Embedded newform 138.3.g.a.29.6

• Label: 138.3.g.a.29.6
• Level: $138$
• Weight: $3$
• Character: 138.29
• Analytic conductor: $3.760$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	138.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76022764817\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			29.6
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(1.85221 - 2.35994i) q^{3} +(1.30972 + 1.51150i) q^{4} +(-1.93323 - 3.00816i) q^{5} +(-3.76914 + 1.94772i) q^{6} +(0.370720 - 2.57842i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(-2.13865 - 8.74221i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28641	−	0.587486i	−0.643207	−	0.293743i
							\(3\)	1.85221	−	2.35994i	0.617403	−	0.786647i
\(5\)	−1.93323	−	3.00816i	−0.386646	−	0.601633i								0.592309	−	0.805711i	\(-0.298217\pi\)
							−0.978955	+	0.204078i	\(0.934580\pi\)
\(7\)	0.370720	−	2.57842i	0.0529601	−	0.368345i	−0.946056	−	0.324002i	\(-0.894971\pi\)
							0.999016	−	0.0443430i	\(-0.0141195\pi\)
\(11\)	−1.30119	+	0.594236i	−0.118290	+	0.0540214i	−0.473681	−	0.880696i	\(-0.657075\pi\)
							0.355391	+	0.934718i	\(0.384348\pi\)
\(13\)	−0.496919	−	3.45615i	−0.0382245	−	0.265857i	0.961743	−	0.273954i	\(-0.0883317\pi\)
							−0.999967	+	0.00809694i	\(0.997423\pi\)
\(17\)	−4.77527	−	4.13779i	−0.280898	−	0.243400i	0.503004	−	0.864284i	\(-0.332228\pi\)
							−0.783902	+	0.620884i	\(0.786774\pi\)
\(19\)	−1.48598	−	1.71491i	−0.0782092	−	0.0902583i	0.715296	−	0.698821i	\(-0.246292\pi\)
							−0.793506	+	0.608563i	\(0.791746\pi\)
\(23\)	−20.5349	−	10.3595i	−0.892821	−	0.450412i
							\(29\)	10.4354	+	9.04232i	0.359841	+	0.311804i	0.815953	−	0.578118i	\(-0.196213\pi\)
−0.456112	+	0.889922i	\(0.650758\pi\)
\(31\)	37.1296	−	10.9022i	1.19773	−	0.351685i	0.378745	−	0.925501i	\(-0.376356\pi\)
							0.818984	+	0.573816i	\(0.194538\pi\)
\(37\)	−8.22010	−	5.28274i	−0.222165	−	0.142777i	0.424824	−	0.905276i	\(-0.360336\pi\)
							−0.646989	+	0.762499i	\(0.723972\pi\)
\(41\)	39.8273	+	61.9725i	0.971398	+	1.51152i	0.855200	+	0.518298i	\(0.173434\pi\)
							0.116198	+	0.993226i	\(0.462929\pi\)
\(43\)	56.0946	+	16.4709i	1.30453	+	0.383043i	0.858885	−	0.512169i	\(-0.171158\pi\)
							0.445640	+	0.895212i	\(0.352976\pi\)
\(47\)			6.08465i			0.129461i	0.997903	+	0.0647303i	\(0.0206187\pi\)
							−0.997903	+	0.0647303i	\(0.979381\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.29.6	✓	160
3.2	odd	2	inner	138.3.g.a.29.12	yes	160
23.4	even	11	inner	138.3.g.a.119.12	yes	160
69.50	odd	22	inner	138.3.g.a.119.6	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.29.6	✓	160	1.1	even	1	trivial
138.3.g.a.29.12	yes	160	3.2	odd	2	inner
138.3.g.a.119.6	yes	160	69.50	odd	22	inner
138.3.g.a.119.12	yes	160	23.4	even	11	inner