LMFDB - Embedded newform 2808.2.cw.b.1585.20

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			1585.20
Character	$\chi$	$=$	2808.1585
Dual form			2808.2.cw.b.2521.20

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0518638 + 0.0299436i) q^{5} +(3.08098 + 1.77880i) q^{7} +(1.46178 + 0.843959i) q^{11} +(-2.26535 + 2.80503i) q^{13} -4.45763 q^{17} -0.259933i q^{19} +(-0.0530409 - 0.0918694i) q^{23} +(-2.49821 + 4.32702i) q^{25} +(-5.01275 + 8.68234i) q^{29} +(-6.75507 + 3.90004i) q^{31} -0.213055 q^{35} -2.32326i q^{37} +(3.88390 - 2.24237i) q^{41} +(2.23994 - 3.87969i) q^{43} +(-2.55586 - 1.47563i) q^{47} +(2.82827 + 4.89871i) q^{49} +6.81558 q^{53} -0.101085 q^{55} +(-10.7981 + 6.23426i) q^{59} +(0.436766 - 0.756501i) q^{61} +(0.0334972 - 0.213312i) q^{65} +(4.54557 - 2.62439i) q^{67} -6.85509i q^{71} -14.4296i q^{73} +(3.00247 + 5.20043i) q^{77} +(-7.70339 + 13.3427i) q^{79} +(-4.76733 - 2.75242i) q^{83} +(0.231190 - 0.133477i) q^{85} +3.55749i q^{89} +(-11.9691 + 4.61261i) q^{91} +(0.00778333 + 0.0134811i) q^{95} +(-5.53546 - 3.19590i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.0518638	+	0.0299436i	−0.0231942	+	0.0133912i	−0.511552	−	0.859252i	$-0.670929\pi$
$5$	−0.0518638	+	0.0299436i	−0.0231942	+	0.0133912i	0.488358	+	0.872643i	$0.337596\pi$
$6$		0			0
$7$	3.08098	+	1.77880i	1.16450	+	0.672324i	0.952378	−	0.304920i	$-0.0986296\pi$
$7$	3.08098	+	1.77880i	1.16450	+	0.672324i	0.212121	+	0.977243i	$0.431963\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.46178	+	0.843959i	0.440743	+	0.254463i	0.703913	−	0.710286i	$-0.251435\pi$
$11$	1.46178	+	0.843959i	0.440743	+	0.254463i	−0.263170	+	0.964750i	$0.584768\pi$
$12$		0			0
$13$	−2.26535	+	2.80503i	−0.628296	+	0.777975i
$13$	−2.26535	+	2.80503i	−0.628296	+	0.777975i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.45763			−1.08113			−0.540567	−	0.841301i	$-0.681790\pi$
$17$	−4.45763			−1.08113			−0.540567	+	0.841301i	$0.681790\pi$
$18$		0			0
$19$		−	0.259933i		−	0.0596327i	−0.999555	−	0.0298164i	$-0.990508\pi$
$19$		−	0.259933i		−	0.0596327i	0.999555	−	0.0298164i	$-0.00949225\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.0530409	−	0.0918694i	−0.0110598	−	0.0191561i	0.860443	−	0.509547i	$-0.170187\pi$
$23$	−0.0530409	−	0.0918694i	−0.0110598	−	0.0191561i	−0.871502	+	0.490391i	$0.836854\pi$
$24$		0			0
$25$	−2.49821	+	4.32702i	−0.499641	+	0.865404i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.01275	+	8.68234i	−0.930844	+	1.61227i	−0.148962	+	0.988843i	$0.547593\pi$
$29$	−5.01275	+	8.68234i	−0.930844	+	1.61227i	−0.781882	+	0.623427i	$0.785740\pi$
$30$		0			0
$31$	−6.75507	+	3.90004i	−1.21325	+	0.700468i	−0.963465	−	0.267834i	$-0.913692\pi$
$31$	−6.75507	+	3.90004i	−1.21325	+	0.700468i	−0.249781	+	0.968302i	$0.580359\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.213055			−0.0360128
$36$		0			0
$37$		−	2.32326i		−	0.381941i	−0.981596	−	0.190970i	$-0.938837\pi$
$37$		−	2.32326i		−	0.381941i	0.981596	−	0.190970i	$-0.0611635\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.88390	−	2.24237i	0.606563	−	0.350199i	−0.165056	−	0.986284i	$-0.552781\pi$
$41$	3.88390	−	2.24237i	0.606563	−	0.350199i	0.771619	+	0.636085i	$0.219447\pi$
$42$		0			0
$43$	2.23994	−	3.87969i	0.341588	−	0.591647i	−0.643140	−	0.765748i	$-0.722369\pi$
$43$	2.23994	−	3.87969i	0.341588	−	0.591647i	0.984728	+	0.174101i	$0.0557021\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.55586	−	1.47563i	−0.372810	−	0.215242i	0.301875	−	0.953347i	$-0.402387\pi$
$47$	−2.55586	−	1.47563i	−0.372810	−	0.215242i	−0.674686	+	0.738105i	$0.735721\pi$
$48$		0			0
$49$	2.82827	+	4.89871i	0.404039	+	0.699816i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.81558			0.936193			0.468096	−	0.883677i	$-0.344940\pi$
$53$	6.81558			0.936193			0.468096	+	0.883677i	$0.344940\pi$
$54$		0			0
$55$	−0.101085			−0.0136302
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.7981	+	6.23426i	−1.40579	+	0.811632i	−0.994978	−	0.100089i	$-0.968087\pi$
$59$	−10.7981	+	6.23426i	−1.40579	+	0.811632i	−0.410810	+	0.911721i	$0.634754\pi$
$60$		0			0
$61$	0.436766	−	0.756501i	0.0559222	−	0.0968601i	−0.836709	−	0.547648i	$-0.815523\pi$
$61$	0.436766	−	0.756501i	0.0559222	−	0.0968601i	0.892631	+	0.450788i	$0.148857\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.0334972	−	0.213312i	0.00415482	−	0.0264581i
$66$		0			0
$67$	4.54557	−	2.62439i	0.555330	−	0.320620i	−0.195939	−	0.980616i	$-0.562775\pi$
$67$	4.54557	−	2.62439i	0.555330	−	0.320620i	0.751269	+	0.659996i	$0.229442\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.85509i		−	0.813550i	−0.913528	−	0.406775i	$-0.866653\pi$
$71$		−	6.85509i		−	0.813550i	0.913528	−	0.406775i	$-0.133347\pi$
$72$		0			0
$73$		−	14.4296i		−	1.68886i	−0.535669	−	0.844428i	$-0.679940\pi$
$73$		−	14.4296i		−	1.68886i	0.535669	−	0.844428i	$-0.320060\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.00247	+	5.20043i	0.342163	+	0.592644i
$78$		0			0
$79$	−7.70339	+	13.3427i	−0.866699	+	1.50117i	−0.00134973	+	0.999999i	$0.500430\pi$
$79$	−7.70339	+	13.3427i	−0.866699	+	1.50117i	−0.865350	+	0.501168i	$0.832904\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.76733	−	2.75242i	−0.523283	−	0.302117i	0.214994	−	0.976615i	$-0.431027\pi$
$83$	−4.76733	−	2.75242i	−0.523283	−	0.302117i	−0.738277	+	0.674498i	$0.764360\pi$
$84$		0			0
$85$	0.231190	−	0.133477i	0.0250760	−	0.0144777i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.55749i			0.377093i	0.982064	+	0.188547i	$0.0603777\pi$
$89$			3.55749i			0.377093i	−0.982064	+	0.188547i	$0.939622\pi$
$90$		0			0
$91$	−11.9691	+	4.61261i	−1.25470	+	0.483532i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.00778333	+	0.0134811i	0.000798553	+	0.00138313i
