Embedded newform 138.5.g.a.29.15

• Label: 138.5.g.a.29.15
• 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			29.15
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(8.82038 + 1.78913i) q^{3} +(5.23889 + 6.04600i) q^{4} +(8.08738 + 12.5842i) q^{5} +(-20.5911 - 14.9668i) q^{6} +(-4.63055 + 32.2062i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(74.5980 + 31.5616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	8.82038	+	1.78913i	0.980042	+	0.198792i
\(5\)	8.08738	+	12.5842i	0.323495	+	0.503369i								0.964470	−	0.264194i	\(-0.0851058\pi\)
							−0.640974	+	0.767562i	\(0.721469\pi\)
\(7\)	−4.63055	+	32.2062i	−0.0945010	+	0.657269i	0.886423	+	0.462876i	\(0.153183\pi\)
							−0.980924	+	0.194393i	\(0.937726\pi\)
\(11\)	−51.8483	+	23.6783i	−0.428499	+	0.195689i	−0.617977	−	0.786196i	\(-0.712048\pi\)
							0.189479	+	0.981885i	\(0.439320\pi\)
\(13\)	35.4856	+	246.808i	0.209974	+	1.46040i	0.773233	+	0.634122i	\(0.218638\pi\)
							−0.563259	+	0.826281i	\(0.690452\pi\)
\(17\)	−344.386	−	298.412i	−1.19165	−	1.03257i	−0.998680	−	0.0513641i	\(-0.983643\pi\)
							−0.192968	−	0.981205i	\(-0.561811\pi\)
\(19\)	−66.7214	−	77.0006i	−0.184824	−	0.213298i	0.655775	−	0.754957i	\(-0.272342\pi\)
							−0.840599	+	0.541658i	\(0.817797\pi\)
\(23\)	−210.985	+	485.104i	−0.398838	+	0.917021i
							\(29\)	896.594	+	776.903i	1.06610	+	0.923785i	0.997270	−	0.0738433i	\(-0.0235265\pi\)
0.0688346	+	0.997628i	\(0.478072\pi\)
\(31\)	−1137.11	+	333.886i	−1.18326	+	0.347436i	−0.813430	−	0.581663i	\(-0.802402\pi\)
							−0.369830	+	0.929099i	\(0.620584\pi\)
\(37\)	−125.365	−	80.5671i	−0.0915740	−	0.0588510i	0.494052	−	0.869432i	\(-0.335515\pi\)
							−0.585626	+	0.810581i	\(0.699151\pi\)
\(41\)	1259.51	+	1959.83i	0.749260	+	1.16587i	0.981172	+	0.193135i	\(0.0618654\pi\)
							−0.231912	+	0.972737i	\(0.574498\pi\)
\(43\)	2557.48	+	750.945i	1.38317	+	0.406136i	0.886872	−	0.462016i	\(-0.152874\pi\)
							0.496299	+	0.868152i	\(0.334692\pi\)
\(47\)			1147.25i			0.519353i	0.965696	+	0.259677i	\(0.0836160\pi\)
							−0.965696	+	0.259677i	\(0.916384\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.15	✓	320
3.2	odd	2	inner	138.5.g.a.29.31	yes	320
23.4	even	11	inner	138.5.g.a.119.31	yes	320
69.50	odd	22	inner	138.5.g.a.119.15	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.15	✓	320	1.1	even	1	trivial
138.5.g.a.29.31	yes	320	3.2	odd	2	inner
138.5.g.a.119.15	yes	320	69.50	odd	22	inner
138.5.g.a.119.31	yes	320	23.4	even	11	inner