LMFDB - Embedded newform 162.2.c.b.55.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,2,Mod(55,162)] 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("162.55"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-1,0,-1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$1.29357651274$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	162.55
Dual form			162.2.c.b.109.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.50000 + 2.59808i) q^{5} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} -3.00000 q^{10} +(-1.50000 + 2.59808i) q^{11} +(2.00000 + 3.46410i) q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +2.00000 q^{19} +(1.50000 - 2.59808i) q^{20} +(-1.50000 - 2.59808i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(-2.00000 + 3.46410i) q^{25} -4.00000 q^{26} -1.00000 q^{28} +(3.00000 - 5.19615i) q^{29} +(-2.50000 - 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{32} +3.00000 q^{35} +2.00000 q^{37} +(-1.00000 + 1.73205i) q^{38} +(1.50000 + 2.59808i) q^{40} +(-3.00000 - 5.19615i) q^{41} +(5.00000 - 8.66025i) q^{43} +3.00000 q^{44} +6.00000 q^{46} +(3.00000 - 5.19615i) q^{47} +(3.00000 + 5.19615i) q^{49} +(-2.00000 - 3.46410i) q^{50} +(2.00000 - 3.46410i) q^{52} -9.00000 q^{53} -9.00000 q^{55} +(0.500000 - 0.866025i) q^{56} +(3.00000 + 5.19615i) q^{58} +(6.00000 + 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +5.00000 q^{62} +1.00000 q^{64} +(-6.00000 + 10.3923i) q^{65} +(-7.00000 - 12.1244i) q^{67} +(-1.50000 + 2.59808i) q^{70} -7.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(1.50000 + 2.59808i) q^{77} +(-4.00000 + 6.92820i) q^{79} -3.00000 q^{80} +6.00000 q^{82} +(-1.50000 + 2.59808i) q^{83} +(5.00000 + 8.66025i) q^{86} +(-1.50000 + 2.59808i) q^{88} +18.0000 q^{89} +4.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(3.00000 + 5.19615i) q^{94} +(3.00000 + 5.19615i) q^{95} +(0.500000 - 0.866025i) q^{97} -6.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 3 q^{5} + q^{7} + 2 q^{8} - 6 q^{10} - 3 q^{11} + 4 q^{13} + q^{14} - q^{16} + 4 q^{19} + 3 q^{20} - 3 q^{22} - 6 q^{23} - 4 q^{25} - 8 q^{26} - 2 q^{28} + 6 q^{29} - 5 q^{31}+ \cdots - 12 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 3 * q^5 + q^7 + 2 * q^8 - 6 * q^10 - 3 * q^11 + 4 * q^13 + q^14 - q^16 + 4 * q^19 + 3 * q^20 - 3 * q^22 - 6 * q^23 - 4 * q^25 - 8 * q^26 - 2 * q^28 + 6 * q^29 - 5 * q^31 - q^32 + 6 * q^35 + 4 * q^37 - 2 * q^38 + 3 * q^40 - 6 * q^41 + 10 * q^43 + 6 * q^44 + 12 * q^46 + 6 * q^47 + 6 * q^49 - 4 * q^50 + 4 * q^52 - 18 * q^53 - 18 * q^55 + q^56 + 6 * q^58 + 12 * q^59 - 8 * q^61 + 10 * q^62 + 2 * q^64 - 12 * q^65 - 14 * q^67 - 3 * q^70 - 14 * q^73 - 2 * q^74 - 2 * q^76 + 3 * q^77 - 8 * q^79 - 6 * q^80 + 12 * q^82 - 3 * q^83 + 10 * q^86 - 3 * q^88 + 36 * q^89 + 8 * q^91 - 6 * q^92 + 6 * q^94 + 6 * q^95 + q^97 - 12 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	0.944911	+	0.327327i	$0.106148\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−3.00000			−0.948683
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	0.997927	+	0.0643593i	$0.0205004\pi$
$13$	2.00000	+	3.46410i	0.554700	+	0.960769i	−0.443227	+	0.896410i	$0.646166\pi$
$14$	0.500000	+	0.866025i	0.133631	+	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$	1.50000	−	2.59808i	0.335410	−	0.580948i
$21$		0			0
$22$	−1.50000	−	2.59808i	−0.319801	−	0.553912i
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$	−4.00000			−0.784465
$27$		0			0
$28$	−1.00000			−0.188982
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		0			0
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	0.549309	−	0.835619i	$-0.314891\pi$
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$		0			0
$35$	3.00000			0.507093
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	−1.00000	+	1.73205i	−0.162221	+	0.280976i
$39$		0			0
$40$	1.50000	+	2.59808i	0.237171	+	0.410792i
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	0.530831	−	0.847477i	$-0.321880\pi$
$41$	−3.00000	−	5.19615i	−0.468521	−	0.811503i	−0.999353	+	0.0359748i	$0.988546\pi$
$42$		0			0
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	−0.179069	−	0.983836i	$-0.557309\pi$
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	0.941562	−	0.336840i	$-0.109358\pi$
$44$	3.00000			0.452267
$45$		0			0
$46$	6.00000			0.884652
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$	−2.00000	−	3.46410i	−0.282843	−	0.489898i
$51$		0			0
$52$	2.00000	−	3.46410i	0.277350	−	0.480384i
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		0			0
$55$	−9.00000			−1.21356
$56$	0.500000	−	0.866025i	0.0668153	−	0.115728i
$57$		0			0
$58$	3.00000	+	5.19615i	0.393919	+	0.682288i
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$	5.00000			0.635001
$63$		0			0
$64$	1.00000			0.125000
$65$	−6.00000	+	10.3923i	−0.744208	+	1.28901i
$66$		0			0
$67$	−7.00000	−	12.1244i	−0.855186	−	1.48123i	−0.876472	−	0.481452i	$-0.840109\pi$
$67$	−7.00000	−	12.1244i	−0.855186	−	1.48123i	0.0212861	−	0.999773i	$-0.493224\pi$
$68$		0			0
$69$		0			0
$70$	−1.50000	+	2.59808i	−0.179284	+	0.310530i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$	−1.00000	+	1.73205i	−0.116248	+	0.201347i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.114708	−	0.198680i
$77$	1.50000	+	2.59808i	0.170941	+	0.296078i
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$	−3.00000			−0.335410
$81$		0			0
$82$	6.00000			0.662589
$83$	−1.50000	+	2.59808i	−0.164646	+	0.285176i	−0.936530	−	0.350588i	$-0.885982\pi$
$83$	−1.50000	+	2.59808i	−0.164646	+	0.285176i	0.771883	+	0.635764i	$0.219315\pi$
$84$		0			0
$85$		0			0
$86$	5.00000	+	8.66025i	0.539164	+	0.933859i
$87$		0			0
$88$	−1.50000	+	2.59808i	−0.159901	+	0.276956i
$89$	18.0000			1.90800			0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000			1.90800			0.953998	+	0.299813i	$0.0969242\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$	−3.00000	+	5.19615i	−0.312772	+	0.541736i
$93$		0			0
$94$	3.00000	+	5.19615i	0.309426	+	0.535942i
$95$	3.00000	+	5.19615i	0.307794	+	0.533114i
$96$		0			0
$97$	0.500000	−	0.866025i	0.0507673	−	0.0879316i	−0.839525	−	0.543321i	$-0.817167\pi$
$97$	0.500000	−	0.866025i	0.0507673	−	0.0879316i	0.890292	+	0.455389i	$0.150500\pi$
$98$	−6.00000			−0.606092
$99$		0			0
$100$	4.00000			0.400000
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	−0.930953	−	0.365140i	$-0.881021\pi$
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	0.781697	+	0.623658i	$0.214354\pi$
