LMFDB - Embedded newform 729.2.a.e.1.3

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.7459857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 $ x^6 - 3x^5 - 6x^4 + 13x^3 + 12x^2 - 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.70506$ 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)

$f(q)$	$=$	\(q-0.172976 q^{2} -1.97008 q^{4} +3.73656 q^{5} +3.03150 q^{7} +0.686728 q^{8} -0.646335 q^{10} +2.49170 q^{11} -0.765139 q^{13} -0.524376 q^{14} +3.82137 q^{16} -4.62278 q^{17} -0.611844 q^{19} -7.36132 q^{20} -0.431003 q^{22} -6.52438 q^{23} +8.96190 q^{25} +0.132351 q^{26} -5.97229 q^{28} +6.55089 q^{29} +6.55043 q^{31} -2.03446 q^{32} +0.799630 q^{34} +11.3274 q^{35} +4.95969 q^{37} +0.105834 q^{38} +2.56600 q^{40} -5.26024 q^{41} +5.57057 q^{43} -4.90884 q^{44} +1.12856 q^{46} -1.10762 q^{47} +2.18998 q^{49} -1.55019 q^{50} +1.50738 q^{52} -8.84310 q^{53} +9.31038 q^{55} +2.08181 q^{56} -1.13315 q^{58} +11.8518 q^{59} +8.18700 q^{61} -1.13307 q^{62} -7.29083 q^{64} -2.85899 q^{65} -1.21234 q^{67} +9.10725 q^{68} -1.95936 q^{70} +4.91946 q^{71} +4.29945 q^{73} -0.857907 q^{74} +1.20538 q^{76} +7.55357 q^{77} -11.7946 q^{79} +14.2788 q^{80} +0.909895 q^{82} -9.01607 q^{83} -17.2733 q^{85} -0.963575 q^{86} +1.71112 q^{88} -7.53885 q^{89} -2.31952 q^{91} +12.8535 q^{92} +0.191591 q^{94} -2.28619 q^{95} -0.948354 q^{97} -0.378814 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 + 9 * q^4 - 3 * q^5 + 6 * q^7 + 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 24 * q^14 + 15 * q^16 - 9 * q^17 + 12 * q^19 - 21 * q^20 + 3 * q^22 - 12 * q^23 + 9 * q^25 + 24 * q^26 + 3 * q^28 + 21 * q^29 + 15 * q^31 + 30 * q^35 + 3 * q^37 + 15 * q^38 + 3 * q^40 - 12 * q^41 + 6 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 12 * q^49 - 24 * q^50 + 3 * q^52 - 9 * q^53 + 15 * q^55 + 12 * q^56 - 15 * q^58 + 6 * q^59 + 24 * q^61 - 30 * q^62 + 6 * q^64 - 15 * q^65 + 15 * q^67 + 36 * q^68 - 15 * q^70 + 12 * q^73 + 24 * q^74 + 9 * q^76 + 15 * q^77 + 24 * q^79 - 21 * q^80 - 21 * q^82 - 6 * q^83 - 18 * q^85 - 30 * q^86 - 21 * q^88 - 9 * q^89 + 18 * q^91 + 6 * q^92 - 6 * q^94 - 33 * q^95 - 21 * q^97 + 18 * 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 $

$2$	−0.172976	−0.122312	−0.0611562	−	0.998128i	$-0.519479\pi$
$2$	−0.172976	−0.122312	−0.0611562	+	0.998128i	$0.519479\pi$
$3$	0	0
$3$	0	0
$4$	−1.97008	−0.985040
$5$	3.73656	1.67104	0.835521	−	0.549459i	$-0.185166\pi$
$5$	3.73656	1.67104	0.835521	+	0.549459i	$0.185166\pi$
$6$	0	0
$7$	3.03150	1.14580	0.572899	−	0.819626i	$-0.305819\pi$
$7$	3.03150	1.14580	0.572899	+	0.819626i	$0.305819\pi$
$8$	0.686728	0.242795
$9$	0	0
$10$	−0.646335	−0.204389
$11$	2.49170	0.751275	0.375637	−	0.926767i	$-0.377424\pi$
$11$	2.49170	0.751275	0.375637	+	0.926767i	$0.377424\pi$
$12$	0	0
$13$	−0.765139	−0.212211	−0.106106	−	0.994355i	$-0.533838\pi$
$13$	−0.765139	−0.212211	−0.106106	+	0.994355i	$0.533838\pi$
$14$	−0.524376	−0.140145
$15$	0	0
$16$	3.82137	0.955343
$17$	−4.62278	−1.12119	−0.560595	−	0.828090i	$-0.689427\pi$
$17$	−4.62278	−1.12119	−0.560595	+	0.828090i	$0.689427\pi$
$18$	0	0
$19$	−0.611844	−0.140367	−0.0701833	−	0.997534i	$-0.522358\pi$
$19$	−0.611844	−0.140367	−0.0701833	+	0.997534i	$0.522358\pi$
$20$	−7.36132	−1.64604
$21$	0	0
$22$	−0.431003	−0.0918902
$23$	−6.52438	−1.36043	−0.680213	−	0.733014i	$-0.738113\pi$
$23$	−6.52438	−1.36043	−0.680213	+	0.733014i	$0.738113\pi$
$24$	0	0
$25$	8.96190	1.79238
$26$	0.132351	0.0259561
$27$	0	0
$28$	−5.97229	−1.12866
$29$	6.55089	1.21647	0.608235	−	0.793757i	$-0.291878\pi$
$29$	6.55089	1.21647	0.608235	+	0.793757i	$0.291878\pi$
$30$	0	0
$31$	6.55043	1.17649	0.588246	−	0.808682i	$-0.299819\pi$
$31$	6.55043	1.17649	0.588246	+	0.808682i	$0.299819\pi$
$32$	−2.03446	−0.359645
$33$	0	0
$34$	0.799630	0.137135
$35$	11.3274	1.91468
$36$	0	0
$37$	4.95969	0.815368	0.407684	−	0.913123i	$-0.366337\pi$
$37$	4.95969	0.815368	0.407684	+	0.913123i	$0.366337\pi$
$38$	0.105834	0.0171686
$39$	0	0
$40$	2.56600	0.405721
$41$	−5.26024	−0.821511	−0.410756	−	0.911745i	$-0.634735\pi$
$41$	−5.26024	−0.821511	−0.410756	+	0.911745i	$0.634735\pi$
$42$	0	0
$43$	5.57057	0.849505	0.424752	−	0.905310i	$-0.360361\pi$
$43$	5.57057	0.849505	0.424752	+	0.905310i	$0.360361\pi$
$44$	−4.90884	−0.740035
$45$	0	0
$46$	1.12856	0.166397
$47$	−1.10762	−0.161562	−0.0807812	−	0.996732i	$-0.525741\pi$
$47$	−1.10762	−0.161562	−0.0807812	+	0.996732i	$0.525741\pi$
$48$	0	0
$49$	2.18998	0.312854
$50$	−1.55019	−0.219230
$51$	0	0
$52$	1.50738	0.209037
$53$	−8.84310	−1.21469	−0.607346	−	0.794437i	$-0.707766\pi$
$53$	−8.84310	−1.21469	−0.607346	+	0.794437i	$0.707766\pi$
$54$	0	0
$55$	9.31038	1.25541
$56$	2.08181	0.278194
$57$	0	0
$58$	−1.13315	−0.148789
$59$	11.8518	1.54297	0.771484	−	0.636249i	$-0.219515\pi$
$59$	11.8518	1.54297	0.771484	+	0.636249i	$0.219515\pi$
$60$	0	0
$61$	8.18700	1.04824	0.524119	−	0.851645i	$-0.324395\pi$
$61$	8.18700	1.04824	0.524119	+	0.851645i	$0.324395\pi$
$62$	−1.13307	−0.143900
$63$	0	0
$64$	−7.29083	−0.911354
$65$	−2.85899	−0.354614
$66$	0	0
$67$	−1.21234	−0.148111	−0.0740553	−	0.997254i	$-0.523594\pi$
$67$	−1.21234	−0.148111	−0.0740553	+	0.997254i	$0.523594\pi$
