Embedded newform 138.5.g.a.29.16

• Label: 138.5.g.a.29.16
• 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			29.16
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(8.99975 - 0.0668439i) q^{3} +(5.23889 + 6.04600i) q^{4} +(22.3598 + 34.7925i) q^{5} +(-23.2334 - 10.4025i) q^{6} +(12.5271 - 87.1280i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(80.9911 - 1.20316i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	8.99975	−	0.0668439i	0.999972	−	0.00742711i
\(5\)	22.3598	+	34.7925i	0.894392	+	1.39170i								0.919954	+	0.392027i	\(0.128226\pi\)
							−0.0255621	+	0.999673i	\(0.508138\pi\)
\(7\)	12.5271	−	87.1280i	0.255655	−	1.77812i	−0.307277	−	0.951620i	\(-0.599418\pi\)
							0.562932	−	0.826503i	\(-0.309673\pi\)
\(11\)	−18.9511	+	8.65467i	−0.156621	+	0.0715262i	−0.492183	−	0.870492i	\(-0.663801\pi\)
							0.335562	+	0.942018i	\(0.391074\pi\)
\(13\)	−31.7218	−	220.630i	−0.187703	−	1.30550i	−0.837936	−	0.545769i	\(-0.816238\pi\)
							0.650233	−	0.759735i	\(-0.274671\pi\)
\(17\)	319.687	+	277.010i	1.10618	+	0.958514i	0.999360	−	0.0357644i	\(-0.0113866\pi\)
							0.106823	+	0.994278i	\(0.465932\pi\)
\(19\)	272.446	+	314.419i	0.754697	+	0.870967i	0.995015	−	0.0997275i	\(-0.0317971\pi\)
							−0.240318	+	0.970694i	\(0.577252\pi\)
\(23\)	−393.507	−	353.544i	−0.743870	−	0.668324i
							\(29\)	−397.078	−	344.070i	−0.472149	−	0.409120i	0.386017	−	0.922492i	\(-0.373851\pi\)
−0.858166	+	0.513372i	\(0.828396\pi\)
\(31\)	581.316	−	170.690i	0.604907	−	0.177617i	0.0350845	−	0.999384i	\(-0.488830\pi\)
							0.569823	+	0.821768i	\(0.307012\pi\)
\(37\)	−86.4068	−	55.5303i	−0.0631168	−	0.0405627i	0.508702	−	0.860943i	\(-0.330126\pi\)
							−0.571819	+	0.820380i	\(0.693762\pi\)
\(41\)	626.914	+	975.497i	0.372941	+	0.580308i	0.976102	−	0.217312i	\(-0.0697288\pi\)
							−0.603161	+	0.797619i	\(0.706092\pi\)
\(43\)	3163.38	+	928.853i	1.71086	+	0.502354i	0.983037	−	0.183410i	\(-0.0587135\pi\)
							0.727824	+	0.685764i	\(0.240532\pi\)
\(47\)			211.514i			0.0957510i	0.998853	+	0.0478755i	\(0.0152451\pi\)
							−0.998853	+	0.0478755i	\(0.984755\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.16	✓	320
3.2	odd	2	inner	138.5.g.a.29.30	yes	320
23.4	even	11	inner	138.5.g.a.119.30	yes	320
69.50	odd	22	inner	138.5.g.a.119.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.16	✓	320	1.1	even	1	trivial
138.5.g.a.29.30	yes	320	3.2	odd	2	inner
138.5.g.a.119.16	yes	320	69.50	odd	22	inner
138.5.g.a.119.30	yes	320	23.4	even	11	inner