LMFDB - Embedded newform 2592.2.s.h.1727.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(863,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2592.863"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1727.1
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2592.1727
Dual form			2592.2.s.h.863.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.275255 - 0.158919i) q^{5} +(-1.25529 - 0.724745i) q^{7} +(-0.548188 + 0.949490i) q^{11} +(-1.44949 - 2.51059i) q^{13} -3.46410i q^{17} +4.89898i q^{19} +(1.41421 + 2.44949i) q^{23} +(-2.44949 + 4.24264i) q^{25} +(-7.89898 - 4.56048i) q^{29} +(6.45145 - 3.72474i) q^{31} -0.460702 q^{35} +4.89898 q^{37} +(-8.44949 + 4.87832i) q^{41} +(-5.97469 - 3.44949i) q^{43} +(-4.56048 + 7.89898i) q^{47} +(-2.44949 - 4.24264i) q^{49} +4.41761i q^{53} +0.348469i q^{55} +(4.56048 + 7.89898i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(-0.797959 - 0.460702i) q^{65} +(-4.41761 + 2.55051i) q^{67} -7.56388 q^{71} +1.89898 q^{73} +(1.37628 - 0.794593i) q^{77} +(1.73205 + 1.00000i) q^{79} +(-7.93709 + 13.7474i) q^{83} +(-0.550510 - 0.953512i) q^{85} -11.9494i q^{89} +4.20204i q^{91} +(0.778539 + 1.34847i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{5} + 8 q^{13} - 24 q^{29} - 48 q^{41} - 16 q^{61} + 72 q^{65} - 24 q^{73} + 60 q^{77} - 24 q^{85} + 20 q^{97}+O(q^{100}) $ 8 * q + 12 * q^5 + 8 * q^13 - 24 * q^29 - 48 * q^41 - 16 * q^61 + 72 * q^65 - 24 * q^73 + 60 * q^77 - 24 * q^85 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.275255	−	0.158919i	0.123098	−	0.0710706i	−0.437187	−	0.899371i	$-0.644025\pi$
$5$	0.275255	−	0.158919i	0.123098	−	0.0710706i	0.560285	+	0.828300i	$0.310692\pi$
$6$		0			0
$7$	−1.25529	−	0.724745i	−0.474457	−	0.273928i	0.243647	−	0.969864i	$-0.421656\pi$
$7$	−1.25529	−	0.724745i	−0.474457	−	0.273928i	−0.718104	+	0.695936i	$0.754990\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.548188	+	0.949490i	−0.165285	+	0.286282i	−0.936756	−	0.349982i	$-0.886188\pi$
$11$	−0.548188	+	0.949490i	−0.165285	+	0.286282i	0.771471	+	0.636264i	$0.219521\pi$
$12$		0			0
$13$	−1.44949	−	2.51059i	−0.402016	−	0.696312i	0.591953	−	0.805972i	$-0.298357\pi$
$13$	−1.44949	−	2.51059i	−0.402016	−	0.696312i	−0.993969	+	0.109660i	$0.965024\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		0			0
$19$			4.89898i			1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$			4.89898i			1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.41421	+	2.44949i	0.294884	+	0.510754i	0.974958	−	0.222390i	$-0.0713857\pi$
$23$	1.41421	+	2.44949i	0.294884	+	0.510754i	−0.680074	+	0.733144i	$0.738052\pi$
$24$		0			0
$25$	−2.44949	+	4.24264i	−0.489898	+	0.848528i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.89898	−	4.56048i	−1.46680	−	0.846859i	−0.467493	−	0.883997i	$-0.654843\pi$
$29$	−7.89898	−	4.56048i	−1.46680	−	0.846859i	−0.999310	+	0.0371370i	$0.988176\pi$
$30$		0			0
$31$	6.45145	−	3.72474i	1.15871	−	0.668984i	0.207719	−	0.978189i	$-0.433396\pi$
$31$	6.45145	−	3.72474i	1.15871	−	0.668984i	0.950996	+	0.309205i	$0.100063\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.460702			−0.0778728
$36$		0			0
$37$	4.89898			0.805387			0.402694	−	0.915335i	$-0.368074\pi$
$37$	4.89898			0.805387			0.402694	+	0.915335i	$0.368074\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.44949	+	4.87832i	−1.31959	+	0.761865i	−0.983662	−	0.180023i	$-0.942383\pi$
$41$	−8.44949	+	4.87832i	−1.31959	+	0.761865i	−0.335926	+	0.941888i	$0.609049\pi$
$42$		0			0
$43$	−5.97469	−	3.44949i	−0.911132	−	0.526042i	−0.0303367	−	0.999540i	$-0.509658\pi$
$43$	−5.97469	−	3.44949i	−0.911132	−	0.526042i	−0.880795	+	0.473497i	$0.842991\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.56048	+	7.89898i	−0.665214	+	1.15218i	0.314013	+	0.949419i	$0.398326\pi$
$47$	−4.56048	+	7.89898i	−0.665214	+	1.15218i	−0.979227	+	0.202766i	$0.935007\pi$
$48$		0			0
$49$	−2.44949	−	4.24264i	−0.349927	−	0.606092i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.41761i			0.606806i	0.952862	+	0.303403i	$0.0981228\pi$
$53$			4.41761i			0.606806i	−0.952862	+	0.303403i	$0.901877\pi$
$54$		0			0
$55$			0.348469i			0.0469876i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.56048	+	7.89898i	0.593724	+	1.02836i	0.993726	+	0.111845i	$0.0356761\pi$
$59$	4.56048	+	7.89898i	0.593724	+	1.02836i	−0.400002	+	0.916514i	$0.630991\pi$
$60$		0			0
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	−0.965187	−	0.261562i	$-0.915762\pi$
$61$	−2.00000	+	3.46410i	−0.256074	+	0.443533i	0.709113	+	0.705095i	$0.249096\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.797959	−	0.460702i	−0.0989746	−	0.0571430i
$66$		0			0
$67$	−4.41761	+	2.55051i	−0.539697	+	0.311594i	−0.744956	−	0.667113i	$-0.767530\pi$
$67$	−4.41761	+	2.55051i	−0.539697	+	0.311594i	0.205259	+	0.978708i	$0.434196\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.56388			−0.897667			−0.448834	−	0.893615i	$-0.648160\pi$
$71$	−7.56388			−0.897667			−0.448834	+	0.893615i	$0.648160\pi$
$72$		0			0
$73$	1.89898			0.222259			0.111129	−	0.993806i	$-0.464553\pi$
$73$	1.89898			0.222259			0.111129	+	0.993806i	$0.464553\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.37628	−	0.794593i	0.156841	−	0.0905523i
$78$		0			0
$79$	1.73205	+	1.00000i	0.194871	+	0.112509i	0.594261	−	0.804272i	$-0.297445\pi$
$79$	1.73205	+	1.00000i	0.194871	+	0.112509i	−0.399390	+	0.916781i	$0.630778\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.93709	+	13.7474i	−0.871209	+	1.50898i	−0.0104623	+	0.999945i	$0.503330\pi$
$83$	−7.93709	+	13.7474i	−0.871209	+	1.50898i	−0.860747	+	0.509033i	$0.830003\pi$
$84$		0			0
