LMFDB - Embedded newform 6048.2.j.d.5615.39

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			5615.39
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.40

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85878 q^{5} +1.00000i q^{7} +O(q^{10})$	$q+2.85878 q^{5} +1.00000i q^{7} +4.98479i q^{11} -2.35304i q^{13} +8.19706i q^{17} -5.90589 q^{19} -4.43848 q^{23} +3.17265 q^{25} +6.41440 q^{29} -3.31678i q^{31} +2.85878i q^{35} +5.51080i q^{37} -3.41549i q^{41} -1.37638 q^{43} -2.60102 q^{47} -1.00000 q^{49} -13.3184 q^{53} +14.2505i q^{55} +10.6485i q^{59} +1.41211i q^{61} -6.72683i q^{65} -0.221252 q^{67} +0.398786 q^{71} -10.7893 q^{73} -4.98479 q^{77} -9.76133i q^{79} -2.06831i q^{83} +23.4336i q^{85} -17.6684i q^{89} +2.35304 q^{91} -16.8837 q^{95} -3.62651 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.85878	1.27849	0.639244	−	0.769004i	$-0.279247\pi$
$5$	2.85878	1.27849	0.639244	+	0.769004i	$0.279247\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.98479i	1.50297i	0.659749	+	0.751486i	$0.270663\pi$
$11$	4.98479i	1.50297i	−0.659749	+	0.751486i	$0.729337\pi$
$12$	0	0
$13$	− 2.35304i	− 0.652615i	−0.945264	−	0.326308i	$-0.894195\pi$
$13$	− 2.35304i	− 0.652615i	0.945264	−	0.326308i	$-0.105805\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.19706i	1.98808i	0.109021	+	0.994039i	$0.465228\pi$
$17$	8.19706i	1.98808i	−0.109021	+	0.994039i	$0.534772\pi$
$18$	0	0
$19$	−5.90589	−1.35491	−0.677453	−	0.735566i	$-0.736916\pi$
$19$	−5.90589	−1.35491	−0.677453	+	0.735566i	$0.736916\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.43848	−0.925487	−0.462744	−	0.886492i	$-0.653135\pi$
$23$	−4.43848	−0.925487	−0.462744	+	0.886492i	$0.653135\pi$
$24$	0	0
$25$	3.17265	0.634530
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.41440	1.19112	0.595562	−	0.803309i	$-0.296929\pi$
$29$	6.41440	1.19112	0.595562	+	0.803309i	$0.296929\pi$
$30$	0	0
$31$	− 3.31678i	− 0.595710i	−0.954611	−	0.297855i	$-0.903729\pi$
$31$	− 3.31678i	− 0.595710i	0.954611	−	0.297855i	$-0.0962713\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.85878i	0.483223i
$36$	0	0
$37$	5.51080i	0.905970i	0.891518	+	0.452985i	$0.149641\pi$
$37$	5.51080i	0.905970i	−0.891518	+	0.452985i	$0.850359\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.41549i	− 0.533410i	−0.963778	−	0.266705i	$-0.914065\pi$
$41$	− 3.41549i	− 0.533410i	0.963778	−	0.266705i	$-0.0859349\pi$
$42$	0	0
$43$	−1.37638	−0.209895	−0.104948	−	0.994478i	$-0.533468\pi$
$43$	−1.37638	−0.209895	−0.104948	+	0.994478i	$0.533468\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.60102	−0.379398	−0.189699	−	0.981842i	$-0.560751\pi$
$47$	−2.60102	−0.379398	−0.189699	+	0.981842i	$0.560751\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.3184	−1.82942	−0.914710	−	0.404110i	$-0.867581\pi$
$53$	−13.3184	−1.82942	−0.914710	+	0.404110i	$0.867581\pi$
$54$	0	0
$55$	14.2505i	1.92153i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.6485i	1.38631i	0.720788	+	0.693156i	$0.243780\pi$
$59$	10.6485i	1.38631i	−0.720788	+	0.693156i	$0.756220\pi$
$60$	0	0
$61$	1.41211i	0.180802i	0.995905	+	0.0904011i	$0.0288149\pi$
$61$	1.41211i	0.180802i	−0.995905	+	0.0904011i	$0.971185\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.72683i	− 0.834361i
$66$	0	0
$67$	−0.221252	−0.0270303	−0.0135151	−	0.999909i	$-0.504302\pi$
$67$	−0.221252	−0.0270303	−0.0135151	+	0.999909i	$0.504302\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.398786	0.0473271	0.0236636	−	0.999720i	$-0.492467\pi$
$71$	0.398786	0.0473271	0.0236636	+	0.999720i	$0.492467\pi$
$72$	0	0
$73$	−10.7893	−1.26279	−0.631397	−	0.775460i	$-0.717518\pi$
$73$	−10.7893	−1.26279	−0.631397	+	0.775460i	$0.717518\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.98479	−0.568070
$78$	0	0
$79$	− 9.76133i	− 1.09824i	−0.835745	−	0.549118i	$-0.814964\pi$
$79$	− 9.76133i	− 1.09824i	0.835745	−	0.549118i	$-0.185036\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.06831i	− 0.227027i	−0.993536	−	0.113513i	$-0.963789\pi$
$83$	− 2.06831i	− 0.227027i	0.993536	−	0.113513i	$-0.0362105\pi$
$84$	0	0
$85$	23.4336i	2.54173i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 17.6684i	− 1.87284i	−0.350875	−	0.936422i	$-0.614116\pi$
$89$	− 17.6684i	− 1.87284i	0.350875	−	0.936422i	$-0.385884\pi$
$90$	0	0
$91$	2.35304	0.246665
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.8837	−1.73223
$96$	0	0
$97$	−3.62651	−0.368217	−0.184108	−	0.982906i	$-0.558940\pi$
