LMFDB - Embedded newform 108.4.i.a.49.1

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			49.1
Character	$\chi$	$=$	108.49
Dual form			108.4.i.a.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.16735 + 0.546343i) q^{3} +(5.21695 - 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +(26.4030 - 5.64629i) q^{9} +O(q^{10})$	\(q+(-5.16735 + 0.546343i) q^{3} +(5.21695 - 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +(26.4030 - 5.64629i) q^{9} +(-67.4972 - 24.5670i) q^{11} +(-63.1985 - 53.0298i) q^{13} +(-25.9204 + 12.6621i) q^{15} +(21.0212 - 36.4097i) q^{17} +(-22.5218 - 39.0089i) q^{19} +(-13.1317 - 45.6891i) q^{21} +(12.1174 - 68.7214i) q^{23} +(-72.1445 + 60.5364i) q^{25} +(-133.349 + 43.6014i) q^{27} +(142.727 - 119.762i) q^{29} +(-46.3221 + 262.706i) q^{31} +(362.204 + 90.0696i) q^{33} +(25.3961 + 43.9873i) q^{35} +(-40.4209 + 70.0111i) q^{37} +(355.541 + 239.496i) q^{39} +(-216.792 - 181.910i) q^{41} +(-249.820 - 90.9271i) q^{43} +(127.022 - 79.5908i) q^{45} +(-10.5496 - 59.8299i) q^{47} +(243.661 - 88.6854i) q^{49} +(-88.7316 + 199.627i) q^{51} +206.241 q^{53} -398.778 q^{55} +(137.690 + 189.268i) q^{57} +(558.381 - 203.234i) q^{59} +(124.554 + 706.379i) q^{61} +(92.8181 + 228.917i) q^{63} +(-430.397 - 156.652i) q^{65} +(23.6459 + 19.8412i) q^{67} +(-25.0696 + 361.728i) q^{69} +(155.414 - 269.185i) q^{71} +(-263.889 - 457.070i) q^{73} +(339.722 - 352.229i) q^{75} +(114.113 - 647.169i) q^{77} +(-500.648 + 420.094i) q^{79} +(665.239 - 298.158i) q^{81} +(-883.108 + 741.015i) q^{83} +(40.5310 - 229.863i) q^{85} +(-672.091 + 696.832i) q^{87} +(240.988 + 417.404i) q^{89} +(377.389 - 653.656i) q^{91} +(95.8352 - 1382.80i) q^{93} +(-191.566 - 160.743i) q^{95} +(-858.230 - 312.370i) q^{97} +(-1920.84 - 267.534i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−5.16735	+	0.546343i	−0.994457	+	0.105144i
$3$	−5.16735	+	0.546343i	−0.994457	+	0.105144i
$4$		0			0
$5$	5.21695	−	1.89881i	0.466618	−	0.169835i	−0.0980017	−	0.995186i	$-0.531245\pi$
$5$	5.21695	−	1.89881i	0.466618	−	0.169835i	0.564620	+	0.825351i	$0.309023\pi$
$6$		0			0
$7$	1.58868	+	9.00985i	0.0857806	+	0.486486i	0.997185	+	0.0749750i	$0.0238877\pi$
$7$	1.58868	+	9.00985i	0.0857806	+	0.486486i	−0.911405	+	0.411511i	$0.865001\pi$
$8$		0			0
$9$	26.4030	−	5.64629i	0.977890	−	0.209122i
$10$		0			0
$11$	−67.4972	−	24.5670i	−1.85011	−	0.673384i	−0.985189	−	0.171472i	$-0.945148\pi$
$11$	−67.4972	−	24.5670i	−1.85011	−	0.673384i	−0.864918	−	0.501912i	$-0.832630\pi$
$12$		0			0
$13$	−63.1985	−	53.0298i	−1.34832	−	1.13137i	−0.979405	−	0.201907i	$-0.935286\pi$
$13$	−63.1985	−	53.0298i	−1.34832	−	1.13137i	−0.368911	−	0.929465i	$-0.620269\pi$
$14$		0			0
$15$	−25.9204	+	12.6621i	−0.446175	+	0.217956i
$16$		0			0
$17$	21.0212	−	36.4097i	0.299905	−	0.519450i	−0.676209	−	0.736710i	$-0.736378\pi$
$17$	21.0212	−	36.4097i	0.299905	−	0.519450i	0.976114	+	0.217259i	$0.0697117\pi$
$18$		0			0
$19$	−22.5218	−	39.0089i	−0.271940	−	0.471013i	0.697419	−	0.716664i	$-0.254332\pi$
$19$	−22.5218	−	39.0089i	−0.271940	−	0.471013i	−0.969358	+	0.245650i	$0.920998\pi$
$20$		0			0
$21$	−13.1317	−	45.6891i	−0.136456	−	0.474770i
$22$		0			0
$23$	12.1174	−	68.7214i	0.109855	−	0.623018i	−0.879315	−	0.476241i	$-0.841999\pi$
$23$	12.1174	−	68.7214i	0.109855	−	0.623018i	0.989170	−	0.146777i	$-0.0468899\pi$
$24$		0			0
$25$	−72.1445	+	60.5364i	−0.577156	+	0.484291i
$26$		0			0
$27$	−133.349	+	43.6014i	−0.950481	+	0.310782i
$28$		0			0
$29$	142.727	−	119.762i	0.913924	−	0.766873i	−0.0589376	−	0.998262i	$-0.518771\pi$
$29$	142.727	−	119.762i	0.913924	−	0.766873i	0.972861	+	0.231388i	$0.0743268\pi$
$30$		0			0
$31$	−46.3221	+	262.706i	−0.268377	+	1.52204i	0.490866	+	0.871235i	$0.336680\pi$
$31$	−46.3221	+	262.706i	−0.268377	+	1.52204i	−0.759243	+	0.650807i	$0.774431\pi$
$32$		0			0
$33$	362.204	+	90.0696i	1.91065	+	0.475124i
$34$		0			0
$35$	25.3961	+	43.9873i	0.122649	+	0.212435i
$36$		0			0
$37$	−40.4209	+	70.0111i	−0.179599	+	0.311074i	−0.941743	−	0.336333i	$-0.890813\pi$
$37$	−40.4209	+	70.0111i	−0.179599	+	0.311074i	0.762144	+	0.647407i	$0.224147\pi$
$38$		0			0
$39$	355.541	+	239.496i	1.45980	+	0.983334i
$40$		0			0
$41$	−216.792	−	181.910i	−0.825787	−	0.692918i	0.128533	−	0.991705i	$-0.458973\pi$
$41$	−216.792	−	181.910i	−0.825787	−	0.692918i	−0.954320	+	0.298788i	$0.903418\pi$
$42$		0			0
$43$	−249.820	−	90.9271i	−0.885982	−	0.322471i	−0.141361	−	0.989958i	$-0.545148\pi$
$43$	−249.820	−	90.9271i	−0.885982	−	0.322471i	−0.744621	+	0.667487i	$0.767370\pi$
$44$		0			0
$45$	127.022	−	79.5908i	0.420785	−	0.263660i
$46$		0			0
$47$	−10.5496	−	59.8299i	−0.0327409	−	0.185683i	0.964051	−	0.265716i	$-0.0856084\pi$
$47$	−10.5496	−	59.8299i	−0.0327409	−	0.185683i	−0.996792	+	0.0800335i	$0.974497\pi$
$48$		0			0
$49$	243.661	−	88.6854i	0.710382	−	0.258558i
$50$		0			0
$51$	−88.7316	+	199.627i	−0.243626	+	0.548104i
$52$		0			0
$53$	206.241			0.534517			0.267258	−	0.963625i	$-0.413882\pi$
$53$	206.241			0.534517			0.267258	+	0.963625i	$0.413882\pi$
$54$		0			0
$55$	−398.778			−0.977658
$56$		0			0
$57$	137.690	+	189.268i	0.319956	+	0.439810i
$58$		0			0
$59$	558.381	−	203.234i	1.23212	−	0.448455i	0.357797	−	0.933800i	$-0.383528\pi$
$59$	558.381	−	203.234i	1.23212	−	0.448455i	0.874323	+	0.485345i	$0.161306\pi$
$60$		0			0
$61$	124.554	+	706.379i	0.261434	+	1.48266i	0.779002	+	0.627022i	$0.215726\pi$
$61$	124.554	+	706.379i	0.261434	+	1.48266i	−0.517568	+	0.855642i	$0.673163\pi$
$62$		0			0
$63$	92.8181	+	228.917i	0.185619	+	0.457791i
$64$		0			0
$65$	−430.397	−	156.652i	−0.821295	−	0.298927i
$66$		0			0
$67$	23.6459	+	19.8412i	0.0431164	+	0.0361790i	0.664091	−	0.747652i	$-0.268819\pi$
$67$	23.6459	+	19.8412i	0.0431164	+	0.0361790i	−0.620975	+	0.783831i	$0.713263\pi$
$68$		0			0
$69$	−25.0696	+	361.728i	−0.0437395	+	0.631115i
$70$		0			0
$71$	155.414	−	269.185i	0.259779	−	0.449950i	−0.706404	−	0.707809i	$-0.749684\pi$
$71$	155.414	−	269.185i	0.259779	−	0.449950i	0.966182	+	0.257859i	$0.0830171\pi$
$72$		0			0
$73$	−263.889	−	457.070i	−0.423094	−	0.732821i	0.573146	−	0.819453i	$-0.305723\pi$
$73$	−263.889	−	457.070i	−0.423094	−	0.732821i	−0.996240	+	0.0866322i	$0.972389\pi$
$74$		0			0
$75$	339.722	−	352.229i	0.523037	−	0.542291i
$76$		0			0
$77$	114.113	−	647.169i	0.168889	−	0.957815i
$78$		0			0
$79$	−500.648	+	420.094i	−0.713004	+	0.598281i	−0.925440	−	0.378893i	$-0.876305\pi$
$79$	−500.648	+	420.094i	−0.713004	+	0.598281i	0.212436	+	0.977175i	$0.431860\pi$
$80$		0			0
$81$	665.239	−	298.158i	0.912536	−	0.408996i
$82$		0			0
$83$	−883.108	+	741.015i	−1.16788	+	0.979964i	−0.999983	−	0.00584137i	$-0.998141\pi$
