LMFDB - Embedded newform 2880.3.e.n.2431.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2880,3,Mod(2431,2880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2880.2431"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2880.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,0,40,0,0,0,-96, 0,0,0,0,0,0,0,112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$78.4743161358$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.1
Root			$-0.535233 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	2880.2431
Dual form			2880.3.e.n.2431.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{5} -13.4003i q^{7} +21.2261i q^{11} -2.66241 q^{13} +19.7577 q^{17} -13.5547i q^{19} -40.1023i q^{23} +5.00000 q^{25} +1.41641 q^{29} +41.1420i q^{31} +29.9641i q^{35} +54.0590 q^{37} +2.74692 q^{41} +22.6732i q^{43} +45.6501i q^{47} -130.569 q^{49} +4.37878 q^{53} -47.4631i q^{55} -30.8060i q^{59} +27.0376 q^{61} +5.95332 q^{65} -103.603i q^{67} -88.3730i q^{71} +36.0216 q^{73} +284.437 q^{77} +23.6019i q^{79} +119.743i q^{83} -44.1796 q^{85} +71.7494 q^{89} +35.6772i q^{91} +30.3092i q^{95} +121.426 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{13} + 16 q^{17} + 40 q^{25} - 96 q^{29} + 112 q^{37} - 112 q^{41} - 184 q^{49} + 224 q^{53} - 80 q^{61} + 160 q^{65} + 272 q^{73} + 256 q^{77} + 160 q^{85} + 48 q^{89} + 528 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 + 16 * q^17 + 40 * q^25 - 96 * q^29 + 112 * q^37 - 112 * q^41 - 184 * q^49 + 224 * q^53 - 80 * q^61 + 160 * q^65 + 272 * q^73 + 256 * q^77 + 160 * q^85 + 48 * q^89 + 528 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	− 13.4003i	− 1.91433i	−0.289536	−	0.957167i	$-0.593501\pi$
$7$	− 13.4003i	− 1.91433i	0.289536	−	0.957167i	$-0.406499\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	21.2261i	1.92965i	0.262895	+	0.964824i	$0.415323\pi$
$11$	21.2261i	1.92965i	−0.262895	+	0.964824i	$0.584677\pi$
$12$	0	0
$13$	−2.66241	−0.204801	−0.102400	−	0.994743i	$-0.532652\pi$
$13$	−2.66241	−0.204801	−0.102400	+	0.994743i	$0.532652\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	19.7577	1.16222	0.581110	−	0.813825i	$-0.302619\pi$
$17$	19.7577	1.16222	0.581110	+	0.813825i	$0.302619\pi$
$18$	0	0
$19$	− 13.5547i	− 0.713404i	−0.934218	−	0.356702i	$-0.883901\pi$
$19$	− 13.5547i	− 0.713404i	0.934218	−	0.356702i	$-0.116099\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 40.1023i	− 1.74358i	−0.489880	−	0.871790i	$-0.662959\pi$
$23$	− 40.1023i	− 1.74358i	0.489880	−	0.871790i	$-0.337041\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.41641	0.0488417	0.0244208	−	0.999702i	$-0.492226\pi$
$29$	1.41641	0.0488417	0.0244208	+	0.999702i	$0.492226\pi$
$30$	0	0
$31$	41.1420i	1.32716i	0.748105	+	0.663581i	$0.230964\pi$
$31$	41.1420i	1.32716i	−0.748105	+	0.663581i	$0.769036\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	29.9641i	0.856116i
$36$	0	0
$37$	54.0590	1.46105	0.730526	−	0.682884i	$-0.239275\pi$
$37$	54.0590	1.46105	0.730526	+	0.682884i	$0.239275\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.74692	0.0669980	0.0334990	−	0.999439i	$-0.489335\pi$
$41$	2.74692	0.0669980	0.0334990	+	0.999439i	$0.489335\pi$
$42$	0	0
$43$	22.6732i	0.527283i	0.964621	+	0.263641i	$0.0849235\pi$
$43$	22.6732i	0.527283i	−0.964621	+	0.263641i	$0.915076\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	45.6501i	0.971278i	0.874159	+	0.485639i	$0.161413\pi$
$47$	45.6501i	0.971278i	−0.874159	+	0.485639i	$0.838587\pi$
$48$	0	0
$49$	−130.569	−2.66468
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.37878	0.0826185	0.0413093	−	0.999146i	$-0.486847\pi$
$53$	4.37878	0.0826185	0.0413093	+	0.999146i	$0.486847\pi$
$54$	0	0
$55$	− 47.4631i	− 0.862965i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 30.8060i	− 0.522136i	−0.965320	−	0.261068i	$-0.915925\pi$
$59$	− 30.8060i	− 0.522136i	0.965320	−	0.261068i	$-0.0840746\pi$
$60$	0	0
$61$	27.0376	0.443240	0.221620	−	0.975133i	$-0.428866\pi$
$61$	27.0376	0.443240	0.221620	+	0.975133i	$0.428866\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.95332	0.0915896
$66$	0	0
$67$	− 103.603i	− 1.54632i	−0.634213	−	0.773159i	$-0.718676\pi$
$67$	− 103.603i	− 1.54632i	0.634213	−	0.773159i	$-0.281324\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 88.3730i	− 1.24469i	−0.782743	−	0.622345i	$-0.786180\pi$
$71$	− 88.3730i	− 1.24469i	0.782743	−	0.622345i	$-0.213820\pi$
$72$	0	0
$73$	36.0216	0.493447	0.246723	−	0.969086i	$-0.420646\pi$
$73$	36.0216	0.493447	0.246723	+	0.969086i	$0.420646\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	284.437	3.69399
$78$	0	0
$79$	23.6019i	0.298758i	0.988780	+	0.149379i	$0.0477274\pi$
