LMFDB - Embedded newform 81.4.c.e.55.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [81,4,Mod(28,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.28"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$4.77915471046$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.4.c.e.28.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.12132 - 3.67423i) q^{2} +(-5.00000 - 8.66025i) q^{4} +(-8.48528 - 14.6969i) q^{5} +(-5.50000 + 9.52628i) q^{7} -8.48528 q^{8} -72.0000 q^{10} +(8.48528 - 14.6969i) q^{11} +(-14.5000 - 25.1147i) q^{13} +(23.3345 + 40.4166i) q^{14} +(22.0000 - 38.1051i) q^{16} +50.9117 q^{17} +29.0000 q^{19} +(-84.8528 + 146.969i) q^{20} +(-36.0000 - 62.3538i) q^{22} +(42.4264 + 73.4847i) q^{23} +(-81.5000 + 141.162i) q^{25} -123.037 q^{26} +110.000 q^{28} +(135.765 - 235.151i) q^{29} +(134.000 + 232.095i) q^{31} +(-127.279 - 220.454i) q^{32} +(108.000 - 187.061i) q^{34} +186.676 q^{35} +83.0000 q^{37} +(61.5183 - 106.553i) q^{38} +(72.0000 + 124.708i) q^{40} +(-135.765 - 235.151i) q^{41} +(116.000 - 200.918i) q^{43} -169.706 q^{44} +360.000 q^{46} +(-195.161 + 338.030i) q^{47} +(111.000 + 192.258i) q^{49} +(345.775 + 598.900i) q^{50} +(-145.000 + 251.147i) q^{52} -305.470 q^{53} -288.000 q^{55} +(46.6690 - 80.8332i) q^{56} +(-576.000 - 997.661i) q^{58} +(144.250 + 249.848i) q^{59} +(-383.500 + 664.241i) q^{61} +1137.03 q^{62} -728.000 q^{64} +(-246.073 + 426.211i) q^{65} +(255.500 + 442.539i) q^{67} +(-254.558 - 440.908i) q^{68} +(396.000 - 685.892i) q^{70} -712.764 q^{71} +137.000 q^{73} +(176.070 - 304.961i) q^{74} +(-145.000 - 251.147i) q^{76} +(93.3381 + 161.666i) q^{77} +(237.500 - 411.362i) q^{79} -746.705 q^{80} -1152.00 q^{82} +(288.500 - 499.696i) q^{83} +(-432.000 - 748.246i) q^{85} +(-492.146 - 852.422i) q^{86} +(-72.0000 + 124.708i) q^{88} +254.558 q^{89} +319.000 q^{91} +(424.264 - 734.847i) q^{92} +(828.000 + 1434.14i) q^{94} +(-246.073 - 426.211i) q^{95} +(-410.500 + 711.007i) q^{97} +941.866 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{4} - 22 q^{7} - 288 q^{10} - 58 q^{13} + 88 q^{16} + 116 q^{19} - 144 q^{22} - 326 q^{25} + 440 q^{28} + 536 q^{31} + 432 q^{34} + 332 q^{37} + 288 q^{40} + 464 q^{43} + 1440 q^{46} + 444 q^{49}+ \cdots - 1642 q^{97}+O(q^{100}) $ 4 * q - 20 * q^4 - 22 * q^7 - 288 * q^10 - 58 * q^13 + 88 * q^16 + 116 * q^19 - 144 * q^22 - 326 * q^25 + 440 * q^28 + 536 * q^31 + 432 * q^34 + 332 * q^37 + 288 * q^40 + 464 * q^43 + 1440 * q^46 + 444 * q^49 - 580 * q^52 - 1152 * q^55 - 2304 * q^58 - 1534 * q^61 - 2912 * q^64 + 1022 * q^67 + 1584 * q^70 + 548 * q^73 - 580 * q^76 + 950 * q^79 - 4608 * q^82 - 1728 * q^85 - 288 * q^88 + 1276 * q^91 + 3312 * q^94 - 1642 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	2.12132	−	3.67423i	0.750000	−	1.29904i	−0.197822	−	0.980238i	$-0.563387\pi$
$2$	2.12132	−	3.67423i	0.750000	−	1.29904i	0.947822	−	0.318800i	$-0.103280\pi$
$3$		0			0
$3$		0			0
$4$	−5.00000	−	8.66025i	−0.625000	−	1.08253i
$5$	−8.48528	−	14.6969i	−0.758947	−	1.31453i	−0.943388	−	0.331691i	$-0.892381\pi$
$5$	−8.48528	−	14.6969i	−0.758947	−	1.31453i	0.184442	−	0.982843i	$-0.440952\pi$
$6$		0			0
$7$	−5.50000	+	9.52628i	−0.296972	+	0.514371i	−0.975442	−	0.220258i	$-0.929310\pi$
$7$	−5.50000	+	9.52628i	−0.296972	+	0.514371i	0.678470	+	0.734628i	$0.262644\pi$
$8$	−8.48528			−0.375000
$9$		0			0
$10$	−72.0000			−2.27684
$11$	8.48528	−	14.6969i	0.232583	−	0.402845i	−0.725985	−	0.687711i	$-0.758616\pi$
$11$	8.48528	−	14.6969i	0.232583	−	0.402845i	0.958567	+	0.284866i	$0.0919491\pi$
$12$		0			0
$13$	−14.5000	−	25.1147i	−0.309352	−	0.535813i	0.668869	−	0.743381i	$-0.266779\pi$
$13$	−14.5000	−	25.1147i	−0.309352	−	0.535813i	−0.978221	+	0.207567i	$0.933445\pi$
$14$	23.3345	+	40.4166i	0.445458	+	0.771556i
$15$		0			0
$16$	22.0000	−	38.1051i	0.343750	−	0.595392i
$17$	50.9117			0.726347			0.363173	−	0.931722i	$-0.381693\pi$
$17$	50.9117			0.726347			0.363173	+	0.931722i	$0.381693\pi$
$18$		0			0
$19$	29.0000			0.350161			0.175080	−	0.984554i	$-0.443981\pi$
$19$	29.0000			0.350161			0.175080	+	0.984554i	$0.443981\pi$
$20$	−84.8528	+	146.969i	−0.948683	+	1.64317i
$21$		0			0
$22$	−36.0000	−	62.3538i	−0.348874	−	0.604267i
$23$	42.4264	+	73.4847i	0.384631	+	0.666201i	0.991718	−	0.128435i	$-0.0409953\pi$
$23$	42.4264	+	73.4847i	0.384631	+	0.666201i	−0.607087	+	0.794636i	$0.707662\pi$
$24$		0			0
$25$	−81.5000	+	141.162i	−0.652000	+	1.12930i
$26$	−123.037			−0.928056
$27$		0			0
$28$	110.000			0.742430
$29$	135.765	−	235.151i	0.869339	−	1.50574i	0.00666596	−	0.999978i	$-0.497878\pi$
$29$	135.765	−	235.151i	0.869339	−	1.50574i	0.862673	−	0.505762i	$-0.168789\pi$
$30$		0			0
$31$	134.000	+	232.095i	0.776358	+	1.34469i	0.934028	+	0.357200i	$0.116269\pi$
$31$	134.000	+	232.095i	0.776358	+	1.34469i	−0.157669	+	0.987492i	$0.550398\pi$
$32$	−127.279	−	220.454i	−0.703125	−	1.21785i
$33$		0			0
$34$	108.000	−	187.061i	0.544760	−	0.943552i
$35$	186.676			0.901544
$36$		0			0
$37$	83.0000			0.368787			0.184393	−	0.982853i	$-0.440968\pi$
$37$	83.0000			0.368787			0.184393	+	0.982853i	$0.440968\pi$
$38$	61.5183	−	106.553i	0.262621	−	0.454872i
$39$		0			0
$40$	72.0000	+	124.708i	0.284605	+	0.492950i
$41$	−135.765	−	235.151i	−0.517143	−	0.895718i	−0.999802	−	0.0199092i	$-0.993662\pi$
$41$	−135.765	−	235.151i	−0.517143	−	0.895718i	0.482659	−	0.875808i	$-0.339671\pi$
$42$		0			0
$43$	116.000	−	200.918i	0.411391	−	0.712551i	−0.583651	−	0.812005i	$-0.698376\pi$
$43$	116.000	−	200.918i	0.411391	−	0.712551i	0.995042	+	0.0994539i	$0.0317096\pi$
$44$	−169.706			−0.581456
$45$		0			0
$46$	360.000			1.15389
$47$	−195.161	+	338.030i	−0.605686	+	1.04908i	0.386257	+	0.922391i	$0.373768\pi$
$47$	−195.161	+	338.030i	−0.605686	+	1.04908i	−0.991943	+	0.126687i	$0.959566\pi$
$48$		0			0
$49$	111.000	+	192.258i	0.323615	+	0.560518i
$50$	345.775	+	598.900i	0.978000	+	1.69395i
$51$		0			0
$52$	−145.000	+	251.147i	−0.386690	+	0.669767i
$53$	−305.470			−0.791690			−0.395845	−	0.918317i	$-0.629548\pi$
$53$	−305.470			−0.791690			−0.395845	+	0.918317i	$0.629548\pi$
$54$		0			0
$55$	−288.000			−0.706071
$56$	46.6690	−	80.8332i	0.111365	−	0.192889i
$57$		0			0
$58$	−576.000	−	997.661i	−1.30401	−	2.25861i
$59$	144.250	+	249.848i	0.318300	+	0.551312i	0.980134	−	0.198339i	$-0.0635546\pi$
$59$	144.250	+	249.848i	0.318300	+	0.551312i	−0.661833	+	0.749651i	$0.730221\pi$
$60$		0			0
$61$	−383.500	+	664.241i	−0.804953	+	1.39422i	0.111369	+	0.993779i	$0.464476\pi$
$61$	−383.500	+	664.241i	−0.804953	+	1.39422i	−0.916323	+	0.400441i	$0.868857\pi$
$62$	1137.03			2.32908
$63$		0			0
$64$	−728.000			−1.42188
$65$	−246.073	+	426.211i	−0.469563	+	0.813308i
$66$		0			0
$67$	255.500	+	442.539i	0.465885	+	0.806936i	0.999241	−	0.0389544i	$-0.0124027\pi$
$67$	255.500	+	442.539i	0.465885	+	0.806936i	−0.533356	+	0.845891i	$0.679069\pi$
$68$	−254.558	−	440.908i	−0.453967	−	0.786294i
$69$		0			0
$70$	396.000	−	685.892i	0.676158	−	1.17114i
$71$	−712.764			−1.19140			−0.595701	−	0.803207i	$-0.703126\pi$
$71$	−712.764			−1.19140			−0.595701	+	0.803207i	$0.703126\pi$
$72$		0			0
