Embedded newform 138.3.g.a.35.4

• Label: 138.3.g.a.35.4
• 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			35.4
Character	\(\chi\)	\(=\)	138.35
Dual form			138.3.g.a.71.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.764582 - 1.18971i) q^{2} +(-0.618173 - 2.93562i) q^{3} +(-0.830830 + 1.81926i) q^{4} +(2.12160 + 7.22551i) q^{5} +(-3.01990 + 2.97997i) q^{6} +(7.03927 + 8.12375i) q^{7} +(2.79964 - 0.402527i) q^{8} +(-8.23573 + 3.62944i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 4 q^{3} + 32 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.618173	−	2.93562i	−0.206058	−	0.978540i
\(5\)	2.12160	+	7.22551i	0.424320	+	1.44510i								0.843466	+	0.537183i	\(0.180512\pi\)
							−0.419145	+	0.907919i	\(0.637670\pi\)
\(7\)	7.03927	+	8.12375i	1.00561	+	1.16054i	0.987002	+	0.160709i	\(0.0513782\pi\)
							0.0186087	+	0.999827i	\(0.494076\pi\)
\(11\)	−7.16803	+	11.1537i	−0.651639	+	1.01397i	0.345503	+	0.938418i	\(0.387708\pi\)
							−0.997142	+	0.0755525i	\(0.975928\pi\)
\(13\)	11.8783	−	13.7083i	0.913713	−	1.05448i	−0.0845995	−	0.996415i	\(-0.526961\pi\)
							0.998313	−	0.0580662i	\(-0.0184935\pi\)
\(17\)	5.57789	−	2.54734i	0.328111	−	0.149843i	−0.244547	−	0.969637i	\(-0.578639\pi\)
							0.572659	+	0.819794i	\(0.305912\pi\)
\(19\)	−8.42361	+	18.4451i	−0.443348	+	0.970797i	0.547624	+	0.836725i	\(0.315533\pi\)
							−0.990972	+	0.134072i	\(0.957195\pi\)
\(23\)	−2.52192	−	22.8613i	−0.109649	−	0.993970i
							\(29\)	23.6439	−	10.7978i	0.815305	−	0.372337i	0.0362892	−	0.999341i	\(-0.488446\pi\)
0.779016	+	0.627004i	\(0.215719\pi\)
\(31\)	7.66667	+	53.3228i	0.247312	+	1.72009i	0.613623	+	0.789599i	\(0.289711\pi\)
							−0.366311	+	0.930492i	\(0.619379\pi\)
\(37\)	11.6302	+	3.41493i	0.314329	+	0.0922954i	0.435092	−	0.900386i	\(-0.356716\pi\)
							−0.120763	+	0.992681i	\(0.538534\pi\)
\(41\)	−13.3937	−	45.6148i	−0.326676	−	1.11256i	−0.945119	−	0.326727i	\(-0.894054\pi\)
							0.618443	−	0.785830i	\(-0.287764\pi\)
\(43\)	6.63977	−	46.1806i	0.154413	−	1.07397i	−0.754295	−	0.656536i	\(-0.772021\pi\)
							0.908708	−	0.417432i	\(-0.137070\pi\)
\(47\)			29.8094i			0.634244i	0.948385	+	0.317122i	\(0.102716\pi\)
							−0.948385	+	0.317122i	\(0.897284\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.3.g.a.35.4	✓	160
3.2	odd	2	inner	138.3.g.a.35.14	yes	160
23.2	even	11	inner	138.3.g.a.71.14	yes	160
69.2	odd	22	inner	138.3.g.a.71.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.3.g.a.35.4	✓	160	1.1	even	1	trivial
138.3.g.a.35.14	yes	160	3.2	odd	2	inner
138.3.g.a.71.4	yes	160	69.2	odd	22	inner
138.3.g.a.71.14	yes	160	23.2	even	11	inner