LMFDB - Embedded newform 6048.2.j.c.5615.24

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			5615.24
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.c.5615.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17792 q^{5} -1.00000i q^{7} +O(q^{10})$	$q+1.17792 q^{5} -1.00000i q^{7} +2.90595i q^{11} +1.05317i q^{13} -6.48837i q^{17} +3.11597 q^{19} +0.812516 q^{23} -3.61251 q^{25} -4.03607 q^{29} -0.108779i q^{31} -1.17792i q^{35} -9.97034i q^{37} -9.44298i q^{41} -10.3446 q^{43} +4.03126 q^{47} -1.00000 q^{49} -2.97772 q^{53} +3.42296i q^{55} +0.868874i q^{59} +5.33629i q^{61} +1.24054i q^{65} +3.97987 q^{67} -0.844302 q^{71} +3.59513 q^{73} +2.90595 q^{77} -9.48320i q^{79} -2.71280i q^{83} -7.64276i q^{85} +2.22575i q^{89} +1.05317 q^{91} +3.67035 q^{95} -3.93019 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.17792	0.526780	0.263390	−	0.964689i	$-0.415159\pi$
$5$	1.17792	0.526780	0.263390	+	0.964689i	$0.415159\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.90595i	0.876176i	0.898932	+	0.438088i	$0.144344\pi$
$11$	2.90595i	0.876176i	−0.898932	+	0.438088i	$0.855656\pi$
$12$	0	0
$13$	1.05317i	0.292096i	0.989278	+	0.146048i	$0.0466554\pi$
$13$	1.05317i	0.292096i	−0.989278	+	0.146048i	$0.953345\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.48837i	− 1.57366i	−0.617169	−	0.786830i	$-0.711721\pi$
$17$	− 6.48837i	− 1.57366i	0.617169	−	0.786830i	$-0.288279\pi$
$18$	0	0
$19$	3.11597	0.714853	0.357426	−	0.933941i	$-0.383654\pi$
$19$	3.11597	0.714853	0.357426	+	0.933941i	$0.383654\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.812516	0.169421	0.0847107	−	0.996406i	$-0.473003\pi$
$23$	0.812516	0.169421	0.0847107	+	0.996406i	$0.473003\pi$
$24$	0	0
$25$	−3.61251	−0.722503
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.03607	−0.749480	−0.374740	−	0.927130i	$-0.622268\pi$
$29$	−4.03607	−0.749480	−0.374740	+	0.927130i	$0.622268\pi$
$30$	0	0
$31$	− 0.108779i	− 0.0195372i	−0.999952	−	0.00976861i	$-0.996891\pi$
$31$	− 0.108779i	− 0.0195372i	0.999952	−	0.00976861i	$-0.00310949\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.17792i	− 0.199104i
$36$	0	0
$37$	− 9.97034i	− 1.63911i	−0.572998	−	0.819557i	$-0.694220\pi$
$37$	− 9.97034i	− 1.63911i	0.572998	−	0.819557i	$-0.305780\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 9.44298i	− 1.47475i	−0.675486	−	0.737373i	$-0.736066\pi$
$41$	− 9.44298i	− 1.47475i	0.675486	−	0.737373i	$-0.263934\pi$
$42$	0	0
$43$	−10.3446	−1.57754	−0.788770	−	0.614688i	$-0.789282\pi$
$43$	−10.3446	−1.57754	−0.788770	+	0.614688i	$0.789282\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.03126	0.588019	0.294010	−	0.955802i	$-0.405010\pi$
$47$	4.03126	0.588019	0.294010	+	0.955802i	$0.405010\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.97772	−0.409022	−0.204511	−	0.978864i	$-0.565560\pi$
$53$	−2.97772	−0.409022	−0.204511	+	0.978864i	$0.565560\pi$
$54$	0	0
$55$	3.42296i	0.461552i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.868874i	0.113118i	0.998399	+	0.0565589i	$0.0180129\pi$
$59$	0.868874i	0.113118i	−0.998399	+	0.0565589i	$0.981987\pi$
$60$	0	0
$61$	5.33629i	0.683242i	0.939838	+	0.341621i	$0.110976\pi$
$61$	5.33629i	0.683242i	−0.939838	+	0.341621i	$0.889024\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.24054i	0.153870i
$66$	0	0
$67$	3.97987	0.486218	0.243109	−	0.969999i	$-0.421833\pi$
$67$	3.97987	0.486218	0.243109	+	0.969999i	$0.421833\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.844302	−0.100200	−0.0501001	−	0.998744i	$-0.515954\pi$
$71$	−0.844302	−0.100200	−0.0501001	+	0.998744i	$0.515954\pi$
$72$	0	0
$73$	3.59513	0.420778	0.210389	−	0.977618i	$-0.432527\pi$
$73$	3.59513	0.420778	0.210389	+	0.977618i	$0.432527\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.90595	0.331164
$78$	0	0
$79$	− 9.48320i	− 1.06694i	−0.845818	−	0.533472i	$-0.820887\pi$
$79$	− 9.48320i	− 1.06694i	0.845818	−	0.533472i	$-0.179113\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.71280i	− 0.297769i	−0.988855	−	0.148884i	$-0.952432\pi$
$83$	− 2.71280i	− 0.297769i	0.988855	−	0.148884i	$-0.0475682\pi$
$84$	0	0
$85$	− 7.64276i	− 0.828973i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.22575i	0.235930i	0.993018	+	0.117965i	$0.0376370\pi$
$89$	2.22575i	0.235930i	−0.993018	+	0.117965i	$0.962363\pi$
$90$	0	0
$91$	1.05317	0.110402
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.67035	0.376570
$96$	0	0
$97$	−3.93019	−0.399050	−0.199525	−	0.979893i	$-0.563940\pi$