$73$	137.000			0.219653			0.109826	−	0.993951i	$-0.464971\pi$
$73$	137.000			0.219653			0.109826	+	0.993951i	$0.464971\pi$
$74$	176.070	−	304.961i	0.276590	−	0.479068i
$75$		0			0
$76$	−145.000	−	251.147i	−0.218851	−	0.379060i
$77$	93.3381	+	161.666i	0.138141	+	0.239267i
$78$		0			0
$79$	237.500	−	411.362i	0.338238	−	0.585846i	−0.645863	−	0.763453i	$-0.723502\pi$
$79$	237.500	−	411.362i	0.338238	−	0.585846i	0.984101	+	0.177607i	$0.0568356\pi$
$80$	−746.705			−1.04355
$81$		0			0
$82$	−1152.00			−1.55143
$83$	288.500	−	499.696i	0.381529	−	0.660828i	−0.609752	−	0.792593i	$-0.708731\pi$
$83$	288.500	−	499.696i	0.381529	−	0.660828i	0.991281	+	0.131764i	$0.0420642\pi$
$84$		0			0
$85$	−432.000	−	748.246i	−0.551259	−	0.954808i
$86$	−492.146	−	852.422i	−0.617087	−	1.06883i
$87$		0			0
$88$	−72.0000	+	124.708i	−0.0872185	+	0.151067i
$89$	254.558			0.303181			0.151591	−	0.988443i	$-0.451560\pi$
$89$	254.558			0.303181			0.151591	+	0.988443i	$0.451560\pi$
$90$		0			0
$91$	319.000			0.367476
$92$	424.264	−	734.847i	0.480789	−	0.832751i
$93$		0			0
$94$	828.000	+	1434.14i	0.908529	+	1.57362i
$95$	−246.073	−	426.211i	−0.265753	−	0.460298i
$96$		0			0
$97$	−410.500	+	711.007i	−0.429690	+	0.744245i	−0.996846	−	0.0793654i	$-0.974711\pi$
$97$	−410.500	+	711.007i	−0.429690	+	0.744245i	0.567155	+	0.823611i	$0.308044\pi$
$98$	941.866			0.970845
$99$		0			0
$100$	1630.00			1.63000
$101$	−271.529	+	470.302i	−0.267506	+	0.463335i	−0.968217	−	0.250111i	$-0.919533\pi$
$101$	−271.529	+	470.302i	−0.267506	+	0.463335i	0.700711	+	0.713445i	$0.252866\pi$
$102$		0			0
$103$	−419.500	−	726.595i	−0.401306	−	0.695083i	0.592577	−	0.805513i	$-0.298110\pi$
$103$	−419.500	−	726.595i	−0.401306	−	0.695083i	−0.993884	+	0.110430i	$0.964777\pi$
$104$	123.037	+	213.106i	0.116007	+	0.200930i
$105$		0			0
$106$	−648.000	+	1122.37i	−0.593767	+	1.02843i
$107$	763.675			0.689975			0.344987	−	0.938607i	$-0.387883\pi$
$107$	763.675			0.689975			0.344987	+	0.938607i	$0.387883\pi$
$108$		0			0
$109$	218.000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	218.000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$	−610.940	+	1058.18i	−0.529553	+	0.917213i
$111$		0			0
$112$	242.000	+	419.156i	0.204168	+	0.353630i
$113$	755.190	+	1308.03i	0.628693	+	1.08893i	0.987814	+	0.155637i	$0.0497431\pi$
$113$	755.190	+	1308.03i	0.628693	+	1.08893i	−0.359121	+	0.933291i	$0.616924\pi$
$114$		0			0
$115$	720.000	−	1247.08i	0.583829	−	1.01122i
$116$	−2715.29			−2.17335
$117$		0			0
$118$	1224.00			0.954901
$119$	−280.014	+	484.999i	−0.215705	+	0.373612i
$120$		0			0
$121$	521.500	+	903.264i	0.391811	+	0.678636i
$122$	1627.05	+	2818.14i	1.20743	+	2.09133i
$123$		0			0
$124$	1340.00	−	2320.95i	0.970448	−	1.68087i
$125$	644.881			0.461440
$126$		0			0
$127$	1244.00			0.869190			0.434595	−	0.900626i	$-0.356891\pi$
$127$	1244.00			0.869190			0.434595	+	0.900626i	$0.356891\pi$
$128$	−526.087	+	911.210i	−0.363281	+	0.629222i
$129$		0			0
$130$	1044.00	+	1808.26i	0.704345	+	1.21996i
$131$	−1255.82	−	2175.15i	−0.837570	−	1.45071i	−0.891921	−	0.452192i	$-0.850642\pi$
$131$	−1255.82	−	2175.15i	−0.837570	−	1.45071i	0.0543510	−	0.998522i	$-0.482691\pi$
$132$		0			0
$133$	−159.500	+	276.262i	−0.103988	+	0.180112i
$134$	2167.99			1.39765
$135$		0			0
$136$	−432.000			−0.272380
$137$	670.337	−	1161.06i	0.418035	−	0.724058i	−0.577707	−	0.816244i	$-0.696052\pi$
$137$	670.337	−	1161.06i	0.418035	−	0.724058i	0.995742	+	0.0921867i	$0.0293857\pi$
$138$		0			0
$139$	−473.500	−	820.126i	−0.288933	−	0.500447i	0.684622	−	0.728898i	$-0.259967\pi$
$139$	−473.500	−	820.126i	−0.288933	−	0.500447i	−0.973555	+	0.228451i	$0.926634\pi$
$140$	−933.381	−	1616.66i	−0.563465	−	0.975950i
$141$		0			0
$142$	−1512.00	+	2618.86i	−0.893551	+	1.54768i
$143$	−492.146			−0.287800
$144$		0			0
$145$	−4608.00			−2.63913
$146$	290.621	−	503.370i	0.164739	−	0.285337i
$147$		0			0
$148$	−415.000	−	718.801i	−0.230492	−	0.399224i
$149$	−288.500	−	499.696i	−0.158623	−	0.274743i	0.775749	−	0.631041i	$-0.217372\pi$
$149$	−288.500	−	499.696i	−0.158623	−	0.274743i	−0.934372	+	0.356298i	$0.884039\pi$
$150$		0			0
$151$	1155.50	−	2001.38i	0.622737	−	1.07861i	−0.366237	−	0.930522i	$-0.619354\pi$
$151$	1155.50	−	2001.38i	0.622737	−	1.07861i	0.988974	−	0.148090i	$-0.0473126\pi$
$152$	−246.073			−0.131310
$153$		0			0
$154$	792.000			0.414423
$155$	2274.06	−	3938.78i	1.17843	−	2.04110i
$156$		0			0
$157$	−811.000	−	1404.69i	−0.412260	−	0.714056i	0.582876	−	0.812561i	$-0.301927\pi$
$157$	−811.000	−	1404.69i	−0.412260	−	0.714056i	−0.995137	+	0.0985053i	$0.968594\pi$
$158$	−1007.63	−	1745.26i	−0.507358	−	0.878769i
$159$		0			0
$160$	−2160.00	+	3741.23i	−1.06727	+	1.84856i
$161$	−933.381			−0.456899
$162$		0			0
$163$	2243.00			1.07782			0.538912	−	0.842362i	$-0.318836\pi$
$163$	2243.00			1.07782			0.538912	+	0.842362i	$0.318836\pi$
$164$	−1357.65	+	2351.51i	−0.646428	+	1.11965i
$165$		0			0
$166$	−1224.00	−	2120.03i	−0.572294	−	0.991242i
$167$	1366.13	+	2366.21i	0.633020	+	1.09642i	0.986931	+	0.161144i	$0.0515182\pi$
$167$	1366.13	+	2366.21i	0.633020	+	1.09642i	−0.353911	+	0.935279i	$0.615148\pi$
$168$		0			0
$169$	678.000	−	1174.33i	0.308603	−	0.534515i
$170$	−3665.64			−1.65378
$171$		0			0
$172$	−2320.00			−1.02848
$173$	−678.823	+	1175.76i	−0.298323	+	0.516711i	−0.975752	−	0.218877i	$-0.929761\pi$
$173$	−678.823	+	1175.76i	−0.298323	+	0.516711i	0.677429	+	0.735588i	$0.263094\pi$
$174$		0			0
$175$	−896.500	−	1552.78i	−0.387252	−	0.670739i
$176$	−373.352	−	646.665i	−0.159901	−	0.276956i
$177$		0			0
$178$	540.000	−	935.307i	0.227386	−	0.393844i
$179$	2341.94			0.977903			0.488952	−	0.872311i	$-0.337379\pi$
$179$	2341.94			0.977903			0.488952	+	0.872311i	$0.337379\pi$
$180$		0			0
$181$	−1591.00			−0.653360			−0.326680	−	0.945135i	$-0.605930\pi$
$181$	−1591.00			−0.653360			−0.326680	+	0.945135i	$0.605930\pi$
$182$	676.701	−	1172.08i	0.275607	−	0.477365i
$183$		0			0
$184$	−360.000	−	623.538i	−0.144237	−	0.249825i
$185$	−704.278	−	1219.85i	−0.279890	−	0.484783i
$186$		0			0
$187$	432.000	−	748.246i	0.168936	−	0.292605i
$188$	3903.23			1.51421
$189$		0			0
$190$	−2088.00			−0.797260
$191$	−2384.36	+	4129.84i	−0.903280	+	1.56453i	−0.0800710	+	0.996789i	$0.525515\pi$
$191$	−2384.36	+	4129.84i	−0.903280	+	1.56453i	−0.823209	+	0.567738i	$0.807819\pi$
$192$		0			0
$193$	1740.50	+	3014.63i	0.649140	+	1.12434i	0.983329	+	0.181837i	$0.0582042\pi$
$193$	1740.50	+	3014.63i	0.649140	+	1.12434i	−0.334189	+	0.942506i	$0.608462\pi$
$194$	1741.60	+	3016.55i	0.644535	+	1.11637i
$195$		0			0
$196$	1110.00	−	1922.58i	0.404519	−	0.700647i
$197$	2392.85			0.865398			0.432699	−	0.901538i	$-0.357561\pi$
$197$	2392.85			0.865398			0.432699	+	0.901538i	$0.357561\pi$
$198$		0			0
$199$	2351.00			0.837477			0.418739	−	0.908107i	$-0.362472\pi$
$199$	2351.00			0.837477			0.418739	+	0.908107i	$0.362472\pi$
$200$	691.550	−	1197.80i	0.244500	−	0.423486i
$201$		0			0
$202$	1152.00	+	1995.32i	0.401260	+	0.695002i
$203$	1493.41	+	2586.66i	0.516339	+	0.894325i
$204$		0			0
$205$	−2304.00	+	3990.65i	−0.784968	+	1.35960i
$206$	−3559.58			−1.20392
$207$		0			0
