Embedded newform 138.5.g.a.29.5

• Label: 138.5.g.a.29.5
• 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			29.5
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(-6.25980 - 6.46644i) q^{3} +(5.23889 + 6.04600i) q^{4} +(3.94347 + 6.13615i) q^{5} +(8.50750 + 23.9921i) q^{6} +(4.85333 - 33.7557i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(-2.62980 + 80.9573i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	−6.25980	−	6.46644i	−0.695533	−	0.718494i
\(5\)	3.94347	+	6.13615i	0.157739	+	0.245446i								0.911123	−	0.412134i	\(-0.135216\pi\)
							−0.753385	+	0.657580i	\(0.771580\pi\)
\(7\)	4.85333	−	33.7557i	0.0990477	−	0.688892i	−0.878433	−	0.477866i	\(-0.841411\pi\)
							0.977481	−	0.211026i	\(-0.0676804\pi\)
\(11\)	−92.0998	+	42.0606i	−0.761155	+	0.347608i	−0.757886	−	0.652387i	\(-0.773768\pi\)
							−0.00326911	+	0.999995i	\(0.501041\pi\)
\(13\)	29.6601	+	206.291i	0.175504	+	1.22066i	0.867012	+	0.498287i	\(0.166037\pi\)
							−0.691508	+	0.722369i	\(0.743053\pi\)
\(17\)	−120.414	−	104.339i	−0.416656	−	0.361035i	0.421161	−	0.906986i	\(-0.361623\pi\)
							−0.837817	+	0.545951i	\(0.816168\pi\)
\(19\)	334.021	+	385.481i	0.925266	+	1.06781i	0.997517	+	0.0704278i	\(0.0224364\pi\)
							−0.0722505	+	0.997387i	\(0.523018\pi\)
\(23\)	528.523	−	22.4664i	0.999098	−	0.0424695i
							\(29\)	−1010.36	−	875.479i	−1.20138	−	1.04100i	−0.998084	−	0.0618791i	\(-0.980291\pi\)
−0.203291	−	0.979118i	\(-0.565164\pi\)
\(31\)	1602.54	−	470.547i	1.66757	−	0.489643i	0.694374	−	0.719614i	\(-0.255681\pi\)
							0.973197	+	0.229971i	\(0.0738632\pi\)
\(37\)	241.057	+	154.918i	0.176082	+	0.113161i	0.625713	−	0.780053i	\(-0.284808\pi\)
							−0.449631	+	0.893215i	\(0.648444\pi\)
\(41\)	519.588	+	808.495i	0.309095	+	0.480960i	0.960697	−	0.277600i	\(-0.0895391\pi\)
							−0.651602	+	0.758561i	\(0.725903\pi\)
\(43\)	1725.01	+	506.508i	0.932940	+	0.273936i	0.712667	−	0.701502i	\(-0.247487\pi\)
							0.220273	+	0.975438i	\(0.429305\pi\)
\(47\)			1760.36i			0.796905i	0.917189	+	0.398452i	\(0.130453\pi\)
							−0.917189	+	0.398452i	\(0.869547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.5	✓	320
3.2	odd	2	inner	138.5.g.a.29.18	yes	320
23.4	even	11	inner	138.5.g.a.119.18	yes	320
69.50	odd	22	inner	138.5.g.a.119.5	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.5	✓	320	1.1	even	1	trivial
138.5.g.a.29.18	yes	320	3.2	odd	2	inner
138.5.g.a.119.5	yes	320	69.50	odd	22	inner
138.5.g.a.119.18	yes	320	23.4	even	11	inner