$97$	−3.93019	−0.399050	−0.199525	+	0.979893i	$0.563940\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.0700	1.79803	0.899016	−	0.437915i	$-0.144283\pi$
$101$	18.0700	1.79803	0.899016	+	0.437915i	$0.144283\pi$
$102$	0	0
$103$	17.8616i	1.75995i	0.475017	+	0.879977i	$0.342442\pi$
$103$	17.8616i	1.75995i	−0.475017	+	0.879977i	$0.657558\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.67720i	− 0.742183i	−0.928596	−	0.371091i	$-0.878984\pi$
$107$	− 7.67720i	− 0.742183i	0.928596	−	0.371091i	$-0.121016\pi$
$108$	0	0
$109$	− 12.6991i	− 1.21635i	−0.793801	−	0.608177i	$-0.791901\pi$
$109$	− 12.6991i	− 1.21635i	0.793801	−	0.608177i	$-0.208099\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 15.4017i	− 1.44887i	−0.689342	−	0.724437i	$-0.742100\pi$
$113$	− 15.4017i	− 1.44887i	0.689342	−	0.724437i	$-0.257900\pi$
$114$	0	0
$115$	0.957076	0.0892478
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.48837	−0.594788
$120$	0	0
$121$	2.55546	0.232315
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.1448	−0.907380
$126$	0	0
$127$	17.1514i	1.52194i	0.648787	+	0.760970i	$0.275276\pi$
$127$	17.1514i	1.52194i	−0.648787	+	0.760970i	$0.724724\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 10.5118i	− 0.918421i	−0.888328	−	0.459210i	$-0.848132\pi$
$131$	− 10.5118i	− 0.918421i	0.888328	−	0.459210i	$-0.151868\pi$
$132$	0	0
$133$	− 3.11597i	− 0.270189i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.6268i	− 1.16422i	−0.813110	−	0.582110i	$-0.802227\pi$
$137$	− 13.6268i	− 1.16422i	0.813110	−	0.582110i	$-0.197773\pi$
$138$	0	0
$139$	5.37620	0.456003	0.228002	−	0.973661i	$-0.426781\pi$
$139$	5.37620	0.456003	0.228002	+	0.973661i	$0.426781\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.06045	−0.255928
$144$	0	0
$145$	−4.75416	−0.394811
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.3187	−1.41880	−0.709401	−	0.704805i	$-0.751035\pi$
$149$	−17.3187	−1.41880	−0.709401	+	0.704805i	$0.751035\pi$
$150$	0	0
$151$	4.50773i	0.366834i	0.983035	+	0.183417i	$0.0587158\pi$
$151$	4.50773i	0.366834i	−0.983035	+	0.183417i	$0.941284\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.128132i	− 0.0102918i
$156$	0	0
$157$	− 8.84696i	− 0.706064i	−0.935611	−	0.353032i	$-0.885151\pi$
$157$	− 8.84696i	− 0.706064i	0.935611	−	0.353032i	$-0.114849\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 0.812516i	− 0.0640352i
$162$	0	0
$163$	8.38238	0.656558	0.328279	−	0.944581i	$-0.393531\pi$
$163$	8.38238	0.656558	0.328279	+	0.944581i	$0.393531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.93563	−0.691460	−0.345730	−	0.938334i	$-0.612369\pi$
$167$	−8.93563	−0.691460	−0.345730	+	0.938334i	$0.612369\pi$
$168$	0	0
$169$	11.8908	0.914680
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.1612	−1.07666	−0.538328	−	0.842736i	$-0.680944\pi$
$173$	−14.1612	−1.07666	−0.538328	+	0.842736i	$0.680944\pi$
$174$	0	0
$175$	3.61251i	0.273080i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 20.3695i	− 1.52249i	−0.648467	−	0.761243i	$-0.724589\pi$
$179$	− 20.3695i	− 1.52249i	0.648467	−	0.761243i	$-0.275411\pi$
$180$	0	0
$181$	− 18.2659i	− 1.35770i	−0.734279	−	0.678848i	$-0.762480\pi$
$181$	− 18.2659i	− 1.35770i	0.734279	−	0.678848i	$-0.237520\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 11.7442i	− 0.863453i
$186$	0	0
$187$	18.8549	1.37880
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.83849	0.567173	0.283587	−	0.958947i	$-0.408476\pi$
$191$	7.83849	0.567173	0.283587	+	0.958947i	$0.408476\pi$
$192$	0	0
$193$	−13.0241	−0.937498	−0.468749	−	0.883331i	$-0.655295\pi$
$193$	−13.0241	−0.937498	−0.468749	+	0.883331i	$0.655295\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.9576	1.06568	0.532842	−	0.846215i	$-0.321124\pi$
$197$	14.9576	1.06568	0.532842	+	0.846215i	$0.321124\pi$
$198$	0	0
$199$	16.9147i	1.19905i	0.800356	+	0.599525i	$0.204644\pi$
$199$	16.9147i	1.19905i	−0.800356	+	0.599525i	$0.795356\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.03607i	0.283277i
$204$	0	0
$205$	− 11.1230i	− 0.776867i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	9.05485i	0.626337i
$210$	0	0
$211$	−5.68386	−0.391293	−0.195647	−	0.980674i	$-0.562681\pi$
$211$	−5.68386	−0.391293	−0.195647	+	0.980674i	$0.562681\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−12.1851	−0.831017
$216$	0	0
$217$	−0.108779	−0.00738437
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.83333	0.459660
$222$	0	0
$223$	− 3.43963i	− 0.230335i	−0.993346	−	0.115167i	$-0.963260\pi$
