LMFDB - Embedded newform 2808.2.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(1,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-3,0,5,0,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.4219928876$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.561553 q^{5} +0.438447 q^{7} -5.56155 q^{11} +1.00000 q^{13} -4.00000 q^{17} +5.12311 q^{19} +6.68466 q^{23} -4.68466 q^{25} -2.68466 q^{29} -6.68466 q^{31} +0.246211 q^{35} +6.68466 q^{37} -12.2462 q^{41} -1.87689 q^{43} +3.68466 q^{47} -6.80776 q^{49} +1.56155 q^{53} -3.12311 q^{55} -2.12311 q^{59} -14.5616 q^{61} +0.561553 q^{65} +10.2462 q^{67} -7.68466 q^{71} -12.0000 q^{73} -2.43845 q^{77} +8.00000 q^{79} -17.2462 q^{83} -2.24621 q^{85} -1.00000 q^{89} +0.438447 q^{91} +2.87689 q^{95} +16.4924 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} + 5 q^{7} - 7 q^{11} + 2 q^{13} - 8 q^{17} + 2 q^{19} + q^{23} + 3 q^{25} + 7 q^{29} - q^{31} - 16 q^{35} + q^{37} - 8 q^{41} - 12 q^{43} - 5 q^{47} + 7 q^{49} - q^{53} + 2 q^{55} + 4 q^{59}+ \cdots + 14 q^{95}+O(q^{100}) $ 2 * q - 3 * q^5 + 5 * q^7 - 7 * q^11 + 2 * q^13 - 8 * q^17 + 2 * q^19 + q^23 + 3 * q^25 + 7 * q^29 - q^31 - 16 * q^35 + q^37 - 8 * q^41 - 12 * q^43 - 5 * q^47 + 7 * q^49 - q^53 + 2 * q^55 + 4 * q^59 - 25 * q^61 - 3 * q^65 + 4 * q^67 - 3 * q^71 - 24 * q^73 - 9 * q^77 + 16 * q^79 - 18 * q^83 + 12 * q^85 - 2 * q^89 + 5 * q^91 + 14 * q^95

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
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	0.438447	0.165717	0.0828587	−	0.996561i	$-0.473595\pi$
$7$	0.438447	0.165717	0.0828587	+	0.996561i	$0.473595\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.56155	−1.67687	−0.838436	−	0.545001i	$-0.816529\pi$
$11$	−5.56155	−1.67687	−0.838436	+	0.545001i	$0.816529\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	5.12311	1.17532	0.587661	−	0.809108i	$-0.300049\pi$
$19$	5.12311	1.17532	0.587661	+	0.809108i	$0.300049\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.68466	1.39385	0.696924	−	0.717145i	$-0.254552\pi$
$23$	6.68466	1.39385	0.696924	+	0.717145i	$0.254552\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.68466	−0.498529	−0.249264	−	0.968436i	$-0.580189\pi$
$29$	−2.68466	−0.498529	−0.249264	+	0.968436i	$0.580189\pi$
$30$	0	0
$31$	−6.68466	−1.20060	−0.600300	−	0.799775i	$-0.704952\pi$
$31$	−6.68466	−1.20060	−0.600300	+	0.799775i	$0.704952\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.246211	0.0416173
$36$	0	0
$37$	6.68466	1.09895	0.549476	−	0.835510i	$-0.314828\pi$
$37$	6.68466	1.09895	0.549476	+	0.835510i	$0.314828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.2462	−1.91254	−0.956268	−	0.292490i	$-0.905516\pi$
$41$	−12.2462	−1.91254	−0.956268	+	0.292490i	$0.905516\pi$
$42$	0	0
$43$	−1.87689	−0.286224	−0.143112	−	0.989707i	$-0.545711\pi$
$43$	−1.87689	−0.286224	−0.143112	+	0.989707i	$0.545711\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.68466	0.537463	0.268731	−	0.963215i	$-0.413396\pi$
$47$	3.68466	0.537463	0.268731	+	0.963215i	$0.413396\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.56155	0.214496	0.107248	−	0.994232i	$-0.465796\pi$
$53$	1.56155	0.214496	0.107248	+	0.994232i	$0.465796\pi$
$54$	0	0
$55$	−3.12311	−0.421119
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.12311	−0.276405	−0.138202	−	0.990404i	$-0.544132\pi$
$59$	−2.12311	−0.276405	−0.138202	+	0.990404i	$0.544132\pi$
$60$	0	0
$61$	−14.5616	−1.86442	−0.932208	−	0.361923i	$-0.882120\pi$
$61$	−14.5616	−1.86442	−0.932208	+	0.361923i	$0.882120\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.561553	0.0696521
$66$	0	0
$67$	10.2462	1.25177	0.625887	−	0.779914i	$-0.284737\pi$
$67$	10.2462	1.25177	0.625887	+	0.779914i	$0.284737\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.68466	−0.912001	−0.456001	−	0.889979i	$-0.650719\pi$
$71$	−7.68466	−0.912001	−0.456001	+	0.889979i	$0.650719\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.43845	−0.277887
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17.2462	−1.89302	−0.946509	−	0.322678i	$-0.895417\pi$
$83$	−17.2462	−1.89302	−0.946509	+	0.322678i	$0.895417\pi$
$84$	0	0
$85$	−2.24621	−0.243636