$85$	−0.550510	−	0.953512i	−0.0597112	−	0.103423i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	11.9494i		−	1.26663i	−0.773893	−	0.633316i	$-0.781693\pi$
$89$		−	11.9494i		−	1.26663i	0.773893	−	0.633316i	$-0.218307\pi$
$90$		0			0
$91$			4.20204i			0.440494i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.778539	+	1.34847i	0.0798764	+	0.138350i
$96$		0			0
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	0.964579	+	0.263795i	$0.0849741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.6237	−	7.86566i	−1.35561	−	0.782663i	−0.366582	−	0.930386i	$-0.619472\pi$
$101$	−13.6237	−	7.86566i	−1.35561	−	0.782663i	−0.989029	+	0.147723i	$0.952806\pi$
$102$		0			0
$103$	8.66025	−	5.00000i	0.853320	−	0.492665i	−0.00844953	−	0.999964i	$-0.502690\pi$
$103$	8.66025	−	5.00000i	0.853320	−	0.492665i	0.861770	+	0.507300i	$0.169356\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.66025			−0.837218			−0.418609	−	0.908166i	$-0.637482\pi$
$107$	−8.66025			−0.837218			−0.418609	+	0.908166i	$0.637482\pi$
$108$		0			0
$109$	−4.89898			−0.469237			−0.234619	−	0.972088i	$-0.575384\pi$
$109$	−4.89898			−0.469237			−0.234619	+	0.972088i	$0.575384\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.2474	+	7.07107i	−1.15214	+	0.665190i	−0.949409	−	0.314044i	$-0.898316\pi$
$113$	−12.2474	+	7.07107i	−1.15214	+	0.665190i	−0.202735	+	0.979234i	$0.564983\pi$
$114$		0			0
$115$	0.778539	+	0.449490i	0.0725991	+	0.0419151i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.51059	+	4.34847i	−0.230145	+	0.398624i
$120$		0			0
$121$	4.89898	+	8.48528i	0.445362	+	0.771389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.14626i			0.281410i
$126$		0			0
$127$		−	15.2474i		−	1.35299i	−0.736446	−	0.676496i	$-0.763498\pi$
$127$		−	15.2474i		−	1.35299i	0.736446	−	0.676496i	$-0.236502\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.50170	−	2.60102i	−0.131204	−	0.227252i	0.792937	−	0.609304i	$-0.208551\pi$
$131$	−1.50170	−	2.60102i	−0.131204	−	0.227252i	−0.924141	+	0.382052i	$0.875218\pi$
$132$		0			0
$133$	3.55051	−	6.14966i	0.307868	−	0.533244i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.24745	−	5.33902i	−0.790063	−	0.456143i	0.0499218	−	0.998753i	$-0.484103\pi$
$137$	−9.24745	−	5.33902i	−0.790063	−	0.456143i	−0.839985	+	0.542610i	$0.817436\pi$
$138$		0			0
$139$	−17.3205	+	10.0000i	−1.46911	+	0.848189i	−0.999400	−	0.0346338i	$-0.988974\pi$
$139$	−17.3205	+	10.0000i	−1.46911	+	0.848189i	−0.469706	+	0.882823i	$0.655640\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.17837			0.265789
$144$		0			0
$145$	−2.89898			−0.240747
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.27526	+	1.89097i	−0.268319	+	0.154914i	−0.628124	−	0.778114i	$-0.716177\pi$
$149$	−3.27526	+	1.89097i	−0.268319	+	0.154914i	0.359804	+	0.933028i	$0.382843\pi$
$150$		0			0
$151$	14.9367	+	8.62372i	1.21553	+	0.701789i	0.963959	−	0.266049i	$-0.0857184\pi$
$151$	14.9367	+	8.62372i	1.21553	+	0.701789i	0.251574	+	0.967838i	$0.419052\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.18386	−	2.05051i	0.0950901	−	0.164701i
$156$		0			0
$157$	6.89898	+	11.9494i	0.550599	+	0.953665i	0.998231	+	0.0594472i	$0.0189338\pi$
$157$	6.89898	+	11.9494i	0.550599	+	0.953665i	−0.447633	+	0.894217i	$0.647733\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	4.09978i		−	0.323108i
$162$		0			0
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.4887	+	19.8990i	0.889021	+	1.53983i	0.841035	+	0.540981i	$0.181947\pi$
$167$	11.4887	+	19.8990i	0.889021	+	1.53983i	0.0479862	+	0.998848i	$0.484720\pi$
$168$		0			0
$169$	2.29796	−	3.98018i	0.176766	−	0.306168i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.62372	−	2.66951i	−0.351535	−	0.202959i	0.313826	−	0.949481i	$-0.398389\pi$
$173$	−4.62372	−	2.66951i	−0.351535	−	0.202959i	−0.665361	+	0.746522i	$0.731722\pi$
$174$		0			0
$175$	6.14966	−	3.55051i	0.464871	−	0.268393i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.66025			0.647298			0.323649	−	0.946177i	$-0.395090\pi$
$179$	8.66025			0.647298			0.323649	+	0.946177i	$0.395090\pi$
$180$		0			0
$181$	−9.79796			−0.728277			−0.364138	−	0.931345i	$-0.618636\pi$
$181$	−9.79796			−0.728277			−0.364138	+	0.931345i	$0.618636\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.34847	−	0.778539i	0.0991414	−	0.0572393i
$186$		0			0
$187$	3.28913	+	1.89898i	0.240525	+	0.138867i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.80312	+	15.2474i	−0.636971	+	1.10327i	0.349123	+	0.937077i	$0.386480\pi$
$191$	−8.80312	+	15.2474i	−0.636971	+	1.10327i	−0.986094	+	0.166190i	$0.946854\pi$
$192$		0			0
$193$	−1.05051	−	1.81954i	−0.0756174	−	0.130973i	0.825737	−	0.564055i	$-0.190759\pi$
$193$	−1.05051	−	1.81954i	−0.0756174	−	0.130973i	−0.901355	+	0.433082i	$0.857426\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.2672i		−	0.874003i	−0.899461	−	0.437002i	$-0.856040\pi$
$197$		−	12.2672i		−	0.874003i	0.899461	−	0.437002i	$-0.143960\pi$
$198$		0			0
$199$		−	20.3485i		−	1.44246i	−0.692693	−	0.721232i	$-0.743576\pi$
$199$		−	20.3485i		−	1.44246i	0.692693	−	0.721232i	$-0.256424\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.61037	+	11.4495i	0.463957	+	0.803597i
$204$		0			0
$205$	−1.55051	+	2.68556i	−0.108292	+	0.187568i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.65153	−	2.68556i	−0.321753	−	0.185764i
$210$		0			0
$211$	−24.6773	+	14.2474i	−1.69886	+	0.980835i	−0.752007	+	0.659155i	$0.770914\pi$
$211$	−24.6773	+	14.2474i	−1.69886	+	0.980835i	−0.946848	+	0.321680i	$0.895752\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.19275			−0.149544
$216$		0			0
$217$	−10.7980			−0.733013
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.69694	+	5.02118i	−0.585019	+	0.337761i
$222$		0			0
$223$	0.174973	+	0.101021i	0.0117170	+	0.00676483i	0.505847	−	0.862623i	$-0.331180\pi$