$223$	− 3.43963i	− 0.230335i	0.993346	−	0.115167i	$-0.0367404\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.86522i	− 0.588406i	−0.955743	−	0.294203i	$-0.904946\pi$
$227$	− 8.86522i	− 0.588406i	0.955743	−	0.294203i	$-0.0950541\pi$
$228$	0	0
$229$	17.3653i	1.14753i	0.819019	+	0.573767i	$0.194518\pi$
$229$	17.3653i	1.14753i	−0.819019	+	0.573767i	$0.805482\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 9.99149i	− 0.654564i	−0.944927	−	0.327282i	$-0.893867\pi$
$233$	− 9.99149i	− 0.654564i	0.944927	−	0.327282i	$-0.106133\pi$
$234$	0	0
$235$	4.74848	0.309757
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.5708	1.58935	0.794677	−	0.607032i	$-0.207640\pi$
$239$	24.5708	1.58935	0.794677	+	0.607032i	$0.207640\pi$
$240$	0	0
$241$	23.1786	1.49306	0.746532	−	0.665349i	$-0.231717\pi$
$241$	23.1786	1.49306	0.746532	+	0.665349i	$0.231717\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.17792	−0.0752543
$246$	0	0
$247$	3.28164i	0.208806i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 21.3670i	− 1.34868i	−0.738423	−	0.674338i	$-0.764429\pi$
$251$	− 21.3670i	− 1.34868i	0.738423	−	0.674338i	$-0.235571\pi$
$252$	0	0
$253$	2.36113i	0.148443i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 29.5594i	− 1.84386i	−0.387352	−	0.921932i	$-0.626610\pi$
$257$	− 29.5594i	− 1.84386i	0.387352	−	0.921932i	$-0.373390\pi$
$258$	0	0
$259$	−9.97034	−0.619527
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.8001	−0.974277	−0.487139	−	0.873325i	$-0.661959\pi$
$263$	−15.8001	−0.974277	−0.487139	+	0.873325i	$0.661959\pi$
$264$	0	0
$265$	−3.50751	−0.215464
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.41502	0.391131	0.195565	−	0.980691i	$-0.437346\pi$
$269$	6.41502	0.391131	0.195565	+	0.980691i	$0.437346\pi$
$270$	0	0
$271$	− 0.0927686i	− 0.00563529i	−0.999996	−	0.00281765i	$-0.999103\pi$
$271$	− 0.0927686i	− 0.00563529i	0.999996	−	0.00281765i	$-0.000896886\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 10.4978i	− 0.633040i
$276$	0	0
$277$	19.7900i	1.18907i	0.804071	+	0.594534i	$0.202663\pi$
$277$	19.7900i	1.18907i	−0.804071	+	0.594534i	$0.797337\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.4404i	0.861443i	0.902485	+	0.430721i	$0.141741\pi$
$281$	14.4404i	0.861443i	−0.902485	+	0.430721i	$0.858259\pi$
$282$	0	0
$283$	−20.3759	−1.21122	−0.605612	−	0.795760i	$-0.707071\pi$
$283$	−20.3759	−1.21122	−0.605612	+	0.795760i	$0.707071\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.44298	−0.557402
$288$	0	0
$289$	−25.0989	−1.47641
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.2919	−0.601262	−0.300631	−	0.953741i	$-0.597197\pi$
$293$	−10.2919	−0.601262	−0.300631	+	0.953741i	$0.597197\pi$
$294$	0	0
$295$	1.02346i	0.0595882i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.855715i	0.0494873i
$300$	0	0
$301$	10.3446i	0.596254i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.28571i	0.359919i
$306$	0	0
$307$	−19.8799	−1.13461	−0.567304	−	0.823508i	$-0.692014\pi$
$307$	−19.8799	−1.13461	−0.567304	+	0.823508i	$0.692014\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.94464	−0.450499	−0.225250	−	0.974301i	$-0.572320\pi$
$311$	−7.94464	−0.450499	−0.225250	+	0.974301i	$0.572320\pi$
$312$	0	0
$313$	−28.2451	−1.59651	−0.798254	−	0.602321i	$-0.794243\pi$
$313$	−28.2451	−1.59651	−0.798254	+	0.602321i	$0.794243\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.8609	−0.778506	−0.389253	−	0.921131i	$-0.627267\pi$
$317$	−13.8609	−0.778506	−0.389253	+	0.921131i	$0.627267\pi$
$318$	0	0
$319$	− 11.7286i	− 0.656677i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 20.2176i	− 1.12494i
$324$	0	0
$325$	− 3.80458i	− 0.211040i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.03126i	− 0.222250i
$330$	0	0
$331$	14.5450	0.799468	0.399734	−	0.916631i	$-0.369103\pi$
$331$	14.5450	0.799468	0.399734	+	0.916631i	$0.369103\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.68795	0.256130
$336$	0	0
$337$	5.26170	0.286623	0.143312	−	0.989678i	$-0.454225\pi$
$337$	5.26170	0.286623	0.143312	+	0.989678i	$0.454225\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.316105	0.0171181
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.5549i	− 0.835029i	−0.908670	−	0.417514i	$-0.862901\pi$
$347$	− 15.5549i	− 0.835029i	0.908670	−	0.417514i	$-0.137099\pi$
$348$	0	0
$349$	17.1607i	0.918594i	0.888283	+	0.459297i	$0.151899\pi$
$349$	17.1607i	0.918594i	−0.888283	+	0.459297i	$0.848101\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.5438i	0.827314i	0.910433	+	0.413657i	$0.135749\pi$
