LMFDB - Embedded newform 2808.2.s.f.1225.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(1153,2808)] 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, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.1153"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,-1,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1225.8
Character	$\chi$	$=$	2808.1225
Dual form			2808.2.s.f.1153.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.766914 + 1.32833i) q^{5} -3.51479 q^{7} +(3.19399 - 5.53215i) q^{11} +(-3.46170 - 1.00830i) q^{13} +(0.550501 - 0.953495i) q^{17} +(-1.13796 + 1.97100i) q^{19} +3.32815 q^{23} +(1.32369 + 2.29269i) q^{25} +(-0.714825 + 1.23811i) q^{29} +(-1.93596 + 3.35318i) q^{31} +(2.69554 - 4.66881i) q^{35} +(1.81670 + 3.14662i) q^{37} -11.9875 q^{41} +12.5528 q^{43} +(1.23298 + 2.13558i) q^{47} +5.35375 q^{49} +6.18395 q^{53} +(4.89903 + 8.48537i) q^{55} +(2.42804 + 4.20550i) q^{59} +3.09918 q^{61} +(3.99418 - 3.82501i) q^{65} -7.60626 q^{67} +(-2.43267 + 4.21351i) q^{71} -5.40814 q^{73} +(-11.2262 + 19.4444i) q^{77} +(2.39185 + 4.14281i) q^{79} +(3.19592 + 5.53550i) q^{83} +(0.844373 + 1.46250i) q^{85} +(4.25286 + 7.36618i) q^{89} +(12.1671 + 3.54395i) q^{91} +(-1.74543 - 3.02318i) q^{95} -3.52623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - q^{5} - 14 q^{7} - 3 q^{13} + q^{17} + 2 q^{19} - 2 q^{23} - 23 q^{25} - 12 q^{29} + 8 q^{31} + 12 q^{35} + 18 q^{37} - 6 q^{41} - 16 q^{43} - 4 q^{47} + 46 q^{49} + 40 q^{53} - 14 q^{55} + 4 q^{59}+ \cdots - 70 q^{97}+O(q^{100}) $ 40 * q - q^5 - 14 * q^7 - 3 * q^13 + q^17 + 2 * q^19 - 2 * q^23 - 23 * q^25 - 12 * q^29 + 8 * q^31 + 12 * q^35 + 18 * q^37 - 6 * q^41 - 16 * q^43 - 4 * q^47 + 46 * q^49 + 40 * q^53 - 14 * q^55 + 4 * q^59 + 6 * q^61 - 7 * q^65 - 54 * q^67 - q^71 - 12 * q^73 + 4 * q^77 + 13 * q^79 + 15 * q^83 + 5 * q^85 - 12 * q^89 - 57 * q^91 - q^95 - 70 * q^97

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.766914	+	1.32833i	−0.342974	+	0.594049i	−0.984984	−	0.172648i	$-0.944768\pi$
$5$	−0.766914	+	1.32833i	−0.342974	+	0.594049i	0.642009	+	0.766697i	$0.278101\pi$
$6$		0			0
$7$	−3.51479			−1.32847			−0.664233	−	0.747526i	$-0.731242\pi$
$7$	−3.51479			−1.32847			−0.664233	+	0.747526i	$0.731242\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.19399	−	5.53215i	0.963024	−	1.66801i	0.248192	−	0.968711i	$-0.420163\pi$
$11$	3.19399	−	5.53215i	0.963024	−	1.66801i	0.714832	−	0.699296i	$-0.246503\pi$
$12$		0			0
$13$	−3.46170	−	1.00830i	−0.960102	−	0.279651i
$13$	−3.46170	−	1.00830i	−0.960102	−	0.279651i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.550501	−	0.953495i	0.133516	−	0.231256i	−0.791514	−	0.611152i	$-0.790707\pi$
$17$	0.550501	−	0.953495i	0.133516	−	0.231256i	0.925030	+	0.379895i	$0.124040\pi$
$18$		0			0
$19$	−1.13796	+	1.97100i	−0.261065	+	0.452179i	−0.966525	−	0.256571i	$-0.917407\pi$
$19$	−1.13796	+	1.97100i	−0.261065	+	0.452179i	0.705460	+	0.708750i	$0.250741\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.32815			0.693968			0.346984	−	0.937871i	$-0.387206\pi$
$23$	3.32815			0.693968			0.346984	+	0.937871i	$0.387206\pi$
$24$		0			0
$25$	1.32369	+	2.29269i	0.264737	+	0.458538i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.714825	+	1.23811i	−0.132740	+	0.229912i	−0.924732	−	0.380620i	$-0.875711\pi$
$29$	−0.714825	+	1.23811i	−0.132740	+	0.229912i	0.791992	+	0.610531i	$0.209044\pi$
$30$		0			0
$31$	−1.93596	+	3.35318i	−0.347708	+	0.602248i	−0.985842	−	0.167677i	$-0.946373\pi$
$31$	−1.93596	+	3.35318i	−0.347708	+	0.602248i	0.638134	+	0.769926i	$0.279707\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.69554	−	4.66881i	0.455630	−	0.789174i
$36$		0			0
$37$	1.81670	+	3.14662i	0.298664	+	0.517302i	0.975831	−	0.218528i	$-0.0701256\pi$
$37$	1.81670	+	3.14662i	0.298664	+	0.517302i	−0.677166	+	0.735830i	$0.736792\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.9875			−1.87213			−0.936065	−	0.351827i	$-0.885561\pi$
$41$	−11.9875			−1.87213			−0.936065	+	0.351827i	$0.885561\pi$
$42$		0			0
$43$	12.5528			1.91428			0.957142	−	0.289619i	$-0.0935287\pi$
$43$	12.5528			1.91428			0.957142	+	0.289619i	$0.0935287\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.23298	+	2.13558i	0.179848	+	0.311506i	0.941828	−	0.336094i	$-0.109106\pi$
$47$	1.23298	+	2.13558i	0.179848	+	0.311506i	−0.761980	+	0.647600i	$0.775773\pi$
$48$		0			0
$49$	5.35375			0.764821
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.18395			0.849431			0.424716	−	0.905327i	$-0.360374\pi$
$53$	6.18395			0.849431			0.424716	+	0.905327i	$0.360374\pi$
$54$		0			0
$55$	4.89903	+	8.48537i	0.660585	+	1.14417i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.42804	+	4.20550i	0.316104	+	0.547509i	0.979672	−	0.200608i	$-0.0642917\pi$
$59$	2.42804	+	4.20550i	0.316104	+	0.547509i	−0.663567	+	0.748117i	$0.730958\pi$
$60$		0			0
$61$	3.09918			0.396809			0.198405	−	0.980120i	$-0.436424\pi$
$61$	3.09918			0.396809			0.198405	+	0.980120i	$0.436424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.99418	−	3.82501i	0.495417	−	0.474434i
$66$		0			0
$67$	−7.60626			−0.929252			−0.464626	−	0.885507i	$-0.653811\pi$
$67$	−7.60626			−0.929252			−0.464626	+	0.885507i	$0.653811\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.43267	+	4.21351i	−0.288705	+	0.500051i	−0.973501	−	0.228683i	$-0.926558\pi$
$71$	−2.43267	+	4.21351i	−0.288705	+	0.500051i	0.684796	+	0.728735i	$0.259891\pi$
$72$		0			0
$73$	−5.40814			−0.632975			−0.316487	−	0.948597i	$-0.602504\pi$
$73$	−5.40814			−0.632975			−0.316487	+	0.948597i	$0.602504\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−11.2262	+	19.4444i	−1.27934	+	2.21589i
$78$		0			0
$79$	2.39185	+	4.14281i	0.269104	+	0.466102i	0.968631	−	0.248505i	$-0.0799391\pi$
