LMFDB - Embedded newform 2808.2.cw.b.1585.17

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)

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);

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")

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			1585.17
Character	$\chi$	$=$	2808.1585
Dual form			2808.2.cw.b.2521.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.834885 + 0.482021i) q^{5} +(-2.67273 - 1.54310i) q^{7} +(-1.06132 - 0.612753i) q^{11} +(1.52066 - 3.26919i) q^{13} +0.546948 q^{17} -1.79737i q^{19} +(2.77669 + 4.80937i) q^{23} +(-2.03531 + 3.52526i) q^{25} +(2.22766 - 3.85842i) q^{29} +(-5.62571 + 3.24801i) q^{31} +2.97523 q^{35} -10.3326i q^{37} +(-2.93292 + 1.69332i) q^{41} +(-4.61987 + 8.00186i) q^{43} +(3.04696 + 1.75916i) q^{47} +(1.26233 + 2.18641i) q^{49} +11.1466 q^{53} +1.18144 q^{55} +(0.294462 - 0.170008i) q^{59} +(-7.25146 + 12.5599i) q^{61} +(0.306243 + 3.46239i) q^{65} +(-5.17608 + 2.98841i) q^{67} +15.2061i q^{71} -0.591132i q^{73} +(1.89108 + 3.27545i) q^{77} +(-1.55716 + 2.69709i) q^{79} +(-4.02026 - 2.32110i) q^{83} +(-0.456638 + 0.263640i) q^{85} +7.95888i q^{89} +(-9.10900 + 6.39113i) q^{91} +(0.866370 + 1.50060i) q^{95} +(-9.64696 - 5.56967i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.834885	+	0.482021i	−0.373372	+	0.215566i	−0.674931	−	0.737881i	$-0.735826\pi$
$5$	−0.834885	+	0.482021i	−0.373372	+	0.215566i	0.301559	+	0.953448i	$0.402493\pi$
$6$		0			0
$7$	−2.67273	−	1.54310i	−1.01020	−	0.583238i	−0.0989479	−	0.995093i	$-0.531548\pi$
$7$	−2.67273	−	1.54310i	−1.01020	−	0.583238i	−0.911249	+	0.411855i	$0.864881\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.06132	−	0.612753i	−0.320000	−	0.184752i	0.331393	−	0.943493i	$-0.392481\pi$
$11$	−1.06132	−	0.612753i	−0.320000	−	0.184752i	−0.651392	+	0.758741i	$0.725815\pi$
$12$		0			0
$13$	1.52066	−	3.26919i	0.421755	−	0.906710i
$13$	1.52066	−	3.26919i	0.421755	−	0.906710i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.546948			0.132654			0.0663272	−	0.997798i	$-0.478872\pi$
$17$	0.546948			0.132654			0.0663272	+	0.997798i	$0.478872\pi$
$18$		0			0
$19$		−	1.79737i		−	0.412345i	−0.978516	−	0.206173i	$-0.933899\pi$
$19$		−	1.79737i		−	0.412345i	0.978516	−	0.206173i	$-0.0661008\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.77669	+	4.80937i	0.578980	+	1.00282i	0.995597	+	0.0937399i	$0.0298822\pi$
$23$	2.77669	+	4.80937i	0.578980	+	1.00282i	−0.416617	+	0.909082i	$0.636784\pi$
$24$		0			0
$25$	−2.03531	+	3.52526i	−0.407062	+	0.705053i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.22766	−	3.85842i	0.413666	−	0.716490i	−0.581621	−	0.813460i	$-0.697581\pi$
$29$	2.22766	−	3.85842i	0.413666	−	0.716490i	0.995287	+	0.0969692i	$0.0309148\pi$
$30$		0			0
$31$	−5.62571	+	3.24801i	−1.01041	+	0.583359i	−0.911311	−	0.411719i	$-0.864929\pi$
$31$	−5.62571	+	3.24801i	−1.01041	+	0.583359i	−0.0990968	+	0.995078i	$0.531595\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.97523			0.502906
$36$		0			0
$37$		−	10.3326i		−	1.69867i	−0.527852	−	0.849336i	$-0.677003\pi$
$37$		−	10.3326i		−	1.69867i	0.527852	−	0.849336i	$-0.322997\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.93292	+	1.69332i	−0.458045	+	0.264452i	−0.711222	−	0.702968i	$-0.751858\pi$
$41$	−2.93292	+	1.69332i	−0.458045	+	0.264452i	0.253177	+	0.967420i	$0.418524\pi$
$42$		0			0
$43$	−4.61987	+	8.00186i	−0.704524	+	1.22027i	0.262339	+	0.964976i	$0.415506\pi$
$43$	−4.61987	+	8.00186i	−0.704524	+	1.22027i	−0.966863	+	0.255296i	$0.917827\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.04696	+	1.75916i	0.444444	+	0.256600i	0.705481	−	0.708729i	$-0.250731\pi$
$47$	3.04696	+	1.75916i	0.444444	+	0.256600i	−0.261037	+	0.965329i	$0.584064\pi$
$48$		0			0
$49$	1.26233	+	2.18641i	0.180332	+	0.312345i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.1466			1.53111			0.765553	−	0.643372i	$-0.222465\pi$
$53$	11.1466			1.53111			0.765553	+	0.643372i	$0.222465\pi$
$54$		0			0
$55$	1.18144			0.159305
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.294462	−	0.170008i	0.0383357	−	0.0221331i	−0.480710	−	0.876880i	$-0.659621\pi$
$59$	0.294462	−	0.170008i	0.0383357	−	0.0221331i	0.519045	+	0.854747i	$0.326288\pi$
$60$		0			0
$61$	−7.25146	+	12.5599i	−0.928455	+	1.60813i	−0.142546	+	0.989788i	$0.545529\pi$
$61$	−7.25146	+	12.5599i	−0.928455	+	1.60813i	−0.785908	+	0.618343i	$0.787804\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.306243	+	3.46239i	0.0379848	+	0.429456i
$66$		0			0
$67$	−5.17608	+	2.98841i	−0.632359	+	0.365093i	−0.781665	−	0.623698i	$-0.785629\pi$
$67$	−5.17608	+	2.98841i	−0.632359	+	0.365093i	0.149306	+	0.988791i	$0.452296\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			15.2061i			1.80463i	0.431075	+	0.902316i	$0.358134\pi$
$71$			15.2061i			1.80463i	−0.431075	+	0.902316i	$0.641866\pi$
$72$		0			0
$73$		−	0.591132i		−	0.0691867i	−0.999401	−	0.0345934i	$-0.988986\pi$
$73$		−	0.591132i		−	0.0691867i	0.999401	−	0.0345934i	$-0.0110136\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.89108	+	3.27545i	0.215509	+	0.373272i
$78$		0			0
$79$	−1.55716	+	2.69709i	−0.175195	+	0.303446i	−0.940229	−	0.340544i	$-0.889389\pi$
$79$	−1.55716	+	2.69709i	−0.175195	+	0.303446i	0.765034	+	0.643990i	$0.222722\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.02026	−	2.32110i	−0.441281	−	0.254773i	0.262860	−	0.964834i	$-0.415334\pi$
$83$	−4.02026	−	2.32110i	−0.441281	−	0.254773i	−0.704141	+	0.710060i	$0.748668\pi$
$84$		0			0
$85$	−0.456638	+	0.263640i	−0.0495294	+	0.0285958i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.95888i			0.843640i	0.906680	+	0.421820i	$0.138609\pi$