$353$	15.5438i	0.827314i	−0.910433	+	0.413657i	$0.864251\pi$
$354$	0	0
$355$	−0.994517	−0.0527835
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.9471	0.788876	0.394438	−	0.918923i	$-0.370939\pi$
$359$	14.9471	0.788876	0.394438	+	0.918923i	$0.370939\pi$
$360$	0	0
$361$	−9.29073	−0.488986
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.23476	0.221657
$366$	0	0
$367$	2.86184i	0.149387i	0.997207	+	0.0746934i	$0.0237978\pi$
$367$	2.86184i	0.149387i	−0.997207	+	0.0746934i	$0.976202\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.97772i	0.154596i
$372$	0	0
$373$	20.7327i	1.07350i	0.843743	+	0.536748i	$0.180347\pi$
$373$	20.7327i	1.07350i	−0.843743	+	0.536748i	$0.819653\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.25066i	− 0.218920i
$378$	0	0
$379$	−28.1561	−1.44628	−0.723141	−	0.690700i	$-0.757302\pi$
$379$	−28.1561	−1.44628	−0.723141	+	0.690700i	$0.757302\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.9733	−1.53156	−0.765781	−	0.643101i	$-0.777648\pi$
$383$	−29.9733	−1.53156	−0.765781	+	0.643101i	$0.777648\pi$
$384$	0	0
$385$	3.42296	0.174450
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.4931	1.49536	0.747679	−	0.664060i	$-0.231168\pi$
$389$	29.4931	1.49536	0.747679	+	0.664060i	$0.231168\pi$
$390$	0	0
$391$	− 5.27190i	− 0.266612i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 11.1704i	− 0.562045i
$396$	0	0
$397$	− 22.0845i	− 1.10839i	−0.832386	−	0.554196i	$-0.813026\pi$
$397$	− 22.0845i	− 1.10839i	0.832386	−	0.554196i	$-0.186974\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 7.42054i	− 0.370564i	−0.982685	−	0.185282i	$-0.940680\pi$
$401$	− 7.42054i	− 0.370564i	0.982685	−	0.185282i	$-0.0593198\pi$
$402$	0	0
$403$	0.114562	0.00570674
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.9733	1.43615
$408$	0	0
$409$	37.3983	1.84923	0.924615	−	0.380904i	$-0.124387\pi$
$409$	37.3983	1.84923	0.924615	+	0.380904i	$0.124387\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.868874	0.0427545
$414$	0	0
$415$	− 3.19546i	− 0.156859i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 8.55960i	− 0.418164i	−0.977898	−	0.209082i	$-0.932952\pi$
$419$	− 8.55960i	− 0.418164i	0.977898	−	0.209082i	$-0.0670475\pi$
$420$	0	0
$421$	− 6.56688i	− 0.320050i	−0.987113	−	0.160025i	$-0.948842\pi$
$421$	− 6.56688i	− 0.320050i	0.987113	−	0.160025i	$-0.0511575\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	23.4393i	1.13697i
$426$	0	0
$427$	5.33629	0.258241
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.61014	−0.270231	−0.135116	−	0.990830i	$-0.543141\pi$
$431$	−5.61014	−0.270231	−0.135116	+	0.990830i	$0.543141\pi$
$432$	0	0
$433$	7.96675	0.382857	0.191429	−	0.981507i	$-0.438688\pi$
$433$	7.96675	0.382857	0.191429	+	0.981507i	$0.438688\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.53178	0.121111
$438$	0	0
$439$	− 33.9176i	− 1.61880i	−0.587258	−	0.809400i	$-0.699792\pi$
$439$	− 33.9176i	− 1.61880i	0.587258	−	0.809400i	$-0.300208\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.31862i	− 0.300207i	−0.988670	−	0.150103i	$-0.952039\pi$
$443$	− 6.31862i	− 0.300207i	0.988670	−	0.150103i	$-0.0479606\pi$
$444$	0	0
$445$	2.62175i	0.124283i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 11.9562i	− 0.564246i	−0.959378	−	0.282123i	$-0.908961\pi$
$449$	− 11.9562i	− 0.564246i	0.959378	−	0.282123i	$-0.0910387\pi$
$450$	0	0
$451$	27.4408	1.29214
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.24054	0.0581575
$456$	0	0
$457$	10.7774	0.504144	0.252072	−	0.967708i	$-0.418888\pi$
$457$	10.7774	0.504144	0.252072	+	0.967708i	$0.418888\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.36887	−0.110329	−0.0551647	−	0.998477i	$-0.517568\pi$
$461$	−2.36887	−0.110329	−0.0551647	+	0.998477i	$0.517568\pi$
$462$	0	0
$463$	− 38.4537i	− 1.78709i	−0.448969	−	0.893547i	$-0.648209\pi$
$463$	− 38.4537i	− 1.78709i	0.448969	−	0.893547i	$-0.351791\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.09381i	− 0.235713i	−0.993031	−	0.117857i	$-0.962398\pi$
$467$	− 5.09381i	− 0.235713i	0.993031	−	0.117857i	$-0.0376023\pi$
$468$	0	0
$469$	− 3.97987i	− 0.183773i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 30.0609i	− 1.38220i
$474$	0	0
$475$	−11.2565	−0.516483
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.8166	−0.951137	−0.475568	−	0.879679i	$-0.657758\pi$
$479$	−20.8166	−0.951137	−0.475568	+	0.879679i	$0.657758\pi$
$480$	0	0
$481$	10.5004	0.478778
