LMFDB - Embedded newform 6048.2.a.be.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q+2.56155 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+2.56155 q^{5} -1.00000 q^{7} +5.12311 q^{11} -2.43845 q^{13} +4.12311 q^{17} +3.12311 q^{19} +8.68466 q^{23} +1.56155 q^{25} +9.56155 q^{29} +1.56155 q^{31} -2.56155 q^{35} -0.561553 q^{37} -7.43845 q^{41} -7.24621 q^{43} +0.561553 q^{47} +1.00000 q^{49} -6.68466 q^{53} +13.1231 q^{55} -8.12311 q^{59} -8.24621 q^{61} -6.24621 q^{65} +3.31534 q^{67} +1.56155 q^{71} -7.12311 q^{73} -5.12311 q^{77} +8.56155 q^{79} +2.56155 q^{83} +10.5616 q^{85} -1.31534 q^{89} +2.43845 q^{91} +8.00000 q^{95} -1.75379 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + q^5 - 2 * q^7	$ 2 q + q^{5} - 2 q^{7} + 2 q^{11} - 9 q^{13} - 2 q^{19} + 5 q^{23} - q^{25} + 15 q^{29} - q^{31} - q^{35} + 3 q^{37} - 19 q^{41} + 2 q^{43} - 3 q^{47} + 2 q^{49} - q^{53} + 18 q^{55} - 8 q^{59} + 4 q^{65} + 19 q^{67} - q^{71} - 6 q^{73} - 2 q^{77} + 13 q^{79} + q^{83} + 17 q^{85} - 15 q^{89} + 9 q^{91} + 16 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q + q^5 - 2 * q^7 + 2 * q^11 - 9 * q^13 - 2 * q^19 + 5 * q^23 - q^25 + 15 * q^29 - q^31 - q^35 + 3 * q^37 - 19 * q^41 + 2 * q^43 - 3 * q^47 + 2 * q^49 - q^53 + 18 * q^55 - 8 * q^59 + 4 * q^65 + 19 * q^67 - q^71 - 6 * q^73 - 2 * q^77 + 13 * q^79 + q^83 + 17 * q^85 - 15 * q^89 + 9 * q^91 + 16 * q^95 - 20 * 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
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	0	0
$13$	−2.43845	−0.676304	−0.338152	−	0.941092i	$-0.609802\pi$
$13$	−2.43845	−0.676304	−0.338152	+	0.941092i	$0.609802\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.12311	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$17$	4.12311	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.68466	1.81088	0.905438	−	0.424478i	$-0.139542\pi$
$23$	8.68466	1.81088	0.905438	+	0.424478i	$0.139542\pi$
$24$	0	0
$25$	1.56155	0.312311
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.56155	1.77554	0.887768	−	0.460291i	$-0.152255\pi$
$29$	9.56155	1.77554	0.887768	+	0.460291i	$0.152255\pi$
$30$	0	0
$31$	1.56155	0.280463	0.140232	−	0.990119i	$-0.455215\pi$
$31$	1.56155	0.280463	0.140232	+	0.990119i	$0.455215\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.56155	−0.432981
$36$	0	0
$37$	−0.561553	−0.0923187	−0.0461594	−	0.998934i	$-0.514698\pi$
$37$	−0.561553	−0.0923187	−0.0461594	+	0.998934i	$0.514698\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.43845	−1.16169	−0.580845	−	0.814014i	$-0.697278\pi$
$41$	−7.43845	−1.16169	−0.580845	+	0.814014i	$0.697278\pi$
$42$	0	0
$43$	−7.24621	−1.10504	−0.552518	−	0.833501i	$-0.686333\pi$
$43$	−7.24621	−1.10504	−0.552518	+	0.833501i	$0.686333\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.561553	0.0819109	0.0409554	−	0.999161i	$-0.486960\pi$
$47$	0.561553	0.0819109	0.0409554	+	0.999161i	$0.486960\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.68466	−0.918208	−0.459104	−	0.888382i	$-0.651830\pi$
$53$	−6.68466	−0.918208	−0.459104	+	0.888382i	$0.651830\pi$
$54$	0	0
$55$	13.1231	1.76952
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.12311	−1.05754	−0.528769	−	0.848766i	$-0.677346\pi$
$59$	−8.12311	−1.05754	−0.528769	+	0.848766i	$0.677346\pi$
$60$	0	0
$61$	−8.24621	−1.05582	−0.527910	−	0.849301i	$-0.677024\pi$
$61$	−8.24621	−1.05582	−0.527910	+	0.849301i	$0.677024\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.24621	−0.774747
$66$	0	0
$67$	3.31534	0.405033	0.202517	−	0.979279i	$-0.435088\pi$
$67$	3.31534	0.405033	0.202517	+	0.979279i	$0.435088\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.56155	0.185322	0.0926611	−	0.995698i	$-0.470463\pi$
$71$	1.56155	0.185322	0.0926611	+	0.995698i	$0.470463\pi$
$72$	0	0
$73$	−7.12311	−0.833696	−0.416848	−	0.908976i	$-0.636865\pi$
$73$	−7.12311	−0.833696	−0.416848	+	0.908976i	$0.636865\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.12311	−0.583832
$78$	0	0
$79$	8.56155	0.963250	0.481625	−	0.876377i	$-0.340047\pi$
$79$	8.56155	0.963250	0.481625	+	0.876377i	$0.340047\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.56155	0.281167	0.140583	−	0.990069i	$-0.455102\pi$
$83$	2.56155	0.281167	0.140583	+	0.990069i	$0.455102\pi$
$84$	0	0