$96$		0			0
$97$	−5.53546	−	3.19590i	−0.562041	−	0.324494i	0.191924	−	0.981410i	$-0.438527\pi$
$97$	−5.53546	−	3.19590i	−0.562041	−	0.324494i	−0.753964	+	0.656916i	$0.771861\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.0638338	−	0.110563i	0.00635170	−	0.0110015i	−0.862832	−	0.505491i	$-0.831312\pi$
$101$	0.0638338	−	0.110563i	0.00635170	−	0.0110015i	0.869184	+	0.494489i	$0.164645\pi$
$102$		0			0
$103$	2.01566	+	3.49123i	0.198609	+	0.344001i	0.948078	−	0.318039i	$-0.103024\pi$
$103$	2.01566	+	3.49123i	0.198609	+	0.344001i	−0.749468	+	0.662040i	$0.769691\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.21253			0.213894			0.106947	−	0.994265i	$-0.465893\pi$
$107$	2.21253			0.213894			0.106947	+	0.994265i	$0.465893\pi$
$108$		0			0
$109$		−	8.24035i		−	0.789283i	−0.918835	−	0.394641i	$-0.870869\pi$
$109$		−	8.24035i		−	0.789283i	0.918835	−	0.394641i	$-0.129131\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.38961	+	11.0671i	0.601084	+	1.04111i	0.992657	+	0.120961i	$0.0385976\pi$
$113$	6.38961	+	11.0671i	0.601084	+	1.04111i	−0.391573	+	0.920147i	$0.628069\pi$
$114$		0			0
$115$	0.00550180	+	0.00317647i	0.000513046	+	0.000296207i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13.7338	−	7.92924i	−1.25898	−	0.726872i
$120$		0			0
$121$	−4.07547	−	7.05892i	−0.370497	−	0.641720i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	0.598657i		−	0.0535455i
$126$		0			0
$127$	−14.5269			−1.28906			−0.644529	−	0.764580i	$-0.722946\pi$
$127$	−14.5269			−1.28906			−0.644529	+	0.764580i	$0.722946\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.46503	+	16.3939i	0.826963	+	1.43234i	0.900409	+	0.435043i	$0.143267\pi$
$131$	9.46503	+	16.3939i	0.826963	+	1.43234i	−0.0734460	+	0.997299i	$0.523400\pi$
$132$		0			0
$133$	0.462369	−	0.800847i	0.0400925	−	0.0694423i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.4386	+	7.18140i	1.06270	+	0.613549i	0.926177	−	0.377088i	$-0.123075\pi$
$137$	12.4386	+	7.18140i	1.06270	+	0.613549i	0.136520	+	0.990637i	$0.456408\pi$
$138$		0			0
$139$	9.82150	+	17.0113i	0.833048	+	1.44288i	0.895610	+	0.444841i	$0.146740\pi$
$139$	9.82150	+	17.0113i	0.833048	+	1.44288i	−0.0625611	+	0.998041i	$0.519927\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.67877	+	2.18847i	−0.474883	+	0.183009i
$144$		0			0
$145$		−	0.600399i		−	0.0498604i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	9.59543	−	5.53993i	0.786089	−	0.453848i	−0.0524951	−	0.998621i	$-0.516717\pi$
$149$	9.59543	−	5.53993i	0.786089	−	0.453848i	0.838584	+	0.544773i	$0.183384\pi$
$150$		0			0
$151$	13.7534	+	7.94054i	1.11924	+	0.646192i	0.941206	−	0.337833i	$-0.109694\pi$
$151$	13.7534	+	7.94054i	1.11924	+	0.646192i	0.178031	+	0.984025i	$0.443027\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.233562	−	0.404542i	0.0187602	−	0.0324936i
$156$		0			0
$157$	9.20507	+	15.9436i	0.734644	+	1.27244i	0.954879	+	0.296994i	$0.0959843\pi$
$157$	9.20507	+	15.9436i	0.734644	+	1.27244i	−0.220235	+	0.975447i	$0.570682\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.377397i		−	0.0297430i
$162$		0			0
$163$			8.14878i			0.638262i	0.947711	+	0.319131i	$0.103391\pi$
$163$			8.14878i			0.638262i	−0.947711	+	0.319131i	$0.896609\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.6694	+	6.16001i	−0.825626	+	0.476676i	−0.852353	−	0.522967i	$-0.824825\pi$
$167$	−10.6694	+	6.16001i	−0.825626	+	0.476676i	0.0267265	+	0.999643i	$0.491492\pi$
$168$		0			0
$169$	−2.73636	−	12.7088i	−0.210489	−	0.977596i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.12447	+	8.87585i	−0.389606	+	0.674818i	−0.992397	−	0.123082i	$-0.960722\pi$
$173$	−5.12447	+	8.87585i	−0.389606	+	0.674818i	0.602790	+	0.797900i	$0.294056\pi$
$174$		0			0
$175$	−15.3938	+	8.88763i	−1.16366	+	0.671842i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.01007			0.374470			0.187235	−	0.982315i	$-0.440047\pi$
$179$	5.01007			0.374470			0.187235	+	0.982315i	$0.440047\pi$
$180$		0			0
$181$	−17.4233			−1.29506			−0.647531	−	0.762039i	$-0.724198\pi$
$181$	−17.4233			−1.29506			−0.647531	+	0.762039i	$0.724198\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.0695666	+	0.120493i	0.00511464	+	0.00885881i
$186$		0			0
$187$	−6.51607	−	3.76206i	−0.476503	−	0.275109i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.12065	+	5.40513i	−0.225803	+	0.391102i	−0.956560	−	0.291536i	$-0.905834\pi$
$191$	−3.12065	+	5.40513i	−0.225803	+	0.391102i	0.730757	+	0.682637i	$0.239167\pi$
$192$		0			0
$193$	0.387344	−	0.223633i	0.0278816	−	0.0160975i	−0.485994	−	0.873962i	$-0.661542\pi$
$193$	0.387344	−	0.223633i	0.0278816	−	0.0160975i	0.513876	+	0.857864i	$0.328209\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.3756i			1.52295i	0.648197	+	0.761473i	$0.275523\pi$
$197$			21.3756i			1.52295i	−0.648197	+	0.761473i	$0.724477\pi$
$198$		0			0
$199$	−18.6246			−1.32026			−0.660131	−	0.751151i	$-0.729499\pi$
$199$	−18.6246			−1.32026			−0.660131	+	0.751151i	$0.729499\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−30.8883	+	17.8334i	−2.16793	+	1.25166i
$204$		0			0
$205$	−0.134289	+	0.232595i	−0.00937916	+	0.0162452i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.219373	−	0.379965i	0.0151743	−	0.0262827i
$210$		0			0
$211$	−5.44748	−	9.43531i	−0.375020	−	0.649553i	0.615310	−	0.788285i	$-0.289031\pi$
$211$	−5.44748	−	9.43531i	−0.375020	−	0.649553i	−0.990330	+	0.138732i	$0.955697\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.268287i			0.0182970i
$216$		0			0
$217$	−27.7496			−1.88377
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.0981	−	12.5038i	0.679272	−	0.841095i