$208$	−1276.00			−0.425359
$209$	246.073	−	426.211i	0.0814413	−	0.141061i
$210$		0			0
$211$	−851.500	−	1474.84i	−0.277818	−	0.481196i	0.693024	−	0.720915i	$-0.256278\pi$
$211$	−851.500	−	1474.84i	−0.277818	−	0.481196i	−0.970842	+	0.239719i	$0.922945\pi$
$212$	1527.35	+	2645.45i	0.494806	+	0.857029i
$213$		0			0
$214$	1620.00	−	2805.92i	0.517481	−	0.896303i
$215$	−3937.17			−1.24890
$216$		0			0
$217$	−2948.00			−0.922227
$218$	462.448	−	800.983i	0.143674	−	0.248851i
$219$		0			0
$220$	1440.00	+	2494.15i	0.441294	+	0.764344i
$221$	−738.219	−	1278.63i	−0.224697	−	0.389186i
$222$		0			0
$223$	−694.000	+	1202.04i	−0.208402	+	0.360963i	−0.951211	−	0.308540i	$-0.900160\pi$
$223$	−694.000	+	1202.04i	−0.208402	+	0.360963i	0.742809	+	0.669503i	$0.233493\pi$
$224$	2800.14			0.835234
$225$		0			0
$226$	6408.00			1.88608
$227$	−2358.91	+	4085.75i	−0.689719	+	1.19463i	0.282210	+	0.959353i	$0.408933\pi$
$227$	−2358.91	+	4085.75i	−0.689719	+	1.19463i	−0.971929	+	0.235276i	$0.924401\pi$
$228$		0			0
$229$	−217.000	−	375.855i	−0.0626191	−	0.108459i	0.833016	−	0.553248i	$-0.186612\pi$
$229$	−217.000	−	375.855i	−0.0626191	−	0.108459i	−0.895635	+	0.444789i	$0.853279\pi$
$230$	−3054.70	−	5290.90i	−0.875744	−	1.51683i
$231$		0			0
$232$	−1152.00	+	1995.32i	−0.326002	+	0.564652i
$233$	−3461.99			−0.973403			−0.486701	−	0.873568i	$-0.661800\pi$
$233$	−3461.99			−0.973403			−0.486701	+	0.873568i	$0.661800\pi$
$234$		0			0
$235$	6624.00			1.83873
$236$	1442.50	−	2498.48i	0.397875	−	0.689141i
$237$		0			0
$238$	1188.00	+	2057.68i	0.323557	+	0.560417i
$239$	−1408.56	−	2439.69i	−0.381222	−	0.660295i	0.610016	−	0.792389i	$-0.291163\pi$
$239$	−1408.56	−	2439.69i	−0.381222	−	0.660295i	−0.991237	+	0.132094i	$0.957830\pi$
$240$		0			0
$241$	1047.50	−	1814.32i	0.279981	−	0.484941i	−0.691399	−	0.722473i	$-0.743005\pi$
$241$	1047.50	−	1814.32i	0.279981	−	0.484941i	0.971380	+	0.237532i	$0.0763385\pi$
$242$	4425.07			1.17543
$243$		0			0
$244$	7670.00			2.01238
$245$	1883.73	−	3262.72i	0.491213	−	0.850806i
$246$		0			0
$247$	−420.500	−	728.327i	−0.108323	−	0.187621i
$248$	−1137.03	−	1969.39i	−0.291134	−	0.504260i
$249$		0			0
$250$	1368.00	−	2369.45i	0.346080	−	0.599428i
$251$	−203.647			−0.0512114			−0.0256057	−	0.999672i	$-0.508151\pi$
$251$	−203.647			−0.0512114			−0.0256057	+	0.999672i	$0.508151\pi$
$252$		0			0
$253$	1440.00			0.357834
$254$	2638.92	−	4570.75i	0.651893	−	1.12911i
$255$		0			0
$256$	−680.000	−	1177.79i	−0.166016	−	0.287547i
$257$	−899.440	−	1557.88i	−0.218309	−	0.378123i	0.735982	−	0.677001i	$-0.236721\pi$
$257$	−899.440	−	1557.88i	−0.218309	−	0.378123i	−0.954291	+	0.298878i	$0.903388\pi$
$258$		0			0
$259$	−456.500	+	790.681i	−0.109519	+	0.189693i
$260$	4921.46			1.17391
$261$		0			0
$262$	−10656.0			−2.51271
$263$	186.676	−	323.333i	0.0437679	−	0.0758082i	−0.843312	−	0.537425i	$-0.819397\pi$
$263$	186.676	−	323.333i	0.0437679	−	0.0758082i	0.887079	+	0.461617i	$0.152730\pi$
$264$		0			0
$265$	2592.00	+	4489.48i	0.600850	+	1.04070i
$266$	676.701	+	1172.08i	0.155982	+	0.270169i
$267$		0			0
$268$	2555.00	−	4425.39i	0.582356	−	1.00867i
$269$	−7484.02			−1.69631			−0.848157	−	0.529744i	$-0.822288\pi$
$269$	−7484.02			−1.69631			−0.848157	+	0.529744i	$0.822288\pi$
$270$		0			0
$271$	−3319.00			−0.743966			−0.371983	−	0.928239i	$-0.621322\pi$
$271$	−3319.00			−0.743966			−0.371983	+	0.928239i	$0.621322\pi$
$272$	1120.06	−	1940.00i	0.249682	−	0.432462i
$273$		0			0
$274$	−2844.00	−	4925.95i	−0.627052	−	1.08609i
$275$	1383.10	+	2395.60i	0.303288	+	0.525310i
$276$		0			0
$277$	−4177.00	+	7234.78i	−0.906035	+	1.56930i	−0.0865124	+	0.996251i	$0.527572\pi$
$277$	−4177.00	+	7234.78i	−0.906035	+	1.56930i	−0.819522	+	0.573047i	$0.805761\pi$
$278$	−4017.78			−0.866800
$279$		0			0
$280$	−1584.00			−0.338079
$281$	−1289.76	+	2233.93i	−0.273811	+	0.474254i	−0.969834	−	0.243765i	$-0.921617\pi$
$281$	−1289.76	+	2233.93i	−0.273811	+	0.474254i	0.696024	+	0.718019i	$0.254951\pi$
$282$		0			0
$283$	3104.00	+	5376.29i	0.651992	+	1.12928i	0.982639	+	0.185528i	$0.0593996\pi$
$283$	3104.00	+	5376.29i	0.651992	+	1.12928i	−0.330647	+	0.943754i	$0.607267\pi$
$284$	3563.82	+	6172.71i	0.744626	+	1.28973i
$285$		0			0
$286$	−1044.00	+	1808.26i	−0.215850	+	0.373863i
$287$	2986.82			0.614308
$288$		0			0
$289$	−2321.00			−0.472420
$290$	−9775.04	+	16930.9i	−1.97935	+	3.42833i
$291$		0			0
$292$	−685.000	−	1186.45i	−0.137283	−	0.237781i
$293$	3097.13	+	5364.38i	0.617529	+	1.06959i	0.989935	+	0.141522i	$0.0451997\pi$
$293$	3097.13	+	5364.38i	0.617529	+	1.06959i	−0.372406	+	0.928070i	$0.621467\pi$
$294$		0			0
$295$	2448.00	−	4240.06i	0.483146	−	0.836833i
$296$	−704.278			−0.138295
$297$		0			0
$298$	−2448.00			−0.475869
$299$	1230.37	−	2131.06i	0.237973	−	0.412181i
$300$		0			0
$301$	1276.00	+	2210.10i	0.244344	+	0.423215i
$302$	−4902.37	−	8491.16i	−0.934105	−	1.61792i
$303$		0			0
$304$	638.000	−	1105.05i	0.120368	−	0.208483i
$305$	13016.4			2.44367
$306$		0			0
$307$	−2320.00			−0.431301			−0.215650	−	0.976471i	$-0.569187\pi$
$307$	−2320.00			−0.431301			−0.215650	+	0.976471i	$0.569187\pi$
$308$	933.381	−	1616.66i	0.172676	−	0.299084i
$309$		0			0
$310$	−9648.00	−	16710.8i	−1.76764	−	3.06165i
$311$	398.808	+	690.756i	0.0727149	+	0.125946i	0.900090	−	0.435703i	$-0.143500\pi$
$311$	398.808	+	690.756i	0.0727149	+	0.125946i	−0.827375	+	0.561649i	$0.810167\pi$
$312$		0			0
$313$	−653.500	+	1131.90i	−0.118013	+	0.204404i	−0.918980	−	0.394304i	$-0.870986\pi$
$313$	−653.500	+	1131.90i	−0.118013	+	0.204404i	0.800967	+	0.598708i	$0.204319\pi$
$314$	−6881.56			−1.23678
$315$		0			0
$316$	−4750.00			−0.845596
$317$	−1035.20	+	1793.03i	−0.183416	+	0.317686i	−0.943042	−	0.332675i	$-0.892049\pi$
$317$	−1035.20	+	1793.03i	−0.183416	+	0.317686i	0.759626	+	0.650361i	$0.225382\pi$
$318$		0			0
$319$	−2304.00	−	3990.65i	−0.404386	−	0.700418i
$320$	6177.28	+	10699.4i	1.07913	+	1.86910i
$321$		0			0
$322$	−1980.00	+	3429.46i	−0.342674	+	0.593529i
$323$	1476.44			0.254338
$324$		0			0
$325$	4727.00			0.806790
$326$	4758.12	−	8241.31i	0.808368	−	1.40013i
$327$		0			0
$328$	1152.00	+	1995.32i	0.193929	+	0.335894i
$329$	−2146.78	−	3718.33i	−0.359743	−	0.623094i
$330$		0			0
$331$	2586.50	−	4479.95i	0.429507	−	0.743928i	−0.567322	−	0.823496i	$-0.692021\pi$
$331$	2586.50	−	4479.95i	0.429507	−	0.743928i	0.996829	+	0.0795675i	$0.0253539\pi$
$332$	−5769.99			−0.953824
$333$		0			0
$334$	11592.0			1.89906
$335$	4335.98	−	7510.14i	0.707164	−	1.22484i
$336$		0			0
$337$	−1310.50	−	2269.85i	−0.211832	−	0.366904i	0.740456	−	0.672105i	$-0.234610\pi$
$337$	−1310.50	−	2269.85i	−0.211832	−	0.366904i	−0.952288	+	0.305201i	$0.901276\pi$
$338$	−2876.51	−	4982.26i	−0.462904	−	0.801773i
$339$		0			0
$340$	−4320.00	+	7482.46i	−0.689073	+	1.19351i
$341$	4548.11			0.722270
$342$		0			0
$343$	−6215.00			−0.978363
$344$	−984.293	+	1704.84i	−0.154272	+	0.267207i
$345$		0			0
$346$	2880.00	+	4988.31i	0.447485	+	0.775067i
$347$	−3852.32	−	6672.41i	−0.595975	−	1.03226i	−0.993409	−	0.114628i	$-0.963432\pi$
$347$	−3852.32	−	6672.41i	−0.595975	−	1.03226i	0.397434	−	0.917631i	$-0.369901\pi$
