LMFDB - Embedded newform 729.2.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [729,2,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	729.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1397493.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 $ x^6 - 3x^5 - 3x^4 + 10x^3 + 3x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.11662$ of defining polynomial
Character	$\chi$	$=$	729.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.415466 q^{2} -1.82739 q^{4} -2.21519 q^{5} -1.31963 q^{7} -1.59015 q^{8} -0.920335 q^{10} +5.21519 q^{11} -0.0180585 q^{13} -0.548261 q^{14} +2.99412 q^{16} +3.13280 q^{17} +0.417352 q^{19} +4.04801 q^{20} +2.16673 q^{22} +1.03439 q^{23} -0.0929475 q^{25} -0.00750270 q^{26} +2.41147 q^{28} +7.80722 q^{29} +3.72966 q^{31} +4.42426 q^{32} +1.30157 q^{34} +2.92322 q^{35} +4.42476 q^{37} +0.173396 q^{38} +3.52248 q^{40} +3.67494 q^{41} -8.30787 q^{43} -9.53017 q^{44} +0.429753 q^{46} +7.09791 q^{47} -5.25858 q^{49} -0.0386165 q^{50} +0.0329999 q^{52} +1.30057 q^{53} -11.5526 q^{55} +2.09841 q^{56} +3.24364 q^{58} +3.70181 q^{59} +6.91424 q^{61} +1.54955 q^{62} -4.15011 q^{64} +0.0400030 q^{65} -11.0268 q^{67} -5.72483 q^{68} +1.21450 q^{70} +6.08428 q^{71} -0.546973 q^{73} +1.83834 q^{74} -0.762665 q^{76} -6.88211 q^{77} +0.489144 q^{79} -6.63254 q^{80} +1.52681 q^{82} -4.61367 q^{83} -6.93973 q^{85} -3.45164 q^{86} -8.29293 q^{88} +3.37307 q^{89} +0.0238305 q^{91} -1.89023 q^{92} +2.94894 q^{94} -0.924513 q^{95} +9.94136 q^{97} -2.18476 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35}+ \cdots - 45 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^8 + 3 * q^10 + 12 * q^11 + 6 * q^14 - 3 * q^16 + 9 * q^17 + 3 * q^19 + 6 * q^20 + 6 * q^22 + 15 * q^23 - 6 * q^25 + 15 * q^26 - 6 * q^28 + 12 * q^29 + 12 * q^35 + 3 * q^37 - 3 * q^38 + 6 * q^40 + 15 * q^41 + 3 * q^44 + 3 * q^46 + 21 * q^47 - 12 * q^49 + 3 * q^50 + 12 * q^52 + 9 * q^53 - 6 * q^55 - 6 * q^56 - 12 * q^58 + 24 * q^59 - 9 * q^61 - 12 * q^62 - 12 * q^64 - 6 * q^65 - 9 * q^67 - 9 * q^68 + 15 * q^70 + 27 * q^71 - 6 * q^73 - 12 * q^74 + 6 * q^76 - 12 * q^77 - 21 * q^80 - 6 * q^82 + 12 * q^83 - 21 * q^86 + 12 * q^88 + 9 * q^89 - 6 * q^91 + 6 * q^92 + 6 * q^94 + 12 * q^95 - 45 * q^98

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 $

$2$	0.415466	0.293779	0.146889	−	0.989153i	$-0.453074\pi$
$2$	0.415466	0.293779	0.146889	+	0.989153i	$0.453074\pi$
$3$	0	0
$3$	0	0
$4$	−1.82739	−0.913694
$5$	−2.21519	−0.990662	−0.495331	−	0.868704i	$-0.664953\pi$
$5$	−2.21519	−0.990662	−0.495331	+	0.868704i	$0.664953\pi$
$6$	0	0
$7$	−1.31963	−0.498773	−0.249386	−	0.968404i	$-0.580229\pi$
$7$	−1.31963	−0.498773	−0.249386	+	0.968404i	$0.580229\pi$
$8$	−1.59015	−0.562203
$9$	0	0
$10$	−0.920335	−0.291036
$11$	5.21519	1.57244	0.786219	−	0.617948i	$-0.212036\pi$
$11$	5.21519	1.57244	0.786219	+	0.617948i	$0.212036\pi$
$12$	0	0
$13$	−0.0180585	−0.00500853	−0.00250427	−	0.999997i	$-0.500797\pi$
$13$	−0.0180585	−0.00500853	−0.00250427	+	0.999997i	$0.500797\pi$
$14$	−0.548261	−0.146529
$15$	0	0
$16$	2.99412	0.748530
$17$	3.13280	0.759814	0.379907	−	0.925025i	$-0.375956\pi$
$17$	3.13280	0.759814	0.379907	+	0.925025i	$0.375956\pi$
$18$	0	0
$19$	0.417352	0.0957472	0.0478736	−	0.998853i	$-0.484756\pi$
$19$	0.417352	0.0957472	0.0478736	+	0.998853i	$0.484756\pi$
$20$	4.04801	0.905162
$21$	0	0
$22$	2.16673	0.461949
$23$	1.03439	0.215684	0.107842	−	0.994168i	$-0.465606\pi$
$23$	1.03439	0.215684	0.107842	+	0.994168i	$0.465606\pi$
$24$	0	0
$25$	−0.0929475	−0.0185895
$26$	−0.00750270	−0.00147140
$27$	0	0
$28$	2.41147	0.455726
$29$	7.80722	1.44976	0.724882	−	0.688873i	$-0.241894\pi$
$29$	7.80722	1.44976	0.724882	+	0.688873i	$0.241894\pi$
$30$	0	0
$31$	3.72966	0.669868	0.334934	−	0.942242i	$-0.391286\pi$
$31$	3.72966	0.669868	0.334934	+	0.942242i	$0.391286\pi$
$32$	4.42426	0.782106
$33$	0	0
$34$	1.30157	0.223218
$35$	2.92322	0.494115
$36$	0	0
$37$	4.42476	0.727426	0.363713	−	0.931511i	$-0.381509\pi$
$37$	4.42476	0.727426	0.363713	+	0.931511i	$0.381509\pi$
$38$	0.173396	0.0281285
$39$	0	0
$40$	3.52248	0.556953
$41$	3.67494	0.573929	0.286965	−	0.957941i	$-0.407354\pi$
$41$	3.67494	0.573929	0.286965	+	0.957941i	$0.407354\pi$
$42$	0	0
$43$	−8.30787	−1.26694	−0.633469	−	0.773768i	$-0.718370\pi$
$43$	−8.30787	−1.26694	−0.633469	+	0.773768i	$0.718370\pi$
$44$	−9.53017	−1.43673
$45$	0	0
$46$	0.429753	0.0633636
$47$	7.09791	1.03534	0.517668	−	0.855581i	$-0.326800\pi$
$47$	7.09791	1.03534	0.517668	+	0.855581i	$0.326800\pi$
$48$	0	0
$49$	−5.25858	−0.751226
$50$	−0.0386165	−0.00546120
$51$	0	0
$52$	0.0329999	0.00457627
$53$	1.30057	0.178648	0.0893238	−	0.996003i	$-0.471529\pi$
$53$	1.30057	0.178648	0.0893238	+	0.996003i	$0.471529\pi$
$54$	0	0
$55$	−11.5526	−1.55775
$56$	2.09841	0.280412
$57$	0	0
$58$	3.24364	0.425910
$59$	3.70181	0.481935	0.240967	−	0.970533i	$-0.422535\pi$
$59$	3.70181	0.481935	0.240967	+	0.970533i	$0.422535\pi$
$60$	0	0
$61$	6.91424	0.885277	0.442639	−	0.896700i	$-0.354042\pi$
$61$	6.91424	0.885277	0.442639	+	0.896700i	$0.354042\pi$
$62$	1.54955	0.196793
$63$	0	0
$64$	−4.15011	−0.518764
$65$	0.0400030	0.00496176
$66$	0	0
$67$	−11.0268	−1.34714	−0.673569	−	0.739125i	$-0.735239\pi$
$67$	−11.0268	−1.34714	−0.673569	+	0.739125i	$0.735239\pi$