$79$	23.6019i	0.298758i	−0.988780	+	0.149379i	$0.952273\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	119.743i	1.44269i	0.692578	+	0.721343i	$0.256475\pi$
$83$	119.743i	1.44269i	−0.692578	+	0.721343i	$0.743525\pi$
$84$	0	0
$85$	−44.1796	−0.519760
$86$	0	0
$87$	0	0
$88$	0	0
$89$	71.7494	0.806172	0.403086	−	0.915162i	$-0.367937\pi$
$89$	71.7494	0.806172	0.403086	+	0.915162i	$0.367937\pi$
$90$	0	0
$91$	35.6772i	0.392057i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	30.3092i	0.319044i
$96$	0	0
$97$	121.426	1.25181	0.625905	−	0.779899i	$-0.284730\pi$
$97$	121.426	1.25181	0.625905	+	0.779899i	$0.284730\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−31.2804	−0.309707	−0.154853	−	0.987937i	$-0.549491\pi$
$101$	−31.2804	−0.309707	−0.154853	+	0.987937i	$0.549491\pi$
$102$	0	0
$103$	− 122.811i	− 1.19234i	−0.802859	−	0.596170i	$-0.796689\pi$
$103$	− 122.811i	− 1.19234i	0.802859	−	0.596170i	$-0.203311\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 60.3613i	− 0.564124i	−0.959396	−	0.282062i	$-0.908982\pi$
$107$	− 60.3613i	− 0.564124i	0.959396	−	0.282062i	$-0.0910184\pi$
$108$	0	0
$109$	−119.167	−1.09327	−0.546637	−	0.837370i	$-0.684092\pi$
$109$	−119.167	−1.09327	−0.546637	+	0.837370i	$0.684092\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.09956	0.0362793	0.0181396	−	0.999835i	$-0.494226\pi$
$113$	4.09956	0.0362793	0.0181396	+	0.999835i	$0.494226\pi$
$114$	0	0
$115$	89.6716i	0.779753i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 264.760i	− 2.22488i
$120$	0	0
$121$	−329.549	−2.72354
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 123.411i	− 0.971741i	−0.874031	−	0.485870i	$-0.838503\pi$
$127$	− 123.411i	− 0.971741i	0.874031	−	0.485870i	$-0.161497\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 55.7181i	− 0.425329i	−0.977125	−	0.212665i	$-0.931786\pi$
$131$	− 55.7181i	− 0.425329i	0.977125	−	0.212665i	$-0.0682142\pi$
$132$	0	0
$133$	−181.637	−1.36569
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−39.1191	−0.285541	−0.142770	−	0.989756i	$-0.545601\pi$
$137$	−39.1191	−0.285541	−0.142770	+	0.989756i	$0.545601\pi$
$138$	0	0
$139$	− 56.9408i	− 0.409646i	−0.978799	−	0.204823i	$-0.934338\pi$
$139$	− 56.9408i	− 0.409646i	0.978799	−	0.204823i	$-0.0656619\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 56.5126i	− 0.395193i
$144$	0	0
$145$	−3.16718	−0.0218427
$146$	0	0
$147$	0	0
$148$	0	0
$149$	149.643	1.00431	0.502156	−	0.864777i	$-0.332540\pi$
$149$	149.643	1.00431	0.502156	+	0.864777i	$0.332540\pi$
$150$	0	0
$151$	− 155.119i	− 1.02728i	−0.858006	−	0.513640i	$-0.828297\pi$
$151$	− 155.119i	− 1.02728i	0.858006	−	0.513640i	$-0.171703\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 91.9963i	− 0.593525i
$156$	0	0
$157$	−58.9293	−0.375346	−0.187673	−	0.982232i	$-0.560095\pi$
$157$	−58.9293	−0.375346	−0.187673	+	0.982232i	$0.560095\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−537.385	−3.33779
$162$	0	0
$163$	− 209.423i	− 1.28481i	−0.766367	−	0.642403i	$-0.777937\pi$
$163$	− 209.423i	− 1.28481i	0.766367	−	0.642403i	$-0.222063\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.65069i	− 0.0278485i	−0.999903	−	0.0139242i	$-0.995568\pi$
$167$	− 4.65069i	− 0.0278485i	0.999903	−	0.0139242i	$-0.00443236\pi$
$168$	0	0
$169$	−161.912	−0.958057
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−140.649	−0.813001	−0.406501	−	0.913651i	$-0.633251\pi$
$173$	−140.649	−0.813001	−0.406501	+	0.913651i	$0.633251\pi$
$174$	0	0
$175$	− 67.0017i	− 0.382867i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	120.713i	0.674374i	0.941438	+	0.337187i	$0.109475\pi$
$179$	120.713i	0.674374i	−0.941438	+	0.337187i	$0.890525\pi$
$180$	0	0
$181$	5.48664	0.0303129	0.0151565	−	0.999885i	$-0.495175\pi$
$181$	5.48664	0.0303129	0.0151565	+	0.999885i	$0.495175\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−120.879	−0.653403
$186$	0	0
$187$	419.380i	2.24268i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.80252i	0.00943726i	0.999989	+	0.00471863i	$0.00150199\pi$
$191$	1.80252i	0.00943726i	−0.999989	+	0.00471863i	$0.998498\pi$
$192$	0	0
$193$	−120.214	−0.622873	−0.311436	−	0.950267i	$-0.600810\pi$
$193$	−120.214	−0.622873	−0.311436	+	0.950267i	$0.600810\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−29.2006	−0.148226	−0.0741132	−	0.997250i	$-0.523613\pi$
$197$	−29.2006	−0.148226	−0.0741132	+	0.997250i	$0.523613\pi$
$198$	0	0
$199$	− 125.025i	− 0.628267i	−0.949379	−	0.314134i	$-0.898286\pi$
$199$	− 125.025i	− 0.628267i	0.949379	−	0.314134i	$-0.101714\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 18.9803i	− 0.0934992i
$204$	0	0
$205$	−6.14230	−0.0299624
$206$	0	0
$207$	0	0
$208$	0	0
