LMFDB - Embedded newform 648.2.l.a.539.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,2,Mod(107,648)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("648.107"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.l (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-8,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$5.17430605098$
Analytic rank:	$1$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			539.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	648.539
Dual form			648.2.l.a.107.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} +(1.22474 - 2.12132i) q^{5} +(-3.00000 + 1.73205i) q^{7} +2.82843i q^{8} +(-3.00000 - 1.73205i) q^{10} +(-2.44949 + 1.41421i) q^{11} +(-3.00000 - 1.73205i) q^{13} +(2.44949 + 4.24264i) q^{14} +4.00000 q^{16} +1.41421i q^{17} -4.00000 q^{19} +(-2.44949 + 4.24264i) q^{20} +(2.00000 + 3.46410i) q^{22} +(-2.44949 + 4.24264i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.44949 + 4.24264i) q^{26} +(6.00000 - 3.46410i) q^{28} +(-1.22474 - 2.12132i) q^{29} +(-3.00000 - 1.73205i) q^{31} -5.65685i q^{32} +2.00000 q^{34} +8.48528i q^{35} +5.65685i q^{38} +(6.00000 + 3.46410i) q^{40} +(-1.22474 - 0.707107i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(4.89898 - 2.82843i) q^{44} +(6.00000 + 3.46410i) q^{46} +(2.44949 + 4.24264i) q^{47} +(2.50000 - 4.33013i) q^{49} +(-1.22474 + 0.707107i) q^{50} +(6.00000 + 3.46410i) q^{52} +7.34847 q^{53} +6.92820i q^{55} +(-4.89898 - 8.48528i) q^{56} +(-3.00000 + 1.73205i) q^{58} +(9.79796 + 5.65685i) q^{59} +(-12.0000 + 6.92820i) q^{61} +(-2.44949 + 4.24264i) q^{62} -8.00000 q^{64} +(-7.34847 + 4.24264i) q^{65} +(2.00000 - 3.46410i) q^{67} -2.82843i q^{68} +12.0000 q^{70} -14.6969 q^{71} -4.00000 q^{73} +8.00000 q^{76} +(4.89898 - 8.48528i) q^{77} +(-3.00000 + 1.73205i) q^{79} +(4.89898 - 8.48528i) q^{80} +(-1.00000 + 1.73205i) q^{82} +(12.2474 - 7.07107i) q^{83} +(3.00000 + 1.73205i) q^{85} +(-9.79796 + 5.65685i) q^{86} +(-4.00000 - 6.92820i) q^{88} -7.07107i q^{89} +12.0000 q^{91} +(4.89898 - 8.48528i) q^{92} +(6.00000 - 3.46410i) q^{94} +(-4.89898 + 8.48528i) q^{95} +(-4.00000 - 6.92820i) q^{97} +(-6.12372 - 3.53553i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 12 q^{7} - 12 q^{10} - 12 q^{13} + 16 q^{16} - 16 q^{19} + 8 q^{22} - 2 q^{25} + 24 q^{28} - 12 q^{31} + 8 q^{34} + 24 q^{40} - 16 q^{43} + 24 q^{46} + 10 q^{49} + 24 q^{52} - 12 q^{58}+ \cdots - 16 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 12 * q^7 - 12 * q^10 - 12 * q^13 + 16 * q^16 - 16 * q^19 + 8 * q^22 - 2 * q^25 + 24 * q^28 - 12 * q^31 + 8 * q^34 + 24 * q^40 - 16 * q^43 + 24 * q^46 + 10 * q^49 + 24 * q^52 - 12 * q^58 - 48 * q^61 - 32 * q^64 + 8 * q^67 + 48 * q^70 - 16 * q^73 + 32 * q^76 - 12 * q^79 - 4 * q^82 + 12 * q^85 - 16 * q^88 + 48 * q^91 + 24 * q^94 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		−	1.41421i		−	1.00000i
$2$		−	1.41421i		−	1.00000i
$3$		0			0
$3$		0			0
$4$	−2.00000			−1.00000
$5$	1.22474	−	2.12132i	0.547723	−	0.948683i	−0.450708	−	0.892672i	$-0.648828\pi$
$5$	1.22474	−	2.12132i	0.547723	−	0.948683i	0.998430	−	0.0560116i	$-0.0178384\pi$
$6$		0			0
$7$	−3.00000	+	1.73205i	−1.13389	+	0.654654i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−3.00000	+	1.73205i	−1.13389	+	0.654654i	−0.188982	+	0.981981i	$0.560519\pi$
$8$			2.82843i			1.00000i
$9$		0			0
$10$	−3.00000	−	1.73205i	−0.948683	−	0.547723i
$11$	−2.44949	+	1.41421i	−0.738549	+	0.426401i	−0.821541	−	0.570149i	$-0.806886\pi$
$11$	−2.44949	+	1.41421i	−0.738549	+	0.426401i	0.0829925	+	0.996550i	$0.473552\pi$
$12$		0			0
$13$	−3.00000	−	1.73205i	−0.832050	−	0.480384i	0.0225039	−	0.999747i	$-0.492836\pi$
$13$	−3.00000	−	1.73205i	−0.832050	−	0.480384i	−0.854554	+	0.519362i	$0.826170\pi$
$14$	2.44949	+	4.24264i	0.654654	+	1.13389i
$15$		0			0
$16$	4.00000			1.00000
$17$			1.41421i			0.342997i	0.985184	+	0.171499i	$0.0548609\pi$
$17$			1.41421i			0.342997i	−0.985184	+	0.171499i	$0.945139\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.44949	+	4.24264i	−0.547723	+	0.948683i
$21$		0			0
$22$	2.00000	+	3.46410i	0.426401	+	0.738549i
$23$	−2.44949	+	4.24264i	−0.510754	+	0.884652i	0.489168	+	0.872189i	$0.337300\pi$
$23$	−2.44949	+	4.24264i	−0.510754	+	0.884652i	−0.999922	+	0.0124624i	$0.996033\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	−2.44949	+	4.24264i	−0.480384	+	0.832050i
$27$		0			0
$28$	6.00000	−	3.46410i	1.13389	−	0.654654i
$29$	−1.22474	−	2.12132i	−0.227429	−	0.393919i	0.729616	−	0.683857i	$-0.239699\pi$
$29$	−1.22474	−	2.12132i	−0.227429	−	0.393919i	−0.957046	+	0.289938i	$0.906365\pi$
$30$		0			0
$31$	−3.00000	−	1.73205i	−0.538816	−	0.311086i	0.205783	−	0.978598i	$-0.434026\pi$
$31$	−3.00000	−	1.73205i	−0.538816	−	0.311086i	−0.744599	+	0.667512i	$0.767359\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$	2.00000			0.342997
$35$			8.48528i			1.43427i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$			5.65685i			0.917663i
$39$		0			0
$40$	6.00000	+	3.46410i	0.948683	+	0.547723i
$41$	−1.22474	−	0.707107i	−0.191273	−	0.110432i	0.401305	−	0.915944i	$-0.368557\pi$
$41$	−1.22474	−	0.707107i	−0.191273	−	0.110432i	−0.592578	+	0.805513i	$0.701890\pi$
$42$		0			0
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−4.00000	−	6.92820i	−0.609994	−	1.05654i	0.381246	−	0.924473i	$-0.375495\pi$
$44$	4.89898	−	2.82843i	0.738549	−	0.426401i
$45$		0			0
$46$	6.00000	+	3.46410i	0.884652	+	0.510754i
$47$	2.44949	+	4.24264i	0.357295	+	0.618853i	0.987508	−	0.157569i	$-0.0503658\pi$
$47$	2.44949	+	4.24264i	0.357295	+	0.618853i	−0.630213	+	0.776422i	$0.717032\pi$
$48$		0			0
$49$	2.50000	−	4.33013i	0.357143	−	0.618590i
$50$	−1.22474	+	0.707107i	−0.173205	+	0.100000i
$51$		0			0
$52$	6.00000	+	3.46410i	0.832050	+	0.480384i
$53$	7.34847			1.00939			0.504695	−	0.863298i	$-0.331605\pi$
$53$	7.34847			1.00939			0.504695	+	0.863298i	$0.331605\pi$
$54$		0			0
$55$			6.92820i			0.934199i
$56$	−4.89898	−	8.48528i	−0.654654	−	1.13389i
$57$		0			0
$58$	−3.00000	+	1.73205i	−0.393919	+	0.227429i
$59$	9.79796	+	5.65685i	1.27559	+	0.736460i	0.976034	−	0.217620i	$-0.0698294\pi$
$59$	9.79796	+	5.65685i	1.27559	+	0.736460i	0.299552	+	0.954080i	$0.403163\pi$
$60$		0			0
$61$	−12.0000	+	6.92820i	−1.53644	+	0.887066i	−0.537400	+	0.843328i	$0.680593\pi$
$61$	−12.0000	+	6.92820i	−1.53644	+	0.887066i	−0.999043	+	0.0437377i	$0.986073\pi$
$62$	−2.44949	+	4.24264i	−0.311086	+	0.538816i
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−7.34847	+	4.24264i	−0.911465	+	0.526235i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		−	2.82843i		−	0.342997i
$69$		0			0
$70$	12.0000			1.43427
$71$	−14.6969			−1.74421			−0.872103	−	0.489323i	$-0.837244\pi$
$71$	−14.6969			−1.74421			−0.872103	+	0.489323i	$0.837244\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$		0			0
$76$	8.00000			0.917663
$77$	4.89898	−	8.48528i	0.558291	−	0.966988i
$78$		0			0
$79$	−3.00000	+	1.73205i	−0.337526	+	0.194871i	−0.659178	−	0.751987i	$-0.729095\pi$
$79$	−3.00000	+	1.73205i	−0.337526	+	0.194871i	0.321651	+	0.946858i	$0.395762\pi$
$80$	4.89898	−	8.48528i	0.547723	−	0.948683i
$81$		0			0
$82$	−1.00000	+	1.73205i	−0.110432	+	0.191273i
