LMFDB - Embedded newform 6048.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(1,6048)]

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([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("6048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.a (trivial)

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:	yes
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.22896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 4x + 3 $ x^4 - 9x^2 - 4x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.398683$ of defining polynomial
Character	$\chi$	$=$	6048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.38773 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+4.38773 q^{5} -1.00000 q^{7} -3.59037 q^{11} +0.797366 q^{13} +0.295993 q^{17} -0.704007 q^{19} +7.86446 q^{23} +14.2522 q^{25} +4.50137 q^{29} +1.50137 q^{31} -4.38773 q^{35} -0.202634 q^{37} -8.66182 q^{41} +2.79737 q^{43} -8.27409 q^{47} +1.00000 q^{49} +4.00000 q^{53} -15.7536 q^{55} -8.27409 q^{59} +11.9562 q^{61} +3.49863 q^{65} -13.3455 q^{67} +6.79300 q^{71} +6.20538 q^{73} +3.59037 q^{77} +14.7536 q^{79} +14.6443 q^{83} +1.29874 q^{85} -4.66182 q^{89} -0.797366 q^{91} -3.08899 q^{95} -8.54818 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^5 - 4 * q^7	$ 4 q + 2 q^{5} - 4 q^{7} - 2 q^{11} + 8 q^{17} + 4 q^{19} - 2 q^{23} + 8 q^{25} + 8 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{37} + 2 q^{41} + 8 q^{43} - 12 q^{47} + 4 q^{49} + 16 q^{53} - 4 q^{55} - 12 q^{59} - 8 q^{61} + 24 q^{65} - 8 q^{67} + 18 q^{71} + 8 q^{73} + 2 q^{77} - 8 q^{85} + 18 q^{89} - 10 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 - 4 * q^7 - 2 * q^11 + 8 * q^17 + 4 * q^19 - 2 * q^23 + 8 * q^25 + 8 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^37 + 2 * q^41 + 8 * q^43 - 12 * q^47 + 4 * q^49 + 16 * q^53 - 4 * q^55 - 12 * q^59 - 8 * q^61 + 24 * q^65 - 8 * q^67 + 18 * q^71 + 8 * q^73 + 2 * q^77 - 8 * q^85 + 18 * q^89 - 10 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	4.38773	1.96225	0.981127	−	0.193367i	$-0.0619407\pi$
$5$	4.38773	1.96225	0.981127	+	0.193367i	$0.0619407\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.59037	−1.08254	−0.541268	−	0.840850i	$-0.682056\pi$
$11$	−3.59037	−1.08254	−0.541268	+	0.840850i	$0.682056\pi$
$12$	0	0
$13$	0.797366	0.221150	0.110575	−	0.993868i	$-0.464731\pi$
$13$	0.797366	0.221150	0.110575	+	0.993868i	$0.464731\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.295993	0.0717889	0.0358944	−	0.999356i	$-0.488572\pi$
$17$	0.295993	0.0717889	0.0358944	+	0.999356i	$0.488572\pi$
$18$	0	0
$19$	−0.704007	−0.161510	−0.0807551	−	0.996734i	$-0.525733\pi$
$19$	−0.704007	−0.161510	−0.0807551	+	0.996734i	$0.525733\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.86446	1.63985	0.819926	−	0.572469i	$-0.194014\pi$
$23$	7.86446	1.63985	0.819926	+	0.572469i	$0.194014\pi$
$24$	0	0
$25$	14.2522	2.85044
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.50137	0.835884	0.417942	−	0.908474i	$-0.362752\pi$
$29$	4.50137	0.835884	0.417942	+	0.908474i	$0.362752\pi$
$30$	0	0
$31$	1.50137	0.269654	0.134827	−	0.990869i	$-0.456952\pi$
$31$	1.50137	0.269654	0.134827	+	0.990869i	$0.456952\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.38773	−0.741662
$36$	0	0
$37$	−0.202634	−0.0333128	−0.0166564	−	0.999861i	$-0.505302\pi$
$37$	−0.202634	−0.0333128	−0.0166564	+	0.999861i	$0.505302\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.66182	−1.35275	−0.676375	−	0.736558i	$-0.736450\pi$
$41$	−8.66182	−1.35275	−0.676375	+	0.736558i	$0.736450\pi$
$42$	0	0
$43$	2.79737	0.426594	0.213297	−	0.976987i	$-0.431580\pi$
$43$	2.79737	0.426594	0.213297	+	0.976987i	$0.431580\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.27409	−1.20690	−0.603450	−	0.797401i	$-0.706208\pi$
$47$	−8.27409	−1.20690	−0.603450	+	0.797401i	$0.706208\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	−15.7536	−2.12421
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.27409	−1.07719	−0.538597	−	0.842563i	$-0.681046\pi$
$59$	−8.27409	−1.07719	−0.538597	+	0.842563i	$0.681046\pi$
$60$	0	0
$61$	11.9562	1.53083	0.765417	−	0.643535i	$-0.222533\pi$
$61$	11.9562	1.53083	0.765417	+	0.643535i	$0.222533\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.49863	0.433951
$66$	0	0
$67$	−13.3455	−1.63042	−0.815209	−	0.579167i	$-0.803378\pi$
$67$	−13.3455	−1.63042	−0.815209	+	0.579167i	$0.803378\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.79300	0.806181	0.403090	−	0.915160i	$-0.367936\pi$
