LMFDB - Embedded newform 6048.2.j.d.5615.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.18
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27600 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-1.27600 q^{5} -1.00000i q^{7} -6.57401i q^{11} -4.05196i q^{13} -6.09788i q^{17} +2.08352 q^{19} +6.85760 q^{23} -3.37182 q^{25} +6.53304 q^{29} -3.26844i q^{31} +1.27600i q^{35} -2.95562i q^{37} +3.35381i q^{41} -10.9901 q^{43} +7.12041 q^{47} -1.00000 q^{49} -2.87278 q^{53} +8.38843i q^{55} +7.75213i q^{59} -12.0251i q^{61} +5.17030i q^{65} +3.01966 q^{67} +3.48158 q^{71} +2.76889 q^{73} -6.57401 q^{77} +0.849261i q^{79} +15.8068i q^{83} +7.78089i q^{85} -4.06349i q^{89} -4.05196 q^{91} -2.65858 q^{95} -4.90438 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) $ 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.27600	−0.570644	−0.285322	−	0.958432i	$-0.592101\pi$
$5$	−1.27600	−0.570644	−0.285322	+	0.958432i	$0.592101\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 6.57401i	− 1.98214i	−0.133353	−	0.991069i	$-0.542574\pi$
$11$	− 6.57401i	− 1.98214i	0.133353	−	0.991069i	$-0.457426\pi$
$12$	0	0
$13$	− 4.05196i	− 1.12381i	−0.827201	−	0.561906i	$-0.810068\pi$
$13$	− 4.05196i	− 1.12381i	0.827201	−	0.561906i	$-0.189932\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.09788i	− 1.47895i	−0.673182	−	0.739477i	$-0.735073\pi$
$17$	− 6.09788i	− 1.47895i	0.673182	−	0.739477i	$-0.264927\pi$
$18$	0	0
$19$	2.08352	0.477993	0.238997	−	0.971020i	$-0.423181\pi$
$19$	2.08352	0.477993	0.238997	+	0.971020i	$0.423181\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.85760	1.42991	0.714954	−	0.699171i	$-0.246448\pi$
$23$	6.85760	1.42991	0.714954	+	0.699171i	$0.246448\pi$
$24$	0	0
$25$	−3.37182	−0.674365
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.53304	1.21316	0.606578	−	0.795024i	$-0.292542\pi$
$29$	6.53304	1.21316	0.606578	+	0.795024i	$0.292542\pi$
$30$	0	0
$31$	− 3.26844i	− 0.587028i	−0.955955	−	0.293514i	$-0.905175\pi$
$31$	− 3.26844i	− 0.587028i	0.955955	−	0.293514i	$-0.0948248\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.27600i	0.215683i
$36$	0	0
$37$	− 2.95562i	− 0.485901i	−0.970039	−	0.242950i	$-0.921885\pi$
$37$	− 2.95562i	− 0.485901i	0.970039	−	0.242950i	$-0.0781152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.35381i	0.523777i	0.965098	+	0.261888i	$0.0843452\pi$
$41$	3.35381i	0.523777i	−0.965098	+	0.261888i	$0.915655\pi$
$42$	0	0
$43$	−10.9901	−1.67598	−0.837991	−	0.545685i	$-0.816270\pi$
$43$	−10.9901	−1.67598	−0.837991	+	0.545685i	$0.816270\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.12041	1.03862	0.519310	−	0.854586i	$-0.326189\pi$
$47$	7.12041	1.03862	0.519310	+	0.854586i	$0.326189\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.87278	−0.394607	−0.197304	−	0.980342i	$-0.563218\pi$
$53$	−2.87278	−0.394607	−0.197304	+	0.980342i	$0.563218\pi$
$54$	0	0
$55$	8.38843i	1.13110i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.75213i	1.00924i	0.863341	+	0.504621i	$0.168368\pi$
$59$	7.75213i	1.00924i	−0.863341	+	0.504621i	$0.831632\pi$
$60$	0	0
$61$	− 12.0251i	− 1.53966i	−0.638249	−	0.769830i	$-0.720341\pi$
$61$	− 12.0251i	− 1.53966i	0.638249	−	0.769830i	$-0.279659\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.17030i	0.641296i
$66$	0	0
$67$	3.01966	0.368910	0.184455	−	0.982841i	$-0.440948\pi$
$67$	3.01966	0.368910	0.184455	+	0.982841i	$0.440948\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.48158	0.413188	0.206594	−	0.978427i	$-0.433762\pi$
$71$	3.48158	0.413188	0.206594	+	0.978427i	$0.433762\pi$
$72$	0	0
$73$	2.76889	0.324075	0.162037	−	0.986785i	$-0.448194\pi$
$73$	2.76889	0.324075	0.162037	+	0.986785i	$0.448194\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.57401	−0.749177
$78$	0	0
$79$	0.849261i	0.0955494i	0.998858	+	0.0477747i	$0.0152129\pi$
$79$	0.849261i	0.0955494i	−0.998858	+	0.0477747i	$0.984787\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.8068i	1.73503i	0.497414	+	0.867513i	$0.334283\pi$
$83$	15.8068i	1.73503i	−0.497414	+	0.867513i	$0.665717\pi$
$84$	0	0
$85$	7.78089i	0.843956i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.06349i	− 0.430729i	−0.976534	−	0.215365i	$-0.930906\pi$
$89$	− 4.06349i	− 0.430729i	0.976534	−	0.215365i	$-0.0690940\pi$
$90$	0	0
$91$	−4.05196	−0.424761
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.65858	−0.272764
$96$	0	0
$97$	−4.90438	−0.497964	−0.248982	−	0.968508i	$-0.580096\pi$
