LMFDB - Embedded newform 2808.2.cw.b.1585.28

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(1585,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.1585"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.cw (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1585.28
Character	$\chi$	$=$	2808.1585
Dual form			2808.2.cw.b.2521.28

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.12193 - 0.647747i) q^{5} +(-1.01499 - 0.586005i) q^{7} +(-3.37140 - 1.94648i) q^{11} +(-0.292273 - 3.59369i) q^{13} +6.56850 q^{17} -4.02682i q^{19} +(-1.49532 - 2.58997i) q^{23} +(-1.66085 + 2.87667i) q^{25} +(-4.62537 + 8.01138i) q^{29} +(6.82747 - 3.94184i) q^{31} -1.51833 q^{35} +7.11245i q^{37} +(-8.68452 + 5.01401i) q^{41} +(2.21816 - 3.84196i) q^{43} +(-9.72175 - 5.61286i) q^{47} +(-2.81320 - 4.87260i) q^{49} -3.94629 q^{53} -5.04331 q^{55} +(-8.74216 + 5.04729i) q^{59} +(-0.267490 + 0.463307i) q^{61} +(-2.65571 - 3.84255i) q^{65} +(0.708619 - 0.409121i) q^{67} -11.3561i q^{71} -9.27918i q^{73} +(2.28130 + 3.95132i) q^{77} +(-3.92094 + 6.79126i) q^{79} +(6.60302 + 3.81226i) q^{83} +(7.36940 - 4.25473i) q^{85} -18.1154i q^{89} +(-1.80926 + 3.81883i) q^{91} +(-2.60836 - 4.51782i) q^{95} +(-5.94315 - 3.43128i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) $ 80 * q + 6 * q^13 + 4 * q^17 - 10 * q^23 + 44 * q^25 + 52 * q^35 - 26 * q^43 + 48 * q^49 - 60 * q^53 - 16 * q^55 - 10 * q^61 + 26 * q^65 - 32 * q^77 + 6 * q^79 - 4 * q^91 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.12193	−	0.647747i	0.501743	−	0.289681i	−0.227690	−	0.973734i	$-0.573117\pi$
$5$	1.12193	−	0.647747i	0.501743	−	0.289681i	0.729433	+	0.684052i	$0.239784\pi$
$6$		0			0
$7$	−1.01499	−	0.586005i	−0.383630	−	0.221489i	0.295766	−	0.955260i	$-0.404425\pi$
$7$	−1.01499	−	0.586005i	−0.383630	−	0.221489i	−0.679397	+	0.733771i	$0.737758\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.37140	−	1.94648i	−1.01652	−	0.586886i	−0.103424	−	0.994637i	$-0.532980\pi$
$11$	−3.37140	−	1.94648i	−1.01652	−	0.586886i	−0.913093	+	0.407751i	$0.866313\pi$
$12$		0			0
$13$	−0.292273	−	3.59369i	−0.0810619	−	0.996709i
$13$	−0.292273	−	3.59369i	−0.0810619	−	0.996709i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.56850			1.59309			0.796547	−	0.604576i	$-0.206657\pi$
$17$	6.56850			1.59309			0.796547	+	0.604576i	$0.206657\pi$
$18$		0			0
$19$		−	4.02682i		−	0.923817i	−0.886928	−	0.461908i	$-0.847165\pi$
$19$		−	4.02682i		−	0.923817i	0.886928	−	0.461908i	$-0.152835\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.49532	−	2.58997i	−0.311796	−	0.540047i	0.666955	−	0.745098i	$-0.267597\pi$
$23$	−1.49532	−	2.58997i	−0.311796	−	0.540047i	−0.978751	+	0.205051i	$0.934264\pi$
$24$		0			0
$25$	−1.66085	+	2.87667i	−0.332170	+	0.575335i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.62537	+	8.01138i	−0.858910	+	1.48768i	0.0140595	+	0.999901i	$0.495525\pi$
$29$	−4.62537	+	8.01138i	−0.858910	+	1.48768i	−0.872970	+	0.487775i	$0.837809\pi$
$30$		0			0
$31$	6.82747	−	3.94184i	1.22625	−	0.707976i	0.260007	−	0.965607i	$-0.416275\pi$
$31$	6.82747	−	3.94184i	1.22625	−	0.707976i	0.966243	+	0.257631i	$0.0829419\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.51833			−0.256645
$36$		0			0
$37$			7.11245i			1.16928i	0.811293	+	0.584640i	$0.198764\pi$
$37$			7.11245i			1.16928i	−0.811293	+	0.584640i	$0.801236\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.68452	+	5.01401i	−1.35629	+	0.783056i	−0.989122	−	0.147097i	$-0.953007\pi$
$41$	−8.68452	+	5.01401i	−1.35629	+	0.783056i	−0.367171	+	0.930153i	$0.619674\pi$
$42$		0			0
$43$	2.21816	−	3.84196i	0.338266	−	0.585894i	−0.645841	−	0.763472i	$-0.723493\pi$
$43$	2.21816	−	3.84196i	0.338266	−	0.585894i	0.984107	+	0.177578i	$0.0568263\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.72175	−	5.61286i	−1.41806	−	0.818719i	−0.421935	−	0.906626i	$-0.638649\pi$
$47$	−9.72175	−	5.61286i	−1.41806	−	0.818719i	−0.996129	+	0.0879068i	$0.971982\pi$
$48$		0			0
$49$	−2.81320	−	4.87260i	−0.401885	−	0.696085i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.94629			−0.542065			−0.271032	−	0.962570i	$-0.587365\pi$
$53$	−3.94629			−0.542065			−0.271032	+	0.962570i	$0.587365\pi$
$54$		0			0
$55$	−5.04331			−0.680040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.74216	+	5.04729i	−1.13813	+	0.657101i	−0.945967	−	0.324262i	$-0.894884\pi$
$59$	−8.74216	+	5.04729i	−1.13813	+	0.657101i	−0.192165	+	0.981363i	$0.561551\pi$
$60$		0			0
$61$	−0.267490	+	0.463307i	−0.0342486	+	0.0593204i	−0.882642	−	0.470046i	$-0.844237\pi$
$61$	−0.267490	+	0.463307i	−0.0342486	+	0.0593204i	0.848393	+	0.529367i	$0.177570\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.65571	−	3.84255i	−0.329400	−	0.476609i
$66$		0			0
$67$	0.708619	−	0.409121i	0.0865716	−	0.0499821i	−0.456089	−	0.889934i	$-0.650750\pi$
$67$	0.708619	−	0.409121i	0.0865716	−	0.0499821i	0.542661	+	0.839952i	$0.317417\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.3561i		−	1.34773i	−0.738856	−	0.673863i	$-0.764634\pi$
$71$		−	11.3561i		−	1.34773i	0.738856	−	0.673863i	$-0.235366\pi$
$72$		0			0
$73$		−	9.27918i		−	1.08605i	−0.839718	−	0.543023i	$-0.817280\pi$
$73$		−	9.27918i		−	1.08605i	0.839718	−	0.543023i	$-0.182720\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.28130	+	3.95132i	0.259978	+	0.450295i
$78$		0			0
$79$	−3.92094	+	6.79126i	−0.441140	+	0.764076i	−0.997774	−	0.0666810i	$-0.978759\pi$
$79$	−3.92094	+	6.79126i	−0.441140	+	0.764076i	0.556635	+	0.830757i	$0.312092\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.60302	+	3.81226i	0.724776	+	0.418449i	0.816508	−	0.577334i	$-0.195907\pi$
$83$	6.60302	+	3.81226i	0.724776	+	0.418449i	−0.0917322	+	0.995784i	$0.529240\pi$
$84$		0			0