$71$	6.79300	0.806181	0.403090	+	0.915160i	$0.367936\pi$
$72$	0	0
$73$	6.20538	0.726285	0.363142	−	0.931734i	$-0.381704\pi$
$73$	6.20538	0.726285	0.363142	+	0.931734i	$0.381704\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.59037	0.409160
$78$	0	0
$79$	14.7536	1.65991	0.829953	−	0.557834i	$-0.188367\pi$
$79$	14.7536	1.65991	0.829953	+	0.557834i	$0.188367\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.6443	1.60742	0.803710	−	0.595022i	$-0.202857\pi$
$83$	14.6443	1.60742	0.803710	+	0.595022i	$0.202857\pi$
$84$	0	0
$85$	1.29874	0.140868
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.66182	−0.494152	−0.247076	−	0.968996i	$-0.579470\pi$
$89$	−4.66182	−0.494152	−0.247076	+	0.968996i	$0.579470\pi$
$90$	0	0
$91$	−0.797366	−0.0835867
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.08899	−0.316924
$96$	0	0
$97$	−8.54818	−0.867936	−0.433968	−	0.900928i	$-0.642887\pi$
$97$	−8.54818	−0.867936	−0.433968	+	0.900928i	$0.642887\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.29599	0.825482	0.412741	−	0.910848i	$-0.364571\pi$
$101$	8.29599	0.825482	0.412741	+	0.910848i	$0.364571\pi$
$102$	0	0
$103$	9.04681	0.891408	0.445704	−	0.895180i	$-0.352953\pi$
$103$	9.04681	0.891408	0.445704	+	0.895180i	$0.352953\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.8847	1.05227	0.526134	−	0.850402i	$-0.323641\pi$
$107$	10.8847	1.05227	0.526134	+	0.850402i	$0.323641\pi$
$108$	0	0
$109$	6.45482	0.618260	0.309130	−	0.951020i	$-0.399962\pi$
$109$	6.45482	0.618260	0.309130	+	0.951020i	$0.399962\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.1807	1.05179	0.525897	−	0.850548i	$-0.323730\pi$
$113$	11.1807	1.05179	0.525897	+	0.850548i	$0.323730\pi$
$114$	0	0
$115$	34.5071	3.21781
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.295993	−0.0271337
$120$	0	0
$121$	1.89072	0.171884
$122$	0	0
$123$	0	0
$124$	0	0
$125$	40.5961	3.63103
$126$	0	0
$127$	−14.3483	−1.27320	−0.636602	−	0.771192i	$-0.719661\pi$
$127$	−14.3483	−1.27320	−0.636602	+	0.771192i	$0.719661\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.2768	−1.50948	−0.754742	−	0.656022i	$-0.772238\pi$
$131$	−17.2768	−1.50948	−0.754742	+	0.656022i	$0.772238\pi$
$132$	0	0
$133$	0.704007	0.0610451
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.31191	0.453827	0.226914	−	0.973915i	$-0.427137\pi$
$137$	5.31191	0.453827	0.226914	+	0.973915i	$0.427137\pi$
$138$	0	0
$139$	10.9534	0.929059	0.464530	−	0.885558i	$-0.346223\pi$
$139$	10.9534	0.929059	0.464530	+	0.885558i	$0.346223\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.86283	−0.239402
$144$	0	0
$145$	19.7508	1.64022
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.3674	1.09510	0.547552	−	0.836772i	$-0.315560\pi$
$149$	13.3674	1.09510	0.547552	+	0.836772i	$0.315560\pi$
$150$	0	0
$151$	9.95619	0.810224	0.405112	−	0.914267i	$-0.367233\pi$
$151$	9.95619	0.810224	0.405112	+	0.914267i	$0.367233\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.58762	0.529130
$156$	0	0
$157$	−9.80011	−0.782134	−0.391067	−	0.920362i	$-0.627894\pi$
$157$	−9.80011	−0.782134	−0.391067	+	0.920362i	$0.627894\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.86446	−0.619806
$162$	0	0
$163$	−5.38935	−0.422127	−0.211063	−	0.977472i	$-0.567693\pi$
$163$	−5.38935	−0.422127	−0.211063	+	0.977472i	$0.567693\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.1807	−1.17472	−0.587360	−	0.809326i	$-0.699833\pi$
$167$	−15.1807	−1.17472	−0.587360	+	0.809326i	$0.699833\pi$
$168$	0	0
$169$	−12.3642	−0.951093
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.6837	0.964326	0.482163	−	0.876082i	$-0.339851\pi$
$173$	12.6837	0.964326	0.482163	+	0.876082i	$0.339851\pi$
$174$	0	0
$175$	−14.2522	−1.07736
$176$	0	0
$177$	0	0
$178$	0	0
$179$	17.0715	1.27598	0.637990	−	0.770044i	$-0.279766\pi$
$179$	17.0715	1.27598	0.637990	+	0.770044i	$0.279766\pi$
$180$	0	0
$181$	−21.3642	−1.58799	−0.793995	−	0.607925i	$-0.792002\pi$
$181$	−21.3642	−1.58799	−0.793995	+	0.607925i	$0.792002\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.889104	−0.0653682
$186$	0	0
$187$	−1.06272	−0.0777141
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.31902	−0.312513	−0.156257	−	0.987716i	$-0.549943\pi$
$191$	−4.31902	−0.312513	−0.156257	+	0.987716i	$0.549943\pi$
$192$	0	0
$193$	−8.14291	−0.586140	−0.293070	−	0.956091i	$-0.594677\pi$
$193$	−8.14291	−0.586140	−0.293070	+	0.956091i	$0.594677\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.32364	−0.664282	−0.332141	−	0.943230i	$-0.607771\pi$
$197$	−9.32364	−0.664282	−0.332141	+	0.943230i	$0.607771\pi$
