Embedded newform 187.2.p.a.4.9

• Label: 187.2.p.a.4.9
• Level: $187$
• Weight: $2$
• Character: 187.4
• 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			4.9
Character	\(\chi\)	\(=\)	187.4
Dual form			187.2.p.a.47.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0756214 + 0.104084i) q^{2} +(1.34406 + 2.63788i) q^{3} +(0.612919 + 1.88637i) q^{4} +(0.0370110 - 0.233679i) q^{5} +(-0.376200 - 0.0595843i) q^{6} +(2.00593 - 3.93685i) q^{7} +(-0.487406 - 0.158368i) q^{8} +(-3.38852 + 4.66390i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0756214	+	0.104084i	−0.0534724	+	0.0735984i	−0.834916	−	0.550378i	\(-0.814484\pi\)
							0.781443	+	0.623976i	\(0.214484\pi\)
\(3\)	1.34406	+	2.63788i	0.775996	+	1.52298i	0.850629	+	0.525767i	\(0.176222\pi\)
							−0.0746323	+	0.997211i	\(0.523778\pi\)
\(5\)	0.0370110	−	0.233679i	0.0165518	−	0.104504i	−0.978029	−	0.208468i	\(-0.933152\pi\)
							0.994581	+	0.103963i	\(0.0331525\pi\)
\(7\)	2.00593	−	3.93685i	0.758169	−	1.48799i	−0.111185	−	0.993800i	\(-0.535465\pi\)
							0.869354	−	0.494190i	\(-0.164535\pi\)
\(11\)	0.152793	−	3.31310i	0.0460689	−	0.998938i
							\(13\)	−0.886214	−	0.643872i	−0.245792	−	0.178578i	0.458068	−	0.888917i	\(-0.348542\pi\)
−0.703860	+	0.710339i	\(0.748542\pi\)
\(17\)	−4.08124	−	0.586088i	−0.989846	−	0.142147i
							\(19\)	−2.33433	−	0.758470i	−0.535532	−	0.174005i	0.0287504	−	0.999587i	\(-0.490847\pi\)
−0.564283	+	0.825582i	\(0.690847\pi\)
\(23\)	−1.85674	−	1.85674i	−0.387157	−	0.387157i	0.486515	−	0.873672i	\(-0.338268\pi\)
							−0.873672	+	0.486515i	\(0.838268\pi\)
\(29\)	−0.0983429	−	0.0501082i	−0.0182618	−	0.00930486i	0.444836	−	0.895612i	\(-0.353262\pi\)
							−0.463098	+	0.886307i	\(0.653262\pi\)
\(31\)	8.30132	−	1.31480i	1.49096	−	0.236145i	0.642862	−	0.765982i	\(-0.277747\pi\)
							0.848099	+	0.529837i	\(0.177747\pi\)
\(37\)	−3.80800	−	1.94027i	−0.626031	−	0.318979i	0.112038	−	0.993704i	\(-0.464262\pi\)
							−0.738069	+	0.674725i	\(0.764262\pi\)
\(41\)	−6.99716	+	3.56523i	−1.09277	+	0.556795i	−0.904998	−	0.425417i	\(-0.860128\pi\)
							−0.187775	+	0.982212i	\(0.560128\pi\)
\(43\)			9.54002i			1.45484i	0.686193	+	0.727419i	\(0.259281\pi\)
							−0.686193	+	0.727419i	\(0.740719\pi\)
\(47\)	−2.86547	+	8.81902i	−0.417972	+	1.28639i	0.491593	+	0.870825i	\(0.336415\pi\)
							−0.909565	+	0.415561i	\(0.863585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.p.a.4.9	✓	128
11.3	even	5	inner	187.2.p.a.157.8	yes	128
17.13	even	4	inner	187.2.p.a.81.8	yes	128
187.47	even	20	inner	187.2.p.a.47.9	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.p.a.4.9	✓	128	1.1	even	1	trivial
187.2.p.a.47.9	yes	128	187.47	even	20	inner
187.2.p.a.81.8	yes	128	17.13	even	4	inner
187.2.p.a.157.8	yes	128	11.3	even	5	inner