$89$			7.95888i			0.843640i	−0.906680	+	0.421820i	$0.861391\pi$
$90$		0			0
$91$	−9.10900	+	6.39113i	−0.954883	+	0.669973i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.866370	+	1.50060i	0.0888877	+	0.153958i
$96$		0			0
$97$	−9.64696	−	5.56967i	−0.979500	−	0.565515i	−0.0773809	−	0.997002i	$-0.524656\pi$
$97$	−9.64696	−	5.56967i	−0.979500	−	0.565515i	−0.902119	+	0.431487i	$0.857989\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.48042	+	9.49236i	−0.545322	+	0.944525i	0.453265	+	0.891376i	$0.350259\pi$
$101$	−5.48042	+	9.49236i	−0.545322	+	0.944525i	−0.998587	+	0.0531491i	$0.983074\pi$
$102$		0			0
$103$	−3.32275	−	5.75517i	−0.327400	−	0.567073i	0.654595	−	0.755980i	$-0.272839\pi$
$103$	−3.32275	−	5.75517i	−0.327400	−	0.567073i	−0.981995	+	0.188906i	$0.939506\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.27194			−0.606331			−0.303165	−	0.952938i	$-0.598043\pi$
$107$	−6.27194			−0.606331			−0.303165	+	0.952938i	$0.598043\pi$
$108$		0			0
$109$			5.56780i			0.533298i	0.963794	+	0.266649i	$0.0859165\pi$
$109$			5.56780i			0.533298i	−0.963794	+	0.266649i	$0.914084\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.49977	+	6.06178i	0.329231	+	0.570244i	0.982359	−	0.187003i	$-0.0598773\pi$
$113$	3.49977	+	6.06178i	0.329231	+	0.570244i	−0.653129	+	0.757247i	$0.726544\pi$
$114$		0			0
$115$	−4.63643	−	2.67684i	−0.432349	−	0.249617i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.46184	−	0.843996i	−0.134007	−	0.0773690i
$120$		0			0
$121$	−4.74907	−	8.22563i	−0.431733	−	0.747784i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	8.74446i		−	0.782128i
$126$		0			0
$127$	−3.82713			−0.339603			−0.169801	−	0.985478i	$-0.554313\pi$
$127$	−3.82713			−0.339603			−0.169801	+	0.985478i	$0.554313\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.76389	+	8.25130i	0.416223	+	0.720920i	0.995556	−	0.0941713i	$-0.0300201\pi$
$131$	4.76389	+	8.25130i	0.416223	+	0.720920i	−0.579333	+	0.815091i	$0.696687\pi$
$132$		0			0
$133$	−2.77353	+	4.80389i	−0.240495	+	0.416550i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.0185	−	5.78417i	−0.855937	−	0.494175i	0.00671288	−	0.999977i	$-0.497863\pi$
$137$	−10.0185	−	5.78417i	−0.855937	−	0.494175i	−0.862649	+	0.505802i	$0.831197\pi$
$138$		0			0
$139$	3.61971	+	6.26953i	0.307020	+	0.531774i	0.977709	−	0.209964i	$-0.0673348\pi$
$139$	3.61971	+	6.26953i	0.307020	+	0.531774i	−0.670689	+	0.741739i	$0.734001\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.61711	+	2.53786i	−0.302478	+	0.212227i
$144$		0			0
$145$			4.29511i			0.356690i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.96224	−	1.13290i	0.160753	−	0.0928108i	−0.417465	−	0.908693i	$-0.637081\pi$
$149$	1.96224	−	1.13290i	0.160753	−	0.0928108i	0.578218	+	0.815882i	$0.303748\pi$
$150$		0			0
$151$	3.02877	+	1.74866i	0.246478	+	0.142304i	0.618151	−	0.786060i	$-0.287882\pi$
$151$	3.02877	+	1.74866i	0.246478	+	0.142304i	−0.371672	+	0.928364i	$0.621216\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.13122	−	5.42342i	0.251505	−	0.435620i
$156$		0			0
$157$	10.2470	+	17.7483i	0.817800	+	1.41647i	0.907300	+	0.420484i	$0.138140\pi$
$157$	10.2470	+	17.7483i	0.817800	+	1.41647i	−0.0894997	+	0.995987i	$0.528527\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	17.1389i		−	1.35073i
$162$		0			0
$163$			13.2244i			1.03581i	0.855438	+	0.517906i	$0.173288\pi$
$163$			13.2244i			1.03581i	−0.855438	+	0.517906i	$0.826712\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−22.0571	+	12.7347i	−1.70683	+	0.985440i	−0.768405	+	0.639964i	$0.778949\pi$
$167$	−22.0571	+	12.7347i	−1.70683	+	0.985440i	−0.938427	+	0.345476i	$0.887717\pi$
$168$		0			0
$169$	−8.37519	−	9.94264i	−0.644246	−	0.764819i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.36419	+	12.7552i	−0.559889	+	0.969756i	0.437616	+	0.899162i	$0.355823\pi$
$173$	−7.36419	+	12.7552i	−0.559889	+	0.969756i	−0.997505	+	0.0705944i	$0.977510\pi$
$174$		0			0
$175$	10.8797	−	6.28139i	0.822426	−	0.474828i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.68762			−0.126138			−0.0630691	−	0.998009i	$-0.520089\pi$
$179$	−1.68762			−0.126138			−0.0630691	+	0.998009i	$0.520089\pi$
$180$		0			0
$181$	20.3976			1.51614			0.758069	−	0.652174i	$-0.226143\pi$
$181$	20.3976			1.51614			0.758069	+	0.652174i	$0.226143\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.98054	+	8.62655i	0.366177	+	0.634236i
$186$		0			0
$187$	−0.580486	−	0.335144i	−0.0424493	−	0.0245081i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.64740	−	8.04953i	0.336274	−	0.582443i	−0.647455	−	0.762104i	$-0.724167\pi$
$191$	4.64740	−	8.04953i	0.336274	−	0.582443i	0.983729	+	0.179660i	$0.0574999\pi$
$192$		0			0
$193$	15.0682	−	8.69961i	1.08463	−	0.626212i	0.152489	−	0.988305i	$-0.451271\pi$
$193$	15.0682	−	8.69961i	1.08463	−	0.626212i	0.932142	+	0.362093i	$0.117938\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			14.5356i			1.03562i	0.855495	+	0.517811i	$0.173253\pi$
$197$			14.5356i			1.03562i	−0.855495	+	0.517811i	$0.826747\pi$
$198$		0			0
$199$	8.75359			0.620526			0.310263	−	0.950651i	$-0.399583\pi$
$199$	8.75359			0.620526			0.310263	+	0.950651i	$0.399583\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.9079	+	6.87501i	−0.835768	+	0.482531i
$204$		0			0
$205$	1.63243	−	2.82746i	0.114014	−	0.197478i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.10134	+	1.90758i	−0.0761815	+	0.131950i
$210$		0			0
$211$	−7.10165	−	12.3004i	−0.488898	−	0.846796i	0.511021	−	0.859568i	$-0.329268\pi$
$211$	−7.10165	−	12.3004i	−0.488898	−	0.846796i	−0.999918	+	0.0127727i	$0.995934\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	8.90750i		−	0.607487i
$216$		0			0
