LMFDB - Embedded newform 6048.2.j.c.5615.16

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:	$32$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.16
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.162025 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-0.162025 q^{5} -1.00000i q^{7} -2.40352i q^{11} -4.76784i q^{13} -2.74335i q^{17} +1.86073 q^{19} -1.32970 q^{23} -4.97375 q^{25} -3.96112 q^{29} +8.01692i q^{31} +0.162025i q^{35} -6.09436i q^{37} -3.16889i q^{41} +1.70927 q^{43} -12.1502 q^{47} -1.00000 q^{49} +8.22731 q^{53} +0.389431i q^{55} -5.45449i q^{59} -4.28553i q^{61} +0.772509i q^{65} +4.61226 q^{67} -13.9767 q^{71} -6.37815 q^{73} -2.40352 q^{77} +9.75258i q^{79} +16.4983i q^{83} +0.444491i q^{85} +2.35048i q^{89} -4.76784 q^{91} -0.301485 q^{95} +14.4569 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) $ 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

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
$5$	−0.162025	−0.0724598	−0.0362299	−	0.999343i	$-0.511535\pi$
$5$	−0.162025	−0.0724598	−0.0362299	+	0.999343i	$0.511535\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.40352i	− 0.724689i	−0.932044	−	0.362345i	$-0.881976\pi$
$11$	− 2.40352i	− 0.724689i	0.932044	−	0.362345i	$-0.118024\pi$
$12$	0	0
$13$	− 4.76784i	− 1.32236i	−0.750227	−	0.661181i	$-0.770056\pi$
$13$	− 4.76784i	− 1.32236i	0.750227	−	0.661181i	$-0.229944\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.74335i	− 0.665359i	−0.943040	−	0.332680i	$-0.892047\pi$
$17$	− 2.74335i	− 0.665359i	0.943040	−	0.332680i	$-0.107953\pi$
$18$	0	0
$19$	1.86073	0.426881	0.213440	−	0.976956i	$-0.431533\pi$
$19$	1.86073	0.426881	0.213440	+	0.976956i	$0.431533\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.32970	−0.277262	−0.138631	−	0.990344i	$-0.544270\pi$
$23$	−1.32970	−0.277262	−0.138631	+	0.990344i	$0.544270\pi$
$24$	0	0
$25$	−4.97375	−0.994750
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.96112	−0.735561	−0.367780	−	0.929913i	$-0.619882\pi$
$29$	−3.96112	−0.735561	−0.367780	+	0.929913i	$0.619882\pi$
$30$	0	0
$31$	8.01692i	1.43988i	0.694036	+	0.719941i	$0.255831\pi$
$31$	8.01692i	1.43988i	−0.694036	+	0.719941i	$0.744169\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.162025i	0.0273872i
$36$	0	0
$37$	− 6.09436i	− 1.00191i	−0.865474	−	0.500953i	$-0.832983\pi$
$37$	− 6.09436i	− 1.00191i	0.865474	−	0.500953i	$-0.167017\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.16889i	− 0.494898i	−0.968901	−	0.247449i	$-0.920408\pi$
$41$	− 3.16889i	− 0.494898i	0.968901	−	0.247449i	$-0.0795922\pi$
$42$	0	0
$43$	1.70927	0.260662	0.130331	−	0.991471i	$-0.458396\pi$
$43$	1.70927	0.260662	0.130331	+	0.991471i	$0.458396\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.1502	−1.77229	−0.886144	−	0.463410i	$-0.846626\pi$
$47$	−12.1502	−1.77229	−0.886144	+	0.463410i	$0.846626\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.22731	1.13011	0.565054	−	0.825054i	$-0.308855\pi$
$53$	8.22731	1.13011	0.565054	+	0.825054i	$0.308855\pi$
$54$	0	0
$55$	0.389431i	0.0525108i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.45449i	− 0.710114i	−0.934845	−	0.355057i	$-0.884461\pi$
$59$	− 5.45449i	− 0.710114i	0.934845	−	0.355057i	$-0.115539\pi$
$60$	0	0
$61$	− 4.28553i	− 0.548706i	−0.961629	−	0.274353i	$-0.911536\pi$
$61$	− 4.28553i	− 0.548706i	0.961629	−	0.274353i	$-0.0884636\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.772509i	0.0958180i
$66$	0	0
$67$	4.61226	0.563477	0.281738	−	0.959491i	$-0.409089\pi$
$67$	4.61226	0.563477	0.281738	+	0.959491i	$0.409089\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.9767	−1.65873	−0.829363	−	0.558710i	$-0.811297\pi$
$71$	−13.9767	−1.65873	−0.829363	+	0.558710i	$0.811297\pi$
$72$	0	0
$73$	−6.37815	−0.746506	−0.373253	−	0.927730i	$-0.621758\pi$
$73$	−6.37815	−0.746506	−0.373253	+	0.927730i	$0.621758\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.40352	−0.273907
$78$	0	0
$79$	9.75258i	1.09725i	0.836068	+	0.548625i	$0.184849\pi$
$79$	9.75258i	1.09725i	−0.836068	+	0.548625i	$0.815151\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.4983i	1.81093i	0.424426	+	0.905463i	$0.360476\pi$
$83$	16.4983i	1.81093i	−0.424426	+	0.905463i	$0.639524\pi$
$84$	0	0
$85$	0.444491i	0.0482118i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.35048i	0.249150i	0.992210	+	0.124575i	$0.0397567\pi$
$89$	2.35048i	0.249150i	−0.992210	+	0.124575i	$0.960243\pi$
$90$	0	0
$91$	−4.76784	−0.499806
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.301485	−0.0309317
