LMFDB - Embedded newform 144.4.k.a.109.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,4,Mod(37,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("144.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$8.49627504083$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - x^{8} + 6x^{7} + 14x^{6} - 80x^{5} + 56x^{4} + 96x^{3} - 64x^{2} - 512x + 1024 $ x^10 - 2x^9 - x^8 + 6x^7 + 14x^6 - 80x^5 + 56x^4 + 96x^3 - 64x^2 - 512x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			109.5
Root			$1.28199 + 1.53509i$ of defining polynomial
Character	$\chi$	$=$	144.109
Dual form			144.4.k.a.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.81708 - 0.253099i) q^{2} +(7.87188 - 1.42600i) q^{4} +(4.66372 - 4.66372i) q^{5} -24.8965i q^{7} +(21.8148 - 6.00953i) q^{8} +(11.9577 - 14.3185i) q^{10} +(-22.3431 + 22.3431i) q^{11} +(-11.2714 - 11.2714i) q^{13} +(-6.30129 - 70.1355i) q^{14} +(59.9330 - 22.4506i) q^{16} +88.4846 q^{17} +(37.8187 + 37.8187i) q^{19} +(30.0618 - 43.3627i) q^{20} +(-57.2873 + 68.5974i) q^{22} -48.1224i q^{23} +81.4994i q^{25} +(-34.6053 - 28.8997i) q^{26} +(-35.5025 - 195.982i) q^{28} +(-10.4432 - 10.4432i) q^{29} -96.9578 q^{31} +(163.154 - 78.4142i) q^{32} +(249.268 - 22.3954i) q^{34} +(-116.110 - 116.110i) q^{35} +(-163.279 + 163.279i) q^{37} +(116.110 + 96.9665i) q^{38} +(73.7114 - 129.765i) q^{40} +360.519i q^{41} +(-100.249 + 100.249i) q^{43} +(-144.021 + 207.744i) q^{44} +(-12.1797 - 135.565i) q^{46} -220.669 q^{47} -276.837 q^{49} +(20.6274 + 229.590i) q^{50} +(-104.800 - 72.6542i) q^{52} +(175.752 - 175.752i) q^{53} +208.404i q^{55} +(-149.616 - 543.113i) q^{56} +(-32.0624 - 26.7761i) q^{58} +(-405.008 + 405.008i) q^{59} +(664.576 + 664.576i) q^{61} +(-273.138 + 24.5399i) q^{62} +(439.771 - 262.193i) q^{64} -105.133 q^{65} +(-107.377 - 107.377i) q^{67} +(696.540 - 126.179i) q^{68} +(-356.480 - 297.705i) q^{70} -215.050i q^{71} -668.587i q^{73} +(-418.644 + 501.296i) q^{74} +(351.634 + 243.775i) q^{76} +(556.266 + 556.266i) q^{77} -822.956 q^{79} +(174.808 - 384.215i) q^{80} +(91.2471 + 1015.61i) q^{82} +(-326.873 - 326.873i) q^{83} +(412.668 - 412.668i) q^{85} +(-257.037 + 307.783i) q^{86} +(-353.139 + 621.682i) q^{88} +262.733i q^{89} +(-280.619 + 280.619i) q^{91} +(-68.6226 - 378.814i) q^{92} +(-621.643 + 55.8512i) q^{94} +352.752 q^{95} -150.801 q^{97} +(-779.871 + 70.0671i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{2} + 8 q^{4} + 2 q^{5} + 44 q^{8} - 68 q^{10} - 18 q^{11} - 2 q^{13} - 188 q^{14} + 280 q^{16} + 4 q^{17} - 26 q^{19} + 196 q^{20} - 588 q^{22} + 264 q^{26} + 280 q^{28} + 202 q^{29} + 368 q^{31}+ \cdots - 314 q^{98}+O(q^{100}) $ 10 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 + 44 * q^8 - 68 * q^10 - 18 * q^11 - 2 * q^13 - 188 * q^14 + 280 * q^16 + 4 * q^17 - 26 * q^19 + 196 * q^20 - 588 * q^22 + 264 * q^26 + 280 * q^28 + 202 * q^29 + 368 * q^31 - 968 * q^32 + 436 * q^34 - 476 * q^35 - 10 * q^37 + 1232 * q^38 - 1336 * q^40 - 838 * q^43 - 868 * q^44 + 1132 * q^46 + 944 * q^47 + 94 * q^49 - 726 * q^50 - 236 * q^52 + 378 * q^53 + 488 * q^56 + 8 * q^58 - 1706 * q^59 + 910 * q^61 + 80 * q^62 + 512 * q^64 + 492 * q^65 + 1942 * q^67 + 880 * q^68 + 160 * q^70 + 452 * q^74 - 1228 * q^76 + 268 * q^77 - 4416 * q^79 + 2648 * q^80 - 704 * q^82 + 2562 * q^83 - 12 * q^85 - 3764 * q^86 + 1528 * q^88 + 3332 * q^91 - 632 * q^92 - 3248 * q^94 - 6900 * q^95 - 4 * q^97 - 314 * q^98

Character values

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

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

Coefficient data

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

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

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

$2$	2.81708	−	0.253099i	0.995988	−	0.0894841i
$2$	2.81708	−	0.253099i	0.995988	−	0.0894841i
$3$		0			0
$3$		0			0
$4$	7.87188	−	1.42600i	0.983985	−	0.178250i
$5$	4.66372	−	4.66372i	0.417136	−	0.417136i	−0.467079	−	0.884215i	$-0.654694\pi$
$5$	4.66372	−	4.66372i	0.417136	−	0.417136i	0.884215	+	0.467079i	$0.154694\pi$
$6$		0			0
$7$		−	24.8965i		−	1.34429i	−0.740422	−	0.672143i	$-0.765374\pi$
$7$		−	24.8965i		−	1.34429i	0.740422	−	0.672143i	$-0.234626\pi$
$8$	21.8148	−	6.00953i	0.964087	−	0.265586i
$9$		0			0
$10$	11.9577	−	14.3185i	0.378136	−	0.452790i
$11$	−22.3431	+	22.3431i	−0.612427	+	0.612427i	−0.943578	−	0.331151i	$-0.892563\pi$
$11$	−22.3431	+	22.3431i	−0.612427	+	0.612427i	0.331151	+	0.943578i	$0.392563\pi$
$12$		0			0
$13$	−11.2714	−	11.2714i	−0.240471	−	0.240471i	0.576574	−	0.817045i	$-0.304389\pi$
$13$	−11.2714	−	11.2714i	−0.240471	−	0.240471i	−0.817045	+	0.576574i	$0.804389\pi$
$14$	−6.30129	−	70.1355i	−0.120292	−	1.33889i
$15$		0			0
$16$	59.9330	−	22.4506i	0.936454	−	0.350791i
$17$	88.4846			1.26239			0.631196	−	0.775623i	$-0.282564\pi$
$17$	88.4846			1.26239			0.631196	+	0.775623i	$0.282564\pi$
$18$		0			0
$19$	37.8187	+	37.8187i	0.456643	+	0.456643i	0.897552	−	0.440909i	$-0.145344\pi$
$19$	37.8187	+	37.8187i	0.456643	+	0.456643i	−0.440909	+	0.897552i	$0.645344\pi$
$20$	30.0618	−	43.3627i	0.336101	−	0.484810i
$21$		0			0
$22$	−57.2873	+	68.5974i	−0.555168	+	0.664773i
$23$		−	48.1224i		−	0.436270i	−0.975919	−	0.218135i	$-0.930003\pi$
$23$		−	48.1224i		−	0.436270i	0.975919	−	0.218135i	$-0.0699973\pi$
$24$		0			0
$25$			81.4994i			0.651995i
$26$	−34.6053	−	28.8997i	−0.261025	−	0.217988i
$27$		0			0
$28$	−35.5025	−	195.982i	−0.239619	−	1.32276i
$29$	−10.4432	−	10.4432i	−0.0668705	−	0.0668705i	0.672881	−	0.739751i	$-0.265057\pi$
$29$	−10.4432	−	10.4432i	−0.0668705	−	0.0668705i	−0.739751	+	0.672881i	$0.765057\pi$
$30$		0			0
$31$	−96.9578			−0.561746			−0.280873	−	0.959745i	$-0.590624\pi$
$31$	−96.9578			−0.561746			−0.280873	+	0.959745i	$0.590624\pi$
$32$	163.154	−	78.4142i	0.901307	−	0.433181i
$33$		0			0
$34$	249.268	−	22.3954i	1.25733	−	0.112964i
$35$	−116.110	−	116.110i	−0.560750	−	0.560750i
$36$		0			0
$37$	−163.279	+	163.279i	−0.725484	+	0.725484i	−0.969717	−	0.244233i	$-0.921464\pi$
$37$	−163.279	+	163.279i	−0.725484	+	0.725484i	0.244233	+	0.969717i	$0.421464\pi$
$38$	116.110	+	96.9665i	0.495673	+	0.413949i
$39$		0			0
$40$	73.7114	−	129.765i	0.291370	−	0.512941i
$41$			360.519i			1.37326i	0.727008	+	0.686629i	$0.240910\pi$
$41$			360.519i			1.37326i	−0.727008	+	0.686629i	$0.759090\pi$
$42$		0			0
$43$	−100.249	+	100.249i	−0.355531	+	0.355531i	−0.862163	−	0.506632i	$-0.830890\pi$
$43$	−100.249	+	100.249i	−0.355531	+	0.355531i	0.506632	+	0.862163i	$0.330890\pi$
$44$	−144.021	+	207.744i	−0.493454	+	0.711785i
$45$		0			0
$46$	−12.1797	−	135.565i	−0.0390392	−	0.434520i
$47$	−220.669			−0.684849			−0.342425	−	0.939545i	$-0.611248\pi$
$47$	−220.669			−0.684849			−0.342425	+	0.939545i	$0.611248\pi$
$48$		0			0
$49$	−276.837			−0.807104
$50$	20.6274	+	229.590i	0.0583432	+	0.649379i
$51$		0			0
$52$	−104.800	−	72.6542i	−0.279484	−	0.193756i
$53$	175.752	−	175.752i	0.455498	−	0.455498i	−0.441676	−	0.897174i	$-0.645616\pi$
$53$	175.752	−	175.752i	0.455498	−	0.455498i	0.897174	+	0.441676i	$0.145616\pi$
$54$		0			0
$55$			208.404i			0.510931i
$56$	−149.616	−	543.113i	−0.357024	−	1.29601i
$57$		0			0
$58$	−32.0624	−	26.7761i	−0.0725861	−	0.0606184i
$59$	−405.008	+	405.008i	−0.893687	+	0.893687i	−0.994868	−	0.101181i	$-0.967738\pi$
$59$	−405.008	+	405.008i	−0.893687	+	0.893687i	0.101181	+	0.994868i	$0.467738\pi$
$60$		0			0
$61$	664.576	+	664.576i	1.39492	+	1.39492i	0.813860	+	0.581061i	$0.197362\pi$