$68$	9.10725	1.10442
$69$	0	0
$70$	−1.95936	−0.234189
$71$	4.91946	0.583833	0.291916	−	0.956444i	$-0.405707\pi$
$71$	4.91946	0.583833	0.291916	+	0.956444i	$0.405707\pi$
$72$	0	0
$73$	4.29945	0.503213	0.251606	−	0.967830i	$-0.419041\pi$
$73$	4.29945	0.503213	0.251606	+	0.967830i	$0.419041\pi$
$74$	−0.857907	−0.0997296
$75$	0	0
$76$	1.20538	0.138267
$77$	7.55357	0.860809
$78$	0	0
$79$	−11.7946	−1.32700	−0.663498	−	0.748178i	$-0.730929\pi$
$79$	−11.7946	−1.32700	−0.663498	+	0.748178i	$0.730929\pi$
$80$	14.2788	1.59642
$81$	0	0
$82$	0.909895	0.100481
$83$	−9.01607	−0.989642	−0.494821	−	0.868995i	$-0.664766\pi$
$83$	−9.01607	−0.989642	−0.494821	+	0.868995i	$0.664766\pi$
$84$	0	0
$85$	−17.2733	−1.87355
$86$	−0.963575	−0.103905
$87$	0	0
$88$	1.71112	0.182406
$89$	−7.53885	−0.799117	−0.399558	−	0.916708i	$-0.630837\pi$
$89$	−7.53885	−0.799117	−0.399558	+	0.916708i	$0.630837\pi$
$90$	0	0
$91$	−2.31952	−0.243151
$92$	12.8535	1.34007
$93$	0	0
$94$	0.191591	0.0197611
$95$	−2.28619	−0.234558
$96$	0	0
$97$	−0.948354	−0.0962908	−0.0481454	−	0.998840i	$-0.515331\pi$
$97$	−0.948354	−0.0962908	−0.0481454	+	0.998840i	$0.515331\pi$
$98$	−0.378814	−0.0382659
$99$	0	0
$100$	−17.6557	−1.76557
$101$	5.60815	0.558032	0.279016	−	0.960286i	$-0.409992\pi$
$101$	5.60815	0.558032	0.279016	+	0.960286i	$0.409992\pi$
$102$	0	0
$103$	−9.42502	−0.928675	−0.464337	−	0.885658i	$-0.653707\pi$
$103$	−9.42502	−0.928675	−0.464337	+	0.885658i	$0.653707\pi$
$104$	−0.525442	−0.0515239
$105$	0	0
$106$	1.52964	0.148572
$107$	1.27825	0.123573	0.0617864	−	0.998089i	$-0.480320\pi$
$107$	1.27825	0.123573	0.0617864	+	0.998089i	$0.480320\pi$
$108$	0	0
$109$	−7.40689	−0.709451	−0.354726	−	0.934970i	$-0.615426\pi$
$109$	−7.40689	−0.709451	−0.354726	+	0.934970i	$0.615426\pi$
$110$	−1.61047	−0.153552
$111$	0	0
$112$	11.5845	1.09463
$113$	−9.35196	−0.879759	−0.439879	−	0.898057i	$-0.644979\pi$
$113$	−9.35196	−0.879759	−0.439879	+	0.898057i	$0.644979\pi$
$114$	0	0
$115$	−24.3787	−2.27333
$116$	−12.9058	−1.19827
$117$	0	0
$118$	−2.05007	−0.188724
$119$	−14.0140	−1.28466
$120$	0	0
$121$	−4.79145	−0.435586
$122$	−1.41615	−0.128213
$123$	0	0
$124$	−12.9049	−1.15889
$125$	14.8039	1.32410
$126$	0	0
$127$	20.7968	1.84542	0.922710	−	0.385496i	$-0.125970\pi$
$127$	20.7968	1.84542	0.922710	+	0.385496i	$0.125970\pi$
$128$	5.33006	0.471115
$129$	0	0
$130$	0.494536	0.0433737
$131$	0.655830	0.0573001	0.0286501	−	0.999590i	$-0.490879\pi$
$131$	0.655830	0.0573001	0.0286501	+	0.999590i	$0.490879\pi$
$132$	0	0
$133$	−1.85480	−0.160832
$134$	0.209705	0.0181158
$135$	0	0
$136$	−3.17460	−0.272219
$137$	−8.58760	−0.733689	−0.366844	−	0.930282i	$-0.619562\pi$
$137$	−8.58760	−0.733689	−0.366844	+	0.930282i	$0.619562\pi$
$138$	0	0
$139$	13.4461	1.14049	0.570243	−	0.821476i	$-0.306849\pi$
$139$	13.4461	1.14049	0.570243	+	0.821476i	$0.306849\pi$
$140$	−22.3158	−1.88603
$141$	0	0
$142$	−0.850949	−0.0714100
$143$	−1.90649	−0.159429
$144$	0	0
$145$	24.4778	2.03277
$146$	−0.743701	−0.0615492
$147$	0	0
$148$	−9.77098	−0.803170
$149$	−9.62207	−0.788270	−0.394135	−	0.919052i	$-0.628956\pi$
$149$	−9.62207	−0.788270	−0.394135	+	0.919052i	$0.628956\pi$
$150$	0	0
$151$	−7.12820	−0.580085	−0.290042	−	0.957014i	$-0.593669\pi$
$151$	−7.12820	−0.580085	−0.290042	+	0.957014i	$0.593669\pi$
$152$	−0.420170	−0.0340803
$153$	0	0
$154$	−1.30659	−0.105288
$155$	24.4761	1.96597
$156$	0	0
$157$	7.68577	0.613391	0.306696	−	0.951808i	$-0.400777\pi$
$157$	7.68577	0.613391	0.306696	+	0.951808i	$0.400777\pi$
$158$	2.04018	0.162308
$159$	0	0
$160$	−7.60189	−0.600982
$161$	−19.7786	−1.55877
$162$	0	0
$163$	1.04750	0.0820465	0.0410232	−	0.999158i	$-0.486938\pi$
$163$	1.04750	0.0820465	0.0410232	+	0.999158i	$0.486938\pi$
$164$	10.3631	0.809221
$165$	0	0
$166$	1.55956	0.121046
$167$	−8.35408	−0.646458	−0.323229	−	0.946321i	$-0.604768\pi$
$167$	−8.35408	−0.646458	−0.323229	+	0.946321i	$0.604768\pi$
$168$	0	0
$169$	−12.4146	−0.954966
$170$	2.98787	0.229159
$171$	0	0
$172$	−10.9745	−0.836796
$173$	−21.8458	−1.66090	−0.830452	−	0.557090i	$-0.811918\pi$
$173$	−21.8458	−1.66090	−0.830452	+	0.557090i	$0.811918\pi$
$174$	0	0
$175$	27.1680	2.05371
$176$	9.52170	0.717725
$177$	0	0
$178$	1.30404	0.0977419
$179$	9.08866	0.679319	0.339659	−	0.940549i	$-0.389688\pi$
$179$	9.08866	0.679319	0.339659	+	0.940549i	$0.389688\pi$
$180$	0	0
$181$	−7.13077	−0.530026	−0.265013	−	0.964245i	$-0.585376\pi$
$181$	−7.13077	−0.530026	−0.265013	+	0.964245i	$0.585376\pi$
$182$	0.401221	0.0297404
$183$	0	0
$184$	−4.48047	−0.330305
$185$	18.5322	1.36251
$186$	0	0
$187$	−11.5186	−0.842321
$188$	2.18209	0.159145
$189$	0	0
$190$	0.395456	0.0286894
$191$	−11.9556	−0.865079	−0.432539	−	0.901615i	$-0.642382\pi$
$191$	−11.9556	−0.865079	−0.432539	+	0.901615i	$0.642382\pi$
$192$	0	0
$193$	−8.87364	−0.638738	−0.319369	−	0.947630i	$-0.603471\pi$
$193$	−8.87364	−0.638738	−0.319369	+	0.947630i	$0.603471\pi$
$194$	0.164042	0.0117776
$195$	0	0
$196$	−4.31443	−0.308174
$197$	−7.39790	−0.527079	−0.263539	−	0.964649i	$-0.584890\pi$
$197$	−7.39790	−0.527079	−0.263539	+	0.964649i	$0.584890\pi$
$198$	0	0
$199$	−10.3837	−0.736084	−0.368042	−	0.929809i	$-0.619972\pi$