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	0.438447	0.0459618
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.87689	0.295163
$96$	0	0
$97$	16.4924	1.67455	0.837276	−	0.546781i	$-0.184147\pi$
$97$	16.4924	1.67455	0.837276	+	0.546781i	$0.184147\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.56155	−0.951410	−0.475705	−	0.879605i	$-0.657807\pi$
$101$	−9.56155	−0.951410	−0.475705	+	0.879605i	$0.657807\pi$
$102$	0	0
$103$	15.9309	1.56972	0.784858	−	0.619676i	$-0.212736\pi$
$103$	15.9309	1.56972	0.784858	+	0.619676i	$0.212736\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.3693	−1.87250	−0.936251	−	0.351331i	$-0.885729\pi$
$107$	−19.3693	−1.87250	−0.936251	+	0.351331i	$0.885729\pi$
$108$	0	0
$109$	−7.12311	−0.682270	−0.341135	−	0.940014i	$-0.610811\pi$
$109$	−7.12311	−0.682270	−0.341135	+	0.940014i	$0.610811\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.246211	−0.0231616	−0.0115808	−	0.999933i	$-0.503686\pi$
$113$	−0.246211	−0.0231616	−0.0115808	+	0.999933i	$0.503686\pi$
$114$	0	0
$115$	3.75379	0.350043
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.75379	−0.160770
$120$	0	0
$121$	19.9309	1.81190
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.43845	−0.486430
$126$	0	0
$127$	12.8078	1.13651	0.568253	−	0.822854i	$-0.307620\pi$
$127$	12.8078	1.13651	0.568253	+	0.822854i	$0.307620\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.12311	0.272867	0.136434	−	0.990649i	$-0.456436\pi$
$131$	3.12311	0.272867	0.136434	+	0.990649i	$0.456436\pi$
$132$	0	0
$133$	2.24621	0.194771
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.43845	0.379202	0.189601	−	0.981861i	$-0.439281\pi$
$137$	4.43845	0.379202	0.189601	+	0.981861i	$0.439281\pi$
$138$	0	0
$139$	−19.6847	−1.66963	−0.834815	−	0.550530i	$-0.814426\pi$
$139$	−19.6847	−1.66963	−0.834815	+	0.550530i	$0.814426\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.56155	−0.465080
$144$	0	0
$145$	−1.50758	−0.125197
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.1771	1.81682	0.908409	−	0.418083i	$-0.137298\pi$
$149$	22.1771	1.81682	0.908409	+	0.418083i	$0.137298\pi$
$150$	0	0
$151$	−12.4924	−1.01662	−0.508309	−	0.861174i	$-0.669729\pi$
$151$	−12.4924	−1.01662	−0.508309	+	0.861174i	$0.669729\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.75379	−0.301512
$156$	0	0
$157$	−5.43845	−0.434035	−0.217018	−	0.976168i	$-0.569633\pi$
$157$	−5.43845	−0.434035	−0.217018	+	0.976168i	$0.569633\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.93087	0.230985
$162$	0	0
$163$	10.2462	0.802545	0.401273	−	0.915959i	$-0.368568\pi$
$163$	10.2462	0.802545	0.401273	+	0.915959i	$0.368568\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.19224	−0.247023	−0.123511	−	0.992343i	$-0.539416\pi$
$167$	−3.19224	−0.247023	−0.123511	+	0.992343i	$0.539416\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.1231	−1.45390	−0.726951	−	0.686689i	$-0.759063\pi$
$173$	−19.1231	−1.45390	−0.726951	+	0.686689i	$0.759063\pi$
$174$	0	0
$175$	−2.05398	−0.155266
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.36932	0.251835	0.125917	−	0.992041i	$-0.459813\pi$
$179$	3.36932	0.251835	0.125917	+	0.992041i	$0.459813\pi$
$180$	0	0
$181$	−21.4384	−1.59351	−0.796754	−	0.604304i	$-0.793451\pi$
$181$	−21.4384	−1.59351	−0.796754	+	0.604304i	$0.793451\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.75379	0.275984
$186$	0	0
$187$	22.2462	1.62680
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.6847	−1.20726	−0.603630	−	0.797265i	$-0.706279\pi$
$191$	−16.6847	−1.20726	−0.603630	+	0.797265i	$0.706279\pi$
$192$	0	0
$193$	−21.3693	−1.53820	−0.769099	−	0.639130i	$-0.779294\pi$
$193$	−21.3693	−1.53820	−0.769099	+	0.639130i	$0.779294\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.68466	0.262521	0.131261	−	0.991348i	$-0.458098\pi$
$197$	3.68466	0.262521	0.131261	+	0.991348i	$0.458098\pi$
$198$	0	0
$199$	15.4384	1.09440	0.547201	−	0.837001i	$-0.315693\pi$
$199$	15.4384	1.09440	0.547201	+	0.837001i	$0.315693\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.17708	−0.0826149
$204$	0	0
$205$	−6.87689	−0.480303
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−28.4924	−1.97086
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.05398	−0.0718805