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.62943	−0.210212
$486$	0	0
$487$	26.1716i	1.18595i	0.805222	+	0.592973i	$0.202046\pi$
$487$	26.1716i	1.18595i	−0.805222	+	0.592973i	$0.797954\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 27.7686i	− 1.25318i	−0.779349	−	0.626590i	$-0.784450\pi$
$491$	− 27.7686i	− 1.25318i	0.779349	−	0.626590i	$-0.215550\pi$
$492$	0	0
$493$	26.1875i	1.17943i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.844302i	0.0378721i
$498$	0	0
$499$	13.7781	0.616791	0.308395	−	0.951258i	$-0.400208\pi$
$499$	13.7781	0.616791	0.308395	+	0.951258i	$0.400208\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	32.5875	1.45300	0.726502	−	0.687165i	$-0.241145\pi$
$503$	32.5875	1.45300	0.726502	+	0.687165i	$0.241145\pi$
$504$	0	0
$505$	21.2850	0.947168
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17.1161	0.758657	0.379328	−	0.925262i	$-0.376155\pi$
$509$	17.1161	0.758657	0.379328	+	0.925262i	$0.376155\pi$
$510$	0	0
$511$	− 3.59513i	− 0.159039i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.0394i	0.927109i
$516$	0	0
$517$	11.7146i	0.515209i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 22.6308i	− 0.991471i	−0.868473	−	0.495736i	$-0.834898\pi$
$521$	− 22.6308i	− 0.991471i	0.868473	−	0.495736i	$-0.165102\pi$
$522$	0	0
$523$	19.0252	0.831914	0.415957	−	0.909384i	$-0.363447\pi$
$523$	19.0252	0.831914	0.415957	+	0.909384i	$0.363447\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.705796	−0.0307450
$528$	0	0
$529$	−22.3398	−0.971296
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.94503	0.430767
$534$	0	0
$535$	− 9.04310i	− 0.390967i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.90595i	− 0.125168i
$540$	0	0
$541$	17.3996i	0.748066i	0.927415	+	0.374033i	$0.122025\pi$
$541$	17.3996i	0.748066i	−0.927415	+	0.374033i	$0.877975\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 14.9585i	− 0.640751i
$546$	0	0
$547$	−4.96571	−0.212318	−0.106159	−	0.994349i	$-0.533855\pi$
$547$	−4.96571	−0.212318	−0.106159	+	0.994349i	$0.533855\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.5763	−0.535768
$552$	0	0
$553$	−9.48320	−0.403267
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.67400	0.240415	0.120208	−	0.992749i	$-0.461644\pi$
$557$	5.67400	0.240415	0.120208	+	0.992749i	$0.461644\pi$
$558$	0	0
$559$	− 10.8946i	− 0.460793i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 25.4752i	− 1.07365i	−0.843693	−	0.536826i	$-0.819623\pi$
$563$	− 25.4752i	− 1.07365i	0.843693	−	0.536826i	$-0.180377\pi$
$564$	0	0
$565$	− 18.1420i	− 0.763238i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.5120i	0.817986i	0.912537	+	0.408993i	$0.134120\pi$
$569$	19.5120i	0.817986i	−0.912537	+	0.408993i	$0.865880\pi$
$570$	0	0
$571$	38.0268	1.59137	0.795686	−	0.605709i	$-0.207111\pi$
$571$	38.0268	1.59137	0.795686	+	0.605709i	$0.207111\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.93522	−0.122407
$576$	0	0
$577$	5.32632	0.221738	0.110869	−	0.993835i	$-0.464637\pi$
$577$	5.32632	0.221738	0.110869	+	0.993835i	$0.464637\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.71280	−0.112546
$582$	0	0
$583$	− 8.65311i	− 0.358375i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.6946i	1.06053i	0.847832	+	0.530265i	$0.177908\pi$
$587$	25.6946i	1.06053i	−0.847832	+	0.530265i	$0.822092\pi$
$588$	0	0
$589$	− 0.338951i	− 0.0139662i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 35.9249i	− 1.47526i	−0.675205	−	0.737630i	$-0.735945\pi$
$593$	− 35.9249i	− 1.47526i	0.675205	−	0.737630i	$-0.264055\pi$
$594$	0	0
$595$	−7.64276	−0.313322
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.08090	0.330177	0.165088	−	0.986279i	$-0.447209\pi$
$599$	8.08090	0.330177	0.165088	+	0.986279i	$0.447209\pi$
$600$	0	0
$601$	39.0492	1.59285	0.796425	−	0.604738i	$-0.206722\pi$
$601$	39.0492	1.59285	0.796425	+	0.604738i	$0.206722\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.01012	0.122379
$606$	0	0
$607$	− 39.0636i	− 1.58554i	−0.609519	−	0.792772i	$-0.708637\pi$
$607$	− 39.0636i	− 1.58554i	0.609519	−	0.792772i	$-0.291363\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.24559i	0.171758i
$612$	0	0
$613$	27.0877i	1.09406i	0.837112	+	0.547031i	$0.184242\pi$
$613$	27.0877i	1.09406i	−0.837112	+	0.547031i	$0.815758\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.2143i	0.411213i	0.978635	+	0.205607i	$0.0659167\pi$
$617$	10.2143i	0.411213i	−0.978635	+	0.205607i	$0.934083\pi$
$618$	0	0
$619$	−28.5400	−1.14712	−0.573560	−	0.819164i	$-0.694438\pi$