$83$	12.2474	−	7.07107i	1.34433	−	0.776151i	0.356892	−	0.934146i	$-0.383836\pi$
$83$	12.2474	−	7.07107i	1.34433	−	0.776151i	0.987440	+	0.157995i	$0.0505030\pi$
$84$		0			0
$85$	3.00000	+	1.73205i	0.325396	+	0.187867i
$86$	−9.79796	+	5.65685i	−1.05654	+	0.609994i
$87$		0			0
$88$	−4.00000	−	6.92820i	−0.426401	−	0.738549i
$89$		−	7.07107i		−	0.749532i	−0.927119	−	0.374766i	$-0.877723\pi$
$89$		−	7.07107i		−	0.749532i	0.927119	−	0.374766i	$-0.122277\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$	4.89898	−	8.48528i	0.510754	−	0.884652i
$93$		0			0
$94$	6.00000	−	3.46410i	0.618853	−	0.357295i
$95$	−4.89898	+	8.48528i	−0.502625	+	0.870572i
$96$		0			0
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	0.588315	−	0.808632i	$-0.299792\pi$
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	−0.994453	+	0.105180i	$0.966458\pi$
$98$	−6.12372	−	3.53553i	−0.618590	−	0.357143i
$99$		0			0
$100$	1.00000	+	1.73205i	0.100000	+	0.173205i
$101$	−1.22474	−	2.12132i	−0.121867	−	0.211079i	0.798637	−	0.601813i	$-0.205555\pi$
$101$	−1.22474	−	2.12132i	−0.121867	−	0.211079i	−0.920504	+	0.390734i	$0.872221\pi$
$102$		0			0
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	0.344865	−	0.938652i	$-0.387925\pi$
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	−0.640464	+	0.767988i	$0.721258\pi$
$104$	4.89898	−	8.48528i	0.480384	−	0.832050i
$105$		0			0
$106$		−	10.3923i		−	1.00939i
$107$		−	11.3137i		−	1.09374i	−0.837218	−	0.546869i	$-0.815820\pi$
$107$		−	11.3137i		−	1.09374i	0.837218	−	0.546869i	$-0.184180\pi$
$108$		0			0
$109$			10.3923i			0.995402i	0.867349	+	0.497701i	$0.165822\pi$
$109$			10.3923i			0.995402i	−0.867349	+	0.497701i	$0.834178\pi$
$110$	9.79796			0.934199
$111$		0			0
$112$	−12.0000	+	6.92820i	−1.13389	+	0.654654i
$113$	−15.9217	−	9.19239i	−1.49779	−	0.864747i	−0.497789	−	0.867298i	$-0.665855\pi$
$113$	−15.9217	−	9.19239i	−1.49779	−	0.864747i	−0.999997	+	0.00255090i	$0.999188\pi$
$114$		0			0
$115$	6.00000	+	10.3923i	0.559503	+	0.969087i
$116$	2.44949	+	4.24264i	0.227429	+	0.393919i
$117$		0			0
$118$	8.00000	−	13.8564i	0.736460	−	1.27559i
$119$	−2.44949	−	4.24264i	−0.224544	−	0.388922i
$120$		0			0
$121$	−1.50000	+	2.59808i	−0.136364	+	0.236189i
$122$	9.79796	+	16.9706i	0.887066	+	1.53644i
$123$		0			0
$124$	6.00000	+	3.46410i	0.538816	+	0.311086i
$125$	9.79796			0.876356
$126$		0			0
$127$		−	10.3923i		−	0.922168i	−0.887357	−	0.461084i	$-0.847461\pi$
$127$		−	10.3923i		−	0.922168i	0.887357	−	0.461084i	$-0.152539\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$	6.00000	+	10.3923i	0.526235	+	0.911465i
$131$	−4.89898	−	2.82843i	−0.428026	−	0.247121i	0.270479	−	0.962726i	$-0.412818\pi$
$131$	−4.89898	−	2.82843i	−0.428026	−	0.247121i	−0.698505	+	0.715605i	$0.746151\pi$
$132$		0			0
$133$	12.0000	−	6.92820i	1.04053	−	0.600751i
$134$	−4.89898	−	2.82843i	−0.423207	−	0.244339i
$135$		0			0
$136$	−4.00000			−0.342997
$137$	8.57321	−	4.94975i	0.732459	−	0.422885i	−0.0868620	−	0.996220i	$-0.527684\pi$
$137$	8.57321	−	4.94975i	0.732459	−	0.422885i	0.819321	+	0.573335i	$0.194351\pi$
$138$		0			0
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$		−	16.9706i		−	1.43427i
$141$		0			0
$142$			20.7846i			1.74421i
$143$	9.79796			0.819346
$144$		0			0
$145$	−6.00000			−0.498273
$146$			5.65685i			0.468165i
$147$		0			0
$148$		0			0
$149$	8.57321	−	14.8492i	0.702345	−	1.21650i	−0.265296	−	0.964167i	$-0.585470\pi$
$149$	8.57321	−	14.8492i	0.702345	−	1.21650i	0.967641	−	0.252330i	$-0.0811970\pi$
$150$		0			0
$151$	−3.00000	+	1.73205i	−0.244137	+	0.140952i	−0.617076	−	0.786903i	$-0.711683\pi$
$151$	−3.00000	+	1.73205i	−0.244137	+	0.140952i	0.372940	+	0.927855i	$0.378350\pi$
$152$		−	11.3137i		−	0.917663i
$153$		0			0
$154$	−12.0000	−	6.92820i	−0.966988	−	0.558291i
$155$	−7.34847	+	4.24264i	−0.590243	+	0.340777i
$156$		0			0
$157$	−12.0000	−	6.92820i	−0.957704	−	0.552931i	−0.0622385	−	0.998061i	$-0.519824\pi$
$157$	−12.0000	−	6.92820i	−0.957704	−	0.552931i	−0.895466	+	0.445130i	$0.853157\pi$
$158$	2.44949	+	4.24264i	0.194871	+	0.337526i
$159$		0			0
$160$	−12.0000	−	6.92820i	−0.948683	−	0.547723i
$161$		−	16.9706i		−	1.33747i
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$	2.44949	+	1.41421i	0.191273	+	0.110432i
$165$		0			0
$166$	−10.0000	−	17.3205i	−0.776151	−	1.34433i
$167$	−9.79796	+	16.9706i	−0.758189	+	1.31322i	0.185584	+	0.982628i	$0.440582\pi$
$167$	−9.79796	+	16.9706i	−0.758189	+	1.31322i	−0.943773	+	0.330593i	$0.892751\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.0384615	−	0.0666173i
$170$	2.44949	−	4.24264i	0.187867	−	0.325396i
$171$		0			0
$172$	8.00000	+	13.8564i	0.609994	+	1.05654i
$173$	−1.22474	−	2.12132i	−0.0931156	−	0.161281i	0.815705	−	0.578468i	$-0.196349\pi$
$173$	−1.22474	−	2.12132i	−0.0931156	−	0.161281i	−0.908821	+	0.417187i	$0.863016\pi$
$174$		0			0
$175$	3.00000	+	1.73205i	0.226779	+	0.130931i
$176$	−9.79796	+	5.65685i	−0.738549	+	0.426401i
$177$		0			0
$178$	−10.0000			−0.749532
$179$			5.65685i			0.422813i	0.977398	+	0.211407i	$0.0678044\pi$
$179$			5.65685i			0.422813i	−0.977398	+	0.211407i	$0.932196\pi$
$180$		0			0
$181$		−	10.3923i		−	0.772454i	−0.922404	−	0.386227i	$-0.873778\pi$
$181$		−	10.3923i		−	0.772454i	0.922404	−	0.386227i	$-0.126222\pi$
$182$		−	16.9706i		−	1.25794i
$183$		0			0
$184$	−12.0000	−	6.92820i	−0.884652	−	0.510754i
$185$		0			0
$186$		0			0
$187$	−2.00000	−	3.46410i	−0.146254	−	0.253320i
$188$	−4.89898	−	8.48528i	−0.357295	−	0.618853i
$189$		0			0
$190$	12.0000	+	6.92820i	0.870572	+	0.502625i
$191$	9.79796	+	16.9706i	0.708955	+	1.22795i	0.965245	+	0.261347i	$0.0841666\pi$
$191$	9.79796	+	16.9706i	0.708955	+	1.22795i	−0.256290	+	0.966600i	$0.582500\pi$
$192$		0			0
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	0.496119	+	0.868255i	$0.334758\pi$
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$	−9.79796	+	5.65685i	−0.703452	+	0.406138i
$195$		0			0
$196$	−5.00000	+	8.66025i	−0.357143	+	0.618590i
$197$	−7.34847			−0.523557			−0.261778	−	0.965128i	$-0.584309\pi$
$197$	−7.34847			−0.523557			−0.261778	+	0.965128i	$0.584309\pi$
$198$		0			0
$199$		−	10.3923i		−	0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$		−	10.3923i		−	0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$	2.44949	−	1.41421i	0.173205	−	0.100000i
$201$		0			0
$202$	−3.00000	+	1.73205i	−0.211079	+	0.121867i
$203$	7.34847	+	4.24264i	0.515761	+	0.297775i
$204$		0			0
$205$	−3.00000	+	1.73205i	−0.209529	+	0.120972i
$206$	−2.44949	+	4.24264i	−0.170664	+	0.295599i
$207$		0			0
$208$	−12.0000	−	6.92820i	−0.832050	−	0.480384i
$209$	9.79796	−	5.65685i	0.677739	−	0.391293i
$210$		0			0
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	0.283918	+	0.958849i	$0.408366\pi$
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	−0.972346	+	0.233544i	$0.924968\pi$
$212$	−14.6969			−1.00939
$213$		0			0
$214$	−16.0000			−1.09374
$215$	−19.5959			−1.33643
$216$		0			0
$217$	12.0000			0.814613
$218$	14.6969			0.995402
$219$		0			0
$220$		−	13.8564i		−	0.934199i
$221$	2.44949	−	4.24264i	0.164771	−	0.285391i
$222$		0			0
$223$	15.0000	−	8.66025i	1.00447	−	0.579934i	0.0949052	−	0.995486i	$-0.469745\pi$
$223$	15.0000	−	8.66025i	1.00447	−	0.579934i	0.909569	+	0.415553i	$0.136412\pi$