$79$	2.39185	+	4.14281i	0.269104	+	0.466102i	−0.699527	+	0.714607i	$0.746606\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.19592	+	5.53550i	0.350798	+	0.607600i	0.986390	−	0.164425i	$-0.0525770\pi$
$83$	3.19592	+	5.53550i	0.350798	+	0.607600i	−0.635591	+	0.772026i	$0.719244\pi$
$84$		0			0
$85$	0.844373	+	1.46250i	0.0915851	+	0.158630i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.25286	+	7.36618i	0.450803	+	0.780813i	0.998436	−	0.0559053i	$-0.0178045\pi$
$89$	4.25286	+	7.36618i	0.450803	+	0.780813i	−0.547633	+	0.836718i	$0.684471\pi$
$90$		0			0
$91$	12.1671	+	3.54395i	1.27546	+	0.371507i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.74543	−	3.02318i	−0.179077	−	0.310171i
$96$		0			0
$97$	−3.52623			−0.358035			−0.179017	−	0.983846i	$-0.557292\pi$
$97$	−3.52623			−0.358035			−0.179017	+	0.983846i	$0.557292\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.43406	+	12.8762i	−0.739717	+	1.28123i	0.212906	+	0.977073i	$0.431707\pi$
$101$	−7.43406	+	12.8762i	−0.739717	+	1.28123i	−0.952623	+	0.304154i	$0.901626\pi$
$102$		0			0
$103$	−5.76469	+	9.98474i	−0.568012	+	0.983826i	0.428750	+	0.903423i	$0.358954\pi$
$103$	−5.76469	+	9.98474i	−0.568012	+	0.983826i	−0.996762	+	0.0804029i	$0.974379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.41074	−	7.63962i	−0.426402	−	0.738550i	0.570148	−	0.821542i	$-0.306886\pi$
$107$	−4.41074	−	7.63962i	−0.426402	−	0.738550i	−0.996550	+	0.0829919i	$0.973552\pi$
$108$		0			0
$109$	−9.26955			−0.887862			−0.443931	−	0.896061i	$-0.646416\pi$
$109$	−9.26955			−0.887862			−0.443931	+	0.896061i	$0.646416\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.36243	−	5.82389i	−0.316310	−	0.547866i	0.663405	−	0.748261i	$-0.269111\pi$
$113$	−3.36243	−	5.82389i	−0.316310	−	0.547866i	−0.979715	+	0.200395i	$0.935777\pi$
$114$		0			0
$115$	−2.55241	+	4.42090i	−0.238013	+	0.412251i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.93489	+	3.35133i	−0.177371	+	0.307216i
$120$		0			0
$121$	−14.9031	−	25.8130i	−1.35483	−	2.34664i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.7298			−1.04914
$126$		0			0
$127$	7.61299	+	13.1861i	0.675544	+	1.17008i	0.976310	+	0.216378i	$0.0694245\pi$
$127$	7.61299	+	13.1861i	0.675544	+	1.17008i	−0.300766	+	0.953698i	$0.597242\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.89951	+	10.2183i	−0.515443	+	0.892773i	0.484396	+	0.874849i	$0.339039\pi$
$131$	−5.89951	+	10.2183i	−0.515443	+	0.892773i	−0.999839	+	0.0179247i	$0.994294\pi$
$132$		0			0
$133$	3.99968	−	6.92765i	0.346817	−	0.600704i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.0053			1.19655			0.598277	−	0.801289i	$-0.295852\pi$
$137$	14.0053			1.19655			0.598277	+	0.801289i	$0.295852\pi$
$138$		0			0
$139$	4.14354	+	7.17682i	0.351450	+	0.608730i	0.986504	−	0.163738i	$-0.0523553\pi$
$139$	4.14354	+	7.17682i	0.351450	+	0.608730i	−0.635054	+	0.772468i	$0.719022\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−16.6347	+	15.9301i	−1.39106	+	1.33215i
$144$		0			0
$145$	−1.09642	−	1.89905i	−0.0910526	−	0.157708i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.96851	+	3.40955i	0.161266	+	0.279321i	0.935323	−	0.353795i	$-0.115109\pi$
$149$	1.96851	+	3.40955i	0.161266	+	0.279321i	−0.774057	+	0.633116i	$0.781776\pi$
$150$		0			0
$151$	10.1395	+	17.5621i	0.825138	+	1.42918i	0.901814	+	0.432125i	$0.142236\pi$
$151$	10.1395	+	17.5621i	0.825138	+	1.42918i	−0.0766753	+	0.997056i	$0.524430\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.96943	−	5.14320i	−0.238510	−	0.413111i
$156$		0			0
$157$	−11.7247	+	20.3078i	−0.935735	+	1.62074i	−0.162416	+	0.986722i	$0.551929\pi$
$157$	−11.7247	+	20.3078i	−0.935735	+	1.62074i	−0.773319	+	0.634017i	$0.781405\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.6978			−0.921913
$162$		0			0
$163$	−1.54310	+	2.67272i	−0.120865	+	0.209344i	−0.920109	−	0.391663i	$-0.871900\pi$
$163$	−1.54310	+	2.67272i	−0.120865	+	0.209344i	0.799244	+	0.601006i	$0.205233\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.5069			1.50949			0.754744	−	0.656019i	$-0.227761\pi$
$167$	19.5069			1.50949			0.754744	+	0.656019i	$0.227761\pi$
$168$		0			0
$169$	10.9667	+	6.98083i	0.843591	+	0.536987i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.7005			0.813543			0.406771	−	0.913530i	$-0.366655\pi$
$173$	10.7005			0.813543			0.406771	+	0.913530i	$0.366655\pi$
$174$		0			0
$175$	−4.65248	−	8.05833i	−0.351694	−	0.609153i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.14067	−	1.97569i	−0.0852574	−	0.147670i	0.820243	−	0.572015i	$-0.193838\pi$
$179$	−1.14067	−	1.97569i	−0.0852574	−	0.147670i	−0.905501	+	0.424344i	$0.860505\pi$
$180$		0			0
$181$	10.2756			0.763780			0.381890	−	0.924208i	$-0.375273\pi$
$181$	10.2756			0.763780			0.381890	+	0.924208i	$0.375273\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.57302			−0.409737
$186$		0			0
$187$	−3.51659	−	6.09091i	−0.257158	−	0.445411i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−23.6279			−1.70965			−0.854826	−	0.518914i	$-0.826336\pi$
$191$	−23.6279			−1.70965			−0.854826	+	0.518914i	$0.826336\pi$
$192$		0			0
$193$	11.0395			0.794640			0.397320	−	0.917680i	$-0.369940\pi$
$193$	11.0395			0.794640			0.397320	+	0.917680i	$0.369940\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.47677	−	7.75399i	−0.318957	−	0.552449i	0.661314	−	0.750109i	$-0.269999\pi$
$197$	−4.47677	−	7.75399i	−0.318957	−	0.552449i	−0.980271	+	0.197660i	$0.936666\pi$
$198$		0			0
$199$	11.6750	−	20.2217i	0.827620	−	1.43348i	−0.0722797	−	0.997384i	$-0.523027\pi$
$199$	11.6750	−	20.2217i	0.827620	−	1.43348i	0.899900	−	0.436096i	$-0.143639\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.51246	−	4.35171i	0.176340	−	0.305430i
$204$		0			0
