Embedded newform 138.3.g.a.29.16

• Label: 138.3.g.a.29.16
• 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			29.16
Character	\(\chi\)	\(=\)	138.29
Dual form			138.3.g.a.119.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28641 + 0.587486i) q^{2} +(2.99208 - 0.217869i) q^{3} +(1.30972 + 1.51150i) q^{4} +(1.08843 + 1.69363i) q^{5} +(3.97705 + 1.47753i) q^{6} +(1.71626 - 11.9369i) q^{7} +(0.796860 + 2.71386i) q^{8} +(8.90507 - 1.30376i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	2.99208	−	0.217869i	0.997359	−	0.0726229i
\(5\)	1.08843	+	1.69363i	0.217686	+	0.338725i								0.932872	−	0.360209i	\(-0.117295\pi\)
							−0.715186	+	0.698934i	\(0.753658\pi\)
\(7\)	1.71626	−	11.9369i	0.245180	−	1.70526i	−0.380167	−	0.924918i	\(-0.624134\pi\)
							0.625347	−	0.780347i	\(-0.284957\pi\)
\(11\)	−17.2269	+	7.86726i	−1.56608	+	0.715206i	−0.994441	−	0.105298i	\(-0.966420\pi\)
							−0.571641	+	0.820504i	\(0.693693\pi\)
\(13\)	2.29531	+	15.9642i	0.176562	+	1.22802i	0.864645	+	0.502383i	\(0.167543\pi\)
							−0.688083	+	0.725632i	\(0.741547\pi\)
\(17\)	−3.90502	−	3.38372i	−0.229707	−	0.199042i	0.532401	−	0.846492i	\(-0.321290\pi\)
							−0.762108	+	0.647450i	\(0.775835\pi\)
\(19\)	−2.20948	−	2.54987i	−0.116288	−	0.134204i	0.694621	−	0.719376i	\(-0.255572\pi\)
							−0.810909	+	0.585172i	\(0.801027\pi\)
\(23\)	−22.3597	−	5.38942i	−0.972159	−	0.234323i
							\(29\)	30.0577	+	26.0452i	1.03647	+	0.898109i	0.994884	−	0.101026i	\(-0.0322124\pi\)
0.0415893	+	0.999135i	\(0.486758\pi\)
\(31\)	−39.8754	+	11.7085i	−1.28630	+	0.377692i	−0.852221	−	0.523182i	\(-0.824745\pi\)
							−0.434081	+	0.900874i	\(0.642927\pi\)
\(37\)	−18.5541	−	11.9240i	−0.501463	−	0.322271i	0.265338	−	0.964155i	\(-0.414516\pi\)
							−0.766801	+	0.641885i	\(0.778153\pi\)
\(41\)	−29.2984	−	45.5892i	−0.714595	−	1.11193i	−0.988653	−	0.150219i	\(-0.952002\pi\)
							0.274057	−	0.961713i	\(-0.411634\pi\)
\(43\)	10.2310	+	3.00409i	0.237930	+	0.0698626i	0.398524	−	0.917158i	\(-0.369523\pi\)
							−0.160594	+	0.987021i	\(0.551341\pi\)
\(47\)		−	21.2552i		−	0.452238i	−0.974100	−	0.226119i	\(-0.927396\pi\)
							0.974100	−	0.226119i	\(-0.0726038\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.29.16	yes	160
3.2	odd	2	inner	138.3.g.a.29.7	✓	160
23.4	even	11	inner	138.3.g.a.119.7	yes	160
69.50	odd	22	inner	138.3.g.a.119.16	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.29.7	✓	160	3.2	odd	2	inner
138.3.g.a.29.16	yes	160	1.1	even	1	trivial
138.3.g.a.119.7	yes	160	23.4	even	11	inner
138.3.g.a.119.16	yes	160	69.50	odd	22	inner