$85$	10.5616	1.14556
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.31534	−0.139426	−0.0697130	−	0.997567i	$-0.522208\pi$
$89$	−1.31534	−0.139426	−0.0697130	+	0.997567i	$0.522208\pi$
$90$	0	0
$91$	2.43845	0.255619
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−1.75379	−0.178070	−0.0890351	−	0.996028i	$-0.528378\pi$
$97$	−1.75379	−0.178070	−0.0890351	+	0.996028i	$0.528378\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	6.43845	0.634399	0.317200	−	0.948359i	$-0.397258\pi$
$103$	6.43845	0.634399	0.317200	+	0.948359i	$0.397258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.4924	1.59438	0.797191	−	0.603727i	$-0.206318\pi$
$107$	16.4924	1.59438	0.797191	+	0.603727i	$0.206318\pi$
$108$	0	0
$109$	−16.8078	−1.60989	−0.804946	−	0.593348i	$-0.797806\pi$
$109$	−16.8078	−1.60989	−0.804946	+	0.593348i	$0.797806\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.12311	−0.481941	−0.240971	−	0.970532i	$-0.577466\pi$
$113$	−5.12311	−0.481941	−0.240971	+	0.970532i	$0.577466\pi$
$114$	0	0
$115$	22.2462	2.07447
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.12311	−0.377964
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.80776	−0.787790
$126$	0	0
$127$	18.5616	1.64707	0.823536	−	0.567264i	$-0.191998\pi$
$127$	18.5616	1.64707	0.823536	+	0.567264i	$0.191998\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.31534	0.289663	0.144831	−	0.989456i	$-0.453736\pi$
$131$	3.31534	0.289663	0.144831	+	0.989456i	$0.453736\pi$
$132$	0	0
$133$	−3.12311	−0.270808
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.12311	−0.608568	−0.304284	−	0.952581i	$-0.598417\pi$
$137$	−7.12311	−0.608568	−0.304284	+	0.952581i	$0.598417\pi$
$138$	0	0
$139$	−18.4924	−1.56851	−0.784253	−	0.620441i	$-0.786954\pi$
$139$	−18.4924	−1.56851	−0.784253	+	0.620441i	$0.786954\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.4924	−1.04467
$144$	0	0
$145$	24.4924	2.03398
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.9309	1.05934	0.529669	−	0.848204i	$-0.322316\pi$
$149$	12.9309	1.05934	0.529669	+	0.848204i	$0.322316\pi$
$150$	0	0
$151$	−0.561553	−0.0456985	−0.0228493	−	0.999739i	$-0.507274\pi$
$151$	−0.561553	−0.0456985	−0.0228493	+	0.999739i	$0.507274\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	18.9309	1.51085	0.755424	−	0.655236i	$-0.227431\pi$
$157$	18.9309	1.51085	0.755424	+	0.655236i	$0.227431\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.68466	−0.684447
$162$	0	0
$163$	14.3693	1.12549	0.562746	−	0.826630i	$-0.309745\pi$
$163$	14.3693	1.12549	0.562746	+	0.826630i	$0.309745\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.315342	−0.0244019	−0.0122009	−	0.999926i	$-0.503884\pi$
$167$	−0.315342	−0.0244019	−0.0122009	+	0.999926i	$0.503884\pi$
$168$	0	0
$169$	−7.05398	−0.542613
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.3693	1.47262	0.736311	−	0.676643i	$-0.236566\pi$
$173$	19.3693	1.47262	0.736311	+	0.676643i	$0.236566\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.8769	−1.11195	−0.555976	−	0.831199i	$-0.687655\pi$
$179$	−14.8769	−1.11195	−0.555976	+	0.831199i	$0.687655\pi$
$180$	0	0
$181$	−19.5616	−1.45400	−0.726999	−	0.686638i	$-0.759086\pi$
$181$	−19.5616	−1.45400	−0.726999	+	0.686638i	$0.759086\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.43845	−0.105757
$186$	0	0
$187$	21.1231	1.54467
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.12311	0.660125	0.330062	−	0.943959i	$-0.392930\pi$
$191$	9.12311	0.660125	0.330062	+	0.943959i	$0.392930\pi$
$192$	0	0
$193$	22.6155	1.62790	0.813951	−	0.580934i	$-0.197313\pi$
$193$	22.6155	1.62790	0.813951	+	0.580934i	$0.197313\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.1231	−1.07748	−0.538738	−	0.842473i	$-0.681099\pi$
$197$	−15.1231	−1.07748	−0.538738	+	0.842473i	$0.681099\pi$
$198$	0	0
$199$	−6.68466	−0.473863	−0.236931	−	0.971526i	$-0.576142\pi$
$199$	−6.68466	−0.473863	−0.236931	+	0.971526i	$0.576142\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.56155	−0.671089
$204$	0	0
$205$	−19.0540	−1.33079
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	26.0540	1.79363	0.896815	−	0.442406i	$-0.145875\pi$
$211$	26.0540	1.79363	0.896815	+	0.442406i	$0.145875\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.5616	−1.26589
$216$	0	0