$97$	−4.90438	−0.497964	−0.248982	+	0.968508i	$0.580096\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−16.5134	−1.64314	−0.821572	−	0.570105i	$-0.806902\pi$
$101$	−16.5134	−1.64314	−0.821572	+	0.570105i	$0.806902\pi$
$102$	0	0
$103$	10.7097i	1.05526i	0.849475	+	0.527629i	$0.176919\pi$
$103$	10.7097i	1.05526i	−0.849475	+	0.527629i	$0.823081\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.41994i	0.910660i	0.890323	+	0.455330i	$0.150479\pi$
$107$	9.41994i	0.910660i	−0.890323	+	0.455330i	$0.849521\pi$
$108$	0	0
$109$	− 7.95370i	− 0.761826i	−0.924611	−	0.380913i	$-0.875610\pi$
$109$	− 7.95370i	− 0.761826i	0.924611	−	0.380913i	$-0.124390\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.56398i	0.147127i	0.997291	+	0.0735636i	$0.0234372\pi$
$113$	1.56398i	0.147127i	−0.997291	+	0.0735636i	$0.976563\pi$
$114$	0	0
$115$	−8.75029	−0.815969
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.09788	−0.558992
$120$	0	0
$121$	−32.2175	−2.92887
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.6824	0.955467
$126$	0	0
$127$	− 19.7895i	− 1.75603i	−0.478629	−	0.878017i	$-0.658866\pi$
$127$	− 19.7895i	− 1.75603i	0.478629	−	0.878017i	$-0.341134\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.6907i	1.19616i	0.801436	+	0.598080i	$0.204069\pi$
$131$	13.6907i	1.19616i	−0.801436	+	0.598080i	$0.795931\pi$
$132$	0	0
$133$	− 2.08352i	− 0.180664i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 20.0099i	− 1.70956i	−0.518990	−	0.854780i	$-0.673692\pi$
$137$	− 20.0099i	− 1.70956i	0.518990	−	0.854780i	$-0.326308\pi$
$138$	0	0
$139$	11.0306	0.935607	0.467804	−	0.883832i	$-0.345045\pi$
$139$	11.0306	0.935607	0.467804	+	0.883832i	$0.345045\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−26.6376	−2.22755
$144$	0	0
$145$	−8.33616	−0.692280
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.391867	0.0321030	0.0160515	−	0.999871i	$-0.494890\pi$
$149$	0.391867	0.0321030	0.0160515	+	0.999871i	$0.494890\pi$
$150$	0	0
$151$	10.3302i	0.840662i	0.907371	+	0.420331i	$0.138086\pi$
$151$	10.3302i	0.840662i	−0.907371	+	0.420331i	$0.861914\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.17052i	0.334984i
$156$	0	0
$157$	− 9.78139i	− 0.780640i	−0.920679	−	0.390320i	$-0.872364\pi$
$157$	− 9.78139i	− 0.780640i	0.920679	−	0.390320i	$-0.127636\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 6.85760i	− 0.540455i
$162$	0	0
$163$	−1.11525	−0.0873533	−0.0436766	−	0.999046i	$-0.513907\pi$
$163$	−1.11525	−0.0873533	−0.0436766	+	0.999046i	$0.513907\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.7973	1.29982	0.649908	−	0.760013i	$-0.274807\pi$
$167$	16.7973	1.29982	0.649908	+	0.760013i	$0.274807\pi$
$168$	0	0
$169$	−3.41837	−0.262951
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.2063	−1.38420	−0.692100	−	0.721802i	$-0.743314\pi$
$173$	−18.2063	−1.38420	−0.692100	+	0.721802i	$0.743314\pi$
$174$	0	0
$175$	3.37182i	0.254886i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.03488i	− 0.600555i	−0.953852	−	0.300278i	$-0.902921\pi$
$179$	− 8.03488i	− 0.600555i	0.953852	−	0.300278i	$-0.0970793\pi$
$180$	0	0
$181$	18.5478i	1.37865i	0.724454	+	0.689323i	$0.242092\pi$
$181$	18.5478i	1.37865i	−0.724454	+	0.689323i	$0.757908\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.77137i	0.277276i
$186$	0	0
$187$	−40.0875	−2.93149
$188$	0	0
$189$	0	0
$190$	0	0
$191$	22.8458	1.65306	0.826532	−	0.562890i	$-0.190311\pi$
$191$	22.8458	1.65306	0.826532	+	0.562890i	$0.190311\pi$
$192$	0	0
$193$	3.49311	0.251439	0.125720	−	0.992066i	$-0.459876\pi$
$193$	3.49311	0.251439	0.125720	+	0.992066i	$0.459876\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.9432	1.56339	0.781693	−	0.623663i	$-0.214356\pi$
$197$	21.9432	1.56339	0.781693	+	0.623663i	$0.214356\pi$
$198$	0	0
$199$	− 23.0870i	− 1.63660i	−0.574793	−	0.818299i	$-0.694918\pi$
$199$	− 23.0870i	− 1.63660i	0.574793	−	0.818299i	$-0.305082\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.53304i	− 0.458529i
$204$	0	0
$205$	− 4.27946i	− 0.298890i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 13.6971i	− 0.947448i
$210$	0	0
$211$	1.71426	0.118014	0.0590072	−	0.998258i	$-0.481207\pi$
$211$	1.71426	0.118014	0.0590072	+	0.998258i	$0.481207\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.0234	0.956389
$216$	0	0
$217$	−3.26844	−0.221876
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.7084	−1.66206
$222$	0	0
$223$	− 0.545891i	− 0.0365556i	−0.999833	−	0.0182778i	$-0.994182\pi$