$216$	0	0
$217$	−2.93087	−0.198960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.31534	−0.551909	−0.275954	−	0.961171i	$-0.588994\pi$
$227$	−8.31534	−0.551909	−0.275954	+	0.961171i	$0.588994\pi$
$228$	0	0
$229$	0.438447	0.0289734	0.0144867	−	0.999895i	$-0.495389\pi$
$229$	0.438447	0.0289734	0.0144867	+	0.999895i	$0.495389\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.2462	−1.45740	−0.728699	−	0.684834i	$-0.759875\pi$
$233$	−22.2462	−1.45740	−0.728699	+	0.684834i	$0.759875\pi$
$234$	0	0
$235$	2.06913	0.134975
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.1231	−0.719494	−0.359747	−	0.933050i	$-0.617137\pi$
$239$	−11.1231	−0.719494	−0.359747	+	0.933050i	$0.617137\pi$
$240$	0	0
$241$	4.87689	0.314148	0.157074	−	0.987587i	$-0.449794\pi$
$241$	4.87689	0.314148	0.157074	+	0.987587i	$0.449794\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.82292	−0.244237
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.1231	1.83823	0.919117	−	0.393985i	$-0.128904\pi$
$251$	29.1231	1.83823	0.919117	+	0.393985i	$0.128904\pi$
$252$	0	0
$253$	−37.1771	−2.33730
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.2462	1.26292	0.631462	−	0.775407i	$-0.282455\pi$
$257$	20.2462	1.26292	0.631462	+	0.775407i	$0.282455\pi$
$258$	0	0
$259$	2.93087	0.182115
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.93087	0.427376	0.213688	−	0.976902i	$-0.431452\pi$
$263$	6.93087	0.427376	0.213688	+	0.976902i	$0.431452\pi$
$264$	0	0
$265$	0.876894	0.0538672
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.4384	−1.24615	−0.623077	−	0.782160i	$-0.714118\pi$
$269$	−20.4384	−1.24615	−0.623077	+	0.782160i	$0.714118\pi$
$270$	0	0
$271$	26.2462	1.59434	0.797172	−	0.603752i	$-0.206328\pi$
$271$	26.2462	1.59434	0.797172	+	0.603752i	$0.206328\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	26.0540	1.57111
$276$	0	0
$277$	−5.68466	−0.341558	−0.170779	−	0.985309i	$-0.554628\pi$
$277$	−5.68466	−0.341558	−0.170779	+	0.985309i	$0.554628\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.00000	−0.417585	−0.208792	−	0.977960i	$-0.566953\pi$
$281$	−7.00000	−0.417585	−0.208792	+	0.977960i	$0.566953\pi$
$282$	0	0
$283$	3.87689	0.230457	0.115229	−	0.993339i	$-0.463240\pi$
$283$	3.87689	0.230457	0.115229	+	0.993339i	$0.463240\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.36932	−0.316941
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.8769	1.10280	0.551400	−	0.834241i	$-0.314094\pi$
$293$	18.8769	1.10280	0.551400	+	0.834241i	$0.314094\pi$
$294$	0	0
$295$	−1.19224	−0.0694147
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.68466	0.386584
$300$	0	0
$301$	−0.822919	−0.0474323
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.17708	−0.468218
$306$	0	0
$307$	−17.6155	−1.00537	−0.502686	−	0.864469i	$-0.667655\pi$
$307$	−17.6155	−1.00537	−0.502686	+	0.864469i	$0.667655\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.5616	1.22264	0.611322	−	0.791382i	$-0.290638\pi$
$311$	21.5616	1.22264	0.611322	+	0.791382i	$0.290638\pi$
$312$	0	0
$313$	−14.6155	−0.826118	−0.413059	−	0.910704i	$-0.635540\pi$
$313$	−14.6155	−0.826118	−0.413059	+	0.910704i	$0.635540\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.7538	−0.660159	−0.330079	−	0.943953i	$-0.607075\pi$
$317$	−11.7538	−0.660159	−0.330079	+	0.943953i	$0.607075\pi$
$318$	0	0
$319$	14.9309	0.835968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.4924	−1.14023
$324$	0	0
$325$	−4.68466	−0.259858
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.61553	0.0890669
$330$	0	0
$331$	10.8769	0.597848	0.298924	−	0.954277i	$-0.403372\pi$
$331$	10.8769	0.597848	0.298924	+	0.954277i	$0.403372\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.75379	0.314363
$336$	0	0
$337$	21.4924	1.17077	0.585383	−	0.810757i	$-0.300944\pi$
$337$	21.4924	1.17077	0.585383	+	0.810757i	$0.300944\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	37.1771	2.01325
$342$	0	0
$343$	−6.05398	−0.326884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.36932	−0.180874	−0.0904372	−	0.995902i	$-0.528826\pi$
$347$	−3.36932	−0.180874	−0.0904372	+	0.995902i	$0.528826\pi$
$348$	0	0
$349$	−7.75379	−0.415051	−0.207525	−	0.978230i	$-0.566541\pi$
