Embedded newform 138.5.g.a.29.13

• Label: 138.5.g.a.29.13
• 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			29.13
Character	\(\chi\)	\(=\)	138.29
Dual form			138.5.g.a.119.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57283 - 1.17497i) q^{2} +(7.44767 - 5.05294i) q^{3} +(5.23889 + 6.04600i) q^{4} +(-5.01473 - 7.80307i) q^{5} +(-25.0986 + 4.24953i) q^{6} +(-6.24276 + 43.4194i) q^{7} +(-6.37488 - 21.7108i) q^{8} +(29.9357 - 75.2652i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 20 q^{3} + 256 q^{4} - 64 q^{6} + 172 q^{9}+O(q^{10}) \)

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
							\(3\)	7.44767	−	5.05294i	0.827519	−	0.561437i
\(5\)	−5.01473	−	7.80307i	−0.200589	−	0.312123i								0.726357	−	0.687318i	\(-0.241212\pi\)
							−0.926946	+	0.375195i	\(0.877576\pi\)
\(7\)	−6.24276	+	43.4194i	−0.127403	+	0.886109i	0.821425	+	0.570317i	\(0.193179\pi\)
							−0.948828	+	0.315793i	\(0.897730\pi\)
\(11\)	63.1770	−	28.8520i	0.522124	−	0.238446i	−0.136877	−	0.990588i	\(-0.543707\pi\)
							0.659001	+	0.752142i	\(0.270979\pi\)
\(13\)	−42.1940	−	293.466i	−0.249669	−	1.73649i	−0.600125	−	0.799906i	\(-0.704883\pi\)
							0.350456	−	0.936579i	\(-0.386027\pi\)
\(17\)	−243.191	−	210.726i	−0.841491	−	0.729156i	0.123245	−	0.992376i	\(-0.460670\pi\)
							−0.964736	+	0.263220i	\(0.915215\pi\)
\(19\)	271.941	+	313.837i	0.753300	+	0.869354i	0.994884	−	0.101026i	\(-0.0322124\pi\)
							−0.241584	+	0.970380i	\(0.577667\pi\)
\(23\)	528.962	−	6.36503i	0.999928	−	0.0120322i
							\(29\)	−994.175	−	861.458i	−1.18213	−	1.02433i	−0.999151	−	0.0412094i	\(-0.986879\pi\)
−0.182984	−	0.983116i	\(-0.558576\pi\)
\(31\)	−869.600	+	255.338i	−0.904891	+	0.265700i	−0.700889	−	0.713270i	\(-0.747213\pi\)
							−0.204002	+	0.978970i	\(0.565395\pi\)
\(37\)	−1415.65	−	909.782i	−1.03408	−	0.664560i	−0.0905606	−	0.995891i	\(-0.528866\pi\)
							−0.943515	+	0.331331i	\(0.892502\pi\)
\(41\)	−342.906	−	533.572i	−0.203989	−	0.317413i	0.724153	−	0.689639i	\(-0.242231\pi\)
							−0.928142	+	0.372226i	\(0.878595\pi\)
\(43\)	1707.71	+	501.429i	0.923587	+	0.271189i	0.708749	−	0.705460i	\(-0.249260\pi\)
							0.214837	+	0.976650i	\(0.431078\pi\)
\(47\)		−	1706.00i		−	0.772295i	−0.922437	−	0.386147i	\(-0.873806\pi\)
							0.922437	−	0.386147i	\(-0.126194\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.g.a.29.13	✓	320
3.2	odd	2	inner	138.5.g.a.29.27	yes	320
23.4	even	11	inner	138.5.g.a.119.27	yes	320
69.50	odd	22	inner	138.5.g.a.119.13	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.g.a.29.13	✓	320	1.1	even	1	trivial
138.5.g.a.29.27	yes	320	3.2	odd	2	inner
138.5.g.a.119.13	yes	320	69.50	odd	22	inner
138.5.g.a.119.27	yes	320	23.4	even	11	inner