$205$	9.19337	−	15.9234i	0.642093	−	1.11214i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.26925	+	12.5907i	0.502825	+	0.870918i
$210$		0			0
$211$	−0.171961			−0.0118383			−0.00591913	−	0.999982i	$-0.501884\pi$
$211$	−0.171961			−0.0118383			−0.00591913	+	0.999982i	$0.501884\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.62691	+	16.6743i	−0.656550	+	1.13718i
$216$		0			0
$217$	6.80449	−	11.7857i	0.461919	−	0.800066i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.86707	+	2.74564i	−0.192860	+	0.184692i
$222$		0			0
$223$	0.813127	−	1.40838i	0.0544510	−	0.0943119i	−0.837515	−	0.546414i	$-0.815992\pi$
$223$	0.813127	−	1.40838i	0.0544510	−	0.0943119i	0.891966	+	0.452102i	$0.149326\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	22.5389			1.49596			0.747979	−	0.663722i	$-0.231024\pi$
$227$	22.5389			1.49596			0.747979	+	0.663722i	$0.231024\pi$
$228$		0			0
$229$	−12.0818	+	20.9263i	−0.798389	+	1.38285i	0.122276	+	0.992496i	$0.460981\pi$
$229$	−12.0818	+	20.9263i	−0.798389	+	1.38285i	−0.920665	+	0.390354i	$0.872353\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.4919			−1.08042			−0.540211	−	0.841530i	$-0.681656\pi$
$233$	−16.4919			−1.08042			−0.540211	+	0.841530i	$0.681656\pi$
$234$		0			0
$235$	−3.78235			−0.246733
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.24014	−	9.07618i	0.338956	−	0.587089i	−0.645280	−	0.763946i	$-0.723259\pi$
$239$	5.24014	−	9.07618i	0.338956	−	0.587089i	0.984237	+	0.176856i	$0.0565928\pi$
$240$		0			0
$241$	23.1800			1.49315			0.746577	−	0.665299i	$-0.231696\pi$
$241$	23.1800			1.49315			0.746577	+	0.665299i	$0.231696\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.10587	+	7.11157i	−0.262314	+	0.454341i
$246$		0			0
$247$	5.92662	−	5.67561i	0.377102	−	0.361130i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.1392	−	17.5616i	0.639979	−	1.10848i	−0.345458	−	0.938434i	$-0.612276\pi$
$251$	10.1392	−	17.5616i	0.639979	−	1.10848i	0.985437	−	0.170042i	$-0.0543904\pi$
$252$		0			0
$253$	10.6301	−	18.4119i	0.668308	−	1.15754i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.725978			0.0452853			0.0226426	−	0.999744i	$-0.492792\pi$
$257$	0.725978			0.0452853			0.0226426	+	0.999744i	$0.492792\pi$
$258$		0			0
$259$	−6.38534	−	11.0597i	−0.396765	−	0.687218i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.81064	+	8.33227i	−0.296637	+	0.513790i	−0.975364	−	0.220600i	$-0.929198\pi$
$263$	−4.81064	+	8.33227i	−0.296637	+	0.513790i	0.678728	+	0.734390i	$0.262532\pi$
$264$		0			0
$265$	−4.74256	+	8.21435i	−0.291333	+	0.504604i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.12405	−	10.6072i	0.373390	−	0.646731i	−0.616695	−	0.787202i	$-0.711529\pi$
$269$	6.12405	−	10.6072i	0.373390	−	0.646731i	0.990085	+	0.140472i	$0.0448620\pi$
$270$		0			0
$271$	2.64462	+	4.58062i	0.160650	+	0.278253i	0.935102	−	0.354379i	$-0.115308\pi$
$271$	2.64462	+	4.58062i	0.160650	+	0.278253i	−0.774452	+	0.632632i	$0.781974\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.9114			1.01979
$276$		0			0
$277$	5.92911			0.356246			0.178123	−	0.984008i	$-0.442998\pi$
$277$	5.92911			0.356246			0.178123	+	0.984008i	$0.442998\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.2879	−	28.2115i	−0.971654	−	1.68295i	−0.690561	−	0.723275i	$-0.742636\pi$
$281$	−16.2879	−	28.2115i	−0.971654	−	1.68295i	−0.281094	−	0.959680i	$-0.590697\pi$
$282$		0			0
$283$	18.5480			1.10256			0.551281	−	0.834320i	$-0.314139\pi$
$283$	18.5480			1.10256			0.551281	+	0.834320i	$0.314139\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	42.1335			2.48706
$288$		0			0
$289$	7.89390	+	13.6726i	0.464347	+	0.804273i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8.99242	−	15.5753i	−0.525343	−	0.909920i	−0.999564	−	0.0295149i	$-0.990604\pi$
$293$	−8.99242	−	15.5753i	−0.525343	−	0.909920i	0.474222	−	0.880406i	$-0.342730\pi$
$294$		0			0
$295$	−7.44840			−0.433663
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11.5211	−	3.35576i	−0.666280	−	0.194069i
$300$		0			0
$301$	−44.1205			−2.54306
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.37680	+	4.11675i	−0.136095	+	0.235724i
$306$		0			0
$307$	14.0831			0.803768			0.401884	−	0.915691i	$-0.368356\pi$
$307$	14.0831			0.803768			0.401884	+	0.915691i	$0.368356\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.3440	+	26.5767i	−0.870081	+	1.50702i	−0.00816904	+	0.999967i	$0.502600\pi$
$311$	−15.3440	+	26.5767i	−0.870081	+	1.50702i	−0.861912	+	0.507058i	$0.830733\pi$
$312$		0			0
$313$	−5.48343	−	9.49757i	−0.309942	−	0.536835i	0.668408	−	0.743795i	$-0.266976\pi$
$313$	−5.48343	−	9.49757i	−0.309942	−	0.536835i	−0.978349	+	0.206961i	$0.933643\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.15353	+	1.99797i	0.0647885	+	0.112217i	0.896600	−	0.442841i	$-0.146029\pi$
$317$	1.15353	+	1.99797i	0.0647885	+	0.112217i	−0.831812	+	0.555058i	$0.812696\pi$
$318$		0			0
$319$	4.56629	+	7.90904i	0.255663	+	0.442821i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.25289	+	2.17007i	0.0697128	+	0.120746i
$324$		0			0
$325$	−2.27049	−	9.27127i	−0.125944	−	0.514278i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.33365	−	7.50611i	−0.238922	−	0.413825i
$330$		0			0
$331$	−33.6498			−1.84956			−0.924780	−	0.380502i	$-0.875751\pi$
$331$	−33.6498			−1.84956			−0.924780	+	0.380502i	$0.875751\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.83334	−	10.1036i	0.318710	−	0.552021i
$336$		0			0
$337$	6.88906	−	11.9322i	0.375271	−	0.649988i	−0.615097	−	0.788452i	$-0.710883\pi$
$337$	6.88906	−	11.9322i	0.375271	−	0.649988i	0.990368	+	0.138464i	$0.0442164\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.3669	+	21.4200i	0.669703	+	1.15996i
$342$		0			0
$343$	5.78622			0.312427
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.16260	+	5.47778i	0.169777	+	0.294063i	0.938341	−	0.345710i	$-0.112362\pi$