$223$	− 0.545891i	− 0.0365556i	0.999833	−	0.0182778i	$-0.00581832\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.60514i	− 0.106537i	−0.998580	−	0.0532685i	$-0.983036\pi$
$227$	− 1.60514i	− 0.106537i	0.998580	−	0.0532685i	$-0.0169639\pi$
$228$	0	0
$229$	1.39412i	0.0921257i	0.998939	+	0.0460629i	$0.0146675\pi$
$229$	1.39412i	0.0921257i	−0.998939	+	0.0460629i	$0.985333\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 10.0130i	− 0.655971i	−0.944683	−	0.327985i	$-0.893630\pi$
$233$	− 10.0130i	− 0.655971i	0.944683	−	0.327985i	$-0.106370\pi$
$234$	0	0
$235$	−9.08565	−0.592682
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.62073	−0.622313	−0.311157	−	0.950359i	$-0.600716\pi$
$239$	−9.62073	−0.622313	−0.311157	+	0.950359i	$0.600716\pi$
$240$	0	0
$241$	13.9788	0.900456	0.450228	−	0.892914i	$-0.351343\pi$
$241$	13.9788	0.900456	0.450228	+	0.892914i	$0.351343\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.27600	0.0815206
$246$	0	0
$247$	− 8.44235i	− 0.537174i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.99814i	− 0.441719i	−0.975306	−	0.220859i	$-0.929114\pi$
$251$	− 6.99814i	− 0.441719i	0.975306	−	0.220859i	$-0.0708862\pi$
$252$	0	0
$253$	− 45.0819i	− 2.83427i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.39346i	− 0.274057i	−0.990567	−	0.137028i	$-0.956245\pi$
$257$	− 4.39346i	− 0.274057i	0.990567	−	0.137028i	$-0.0437551\pi$
$258$	0	0
$259$	−2.95562	−0.183653
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.54094	0.526657	0.263328	−	0.964706i	$-0.415180\pi$
$263$	8.54094	0.526657	0.263328	+	0.964706i	$0.415180\pi$
$264$	0	0
$265$	3.66567	0.225180
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.7389	0.898647	0.449324	−	0.893369i	$-0.351665\pi$
$269$	14.7389	0.898647	0.449324	+	0.893369i	$0.351665\pi$
$270$	0	0
$271$	− 4.21696i	− 0.256162i	−0.991764	−	0.128081i	$-0.959118\pi$
$271$	− 4.21696i	− 0.256162i	0.991764	−	0.128081i	$-0.0408817\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.1664i	1.33668i
$276$	0	0
$277$	2.73096i	0.164088i	0.996629	+	0.0820438i	$0.0261447\pi$
$277$	2.73096i	0.164088i	−0.996629	+	0.0820438i	$0.973855\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 9.32789i	− 0.556455i	−0.960515	−	0.278228i	$-0.910253\pi$
$281$	− 9.32789i	− 0.556455i	0.960515	−	0.278228i	$-0.0897469\pi$
$282$	0	0
$283$	−20.2355	−1.20288	−0.601438	−	0.798919i	$-0.705405\pi$
$283$	−20.2355	−1.20288	−0.601438	+	0.798919i	$0.705405\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.35381	0.197969
$288$	0	0
$289$	−20.1841	−1.18730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.259973	0.0151878	0.00759391	−	0.999971i	$-0.497583\pi$
$293$	0.259973	0.0151878	0.00759391	+	0.999971i	$0.497583\pi$
$294$	0	0
$295$	− 9.89172i	− 0.575918i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 27.7867i	− 1.60695i
$300$	0	0
$301$	10.9901i	0.633461i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.3441i	0.878598i
$306$	0	0
$307$	−25.8255	−1.47394	−0.736969	−	0.675927i	$-0.763743\pi$
$307$	−25.8255	−1.47394	−0.736969	+	0.675927i	$0.763743\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.8253	−1.12419	−0.562094	−	0.827074i	$-0.690004\pi$
$311$	−19.8253	−1.12419	−0.562094	+	0.827074i	$0.690004\pi$
$312$	0	0
$313$	10.2201	0.577673	0.288837	−	0.957378i	$-0.406732\pi$
$313$	10.2201	0.577673	0.288837	+	0.957378i	$0.406732\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.21191	0.180399	0.0901995	−	0.995924i	$-0.471250\pi$
$317$	3.21191	0.180399	0.0901995	+	0.995924i	$0.471250\pi$
$318$	0	0
$319$	− 42.9482i	− 2.40464i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12.7051i	− 0.706930i
$324$	0	0
$325$	13.6625i	0.757859i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 7.12041i	− 0.392561i
$330$	0	0
$331$	29.8260	1.63939	0.819694	−	0.572802i	$-0.194144\pi$
$331$	29.8260	1.63939	0.819694	+	0.572802i	$0.194144\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.85308	−0.210516
$336$	0	0
$337$	−26.9588	−1.46854	−0.734271	−	0.678857i	$-0.762476\pi$
$337$	−26.9588	−1.46854	−0.734271	+	0.678857i	$0.762476\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−21.4867	−1.16357
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.73769i	− 0.0932843i	−0.998912	−	0.0466421i	$-0.985148\pi$
$347$	− 1.73769i	− 0.0932843i	0.998912	−	0.0466421i	$-0.0148520\pi$
$348$	0	0
$349$	16.8981i	0.904536i	0.891882	+	0.452268i	$0.149385\pi$
$349$	16.8981i	0.904536i	−0.891882	+	0.452268i	$0.850615\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.2774i	0.653459i	0.945118	+	0.326730i	$0.105947\pi$