$224$	9.79796	+	16.9706i	0.654654	+	1.13389i
$225$		0			0
$226$	−13.0000	+	22.5167i	−0.864747	+	1.49779i
$227$	−2.44949	+	1.41421i	−0.162578	+	0.0938647i	−0.579082	−	0.815270i	$-0.696589\pi$
$227$	−2.44949	+	1.41421i	−0.162578	+	0.0938647i	0.416503	+	0.909134i	$0.363255\pi$
$228$		0			0
$229$	15.0000	+	8.66025i	0.991228	+	0.572286i	0.905641	−	0.424045i	$-0.139390\pi$
$229$	15.0000	+	8.66025i	0.991228	+	0.572286i	0.0855868	+	0.996331i	$0.472724\pi$
$230$	14.6969	−	8.48528i	0.969087	−	0.559503i
$231$		0			0
$232$	6.00000	−	3.46410i	0.393919	−	0.227429i
$233$			1.41421i			0.0926482i	0.998926	+	0.0463241i	$0.0147507\pi$
$233$			1.41421i			0.0926482i	−0.998926	+	0.0463241i	$0.985249\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$	−19.5959	−	11.3137i	−1.27559	−	0.736460i
$237$		0			0
$238$	−6.00000	+	3.46410i	−0.388922	+	0.224544i
$239$	4.89898	−	8.48528i	0.316889	−	0.548867i	−0.662949	−	0.748665i	$-0.730695\pi$
$239$	4.89898	−	8.48528i	0.316889	−	0.548867i	0.979837	+	0.199798i	$0.0640285\pi$
$240$		0			0
$241$	14.0000	+	24.2487i	0.901819	+	1.56200i	0.825131	+	0.564942i	$0.191101\pi$
$241$	14.0000	+	24.2487i	0.901819	+	1.56200i	0.0766885	+	0.997055i	$0.475565\pi$
$242$	3.67423	+	2.12132i	0.236189	+	0.136364i
$243$		0			0
$244$	24.0000	−	13.8564i	1.53644	−	0.887066i
$245$	−6.12372	−	10.6066i	−0.391230	−	0.677631i
$246$		0			0
$247$	12.0000	+	6.92820i	0.763542	+	0.440831i
$248$	4.89898	−	8.48528i	0.311086	−	0.538816i
$249$		0			0
$250$		−	13.8564i		−	0.876356i
$251$			14.1421i			0.892644i	0.894873	+	0.446322i	$0.147266\pi$
$251$			14.1421i			0.892644i	−0.894873	+	0.446322i	$0.852734\pi$
$252$		0			0
$253$		−	13.8564i		−	0.871145i
$254$	−14.6969			−0.922168
$255$		0			0
$256$	16.0000			1.00000
$257$	20.8207	+	12.0208i	1.29876	+	0.749838i	0.980189	−	0.198062i	$-0.0634648\pi$
$257$	20.8207	+	12.0208i	1.29876	+	0.749838i	0.318568	+	0.947900i	$0.396798\pi$
$258$		0			0
$259$		0			0
$260$	14.6969	−	8.48528i	0.911465	−	0.526235i
$261$		0			0
$262$	−4.00000	+	6.92820i	−0.247121	+	0.428026i
$263$	−4.89898	−	8.48528i	−0.302084	−	0.523225i	0.674524	−	0.738253i	$-0.264349\pi$
$263$	−4.89898	−	8.48528i	−0.302084	−	0.523225i	−0.976608	+	0.215028i	$0.931016\pi$
$264$		0			0
$265$	9.00000	−	15.5885i	0.552866	−	0.957591i
$266$	−9.79796	−	16.9706i	−0.600751	−	1.04053i
$267$		0			0
$268$	−4.00000	+	6.92820i	−0.244339	+	0.423207i
$269$	7.34847			0.448044			0.224022	−	0.974584i	$-0.428081\pi$
$269$	7.34847			0.448044			0.224022	+	0.974584i	$0.428081\pi$
$270$		0			0
$271$			31.1769i			1.89386i	0.321436	+	0.946931i	$0.395835\pi$
$271$			31.1769i			1.89386i	−0.321436	+	0.946931i	$0.604165\pi$
$272$			5.65685i			0.342997i
$273$		0			0
$274$	−7.00000	−	12.1244i	−0.422885	−	0.732459i
$275$	2.44949	+	1.41421i	0.147710	+	0.0852803i
$276$		0			0
$277$	15.0000	−	8.66025i	0.901263	−	0.520344i	0.0236530	−	0.999720i	$-0.492470\pi$
$277$	15.0000	−	8.66025i	0.901263	−	0.520344i	0.877610	+	0.479376i	$0.159137\pi$
$278$	−4.89898	−	2.82843i	−0.293821	−	0.169638i
$279$		0			0
$280$	−24.0000			−1.43427
$281$	−20.8207	+	12.0208i	−1.24206	+	0.717102i	−0.969513	−	0.245042i	$-0.921198\pi$
$281$	−20.8207	+	12.0208i	−1.24206	+	0.717102i	−0.272544	+	0.962143i	$0.587865\pi$
$282$		0			0
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	−0.524057	−	0.851683i	$-0.675582\pi$
$283$	8.00000	−	13.8564i	0.475551	−	0.823678i	0.999608	+	0.0280052i	$0.00891551\pi$
$284$	29.3939			1.74421
$285$		0			0
$286$		−	13.8564i		−	0.819346i
$287$	4.89898			0.289178
$288$		0			0
$289$	15.0000			0.882353
$290$			8.48528i			0.498273i
$291$		0			0
$292$	8.00000			0.468165
$293$	−13.4722	+	23.3345i	−0.787054	+	1.36322i	0.140710	+	0.990051i	$0.455061\pi$
$293$	−13.4722	+	23.3345i	−0.787054	+	1.36322i	−0.927764	+	0.373167i	$0.878272\pi$
$294$		0			0
$295$	24.0000	−	13.8564i	1.39733	−	0.806751i
$296$		0			0
$297$		0			0
$298$	−21.0000	−	12.1244i	−1.21650	−	0.702345i
$299$	14.6969	−	8.48528i	0.849946	−	0.490716i
$300$		0			0
$301$	24.0000	+	13.8564i	1.38334	+	0.798670i
$302$	2.44949	+	4.24264i	0.140952	+	0.244137i
$303$		0			0
$304$	−16.0000			−0.917663
$305$			33.9411i			1.94346i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$	−9.79796	+	16.9706i	−0.558291	+	0.966988i
$309$		0			0
$310$	6.00000	+	10.3923i	0.340777	+	0.590243i
$311$	4.89898	−	8.48528i	0.277796	−	0.481156i	−0.693041	−	0.720898i	$-0.743730\pi$
$311$	4.89898	−	8.48528i	0.277796	−	0.481156i	0.970837	+	0.239742i	$0.0770629\pi$
$312$		0			0
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	−0.689412	+	0.724370i	$0.742131\pi$
$314$	−9.79796	+	16.9706i	−0.552931	+	0.957704i
$315$		0			0
$316$	6.00000	−	3.46410i	0.337526	−	0.194871i
$317$	6.12372	+	10.6066i	0.343943	+	0.595726i	0.985161	−	0.171632i	$-0.0549041\pi$
$317$	6.12372	+	10.6066i	0.343943	+	0.595726i	−0.641218	+	0.767358i	$0.721571\pi$
$318$		0			0
$319$	6.00000	+	3.46410i	0.335936	+	0.193952i
$320$	−9.79796	+	16.9706i	−0.547723	+	0.948683i
$321$		0			0
$322$	−24.0000			−1.33747
$323$		−	5.65685i		−	0.314756i
$324$		0			0
$325$			3.46410i			0.192154i
$326$			22.6274i			1.25322i
$327$		0			0
$328$	2.00000	−	3.46410i	0.110432	−	0.191273i
$329$	−14.6969	−	8.48528i	−0.810268	−	0.467809i
$330$		0			0
$331$	2.00000	+	3.46410i	0.109930	+	0.190404i	0.915742	−	0.401768i	$-0.131604\pi$
$331$	2.00000	+	3.46410i	0.109930	+	0.190404i	−0.805812	+	0.592172i	$0.798271\pi$
$332$	−24.4949	+	14.1421i	−1.34433	+	0.776151i
$333$		0			0
$334$	24.0000	+	13.8564i	1.31322	+	0.758189i
$335$	−4.89898	−	8.48528i	−0.267660	−	0.463600i
$336$		0			0
$337$	2.00000	−	3.46410i	0.108947	−	0.188702i	−0.806397	−	0.591375i	$-0.798585\pi$
$337$	2.00000	−	3.46410i	0.108947	−	0.188702i	0.915344	+	0.402673i	$0.131919\pi$
$338$	−1.22474	+	0.707107i	−0.0666173	+	0.0384615i
$339$		0			0
$340$	−6.00000	−	3.46410i	−0.325396	−	0.187867i
$341$	9.79796			0.530589
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$	19.5959	−	11.3137i	1.05654	−	0.609994i
$345$		0			0
$346$	−3.00000	+	1.73205i	−0.161281	+	0.0931156i
$347$	−12.2474	−	7.07107i	−0.657477	−	0.379595i	0.133838	−	0.991003i	$-0.457270\pi$
$347$	−12.2474	−	7.07107i	−0.657477	−	0.379595i	−0.791315	+	0.611408i	$0.790603\pi$
$348$		0			0
$349$	24.0000	−	13.8564i	1.28469	−	0.741716i	0.306988	−	0.951713i	$-0.400679\pi$
$349$	24.0000	−	13.8564i	1.28469	−	0.741716i	0.977702	+	0.209997i	$0.0673454\pi$
$350$	2.44949	−	4.24264i	0.130931	−	0.226779i
$351$		0			0
$352$	8.00000	+	13.8564i	0.426401	+	0.738549i
$353$	−13.4722	+	7.77817i	−0.717053	+	0.413990i	−0.813667	−	0.581331i	$-0.802532\pi$
$353$	−13.4722	+	7.77817i	−0.717053	+	0.413990i	0.0966144	+	0.995322i	$0.469199\pi$
$354$		0			0
$355$	−18.0000	+	31.1769i	−0.955341	+	1.65470i
$356$			14.1421i			0.749532i
$357$		0			0
$358$	8.00000			0.422813
$359$	14.6969			0.775675			0.387837	−	0.921728i	$-0.373222\pi$
$359$	14.6969			0.775675			0.387837	+	0.921728i	$0.373222\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	−14.6969			−0.772454
$363$		0			0
$364$	−24.0000			−1.25794
$365$	−4.89898	+	8.48528i	−0.256424	+	0.444140i
$366$		0			0
$367$	−21.0000	+	12.1244i	−1.09619	+	0.632886i	−0.935218	−	0.354073i	$-0.884797\pi$
$367$	−21.0000	+	12.1244i	−1.09619	+	0.632886i	−0.160973	+	0.986959i	$0.551463\pi$
