LMFDB - Embedded newform 243.2.c.c.82.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,2,Mod(82,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("243.82");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 243 = 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	243.c (of order $3$, degree $2$, not 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:	$1.94036476912$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			82.2
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	243.82
Dual form			243.2.c.c.163.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 - 1.73205i) q^{7} -4.89898 q^{8} +O(q^{10})$	\(q+(1.22474 + 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 - 1.73205i) q^{7} -4.89898 q^{8} -6.00000 q^{10} +(1.22474 + 2.12132i) q^{11} +(0.500000 - 0.866025i) q^{13} +(2.44949 - 4.24264i) q^{14} +(-2.00000 - 3.46410i) q^{16} +7.34847 q^{17} -1.00000 q^{19} +(-4.89898 - 8.48528i) q^{20} +(-3.00000 + 5.19615i) q^{22} +(-1.22474 + 2.12132i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.44949 q^{26} +8.00000 q^{28} +(-2.44949 - 4.24264i) q^{29} +(0.500000 - 0.866025i) q^{31} +(9.00000 + 15.5885i) q^{34} +4.89898 q^{35} +8.00000 q^{37} +(-1.22474 - 2.12132i) q^{38} +(6.00000 - 10.3923i) q^{40} +(2.44949 - 4.24264i) q^{41} +(-5.50000 - 9.52628i) q^{43} -9.79796 q^{44} -6.00000 q^{46} +(4.89898 + 8.48528i) q^{47} +(1.50000 - 2.59808i) q^{49} +(1.22474 - 2.12132i) q^{50} +(2.00000 + 3.46410i) q^{52} -7.34847 q^{53} -6.00000 q^{55} +(4.89898 + 8.48528i) q^{56} +(6.00000 - 10.3923i) q^{58} +(-1.22474 + 2.12132i) q^{59} +(-2.50000 - 4.33013i) q^{61} +2.44949 q^{62} -8.00000 q^{64} +(1.22474 + 2.12132i) q^{65} +(3.50000 - 6.06218i) q^{67} +(-14.6969 + 25.4558i) q^{68} +(6.00000 + 10.3923i) q^{70} -7.34847 q^{71} +11.0000 q^{73} +(9.79796 + 16.9706i) q^{74} +(2.00000 - 3.46410i) q^{76} +(2.44949 - 4.24264i) q^{77} +(3.50000 + 6.06218i) q^{79} +9.79796 q^{80} +12.0000 q^{82} +(-6.12372 - 10.6066i) q^{83} +(-9.00000 + 15.5885i) q^{85} +(13.4722 - 23.3345i) q^{86} +(-6.00000 - 10.3923i) q^{88} -2.00000 q^{91} +(-4.89898 - 8.48528i) q^{92} +(-12.0000 + 20.7846i) q^{94} +(1.22474 - 2.12132i) q^{95} +(3.50000 + 6.06218i) q^{97} +7.34847 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^7	$ 4 q - 8 q^{4} - 4 q^{7} - 24 q^{10} + 2 q^{13} - 8 q^{16} - 4 q^{19} - 12 q^{22} - 2 q^{25} + 32 q^{28} + 2 q^{31} + 36 q^{34} + 32 q^{37} + 24 q^{40} - 22 q^{43} - 24 q^{46} + 6 q^{49} + 8 q^{52} - 24 q^{55} + 24 q^{58} - 10 q^{61} - 32 q^{64} + 14 q^{67} + 24 q^{70} + 44 q^{73} + 8 q^{76} + 14 q^{79} + 48 q^{82} - 36 q^{85} - 24 q^{88} - 8 q^{91} - 48 q^{94} + 14 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^7 - 24 * q^10 + 2 * q^13 - 8 * q^16 - 4 * q^19 - 12 * q^22 - 2 * q^25 + 32 * q^28 + 2 * q^31 + 36 * q^34 + 32 * q^37 + 24 * q^40 - 22 * q^43 - 24 * q^46 + 6 * q^49 + 8 * q^52 - 24 * q^55 + 24 * q^58 - 10 * q^61 - 32 * q^64 + 14 * q^67 + 24 * q^70 + 44 * q^73 + 8 * q^76 + 14 * q^79 + 48 * q^82 - 36 * q^85 - 24 * q^88 - 8 * q^91 - 48 * q^94 + 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.22474	+	2.12132i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$2$	1.22474	+	2.12132i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−1.00000	+	1.73205i
$5$	−1.22474	+	2.12132i	−0.547723	+	0.948683i	0.450708	+	0.892672i	$0.351172\pi$
$5$	−1.22474	+	2.12132i	−0.547723	+	0.948683i	−0.998430	+	0.0560116i	$0.982162\pi$
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$	−4.89898			−1.73205
$9$		0			0
$10$	−6.00000			−1.89737
$11$	1.22474	+	2.12132i	0.369274	+	0.639602i	0.989452	−	0.144859i	$-0.0462729\pi$
$11$	1.22474	+	2.12132i	0.369274	+	0.639602i	−0.620178	+	0.784461i	$0.712940\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$	2.44949	−	4.24264i	0.654654	−	1.13389i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	7.34847			1.78227			0.891133	−	0.453743i	$-0.149911\pi$
$17$	7.34847			1.78227			0.891133	+	0.453743i	$0.149911\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$	−4.89898	−	8.48528i	−1.09545	−	1.89737i
$21$		0			0
$22$	−3.00000	+	5.19615i	−0.639602	+	1.10782i
$23$	−1.22474	+	2.12132i	−0.255377	+	0.442326i	−0.964998	−	0.262258i	$-0.915533\pi$
$23$	−1.22474	+	2.12132i	−0.255377	+	0.442326i	0.709621	+	0.704584i	$0.248866\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	2.44949			0.480384
$27$		0			0
$28$	8.00000			1.51186
$29$	−2.44949	−	4.24264i	−0.454859	−	0.787839i	0.543821	−	0.839201i	$-0.316977\pi$
$29$	−2.44949	−	4.24264i	−0.454859	−	0.787839i	−0.998680	+	0.0513625i	$0.983644\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	−0.817625	−	0.575751i	$-0.804710\pi$
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	0.907428	+	0.420208i	$0.138043\pi$
$32$		0			0
$33$		0			0
$34$	9.00000	+	15.5885i	1.54349	+	2.67340i
$35$	4.89898			0.828079
$36$		0			0
$37$	8.00000			1.31519			0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000			1.31519			0.657596	+	0.753371i	$0.271573\pi$
$38$	−1.22474	−	2.12132i	−0.198680	−	0.344124i
$39$		0			0
$40$	6.00000	−	10.3923i	0.948683	−	1.64317i
$41$	2.44949	−	4.24264i	0.382546	−	0.662589i	−0.608879	−	0.793263i	$-0.708381\pi$
$41$	2.44949	−	4.24264i	0.382546	−	0.662589i	0.991425	+	0.130674i	$0.0417140\pi$
$42$		0			0
$43$	−5.50000	−	9.52628i	−0.838742	−	1.45274i	−0.890947	−	0.454108i	$-0.849958\pi$
$43$	−5.50000	−	9.52628i	−0.838742	−	1.45274i	0.0522047	−	0.998636i	$-0.483375\pi$
$44$	−9.79796			−1.47710
$45$		0			0
$46$	−6.00000			−0.884652
$47$	4.89898	+	8.48528i	0.714590	+	1.23771i	0.963118	+	0.269081i	$0.0867199\pi$
$47$	4.89898	+	8.48528i	0.714590	+	1.23771i	−0.248528	+	0.968625i	$0.579947\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$	1.22474	−	2.12132i	0.173205	−	0.300000i
$51$		0			0
$52$	2.00000	+	3.46410i	0.277350	+	0.480384i
$53$	−7.34847			−1.00939			−0.504695	−	0.863298i	$-0.668395\pi$
$53$	−7.34847			−1.00939			−0.504695	+	0.863298i	$0.668395\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$	4.89898	+	8.48528i	0.654654	+	1.13389i
$57$		0			0
$58$	6.00000	−	10.3923i	0.787839	−	1.36458i
$59$	−1.22474	+	2.12132i	−0.159448	+	0.276172i	−0.934670	−	0.355517i	$-0.884305\pi$
$59$	−1.22474	+	2.12132i	−0.159448	+	0.276172i	0.775222	+	0.631689i	$0.217638\pi$
$60$		0			0
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	0.660415	−	0.750901i	$-0.270381\pi$
$61$	−2.50000	−	4.33013i	−0.320092	−	0.554416i	−0.980507	+	0.196485i	$0.937047\pi$
$62$	2.44949			0.311086
$63$		0			0
$64$	−8.00000			−1.00000
$65$	1.22474	+	2.12132i	0.151911	+	0.263117i
$66$		0			0
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	−0.569066	−	0.822292i	$-0.692695\pi$
$67$	3.50000	−	6.06218i	0.427593	−	0.740613i	0.996659	+	0.0816792i	$0.0260283\pi$
$68$	−14.6969	+	25.4558i	−1.78227	+	3.08697i
$69$		0			0
$70$	6.00000	+	10.3923i	0.717137	+	1.24212i
$71$	−7.34847			−0.872103			−0.436051	−	0.899922i	$-0.643623\pi$
$71$	−7.34847			−0.872103			−0.436051	+	0.899922i	$0.643623\pi$
$72$		0			0