$353$	12.2774i	0.653459i	−0.945118	+	0.326730i	$0.894053\pi$
$354$	0	0
$355$	−4.44249	−0.235783
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.77380	0.199173	0.0995867	−	0.995029i	$-0.468248\pi$
$359$	3.77380	0.199173	0.0995867	+	0.995029i	$0.468248\pi$
$360$	0	0
$361$	−14.6589	−0.771522
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.53311	−0.184931
$366$	0	0
$367$	28.6703i	1.49658i	0.663372	+	0.748289i	$0.269125\pi$
$367$	28.6703i	1.49658i	−0.663372	+	0.748289i	$0.730875\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.87278i	0.149147i
$372$	0	0
$373$	27.3540i	1.41634i	0.706043	+	0.708169i	$0.250478\pi$
$373$	27.3540i	1.41634i	−0.706043	+	0.708169i	$0.749522\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 26.4716i	− 1.36336i
$378$	0	0
$379$	18.6336	0.957146	0.478573	−	0.878048i	$-0.341154\pi$
$379$	18.6336	0.957146	0.478573	+	0.878048i	$0.341154\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−27.8368	−1.42240	−0.711198	−	0.702992i	$-0.751847\pi$
$383$	−27.8368	−1.42240	−0.711198	+	0.702992i	$0.751847\pi$
$384$	0	0
$385$	8.38843	0.427514
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7.10860	−0.360420	−0.180210	−	0.983628i	$-0.557678\pi$
$389$	−7.10860	−0.360420	−0.180210	+	0.983628i	$0.557678\pi$
$390$	0	0
$391$	− 41.8168i	− 2.11477i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.08366i	− 0.0545247i
$396$	0	0
$397$	32.8312i	1.64775i	0.566771	+	0.823875i	$0.308192\pi$
$397$	32.8312i	1.64775i	−0.566771	+	0.823875i	$0.691808\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.0585i	1.45111i	0.688164	+	0.725555i	$0.258417\pi$
$401$	29.0585i	1.45111i	−0.688164	+	0.725555i	$0.741583\pi$
$402$	0	0
$403$	−13.2436	−0.659709
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−19.4303	−0.963122
$408$	0	0
$409$	−5.68368	−0.281040	−0.140520	−	0.990078i	$-0.544877\pi$
$409$	−5.68368	−0.281040	−0.140520	+	0.990078i	$0.544877\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.75213	0.381458
$414$	0	0
$415$	− 20.1695i	− 0.990083i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.00749i	− 0.0980722i	−0.998797	−	0.0490361i	$-0.984385\pi$
$419$	− 2.00749i	− 0.0980722i	0.998797	−	0.0490361i	$-0.0156149\pi$
$420$	0	0
$421$	− 16.4735i	− 0.802871i	−0.915887	−	0.401435i	$-0.868511\pi$
$421$	− 16.4735i	− 0.802871i	0.915887	−	0.401435i	$-0.131489\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	20.5610i	0.997354i
$426$	0	0
$427$	−12.0251	−0.581936
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.2399	0.830417	0.415209	−	0.909726i	$-0.363709\pi$
$431$	17.2399	0.830417	0.415209	+	0.909726i	$0.363709\pi$
$432$	0	0
$433$	−38.1465	−1.83321	−0.916603	−	0.399799i	$-0.869080\pi$
$433$	−38.1465	−1.83321	−0.916603	+	0.399799i	$0.869080\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.2880	0.683487
$438$	0	0
$439$	− 7.30831i	− 0.348806i	−0.984674	−	0.174403i	$-0.944200\pi$
$439$	− 7.30831i	− 0.348806i	0.984674	−	0.174403i	$-0.0557996\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 21.7787i	− 1.03474i	−0.855763	−	0.517368i	$-0.826912\pi$
$443$	− 21.7787i	− 1.03474i	0.855763	−	0.517368i	$-0.173088\pi$
$444$	0	0
$445$	5.18501i	0.245793i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.30200i	− 0.0614451i	−0.999528	−	0.0307225i	$-0.990219\pi$
$449$	− 1.30200i	− 0.0614451i	0.999528	−	0.0307225i	$-0.00978083\pi$
$450$	0	0
$451$	22.0479	1.03820
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.17030	0.242387
$456$	0	0
$457$	3.17657	0.148594	0.0742968	−	0.997236i	$-0.476329\pi$
$457$	3.17657	0.148594	0.0742968	+	0.997236i	$0.476329\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.56764	0.399035	0.199517	−	0.979894i	$-0.436063\pi$
$461$	8.56764	0.399035	0.199517	+	0.979894i	$0.436063\pi$
$462$	0	0
$463$	19.9440i	0.926877i	0.886129	+	0.463438i	$0.153385\pi$
$463$	19.9440i	0.926877i	−0.886129	+	0.463438i	$0.846615\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11.5091i	− 0.532579i	−0.963893	−	0.266290i	$-0.914202\pi$
$467$	− 11.5091i	− 0.532579i	0.963893	−	0.266290i	$-0.0857978\pi$
$468$	0	0
$469$	− 3.01966i	− 0.139435i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	72.2493i	3.32203i
$474$	0	0
$475$	−7.02528	−0.322342
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.91391	0.270213	0.135107	−	0.990831i	$-0.456862\pi$
$479$	5.91391	0.270213	0.135107	+	0.990831i	$0.456862\pi$
$480$	0	0
$481$	−11.9760	−0.546061
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.25798	0.284160
$486$	0	0
