LMFDB - Embedded newform 6048.2.h.i.2591.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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, 0, 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("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, 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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.17
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.i.2591.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76574i q^{5} -1.00000i q^{7} +O(q^{10})$	$q-2.76574i q^{5} -1.00000i q^{7} -2.55543 q^{11} -2.32854 q^{13} -7.36547i q^{17} -1.78261i q^{19} -3.87170 q^{23} -2.64930 q^{25} -7.32622i q^{29} -2.17647i q^{31} -2.76574 q^{35} -1.24968 q^{37} +3.52679i q^{41} +5.28762i q^{43} +5.77479 q^{47} -1.00000 q^{49} +7.29178i q^{53} +7.06766i q^{55} -4.98108 q^{59} +3.15465 q^{61} +6.44014i q^{65} +7.72509i q^{67} +7.98532 q^{71} -10.7548 q^{73} +2.55543i q^{77} +2.10858i q^{79} -12.1458 q^{83} -20.3710 q^{85} -9.71061i q^{89} +2.32854i q^{91} -4.93024 q^{95} +15.6053 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{13} - 24 q^{25} + 48 q^{37} - 24 q^{49} - 48 q^{61} - 32 q^{73} - 80 q^{85} + 64 q^{97}+O(q^{100}) $ 24 * q - 8 * q^13 - 24 * q^25 + 48 * q^37 - 24 * q^49 - 48 * q^61 - 32 * q^73 - 80 * q^85 + 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
$5$	− 2.76574i	− 1.23688i	−0.785834	−	0.618438i	$-0.787766\pi$
$5$	− 2.76574i	− 1.23688i	0.785834	−	0.618438i	$-0.212234\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.55543	−0.770493	−0.385246	−	0.922814i	$-0.625884\pi$
$11$	−2.55543	−0.770493	−0.385246	+	0.922814i	$0.625884\pi$
$12$	0	0
$13$	−2.32854	−0.645822	−0.322911	−	0.946429i	$-0.604661\pi$
$13$	−2.32854	−0.645822	−0.322911	+	0.946429i	$0.604661\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.36547i	− 1.78639i	−0.449669	−	0.893195i	$-0.648458\pi$
$17$	− 7.36547i	− 1.78639i	0.449669	−	0.893195i	$-0.351542\pi$
$18$	0	0
$19$	− 1.78261i	− 0.408959i	−0.978871	−	0.204480i	$-0.934450\pi$
$19$	− 1.78261i	− 0.408959i	0.978871	−	0.204480i	$-0.0655503\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.87170	−0.807305	−0.403653	−	0.914912i	$-0.632260\pi$
$23$	−3.87170	−0.807305	−0.403653	+	0.914912i	$0.632260\pi$
$24$	0	0
$25$	−2.64930	−0.529861
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.32622i	− 1.36044i	−0.733006	−	0.680222i	$-0.761883\pi$
$29$	− 7.32622i	− 1.36044i	0.733006	−	0.680222i	$-0.238117\pi$
$30$	0	0
$31$	− 2.17647i	− 0.390905i	−0.980713	−	0.195453i	$-0.937382\pi$
$31$	− 2.17647i	− 0.390905i	0.980713	−	0.195453i	$-0.0626176\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.76574	−0.467495
$36$	0	0
$37$	−1.24968	−0.205445	−0.102723	−	0.994710i	$-0.532755\pi$
$37$	−1.24968	−0.205445	−0.102723	+	0.994710i	$0.532755\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.52679i	0.550792i	0.961331	+	0.275396i	$0.0888090\pi$
$41$	3.52679i	0.550792i	−0.961331	+	0.275396i	$0.911191\pi$
$42$	0	0
$43$	5.28762i	0.806355i	0.915122	+	0.403178i	$0.132094\pi$
$43$	5.28762i	0.806355i	−0.915122	+	0.403178i	$0.867906\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.77479	0.842340	0.421170	−	0.906982i	$-0.361620\pi$
$47$	5.77479	0.842340	0.421170	+	0.906982i	$0.361620\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.29178i	1.00160i	0.865562	+	0.500801i	$0.166961\pi$
$53$	7.29178i	1.00160i	−0.865562	+	0.500801i	$0.833039\pi$
$54$	0	0
$55$	7.06766i	0.953003i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.98108	−0.648482	−0.324241	−	0.945975i	$-0.605109\pi$
$59$	−4.98108	−0.648482	−0.324241	+	0.945975i	$0.605109\pi$
$60$	0	0
$61$	3.15465	0.403912	0.201956	−	0.979395i	$-0.435270\pi$
$61$	3.15465	0.403912	0.201956	+	0.979395i	$0.435270\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.44014i	0.798801i
$66$	0	0
$67$	7.72509i	0.943770i	0.881660	+	0.471885i	$0.156426\pi$
$67$	7.72509i	0.943770i	−0.881660	+	0.471885i	$0.843574\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.98532	0.947683	0.473842	−	0.880610i	$-0.342867\pi$
$71$	7.98532	0.947683	0.473842	+	0.880610i	$0.342867\pi$
$72$	0	0
$73$	−10.7548	−1.25875	−0.629377	−	0.777100i	$-0.716690\pi$
$73$	−10.7548	−1.25875	−0.629377	+	0.777100i	$0.716690\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.55543i	0.291219i
$78$	0	0
$79$	2.10858i	0.237234i	0.992940	+	0.118617i	$0.0378461\pi$
$79$	2.10858i	0.237234i	−0.992940	+	0.118617i	$0.962154\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.1458	−1.33318	−0.666590	−	0.745425i	$-0.732247\pi$
