Embedded newform 138.4.e.c.85.2

• Label: 138.4.e.c.85.2
• Level: $138$
• Weight: $4$
• Character: 138.85
• 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			85.2
Character	\(\chi\)	\(=\)	138.85
Dual form			138.4.e.c.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30972 - 1.51150i) q^{2} +(2.52376 - 1.62192i) q^{3} +(-0.569259 - 3.95929i) q^{4} +(-3.70472 - 8.11221i) q^{5} +(0.853889 - 5.93893i) q^{6} +(-24.4006 + 7.16466i) q^{7} +(-6.73003 - 4.32513i) q^{8} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{11}\right)\)

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
\(5\)	−3.70472	−	8.11221i	−0.331360	−	0.725578i								0.668475	−	0.743735i	\(-0.266948\pi\)
							−0.999835	+	0.0181566i	\(0.994220\pi\)
\(7\)	−24.4006	+	7.16466i	−1.31751	+	0.386855i	−0.863593	−	0.504190i	\(-0.831791\pi\)
							−0.453915	+	0.891045i	\(0.649973\pi\)
\(11\)	−47.4686	−	54.7817i	−1.30112	−	1.50157i	−0.737033	−	0.675857i	\(-0.763774\pi\)
							−0.564087	−	0.825715i	\(-0.690772\pi\)
\(13\)	64.0607	+	18.8099i	1.36671	+	0.401302i	0.881124	−	0.472886i	\(-0.156788\pi\)
							0.485587	+	0.874188i	\(0.338606\pi\)
\(17\)	−6.65905	+	46.3147i	−0.0950033	+	0.660763i	0.885555	+	0.464535i	\(0.153778\pi\)
							−0.980558	+	0.196228i	\(0.937131\pi\)
\(19\)	−7.81172	−	54.3317i	−0.0943227	−	0.656029i	−0.981053	−	0.193741i	\(-0.937938\pi\)
							0.886730	−	0.462288i	\(-0.152971\pi\)
\(23\)	106.965	−	26.9350i	0.969728	−	0.244188i
							\(29\)	22.2405	−	154.686i	0.142413	−	0.990501i	−0.785808	−	0.618471i	\(-0.787753\pi\)
0.928221	−	0.372030i	\(-0.121338\pi\)
\(31\)	−97.8480	−	62.8831i	−0.566904	−	0.364327i	0.225576	−	0.974226i	\(-0.427574\pi\)
							−0.792479	+	0.609899i	\(0.791210\pi\)
\(37\)	−172.479	+	377.677i	−0.766362	+	1.67810i	−0.0318605	+	0.999492i	\(0.510143\pi\)
							−0.734502	+	0.678607i	\(0.762584\pi\)
\(41\)	−78.6394	−	172.196i	−0.299547	−	0.655916i	0.698681	−	0.715434i	\(-0.253771\pi\)
							−0.998227	+	0.0595181i	\(0.981044\pi\)
\(43\)	242.342	−	155.743i	0.859459	−	0.552341i	−0.0350526	−	0.999385i	\(-0.511160\pi\)
							0.894512	+	0.447045i	\(0.147524\pi\)
\(47\)	46.9121			0.145592			0.0727961	−	0.997347i	\(-0.476808\pi\)
							0.0727961	+	0.997347i	\(0.476808\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.85.2	yes	30
23.13	even	11	inner	138.4.e.c.13.2	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.13.2	✓	30	23.13	even	11	inner
138.4.e.c.85.2	yes	30	1.1	even	1	trivial