$209$	287.713	1.37662
$210$	0	0
$211$	− 165.547i	− 0.784582i	−0.919841	−	0.392291i	$-0.871683\pi$
$211$	− 165.547i	− 0.784582i	0.919841	−	0.392291i	$-0.128317\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 50.6987i	− 0.235808i
$216$	0	0
$217$	551.317	2.54063
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−52.6031	−0.238023
$222$	0	0
$223$	− 367.671i	− 1.64875i	−0.566044	−	0.824375i	$-0.691527\pi$
$223$	− 367.671i	− 1.64875i	0.566044	−	0.824375i	$-0.308473\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 218.397i	− 0.962102i	−0.876693	−	0.481051i	$-0.840255\pi$
$227$	− 218.397i	− 0.962102i	0.876693	−	0.481051i	$-0.159745\pi$
$228$	0	0
$229$	163.566	0.714260	0.357130	−	0.934055i	$-0.383755\pi$
$229$	163.566	0.714260	0.357130	+	0.934055i	$0.383755\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−253.285	−1.08706	−0.543530	−	0.839390i	$-0.682913\pi$
$233$	−253.285	−1.08706	−0.543530	+	0.839390i	$0.682913\pi$
$234$	0	0
$235$	− 102.077i	− 0.434369i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	115.321i	0.482516i	0.970461	+	0.241258i	$0.0775600\pi$
$239$	115.321i	0.482516i	−0.970461	+	0.241258i	$0.922440\pi$
$240$	0	0
$241$	127.121	0.527473	0.263737	−	0.964595i	$-0.415045\pi$
$241$	127.121	0.527473	0.263737	+	0.964595i	$0.415045\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	291.961	1.19168
$246$	0	0
$247$	36.0881i	0.146106i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.18165i	0.0325962i	0.999867	+	0.0162981i	$0.00518808\pi$
$251$	8.18165i	0.0325962i	−0.999867	+	0.0162981i	$0.994812\pi$
$252$	0	0
$253$	851.218	3.36450
$254$	0	0
$255$	0	0
$256$	0	0
$257$	230.691	0.897630	0.448815	−	0.893625i	$-0.351846\pi$
$257$	230.691	0.897630	0.448815	+	0.893625i	$0.351846\pi$
$258$	0	0
$259$	− 724.408i	− 2.79694i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 0.418350i	− 0.00159068i	−1.00000		0.000795342i	$-0.999747\pi$
$263$	− 0.418350i	− 0.00159068i	1.00000		0.000795342i	$-0.000253165\pi$
$264$	0	0
$265$	−9.79125	−0.0369481
$266$	0	0
$267$	0	0
$268$	0	0
$269$	79.2328	0.294546	0.147273	−	0.989096i	$-0.452950\pi$
$269$	79.2328	0.294546	0.147273	+	0.989096i	$0.452950\pi$
$270$	0	0
$271$	− 273.411i	− 1.00890i	−0.863441	−	0.504449i	$-0.831696\pi$
$271$	− 273.411i	− 1.00890i	0.863441	−	0.504449i	$-0.168304\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	106.131i	0.385930i
$276$	0	0
$277$	−119.347	−0.430855	−0.215427	−	0.976520i	$-0.569114\pi$
$277$	−119.347	−0.430855	−0.215427	+	0.976520i	$0.569114\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−150.779	−0.536578	−0.268289	−	0.963338i	$-0.586458\pi$
$281$	−150.779	−0.536578	−0.268289	+	0.963338i	$0.586458\pi$
$282$	0	0
$283$	− 311.820i	− 1.10184i	−0.834558	−	0.550919i	$-0.814277\pi$
$283$	− 311.820i	− 1.10184i	0.834558	−	0.550919i	$-0.185723\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 36.8096i	− 0.128257i
$288$	0	0
$289$	101.368	0.350754
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−181.094	−0.618067	−0.309034	−	0.951051i	$-0.600006\pi$
$293$	−181.094	−0.618067	−0.309034	+	0.951051i	$0.600006\pi$
$294$	0	0
$295$	68.8843i	0.233506i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	106.769i	0.357086i
$300$	0	0
$301$	303.828	1.00940
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−60.4580	−0.198223
$306$	0	0
$307$	− 306.837i	− 0.999468i	−0.866179	−	0.499734i	$-0.833431\pi$
$307$	− 306.837i	− 0.999468i	0.866179	−	0.499734i	$-0.166569\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 525.917i	− 1.69105i	−0.533934	−	0.845526i	$-0.679287\pi$
$311$	− 525.917i	− 1.69105i	0.533934	−	0.845526i	$-0.320713\pi$
$312$	0	0
$313$	384.508	1.22846	0.614230	−	0.789127i	$-0.289467\pi$
$313$	384.508	1.22846	0.614230	+	0.789127i	$0.289467\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	46.6842	0.147269	0.0736345	−	0.997285i	$-0.476540\pi$
$317$	46.6842	0.147269	0.0736345	+	0.997285i	$0.476540\pi$
$318$	0	0
$319$	30.0649i	0.0942472i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 267.810i	− 0.829132i
$324$	0	0
$325$	−13.3120	−0.0409601
$326$	0	0
$327$	0	0
$328$	0	0
$329$	611.726	1.85935
$330$	0	0
$331$	377.349i	1.14003i	0.821635	+	0.570014i	$0.193062\pi$
$331$	377.349i	1.14003i	−0.821635	+	0.570014i	$0.806938\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	231.664i	0.691534i
$336$	0	0
$337$	109.117	0.323788	0.161894	−	0.986808i	$-0.448240\pi$
$337$	109.117	0.323788	0.161894	+	0.986808i	$0.448240\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−873.286	−2.56096
$342$	0	0
$343$	1093.05i	3.18674i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 177.296i	− 0.510940i	−0.966817	−	0.255470i	$-0.917770\pi$