$487$	20.4470i	0.926541i	0.886217	+	0.463271i	$0.153324\pi$
$487$	20.4470i	0.926541i	−0.886217	+	0.463271i	$0.846676\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	24.0391i	1.08487i	0.840098	+	0.542434i	$0.182497\pi$
$491$	24.0391i	1.08487i	−0.840098	+	0.542434i	$0.817503\pi$
$492$	0	0
$493$	− 39.8377i	− 1.79420i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.48158i	− 0.156170i
$498$	0	0
$499$	−29.1375	−1.30438	−0.652188	−	0.758057i	$-0.726149\pi$
$499$	−29.1375	−1.30438	−0.652188	+	0.758057i	$0.726149\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.70604	0.432771	0.216386	−	0.976308i	$-0.430573\pi$
$503$	9.70604	0.432771	0.216386	+	0.976308i	$0.430573\pi$
$504$	0	0
$505$	21.0711	0.937651
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.5042	−0.687213	−0.343606	−	0.939114i	$-0.611649\pi$
$509$	−15.5042	−0.687213	−0.343606	+	0.939114i	$0.611649\pi$
$510$	0	0
$511$	− 2.76889i	− 0.122489i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 13.6656i	− 0.602177i
$516$	0	0
$517$	− 46.8096i	− 2.05869i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 6.22420i	− 0.272687i	−0.990662	−	0.136344i	$-0.956465\pi$
$521$	− 6.22420i	− 0.272687i	0.990662	−	0.136344i	$-0.0435351\pi$
$522$	0	0
$523$	−33.3752	−1.45939	−0.729697	−	0.683770i	$-0.760339\pi$
$523$	−33.3752	−1.45939	−0.729697	+	0.683770i	$0.760339\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.9305	−0.868188
$528$	0	0
$529$	24.0267	1.04464
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.5895	0.588626
$534$	0	0
$535$	− 12.0198i	− 0.519663i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.57401i	0.283162i
$540$	0	0
$541$	4.04207i	0.173782i	0.996218	+	0.0868911i	$0.0276932\pi$
$541$	4.04207i	0.173782i	−0.996218	+	0.0868911i	$0.972307\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.1489i	0.434732i
$546$	0	0
$547$	19.7497	0.844436	0.422218	−	0.906494i	$-0.361252\pi$
$547$	19.7497	0.844436	0.422218	+	0.906494i	$0.361252\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.6117	0.579880
$552$	0	0
$553$	0.849261	0.0361143
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.2192	0.475373	0.237687	−	0.971342i	$-0.423611\pi$
$557$	11.2192	0.475373	0.237687	+	0.971342i	$0.423611\pi$
$558$	0	0
$559$	44.5316i	1.88349i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.66666i	0.365256i	0.983182	+	0.182628i	$0.0584604\pi$
$563$	8.66666i	0.365256i	−0.983182	+	0.182628i	$0.941540\pi$
$564$	0	0
$565$	− 1.99564i	− 0.0839573i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	27.0778i	1.13516i	0.823317	+	0.567581i	$0.192121\pi$
$569$	27.0778i	1.13516i	−0.823317	+	0.567581i	$0.807879\pi$
$570$	0	0
$571$	−19.5104	−0.816485	−0.408243	−	0.912873i	$-0.633858\pi$
$571$	−19.5104	−0.816485	−0.408243	+	0.912873i	$0.633858\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−23.1226	−0.964280
$576$	0	0
$577$	−23.4544	−0.976421	−0.488211	−	0.872726i	$-0.662350\pi$
$577$	−23.4544	−0.976421	−0.488211	+	0.872726i	$0.662350\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.8068	0.655778
$582$	0	0
$583$	18.8857i	0.782165i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.29331i	0.342302i	0.985245	+	0.171151i	$0.0547485\pi$
$587$	8.29331i	0.342302i	−0.985245	+	0.171151i	$0.945251\pi$
$588$	0	0
$589$	− 6.80987i	− 0.280596i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.99359i	0.246127i	0.992399	+	0.123064i	$0.0392719\pi$
$593$	5.99359i	0.246127i	−0.992399	+	0.123064i	$0.960728\pi$
$594$	0	0
$595$	7.78089	0.318986
$596$	0	0
$597$	0	0
$598$	0	0
$599$	41.0912	1.67894	0.839470	−	0.543405i	$-0.182865\pi$
$599$	41.0912	1.67894	0.839470	+	0.543405i	$0.182865\pi$
$600$	0	0
$601$	−14.9084	−0.608128	−0.304064	−	0.952652i	$-0.598344\pi$
$601$	−14.9084	−0.608128	−0.304064	+	0.952652i	$0.598344\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	41.1096	1.67134
$606$	0	0
$607$	21.1672i	0.859152i	0.903031	+	0.429576i	$0.141337\pi$
$607$	21.1672i	0.859152i	−0.903031	+	0.429576i	$0.858663\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 28.8516i	− 1.16721i
$612$	0	0
$613$	− 21.0293i	− 0.849367i	−0.905342	−	0.424683i	$-0.860385\pi$
$613$	− 21.0293i	− 0.849367i	0.905342	−	0.424683i	$-0.139615\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 29.2848i	− 1.17896i	−0.807783	−	0.589480i	$-0.799333\pi$
$617$	− 29.2848i	− 1.17896i	0.807783	−	0.589480i	$-0.200667\pi$
$618$	0	0
$619$	−43.6407	−1.75407	−0.877034	−	0.480428i	$-0.840481\pi$