$368$	−9.79796	+	16.9706i	−0.510754	+	0.884652i
$369$		0			0
$370$		0			0
$371$	−22.0454	+	12.7279i	−1.14454	+	0.660801i
$372$		0			0
$373$	−12.0000	−	6.92820i	−0.621336	−	0.358729i	0.156053	−	0.987749i	$-0.450123\pi$
$373$	−12.0000	−	6.92820i	−0.621336	−	0.358729i	−0.777389	+	0.629020i	$0.783456\pi$
$374$	−4.89898	+	2.82843i	−0.253320	+	0.146254i
$375$		0			0
$376$	−12.0000	+	6.92820i	−0.618853	+	0.357295i
$377$			8.48528i			0.437014i
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$	9.79796	−	16.9706i	0.502625	−	0.870572i
$381$		0			0
$382$	24.0000	−	13.8564i	1.22795	−	0.708955i
$383$	4.89898	−	8.48528i	0.250326	−	0.433578i	−0.713289	−	0.700870i	$-0.752795\pi$
$383$	4.89898	−	8.48528i	0.250326	−	0.433578i	0.963616	+	0.267292i	$0.0861288\pi$
$384$		0			0
$385$	−12.0000	−	20.7846i	−0.611577	−	1.05928i
$386$	17.1464	+	9.89949i	0.872730	+	0.503871i
$387$		0			0
$388$	8.00000	+	13.8564i	0.406138	+	0.703452i
$389$	13.4722	+	23.3345i	0.683067	+	1.18311i	0.974040	+	0.226376i	$0.0726878\pi$
$389$	13.4722	+	23.3345i	0.683067	+	1.18311i	−0.290973	+	0.956731i	$0.593979\pi$
$390$		0			0
$391$	−6.00000	−	3.46410i	−0.303433	−	0.175187i
$392$	12.2474	+	7.07107i	0.618590	+	0.357143i
$393$		0			0
$394$			10.3923i			0.523557i
$395$			8.48528i			0.426941i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−14.6969			−0.736691
$399$		0			0
$400$	−2.00000	−	3.46410i	−0.100000	−	0.173205i
$401$	−1.22474	−	0.707107i	−0.0611608	−	0.0353112i	0.469108	−	0.883141i	$-0.344576\pi$
$401$	−1.22474	−	0.707107i	−0.0611608	−	0.0353112i	−0.530269	+	0.847830i	$0.677909\pi$
$402$		0			0
$403$	6.00000	+	10.3923i	0.298881	+	0.517678i
$404$	2.44949	+	4.24264i	0.121867	+	0.211079i
$405$		0			0
$406$	6.00000	−	10.3923i	0.297775	−	0.515761i
$407$		0			0
$408$		0			0
$409$	−16.0000	+	27.7128i	−0.791149	+	1.37031i	0.134107	+	0.990967i	$0.457183\pi$
$409$	−16.0000	+	27.7128i	−0.791149	+	1.37031i	−0.925256	+	0.379344i	$0.876150\pi$
$410$	2.44949	+	4.24264i	0.120972	+	0.209529i
$411$		0			0
$412$	6.00000	+	3.46410i	0.295599	+	0.170664i
$413$	−39.1918			−1.92850
$414$		0			0
$415$		−	34.6410i		−	1.70046i
$416$	−9.79796	+	16.9706i	−0.480384	+	0.832050i
$417$		0			0
$418$	−8.00000	−	13.8564i	−0.391293	−	0.677739i
$419$	17.1464	+	9.89949i	0.837658	+	0.483622i	0.856467	−	0.516201i	$-0.172654\pi$
$419$	17.1464	+	9.89949i	0.837658	+	0.483622i	−0.0188096	+	0.999823i	$0.505988\pi$
$420$		0			0
$421$	−21.0000	+	12.1244i	−1.02348	+	0.590905i	−0.915109	−	0.403206i	$-0.867896\pi$
$421$	−21.0000	+	12.1244i	−1.02348	+	0.590905i	−0.108368	+	0.994111i	$0.534563\pi$
$422$	24.4949	+	14.1421i	1.19239	+	0.688428i
$423$		0			0
$424$			20.7846i			1.00939i
$425$	1.22474	−	0.707107i	0.0594089	−	0.0342997i
$426$		0			0
$427$	24.0000	−	41.5692i	1.16144	−	2.01168i
$428$			22.6274i			1.09374i
$429$		0			0
$430$			27.7128i			1.33643i
$431$	14.6969			0.707927			0.353963	−	0.935259i	$-0.384834\pi$
$431$	14.6969			0.707927			0.353963	+	0.935259i	$0.384834\pi$
$432$		0			0
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$		−	16.9706i		−	0.814613i
$435$		0			0
$436$		−	20.7846i		−	0.995402i
$437$	9.79796	−	16.9706i	0.468700	−	0.811812i
$438$		0			0
$439$	−3.00000	+	1.73205i	−0.143182	+	0.0826663i	−0.569880	−	0.821728i	$-0.693010\pi$
$439$	−3.00000	+	1.73205i	−0.143182	+	0.0826663i	0.426698	+	0.904394i	$0.359677\pi$
$440$	−19.5959			−0.934199
$441$		0			0
$442$	−6.00000	−	3.46410i	−0.285391	−	0.164771i
$443$	−2.44949	+	1.41421i	−0.116379	+	0.0671913i	−0.557059	−	0.830473i	$-0.688070\pi$
$443$	−2.44949	+	1.41421i	−0.116379	+	0.0671913i	0.440681	+	0.897664i	$0.354737\pi$
$444$		0			0
$445$	−15.0000	−	8.66025i	−0.711068	−	0.410535i
$446$	−12.2474	−	21.2132i	−0.579934	−	1.00447i
$447$		0			0
$448$	24.0000	−	13.8564i	1.13389	−	0.654654i
$449$		−	24.0416i		−	1.13459i	−0.823513	−	0.567297i	$-0.807989\pi$
$449$		−	24.0416i		−	1.13459i	0.823513	−	0.567297i	$-0.192011\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$	31.8434	+	18.3848i	1.49779	+	0.864747i
$453$		0			0
$454$	2.00000	+	3.46410i	0.0938647	+	0.162578i
$455$	14.6969	−	25.4558i	0.689003	−	1.19339i
$456$		0			0
$457$	−4.00000	−	6.92820i	−0.187112	−	0.324088i	0.757174	−	0.653213i	$-0.226579\pi$
$457$	−4.00000	−	6.92820i	−0.187112	−	0.324088i	−0.944286	+	0.329125i	$0.893246\pi$
$458$	12.2474	−	21.2132i	0.572286	−	0.991228i
$459$		0			0
$460$	−12.0000	−	20.7846i	−0.559503	−	0.969087i
$461$	−8.57321	−	14.8492i	−0.399294	−	0.691598i	0.594345	−	0.804210i	$-0.297412\pi$
$461$	−8.57321	−	14.8492i	−0.399294	−	0.691598i	−0.993639	+	0.112612i	$0.964078\pi$
$462$		0			0
$463$	15.0000	+	8.66025i	0.697109	+	0.402476i	0.806270	−	0.591548i	$-0.201483\pi$
$463$	15.0000	+	8.66025i	0.697109	+	0.402476i	−0.109161	+	0.994024i	$0.534816\pi$
$464$	−4.89898	−	8.48528i	−0.227429	−	0.393919i
$465$		0			0
$466$	2.00000			0.0926482
$467$		−	36.7696i		−	1.70149i	−0.525577	−	0.850746i	$-0.676151\pi$
$467$		−	36.7696i		−	1.70149i	0.525577	−	0.850746i	$-0.323849\pi$
$468$		0			0
$469$			13.8564i			0.639829i
$470$		−	16.9706i		−	0.782794i
$471$		0			0
$472$	−16.0000	+	27.7128i	−0.736460	+	1.27559i
$473$	19.5959	+	11.3137i	0.901021	+	0.520205i
$474$		0			0
$475$	2.00000	+	3.46410i	0.0917663	+	0.158944i
$476$	4.89898	+	8.48528i	0.224544	+	0.388922i
$477$		0			0
$478$	−12.0000	−	6.92820i	−0.548867	−	0.316889i
$479$	−12.2474	−	21.2132i	−0.559600	−	0.969256i	−0.997530	−	0.0702467i	$-0.977621\pi$
$479$	−12.2474	−	21.2132i	−0.559600	−	0.969256i	0.437929	−	0.899009i	$-0.355712\pi$
$480$		0			0
$481$		0			0
$482$	34.2929	−	19.7990i	1.56200	−	0.901819i
$483$		0			0
$484$	3.00000	−	5.19615i	0.136364	−	0.236189i
$485$	−19.5959			−0.889805
$486$		0			0
$487$		−	10.3923i		−	0.470920i	−0.971884	−	0.235460i	$-0.924340\pi$
$487$		−	10.3923i		−	0.470920i	0.971884	−	0.235460i	$-0.0756597\pi$
$488$	−19.5959	−	33.9411i	−0.887066	−	1.53644i
$489$		0			0
$490$	−15.0000	+	8.66025i	−0.677631	+	0.391230i
$491$	−34.2929	−	19.7990i	−1.54761	−	0.893516i	−0.998323	−	0.0578852i	$-0.981564\pi$
$491$	−34.2929	−	19.7990i	−1.54761	−	0.893516i	−0.549292	−	0.835631i	$-0.685102\pi$
$492$		0			0
$493$	3.00000	−	1.73205i	0.135113	−	0.0780076i
$494$	9.79796	−	16.9706i	0.440831	−	0.763542i
$495$		0			0
$496$	−12.0000	−	6.92820i	−0.538816	−	0.311086i
$497$	44.0908	−	25.4558i	1.97774	−	1.14185i
$498$		0			0
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	0.246214	+	0.969216i	$0.420813\pi$
$499$	−16.0000	+	27.7128i	−0.716258	+	1.24060i	−0.962472	+	0.271380i	$0.912520\pi$
$500$	−19.5959			−0.876356
$501$		0			0
$502$	20.0000			0.892644
$503$	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
$505$	−6.00000			−0.266996
$506$	−19.5959			−0.871145
$507$		0			0
$508$			20.7846i			0.922168i
$509$	−20.8207	+	36.0624i	−0.922860	+	1.59844i	−0.127892	+	0.991788i	$0.540821\pi$
$509$	−20.8207	+	36.0624i	−0.922860	+	1.59844i	−0.794968	+	0.606652i	$0.792512\pi$
$510$		0			0
$511$	12.0000	−	6.92820i	0.530849	−	0.306486i
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$	17.0000	−	29.4449i	0.749838	−	1.29876i
$515$	−7.34847	+	4.24264i	−0.323812	+	0.186953i
$516$		0			0
$517$	−12.0000	−	6.92820i	−0.527759	−	0.304702i
$518$		0			0
$519$		0			0
