Embedded newform 187.2.p.a.4.8

• Label: 187.2.p.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	187.4
Dual form			187.2.p.a.47.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.212815 + 0.292914i) q^{2} +(-1.13391 - 2.22542i) q^{3} +(0.577525 + 1.77744i) q^{4} +(-0.417832 + 2.63808i) q^{5} +(0.893168 + 0.141464i) q^{6} +(-1.09590 + 2.15082i) q^{7} +(-1.33223 - 0.432866i) q^{8} +(-1.90338 + 2.61978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.212815	+	0.292914i	−0.150483	+	0.207122i	−0.877603	−	0.479389i	\(-0.840858\pi\)
							0.727120	+	0.686510i	\(0.240858\pi\)
\(3\)	−1.13391	−	2.22542i	−0.654661	−	1.28485i	−0.944733	−	0.327840i	\(-0.893679\pi\)
							0.290072	−	0.957005i	\(-0.406321\pi\)
\(5\)	−0.417832	+	2.63808i	−0.186860	+	1.17979i	0.698753	+	0.715363i	\(0.253739\pi\)
							−0.885613	+	0.464424i	\(0.846261\pi\)
\(7\)	−1.09590	+	2.15082i	−0.414211	+	0.812934i	0.585786	+	0.810466i	\(0.300786\pi\)
							−0.999997	+	0.00246882i	\(0.999214\pi\)
\(11\)	3.14013	+	1.06751i	0.946785	+	0.321867i
							\(13\)	0.246681	+	0.179224i	0.0684169	+	0.0497078i	0.621468	−	0.783440i	\(-0.286537\pi\)
−0.553051	+	0.833147i	\(0.686537\pi\)
\(17\)	−0.0832945	+	4.12226i	−0.0202019	+	0.999796i
							\(19\)	4.79068	+	1.55659i	1.09906	+	0.357105i	0.801739	−	0.597674i	\(-0.203908\pi\)
0.297317	+	0.954779i	\(0.403908\pi\)
\(23\)	−3.58870	−	3.58870i	−0.748295	−	0.748295i	0.225864	−	0.974159i	\(-0.427479\pi\)
							−0.974159	+	0.225864i	\(0.927479\pi\)
\(29\)	−7.38127	−	3.76094i	−1.37067	−	0.698390i	−0.395212	−	0.918590i	\(-0.629329\pi\)
							−0.975455	+	0.220200i	\(0.929329\pi\)
\(31\)	9.60223	−	1.52084i	1.72461	−	0.273152i	0.786025	−	0.618195i	\(-0.212136\pi\)
							0.938587	+	0.345044i	\(0.112136\pi\)
\(37\)	−1.47213	−	0.750087i	−0.242017	−	0.123314i	0.328776	−	0.944408i	\(-0.393364\pi\)
							−0.570792	+	0.821094i	\(0.693364\pi\)
\(41\)	−4.57680	+	2.33199i	−0.714775	+	0.364196i	−0.773257	−	0.634092i	\(-0.781374\pi\)
							0.0584821	+	0.998288i	\(0.481374\pi\)
\(43\)			4.34129i			0.662040i	0.943624	+	0.331020i	\(0.107393\pi\)
							−0.943624	+	0.331020i	\(0.892607\pi\)
\(47\)	0.901542	−	2.77466i	0.131503	−	0.404726i	−0.863526	−	0.504304i	\(-0.831749\pi\)
							0.995030	+	0.0995778i	\(0.0317492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.p.a.4.8	✓	128
11.3	even	5	inner	187.2.p.a.157.9	yes	128
17.13	even	4	inner	187.2.p.a.81.9	yes	128
187.47	even	20	inner	187.2.p.a.47.8	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.p.a.4.8	✓	128	1.1	even	1	trivial
187.2.p.a.47.8	yes	128	187.47	even	20	inner
187.2.p.a.81.9	yes	128	17.13	even	4	inner
187.2.p.a.157.9	yes	128	11.3	even	5	inner