$83$	−12.1458	−1.33318	−0.666590	+	0.745425i	$0.732247\pi$
$84$	0	0
$85$	−20.3710	−2.20954
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 9.71061i	− 1.02932i	−0.857394	−	0.514661i	$-0.827918\pi$
$89$	− 9.71061i	− 1.02932i	0.857394	−	0.514661i	$-0.172082\pi$
$90$	0	0
$91$	2.32854i	0.244098i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.93024	−0.505832
$96$	0	0
$97$	15.6053	1.58448	0.792239	−	0.610211i	$-0.208915\pi$
$97$	15.6053	1.58448	0.792239	+	0.610211i	$0.208915\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 8.43842i	− 0.839654i	−0.907604	−	0.419827i	$-0.862091\pi$
$101$	− 8.43842i	− 0.839654i	0.907604	−	0.419827i	$-0.137909\pi$
$102$	0	0
$103$	1.53472i	0.151221i	0.997137	+	0.0756105i	$0.0240905\pi$
$103$	1.53472i	0.151221i	−0.997137	+	0.0756105i	$0.975909\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.42794	0.524739	0.262369	−	0.964967i	$-0.415496\pi$
$107$	5.42794	0.524739	0.262369	+	0.964967i	$0.415496\pi$
$108$	0	0
$109$	−7.47199	−0.715687	−0.357843	−	0.933782i	$-0.616488\pi$
$109$	−7.47199	−0.715687	−0.357843	+	0.933782i	$0.616488\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.64780i	0.249084i	0.992214	+	0.124542i	$0.0397461\pi$
$113$	2.64780i	0.249084i	−0.992214	+	0.124542i	$0.960254\pi$
$114$	0	0
$115$	10.7081i	0.998536i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.36547	−0.675192
$120$	0	0
$121$	−4.46975	−0.406341
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 6.50141i	− 0.581504i
$126$	0	0
$127$	− 19.2079i	− 1.70443i	−0.523195	−	0.852213i	$-0.675260\pi$
$127$	− 19.2079i	− 1.70443i	0.523195	−	0.852213i	$-0.324740\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.182412	−0.0159374	−0.00796869	−	0.999968i	$-0.502537\pi$
$131$	−0.182412	−0.0159374	−0.00796869	+	0.999968i	$0.502537\pi$
$132$	0	0
$133$	−1.78261	−0.154572
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.55207i	0.474345i	0.971467	+	0.237173i	$0.0762207\pi$
$137$	5.55207i	0.474345i	−0.971467	+	0.237173i	$0.923779\pi$
$138$	0	0
$139$	18.7949i	1.59416i	0.603873	+	0.797080i	$0.293623\pi$
$139$	18.7949i	1.59416i	−0.603873	+	0.797080i	$0.706377\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.95044	0.497601
$144$	0	0
$145$	−20.2624	−1.68270
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.82804i	0.805144i	0.915388	+	0.402572i	$0.131884\pi$
$149$	9.82804i	0.805144i	−0.915388	+	0.402572i	$0.868116\pi$
$150$	0	0
$151$	8.96087i	0.729225i	0.931159	+	0.364613i	$0.118799\pi$
$151$	8.96087i	0.729225i	−0.931159	+	0.364613i	$0.881201\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.01954	−0.483501
$156$	0	0
$157$	14.4314	1.15175	0.575876	−	0.817537i	$-0.304661\pi$
$157$	14.4314	1.15175	0.575876	+	0.817537i	$0.304661\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.87170i	0.305133i
$162$	0	0
$163$	14.4908i	1.13500i	0.823372	+	0.567502i	$0.192090\pi$
$163$	14.4908i	1.13500i	−0.823372	+	0.567502i	$0.807910\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.88572	0.610215	0.305108	−	0.952318i	$-0.401308\pi$
$167$	7.88572	0.610215	0.305108	+	0.952318i	$0.401308\pi$
$168$	0	0
$169$	−7.57788	−0.582914
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 15.8802i	− 1.20735i	−0.797230	−	0.603676i	$-0.793702\pi$
$173$	− 15.8802i	− 1.20735i	0.797230	−	0.603676i	$-0.206298\pi$
$174$	0	0
$175$	2.64930i	0.200269i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.2503	−1.36409	−0.682046	−	0.731309i	$-0.738910\pi$
$179$	−18.2503	−1.36409	−0.682046	+	0.731309i	$0.738910\pi$
$180$	0	0
$181$	−24.1717	−1.79667	−0.898335	−	0.439311i	$-0.855223\pi$
$181$	−24.1717	−1.79667	−0.898335	+	0.439311i	$0.855223\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.45628i	0.254110i
$186$	0	0
$187$	18.8220i	1.37640i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.6845	1.35197	0.675983	−	0.736917i	$-0.263719\pi$
$191$	18.6845	1.35197	0.675983	+	0.736917i	$0.263719\pi$
$192$	0	0
$193$	6.89411	0.496249	0.248124	−	0.968728i	$-0.420186\pi$
$193$	6.89411	0.496249	0.248124	+	0.968728i	$0.420186\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.10853i	0.648956i	0.945893	+	0.324478i	$0.105189\pi$
$197$	9.10853i	0.648956i	−0.945893	+	0.324478i	$0.894811\pi$
$198$	0	0
$199$	− 9.53988i	− 0.676264i	−0.941099	−	0.338132i	$-0.890205\pi$
$199$	− 9.53988i	− 0.676264i	0.941099	−	0.338132i	$-0.109795\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.32622	−0.514200
$204$	0	0
$205$	9.75418	0.681261
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.55535i	0.315100i
$210$	0	0