$96$	0	0
$97$	14.4569	1.46788	0.733938	−	0.679217i	$-0.237680\pi$
$97$	14.4569	1.46788	0.733938	+	0.679217i	$0.237680\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.453215	−0.0450966	−0.0225483	−	0.999746i	$-0.507178\pi$
$101$	−0.453215	−0.0450966	−0.0225483	+	0.999746i	$0.507178\pi$
$102$	0	0
$103$	8.93542i	0.880434i	0.897892	+	0.440217i	$0.145098\pi$
$103$	8.93542i	0.880434i	−0.897892	+	0.440217i	$0.854902\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.9379i	1.44411i	0.691838	+	0.722053i	$0.256801\pi$
$107$	14.9379i	1.44411i	−0.691838	+	0.722053i	$0.743199\pi$
$108$	0	0
$109$	4.33798i	0.415503i	0.978182	+	0.207751i	$0.0666145\pi$
$109$	4.33798i	0.415503i	−0.978182	+	0.207751i	$0.933385\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.451095i	0.0424354i	0.999775	+	0.0212177i	$0.00675431\pi$
$113$	0.451095i	0.0424354i	−0.999775	+	0.0212177i	$0.993246\pi$
$114$	0	0
$115$	0.215445	0.0200903
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.74335	−0.251482
$120$	0	0
$121$	5.22308	0.474826
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.61600	0.144539
$126$	0	0
$127$	− 0.196751i	− 0.0174589i	−0.999962	−	0.00872943i	$-0.997221\pi$
$127$	− 0.196751i	− 0.0174589i	0.999962	−	0.00872943i	$-0.00277870\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 4.69097i	− 0.409852i	−0.978778	−	0.204926i	$-0.934305\pi$
$131$	− 4.69097i	− 0.409852i	0.978778	−	0.204926i	$-0.0656953\pi$
$132$	0	0
$133$	− 1.86073i	− 0.161346i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.1485i	0.952479i	0.879316	+	0.476240i	$0.158001\pi$
$137$	11.1485i	0.952479i	−0.879316	+	0.476240i	$0.841999\pi$
$138$	0	0
$139$	−6.93228	−0.587988	−0.293994	−	0.955807i	$-0.594985\pi$
$139$	−6.93228	−0.587988	−0.293994	+	0.955807i	$0.594985\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−11.4596	−0.958301
$144$	0	0
$145$	0.641800	0.0532986
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.2542	−1.33160	−0.665799	−	0.746132i	$-0.731909\pi$
$149$	−16.2542	−1.33160	−0.665799	+	0.746132i	$0.731909\pi$
$150$	0	0
$151$	0.426011i	0.0346683i	0.999850	+	0.0173341i	$0.00551791\pi$
$151$	0.426011i	0.0346683i	−0.999850	+	0.0173341i	$0.994482\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.29894i	− 0.104333i
$156$	0	0
$157$	− 14.0451i	− 1.12092i	−0.828180	−	0.560462i	$-0.810624\pi$
$157$	− 14.0451i	− 1.12092i	0.828180	−	0.560462i	$-0.189376\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.32970i	0.104795i
$162$	0	0
$163$	4.72962	0.370453	0.185226	−	0.982696i	$-0.440698\pi$
$163$	4.72962	0.370453	0.185226	+	0.982696i	$0.440698\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.7344	0.908036	0.454018	−	0.890993i	$-0.349990\pi$
$167$	11.7344	0.908036	0.454018	+	0.890993i	$0.349990\pi$
$168$	0	0
$169$	−9.73231	−0.748639
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.2577	−1.38810	−0.694052	−	0.719925i	$-0.744176\pi$
$173$	−18.2577	−1.38810	−0.694052	+	0.719925i	$0.744176\pi$
$174$	0	0
$175$	4.97375i	0.375980i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 14.5950i	− 1.09088i	−0.838148	−	0.545442i	$-0.816362\pi$
$179$	− 14.5950i	− 1.09088i	0.838148	−	0.545442i	$-0.183638\pi$
$180$	0	0
$181$	3.82026i	0.283958i	0.989870	+	0.141979i	$0.0453465\pi$
$181$	3.82026i	0.283958i	−0.989870	+	0.141979i	$0.954653\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.987438i	0.0725979i
$186$	0	0
$187$	−6.59369	−0.482178
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.4447	−1.55168	−0.775840	−	0.630929i	$-0.782674\pi$
$191$	−21.4447	−1.55168	−0.775840	+	0.630929i	$0.782674\pi$
$192$	0	0
$193$	−17.5991	−1.26681	−0.633407	−	0.773819i	$-0.718344\pi$
$193$	−17.5991	−1.26681	−0.633407	+	0.773819i	$0.718344\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.1329	−1.36316	−0.681581	−	0.731743i	$-0.738707\pi$
$197$	−19.1329	−1.36316	−0.681581	+	0.731743i	$0.738707\pi$
$198$	0	0
$199$	− 13.3125i	− 0.943699i	−0.881679	−	0.471850i	$-0.843586\pi$
$199$	− 13.3125i	− 0.943699i	0.881679	−	0.471850i	$-0.156414\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.96112i	0.278016i
$204$	0	0
$205$	0.513440i	0.0358602i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.47230i	− 0.309356i
$210$	0	0
$211$	23.9337	1.64766	0.823830	−	0.566836i	$-0.191833\pi$
$211$	23.9337	1.64766	0.823830	+	0.566836i	$0.191833\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.276945	−0.0188875
$216$	0	0
$217$	8.01692	0.544224
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.0798	−0.879845
$222$	0	0