$73$	11.0000			1.28745			0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000			1.28745			0.643726	+	0.765256i	$0.277388\pi$
$74$	9.79796	+	16.9706i	1.13899	+	1.97279i
$75$		0			0
$76$	2.00000	−	3.46410i	0.229416	−	0.397360i
$77$	2.44949	−	4.24264i	0.279145	−	0.483494i
$78$		0			0
$79$	3.50000	+	6.06218i	0.393781	+	0.682048i	0.992945	−	0.118578i	$-0.0378336\pi$
$79$	3.50000	+	6.06218i	0.393781	+	0.682048i	−0.599164	+	0.800626i	$0.704500\pi$
$80$	9.79796			1.09545
$81$		0			0
$82$	12.0000			1.32518
$83$	−6.12372	−	10.6066i	−0.672166	−	1.16423i	−0.977289	−	0.211913i	$-0.932031\pi$
$83$	−6.12372	−	10.6066i	−0.672166	−	1.16423i	0.305123	−	0.952313i	$-0.401303\pi$
$84$		0			0
$85$	−9.00000	+	15.5885i	−0.976187	+	1.69081i
$86$	13.4722	−	23.3345i	1.45274	−	2.51623i
$87$		0			0
$88$	−6.00000	−	10.3923i	−0.639602	−	1.10782i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−2.00000			−0.209657
$92$	−4.89898	−	8.48528i	−0.510754	−	0.884652i
$93$		0			0
$94$	−12.0000	+	20.7846i	−1.23771	+	2.14377i
$95$	1.22474	−	2.12132i	0.125656	−	0.217643i
$96$		0			0
$97$	3.50000	+	6.06218i	0.355371	+	0.615521i	0.987181	−	0.159602i	$-0.0510211\pi$
$97$	3.50000	+	6.06218i	0.355371	+	0.615521i	−0.631810	+	0.775123i	$0.717688\pi$
$98$	7.34847			0.742307
$99$		0			0
$100$	4.00000			0.400000
$101$	−2.44949	−	4.24264i	−0.243733	−	0.422159i	0.718041	−	0.696000i	$-0.245039\pi$
$101$	−2.44949	−	4.24264i	−0.243733	−	0.422159i	−0.961775	+	0.273842i	$0.911706\pi$
$102$		0			0
$103$	3.50000	−	6.06218i	0.344865	−	0.597324i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	3.50000	−	6.06218i	0.344865	−	0.597324i	0.985329	+	0.170664i	$0.0545913\pi$
$104$	−2.44949	+	4.24264i	−0.240192	+	0.416025i
$105$		0			0
$106$	−9.00000	−	15.5885i	−0.874157	−	1.51408i
$107$	−14.6969			−1.42081			−0.710403	−	0.703795i	$-0.751487\pi$
$107$	−14.6969			−1.42081			−0.710403	+	0.703795i	$0.751487\pi$
$108$		0			0
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$	−7.34847	−	12.7279i	−0.700649	−	1.21356i
$111$		0			0
$112$	−4.00000	+	6.92820i	−0.377964	+	0.654654i
$113$	−4.89898	+	8.48528i	−0.460857	+	0.798228i	−0.999004	−	0.0446231i	$-0.985791\pi$
$113$	−4.89898	+	8.48528i	−0.460857	+	0.798228i	0.538147	+	0.842851i	$0.319125\pi$
$114$		0			0
$115$	−3.00000	−	5.19615i	−0.279751	−	0.484544i
$116$	19.5959			1.81944
$117$		0			0
$118$	−6.00000			−0.552345
$119$	−7.34847	−	12.7279i	−0.673633	−	1.16677i
$120$		0			0
$121$	2.50000	−	4.33013i	0.227273	−	0.393648i
$122$	6.12372	−	10.6066i	0.554416	−	0.960277i
$123$		0			0
$124$	2.00000	+	3.46410i	0.179605	+	0.311086i
$125$	−9.79796			−0.876356
$126$		0			0
$127$	−19.0000			−1.68598			−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000			−1.68598			−0.842989	+	0.537931i	$0.819206\pi$
$128$	−9.79796	−	16.9706i	−0.866025	−	1.50000i
$129$		0			0
$130$	−3.00000	+	5.19615i	−0.263117	+	0.455733i
$131$	6.12372	−	10.6066i	0.535032	−	0.926703i	−0.464130	−	0.885767i	$-0.653633\pi$
$131$	6.12372	−	10.6066i	0.535032	−	0.926703i	0.999162	−	0.0409357i	$-0.0130339\pi$
$132$		0			0
$133$	1.00000	+	1.73205i	0.0867110	+	0.150188i
$134$	17.1464			1.48123
$135$		0			0
$136$	−36.0000			−3.08697
$137$	4.89898	+	8.48528i	0.418548	+	0.724947i	0.995794	−	0.0916241i	$-0.0292058\pi$
$137$	4.89898	+	8.48528i	0.418548	+	0.724947i	−0.577246	+	0.816571i	$0.695872\pi$
$138$		0			0
$139$	5.00000	−	8.66025i	0.424094	−	0.734553i	−0.572241	−	0.820086i	$-0.693926\pi$
$139$	5.00000	−	8.66025i	0.424094	−	0.734553i	0.996335	+	0.0855324i	$0.0272591\pi$
$140$	−9.79796	+	16.9706i	−0.828079	+	1.43427i
$141$		0			0
$142$	−9.00000	−	15.5885i	−0.755263	−	1.30815i
$143$	2.44949			0.204837
$144$		0			0
$145$	12.0000			0.996546
$146$	13.4722	+	23.3345i	1.11497	+	1.93118i
$147$		0			0
$148$	−16.0000	+	27.7128i	−1.31519	+	2.27798i
$149$	6.12372	−	10.6066i	0.501675	−	0.868927i	−0.498323	−	0.866991i	$-0.666051\pi$
$149$	6.12372	−	10.6066i	0.501675	−	0.868927i	0.999998	−	0.00193526i	$-0.000616012\pi$
$150$		0			0
$151$	−2.50000	−	4.33013i	−0.203447	−	0.352381i	0.746190	−	0.665733i	$-0.231881\pi$
$151$	−2.50000	−	4.33013i	−0.203447	−	0.352381i	−0.949637	+	0.313353i	$0.898548\pi$
$152$	4.89898			0.397360
$153$		0			0
$154$	12.0000			0.966988
$155$	1.22474	+	2.12132i	0.0983739	+	0.170389i
$156$		0			0
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	0.297097	+	0.954847i	$0.403982\pi$
$157$	−8.50000	+	14.7224i	−0.678374	+	1.17498i	−0.975470	+	0.220131i	$0.929352\pi$
$158$	−8.57321	+	14.8492i	−0.682048	+	1.18134i
$159$		0			0
$160$		0			0
$161$	4.89898			0.386094
$162$		0			0
$163$	−10.0000			−0.783260			−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000			−0.783260			−0.391630	+	0.920123i	$0.628089\pi$
$164$	9.79796	+	16.9706i	0.765092	+	1.32518i
$165$		0			0
$166$	15.0000	−	25.9808i	1.16423	−	2.01650i
$167$	2.44949	−	4.24264i	0.189547	−	0.328305i	−0.755552	−	0.655089i	$-0.772631\pi$
$167$	2.44949	−	4.24264i	0.189547	−	0.328305i	0.945099	+	0.326783i	$0.105965\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$	−44.0908			−3.38161
$171$		0			0
$172$	44.0000			3.35497
$173$	4.89898	+	8.48528i	0.372463	+	0.645124i	0.989944	−	0.141462i	$-0.0451802\pi$
$173$	4.89898	+	8.48528i	0.372463	+	0.645124i	−0.617481	+	0.786586i	$0.711847\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$	4.89898	−	8.48528i	0.369274	−	0.639602i
$177$		0			0
$178$		0			0
$179$	14.6969			1.09850			0.549250	−	0.835658i	$-0.314913\pi$
$179$	14.6969			1.09850			0.549250	+	0.835658i	$0.314913\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$	−2.44949	−	4.24264i	−0.181568	−	0.314485i
$183$		0			0
$184$	6.00000	−	10.3923i	0.442326	−	0.766131i
$185$	−9.79796	+	16.9706i	−0.720360	+	1.24770i
$186$		0			0
$187$	9.00000	+	15.5885i	0.658145	+	1.13994i
$188$	−39.1918			−2.85836
$189$		0			0
$190$	6.00000			0.435286
$191$	4.89898	+	8.48528i	0.354478	+	0.613973i	0.987028	−	0.160546i	$-0.0513253\pi$
$191$	4.89898	+	8.48528i	0.354478	+	0.613973i	−0.632551	+	0.774519i	$0.717992\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$	−8.57321	+	14.8492i	−0.615521	+	1.06611i
$195$		0			0
$196$	6.00000	+	10.3923i	0.428571	+	0.742307i
$197$	−14.6969			−1.04711			−0.523557	−	0.851991i	$-0.675395\pi$
$197$	−14.6969			−1.04711			−0.523557	+	0.851991i	$0.675395\pi$
$198$		0			0
$199$	−1.00000			−0.0708881			−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000			−0.0708881			−0.0354441	+	0.999372i	$0.511285\pi$
$200$	2.44949	+	4.24264i	0.173205	+	0.300000i
$201$		0			0
$202$	6.00000	−	10.3923i	0.422159	−	0.731200i
$203$	−4.89898	+	8.48528i	−0.343841	+	0.595550i
$204$		0			0
$205$	6.00000	+	10.3923i	0.419058	+	0.725830i
$206$	17.1464			1.19465
$207$		0			0
$208$	−4.00000			−0.277350
$209$	−1.22474	−	2.12132i	−0.0847174	−	0.146735i
$210$		0			0
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	−0.848301	−	0.529514i	$-0.822374\pi$
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	0.882723	+	0.469894i	$0.155708\pi$
$212$	14.6969	−	25.4558i	1.00939	−	1.74831i
$213$		0			0
$214$	−18.0000	−	31.1769i	−1.23045	−	2.13121i