$223$	0.174973	+	0.101021i	0.0117170	+	0.00676483i	−0.494130	+	0.869388i	$0.664513\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.83183	+	10.1010i	−0.387072	+	0.670428i	−0.992054	−	0.125811i	$-0.959847\pi$
$227$	−5.83183	+	10.1010i	−0.387072	+	0.670428i	0.604982	+	0.796239i	$0.293180\pi$
$228$		0			0
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	0.831092	−	0.556136i	$-0.187717\pi$
$229$	−1.00000	−	1.73205i	−0.0660819	−	0.114457i	−0.897173	+	0.441679i	$0.854383\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			13.2207i			0.866119i	0.901365	+	0.433059i	$0.142566\pi$
$233$			13.2207i			0.866119i	−0.901365	+	0.433059i	$0.857434\pi$
$234$		0			0
$235$			2.89898i			0.189109i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.5352	−	18.2474i	−0.681463	−	1.18033i	−0.974534	−	0.224239i	$-0.928011\pi$
$239$	−10.5352	−	18.2474i	−0.681463	−	1.18033i	0.293071	−	0.956091i	$-0.405323\pi$
$240$		0			0
$241$	9.89898	−	17.1455i	0.637649	−	1.10444i	−0.348298	−	0.937384i	$-0.613240\pi$
$241$	9.89898	−	17.1455i	0.637649	−	1.10444i	0.985947	−	0.167057i	$-0.0534264\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.34847	−	0.778539i	−0.0861505	−	0.0497390i
$246$		0			0
$247$	12.2993	−	7.10102i	0.782588	−	0.451827i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−17.3205			−1.09326			−0.546630	−	0.837374i	$-0.684090\pi$
$251$	−17.3205			−1.09326			−0.546630	+	0.837374i	$0.684090\pi$
$252$		0			0
$253$	−3.10102			−0.194959
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.1010	−	5.83183i	0.630084	−	0.363779i	−0.150700	−	0.988579i	$-0.548153\pi$
$257$	10.1010	−	5.83183i	0.630084	−	0.363779i	0.780785	+	0.624800i	$0.214820\pi$
$258$		0			0
$259$	−6.14966	−	3.55051i	−0.382122	−	0.220618i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.4422	−	21.5505i	0.767218	−	1.32886i	−0.171847	−	0.985124i	$-0.554974\pi$
$263$	12.4422	−	21.5505i	0.767218	−	1.32886i	0.939066	−	0.343738i	$-0.111693\pi$
$264$		0			0
$265$	0.702041	+	1.21597i	0.0431260	+	0.0746965i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	10.3923i		−	0.633630i	−0.948487	−	0.316815i	$-0.897387\pi$
$269$		−	10.3923i		−	0.633630i	0.948487	−	0.316815i	$-0.102613\pi$
$270$		0			0
$271$		−	12.5505i		−	0.762389i	−0.924495	−	0.381195i	$-0.875513\pi$
$271$		−	12.5505i		−	0.762389i	0.924495	−	0.381195i	$-0.124487\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.68556	−	4.65153i	−0.161946	−	0.280498i
$276$		0			0
$277$	6.89898	−	11.9494i	0.414520	−	0.717969i	−0.580858	−	0.814005i	$-0.697283\pi$
$277$	6.89898	−	11.9494i	0.414520	−	0.717969i	0.995378	+	0.0960358i	$0.0306163\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.797959	+	0.460702i	0.0476022	+	0.0274832i	0.523612	−	0.851957i	$-0.324584\pi$
$281$	0.797959	+	0.460702i	0.0476022	+	0.0274832i	−0.476010	+	0.879440i	$0.657917\pi$
$282$		0			0
$283$	−2.68556	+	1.55051i	−0.159640	+	0.0921683i	−0.577692	−	0.816255i	$-0.696046\pi$
$283$	−2.68556	+	1.55051i	−0.159640	+	0.0921683i	0.418052	+	0.908423i	$0.362713\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.1421			0.834784
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5.20204	+	3.00340i	−0.303906	+	0.175460i	−0.644196	−	0.764860i	$-0.722808\pi$
$293$	−5.20204	+	3.00340i	−0.303906	+	0.175460i	0.340290	+	0.940321i	$0.389475\pi$
$294$		0			0
$295$	2.51059	+	1.44949i	0.146172	+	0.0843926i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.09978	−	7.10102i	0.237096	−	0.410663i
$300$		0			0
$301$	5.00000	+	8.66025i	0.288195	+	0.499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.27135i			0.0727972i
$306$		0			0
$307$			18.0000i			1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$			18.0000i			1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.04989	−	3.55051i	−0.116238	−	0.201331i	0.802036	−	0.597276i	$-0.203750\pi$
$311$	−2.04989	−	3.55051i	−0.116238	−	0.201331i	−0.918274	+	0.395945i	$0.870417\pi$
$312$		0			0
$313$	−14.7474	+	25.5433i	−0.833575	+	1.44379i	0.0616102	+	0.998100i	$0.480376\pi$
$313$	−14.7474	+	25.5433i	−0.833575	+	1.44379i	−0.895185	+	0.445694i	$0.852957\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	25.6237	+	14.7939i	1.43917	+	0.830906i	0.997792	−	0.0664143i	$-0.0211559\pi$
$317$	25.6237	+	14.7939i	1.43917	+	0.830906i	0.441380	+	0.897320i	$0.354489\pi$
$318$		0			0
$319$	8.66025	−	5.00000i	0.484881	−	0.279946i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.9706			0.944267
$324$		0			0
$325$	14.2020			0.787787
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.4495	−	6.61037i	0.631231	−	0.364441i
$330$		0			0
$331$	−0.603566	−	0.348469i	−0.0331750	−	0.0191536i	0.483321	−	0.875443i	$-0.339430\pi$
$331$	−0.603566	−	0.348469i	−0.0331750	−	0.0191536i	−0.516496	+	0.856290i	$0.672764\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.810647	+	1.40408i	−0.0442904	+	0.0767132i
$336$		0			0
$337$	−2.10102	−	3.63907i	−0.114450	−	0.198233i	0.803110	−	0.595831i	$-0.203177\pi$
$337$	−2.10102	−	3.63907i	−0.114450	−	0.198233i	−0.917560	+	0.397598i	$0.869844\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			8.16744i			0.442292i
$342$		0			0
$343$			17.2474i			0.931275i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.74094	−	4.74745i	−0.147141	−	0.254856i	0.783028	−	0.621986i	$-0.213674\pi$
$347$	−2.74094	−	4.74745i	−0.147141	−	0.254856i	−0.930170	+	0.367130i	$0.880341\pi$
$348$		0			0
$349$	−12.4495	+	21.5631i	−0.666406	+	1.15425i	0.312496	+	0.949919i	$0.398835\pi$
$349$	−12.4495	+	21.5631i	−0.666406	+	1.15425i	−0.978902	+	0.204330i	$0.934498\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.2474	−	7.07107i	−0.651866	−	0.376355i	0.137305	−	0.990529i	$-0.456156\pi$