$223$	13.9928i	0.937024i	0.883457	+	0.468512i	$0.155210\pi$
$223$	13.9928i	0.937024i	−0.883457	+	0.468512i	$0.844790\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.8975i	− 0.723292i	−0.932316	−	0.361646i	$-0.882215\pi$
$227$	− 10.8975i	− 0.723292i	0.932316	−	0.361646i	$-0.117785\pi$
$228$	0	0
$229$	− 11.1299i	− 0.735485i	−0.929928	−	0.367743i	$-0.880131\pi$
$229$	− 11.1299i	− 0.735485i	0.929928	−	0.367743i	$-0.119869\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 13.7778i	− 0.902613i	−0.892369	−	0.451307i	$-0.850958\pi$
$233$	− 13.7778i	− 0.902613i	0.892369	−	0.451307i	$-0.149042\pi$
$234$	0	0
$235$	1.96863	0.128420
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.99259	0.516998	0.258499	−	0.966012i	$-0.416772\pi$
$239$	7.99259	0.516998	0.258499	+	0.966012i	$0.416772\pi$
$240$	0	0
$241$	−5.18318	−0.333878	−0.166939	−	0.985967i	$-0.553388\pi$
$241$	−5.18318	−0.333878	−0.166939	+	0.985967i	$0.553388\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.162025	0.0103514
$246$	0	0
$247$	− 8.87166i	− 0.564490i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 7.88280i	− 0.497558i	−0.968560	−	0.248779i	$-0.919971\pi$
$251$	− 7.88280i	− 0.497558i	0.968560	−	0.248779i	$-0.0800293\pi$
$252$	0	0
$253$	3.19597i	0.200929i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.1394i	1.75529i	0.479312	+	0.877645i	$0.340886\pi$
$257$	28.1394i	1.75529i	−0.479312	+	0.877645i	$0.659114\pi$
$258$	0	0
$259$	−6.09436	−0.378685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.5298	−1.88255	−0.941274	−	0.337645i	$-0.890370\pi$
$263$	−30.5298	−1.88255	−0.941274	+	0.337645i	$0.890370\pi$
$264$	0	0
$265$	−1.33303	−0.0818874
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.2495	−0.807839	−0.403920	−	0.914795i	$-0.632352\pi$
$269$	−13.2495	−0.807839	−0.403920	+	0.914795i	$0.632352\pi$
$270$	0	0
$271$	− 7.06832i	− 0.429370i	−0.976683	−	0.214685i	$-0.931128\pi$
$271$	− 7.06832i	− 0.429370i	0.976683	−	0.214685i	$-0.0688725\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.9545i	0.720884i
$276$	0	0
$277$	− 10.3435i	− 0.621481i	−0.950495	−	0.310740i	$-0.899423\pi$
$277$	− 10.3435i	− 0.621481i	0.950495	−	0.310740i	$-0.100577\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 27.0638i	− 1.61449i	−0.590217	−	0.807245i	$-0.700958\pi$
$281$	− 27.0638i	− 1.61449i	0.590217	−	0.807245i	$-0.299042\pi$
$282$	0	0
$283$	−17.2246	−1.02390	−0.511948	−	0.859017i	$-0.671076\pi$
$283$	−17.2246	−1.02390	−0.511948	+	0.859017i	$0.671076\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.16889	−0.187054
$288$	0	0
$289$	9.47405	0.557297
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.199121	0.0116328	0.00581639	−	0.999983i	$-0.498149\pi$
$293$	0.199121	0.0116328	0.00581639	+	0.999983i	$0.498149\pi$
$294$	0	0
$295$	0.883763i	0.0514547i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.33981i	0.366641i
$300$	0	0
$301$	− 1.70927i	− 0.0985209i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.694363i	0.0397591i
$306$	0	0
$307$	−10.0504	−0.573606	−0.286803	−	0.957990i	$-0.592593\pi$
$307$	−10.0504	−0.573606	−0.286803	+	0.957990i	$0.592593\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.74569	0.552627	0.276314	−	0.961068i	$-0.410887\pi$
$311$	9.74569	0.552627	0.276314	+	0.961068i	$0.410887\pi$
$312$	0	0
$313$	3.46199	0.195683	0.0978415	−	0.995202i	$-0.468806\pi$
$313$	3.46199	0.195683	0.0978415	+	0.995202i	$0.468806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.5983	−1.55008	−0.775039	−	0.631914i	$-0.782270\pi$
$317$	−27.5983	−1.55008	−0.775039	+	0.631914i	$0.782270\pi$
$318$	0	0
$319$	9.52063i	0.533053i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.10462i	− 0.284029i
$324$	0	0
$325$	23.7140i	1.31542i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.1502i	0.669862i
$330$	0	0
$331$	13.5802	0.746434	0.373217	−	0.927744i	$-0.378255\pi$
$331$	13.5802	0.746434	0.373217	+	0.927744i	$0.378255\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.747301	−0.0408294
$336$	0	0
$337$	−0.0989875	−0.00539219	−0.00269610	−	0.999996i	$-0.500858\pi$
$337$	−0.0989875	−0.00539219	−0.00269610	+	0.999996i	$0.500858\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	19.2688	1.04347
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.82609i	0.312761i	0.987697	+	0.156380i	$0.0499826\pi$
$347$	5.82609i	0.312761i	−0.987697	+	0.156380i	$0.950017\pi$
$348$	0	0
$349$	− 30.1726i	− 1.61510i	−0.589798	−	0.807551i	$-0.700793\pi$
$349$	− 30.1726i	− 1.61510i	0.589798	−	0.807551i	$-0.299207\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.0333i	0.853364i	0.904402	+	0.426682i	$0.140318\pi$