$347$	− 177.296i	− 0.510940i	0.966817	−	0.255470i	$-0.0822302\pi$
$348$	0	0
$349$	−52.4982	−0.150425	−0.0752123	−	0.997168i	$-0.523963\pi$
$349$	−52.4982	−0.150425	−0.0752123	+	0.997168i	$0.523963\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	163.591	0.463431	0.231715	−	0.972784i	$-0.425566\pi$
$353$	163.591	0.463431	0.231715	+	0.972784i	$0.425566\pi$
$354$	0	0
$355$	197.608i	0.556642i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 550.055i	− 1.53219i	−0.642729	−	0.766094i	$-0.722198\pi$
$359$	− 550.055i	− 1.53219i	0.642729	−	0.766094i	$-0.277802\pi$
$360$	0	0
$361$	177.271	0.491054
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−80.5468	−0.220676
$366$	0	0
$367$	128.260i	0.349481i	0.984615	+	0.174740i	$0.0559086\pi$
$367$	128.260i	0.349481i	−0.984615	+	0.174740i	$0.944091\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 58.6772i	− 0.158159i
$372$	0	0
$373$	345.575	0.926474	0.463237	−	0.886234i	$-0.346688\pi$
$373$	345.575	0.926474	0.463237	+	0.886234i	$0.346688\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.77105	−0.0100028
$378$	0	0
$379$	− 132.846i	− 0.350516i	−0.984523	−	0.175258i	$-0.943924\pi$
$379$	− 132.846i	− 0.350516i	0.984523	−	0.175258i	$-0.0560760\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 409.707i	− 1.06973i	−0.844937	−	0.534866i	$-0.820362\pi$
$383$	− 409.707i	− 1.06973i	0.844937	−	0.534866i	$-0.179638\pi$
$384$	0	0
$385$	−636.021	−1.65200
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−257.665	−0.662378	−0.331189	−	0.943564i	$-0.607450\pi$
$389$	−257.665	−0.662378	−0.331189	+	0.943564i	$0.607450\pi$
$390$	0	0
$391$	− 792.331i	− 2.02642i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 52.7754i	− 0.133609i
$396$	0	0
$397$	−583.420	−1.46957	−0.734786	−	0.678299i	$-0.762717\pi$
$397$	−583.420	−1.46957	−0.734786	+	0.678299i	$0.762717\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	530.329	1.32252	0.661259	−	0.750158i	$-0.270023\pi$
$401$	530.329	1.32252	0.661259	+	0.750158i	$0.270023\pi$
$402$	0	0
$403$	− 109.537i	− 0.271803i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1147.46i	2.81932i
$408$	0	0
$409$	215.526	0.526958	0.263479	−	0.964665i	$-0.415130\pi$
$409$	215.526	0.526958	0.263479	+	0.964665i	$0.415130\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−412.811	−0.999542
$414$	0	0
$415$	− 267.753i	− 0.645189i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 190.248i	− 0.454052i	−0.973889	−	0.227026i	$-0.927100\pi$
$419$	− 190.248i	− 0.454052i	0.973889	−	0.227026i	$-0.0729002\pi$
$420$	0	0
$421$	442.383	1.05079	0.525395	−	0.850858i	$-0.323917\pi$
$421$	442.383	1.05079	0.525395	+	0.850858i	$0.323917\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	98.7886	0.232444
$426$	0	0
$427$	− 362.313i	− 0.848509i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	99.5817i	0.231048i	0.993305	+	0.115524i	$0.0368547\pi$
$431$	99.5817i	0.231048i	−0.993305	+	0.115524i	$0.963145\pi$
$432$	0	0
$433$	269.237	0.621795	0.310897	−	0.950444i	$-0.399370\pi$
$433$	269.237	0.621795	0.310897	+	0.950444i	$0.399370\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−543.574	−1.24388
$438$	0	0
$439$	494.556i	1.12655i	0.826269	+	0.563275i	$0.190459\pi$
$439$	494.556i	1.12655i	−0.826269	+	0.563275i	$0.809541\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 469.538i	− 1.05991i	−0.848027	−	0.529953i	$-0.822210\pi$
$443$	− 469.538i	− 1.05991i	0.848027	−	0.529953i	$-0.177790\pi$
$444$	0	0
$445$	−160.436	−0.360531
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−210.693	−0.469250	−0.234625	−	0.972086i	$-0.575386\pi$
$449$	−210.693	−0.469250	−0.234625	+	0.972086i	$0.575386\pi$
$450$	0	0
$451$	58.3065i	0.129283i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 79.7765i	− 0.175333i
$456$	0	0
$457$	−125.599	−0.274833	−0.137416	−	0.990513i	$-0.543880\pi$
$457$	−125.599	−0.274833	−0.137416	+	0.990513i	$0.543880\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	320.080	0.694316	0.347158	−	0.937807i	$-0.387147\pi$
$461$	320.080	0.694316	0.347158	+	0.937807i	$0.387147\pi$
$462$	0	0
$463$	− 589.987i	− 1.27427i	−0.770752	−	0.637135i	$-0.780119\pi$
$463$	− 589.987i	− 1.27427i	0.770752	−	0.637135i	$-0.219881\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	841.638i	1.80222i	0.433589	+	0.901111i	$0.357247\pi$
$467$	841.638i	1.80222i	−0.433589	+	0.901111i	$0.642753\pi$
$468$	0	0
$469$	−1388.32	−2.96017
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−481.263	−1.01747
$474$	0	0
$475$	− 67.7734i	− 0.142681i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	306.414i	0.639695i	0.947469	+	0.319847i	$0.103632\pi$
$479$	306.414i	0.639695i	−0.947469	+	0.319847i	$0.896368\pi$