$520$	−12.0000	−	20.7846i	−0.526235	−	0.911465i
$521$			26.8701i			1.17720i	0.808425	+	0.588599i	$0.200320\pi$
$521$			26.8701i			1.17720i	−0.808425	+	0.588599i	$0.799680\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$	9.79796	+	5.65685i	0.428026	+	0.247121i
$525$		0			0
$526$	−12.0000	+	6.92820i	−0.523225	+	0.302084i
$527$	2.44949	−	4.24264i	0.106701	−	0.184812i
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.0217391	−	0.0376533i
$530$	−22.0454	−	12.7279i	−0.957591	−	0.552866i
$531$		0			0
$532$	−24.0000	+	13.8564i	−1.04053	+	0.600751i
$533$	2.44949	+	4.24264i	0.106099	+	0.183769i
$534$		0			0
$535$	−24.0000	−	13.8564i	−1.03761	−	0.599065i
$536$	9.79796	+	5.65685i	0.423207	+	0.244339i
$537$		0			0
$538$		−	10.3923i		−	0.448044i
$539$			14.1421i			0.609145i
$540$		0			0
$541$			10.3923i			0.446800i	0.974727	+	0.223400i	$0.0717156\pi$
$541$			10.3923i			0.446800i	−0.974727	+	0.223400i	$0.928284\pi$
$542$	44.0908			1.89386
$543$		0			0
$544$	8.00000			0.342997
$545$	22.0454	+	12.7279i	0.944322	+	0.545204i
$546$		0			0
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	0.767752	−	0.640747i	$-0.221375\pi$
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	−0.938779	+	0.344519i	$0.888042\pi$
$548$	−17.1464	+	9.89949i	−0.732459	+	0.422885i
$549$		0			0
$550$	2.00000	−	3.46410i	0.0852803	−	0.147710i
$551$	4.89898	+	8.48528i	0.208704	+	0.361485i
$552$		0			0
$553$	6.00000	−	10.3923i	0.255146	−	0.441926i
$554$	−12.2474	−	21.2132i	−0.520344	−	0.901263i
$555$		0			0
$556$	−4.00000	+	6.92820i	−0.169638	+	0.293821i
$557$	7.34847			0.311365			0.155682	−	0.987807i	$-0.450242\pi$
$557$	7.34847			0.311365			0.155682	+	0.987807i	$0.450242\pi$
$558$		0			0
$559$			27.7128i			1.17213i
$560$			33.9411i			1.43427i
$561$		0			0
$562$	17.0000	+	29.4449i	0.717102	+	1.24206i
$563$	−12.2474	−	7.07107i	−0.516168	−	0.298010i	0.219197	−	0.975681i	$-0.429656\pi$
$563$	−12.2474	−	7.07107i	−0.516168	−	0.298010i	−0.735366	+	0.677671i	$0.762990\pi$
$564$		0			0
$565$	−39.0000	+	22.5167i	−1.64074	+	0.947283i
$566$	−19.5959	−	11.3137i	−0.823678	−	0.475551i
$567$		0			0
$568$		−	41.5692i		−	1.74421i
$569$	8.57321	−	4.94975i	0.359408	−	0.207504i	−0.309413	−	0.950928i	$-0.600133\pi$
$569$	8.57321	−	4.94975i	0.359408	−	0.207504i	0.668821	+	0.743423i	$0.266799\pi$
$570$		0			0
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	−0.0554391	−	0.998462i	$-0.517656\pi$
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	0.892413	−	0.451219i	$-0.149011\pi$
$572$	−19.5959			−0.819346
$573$		0			0
$574$		−	6.92820i		−	0.289178i
$575$	4.89898			0.204302
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		−	21.2132i		−	0.882353i
$579$		0			0
$580$	12.0000			0.498273
$581$	−24.4949	+	42.4264i	−1.01622	+	1.76014i
$582$		0			0
$583$	−18.0000	+	10.3923i	−0.745484	+	0.430405i
$584$		−	11.3137i		−	0.468165i
$585$		0			0
$586$	33.0000	+	19.0526i	1.36322	+	0.787054i
$587$	19.5959	−	11.3137i	0.808810	−	0.466967i	−0.0377324	−	0.999288i	$-0.512013\pi$
$587$	19.5959	−	11.3137i	0.808810	−	0.466967i	0.846542	+	0.532321i	$0.178680\pi$
$588$		0			0
$589$	12.0000	+	6.92820i	0.494451	+	0.285472i
$590$	−19.5959	−	33.9411i	−0.806751	−	1.39733i
$591$		0			0
$592$		0			0
$593$		−	7.07107i		−	0.290374i	−0.989404	−	0.145187i	$-0.953622\pi$
$593$		−	7.07107i		−	0.290374i	0.989404	−	0.145187i	$-0.0463784\pi$
$594$		0			0
$595$	−12.0000			−0.491952
$596$	−17.1464	+	29.6985i	−0.702345	+	1.21650i
$597$		0			0
$598$	−12.0000	−	20.7846i	−0.490716	−	0.849946i
$599$	−2.44949	+	4.24264i	−0.100083	+	0.173350i	−0.911719	−	0.410815i	$-0.865244\pi$
$599$	−2.44949	+	4.24264i	−0.100083	+	0.173350i	0.811635	+	0.584164i	$0.198578\pi$
$600$		0			0
$601$	5.00000	+	8.66025i	0.203954	+	0.353259i	0.949799	−	0.312861i	$-0.101287\pi$
$601$	5.00000	+	8.66025i	0.203954	+	0.353259i	−0.745845	+	0.666120i	$0.767954\pi$
$602$	19.5959	−	33.9411i	0.798670	−	1.38334i
$603$		0			0
$604$	6.00000	−	3.46410i	0.244137	−	0.140952i
$605$	3.67423	+	6.36396i	0.149379	+	0.258732i
$606$		0			0
$607$	−39.0000	−	22.5167i	−1.58296	−	0.913923i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−39.0000	−	22.5167i	−1.58296	−	0.913923i	−0.588537	−	0.808470i	$-0.700296\pi$
$608$			22.6274i			0.917663i
$609$		0			0
$610$	48.0000			1.94346
$611$		−	16.9706i		−	0.686555i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			5.65685i			0.228292i
$615$		0			0
$616$	24.0000	+	13.8564i	0.966988	+	0.558291i
$617$	20.8207	+	12.0208i	0.838208	+	0.483940i	0.856655	−	0.515890i	$-0.172539\pi$
$617$	20.8207	+	12.0208i	0.838208	+	0.483940i	−0.0184465	+	0.999830i	$0.505872\pi$
$618$		0			0
$619$	−4.00000	−	6.92820i	−0.160774	−	0.278468i	0.774373	−	0.632730i	$-0.218066\pi$
$619$	−4.00000	−	6.92820i	−0.160774	−	0.278468i	−0.935146	+	0.354262i	$0.884732\pi$
$620$	14.6969	−	8.48528i	0.590243	−	0.340777i
$621$		0			0
$622$	−12.0000	−	6.92820i	−0.481156	−	0.277796i
$623$	12.2474	+	21.2132i	0.490684	+	0.849889i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	12.2474	−	7.07107i	0.489506	−	0.282617i
$627$		0			0
$628$	24.0000	+	13.8564i	0.957704	+	0.552931i
$629$		0			0
$630$		0			0
$631$			31.1769i			1.24113i	0.784154	+	0.620567i	$0.213097\pi$
$631$			31.1769i			1.24113i	−0.784154	+	0.620567i	$0.786903\pi$
$632$	−4.89898	−	8.48528i	−0.194871	−	0.337526i
$633$		0			0
$634$	15.0000	−	8.66025i	0.595726	−	0.343943i
$635$	−22.0454	−	12.7279i	−0.874845	−	0.505092i
$636$		0			0
$637$	−15.0000	+	8.66025i	−0.594322	+	0.343132i
$638$	4.89898	−	8.48528i	0.193952	−	0.335936i
$639$		0			0
$640$	24.0000	+	13.8564i	0.948683	+	0.547723i
$641$	−13.4722	+	7.77817i	−0.532120	+	0.307219i	−0.741879	−	0.670534i	$-0.766065\pi$
$641$	−13.4722	+	7.77817i	−0.532120	+	0.307219i	0.209760	+	0.977753i	$0.432732\pi$
$642$		0			0
$643$	−16.0000	+	27.7128i	−0.630978	+	1.09289i	0.356374	+	0.934344i	$0.384013\pi$
$643$	−16.0000	+	27.7128i	−0.630978	+	1.09289i	−0.987352	+	0.158543i	$0.949320\pi$
$644$			33.9411i			1.33747i
$645$		0			0
$646$	−8.00000			−0.314756
$647$	14.6969			0.577796			0.288898	−	0.957360i	$-0.406711\pi$
$647$	14.6969			0.577796			0.288898	+	0.957360i	$0.406711\pi$
$648$		0			0
$649$	−32.0000			−1.25611
$650$	4.89898			0.192154
$651$		0			0
$652$	32.0000			1.25322
$653$	8.57321	−	14.8492i	0.335496	−	0.581096i	−0.648084	−	0.761569i	$-0.724430\pi$
$653$	8.57321	−	14.8492i	0.335496	−	0.581096i	0.983580	+	0.180473i	$0.0577628\pi$
$654$		0			0
$655$	−12.0000	+	6.92820i	−0.468879	+	0.270707i
$656$	−4.89898	−	2.82843i	−0.191273	−	0.110432i
$657$		0			0
$658$	−12.0000	+	20.7846i	−0.467809	+	0.810268i
$659$	−9.79796	+	5.65685i	−0.381674	+	0.220360i	−0.678546	−	0.734557i	$-0.737390\pi$
$659$	−9.79796	+	5.65685i	−0.381674	+	0.220360i	0.296872	+	0.954917i	$0.404056\pi$
$660$		0			0
$661$	−12.0000	−	6.92820i	−0.466746	−	0.269476i	0.248131	−	0.968727i	$-0.420184\pi$
$661$	−12.0000	−	6.92820i	−0.466746	−	0.269476i	−0.714877	+	0.699251i	$0.753517\pi$
$662$	4.89898	−	2.82843i	0.190404	−	0.109930i
$663$		0			0
$664$	20.0000	+	34.6410i	0.776151	+	1.34433i
$665$		−	33.9411i		−	1.31618i
$666$		0			0
$667$	12.0000			0.464642
$668$	19.5959	−	33.9411i	0.758189	−	1.31322i
$669$		0			0
$670$	−12.0000	+	6.92820i	−0.463600	+	0.267660i