$68$	−5.72483	−0.694238
$69$	0	0
$70$	1.21450	0.145161
$71$	6.08428	0.722071	0.361035	−	0.932552i	$-0.382423\pi$
$71$	6.08428	0.722071	0.361035	+	0.932552i	$0.382423\pi$
$72$	0	0
$73$	−0.546973	−0.0640183	−0.0320092	−	0.999488i	$-0.510191\pi$
$73$	−0.546973	−0.0640183	−0.0320092	+	0.999488i	$0.510191\pi$
$74$	1.83834	0.213702
$75$	0	0
$76$	−0.762665	−0.0874836
$77$	−6.88211	−0.784289
$78$	0	0
$79$	0.489144	0.0550330	0.0275165	−	0.999621i	$-0.491240\pi$
$79$	0.489144	0.0550330	0.0275165	+	0.999621i	$0.491240\pi$
$80$	−6.63254	−0.741540
$81$	0	0
$82$	1.52681	0.168608
$83$	−4.61367	−0.506416	−0.253208	−	0.967412i	$-0.581486\pi$
$83$	−4.61367	−0.506416	−0.253208	+	0.967412i	$0.581486\pi$
$84$	0	0
$85$	−6.93973	−0.752719
$86$	−3.45164	−0.372200
$87$	0	0
$88$	−8.29293	−0.884029
$89$	3.37307	0.357544	0.178772	−	0.983891i	$-0.442788\pi$
$89$	3.37307	0.357544	0.178772	+	0.983891i	$0.442788\pi$
$90$	0	0
$91$	0.0238305	0.00249812
$92$	−1.89023	−0.197070
$93$	0	0
$94$	2.94894	0.304160
$95$	−0.924513	−0.0948531
$96$	0	0
$97$	9.94136	1.00939	0.504696	−	0.863297i	$-0.331605\pi$
$97$	9.94136	1.00939	0.504696	+	0.863297i	$0.331605\pi$
$98$	−2.18476	−0.220694
$99$	0	0
$100$	0.169851	0.0169851
$101$	−13.7995	−1.37310	−0.686550	−	0.727082i	$-0.740876\pi$
$101$	−13.7995	−1.37310	−0.686550	+	0.727082i	$0.740876\pi$
$102$	0	0
$103$	4.56512	0.449815	0.224907	−	0.974380i	$-0.427792\pi$
$103$	4.56512	0.449815	0.224907	+	0.974380i	$0.427792\pi$
$104$	0.0287158	0.00281581
$105$	0	0
$106$	0.540345	0.0524829
$107$	11.2965	1.09207	0.546035	−	0.837762i	$-0.316136\pi$
$107$	11.2965	1.09207	0.546035	+	0.837762i	$0.316136\pi$
$108$	0	0
$109$	14.5032	1.38915	0.694577	−	0.719419i	$-0.255592\pi$
$109$	14.5032	1.38915	0.694577	+	0.719419i	$0.255592\pi$
$110$	−4.79972	−0.457635
$111$	0	0
$112$	−3.95113	−0.373347
$113$	−12.5584	−1.18140	−0.590699	−	0.806892i	$-0.701148\pi$
$113$	−12.5584	−1.18140	−0.590699	+	0.806892i	$0.701148\pi$
$114$	0	0
$115$	−2.29136	−0.213670
$116$	−14.2668	−1.32464
$117$	0	0
$118$	1.53798	0.141582
$119$	−4.13413	−0.378975
$120$	0	0
$121$	16.1982	1.47256
$122$	2.87263	0.260076
$123$	0	0
$124$	−6.81554	−0.612054
$125$	11.2818	1.00908
$126$	0	0
$127$	−8.39499	−0.744935	−0.372467	−	0.928045i	$-0.621488\pi$
$127$	−8.39499	−0.744935	−0.372467	+	0.928045i	$0.621488\pi$
$128$	−10.5727	−0.934508
$129$	0	0
$130$	0.0166199	0.00145766
$131$	15.5349	1.35729	0.678645	−	0.734466i	$-0.262567\pi$
$131$	15.5349	1.35729	0.678645	+	0.734466i	$0.262567\pi$
$132$	0	0
$133$	−0.550750	−0.0477561
$134$	−4.58126	−0.395761
$135$	0	0
$136$	−4.98162	−0.427170
$137$	−12.0074	−1.02586	−0.512930	−	0.858430i	$-0.671440\pi$
$137$	−12.0074	−1.02586	−0.512930	+	0.858430i	$0.671440\pi$
$138$	0	0
$139$	6.14512	0.521222	0.260611	−	0.965444i	$-0.416076\pi$
$139$	6.14512	0.521222	0.260611	+	0.965444i	$0.416076\pi$
$140$	−5.34187	−0.451470
$141$	0	0
$142$	2.52781	0.212129
$143$	−0.0941785	−0.00787561
$144$	0	0
$145$	−17.2945	−1.43623
$146$	−0.227249	−0.0188072
$147$	0	0
$148$	−8.08575	−0.664644
$149$	−0.882820	−0.0723235	−0.0361617	−	0.999346i	$-0.511513\pi$
$149$	−0.882820	−0.0723235	−0.0361617	+	0.999346i	$0.511513\pi$
$150$	0	0
$151$	8.22547	0.669379	0.334690	−	0.942328i	$-0.391368\pi$
$151$	8.22547	0.669379	0.334690	+	0.942328i	$0.391368\pi$
$152$	−0.663653	−0.0538294
$153$	0	0
$154$	−2.85929	−0.230408
$155$	−8.26190	−0.663612
$156$	0	0
$157$	12.5598	1.00238	0.501192	−	0.865336i	$-0.332895\pi$
$157$	12.5598	1.00238	0.501192	+	0.865336i	$0.332895\pi$
$158$	0.203223	0.0161675
$159$	0	0
$160$	−9.80056	−0.774802
$161$	−1.36501	−0.107578
$162$	0	0
$163$	3.31466	0.259624	0.129812	−	0.991539i	$-0.458563\pi$
$163$	3.31466	0.259624	0.129812	+	0.991539i	$0.458563\pi$
$164$	−6.71554	−0.524396
$165$	0	0
$166$	−1.91682	−0.148774
$167$	20.5630	1.59121	0.795606	−	0.605815i	$-0.207153\pi$
$167$	20.5630	1.59121	0.795606	+	0.605815i	$0.207153\pi$
$168$	0	0
$169$	−12.9997	−0.999975
$170$	−2.88322	−0.221133
$171$	0	0
$172$	15.1817	1.15759
$173$	−14.0333	−1.06693	−0.533465	−	0.845822i	$-0.679110\pi$
$173$	−14.0333	−1.06693	−0.533465	+	0.845822i	$0.679110\pi$
$174$	0	0
$175$	0.122656	0.00927194
$176$	15.6149	1.17702
$177$	0	0
$178$	1.40139	0.105039
$179$	−10.1900	−0.761636	−0.380818	−	0.924650i	$-0.624358\pi$
$179$	−10.1900	−0.761636	−0.380818	+	0.924650i	$0.624358\pi$
$180$	0	0
$181$	24.0547	1.78797	0.893987	−	0.448093i	$-0.147897\pi$
$181$	24.0547	1.78797	0.893987	+	0.448093i	$0.147897\pi$
$182$	0.00990079	0.000733895 0
$183$	0	0
$184$	−1.64483	−0.121258
$185$	−9.80166	−0.720633
$186$	0	0
$187$	16.3381	1.19476
$188$	−12.9706	−0.945981
$189$	0	0
$190$	−0.384104	−0.0278658
$191$	−10.9464	−0.792052	−0.396026	−	0.918239i	$-0.629611\pi$
$191$	−10.9464	−0.792052	−0.396026	+	0.918239i	$0.629611\pi$
$192$	0	0
$193$	−10.8060	−0.777830	−0.388915	−	0.921274i	$-0.627150\pi$
$193$	−10.8060	−0.777830	−0.388915	+	0.921274i	$0.627150\pi$
$194$	4.13030	0.296538
$195$	0	0
$196$	9.60946	0.686390
$197$	22.0734	1.57266	0.786331	−	0.617806i	$-0.211978\pi$
$197$	22.0734	1.57266	0.786331	+	0.617806i	$0.211978\pi$
$198$	0	0
$199$	12.8868	0.913518	0.456759	−	0.889590i	$-0.349010\pi$
$199$	12.8868	0.913518	0.456759	+	0.889590i	$0.349010\pi$
$200$	0.147801	0.0104511
$201$	0	0
$202$	−5.73322	−0.403388