$61$	664.576	+	664.576i	1.39492	+	1.39492i	0.581061	+	0.813860i	$0.302638\pi$
$62$	−273.138	+	24.5399i	−0.559493	+	0.0502674i
$63$		0			0
$64$	439.771	−	262.193i	0.858928	−	0.512096i
$65$	−105.133			−0.200619
$66$		0			0
$67$	−107.377	−	107.377i	−0.195794	−	0.195794i	0.602400	−	0.798194i	$-0.294211\pi$
$67$	−107.377	−	107.377i	−0.195794	−	0.195794i	−0.798194	+	0.602400i	$0.794211\pi$
$68$	696.540	−	126.179i	1.24218	−	0.225022i
$69$		0			0
$70$	−356.480	−	297.705i	−0.608679	−	0.508322i
$71$		−	215.050i		−	0.359461i	−0.983716	−	0.179731i	$-0.942477\pi$
$71$		−	215.050i		−	0.359461i	0.983716	−	0.179731i	$-0.0575226\pi$
$72$		0			0
$73$		−	668.587i		−	1.07195i	−0.844235	−	0.535974i	$-0.819945\pi$
$73$		−	668.587i		−	1.07195i	0.844235	−	0.535974i	$-0.180055\pi$
$74$	−418.644	+	501.296i	−0.657654	+	0.787492i
$75$		0			0
$76$	351.634	+	243.775i	0.530726	+	0.367933i
$77$	556.266	+	556.266i	0.823277	+	0.823277i
$78$		0			0
$79$	−822.956			−1.17202			−0.586012	−	0.810303i	$-0.699303\pi$
$79$	−822.956			−1.17202			−0.586012	+	0.810303i	$0.699303\pi$
$80$	174.808	−	384.215i	0.244301	−	0.536956i
$81$		0			0
$82$	91.2471	+	1015.61i	0.122885	+	1.36775i
$83$	−326.873	−	326.873i	−0.432277	−	0.432277i	0.457125	−	0.889402i	$-0.348879\pi$
$83$	−326.873	−	326.873i	−0.432277	−	0.432277i	−0.889402	+	0.457125i	$0.848879\pi$
$84$		0			0
$85$	412.668	−	412.668i	0.526589	−	0.526589i
$86$	−257.037	+	307.783i	−0.322290	+	0.385919i
$87$		0			0
$88$	−353.139	+	621.682i	−0.427781	+	0.753086i
$89$			262.733i			0.312918i	0.987684	+	0.156459i	$0.0500079\pi$
$89$			262.733i			0.312918i	−0.987684	+	0.156459i	$0.949992\pi$
$90$		0			0
$91$	−280.619	+	280.619i	−0.323262	+	0.323262i
$92$	−68.6226	−	378.814i	−0.0777652	−	0.429283i
$93$		0			0
$94$	−621.643	+	55.8512i	−0.682102	+	0.0612831i
$95$	352.752			0.380964
$96$		0			0
$97$	−150.801			−0.157850			−0.0789251	−	0.996881i	$-0.525149\pi$
$97$	−150.801			−0.157850			−0.0789251	+	0.996881i	$0.525149\pi$
$98$	−779.871	+	70.0671i	−0.803866	+	0.0722229i
$99$		0			0
$100$	116.218	+	641.553i	0.116218	+	0.641553i
$101$	−487.985	+	487.985i	−0.480755	+	0.480755i	−0.905373	−	0.424617i	$-0.860409\pi$
$101$	−487.985	+	487.985i	−0.480755	+	0.480755i	0.424617	+	0.905373i	$0.360409\pi$
$102$		0			0
$103$		−	1840.58i		−	1.76075i	−0.474275	−	0.880377i	$-0.657290\pi$
$103$		−	1840.58i		−	1.76075i	0.474275	−	0.880377i	$-0.342710\pi$
$104$	−313.620	−	178.148i	−0.295701	−	0.167970i
$105$		0			0
$106$	450.625	−	539.590i	0.412911	−	0.494431i
$107$	79.4098	−	79.4098i	0.0717461	−	0.0717461i	−0.670323	−	0.742069i	$-0.733845\pi$
$107$	79.4098	−	79.4098i	0.0717461	−	0.0717461i	0.742069	+	0.670323i	$0.233845\pi$
$108$		0			0
$109$	952.979	+	952.979i	0.837421	+	0.837421i	0.988519	−	0.151098i	$-0.0482809\pi$
$109$	952.979	+	952.979i	0.837421	+	0.837421i	−0.151098	+	0.988519i	$0.548281\pi$
$110$	52.7469	+	587.091i	0.0457202	+	0.508881i
$111$		0			0
$112$	−558.942	−	1492.12i	−0.471563	−	1.25886i
$113$	720.469			0.599788			0.299894	−	0.953973i	$-0.403049\pi$
$113$	720.469			0.599788			0.299894	+	0.953973i	$0.403049\pi$
$114$		0			0
$115$	−224.430	−	224.430i	−0.181984	−	0.181984i
$116$	−97.0993	−	67.3153i	−0.0777193	−	0.0538799i
$117$		0			0
$118$	−1038.43	+	1243.45i	−0.810131	+	0.970072i
$119$		−	2202.96i		−	1.69702i
$120$		0			0
$121$			332.571i			0.249865i
$122$	2040.37	+	1703.96i	1.51415	+	1.26450i
$123$		0			0
$124$	−763.240	+	138.262i	−0.552750	+	0.100131i
$125$	963.056	+	963.056i	0.689107	+	0.689107i
$126$		0			0
$127$	2622.35			1.83225			0.916124	−	0.400895i	$-0.131301\pi$
$127$	2622.35			1.83225			0.916124	+	0.400895i	$0.131301\pi$
$128$	1172.51	−	849.925i	0.809658	−	0.586902i
$129$		0			0
$130$	−296.169	+	26.6092i	−0.199814	+	0.0179522i
$131$	−657.574	−	657.574i	−0.438569	−	0.438569i	0.452961	−	0.891530i	$-0.350368\pi$
$131$	−657.574	−	657.574i	−0.438569	−	0.438569i	−0.891530	+	0.452961i	$0.850368\pi$
$132$		0			0
$133$	941.555	−	941.555i	0.613858	−	0.613858i
$134$	−329.666	−	275.312i	−0.212529	−	0.177488i
$135$		0			0
$136$	1930.27	−	531.751i	1.21706	−	0.335274i
$137$		−	2511.52i		−	1.56623i	−0.621874	−	0.783117i	$-0.713629\pi$
$137$		−	2511.52i		−	1.56623i	0.621874	−	0.783117i	$-0.286371\pi$
$138$		0			0
$139$	−1086.02	+	1086.02i	−0.662697	+	0.662697i	−0.956015	−	0.293318i	$-0.905241\pi$
$139$	−1086.02	+	1086.02i	−0.662697	+	0.662697i	0.293318	+	0.956015i	$0.405241\pi$
$140$	−1079.58	−	748.434i	−0.651723	−	0.451816i
$141$		0			0
$142$	−54.4290	−	605.813i	−0.0321661	−	0.358019i
$143$	503.677			0.294543
$144$		0			0
$145$	−97.4080			−0.0557882
$146$	−169.219	−	1883.46i	−0.0959222	−	1.06765i
$147$		0			0
$148$	−1052.48	+	1518.15i	−0.584547	+	0.843183i
$149$	2284.63	−	2284.63i	1.25614	−	1.25614i	0.303214	−	0.952922i	$-0.401940\pi$
$149$	2284.63	−	2284.63i	1.25614	−	1.25614i	0.952922	−	0.303214i	$-0.0980598\pi$
$150$		0			0
$151$		−	2814.39i		−	1.51677i	−0.651809	−	0.758383i	$-0.725990\pi$
$151$		−	2814.39i		−	1.51677i	0.651809	−	0.758383i	$-0.274010\pi$
$152$	1052.28	+	597.735i	0.561521	+	0.318965i
$153$		0			0
$154$	1707.84	+	1426.25i	0.893645	+	0.746304i
$155$	−452.184	+	452.184i	−0.234325	+	0.234325i
$156$		0			0
$157$	−906.308	−	906.308i	−0.460709	−	0.460709i	0.438179	−	0.898888i	$-0.355624\pi$
$157$	−906.308	−	906.308i	−0.460709	−	0.460709i	−0.898888	+	0.438179i	$0.855624\pi$
$158$	−2318.33	+	208.290i	−1.16732	+	0.104877i
$159$		0			0
$160$	395.203	−	1126.61i	0.195272	−	0.556663i
$161$	−1198.08			−0.586472
$162$		0			0
$163$	1392.36	+	1392.36i	0.669067	+	0.669067i	0.957500	−	0.288433i	$-0.0931342\pi$
$163$	1392.36	+	1392.36i	0.669067	+	0.669067i	−0.288433	+	0.957500i	$0.593134\pi$
$164$	514.101	+	2837.96i	0.244784	+	1.35127i
$165$		0			0
$166$	−1003.56	−	838.097i	−0.469225	−	0.391861i
$167$			1221.66i			0.566075i	0.959109	+	0.283038i	$0.0913421\pi$
$167$			1221.66i			0.566075i	−0.959109	+	0.283038i	$0.908658\pi$
$168$		0			0
$169$		−	1942.91i		−	0.884347i
$170$	1058.07	−	1266.96i	0.477356	−	0.571598i
$171$		0			0
$172$	−646.193	+	932.104i	−0.286464	+	0.413211i
$173$	563.418	+	563.418i	0.247606	+	0.247606i	0.819988	−	0.572381i	$-0.193980\pi$
$173$	563.418	+	563.418i	0.247606	+	0.247606i	−0.572381	+	0.819988i	$0.693980\pi$
$174$		0			0
$175$	2029.05			0.876468
$176$	−837.474	+	1840.71i	−0.358676	+	0.788344i
$177$		0			0
$178$	66.4976	+	740.141i	0.0280012	+	0.311662i
$179$	2202.23	+	2202.23i	0.919565	+	0.919565i	0.996998	−	0.0774329i	$-0.0246724\pi$
$179$	2202.23	+	2202.23i	0.919565	+	0.919565i	−0.0774329	+	0.996998i	$0.524672\pi$
$180$		0			0
$181$	121.294	−	121.294i	0.0498104	−	0.0498104i	−0.681763	−	0.731573i	$-0.738786\pi$
$181$	121.294	−	121.294i	0.0498104	−	0.0498104i	0.731573	+	0.681763i	$0.238786\pi$
$182$	−719.502	+	861.551i	−0.293039	+	0.350892i
$183$		0			0
$184$	−289.193	−	1049.78i	−0.115867	−	0.420602i
$185$			1522.98i			0.605251i
$186$		0			0
$187$	−1977.02	+	1977.02i	−0.773124	+	0.773124i
$188$	−1737.08	+	314.675i	−0.673882	+	0.122074i
$189$		0			0
$190$	993.731	−	89.2813i	0.379436	−	0.0340902i
$191$	−3927.65			−1.48793			−0.743966	−	0.668218i	$-0.767058\pi$
$191$	−3927.65			−1.48793			−0.743966	+	0.668218i	$0.767058\pi$
$192$		0			0
$193$	−3249.02			−1.21176			−0.605880	−	0.795556i	$-0.707179\pi$
$193$	−3249.02			−1.21176			−0.605880	+	0.795556i	$0.707179\pi$
$194$	−424.817	+	38.1675i	−0.157217	+	0.0141251i
$195$		0			0
$196$	−2179.23	+	394.769i	−0.794178	+	0.143866i