$619$	−28.5400	−1.14712	−0.573560	+	0.819164i	$0.694438\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.22575	0.0891730
$624$	0	0
$625$	6.11281	0.244512
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−64.6912	−2.57941
$630$	0	0
$631$	17.9037i	0.712734i	0.934346	+	0.356367i	$0.115985\pi$
$631$	17.9037i	0.712734i	−0.934346	+	0.356367i	$0.884015\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	20.2029i	0.801728i
$636$	0	0
$637$	− 1.05317i	− 0.0417280i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.5576i	1.44394i	0.691925	+	0.721969i	$0.256763\pi$
$641$	36.5576i	1.44394i	−0.691925	+	0.721969i	$0.743237\pi$
$642$	0	0
$643$	−28.5306	−1.12514	−0.562568	−	0.826751i	$-0.690187\pi$
$643$	−28.5306	−1.12514	−0.562568	+	0.826751i	$0.690187\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.0326	0.984134	0.492067	−	0.870557i	$-0.336242\pi$
$647$	25.0326	0.984134	0.492067	+	0.870557i	$0.336242\pi$
$648$	0	0
$649$	−2.52490	−0.0991112
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−43.1196	−1.68740	−0.843699	−	0.536816i	$-0.819627\pi$
$653$	−43.1196	−1.68740	−0.843699	+	0.536816i	$0.819627\pi$
$654$	0	0
$655$	− 12.3820i	− 0.483806i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.36352i	0.325796i	0.986643	+	0.162898i	$0.0520842\pi$
$659$	8.36352i	0.325796i	−0.986643	+	0.162898i	$0.947916\pi$
$660$	0	0
$661$	− 8.48754i	− 0.330127i	−0.986283	−	0.165064i	$-0.947217\pi$
$661$	− 8.48754i	− 0.330127i	0.986283	−	0.165064i	$-0.0527829\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 3.67035i	− 0.142330i
$666$	0	0
$667$	−3.27937	−0.126978
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.5070	−0.598641
$672$	0	0
$673$	−25.4287	−0.980206	−0.490103	−	0.871664i	$-0.663041\pi$
$673$	−25.4287	−0.980206	−0.490103	+	0.871664i	$0.663041\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.0820	−0.656515	−0.328258	−	0.944588i	$-0.606461\pi$
$677$	−17.0820	−0.656515	−0.328258	+	0.944588i	$0.606461\pi$
$678$	0	0
$679$	3.93019i	0.150827i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.0552i	1.68573i	0.538128	+	0.842863i	$0.319132\pi$
$683$	44.0552i	1.68573i	−0.538128	+	0.842863i	$0.680868\pi$
$684$	0	0
$685$	− 16.0513i	− 0.613288i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.13604i	− 0.119474i
$690$	0	0
$691$	18.3483	0.698002	0.349001	−	0.937122i	$-0.386521\pi$
$691$	18.3483	0.698002	0.349001	+	0.937122i	$0.386521\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.33271	0.240213
$696$	0	0
$697$	−61.2695	−2.32075
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.6724	1.27179	0.635894	−	0.771777i	$-0.280632\pi$
$701$	33.6724	1.27179	0.635894	+	0.771777i	$0.280632\pi$
$702$	0	0
$703$	− 31.0673i	− 1.17172i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.0700i	− 0.679592i
$708$	0	0
$709$	15.9147i	0.597689i	0.954302	+	0.298845i	$0.0966013\pi$
$709$	15.9147i	0.597689i	−0.954302	+	0.298845i	$0.903399\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 0.0883844i	− 0.00331002i
$714$	0	0
$715$	−3.60495	−0.134818
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.6441	−1.29201	−0.646004	−	0.763334i	$-0.723561\pi$
$719$	−34.6441	−1.29201	−0.646004	+	0.763334i	$0.723561\pi$
$720$	0	0
$721$	17.8616	0.665200
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14.5804	0.541501
$726$	0	0
$727$	38.6477i	1.43336i	0.697400	+	0.716682i	$0.254340\pi$
$727$	38.6477i	1.43336i	−0.697400	+	0.716682i	$0.745660\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	67.1197i	2.48251i
$732$	0	0
$733$	− 4.45768i	− 0.164648i	−0.996606	−	0.0823240i	$-0.973766\pi$
$733$	− 4.45768i	− 0.164648i	0.996606	−	0.0823240i	$-0.0262342\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.5653i	0.426013i
$738$	0	0
$739$	32.0619	1.17942	0.589709	−	0.807616i	$-0.299243\pi$
$739$	32.0619	1.17942	0.589709	+	0.807616i	$0.299243\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.9201	1.64796	0.823979	−	0.566620i	$-0.191749\pi$
$743$	44.9201	1.64796	0.823979	+	0.566620i	$0.191749\pi$
$744$	0	0
$745$	−20.4000	−0.747397
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.67720	−0.280519
$750$	0	0
$751$	20.4697i	0.746951i	0.927640	+	0.373475i	$0.121834\pi$
$751$	20.4697i	0.746951i	−0.927640	+	0.373475i	$0.878166\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.30973i	0.193241i
$756$	0	0
$757$	− 11.4003i	− 0.414352i	−0.978304	−	0.207176i	$-0.933573\pi$
