Embedded newform 138.3.g.a.35.3

• Label: 138.3.g.a.35.3
• Level: $138$
• Weight: $3$
• Character: 138.35
• 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			35.3
Character	\(\chi\)	\(=\)	138.35
Dual form			138.3.g.a.71.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.764582 - 1.18971i) q^{2} +(-0.894326 + 2.86360i) q^{3} +(-0.830830 + 1.81926i) q^{4} +(-0.674627 - 2.29757i) q^{5} +(4.09064 - 1.12546i) q^{6} +(1.86579 + 2.15324i) q^{7} +(2.79964 - 0.402527i) q^{8} +(-7.40036 - 5.12197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.764582	−	1.18971i	−0.382291	−	0.594856i
							\(3\)	−0.894326	+	2.86360i	−0.298109	+	0.954532i
\(5\)	−0.674627	−	2.29757i	−0.134925	−	0.459514i								0.864116	−	0.503293i	\(-0.167878\pi\)
							−0.999041	+	0.0437792i	\(0.986060\pi\)
\(7\)	1.86579	+	2.15324i	0.266541	+	0.307605i	0.873205	−	0.487354i	\(-0.162038\pi\)
							−0.606663	+	0.794959i	\(0.707492\pi\)
\(11\)	−7.66071	+	11.9203i	−0.696429	+	1.08366i	0.295312	+	0.955401i	\(0.404577\pi\)
							−0.991740	+	0.128263i	\(0.959060\pi\)
\(13\)	−13.6124	+	15.7096i	−1.04711	+	1.20843i	−0.0695902	+	0.997576i	\(0.522169\pi\)
							−0.977518	+	0.210852i	\(0.932376\pi\)
\(17\)	−18.9028	+	8.63260i	−1.11193	+	0.507800i	−0.884756	−	0.466055i	\(-0.845675\pi\)
							−0.227171	+	0.973855i	\(0.572948\pi\)
\(19\)	−3.01139	+	6.59403i	−0.158494	+	0.347054i	−0.972174	−	0.234259i	\(-0.924734\pi\)
							0.813680	+	0.581313i	\(0.197461\pi\)
\(23\)	12.2823	−	19.4459i	0.534014	−	0.845476i
							\(29\)	−32.3124	+	14.7566i	−1.11422	+	0.508847i	−0.885497	−	0.464645i	\(-0.846182\pi\)
−0.228722	+	0.973492i	\(0.573455\pi\)
\(31\)	4.28352	+	29.7926i	0.138178	+	0.961050i	0.934446	+	0.356106i	\(0.115896\pi\)
							−0.796268	+	0.604945i	\(0.793195\pi\)
\(37\)	53.0129	+	15.5660i	1.43278	+	0.420702i	0.903809	−	0.427937i	\(-0.140759\pi\)
							0.528972	+	0.848639i	\(0.322578\pi\)
\(41\)	2.14681	+	7.31138i	0.0523613	+	0.178326i	0.981522	−	0.191350i	\(-0.0612867\pi\)
							−0.929160	+	0.369677i	\(0.879468\pi\)
\(43\)	2.66972	−	18.5683i	0.0620866	−	0.431822i	−0.934943	−	0.354799i	\(-0.884549\pi\)
							0.997029	−	0.0770230i	\(-0.0245415\pi\)
\(47\)		−	38.6020i		−	0.821319i	−0.911789	−	0.410660i	\(-0.865298\pi\)
							0.911789	−	0.410660i	\(-0.134702\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.35.3	✓	160
3.2	odd	2	inner	138.3.g.a.35.13	yes	160
23.2	even	11	inner	138.3.g.a.71.13	yes	160
69.2	odd	22	inner	138.3.g.a.71.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.35.3	✓	160	1.1	even	1	trivial
138.3.g.a.35.13	yes	160	3.2	odd	2	inner
138.3.g.a.71.3	yes	160	69.2	odd	22	inner
138.3.g.a.71.13	yes	160	23.2	even	11	inner