$222$		0			0
$223$	−8.90104	−	5.13902i	−0.596057	−	0.344134i	0.171432	−	0.985196i	$-0.445161\pi$
$223$	−8.90104	−	5.13902i	−0.596057	−	0.344134i	−0.767489	+	0.641062i	$0.778494\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	15.9627	+	9.21609i	1.05948	+	0.611694i	0.925290	−	0.379261i	$-0.123822\pi$
$227$	15.9627	+	9.21609i	1.05948	+	0.611694i	0.134195	+	0.990955i	$0.457155\pi$
$228$		0			0
$229$	17.1859	−	9.92226i	1.13567	−	0.655681i	0.190318	−	0.981723i	$-0.439048\pi$
$229$	17.1859	−	9.92226i	1.13567	−	0.655681i	0.945356	+	0.326041i	$0.105715\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.7692			0.705511			0.352755	−	0.935716i	$-0.385245\pi$
$233$	10.7692			0.705511			0.352755	+	0.935716i	$0.385245\pi$
$234$		0			0
$235$	0.176742			0.0115294
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.3399	−	12.3206i	1.38036	−	0.796952i	0.388160	−	0.921592i	$-0.373111\pi$
$239$	21.3399	−	12.3206i	1.38036	−	0.796952i	0.992202	+	0.124640i	$0.0397776\pi$
$240$		0			0
$241$	−7.69000	−	4.43983i	−0.495357	−	0.285994i	0.231437	−	0.972850i	$-0.425657\pi$
$241$	−7.69000	−	4.43983i	−0.495357	−	0.285994i	−0.726794	+	0.686855i	$0.758991\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.293370	−	0.169377i	−0.0187427	−	0.0108211i
$246$		0			0
$247$	0.729119	+	0.588840i	0.0463928	+	0.0374670i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.04133			−0.381325			−0.190663	−	0.981656i	$-0.561064\pi$
$251$	−6.04133			−0.381325			−0.190663	+	0.981656i	$0.561064\pi$
$252$		0			0
$253$		−	0.179057i		−	0.0112572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.0273	+	17.3678i	0.625485	+	1.08337i	0.988447	+	0.151568i	$0.0484324\pi$
$257$	10.0273	+	17.3678i	0.625485	+	1.08337i	−0.362961	+	0.931804i	$0.618234\pi$
$258$		0			0
$259$	4.13261	−	7.15789i	0.256788	−	0.444770i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.659359	−	1.14204i	0.0406578	−	0.0704214i	−0.844980	−	0.534797i	$-0.820388\pi$
$263$	0.659359	−	1.14204i	0.0406578	−	0.0704214i	0.885638	+	0.464376i	$0.153721\pi$
$264$		0			0
$265$	−0.353482	+	0.204083i	−0.0217142	+	0.0125367i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	30.5766			1.86429			0.932143	−	0.362090i	$-0.117937\pi$
$269$	30.5766			1.86429			0.932143	+	0.362090i	$0.117937\pi$
$270$		0			0
$271$			17.8738i			1.08575i	0.839812	+	0.542877i	$0.182665\pi$
$271$			17.8738i			1.08575i	−0.839812	+	0.542877i	$0.817335\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.30366	+	4.21677i	−0.440427	+	0.254281i
$276$		0			0
$277$	−3.12403	+	5.41098i	−0.187705	+	0.325115i	−0.944485	−	0.328555i	$-0.893438\pi$
$277$	−3.12403	+	5.41098i	−0.187705	+	0.325115i	0.756780	+	0.653670i	$0.226772\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−23.3573	−	13.4854i	−1.39338	−	0.804470i	−0.399694	−	0.916648i	$-0.630884\pi$
$281$	−23.3573	−	13.4854i	−1.39338	−	0.804470i	−0.993688	+	0.112179i	$0.964217\pi$
$282$		0			0
$283$	−1.05980	−	1.83562i	−0.0629984	−	0.109116i	0.832806	−	0.553565i	$-0.186733\pi$
$283$	−1.05980	−	1.83562i	−0.0629984	−	0.109116i	−0.895804	+	0.444449i	$0.853400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.9549			0.941789
$288$		0			0
$289$	2.87047			0.168851
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.8842	−	9.17076i	0.927967	−	0.535762i	0.0417987	−	0.999126i	$-0.486691\pi$
$293$	15.8842	−	9.17076i	0.927967	−	0.535762i	0.886168	+	0.463364i	$0.153358\pi$
$294$		0			0
$295$	0.373352	−	0.646665i	0.0217374	−	0.0376503i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.377853	+	0.0593357i	0.0218518	+	0.00343147i
$300$		0			0
$301$	13.8024	−	7.96882i	0.795557	−	0.459315i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			0.0523134i			0.00299546i
$306$		0			0
$307$		−	7.79429i		−	0.444844i	−0.974951	−	0.222422i	$-0.928604\pi$
$307$		−	7.79429i		−	0.444844i	0.974951	−	0.222422i	$-0.0713962\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.44985	−	5.97531i	−0.195623	−	0.338829i	0.751482	−	0.659754i	$-0.229340\pi$
$311$	−3.44985	−	5.97531i	−0.195623	−	0.338829i	−0.947105	+	0.320925i	$0.896006\pi$
$312$		0			0
$313$	7.72605	−	13.3819i	0.436702	−	0.756390i	−0.560731	−	0.827998i	$-0.689480\pi$
$313$	7.72605	−	13.3819i	0.436702	−	0.756390i	0.997433	+	0.0716082i	$0.0228131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.8281	+	6.25159i	0.608165	+	0.351124i	0.772247	−	0.635322i	$-0.219133\pi$
$317$	10.8281	+	6.25159i	0.608165	+	0.351124i	−0.164082	+	0.986447i	$0.552466\pi$
$318$		0			0
$319$	−14.6551	+	8.46111i	−0.820527	+	0.473731i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.15869i			0.0644710i
$324$		0			0
$325$	−6.47809	−	16.8098i	−0.359340	−	0.932438i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.24969	−	9.09273i	−0.289425	−	0.501299i
$330$		0			0
$331$	7.03745	+	4.06308i	0.386813	+	0.223327i	0.680779	−	0.732489i	$-0.261642\pi$
$331$	7.03745	+	4.06308i	0.386813	+	0.223327i	−0.293965	+	0.955816i	$0.594975\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.157167	+	0.272221i	−0.00858696	+	0.0148730i
$336$		0			0
$337$	11.5854	+	20.0666i	0.631099	+	1.09310i	0.987327	+	0.158697i	$0.0507293\pi$
$337$	11.5854	+	20.0666i	0.631099	+	1.09310i	−0.356228	+	0.934399i	$0.615937\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−13.1659			−0.712973
$342$		0			0
$343$		−	4.77949i		−	0.258068i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.17177	−	12.4219i	−0.385001	−	0.666841i	0.606768	−	0.794879i	$-0.292466\pi$
$347$	−7.17177	−	12.4219i	−0.385001	−	0.666841i	−0.991769	+	0.128037i	$0.959132\pi$
$348$		0			0
$349$	18.7601	+	10.8311i	1.00421	+	0.579778i	0.909490	−	0.415726i	$-0.136473\pi$