$83$	−883.108	+	741.015i	−1.16788	+	0.979964i	−0.167893	+	0.985805i	$0.553696\pi$
$84$		0			0
$85$	40.5310	−	229.863i	0.0517201	−	0.293319i
$86$		0			0
$87$	−672.091	+	696.832i	−0.828226	+	0.858716i
$88$		0			0
$89$	240.988	+	417.404i	0.287019	+	0.497132i	0.973097	−	0.230397i	$-0.0740023\pi$
$89$	240.988	+	417.404i	0.287019	+	0.497132i	−0.686078	+	0.727528i	$0.740669\pi$
$90$		0			0
$91$	377.389	−	653.656i	0.434737	−	0.752987i
$92$		0			0
$93$	95.8352	−	1382.80i	0.106856	−	1.54182i
$94$		0			0
$95$	−191.566	−	160.743i	−0.206887	−	0.173598i
$96$		0			0
$97$	−858.230	−	312.370i	−0.898351	−	0.326973i	−0.148759	−	0.988873i	$-0.547528\pi$
$97$	−858.230	−	312.370i	−0.898351	−	0.326973i	−0.749592	+	0.661901i	$0.769750\pi$
$98$		0			0
$99$	−1920.84	−	267.534i	−1.95002	−	0.271598i
$100$		0			0
$101$	−92.4446	−	524.279i	−0.0910750	−	0.516512i	−0.995880	−	0.0906852i	$-0.971094\pi$
$101$	−92.4446	−	524.279i	−0.0910750	−	0.516512i	0.904805	−	0.425827i	$-0.140017\pi$
$102$		0			0
$103$	303.355	−	110.412i	0.290199	−	0.105624i	−0.192819	−	0.981234i	$-0.561763\pi$
$103$	303.355	−	110.412i	0.290199	−	0.105624i	0.483017	+	0.875611i	$0.339541\pi$
$104$		0			0
$105$	−155.263	−	213.423i	−0.144305	−	0.198361i
$106$		0			0
$107$	−209.423			−0.189212			−0.0946062	−	0.995515i	$-0.530159\pi$
$107$	−209.423			−0.189212			−0.0946062	+	0.995515i	$0.530159\pi$
$108$		0			0
$109$	−196.115			−0.172334			−0.0861672	−	0.996281i	$-0.527462\pi$
$109$	−196.115			−0.172334			−0.0861672	+	0.996281i	$0.527462\pi$
$110$		0			0
$111$	170.619	−	383.855i	0.145896	−	0.328234i
$112$		0			0
$113$	203.446	−	74.0482i	0.169368	−	0.0616448i	−0.255945	−	0.966691i	$-0.582386\pi$
$113$	203.446	−	74.0482i	0.169368	−	0.0616448i	0.425312	+	0.905047i	$0.360164\pi$
$114$		0			0
$115$	−67.2731	−	381.525i	−0.0545500	−	0.309368i
$116$		0			0
$117$	−1968.05	−	1043.31i	−1.55510	−	0.824394i
$118$		0			0
$119$	361.442	+	131.554i	0.278431	+	0.101341i
$120$		0			0
$121$	2932.73	+	2460.86i	2.20341	+	1.84888i
$122$		0			0
$123$	1219.63	+	821.552i	0.894066	+	0.602251i
$124$		0			0
$125$	−608.412	+	1053.80i	−0.435344	+	0.754038i
$126$		0			0
$127$	−611.723	−	1059.53i	−0.427414	−	0.740303i	0.569228	−	0.822180i	$-0.307242\pi$
$127$	−611.723	−	1059.53i	−0.427414	−	0.740303i	−0.996642	+	0.0818763i	$0.973909\pi$
$128$		0			0
$129$	1340.59	+	333.365i	0.914977	+	0.227528i
$130$		0			0
$131$	194.741	−	1104.43i	0.129882	−	0.736599i	−0.848405	−	0.529347i	$-0.822437\pi$
$131$	194.741	−	1104.43i	0.129882	−	0.736599i	0.978288	−	0.207252i	$-0.0664519\pi$
$132$		0			0
$133$	315.684	−	264.891i	0.205814	−	0.172699i
$134$		0			0
$135$	−612.883	+	480.671i	−0.390730	+	0.306441i
$136$		0			0
$137$	−1967.06	+	1650.56i	−1.22670	+	1.02932i	−0.228252	+	0.973602i	$0.573301\pi$
$137$	−1967.06	+	1650.56i	−1.22670	+	1.02932i	−0.998446	+	0.0557200i	$0.982255\pi$
$138$		0			0
$139$	434.934	−	2466.63i	0.265400	−	1.50516i	−0.502494	−	0.864581i	$-0.667584\pi$
$139$	434.934	−	2466.63i	0.265400	−	1.50516i	0.767894	−	0.640577i	$-0.221305\pi$
$140$		0			0
$141$	87.2012	+	303.398i	0.0520827	+	0.181211i
$142$		0			0
$143$	2962.94	+	5131.96i	1.73268	+	3.00109i
$144$		0			0
$145$	517.194	−	895.807i	0.296211	−	0.513053i
$146$		0			0
$147$	−1210.63	+	591.391i	−0.679259	+	0.331817i
$148$		0			0
$149$	88.5089	+	74.2678i	0.0486640	+	0.0408339i	0.666795	−	0.745241i	$-0.267666\pi$
$149$	88.5089	+	74.2678i	0.0486640	+	0.0408339i	−0.618131	+	0.786075i	$0.712110\pi$
$150$		0			0
$151$	1464.39	+	532.995i	0.789209	+	0.287249i	0.705007	−	0.709200i	$-0.250944\pi$
$151$	1464.39	+	532.995i	0.789209	+	0.287249i	0.0842019	+	0.996449i	$0.473166\pi$
$152$		0			0
$153$	349.443	−	1080.02i	0.184645	−	0.570682i
$154$		0			0
$155$	257.169	+	1458.48i	0.133267	+	0.755792i
$156$		0			0
$157$	1712.19	−	623.186i	0.870367	−	0.316788i	0.132052	−	0.991243i	$-0.457844\pi$
$157$	1712.19	−	623.186i	0.870367	−	0.316788i	0.738316	+	0.674455i	$0.235621\pi$
$158$		0			0
$159$	−1065.72	+	112.678i	−0.531554	+	0.0562011i
$160$		0			0
$161$	638.420			0.312513
$162$		0			0
$163$	1785.32			0.857897			0.428948	−	0.903329i	$-0.358884\pi$
$163$	1785.32			0.857897			0.428948	+	0.903329i	$0.358884\pi$
$164$		0			0
$165$	2060.62	−	217.869i	0.972239	−	0.102795i
$166$		0			0
$167$	1097.06	−	399.298i	0.508343	−	0.185022i	−0.0751000	−	0.997176i	$-0.523928\pi$
$167$	1097.06	−	399.298i	0.508343	−	0.185022i	0.583442	+	0.812154i	$0.301705\pi$
$168$		0			0
$169$	800.382	+	4539.19i	0.364307	+	2.06609i
$170$		0			0
$171$	−814.899	−	902.788i	−0.364426	−	0.403731i
$172$		0			0
$173$	−3928.51	−	1429.86i	−1.72647	−	0.628383i	−0.728100	−	0.685471i	$-0.759596\pi$
$173$	−3928.51	−	1429.86i	−1.72647	−	0.628383i	−0.998369	+	0.0570873i	$0.981819\pi$
$174$		0			0
$175$	−660.038	−	553.838i	−0.285110	−	0.239235i
$176$		0			0
$177$	−2774.32	+	1355.25i	−1.17814	+	0.575519i
$178$		0			0
$179$	918.530	−	1590.94i	0.383543	−	0.664316i	−0.608023	−	0.793919i	$-0.708037\pi$
$179$	918.530	−	1590.94i	0.383543	−	0.664316i	0.991566	+	0.129604i	$0.0413705\pi$
$180$		0			0
$181$	−1157.45	−	2004.76i	−0.475318	−	0.823275i	0.524282	−	0.851544i	$-0.324334\pi$
$181$	−1157.45	−	2004.76i	−0.475318	−	0.823275i	−0.999600	+	0.0282697i	$0.991000\pi$
$182$		0			0
$183$	−1029.54	−	3582.06i	−0.415877	−	1.44696i
$184$		0			0
$185$	−77.9358	+	441.996i	−0.0309727	+	0.175655i
$186$		0			0
$187$	−2313.35	+	1941.13i	−0.904646	+	0.759088i
$188$		0			0
$189$	−604.691	−	1132.18i	−0.232724	−	0.435737i
$190$		0			0
$191$	1309.35	−	1098.67i	0.496027	−	0.416216i	−0.360153	−	0.932893i	$-0.617276\pi$
$191$	1309.35	−	1098.67i	0.496027	−	0.416216i	0.856180	+	0.516677i	$0.172831\pi$
$192$		0			0
$193$	−538.413	+	3053.49i	−0.200808	+	1.13884i	0.703095	+	0.711096i	$0.251801\pi$
$193$	−538.413	+	3053.49i	−0.200808	+	1.13884i	−0.903902	+	0.427740i	$0.859310\pi$
$194$		0			0
$195$	2309.60	+	574.330i	0.848173	+	0.210916i
$196$		0			0
$197$	−269.298	−	466.438i	−0.0973943	−	0.168692i	0.813211	−	0.581969i	$-0.197717\pi$
$197$	−269.298	−	466.438i	−0.0973943	−	0.168692i	−0.910605	+	0.413277i	$0.864384\pi$
$198$		0			0
$199$	1847.26	−	3199.55i	0.658035	−	1.13975i	−0.323089	−	0.946369i	$-0.604721\pi$
$199$	1847.26	−	3199.55i	0.658035	−	1.13975i	0.981124	−	0.193381i	$-0.0619454\pi$
$200$		0			0
$201$	−133.027	−	89.6078i	−0.0466814	−	0.0314450i
$202$		0			0
$203$	1305.79	+	1095.69i	0.451470	+	0.378828i
$204$		0			0
$205$	−1476.41	−	537.369i	−0.503009	−	0.183080i
$206$		0			0
$207$	−68.0839	−	1882.87i	−0.0228607	−	0.632216i
$208$		0			0
$209$	561.828	+	3186.28i	0.185945	+	1.05455i
$210$		0			0
$211$	3344.47	−	1217.29i	1.09120	−	0.397164i	0.267134	−	0.963660i	$-0.413924\pi$
$211$	3344.47	−	1217.29i	1.09120	−	0.397164i	0.824065	+	0.566496i	$0.191701\pi$