$619$	−43.6407	−1.75407	−0.877034	+	0.480428i	$0.840481\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.06349	−0.162800
$624$	0	0
$625$	3.22833	0.129133
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.0230	−0.718624
$630$	0	0
$631$	− 33.7501i	− 1.34357i	−0.740747	−	0.671784i	$-0.765528\pi$
$631$	− 33.7501i	− 1.34357i	0.740747	−	0.671784i	$-0.234472\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	25.2514i	1.00207i
$636$	0	0
$637$	4.05196i	0.160544i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 13.1478i	− 0.519306i	−0.965702	−	0.259653i	$-0.916392\pi$
$641$	− 13.1478i	− 0.519306i	0.965702	−	0.259653i	$-0.0836083\pi$
$642$	0	0
$643$	26.7796	1.05608	0.528042	−	0.849218i	$-0.322926\pi$
$643$	26.7796	1.05608	0.528042	+	0.849218i	$0.322926\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.7850	0.541944	0.270972	−	0.962587i	$-0.412655\pi$
$647$	13.7850	0.541944	0.270972	+	0.962587i	$0.412655\pi$
$648$	0	0
$649$	50.9626	2.00046
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.5245	−1.35105	−0.675524	−	0.737338i	$-0.736083\pi$
$653$	−34.5245	−1.35105	−0.675524	+	0.737338i	$0.736083\pi$
$654$	0	0
$655$	− 17.4693i	− 0.682582i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 31.8035i	− 1.23889i	−0.785040	−	0.619445i	$-0.787358\pi$
$659$	− 31.8035i	− 1.23889i	0.785040	−	0.619445i	$-0.212642\pi$
$660$	0	0
$661$	− 14.7156i	− 0.572370i	−0.958174	−	0.286185i	$-0.907613\pi$
$661$	− 14.7156i	− 0.572370i	0.958174	−	0.286185i	$-0.0923873\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.65858i	0.103095i
$666$	0	0
$667$	44.8010	1.73470
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−79.0532	−3.05182
$672$	0	0
$673$	14.1705	0.546233	0.273116	−	0.961981i	$-0.411946\pi$
$673$	14.1705	0.546233	0.273116	+	0.961981i	$0.411946\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.9994	1.30670	0.653351	−	0.757055i	$-0.273362\pi$
$677$	33.9994	1.30670	0.653351	+	0.757055i	$0.273362\pi$
$678$	0	0
$679$	4.90438i	0.188213i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 24.7265i	− 0.946132i	−0.881027	−	0.473066i	$-0.843147\pi$
$683$	− 24.7265i	− 0.946132i	0.881027	−	0.473066i	$-0.156853\pi$
$684$	0	0
$685$	25.5326i	0.975551i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.6404i	0.443464i
$690$	0	0
$691$	19.2490	0.732267	0.366134	−	0.930562i	$-0.380681\pi$
$691$	19.2490	0.732267	0.366134	+	0.930562i	$0.380681\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.0751	−0.533899
$696$	0	0
$697$	20.4511	0.774641
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.4914	0.509563	0.254781	−	0.966999i	$-0.417996\pi$
$701$	13.4914	0.509563	0.254781	+	0.966999i	$0.417996\pi$
$702$	0	0
$703$	− 6.15810i	− 0.232257i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.5134i	0.621050i
$708$	0	0
$709$	− 12.1839i	− 0.457575i	−0.973476	−	0.228788i	$-0.926524\pi$
$709$	− 12.1839i	− 0.457575i	0.973476	−	0.228788i	$-0.0734761\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 22.4136i	− 0.839397i
$714$	0	0
$715$	33.9896	1.27114
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.14139	0.0798603	0.0399301	−	0.999202i	$-0.487286\pi$
$719$	2.14139	0.0798603	0.0399301	+	0.999202i	$0.487286\pi$
$720$	0	0
$721$	10.7097	0.398850
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−22.0283	−0.818109
$726$	0	0
$727$	36.8865i	1.36804i	0.729462	+	0.684022i	$0.239771\pi$
$727$	36.8865i	1.36804i	−0.729462	+	0.684022i	$0.760229\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	67.0166i	2.47870i
$732$	0	0
$733$	28.2007i	1.04162i	0.853674	+	0.520808i	$0.174369\pi$
$733$	28.2007i	1.04162i	−0.853674	+	0.520808i	$0.825631\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19.8513i	− 0.731230i
$738$	0	0
$739$	13.9006	0.511341	0.255670	−	0.966764i	$-0.417704\pi$
$739$	13.9006	0.511341	0.255670	+	0.966764i	$0.417704\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.81892	0.213476	0.106738	−	0.994287i	$-0.465959\pi$
$743$	5.81892	0.213476	0.106738	+	0.994287i	$0.465959\pi$
$744$	0	0
$745$	−0.500022	−0.0183194
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.41994	0.344197
$750$	0	0
$751$	36.7860i	1.34234i	0.741304	+	0.671170i	$0.234208\pi$
$751$	36.7860i	1.34234i	−0.741304	+	0.671170i	$0.765792\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 13.1814i	− 0.479719i
$756$	0	0
$757$	− 16.6555i	− 0.605356i	−0.953093	−	0.302678i	$-0.902119\pi$
$757$	− 16.6555i	− 0.605356i	0.953093	−	0.302678i	$-0.0978806\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.2510i	1.38660i	0.720651	+	0.693298i	$0.243843\pi$