$85$	7.36940	−	4.25473i	0.799324	−	0.461490i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	18.1154i		−	1.92023i	−0.279606	−	0.960115i	$-0.590204\pi$
$89$		−	18.1154i		−	1.92023i	0.279606	−	0.960115i	$-0.409796\pi$
$90$		0			0
$91$	−1.80926	+	3.81883i	−0.189662	+	0.400322i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.60836	−	4.51782i	−0.267612	−	0.463518i
$96$		0			0
$97$	−5.94315	−	3.43128i	−0.603435	−	0.348393i	0.166957	−	0.985964i	$-0.446606\pi$
$97$	−5.94315	−	3.43128i	−0.603435	−	0.348393i	−0.770392	+	0.637571i	$0.779939\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.12658	+	14.0757i	−0.808625	+	1.40058i	0.105191	+	0.994452i	$0.466455\pi$
$101$	−8.12658	+	14.0757i	−0.808625	+	1.40058i	−0.913816	+	0.406128i	$0.866879\pi$
$102$		0			0
$103$	−3.30921	−	5.73172i	−0.326066	−	0.564763i	0.655661	−	0.755055i	$-0.272390\pi$
$103$	−3.30921	−	5.73172i	−0.326066	−	0.564763i	−0.981728	+	0.190292i	$0.939057\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.32645			−0.901622			−0.450811	−	0.892619i	$-0.648865\pi$
$107$	−9.32645			−0.901622			−0.450811	+	0.892619i	$0.648865\pi$
$108$		0			0
$109$		−	19.9089i		−	1.90693i	−0.301504	−	0.953465i	$-0.597489\pi$
$109$		−	19.9089i		−	1.90693i	0.301504	−	0.953465i	$-0.402511\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	2.78841	+	4.82966i	0.262311	+	0.454336i	0.966856	−	0.255324i	$-0.0821820\pi$
$113$	2.78841	+	4.82966i	0.262311	+	0.454336i	−0.704545	+	0.709660i	$0.748849\pi$
$114$		0			0
$115$	−3.35530	−	1.93718i	−0.312883	−	0.180643i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.66697	−	3.84917i	−0.611160	−	0.352853i
$120$		0			0
$121$	2.07758	+	3.59847i	0.188871	+	0.327134i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.7807i			0.964256i
$126$		0			0
$127$	14.6816			1.30278			0.651391	−	0.758742i	$-0.274186\pi$
$127$	14.6816			1.30278			0.651391	+	0.758742i	$0.274186\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.50338	+	7.80009i	0.393462	+	0.681497i	0.992904	−	0.118922i	$-0.0379438\pi$
$131$	4.50338	+	7.80009i	0.393462	+	0.681497i	−0.599441	+	0.800419i	$0.704610\pi$
$132$		0			0
$133$	−2.35974	+	4.08719i	−0.204615	+	0.354404i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.20317	+	0.694649i	0.102794	+	0.0593479i	0.550515	−	0.834825i	$-0.314431\pi$
$137$	1.20317	+	0.694649i	0.102794	+	0.0593479i	−0.447722	+	0.894173i	$0.647765\pi$
$138$		0			0
$139$	−9.03975	−	15.6573i	−0.766742	−	1.32804i	−0.939321	−	0.343040i	$-0.888543\pi$
$139$	−9.03975	−	15.6573i	−0.766742	−	1.32804i	0.172579	−	0.984996i	$-0.444790\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.00967	+	12.6847i	−0.502554	+	1.06075i
$144$		0			0
$145$			11.9843i			0.995241i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.89426	−	4.55775i	0.646723	−	0.373386i	−0.140477	−	0.990084i	$-0.544864\pi$
$149$	7.89426	−	4.55775i	0.646723	−	0.373386i	0.787200	+	0.616698i	$0.211530\pi$
$150$		0			0
$151$	−2.38477	−	1.37685i	−0.194070	−	0.112046i	0.399817	−	0.916595i	$-0.369074\pi$
$151$	−2.38477	−	1.37685i	−0.194070	−	0.112046i	−0.593886	+	0.804549i	$0.702407\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.10663	−	8.84495i	0.410175	−	0.710443i
$156$		0			0
$157$	10.3386	+	17.9069i	0.825107	+	1.42913i	0.901837	+	0.432076i	$0.142219\pi$
$157$	10.3386	+	17.9069i	0.825107	+	1.42913i	−0.0767297	+	0.997052i	$0.524448\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.50507i			0.276238i
$162$		0			0
$163$		−	10.1790i		−	0.797284i	−0.917107	−	0.398642i	$-0.869482\pi$
$163$		−	10.1790i		−	0.797284i	0.917107	−	0.398642i	$-0.130518\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.66878	+	0.963473i	−0.129134	+	0.0745558i	−0.563176	−	0.826337i	$-0.690421\pi$
$167$	−1.66878	+	0.963473i	−0.129134	+	0.0745558i	0.434041	+	0.900893i	$0.357087\pi$
$168$		0			0
$169$	−12.8292	+	2.10067i	−0.986858	+	0.161590i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.346784	−	0.600648i	0.0263655	−	0.0456664i	−0.852542	−	0.522659i	$-0.824940\pi$
$173$	0.346784	−	0.600648i	0.0263655	−	0.0456664i	0.878907	+	0.476993i	$0.158273\pi$
$174$		0			0
$175$	3.37149	−	1.94653i	0.254861	−	0.147144i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	16.1961			1.21055			0.605275	−	0.796016i	$-0.293063\pi$
$179$	16.1961			1.21055			0.605275	+	0.796016i	$0.293063\pi$
$180$		0			0
$181$	−7.12194			−0.529370			−0.264685	−	0.964335i	$-0.585268\pi$
$181$	−7.12194			−0.529370			−0.264685	+	0.964335i	$0.585268\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.60707	+	7.97967i	0.338718	+	0.586677i
$186$		0			0
$187$	−22.1451	−	12.7855i	−1.61941	−	0.934965i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.63623	−	11.4943i	0.480181	−	0.831698i	−0.519561	−	0.854434i	$-0.673904\pi$
$191$	6.63623	−	11.4943i	0.480181	−	0.831698i	0.999742	+	0.0227360i	$0.00723772\pi$
$192$		0			0
$193$	16.7689	−	9.68153i	1.20705	−	0.696892i	0.244938	−	0.969539i	$-0.421233\pi$
$193$	16.7689	−	9.68153i	1.20705	−	0.696892i	0.962114	+	0.272647i	$0.0878992\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	2.07336i		−	0.147720i	−0.997269	−	0.0738602i	$-0.976468\pi$
$197$		−	2.07336i		−	0.147720i	0.997269	−	0.0738602i	$-0.0235319\pi$
$198$		0			0
$199$	−9.20616			−0.652607			−0.326304	−	0.945265i	$-0.605803\pi$
$199$	−9.20616			−0.652607			−0.326304	+	0.945265i	$0.605803\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.38942	−	5.42098i	0.659008	−	0.380479i
$204$		0			0
$205$	−6.49562	+	11.2507i	−0.453674	+	0.785786i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.83814	+	13.5760i	−0.542175	+	0.939075i
$210$		0			0
$211$	8.43931	+	14.6173i	0.580986	+	1.00630i	0.995363	+	0.0961921i	$0.0306663\pi$