$203$	−10.3026	−0.723103
$204$	0	0
$205$	−8.14068	−0.568570
$206$	1.89665	0.132146
$207$	0	0
$208$	−0.0540694	−0.00374904
$209$	2.17657	0.150557
$210$	0	0
$211$	−23.9956	−1.65193	−0.825964	−	0.563723i	$-0.809368\pi$
$211$	−23.9956	−1.65193	−0.825964	+	0.563723i	$0.809368\pi$
$212$	−2.37665	−0.163229
$213$	0	0
$214$	4.69330	0.320827
$215$	18.4035	1.25511
$216$	0	0
$217$	−4.92177	−0.334112
$218$	6.02558	0.408104
$219$	0	0
$220$	21.1111	1.42331
$221$	−0.0565736	−0.00380555
$222$	0	0
$223$	21.6622	1.45061	0.725303	−	0.688430i	$-0.241700\pi$
$223$	21.6622	1.45061	0.725303	+	0.688430i	$0.241700\pi$
$224$	−5.83838	−0.390093
$225$	0	0
$226$	−5.21760	−0.347070
$227$	−21.6419	−1.43642	−0.718211	−	0.695826i	$-0.755039\pi$
$227$	−21.6419	−1.43642	−0.718211	+	0.695826i	$0.755039\pi$
$228$	0	0
$229$	−10.8054	−0.714038	−0.357019	−	0.934097i	$-0.616207\pi$
$229$	−10.8054	−0.714038	−0.357019	+	0.934097i	$0.616207\pi$
$230$	−0.951982	−0.0627719
$231$	0	0
$232$	−12.4147	−0.815062
$233$	−7.63900	−0.500447	−0.250224	−	0.968188i	$-0.580504\pi$
$233$	−7.63900	−0.500447	−0.250224	+	0.968188i	$0.580504\pi$
$234$	0	0
$235$	−15.7232	−1.02567
$236$	−6.76465	−0.440341
$237$	0	0
$238$	−1.71759	−0.111335
$239$	3.23149	0.209028	0.104514	−	0.994523i	$-0.466671\pi$
$239$	3.23149	0.209028	0.104514	+	0.994523i	$0.466671\pi$
$240$	0	0
$241$	−26.5449	−1.70991	−0.854955	−	0.518702i	$-0.826415\pi$
$241$	−26.5449	−1.70991	−0.854955	+	0.518702i	$0.826415\pi$
$242$	6.72979	0.432608
$243$	0	0
$244$	−12.6350	−0.808873
$245$	11.6487	0.744210
$246$	0	0
$247$	−0.00753676	−0.000479553 0
$248$	−5.93073	−0.376602
$249$	0	0
$250$	4.68722	0.296446
$251$	4.49930	0.283993	0.141997	−	0.989867i	$-0.454648\pi$
$251$	4.49930	0.283993	0.141997	+	0.989867i	$0.454648\pi$
$252$	0	0
$253$	5.39452	0.339150
$254$	−3.48783	−0.218846
$255$	0	0
$256$	3.90761	0.244226
$257$	−13.7354	−0.856792	−0.428396	−	0.903591i	$-0.640921\pi$
$257$	−13.7354	−0.856792	−0.428396	+	0.903591i	$0.640921\pi$
$258$	0	0
$259$	−5.83904	−0.362820
$260$	−0.0731010	−0.00453353
$261$	0	0
$262$	6.45423	0.398743
$263$	24.2026	1.49239	0.746197	−	0.665725i	$-0.231878\pi$
$263$	24.2026	1.49239	0.746197	+	0.665725i	$0.231878\pi$
$264$	0	0
$265$	−2.88101	−0.176979
$266$	−0.228818	−0.0140297
$267$	0	0
$268$	20.1502	1.23087
$269$	−12.0062	−0.732032	−0.366016	−	0.930609i	$-0.619278\pi$
$269$	−12.0062	−0.732032	−0.366016	+	0.930609i	$0.619278\pi$
$270$	0	0
$271$	3.71777	0.225839	0.112919	−	0.993604i	$-0.463980\pi$
$271$	3.71777	0.225839	0.112919	+	0.993604i	$0.463980\pi$
$272$	9.37997	0.568744
$273$	0	0
$274$	−4.98867	−0.301376
$275$	−0.484739	−0.0292308
$276$	0	0
$277$	−23.4831	−1.41096	−0.705482	−	0.708728i	$-0.749269\pi$
$277$	−23.4831	−1.41096	−0.705482	+	0.708728i	$0.749269\pi$
$278$	2.55309	0.153124
$279$	0	0
$280$	−4.64837	−0.277793
$281$	20.3717	1.21528	0.607638	−	0.794214i	$-0.292117\pi$
$281$	20.3717	1.21528	0.607638	+	0.794214i	$0.292117\pi$
$282$	0	0
$283$	11.5999	0.689545	0.344772	−	0.938686i	$-0.387956\pi$
$283$	11.5999	0.689545	0.344772	+	0.938686i	$0.387956\pi$
$284$	−11.1183	−0.659752
$285$	0	0
$286$	−0.0391280	−0.00231369
$287$	−4.84956	−0.286260
$288$	0	0
$289$	−7.18559	−0.422682
$290$	−7.18526	−0.421933
$291$	0	0
$292$	0.999532	0.0584932
$293$	−31.5742	−1.84458	−0.922291	−	0.386496i	$-0.873685\pi$
$293$	−31.5742	−1.84458	−0.922291	+	0.386496i	$0.873685\pi$
$294$	0	0
$295$	−8.20020	−0.477434
$296$	−7.03603	−0.408961
$297$	0	0
$298$	−0.366782	−0.0212471
$299$	−0.0186795	−0.00108026
$300$	0	0
$301$	10.9633	0.631915
$302$	3.41741	0.196650
$303$	0	0
$304$	1.24960	0.0716697
$305$	−15.3163	−0.877010
$306$	0	0
$307$	−8.12054	−0.463464	−0.231732	−	0.972780i	$-0.574439\pi$
$307$	−8.12054	−0.463464	−0.231732	+	0.972780i	$0.574439\pi$
$308$	12.5763	0.716601
$309$	0	0
$310$	−3.43254	−0.194955
$311$	23.8486	1.35233	0.676164	−	0.736751i	$-0.263641\pi$
$311$	23.8486	1.35233	0.676164	+	0.736751i	$0.263641\pi$
$312$	0	0
$313$	26.9105	1.52107	0.760535	−	0.649297i	$-0.224937\pi$
$313$	26.9105	1.52107	0.760535	+	0.649297i	$0.224937\pi$
$314$	5.21818	0.294479
$315$	0	0
$316$	−0.893856	−0.0502833
$317$	8.33233	0.467990	0.233995	−	0.972238i	$-0.424820\pi$
$317$	8.33233	0.467990	0.233995	+	0.972238i	$0.424820\pi$
$318$	0	0
$319$	40.7161	2.27967
$320$	9.19328	0.513920
$321$	0	0
$322$	−0.567114	−0.0316040
$323$	1.30748	0.0727501
$324$	0	0
$325$	0.00167849	9.31061e−5 0
$326$	1.37713	0.0762721
$327$	0	0
$328$	−5.84371	−0.322665
$329$	−9.36661	−0.516398
$330$	0	0
$331$	−6.42026	−0.352889	−0.176445	−	0.984311i	$-0.556460\pi$
$331$	−6.42026	−0.352889	−0.176445	+	0.984311i	$0.556460\pi$
$332$	8.43096	0.462709
$333$	0	0
$334$	8.54322	0.467464
$335$	24.4264	1.33456
$336$	0	0
$337$	7.47489	0.407183	0.203592	−	0.979056i	$-0.434739\pi$
$337$	7.47489	0.407183	0.203592	+	0.979056i	$0.434739\pi$
$338$	−5.40093	−0.293772
$339$	0	0
$340$	12.6816	0.687755
$341$	19.4509	1.05333
$342$	0	0
$343$	16.1768	0.873464
$344$	13.2108	0.712277
$345$	0	0
$346$	−5.83035	−0.313442
$347$	31.4545	1.68857	0.844283	−	0.535898i	$-0.180027\pi$
$347$	31.4545	1.68857	0.844283	+	0.535898i	$0.180027\pi$
$348$	0	0
$349$	11.8529	0.634474	0.317237	−	0.948346i	$-0.397245\pi$
$349$	11.8529	0.634474	0.317237	+	0.948346i	$0.397245\pi$
$350$	0.0509595	0.00272390