$349$	18.7601	+	10.8311i	1.00421	+	0.579778i	0.0947153	+	0.995504i	$0.469806\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−29.0176	−	16.7533i	−1.54445	−	0.891689i	−0.998550	−	0.0538398i	$-0.982854\pi$
$353$	−29.0176	−	16.7533i	−1.54445	−	0.891689i	−0.545901	−	0.837849i	$-0.683813\pi$
$354$		0			0
$355$	0.205266	+	0.355531i	0.0108944	+	0.0188696i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			7.79779i			0.411552i	0.978599	+	0.205776i	$0.0659718\pi$
$359$			7.79779i			0.411552i	−0.978599	+	0.205776i	$0.934028\pi$
$360$		0			0
$361$	18.9324			0.996444
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.432074	+	0.748374i	0.0226158	+	0.0391717i
$366$		0			0
$367$	−0.560705	+	0.971170i	−0.0292686	+	0.0506946i	−0.880289	−	0.474438i	$-0.842651\pi$
$367$	−0.560705	+	0.971170i	−0.0292686	+	0.0506946i	0.851020	+	0.525133i	$0.175984\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	20.9986	+	12.1236i	1.09020	+	0.629425i
$372$		0			0
$373$	11.7643	+	20.3764i	0.609135	+	1.05505i	0.991383	+	0.130993i	$0.0418164\pi$
$373$	11.7643	+	20.3764i	0.609135	+	1.05505i	−0.382249	+	0.924059i	$0.624850\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.9985	−	33.7295i	−0.669459	−	1.73716i
$378$		0			0
$379$		−	27.3824i		−	1.40654i	−0.710924	−	0.703269i	$-0.751723\pi$
$379$		−	27.3824i		−	1.40654i	0.710924	−	0.703269i	$-0.248277\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−26.4482	+	15.2699i	−1.35144	+	0.780255i	−0.988451	−	0.151538i	$-0.951577\pi$
$383$	−26.4482	+	15.2699i	−1.35144	+	0.780255i	−0.362990	+	0.931793i	$0.618244\pi$
$384$		0			0
$385$	−0.311439	−	0.179809i	−0.0158724	−	0.00916394i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.49704	−	2.59294i	0.0759028	−	0.131467i	−0.825576	−	0.564291i	$-0.809149\pi$
$389$	1.49704	−	2.59294i	0.0759028	−	0.131467i	0.901478	+	0.432824i	$0.142483\pi$
$390$		0			0
$391$	0.236437	+	0.409520i	0.0119571	+	0.0207103i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	0.922669i		−	0.0464245i
$396$		0			0
$397$			12.2130i			0.612951i	0.951879	+	0.306475i	$0.0991497\pi$
$397$			12.2130i			0.612951i	−0.951879	+	0.306475i	$0.900850\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.88631	+	1.66641i	−0.144136	+	0.0832167i	−0.570334	−	0.821413i	$-0.693186\pi$
$401$	−2.88631	+	1.66641i	−0.144136	+	0.0832167i	0.426198	+	0.904630i	$0.359853\pi$
$402$		0			0
$403$	4.36289	−	27.7831i	0.217331	−	1.38398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.96073	−	3.39609i	0.0971899	−	0.168338i
$408$		0			0
$409$	−10.9025	+	6.29455i	−0.539093	+	0.311245i	−0.744711	−	0.667387i	$-0.767413\pi$
$409$	−10.9025	+	6.29455i	−0.539093	+	0.311245i	0.205618	+	0.978632i	$0.434079\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−44.3581			−2.18272
$414$		0			0
$415$	0.329669			0.0161828
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.3192	−	19.6054i	−0.552979	−	0.957789i	−0.998058	−	0.0622968i	$-0.980157\pi$
$419$	−11.3192	−	19.6054i	−0.552979	−	0.957789i	0.445078	−	0.895492i	$-0.353176\pi$
$420$		0			0
$421$	14.9600	+	8.63718i	0.729107	+	0.420950i	0.818095	−	0.575082i	$-0.195030\pi$
$421$	14.9600	+	8.63718i	0.729107	+	0.420950i	−0.0889882	+	0.996033i	$0.528363\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.1361	−	19.2883i	0.540179	−	0.935618i
$426$		0			0
$427$	2.69133	−	1.55384i	0.130243	−	0.0751956i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	35.0943i		−	1.69043i	−0.534426	−	0.845215i	$-0.679472\pi$
$431$		−	35.0943i		−	1.69043i	0.534426	−	0.845215i	$-0.320528\pi$
$432$		0			0
$433$	11.4234			0.548971			0.274486	−	0.961591i	$-0.411492\pi$
$433$	11.4234			0.548971			0.274486	+	0.961591i	$0.411492\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.0238799	+	0.0137871i	−0.00114233	+	0.000659525i
$438$		0			0
$439$	9.65550	−	16.7238i	0.460832	−	0.798184i	−0.538171	−	0.842836i	$-0.680885\pi$
$439$	9.65550	−	16.7238i	0.460832	−	0.798184i	0.999003	+	0.0446518i	$0.0142179\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.54814	−	2.68146i	0.0735544	−	0.127400i	−0.826902	−	0.562346i	$-0.809899\pi$
$443$	1.54814	−	2.68146i	0.0735544	−	0.127400i	0.900457	+	0.434946i	$0.143232\pi$
$444$		0			0
$445$	−0.106524	−	0.184505i	−0.00504972	−	0.00874637i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			7.31689i			0.345305i	0.984983	+	0.172653i	$0.0552338\pi$
$449$			7.31689i			0.345305i	−0.984983	+	0.172653i	$0.944766\pi$
$450$		0			0
$451$	7.56987			0.356451
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.482644	−	0.597624i	0.0226267	−	0.0280171i
$456$		0			0
$457$	21.2881	+	12.2907i	0.995817	+	0.574935i	0.907008	−	0.421114i	$-0.138361\pi$
$457$	21.2881	+	12.2907i	0.995817	+	0.574935i	0.0888088	+	0.996049i	$0.471694\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.6875	+	7.90250i	0.637492	+	0.368056i	0.783648	−	0.621206i	$-0.213357\pi$
$461$	13.6875	+	7.90250i	0.637492	+	0.368056i	−0.146156	+	0.989262i	$0.546690\pi$
$462$		0			0
$463$	15.1784	−	8.76324i	0.705399	−	0.407262i	−0.103956	−	0.994582i	$-0.533150\pi$
$463$	15.1784	−	8.76324i	0.705399	−	0.407262i	0.809355	+	0.587320i	$0.199817\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.7626			−0.498034			−0.249017	−	0.968499i	$-0.580107\pi$
$467$	−10.7626			−0.498034			−0.249017	+	0.968499i	$0.580107\pi$
$468$		0			0
$469$	18.6731			0.862242
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.54860	−	3.78083i	0.301105	−	0.173843i
$474$		0			0
$475$	1.12474	+	0.649367i	0.0516064	+	0.0297950i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.83048	+	2.78888i	0.220710	+	0.127427i	0.606279	−	0.795252i	$-0.292661\pi$
$479$	4.83048	+	2.78888i	0.220710	+	0.127427i	−0.385569	+	0.922679i	$0.625995\pi$
$480$		0			0