$211$	8.43931	+	14.6173i	0.580986	+	1.00630i	−0.414377	+	0.910106i	$0.636000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	5.74722i		−	0.391957i
$216$		0			0
$217$	−9.23976			−0.627236
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.91979	−	23.6051i	−0.129139	−	1.58785i
$222$		0			0
$223$	−11.6114	−	6.70384i	−0.777557	−	0.448923i	0.0580070	−	0.998316i	$-0.481525\pi$
$223$	−11.6114	−	6.70384i	−0.777557	−	0.448923i	−0.835564	+	0.549394i	$0.814859\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−8.22562	−	4.74907i	−0.545954	−	0.315207i	0.201535	−	0.979481i	$-0.435407\pi$
$227$	−8.22562	−	4.74907i	−0.545954	−	0.315207i	−0.747488	+	0.664275i	$0.768740\pi$
$228$		0			0
$229$	6.81996	−	3.93751i	0.450676	−	0.260198i	−0.257440	−	0.966294i	$-0.582879\pi$
$229$	6.81996	−	3.93751i	0.450676	−	0.260198i	0.708116	+	0.706097i	$0.249546\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.61925			−0.499154			−0.249577	−	0.968355i	$-0.580292\pi$
$233$	−7.61925			−0.499154			−0.249577	+	0.968355i	$0.580292\pi$
$234$		0			0
$235$	−14.5428			−0.948671
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.0862	+	8.70999i	−0.975842	+	0.563403i	−0.901012	−	0.433794i	$-0.857175\pi$
$239$	−15.0862	+	8.70999i	−0.975842	+	0.563403i	−0.0748299	+	0.997196i	$0.523841\pi$
$240$		0			0
$241$	−13.7382	−	7.93173i	−0.884953	−	0.510928i	−0.0126645	−	0.999920i	$-0.504031\pi$
$241$	−13.7382	−	7.93173i	−0.884953	−	0.510928i	−0.872288	+	0.488992i	$0.837365\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.31242	−	3.64448i	−0.403286	−	0.232837i
$246$		0			0
$247$	−14.4711	+	1.17693i	−0.920776	+	0.0748864i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.81440			−0.619479			−0.309740	−	0.950821i	$-0.600242\pi$
$251$	−9.81440			−0.619479			−0.309740	+	0.950821i	$0.600242\pi$
$252$		0			0
$253$			11.6425i			0.731956i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.24272	−	3.88451i	−0.139897	−	0.242309i	0.787560	−	0.616238i	$-0.211344\pi$
$257$	−2.24272	−	3.88451i	−0.139897	−	0.242309i	−0.927458	+	0.373928i	$0.878011\pi$
$258$		0			0
$259$	4.16793	−	7.21907i	0.258983	−	0.448571i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	11.0782	−	19.1881i	0.683113	−	1.18319i	−0.290912	−	0.956750i	$-0.593959\pi$
$263$	11.0782	−	19.1881i	0.683113	−	1.18319i	0.974026	−	0.226437i	$-0.0727078\pi$
$264$		0			0
$265$	−4.42747	+	2.55620i	−0.271977	+	0.157026i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.40882			0.0858975			0.0429488	−	0.999077i	$-0.486325\pi$
$269$	1.40882			0.0858975			0.0429488	+	0.999077i	$0.486325\pi$
$270$		0			0
$271$			13.0804i			0.794580i	0.917693	+	0.397290i	$0.130049\pi$
$271$			13.0804i			0.794580i	−0.917693	+	0.397290i	$0.869951\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.1988	−	6.46562i	0.675312	−	0.389891i
$276$		0			0
$277$	9.34430	−	16.1848i	0.561444	−	0.972450i	−0.435926	−	0.899982i	$-0.643579\pi$
$277$	9.34430	−	16.1848i	0.561444	−	0.972450i	0.997371	−	0.0724677i	$-0.0230874\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.11062	−	3.52797i	−0.364529	−	0.210461i	0.306537	−	0.951859i	$-0.400830\pi$
$281$	−6.11062	−	3.52797i	−0.364529	−	0.210461i	−0.671066	+	0.741398i	$0.734163\pi$
$282$		0			0
$283$	−13.2306	−	22.9160i	−0.786476	−	1.36222i	−0.928113	−	0.372299i	$-0.878570\pi$
$283$	−13.2306	−	22.9160i	−0.786476	−	1.36222i	0.141637	−	0.989919i	$-0.454764\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.7529			0.693754
$288$		0			0
$289$	26.1452			1.53795
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.21621	+	3.58893i	−0.363155	+	0.209668i	−0.670464	−	0.741942i	$-0.733905\pi$
$293$	−6.21621	+	3.58893i	−0.363155	+	0.209668i	0.307309	+	0.951610i	$0.400571\pi$
$294$		0			0
$295$	−6.53873	+	11.3254i	−0.380700	+	0.659391i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.87051	+	6.13070i	−0.512995	+	0.354547i
$300$		0			0
$301$	−4.50282	+	2.59970i	−0.259538	+	0.149844i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			0.693065i			0.0396848i
$306$		0			0
$307$			22.7606i			1.29901i	0.760356	+	0.649507i	$0.225025\pi$
$307$			22.7606i			1.29901i	−0.760356	+	0.649507i	$0.774975\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.186725	+	0.323417i	0.0105882	+	0.0183393i	0.871271	−	0.490802i	$-0.163296\pi$
$311$	0.186725	+	0.323417i	0.0105882	+	0.0183393i	−0.860683	+	0.509142i	$0.829963\pi$
$312$		0			0
$313$	0.282788	−	0.489804i	0.0159841	−	0.0276853i	−0.857923	−	0.513779i	$-0.828245\pi$
$313$	0.282788	−	0.489804i	0.0159841	−	0.0276853i	0.873907	+	0.486093i	$0.161579\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.95778	+	1.70767i	0.166125	+	0.0959124i	0.580757	−	0.814077i	$-0.302756\pi$
$317$	2.95778	+	1.70767i	0.166125	+	0.0959124i	−0.414632	+	0.909989i	$0.636090\pi$
$318$		0			0
$319$	31.1880	−	18.0064i	1.74619	−	1.00816i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	26.4502i		−	1.47173i
$324$		0			0
$325$	10.8233	+	5.12779i	0.600367	+	0.284439i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.57833	+	11.3940i	0.362675	+	0.628171i
$330$		0			0
$331$	2.45863	+	1.41949i	0.135139	+	0.0780223i	0.566045	−	0.824374i	$-0.308473\pi$
$331$	2.45863	+	1.41949i	0.135139	+	0.0780223i	−0.430906	+	0.902397i	$0.641806\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.530014	−	0.918011i	0.0289578	−	0.0501563i
$336$		0			0
$337$	−3.43582	−	5.95101i	−0.187161	−	0.324172i	0.757142	−	0.653251i	$-0.226595\pi$
$337$	−3.43582	−	5.95101i	−0.187161	−	0.324172i	−0.944303	+	0.329079i	$0.893262\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−30.6909			−1.66200
$342$		0			0
$343$			14.7983i			0.799031i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.87270	+	10.1718i	0.315263	+	0.546051i	0.979493	−	0.201477i	$-0.0645740\pi$