$349$	−7.75379	−0.415051	−0.207525	+	0.978230i	$0.566541\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	0	0
$355$	−4.31534	−0.229035
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.8078	1.09819	0.549096	−	0.835759i	$-0.314972\pi$
$359$	20.8078	1.09819	0.549096	+	0.835759i	$0.314972\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.73863	−0.352716
$366$	0	0
$367$	−3.12311	−0.163025	−0.0815124	−	0.996672i	$-0.525975\pi$
$367$	−3.12311	−0.163025	−0.0815124	+	0.996672i	$0.525975\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.684658	0.0355457
$372$	0	0
$373$	20.8078	1.07739	0.538693	−	0.842502i	$-0.318918\pi$
$373$	20.8078	1.07739	0.538693	+	0.842502i	$0.318918\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.68466	−0.138267
$378$	0	0
$379$	11.7538	0.603752	0.301876	−	0.953347i	$-0.402387\pi$
$379$	11.7538	0.603752	0.301876	+	0.953347i	$0.402387\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.8617	−1.52586	−0.762932	−	0.646479i	$-0.776241\pi$
$383$	−29.8617	−1.52586	−0.762932	+	0.646479i	$0.776241\pi$
$384$	0	0
$385$	−1.36932	−0.0697869
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.7386	−1.65991	−0.829957	−	0.557827i	$-0.811635\pi$
$389$	−32.7386	−1.65991	−0.829957	+	0.557827i	$0.811635\pi$
$390$	0	0
$391$	−26.7386	−1.35223
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.49242	0.226038
$396$	0	0
$397$	11.3153	0.567901	0.283950	−	0.958839i	$-0.408355\pi$
$397$	11.3153	0.567901	0.283950	+	0.958839i	$0.408355\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.1231	1.50428	0.752138	−	0.659006i	$-0.229023\pi$
$401$	30.1231	1.50428	0.752138	+	0.659006i	$0.229023\pi$
$402$	0	0
$403$	−6.68466	−0.332987
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−37.1771	−1.84280
$408$	0	0
$409$	6.87689	0.340041	0.170020	−	0.985441i	$-0.445617\pi$
$409$	6.87689	0.340041	0.170020	+	0.985441i	$0.445617\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.930870	−0.0458051
$414$	0	0
$415$	−9.68466	−0.475401
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.8769	1.01990	0.509952	−	0.860203i	$-0.329663\pi$
$419$	20.8769	1.01990	0.509952	+	0.860203i	$0.329663\pi$
$420$	0	0
$421$	−32.7386	−1.59558	−0.797792	−	0.602933i	$-0.793999\pi$
$421$	−32.7386	−1.59558	−0.797792	+	0.602933i	$0.793999\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.7386	0.908957
$426$	0	0
$427$	−6.38447	−0.308966
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.43845	−0.261961	−0.130980	−	0.991385i	$-0.541812\pi$
$431$	−5.43845	−0.261961	−0.130980	+	0.991385i	$0.541812\pi$
$432$	0	0
$433$	0.930870	0.0447347	0.0223674	−	0.999750i	$-0.492880\pi$
$433$	0.930870	0.0447347	0.0223674	+	0.999750i	$0.492880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	34.2462	1.63822
$438$	0	0
$439$	−7.19224	−0.343267	−0.171633	−	0.985161i	$-0.554904\pi$
$439$	−7.19224	−0.343267	−0.171633	+	0.985161i	$0.554904\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.2462	0.961927	0.480963	−	0.876741i	$-0.340287\pi$
$443$	20.2462	0.961927	0.480963	+	0.876741i	$0.340287\pi$
$444$	0	0
$445$	−0.561553	−0.0266202
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.3153	0.675583	0.337791	−	0.941221i	$-0.390320\pi$
$449$	14.3153	0.675583	0.337791	+	0.941221i	$0.390320\pi$
$450$	0	0
$451$	68.1080	3.20708
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.246211	0.0115426
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	10.8078	0.503368	0.251684	−	0.967809i	$-0.419016\pi$
$461$	10.8078	0.503368	0.251684	+	0.967809i	$0.419016\pi$
$462$	0	0
$463$	40.3002	1.87291	0.936454	−	0.350790i	$-0.114087\pi$
$463$	40.3002	1.87291	0.936454	+	0.350790i	$0.114087\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−31.1231	−1.44021	−0.720103	−	0.693867i	$-0.755905\pi$
$467$	−31.1231	−1.44021	−0.720103	+	0.693867i	$0.755905\pi$
$468$	0	0
$469$	4.49242	0.207441
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.4384	0.479960
$474$	0	0
$475$	−24.0000	−1.10120
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15.4384	0.705401	0.352700	−	0.935736i	$-0.385264\pi$
$479$	15.4384	0.705401	0.352700	+	0.935736i	$0.385264\pi$
$480$	0	0
$481$	6.68466	0.304794