$347$	3.16260	+	5.47778i	0.169777	+	0.294063i	−0.768564	+	0.639773i	$0.779029\pi$
$348$		0			0
$349$	−6.66685	+	11.5473i	−0.356868	+	0.618114i	−0.987436	−	0.158021i	$-0.949489\pi$
$349$	−6.66685	+	11.5473i	−0.356868	+	0.618114i	0.630568	+	0.776134i	$0.282822\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.6247	+	21.8666i	−0.671945	+	1.16384i	0.305407	+	0.952222i	$0.401208\pi$
$353$	−12.6247	+	21.8666i	−0.671945	+	1.16384i	−0.977352	+	0.211621i	$0.932126\pi$
$354$		0			0
$355$	−3.73130	−	6.46279i	−0.198037	−	0.343010i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.22423			−0.0646123			−0.0323062	−	0.999478i	$-0.510285\pi$
$359$	−1.22423			−0.0646123			−0.0323062	+	0.999478i	$0.510285\pi$
$360$		0			0
$361$	6.91010	+	11.9687i	0.363690	+	0.629929i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.14758	−	7.18381i	0.217094	−	0.376018i
$366$		0			0
$367$	7.09999	−	12.2975i	0.370617	−	0.641927i	−0.619044	−	0.785356i	$-0.712480\pi$
$367$	7.09999	−	12.2975i	0.370617	−	0.641927i	0.989661	+	0.143430i	$0.0458131\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−21.7353			−1.12844
$372$		0			0
$373$	8.38836	+	14.5291i	0.434333	+	0.752287i	0.997241	−	0.0742329i	$-0.0236508\pi$
$373$	8.38836	+	14.5291i	0.434333	+	0.752287i	−0.562908	+	0.826520i	$0.690317\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.72289	−	3.56522i	0.191739	−	0.183618i
$378$		0			0
$379$	18.3934	+	31.8583i	0.944807	+	1.63645i	0.756138	+	0.654413i	$0.227084\pi$
$379$	18.3934	+	31.8583i	0.944807	+	1.63645i	0.188669	+	0.982041i	$0.439583\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.66299	+	13.2727i	0.391561	+	0.678203i	0.992656	−	0.120975i	$-0.0386020\pi$
$383$	7.66299	+	13.2727i	0.391561	+	0.678203i	−0.601095	+	0.799178i	$0.705269\pi$
$384$		0			0
$385$	−17.2191	−	29.8243i	−0.877565	−	1.51999i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.10671	−	7.11304i	−0.208219	−	0.360645i	0.742935	−	0.669364i	$-0.233433\pi$
$389$	−4.10671	−	7.11304i	−0.208219	−	0.360645i	−0.951153	+	0.308718i	$0.900100\pi$
$390$		0			0
$391$	1.83215	−	3.17338i	0.0926558	−	0.160485i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.33737			−0.369183
$396$		0			0
$397$	−12.3499	+	21.3906i	−0.619822	+	1.07356i	0.369695	+	0.929153i	$0.379462\pi$
$397$	−12.3499	+	21.3906i	−0.619822	+	1.07356i	−0.989518	+	0.144411i	$0.953871\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.8473			−0.691501			−0.345750	−	0.938327i	$-0.612376\pi$
$401$	−13.8473			−0.691501			−0.345750	+	0.938327i	$0.612376\pi$
$402$		0			0
$403$	10.0827	−	9.65566i	0.502255	−	0.480983i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.2101			1.15048
$408$		0			0
$409$	−9.32539	−	16.1521i	−0.461111	−	0.798668i	0.537906	−	0.843005i	$-0.319216\pi$
$409$	−9.32539	−	16.1521i	−0.461111	−	0.798668i	−0.999017	+	0.0443375i	$0.985882\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.53407	−	14.7814i	−0.419934	−	0.727347i
$414$		0			0
$415$	−9.80399			−0.481259
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15.7552			0.769693			0.384846	−	0.922981i	$-0.374254\pi$
$419$	15.7552			0.769693			0.384846	+	0.922981i	$0.374254\pi$
$420$		0			0
$421$	−7.16989	−	12.4186i	−0.349439	−	0.605246i	0.636711	−	0.771103i	$-0.280295\pi$
$421$	−7.16989	−	12.4186i	−0.349439	−	0.605246i	−0.986150	+	0.165856i	$0.946961\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.91476			0.141387
$426$		0			0
$427$	−10.8930			−0.527148
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.49274	−	7.78165i	−0.216408	−	0.374829i	0.737300	−	0.675566i	$-0.236101\pi$
$431$	−4.49274	−	7.78165i	−0.216408	−	0.374829i	−0.953707	+	0.300737i	$0.902767\pi$
$432$		0			0
$433$	−13.8669	+	24.0182i	−0.666401	+	1.15424i	0.312503	+	0.949917i	$0.398833\pi$
$433$	−13.8669	+	24.0182i	−0.666401	+	1.15424i	−0.978904	+	0.204323i	$0.934501\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.78730	+	6.55979i	−0.181171	+	0.313797i
$438$		0			0
$439$	−2.97597	+	5.15453i	−0.142035	+	0.246012i	−0.928263	−	0.371925i	$-0.878698\pi$
$439$	−2.97597	+	5.15453i	−0.142035	+	0.246012i	0.786228	+	0.617937i	$0.212031\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.66685	+	9.81526i	0.269240	+	0.466337i	0.968666	−	0.248368i	$-0.0798942\pi$
$443$	5.66685	+	9.81526i	0.269240	+	0.466337i	−0.699426	+	0.714705i	$0.746561\pi$
$444$		0			0
$445$	−13.0463			−0.618455
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.47394	+	11.2132i	−0.305524	+	0.529183i	−0.977378	−	0.211501i	$-0.932165\pi$
$449$	−6.47394	+	11.2132i	−0.305524	+	0.529183i	0.671854	+	0.740684i	$0.265498\pi$
$450$		0			0
$451$	−38.2879	+	66.3166i	−1.80291	+	3.12273i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−14.0387	+	13.4441i	−0.658144	+	0.630270i
$456$		0			0
$457$	9.67396	−	16.7558i	0.452529	−	0.783803i	−0.546013	−	0.837776i	$-0.683855\pi$
$457$	9.67396	−	16.7558i	0.452529	−	0.783803i	0.998542	+	0.0539733i	$0.0171886\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−34.7972			−1.62067			−0.810334	−	0.585969i	$-0.800714\pi$
$461$	−34.7972			−1.62067			−0.810334	+	0.585969i	$0.800714\pi$
$462$		0			0
$463$	13.0277	−	22.5647i	0.605450	−	1.04867i	−0.386530	−	0.922277i	$-0.626326\pi$
$463$	13.0277	−	22.5647i	0.605450	−	1.04867i	0.991980	−	0.126394i	$-0.0403402\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	39.3170			1.81937			0.909686	−	0.415296i	$-0.136322\pi$
$467$	39.3170			1.81937			0.909686	+	0.415296i	$0.136322\pi$
$468$		0			0
$469$	26.7344			1.23448
$470$		0			0
$471$		0			0
$472$		0			0
$473$	40.0935	−	69.4440i	1.84350	−	3.19304i
$474$		0			0
$475$	−6.02520			−0.276455
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.3647	−	21.4163i	0.564957	−	0.978534i	−0.432097	−	0.901827i	$-0.642226\pi$