$348$		0			0
$349$	−977.500	+	1693.08i	−0.149927	+	0.259680i	−0.931200	−	0.364508i	$-0.881237\pi$
$349$	−977.500	+	1693.08i	−0.149927	+	0.259680i	0.781274	+	0.624189i	$0.214570\pi$
$350$	−7607.05			−1.16175
$351$		0			0
$352$	−4320.00			−0.654139
$353$	593.970	−	1028.79i	0.0895576	−	0.155118i	−0.817767	−	0.575550i	$-0.804788\pi$
$353$	593.970	−	1028.79i	0.0895576	−	0.155118i	0.907324	+	0.420432i	$0.138121\pi$
$354$		0			0
$355$	6048.00	+	10475.4i	0.904210	+	1.56614i
$356$	−1272.79	−	2204.54i	−0.189488	−	0.328203i
$357$		0			0
$358$	4968.00	−	8604.83i	0.733427	−	1.27033i
$359$	560.029			0.0823320			0.0411660	−	0.999152i	$-0.486893\pi$
$359$	560.029			0.0823320			0.0411660	+	0.999152i	$0.486893\pi$
$360$		0			0
$361$	−6018.00			−0.877387
$362$	−3375.02	+	5845.71i	−0.490020	+	0.848739i
$363$		0			0
$364$	−1595.00	−	2762.62i	−0.229672	−	0.397804i
$365$	−1162.48	−	2013.48i	−0.166705	−	0.288741i
$366$		0			0
$367$	−3947.50	+	6837.27i	−0.561465	+	0.972487i	0.435903	+	0.899993i	$0.356429\pi$
$367$	−3947.50	+	6837.27i	−0.561465	+	0.972487i	−0.997369	+	0.0724933i	$0.976904\pi$
$368$	3733.52			0.528868
$369$		0			0
$370$	−5976.00			−0.839669
$371$	1680.09	−	2909.99i	0.235110	−	0.407222i
$372$		0			0
$373$	−4901.50	−	8489.65i	−0.680402	−	1.17849i	−0.974858	−	0.222826i	$-0.928472\pi$
$373$	−4901.50	−	8489.65i	−0.680402	−	1.17849i	0.294456	−	0.955665i	$-0.404862\pi$
$374$	−1832.82	−	3174.54i	−0.253403	−	0.438908i
$375$		0			0
$376$	1656.00	−	2868.28i	0.227132	−	0.393404i
$377$	−7874.34			−1.07573
$378$		0			0
$379$	10505.0			1.42376			0.711881	−	0.702300i	$-0.247844\pi$
$379$	10505.0			1.42376			0.711881	+	0.702300i	$0.247844\pi$
$380$	−2460.73	+	4262.11i	−0.332192	+	0.575373i
$381$		0			0
$382$	10116.0	+	17521.4i	1.35492	+	2.34679i
$383$	−543.058	−	940.604i	−0.0724516	−	0.125490i	0.827524	−	0.561431i	$-0.189749\pi$
$383$	−543.058	−	940.604i	−0.0724516	−	0.125490i	−0.899975	+	0.435941i	$0.856416\pi$
$384$		0			0
$385$	1584.00	−	2743.57i	0.209683	−	0.363182i
$386$	14768.6			1.94742
$387$		0			0
$388$	8210.00			1.07423
$389$	1077.63	−	1866.51i	0.140458	−	0.243280i	−0.787211	−	0.616683i	$-0.788476\pi$
$389$	1077.63	−	1866.51i	0.140458	−	0.243280i	0.927669	+	0.373403i	$0.121809\pi$
$390$		0			0
$391$	2160.00	+	3741.23i	0.279376	+	0.483893i
$392$	−941.866	−	1631.36i	−0.121356	−	0.210194i
$393$		0			0
$394$	5076.00	−	8791.89i	0.649049	−	1.12419i
$395$	−8061.02			−1.02682
$396$		0			0
$397$	12422.0			1.57038			0.785192	−	0.619253i	$-0.212564\pi$
$397$	12422.0			1.57038			0.785192	+	0.619253i	$0.212564\pi$
$398$	4987.22	−	8638.13i	0.628108	−	1.08791i
$399$		0			0
$400$	3586.00	+	6211.13i	0.448250	+	0.776392i
$401$	7755.55	+	13433.0i	0.965819	+	1.67285i	0.707398	+	0.706816i	$0.249869\pi$
$401$	7755.55	+	13433.0i	0.965819	+	1.67285i	0.258422	+	0.966032i	$0.416798\pi$
$402$		0			0
$403$	3886.00	−	6730.75i	0.480336	−	0.831967i
$404$	5430.58			0.668766
$405$		0			0
$406$	12672.0			1.54902
$407$	704.278	−	1219.85i	0.0857734	−	0.148564i
$408$		0			0
$409$	−3632.50	−	6291.67i	−0.439158	−	0.760644i	0.558467	−	0.829527i	$-0.311390\pi$
$409$	−3632.50	−	6291.67i	−0.439158	−	0.760644i	−0.997625	+	0.0688831i	$0.978056\pi$
$410$	9775.04	+	16930.9i	1.17745	+	2.03941i
$411$		0			0
$412$	−4195.00	+	7265.95i	−0.501633	+	0.868854i
$413$	−3173.50			−0.378105
$414$		0			0
$415$	−9792.00			−1.15824
$416$	−3691.10	+	6393.17i	−0.435026	+	0.753488i
$417$		0			0
$418$	−1044.00	−	1808.26i	−0.122162	−	0.211591i
$419$	−1586.75	−	2748.33i	−0.185006	−	0.320441i	0.758572	−	0.651589i	$-0.225897\pi$
$419$	−1586.75	−	2748.33i	−0.185006	−	0.320441i	−0.943579	+	0.331148i	$0.892564\pi$
$420$		0			0
$421$	−1706.50	+	2955.74i	−0.197553	+	0.342171i	−0.947734	−	0.319060i	$-0.896633\pi$
$421$	−1706.50	+	2955.74i	−0.197553	+	0.342171i	0.750182	+	0.661232i	$0.229966\pi$
$422$	−7225.22			−0.833455
$423$		0			0
$424$	2592.00			0.296884
$425$	−4149.30	+	7186.80i	−0.473578	+	0.820262i
$426$		0			0
$427$	−4218.50	−	7306.66i	−0.478097	−	0.828089i
$428$	−3818.38	−	6613.62i	−0.431234	−	0.746919i
$429$		0			0
$430$	−8352.00	+	14466.1i	−0.936673	+	1.62236i
$431$	12677.0			1.41678			0.708388	−	0.705824i	$-0.249423\pi$
$431$	12677.0			1.41678			0.708388	+	0.705824i	$0.249423\pi$
$432$		0			0
$433$	8642.00			0.959141			0.479570	−	0.877503i	$-0.340792\pi$
$433$	8642.00			0.959141			0.479570	+	0.877503i	$0.340792\pi$
$434$	−6253.65	+	10831.6i	−0.691670	+	1.19801i
$435$		0			0
$436$	−1090.00	−	1887.94i	−0.119728	−	0.207375i
$437$	1230.37	+	2131.06i	0.134683	+	0.233277i
$438$		0			0
$439$	−262.000	+	453.797i	−0.0284842	+	0.0493361i	−0.879916	−	0.475129i	$-0.842401\pi$
$439$	−262.000	+	453.797i	−0.0284842	+	0.0493361i	0.851432	+	0.524465i	$0.175735\pi$
$440$	2443.76			0.264777
$441$		0			0
$442$	−6264.00			−0.674091
$443$	−9079.25	+	15725.7i	−0.973743	+	1.68657i	−0.289722	+	0.957111i	$0.593563\pi$
$443$	−9079.25	+	15725.7i	−0.973743	+	1.68657i	−0.684022	+	0.729462i	$0.739771\pi$
$444$		0			0
$445$	−2160.00	−	3741.23i	−0.230098	−	0.398542i
$446$	2944.39	+	5099.84i	0.312603	+	0.541445i
$447$		0			0
$448$	4004.00	−	6935.13i	0.422257	−	0.731371i
$449$	−3309.26			−0.347825			−0.173913	−	0.984761i	$-0.555641\pi$
$449$	−3309.26			−0.347825			−0.173913	+	0.984761i	$0.555641\pi$
$450$		0			0
$451$	−4608.00			−0.481114
$452$	7551.90	−	13080.3i	0.785866	−	1.36116i
$453$		0			0
$454$	10008.0	+	17334.4i	1.03458	+	1.79194i
$455$	−2706.80	−	4688.32i	−0.278894	−	0.483059i
$456$		0			0
$457$	4733.00	−	8197.80i	0.484465	−	0.839118i	−0.515376	−	0.856964i	$-0.672348\pi$
$457$	4733.00	−	8197.80i	0.484465	−	0.839118i	0.999841	+	0.0178466i	$0.00568105\pi$
$458$	−1841.31			−0.187857
$459$		0			0
$460$	−14400.0			−1.45957
$461$	−1620.69	+	2807.12i	−0.163738	+	0.283602i	−0.936206	−	0.351451i	$-0.885688\pi$
$461$	−1620.69	+	2807.12i	−0.163738	+	0.283602i	0.772469	+	0.635053i	$0.219022\pi$
$462$		0			0
$463$	−5657.50	−	9799.08i	−0.567875	−	0.983589i	−0.996776	−	0.0802373i	$-0.974432\pi$
$463$	−5657.50	−	9799.08i	−0.567875	−	0.983589i	0.428900	−	0.903352i	$-0.358901\pi$
$464$	−5973.64	−	10346.6i	−0.597671	−	1.03520i
$465$		0			0
$466$	−7344.00	+	12720.2i	−0.730052	+	1.26449i
$467$	−17462.7			−1.73036			−0.865180	−	0.501462i	$-0.832796\pi$
$467$	−17462.7			−1.73036			−0.865180	+	0.501462i	$0.832796\pi$
$468$		0			0
$469$	−5621.00			−0.553419
$470$	14051.6	−	24338.1i	1.37905	−	2.38858i
$471$		0			0
$472$	−1224.00	−	2120.03i	−0.119363	−	0.206742i
$473$	−1968.59	−	3409.69i	−0.191365	−	0.331454i
$474$		0			0
$475$	−2363.50	+	4093.70i	−0.228305	+	0.395436i
$476$	5600.29			0.539262
$477$		0			0
$478$	−11952.0			−1.14366
$479$	4463.26	−	7730.59i	0.425744	−	0.737411i	−0.570745	−	0.821127i	$-0.693346\pi$
$479$	4463.26	−	7730.59i	0.425744	−	0.737411i	0.996490	+	0.0837165i	$0.0266790\pi$
$480$		0			0
$481$	−1203.50	−	2084.52i	−0.114085	−	0.197601i
$482$	−4444.17	−	7697.52i	−0.419971	−	0.727412i
$483$		0			0
$484$	5215.00	−	9032.64i	0.489763	−	0.848295i
$485$	13932.8			1.30445
$486$		0			0
$487$	−18493.0			−1.72073			−0.860367	−	0.509674i	$-0.829766\pi$
$487$	−18493.0			−1.72073			−0.860367	+	0.509674i	$0.829766\pi$