$198$	0	0
$199$	17.3483	1.22979	0.614894	−	0.788610i	$-0.289199\pi$
$199$	17.3483	1.22979	0.614894	+	0.788610i	$0.289199\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.50137	−0.315934
$204$	0	0
$205$	−38.0057	−2.65444
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.52764	0.174841
$210$	0	0
$211$	−22.9589	−1.58056	−0.790279	−	0.612747i	$-0.790065\pi$
$211$	−22.9589	−1.58056	−0.790279	+	0.612747i	$0.790065\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.2741	0.837086
$216$	0	0
$217$	−1.50137	−0.101920
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.236015	0.0158761
$222$	0	0
$223$	1.79737	0.120361	0.0601803	−	0.998188i	$-0.480832\pi$
$223$	1.79737	0.120361	0.0601803	+	0.998188i	$0.480832\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.1835	0.808646	0.404323	−	0.914616i	$-0.367507\pi$
$227$	12.1835	0.808646	0.404323	+	0.914616i	$0.367507\pi$
$228$	0	0
$229$	−23.1457	−1.52951	−0.764754	−	0.644322i	$-0.777140\pi$
$229$	−23.1457	−1.52951	−0.764754	+	0.644322i	$0.777140\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.7289	−0.768387	−0.384193	−	0.923253i	$-0.625520\pi$
$233$	−11.7289	−0.768387	−0.384193	+	0.923253i	$0.625520\pi$
$234$	0	0
$235$	−36.3045	−2.36824
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.4822	1.13083	0.565415	−	0.824806i	$-0.308716\pi$
$239$	17.4822	1.13083	0.565415	+	0.824806i	$0.308716\pi$
$240$	0	0
$241$	21.7508	1.40109	0.700547	−	0.713607i	$-0.252940\pi$
$241$	21.7508	1.40109	0.700547	+	0.713607i	$0.252940\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.38773	0.280322
$246$	0	0
$247$	−0.561351	−0.0357179
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.5975	0.668907	0.334453	−	0.942412i	$-0.391448\pi$
$251$	10.5975	0.668907	0.334453	+	0.942412i	$0.391448\pi$
$252$	0	0
$253$	−28.2363	−1.77520
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.9578	1.30731	0.653656	−	0.756792i	$-0.273234\pi$
$257$	20.9578	1.30731	0.653656	+	0.756792i	$0.273234\pi$
$258$	0	0
$259$	0.202634	0.0125911
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.27846	−0.140496	−0.0702478	−	0.997530i	$-0.522379\pi$
$263$	−2.27846	−0.140496	−0.0702478	+	0.997530i	$0.522379\pi$
$264$	0	0
$265$	17.5509	1.07814
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.8672	1.02841	0.514206	−	0.857667i	$-0.328087\pi$
$269$	16.8672	1.02841	0.514206	+	0.857667i	$0.328087\pi$
$270$	0	0
$271$	5.81328	0.353132	0.176566	−	0.984289i	$-0.443501\pi$
$271$	5.81328	0.353132	0.176566	+	0.984289i	$0.443501\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−51.1706	−3.08570
$276$	0	0
$277$	−16.1588	−0.970890	−0.485445	−	0.874267i	$-0.661342\pi$
$277$	−16.1588	−0.970890	−0.485445	+	0.874267i	$0.661342\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.86608	0.290286	0.145143	−	0.989411i	$-0.453636\pi$
$281$	4.86608	0.290286	0.145143	+	0.989411i	$0.453636\pi$
$282$	0	0
$283$	21.5509	1.28107	0.640535	−	0.767929i	$-0.278713\pi$
$283$	21.5509	1.28107	0.640535	+	0.767929i	$0.278713\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.66182	0.511291
$288$	0	0
$289$	−16.9124	−0.994846
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−23.3978	−1.36692	−0.683458	−	0.729990i	$-0.739525\pi$
$293$	−23.3978	−1.36692	−0.683458	+	0.729990i	$0.739525\pi$
$294$	0	0
$295$	−36.3045	−2.11373
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.27085	0.362653
$300$	0	0
$301$	−2.79737	−0.161237
$302$	0	0
$303$	0	0
$304$	0	0
$305$	52.4606	3.00388
$306$	0	0
$307$	−34.0189	−1.94156	−0.970781	−	0.239967i	$-0.922863\pi$
$307$	−34.0189	−1.94156	−0.970781	+	0.239967i	$0.922863\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.5977	1.22470	0.612348	−	0.790589i	$-0.290225\pi$
$311$	21.5977	1.22470	0.612348	+	0.790589i	$0.290225\pi$
$312$	0	0
$313$	−8.34280	−0.471563	−0.235781	−	0.971806i	$-0.575765\pi$
$313$	−8.34280	−0.471563	−0.235781	+	0.971806i	$0.575765\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.0901	1.29687	0.648435	−	0.761270i	$-0.275424\pi$
$317$	23.0901	1.29687	0.648435	+	0.761270i	$0.275424\pi$
$318$	0	0
$319$	−16.1616	−0.904874
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.208381	−0.0115946
$324$	0	0
$325$	11.3642	0.630373
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.27409	0.456165
$330$	0	0
$331$	30.0991	1.65440	0.827198	−	0.561910i	$-0.189933\pi$
$331$	30.0991	1.65440	0.827198	+	0.561910i	$0.189933\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−58.5567	−3.19929
$336$	0	0
$337$	25.3615	1.38153	0.690763	−	0.723081i	$-0.257275\pi$