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.26137	0.420537
$486$	0	0
$487$	−2.19224	−0.0993397	−0.0496698	−	0.998766i	$-0.515817\pi$
$487$	−2.19224	−0.0993397	−0.0496698	+	0.998766i	$0.515817\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	10.7386	0.483644
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.36932	−0.151135
$498$	0	0
$499$	19.1231	0.856068	0.428034	−	0.903763i	$-0.359206\pi$
$499$	19.1231	0.856068	0.428034	+	0.903763i	$0.359206\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.0540	0.804987	0.402493	−	0.915423i	$-0.368144\pi$
$503$	18.0540	0.804987	0.402493	+	0.915423i	$0.368144\pi$
$504$	0	0
$505$	−5.36932	−0.238931
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.68466	−0.0746712	−0.0373356	−	0.999303i	$-0.511887\pi$
$509$	−1.68466	−0.0746712	−0.0373356	+	0.999303i	$0.511887\pi$
$510$	0	0
$511$	−5.26137	−0.232749
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.94602	0.394209
$516$	0	0
$517$	−20.4924	−0.901256
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	0	0
$523$	17.3002	0.756484	0.378242	−	0.925707i	$-0.376529\pi$
$523$	17.3002	0.756484	0.378242	+	0.925707i	$0.376529\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.7386	1.16475
$528$	0	0
$529$	21.6847	0.942811
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.2462	−0.530442
$534$	0	0
$535$	−10.8769	−0.470249
$536$	0	0
$537$	0	0
$538$	0	0
$539$	37.8617	1.63082
$540$	0	0
$541$	−32.6847	−1.40522	−0.702612	−	0.711574i	$-0.747983\pi$
$541$	−32.6847	−1.40522	−0.702612	+	0.711574i	$0.747983\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−21.1771	−0.905467	−0.452733	−	0.891646i	$-0.649551\pi$
$547$	−21.1771	−0.905467	−0.452733	+	0.891646i	$0.649551\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−13.7538	−0.585931
$552$	0	0
$553$	3.50758	0.149157
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.8617	1.35003	0.675013	−	0.737806i	$-0.264138\pi$
$557$	31.8617	1.35003	0.675013	+	0.737806i	$0.264138\pi$
$558$	0	0
$559$	−1.87689	−0.0793842
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	0	0
$565$	−0.138261	−0.00581667
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	−32.6695	−1.36718	−0.683588	−	0.729868i	$-0.739581\pi$
$571$	−32.6695	−1.36718	−0.683588	+	0.729868i	$0.739581\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−31.3153	−1.30594
$576$	0	0
$577$	−12.2462	−0.509816	−0.254908	−	0.966965i	$-0.582045\pi$
$577$	−12.2462	−0.509816	−0.254908	+	0.966965i	$0.582045\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.56155	−0.313706
$582$	0	0
$583$	−8.68466	−0.359682
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.8078	−0.817554	−0.408777	−	0.912634i	$-0.634045\pi$
$587$	−19.8078	−0.817554	−0.408777	+	0.912634i	$0.634045\pi$
$588$	0	0
$589$	−34.2462	−1.41109
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.6695	1.62903	0.814516	−	0.580142i	$-0.197003\pi$
$593$	39.6695	1.62903	0.814516	+	0.580142i	$0.197003\pi$
$594$	0	0
$595$	−0.984845	−0.0403747
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.7386	0.438769	0.219384	−	0.975639i	$-0.429595\pi$
$599$	10.7386	0.438769	0.219384	+	0.975639i	$0.429595\pi$
$600$	0	0
$601$	10.7538	0.438656	0.219328	−	0.975651i	$-0.429613\pi$
$601$	10.7538	0.438656	0.219328	+	0.975651i	$0.429613\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.1922	0.455029
$606$	0	0
$607$	−18.7386	−0.760578	−0.380289	−	0.924868i	$-0.624175\pi$
$607$	−18.7386	−0.760578	−0.380289	+	0.924868i	$0.624175\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.68466	0.149065
$612$	0	0
$613$	−38.0540	−1.53699	−0.768493	−	0.639858i	$-0.778993\pi$
$613$	−38.0540	−1.53699	−0.768493	+	0.639858i	$0.778993\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.19224	0.0479976	0.0239988	−	0.999712i	$-0.492360\pi$
$617$	1.19224	0.0479976	0.0239988	+	0.999712i	$0.492360\pi$
$618$	0	0
$619$	39.6155	1.59228	0.796141	−	0.605111i	$-0.206871\pi$
$619$	39.6155	1.59228	0.796141	+	0.605111i	$0.206871\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.438447	−0.0175660
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−26.7386	−1.06614
$630$	0	0
$631$	−45.4233	−1.80827	−0.904136	−	0.427244i	$-0.859485\pi$
