	−0.760728	+	0.276882i	−0.693113	−	0.720829i	$-0.743761\pi$
$211$	−2331.59	+	848.631i	−0.760728	+	0.276882i	−0.0676152	+	0.997711i	$0.521539\pi$
$212$		0			0
$213$	−432.748	−	3439.95i	−0.139208	−	1.10658i
$214$		0			0
$215$	1305.81			0.414210
$216$		0			0
$217$	−4142.69			−1.29596
$218$		0			0
$219$	1023.44	+	430.842i	0.315789	+	0.132939i
$220$		0			0
$221$	5190.53	−	1889.20i	1.57988	−	0.575028i
$222$		0			0
$223$	−80.0661	−	454.077i	−0.0240432	−	0.136355i	0.970423	−	0.241409i	$-0.0776096\pi$
$223$	−80.0661	−	454.077i	−0.0240432	−	0.136355i	−0.994467	+	0.105054i	$0.966499\pi$
$224$		0			0
$225$	1160.98	+	2566.46i	0.343994	+	0.760433i
$226$		0			0
$227$	−1280.18	−	465.948i	−0.374311	−	0.136238i	0.148012	−	0.988986i	$-0.452712\pi$
$227$	−1280.18	−	465.948i	−0.374311	−	0.136238i	−0.522324	+	0.852747i	$0.674935\pi$
$228$		0			0
$229$	1253.28	+	1051.62i	0.361654	+	0.303464i	0.805450	−	0.592664i	$-0.201924\pi$
$229$	1253.28	+	1051.62i	0.361654	+	0.303464i	−0.443796	+	0.896128i	$0.646368\pi$
$230$		0			0
$231$	3478.95	+	6777.99i	0.990902	+	1.93056i
$232$		0			0
$233$	2293.06	−	3971.70i	0.644736	−	1.11672i	−0.339627	−	0.940560i	$-0.610301\pi$
$233$	2293.06	−	3971.70i	0.644736	−	1.11672i	0.984362	−	0.176155i	$-0.0563659\pi$
$234$		0			0
$235$	636.500	+	1102.45i	0.176684	+	0.306025i
$236$		0			0
$237$	634.336	−	2785.33i	0.173859	−	0.763403i
$238$		0			0
$239$	285.319	−	1618.12i	0.0772206	−	0.437940i	−0.921545	−	0.388271i	$-0.873072\pi$
$239$	285.319	−	1618.12i	0.0772206	−	0.437940i	0.998766	−	0.0496686i	$-0.0158165\pi$
$240$		0			0
$241$	−4699.71	+	3943.52i	−1.25616	+	1.05404i	−0.260080	+	0.965587i	$0.583749\pi$
$241$	−4699.71	+	3943.52i	−1.25616	+	1.05404i	−0.996080	+	0.0884567i	$0.971807\pi$
$242$		0			0
$243$	3768.26	−	386.146i	0.994791	−	0.101940i
$244$		0			0
$245$	1352.21	−	1134.63i	0.352609	−	0.295874i
$246$		0			0
$247$	1322.50	−	7500.29i	0.340684	−	1.93211i
$248$		0			0
$249$	−467.112	+	2051.06i	−0.118884	+	0.522010i
$250$		0			0
$251$	2445.01	+	4234.88i	0.614851	+	1.06495i	0.990411	+	0.138154i	$0.0441170\pi$
$251$	2445.01	+	4234.88i	0.614851	+	1.06495i	−0.375560	+	0.926798i	$0.622550\pi$
$252$		0			0
$253$	3973.07	−	6881.55i	0.987291	−	1.71004i
$254$		0			0
$255$	−1065.57	−	2076.04i	−0.261681	−	0.509829i
$256$		0			0
$257$	−904.386	−	758.870i	−0.219510	−	0.184191i	0.526401	−	0.850236i	$-0.323541\pi$
$257$	−904.386	−	758.870i	−0.219510	−	0.184191i	−0.745911	+	0.666046i	$0.767986\pi$
$258$		0			0
$259$	−598.395	−	217.798i	−0.143562	−	0.0522521i
$260$		0			0
$261$	546.421	+	1207.92i	0.129589	+	0.286469i
$262$		0			0
$263$	1187.46	+	6734.40i	0.278410	+	1.57894i	0.727919	+	0.685664i	$0.240488\pi$
$263$	1187.46	+	6734.40i	0.278410	+	1.57894i	−0.449509	+	0.893276i	$0.648401\pi$
$264$		0			0
$265$	−1061.68	+	386.418i	−0.246106	+	0.0895754i
$266$		0			0
$267$	−4835.01	−	2035.41i	−1.10823	−	0.466536i
$268$		0			0
$269$	343.601			0.0778801			0.0389401	−	0.999242i	$-0.487602\pi$
$269$	343.601			0.0778801			0.0389401	+	0.999242i	$0.487602\pi$
$270$		0			0
$271$	2607.43			0.584466			0.292233	−	0.956347i	$-0.405602\pi$
$271$	2607.43			0.584466			0.292233	+	0.956347i	$0.405602\pi$
$272$		0			0
$273$	980.782	+	7796.33i	0.217435	+	1.72841i
$274$		0			0
$275$	−5315.63	+	1934.73i	−1.16562	+	0.424249i
$276$		0			0
$277$	0.0627023	+	0.355603i	1.36008e−5	+	7.71339e-5i	0.984815	−	0.173610i	$-0.0555431\pi$
$277$	0.0627023	+	0.355603i	1.36008e−5	+	7.71339e-5i	−0.984801	+	0.173687i	$0.944432\pi$
$278$		0			0
$279$	−3728.44	−	1791.14i	−0.800057	−	0.384348i
$280$		0			0
$281$	4627.50	+	1684.27i	0.982396	+	0.357563i	0.782771	−	0.622310i	$-0.213806\pi$
$281$	4627.50	+	1684.27i	0.982396	+	0.357563i	0.199625	+	0.979872i	$0.436028\pi$
$282$		0			0
$283$	−4317.90	−	3623.15i	−0.906971	−	0.761039i	0.0645695	−	0.997913i	$-0.479433\pi$
$283$	−4317.90	−	3623.15i	−0.906971	−	0.761039i	−0.971540	+	0.236874i	$0.923877\pi$
$284$		0			0
$285$	−3213.55	−	158.846i	−0.667909	−	0.0330148i
$286$		0			0
$287$	5600.43	−	9700.22i	1.15186	−	1.99507i
$288$		0			0
$289$	−2421.49	−	4194.15i	−0.492874	−	0.853684i
$290$		0			0
$291$	146.157	−	45.1597i	0.0294428	−	0.00909727i
$292$		0			0
$293$	−397.381	+	2253.66i	−0.0792328	+	0.449352i	0.919220	+	0.393745i	$0.128821\pi$
$293$	−397.381	+	2253.66i	−0.0792328	+	0.449352i	−0.998453	+	0.0556072i	$0.982291\pi$
$294$		0			0
$295$	−902.692	+	757.448i	−0.178158	+	0.149493i
$296$		0			0
$297$	200.525	+	7604.39i	0.0391773	+	1.48570i
$298$		0			0
$299$	6278.12	−	5267.97i	1.21429	−	1.01891i
$300$		0			0
$301$	−1348.59	+	7648.23i	−0.258244	+	1.46457i
$302$		0			0
$303$	352.035	+	326.329i	0.0667455	+	0.0618716i
$304$		0			0
$305$	753.528	+	1305.15i	0.141465	+	0.245025i
$306$		0			0
$307$	−3242.45	+	5616.10i	−0.602790	+	1.04406i	0.389606	+	0.920982i	$0.372611\pi$
$307$	−3242.45	+	5616.10i	−0.602790	+	1.04406i	−0.992397	+	0.123082i	$0.960722\pi$
$308$		0			0
$309$	−5747.44	+	8908.09i	−1.05812	+	1.64001i
$310$		0			0
$311$	2064.12	+	1732.01i	0.376353	+	0.315797i	0.811269	−	0.584674i	$-0.198777\pi$
$311$	2064.12	+	1732.01i	0.376353	+	0.315797i	−0.434916	+	0.900471i	$0.643222\pi$
$312$		0			0