$479$	12.3647	−	21.4163i	0.564957	−	0.978534i	0.997054	−	0.0767069i	$-0.0244406\pi$
$480$		0			0
$481$	−3.11615	−	12.7244i	−0.142084	−	0.580184i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.70432	−	4.68401i	0.122797	−	0.212690i
$486$		0			0
$487$	15.5768	−	26.9798i	0.705853	−	1.22257i	−0.260530	−	0.965466i	$-0.583897\pi$
$487$	15.5768	−	26.9798i	0.705853	−	1.22257i	0.966383	−	0.257107i	$-0.0827694\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.64887			0.0744124			0.0372062	−	0.999308i	$-0.488154\pi$
$491$	1.64887			0.0744124			0.0372062	+	0.999308i	$0.488154\pi$
$492$		0			0
$493$	0.787023	+	1.36316i	0.0354457	+	0.0613938i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.55032	−	14.8096i	0.383534	−	0.664301i
$498$		0			0
$499$	−21.3882	+	37.0455i	−0.957468	+	1.65838i	−0.228850	+	0.973462i	$0.573496\pi$
$499$	−21.3882	+	37.0455i	−0.957468	+	1.65838i	−0.728618	+	0.684921i	$0.759837\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.6416	−	34.0203i	0.875777	−	1.51689i	0.0198439	−	0.999803i	$-0.493683\pi$
$503$	19.6416	−	34.0203i	0.875777	−	1.51689i	0.855933	−	0.517087i	$-0.172984\pi$
$504$		0			0
$505$	−11.4026	−	19.7498i	−0.507408	−	0.878856i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.0577			−0.534448			−0.267224	−	0.963634i	$-0.586106\pi$
$509$	−12.0577			−0.534448			−0.267224	+	0.963634i	$0.586106\pi$
$510$		0			0
$511$	19.0085			0.840885
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.84205	−	15.3149i	−0.389627	−	0.674854i
$516$		0			0
$517$	15.7525			0.692792
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.6196			−0.728116			−0.364058	−	0.931376i	$-0.618609\pi$
$521$	−16.6196			−0.728116			−0.364058	+	0.931376i	$0.618609\pi$
$522$		0			0
$523$	−0.621458	−	1.07640i	−0.0271745	−	0.0470676i	0.852118	−	0.523349i	$-0.175318\pi$
$523$	−0.621458	−	1.07640i	−0.0271745	−	0.0470676i	−0.879293	+	0.476282i	$0.841984\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.13149	+	3.69185i	0.0928492	+	0.160820i
$528$		0			0
$529$	−11.9234			−0.518409
$530$		0			0
$531$		0			0
$532$		0			0
$533$	41.4970	+	12.0869i	1.79744	+	0.523543i
$534$		0			0
$535$	13.5306			0.584980
$536$		0			0
$537$		0			0
$538$		0			0
$539$	17.0998	−	29.6178i	0.736542	−	1.27573i
$540$		0			0
$541$	−41.8105			−1.79757			−0.898787	−	0.438386i	$-0.855550\pi$
$541$	−41.8105			−1.79757			−0.898787	+	0.438386i	$0.855550\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.10895	−	12.3131i	0.304514	−	0.527433i
$546$		0			0
$547$	−0.266912	−	0.462305i	−0.0114123	−	0.0197668i	0.860263	−	0.509851i	$-0.170299\pi$
$547$	−0.266912	−	0.462305i	−0.0114123	−	0.0197668i	−0.871675	+	0.490084i	$0.836966\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.62688	−	2.81784i	−0.0693075	−	0.120044i
$552$		0			0
$553$	−8.40685	−	14.5611i	−0.357496	−	0.619201i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.96368	−	6.86530i	−0.167947	−	0.290892i	0.769751	−	0.638344i	$-0.220380\pi$
$557$	−3.96368	−	6.86530i	−0.167947	−	0.290892i	−0.937698	+	0.347452i	$0.887047\pi$
$558$		0			0
$559$	−43.4540	−	12.6569i	−1.83791	−	0.535331i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.5866	+	26.9968i	0.656897	+	1.13778i	0.981415	+	0.191899i	$0.0614647\pi$
$563$	15.5866	+	26.9968i	0.656897	+	1.13778i	−0.324518	+	0.945880i	$0.605202\pi$
$564$		0			0
$565$	10.3148			0.433945
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.5253	−	18.2304i	0.441244	−	0.764257i	−0.556538	−	0.830822i	$-0.687871\pi$
$569$	10.5253	−	18.2304i	0.441244	−	0.764257i	0.997782	+	0.0665652i	$0.0212040\pi$
$570$		0			0
$571$	−2.80025	+	4.85017i	−0.117187	+	0.202973i	−0.918652	−	0.395068i	$-0.870721\pi$
$571$	−2.80025	+	4.85017i	−0.117187	+	0.202973i	0.801465	+	0.598042i	$0.204054\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.40543	+	7.63043i	0.183719	+	0.318211i
$576$		0			0
$577$	−30.6676			−1.27671			−0.638355	−	0.769742i	$-0.720385\pi$
$577$	−30.6676			−1.27671			−0.638355	+	0.769742i	$0.720385\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−11.2330	−	19.4561i	−0.466023	−	0.807176i
$582$		0			0
$583$	19.7515	−	34.2106i	0.818023	−	1.41686i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15.7329	+	27.2501i	−0.649364	+	1.12473i	0.333911	+	0.942605i	$0.391632\pi$
$587$	−15.7329	+	27.2501i	−0.649364	+	1.12473i	−0.983275	+	0.182127i	$0.941702\pi$
$588$		0			0
$589$	−4.40608	−	7.63155i	−0.181549	−	0.314452i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.583862			−0.0239763			−0.0119882	−	0.999928i	$-0.503816\pi$
$593$	−0.583862			−0.0239763			−0.0119882	+	0.999928i	$0.503816\pi$
$594$		0			0
$595$	−2.96779	−	5.14037i	−0.121668	−	0.210735i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.5346	−	21.7106i	0.512150	−	0.887069i	−0.487751	−	0.872983i	$-0.662183\pi$
$599$	12.5346	−	21.7106i	0.512150	−	0.887069i	0.999901	−	0.0140866i	$-0.00448405\pi$
$600$		0			0
$601$	−7.76215	+	13.4444i	−0.316625	+	0.548410i	−0.979782	−	0.200071i	$-0.935883\pi$
$601$	−7.76215	+	13.4444i	−0.316625	+	0.548410i	0.663157	+	0.748480i	$0.269216\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	45.7177			1.85869
$606$		0			0
$607$	−15.4441	−	26.7500i	−0.626858	−	1.08575i	−0.988178	−	0.153308i	$-0.951007\pi$
$607$	−15.4441	−	26.7500i	−0.626858	−	1.08575i	0.361321	−	0.932442i	$-0.382326\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.11490	−	8.63592i	−0.0855595	−	0.349372i
$612$		0			0
$613$	15.1558	+	26.2506i	0.612138	+	1.06025i	0.990879	+	0.134751i	$0.0430235\pi$
$613$	15.1558	+	26.2506i	0.612138	+	1.06025i	−0.378742	+	0.925502i	$0.623643\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	5.97492	+	10.3489i	0.240541	+	0.416630i	0.960869	−	0.277005i	$-0.0893417\pi$
$617$	5.97492	+	10.3489i	0.240541	+	0.416630i	−0.720327	+	0.693634i	$0.756008\pi$
$618$		0			0