$353$	16.0333i	0.853364i	−0.904402	+	0.426682i	$0.859682\pi$
$354$	0	0
$355$	2.26457	0.120191
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.0001	0.844451	0.422225	−	0.906491i	$-0.361249\pi$
$359$	16.0001	0.844451	0.422225	+	0.906491i	$0.361249\pi$
$360$	0	0
$361$	−15.5377	−0.817773
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.03342	0.0540917
$366$	0	0
$367$	13.0471i	0.681053i	0.940235	+	0.340527i	$0.110605\pi$
$367$	13.0471i	0.681053i	−0.940235	+	0.340527i	$0.889395\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 8.22731i	− 0.427141i
$372$	0	0
$373$	0.319142i	0.0165245i	0.999966	+	0.00826227i	$0.00262999\pi$
$373$	0.319142i	0.0165245i	−0.999966	+	0.00826227i	$0.997370\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	18.8860i	0.972677i
$378$	0	0
$379$	−13.4802	−0.692431	−0.346215	−	0.938155i	$-0.612533\pi$
$379$	−13.4802	−0.692431	−0.346215	+	0.938155i	$0.612533\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.80894	−0.296823	−0.148411	−	0.988926i	$-0.547416\pi$
$383$	−5.80894	−0.296823	−0.148411	+	0.988926i	$0.547416\pi$
$384$	0	0
$385$	0.389431	0.0198472
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.7958	−1.66282	−0.831408	−	0.555663i	$-0.812465\pi$
$389$	−32.7958	−1.66282	−0.831408	+	0.555663i	$0.812465\pi$
$390$	0	0
$391$	3.64783i	0.184479i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.58016i	− 0.0795066i
$396$	0	0
$397$	26.2294i	1.31641i	0.752837	+	0.658207i	$0.228685\pi$
$397$	26.2294i	1.31641i	−0.752837	+	0.658207i	$0.771315\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 20.9069i	− 1.04404i	−0.852933	−	0.522020i	$-0.825179\pi$
$401$	− 20.9069i	− 1.04404i	0.852933	−	0.522020i	$-0.174821\pi$
$402$	0	0
$403$	38.2234	1.90404
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−14.6479	−0.726071
$408$	0	0
$409$	33.7856	1.67059	0.835295	−	0.549801i	$-0.185297\pi$
$409$	33.7856	1.67059	0.835295	+	0.549801i	$0.185297\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.45449	−0.268398
$414$	0	0
$415$	− 2.67314i	− 0.131219i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.4203i	0.606771i	0.952868	+	0.303385i	$0.0981170\pi$
$419$	12.4203i	0.606771i	−0.952868	+	0.303385i	$0.901883\pi$
$420$	0	0
$421$	− 21.3631i	− 1.04118i	−0.853808	−	0.520588i	$-0.825713\pi$
$421$	− 21.3631i	− 1.04118i	0.853808	−	0.520588i	$-0.174287\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.6447i	0.661866i
$426$	0	0
$427$	−4.28553	−0.207391
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.3649	1.02911	0.514557	−	0.857456i	$-0.327957\pi$
$431$	21.3649	1.02911	0.514557	+	0.857456i	$0.327957\pi$
$432$	0	0
$433$	21.5834	1.03723	0.518617	−	0.855007i	$-0.326447\pi$
$433$	21.5834	1.03723	0.518617	+	0.855007i	$0.326447\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.47422	−0.118358
$438$	0	0
$439$	13.4860i	0.643650i	0.946799	+	0.321825i	$0.104296\pi$
$439$	13.4860i	0.643650i	−0.946799	+	0.321825i	$0.895704\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.29225i	0.203931i	0.994788	+	0.101965i	$0.0325131\pi$
$443$	4.29225i	0.203931i	−0.994788	+	0.101965i	$0.967487\pi$
$444$	0	0
$445$	− 0.380836i	− 0.0180534i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 33.7152i	− 1.59112i	−0.605876	−	0.795559i	$-0.707177\pi$
$449$	− 33.7152i	− 1.59112i	0.605876	−	0.795559i	$-0.292823\pi$
$450$	0	0
$451$	−7.61650	−0.358647
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.772509	0.0362158
$456$	0	0
$457$	−31.2919	−1.46377	−0.731886	−	0.681427i	$-0.761360\pi$
$457$	−31.2919	−1.46377	−0.731886	+	0.681427i	$0.761360\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.9245	−1.44030	−0.720149	−	0.693820i	$-0.755927\pi$
$461$	−30.9245	−1.44030	−0.720149	+	0.693820i	$0.755927\pi$
$462$	0	0
$463$	− 12.0160i	− 0.558432i	−0.960228	−	0.279216i	$-0.909925\pi$
$463$	− 12.0160i	− 0.558432i	0.960228	−	0.279216i	$-0.0900746\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.24380i	0.335203i	0.985855	+	0.167601i	$0.0536022\pi$
$467$	7.24380i	0.335203i	−0.985855	+	0.167601i	$0.946398\pi$
$468$	0	0
$469$	− 4.61226i	− 0.212974i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4.10828i	− 0.188899i
$474$	0	0
$475$	−9.25480	−0.424639
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−31.8562	−1.45555	−0.727773	−	0.685818i	$-0.759445\pi$
$479$	−31.8562	−1.45555	−0.727773	+	0.685818i	$0.759445\pi$
$480$	0	0
$481$	−29.0569	−1.32488
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.34238	−0.106362
$486$	0	0