$313$	−4366.05	−	1589.11i	−0.788446	−	0.286971i	−0.0837560	−	0.996486i	$-0.526692\pi$
$313$	−4366.05	−	1589.11i	−0.788446	−	0.286971i	−0.704690	+	0.709515i	$0.748914\pi$
$314$		0			0
$315$	3216.18	−	822.207i	0.575274	−	0.147067i
$316$		0			0
$317$	860.106	+	4877.90i	0.152392	+	0.864260i	0.961131	+	0.276092i	$0.0890394\pi$
$317$	860.106	+	4877.90i	0.152392	+	0.864260i	−0.808739	+	0.588168i	$0.799849\pi$
$318$		0			0
$319$	−2501.83	+	910.591i	−0.439108	+	0.159822i
$320$		0			0
$321$	1453.68	−	1102.21i	0.252762	−	0.191650i
$322$		0			0
$323$	−13451.5			−2.31723
$324$		0			0
$325$	−5834.30			−0.995781
$326$		0			0
$327$	−2720.03	+	2062.39i	−0.459994	+	0.348777i
$328$		0			0
$329$	−7114.53	+	2589.48i	−1.19221	+	0.433928i
$330$		0			0
$331$	−307.323	−	1742.92i	−0.0510333	−	0.289424i	0.948601	−	0.316475i	$-0.102499\pi$
$331$	−307.323	−	1742.92i	−0.0510333	−	0.289424i	−0.999634	+	0.0270512i	$0.991388\pi$
$332$		0			0
$333$	−444.391	−	454.743i	−0.0731305	−	0.0748341i
$334$		0			0
$335$	−399.074	−	145.251i	−0.0650859	−	0.0236893i
$336$		0			0
$337$	−625.914	−	525.204i	−0.101174	−	0.0848952i	0.590797	−	0.806820i	$-0.298813\pi$
$337$	−625.914	−	525.204i	−0.101174	−	0.0848952i	−0.691972	+	0.721925i	$0.743258\pi$
$338$		0			0
$339$	359.694	−	557.498i	0.0576280	−	0.0893190i
$340$		0			0
$341$	4153.31	−	7193.75i	0.659573	−	1.14241i
$342$		0			0
$343$	611.577	+	1059.28i	0.0962743	+	0.166752i
$344$		0			0
$345$	−2539.17	−	2353.75i	−0.396244	−	0.367309i
$346$		0			0
$347$	1301.07	−	7378.71i	0.201282	−	1.14153i	−0.701902	−	0.712274i	$-0.747665\pi$
$347$	1301.07	−	7378.71i	0.201282	−	1.14153i	0.903184	−	0.429254i	$-0.141223\pi$
$348$		0			0
$349$	9295.98	−	7800.25i	1.42580	−	1.19638i	0.477648	−	0.878551i	$-0.341489\pi$
$349$	9295.98	−	7800.25i	1.42580	−	1.19638i	0.948147	−	0.317833i	$-0.102955\pi$
$350$		0			0
$351$	−2488.13	+	7440.79i	−0.378367	+	1.13151i
$352$		0			0
$353$	−1902.32	+	1596.24i	−0.286828	+	0.240678i	−0.774837	−	0.632161i	$-0.782168\pi$
$353$	−1902.32	+	1596.24i	−0.286828	+	0.240678i	0.488008	+	0.872839i	$0.337724\pi$
$354$		0			0
$355$	526.802	−	2987.64i	0.0787598	−	0.446669i
$356$		0			0
$357$	13260.0	−	4097.11i	1.96582	−	0.607400i
$358$		0			0
$359$	5053.00	+	8752.06i	0.742862	+	1.28667i	0.951187	+	0.308614i	$0.0998653\pi$
$359$	5053.00	+	8752.06i	0.742862	+	1.28667i	−0.208326	+	0.978060i	$0.566801\pi$
$360$		0			0
$361$	−5843.98	+	10122.1i	−0.852017	+	1.47574i
$362$		0			0
$363$	−8350.15	−	412.748i	−1.20735	−	0.0596794i
$364$		0			0
$365$	744.320	+	624.559i	0.106738	+	0.0895641i
$366$		0			0
$367$	5169.49	+	1881.54i	0.735273	+	0.267618i	0.682395	−	0.730983i	$-0.260938\pi$
$367$	5169.49	+	1881.54i	0.735273	+	0.267618i	0.0528780	+	0.998601i	$0.483161\pi$
$368$		0			0
$369$	9234.43	−	6308.83i	1.30278	−	0.890040i
$370$		0			0
$371$	−1166.83	−	6617.42i	−0.163285	−	0.926036i
$372$		0			0
$373$	8111.81	−	2952.46i	1.12604	−	0.409845i	0.289188	−	0.957272i	$-0.406615\pi$
$373$	8111.81	−	2952.46i	1.12604	−	0.409845i	0.836853	+	0.547427i	$0.184393\pi$
$374$		0			0
$375$	676.250	+	5375.57i	0.0931237	+	0.740249i
$376$		0			0
$377$	−2745.95			−0.375128
$378$		0			0
$379$	1742.39			0.236149			0.118075	−	0.993005i	$-0.462328\pi$
$379$	1742.39			0.236149			0.118075	+	0.993005i	$0.462328\pi$
$380$		0			0
$381$	6216.49	+	2616.98i	0.835906	+	0.351895i
$382$		0			0
$383$	2574.27	−	936.958i	0.343444	−	0.125003i	−0.164539	−	0.986371i	$-0.552614\pi$
$383$	2574.27	−	936.958i	0.343444	−	0.125003i	0.507983	+	0.861367i	$0.330391\pi$
$384$		0			0
$385$	1157.62	+	6565.16i	0.153240	+	0.869069i
$386$		0			0
$387$	−4520.55	+	6300.37i	−0.593779	+	0.827560i
$388$		0			0
$389$	6652.69	+	2421.38i	0.867108	+	0.315601i	0.736995	−	0.675898i	$-0.236244\pi$
$389$	6652.69	+	2421.38i	0.867108	+	0.315601i	0.130112	+	0.991499i	$0.458466\pi$
$390$		0			0
$391$	−11088.6	−	9304.41i	−1.43420	−	1.20344i
$392$		0			0
$393$	−2679.88	−	5221.16i	−0.343975	−	0.670160i
$394$		0			0
$395$	1249.80	−	2164.72i	0.159201	−	0.275744i
$396$		0			0
$397$	2624.35	+	4545.52i	0.331770	+	0.574642i	0.982859	−	0.184359i	$-0.0590210\pi$
$397$	2624.35	+	4545.52i	0.331770	+	0.574642i	−0.651089	+	0.759001i	$0.725688\pi$
$398$		0			0
$399$	4249.22	−	18658.0i	0.533150	−	2.34103i
$400$		0			0
$401$	−949.429	+	5384.48i	−0.118235	+	0.670544i	0.866863	+	0.498547i	$0.166133\pi$
$401$	−949.429	+	5384.48i	−0.118235	+	0.670544i	−0.985098	+	0.171997i	$0.944978\pi$
$402$		0			0
$403$	6562.94	−	5506.96i	0.811224	−	0.680698i
$404$		0			0
$405$	3250.07	+	650.568i	0.398759	+	0.0798197i
$406$		0			0
$407$	978.133	−	820.751i	0.119126	−	0.0999585i
$408$		0			0
$409$	689.848	−	3912.32i	0.0834005	−	0.472987i	−0.914290	−	0.405061i	$-0.867250\pi$
$409$	689.848	−	3912.32i	0.0834005	−	0.472987i	0.997690	−	0.0679269i	$-0.0216385\pi$
$410$		0			0
$411$	2980.27	−	13086.2i	0.357679	−	1.57054i
$412$		0			0
$413$	−3504.18	−	6069.42i	−0.417505	−	0.723140i
$414$		0			0
$415$	−920.329	+	1594.06i	−0.108861	+	0.188552i
$416$		0			0
$417$	−6660.48	−	12976.5i	−0.782170	−	1.52389i
$418$		0			0
$419$	11193.9	+	9392.79i	1.30515	+	1.09515i	0.989231	+	0.146363i	$0.0467566\pi$