$217$	−1.56155	−0.106005
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.0540	−0.676304
$222$	0	0
$223$	10.2462	0.686137	0.343069	−	0.939310i	$-0.388534\pi$
$223$	10.2462	0.686137	0.343069	+	0.939310i	$0.388534\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.8078	−1.58018	−0.790088	−	0.612993i	$-0.789965\pi$
$227$	−23.8078	−1.58018	−0.790088	+	0.612993i	$0.789965\pi$
$228$	0	0
$229$	−26.4924	−1.75067	−0.875334	−	0.483518i	$-0.839359\pi$
$229$	−26.4924	−1.75067	−0.875334	+	0.483518i	$0.839359\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.75379	−0.376943	−0.188472	−	0.982079i	$-0.560353\pi$
$233$	−5.75379	−0.376943	−0.188472	+	0.982079i	$0.560353\pi$
$234$	0	0
$235$	1.43845	0.0938339
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.87689	−0.444829	−0.222415	−	0.974952i	$-0.571394\pi$
$239$	−6.87689	−0.444829	−0.222415	+	0.974952i	$0.571394\pi$
$240$	0	0
$241$	0.876894	0.0564857	0.0282429	−	0.999601i	$-0.491009\pi$
$241$	0.876894	0.0564857	0.0282429	+	0.999601i	$0.491009\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.56155	0.163652
$246$	0	0
$247$	−7.61553	−0.484564
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.3153	0.777337	0.388669	−	0.921378i	$-0.372935\pi$
$251$	12.3153	0.777337	0.388669	+	0.921378i	$0.372935\pi$
$252$	0	0
$253$	44.4924	2.79721
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	0.561553	0.0348932
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.1771	−0.812534	−0.406267	−	0.913754i	$-0.633170\pi$
$263$	−13.1771	−0.812534	−0.406267	+	0.913754i	$0.633170\pi$
$264$	0	0
$265$	−17.1231	−1.05186
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.3002	1.29870	0.649348	−	0.760492i	$-0.275042\pi$
$269$	21.3002	1.29870	0.649348	+	0.760492i	$0.275042\pi$
$270$	0	0
$271$	−20.0540	−1.21819	−0.609096	−	0.793096i	$-0.708468\pi$
$271$	−20.0540	−1.21819	−0.609096	+	0.793096i	$0.708468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.00000	0.482418
$276$	0	0
$277$	2.56155	0.153909	0.0769544	−	0.997035i	$-0.475480\pi$
$277$	2.56155	0.153909	0.0769544	+	0.997035i	$0.475480\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.3693	−1.75203	−0.876013	−	0.482287i	$-0.839806\pi$
$281$	−29.3693	−1.75203	−0.876013	+	0.482287i	$0.839806\pi$
$282$	0	0
$283$	21.6155	1.28491	0.642455	−	0.766324i	$-0.277916\pi$
$283$	21.6155	1.28491	0.642455	+	0.766324i	$0.277916\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.43845	0.439078
$288$	0	0
$289$	0	0
$290$	0	0
$291$	0	0
$292$	0	0
$293$	24.5616	1.43490	0.717451	−	0.696609i	$-0.245309\pi$
$293$	24.5616	1.43490	0.717451	+	0.696609i	$0.245309\pi$
$294$	0	0
$295$	−20.8078	−1.21147
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.1771	−1.22470
$300$	0	0
$301$	7.24621	0.417665
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−21.1231	−1.20951
$306$	0	0
$307$	12.8769	0.734923	0.367462	−	0.930039i	$-0.380227\pi$
$307$	12.8769	0.734923	0.367462	+	0.930039i	$0.380227\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.6847	0.889395	0.444698	−	0.895681i	$-0.353311\pi$
$311$	15.6847	0.889395	0.444698	+	0.895681i	$0.353311\pi$
$312$	0	0
$313$	−24.2462	−1.37048	−0.685238	−	0.728319i	$-0.740302\pi$
$313$	−24.2462	−1.37048	−0.685238	+	0.728319i	$0.740302\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.1231	1.74805	0.874024	−	0.485883i	$-0.161502\pi$
$317$	31.1231	1.74805	0.874024	+	0.485883i	$0.161502\pi$
$318$	0	0
$319$	48.9848	2.74262
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.8769	0.716490
$324$	0	0
$325$	−3.80776	−0.211217
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.561553	−0.0309594
$330$	0	0
$331$	22.1231	1.21600	0.607998	−	0.793939i	$-0.291973\pi$
$331$	22.1231	1.21600	0.607998	+	0.793939i	$0.291973\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.49242	0.463991
$336$	0	0
$337$	0.753789	0.0410615	0.0205307	−	0.999789i	$-0.493464\pi$
$337$	0.753789	0.0410615	0.0205307	+	0.999789i	$0.493464\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−29.8617	−1.60306	−0.801531	−	0.597953i	$-0.795981\pi$
$347$	−29.8617	−1.60306	−0.801531	+	0.597953i	$0.795981\pi$
$348$	0	0
$349$	29.1771	1.56181	0.780907	−	0.624648i	$-0.214757\pi$
$349$	29.1771	1.56181	0.780907	+	0.624648i	$0.214757\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	33.9848	1.80883	0.904415	−	0.426653i	$-0.140307\pi$