$347$	5.87270	+	10.1718i	0.315263	+	0.546051i	−0.664231	+	0.747528i	$0.731241\pi$
$348$		0			0
$349$	21.5356	+	12.4336i	1.15277	+	0.665555i	0.949562	−	0.313580i	$-0.101528\pi$
$349$	21.5356	+	12.4336i	1.15277	+	0.665555i	0.203213	+	0.979135i	$0.434862\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.9770	+	16.7299i	1.54229	+	0.890440i	0.998694	+	0.0510923i	$0.0162703\pi$
$353$	28.9770	+	16.7299i	1.54229	+	0.890440i	0.543594	+	0.839348i	$0.317063\pi$
$354$		0			0
$355$	−7.35591	−	12.7408i	−0.390411	−	0.676212i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			20.9981i			1.10824i	0.832438	+	0.554118i	$0.186944\pi$
$359$			20.9981i			1.10824i	−0.832438	+	0.554118i	$0.813056\pi$
$360$		0			0
$361$	2.78470			0.146563
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.01056	−	10.4106i	−0.314607	−	0.544916i
$366$		0			0
$367$	−8.49086	+	14.7066i	−0.443219	+	0.767678i	−0.997926	−	0.0643677i	$-0.979497\pi$
$367$	−8.49086	+	14.7066i	−0.443219	+	0.767678i	0.554707	+	0.832046i	$0.312830\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.00545	+	2.31255i	0.207953	+	0.120062i
$372$		0			0
$373$	10.0047	+	17.3286i	0.518023	+	0.897241i	0.999781	+	0.0209372i	$0.00666500\pi$
$373$	10.0047	+	17.3286i	0.518023	+	0.897241i	−0.481758	+	0.876304i	$0.660002\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	30.1423	+	14.2806i	1.55240	+	0.735490i
$378$		0			0
$379$			17.6142i			0.904780i	0.891820	+	0.452390i	$0.149429\pi$
$379$			17.6142i			0.904780i	−0.891820	+	0.452390i	$0.850571\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.07133	+	2.35058i	−0.208035	+	0.120109i	−0.600398	−	0.799701i	$-0.704991\pi$
$383$	−4.07133	+	2.35058i	−0.208035	+	0.120109i	0.392363	+	0.919811i	$0.371658\pi$
$384$		0			0
$385$	5.11891	+	2.95541i	0.260884	+	0.150621i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.06849	+	5.31479i	−0.155579	+	0.269470i	−0.933270	−	0.359177i	$-0.883058\pi$
$389$	−3.06849	+	5.31479i	−0.155579	+	0.269470i	0.777691	+	0.628647i	$0.216391\pi$
$390$		0			0
$391$	−9.82202	−	17.0122i	−0.496721	−	0.860346i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			10.1591i			0.511160i
$396$		0			0
$397$		−	23.3075i		−	1.16977i	−0.811116	−	0.584885i	$-0.801140\pi$
$397$		−	23.3075i		−	1.16977i	0.811116	−	0.584885i	$-0.198860\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.5604	−	7.25174i	0.627236	−	0.362135i	−0.152445	−	0.988312i	$-0.548715\pi$
$401$	12.5604	−	7.25174i	0.627236	−	0.362135i	0.779681	+	0.626177i	$0.215381\pi$
$402$		0			0
$403$	−16.1612	−	23.3837i	−0.805048	−	1.16482i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.8442	−	23.9789i	0.686234	−	1.18859i
$408$		0			0
$409$	14.3903	−	8.30824i	0.711555	−	0.410816i	−0.100082	−	0.994979i	$-0.531910\pi$
$409$	14.3903	−	8.30824i	0.711555	−	0.410816i	0.811636	+	0.584163i	$0.198577\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	11.8310			0.582163
$414$		0			0
$415$	9.87751			0.484868
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.50413	−	6.06932i	−0.171188	−	0.296506i	0.767648	−	0.640872i	$-0.221427\pi$
$419$	−3.50413	−	6.06932i	−0.171188	−	0.296506i	−0.938835	+	0.344366i	$0.888094\pi$
$420$		0			0
$421$	3.35071	+	1.93453i	0.163304	+	0.0942834i	0.579425	−	0.815026i	$-0.303277\pi$
$421$	3.35071	+	1.93453i	0.163304	+	0.0942834i	−0.416121	+	0.909309i	$0.636611\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.9093	+	18.8954i	−0.529178	+	0.916562i
$426$		0			0
$427$	0.543001	−	0.313502i	0.0262776	−	0.0151714i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.94334i		−	0.286281i	−0.989702	−	0.143140i	$-0.954280\pi$
$431$		−	5.94334i		−	0.286281i	0.989702	−	0.143140i	$-0.0457201\pi$
$432$		0			0
$433$	25.3808			1.21973			0.609863	−	0.792507i	$-0.291225\pi$
$433$	25.3808			1.21973			0.609863	+	0.792507i	$0.291225\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.4294	+	6.02140i	−0.498904	+	0.288042i
$438$		0			0
$439$	3.20445	−	5.55026i	0.152940	−	0.264900i	−0.779367	−	0.626568i	$-0.784459\pi$
$439$	3.20445	−	5.55026i	0.152940	−	0.264900i	0.932307	+	0.361668i	$0.117793\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.08746	−	1.88354i	0.0516668	−	0.0894895i	−0.839035	−	0.544077i	$-0.816880\pi$
$443$	1.08746	−	1.88354i	0.0516668	−	0.0894895i	0.890702	+	0.454587i	$0.150213\pi$
$444$		0			0
$445$	−11.7342	−	20.3242i	−0.556255	−	0.963461i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3.54524i		−	0.167310i	−0.996495	−	0.0836550i	$-0.973341\pi$
$449$		−	3.54524i		−	0.167310i	0.996495	−	0.0836550i	$-0.0266594\pi$
$450$		0			0
$451$	39.0387			1.83826
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.443768	+	5.45641i	0.0208041	+	0.255800i
$456$		0			0
$457$	−14.7188	−	8.49788i	−0.688514	−	0.397514i	0.114541	−	0.993419i	$-0.463460\pi$
$457$	−14.7188	−	8.49788i	−0.688514	−	0.397514i	−0.803055	+	0.595905i	$0.796794\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.29670	+	4.79010i	0.386416	+	0.223097i	0.680606	−	0.732650i	$-0.261717\pi$
$461$	8.29670	+	4.79010i	0.386416	+	0.223097i	−0.294190	+	0.955747i	$0.595050\pi$
$462$		0			0
$463$	−14.5135	+	8.37937i	−0.674499	+	0.389422i	−0.797779	−	0.602950i	$-0.793992\pi$
$463$	−14.5135	+	8.37937i	−0.674499	+	0.389422i	0.123280	+	0.992372i	$0.460659\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.8428			−0.640567			−0.320283	−	0.947322i	$-0.603778\pi$
$467$	−13.8428			−0.640567			−0.320283	+	0.947322i	$0.603778\pi$
$468$		0			0
$469$	−0.958989			−0.0442820
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.9566	+	8.63521i	−0.687706	+	0.397047i
$474$		0			0
$475$	11.5839	+	6.68794i	0.531504	+	0.306864i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.5215	−	13.0028i	−1.02903	−	0.594112i	−0.112324	−	0.993672i	$-0.535830\pi$