$487$	− 37.3634i	− 1.69310i	−0.532313	−	0.846548i	$-0.678677\pi$
$487$	− 37.3634i	− 1.69310i	0.532313	−	0.846548i	$-0.321323\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 23.8389i	− 1.07583i	−0.842998	−	0.537917i	$-0.819211\pi$
$491$	− 23.8389i	− 1.07583i	0.842998	−	0.537917i	$-0.180789\pi$
$492$	0	0
$493$	10.8667i	0.489412i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.9767i	0.626940i
$498$	0	0
$499$	−13.0105	−0.582428	−0.291214	−	0.956658i	$-0.594059\pi$
$499$	−13.0105	−0.582428	−0.291214	+	0.956658i	$0.594059\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−34.4474	−1.53593	−0.767966	−	0.640491i	$-0.778731\pi$
$503$	−34.4474	−1.53593	−0.767966	+	0.640491i	$0.778731\pi$
$504$	0	0
$505$	0.0734321	0.00326769
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.29166	0.323197	0.161599	−	0.986857i	$-0.448335\pi$
$509$	7.29166	0.323197	0.161599	+	0.986857i	$0.448335\pi$
$510$	0	0
$511$	6.37815i	0.282153i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.44776i	− 0.0637960i
$516$	0	0
$517$	29.2033i	1.28436i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.0013i	0.876274i	0.898908	+	0.438137i	$0.144362\pi$
$521$	20.0013i	0.876274i	−0.898908	+	0.438137i	$0.855638\pi$
$522$	0	0
$523$	14.7421	0.644629	0.322314	−	0.946633i	$-0.395539\pi$
$523$	14.7421	0.644629	0.322314	+	0.946633i	$0.395539\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.9932	0.958038
$528$	0	0
$529$	−21.2319	−0.923126
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.1088	−0.654433
$534$	0	0
$535$	− 2.42032i	− 0.104640i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.40352i	0.103527i
$540$	0	0
$541$	36.6011i	1.57360i	0.617206	+	0.786802i	$0.288265\pi$
$541$	36.6011i	1.57360i	−0.617206	+	0.786802i	$0.711735\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 0.702861i	− 0.0301073i
$546$	0	0
$547$	−41.4335	−1.77157	−0.885783	−	0.464099i	$-0.846378\pi$
$547$	−41.4335	−1.77157	−0.885783	+	0.464099i	$0.846378\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.37057	−0.313997
$552$	0	0
$553$	9.75258	0.414722
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.2128	0.983558	0.491779	−	0.870720i	$-0.336347\pi$
$557$	23.2128	0.983558	0.491779	+	0.870720i	$0.336347\pi$
$558$	0	0
$559$	− 8.14954i	− 0.344689i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 36.2689i	− 1.52855i	−0.644890	−	0.764276i	$-0.723097\pi$
$563$	− 36.2689i	− 1.52855i	0.644890	−	0.764276i	$-0.276903\pi$
$564$	0	0
$565$	− 0.0730886i	− 0.00307486i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.0163i	0.671439i	0.941962	+	0.335719i	$0.108979\pi$
$569$	16.0163i	0.671439i	−0.941962	+	0.335719i	$0.891021\pi$
$570$	0	0
$571$	32.8412	1.37436	0.687180	−	0.726487i	$-0.258848\pi$
$571$	32.8412	1.37436	0.687180	+	0.726487i	$0.258848\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.61360	0.275806
$576$	0	0
$577$	10.0637	0.418956	0.209478	−	0.977813i	$-0.432823\pi$
$577$	10.0637	0.418956	0.209478	+	0.977813i	$0.432823\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.4983	0.684465
$582$	0	0
$583$	− 19.7745i	− 0.818977i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.8278i	1.23112i	0.788088	+	0.615562i	$0.211071\pi$
$587$	29.8278i	1.23112i	−0.788088	+	0.615562i	$0.788929\pi$
$588$	0	0
$589$	14.9173i	0.614657i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.8440i	0.938091i	0.883174	+	0.469046i	$0.155402\pi$
$593$	22.8440i	0.938091i	−0.883174	+	0.469046i	$0.844598\pi$
$594$	0	0
$595$	0.444491	0.0182223
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.07805	0.330060	0.165030	−	0.986289i	$-0.447228\pi$
$599$	8.07805	0.330060	0.165030	+	0.986289i	$0.447228\pi$
$600$	0	0
$601$	30.9036	1.26058	0.630292	−	0.776358i	$-0.282935\pi$
$601$	30.9036	1.26058	0.630292	+	0.776358i	$0.282935\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.846270	−0.0344058
$606$	0	0
$607$	− 21.2157i	− 0.861121i	−0.902562	−	0.430560i	$-0.858316\pi$
$607$	− 21.2157i	− 0.861121i	0.902562	−	0.430560i	$-0.141684\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	57.9302i	2.34360i
$612$	0	0
$613$	22.9612i	0.927392i	0.885994	+	0.463696i	$0.153477\pi$
$613$	22.9612i	0.927392i	−0.885994	+	0.463696i	$0.846523\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 14.9091i	− 0.600217i	−0.953905	−	0.300108i	$-0.902977\pi$
$617$	− 14.9091i	− 0.600217i	0.953905	−	0.300108i	$-0.0970228\pi$
$618$	0	0
$619$	24.6832	0.992100	0.496050	−	0.868294i	$-0.334783\pi$
$619$	24.6832	0.992100	0.496050	+	0.868294i	$0.334783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.35048	0.0941699