$211$	16.0066i	1.10194i	0.834525	+	0.550970i	$0.185742\pi$
$211$	16.0066i	1.10194i	−0.834525	+	0.550970i	$0.814258\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.6242	0.997361
$216$	0	0
$217$	−2.17647	−0.147748
$218$	0	0
$219$	0	0
$220$	0	0
$221$	17.1508i	1.15369i
$222$	0	0
$223$	− 3.28527i	− 0.219998i	−0.993932	−	0.109999i	$-0.964915\pi$
$223$	− 3.28527i	− 0.219998i	0.993932	−	0.109999i	$-0.0350848\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.2772	0.881238	0.440619	−	0.897694i	$-0.354759\pi$
$227$	13.2772	0.881238	0.440619	+	0.897694i	$0.354759\pi$
$228$	0	0
$229$	5.23836	0.346160	0.173080	−	0.984908i	$-0.444628\pi$
$229$	5.23836	0.346160	0.173080	+	0.984908i	$0.444628\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.32608i	− 0.414435i	−0.978295	−	0.207218i	$-0.933559\pi$
$233$	− 6.32608i	− 0.414435i	0.978295	−	0.207218i	$-0.0664408\pi$
$234$	0	0
$235$	− 15.9716i	− 1.04187i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−17.2297	−1.11450	−0.557249	−	0.830346i	$-0.688143\pi$
$239$	−17.2297	−1.11450	−0.557249	+	0.830346i	$0.688143\pi$
$240$	0	0
$241$	−27.6743	−1.78266	−0.891328	−	0.453358i	$-0.850226\pi$
$241$	−27.6743	−1.78266	−0.891328	+	0.453358i	$0.850226\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.76574i	0.176696i
$246$	0	0
$247$	4.15089i	0.264115i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.3083	1.02937	0.514685	−	0.857380i	$-0.327909\pi$
$251$	16.3083	1.02937	0.514685	+	0.857380i	$0.327909\pi$
$252$	0	0
$253$	9.89388	0.622023
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.98554i	− 0.123855i	−0.998081	−	0.0619274i	$-0.980275\pi$
$257$	− 1.98554i	− 0.123855i	0.998081	−	0.0619274i	$-0.0197247\pi$
$258$	0	0
$259$	1.24968i	0.0776511i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.93854	0.366186	0.183093	−	0.983096i	$-0.441389\pi$
$263$	5.93854	0.366186	0.183093	+	0.983096i	$0.441389\pi$
$264$	0	0
$265$	20.1671	1.23886
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 11.1048i	− 0.677073i	−0.940953	−	0.338537i	$-0.890068\pi$
$269$	− 11.1048i	− 0.677073i	0.940953	−	0.338537i	$-0.109932\pi$
$270$	0	0
$271$	23.6992i	1.43962i	0.694170	+	0.719811i	$0.255772\pi$
$271$	23.6992i	1.43962i	−0.694170	+	0.719811i	$0.744228\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.77012	0.408254
$276$	0	0
$277$	−23.7917	−1.42951	−0.714753	−	0.699377i	$-0.753461\pi$
$277$	−23.7917	−1.42951	−0.714753	+	0.699377i	$0.753461\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.1589i	1.32189i	0.750434	+	0.660945i	$0.229844\pi$
$281$	22.1589i	1.32189i	−0.750434	+	0.660945i	$0.770156\pi$
$282$	0	0
$283$	6.58388i	0.391371i	0.980667	+	0.195686i	$0.0626932\pi$
$283$	6.58388i	0.391371i	−0.980667	+	0.195686i	$0.937307\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.52679	0.208180
$288$	0	0
$289$	−37.2502	−2.19119
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.1727i	1.35376i	0.736091	+	0.676882i	$0.236669\pi$
$293$	23.1727i	1.35376i	−0.736091	+	0.676882i	$0.763331\pi$
$294$	0	0
$295$	13.7764i	0.802091i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.01543	0.521376
$300$	0	0
$301$	5.28762	0.304774
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.72494i	− 0.499589i
$306$	0	0
$307$	12.7944i	0.730216i	0.930965	+	0.365108i	$0.118968\pi$
$307$	12.7944i	0.730216i	−0.930965	+	0.365108i	$0.881032\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	29.6695	1.68240	0.841202	−	0.540720i	$-0.181848\pi$
$311$	29.6695	1.68240	0.841202	+	0.540720i	$0.181848\pi$
$312$	0	0
$313$	4.54381	0.256831	0.128416	−	0.991720i	$-0.459011\pi$
$313$	4.54381	0.256831	0.128416	+	0.991720i	$0.459011\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.93868i	− 0.558212i	−0.960260	−	0.279106i	$-0.909962\pi$
$317$	− 9.93868i	− 0.558212i	0.960260	−	0.279106i	$-0.0900381\pi$
$318$	0	0
$319$	18.7217i	1.04821i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.1298	−0.730561
$324$	0	0
$325$	6.16902	0.342196
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.77479i	− 0.318375i
$330$	0	0
$331$	− 19.7766i	− 1.08702i	−0.839403	−	0.543509i	$-0.817095\pi$
$331$	− 19.7766i	− 1.08702i	0.839403	−	0.543509i	$-0.182905\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	21.3656	1.16733
$336$	0	0
$337$	−30.5056	−1.66175	−0.830873	−	0.556462i	$-0.812158\pi$
$337$	−30.5056	−1.66175	−0.830873	+	0.556462i	$0.812158\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.56182i	0.301189i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.0080	−1.50355	−0.751773	−	0.659422i	$-0.770801\pi$
