LMFDB - Embedded newform 729.2.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$5.82109430735$
Analytic rank:	$1$
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			1.4
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	$-0.546926\pi$
$2$	−0.415466	−0.293779	−0.146889	+	0.989153i	$0.546926\pi$
$3$	0	0
$3$	0	0
$4$	−1.82739	−0.913694
$5$	2.21519	0.990662	0.495331	−	0.868704i	$-0.335047\pi$
$5$	2.21519	0.990662	0.495331	+	0.868704i	$0.335047\pi$
$6$	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	$-0.787964\pi$
$11$	−5.21519	−1.57244	−0.786219	+	0.617948i	$0.787964\pi$
$12$	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	$-0.624044\pi$
$17$	−3.13280	−0.759814	−0.379907	+	0.925025i	$0.624044\pi$
$18$	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	$-0.534394\pi$
$23$	−1.03439	−0.215684	−0.107842	+	0.994168i	$0.534394\pi$
$24$	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	$-0.758106\pi$
$29$	−7.80722	−1.44976	−0.724882	+	0.688873i	$0.758106\pi$
$30$	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	$-0.592646\pi$
$41$	−3.67494	−0.573929	−0.286965	+	0.957941i	$0.592646\pi$
$42$	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	$-0.673200\pi$
$47$	−7.09791	−1.03534	−0.517668	+	0.855581i	$0.673200\pi$
$48$	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	$-0.528471\pi$
$53$	−1.30057	−0.178648	−0.0893238	+	0.996003i	$0.528471\pi$
$54$	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	$-0.577465\pi$
$59$	−3.70181	−0.481935	−0.240967	+	0.970533i	$0.577465\pi$
$60$	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	$-0.617577\pi$
$71$	−6.08428	−0.722071	−0.361035	+	0.932552i	$0.617577\pi$
$72$	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	$-0.418514\pi$
$83$	4.61367	0.506416	0.253208	+	0.967412i	$0.418514\pi$
$84$	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	$-0.557212\pi$
$89$	−3.37307	−0.357544	−0.178772	+	0.983891i	$0.557212\pi$
$90$	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	$-0.259124\pi$
$101$	13.7995	1.37310	0.686550	+	0.727082i	$0.259124\pi$
$102$	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	$-0.683864\pi$
$107$	−11.2965	−1.09207	−0.546035	+	0.837762i	$0.683864\pi$
$108$	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	$-0.298852\pi$
$113$	12.5584	1.18140	0.590699	+	0.806892i	$0.298852\pi$
$114$	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	$-0.737433\pi$
$131$	−15.5349	−1.35729	−0.678645	+	0.734466i	$0.737433\pi$
$132$	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	$-0.328560\pi$
$137$	12.0074	1.02586	0.512930	+	0.858430i	$0.328560\pi$
$138$	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	$-0.488487\pi$
$149$	0.882820	0.0723235	0.0361617	+	0.999346i	$0.488487\pi$
$150$	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	$-0.792847\pi$
$167$	−20.5630	−1.59121	−0.795606	+	0.605815i	$0.792847\pi$
$168$	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	$-0.320890\pi$
$173$	14.0333	1.06693	0.533465	+	0.845822i	$0.320890\pi$
$174$	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	$-0.375642\pi$
$179$	10.1900	0.761636	0.380818	+	0.924650i	$0.375642\pi$
$180$	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	$-0.370389\pi$
$191$	10.9464	0.792052	0.396026	+	0.918239i	$0.370389\pi$
$192$	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	$-0.788022\pi$
$197$	−22.0734	−1.57266	−0.786331	+	0.617806i	$0.788022\pi$
$198$	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	$-0.244961\pi$
$227$	21.6419	1.43642	0.718211	+	0.695826i	$0.244961\pi$
$228$	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	$-0.419496\pi$
$233$	7.63900	0.500447	0.250224	+	0.968188i	$0.419496\pi$
$234$	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	$-0.533329\pi$
$239$	−3.23149	−0.209028	−0.104514	+	0.994523i	$0.533329\pi$
$240$	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	$-0.545352\pi$
$251$	−4.49930	−0.283993	−0.141997	+	0.989867i	$0.545352\pi$
$252$	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	$-0.359079\pi$
$257$	13.7354	0.856792	0.428396	+	0.903591i	$0.359079\pi$
$258$	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	$-0.768122\pi$
$263$	−24.2026	−1.49239	−0.746197	+	0.665725i	$0.768122\pi$
$264$	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	$-0.380722\pi$
$269$	12.0062	0.732032	0.366016	+	0.930609i	$0.380722\pi$
$270$	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	$-0.707883\pi$
$281$	−20.3717	−1.21528	−0.607638	+	0.794214i	$0.707883\pi$
$282$	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	$-0.126315\pi$
$293$	31.5742	1.84458	0.922291	+	0.386496i	$0.126315\pi$