$353$	33.9848	1.80883	0.904415	+	0.426653i	$0.140307\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.8078	−1.36208	−0.681041	−	0.732245i	$-0.738472\pi$
$359$	−25.8078	−1.36208	−0.681041	+	0.732245i	$0.738472\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18.2462	−0.955050
$366$	0	0
$367$	−0.0539753	−0.00281749	−0.00140874	−	0.999999i	$-0.500448\pi$
$367$	−0.0539753	−0.00281749	−0.00140874	+	0.999999i	$0.500448\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.68466	0.347050
$372$	0	0
$373$	2.80776	0.145381	0.0726903	−	0.997355i	$-0.476842\pi$
$373$	2.80776	0.145381	0.0726903	+	0.997355i	$0.476842\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.3153	−1.20080
$378$	0	0
$379$	4.31534	0.221664	0.110832	−	0.993839i	$-0.464648\pi$
$379$	4.31534	0.221664	0.110832	+	0.993839i	$0.464648\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.0540	−0.973613	−0.486806	−	0.873510i	$-0.661838\pi$
$383$	−19.0540	−0.973613	−0.486806	+	0.873510i	$0.661838\pi$
$384$	0	0
$385$	−13.1231	−0.668815
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.2462	1.63495	0.817474	−	0.575966i	$-0.195374\pi$
$389$	32.2462	1.63495	0.817474	+	0.575966i	$0.195374\pi$
$390$	0	0
$391$	35.8078	1.81088
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.9309	1.10346
$396$	0	0
$397$	7.61553	0.382212	0.191106	−	0.981569i	$-0.438793\pi$
$397$	7.61553	0.382212	0.191106	+	0.981569i	$0.438793\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.4924	0.723717	0.361859	−	0.932233i	$-0.382142\pi$
$401$	14.4924	0.723717	0.361859	+	0.932233i	$0.382142\pi$
$402$	0	0
$403$	−3.80776	−0.189678
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.87689	−0.142602
$408$	0	0
$409$	−5.36932	−0.265496	−0.132748	−	0.991150i	$-0.542380\pi$
$409$	−5.36932	−0.265496	−0.132748	+	0.991150i	$0.542380\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.12311	0.399712
$414$	0	0
$415$	6.56155	0.322094
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.75379	−0.232238	−0.116119	−	0.993235i	$-0.537045\pi$
$419$	−4.75379	−0.232238	−0.116119	+	0.993235i	$0.537045\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.43845	0.312311
$426$	0	0
$427$	8.24621	0.399062
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	−38.1080	−1.83135	−0.915676	−	0.401918i	$-0.868344\pi$
$433$	−38.1080	−1.83135	−0.915676	+	0.401918i	$0.868344\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.1231	1.29747
$438$	0	0
$439$	16.0540	0.766214	0.383107	−	0.923704i	$-0.374854\pi$
$439$	16.0540	0.766214	0.383107	+	0.923704i	$0.374854\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.876894	−0.0416625	−0.0208313	−	0.999783i	$-0.506631\pi$
$443$	−0.876894	−0.0416625	−0.0208313	+	0.999783i	$0.506631\pi$
$444$	0	0
$445$	−3.36932	−0.159721
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−32.4924	−1.53341	−0.766706	−	0.641998i	$-0.778106\pi$
$449$	−32.4924	−1.53341	−0.766706	+	0.641998i	$0.778106\pi$
$450$	0	0
$451$	−38.1080	−1.79443
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.24621	0.292827
$456$	0	0
$457$	−29.8078	−1.39435	−0.697174	−	0.716902i	$-0.745560\pi$
$457$	−29.8078	−1.39435	−0.697174	+	0.716902i	$0.745560\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.4233	−1.51010	−0.755052	−	0.655665i	$-0.772388\pi$
$461$	−32.4233	−1.51010	−0.755052	+	0.655665i	$0.772388\pi$
$462$	0	0
$463$	11.4384	0.531590	0.265795	−	0.964030i	$-0.414366\pi$
$463$	11.4384	0.531590	0.265795	+	0.964030i	$0.414366\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.24621	0.103942	0.0519711	−	0.998649i	$-0.483450\pi$
$467$	2.24621	0.103942	0.0519711	+	0.998649i	$0.483450\pi$
$468$	0	0
$469$	−3.31534	−0.153088
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−37.1231	−1.70692
$474$	0	0
$475$	4.87689	0.223767
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.0540	0.870598	0.435299	−	0.900286i	$-0.356643\pi$
$479$	19.0540	0.870598	0.435299	+	0.900286i	$0.356643\pi$
$480$	0	0
$481$	1.36932	0.0624355
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.49242	−0.203990
$486$	0	0
$487$	6.63068	0.300465	0.150232	−	0.988651i	$-0.451998\pi$
$487$	6.63068	0.300465	0.150232	+	0.988651i	$0.451998\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.75379	−0.0791474	−0.0395737	−	0.999217i	$-0.512600\pi$