$197$	−2420.90	+	2420.90i	−0.875545	+	0.875545i	−0.993070	−	0.117525i	$-0.962504\pi$
$197$	−2420.90	+	2420.90i	−0.875545	+	0.875545i	0.117525	+	0.993070i	$0.462504\pi$
$198$		0			0
$199$			1371.30i			0.488488i	0.969714	+	0.244244i	$0.0785397\pi$
$199$			1371.30i			0.488488i	−0.969714	+	0.244244i	$0.921460\pi$
$200$	489.773	+	1777.89i	0.173161	+	0.628580i
$201$		0			0
$202$	−1251.18	+	1498.20i	−0.435807	+	0.521847i
$203$	−259.998	+	259.998i	−0.0898931	+	0.0898931i
$204$		0			0
$205$	1681.36	+	1681.36i	0.572836	+	0.572836i
$206$	−465.849	−	5185.06i	−0.157559	−	1.75369i
$207$		0			0
$208$	−928.580	−	422.480i	−0.309546	−	0.140835i
$209$	−1689.98			−0.559321
$210$		0			0
$211$	−1620.50	−	1620.50i	−0.528719	−	0.528719i	0.391471	−	0.920190i	$-0.371966\pi$
$211$	−1620.50	−	1620.50i	−0.528719	−	0.528719i	−0.920190	+	0.391471i	$0.871966\pi$
$212$	1132.88	−	1634.12i	0.367011	−	0.529396i
$213$		0			0
$214$	203.605	−	243.802i	0.0650382	−	0.0778784i
$215$			935.068i			0.296610i
$216$		0			0
$217$			2413.91i			0.755148i
$218$	2925.82	+	2443.42i	0.908997	+	0.759125i
$219$		0			0
$220$	297.185	+	1640.53i	0.0910736	+	0.502749i
$221$	−997.347	−	997.347i	−0.303569	−	0.303569i
$222$		0			0
$223$	−419.617			−0.126007			−0.0630036	−	0.998013i	$-0.520068\pi$
$223$	−419.617			−0.126007			−0.0630036	+	0.998013i	$0.520068\pi$
$224$	−1952.24	−	4061.97i	−0.582320	−	1.21161i
$225$		0			0
$226$	2029.62	−	182.350i	0.597381	−	0.0536714i
$227$	2133.64	+	2133.64i	0.623853	+	0.623853i	0.946515	−	0.322661i	$-0.104577\pi$
$227$	2133.64	+	2133.64i	0.623853	+	0.623853i	−0.322661	+	0.946515i	$0.604577\pi$
$228$		0			0
$229$	−1574.42	+	1574.42i	−0.454325	+	0.454325i	−0.896787	−	0.442462i	$-0.854105\pi$
$229$	−1574.42	+	1574.42i	−0.454325	+	0.454325i	0.442462	+	0.896787i	$0.354105\pi$
$230$	−689.039	−	575.433i	−0.197539	−	0.164969i
$231$		0			0
$232$	−290.574	−	165.057i	−0.0822289	−	0.0467091i
$233$		−	1194.86i		−	0.335957i	−0.985791	−	0.167978i	$-0.946276\pi$
$233$		−	1194.86i		−	0.335957i	0.985791	−	0.167978i	$-0.0537239\pi$
$234$		0			0
$235$	−1029.14	+	1029.14i	−0.285675	+	0.285675i
$236$	−2610.63	+	3765.71i	−0.720075	+	1.03867i
$237$		0			0
$238$	−557.567	−	6205.91i	−0.151856	−	1.69021i
$239$	4241.03			1.14782			0.573911	−	0.818917i	$-0.305425\pi$
$239$	4241.03			1.14782			0.573911	+	0.818917i	$0.305425\pi$
$240$		0			0
$241$	−5571.19			−1.48910			−0.744548	−	0.667569i	$-0.767335\pi$
$241$	−5571.19			−1.48910			−0.744548	+	0.667569i	$0.767335\pi$
$242$	84.1734	+	936.878i	0.0223590	+	0.248863i
$243$		0			0
$244$	6179.15	+	4283.77i	1.62123	+	1.12394i
$245$	−1291.09	+	1291.09i	−0.336672	+	0.336672i
$246$		0			0
$247$		−	852.541i		−	0.219619i
$248$	−2115.12	+	582.671i	−0.541572	+	0.149192i
$249$		0			0
$250$	2956.75	+	2469.26i	0.748006	+	0.624678i
$251$	482.728	−	482.728i	0.121393	−	0.121393i	−0.643801	−	0.765193i	$-0.722643\pi$
$251$	482.728	−	482.728i	0.121393	−	0.121393i	0.765193	+	0.643801i	$0.222643\pi$
$252$		0			0
$253$	1075.20	+	1075.20i	0.267184	+	0.267184i
$254$	7387.36	−	663.713i	1.82490	−	0.163957i
$255$		0			0
$256$	3087.94	−	2691.07i	0.753891	−	0.656999i
$257$	−8093.12			−1.96434			−0.982169	−	0.188002i	$-0.939799\pi$
$257$	−8093.12			−1.96434			−0.982169	+	0.188002i	$0.939799\pi$
$258$		0			0
$259$	4065.08	+	4065.08i	0.975257	+	0.975257i
$260$	−827.598	+	149.921i	−0.197406	+	0.0357603i
$261$		0			0
$262$	−2018.87	−	1686.01i	−0.476054	−	0.397564i
$263$			410.300i			0.0961984i	0.998843	+	0.0480992i	$0.0153164\pi$
$263$			410.300i			0.0961984i	−0.998843	+	0.0480992i	$0.984684\pi$
$264$		0			0
$265$		−	1639.32i		−	0.380009i
$266$	2414.13	−	2890.74i	0.556465	−	0.666326i
$267$		0			0
$268$	−998.378	−	692.139i	−0.227558	−	0.157758i
$269$	4.77962	+	4.77962i	0.00108334	+	0.00108334i	0.707648	−	0.706565i	$-0.249756\pi$
$269$	4.77962	+	4.77962i	0.00108334	+	0.00108334i	−0.706565	+	0.707648i	$0.749756\pi$
$270$		0			0
$271$	−2833.98			−0.635247			−0.317623	−	0.948217i	$-0.602885\pi$
$271$	−2833.98			−0.635247			−0.317623	+	0.948217i	$0.602885\pi$
$272$	5303.15	−	1986.54i	1.18217	−	0.442836i
$273$		0			0
$274$	−635.665	−	7075.17i	−0.140153	−	1.55995i
$275$	−1820.95	−	1820.95i	−0.399300	−	0.399300i
$276$		0			0
$277$	1525.80	−	1525.80i	0.330962	−	0.330962i	−0.521989	−	0.852952i	$-0.674810\pi$
$277$	1525.80	−	1525.80i	0.330962	−	0.330962i	0.852952	+	0.521989i	$0.174810\pi$
$278$	−2784.53	+	3334.27i	−0.600738	+	0.719339i
$279$		0			0
$280$	−3230.70	−	1835.16i	−0.689539	−	0.391684i
$281$		−	4750.23i		−	1.00845i	−0.863572	−	0.504226i	$-0.831778\pi$
$281$		−	4750.23i		−	1.00845i	0.863572	−	0.504226i	$-0.168222\pi$
$282$		0			0
$283$	644.104	−	644.104i	0.135293	−	0.135293i	−0.636217	−	0.771510i	$-0.719502\pi$
$283$	644.104	−	644.104i	0.135293	−	0.135293i	0.771510	+	0.636217i	$0.219502\pi$
$284$	−306.662	−	1692.85i	−0.0640740	−	0.353705i
$285$		0			0
$286$	1418.90	−	127.480i	0.293361	−	0.0263569i
$287$	8975.67			1.84605
$288$		0			0
$289$	2916.53			0.593635
$290$	−274.406	+	24.6539i	−0.0555644	+	0.00499216i
$291$		0			0
$292$	−953.405	−	5263.03i	−0.191075	−	1.05478i
$293$	1433.16	−	1433.16i	0.285755	−	0.285755i	−0.549644	−	0.835399i	$-0.685237\pi$
$293$	1433.16	−	1433.16i	0.285755	−	0.285755i	0.835399	+	0.549644i	$0.185237\pi$
$294$		0			0
$295$			3777.69i			0.745578i
$296$	−2580.67	+	4543.13i	−0.506751	+	0.892108i
$297$		0			0
$298$	5857.75	−	7014.23i	1.13869	−	1.36350i
$299$	−542.407	+	542.407i	−0.104910	+	0.104910i
$300$		0			0
$301$	2495.85	+	2495.85i	0.477935	+	0.477935i
$302$	−712.319	−	7928.36i	−0.135726	−	1.51068i
$303$		0			0
$304$	3115.65	+	1417.54i	0.587811	+	0.267439i
$305$	6198.79			1.16374
$306$		0			0
$307$	−231.211	−	231.211i	−0.0429834	−	0.0429834i	0.685288	−	0.728272i	$-0.259676\pi$
$307$	−231.211	−	231.211i	−0.0429834	−	0.0429834i	−0.728272	+	0.685288i	$0.759676\pi$
$308$	5172.09	+	3585.62i	0.956842	+	0.663343i
$309$		0			0
$310$	−1159.39	+	1388.29i	−0.212416	+	0.254353i
$311$		−	871.410i		−	0.158885i	−0.996839	−	0.0794423i	$-0.974686\pi$
$311$		−	871.410i		−	0.158885i	0.996839	−	0.0794423i	$-0.0253140\pi$
$312$		0			0
$313$		−	3515.02i		−	0.634762i	−0.948298	−	0.317381i	$-0.897197\pi$
$313$		−	3515.02i		−	0.634762i	0.948298	−	0.317381i	$-0.102803\pi$
$314$	−2782.53	−	2323.76i	−0.500087	−	0.417634i
$315$		0			0
$316$	−6478.22	+	1173.54i	−1.15325	+	0.208913i
$317$	−4723.77	−	4723.77i	−0.836951	−	0.836951i	0.151506	−	0.988456i	$-0.451588\pi$
$317$	−4723.77	−	4723.77i	−0.836951	−	0.836951i	−0.988456	+	0.151506i	$0.951588\pi$
$318$		0			0
$319$	466.665			0.0819067
$320$	828.174	−	3273.77i	0.144676	−	0.571904i
$321$		0			0
$322$	−3375.09	+	303.233i	−0.584119	+	0.0524799i
$323$	3346.38	+	3346.38i	0.576462	+	0.576462i
$324$		0			0
$325$	918.613	−	918.613i	0.156786	−	0.156786i
$326$	4274.79	+	3569.98i	0.726254	+	0.606512i
$327$		0			0
$328$	2166.55	+	7864.65i	0.364718	+	1.32394i
$329$			5493.90i			0.920633i
$330$		0			0
$331$	−1820.26	+	1820.26i	−0.302268	+	0.302268i	−0.841901	−	0.539633i	$-0.818563\pi$
$331$	−1820.26	+	1820.26i	−0.302268	+	0.302268i	0.539633	+	0.841901i	$0.318563\pi$
$332$	−3039.23	−	2106.99i	−0.502408	−	0.348301i
$333$		0			0
$334$	309.200	+	3441.50i	0.0506547	+	0.563804i
$335$	−1001.55			−0.163345
$336$		0			0
$337$	74.0970			0.0119772			0.00598861	−	0.999982i	$-0.498094\pi$
$337$	74.0970			0.0119772			0.00598861	+	0.999982i	$0.498094\pi$
$338$	−491.749	−	5473.33i	−0.0791350	−	0.880799i
$339$		0			0
$340$	2660.01	−	3836.94i	0.424292	−	0.612021i