$671$	19.5959	−	33.9411i	0.756492	−	1.31028i
$672$		0			0
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	0.698988	−	0.715134i	$-0.253634\pi$
$673$	−7.00000	−	12.1244i	−0.269830	−	0.467360i	−0.968818	+	0.247774i	$0.920301\pi$
$674$	−4.89898	−	2.82843i	−0.188702	−	0.108947i
$675$		0			0
$676$	1.00000	+	1.73205i	0.0384615	+	0.0666173i
$677$	−8.57321	−	14.8492i	−0.329495	−	0.570703i	0.652916	−	0.757430i	$-0.273545\pi$
$677$	−8.57321	−	14.8492i	−0.329495	−	0.570703i	−0.982412	+	0.186727i	$0.940212\pi$
$678$		0			0
$679$	24.0000	+	13.8564i	0.921035	+	0.531760i
$680$	−4.89898	+	8.48528i	−0.187867	+	0.325396i
$681$		0			0
$682$		−	13.8564i		−	0.530589i
$683$		−	36.7696i		−	1.40695i	−0.710721	−	0.703474i	$-0.751631\pi$
$683$		−	36.7696i		−	1.40695i	0.710721	−	0.703474i	$-0.248369\pi$
$684$		0			0
$685$		−	24.2487i		−	0.926496i
$686$	−9.79796			−0.374088
$687$		0			0
$688$	−16.0000	−	27.7128i	−0.609994	−	1.05654i
$689$	−22.0454	−	12.7279i	−0.839863	−	0.484895i
$690$		0			0
$691$	−16.0000	−	27.7128i	−0.608669	−	1.05425i	−0.991460	−	0.130410i	$-0.958371\pi$
$691$	−16.0000	−	27.7128i	−0.608669	−	1.05425i	0.382791	−	0.923835i	$-0.374963\pi$
$692$	2.44949	+	4.24264i	0.0931156	+	0.161281i
$693$		0			0
$694$	−10.0000	+	17.3205i	−0.379595	+	0.657477i
$695$	−4.89898	−	8.48528i	−0.185829	−	0.321865i
$696$		0			0
$697$	1.00000	−	1.73205i	0.0378777	−	0.0656061i
$698$	−19.5959	−	33.9411i	−0.741716	−	1.28469i
$699$		0			0
$700$	−6.00000	−	3.46410i	−0.226779	−	0.130931i
$701$	−7.34847			−0.277548			−0.138774	−	0.990324i	$-0.544316\pi$
$701$	−7.34847			−0.277548			−0.138774	+	0.990324i	$0.544316\pi$
$702$		0			0
$703$		0			0
$704$	19.5959	−	11.3137i	0.738549	−	0.426401i
$705$		0			0
$706$	11.0000	+	19.0526i	0.413990	+	0.717053i
$707$	7.34847	+	4.24264i	0.276368	+	0.159561i
$708$		0			0
$709$	−3.00000	+	1.73205i	−0.112667	+	0.0650485i	−0.555275	−	0.831667i	$-0.687387\pi$
$709$	−3.00000	+	1.73205i	−0.112667	+	0.0650485i	0.442607	+	0.896716i	$0.354054\pi$
$710$	44.0908	+	25.4558i	1.65470	+	0.955341i
$711$		0			0
$712$	20.0000			0.749532
$713$	14.6969	−	8.48528i	0.550405	−	0.317776i
$714$		0			0
$715$	12.0000	−	20.7846i	0.448775	−	0.777300i
$716$		−	11.3137i		−	0.422813i
$717$		0			0
$718$		−	20.7846i		−	0.775675i
$719$	−44.0908			−1.64431			−0.822155	−	0.569264i	$-0.807228\pi$
$719$	−44.0908			−1.64431			−0.822155	+	0.569264i	$0.807228\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$			4.24264i			0.157895i
$723$		0			0
$724$			20.7846i			0.772454i
$725$	−1.22474	+	2.12132i	−0.0454859	+	0.0787839i
$726$		0			0
$727$	−21.0000	+	12.1244i	−0.778847	+	0.449667i	−0.836021	−	0.548697i	$-0.815124\pi$
$727$	−21.0000	+	12.1244i	−0.778847	+	0.449667i	0.0571746	+	0.998364i	$0.481791\pi$
$728$			33.9411i			1.25794i
$729$		0			0
$730$	12.0000	+	6.92820i	0.444140	+	0.256424i
$731$	9.79796	−	5.65685i	0.362391	−	0.209226i
$732$		0			0
$733$	33.0000	+	19.0526i	1.21888	+	0.703722i	0.964679	−	0.263428i	$-0.0848533\pi$
$733$	33.0000	+	19.0526i	1.21888	+	0.703722i	0.254204	+	0.967151i	$0.418187\pi$
$734$	17.1464	+	29.6985i	0.632886	+	1.09619i
$735$		0			0
$736$	24.0000	+	13.8564i	0.884652	+	0.510754i
$737$			11.3137i			0.416746i
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$		0			0
$741$		0			0
$742$	18.0000	+	31.1769i	0.660801	+	1.14454i
$743$	19.5959	−	33.9411i	0.718905	−	1.24518i	−0.242530	−	0.970144i	$-0.577977\pi$
$743$	19.5959	−	33.9411i	0.718905	−	1.24518i	0.961434	−	0.275035i	$-0.0886895\pi$
$744$		0			0
$745$	−21.0000	−	36.3731i	−0.769380	−	1.33261i
$746$	−9.79796	+	16.9706i	−0.358729	+	0.621336i
$747$		0			0
$748$	4.00000	+	6.92820i	0.146254	+	0.253320i
$749$	19.5959	+	33.9411i	0.716019	+	1.24018i
$750$		0			0
$751$	−3.00000	−	1.73205i	−0.109472	−	0.0632034i	0.444265	−	0.895896i	$-0.353465\pi$
$751$	−3.00000	−	1.73205i	−0.109472	−	0.0632034i	−0.553736	+	0.832692i	$0.686798\pi$
$752$	9.79796	+	16.9706i	0.357295	+	0.618853i
$753$		0			0
$754$	12.0000			0.437014
$755$			8.48528i			0.308811i
$756$		0			0
$757$			10.3923i			0.377715i	0.982005	+	0.188857i	$0.0604784\pi$
$757$			10.3923i			0.377715i	−0.982005	+	0.188857i	$0.939522\pi$
$758$			5.65685i			0.205466i
$759$		0			0
$760$	−24.0000	−	13.8564i	−0.870572	−	0.502625i
$761$	−15.9217	−	9.19239i	−0.577161	−	0.333224i	0.182844	−	0.983142i	$-0.441470\pi$
$761$	−15.9217	−	9.19239i	−0.577161	−	0.333224i	−0.760004	+	0.649918i	$0.774803\pi$
$762$		0			0
$763$	−18.0000	−	31.1769i	−0.651644	−	1.12868i
$764$	−19.5959	−	33.9411i	−0.708955	−	1.22795i
$765$		0			0
$766$	−12.0000	−	6.92820i	−0.433578	−	0.250326i
$767$	−19.5959	−	33.9411i	−0.707568	−	1.22554i
$768$		0			0
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	0.847432	+	0.530904i	$0.178148\pi$
$770$	−29.3939	+	16.9706i	−1.05928	+	0.611577i
$771$		0			0
$772$	14.0000	−	24.2487i	0.503871	−	0.872730i
$773$	−22.0454			−0.792918			−0.396459	−	0.918052i	$-0.629761\pi$
$773$	−22.0454			−0.792918			−0.396459	+	0.918052i	$0.629761\pi$
$774$		0			0
$775$			3.46410i			0.124434i
$776$	19.5959	−	11.3137i	0.703452	−	0.406138i
$777$		0			0
$778$	33.0000	−	19.0526i	1.18311	−	0.683067i
$779$	4.89898	+	2.82843i	0.175524	+	0.101339i
$780$		0			0
$781$	36.0000	−	20.7846i	1.28818	−	0.743732i
$782$	−4.89898	+	8.48528i	−0.175187	+	0.303433i
$783$		0			0
$784$	10.0000	−	17.3205i	0.357143	−	0.618590i
$785$	−29.3939	+	16.9706i	−1.04911	+	0.605705i
$786$		0			0
$787$	−10.0000	+	17.3205i	−0.356462	+	0.617409i	−0.987367	−	0.158450i	$-0.949350\pi$
$787$	−10.0000	+	17.3205i	−0.356462	+	0.617409i	0.630905	+	0.775860i	$0.282684\pi$
$788$	14.6969			0.523557
$789$		0			0
$790$	12.0000			0.426941
$791$	63.6867			2.26444
$792$		0			0
$793$	48.0000			1.70453
$794$		0			0
$795$		0			0
$796$			20.7846i			0.736691i
$797$	−6.12372	+	10.6066i	−0.216913	+	0.375705i	−0.953863	−	0.300243i	$-0.902932\pi$
$797$	−6.12372	+	10.6066i	−0.216913	+	0.375705i	0.736949	+	0.675948i	$0.236266\pi$
$798$		0			0
$799$	−6.00000	+	3.46410i	−0.212265	+	0.122551i
$800$	−4.89898	+	2.82843i	−0.173205	+	0.100000i
$801$		0			0
$802$	−1.00000	+	1.73205i	−0.0353112	+	0.0611608i
$803$	9.79796	−	5.65685i	0.345762	−	0.199626i
$804$		0			0
$805$	−36.0000	−	20.7846i	−1.26883	−	0.732561i
$806$	14.6969	−	8.48528i	0.517678	−	0.298881i
$807$		0			0
$808$	6.00000	−	3.46410i	0.211079	−	0.121867i
$809$			1.41421i			0.0497211i	0.999691	+	0.0248606i	$0.00791417\pi$
$809$			1.41421i			0.0497211i	−0.999691	+	0.0248606i	$0.992086\pi$
$810$		0			0
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$	−14.6969	−	8.48528i	−0.515761	−	0.297775i
$813$		0			0
$814$		0			0
$815$	−19.5959	+	33.9411i	−0.686415	+	1.18891i
$816$		0			0
$817$	16.0000	+	27.7128i	0.559769	+	0.969549i
$818$	39.1918	+	22.6274i	1.37031	+	0.791149i
$819$		0			0
$820$	6.00000	−	3.46410i	0.209529	−	0.120972i
$821$	−1.22474	−	2.12132i	−0.0427439	−	0.0740346i	0.843862	−	0.536560i	$-0.180277\pi$
$821$	−1.22474	−	2.12132i	−0.0427439	−	0.0740346i	−0.886606	+	0.462526i	$0.846943\pi$
$822$		0			0
$823$	15.0000	+	8.66025i	0.522867	+	0.301877i	0.738107	−	0.674684i	$-0.235720\pi$
$823$	15.0000	+	8.66025i	0.522867	+	0.301877i	−0.215240	+	0.976561i	$0.569053\pi$
$824$	4.89898	−	8.48528i	0.170664	−	0.295599i
$825$		0			0
$826$			55.4256i			1.92850i
