Embedded newform 138.4.e.b.13.2

• Label: 138.4.e.b.13.2
• 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			13.2
Character	\(\chi\)	\(=\)	138.13
Dual form			138.4.e.b.85.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30972 - 1.51150i) q^{2} +(-2.52376 - 1.62192i) q^{3} +(-0.569259 + 3.95929i) q^{4} +(2.02822 - 4.44117i) q^{5} +(0.853889 + 5.93893i) q^{6} +(-16.2703 - 4.77739i) q^{7} +(6.73003 - 4.32513i) q^{8} +(3.73874 + 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

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
\(5\)	2.02822	−	4.44117i	0.181409	−	0.397231i								−0.796979	−	0.604007i	\(-0.793570\pi\)
							0.978388	+	0.206776i	\(0.0662972\pi\)
\(7\)	−16.2703	−	4.77739i	−0.878514	−	0.257955i	−0.188781	−	0.982019i	\(-0.560454\pi\)
							−0.689733	+	0.724064i	\(0.742272\pi\)
\(11\)	−9.92909	+	11.4588i	−0.272158	+	0.314087i	−0.875332	−	0.483523i	\(-0.839357\pi\)
							0.603174	+	0.797610i	\(0.293902\pi\)
\(13\)	−17.3207	+	5.08581i	−0.369530	+	0.108504i	−0.461225	−	0.887283i	\(-0.652590\pi\)
							0.0916947	+	0.995787i	\(0.470772\pi\)
\(17\)	11.0609	+	76.9300i	0.157803	+	1.09754i	0.902671	+	0.430332i	\(0.141604\pi\)
							−0.744867	+	0.667212i	\(0.767487\pi\)
\(19\)	−0.468800	+	3.26058i	−0.00566053	+	0.0393699i	−0.992456	−	0.122602i	\(-0.960876\pi\)
							0.986795	+	0.161972i	\(0.0517853\pi\)
\(23\)	−7.59155	+	110.043i	−0.0688238	+	0.997629i
							\(29\)	31.7107	+	220.553i	0.203053	+	1.41226i	0.795158	+	0.606402i	\(0.207388\pi\)
−0.592105	+	0.805860i	\(0.701703\pi\)
\(31\)	11.9199	−	7.66048i	0.0690607	−	0.0443826i	−0.505655	−	0.862736i	\(-0.668749\pi\)
							0.574716	+	0.818353i	\(0.305113\pi\)
\(37\)	−36.4857	−	79.8925i	−0.162114	−	0.354980i	0.811091	−	0.584920i	\(-0.198874\pi\)
							−0.973205	+	0.229940i	\(0.926147\pi\)
\(41\)	−29.8499	+	65.3622i	−0.113702	+	0.248972i	−0.957924	−	0.287020i	\(-0.907335\pi\)
							0.844223	+	0.535993i	\(0.180063\pi\)
\(43\)	−268.291	−	172.420i	−0.951490	−	0.611485i	−0.0298595	−	0.999554i	\(-0.509506\pi\)
							−0.921630	+	0.388069i	\(0.873142\pi\)
\(47\)	−416.953			−1.29402			−0.647009	−	0.762482i	\(-0.723980\pi\)
							−0.647009	+	0.762482i	\(0.723980\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.b.13.2	✓	30	1.1	even	1	trivial
138.4.e.b.85.2	yes	30	23.16	even	11	inner