$212$		0			0
$213$	−656.012	+	1475.88i	−0.211029	+	0.474770i
$214$		0			0
$215$	−1475.95			−0.468182
$216$		0			0
$217$	−2440.53			−0.763474
$218$		0			0
$219$	1613.32	+	2217.66i	0.497801	+	0.684273i
$220$		0			0
$221$	−3259.31	+	1186.29i	−0.992058	+	0.361080i
$222$		0			0
$223$	−932.642	−	5289.28i	−0.280064	−	1.58832i	−0.722401	−	0.691474i	$-0.756962\pi$
$223$	−932.642	−	5289.28i	−0.280064	−	1.58832i	0.442337	−	0.896849i	$-0.354150\pi$
$224$		0			0
$225$	−1563.03	+	2005.69i	−0.463119	+	0.594279i
$226$		0			0
$227$	−1788.67	−	651.023i	−0.522988	−	0.190352i	0.0670166	−	0.997752i	$-0.478652\pi$
$227$	−1788.67	−	651.023i	−0.522988	−	0.190352i	−0.590005	+	0.807400i	$0.700874\pi$
$228$		0			0
$229$	−1969.18	−	1652.34i	−0.568240	−	0.476810i	0.312821	−	0.949812i	$-0.398726\pi$
$229$	−1969.18	−	1652.34i	−0.568240	−	0.476810i	−0.881061	+	0.473002i	$0.843170\pi$
$230$		0			0
$231$	−236.088	+	3406.49i	−0.0672443	+	0.970263i
$232$		0			0
$233$	1769.64	−	3065.11i	0.497567	−	0.861811i	−0.502429	−	0.864618i	$-0.667560\pi$
$233$	1769.64	−	3065.11i	0.497567	−	0.861811i	0.999996	+	0.00280750i	$0.000893656\pi$
$234$		0			0
$235$	−168.643	−	292.098i	−0.0468129	−	0.0810823i
$236$		0			0
$237$	2357.51	−	2444.30i	0.646146	−	0.669933i
$238$		0			0
$239$	1060.55	−	6014.68i	0.287035	−	1.62786i	−0.410890	−	0.911685i	$-0.634782\pi$
$239$	1060.55	−	6014.68i	0.287035	−	1.62786i	0.697925	−	0.716171i	$-0.254107\pi$
$240$		0			0
$241$	−2459.06	+	2063.40i	−0.657270	+	0.551515i	−0.909267	−	0.416213i	$-0.863357\pi$
$241$	−2459.06	+	2063.40i	−0.657270	+	0.551515i	0.251997	+	0.967728i	$0.418913\pi$
$242$		0			0
$243$	−3274.63	+	1904.14i	−0.864475	+	0.502676i
$244$		0			0
$245$	1102.77	−	925.334i	0.287565	−	0.241296i
$246$		0			0
$247$	−645.292	+	3659.63i	−0.166230	+	0.942740i
$248$		0			0
$249$	4158.48	−	4311.57i	1.05837	−	1.09733i
$250$		0			0
$251$	1804.49	+	3125.47i	0.453778	+	0.785967i	0.998617	−	0.0525736i	$-0.0167424\pi$
$251$	1804.49	+	3125.47i	0.453778	+	0.785967i	−0.544839	+	0.838541i	$0.683409\pi$
$252$		0			0
$253$	−2506.17	+	4340.82i	−0.622773	+	1.07868i
$254$		0			0
$255$	−83.8542	+	1209.93i	−0.0205928	+	0.297131i
$256$		0			0
$257$	−4944.99	−	4149.34i	−1.20023	−	1.00712i	−0.999623	−	0.0274439i	$-0.991263\pi$
$257$	−4944.99	−	4149.34i	−1.20023	−	1.00712i	−0.200610	−	0.979671i	$-0.564292\pi$
$258$		0			0
$259$	−695.005	−	252.961i	−0.166739	−	0.0606882i
$260$		0			0
$261$	3092.22	−	3967.97i	0.733347	−	0.941039i
$262$		0			0
$263$	−1324.95	−	7514.17i	−0.310646	−	1.76176i	−0.595657	−	0.803239i	$-0.703108\pi$
$263$	−1324.95	−	7514.17i	−0.310646	−	1.76176i	0.285011	−	0.958524i	$-0.408003\pi$
$264$		0			0
$265$	1075.95	−	391.613i	0.249415	−	0.0907797i
$266$		0			0
$267$	−1473.32	−	2025.21i	−0.337698	−	0.464198i
$268$		0			0
$269$	−7418.04			−1.68136			−0.840681	−	0.541531i	$-0.817845\pi$
$269$	−7418.04			−1.68136			−0.840681	+	0.541531i	$0.817845\pi$
$270$		0			0
$271$	−5946.94			−1.33303			−0.666515	−	0.745492i	$-0.732215\pi$
$271$	−5946.94			−1.33303			−0.666515	+	0.745492i	$0.732215\pi$
$272$		0			0
$273$	−1592.98	+	3583.85i	−0.353156	+	0.794523i
$274$		0			0
$275$	6356.75	−	2313.67i	1.39391	−	0.507343i
$276$		0			0
$277$	105.510	+	598.379i	0.0228863	+	0.129795i	0.994110	−	0.108373i	$-0.0345642\pi$
$277$	105.510	+	598.379i	0.0228863	+	0.129795i	−0.971224	+	0.238168i	$0.923453\pi$
$278$		0			0
$279$	260.268	+	7197.77i	0.0558489	+	1.54451i
$280$		0			0
$281$	6387.32	+	2324.79i	1.35600	+	0.493543i	0.914815	−	0.403874i	$-0.132336\pi$
$281$	6387.32	+	2324.79i	1.35600	+	0.493543i	0.441184	+	0.897417i	$0.354559\pi$
$282$		0			0
$283$	−56.9781	−	47.8103i	−0.0119682	−	0.0100425i	0.636784	−	0.771042i	$-0.280264\pi$
$283$	−56.9781	−	47.8103i	−0.0119682	−	0.0100425i	−0.648752	+	0.761000i	$0.724709\pi$
$284$		0			0
$285$	1077.71	+	725.953i	0.223993	+	0.150883i
$286$		0			0
$287$	1294.57	−	2242.26i	0.266258	−	0.461173i
$288$		0			0
$289$	1572.72	+	2724.03i	0.320114	+	0.554454i
$290$		0			0
$291$	4605.43	+	1145.24i	0.927750	+	0.230705i
$292$		0			0
$293$	−17.3041	+	98.1364i	−0.00345023	+	0.0195672i	−0.986484	−	0.163856i	$-0.947607\pi$
$293$	−17.3041	+	98.1364i	−0.00345023	+	0.0195672i	0.983034	+	0.183423i	$0.0587179\pi$
$294$		0			0
$295$	2527.14	−	2120.52i	0.498766	−	0.418514i
$296$		0			0
$297$	10071.8	+	333.002i	1.96777	+	0.0650598i
$298$		0			0
$299$	−4410.09	+	3700.50i	−0.852983	+	0.715738i
$300$		0			0
$301$	422.355	−	2395.30i	0.0808776	−	0.458680i
$302$		0			0
$303$	764.130	+	2658.63i	0.144878	+	0.504073i
$304$		0			0
$305$	1991.07	+	3448.64i	0.373798	+	0.647437i
$306$		0			0
$307$	−2496.99	+	4324.91i	−0.464204	+	0.804024i	−0.999165	−	0.0408521i	$-0.986993\pi$
$307$	−2496.99	+	4324.91i	−0.464204	+	0.804024i	0.534962	+	0.844876i	$0.320326\pi$
$308$		0			0
$309$	−1507.22	+	736.274i	−0.277484	+	0.135551i
$310$		0			0
$311$	5733.38	+	4810.88i	1.04537	+	0.877170i	0.992599	−	0.121437i	$-0.0387502\pi$
$311$	5733.38	+	4810.88i	1.04537	+	0.877170i	0.0527712	+	0.998607i	$0.483195\pi$
$312$		0			0
$313$	4171.87	+	1518.44i	0.753380	+	0.274208i	0.690027	−	0.723783i	$-0.257599\pi$
$313$	4171.87	+	1518.44i	0.753380	+	0.274208i	0.0633524	+	0.997991i	$0.479821\pi$
$314$		0			0
$315$	918.898	+	1018.00i	0.164362	+	0.182089i
$316$		0			0
$317$	−299.891	−	1700.77i	−0.0531342	−	0.301339i	0.946647	−	0.322273i	$-0.104447\pi$
$317$	−299.891	−	1700.77i	−0.0531342	−	0.301339i	−0.999781	+	0.0209343i	$0.993336\pi$
$318$		0			0
$319$	−12575.9	+	4577.25i	−2.20726	+	0.803376i
$320$		0			0
$321$	1082.16	−	114.417i	0.188164	−	0.0198945i
$322$		0			0
$323$	−1893.74			−0.326224
$324$		0			0
$325$	7769.66			1.32610
$326$		0			0
$327$	1013.40	−	107.146i	0.171379	−	0.0181199i
$328$		0			0
$329$	522.298	−	190.101i	0.0875235	−	0.0318559i
$330$		0			0
$331$	−146.572	−	831.251i	−0.0243393	−	0.138035i	0.970217	−	0.242239i	$-0.0778818\pi$
$331$	−146.572	−	831.251i	−0.0243393	−	0.138035i	−0.994556	+	0.104204i	$0.966771\pi$
$332$		0			0
$333$	−671.932	+	2076.73i	−0.110575	+	0.341754i
$334$		0			0
$335$	161.034	+	58.6116i	0.0262634	+	0.00955908i
$336$		0			0
$337$	25.8927	+	21.7265i	0.00418536	+	0.00351193i	0.644878	−	0.764286i	$-0.276908\pi$
$337$	25.8927	+	21.7265i	0.00418536	+	0.00351193i	−0.640693	+	0.767798i	$0.721353\pi$
$338$		0			0
$339$	−1010.82	+	493.784i	−0.161947	+	0.0791111i
$340$		0			0
$341$	9580.49	−	16593.9i	1.52145	−	2.63522i
$342$		0			0
$343$	2755.17	+	4772.09i	0.433717	+	0.751220i
$344$		0			0
$345$	556.067	+	1934.72i	0.0867758	+	0.301918i
$346$		0			0
$347$	−1785.27	+	10124.8i	−0.276191	+	1.56636i	0.458963	+	0.888455i	$0.348221\pi$
