Embedded newform 138.3.g.a.29.12

• Label: 138.3.g.a.29.12
• 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			29.12
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(0.282296 + 2.98669i) q^{3} +(1.30972 + 1.51150i) q^{4} +(1.93323 + 3.00816i) q^{5} +(-1.39149 + 4.00796i) q^{6} +(0.370720 - 2.57842i) q^{7} +(0.796860 + 2.71386i) q^{8} +(-8.84062 + 1.68626i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	0.282296	+	2.98669i	0.0940988	+	0.995563i
\(5\)	1.93323	+	3.00816i	0.386646	+	0.601633i								0.978955	−	0.204078i	\(-0.0654198\pi\)
							−0.592309	+	0.805711i	\(0.701783\pi\)
\(7\)	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	−0.355391	−	0.934718i	\(-0.615652\pi\)
							0.473681	+	0.880696i	\(0.342925\pi\)
\(13\)	−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	0.783902	−	0.620884i	\(-0.213226\pi\)
							−0.503004	+	0.864284i	\(0.667772\pi\)
\(19\)	−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	0.456112	−	0.889922i	\(-0.349242\pi\)
−0.815953	+	0.578118i	\(0.803787\pi\)
\(31\)	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	\(-0.826566\pi\)
							−0.116198	−	0.993226i	\(-0.537071\pi\)
\(43\)	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	\(-0.979381\pi\)
							0.997903	−	0.0647303i	\(-0.0206187\pi\)

(See \(a_n\) instead)

Twists

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

    

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