$480$	0	0
$481$	−143.927	−0.299224
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−271.516	−0.559827
$486$	0	0
$487$	323.935i	0.665164i	0.943074	+	0.332582i	$0.107920\pi$
$487$	323.935i	0.665164i	−0.943074	+	0.332582i	$0.892080\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	707.651i	1.44124i	0.693328	+	0.720622i	$0.256144\pi$
$491$	707.651i	1.44124i	−0.693328	+	0.720622i	$0.743856\pi$
$492$	0	0
$493$	27.9850	0.0567647
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1184.23	−2.38275
$498$	0	0
$499$	677.962i	1.35864i	0.733841	+	0.679321i	$0.237726\pi$
$499$	677.962i	1.35864i	−0.733841	+	0.679321i	$0.762274\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	816.510i	1.62328i	0.584158	+	0.811640i	$0.301425\pi$
$503$	816.510i	1.62328i	−0.584158	+	0.811640i	$0.698575\pi$
$504$	0	0
$505$	69.9451	0.138505
$506$	0	0
$507$	0	0
$508$	0	0
$509$	460.885	0.905472	0.452736	−	0.891645i	$-0.350448\pi$
$509$	460.885	0.905472	0.452736	+	0.891645i	$0.350448\pi$
$510$	0	0
$511$	− 482.702i	− 0.944622i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	274.614i	0.533230i
$516$	0	0
$517$	−968.975	−1.87423
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−190.738	−0.366100	−0.183050	−	0.983104i	$-0.558597\pi$
$521$	−190.738	−0.366100	−0.183050	+	0.983104i	$0.558597\pi$
$522$	0	0
$523$	− 500.450i	− 0.956884i	−0.878119	−	0.478442i	$-0.841202\pi$
$523$	− 500.450i	− 0.956884i	0.878119	−	0.478442i	$-0.158798\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	812.873i	1.54245i
$528$	0	0
$529$	−1079.20	−2.04007
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.31341	−0.0137212
$534$	0	0
$535$	134.972i	0.252284i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2771.48i	− 5.14189i
$540$	0	0
$541$	261.127	0.482675	0.241338	−	0.970441i	$-0.422414\pi$
$541$	261.127	0.482675	0.241338	+	0.970441i	$0.422414\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	266.465	0.488927
$546$	0	0
$547$	− 980.023i	− 1.79163i	−0.444425	−	0.895816i	$-0.646592\pi$
$547$	− 980.023i	− 1.79163i	0.444425	−	0.895816i	$-0.353408\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 19.1990i	− 0.0348438i
$552$	0	0
$553$	316.273	0.571922
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1024.43	−1.83918	−0.919592	−	0.392874i	$-0.871481\pi$
$557$	−1024.43	−1.83918	−0.919592	+	0.392874i	$0.871481\pi$
$558$	0	0
$559$	− 60.3652i	− 0.107988i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 734.755i	− 1.30507i	−0.757758	−	0.652536i	$-0.773705\pi$
$563$	− 734.755i	− 1.30507i	0.757758	−	0.652536i	$-0.226295\pi$
$564$	0	0
$565$	−9.16689	−0.0162246
$566$	0	0
$567$	0	0
$568$	0	0
$569$	938.497	1.64938	0.824690	−	0.565585i	$-0.191350\pi$
$569$	938.497	1.64938	0.824690	+	0.565585i	$0.191350\pi$
$570$	0	0
$571$	560.946i	0.982393i	0.871049	+	0.491196i	$0.163440\pi$
$571$	560.946i	0.982393i	−0.871049	+	0.491196i	$0.836560\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 200.512i	− 0.348716i
$576$	0	0
$577$	−576.150	−0.998527	−0.499264	−	0.866450i	$-0.666396\pi$
$577$	−576.150	−0.998527	−0.499264	+	0.866450i	$0.666396\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1604.60	2.76178
$582$	0	0
$583$	92.9446i	0.159425i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 251.493i	− 0.428438i	−0.976786	−	0.214219i	$-0.931279\pi$
$587$	− 251.493i	− 0.428438i	0.976786	−	0.214219i	$-0.0687207\pi$
$588$	0	0
$589$	557.667	0.946802
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−491.708	−0.829187	−0.414594	−	0.910007i	$-0.636076\pi$
$593$	−491.708	−0.829187	−0.414594	+	0.910007i	$0.636076\pi$
$594$	0	0
$595$	592.022i	0.994995i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	139.629i	0.233103i	0.993185	+	0.116552i	$0.0371840\pi$
$599$	139.629i	0.233103i	−0.993185	+	0.116552i	$0.962816\pi$
$600$	0	0
$601$	510.946	0.850160	0.425080	−	0.905156i	$-0.360246\pi$
$601$	510.946	0.850160	0.425080	+	0.905156i	$0.360246\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	736.894	1.21801
$606$	0	0
$607$	438.309i	0.722091i	0.932548	+	0.361045i	$0.117580\pi$
$607$	438.309i	0.722091i	−0.932548	+	0.361045i	$0.882420\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 121.539i	− 0.198918i
$612$	0	0
$613$	−1091.72	−1.78094	−0.890470	−	0.455041i	$-0.849624\pi$
$613$	−1091.72	−1.78094	−0.890470	+	0.455041i	$0.849624\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	902.138	1.46214	0.731068	−	0.682304i	$-0.239022\pi$
$617$	902.138	1.46214	0.731068	+	0.682304i	$0.239022\pi$
$618$	0	0
$619$	− 466.462i	− 0.753573i	−0.926300	−	0.376786i	$-0.877029\pi$
$619$	− 466.462i	− 0.753573i	0.926300	−	0.376786i	$-0.122971\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 961.466i	− 1.54328i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1068.08	1.69806