$294$	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	$-0.736359\pi$
$311$	−23.8486	−1.35233	−0.676164	+	0.736751i	$0.736359\pi$
$312$	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	$-0.575180\pi$
$317$	−8.33233	−0.467990	−0.233995	+	0.972238i	$0.575180\pi$
$318$	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	$-0.819973\pi$
$347$	−31.4545	−1.68857	−0.844283	+	0.535898i	$0.819973\pi$
$348$	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	$-0.570087\pi$
$353$	−8.20708	−0.436819	−0.218409	+	0.975857i	$0.570087\pi$
$354$	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	$-0.345045\pi$
$359$	17.7273	0.935611	0.467806	+	0.883831i	$0.345045\pi$
$360$	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	$-0.461274\pi$
$383$	4.75018	0.242723	0.121362	+	0.992608i	$0.461274\pi$
$384$	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	$-0.686513\pi$
$389$	−21.8133	−1.10598	−0.552990	+	0.833188i	$0.686513\pi$
$390$	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	$-0.655769\pi$
$401$	−18.8261	−0.940130	−0.470065	+	0.882632i	$0.655769\pi$
$402$	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	$-0.702454\pi$
$419$	−24.3180	−1.18801	−0.594005	+	0.804461i	$0.702454\pi$
$420$	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	$-0.423593\pi$
$431$	9.87124	0.475481	0.237740	+	0.971329i	$0.423593\pi$
$432$	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	$-0.505466\pi$
$443$	−0.722793	−0.0343410	−0.0171705	+	0.999853i	$0.505466\pi$
$444$	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	$-0.487465\pi$
$449$	1.66845	0.0787389	0.0393695	+	0.999225i	$0.487465\pi$
$450$	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	$-0.670217\pi$
$461$	−21.8844	−1.01926	−0.509629	+	0.860394i	$0.670217\pi$
$462$	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	$-0.588294\pi$
$467$	−11.8355	−0.547683	−0.273842	+	0.961775i	$0.588294\pi$
$468$	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	$-0.478988\pi$
$479$	2.88735	0.131926	0.0659632	+	0.997822i	$0.478988\pi$
$480$	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	$-0.670115\pi$
$491$	−22.5730	−1.01871	−0.509354	+	0.860557i	$0.670115\pi$
$492$	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	$-0.473399\pi$
$503$	3.74414	0.166943	0.0834714	+	0.996510i	$0.473399\pi$
$504$	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	$-0.318559\pi$
$509$	24.3499	1.07929	0.539645	+	0.841893i	$0.318559\pi$
$510$	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	$-0.358583\pi$
$521$	19.6209	0.859608	0.429804	+	0.902922i	$0.358583\pi$
$522$	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	$-0.219128\pi$
$557$	36.4518	1.54451	0.772256	+	0.635311i	$0.219128\pi$
$558$	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	$-0.311276\pi$
$563$	26.5162	1.11753	0.558763	+	0.829327i	$0.311276\pi$
$564$	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	$-0.340121\pi$
$569$	22.9674	0.962844	0.481422	+	0.876489i	$0.340121\pi$
$570$	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	$-0.168524\pi$
$587$	41.8222	1.72619	0.863094	+	0.505044i	$0.168524\pi$
$588$	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	$-0.724200\pi$
$593$	−31.5370	−1.29507	−0.647536	+	0.762035i	$0.724200\pi$
$594$	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	$-0.583073\pi$
$599$	−12.6303	−0.516060	−0.258030	+	0.966137i	$0.583073\pi$
$600$	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	$-0.545912\pi$
$617$	−7.14078	−0.287477	−0.143739	+	0.989616i	$0.545912\pi$
$618$	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	$-0.531553\pi$
$641$	−5.01121	−0.197931	−0.0989655	+	0.995091i	$0.531553\pi$
$642$	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	$-0.262836\pi$
$647$	34.4927	1.35605	0.678024	+	0.735040i	$0.262836\pi$
$648$	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	$-0.225977\pi$
$653$	38.7606	1.51682	0.758410	+	0.651778i	$0.225977\pi$
$654$	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	$-0.441442\pi$
$659$	9.39192	0.365857	0.182929	+	0.983126i	$0.441442\pi$
$660$	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	$-0.296360\pi$
$677$	31.0668	1.19400	0.596998	+	0.802243i	$0.296360\pi$
$678$	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	$-0.239697\pi$
$683$	38.1361	1.45924	0.729619	+	0.683854i	$0.239697\pi$
$684$	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	$-0.513873\pi$
$701$	−2.30710	−0.0871381	−0.0435690	+	0.999050i	$0.513873\pi$
$702$	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	$-0.704781\pi$
$719$	−32.1700	−1.19974	−0.599869	+	0.800098i	$0.704781\pi$
