Embedded newform 187.2.p.a.4.13

• Label: 187.2.p.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	187.4
Dual form			187.2.p.a.47.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.827467 - 1.13891i) q^{2} +(0.330414 + 0.648475i) q^{3} +(0.00561864 + 0.0172924i) q^{4} +(-0.427401 + 2.69851i) q^{5} +(1.01196 + 0.160279i) q^{6} +(-1.38515 + 2.71850i) q^{7} +(2.70208 + 0.877960i) q^{8} +(1.45201 - 1.99852i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.827467	−	1.13891i	0.585107	−	0.805331i	−0.409136	−	0.912473i	\(-0.634170\pi\)
							0.994244	+	0.107142i	\(0.0341700\pi\)
\(3\)	0.330414	+	0.648475i	0.190765	+	0.374397i	0.966502	−	0.256659i	\(-0.0826218\pi\)
							−0.775737	+	0.631056i	\(0.782622\pi\)
\(5\)	−0.427401	+	2.69851i	−0.191140	+	1.20681i	0.686371	+	0.727251i	\(0.259203\pi\)
							−0.877511	+	0.479557i	\(0.840797\pi\)
\(7\)	−1.38515	+	2.71850i	−0.523536	+	1.02750i	0.466212	+	0.884673i	\(0.345618\pi\)
							−0.989748	+	0.142825i	\(0.954382\pi\)
\(11\)	1.55817	−	2.92782i	0.469806	−	0.882770i
							\(13\)	−4.54012	−	3.29859i	−1.25920	−	0.914864i	−0.260484	−	0.965478i	\(-0.583882\pi\)
−0.998718	+	0.0506138i	\(0.983882\pi\)
\(17\)	−2.32765	−	3.40324i	−0.564539	−	0.825407i
							\(19\)	0.409221	+	0.132964i	0.0938818	+	0.0305040i	0.355581	−	0.934645i	\(-0.384283\pi\)
−0.261699	+	0.965149i	\(0.584283\pi\)
\(23\)	−1.11081	−	1.11081i	−0.231619	−	0.231619i	0.581749	−	0.813368i	\(-0.302369\pi\)
							−0.813368	+	0.581749i	\(0.802369\pi\)
\(29\)	6.29931	+	3.20966i	1.16975	+	0.596019i	0.927364	−	0.374160i	\(-0.122069\pi\)
							0.242389	+	0.970179i	\(0.422069\pi\)
\(31\)	5.86059	−	0.928226i	1.05259	−	0.166714i	0.393920	−	0.919145i	\(-0.371119\pi\)
							0.658672	+	0.752430i	\(0.271119\pi\)
\(37\)	0.613678	+	0.312685i	0.100888	+	0.0514050i	0.503706	−	0.863875i	\(-0.331969\pi\)
							−0.402818	+	0.915280i	\(0.631969\pi\)
\(41\)	−8.12599	+	4.14040i	−1.26907	+	0.646622i	−0.953245	−	0.302197i	\(-0.902280\pi\)
							−0.315821	+	0.948819i	\(0.602280\pi\)
\(43\)		−	10.7229i		−	1.63523i	−0.575768	−	0.817613i	\(-0.695297\pi\)
							0.575768	−	0.817613i	\(-0.304703\pi\)
\(47\)	0.0244911	−	0.0753757i	0.00357239	−	0.0109947i	−0.949254	−	0.314509i	\(-0.898160\pi\)
							0.952827	+	0.303514i	\(0.0981601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.p.a.4.13	✓	128
11.3	even	5	inner	187.2.p.a.157.4	yes	128
17.13	even	4	inner	187.2.p.a.81.4	yes	128
187.47	even	20	inner	187.2.p.a.47.13	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.p.a.4.13	✓	128	1.1	even	1	trivial
187.2.p.a.47.13	yes	128	187.47	even	20	inner
187.2.p.a.81.4	yes	128	17.13	even	4	inner
187.2.p.a.157.4	yes	128	11.3	even	5	inner