$419$	11193.9	+	9392.79i	1.30515	+	1.09515i	0.315917	+	0.948787i	$0.397688\pi$
$420$		0			0
$421$	13068.9	+	4756.68i	1.51292	+	0.550657i	0.959367	−	0.282160i	$-0.0910509\pi$
$421$	13068.9	+	4756.68i	1.51292	+	0.550657i	0.553549	+	0.832816i	$0.313273\pi$
$422$		0			0
$423$	−7522.70	−	745.516i	−0.864696	−	0.0856932i
$424$		0			0
$425$	1789.39	+	10148.1i	0.204231	+	1.15825i
$426$		0			0
$427$	−8422.61	+	3065.58i	−0.954563	+	0.347433i
$428$		0			0
$429$	−14521.6	−	6113.20i	−1.63428	−	0.687990i
$430$		0			0
$431$	−14619.4			−1.63385			−0.816927	−	0.576741i	$-0.804324\pi$
$431$	−14619.4			−1.63385			−0.816927	+	0.576741i	$0.804324\pi$
$432$		0			0
$433$	14257.6			1.58239			0.791197	−	0.611561i	$-0.209458\pi$
$433$	14257.6			1.58239			0.791197	+	0.611561i	$0.209458\pi$
$434$		0			0
$435$	144.795	+	1150.99i	0.0159595	+	0.126864i
$436$		0			0
$437$	−18754.6	+	6826.12i	−2.05299	+	0.747226i
$438$		0			0
$439$	−2645.69	−	15004.5i	−0.287635	−	1.63126i	−0.695717	−	0.718316i	$-0.744913\pi$
$439$	−2645.69	−	15004.5i	−0.287635	−	1.63126i	0.408081	−	0.912946i	$-0.366198\pi$
$440$		0			0
$441$	793.309	+	10452.2i	0.0856613	+	1.12863i
$442$		0			0
$443$	−4770.78	−	1736.42i	−0.511663	−	0.186230i	0.0732692	−	0.997312i	$-0.476657\pi$
$443$	−4770.78	−	1736.42i	−0.511663	−	0.186230i	−0.584932	+	0.811082i	$0.698879\pi$
$444$		0			0
$445$	−3516.37	−	2950.58i	−0.374588	−	0.314317i
$446$		0			0
$447$	14383.1	+	710.956i	1.52192	+	0.0752283i
$448$		0			0
$449$	7046.08	−	12204.2i	0.740591	−	1.28274i	−0.211636	−	0.977349i	$-0.567879\pi$
$449$	7046.08	−	12204.2i	0.740591	−	1.28274i	0.952227	−	0.305392i	$-0.0987875\pi$
$450$		0			0
$451$	11229.6	+	19450.2i	1.17246	+	2.03076i
$452$		0			0
$453$	3759.45	−	1161.60i	0.389921	−	0.120478i
$454$		0			0
$455$	−1193.95	+	6771.20i	−0.123018	+	0.697668i
$456$		0			0
$457$	−949.074	+	796.368i	−0.0971462	+	0.0815153i	−0.690067	−	0.723745i	$-0.742419\pi$
$457$	−949.074	+	796.368i	−0.0971462	+	0.0815153i	0.592921	+	0.805261i	$0.297975\pi$
$458$		0			0
$459$	13705.5	+	2045.73i	1.39373	+	0.208032i
$460$		0			0
$461$	−2354.96	+	1976.04i	−0.237920	+	0.199639i	−0.753950	−	0.656932i	$-0.771854\pi$
$461$	−2354.96	+	1976.04i	−0.237920	+	0.199639i	0.516030	+	0.856571i	$0.327409\pi$
$462$		0			0
$463$	−1637.88	+	9288.87i	−0.164403	+	0.932377i	0.785274	+	0.619148i	$0.212522\pi$
$463$	−1637.88	+	9288.87i	−0.164403	+	0.932377i	−0.949678	+	0.313229i	$0.898589\pi$
$464$		0			0
$465$	−2654.36	−	2460.53i	−0.264716	−	0.245386i
$466$		0			0
$467$	−5454.27	−	9447.08i	−0.540457	−	0.936100i	−0.998878	−	0.0473641i	$-0.984918\pi$
$467$	−5454.27	−	9447.08i	−0.540457	−	0.936100i	0.458420	−	0.888736i	$-0.348415\pi$
$468$		0			0
$469$	1262.90	−	2187.41i	0.124340	−	0.215363i
$470$		0			0
$471$	−3313.72	+	5136.01i	−0.324179	+	0.502452i
$472$		0			0
$473$	−11929.0	−	10009.7i	−1.15962	−	0.973033i
$474$		0			0
$475$	13351.2	+	4859.45i	1.28968	+	0.469404i
$476$		0			0
$477$	1810.97	−	6460.21i	0.173834	−	0.620110i
$478$		0			0
$479$	−2009.10	−	11394.2i	−0.191645	−	1.08687i	−0.917116	−	0.398621i	$-0.869489\pi$
$479$	−2009.10	−	11394.2i	−0.191645	−	1.08687i	0.725470	−	0.688253i	$-0.241622\pi$
$480$		0			0
$481$	1237.51	−	450.418i	0.117309	−	0.0426971i
$482$		0			0
$483$	16408.5	−	12441.3i	1.54578	−	1.17205i
$484$		0			0
$485$	133.855			0.0125320
$486$		0			0
$487$	−15337.4			−1.42711			−0.713554	−	0.700600i	$-0.752916\pi$
$487$	−15337.4			−1.42711			−0.713554	+	0.700600i	$0.752916\pi$
$488$		0			0
$489$	−9620.29	+	7294.31i	−0.889661	+	0.674560i
$490$		0			0
$491$	−6647.78	+	2419.59i	−0.611018	+	0.222392i	−0.628949	−	0.777447i	$-0.716514\pi$
$491$	−6647.78	+	2419.59i	−0.611018	+	0.222392i	0.0179304	+	0.999839i	$0.494292\pi$
$492$		0			0
$493$	842.185	+	4776.27i	0.0769374	+	0.436334i
$494$		0			0
$495$	−1796.67	+	6409.19i	−0.163140	+	0.581963i
$496$		0			0
$497$	16954.9	+	6171.06i	1.53024	+	0.556962i
$498$		0			0
$499$	10522.3	+	8829.24i	0.943972	+	0.792087i	0.978272	−	0.207325i	$-0.0664758\pi$
$499$	10522.3	+	8829.24i	0.943972	+	0.792087i	−0.0343001	+	0.999412i	$0.510920\pi$
$500$		0			0
$501$	−10124.3	+	15691.9i	−0.902833	+	1.39932i
$502$		0			0
$503$	1396.87	−	2419.45i	0.123824	−	0.214469i	−0.797449	−	0.603387i	$-0.793818\pi$
$503$	1396.87	−	2419.45i	0.123824	−	0.214469i	0.921273	+	0.388918i	$0.127151\pi$
$504$		0			0
$505$	210.012	+	363.751i	0.0185057	+	0.0320529i
$506$		0			0
$507$	−3545.41	−	3286.51i	−0.310566	−	0.287888i
$508$		0			0
$509$	−1433.78	+	8131.38i	−0.124855	+	0.708089i	0.856539	+	0.516083i	$0.172610\pi$
$509$	−1433.78	+	8131.38i	−0.124855	+	0.708089i	−0.981394	+	0.192006i	$0.938501\pi$
$510$		0			0
$511$	−4426.81	+	3714.54i	−0.383230	+	0.321568i
$512$		0			0
$513$	11891.4	−	14955.1i	1.02342	−	1.28710i
$514$		0			0
$515$	−7106.03	+	5962.66i	−0.608017	+	0.510187i
$516$		0			0
$517$	2636.16	−	14950.4i	0.224252	−	1.27180i
$518$		0			0
$519$	1665.07	−	514.474i	0.140825	−	0.0435123i
$520$		0			0
$521$	10192.6	+	17654.1i	0.857094	+	1.48453i	0.874689	+	0.484685i	$0.161066\pi$
$521$	10192.6	+	17654.1i	0.857094	+	1.48453i	−0.0175944	+	0.999845i	$0.505601\pi$
$522$		0			0
$523$	−7162.07	+	12405.1i	−0.598806	+	1.03716i	0.394192	+	0.919028i	$0.371025\pi$