$337$	25.3615	1.38153	0.690763	+	0.723081i	$0.257275\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.39048	−0.291911
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.43454	0.452790	0.226395	−	0.974036i	$-0.427306\pi$
$347$	8.43454	0.452790	0.226395	+	0.974036i	$0.427306\pi$
$348$	0	0
$349$	−19.5947	−1.04888	−0.524441	−	0.851447i	$-0.675726\pi$
$349$	−19.5947	−1.04888	−0.524441	+	0.851447i	$0.675726\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.7111	0.995892	0.497946	−	0.867208i	$-0.334088\pi$
$353$	18.7111	0.995892	0.497946	+	0.867208i	$0.334088\pi$
$354$	0	0
$355$	29.8059	1.58193
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.02216	−0.370615	−0.185308	−	0.982681i	$-0.559328\pi$
$359$	−7.02216	−0.370615	−0.185308	+	0.982681i	$0.559328\pi$
$360$	0	0
$361$	−18.5044	−0.973914
$362$	0	0
$363$	0	0
$364$	0	0
$365$	27.2275	1.42515
$366$	0	0
$367$	19.4912	1.01743	0.508716	−	0.860934i	$-0.330120\pi$
$367$	19.4912	1.01743	0.508716	+	0.860934i	$0.330120\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	25.0085	1.29489	0.647445	−	0.762112i	$-0.275837\pi$
$373$	25.0085	1.29489	0.647445	+	0.762112i	$0.275837\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.58924	0.184855
$378$	0	0
$379$	−16.3921	−0.842005	−0.421003	−	0.907059i	$-0.638322\pi$
$379$	−16.3921	−0.842005	−0.421003	+	0.907059i	$0.638322\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.2798	0.678568	0.339284	−	0.940684i	$-0.389815\pi$
$383$	13.2798	0.678568	0.339284	+	0.940684i	$0.389815\pi$
$384$	0	0
$385$	15.7536	0.802876
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.5537	−1.14352	−0.571758	−	0.820422i	$-0.693738\pi$
$389$	−22.5537	−1.14352	−0.571758	+	0.820422i	$0.693738\pi$
$390$	0	0
$391$	2.32783	0.117723
$392$	0	0
$393$	0	0
$394$	0	0
$395$	64.7347	3.25715
$396$	0	0
$397$	−35.3149	−1.77240	−0.886202	−	0.463299i	$-0.846666\pi$
$397$	−35.3149	−1.77240	−0.886202	+	0.463299i	$0.846666\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.8218	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$401$	35.8218	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$402$	0	0
$403$	1.19714	0.0596340
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.727531	0.0360624
$408$	0	0
$409$	−21.9430	−1.08501	−0.542506	−	0.840052i	$-0.682525\pi$
$409$	−21.9430	−1.08501	−0.542506	+	0.840052i	$0.682525\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.27409	0.407141
$414$	0	0
$415$	64.2552	3.15416
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.67636	−0.130749	−0.0653743	−	0.997861i	$-0.520824\pi$
$419$	−2.67636	−0.130749	−0.0653743	+	0.997861i	$0.520824\pi$
$420$	0	0
$421$	7.93479	0.386718	0.193359	−	0.981128i	$-0.438062\pi$
$421$	7.93479	0.386718	0.193359	+	0.981128i	$0.438062\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.21855	0.204630
$426$	0	0
$427$	−11.9562	−0.578601
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.13280	0.150902	0.0754508	−	0.997150i	$-0.475960\pi$
$431$	3.13280	0.150902	0.0754508	+	0.997150i	$0.475960\pi$
$432$	0	0
$433$	26.1561	1.25698	0.628491	−	0.777817i	$-0.283673\pi$
$433$	26.1561	1.25698	0.628491	+	0.777817i	$0.283673\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.53663	−0.264853
$438$	0	0
$439$	−9.18397	−0.438327	−0.219164	−	0.975688i	$-0.570333\pi$
$439$	−9.18397	−0.438327	−0.219164	+	0.975688i	$0.570333\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.1353	−1.81186	−0.905931	−	0.423424i	$-0.860828\pi$
$443$	−38.1353	−1.81186	−0.905931	+	0.423424i	$0.860828\pi$
$444$	0	0
$445$	−20.4548	−0.969652
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.4140	0.491467	0.245734	−	0.969337i	$-0.420971\pi$
$449$	10.4140	0.491467	0.245734	+	0.969337i	$0.420971\pi$
$450$	0	0
$451$	31.0991	1.46440
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.49863	−0.164018
$456$	0	0
$457$	−14.1457	−0.661706	−0.330853	−	0.943682i	$-0.607336\pi$
$457$	−14.1457	−0.661706	−0.330853	+	0.943682i	$0.607336\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.91101	−0.228728	−0.114364	−	0.993439i	$-0.536483\pi$
$461$	−4.91101	−0.228728	−0.114364	+	0.993439i	$0.536483\pi$
$462$	0	0
$463$	−3.59473	−0.167061	−0.0835307	−	0.996505i	$-0.526620\pi$
$463$	−3.59473	−0.167061	−0.0835307	+	0.996505i	$0.526620\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−23.0463	−1.06646	−0.533228	−	0.845972i	$-0.679021\pi$
$467$	−23.0463	−1.06646	−0.533228	+	0.845972i	$0.679021\pi$
$468$	0	0
$469$	13.3455	0.616240
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.0436	−0.461804
$474$	0	0
$475$	−10.0336	−0.460375