$720$	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	$-0.452458\pi$
$743$	8.11221	0.297608	0.148804	+	0.988867i	$0.452458\pi$
$744$	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	$-0.646759\pi$
$761$	−24.5459	−0.889789	−0.444894	+	0.895583i	$0.646759\pi$
$762$	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	$-0.327274\pi$
$773$	28.7145	1.03279	0.516395	+	0.856351i	$0.327274\pi$
$774$	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	$-0.522762\pi$
$797$	−4.03410	−0.142895	−0.0714476	+	0.997444i	$0.522762\pi$
$798$	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	$-0.676463\pi$
$809$	−29.9454	−1.05283	−0.526413	+	0.850229i	$0.676463\pi$
$810$	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	$-0.344633\pi$
$821$	26.8736	0.937896	0.468948	+	0.883226i	$0.344633\pi$
$822$	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	$-0.528315\pi$
$827$	−5.10953	−0.177676	−0.0888378	+	0.996046i	$0.528315\pi$
$828$	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	$-0.924492\pi$
$839$	−56.3087	−1.94399	−0.971996	+	0.234997i	$0.924492\pi$
$840$	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	$-0.596556\pi$
$857$	−17.4892	−0.597419	−0.298709	+	0.954344i	$0.596556\pi$
$858$	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	$-0.474761\pi$
$863$	4.65373	0.158415	0.0792073	+	0.996858i	$0.474761\pi$
$864$	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	$-0.276777\pi$
$881$	38.3008	1.29039	0.645193	+	0.764020i	$0.276777\pi$
$882$	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	$-0.510137\pi$
$887$	−1.89656	−0.0636802	−0.0318401	+	0.999493i	$0.510137\pi$
$888$	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	$-0.752638\pi$
$911$	−43.0371	−1.42588	−0.712942	+	0.701223i	$0.752638\pi$
$912$	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	$-0.560943\pi$
$929$	−11.6000	−0.380584	−0.190292	+	0.981728i	$0.560943\pi$
$930$	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	$-0.441245\pi$
$941$	11.2604	0.367077	0.183538	+	0.983013i	$0.441245\pi$
$942$	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	$-0.461990\pi$
$947$	7.33189	0.238255	0.119127	+	0.992879i	$0.461990\pi$
$948$	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	$-0.631996\pi$
$953$	−24.8753	−0.805791	−0.402895	+	0.915246i	$0.631996\pi$
$954$	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	$-0.685193\pi$
$971$	−34.2476	−1.09906	−0.549530	+	0.835474i	$0.685193\pi$
$972$	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	$-0.622218\pi$
$977$	−23.4173	−0.749186	−0.374593	+	0.927189i	$0.622218\pi$
$978$	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	$-0.677623\pi$
$983$	−33.2031	−1.05902	−0.529508	+	0.848305i	$0.677623\pi$
$984$	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.a.1.4		6
3.2	odd	2		729.2.a.d.1.3		6
9.2	odd	6		729.2.c.b.244.4		12
9.4	even	3		729.2.c.e.487.3		12
9.5	odd	6		729.2.c.b.487.4		12
9.7	even	3		729.2.c.e.244.3		12
27.2	odd	18		243.2.e.b.190.2		12
27.4	even	9		27.2.e.a.16.2	✓	12
27.5	odd	18		243.2.e.a.217.1		12
27.7	even	9		27.2.e.a.22.2	yes	12
27.11	odd	18		243.2.e.a.28.1		12
27.13	even	9		243.2.e.c.55.1		12
27.14	odd	18		243.2.e.b.55.2		12
27.16	even	9		243.2.e.d.28.2		12
27.20	odd	18		81.2.e.a.37.1		12
27.22	even	9		243.2.e.d.217.2		12
27.23	odd	18		81.2.e.a.46.1		12
27.25	even	9		243.2.e.c.190.1		12
108.7	odd	18		432.2.u.c.49.1		12
108.31	odd	18		432.2.u.c.97.1		12
135.4	even	18		675.2.l.c.151.1		12
135.7	odd	36		675.2.u.b.49.2		24
135.34	even	18		675.2.l.c.76.1		12
135.58	odd	36		675.2.u.b.124.2		24
135.88	odd	36		675.2.u.b.49.3		24
135.112	odd	36		675.2.u.b.124.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.16.2	✓	12	27.4	even	9
27.2.e.a.22.2	yes	12	27.7	even	9
81.2.e.a.37.1		12	27.20	odd	18
81.2.e.a.46.1		12	27.23	odd	18
243.2.e.a.28.1		12	27.11	odd	18
243.2.e.a.217.1		12	27.5	odd	18
243.2.e.b.55.2		12	27.14	odd	18
243.2.e.b.190.2		12	27.2	odd	18
243.2.e.c.55.1		12	27.13	even	9
243.2.e.c.190.1		12	27.25	even	9
243.2.e.d.28.2		12	27.16	even	9
243.2.e.d.217.2		12	27.22	even	9
432.2.u.c.49.1		12	108.7	odd	18
432.2.u.c.97.1		12	108.31	odd	18
675.2.l.c.76.1		12	135.34	even	18
675.2.l.c.151.1		12	135.4	even	18
675.2.u.b.49.2		24	135.7	odd	36
675.2.u.b.49.3		24	135.88	odd	36
675.2.u.b.124.2		24	135.58	odd	36
675.2.u.b.124.3		24	135.112	odd	36
729.2.a.a.1.4		6	1.1	even	1	trivial
729.2.a.d.1.3		6	3.2	odd	2
729.2.c.b.244.4		12	9.2	odd	6
729.2.c.b.487.4		12	9.5	odd	6
729.2.c.e.244.3		12	9.7	even	3
729.2.c.e.487.3		12	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
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

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