$347$	−28.0080	−1.50355	−0.751773	+	0.659422i	$0.770801\pi$
$348$	0	0
$349$	−6.47317	−0.346501	−0.173251	−	0.984878i	$-0.555427\pi$
$349$	−6.47317	−0.346501	−0.173251	+	0.984878i	$0.555427\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 33.5190i	− 1.78404i	−0.452000	−	0.892018i	$-0.649289\pi$
$353$	− 33.5190i	− 1.78404i	0.452000	−	0.892018i	$-0.350711\pi$
$354$	0	0
$355$	− 22.0853i	− 1.17217i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	33.2112	1.75282	0.876410	−	0.481565i	$-0.159931\pi$
$359$	33.2112	1.75282	0.876410	+	0.481565i	$0.159931\pi$
$360$	0	0
$361$	15.8223	0.832752
$362$	0	0
$363$	0	0
$364$	0	0
$365$	29.7450i	1.55692i
$366$	0	0
$367$	15.5534i	0.811883i	0.913899	+	0.405941i	$0.133056\pi$
$367$	15.5534i	0.811883i	−0.913899	+	0.405941i	$0.866944\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.29178	0.378570
$372$	0	0
$373$	−7.84450	−0.406173	−0.203086	−	0.979161i	$-0.565097\pi$
$373$	−7.84450	−0.406173	−0.203086	+	0.979161i	$0.565097\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.0594i	0.878605i
$378$	0	0
$379$	− 34.6551i	− 1.78011i	−0.455851	−	0.890056i	$-0.650665\pi$
$379$	− 34.6551i	− 1.78011i	0.455851	−	0.890056i	$-0.349335\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.57082	−0.335753	−0.167877	−	0.985808i	$-0.553691\pi$
$383$	−6.57082	−0.335753	−0.167877	+	0.985808i	$0.553691\pi$
$384$	0	0
$385$	7.06766	0.360201
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.56366i	0.383493i	0.981444	+	0.191746i	$0.0614151\pi$
$389$	7.56366i	0.383493i	−0.981444	+	0.191746i	$0.938585\pi$
$390$	0	0
$391$	28.5169i	1.44216i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.83178	0.293429
$396$	0	0
$397$	9.26100	0.464796	0.232398	−	0.972621i	$-0.425343\pi$
$397$	9.26100	0.464796	0.232398	+	0.972621i	$0.425343\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.158397i	0.00790995i	0.999992	+	0.00395497i	$0.00125891\pi$
$401$	0.158397i	0.00790995i	−0.999992	+	0.00395497i	$0.998741\pi$
$402$	0	0
$403$	5.06800i	0.252455i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.19347	0.158294
$408$	0	0
$409$	−6.79241	−0.335863	−0.167932	−	0.985799i	$-0.553709\pi$
$409$	−6.79241	−0.335863	−0.167932	+	0.985799i	$0.553709\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.98108i	0.245103i
$414$	0	0
$415$	33.5922i	1.64898i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−35.2185	−1.72054	−0.860269	−	0.509841i	$-0.829704\pi$
$419$	−35.2185	−1.72054	−0.860269	+	0.509841i	$0.829704\pi$
$420$	0	0
$421$	−28.1907	−1.37393	−0.686966	−	0.726689i	$-0.741058\pi$
$421$	−28.1907	−1.37393	−0.686966	+	0.726689i	$0.741058\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	19.5134i	0.946538i
$426$	0	0
$427$	− 3.15465i	− 0.152664i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.5840	0.895160	0.447580	−	0.894244i	$-0.352286\pi$
$431$	18.5840	0.895160	0.447580	+	0.894244i	$0.352286\pi$
$432$	0	0
$433$	−10.1063	−0.485680	−0.242840	−	0.970066i	$-0.578079\pi$
$433$	−10.1063	−0.485680	−0.242840	+	0.970066i	$0.578079\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.90174i	0.330155i
$438$	0	0
$439$	33.9732i	1.62145i	0.585426	+	0.810726i	$0.300927\pi$
$439$	33.9732i	1.62145i	−0.585426	+	0.810726i	$0.699073\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−18.6709	−0.887082	−0.443541	−	0.896254i	$-0.646278\pi$
$443$	−18.6709	−0.887082	−0.443541	+	0.896254i	$0.646278\pi$
$444$	0	0
$445$	−26.8570	−1.27314
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18.2721i	0.862316i	0.902276	+	0.431158i	$0.141895\pi$
$449$	18.2721i	0.862316i	−0.902276	+	0.431158i	$0.858105\pi$
$450$	0	0
$451$	− 9.01249i	− 0.424381i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.44014	0.301918
$456$	0	0
$457$	23.1665	1.08369	0.541843	−	0.840480i	$-0.317727\pi$
$457$	23.1665	1.08369	0.541843	+	0.840480i	$0.317727\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 27.0478i	− 1.25974i	−0.776700	−	0.629870i	$-0.783108\pi$
$461$	− 27.0478i	− 1.25974i	0.776700	−	0.629870i	$-0.216892\pi$
$462$	0	0
$463$	− 28.2067i	− 1.31088i	−0.755249	−	0.655438i	$-0.772484\pi$
$463$	− 28.2067i	− 1.31088i	0.755249	−	0.655438i	$-0.227516\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.3299	−0.987029	−0.493514	−	0.869738i	$-0.664288\pi$
$467$	−21.3299	−0.987029	−0.493514	+	0.869738i	$0.664288\pi$
$468$	0	0
$469$	7.72509	0.356711
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 13.5122i	− 0.621291i
$474$	0	0
$475$	4.72268i	0.216692i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.5415	1.02995	0.514973	−	0.857207i	$-0.327802\pi$