$341$	2166.34	−	2166.34i	0.344029	−	0.344029i
$342$		0			0
$343$		−	1647.24i		−	0.259307i
$344$	−1584.46	+	2789.36i	−0.248339	+	0.437187i
$345$		0			0
$346$	1729.80	+	1444.59i	0.268770	+	0.224456i
$347$	−2102.73	+	2102.73i	−0.325305	+	0.325305i	−0.850798	−	0.525493i	$-0.823881\pi$
$347$	−2102.73	+	2102.73i	−0.325305	+	0.325305i	0.525493	+	0.850798i	$0.323881\pi$
$348$		0			0
$349$	−6612.85	−	6612.85i	−1.01426	−	1.01426i	−0.999897	−	0.0143661i	$-0.995427\pi$
$349$	−6612.85	−	6612.85i	−1.01426	−	1.01426i	−0.0143661	−	0.999897i	$-0.504573\pi$
$350$	5716.00	−	513.551i	0.872951	−	0.0784299i
$351$		0			0
$352$	−1893.35	+	5397.38i	−0.286693	+	0.817277i
$353$	−2216.90			−0.334259			−0.167130	−	0.985935i	$-0.553450\pi$
$353$	−2216.90			−0.334259			−0.167130	+	0.985935i	$0.553450\pi$
$354$		0			0
$355$	−1002.93	−	1002.93i	−0.149944	−	0.149944i
$356$	374.658	+	2068.21i	0.0557776	+	0.307906i
$357$		0			0
$358$	6761.23	+	5646.46i	0.998162	+	0.833589i
$359$			2082.23i			0.306117i	0.988217	+	0.153059i	$0.0489123\pi$
$359$			2082.23i			0.306117i	−0.988217	+	0.153059i	$0.951088\pi$
$360$		0			0
$361$		−	3998.49i		−	0.582955i
$362$	310.995	−	372.393i	0.0451533	−	0.0540678i
$363$		0			0
$364$	−1808.84	+	2609.16i	−0.260464	+	0.375707i
$365$	−3118.10	−	3118.10i	−0.447148	−	0.447148i
$366$		0			0
$367$	4509.22			0.641360			0.320680	−	0.947188i	$-0.396089\pi$
$367$	4509.22			0.641360			0.320680	+	0.947188i	$0.396089\pi$
$368$	−1080.38	−	2884.12i	−0.153040	−	0.408547i
$369$		0			0
$370$	385.464	+	4290.34i	0.0541603	+	0.602823i
$371$	−4375.61	−	4375.61i	−0.612320	−	0.612320i
$372$		0			0
$373$	−8661.56	+	8661.56i	−1.20236	+	1.20236i	−0.228908	+	0.973448i	$0.573515\pi$
$373$	−8661.56	+	8661.56i	−1.20236	+	1.20236i	−0.973448	+	0.228908i	$0.926485\pi$
$374$	−5069.05	+	6069.81i	−0.700840	+	0.839205i
$375$		0			0
$376$	−4813.86	+	1326.12i	−0.660254	+	0.181886i
$377$			235.418i			0.0321609i
$378$		0			0
$379$	3522.46	−	3522.46i	0.477405	−	0.477405i	−0.426896	−	0.904301i	$-0.640393\pi$
$379$	3522.46	−	3522.46i	0.477405	−	0.477405i	0.904301	+	0.426896i	$0.140393\pi$
$380$	2776.82	−	503.025i	0.374863	−	0.0679069i
$381$		0			0
$382$	−11064.5	+	994.086i	−1.48196	+	0.133146i
$383$	−3044.88			−0.406229			−0.203115	−	0.979155i	$-0.565106\pi$
$383$	−3044.88			−0.406229			−0.203115	+	0.979155i	$0.565106\pi$
$384$		0			0
$385$	5188.54			0.686837
$386$	−9152.75	+	822.325i	−1.20690	+	0.108433i
$387$		0			0
$388$	−1187.08	+	215.042i	−0.155322	+	0.0281368i
$389$	−1932.73	+	1932.73i	−0.251911	+	0.251911i	−0.821754	−	0.569843i	$-0.807004\pi$
$389$	−1932.73	+	1932.73i	−0.251911	+	0.251911i	0.569843	+	0.821754i	$0.307004\pi$
$390$		0			0
$391$		−	4258.09i		−	0.550744i
$392$	−6039.14	+	1663.66i	−0.778119	+	0.214356i
$393$		0			0
$394$	−6207.15	+	7432.61i	−0.793685	+	0.950379i
$395$	−3838.04	+	3838.04i	−0.488893	+	0.488893i
$396$		0			0
$397$	−672.457	−	672.457i	−0.0850117	−	0.0850117i	0.663322	−	0.748334i	$-0.269146\pi$
$397$	−672.457	−	672.457i	−0.0850117	−	0.0850117i	−0.748334	+	0.663322i	$0.769146\pi$
$398$	347.075	+	3863.07i	0.0437119	+	0.486528i
$399$		0			0
$400$	1829.71	+	4884.51i	0.228714	+	0.610563i
$401$	7606.74			0.947288			0.473644	−	0.880716i	$-0.342938\pi$
$401$	7606.74			0.947288			0.473644	+	0.880716i	$0.342938\pi$
$402$		0			0
$403$	1092.85	+	1092.85i	0.135084	+	0.135084i
$404$	−3145.49	+	4537.23i	−0.387361	+	0.558751i
$405$		0			0
$406$	−666.631	+	798.241i	−0.0814885	+	0.0975765i
$407$		−	7296.32i		−	0.888612i
$408$		0			0
$409$			4981.58i			0.602257i	0.953584	+	0.301129i	$0.0973634\pi$
$409$			4981.58i			0.602257i	−0.953584	+	0.301129i	$0.902637\pi$
$410$	5162.08	+	4310.98i	0.621797	+	0.519278i
$411$		0			0
$412$	−2624.67	−	14488.8i	−0.313855	−	1.73256i
$413$	10083.3	+	10083.3i	1.20137	+	1.20137i
$414$		0			0
$415$	−3048.89			−0.360637
$416$	−2722.81	−	955.136i	−0.320906	−	0.112571i
$417$		0			0
$418$	−4760.80	+	427.732i	−0.557077	+	0.0500503i
$419$	3433.38	+	3433.38i	0.400314	+	0.400314i	0.878344	−	0.478030i	$-0.158649\pi$
$419$	3433.38	+	3433.38i	0.400314	+	0.400314i	−0.478030	+	0.878344i	$0.658649\pi$
$420$		0			0
$421$	7973.72	−	7973.72i	0.923077	−	0.923077i	−0.0741686	−	0.997246i	$-0.523630\pi$
$421$	7973.72	−	7973.72i	0.923077	−	0.923077i	0.997246	+	0.0741686i	$0.0236303\pi$
$422$	−4975.22	−	4154.93i	−0.573910	−	0.479286i
$423$		0			0
$424$	2777.81	−	4890.18i	0.318166	−	0.560114i
$425$			7211.44i			0.823074i
$426$		0			0
$427$	16545.6	−	16545.6i	1.87517	−	1.87517i
$428$	511.866	−	738.343i	0.0578084	−	0.0833859i
$429$		0			0
$430$	236.665	+	2634.16i	0.0265418	+	0.295420i
$431$	−4800.16			−0.536463			−0.268232	−	0.963354i	$-0.586439\pi$
$431$	−4800.16			−0.536463			−0.268232	+	0.963354i	$0.586439\pi$
$432$		0			0
$433$	6242.32			0.692810			0.346405	−	0.938085i	$-0.387402\pi$
$433$	6242.32			0.692810			0.346405	+	0.938085i	$0.387402\pi$
$434$	610.959	+	6800.18i	0.0675737	+	0.752118i
$435$		0			0
$436$	8860.69	+	6142.79i	0.973280	+	0.674739i
$437$	1819.93	−	1819.93i	0.199220	−	0.199220i
$438$		0			0
$439$			4929.27i			0.535903i	0.963432	+	0.267951i	$0.0863466\pi$
$439$			4929.27i			0.535903i	−0.963432	+	0.267951i	$0.913653\pi$
$440$	1252.41	+	4546.30i	0.135696	+	0.492582i
$441$		0			0
$442$	−3062.03	−	2557.18i	−0.329516	−	0.275187i
$443$	7670.67	−	7670.67i	0.822674	−	0.822674i	−0.163817	−	0.986491i	$-0.552381\pi$
$443$	7670.67	−	7670.67i	0.822674	−	0.822674i	0.986491	+	0.163817i	$0.0523807\pi$
$444$		0			0
$445$	1225.32	+	1225.32i	0.130529	+	0.130529i
$446$	−1182.09	+	106.205i	−0.125502	+	0.0112756i
$447$		0			0
$448$	−6527.70	−	10948.8i	−0.688404	−	1.15464i
$449$	11515.2			1.21032			0.605162	−	0.796102i	$-0.293108\pi$
$449$	11515.2			1.21032			0.605162	+	0.796102i	$0.293108\pi$
$450$		0			0
$451$	−8055.12	−	8055.12i	−0.841021	−	0.841021i
$452$	5671.44	−	1027.39i	0.590182	−	0.106912i
$453$		0			0
$454$	6550.66	+	5470.61i	0.677176	+	0.565526i
$455$			2617.46i			0.269689i
$456$		0			0
$457$			4829.89i			0.494383i	0.968967	+	0.247191i	$0.0795076\pi$
$457$			4829.89i			0.494383i	−0.968967	+	0.247191i	$0.920492\pi$
$458$	−4036.78	+	4833.74i	−0.411848	+	0.493157i
$459$		0			0
$460$	−2086.72	−	1446.65i	−0.211508	−	0.146631i
$461$	−8265.79	−	8265.79i	−0.835090	−	0.835090i	0.153118	−	0.988208i	$-0.451069\pi$
$461$	−8265.79	−	8265.79i	−0.835090	−	0.835090i	−0.988208	+	0.153118i	$0.951069\pi$
$462$		0			0
$463$	−5043.86			−0.506281			−0.253141	−	0.967430i	$-0.581464\pi$
$463$	−5043.86			−0.506281			−0.253141	+	0.967430i	$0.581464\pi$
$464$	−860.346	−	391.435i	−0.0860788	−	0.0391636i
$465$		0			0
$466$	−302.418	−	3366.02i	−0.0300628	−	0.334609i
$467$	−12438.8	−	12438.8i	−1.23255	−	1.23255i	−0.962982	−	0.269564i	$-0.913120\pi$
$467$	−12438.8	−	12438.8i	−1.23255	−	1.23255i	−0.269564	−	0.962982i	$-0.586880\pi$
$468$		0			0
$469$	−2673.31	+	2673.31i	−0.263203	+	0.263203i
$470$	−2638.70	+	3159.64i	−0.258966	+	0.310093i
$471$		0			0
$472$	−6401.26	+	11269.1i	−0.624241	+	1.09894i
$473$		−	4479.75i		−	0.435474i
$474$		0			0
$475$	−3082.20	+	3082.20i	−0.297729	+	0.297729i
$476$	−3141.42	−	17341.4i	−0.302493	−	1.66984i
$477$		0			0
$478$	11947.3	−	1073.40i	1.14322	−	0.102712i
$479$	−13059.7			−1.24575			−0.622875	−	0.782321i	$-0.714036\pi$
$479$	−13059.7			−1.24575			−0.622875	+	0.782321i	$0.714036\pi$
$480$		0			0
$481$	3680.77			0.348916
$482$	−15694.5	+	1410.06i	−1.48312	+	0.133250i
$483$		0			0
$484$	474.246	+	2617.96i	0.0445385	+	0.245864i
$485$	−703.292	+	703.292i	−0.0658450	+	0.0658450i
