Embedded newform 187.2.p.a.38.2

• Label: 187.2.p.a.38.2
• Level: $187$
• Weight: $2$
• Character: 187.38
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.p (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			38.2
Character	\(\chi\)	\(=\)	187.38
Dual form			187.2.p.a.64.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96554 + 0.638643i) q^{2} +(-1.63407 + 0.258811i) q^{3} +(1.83745 - 1.33498i) q^{4} +(0.781962 + 1.53469i) q^{5} +(3.04654 - 1.55229i) q^{6} +(3.21586 + 0.509342i) q^{7} +(-0.329459 + 0.453461i) q^{8} +(-0.249973 + 0.0812212i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96554	+	0.638643i	−1.38985	+	0.451588i	−0.905894	−	0.423506i	\(-0.860799\pi\)
							−0.483953	+	0.875094i	\(0.660799\pi\)
\(3\)	−1.63407	+	0.258811i	−0.943430	+	0.149425i	−0.609155	−	0.793051i	\(-0.708491\pi\)
							−0.334275	+	0.942476i	\(0.608491\pi\)
\(5\)	0.781962	+	1.53469i	0.349704	+	0.686333i	0.997123	−	0.0758014i	\(-0.0241515\pi\)
							−0.647419	+	0.762134i	\(0.724151\pi\)
\(7\)	3.21586	+	0.509342i	1.21548	+	0.192513i	0.731055	−	0.682319i	\(-0.239029\pi\)
							0.484426	+	0.874832i	\(0.339029\pi\)
\(11\)	−3.22188	+	0.787074i	−0.971434	+	0.237312i
							\(13\)	0.664548	+	2.04527i	0.184312	+	0.567255i	0.999936	−	0.0113278i	\(-0.00360582\pi\)
−0.815623	+	0.578583i	\(0.803606\pi\)
\(17\)	−4.00606	+	0.975434i	−0.971613	+	0.236577i
							\(19\)	−2.64979	+	3.64712i	−0.607904	+	0.836708i	−0.996403	−	0.0847413i	\(-0.972994\pi\)
0.388499	+	0.921449i	\(0.372994\pi\)
\(23\)	−2.83755	−	2.83755i	−0.591671	−	0.591671i	0.346412	−	0.938083i	\(-0.387400\pi\)
							−0.938083	+	0.346412i	\(0.887400\pi\)
\(29\)	−1.65212	+	10.4311i	−0.306791	+	1.93700i	0.0403431	+	0.999186i	\(0.487155\pi\)
							−0.347134	+	0.937816i	\(0.612845\pi\)
\(31\)	−4.31072	−	2.19642i	−0.774228	−	0.394489i	0.0217893	−	0.999763i	\(-0.493064\pi\)
							−0.796017	+	0.605274i	\(0.793064\pi\)
\(37\)	−0.386442	+	2.43990i	−0.0635307	+	0.401117i	0.935346	+	0.353734i	\(0.115088\pi\)
							−0.998877	+	0.0473830i	\(0.984912\pi\)
\(41\)	1.25493	+	7.92328i	0.195986	+	1.23741i	0.867885	+	0.496766i	\(0.165479\pi\)
							−0.671898	+	0.740643i	\(0.734521\pi\)
\(43\)		−	1.75864i		−	0.268190i	−0.990968	−	0.134095i	\(-0.957187\pi\)
							0.990968	−	0.134095i	\(-0.0428127\pi\)
\(47\)	1.27188	+	0.924074i	0.185523	+	0.134790i	0.676670	−	0.736286i	\(-0.263422\pi\)
							−0.491147	+	0.871076i	\(0.663422\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.p.a.38.2	✓	128
11.9	even	5	inner	187.2.p.a.174.15	yes	128
17.13	even	4	inner	187.2.p.a.115.15	yes	128
187.64	even	20	inner	187.2.p.a.64.2	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.p.a.38.2	✓	128	1.1	even	1	trivial
187.2.p.a.64.2	yes	128	187.64	even	20	inner
187.2.p.a.115.15	yes	128	17.13	even	4	inner
187.2.p.a.174.15	yes	128	11.9	even	5	inner