$353$	−12.2474	−	7.07107i	−0.651866	−	0.376355i	−0.789171	+	0.614174i	$0.789489\pi$
$354$		0			0
$355$	−2.08200	+	1.20204i	−0.110501	+	0.0637977i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−34.2911			−1.80981			−0.904907	−	0.425610i	$-0.860060\pi$
$359$	−34.2911			−1.80981			−0.904907	+	0.425610i	$0.860060\pi$
$360$		0			0
$361$	−5.00000			−0.263158
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.522704	−	0.301783i	0.0273596	−	0.0157961i
$366$		0			0
$367$	7.40496	+	4.27526i	0.386536	+	0.223167i	0.680658	−	0.732601i	$-0.261694\pi$
$367$	7.40496	+	4.27526i	0.386536	+	0.223167i	−0.294122	+	0.955768i	$0.595027\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.20164	−	5.54541i	0.166221	−	0.287903i
$372$		0			0
$373$	−17.2474	−	29.8735i	−0.893039	−	1.54679i	−0.836213	−	0.548405i	$-0.815235\pi$
$373$	−17.2474	−	29.8735i	−0.893039	−	1.54679i	−0.0568261	−	0.998384i	$-0.518098\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			26.4415i			1.36180i
$378$		0			0
$379$			8.00000i			0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$			8.00000i			0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.87492	+	3.24745i	0.0958037	+	0.165937i	0.909944	−	0.414732i	$-0.136125\pi$
$383$	1.87492	+	3.24745i	0.0958037	+	0.165937i	−0.814140	+	0.580669i	$0.802791\pi$
$384$		0			0
$385$	0.252551	−	0.437432i	0.0128712	−	0.0222936i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.7702	+	16.0331i	1.40800	+	0.812911i	0.995195	−	0.0979078i	$-0.0312150\pi$
$389$	27.7702	+	16.0331i	1.40800	+	0.812911i	0.412807	+	0.910818i	$0.364548\pi$
$390$		0			0
$391$	8.48528	−	4.89898i	0.429119	−	0.247752i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.635674			0.0319843
$396$		0			0
$397$	8.69694			0.436487			0.218243	−	0.975894i	$-0.429967\pi$
$397$	8.69694			0.436487			0.218243	+	0.975894i	$0.429967\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	19.5959	−	11.3137i	0.978573	−	0.564980i	0.0767343	−	0.997052i	$-0.475551\pi$
$401$	19.5959	−	11.3137i	0.978573	−	0.564980i	0.901839	+	0.432072i	$0.142217\pi$
$402$		0			0
$403$	−18.7026	−	10.7980i	−0.931644	−	0.537885i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.68556	+	4.65153i	−0.133118	+	0.230568i
$408$		0			0
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	0.939490	−	0.342578i	$-0.111300\pi$
$409$	3.50000	+	6.06218i	0.173064	+	0.299755i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	13.2207i		−	0.650550i
$414$		0			0
$415$			5.04541i			0.247669i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.0458	−	22.5959i	−0.637327	−	1.10388i	−0.986017	−	0.166645i	$-0.946707\pi$
$419$	−13.0458	−	22.5959i	−0.637327	−	1.10388i	0.348690	−	0.937238i	$-0.386627\pi$
$420$		0			0
$421$	14.8990	−	25.8058i	0.726132	−	1.25770i	−0.232375	−	0.972626i	$-0.574650\pi$
$421$	14.8990	−	25.8058i	0.726132	−	1.25770i	0.958506	−	0.285071i	$-0.0920171\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14.6969	+	8.48528i	0.712906	+	0.411597i
$426$		0			0
$427$	5.02118	−	2.89898i	0.242992	−	0.140291i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	27.3629			1.31802			0.659011	−	0.752133i	$-0.270975\pi$
$431$	27.3629			1.31802			0.659011	+	0.752133i	$0.270975\pi$
$432$		0			0
$433$	32.3939			1.55675			0.778375	−	0.627799i	$-0.216044\pi$
$433$	32.3939			1.55675			0.778375	+	0.627799i	$0.216044\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	+	6.92820i	−0.574038	+	0.331421i
$438$		0			0
$439$	−16.6688	−	9.62372i	−0.795557	−	0.459315i	0.0463579	−	0.998925i	$-0.485239\pi$
$439$	−16.6688	−	9.62372i	−0.795557	−	0.459315i	−0.841915	+	0.539610i	$0.818572\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.3171	−	24.7980i	0.680226	−	1.17819i	−0.294685	−	0.955594i	$-0.595215\pi$
$443$	14.3171	−	24.7980i	0.680226	−	1.17819i	0.974912	−	0.222592i	$-0.0714518\pi$
$444$		0			0
$445$	−1.89898	−	3.28913i	−0.0900203	−	0.155920i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			22.3417i			1.05437i	0.849751	+	0.527185i	$0.176752\pi$
$449$			22.3417i			1.05437i	−0.849751	+	0.527185i	$0.823248\pi$
$450$		0			0
$451$		−	10.6969i		−	0.503699i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.667783	+	1.15663i	0.0313061	+	0.0542238i
$456$		0			0
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	0.410184	+	0.912003i	$0.365464\pi$
$457$	−12.5000	+	21.6506i	−0.584725	+	1.01277i	−0.994910	+	0.100771i	$0.967869\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.62372	+	0.937458i	0.0756244	+	0.0436618i	0.537335	−	0.843369i	$-0.319431\pi$
$461$	1.62372	+	0.937458i	0.0756244	+	0.0436618i	−0.461711	+	0.887030i	$0.652764\pi$
$462$		0			0
$463$	23.4220	−	13.5227i	1.08851	−	0.628453i	0.155333	−	0.987862i	$-0.450355\pi$
$463$	23.4220	−	13.5227i	1.08851	−	0.628453i	0.933180	+	0.359409i	$0.117022\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.174973			0.00809677			0.00404838	−	0.999992i	$-0.498711\pi$
$467$	0.174973			0.00809677			0.00404838	+	0.999992i	$0.498711\pi$
$468$		0			0
$469$	7.39388			0.341418
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.55051	−	3.78194i	0.301193	−	0.173894i
$474$		0			0
$475$	−20.7846	−	12.0000i	−0.953663	−	0.550598i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.0458	−	22.5959i	0.596076	−	1.03243i	−0.397318	−	0.917681i	$-0.630059\pi$
$479$	13.0458	−	22.5959i	0.596076	−	1.03243i	0.993394	−	0.114753i	$-0.0366076\pi$
$480$		0			0
$481$	−7.10102	−	12.2993i	−0.323779	−	0.560801i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	1.58919i		−	0.0721612i
$486$		0			0
$487$		−	3.79796i		−	0.172102i	−0.996291	−	0.0860510i	$-0.972575\pi$
$487$		−	3.79796i		−	0.172102i	0.996291	−	0.0860510i	$-0.0274248\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.15855	+	12.3990i	0.323061	+	0.559558i	0.981118	−	0.193410i	$-0.0619548\pi$