$481$	6.51679	+	5.26299i	0.297140	+	0.239972i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.382786			0.0173814
$486$		0			0
$487$			29.3212i			1.32867i	0.747436	+	0.664334i	$0.231285\pi$
$487$			29.3212i			1.32867i	−0.747436	+	0.664334i	$0.768715\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.10138	−	12.3000i	−0.320481	−	0.555089i	0.660106	−	0.751172i	$-0.270511\pi$
$491$	−7.10138	−	12.3000i	−0.320481	−	0.555089i	−0.980587	+	0.196083i	$0.937178\pi$
$492$		0			0
$493$	22.3450	−	38.7027i	1.00637	−	1.74308i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.1939	−	21.1204i	0.546969	−	0.947378i
$498$		0			0
$499$	−32.9511	+	19.0243i	−1.47509	+	0.851646i	−0.999606	−	0.0280752i	$-0.991062\pi$
$499$	−32.9511	+	19.0243i	−1.47509	+	0.851646i	−0.475489	+	0.879722i	$0.657729\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−14.5101			−0.646975			−0.323488	−	0.946232i	$-0.604855\pi$
$503$	−14.5101			−0.646975			−0.323488	+	0.946232i	$0.604855\pi$
$504$		0			0
$505$			0.00764565i			0.000340227i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.68729	−	4.43826i	0.340733	−	0.196722i	−0.319863	−	0.947464i	$-0.603637\pi$
$509$	7.68729	−	4.43826i	0.340733	−	0.196722i	0.660596	+	0.750741i	$0.270304\pi$
$510$		0			0
$511$	25.6674	−	44.4572i	1.13546	−	1.96667i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.209080	−	0.120712i	−0.00921317	−	0.00531923i
$516$		0			0
$517$	−2.49073	−	4.31408i	−0.109542	−	0.189733i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.7656			−1.26025			−0.630123	−	0.776496i	$-0.716995\pi$
$521$	−28.7656			−1.26025			−0.630123	+	0.776496i	$0.716995\pi$
$522$		0			0
$523$	3.92567			0.171658			0.0858289	−	0.996310i	$-0.472646\pi$
$523$	3.92567			0.171658			0.0858289	+	0.996310i	$0.472646\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	30.1116	−	17.3849i	1.31168	−	0.757300i
$528$		0			0
$529$	11.4944	−	19.9088i	0.499755	−	0.865602i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.50849	+	15.9742i	−0.108655	+	0.691919i
$534$		0			0
$535$	−0.114750	+	0.0662512i	−0.00496110	+	0.00286429i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			9.54778i			0.411252i
$540$		0			0
$541$		−	22.8833i		−	0.983829i	−0.870644	−	0.491914i	$-0.836297\pi$
$541$		−	22.8833i		−	0.983829i	0.870644	−	0.491914i	$-0.163703\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.246746	+	0.427376i	0.0105694	+	0.0183068i
$546$		0			0
$547$	−9.96461	+	17.2592i	−0.426056	+	0.737950i	−0.996518	−	0.0833732i	$-0.973431\pi$
$547$	−9.96461	+	17.2592i	−0.426056	+	0.737950i	0.570463	+	0.821324i	$0.306764\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.25683	+	1.30298i	0.0961441	+	0.0555088i
$552$		0			0
$553$	−47.4679	+	27.4056i	−2.01854	+	1.16541i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			30.6804i			1.29997i	0.759946	+	0.649986i	$0.225225\pi$
$557$			30.6804i			1.29997i	−0.759946	+	0.649986i	$0.774775\pi$
$558$		0			0
$559$	5.80838	+	15.0720i	0.245668	+	0.637476i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.2617	−	29.8982i	−0.727495	−	1.26006i	−0.957939	−	0.286973i	$-0.907351\pi$
$563$	−17.2617	−	29.8982i	−0.727495	−	1.26006i	0.230444	−	0.973086i	$-0.425982\pi$
$564$		0			0
$565$	−0.662779	−	0.382656i	−0.0278833	−	0.0160984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.70622	−	6.41935i	0.155373	−	0.269113i	−0.777822	−	0.628485i	$-0.783676\pi$
$569$	3.70622	−	6.41935i	0.155373	−	0.269113i	0.933195	+	0.359371i	$0.117009\pi$
$570$		0			0
$571$	15.8005	+	27.3673i	0.661230	+	1.14528i	0.980293	+	0.197551i	$0.0632988\pi$
$571$	15.8005	+	27.3673i	0.661230	+	1.14528i	−0.319062	+	0.947734i	$0.603368\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.530028			0.0221037
$576$		0			0
$577$			31.8254i			1.32491i	0.749102	+	0.662455i	$0.230485\pi$
$577$			31.8254i			1.32491i	−0.749102	+	0.662455i	$0.769515\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.79202	−	16.9603i	−0.406241	−	0.703631i
$582$		0			0
$583$	9.96288	+	5.75207i	0.412620	+	0.238227i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.20029	+	3.00239i	0.214639	+	0.123922i	0.603465	−	0.797389i	$-0.293786\pi$
$587$	5.20029	+	3.00239i	0.214639	+	0.123922i	−0.388827	+	0.921311i	$0.627119\pi$
$588$		0			0
$589$	1.01375	+	1.75587i	0.0417708	+	0.0723492i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	21.7811i		−	0.894442i	−0.894424	−	0.447221i	$-0.852414\pi$
$593$		−	21.7811i		−	0.894442i	0.894424	−	0.447221i	$-0.147586\pi$
$594$		0			0
$595$	0.949719			0.0389347
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−8.42383	−	14.5905i	−0.344188	−	0.596152i	0.641018	−	0.767526i	$-0.278513\pi$
$599$	−8.42383	−	14.5905i	−0.344188	−	0.596152i	−0.985206	+	0.171375i	$0.945179\pi$
$600$		0			0
$601$	−5.67287	+	9.82571i	−0.231401	+	0.400799i	−0.958221	−	0.286030i	$-0.907664\pi$
$601$	−5.67287	+	9.82571i	−0.231401	+	0.400799i	0.726819	+	0.686829i	$0.240998\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.422738	+	0.244068i	0.0171868	+	0.00992278i
$606$		0			0
$607$	7.03033	+	12.1769i	0.285352	+	0.494245i	0.972695	−	0.232089i	$-0.0745560\pi$
$607$	7.03033	+	12.1769i	0.285352	+	0.494245i	−0.687342	+	0.726334i	$0.741223\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.92909	−	3.82644i	0.401688	−	0.154801i
$612$		0			0
$613$		−	32.0837i		−	1.29585i	−0.761704	−	0.647925i	$-0.775637\pi$
$613$		−	32.0837i		−	1.29585i	0.761704	−	0.647925i	$-0.224363\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.1022	−	16.8022i	1.17161	−	0.676431i	0.217553	−	0.976049i	$-0.430193\pi$
$617$	29.1022	−	16.8022i	1.17161	−	0.676431i	0.954059	+	0.299618i	$0.0968592\pi$
$618$		0			0
$619$	34.0448	+	19.6558i	1.36838	+	0.790033i	0.990721	−	0.135914i	$-0.0433969\pi$