$827$			5.65685i			0.196708i	0.995151	+	0.0983540i	$0.0313578\pi$
$827$			5.65685i			0.196708i	−0.995151	+	0.0983540i	$0.968642\pi$
$828$		0			0
$829$			10.3923i			0.360940i	0.983581	+	0.180470i	$0.0577618\pi$
$829$			10.3923i			0.360940i	−0.983581	+	0.180470i	$0.942238\pi$
$830$	−48.9898			−1.70046
$831$		0			0
$832$	24.0000	+	13.8564i	0.832050	+	0.480384i
$833$	6.12372	+	3.53553i	0.212174	+	0.122499i
$834$		0			0
$835$	24.0000	+	41.5692i	0.830554	+	1.43856i
$836$	−19.5959	+	11.3137i	−0.677739	+	0.391293i
$837$		0			0
$838$	14.0000	−	24.2487i	0.483622	−	0.837658i
$839$	2.44949	+	4.24264i	0.0845658	+	0.146472i	0.905206	−	0.424973i	$-0.139716\pi$
$839$	2.44949	+	4.24264i	0.0845658	+	0.146472i	−0.820640	+	0.571445i	$0.806383\pi$
$840$		0			0
$841$	11.5000	−	19.9186i	0.396552	−	0.686848i
$842$	17.1464	+	29.6985i	0.590905	+	1.02348i
$843$		0			0
$844$	20.0000	−	34.6410i	0.688428	−	1.19239i
$845$	−2.44949			−0.0842650
$846$		0			0
$847$		−	10.3923i		−	0.357084i
$848$	29.3939			1.00939
$849$		0			0
$850$	−1.00000	−	1.73205i	−0.0342997	−	0.0594089i
$851$		0			0
$852$		0			0
$853$	−12.0000	+	6.92820i	−0.410872	+	0.237217i	−0.691164	−	0.722698i	$-0.742902\pi$
$853$	−12.0000	+	6.92820i	−0.410872	+	0.237217i	0.280292	+	0.959915i	$0.409569\pi$
$854$	−58.7878	−	33.9411i	−2.01168	−	1.16144i
$855$		0			0
$856$	32.0000			1.09374
$857$	1.22474	−	0.707107i	0.0418365	−	0.0241543i	−0.478936	−	0.877850i	$-0.658977\pi$
$857$	1.22474	−	0.707107i	0.0418365	−	0.0241543i	0.520772	+	0.853696i	$0.325644\pi$
$858$		0			0
$859$	20.0000	−	34.6410i	0.682391	−	1.18194i	−0.291858	−	0.956462i	$-0.594273\pi$
$859$	20.0000	−	34.6410i	0.682391	−	1.18194i	0.974249	−	0.225475i	$-0.0723932\pi$
$860$	39.1918			1.33643
$861$		0			0
$862$		−	20.7846i		−	0.707927i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−6.00000			−0.204006
$866$			31.1127i			1.05725i
$867$		0			0
$868$	−24.0000			−0.814613
$869$	4.89898	−	8.48528i	0.166186	−	0.287843i
$870$		0			0
$871$	−12.0000	+	6.92820i	−0.406604	+	0.234753i
$872$	−29.3939			−0.995402
$873$		0			0
$874$	−24.0000	−	13.8564i	−0.811812	−	0.468700i
$875$	−29.3939	+	16.9706i	−0.993694	+	0.573710i
$876$		0			0
$877$	−48.0000	−	27.7128i	−1.62084	−	0.935795i	−0.986694	−	0.162585i	$-0.948017\pi$
$877$	−48.0000	−	27.7128i	−1.62084	−	0.935795i	−0.634150	−	0.773210i	$-0.718650\pi$
$878$	2.44949	+	4.24264i	0.0826663	+	0.143182i
$879$		0			0
$880$			27.7128i			0.934199i
$881$			43.8406i			1.47703i	0.674238	+	0.738514i	$0.264472\pi$
$881$			43.8406i			1.47703i	−0.674238	+	0.738514i	$0.735528\pi$
$882$		0			0
$883$	−40.0000			−1.34611			−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000			−1.34611			−0.673054	+	0.739594i	$0.735018\pi$
$884$	−4.89898	+	8.48528i	−0.164771	+	0.285391i
$885$		0			0
$886$	2.00000	+	3.46410i	0.0671913	+	0.116379i
$887$	4.89898	−	8.48528i	0.164492	−	0.284908i	−0.771983	−	0.635643i	$-0.780735\pi$
$887$	4.89898	−	8.48528i	0.164492	−	0.284908i	0.936475	+	0.350735i	$0.114068\pi$
$888$		0			0
$889$	18.0000	+	31.1769i	0.603701	+	1.04564i
$890$	−12.2474	+	21.2132i	−0.410535	+	0.711068i
$891$		0			0
$892$	−30.0000	+	17.3205i	−1.00447	+	0.579934i
$893$	−9.79796	−	16.9706i	−0.327876	−	0.567898i
$894$		0			0
$895$	12.0000	+	6.92820i	0.401116	+	0.231584i
$896$	−19.5959	−	33.9411i	−0.654654	−	1.13389i
$897$		0			0
$898$	−34.0000			−1.13459
$899$			8.48528i			0.283000i
$900$		0			0
$901$			10.3923i			0.346218i
$902$		−	5.65685i		−	0.188353i
$903$		0			0
$904$	26.0000	−	45.0333i	0.864747	−	1.49779i
$905$	−22.0454	−	12.7279i	−0.732814	−	0.423090i
$906$		0			0
$907$	−4.00000	−	6.92820i	−0.132818	−	0.230047i	0.791944	−	0.610594i	$-0.209069\pi$
$907$	−4.00000	−	6.92820i	−0.132818	−	0.230047i	−0.924762	+	0.380547i	$0.875736\pi$
$908$	4.89898	−	2.82843i	0.162578	−	0.0938647i
$909$		0			0
$910$	−36.0000	−	20.7846i	−1.19339	−	0.689003i
$911$	−19.5959	−	33.9411i	−0.649242	−	1.12452i	−0.983304	−	0.181969i	$-0.941753\pi$
$911$	−19.5959	−	33.9411i	−0.649242	−	1.12452i	0.334063	−	0.942551i	$-0.391580\pi$
$912$		0			0
$913$	−20.0000	+	34.6410i	−0.661903	+	1.14645i
$914$	−9.79796	+	5.65685i	−0.324088	+	0.187112i
$915$		0			0
$916$	−30.0000	−	17.3205i	−0.991228	−	0.572286i
$917$	19.5959			0.647114
$918$		0			0
$919$			10.3923i			0.342811i	0.985201	+	0.171405i	$0.0548307\pi$
$919$			10.3923i			0.342811i	−0.985201	+	0.171405i	$0.945169\pi$
$920$	−29.3939	+	16.9706i	−0.969087	+	0.559503i
$921$		0			0
$922$	−21.0000	+	12.1244i	−0.691598	+	0.399294i
$923$	44.0908	+	25.4558i	1.45127	+	0.837889i
$924$		0			0
$925$		0			0
$926$	12.2474	−	21.2132i	0.402476	−	0.697109i
$927$		0			0
$928$	−12.0000	+	6.92820i	−0.393919	+	0.227429i
$929$	1.22474	−	0.707107i	0.0401826	−	0.0231994i	−0.479774	−	0.877392i	$-0.659281\pi$
$929$	1.22474	−	0.707107i	0.0401826	−	0.0231994i	0.519957	+	0.854193i	$0.325948\pi$
$930$		0			0
$931$	−10.0000	+	17.3205i	−0.327737	+	0.567657i
$932$		−	2.82843i		−	0.0926482i
$933$		0			0
$934$	−52.0000			−1.70149
$935$	−9.79796			−0.320428
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$	19.5959			0.639829
$939$		0			0
$940$	−24.0000			−0.782794
$941$	1.22474	−	2.12132i	0.0399255	−	0.0691531i	−0.845372	−	0.534178i	$-0.820621\pi$
$941$	1.22474	−	2.12132i	0.0399255	−	0.0691531i	0.885298	+	0.465025i	$0.153955\pi$
$942$		0			0
$943$	6.00000	−	3.46410i	0.195387	−	0.112807i
$944$	39.1918	+	22.6274i	1.27559	+	0.736460i
$945$		0			0
$946$	16.0000	−	27.7128i	0.520205	−	0.901021i
$947$	19.5959	−	11.3137i	0.636782	−	0.367646i	−0.146592	−	0.989197i	$-0.546830\pi$
$947$	19.5959	−	11.3137i	0.636782	−	0.367646i	0.783374	+	0.621551i	$0.213497\pi$
$948$		0			0
$949$	12.0000	+	6.92820i	0.389536	+	0.224899i
$950$	4.89898	−	2.82843i	0.158944	−	0.0917663i
$951$		0			0
$952$	12.0000	−	6.92820i	0.388922	−	0.224544i
$953$			26.8701i			0.870407i	0.900332	+	0.435203i	$0.143323\pi$
$953$			26.8701i			0.870407i	−0.900332	+	0.435203i	$0.856677\pi$
$954$		0			0
$955$	48.0000			1.55324
$956$	−9.79796	+	16.9706i	−0.316889	+	0.548867i
$957$		0			0
$958$	−30.0000	+	17.3205i	−0.969256	+	0.559600i
$959$	−17.1464	+	29.6985i	−0.553687	+	0.959014i
$960$		0			0
$961$	−9.50000	−	16.4545i	−0.306452	−	0.530790i
$962$		0			0
$963$		0			0
$964$	−28.0000	−	48.4974i	−0.901819	−	1.56200i
$965$	17.1464	+	29.6985i	0.551963	+	0.956028i
$966$		0			0
$967$	−39.0000	−	22.5167i	−1.25416	−	0.724087i	−0.282223	−	0.959349i	$-0.591072\pi$
$967$	−39.0000	−	22.5167i	−1.25416	−	0.724087i	−0.971932	+	0.235262i	$0.924405\pi$
$968$	−7.34847	−	4.24264i	−0.236189	−	0.136364i
$969$		0			0
$970$			27.7128i			0.889805i
$971$			31.1127i			0.998454i	0.866471	+	0.499227i	$0.166383\pi$
$971$			31.1127i			0.998454i	−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$			13.8564i			0.444216i
$974$	−14.6969			−0.470920
$975$		0			0
$976$	−48.0000	+	27.7128i	−1.53644	+	0.887066i
$977$	20.8207	+	12.0208i	0.666112	+	0.384580i	0.794602	−	0.607131i	$-0.207680\pi$
$977$	20.8207	+	12.0208i	0.666112	+	0.384580i	−0.128490	+	0.991711i	$0.541013\pi$
$978$		0			0
$979$	10.0000	+	17.3205i	0.319601	+	0.553566i
$980$	12.2474	+	21.2132i	0.391230	+	0.677631i
$981$		0			0
$982$	−28.0000	+	48.4974i	−0.893516	+	1.54761i
$983$	24.4949	+	42.4264i	0.781266	+	1.35319i	0.931205	+	0.364497i	$0.118759\pi$