$631$	−45.4233	−1.80827	−0.904136	+	0.427244i	$0.859485\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.19224	0.285415
$636$	0	0
$637$	−6.80776	−0.269733
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.61553	0.221800	0.110900	−	0.993832i	$-0.464627\pi$
$641$	5.61553	0.221800	0.110900	+	0.993832i	$0.464627\pi$
$642$	0	0
$643$	−10.2462	−0.404071	−0.202036	−	0.979378i	$-0.564756\pi$
$643$	−10.2462	−0.404071	−0.202036	+	0.979378i	$0.564756\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.9309	0.822877	0.411439	−	0.911437i	$-0.365026\pi$
$647$	20.9309	0.822877	0.411439	+	0.911437i	$0.365026\pi$
$648$	0	0
$649$	11.8078	0.463495
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.87689	0.190848	0.0954238	−	0.995437i	$-0.469579\pi$
$653$	4.87689	0.190848	0.0954238	+	0.995437i	$0.469579\pi$
$654$	0	0
$655$	1.75379	0.0685262
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22.4924	−0.876180	−0.438090	−	0.898931i	$-0.644345\pi$
$659$	−22.4924	−0.876180	−0.438090	+	0.898931i	$0.644345\pi$
$660$	0	0
$661$	−23.3153	−0.906862	−0.453431	−	0.891291i	$-0.649800\pi$
$661$	−23.3153	−0.906862	−0.453431	+	0.891291i	$0.649800\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.26137	0.0489137
$666$	0	0
$667$	−17.9460	−0.694873
$668$	0	0
$669$	0	0
$670$	0	0
$671$	80.9848	3.12639
$672$	0	0
$673$	14.8078	0.570797	0.285399	−	0.958409i	$-0.407874\pi$
$673$	14.8078	0.570797	0.285399	+	0.958409i	$0.407874\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.3693	0.513825	0.256912	−	0.966435i	$-0.417295\pi$
$677$	13.3693	0.513825	0.256912	+	0.966435i	$0.417295\pi$
$678$	0	0
$679$	7.23106	0.277502
$680$	0	0
$681$	0	0
$682$	0	0
$683$	23.6307	0.904203	0.452101	−	0.891967i	$-0.350674\pi$
$683$	23.6307	0.904203	0.452101	+	0.891967i	$0.350674\pi$
$684$	0	0
$685$	2.49242	0.0952306
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.56155	0.0594904
$690$	0	0
$691$	18.8769	0.718111	0.359055	−	0.933316i	$-0.383099\pi$
$691$	18.8769	0.718111	0.359055	+	0.933316i	$0.383099\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.0540	−0.419301
$696$	0	0
$697$	48.9848	1.85543
$698$	0	0
$699$	0	0
$700$	0	0
$701$	32.1080	1.21270	0.606350	−	0.795198i	$-0.292633\pi$
$701$	32.1080	1.21270	0.606350	+	0.795198i	$0.292633\pi$
$702$	0	0
$703$	34.2462	1.29162
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.19224	−0.157665
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−44.6847	−1.67345
$714$	0	0
$715$	−3.12311	−0.116798
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.3002	−1.05542	−0.527709	−	0.849425i	$-0.676949\pi$
$719$	−28.3002	−1.05542	−0.527709	+	0.849425i	$0.676949\pi$
$720$	0	0
$721$	6.98485	0.260129
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.5767	0.467087
$726$	0	0
$727$	28.1080	1.04247	0.521233	−	0.853414i	$-0.325472\pi$
$727$	28.1080	1.04247	0.521233	+	0.853414i	$0.325472\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.50758	0.277678
$732$	0	0
$733$	48.3002	1.78401	0.892004	−	0.452027i	$-0.149299\pi$
$733$	48.3002	1.78401	0.892004	+	0.452027i	$0.149299\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−56.9848	−2.09906
$738$	0	0
$739$	19.7538	0.726655	0.363327	−	0.931662i	$-0.381641\pi$
$739$	19.7538	0.726655	0.363327	+	0.931662i	$0.381641\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.9309	1.17143	0.585715	−	0.810517i	$-0.300814\pi$
$743$	31.9309	1.17143	0.585715	+	0.810517i	$0.300814\pi$
$744$	0	0
$745$	12.4536	0.456265
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.49242	−0.310306
$750$	0	0
$751$	−3.12311	−0.113964	−0.0569819	−	0.998375i	$-0.518148\pi$
$751$	−3.12311	−0.113964	−0.0569819	+	0.998375i	$0.518148\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.01515	−0.255308
$756$	0	0
$757$	−9.61553	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$757$	−9.61553	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.1231	−0.366962	−0.183481	−	0.983023i	$-0.558737\pi$
$761$	−10.1231	−0.366962	−0.183481	+	0.983023i	$0.558737\pi$
$762$	0	0
$763$	−3.12311	−0.113064
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.12311	−0.0766609
$768$	0	0
$769$	1.12311	0.0405002	0.0202501	−	0.999795i	$-0.493554\pi$
