LMFDB - Embedded newform 2160.4.a.bp.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(1,2160)] 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("2160.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,0,0,0,0,27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$127.444125612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1765.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 16 $ x^3 - x^2 - 11*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.59024$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +19.1841 q^{7} -18.3867 q^{11} +14.0829 q^{13} -63.8746 q^{17} -101.409 q^{19} +155.528 q^{23} +25.0000 q^{25} +58.2428 q^{29} -136.467 q^{31} +95.9207 q^{35} -117.924 q^{37} +276.802 q^{41} +451.406 q^{43} -15.2922 q^{47} +25.0310 q^{49} +266.666 q^{53} -91.9333 q^{55} +585.668 q^{59} -392.105 q^{61} +70.4144 q^{65} +1031.46 q^{67} -657.723 q^{71} +125.680 q^{73} -352.732 q^{77} +962.698 q^{79} +721.338 q^{83} -319.373 q^{85} -122.234 q^{89} +270.168 q^{91} -507.043 q^{95} +153.724 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 27 q^{11} - 3 q^{13} - 15 q^{17} + 78 q^{19} + 105 q^{23} + 75 q^{25} - 117 q^{29} + 207 q^{31} - 120 q^{37} - 300 q^{41} + 483 q^{43} + 303 q^{47} - 15 q^{49} + 492 q^{53} + 135 q^{55}+ \cdots - 1392 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 27 * q^11 - 3 * q^13 - 15 * q^17 + 78 * q^19 + 105 * q^23 + 75 * q^25 - 117 * q^29 + 207 * q^31 - 120 * q^37 - 300 * q^41 + 483 * q^43 + 303 * q^47 - 15 * q^49 + 492 * q^53 + 135 * q^55 + 240 * q^59 - 444 * q^61 - 15 * q^65 + 522 * q^67 - 168 * q^71 - 876 * q^73 + 150 * q^77 + 2103 * q^79 - 42 * q^83 - 75 * q^85 + 2268 * q^89 + 1596 * q^91 + 390 * q^95 - 1392 * q^97

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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	19.1841	1.03585	0.517923	−	0.855427i	$-0.326705\pi$
$7$	19.1841	1.03585	0.517923	+	0.855427i	$0.326705\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−18.3867	−0.503980	−0.251990	−	0.967730i	$-0.581085\pi$
$11$	−18.3867	−0.503980	−0.251990	+	0.967730i	$0.581085\pi$
$12$	0	0
$13$	14.0829	0.300453	0.150226	−	0.988652i	$-0.452000\pi$
$13$	14.0829	0.300453	0.150226	+	0.988652i	$0.452000\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−63.8746	−0.911286	−0.455643	−	0.890163i	$-0.650591\pi$
$17$	−63.8746	−0.911286	−0.455643	+	0.890163i	$0.650591\pi$
$18$	0	0
$19$	−101.409	−1.22446	−0.612230	−	0.790680i	$-0.709727\pi$
$19$	−101.409	−1.22446	−0.612230	+	0.790680i	$0.709727\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	155.528	1.40999	0.704997	−	0.709210i	$-0.250948\pi$
$23$	155.528	1.40999	0.704997	+	0.709210i	$0.250948\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	58.2428	0.372946	0.186473	−	0.982460i	$-0.440294\pi$
$29$	58.2428	0.372946	0.186473	+	0.982460i	$0.440294\pi$
$30$	0	0
$31$	−136.467	−0.790653	−0.395327	−	0.918541i	$-0.629369\pi$
$31$	−136.467	−0.790653	−0.395327	+	0.918541i	$0.629369\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	95.9207	0.463244
$36$	0	0
$37$	−117.924	−0.523962	−0.261981	−	0.965073i	$-0.584376\pi$
$37$	−117.924	−0.523962	−0.261981	+	0.965073i	$0.584376\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	276.802	1.05437	0.527186	−	0.849750i	$-0.323247\pi$
$41$	276.802	1.05437	0.527186	+	0.849750i	$0.323247\pi$
$42$	0	0
$43$	451.406	1.60090	0.800451	−	0.599398i	$-0.204593\pi$
$43$	451.406	1.60090	0.800451	+	0.599398i	$0.204593\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−15.2922	−0.0474596	−0.0237298	−	0.999718i	$-0.507554\pi$
$47$	−15.2922	−0.0474596	−0.0237298	+	0.999718i	$0.507554\pi$
$48$	0	0
$49$	25.0310	0.0729767
$50$	0	0
$51$	0	0
$52$	0	0
$53$	266.666	0.691121	0.345561	−	0.938396i	$-0.387689\pi$
$53$	266.666	0.691121	0.345561	+	0.938396i	$0.387689\pi$
$54$	0	0
$55$	−91.9333	−0.225387
$56$	0	0
$57$	0	0
$58$	0	0
$59$	585.668	1.29233	0.646165	−	0.763198i	$-0.276372\pi$
$59$	585.668	1.29233	0.646165	+	0.763198i	$0.276372\pi$
$60$	0	0
$61$	−392.105	−0.823015	−0.411507	−	0.911406i	$-0.634998\pi$
$61$	−392.105	−0.823015	−0.411507	+	0.911406i	$0.634998\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	70.4144	0.134367
$66$	0	0
$67$	1031.46	1.88079	0.940393	−	0.340089i	$-0.110457\pi$
$67$	1031.46	1.88079	0.940393	+	0.340089i	$0.110457\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−657.723	−1.09940	−0.549700	−	0.835362i	$-0.685258\pi$
$71$	−657.723	−1.09940	−0.549700	+	0.835362i	$0.685258\pi$
$72$	0	0
$73$	125.680	0.201503	0.100752	−	0.994912i	$-0.467875\pi$