$97$	−3.62651	−0.368217	−0.184108	+	0.982906i	$0.558940\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.4729	−1.04209	−0.521047	−	0.853528i	$-0.674459\pi$
$101$	−10.4729	−1.04209	−0.521047	+	0.853528i	$0.674459\pi$
$102$	0	0
$103$	− 4.50892i	− 0.444277i	−0.975015	−	0.222139i	$-0.928696\pi$
$103$	− 4.50892i	− 0.444277i	0.975015	−	0.222139i	$-0.0713037\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.82640i	0.659933i	0.943993	+	0.329966i	$0.107037\pi$
$107$	6.82640i	0.659933i	−0.943993	+	0.329966i	$0.892963\pi$
$108$	0	0
$109$	13.1273i	1.25737i	0.777660	+	0.628685i	$0.216407\pi$
$109$	13.1273i	1.25737i	−0.777660	+	0.628685i	$0.783593\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5.30759i	− 0.499297i	−0.968337	−	0.249648i	$-0.919685\pi$
$113$	− 5.30759i	− 0.499297i	0.968337	−	0.249648i	$-0.0803150\pi$
$114$	0	0
$115$	−12.6887	−1.18322
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.19706	−0.751423
$120$	0	0
$121$	−13.8482	−1.25892
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.22400	−0.467248
$126$	0	0
$127$	− 1.80347i	− 0.160032i	−0.996794	−	0.0800160i	$-0.974503\pi$
$127$	− 1.80347i	− 0.160032i	0.996794	−	0.0800160i	$-0.0254972\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.93535i	0.868056i	0.900899	+	0.434028i	$0.142908\pi$
$131$	9.93535i	0.868056i	−0.900899	+	0.434028i	$0.857092\pi$
$132$	0	0
$133$	− 5.90589i	− 0.512106i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.16258i	− 0.0993261i	−0.998766	−	0.0496630i	$-0.984185\pi$
$137$	− 1.16258i	− 0.0993261i	0.998766	−	0.0496630i	$-0.0158147\pi$
$138$	0	0
$139$	−17.0996	−1.45037	−0.725186	−	0.688553i	$-0.758246\pi$
$139$	−17.0996	−1.45037	−0.725186	+	0.688553i	$0.758246\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.7294	0.980863
$144$	0	0
$145$	18.3374	1.52284
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.1689	1.89807	0.949034	−	0.315173i	$-0.102063\pi$
$149$	23.1689	1.89807	0.949034	+	0.315173i	$0.102063\pi$
$150$	0	0
$151$	1.67795i	0.136549i	0.997667	+	0.0682747i	$0.0217494\pi$
$151$	1.67795i	0.136549i	−0.997667	+	0.0682747i	$0.978251\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9.48195i	− 0.761608i
$156$	0	0
$157$	− 2.22459i	− 0.177542i	−0.996052	−	0.0887710i	$-0.971706\pi$
$157$	− 2.22459i	− 0.177542i	0.996052	−	0.0887710i	$-0.0282939\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 4.43848i	− 0.349801i
$162$	0	0
$163$	17.3358	1.35785	0.678924	−	0.734209i	$-0.262447\pi$
$163$	17.3358	1.35785	0.678924	+	0.734209i	$0.262447\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.4429	1.65930	0.829650	−	0.558284i	$-0.188540\pi$
$167$	21.4429	1.65930	0.829650	+	0.558284i	$0.188540\pi$
$168$	0	0
$169$	7.46321	0.574093
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.70545	0.737892	0.368946	−	0.929451i	$-0.379719\pi$
$173$	9.70545	0.737892	0.368946	+	0.929451i	$0.379719\pi$
$174$	0	0
$175$	3.17265i	0.239830i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 14.7350i	− 1.10134i	−0.834722	−	0.550671i	$-0.814372\pi$
$179$	− 14.7350i	− 1.10134i	0.834722	−	0.550671i	$-0.185628\pi$
$180$	0	0
$181$	17.7055i	1.31604i	0.753001	+	0.658019i	$0.228605\pi$
$181$	17.7055i	1.31604i	−0.753001	+	0.658019i	$0.771395\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15.7542i	1.15827i
$186$	0	0
$187$	−40.8606	−2.98803
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.9941	1.30201	0.651003	−	0.759075i	$-0.274348\pi$
$191$	17.9941	1.30201	0.651003	+	0.759075i	$0.274348\pi$
$192$	0	0
$193$	−24.8594	−1.78942	−0.894709	−	0.446649i	$-0.852617\pi$
$193$	−24.8594	−1.78942	−0.894709	+	0.446649i	$0.852617\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.4145	−1.24073	−0.620367	−	0.784312i	$-0.713016\pi$
$197$	−17.4145	−1.24073	−0.620367	+	0.784312i	$0.713016\pi$
$198$	0	0
$199$	20.0065i	1.41823i	0.705095	+	0.709113i	$0.250904\pi$
$199$	20.0065i	1.41823i	−0.705095	+	0.709113i	$0.749096\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.41440i	0.450203i
$204$	0	0
$205$	− 9.76415i	− 0.681958i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 29.4397i	− 2.03638i
$210$	0	0
$211$	15.0542	1.03637	0.518186	−	0.855268i	$-0.326608\pi$
$211$	15.0542	1.03637	0.518186	+	0.855268i	$0.326608\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.93476	−0.268349
$216$	0	0
$217$	3.31678	0.225157
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.2880	1.29745
$222$	0	0
$223$	− 12.7615i	− 0.854570i	−0.904117	−	0.427285i	$-0.859470\pi$
