LMFDB - Embedded newform 729.2.a.b.1.4

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,-3,0,9] 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:	6.6.7459857.1
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			1.4
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 $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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	$-0.480521\pi$
$2$	0.172976	0.122312	0.0611562	+	0.998128i	$0.480521\pi$
$3$	0	0
$3$	0	0
$4$	−1.97008	−0.985040
$5$	−3.73656	−1.67104	−0.835521	−	0.549459i	$-0.814834\pi$
$5$	−3.73656	−1.67104	−0.835521	+	0.549459i	$0.814834\pi$
$6$	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	$-0.622576\pi$
$11$	−2.49170	−0.751275	−0.375637	+	0.926767i	$0.622576\pi$
$12$	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	$-0.310573\pi$
$17$	4.62278	1.12119	0.560595	+	0.828090i	$0.310573\pi$
$18$	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	$-0.261887\pi$
$23$	6.52438	1.36043	0.680213	+	0.733014i	$0.261887\pi$
$24$	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	$-0.708122\pi$
$29$	−6.55089	−1.21647	−0.608235	+	0.793757i	$0.708122\pi$
$30$	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	$-0.365265\pi$
$41$	5.26024	0.821511	0.410756	+	0.911745i	$0.365265\pi$
$42$	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	$-0.474259\pi$
$47$	1.10762	0.161562	0.0807812	+	0.996732i	$0.474259\pi$
$48$	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	$-0.292234\pi$
$53$	8.84310	1.21469	0.607346	+	0.794437i	$0.292234\pi$
$54$	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	$-0.780485\pi$
$59$	−11.8518	−1.54297	−0.771484	+	0.636249i	$0.780485\pi$
$60$	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	$-0.594293\pi$
$71$	−4.91946	−0.583833	−0.291916	+	0.956444i	$0.594293\pi$
$72$	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	$-0.335234\pi$
$83$	9.01607	0.989642	0.494821	+	0.868995i	$0.335234\pi$
$84$	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	$-0.369163\pi$
$89$	7.53885	0.799117	0.399558	+	0.916708i	$0.369163\pi$
$90$	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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

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

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

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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

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