Embedded newform 138.5.g.a.29.4

• Label: 138.5.g.a.29.4
• 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			29.4
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(-7.78039 + 4.52388i) q^{3} +(5.23889 + 6.04600i) q^{4} +(-1.76067 - 2.73965i) q^{5} +(25.3330 - 2.49742i) q^{6} +(1.63820 - 11.3939i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(40.0690 - 70.3951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	−7.78039	+	4.52388i	−0.864488	+	0.502653i
\(5\)	−1.76067	−	2.73965i	−0.0704267	−	0.109586i								0.804254	−	0.594286i	\(-0.202565\pi\)
							−0.874681	+	0.484700i	\(0.838929\pi\)
\(7\)	1.63820	−	11.3939i	0.0334326	−	0.232529i	−0.966253	−	0.257594i	\(-0.917070\pi\)
							0.999686	+	0.0250651i	\(0.00797930\pi\)
\(11\)	−204.744	+	93.5036i	−1.69210	+	0.772757i	−0.693470	+	0.720486i	\(0.743919\pi\)
							−0.998633	+	0.0522715i	\(0.983354\pi\)
\(13\)	−32.2773	−	224.494i	−0.190990	−	1.32836i	−0.829394	−	0.558663i	\(-0.811314\pi\)
							0.638404	−	0.769701i	\(-0.279595\pi\)
\(17\)	261.697	+	226.762i	0.905525	+	0.784642i	0.977097	−	0.212793i	\(-0.0682560\pi\)
							−0.0715718	+	0.997435i	\(0.522802\pi\)
\(19\)	−37.8935	−	43.7314i	−0.104968	−	0.121140i	0.700834	−	0.713324i	\(-0.252811\pi\)
							−0.805803	+	0.592184i	\(0.798266\pi\)
\(23\)	395.064	+	351.803i	0.746813	+	0.665035i
							\(29\)	412.692	+	357.600i	0.490716	+	0.425208i	0.864747	−	0.502208i	\(-0.167479\pi\)
−0.374030	+	0.927416i	\(0.622024\pi\)
\(31\)	242.635	−	71.2441i	0.252482	−	0.0741354i	−0.153042	−	0.988220i	\(-0.548907\pi\)
							0.405524	+	0.914084i	\(0.367089\pi\)
\(37\)	1778.13	+	1142.73i	1.29885	+	0.834722i	0.993087	−	0.117384i	\(-0.0374508\pi\)
							0.305767	+	0.952106i	\(0.401087\pi\)
\(41\)	691.117	+	1075.40i	0.411135	+	0.639738i	0.983636	−	0.180165i	\(-0.0576633\pi\)
							−0.572502	+	0.819904i	\(0.694027\pi\)
\(43\)	1171.01	+	343.839i	0.633320	+	0.185960i	0.582608	−	0.812754i	\(-0.302032\pi\)
							0.0507128	+	0.998713i	\(0.483851\pi\)
\(47\)		−	1003.08i		−	0.454088i	−0.973885	−	0.227044i	\(-0.927094\pi\)
							0.973885	−	0.227044i	\(-0.0729061\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.4	✓	320
3.2	odd	2	inner	138.5.g.a.29.23	yes	320
23.4	even	11	inner	138.5.g.a.119.23	yes	320
69.50	odd	22	inner	138.5.g.a.119.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.4	✓	320	1.1	even	1	trivial
138.5.g.a.29.23	yes	320	3.2	odd	2	inner
138.5.g.a.119.4	yes	320	69.50	odd	22	inner
138.5.g.a.119.23	yes	320	23.4	even	11	inner