$619$	−11.3093	−	19.5884i	−0.454561	−	0.787323i	0.544102	−	0.839019i	$-0.316870\pi$
$619$	−11.3093	−	19.5884i	−0.454561	−	0.787323i	−0.998663	+	0.0516965i	$0.983537\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.9479	−	25.8906i	−0.598876	−	1.03728i
$624$		0			0
$625$	2.37728	−	4.11756i	0.0950911	−	0.164703i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.00039			0.159506
$630$		0			0
$631$	2.92924	−	5.07359i	0.116611	−	0.201976i	−0.801811	−	0.597577i	$-0.796130\pi$
$631$	2.92924	−	5.07359i	0.116611	−	0.201976i	0.918423	+	0.395601i	$0.129464\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−23.3540			−0.926777
$636$		0			0
$637$	−18.5331	−	5.39816i	−0.734306	−	0.213883i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.8853			0.548437			0.274218	−	0.961667i	$-0.411581\pi$
$641$	13.8853			0.548437			0.274218	+	0.961667i	$0.411581\pi$
$642$		0			0
$643$	−13.9906	−	24.2324i	−0.551734	−	0.955632i	−0.998150	−	0.0608059i	$-0.980633\pi$
$643$	−13.9906	−	24.2324i	−0.551734	−	0.955632i	0.446415	−	0.894826i	$-0.352700\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13.4477	−	23.2920i	−0.528682	−	0.915705i	−0.999441	−	0.0334425i	$-0.989353\pi$
$647$	−13.4477	−	23.2920i	−0.528682	−	0.915705i	0.470758	−	0.882262i	$-0.343980\pi$
$648$		0			0
$649$	31.0206			1.21767
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.3964			0.759039			0.379519	−	0.925184i	$-0.376089\pi$
$653$	19.3964			0.759039			0.379519	+	0.925184i	$0.376089\pi$
$654$		0			0
$655$	−9.04884	−	15.6730i	−0.353567	−	0.612397i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−39.8501			−1.55234			−0.776171	−	0.630523i	$-0.782841\pi$
$659$	−39.8501			−1.55234			−0.776171	+	0.630523i	$0.782841\pi$
$660$		0			0
$661$	11.4745			0.446306			0.223153	−	0.974783i	$-0.428365\pi$
$661$	11.4745			0.446306			0.223153	+	0.974783i	$0.428365\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.13482	+	10.6258i	0.237898	+	0.412052i
$666$		0			0
$667$	−2.37905	+	4.12063i	−0.0921171	+	0.159551i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9.89875	−	17.1451i	0.382137	−	0.661881i
$672$		0			0
$673$	18.3543	−	31.7906i	0.707506	−	1.22544i	−0.258273	−	0.966072i	$-0.583154\pi$
$673$	18.3543	−	31.7906i	0.707506	−	1.22544i	0.965779	−	0.259365i	$-0.0835132\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.57943	+	6.19976i	0.137569	+	0.238276i	0.926576	−	0.376108i	$-0.122738\pi$
$677$	3.57943	+	6.19976i	0.137569	+	0.238276i	−0.789007	+	0.614384i	$0.789405\pi$
$678$		0			0
$679$	12.3940			0.475637
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−13.2982	+	23.0331i	−0.508841	+	0.881338i	0.491107	+	0.871099i	$0.336592\pi$
$683$	−13.2982	+	23.0331i	−0.508841	+	0.881338i	−0.999948	+	0.0102387i	$0.996741\pi$
$684$		0			0
$685$	−10.7409	+	18.6037i	−0.410387	+	0.710812i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−21.4070	−	6.23525i	−0.815540	−	0.237544i
$690$		0			0
$691$	15.5581	−	26.9474i	0.591858	−	1.02513i	−0.402124	−	0.915585i	$-0.631728\pi$
$691$	15.5581	−	26.9474i	0.591858	−	1.02513i	0.993982	−	0.109543i	$-0.0349387\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.7109			−0.482154
$696$		0			0
$697$	−6.59912	+	11.4300i	−0.249959	+	0.432942i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.5419			1.45571			0.727854	−	0.685732i	$-0.240518\pi$
$701$	38.5419			1.45571			0.727854	+	0.685732i	$0.240518\pi$
$702$		0			0
$703$	−8.26933			−0.311884
$704$		0			0
$705$		0			0
$706$		0			0
$707$	26.1292	−	45.2570i	0.982688	−	1.70207i
$708$		0			0
$709$	1.84079			0.0691325			0.0345662	−	0.999402i	$-0.488995\pi$
$709$	1.84079			0.0691325			0.0345662	+	0.999402i	$0.488995\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.44316	+	11.1599i	−0.241298	+	0.417941i
$714$		0			0
$715$	−8.40319	−	34.3134i	−0.314262	−	1.28325i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.15519	+	15.8573i	−0.341431	+	0.591376i	−0.984699	−	0.174265i	$-0.944245\pi$
$719$	−9.15519	+	15.8573i	−0.341431	+	0.591376i	0.643268	+	0.765641i	$0.277578\pi$
$720$		0			0
$721$	20.2617	−	35.0943i	0.754585	−	1.30698i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.78482			−0.140565
$726$		0			0
$727$	−11.6183	−	20.1235i	−0.430900	−	0.746340i	0.566051	−	0.824370i	$-0.308470\pi$
$727$	−11.6183	−	20.1235i	−0.430900	−	0.746340i	−0.996951	+	0.0780300i	$0.975137\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.91032	−	11.9690i	0.255587	−	0.442691i
$732$		0			0
$733$	−9.62013	+	16.6626i	−0.355328	+	0.615446i	−0.987174	−	0.159648i	$-0.948964\pi$
$733$	−9.62013	+	16.6626i	−0.355328	+	0.615446i	0.631846	+	0.775094i	$0.282297\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.2943	+	42.0790i	−0.894892	+	1.55000i
$738$		0			0
$739$	−1.98535	−	3.43873i	−0.0730324	−	0.126496i	0.827197	−	0.561913i	$-0.189934\pi$
$739$	−1.98535	−	3.43873i	−0.0730324	−	0.126496i	−0.900229	+	0.435417i	$0.856601\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.75177			0.211012			0.105506	−	0.994419i	$-0.466354\pi$
$743$	5.75177			0.211012			0.105506	+	0.994419i	$0.466354\pi$
$744$		0			0
$745$	−6.03870			−0.221241
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.5028	+	26.8517i	0.566460	+	0.981138i
$750$		0			0
$751$	−45.3603			−1.65522			−0.827610	−	0.561304i	$-0.810300\pi$
$751$	−45.3603			−1.65522			−0.827610	+	0.561304i	$0.810300\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.1044			−1.13200
$756$		0			0
$757$	6.54346	+	11.3336i	0.237826	+	0.411927i	0.960090	−	0.279691i	$-0.0902319\pi$
$757$	6.54346	+	11.3336i	0.237826	+	0.411927i	−0.722264	+	0.691617i	$0.756899\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.5091	−	23.3985i	−0.489706	−	0.848195i	0.510224	−	0.860041i	$-0.329562\pi$
$761$	−13.5091	−	23.3985i	−0.489706	−	0.848195i	−0.999930	+	0.0118465i	$0.996229\pi$