$479$	22.5415	1.02995	0.514973	+	0.857207i	$0.327802\pi$
$480$	0	0
$481$	2.90993	0.132681
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 43.1602i	− 1.95980i
$486$	0	0
$487$	− 4.51090i	− 0.204408i	−0.994763	−	0.102204i	$-0.967410\pi$
$487$	− 4.51090i	− 0.204408i	0.994763	−	0.102204i	$-0.0325895\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−30.4636	−1.37480	−0.687401	−	0.726278i	$-0.741248\pi$
$491$	−30.4636	−1.37480	−0.687401	+	0.726278i	$0.741248\pi$
$492$	0	0
$493$	−53.9611	−2.43028
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 7.98532i	− 0.358191i
$498$	0	0
$499$	11.7384i	0.525482i	0.964866	+	0.262741i	$0.0846265\pi$
$499$	11.7384i	0.525482i	−0.964866	+	0.262741i	$0.915373\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.7589	−0.836419	−0.418210	−	0.908351i	$-0.637342\pi$
$503$	−18.7589	−0.836419	−0.418210	+	0.908351i	$0.637342\pi$
$504$	0	0
$505$	−23.3385	−1.03855
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.97172i	0.309016i	0.987992	+	0.154508i	$0.0493792\pi$
$509$	6.97172i	0.309016i	−0.987992	+	0.154508i	$0.950621\pi$
$510$	0	0
$511$	10.7548i	0.475764i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.24465	0.187041
$516$	0	0
$517$	−14.7571	−0.649017
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 40.2146i	− 1.76183i	−0.473270	−	0.880917i	$-0.656926\pi$
$521$	− 40.2146i	− 1.76183i	0.473270	−	0.880917i	$-0.343074\pi$
$522$	0	0
$523$	29.6607i	1.29697i	0.761226	+	0.648486i	$0.224598\pi$
$523$	29.6607i	1.29697i	−0.761226	+	0.648486i	$0.775402\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.0307	−0.698309
$528$	0	0
$529$	−8.00993	−0.348258
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 8.21229i	− 0.355714i
$534$	0	0
$535$	− 15.0123i	− 0.649037i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.55543	0.110070
$540$	0	0
$541$	40.6687	1.74849	0.874243	−	0.485489i	$-0.161359\pi$
$541$	40.6687	1.74849	0.874243	+	0.485489i	$0.161359\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	20.6656i	0.885215i
$546$	0	0
$547$	− 18.0822i	− 0.773137i	−0.922261	−	0.386569i	$-0.873660\pi$
$547$	− 18.0822i	− 0.773137i	0.922261	−	0.386569i	$-0.126340\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−13.0598	−0.556366
$552$	0	0
$553$	2.10858	0.0896660
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.7467i	− 0.921436i	−0.887547	−	0.460718i	$-0.847592\pi$
$557$	− 21.7467i	− 0.921436i	0.887547	−	0.460718i	$-0.152408\pi$
$558$	0	0
$559$	− 12.3125i	− 0.520762i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.1061	−0.805228	−0.402614	−	0.915370i	$-0.631898\pi$
$563$	−19.1061	−0.805228	−0.402614	+	0.915370i	$0.631898\pi$
$564$	0	0
$565$	7.32311	0.308086
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.98917i	− 0.0833903i	−0.999130	−	0.0416952i	$-0.986724\pi$
$569$	− 1.98917i	− 0.0833903i	0.999130	−	0.0416952i	$-0.0132758\pi$
$570$	0	0
$571$	− 25.5278i	− 1.06831i	−0.845388	−	0.534153i	$-0.820631\pi$
$571$	− 25.5278i	− 1.06831i	0.845388	−	0.534153i	$-0.179369\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.2573	0.427759
$576$	0	0
$577$	16.8837	0.702879	0.351439	−	0.936211i	$-0.385692\pi$
$577$	16.8837	0.702879	0.351439	+	0.936211i	$0.385692\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.1458i	0.503894i
$582$	0	0
$583$	− 18.6337i	− 0.771727i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.7731	−1.72416	−0.862080	−	0.506773i	$-0.830838\pi$
$587$	−41.7731	−1.72416	−0.862080	+	0.506773i	$0.830838\pi$
$588$	0	0
$589$	−3.87980	−0.159864
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 29.6707i	− 1.21843i	−0.793006	−	0.609214i	$-0.791485\pi$
$593$	− 29.6707i	− 1.21843i	0.793006	−	0.609214i	$-0.208515\pi$
$594$	0	0
$595$	20.3710i	0.835128i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.1882	−1.27432	−0.637158	−	0.770733i	$-0.719890\pi$
$599$	−31.1882	−1.27432	−0.637158	+	0.770733i	$0.719890\pi$
$600$	0	0
$601$	34.4671	1.40594	0.702972	−	0.711218i	$-0.251856\pi$
$601$	34.4671	1.40594	0.702972	+	0.711218i	$0.251856\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.3622i	0.502593i
$606$	0	0
$607$	− 11.7335i	− 0.476249i	−0.971235	−	0.238124i	$-0.923467\pi$
$607$	− 11.7335i	− 0.476249i	0.971235	−	0.238124i	$-0.0765326\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.4469	−0.544002
$612$	0	0
$613$	0.772139	0.0311864	0.0155932	−	0.999878i	$-0.495036\pi$
$613$	0.772139	0.0311864	0.0155932	+	0.999878i	$0.495036\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.9376i	0.641625i	0.947143	+	0.320812i	$0.103956\pi$