$486$		0			0
$487$			15549.3i			1.44683i	0.690414	+	0.723414i	$0.257428\pi$
$487$			15549.3i			1.44683i	−0.690414	+	0.723414i	$0.742572\pi$
$488$	18491.4	+	10503.8i	1.71530	+	0.974354i
$489$		0			0
$490$	−3310.33	+	3963.88i	−0.305195	+	0.365448i
$491$	8628.34	−	8628.34i	0.793058	−	0.793058i	−0.188932	−	0.981990i	$-0.560503\pi$
$491$	8628.34	−	8628.34i	0.793058	−	0.793058i	0.981990	+	0.188932i	$0.0605027\pi$
$492$		0			0
$493$	−924.059	−	924.059i	−0.0844169	−	0.0844169i
$494$	−215.777	−	2401.68i	−0.0196524	−	0.218738i
$495$		0			0
$496$	−5810.98	+	2176.76i	−0.526049	+	0.197056i
$497$	−5354.00			−0.483219
$498$		0			0
$499$	1732.42	+	1732.42i	0.155418	+	0.155418i	0.780533	−	0.625115i	$-0.214948\pi$
$499$	1732.42	+	1732.42i	0.155418	+	0.155418i	−0.625115	+	0.780533i	$0.714948\pi$
$500$	8954.38	+	6207.74i	0.800904	+	0.555237i
$501$		0			0
$502$	1237.71	−	1482.06i	0.110043	−	0.131768i
$503$			16579.6i			1.46968i	0.678241	+	0.734839i	$0.262742\pi$
$503$			16579.6i			1.46968i	−0.678241	+	0.734839i	$0.737258\pi$
$504$		0			0
$505$			4551.65i			0.401081i
$506$	3301.07	+	2756.80i	0.290021	+	0.242203i
$507$		0			0
$508$	20642.8	−	3739.47i	1.80291	−	0.326599i
$509$	7830.92	+	7830.92i	0.681924	+	0.681924i	0.960434	−	0.278509i	$-0.0898403\pi$
$509$	7830.92	+	7830.92i	0.681924	+	0.681924i	−0.278509	+	0.960434i	$0.589840\pi$
$510$		0			0
$511$	−16645.5			−1.44100
$512$	8017.86	−	8362.51i	0.692076	−	0.721825i
$513$		0			0
$514$	−22799.0	+	2048.36i	−1.95646	+	0.175777i
$515$	−8583.95	−	8583.95i	−0.734474	−	0.734474i
$516$		0			0
$517$	4930.44	−	4930.44i	0.419420	−	0.419420i
$518$	12480.5	+	10422.8i	1.05861	+	0.884075i
$519$		0			0
$520$	−2293.47	+	631.803i	−0.193414	+	0.0532815i
$521$			3400.02i			0.285907i	0.989729	+	0.142953i	$0.0456599\pi$
$521$			3400.02i			0.285907i	−0.989729	+	0.142953i	$0.954340\pi$
$522$		0			0
$523$	−2019.50	+	2019.50i	−0.168847	+	0.168847i	−0.786472	−	0.617626i	$-0.788095\pi$
$523$	−2019.50	+	2019.50i	−0.168847	+	0.168847i	0.617626	+	0.786472i	$0.288095\pi$
$524$	−6114.05	−	4238.64i	−0.509720	−	0.353370i
$525$		0			0
$526$	103.847	+	1155.85i	0.00860822	+	0.0958124i
$527$	−8579.28			−0.709144
$528$		0			0
$529$	9851.23			0.809668
$530$	−414.910	−	4618.09i	−0.0340048	−	0.378485i
$531$		0			0
$532$	6069.15	−	8754.47i	0.494607	−	0.713448i
$533$	4063.56	−	4063.56i	0.330229	−	0.330229i
$534$		0			0
$535$		−	740.691i		−	0.0598558i
$536$	−2987.69	−	1697.12i	−0.240762	−	0.136762i
$537$		0			0
$538$	14.6743	+	12.2548i	0.00117594	+	0.000982052i
$539$	6185.39	−	6185.39i	0.494293	−	0.494293i
$540$		0			0
$541$	−13432.6	−	13432.6i	−1.06749	−	1.06749i	−0.997551	−	0.0699388i	$-0.977720\pi$
$541$	−13432.6	−	13432.6i	−1.06749	−	1.06749i	−0.0699388	−	0.997551i	$-0.522280\pi$
$542$	−7983.54	+	717.278i	−0.632699	+	0.0568445i
$543$		0			0
$544$	14436.6	−	6938.45i	1.13780	−	0.546845i
$545$	8888.86			0.698637
$546$		0			0
$547$	−5376.46	−	5376.46i	−0.420257	−	0.420257i	0.465035	−	0.885292i	$-0.346042\pi$
$547$	−5376.46	−	5376.46i	−0.420257	−	0.420257i	−0.885292	+	0.465035i	$0.846042\pi$
$548$	−3581.44	−	19770.4i	−0.279181	−	1.54115i
$549$		0			0
$550$	−5590.64	−	4668.88i	−0.433429	−	0.361967i
$551$		−	789.894i		−	0.0610719i
$552$		0			0
$553$			20488.7i			1.57553i
$554$	3912.13	−	4684.49i	0.300019	−	0.359251i
$555$		0			0
$556$	−7000.34	+	10097.7i	−0.533958	+	0.770210i
$557$	2076.28	+	2076.28i	0.157944	+	0.157944i	0.781655	−	0.623711i	$-0.214376\pi$
$557$	2076.28	+	2076.28i	0.157944	+	0.157944i	−0.623711	+	0.781655i	$0.714376\pi$
$558$		0			0
$559$	2259.90			0.170990
$560$	−9565.61	−	4352.10i	−0.721823	−	0.328410i
$561$		0			0
$562$	−1202.28	−	13381.8i	−0.0902405	−	1.00441i
$563$	16643.1	+	16643.1i	1.24587	+	1.24587i	0.957528	+	0.288339i	$0.0931031\pi$
$563$	16643.1	+	16643.1i	1.24587	+	1.24587i	0.288339	+	0.957528i	$0.406897\pi$
$564$		0			0
$565$	3360.07	−	3360.07i	0.250193	−	0.250193i
$566$	1651.47	−	1977.51i	0.122644	−	0.146857i
$567$		0			0
$568$	−1292.35	−	4691.28i	−0.0954679	−	0.346552i
$569$		−	5659.60i		−	0.416982i	−0.978024	−	0.208491i	$-0.933145\pi$
$569$		−	5659.60i		−	0.416982i	0.978024	−	0.208491i	$-0.0668551\pi$
$570$		0			0
$571$	4872.67	−	4872.67i	0.357119	−	0.357119i	−0.505631	−	0.862750i	$-0.668740\pi$
$571$	4872.67	−	4872.67i	0.357119	−	0.357119i	0.862750	+	0.505631i	$0.168740\pi$
$572$	3964.89	−	718.244i	0.289825	−	0.0525023i
$573$		0			0
$574$	25285.2	−	2271.73i	1.83865	−	0.165192i
$575$	3921.95			0.284446
$576$		0			0
$577$	10652.2			0.768556			0.384278	−	0.923217i	$-0.374450\pi$
$577$	10652.2			0.768556			0.384278	+	0.923217i	$0.374450\pi$
$578$	8216.10	−	738.171i	0.591254	−	0.0531209i
$579$		0			0
$580$	−766.784	+	138.904i	−0.0548948	+	0.00994426i
$581$	−8138.01	+	8138.01i	−0.581104	+	0.581104i
$582$		0			0
$583$			7853.69i			0.557919i
$584$	−4017.89	−	14585.1i	−0.284694	−	1.03345i
$585$		0			0
$586$	3674.61	−	4400.07i	0.259039	−	0.310180i
$587$	−9903.95	+	9903.95i	−0.696388	+	0.696388i	−0.963630	−	0.267242i	$-0.913888\pi$
$587$	−9903.95	+	9903.95i	−0.696388	+	0.696388i	0.267242	+	0.963630i	$0.413888\pi$
$588$		0			0
$589$	−3666.82	−	3666.82i	−0.256517	−	0.256517i
$590$	956.129	+	10642.0i	0.0667173	+	0.742587i
$591$		0			0
$592$	−6120.09	+	13451.5i	−0.424889	+	0.933875i
$593$	3528.04			0.244316			0.122158	−	0.992511i	$-0.461019\pi$
$593$	3528.04			0.244316			0.122158	+	0.992511i	$0.461019\pi$
$594$		0			0
$595$	−10274.0	−	10274.0i	−0.707887	−	0.707887i
$596$	14726.5	−	21242.2i	1.01211	−	1.45993i
$597$		0			0
$598$	−1390.72	+	1665.29i	−0.0951018	+	0.113877i
$599$		−	19024.9i		−	1.29772i	−0.760907	−	0.648861i	$-0.775246\pi$
$599$		−	19024.9i		−	1.29772i	0.760907	−	0.648861i	$-0.224754\pi$
$600$		0			0
$601$			1065.92i			0.0723460i	0.999346	+	0.0361730i	$0.0115167\pi$
$601$			1065.92i			0.0723460i	−0.999346	+	0.0361730i	$0.988483\pi$
$602$	7662.72	+	6399.32i	0.518786	+	0.433250i
$603$		0			0
$604$	−4013.32	−	22154.5i	−0.270364	−	1.49248i
$605$	1551.02	+	1551.02i	0.104228	+	0.104228i
$606$		0			0
$607$	12909.7			0.863242			0.431621	−	0.902055i	$-0.357942\pi$
$607$	12909.7			0.863242			0.431621	+	0.902055i	$0.357942\pi$
$608$	9135.80	+	3204.75i	0.609384	+	0.213766i
$609$		0			0
$610$	17462.5	−	1568.91i	1.15908	−	0.104137i
$611$	2487.25	+	2487.25i	0.164687	+	0.164687i
$612$		0			0
$613$	−8763.41	+	8763.41i	−0.577408	+	0.577408i	−0.934188	−	0.356781i	$-0.883874\pi$
$613$	−8763.41	+	8763.41i	−0.577408	+	0.577408i	0.356781	+	0.934188i	$0.383874\pi$
$614$	−709.859	−	592.820i	−0.0466573	−	0.0389646i
$615$		0			0
$616$	15477.7	+	8791.93i	1.01236	+	0.575060i
$617$			18921.2i			1.23459i	0.786733	+	0.617293i	$0.211771\pi$
$617$			18921.2i			1.23459i	−0.786733	+	0.617293i	$0.788229\pi$
$618$		0			0
$619$	−15116.6	+	15116.6i	−0.981565	+	0.981565i	−0.999833	−	0.0182677i	$-0.994185\pi$
$619$	−15116.6	+	15116.6i	−0.981565	+	0.981565i	0.0182677	+	0.999833i	$0.494185\pi$
$620$	−2914.73	+	4204.36i	−0.188804	+	0.272340i
$621$		0			0
$622$	−220.553	−	2454.83i	−0.0142176	−	0.158247i
$623$	6541.15			0.420651
$624$		0			0
$625$	−1204.57			−0.0770926
$626$	−889.648	−	9902.09i	−0.0568011	−	0.632216i
$627$		0			0
$628$	−8426.75	−	5841.95i	−0.535452	−	0.371209i
$629$	−14447.7	+	14447.7i	−0.915845	+	0.915845i
$630$		0			0
$631$		−	9602.80i		−	0.605834i	−0.953017	−	0.302917i	$-0.902039\pi$
$631$		−	9602.80i		−	0.605834i	0.953017	−	0.302917i	$-0.0979605\pi$
$632$	−17952.6	+	4945.58i	−1.12993	+	0.311273i
$633$		0			0