$351$	0	0
$352$	23.0733	1.22981
$353$	8.20708	0.436819	0.218409	−	0.975857i	$-0.429913\pi$
$353$	8.20708	0.436819	0.218409	+	0.975857i	$0.429913\pi$
$354$	0	0
$355$	−13.4778	−0.715328
$356$	−6.16390	−0.326686
$357$	0	0
$358$	−4.23360	−0.223753
$359$	−17.7273	−0.935611	−0.467806	−	0.883831i	$-0.654955\pi$
$359$	−17.7273	−0.935611	−0.467806	+	0.883831i	$0.654955\pi$
$360$	0	0
$361$	−18.8258	−0.990832
$362$	9.99393	0.525269
$363$	0	0
$364$	−0.0435477	−0.00228252
$365$	1.21165	0.0634205
$366$	0	0
$367$	−20.3195	−1.06067	−0.530335	−	0.847788i	$-0.677934\pi$
$367$	−20.3195	−1.06067	−0.530335	+	0.847788i	$0.677934\pi$
$368$	3.09708	0.161446
$369$	0	0
$370$	−4.07226	−0.211707
$371$	−1.71628	−0.0891046
$372$	0	0
$373$	−9.68144	−0.501286	−0.250643	−	0.968080i	$-0.580642\pi$
$373$	−9.68144	−0.501286	−0.250643	+	0.968080i	$0.580642\pi$
$374$	6.78793	0.350996
$375$	0	0
$376$	−11.2867	−0.582069
$377$	−0.140987	−0.00726119
$378$	0	0
$379$	−4.12905	−0.212095	−0.106048	−	0.994361i	$-0.533820\pi$
$379$	−4.12905	−0.212095	−0.106048	+	0.994361i	$0.533820\pi$
$380$	1.68944	0.0866667
$381$	0	0
$382$	−4.54785	−0.232688
$383$	−4.75018	−0.242723	−0.121362	−	0.992608i	$-0.538726\pi$
$383$	−4.75018	−0.242723	−0.121362	+	0.992608i	$0.538726\pi$
$384$	0	0
$385$	15.2452	0.776966
$386$	−4.48951	−0.228510
$387$	0	0
$388$	−18.1667	−0.922275
$389$	21.8133	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$389$	21.8133	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$390$	0	0
$391$	3.24052	0.163880
$392$	8.36193	0.422341
$393$	0	0
$394$	9.17074	0.462015
$395$	−1.08355	−0.0545191
$396$	0	0
$397$	−34.8490	−1.74902	−0.874512	−	0.485005i	$-0.838818\pi$
$397$	−34.8490	−1.74902	−0.874512	+	0.485005i	$0.838818\pi$
$398$	5.35401	0.268373
$399$	0	0
$400$	−0.278296	−0.0139148
$401$	18.8261	0.940130	0.470065	−	0.882632i	$-0.344231\pi$
$401$	18.8261	0.940130	0.470065	+	0.882632i	$0.344231\pi$
$402$	0	0
$403$	−0.0673522	−0.00335505
$404$	25.2170	1.25459
$405$	0	0
$406$	−4.28040	−0.212433
$407$	23.0759	1.14383
$408$	0	0
$409$	−6.35996	−0.314480	−0.157240	−	0.987560i	$-0.550260\pi$
$409$	−6.35996	−0.314480	−0.157240	+	0.987560i	$0.550260\pi$
$410$	−3.38218	−0.167034
$411$	0	0
$412$	−8.34224	−0.410993
$413$	−4.88502	−0.240376
$414$	0	0
$415$	10.2201	0.501687
$416$	−0.0798955	−0.00391720
$417$	0	0
$418$	0.904291	0.0442303
$419$	24.3180	1.18801	0.594005	−	0.804461i	$-0.297546\pi$
$419$	24.3180	1.18801	0.594005	+	0.804461i	$0.297546\pi$
$420$	0	0
$421$	−7.99004	−0.389411	−0.194705	−	0.980862i	$-0.562375\pi$
$421$	−7.99004	−0.389411	−0.194705	+	0.980862i	$0.562375\pi$
$422$	−9.96937	−0.485302
$423$	0	0
$424$	−2.06811	−0.100436
$425$	−0.291185	−0.0141246
$426$	0	0
$427$	−9.12423	−0.441552
$428$	−20.6430	−0.997818
$429$	0	0
$430$	7.64603	0.368724
$431$	−9.87124	−0.475481	−0.237740	−	0.971329i	$-0.576407\pi$
$431$	−9.87124	−0.475481	−0.237740	+	0.971329i	$0.576407\pi$
$432$	0	0
$433$	−6.10369	−0.293325	−0.146662	−	0.989187i	$-0.546853\pi$
$433$	−6.10369	−0.293325	−0.146662	+	0.989187i	$0.546853\pi$
$434$	−2.04483	−0.0981550
$435$	0	0
$436$	−26.5029	−1.26926
$437$	0.431704	0.0206512
$438$	0	0
$439$	−15.1340	−0.722308	−0.361154	−	0.932506i	$-0.617617\pi$
$439$	−15.1340	−0.722308	−0.361154	+	0.932506i	$0.617617\pi$
$440$	18.3704	0.875774
$441$	0	0
$442$	−0.0235044	−0.00111799
$443$	0.722793	0.0343410	0.0171705	−	0.999853i	$-0.494534\pi$
$443$	0.722793	0.0343410	0.0171705	+	0.999853i	$0.494534\pi$
$444$	0	0
$445$	−7.47197	−0.354205
$446$	8.99990	0.426158
$447$	0	0
$448$	5.47661	0.258746
$449$	−1.66845	−0.0787389	−0.0393695	−	0.999225i	$-0.512535\pi$
$449$	−1.66845	−0.0787389	−0.0393695	+	0.999225i	$0.512535\pi$
$450$	0	0
$451$	19.1655	0.902468
$452$	22.9491	1.07944
$453$	0	0
$454$	−8.99147	−0.421991
$455$	−0.0527891	−0.00247479
$456$	0	0
$457$	11.0834	0.518462	0.259231	−	0.965815i	$-0.416531\pi$
$457$	11.0834	0.518462	0.259231	+	0.965815i	$0.416531\pi$
$458$	−4.48926	−0.209769
$459$	0	0
$460$	4.18720	0.195229
$461$	21.8844	1.01926	0.509629	−	0.860394i	$-0.329783\pi$
$461$	21.8844	1.01926	0.509629	+	0.860394i	$0.329783\pi$
$462$	0	0
$463$	24.8517	1.15496	0.577479	−	0.816406i	$-0.304037\pi$
$463$	24.8517	1.15496	0.577479	+	0.816406i	$0.304037\pi$
$464$	23.3758	1.08519
$465$	0	0
$466$	−3.17375	−0.147021
$467$	11.8355	0.547683	0.273842	−	0.961775i	$-0.411706\pi$
$467$	11.8355	0.547683	0.273842	+	0.961775i	$0.411706\pi$
$468$	0	0
$469$	14.5513	0.671916
$470$	−6.53246	−0.301320
$471$	0	0
$472$	−5.88644	−0.270945
$473$	−43.3271	−1.99218
$474$	0	0
$475$	−0.0387919	−0.00177989
$476$	7.55465	0.346267
$477$	0	0
$478$	1.34258	0.0614080
$479$	−2.88735	−0.131926	−0.0659632	−	0.997822i	$-0.521012\pi$
$479$	−2.88735	−0.131926	−0.0659632	+	0.997822i	$0.521012\pi$
$480$	0	0
$481$	−0.0799046	−0.00364333
$482$	−11.0285	−0.502336
$483$	0	0
$484$	−29.6003	−1.34547
$485$	−22.0220	−0.999966
$486$	0	0
$487$	8.75903	0.396910	0.198455	−	0.980110i	$-0.436408\pi$
$487$	8.75903	0.396910	0.198455	+	0.980110i	$0.436408\pi$
$488$	−10.9947	−0.497706
$489$	0	0
$490$	4.83966	0.218633
$491$	22.5730	1.01871	0.509354	−	0.860557i	$-0.329885\pi$
$491$	22.5730	1.01871	0.509354	+	0.860557i	$0.329885\pi$
$492$	0	0
$493$	24.4584	1.10155
$494$	−0.00313127	−0.000140883 0
$495$	0	0
$496$	11.1671	0.501416
