Embedded newform 138.3.g.a.29.9

• Label: 138.3.g.a.29.9
• 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			29.9
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(-2.99735 + 0.126139i) q^{3} +(1.30972 + 1.51150i) q^{4} +(-2.91988 - 4.54342i) q^{5} +(-3.92993 - 1.59863i) q^{6} +(1.52608 - 10.6142i) q^{7} +(0.796860 + 2.71386i) q^{8} +(8.96818 - 0.756167i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.99735	+	0.126139i	−0.999116	+	0.0420465i
\(5\)	−2.91988	−	4.54342i	−0.583976	−	0.908684i								0.416024	−	0.909354i	\(-0.363423\pi\)
							−1.00000		0.000669622i	\(0.999787\pi\)
\(7\)	1.52608	−	10.6142i	0.218012	−	1.51631i	−0.527350	−	0.849648i	\(-0.676814\pi\)
							0.745362	−	0.666660i	\(-0.232276\pi\)
\(11\)	−0.358486	+	0.163715i	−0.0325896	+	0.0148832i	−0.431643	−	0.902044i	\(-0.642066\pi\)
							0.399054	+	0.916928i	\(0.369339\pi\)
\(13\)	−2.10755	−	14.6583i	−0.162119	−	1.12756i	−0.894630	−	0.446807i	\(-0.852561\pi\)
							0.732511	−	0.680755i	\(-0.238348\pi\)
\(17\)	12.5527	+	10.8770i	0.738395	+	0.639823i	0.940599	−	0.339521i	\(-0.110265\pi\)
							−0.202204	+	0.979343i	\(0.564810\pi\)
\(19\)	−17.2378	−	19.8934i	−0.907250	−	1.04702i	−0.998688	−	0.0512120i	\(-0.983692\pi\)
							0.0914373	−	0.995811i	\(-0.470854\pi\)
\(23\)	−4.88536	+	22.4752i	−0.212407	+	0.977181i
							\(29\)	−30.0536	−	26.0416i	−1.03633	−	0.897987i	−0.0414614	−	0.999140i	\(-0.513201\pi\)
−0.994871	+	0.101153i	\(0.967747\pi\)
\(31\)	3.53241	−	1.03721i	0.113949	−	0.0334583i	−0.224261	−	0.974529i	\(-0.571997\pi\)
							0.338210	+	0.941071i	\(0.390179\pi\)
\(37\)	48.6290	+	31.2520i	1.31430	+	0.844648i	0.994691	−	0.102903i	\(-0.0328131\pi\)
							0.319606	+	0.947551i	\(0.396449\pi\)
\(41\)	33.1242	+	51.5422i	0.807906	+	1.25713i	0.963073	+	0.269239i	\(0.0867721\pi\)
							−0.155167	+	0.987888i	\(0.549592\pi\)
\(43\)	12.8834	+	3.78290i	0.299614	+	0.0879745i	0.428084	−	0.903739i	\(-0.359189\pi\)
							−0.128471	+	0.991713i	\(0.541007\pi\)
\(47\)		−	43.9125i		−	0.934308i	−0.884176	−	0.467154i	\(-0.845279\pi\)
							0.884176	−	0.467154i	\(-0.154721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.29.9	yes	160
3.2	odd	2	inner	138.3.g.a.29.2	✓	160
23.4	even	11	inner	138.3.g.a.119.2	yes	160
69.50	odd	22	inner	138.3.g.a.119.9	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.29.2	✓	160	3.2	odd	2	inner
138.3.g.a.29.9	yes	160	1.1	even	1	trivial
138.3.g.a.119.2	yes	160	23.4	even	11	inner
138.3.g.a.119.9	yes	160	69.50	odd	22	inner