$479$	−22.5215	−	13.0028i	−1.02903	−	0.594112i	−0.916707	+	0.399560i	$0.869163\pi$
$480$		0			0
$481$	25.5599	−	2.07878i	1.16543	−	0.0947840i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.89040			−0.403692
$486$		0			0
$487$			27.2672i			1.23560i	0.786337	+	0.617798i	$0.211975\pi$
$487$			27.2672i			1.23560i	−0.786337	+	0.617798i	$0.788025\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.57007	+	2.71945i	0.0708565	+	0.122727i	0.899277	−	0.437380i	$-0.144093\pi$
$491$	1.57007	+	2.71945i	0.0708565	+	0.122727i	−0.828420	+	0.560107i	$0.810760\pi$
$492$		0			0
$493$	−30.3817	+	52.6227i	−1.36833	+	2.37001i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.65476	+	11.5264i	−0.298507	+	0.517029i
$498$		0			0
$499$	−2.63431	+	1.52092i	−0.117928	+	0.0680858i	−0.557804	−	0.829973i	$-0.688356\pi$
$499$	−2.63431	+	1.52092i	−0.117928	+	0.0680858i	0.439876	+	0.898059i	$0.355022\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.7638			0.479933			0.239967	−	0.970781i	$-0.422864\pi$
$503$	10.7638			0.479933			0.239967	+	0.970781i	$0.422864\pi$
$504$		0			0
$505$			21.0559i			0.936974i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−29.0244	+	16.7573i	−1.28649	+	0.742752i	−0.978026	−	0.208485i	$-0.933147\pi$
$509$	−29.0244	+	16.7573i	−1.28649	+	0.742752i	−0.308459	+	0.951237i	$0.599813\pi$
$510$		0			0
$511$	−5.43765	+	9.41828i	−0.240547	+	0.416640i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.42541	−	4.28706i	−0.327203	−	0.188911i
$516$		0			0
$517$	21.8506	+	37.8464i	0.960990	+	1.66448i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.80816			0.254460			0.127230	−	0.991873i	$-0.459391\pi$
$521$	5.80816			0.254460			0.127230	+	0.991873i	$0.459391\pi$
$522$		0			0
$523$	−39.1530			−1.71204			−0.856020	−	0.516943i	$-0.827070\pi$
$523$	−39.1530			−1.71204			−0.856020	+	0.516943i	$0.827070\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	44.8462	−	25.8920i	1.95353	−	1.12787i
$528$		0			0
$529$	7.02802	−	12.1729i	0.305566	−	0.529256i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	20.5570	+	29.7440i	0.890423	+	1.28835i
$534$		0			0
$535$	−10.4636	+	6.04118i	−0.452382	+	0.261183i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			21.9033i			0.943443i
$540$		0			0
$541$			46.0331i			1.97912i	0.144130	+	0.989559i	$0.453962\pi$
$541$			46.0331i			1.97912i	−0.144130	+	0.989559i	$0.546038\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.8960	−	22.3364i	−0.552402	−	0.956788i
$546$		0			0
$547$	21.8241	−	37.8005i	0.933131	−	1.61623i	0.155198	−	0.987883i	$-0.450398\pi$
$547$	21.8241	−	37.8005i	0.933131	−	1.61623i	0.777933	−	0.628347i	$-0.216268\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	32.2604	+	18.6256i	1.37434	+	0.793475i
$552$		0			0
$553$	7.95943	−	4.59538i	0.338469	−	0.195415i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	25.5959i		−	1.08453i	−0.840206	−	0.542267i	$-0.817566\pi$
$557$		−	25.5959i		−	1.08453i	0.840206	−	0.542267i	$-0.182434\pi$
$558$		0			0
$559$	−14.4551	−	6.84846i	−0.611386	−	0.289659i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.6636	+	28.8622i	0.702286	+	1.21639i	0.967662	+	0.252250i	$0.0811705\pi$
$563$	16.6636	+	28.8622i	0.702286	+	1.21639i	−0.265376	+	0.964145i	$0.585496\pi$
$564$		0			0
$565$	6.25680	+	3.61236i	0.263225	+	0.151973i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	19.0371	−	32.9733i	0.798078	−	1.38231i	−0.122789	−	0.992433i	$-0.539184\pi$
$569$	19.0371	−	32.9733i	0.798078	−	1.38231i	0.920866	−	0.389878i	$-0.127483\pi$
$570$		0			0
$571$	0.538092	+	0.932002i	0.0225184	+	0.0390031i	0.877065	−	0.480372i	$-0.159498\pi$
$571$	0.538092	+	0.932002i	0.0225184	+	0.0390031i	−0.854547	+	0.519375i	$0.826165\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	9.93401			0.414277
$576$		0			0
$577$			1.08287i			0.0450804i	0.999746	+	0.0225402i	$0.00717537\pi$
$577$			1.08287i			0.0450804i	−0.999746	+	0.0225402i	$0.992825\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.46800	−	7.73881i	−0.185364	−	0.321060i
$582$		0			0
$583$	13.3045	+	7.68138i	0.551018	+	0.318130i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.75430	+	3.89960i	0.278780	+	0.160954i	0.632871	−	0.774257i	$-0.281876\pi$
$587$	6.75430	+	3.89960i	0.278780	+	0.160954i	−0.354091	+	0.935211i	$0.615210\pi$
$588$		0			0
$589$	−15.8731	−	27.4930i	−0.654040	−	1.13283i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	27.6766i		−	1.13654i	−0.822841	−	0.568271i	$-0.807612\pi$
$593$		−	27.6766i		−	1.13654i	0.822841	−	0.568271i	$-0.192388\pi$
$594$		0			0
$595$	−9.97316			−0.408860
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.164479	−	0.284885i	−0.00672041	−	0.0116401i	0.862646	−	0.505809i	$-0.168806\pi$
$599$	−0.164479	−	0.284885i	−0.00672041	−	0.0116401i	−0.869366	+	0.494169i	$0.835473\pi$
$600$		0			0
$601$	11.4590	−	19.8476i	0.467424	−	0.809602i	−0.531883	−	0.846818i	$-0.678516\pi$
$601$	11.4590	−	19.8476i	0.467424	−	0.809602i	0.999307	+	0.0372156i	$0.0118488\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.66180	+	2.69149i	0.189529	+	0.109425i
$606$		0			0
$607$	7.75554	+	13.4330i	0.314788	+	0.545228i	0.979392	−	0.201967i	$-0.0647333\pi$
$607$	7.75554	+	13.4330i	0.314788	+	0.545228i	−0.664605	+	0.747195i	$0.731400\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−17.3294	+	36.5774i	−0.701074	+	1.47976i
$612$		0			0
$613$			34.7194i			1.40230i	0.713013	+	0.701151i	$0.247330\pi$
$613$			34.7194i			1.40230i	−0.713013	+	0.701151i	$0.752670\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.04094	−	2.33304i	0.162682	−	0.0939246i	−0.416449	−	0.909159i	$-0.636725\pi$
$617$	4.04094	−	2.33304i	0.162682	−	0.0939246i	0.579131	+	0.815235i	$0.303392\pi$
$618$		0			0