$102$		0			0
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	0.947576	−	0.319531i	$-0.103525\pi$
$103$	2.00000	+	3.46410i	0.197066	+	0.341328i	−0.750510	+	0.660859i	$0.770192\pi$
$104$	2.00000	+	3.46410i	0.196116	+	0.339683i
$105$		0			0
$106$	4.50000	−	7.79423i	0.437079	−	0.757042i
$107$	−9.00000			−0.870063			−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000			−0.870063			−0.435031	+	0.900415i	$0.643263\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$	4.50000	−	7.79423i	0.429058	−	0.743151i
$111$		0			0
$112$	0.500000	+	0.866025i	0.0472456	+	0.0818317i
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	0.689714	−	0.724082i	$-0.257736\pi$
$113$	−3.00000	−	5.19615i	−0.282216	−	0.488813i	−0.971930	+	0.235269i	$0.924403\pi$
$114$		0			0
$115$	9.00000	−	15.5885i	0.839254	−	1.45363i
$116$	−6.00000			−0.557086
$117$		0			0
$118$	−12.0000			−1.10469
$119$		0			0
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$	−4.00000	−	6.92820i	−0.362143	−	0.627250i
$123$		0			0
$124$	−2.50000	+	4.33013i	−0.224507	+	0.388857i
$125$	3.00000			0.268328
$126$		0			0
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−6.00000	−	10.3923i	−0.526235	−	0.911465i
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	−0.981824	−	0.189794i	$-0.939218\pi$
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	0.326546	−	0.945181i	$-0.394115\pi$
$132$		0			0
$133$	1.00000	−	1.73205i	0.0867110	−	0.150188i
$134$	14.0000			1.20942
$135$		0			0
$136$		0			0
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	0.938293	−	0.345843i	$-0.112407\pi$
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	−0.768655	+	0.639664i	$0.779074\pi$
$140$	−1.50000	−	2.59808i	−0.126773	−	0.219578i
$141$		0			0
$142$		0			0
$143$	−12.0000			−1.00349
$144$		0			0
$145$	18.0000			1.49482
$146$	3.50000	−	6.06218i	0.289662	−	0.501709i
$147$		0			0
$148$	−1.00000	−	1.73205i	−0.0821995	−	0.142374i
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	0.920904	−	0.389789i	$-0.127452\pi$
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	−0.798019	+	0.602632i	$0.794119\pi$
$150$		0			0
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	0.279554	+	0.960130i	$0.409814\pi$
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	−0.971274	+	0.237964i	$0.923520\pi$
$152$	2.00000			0.162221
$153$		0			0
$154$	−3.00000			−0.241747
$155$	7.50000	−	12.9904i	0.602414	−	1.04341i
$156$		0			0
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	0.934731	−	0.355357i	$-0.115641\pi$
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	−0.775113	+	0.631822i	$0.782307\pi$
$158$	−4.00000	−	6.92820i	−0.318223	−	0.551178i
$159$		0			0
$160$	1.50000	−	2.59808i	0.118585	−	0.205396i
$161$	−6.00000			−0.472866
$162$		0			0
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$	−3.00000	+	5.19615i	−0.234261	+	0.405751i
$165$		0			0
$166$	−1.50000	−	2.59808i	−0.116423	−	0.201650i
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	0.726293	−	0.687386i	$-0.241242\pi$
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	−0.958440	+	0.285295i	$0.907908\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$	−10.0000			−0.762493
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	−0.426329	−	0.904568i	$-0.640193\pi$
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	0.996544	−	0.0830722i	$-0.0264732\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$	−1.50000	−	2.59808i	−0.113067	−	0.195837i
$177$		0			0
$178$	−9.00000	+	15.5885i	−0.674579	+	1.16840i
$179$	9.00000			0.672692			0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000			0.672692			0.336346	+	0.941739i	$0.390809\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$	−2.00000	+	3.46410i	−0.148250	+	0.256776i
$183$		0			0
$184$	−3.00000	−	5.19615i	−0.221163	−	0.383065i
$185$	3.00000	+	5.19615i	0.220564	+	0.382029i
$186$		0			0
$187$		0			0
$188$	−6.00000			−0.437595
$189$		0			0
$190$	−6.00000			−0.435286
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	−0.997225	−	0.0744412i	$-0.976283\pi$
$191$	−6.00000	+	10.3923i	−0.434145	+	0.751961i	0.563081	+	0.826402i	$0.309616\pi$
$192$		0			0
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	0.761911	−	0.647682i	$-0.224262\pi$
$193$	−2.50000	−	4.33013i	−0.179954	−	0.311689i	−0.941865	+	0.335993i	$0.890928\pi$
$194$	0.500000	+	0.866025i	0.0358979	+	0.0621770i
$195$		0			0
$196$	3.00000	−	5.19615i	0.214286	−	0.371154i
$197$	−9.00000			−0.641223			−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000			−0.641223			−0.320612	+	0.947211i	$0.603888\pi$
$198$		0			0
$199$	−7.00000			−0.496217			−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000			−0.496217			−0.248108	+	0.968732i	$0.579809\pi$
$200$	−2.00000	+	3.46410i	−0.141421	+	0.244949i
$201$		0			0
$202$	−1.50000	−	2.59808i	−0.105540	−	0.182800i
$203$	−3.00000	−	5.19615i	−0.210559	−	0.364698i
$204$		0			0
$205$	9.00000	−	15.5885i	0.628587	−	1.08875i
$206$	−4.00000			−0.278693
$207$		0			0
$208$	−4.00000			−0.277350
$209$	−3.00000	+	5.19615i	−0.207514	+	0.359425i
$210$		0			0
$211$	11.0000	+	19.0526i	0.757271	+	1.31163i	0.944237	+	0.329266i	$0.106801\pi$
$211$	11.0000	+	19.0526i	0.757271	+	1.31163i	−0.186966	+	0.982366i	$0.559865\pi$
$212$	4.50000	+	7.79423i	0.309061	+	0.535310i
$213$		0			0
$214$	4.50000	−	7.79423i	0.307614	−	0.532803i
$215$	30.0000			2.04598
$216$		0			0
$217$	−5.00000			−0.339422
$218$	−1.00000	+	1.73205i	−0.0677285	+	0.117309i
$219$		0			0
$220$	4.50000	+	7.79423i	0.303390	+	0.525487i
$221$		0			0
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$	6.00000			0.399114
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	−0.993508	−	0.113761i	$-0.963710\pi$
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	0.595274	+	0.803523i	$0.297043\pi$
$228$		0			0
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	0.536515	−	0.843891i	$-0.319740\pi$
$229$	−7.00000	−	12.1244i	−0.462573	−	0.801200i	−0.999088	+	0.0426906i	$0.986407\pi$
$230$	9.00000	+	15.5885i	0.593442	+	1.02787i
$231$		0			0
$232$	3.00000	−	5.19615i	0.196960	−	0.341144i
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	18.0000			1.17419
$236$	6.00000	−	10.3923i	0.390567	−	0.676481i
$237$		0			0
$238$		0			0
$239$	15.0000	+	25.9808i	0.970269	+	1.68056i	0.694737	+	0.719264i	$0.255521\pi$
$239$	15.0000	+	25.9808i	0.970269	+	1.68056i	0.275533	+	0.961292i	$0.411146\pi$
$240$		0			0