$634$	−14502.8	−	12111.7i	−0.908487	−	0.758699i
$635$	12229.9	−	12229.9i	0.764297	−	0.764297i
$636$		0			0
$637$	3120.34	+	3120.34i	0.194085	+	0.194085i
$638$	1314.63	−	118.113i	0.0815781	−	0.00732935i
$639$		0			0
$640$	1504.44	−	9432.08i	0.0929194	−	0.582556i
$641$	−4450.84			−0.274256			−0.137128	−	0.990553i	$-0.543787\pi$
$641$	−4450.84			−0.274256			−0.137128	+	0.990553i	$0.543787\pi$
$642$		0			0
$643$	6491.27	+	6491.27i	0.398119	+	0.398119i	0.877569	−	0.479450i	$-0.159164\pi$
$643$	6491.27	+	6491.27i	0.398119	+	0.398119i	−0.479450	+	0.877569i	$0.659164\pi$
$644$	−9431.15	+	1708.46i	−0.577080	+	0.104539i
$645$		0			0
$646$	10274.0	+	8580.05i	0.625734	+	0.522566i
$647$			5546.17i			0.337005i	0.985701	+	0.168503i	$0.0538932\pi$
$647$			5546.17i			0.337005i	−0.985701	+	0.168503i	$0.946107\pi$
$648$		0			0
$649$		−	18098.3i		−	1.09464i
$650$	2355.31	−	2820.31i	0.142127	−	0.170187i
$651$		0			0
$652$	12946.0	+	8974.98i	0.777613	+	0.539091i
$653$	−6327.58	−	6327.58i	−0.379200	−	0.379200i	0.491614	−	0.870813i	$-0.336407\pi$
$653$	−6327.58	−	6327.58i	−0.379200	−	0.379200i	−0.870813	+	0.491614i	$0.836407\pi$
$654$		0			0
$655$	−6133.49			−0.365886
$656$	8093.88	+	21607.0i	0.481727	+	1.28599i
$657$		0			0
$658$	1390.50	+	15476.7i	0.0823820	+	0.916940i
$659$	−15135.2	−	15135.2i	−0.894663	−	0.894663i	0.100294	−	0.994958i	$-0.468022\pi$
$659$	−15135.2	−	15135.2i	−0.894663	−	0.894663i	−0.994958	+	0.100294i	$0.968022\pi$
$660$		0			0
$661$	−23460.1	+	23460.1i	−1.38047	+	1.38047i	−0.536694	+	0.843777i	$0.680327\pi$
$661$	−23460.1	+	23460.1i	−1.38047	+	1.38047i	−0.843777	+	0.536694i	$0.819673\pi$
$662$	−4667.12	+	5588.53i	−0.274007	+	0.328103i
$663$		0			0
$664$	−9095.03	−	5166.32i	−0.531560	−	0.301946i
$665$		−	8782.30i		−	0.512125i
$666$		0			0
$667$	−502.550	+	502.550i	−0.0291736	+	0.0291736i
$668$	1742.08	+	9616.73i	0.100903	+	0.557009i
$669$		0			0
$670$	−2821.45	+	253.492i	−0.162690	+	0.0146168i
$671$	−29697.4			−1.70858
$672$		0			0
$673$	30638.5			1.75487			0.877436	−	0.479694i	$-0.159252\pi$
$673$	30638.5			1.75487			0.877436	+	0.479694i	$0.159252\pi$
$674$	208.737	−	18.7539i	0.0119292	−	0.00107177i
$675$		0			0
$676$	−2770.59	−	15294.4i	−0.157635	−	0.870184i
$677$	12468.9	−	12468.9i	0.707855	−	0.707855i	−0.258229	−	0.966084i	$-0.583139\pi$
$677$	12468.9	−	12468.9i	0.707855	−	0.707855i	0.966084	+	0.258229i	$0.0831388\pi$
$678$		0			0
$679$			3754.41i			0.212196i
$680$	6522.33	−	11482.2i	0.367823	−	0.647533i
$681$		0			0
$682$	5554.45	−	6651.05i	0.311864	−	0.373434i
$683$	−13838.5	+	13838.5i	−0.775280	+	0.775280i	−0.979024	−	0.203744i	$-0.934689\pi$
$683$	−13838.5	+	13838.5i	−0.775280	+	0.775280i	0.203744	+	0.979024i	$0.434689\pi$
$684$		0			0
$685$	−11713.1	−	11713.1i	−0.653333	−	0.653333i
$686$	−416.914	−	4640.40i	−0.0232039	−	0.258267i
$687$		0			0
$688$	−3757.58	+	8258.89i	−0.208221	+	0.457656i
$689$	−3961.95			−0.219068
$690$		0			0
$691$	−106.012	−	106.012i	−0.00583628	−	0.00583628i	0.704183	−	0.710019i	$-0.251314\pi$
$691$	−106.012	−	106.012i	−0.00583628	−	0.00583628i	−0.710019	+	0.704183i	$0.751314\pi$
$692$	5238.60	+	3631.73i	0.287777	+	0.199505i
$693$		0			0
$694$	−5391.37	+	6455.77i	−0.294890	+	0.353109i
$695$			10129.8i			0.552870i
$696$		0			0
$697$			31900.4i			1.73359i
$698$	−20302.6	−	16955.2i	−1.10095	−	0.919434i
$699$		0			0
$700$	15972.4	−	2893.43i	0.862431	−	0.156231i
$701$	−7839.35	−	7839.35i	−0.422380	−	0.422380i	0.463643	−	0.886022i	$-0.346542\pi$
$701$	−7839.35	−	7839.35i	−0.422380	−	0.422380i	−0.886022	+	0.463643i	$0.846542\pi$
$702$		0			0
$703$	−12350.0			−0.662574
$704$	−3967.64	+	15684.1i	−0.212409	+	0.839653i
$705$		0			0
$706$	−6245.17	+	561.095i	−0.332918	+	0.0299109i
$707$	12149.1	+	12149.1i	0.646273	+	0.646273i
$708$		0			0
$709$	2728.30	−	2728.30i	0.144518	−	0.144518i	−0.631146	−	0.775664i	$-0.717415\pi$
$709$	2728.30	−	2728.30i	0.144518	−	0.144518i	0.775664	+	0.631146i	$0.217415\pi$
$710$	−3079.19	−	2571.50i	−0.162760	−	0.135925i
$711$		0			0
$712$	1578.90	+	5731.48i	0.0831066	+	0.301680i
$713$			4665.84i			0.245073i
$714$		0			0
$715$	2349.01	−	2349.01i	0.122864	−	0.122864i
$716$	20476.0	+	14195.3i	1.06875	+	0.740925i
$717$		0			0
$718$	527.011	+	5865.82i	0.0273926	+	0.304889i
$719$	32717.8			1.69704			0.848519	−	0.529166i	$-0.177495\pi$
$719$	32717.8			1.69704			0.848519	+	0.529166i	$0.177495\pi$
$720$		0			0
$721$	−45824.0			−2.36696
$722$	−1012.01	−	11264.1i	−0.0521652	−	0.580616i
$723$		0			0
$724$	781.844	−	1127.77i	0.0401340	−	0.0578914i
$725$	851.111	−	851.111i	0.0435993	−	0.0435993i
$726$		0			0
$727$			25847.2i			1.31859i	0.751883	+	0.659297i	$0.229146\pi$
$727$			25847.2i			1.31859i	−0.751883	+	0.659297i	$0.770854\pi$
$728$	−4435.26	+	7808.03i	−0.225799	+	0.397507i
$729$		0			0
$730$	−9573.13	−	7994.75i	−0.485366	−	0.405341i
$731$	−8870.50	+	8870.50i	−0.448820	+	0.448820i
$732$		0			0
$733$	10131.9	+	10131.9i	0.510547	+	0.510547i	0.914694	−	0.404147i	$-0.132432\pi$
$733$	10131.9	+	10131.9i	0.510547	+	0.510547i	−0.404147	+	0.914694i	$0.632432\pi$
$734$	12702.8	−	1141.28i	0.638787	−	0.0573915i
$735$		0			0
$736$	−3773.48	−	7851.36i	−0.188984	−	0.393213i
$737$	4798.27			0.239819
$738$		0			0
$739$	19163.7	+	19163.7i	0.953921	+	0.953921i	0.998984	−	0.0450633i	$-0.0143490\pi$
$739$	19163.7	+	19163.7i	0.953921	+	0.953921i	−0.0450633	+	0.998984i	$0.514349\pi$
$740$	2171.76	+	11988.7i	0.107886	+	0.595558i
$741$		0			0
$742$	−13433.9	−	11219.0i	−0.664656	−	0.555070i
$743$			23322.1i			1.15155i	0.817607	+	0.575777i	$0.195301\pi$
$743$			23322.1i			1.15155i	−0.817607	+	0.575777i	$0.804699\pi$
$744$		0			0
$745$		−	21309.8i		−	1.04796i
$746$	−22208.1	+	26592.5i	−1.08994	+	1.30512i
$747$		0			0
$748$	−12743.6	+	18382.1i	−0.622933	+	0.898552i
$749$	−1977.03	−	1977.03i	−0.0964473	−	0.0964473i
$750$		0			0
$751$	25994.0			1.26303			0.631515	−	0.775364i	$-0.282434\pi$
$751$	25994.0			1.26303			0.631515	+	0.775364i	$0.282434\pi$
$752$	−13225.4	+	4954.16i	−0.641330	+	0.240239i
$753$		0			0
$754$	59.5842	+	663.192i	0.00287789	+	0.0320319i
$755$	−13125.5	−	13125.5i	−0.632698	−	0.632698i
$756$		0			0
$757$	22145.0	−	22145.0i	1.06324	−	1.06324i	0.0653808	−	0.997860i	$-0.479174\pi$
$757$	22145.0	−	22145.0i	1.06324	−	1.06324i	0.997860	−	0.0653808i	$-0.0208262\pi$
$758$	9031.52	−	10814.6i	0.432770	−	0.518210i
$759$		0			0
$760$	7695.22	−	2119.87i	0.367283	−	0.101179i
$761$		−	16497.5i		−	0.785853i	−0.919570	−	0.392926i	$-0.871463\pi$
$761$		−	16497.5i		−	0.785853i	0.919570	−	0.392926i	$-0.128537\pi$
$762$		0			0
$763$	23725.9	−	23725.9i	1.12573	−	1.12573i
$764$	−30918.0	+	5600.84i	−1.46410	+	0.265224i
$765$		0			0
$766$	−8577.66	+	770.656i	−0.404600	+	0.0363511i
$767$	9130.02			0.429812
$768$		0			0
$769$	−24867.3			−1.16611			−0.583055	−	0.812433i	$-0.698143\pi$
$769$	−24867.3			−1.16611			−0.583055	+	0.812433i	$0.698143\pi$
$770$	14616.5	−	1313.21i	0.684082	−	0.0614610i
$771$		0			0
$772$	−25575.9	+	4633.11i	−1.19235	+	0.215996i
$773$	−1881.72	+	1881.72i	−0.0875559	+	0.0875559i	−0.749528	−	0.661972i	$-0.769720\pi$
$773$	−1881.72	+	1881.72i	−0.0875559	+	0.0875559i	0.661972	+	0.749528i	$0.269720\pi$
$774$		0			0
$775$		−	7902.00i		−	0.366256i
$776$	−3289.68	+	906.240i	−0.152181	+	0.0419228i
$777$		0			0
$778$	−4955.49	+	5933.84i	−0.228359	+	0.273443i
$779$	−13634.4	+	13634.4i	−0.627089	+	0.627089i
$780$		0			0
$781$	4804.89	+	4804.89i	0.220144	+	0.220144i
$782$	−1077.72	−	11995.4i	−0.0492828	−	0.548535i