$491$	7.15855	+	12.3990i	0.323061	+	0.559558i	−0.658057	+	0.752968i	$0.728621\pi$
$492$		0			0
$493$	−15.7980	+	27.3629i	−0.711504	+	1.23236i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.49490	+	5.48188i	0.425904	+	0.245896i
$498$		0			0
$499$	12.9029	−	7.44949i	0.577613	−	0.333485i	−0.182571	−	0.983193i	$-0.558442\pi$
$499$	12.9029	−	7.44949i	0.577613	−	0.333485i	0.760184	+	0.649708i	$0.225109\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	31.1127			1.38725			0.693623	−	0.720338i	$-0.256013\pi$
$503$	31.1127			1.38725			0.693623	+	0.720338i	$0.256013\pi$
$504$		0			0
$505$	−5.00000			−0.222497
$506$		0			0
$507$		0			0
$508$		0			0
$509$	23.1186	−	13.3475i	1.02471	−	0.591619i	0.109249	−	0.994014i	$-0.465155\pi$
$509$	23.1186	−	13.3475i	1.02471	−	0.591619i	0.915466	+	0.402395i	$0.131822\pi$
$510$		0			0
$511$	−2.38378	−	1.37628i	−0.105452	−	0.0608828i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.58919	−	2.75255i	0.0700279	−	0.121292i
$516$		0			0
$517$	−5.00000	−	8.66025i	−0.219900	−	0.380878i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	21.7060i		−	0.950958i	−0.879727	−	0.475479i	$-0.842275\pi$
$521$		−	21.7060i		−	0.950958i	0.879727	−	0.475479i	$-0.157725\pi$
$522$		0			0
$523$			27.3939i			1.19785i	0.800805	+	0.598925i	$0.204405\pi$
$523$			27.3939i			1.19785i	−0.800805	+	0.598925i	$0.795595\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.9029	−	22.3485i	−0.562059	−	0.973515i
$528$		0			0
$529$	7.50000	−	12.9904i	0.326087	−	0.564799i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	24.4949	+	14.1421i	1.06099	+	0.612564i
$534$		0			0
$535$	−2.38378	+	1.37628i	−0.103060	+	0.0595016i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.37113			0.231351
$540$		0			0
$541$	−20.0000			−0.859867			−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000			−0.859867			−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.34847	+	0.778539i	−0.0577621	+	0.0333489i
$546$		0			0
$547$	20.6096	+	11.8990i	0.881204	+	0.508764i	0.871055	−	0.491185i	$-0.163436\pi$
$547$	20.6096	+	11.8990i	0.881204	+	0.508764i	0.0101491	+	0.999948i	$0.496769\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.3417	−	38.6969i	0.951788	−	1.64855i
$552$		0			0
$553$	−1.44949	−	2.51059i	−0.0616386	−	0.106761i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			12.5529i			0.531886i	0.963989	+	0.265943i	$0.0856832\pi$
$557$			12.5529i			0.531886i	−0.963989	+	0.265943i	$0.914317\pi$
$558$		0			0
$559$			20.0000i			0.845910i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11.4012	−	19.7474i	−0.480503	−	0.832256i	0.519247	−	0.854624i	$-0.326213\pi$
$563$	−11.4012	−	19.7474i	−0.480503	−	0.832256i	−0.999750	+	0.0223687i	$0.992879\pi$
$564$		0			0
$565$	−2.24745	+	3.89270i	−0.0945509	+	0.163767i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.10102	−	4.09978i	−0.297690	−	0.171872i	0.343715	−	0.939074i	$-0.388315\pi$
$569$	−7.10102	−	4.09978i	−0.297690	−	0.171872i	−0.641405	+	0.767203i	$0.721648\pi$
$570$		0			0
$571$	−17.1455	+	9.89898i	−0.717518	+	0.414259i	−0.813839	−	0.581091i	$-0.802626\pi$
$571$	−17.1455	+	9.89898i	−0.717518	+	0.414259i	0.0963203	+	0.995350i	$0.469293\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−13.8564			−0.577852
$576$		0			0
$577$	−3.79796			−0.158111			−0.0790556	−	0.996870i	$-0.525190\pi$
$577$	−3.79796			−0.158111			−0.0790556	+	0.996870i	$0.525190\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	19.9268	−	11.5047i	0.826702	−	0.477297i
$582$		0			0
$583$	−4.19448	−	2.42168i	−0.173718	−	0.100296i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11.5441	−	19.9949i	0.476474	−	0.825278i	−0.523162	−	0.852233i	$-0.675248\pi$
$587$	11.5441	−	19.9949i	0.476474	−	0.825278i	0.999637	+	0.0269553i	$0.00858119\pi$
$588$		0			0
$589$	18.2474	+	31.6055i	0.751873	+	1.30228i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			8.13534i			0.334078i	0.985950	+	0.167039i	$0.0534206\pi$
$593$			8.13534i			0.334078i	−0.985950	+	0.167039i	$0.946579\pi$
$594$		0			0
$595$			1.59592i			0.0654263i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.7600	−	22.1010i	−0.521361	−	0.903023i	−0.999691	−	0.0248434i	$-0.992091\pi$
$599$	−12.7600	−	22.1010i	−0.521361	−	0.903023i	0.478331	−	0.878180i	$-0.341242\pi$
$600$		0			0
$601$	−5.15153	+	8.92271i	−0.210135	+	0.363965i	−0.951757	−	0.306854i	$-0.900724\pi$
$601$	−5.15153	+	8.92271i	−0.210135	+	0.363965i	0.741621	+	0.670819i	$0.234057\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.69694	+	1.55708i	0.109646	+	0.0633042i
$606$		0			0
$607$	−20.6096	+	11.8990i	−0.836519	+	0.482965i	−0.856080	−	0.516844i	$-0.827107\pi$
$607$	−20.6096	+	11.8990i	−0.836519	+	0.482965i	0.0195602	+	0.999809i	$0.493773\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	26.4415			1.06971
$612$		0			0
$613$	30.0000			1.21169			0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000			1.21169			0.605844	+	0.795583i	$0.292835\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.2474	−	13.9993i	0.976166	−	0.563589i	0.0750552	−	0.997179i	$-0.476087\pi$
$617$	24.2474	−	13.9993i	0.976166	−	0.563589i	0.901110	+	0.433590i	$0.142753\pi$
$618$		0			0
$619$	18.7026	+	10.7980i	0.751722	+	0.434007i	0.826316	−	0.563207i	$-0.190433\pi$
$619$	18.7026	+	10.7980i	0.751722	+	0.434007i	−0.0745941	+	0.997214i	$0.523766\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.66025	+	15.0000i	−0.346966	+	0.600962i
$624$		0			0
$625$	−11.7474	−	20.3472i	−0.469898	−	0.813887i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	16.9706i		−	0.676661i
$630$		0			0
$631$		−	45.7423i		−	1.82097i	−0.413538	−	0.910487i	$-0.635707\pi$
