Embedded newform 187.2.j.a.135.2

• Label: 187.2.j.a.135.2
• Level: $187$
• Weight: $2$
• Character: 187.135
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			135.2
Character	\(\chi\)	\(=\)	187.135
Dual form			187.2.j.a.169.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.20158 + 1.59954i) q^{2} +(2.67663 - 0.869690i) q^{3} +(1.67040 - 5.14095i) q^{4} +(-0.525344 + 0.723074i) q^{5} +(-4.50172 + 6.19609i) q^{6} +(-1.60348 - 0.521001i) q^{7} +(2.86380 + 8.81387i) q^{8} +(3.98094 - 2.89232i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 8 q^{2} - 20 q^{4} - 4 q^{8} + 30 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/187\mathbb{Z}\right)^\times\).

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.20158	+	1.59954i	−1.55676	+	1.13105i	−0.618148	+	0.786062i	\(0.712117\pi\)
							−0.938608	+	0.344987i	\(0.887883\pi\)
\(3\)	2.67663	−	0.869690i	1.54535	−	0.502116i	0.592506	−	0.805566i	\(-0.298139\pi\)
							0.952847	+	0.303450i	\(0.0981386\pi\)
\(5\)	−0.525344	+	0.723074i	−0.234941	+	0.323369i	−0.910166	−	0.414243i	\(-0.864046\pi\)
							0.675225	+	0.737612i	\(0.264046\pi\)
\(7\)	−1.60348	−	0.521001i	−0.606057	−	0.196920i	−0.0101173	−	0.999949i	\(-0.503220\pi\)
							−0.595940	+	0.803029i	\(0.703220\pi\)
\(11\)	3.01526	+	1.38139i	0.909134	+	0.416503i
							\(13\)	3.39946	−	2.46985i	0.942840	−	0.685014i	−0.00626239	−	0.999980i	\(-0.501993\pi\)
0.949103	+	0.314967i	\(0.101993\pi\)
\(17\)	3.77203	−	1.66487i	0.914852	−	0.403789i
							\(19\)	0.621965	+	1.91421i	0.142688	+	0.439150i	0.996707	−	0.0810935i	\(-0.0258412\pi\)
−0.854018	+	0.520243i	\(0.825841\pi\)
\(23\)		−	0.224129i		−	0.0467341i	−0.999727	−	0.0233670i	\(-0.992561\pi\)
							0.999727	−	0.0233670i	\(-0.00743864\pi\)
\(29\)	−7.72540	−	2.51014i	−1.43457	−	0.466120i	−0.514371	−	0.857568i	\(-0.671974\pi\)
							−0.920200	+	0.391447i	\(0.871974\pi\)
\(31\)	−2.95114	−	4.06190i	−0.530041	−	0.729539i	0.457096	−	0.889418i	\(-0.348890\pi\)
							−0.987137	+	0.159879i	\(0.948890\pi\)
\(37\)	0.147419	+	0.0478995i	0.0242356	+	0.00787462i	0.321110	−	0.947042i	\(-0.395944\pi\)
							−0.296874	+	0.954917i	\(0.595944\pi\)
\(41\)	−10.1359	+	3.29336i	−1.58297	+	0.514337i	−0.962819	−	0.270148i	\(-0.912927\pi\)
							−0.620148	+	0.784485i	\(0.712927\pi\)
\(43\)	−2.05890			−0.313979			−0.156989	−	0.987600i	\(-0.550179\pi\)
							−0.156989	+	0.987600i	\(0.550179\pi\)
\(47\)	2.14137	+	6.59047i	0.312351	+	0.961318i	0.976831	+	0.214012i	\(0.0686532\pi\)
							−0.664480	+	0.747306i	\(0.731347\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.j.a.135.2	yes	64
11.4	even	5	inner	187.2.j.a.169.1	yes	64
17.16	even	2	inner	187.2.j.a.135.1	✓	64
187.169	even	10	inner	187.2.j.a.169.2	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.j.a.135.1	✓	64	17.16	even	2	inner
187.2.j.a.135.2	yes	64	1.1	even	1	trivial
187.2.j.a.169.1	yes	64	11.4	even	5	inner
187.2.j.a.169.2	yes	64	187.169	even	10	inner