$215$	26.9444			1.83759
$216$		0			0
$217$	−2.00000			−0.135769
$218$	−1.22474	−	2.12132i	−0.0829502	−	0.143674i
$219$		0			0
$220$	12.0000	−	20.7846i	0.809040	−	1.40130i
$221$	3.67423	−	6.36396i	0.247156	−	0.428086i
$222$		0			0
$223$	3.50000	+	6.06218i	0.234377	+	0.405953i	0.959092	−	0.283096i	$-0.0913615\pi$
$223$	3.50000	+	6.06218i	0.234377	+	0.405953i	−0.724714	+	0.689050i	$0.758028\pi$
$224$		0			0
$225$		0			0
$226$	−24.0000			−1.59646
$227$	4.89898	+	8.48528i	0.325157	+	0.563188i	0.981544	−	0.191236i	$-0.0612496\pi$
$227$	4.89898	+	8.48528i	0.325157	+	0.563188i	−0.656387	+	0.754424i	$0.727916\pi$
$228$		0			0
$229$	0.500000	−	0.866025i	0.0330409	−	0.0572286i	−0.849032	−	0.528341i	$-0.822814\pi$
$229$	0.500000	−	0.866025i	0.0330409	−	0.0572286i	0.882073	+	0.471113i	$0.156147\pi$
$230$	7.34847	−	12.7279i	0.484544	−	0.839254i
$231$		0			0
$232$	12.0000	+	20.7846i	0.787839	+	1.36458i
$233$	7.34847			0.481414			0.240707	−	0.970598i	$-0.422621\pi$
$233$	7.34847			0.481414			0.240707	+	0.970598i	$0.422621\pi$
$234$		0			0
$235$	−24.0000			−1.56559
$236$	−4.89898	−	8.48528i	−0.318896	−	0.552345i
$237$		0			0
$238$	18.0000	−	31.1769i	1.16677	−	2.02090i
$239$	−1.22474	+	2.12132i	−0.0792222	+	0.137217i	−0.902915	−	0.429820i	$-0.858577\pi$
$239$	−1.22474	+	2.12132i	−0.0792222	+	0.137217i	0.823692	+	0.567037i	$0.191910\pi$
$240$		0			0
$241$	8.00000	+	13.8564i	0.515325	+	0.892570i	0.999842	+	0.0177875i	$0.00566223\pi$
$241$	8.00000	+	13.8564i	0.515325	+	0.892570i	−0.484516	+	0.874782i	$0.661004\pi$
$242$	12.2474			0.787296
$243$		0			0
$244$	20.0000			1.28037
$245$	3.67423	+	6.36396i	0.234738	+	0.406579i
$246$		0			0
$247$	−0.500000	+	0.866025i	−0.0318142	+	0.0551039i
$248$	−2.44949	+	4.24264i	−0.155543	+	0.269408i
$249$		0			0
$250$	−12.0000	−	20.7846i	−0.758947	−	1.31453i
$251$	7.34847			0.463831			0.231916	−	0.972736i	$-0.425501\pi$
$251$	7.34847			0.463831			0.231916	+	0.972736i	$0.425501\pi$
$252$		0			0
$253$	−6.00000			−0.377217
$254$	−23.2702	−	40.3051i	−1.46010	−	2.52897i
$255$		0			0
$256$	16.0000	−	27.7128i	1.00000	−	1.73205i
$257$	−8.57321	+	14.8492i	−0.534782	+	0.926270i	0.464391	+	0.885630i	$0.346273\pi$
$257$	−8.57321	+	14.8492i	−0.534782	+	0.926270i	−0.999174	+	0.0406402i	$0.987060\pi$
$258$		0			0
$259$	−8.00000	−	13.8564i	−0.497096	−	0.860995i
$260$	−9.79796			−0.607644
$261$		0			0
$262$	30.0000			1.85341
$263$	−13.4722	−	23.3345i	−0.830731	−	1.43887i	−0.897459	−	0.441097i	$-0.854589\pi$
$263$	−13.4722	−	23.3345i	−0.830731	−	1.43887i	0.0667283	−	0.997771i	$-0.478744\pi$
$264$		0			0
$265$	9.00000	−	15.5885i	0.552866	−	0.957591i
$266$	−2.44949	+	4.24264i	−0.150188	+	0.260133i
$267$		0			0
$268$	14.0000	+	24.2487i	0.855186	+	1.48123i
$269$	22.0454			1.34413			0.672066	−	0.740491i	$-0.265407\pi$
$269$	22.0454			1.34413			0.672066	+	0.740491i	$0.265407\pi$
$270$		0			0
$271$	−7.00000			−0.425220			−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000			−0.425220			−0.212610	+	0.977137i	$0.568196\pi$
$272$	−14.6969	−	25.4558i	−0.891133	−	1.54349i
$273$		0			0
$274$	−12.0000	+	20.7846i	−0.724947	+	1.25564i
$275$	1.22474	−	2.12132i	0.0738549	−	0.127920i
$276$		0			0
$277$	−5.50000	−	9.52628i	−0.330463	−	0.572379i	0.652140	−	0.758099i	$-0.273872\pi$
$277$	−5.50000	−	9.52628i	−0.330463	−	0.572379i	−0.982603	+	0.185720i	$0.940538\pi$
$278$	24.4949			1.46911
$279$		0			0
$280$	−24.0000			−1.43427
$281$	−6.12372	−	10.6066i	−0.365311	−	0.632737i	0.623515	−	0.781811i	$-0.285704\pi$
$281$	−6.12372	−	10.6066i	−0.365311	−	0.632737i	−0.988826	+	0.149074i	$0.952371\pi$
$282$		0			0
$283$	−8.50000	+	14.7224i	−0.505273	+	0.875158i	0.494709	+	0.869059i	$0.335275\pi$
$283$	−8.50000	+	14.7224i	−0.505273	+	0.875158i	−0.999981	+	0.00609896i	$0.998059\pi$
$284$	14.6969	−	25.4558i	0.872103	−	1.51053i
$285$		0			0
$286$	3.00000	+	5.19615i	0.177394	+	0.307255i
$287$	−9.79796			−0.578355
$288$		0			0
$289$	37.0000			2.17647
$290$	14.6969	+	25.4558i	0.863034	+	1.49482i
$291$		0			0
$292$	−22.0000	+	38.1051i	−1.28745	+	2.22993i
$293$	2.44949	−	4.24264i	0.143101	−	0.247858i	−0.785562	−	0.618783i	$-0.787626\pi$
$293$	2.44949	−	4.24264i	0.143101	−	0.247858i	0.928663	+	0.370925i	$0.120959\pi$
$294$		0			0
$295$	−3.00000	−	5.19615i	−0.174667	−	0.302532i
$296$	−39.1918			−2.27798
$297$		0			0
$298$	30.0000			1.73785
$299$	1.22474	+	2.12132i	0.0708288	+	0.122679i
$300$		0			0
$301$	−11.0000	+	19.0526i	−0.634029	+	1.09817i
$302$	6.12372	−	10.6066i	0.352381	−	0.610341i
$303$		0			0
$304$	2.00000	+	3.46410i	0.114708	+	0.198680i
$305$	12.2474			0.701287
$306$		0			0
$307$	2.00000			0.114146			0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000			0.114146			0.0570730	+	0.998370i	$0.481823\pi$
$308$	9.79796	+	16.9706i	0.558291	+	0.966988i
$309$		0			0
$310$	−3.00000	+	5.19615i	−0.170389	+	0.295122i
$311$	−12.2474	+	21.2132i	−0.694489	+	1.20289i	0.275864	+	0.961197i	$0.411036\pi$
$311$	−12.2474	+	21.2132i	−0.694489	+	1.20289i	−0.970353	+	0.241694i	$0.922297\pi$
$312$		0			0
$313$	8.00000	+	13.8564i	0.452187	+	0.783210i	0.998522	−	0.0543564i	$-0.0173107\pi$
$313$	8.00000	+	13.8564i	0.452187	+	0.783210i	−0.546335	+	0.837567i	$0.683977\pi$
$314$	−41.6413			−2.34996
$315$		0			0
$316$	−28.0000			−1.57512
$317$	4.89898	+	8.48528i	0.275154	+	0.476581i	0.970174	−	0.242410i	$-0.0779378\pi$
$317$	4.89898	+	8.48528i	0.275154	+	0.476581i	−0.695020	+	0.718991i	$0.744604\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$	9.79796	−	16.9706i	0.547723	−	0.948683i
$321$		0			0
$322$	6.00000	+	10.3923i	0.334367	+	0.579141i
$323$	−7.34847			−0.408880
$324$		0			0
$325$	−1.00000			−0.0554700
$326$	−12.2474	−	21.2132i	−0.678323	−	1.17489i
$327$		0			0
$328$	−12.0000	+	20.7846i	−0.662589	+	1.14764i
$329$	9.79796	−	16.9706i	0.540179	−	0.935617i
$330$		0			0
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	0.946038	−	0.324057i	$-0.105047\pi$
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	−0.753660	+	0.657264i	$0.771714\pi$
$332$	48.9898			2.68866
$333$		0			0
$334$	12.0000			0.656611
$335$	8.57321	+	14.8492i	0.468405	+	0.811301i
$336$		0			0
$337$	14.0000	−	24.2487i	0.762629	−	1.32091i	−0.178863	−	0.983874i	$-0.557242\pi$
$337$	14.0000	−	24.2487i	0.762629	−	1.32091i	0.941491	−	0.337037i	$-0.109425\pi$
$338$	−14.6969	+	25.4558i	−0.799408	+	1.38462i
$339$		0			0
$340$	−36.0000	−	62.3538i	−1.95237	−	3.38161i
$341$	2.44949			0.132647
$342$		0			0
$343$	−20.0000			−1.07990
$344$	26.9444	+	46.6690i	1.45274	+	2.51623i
$345$		0			0
$346$	−12.0000	+	20.7846i	−0.645124	+	1.11739i
$347$	−12.2474	+	21.2132i	−0.657477	+	1.13878i	0.323789	+	0.946129i	$0.395043\pi$
$347$	−12.2474	+	21.2132i	−0.657477	+	1.13878i	−0.981267	+	0.192655i	$0.938290\pi$
$348$		0			0
$349$	−10.0000	−	17.3205i	−0.535288	−	0.927146i	−0.999149	−	0.0412379i	$-0.986870\pi$
$349$	−10.0000	−	17.3205i	−0.535288	−	0.927146i	0.463862	−	0.885908i	$-0.346463\pi$
$350$	−4.89898			−0.261861
$351$		0			0
$352$		0			0
$353$	1.22474	+	2.12132i	0.0651866	+	0.112906i	0.896777	−	0.442483i	$-0.145902\pi$