$631$		−	45.7423i		−	1.82097i	0.413538	−	0.910487i	$-0.364293\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.42310	−	4.19694i	−0.0961579	−	0.166550i
$636$		0			0
$637$	−7.10102	+	12.2993i	−0.281353	+	0.487317i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	42.2474	+	24.3916i	1.66867	+	0.963409i	0.968354	+	0.249580i	$0.0802926\pi$
$641$	42.2474	+	24.3916i	1.66867	+	0.963409i	0.700320	+	0.713829i	$0.253041\pi$
$642$		0			0
$643$	32.3840	−	18.6969i	1.27710	−	0.737335i	0.300788	−	0.953691i	$-0.402750\pi$
$643$	32.3840	−	18.6969i	1.27710	−	0.737335i	0.976315	+	0.216356i	$0.0694171\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20.7846			−0.817127			−0.408564	−	0.912730i	$-0.633970\pi$
$647$	−20.7846			−0.817127			−0.408564	+	0.912730i	$0.633970\pi$
$648$		0			0
$649$	−10.0000			−0.392534
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−16.0732	+	9.27987i	−0.628993	+	0.363150i	−0.780362	−	0.625328i	$-0.784965\pi$
$653$	−16.0732	+	9.27987i	−0.628993	+	0.363150i	0.151369	+	0.988477i	$0.451632\pi$
$654$		0			0
$655$	−0.826701	−	0.477296i	−0.0323019	−	0.0186495i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.9118	−	24.0959i	0.541926	−	0.938644i	−0.456867	−	0.889535i	$-0.651029\pi$
$659$	13.9118	−	24.0959i	0.541926	−	0.938644i	0.998793	−	0.0491088i	$-0.0156381\pi$
$660$		0			0
$661$	2.24745	+	3.89270i	0.0874156	+	0.151408i	0.906418	−	0.422382i	$-0.138806\pi$
$661$	2.24745	+	3.89270i	0.0874156	+	0.151408i	−0.819002	+	0.573790i	$0.805473\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2.25697i		−	0.0875215i
$666$		0			0
$667$		−	25.7980i		−	0.998901i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.19275	−	3.79796i	−0.0846503	−	0.146619i
$672$		0			0
$673$	18.2980	−	31.6930i	0.705334	−	1.22168i	−0.261236	−	0.965275i	$-0.584130\pi$
$673$	18.2980	−	31.6930i	0.705334	−	1.22168i	0.966571	−	0.256400i	$-0.0825365\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9.49490	−	5.48188i	−0.364919	−	0.210686i	0.306318	−	0.951929i	$-0.400903\pi$
$677$	−9.49490	−	5.48188i	−0.364919	−	0.210686i	−0.671236	+	0.741244i	$0.734236\pi$
$678$		0			0
$679$	−6.27647	+	3.62372i	−0.240869	+	0.139066i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.3923			−0.397650			−0.198825	−	0.980035i	$-0.563713\pi$
$683$	−10.3923			−0.397650			−0.198825	+	0.980035i	$0.563713\pi$
$684$		0			0
$685$	−3.39388			−0.129673
$686$		0			0
$687$		0			0
$688$		0			0
$689$	11.0908	−	6.40329i	0.422526	−	0.243946i
$690$		0			0
$691$	3.46410	+	2.00000i	0.131781	+	0.0760836i	0.564441	−	0.825473i	$-0.309092\pi$
$691$	3.46410	+	2.00000i	0.131781	+	0.0760836i	−0.432660	+	0.901557i	$0.642425\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.17837	+	5.50510i	−0.120563	+	0.208820i
$696$		0			0
$697$	16.8990	+	29.2699i	0.640094	+	1.10868i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			38.7087i			1.46201i	0.682374	+	0.731003i	$0.260948\pi$
$701$			38.7087i			1.46201i	−0.682374	+	0.731003i	$0.739052\pi$
$702$		0			0
$703$			24.0000i			0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	11.4012	+	19.7474i	0.428786	+	0.742679i
$708$		0			0
$709$	20.0000	−	34.6410i	0.751116	−	1.30097i	−0.196167	−	0.980571i	$-0.562849\pi$
$709$	20.0000	−	34.6410i	0.751116	−	1.30097i	0.947282	−	0.320400i	$-0.103817\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.2474	+	10.5352i	0.683372	+	0.394545i
$714$		0			0
$715$	0.874863	−	0.505103i	0.0327180	−	0.0188898i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.92820			0.258378			0.129189	−	0.991620i	$-0.458763\pi$
$719$	6.92820			0.258378			0.129189	+	0.991620i	$0.458763\pi$
$720$		0			0
$721$	−14.4949			−0.539818
$722$		0			0
$723$		0			0
$724$		0			0
$725$	38.6969	−	22.3417i	1.43717	−	0.829749i
$726$		0			0
$727$	−6.27647	−	3.62372i	−0.232782	−	0.134396i	0.379073	−	0.925367i	$-0.376243\pi$
$727$	−6.27647	−	3.62372i	−0.232782	−	0.134396i	−0.611855	+	0.790970i	$0.709576\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−11.9494	+	20.6969i	−0.441964	+	0.765504i
$732$		0			0
$733$	−2.24745	−	3.89270i	−0.0830114	−	0.143780i	0.821531	−	0.570164i	$-0.193120\pi$
$733$	−2.24745	−	3.89270i	−0.0830114	−	0.143780i	−0.904542	+	0.426384i	$0.859787\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	5.59264i		−	0.206007i
$738$		0			0
$739$		−	19.3939i		−	0.713415i	−0.934216	−	0.356708i	$-0.883899\pi$
$739$		−	19.3939i		−	0.713415i	0.934216	−	0.356708i	$-0.116101\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	15.7313	+	27.2474i	0.577126	+	0.999612i	0.995807	+	0.0914785i	$0.0291593\pi$
$743$	15.7313	+	27.2474i	0.577126	+	0.999612i	−0.418681	+	0.908133i	$0.637507\pi$
$744$		0			0
$745$	−0.601021	+	1.04100i	−0.0220197	+	0.0381392i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10.8712	+	6.27647i	0.397224	+	0.229337i
$750$		0			0
$751$	−41.3461	+	23.8712i	−1.50874	+	0.871071i	−0.508792	+	0.860890i	$0.669908\pi$
$751$	−41.3461	+	23.8712i	−1.50874	+	0.871071i	−0.999948	+	0.0101819i	$0.996759\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.48188			0.199506
$756$		0			0
$757$	−39.3939			−1.43179			−0.715897	−	0.698205i	$-0.753982\pi$
$757$	−39.3939			−1.43179			−0.715897	+	0.698205i	$0.753982\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−45.7980	+	26.4415i	−1.66017	+	0.958502i	−0.687545	+	0.726142i	$0.741312\pi$
$761$	−45.7980	+	26.4415i	−1.66017	+	0.958502i	−0.972630	+	0.232361i	$0.925355\pi$
$762$		0			0
$763$	6.14966	+	3.55051i	0.222633	+	0.128537i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	13.2207	−	22.8990i	0.477373	−	0.826834i
$768$		0			0
$769$	−5.74745	−	9.95487i	−0.207258	−	0.358982i	0.743592	−	0.668634i	$-0.233121\pi$