$619$	30.4860	+	17.6011i	1.22534	+	0.707448i	0.966051	−	0.258352i	$-0.0831795\pi$
$619$	30.4860	+	17.6011i	1.22534	+	0.707448i	0.259286	+	0.965801i	$0.416513\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.6157	+	18.3870i	−0.425310	+	0.736659i
$624$		0			0
$625$	−1.32107	−	2.28816i	−0.0528427	−	0.0915263i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			46.7181i			1.86277i
$630$		0			0
$631$		−	14.5335i		−	0.578569i	−0.957243	−	0.289284i	$-0.906583\pi$
$631$		−	14.5335i		−	0.578569i	0.957243	−	0.289284i	$-0.0934174\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	16.4718	−	9.50997i	0.653662	−	0.377392i
$636$		0			0
$637$	−16.6884	+	11.5339i	−0.661217	+	0.456989i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.17292	−	14.1559i	0.322811	−	0.559125i	−0.658256	−	0.752794i	$-0.728706\pi$
$641$	8.17292	−	14.1559i	0.322811	−	0.559125i	0.981067	+	0.193669i	$0.0620389\pi$
$642$		0			0
$643$	25.7847	−	14.8868i	1.01685	−	0.587079i	0.103661	−	0.994613i	$-0.466944\pi$
$643$	25.7847	−	14.8868i	1.01685	−	0.587079i	0.913190	+	0.407534i	$0.133611\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.87304			−0.270207			−0.135103	−	0.990831i	$-0.543137\pi$
$647$	−6.87304			−0.270207			−0.135103	+	0.990831i	$0.543137\pi$
$648$		0			0
$649$	39.2978			1.54257
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.01601	+	1.75979i	0.0397597	+	0.0688658i	0.885220	−	0.465172i	$-0.154007\pi$
$653$	1.01601	+	1.75979i	0.0397597	+	0.0688658i	−0.845461	+	0.534038i	$0.820674\pi$
$654$		0			0
$655$	10.1050	+	5.83411i	0.394834	+	0.227957i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	21.1157	−	36.5735i	0.822551	−	1.42470i	−0.0812254	−	0.996696i	$-0.525883\pi$
$659$	21.1157	−	36.5735i	0.822551	−	1.42470i	0.903777	−	0.428005i	$-0.140783\pi$
$660$		0			0
$661$	24.5947	−	14.1998i	0.956625	−	0.552308i	0.0614922	−	0.998108i	$-0.480414\pi$
$661$	24.5947	−	14.1998i	0.956625	−	0.552308i	0.895133	+	0.445800i	$0.147081\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			6.11406i			0.237093i
$666$		0			0
$667$	27.6657			1.07122
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.80364	−	1.04133i	0.0696286	−	0.0402001i
$672$		0			0
$673$	18.9395	−	32.8042i	0.730065	−	1.26451i	−0.226789	−	0.973944i	$-0.572823\pi$
$673$	18.9395	−	32.8042i	0.730065	−	1.26451i	0.956855	−	0.290567i	$-0.0938438\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.86904	−	3.23726i	0.0718329	−	0.124418i	−0.827872	−	0.560917i	$-0.810449\pi$
$677$	1.86904	−	3.23726i	0.0718329	−	0.124418i	0.899705	+	0.436499i	$0.143782\pi$
$678$		0			0
$679$	4.02149	+	6.96543i	0.154331	+	0.267309i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	17.5633i		−	0.672042i	−0.941854	−	0.336021i	$-0.890919\pi$
$683$		−	17.5633i		−	0.672042i	0.941854	−	0.336021i	$-0.109081\pi$
$684$		0			0
$685$	1.79983			0.0687679
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.15339	+	14.1817i	0.0439408	+	0.540281i
$690$		0			0
$691$	17.5312	+	10.1216i	0.666918	+	0.385045i	0.794908	−	0.606730i	$-0.207519\pi$
$691$	17.5312	+	10.1216i	0.666918	+	0.385045i	−0.127990	+	0.991776i	$0.540852\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−20.2839	−	11.7109i	−0.769414	−	0.444221i
$696$		0			0
$697$	−57.0442	+	32.9345i	−2.16070	+	1.24748i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.7160			0.820202			0.410101	−	0.912040i	$-0.365493\pi$
$701$	21.7160			0.820202			0.410101	+	0.912040i	$0.365493\pi$
$702$		0			0
$703$	28.6406			1.08020
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.4968	−	9.52444i	0.620427	−	0.358203i
$708$		0			0
$709$	12.5657	+	7.25481i	0.471915	+	0.272460i	0.717041	−	0.697031i	$-0.245496\pi$
$709$	12.5657	+	7.25481i	0.471915	+	0.272460i	−0.245126	+	0.969491i	$0.578829\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−20.4185	−	11.7886i	−0.764680	−	0.441488i
$714$		0			0
$715$	1.47402	+	18.1241i	0.0551253	+	0.677802i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.4539			−0.762801			−0.381400	−	0.924410i	$-0.624558\pi$
$719$	−20.4539			−0.762801			−0.381400	+	0.924410i	$0.624558\pi$
$720$		0			0
$721$			7.75686i			0.288881i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.3641	−	26.6114i	−0.570607	−	0.988321i
$726$		0			0
$727$	4.66361	−	8.07761i	0.172964	−	0.299582i	−0.766491	−	0.642255i	$-0.777999\pi$
$727$	4.66361	−	8.07761i	0.172964	−	0.299582i	0.939455	+	0.342673i	$0.111332\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14.5700	−	25.2359i	0.538890	−	0.933384i
$732$		0			0
$733$	−8.45225	+	4.87991i	−0.312191	+	0.180243i	−0.647906	−	0.761720i	$-0.724355\pi$
$733$	−8.45225	+	4.87991i	−0.312191	+	0.180243i	0.335716	+	0.941963i	$0.391022\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.18539			−0.117335
$738$		0			0
$739$			14.3579i			0.528164i	0.964500	+	0.264082i	$0.0850690\pi$
$739$			14.3579i			0.528164i	−0.964500	+	0.264082i	$0.914931\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17.0494	+	9.84349i	−0.625483	+	0.361123i	−0.779001	−	0.627023i	$-0.784273\pi$
$743$	−17.0494	+	9.84349i	−0.625483	+	0.361123i	0.153518	+	0.988146i	$0.450940\pi$
$744$		0			0
$745$	5.90454	−	10.2270i	0.216326	−	0.374687i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.46626	+	5.46535i	0.345890	+	0.199700i
$750$		0			0
$751$	−2.92800	−	5.07145i	−0.106844	−	0.185060i	0.807646	−	0.589668i	$-0.200741\pi$
$751$	−2.92800	−	5.07145i	−0.106844	−	0.185060i	−0.914490	+	0.404608i	$0.867408\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.56740			−0.129831
$756$		0			0
$757$	−10.4064			−0.378228			−0.189114	−	0.981955i	$-0.560562\pi$
$757$	−10.4064			−0.378228			−0.189114	+	0.981955i	$0.560562\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	40.2552	−	23.2414i	1.45925	−	0.842499i	0.460276	−	0.887776i	$-0.347750\pi$