$497$	−8.02899	−0.360149
$498$	0	0
$499$	−25.3328	−1.13405	−0.567026	−	0.823700i	$-0.691906\pi$
$499$	−25.3328	−1.13405	−0.567026	+	0.823700i	$0.691906\pi$
$500$	−20.6163	−0.921988
$501$	0	0
$502$	1.86931	0.0834312
$503$	−3.74414	−0.166943	−0.0834714	−	0.996510i	$-0.526601\pi$
$503$	−3.74414	−0.166943	−0.0834714	+	0.996510i	$0.526601\pi$
$504$	0	0
$505$	30.5684	1.36028
$506$	2.24124	0.0996353
$507$	0	0
$508$	15.3409	0.680642
$509$	−24.3499	−1.07929	−0.539645	−	0.841893i	$-0.681441\pi$
$509$	−24.3499	−1.07929	−0.539645	+	0.841893i	$0.681441\pi$
$510$	0	0
$511$	0.721801	0.0319306
$512$	22.7690	1.00626
$513$	0	0
$514$	−5.70660	−0.251707
$515$	−10.1126	−0.445614
$516$	0	0
$517$	37.0169	1.62800
$518$	−2.42592	−0.106589
$519$	0	0
$520$	−0.0636108	−0.00278952
$521$	−19.6209	−0.859608	−0.429804	−	0.902922i	$-0.641417\pi$
$521$	−19.6209	−0.859608	−0.429804	+	0.902922i	$0.641417\pi$
$522$	0	0
$523$	20.8154	0.910194	0.455097	−	0.890442i	$-0.349605\pi$
$523$	20.8154	0.910194	0.455097	+	0.890442i	$0.349605\pi$
$524$	−28.3883	−1.24015
$525$	0	0
$526$	10.0553	0.438434
$527$	11.6843	0.508975
$528$	0	0
$529$	−21.9300	−0.953480
$530$	−1.19696	−0.0519928
$531$	0	0
$532$	1.00643	0.0436345
$533$	−0.0663640	−0.00287454
$534$	0	0
$535$	−25.0238	−1.08187
$536$	17.5343	0.757365
$537$	0	0
$538$	−4.98818	−0.215056
$539$	−27.4245	−1.18126
$540$	0	0
$541$	−30.6272	−1.31676	−0.658382	−	0.752684i	$-0.728759\pi$
$541$	−30.6272	−1.31676	−0.658382	+	0.752684i	$0.728759\pi$
$542$	1.54461	0.0663467
$543$	0	0
$544$	13.8603	0.594255
$545$	−32.1273	−1.37618
$546$	0	0
$547$	22.6477	0.968345	0.484172	−	0.874973i	$-0.339121\pi$
$547$	22.6477	0.968345	0.484172	+	0.874973i	$0.339121\pi$
$548$	21.9422	0.937323
$549$	0	0
$550$	−0.201393	−0.00858741
$551$	3.25836	0.138811
$552$	0	0
$553$	−0.645489	−0.0274490
$554$	−9.75644	−0.414512
$555$	0	0
$556$	−11.2295	−0.476237
$557$	−36.4518	−1.54451	−0.772256	−	0.635311i	$-0.780872\pi$
$557$	−36.4518	−1.54451	−0.772256	+	0.635311i	$0.780872\pi$
$558$	0	0
$559$	0.150028	0.00634550
$560$	8.75249	0.369860
$561$	0	0
$562$	8.46377	0.357023
$563$	−26.5162	−1.11753	−0.558763	−	0.829327i	$-0.688724\pi$
$563$	−26.5162	−1.11753	−0.558763	+	0.829327i	$0.688724\pi$
$564$	0	0
$565$	27.8193	1.17037
$566$	4.81938	0.202574
$567$	0	0
$568$	−9.67492	−0.405950
$569$	−22.9674	−0.962844	−0.481422	−	0.876489i	$-0.659879\pi$
$569$	−22.9674	−0.962844	−0.481422	+	0.876489i	$0.659879\pi$
$570$	0	0
$571$	−4.79801	−0.200790	−0.100395	−	0.994948i	$-0.532011\pi$
$571$	−4.79801	−0.200790	−0.100395	+	0.994948i	$0.532011\pi$
$572$	0.172101	0.00719589
$573$	0	0
$574$	−2.01483	−0.0840973
$575$	−0.0961436	−0.00400947
$576$	0	0
$577$	−4.31333	−0.179566	−0.0897831	−	0.995961i	$-0.528617\pi$
$577$	−4.31333	−0.179566	−0.0897831	+	0.995961i	$0.528617\pi$
$578$	−2.98537	−0.124175
$579$	0	0
$580$	31.6037	1.31227
$581$	6.08833	0.252587
$582$	0	0
$583$	6.78274	0.280912
$584$	0.869769	0.0359913
$585$	0	0
$586$	−13.1180	−0.541899
$587$	−41.8222	−1.72619	−0.863094	−	0.505044i	$-0.831476\pi$
$587$	−41.8222	−1.72619	−0.863094	+	0.505044i	$0.831476\pi$
$588$	0	0
$589$	1.55658	0.0641379
$590$	−3.40691	−0.140260
$591$	0	0
$592$	13.2483	0.544500
$593$	31.5370	1.29507	0.647536	−	0.762035i	$-0.275800\pi$
$593$	31.5370	1.29507	0.647536	+	0.762035i	$0.275800\pi$
$594$	0	0
$595$	9.15786	0.375436
$596$	1.61326	0.0660815
$597$	0	0
$598$	−0.00776070	−0.000317358 0
$599$	12.6303	0.516060	0.258030	−	0.966137i	$-0.416927\pi$
$599$	12.6303	0.516060	0.258030	+	0.966137i	$0.416927\pi$
$600$	0	0
$601$	20.5430	0.837965	0.418983	−	0.907994i	$-0.362387\pi$
$601$	20.5430	0.837965	0.418983	+	0.907994i	$0.362387\pi$
$602$	4.55489	0.185643
$603$	0	0
$604$	−15.0311	−0.611608
$605$	−35.8820	−1.45881
$606$	0	0
$607$	12.9126	0.524105	0.262052	−	0.965054i	$-0.415601\pi$
$607$	12.9126	0.524105	0.262052	+	0.965054i	$0.415601\pi$
$608$	1.84647	0.0748844
$609$	0	0
$610$	−6.36342	−0.257647
$611$	−0.128178	−0.00518552
$612$	0	0
$613$	−31.1598	−1.25853	−0.629265	−	0.777191i	$-0.716644\pi$
$613$	−31.1598	−1.25853	−0.629265	+	0.777191i	$0.716644\pi$
$614$	−3.37381	−0.136156
$615$	0	0
$616$	10.9436	0.440930
$617$	7.14078	0.287477	0.143739	−	0.989616i	$-0.454088\pi$
$617$	7.14078	0.287477	0.143739	+	0.989616i	$0.454088\pi$
$618$	0	0
$619$	−10.0309	−0.403176	−0.201588	−	0.979470i	$-0.564610\pi$
$619$	−10.0309	−0.403176	−0.201588	+	0.979470i	$0.564610\pi$
$620$	15.0977	0.606338
$621$	0	0
$622$	9.90827	0.397285
$623$	−4.45119	−0.178333
$624$	0	0
$625$	−24.5266	−0.981065
$626$	11.1804	0.446858
$627$	0	0
$628$	−22.9517	−0.915871
$629$	13.8619	0.552708
$630$	0	0
$631$	−7.07560	−0.281675	−0.140838	−	0.990033i	$-0.544980\pi$
$631$	−7.07560	−0.281675	−0.140838	+	0.990033i	$0.544980\pi$
$632$	−0.777813	−0.0309397
$633$	0	0
$634$	3.46180	0.137486
$635$	18.5965	0.737978
$636$	0	0
$637$	0.0949621	0.00376254
$638$	16.9162	0.669718
$639$	0	0
$640$	23.4206	0.925781
$641$	5.01121	0.197931	0.0989655	−	0.995091i	$-0.468447\pi$
$641$	5.01121	0.197931	0.0989655	+	0.995091i	$0.468447\pi$
$642$	0	0
$643$	1.63840	0.0646123	0.0323062	−	0.999478i	$-0.489715\pi$
$643$	1.63840	0.0646123	0.0323062	+	0.999478i	$0.489715\pi$
$644$	2.49440	0.0982930
$645$	0	0
$646$	0.543213	0.0213724
$647$	−34.4927	−1.35605	−0.678024	−	0.735040i	$-0.737164\pi$