$199$	−10.3837	−0.736084	−0.368042	+	0.929809i	$0.619972\pi$
$200$	6.15439	0.435181
$201$	0	0
$202$	−0.970076	−0.0682543
$203$	19.8590	1.39383
$204$	0	0
$205$	−19.6552	−1.37278
$206$	1.63030	0.113588
$207$	0	0
$208$	−2.92388	−0.202735
$209$	−1.52453	−0.105454
$210$	0	0
$211$	20.8611	1.43614	0.718070	−	0.695971i	$-0.245026\pi$
$211$	20.8611	1.43614	0.718070	+	0.695971i	$0.245026\pi$
$212$	17.4216	1.19652
$213$	0	0
$214$	−0.221106	−0.0151145
$215$	20.8148	1.41956
$216$	0	0
$217$	19.8576	1.34802
$218$	1.28121	0.0867747
$219$	0	0
$220$	−18.3422	−1.23663
$221$	3.53707	0.237929
$222$	0	0
$223$	23.5785	1.57893	0.789466	−	0.613794i	$-0.210357\pi$
$223$	23.5785	1.57893	0.789466	+	0.613794i	$0.210357\pi$
$224$	−6.16747	−0.412081
$225$	0	0
$226$	1.61766	0.107605
$227$	10.4841	0.695856	0.347928	−	0.937521i	$-0.386885\pi$
$227$	10.4841	0.695856	0.347928	+	0.937521i	$0.386885\pi$
$228$	0	0
$229$	−13.8824	−0.917376	−0.458688	−	0.888597i	$-0.651680\pi$
$229$	−13.8824	−0.917376	−0.458688	+	0.888597i	$0.651680\pi$
$230$	4.21694	0.278056
$231$	0	0
$232$	4.49868	0.295353
$233$	−7.59964	−0.497869	−0.248935	−	0.968520i	$-0.580080\pi$
$233$	−7.59964	−0.497869	−0.248935	+	0.968520i	$0.580080\pi$
$234$	0	0
$235$	−4.13868	−0.269977
$236$	−23.3489	−1.51988
$237$	0	0
$238$	2.42408	0.157130
$239$	−16.5587	−1.07109	−0.535546	−	0.844506i	$-0.679894\pi$
$239$	−16.5587	−1.07109	−0.535546	+	0.844506i	$0.679894\pi$
$240$	0	0
$241$	−14.5185	−0.935218	−0.467609	−	0.883935i	$-0.654884\pi$
$241$	−14.5185	−0.935218	−0.467609	+	0.883935i	$0.654884\pi$
$242$	0.828806	0.0532776
$243$	0	0
$244$	−16.1290	−1.03256
$245$	8.18299	0.522792
$246$	0	0
$247$	0.468145	0.0297874
$248$	4.49836	0.285646
$249$	0	0
$250$	−2.56072	−0.161954
$251$	−9.05181	−0.571345	−0.285673	−	0.958327i	$-0.592217\pi$
$251$	−9.05181	−0.571345	−0.285673	+	0.958327i	$0.592217\pi$
$252$	0	0
$253$	−16.2568	−1.02205
$254$	−3.59735	−0.225718
$255$	0	0
$256$	13.6597	0.853730
$257$	−9.69988	−0.605062	−0.302531	−	0.953140i	$-0.597832\pi$
$257$	−9.69988	−0.605062	−0.302531	+	0.953140i	$0.597832\pi$
$258$	0	0
$259$	15.0353	0.934247
$260$	5.63243	0.349309
$261$	0	0
$262$	−0.113443	−0.00700852
$263$	26.8552	1.65597	0.827983	−	0.560754i	$-0.189489\pi$
$263$	26.8552	1.65597	0.827983	+	0.560754i	$0.189489\pi$
$264$	0	0
$265$	−33.0428	−2.02980
$266$	0.320836	0.0196717
$267$	0	0
$268$	2.38840	0.145895
$269$	−11.7388	−0.715729	−0.357865	−	0.933774i	$-0.616495\pi$
$269$	−11.7388	−0.715729	−0.357865	+	0.933774i	$0.616495\pi$
$270$	0	0
$271$	0.144576	0.00878238	0.00439119	−	0.999990i	$-0.498602\pi$
$271$	0.144576	0.00878238	0.00439119	+	0.999990i	$0.498602\pi$
$272$	−17.6654	−1.07112
$273$	0	0
$274$	1.48545	0.0897392
$275$	22.3303	1.34657
$276$	0	0
$277$	−1.01442	−0.0609509	−0.0304754	−	0.999536i	$-0.509702\pi$
$277$	−1.01442	−0.0609509	−0.0304754	+	0.999536i	$0.509702\pi$
$278$	−2.32586	−0.139496
$279$	0	0
$280$	7.77883	0.464874
$281$	27.5295	1.64227	0.821135	−	0.570734i	$-0.193341\pi$
$281$	27.5295	1.64227	0.821135	+	0.570734i	$0.193341\pi$
$282$	0	0
$283$	−26.8029	−1.59327	−0.796633	−	0.604463i	$-0.793388\pi$
$283$	−26.8029	−1.59327	−0.796633	+	0.604463i	$0.793388\pi$
$284$	−9.69173	−0.575098
$285$	0	0
$286$	0.329777	0.0195001
$287$	−15.9464	−0.941286
$288$	0	0
$289$	4.37012	0.257066
$290$	−4.23407	−0.248633
$291$	0	0
$292$	−8.47026	−0.495684
$293$	−18.6573	−1.08997	−0.544984	−	0.838446i	$-0.683464\pi$
$293$	−18.6573	−1.08997	−0.544984	+	0.838446i	$0.683464\pi$
$294$	0	0
$295$	44.2848	2.57836
$296$	3.40596	0.197967
$297$	0	0
$298$	1.66439	0.0964153
$299$	4.99205	0.288698
$300$	0	0
$301$	16.8872	0.973361
$302$	1.23301	0.0709516
$303$	0	0
$304$	−2.33808	−0.134098
$305$	30.5913	1.75165
$306$	0	0
$307$	33.7893	1.92845	0.964227	−	0.265077i	$-0.0853973\pi$
$307$	33.7893	1.92845	0.964227	+	0.265077i	$0.0853973\pi$
$308$	−14.8811	−0.847931
$309$	0	0
$310$	−4.23377	−0.240462
$311$	−34.6866	−1.96690	−0.983448	−	0.181193i	$-0.942004\pi$
$311$	−34.6866	−1.96690	−0.983448	+	0.181193i	$0.942004\pi$
$312$	0	0
$313$	3.34038	0.188809	0.0944047	−	0.995534i	$-0.469905\pi$
$313$	3.34038	0.188809	0.0944047	+	0.995534i	$0.469905\pi$
$314$	−1.32945	−0.0750254
$315$	0	0
$316$	23.2363	1.30714
$317$	−31.0328	−1.74298	−0.871488	−	0.490417i	$-0.836844\pi$
$317$	−31.0328	−1.74298	−0.871488	+	0.490417i	$0.836844\pi$
$318$	0	0
$319$	16.3228	0.913903
$320$	−27.2426	−1.52291
$321$	0	0
$322$	3.42123	0.190658
$323$	2.82842	0.157378
$324$	0	0
$325$	−6.85710	−0.380363
$326$	−0.181192	−0.0100353
$327$	0	0
$328$	−3.61235	−0.199459
$329$	−3.35773	−0.185118
$330$	0	0
$331$	3.27168	0.179828	0.0899138	−	0.995950i	$-0.471341\pi$
$331$	3.27168	0.179828	0.0899138	+	0.995950i	$0.471341\pi$
$332$	17.7624	0.974837
$333$	0	0
$334$	1.44505	0.0790699
$335$	−4.52998	−0.247499
$336$	0	0
$337$	6.36581	0.346768	0.173384	−	0.984854i	$-0.444530\pi$
$337$	6.36581	0.346768	0.173384	+	0.984854i	$0.444530\pi$
$338$	2.14742	0.116804
$339$	0	0
$340$	34.0298	1.84553
$341$	16.3217	0.883868
$342$	0	0
$343$	−14.5816	−0.787331
$344$	3.82547	0.206256
$345$	0	0
$346$	3.77880	0.203149
$347$	−8.79241	−0.472001	−0.236001	−	0.971753i	$-0.575837\pi$