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$	−2.00000			−0.128565
$243$		0			0
$244$	8.00000			0.512148
$245$	−9.00000	+	15.5885i	−0.574989	+	0.995910i
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$	−2.50000	−	4.33013i	−0.158750	−	0.274963i
$249$		0			0
$250$	−1.50000	+	2.59808i	−0.0948683	+	0.164317i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$	3.50000	−	6.06218i	0.219610	−	0.380375i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	0.990217	−	0.139533i	$-0.0445601\pi$
$257$	6.00000	+	10.3923i	0.374270	+	0.648254i	−0.615948	+	0.787787i	$0.711227\pi$
$258$		0			0
$259$	1.00000	−	1.73205i	0.0621370	−	0.107624i
$260$	12.0000			0.744208
$261$		0			0
$262$	15.0000			0.926703
$263$	−15.0000	+	25.9808i	−0.924940	+	1.60204i	−0.133281	+	0.991078i	$0.542551\pi$
$263$	−15.0000	+	25.9808i	−0.924940	+	1.60204i	−0.791658	+	0.610964i	$0.790782\pi$
$264$		0			0
$265$	−13.5000	−	23.3827i	−0.829298	−	1.43639i
$266$	1.00000	+	1.73205i	0.0613139	+	0.106199i
$267$		0			0
$268$	−7.00000	+	12.1244i	−0.427593	+	0.740613i
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	−25.0000			−1.51864			−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000			−1.51864			−0.759321	+	0.650716i	$0.774469\pi$
$272$		0			0
$273$		0			0
$274$	3.00000	+	5.19615i	0.181237	+	0.313911i
$275$	−6.00000	−	10.3923i	−0.361814	−	0.626680i
$276$		0			0
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	−0.960810	−	0.277207i	$-0.910591\pi$
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	0.720473	+	0.693482i	$0.243925\pi$
$278$	−4.00000			−0.239904
$279$		0			0
$280$	3.00000			0.179284
$281$	12.0000	−	20.7846i	0.715860	−	1.23991i	−0.246767	−	0.969075i	$-0.579368\pi$
$281$	12.0000	−	20.7846i	0.715860	−	1.23991i	0.962627	−	0.270831i	$-0.0872985\pi$
$282$		0			0
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	0.579437	−	0.815017i	$-0.303272\pi$
$283$	−7.00000	−	12.1244i	−0.416107	−	0.720718i	−0.995544	+	0.0942988i	$0.969939\pi$
$284$		0			0
$285$		0			0
$286$	6.00000	−	10.3923i	0.354787	−	0.614510i
$287$	−6.00000			−0.354169
$288$		0			0
$289$	−17.0000			−1.00000
$290$	−9.00000	+	15.5885i	−0.528498	+	0.915386i
$291$		0			0
$292$	3.50000	+	6.06218i	0.204822	+	0.354762i
$293$	−3.00000	−	5.19615i	−0.175262	−	0.303562i	0.764990	−	0.644042i	$-0.222744\pi$
$293$	−3.00000	−	5.19615i	−0.175262	−	0.303562i	−0.940252	+	0.340480i	$0.889411\pi$
$294$		0			0
$295$	−18.0000	+	31.1769i	−1.04800	+	1.81519i
$296$	2.00000			0.116248
$297$		0			0
$298$	−3.00000			−0.173785
$299$	12.0000	−	20.7846i	0.693978	−	1.20201i
$300$		0			0
$301$	−5.00000	−	8.66025i	−0.288195	−	0.499169i
$302$	−8.50000	−	14.7224i	−0.489120	−	0.847181i
$303$		0			0
$304$	−1.00000	+	1.73205i	−0.0573539	+	0.0993399i
$305$	−24.0000			−1.37424
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$	1.50000	−	2.59808i	0.0854704	−	0.148039i
$309$		0			0
$310$	7.50000	+	12.9904i	0.425971	+	0.737804i
$311$	−3.00000	−	5.19615i	−0.170114	−	0.294647i	0.768345	−	0.640036i	$-0.221080\pi$
$311$	−3.00000	−	5.19615i	−0.170114	−	0.294647i	−0.938460	+	0.345389i	$0.887747\pi$
$312$		0			0
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	−0.462093	−	0.886831i	$-0.652902\pi$
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	0.999065	−	0.0432311i	$-0.0137652\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	8.00000			0.450035
$317$	−1.50000	+	2.59808i	−0.0842484	+	0.145922i	−0.905071	−	0.425261i	$-0.860182\pi$
$317$	−1.50000	+	2.59808i	−0.0842484	+	0.145922i	0.820822	+	0.571184i	$0.193516\pi$
$318$		0			0
$319$	9.00000	+	15.5885i	0.503903	+	0.872786i
$320$	1.50000	+	2.59808i	0.0838525	+	0.145237i
$321$		0			0
$322$	3.00000	−	5.19615i	0.167183	−	0.289570i
$323$		0			0
$324$		0			0
$325$	−16.0000			−0.887520
$326$	−10.0000	+	17.3205i	−0.553849	+	0.959294i
$327$		0			0
$328$	−3.00000	−	5.19615i	−0.165647	−	0.286910i
$329$	−3.00000	−	5.19615i	−0.165395	−	0.286473i
$330$		0			0
$331$	5.00000	−	8.66025i	0.274825	−	0.476011i	−0.695266	−	0.718752i	$-0.744713\pi$
$331$	5.00000	−	8.66025i	0.274825	−	0.476011i	0.970091	+	0.242742i	$0.0780468\pi$
$332$	3.00000			0.164646
$333$		0			0
$334$	6.00000			0.328305
$335$	21.0000	−	36.3731i	1.14735	−	1.98727i
$336$		0			0
$337$	11.0000	+	19.0526i	0.599208	+	1.03786i	0.992938	+	0.118633i	$0.0378512\pi$
$337$	11.0000	+	19.0526i	0.599208	+	1.03786i	−0.393730	+	0.919226i	$0.628816\pi$
$338$	−1.50000	−	2.59808i	−0.0815892	−	0.141317i
$339$		0			0
$340$		0			0
$341$	15.0000			0.812296
$342$		0			0
$343$	13.0000			0.701934
$344$	5.00000	−	8.66025i	0.269582	−	0.466930i
$345$		0			0
$346$	7.50000	+	12.9904i	0.403202	+	0.698367i
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	0.903475	−	0.428640i	$-0.141007\pi$
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	−0.822951	+	0.568112i	$0.807674\pi$
$348$		0			0
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	−0.700609	−	0.713545i	$-0.747088\pi$
$349$	5.00000	−	8.66025i	0.267644	−	0.463573i	0.968253	+	0.249973i	$0.0804216\pi$
$350$	−4.00000			−0.213809
$351$		0			0
$352$	3.00000			0.159901
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	−0.775077	−	0.631867i	$-0.782289\pi$
$353$	3.00000	−	5.19615i	0.159674	−	0.276563i	0.934751	+	0.355303i	$0.115622\pi$
$354$		0			0
$355$		0			0
$356$	−9.00000	−	15.5885i	−0.476999	−	0.826187i
$357$		0			0
$358$	−4.50000	+	7.79423i	−0.237832	+	0.411938i
$359$	−18.0000			−0.950004			−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000			−0.950004			−0.475002	+	0.879985i	$0.657553\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$	8.00000	−	13.8564i	0.420471	−	0.728277i
$363$		0			0
$364$	−2.00000	−	3.46410i	−0.104828	−	0.181568i
$365$	−10.5000	−	18.1865i	−0.549595	−	0.951927i
$366$		0			0
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	−0.997960	−	0.0638362i	$-0.979666\pi$
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	0.554264	+	0.832341i	$0.313000\pi$
$368$	6.00000			0.312772
$369$		0			0
$370$	−6.00000			−0.311925
$371$	−4.50000	+	7.79423i	−0.233628	+	0.404656i
$372$		0			0
$373$	−16.0000	−	27.7128i	−0.828449	−	1.43492i	−0.899255	−	0.437425i	$-0.855891\pi$
$373$	−16.0000	−	27.7128i	−0.828449	−	1.43492i	0.0708063	−	0.997490i	$-0.477443\pi$
$374$		0			0
$375$		0			0
$376$	3.00000	−	5.19615i	0.154713	−	0.267971i
$377$	24.0000			1.23606
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$	3.00000	−	5.19615i	0.153897	−	0.266557i
$381$		0			0
$382$	−6.00000	−	10.3923i	−0.306987	−	0.531717i
