Embedded newform 138.5.g.a.29.20

• Label: 138.5.g.a.29.20
• 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			29.20
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.57283 + 1.17497i) q^{2} +(-6.41693 + 6.31055i) q^{3} +(5.23889 + 6.04600i) q^{4} +(19.3206 + 30.0634i) q^{5} +(-23.9244 + 8.69625i) q^{6} +(-7.22090 + 50.2225i) q^{7} +(6.37488 + 21.7108i) q^{8} +(1.35397 - 80.9887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	−6.41693	+	6.31055i	−0.712992	+	0.701172i
\(5\)	19.3206	+	30.0634i	0.772823	+	1.20254i								0.974786	+	0.223143i	\(0.0716318\pi\)
							−0.201962	+	0.979393i	\(0.564732\pi\)
\(7\)	−7.22090	+	50.2225i	−0.147365	+	1.02495i	0.773145	+	0.634229i	\(0.218683\pi\)
							−0.920510	+	0.390719i	\(0.872227\pi\)
\(11\)	49.1468	−	22.4446i	0.406172	−	0.185492i	−0.201844	−	0.979418i	\(-0.564693\pi\)
							0.608015	+	0.793925i	\(0.291966\pi\)
\(13\)	1.66394	+	11.5729i	0.00984579	+	0.0684790i	0.994152	−	0.107990i	\(-0.0344415\pi\)
							−0.984306	+	0.176469i	\(0.943532\pi\)
\(17\)	−75.4259	−	65.3570i	−0.260989	−	0.226149i	0.514530	−	0.857472i	\(-0.327966\pi\)
							−0.775519	+	0.631324i	\(0.782512\pi\)
\(19\)	−77.2941	−	89.2021i	−0.214111	−	0.247097i	0.638527	−	0.769599i	\(-0.279544\pi\)
							−0.852638	+	0.522502i	\(0.824999\pi\)
\(23\)	4.20074	+	528.983i	0.00794091	+	0.999968i
							\(29\)	−999.535	−	866.102i	−1.18851	−	1.02985i	−0.998847	−	0.0479979i	\(-0.984716\pi\)
−0.189660	−	0.981850i	\(-0.560739\pi\)
\(31\)	−611.608	+	179.584i	−0.636429	+	0.186872i	−0.584002	−	0.811752i	\(-0.698514\pi\)
							−0.0524271	+	0.998625i	\(0.516696\pi\)
\(37\)	626.583	+	402.680i	0.457694	+	0.294142i	0.749099	−	0.662458i	\(-0.230487\pi\)
							−0.291405	+	0.956600i	\(0.594123\pi\)
\(41\)	−406.646	−	632.754i	−0.241907	−	0.376415i	0.698976	−	0.715145i	\(-0.253639\pi\)
							−0.940884	+	0.338729i	\(0.890003\pi\)
\(43\)	2285.06	+	670.953i	1.23583	+	0.362874i	0.833449	−	0.552597i	\(-0.186363\pi\)
							0.402385	+	0.915470i	\(0.368181\pi\)
\(47\)		−	1379.76i		−	0.624609i	−0.949982	−	0.312305i	\(-0.898899\pi\)
							0.949982	−	0.312305i	\(-0.101101\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.20	yes	320
3.2	odd	2	inner	138.5.g.a.29.8	✓	320
23.4	even	11	inner	138.5.g.a.119.8	yes	320
69.50	odd	22	inner	138.5.g.a.119.20	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.8	✓	320	3.2	odd	2	inner
138.5.g.a.29.20	yes	320	1.1	even	1	trivial
138.5.g.a.119.8	yes	320	23.4	even	11	inner
138.5.g.a.119.20	yes	320	69.50	odd	22	inner