$347$	−8.79241	−0.472001	−0.236001	+	0.971753i	$0.575837\pi$
$348$	0	0
$349$	14.4002	0.770826	0.385413	−	0.922744i	$-0.374059\pi$
$349$	14.4002	0.770826	0.385413	+	0.922744i	$0.374059\pi$
$350$	−4.69941	−0.251194
$351$	0	0
$352$	−5.06926	−0.270192
$353$	33.2005	1.76708	0.883542	−	0.468353i	$-0.155152\pi$
$353$	33.2005	1.76708	0.883542	+	0.468353i	$0.155152\pi$
$354$	0	0
$355$	18.3819	0.975609
$356$	14.8521	0.787162
$357$	0	0
$358$	−1.57212	−0.0830891
$359$	4.94514	0.260995	0.130497	−	0.991449i	$-0.458343\pi$
$359$	4.94514	0.260995	0.130497	+	0.991449i	$0.458343\pi$
$360$	0	0
$361$	−18.6256	−0.980297
$362$	1.23345	0.0648288
$363$	0	0
$364$	4.56963	0.239514
$365$	16.0652	0.840889
$366$	0	0
$367$	−2.49245	−0.130105	−0.0650525	−	0.997882i	$-0.520721\pi$
$367$	−2.49245	−0.130105	−0.0650525	+	0.997882i	$0.520721\pi$
$368$	−24.9321	−1.29967
$369$	0	0
$370$	−3.20562	−0.166652
$371$	−26.8078	−1.39179
$372$	0	0
$373$	−28.0476	−1.45225	−0.726124	−	0.687563i	$-0.758680\pi$
$373$	−28.0476	−1.45225	−0.726124	+	0.687563i	$0.758680\pi$
$374$	1.99244	0.103026
$375$	0	0
$376$	−0.760631	−0.0392265
$377$	−5.01234	−0.258149
$378$	0	0
$379$	5.13991	0.264019	0.132010	−	0.991248i	$-0.457857\pi$
$379$	5.13991	0.264019	0.132010	+	0.991248i	$0.457857\pi$
$380$	4.50398	0.231049
$381$	0	0
$382$	2.06804	0.105810
$383$	−0.0446729	−0.00228268	−0.00114134	−	0.999999i	$-0.500363\pi$
$383$	−0.0446729	−0.00228268	−0.00114134	+	0.999999i	$0.500363\pi$
$384$	0	0
$385$	28.2244	1.43845
$386$	1.53493	0.0781256
$387$	0	0
$388$	1.86833	0.0948502
$389$	20.9823	1.06384	0.531921	−	0.846794i	$-0.321470\pi$
$389$	20.9823	1.06384	0.531921	+	0.846794i	$0.321470\pi$
$390$	0	0
$391$	30.1608	1.52530
$392$	1.50392	0.0759594
$393$	0	0
$394$	1.27966	0.0644683
$395$	−44.0713	−2.21747
$396$	0	0
$397$	−0.00245641	−0.000123284 0	−6.16419e−5	−	1.00000i	$-0.500020\pi$
$397$	−0.00245641	−0.000123284 0	−6.16419e−5		1.00000i	$0.500020\pi$
$398$	1.79614	0.0900323
$399$	0	0
$400$	34.2467	1.71234
$401$	25.2563	1.26124	0.630620	−	0.776091i	$-0.282801\pi$
$401$	25.2563	1.26124	0.630620	+	0.776091i	$0.282801\pi$
$402$	0	0
$403$	−5.01198	−0.249665
$404$	−11.0485	−0.549684
$405$	0	0
$406$	−3.43513	−0.170483
$407$	12.3580	0.612565
$408$	0	0
$409$	−23.2885	−1.15154	−0.575772	−	0.817610i	$-0.695298\pi$
$409$	−23.2885	−1.15154	−0.575772	+	0.817610i	$0.695298\pi$
$410$	3.39988	0.167908
$411$	0	0
$412$	18.5680	0.914781
$413$	35.9286	1.76793
$414$	0	0
$415$	−33.6891	−1.65373
$416$	1.55665	0.0763208
$417$	0	0
$418$	0.263707	0.0128983
$419$	31.2884	1.52854	0.764268	−	0.644898i	$-0.223100\pi$
$419$	31.2884	1.52854	0.764268	+	0.644898i	$0.223100\pi$
$420$	0	0
$421$	30.5296	1.48792	0.743960	−	0.668224i	$-0.232945\pi$
$421$	30.5296	1.48792	0.743960	+	0.668224i	$0.232945\pi$
$422$	−3.60848	−0.175658
$423$	0	0
$424$	−6.07280	−0.294921
$425$	−41.4289	−2.00960
$426$	0	0
$427$	24.8189	1.20107
$428$	−2.51825	−0.121724
$429$	0	0
$430$	−3.60046	−0.173630
$431$	12.4246	0.598474	0.299237	−	0.954179i	$-0.403268\pi$
$431$	12.4246	0.598474	0.299237	+	0.954179i	$0.403268\pi$
$432$	0	0
$433$	−0.760649	−0.0365545	−0.0182772	−	0.999833i	$-0.505818\pi$
$433$	−0.760649	−0.0365545	−0.0182772	+	0.999833i	$0.505818\pi$
$434$	−3.43489	−0.164880
$435$	0	0
$436$	14.5922	0.698838
$437$	3.99190	0.190958
$438$	0	0
$439$	30.1094	1.43704	0.718521	−	0.695506i	$-0.244820\pi$
$439$	30.1094	1.43704	0.718521	+	0.695506i	$0.244820\pi$
$440$	6.39370	0.304808
$441$	0	0
$442$	−0.611828	−0.0291017
$443$	−13.6616	−0.649081	−0.324541	−	0.945872i	$-0.605210\pi$
$443$	−13.6616	−0.649081	−0.324541	+	0.945872i	$0.605210\pi$
$444$	0	0
$445$	−28.1694	−1.33536
$446$	−4.07851	−0.193123
$447$	0	0
$448$	−22.1021	−1.04423
$449$	21.9989	1.03819	0.519097	−	0.854715i	$-0.326268\pi$
$449$	21.9989	1.03819	0.519097	+	0.854715i	$0.326268\pi$
$450$	0	0
$451$	−13.1069	−0.617181
$452$	18.4241	0.866597
$453$	0	0
$454$	−1.81350	−0.0851119
$455$	−8.66702	−0.406316
$456$	0	0
$457$	−1.48883	−0.0696444	−0.0348222	−	0.999394i	$-0.511086\pi$
$457$	−1.48883	−0.0696444	−0.0348222	+	0.999394i	$0.511086\pi$
$458$	2.40132	0.112206
$459$	0	0
$460$	48.0281	2.23932
$461$	7.11334	0.331301	0.165651	−	0.986185i	$-0.447028\pi$
$461$	7.11334	0.331301	0.165651	+	0.986185i	$0.447028\pi$
$462$	0	0
$463$	−26.5407	−1.23345	−0.616726	−	0.787178i	$-0.711541\pi$
$463$	−26.5407	−1.23345	−0.616726	+	0.787178i	$0.711541\pi$
$464$	25.0334	1.16215
$465$	0	0
$466$	1.31456	0.0608956
$467$	−26.1519	−1.21017	−0.605084	−	0.796162i	$-0.706860\pi$
$467$	−26.1519	−1.21017	−0.605084	+	0.796162i	$0.706860\pi$
$468$	0	0
$469$	−3.67520	−0.169705
$470$	0.715891	0.0330216
$471$	0	0
$472$	8.13894	0.374625
$473$	13.8802	0.638211
$474$	0	0
$475$	−5.48328	−0.251590
$476$	27.6086	1.26544
$477$	0	0
$478$	2.86425	0.131008
$479$	10.4065	0.475487	0.237744	−	0.971328i	$-0.423592\pi$
$479$	10.4065	0.475487	0.237744	+	0.971328i	$0.423592\pi$
$480$	0	0
$481$	−3.79485	−0.173030
$482$	2.51135	0.114389
$483$	0	0
$484$	9.43954	0.429070
$485$	−3.54358	−0.160906
$486$	0	0
$487$	−18.4664	−0.836791	−0.418396	−	0.908265i	$-0.637407\pi$
$487$	−18.4664	−0.836791	−0.418396	+	0.908265i	$0.637407\pi$
$488$	5.62225	0.254507