$488$	3254.11	−	5636.28i	0.301857	−	0.522832i
$489$		0			0
$490$	−7992.00	−	13842.6i	−0.736820	−	1.27621i
$491$	6406.39	+	11096.2i	0.588831	+	1.01989i	0.994386	+	0.105815i	$0.0337451\pi$
$491$	6406.39	+	11096.2i	0.588831	+	1.01989i	−0.405554	+	0.914071i	$0.632922\pi$
$492$		0			0
$493$	6912.00	−	11971.9i	0.631442	−	1.09369i
$494$	−3568.06			−0.324969
$495$		0			0
$496$	11792.0			1.06749
$497$	3920.20	−	6789.99i	0.353813	−	0.612822i
$498$		0			0
$499$	6128.00	+	10614.0i	0.549753	+	0.952201i	0.998291	+	0.0584369i	$0.0186117\pi$
$499$	6128.00	+	10614.0i	0.549753	+	0.952201i	−0.448538	+	0.893764i	$0.648055\pi$
$500$	−3224.41	−	5584.84i	−0.288400	−	0.499523i
$501$		0			0
$502$	−432.000	+	748.246i	−0.0384086	+	0.0665256i
$503$	7382.19			0.654385			0.327193	−	0.944958i	$-0.393897\pi$
$503$	7382.19			0.654385			0.327193	+	0.944958i	$0.393897\pi$
$504$		0			0
$505$	9216.00			0.812092
$506$	3054.70	−	5290.90i	0.268376	−	0.464840i
$507$		0			0
$508$	−6220.00	−	10773.4i	−0.543244	−	0.940926i
$509$	−10496.3	−	18180.1i	−0.914028	−	1.58314i	−0.808319	−	0.588745i	$-0.799622\pi$
$509$	−10496.3	−	18180.1i	−0.914028	−	1.58314i	−0.105709	−	0.994397i	$-0.533711\pi$
$510$		0			0
$511$	−753.500	+	1305.10i	−0.0652307	+	0.112983i
$512$	−14187.4			−1.22461
$513$		0			0
$514$	−7632.00			−0.654928
$515$	−7119.15	+	12330.7i	−0.609140	+	1.05506i
$516$		0			0
$517$	3312.00	+	5736.55i	0.281744	+	0.487995i
$518$	1936.77	+	3354.58i	0.164279	+	0.284540i
$519$		0			0
$520$	2088.00	−	3616.52i	0.176086	−	0.304990i
$521$	−12269.7			−1.03176			−0.515879	−	0.856661i	$-0.672535\pi$
$521$	−12269.7			−1.03176			−0.515879	+	0.856661i	$0.672535\pi$
$522$		0			0
$523$	6833.00			0.571293			0.285646	−	0.958335i	$-0.407792\pi$
$523$	6833.00			0.571293			0.285646	+	0.958335i	$0.407792\pi$
$524$	−12558.2	+	21751.5i	−1.04696	+	1.81339i
$525$		0			0
$526$	−792.000	−	1371.78i	−0.0656518	−	0.113712i
$527$	6822.17	+	11816.3i	0.563906	+	0.976713i
$528$		0			0
$529$	2483.50	−	4301.55i	0.204118	−	0.353542i
$530$	21993.8			1.80255
$531$		0			0
$532$	3190.00			0.259970
$533$	−3937.17	+	6819.38i	−0.319958	+	0.554184i
$534$		0			0
$535$	−6480.00	−	11223.7i	−0.523654	−	0.906995i
$536$	−2167.99	−	3755.07i	−0.174707	−	0.302601i
$537$		0			0
$538$	−15876.0	+	27498.0i	−1.27224	+	2.20358i
$539$	3767.46			0.301069
$540$		0			0
$541$	−10555.0			−0.838808			−0.419404	−	0.907800i	$-0.637761\pi$
$541$	−10555.0			−0.838808			−0.419404	+	0.907800i	$0.637761\pi$
$542$	−7040.66	+	12194.8i	−0.557975	+	0.966441i
$543$		0			0
$544$	−6480.00	−	11223.7i	−0.510713	−	0.884580i
$545$	−1849.79	−	3203.93i	−0.145388	−	0.251819i
$546$		0			0
$547$	−8645.50	+	14974.4i	−0.675786	+	1.17050i	0.300453	+	0.953797i	$0.402862\pi$
$547$	−8645.50	+	14974.4i	−0.675786	+	1.17050i	−0.976239	+	0.216699i	$0.930471\pi$
$548$	−13406.7			−1.04509
$549$		0			0
$550$	11736.0			0.909863
$551$	3937.17	−	6819.38i	0.304409	−	0.527251i
$552$		0			0
$553$	2612.50	+	4524.98i	0.200895	+	0.347960i
$554$	17721.5	+	30694.6i	1.35905	+	2.35395i
$555$		0			0
$556$	−4735.00	+	8201.26i	−0.361167	+	0.625559i
$557$	−10335.1			−0.786196			−0.393098	−	0.919497i	$-0.628597\pi$
$557$	−10335.1			−0.786196			−0.393098	+	0.919497i	$0.628597\pi$
$558$		0			0
$559$	−6728.00			−0.509059
$560$	4106.88	−	7113.32i	0.309906	−	0.536772i
$561$		0			0
$562$	5472.00	+	9477.78i	0.410716	+	0.711381i
$563$	8213.75	+	14226.6i	0.614864	+	1.06498i	0.990408	+	0.138171i	$0.0441224\pi$
$563$	8213.75	+	14226.6i	0.614864	+	1.06498i	−0.375544	+	0.926804i	$0.622544\pi$
$564$		0			0
$565$	12816.0	−	22198.0i	0.954289	−	1.65288i
$566$	26338.3			1.95598
$567$		0			0
$568$	6048.00			0.446775
$569$	7848.89	−	13594.7i	0.578282	−	1.00161i	−0.417395	−	0.908725i	$-0.637057\pi$
$569$	7848.89	−	13594.7i	0.578282	−	1.00161i	0.995677	−	0.0928883i	$-0.0296099\pi$
$570$		0			0
$571$	2037.50	+	3529.05i	0.149329	+	0.258645i	0.930980	−	0.365071i	$-0.118955\pi$
$571$	2037.50	+	3529.05i	0.149329	+	0.258645i	−0.781651	+	0.623716i	$0.785622\pi$
$572$	2460.73	+	4262.11i	0.179875	+	0.311552i
$573$		0			0
$574$	6336.00	−	10974.3i	0.460731	−	0.798009i
$575$	−13831.0			−1.00312
$576$		0			0
$577$	6995.00			0.504689			0.252345	−	0.967637i	$-0.418798\pi$
$577$	6995.00			0.504689			0.252345	+	0.967637i	$0.418798\pi$
$578$	−4923.58	+	8527.90i	−0.354315	+	0.613692i
$579$		0			0
$580$	23040.0	+	39906.5i	1.64946	+	2.85694i
$581$	3173.50	+	5496.65i	0.226607	+	0.392495i
$582$		0			0
$583$	−2592.00	+	4489.48i	−0.184133	+	0.318928i
$584$	−1162.48			−0.0823697
$585$		0			0
$586$	26280.0			1.85259
$587$	−2791.66	+	4835.29i	−0.196293	+	0.339990i	−0.947324	−	0.320278i	$-0.896224\pi$
$587$	−2791.66	+	4835.29i	−0.196293	+	0.339990i	0.751031	+	0.660267i	$0.229557\pi$
$588$		0			0
$589$	3886.00	+	6730.75i	0.271850	+	0.470859i
$590$	−10386.0	−	17989.1i	−0.724719	−	1.25525i
$591$		0			0
$592$	1826.00	−	3162.72i	0.126771	−	0.219573i
$593$	14968.0			1.03653			0.518266	−	0.855219i	$-0.326578\pi$
$593$	14968.0			1.03653			0.518266	+	0.855219i	$0.326578\pi$
$594$		0			0
$595$	9504.00			0.654834
$596$	−2885.00	+	4996.96i	−0.198279	+	0.343429i
$597$		0			0
$598$	−5220.00	−	9041.31i	−0.356959	−	0.618272i
$599$	−9096.22	−	15755.1i	−0.620470	−	1.07469i	−0.989398	−	0.145228i	$-0.953609\pi$
$599$	−9096.22	−	15755.1i	−0.620470	−	1.07469i	0.368928	−	0.929458i	$-0.379725\pi$
$600$		0			0
$601$	3275.00	−	5672.47i	0.222280	−	0.385000i	−0.733220	−	0.679991i	$-0.761984\pi$
$601$	3275.00	−	5672.47i	0.222280	−	0.385000i	0.955500	+	0.294992i	$0.0953169\pi$
$602$	10827.2			0.733031
$603$		0			0
$604$	−23110.0			−1.55684
$605$	8850.15	−	15328.9i	0.594727	−	1.03010i
$606$		0			0
$607$	−6413.50	−	11108.5i	−0.428857	−	0.742801i	0.567915	−	0.823087i	$-0.307750\pi$
$607$	−6413.50	−	11108.5i	−0.428857	−	0.742801i	−0.996772	+	0.0802856i	$0.974417\pi$
$608$	−3691.10	−	6393.17i	−0.246207	−	0.426443i
$609$		0			0
$610$	27612.0	−	47825.4i	1.83275	−	3.17442i
$611$	11319.4			0.749480
$612$		0			0
$613$	18767.0			1.23653			0.618264	−	0.785970i	$-0.287836\pi$
$613$	18767.0			1.23653			0.618264	+	0.785970i	$0.287836\pi$
$614$	−4921.46	+	8524.22i	−0.323476	+	0.560276i
$615$		0			0
$616$	−792.000	−	1371.78i	−0.0518029	−	0.0897253i
$617$	−3775.95	−	6540.14i	−0.246376	−	0.426736i	0.716142	−	0.697955i	$-0.245906\pi$
$617$	−3775.95	−	6540.14i	−0.246376	−	0.426736i	−0.962518	+	0.271219i	$0.912573\pi$
$618$		0			0
$619$	−12290.5	+	21287.8i	−0.798056	+	1.38227i	0.122824	+	0.992428i	$0.460805\pi$
$619$	−12290.5	+	21287.8i	−0.798056	+	1.38227i	−0.920880	+	0.389846i	$0.872528\pi$
$620$	−45481.1			−2.94607
$621$		0			0
$622$	3384.00			0.218145
$623$	−1400.07	+	2424.99i	−0.0900364	+	0.155948i
$624$		0			0
$625$	4715.50	+	8167.49i	0.301792	+	0.522719i
$626$	2772.57	+	4802.22i	0.177019	+	0.306606i
$627$		0			0
$628$	−8110.00	+	14046.9i	−0.515325	+	0.892569i
$629$	4225.67			0.267867
$630$		0			0
$631$	−18223.0			−1.14968			−0.574838	−	0.818267i	$-0.694935\pi$
$631$	−18223.0			−1.14968			−0.574838	+	0.818267i	$0.694935\pi$
$632$	−2015.25	+	3490.52i	−0.126839	+	0.219692i
$633$		0			0
$634$	4392.00	+	7607.17i	0.275124	+	0.476529i