$353$	1.22474	+	2.12132i	0.0651866	+	0.112906i	−0.831590	+	0.555390i	$0.812569\pi$
$354$		0			0
$355$	9.00000	−	15.5885i	0.477670	−	0.827349i
$356$		0			0
$357$		0			0
$358$	18.0000	+	31.1769i	0.951330	+	1.64775i
$359$	−29.3939			−1.55135			−0.775675	−	0.631133i	$-0.782590\pi$
$359$	−29.3939			−1.55135			−0.775675	+	0.631133i	$0.782590\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$	9.79796	+	16.9706i	0.514969	+	0.891953i
$363$		0			0
$364$	4.00000	−	6.92820i	0.209657	−	0.363137i
$365$	−13.4722	+	23.3345i	−0.705167	+	1.22138i
$366$		0			0
$367$	−2.50000	−	4.33013i	−0.130499	−	0.226031i	0.793370	−	0.608740i	$-0.208325\pi$
$367$	−2.50000	−	4.33013i	−0.130499	−	0.226031i	−0.923869	+	0.382709i	$0.874991\pi$
$368$	9.79796			0.510754
$369$		0			0
$370$	−48.0000			−2.49540
$371$	7.34847	+	12.7279i	0.381514	+	0.660801i
$372$		0			0
$373$	−17.5000	+	30.3109i	−0.906116	+	1.56944i	−0.0867031	+	0.996234i	$0.527633\pi$
$373$	−17.5000	+	30.3109i	−0.906116	+	1.56944i	−0.819413	+	0.573204i	$0.805700\pi$
$374$	−22.0454	+	38.1838i	−1.13994	+	1.97444i
$375$		0			0
$376$	−24.0000	−	41.5692i	−1.23771	−	2.14377i
$377$	−4.89898			−0.252310
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$	4.89898	+	8.48528i	0.251312	+	0.435286i
$381$		0			0
$382$	−12.0000	+	20.7846i	−0.613973	+	1.06343i
$383$	17.1464	−	29.6985i	0.876142	−	1.51752i	0.0205998	−	0.999788i	$-0.493442\pi$
$383$	17.1464	−	29.6985i	0.876142	−	1.51752i	0.855542	−	0.517734i	$-0.173224\pi$
$384$		0			0
$385$	6.00000	+	10.3923i	0.305788	+	0.529641i
$386$	−26.9444			−1.37143
$387$		0			0
$388$	−28.0000			−1.42148
$389$	−13.4722	−	23.3345i	−0.683067	−	1.18311i	−0.974040	−	0.226376i	$-0.927312\pi$
$389$	−13.4722	−	23.3345i	−0.683067	−	1.18311i	0.290973	−	0.956731i	$-0.406021\pi$
$390$		0			0
$391$	−9.00000	+	15.5885i	−0.455150	+	0.788342i
$392$	−7.34847	+	12.7279i	−0.371154	+	0.642857i
$393$		0			0
$394$	−18.0000	−	31.1769i	−0.906827	−	1.57067i
$395$	−17.1464			−0.862730
$396$		0			0
$397$	−1.00000			−0.0501886			−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−1.00000			−0.0501886			−0.0250943	+	0.999685i	$0.507989\pi$
$398$	−1.22474	−	2.12132i	−0.0613909	−	0.106332i
$399$		0			0
$400$	−2.00000	+	3.46410i	−0.100000	+	0.173205i
$401$	17.1464	−	29.6985i	0.856252	−	1.48307i	−0.0192271	−	0.999815i	$-0.506121\pi$
$401$	17.1464	−	29.6985i	0.856252	−	1.48307i	0.875479	−	0.483256i	$-0.160546\pi$
$402$		0			0
$403$	−0.500000	−	0.866025i	−0.0249068	−	0.0431398i
$404$	19.5959			0.974933
$405$		0			0
$406$	−24.0000			−1.19110
$407$	9.79796	+	16.9706i	0.485667	+	0.841200i
$408$		0			0
$409$	14.0000	−	24.2487i	0.692255	−	1.19902i	−0.278842	−	0.960337i	$-0.589950\pi$
$409$	14.0000	−	24.2487i	0.692255	−	1.19902i	0.971097	−	0.238685i	$-0.0767162\pi$
$410$	−14.6969	+	25.4558i	−0.725830	+	1.25717i
$411$		0			0
$412$	14.0000	+	24.2487i	0.689730	+	1.19465i
$413$	4.89898			0.241063
$414$		0			0
$415$	30.0000			1.47264
$416$		0			0
$417$		0			0
$418$	3.00000	−	5.19615i	0.146735	−	0.254152i
$419$	17.1464	−	29.6985i	0.837658	−	1.45087i	−0.0541901	−	0.998531i	$-0.517258\pi$
$419$	17.1464	−	29.6985i	0.837658	−	1.45087i	0.891848	−	0.452335i	$-0.149409\pi$
$420$		0			0
$421$	−1.00000	−	1.73205i	−0.0487370	−	0.0844150i	0.840628	−	0.541613i	$-0.182186\pi$
$421$	−1.00000	−	1.73205i	−0.0487370	−	0.0844150i	−0.889365	+	0.457198i	$0.848853\pi$
$422$	2.44949			0.119239
$423$		0			0
$424$	36.0000			1.74831
$425$	−3.67423	−	6.36396i	−0.178227	−	0.308697i
$426$		0			0
$427$	−5.00000	+	8.66025i	−0.241967	+	0.419099i
$428$	29.3939	−	50.9117i	1.42081	−	2.46091i
$429$		0			0
$430$	33.0000	+	57.1577i	1.59140	+	2.75639i
$431$	7.34847			0.353963			0.176982	−	0.984214i	$-0.443367\pi$
$431$	7.34847			0.353963			0.176982	+	0.984214i	$0.443367\pi$
$432$		0			0
$433$	17.0000			0.816968			0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000			0.816968			0.408484	+	0.912766i	$0.366058\pi$
$434$	−2.44949	−	4.24264i	−0.117579	−	0.203653i
$435$		0			0
$436$	2.00000	−	3.46410i	0.0957826	−	0.165900i
$437$	1.22474	−	2.12132i	0.0585875	−	0.101477i
$438$		0			0
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	0.649218	−	0.760602i	$-0.275096\pi$
$439$	−7.00000	−	12.1244i	−0.334092	−	0.578664i	−0.983310	+	0.181938i	$0.941763\pi$
$440$	29.3939			1.40130
$441$		0			0
$442$	18.0000			0.856173
$443$	−6.12372	−	10.6066i	−0.290947	−	0.503935i	0.683087	−	0.730337i	$-0.260637\pi$
$443$	−6.12372	−	10.6066i	−0.290947	−	0.503935i	−0.974034	+	0.226402i	$0.927304\pi$
$444$		0			0
$445$		0			0
$446$	−8.57321	+	14.8492i	−0.405953	+	0.703132i
$447$		0			0
$448$	8.00000	+	13.8564i	0.377964	+	0.654654i
$449$	−22.0454			−1.04039			−0.520194	−	0.854048i	$-0.674140\pi$
$449$	−22.0454			−1.04039			−0.520194	+	0.854048i	$0.674140\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$	−19.5959	−	33.9411i	−0.921714	−	1.59646i
$453$		0			0
$454$	−12.0000	+	20.7846i	−0.563188	+	0.975470i
$455$	2.44949	−	4.24264i	0.114834	−	0.198898i
$456$		0			0
$457$	−14.5000	−	25.1147i	−0.678281	−	1.17482i	−0.975498	−	0.220008i	$-0.929392\pi$
$457$	−14.5000	−	25.1147i	−0.678281	−	1.17482i	0.297217	−	0.954810i	$-0.403942\pi$
$458$	2.44949			0.114457
$459$		0			0
$460$	24.0000			1.11901
$461$	−13.4722	−	23.3345i	−0.627463	−	1.08680i	−0.988059	−	0.154075i	$-0.950760\pi$
$461$	−13.4722	−	23.3345i	−0.627463	−	1.08680i	0.360597	−	0.932722i	$-0.382573\pi$
$462$		0			0
$463$	9.50000	−	16.4545i	0.441502	−	0.764705i	−0.556299	−	0.830982i	$-0.687779\pi$
$463$	9.50000	−	16.4545i	0.441502	−	0.764705i	0.997801	+	0.0662777i	$0.0211123\pi$
$464$	−9.79796	+	16.9706i	−0.454859	+	0.787839i
$465$		0			0
$466$	9.00000	+	15.5885i	0.416917	+	0.722121i
$467$	−14.6969			−0.680093			−0.340047	−	0.940409i	$-0.610443\pi$
$467$	−14.6969			−0.680093			−0.340047	+	0.940409i	$0.610443\pi$
$468$		0			0
$469$	−14.0000			−0.646460
$470$	−29.3939	−	50.9117i	−1.35584	−	2.34838i
$471$		0			0
$472$	6.00000	−	10.3923i	0.276172	−	0.478345i
$473$	13.4722	−	23.3345i	0.619452	−	1.07292i
$474$		0			0
$475$	0.500000	+	0.866025i	0.0229416	+	0.0397360i
$476$	58.7878			2.69453
$477$		0			0
$478$	−6.00000			−0.274434
$479$	−13.4722	−	23.3345i	−0.615560	−	1.06618i	−0.990286	−	0.139046i	$-0.955596\pi$
$479$	−13.4722	−	23.3345i	−0.615560	−	1.06618i	0.374726	−	0.927136i	$-0.377737\pi$
$480$		0			0
$481$	4.00000	−	6.92820i	0.182384	−	0.315899i
$482$	−19.5959	+	33.9411i	−0.892570	+	1.54598i
$483$		0			0
$484$	10.0000	+	17.3205i	0.454545	+	0.787296i
$485$	−17.1464			−0.778579
$486$		0			0
$487$	35.0000			1.58600			0.793001	−	0.609221i	$-0.208518\pi$
$487$	35.0000			1.58600			0.793001	+	0.609221i	$0.208518\pi$
$488$	12.2474	+	21.2132i	0.554416	+	0.960277i
$489$		0			0
$490$	−9.00000	+	15.5885i	−0.406579	+	0.704215i
$491$	−19.5959	+	33.9411i	−0.884351	+	1.53174i	−0.0378961	+	0.999282i	$0.512066\pi$
$491$	−19.5959	+	33.9411i	−0.884351	+	1.53174i	−0.846455	+	0.532460i	$0.821268\pi$
$492$		0			0
$493$	−18.0000	−	31.1769i	−0.810679	−	1.40414i
$494$	−2.44949			−0.110208