$489$	0	0
$490$	−1.41546	−0.0639440
$491$	16.9739	0.766021	0.383011	−	0.923744i	$-0.374887\pi$
$491$	16.9739	0.766021	0.383011	+	0.923744i	$0.374887\pi$
$492$	0	0
$493$	−30.2834	−1.36389
$494$	−0.0809779	−0.00364337
$495$	0	0
$496$	25.0316	1.12395
$497$	14.9133	0.668955
$498$	0	0
$499$	24.6462	1.10331	0.551657	−	0.834071i	$-0.313996\pi$
$499$	24.6462	1.10331	0.551657	+	0.834071i	$0.313996\pi$
$500$	−29.1648	−1.30429
$501$	0	0
$502$	1.56575	0.0698826
$503$	−40.1137	−1.78858	−0.894291	−	0.447485i	$-0.852320\pi$
$503$	−40.1137	−1.78858	−0.894291	+	0.447485i	$0.852320\pi$
$504$	0	0
$505$	20.9552	0.932495
$506$	2.81203	0.125010
$507$	0	0
$508$	−40.9714	−1.81781
$509$	−4.94852	−0.219339	−0.109670	−	0.993968i	$-0.534979\pi$
$509$	−4.94852	−0.219339	−0.109670	+	0.993968i	$0.534979\pi$
$510$	0	0
$511$	13.0338	0.576580
$512$	−13.0229	−0.575537
$513$	0	0
$514$	1.67785	0.0740066
$515$	−35.2172	−1.55185
$516$	0	0
$517$	−2.75984	−0.121378
$518$	−2.60074	−0.114270
$519$	0	0
$520$	−1.96335	−0.0860985
$521$	−7.73958	−0.339077	−0.169539	−	0.985524i	$-0.554228\pi$
$521$	−7.73958	−0.339077	−0.169539	+	0.985524i	$0.554228\pi$
$522$	0	0
$523$	36.0140	1.57478	0.787391	−	0.616453i	$-0.211431\pi$
$523$	36.0140	1.57478	0.787391	+	0.616453i	$0.211431\pi$
$524$	−1.29204	−0.0564429
$525$	0	0
$526$	−4.64531	−0.202545
$527$	−30.2812	−1.31907
$528$	0	0
$529$	19.5675	0.850760
$530$	5.71561	0.248270
$531$	0	0
$532$	3.65411	0.158426
$533$	4.02481	0.174334
$534$	0	0
$535$	4.77625	0.206495
$536$	−0.832547	−0.0359605
$537$	0	0
$538$	2.03053	0.0875426
$539$	5.45676	0.235039
$540$	0	0
$541$	−24.4147	−1.04967	−0.524834	−	0.851204i	$-0.675873\pi$
$541$	−24.4147	−1.04967	−0.524834	+	0.851204i	$0.675873\pi$
$542$	−0.0250082	−0.00107419
$543$	0	0
$544$	9.40487	0.403231
$545$	−27.6763	−1.18552
$546$	0	0
$547$	−28.3618	−1.21266	−0.606331	−	0.795212i	$-0.707359\pi$
$547$	−28.3618	−1.21266	−0.606331	+	0.795212i	$0.707359\pi$
$548$	16.9183	0.722712
$549$	0	0
$550$	−3.86261	−0.164702
$551$	−4.00812	−0.170752
$552$	0	0
$553$	−35.7553	−1.52047
$554$	0.175471	0.00745505
$555$	0	0
$556$	−26.4900	−1.12342
$557$	36.9373	1.56508	0.782542	−	0.622598i	$-0.213923\pi$
$557$	36.9373	1.56508	0.782542	+	0.622598i	$0.213923\pi$
$558$	0	0
$559$	−4.26226	−0.180274
$560$	43.2861	1.82917
$561$	0	0
$562$	−4.76193	−0.200870
$563$	−22.7754	−0.959869	−0.479935	−	0.877304i	$-0.659340\pi$
$563$	−22.7754	−0.959869	−0.479935	+	0.877304i	$0.659340\pi$
$564$	0	0
$565$	−34.9442	−1.47011
$566$	4.63625	0.194876
$567$	0	0
$568$	3.37833	0.141752
$569$	−30.8688	−1.29409	−0.647043	−	0.762454i	$-0.723995\pi$
$569$	−30.8688	−1.29409	−0.647043	+	0.762454i	$0.723995\pi$
$570$	0	0
$571$	12.8205	0.536521	0.268260	−	0.963346i	$-0.413551\pi$
$571$	12.8205	0.536521	0.268260	+	0.963346i	$0.413551\pi$
$572$	3.75594	0.157044
$573$	0	0
$574$	2.75834	0.115131
$575$	−58.4708	−2.43840
$576$	0	0
$577$	−23.5264	−0.979417	−0.489708	−	0.871886i	$-0.662897\pi$
$577$	−23.5264	−0.979417	−0.489708	+	0.871886i	$0.662897\pi$
$578$	−0.755926	−0.0314424
$579$	0	0
$580$	−48.2232	−2.00236
$581$	−27.3322	−1.13393
$582$	0	0
$583$	−22.0343	−0.912568
$584$	2.95255	0.122178
$585$	0	0
$586$	3.22726	0.133317
$587$	−11.3874	−0.470010	−0.235005	−	0.971994i	$-0.575511\pi$
$587$	−11.3874	−0.470010	−0.235005	+	0.971994i	$0.575511\pi$
$588$	0	0
$589$	−4.00784	−0.165140
$590$	−7.66021	−0.315366
$591$	0	0
$592$	18.9528	0.778956
$593$	37.7324	1.54948	0.774742	−	0.632277i	$-0.217880\pi$
$593$	37.7324	1.54948	0.774742	+	0.632277i	$0.217880\pi$
$594$	0	0
$595$	−52.3640	−2.14672
$596$	18.9562	0.776478
$597$	0	0
$598$	−0.863505	−0.0353113
$599$	−47.3582	−1.93500	−0.967502	−	0.252865i	$-0.918627\pi$
$599$	−47.3582	−1.93500	−0.967502	+	0.252865i	$0.918627\pi$
$600$	0	0
$601$	−31.1074	−1.26890	−0.634449	−	0.772964i	$-0.718773\pi$
$601$	−31.1074	−1.26890	−0.634449	+	0.772964i	$0.718773\pi$
$602$	−2.92108	−0.119054
$603$	0	0
$604$	14.0431	0.571407
$605$	−17.9036	−0.727883
$606$	0	0
$607$	29.4864	1.19682	0.598409	−	0.801191i	$-0.295800\pi$
$607$	29.4864	1.19682	0.598409	+	0.801191i	$0.295800\pi$
$608$	1.24477	0.0504822
$609$	0	0
$610$	−5.29155	−0.214249
$611$	0.847480	0.0342854
$612$	0	0
$613$	−6.10428	−0.246550	−0.123275	−	0.992373i	$-0.539340\pi$
$613$	−6.10428	−0.246550	−0.123275	+	0.992373i	$0.539340\pi$
$614$	−5.84473	−0.235874
$615$	0	0
$616$	5.18725	0.209000
$617$	19.1201	0.769747	0.384873	−	0.922969i	$-0.374245\pi$
$617$	19.1201	0.769747	0.384873	+	0.922969i	$0.374245\pi$
$618$	0	0
$619$	−6.75385	−0.271460	−0.135730	−	0.990746i	$-0.543338\pi$
$619$	−6.75385	−0.271460	−0.135730	+	0.990746i	$0.543338\pi$
$620$	−48.2198	−1.93655
$621$	0	0
$622$	5.99994	0.240576
$623$	−22.8540	−0.915627
$624$	0	0
$625$	10.5061	0.420246
$626$	−0.577805	−0.0230937
$627$	0	0
$628$	−15.1416	−0.604215
$629$	−22.9276	−0.914182
$630$	0	0
$631$	−0.456907	−0.0181892	−0.00909458	−	0.999959i	$-0.502895\pi$
$631$	−0.456907	−0.0181892	−0.00909458	+	0.999959i	$0.502895\pi$
$632$	−8.09969	−0.322188
$633$	0	0
$634$	5.36793	0.213188
$635$	77.7086	3.08377
$636$	0	0
$637$	−1.67564	−0.0663912
$638$	−2.82346	−0.111782
$639$	0	0
$640$	19.9161	0.787253
