Embedded newform 138.4.e.b.13.3

• Label: 138.4.e.b.13.3
• Level: $138$
• Weight: $4$
• Character: 138.13
• Analytic conductor: $8.142$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.14226358079\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(3\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	138.13
Dual form			138.4.e.b.85.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30972 - 1.51150i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(-0.569259 + 3.95929i) q^{4} +(6.04376 - 13.2340i) q^{5} +(0.853889 + 5.93893i) q^{6} +(29.2734 + 8.59546i) q^{7} +(6.73003 - 4.32513i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30972	−	1.51150i	−0.463056	−	0.534396i
							\(3\)	−2.52376	−	1.62192i	−0.485698	−	0.312139i
\(5\)	6.04376	−	13.2340i	0.540571	−	1.18368i								−0.420477	−	0.907303i	\(-0.638137\pi\)
							0.961048	−	0.276382i	\(-0.0891354\pi\)
\(7\)	29.2734	+	8.59546i	1.58062	+	0.464111i	0.950069	−	0.312040i	\(-0.101012\pi\)
							0.630548	+	0.776151i	\(0.282830\pi\)
\(11\)	−31.1413	+	35.9390i	−0.853587	+	0.985092i	−0.999991	−	0.00413323i	\(-0.998684\pi\)
							0.146405	+	0.989225i	\(0.453230\pi\)
\(13\)	68.3421	−	20.0671i	1.45805	−	0.428123i	0.545857	−	0.837878i	\(-0.316204\pi\)
							0.912196	+	0.409755i	\(0.134386\pi\)
\(17\)	−18.4228	−	128.133i	−0.262834	−	1.82805i	−0.511288	−	0.859409i	\(-0.670831\pi\)
							0.248454	−	0.968644i	\(-0.420078\pi\)
\(19\)	6.80151	−	47.3055i	0.0821249	−	0.571191i	−0.906662	−	0.421858i	\(-0.861378\pi\)
							0.988787	−	0.149334i	\(-0.0477128\pi\)
\(23\)	3.92306	−	110.234i	0.0355658	−	0.999367i
							\(29\)	18.3067	+	127.326i	0.117223	+	0.815303i	0.960591	+	0.277967i	\(0.0896604\pi\)
−0.843368	+	0.537337i	\(0.819430\pi\)
\(31\)	−148.185	+	95.2330i	−0.858545	+	0.551753i	−0.894229	−	0.447610i	\(-0.852275\pi\)
							0.0356840	+	0.999363i	\(0.488639\pi\)
\(37\)	−84.2049	−	184.383i	−0.374140	−	0.819253i	−0.999250	−	0.0387165i	\(-0.987673\pi\)
							0.625110	−	0.780537i	\(-0.285054\pi\)
\(41\)	0.354387	−	0.776000i	0.00134990	−	0.00295587i	−0.908956	−	0.416892i	\(-0.863119\pi\)
							0.910306	+	0.413937i	\(0.135846\pi\)
\(43\)	−53.3956	−	34.3153i	−0.189366	−	0.121698i	0.442522	−	0.896757i	\(-0.354084\pi\)
							−0.631889	+	0.775059i	\(0.717720\pi\)
\(47\)	387.105			1.20138			0.600692	−	0.799480i	\(-0.294892\pi\)
							0.600692	+	0.799480i	\(0.294892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.b.13.3	✓	30
23.16	even	11	inner	138.4.e.b.85.3	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.13.3	✓	30	1.1	even	1	trivial
138.4.e.b.85.3	yes	30	23.16	even	11	inner