$347$	−1785.27	+	10124.8i	−0.276191	+	1.56636i	−0.735154	+	0.677900i	$0.762890\pi$
$348$		0			0
$349$	−5099.40	+	4278.91i	−0.782134	+	0.656289i	−0.943785	−	0.330559i	$-0.892763\pi$
$349$	−5099.40	+	4278.91i	−0.782134	+	0.656289i	0.161651	+	0.986848i	$0.448318\pi$
$350$		0			0
$351$	10739.6	+	4315.92i	1.63316	+	0.656316i
$352$		0			0
$353$	6159.61	−	5168.53i	0.928734	−	0.779300i	−0.0468557	−	0.998902i	$-0.514920\pi$
$353$	6159.61	−	5168.53i	0.928734	−	0.779300i	0.975590	+	0.219601i	$0.0704756\pi$
$354$		0			0
$355$	299.655	−	1699.43i	0.0448001	−	0.254074i
$356$		0			0
$357$	−1939.57	−	482.315i	−0.287543	−	0.0715037i
$358$		0			0
$359$	−1712.40	−	2965.97i	−0.251747	−	0.436039i	0.712260	−	0.701916i	$-0.247672\pi$
$359$	−1712.40	−	2965.97i	−0.251747	−	0.436039i	−0.964007	+	0.265877i	$0.914338\pi$
$360$		0			0
$361$	2415.04	−	4182.97i	0.352098	−	0.609851i
$362$		0			0
$363$	−16498.9	−	11113.8i	−2.38559	−	1.60696i
$364$		0			0
$365$	−2244.59	−	1883.43i	−0.321882	−	0.270091i
$366$		0			0
$367$	−3402.99	−	1238.59i	−0.484018	−	0.176168i	0.0884742	−	0.996078i	$-0.471801\pi$
$367$	−3402.99	−	1238.59i	−0.484018	−	0.176168i	−0.572492	+	0.819910i	$0.694023\pi$
$368$		0			0
$369$	−6751.09	−	3578.91i	−0.952433	−	0.504907i
$370$		0			0
$371$	327.651	+	1858.20i	0.0458512	+	0.260035i
$372$		0			0
$373$	−1812.31	+	659.627i	−0.251576	+	0.0915662i	−0.464730	−	0.885452i	$-0.653849\pi$
$373$	−1812.31	+	659.627i	−0.251576	+	0.0915662i	0.213154	+	0.977019i	$0.431626\pi$
$374$		0			0
$375$	2568.14	−	5777.76i	0.353649	−	0.795632i
$376$		0			0
$377$	−15371.1			−2.09988
$378$		0			0
$379$	4403.13			0.596763			0.298382	−	0.954447i	$-0.403553\pi$
$379$	4403.13			0.596763			0.298382	+	0.954447i	$0.403553\pi$
$380$		0			0
$381$	3739.85	+	5140.78i	0.502883	+	0.691260i
$382$		0			0
$383$	−10231.3	+	3723.88i	−1.36500	+	0.496819i	−0.917596	−	0.397515i	$-0.869873\pi$
$383$	−10231.3	+	3723.88i	−1.36500	+	0.496819i	−0.447402	+	0.894333i	$0.647651\pi$
$384$		0			0
$385$	−633.530	−	3592.93i	−0.0838641	−	0.475617i
$386$		0			0
$387$	−7109.41	−	990.194i	−0.933828	−	0.130063i
$388$		0			0
$389$	7053.47	+	2567.25i	0.919345	+	0.334614i	0.757978	−	0.652280i	$-0.226187\pi$
$389$	7053.47	+	2567.25i	0.919345	+	0.334614i	0.161367	+	0.986894i	$0.448410\pi$
$390$		0			0
$391$	−2247.41	−	1885.80i	−0.290681	−	0.243910i
$392$		0			0
$393$	−402.897	+	5813.37i	−0.0517136	+	0.746172i
$394$		0			0
$395$	−1814.18	+	3142.25i	−0.231091	+	0.400262i
$396$		0			0
$397$	−3635.79	−	6297.37i	−0.459635	−	0.796111i	0.539307	−	0.842110i	$-0.318686\pi$
$397$	−3635.79	−	6297.37i	−0.459635	−	0.796111i	−0.998942	+	0.0459984i	$0.985353\pi$
$398$		0			0
$399$	−1486.53	+	1541.25i	−0.186515	+	0.193381i
$400$		0			0
$401$	−2014.16	+	11422.9i	−0.250828	+	1.42252i	0.555729	+	0.831363i	$0.312439\pi$
$401$	−2014.16	+	11422.9i	−0.250828	+	1.42252i	−0.806558	+	0.591155i	$0.798672\pi$
$402$		0			0
$403$	16858.7	−	14146.1i	2.08385	−	1.74856i
$404$		0			0
$405$	2904.37	−	2818.64i	0.356344	−	0.345826i
$406$		0			0
$407$	4448.26	−	3732.53i	0.541750	−	0.454582i
$408$		0			0
$409$	1420.37	−	8055.31i	0.171718	−	0.973862i	−0.770146	−	0.637868i	$-0.779817\pi$
$409$	1420.37	−	8055.31i	0.171718	−	0.973862i	0.941864	−	0.335994i	$-0.109072\pi$
$410$		0			0
$411$	9262.74	−	9603.73i	1.11167	−	1.15260i
$412$		0			0
$413$	2718.20	+	4708.06i	0.323859	+	0.560940i
$414$		0			0
$415$	−3200.08	+	5542.70i	−0.378520	+	0.655615i
$416$		0			0
$417$	−899.829	+	12983.6i	−0.105671	+	1.52472i
$418$		0			0
$419$	−6454.20	−	5415.71i	−0.752525	−	0.631444i	0.183644	−	0.982993i	$-0.441211\pi$
$419$	−6454.20	−	5415.71i	−0.752525	−	0.631444i	−0.936169	+	0.351549i	$0.885655\pi$
$420$		0			0
$421$	4401.68	+	1602.08i	0.509560	+	0.185465i	0.583989	−	0.811762i	$-0.301491\pi$
$421$	4401.68	+	1602.08i	0.509560	+	0.185465i	−0.0744290	+	0.997226i	$0.523713\pi$
$422$		0			0
$423$	−616.358	−	1520.12i	−0.0708472	−	0.174730i
$424$		0			0
$425$	687.553	+	3899.31i	0.0784735	+	0.445045i
$426$		0			0
$427$	−6166.49	+	2244.42i	−0.698869	+	0.254368i
$428$		0			0
$429$	−18114.4	−	24899.9i	−2.03862	−	2.80228i
$430$		0			0
$431$	−8784.41			−0.981740			−0.490870	−	0.871233i	$-0.663321\pi$
$431$	−8784.41			−0.981740			−0.490870	+	0.871233i	$0.663321\pi$
$432$		0			0
$433$	−6469.85			−0.718063			−0.359031	−	0.933325i	$-0.616893\pi$
$433$	−6469.85			−0.718063			−0.359031	+	0.933325i	$0.616893\pi$
$434$		0			0
$435$	−2183.11	+	4911.51i	−0.240625	+	0.541354i
$436$		0			0
$437$	−2953.65	+	1075.04i	−0.323323	+	0.117680i
$438$		0			0
$439$	−148.990	−	844.963i	−0.0161979	−	0.0918631i	0.975637	−	0.219391i	$-0.0704070\pi$
$439$	−148.990	−	844.963i	−0.0161979	−	0.0918631i	−0.991835	+	0.127528i	$0.959296\pi$
$440$		0			0
$441$	5932.65	−	3717.34i	0.640605	−	0.401398i
$442$		0			0
$443$	412.777	+	150.239i	0.0442701	+	0.0161130i	0.364060	−	0.931375i	$-0.381390\pi$
$443$	412.777	+	150.239i	0.0442701	+	0.0161130i	−0.319790	+	0.947488i	$0.603612\pi$
$444$		0			0
$445$	2049.80	+	1719.98i	0.218359	+	0.183225i
$446$		0			0
$447$	−497.932	−	335.412i	−0.0526877	−	0.0354909i
$448$		0			0
$449$	1510.09	−	2615.54i	0.158720	−	0.274911i	−0.775687	−	0.631117i	$-0.782597\pi$
$449$	1510.09	−	2615.54i	0.158720	−	0.274911i	0.934407	+	0.356206i	$0.115930\pi$
$450$		0			0
$451$	10163.9	+	17604.4i	1.06120	+	1.83804i
$452$		0			0
$453$	−7858.23	−	1954.11i	−0.815037	−	0.202676i
$454$		0			0
$455$	727.645	−	4126.68i	0.0749726	−	0.425191i
$456$		0			0
$457$	5728.39	−	4806.69i	0.586352	−	0.492008i	−0.300674	−	0.953727i	$-0.597212\pi$
$457$	5728.39	−	4806.69i	0.586352	−	0.492008i	0.887026	+	0.461719i	$0.152767\pi$
$458$		0			0
$459$	−1215.63	+	5771.75i	−0.123618	+	0.586933i
$460$		0			0
$461$	4559.80	−	3826.13i	0.460675	−	0.386552i	−0.382705	−	0.923871i	$-0.625007\pi$
$461$	4559.80	−	3826.13i	0.460675	−	0.386552i	0.843379	+	0.537319i	$0.180563\pi$
$462$		0			0
$463$	299.450	−	1698.27i	0.0300575	−	0.170465i	−0.966084	−	0.258229i	$-0.916861\pi$
$463$	299.450	−	1698.27i	0.0300575	−	0.170465i	0.996141	+	0.0877642i	$0.0279722\pi$
$464$		0			0
$465$	−2125.71	−	7395.97i	−0.211995	−	0.737591i
$466$		0			0
$467$	7069.94	+	12245.5i	0.700552	+	1.21339i	0.968273	+	0.249895i	$0.0803961\pi$
$467$	7069.94	+	12245.5i	0.700552	+	1.21339i	−0.267721	+	0.963496i	$0.586271\pi$
$468$		0			0
$469$	−141.201	+	244.567i	−0.0139020	+	0.0240790i
$470$		0			0
$471$	−8507.02	+	4155.67i	−0.832235	+	0.406546i
$472$		0			0
$473$	14628.4	+	12274.7i	1.42202	+	1.19321i
$474$		0			0
$475$	3986.28	+	1450.89i	0.385059	+	0.140150i
$476$		0			0
$477$	5445.39	−	1164.50i	0.522698	−	0.111779i
$478$		0			0
$479$	467.551	+	2651.62i	0.0445991	+	0.252934i	0.998953	−	0.0457434i	$-0.0145656\pi$
