Embedded newform 138.3.g.a.29.10

• Label: 138.3.g.a.29.10
• Level: $138$
• Weight: $3$
• Character: 138.29
• Analytic conductor: $3.760$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76022764817\)
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			29.10
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(-2.69091 - 1.32627i) q^{3} +(1.30972 + 1.51150i) q^{4} +(3.03728 + 4.72610i) q^{5} +(-2.68246 - 3.28701i) q^{6} +(-0.130023 + 0.904330i) q^{7} +(0.796860 + 2.71386i) q^{8} +(5.48199 + 7.13777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28641	+	0.587486i	0.643207	+	0.293743i
							\(3\)	−2.69091	−	1.32627i	−0.896970	−	0.442092i
\(5\)	3.03728	+	4.72610i	0.607456	+	0.945220i								0.999679	+	0.0253500i	\(0.00807001\pi\)
							−0.392222	+	0.919870i	\(0.628294\pi\)
\(7\)	−0.130023	+	0.904330i	−0.0185747	+	0.129190i	−0.996999	−	0.0774165i	\(-0.975333\pi\)
							0.978424	+	0.206606i	\(0.0662420\pi\)
\(11\)	8.83358	−	4.03416i	0.803053	−	0.366742i	0.0287733	−	0.999586i	\(-0.490840\pi\)
							0.774279	+	0.632844i	\(0.218113\pi\)
\(13\)	3.17528	+	22.0845i	0.244252	+	1.69881i	0.630316	+	0.776338i	\(0.282925\pi\)
							−0.386064	+	0.922472i	\(0.626166\pi\)
\(17\)	−6.38489	−	5.53254i	−0.375582	−	0.325443i	0.446531	−	0.894768i	\(-0.352659\pi\)
							−0.822113	+	0.569325i	\(0.807205\pi\)
\(19\)	6.70716	+	7.74047i	0.353008	+	0.407393i	0.904285	−	0.426929i	\(-0.140405\pi\)
							−0.551277	+	0.834322i	\(0.685859\pi\)
\(23\)	12.0663	−	19.5807i	0.524621	−	0.851336i
							\(29\)	−27.4447	−	23.7810i	−0.946369	−	0.820034i	0.0374313	−	0.999299i	\(-0.488082\pi\)
−0.983801	+	0.179265i	\(0.942628\pi\)
\(31\)	−9.25280	+	2.71687i	−0.298477	+	0.0876409i	−0.427543	−	0.903995i	\(-0.640621\pi\)
							0.129065	+	0.991636i	\(0.458802\pi\)
\(37\)	−45.2398	−	29.0738i	−1.22270	−	0.785779i	−0.239959	−	0.970783i	\(-0.577134\pi\)
							−0.982738	+	0.185004i	\(0.940770\pi\)
\(41\)	5.06996	+	7.88902i	0.123658	+	0.192415i	0.897563	−	0.440886i	\(-0.145336\pi\)
							−0.773905	+	0.633301i	\(0.781699\pi\)
\(43\)	−29.7583	−	8.73781i	−0.692053	−	0.203205i	−0.0832523	−	0.996528i	\(-0.526531\pi\)
							−0.608800	+	0.793324i	\(0.708349\pi\)
\(47\)		−	39.3439i		−	0.837104i	−0.908193	−	0.418552i	\(-0.862538\pi\)
							0.908193	−	0.418552i	\(-0.137462\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.29.10	yes	160
3.2	odd	2	inner	138.3.g.a.29.1	✓	160
23.4	even	11	inner	138.3.g.a.119.1	yes	160
69.50	odd	22	inner	138.3.g.a.119.10	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.29.1	✓	160	3.2	odd	2	inner
138.3.g.a.29.10	yes	160	1.1	even	1	trivial
138.3.g.a.119.1	yes	160	23.4	even	11	inner
138.3.g.a.119.10	yes	160	69.50	odd	22	inner