$619$	34.0448	+	19.6558i	1.36838	+	0.790033i	0.377656	+	0.925946i	$0.376730\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.32807	+	10.9605i	−0.253529	+	0.439125i
$624$		0			0
$625$	−12.4731	−	21.6041i	−0.498924	−	0.864162i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			10.3562i			0.412929i
$630$		0			0
$631$			14.4972i			0.577124i	0.957461	+	0.288562i	$0.0931771\pi$
$631$			14.4972i			0.577124i	−0.957461	+	0.288562i	$0.906823\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.753422	−	0.434989i	0.0298986	−	0.0172620i
$636$		0			0
$637$	−20.1480	−	3.16393i	−0.798295	−	0.125359i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.89387	−	15.4046i	0.351287	−	0.608447i	−0.635188	−	0.772357i	$-0.719078\pi$
$641$	8.89387	−	15.4046i	0.351287	−	0.608447i	0.986475	+	0.163911i	$0.0524109\pi$
$642$		0			0
$643$	34.3076	−	19.8075i	1.35296	−	0.781131i	0.364296	−	0.931283i	$-0.381310\pi$
$643$	34.3076	−	19.8075i	1.35296	−	0.781131i	0.988663	+	0.150152i	$0.0479764\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	44.2481			1.73957			0.869786	−	0.493429i	$-0.164257\pi$
$647$	44.2481			1.73957			0.869786	+	0.493429i	$0.164257\pi$
$648$		0			0
$649$	−21.0459			−0.826122
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.3543	+	17.9342i	0.405195	+	0.701818i	0.994344	−	0.106206i	$-0.0338704\pi$
$653$	10.3543	+	17.9342i	0.405195	+	0.701818i	−0.589150	+	0.808024i	$0.700537\pi$
$654$		0			0
$655$	−0.981785	−	0.566834i	−0.0383615	−	0.0221480i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15.5524	+	26.9376i	−0.605836	+	1.04934i	0.386083	+	0.922464i	$0.373828\pi$
$659$	−15.5524	+	26.9376i	−0.605836	+	1.04934i	−0.991919	+	0.126874i	$0.959505\pi$
$660$		0			0
$661$	37.7135	−	21.7739i	1.46689	−	0.846908i	0.467573	−	0.883954i	$-0.345128\pi$
$661$	37.7135	−	21.7739i	1.46689	−	0.846908i	0.999314	+	0.0370468i	$0.0117951\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			0.0553800i			0.00214754i
$666$		0			0
$667$	1.06352			0.0411797
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.27691	−	0.737226i	0.0492946	−	0.0284603i
$672$		0			0
$673$	3.08342	−	5.34065i	0.118857	−	0.205867i	−0.800458	−	0.599389i	$-0.795410\pi$
$673$	3.08342	−	5.34065i	0.118857	−	0.205867i	0.919315	+	0.393522i	$0.128744\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13.8360	−	23.9646i	0.531760	−	0.921035i	−0.467553	−	0.883965i	$-0.654864\pi$
$677$	13.8360	−	23.9646i	0.531760	−	0.921035i	0.999313	−	0.0370696i	$-0.0118023\pi$
$678$		0			0
$679$	−11.3697	−	19.6930i	−0.436330	−	0.755747i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.42967i			0.169497i	0.996402	+	0.0847483i	$0.0270086\pi$
$683$			4.42967i			0.169497i	−0.996402	+	0.0847483i	$0.972991\pi$
$684$		0			0
$685$	−0.860148			−0.0328646
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.4397	+	19.1179i	−0.588206	+	0.728334i
$690$		0			0
$691$	−12.9164	−	7.45729i	−0.491363	−	0.283689i	0.233777	−	0.972290i	$-0.424892\pi$
$691$	−12.9164	−	7.45729i	−0.491363	−	0.283689i	−0.725140	+	0.688602i	$0.758225\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.01876	−	0.588181i	−0.0386438	−	0.0223110i
$696$		0			0
$697$	−17.3130	+	9.99565i	−0.655776	+	0.378612i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.72229			0.140589			0.0702945	−	0.997526i	$-0.477606\pi$
$701$	3.72229			0.140589			0.0702945	+	0.997526i	$0.477606\pi$
$702$		0			0
$703$	−0.603891			−0.0227762
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.393341	−	0.227095i	0.0147931	−	0.00854080i
$708$		0			0
$709$	40.4480	+	23.3526i	1.51906	+	0.877027i	0.999748	+	0.0224395i	$0.00714332\pi$
$709$	40.4480	+	23.3526i	1.51906	+	0.877027i	0.519307	+	0.854588i	$0.326190\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.716589	+	0.413723i	0.0268365	+	0.0154940i
$714$		0			0
$715$	0.228992	−	0.283545i	0.00856383	−	0.0106040i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.67210			−0.136946			−0.0684731	−	0.997653i	$-0.521813\pi$
$719$	−3.67210			−0.136946			−0.0684731	+	0.997653i	$0.521813\pi$
$720$		0			0
$721$			14.3419i			0.534119i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−25.0458	−	43.3805i	−0.930177	−	1.61111i
$726$		0			0
$727$	3.61026	−	6.25316i	0.133897	−	0.231917i	−0.791278	−	0.611456i	$-0.790584\pi$
$727$	3.61026	−	6.25316i	0.133897	−	0.231917i	0.925176	+	0.379539i	$0.123917\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9.98482	+	17.2942i	−0.369302	+	0.639650i
$732$		0			0
$733$	−21.8645	+	12.6235i	−0.807583	+	0.466258i	−0.846116	−	0.532999i	$-0.821065\pi$
$733$	−21.8645	+	12.6235i	−0.807583	+	0.466258i	0.0385326	+	0.999257i	$0.487732\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.85950			0.326344
$738$		0			0
$739$		−	10.5123i		−	0.386700i	−0.981130	−	0.193350i	$-0.938065\pi$
$739$		−	10.5123i		−	0.386700i	0.981130	−	0.193350i	$-0.0619353\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17.7866	+	10.2691i	−0.652527	+	0.376737i	−0.789424	−	0.613849i	$-0.789621\pi$
$743$	−17.7866	+	10.2691i	−0.652527	+	0.376737i	0.136897	+	0.990585i	$0.456287\pi$
$744$		0			0
$745$	−0.331770	+	0.574643i	−0.0121551	+	0.0210533i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.81677	+	3.93566i	0.249079	+	0.143806i
$750$		0			0
$751$	−7.92121	−	13.7199i	−0.289049	−	0.500648i	0.684534	−	0.728981i	$-0.260006\pi$
$751$	−7.92121	−	13.7199i	−0.289049	−	0.500648i	−0.973583	+	0.228333i	$0.926672\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.951073			−0.0346131
$756$		0			0
$757$	9.27761			0.337200			0.168600	−	0.985685i	$-0.446075\pi$
$757$	9.27761			0.337200			0.168600	+	0.985685i	$0.446075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.4846	−	13.5589i	0.851318	−	0.491508i	−0.00977762	−	0.999952i	$-0.503112\pi$