$769$	−5.74745	−	9.95487i	−0.207258	−	0.358982i	−0.950850	+	0.309652i	$0.899787\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			34.2911i			1.23336i	0.787212	+	0.616682i	$0.211524\pi$
$773$			34.2911i			1.23336i	−0.787212	+	0.616682i	$0.788476\pi$
$774$		0			0
$775$			36.4949i			1.31094i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−23.8988	−	41.3939i	−0.856262	−	1.48309i
$780$		0			0
$781$	4.14643	−	7.18182i	0.148371	−	0.256986i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.79796	+	2.19275i	0.135555	+	0.0782627i
$786$		0			0
$787$	−30.4770	+	17.5959i	−1.08639	+	0.627227i	−0.932613	−	0.360879i	$-0.882477\pi$
$787$	−30.4770	+	17.5959i	−1.08639	+	0.627227i	−0.153776	+	0.988106i	$0.549143\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.4989			0.728856
$792$		0			0
$793$	11.5959			0.411783
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.27526	+	3.62302i	−0.222281	+	0.128334i	−0.607006	−	0.794697i	$-0.707630\pi$
$797$	−6.27526	+	3.62302i	−0.222281	+	0.128334i	0.384725	+	0.923031i	$0.374296\pi$
$798$		0			0
$799$	27.3629	+	15.7980i	0.968029	+	0.558892i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.04100	+	1.80306i	−0.0367360	+	0.0636287i
$804$		0			0
$805$	−0.651531	−	1.12848i	−0.0229634	−	0.0397738i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	11.9494i		−	0.420118i	−0.977689	−	0.210059i	$-0.932634\pi$
$809$		−	11.9494i		−	0.420118i	0.977689	−	0.210059i	$-0.0673656\pi$
$810$		0			0
$811$		−	24.4949i		−	0.860132i	−0.902797	−	0.430066i	$-0.858490\pi$
$811$		−	24.4949i		−	0.860132i	0.902797	−	0.430066i	$-0.141510\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.953512	+	1.65153i	0.0334001	+	0.0578506i
$816$		0			0
$817$	16.8990	−	29.2699i	0.591220	−	1.02402i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.8990	−	11.4887i	−0.694479	−	0.400958i	0.110809	−	0.993842i	$-0.464656\pi$
$821$	−19.8990	−	11.4887i	−0.694479	−	0.400958i	−0.805288	+	0.592884i	$0.797989\pi$
$822$		0			0
$823$	−25.6790	+	14.8258i	−0.895113	+	0.516794i	−0.875611	−	0.483016i	$-0.839541\pi$
$823$	−25.6790	+	14.8258i	−0.895113	+	0.516794i	−0.0195015	+	0.999810i	$0.506208\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.6206			0.577955			0.288978	−	0.957336i	$-0.406685\pi$
$827$	16.6206			0.577955			0.288978	+	0.957336i	$0.406685\pi$
$828$		0			0
$829$	−37.5959			−1.30576			−0.652880	−	0.757461i	$-0.726439\pi$
$829$	−37.5959			−1.30576			−0.652880	+	0.757461i	$0.726439\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.6969	+	8.48528i	−0.509219	+	0.293998i
$834$		0			0
$835$	6.32464	+	3.65153i	0.218873	+	0.126366i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.58919	−	2.75255i	0.0548648	−	0.0950286i	−0.837289	−	0.546761i	$-0.815861\pi$
$839$	1.58919	−	2.75255i	0.0548648	−	0.0950286i	0.892153	+	0.451733i	$0.149194\pi$
$840$		0			0
$841$	27.0959	+	46.9315i	0.934342	+	1.61833i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1.46075i		−	0.0502515i
$846$		0			0
$847$		−	14.2020i		−	0.487988i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.92820	+	12.0000i	0.237496	+	0.411355i
$852$		0			0
$853$	−5.00000	+	8.66025i	−0.171197	+	0.296521i	−0.938839	−	0.344358i	$-0.888097\pi$
$853$	−5.00000	+	8.66025i	−0.171197	+	0.296521i	0.767642	+	0.640879i	$0.221430\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.1010	+	5.83183i	0.345044	+	0.199211i	0.662500	−	0.749062i	$-0.269495\pi$
$857$	10.1010	+	5.83183i	0.345044	+	0.199211i	−0.317456	+	0.948273i	$0.602829\pi$
$858$		0			0
$859$	34.2911	−	19.7980i	1.17000	−	0.675498i	0.216317	−	0.976323i	$-0.430596\pi$
$859$	34.2911	−	19.7980i	1.17000	−	0.675498i	0.953679	+	0.300826i	$0.0972622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.92820			−0.235839			−0.117919	−	0.993023i	$-0.537622\pi$
$863$	−6.92820			−0.235839			−0.117919	+	0.993023i	$0.537622\pi$
$864$		0			0
$865$	−1.69694			−0.0576976
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.89898	+	1.09638i	−0.0644185	+	0.0371920i
$870$		0			0
$871$	12.8066	+	7.39388i	0.433934	+	0.250532i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.28024	−	3.94949i	0.0770861	−	0.133517i
$876$		0			0
$877$	8.10102	+	14.0314i	0.273552	+	0.473806i	0.969769	−	0.244026i	$-0.0784681\pi$
$877$	8.10102	+	14.0314i	0.273552	+	0.473806i	−0.696217	+	0.717832i	$0.745135\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	7.27815i		−	0.245207i	−0.992456	−	0.122604i	$-0.960876\pi$
$881$		−	7.27815i		−	0.245207i	0.992456	−	0.122604i	$-0.0391244\pi$
$882$		0			0
$883$		−	52.8990i		−	1.78019i	−0.455773	−	0.890096i	$-0.650637\pi$
$883$		−	52.8990i		−	1.78019i	0.455773	−	0.890096i	$-0.349363\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22.6595	−	39.2474i	−0.760832	−	1.31780i	−0.942422	−	0.334426i	$-0.891458\pi$
$887$	−22.6595	−	39.2474i	−0.760832	−	1.31780i	0.181590	−	0.983374i	$-0.441876\pi$
$888$		0			0
$889$	−11.0505	+	19.1400i	−0.370622	+	0.641937i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−38.6969	−	22.3417i	−1.29494	−	0.747636i
$894$		0			0
$895$	2.38378	−	1.37628i	0.0796810	−	0.0460038i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−67.9465			−2.26614
$900$		0			0
$901$	15.3031			0.509819
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.69694	+	1.55708i	−0.0896493	+	0.0517590i
$906$		0			0
$907$	3.89270	+	2.24745i	0.129255	+	0.0746253i	0.563233	−	0.826298i	$-0.309557\pi$
$907$	3.89270	+	2.24745i	0.129255	+	0.0746253i	−0.433978	+	0.900923i	$0.642891\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.460702	+	0.797959i	−0.0152637	+	0.0264376i	−0.873556	−	0.486723i	$-0.838192\pi$
$911$	−0.460702	+	0.797959i	−0.0152637	+	0.0264376i	0.858293	+	0.513161i	$0.171525\pi$
$912$		0			0