$217$	20.0480			1.36095
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.831721	−	1.78808i	0.0559476	−	0.120279i
$222$		0			0
$223$	−5.32632	−	3.07515i	−0.356677	−	0.205927i	0.310945	−	0.950428i	$-0.399354\pi$
$223$	−5.32632	−	3.07515i	−0.356677	−	0.205927i	−0.667622	+	0.744500i	$0.732688\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−20.9114	−	12.0732i	−1.38794	−	0.801328i	−0.394858	−	0.918742i	$-0.629206\pi$
$227$	−20.9114	−	12.0732i	−1.38794	−	0.801328i	−0.993083	+	0.117414i	$0.962540\pi$
$228$		0			0
$229$	0.190914	−	0.110224i	0.0126160	−	0.00728384i	−0.493679	−	0.869644i	$-0.664348\pi$
$229$	0.190914	−	0.110224i	0.0126160	−	0.00728384i	0.506295	+	0.862361i	$0.331015\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.6872			0.765651			0.382825	−	0.923821i	$-0.374951\pi$
$233$	11.6872			0.765651			0.382825	+	0.923821i	$0.374951\pi$
$234$		0			0
$235$	−3.39181			−0.221257
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.1577	−	6.44188i	0.721729	−	0.416690i	−0.0936597	−	0.995604i	$-0.529857\pi$
$239$	11.1577	−	6.44188i	0.721729	−	0.416690i	0.815389	+	0.578914i	$0.196523\pi$
$240$		0			0
$241$	−10.4905	−	6.05669i	−0.675752	−	0.390146i	0.122501	−	0.992468i	$-0.460909\pi$
$241$	−10.4905	−	6.05669i	−0.675752	−	0.390146i	−0.798253	+	0.602323i	$0.794242\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.10779	−	1.21694i	−0.134662	−	0.0777472i
$246$		0			0
$247$	−5.87594	−	2.73319i	−0.373877	−	0.173909i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.2460			−1.34103			−0.670517	−	0.741894i	$-0.733928\pi$
$251$	−21.2460			−1.34103			−0.670517	+	0.741894i	$0.733928\pi$
$252$		0			0
$253$		−	6.80569i		−	0.427870i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.10826	−	1.91957i	−0.0691315	−	0.119739i	0.829388	−	0.558673i	$-0.188689\pi$
$257$	−1.10826	−	1.91957i	−0.0691315	−	0.119739i	−0.898519	+	0.438934i	$0.855356\pi$
$258$		0			0
$259$	−15.9443	+	27.6163i	−0.990730	+	1.71599i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.6562	+	18.4570i	−0.657088	+	1.13811i	0.324278	+	0.945962i	$0.394879\pi$
$263$	−10.6562	+	18.4570i	−0.657088	+	1.13811i	−0.981366	+	0.192147i	$0.938455\pi$
$264$		0			0
$265$	−9.30615	+	5.37291i	−0.571672	+	0.330055i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.33035			0.507911			0.253955	−	0.967216i	$-0.418268\pi$
$269$	8.33035			0.507911			0.253955	+	0.967216i	$0.418268\pi$
$270$		0			0
$271$		−	8.21173i		−	0.498827i	−0.968397	−	0.249414i	$-0.919762\pi$
$271$		−	8.21173i		−	0.498827i	0.968397	−	0.249414i	$-0.0802379\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.32023	−	2.49429i	0.260520	−	0.150411i
$276$		0			0
$277$	2.78690	−	4.82705i	0.167449	−	0.290030i	−0.770073	−	0.637955i	$-0.779780\pi$
$277$	2.78690	−	4.82705i	0.167449	−	0.290030i	0.937522	+	0.347926i	$0.113114\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.159791	+	0.0922553i	0.00953232	+	0.00550349i	0.504759	−	0.863261i	$-0.331582\pi$
$281$	0.159791	+	0.0922553i	0.00953232	+	0.00550349i	−0.495226	+	0.868764i	$0.664915\pi$
$282$		0			0
$283$	2.75369	+	4.76953i	0.163690	+	0.283519i	0.936189	−	0.351496i	$-0.114327\pi$
$283$	2.75369	+	4.76953i	0.163690	+	0.283519i	−0.772499	+	0.635015i	$0.780994\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.4519			0.616954
$288$		0			0
$289$	−16.7008			−0.982403
$290$		0			0
$291$		0			0
$292$		0			0
$293$	15.6474	−	9.03401i	0.914129	−	0.527773i	0.0323715	−	0.999476i	$-0.489694\pi$
$293$	15.6474	−	9.03401i	0.914129	−	0.527773i	0.881757	+	0.471703i	$0.156361\pi$
$294$		0			0
$295$	−0.163895	+	0.283874i	−0.00954231	+	0.0165278i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	19.9451	−	1.76412i	1.15346	−	0.102021i
$300$		0			0
$301$	24.6954	−	14.2579i	1.42342	−	0.821810i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	13.9814i		−	0.800575i
$306$		0			0
$307$			24.5251i			1.39972i	0.714280	+	0.699860i	$0.246754\pi$
$307$			24.5251i			1.39972i	−0.714280	+	0.699860i	$0.753246\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7.44695	−	12.8985i	−0.422278	−	0.731406i	0.573884	−	0.818936i	$-0.305436\pi$
$311$	−7.44695	−	12.8985i	−0.422278	−	0.731406i	−0.996162	+	0.0875301i	$0.972103\pi$
$312$		0			0
$313$	−2.63501	+	4.56397i	−0.148940	+	0.257971i	−0.930836	−	0.365438i	$-0.880919\pi$
$313$	−2.63501	+	4.56397i	−0.148940	+	0.257971i	0.781896	+	0.623409i	$0.214253\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	21.3942	+	12.3520i	1.20162	+	0.693755i	0.960915	−	0.276844i	$-0.0892885\pi$
$317$	21.3942	+	12.3520i	1.20162	+	0.693755i	0.240704	+	0.970599i	$0.422622\pi$
$318$		0			0
$319$	−4.72851	+	2.73001i	−0.264746	+	0.152851i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	0.983068i		−	0.0546994i
$324$		0			0
$325$	8.42974	+	12.0145i	0.467598	+	0.666447i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.42913	−	9.40352i	−0.299317	−	0.518433i
$330$		0			0
$331$	−2.18811	−	1.26330i	−0.120269	−	0.0694375i	0.438659	−	0.898654i	$-0.355454\pi$
$331$	−2.18811	−	1.26330i	−0.120269	−	0.0694375i	−0.558928	+	0.829216i	$0.688787\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.88096	−	4.98996i	0.157403	−	0.272631i
$336$		0			0
$337$	−9.55590	−	16.5513i	−0.520543	−	0.901607i	−0.999715	−	0.0238855i	$-0.992396\pi$
$337$	−9.55590	−	16.5513i	−0.520543	−	0.901607i	0.479172	−	0.877721i	$-0.340937\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	7.96090			0.431107
$342$		0			0
$343$			13.8118i			0.745769i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.68927	+	8.12206i	0.251733	+	0.436015i	0.964003	−	0.265891i	$-0.0856661\pi$
$347$	4.68927	+	8.12206i	0.251733	+	0.436015i	−0.712270	+	0.701906i	$0.752333\pi$
$348$		0			0