$495$		0			0
$496$	−4.00000			−0.179605
$497$	7.34847	+	12.7279i	0.329624	+	0.570925i
$498$		0			0
$499$	−1.00000	+	1.73205i	−0.0447661	+	0.0775372i	−0.887540	−	0.460730i	$-0.847588\pi$
$499$	−1.00000	+	1.73205i	−0.0447661	+	0.0775372i	0.842774	+	0.538267i	$0.180921\pi$
$500$	19.5959	−	33.9411i	0.876356	−	1.51789i
$501$		0			0
$502$	9.00000	+	15.5885i	0.401690	+	0.695747i
$503$	14.6969			0.655304			0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969			0.655304			0.327652	+	0.944798i	$0.393743\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$	−7.34847	−	12.7279i	−0.326679	−	0.565825i
$507$		0			0
$508$	38.0000	−	65.8179i	1.68598	−	2.92020i
$509$	−4.89898	+	8.48528i	−0.217143	+	0.376103i	−0.953934	−	0.300018i	$-0.903007\pi$
$509$	−4.89898	+	8.48528i	−0.217143	+	0.376103i	0.736790	+	0.676122i	$0.236341\pi$
$510$		0			0
$511$	−11.0000	−	19.0526i	−0.486611	−	0.842836i
$512$	39.1918			1.73205
$513$		0			0
$514$	−42.0000			−1.85254
$515$	8.57321	+	14.8492i	0.377781	+	0.654336i
$516$		0			0
$517$	−12.0000	+	20.7846i	−0.527759	+	0.914106i
$518$	19.5959	−	33.9411i	0.860995	−	1.49129i
$519$		0			0
$520$	−6.00000	−	10.3923i	−0.263117	−	0.455733i
$521$	−22.0454			−0.965827			−0.482913	−	0.875668i	$-0.660421\pi$
$521$	−22.0454			−0.965827			−0.482913	+	0.875668i	$0.660421\pi$
$522$		0			0
$523$	−25.0000			−1.09317			−0.546587	−	0.837402i	$-0.684073\pi$
$523$	−25.0000			−1.09317			−0.546587	+	0.837402i	$0.684073\pi$
$524$	24.4949	+	42.4264i	1.07006	+	1.85341i
$525$		0			0
$526$	33.0000	−	57.1577i	1.43887	−	2.49219i
$527$	3.67423	−	6.36396i	0.160052	−	0.277218i
$528$		0			0
$529$	8.50000	+	14.7224i	0.369565	+	0.640106i
$530$	44.0908			1.91518
$531$		0			0
$532$	−8.00000			−0.346844
$533$	−2.44949	−	4.24264i	−0.106099	−	0.183769i
$534$		0			0
$535$	18.0000	−	31.1769i	0.778208	−	1.34790i
$536$	−17.1464	+	29.6985i	−0.740613	+	1.28278i
$537$		0			0
$538$	27.0000	+	46.7654i	1.16405	+	2.01620i
$539$	7.34847			0.316521
$540$		0			0
$541$	−28.0000			−1.20381			−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000			−1.20381			−0.601907	+	0.798566i	$0.705592\pi$
$542$	−8.57321	−	14.8492i	−0.368251	−	0.637830i
$543$		0			0
$544$		0			0
$545$	1.22474	−	2.12132i	0.0524623	−	0.0908674i
$546$		0			0
$547$	6.50000	+	11.2583i	0.277920	+	0.481371i	0.970868	−	0.239616i	$-0.0770217\pi$
$547$	6.50000	+	11.2583i	0.277920	+	0.481371i	−0.692948	+	0.720988i	$0.743688\pi$
$548$	−39.1918			−1.67419
$549$		0			0
$550$	6.00000			0.255841
$551$	2.44949	+	4.24264i	0.104352	+	0.180743i
$552$		0			0
$553$	7.00000	−	12.1244i	0.297670	−	0.515580i
$554$	13.4722	−	23.3345i	0.572379	−	0.991389i
$555$		0			0
$556$	20.0000	+	34.6410i	0.848189	+	1.46911i
$557$	−7.34847			−0.311365			−0.155682	−	0.987807i	$-0.549758\pi$
$557$	−7.34847			−0.311365			−0.155682	+	0.987807i	$0.549758\pi$
$558$		0			0
$559$	−11.0000			−0.465250
$560$	−9.79796	−	16.9706i	−0.414039	−	0.717137i
$561$		0			0
$562$	15.0000	−	25.9808i	0.632737	−	1.09593i
$563$	6.12372	−	10.6066i	0.258084	−	0.447015i	−0.707644	−	0.706569i	$-0.750242\pi$
$563$	6.12372	−	10.6066i	0.258084	−	0.447015i	0.965729	+	0.259554i	$0.0835755\pi$
$564$		0			0
$565$	−12.0000	−	20.7846i	−0.504844	−	0.874415i
$566$	−41.6413			−1.75032
$567$		0			0
$568$	36.0000			1.51053
$569$	−6.12372	−	10.6066i	−0.256720	−	0.444652i	0.708641	−	0.705569i	$-0.249308\pi$
$569$	−6.12372	−	10.6066i	−0.256720	−	0.444652i	−0.965361	+	0.260917i	$0.915975\pi$
$570$		0			0
$571$	5.00000	−	8.66025i	0.209243	−	0.362420i	−0.742233	−	0.670142i	$-0.766233\pi$
$571$	5.00000	−	8.66025i	0.209243	−	0.362420i	0.951476	+	0.307722i	$0.0995665\pi$
$572$	−4.89898	+	8.48528i	−0.204837	+	0.354787i
$573$		0			0
$574$	−12.0000	−	20.7846i	−0.500870	−	0.867533i
$575$	2.44949			0.102151
$576$		0			0
$577$	−25.0000			−1.04076			−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000			−1.04076			−0.520382	+	0.853934i	$0.674210\pi$
$578$	45.3156	+	78.4889i	1.88488	+	3.26471i
$579$		0			0
$580$	−24.0000	+	41.5692i	−0.996546	+	1.72607i
$581$	−12.2474	+	21.2132i	−0.508110	+	0.880072i
$582$		0			0
$583$	−9.00000	−	15.5885i	−0.372742	−	0.645608i
$584$	−53.8888			−2.22993
$585$		0			0
$586$	12.0000			0.495715
$587$	1.22474	+	2.12132i	0.0505506	+	0.0875563i	0.890194	−	0.455583i	$-0.150569\pi$
$587$	1.22474	+	2.12132i	0.0505506	+	0.0875563i	−0.839643	+	0.543139i	$0.817236\pi$
$588$		0			0
$589$	−0.500000	+	0.866025i	−0.0206021	+	0.0356840i
$590$	7.34847	−	12.7279i	0.302532	−	0.524000i
$591$		0			0
$592$	−16.0000	−	27.7128i	−0.657596	−	1.13899i
$593$	−7.34847			−0.301765			−0.150883	−	0.988552i	$-0.548212\pi$
$593$	−7.34847			−0.301765			−0.150883	+	0.988552i	$0.548212\pi$
$594$		0			0
$595$	36.0000			1.47586
$596$	24.4949	+	42.4264i	1.00335	+	1.73785i
$597$		0			0
$598$	−3.00000	+	5.19615i	−0.122679	+	0.212486i
$599$	−19.5959	+	33.9411i	−0.800668	+	1.38680i	0.118510	+	0.992953i	$0.462188\pi$
$599$	−19.5959	+	33.9411i	−0.800668	+	1.38680i	−0.919177	+	0.393844i	$0.871145\pi$
$600$		0			0
$601$	3.50000	+	6.06218i	0.142768	+	0.247281i	0.928538	−	0.371237i	$-0.121066\pi$
$601$	3.50000	+	6.06218i	0.142768	+	0.247281i	−0.785770	+	0.618519i	$0.787733\pi$
$602$	−53.8888			−2.19634
$603$		0			0
$604$	20.0000			0.813788
$605$	6.12372	+	10.6066i	0.248965	+	0.431220i
$606$		0			0
$607$	−22.0000	+	38.1051i	−0.892952	+	1.54664i	−0.0566340	+	0.998395i	$0.518037\pi$
$607$	−22.0000	+	38.1051i	−0.892952	+	1.54664i	−0.836318	+	0.548244i	$0.815297\pi$
$608$		0			0
$609$		0			0
$610$	15.0000	+	25.9808i	0.607332	+	1.05193i
$611$	9.79796			0.396383
$612$		0			0
$613$	11.0000			0.444286			0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000			0.444286			0.222143	+	0.975014i	$0.428695\pi$
$614$	2.44949	+	4.24264i	0.0988534	+	0.171219i
$615$		0			0
$616$	−12.0000	+	20.7846i	−0.483494	+	0.837436i
$617$	−12.2474	+	21.2132i	−0.493064	+	0.854011i	−0.999968	−	0.00799089i	$-0.997456\pi$
$617$	−12.2474	+	21.2132i	−0.493064	+	0.854011i	0.506904	+	0.862002i	$0.330790\pi$
$618$		0			0
$619$	24.5000	+	42.4352i	0.984738	+	1.70562i	0.643094	+	0.765787i	$0.277650\pi$
$619$	24.5000	+	42.4352i	0.984738	+	1.70562i	0.341644	+	0.939829i	$0.389016\pi$
$620$	−9.79796			−0.393496
$621$		0			0
$622$	−60.0000			−2.40578
$623$		0			0
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	−19.5959	+	33.9411i	−0.783210	+	1.35656i
$627$		0			0
$628$	−34.0000	−	58.8897i	−1.35675	−	2.34996i
$629$	58.7878			2.34402
$630$		0			0
$631$	44.0000			1.75161			0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000			1.75161			0.875806	+	0.482663i	$0.160330\pi$
$632$	−17.1464	−	29.6985i	−0.682048	−	1.18134i
$633$		0			0
$634$	−12.0000	+	20.7846i	−0.476581	+	0.825462i
$635$	23.2702	−	40.3051i	0.923448	−	1.59946i
$636$		0			0
$637$	−1.50000	−	2.59808i	−0.0594322	−	0.102940i
$638$	29.3939			1.16371
$639$		0			0
$640$	48.0000			1.89737
$641$	8.57321	+	14.8492i	0.338622	+	0.586510i	0.984174	−	0.177206i	$-0.0567060\pi$
$641$	8.57321	+	14.8492i	0.338622	+	0.586510i	−0.645552	+	0.763716i	$0.723373\pi$
$642$		0			0