$491$	−1.75379	−0.0791474	−0.0395737	+	0.999217i	$0.512600\pi$
$492$	0	0
$493$	39.4233	1.77554
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.56155	−0.0700452
$498$	0	0
$499$	−25.9309	−1.16083	−0.580413	−	0.814323i	$-0.697109\pi$
$499$	−25.9309	−1.16083	−0.580413	+	0.814323i	$0.697109\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.19224	0.320686	0.160343	−	0.987061i	$-0.448740\pi$
$503$	7.19224	0.320686	0.160343	+	0.987061i	$0.448740\pi$
$504$	0	0
$505$	25.6155	1.13988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.0540	−0.755904	−0.377952	−	0.925825i	$-0.623372\pi$
$509$	−17.0540	−0.755904	−0.377952	+	0.925825i	$0.623372\pi$
$510$	0	0
$511$	7.12311	0.315108
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.4924	0.726743
$516$	0	0
$517$	2.87689	0.126526
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−40.3693	−1.76861	−0.884306	−	0.466908i	$-0.845368\pi$
$521$	−40.3693	−1.76861	−0.884306	+	0.466908i	$0.845368\pi$
$522$	0	0
$523$	4.63068	0.202486	0.101243	−	0.994862i	$-0.467718\pi$
$523$	4.63068	0.202486	0.101243	+	0.994862i	$0.467718\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.43845	0.280463
$528$	0	0
$529$	52.4233	2.27927
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18.1383	0.785655
$534$	0	0
$535$	42.2462	1.82646
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.12311	0.220668
$540$	0	0
$541$	−11.1922	−0.481192	−0.240596	−	0.970625i	$-0.577343\pi$
$541$	−11.1922	−0.481192	−0.240596	+	0.970625i	$0.577343\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−43.0540	−1.84423
$546$	0	0
$547$	31.6847	1.35474	0.677369	−	0.735643i	$-0.263120\pi$
$547$	31.6847	1.35474	0.677369	+	0.735643i	$0.263120\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	29.8617	1.27215
$552$	0	0
$553$	−8.56155	−0.364074
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.1922	−0.516602	−0.258301	−	0.966065i	$-0.583163\pi$
$557$	−12.1922	−0.516602	−0.258301	+	0.966065i	$0.583163\pi$
$558$	0	0
$559$	17.6695	0.747340
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.9309	0.460681	0.230341	−	0.973110i	$-0.426016\pi$
$563$	10.9309	0.460681	0.230341	+	0.973110i	$0.426016\pi$
$564$	0	0
$565$	−13.1231	−0.552093
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.2462	0.429544	0.214772	−	0.976664i	$-0.431099\pi$
$569$	10.2462	0.429544	0.214772	+	0.976664i	$0.431099\pi$
$570$	0	0
$571$	−36.1231	−1.51170	−0.755852	−	0.654742i	$-0.772777\pi$
$571$	−36.1231	−1.51170	−0.755852	+	0.654742i	$0.772777\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.5616	0.565556
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.56155	−0.106271
$582$	0	0
$583$	−34.2462	−1.41833
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.6847	1.18394	0.591971	−	0.805959i	$-0.298350\pi$
$587$	28.6847	1.18394	0.591971	+	0.805959i	$0.298350\pi$
$588$	0	0
$589$	4.87689	0.200949
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−41.0540	−1.68588	−0.842942	−	0.538004i	$-0.819179\pi$
$593$	−41.0540	−1.68588	−0.842942	+	0.538004i	$0.819179\pi$
$594$	0	0
$595$	−10.5616	−0.432981
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.9309	1.50895	0.754477	−	0.656326i	$-0.227890\pi$
$599$	36.9309	1.50895	0.754477	+	0.656326i	$0.227890\pi$
$600$	0	0
$601$	16.4924	0.672740	0.336370	−	0.941730i	$-0.390801\pi$
$601$	16.4924	0.672740	0.336370	+	0.941730i	$0.390801\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	39.0540	1.58777
$606$	0	0
$607$	4.43845	0.180151	0.0900755	−	0.995935i	$-0.471289\pi$
$607$	4.43845	0.180151	0.0900755	+	0.995935i	$0.471289\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.36932	−0.0553966
$612$	0	0
$613$	−8.73863	−0.352950	−0.176475	−	0.984305i	$-0.556469\pi$
$613$	−8.73863	−0.352950	−0.176475	+	0.984305i	$0.556469\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7.12311	−0.286765	−0.143383	−	0.989667i	$-0.545798\pi$
$617$	−7.12311	−0.286765	−0.143383	+	0.989667i	$0.545798\pi$
$618$	0	0
$619$	−31.6155	−1.27074	−0.635368	−	0.772210i	$-0.719151\pi$
$619$	−31.6155	−1.27074	−0.635368	+	0.772210i	$0.719151\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.31534	0.0526980
$624$	0	0
$625$	−30.3693	−1.21477
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.31534	−0.0923187
