Embedded newform 138.5.g.a.29.2

• Label: 138.5.g.a.29.2
• 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			29.2
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(-8.69255 + 2.33227i) q^{3} +(5.23889 + 6.04600i) q^{4} +(-23.5192 - 36.5966i) q^{5} +(25.1048 + 4.21297i) q^{6} +(6.58071 - 45.7699i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(70.1210 - 40.5468i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	−8.69255	+	2.33227i	−0.965839	+	0.259141i
\(5\)	−23.5192	−	36.5966i	−0.940767	−	1.46386i								−0.885094	−	0.465413i	\(-0.845906\pi\)
							−0.0556736	−	0.998449i	\(-0.517731\pi\)
\(7\)	6.58071	−	45.7699i	0.134300	−	0.934079i	−0.805559	−	0.592516i	\(-0.798135\pi\)
							0.939859	−	0.341563i	\(-0.110956\pi\)
\(11\)	128.162	−	58.5294i	1.05919	−	0.483714i	0.191843	−	0.981426i	\(-0.438553\pi\)
							0.867343	+	0.497711i	\(0.165826\pi\)
\(13\)	0.611041	+	4.24988i	0.00361563	+	0.0251472i	0.991549	−	0.129730i	\(-0.0414110\pi\)
							−0.987934	+	0.154877i	\(0.950502\pi\)
\(17\)	−388.788	−	336.887i	−1.34529	−	1.16570i	−0.971189	−	0.238309i	\(-0.923407\pi\)
							−0.374098	−	0.927389i	\(-0.622048\pi\)
\(19\)	23.7806	+	27.4443i	0.0658743	+	0.0760230i	0.787730	−	0.616020i	\(-0.211256\pi\)
							−0.721856	+	0.692043i	\(0.756711\pi\)
\(23\)	−70.6105	−	524.266i	−0.133479	−	0.991052i
							\(29\)	−714.673	−	619.268i	−0.849789	−	0.736347i	0.116669	−	0.993171i	\(-0.462778\pi\)
−0.966458	+	0.256824i	\(0.917324\pi\)
\(31\)	−236.286	+	69.3799i	−0.245875	+	0.0721955i	−0.402348	−	0.915487i	\(-0.631806\pi\)
							0.156473	+	0.987682i	\(0.449988\pi\)
\(37\)	1595.67	+	1025.47i	1.16557	+	0.749066i	0.972679	−	0.232154i	\(-0.0745773\pi\)
							0.192891	+	0.981220i	\(0.438214\pi\)
\(41\)	1390.21	+	2163.21i	0.827016	+	1.28686i	0.955453	+	0.295144i	\(0.0953676\pi\)
							−0.128437	+	0.991718i	\(0.540996\pi\)
\(43\)	−2173.27	−	638.130i	−1.17538	−	0.345122i	−0.364987	−	0.931013i	\(-0.618927\pi\)
							−0.810390	+	0.585891i	\(0.800745\pi\)
\(47\)			1450.78i			0.656761i	0.944546	+	0.328380i	\(0.106503\pi\)
							−0.944546	+	0.328380i	\(0.893497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.2	✓	320
3.2	odd	2	inner	138.5.g.a.29.21	yes	320
23.4	even	11	inner	138.5.g.a.119.21	yes	320
69.50	odd	22	inner	138.5.g.a.119.2	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.2	✓	320	1.1	even	1	trivial
138.5.g.a.29.21	yes	320	3.2	odd	2	inner
138.5.g.a.119.2	yes	320	69.50	odd	22	inner
138.5.g.a.119.21	yes	320	23.4	even	11	inner