$762$		0			0
$763$	32.5805			1.17949
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.16477	−	17.0063i	−0.150381	−	0.614063i
$768$		0			0
$769$	46.2575			1.66809			0.834045	−	0.551697i	$-0.186019\pi$
$769$	46.2575			1.66809			0.834045	+	0.551697i	$0.186019\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.9744	+	36.3287i	−0.754397	+	1.30665i	0.191276	+	0.981536i	$0.438737\pi$
$773$	−20.9744	+	36.3287i	−0.754397	+	1.30665i	−0.945673	+	0.325118i	$0.894596\pi$
$774$		0			0
$775$	−10.2504			−0.368205
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.6413	−	23.6273i	0.488749	−	0.846537i
$780$		0			0
$781$	15.5398	+	26.9158i	0.556059	+	0.963123i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.9837	−	31.1487i	−0.641866	−	1.11174i
$786$		0			0
$787$	−20.5108	−	35.5257i	−0.731131	−	1.26636i	−0.956400	−	0.292059i	$-0.905660\pi$
$787$	−20.5108	−	35.5257i	−0.731131	−	1.26636i	0.225270	−	0.974296i	$-0.427674\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.8182	+	20.4698i	0.420207	+	0.727821i
$792$		0			0
$793$	−10.7284	−	3.12489i	−0.380977	−	0.110968i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.28106	+	14.3432i	0.293330	+	0.508063i	0.974595	−	0.223974i	$-0.0719031\pi$
$797$	8.28106	+	14.3432i	0.293330	+	0.508063i	−0.681265	+	0.732037i	$0.738570\pi$
$798$		0			0
$799$	2.71502			0.0960504
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.2735	+	29.9187i	−0.609570	+	1.05581i
$804$		0			0
$805$	8.97117	−	15.5385i	0.316192	−	0.547661i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.0672	+	43.4176i	0.881314	+	1.52648i	0.849880	+	0.526976i	$0.176674\pi$
$809$	25.0672	+	43.4176i	0.881314	+	1.52648i	0.0314339	+	0.999506i	$0.489993\pi$
$810$		0			0
$811$	−15.2618			−0.535914			−0.267957	−	0.963431i	$-0.586349\pi$
$811$	−15.2618			−0.535914			−0.267957	+	0.963431i	$0.586349\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.36684	−	4.09950i	−0.0829070	−	0.143599i
$816$		0			0
$817$	−14.2846	+	24.7416i	−0.499753	+	0.865598i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.56075	+	4.43535i	−0.0893708	+	0.154795i	−0.907245	−	0.420602i	$-0.861819\pi$
$821$	−2.56075	+	4.43535i	−0.0893708	+	0.154795i	0.817875	+	0.575397i	$0.195152\pi$
$822$		0			0
$823$	7.45156	+	12.9065i	0.259745	+	0.449891i	0.966174	−	0.257893i	$-0.0830282\pi$
$823$	7.45156	+	12.9065i	0.259745	+	0.449891i	−0.706429	+	0.707784i	$0.749695\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.5879			0.646364			0.323182	−	0.946337i	$-0.395247\pi$
$827$	18.5879			0.646364			0.323182	+	0.946337i	$0.395247\pi$
$828$		0			0
$829$	−10.4298	−	18.0650i	−0.362243	−	0.627422i	0.626087	−	0.779753i	$-0.284655\pi$
$829$	−10.4298	−	18.0650i	−0.362243	−	0.627422i	−0.988330	+	0.152331i	$0.951322\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.94724	−	5.10477i	0.102116	−	0.176870i
$834$		0			0
$835$	−14.9601	+	25.9117i	−0.517716	+	0.896710i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36.0457			−1.24444			−0.622219	−	0.782843i	$-0.713769\pi$
$839$	−36.0457			−1.24444			−0.622219	+	0.782843i	$0.713769\pi$
$840$		0			0
$841$	13.4781	+	23.3447i	0.464760	+	0.804989i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.6834	+	9.21372i	−0.608326	+	0.316962i
$846$		0			0
$847$	52.3814	+	90.7273i	1.79985	+	3.11743i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.04627	+	10.4724i	0.207263	+	0.358991i
$852$		0			0
$853$	6.97776	+	12.0858i	0.238914	+	0.413811i	0.960403	−	0.278615i	$-0.0898753\pi$
$853$	6.97776	+	12.0858i	0.238914	+	0.413811i	−0.721489	+	0.692426i	$0.756542\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.73939	−	15.1371i	−0.298532	−	0.517073i	0.677268	−	0.735736i	$-0.263164\pi$
$857$	−8.73939	−	15.1371i	−0.298532	−	0.517073i	−0.975800	+	0.218664i	$0.929830\pi$
$858$		0			0
$859$	−4.17576	+	7.23263i	−0.142475	+	0.246774i	−0.928428	−	0.371512i	$-0.878839\pi$
$859$	−4.17576	+	7.23263i	−0.142475	+	0.246774i	0.785953	+	0.618286i	$0.212173\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.11501			−0.140077			−0.0700384	−	0.997544i	$-0.522312\pi$
$863$	−4.11501			−0.140077			−0.0700384	+	0.997544i	$0.522312\pi$
$864$		0			0
$865$	−8.20635	+	14.2138i	−0.279024	+	0.483284i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	30.5582			1.03662
$870$		0			0
$871$	26.3305	+	7.66935i	0.892176	+	0.259866i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	41.2276			1.39375
$876$		0			0
$877$	−5.30991	−	9.19704i	−0.179303	−	0.310562i	0.762339	−	0.647178i	$-0.224051\pi$
$877$	−5.30991	−	9.19704i	−0.179303	−	0.310562i	−0.941642	+	0.336616i	$0.890718\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.0048	−	25.9891i	−0.505525	−	0.875595i	−0.999980	−	0.00639173i	$-0.997965\pi$
$881$	−15.0048	−	25.9891i	−0.505525	−	0.875595i	0.494454	−	0.869204i	$-0.335368\pi$
$882$		0			0
$883$	21.3674			0.719072			0.359536	−	0.933131i	$-0.382935\pi$
$883$	21.3674			0.719072			0.359536	+	0.933131i	$0.382935\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−40.3161			−1.35368			−0.676841	−	0.736129i	$-0.736652\pi$
$887$	−40.3161			−1.35368			−0.676841	+	0.736129i	$0.736652\pi$
$888$		0			0
$889$	−26.7581	−	46.3464i	−0.897437	−	1.55441i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.61230			−0.187808
$894$		0			0
$895$	3.49917			0.116964
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.76774	−	4.79387i	−0.0923094	−	0.159884i
$900$		0			0
$901$	3.40427	−	5.89637i	0.113413	−	0.196436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.88051	+	13.6494i	−0.261957	+	0.453723i
$906$		0			0
$907$	−7.39556	+	12.8095i	−0.245566	+	0.425332i	−0.962290	−	0.272024i	$-0.912307\pi$
$907$	−7.39556	+	12.8095i	−0.245566	+	0.425332i	0.716725	+	0.697356i	$0.245640\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.1529	+	27.9777i	0.535170	+	0.926942i	0.999155	+	0.0410987i	$0.0130858\pi$