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	−0.990702	−	0.136047i	$-0.956560\pi$
$383$	−12.0000	−	20.7846i	−0.613171	−	1.06204i	0.377531	−	0.925997i	$-0.376773\pi$
$384$		0			0
$385$	−4.50000	+	7.79423i	−0.229341	+	0.397231i
$386$	5.00000			0.254493
$387$		0			0
$388$	−1.00000			−0.0507673
$389$	−10.5000	+	18.1865i	−0.532371	+	0.922094i	0.466915	+	0.884302i	$0.345366\pi$
$389$	−10.5000	+	18.1865i	−0.532371	+	0.922094i	−0.999286	+	0.0377914i	$0.987968\pi$
$390$		0			0
$391$		0			0
$392$	3.00000	+	5.19615i	0.151523	+	0.262445i
$393$		0			0
$394$	4.50000	−	7.79423i	0.226707	−	0.392668i
$395$	−24.0000			−1.20757
$396$		0			0
$397$	20.0000			1.00377			0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000			1.00377			0.501886	+	0.864934i	$0.332640\pi$
$398$	3.50000	−	6.06218i	0.175439	−	0.303870i
$399$		0			0
$400$	−2.00000	−	3.46410i	−0.100000	−	0.173205i
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	0.976050	−	0.217545i	$-0.0698049\pi$
$401$	6.00000	+	10.3923i	0.299626	+	0.518967i	−0.676425	+	0.736512i	$0.736472\pi$
$402$		0			0
$403$	10.0000	−	17.3205i	0.498135	−	0.862796i
$404$	3.00000			0.149256
$405$		0			0
$406$	6.00000			0.297775
$407$	−3.00000	+	5.19615i	−0.148704	+	0.257564i
$408$		0			0
$409$	−11.5000	−	19.9186i	−0.568638	−	0.984911i	−0.996701	−	0.0811615i	$-0.974137\pi$
$409$	−11.5000	−	19.9186i	−0.568638	−	0.984911i	0.428063	−	0.903749i	$-0.359196\pi$
$410$	9.00000	+	15.5885i	0.444478	+	0.769859i
$411$		0			0
$412$	2.00000	−	3.46410i	0.0985329	−	0.170664i
$413$	12.0000			0.590481
$414$		0			0
$415$	−9.00000			−0.441793
$416$	2.00000	−	3.46410i	0.0980581	−	0.169842i
$417$		0			0
$418$	−3.00000	−	5.19615i	−0.146735	−	0.254152i
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	0.974546	−	0.224189i	$-0.0719734\pi$
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	−0.681426	+	0.731887i	$0.738640\pi$
$420$		0			0
$421$	−4.00000	+	6.92820i	−0.194948	+	0.337660i	−0.946883	−	0.321577i	$-0.895787\pi$
$421$	−4.00000	+	6.92820i	−0.194948	+	0.337660i	0.751935	+	0.659237i	$0.229121\pi$
$422$	−22.0000			−1.07094
$423$		0			0
$424$	−9.00000			−0.437079
$425$		0			0
$426$		0			0
$427$	4.00000	+	6.92820i	0.193574	+	0.335279i
$428$	4.50000	+	7.79423i	0.217516	+	0.376748i
$429$		0			0
$430$	−15.0000	+	25.9808i	−0.723364	+	1.25290i
$431$	−18.0000			−0.867029			−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000			−0.867029			−0.433515	+	0.901146i	$0.642727\pi$
$432$		0			0
$433$	29.0000			1.39365			0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000			1.39365			0.696826	+	0.717241i	$0.254595\pi$
$434$	2.50000	−	4.33013i	0.120004	−	0.207853i
$435$		0			0
$436$	−1.00000	−	1.73205i	−0.0478913	−	0.0829502i
$437$	−6.00000	−	10.3923i	−0.287019	−	0.497131i
$438$		0			0
$439$	9.50000	−	16.4545i	0.453410	−	0.785330i	−0.545185	−	0.838316i	$-0.683541\pi$
$439$	9.50000	−	16.4545i	0.453410	−	0.785330i	0.998595	+	0.0529862i	$0.0168739\pi$
$440$	−9.00000			−0.429058
$441$		0			0
$442$		0			0
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	−0.972626	−	0.232377i	$-0.925350\pi$
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	0.687557	+	0.726130i	$0.258683\pi$
$444$		0			0
$445$	27.0000	+	46.7654i	1.27992	+	2.21689i
$446$	−4.00000	−	6.92820i	−0.189405	−	0.328060i
$447$		0			0
$448$	0.500000	−	0.866025i	0.0236228	−	0.0409159i
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$	18.0000			0.847587
$452$	−3.00000	+	5.19615i	−0.141108	+	0.244406i
$453$		0			0
$454$	−6.00000	−	10.3923i	−0.281594	−	0.487735i
$455$	6.00000	+	10.3923i	0.281284	+	0.487199i
$456$		0			0
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	−0.854094	−	0.520119i	$-0.825888\pi$
$457$	0.500000	−	0.866025i	0.0233890	−	0.0405110i	0.877483	+	0.479608i	$0.159221\pi$
$458$	14.0000			0.654177
$459$		0			0
$460$	−18.0000			−0.839254
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	−0.999920	−	0.0126168i	$-0.995984\pi$
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	0.510887	+	0.859648i	$0.329317\pi$
$462$		0			0
$463$	6.50000	+	11.2583i	0.302081	+	0.523219i	0.976607	−	0.215032i	$-0.0689855\pi$
$463$	6.50000	+	11.2583i	0.302081	+	0.523219i	−0.674526	+	0.738251i	$0.735652\pi$
$464$	3.00000	+	5.19615i	0.139272	+	0.241225i
$465$		0			0
$466$	9.00000	−	15.5885i	0.416917	−	0.722121i
$467$	27.0000			1.24941			0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000			1.24941			0.624705	+	0.780860i	$0.285219\pi$
$468$		0			0
$469$	−14.0000			−0.646460
$470$	−9.00000	+	15.5885i	−0.415139	+	0.719042i
$471$		0			0
$472$	6.00000	+	10.3923i	0.276172	+	0.478345i
$473$	15.0000	+	25.9808i	0.689701	+	1.19460i
$474$		0			0
$475$	−4.00000	+	6.92820i	−0.183533	+	0.317888i
$476$		0			0
$477$		0			0
$478$	−30.0000			−1.37217
$479$	3.00000	−	5.19615i	0.137073	−	0.237418i	−0.789314	−	0.613990i	$-0.789564\pi$
$479$	3.00000	−	5.19615i	0.137073	−	0.237418i	0.926388	+	0.376571i	$0.122897\pi$
$480$		0			0
$481$	4.00000	+	6.92820i	0.182384	+	0.315899i
$482$	5.00000	+	8.66025i	0.227744	+	0.394464i
$483$		0			0
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$	3.00000			0.136223
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	−4.00000	+	6.92820i	−0.181071	+	0.313625i
$489$		0			0
$490$	−9.00000	−	15.5885i	−0.406579	−	0.704215i
$491$	19.5000	+	33.7750i	0.880023	+	1.52424i	0.851314	+	0.524656i	$0.175806\pi$
$491$	19.5000	+	33.7750i	0.880023	+	1.52424i	0.0287085	+	0.999588i	$0.490861\pi$
$492$		0			0
$493$		0			0
$494$	−8.00000			−0.359937
$495$		0			0
$496$	5.00000			0.224507
$497$		0			0
$498$		0			0
$499$	−7.00000	−	12.1244i	−0.313363	−	0.542761i	0.665725	−	0.746197i	$-0.268122\pi$
$499$	−7.00000	−	12.1244i	−0.313363	−	0.542761i	−0.979088	+	0.203436i	$0.934789\pi$
$500$	−1.50000	−	2.59808i	−0.0670820	−	0.116190i
$501$		0			0
$502$		0			0
$503$	18.0000			0.802580			0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000			0.802580			0.401290	+	0.915951i	$0.368562\pi$
$504$		0			0
$505$	−9.00000			−0.400495
$506$	−9.00000	+	15.5885i	−0.400099	+	0.692991i
$507$		0			0
$508$	3.50000	+	6.06218i	0.155287	+	0.268966i
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	0.650556	−	0.759458i	$-0.274536\pi$
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	−0.982988	+	0.183669i	$0.941202\pi$
$510$		0			0
$511$	−3.50000	+	6.06218i	−0.154831	+	0.268175i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−12.0000			−0.529297
$515$	−6.00000	+	10.3923i	−0.264392	+	0.457940i