$73$	125.680	0.201503	0.100752	+	0.994912i	$0.467875\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−352.732	−0.522046
$78$	0	0
$79$	962.698	1.37104	0.685519	−	0.728055i	$-0.259576\pi$
$79$	962.698	1.37104	0.685519	+	0.728055i	$0.259576\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	721.338	0.953942	0.476971	−	0.878919i	$-0.341735\pi$
$83$	721.338	0.953942	0.476971	+	0.878919i	$0.341735\pi$
$84$	0	0
$85$	−319.373	−0.407539
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−122.234	−0.145582	−0.0727910	−	0.997347i	$-0.523191\pi$
$89$	−122.234	−0.145582	−0.0727910	+	0.997347i	$0.523191\pi$
$90$	0	0
$91$	270.168	0.311223
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−507.043	−0.547595
$96$	0	0
$97$	153.724	0.160910	0.0804552	−	0.996758i	$-0.474363\pi$
$97$	153.724	0.160910	0.0804552	+	0.996758i	$0.474363\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	254.759	0.250985	0.125493	−	0.992095i	$-0.459949\pi$
$101$	254.759	0.250985	0.125493	+	0.992095i	$0.459949\pi$
$102$	0	0
$103$	−729.969	−0.698311	−0.349155	−	0.937065i	$-0.613531\pi$
$103$	−729.969	−0.698311	−0.349155	+	0.937065i	$0.613531\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	516.799	0.466924	0.233462	−	0.972366i	$-0.424995\pi$
$107$	516.799	0.466924	0.233462	+	0.972366i	$0.424995\pi$
$108$	0	0
$109$	−1040.20	−0.914067	−0.457034	−	0.889449i	$-0.651088\pi$
$109$	−1040.20	−0.914067	−0.457034	+	0.889449i	$0.651088\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1491.03	1.24128	0.620639	−	0.784096i	$-0.286873\pi$
$113$	1491.03	1.24128	0.620639	+	0.784096i	$0.286873\pi$
$114$	0	0
$115$	777.641	0.630569
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1225.38	−0.943952
$120$	0	0
$121$	−992.931	−0.746004
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1272.29	0.888954	0.444477	−	0.895790i	$-0.353390\pi$
$127$	1272.29	0.888954	0.444477	+	0.895790i	$0.353390\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	490.988	0.327464	0.163732	−	0.986505i	$-0.447647\pi$
$131$	490.988	0.327464	0.163732	+	0.986505i	$0.447647\pi$
$132$	0	0
$133$	−1945.44	−1.26835
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2295.08	−1.43125	−0.715627	−	0.698483i	$-0.753859\pi$
$137$	−2295.08	−1.43125	−0.715627	+	0.698483i	$0.753859\pi$
$138$	0	0
$139$	−1514.52	−0.924173	−0.462086	−	0.886835i	$-0.652899\pi$
$139$	−1514.52	−0.924173	−0.462086	+	0.886835i	$0.652899\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−258.937	−0.151422
$144$	0	0
$145$	291.214	0.166786
$146$	0	0
$147$	0	0
$148$	0	0
$149$	58.6349	0.0322386	0.0161193	−	0.999870i	$-0.494869\pi$
$149$	58.6349	0.0322386	0.0161193	+	0.999870i	$0.494869\pi$
$150$	0	0
$151$	354.303	0.190945	0.0954727	−	0.995432i	$-0.469564\pi$
$151$	354.303	0.190945	0.0954727	+	0.995432i	$0.469564\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−682.336	−0.353591
$156$	0	0
$157$	1515.71	0.770488	0.385244	−	0.922815i	$-0.374117\pi$
$157$	1515.71	0.770488	0.385244	+	0.922815i	$0.374117\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2983.67	1.46054
$162$	0	0
$163$	998.554	0.479833	0.239916	−	0.970794i	$-0.422880\pi$
$163$	998.554	0.479833	0.239916	+	0.970794i	$0.422880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3085.21	1.42958	0.714792	−	0.699337i	$-0.246521\pi$
$167$	3085.21	1.42958	0.714792	+	0.699337i	$0.246521\pi$
$168$	0	0
$169$	−1998.67	−0.909728
$170$	0	0
$171$	0	0
$172$	0	0
$173$	911.357	0.400516	0.200258	−	0.979743i	$-0.435822\pi$
$173$	911.357	0.400516	0.200258	+	0.979743i	$0.435822\pi$
$174$	0	0
$175$	479.603	0.207169
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2443.08	1.02013	0.510067	−	0.860134i	$-0.329620\pi$
$179$	2443.08	1.02013	0.510067	+	0.860134i	$0.329620\pi$
$180$	0	0
$181$	−3175.54	−1.30407	−0.652034	−	0.758190i	$-0.726084\pi$
$181$	−3175.54	−1.30407	−0.652034	+	0.758190i	$0.726084\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−589.620	−0.234323
$186$	0	0
$187$	1174.44	0.459270
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1335.56	0.505956	0.252978	−	0.967472i	$-0.418590\pi$
$191$	1335.56	0.505956	0.252978	+	0.967472i	$0.418590\pi$
$192$	0	0
$193$	−2041.12	−0.761261	−0.380630	−	0.924727i	$-0.624293\pi$
$193$	−2041.12	−0.761261	−0.380630	+	0.924727i	$0.624293\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2565.55	0.927858	0.463929	−	0.885872i	$-0.346439\pi$
$197$	2565.55	0.927858	0.463929	+	0.885872i	$0.346439\pi$
$198$	0	0
$199$	2650.29	0.944091	0.472046	−	0.881574i	$-0.343516\pi$
