Embedded newform 138.5.h.a.19.4

• Label: 138.5.h.a.19.4
• Level: $138$
• Weight: $5$
• Character: 138.19
• 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			19.4
Character	\(\chi\)	\(=\)	138.19
Dual form			138.5.h.a.109.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17497 - 2.57283i) q^{2} +(-4.98567 + 1.46393i) q^{3} +(-5.23889 + 6.04600i) q^{4} +(19.8768 - 30.9289i) q^{5} +(9.62445 + 11.1072i) q^{6} +(-20.0835 + 2.88758i) q^{7} +(21.7108 + 6.37488i) q^{8} +(22.7138 - 14.5973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17497	−	2.57283i	−0.293743	−	0.643207i
							\(3\)	−4.98567	+	1.46393i	−0.553964	+	0.162658i
\(5\)	19.8768	−	30.9289i	0.795071	−	1.23715i								−0.172613	−	0.984990i	\(-0.555221\pi\)
							0.967684	−	0.252165i	\(-0.0811426\pi\)
\(7\)	−20.0835	+	2.88758i	−0.409868	+	0.0589301i	−0.344165	−	0.938909i	\(-0.611838\pi\)
							−0.0657027	+	0.997839i	\(0.520929\pi\)
\(11\)	119.942	+	54.7757i	0.991257	+	0.452692i	0.843963	−	0.536401i	\(-0.180217\pi\)
							0.147294	+	0.989093i	\(0.452944\pi\)
\(13\)	32.1413	−	223.547i	0.190185	−	1.32277i	−0.641331	−	0.767264i	\(-0.721617\pi\)
							0.831516	−	0.555501i	\(-0.187473\pi\)
\(17\)	−59.4525	+	51.5159i	−0.205718	+	0.178256i	−0.751626	−	0.659589i	\(-0.770730\pi\)
							0.545908	+	0.837845i	\(0.316185\pi\)
\(19\)	−100.789	−	87.3344i	−0.279195	−	0.241924i	0.503995	−	0.863707i	\(-0.331863\pi\)
							−0.783189	+	0.621783i	\(0.786409\pi\)
\(23\)	−520.845	−	92.5309i	−0.984583	−	0.174917i
							\(29\)	−780.760	−	901.045i	−0.928371	−	1.07140i	−0.997275	−	0.0737735i	\(-0.976496\pi\)
0.0689044	−	0.997623i	\(-0.478050\pi\)
\(31\)	−1185.15	−	347.992i	−1.23325	−	0.362115i	−0.400775	−	0.916177i	\(-0.631259\pi\)
							−0.832476	+	0.554062i	\(0.813077\pi\)
\(37\)	335.430	+	521.939i	0.245018	+	0.381255i	0.941880	−	0.335950i	\(-0.109057\pi\)
							−0.696862	+	0.717205i	\(0.745421\pi\)
\(41\)	−1466.86	−	942.693i	−0.872611	−	0.560793i	0.0259394	−	0.999664i	\(-0.491742\pi\)
							−0.898550	+	0.438871i	\(0.855379\pi\)
\(43\)	357.956	+	1219.09i	0.193594	+	0.659322i	0.997880	+	0.0650794i	\(0.0207301\pi\)
							−0.804286	+	0.594243i	\(0.797452\pi\)
\(47\)	832.907			0.377052			0.188526	−	0.982068i	\(-0.439629\pi\)
							0.188526	+	0.982068i	\(0.439629\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.h.a.19.4	✓	160
23.17	odd	22	inner	138.5.h.a.109.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.h.a.19.4	✓	160	1.1	even	1	trivial
138.5.h.a.109.4	yes	160	23.17	odd	22	inner