$635$	−10555.7	−	18283.0i	−0.659669	−	1.14258i
$636$		0			0
$637$	3219.00	−	5575.47i	0.200222	−	0.346795i
$638$	−19550.1			−1.21316
$639$		0			0
$640$	17856.0			1.10284
$641$	797.616	−	1381.51i	0.0491481	−	0.0851271i	−0.840405	−	0.541959i	$-0.817683\pi$
$641$	797.616	−	1381.51i	0.0491481	−	0.0851271i	0.889553	+	0.456832i	$0.151016\pi$
$642$		0			0
$643$	13148.0	+	22773.0i	0.806386	+	1.39670i	0.915351	+	0.402657i	$0.131913\pi$
$643$	13148.0	+	22773.0i	0.806386	+	1.39670i	−0.108965	+	0.994046i	$0.534754\pi$
$644$	4666.90	+	8083.32i	0.285562	+	0.494608i
$645$		0			0
$646$	3132.00	−	5424.78i	0.190754	−	0.330395i
$647$	25659.5			1.55916			0.779582	−	0.626301i	$-0.215432\pi$
$647$	25659.5			1.55916			0.779582	+	0.626301i	$0.215432\pi$
$648$		0			0
$649$	4896.00			0.296125
$650$	10027.5	−	17368.1i	0.605093	−	1.04805i
$651$		0			0
$652$	−11215.0	−	19424.9i	−0.673640	−	1.16678i
$653$	220.617	+	382.120i	0.0132212	+	0.0228997i	0.872560	−	0.488506i	$-0.162458\pi$
$653$	220.617	+	382.120i	0.0132212	+	0.0228997i	−0.859339	+	0.511406i	$0.829125\pi$
$654$		0			0
$655$	−21312.0	+	36913.5i	−1.27134	+	2.20203i
$656$	−11947.3			−0.711071
$657$		0			0
$658$	−18216.0			−1.07923
$659$	13525.5	−	23426.9i	0.799515	−	1.38480i	−0.120418	−	0.992723i	$-0.538423\pi$
$659$	13525.5	−	23426.9i	0.799515	−	1.38480i	0.919933	−	0.392077i	$-0.128243\pi$
$660$		0			0
$661$	−311.500	−	539.534i	−0.0183297	−	0.0317480i	0.856715	−	0.515790i	$-0.172502\pi$
$661$	−311.500	−	539.534i	−0.0183297	−	0.0317480i	−0.875045	+	0.484042i	$0.839168\pi$
$662$	−10973.6	−	19006.8i	−0.644261	−	1.11589i
$663$		0			0
$664$	−2448.00	+	4240.06i	−0.143074	+	0.247811i
$665$	5413.61			0.315685
$666$		0			0
$667$	23040.0			1.33750
$668$	13661.3	−	23662.1i	0.791275	−	1.37053i
$669$		0			0
$670$	−18396.0	−	31862.8i	−1.06075	−	1.83727i
$671$	6508.21	+	11272.6i	0.374436	+	0.648543i
$672$		0			0
$673$	−13937.5	+	24140.5i	−0.798293	+	1.38268i	0.122434	+	0.992477i	$0.460930\pi$
$673$	−13937.5	+	24140.5i	−0.798293	+	1.38268i	−0.920727	+	0.390207i	$0.872403\pi$
$674$	−11120.0			−0.635497
$675$		0			0
$676$	−13560.0			−0.771507
$677$	−3300.77	+	5717.11i	−0.187384	+	0.324559i	−0.944377	−	0.328864i	$-0.893334\pi$
$677$	−3300.77	+	5717.11i	−0.187384	+	0.324559i	0.756993	+	0.653423i	$0.226668\pi$
$678$		0			0
$679$	−4515.50	−	7821.08i	−0.255212	−	0.442040i
$680$	3665.64	+	6349.08i	0.206722	+	0.358053i
$681$		0			0
$682$	9648.00	−	16710.8i	0.541702	−	0.938256i
$683$	15680.8			0.878491			0.439245	−	0.898367i	$-0.355246\pi$
$683$	15680.8			0.878491			0.439245	+	0.898367i	$0.355246\pi$
$684$		0			0
$685$	−22752.0			−1.26906
$686$	−13184.0	+	22835.4i	−0.733772	+	1.27093i
$687$		0			0
$688$	−5104.00	−	8840.39i	−0.282832	−	0.489879i
$689$	4429.32	+	7671.80i	0.244911	+	0.424198i
$690$		0			0
$691$	5300.00	−	9179.87i	0.291782	−	0.505382i	−0.682449	−	0.730933i	$-0.739085\pi$
$691$	5300.00	−	9179.87i	0.291782	−	0.505382i	0.974231	+	0.225552i	$0.0724184\pi$
$692$	13576.5			0.745808
$693$		0			0
$694$	−32688.0			−1.78792
$695$	−8035.56	+	13918.0i	−0.438570	+	0.759626i
$696$		0			0
$697$	−6912.00	−	11971.9i	−0.375625	−	0.650602i
$698$	4147.18	+	7183.13i	0.224890	+	0.389521i
$699$		0			0
$700$	−8965.00	+	15527.8i	−0.484065	+	0.838424i
$701$	−13593.4			−0.732406			−0.366203	−	0.930535i	$-0.619342\pi$
$701$	−13593.4			−0.732406			−0.366203	+	0.930535i	$0.619342\pi$
$702$		0			0
$703$	2407.00			0.129135
$704$	−6177.28	+	10699.4i	−0.330703	+	0.572795i
$705$		0			0
$706$	−2520.00	−	4364.77i	−0.134336	−	0.232677i
$707$	−2986.82	−	5173.32i	−0.158884	−	0.275195i
$708$		0			0
$709$	16761.5	−	29031.8i	0.887858	−	1.53782i	0.0454556	−	0.998966i	$-0.485526\pi$
$709$	16761.5	−	29031.8i	0.887858	−	1.53782i	0.842402	−	0.538849i	$-0.181141\pi$
$710$	51319.0			2.71263
$711$		0			0
$712$	−2160.00			−0.113693
$713$	−11370.3	+	19693.9i	−0.597223	+	1.03442i
$714$		0			0
$715$	4176.00	+	7233.04i	0.218425	+	0.378322i
$716$	−11709.7	−	20281.8i	−0.611189	−	1.05861i
$717$		0			0
$718$	1188.00	−	2057.68i	0.0617490	−	0.106952i
$719$	−31870.7			−1.65310			−0.826549	−	0.562865i	$-0.809699\pi$
$719$	−31870.7			−1.65310			−0.826549	+	0.562865i	$0.809699\pi$
$720$		0			0
$721$	9229.00			0.476707
$722$	−12766.1	+	22111.5i	−0.658041	+	1.13976i
$723$		0			0
$724$	7955.00	+	13778.5i	0.408350	+	0.707283i
$725$	22129.6	+	38329.6i	1.13362	+	1.96348i
$726$		0			0
$727$	6542.00	−	11331.1i	0.333741	−	0.578056i	−0.649502	−	0.760360i	$-0.725022\pi$
$727$	6542.00	−	11331.1i	0.333741	−	0.578056i	0.983242	+	0.182305i	$0.0583557\pi$
$728$	−2706.80			−0.137803
$729$		0			0
$730$	−9864.00			−0.500114
$731$	5905.76	−	10229.1i	0.298813	−	0.517559i
$732$		0			0
$733$	9611.00	+	16646.7i	0.484298	+	0.838829i	0.999837	−	0.0180373i	$-0.00574175\pi$
$733$	9611.00	+	16646.7i	0.484298	+	0.838829i	−0.515539	+	0.856866i	$0.672408\pi$
$734$	16747.8	+	29008.1i	0.842198	+	1.45873i
$735$		0			0
$736$	10800.0	−	18706.1i	0.540888	−	0.936845i
$737$	8671.96			0.433427
$738$		0			0
$739$	6320.00			0.314594			0.157297	−	0.987551i	$-0.449722\pi$
$739$	6320.00			0.314594			0.157297	+	0.987551i	$0.449722\pi$
$740$	−7042.78	+	12198.5i	−0.349862	+	0.605979i
$741$		0			0
$742$	−7128.00	−	12346.1i	−0.352665	−	0.610833i
$743$	2078.89	+	3600.75i	0.102648	+	0.177791i	0.912775	−	0.408463i	$-0.133935\pi$
$743$	2078.89	+	3600.75i	0.102648	+	0.177791i	−0.810127	+	0.586254i	$0.800602\pi$
$744$		0			0
$745$	−4896.00	+	8480.12i	−0.240773	+	0.417030i
$746$	−41590.6			−2.04121
$747$		0			0
$748$	−8640.00			−0.422339
$749$	−4200.21	+	7274.98i	−0.204903	+	0.354903i
$750$		0			0
$751$	−10166.5	−	17608.9i	−0.493982	−	0.855603i	0.505993	−	0.862537i	$-0.331126\pi$
$751$	−10166.5	−	17608.9i	−0.493982	−	0.855603i	−0.999976	+	0.00693453i	$0.997793\pi$
$752$	8587.10	+	14873.3i	0.416409	+	0.721241i
$753$		0			0
$754$	−16704.0	+	28932.2i	−0.806795	+	1.39741i
$755$	−39219.0			−1.89050
$756$		0			0
$757$	−14011.0			−0.672706			−0.336353	−	0.941736i	$-0.609194\pi$
$757$	−14011.0			−0.672706			−0.336353	+	0.941736i	$0.609194\pi$
$758$	22284.5	−	38597.8i	1.06782	−	1.84952i
$759$		0			0
$760$	2088.00	+	3616.52i	0.0996575	+	0.172612i
$761$	−12991.0	−	22501.0i	−0.618820	−	1.07183i	−0.989701	−	0.143148i	$-0.954278\pi$
$761$	−12991.0	−	22501.0i	−0.618820	−	1.07183i	0.370881	−	0.928680i	$-0.379056\pi$
$762$		0			0
$763$	−1199.00	+	2076.73i	−0.0568895	+	0.0985356i
$764$	47687.3			2.25820
$765$		0			0
$766$	−4608.00			−0.217355
$767$	4183.24	−	7245.59i	0.196934	−	0.341099i
$768$		0			0
$769$	3144.50	+	5446.43i	0.147456	+	0.255401i	0.930287	−	0.366834i	$-0.119558\pi$
$769$	3144.50	+	5446.43i	0.147456	+	0.255401i	−0.782831	+	0.622235i	$0.786225\pi$
$770$	−6720.34	−	11640.0i	−0.314525	−	0.544773i
$771$		0			0
$772$	17405.0	−	30146.3i	0.811424	−	1.40543i
$773$	7229.46			0.336385			0.168192	−	0.985754i	$-0.446207\pi$
$773$	7229.46			0.336385			0.168192	+	0.985754i	$0.446207\pi$
$774$		0			0
$775$	−43684.0			−2.02474
$776$	3483.21	−	6033.09i	0.161134	−	0.279092i
$777$		0			0
$778$	−4572.00	−	7918.94i	−0.210687	−	0.364920i
$779$	−3937.17	−	6819.38i	−0.181083	−	0.313645i
$780$		0			0
