Embedded newform 138.5.g.a.29.9

• Label: 138.5.g.a.29.9
• 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			29.9
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(0.378009 - 8.99206i) q^{3} +(5.23889 + 6.04600i) q^{4} +(21.0690 + 32.7840i) q^{5} +(-11.5380 + 22.6909i) q^{6} +(-4.03906 + 28.0923i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(-80.7142 - 6.79816i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	0.378009	−	8.99206i	0.0420010	−	0.999118i
\(5\)	21.0690	+	32.7840i	0.842760	+	1.31136i								0.948439	+	0.316960i	\(0.102662\pi\)
							−0.105679	+	0.994400i	\(0.533702\pi\)
\(7\)	−4.03906	+	28.0923i	−0.0824298	+	0.573312i	0.906189	+	0.422872i	\(0.138978\pi\)
							−0.988619	+	0.150440i	\(0.951931\pi\)
\(11\)	−135.888	+	62.0578i	−1.12304	+	0.512875i	−0.888337	−	0.459193i	\(-0.848139\pi\)
							−0.234702	+	0.972067i	\(0.575412\pi\)
\(13\)	−28.6747	−	199.437i	−0.169673	−	1.18010i	−0.879561	−	0.475786i	\(-0.842164\pi\)
							0.709889	−	0.704314i	\(-0.248745\pi\)
\(17\)	−202.084	−	175.107i	−0.699252	−	0.605905i	0.230945	−	0.972967i	\(-0.425818\pi\)
							−0.930197	+	0.367062i	\(0.880364\pi\)
\(19\)	−441.902	−	509.982i	−1.22410	−	1.41269i	−0.880813	−	0.473463i	\(-0.843004\pi\)
							−0.343291	−	0.939229i	\(-0.611542\pi\)
\(23\)	−380.015	+	368.008i	−0.718364	+	0.695667i
							\(29\)	−407.578	−	353.168i	−0.484635	−	0.419939i	0.377969	−	0.925818i	\(-0.376623\pi\)
−0.862604	+	0.505880i	\(0.831168\pi\)
\(31\)	1143.60	−	335.791i	1.19001	−	0.349418i	0.373983	−	0.927435i	\(-0.377992\pi\)
							0.816024	+	0.578018i	\(0.196173\pi\)
\(37\)	−1480.43	−	951.414i	−1.08139	−	0.694970i	−0.126515	−	0.991965i	\(-0.540379\pi\)
							−0.954879	+	0.296995i	\(0.904016\pi\)
\(41\)	387.274	+	602.611i	0.230383	+	0.358483i	0.937128	−	0.348985i	\(-0.113474\pi\)
							−0.706745	+	0.707468i	\(0.749837\pi\)
\(43\)	−3392.73	−	996.196i	−1.83490	−	0.538776i	−0.834968	−	0.550299i	\(-0.814514\pi\)
							−0.999934	+	0.0115236i	\(0.996332\pi\)
\(47\)			2267.33i			1.02640i	0.858268	+	0.513202i	\(0.171541\pi\)
							−0.858268	+	0.513202i	\(0.828459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.9	✓	320
3.2	odd	2	inner	138.5.g.a.29.22	yes	320
23.4	even	11	inner	138.5.g.a.119.22	yes	320
69.50	odd	22	inner	138.5.g.a.119.9	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.9	✓	320	1.1	even	1	trivial
138.5.g.a.29.22	yes	320	3.2	odd	2	inner
138.5.g.a.119.9	yes	320	69.50	odd	22	inner
138.5.g.a.119.22	yes	320	23.4	even	11	inner