$199$	2650.29	0.944091	0.472046	+	0.881574i	$0.343516\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1117.34	0.386314
$204$	0	0
$205$	1384.01	0.471530
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1864.56	0.617104
$210$	0	0
$211$	3078.89	1.00455	0.502274	−	0.864708i	$-0.332497\pi$
$211$	3078.89	1.00455	0.502274	+	0.864708i	$0.332497\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2257.03	0.715946
$216$	0	0
$217$	−2618.01	−0.818995
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−899.537	−0.273798
$222$	0	0
$223$	3855.41	1.15775	0.578873	−	0.815417i	$-0.303493\pi$
$223$	3855.41	1.15775	0.578873	+	0.815417i	$0.303493\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2003.70	0.585859	0.292929	−	0.956134i	$-0.405370\pi$
$227$	2003.70	0.585859	0.292929	+	0.956134i	$0.405370\pi$
$228$	0	0
$229$	−1861.72	−0.537230	−0.268615	−	0.963248i	$-0.586566\pi$
$229$	−1861.72	−0.537230	−0.268615	+	0.963248i	$0.586566\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5152.83	−1.44881	−0.724406	−	0.689373i	$-0.757886\pi$
$233$	−5152.83	−1.44881	−0.724406	+	0.689373i	$0.757886\pi$
$234$	0	0
$235$	−76.4611	−0.0212246
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2866.90	−0.775919	−0.387959	−	0.921676i	$-0.626820\pi$
$239$	−2866.90	−0.775919	−0.387959	+	0.921676i	$0.626820\pi$
$240$	0	0
$241$	−336.196	−0.0898600	−0.0449300	−	0.998990i	$-0.514306\pi$
$241$	−336.196	−0.0898600	−0.0449300	+	0.998990i	$0.514306\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	125.155	0.0326362
$246$	0	0
$247$	−1428.12	−0.367892
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1897.62	0.477198	0.238599	−	0.971118i	$-0.423312\pi$
$251$	1897.62	0.477198	0.238599	+	0.971118i	$0.423312\pi$
$252$	0	0
$253$	−2859.64	−0.710610
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5125.56	1.24406	0.622030	−	0.782993i	$-0.286308\pi$
$257$	5125.56	1.24406	0.622030	+	0.782993i	$0.286308\pi$
$258$	0	0
$259$	−2262.27	−0.542743
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−724.470	−0.169858	−0.0849291	−	0.996387i	$-0.527066\pi$
$263$	−724.470	−0.169858	−0.0849291	+	0.996387i	$0.527066\pi$
$264$	0	0
$265$	1333.33	0.309079
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8315.15	1.88470	0.942349	−	0.334632i	$-0.108612\pi$
$269$	8315.15	1.88470	0.942349	+	0.334632i	$0.108612\pi$
$270$	0	0
$271$	−4207.73	−0.943178	−0.471589	−	0.881819i	$-0.656319\pi$
$271$	−4207.73	−0.943178	−0.471589	+	0.881819i	$0.656319\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−459.666	−0.100796
$276$	0	0
$277$	−4105.54	−0.890534	−0.445267	−	0.895398i	$-0.646891\pi$
$277$	−4105.54	−0.890534	−0.445267	+	0.895398i	$0.646891\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5166.52	1.09683	0.548414	−	0.836207i	$-0.315232\pi$
$281$	5166.52	1.09683	0.548414	+	0.836207i	$0.315232\pi$
$282$	0	0
$283$	−5575.42	−1.17111	−0.585556	−	0.810632i	$-0.699124\pi$
$283$	−5575.42	−1.17111	−0.585556	+	0.810632i	$0.699124\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5310.21	1.09217
$288$	0	0
$289$	−833.039	−0.169558
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8071.59	1.60938	0.804689	−	0.593697i	$-0.202332\pi$
$293$	8071.59	1.60938	0.804689	+	0.593697i	$0.202332\pi$
$294$	0	0
$295$	2928.34	0.577947
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2190.28	0.423637
$300$	0	0
$301$	8659.84	1.65829
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1960.53	−0.368063
$306$	0	0
$307$	3972.33	0.738478	0.369239	−	0.929335i	$-0.379618\pi$
$307$	3972.33	0.738478	0.369239	+	0.929335i	$0.379618\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4668.82	0.851269	0.425634	−	0.904895i	$-0.360051\pi$
$311$	4668.82	0.851269	0.425634	+	0.904895i	$0.360051\pi$
$312$	0	0
$313$	3161.61	0.570941	0.285470	−	0.958388i	$-0.407850\pi$
$313$	3161.61	0.570941	0.285470	+	0.958388i	$0.407850\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−91.1973	−0.0161582	−0.00807910	−	0.999967i	$-0.502572\pi$
$317$	−91.1973	−0.0161582	−0.00807910	+	0.999967i	$0.502572\pi$
$318$	0	0
$319$	−1070.89	−0.187957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6477.43	1.11583
$324$	0	0
$325$	352.072	0.0600906
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−293.368	−0.0491608
$330$	0	0
$331$	7496.53	1.24485	0.622426	−	0.782678i	$-0.286147\pi$
$331$	7496.53	1.24485	0.622426	+	0.782678i	$0.286147\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5157.29	0.841113
$336$	0	0
$337$	11689.4	1.88949	0.944747	−	0.327799i	$-0.106307\pi$