$624$	0	0
$625$	24.6069	0.984276
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.7189	−0.666627
$630$	0	0
$631$	− 11.1068i	− 0.442154i	−0.975256	−	0.221077i	$-0.929043\pi$
$631$	− 11.1068i	− 0.442154i	0.975256	−	0.221077i	$-0.0709573\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.0318786i	0.00126507i
$636$	0	0
$637$	4.76784i	0.188909i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 35.4613i	− 1.40064i	−0.713831	−	0.700318i	$-0.753041\pi$
$641$	− 35.4613i	− 1.40064i	0.713831	−	0.700318i	$-0.246959\pi$
$642$	0	0
$643$	−27.6728	−1.09131	−0.545654	−	0.838011i	$-0.683719\pi$
$643$	−27.6728	−1.09131	−0.545654	+	0.838011i	$0.683719\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.7035	−0.499426	−0.249713	−	0.968320i	$-0.580336\pi$
$647$	−12.7035	−0.499426	−0.249713	+	0.968320i	$0.580336\pi$
$648$	0	0
$649$	−13.1100	−0.514612
$650$	0	0
$651$	0	0
$652$	0	0
$653$	42.8339	1.67622	0.838110	−	0.545501i	$-0.183660\pi$
$653$	42.8339	1.67622	0.838110	+	0.545501i	$0.183660\pi$
$654$	0	0
$655$	0.760054i	0.0296978i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 35.5914i	− 1.38645i	−0.720723	−	0.693223i	$-0.756190\pi$
$659$	− 35.5914i	− 1.38645i	0.720723	−	0.693223i	$-0.243810\pi$
$660$	0	0
$661$	− 0.388331i	− 0.0151043i	−0.999971	−	0.00755217i	$-0.997596\pi$
$661$	− 0.388331i	− 0.0151043i	0.999971	−	0.00755217i	$-0.00240395\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.301485i	0.0116911i
$666$	0	0
$667$	5.26711	0.203943
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.3004	−0.397641
$672$	0	0
$673$	37.9424	1.46257	0.731286	−	0.682070i	$-0.238920\pi$
$673$	37.9424	1.46257	0.731286	+	0.682070i	$0.238920\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.1764	−0.929173	−0.464586	−	0.885528i	$-0.653797\pi$
$677$	−24.1764	−0.929173	−0.464586	+	0.885528i	$0.653797\pi$
$678$	0	0
$679$	− 14.4569i	− 0.554805i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 17.3036i	− 0.662104i	−0.943613	−	0.331052i	$-0.892596\pi$
$683$	− 17.3036i	− 0.662104i	0.943613	−	0.331052i	$-0.107404\pi$
$684$	0	0
$685$	− 1.80633i	− 0.0690164i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 39.2265i	− 1.49441i
$690$	0	0
$691$	−14.6123	−0.555877	−0.277938	−	0.960599i	$-0.589651\pi$
$691$	−14.6123	−0.555877	−0.277938	+	0.960599i	$0.589651\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.12320	0.0426055
$696$	0	0
$697$	−8.69336	−0.329285
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.3532	1.07089	0.535443	−	0.844572i	$-0.320145\pi$
$701$	28.3532	1.07089	0.535443	+	0.844572i	$0.320145\pi$
$702$	0	0
$703$	− 11.3400i	− 0.427694i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.453215i	0.0170449i
$708$	0	0
$709$	− 22.7293i	− 0.853618i	−0.904342	−	0.426809i	$-0.859638\pi$
$709$	− 22.7293i	− 0.853618i	0.904342	−	0.426809i	$-0.140362\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 10.6601i	− 0.399224i
$714$	0	0
$715$	1.85674	0.0694383
$716$	0	0
$717$	0	0
$718$	0	0
$719$	27.7959	1.03661	0.518306	−	0.855195i	$-0.326563\pi$
$719$	27.7959	1.03661	0.518306	+	0.855195i	$0.326563\pi$
$720$	0	0
$721$	8.93542	0.332773
$722$	0	0
$723$	0	0
$724$	0	0
$725$	19.7016	0.731699
$726$	0	0
$727$	− 24.4253i	− 0.905886i	−0.891540	−	0.452943i	$-0.850374\pi$
$727$	− 24.4253i	− 0.905886i	0.891540	−	0.452943i	$-0.149626\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 4.68913i	− 0.173434i
$732$	0	0
$733$	17.1266i	0.632585i	0.948662	+	0.316292i	$0.102438\pi$
$733$	17.1266i	0.632585i	−0.948662	+	0.316292i	$0.897562\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.0857i	− 0.408346i
$738$	0	0
$739$	7.60957	0.279923	0.139961	−	0.990157i	$-0.455302\pi$
$739$	7.60957	0.279923	0.139961	+	0.990157i	$0.455302\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.75152	0.137630	0.0688150	−	0.997629i	$-0.478078\pi$
$743$	3.75152	0.137630	0.0688150	+	0.997629i	$0.478078\pi$
$744$	0	0
$745$	2.63359	0.0964872
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.9379	0.545821
$750$	0	0
$751$	39.9652i	1.45835i	0.684326	+	0.729176i	$0.260096\pi$
$751$	39.9652i	1.45835i	−0.684326	+	0.729176i	$0.739904\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 0.0690244i	− 0.00251206i
$756$	0	0
$757$	26.4627i	0.961802i	0.876775	+	0.480901i	$0.159690\pi$
$757$	26.4627i	0.961802i	−0.876775	+	0.480901i	$0.840310\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 3.50130i	− 0.126922i	−0.997984	−	0.0634610i	$-0.979786\pi$
$761$	− 3.50130i	− 0.126922i	0.997984	−	0.0634610i	$-0.0202139\pi$
$762$	0	0