$516$		0			0
$517$	9.00000	+	15.5885i	0.395820	+	0.685580i
$518$	1.00000	+	1.73205i	0.0439375	+	0.0761019i
$519$		0			0
$520$	−6.00000	+	10.3923i	−0.263117	+	0.455733i
$521$	−36.0000			−1.57719			−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000			−1.57719			−0.788594	+	0.614914i	$0.789191\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$	−7.50000	+	12.9904i	−0.327639	+	0.567487i
$525$		0			0
$526$	−15.0000	−	25.9808i	−0.654031	−	1.13282i
$527$		0			0
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$	27.0000			1.17281
$531$		0			0
$532$	−2.00000			−0.0867110
$533$	12.0000	−	20.7846i	0.519778	−	0.900281i
$534$		0			0
$535$	−13.5000	−	23.3827i	−0.583656	−	1.01092i
$536$	−7.00000	−	12.1244i	−0.302354	−	0.523692i
$537$		0			0
$538$	−9.00000	+	15.5885i	−0.388018	+	0.672066i
$539$	−18.0000			−0.775315
$540$		0			0
$541$	20.0000			0.859867			0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000			0.859867			0.429934	+	0.902861i	$0.358537\pi$
$542$	12.5000	−	21.6506i	0.536921	−	0.929974i
$543$		0			0
$544$		0			0
$545$	3.00000	+	5.19615i	0.128506	+	0.222579i
$546$		0			0
$547$	−4.00000	+	6.92820i	−0.171028	+	0.296229i	−0.938779	−	0.344519i	$-0.888042\pi$
$547$	−4.00000	+	6.92820i	−0.171028	+	0.296229i	0.767752	+	0.640747i	$0.221375\pi$
$548$	−6.00000			−0.256307
$549$		0			0
$550$	12.0000			0.511682
$551$	6.00000	−	10.3923i	0.255609	−	0.442727i
$552$		0			0
$553$	4.00000	+	6.92820i	0.170097	+	0.294617i
$554$	−4.00000	−	6.92820i	−0.169944	−	0.294351i
$555$		0			0
$556$	2.00000	−	3.46410i	0.0848189	−	0.146911i
$557$	−27.0000			−1.14403			−0.572013	−	0.820244i	$-0.693837\pi$
$557$	−27.0000			−1.14403			−0.572013	+	0.820244i	$0.693837\pi$
$558$		0			0
$559$	40.0000			1.69182
$560$	−1.50000	+	2.59808i	−0.0633866	+	0.109789i
$561$		0			0
$562$	12.0000	+	20.7846i	0.506189	+	0.876746i
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	0.895902	−	0.444252i	$-0.146530\pi$
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	−0.832684	+	0.553748i	$0.813197\pi$
$564$		0			0
$565$	9.00000	−	15.5885i	0.378633	−	0.655811i
$566$	14.0000			0.588464
$567$		0			0
$568$		0			0
$569$	−6.00000	+	10.3923i	−0.251533	+	0.435668i	−0.963948	−	0.266090i	$-0.914268\pi$
$569$	−6.00000	+	10.3923i	−0.251533	+	0.435668i	0.712415	+	0.701758i	$0.247601\pi$
$570$		0			0
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	0.904835	−	0.425762i	$-0.139994\pi$
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	−0.821138	+	0.570730i	$0.806660\pi$
$572$	6.00000	+	10.3923i	0.250873	+	0.434524i
$573$		0			0
$574$	3.00000	−	5.19615i	0.125218	−	0.216883i
$575$	24.0000			1.00087
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$	8.50000	−	14.7224i	0.353553	−	0.612372i
$579$		0			0
$580$	−9.00000	−	15.5885i	−0.373705	−	0.647275i
$581$	1.50000	+	2.59808i	0.0622305	+	0.107786i
$582$		0			0
$583$	13.5000	−	23.3827i	0.559113	−	0.968412i
$584$	−7.00000			−0.289662
$585$		0			0
$586$	6.00000			0.247858
$587$	−1.50000	+	2.59808i	−0.0619116	+	0.107234i	−0.895320	−	0.445424i	$-0.853053\pi$
$587$	−1.50000	+	2.59808i	−0.0619116	+	0.107234i	0.833408	+	0.552658i	$0.186386\pi$
$588$		0			0
$589$	−5.00000	−	8.66025i	−0.206021	−	0.356840i
$590$	−18.0000	−	31.1769i	−0.741048	−	1.28353i
$591$		0			0
$592$	−1.00000	+	1.73205i	−0.0410997	+	0.0711868i
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$	1.50000	−	2.59808i	0.0614424	−	0.106421i
$597$		0			0
$598$	12.0000	+	20.7846i	0.490716	+	0.849946i
$599$	−21.0000	−	36.3731i	−0.858037	−	1.48616i	−0.873799	−	0.486287i	$-0.838351\pi$
$599$	−21.0000	−	36.3731i	−0.858037	−	1.48616i	0.0157622	−	0.999876i	$-0.494983\pi$
$600$		0			0
$601$	−17.5000	+	30.3109i	−0.713840	+	1.23641i	0.249565	+	0.968358i	$0.419712\pi$
$601$	−17.5000	+	30.3109i	−0.713840	+	1.23641i	−0.963405	+	0.268049i	$0.913621\pi$
$602$	10.0000			0.407570
$603$		0			0
$604$	17.0000			0.691720
$605$	−3.00000	+	5.19615i	−0.121967	+	0.211254i
$606$		0			0
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	−0.983262	−	0.182199i	$-0.941678\pi$
$607$	−16.0000	−	27.7128i	−0.649420	−	1.12483i	0.333842	−	0.942629i	$-0.391655\pi$
$608$	−1.00000	−	1.73205i	−0.0405554	−	0.0702439i
$609$		0			0
$610$	12.0000	−	20.7846i	0.485866	−	0.841544i
$611$	24.0000			0.970936
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$	8.00000	−	13.8564i	0.322854	−	0.559199i
$615$		0			0
$616$	1.50000	+	2.59808i	0.0604367	+	0.104679i
$617$	−21.0000	−	36.3731i	−0.845428	−	1.46432i	−0.885249	−	0.465118i	$-0.846012\pi$
$617$	−21.0000	−	36.3731i	−0.845428	−	1.46432i	0.0398207	−	0.999207i	$-0.487321\pi$
$618$		0			0
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	−0.434551	−	0.900647i	$-0.643093\pi$
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	0.997259	−	0.0739910i	$-0.0235736\pi$
$620$	−15.0000			−0.602414
$621$		0			0
$622$	6.00000			0.240578
$623$	9.00000	−	15.5885i	0.360577	−	0.624538i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$	9.50000	+	16.4545i	0.379696	+	0.657653i
$627$		0			0
$628$	2.00000	−	3.46410i	0.0798087	−	0.138233i
$629$		0			0
$630$		0			0
$631$	−25.0000			−0.995234			−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000			−0.995234			−0.497617	+	0.867397i	$0.665792\pi$
$632$	−4.00000	+	6.92820i	−0.159111	+	0.275589i
$633$		0			0
$634$	−1.50000	−	2.59808i	−0.0595726	−	0.103183i
$635$	−10.5000	−	18.1865i	−0.416680	−	0.721711i
$636$		0			0
$637$	−12.0000	+	20.7846i	−0.475457	+	0.823516i
$638$	−18.0000			−0.712627
$639$		0			0
$640$	−3.00000			−0.118585
$641$	21.0000	−	36.3731i	0.829450	−	1.43665i	−0.0690201	−	0.997615i	$-0.521987\pi$
$641$	21.0000	−	36.3731i	0.829450	−	1.43665i	0.898470	−	0.439034i	$-0.144679\pi$
$642$		0			0
$643$	2.00000	+	3.46410i	0.0788723	+	0.136611i	0.902764	−	0.430137i	$-0.141535\pi$
$643$	2.00000	+	3.46410i	0.0788723	+	0.136611i	−0.823891	+	0.566748i	$0.808201\pi$
$644$	3.00000	+	5.19615i	0.118217	+	0.204757i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−36.0000			−1.41312
$650$	8.00000	−	13.8564i	0.313786	−	0.543493i
$651$		0			0
$652$	−10.0000	−	17.3205i	−0.391630	−	0.678323i
$653$	19.5000	+	33.7750i	0.763094	+	1.32172i	0.941248	+	0.337715i	$0.109654\pi$
$653$	19.5000	+	33.7750i	0.763094	+	1.32172i	−0.178154	+	0.984003i	$0.557013\pi$
$654$		0			0
$655$	22.5000	−	38.9711i	0.879148	−	1.52273i
$656$	6.00000			0.234261
$657$		0			0
$658$	6.00000			0.233904
$659$	−10.5000	+	18.1865i	−0.409022	+	0.708447i	−0.994780	−	0.102039i	$-0.967463\pi$