$761$	40.2552	−	23.2414i	1.45925	−	0.842499i	0.998974	+	0.0452769i	$0.0144170\pi$
$762$		0			0
$763$	−11.6667	+	20.2074i	−0.422364	+	0.731556i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.6935	+	29.9414i	0.747198	+	1.08112i
$768$		0			0
$769$	−19.1017	+	11.0283i	−0.688823	+	0.397692i	−0.803171	−	0.595748i	$-0.796856\pi$
$769$	−19.1017	+	11.0283i	−0.688823	+	0.397692i	0.114348	+	0.993441i	$0.463522\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	3.52236i		−	0.126690i	−0.997992	−	0.0633452i	$-0.979823\pi$
$773$		−	3.52236i		−	0.126690i	0.997992	−	0.0633452i	$-0.0201769\pi$
$774$		0			0
$775$			26.1872i			0.940672i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.1905	+	34.9710i	0.723400	+	1.25297i
$780$		0			0
$781$	−22.1045	+	38.2862i	−0.790962	+	1.36999i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	23.1983	+	13.3936i	0.827983	+	0.478036i
$786$		0			0
$787$	−20.3257	+	11.7350i	−0.724531	+	0.418308i	−0.816418	−	0.577461i	$-0.804044\pi$
$787$	−20.3257	+	11.7350i	−0.724531	+	0.418308i	0.0918869	+	0.995769i	$0.470710\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	6.53608i		−	0.232396i
$792$		0			0
$793$	1.74316	+	0.825864i	0.0619014	+	0.0293273i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.665795	−	1.15319i	−0.0235837	−	0.0408481i	0.853993	−	0.520285i	$-0.174174\pi$
$797$	−0.665795	−	1.15319i	−0.0235837	−	0.0408481i	−0.877576	+	0.479437i	$0.840841\pi$
$798$		0			0
$799$	−63.8573	−	36.8680i	−2.25911	−	1.30430i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.0618	+	31.2839i	−0.637385	+	1.10398i
$804$		0			0
$805$	2.27040	+	3.93244i	0.0800209	+	0.138600i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	51.9672			1.82707			0.913535	−	0.406760i	$-0.133342\pi$
$809$	51.9672			1.82707			0.913535	+	0.406760i	$0.133342\pi$
$810$		0			0
$811$		−	35.6357i		−	1.25134i	−0.780088	−	0.625670i	$-0.784826\pi$
$811$		−	35.6357i		−	1.25134i	0.780088	−	0.625670i	$-0.215174\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.59344	−	11.4202i	−0.230958	−	0.400031i
$816$		0			0
$817$	−15.4709	−	8.93213i	−0.541258	−	0.312496i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.92925	+	5.15531i	0.311633	+	0.179921i	0.647657	−	0.761932i	$-0.275749\pi$
$821$	8.92925	+	5.15531i	0.311633	+	0.179921i	−0.336024	+	0.941853i	$0.609082\pi$
$822$		0			0
$823$	20.9553	+	36.2957i	0.730456	+	1.26519i	0.956688	+	0.291114i	$0.0940259\pi$
$823$	20.9553	+	36.2957i	0.730456	+	1.26519i	−0.226232	+	0.974073i	$0.572641\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	14.6304i		−	0.508748i	−0.967106	−	0.254374i	$-0.918131\pi$
$827$		−	14.6304i		−	0.508748i	0.967106	−	0.254374i	$-0.0818694\pi$
$828$		0			0
$829$	−5.45165			−0.189344			−0.0946719	−	0.995509i	$-0.530180\pi$
$829$	−5.45165			−0.189344			−0.0946719	+	0.995509i	$0.530180\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.4785	−	32.0057i	−0.640241	−	1.10893i
$834$		0			0
$835$	−1.24817	+	2.16190i	−0.0431948	+	0.0748156i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.64378	−	3.25844i	−0.194845	−	0.112494i	0.399404	−	0.916775i	$-0.369217\pi$
$839$	−5.64378	−	3.25844i	−0.194845	−	0.112494i	−0.594249	+	0.804281i	$0.702551\pi$
$840$		0			0
$841$	−28.2881	−	48.9965i	−0.975453	−	1.68953i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−13.0327	+	10.6669i	−0.448339	+	0.366951i
$846$		0			0
$847$		−	4.86989i		−	0.167331i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	18.4211	−	10.6354i	0.631466	−	0.364577i
$852$		0			0
$853$	−27.5422	−	15.9015i	−0.943029	−	0.544458i	−0.0521202	−	0.998641i	$-0.516598\pi$
$853$	−27.5422	−	15.9015i	−0.943029	−	0.544458i	−0.890908	+	0.454183i	$0.849931\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.82625	−	3.16316i	0.0623836	−	0.108052i	−0.833147	−	0.553052i	$-0.813463\pi$
$857$	1.82625	−	3.16316i	0.0623836	−	0.108052i	0.895530	+	0.445000i	$0.146796\pi$
$858$		0			0
$859$	0.814178	+	1.41020i	0.0277794	+	0.0481153i	0.879581	−	0.475749i	$-0.157823\pi$
$859$	0.814178	+	1.41020i	0.0277794	+	0.0481153i	−0.851801	+	0.523865i	$0.824490\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.90203i			0.0647458i	0.999476	+	0.0323729i	$0.0103064\pi$
$863$			1.90203i			0.0647458i	−0.999476	+	0.0323729i	$0.989694\pi$
$864$		0			0
$865$		−	0.898513i		−	0.0305504i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	26.4381	−	15.2641i	0.896852	−	0.517798i
$870$		0			0
$871$	−1.67736	−	2.42698i	−0.0568353	−	0.0822350i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.31755	−	10.9423i	0.213572	−	0.369918i
$876$		0			0
$877$	12.4613	−	7.19456i	0.420790	−	0.242943i	−0.274625	−	0.961551i	$-0.588554\pi$
$877$	12.4613	−	7.19456i	0.420790	−	0.242943i	0.695415	+	0.718608i	$0.255221\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	47.2375			1.59147			0.795736	−	0.605644i	$-0.207085\pi$
$881$	47.2375			1.59147			0.795736	+	0.605644i	$0.207085\pi$
$882$		0			0
$883$	10.5214			0.354073			0.177036	−	0.984204i	$-0.443349\pi$
$883$	10.5214			0.354073			0.177036	+	0.984204i	$0.443349\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.5854	−	32.1908i	−0.624036	−	1.08086i	−0.988726	−	0.149733i	$-0.952159\pi$
$887$	−18.5854	−	32.1908i	−0.624036	−	1.08086i	0.364691	−	0.931129i	$-0.381175\pi$
$888$		0			0
$889$	−14.9017	−	8.60350i	−0.499787	−	0.288552i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−22.6020	+	39.1478i	−0.756347	+	1.31003i
$894$		0			0
$895$	18.1709	−	10.4909i	0.607385	−	0.350674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			72.9300i			2.43235i
$900$		0			0
$901$	−25.9212			−0.863561
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.99033	+	4.61322i	−0.265607	+	0.153349i
$906$		0			0