$337$	11689.4	1.88949	0.944747	+	0.327799i	$0.106307\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2509.18	0.398474
$342$	0	0
$343$	−6099.96	−0.960253
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3958.51	0.612403	0.306202	−	0.951967i	$-0.400942\pi$
$347$	3958.51	0.612403	0.306202	+	0.951967i	$0.400942\pi$
$348$	0	0
$349$	−5997.04	−0.919811	−0.459906	−	0.887968i	$-0.652117\pi$
$349$	−5997.04	−0.919811	−0.459906	+	0.887968i	$0.652117\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8029.56	1.21068	0.605340	−	0.795967i	$-0.293037\pi$
$353$	8029.56	1.21068	0.605340	+	0.795967i	$0.293037\pi$
$354$	0	0
$355$	−3288.62	−0.491667
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5454.65	−0.801909	−0.400955	−	0.916098i	$-0.631322\pi$
$359$	−5454.65	−0.801909	−0.400955	+	0.916098i	$0.631322\pi$
$360$	0	0
$361$	3424.70	0.499300
$362$	0	0
$363$	0	0
$364$	0	0
$365$	628.401	0.0901151
$366$	0	0
$367$	3946.72	0.561354	0.280677	−	0.959802i	$-0.409441\pi$
$367$	3946.72	0.561354	0.280677	+	0.959802i	$0.409441\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5115.76	0.715895
$372$	0	0
$373$	2114.12	0.293472	0.146736	−	0.989176i	$-0.453123\pi$
$373$	2114.12	0.293472	0.146736	+	0.989176i	$0.453123\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	820.227	0.112053
$378$	0	0
$379$	−426.559	−0.0578123	−0.0289062	−	0.999582i	$-0.509202\pi$
$379$	−426.559	−0.0578123	−0.0289062	+	0.999582i	$0.509202\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10172.3	−1.35713	−0.678565	−	0.734541i	$-0.737398\pi$
$383$	−10172.3	−1.35713	−0.678565	+	0.734541i	$0.737398\pi$
$384$	0	0
$385$	−1763.66	−0.233466
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3696.72	0.481828	0.240914	−	0.970546i	$-0.422553\pi$
$389$	3696.72	0.481828	0.240914	+	0.970546i	$0.422553\pi$
$390$	0	0
$391$	−9934.30	−1.28491
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4813.49	0.613146
$396$	0	0
$397$	10586.5	1.33834	0.669169	−	0.743110i	$-0.266650\pi$
$397$	10586.5	1.33834	0.669169	+	0.743110i	$0.266650\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14940.5	−1.86058	−0.930292	−	0.366821i	$-0.880446\pi$
$401$	−14940.5	−1.86058	−0.930292	+	0.366821i	$0.880446\pi$
$402$	0	0
$403$	−1921.85	−0.237554
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2168.23	0.264066
$408$	0	0
$409$	14189.2	1.71544	0.857718	−	0.514121i	$-0.171882\pi$
$409$	14189.2	1.71544	0.857718	+	0.514121i	$0.171882\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11235.5	1.33865
$414$	0	0
$415$	3606.69	0.426616
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5880.89	−0.685681	−0.342840	−	0.939394i	$-0.611389\pi$
$419$	−5880.89	−0.685681	−0.342840	+	0.939394i	$0.611389\pi$
$420$	0	0
$421$	−7490.46	−0.867133	−0.433566	−	0.901122i	$-0.642745\pi$
$421$	−7490.46	−0.867133	−0.433566	+	0.901122i	$0.642745\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1596.86	−0.182257
$426$	0	0
$427$	−7522.20	−0.852517
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4049.42	0.452561	0.226281	−	0.974062i	$-0.427343\pi$
$431$	4049.42	0.452561	0.226281	+	0.974062i	$0.427343\pi$
$432$	0	0
$433$	−6088.93	−0.675785	−0.337893	−	0.941185i	$-0.609714\pi$
$433$	−6088.93	−0.675785	−0.337893	+	0.941185i	$0.609714\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15771.9	−1.72648
$438$	0	0
$439$	3087.19	0.335634	0.167817	−	0.985818i	$-0.446328\pi$
$439$	3087.19	0.335634	0.167817	+	0.985818i	$0.446328\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2650.74	−0.284290	−0.142145	−	0.989846i	$-0.545400\pi$
$443$	−2650.74	−0.284290	−0.142145	+	0.989846i	$0.545400\pi$
$444$	0	0
$445$	−611.171	−0.0651062
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10985.3	1.15463	0.577316	−	0.816521i	$-0.304100\pi$
$449$	10985.3	1.15463	0.577316	+	0.816521i	$0.304100\pi$
$450$	0	0
$451$	−5089.47	−0.531383
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1350.84	0.139183
$456$	0	0
$457$	−12928.5	−1.32335	−0.661674	−	0.749792i	$-0.730154\pi$
$457$	−12928.5	−1.32335	−0.661674	+	0.749792i	$0.730154\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−664.298	−0.0671138	−0.0335569	−	0.999437i	$-0.510683\pi$
$461$	−664.298	−0.0671138	−0.0335569	+	0.999437i	$0.510683\pi$
$462$	0	0
$463$	−8395.26	−0.842680	−0.421340	−	0.906903i	$-0.638440\pi$
$463$	−8395.26	−0.842680	−0.421340	+	0.906903i	$0.638440\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3022.18	0.299464	0.149732	−	0.988727i	$-0.452159\pi$
$467$	3022.18	0.299464	0.149732	+	0.988727i	$0.452159\pi$
$468$	0	0
$469$	19787.6	1.94820