$643$	−19.0000	+	32.9090i	−0.749287	+	1.29780i	0.198878	+	0.980024i	$0.436270\pi$
$643$	−19.0000	+	32.9090i	−0.749287	+	1.29780i	−0.948165	+	0.317779i	$0.897063\pi$
$644$	−9.79796	+	16.9706i	−0.386094	+	0.668734i
$645$		0			0
$646$	−9.00000	−	15.5885i	−0.354100	−	0.613320i
$647$	−36.7423			−1.44449			−0.722245	−	0.691637i	$-0.756890\pi$
$647$	−36.7423			−1.44449			−0.722245	+	0.691637i	$0.756890\pi$
$648$		0			0
$649$	−6.00000			−0.235521
$650$	−1.22474	−	2.12132i	−0.0480384	−	0.0832050i
$651$		0			0
$652$	20.0000	−	34.6410i	0.783260	−	1.35665i
$653$	−4.89898	+	8.48528i	−0.191712	+	0.332055i	−0.945818	−	0.324698i	$-0.894737\pi$
$653$	−4.89898	+	8.48528i	−0.191712	+	0.332055i	0.754106	+	0.656753i	$0.228071\pi$
$654$		0			0
$655$	15.0000	+	25.9808i	0.586098	+	1.01515i
$656$	−19.5959			−0.765092
$657$		0			0
$658$	48.0000			1.87123
$659$	−9.79796	−	16.9706i	−0.381674	−	0.661079i	0.609627	−	0.792688i	$-0.291319\pi$
$659$	−9.79796	−	16.9706i	−0.381674	−	0.661079i	−0.991302	+	0.131609i	$0.957986\pi$
$660$		0			0
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	0.739014	+	0.673690i	$0.235292\pi$
$662$	−8.57321	+	14.8492i	−0.333207	+	0.577132i
$663$		0			0
$664$	30.0000	+	51.9615i	1.16423	+	2.01650i
$665$	−4.89898			−0.189974
$666$		0			0
$667$	12.0000			0.464642
$668$	9.79796	+	16.9706i	0.379094	+	0.656611i
$669$		0			0
$670$	−21.0000	+	36.3731i	−0.811301	+	1.40521i
$671$	6.12372	−	10.6066i	0.236404	−	0.409463i
$672$		0			0
$673$	−14.5000	−	25.1147i	−0.558934	−	0.968102i	−0.997586	−	0.0694449i	$-0.977877\pi$
$673$	−14.5000	−	25.1147i	−0.558934	−	0.968102i	0.438652	−	0.898657i	$-0.355456\pi$
$674$	68.5857			2.64182
$675$		0			0
$676$	−48.0000			−1.84615
$677$	23.2702	+	40.3051i	0.894345	+	1.54905i	0.834614	+	0.550836i	$0.185691\pi$
$677$	23.2702	+	40.3051i	0.894345	+	1.54905i	0.0597310	+	0.998215i	$0.480976\pi$
$678$		0			0
$679$	7.00000	−	12.1244i	0.268635	−	0.465290i
$680$	44.0908	−	76.3675i	1.69081	−	2.92856i
$681$		0			0
$682$	3.00000	+	5.19615i	0.114876	+	0.198971i
$683$	22.0454			0.843544			0.421772	−	0.906702i	$-0.361408\pi$
$683$	22.0454			0.843544			0.421772	+	0.906702i	$0.361408\pi$
$684$		0			0
$685$	−24.0000			−0.916993
$686$	−24.4949	−	42.4264i	−0.935220	−	1.61985i
$687$		0			0
$688$	−22.0000	+	38.1051i	−0.838742	+	1.45274i
$689$	−3.67423	+	6.36396i	−0.139977	+	0.242448i
$690$		0			0
$691$	−23.5000	−	40.7032i	−0.893982	−	1.54842i	−0.835059	−	0.550160i	$-0.814567\pi$
$691$	−23.5000	−	40.7032i	−0.893982	−	1.54842i	−0.0589228	−	0.998263i	$-0.518767\pi$
$692$	−39.1918			−1.48985
$693$		0			0
$694$	−60.0000			−2.27757
$695$	12.2474	+	21.2132i	0.464572	+	0.804663i
$696$		0			0
$697$	18.0000	−	31.1769i	0.681799	−	1.18091i
$698$	24.4949	−	42.4264i	0.927146	−	1.60586i
$699$		0			0
$700$	−4.00000	−	6.92820i	−0.151186	−	0.261861i
$701$	14.6969			0.555096			0.277548	−	0.960712i	$-0.410478\pi$
$701$	14.6969			0.555096			0.277548	+	0.960712i	$0.410478\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$	−9.79796	−	16.9706i	−0.369274	−	0.639602i
$705$		0			0
$706$	−3.00000	+	5.19615i	−0.112906	+	0.195560i
$707$	−4.89898	+	8.48528i	−0.184245	+	0.319122i
$708$		0			0
$709$	3.50000	+	6.06218i	0.131445	+	0.227670i	0.924234	−	0.381827i	$-0.124705\pi$
$709$	3.50000	+	6.06218i	0.131445	+	0.227670i	−0.792789	+	0.609497i	$0.791372\pi$
$710$	44.0908			1.65470
$711$		0			0
$712$		0			0
$713$	1.22474	+	2.12132i	0.0458671	+	0.0794441i
$714$		0			0
$715$	−3.00000	+	5.19615i	−0.112194	+	0.194325i
$716$	−29.3939	+	50.9117i	−1.09850	+	1.90266i
$717$		0			0
$718$	−36.0000	−	62.3538i	−1.34351	−	2.32702i
$719$	36.7423			1.37026			0.685129	−	0.728422i	$-0.259746\pi$
$719$	36.7423			1.37026			0.685129	+	0.728422i	$0.259746\pi$
$720$		0			0
$721$	−14.0000			−0.521387
$722$	−22.0454	−	38.1838i	−0.820445	−	1.42105i
$723$		0			0
$724$	−16.0000	+	27.7128i	−0.594635	+	1.02994i
$725$	−2.44949	+	4.24264i	−0.0909718	+	0.157568i
$726$		0			0
$727$	−7.00000	−	12.1244i	−0.259616	−	0.449667i	0.706523	−	0.707690i	$-0.250263\pi$
$727$	−7.00000	−	12.1244i	−0.259616	−	0.449667i	−0.966139	+	0.258022i	$0.916929\pi$
$728$	9.79796			0.363137
$729$		0			0
$730$	−66.0000			−2.44277
$731$	−40.4166	−	70.0036i	−1.49486	−	2.58918i
$732$		0			0
$733$	−8.50000	+	14.7224i	−0.313955	+	0.543785i	−0.979215	−	0.202826i	$-0.934987\pi$
$733$	−8.50000	+	14.7224i	−0.313955	+	0.543785i	0.665260	+	0.746612i	$0.268321\pi$
$734$	6.12372	−	10.6066i	0.226031	−	0.391497i
$735$		0			0
$736$		0			0
$737$	17.1464			0.631597
$738$		0			0
$739$	−1.00000			−0.0367856			−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000			−0.0367856			−0.0183928	+	0.999831i	$0.505855\pi$
$740$	−39.1918	−	67.8823i	−1.44072	−	2.49540i
$741$		0			0
$742$	−18.0000	+	31.1769i	−0.660801	+	1.14454i
$743$	−15.9217	+	27.5772i	−0.584110	+	1.01171i	0.410876	+	0.911691i	$0.365223\pi$
$743$	−15.9217	+	27.5772i	−0.584110	+	1.01171i	−0.994986	+	0.100017i	$0.968110\pi$
$744$		0			0
$745$	15.0000	+	25.9808i	0.549557	+	0.951861i
$746$	−85.7321			−3.13888
$747$		0			0
$748$	−72.0000			−2.63258
$749$	14.6969	+	25.4558i	0.537014	+	0.930136i
$750$		0			0
$751$	−13.0000	+	22.5167i	−0.474377	+	0.821645i	−0.999570	−	0.0293387i	$-0.990660\pi$
$751$	−13.0000	+	22.5167i	−0.474377	+	0.821645i	0.525193	+	0.850983i	$0.323993\pi$
$752$	19.5959	−	33.9411i	0.714590	−	1.23771i
$753$		0			0
$754$	−6.00000	−	10.3923i	−0.218507	−	0.378465i
$755$	12.2474			0.445730
$756$		0			0
$757$	−7.00000			−0.254419			−0.127210	−	0.991876i	$-0.540602\pi$
$757$	−7.00000			−0.254419			−0.127210	+	0.991876i	$0.540602\pi$
$758$	9.79796	+	16.9706i	0.355878	+	0.616399i
$759$		0			0
$760$	−6.00000	+	10.3923i	−0.217643	+	0.376969i
$761$	−1.22474	+	2.12132i	−0.0443970	+	0.0768978i	−0.887370	−	0.461058i	$-0.847470\pi$
$761$	−1.22474	+	2.12132i	−0.0443970	+	0.0768978i	0.842973	+	0.537956i	$0.180803\pi$
$762$		0			0
$763$	1.00000	+	1.73205i	0.0362024	+	0.0627044i
$764$	−39.1918			−1.41791
$765$		0			0
$766$	84.0000			3.03504
$767$	1.22474	+	2.12132i	0.0442230	+	0.0765964i
$768$		0			0
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	−0.311577	−	0.950221i	$-0.600857\pi$
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	0.978704	−	0.205277i	$-0.0658095\pi$
$770$	−14.6969	+	25.4558i	−0.529641	+	0.917365i
$771$		0			0
$772$	−22.0000	−	38.1051i	−0.791797	−	1.37143i
$773$	44.0908			1.58584			0.792918	−	0.609328i	$-0.208561\pi$
$773$	44.0908			1.58584			0.792918	+	0.609328i	$0.208561\pi$
$774$		0			0
$775$	−1.00000			−0.0359211
$776$	−17.1464	−	29.6985i	−0.615521	−	1.06611i
$777$		0			0
$778$	33.0000	−	57.1577i	1.18311	−	2.04920i
$779$	−2.44949	+	4.24264i	−0.0877621	+	0.152008i
$780$		0			0
$781$	−9.00000	−	15.5885i	−0.322045	−	0.557799i
$782$	−44.0908			−1.57668
$783$		0			0
$784$	−12.0000			−0.428571
$785$	−20.8207	−	36.0624i	−0.743121	−	1.28712i
$786$		0			0
$787$	12.5000	−	21.6506i	0.445577	−	0.771762i	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	12.5000	−	21.6506i	0.445577	−	0.771762i	0.998092	+	0.0617409i	$0.0196653\pi$
