Embedded newform 138.4.e.b.85.2

• Label: 138.4.e.b.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.b.13.2

$q$-expansion

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

Character values

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

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

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
\(5\)	2.02822	+	4.44117i	0.181409	+	0.397231i								0.978388	−	0.206776i	\(-0.0662972\pi\)
							−0.796979	+	0.604007i	\(0.793570\pi\)
\(7\)	−16.2703	+	4.77739i	−0.878514	+	0.257955i	−0.689733	−	0.724064i	\(-0.742272\pi\)
							−0.188781	+	0.982019i	\(0.560454\pi\)
\(11\)	−9.92909	−	11.4588i	−0.272158	−	0.314087i	0.603174	−	0.797610i	\(-0.293902\pi\)
							−0.875332	+	0.483523i	\(0.839357\pi\)
\(13\)	−17.3207	−	5.08581i	−0.369530	−	0.108504i	0.0916947	−	0.995787i	\(-0.470772\pi\)
							−0.461225	+	0.887283i	\(0.652590\pi\)
\(17\)	11.0609	−	76.9300i	0.157803	−	1.09754i	−0.744867	−	0.667212i	\(-0.767487\pi\)
							0.902671	−	0.430332i	\(-0.141604\pi\)
\(19\)	−0.468800	−	3.26058i	−0.00566053	−	0.0393699i	0.986795	−	0.161972i	\(-0.0517853\pi\)
							−0.992456	+	0.122602i	\(0.960876\pi\)
\(23\)	−7.59155	−	110.043i	−0.0688238	−	0.997629i
							\(29\)	31.7107	−	220.553i	0.203053	−	1.41226i	−0.592105	−	0.805860i	\(-0.701703\pi\)
0.795158	−	0.606402i	\(-0.207388\pi\)
\(31\)	11.9199	+	7.66048i	0.0690607	+	0.0443826i	0.574716	−	0.818353i	\(-0.305113\pi\)
							−0.505655	+	0.862736i	\(0.668749\pi\)
\(37\)	−36.4857	+	79.8925i	−0.162114	+	0.354980i	−0.973205	−	0.229940i	\(-0.926147\pi\)
							0.811091	+	0.584920i	\(0.198874\pi\)
\(41\)	−29.8499	−	65.3622i	−0.113702	−	0.248972i	0.844223	−	0.535993i	\(-0.180063\pi\)
							−0.957924	+	0.287020i	\(0.907335\pi\)
\(43\)	−268.291	+	172.420i	−0.951490	+	0.611485i	−0.921630	−	0.388069i	\(-0.873142\pi\)
							−0.0298595	+	0.999554i	\(0.509506\pi\)
\(47\)	−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.85.2	yes	30
23.13	even	11	inner	138.4.e.b.13.2	✓	30

    

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