$763$	4.33798	0.157045
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.0061	−0.939027
$768$	0	0
$769$	−26.3596	−0.950551	−0.475275	−	0.879837i	$-0.657652\pi$
$769$	−26.3596	−0.950551	−0.475275	+	0.879837i	$0.657652\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.0363	0.792592	0.396296	−	0.918123i	$-0.370295\pi$
$773$	22.0363	0.792592	0.396296	+	0.918123i	$0.370295\pi$
$774$	0	0
$775$	− 39.8741i	− 1.43232i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.89645i	− 0.211262i
$780$	0	0
$781$	33.5933i	1.20206i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.27566i	0.0812218i
$786$	0	0
$787$	−44.0988	−1.57195	−0.785976	−	0.618257i	$-0.787839\pi$
$787$	−44.0988	−1.57195	−0.785976	+	0.618257i	$0.787839\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.451095	0.0160391
$792$	0	0
$793$	−20.4327	−0.725587
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17.4271	0.617298	0.308649	−	0.951176i	$-0.400123\pi$
$797$	17.4271	0.617298	0.308649	+	0.951176i	$0.400123\pi$
$798$	0	0
$799$	33.3322i	1.17921i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15.3300i	0.540985i
$804$	0	0
$805$	− 0.215445i	− 0.00759344i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1.23533i	− 0.0434319i	−0.999764	−	0.0217159i	$-0.993087\pi$
$809$	− 1.23533i	− 0.0434319i	0.999764	−	0.0217159i	$-0.00691294\pi$
$810$	0	0
$811$	−40.7818	−1.43204	−0.716021	−	0.698078i	$-0.754039\pi$
$811$	−40.7818	−1.43204	−0.716021	+	0.698078i	$0.754039\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.766317	−0.0268429
$816$	0	0
$817$	3.18050	0.111271
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.2037	0.879614	0.439807	−	0.898092i	$-0.355047\pi$
$821$	25.2037	0.879614	0.439807	+	0.898092i	$0.355047\pi$
$822$	0	0
$823$	− 46.4360i	− 1.61866i	−0.587356	−	0.809329i	$-0.699831\pi$
$823$	− 46.4360i	− 1.61866i	0.587356	−	0.809329i	$-0.300169\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 24.6612i	− 0.857554i	−0.903410	−	0.428777i	$-0.858945\pi$
$827$	− 24.6612i	− 0.857554i	0.903410	−	0.428777i	$-0.141055\pi$
$828$	0	0
$829$	− 29.3304i	− 1.01869i	−0.860564	−	0.509343i	$-0.829889\pi$
$829$	− 29.3304i	− 1.01869i	0.860564	−	0.509343i	$-0.170111\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.74335i	0.0950513i
$834$	0	0
$835$	−1.90127	−0.0657961
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−10.5258	−0.363390	−0.181695	−	0.983355i	$-0.558158\pi$
$839$	−10.5258	−0.363390	−0.181695	+	0.983355i	$0.558158\pi$
$840$	0	0
$841$	−13.3096	−0.458950
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.57688	0.0542462
$846$	0	0
$847$	− 5.22308i	− 0.179467i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.10368i	0.277791i
$852$	0	0
$853$	36.6192i	1.25382i	0.779093	+	0.626908i	$0.215680\pi$
$853$	36.6192i	1.25382i	−0.779093	+	0.626908i	$0.784320\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.24273i	− 0.315726i	−0.987461	−	0.157863i	$-0.949540\pi$
$857$	− 9.24273i	− 0.315726i	0.987461	−	0.157863i	$-0.0504604\pi$
$858$	0	0
$859$	49.5357	1.69014	0.845068	−	0.534659i	$-0.179560\pi$
$859$	49.5357	1.69014	0.845068	+	0.534659i	$0.179560\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.0407	0.988557	0.494278	−	0.869304i	$-0.335432\pi$
$863$	29.0407	0.988557	0.494278	+	0.869304i	$0.335432\pi$
$864$	0	0
$865$	2.95820	0.100582
$866$	0	0
$867$	0	0
$868$	0	0
$869$	23.4405	0.795166
$870$	0	0
$871$	− 21.9905i	− 0.745120i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.61600i	− 0.0546307i
$876$	0	0
$877$	− 15.3569i	− 0.518564i	−0.965802	−	0.259282i	$-0.916514\pi$
$877$	− 15.3569i	− 0.518564i	0.965802	−	0.259282i	$-0.0834859\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	52.3468i	1.76361i	0.471616	+	0.881804i	$0.343671\pi$
$881$	52.3468i	1.76361i	−0.471616	+	0.881804i	$0.656329\pi$
$882$	0	0
$883$	−7.73223	−0.260210	−0.130105	−	0.991500i	$-0.541531\pi$
$883$	−7.73223	−0.260210	−0.130105	+	0.991500i	$0.541531\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.43973	−0.249802	−0.124901	−	0.992169i	$-0.539861\pi$
$887$	−7.43973	−0.249802	−0.124901	+	0.992169i	$0.539861\pi$
$888$	0	0
$889$	−0.196751	−0.00659883
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−22.6082	−0.756555
$894$	0	0
$895$	2.36476i	0.0790452i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 31.7560i	− 1.05912i
$900$	0	0
$901$	− 22.5704i	− 0.751928i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 0.618978i	− 0.0205755i
$906$	0	0
$907$	31.5786	1.04855	0.524276	−	0.851549i	$-0.324336\pi$