$761$	38.2510i	1.38660i	−0.720651	+	0.693298i	$0.756157\pi$
$762$	0	0
$763$	−7.95370	−0.287943
$764$	0	0
$765$	0	0
$766$	0	0
$767$	31.4113	1.13420
$768$	0	0
$769$	−40.1021	−1.44612	−0.723059	−	0.690787i	$-0.757264\pi$
$769$	−40.1021	−1.44612	−0.723059	+	0.690787i	$0.757264\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.84516	0.0663659	0.0331830	−	0.999449i	$-0.489436\pi$
$773$	1.84516	0.0663659	0.0331830	+	0.999449i	$0.489436\pi$
$774$	0	0
$775$	11.0206i	0.395871i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.98774i	0.250362i
$780$	0	0
$781$	− 22.8879i	− 0.818994i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.4810i	0.445468i
$786$	0	0
$787$	40.4160	1.44068	0.720338	−	0.693623i	$-0.243987\pi$
$787$	40.4160	1.44068	0.720338	+	0.693623i	$0.243987\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.56398	0.0556088
$792$	0	0
$793$	−48.7253	−1.73029
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.9867	1.69978	0.849889	−	0.526962i	$-0.176669\pi$
$797$	47.9867	1.69978	0.849889	+	0.526962i	$0.176669\pi$
$798$	0	0
$799$	− 43.4194i	− 1.53607i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 18.2027i	− 0.642360i
$804$	0	0
$805$	8.75029i	0.308407i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 14.1754i	− 0.498379i	−0.968455	−	0.249190i	$-0.919836\pi$
$809$	− 14.1754i	− 0.498379i	0.968455	−	0.249190i	$-0.0801643\pi$
$810$	0	0
$811$	5.80289	0.203767	0.101884	−	0.994796i	$-0.467513\pi$
$811$	5.80289	0.203767	0.101884	+	0.994796i	$0.467513\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.42306	0.0498477
$816$	0	0
$817$	−22.8982	−0.801108
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.67601	0.302795	0.151397	−	0.988473i	$-0.451623\pi$
$821$	8.67601	0.302795	0.151397	+	0.988473i	$0.451623\pi$
$822$	0	0
$823$	− 0.199240i	− 0.00694506i	−0.999994	−	0.00347253i	$-0.998895\pi$
$823$	− 0.199240i	− 0.00694506i	0.999994	−	0.00347253i	$-0.00110534\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 42.2411i	− 1.46887i	−0.678681	−	0.734433i	$-0.737448\pi$
$827$	− 42.2411i	− 1.46887i	0.678681	−	0.734433i	$-0.262552\pi$
$828$	0	0
$829$	− 10.9282i	− 0.379554i	−0.981827	−	0.189777i	$-0.939224\pi$
$829$	− 10.9282i	− 0.379554i	0.981827	−	0.189777i	$-0.0607765\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.09788i	0.211279i
$834$	0	0
$835$	−21.4334	−0.741733
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.6038	−1.43632	−0.718161	−	0.695877i	$-0.755016\pi$
$839$	−41.6038	−1.43632	−0.718161	+	0.695877i	$0.755016\pi$
$840$	0	0
$841$	13.6806	0.471745
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.36184	0.150052
$846$	0	0
$847$	32.2175i	1.10701i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 20.2684i	− 0.694793i
$852$	0	0
$853$	10.8964i	0.373084i	0.982447	+	0.186542i	$0.0597281\pi$
$853$	10.8964i	0.373084i	−0.982447	+	0.186542i	$0.940272\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 11.9467i	− 0.408093i	−0.978961	−	0.204046i	$-0.934591\pi$
$857$	− 11.9467i	− 0.408093i	0.978961	−	0.204046i	$-0.0654093\pi$
$858$	0	0
$859$	15.8684	0.541423	0.270711	−	0.962661i	$-0.412741\pi$
$859$	15.8684	0.541423	0.270711	+	0.962661i	$0.412741\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.7712	−1.25171	−0.625853	−	0.779941i	$-0.715249\pi$
$863$	−36.7712	−1.25171	−0.625853	+	0.779941i	$0.715249\pi$
$864$	0	0
$865$	23.2312	0.789886
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.58305	0.189392
$870$	0	0
$871$	− 12.2355i	− 0.414585i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 10.6824i	− 0.361133i
$876$	0	0
$877$	48.2439i	1.62908i	0.580106	+	0.814541i	$0.303011\pi$
$877$	48.2439i	1.62908i	−0.580106	+	0.814541i	$0.696989\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21.4780i	0.723611i	0.932254	+	0.361805i	$0.117840\pi$
$881$	21.4780i	0.723611i	−0.932254	+	0.361805i	$0.882160\pi$
$882$	0	0
$883$	2.78771	0.0938139	0.0469070	−	0.998899i	$-0.485064\pi$
$883$	2.78771	0.0938139	0.0469070	+	0.998899i	$0.485064\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.3399	1.25375	0.626875	−	0.779120i	$-0.284334\pi$
$887$	37.3399	1.25375	0.626875	+	0.779120i	$0.284334\pi$
$888$	0	0
$889$	−19.7895	−0.663719
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.8356	0.496453
$894$	0	0
$895$	10.2525i	0.342703i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 21.3528i	− 0.712156i
$900$	0	0
$901$	17.5179i	0.583605i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 23.6670i	− 0.786717i
$906$	0	0