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.7404	−0.582123	−0.291062	−	0.956704i	$-0.594008\pi$
$479$	−12.7404	−0.582123	−0.291062	+	0.956704i	$0.594008\pi$
$480$	0	0
$481$	−0.161574	−0.00736712
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−37.5071	−1.70311
$486$	0	0
$487$	−29.1150	−1.31933	−0.659664	−	0.751561i	$-0.729301\pi$
$487$	−29.1150	−1.31933	−0.659664	+	0.751561i	$0.729301\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−31.2155	−1.40874	−0.704368	−	0.709835i	$-0.748769\pi$
$491$	−31.2155	−1.40874	−0.704368	+	0.709835i	$0.748769\pi$
$492$	0	0
$493$	1.33238	0.0600072
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.79300	−0.304708
$498$	0	0
$499$	−29.1588	−1.30533	−0.652664	−	0.757647i	$-0.726349\pi$
$499$	−29.1588	−1.30533	−0.652664	+	0.757647i	$0.726349\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.8337	−0.795168	−0.397584	−	0.917566i	$-0.630151\pi$
$503$	−17.8337	−0.795168	−0.397584	+	0.917566i	$0.630151\pi$
$504$	0	0
$505$	36.4006	1.61980
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.65471	−0.250641	−0.125320	−	0.992116i	$-0.539996\pi$
$509$	−5.65471	−0.250641	−0.125320	+	0.992116i	$0.539996\pi$
$510$	0	0
$511$	−6.20538	−0.274510
$512$	0	0
$513$	0	0
$514$	0	0
$515$	39.6950	1.74917
$516$	0	0
$517$	29.7070	1.30651
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.00162	−0.394368	−0.197184	−	0.980366i	$-0.563180\pi$
$521$	−9.00162	−0.394368	−0.197184	+	0.980366i	$0.563180\pi$
$522$	0	0
$523$	13.9589	0.610382	0.305191	−	0.952291i	$-0.401280\pi$
$523$	13.9589	0.610382	0.305191	+	0.952291i	$0.401280\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.444396	0.0193582
$528$	0	0
$529$	38.8497	1.68912
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.90664	−0.299160
$534$	0	0
$535$	47.7593	2.06481
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.59037	−0.154648
$540$	0	0
$541$	−1.24919	−0.0537067	−0.0268533	−	0.999639i	$-0.508549\pi$
$541$	−1.24919	−0.0537067	−0.0268533	+	0.999639i	$0.508549\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	28.3220	1.21318
$546$	0	0
$547$	−18.6911	−0.799173	−0.399587	−	0.916695i	$-0.630846\pi$
$547$	−18.6911	−0.799173	−0.399587	+	0.916695i	$0.630846\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.16900	−0.135004
$552$	0	0
$553$	−14.7536	−0.627385
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−42.2388	−1.78971	−0.894857	−	0.446353i	$-0.852723\pi$
$557$	−42.2388	−1.78971	−0.894857	+	0.446353i	$0.852723\pi$
$558$	0	0
$559$	2.23052	0.0943411
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.0030	−0.843026	−0.421513	−	0.906822i	$-0.638501\pi$
$563$	−20.0030	−0.843026	−0.421513	+	0.906822i	$0.638501\pi$
$564$	0	0
$565$	49.0580	2.06389
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.36470	0.350667	0.175333	−	0.984509i	$-0.443900\pi$
$569$	8.36470	0.350667	0.175333	+	0.984509i	$0.443900\pi$
$570$	0	0
$571$	19.7695	0.827327	0.413663	−	0.910430i	$-0.364249\pi$
$571$	19.7695	0.827327	0.413663	+	0.910430i	$0.364249\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	112.086	4.67430
$576$	0	0
$577$	12.1867	0.507340	0.253670	−	0.967291i	$-0.418362\pi$
$577$	12.1867	0.507340	0.253670	+	0.967291i	$0.418362\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.6443	−0.607547
$582$	0	0
$583$	−14.3615	−0.594791
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.54219	0.393848	0.196924	−	0.980419i	$-0.436905\pi$
$587$	9.54219	0.393848	0.196924	+	0.980419i	$0.436905\pi$
$588$	0	0
$589$	−1.05698	−0.0435520
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.1002	1.23607	0.618034	−	0.786151i	$-0.287929\pi$
$593$	30.1002	1.23607	0.618034	+	0.786151i	$0.287929\pi$
$594$	0	0
$595$	−1.29874	−0.0532431
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.0101	−1.59391	−0.796955	−	0.604039i	$-0.793557\pi$
$599$	−39.0101	−1.59391	−0.796955	+	0.604039i	$0.793557\pi$
$600$	0	0
$601$	3.20263	0.130638	0.0653191	−	0.997864i	$-0.479193\pi$
$601$	3.20263	0.130638	0.0653191	+	0.997864i	$0.479193\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.29599	0.337280
$606$	0	0
$607$	−22.1429	−0.898753	−0.449377	−	0.893342i	$-0.648354\pi$
$607$	−22.1429	−0.898753	−0.449377	+	0.893342i	$0.648354\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.59748	−0.266905
$612$	0	0
$613$	1.50137	0.0606399	0.0303199	−	0.999540i	$-0.490347\pi$
$613$	1.50137	0.0606399	0.0303199	+	0.999540i	$0.490347\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.2275	−0.452004	−0.226002	−	0.974127i	$-0.572566\pi$