$769$	1.12311	0.0405002	0.0202501	+	0.999795i	$0.493554\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.80776	−0.244858	−0.122429	−	0.992477i	$-0.539068\pi$
$773$	−6.80776	−0.244858	−0.122429	+	0.992477i	$0.539068\pi$
$774$	0	0
$775$	31.3153	1.12488
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−62.7386	−2.24784
$780$	0	0
$781$	42.7386	1.52931
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.05398	−0.109001
$786$	0	0
$787$	−37.3693	−1.33207	−0.666036	−	0.745919i	$-0.732010\pi$
$787$	−37.3693	−1.33207	−0.666036	+	0.745919i	$0.732010\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.107951	−0.00383828
$792$	0	0
$793$	−14.5616	−0.517096
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.3002	0.364851	0.182426	−	0.983220i	$-0.441605\pi$
$797$	10.3002	0.364851	0.182426	+	0.983220i	$0.441605\pi$
$798$	0	0
$799$	−14.7386	−0.521415
$800$	0	0
$801$	0	0
$802$	0	0
$803$	66.7386	2.35516
$804$	0	0
$805$	1.64584	0.0580082
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.2462	−0.922768	−0.461384	−	0.887201i	$-0.652647\pi$
$809$	−26.2462	−0.922768	−0.461384	+	0.887201i	$0.652647\pi$
$810$	0	0
$811$	−6.63068	−0.232835	−0.116417	−	0.993200i	$-0.537141\pi$
$811$	−6.63068	−0.232835	−0.116417	+	0.993200i	$0.537141\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.75379	0.201546
$816$	0	0
$817$	−9.61553	−0.336405
$818$	0	0
$819$	0	0
$820$	0	0
$821$	52.2462	1.82341	0.911703	−	0.410851i	$-0.134768\pi$
$821$	52.2462	1.82341	0.911703	+	0.410851i	$0.134768\pi$
$822$	0	0
$823$	−13.9309	−0.485600	−0.242800	−	0.970076i	$-0.578066\pi$
$823$	−13.9309	−0.485600	−0.242800	+	0.970076i	$0.578066\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.8078	0.549690	0.274845	−	0.961489i	$-0.411373\pi$
$827$	15.8078	0.549690	0.274845	+	0.961489i	$0.411373\pi$
$828$	0	0
$829$	−13.1922	−0.458185	−0.229093	−	0.973405i	$-0.573576\pi$
$829$	−13.1922	−0.458185	−0.229093	+	0.973405i	$0.573576\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	27.2311	0.943500
$834$	0	0
$835$	−1.79261	−0.0620358
$836$	0	0
$837$	0	0
$838$	0	0
$839$	45.0540	1.55544	0.777718	−	0.628613i	$-0.216377\pi$
$839$	45.0540	1.55544	0.777718	+	0.628613i	$0.216377\pi$
$840$	0	0
$841$	−21.7926	−0.751469
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.561553	0.0193180
$846$	0	0
$847$	8.73863	0.300263
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.6847	1.53177
$852$	0	0
$853$	9.31534	0.318951	0.159476	−	0.987202i	$-0.449020\pi$
$853$	9.31534	0.318951	0.159476	+	0.987202i	$0.449020\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.73863	0.298506	0.149253	−	0.988799i	$-0.452313\pi$
$857$	8.73863	0.298506	0.149253	+	0.988799i	$0.452313\pi$
$858$	0	0
$859$	−41.1771	−1.40494	−0.702472	−	0.711711i	$-0.747920\pi$
$859$	−41.1771	−1.40494	−0.702472	+	0.711711i	$0.747920\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.17708	−0.210270	−0.105135	−	0.994458i	$-0.533528\pi$
$863$	−6.17708	−0.210270	−0.105135	+	0.994458i	$0.533528\pi$
$864$	0	0
$865$	−10.7386	−0.365125
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−44.4924	−1.50930
$870$	0	0
$871$	10.2462	0.347180
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.38447	−0.0806099
$876$	0	0
$877$	−2.05398	−0.0693578	−0.0346789	−	0.999399i	$-0.511041\pi$
$877$	−2.05398	−0.0693578	−0.0346789	+	0.999399i	$0.511041\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	37.7538	1.27196	0.635979	−	0.771707i	$-0.280597\pi$
$881$	37.7538	1.27196	0.635979	+	0.771707i	$0.280597\pi$
$882$	0	0
$883$	3.31534	0.111570	0.0557851	−	0.998443i	$-0.482234\pi$
$883$	3.31534	0.111570	0.0557851	+	0.998443i	$0.482234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.0540	−1.00911	−0.504557	−	0.863378i	$-0.668344\pi$
$887$	−30.0540	−1.00911	−0.504557	+	0.863378i	$0.668344\pi$
$888$	0	0
$889$	5.61553	0.188339
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.8769	0.631691
$894$	0	0
$895$	1.89205	0.0632442
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.9460	0.598533
$900$	0	0
$901$	−6.24621	−0.208091
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.0388	−0.400184
$906$	0	0
$907$	9.93087	0.329749	0.164875	−	0.986315i	$-0.447278\pi$