$641$	−2.87103	−0.113399	−0.0566994	−	0.998391i	$-0.518058\pi$
$641$	−2.87103	−0.113399	−0.0566994	+	0.998391i	$0.518058\pi$
$642$	0	0
$643$	−1.70284	−0.0671536	−0.0335768	−	0.999436i	$-0.510690\pi$
$643$	−1.70284	−0.0671536	−0.0335768	+	0.999436i	$0.510690\pi$
$644$	38.9655	1.53545
$645$	0	0
$646$	−0.489249	−0.0192492
$647$	36.1004	1.41925	0.709626	−	0.704579i	$-0.248864\pi$
$647$	36.1004	1.41925	0.709626	+	0.704579i	$0.248864\pi$
$648$	0	0
$649$	29.5310	1.15919
$650$	1.18611	0.0465232
$651$	0	0
$652$	−2.06366	−0.0808191
$653$	−43.5680	−1.70495	−0.852473	−	0.522771i	$-0.824898\pi$
$653$	−43.5680	−1.70495	−0.852473	+	0.522771i	$0.824898\pi$
$654$	0	0
$655$	2.45055	0.0957509
$656$	−20.1013	−0.784825
$657$	0	0
$658$	0.580807	0.0226422
$659$	25.4810	0.992598	0.496299	−	0.868152i	$-0.334692\pi$
$659$	25.4810	0.992598	0.496299	+	0.868152i	$0.334692\pi$
$660$	0	0
$661$	34.1672	1.32895	0.664475	−	0.747310i	$-0.268655\pi$
$661$	34.1672	1.32895	0.664475	+	0.747310i	$0.268655\pi$
$662$	−0.565921	−0.0219952
$663$	0	0
$664$	−6.19159	−0.240280
$665$	−6.93059	−0.268757
$666$	0	0
$667$	−42.7405	−1.65492
$668$	16.4582	0.636787
$669$	0	0
$670$	0.783577	0.0302722
$671$	20.3995	0.787515
$672$	0	0
$673$	−29.5437	−1.13883	−0.569413	−	0.822051i	$-0.692830\pi$
$673$	−29.5437	−1.13883	−0.569413	+	0.822051i	$0.692830\pi$
$674$	−1.10113	−0.0424140
$675$	0	0
$676$	24.4577	0.940680
$677$	40.7802	1.56731	0.783656	−	0.621195i	$-0.213353\pi$
$677$	40.7802	1.56731	0.783656	+	0.621195i	$0.213353\pi$
$678$	0	0
$679$	−2.87493	−0.110330
$680$	−11.8621	−0.454890
$681$	0	0
$682$	−2.82326	−0.108108
$683$	31.6426	1.21077	0.605384	−	0.795933i	$-0.293019\pi$
$683$	31.6426	1.21077	0.605384	+	0.795933i	$0.293019\pi$
$684$	0	0
$685$	−32.0881	−1.22602
$686$	2.52226	0.0963004
$687$	0	0
$688$	21.2872	0.811568
$689$	6.76620	0.257772
$690$	0	0
$691$	−28.5848	−1.08742	−0.543708	−	0.839275i	$-0.682980\pi$
$691$	−28.5848	−1.08742	−0.543708	+	0.839275i	$0.682980\pi$
$692$	43.0379	1.63606
$693$	0	0
$694$	1.52088	0.0577316
$695$	50.2423	1.90580
$696$	0	0
$697$	24.3169	0.921070
$698$	−2.49089	−0.0942817
$699$	0	0
$700$	−53.5231	−2.02298
$701$	−7.52982	−0.284397	−0.142199	−	0.989838i	$-0.545417\pi$
$701$	−7.52982	−0.284397	−0.142199	+	0.989838i	$0.545417\pi$
$702$	0	0
$703$	−3.03455	−0.114450
$704$	−18.1665	−0.684677
$705$	0	0
$706$	−5.74288	−0.216136
$707$	17.0011	0.639392
$708$	0	0
$709$	8.01399	0.300972	0.150486	−	0.988612i	$-0.451916\pi$
$709$	8.01399	0.300972	0.150486	+	0.988612i	$0.451916\pi$
$710$	−3.17962	−0.119329
$711$	0	0
$712$	−5.17714	−0.194022
$713$	−42.7374	−1.60053
$714$	0	0
$715$	−7.12373	−0.266412
$716$	−17.9054	−0.669156
$717$	0	0
$718$	−0.855391	−0.0319229
$719$	26.9826	1.00628	0.503140	−	0.864205i	$-0.332178\pi$
$719$	26.9826	1.00628	0.503140	+	0.864205i	$0.332178\pi$
$720$	0	0
$721$	−28.5719	−1.06407
$722$	3.22179	0.119903
$723$	0	0
$724$	14.0482	0.522097
$725$	58.7084	2.18038
$726$	0	0
$727$	−14.6943	−0.544983	−0.272491	−	0.962158i	$-0.587848\pi$
$727$	−14.6943	−0.544983	−0.272491	+	0.962158i	$0.587848\pi$
$728$	−1.59288	−0.0590360
$729$	0	0
$730$	−2.77889	−0.102851
$731$	−25.7516	−0.952456
$732$	0	0
$733$	−31.3438	−1.15771	−0.578854	−	0.815431i	$-0.696500\pi$
$733$	−31.3438	−1.15771	−0.578854	+	0.815431i	$0.696500\pi$
$734$	0.431134	0.0159135
$735$	0	0
$736$	13.2736	0.489271
$737$	−3.02078	−0.111272
$738$	0	0
$739$	0.482909	0.0177641	0.00888205	−	0.999961i	$-0.497173\pi$
$739$	0.482909	0.0177641	0.00888205	+	0.999961i	$0.497173\pi$
$740$	−36.5099	−1.34213
$741$	0	0
$742$	4.63711	0.170234
$743$	43.0507	1.57938	0.789689	−	0.613507i	$-0.210242\pi$
$743$	43.0507	1.57938	0.789689	+	0.613507i	$0.210242\pi$
$744$	0	0
$745$	−35.9535	−1.31723
$746$	4.85156	0.177628
$747$	0	0
$748$	22.6925	0.829720
$749$	3.87500	0.141589
$750$	0	0
$751$	43.9216	1.60272	0.801361	−	0.598181i	$-0.204109\pi$
$751$	43.9216	1.60272	0.801361	+	0.598181i	$0.204109\pi$
$752$	−4.23261	−0.154347
$753$	0	0
$754$	0.867014	0.0315748
$755$	−26.6350	−0.969346
$756$	0	0
$757$	22.4143	0.814661	0.407331	−	0.913281i	$-0.366460\pi$
$757$	22.4143	0.814661	0.407331	+	0.913281i	$0.366460\pi$
$758$	−0.889081	−0.0322929
$759$	0	0
$760$	−1.56999	−0.0569496
$761$	−9.99674	−0.362382	−0.181191	−	0.983448i	$-0.557995\pi$
$761$	−9.99674	−0.362382	−0.181191	+	0.983448i	$0.557995\pi$
$762$	0	0
$763$	−22.4540	−0.812888
$764$	23.5535	0.852137
$765$	0	0
$766$	0.00772733	0.000279200 0
$767$	−9.06824	−0.327435
$768$	0	0
$769$	−7.49619	−0.270320	−0.135160	−	0.990824i	$-0.543155\pi$
$769$	−7.49619	−0.270320	−0.135160	+	0.990824i	$0.543155\pi$
$770$	−4.88214	−0.175940
$771$	0	0
$772$	17.4818	0.629183
$773$	−19.8391	−0.713562	−0.356781	−	0.934188i	$-0.616126\pi$
$773$	−19.8391	−0.713562	−0.356781	+	0.934188i	$0.616126\pi$
$774$	0	0
$775$	58.7043	2.10872
$776$	−0.651262	−0.0233789
$777$	0	0
$778$	−3.62943	−0.130121
$779$	3.21845	0.115313
$780$	0	0
$781$	12.2578	0.438619
$782$	−5.21709	−0.186563
$783$	0	0
$784$	8.36872	0.298883
$785$	28.7184	1.02500
$786$	0	0
$787$	39.7283	1.41616	0.708080	−	0.706133i	$-0.249562\pi$
$787$	39.7283	1.41616	0.708080	+	0.706133i	$0.249562\pi$
$788$	14.5745	0.519193
$789$	0	0