$617$	−11.2275	−0.452004	−0.226002	+	0.974127i	$0.572566\pi$
$618$	0	0
$619$	−7.09911	−0.285337	−0.142669	−	0.989771i	$-0.545568\pi$
$619$	−7.09911	−0.285337	−0.142669	+	0.989771i	$0.545568\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.66182	0.186772
$624$	0	0
$625$	106.864	4.27456
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.0599783	−0.00239149
$630$	0	0
$631$	46.7098	1.85949	0.929743	−	0.368209i	$-0.120029\pi$
$631$	46.7098	1.85949	0.929743	+	0.368209i	$0.120029\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−62.9565	−2.49835
$636$	0	0
$637$	0.797366	0.0315928
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.13693	0.123901	0.0619505	−	0.998079i	$-0.480268\pi$
$641$	3.13693	0.123901	0.0619505	+	0.998079i	$0.480268\pi$
$642$	0	0
$643$	−29.2522	−1.15359	−0.576797	−	0.816888i	$-0.695698\pi$
$643$	−29.2522	−1.15359	−0.576797	+	0.816888i	$0.695698\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.37320	−0.250556	−0.125278	−	0.992122i	$-0.539982\pi$
$647$	−6.37320	−0.250556	−0.125278	+	0.992122i	$0.539982\pi$
$648$	0	0
$649$	29.7070	1.16610
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.0901	−0.433990	−0.216995	−	0.976173i	$-0.569626\pi$
$653$	−11.0901	−0.433990	−0.216995	+	0.976173i	$0.569626\pi$
$654$	0	0
$655$	−75.8061	−2.96199
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.9633	1.36198	0.680989	−	0.732294i	$-0.261550\pi$
$659$	34.9633	1.36198	0.680989	+	0.732294i	$0.261550\pi$
$660$	0	0
$661$	−35.2524	−1.37116	−0.685581	−	0.727997i	$-0.740452\pi$
$661$	−35.2524	−1.37116	−0.685581	+	0.727997i	$0.740452\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.08899	0.119786
$666$	0	0
$667$	35.4008	1.37073
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−42.9271	−1.65718
$672$	0	0
$673$	−4.14291	−0.159698	−0.0798488	−	0.996807i	$-0.525444\pi$
$673$	−4.14291	−0.159698	−0.0798488	+	0.996807i	$0.525444\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.5306	−0.558458	−0.279229	−	0.960225i	$-0.590079\pi$
$677$	−14.5306	−0.558458	−0.279229	+	0.960225i	$0.590079\pi$
$678$	0	0
$679$	8.54818	0.328049
$680$	0	0
$681$	0	0
$682$	0	0
$683$	23.0422	0.881685	0.440842	−	0.897585i	$-0.354680\pi$
$683$	23.0422	0.881685	0.440842	+	0.897585i	$0.354680\pi$
$684$	0	0
$685$	23.3072	0.890524
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.18946	0.121509
$690$	0	0
$691$	25.3587	0.964691	0.482346	−	0.875981i	$-0.339785\pi$
$691$	25.3587	0.964691	0.482346	+	0.875981i	$0.339785\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	48.0608	1.82305
$696$	0	0
$697$	−2.56384	−0.0971124
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34.1459	1.28967	0.644837	−	0.764320i	$-0.276925\pi$
$701$	34.1459	1.28967	0.644837	+	0.764320i	$0.276925\pi$
$702$	0	0
$703$	0.142656	0.00538037
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.29599	−0.312003
$708$	0	0
$709$	−39.9301	−1.49961	−0.749803	−	0.661661i	$-0.769852\pi$
$709$	−39.9301	−1.49961	−0.749803	+	0.661661i	$0.769852\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11.8075	0.442194
$714$	0	0
$715$	−12.5614	−0.469768
$716$	0	0
$717$	0	0
$718$	0	0
$719$	5.49539	0.204943	0.102472	−	0.994736i	$-0.467325\pi$
$719$	5.49539	0.204943	0.102472	+	0.994736i	$0.467325\pi$
$720$	0	0
$721$	−9.04681	−0.336921
$722$	0	0
$723$	0	0
$724$	0	0
$725$	64.1544	2.38263
$726$	0	0
$727$	−35.5564	−1.31871	−0.659357	−	0.751830i	$-0.729172\pi$
$727$	−35.5564	−1.31871	−0.659357	+	0.751830i	$0.729172\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.828001	0.0306247
$732$	0	0
$733$	−34.7536	−1.28365	−0.641826	−	0.766850i	$-0.721823\pi$
$733$	−34.7536	−1.28365	−0.641826	+	0.766850i	$0.721823\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	47.9154	1.76499
$738$	0	0
$739$	47.8367	1.75970	0.879852	−	0.475248i	$-0.157642\pi$
$739$	47.8367	1.75970	0.879852	+	0.475248i	$0.157642\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.5717	0.644643	0.322322	−	0.946630i	$-0.395537\pi$
$743$	17.5717	0.644643	0.322322	+	0.946630i	$0.395537\pi$
$744$	0	0
$745$	58.6528	2.14887
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.8847	−0.397720
$750$	0	0
$751$	25.7508	0.939661	0.469830	−	0.882757i	$-0.344315\pi$
$751$	25.7508	0.939661	0.469830	+	0.882757i	$0.344315\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	43.6851	1.58986
$756$	0	0
$757$	5.21580	0.189572	0.0947858	−	0.995498i	$-0.469783\pi$
$757$	5.21580	0.189572	0.0947858	+	0.995498i	$0.469783\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26.0655	−0.944873	−0.472436	−	0.881365i	$-0.656625\pi$