$617$	15.9376i	0.641625i	−0.947143	+	0.320812i	$0.896044\pi$
$618$	0	0
$619$	8.41774i	0.338337i	0.985587	+	0.169169i	$0.0541083\pi$
$619$	8.41774i	0.338337i	−0.985587	+	0.169169i	$0.945892\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.71061	−0.389047
$624$	0	0
$625$	−31.2277	−1.24911
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.20446i	0.367006i
$630$	0	0
$631$	35.0824i	1.39661i	0.715800	+	0.698305i	$0.246062\pi$
$631$	35.0824i	1.39661i	−0.715800	+	0.698305i	$0.753938\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−53.1240	−2.10816
$636$	0	0
$637$	2.32854	0.0922603
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 7.02263i	− 0.277377i	−0.990336	−	0.138689i	$-0.955711\pi$
$641$	− 7.02263i	− 0.277377i	0.990336	−	0.138689i	$-0.0442887\pi$
$642$	0	0
$643$	− 19.4292i	− 0.766214i	−0.923704	−	0.383107i	$-0.874854\pi$
$643$	− 19.4292i	− 0.766214i	0.923704	−	0.383107i	$-0.125146\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.3921	0.447869	0.223935	−	0.974604i	$-0.428110\pi$
$647$	11.3921	0.447869	0.223935	+	0.974604i	$0.428110\pi$
$648$	0	0
$649$	12.7288	0.499650
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 37.4801i	− 1.46671i	−0.679845	−	0.733356i	$-0.737953\pi$
$653$	− 37.4801i	− 1.46671i	0.679845	−	0.733356i	$-0.262047\pi$
$654$	0	0
$655$	0.504503i	0.0197126i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−31.6058	−1.23119	−0.615594	−	0.788063i	$-0.711084\pi$
$659$	−31.6058	−1.23119	−0.615594	+	0.788063i	$0.711084\pi$
$660$	0	0
$661$	−38.7889	−1.50871	−0.754356	−	0.656466i	$-0.772051\pi$
$661$	−38.7889	−1.50871	−0.754356	+	0.656466i	$0.772051\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.93024i	0.191186i
$666$	0	0
$667$	28.3649i	1.09829i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.06151	−0.311211
$672$	0	0
$673$	−12.7203	−0.490331	−0.245166	−	0.969481i	$-0.578842\pi$
$673$	−12.7203	−0.490331	−0.245166	+	0.969481i	$0.578842\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.8409i	0.839414i	0.907660	+	0.419707i	$0.137867\pi$
$677$	21.8409i	0.839414i	−0.907660	+	0.419707i	$0.862133\pi$
$678$	0	0
$679$	− 15.6053i	− 0.598876i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.5781	1.09351	0.546755	−	0.837293i	$-0.315863\pi$
$683$	28.5781	1.09351	0.546755	+	0.837293i	$0.315863\pi$
$684$	0	0
$685$	15.3556	0.586706
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 16.9792i	− 0.646857i
$690$	0	0
$691$	− 12.4404i	− 0.473254i	−0.971601	−	0.236627i	$-0.923958\pi$
$691$	− 12.4404i	− 0.473254i	0.971601	−	0.236627i	$-0.0760420\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	51.9817	1.97178
$696$	0	0
$697$	25.9765	0.983930
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 30.1067i	− 1.13711i	−0.822644	−	0.568557i	$-0.807502\pi$
$701$	− 30.1067i	− 1.13711i	0.822644	−	0.568557i	$-0.192498\pi$
$702$	0	0
$703$	2.22769i	0.0840189i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.43842	−0.317359
$708$	0	0
$709$	50.3032	1.88918	0.944589	−	0.328256i	$-0.106461\pi$
$709$	50.3032	1.88918	0.944589	+	0.328256i	$0.106461\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.42663i	0.315580i
$714$	0	0
$715$	− 16.4574i	− 0.615470i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.3376	0.646583	0.323291	−	0.946299i	$-0.395211\pi$
$719$	17.3376	0.646583	0.323291	+	0.946299i	$0.395211\pi$
$720$	0	0
$721$	1.53472	0.0571561
$722$	0	0
$723$	0	0
$724$	0	0
$725$	19.4094i	0.720846i
$726$	0	0
$727$	− 42.4708i	− 1.57515i	−0.616217	−	0.787576i	$-0.711336\pi$
$727$	− 42.4708i	− 1.57515i	0.616217	−	0.787576i	$-0.288664\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	38.9459	1.44046
$732$	0	0
$733$	20.4759	0.756294	0.378147	−	0.925746i	$-0.376561\pi$
$733$	20.4759	0.756294	0.378147	+	0.925746i	$0.376561\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 19.7410i	− 0.727168i
$738$	0	0
$739$	5.95637i	0.219109i	0.993981	+	0.109554i	$0.0349424\pi$
$739$	5.95637i	0.219109i	−0.993981	+	0.109554i	$0.965058\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.66621	0.0611274	0.0305637	−	0.999533i	$-0.490270\pi$
$743$	1.66621	0.0611274	0.0305637	+	0.999533i	$0.490270\pi$
$744$	0	0
$745$	27.1818	0.995863
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 5.42794i	− 0.198333i
$750$	0	0
$751$	49.1354i	1.79298i	0.443068	+	0.896488i	$0.353890\pi$
$751$	49.1354i	1.79298i	−0.443068	+	0.896488i	$0.646110\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24.7834	0.901961
$756$	0	0
$757$	36.9665	1.34357	0.671785	−	0.740747i	$-0.265528\pi$