$630$	0	0
$631$	1252.04i	1.98422i	0.125378	+	0.992109i	$0.459986\pi$
$631$	1252.04i	1.98422i	−0.125378	+	0.992109i	$0.540014\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	275.956i	0.434576i
$636$	0	0
$637$	347.628	0.545727
$638$	0	0
$639$	0	0
$640$	0	0
$641$	321.815	0.502052	0.251026	−	0.967980i	$-0.419232\pi$
$641$	321.815	0.502052	0.251026	+	0.967980i	$0.419232\pi$
$642$	0	0
$643$	1096.83i	1.70580i	0.522078	+	0.852898i	$0.325157\pi$
$643$	1096.83i	1.70580i	−0.522078	+	0.852898i	$0.674843\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 446.564i	− 0.690207i	−0.938565	−	0.345103i	$-0.887844\pi$
$647$	− 446.564i	− 0.690207i	0.938565	−	0.345103i	$-0.112156\pi$
$648$	0	0
$649$	653.892	1.00754
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−387.067	−0.592752	−0.296376	−	0.955071i	$-0.595778\pi$
$653$	−387.067	−0.592752	−0.296376	+	0.955071i	$0.595778\pi$
$654$	0	0
$655$	124.590i	0.190213i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1294.49i	1.96433i	0.188019	+	0.982165i	$0.439793\pi$
$659$	1294.49i	1.96433i	−0.188019	+	0.982165i	$0.560207\pi$
$660$	0	0
$661$	339.882	0.514193	0.257097	−	0.966386i	$-0.417234\pi$
$661$	339.882	0.514193	0.257097	+	0.966386i	$0.417234\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	406.153	0.610757
$666$	0	0
$667$	− 56.8013i	− 0.0851593i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	573.904i	0.855297i
$672$	0	0
$673$	1144.40	1.70045	0.850225	−	0.526420i	$-0.176466\pi$
$673$	1144.40	1.70045	0.850225	+	0.526420i	$0.176466\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−687.908	−1.01611	−0.508056	−	0.861324i	$-0.669636\pi$
$677$	−687.908	−1.01611	−0.508056	+	0.861324i	$0.669636\pi$
$678$	0	0
$679$	− 1627.14i	− 2.39638i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	513.570i	0.751932i	0.926633	+	0.375966i	$0.122689\pi$
$683$	513.570i	0.751932i	−0.926633	+	0.375966i	$0.877311\pi$
$684$	0	0
$685$	87.4729	0.127698
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11.6581	−0.0169203
$690$	0	0
$691$	188.414i	0.272668i	0.990663	+	0.136334i	$0.0435320\pi$
$691$	188.414i	0.272668i	−0.990663	+	0.136334i	$0.956468\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	127.324i	0.183199i
$696$	0	0
$697$	54.2729	0.0778664
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1170.06	1.66913	0.834564	−	0.550911i	$-0.185720\pi$
$701$	1170.06	1.66913	0.834564	+	0.550911i	$0.185720\pi$
$702$	0	0
$703$	− 732.752i	− 1.04232i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	419.168i	0.592883i
$708$	0	0
$709$	−847.689	−1.19561	−0.597806	−	0.801641i	$-0.703961\pi$
$709$	−847.689	−1.19561	−0.597806	+	0.801641i	$0.703961\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1649.89	2.31401
$714$	0	0
$715$	126.366i	0.176736i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	437.962i	0.609127i	0.952492	+	0.304563i	$0.0985106\pi$
$719$	437.962i	0.609127i	−0.952492	+	0.304563i	$0.901489\pi$
$720$	0	0
$721$	−1645.71	−2.28254
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.08204	0.00976833
$726$	0	0
$727$	684.271i	0.941225i	0.882340	+	0.470613i	$0.155967\pi$
$727$	684.271i	0.941225i	−0.882340	+	0.470613i	$0.844033\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	447.970i	0.612818i
$732$	0	0
$733$	−870.642	−1.18778	−0.593889	−	0.804547i	$-0.702408\pi$
$733$	−870.642	−1.18778	−0.593889	+	0.804547i	$0.702408\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2199.10	2.98385
$738$	0	0
$739$	− 482.490i	− 0.652896i	−0.945215	−	0.326448i	$-0.894148\pi$
$739$	− 482.490i	− 0.652896i	0.945215	−	0.326448i	$-0.105852\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 93.4856i	− 0.125822i	−0.998019	−	0.0629109i	$-0.979962\pi$
$743$	− 93.4856i	− 0.125822i	0.998019	−	0.0629109i	$-0.0200384\pi$
$744$	0	0
$745$	−334.611	−0.449142
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−808.862	−1.07992
$750$	0	0
$751$	− 621.625i	− 0.827729i	−0.910338	−	0.413865i	$-0.864179\pi$
$751$	− 621.625i	− 0.827729i	0.910338	−	0.413865i	$-0.135821\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	346.857i	0.459414i
$756$	0	0
$757$	215.708	0.284951	0.142476	−	0.989798i	$-0.454494\pi$
$757$	215.708	0.284951	0.142476	+	0.989798i	$0.454494\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−85.6655	−0.112570	−0.0562848	−	0.998415i	$-0.517925\pi$
$761$	−85.6655	−0.112570	−0.0562848	+	0.998415i	$0.517925\pi$
$762$	0	0
$763$	1596.88i	2.09289i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	82.0181i	0.106934i
$768$	0	0
$769$	1303.45	1.69499	0.847496	−	0.530802i	$-0.178109\pi$