$630$	0	0
$631$	−27.1922	−1.08251	−0.541253	−	0.840860i	$-0.682050\pi$
$631$	−27.1922	−1.08251	−0.541253	+	0.840860i	$0.682050\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	47.5464	1.88682
$636$	0	0
$637$	−2.43845	−0.0966148
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.6307	0.656872	0.328436	−	0.944526i	$-0.393478\pi$
$641$	16.6307	0.656872	0.328436	+	0.944526i	$0.393478\pi$
$642$	0	0
$643$	−35.3693	−1.39483	−0.697415	−	0.716668i	$-0.745666\pi$
$643$	−35.3693	−1.39483	−0.697415	+	0.716668i	$0.745666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.8769	0.506243	0.253121	−	0.967435i	$-0.418543\pi$
$647$	12.8769	0.506243	0.253121	+	0.967435i	$0.418543\pi$
$648$	0	0
$649$	−41.6155	−1.63355
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.05398	0.158644	0.0793222	−	0.996849i	$-0.474724\pi$
$653$	4.05398	0.158644	0.0793222	+	0.996849i	$0.474724\pi$
$654$	0	0
$655$	8.49242	0.331826
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.8769	0.891157	0.445579	−	0.895243i	$-0.352998\pi$
$659$	22.8769	0.891157	0.445579	+	0.895243i	$0.352998\pi$
$660$	0	0
$661$	−2.49242	−0.0969440	−0.0484720	−	0.998825i	$-0.515435\pi$
$661$	−2.49242	−0.0969440	−0.0484720	+	0.998825i	$0.515435\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	83.0388	3.21528
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−42.2462	−1.63090
$672$	0	0
$673$	30.1922	1.16383	0.581913	−	0.813251i	$-0.302305\pi$
$673$	30.1922	1.16383	0.581913	+	0.813251i	$0.302305\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.8769	1.18670	0.593348	−	0.804946i	$-0.297806\pi$
$677$	30.8769	1.18670	0.593348	+	0.804946i	$0.297806\pi$
$678$	0	0
$679$	1.75379	0.0673042
$680$	0	0
$681$	0	0
$682$	0	0
$683$	4.49242	0.171898	0.0859489	−	0.996300i	$-0.472608\pi$
$683$	4.49242	0.171898	0.0859489	+	0.996300i	$0.472608\pi$
$684$	0	0
$685$	−18.2462	−0.697152
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16.3002	0.620988
$690$	0	0
$691$	5.61553	0.213625	0.106812	−	0.994279i	$-0.465936\pi$
$691$	5.61553	0.213625	0.106812	+	0.994279i	$0.465936\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−47.3693	−1.79682
$696$	0	0
$697$	−30.6695	−1.16169
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	−1.75379	−0.0661454
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.0000	−0.376089
$708$	0	0
$709$	9.93087	0.372962	0.186481	−	0.982459i	$-0.440292\pi$
$709$	9.93087	0.372962	0.186481	+	0.982459i	$0.440292\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	13.5616	0.507884
$714$	0	0
$715$	−32.0000	−1.19673
$716$	0	0
$717$	0	0
$718$	0	0
$719$	44.1771	1.64753	0.823764	−	0.566934i	$-0.191870\pi$
$719$	44.1771	1.64753	0.823764	+	0.566934i	$0.191870\pi$
$720$	0	0
$721$	−6.43845	−0.239780
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14.9309	0.554519
$726$	0	0
$727$	−10.4384	−0.387141	−0.193570	−	0.981086i	$-0.562007\pi$
$727$	−10.4384	−0.387141	−0.193570	+	0.981086i	$0.562007\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.8769	−1.10504
$732$	0	0
$733$	−10.1922	−0.376459	−0.188229	−	0.982125i	$-0.560275\pi$
$733$	−10.1922	−0.376459	−0.188229	+	0.982125i	$0.560275\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.9848	0.625645
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.5616	−1.08451	−0.542254	−	0.840215i	$-0.682429\pi$
$743$	−29.5616	−1.08451	−0.542254	+	0.840215i	$0.682429\pi$
$744$	0	0
$745$	33.1231	1.21354
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−16.4924	−0.602620
$750$	0	0
$751$	−52.1080	−1.90145	−0.950723	−	0.310041i	$-0.899657\pi$
$751$	−52.1080	−1.90145	−0.950723	+	0.310041i	$0.899657\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.43845	−0.0523505
$756$	0	0
$757$	23.1922	0.842936	0.421468	−	0.906843i	$-0.361515\pi$
$757$	23.1922	0.842936	0.421468	+	0.906843i	$0.361515\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29.8769	−1.08304	−0.541518	−	0.840689i	$-0.682150\pi$
$761$	−29.8769	−1.08304	−0.541518	+	0.840689i	$0.682150\pi$
$762$	0	0
$763$	16.8078	0.608482
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.8078	0.715217
$768$	0	0
$769$	−39.8617	−1.43745	−0.718726	−	0.695294i	$-0.755274\pi$
$769$	−39.8617	−1.43745	−0.718726	+	0.695294i	$0.755274\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.43845	0.339477	0.169739	−	0.985489i	$-0.445708\pi$