$761$	−26.0655	−0.944873	−0.472436	+	0.881365i	$0.656625\pi$
$762$	0	0
$763$	−6.45482	−0.233680
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.59748	−0.238221
$768$	0	0
$769$	16.9896	0.612660	0.306330	−	0.951925i	$-0.400899\pi$
$769$	16.9896	0.612660	0.306330	+	0.951925i	$0.400899\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15.8396	−0.569709	−0.284855	−	0.958571i	$-0.591945\pi$
$773$	−15.8396	−0.569709	−0.284855	+	0.958571i	$0.591945\pi$
$774$	0	0
$775$	21.3978	0.768633
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.09798	0.218483
$780$	0	0
$781$	−24.3894	−0.872720
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−43.0003	−1.53474
$786$	0	0
$787$	27.7376	0.988740	0.494370	−	0.869252i	$-0.335399\pi$
$787$	27.7376	0.988740	0.494370	+	0.869252i	$0.335399\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.1807	−0.397541
$792$	0	0
$793$	9.53346	0.338543
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.5192	0.762248	0.381124	−	0.924524i	$-0.375537\pi$
$797$	21.5192	0.762248	0.381124	+	0.924524i	$0.375537\pi$
$798$	0	0
$799$	−2.44907	−0.0866420
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−22.2796	−0.786229
$804$	0	0
$805$	−34.5071	−1.21622
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.90939	0.278079	0.139040	−	0.990287i	$-0.455598\pi$
$809$	7.90939	0.278079	0.139040	+	0.990287i	$0.455598\pi$
$810$	0	0
$811$	−26.5947	−0.933867	−0.466934	−	0.884292i	$-0.654641\pi$
$811$	−26.5947	−0.933867	−0.466934	+	0.884292i	$0.654641\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.6470	−0.828319
$816$	0	0
$817$	−1.96936	−0.0688993
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.7757	−1.24858	−0.624291	−	0.781192i	$-0.714612\pi$
$821$	−35.7757	−1.24858	−0.624291	+	0.781192i	$0.714612\pi$
$822$	0	0
$823$	−8.00549	−0.279054	−0.139527	−	0.990218i	$-0.544558\pi$
$823$	−8.00549	−0.279054	−0.139527	+	0.990218i	$0.544558\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.60927	0.264600	0.132300	−	0.991210i	$-0.457764\pi$
$827$	7.60927	0.264600	0.132300	+	0.991210i	$0.457764\pi$
$828$	0	0
$829$	−23.2081	−0.806051	−0.403026	−	0.915189i	$-0.632041\pi$
$829$	−23.2081	−0.806051	−0.403026	+	0.915189i	$0.632041\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.295993	0.0102556
$834$	0	0
$835$	−66.6090	−2.30510
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24.9121	−0.860062	−0.430031	−	0.902814i	$-0.641497\pi$
$839$	−24.9121	−0.860062	−0.430031	+	0.902814i	$0.641497\pi$
$840$	0	0
$841$	−8.73764	−0.301298
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−54.2508	−1.86629
$846$	0	0
$847$	−1.89072	−0.0649661
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.59361	−0.0546281
$852$	0	0
$853$	−19.7376	−0.675804	−0.337902	−	0.941181i	$-0.609717\pi$
$853$	−19.7376	−0.675804	−0.337902	+	0.941181i	$0.609717\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.2916	0.761467	0.380734	−	0.924685i	$-0.375671\pi$
$857$	22.2916	0.761467	0.380734	+	0.924685i	$0.375671\pi$
$858$	0	0
$859$	−25.1646	−0.858604	−0.429302	−	0.903161i	$-0.641240\pi$
$859$	−25.1646	−0.858604	−0.429302	+	0.903161i	$0.641240\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.7625	1.76202	0.881009	−	0.473100i	$-0.156865\pi$
$863$	51.7625	1.76202	0.881009	+	0.473100i	$0.156865\pi$
$864$	0	0
$865$	55.6528	1.89225
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−52.9707	−1.79691
$870$	0	0
$871$	−10.6413	−0.360566
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−40.5961	−1.37240
$876$	0	0
$877$	58.7464	1.98372	0.991862	−	0.127315i	$-0.0406358\pi$
$877$	58.7464	1.98372	0.991862	+	0.127315i	$0.0406358\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−56.1284	−1.89101	−0.945507	−	0.325603i	$-0.894433\pi$
$881$	−56.1284	−1.89101	−0.945507	+	0.325603i	$0.894433\pi$
$882$	0	0
$883$	−9.14017	−0.307591	−0.153796	−	0.988103i	$-0.549150\pi$
$883$	−9.14017	−0.307591	−0.153796	+	0.988103i	$0.549150\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.4083	−1.55824	−0.779119	−	0.626877i	$-0.784333\pi$
$887$	−46.4083	−1.55824	−0.779119	+	0.626877i	$0.784333\pi$
$888$	0	0
$889$	14.3483	0.481226
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.82502	0.194927
$894$	0	0
$895$	74.9050	2.50380
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.75824	0.225400
$900$	0	0
$901$	1.18397	0.0394438
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−93.7404	−3.11604
$906$	0	0
$907$	28.1867	0.935925	0.467962	−	0.883748i	$-0.344988\pi$