$659$	−10.5000	+	18.1865i	−0.409022	+	0.708447i	0.585758	+	0.810486i	$0.300797\pi$
$660$		0			0
$661$	−7.00000	−	12.1244i	−0.272268	−	0.471583i	0.697174	−	0.716902i	$-0.254441\pi$
$661$	−7.00000	−	12.1244i	−0.272268	−	0.471583i	−0.969442	+	0.245319i	$0.921107\pi$
$662$	5.00000	+	8.66025i	0.194331	+	0.336590i
$663$		0			0
$664$	−1.50000	+	2.59808i	−0.0582113	+	0.100825i
$665$	6.00000			0.232670
$666$		0			0
$667$	−36.0000			−1.39393
$668$	−3.00000	+	5.19615i	−0.116073	+	0.201045i
$669$		0			0
$670$	21.0000	+	36.3731i	0.811301	+	1.40521i
$671$	−12.0000	−	20.7846i	−0.463255	−	0.802381i
$672$		0			0
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	−0.622770	−	0.782405i	$-0.713993\pi$
$673$	9.50000	−	16.4545i	0.366198	−	0.634274i	0.988968	+	0.148132i	$0.0473259\pi$
$674$	−22.0000			−0.847408
$675$		0			0
$676$	3.00000			0.115385
$677$	21.0000	−	36.3731i	0.807096	−	1.39793i	−0.107772	−	0.994176i	$-0.534372\pi$
$677$	21.0000	−	36.3731i	0.807096	−	1.39793i	0.914867	−	0.403755i	$-0.132295\pi$
$678$		0			0
$679$	−0.500000	−	0.866025i	−0.0191882	−	0.0332350i
$680$		0			0
$681$		0			0
$682$	−7.50000	+	12.9904i	−0.287190	+	0.497427i
$683$	36.0000			1.37750			0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$	−6.50000	+	11.2583i	−0.248171	+	0.429845i
$687$		0			0
$688$	5.00000	+	8.66025i	0.190623	+	0.330169i
$689$	−18.0000	−	31.1769i	−0.685745	−	1.18775i
$690$		0			0
$691$	−22.0000	+	38.1051i	−0.836919	+	1.44959i	0.0555386	+	0.998457i	$0.482312\pi$
$691$	−22.0000	+	38.1051i	−0.836919	+	1.44959i	−0.892458	+	0.451130i	$0.851021\pi$
$692$	−15.0000			−0.570214
$693$		0			0
$694$	−3.00000			−0.113878
$695$	−6.00000	+	10.3923i	−0.227593	+	0.394203i
$696$		0			0
$697$		0			0
$698$	5.00000	+	8.66025i	0.189253	+	0.327795i
$699$		0			0
$700$	2.00000	−	3.46410i	0.0755929	−	0.130931i
$701$	−9.00000			−0.339925			−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000			−0.339925			−0.169963	+	0.985451i	$0.554365\pi$
$702$		0			0
$703$	4.00000			0.150863
$704$	−1.50000	+	2.59808i	−0.0565334	+	0.0979187i
$705$		0			0
$706$	3.00000	+	5.19615i	0.112906	+	0.195560i
$707$	1.50000	+	2.59808i	0.0564133	+	0.0977107i
$708$		0			0
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	0.0747503	+	0.997202i	$0.476184\pi$
$709$	−22.0000	+	38.1051i	−0.826227	+	1.43107i	−0.900978	+	0.433865i	$0.857149\pi$
$710$		0			0
$711$		0			0
$712$	18.0000			0.674579
$713$	−15.0000	+	25.9808i	−0.561754	+	0.972987i
$714$		0			0
$715$	−18.0000	−	31.1769i	−0.673162	−	1.16595i
$716$	−4.50000	−	7.79423i	−0.168173	−	0.291284i
$717$		0			0
$718$	9.00000	−	15.5885i	0.335877	−	0.581756i
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$	7.50000	−	12.9904i	0.279121	−	0.483452i
$723$		0			0
$724$	8.00000	+	13.8564i	0.297318	+	0.514969i
$725$	12.0000	+	20.7846i	0.445669	+	0.771921i
$726$		0			0
$727$	0.500000	−	0.866025i	0.0185440	−	0.0321191i	−0.856605	−	0.515974i	$-0.827430\pi$
$727$	0.500000	−	0.866025i	0.0185440	−	0.0321191i	0.875148	+	0.483854i	$0.160764\pi$
$728$	4.00000			0.148250
$729$		0			0
$730$	21.0000			0.777245
$731$		0			0
$732$		0			0
$733$	11.0000	+	19.0526i	0.406294	+	0.703722i	0.994471	−	0.105010i	$-0.0334875\pi$
$733$	11.0000	+	19.0526i	0.406294	+	0.703722i	−0.588177	+	0.808732i	$0.700154\pi$
$734$	−8.50000	−	14.7224i	−0.313741	−	0.543415i
$735$		0			0
$736$	−3.00000	+	5.19615i	−0.110581	+	0.191533i
$737$	42.0000			1.54709
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$	3.00000	−	5.19615i	0.110282	−	0.191014i
$741$		0			0
$742$	−4.50000	−	7.79423i	−0.165200	−	0.286135i
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	0.954844	−	0.297108i	$-0.0960222\pi$
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	−0.734725	+	0.678365i	$0.762689\pi$
$744$		0			0
$745$	−4.50000	+	7.79423i	−0.164867	+	0.285558i
$746$	32.0000			1.17160
$747$		0			0
$748$		0			0
$749$	−4.50000	+	7.79423i	−0.164426	+	0.284795i
$750$		0			0
$751$	−20.5000	−	35.5070i	−0.748056	−	1.29567i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−20.5000	−	35.5070i	−0.748056	−	1.29567i	0.200698	−	0.979653i	$-0.435679\pi$
$752$	3.00000	+	5.19615i	0.109399	+	0.189484i
$753$		0			0
$754$	−12.0000	+	20.7846i	−0.437014	+	0.756931i
$755$	−51.0000			−1.85608
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	−10.0000	+	17.3205i	−0.363216	+	0.629109i
$759$		0			0
$760$	3.00000	+	5.19615i	0.108821	+	0.188484i
$761$	24.0000	+	41.5692i	0.869999	+	1.50688i	0.861996	+	0.506915i	$0.169214\pi$
$761$	24.0000	+	41.5692i	0.869999	+	1.50688i	0.00800331	+	0.999968i	$0.497452\pi$
$762$		0			0
$763$	1.00000	−	1.73205i	0.0362024	−	0.0627044i
$764$	12.0000			0.434145
$765$		0			0
$766$	24.0000			0.867155
$767$	−24.0000	+	41.5692i	−0.866590	+	1.50098i
$768$		0			0
$769$	15.5000	+	26.8468i	0.558944	+	0.968120i	0.997585	+	0.0694574i	$0.0221268\pi$
$769$	15.5000	+	26.8468i	0.558944	+	0.968120i	−0.438641	+	0.898663i	$0.644540\pi$
$770$	−4.50000	−	7.79423i	−0.162169	−	0.280885i
$771$		0			0
$772$	−2.50000	+	4.33013i	−0.0899770	+	0.155845i
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$	20.0000			0.718421
$776$	0.500000	−	0.866025i	0.0179490	−	0.0310885i
$777$		0			0
$778$	−10.5000	−	18.1865i	−0.376443	−	0.652019i
$779$	−6.00000	−	10.3923i	−0.214972	−	0.372343i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−6.00000			−0.214286
$785$	−6.00000	+	10.3923i	−0.214149	+	0.370917i
$786$		0			0
$787$	−16.0000	−	27.7128i	−0.570338	−	0.987855i	−0.996531	−	0.0832226i	$-0.973479\pi$
$787$	−16.0000	−	27.7128i	−0.570338	−	0.987855i	0.426193	−	0.904632i	$-0.359855\pi$
$788$	4.50000	+	7.79423i	0.160306	+	0.277658i
$789$		0			0
$790$	12.0000	−	20.7846i	0.426941	−	0.739483i
$791$	−6.00000			−0.213335
$792$		0			0
$793$	−32.0000			−1.13635
$794$	−10.0000	+	17.3205i	−0.354887	+	0.614682i
$795$		0			0
$796$	3.50000	+	6.06218i	0.124054	+	0.214868i
$797$	19.5000	+	33.7750i	0.690725	+	1.19637i	0.971601	+	0.236627i	$0.0760420\pi$
$797$	19.5000	+	33.7750i	0.690725	+	1.19637i	−0.280875	+	0.959744i	$0.590625\pi$
$798$		0			0
$799$		0			0
$800$	4.00000			0.141421
$801$		0			0
$802$	−12.0000			−0.423735
$803$	10.5000	−	18.1865i	0.370537	−	0.641789i
$804$		0			0
$805$	−9.00000	−	15.5885i	−0.317208	−	0.549421i
$806$	10.0000	+	17.3205i	0.352235	+	0.610089i
$807$		0			0
$808$	−1.50000	+	2.59808i	−0.0527698	+	0.0914000i