$223$	− 12.7615i	− 0.854570i	0.904117	−	0.427285i	$-0.140530\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.7167i	1.70688i	0.521194	+	0.853438i	$0.325487\pi$
$227$	25.7167i	1.70688i	−0.521194	+	0.853438i	$0.674513\pi$
$228$	0	0
$229$	− 10.7033i	− 0.707293i	−0.935379	−	0.353647i	$-0.884942\pi$
$229$	− 10.7033i	− 0.707293i	0.935379	−	0.353647i	$-0.115058\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5.55917i	− 0.364194i	−0.983281	−	0.182097i	$-0.941712\pi$
$233$	− 5.55917i	− 0.364194i	0.983281	−	0.182097i	$-0.0582884\pi$
$234$	0	0
$235$	−7.43577	−0.485056
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.7187	1.34018	0.670092	−	0.742278i	$-0.266255\pi$
$239$	20.7187	1.34018	0.670092	+	0.742278i	$0.266255\pi$
$240$	0	0
$241$	6.03091	0.388485	0.194242	−	0.980954i	$-0.437775\pi$
$241$	6.03091	0.388485	0.194242	+	0.980954i	$0.437775\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.85878	−0.182641
$246$	0	0
$247$	13.8968i	0.884232i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.01117i	0.505660i	0.967511	+	0.252830i	$0.0813614\pi$
$251$	8.01117i	0.505660i	−0.967511	+	0.252830i	$0.918639\pi$
$252$	0	0
$253$	− 22.1249i	− 1.39098i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.1009i	0.879590i	0.898098	+	0.439795i	$0.144949\pi$
$257$	14.1009i	0.879590i	−0.898098	+	0.439795i	$0.855051\pi$
$258$	0	0
$259$	−5.51080	−0.342424
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.6348	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$263$	−24.6348	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$264$	0	0
$265$	−38.0744	−2.33889
$266$	0	0
$267$	0	0
$268$	0	0
$269$	9.64498	0.588065	0.294032	−	0.955795i	$-0.405003\pi$
$269$	9.64498	0.588065	0.294032	+	0.955795i	$0.405003\pi$
$270$	0	0
$271$	13.3466i	0.810751i	0.914150	+	0.405375i	$0.132859\pi$
$271$	13.3466i	0.810751i	−0.914150	+	0.405375i	$0.867141\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.8150i	0.953681i
$276$	0	0
$277$	18.5217i	1.11286i	0.830895	+	0.556429i	$0.187829\pi$
$277$	18.5217i	1.11286i	−0.830895	+	0.556429i	$0.812171\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 18.7455i	− 1.11826i	−0.829079	−	0.559131i	$-0.811135\pi$
$281$	− 18.7455i	− 1.11826i	0.829079	−	0.559131i	$-0.188865\pi$
$282$	0	0
$283$	2.32063	0.137947	0.0689736	−	0.997618i	$-0.478028\pi$
$283$	2.32063	0.137947	0.0689736	+	0.997618i	$0.478028\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.41549	0.201610
$288$	0	0
$289$	−50.1918	−2.95246
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.61125	0.210972	0.105486	−	0.994421i	$-0.466360\pi$
$293$	3.61125	0.210972	0.105486	+	0.994421i	$0.466360\pi$
$294$	0	0
$295$	30.4417i	1.77238i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.4439i	0.603987i
$300$	0	0
$301$	− 1.37638i	− 0.0793330i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.03692i	0.231153i
$306$	0	0
$307$	19.9273	1.13731	0.568656	−	0.822576i	$-0.307464\pi$
$307$	19.9273	1.13731	0.568656	+	0.822576i	$0.307464\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.68722	0.265788	0.132894	−	0.991130i	$-0.457573\pi$
$311$	4.68722	0.265788	0.132894	+	0.991130i	$0.457573\pi$
$312$	0	0
$313$	−13.2179	−0.747118	−0.373559	−	0.927606i	$-0.621863\pi$
$313$	−13.2179	−0.747118	−0.373559	+	0.927606i	$0.621863\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.8162	0.775995	0.387997	−	0.921660i	$-0.373167\pi$
$317$	13.8162	0.775995	0.387997	+	0.921660i	$0.373167\pi$
$318$	0	0
$319$	31.9745i	1.79023i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 48.4110i	− 2.69366i
$324$	0	0
$325$	− 7.46537i	− 0.414104i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 2.60102i	− 0.143399i
$330$	0	0
$331$	−9.70429	−0.533396	−0.266698	−	0.963780i	$-0.585933\pi$
$331$	−9.70429	−0.533396	−0.266698	+	0.963780i	$0.585933\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.632513	−0.0345579
$336$	0	0
$337$	5.04484	0.274810	0.137405	−	0.990515i	$-0.456124\pi$
$337$	5.04484	0.274810	0.137405	+	0.990515i	$0.456124\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.5334	0.895336
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.01718i	− 0.108288i	−0.998533	−	0.0541440i	$-0.982757\pi$
$347$	− 2.01718i	− 0.108288i	0.998533	−	0.0541440i	$-0.0172430\pi$
$348$	0	0
$349$	− 0.528557i	− 0.0282930i	−0.999900	−	0.0141465i	$-0.995497\pi$
$349$	− 0.528557i	− 0.0282930i	0.999900	−	0.0141465i	$-0.00450312\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3448i	0.550600i	0.961358	+	0.275300i	$0.0887772\pi$
$353$	10.3448i	0.550600i	−0.961358	+	0.275300i	$0.911223\pi$