$523$	−7162.07	+	12405.1i	−0.598806	+	1.03716i	−0.992998	+	0.118134i	$0.962309\pi$
$524$		0			0
$525$	−14641.3	−	723.718i	−1.21714	−	0.0601631i
$526$		0			0
$527$	−11591.6	−	9726.52i	−0.958138	−	0.803973i
$528$		0			0
$529$	−8748.46	−	3184.18i	−0.719032	−	0.261706i
$530$		0			0
$531$	−529.589	−	6977.59i	−0.0432810	−	0.570248i
$532$		0			0
$533$	4022.38	+	22812.1i	0.326883	+	1.85385i
$534$		0			0
$535$	1500.02	−	545.962i	0.121218	−	0.0441196i
$536$		0			0
$537$	−2384.81	−	18957.0i	−0.191642	−	1.52338i
$538$		0			0
$539$	−21050.5			−1.68221
$540$		0			0
$541$	−9772.64			−0.776633			−0.388317	−	0.921526i	$-0.626943\pi$
$541$	−9772.64			−0.776633			−0.388317	+	0.921526i	$0.626943\pi$
$542$		0			0
$543$	1995.77	+	840.166i	0.157729	+	0.0663996i
$544$		0			0
$545$	−2806.73	+	1021.57i	−0.220600	+	0.0802919i
$546$		0			0
$547$	2487.70	+	14108.5i	0.194454	+	1.10280i	0.913194	+	0.407525i	$0.133608\pi$
$547$	2487.70	+	14108.5i	0.194454	+	1.10280i	−0.718740	+	0.695279i	$0.755281\pi$
$548$		0			0
$549$	−8905.83	−	882.587i	−0.692334	−	0.0686118i
$550$		0			0
$551$	6283.83	+	2287.13i	0.485844	+	0.176833i
$552$		0			0
$553$	11388.2	+	9555.87i	0.875728	+	0.734823i
$554$		0			0
$555$	−254.051	−	494.964i	−0.0194304	−	0.0378560i
$556$		0			0
$557$	7088.20	−	12277.1i	0.539204	−	0.933929i	−0.459743	−	0.888052i	$-0.652058\pi$
$557$	7088.20	−	12277.1i	0.539204	−	0.933929i	0.998947	−	0.0458770i	$-0.0146082\pi$
$558$		0			0
$559$	−8030.49	−	13909.2i	−0.607609	−	1.05241i
$560$		0			0
$561$	−6179.45	+	27133.6i	−0.465056	+	2.04203i
$562$		0			0
$563$	−2412.55	+	13682.2i	−0.180598	+	1.02422i	0.750884	+	0.660435i	$0.229628\pi$
$563$	−2412.55	+	13682.2i	−0.180598	+	1.02422i	−0.931482	+	0.363788i	$0.881483\pi$
$564$		0			0
$565$	444.719	−	373.164i	0.0331141	−	0.0277860i
$566$		0			0
$567$	−7167.00	+	18364.1i	−0.530839	+	1.36018i
$568$		0			0
$569$	1899.81	−	1594.13i	0.139972	−	0.117451i	−0.570113	−	0.821566i	$-0.693101\pi$
$569$	1899.81	−	1594.13i	0.139972	−	0.117451i	0.710086	+	0.704115i	$0.248656\pi$
$570$		0			0
$571$	680.576	−	3859.74i	0.0498795	−	0.282881i	−0.949658	−	0.313288i	$-0.898569\pi$
$571$	680.576	−	3859.74i	0.0498795	−	0.282881i	0.999538	+	0.0304075i	$0.00968049\pi$
$572$		0			0
$573$	2123.06	−	9322.24i	0.154786	−	0.679654i
$574$		0			0
$575$	7644.60	+	13240.8i	0.554438	+	0.960315i
$576$		0			0
$577$	9634.98	−	16688.3i	0.695164	−	1.20406i	−0.274962	−	0.961455i	$-0.588665\pi$
$577$	9634.98	−	16688.3i	0.695164	−	1.20406i	0.970125	−	0.242604i	$-0.0780016\pi$
$578$		0			0
$579$	747.227	+	1455.81i	0.0536333	+	0.104493i
$580$		0			0
$581$	−8386.07	−	7036.74i	−0.598817	−	0.502467i
$582$		0			0
$583$	12660.9	+	4608.20i	0.899420	+	0.327362i
$584$		0			0
$585$	−4002.17	+	5577.90i	−0.282854	+	0.394218i
$586$		0			0
$587$	−2051.32	−	11633.6i	−0.144237	−	0.818007i	−0.967977	−	0.251040i	$-0.919228\pi$
$587$	−2051.32	−	11633.6i	−0.144237	−	0.818007i	0.823740	−	0.566968i	$-0.191884\pi$
$588$		0			0
$589$	−19605.5	+	7135.80i	−1.37153	+	0.499194i
$590$		0			0
$591$	−8038.15	−	3383.85i	−0.559468	−	0.235521i
$592$		0			0
$593$	−13127.9			−0.909101			−0.454551	−	0.890721i	$-0.650200\pi$
$593$	−13127.9			−0.909101			−0.454551	+	0.890721i	$0.650200\pi$
$594$		0			0
$595$	12143.9			0.836728
$596$		0			0
$597$	−331.698	−	2636.70i	−0.0227395	−	0.180759i
$598$		0			0
$599$	26377.9	−	9600.77i	1.79928	−	0.654886i	0.800855	−	0.598858i	$-0.204378\pi$
$599$	26377.9	−	9600.77i	1.79928	−	0.654886i	0.998429	−	0.0560284i	$-0.0178437\pi$
$600$		0			0
$601$	−2662.66	−	15100.7i	−0.180719	−	1.02491i	−0.931333	−	0.364167i	$-0.881354\pi$
$601$	−2662.66	−	15100.7i	−0.180719	−	1.02491i	0.750615	−	0.660740i	$-0.229758\pi$
$602$		0			0
$603$	2082.37	−	1422.65i	0.140631	−	0.0960774i
$604$		0			0
$605$	−6874.24	−	2502.02i	−0.461946	−	0.168135i
$606$		0			0
$607$	−20854.2	−	17498.7i	−1.39447	−	1.17010i	−0.963491	−	0.267740i	$-0.913723\pi$
$607$	−20854.2	−	17498.7i	−1.39447	−	1.17010i	−0.430981	−	0.902361i	$-0.641832\pi$
$608$		0			0
$609$	−6890.99	−	340.622i	−0.458517	−	0.0226645i
$610$		0			0
$611$	7828.75	−	13559.8i	0.518359	−	0.897824i
$612$		0			0
$613$	2651.31	+	4592.20i	0.174690	+	0.302573i	0.940054	−	0.341025i	$-0.110774\pi$
$613$	2651.31	+	4592.20i	0.174690	+	0.302573i	−0.765364	+	0.643598i	$0.777441\pi$
$614$		0			0
$615$	9349.78	−	2888.91i	0.613040	−	0.189418i
$616$		0			0
$617$	−1200.77	+	6809.92i	−0.0783489	+	0.444339i	0.920246	+	0.391341i	$0.127989\pi$
$617$	−1200.77	+	6809.92i	−0.0783489	+	0.444339i	−0.998595	+	0.0529978i	$0.983122\pi$
$618$		0			0
$619$	1564.51	−	1312.78i	0.101588	−	0.0852424i	−0.590579	−	0.806980i	$-0.701101\pi$
$619$	1564.51	−	1312.78i	0.101588	−	0.0852424i	0.692167	+	0.721737i	$0.256656\pi$
$620$		0			0
$621$	20146.9	−	4102.78i	1.30188	−	0.265119i
$622$		0			0
$623$	20913.4	−	17548.5i	1.34491	−	1.12851i
$624$		0			0
$625$	1441.31	−	8174.07i	0.0922438	−	0.523140i
$626$		0			0
$627$	28139.4	+	26084.6i	1.79231	+	1.66143i
$628$		0			0
$629$	−1163.00	−	2014.38i	−0.0737231	−	0.127692i
$630$		0			0
$631$	−7978.31	+	13818.8i	−0.503346	+	0.871821i	0.496646	+	0.867953i	$0.334565\pi$