$809$	−18.0000			−0.632846			−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000			−0.632846			−0.316423	+	0.948618i	$0.602482\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$	−3.00000	+	5.19615i	−0.105279	+	0.182349i
$813$		0			0
$814$	−3.00000	−	5.19615i	−0.105150	−	0.182125i
$815$	30.0000	+	51.9615i	1.05085	+	1.82013i
$816$		0			0
$817$	10.0000	−	17.3205i	0.349856	−	0.605968i
$818$	23.0000			0.804176
$819$		0			0
$820$	−18.0000			−0.628587
$821$	3.00000	−	5.19615i	0.104701	−	0.181347i	−0.808915	−	0.587925i	$-0.799945\pi$
$821$	3.00000	−	5.19615i	0.104701	−	0.181347i	0.913616	+	0.406578i	$0.133278\pi$
$822$		0			0
$823$	15.5000	+	26.8468i	0.540296	+	0.935820i	0.998887	+	0.0471726i	$0.0150211\pi$
$823$	15.5000	+	26.8468i	0.540296	+	0.935820i	−0.458591	+	0.888648i	$0.651646\pi$
$824$	2.00000	+	3.46410i	0.0696733	+	0.120678i
$825$		0			0
$826$	−6.00000	+	10.3923i	−0.208767	+	0.361595i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$	4.50000	−	7.79423i	0.156197	−	0.270542i
$831$		0			0
$832$	2.00000	+	3.46410i	0.0693375	+	0.120096i
$833$		0			0
$834$		0			0
$835$	9.00000	−	15.5885i	0.311458	−	0.539461i
$836$	6.00000			0.207514
$837$		0			0
$838$	−12.0000			−0.414533
$839$	12.0000	−	20.7846i	0.414286	−	0.717564i	−0.581067	−	0.813856i	$-0.697365\pi$
$839$	12.0000	−	20.7846i	0.414286	−	0.717564i	0.995353	+	0.0962912i	$0.0306980\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$	−4.00000	−	6.92820i	−0.137849	−	0.238762i
$843$		0			0
$844$	11.0000	−	19.0526i	0.378636	−	0.655816i
$845$	−9.00000			−0.309609
$846$		0			0
$847$	2.00000			0.0687208
$848$	4.50000	−	7.79423i	0.154531	−	0.267655i
$849$		0			0
$850$		0			0
$851$	−6.00000	−	10.3923i	−0.205677	−	0.356244i
$852$		0			0
$853$	5.00000	−	8.66025i	0.171197	−	0.296521i	−0.767642	−	0.640879i	$-0.778570\pi$
$853$	5.00000	−	8.66025i	0.171197	−	0.296521i	0.938839	+	0.344358i	$0.111903\pi$
$854$	−8.00000			−0.273754
$855$		0			0
$856$	−9.00000			−0.307614
$857$	−24.0000	+	41.5692i	−0.819824	+	1.41998i	0.0859870	+	0.996296i	$0.472596\pi$
$857$	−24.0000	+	41.5692i	−0.819824	+	1.41998i	−0.905811	+	0.423681i	$0.860738\pi$
$858$		0			0
$859$	20.0000	+	34.6410i	0.682391	+	1.18194i	0.974249	+	0.225475i	$0.0723932\pi$
$859$	20.0000	+	34.6410i	0.682391	+	1.18194i	−0.291858	+	0.956462i	$0.594273\pi$
$860$	−15.0000	−	25.9808i	−0.511496	−	0.885937i
$861$		0			0
$862$	9.00000	−	15.5885i	0.306541	−	0.530945i
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$	45.0000			1.53005
$866$	−14.5000	+	25.1147i	−0.492730	+	0.853433i
$867$		0			0
$868$	2.50000	+	4.33013i	0.0848555	+	0.146974i
$869$	−12.0000	−	20.7846i	−0.407072	−	0.705070i
$870$		0			0
$871$	28.0000	−	48.4974i	0.948744	−	1.64327i
$872$	2.00000			0.0677285
$873$		0			0
$874$	12.0000			0.405906
$875$	1.50000	−	2.59808i	0.0507093	−	0.0878310i
$876$		0			0
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	0.989788	−	0.142548i	$-0.0455296\pi$
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	−0.618344	+	0.785907i	$0.712196\pi$
$878$	9.50000	+	16.4545i	0.320609	+	0.555312i
$879$		0			0
$880$	4.50000	−	7.79423i	0.151695	−	0.262743i
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$	2.00000			0.0673054			0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000			0.0673054			0.0336527	+	0.999434i	$0.489286\pi$
$884$		0			0
$885$		0			0
$886$	−6.00000	−	10.3923i	−0.201574	−	0.349136i
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	0.948999	−	0.315279i	$-0.102098\pi$
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	−0.747539	+	0.664218i	$0.768765\pi$
$888$		0			0
$889$	−3.50000	+	6.06218i	−0.117386	+	0.203319i
$890$	−54.0000			−1.81008
$891$		0			0
$892$	8.00000			0.267860
$893$	6.00000	−	10.3923i	0.200782	−	0.347765i
$894$		0			0
$895$	13.5000	+	23.3827i	0.451255	+	0.781597i
$896$	0.500000	+	0.866025i	0.0167038	+	0.0289319i
$897$		0			0
$898$	−9.00000	+	15.5885i	−0.300334	+	0.520194i
$899$	−30.0000			−1.00056
$900$		0			0
$901$		0			0
$902$	−9.00000	+	15.5885i	−0.299667	+	0.519039i
$903$		0			0
$904$	−3.00000	−	5.19615i	−0.0997785	−	0.172821i
$905$	−24.0000	−	41.5692i	−0.797787	−	1.38181i
$906$		0			0
$907$	−13.0000	+	22.5167i	−0.431658	+	0.747653i	−0.997016	−	0.0771920i	$-0.975405\pi$
$907$	−13.0000	+	22.5167i	−0.431658	+	0.747653i	0.565358	+	0.824845i	$0.308738\pi$
$908$	12.0000			0.398234
$909$		0			0
$910$	−12.0000			−0.397796
$911$	3.00000	−	5.19615i	0.0993944	−	0.172156i	−0.812040	−	0.583602i	$-0.801643\pi$
$911$	3.00000	−	5.19615i	0.0993944	−	0.172156i	0.911434	+	0.411446i	$0.134976\pi$
$912$		0			0
$913$	−4.50000	−	7.79423i	−0.148928	−	0.257951i
$914$	0.500000	+	0.866025i	0.0165385	+	0.0286456i
$915$		0			0
$916$	−7.00000	+	12.1244i	−0.231287	+	0.400600i
$917$	−15.0000			−0.495344
$918$		0			0
$919$	29.0000			0.956622			0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000			0.956622			0.478311	+	0.878191i	$0.341249\pi$
$920$	9.00000	−	15.5885i	0.296721	−	0.513936i
$921$		0			0
$922$	−10.5000	−	18.1865i	−0.345799	−	0.598942i
$923$		0			0
$924$		0			0
$925$	−4.00000	+	6.92820i	−0.131519	+	0.227798i
$926$	−13.0000			−0.427207
$927$		0			0
$928$	−6.00000			−0.196960
$929$	−6.00000	+	10.3923i	−0.196854	+	0.340960i	−0.947507	−	0.319736i	$-0.896406\pi$
$929$	−6.00000	+	10.3923i	−0.196854	+	0.340960i	0.750653	+	0.660697i	$0.229739\pi$
$930$		0			0
$931$	6.00000	+	10.3923i	0.196642	+	0.340594i
$932$	9.00000	+	15.5885i	0.294805	+	0.510617i
$933$		0			0
$934$	−13.5000	+	23.3827i	−0.441733	+	0.765105i
$935$		0			0
$936$		0			0
$937$	11.0000			0.359354			0.179677	−	0.983726i	$-0.442495\pi$
$937$	11.0000			0.359354			0.179677	+	0.983726i	$0.442495\pi$
$938$	7.00000	−	12.1244i	0.228558	−	0.395874i
$939$		0			0
$940$	−9.00000	−	15.5885i	−0.293548	−	0.508439i
$941$	−16.5000	−	28.5788i	−0.537885	−	0.931644i	−0.999018	−	0.0443125i	$-0.985890\pi$
$941$	−16.5000	−	28.5788i	−0.537885	−	0.931644i	0.461133	−	0.887331i	$-0.347443\pi$
$942$		0			0
$943$	−18.0000	+	31.1769i	−0.586161	+	1.01526i
$944$	−12.0000			−0.390567
$945$		0			0
$946$	−30.0000			−0.975384
$947$	25.5000	−	44.1673i	0.828639	−	1.43524i	−0.0704677	−	0.997514i	$-0.522449\pi$
$947$	25.5000	−	44.1673i	0.828639	−	1.43524i	0.899106	−	0.437730i	$-0.144217\pi$
$948$		0			0
$949$	−14.0000	−	24.2487i	−0.454459	−	0.787146i
$950$	−4.00000	−	6.92820i	−0.129777	−	0.224781i
$951$		0			0
$952$		0			0
$953$	36.0000			1.16615			0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000			1.16615			0.583077	+	0.812417i	$0.301849\pi$
$954$		0			0
$955$	−36.0000			−1.16493