$354$	0	0
$355$	1.14004	0.0605072
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.62306	0.0856618	0.0428309	−	0.999082i	$-0.486362\pi$
$359$	1.62306	0.0856618	0.0428309	+	0.999082i	$0.486362\pi$
$360$	0	0
$361$	15.8796	0.835768
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−30.8443	−1.61447
$366$	0	0
$367$	33.5635i	1.75200i	0.482312	+	0.875999i	$0.339797\pi$
$367$	33.5635i	1.75200i	−0.482312	+	0.875999i	$0.660203\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 13.3184i	− 0.691456i
$372$	0	0
$373$	20.4199i	1.05730i	0.848840	+	0.528650i	$0.177302\pi$
$373$	20.4199i	1.05730i	−0.848840	+	0.528650i	$0.822698\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.0933i	− 0.777346i
$378$	0	0
$379$	−29.6529	−1.52317	−0.761583	−	0.648068i	$-0.775577\pi$
$379$	−29.6529	−1.52317	−0.761583	+	0.648068i	$0.775577\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.82347	0.246468	0.123234	−	0.992378i	$-0.460673\pi$
$383$	4.82347	0.246468	0.123234	+	0.992378i	$0.460673\pi$
$384$	0	0
$385$	−14.2505	−0.726270
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.5242	0.939214	0.469607	−	0.882876i	$-0.344396\pi$
$389$	18.5242	0.939214	0.469607	+	0.882876i	$0.344396\pi$
$390$	0	0
$391$	− 36.3825i	− 1.83994i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 27.9056i	− 1.40408i
$396$	0	0
$397$	2.91585i	0.146342i	0.997319	+	0.0731712i	$0.0233119\pi$
$397$	2.91585i	0.146342i	−0.997319	+	0.0731712i	$0.976688\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2.62856i	− 0.131264i	−0.997844	−	0.0656319i	$-0.979094\pi$
$401$	− 2.62856i	− 0.131264i	0.997844	−	0.0656319i	$-0.0209063\pi$
$402$	0	0
$403$	−7.80450	−0.388770
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−27.4702	−1.36165
$408$	0	0
$409$	26.0556	1.28836	0.644182	−	0.764872i	$-0.277198\pi$
$409$	26.0556	1.28836	0.644182	+	0.764872i	$0.277198\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.6485	−0.523977
$414$	0	0
$415$	− 5.91286i	− 0.290251i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 9.43185i	− 0.460776i	−0.973099	−	0.230388i	$-0.926000\pi$
$419$	− 9.43185i	− 0.460776i	0.973099	−	0.230388i	$-0.0739995\pi$
$420$	0	0
$421$	22.3767i	1.09057i	0.838249	+	0.545287i	$0.183579\pi$
$421$	22.3767i	1.09057i	−0.838249	+	0.545287i	$0.816421\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	26.0064i	1.26150i
$426$	0	0
$427$	−1.41211	−0.0683368
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.1600	−0.682061	−0.341030	−	0.940052i	$-0.610776\pi$
$431$	−14.1600	−0.682061	−0.341030	+	0.940052i	$0.610776\pi$
$432$	0	0
$433$	9.47617	0.455396	0.227698	−	0.973732i	$-0.426880\pi$
$433$	9.47617	0.455396	0.227698	+	0.973732i	$0.426880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.2132	1.25395
$438$	0	0
$439$	5.40493i	0.257963i	0.991647	+	0.128982i	$0.0411709\pi$
$439$	5.40493i	0.257963i	−0.991647	+	0.128982i	$0.958829\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 28.2417i	− 1.34180i	−0.741546	−	0.670902i	$-0.765907\pi$
$443$	− 28.2417i	− 1.34180i	0.741546	−	0.670902i	$-0.234093\pi$
$444$	0	0
$445$	− 50.5101i	− 2.39441i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.0885i	1.46716i	0.679605	+	0.733578i	$0.262151\pi$
$449$	31.0885i	1.46716i	−0.679605	+	0.733578i	$0.737849\pi$
$450$	0	0
$451$	17.0255	0.801700
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.72683	0.315359
$456$	0	0
$457$	15.4214	0.721383	0.360691	−	0.932685i	$-0.382541\pi$
$457$	15.4214	0.721383	0.360691	+	0.932685i	$0.382541\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.7699	−1.01392	−0.506962	−	0.861968i	$-0.669232\pi$
$461$	−21.7699	−1.01392	−0.506962	+	0.861968i	$0.669232\pi$
$462$	0	0
$463$	33.1540i	1.54080i	0.637563	+	0.770398i	$0.279943\pi$
$463$	33.1540i	1.54080i	−0.637563	+	0.770398i	$0.720057\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.3233i	1.07927i	0.841898	+	0.539637i	$0.181439\pi$
$467$	23.3233i	1.07927i	−0.841898	+	0.539637i	$0.818561\pi$
$468$	0	0
$469$	− 0.221252i	− 0.0102165i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6.86095i	− 0.315467i
$474$	0	0
$475$	−18.7373	−0.859728
$476$	0	0
$477$	0	0
$478$	0	0
$479$	29.7837	1.36085	0.680426	−	0.732817i	$-0.261795\pi$
$479$	29.7837	1.36085	0.680426	+	0.732817i	$0.261795\pi$
$480$	0	0
$481$	12.9671	0.591250
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.3674	−0.470761
$486$	0	0
$487$	− 7.61251i	− 0.344956i	−0.985013	−	0.172478i	$-0.944823\pi$