$783$		0			0
$784$	−16591.7	+	6215.16i	−0.755815	+	0.283125i
$785$	−8453.54			−0.384356
$786$		0			0
$787$	−7790.94	−	7790.94i	−0.352881	−	0.352881i	0.508300	−	0.861180i	$-0.330274\pi$
$787$	−7790.94	−	7790.94i	−0.352881	−	0.352881i	−0.861180	+	0.508300i	$0.830274\pi$
$788$	−15604.9	+	22509.3i	−0.705457	+	1.01759i
$789$		0			0
$790$	−9840.66	+	11783.5i	−0.443184	+	0.530680i
$791$		−	17937.2i		−	0.806286i
$792$		0			0
$793$		−	14981.4i		−	0.670877i
$794$	−2064.56	−	1724.17i	−0.0922778	−	0.0770634i
$795$		0			0
$796$	1955.48	+	10794.7i	0.0870730	+	0.480665i
$797$	4209.84	+	4209.84i	0.187102	+	0.187102i	0.794442	−	0.607340i	$-0.207763\pi$
$797$	4209.84	+	4209.84i	0.187102	+	0.187102i	−0.607340	+	0.794442i	$0.707763\pi$
$798$		0			0
$799$	−19525.8			−0.864549
$800$	6390.71	+	13296.9i	0.282432	+	0.587648i
$801$		0			0
$802$	21428.8	−	1925.26i	0.943488	−	0.0847672i
$803$	14938.3	+	14938.3i	0.656490	+	0.656490i
$804$		0			0
$805$	−5587.51	+	5587.51i	−0.244639	+	0.244639i
$806$	3355.25	+	2802.05i	0.146630	+	0.122454i
$807$		0			0
$808$	−7712.73	+	13577.8i	−0.335808	+	0.591172i
$809$		−	27554.3i		−	1.19747i	−0.800946	−	0.598737i	$-0.795670\pi$
$809$		−	27554.3i		−	1.19747i	0.800946	−	0.598737i	$-0.204330\pi$
$810$		0			0
$811$	3406.54	−	3406.54i	0.147497	−	0.147497i	−0.629502	−	0.776999i	$-0.716741\pi$
$811$	3406.54	−	3406.54i	0.147497	−	0.147497i	0.776999	+	0.629502i	$0.216741\pi$
$812$	−1675.92	+	2417.43i	−0.0724300	+	0.104477i
$813$		0			0
$814$	−1846.69	−	20554.3i	−0.0795166	−	0.885047i
$815$	12987.1			0.558184
$816$		0			0
$817$	−7582.59			−0.324701
$818$	1260.83	+	14033.5i	0.0538924	+	0.599841i
$819$		0			0
$820$	15633.1	+	10837.8i	0.665770	+	0.461554i
$821$	9839.14	−	9839.14i	0.418256	−	0.418256i	−0.466346	−	0.884602i	$-0.654430\pi$
$821$	9839.14	−	9839.14i	0.418256	−	0.418256i	0.884602	+	0.466346i	$0.154430\pi$
$822$		0			0
$823$		−	36653.5i		−	1.55244i	−0.630461	−	0.776221i	$-0.717134\pi$
$823$		−	36653.5i		−	1.55244i	0.630461	−	0.776221i	$-0.282866\pi$
$824$	−11061.0	−	40151.9i	−0.467632	−	1.69752i
$825$		0			0
$826$	30957.5	+	25853.3i	1.30405	+	1.08905i
$827$	22223.2	−	22223.2i	0.934431	−	0.934431i	−0.0635475	−	0.997979i	$-0.520241\pi$
$827$	22223.2	−	22223.2i	0.934431	−	0.934431i	0.997979	+	0.0635475i	$0.0202414\pi$
$828$		0			0
$829$	14715.5	+	14715.5i	0.616516	+	0.616516i	0.944636	−	0.328120i	$-0.106415\pi$
$829$	14715.5	+	14715.5i	0.616516	+	0.616516i	−0.328120	+	0.944636i	$0.606415\pi$
$830$	−8588.98	+	771.672i	−0.359190	+	0.0322713i
$831$		0			0
$832$	−7912.13	−	2001.55i	−0.329692	−	0.0834031i
$833$	−24495.8			−1.01888
$834$		0			0
$835$	5697.46	+	5697.46i	0.236130	+	0.236130i
$836$	−13303.3	+	2409.91i	−0.550364	+	0.0996991i
$837$		0			0
$838$	10541.1	+	8803.12i	0.434530	+	0.362886i
$839$			11010.0i			0.453050i	0.974005	+	0.226525i	$0.0727365\pi$
$839$			11010.0i			0.453050i	−0.974005	+	0.226525i	$0.927263\pi$
$840$		0			0
$841$		−	24170.9i		−	0.991057i
$842$	20444.5	−	24480.8i	0.836773	−	1.00197i
$843$		0			0
$844$	−15067.2	−	10445.5i	−0.614496	−	0.426007i
$845$	−9061.20	−	9061.20i	−0.368893	−	0.368893i
$846$		0			0
$847$	8279.85			0.335890
$848$	6587.61	−	14479.1i	0.266768	−	0.586338i
$849$		0			0
$850$	1825.21	+	20315.2i	0.0736520	+	0.819772i
$851$	7857.37	+	7857.37i	0.316507	+	0.316507i
$852$		0			0
$853$	12809.9	−	12809.9i	0.514190	−	0.514190i	−0.401617	−	0.915808i	$-0.631552\pi$
$853$	12809.9	−	12809.9i	0.514190	−	0.514190i	0.915808	+	0.401617i	$0.131552\pi$
$854$	42422.7	−	50798.0i	1.69985	−	2.03545i
$855$		0			0
$856$	1255.09	−	2209.52i	0.0501147	−	0.0882243i
$857$			38510.8i			1.53501i	0.641043	+	0.767505i	$0.278502\pi$
$857$			38510.8i			1.53501i	−0.641043	+	0.767505i	$0.721498\pi$
$858$		0			0
$859$	23234.6	−	23234.6i	0.922882	−	0.922882i	−0.0743503	−	0.997232i	$-0.523688\pi$
$859$	23234.6	−	23234.6i	0.922882	−	0.922882i	0.997232	+	0.0743503i	$0.0236883\pi$
$860$	1333.41	+	7360.74i	0.0528707	+	0.291860i
$861$		0			0
$862$	−13522.4	+	1214.92i	−0.534311	+	0.0480049i
$863$	22079.5			0.870911			0.435456	−	0.900210i	$-0.356587\pi$
$863$	22079.5			0.870911			0.435456	+	0.900210i	$0.356587\pi$
$864$		0			0
$865$	5255.26			0.206571
$866$	17585.1	−	1579.93i	0.690031	−	0.0619955i
$867$		0			0
$868$	3442.24	+	19002.0i	0.134605	+	0.743054i
$869$	18387.4	−	18387.4i	0.717779	−	0.717779i
$870$		0			0
$871$			2420.58i			0.0941656i
$872$	26516.0	+	15062.1i	1.02975	+	0.584939i
$873$		0			0
$874$	4666.26	−	5587.51i	0.180593	−	0.216247i
$875$	23976.7	−	23976.7i	0.926356	−	0.926356i
$876$		0			0
$877$	4082.13	+	4082.13i	0.157176	+	0.157176i	0.781314	−	0.624138i	$-0.214550\pi$
$877$	4082.13	+	4082.13i	0.157176	+	0.157176i	−0.624138	+	0.781314i	$0.714550\pi$
$878$	1247.59	+	13886.2i	0.0479547	+	0.533753i
$879$		0			0
$880$	4678.80	+	12490.3i	0.179230	+	0.478463i
$881$	−7132.59			−0.272762			−0.136381	−	0.990656i	$-0.543547\pi$
$881$	−7132.59			−0.272762			−0.136381	+	0.990656i	$0.543547\pi$
$882$		0			0
$883$	19170.0	+	19170.0i	0.730601	+	0.730601i	0.970739	−	0.240138i	$-0.0771926\pi$
$883$	19170.0	+	19170.0i	0.730601	+	0.730601i	−0.240138	+	0.970739i	$0.577193\pi$
$884$	−9273.21	−	6428.78i	−0.352819	−	0.244596i
$885$		0			0
$886$	19667.4	−	23550.3i	0.745757	−	0.892989i
$887$		−	45045.7i		−	1.70517i	−0.522589	−	0.852585i	$-0.675034\pi$
$887$		−	45045.7i		−	1.70517i	0.522589	−	0.852585i	$-0.324966\pi$
$888$		0			0
$889$		−	65287.3i		−	2.46306i
$890$	3761.94	+	3141.69i	0.141686	+	0.118325i
$891$		0			0
$892$	−3303.17	+	598.374i	−0.123989	+	0.0224608i
$893$	−8345.43	−	8345.43i	−0.312731	−	0.312731i
$894$		0			0
$895$	20541.1			0.767167
$896$	−21160.2	−	29191.4i	−0.788964	−	1.08841i
$897$		0			0
$898$	32439.2	−	2914.49i	1.20547	−	0.108305i
$899$	1012.55	+	1012.55i	0.0375643	+	0.0375643i
$900$		0			0
$901$	15551.4	−	15551.4i	0.575017	−	0.575017i
$902$	−24730.6	−	20653.2i	−0.912905	−	0.762389i
$903$		0			0
$904$	15716.9	−	4329.68i	0.578247	−	0.159295i
$905$		−	1131.36i		−	0.0415554i
$906$		0			0
$907$	−33658.2	+	33658.2i	−1.23220	+	1.23220i	−0.269079	+	0.963118i	$0.586719\pi$
$907$	−33658.2	+	33658.2i	−1.23220	+	1.23220i	−0.963118	+	0.269079i	$0.913281\pi$
$908$	19838.3	+	13753.2i	0.725065	+	0.502661i
$909$		0			0
$910$	662.476	+	7373.59i	0.0241328	+	0.268607i
$911$	42503.8			1.54579			0.772895	−	0.634534i	$-0.218808\pi$
$911$	42503.8			1.54579			0.772895	+	0.634534i	$0.218808\pi$
$912$		0			0
$913$	14606.7			0.529477
$914$	1222.44	+	13606.2i	0.0442394	+	0.492399i
$915$		0			0
$916$	−10148.5	+	14638.7i	−0.366066	+	0.528033i
$917$	−16371.3	+	16371.3i	−0.589562	+	0.589562i
$918$		0			0
$919$			8819.41i			0.316568i	0.987394	+	0.158284i	$0.0505961\pi$
$919$			8819.41i			0.316568i	−0.987394	+	0.158284i	$0.949404\pi$
$920$	−6244.60	−	3547.17i	−0.223781	−	0.127116i
$921$		0			0
$922$	−25377.5	−	21193.3i	−0.906467	−	0.757012i
$923$	−2423.92	+	2423.92i	−0.0864402	+	0.0864402i
$924$		0			0
$925$	−13307.1	−	13307.1i	−0.473012	−	0.473012i
$926$	−14209.0	+	1276.60i	−0.504250	+	0.0453041i
$927$		0			0
$928$	−2522.73	−	884.950i	−0.0892380	−	0.0313038i
$929$	14155.6			0.499925			0.249963	−	0.968256i	$-0.419582\pi$
$929$	14155.6			0.499925			0.249963	+	0.968256i	$0.419582\pi$
$930$		0			0
$931$	−10469.6	−	10469.6i	−0.368558	−	0.368558i
$932$	−1703.87	−	9405.80i	−0.0598844	−	0.330577i
$933$		0			0
$934$	−38189.4	−	31892.9i	−1.33790	−	1.11731i
$935$			18440.6i			0.644996i
$936$		0			0
$937$			38518.7i			1.34296i	0.741023	+	0.671479i	$0.234341\pi$