$907$	28.1867	0.935925	0.467962	+	0.883748i	$0.344988\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	40.1235	1.32935	0.664675	−	0.747132i	$-0.268570\pi$
$911$	40.1235	1.32935	0.664675	+	0.747132i	$0.268570\pi$
$912$	0	0
$913$	−52.5783	−1.74009
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.2768	0.570531
$918$	0	0
$919$	−17.6445	−0.582040	−0.291020	−	0.956717i	$-0.593995\pi$
$919$	−17.6445	−0.582040	−0.291020	+	0.956717i	$0.593995\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.41651	0.178286
$924$	0	0
$925$	−2.88798	−0.0949562
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.3924	0.701860	0.350930	−	0.936402i	$-0.385865\pi$
$929$	21.3924	0.701860	0.350930	+	0.936402i	$0.385865\pi$
$930$	0	0
$931$	−0.704007	−0.0230729
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.66295	−0.152495
$936$	0	0
$937$	−47.0580	−1.53732	−0.768660	−	0.639658i	$-0.779076\pi$
$937$	−47.0580	−1.53732	−0.768660	+	0.639658i	$0.779076\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−16.0668	−0.523764	−0.261882	−	0.965100i	$-0.584343\pi$
$941$	−16.0668	−0.523764	−0.261882	+	0.965100i	$0.584343\pi$
$942$	0	0
$943$	−68.1205	−2.21831
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.4838	−0.860609	−0.430305	−	0.902684i	$-0.641594\pi$
$947$	−26.4838	−0.860609	−0.430305	+	0.902684i	$0.641594\pi$
$948$	0	0
$949$	4.94796	0.160618
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.8337	−0.577692	−0.288846	−	0.957376i	$-0.593272\pi$
$953$	−17.8337	−0.577692	−0.288846	+	0.957376i	$0.593272\pi$
$954$	0	0
$955$	−18.9507	−0.613230
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.31191	−0.171530
$960$	0	0
$961$	−28.7459	−0.927286
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−35.7289	−1.15015
$966$	0	0
$967$	−28.4113	−0.913645	−0.456822	−	0.889558i	$-0.651013\pi$
$967$	−28.4113	−0.913645	−0.456822	+	0.889558i	$0.651013\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.0550	0.611505	0.305753	−	0.952111i	$-0.401092\pi$
$971$	19.0550	0.611505	0.305753	+	0.952111i	$0.401092\pi$
$972$	0	0
$973$	−10.9534	−0.351151
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44.1046	−1.41103	−0.705516	−	0.708694i	$-0.749284\pi$
$977$	−44.1046	−1.41103	−0.705516	+	0.708694i	$0.749284\pi$
$978$	0	0
$979$	16.7376	0.534937
$980$	0	0
$981$	0	0
$982$	0	0
$983$	47.7867	1.52416	0.762080	−	0.647483i	$-0.224179\pi$
$983$	47.7867	1.52416	0.762080	+	0.647483i	$0.224179\pi$
$984$	0	0
$985$	−40.9096	−1.30349
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.9998	0.699552
$990$	0	0
$991$	20.6161	0.654893	0.327447	−	0.944870i	$-0.393812\pi$
$991$	20.6161	0.654893	0.327447	+	0.944870i	$0.393812\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	76.1196	2.41315
$996$	0	0
$997$	16.8160	0.532569	0.266284	−	0.963894i	$-0.414204\pi$
$997$	16.8160	0.532569	0.266284	+	0.963894i	$0.414204\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.br.1.4	yes	4
3.2	odd	2		6048.2.a.bm.1.1	✓	4
4.3	odd	2		6048.2.a.bv.1.4	yes	4
12.11	even	2		6048.2.a.bo.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bm.1.1	✓	4	3.2	odd	2
6048.2.a.bo.1.1	yes	4	12.11	even	2
6048.2.a.br.1.4	yes	4	1.1	even	1	trivial
6048.2.a.bv.1.4	yes	4	4.3	odd	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q+4.38773 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+4.38773 q^{5} -1.00000 q^{7} -3.59037 q^{11} +0.797366 q^{13} +0.295993 q^{17} -0.704007 q^{19} +7.86446 q^{23} +14.2522 q^{25} +4.50137 q^{29} +1.50137 q^{31} -4.38773 q^{35} -0.202634 q^{37} -8.66182 q^{41} +2.79737 q^{43} -8.27409 q^{47} +1.00000 q^{49} +4.00000 q^{53} -15.7536 q^{55} -8.27409 q^{59} +11.9562 q^{61} +3.49863 q^{65} -13.3455 q^{67} +6.79300 q^{71} +6.20538 q^{73} +3.59037 q^{77} +14.7536 q^{79} +14.6443 q^{83} +1.29874 q^{85} -4.66182 q^{89} -0.797366 q^{91} -3.08899 q^{95} -8.54818 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 - 4 * q^7	\( 4 q + 2 q^{5} - 4 q^{7} - 2 q^{11} + 8 q^{17} + 4 q^{19} - 2 q^{23} + 8 q^{25} + 8 q^{29} - 4 q^{31} - 2 q^{35} - 4 q^{37} + 2 q^{41} + 8 q^{43} - 12 q^{47} + 4 q^{49} + 16 q^{53} - 4 q^{55} - 12 q^{59} - 8 q^{61} + 24 q^{65} - 8 q^{67} + 18 q^{71} + 8 q^{73} + 2 q^{77} - 8 q^{85} + 18 q^{89} - 10 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 - 4 * q^7 - 2 * q^11 + 8 * q^17 + 4 * q^19 - 2 * q^23 + 8 * q^25 + 8 * q^29 - 4 * q^31 - 2 * q^35 - 4 * q^37 + 2 * q^41 + 8 * q^43 - 12 * q^47 + 4 * q^49 + 16 * q^53 - 4 * q^55 - 12 * q^59 - 8 * q^61 + 24 * q^65 - 8 * q^67 + 18 * q^71 + 8 * q^73 + 2 * q^77 - 8 * q^85 + 18 * q^89 - 10 * q^95 + 8 * q^97

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