$479$	467.551	+	2651.62i	0.0445991	+	0.252934i	−0.954354	+	0.298677i	$0.903455\pi$
$480$		0			0
$481$	6267.22	−	2281.08i	0.594097	−	0.216234i
$482$		0			0
$483$	−3298.94	+	348.796i	−0.310781	+	0.0328587i
$484$		0			0
$485$	−5070.47			−0.474718
$486$		0			0
$487$	12687.7			1.18056			0.590280	−	0.807199i	$-0.299017\pi$
$487$	12687.7			1.18056			0.590280	+	0.807199i	$0.299017\pi$
$488$		0			0
$489$	−9225.38	+	975.397i	−0.853142	+	0.0902024i
$490$		0			0
$491$	−5094.55	+	1854.26i	−0.468256	+	0.170431i	−0.565362	−	0.824843i	$-0.691264\pi$
$491$	−5094.55	+	1854.26i	−0.468256	+	0.170431i	0.0971061	+	0.995274i	$0.469041\pi$
$492$		0			0
$493$	−1360.22	−	7714.21i	−0.124262	−	0.704727i
$494$		0			0
$495$	−10528.9	+	2251.61i	−0.956041	+	0.204450i
$496$		0			0
$497$	2672.22	+	972.609i	0.241178	+	0.0877817i
$498$		0			0
$499$	−10451.5	−	8769.82i	−0.937619	−	0.786756i	0.0395506	−	0.999218i	$-0.487407\pi$
$499$	−10451.5	−	8769.82i	−0.937619	−	0.786756i	−0.977169	+	0.212462i	$0.931852\pi$
$500$		0			0
$501$	−5450.75	+	2662.68i	−0.486071	+	0.237445i
$502$		0			0
$503$	3966.78	−	6870.66i	0.351630	−	0.609041i	−0.634905	−	0.772590i	$-0.718961\pi$
$503$	3966.78	−	6870.66i	0.351630	−	0.609041i	0.986535	+	0.163549i	$0.0522942\pi$
$504$		0			0
$505$	−1477.79	−	2559.60i	−0.130219	−	0.225546i
$506$		0			0
$507$	−6615.80	−	23018.3i	−0.579523	−	2.01633i
$508$		0			0
$509$	−3183.90	+	18056.8i	−0.277257	+	1.57240i	0.454443	+	0.890776i	$0.349838\pi$
$509$	−3183.90	+	18056.8i	−0.277257	+	1.57240i	−0.731700	+	0.681627i	$0.761273\pi$
$510$		0			0
$511$	3698.89	−	3103.74i	0.320214	−	0.268691i
$512$		0			0
$513$	4704.10	+	4219.81i	0.404856	+	0.363176i
$514$		0			0
$515$	1372.94	−	1152.03i	0.117473	−	0.0985718i
$516$		0			0
$517$	−757.769	+	4297.52i	−0.0644616	+	0.365580i
$518$		0			0
$519$	21081.2	+	5242.28i	1.78297	+	0.443373i
$520$		0			0
$521$	−3020.36	−	5231.43i	−0.253982	−	0.439910i	0.710637	−	0.703559i	$-0.248407\pi$
$521$	−3020.36	−	5231.43i	−0.253982	−	0.439910i	−0.964618	+	0.263650i	$0.915074\pi$
$522$		0			0
$523$	2909.21	−	5038.90i	0.243233	−	0.421292i	−0.718400	−	0.695630i	$-0.755125\pi$
$523$	2909.21	−	5038.90i	0.243233	−	0.421292i	0.961633	+	0.274338i	$0.0884586\pi$
$524$		0			0
$525$	3713.23	+	2501.27i	0.308684	+	0.207932i
$526$		0			0
$527$	8591.29	+	7208.95i	0.710138	+	0.595876i
$528$		0			0
$529$	6857.44	+	2495.90i	0.563610	+	0.205137i
$530$		0			0
$531$	13595.4	−	8518.78i	1.11110	−	0.696202i
$532$		0			0
$533$	4054.27	+	22992.9i	0.329475	+	1.86854i
$534$		0			0
$535$	−1092.55	+	397.656i	−0.0882899	+	0.0321349i
$536$		0			0
$537$	−3877.17	+	8722.78i	−0.311568	+	0.700960i
$538$		0			0
$539$	−18625.2			−1.48839
$540$		0			0
$541$	6742.32			0.535813			0.267907	−	0.963445i	$-0.413668\pi$
$541$	6742.32			0.535813			0.267907	+	0.963445i	$0.413668\pi$
$542$		0			0
$543$	7076.23	+	9726.94i	0.559245	+	0.768735i
$544$		0			0
$545$	−1023.12	+	372.387i	−0.0804144	+	0.0292684i
$546$		0			0
$547$	996.065	+	5648.96i	0.0778586	+	0.441558i	0.998670	+	0.0515507i	$0.0164164\pi$
$547$	996.065	+	5648.96i	0.0778586	+	0.441558i	−0.920812	+	0.390007i	$0.872473\pi$
$548$		0			0
$549$	7277.01	+	17947.3i	0.565711	+	1.39521i
$550$		0			0
$551$	−7886.27	−	2870.37i	−0.609740	−	0.221927i
$552$		0			0
$553$	−4580.35	−	3843.37i	−0.352217	−	0.295546i
$554$		0			0
$555$	161.240	−	2326.53i	0.0123320	−	0.177938i
$556$		0			0
$557$	11916.4	−	20639.9i	0.906492	−	1.57009i	0.0875890	−	0.996157i	$-0.472084\pi$
$557$	11916.4	−	20639.9i	0.906492	−	1.57009i	0.818902	−	0.573933i	$-0.194583\pi$
$558$		0			0
$559$	10966.4	+	18994.4i	0.829749	+	1.43717i
$560$		0			0
$561$	10893.4	−	11294.4i	0.819818	−	0.849998i
$562$		0			0
$563$	143.937	−	816.308i	0.0107748	−	0.0611070i	−0.978946	−	0.204118i	$-0.934567\pi$
$563$	143.937	−	816.308i	0.0107748	−	0.0611070i	0.989721	+	0.143011i	$0.0456785\pi$
$564$		0			0
$565$	920.762	−	772.611i	0.0685606	−	0.0575292i
$566$		0			0
$567$	3743.21	+	5520.02i	0.277249	+	0.408852i
$568$		0			0
$569$	809.884	−	679.573i	0.0596698	−	0.0500689i	−0.612465	−	0.790498i	$-0.709822\pi$
$569$	809.884	−	679.573i	0.0596698	−	0.0500689i	0.672135	+	0.740429i	$0.265378\pi$
$570$		0			0
$571$	−4496.33	+	25499.9i	−0.329537	+	1.86890i	0.146124	+	0.989266i	$0.453320\pi$
$571$	−4496.33	+	25499.9i	−0.329537	+	1.86890i	−0.475661	+	0.879629i	$0.657791\pi$
$572$		0			0
$573$	−6165.62	+	6392.59i	−0.449515	+	0.466063i
$574$		0			0
$575$	3285.94	+	5691.42i	0.238319	+	0.412780i
$576$		0			0
$577$	−1489.92	+	2580.62i	−0.107498	+	0.186192i	−0.914756	−	0.404007i	$-0.867617\pi$
$577$	−1489.92	+	2580.62i	−0.107498	+	0.186192i	0.807258	+	0.590199i	$0.200951\pi$
$578$		0			0
$579$	1113.92	−	16072.6i	0.0799530	−	1.15364i
$580$		0			0
$581$	−8079.41	−	6779.43i	−0.576920	−	0.484093i
$582$		0			0
$583$	−13920.7	−	5066.72i	−0.988913	−	0.359935i
$584$		0			0
$585$	−12248.3	−	1705.93i	−0.865648	−	0.120567i
$586$		0			0
$587$	−3085.89	−	17501.0i	−0.216982	−	1.23057i	−0.877434	−	0.479698i	$-0.840746\pi$
$587$	−3085.89	−	17501.0i	−0.216982	−	1.23057i	0.660452	−	0.750868i	$-0.270365\pi$
$588$		0			0
$589$	11291.1	−	4109.63i	0.789884	−	0.287494i
$590$		0			0
$591$	1646.39	+	2263.12i	0.114591	+	0.157516i
$592$		0			0
$593$	−3329.70			−0.230581			−0.115290	−	0.993332i	$-0.536780\pi$
$593$	−3329.70			−0.230581			−0.115290	+	0.993332i	$0.536780\pi$
$594$		0			0
$595$	2135.42			0.147132
$596$		0			0
$597$	−7797.40	+	17542.4i	−0.534550	+	1.20262i
$598$		0			0
$599$	12799.7	−	4658.71i	0.873090	−	0.317779i	0.133672	−	0.991026i	$-0.457323\pi$
$599$	12799.7	−	4658.71i	0.873090	−	0.317779i	0.739418	+	0.673247i	$0.235101\pi$
$600$		0			0
$601$	447.186	+	2536.12i	0.0303513	+	0.172131i	0.996215	−	0.0869185i	$-0.0277020\pi$
$601$	447.186	+	2536.12i	0.0303513	+	0.172131i	−0.965864	+	0.259049i	$0.916591\pi$
$602$		0			0
$603$	736.351	+	390.357i	0.0497289	+	0.0263625i
$604$		0			0
$605$	19972.6	+	7269.45i	1.34215	+	0.488504i
$606$		0			0
$607$	−8011.80	−	6722.70i	−0.535731	−	0.449532i	0.334344	−	0.942451i	$-0.391485\pi$
$607$	−8011.80	−	6722.70i	−0.535731	−	0.449532i	−0.870075	+	0.492919i	$0.835930\pi$
$608$		0			0
$609$	−7346.09	−	4948.39i	−0.488799	−	0.329259i
$610$		0			0
$611$	−2506.05	+	4340.60i	−0.165931	+	0.287401i
$612$		0			0
$613$	−12357.4	−	21403.7i	−0.814211	−	1.41025i	−0.909893	−	0.414842i	$-0.863837\pi$
$613$	−12357.4	−	21403.7i	−0.814211	−	1.41025i	0.0956826	−	0.995412i	$-0.469497\pi$
$614$		0			0
$615$	7922.71	+	1970.15i	0.519470	+	0.129177i
$616$		0			0
$617$	1578.77	−	8953.67i	0.103013	−	0.584216i	−0.888982	−	0.457942i	$-0.848587\pi$
$617$	1578.77	−	8953.67i	0.103013	−	0.584216i	0.991995	−	0.126274i	$-0.0403020\pi$
$618$		0			0