$956$	15.0000	−	25.9808i	0.485135	−	0.840278i
$957$		0			0
$958$	3.00000	+	5.19615i	0.0969256	+	0.167880i
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$	−8.00000			−0.257930
$963$		0			0
$964$	−10.0000			−0.322078
$965$	7.50000	−	12.9904i	0.241434	−	0.418175i
$966$		0			0
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	0.619721	−	0.784822i	$-0.287246\pi$
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	−0.989536	+	0.144283i	$0.953912\pi$
$968$	1.00000	+	1.73205i	0.0321412	+	0.0556702i
$969$		0			0
$970$	−1.50000	+	2.59808i	−0.0481621	+	0.0834192i
$971$	27.0000			0.866471			0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000			0.866471			0.433236	+	0.901281i	$0.357372\pi$
$972$		0			0
$973$	4.00000			0.128234
$974$	8.00000	−	13.8564i	0.256337	−	0.443988i
$975$		0			0
$976$	−4.00000	−	6.92820i	−0.128037	−	0.221766i
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	−0.977379	−	0.211495i	$-0.932167\pi$
$977$	−21.0000	−	36.3731i	−0.671850	−	1.16368i	0.305530	−	0.952183i	$-0.401167\pi$
$978$		0			0
$979$	−27.0000	+	46.7654i	−0.862924	+	1.49463i
$980$	18.0000			0.574989
$981$		0			0
$982$	−39.0000			−1.24454
$983$	3.00000	−	5.19615i	0.0956851	−	0.165732i	−0.814209	−	0.580572i	$-0.802829\pi$
$983$	3.00000	−	5.19615i	0.0956851	−	0.165732i	0.909894	+	0.414840i	$0.136162\pi$
$984$		0			0
$985$	−13.5000	−	23.3827i	−0.430146	−	0.745034i
$986$		0			0
$987$		0			0
$988$	4.00000	−	6.92820i	0.127257	−	0.220416i
$989$	−60.0000			−1.90789
$990$		0			0
$991$	47.0000			1.49300			0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000			1.49300			0.746502	+	0.665383i	$0.231732\pi$
$992$	−2.50000	+	4.33013i	−0.0793751	+	0.137482i
$993$		0			0
$994$		0			0
$995$	−10.5000	−	18.1865i	−0.332872	−	0.576552i
$996$		0			0
$997$	14.0000	−	24.2487i	0.443384	−	0.767964i	−0.554554	−	0.832148i	$-0.687111\pi$
$997$	14.0000	−	24.2487i	0.443384	−	0.767964i	0.997938	+	0.0641836i	$0.0204443\pi$
$998$	14.0000			0.443162
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.2.c.b.55.1		2
3.2	odd	2		162.2.c.c.55.1		2
4.3	odd	2		1296.2.i.o.865.1		2
9.2	odd	6		54.2.a.a.1.1	✓	1
9.4	even	3	inner	162.2.c.b.109.1		2
9.5	odd	6		162.2.c.c.109.1		2
9.7	even	3		54.2.a.b.1.1	yes	1
12.11	even	2		1296.2.i.c.865.1		2
36.7	odd	6		432.2.a.b.1.1		1
36.11	even	6		432.2.a.g.1.1		1
36.23	even	6		1296.2.i.c.433.1		2
36.31	odd	6		1296.2.i.o.433.1		2
45.2	even	12		1350.2.c.b.649.1		2
45.7	odd	12		1350.2.c.k.649.2		2
45.29	odd	6		1350.2.a.r.1.1		1
45.34	even	6		1350.2.a.h.1.1		1
45.38	even	12		1350.2.c.b.649.2		2
45.43	odd	12		1350.2.c.k.649.1		2
63.20	even	6		2646.2.a.a.1.1		1
63.34	odd	6		2646.2.a.bd.1.1		1
72.11	even	6		1728.2.a.d.1.1		1
72.29	odd	6		1728.2.a.c.1.1		1
72.43	odd	6		1728.2.a.z.1.1		1
72.61	even	6		1728.2.a.y.1.1		1
99.43	odd	6		6534.2.a.b.1.1		1
99.65	even	6		6534.2.a.bc.1.1		1
117.25	even	6		9126.2.a.r.1.1		1
117.38	odd	6		9126.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.a.a.1.1	✓	1	9.2	odd	6
54.2.a.b.1.1	yes	1	9.7	even	3
162.2.c.b.55.1		2	1.1	even	1	trivial
162.2.c.b.109.1		2	9.4	even	3	inner
162.2.c.c.55.1		2	3.2	odd	2
162.2.c.c.109.1		2	9.5	odd	6
432.2.a.b.1.1		1	36.7	odd	6
432.2.a.g.1.1		1	36.11	even	6
1296.2.i.c.433.1		2	36.23	even	6
1296.2.i.c.865.1		2	12.11	even	2
1296.2.i.o.433.1		2	36.31	odd	6
1296.2.i.o.865.1		2	4.3	odd	2
1350.2.a.h.1.1		1	45.34	even	6
1350.2.a.r.1.1		1	45.29	odd	6
1350.2.c.b.649.1		2	45.2	even	12
1350.2.c.b.649.2		2	45.38	even	12
1350.2.c.k.649.1		2	45.43	odd	12
1350.2.c.k.649.2		2	45.7	odd	12
1728.2.a.c.1.1		1	72.29	odd	6
1728.2.a.d.1.1		1	72.11	even	6
1728.2.a.y.1.1		1	72.61	even	6
1728.2.a.z.1.1		1	72.43	odd	6
2646.2.a.a.1.1		1	63.20	even	6
2646.2.a.bd.1.1		1	63.34	odd	6
6534.2.a.b.1.1		1	99.43	odd	6
6534.2.a.bc.1.1		1	99.65	even	6
9126.2.a.r.1.1		1	117.25	even	6
9126.2.a.u.1.1		1	117.38	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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.50000 + 2.59808i) q^{5} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} -3.00000 q^{10} +(-1.50000 + 2.59808i) q^{11} +(2.00000 + 3.46410i) q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +2.00000 q^{19} +(1.50000 - 2.59808i) q^{20} +(-1.50000 - 2.59808i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(-2.00000 + 3.46410i) q^{25} -4.00000 q^{26} -1.00000 q^{28} +(3.00000 - 5.19615i) q^{29} +(-2.50000 - 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{32} +3.00000 q^{35} +2.00000 q^{37} +(-1.00000 + 1.73205i) q^{38} +(1.50000 + 2.59808i) q^{40} +(-3.00000 - 5.19615i) q^{41} +(5.00000 - 8.66025i) q^{43} +3.00000 q^{44} +6.00000 q^{46} +(3.00000 - 5.19615i) q^{47} +(3.00000 + 5.19615i) q^{49} +(-2.00000 - 3.46410i) q^{50} +(2.00000 - 3.46410i) q^{52} -9.00000 q^{53} -9.00000 q^{55} +(0.500000 - 0.866025i) q^{56} +(3.00000 + 5.19615i) q^{58} +(6.00000 + 10.3923i) q^{59} +(-4.00000 + 6.92820i) q^{61} +5.00000 q^{62} +1.00000 q^{64} +(-6.00000 + 10.3923i) q^{65} +(-7.00000 - 12.1244i) q^{67} +(-1.50000 + 2.59808i) q^{70} -7.00000 q^{73} +(-1.00000 + 1.73205i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(1.50000 + 2.59808i) q^{77} +(-4.00000 + 6.92820i) q^{79} -3.00000 q^{80} +6.00000 q^{82} +(-1.50000 + 2.59808i) q^{83} +(5.00000 + 8.66025i) q^{86} +(-1.50000 + 2.59808i) q^{88} +18.0000 q^{89} +4.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(3.00000 + 5.19615i) q^{94} +(3.00000 + 5.19615i) q^{95} +(0.500000 - 0.866025i) q^{97} -6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 3 q^{5} + q^{7} + 2 q^{8} - 6 q^{10} - 3 q^{11} + 4 q^{13} + q^{14} - q^{16} + 4 q^{19} + 3 q^{20} - 3 q^{22} - 6 q^{23} - 4 q^{25} - 8 q^{26} - 2 q^{28} + 6 q^{29} - 5 q^{31}+ \cdots - 12 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 3 * q^5 + q^7 + 2 * q^8 - 6 * q^10 - 3 * q^11 + 4 * q^13 + q^14 - q^16 + 4 * q^19 + 3 * q^20 - 3 * q^22 - 6 * q^23 - 4 * q^25 - 8 * q^26 - 2 * q^28 + 6 * q^29 - 5 * q^31 - q^32 + 6 * q^35 + 4 * q^37 - 2 * q^38 + 3 * q^40 - 6 * q^41 + 10 * q^43 + 6 * q^44 + 12 * q^46 + 6 * q^47 + 6 * q^49 - 4 * q^50 + 4 * q^52 - 18 * q^53 - 18 * q^55 + q^56 + 6 * q^58 + 12 * q^59 - 8 * q^61 + 10 * q^62 + 2 * q^64 - 12 * q^65 - 14 * q^67 - 3 * q^70 - 14 * q^73 - 2 * q^74 - 2 * q^76 + 3 * q^77 - 8 * q^79 - 6 * q^80 + 12 * q^82 - 3 * q^83 + 10 * q^86 - 3 * q^88 + 36 * q^89 + 8 * q^91 - 6 * q^92 + 6 * q^94 + 6 * q^95 + q^97 - 12 * q^98

Introduction

L-functions

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