Embedded newform 187.2.r.a.9.10

• Label: 187.2.r.a.9.10
• Level: $187$
• Weight: $2$
• Character: 187.9
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.r (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			9.10
Character	\(\chi\)	\(=\)	187.9
Dual form			187.2.r.a.104.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.135448 + 0.265831i) q^{2} +(-3.01634 + 0.724159i) q^{3} +(1.12325 - 1.54602i) q^{4} +(1.25823 + 0.0990246i) q^{5} +(-0.601061 - 0.703752i) q^{6} +(-0.700911 + 2.91950i) q^{7} +(1.15248 + 0.182534i) q^{8} +(5.90088 - 3.00665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.135448	+	0.265831i	0.0957760	+	0.187971i	0.933924	−	0.357472i	\(-0.116361\pi\)
							−0.838148	+	0.545443i	\(0.816361\pi\)
\(3\)	−3.01634	+	0.724159i	−1.74148	+	0.418093i	−0.974647	−	0.223748i	\(-0.928171\pi\)
							−0.766837	+	0.641841i	\(0.778171\pi\)
\(5\)	1.25823	+	0.0990246i	0.562696	+	0.0442852i	0.356618	−	0.934250i	\(-0.383930\pi\)
							0.206078	+	0.978535i	\(0.433930\pi\)
\(7\)	−0.700911	+	2.91950i	−0.264919	+	1.10347i	0.668223	+	0.743961i	\(0.267055\pi\)
							−0.933142	+	0.359507i	\(0.882945\pi\)
\(11\)	1.83393	+	2.76346i	0.552951	+	0.833214i
							\(13\)	3.69281	+	1.19987i	1.02420	+	0.332784i	0.772496	−	0.635020i	\(-0.219008\pi\)
0.251707	+	0.967804i	\(0.419008\pi\)
\(17\)	0.560947	+	4.08477i	0.136050	+	0.990702i
							\(19\)	−0.130731	+	0.825402i	−0.0299917	+	0.189360i	−0.998136	−	0.0610340i	\(-0.980560\pi\)
0.968144	+	0.250394i	\(0.0805602\pi\)
\(23\)	1.32100	−	0.547178i	0.275449	−	0.114095i	−0.240683	−	0.970604i	\(-0.577371\pi\)
							0.516132	+	0.856509i	\(0.327371\pi\)
\(29\)	3.23987	−	1.98539i	0.601628	−	0.368678i	−0.188087	−	0.982152i	\(-0.560229\pi\)
							0.789715	+	0.613474i	\(0.210229\pi\)
\(31\)	2.03049	−	2.37740i	0.364687	−	0.426994i	−0.547476	−	0.836822i	\(-0.684411\pi\)
							0.912163	+	0.409828i	\(0.134411\pi\)
\(37\)	−4.25913	−	6.95027i	−0.700197	−	1.14262i	−0.983029	−	0.183448i	\(-0.941274\pi\)
							0.282833	−	0.959169i	\(-0.408726\pi\)
\(41\)	−1.58994	−	0.974319i	−0.248308	−	0.152163i	0.392776	−	0.919634i	\(-0.371515\pi\)
							−0.641084	+	0.767471i	\(0.721515\pi\)
\(43\)	−4.70248	−	4.70248i	−0.717121	−	0.717121i	0.250893	−	0.968015i	\(-0.419276\pi\)
							−0.968015	+	0.250893i	\(0.919276\pi\)
\(47\)	−2.24118	−	3.08472i	−0.326910	−	0.449953i	0.613651	−	0.789577i	\(-0.289700\pi\)
							−0.940561	+	0.339624i	\(0.889700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.9.10	✓	256
11.5	even	5	inner	187.2.r.a.60.7	yes	256
17.2	even	8	inner	187.2.r.a.53.7	yes	256
187.104	even	40	inner	187.2.r.a.104.10	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.9.10	✓	256	1.1	even	1	trivial
187.2.r.a.53.7	yes	256	17.2	even	8	inner
187.2.r.a.60.7	yes	256	11.5	even	5	inner
187.2.r.a.104.10	yes	256	187.104	even	40	inner