$619$	7255.84	−	6088.37i	0.471142	−	0.395335i	−0.376069	−	0.926592i	$-0.622725\pi$
$619$	7255.84	−	6088.37i	0.471142	−	0.395335i	0.847211	+	0.531257i	$0.178280\pi$
$620$		0			0
$621$	1380.51	+	9692.26i	0.0892074	+	0.626308i
$622$		0			0
$623$	−3377.89	+	2834.39i	−0.217227	+	0.182275i
$624$		0			0
$625$	871.146	−	4940.52i	0.0557534	−	0.316193i
$626$		0			0
$627$	−4643.97	−	16157.7i	−0.295793	−	1.02915i
$628$		0			0
$629$	1699.39	+	2943.43i	0.107725	+	0.186585i
$630$		0			0
$631$	−7345.25	+	12722.3i	−0.463407	+	0.802644i	−0.999128	−	0.0417505i	$-0.986707\pi$
$631$	−7345.25	+	12722.3i	−0.463407	+	0.802644i	0.535721	+	0.844395i	$0.320040\pi$
$632$		0			0
$633$	−16617.0	+	8117.38i	−1.04339	+	0.509695i
$634$		0			0
$635$	−5203.19	−	4365.99i	−0.325169	−	0.272849i
$636$		0			0
$637$	−20102.0	−	7316.53i	−1.25035	−	0.455088i
$638$		0			0
$639$	2583.51	−	7984.82i	0.159940	−	0.494326i
$640$		0			0
$641$	−4526.38	−	25670.4i	−0.278910	−	1.58178i	−0.726261	−	0.687419i	$-0.758744\pi$
$641$	−4526.38	−	25670.4i	−0.278910	−	1.58178i	0.447351	−	0.894359i	$-0.352368\pi$
$642$		0			0
$643$	2079.34	−	756.817i	0.127529	−	0.0464167i	−0.277467	−	0.960735i	$-0.589495\pi$
$643$	2079.34	−	756.817i	0.127529	−	0.0464167i	0.404996	+	0.914318i	$0.367273\pi$
$644$		0			0
$645$	7626.77	−	806.376i	0.465587	−	0.0492264i
$646$		0			0
$647$	−13758.7			−0.836027			−0.418013	−	0.908441i	$-0.637273\pi$
$647$	−13758.7			−0.836027			−0.418013	+	0.908441i	$0.637273\pi$
$648$		0			0
$649$	−42682.0			−2.58154
$650$		0			0
$651$	12611.1	−	1333.36i	0.759242	−	0.0802744i
$652$		0			0
$653$	−15288.2	+	5564.44i	−0.916191	+	0.333466i	−0.756722	−	0.653737i	$-0.773200\pi$
$653$	−15288.2	+	5564.44i	−0.916191	+	0.333466i	−0.159469	+	0.987203i	$0.550978\pi$
$654$		0			0
$655$	−1081.15	−	6131.53i	−0.0644949	−	0.365769i
$656$		0			0
$657$	−9548.22	−	10578.0i	−0.566988	−	0.628140i
$658$		0			0
$659$	−10222.0	−	3720.52i	−0.604240	−	0.219925i	0.0217411	−	0.999764i	$-0.493079\pi$
$659$	−10222.0	−	3720.52i	−0.604240	−	0.219925i	−0.625981	+	0.779838i	$0.715301\pi$
$660$		0			0
$661$	316.290	+	265.399i	0.0186116	+	0.0156170i	0.652046	−	0.758179i	$-0.273911\pi$
$661$	316.290	+	265.399i	0.0186116	+	0.0156170i	−0.633434	+	0.773796i	$0.718355\pi$
$662$		0			0
$663$	16193.9	−	7910.68i	0.948594	−	0.463387i
$664$		0			0
$665$	1143.93	−	1981.35i	0.0667063	−	0.115539i
$666$		0			0
$667$	−6500.75	−	11259.6i	−0.377377	−	0.653635i
$668$		0			0
$669$	7709.04	+	26822.0i	0.445514	+	1.55007i
$670$		0			0
$671$	8946.57	−	50738.5i	0.514722	−	2.91913i
$672$		0			0
$673$	−18608.4	+	15614.3i	−1.06582	+	0.894333i	−0.994668	−	0.103133i	$-0.967113\pi$
$673$	−18608.4	+	15614.3i	−1.06582	+	0.894333i	−0.0711564	+	0.997465i	$0.522669\pi$
$674$		0			0
$675$	6980.91	−	11218.1i	0.398067	−	0.639679i
$676$		0			0
$677$	−8328.43	+	6988.38i	−0.472803	+	0.396729i	−0.847816	−	0.530291i	$-0.822083\pi$
$677$	−8328.43	+	6988.38i	−0.472803	+	0.396729i	0.375013	+	0.927020i	$0.377638\pi$
$678$		0			0
$679$	1450.95	−	8228.77i	0.0820067	−	0.465083i
$680$		0			0
$681$	9598.38	+	2386.84i	0.540104	+	0.134308i
$682$		0			0
$683$	6279.52	+	10876.5i	0.351800	+	0.609335i	0.986565	−	0.163370i	$-0.0522365\pi$
$683$	6279.52	+	10876.5i	0.351800	+	0.609335i	−0.634765	+	0.772705i	$0.718903\pi$
$684$		0			0
$685$	−7127.96	+	12346.0i	−0.397585	+	0.688637i
$686$		0			0
$687$	11078.2	+	7462.36i	0.615224	+	0.414420i
$688$		0			0
$689$	−13034.1	−	10936.9i	−0.720698	−	0.604737i
$690$		0			0
$691$	16718.3	+	6084.96i	0.920396	+	0.334997i	0.758396	−	0.651794i	$-0.225983\pi$
$691$	16718.3	+	6084.96i	0.920396	+	0.334997i	0.162000	+	0.986791i	$0.448206\pi$
$692$		0			0
$693$	−641.165	−	17731.5i	−0.0351455	−	0.971955i
$694$		0			0
$695$	−2414.65	−	13694.2i	−0.131788	−	0.747408i
$696$		0			0
$697$	−11180.5	+	4069.38i	−0.607594	+	0.221146i
$698$		0			0
$699$	−7469.76	+	16805.3i	−0.404195	+	0.909350i
$700$		0			0
$701$	21646.4			1.16629			0.583147	−	0.812367i	$-0.301821\pi$
$701$	21646.4			1.16629			0.583147	+	0.812367i	$0.301821\pi$
$702$		0			0
$703$	3641.41			0.195360
$704$		0			0
$705$	1031.02	+	1417.23i	0.0550787	+	0.0757108i
$706$		0			0
$707$	4576.81	−	1665.82i	0.243463	−	0.0886135i
$708$		0			0
$709$	3459.28	+	19618.5i	0.183238	+	1.03920i	0.928198	+	0.372086i	$0.121357\pi$
$709$	3459.28	+	19618.5i	0.183238	+	1.03920i	−0.744960	+	0.667109i	$0.767532\pi$
$710$		0			0
$711$	−10846.7	+	13918.5i	−0.572126	+	0.734158i
$712$		0			0
$713$	17492.2	+	6366.64i	0.918777	+	0.334407i
$714$		0			0
$715$	25202.2	+	21147.1i	1.31819	+	1.10609i
$716$		0			0
$717$	−2194.16	+	31659.4i	−0.114285	+	1.64901i
$718$		0			0
$719$	10932.0	−	18934.8i	0.567031	−	0.982126i	−0.429827	−	0.902911i	$-0.641425\pi$
$719$	10932.0	−	18934.8i	0.567031	−	0.982126i	0.996858	−	0.0792148i	$-0.0252413\pi$
$720$		0			0
$721$	1476.73	+	2557.77i	0.0762779	+	0.132117i
$722$		0			0
$723$	11579.5	−	12005.8i	0.595639	−	0.617566i
$724$		0			0
$725$	−3047.00	+	17280.4i	−0.156087	+	0.885211i
$726$		0			0
$727$	−25330.4	+	21254.7i	−1.29223	+	1.08431i	−0.300799	+	0.953688i	$0.597253\pi$
$727$	−25330.4	+	21254.7i	−1.29223	+	1.08431i	−0.991432	+	0.130623i	$0.958302\pi$
$728$		0			0
$729$	15880.8	−	11628.4i	0.806830	−	0.590784i
$730$		0			0
$731$	−8562.15	+	7184.49i	−0.433218	+	0.363513i
$732$		0			0
$733$	2985.32	−	16930.6i	0.150430	−	0.853131i	−0.812416	−	0.583079i	$-0.801848\pi$
$733$	2985.32	−	16930.6i	0.150430	−	0.853131i	0.962846	−	0.270052i	$-0.0870410\pi$
$734$		0			0
$735$	−5192.85	+	5384.02i	−0.260600	+	0.270194i
$736$		0			0
$737$	−1108.59	−	1920.14i	−0.0554077	−	0.0959689i
$738$		0			0
$739$	9649.35	−	16713.2i	0.480321	−	0.831940i	−0.519424	−	0.854516i	$-0.673854\pi$
$739$	9649.35	−	16713.2i	0.480321	−	0.831940i	0.999745	+	0.0225766i	$0.00718696\pi$
$740$		0			0
$741$	1335.04	−	19263.1i	0.0661859	−	0.954992i
$742$		0			0
$743$	20573.9	+	17263.5i	1.01586	+	0.852405i	0.989101	−	0.147237i	$-0.0470380\pi$
$743$	20573.9	+	17263.5i	1.01586	+	0.852405i	0.0267555	+	0.999642i	$0.491482\pi$
$744$		0			0
$745$	602.767	+	219.389i	0.0296425	+	0.0107890i
$746$		0			0
$747$	−19132.7	+	24551.3i	−0.937122	+	1.20252i
$748$		0			0
$749$	−332.707	−	1886.87i	−0.0162308	−	0.0920492i
$750$		0			0
$751$	−27645.8	+	10062.2i	−1.34329	+	0.488916i	−0.910846	−	0.412746i	$-0.864570\pi$
$751$	−27645.8	+	10062.2i	−1.34329	+	0.488916i	−0.432440	+	0.901663i	$0.642347\pi$
$752$		0			0
$753$	−11032.0	−	15164.5i	−0.533903	−	0.733899i
$754$		0			0
$755$	8651.72			0.417044
$756$		0			0
$757$	24191.8			1.16151			0.580756	−	0.814077i	$-0.302757\pi$
$757$	24191.8			1.16151			0.580756	+	0.814077i	$0.302757\pi$
$758$		0			0
$759$	10578.7	−	23799.7i	0.505905	−	1.13818i
$760$		0			0