$781$	−6048.00	+	10475.4i	−0.277099	+	0.479950i
$782$	18328.2			0.838127
$783$		0			0
$784$	9768.00			0.444971
$785$	−13763.1	+	23838.4i	−0.625767	+	1.08386i
$786$		0			0
$787$	12837.5	+	22235.2i	0.581458	+	1.00711i	0.995307	+	0.0967692i	$0.0308509\pi$
$787$	12837.5	+	22235.2i	0.581458	+	1.00711i	−0.413849	+	0.910346i	$0.635816\pi$
$788$	−11964.2	−	20722.7i	−0.540874	−	0.936821i
$789$		0			0
$790$	−17100.0	+	29618.1i	−0.770115	+	1.33388i
$791$	−16614.2			−0.746817
$792$		0			0
$793$	22243.0			0.996056
$794$	26351.0	−	45641.3i	1.17779	−	2.03999i
$795$		0			0
$796$	−11755.0	−	20360.3i	−0.523423	−	0.906596i
$797$	−1663.12	−	2880.60i	−0.0739154	−	0.128025i	0.826699	−	0.562645i	$-0.190216\pi$
$797$	−1663.12	−	2880.60i	−0.0739154	−	0.128025i	−0.900614	+	0.434620i	$0.856883\pi$
$798$		0			0
$799$	−9936.00	+	17209.7i	−0.439938	+	0.761995i
$800$	41493.0			1.83375
$801$		0			0
$802$	65808.0			2.89746
$803$	1162.48	−	2013.48i	0.0510874	−	0.0884859i
$804$		0			0
$805$	7920.00	+	13717.8i	0.346762	+	0.600609i
$806$	−16486.9	−	28556.2i	−0.720504	−	1.24795i
$807$		0			0
$808$	2304.00	−	3990.65i	0.100315	−	0.173750i
$809$	−10080.5			−0.438087			−0.219043	−	0.975715i	$-0.570294\pi$
$809$	−10080.5			−0.438087			−0.219043	+	0.975715i	$0.570294\pi$
$810$		0			0
$811$	14312.0			0.619682			0.309841	−	0.950788i	$-0.399724\pi$
$811$	14312.0			0.619682			0.309841	+	0.950788i	$0.399724\pi$
$812$	14934.1	−	25866.6i	0.645424	−	1.11791i
$813$		0			0
$814$	−2988.00	−	5175.37i	−0.128660	−	0.222846i
$815$	−19032.5	−	32965.2i	−0.818011	−	1.41684i
$816$		0			0
$817$	3364.00	−	5826.62i	0.144053	−	0.249507i
$818$	−30822.8			−1.31747
$819$		0			0
$820$	46080.0			1.96242
$821$	1383.10	−	2395.60i	0.0587948	−	0.101836i	−0.835130	−	0.550053i	$-0.814608\pi$
$821$	1383.10	−	2395.60i	0.0587948	−	0.101836i	0.893925	+	0.448217i	$0.147941\pi$
$822$		0			0
$823$	16671.5	+	28875.9i	0.706114	+	1.22303i	0.966288	+	0.257464i	$0.0828868\pi$
$823$	16671.5	+	28875.9i	0.706114	+	1.22303i	−0.260174	+	0.965562i	$0.583780\pi$
$824$	3559.58	+	6165.37i	0.150490	+	0.260656i
$825$		0			0
$826$	−6732.00	+	11660.2i	−0.283579	+	0.491173i
$827$	18379.1			0.772799			0.386399	−	0.922332i	$-0.373719\pi$
$827$	18379.1			0.772799			0.386399	+	0.922332i	$0.373719\pi$
$828$		0			0
$829$	3593.00			0.150531			0.0752654	−	0.997164i	$-0.476020\pi$
$829$	3593.00			0.150531			0.0752654	+	0.997164i	$0.476020\pi$
$830$	−20772.0	+	35978.1i	−0.868681	+	1.50460i
$831$		0			0
$832$	10556.0	+	18283.5i	0.439860	+	0.761860i
$833$	5651.20	+	9788.16i	0.235057	+	0.407130i
$834$		0			0
$835$	23184.0	−	40155.9i	0.960857	−	1.66425i
$836$	−4921.46			−0.203603
$837$		0			0
$838$	−13464.0			−0.555019
$839$	−8570.13	+	14843.9i	−0.352651	+	0.610809i	−0.986713	−	0.162473i	$-0.948053\pi$
$839$	−8570.13	+	14843.9i	−0.352651	+	0.610809i	0.634062	+	0.773282i	$0.281386\pi$
$840$		0			0
$841$	−24669.5	−	42728.8i	−1.01150	−	1.75197i
$842$	7240.07	+	12540.2i	0.296329	+	0.513257i
$843$		0			0
$844$	−8515.00	+	14748.4i	−0.347273	+	0.601494i
$845$	−23012.1			−0.936852
$846$		0			0
$847$	−11473.0			−0.465427
$848$	−6720.34	+	11640.0i	−0.272143	+	0.471366i
$849$		0			0
$850$	17604.0	+	30491.0i	0.710367	+	1.23039i
$851$	3521.39	+	6099.23i	0.141847	+	0.245686i
$852$		0			0
$853$	2370.50	−	4105.83i	0.0951517	−	0.164808i	−0.814520	−	0.580135i	$-0.803000\pi$
$853$	2370.50	−	4105.83i	0.0951517	−	0.164808i	0.909672	+	0.415328i	$0.136333\pi$
$854$	−35795.2			−1.43429
$855$		0			0
$856$	−6480.00			−0.258740
$857$	5990.61	−	10376.0i	0.238781	−	0.413581i	−0.721584	−	0.692327i	$-0.756586\pi$
$857$	5990.61	−	10376.0i	0.238781	−	0.413581i	0.960365	+	0.278746i	$0.0899189\pi$
$858$		0			0
$859$	−3443.50	−	5964.32i	−0.136776	−	0.236903i	0.789498	−	0.613753i	$-0.210341\pi$
$859$	−3443.50	−	5964.32i	−0.136776	−	0.236903i	−0.926275	+	0.376849i	$0.877007\pi$
$860$	19685.9	+	34096.9i	0.780560	+	1.35197i
$861$		0			0
$862$	26892.0	−	46578.3i	1.06258	−	1.84044i
$863$	8400.43			0.331349			0.165674	−	0.986181i	$-0.447020\pi$
$863$	8400.43			0.331349			0.165674	+	0.986181i	$0.447020\pi$
$864$		0			0
$865$	23040.0			0.905646
$866$	18332.5	−	31752.7i	0.719356	−	1.24596i
$867$		0			0
$868$	14740.0	+	25530.4i	0.576392	+	0.998340i
$869$	−4030.51	−	6981.05i	−0.157337	−	0.272515i
$870$		0			0
$871$	7409.50	−	12833.6i	0.288245	−	0.499255i
$872$	−1849.79			−0.0718370
$873$		0			0
$874$	10440.0			0.404048
$875$	−3546.85	+	6143.32i	−0.137035	+	0.237351i
$876$		0			0
$877$	−6737.50	−	11669.7i	−0.259418	−	0.449324i	0.706668	−	0.707545i	$-0.250197\pi$
$877$	−6737.50	−	11669.7i	−0.259418	−	0.449324i	−0.966086	+	0.258220i	$0.916864\pi$
$878$	1111.57	+	1925.30i	0.0427263	+	0.0740042i
$879$		0			0
$880$	−6336.00	+	10974.3i	−0.242712	+	0.420389i
$881$	5243.90			0.200535			0.100268	−	0.994961i	$-0.468030\pi$
$881$	5243.90			0.200535			0.100268	+	0.994961i	$0.468030\pi$
$882$		0			0
$883$	−7909.00			−0.301426			−0.150713	−	0.988578i	$-0.548157\pi$
$883$	−7909.00			−0.301426			−0.150713	+	0.988578i	$0.548157\pi$
$884$	−7382.19	+	12786.3i	−0.280871	+	0.486483i
$885$		0			0
$886$	38520.0	+	66718.6i	1.46061	+	2.52986i
$887$	17836.1	+	30893.0i	0.675171	+	1.16943i	0.976419	+	0.215884i	$0.0692633\pi$
$887$	17836.1	+	30893.0i	0.675171	+	1.16943i	−0.301248	+	0.953546i	$0.597403\pi$
$888$		0			0
$889$	−6842.00	+	11850.7i	−0.258125	+	0.447086i
$890$	−18328.2			−0.690295
$891$		0			0
$892$	13880.0			0.521005
$893$	−5659.68	+	9802.86i	−0.212087	+	0.367346i
$894$		0			0
$895$	−19872.0	−	34419.3i	−0.742176	−	1.28549i
$896$	−5786.96	−	10023.3i	−0.215769	−	0.373722i
$897$		0			0
$898$	−7020.00	+	12159.0i	−0.260869	+	0.451839i
$899$	72769.8			2.69967
$900$		0			0
$901$	−15552.0			−0.575041
$902$	−9775.04	+	16930.9i	−0.360835	+	0.624985i
$903$		0			0
$904$	−6408.00	−	11099.0i	−0.235760	−	0.408348i
$905$	13500.1	+	23382.8i	0.495865	+	0.858864i
$906$		0			0
$907$	8499.50	−	14721.6i	0.311159	−	0.538943i	−0.667455	−	0.744651i	$-0.732616\pi$
$907$	8499.50	−	14721.6i	0.311159	−	0.538943i	0.978614	+	0.205707i	$0.0659495\pi$
$908$	47178.2			1.72430
$909$		0			0
$910$	−22968.0			−0.836683
$911$	−19516.1	+	33803.0i	−0.709768	+	1.22935i	0.255175	+	0.966895i	$0.417867\pi$
$911$	−19516.1	+	33803.0i	−0.709768	+	1.22935i	−0.964943	+	0.262460i	$0.915466\pi$
$912$		0			0
$913$	−4896.00	−	8480.12i	−0.177474	−	0.307394i
$914$	−20080.4	−	34780.3i	−0.726697	−	1.25868i
$915$		0			0
$916$	−2170.00	+	3758.55i	−0.0782738	+	0.135574i
$917$	27628.1			0.994939
$918$		0			0
$919$	−28348.0			−1.01753			−0.508767	−	0.860904i	$-0.669899\pi$
$919$	−28348.0			−1.01753			−0.508767	+	0.860904i	$0.669899\pi$
$920$	−6109.40	+	10581.8i	−0.218936	+	0.379208i
$921$		0			0
$922$	6876.00	+	11909.6i	0.245606	+	0.425403i
$923$	10335.1	+	17900.9i	0.368562	+	0.638369i
$924$		0			0
$925$	−6764.50	+	11716.5i	−0.240449	+	0.416470i
$926$	−48005.5			−1.70363
$927$		0			0
$928$	−69120.0			−2.44502
$929$	16580.2	−	28717.8i	0.585554	−	1.01421i	−0.409252	−	0.912422i	$-0.634210\pi$
$929$	16580.2	−	28717.8i	0.585554	−	1.01421i	0.994806	−	0.101788i	$-0.0324565\pi$
$930$		0			0
$931$	3219.00	+	5575.47i	0.113317	+	0.196271i