$769$	1303.45	1.69499	0.847496	+	0.530802i	$0.178109\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	95.5749	0.123641	0.0618207	−	0.998087i	$-0.480309\pi$
$773$	95.5749	0.123641	0.0618207	+	0.998087i	$0.480309\pi$
$774$	0	0
$775$	205.710i	0.265432i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 37.2336i	− 0.0477967i
$780$	0	0
$781$	1875.82	2.40181
$782$	0	0
$783$	0	0
$784$	0	0
$785$	131.770	0.167860
$786$	0	0
$787$	758.917i	0.964317i	0.876084	+	0.482158i	$0.160147\pi$
$787$	758.917i	0.964317i	−0.876084	+	0.482158i	$0.839853\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 54.9354i	− 0.0694506i
$792$	0	0
$793$	−71.9852	−0.0907757
$794$	0	0
$795$	0	0
$796$	0	0
$797$	415.809	0.521717	0.260859	−	0.965377i	$-0.415994\pi$
$797$	415.809	0.521717	0.260859	+	0.965377i	$0.415994\pi$
$798$	0	0
$799$	901.942i	1.12884i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	764.600i	0.952179i
$804$	0	0
$805$	1201.63	1.49271
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−874.754	−1.08128	−0.540639	−	0.841255i	$-0.681817\pi$
$809$	−874.754	−1.08128	−0.540639	+	0.841255i	$0.681817\pi$
$810$	0	0
$811$	− 644.197i	− 0.794324i	−0.917748	−	0.397162i	$-0.869995\pi$
$811$	− 644.197i	− 0.794324i	0.917748	−	0.397162i	$-0.130005\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	468.285i	0.574583i
$816$	0	0
$817$	307.327	0.376166
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−224.881	−0.273911	−0.136956	−	0.990577i	$-0.543732\pi$
$821$	−224.881	−0.273911	−0.136956	+	0.990577i	$0.543732\pi$
$822$	0	0
$823$	378.145i	0.459472i	0.973253	+	0.229736i	$0.0737862\pi$
$823$	378.145i	0.459472i	−0.973253	+	0.229736i	$0.926214\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	989.341i	1.19630i	0.801384	+	0.598151i	$0.204098\pi$
$827$	989.341i	1.19630i	−0.801384	+	0.598151i	$0.795902\pi$
$828$	0	0
$829$	−1441.57	−1.73893	−0.869466	−	0.493993i	$-0.835537\pi$
$829$	−1441.57	−1.73893	−0.869466	+	0.493993i	$0.835537\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2579.75	−3.09694
$834$	0	0
$835$	10.3993i	0.0124542i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1261.59i	− 1.50369i	−0.659343	−	0.751843i	$-0.729165\pi$
$839$	− 1261.59i	− 1.50369i	0.659343	−	0.751843i	$-0.270835\pi$
$840$	0	0
$841$	−838.994	−0.997614
$842$	0	0
$843$	0	0
$844$	0	0
$845$	362.045	0.428456
$846$	0	0
$847$	4416.07i	5.21377i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2167.89i	− 2.54746i
$852$	0	0
$853$	854.061	1.00124	0.500622	−	0.865666i	$-0.333105\pi$
$853$	854.061	1.00124	0.500622	+	0.865666i	$0.333105\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1585.26	−1.84978	−0.924891	−	0.380231i	$-0.875844\pi$
$857$	−1585.26	−1.84978	−0.924891	+	0.380231i	$0.875844\pi$
$858$	0	0
$859$	− 694.468i	− 0.808460i	−0.914657	−	0.404230i	$-0.867539\pi$
$859$	− 694.468i	− 0.808460i	0.914657	−	0.404230i	$-0.132461\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 698.741i	− 0.809665i	−0.914391	−	0.404832i	$-0.867330\pi$
$863$	− 698.741i	− 0.809665i	0.914391	−	0.404832i	$-0.132670\pi$
$864$	0	0
$865$	314.501	0.363585
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−500.976	−0.576498
$870$	0	0
$871$	275.834i	0.316687i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	149.820i	0.171223i
$876$	0	0
$877$	1300.63	1.48304	0.741520	−	0.670930i	$-0.234105\pi$
$877$	1300.63	1.48304	0.741520	+	0.670930i	$0.234105\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−629.171	−0.714156	−0.357078	−	0.934075i	$-0.616227\pi$
$881$	−629.171	−0.714156	−0.357078	+	0.934075i	$0.616227\pi$
$882$	0	0
$883$	− 940.453i	− 1.06507i	−0.846409	−	0.532533i	$-0.821240\pi$
$883$	− 940.453i	− 1.06507i	0.846409	−	0.532533i	$-0.178760\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 660.503i	− 0.744649i	−0.928103	−	0.372324i	$-0.878561\pi$
$887$	− 660.503i	− 0.744649i	0.928103	−	0.372324i	$-0.121439\pi$
$888$	0	0
$889$	−1653.75	−1.86024
$890$	0	0
$891$	0	0
$892$	0	0
$893$	618.772	0.692914
$894$	0	0
$895$	− 269.922i	− 0.301589i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	58.2739i	0.0648207i
$900$	0	0
$901$	86.5148	0.0960208
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12.2685	−0.0135563
$906$	0	0
$907$	412.583i	0.454888i	0.973791	+	0.227444i	$0.0730369\pi$
$907$	412.583i	0.454888i	−0.973791	+	0.227444i	$0.926963\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 160.140i	− 0.175785i	−0.996130	−	0.0878927i	$-0.971987\pi$
$911$	− 160.140i	− 0.175785i	0.996130	−	0.0878927i	$-0.0280133\pi$
$912$	0	0
$913$	−2541.68	−2.78388
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−746.642	−0.814222
$918$	0	0
$919$	− 655.200i	− 0.712949i	−0.934305	−	0.356474i	$-0.883979\pi$