$761$	23.4846	−	13.5589i	0.851318	−	0.491508i	0.861095	+	0.508444i	$0.169779\pi$
$762$		0			0
$763$	14.6580	−	25.3883i	0.530654	−	0.919119i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.97414	−	44.4117i	0.251822	−	1.60361i
$768$		0			0
$769$	16.5174	−	9.53633i	0.595633	−	0.343889i	−0.171689	−	0.985151i	$-0.554922\pi$
$769$	16.5174	−	9.53633i	0.595633	−	0.343889i	0.767322	+	0.641263i	$0.221589\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			19.3989i			0.697730i	0.937173	+	0.348865i	$0.113433\pi$
$773$			19.3989i			0.697730i	−0.937173	+	0.348865i	$0.886567\pi$
$774$		0			0
$775$		−	38.9724i		−	1.39993i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.582866	−	1.00955i	−0.0208833	−	0.0361710i
$780$		0			0
$781$	5.78542	−	10.0206i	0.207018	−	0.358567i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.954819	−	0.551265i	−0.0340790	−	0.0196755i
$786$		0			0
$787$	−44.4615	+	25.6698i	−1.58488	+	0.915031i	−0.590748	+	0.806856i	$0.701167\pi$
$787$	−44.4615	+	25.6698i	−1.58488	+	0.915031i	−0.994132	+	0.108175i	$0.965499\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			45.4634i			1.61649i
$792$		0			0
$793$	1.13258	+	2.93888i	0.0402190	+	0.104363i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	21.7117	+	37.6058i	0.769068	+	1.33207i	0.938069	+	0.346450i	$0.112613\pi$
$797$	21.7117	+	37.6058i	0.769068	+	1.33207i	−0.169000	+	0.985616i	$0.554054\pi$
$798$		0			0
$799$	11.3931	+	6.57779i	0.403058	+	0.232706i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.1780	−	21.0929i	0.429752	−	0.744352i
$804$		0			0
$805$	0.0113006	+	0.0195732i	0.000398294	+	0.000689865i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.73990			0.0611717			0.0305859	−	0.999532i	$-0.490263\pi$
$809$	1.73990			0.0611717			0.0305859	+	0.999532i	$0.490263\pi$
$810$		0			0
$811$		−	12.9989i		−	0.456454i	−0.973608	−	0.228227i	$-0.926707\pi$
$811$		−	12.9989i		−	0.456454i	0.973608	−	0.228227i	$-0.0732928\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.244004	−	0.422627i	−0.00854707	−	0.0148040i
$816$		0			0
$817$	−1.00846	−	0.582234i	−0.0352815	−	0.0203698i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.9133	−	9.18756i	−0.555378	−	0.320648i	0.195910	−	0.980622i	$-0.437234\pi$
$821$	−15.9133	−	9.18756i	−0.555378	−	0.320648i	−0.751288	+	0.659974i	$0.770567\pi$
$822$		0			0
$823$	21.5812	+	37.3797i	0.752273	+	1.30298i	0.946719	+	0.322062i	$0.104376\pi$
$823$	21.5812	+	37.3797i	0.752273	+	1.30298i	−0.194445	+	0.980913i	$0.562291\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	30.3552i		−	1.05555i	−0.849383	−	0.527776i	$-0.823026\pi$
$827$		−	30.3552i		−	1.05555i	0.849383	−	0.527776i	$-0.176974\pi$
$828$		0			0
$829$	19.7899			0.687332			0.343666	−	0.939092i	$-0.388331\pi$
$829$	19.7899			0.687332			0.343666	+	0.939092i	$0.388331\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.6074	−	21.8366i	−0.436820	−	0.756595i
$834$		0			0
$835$	0.368905	−	0.638963i	0.0127665	−	0.0221122i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.6346	+	24.6151i	1.47191	+	0.849807i	0.999501	−	0.0315737i	$-0.0100519\pi$
$839$	42.6346	+	24.6151i	1.47191	+	0.849807i	0.472407	+	0.881380i	$0.343385\pi$
$840$		0			0
$841$	−35.7553	−	61.9300i	−1.23294	−	2.13552i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.522463	+	0.577188i	0.0179733	+	0.0198559i
$846$		0			0
$847$		−	28.9978i		−	0.996376i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.213436	+	0.123227i	−0.00731650	+	0.00422418i
$852$		0			0
$853$	−7.40541	−	4.27552i	−0.253556	−	0.146391i	0.367835	−	0.929891i	$-0.380099\pi$
$853$	−7.40541	−	4.27552i	−0.253556	−	0.146391i	−0.621392	+	0.783500i	$0.713432\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.8253	−	25.6782i	0.506423	−	0.877150i	−0.493550	−	0.869718i	$-0.664301\pi$
$857$	14.8253	−	25.6782i	0.506423	−	0.877150i	0.999972	−	0.00743241i	$-0.00236583\pi$
$858$		0			0
$859$	−14.2933	−	24.7568i	−0.487682	−	0.844690i	0.512217	−	0.858856i	$-0.328824\pi$
$859$	−14.2933	−	24.7568i	−0.487682	−	0.844690i	−0.999900	+	0.0141653i	$0.995491\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	31.7886i		−	1.08210i	−0.840991	−	0.541049i	$-0.818027\pi$
$863$		−	31.7886i		−	1.08210i	0.840991	−	0.541049i	$-0.181973\pi$
$864$		0			0
$865$		−	0.613780i		−	0.0208692i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−22.5213	+	13.0027i	−0.763984	+	0.441086i
$870$		0			0
$871$	−2.93585	+	18.6956i	−0.0994773	+	0.633477i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.06489	−	1.84445i	0.0359999	−	0.0623537i
$876$		0			0
$877$	24.2939	−	14.0261i	0.820348	−	0.473628i	−0.0301886	−	0.999544i	$-0.509611\pi$
$877$	24.2939	−	14.0261i	0.820348	−	0.473628i	0.850536	+	0.525916i	$0.176277\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.31849			0.0781119			0.0390559	−	0.999237i	$-0.487565\pi$
$881$	2.31849			0.0781119			0.0390559	+	0.999237i	$0.487565\pi$
$882$		0			0
$883$	−24.4927			−0.824244			−0.412122	−	0.911129i	$-0.635212\pi$
$883$	−24.4927			−0.824244			−0.412122	+	0.911129i	$0.635212\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.1260	−	17.5387i	−0.339997	−	0.588892i	0.644435	−	0.764659i	$-0.277093\pi$
$887$	−10.1260	−	17.5387i	−0.339997	−	0.588892i	−0.984432	+	0.175767i	$0.943759\pi$
$888$		0			0
$889$	−44.7571	−	25.8405i	−1.50111	−	0.866664i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.383564	+	0.664352i	−0.0128355	+	0.0222317i
$894$		0			0
$895$	−0.259841	+	0.150019i	−0.00868553	+	0.00501459i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	78.1997i		−	2.60811i
$900$		0			0
$901$	−30.3814			−1.01215