$983$	24.4949	+	42.4264i	0.781266	+	1.35319i	−0.149939	+	0.988695i	$0.547908\pi$
$984$		0			0
$985$	−9.00000	+	15.5885i	−0.286764	+	0.496690i
$986$	−2.44949	−	4.24264i	−0.0780076	−	0.135113i
$987$		0			0
$988$	−24.0000	−	13.8564i	−0.763542	−	0.440831i
$989$	39.1918			1.24623
$990$		0			0
$991$		−	51.9615i		−	1.65061i	−0.564686	−	0.825306i	$-0.691003\pi$
$991$		−	51.9615i		−	1.65061i	0.564686	−	0.825306i	$-0.308997\pi$
$992$	−9.79796	+	16.9706i	−0.311086	+	0.538816i
$993$		0			0
$994$	−36.0000	−	62.3538i	−1.14185	−	1.97774i
$995$	−22.0454	−	12.7279i	−0.698886	−	0.403502i
$996$		0			0
$997$	−12.0000	+	6.92820i	−0.380044	+	0.219418i	−0.677837	−	0.735212i	$-0.737083\pi$
$997$	−12.0000	+	6.92820i	−0.380044	+	0.219418i	0.297794	+	0.954630i	$0.403749\pi$
$998$	39.1918	+	22.6274i	1.24060	+	0.716258i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.l.a.539.1		4
3.2	odd	2	inner	648.2.l.a.539.2		4
4.3	odd	2		2592.2.p.c.2159.2		4
8.3	odd	2		648.2.l.c.539.2		4
8.5	even	2		2592.2.p.a.2159.1		4
9.2	odd	6		648.2.l.c.107.2		4
9.4	even	3		72.2.f.a.35.4	yes	4
9.5	odd	6		72.2.f.a.35.1	✓	4
9.7	even	3		648.2.l.c.107.1		4
12.11	even	2		2592.2.p.c.2159.1		4
24.5	odd	2		2592.2.p.a.2159.2		4
24.11	even	2		648.2.l.c.539.1		4
36.7	odd	6		2592.2.p.a.431.2		4
36.11	even	6		2592.2.p.a.431.1		4
36.23	even	6		288.2.f.a.143.4		4
36.31	odd	6		288.2.f.a.143.2		4
45.4	even	6		1800.2.b.c.251.1		4
45.13	odd	12		1800.2.m.c.899.5		8
45.14	odd	6		1800.2.b.c.251.4		4
45.22	odd	12		1800.2.m.c.899.4		8
45.23	even	12		1800.2.m.c.899.3		8
45.32	even	12		1800.2.m.c.899.6		8
72.5	odd	6		288.2.f.a.143.1		4
72.11	even	6	inner	648.2.l.a.107.2		4
72.13	even	6		288.2.f.a.143.3		4
72.29	odd	6		2592.2.p.c.431.2		4
72.43	odd	6	inner	648.2.l.a.107.1		4
72.59	even	6		72.2.f.a.35.3	yes	4
72.61	even	6		2592.2.p.c.431.1		4
72.67	odd	6		72.2.f.a.35.2	yes	4
144.5	odd	12		2304.2.c.i.2303.8		8
144.13	even	12		2304.2.c.i.2303.7		8
144.59	even	12		2304.2.c.i.2303.6		8
144.67	odd	12		2304.2.c.i.2303.5		8
144.77	odd	12		2304.2.c.i.2303.3		8
144.85	even	12		2304.2.c.i.2303.4		8
144.131	even	12		2304.2.c.i.2303.1		8
144.139	odd	12		2304.2.c.i.2303.2		8
180.23	odd	12		7200.2.m.c.3599.5		8
180.59	even	6		7200.2.b.c.4751.1		4
180.67	even	12		7200.2.m.c.3599.3		8
180.103	even	12		7200.2.m.c.3599.8		8
180.139	odd	6		7200.2.b.c.4751.2		4
180.167	odd	12		7200.2.m.c.3599.2		8
360.13	odd	12		7200.2.m.c.3599.4		8
360.59	even	6		1800.2.b.c.251.2		4
360.67	even	12		1800.2.m.c.899.1		8
360.77	even	12		7200.2.m.c.3599.6		8
360.139	odd	6		1800.2.b.c.251.3		4
360.149	odd	6		7200.2.b.c.4751.3		4
360.157	odd	12		7200.2.m.c.3599.7		8
360.203	odd	12		1800.2.m.c.899.2		8
360.229	even	6		7200.2.b.c.4751.4		4
360.283	even	12		1800.2.m.c.899.8		8
360.293	even	12		7200.2.m.c.3599.1		8
360.347	odd	12		1800.2.m.c.899.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.f.a.35.1	✓	4	9.5	odd	6
72.2.f.a.35.2	yes	4	72.67	odd	6
72.2.f.a.35.3	yes	4	72.59	even	6
72.2.f.a.35.4	yes	4	9.4	even	3
288.2.f.a.143.1		4	72.5	odd	6
288.2.f.a.143.2		4	36.31	odd	6
288.2.f.a.143.3		4	72.13	even	6
288.2.f.a.143.4		4	36.23	even	6
648.2.l.a.107.1		4	72.43	odd	6	inner
648.2.l.a.107.2		4	72.11	even	6	inner
648.2.l.a.539.1		4	1.1	even	1	trivial
648.2.l.a.539.2		4	3.2	odd	2	inner
648.2.l.c.107.1		4	9.7	even	3
648.2.l.c.107.2		4	9.2	odd	6
648.2.l.c.539.1		4	24.11	even	2
648.2.l.c.539.2		4	8.3	odd	2
1800.2.b.c.251.1		4	45.4	even	6
1800.2.b.c.251.2		4	360.59	even	6
1800.2.b.c.251.3		4	360.139	odd	6
1800.2.b.c.251.4		4	45.14	odd	6
1800.2.m.c.899.1		8	360.67	even	12
1800.2.m.c.899.2		8	360.203	odd	12
1800.2.m.c.899.3		8	45.23	even	12
1800.2.m.c.899.4		8	45.22	odd	12
1800.2.m.c.899.5		8	45.13	odd	12
1800.2.m.c.899.6		8	45.32	even	12
1800.2.m.c.899.7		8	360.347	odd	12
1800.2.m.c.899.8		8	360.283	even	12
2304.2.c.i.2303.1		8	144.131	even	12
2304.2.c.i.2303.2		8	144.139	odd	12
2304.2.c.i.2303.3		8	144.77	odd	12
2304.2.c.i.2303.4		8	144.85	even	12
2304.2.c.i.2303.5		8	144.67	odd	12
2304.2.c.i.2303.6		8	144.59	even	12
2304.2.c.i.2303.7		8	144.13	even	12
2304.2.c.i.2303.8		8	144.5	odd	12
2592.2.p.a.431.1		4	36.11	even	6
2592.2.p.a.431.2		4	36.7	odd	6
2592.2.p.a.2159.1		4	8.5	even	2
2592.2.p.a.2159.2		4	24.5	odd	2
2592.2.p.c.431.1		4	72.61	even	6
2592.2.p.c.431.2		4	72.29	odd	6
2592.2.p.c.2159.1		4	12.11	even	2
2592.2.p.c.2159.2		4	4.3	odd	2
7200.2.b.c.4751.1		4	180.59	even	6
7200.2.b.c.4751.2		4	180.139	odd	6
7200.2.b.c.4751.3		4	360.149	odd	6
7200.2.b.c.4751.4		4	360.229	even	6
7200.2.m.c.3599.1		8	360.293	even	12
7200.2.m.c.3599.2		8	180.167	odd	12
7200.2.m.c.3599.3		8	180.67	even	12
7200.2.m.c.3599.4		8	360.13	odd	12
7200.2.m.c.3599.5		8	180.23	odd	12
7200.2.m.c.3599.6		8	360.77	even	12
7200.2.m.c.3599.7		8	360.157	odd	12
7200.2.m.c.3599.8		8	180.103	even	12

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.l (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +(1.22474 - 2.12132i) q^{5} +(-3.00000 + 1.73205i) q^{7} +2.82843i q^{8} +(-3.00000 - 1.73205i) q^{10} +(-2.44949 + 1.41421i) q^{11} +(-3.00000 - 1.73205i) q^{13} +(2.44949 + 4.24264i) q^{14} +4.00000 q^{16} +1.41421i q^{17} -4.00000 q^{19} +(-2.44949 + 4.24264i) q^{20} +(2.00000 + 3.46410i) q^{22} +(-2.44949 + 4.24264i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-2.44949 + 4.24264i) q^{26} +(6.00000 - 3.46410i) q^{28} +(-1.22474 - 2.12132i) q^{29} +(-3.00000 - 1.73205i) q^{31} -5.65685i q^{32} +2.00000 q^{34} +8.48528i q^{35} +5.65685i q^{38} +(6.00000 + 3.46410i) q^{40} +(-1.22474 - 0.707107i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(4.89898 - 2.82843i) q^{44} +(6.00000 + 3.46410i) q^{46} +(2.44949 + 4.24264i) q^{47} +(2.50000 - 4.33013i) q^{49} +(-1.22474 + 0.707107i) q^{50} +(6.00000 + 3.46410i) q^{52} +7.34847 q^{53} +6.92820i q^{55} +(-4.89898 - 8.48528i) q^{56} +(-3.00000 + 1.73205i) q^{58} +(9.79796 + 5.65685i) q^{59} +(-12.0000 + 6.92820i) q^{61} +(-2.44949 + 4.24264i) q^{62} -8.00000 q^{64} +(-7.34847 + 4.24264i) q^{65} +(2.00000 - 3.46410i) q^{67} -2.82843i q^{68} +12.0000 q^{70} -14.6969 q^{71} -4.00000 q^{73} +8.00000 q^{76} +(4.89898 - 8.48528i) q^{77} +(-3.00000 + 1.73205i) q^{79} +(4.89898 - 8.48528i) q^{80} +(-1.00000 + 1.73205i) q^{82} +(12.2474 - 7.07107i) q^{83} +(3.00000 + 1.73205i) q^{85} +(-9.79796 + 5.65685i) q^{86} +(-4.00000 - 6.92820i) q^{88} -7.07107i q^{89} +12.0000 q^{91} +(4.89898 - 8.48528i) q^{92} +(6.00000 - 3.46410i) q^{94} +(-4.89898 + 8.48528i) q^{95} +(-4.00000 - 6.92820i) q^{97} +(-6.12372 - 3.53553i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 12 q^{7} - 12 q^{10} - 12 q^{13} + 16 q^{16} - 16 q^{19} + 8 q^{22} - 2 q^{25} + 24 q^{28} - 12 q^{31} + 8 q^{34} + 24 q^{40} - 16 q^{43} + 24 q^{46} + 10 q^{49} + 24 q^{52} - 12 q^{58}+ \cdots - 16 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 12 * q^7 - 12 * q^10 - 12 * q^13 + 16 * q^16 - 16 * q^19 + 8 * q^22 - 2 * q^25 + 24 * q^28 - 12 * q^31 + 8 * q^34 + 24 * q^40 - 16 * q^43 + 24 * q^46 + 10 * q^49 + 24 * q^52 - 12 * q^58 - 48 * q^61 - 32 * q^64 + 8 * q^67 + 48 * q^70 - 16 * q^73 + 32 * q^76 - 12 * q^79 - 4 * q^82 + 12 * q^85 - 16 * q^88 + 48 * q^91 + 24 * q^94 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more