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8299.85	−0.806824
$474$	0	0
$475$	−2535.21	−0.244892
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5507.36	0.525339	0.262670	−	0.964886i	$-0.415397\pi$
$479$	5507.36	0.525339	0.262670	+	0.964886i	$0.415397\pi$
$480$	0	0
$481$	−1660.71	−0.157426
$482$	0	0
$483$	0	0
$484$	0	0
$485$	768.620	0.0719613
$486$	0	0
$487$	−2170.87	−0.201995	−0.100998	−	0.994887i	$-0.532203\pi$
$487$	−2170.87	−0.201995	−0.100998	+	0.994887i	$0.532203\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−14007.5	−1.28747	−0.643735	−	0.765248i	$-0.722616\pi$
$491$	−14007.5	−1.28747	−0.643735	+	0.765248i	$0.722616\pi$
$492$	0	0
$493$	−3720.24	−0.339860
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12617.9	−1.13881
$498$	0	0
$499$	−9227.59	−0.827823	−0.413911	−	0.910317i	$-0.635838\pi$
$499$	−9227.59	−0.827823	−0.413911	+	0.910317i	$0.635838\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17747.2	1.57318	0.786591	−	0.617475i	$-0.211844\pi$
$503$	17747.2	1.57318	0.786591	+	0.617475i	$0.211844\pi$
$504$	0	0
$505$	1273.80	0.112244
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15368.2	1.33828	0.669140	−	0.743137i	$-0.266663\pi$
$509$	15368.2	1.33828	0.669140	+	0.743137i	$0.266663\pi$
$510$	0	0
$511$	2411.07	0.208727
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3649.85	−0.312294
$516$	0	0
$517$	281.173	0.0239187
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3932.31	0.330667	0.165333	−	0.986238i	$-0.447130\pi$
$521$	3932.31	0.330667	0.165333	+	0.986238i	$0.447130\pi$
$522$	0	0
$523$	2709.21	0.226511	0.113256	−	0.993566i	$-0.463872\pi$
$523$	2709.21	0.226511	0.113256	+	0.993566i	$0.463872\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8716.79	0.720511
$528$	0	0
$529$	12022.0	0.988085
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3898.17	0.316789
$534$	0	0
$535$	2584.00	0.208815
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−460.236	−0.0367788
$540$	0	0
$541$	−14223.8	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$541$	−14223.8	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5201.01	−0.408783
$546$	0	0
$547$	−11158.6	−0.872222	−0.436111	−	0.899893i	$-0.643644\pi$
$547$	−11158.6	−0.872222	−0.436111	+	0.899893i	$0.643644\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5906.32	−0.456657
$552$	0	0
$553$	18468.5	1.42018
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11127.4	0.846472	0.423236	−	0.906019i	$-0.360894\pi$
$557$	11127.4	0.846472	0.423236	+	0.906019i	$0.360894\pi$
$558$	0	0
$559$	6357.10	0.480996
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15868.5	−1.18788	−0.593941	−	0.804508i	$-0.702429\pi$
$563$	−15868.5	−1.18788	−0.593941	+	0.804508i	$0.702429\pi$
$564$	0	0
$565$	7455.16	0.555117
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12406.3	0.914061	0.457030	−	0.889451i	$-0.348913\pi$
$569$	12406.3	0.914061	0.457030	+	0.889451i	$0.348913\pi$
$570$	0	0
$571$	−21374.3	−1.56652	−0.783262	−	0.621692i	$-0.786446\pi$
$571$	−21374.3	−1.56652	−0.783262	+	0.621692i	$0.786446\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3888.21	0.281999
$576$	0	0
$577$	8980.90	0.647972	0.323986	−	0.946062i	$-0.394977\pi$
$577$	8980.90	0.647972	0.323986	+	0.946062i	$0.394977\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13838.3	0.988137
$582$	0	0
$583$	−4903.10	−0.348312
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18682.7	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$587$	18682.7	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$588$	0	0
$589$	13839.0	0.968123
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1890.65	0.130927	0.0654634	−	0.997855i	$-0.479147\pi$
$593$	1890.65	0.130927	0.0654634	+	0.997855i	$0.479147\pi$
$594$	0	0
$595$	−6126.89	−0.422148
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−482.005	−0.0328784	−0.0164392	−	0.999865i	$-0.505233\pi$
$599$	−482.005	−0.0328784	−0.0164392	+	0.999865i	$0.505233\pi$
$600$	0	0
$601$	16119.3	1.09404	0.547020	−	0.837120i	$-0.315762\pi$
$601$	16119.3	1.09404	0.547020	+	0.837120i	$0.315762\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4964.65	−0.333623
$606$	0	0
$607$	−10459.0	−0.699368	−0.349684	−	0.936868i	$-0.613711\pi$
$607$	−10459.0	−0.699368	−0.349684	+	0.936868i	$0.613711\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−215.358	−0.0142594
$612$	0	0
$613$	10833.9	0.713826	0.356913	−	0.934138i	$-0.383829\pi$
$613$	10833.9	0.713826	0.356913	+	0.934138i	$0.383829\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27174.2	−1.77308	−0.886540	−	0.462652i	$-0.846898\pi$