$487$	− 7.61251i	− 0.344956i	0.985013	−	0.172478i	$-0.0551773\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.4249i	0.470471i	0.971938	+	0.235236i	$0.0755862\pi$
$491$	10.4249i	0.470471i	−0.971938	+	0.235236i	$0.924414\pi$
$492$	0	0
$493$	52.5792i	2.36805i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.398786i	0.0178880i
$498$	0	0
$499$	24.3685	1.09088	0.545441	−	0.838149i	$-0.316362\pi$
$499$	24.3685	1.09088	0.545441	+	0.838149i	$0.316362\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−38.3668	−1.71069	−0.855345	−	0.518058i	$-0.826655\pi$
$503$	−38.3668	−1.71069	−0.855345	+	0.518058i	$0.826655\pi$
$504$	0	0
$505$	−29.9398	−1.33231
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.6708	−0.517298	−0.258649	−	0.965971i	$-0.583277\pi$
$509$	−11.6708	−0.517298	−0.258649	+	0.965971i	$0.583277\pi$
$510$	0	0
$511$	− 10.7893i	− 0.477291i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 12.8900i	− 0.568003i
$516$	0	0
$517$	− 12.9656i	− 0.570225i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.09319i	− 0.354569i	−0.984160	−	0.177284i	$-0.943269\pi$
$521$	− 8.09319i	− 0.354569i	0.984160	−	0.177284i	$-0.0567313\pi$
$522$	0	0
$523$	12.5864	0.550363	0.275181	−	0.961392i	$-0.411262\pi$
$523$	12.5864	0.550363	0.275181	+	0.961392i	$0.411262\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.1878	1.18432
$528$	0	0
$529$	−3.29989	−0.143473
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.03678	−0.348112
$534$	0	0
$535$	19.5152i	0.843716i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.98479i	− 0.214710i
$540$	0	0
$541$	33.6824i	1.44812i	0.689738	+	0.724059i	$0.257726\pi$
$541$	33.6824i	1.44812i	−0.689738	+	0.724059i	$0.742274\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	37.5282i	1.60753i
$546$	0	0
$547$	31.7087	1.35576	0.677882	−	0.735170i	$-0.262898\pi$
$547$	31.7087	1.35576	0.677882	+	0.735170i	$0.262898\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−37.8828	−1.61386
$552$	0	0
$553$	9.76133	0.415094
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.0418	1.23054	0.615270	−	0.788316i	$-0.289047\pi$
$557$	29.0418	1.23054	0.615270	+	0.788316i	$0.289047\pi$
$558$	0	0
$559$	3.23867i	0.136981i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.5955i	1.45803i	0.684499	+	0.729013i	$0.260021\pi$
$563$	34.5955i	1.45803i	−0.684499	+	0.729013i	$0.739979\pi$
$564$	0	0
$565$	− 15.1733i	− 0.638344i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.7913i	1.87775i	0.344258	+	0.938875i	$0.388131\pi$
$569$	44.7913i	1.87775i	−0.344258	+	0.938875i	$0.611869\pi$
$570$	0	0
$571$	13.8182	0.578276	0.289138	−	0.957287i	$-0.406631\pi$
$571$	13.8182	0.578276	0.289138	+	0.957287i	$0.406631\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−14.0818	−0.587250
$576$	0	0
$577$	−2.53927	−0.105711	−0.0528557	−	0.998602i	$-0.516832\pi$
$577$	−2.53927	−0.105711	−0.0528557	+	0.998602i	$0.516832\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.06831	0.0858081
$582$	0	0
$583$	− 66.3894i	− 2.74957i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.23636i	0.0510300i	0.999674	+	0.0255150i	$0.00812256\pi$
$587$	1.23636i	0.0510300i	−0.999674	+	0.0255150i	$0.991877\pi$
$588$	0	0
$589$	19.5885i	0.807131i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 2.47202i	− 0.101514i	−0.998711	−	0.0507568i	$-0.983837\pi$
$593$	− 2.47202i	− 0.101514i	0.998711	−	0.0507568i	$-0.0161633\pi$
$594$	0	0
$595$	−23.4336	−0.960685
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.80798	−0.278166	−0.139083	−	0.990281i	$-0.544416\pi$
$599$	−6.80798	−0.278166	−0.139083	+	0.990281i	$0.544416\pi$
$600$	0	0
$601$	−0.857050	−0.0349598	−0.0174799	−	0.999847i	$-0.505564\pi$
$601$	−0.857050	−0.0349598	−0.0174799	+	0.999847i	$0.505564\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−39.5889	−1.60952
$606$	0	0
$607$	− 22.9929i	− 0.933252i	−0.884455	−	0.466626i	$-0.845469\pi$
$607$	− 22.9929i	− 0.933252i	0.884455	−	0.466626i	$-0.154531\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.12031i	0.247601i
$612$	0	0
$613$	− 10.9605i	− 0.442691i	−0.975195	−	0.221346i	$-0.928955\pi$
$613$	− 10.9605i	− 0.442691i	0.975195	−	0.221346i	$-0.0710449\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.9140i	0.640675i	0.947304	+	0.320337i	$0.103796\pi$
$617$	15.9140i	0.640675i	−0.947304	+	0.320337i	$0.896204\pi$
$618$	0	0
$619$	−24.1115	−0.969124	−0.484562	−	0.874757i	$-0.661021\pi$
$619$	−24.1115	−0.969124	−0.484562	+	0.874757i	$0.661021\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	17.6684	0.707869
$624$	0	0