$907$	9.93087	0.329749	0.164875	+	0.986315i	$0.447278\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.31534	−0.242368	−0.121184	−	0.992630i	$-0.538669\pi$
$911$	−7.31534	−0.242368	−0.121184	+	0.992630i	$0.538669\pi$
$912$	0	0
$913$	95.9157	3.17435
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.36932	0.0452188
$918$	0	0
$919$	−28.8078	−0.950280	−0.475140	−	0.879910i	$-0.657603\pi$
$919$	−28.8078	−0.950280	−0.475140	+	0.879910i	$0.657603\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.68466	−0.252944
$924$	0	0
$925$	−31.3153	−1.02964
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.8229	−0.420706	−0.210353	−	0.977625i	$-0.567461\pi$
$929$	−12.8229	−0.420706	−0.210353	+	0.977625i	$0.567461\pi$
$930$	0	0
$931$	−34.8769	−1.14304
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.4924	0.408546
$936$	0	0
$937$	9.73863	0.318147	0.159074	−	0.987267i	$-0.449149\pi$
$937$	9.73863	0.318147	0.159074	+	0.987267i	$0.449149\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−81.8617	−2.66579
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.3693	−0.921879	−0.460939	−	0.887432i	$-0.652487\pi$
$947$	−28.3693	−0.921879	−0.460939	+	0.887432i	$0.652487\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.3542	1.43677	0.718386	−	0.695645i	$-0.244881\pi$
$953$	44.3542	1.43677	0.718386	+	0.695645i	$0.244881\pi$
$954$	0	0
$955$	−9.36932	−0.303184
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.94602	0.0628404
$960$	0	0
$961$	13.6847	0.441441
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−27.8078	−0.894237	−0.447119	−	0.894475i	$-0.647550\pi$
$967$	−27.8078	−0.894237	−0.447119	+	0.894475i	$0.647550\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.63068	−0.276972	−0.138486	−	0.990364i	$-0.544224\pi$
$971$	−8.63068	−0.276972	−0.138486	+	0.990364i	$0.544224\pi$
$972$	0	0
$973$	−8.63068	−0.276687
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.61553	0.211649	0.105825	−	0.994385i	$-0.466252\pi$
$977$	6.61553	0.211649	0.105825	+	0.994385i	$0.466252\pi$
$978$	0	0
$979$	5.56155	0.177748
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.6155	−1.00838	−0.504189	−	0.863593i	$-0.668209\pi$
$983$	−31.6155	−1.00838	−0.504189	+	0.863593i	$0.668209\pi$
$984$	0	0
$985$	2.06913	0.0659280
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.5464	−0.398952
$990$	0	0
$991$	−12.0691	−0.383389	−0.191694	−	0.981455i	$-0.561398\pi$
$991$	−12.0691	−0.383389	−0.191694	+	0.981455i	$0.561398\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.66950	0.274842
$996$	0	0
$997$	−14.1771	−0.448993	−0.224496	−	0.974475i	$-0.572074\pi$
$997$	−14.1771	−0.448993	−0.224496	+	0.974475i	$0.572074\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.a.p.1.2	✓	2
3.2	odd	2		2808.2.a.v.1.1	yes	2
4.3	odd	2		5616.2.a.bj.1.2		2
12.11	even	2		5616.2.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2808.2.a.p.1.2	✓	2	1.1	even	1	trivial
2808.2.a.v.1.1	yes	2	3.2	odd	2
5616.2.a.bj.1.2		2	4.3	odd	2
5616.2.a.bv.1.1		2	12.11	even	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q+0.561553 q^{5} +0.438447 q^{7} -5.56155 q^{11} +1.00000 q^{13} -4.00000 q^{17} +5.12311 q^{19} +6.68466 q^{23} -4.68466 q^{25} -2.68466 q^{29} -6.68466 q^{31} +0.246211 q^{35} +6.68466 q^{37} -12.2462 q^{41} -1.87689 q^{43} +3.68466 q^{47} -6.80776 q^{49} +1.56155 q^{53} -3.12311 q^{55} -2.12311 q^{59} -14.5616 q^{61} +0.561553 q^{65} +10.2462 q^{67} -7.68466 q^{71} -12.0000 q^{73} -2.43845 q^{77} +8.00000 q^{79} -17.2462 q^{83} -2.24621 q^{85} -1.00000 q^{89} +0.438447 q^{91} +2.87689 q^{95} +16.4924 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} + 5 q^{7} - 7 q^{11} + 2 q^{13} - 8 q^{17} + 2 q^{19} + q^{23} + 3 q^{25} + 7 q^{29} - q^{31} - 16 q^{35} + q^{37} - 8 q^{41} - 12 q^{43} - 5 q^{47} + 7 q^{49} - q^{53} + 2 q^{55} + 4 q^{59}+ \cdots + 14 q^{95}+O(q^{100}) \) 2 * q - 3 * q^5 + 5 * q^7 - 7 * q^11 + 2 * q^13 - 8 * q^17 + 2 * q^19 + q^23 + 3 * q^25 + 7 * q^29 - q^31 - 16 * q^35 + q^37 - 8 * q^41 - 12 * q^43 - 5 * q^47 + 7 * q^49 - q^53 + 2 * q^55 + 4 * q^59 - 25 * q^61 - 3 * q^65 + 4 * q^67 - 3 * q^71 - 24 * q^73 - 9 * q^77 + 16 * q^79 - 18 * q^83 + 12 * q^85 - 2 * q^89 + 5 * q^91 + 14 * q^95

Introduction

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