$907$	−38.2433	−1.26985	−0.634924	−	0.772574i	$-0.718969\pi$
$907$	−38.2433	−1.26985	−0.634924	+	0.772574i	$0.718969\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.40878	−0.0798063	−0.0399032	−	0.999204i	$-0.512705\pi$
$911$	−2.40878	−0.0798063	−0.0399032	+	0.999204i	$0.512705\pi$
$912$	0	0
$913$	103.914	3.43906
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.6907	0.452106
$918$	0	0
$919$	21.5153i	0.709723i	0.934919	+	0.354862i	$0.115472\pi$
$919$	21.5153i	0.709723i	−0.934919	+	0.354862i	$0.884528\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 14.1072i	− 0.464345i
$924$	0	0
$925$	9.96583i	0.327674i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 21.9659i	− 0.720677i	−0.932822	−	0.360339i	$-0.882661\pi$
$929$	− 21.9659i	− 0.720677i	0.932822	−	0.360339i	$-0.117339\pi$
$930$	0	0
$931$	−2.08352	−0.0682848
$932$	0	0
$933$	0	0
$934$	0	0
$935$	51.1516	1.67284
$936$	0	0
$937$	−23.9371	−0.781991	−0.390995	−	0.920393i	$-0.627869\pi$
$937$	−23.9371	−0.781991	−0.390995	+	0.920393i	$0.627869\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−9.56414	−0.311782	−0.155891	−	0.987774i	$-0.549825\pi$
$941$	−9.56414	−0.311782	−0.155891	+	0.987774i	$0.549825\pi$
$942$	0	0
$943$	22.9991i	0.748953i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 35.0824i	− 1.14003i	−0.821636	−	0.570013i	$-0.806938\pi$
$947$	− 35.0824i	− 1.14003i	0.821636	−	0.570013i	$-0.193062\pi$
$948$	0	0
$949$	− 11.2194i	− 0.364199i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 14.3190i	− 0.463837i	−0.972735	−	0.231919i	$-0.925500\pi$
$953$	− 14.3190i	− 0.463837i	0.972735	−	0.231919i	$-0.0745003\pi$
$954$	0	0
$955$	−29.1512	−0.943311
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.0099	−0.646153
$960$	0	0
$961$	20.3173	0.655398
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.45720	−0.143482
$966$	0	0
$967$	− 48.7056i	− 1.56627i	−0.621855	−	0.783133i	$-0.713621\pi$
$967$	− 48.7056i	− 1.56627i	0.621855	−	0.783133i	$-0.286379\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.39723i	0.0769308i	0.999260	+	0.0384654i	$0.0122469\pi$
$971$	2.39723i	0.0769308i	−0.999260	+	0.0384654i	$0.987753\pi$
$972$	0	0
$973$	− 11.0306i	− 0.353626i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.33844i	0.202785i	0.994847	+	0.101392i	$0.0323297\pi$
$977$	6.33844i	0.202785i	−0.994847	+	0.101392i	$0.967670\pi$
$978$	0	0
$979$	−26.7134	−0.853764
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13.6385	0.434999	0.217500	−	0.976060i	$-0.430210\pi$
$983$	13.6385	0.434999	0.217500	+	0.976060i	$0.430210\pi$
$984$	0	0
$985$	−27.9995	−0.892138
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−75.3660	−2.39650
$990$	0	0
$991$	24.8997i	0.790964i	0.918474	+	0.395482i	$0.129422\pi$
$991$	24.8997i	0.790964i	−0.918474	+	0.395482i	$0.870578\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	29.4591i	0.933915i
$996$	0	0
$997$	− 25.3869i	− 0.804011i	−0.915637	−	0.402005i	$-0.868313\pi$
$997$	− 25.3869i	− 0.804011i	0.915637	−	0.402005i	$-0.131687\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.d.5615.18		48
3.2	odd	2	inner	6048.2.j.d.5615.31		48
4.3	odd	2		1512.2.j.d.323.8	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.32		48
8.5	even	2		1512.2.j.d.323.42	yes	48
12.11	even	2		1512.2.j.d.323.41	yes	48
24.5	odd	2		1512.2.j.d.323.7	✓	48
24.11	even	2	inner	6048.2.j.d.5615.17		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.7	✓	48	24.5	odd	2
1512.2.j.d.323.8	yes	48	4.3	odd	2
1512.2.j.d.323.41	yes	48	12.11	even	2
1512.2.j.d.323.42	yes	48	8.5	even	2
6048.2.j.d.5615.17		48	24.11	even	2	inner
6048.2.j.d.5615.18		48	1.1	even	1	trivial
6048.2.j.d.5615.31		48	3.2	odd	2	inner
6048.2.j.d.5615.32		48	8.3	odd	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.27600 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-1.27600 q^{5} -1.00000i q^{7} -6.57401i q^{11} -4.05196i q^{13} -6.09788i q^{17} +2.08352 q^{19} +6.85760 q^{23} -3.37182 q^{25} +6.53304 q^{29} -3.26844i q^{31} +1.27600i q^{35} -2.95562i q^{37} +3.35381i q^{41} -10.9901 q^{43} +7.12041 q^{47} -1.00000 q^{49} -2.87278 q^{53} +8.38843i q^{55} +7.75213i q^{59} -12.0251i q^{61} +5.17030i q^{65} +3.01966 q^{67} +3.48158 q^{71} +2.76889 q^{73} -6.57401 q^{77} +0.849261i q^{79} +15.8068i q^{83} +7.78089i q^{85} -4.06349i q^{89} -4.05196 q^{91} -2.65858 q^{95} -4.90438 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) \) 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

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