$625$	−30.7975	−1.23190
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−45.1723	−1.80114
$630$	0	0
$631$	− 19.0647i	− 0.758955i	−0.925201	−	0.379477i	$-0.876104\pi$
$631$	− 19.0647i	− 0.758955i	0.925201	−	0.379477i	$-0.123896\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 5.15573i	− 0.204599i
$636$	0	0
$637$	2.35304i	0.0932308i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 16.6996i	− 0.659595i	−0.944052	−	0.329797i	$-0.893020\pi$
$641$	− 16.6996i	− 0.659595i	0.944052	−	0.329797i	$-0.106980\pi$
$642$	0	0
$643$	−15.2897	−0.602966	−0.301483	−	0.953472i	$-0.597482\pi$
$643$	−15.2897	−0.602966	−0.301483	+	0.953472i	$0.597482\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.6300	1.67596	0.837979	−	0.545702i	$-0.183737\pi$
$647$	42.6300	1.67596	0.837979	+	0.545702i	$0.183737\pi$
$648$	0	0
$649$	−53.0804	−2.08359
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.14973	−0.201524	−0.100762	−	0.994911i	$-0.532128\pi$
$653$	−5.14973	−0.201524	−0.100762	+	0.994911i	$0.532128\pi$
$654$	0	0
$655$	28.4030i	1.10980i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 0.965465i	− 0.0376092i	−0.999823	−	0.0188046i	$-0.994014\pi$
$659$	− 0.965465i	− 0.0376092i	0.999823	−	0.0188046i	$-0.00598604\pi$
$660$	0	0
$661$	− 8.54386i	− 0.332318i	−0.986099	−	0.166159i	$-0.946864\pi$
$661$	− 8.54386i	− 0.332318i	0.986099	−	0.166159i	$-0.0531365\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 16.8837i	− 0.654721i
$666$	0	0
$667$	−28.4702	−1.10237
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.03908	−0.271741
$672$	0	0
$673$	35.8364	1.38139	0.690695	−	0.723146i	$-0.257305\pi$
$673$	35.8364	1.38139	0.690695	+	0.723146i	$0.257305\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.3770	−0.590985	−0.295493	−	0.955345i	$-0.595484\pi$
$677$	−15.3770	−0.590985	−0.295493	+	0.955345i	$0.595484\pi$
$678$	0	0
$679$	− 3.62651i	− 0.139173i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 31.3038i	− 1.19781i	−0.800822	−	0.598903i	$-0.795604\pi$
$683$	− 31.3038i	− 1.19781i	0.800822	−	0.598903i	$-0.204396\pi$
$684$	0	0
$685$	− 3.32357i	− 0.126987i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	31.3387i	1.19391i
$690$	0	0
$691$	−38.1921	−1.45289	−0.726447	−	0.687222i	$-0.758830\pi$
$691$	−38.1921	−1.45289	−0.726447	+	0.687222i	$0.758830\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−48.8842	−1.85428
$696$	0	0
$697$	27.9970	1.06046
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−45.5040	−1.71866	−0.859331	−	0.511419i	$-0.829120\pi$
$701$	−45.5040	−1.71866	−0.859331	+	0.511419i	$0.829120\pi$
$702$	0	0
$703$	− 32.5462i	− 1.22750i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.4729i	− 0.393875i
$708$	0	0
$709$	− 17.7256i	− 0.665698i	−0.942980	−	0.332849i	$-0.891990\pi$
$709$	− 17.7256i	− 0.665698i	0.942980	−	0.332849i	$-0.108010\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.7214i	0.551322i
$714$	0	0
$715$	33.5319	1.25402
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.0561	−1.15820	−0.579098	−	0.815258i	$-0.696595\pi$
$719$	−31.0561	−1.15820	−0.579098	+	0.815258i	$0.696595\pi$
$720$	0	0
$721$	4.50892	0.167921
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20.3507	0.755804
$726$	0	0
$727$	7.88096i	0.292289i	0.989263	+	0.146144i	$0.0466864\pi$
$727$	7.88096i	0.292289i	−0.989263	+	0.146144i	$0.953314\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 11.2822i	− 0.417288i
$732$	0	0
$733$	31.2981i	1.15602i	0.816029	+	0.578011i	$0.196171\pi$
$733$	31.2981i	1.15602i	−0.816029	+	0.578011i	$0.803829\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.10290i	− 0.0406257i
$738$	0	0
$739$	−9.91989	−0.364909	−0.182455	−	0.983214i	$-0.558404\pi$
$739$	−9.91989	−0.364909	−0.182455	+	0.983214i	$0.558404\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	21.2085	0.778065	0.389032	−	0.921224i	$-0.372809\pi$
$743$	21.2085	0.778065	0.389032	+	0.921224i	$0.372809\pi$
$744$	0	0
$745$	66.2348	2.42666
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6.82640	−0.249431
$750$	0	0
$751$	31.2986i	1.14210i	0.820915	+	0.571051i	$0.193464\pi$
$751$	31.2986i	1.14210i	−0.820915	+	0.571051i	$0.806536\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.79689i	0.174577i
$756$	0	0
$757$	50.1185i	1.82159i	0.412863	+	0.910793i	$0.364529\pi$
$757$	50.1185i	1.82159i	−0.412863	+	0.910793i	$0.635471\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.4522i	1.21264i	0.795220	+	0.606321i	$0.207355\pi$
$761$	33.4522i	1.21264i	−0.795220	+	0.606321i	$0.792645\pi$