$631$	−7978.31	+	13818.8i	−0.503346	+	0.871821i	−0.999993	+	0.00386809i	$0.998769\pi$
$632$		0			0
$633$	6989.81	−	10833.7i	0.438894	−	0.680252i
$634$		0			0
$635$	4521.08	+	3793.64i	0.282541	+	0.237080i
$636$		0			0
$637$	−20401.8	−	7425.64i	−1.26899	−	0.461875i
$638$		0			0
$639$	12591.3	+	12884.6i	0.779507	+	0.797666i
$640$		0			0
$641$	95.6010	+	542.180i	0.00589081	+	0.0334085i	0.987611	−	0.156919i	$-0.0501561\pi$
$641$	95.6010	+	542.180i	0.00589081	+	0.0334085i	−0.981721	+	0.190327i	$0.939045\pi$
$642$		0			0
$643$	−10646.4	+	3874.97i	−0.652959	+	0.237658i	−0.647194	−	0.762326i	$-0.724058\pi$
$643$	−10646.4	+	3874.97i	−0.652959	+	0.237658i	−0.00576568	+	0.999983i	$0.501835\pi$
$644$		0			0
$645$	−5406.72	+	4099.49i	−0.330061	+	0.250259i
$646$		0			0
$647$	21233.5			1.29022			0.645112	−	0.764088i	$-0.276811\pi$
$647$	21233.5			1.29022			0.645112	+	0.764088i	$0.276811\pi$
$648$		0			0
$649$	14052.7			0.849947
$650$		0			0
$651$	17152.9	−	13005.7i	1.03268	−	0.783001i
$652$		0			0
$653$	21181.2	−	7709.31i	1.26935	−	0.462004i	0.382451	−	0.923976i	$-0.375080\pi$
$653$	21181.2	−	7709.31i	1.26935	−	0.462004i	0.886895	+	0.461971i	$0.152858\pi$
$654$		0			0
$655$	−891.725	−	5057.22i	−0.0531948	−	0.301682i
$656$		0			0
$657$	−5590.18	+	1429.11i	−0.331954	+	0.0848630i
$658$		0			0
$659$	−12762.0	−	4644.99i	−0.754381	−	0.274572i	−0.0639327	−	0.997954i	$-0.520364\pi$
$659$	−12762.0	−	4644.99i	−0.754381	−	0.274572i	−0.690448	+	0.723382i	$0.742587\pi$
$660$		0			0
$661$	1688.35	+	1416.69i	0.0993481	+	0.0833630i	0.691109	−	0.722751i	$-0.257123\pi$
$661$	1688.35	+	1416.69i	0.0993481	+	0.0833630i	−0.591761	+	0.806114i	$0.701567\pi$
$662$		0			0
$663$	−15560.5	+	24117.6i	−0.911492	+	1.41274i
$664$		0			0
$665$	8372.03	−	14500.8i	0.488201	−	0.845588i
$666$		0			0
$667$	3597.97	+	6231.87i	0.208867	+	0.361768i
$668$		0			0
$669$	1757.06	+	1628.76i	0.101543	+	0.0941276i
$670$		0			0
$671$	3120.85	−	17699.2i	0.179552	−	1.01829i
$672$		0			0
$673$	−468.396	+	393.031i	−0.0268281	+	0.0225115i	−0.656103	−	0.754671i	$-0.727796\pi$
$673$	−468.396	+	393.031i	−0.0268281	+	0.0225115i	0.629275	+	0.777183i	$0.283352\pi$
$674$		0			0
$675$	−12864.3	−	6981.69i	−0.733551	−	0.398112i
$676$		0			0
$677$	10915.1	−	9158.89i	0.619650	−	0.519948i	−0.278044	−	0.960568i	$-0.589686\pi$
$677$	10915.1	−	9158.89i	0.619650	−	0.519948i	0.897693	+	0.440621i	$0.145242\pi$
$678$		0			0
$679$	−138.240	+	784.000i	−0.00781322	+	0.0443110i
$680$		0			0
$681$	6763.44	−	2089.78i	0.380581	−	0.117592i
$682$		0			0
$683$	7351.56	+	12733.3i	0.411859	+	0.713360i	0.995093	−	0.0989436i	$-0.0315464\pi$
$683$	7351.56	+	12733.3i	0.411859	+	0.713360i	−0.583234	+	0.812304i	$0.698213\pi$
$684$		0			0
$685$	5871.89	−	10170.4i	0.327523	−	0.567286i
$686$		0			0
$687$	−8490.72	−	419.696i	−0.471530	−	0.0233077i
$688$		0			0
$689$	10645.2	+	8932.37i	0.588606	+	0.493899i
$690$		0			0
$691$	−17454.4	−	6352.88i	−0.960921	−	0.349747i	−0.186527	−	0.982450i	$-0.559723\pi$
$691$	−17454.4	−	6352.88i	−0.960921	−	0.349747i	−0.774394	+	0.632703i	$0.781945\pi$
$692$		0			0
$693$	−35683.7	−	17142.5i	−1.95601	−	0.939667i
$694$		0			0
$695$	−2216.26	−	12569.0i	−0.120961	−	0.686002i
$696$		0			0
$697$	38445.4	−	13993.0i	2.08927	−	0.760433i
$698$		0			0
$699$	2974.41	+	23643.8i	0.160948	+	1.27939i
$700$		0			0
$701$	3749.76			0.202035			0.101017	−	0.994885i	$-0.467790\pi$
$701$	3749.76			0.202035			0.101017	+	0.994885i	$0.467790\pi$
$702$		0			0
$703$	−3207.09			−0.172059
$704$		0			0
$705$	−6096.52	−	2566.48i	−0.325685	−	0.137105i
$706$		0			0
$707$	−2347.42	+	854.391i	−0.124871	+	0.0454494i
$708$		0			0
$709$	101.160	+	573.705i	0.00535843	+	0.0303892i	0.987370	−	0.158433i	$-0.0506442\pi$
$709$	101.160	+	573.705i	0.00535843	+	0.0303892i	−0.982011	+	0.188822i	$0.939533\pi$
$710$		0			0
$711$	6117.87	+	13524.2i	0.322698	+	0.713357i
$712$		0			0
$713$	−21097.3	−	7678.78i	−1.10813	−	0.403328i
$714$		0			0
$715$	−10561.1	−	8861.84i	−0.552397	−	0.463516i
$716$		0			0
$717$	3898.62	+	7595.61i	0.203064	+	0.395625i
$718$		0			0
$719$	−13076.5	+	22649.2i	−0.678264	+	1.17479i	0.297239	+	0.954803i	$0.403934\pi$
$719$	−13076.5	+	22649.2i	−0.678264	+	1.17479i	−0.975503	+	0.219985i	$0.929399\pi$
$720$		0			0
$721$	−27585.1	−	47778.8i	−1.42486	−	2.46793i
$722$		0			0
$723$	7078.83	−	31082.7i	0.364128	−	1.59886i
$724$		0			0
$725$	889.551	−	5044.90i	0.0455684	−	0.258431i
$726$		0			0
$727$	8156.67	−	6844.26i	0.416113	−	0.349160i	−0.410569	−	0.911829i	$-0.634670\pi$
$727$	8156.67	−	6844.26i	0.416113	−	0.349160i	0.826682	+	0.562669i	$0.190226\pi$
$728$		0			0
$729$	−14390.3	+	13429.1i	−0.731103	+	0.682267i
$730$		0			0
$731$	−21730.6	+	18234.1i	−1.09950	+	0.922590i
$732$		0			0
$733$	−3183.40	+	18053.9i	−0.160411	+	0.909738i	0.793259	+	0.608884i	$0.208383\pi$
$733$	−3183.40	+	18053.9i	−0.160411	+	0.909738i	−0.953670	+	0.300853i	$0.902729\pi$
$734$		0			0
$735$	−2036.73	+	8943.14i	−0.102212	+	0.448807i
$736$		0			0
$737$	2532.28	+	4386.04i	0.126564	+	0.219215i
$738$		0			0
$739$	−4580.62	+	7933.86i	−0.228012	+	0.394928i	−0.957219	−	0.289365i	$-0.906556\pi$
$739$	−4580.62	+	7933.86i	−0.228012	+	0.394928i	0.729207	+	0.684293i	$0.239889\pi$