$757$	36.9665	1.34357	0.671785	+	0.740747i	$0.265528\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.5213i	1.32390i	0.749550	+	0.661948i	$0.230270\pi$
$761$	36.5213i	1.32390i	−0.749550	+	0.661948i	$0.769730\pi$
$762$	0	0
$763$	7.47199i	0.270504i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.5987	0.418804
$768$	0	0
$769$	27.9601	1.00827	0.504133	−	0.863626i	$-0.331812\pi$
$769$	27.9601	1.00827	0.504133	+	0.863626i	$0.331812\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 13.1835i	− 0.474178i	−0.971488	−	0.237089i	$-0.923807\pi$
$773$	− 13.1835i	− 0.474178i	0.971488	−	0.237089i	$-0.0761933\pi$
$774$	0	0
$775$	5.76612i	0.207125i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.28690	0.225252
$780$	0	0
$781$	−20.4060	−0.730183
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 39.9135i	− 1.42457i
$786$	0	0
$787$	− 36.1472i	− 1.28851i	−0.764812	−	0.644254i	$-0.777168\pi$
$787$	− 36.1472i	− 1.28851i	0.764812	−	0.644254i	$-0.222832\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.64780	0.0941448
$792$	0	0
$793$	−7.34575	−0.260855
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.63891i	0.0580531i	0.999579	+	0.0290266i	$0.00924074\pi$
$797$	1.63891i	0.0580531i	−0.999579	+	0.0290266i	$0.990759\pi$
$798$	0	0
$799$	− 42.5341i	− 1.50475i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27.4832	0.969861
$804$	0	0
$805$	10.7081	0.377411
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.2010i	1.13213i	0.824362	+	0.566063i	$0.191534\pi$
$809$	32.2010i	1.13213i	−0.824362	+	0.566063i	$0.808466\pi$
$810$	0	0
$811$	18.6552i	0.655072i	0.944839	+	0.327536i	$0.106218\pi$
$811$	18.6552i	0.655072i	−0.944839	+	0.327536i	$0.893782\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	40.0776	1.40386
$816$	0	0
$817$	9.42579	0.329767
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 6.85032i	− 0.239078i	−0.992830	−	0.119539i	$-0.961858\pi$
$821$	− 6.85032i	− 0.239078i	0.992830	−	0.119539i	$-0.0381416\pi$
$822$	0	0
$823$	− 20.8948i	− 0.728348i	−0.931331	−	0.364174i	$-0.881351\pi$
$823$	− 20.8948i	− 0.728348i	0.931331	−	0.364174i	$-0.118649\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.4714	1.37255	0.686277	−	0.727341i	$-0.259244\pi$
$827$	39.4714	1.37255	0.686277	+	0.727341i	$0.259244\pi$
$828$	0	0
$829$	7.23657	0.251336	0.125668	−	0.992072i	$-0.459893\pi$
$829$	7.23657	0.251336	0.125668	+	0.992072i	$0.459893\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.36547i	0.255199i
$834$	0	0
$835$	− 21.8098i	− 0.754760i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.32006	−0.218193	−0.109096	−	0.994031i	$-0.534796\pi$
$839$	−6.32006	−0.218193	−0.109096	+	0.994031i	$0.534796\pi$
$840$	0	0
$841$	−24.6734	−0.850808
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20.9584i	0.720992i
$846$	0	0
$847$	4.46975i	0.153583i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.83837	0.165857
$852$	0	0
$853$	−28.4940	−0.975615	−0.487807	−	0.872951i	$-0.662203\pi$
$853$	−28.4940	−0.975615	−0.487807	+	0.872951i	$0.662203\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.4708i	0.870067i	0.900414	+	0.435034i	$0.143264\pi$
$857$	25.4708i	0.870067i	−0.900414	+	0.435034i	$0.856736\pi$
$858$	0	0
$859$	48.0030i	1.63784i	0.573907	+	0.818921i	$0.305427\pi$
$859$	48.0030i	1.63784i	−0.573907	+	0.818921i	$0.694573\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34.5532	−1.17621	−0.588103	−	0.808786i	$-0.700125\pi$
$863$	−34.5532	−1.17621	−0.588103	+	0.808786i	$0.700125\pi$
$864$	0	0
$865$	−43.9205	−1.49334
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 5.38834i	− 0.182787i
$870$	0	0
$871$	− 17.9882i	− 0.609507i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−6.50141	−0.219788
$876$	0	0
$877$	−4.46779	−0.150867	−0.0754333	−	0.997151i	$-0.524034\pi$
$877$	−4.46779	−0.150867	−0.0754333	+	0.997151i	$0.524034\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 47.9230i	− 1.61457i	−0.590164	−	0.807283i	$-0.700937\pi$
$881$	− 47.9230i	− 1.61457i	0.590164	−	0.807283i	$-0.299063\pi$
$882$	0	0
$883$	− 7.50033i	− 0.252406i	−0.992004	−	0.126203i	$-0.959721\pi$
$883$	− 7.50033i	− 0.252406i	0.992004	−	0.126203i	$-0.0402791\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.8246	0.397030	0.198515	−	0.980098i	$-0.436388\pi$
$887$	11.8246	0.397030	0.198515	+	0.980098i	$0.436388\pi$
$888$	0	0
$889$	−19.2079	−0.644213
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10.2942i	− 0.344483i
$894$	0	0
$895$	50.4755i	1.68721i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.9453	−0.531804
$900$	0	0
$901$	53.7074	1.78925
$902$	0	0
$903$	0	0