$757$	− 11.4003i	− 0.414352i	0.978304	−	0.207176i	$-0.0664273\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.5142i	1.17864i	0.807901	+	0.589319i	$0.200604\pi$
$761$	32.5142i	1.17864i	−0.807901	+	0.589319i	$0.799396\pi$
$762$	0	0
$763$	−12.6991	−0.459739
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.915070	−0.0330413
$768$	0	0
$769$	25.1134	0.905614	0.452807	−	0.891609i	$-0.350423\pi$
$769$	25.1134	0.905614	0.452807	+	0.891609i	$0.350423\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42.4890	1.52822	0.764112	−	0.645083i	$-0.223177\pi$
$773$	42.4890	1.52822	0.764112	+	0.645083i	$0.223177\pi$
$774$	0	0
$775$	0.392964i	0.0141157i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 29.4240i	− 1.05423i
$780$	0	0
$781$	− 2.45350i	− 0.0877930i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 10.4210i	− 0.371941i
$786$	0	0
$787$	−19.1415	−0.682322	−0.341161	−	0.940005i	$-0.610820\pi$
$787$	−19.1415	−0.682322	−0.341161	+	0.940005i	$0.610820\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.4017	−0.547623
$792$	0	0
$793$	−5.62001	−0.199572
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.9802	0.530625	0.265312	−	0.964162i	$-0.414525\pi$
$797$	14.9802	0.530625	0.265312	+	0.964162i	$0.414525\pi$
$798$	0	0
$799$	− 26.1563i	− 0.925343i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.4473i	0.368675i
$804$	0	0
$805$	− 0.957076i	− 0.0337325i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 20.4899i	− 0.720388i	−0.932877	−	0.360194i	$-0.882711\pi$
$809$	− 20.4899i	− 0.720388i	0.932877	−	0.360194i	$-0.117289\pi$
$810$	0	0
$811$	−19.0258	−0.668087	−0.334043	−	0.942558i	$-0.608413\pi$
$811$	−19.0258	−0.668087	−0.334043	+	0.942558i	$0.608413\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.87374	0.345862
$816$	0	0
$817$	−32.2335	−1.12771
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.0750	−1.36373	−0.681863	−	0.731480i	$-0.738830\pi$
$821$	−39.0750	−1.36373	−0.681863	+	0.731480i	$0.738830\pi$
$822$	0	0
$823$	− 38.1369i	− 1.32937i	−0.747124	−	0.664684i	$-0.768566\pi$
$823$	− 38.1369i	− 1.32937i	0.747124	−	0.664684i	$-0.231434\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.8842i	0.482802i	0.970425	+	0.241401i	$0.0776070\pi$
$827$	13.8842i	0.482802i	−0.970425	+	0.241401i	$0.922393\pi$
$828$	0	0
$829$	0.705296i	0.0244960i	0.999925	+	0.0122480i	$0.00389875\pi$
$829$	0.705296i	0.0244960i	−0.999925	+	0.0122480i	$0.996101\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.48837i	0.224809i
$834$	0	0
$835$	−10.5254	−0.364248
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−50.4627	−1.74217	−0.871083	−	0.491136i	$-0.836582\pi$
$839$	−50.4627	−1.74217	−0.871083	+	0.491136i	$0.836582\pi$
$840$	0	0
$841$	−12.7101	−0.438280
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.0064	0.481835
$846$	0	0
$847$	− 2.55546i	− 0.0878068i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 8.10106i	− 0.277701i
$852$	0	0
$853$	10.7370i	0.367628i	0.982961	+	0.183814i	$0.0588444\pi$
$853$	10.7370i	0.367628i	−0.982961	+	0.183814i	$0.941156\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.75153i	− 0.333106i	−0.986033	−	0.166553i	$-0.946736\pi$
$857$	− 9.75153i	− 0.333106i	0.986033	−	0.166553i	$-0.0532637\pi$
$858$	0	0
$859$	−28.9169	−0.986630	−0.493315	−	0.869851i	$-0.664215\pi$
$859$	−28.9169	−0.986630	−0.493315	+	0.869851i	$0.664215\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.0259	−0.545528	−0.272764	−	0.962081i	$-0.587938\pi$
$863$	−16.0259	−0.545528	−0.272764	+	0.962081i	$0.587938\pi$
$864$	0	0
$865$	−16.6807	−0.567161
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.5577	0.934831
$870$	0	0
$871$	4.19146i	0.142022i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.1448i	0.342958i
$876$	0	0
$877$	2.24225i	0.0757154i	0.999283	+	0.0378577i	$0.0120534\pi$
$877$	2.24225i	0.0757154i	−0.999283	+	0.0378577i	$0.987947\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.97799i	0.336167i	0.985773	+	0.168083i	$0.0537578\pi$
$881$	9.97799i	0.336167i	−0.985773	+	0.168083i	$0.946242\pi$
$882$	0	0
$883$	−19.8382	−0.667609	−0.333805	−	0.942642i	$-0.608333\pi$
$883$	−19.8382	−0.667609	−0.333805	+	0.942642i	$0.608333\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.8500	−1.17015	−0.585074	−	0.810980i	$-0.698935\pi$
$887$	−34.8500	−1.17015	−0.585074	+	0.810980i	$0.698935\pi$
$888$	0	0
$889$	17.1514	0.575239
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.5613	0.420347
$894$	0	0
$895$	− 23.9935i	− 0.802016i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.439039i	0.0146428i
$900$	0	0