$647$	−34.4927	−1.35605	−0.678024	+	0.735040i	$0.737164\pi$
$648$	0	0
$649$	19.3056	0.757813
$650$	0.000697358 0	2.73526e−5 0
$651$	0	0
$652$	−6.05717	−0.237217
$653$	−38.7606	−1.51682	−0.758410	−	0.651778i	$-0.774023\pi$
$653$	−38.7606	−1.51682	−0.758410	+	0.651778i	$0.774023\pi$
$654$	0	0
$655$	−34.4127	−1.34462
$656$	11.0032	0.429604
$657$	0	0
$658$	−3.89151	−0.151707
$659$	−9.39192	−0.365857	−0.182929	−	0.983126i	$-0.558558\pi$
$659$	−9.39192	−0.365857	−0.182929	+	0.983126i	$0.558558\pi$
$660$	0	0
$661$	−24.1474	−0.939226	−0.469613	−	0.882872i	$-0.655607\pi$
$661$	−24.1474	−0.939226	−0.469613	+	0.882872i	$0.655607\pi$
$662$	−2.66740	−0.103671
$663$	0	0
$664$	7.33643	0.284709
$665$	1.22001	0.0473101
$666$	0	0
$667$	8.07569	0.312692
$668$	−37.5765	−1.45388
$669$	0	0
$670$	10.1483	0.392065
$671$	36.0590	1.39204
$672$	0	0
$673$	−26.4661	−1.02019	−0.510097	−	0.860117i	$-0.670391\pi$
$673$	−26.4661	−1.02019	−0.510097	+	0.860117i	$0.670391\pi$
$674$	3.10557	0.119622
$675$	0	0
$676$	23.7554	0.913671
$677$	−31.0668	−1.19400	−0.596998	−	0.802243i	$-0.703640\pi$
$677$	−31.0668	−1.19400	−0.596998	+	0.802243i	$0.703640\pi$
$678$	0	0
$679$	−13.1189	−0.503457
$680$	11.0352	0.423181
$681$	0	0
$682$	8.08119	0.309445
$683$	−38.1361	−1.45924	−0.729619	−	0.683854i	$-0.760303\pi$
$683$	−38.1361	−1.45924	−0.729619	+	0.683854i	$0.760303\pi$
$684$	0	0
$685$	26.5986	1.01628
$686$	6.72090	0.256605
$687$	0	0
$688$	−24.8748	−0.948342
$689$	−0.0234864	−0.000894762 0
$690$	0	0
$691$	32.9295	1.25270	0.626349	−	0.779543i	$-0.284548\pi$
$691$	32.9295	1.25270	0.626349	+	0.779543i	$0.284548\pi$
$692$	25.6442	0.974847
$693$	0	0
$694$	13.0683	0.496065
$695$	−13.6126	−0.516355
$696$	0	0
$697$	11.5128	0.436080
$698$	4.92450	0.186395
$699$	0	0
$700$	−0.224140	−0.00847171
$701$	2.30710	0.0871381	0.0435690	−	0.999050i	$-0.486127\pi$
$701$	2.30710	0.0871381	0.0435690	+	0.999050i	$0.486127\pi$
$702$	0	0
$703$	1.84668	0.0696490
$704$	−21.6436	−0.815725
$705$	0	0
$706$	3.40977	0.128328
$707$	18.2102	0.684865
$708$	0	0
$709$	−11.1521	−0.418825	−0.209412	−	0.977827i	$-0.567155\pi$
$709$	−11.1521	−0.418825	−0.209412	+	0.977827i	$0.567155\pi$
$710$	−5.59958	−0.210148
$711$	0	0
$712$	−5.36368	−0.201012
$713$	3.85791	0.144480
$714$	0	0
$715$	0.208623	0.00780206
$716$	18.6211	0.695902
$717$	0	0
$718$	−7.36510	−0.274863
$719$	32.1700	1.19974	0.599869	−	0.800098i	$-0.295219\pi$
$719$	32.1700	1.19974	0.599869	+	0.800098i	$0.295219\pi$
$720$	0	0
$721$	−6.02426	−0.224355
$722$	−7.82149	−0.291086
$723$	0	0
$724$	−43.9573	−1.63366
$725$	−0.725662	−0.0269504
$726$	0	0
$727$	−5.36551	−0.198996	−0.0994979	−	0.995038i	$-0.531724\pi$
$727$	−5.36551	−0.198996	−0.0994979	+	0.995038i	$0.531724\pi$
$728$	−0.0378942	−0.00140445
$729$	0	0
$730$	0.503399	0.0186316
$731$	−26.0269	−0.962638
$732$	0	0
$733$	−14.5964	−0.539129	−0.269564	−	0.962982i	$-0.586880\pi$
$733$	−14.5964	−0.539129	−0.269564	+	0.962982i	$0.586880\pi$
$734$	−8.44208	−0.311603
$735$	0	0
$736$	4.57639	0.168688
$737$	−57.5068	−2.11829
$738$	0	0
$739$	−43.2165	−1.58975	−0.794873	−	0.606776i	$-0.792463\pi$
$739$	−43.2165	−1.58975	−0.794873	+	0.606776i	$0.792463\pi$
$740$	17.9114	0.658438
$741$	0	0
$742$	−0.713055	−0.0261771
$743$	−8.11221	−0.297608	−0.148804	−	0.988867i	$-0.547542\pi$
$743$	−8.11221	−0.297608	−0.148804	+	0.988867i	$0.547542\pi$
$744$	0	0
$745$	1.95561	0.0716481
$746$	−4.02231	−0.147267
$747$	0	0
$748$	−29.8561	−1.09165
$749$	−14.9071	−0.544695
$750$	0	0
$751$	8.75545	0.319491	0.159746	−	0.987158i	$-0.448933\pi$
$751$	8.75545	0.319491	0.159746	+	0.987158i	$0.448933\pi$
$752$	21.2520	0.774981
$753$	0	0
$754$	−0.0585753	−0.00213319
$755$	−18.2210	−0.663129
$756$	0	0
$757$	−32.1511	−1.16855	−0.584276	−	0.811555i	$-0.698622\pi$
$757$	−32.1511	−1.16855	−0.584276	+	0.811555i	$0.698622\pi$
$758$	−1.71548	−0.0623091
$759$	0	0
$760$	1.47012	0.0533267
$761$	24.5459	0.889789	0.444894	−	0.895583i	$-0.353241\pi$
$761$	24.5459	0.889789	0.444894	+	0.895583i	$0.353241\pi$
$762$	0	0
$763$	−19.1388	−0.692872
$764$	20.0033	0.723693
$765$	0	0
$766$	−1.97354	−0.0713069
$767$	−0.0668492	−0.00241379
$768$	0	0
$769$	31.4144	1.13283	0.566416	−	0.824119i	$-0.308330\pi$
$769$	31.4144	1.13283	0.566416	+	0.824119i	$0.308330\pi$
$770$	6.33385	0.228256
$771$	0	0
$772$	19.7467	0.710699
$773$	−28.7145	−1.03279	−0.516395	−	0.856351i	$-0.672726\pi$
$773$	−28.7145	−1.03279	−0.516395	+	0.856351i	$0.672726\pi$
$774$	0	0
$775$	−0.346663	−0.0124525
$776$	−15.8083	−0.567483
$777$	0	0
$778$	9.06270	0.324913
$779$	1.53374	0.0549521
$780$	0	0
$781$	31.7306	1.13541
$782$	1.34633	0.0481446
$783$	0	0
$784$	−15.7448	−0.562315
$785$	−27.8224	−0.993022
$786$	0	0
$787$	−38.8160	−1.38364	−0.691821	−	0.722069i	$-0.743191\pi$
$787$	−38.8160	−1.38364	−0.691821	+	0.722069i	$0.743191\pi$
$788$	−40.3366	−1.43693
$789$	0	0
$790$	−0.450177	−0.0160166
$791$	16.5725	0.589249
$792$	0	0
$793$	−0.124861	−0.00443394
$794$	−14.4786	−0.513826
$795$	0	0
$796$	−23.5491	−0.834676
$797$	4.03410	0.142895	0.0714476	−	0.997444i	$-0.477238\pi$
$797$	4.03410	0.142895	0.0714476	+	0.997444i	$0.477238\pi$
$798$	0	0
$799$	22.2363	0.786664
$800$	−0.411224	−0.0145390
$801$	0	0
$802$	7.82160	0.276190
$803$	−2.85257	−0.100665
$804$	0	0
$805$	3.02374	0.106573