$740$		0			0
$741$	18070.8	+	35207.0i	0.895880	+	1.74543i
$742$		0			0
$743$	19612.0	+	16456.4i	0.968362	+	0.812552i	0.982293	−	0.187351i	$-0.0599902\pi$
$743$	19612.0	+	16456.4i	0.968362	+	0.812552i	−0.0139313	+	0.999903i	$0.504435\pi$
$744$		0			0
$745$	11840.8	+	4309.71i	0.582302	+	0.211940i
$746$		0			0
$747$	−4505.07	−	9958.93i	−0.220659	−	0.487789i
$748$		0			0
$749$	1648.59	+	9349.61i	0.0804247	+	0.456111i
$750$		0			0
$751$	−11820.2	+	4302.20i	−0.574334	+	0.209040i	−0.612825	−	0.790218i	$-0.709967\pi$
$751$	−11820.2	+	4302.20i	−0.574334	+	0.209040i	0.0384913	+	0.999259i	$0.487745\pi$
$752$		0			0
$753$	−23418.7	−	9858.68i	−1.13337	−	0.477118i
$754$		0			0
$755$	3443.01			0.165966
$756$		0			0
$757$	−31497.3			−1.51227			−0.756134	−	0.654416i	$-0.772914\pi$
$757$	−31497.3			−1.51227			−0.756134	+	0.654416i	$0.772914\pi$
$758$		0			0
$759$	5153.60	+	40966.4i	0.246461	+	1.95914i
$760$		0			0
$761$	22625.0	−	8234.84i	1.07774	−	0.392264i	0.258672	−	0.965965i	$-0.416715\pi$
$761$	22625.0	−	8234.84i	1.07774	−	0.392264i	0.819064	+	0.573702i	$0.194493\pi$
$762$		0			0
$763$	−3084.73	−	17494.4i	−0.146362	−	0.830063i
$764$		0			0
$765$	10929.6	+	5250.59i	0.516550	+	0.248151i
$766$		0			0
$767$	13619.6	+	4957.13i	0.641168	+	0.233366i
$768$		0			0
$769$	17213.9	+	14444.1i	0.807214	+	0.677333i	0.949941	−	0.312429i	$-0.101143\pi$
$769$	17213.9	+	14444.1i	0.807214	+	0.677333i	−0.142727	+	0.989762i	$0.545587\pi$
$770$		0			0
$771$	6127.05	+	302.860i	0.286200	+	0.0141469i
$772$		0			0
$773$	−16376.9	+	28365.5i	−0.762011	+	1.31984i	0.179801	+	0.983703i	$0.442455\pi$
$773$	−16376.9	+	28365.5i	−0.762011	+	1.31984i	−0.941812	+	0.336139i	$0.890879\pi$
$774$		0			0
$775$	7991.41	+	13841.5i	0.370400	+	0.641551i
$776$		0			0
$777$	3161.43	−	976.823i	0.145966	−	0.0451008i
$778$		0			0
$779$	9795.57	−	55553.5i	0.450530	−	2.55508i
$780$		0			0
$781$	−27714.3	+	23255.1i	−1.26978	+	1.06547i
$782$		0			0
$783$	−6054.65	−	3285.97i	−0.276342	−	0.149976i
$784$		0			0
$785$	−4097.02	+	3437.81i	−0.186279	+	0.156307i
$786$		0			0
$787$	−5068.73	+	28746.2i	−0.229582	+	1.30202i	0.624148	+	0.781306i	$0.285446\pi$
$787$	−5068.73	+	28746.2i	−0.229582	+	1.30202i	−0.853730	+	0.520716i	$0.825665\pi$
$788$		0			0
$789$	−26058.9	−	24156.0i	−1.17582	−	1.08996i
$790$		0			0
$791$	1726.37	+	2990.16i	0.0776012	+	0.134409i
$792$		0			0
$793$	9268.14	−	16052.9i	0.415033	−	0.718859i
$794$		0			0
$795$	3182.76	−	4933.03i	0.141988	−	0.220071i
$796$		0			0
$797$	−12766.2	−	10712.1i	−0.567381	−	0.476089i	0.313395	−	0.949623i	$-0.398534\pi$
$797$	−12766.2	−	10712.1i	−0.567381	−	0.476089i	−0.880776	+	0.473534i	$0.842978\pi$
$798$		0			0
$799$	−25986.9	−	9458.44i	−1.15062	−	0.418793i
$800$		0			0
$801$	26409.5	−	6751.51i	1.16496	−	0.297819i
$802$		0			0
$803$	−2012.10	−	11411.2i	−0.0884252	−	0.501485i
$804$		0			0
$805$	16931.5	−	6162.57i	0.741314	−	0.269816i
$806$		0			0
$807$	−1422.69	+	1078.71i	−0.0620583	+	0.0470539i
$808$		0			0
$809$	−5273.69			−0.229188			−0.114594	−	0.993412i	$-0.536557\pi$
$809$	−5273.69			−0.229188			−0.114594	+	0.993412i	$0.536557\pi$
$810$		0			0
$811$	2370.93			0.102657			0.0513283	−	0.998682i	$-0.483654\pi$
$811$	2370.93			0.102657			0.0513283	+	0.998682i	$0.483654\pi$
$812$		0			0
$813$	−10796.1	+	8185.86i	−0.465728	+	0.353125i
$814$		0			0
$815$	−9926.93	+	3613.11i	−0.426657	+	0.155290i
$816$		0			0
$817$	6791.86	+	38518.5i	0.290841	+	1.64944i
$818$		0			0
$819$	−28537.0	−	29201.8i	−1.21754	−	1.24590i
$820$		0			0
$821$	9376.22	+	3412.67i	0.398578	+	0.145070i	0.533529	−	0.845782i	$-0.320866\pi$
$821$	9376.22	+	3412.67i	0.398578	+	0.145070i	−0.134951	+	0.990852i	$0.543088\pi$
$822$		0			0
$823$	10088.6	+	8465.38i	0.427300	+	0.358548i	0.830932	−	0.556374i	$-0.187808\pi$
$823$	10088.6	+	8465.38i	0.427300	+	0.358548i	−0.403632	+	0.914922i	$0.632252\pi$
$824$		0			0
$825$	15935.5	−	24698.9i	0.672490	−	1.04231i
$826$		0			0
$827$	−8360.67	+	14481.1i	−0.351547	+	0.608897i	−0.986521	−	0.163637i	$-0.947677\pi$
$827$	−8360.67	+	14481.1i	−0.351547	+	0.608897i	0.634974	+	0.772533i	$0.281011\pi$
$828$		0			0
$829$	−1210.13	−	2096.00i	−0.0506990	−	0.0878132i	0.839562	−	0.543264i	$-0.182812\pi$
$829$	−1210.13	−	2096.00i	−0.0506990	−	0.0878132i	−0.890261	+	0.455450i	$0.849478\pi$
$830$		0			0
$831$	−1.37601	−	1.27553i	−5.74408e−5	−	5.32463e-5i
$832$		0			0
$833$	−6658.83	+	37764.1i	−0.276969	+	1.57077i
$834$		0			0
$835$	−12517.5	+	10503.4i	−0.518784	+	0.435312i
$836$		0			0
$837$	21060.9	−	4288.91i	0.869738	−	0.177117i
$838$		0			0
$839$	−10927.3	+	9169.08i	−0.449645	+	0.377297i	−0.839304	−	0.543662i	$-0.817037\pi$
$839$	−10927.3	+	9169.08i	−0.449645	+	0.377297i	0.389659	+	0.920959i	$0.372593\pi$
$840$		0			0
$841$	−3816.43	+	21644.1i	−0.156482	+	0.887452i
$842$		0			0
$843$	−24447.9	+	7553.95i	−0.998851	+	0.308626i
$844$		0			0
$845$	−2115.07	−	3663.40i	−0.0861071	−	0.149142i
$846$		0			0
$847$	21754.0	−	37679.1i	0.882500	−	1.52854i
$848$		0			0
$849$	29253.0	+	1445.98i	1.18252	+	0.0584521i
$850$		0			0
$851$	−2643.71	−	2218.34i	−0.106493	−	0.0893580i
$852$		0			0