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.903637	−	0.521715i	0.0300379	−	0.0173424i
$906$		0			0
$907$	−24.4963	+	42.4288i	−0.813386	+	1.40883i	0.0970944	+	0.995275i	$0.469045\pi$
$907$	−24.4963	+	42.4288i	−0.813386	+	1.40883i	−0.910481	+	0.413551i	$0.864288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.48302	+	12.9610i	−0.247924	+	0.429416i	−0.962950	−	0.269682i	$-0.913082\pi$
$911$	−7.48302	+	12.9610i	−0.247924	+	0.429416i	0.715026	+	0.699098i	$0.246415\pi$
$912$		0			0
$913$	−4.64586	−	8.04686i	−0.153756	−	0.266312i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			67.3456i			2.22395i
$918$		0			0
$919$	31.2552			1.03101			0.515506	−	0.856886i	$-0.327604\pi$
$919$	31.2552			1.03101			0.515506	+	0.856886i	$0.327604\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	19.2287	+	15.5292i	0.632921	+	0.511150i
$924$		0			0
$925$	10.0528	+	5.80397i	0.330533	+	0.190833i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−47.4285	−	27.3829i	−1.55608	−	0.898403i	−0.997626	−	0.0688671i	$-0.978062\pi$
$929$	−47.4285	−	27.3829i	−1.55608	−	0.898403i	−0.558454	−	0.829536i	$-0.688605\pi$
$930$		0			0
$931$	1.27334	−	0.735161i	0.0417319	−	0.0240939i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.450598			0.0147361
$936$		0			0
$937$	−6.18261			−0.201977			−0.100988	−	0.994888i	$-0.532201\pi$
$937$	−6.18261			−0.201977			−0.100988	+	0.994888i	$0.532201\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25.1961	−	14.5470i	0.821370	−	0.474218i	−0.0295189	−	0.999564i	$-0.509398\pi$
$941$	25.1961	−	14.5470i	0.821370	−	0.474218i	0.850889	+	0.525346i	$0.176064\pi$
$942$		0			0
$943$	−0.412010	−	0.237874i	−0.0134169	−	0.00774625i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.07292	−	2.92885i	−0.164848	−	0.0951750i	0.415306	−	0.909682i	$-0.363674\pi$
$947$	−5.07292	−	2.92885i	−0.164848	−	0.0951750i	−0.580154	+	0.814507i	$0.697008\pi$
$948$		0			0
$949$	40.4754	+	32.6881i	1.31389	+	1.06110i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	27.6005			0.894067			0.447034	−	0.894517i	$-0.352480\pi$
$953$	27.6005			0.894067			0.447034	+	0.894517i	$0.352480\pi$
$954$		0			0
$955$		−	0.373774i		−	0.0120950i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	25.5486	+	44.2515i	0.825007	+	1.42895i
$960$		0			0
$961$	14.9206	−	25.8433i	0.481311	−	0.833654i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.0133927	+	0.0231969i	−0.000431128	+	0.000746735i
$966$		0			0
$967$	26.8101	−	15.4788i	0.862156	−	0.497766i	−0.00257789	−	0.999997i	$-0.500821\pi$
$967$	26.8101	−	15.4788i	0.862156	−	0.497766i	0.864734	+	0.502231i	$0.167487\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−50.0574			−1.60642			−0.803209	−	0.595697i	$-0.796876\pi$
$971$	−50.0574			−1.60642			−0.803209	+	0.595697i	$0.796876\pi$
$972$		0			0
$973$			69.8820i			2.24031i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−44.2147	+	25.5274i	−1.41455	+	0.816692i	−0.995813	−	0.0914138i	$-0.970861\pi$
$977$	−44.2147	+	25.5274i	−1.41455	+	0.816692i	−0.418740	+	0.908106i	$0.637528\pi$
$978$		0			0
$979$	−3.00238	+	5.20027i	−0.0959564	+	0.166201i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−9.28036	−	5.35802i	−0.295997	−	0.170894i	0.344646	−	0.938733i	$-0.387999\pi$
$983$	−9.28036	−	5.35802i	−0.295997	−	0.170894i	−0.640643	+	0.767839i	$0.721332\pi$
$984$		0			0
$985$	−0.640061	−	1.10862i	−0.0203940	−	0.0353235i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.475233			−0.0151115
$990$		0			0
$991$	48.7981			1.55012			0.775062	−	0.631885i	$-0.217719\pi$
$991$	48.7981			1.55012			0.775062	+	0.631885i	$0.217719\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.965942	−	0.557687i	0.0306224	−	0.0176799i
$996$		0			0
$997$	−1.13441	+	1.96485i	−0.0359270	+	0.0622273i	−0.883430	−	0.468564i	$-0.844772\pi$
$997$	−1.13441	+	1.96485i	−0.0359270	+	0.0622273i	0.847503	+	0.530791i	$0.178105\pi$
$998$		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.1585.20		80
3.2	odd	2		936.2.cw.b.25.38	yes	80
9.4	even	3	inner	2808.2.cw.b.2521.21		80
9.5	odd	6		936.2.cw.b.337.37	yes	80
13.12	even	2	inner	2808.2.cw.b.1585.21		80
39.38	odd	2		936.2.cw.b.25.37	✓	80
117.77	odd	6		936.2.cw.b.337.38	yes	80
117.103	even	6	inner	2808.2.cw.b.2521.20		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.37	✓	80	39.38	odd	2
936.2.cw.b.25.38	yes	80	3.2	odd	2
936.2.cw.b.337.37	yes	80	9.5	odd	6
936.2.cw.b.337.38	yes	80	117.77	odd	6
2808.2.cw.b.1585.20		80	1.1	even	1	trivial
2808.2.cw.b.1585.21		80	13.12	even	2	inner
2808.2.cw.b.2521.20		80	117.103	even	6	inner
2808.2.cw.b.2521.21		80	9.4	even	3	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)

\(f(q)\)	\(=\)	\(q+(-0.0518638 + 0.0299436i) q^{5} +(3.08098 + 1.77880i) q^{7} +(1.46178 + 0.843959i) q^{11} +(-2.26535 + 2.80503i) q^{13} -4.45763 q^{17} -0.259933i q^{19} +(-0.0530409 - 0.0918694i) q^{23} +(-2.49821 + 4.32702i) q^{25} +(-5.01275 + 8.68234i) q^{29} +(-6.75507 + 3.90004i) q^{31} -0.213055 q^{35} -2.32326i q^{37} +(3.88390 - 2.24237i) q^{41} +(2.23994 - 3.87969i) q^{43} +(-2.55586 - 1.47563i) q^{47} +(2.82827 + 4.89871i) q^{49} +6.81558 q^{53} -0.101085 q^{55} +(-10.7981 + 6.23426i) q^{59} +(0.436766 - 0.756501i) q^{61} +(0.0334972 - 0.213312i) q^{65} +(4.54557 - 2.62439i) q^{67} -6.85509i q^{71} -14.4296i q^{73} +(3.00247 + 5.20043i) q^{77} +(-7.70339 + 13.3427i) q^{79} +(-4.76733 - 2.75242i) q^{83} +(0.231190 - 0.133477i) q^{85} +3.55749i q^{89} +(-11.9691 + 4.61261i) q^{91} +(0.00778333 + 0.0134811i) q^{95} +(-5.53546 - 3.19590i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 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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