$617$	−27174.2	−1.77308	−0.886540	+	0.462652i	$0.846898\pi$
$618$	0	0
$619$	27200.7	1.76622	0.883109	−	0.469169i	$-0.155446\pi$
$619$	27200.7	1.76622	0.883109	+	0.469169i	$0.155446\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2344.96	−0.150800
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7532.34	0.477479
$630$	0	0
$631$	−15341.6	−0.967890	−0.483945	−	0.875098i	$-0.660796\pi$
$631$	−15341.6	−0.967890	−0.483945	+	0.875098i	$0.660796\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6361.43	0.397552
$636$	0	0
$637$	352.508	0.0219260
$638$	0	0
$639$	0	0
$640$	0	0
$641$	22072.0	1.36005	0.680024	−	0.733190i	$-0.261970\pi$
$641$	22072.0	1.36005	0.680024	+	0.733190i	$0.261970\pi$
$642$	0	0
$643$	−14931.1	−0.915746	−0.457873	−	0.889018i	$-0.651389\pi$
$643$	−14931.1	−0.915746	−0.457873	+	0.889018i	$0.651389\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22095.3	1.34259	0.671297	−	0.741189i	$-0.265738\pi$
$647$	22095.3	1.34259	0.671297	+	0.741189i	$0.265738\pi$
$648$	0	0
$649$	−10768.5	−0.651309
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3457.94	−0.207228	−0.103614	−	0.994618i	$-0.533041\pi$
$653$	−3457.94	−0.207228	−0.103614	+	0.994618i	$0.533041\pi$
$654$	0	0
$655$	2454.94	0.146446
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16748.0	−0.990001	−0.495001	−	0.868893i	$-0.664832\pi$
$659$	−16748.0	−0.990001	−0.495001	+	0.868893i	$0.664832\pi$
$660$	0	0
$661$	12355.4	0.727031	0.363516	−	0.931588i	$-0.381576\pi$
$661$	12355.4	0.727031	0.363516	+	0.931588i	$0.381576\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9727.18	−0.567224
$666$	0	0
$667$	9058.41	0.525851
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7209.50	0.414783
$672$	0	0
$673$	−33472.3	−1.91718	−0.958589	−	0.284793i	$-0.908075\pi$
$673$	−33472.3	−1.91718	−0.958589	+	0.284793i	$0.908075\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27977.0	−1.58825	−0.794124	−	0.607756i	$-0.792070\pi$
$677$	−27977.0	−1.58825	−0.794124	+	0.607756i	$0.792070\pi$
$678$	0	0
$679$	2949.06	0.166678
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−17231.4	−0.965361	−0.482681	−	0.875796i	$-0.660337\pi$
$683$	−17231.4	−0.965361	−0.482681	+	0.875796i	$0.660337\pi$
$684$	0	0
$685$	−11475.4	−0.640076
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3755.43	0.207649
$690$	0	0
$691$	15600.0	0.858832	0.429416	−	0.903107i	$-0.358719\pi$
$691$	15600.0	0.858832	0.429416	+	0.903107i	$0.358719\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7572.61	−0.413303
$696$	0	0
$697$	−17680.6	−0.960834
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27816.5	1.49874	0.749369	−	0.662153i	$-0.230357\pi$
$701$	27816.5	1.49874	0.749369	+	0.662153i	$0.230357\pi$
$702$	0	0
$703$	11958.5	0.641570
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4887.34	0.259982
$708$	0	0
$709$	−12257.3	−0.649269	−0.324634	−	0.945840i	$-0.605241\pi$
$709$	−12257.3	−0.649269	−0.324634	+	0.945840i	$0.605241\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21224.5	−1.11482
$714$	0	0
$715$	−1294.68	−0.0677181
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21906.3	−1.13625	−0.568127	−	0.822941i	$-0.692332\pi$
$719$	−21906.3	−1.13625	−0.568127	+	0.822941i	$0.692332\pi$
$720$	0	0
$721$	−14003.8	−0.723342
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1456.07	0.0745891
$726$	0	0
$727$	−33924.8	−1.73067	−0.865337	−	0.501190i	$-0.832896\pi$
$727$	−33924.8	−1.73067	−0.865337	+	0.501190i	$0.832896\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−28833.4	−1.45888
$732$	0	0
$733$	−2784.84	−0.140328	−0.0701639	−	0.997535i	$-0.522352\pi$
$733$	−2784.84	−0.140328	−0.0701639	+	0.997535i	$0.522352\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18965.1	−0.947880
$738$	0	0
$739$	27327.4	1.36029	0.680144	−	0.733078i	$-0.261917\pi$
$739$	27327.4	1.36029	0.680144	+	0.733078i	$0.261917\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4088.32	−0.201865	−0.100933	−	0.994893i	$-0.532183\pi$
$743$	−4088.32	−0.201865	−0.100933	+	0.994893i	$0.532183\pi$
$744$	0	0
$745$	293.175	0.0144176
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9914.34	0.483661
$750$	0	0
$751$	−3058.73	−0.148622	−0.0743108	−	0.997235i	$-0.523676\pi$
$751$	−3058.73	−0.148622	−0.0743108	+	0.997235i	$0.523676\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1771.51	0.0853934
$756$	0	0
$757$	−19443.0	−0.933510	−0.466755	−	0.884387i	$-0.654577\pi$