$911$	16.1529	+	27.9777i	0.535170	+	0.926942i	−0.463985	+	0.885843i	$0.653581\pi$
$912$		0			0
$913$	40.8310			1.35131
$914$		0			0
$915$		0			0
$916$		0			0
$917$	20.7356	−	35.9150i	0.684748	−	1.18602i
$918$		0			0
$919$	18.5950	−	32.2075i	0.613392	−	1.06243i	−0.377272	−	0.926103i	$-0.623138\pi$
$919$	18.5950	−	32.2075i	0.613392	−	1.06243i	0.990664	−	0.136324i	$-0.0435289\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.6696	−	12.1330i	0.417026	−	0.399364i
$924$		0			0
$925$	−4.80949	+	8.33029i	−0.158135	+	0.273898i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−15.2044			−0.498841			−0.249421	−	0.968395i	$-0.580240\pi$
$929$	−15.2044			−0.498841			−0.249421	+	0.968395i	$0.580240\pi$
$930$		0			0
$931$	−6.09234	+	10.5522i	−0.199668	+	0.345836i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	10.7877			0.352795
$936$		0			0
$937$	56.1934			1.83576			0.917880	−	0.396859i	$-0.129900\pi$
$937$	56.1934			1.83576			0.917880	+	0.396859i	$0.129900\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.6111	+	27.0393i	−0.508909	+	0.881455i	0.491038	+	0.871138i	$0.336618\pi$
$941$	−15.6111	+	27.0393i	−0.508909	+	0.881455i	−0.999947	+	0.0103174i	$0.996716\pi$
$942$		0			0
$943$	−39.8962			−1.29920
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.3364	+	35.2236i	−0.660843	+	1.14461i	0.319552	+	0.947569i	$0.396468\pi$
$947$	−20.3364	+	35.2236i	−0.660843	+	1.14461i	−0.980395	+	0.197044i	$0.936866\pi$
$948$		0			0
$949$	18.7213	+	5.45300i	0.607720	+	0.177012i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.35525	+	2.34736i	−0.0439008	+	0.0760384i	−0.887141	−	0.461499i	$-0.847312\pi$
$953$	−1.35525	+	2.34736i	−0.0439008	+	0.0760384i	0.843240	+	0.537537i	$0.180645\pi$
$954$		0			0
$955$	18.1205	−	31.3857i	0.586367	−	1.01562i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−49.2257			−1.58958
$960$		0			0
$961$	8.00414	+	13.8636i	0.258198	+	0.447212i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.46634	+	14.6641i	−0.272541	+	0.472055i
$966$		0			0
$967$	−2.04491	+	3.54189i	−0.0657600	+	0.113900i	−0.897031	−	0.441968i	$-0.854280\pi$
$967$	−2.04491	+	3.54189i	−0.0657600	+	0.113900i	0.831271	+	0.555868i	$0.187614\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.04999	−	10.4789i	0.194153	−	0.336283i	−0.752469	−	0.658627i	$-0.771137\pi$
$971$	6.04999	−	10.4789i	0.194153	−	0.336283i	0.946623	+	0.322344i	$0.104471\pi$
$972$		0			0
$973$	−14.5637	−	25.2250i	−0.466890	−	0.808677i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−7.64951			−0.244729			−0.122365	−	0.992485i	$-0.539048\pi$
$977$	−7.64951			−0.244729			−0.122365	+	0.992485i	$0.539048\pi$
$978$		0			0
$979$	54.3344			1.73654
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.0980	−	43.4711i	−0.800503	−	1.38651i	−0.919286	−	0.393591i	$-0.871233\pi$
$983$	−25.0980	−	43.4711i	−0.800503	−	1.38651i	0.118783	−	0.992920i	$-0.462101\pi$
$984$		0			0
$985$	13.7332			0.437576
$986$		0			0
$987$		0			0
$988$		0			0
$989$	41.7776			1.32845
$990$		0			0
$991$	−21.7547	−	37.6803i	−0.691062	−	1.19695i	−0.971490	−	0.237079i	$-0.923810\pi$
$991$	−21.7547	−	37.6803i	−0.691062	−	1.19695i	0.280428	−	0.959875i	$-0.409523\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	17.9075	+	31.0167i	0.567705	+	0.983294i
$996$		0			0
$997$	−30.2028			−0.956532			−0.478266	−	0.878215i	$-0.658735\pi$
$997$	−30.2028			−0.956532			−0.478266	+	0.878215i	$0.658735\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.s.f.1225.8		40
3.2	odd	2		936.2.s.f.913.17	yes	40
9.2	odd	6		936.2.r.f.601.11	✓	40
9.7	even	3		2808.2.r.f.289.8		40
13.9	even	3		2808.2.r.f.2089.8		40
39.35	odd	6		936.2.r.f.841.11	yes	40
117.61	even	3	inner	2808.2.s.f.1153.8		40
117.74	odd	6		936.2.s.f.529.17	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.r.f.601.11	✓	40	9.2	odd	6
936.2.r.f.841.11	yes	40	39.35	odd	6
936.2.s.f.529.17	yes	40	117.74	odd	6
936.2.s.f.913.17	yes	40	3.2	odd	2
2808.2.r.f.289.8		40	9.7	even	3
2808.2.r.f.2089.8		40	13.9	even	3
2808.2.s.f.1153.8		40	117.61	even	3	inner
2808.2.s.f.1225.8		40	1.1	even	1	trivial

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.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.766914 + 1.32833i) q^{5} -3.51479 q^{7} +(3.19399 - 5.53215i) q^{11} +(-3.46170 - 1.00830i) q^{13} +(0.550501 - 0.953495i) q^{17} +(-1.13796 + 1.97100i) q^{19} +3.32815 q^{23} +(1.32369 + 2.29269i) q^{25} +(-0.714825 + 1.23811i) q^{29} +(-1.93596 + 3.35318i) q^{31} +(2.69554 - 4.66881i) q^{35} +(1.81670 + 3.14662i) q^{37} -11.9875 q^{41} +12.5528 q^{43} +(1.23298 + 2.13558i) q^{47} +5.35375 q^{49} +6.18395 q^{53} +(4.89903 + 8.48537i) q^{55} +(2.42804 + 4.20550i) q^{59} +3.09918 q^{61} +(3.99418 - 3.82501i) q^{65} -7.60626 q^{67} +(-2.43267 + 4.21351i) q^{71} -5.40814 q^{73} +(-11.2262 + 19.4444i) q^{77} +(2.39185 + 4.14281i) q^{79} +(3.19592 + 5.53550i) q^{83} +(0.844373 + 1.46250i) q^{85} +(4.25286 + 7.36618i) q^{89} +(12.1671 + 3.54395i) q^{91} +(-1.74543 - 3.02318i) q^{95} -3.52623 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - q^{5} - 14 q^{7} - 3 q^{13} + q^{17} + 2 q^{19} - 2 q^{23} - 23 q^{25} - 12 q^{29} + 8 q^{31} + 12 q^{35} + 18 q^{37} - 6 q^{41} - 16 q^{43} - 4 q^{47} + 46 q^{49} + 40 q^{53} - 14 q^{55} + 4 q^{59}+ \cdots - 70 q^{97}+O(q^{100}) \) 40 * q - q^5 - 14 * q^7 - 3 * q^13 + q^17 + 2 * q^19 - 2 * q^23 - 23 * q^25 - 12 * q^29 + 8 * q^31 + 12 * q^35 + 18 * q^37 - 6 * q^41 - 16 * q^43 - 4 * q^47 + 46 * q^49 + 40 * q^53 - 14 * q^55 + 4 * q^59 + 6 * q^61 - 7 * q^65 - 54 * q^67 - q^71 - 12 * q^73 + 4 * q^77 + 13 * q^79 + 15 * q^83 + 5 * q^85 - 12 * q^89 - 57 * q^91 - q^95 - 70 * q^97

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