$904$	0	0
$905$	66.8527i	2.22226i
$906$	0	0
$907$	− 34.2557i	− 1.13744i	−0.822531	−	0.568721i	$-0.807438\pi$
$907$	− 34.2557i	− 1.13744i	0.822531	−	0.568721i	$-0.192562\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.8731	−0.459638	−0.229819	−	0.973233i	$-0.573813\pi$
$911$	−13.8731	−0.459638	−0.229819	+	0.973233i	$0.573813\pi$
$912$	0	0
$913$	31.0379	1.02720
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.182412i	0.00602376i
$918$	0	0
$919$	− 19.0500i	− 0.628402i	−0.949356	−	0.314201i	$-0.898263\pi$
$919$	− 19.0500i	− 0.628402i	0.949356	−	0.314201i	$-0.101737\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.5942	−0.612035
$924$	0	0
$925$	3.31077	0.108857
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 6.93319i	− 0.227470i	−0.993511	−	0.113735i	$-0.963718\pi$
$929$	− 6.93319i	− 0.227470i	0.993511	−	0.113735i	$-0.0362816\pi$
$930$	0	0
$931$	1.78261i	0.0584228i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	52.0567	1.70244
$936$	0	0
$937$	16.7279	0.546475	0.273238	−	0.961947i	$-0.411905\pi$
$937$	16.7279	0.546475	0.273238	+	0.961947i	$0.411905\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13.5223i	0.440814i	0.975408	+	0.220407i	$0.0707386\pi$
$941$	13.5223i	0.440814i	−0.975408	+	0.220407i	$0.929261\pi$
$942$	0	0
$943$	− 13.6547i	− 0.444658i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.4695	−0.502692	−0.251346	−	0.967897i	$-0.580873\pi$
$947$	−15.4695	−0.502692	−0.251346	+	0.967897i	$0.580873\pi$
$948$	0	0
$949$	25.0430	0.812931
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 16.4797i	− 0.533831i	−0.963720	−	0.266915i	$-0.913996\pi$
$953$	− 16.4797i	− 0.533831i	0.963720	−	0.266915i	$-0.0860044\pi$
$954$	0	0
$955$	− 51.6765i	− 1.67221i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.55207	0.179286
$960$	0	0
$961$	26.2630	0.847193
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 19.0673i	− 0.613798i
$966$	0	0
$967$	− 43.1895i	− 1.38888i	−0.719550	−	0.694441i	$-0.755652\pi$
$967$	− 43.1895i	− 1.38888i	0.719550	−	0.694441i	$-0.244348\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.5954	−0.693029	−0.346515	−	0.938045i	$-0.612635\pi$
$971$	−21.5954	−0.693029	−0.346515	+	0.938045i	$0.612635\pi$
$972$	0	0
$973$	18.7949	0.602536
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.08900i	0.290783i	0.989374	+	0.145391i	$0.0464442\pi$
$977$	9.08900i	0.290783i	−0.989374	+	0.145391i	$0.953556\pi$
$978$	0	0
$979$	24.8148i	0.793085i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.7430	−0.725388	−0.362694	−	0.931908i	$-0.618143\pi$
$983$	−22.7430	−0.725388	−0.362694	+	0.931908i	$0.618143\pi$
$984$	0	0
$985$	25.1918	0.802678
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 20.4721i	− 0.650975i
$990$	0	0
$991$	− 5.96925i	− 0.189619i	−0.995495	−	0.0948097i	$-0.969776\pi$
$991$	− 5.96925i	− 0.189619i	0.995495	−	0.0948097i	$-0.0302243\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−26.3848	−0.836454
$996$	0	0
$997$	−42.1258	−1.33414	−0.667069	−	0.744996i	$-0.732451\pi$
$997$	−42.1258	−1.33414	−0.667069	+	0.744996i	$0.732451\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.i.2591.17	yes	24
3.2	odd	2	inner	6048.2.h.i.2591.8	yes	24
4.3	odd	2	inner	6048.2.h.i.2591.7	✓	24
12.11	even	2	inner	6048.2.h.i.2591.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.i.2591.7	✓	24	4.3	odd	2	inner
6048.2.h.i.2591.8	yes	24	3.2	odd	2	inner
6048.2.h.i.2591.17	yes	24	1.1	even	1	trivial
6048.2.h.i.2591.18	yes	24	12.11	even	2	inner

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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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.76574i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-2.76574i q^{5} -1.00000i q^{7} -2.55543 q^{11} -2.32854 q^{13} -7.36547i q^{17} -1.78261i q^{19} -3.87170 q^{23} -2.64930 q^{25} -7.32622i q^{29} -2.17647i q^{31} -2.76574 q^{35} -1.24968 q^{37} +3.52679i q^{41} +5.28762i q^{43} +5.77479 q^{47} -1.00000 q^{49} +7.29178i q^{53} +7.06766i q^{55} -4.98108 q^{59} +3.15465 q^{61} +6.44014i q^{65} +7.72509i q^{67} +7.98532 q^{71} -10.7548 q^{73} +2.55543i q^{77} +2.10858i q^{79} -12.1458 q^{83} -20.3710 q^{85} -9.71061i q^{89} +2.32854i q^{91} -4.93024 q^{95} +15.6053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{13} - 24 q^{25} + 48 q^{37} - 24 q^{49} - 48 q^{61} - 32 q^{73} - 80 q^{85} + 64 q^{97}+O(q^{100}) \) 24 * q - 8 * q^13 - 24 * q^25 + 48 * q^37 - 24 * q^49 - 48 * q^61 - 32 * q^73 - 80 * q^85 + 64 * q^97

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