$762$	0	0
$763$	−13.1273	−0.475241
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.0562	0.904728
$768$	0	0
$769$	−34.8586	−1.25703	−0.628516	−	0.777797i	$-0.716337\pi$
$769$	−34.8586	−1.25703	−0.628516	+	0.777797i	$0.716337\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.819156	0.0294630	0.0147315	−	0.999891i	$-0.495311\pi$
$773$	0.819156	0.0294630	0.0147315	+	0.999891i	$0.495311\pi$
$774$	0	0
$775$	− 10.5230i	− 0.377996i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.1715i	0.722720i
$780$	0	0
$781$	1.98786i	0.0711314i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 6.35964i	− 0.226985i
$786$	0	0
$787$	−21.1948	−0.755514	−0.377757	−	0.925905i	$-0.623305\pi$
$787$	−21.1948	−0.755514	−0.377757	+	0.925905i	$0.623305\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.30759	0.188716
$792$	0	0
$793$	3.32275	0.117994
$794$	0	0
$795$	0	0
$796$	0	0
$797$	10.4236	0.369221	0.184611	−	0.982812i	$-0.440898\pi$
$797$	10.4236	0.369221	0.184611	+	0.982812i	$0.440898\pi$
$798$	0	0
$799$	− 21.3207i	− 0.754274i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 53.7825i	− 1.89794i
$804$	0	0
$805$	− 12.6887i	− 0.447217i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 6.79289i	− 0.238825i	−0.992845	−	0.119413i	$-0.961899\pi$
$809$	− 6.79289i	− 0.238825i	0.992845	−	0.119413i	$-0.0381012\pi$
$810$	0	0
$811$	−15.6600	−0.549897	−0.274949	−	0.961459i	$-0.588661\pi$
$811$	−15.6600	−0.549897	−0.274949	+	0.961459i	$0.588661\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	49.5594	1.73599
$816$	0	0
$817$	8.12873	0.284388
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.83489	0.343240	0.171620	−	0.985163i	$-0.445100\pi$
$821$	9.83489	0.343240	0.171620	+	0.985163i	$0.445100\pi$
$822$	0	0
$823$	− 16.9492i	− 0.590811i	−0.955372	−	0.295406i	$-0.904545\pi$
$823$	− 16.9492i	− 0.590811i	0.955372	−	0.295406i	$-0.0954548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.3499i	1.88993i	0.327169	+	0.944966i	$0.393905\pi$
$827$	54.3499i	1.88993i	−0.327169	+	0.944966i	$0.606095\pi$
$828$	0	0
$829$	− 0.679562i	− 0.0236022i	−0.999930	−	0.0118011i	$-0.996244\pi$
$829$	− 0.679562i	− 0.0236022i	0.999930	−	0.0118011i	$-0.00375649\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 8.19706i	− 0.284011i
$834$	0	0
$835$	61.3006	2.12139
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46.0696	1.59050	0.795249	−	0.606283i	$-0.207340\pi$
$839$	46.0696	1.59050	0.795249	+	0.606283i	$0.207340\pi$
$840$	0	0
$841$	12.1445	0.418777
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21.3357	0.733971
$846$	0	0
$847$	− 13.8482i	− 0.475828i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 24.4596i	− 0.838463i
$852$	0	0
$853$	− 35.6910i	− 1.22204i	−0.791617	−	0.611018i	$-0.790760\pi$
$853$	− 35.6910i	− 1.22204i	0.791617	−	0.611018i	$-0.209240\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 27.4542i	− 0.937818i	−0.883247	−	0.468909i	$-0.844647\pi$
$857$	− 27.4542i	− 0.937818i	0.883247	−	0.468909i	$-0.155353\pi$
$858$	0	0
$859$	20.8392	0.711025	0.355512	−	0.934672i	$-0.384306\pi$
$859$	20.8392	0.711025	0.355512	+	0.934672i	$0.384306\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.8505	−0.879960	−0.439980	−	0.898008i	$-0.645014\pi$
$863$	−25.8505	−0.879960	−0.439980	+	0.898008i	$0.645014\pi$
$864$	0	0
$865$	27.7458	0.943385
$866$	0	0
$867$	0	0
$868$	0	0
$869$	48.6582	1.65062
$870$	0	0
$871$	0.520615i	0.0176404i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 5.22400i	− 0.176603i
$876$	0	0
$877$	12.7819i	0.431613i	0.976436	+	0.215806i	$0.0692380\pi$
$877$	12.7819i	0.431613i	−0.976436	+	0.215806i	$0.930762\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 9.54473i	− 0.321570i	−0.986989	−	0.160785i	$-0.948597\pi$
$881$	− 9.54473i	− 0.321570i	0.986989	−	0.160785i	$-0.0514026\pi$
$882$	0	0
$883$	18.8835	0.635479	0.317740	−	0.948178i	$-0.397076\pi$
$883$	18.8835	0.635479	0.317740	+	0.948178i	$0.397076\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.9738	0.670654	0.335327	−	0.942102i	$-0.391153\pi$
$887$	19.9738	0.670654	0.335327	+	0.942102i	$0.391153\pi$
$888$	0	0
$889$	1.80347	0.0604864
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.3614	0.514049
$894$	0	0
$895$	− 42.1241i	− 1.40805i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 21.2751i	− 0.709565i
$900$	0	0
$901$	− 109.172i	− 3.63703i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	50.6162i	1.68254i
$906$	0	0
$907$	−7.50951	−0.249349	−0.124675	−	0.992198i	$-0.539789\pi$