$919$	− 655.200i	− 0.712949i	0.934305	−	0.356474i	$-0.116021\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	235.285i	0.254913i
$924$	0	0
$925$	270.295	0.292211
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1502.84	−1.61770	−0.808851	−	0.588014i	$-0.799910\pi$
$929$	−1502.84	−1.61770	−0.808851	+	0.588014i	$0.799910\pi$
$930$	0	0
$931$	1769.82i	1.90099i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 937.763i	− 1.00295i
$936$	0	0
$937$	1559.91	1.66479	0.832395	−	0.554182i	$-0.186969\pi$
$937$	1559.91	1.66479	0.832395	+	0.554182i	$0.186969\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−101.791	−0.108173	−0.0540865	−	0.998536i	$-0.517225\pi$
$941$	−101.791	−0.108173	−0.0540865	+	0.998536i	$0.517225\pi$
$942$	0	0
$943$	− 110.158i	− 0.116816i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1567.59i	1.65532i	0.561228	+	0.827661i	$0.310329\pi$
$947$	1567.59i	1.65532i	−0.561228	+	0.827661i	$0.689671\pi$
$948$	0	0
$949$	−95.9042	−0.101058
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1469.65	1.54213	0.771063	−	0.636759i	$-0.219726\pi$
$953$	1469.65	1.54213	0.771063	+	0.636759i	$0.219726\pi$
$954$	0	0
$955$	− 4.03055i	− 0.00422047i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	524.209i	0.546620i
$960$	0	0
$961$	−731.664	−0.761357
$962$	0	0
$963$	0	0
$964$	0	0
$965$	268.808	0.278557
$966$	0	0
$967$	− 218.682i	− 0.226145i	−0.993587	−	0.113073i	$-0.963931\pi$
$967$	− 218.682i	− 0.226145i	0.993587	−	0.113073i	$-0.0360693\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1602.56i	1.65042i	0.564823	+	0.825212i	$0.308944\pi$
$971$	1602.56i	1.65042i	−0.564823	+	0.825212i	$0.691056\pi$
$972$	0	0
$973$	−763.027	−0.784200
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−170.494	−0.174508	−0.0872538	−	0.996186i	$-0.527809\pi$
$977$	−170.494	−0.174508	−0.0872538	+	0.996186i	$0.527809\pi$
$978$	0	0
$979$	1522.96i	1.55563i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1780.88i	1.81168i	0.423622	+	0.905839i	$0.360758\pi$
$983$	1780.88i	1.81168i	−0.423622	+	0.905839i	$0.639242\pi$
$984$	0	0
$985$	65.2946	0.0662889
$986$	0	0
$987$	0	0
$988$	0	0
$989$	909.247	0.919359
$990$	0	0
$991$	− 1174.41i	− 1.18508i	−0.805541	−	0.592539i	$-0.798125\pi$
$991$	− 1174.41i	− 1.18508i	0.805541	−	0.592539i	$-0.201875\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	279.565i	0.280970i
$996$	0	0
$997$	−279.730	−0.280572	−0.140286	−	0.990111i	$-0.544802\pi$
$997$	−279.730	−0.280572	−0.140286	+	0.990111i	$0.544802\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.3.e.n.2431.1		8
3.2	odd	2		960.3.e.d.511.3		8
4.3	odd	2	inner	2880.3.e.n.2431.4		8
8.3	odd	2		1440.3.e.e.991.8		8
8.5	even	2		1440.3.e.e.991.5		8
12.11	even	2		960.3.e.d.511.8		8
24.5	odd	2		480.3.e.a.31.5	yes	8
24.11	even	2		480.3.e.a.31.2	✓	8
120.29	odd	2		2400.3.e.g.1951.4		8
120.53	even	4		2400.3.j.d.799.2		8
120.59	even	2		2400.3.e.g.1951.5		8
120.77	even	4		2400.3.j.j.799.8		8
120.83	odd	4		2400.3.j.j.799.7		8
120.107	odd	4		2400.3.j.d.799.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.e.a.31.2	✓	8	24.11	even	2
480.3.e.a.31.5	yes	8	24.5	odd	2
960.3.e.d.511.3		8	3.2	odd	2
960.3.e.d.511.8		8	12.11	even	2
1440.3.e.e.991.5		8	8.5	even	2
1440.3.e.e.991.8		8	8.3	odd	2
2400.3.e.g.1951.4		8	120.29	odd	2
2400.3.e.g.1951.5		8	120.59	even	2
2400.3.j.d.799.1		8	120.107	odd	4
2400.3.j.d.799.2		8	120.53	even	4
2400.3.j.j.799.7		8	120.83	odd	4
2400.3.j.j.799.8		8	120.77	even	4
2880.3.e.n.2431.1		8	1.1	even	1	trivial
2880.3.e.n.2431.4		8	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2880.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607 q^{5} -13.4003i q^{7} +21.2261i q^{11} -2.66241 q^{13} +19.7577 q^{17} -13.5547i q^{19} -40.1023i q^{23} +5.00000 q^{25} +1.41641 q^{29} +41.1420i q^{31} +29.9641i q^{35} +54.0590 q^{37} +2.74692 q^{41} +22.6732i q^{43} +45.6501i q^{47} -130.569 q^{49} +4.37878 q^{53} -47.4631i q^{55} -30.8060i q^{59} +27.0376 q^{61} +5.95332 q^{65} -103.603i q^{67} -88.3730i q^{71} +36.0216 q^{73} +284.437 q^{77} +23.6019i q^{79} +119.743i q^{83} -44.1796 q^{85} +71.7494 q^{89} +35.6772i q^{91} +30.3092i q^{95} +121.426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{13} + 16 q^{17} + 40 q^{25} - 96 q^{29} + 112 q^{37} - 112 q^{41} - 184 q^{49} + 224 q^{53} - 80 q^{61} + 160 q^{65} + 272 q^{73} + 256 q^{77} + 160 q^{85} + 48 q^{89} + 528 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 + 16 * q^17 + 40 * q^25 - 96 * q^29 + 112 * q^37 - 112 * q^41 - 184 * q^49 + 224 * q^53 - 80 * q^61 + 160 * q^65 + 272 * q^73 + 256 * q^77 + 160 * q^85 + 48 * q^89 + 528 * q^97

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