$349$	−20.5021	−	11.8369i	−1.09745	−	0.633615i	−0.161902	−	0.986807i	$-0.551763\pi$
$349$	−20.5021	−	11.8369i	−1.09745	−	0.633615i	−0.935551	+	0.353192i	$0.885096\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	31.4939	+	18.1830i	1.67625	+	0.967783i	0.964018	+	0.265837i	$0.0856483\pi$
$353$	31.4939	+	18.1830i	1.67625	+	0.967783i	0.712231	+	0.701945i	$0.247685\pi$
$354$		0			0
$355$	−7.32966	−	12.6953i	−0.389018	−	0.673799i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			36.6326i			1.93340i	0.255917	+	0.966699i	$0.417622\pi$
$359$			36.6326i			1.93340i	−0.255917	+	0.966699i	$0.582378\pi$
$360$		0			0
$361$	15.7695			0.829972
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.284938	+	0.493527i	0.0149143	+	0.0258324i
$366$		0			0
$367$	11.1385	−	19.2925i	0.581426	−	1.00706i	−0.413885	−	0.910329i	$-0.635828\pi$
$367$	11.1385	−	19.2925i	0.581426	−	1.00706i	0.995311	−	0.0967299i	$-0.0308383\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−29.7919	−	17.2004i	−1.54672	−	0.892999i
$372$		0			0
$373$	−5.79906	−	10.0443i	−0.300264	−	0.520072i	0.675932	−	0.736964i	$-0.263741\pi$
$373$	−5.79906	−	10.0443i	−0.300264	−	0.520072i	−0.976196	+	0.216892i	$0.930408\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9.22639	−	13.1500i	−0.475183	−	0.677258i
$378$		0			0
$379$			9.90166i			0.508614i	0.967124	+	0.254307i	$0.0818474\pi$
$379$			9.90166i			0.508614i	−0.967124	+	0.254307i	$0.918153\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.823581	−	0.475495i	0.0420830	−	0.0242967i	−0.478811	−	0.877918i	$-0.658932\pi$
$383$	0.823581	−	0.475495i	0.0420830	−	0.0242967i	0.520894	+	0.853621i	$0.325599\pi$
$384$		0			0
$385$	−3.15767	−	1.82308i	−0.160930	−	0.0929128i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.41896	+	16.3141i	−0.477560	+	0.827159i	−0.999669	−	0.0257200i	$-0.991812\pi$
$389$	−9.41896	+	16.3141i	−0.477560	+	0.827159i	0.522109	+	0.852879i	$0.325146\pi$
$390$		0			0
$391$	1.51870	+	2.63047i	0.0768041	+	0.133029i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	3.00234i		−	0.151064i
$396$		0			0
$397$		−	12.5330i		−	0.629015i	−0.949255	−	0.314507i	$-0.898161\pi$
$397$		−	12.5330i		−	0.629015i	0.949255	−	0.314507i	$-0.101839\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−21.6897	+	12.5225i	−1.08313	+	0.625345i	−0.931739	−	0.363129i	$-0.881708\pi$
$401$	−21.6897	+	12.5225i	−1.08313	+	0.625345i	−0.151391	+	0.988474i	$0.548375\pi$
$402$		0			0
$403$	2.06356	+	23.3306i	0.102793	+	1.16218i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.33134	+	10.9662i	−0.313833	+	0.543575i
$408$		0			0
$409$	−3.94504	+	2.27767i	−0.195070	+	0.112624i	−0.594354	−	0.804204i	$-0.702592\pi$
$409$	−3.94504	+	2.27767i	−0.195070	+	0.112624i	0.399284	+	0.916827i	$0.369259\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.04936			−0.0516355
$414$		0			0
$415$	4.47527			0.219682
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.42799	−	12.8657i	−0.362881	−	0.628528i	0.625553	−	0.780182i	$-0.284874\pi$
$419$	−7.42799	−	12.8657i	−0.362881	−	0.628528i	−0.988434	+	0.151654i	$0.951540\pi$
$420$		0			0
$421$	8.64149	+	4.98917i	0.421160	+	0.243157i	0.695574	−	0.718455i	$-0.255150\pi$
$421$	8.64149	+	4.98917i	0.421160	+	0.243157i	−0.274413	+	0.961612i	$0.588484\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.11321	+	1.92813i	−0.0539986	+	0.0935283i
$426$		0			0
$427$	38.7624	−	22.3795i	1.87585	−	1.08302i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			9.39892i			0.452730i	0.974043	+	0.226365i	$0.0726842\pi$
$431$			9.39892i			0.452730i	−0.974043	+	0.226365i	$0.927316\pi$
$432$		0			0
$433$	−12.6379			−0.607338			−0.303669	−	0.952778i	$-0.598212\pi$
$433$	−12.6379			−0.607338			−0.303669	+	0.952778i	$0.598212\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.64421	−	4.99074i	0.413509	−	0.238739i
$438$		0			0
$439$	−11.5221	+	19.9568i	−0.549918	+	0.952485i	0.448362	+	0.893852i	$0.352008\pi$
$439$	−11.5221	+	19.9568i	−0.549918	+	0.952485i	−0.998280	+	0.0586333i	$0.981326\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.9553	−	29.3675i	0.805572	−	1.39529i	−0.110332	−	0.993895i	$-0.535192\pi$
$443$	16.9553	−	29.3675i	0.805572	−	1.39529i	0.915904	−	0.401397i	$-0.131475\pi$
$444$		0			0
$445$	−3.83635	−	6.64475i	−0.181860	−	0.314991i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	39.2398i		−	1.85184i	−0.377718	−	0.925921i	$-0.623291\pi$
$449$		−	39.2398i		−	1.85184i	0.377718	−	0.925921i	$-0.376709\pi$
$450$		0			0
$451$	4.15035			0.195432
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.52431	−	9.72659i	0.212103	−	0.455990i
$456$		0			0
$457$	13.9019	+	8.02629i	0.650305	+	0.375454i	0.788573	−	0.614941i	$-0.210820\pi$
$457$	13.9019	+	8.02629i	0.650305	+	0.375454i	−0.138268	+	0.990395i	$0.544153\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	23.1433	+	13.3618i	1.07789	+	0.622322i	0.930327	−	0.366730i	$-0.119523\pi$
$461$	23.1433	+	13.3618i	1.07789	+	0.622322i	0.147566	+	0.989052i	$0.452856\pi$
$462$		0			0
$463$	6.70568	−	3.87153i	0.311639	−	0.179925i	−0.336020	−	0.941855i	$-0.609081\pi$
$463$	6.70568	−	3.87153i	0.311639	−	0.179925i	0.647660	+	0.761930i	$0.275748\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.2237			0.935840			0.467920	−	0.883771i	$-0.345004\pi$
$467$	20.2237			0.935840			0.467920	+	0.883771i	$0.345004\pi$
$468$		0			0
$469$	18.4457			0.851743
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.80632	−	5.66168i	0.450895	−	0.260324i
$474$		0			0
$475$	6.33620	+	3.65821i	0.290725	+	0.167850i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.76117	−	1.59416i	−0.126161	−	0.0728391i	0.435591	−	0.900145i	$-0.356539\pi$
