Embedded newform 138.5.h.a.7.16

• Label: 138.5.h.a.7.16
• Level: $138$
• Weight: $5$
• Character: 138.7
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.16
Character	\(\chi\)	\(=\)	138.7
Dual form			138.5.h.a.79.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.85223 - 2.13758i) q^{2} +(4.37128 - 2.80925i) q^{3} +(-1.13852 - 7.91857i) q^{4} +(24.7936 - 11.3229i) q^{5} +(2.09159 - 14.5473i) q^{6} +(-17.6937 - 60.2593i) q^{7} +(-19.0354 - 12.2333i) q^{8} +(11.2162 - 24.5601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.85223	−	2.13758i	0.463056	−	0.534396i
							\(3\)	4.37128	−	2.80925i	0.485698	−	0.312139i
\(5\)	24.7936	−	11.3229i	0.991746	−	0.452915i								0.147611	−	0.989046i	\(-0.452842\pi\)
							0.844135	+	0.536130i	\(0.180114\pi\)
\(7\)	−17.6937	−	60.2593i	−0.361096	−	1.22978i	−0.917119	−	0.398614i	\(-0.869491\pi\)
							0.556022	−	0.831167i	\(-0.312327\pi\)
\(11\)	−88.5762	+	76.7517i	−0.732035	+	0.634312i	−0.938953	−	0.344047i	\(-0.888202\pi\)
							0.206918	+	0.978358i	\(0.433657\pi\)
\(13\)	62.5325	+	18.3612i	0.370015	+	0.108646i	0.461453	−	0.887165i	\(-0.347328\pi\)
							−0.0914384	+	0.995811i	\(0.529146\pi\)
\(17\)	0.204428	+	0.0293923i	0.000707363	+	0.000101703i	0.142669	−	0.989771i	\(-0.454432\pi\)
							−0.141961	+	0.989872i	\(0.545341\pi\)
\(19\)	351.671	−	50.5626i	0.974157	−	0.140063i	0.363187	−	0.931716i	\(-0.381689\pi\)
							0.610970	+	0.791654i	\(0.290779\pi\)
\(23\)	−71.1394	−	524.195i	−0.134479	−	0.990916i
							\(29\)	−9.41608	+	65.4903i	−0.0111963	+	0.0778719i	−0.994653	−	0.103269i	\(-0.967070\pi\)
0.983457	+	0.181141i	\(0.0579789\pi\)
\(31\)	−995.684	−	639.887i	−1.03609	−	0.665856i	−0.0920747	−	0.995752i	\(-0.529350\pi\)
							−0.944017	+	0.329896i	\(0.892986\pi\)
\(37\)	546.652	+	249.648i	0.399308	+	0.182358i	0.604939	−	0.796272i	\(-0.293197\pi\)
							−0.205631	+	0.978630i	\(0.565925\pi\)
\(41\)	879.388	+	1925.59i	0.523134	+	1.14550i	0.968239	+	0.250026i	\(0.0804390\pi\)
							−0.445105	+	0.895478i	\(0.646834\pi\)
\(43\)	84.1733	+	130.976i	0.0455237	+	0.0708363i	0.863269	−	0.504743i	\(-0.168413\pi\)
							−0.817746	+	0.575580i	\(0.804776\pi\)
\(47\)	2556.23			1.15719			0.578593	−	0.815616i	\(-0.303602\pi\)
							0.578593	+	0.815616i	\(0.303602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.7.16	✓	160
23.10	odd	22	inner	138.5.h.a.79.16	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.7.16	✓	160	1.1	even	1	trivial
138.5.h.a.79.16	yes	160	23.10	odd	22	inner