$901$	19.3206i	0.643661i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 21.5157i	− 0.715207i
$906$	0	0
$907$	3.69626	0.122732	0.0613662	−	0.998115i	$-0.480454\pi$
$907$	3.69626	0.122732	0.0613662	+	0.998115i	$0.480454\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.1975	1.69625	0.848125	−	0.529796i	$-0.177732\pi$
$911$	51.1975	1.69625	0.848125	+	0.529796i	$0.177732\pi$
$912$	0	0
$913$	7.88327	0.260898
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.5118	−0.347130
$918$	0	0
$919$	50.4632i	1.66463i	0.554306	+	0.832313i	$0.312984\pi$
$919$	50.4632i	1.66463i	−0.554306	+	0.832313i	$0.687016\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 0.889190i	− 0.0292681i
$924$	0	0
$925$	36.0180i	1.18426i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8.22557i	− 0.269872i	−0.990854	−	0.134936i	$-0.956917\pi$
$929$	− 8.22557i	− 0.269872i	0.990854	−	0.134936i	$-0.0430829\pi$
$930$	0	0
$931$	−3.11597	−0.102122
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.2095	0.726327
$936$	0	0
$937$	0.968784	0.0316488	0.0158244	−	0.999875i	$-0.494963\pi$
$937$	0.968784	0.0316488	0.0158244	+	0.999875i	$0.494963\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.2805	−0.367734	−0.183867	−	0.982951i	$-0.558862\pi$
$941$	−11.2805	−0.367734	−0.183867	+	0.982951i	$0.558862\pi$
$942$	0	0
$943$	− 7.67257i	− 0.249853i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57.8442i	1.87968i	0.341609	+	0.939842i	$0.389028\pi$
$947$	57.8442i	1.87968i	−0.341609	+	0.939842i	$0.610972\pi$
$948$	0	0
$949$	3.78627i	0.122907i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 23.4368i	− 0.759193i	−0.925152	−	0.379596i	$-0.876063\pi$
$953$	− 23.4368i	− 0.759193i	0.925152	−	0.379596i	$-0.123937\pi$
$954$	0	0
$955$	9.23309	0.298776
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.6268	−0.440034
$960$	0	0
$961$	30.9882	0.999618
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.3413	−0.493855
$966$	0	0
$967$	44.8294i	1.44162i	0.693135	+	0.720808i	$0.256229\pi$
$967$	44.8294i	1.44162i	−0.693135	+	0.720808i	$0.743771\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.3436i	0.428216i	0.976810	+	0.214108i	$0.0686844\pi$
$971$	13.3436i	0.428216i	−0.976810	+	0.214108i	$0.931316\pi$
$972$	0	0
$973$	− 5.37620i	− 0.172353i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11.0654i	− 0.354015i	−0.984210	−	0.177007i	$-0.943358\pi$
$977$	− 11.0654i	− 0.354015i	0.984210	−	0.177007i	$-0.0566416\pi$
$978$	0	0
$979$	−6.46793	−0.206716
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48.7128	1.55370	0.776849	−	0.629687i	$-0.216817\pi$
$983$	48.7128	1.55370	0.776849	+	0.629687i	$0.216817\pi$
$984$	0	0
$985$	17.6188	0.561381
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.40517	−0.267269
$990$	0	0
$991$	− 44.6581i	− 1.41861i	−0.704901	−	0.709306i	$-0.749009\pi$
$991$	− 44.6581i	− 1.41861i	0.704901	−	0.709306i	$-0.250991\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	19.9241i	0.631636i
$996$	0	0
$997$	− 20.5408i	− 0.650534i	−0.945622	−	0.325267i	$-0.894546\pi$
$997$	− 20.5408i	− 0.650534i	0.945622	−	0.325267i	$-0.105454\pi$
$998$	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.5615.24		32
3.2	odd	2	inner	6048.2.j.c.5615.9		32
4.3	odd	2		1512.2.j.c.323.1	✓	32
8.3	odd	2	inner	6048.2.j.c.5615.10		32
8.5	even	2		1512.2.j.c.323.31	yes	32
12.11	even	2		1512.2.j.c.323.32	yes	32
24.5	odd	2		1512.2.j.c.323.2	yes	32
24.11	even	2	inner	6048.2.j.c.5615.23		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.c.323.1	✓	32	4.3	odd	2
1512.2.j.c.323.2	yes	32	24.5	odd	2
1512.2.j.c.323.31	yes	32	8.5	even	2
1512.2.j.c.323.32	yes	32	12.11	even	2
6048.2.j.c.5615.9		32	3.2	odd	2	inner
6048.2.j.c.5615.10		32	8.3	odd	2	inner
6048.2.j.c.5615.23		32	24.11	even	2	inner
6048.2.j.c.5615.24		32	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.17792 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+1.17792 q^{5} -1.00000i q^{7} +2.90595i q^{11} +1.05317i q^{13} -6.48837i q^{17} +3.11597 q^{19} +0.812516 q^{23} -3.61251 q^{25} -4.03607 q^{29} -0.108779i q^{31} -1.17792i q^{35} -9.97034i q^{37} -9.44298i q^{41} -10.3446 q^{43} +4.03126 q^{47} -1.00000 q^{49} -2.97772 q^{53} +3.42296i q^{55} +0.868874i q^{59} +5.33629i q^{61} +1.24054i q^{65} +3.97987 q^{67} -0.844302 q^{71} +3.59513 q^{73} +2.90595 q^{77} -9.48320i q^{79} -2.71280i q^{83} -7.64276i q^{85} +2.22575i q^{89} +1.05317 q^{91} +3.67035 q^{95} -3.93019 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more