$479$	−2.76117	−	1.59416i	−0.126161	−	0.0728391i	−0.561752	+	0.827305i	$0.689873\pi$
$480$		0			0
$481$	−33.7793	−	15.7124i	−1.54020	−	0.716423i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.7388			0.487624
$486$		0			0
$487$			10.3517i			0.469080i	0.972106	+	0.234540i	$0.0753584\pi$
$487$			10.3517i			0.469080i	−0.972106	+	0.234540i	$0.924642\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.1824	−	19.3684i	−0.504654	−	0.874086i	−0.999986	−	0.00538181i	$-0.998287\pi$
$491$	−11.1824	−	19.3684i	−0.504654	−	0.874086i	0.495332	−	0.868704i	$-0.335046\pi$
$492$		0			0
$493$	1.21841	−	2.11035i	0.0548746	−	0.0950455i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	23.4646	−	40.6418i	1.05253	−	1.82303i
$498$		0			0
$499$	−17.4174	+	10.0559i	−0.779710	+	0.450166i	−0.836327	−	0.548230i	$-0.815302\pi$
$499$	−17.4174	+	10.0559i	−0.779710	+	0.450166i	0.0566178	+	0.998396i	$0.481968\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.8457			−1.28617			−0.643084	−	0.765796i	$-0.722345\pi$
$503$	−28.8457			−1.28617			−0.643084	+	0.765796i	$0.722345\pi$
$504$		0			0
$505$		−	10.5667i		−	0.470212i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.80754	−	3.35298i	0.257415	−	0.148618i	−0.365740	−	0.930717i	$-0.619184\pi$
$509$	5.80754	−	3.35298i	0.257415	−	0.148618i	0.623155	+	0.782099i	$0.285851\pi$
$510$		0			0
$511$	−0.912176	+	1.57994i	−0.0403523	+	0.0698923i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.54822	+	3.20327i	0.244484	+	0.141153i
$516$		0			0
$517$	−2.15586	−	3.73406i	−0.0948147	−	0.164224i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.19549			−0.402862			−0.201431	−	0.979503i	$-0.564559\pi$
$521$	−9.19549			−0.402862			−0.201431	+	0.979503i	$0.564559\pi$
$522$		0			0
$523$	8.89040			0.388750			0.194375	−	0.980927i	$-0.437732\pi$
$523$	8.89040			0.388750			0.194375	+	0.980927i	$0.437732\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.07697	+	1.77649i	−0.134035	+	0.0773851i
$528$		0			0
$529$	−3.92000	+	6.78963i	−0.170435	+	0.295201i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1.07582	+	12.1632i	0.0465989	+	0.526848i
$534$		0			0
$535$	5.23635	−	3.02321i	0.226387	−	0.130705i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	3.09398i		−	0.133267i
$540$		0			0
$541$		−	41.7785i		−	1.79620i	−0.439792	−	0.898099i	$-0.644948\pi$
$541$		−	41.7785i		−	1.79620i	0.439792	−	0.898099i	$-0.355052\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.68379	−	4.64847i	−0.114961	−	0.199119i
$546$		0			0
$547$	8.21994	−	14.2374i	0.351459	−	0.608745i	−0.635046	−	0.772474i	$-0.719019\pi$
$547$	8.21994	−	14.2374i	0.351459	−	0.608745i	0.986505	+	0.163729i	$0.0523523\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.93501	−	4.00393i	−0.295441	−	0.170573i
$552$		0			0
$553$	8.32376	−	4.80573i	0.353962	−	0.204360i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			19.2254i			0.814608i	0.913293	+	0.407304i	$0.133531\pi$
$557$			19.2254i			0.814608i	−0.913293	+	0.407304i	$0.866469\pi$
$558$		0			0
$559$	19.1343	+	27.2713i	0.809296	+	1.15345i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.35474	+	14.4708i	0.352110	+	0.609873i	0.986619	−	0.163043i	$-0.0521309\pi$
$563$	8.35474	+	14.4708i	0.352110	+	0.609873i	−0.634509	+	0.772916i	$0.718798\pi$
$564$		0			0
$565$	−5.84381	−	3.37393i	−0.245851	−	0.141942i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.02623	−	8.70569i	0.210711	−	0.364961i	−0.741227	−	0.671255i	$-0.765756\pi$
$569$	5.02623	−	8.70569i	0.210711	−	0.364961i	0.951937	+	0.306294i	$0.0990889\pi$
$570$		0			0
$571$	−13.7325	−	23.7854i	−0.574687	−	0.995388i	−0.996076	−	0.0885074i	$-0.971790\pi$
$571$	−13.7325	−	23.7854i	−0.574687	−	0.995388i	0.421388	−	0.906880i	$-0.361543\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22.6057			−0.942723
$576$		0			0
$577$			16.9542i			0.705813i	0.935659	+	0.352907i	$0.114807\pi$
$577$			16.9542i			0.705813i	−0.935659	+	0.352907i	$0.885193\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.16338	+	12.4073i	0.297187	+	0.514743i
$582$		0			0
$583$	−11.8301	−	6.83013i	−0.489954	−	0.282875i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.99432	−	3.46082i	−0.247412	−	0.142843i	0.371167	−	0.928566i	$-0.378958\pi$
$587$	−5.99432	−	3.46082i	−0.247412	−	0.142843i	−0.618579	+	0.785723i	$0.712291\pi$
$588$		0			0
$589$	5.83787	+	10.1115i	0.240545	+	0.416637i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	2.91351i		−	0.119643i	−0.998209	−	0.0598217i	$-0.980947\pi$
$593$		−	2.91351i		−	0.119643i	0.998209	−	0.0598217i	$-0.0190532\pi$
$594$		0			0
$595$	1.62730			0.0667126
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.7811	−	34.2619i	−0.808234	−	1.39990i	−0.914086	−	0.405521i	$-0.867090\pi$
$599$	−19.7811	−	34.2619i	−0.808234	−	1.39990i	0.105851	−	0.994382i	$-0.466243\pi$
$600$		0			0
$601$	8.81624	−	15.2702i	0.359622	−	0.622883i	−0.628276	−	0.777991i	$-0.716239\pi$
$601$	8.81624	−	15.2702i	0.359622	−	0.622883i	0.987898	+	0.155108i	$0.0495724\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.92985	+	4.57830i	0.322394	+	0.186134i
$606$		0			0
$607$	−11.2097	−	19.4158i	−0.454989	−	0.788064i	0.543699	−	0.839280i	$-0.317023\pi$
$607$	−11.2097	−	19.4158i	−0.454989	−	0.788064i	−0.998688	+	0.0512168i	$0.983690\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	10.3844	−	7.28599i	0.420108	−	0.294760i
$612$		0			0
$613$			38.2173i			1.54358i	0.635877	+	0.771791i	$0.280639\pi$
$613$			38.2173i			1.54358i	−0.635877	+	0.771791i	$0.719361\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−26.6211	+	15.3697i	−1.07172	+	0.618761i	−0.928652	−	0.370952i	$-0.879032\pi$