$913$	−8.70204	−	15.0724i	−0.287996	−	0.498823i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			4.35340i			0.143762i
$918$		0			0
$919$			2.75255i			0.0907983i	0.998969	+	0.0453991i	$0.0144560\pi$
$919$			2.75255i			0.0907983i	−0.998969	+	0.0453991i	$0.985544\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.9638	+	18.9898i	0.360877	+	0.625057i
$924$		0			0
$925$	−12.0000	+	20.7846i	−0.394558	+	0.683394i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	13.1010	+	7.56388i	0.429831	+	0.248163i	0.699274	−	0.714853i	$-0.253507\pi$
$929$	13.1010	+	7.56388i	0.429831	+	0.248163i	−0.269444	+	0.963016i	$0.586840\pi$
$930$		0			0
$931$	20.7846	−	12.0000i	0.681188	−	0.393284i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.20713			0.0394775
$936$		0			0
$937$	−15.0000			−0.490029			−0.245014	−	0.969519i	$-0.578793\pi$
$937$	−15.0000			−0.490029			−0.245014	+	0.969519i	$0.578793\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	35.9722	−	20.7686i	1.17266	−	0.677036i	0.218355	−	0.975869i	$-0.429931\pi$
$941$	35.9722	−	20.7686i	1.17266	−	0.677036i	0.954305	+	0.298834i	$0.0965976\pi$
$942$		0			0
$943$	−23.8988	−	13.7980i	−0.778251	−	0.449323i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−11.5441	+	19.9949i	−0.375132	+	0.649747i	−0.990347	−	0.138612i	$-0.955736\pi$
$947$	−11.5441	+	19.9949i	−0.375132	+	0.649747i	0.615215	+	0.788359i	$0.289069\pi$
$948$		0			0
$949$	−2.75255	−	4.76756i	−0.0893516	−	0.154762i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	10.7423i		−	0.347976i	−0.984748	−	0.173988i	$-0.944335\pi$
$953$		−	10.7423i		−	0.347976i	0.984748	−	0.173988i	$-0.0556653\pi$
$954$		0			0
$955$			5.59592i			0.181080i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.73885	+	13.4041i	0.249901	+	0.432840i
$960$		0			0
$961$	12.2474	−	21.2132i	0.395079	−	0.684297i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.578317	−	0.333891i	−0.0186167	−	0.0107483i
$966$		0			0
$967$	−42.6495	+	24.6237i	−1.37152	+	0.791846i	−0.991119	−	0.132977i	$-0.957546\pi$
$967$	−42.6495	+	24.6237i	−1.37152	+	0.791846i	−0.380398	+	0.924823i	$0.624213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	19.3383			0.620595			0.310298	−	0.950639i	$-0.399571\pi$
$971$	19.3383			0.620595			0.310298	+	0.950639i	$0.399571\pi$
$972$		0			0
$973$	28.9898			0.929370
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−27.2474	+	15.7313i	−0.871723	+	0.503290i	−0.867920	−	0.496703i	$-0.834544\pi$
$977$	−27.2474	+	15.7313i	−0.871723	+	0.503290i	−0.00380265	+	0.999993i	$0.501210\pi$
$978$		0			0
$979$	11.3458	+	6.55051i	0.362614	+	0.209355i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−26.2665	+	45.4949i	−0.837771	+	1.45106i	0.0539833	+	0.998542i	$0.482808\pi$
$983$	−26.2665	+	45.4949i	−0.837771	+	1.45106i	−0.891754	+	0.452520i	$0.850525\pi$
$984$		0			0
$985$	−1.94949	−	3.37662i	−0.0621159	−	0.107588i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	19.5133i		−	0.620486i
$990$		0			0
$991$			8.75255i			0.278034i	0.990290	+	0.139017i	$0.0443943\pi$
$991$			8.75255i			0.278034i	−0.990290	+	0.139017i	$0.955606\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.23375	−	5.60102i	−0.102517	−	0.177564i
$996$		0			0
$997$	30.3485	−	52.5651i	0.961146	−	1.66475i	0.241514	−	0.970397i	$-0.422356\pi$
$997$	30.3485	−	52.5651i	0.961146	−	1.66475i	0.719632	−	0.694356i	$-0.244311\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.s.h.1727.1		8
3.2	odd	2		2592.2.s.a.1727.3		8
4.3	odd	2	inner	2592.2.s.h.1727.2		8
9.2	odd	6		864.2.c.a.863.4	yes	8
9.4	even	3		2592.2.s.a.863.4		8
9.5	odd	6	inner	2592.2.s.h.863.2		8
9.7	even	3		864.2.c.a.863.6	yes	8
12.11	even	2		2592.2.s.a.1727.4		8
36.7	odd	6		864.2.c.a.863.5	yes	8
36.11	even	6		864.2.c.a.863.3	✓	8
36.23	even	6	inner	2592.2.s.h.863.1		8
36.31	odd	6		2592.2.s.a.863.3		8
72.11	even	6		1728.2.c.g.1727.5		8
72.29	odd	6		1728.2.c.g.1727.6		8
72.43	odd	6		1728.2.c.g.1727.3		8
72.61	even	6		1728.2.c.g.1727.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.c.a.863.3	✓	8	36.11	even	6
864.2.c.a.863.4	yes	8	9.2	odd	6
864.2.c.a.863.5	yes	8	36.7	odd	6
864.2.c.a.863.6	yes	8	9.7	even	3
1728.2.c.g.1727.3		8	72.43	odd	6
1728.2.c.g.1727.4		8	72.61	even	6
1728.2.c.g.1727.5		8	72.11	even	6
1728.2.c.g.1727.6		8	72.29	odd	6
2592.2.s.a.863.3		8	36.31	odd	6
2592.2.s.a.863.4		8	9.4	even	3
2592.2.s.a.1727.3		8	3.2	odd	2
2592.2.s.a.1727.4		8	12.11	even	2
2592.2.s.h.863.1		8	36.23	even	6	inner
2592.2.s.h.863.2		8	9.5	odd	6	inner
2592.2.s.h.1727.1		8	1.1	even	1	trivial
2592.2.s.h.1727.2		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.275255 - 0.158919i) q^{5} +(-1.25529 - 0.724745i) q^{7} +(-0.548188 + 0.949490i) q^{11} +(-1.44949 - 2.51059i) q^{13} -3.46410i q^{17} +4.89898i q^{19} +(1.41421 + 2.44949i) q^{23} +(-2.44949 + 4.24264i) q^{25} +(-7.89898 - 4.56048i) q^{29} +(6.45145 - 3.72474i) q^{31} -0.460702 q^{35} +4.89898 q^{37} +(-8.44949 + 4.87832i) q^{41} +(-5.97469 - 3.44949i) q^{43} +(-4.56048 + 7.89898i) q^{47} +(-2.44949 - 4.24264i) q^{49} +4.41761i q^{53} +0.348469i q^{55} +(4.56048 + 7.89898i) q^{59} +(-2.00000 + 3.46410i) q^{61} +(-0.797959 - 0.460702i) q^{65} +(-4.41761 + 2.55051i) q^{67} -7.56388 q^{71} +1.89898 q^{73} +(1.37628 - 0.794593i) q^{77} +(1.73205 + 1.00000i) q^{79} +(-7.93709 + 13.7474i) q^{83} +(-0.550510 - 0.953512i) q^{85} -11.9494i q^{89} +4.20204i q^{91} +(0.778539 + 1.34847i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{5} + 8 q^{13} - 24 q^{29} - 48 q^{41} - 16 q^{61} + 72 q^{65} - 24 q^{73} + 60 q^{77} - 24 q^{85} + 20 q^{97}+O(q^{100}) \) 8 * q + 12 * q^5 + 8 * q^13 - 24 * q^29 - 48 * q^41 - 16 * q^61 + 72 * q^65 - 24 * q^73 + 60 * q^77 - 24 * q^85 + 20 * q^97

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