$806$	−0.0279826	−0.000985644 0
$807$	0	0
$808$	21.9433	0.771961
$809$	29.9454	1.05283	0.526413	−	0.850229i	$-0.323537\pi$
$809$	29.9454	1.05283	0.526413	+	0.850229i	$0.323537\pi$
$810$	0	0
$811$	20.2173	0.709927	0.354963	−	0.934880i	$-0.384493\pi$
$811$	20.2173	0.709927	0.354963	+	0.934880i	$0.384493\pi$
$812$	18.8269	0.660695
$813$	0	0
$814$	9.58727	0.336034
$815$	−7.34259	−0.257200
$816$	0	0
$817$	−3.46731	−0.121306
$818$	−2.64235	−0.0923875
$819$	0	0
$820$	14.8762	0.519499
$821$	−26.8736	−0.937896	−0.468948	−	0.883226i	$-0.655367\pi$
$821$	−26.8736	−0.937896	−0.468948	+	0.883226i	$0.655367\pi$
$822$	0	0
$823$	23.0543	0.803623	0.401812	−	0.915722i	$-0.368381\pi$
$823$	23.0543	0.803623	0.401812	+	0.915722i	$0.368381\pi$
$824$	−7.25923	−0.252887
$825$	0	0
$826$	−2.02956	−0.0706174
$827$	5.10953	0.177676	0.0888378	−	0.996046i	$-0.471685\pi$
$827$	5.10953	0.177676	0.0888378	+	0.996046i	$0.471685\pi$
$828$	0	0
$829$	−30.5982	−1.06272	−0.531360	−	0.847146i	$-0.678319\pi$
$829$	−30.5982	−1.06272	−0.531360	+	0.847146i	$0.678319\pi$
$830$	4.24612	0.147385
$831$	0	0
$832$	0.0749449	0.00259825
$833$	−16.4741	−0.570792
$834$	0	0
$835$	−45.5508	−1.57635
$836$	−3.97744	−0.137563
$837$	0	0
$838$	10.1033	0.349013
$839$	56.3087	1.94399	0.971996	−	0.234997i	$-0.0755081\pi$
$839$	56.3087	1.94399	0.971996	+	0.234997i	$0.0755081\pi$
$840$	0	0
$841$	31.9527	1.10182
$842$	−3.31959	−0.114401
$843$	0	0
$844$	43.8493	1.50936
$845$	28.7967	0.990637
$846$	0	0
$847$	−21.3756	−0.734474
$848$	3.89408	0.133723
$849$	0	0
$850$	−0.120978	−0.00414950
$851$	4.57691	0.156894
$852$	0	0
$853$	−45.5450	−1.55943	−0.779715	−	0.626134i	$-0.784636\pi$
$853$	−45.5450	−1.55943	−0.779715	+	0.626134i	$0.784636\pi$
$854$	−3.79081	−0.129719
$855$	0	0
$856$	−17.9631	−0.613966
$857$	17.4892	0.597419	0.298709	−	0.954344i	$-0.403444\pi$
$857$	17.4892	0.597419	0.298709	+	0.954344i	$0.403444\pi$
$858$	0	0
$859$	18.3460	0.625958	0.312979	−	0.949760i	$-0.398673\pi$
$859$	18.3460	0.625958	0.312979	+	0.949760i	$0.398673\pi$
$860$	−33.6303	−1.14678
$861$	0	0
$862$	−4.10117	−0.139686
$863$	−4.65373	−0.158415	−0.0792073	−	0.996858i	$-0.525239\pi$
$863$	−4.65373	−0.158415	−0.0792073	+	0.996858i	$0.525239\pi$
$864$	0	0
$865$	31.0863	1.05697
$866$	−2.53588	−0.0861726
$867$	0	0
$868$	8.99399	0.305276
$869$	2.55098	0.0865360
$870$	0	0
$871$	0.199128	0.00674718
$872$	−23.0622	−0.780986
$873$	0	0
$874$	0.179358	0.00606688
$875$	−14.8878	−0.503301
$876$	0	0
$877$	−3.66710	−0.123829	−0.0619145	−	0.998081i	$-0.519721\pi$
$877$	−3.66710	−0.123829	−0.0619145	+	0.998081i	$0.519721\pi$
$878$	−6.28768	−0.212199
$879$	0	0
$880$	−34.5899	−1.16603
$881$	−38.3008	−1.29039	−0.645193	−	0.764020i	$-0.723223\pi$
$881$	−38.3008	−1.29039	−0.645193	+	0.764020i	$0.723223\pi$
$882$	0	0
$883$	22.6142	0.761027	0.380513	−	0.924775i	$-0.375747\pi$
$883$	22.6142	0.761027	0.380513	+	0.924775i	$0.375747\pi$
$884$	0.103382	0.00347711
$885$	0	0
$886$	0.300296	0.0100887
$887$	1.89656	0.0636802	0.0318401	−	0.999493i	$-0.489863\pi$
$887$	1.89656	0.0636802	0.0318401	+	0.999493i	$0.489863\pi$
$888$	0	0
$889$	11.0783	0.371553
$890$	−3.10435	−0.104058
$891$	0	0
$892$	−39.5852	−1.32541
$893$	2.96233	0.0991306
$894$	0	0
$895$	22.5727	0.754523
$896$	13.9521	0.466107
$897$	0	0
$898$	−0.693184	−0.0231318
$899$	29.1183	0.971150
$900$	0	0
$901$	4.07443	0.135739
$902$	7.96262	0.265126
$903$	0	0
$904$	19.9698	0.664185
$905$	−53.2857	−1.77128
$906$	0	0
$907$	6.53094	0.216856	0.108428	−	0.994104i	$-0.465418\pi$
$907$	6.53094	0.216856	0.108428	+	0.994104i	$0.465418\pi$
$908$	39.5481	1.31245
$909$	0	0
$910$	−0.0219321	−0.000727042 0
$911$	43.0371	1.42588	0.712942	−	0.701223i	$-0.247362\pi$
$911$	43.0371	1.42588	0.712942	+	0.701223i	$0.247362\pi$
$912$	0	0
$913$	−24.0612	−0.796308
$914$	4.60480	0.152313
$915$	0	0
$916$	19.7456	0.652412
$917$	−20.5003	−0.676980
$918$	0	0
$919$	−16.7911	−0.553887	−0.276943	−	0.960886i	$-0.589321\pi$
$919$	−16.7911	−0.553887	−0.276943	+	0.960886i	$0.589321\pi$
$920$	3.64361	0.120126
$921$	0	0
$922$	9.09222	0.299436
$923$	−0.109873	−0.00361652
$924$	0	0
$925$	−0.411270	−0.0135225
$926$	10.3251	0.339302
$927$	0	0
$928$	34.5412	1.13387
$929$	11.6000	0.380584	0.190292	−	0.981728i	$-0.439057\pi$
$929$	11.6000	0.380584	0.190292	+	0.981728i	$0.439057\pi$
$930$	0	0
$931$	−2.19468	−0.0719277
$932$	13.9594	0.457256
$933$	0	0
$934$	4.91727	0.160898
$935$	−36.1920	−1.18360
$936$	0	0
$937$	47.7953	1.56140	0.780702	−	0.624904i	$-0.214862\pi$
$937$	47.7953	1.56140	0.780702	+	0.624904i	$0.214862\pi$
$938$	6.04557	0.197395
$939$	0	0
$940$	28.7324	0.937147
$941$	−11.2604	−0.367077	−0.183538	−	0.983013i	$-0.558755\pi$
$941$	−11.2604	−0.367077	−0.183538	+	0.983013i	$0.558755\pi$
$942$	0	0
$943$	3.80131	0.123788
$944$	11.0837	0.360743
$945$	0	0
$946$	−18.0010	−0.585261
$947$	−7.33189	−0.238255	−0.119127	−	0.992879i	$-0.538010\pi$
$947$	−7.33189	−0.238255	−0.119127	+	0.992879i	$0.538010\pi$
$948$	0	0
$949$	0.00987752	0.000320638 0
$950$	−0.0161167	−0.000522895 0
$951$	0	0
$952$	6.57388	0.213061
$953$	24.8753	0.805791	0.402895	−	0.915246i	$-0.368004\pi$
$953$	24.8753	0.805791	0.402895	+	0.915246i	$0.368004\pi$
$954$	0	0
$955$	24.2483	0.784655
$956$	−5.90519	−0.190987
$957$	0	0
$958$	−1.19960	−0.0387572
$959$	15.8453	0.511672