$617$	−26.6211	+	15.3697i	−1.07172	+	0.618761i	−0.143073	+	0.989712i	$0.545698\pi$
$618$		0			0
$619$	31.3023	+	18.0724i	1.25815	+	0.726390i	0.972714	−	0.232009i	$-0.0745298\pi$
$619$	31.3023	+	18.0724i	1.25815	+	0.726390i	0.285431	+	0.958399i	$0.407863\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.2814	−	21.2719i	0.492042	−	0.852243i
$624$		0			0
$625$	−5.96154	−	10.3257i	−0.238462	−	0.413028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	5.65140i		−	0.225336i
$630$		0			0
$631$		−	48.0242i		−	1.91181i	−0.293673	−	0.955906i	$-0.594878\pi$
$631$		−	48.0242i		−	1.91181i	0.293673	−	0.955906i	$-0.405122\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.19521	−	1.84476i	0.126798	−	0.0732069i
$636$		0			0
$637$	9.06737	−	0.801995i	0.359262	−	0.0317762i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.4738	−	18.1412i	0.413691	−	0.716533i	−0.581599	−	0.813475i	$-0.697573\pi$
$641$	10.4738	−	18.1412i	0.413691	−	0.716533i	0.995290	+	0.0969423i	$0.0309062\pi$
$642$		0			0
$643$	11.6982	−	6.75399i	0.461334	−	0.266351i	−0.251271	−	0.967917i	$-0.580849\pi$
$643$	11.6982	−	6.75399i	0.461334	−	0.266351i	0.712605	+	0.701566i	$0.247515\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	15.1084			0.593972			0.296986	−	0.954882i	$-0.404019\pi$
$647$	15.1084			0.593972			0.296986	+	0.954882i	$0.404019\pi$
$648$		0			0
$649$	−0.416691			−0.0163565
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.7246	−	34.1640i	−0.771883	−	1.33694i	−0.936530	−	0.350588i	$-0.885982\pi$
$653$	−19.7246	−	34.1640i	−0.771883	−	1.33694i	0.164647	−	0.986353i	$-0.447351\pi$
$654$		0			0
$655$	−7.95461	−	4.59259i	−0.310812	−	0.179447i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.60018	−	16.6280i	0.373970	−	0.647735i	−0.616202	−	0.787588i	$-0.711330\pi$
$659$	9.60018	−	16.6280i	0.373970	−	0.647735i	0.990172	+	0.139853i	$0.0446630\pi$
$660$		0			0
$661$	25.4024	−	14.6661i	0.988039	−	0.570445i	0.0833514	−	0.996520i	$-0.473438\pi$
$661$	25.4024	−	14.6661i	0.988039	−	0.570445i	0.904688	+	0.426076i	$0.140104\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	5.34759i		−	0.207371i
$666$		0			0
$667$	24.7421			0.958016
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.3922	−	8.88671i	0.594211	−	0.343068i
$672$		0			0
$673$	−1.04790	+	1.81501i	−0.0403934	+	0.0699635i	−0.885515	−	0.464610i	$-0.846194\pi$
$673$	−1.04790	+	1.81501i	−0.0403934	+	0.0699635i	0.845122	+	0.534574i	$0.179528\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.3251	+	38.6683i	−0.858025	+	1.48614i	0.0157850	+	0.999875i	$0.494975\pi$
$677$	−22.3251	+	38.6683i	−0.858025	+	1.48614i	−0.873810	+	0.486268i	$0.838358\pi$
$678$		0			0
$679$	17.1891	+	29.7725i	0.659659	+	1.14256i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			27.7927i			1.06346i	0.846914	+	0.531729i	$0.178458\pi$
$683$			27.7927i			1.06346i	−0.846914	+	0.531729i	$0.821542\pi$
$684$		0			0
$685$	11.1524			0.426110
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.9502	−	36.4404i	0.645752	−	1.38827i
$690$		0			0
$691$	−3.11860	−	1.80053i	−0.118637	−	0.0684952i	0.439507	−	0.898239i	$-0.355153\pi$
$691$	−3.11860	−	1.80053i	−0.118637	−	0.0684952i	−0.558144	+	0.829744i	$0.688486\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.04409	−	3.48955i	−0.229265	−	0.132366i
$696$		0			0
$697$	−1.60415	+	0.926158i	−0.0607616	+	0.0350807i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−39.7625			−1.50181			−0.750904	−	0.660411i	$-0.770382\pi$
$701$	−39.7625			−1.50181			−0.750904	+	0.660411i	$0.770382\pi$
$702$		0			0
$703$	−18.5715			−0.700439
$704$		0			0
$705$		0			0
$706$		0			0
$707$	29.2954	−	16.9137i	1.10177	−	0.636104i
$708$		0			0
$709$	−9.34209	−	5.39366i	−0.350849	−	0.202563i	0.314210	−	0.949354i	$-0.398260\pi$
$709$	−9.34209	−	5.39366i	−0.350849	−	0.202563i	−0.665059	+	0.746791i	$0.731594\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−31.2417	−	18.0374i	−1.17001	−	0.675506i
$714$		0			0
$715$	1.79657	−	3.86235i	0.0671877	−	0.144444i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12.8343			−0.478640			−0.239320	−	0.970941i	$-0.576924\pi$
$719$	−12.8343			−0.478640			−0.239320	+	0.970941i	$0.576924\pi$
$720$		0			0
$721$			20.5093i			0.763808i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9.06796	+	15.7062i	0.336776	+	0.583312i
$726$		0			0
$727$	8.18448	−	14.1759i	0.303545	−	0.525756i	−0.673391	−	0.739287i	$-0.735163\pi$
$727$	8.18448	−	14.1759i	0.303545	−	0.525756i	0.976936	+	0.213530i	$0.0684962\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.52683	+	4.37660i	−0.0934581	+	0.161874i
$732$		0			0
$733$	19.7182	−	11.3843i	0.728310	−	0.420490i	−0.0894936	−	0.995987i	$-0.528525\pi$
$733$	19.7182	−	11.3843i	0.728310	−	0.420490i	0.817804	+	0.575497i	$0.195192\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.32463			0.269806
$738$		0			0
$739$		−	40.5277i		−	1.49084i	−0.666597	−	0.745418i	$-0.732250\pi$
$739$		−	40.5277i		−	1.49084i	0.666597	−	0.745418i	$-0.267750\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	33.4496	−	19.3122i	1.22715	−	0.708494i	0.260715	−	0.965416i	$-0.416042\pi$
$743$	33.4496	−	19.3122i	1.22715	−	0.708494i	0.966432	+	0.256922i	$0.0827082\pi$
$744$		0			0
$745$	−1.09216	+	1.89168i	−0.0400138	+	0.0693059i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	16.7632	+	9.67824i	0.612514	+	0.353635i
$750$		0			0
$751$	−15.8209	−	27.4027i	−0.577314	−	0.999938i	−0.995786	−	0.0917079i	$-0.970767\pi$
$751$	−15.8209	−	27.4027i	−0.577314	−	0.999938i	0.418472	−	0.908230i	$-0.362566\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.37157			−0.122704
$756$		0			0
$757$	−36.1531			−1.31401			−0.657003	−	0.753888i	$-0.728176\pi$
