Embedded newform 81.2.g.a.49.8

• Label: 81.2.g.a.49.8
• Level: $81$
• Weight: $2$
• Character: 81.49
• Analytic conductor: $0.647$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	81.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.646788256372\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(8\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			49.8
Character	\(\chi\)	\(=\)	81.49
Dual form			81.2.g.a.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.761699 - 2.54425i) q^{2} +(1.50768 + 0.852593i) q^{3} +(-4.22205 - 2.77689i) q^{4} +(-1.12810 + 2.61524i) q^{5} +(3.31761 - 3.18649i) q^{6} +(-0.111224 - 1.90965i) q^{7} +(-6.21207 + 5.21255i) q^{8} +(1.54617 + 2.57087i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{16}{27}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.761699	−	2.54425i	0.538602	−	1.79906i	−0.0602322	−	0.998184i	\(-0.519184\pi\)
							0.598835	−	0.800873i	\(-0.295631\pi\)
\(3\)	1.50768	+	0.852593i	0.870457	+	0.492245i
							\(5\)	−1.12810	+	2.61524i	−0.504504	+	1.16957i	0.455666	+	0.890151i	\(0.349401\pi\)
−0.960169	+	0.279420i	\(0.909858\pi\)
\(7\)	−0.111224	−	1.90965i	−0.0420389	−	0.721779i	−0.951814	−	0.306677i	\(-0.900783\pi\)
							0.909775	−	0.415102i	\(-0.136254\pi\)
\(11\)	−1.10623	+	1.48593i	−0.333541	+	0.448024i	−0.936979	−	0.349387i	\(-0.886390\pi\)
							0.603437	+	0.797411i	\(0.293797\pi\)
\(13\)	0.0925649	−	0.0981130i	0.0256729	−	0.0272117i	−0.714410	−	0.699727i	\(-0.753305\pi\)
							0.740083	+	0.672516i	\(0.234786\pi\)
\(17\)	0.193799	−	1.09909i	0.0470031	−	0.266568i	−0.952245	−	0.305334i	\(-0.901232\pi\)
							0.999248	+	0.0387665i	\(0.0123428\pi\)
\(19\)	−1.05411	−	5.97816i	−0.241830	−	1.37148i	−0.827741	−	0.561110i	\(-0.810375\pi\)
							0.585912	−	0.810375i	\(-0.300737\pi\)
\(23\)	−0.0271279	+	0.465768i	−0.00565655	+	0.0971192i	−0.999955	−	0.00951638i	\(-0.996971\pi\)
							0.994298	+	0.106636i	\(0.0340078\pi\)
\(29\)	0.426645	−	0.101117i	0.0792259	−	0.0187769i	−0.190812	−	0.981627i	\(-0.561112\pi\)
							0.270038	+	0.962850i	\(0.412964\pi\)
\(31\)	2.52937	−	1.27030i	0.454289	−	0.228152i	−0.206915	−	0.978359i	\(-0.566342\pi\)
							0.661204	+	0.750207i	\(0.270046\pi\)
\(37\)	−7.77306	+	2.82916i	−1.27788	+	0.465112i	−0.889732	−	0.456484i	\(-0.849109\pi\)
							−0.388152	+	0.921595i	\(0.626886\pi\)
\(41\)	−1.59594	−	5.33081i	−0.249244	−	0.832533i	−0.987446	−	0.157954i	\(-0.949510\pi\)
							0.738202	−	0.674579i	\(-0.235675\pi\)
\(43\)	7.45009	+	0.870791i	1.13613	+	0.132794i	0.663295	−	0.748358i	\(-0.269157\pi\)
							0.472833	+	0.881152i	\(0.343231\pi\)
\(47\)	−7.62373	−	3.82878i	−1.11204	−	0.558485i	−0.204671	−	0.978831i	\(-0.565613\pi\)
							−0.907364	+	0.420345i	\(0.861909\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.2.g.a.49.8	yes	144
3.2	odd	2		243.2.g.a.37.1		144
9.2	odd	6		729.2.g.a.595.8		144
9.4	even	3		729.2.g.c.109.8		144
9.5	odd	6		729.2.g.b.109.1		144
9.7	even	3		729.2.g.d.595.1		144
81.11	odd	54		729.2.g.a.136.8		144
81.16	even	27		729.2.g.c.622.8		144
81.23	odd	54		6561.2.a.d.1.69		72
81.38	odd	54		243.2.g.a.46.1		144
81.43	even	27	inner	81.2.g.a.43.8	✓	144
81.58	even	27		6561.2.a.c.1.4		72
81.65	odd	54		729.2.g.b.622.1		144
81.70	even	27		729.2.g.d.136.1		144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.g.a.43.8	✓	144	81.43	even	27	inner
81.2.g.a.49.8	yes	144	1.1	even	1	trivial
243.2.g.a.37.1		144	3.2	odd	2
243.2.g.a.46.1		144	81.38	odd	54
729.2.g.a.136.8		144	81.11	odd	54
729.2.g.a.595.8		144	9.2	odd	6
729.2.g.b.109.1		144	9.5	odd	6
729.2.g.b.622.1		144	81.65	odd	54
729.2.g.c.109.8		144	9.4	even	3
729.2.g.c.622.8		144	81.16	even	27
729.2.g.d.136.1		144	81.70	even	27
729.2.g.d.595.1		144	9.7	even	3
6561.2.a.c.1.4		72	81.58	even	27
6561.2.a.d.1.69		72	81.23	odd	54