$937$			38518.7i			1.34296i	−0.741023	+	0.671479i	$0.765659\pi$
$938$	−6854.32	+	8207.55i	−0.238594	+	0.285699i
$939$		0			0
$940$	−6633.71	+	9568.82i	−0.230179	+	0.332022i
$941$	−27998.2	−	27998.2i	−0.969942	−	0.969942i	0.0296196	−	0.999561i	$-0.490570\pi$
$941$	−27998.2	−	27998.2i	−0.969942	−	0.969942i	−0.999561	+	0.0296196i	$0.990570\pi$
$942$		0			0
$943$	17349.0			0.599112
$944$	−15180.7	+	33366.0i	−0.523399	+	1.15039i
$945$		0			0
$946$	−1133.82	−	12619.8i	−0.0389680	−	0.433727i
$947$	32839.5	+	32839.5i	1.12687	+	1.12687i	0.990684	+	0.136181i	$0.0434829\pi$
$947$	32839.5	+	32839.5i	1.12687	+	1.12687i	0.136181	+	0.990684i	$0.456517\pi$
$948$		0			0
$949$	−7535.92	+	7535.92i	−0.257773	+	0.257773i
$950$	−7902.71	+	9462.92i	−0.269892	+	0.323176i
$951$		0			0
$952$	−13238.7	−	48057.1i	−0.450704	−	1.63607i
$953$		−	20600.1i		−	0.700211i	−0.936710	−	0.350106i	$-0.886146\pi$
$953$		−	20600.1i		−	0.700211i	0.936710	−	0.350106i	$-0.113854\pi$
$954$		0			0
$955$	−18317.5	+	18317.5i	−0.620670	+	0.620670i
$956$	33384.9	−	6047.72i	1.12944	−	0.204600i
$957$		0			0
$958$	−36790.3	+	3305.41i	−1.24075	+	0.111475i
$959$	−62528.2			−2.10547
$960$		0			0
$961$	−20390.2			−0.684441
$962$	10369.0	−	931.599i	0.347516	−	0.0312224i
$963$		0			0
$964$	−43855.8	+	7944.53i	−1.46525	+	0.265432i
$965$	−15152.5	+	15152.5i	−0.505469	+	0.505469i
$966$		0			0
$967$		−	38210.9i		−	1.27071i	−0.772219	−	0.635356i	$-0.780853\pi$
$967$		−	38210.9i		−	1.27071i	0.772219	−	0.635356i	$-0.219147\pi$
$968$	1998.59	+	7254.96i	0.0663607	+	0.240892i
$969$		0			0
$970$	−1803.23	+	2159.23i	−0.0596888	+	0.0714730i
$971$	−37224.9	+	37224.9i	−1.23028	+	1.23028i	−0.266427	+	0.963855i	$0.585843\pi$
$971$	−37224.9	+	37224.9i	−1.23028	+	1.23028i	−0.963855	+	0.266427i	$0.914157\pi$
$972$		0			0
$973$	27038.1	+	27038.1i	0.890854	+	0.890854i
$974$	3935.51	+	43803.6i	0.129468	+	1.44102i
$975$		0			0
$976$	54750.2	+	24909.9i	1.79560	+	0.816953i
$977$	−7985.95			−0.261508			−0.130754	−	0.991415i	$-0.541740\pi$
$977$	−7985.95			−0.261508			−0.130754	+	0.991415i	$0.541740\pi$
$978$		0			0
$979$	−5870.28	−	5870.28i	−0.191639	−	0.191639i
$980$	−8322.21	+	12004.4i	−0.271269	+	0.391292i
$981$		0			0
$982$	22122.9	−	26490.5i	0.718910	−	0.860842i
$983$		−	10703.1i		−	0.347279i	−0.984809	−	0.173639i	$-0.944447\pi$
$983$		−	10703.1i		−	0.347279i	0.984809	−	0.173639i	$-0.0555527\pi$
$984$		0			0
$985$			22580.9i			0.730442i
$986$	−2837.03	−	2369.27i	−0.0916322	−	0.0765243i
$987$		0			0
$988$	−1215.72	−	6711.10i	−0.0391471	−	0.216102i
$989$	4824.23	+	4824.23i	0.155108	+	0.155108i
$990$		0			0
$991$	−23945.4			−0.767558			−0.383779	−	0.923425i	$-0.625377\pi$
$991$	−23945.4			−0.767558			−0.383779	+	0.923425i	$0.625377\pi$
$992$	−15819.0	+	7602.87i	−0.506306	+	0.243338i
$993$		0			0
$994$	−15082.6	+	1355.09i	−0.481280	+	0.0432404i
$995$	6395.37	+	6395.37i	0.203766	+	0.203766i
$996$		0			0
$997$	−14292.5	+	14292.5i	−0.454010	+	0.454010i	−0.896683	−	0.442673i	$-0.854030\pi$
$997$	−14292.5	+	14292.5i	−0.454010	+	0.454010i	0.442673	+	0.896683i	$0.354030\pi$
$998$	5318.83	+	4441.88i	0.168702	+	0.140887i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.4.k.a.109.5		10
3.2	odd	2		16.4.e.a.13.1	yes	10
4.3	odd	2		576.4.k.a.145.4		10
12.11	even	2		64.4.e.a.17.1		10
16.5	even	4	inner	144.4.k.a.37.5		10
16.11	odd	4		576.4.k.a.433.4		10
24.5	odd	2		128.4.e.b.33.1		10
24.11	even	2		128.4.e.a.33.5		10
48.5	odd	4		16.4.e.a.5.1	✓	10
48.11	even	4		64.4.e.a.49.1		10
48.29	odd	4		128.4.e.b.97.1		10
48.35	even	4		128.4.e.a.97.5		10
96.5	odd	8		1024.4.a.n.1.2		10
96.11	even	8		1024.4.a.m.1.2		10
96.29	odd	8		1024.4.b.j.513.9		10
96.35	even	8		1024.4.b.k.513.2		10
96.53	odd	8		1024.4.a.n.1.9		10
96.59	even	8		1024.4.a.m.1.9		10
96.77	odd	8		1024.4.b.j.513.2		10
96.83	even	8		1024.4.b.k.513.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.4.e.a.5.1	✓	10	48.5	odd	4
16.4.e.a.13.1	yes	10	3.2	odd	2
64.4.e.a.17.1		10	12.11	even	2
64.4.e.a.49.1		10	48.11	even	4
128.4.e.a.33.5		10	24.11	even	2
128.4.e.a.97.5		10	48.35	even	4
128.4.e.b.33.1		10	24.5	odd	2
128.4.e.b.97.1		10	48.29	odd	4
144.4.k.a.37.5		10	16.5	even	4	inner
144.4.k.a.109.5		10	1.1	even	1	trivial
576.4.k.a.145.4		10	4.3	odd	2
576.4.k.a.433.4		10	16.11	odd	4
1024.4.a.m.1.2		10	96.11	even	8
1024.4.a.m.1.9		10	96.59	even	8
1024.4.a.n.1.2		10	96.5	odd	8
1024.4.a.n.1.9		10	96.53	odd	8
1024.4.b.j.513.2		10	96.77	odd	8
1024.4.b.j.513.9		10	96.29	odd	8
1024.4.b.k.513.2		10	96.35	even	8
1024.4.b.k.513.9		10	96.83	even	8

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

\(f(q)\)	\(=\)	\(q+(2.81708 - 0.253099i) q^{2} +(7.87188 - 1.42600i) q^{4} +(4.66372 - 4.66372i) q^{5} -24.8965i q^{7} +(21.8148 - 6.00953i) q^{8} +(11.9577 - 14.3185i) q^{10} +(-22.3431 + 22.3431i) q^{11} +(-11.2714 - 11.2714i) q^{13} +(-6.30129 - 70.1355i) q^{14} +(59.9330 - 22.4506i) q^{16} +88.4846 q^{17} +(37.8187 + 37.8187i) q^{19} +(30.0618 - 43.3627i) q^{20} +(-57.2873 + 68.5974i) q^{22} -48.1224i q^{23} +81.4994i q^{25} +(-34.6053 - 28.8997i) q^{26} +(-35.5025 - 195.982i) q^{28} +(-10.4432 - 10.4432i) q^{29} -96.9578 q^{31} +(163.154 - 78.4142i) q^{32} +(249.268 - 22.3954i) q^{34} +(-116.110 - 116.110i) q^{35} +(-163.279 + 163.279i) q^{37} +(116.110 + 96.9665i) q^{38} +(73.7114 - 129.765i) q^{40} +360.519i q^{41} +(-100.249 + 100.249i) q^{43} +(-144.021 + 207.744i) q^{44} +(-12.1797 - 135.565i) q^{46} -220.669 q^{47} -276.837 q^{49} +(20.6274 + 229.590i) q^{50} +(-104.800 - 72.6542i) q^{52} +(175.752 - 175.752i) q^{53} +208.404i q^{55} +(-149.616 - 543.113i) q^{56} +(-32.0624 - 26.7761i) q^{58} +(-405.008 + 405.008i) q^{59} +(664.576 + 664.576i) q^{61} +(-273.138 + 24.5399i) q^{62} +(439.771 - 262.193i) q^{64} -105.133 q^{65} +(-107.377 - 107.377i) q^{67} +(696.540 - 126.179i) q^{68} +(-356.480 - 297.705i) q^{70} -215.050i q^{71} -668.587i q^{73} +(-418.644 + 501.296i) q^{74} +(351.634 + 243.775i) q^{76} +(556.266 + 556.266i) q^{77} -822.956 q^{79} +(174.808 - 384.215i) q^{80} +(91.2471 + 1015.61i) q^{82} +(-326.873 - 326.873i) q^{83} +(412.668 - 412.668i) q^{85} +(-257.037 + 307.783i) q^{86} +(-353.139 + 621.682i) q^{88} +262.733i q^{89} +(-280.619 + 280.619i) q^{91} +(-68.6226 - 378.814i) q^{92} +(-621.643 + 55.8512i) q^{94} +352.752 q^{95} -150.801 q^{97} +(-779.871 + 70.0671i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} + 8 q^{4} + 2 q^{5} + 44 q^{8} - 68 q^{10} - 18 q^{11} - 2 q^{13} - 188 q^{14} + 280 q^{16} + 4 q^{17} - 26 q^{19} + 196 q^{20} - 588 q^{22} + 264 q^{26} + 280 q^{28} + 202 q^{29} + 368 q^{31}+ \cdots - 314 q^{98}+O(q^{100}) \) 10 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 + 44 * q^8 - 68 * q^10 - 18 * q^11 - 2 * q^13 - 188 * q^14 + 280 * q^16 + 4 * q^17 - 26 * q^19 + 196 * q^20 - 588 * q^22 + 264 * q^26 + 280 * q^28 + 202 * q^29 + 368 * q^31 - 968 * q^32 + 436 * q^34 - 476 * q^35 - 10 * q^37 + 1232 * q^38 - 1336 * q^40 - 838 * q^43 - 868 * q^44 + 1132 * q^46 + 944 * q^47 + 94 * q^49 - 726 * q^50 - 236 * q^52 + 378 * q^53 + 488 * q^56 + 8 * q^58 - 1706 * q^59 + 910 * q^61 + 80 * q^62 + 512 * q^64 + 492 * q^65 + 1942 * q^67 + 880 * q^68 + 160 * q^70 + 452 * q^74 - 1228 * q^76 + 268 * q^77 - 4416 * q^79 + 2648 * q^80 - 704 * q^82 + 2562 * q^83 - 12 * q^85 - 3764 * q^86 + 1528 * q^88 + 3332 * q^91 - 632 * q^92 - 3248 * q^94 - 6900 * q^95 - 4 * q^97 - 314 * q^98

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