$757$	−36.1531			−1.31401			−0.657003	+	0.753888i	$0.728176\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−41.8216	+	24.1457i	−1.51603	+	0.875281i	−0.516209	+	0.856463i	$0.672657\pi$
$761$	−41.8216	+	24.1457i	−1.51603	+	0.875281i	−0.999823	+	0.0188184i	$0.994010\pi$
$762$		0			0
$763$	8.59168	−	14.8812i	0.311040	−	0.538736i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.108011	−	1.22118i	−0.00390006	−	0.0440941i
$768$		0			0
$769$	−44.0990	+	25.4606i	−1.59025	+	0.918131i	−0.596987	+	0.802251i	$0.703635\pi$
$769$	−44.0990	+	25.4606i	−1.59025	+	0.918131i	−0.993263	+	0.115880i	$0.963031\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	32.8060i		−	1.17995i	−0.807421	−	0.589976i	$-0.799137\pi$
$773$		−	32.8060i		−	1.17995i	0.807421	−	0.589976i	$-0.200863\pi$
$774$		0			0
$775$		−	26.4428i		−	0.949854i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.04352	+	5.27154i	0.109046	+	0.188872i
$780$		0			0
$781$	9.31758	−	16.1385i	0.333409	−	0.577482i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.1101	−	9.87854i	−0.610687	−	0.352580i
$786$		0			0
$787$	−18.0928	+	10.4459i	−0.644939	+	0.372356i	−0.786515	−	0.617572i	$-0.788117\pi$
$787$	−18.0928	+	10.4459i	−0.644939	+	0.372356i	0.141575	+	0.989927i	$0.454783\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	21.6020i		−	0.768079i
$792$		0			0
$793$	30.0337	+	42.8057i	1.06653	+	1.52008i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−11.9825	−	20.7543i	−0.424441	−	0.735154i	0.571927	−	0.820305i	$-0.306196\pi$
$797$	−11.9825	−	20.7543i	−0.424441	−	0.735154i	−0.996368	+	0.0851508i	$0.972863\pi$
$798$		0			0
$799$	1.66653	+	0.962169i	0.0589574	+	0.0340391i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.362218	+	0.627379i	−0.0127824	+	0.0221397i
$804$		0			0
$805$	8.26129	+	14.3090i	0.291172	+	0.504325i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.0779			0.600428			0.300214	−	0.953872i	$-0.402942\pi$
$809$	17.0779			0.600428			0.300214	+	0.953872i	$0.402942\pi$
$810$		0			0
$811$			40.7960i			1.43254i	0.697823	+	0.716270i	$0.254152\pi$
$811$			40.7960i			1.43254i	−0.697823	+	0.716270i	$0.745848\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.37442	−	11.0408i	−0.223286	−	0.386743i
$816$		0			0
$817$	14.3823	+	8.30362i	0.503173	+	0.290507i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	28.0370	+	16.1872i	0.978498	+	0.564936i	0.901816	−	0.432120i	$-0.142234\pi$
$821$	28.0370	+	16.1872i	0.978498	+	0.564936i	0.0766814	+	0.997056i	$0.475568\pi$
$822$		0			0
$823$	12.0337	+	20.8430i	0.419469	+	0.726542i	0.995886	−	0.0906136i	$-0.0288828\pi$
$823$	12.0337	+	20.8430i	0.419469	+	0.726542i	−0.576417	+	0.817156i	$0.695549\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			47.6352i			1.65644i	0.560404	+	0.828219i	$0.310646\pi$
$827$			47.6352i			1.65644i	−0.560404	+	0.828219i	$0.689354\pi$
$828$		0			0
$829$	17.2774			0.600069			0.300034	−	0.953928i	$-0.403002\pi$
$829$	17.2774			0.600069			0.300034	+	0.953928i	$0.403002\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.690427	+	1.19585i	0.0239219	+	0.0414339i
$834$		0			0
$835$	12.2768	−	21.2640i	0.424856	−	0.735871i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.80893	+	3.93114i	0.235070	+	0.135718i	0.612909	−	0.790154i	$-0.289999\pi$
$839$	6.80893	+	3.93114i	0.235070	+	0.135718i	−0.377839	+	0.925871i	$0.623333\pi$
$840$		0			0
$841$	4.57507	+	7.92425i	0.157761	+	0.273250i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.7849	+	4.26394i	0.405412	+	0.146684i
$846$		0			0
$847$			29.3132i			1.00721i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	49.6933	−	28.6905i	1.70347	−	0.983496i
$852$		0			0
$853$	37.8803	+	21.8702i	1.29700	+	0.748822i	0.979884	−	0.199566i	$-0.0639533\pi$
$853$	37.8803	+	21.8702i	1.29700	+	0.748822i	0.317113	+	0.948388i	$0.397287\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.16998	+	15.8829i	−0.313240	+	0.542548i	−0.979062	−	0.203563i	$-0.934748\pi$
$857$	−9.16998	+	15.8829i	−0.313240	+	0.542548i	0.665822	+	0.746111i	$0.268081\pi$
$858$		0			0
$859$	−21.6708	−	37.5349i	−0.739397	−	1.28067i	−0.952767	−	0.303702i	$-0.901777\pi$
$859$	−21.6708	−	37.5349i	−0.739397	−	1.28067i	0.213370	−	0.976971i	$-0.431556\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	44.7839i		−	1.52446i	−0.647305	−	0.762231i	$-0.724104\pi$
$863$		−	44.7839i		−	1.52446i	0.647305	−	0.762231i	$-0.275896\pi$
$864$		0			0
$865$		−	14.1988i		−	0.482773i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.30530	−	1.90831i	0.112125	−	0.0647351i
$870$		0			0
$871$	1.89863	+	21.4659i	0.0643327	+	0.727346i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−13.4936	+	23.3716i	−0.456167	+	0.790104i
$876$		0			0
$877$	20.0726	−	11.5889i	0.677802	−	0.391329i	−0.121224	−	0.992625i	$-0.538682\pi$
$877$	20.0726	−	11.5889i	0.677802	−	0.391329i	0.799026	+	0.601296i	$0.205349\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−50.2918			−1.69438			−0.847188	−	0.531294i	$-0.821706\pi$
$881$	−50.2918			−1.69438			−0.847188	+	0.531294i	$0.821706\pi$
$882$		0			0
$883$	39.1343			1.31697			0.658487	−	0.752592i	$-0.271197\pi$
$883$	39.1343			1.31697			0.658487	+	0.752592i	$0.271197\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	26.8545	+	46.5133i	0.901685	+	1.56176i	0.825307	+	0.564684i	$0.191002\pi$
$887$	26.8545	+	46.5133i	0.901685	+	1.56176i	0.0763777	+	0.997079i	$0.475665\pi$
$888$		0			0
$889$	10.2289	+	5.90565i	0.343066	+	0.198069i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.16186	−	5.47651i	0.105808	−	0.183264i
$894$		0			0
$895$	1.40896	−	0.813466i	0.0470965	−	0.0271912i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			28.9418i			0.965263i