$960$	0	0
$961$	−17.0896	−0.551277
$962$	−0.0331976	−0.00107034
$963$	0	0
$964$	48.5079	1.56233
$965$	23.9372	0.770567
$966$	0	0
$967$	−34.0300	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$967$	−34.0300	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$968$	−25.7575	−0.827878
$969$	0	0
$970$	−9.14938	−0.293769
$971$	34.2476	1.09906	0.549530	−	0.835474i	$-0.314807\pi$
$971$	34.2476	1.09906	0.549530	+	0.835474i	$0.314807\pi$
$972$	0	0
$973$	−8.10928	−0.259971
$974$	3.63908	0.116604
$975$	0	0
$976$	20.7021	0.662657
$977$	23.4173	0.749186	0.374593	−	0.927189i	$-0.377782\pi$
$977$	23.4173	0.749186	0.374593	+	0.927189i	$0.377782\pi$
$978$	0	0
$979$	17.5912	0.562216
$980$	−21.2868	−0.679980
$981$	0	0
$982$	9.37834	0.299275
$983$	33.2031	1.05902	0.529508	−	0.848305i	$-0.322377\pi$
$983$	33.2031	1.05902	0.529508	+	0.848305i	$0.322377\pi$
$984$	0	0
$985$	−48.8966	−1.55798
$986$	10.1617	0.323613
$987$	0	0
$988$	0.0137726	0.000438165 0
$989$	−8.59355	−0.273259
$990$	0	0
$991$	−28.1806	−0.895187	−0.447594	−	0.894237i	$-0.647719\pi$
$991$	−28.1806	−0.895187	−0.447594	+	0.894237i	$0.647719\pi$
$992$	16.5010	0.523907
$993$	0	0
$994$	−3.33577	−0.105804
$995$	−28.5466	−0.904988
$996$	0	0
$997$	−44.9507	−1.42360	−0.711802	−	0.702381i	$-0.752120\pi$
$997$	−44.9507	−1.42360	−0.711802	+	0.702381i	$0.752120\pi$
$998$	−10.5249	−0.333161
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.d.1.3		6
3.2	odd	2		729.2.a.a.1.4		6
9.2	odd	6		729.2.c.e.244.3		12
9.4	even	3		729.2.c.b.487.4		12
9.5	odd	6		729.2.c.e.487.3		12
9.7	even	3		729.2.c.b.244.4		12
27.2	odd	18		243.2.e.c.190.1		12
27.4	even	9		81.2.e.a.46.1		12
27.5	odd	18		243.2.e.d.217.2		12
27.7	even	9		81.2.e.a.37.1		12
27.11	odd	18		243.2.e.d.28.2		12
27.13	even	9		243.2.e.b.55.2		12
27.14	odd	18		243.2.e.c.55.1		12
27.16	even	9		243.2.e.a.28.1		12
27.20	odd	18		27.2.e.a.22.2	yes	12
27.22	even	9		243.2.e.a.217.1		12
27.23	odd	18		27.2.e.a.16.2	✓	12
27.25	even	9		243.2.e.b.190.2		12
108.23	even	18		432.2.u.c.97.1		12
108.47	even	18		432.2.u.c.49.1		12
135.23	even	36		675.2.u.b.124.2		24
135.47	even	36		675.2.u.b.49.2		24
135.74	odd	18		675.2.l.c.76.1		12
135.77	even	36		675.2.u.b.124.3		24
135.104	odd	18		675.2.l.c.151.1		12
135.128	even	36		675.2.u.b.49.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.16.2	✓	12	27.23	odd	18
27.2.e.a.22.2	yes	12	27.20	odd	18
81.2.e.a.37.1		12	27.7	even	9
81.2.e.a.46.1		12	27.4	even	9
243.2.e.a.28.1		12	27.16	even	9
243.2.e.a.217.1		12	27.22	even	9
243.2.e.b.55.2		12	27.13	even	9
243.2.e.b.190.2		12	27.25	even	9
243.2.e.c.55.1		12	27.14	odd	18
243.2.e.c.190.1		12	27.2	odd	18
243.2.e.d.28.2		12	27.11	odd	18
243.2.e.d.217.2		12	27.5	odd	18
432.2.u.c.49.1		12	108.47	even	18
432.2.u.c.97.1		12	108.23	even	18
675.2.l.c.76.1		12	135.74	odd	18
675.2.l.c.151.1		12	135.104	odd	18
675.2.u.b.49.2		24	135.47	even	36
675.2.u.b.49.3		24	135.128	even	36
675.2.u.b.124.2		24	135.23	even	36
675.2.u.b.124.3		24	135.77	even	36
729.2.a.a.1.4		6	3.2	odd	2
729.2.a.d.1.3		6	1.1	even	1	trivial
729.2.c.b.244.4		12	9.7	even	3
729.2.c.b.487.4		12	9.4	even	3
729.2.c.e.244.3		12	9.2	odd	6
729.2.c.e.487.3		12	9.5	odd	6

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.a (trivial)

\(f(q)\)	\(=\)	\(q+0.415466 q^{2} -1.82739 q^{4} -2.21519 q^{5} -1.31963 q^{7} -1.59015 q^{8} -0.920335 q^{10} +5.21519 q^{11} -0.0180585 q^{13} -0.548261 q^{14} +2.99412 q^{16} +3.13280 q^{17} +0.417352 q^{19} +4.04801 q^{20} +2.16673 q^{22} +1.03439 q^{23} -0.0929475 q^{25} -0.00750270 q^{26} +2.41147 q^{28} +7.80722 q^{29} +3.72966 q^{31} +4.42426 q^{32} +1.30157 q^{34} +2.92322 q^{35} +4.42476 q^{37} +0.173396 q^{38} +3.52248 q^{40} +3.67494 q^{41} -8.30787 q^{43} -9.53017 q^{44} +0.429753 q^{46} +7.09791 q^{47} -5.25858 q^{49} -0.0386165 q^{50} +0.0329999 q^{52} +1.30057 q^{53} -11.5526 q^{55} +2.09841 q^{56} +3.24364 q^{58} +3.70181 q^{59} +6.91424 q^{61} +1.54955 q^{62} -4.15011 q^{64} +0.0400030 q^{65} -11.0268 q^{67} -5.72483 q^{68} +1.21450 q^{70} +6.08428 q^{71} -0.546973 q^{73} +1.83834 q^{74} -0.762665 q^{76} -6.88211 q^{77} +0.489144 q^{79} -6.63254 q^{80} +1.52681 q^{82} -4.61367 q^{83} -6.93973 q^{85} -3.45164 q^{86} -8.29293 q^{88} +3.37307 q^{89} +0.0238305 q^{91} -1.89023 q^{92} +2.94894 q^{94} -0.924513 q^{95} +9.94136 q^{97} -2.18476 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8} + 3 q^{10} + 12 q^{11} + 6 q^{14} - 3 q^{16} + 9 q^{17} + 3 q^{19} + 6 q^{20} + 6 q^{22} + 15 q^{23} - 6 q^{25} + 15 q^{26} - 6 q^{28} + 12 q^{29} + 12 q^{35}+ \cdots - 45 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 6 * q^8 + 3 * q^10 + 12 * q^11 + 6 * q^14 - 3 * q^16 + 9 * q^17 + 3 * q^19 + 6 * q^20 + 6 * q^22 + 15 * q^23 - 6 * q^25 + 15 * q^26 - 6 * q^28 + 12 * q^29 + 12 * q^35 + 3 * q^37 - 3 * q^38 + 6 * q^40 + 15 * q^41 + 3 * q^44 + 3 * q^46 + 21 * q^47 - 12 * q^49 + 3 * q^50 + 12 * q^52 + 9 * q^53 - 6 * q^55 - 6 * q^56 - 12 * q^58 + 24 * q^59 - 9 * q^61 - 12 * q^62 - 12 * q^64 - 6 * q^65 - 9 * q^67 - 9 * q^68 + 15 * q^70 + 27 * q^71 - 6 * q^73 - 12 * q^74 + 6 * q^76 - 12 * q^77 - 21 * q^80 - 6 * q^82 + 12 * q^83 - 21 * q^86 + 12 * q^88 + 9 * q^89 - 6 * q^91 + 6 * q^92 + 6 * q^94 + 12 * q^95 - 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more