$907$	31.5786	1.04855	0.524276	+	0.851549i	$0.324336\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.2059	−0.868240	−0.434120	−	0.900855i	$-0.642941\pi$
$911$	−26.2059	−0.868240	−0.434120	+	0.900855i	$0.642941\pi$
$912$	0	0
$913$	39.6541	1.31236
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.69097	−0.154909
$918$	0	0
$919$	− 37.9577i	− 1.25211i	−0.779779	−	0.626055i	$-0.784668\pi$
$919$	− 37.9577i	− 1.25211i	0.779779	−	0.626055i	$-0.215332\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	66.6386i	2.19344i
$924$	0	0
$925$	30.3118i	0.996646i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	33.3343i	1.09366i	0.837243	+	0.546831i	$0.184166\pi$
$929$	33.3343i	1.09366i	−0.837243	+	0.546831i	$0.815834\pi$
$930$	0	0
$931$	−1.86073	−0.0609829
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.06834	0.0349385
$936$	0	0
$937$	8.18384	0.267355	0.133677	−	0.991025i	$-0.457321\pi$
$937$	8.18384	0.267355	0.133677	+	0.991025i	$0.457321\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	43.9315	1.43213	0.716064	−	0.698035i	$-0.245942\pi$
$941$	43.9315	1.43213	0.716064	+	0.698035i	$0.245942\pi$
$942$	0	0
$943$	4.21368i	0.137216i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.0323i	1.46335i	0.681652	+	0.731676i	$0.261262\pi$
$947$	45.0323i	1.46335i	−0.681652	+	0.731676i	$0.738738\pi$
$948$	0	0
$949$	30.4100i	0.987150i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 60.2109i	− 1.95042i	−0.221278	−	0.975211i	$-0.571023\pi$
$953$	− 60.2109i	− 1.95042i	0.221278	−	0.975211i	$-0.428977\pi$
$954$	0	0
$955$	3.47457	0.112434
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.1485	0.360003
$960$	0	0
$961$	−33.2710	−1.07326
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.85150	0.0917931
$966$	0	0
$967$	− 1.96099i	− 0.0630613i	−0.999503	−	0.0315307i	$-0.989962\pi$
$967$	− 1.96099i	− 0.0630613i	0.999503	−	0.0315307i	$-0.0100382\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.5960i	1.17442i	0.809434	+	0.587210i	$0.199774\pi$
$971$	36.5960i	1.17442i	−0.809434	+	0.587210i	$0.800226\pi$
$972$	0	0
$973$	6.93228i	0.222239i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0891i	0.578721i	0.957220	+	0.289361i	$0.0934427\pi$
$977$	18.0891i	0.578721i	−0.957220	+	0.289361i	$0.906557\pi$
$978$	0	0
$979$	5.64942	0.180556
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.7144	0.405525	0.202763	−	0.979228i	$-0.435008\pi$
$983$	12.7144	0.405525	0.202763	+	0.979228i	$0.435008\pi$
$984$	0	0
$985$	3.10001	0.0987744
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.27282	−0.0722716
$990$	0	0
$991$	58.4330i	1.85619i	0.372348	+	0.928093i	$0.378553\pi$
$991$	58.4330i	1.85619i	−0.372348	+	0.928093i	$0.621447\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.15696i	0.0683803i
$996$	0	0
$997$	− 33.7142i	− 1.06774i	−0.845567	−	0.533869i	$-0.820737\pi$
$997$	− 33.7142i	− 1.06774i	0.845567	−	0.533869i	$-0.179263\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.c.5615.16		32
3.2	odd	2	inner	6048.2.j.c.5615.17		32
4.3	odd	2		1512.2.j.c.323.16	yes	32
8.3	odd	2	inner	6048.2.j.c.5615.18		32
8.5	even	2		1512.2.j.c.323.18	yes	32
12.11	even	2		1512.2.j.c.323.17	yes	32
24.5	odd	2		1512.2.j.c.323.15	✓	32
24.11	even	2	inner	6048.2.j.c.5615.15		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.15	✓	32	24.5	odd	2
1512.2.j.c.323.16	yes	32	4.3	odd	2
1512.2.j.c.323.17	yes	32	12.11	even	2
1512.2.j.c.323.18	yes	32	8.5	even	2
6048.2.j.c.5615.15		32	24.11	even	2	inner
6048.2.j.c.5615.16		32	1.1	even	1	trivial
6048.2.j.c.5615.17		32	3.2	odd	2	inner
6048.2.j.c.5615.18		32	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)

\(f(q)\)	\(=\)	\(q-0.162025 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-0.162025 q^{5} -1.00000i q^{7} -2.40352i q^{11} -4.76784i q^{13} -2.74335i q^{17} +1.86073 q^{19} -1.32970 q^{23} -4.97375 q^{25} -3.96112 q^{29} +8.01692i q^{31} +0.162025i q^{35} -6.09436i q^{37} -3.16889i q^{41} +1.70927 q^{43} -12.1502 q^{47} -1.00000 q^{49} +8.22731 q^{53} +0.389431i q^{55} -5.45449i q^{59} -4.28553i q^{61} +0.772509i q^{65} +4.61226 q^{67} -13.9767 q^{71} -6.37815 q^{73} -2.40352 q^{77} +9.75258i q^{79} +16.4983i q^{83} +0.444491i q^{85} +2.35048i q^{89} -4.76784 q^{91} -0.301485 q^{95} +14.4569 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 64 q^{19} - 16 q^{25} + 48 q^{43} - 32 q^{49} - 16 q^{67} - 16 q^{73} - 16 q^{91} + 64 q^{97}+O(q^{100}) \) 32 * q + 64 * q^19 - 16 * q^25 + 48 * q^43 - 32 * q^49 - 16 * q^67 - 16 * q^73 - 16 * q^91 + 64 * q^97

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