$853$	−13748.6	−	5004.09i	−0.551869	−	0.200864i	0.0510079	−	0.998698i	$-0.483757\pi$
$853$	−13748.6	−	5004.09i	−0.551869	−	0.200864i	−0.602877	+	0.797834i	$0.705979\pi$
$854$		0			0
$855$	13804.5	−	9431.02i	0.552167	−	0.377233i
$856$		0			0
$857$	−33.9693	−	192.649i	−0.00135399	−	0.00767885i	0.984123	−	0.177486i	$-0.0567965\pi$
$857$	−33.9693	−	192.649i	−0.00135399	−	0.00767885i	−0.985477	+	0.169807i	$0.945685\pi$
$858$		0			0
$859$	24809.7	−	9030.01i	0.985445	−	0.358673i	0.201490	−	0.979490i	$-0.435422\pi$
$859$	24809.7	−	9030.01i	0.985445	−	0.358673i	0.783955	+	0.620818i	$0.213199\pi$
$860$		0			0
$861$	7264.50	+	57746.2i	0.287542	+	2.28570i
$862$		0			0
$863$	−2803.15			−0.110568			−0.0552840	−	0.998471i	$-0.517606\pi$
$863$	−2803.15			−0.110568			−0.0552840	+	0.998471i	$0.517606\pi$
$864$		0			0
$865$	1524.92			0.0599407
$866$		0			0
$867$	23193.5	+	9763.86i	0.908527	+	0.382466i
$868$		0			0
$869$	−28011.1	+	10195.2i	−1.09346	+	0.397985i
$870$		0			0
$871$	907.051	+	5144.14i	0.0352862	+	0.200118i
$872$		0			0
$873$	−463.389	+	645.834i	−0.0179649	+	0.0250380i
$874$		0			0
$875$	−26495.1	−	9643.44i	−1.02366	−	0.372580i
$876$		0			0
$877$	15359.2	+	12887.9i	0.591385	+	0.496231i	0.888663	−	0.458560i	$-0.151635\pi$
$877$	15359.2	+	12887.9i	0.591385	+	0.496231i	−0.297279	+	0.954791i	$0.596079\pi$
$878$		0			0
$879$	−5429.84	−	10578.9i	−0.208355	−	0.405935i
$880$		0			0
$881$	18708.5	−	32404.1i	0.715443	−	1.23918i	−0.247345	−	0.968927i	$-0.579558\pi$
$881$	18708.5	−	32404.1i	0.715443	−	1.23918i	0.962788	−	0.270256i	$-0.0871085\pi$
$882$		0			0
$883$	3348.08	+	5799.04i	0.127601	+	0.221012i	0.922747	−	0.385407i	$-0.125939\pi$
$883$	3348.08	+	5799.04i	0.127601	+	0.221012i	−0.795146	+	0.606419i	$0.792606\pi$
$884$		0			0
$885$	1359.66	−	5970.17i	0.0516434	−	0.226763i
$886$		0			0
$887$	5380.75	−	30515.7i	0.203684	−	1.15515i	−0.695813	−	0.718223i	$-0.744956\pi$
$887$	5380.75	−	30515.7i	0.203684	−	1.15515i	0.899497	−	0.436927i	$-0.143933\pi$
$888$		0			0
$889$	−26888.9	+	22562.5i	−1.01443	+	0.851206i
$890$		0			0
$891$	−24703.8	−	30856.6i	−0.928852	−	1.16020i
$892$		0			0
$893$	−29209.4	+	24509.6i	−1.09457	+	0.918457i
$894$		0			0
$895$	2903.12	−	16464.4i	0.108425	−	0.614910i
$896$		0			0
$897$	−9456.28	+	41521.9i	−0.351991	+	1.54557i
$898$		0			0
$899$	3761.20	+	6514.59i	0.139536	+	0.241684i
$900$		0			0
$901$	12272.0	−	21255.7i	0.453762	−	0.785938i
$902$		0			0
$903$	−18427.3	−	35901.5i	−0.679093	−	1.32307i
$904$		0			0
$905$	1451.47	+	1217.93i	0.0533131	+	0.0447350i
$906$		0			0
$907$	−1562.19	−	568.589i	−0.0571902	−	0.0208155i	0.313267	−	0.949665i	$-0.398577\pi$
$907$	−1562.19	−	568.589i	−0.0571902	−	0.0208155i	−0.370457	+	0.928850i	$0.620799\pi$
$908$		0			0
$909$	−2482.10	−	245.981i	−0.0905676	−	0.00897544i
$910$		0			0
$911$	−5763.36	−	32685.6i	−0.209603	−	1.18872i	−0.890030	−	0.455903i	$-0.849317\pi$
$911$	−5763.36	−	32685.6i	−0.209603	−	1.18872i	0.680426	−	0.732816i	$-0.261795\pi$
$912$		0			0
$913$	20626.8	−	7507.55i	0.747698	−	0.272140i
$914$		0			0
$915$	−7217.43	−	3038.35i	−0.260766	−	0.109776i
$916$		0			0
$917$	30541.6			1.09986
$918$		0			0
$919$	20905.7			0.750397			0.375199	−	0.926944i	$-0.377575\pi$
$919$	20905.7			0.750397			0.375199	+	0.926944i	$0.377575\pi$
$920$		0			0
$921$	−4205.90	−	33433.1i	−0.150477	−	1.19615i
$922$		0			0
$923$	−35063.6	+	12762.1i	−1.25041	+	0.455113i
$924$		0			0
$925$	426.622	+	2419.49i	0.0151646	+	0.0860027i
$926$		0			0
$927$	−4168.95	−	54927.9i	−0.147709	−	1.94614i
$928$		0			0
$929$	−20083.7	−	7309.88i	−0.709285	−	0.258159i	−0.0379151	−	0.999281i	$-0.512072\pi$
$929$	−20083.7	−	7309.88i	−0.709285	−	0.258159i	−0.671370	+	0.741122i	$0.734294\pi$
$930$		0			0
$931$	40502.6	+	33985.7i	1.42580	+	1.19639i
$932$		0			0
$933$	−13984.1	−	691.233i	−0.490694	−	0.0242550i
$934$		0			0
$935$	−12175.1	+	21087.9i	−0.425848	+	0.737590i
$936$		0			0
$937$	704.534	+	1220.29i	0.0245636	+	0.0425454i	0.878046	−	0.478576i	$-0.158847\pi$
$937$	704.534	+	1220.29i	0.0245636	+	0.0425454i	−0.853482	+	0.521122i	$0.825514\pi$
$938$		0			0
$939$	23066.6	−	7127.16i	0.801652	−	0.247696i
$940$		0			0
$941$	3094.69	−	17550.9i	0.107209	−	0.608015i	−0.883105	−	0.469175i	$-0.844551\pi$
$941$	3094.69	−	17550.9i	0.107209	−	0.608015i	0.990315	−	0.138840i	$-0.0443374\pi$
$942$		0			0
$943$	46501.1	−	39019.0i	1.60582	−	1.34744i
$944$		0			0
$945$	−10735.4	+	13501.4i	−0.369549	+	0.464761i
$946$		0			0
$947$	−27667.7	+	23216.0i	−0.949399	+	0.796640i	−0.979196	−	0.202916i	$-0.934958\pi$
$947$	−27667.7	+	23216.0i	−0.949399	+	0.796640i	0.0297975	+	0.999556i	$0.490514\pi$
$948$		0			0
$949$	2075.25	−	11769.3i	0.0709856	−	0.402579i
$950$		0			0
$951$	−18875.1	−	17496.8i	−0.643605	−	0.596608i
$952$		0			0
$953$	22472.0	+	38922.7i	0.763841	+	1.32301i	0.940857	+	0.338804i	$0.110022\pi$
$953$	22472.0	+	38922.7i	0.763841	+	1.32301i	−0.177016	+	0.984208i	$0.556644\pi$
$954$		0			0
$955$	4182.97	−	7245.12i	0.141736	−	0.245494i
$956$		0			0
$957$	7500.15	−	11624.7i	0.253339	−	0.392656i
$958$		0			0
$959$	53504.8	+	44895.9i	1.80163	+	1.51175i
$960$		0			0
$961$	5940.00	+	2161.98i	0.199389	+	0.0725717i