$773$	9.43845	0.339477	0.169739	+	0.985489i	$0.445708\pi$
$774$	0	0
$775$	2.43845	0.0875916
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−23.2311	−0.832339
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.4924	1.73077
$786$	0	0
$787$	−29.3693	−1.04690	−0.523452	−	0.852055i	$-0.675356\pi$
$787$	−29.3693	−1.04690	−0.523452	+	0.852055i	$0.675356\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.12311	0.182157
$792$	0	0
$793$	20.1080	0.714054
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.9848	0.814165	0.407082	−	0.913391i	$-0.366546\pi$
$797$	22.9848	0.814165	0.407082	+	0.913391i	$0.366546\pi$
$798$	0	0
$799$	2.31534	0.0819109
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−36.4924	−1.28779
$804$	0	0
$805$	−22.2462	−0.784076
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.3693	0.680989	0.340494	−	0.940247i	$-0.389406\pi$
$809$	19.3693	0.680989	0.340494	+	0.940247i	$0.389406\pi$
$810$	0	0
$811$	−34.4924	−1.21119	−0.605596	−	0.795772i	$-0.707065\pi$
$811$	−34.4924	−1.21119	−0.605596	+	0.795772i	$0.707065\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	36.8078	1.28932
$816$	0	0
$817$	−22.6307	−0.791747
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.9309	1.35870	0.679348	−	0.733816i	$-0.262263\pi$
$821$	38.9309	1.35870	0.679348	+	0.733816i	$0.262263\pi$
$822$	0	0
$823$	−4.56155	−0.159006	−0.0795029	−	0.996835i	$-0.525333\pi$
$823$	−4.56155	−0.159006	−0.0795029	+	0.996835i	$0.525333\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.4924	−1.19942	−0.599710	−	0.800218i	$-0.704717\pi$
$827$	−34.4924	−1.19942	−0.599710	+	0.800218i	$0.704717\pi$
$828$	0	0
$829$	43.1231	1.49773	0.748864	−	0.662724i	$-0.230600\pi$
$829$	43.1231	1.49773	0.748864	+	0.662724i	$0.230600\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.12311	0.142857
$834$	0	0
$835$	−0.807764	−0.0279538
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44.9157	−1.55066	−0.775331	−	0.631555i	$-0.782417\pi$
$839$	−44.9157	−1.55066	−0.775331	+	0.631555i	$0.782417\pi$
$840$	0	0
$841$	62.4233	2.15253
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−18.0691	−0.621597
$846$	0	0
$847$	−15.2462	−0.523866
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.87689	−0.167178
$852$	0	0
$853$	36.0540	1.23446	0.617232	−	0.786781i	$-0.288254\pi$
$853$	36.0540	1.23446	0.617232	+	0.786781i	$0.288254\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.1231	1.43890	0.719449	−	0.694545i	$-0.244394\pi$
$857$	42.1231	1.43890	0.719449	+	0.694545i	$0.244394\pi$
$858$	0	0
$859$	−13.5076	−0.460873	−0.230436	−	0.973087i	$-0.574015\pi$
$859$	−13.5076	−0.460873	−0.230436	+	0.973087i	$0.574015\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.8078	−1.01467	−0.507334	−	0.861749i	$-0.669369\pi$
$863$	−29.8078	−1.01467	−0.507334	+	0.861749i	$0.669369\pi$
$864$	0	0
$865$	49.6155	1.68698
$866$	0	0
$867$	0	0
$868$	0	0
$869$	43.8617	1.48791
$870$	0	0
$871$	−8.08429	−0.273926
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.80776	0.297757
$876$	0	0
$877$	−31.6847	−1.06991	−0.534957	−	0.844879i	$-0.679672\pi$
$877$	−31.6847	−1.06991	−0.534957	+	0.844879i	$0.679672\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.4233	−1.12606	−0.563030	−	0.826437i	$-0.690364\pi$
$881$	−33.4233	−1.12606	−0.563030	+	0.826437i	$0.690364\pi$
$882$	0	0
$883$	48.2311	1.62310	0.811552	−	0.584280i	$-0.198623\pi$
$883$	48.2311	1.62310	0.811552	+	0.584280i	$0.198623\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.4384	0.719833	0.359916	−	0.932985i	$-0.382805\pi$
$887$	21.4384	0.719833	0.359916	+	0.932985i	$0.382805\pi$
$888$	0	0
$889$	−18.5616	−0.622535
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.75379	0.0586883
$894$	0	0
$895$	−38.1080	−1.27381
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.9309	0.497972
$900$	0	0
$901$	−27.5616	−0.918208
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−50.1080	−1.66564
$906$	0	0
$907$	26.5616	0.881962	0.440981	−	0.897516i	$-0.354631\pi$
$907$	26.5616	0.881962	0.440981	+	0.897516i	$0.354631\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0.630683	0.0208955	0.0104477	−	0.999945i	$-0.496674\pi$
$911$	0.630683	0.0208955	0.0104477	+	0.999945i	$0.496674\pi$