$907$	−7.50951	−0.249349	−0.124675	+	0.992198i	$0.539789\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.1774	1.09922	0.549609	−	0.835422i	$-0.314777\pi$
$911$	33.1774	1.09922	0.549609	+	0.835422i	$0.314777\pi$
$912$	0	0
$913$	10.3101	0.341215
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.93535	−0.328094
$918$	0	0
$919$	− 13.2043i	− 0.435568i	−0.975997	−	0.217784i	$-0.930117\pi$
$919$	− 13.2043i	− 0.435568i	0.975997	−	0.217784i	$-0.0698829\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 0.938358i	− 0.0308864i
$924$	0	0
$925$	17.4838i	0.574865i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.175426i	0.00575553i	0.999996	+	0.00287776i	$0.000916022\pi$
$929$	0.175426i	0.00575553i	−0.999996	+	0.00287776i	$0.999084\pi$
$930$	0	0
$931$	5.90589	0.193558
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−116.812	−3.82015
$936$	0	0
$937$	−10.2203	−0.333884	−0.166942	−	0.985967i	$-0.553389\pi$
$937$	−10.2203	−0.333884	−0.166942	+	0.985967i	$0.553389\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	18.9676	0.618327	0.309163	−	0.951009i	$-0.399951\pi$
$941$	18.9676	0.618327	0.309163	+	0.951009i	$0.399951\pi$
$942$	0	0
$943$	15.1596i	0.493664i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.01612i	0.195497i	0.995211	+	0.0977487i	$0.0311642\pi$
$947$	6.01612i	0.195497i	−0.995211	+	0.0977487i	$0.968836\pi$
$948$	0	0
$949$	25.3877i	0.824119i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 18.9690i	− 0.614468i	−0.951634	−	0.307234i	$-0.900597\pi$
$953$	− 18.9690i	− 0.614468i	0.951634	−	0.307234i	$-0.0994034\pi$
$954$	0	0
$955$	51.4412	1.66460
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.16258	0.0375417
$960$	0	0
$961$	19.9990	0.645129
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−71.0677	−2.28775
$966$	0	0
$967$	28.6527i	0.921408i	0.887554	+	0.460704i	$0.152403\pi$
$967$	28.6527i	0.921408i	−0.887554	+	0.460704i	$0.847597\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 47.4986i	− 1.52430i	−0.647399	−	0.762151i	$-0.724143\pi$
$971$	− 47.4986i	− 1.52430i	0.647399	−	0.762151i	$-0.275857\pi$
$972$	0	0
$973$	− 17.0996i	− 0.548189i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.0015i	1.27976i	0.768474	+	0.639881i	$0.221016\pi$
$977$	40.0015i	1.27976i	−0.768474	+	0.639881i	$0.778984\pi$
$978$	0	0
$979$	88.0732	2.81483
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.0829	−0.704334	−0.352167	−	0.935937i	$-0.614555\pi$
$983$	−22.0829	−0.704334	−0.352167	+	0.935937i	$0.614555\pi$
$984$	0	0
$985$	−49.7844	−1.58626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.10902	0.194255
$990$	0	0
$991$	− 17.2351i	− 0.547490i	−0.961802	−	0.273745i	$-0.911738\pi$
$991$	− 17.2351i	− 0.547490i	0.961802	−	0.273745i	$-0.0882625\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	57.1944i	1.81318i
$996$	0	0
$997$	− 21.9872i	− 0.696342i	−0.937431	−	0.348171i	$-0.886803\pi$
$997$	− 21.9872i	− 0.696342i	0.937431	−	0.348171i	$-0.113197\pi$
$998$	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.5615.39		48
3.2	odd	2	inner	6048.2.j.d.5615.10		48
4.3	odd	2		1512.2.j.d.323.40	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.9		48
8.5	even	2		1512.2.j.d.323.10	yes	48
12.11	even	2		1512.2.j.d.323.9	✓	48
24.5	odd	2		1512.2.j.d.323.39	yes	48
24.11	even	2	inner	6048.2.j.d.5615.40		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.9	✓	48	12.11	even	2
1512.2.j.d.323.10	yes	48	8.5	even	2
1512.2.j.d.323.39	yes	48	24.5	odd	2
1512.2.j.d.323.40	yes	48	4.3	odd	2
6048.2.j.d.5615.9		48	8.3	odd	2	inner
6048.2.j.d.5615.10		48	3.2	odd	2	inner
6048.2.j.d.5615.39		48	1.1	even	1	trivial
6048.2.j.d.5615.40		48	24.11	even	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)

\(f(q)\)	\(=\)	\(q+2.85878 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+2.85878 q^{5} +1.00000i q^{7} +4.98479i q^{11} -2.35304i q^{13} +8.19706i q^{17} -5.90589 q^{19} -4.43848 q^{23} +3.17265 q^{25} +6.41440 q^{29} -3.31678i q^{31} +2.85878i q^{35} +5.51080i q^{37} -3.41549i q^{41} -1.37638 q^{43} -2.60102 q^{47} -1.00000 q^{49} -13.3184 q^{53} +14.2505i q^{55} +10.6485i q^{59} +1.41211i q^{61} -6.72683i q^{65} -0.221252 q^{67} +0.398786 q^{71} -10.7893 q^{73} -4.98479 q^{77} -9.76133i q^{79} -2.06831i q^{83} +23.4336i q^{85} -17.6684i q^{89} +2.35304 q^{91} -16.8837 q^{95} -3.62651 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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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