$790$	7.62327	0.271224
$791$	−28.3505	−1.00803
$792$	0	0
$793$	−6.26419	−0.222448
$794$	0.000424900 0	1.50791e−5 0
$795$	0	0
$796$	20.4568	0.725072
$797$	−9.10595	−0.322549	−0.161275	−	0.986910i	$-0.551561\pi$
$797$	−9.10595	−0.322549	−0.161275	+	0.986910i	$0.551561\pi$
$798$	0	0
$799$	5.12027	0.181142
$800$	−18.2326	−0.644621
$801$	0	0
$802$	−4.36874	−0.154265
$803$	10.7129	0.378051
$804$	0	0
$805$	−73.9041	−2.60478
$806$	0.866953	0.0305371
$807$	0	0
$808$	3.85128	0.135487
$809$	3.01910	0.106146	0.0530730	−	0.998591i	$-0.483098\pi$
$809$	3.01910	0.106146	0.0530730	+	0.998591i	$0.483098\pi$
$810$	0	0
$811$	33.1722	1.16483	0.582416	−	0.812891i	$-0.302107\pi$
$811$	33.1722	1.16483	0.582416	+	0.812891i	$0.302107\pi$
$812$	−39.1238	−1.37298
$813$	0	0
$814$	−2.13764	−0.0749243
$815$	3.91405	0.137103
$816$	0	0
$817$	−3.40832	−0.119242
$818$	4.02836	0.140848
$819$	0	0
$820$	38.7223	1.35224
$821$	42.7620	1.49240	0.746201	−	0.665720i	$-0.231876\pi$
$821$	42.7620	1.49240	0.746201	+	0.665720i	$0.231876\pi$
$822$	0	0
$823$	−35.2289	−1.22800	−0.614000	−	0.789306i	$-0.710441\pi$
$823$	−35.2289	−1.22800	−0.614000	+	0.789306i	$0.710441\pi$
$824$	−6.47243	−0.225478
$825$	0	0
$826$	−6.21478	−0.216240
$827$	26.0380	0.905429	0.452714	−	0.891656i	$-0.350456\pi$
$827$	26.0380	0.905429	0.452714	+	0.891656i	$0.350456\pi$
$828$	0	0
$829$	−7.90268	−0.274471	−0.137236	−	0.990538i	$-0.543822\pi$
$829$	−7.90268	−0.274471	−0.137236	+	0.990538i	$0.543822\pi$
$830$	5.82741	0.202272
$831$	0	0
$832$	5.57850	0.193400
$833$	−10.1238	−0.350769
$834$	0	0
$835$	−31.2155	−1.08026
$836$	3.00344	0.103876
$837$	0	0
$838$	−5.41213	−0.186959
$839$	5.14839	0.177742	0.0888711	−	0.996043i	$-0.471674\pi$
$839$	5.14839	0.177742	0.0888711	+	0.996043i	$0.471674\pi$
$840$	0	0
$841$	13.9142	0.479800
$842$	−5.28088	−0.181991
$843$	0	0
$844$	−41.0981	−1.41466
$845$	−46.3878	−1.59579
$846$	0	0
$847$	−14.5253	−0.499094
$848$	−33.7928	−1.16045
$849$	0	0
$850$	7.16621	0.245799
$851$	−32.3589	−1.10925
$852$	0	0
$853$	25.9905	0.889896	0.444948	−	0.895556i	$-0.353222\pi$
$853$	25.9905	0.889896	0.444948	+	0.895556i	$0.353222\pi$
$854$	−4.29307	−0.146906
$855$	0	0
$856$	0.877808	0.0300028
$857$	−4.11913	−0.140707	−0.0703535	−	0.997522i	$-0.522413\pi$
$857$	−4.11913	−0.140707	−0.0703535	+	0.997522i	$0.522413\pi$
$858$	0	0
$859$	−6.42943	−0.219369	−0.109685	−	0.993966i	$-0.534984\pi$
$859$	−6.42943	−0.219369	−0.109685	+	0.993966i	$0.534984\pi$
$860$	−41.0068	−1.39832
$861$	0	0
$862$	−2.14916	−0.0732008
$863$	29.6195	1.00826	0.504129	−	0.863628i	$-0.331813\pi$
$863$	29.6195	1.00826	0.504129	+	0.863628i	$0.331813\pi$
$864$	0	0
$865$	−81.6282	−2.77544
$866$	0.131574	0.00447107
$867$	0	0
$868$	−39.1210	−1.32785
$869$	−29.3886	−0.996939
$870$	0	0
$871$	0.927607	0.0314308
$872$	−5.08652	−0.172251
$873$	0	0
$874$	−0.690503	−0.0233566
$875$	44.8779	1.51715
$876$	0	0
$877$	−31.8677	−1.07610	−0.538048	−	0.842914i	$-0.680838\pi$
$877$	−31.8677	−1.07610	−0.538048	+	0.842914i	$0.680838\pi$
$878$	−5.20819	−0.175768
$879$	0	0
$880$	35.5784	1.19935
$881$	34.7864	1.17198	0.585991	−	0.810317i	$-0.300705\pi$
$881$	34.7864	1.17198	0.585991	+	0.810317i	$0.300705\pi$
$882$	0	0
$883$	−30.3764	−1.02225	−0.511124	−	0.859507i	$-0.670771\pi$
$883$	−30.3764	−1.02225	−0.511124	+	0.859507i	$0.670771\pi$
$884$	−6.96831	−0.234370
$885$	0	0
$886$	2.36312	0.0793907
$887$	−50.0276	−1.67976	−0.839882	−	0.542769i	$-0.817376\pi$
$887$	−50.0276	−1.67976	−0.839882	+	0.542769i	$0.817376\pi$
$888$	0	0
$889$	63.0455	2.11448
$890$	4.87263	0.163331
$891$	0	0
$892$	−46.4515	−1.55531
$893$	0.677688	0.0226780
$894$	0	0
$895$	33.9604	1.13517
$896$	16.1581	0.539803
$897$	0	0
$898$	−3.80529	−0.126984
$899$	42.9111	1.43117
$900$	0	0
$901$	40.8797	1.36190
$902$	2.26718	0.0754889
$903$	0	0
$904$	−6.42226	−0.213601
$905$	−26.6446	−0.885696
$906$	0	0
$907$	−44.0643	−1.46313	−0.731566	−	0.681771i	$-0.761210\pi$
$907$	−44.0643	−1.46313	−0.731566	+	0.681771i	$0.761210\pi$
$908$	−20.6546	−0.685446
$909$	0	0
$910$	1.49919	0.0496975
$911$	−37.1783	−1.23177	−0.615885	−	0.787836i	$-0.711202\pi$
$911$	−37.1783	−1.23177	−0.615885	+	0.787836i	$0.711202\pi$
$912$	0	0
$913$	−22.4653	−0.743493
$914$	0.257531	0.00851838
$915$	0	0
$916$	27.3495	0.903652
$917$	1.98815	0.0656544
$918$	0	0
$919$	12.3976	0.408958	0.204479	−	0.978871i	$-0.434450\pi$
$919$	12.3976	0.408958	0.204479	+	0.978871i	$0.434450\pi$
$920$	−16.7416	−0.551953
$921$	0	0
$922$	−1.23044	−0.0405223
$923$	−3.76407	−0.123896
$924$	0	0
$925$	44.4482	1.46145
$926$	4.59091	0.150867
$927$	0	0
$928$	−13.3275	−0.437498
$929$	33.3882	1.09543	0.547716	−	0.836664i	$-0.315498\pi$
$929$	33.3882	1.09543	0.547716	+	0.836664i	$0.315498\pi$
$930$	0	0
$931$	−1.33993	−0.0439143
$932$	14.9719	0.490421
$933$	0	0
$934$	4.52366	0.148019
$935$	−43.0399	−1.40755
$936$	0	0
$937$	−24.8441	−0.811620	−0.405810	−	0.913957i	$-0.633011\pi$
$937$	−24.8441	−0.811620	−0.405810	+	0.913957i	$0.633011\pi$
$938$	0.635721	0.0207570
$939$	0	0
$940$	8.15352	0.265938
$941$	25.4299	0.828990	0.414495	−	0.910051i	$-0.363958\pi$
$941$	25.4299	0.828990	0.414495	+	0.910051i	$0.363958\pi$
$942$	0	0
$943$	34.3198	1.11761