$900$		0			0
$901$	6.09662			0.203108
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17.0296	+	9.83205i	−0.566084	+	0.326829i
$906$		0			0
$907$	7.93300	−	13.7404i	0.263411	−	0.456241i	−0.703735	−	0.710463i	$-0.748486\pi$
$907$	7.93300	−	13.7404i	0.263411	−	0.456241i	0.967146	+	0.254221i	$0.0818191\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11.9973	+	20.7799i	−0.397487	+	0.688467i	−0.993415	−	0.114570i	$-0.963451\pi$
$911$	−11.9973	+	20.7799i	−0.397487	+	0.688467i	0.595928	+	0.803038i	$0.296784\pi$
$912$		0			0
$913$	2.84452	+	4.92685i	0.0941398	+	0.163055i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	29.4047i		−	0.971028i
$918$		0			0
$919$	28.3186			0.934144			0.467072	−	0.884219i	$-0.345309\pi$
$919$	28.3186			0.934144			0.467072	+	0.884219i	$0.345309\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	49.7116	+	23.1233i	1.63628	+	0.761112i
$924$		0			0
$925$	36.4252	+	21.0301i	1.19765	+	0.691465i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.3419	−	17.5179i	−0.995485	−	0.574743i	−0.0885755	−	0.996069i	$-0.528231\pi$
$929$	−30.3419	−	17.5179i	−0.995485	−	0.574743i	−0.906909	+	0.421326i	$0.861565\pi$
$930$		0			0
$931$	3.92980	−	2.26887i	0.128794	−	0.0743592i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	0.646185			0.0211325
$936$		0			0
$937$	−12.2708			−0.400870			−0.200435	−	0.979707i	$-0.564235\pi$
$937$	−12.2708			−0.400870			−0.200435	+	0.979707i	$0.564235\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8.71346	+	5.03072i	−0.284051	+	0.163997i	−0.635256	−	0.772302i	$-0.719105\pi$
$941$	−8.71346	+	5.03072i	−0.284051	+	0.163997i	0.351205	+	0.936299i	$0.385772\pi$
$942$		0			0
$943$	−16.2876	−	9.40365i	−0.530397	−	0.306225i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−37.1040	−	21.4220i	−1.20572	−	0.696122i	−0.243898	−	0.969801i	$-0.578426\pi$
$947$	−37.1040	−	21.4220i	−1.20572	−	0.696122i	−0.961821	+	0.273679i	$0.911760\pi$
$948$		0			0
$949$	−1.93252	−	0.898910i	−0.0627323	−	0.0291798i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.5949			1.47696			0.738482	−	0.674273i	$-0.235543\pi$
$953$	45.5949			1.47696			0.738482	+	0.674273i	$0.235543\pi$
$954$		0			0
$955$			8.96057i			0.289957i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.8511	+	30.9191i	0.576443	+	0.998429i
$960$		0			0
$961$	5.59909	−	9.69791i	0.180616	−	0.312836i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.38679	+	14.5264i	−0.269980	+	0.467620i
$966$		0			0
$967$	−8.87129	+	5.12184i	−0.285282	+	0.164707i	−0.635812	−	0.771844i	$-0.719335\pi$
$967$	−8.87129	+	5.12184i	−0.285282	+	0.164707i	0.350530	+	0.936551i	$0.386001\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−53.0601			−1.70278			−0.851389	−	0.524534i	$-0.824239\pi$
$971$	−53.0601			−1.70278			−0.851389	+	0.524534i	$0.824239\pi$
$972$		0			0
$973$		−	22.3423i		−	0.716262i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−37.8301	+	21.8412i	−1.21029	+	0.698762i	−0.962823	−	0.270134i	$-0.912932\pi$
$977$	−37.8301	+	21.8412i	−1.21029	+	0.698762i	−0.247469	+	0.968896i	$0.579599\pi$
$978$		0			0
$979$	4.87683	−	8.44691i	0.155864	−	0.269964i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	39.1820	+	22.6217i	1.24971	+	0.721521i	0.971052	−	0.238868i	$-0.0767765\pi$
$983$	39.1820	+	22.6217i	1.24971	+	0.721521i	0.278660	+	0.960390i	$0.410110\pi$
$984$		0			0
$985$	−7.00648	−	12.1356i	−0.223245	−	0.386672i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−51.3118			−1.63162
$990$		0			0
$991$	−4.23866			−0.134645			−0.0673227	−	0.997731i	$-0.521446\pi$
$991$	−4.23866			−0.134645			−0.0673227	+	0.997731i	$0.521446\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.30824	+	4.21942i	−0.231687	+	0.133764i
$996$		0			0
$997$	−21.1207	+	36.5821i	−0.668898	+	1.15857i	0.309314	+	0.950960i	$0.399901\pi$
$997$	−21.1207	+	36.5821i	−0.668898	+	1.15857i	−0.978212	+	0.207606i	$0.933433\pi$
$998$		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.1585.17		80
3.2	odd	2		936.2.cw.b.25.6	yes	80
9.4	even	3	inner	2808.2.cw.b.2521.24		80
9.5	odd	6		936.2.cw.b.337.5	yes	80
13.12	even	2	inner	2808.2.cw.b.1585.24		80
39.38	odd	2		936.2.cw.b.25.5	✓	80
117.77	odd	6		936.2.cw.b.337.6	yes	80
117.103	even	6	inner	2808.2.cw.b.2521.17		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.5	✓	80	39.38	odd	2
936.2.cw.b.25.6	yes	80	3.2	odd	2
936.2.cw.b.337.5	yes	80	9.5	odd	6
936.2.cw.b.337.6	yes	80	117.77	odd	6
2808.2.cw.b.1585.17		80	1.1	even	1	trivial
2808.2.cw.b.1585.24		80	13.12	even	2	inner
2808.2.cw.b.2521.17		80	117.103	even	6	inner
2808.2.cw.b.2521.24		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)

\(f(q)\)	\(=\)	\(q+(-0.834885 + 0.482021i) q^{5} +(-2.67273 - 1.54310i) q^{7} +(-1.06132 - 0.612753i) q^{11} +(1.52066 - 3.26919i) q^{13} +0.546948 q^{17} -1.79737i q^{19} +(2.77669 + 4.80937i) q^{23} +(-2.03531 + 3.52526i) q^{25} +(2.22766 - 3.85842i) q^{29} +(-5.62571 + 3.24801i) q^{31} +2.97523 q^{35} -10.3326i q^{37} +(-2.93292 + 1.69332i) q^{41} +(-4.61987 + 8.00186i) q^{43} +(3.04696 + 1.75916i) q^{47} +(1.26233 + 2.18641i) q^{49} +11.1466 q^{53} +1.18144 q^{55} +(0.294462 - 0.170008i) q^{59} +(-7.25146 + 12.5599i) q^{61} +(0.306243 + 3.46239i) q^{65} +(-5.17608 + 2.98841i) q^{67} +15.2061i q^{71} -0.591132i q^{73} +(1.89108 + 3.27545i) q^{77} +(-1.55716 + 2.69709i) q^{79} +(-4.02026 - 2.32110i) q^{83} +(-0.456638 + 0.263640i) q^{85} +7.95888i q^{89} +(-9.10900 + 6.39113i) q^{91} +(0.866370 + 1.50060i) q^{95} +(-9.64696 - 5.56967i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 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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