$788$	29.3939	−	50.9117i	1.04711	−	1.81365i
$789$		0			0
$790$	−21.0000	−	36.3731i	−0.747146	−	1.29410i
$791$	19.5959			0.696751
$792$		0			0
$793$	−5.00000			−0.177555
$794$	−1.22474	−	2.12132i	−0.0434646	−	0.0752828i
$795$		0			0
$796$	2.00000	−	3.46410i	0.0708881	−	0.122782i
$797$	20.8207	−	36.0624i	0.737506	−	1.27740i	−0.216110	−	0.976369i	$-0.569337\pi$
$797$	20.8207	−	36.0624i	0.737506	−	1.27740i	0.953615	−	0.301028i	$-0.0973298\pi$
$798$		0			0
$799$	36.0000	+	62.3538i	1.27359	+	2.20592i
$800$		0			0
$801$		0			0
$802$	84.0000			2.96614
$803$	13.4722	+	23.3345i	0.475423	+	0.823457i
$804$		0			0
$805$	−6.00000	+	10.3923i	−0.211472	+	0.366281i
$806$	1.22474	−	2.12132i	0.0431398	−	0.0747203i
$807$		0			0
$808$	12.0000	+	20.7846i	0.422159	+	0.731200i
$809$	−22.0454			−0.775075			−0.387538	−	0.921854i	$-0.626674\pi$
$809$	−22.0454			−0.775075			−0.387538	+	0.921854i	$0.626674\pi$
$810$		0			0
$811$	35.0000			1.22902			0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000			1.22902			0.614508	+	0.788911i	$0.289355\pi$
$812$	−19.5959	−	33.9411i	−0.687682	−	1.19110i
$813$		0			0
$814$	−24.0000	+	41.5692i	−0.841200	+	1.45700i
$815$	12.2474	−	21.2132i	0.429009	−	0.743066i
$816$		0			0
$817$	5.50000	+	9.52628i	0.192421	+	0.333282i
$818$	68.5857			2.39804
$819$		0			0
$820$	−48.0000			−1.67623
$821$	19.5959	+	33.9411i	0.683902	+	1.18455i	0.973780	+	0.227490i	$0.0730519\pi$
$821$	19.5959	+	33.9411i	0.683902	+	1.18455i	−0.289878	+	0.957064i	$0.593615\pi$
$822$		0			0
$823$	−17.5000	+	30.3109i	−0.610012	+	1.05657i	0.381226	+	0.924482i	$0.375502\pi$
$823$	−17.5000	+	30.3109i	−0.610012	+	1.05657i	−0.991238	+	0.132089i	$0.957831\pi$
$824$	−17.1464	+	29.6985i	−0.597324	+	1.03460i
$825$		0			0
$826$	6.00000	+	10.3923i	0.208767	+	0.361595i
$827$	22.0454			0.766594			0.383297	−	0.923625i	$-0.374789\pi$
$827$	22.0454			0.766594			0.383297	+	0.923625i	$0.374789\pi$
$828$		0			0
$829$	−37.0000			−1.28506			−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000			−1.28506			−0.642532	+	0.766259i	$0.722116\pi$
$830$	36.7423	+	63.6396i	1.27535	+	2.20896i
$831$		0			0
$832$	−4.00000	+	6.92820i	−0.138675	+	0.240192i
$833$	11.0227	−	19.0919i	0.381914	−	0.661495i
$834$		0			0
$835$	6.00000	+	10.3923i	0.207639	+	0.359641i
$836$	9.79796			0.338869
$837$		0			0
$838$	84.0000			2.90173
$839$	−2.44949	−	4.24264i	−0.0845658	−	0.146472i	0.820640	−	0.571445i	$-0.193617\pi$
$839$	−2.44949	−	4.24264i	−0.0845658	−	0.146472i	−0.905206	+	0.424973i	$0.860284\pi$
$840$		0			0
$841$	2.50000	−	4.33013i	0.0862069	−	0.149315i
$842$	2.44949	−	4.24264i	0.0844150	−	0.146211i
$843$		0			0
$844$	2.00000	+	3.46410i	0.0688428	+	0.119239i
$845$	−29.3939			−1.01118
$846$		0			0
$847$	−10.0000			−0.343604
$848$	14.6969	+	25.4558i	0.504695	+	0.874157i
$849$		0			0
$850$	9.00000	−	15.5885i	0.308697	−	0.534680i
$851$	−9.79796	+	16.9706i	−0.335870	+	0.581743i
$852$		0			0
$853$	6.50000	+	11.2583i	0.222556	+	0.385478i	0.955583	−	0.294721i	$-0.0952267\pi$
$853$	6.50000	+	11.2583i	0.222556	+	0.385478i	−0.733028	+	0.680199i	$0.761893\pi$
$854$	−24.4949			−0.838198
$855$		0			0
$856$	72.0000			2.46091
$857$	12.2474	+	21.2132i	0.418365	+	0.724629i	0.995775	−	0.0918249i	$-0.0292700\pi$
$857$	12.2474	+	21.2132i	0.418365	+	0.724629i	−0.577410	+	0.816454i	$0.695937\pi$
$858$		0			0
$859$	−13.0000	+	22.5167i	−0.443554	+	0.768259i	−0.997950	−	0.0639945i	$-0.979616\pi$
$859$	−13.0000	+	22.5167i	−0.443554	+	0.768259i	0.554396	+	0.832253i	$0.312949\pi$
$860$	−53.8888	+	93.3381i	−1.83759	+	3.18280i
$861$		0			0
$862$	9.00000	+	15.5885i	0.306541	+	0.530945i
$863$	7.34847			0.250145			0.125072	−	0.992148i	$-0.460084\pi$
$863$	7.34847			0.250145			0.125072	+	0.992148i	$0.460084\pi$
$864$		0			0
$865$	−24.0000			−0.816024
$866$	20.8207	+	36.0624i	0.707515	+	1.22545i
$867$		0			0
$868$	4.00000	−	6.92820i	0.135769	−	0.235159i
$869$	−8.57321	+	14.8492i	−0.290826	+	0.503726i
$870$		0			0
$871$	−3.50000	−	6.06218i	−0.118593	−	0.205409i
$872$	4.89898			0.165900
$873$		0			0
$874$	6.00000			0.202953
$875$	9.79796	+	16.9706i	0.331231	+	0.573710i
$876$		0			0
$877$	−4.00000	+	6.92820i	−0.135070	+	0.233949i	−0.925624	−	0.378444i	$-0.876459\pi$
$877$	−4.00000	+	6.92820i	−0.135070	+	0.233949i	0.790554	+	0.612392i	$0.209793\pi$
$878$	17.1464	−	29.6985i	0.578664	−	1.00228i
$879$		0			0
$880$	12.0000	+	20.7846i	0.404520	+	0.700649i
$881$	−36.7423			−1.23788			−0.618941	−	0.785438i	$-0.712438\pi$
$881$	−36.7423			−1.23788			−0.618941	+	0.785438i	$0.712438\pi$
$882$		0			0
$883$	17.0000			0.572096			0.286048	−	0.958215i	$-0.407658\pi$
$883$	17.0000			0.572096			0.286048	+	0.958215i	$0.407658\pi$
$884$	14.6969	+	25.4558i	0.494312	+	0.856173i
$885$		0			0
$886$	15.0000	−	25.9808i	0.503935	−	0.872841i
$887$	−8.57321	+	14.8492i	−0.287860	+	0.498589i	−0.973299	−	0.229542i	$-0.926277\pi$
$887$	−8.57321	+	14.8492i	−0.287860	+	0.498589i	0.685438	+	0.728131i	$0.259611\pi$
$888$		0			0
$889$	19.0000	+	32.9090i	0.637240	+	1.10373i
$890$		0			0
$891$		0			0
$892$	−28.0000			−0.937509
$893$	−4.89898	−	8.48528i	−0.163938	−	0.283949i
$894$		0			0
$895$	−18.0000	+	31.1769i	−0.601674	+	1.04213i
$896$	−19.5959	+	33.9411i	−0.654654	+	1.13389i
$897$		0			0
$898$	−27.0000	−	46.7654i	−0.901002	−	1.56058i
$899$	−4.89898			−0.163390
$900$		0			0
$901$	−54.0000			−1.79900
$902$	14.6969	+	25.4558i	0.489355	+	0.847587i
$903$		0			0
$904$	24.0000	−	41.5692i	0.798228	−	1.38257i
$905$	−9.79796	+	16.9706i	−0.325695	+	0.564121i
$906$		0			0
$907$	3.50000	+	6.06218i	0.116216	+	0.201291i	0.918265	−	0.395966i	$-0.129590\pi$
$907$	3.50000	+	6.06218i	0.116216	+	0.201291i	−0.802049	+	0.597258i	$0.796257\pi$
$908$	−39.1918			−1.30063
$909$		0			0
$910$	12.0000			0.397796
$911$	−6.12372	−	10.6066i	−0.202888	−	0.351412i	0.746570	−	0.665307i	$-0.231699\pi$
$911$	−6.12372	−	10.6066i	−0.202888	−	0.351412i	−0.949458	+	0.313895i	$0.898366\pi$
$912$		0			0
$913$	15.0000	−	25.9808i	0.496428	−	0.859838i
$914$	35.5176	−	61.5183i	1.17482	−	2.03484i
$915$		0			0
$916$	2.00000	+	3.46410i	0.0660819	+	0.114457i
$917$	−24.4949			−0.808893
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$	14.6969	+	25.4558i	0.484544	+	0.839254i
$921$		0			0
$922$	33.0000	−	57.1577i	1.08680	−	1.88239i
$923$	−3.67423	+	6.36396i	−0.120939	+	0.209472i
$924$		0			0
$925$	−4.00000	−	6.92820i	−0.131519	−	0.227798i
$926$	46.5403			1.52941
$927$		0			0
$928$		0			0
$929$	−13.4722	−	23.3345i	−0.442008	−	0.765581i	0.555830	−	0.831296i	$-0.312401\pi$
$929$	−13.4722	−	23.3345i	−0.442008	−	0.765581i	−0.997838	+	0.0657150i	$0.979067\pi$
$930$		0			0
$931$	−1.50000	+	2.59808i	−0.0491605	+	0.0851485i
$932$	−14.6969	+	25.4558i	−0.481414	+	0.833834i
$933$		0			0
$934$	−18.0000	−	31.1769i	−0.588978	−	1.02014i
$935$	−44.0908			−1.44192
$936$		0			0
$937$	8.00000			0.261349			0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000			0.261349			0.130674	+	0.991425i	$0.458286\pi$
$938$	−17.1464	−	29.6985i	−0.559851	−	0.969690i
$939$		0			0
$940$	48.0000	−	83.1384i	1.56559	−	2.71168i