$944$	45.2900	1.47406
$945$	0	0
$946$	−2.40094	−0.0780612
$947$	16.6301	0.540407	0.270204	−	0.962803i	$-0.412909\pi$
$947$	16.6301	0.540407	0.270204	+	0.962803i	$0.412909\pi$
$948$	0	0
$949$	−3.28968	−0.106787
$950$	0.948476	0.0307726
$951$	0	0
$952$	−9.62378	−0.311908
$953$	14.2671	0.462158	0.231079	−	0.972935i	$-0.425774\pi$
$953$	14.2671	0.462158	0.231079	+	0.972935i	$0.425774\pi$
$954$	0	0
$955$	−44.6730	−1.44558
$956$	32.6219	1.05507
$957$	0	0
$958$	−1.80008	−0.0581580
$959$	−26.0333	−0.840659
$960$	0	0
$961$	11.9081	0.384131
$962$	0.656418	0.0211638
$963$	0	0
$964$	28.6026	0.921227
$965$	−33.1569	−1.06736
$966$	0	0
$967$	−39.0848	−1.25688	−0.628440	−	0.777858i	$-0.716306\pi$
$967$	−39.0848	−1.25688	−0.628440	+	0.777858i	$0.716306\pi$
$968$	−3.29042	−0.105758
$969$	0	0
$970$	0.612955	0.0196808
$971$	−4.40370	−0.141321	−0.0706607	−	0.997500i	$-0.522511\pi$
$971$	−4.40370	−0.141321	−0.0706607	+	0.997500i	$0.522511\pi$
$972$	0	0
$973$	40.7619	1.30677
$974$	3.19424	0.102350
$975$	0	0
$976$	31.2856	1.00143
$977$	42.1266	1.34775	0.673874	−	0.738846i	$-0.264629\pi$
$977$	42.1266	1.34775	0.673874	+	0.738846i	$0.264629\pi$
$978$	0	0
$979$	−18.7845	−0.600356
$980$	−16.1211	−0.514971
$981$	0	0
$982$	−2.93608	−0.0936939
$983$	−20.4261	−0.651493	−0.325746	−	0.945457i	$-0.605616\pi$
$983$	−20.4261	−0.651493	−0.325746	+	0.945457i	$0.605616\pi$
$984$	0	0
$985$	−27.6427	−0.880770
$986$	5.23829	0.166821
$987$	0	0
$988$	−0.922284	−0.0293418
$989$	−36.3445	−1.15569
$990$	0	0
$991$	0.0680712	0.00216235	0.00108118	−	0.999999i	$-0.499656\pi$
$991$	0.0680712	0.00216235	0.00108118	+	0.999999i	$0.499656\pi$
$992$	−13.3266	−0.423120
$993$	0	0
$994$	−2.57965	−0.0818215
$995$	−38.7995	−1.23003
$996$	0	0
$997$	10.6461	0.337166	0.168583	−	0.985688i	$-0.446081\pi$
$997$	10.6461	0.337166	0.168583	+	0.985688i	$0.446081\pi$
$998$	−4.26319	−0.134949
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.a.e.1.3	yes	6
3.2	odd	2		729.2.a.b.1.4	✓	6
9.2	odd	6		729.2.c.d.244.3		12
9.4	even	3		729.2.c.a.487.4		12
9.5	odd	6		729.2.c.d.487.3		12
9.7	even	3		729.2.c.a.244.4		12
27.2	odd	18		729.2.e.t.568.1		12
27.4	even	9		729.2.e.u.406.1		12
27.5	odd	18		729.2.e.s.649.2		12
27.7	even	9		729.2.e.u.325.1		12
27.11	odd	18		729.2.e.s.82.2		12
27.13	even	9		729.2.e.k.163.2		12
27.14	odd	18		729.2.e.t.163.1		12
27.16	even	9		729.2.e.l.82.1		12
27.20	odd	18		729.2.e.j.325.2		12
27.22	even	9		729.2.e.l.649.1		12
27.23	odd	18		729.2.e.j.406.2		12
27.25	even	9		729.2.e.k.568.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.a.b.1.4	✓	6	3.2	odd	2
729.2.a.e.1.3	yes	6	1.1	even	1	trivial
729.2.c.a.244.4		12	9.7	even	3
729.2.c.a.487.4		12	9.4	even	3
729.2.c.d.244.3		12	9.2	odd	6
729.2.c.d.487.3		12	9.5	odd	6
729.2.e.j.325.2		12	27.20	odd	18
729.2.e.j.406.2		12	27.23	odd	18
729.2.e.k.163.2		12	27.13	even	9
729.2.e.k.568.2		12	27.25	even	9
729.2.e.l.82.1		12	27.16	even	9
729.2.e.l.649.1		12	27.22	even	9
729.2.e.s.82.2		12	27.11	odd	18
729.2.e.s.649.2		12	27.5	odd	18
729.2.e.t.163.1		12	27.14	odd	18
729.2.e.t.568.1		12	27.2	odd	18
729.2.e.u.325.1		12	27.7	even	9
729.2.e.u.406.1		12	27.4	even	9

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)

\(f(q)\)	\(=\)	\(q-0.172976 q^{2} -1.97008 q^{4} +3.73656 q^{5} +3.03150 q^{7} +0.686728 q^{8} -0.646335 q^{10} +2.49170 q^{11} -0.765139 q^{13} -0.524376 q^{14} +3.82137 q^{16} -4.62278 q^{17} -0.611844 q^{19} -7.36132 q^{20} -0.431003 q^{22} -6.52438 q^{23} +8.96190 q^{25} +0.132351 q^{26} -5.97229 q^{28} +6.55089 q^{29} +6.55043 q^{31} -2.03446 q^{32} +0.799630 q^{34} +11.3274 q^{35} +4.95969 q^{37} +0.105834 q^{38} +2.56600 q^{40} -5.26024 q^{41} +5.57057 q^{43} -4.90884 q^{44} +1.12856 q^{46} -1.10762 q^{47} +2.18998 q^{49} -1.55019 q^{50} +1.50738 q^{52} -8.84310 q^{53} +9.31038 q^{55} +2.08181 q^{56} -1.13315 q^{58} +11.8518 q^{59} +8.18700 q^{61} -1.13307 q^{62} -7.29083 q^{64} -2.85899 q^{65} -1.21234 q^{67} +9.10725 q^{68} -1.95936 q^{70} +4.91946 q^{71} +4.29945 q^{73} -0.857907 q^{74} +1.20538 q^{76} +7.55357 q^{77} -11.7946 q^{79} +14.2788 q^{80} +0.909895 q^{82} -9.01607 q^{83} -17.2733 q^{85} -0.963575 q^{86} +1.71112 q^{88} -7.53885 q^{89} -2.31952 q^{91} +12.8535 q^{92} +0.191591 q^{94} -2.28619 q^{95} -0.948354 q^{97} -0.378814 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28}+ \cdots + 18 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 + 9 * q^4 - 3 * q^5 + 6 * q^7 + 6 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 + 24 * q^14 + 15 * q^16 - 9 * q^17 + 12 * q^19 - 21 * q^20 + 3 * q^22 - 12 * q^23 + 9 * q^25 + 24 * q^26 + 3 * q^28 + 21 * q^29 + 15 * q^31 + 30 * q^35 + 3 * q^37 + 15 * q^38 + 3 * q^40 - 12 * q^41 + 6 * q^43 - 33 * q^44 - 3 * q^46 - 15 * q^47 + 12 * q^49 - 24 * q^50 + 3 * q^52 - 9 * q^53 + 15 * q^55 + 12 * q^56 - 15 * q^58 + 6 * q^59 + 24 * q^61 - 30 * q^62 + 6 * q^64 - 15 * q^65 + 15 * q^67 + 36 * q^68 - 15 * q^70 + 12 * q^73 + 24 * q^74 + 9 * q^76 + 15 * q^77 + 24 * q^79 - 21 * q^80 - 21 * q^82 - 6 * q^83 - 18 * q^85 - 30 * q^86 - 21 * q^88 - 9 * q^89 + 18 * q^91 + 6 * q^92 - 6 * q^94 - 33 * q^95 - 21 * q^97 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more