Embedded newform 138.3.g.a.29.8

• Label: 138.3.g.a.29.8
• 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			29.8
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28641 - 0.587486i) q^{2} +(2.89781 + 0.776326i) q^{3} +(1.30972 + 1.51150i) q^{4} +(1.55712 + 2.42292i) q^{5} +(-3.27171 - 2.70110i) q^{6} +(-1.17275 + 8.15664i) q^{7} +(-0.796860 - 2.71386i) q^{8} +(7.79464 + 4.49930i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	2.89781	+	0.776326i	0.965938	+	0.258775i
\(5\)	1.55712	+	2.42292i	0.311423	+	0.484584i								0.961318	−	0.275440i	\(-0.0888235\pi\)
							−0.649895	+	0.760024i	\(0.725187\pi\)
\(7\)	−1.17275	+	8.15664i	−0.167535	+	1.16523i	0.716422	+	0.697667i	\(0.245778\pi\)
							−0.883958	+	0.467567i	\(0.845131\pi\)
\(11\)	−10.4613	+	4.77752i	−0.951028	+	0.434320i	−0.829648	−	0.558286i	\(-0.811459\pi\)
							−0.121379	+	0.992606i	\(0.538732\pi\)
\(13\)	−1.26057	−	8.76743i	−0.0969667	−	0.674418i	−0.979094	−	0.203407i	\(-0.934798\pi\)
							0.882128	−	0.471011i	\(-0.156111\pi\)
\(17\)	15.6615	+	13.5708i	0.921264	+	0.798279i	0.979795	−	0.200006i	\(-0.0640962\pi\)
							−0.0585311	+	0.998286i	\(0.518642\pi\)
\(19\)	−3.35868	−	3.87612i	−0.176772	−	0.204006i	0.660448	−	0.750872i	\(-0.270366\pi\)
							−0.837220	+	0.546865i	\(0.815821\pi\)
\(23\)	18.5225	−	13.6352i	0.805325	−	0.592834i
							\(29\)	26.0439	+	22.5672i	0.898065	+	0.778178i	0.975770	−	0.218799i	\(-0.0702138\pi\)
−0.0777049	+	0.996976i	\(0.524759\pi\)
\(31\)	23.2685	−	6.83225i	0.750597	−	0.220395i	0.116012	−	0.993248i	\(-0.462989\pi\)
							0.634585	+	0.772853i	\(0.281171\pi\)
\(37\)	−33.6235	−	21.6085i	−0.908744	−	0.584015i	0.000627006	−	1.00000i	\(-0.499800\pi\)
							−0.909371	+	0.415985i	\(0.863437\pi\)
\(41\)	−34.4192	−	53.5574i	−0.839494	−	1.30628i	−0.949952	−	0.312397i	\(-0.898868\pi\)
							0.110458	−	0.993881i	\(-0.464768\pi\)
\(43\)	−64.7785	−	19.0207i	−1.50648	−	0.442342i	−0.578720	−	0.815526i	\(-0.696448\pi\)
							−0.927757	+	0.373184i	\(0.878266\pi\)
\(47\)		−	26.1802i		−	0.557025i	−0.960433	−	0.278512i	\(-0.910159\pi\)
							0.960433	−	0.278512i	\(-0.0898413\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.29.8	✓	160
3.2	odd	2	inner	138.3.g.a.29.15	yes	160
23.4	even	11	inner	138.3.g.a.119.15	yes	160
69.50	odd	22	inner	138.3.g.a.119.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.29.8	✓	160	1.1	even	1	trivial
138.3.g.a.29.15	yes	160	3.2	odd	2	inner
138.3.g.a.119.8	yes	160	69.50	odd	22	inner
138.3.g.a.119.15	yes	160	23.4	even	11	inner