$761$	12610.5	−	4589.85i	0.600698	−	0.218636i	−0.0237304	−	0.999718i	$-0.507554\pi$
$761$	12610.5	−	4589.85i	0.600698	−	0.218636i	0.624428	+	0.781082i	$0.285332\pi$
$762$		0			0
$763$	−311.565	−	1766.97i	−0.0147830	−	0.0838383i
$764$		0			0
$765$	−227.730	−	6297.93i	−0.0107629	−	0.297650i
$766$		0			0
$767$	−46066.3	−	16766.8i	−2.16866	−	0.789326i
$768$		0			0
$769$	16690.9	+	14005.3i	0.782689	+	0.656754i	0.943924	−	0.330162i	$-0.107103\pi$
$769$	16690.9	+	14005.3i	0.782689	+	0.656754i	−0.161235	+	0.986916i	$0.551548\pi$
$770$		0			0
$771$	27819.4	+	18739.4i	1.29947	+	0.875336i
$772$		0			0
$773$	−670.497	+	1161.34i	−0.0311981	+	0.0540367i	−0.881203	−	0.472738i	$-0.843266\pi$
$773$	−670.497	+	1161.34i	−0.0311981	+	0.0540367i	0.850005	+	0.526775i	$0.176599\pi$
$774$		0			0
$775$	−12561.4	−	21756.9i	−0.582216	−	1.00843i
$776$		0			0
$777$	3729.54	+	927.428i	0.172196	+	0.0428202i
$778$		0			0
$779$	−2213.57	+	12553.8i	−0.101809	+	0.577389i
$780$		0			0
$781$	−17103.1	+	14351.2i	−0.783607	+	0.657524i
$782$		0			0
$783$	−13810.7	+	22193.3i	−0.630338	+	1.01293i
$784$		0			0
$785$	7749.10	−	6502.26i	0.352327	−	0.295638i
$786$		0			0
$787$	−508.621	+	2884.53i	−0.0230373	+	0.130651i	−0.994157	−	0.107940i	$-0.965575\pi$
$787$	−508.621	+	2884.53i	−0.0230373	+	0.130651i	0.971120	+	0.238591i	$0.0766856\pi$
$788$		0			0
$789$	10951.8	+	38104.5i	0.494163	+	1.71934i
$790$		0			0
$791$	990.373	+	1715.38i	0.0445178	+	0.0771071i
$792$		0			0
$793$	29587.5	−	51247.1i	1.32495	−	2.29488i
$794$		0			0
$795$	−5345.85	+	2611.44i	−0.238488	+	0.116501i
$796$		0			0
$797$	−21377.1	−	17937.5i	−0.950082	−	0.797213i	0.0292296	−	0.999573i	$-0.490695\pi$
$797$	−21377.1	−	17937.5i	−0.950082	−	0.797213i	−0.979311	+	0.202359i	$0.935139\pi$
$798$		0			0
$799$	−2400.15	−	873.585i	−0.106272	−	0.0386799i
$800$		0			0
$801$	8719.60	+	9660.03i	0.384634	+	0.426118i
$802$		0			0
$803$	6582.97	+	37333.9i	0.289300	+	1.64070i
$804$		0			0
$805$	3330.61	−	1212.24i	0.145824	−	0.0530756i
$806$		0			0
$807$	38331.6	−	4052.79i	1.67204	−	0.176785i
$808$		0			0
$809$	12704.1			0.552104			0.276052	−	0.961143i	$-0.410974\pi$
$809$	12704.1			0.552104			0.276052	+	0.961143i	$0.410974\pi$
$810$		0			0
$811$	5012.10			0.217014			0.108507	−	0.994096i	$-0.465393\pi$
$811$	5012.10			0.217014			0.108507	+	0.994096i	$0.465393\pi$
$812$		0			0
$813$	30729.9	−	3249.07i	1.32564	−	0.140160i
$814$		0			0
$815$	9313.93	−	3389.99i	0.400310	−	0.145701i
$816$		0			0
$817$	2079.43	+	11793.1i	0.0890455	+	0.505002i
$818$		0			0
$819$	6273.47	−	19389.3i	0.267659	−	0.827251i
$820$		0			0
$821$	−13568.4	−	4938.51i	−0.576787	−	0.209933i	0.0371213	−	0.999311i	$-0.488181\pi$
$821$	−13568.4	−	4938.51i	−0.576787	−	0.209933i	−0.613908	+	0.789378i	$0.710403\pi$
$822$		0			0
$823$	677.888	+	568.816i	0.0287117	+	0.0240919i	0.657031	−	0.753864i	$-0.271812\pi$
$823$	677.888	+	568.816i	0.0287117	+	0.0240919i	−0.628319	+	0.777956i	$0.716257\pi$
$824$		0			0
$825$	−31583.5	+	15428.5i	−1.33284	+	0.651093i
$826$		0			0
$827$	−16997.2	+	29440.1i	−0.714693	+	1.23789i	0.248384	+	0.968662i	$0.420100\pi$
$827$	−16997.2	+	29440.1i	−0.714693	+	1.23789i	−0.963078	+	0.269224i	$0.913233\pi$
$828$		0			0
$829$	2235.23	+	3871.54i	0.0936463	+	0.162200i	0.909043	−	0.416703i	$-0.136814\pi$
$829$	2235.23	+	3871.54i	0.0936463	+	0.162200i	−0.815397	+	0.578903i	$0.803481\pi$
$830$		0			0
$831$	−872.129	−	3034.39i	−0.0364065	−	0.126669i
$832$		0			0
$833$	1893.03	−	10735.9i	0.0787390	−	0.446551i
$834$		0			0
$835$	4965.12	−	4166.23i	0.205779	−	0.172669i
$836$		0			0
$837$	−5277.34	−	37051.2i	−0.217935	−	1.53008i
$838$		0			0
$839$	−4716.06	+	3957.24i	−0.194060	+	0.162836i	−0.734640	−	0.678457i	$-0.762649\pi$
$839$	−4716.06	+	3957.24i	−0.194060	+	0.162836i	0.540580	+	0.841292i	$0.318205\pi$
$840$		0			0
$841$	1792.94	−	10168.3i	0.0735142	−	0.416920i
$842$		0			0
$843$	−34275.7	−	8523.36i	−1.40038	−	0.348233i
$844$		0			0
$845$	12794.6	+	22160.9i	0.520886	+	0.902201i
$846$		0			0
$847$	−17512.8	+	30333.0i	−0.710444	+	1.23052i
$848$		0			0
$849$	320.547	+	215.923i	0.0129578	+	0.00872846i
$850$		0			0
$851$	4321.46	+	3626.14i	0.174075	+	0.146066i
$852$		0			0
$853$	8837.84	+	3216.71i	0.354750	+	0.129119i	0.513247	−	0.858241i	$-0.328442\pi$
$853$	8837.84	+	3216.71i	0.354750	+	0.129119i	−0.158497	+	0.987360i	$0.550665\pi$
$854$		0			0
$855$	−5965.51	−	3162.46i	−0.238615	−	0.126496i
$856$		0			0
$857$	2250.77	+	12764.8i	0.0897141	+	0.508794i	0.996239	+	0.0866426i	$0.0276138\pi$
$857$	2250.77	+	12764.8i	0.0897141	+	0.508794i	−0.906525	+	0.422151i	$0.861275\pi$
$858$		0			0
$859$	22836.3	−	8311.73i	0.907059	−	0.330143i	0.153981	−	0.988074i	$-0.450791\pi$
$859$	22836.3	−	8311.73i	0.907059	−	0.330143i	0.753078	+	0.657931i	$0.228568\pi$
$860$		0			0
$861$	−5464.46	+	12293.8i	−0.216293	+	0.486612i
$862$		0			0
$863$	21125.6			0.833285			0.416643	−	0.909070i	$-0.363207\pi$
$863$	21125.6			0.833285			0.416643	+	0.909070i	$0.363207\pi$
$864$		0			0
$865$	−23209.9			−0.912323
$866$		0			0
$867$	−9615.06	−	13216.8i	−0.376637	−	0.517723i
$868$		0			0
$869$	44112.8	−	16055.7i	1.72201	−	0.626759i
$870$		0			0
$871$	−442.205	−	2507.87i	−0.0172027	−	0.0975614i
$872$		0			0
$873$	−24423.6	−	3401.70i	−0.946865	−	0.131879i
$874$		0			0
$875$	−10461.1	−	3807.55i	−0.404173	−	0.147107i
$876$		0			0
$877$	22826.9	+	19154.0i	0.878915	+	0.737498i	0.965956	−	0.258707i	$-0.0832965\pi$
$877$	22826.9	+	19154.0i	0.878915	+	0.737498i	−0.0870404	+	0.996205i	$0.527741\pi$
$878$		0			0
$879$	35.8002	−	516.559i	0.00137373	−	0.0198215i
$880$		0			0
$881$	−13196.2	+	22856.4i	−0.504643	+	0.874066i	0.495343	+	0.868697i	$0.335042\pi$
$881$	−13196.2	+	22856.4i	−0.504643	+	0.874066i	−0.999986	+	0.00536903i	$0.998291\pi$
$882$		0			0
$883$	8184.93	+	14176.7i	0.311942	+	0.540299i	0.978783	−	0.204901i	$-0.0656872\pi$
$883$	8184.93	+	14176.7i	0.311942	+	0.540299i	−0.666841	+	0.745200i	$0.732354\pi$
$884$		0			0
$885$	−11900.1	+	12338.2i	−0.451997	+	0.468637i
$886$		0			0
$887$	4816.38	−	27315.0i	0.182320	−	1.03399i	−0.747030	−	0.664791i	$-0.768521\pi$
$887$	4816.38	−	27315.0i	0.182320	−	1.03399i	0.929350	−	0.369200i	$-0.120368\pi$
$888$		0			0
$889$	8574.42	−	7194.79i	0.323483	−	0.271435i
$890$		0			0
$891$	−52226.6	+	3781.93i	−1.96370	+	0.142199i
$892$		0			0
$893$	−2096.30	+	1759.01i	−0.0785555	+	0.0659159i
$894$		0			0
$895$	1771.02	−	10044.0i	0.0661439	−	0.375121i
$896$		0			0
$897$	20766.7	−	21531.2i	0.773000	−	0.801457i
$898$		0			0
$899$	24850.8	+	43042.9i	0.921937	+	1.59684i
$900$		0			0
$901$	4335.43	−	7509.18i	0.160304	−	0.277655i
$902$		0			0
$903$	−873.806	+	12608.1i	−0.0322020	+	0.464641i
$904$		0			0