$757$	−19443.0	−0.933510	−0.466755	+	0.884387i	$0.654577\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20461.5	−0.974677	−0.487338	−	0.873213i	$-0.662032\pi$
$761$	−20461.5	−0.974677	−0.487338	+	0.873213i	$0.662032\pi$
$762$	0	0
$763$	−19955.4	−0.946833
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8247.89	0.388284
$768$	0	0
$769$	6375.33	0.298960	0.149480	−	0.988765i	$-0.452240\pi$
$769$	6375.33	0.298960	0.149480	+	0.988765i	$0.452240\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40557.7	1.88714	0.943569	−	0.331177i	$-0.107446\pi$
$773$	40557.7	1.88714	0.943569	+	0.331177i	$0.107446\pi$
$774$	0	0
$775$	−3411.68	−0.158131
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−28070.1	−1.29104
$780$	0	0
$781$	12093.3	0.554076
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7578.53	0.344573
$786$	0	0
$787$	12638.6	0.572449	0.286225	−	0.958163i	$-0.407600\pi$
$787$	12638.6	0.572449	0.286225	+	0.958163i	$0.407600\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	28604.2	1.28577
$792$	0	0
$793$	−5521.97	−0.247277
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6002.79	−0.266787	−0.133394	−	0.991063i	$-0.542587\pi$
$797$	−6002.79	−0.266787	−0.133394	+	0.991063i	$0.542587\pi$
$798$	0	0
$799$	976.784	0.0432492
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2310.84	−0.101554
$804$	0	0
$805$	14918.4	0.653172
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22138.8	−0.962123	−0.481061	−	0.876687i	$-0.659749\pi$
$809$	−22138.8	−0.962123	−0.481061	+	0.876687i	$0.659749\pi$
$810$	0	0
$811$	36472.9	1.57921	0.789604	−	0.613617i	$-0.210286\pi$
$811$	36472.9	1.57921	0.789604	+	0.613617i	$0.210286\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4992.77	0.214588
$816$	0	0
$817$	−45776.5	−1.96024
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28561.5	−1.21413	−0.607065	−	0.794652i	$-0.707653\pi$
$821$	−28561.5	−1.21413	−0.607065	+	0.794652i	$0.707653\pi$
$822$	0	0
$823$	40572.2	1.71842	0.859209	−	0.511624i	$-0.170956\pi$
$823$	40572.2	1.71842	0.859209	+	0.511624i	$0.170956\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45659.5	1.91987	0.959937	−	0.280215i	$-0.0904057\pi$
$827$	45659.5	1.91987	0.959937	+	0.280215i	$0.0904057\pi$
$828$	0	0
$829$	29623.2	1.24108	0.620540	−	0.784175i	$-0.286913\pi$
$829$	29623.2	1.24108	0.620540	+	0.784175i	$0.286913\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1598.84	−0.0665026
$834$	0	0
$835$	15426.0	0.639329
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−17564.0	−0.722738	−0.361369	−	0.932423i	$-0.617691\pi$
$839$	−17564.0	−0.722738	−0.361369	+	0.932423i	$0.617691\pi$
$840$	0	0
$841$	−20996.8	−0.860912
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9993.36	−0.406843
$846$	0	0
$847$	−19048.5	−0.772745
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−18340.5	−0.738783
$852$	0	0
$853$	−15867.2	−0.636910	−0.318455	−	0.947938i	$-0.603164\pi$
$853$	−15867.2	−0.636910	−0.318455	+	0.947938i	$0.603164\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46932.9	−1.87071	−0.935354	−	0.353714i	$-0.884919\pi$
$857$	−46932.9	−1.87071	−0.935354	+	0.353714i	$0.884919\pi$
$858$	0	0
$859$	24569.7	0.975910	0.487955	−	0.872869i	$-0.337743\pi$
$859$	24569.7	0.975910	0.487955	+	0.872869i	$0.337743\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24918.5	−0.982891	−0.491445	−	0.870908i	$-0.663531\pi$
$863$	−24918.5	−0.982891	−0.491445	+	0.870908i	$0.663531\pi$
$864$	0	0
$865$	4556.79	0.179116
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−17700.8	−0.690976
$870$	0	0
$871$	14525.9	0.565088
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2398.02	0.0926489
$876$	0	0
$877$	17022.5	0.655426	0.327713	−	0.944777i	$-0.393722\pi$
$877$	17022.5	0.655426	0.327713	+	0.944777i	$0.393722\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41291.8	−1.57906	−0.789532	−	0.613710i	$-0.789677\pi$
$881$	−41291.8	−1.57906	−0.789532	+	0.613710i	$0.789677\pi$
$882$	0	0
$883$	−42280.0	−1.61136	−0.805682	−	0.592348i	$-0.798201\pi$
$883$	−42280.0	−1.61136	−0.805682	+	0.592348i	$0.798201\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39376.4	−1.49056	−0.745282	−	0.666750i	$-0.767685\pi$
$887$	−39376.4	−1.49056	−0.745282	+	0.666750i	$0.767685\pi$
$888$	0	0
$889$	24407.7	0.920819
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1550.76	0.0581123
$894$	0	0
$895$	12215.4	0.456218
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7948.24	−0.294871
$900$	0	0
$901$	−17033.2	−0.629809
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15877.7	−0.583197