$962$		0			0
$963$	−2558.69	+	9127.48i	−0.0856204	+	0.305430i
$964$		0			0
$965$	248.638	+	1410.10i	0.00829425	+	0.0470390i
$966$		0			0
$967$	−35745.8	+	13010.4i	−1.18873	+	0.432664i	−0.859279	−	0.511507i	$-0.829087\pi$
$967$	−35745.8	+	13010.4i	−1.18873	+	0.432664i	−0.329455	+	0.944171i	$0.606865\pi$
$968$		0			0
$969$	55696.4	−	42230.2i	1.84647	−	1.40003i
$970$		0			0
$971$	−885.777			−0.0292749			−0.0146374	−	0.999893i	$-0.504659\pi$
$971$	−885.777			−0.0292749			−0.0146374	+	0.999893i	$0.504659\pi$
$972$		0			0
$973$	75907.1			2.50100
$974$		0			0
$975$	24157.1	−	18316.4i	0.793482	−	0.601635i
$976$		0			0
$977$	48395.1	−	17614.4i	1.58475	−	0.576801i	0.608518	−	0.793540i	$-0.291764\pi$
$977$	48395.1	−	17614.4i	1.58475	−	0.576801i	0.976229	+	0.216740i	$0.0695422\pi$
$978$		0			0
$979$	9505.69	+	53909.5i	0.310320	+	1.75991i
$980$		0			0
$981$	4787.64	−	17078.7i	0.155818	−	0.555843i
$982$		0			0
$983$	−17316.6	−	6302.71i	−0.561864	−	0.204502i	0.0454457	−	0.998967i	$-0.485529\pi$
$983$	−17316.6	−	6302.71i	−0.561864	−	0.204502i	−0.607310	+	0.794465i	$0.707751\pi$
$984$		0			0
$985$	−5845.93	−	4905.31i	−0.189103	−	0.158676i
$986$		0			0
$987$	21328.4	−	33057.4i	0.687832	−	1.06609i
$988$		0			0
$989$	−21044.5	+	36450.1i	−0.676618	+	1.17194i
$990$		0			0
$991$	−10476.8	−	18146.4i	−0.335829	−	0.581673i	0.647815	−	0.761798i	$-0.275683\pi$
$991$	−10476.8	−	18146.4i	−0.335829	−	0.581673i	−0.983644	+	0.180125i	$0.942350\pi$
$992$		0			0
$993$	6744.25	+	6251.77i	0.215531	+	0.199792i
$994$		0			0
$995$	403.789	−	2290.00i	0.0128653	−	0.0729628i
$996$		0			0
$997$	1.20726	−	1.01301i	3.83495e−5	−	3.21790e-5i	−0.642768	−	0.766061i	$-0.722214\pi$
$997$	1.20726	−	1.01301i	3.83495e−5	−	3.21790e-5i	0.642807	+	0.766028i	$0.277770\pi$
$998$		0			0
$999$	3267.65	+	487.740i	0.103487	+	0.0154468i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.49.2	✓	54
3.2	odd	2		324.4.i.a.37.5		54
27.11	odd	18		324.4.i.a.289.5		54
27.16	even	9	inner	108.4.i.a.97.2	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.2	✓	54	1.1	even	1	trivial
108.4.i.a.97.2	yes	54	27.16	even	9	inner
324.4.i.a.37.5		54	3.2	odd	2
324.4.i.a.289.5		54	27.11	odd	18

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.14053 + 3.13943i) q^{3} +(-4.27250 + 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +(7.28791 - 25.9978i) q^{9} +O(q^{10})\)	\(q+(-4.14053 + 3.13943i) q^{3} +(-4.27250 + 1.55506i) q^{5} +(-4.69568 - 26.6305i) q^{7} +(7.28791 - 25.9978i) q^{9} +(50.9513 + 18.5448i) q^{11} +(42.8395 + 35.9466i) q^{13} +(12.8084 - 19.8520i) q^{15} +(49.3862 - 85.5394i) q^{17} +(-68.0936 - 117.942i) q^{19} +(103.047 + 95.5225i) q^{21} +(25.4482 - 144.324i) q^{23} +(-79.9195 + 67.0604i) q^{25} +(51.4427 + 130.525i) q^{27} +(-37.6146 + 31.5624i) q^{29} +(26.6026 - 150.871i) q^{31} +(-269.185 + 83.1733i) q^{33} +(61.4744 + 106.477i) q^{35} +(11.7745 - 20.3941i) q^{37} +(-290.230 - 14.3461i) q^{39} +(317.305 + 266.251i) q^{41} +(-269.878 - 98.2276i) q^{43} +(9.29066 + 122.409i) q^{45} +(-48.6186 - 275.730i) q^{47} +(-364.820 + 132.783i) q^{49} +(64.0605 + 509.223i) q^{51} +248.490 q^{53} -246.528 q^{55} +(652.213 + 274.565i) q^{57} +(243.543 - 88.6422i) q^{59} +(-57.5577 - 326.426i) q^{61} +(-726.556 - 72.0033i) q^{63} +(-238.931 - 86.9637i) q^{65} +(71.5526 + 60.0398i) q^{67} +(347.726 + 677.469i) q^{69} +(-333.619 + 577.845i) q^{71} +(-106.851 - 185.072i) q^{73} +(120.377 - 528.567i) q^{75} +(254.606 - 1443.94i) q^{77} +(-421.142 + 353.380i) q^{79} +(-622.773 - 378.939i) q^{81} +(310.120 - 260.222i) q^{83} +(-77.9834 + 442.266i) q^{85} +(56.6561 - 248.773i) q^{87} +(504.793 + 874.328i) q^{89} +(756.115 - 1309.63i) q^{91} +(363.501 + 708.203i) q^{93} +(474.337 + 398.016i) q^{95} +(-27.6645 - 10.0690i) q^{97} +(853.452 - 1189.47i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 54 * q + 12 * q^5 - 48 * q^9	\( 54 q + 12 q^{5} - 48 q^{9} - 87 q^{11} + 234 q^{15} + 204 q^{17} - 12 q^{21} + 96 q^{23} - 216 q^{25} + 27 q^{27} + 318 q^{29} - 54 q^{31} + 63 q^{33} + 6 q^{35} + 66 q^{39} + 867 q^{41} - 513 q^{43} - 306 q^{45} - 1548 q^{47} + 594 q^{49} - 1368 q^{51} - 1068 q^{53} - 1269 q^{57} - 1218 q^{59} - 54 q^{61} + 30 q^{63} + 96 q^{65} - 2997 q^{67} + 1476 q^{69} - 120 q^{71} - 216 q^{73} + 732 q^{75} + 3480 q^{77} + 2808 q^{79} + 3348 q^{81} + 4464 q^{83} + 2160 q^{85} + 4824 q^{87} + 4029 q^{89} + 270 q^{91} + 1164 q^{93} - 1650 q^{95} - 3483 q^{97} - 5076 q^{99}+O(q^{100}) \) 54 * q + 12 * q^5 - 48 * q^9 - 87 * q^11 + 234 * q^15 + 204 * q^17 - 12 * q^21 + 96 * q^23 - 216 * q^25 + 27 * q^27 + 318 * q^29 - 54 * q^31 + 63 * q^33 + 6 * q^35 + 66 * q^39 + 867 * q^41 - 513 * q^43 - 306 * q^45 - 1548 * q^47 + 594 * q^49 - 1368 * q^51 - 1068 * q^53 - 1269 * q^57 - 1218 * q^59 - 54 * q^61 + 30 * q^63 + 96 * q^65 - 2997 * q^67 + 1476 * q^69 - 120 * q^71 - 216 * q^73 + 732 * q^75 + 3480 * q^77 + 2808 * q^79 + 3348 * q^81 + 4464 * q^83 + 2160 * q^85 + 4824 * q^87 + 4029 * q^89 + 270 * q^91 + 1164 * q^93 - 1650 * q^95 - 3483 * q^97 - 5076 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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