$905$	−9845.02	−	8260.96i	−0.361613	−	0.303429i
$906$		0			0
$907$	15389.5	+	5601.34i	0.563397	+	0.205060i	0.607989	−	0.793945i	$-0.291976\pi$
$907$	15389.5	+	5601.34i	0.563397	+	0.205060i	−0.0445917	+	0.999005i	$0.514199\pi$
$908$		0			0
$909$	−5401.05	−	13320.6i	−0.197075	−	0.486046i
$910$		0			0
$911$	1424.52	+	8078.83i	0.0518071	+	0.293813i	0.999692	−	0.0247988i	$-0.00789451\pi$
$911$	1424.52	+	8078.83i	0.0518071	+	0.293813i	−0.947885	+	0.318612i	$0.896783\pi$
$912$		0			0
$913$	77811.8	−	28321.2i	2.82059	−	1.02661i
$914$		0			0
$915$	−12172.7	−	16732.5i	−0.439800	−	0.604546i
$916$		0			0
$917$	10260.1			0.369486
$918$		0			0
$919$	−33771.8			−1.21222			−0.606109	−	0.795381i	$-0.707271\pi$
$919$	−33771.8			−1.21222			−0.606109	+	0.795381i	$0.707271\pi$
$920$		0			0
$921$	10539.9	−	23712.5i	0.377093	−	0.848376i
$922$		0			0
$923$	−24096.8	+	8770.52i	−0.859324	+	0.312768i
$924$		0			0
$925$	−1322.07	−	7497.85i	−0.0469941	−	0.266517i
$926$		0			0
$927$	7386.07	−	4628.05i	0.261694	−	0.163975i
$928$		0			0
$929$	−45473.2	−	16550.9i	−1.60595	−	0.584518i	−0.625317	−	0.780371i	$-0.715030\pi$
$929$	−45473.2	−	16550.9i	−1.60595	−	0.584518i	−0.980633	+	0.195853i	$0.937253\pi$
$930$		0			0
$931$	−8947.20	−	7507.60i	−0.314965	−	0.264287i
$932$		0			0
$933$	−32254.8	−	21727.1i	−1.13180	−	0.762394i
$934$		0			0
$935$	−8382.77	+	14519.4i	−0.293204	+	0.507845i
$936$		0			0
$937$	−20974.0	−	36328.0i	−0.731258	−	1.26658i	−0.956346	−	0.292238i	$-0.905600\pi$
$937$	−20974.0	−	36328.0i	−0.731258	−	1.26658i	0.225087	−	0.974339i	$-0.427733\pi$
$938$		0			0
$939$	−22387.1	−	5567.02i	−0.778035	−	0.193475i
$940$		0			0
$941$	−2249.89	+	12759.8i	−0.0779431	+	0.442037i	0.920714	+	0.390237i	$0.127607\pi$
$941$	−2249.89	+	12759.8i	−0.0779431	+	0.442037i	−0.998657	+	0.0518001i	$0.983504\pi$
$942$		0			0
$943$	−15128.1	+	12694.0i	−0.522417	+	0.438360i
$944$		0			0
$945$	−5304.45	−	4758.35i	−0.182596	−	0.163798i
$946$		0			0
$947$	−16485.1	+	13832.7i	−0.565675	+	0.474658i	−0.880208	−	0.474589i	$-0.842597\pi$
$947$	−16485.1	+	13832.7i	−0.565675	+	0.474658i	0.314532	+	0.949247i	$0.398152\pi$
$948$		0			0
$949$	−7560.92	+	42880.1i	−0.258628	+	1.46675i
$950$		0			0
$951$	2478.84	+	8624.61i	0.0845236	+	0.294082i
$952$		0			0
$953$	−17377.4	−	30098.5i	−0.590669	−	1.02307i	−0.994142	−	0.108078i	$-0.965531\pi$
$953$	−17377.4	−	30098.5i	−0.590669	−	1.02307i	0.403473	−	0.914991i	$-0.367803\pi$
$954$		0			0
$955$	4744.63	−	8217.94i	0.160767	−	0.278457i
$956$		0			0
$957$	62483.3	−	30523.0i	2.11055	−	1.03100i
$958$		0			0
$959$	−17996.4	−	15100.7i	−0.605978	−	0.508476i
$960$		0			0
$961$	−38874.1	−	14149.0i	−1.30489	−	0.474942i
$962$		0			0
$963$	−5529.41	+	1182.46i	−0.185029	+	0.0395684i
$964$		0			0
$965$	2989.14	+	16952.3i	0.0997139	+	0.565506i
$966$		0			0
$967$	17056.1	−	6207.92i	0.567205	−	0.206446i	−0.0424690	−	0.999098i	$-0.513522\pi$
$967$	17056.1	−	6207.92i	0.567205	−	0.206446i	0.609675	+	0.792652i	$0.291300\pi$
$968$		0			0
$969$	9785.61	−	1034.63i	0.324416	−	0.0343004i
$970$		0			0
$971$	−32473.1			−1.07324			−0.536618	−	0.843825i	$-0.680298\pi$
$971$	−32473.1			−1.07324			−0.536618	+	0.843825i	$0.680298\pi$
$972$		0			0
$973$	22914.9			0.755005
$974$		0			0
$975$	−40148.6	+	4244.90i	−1.31875	+	0.139431i
$976$		0			0
$977$	2618.73	−	953.139i	0.0857529	−	0.0312115i	−0.298787	−	0.954320i	$-0.596582\pi$
$977$	2618.73	−	953.139i	0.0857529	−	0.0312115i	0.384540	+	0.923108i	$0.374360\pi$
$978$		0			0
$979$	−6011.69	−	34094.0i	−0.196256	−	1.11302i
$980$		0			0
$981$	−5178.04	+	1107.32i	−0.168524	+	0.0360389i
$982$		0			0
$983$	54587.2	+	19868.1i	1.77117	+	0.644654i	0.999967	+	0.00806713i	$0.00256788\pi$
$983$	54587.2	+	19868.1i	1.77117	+	0.644654i	0.771205	+	0.636587i	$0.219654\pi$
$984$		0			0
$985$	−2290.59	−	1922.04i	−0.0740958	−	0.0621737i
$986$		0			0
$987$	−2595.04	+	1267.67i	−0.0836889	+	0.0408819i
$988$		0			0
$989$	−9275.82	+	16066.2i	−0.298235	+	0.516557i
$990$		0			0
$991$	−16680.9	−	28892.1i	−0.534698	−	0.926124i	−0.999178	−	0.0405401i	$-0.987092\pi$
$991$	−16680.9	−	28892.1i	−0.534698	−	0.926124i	0.464480	−	0.885584i	$-0.346241\pi$
$992$		0			0
$993$	1211.54	+	4215.28i	0.0387180	+	0.134711i
$994$		0			0
$995$	3561.72	−	20199.5i	0.113481	−	0.643585i
$996$		0			0
$997$	21033.1	−	17648.8i	0.668128	−	0.560626i	−0.244383	−	0.969679i	$-0.578585\pi$
$997$	21033.1	−	17648.8i	0.668128	−	0.560626i	0.912511	+	0.409053i	$0.134141\pi$
$998$		0			0
$999$	2337.50	−	11098.3i	0.0740292	−	0.351486i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.i.a.49.1	✓	54
3.2	odd	2		324.4.i.a.37.4		54
27.11	odd	18		324.4.i.a.289.4		54
27.16	even	9	inner	108.4.i.a.97.1	yes	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.49.1	✓	54	1.1	even	1	trivial
108.4.i.a.97.1	yes	54	27.16	even	9	inner
324.4.i.a.37.4		54	3.2	odd	2
324.4.i.a.289.4		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)

\(f(q)\)	\(=\)	\(q+(-5.16735 + 0.546343i) q^{3} +(5.21695 - 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +(26.4030 - 5.64629i) q^{9} +O(q^{10})\)	\(q+(-5.16735 + 0.546343i) q^{3} +(5.21695 - 1.89881i) q^{5} +(1.58868 + 9.00985i) q^{7} +(26.4030 - 5.64629i) q^{9} +(-67.4972 - 24.5670i) q^{11} +(-63.1985 - 53.0298i) q^{13} +(-25.9204 + 12.6621i) q^{15} +(21.0212 - 36.4097i) q^{17} +(-22.5218 - 39.0089i) q^{19} +(-13.1317 - 45.6891i) q^{21} +(12.1174 - 68.7214i) q^{23} +(-72.1445 + 60.5364i) q^{25} +(-133.349 + 43.6014i) q^{27} +(142.727 - 119.762i) q^{29} +(-46.3221 + 262.706i) q^{31} +(362.204 + 90.0696i) q^{33} +(25.3961 + 43.9873i) q^{35} +(-40.4209 + 70.0111i) q^{37} +(355.541 + 239.496i) q^{39} +(-216.792 - 181.910i) q^{41} +(-249.820 - 90.9271i) q^{43} +(127.022 - 79.5908i) q^{45} +(-10.5496 - 59.8299i) q^{47} +(243.661 - 88.6854i) q^{49} +(-88.7316 + 199.627i) q^{51} +206.241 q^{53} -398.778 q^{55} +(137.690 + 189.268i) q^{57} +(558.381 - 203.234i) q^{59} +(124.554 + 706.379i) q^{61} +(92.8181 + 228.917i) q^{63} +(-430.397 - 156.652i) q^{65} +(23.6459 + 19.8412i) q^{67} +(-25.0696 + 361.728i) q^{69} +(155.414 - 269.185i) q^{71} +(-263.889 - 457.070i) q^{73} +(339.722 - 352.229i) q^{75} +(114.113 - 647.169i) q^{77} +(-500.648 + 420.094i) q^{79} +(665.239 - 298.158i) q^{81} +(-883.108 + 741.015i) q^{83} +(40.5310 - 229.863i) q^{85} +(-672.091 + 696.832i) q^{87} +(240.988 + 417.404i) q^{89} +(377.389 - 653.656i) q^{91} +(95.8352 - 1382.80i) q^{93} +(-191.566 - 160.743i) q^{95} +(-858.230 - 312.370i) q^{97} +(-1920.84 - 267.534i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 12 q^{5} - 48 q^{9}+O(q^{10}) \) 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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