$941$	−4.89898	+	8.48528i	−0.159702	+	0.276612i	−0.934761	−	0.355276i	$-0.884387\pi$
$941$	−4.89898	+	8.48528i	−0.159702	+	0.276612i	0.775059	+	0.631889i	$0.217720\pi$
$942$		0			0
$943$	6.00000	+	10.3923i	0.195387	+	0.338420i
$944$	9.79796			0.318896
$945$		0			0
$946$	66.0000			2.14585
$947$	12.2474	+	21.2132i	0.397989	+	0.689336i	0.993478	−	0.114027i	$-0.0363750\pi$
$947$	12.2474	+	21.2132i	0.397989	+	0.689336i	−0.595489	+	0.803363i	$0.703042\pi$
$948$		0			0
$949$	5.50000	−	9.52628i	0.178538	−	0.309236i
$950$	−1.22474	+	2.12132i	−0.0397360	+	0.0688247i
$951$		0			0
$952$	36.0000	+	62.3538i	1.16677	+	2.02090i
$953$	−29.3939			−0.952161			−0.476081	−	0.879402i	$-0.657943\pi$
$953$	−29.3939			−0.952161			−0.476081	+	0.879402i	$0.657943\pi$
$954$		0			0
$955$	−24.0000			−0.776622
$956$	−4.89898	−	8.48528i	−0.158444	−	0.274434i
$957$		0			0
$958$	33.0000	−	57.1577i	1.06618	−	1.84668i
$959$	9.79796	−	16.9706i	0.316393	−	0.548008i
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$	19.5959			0.631798
$963$		0			0
$964$	−64.0000			−2.06130
$965$	−13.4722	−	23.3345i	−0.433685	−	0.751165i
$966$		0			0
$967$	3.50000	−	6.06218i	0.112552	−	0.194946i	−0.804246	−	0.594296i	$-0.797431\pi$
$967$	3.50000	−	6.06218i	0.112552	−	0.194946i	0.916799	+	0.399350i	$0.130764\pi$
$968$	−12.2474	+	21.2132i	−0.393648	+	0.681818i
$969$		0			0
$970$	−21.0000	−	36.3731i	−0.674269	−	1.16787i
$971$	−29.3939			−0.943294			−0.471647	−	0.881787i	$-0.656340\pi$
$971$	−29.3939			−0.943294			−0.471647	+	0.881787i	$0.656340\pi$
$972$		0			0
$973$	−20.0000			−0.641171
$974$	42.8661	+	74.2462i	1.37352	+	2.37900i
$975$		0			0
$976$	−10.0000	+	17.3205i	−0.320092	+	0.554416i
$977$	−12.2474	+	21.2132i	−0.391831	+	0.678671i	−0.992691	−	0.120683i	$-0.961491\pi$
$977$	−12.2474	+	21.2132i	−0.391831	+	0.678671i	0.600860	+	0.799354i	$0.294825\pi$
$978$		0			0
$979$		0			0
$980$	−29.3939			−0.938953
$981$		0			0
$982$	−96.0000			−3.06348
$983$	−2.44949	−	4.24264i	−0.0781266	−	0.135319i	0.824315	−	0.566131i	$-0.191561\pi$
$983$	−2.44949	−	4.24264i	−0.0781266	−	0.135319i	−0.902442	+	0.430812i	$0.858227\pi$
$984$		0			0
$985$	18.0000	−	31.1769i	0.573528	−	0.993379i
$986$	44.0908	−	76.3675i	1.40414	−	2.43204i
$987$		0			0
$988$	−2.00000	−	3.46410i	−0.0636285	−	0.110208i
$989$	26.9444			0.856782
$990$		0			0
$991$	−7.00000			−0.222362			−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000			−0.222362			−0.111181	+	0.993800i	$0.535463\pi$
$992$		0			0
$993$		0			0
$994$	−18.0000	+	31.1769i	−0.570925	+	0.988872i
$995$	1.22474	−	2.12132i	0.0388270	−	0.0672504i
$996$		0			0
$997$	−25.0000	−	43.3013i	−0.791758	−	1.37136i	−0.924878	−	0.380265i	$-0.875833\pi$
$997$	−25.0000	−	43.3013i	−0.791758	−	1.37136i	0.133120	−	0.991100i	$-0.457501\pi$
$998$	−4.89898			−0.155074
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	243.2.c.c.82.2		4
3.2	odd	2	inner	243.2.c.c.82.1		4
9.2	odd	6	inner	243.2.c.c.163.1		4
9.4	even	3		243.2.a.d.1.1	✓	2
9.5	odd	6		243.2.a.d.1.2	yes	2
9.7	even	3	inner	243.2.c.c.163.2		4
27.2	odd	18		729.2.e.p.82.2		12
27.4	even	9		729.2.e.p.649.1		12
27.5	odd	18		729.2.e.p.163.1		12
27.7	even	9		729.2.e.p.568.2		12
27.11	odd	18		729.2.e.p.325.2		12
27.13	even	9		729.2.e.p.406.1		12
27.14	odd	18		729.2.e.p.406.2		12
27.16	even	9		729.2.e.p.325.1		12
27.20	odd	18		729.2.e.p.568.1		12
27.22	even	9		729.2.e.p.163.2		12
27.23	odd	18		729.2.e.p.649.2		12
27.25	even	9		729.2.e.p.82.1		12
36.23	even	6		3888.2.a.z.1.1		2
36.31	odd	6		3888.2.a.z.1.2		2
45.4	even	6		6075.2.a.bn.1.2		2
45.14	odd	6		6075.2.a.bn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
243.2.a.d.1.1	✓	2	9.4	even	3
243.2.a.d.1.2	yes	2	9.5	odd	6
243.2.c.c.82.1		4	3.2	odd	2	inner
243.2.c.c.82.2		4	1.1	even	1	trivial
243.2.c.c.163.1		4	9.2	odd	6	inner
243.2.c.c.163.2		4	9.7	even	3	inner
729.2.e.p.82.1		12	27.25	even	9
729.2.e.p.82.2		12	27.2	odd	18
729.2.e.p.163.1		12	27.5	odd	18
729.2.e.p.163.2		12	27.22	even	9
729.2.e.p.325.1		12	27.16	even	9
729.2.e.p.325.2		12	27.11	odd	18
729.2.e.p.406.1		12	27.13	even	9
729.2.e.p.406.2		12	27.14	odd	18
729.2.e.p.568.1		12	27.20	odd	18
729.2.e.p.568.2		12	27.7	even	9
729.2.e.p.649.1		12	27.4	even	9
729.2.e.p.649.2		12	27.23	odd	18
3888.2.a.z.1.1		2	36.23	even	6
3888.2.a.z.1.2		2	36.31	odd	6
6075.2.a.bn.1.1		2	45.14	odd	6
6075.2.a.bn.1.2		2	45.4	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	243.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 - 1.73205i) q^{7} -4.89898 q^{8} +O(q^{10})\)	\(q+(1.22474 + 2.12132i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 - 1.73205i) q^{7} -4.89898 q^{8} -6.00000 q^{10} +(1.22474 + 2.12132i) q^{11} +(0.500000 - 0.866025i) q^{13} +(2.44949 - 4.24264i) q^{14} +(-2.00000 - 3.46410i) q^{16} +7.34847 q^{17} -1.00000 q^{19} +(-4.89898 - 8.48528i) q^{20} +(-3.00000 + 5.19615i) q^{22} +(-1.22474 + 2.12132i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.44949 q^{26} +8.00000 q^{28} +(-2.44949 - 4.24264i) q^{29} +(0.500000 - 0.866025i) q^{31} +(9.00000 + 15.5885i) q^{34} +4.89898 q^{35} +8.00000 q^{37} +(-1.22474 - 2.12132i) q^{38} +(6.00000 - 10.3923i) q^{40} +(2.44949 - 4.24264i) q^{41} +(-5.50000 - 9.52628i) q^{43} -9.79796 q^{44} -6.00000 q^{46} +(4.89898 + 8.48528i) q^{47} +(1.50000 - 2.59808i) q^{49} +(1.22474 - 2.12132i) q^{50} +(2.00000 + 3.46410i) q^{52} -7.34847 q^{53} -6.00000 q^{55} +(4.89898 + 8.48528i) q^{56} +(6.00000 - 10.3923i) q^{58} +(-1.22474 + 2.12132i) q^{59} +(-2.50000 - 4.33013i) q^{61} +2.44949 q^{62} -8.00000 q^{64} +(1.22474 + 2.12132i) q^{65} +(3.50000 - 6.06218i) q^{67} +(-14.6969 + 25.4558i) q^{68} +(6.00000 + 10.3923i) q^{70} -7.34847 q^{71} +11.0000 q^{73} +(9.79796 + 16.9706i) q^{74} +(2.00000 - 3.46410i) q^{76} +(2.44949 - 4.24264i) q^{77} +(3.50000 + 6.06218i) q^{79} +9.79796 q^{80} +12.0000 q^{82} +(-6.12372 - 10.6066i) q^{83} +(-9.00000 + 15.5885i) q^{85} +(13.4722 - 23.3345i) q^{86} +(-6.00000 - 10.3923i) q^{88} -2.00000 q^{91} +(-4.89898 - 8.48528i) q^{92} +(-12.0000 + 20.7846i) q^{94} +(1.22474 - 2.12132i) q^{95} +(3.50000 + 6.06218i) q^{97} +7.34847 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^7	\( 4 q - 8 q^{4} - 4 q^{7} - 24 q^{10} + 2 q^{13} - 8 q^{16} - 4 q^{19} - 12 q^{22} - 2 q^{25} + 32 q^{28} + 2 q^{31} + 36 q^{34} + 32 q^{37} + 24 q^{40} - 22 q^{43} - 24 q^{46} + 6 q^{49} + 8 q^{52} - 24 q^{55} + 24 q^{58} - 10 q^{61} - 32 q^{64} + 14 q^{67} + 24 q^{70} + 44 q^{73} + 8 q^{76} + 14 q^{79} + 48 q^{82} - 36 q^{85} - 24 q^{88} - 8 q^{91} - 48 q^{94} + 14 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^7 - 24 * q^10 + 2 * q^13 - 8 * q^16 - 4 * q^19 - 12 * q^22 - 2 * q^25 + 32 * q^28 + 2 * q^31 + 36 * q^34 + 32 * q^37 + 24 * q^40 - 22 * q^43 - 24 * q^46 + 6 * q^49 + 8 * q^52 - 24 * q^55 + 24 * q^58 - 10 * q^61 - 32 * q^64 + 14 * q^67 + 24 * q^70 + 44 * q^73 + 8 * q^76 + 14 * q^79 + 48 * q^82 - 36 * q^85 - 24 * q^88 - 8 * q^91 - 48 * q^94 + 14 * q^97

Introduction

L-functions

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