$906$	0	0
$907$	20198.4	0.739445	0.369722	−	0.929142i	$-0.379453\pi$
$907$	20198.4	0.739445	0.369722	+	0.929142i	$0.379453\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−51503.4	−1.87309	−0.936544	−	0.350551i	$-0.885994\pi$
$911$	−51503.4	−1.87309	−0.936544	+	0.350551i	$0.885994\pi$
$912$	0	0
$913$	−13263.0	−0.480768
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9419.17	0.339202
$918$	0	0
$919$	15219.5	0.546295	0.273147	−	0.961972i	$-0.411935\pi$
$919$	15219.5	0.546295	0.273147	+	0.961972i	$0.411935\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9262.63	−0.330318
$924$	0	0
$925$	−2948.10	−0.104792
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34214.8	−1.20834	−0.604172	−	0.796854i	$-0.706496\pi$
$929$	−34214.8	−1.20834	−0.604172	+	0.796854i	$0.706496\pi$
$930$	0	0
$931$	−2538.36	−0.0893569
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5872.20	0.205392
$936$	0	0
$937$	9712.23	0.338618	0.169309	−	0.985563i	$-0.445846\pi$
$937$	9712.23	0.338618	0.169309	+	0.985563i	$0.445846\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36714.3	1.27189	0.635947	−	0.771733i	$-0.280610\pi$
$941$	36714.3	1.27189	0.635947	+	0.771733i	$0.280610\pi$
$942$	0	0
$943$	43050.6	1.48666
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43936.5	1.50765	0.753825	−	0.657076i	$-0.228207\pi$
$947$	43936.5	1.50765	0.753825	+	0.657076i	$0.228207\pi$
$948$	0	0
$949$	1769.94	0.0605423
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15138.8	0.514579	0.257290	−	0.966334i	$-0.417171\pi$
$953$	15138.8	0.514579	0.257290	+	0.966334i	$0.417171\pi$
$954$	0	0
$955$	6677.79	0.226270
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−44029.1	−1.48256
$960$	0	0
$961$	−11167.7	−0.374868
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10205.6	−0.340446
$966$	0	0
$967$	8364.88	0.278176	0.139088	−	0.990280i	$-0.455583\pi$
$967$	8364.88	0.278176	0.139088	+	0.990280i	$0.455583\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27702.4	0.915562	0.457781	−	0.889065i	$-0.348644\pi$
$971$	27702.4	0.915562	0.457781	+	0.889065i	$0.348644\pi$
$972$	0	0
$973$	−29054.8	−0.957301
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45969.7	−1.50532	−0.752662	−	0.658408i	$-0.771230\pi$
$977$	−45969.7	−1.50532	−0.752662	+	0.658408i	$0.771230\pi$
$978$	0	0
$979$	2247.48	0.0733705
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−34555.2	−1.12120	−0.560601	−	0.828086i	$-0.689430\pi$
$983$	−34555.2	−1.12120	−0.560601	+	0.828086i	$0.689430\pi$
$984$	0	0
$985$	12827.8	0.414951
$986$	0	0
$987$	0	0
$988$	0	0
$989$	70206.4	2.25726
$990$	0	0
$991$	−28800.3	−0.923181	−0.461591	−	0.887093i	$-0.652721\pi$
$991$	−28800.3	−0.923181	−0.461591	+	0.887093i	$0.652721\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13251.5	0.422210
$996$	0	0
$997$	3103.42	0.0985819	0.0492910	−	0.998784i	$-0.484304\pi$
$997$	3103.42	0.0985819	0.0492910	+	0.998784i	$0.484304\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bp.1.3		3
3.2	odd	2		2160.4.a.bh.1.3		3
4.3	odd	2		1080.4.a.l.1.1	yes	3
12.11	even	2		1080.4.a.f.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.f.1.1	✓	3	12.11	even	2
1080.4.a.l.1.1	yes	3	4.3	odd	2
2160.4.a.bh.1.3		3	3.2	odd	2
2160.4.a.bp.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +19.1841 q^{7} -18.3867 q^{11} +14.0829 q^{13} -63.8746 q^{17} -101.409 q^{19} +155.528 q^{23} +25.0000 q^{25} +58.2428 q^{29} -136.467 q^{31} +95.9207 q^{35} -117.924 q^{37} +276.802 q^{41} +451.406 q^{43} -15.2922 q^{47} +25.0310 q^{49} +266.666 q^{53} -91.9333 q^{55} +585.668 q^{59} -392.105 q^{61} +70.4144 q^{65} +1031.46 q^{67} -657.723 q^{71} +125.680 q^{73} -352.732 q^{77} +962.698 q^{79} +721.338 q^{83} -319.373 q^{85} -122.234 q^{89} +270.168 q^{91} -507.043 q^{95} +153.724 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 27 q^{11} - 3 q^{13} - 15 q^{17} + 78 q^{19} + 105 q^{23} + 75 q^{25} - 117 q^{29} + 207 q^{31} - 120 q^{37} - 300 q^{41} + 483 q^{43} + 303 q^{47} - 15 q^{49} + 492 q^{53} + 135 q^{55}+ \cdots - 1392 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 27 * q^11 - 3 * q^13 - 15 * q^17 + 78 * q^19 + 105 * q^23 + 75 * q^25 - 117 * q^29 + 207 * q^31 - 120 * q^37 - 300 * q^41 + 483 * q^43 + 303 * q^47 - 15 * q^49 + 492 * q^53 + 135 * q^55 + 240 * q^59 - 444 * q^61 - 15 * q^65 + 522 * q^67 - 168 * q^71 - 876 * q^73 + 150 * q^77 + 2103 * q^79 - 42 * q^83 - 75 * q^85 + 2268 * q^89 + 1596 * q^91 + 390 * q^95 - 1392 * q^97

Introduction

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