Embedded newform 138.5.g.a.29.19

• Label: 138.5.g.a.29.19
• Level: $138$
• Weight: $5$
• Character: 138.29
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(32\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			29.19
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57283 + 1.17497i) q^{2} +(-7.30575 - 5.25605i) q^{3} +(5.23889 + 6.04600i) q^{4} +(-23.8357 - 37.0891i) q^{5} +(-12.6207 - 22.1069i) q^{6} +(-6.99678 + 48.6637i) q^{7} +(6.37488 + 21.7108i) q^{8} +(25.7480 + 76.7987i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.57283	+	1.17497i	0.643207	+	0.293743i
							\(3\)	−7.30575	−	5.25605i	−0.811750	−	0.584005i
\(5\)	−23.8357	−	37.0891i	−0.953430	−	1.48357i								−0.873534	−	0.486762i	\(-0.838178\pi\)
							−0.0798954	−	0.996803i	\(-0.525459\pi\)
\(7\)	−6.99678	+	48.6637i	−0.142791	+	0.993137i	0.784856	+	0.619679i	\(0.212737\pi\)
							−0.927647	+	0.373458i	\(0.878172\pi\)
\(11\)	−19.4627	+	8.88831i	−0.160849	+	0.0734571i	−0.494213	−	0.869341i	\(-0.664544\pi\)
							0.333364	+	0.942798i	\(0.391816\pi\)
\(13\)	−0.701091	−	4.87619i	−0.00414846	−	0.0288532i	0.987642	−	0.156726i	\(-0.0500941\pi\)
							−0.991791	+	0.127873i	\(0.959185\pi\)
\(17\)	139.199	+	120.616i	0.481656	+	0.417357i	0.861550	−	0.507673i	\(-0.169494\pi\)
							−0.379894	+	0.925030i	\(0.624040\pi\)
\(19\)	456.581	+	526.922i	1.26477	+	1.45962i	0.828689	+	0.559709i	\(0.189087\pi\)
							0.436077	+	0.899909i	\(0.356367\pi\)
\(23\)	146.725	+	508.245i	0.277363	+	0.960765i
							\(29\)	−1120.79	−	971.166i	−1.33268	−	1.15478i	−0.975317	−	0.220811i	\(-0.929130\pi\)
−0.357365	−	0.933965i	\(-0.616325\pi\)
\(31\)	−1676.58	+	492.289i	−1.74462	+	0.512268i	−0.989652	−	0.143490i	\(-0.954167\pi\)
							−0.754971	+	0.655758i	\(0.772349\pi\)
\(37\)	315.529	+	202.778i	0.230481	+	0.148121i	0.650785	−	0.759262i	\(-0.274440\pi\)
							−0.420303	+	0.907384i	\(0.638076\pi\)
\(41\)	655.731	+	1020.34i	0.390084	+	0.606982i	0.979644	−	0.200742i	\(-0.0643353\pi\)
							−0.589560	+	0.807724i	\(0.700699\pi\)
\(43\)	−1888.66	−	554.561i	−1.02145	−	0.299925i	−0.272220	−	0.962235i	\(-0.587758\pi\)
							−0.749229	+	0.662311i	\(0.769576\pi\)
\(47\)			1485.44i			0.672451i	0.941781	+	0.336226i	\(0.109150\pi\)
							−0.941781	+	0.336226i	\(0.890850\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.19	yes	320
3.2	odd	2	inner	138.5.g.a.29.1	✓	320
23.4	even	11	inner	138.5.g.a.119.1	yes	320
69.50	odd	22	inner	138.5.g.a.119.19	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.1	✓	320	3.2	odd	2	inner
138.5.g.a.29.19	yes	320	1.1	even	1	trivial
138.5.g.a.119.1	yes	320	23.4	even	11	inner
138.5.g.a.119.19	yes	320	69.50	odd	22	inner