$912$	0	0
$913$	13.1231	0.434311
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.31534	−0.109482
$918$	0	0
$919$	30.1771	0.995450	0.497725	−	0.867335i	$-0.334169\pi$
$919$	30.1771	0.995450	0.497725	+	0.867335i	$0.334169\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.80776	−0.125334
$924$	0	0
$925$	−0.876894	−0.0288321
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.56155	0.149660	0.0748298	−	0.997196i	$-0.476159\pi$
$929$	4.56155	0.149660	0.0748298	+	0.997196i	$0.476159\pi$
$930$	0	0
$931$	3.12311	0.102356
$932$	0	0
$933$	0	0
$934$	0	0
$935$	54.1080	1.76952
$936$	0	0
$937$	9.75379	0.318642	0.159321	−	0.987227i	$-0.449069\pi$
$937$	9.75379	0.318642	0.159321	+	0.987227i	$0.449069\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.8078	−1.00430	−0.502152	−	0.864779i	$-0.667458\pi$
$941$	−30.8078	−1.00430	−0.502152	+	0.864779i	$0.667458\pi$
$942$	0	0
$943$	−64.6004	−2.10368
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.6307	0.475433	0.237717	−	0.971335i	$-0.423601\pi$
$947$	14.6307	0.475433	0.237717	+	0.971335i	$0.423601\pi$
$948$	0	0
$949$	17.3693	0.563832
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23.5076	−0.761485	−0.380743	−	0.924681i	$-0.624332\pi$
$953$	−23.5076	−0.761485	−0.380743	+	0.924681i	$0.624332\pi$
$954$	0	0
$955$	23.3693	0.756213
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.12311	0.230017
$960$	0	0
$961$	−28.5616	−0.921340
$962$	0	0
$963$	0	0
$964$	0	0
$965$	57.9309	1.86486
$966$	0	0
$967$	−8.98485	−0.288933	−0.144467	−	0.989510i	$-0.546147\pi$
$967$	−8.98485	−0.288933	−0.144467	+	0.989510i	$0.546147\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30.1231	−0.966696	−0.483348	−	0.875428i	$-0.660579\pi$
$971$	−30.1231	−0.966696	−0.483348	+	0.875428i	$0.660579\pi$
$972$	0	0
$973$	18.4924	0.592840
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	0	0
$979$	−6.73863	−0.215368
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.8078	−0.408504	−0.204252	−	0.978918i	$-0.565476\pi$
$983$	−12.8078	−0.408504	−0.204252	+	0.978918i	$0.565476\pi$
$984$	0	0
$985$	−38.7386	−1.23432
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−62.9309	−2.00109
$990$	0	0
$991$	−56.4233	−1.79234	−0.896172	−	0.443706i	$-0.853663\pi$
$991$	−56.4233	−1.79234	−0.896172	+	0.443706i	$0.853663\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−17.1231	−0.542839
$996$	0	0
$997$	−45.6695	−1.44637	−0.723184	−	0.690656i	$-0.757322\pi$
$997$	−45.6695	−1.44637	−0.723184	+	0.690656i	$0.757322\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.a.be.1.2	yes	2
3.2	odd	2		6048.2.a.bc.1.1	✓	2
4.3	odd	2		6048.2.a.bf.1.2	yes	2
12.11	even	2		6048.2.a.bd.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.a.bc.1.1	✓	2	3.2	odd	2
6048.2.a.bd.1.1	yes	2	12.11	even	2
6048.2.a.be.1.2	yes	2	1.1	even	1	trivial
6048.2.a.bf.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q+2.56155 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+2.56155 q^{5} -1.00000 q^{7} +5.12311 q^{11} -2.43845 q^{13} +4.12311 q^{17} +3.12311 q^{19} +8.68466 q^{23} +1.56155 q^{25} +9.56155 q^{29} +1.56155 q^{31} -2.56155 q^{35} -0.561553 q^{37} -7.43845 q^{41} -7.24621 q^{43} +0.561553 q^{47} +1.00000 q^{49} -6.68466 q^{53} +13.1231 q^{55} -8.12311 q^{59} -8.24621 q^{61} -6.24621 q^{65} +3.31534 q^{67} +1.56155 q^{71} -7.12311 q^{73} -5.12311 q^{77} +8.56155 q^{79} +2.56155 q^{83} +10.5616 q^{85} -1.31534 q^{89} +2.43845 q^{91} +8.00000 q^{95} -1.75379 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + q^5 - 2 * q^7	\( 2 q + q^{5} - 2 q^{7} + 2 q^{11} - 9 q^{13} - 2 q^{19} + 5 q^{23} - q^{25} + 15 q^{29} - q^{31} - q^{35} + 3 q^{37} - 19 q^{41} + 2 q^{43} - 3 q^{47} + 2 q^{49} - q^{53} + 18 q^{55} - 8 q^{59} + 4 q^{65} + 19 q^{67} - q^{71} - 6 q^{73} - 2 q^{77} + 13 q^{79} + q^{83} + 17 q^{85} - 15 q^{89} + 9 q^{91} + 16 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q + q^5 - 2 * q^7 + 2 * q^11 - 9 * q^13 - 2 * q^19 + 5 * q^23 - q^25 + 15 * q^29 - q^31 - q^35 + 3 * q^37 - 19 * q^41 + 2 * q^43 - 3 * q^47 + 2 * q^49 - q^53 + 18 * q^55 - 8 * q^59 + 4 * q^65 + 19 * q^67 - q^71 - 6 * q^73 - 2 * q^77 + 13 * q^79 + q^83 + 17 * q^85 - 15 * q^89 + 9 * q^91 + 16 * q^95 - 20 * q^97

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