$932$	17310.0	+	29981.8i	0.608377	+	1.05374i
$933$		0			0
$934$	−37044.0	+	64162.1i	−1.29777	+	2.24780i
$935$	−14662.6			−0.512853
$936$		0			0
$937$	−133.000			−0.00463706			−0.00231853	−	0.999997i	$-0.500738\pi$
$937$	−133.000			−0.00463706			−0.00231853	+	0.999997i	$0.500738\pi$
$938$	−11923.9	+	20652.9i	−0.415064	+	0.718913i
$939$		0			0
$940$	−33120.0	−	57365.5i	−1.14921	−	1.99049i
$941$	−24395.2	−	42253.7i	−0.845122	−	1.46380i	−0.885515	−	0.464611i	$-0.846194\pi$
$941$	−24395.2	−	42253.7i	−0.845122	−	1.46380i	0.0403923	−	0.999184i	$-0.487139\pi$
$942$		0			0
$943$	11520.0	−	19953.2i	0.397818	−	0.689042i
$944$	12694.0			0.437663
$945$		0			0
$946$	−16704.0			−0.574095
$947$	22664.2	−	39255.5i	0.777705	−	1.34703i	−0.155556	−	0.987827i	$-0.549717\pi$
$947$	22664.2	−	39255.5i	0.777705	−	1.34703i	0.933261	−	0.359198i	$-0.116950\pi$
$948$		0			0
$949$	−1986.50	−	3440.72i	−0.0679500	−	0.117693i
$950$	10027.5	+	17368.1i	0.342457	+	0.593153i
$951$		0			0
$952$	2376.00	−	4115.35i	0.0808893	−	0.140104i
$953$	−43122.2			−1.46576			−0.732878	−	0.680360i	$-0.761823\pi$
$953$	−43122.2			−1.46576			−0.732878	+	0.680360i	$0.761823\pi$
$954$		0			0
$955$	80928.0			2.74217
$956$	−14085.6	+	24396.9i	−0.476527	+	0.825369i
$957$		0			0
$958$	−18936.0	−	32798.1i	−0.638616	−	1.10612i
$959$	7373.71	+	12771.6i	0.248289	+	0.430050i
$960$		0			0
$961$	−21016.5	+	36401.6i	−0.705465	+	1.22190i
$962$	−10212.0			−0.342255
$963$		0			0
$964$	−20950.0			−0.699952
$965$	29537.3	−	51160.0i	0.985325	−	1.70663i
$966$		0			0
$967$	−11030.5	−	19105.4i	−0.366822	−	0.635355i	0.622245	−	0.782823i	$-0.286221\pi$
$967$	−11030.5	−	19105.4i	−0.366822	−	0.635355i	−0.989067	+	0.147468i	$0.952888\pi$
$968$	−4425.07	−	7664.45i	−0.146929	−	0.254488i
$969$		0			0
$970$	29556.0	−	51192.5i	0.978336	−	1.69453i
$971$	−33449.0			−1.10549			−0.552744	−	0.833351i	$-0.686419\pi$
$971$	−33449.0			−1.10549			−0.552744	+	0.833351i	$0.686419\pi$
$972$		0			0
$973$	10417.0			0.343221
$974$	−39229.6	+	67947.6i	−1.29055	+	2.23530i
$975$		0			0
$976$	16874.0	+	29226.6i	0.553405	+	0.958526i
$977$	10301.1	+	17842.1i	0.337321	+	0.584257i	0.983928	−	0.178567i	$-0.0571460\pi$
$977$	10301.1	+	17842.1i	0.337321	+	0.584257i	−0.646607	+	0.762823i	$0.723813\pi$
$978$		0			0
$979$	2160.00	−	3741.23i	0.0705147	−	0.122135i
$980$	−37674.6			−1.22803
$981$		0			0
$982$	54360.0			1.76649
$983$	5965.15	−	10331.9i	0.193549	−	0.335237i	−0.752875	−	0.658164i	$-0.771333\pi$
$983$	5965.15	−	10331.9i	0.193549	−	0.335237i	0.946424	+	0.322927i	$0.104667\pi$
$984$		0			0
$985$	−20304.0	−	35167.6i	−0.656791	−	1.13760i
$986$	−29325.1	−	50792.6i	−0.947163	−	1.64053i
$987$		0			0
$988$	−4205.00	+	7283.27i	−0.135404	+	0.234526i
$989$	19685.9			0.632936
$990$		0			0
$991$	−35017.0			−1.12245			−0.561227	−	0.827662i	$-0.689670\pi$
$991$	−35017.0			−1.12245			−0.561227	+	0.827662i	$0.689670\pi$
$992$	34110.8	−	59081.7i	1.09175	−	1.89097i
$993$		0			0
$994$	−16632.0	−	28807.5i	−0.530719	−	0.919233i
$995$	−19948.9	−	34552.5i	−0.635601	−	1.10089i
$996$		0			0
$997$	−6823.00	+	11817.8i	−0.216737	+	0.375399i	−0.953808	−	0.300415i	$-0.902875\pi$
$997$	−6823.00	+	11817.8i	−0.216737	+	0.375399i	0.737072	+	0.675815i	$0.236208\pi$
$998$	51997.8			1.64926
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.4.c.e.55.2		4
3.2	odd	2	inner	81.4.c.e.55.1		4
9.2	odd	6		27.4.a.c.1.2	yes	2
9.4	even	3	inner	81.4.c.e.28.2		4
9.5	odd	6	inner	81.4.c.e.28.1		4
9.7	even	3		27.4.a.c.1.1	✓	2
36.7	odd	6		432.4.a.q.1.2		2
36.11	even	6		432.4.a.q.1.1		2
45.2	even	12		675.4.b.i.649.4		4
45.7	odd	12		675.4.b.i.649.2		4
45.29	odd	6		675.4.a.n.1.1		2
45.34	even	6		675.4.a.n.1.2		2
45.38	even	12		675.4.b.i.649.1		4
45.43	odd	12		675.4.b.i.649.3		4
63.20	even	6		1323.4.a.t.1.2		2
63.34	odd	6		1323.4.a.t.1.1		2
72.11	even	6		1728.4.a.bk.1.2		2
72.29	odd	6		1728.4.a.bp.1.2		2
72.43	odd	6		1728.4.a.bk.1.1		2
72.61	even	6		1728.4.a.bp.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.c.1.1	✓	2	9.7	even	3
27.4.a.c.1.2	yes	2	9.2	odd	6
81.4.c.e.28.1		4	9.5	odd	6	inner
81.4.c.e.28.2		4	9.4	even	3	inner
81.4.c.e.55.1		4	3.2	odd	2	inner
81.4.c.e.55.2		4	1.1	even	1	trivial
432.4.a.q.1.1		2	36.11	even	6
432.4.a.q.1.2		2	36.7	odd	6
675.4.a.n.1.1		2	45.29	odd	6
675.4.a.n.1.2		2	45.34	even	6
675.4.b.i.649.1		4	45.38	even	12
675.4.b.i.649.2		4	45.7	odd	12
675.4.b.i.649.3		4	45.43	odd	12
675.4.b.i.649.4		4	45.2	even	12
1323.4.a.t.1.1		2	63.34	odd	6
1323.4.a.t.1.2		2	63.20	even	6
1728.4.a.bk.1.1		2	72.43	odd	6
1728.4.a.bk.1.2		2	72.11	even	6
1728.4.a.bp.1.1		2	72.61	even	6
1728.4.a.bp.1.2		2	72.29	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.12132 - 3.67423i) q^{2} +(-5.00000 - 8.66025i) q^{4} +(-8.48528 - 14.6969i) q^{5} +(-5.50000 + 9.52628i) q^{7} -8.48528 q^{8} -72.0000 q^{10} +(8.48528 - 14.6969i) q^{11} +(-14.5000 - 25.1147i) q^{13} +(23.3345 + 40.4166i) q^{14} +(22.0000 - 38.1051i) q^{16} +50.9117 q^{17} +29.0000 q^{19} +(-84.8528 + 146.969i) q^{20} +(-36.0000 - 62.3538i) q^{22} +(42.4264 + 73.4847i) q^{23} +(-81.5000 + 141.162i) q^{25} -123.037 q^{26} +110.000 q^{28} +(135.765 - 235.151i) q^{29} +(134.000 + 232.095i) q^{31} +(-127.279 - 220.454i) q^{32} +(108.000 - 187.061i) q^{34} +186.676 q^{35} +83.0000 q^{37} +(61.5183 - 106.553i) q^{38} +(72.0000 + 124.708i) q^{40} +(-135.765 - 235.151i) q^{41} +(116.000 - 200.918i) q^{43} -169.706 q^{44} +360.000 q^{46} +(-195.161 + 338.030i) q^{47} +(111.000 + 192.258i) q^{49} +(345.775 + 598.900i) q^{50} +(-145.000 + 251.147i) q^{52} -305.470 q^{53} -288.000 q^{55} +(46.6690 - 80.8332i) q^{56} +(-576.000 - 997.661i) q^{58} +(144.250 + 249.848i) q^{59} +(-383.500 + 664.241i) q^{61} +1137.03 q^{62} -728.000 q^{64} +(-246.073 + 426.211i) q^{65} +(255.500 + 442.539i) q^{67} +(-254.558 - 440.908i) q^{68} +(396.000 - 685.892i) q^{70} -712.764 q^{71} +137.000 q^{73} +(176.070 - 304.961i) q^{74} +(-145.000 - 251.147i) q^{76} +(93.3381 + 161.666i) q^{77} +(237.500 - 411.362i) q^{79} -746.705 q^{80} -1152.00 q^{82} +(288.500 - 499.696i) q^{83} +(-432.000 - 748.246i) q^{85} +(-492.146 - 852.422i) q^{86} +(-72.0000 + 124.708i) q^{88} +254.558 q^{89} +319.000 q^{91} +(424.264 - 734.847i) q^{92} +(828.000 + 1434.14i) q^{94} +(-246.073 - 426.211i) q^{95} +(-410.500 + 711.007i) q^{97} +941.866 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{4} - 22 q^{7} - 288 q^{10} - 58 q^{13} + 88 q^{16} + 116 q^{19} - 144 q^{22} - 326 q^{25} + 440 q^{28} + 536 q^{31} + 432 q^{34} + 332 q^{37} + 288 q^{40} + 464 q^{43} + 1440 q^{46} + 444 q^{49}+ \cdots - 1642 q^{97}+O(q^{100}) \) 4 * q - 20 * q^4 - 22 * q^7 - 288 * q^10 - 58 * q^13 + 88 * q^16 + 116 * q^19 - 144 * q^22 - 326 * q^25 + 440 * q^28 + 536 * q^31 + 432 * q^34 + 332 * q^37 + 288 * q^40 + 464 * q^43 + 1440 * q^46 + 444 * q^49 - 580 * q^52 - 1152 * q^55 - 2304 * q^58 - 1534 * q^61 - 2912 * q^64 + 1022 * q^67 + 1584 * q^70 + 548 * q^73 - 580 * q^76 + 950 * q^79 - 4608 * q^82 - 1728 * q^85 - 288 * q^88 + 1276 * q^91 + 3312 * q^94 - 1642 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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