$907$	−4.73223	+	8.19647i	−0.157131	+	0.272159i	−0.933833	−	0.357709i	$-0.883558\pi$
$907$	−4.73223	+	8.19647i	−0.157131	+	0.272159i	0.776702	+	0.629868i	$0.216891\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.3065	−	24.7795i	0.473995	−	0.820983i	−0.525562	−	0.850755i	$-0.676145\pi$
$911$	14.3065	−	24.7795i	0.473995	−	0.820983i	0.999557	+	0.0297724i	$0.00947826\pi$
$912$		0			0
$913$	−14.8410	−	25.7053i	−0.491164	−	0.850722i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	10.5560i		−	0.348591i
$918$		0			0
$919$	−12.3995			−0.409021			−0.204510	−	0.978864i	$-0.565560\pi$
$919$	−12.3995			−0.409021			−0.204510	+	0.978864i	$0.565560\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−40.8104	+	3.31910i	−1.34329	+	0.109249i
$924$		0			0
$925$	−20.4602	−	11.8127i	−0.672727	−	0.388399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.0132	+	9.82260i	0.558186	+	0.322269i	0.752417	−	0.658687i	$-0.228888\pi$
$929$	17.0132	+	9.82260i	0.558186	+	0.322269i	−0.194231	+	0.980956i	$0.562221\pi$
$930$		0			0
$931$	−19.6211	+	11.3282i	−0.643055	+	0.371268i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−33.1270			−1.08337
$936$		0			0
$937$	−45.1494			−1.47497			−0.737483	−	0.675366i	$-0.763986\pi$
$937$	−45.1494			−1.47497			−0.737483	+	0.675366i	$0.763986\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10.1924	−	5.88460i	0.332263	−	0.191832i	−0.324582	−	0.945857i	$-0.605224\pi$
$941$	10.1924	−	5.88460i	0.332263	−	0.191832i	0.656846	+	0.754025i	$0.271890\pi$
$942$		0			0
$943$	25.9723	+	14.9951i	0.845774	+	0.488308i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.4450	+	7.76245i	0.436902	+	0.252246i	0.702283	−	0.711898i	$-0.252164\pi$
$947$	13.4450	+	7.76245i	0.436902	+	0.252246i	−0.265380	+	0.964144i	$0.585498\pi$
$948$		0			0
$949$	−33.3465	+	2.71205i	−1.08247	+	0.0880370i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	40.6782			1.31770			0.658848	−	0.752276i	$-0.271044\pi$
$953$	40.6782			1.31770			0.658848	+	0.752276i	$0.271044\pi$
$954$		0			0
$955$		−	17.1944i		−	0.556398i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.814136	−	1.41013i	−0.0262898	−	0.0455353i
$960$		0			0
$961$	15.5762	−	26.9788i	0.502459	−	0.870285i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	12.5424	−	21.7240i	0.403753	−	0.699320i
$966$		0			0
$967$	16.3248	−	9.42513i	0.524970	−	0.303092i	−0.213996	−	0.976835i	$-0.568648\pi$
$967$	16.3248	−	9.42513i	0.524970	−	0.303092i	0.738966	+	0.673743i	$0.235314\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25.4536			−0.816845			−0.408423	−	0.912793i	$-0.633921\pi$
$971$	−25.4536			−0.816845			−0.408423	+	0.912793i	$0.633921\pi$
$972$		0			0
$973$			21.1894i			0.679300i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.81800	+	2.20432i	−0.122149	+	0.0705225i	−0.559829	−	0.828608i	$-0.689133\pi$
$977$	−3.81800	+	2.20432i	−0.122149	+	0.0705225i	0.437681	+	0.899130i	$0.355800\pi$
$978$		0			0
$979$	−35.2613	+	61.0744i	−1.12696	+	1.95195i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	31.7489	+	18.3303i	1.01263	+	0.584644i	0.911962	−	0.410275i	$-0.134567\pi$
$983$	31.7489	+	18.3303i	1.01263	+	0.584644i	0.100672	+	0.994920i	$0.467901\pi$
$984$		0			0
$985$	−1.34301	−	2.32616i	−0.0427918	−	0.0741176i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−13.2674			−0.421880
$990$		0			0
$991$	21.7867			0.692078			0.346039	−	0.938220i	$-0.387526\pi$
$991$	21.7867			0.692078			0.346039	+	0.938220i	$0.387526\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.3287	+	5.96326i	−0.327441	+	0.189048i
$996$		0			0
$997$	−0.925027	+	1.60219i	−0.0292959	+	0.0507420i	−0.880302	−	0.474414i	$-0.842660\pi$
$997$	−0.925027	+	1.60219i	−0.0292959	+	0.0507420i	0.851006	+	0.525156i	$0.175993\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.cw.b.1585.28		80
3.2	odd	2		936.2.cw.b.25.11	✓	80
9.4	even	3	inner	2808.2.cw.b.2521.13		80
9.5	odd	6		936.2.cw.b.337.12	yes	80
13.12	even	2	inner	2808.2.cw.b.1585.13		80
39.38	odd	2		936.2.cw.b.25.12	yes	80
117.77	odd	6		936.2.cw.b.337.11	yes	80
117.103	even	6	inner	2808.2.cw.b.2521.28		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.11	✓	80	3.2	odd	2
936.2.cw.b.25.12	yes	80	39.38	odd	2
936.2.cw.b.337.11	yes	80	117.77	odd	6
936.2.cw.b.337.12	yes	80	9.5	odd	6
2808.2.cw.b.1585.13		80	13.12	even	2	inner
2808.2.cw.b.1585.28		80	1.1	even	1	trivial
2808.2.cw.b.2521.13		80	9.4	even	3	inner
2808.2.cw.b.2521.28		80	117.103	even	6	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:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.cw (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.12193 - 0.647747i) q^{5} +(-1.01499 - 0.586005i) q^{7} +(-3.37140 - 1.94648i) q^{11} +(-0.292273 - 3.59369i) q^{13} +6.56850 q^{17} -4.02682i q^{19} +(-1.49532 - 2.58997i) q^{23} +(-1.66085 + 2.87667i) q^{25} +(-4.62537 + 8.01138i) q^{29} +(6.82747 - 3.94184i) q^{31} -1.51833 q^{35} +7.11245i q^{37} +(-8.68452 + 5.01401i) q^{41} +(2.21816 - 3.84196i) q^{43} +(-9.72175 - 5.61286i) q^{47} +(-2.81320 - 4.87260i) q^{49} -3.94629 q^{53} -5.04331 q^{55} +(-8.74216 + 5.04729i) q^{59} +(-0.267490 + 0.463307i) q^{61} +(-2.65571 - 3.84255i) q^{65} +(0.708619 - 0.409121i) q^{67} -11.3561i q^{71} -9.27918i q^{73} +(2.28130 + 3.95132i) q^{77} +(-3.92094 + 6.79126i) q^{79} +(6.60302 + 3.81226i) q^{83} +(7.36940 - 4.25473i) q^{85} -18.1154i q^{89} +(-1.80926 + 3.81883i) q^{91} +(-2.60836 - 4.51782i) q^{95} +(-5.94315 - 3.43128i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 80 * q + 6 * q^13 + 4 * q^17 - 10 * q^23 + 44 * q^25 + 52 * q^35 - 26 * q^43 + 48 * q^49 - 60 * q^53 - 16 * q^55 - 10 * q^61 + 26 * q^65 - 32 * q^77 + 6 * q^79 - 4 * q^91 - 8 * q^95

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