Embedded newform 187.2.p.a.38.4

• Label: 187.2.p.a.38.4
• 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			38.4
Character	\(\chi\)	\(=\)	187.38
Dual form			187.2.p.a.64.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.61464 + 0.524629i) q^{2} +(0.410158 - 0.0649626i) q^{3} +(0.713800 - 0.518606i) q^{4} +(-0.824518 - 1.61821i) q^{5} +(-0.628176 + 0.320072i) q^{6} +(-0.362922 - 0.0574812i) q^{7} +(1.11535 - 1.53515i) q^{8} +(-2.68916 + 0.873761i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.61464	+	0.524629i	−1.14172	+	0.370969i	−0.818020	−	0.575190i	\(-0.804928\pi\)
							−0.323704	+	0.946158i	\(0.604928\pi\)
\(3\)	0.410158	−	0.0649626i	0.236805	−	0.0375062i	−0.0369041	−	0.999319i	\(-0.511750\pi\)
							0.273709	+	0.961813i	\(0.411750\pi\)
\(5\)	−0.824518	−	1.61821i	−0.368736	−	0.723685i	0.629857	−	0.776711i	\(-0.283113\pi\)
							−0.998593	+	0.0530260i	\(0.983113\pi\)
\(7\)	−0.362922	−	0.0574812i	−0.137172	−	0.0217259i	0.0874710	−	0.996167i	\(-0.472121\pi\)
							−0.224643	+	0.974441i	\(0.572121\pi\)
\(11\)	−0.829714	−	3.21116i	−0.250168	−	0.968202i
							\(13\)	−0.970018	−	2.98541i	−0.269035	−	0.828003i	−0.990736	−	0.135799i	\(-0.956640\pi\)
0.721702	−	0.692204i	\(-0.243360\pi\)
\(17\)	3.01466	−	2.81280i	0.731161	−	0.682205i
							\(19\)	1.30042	−	1.78987i	0.298336	−	0.410624i	−0.633363	−	0.773855i	\(-0.718326\pi\)
0.931699	+	0.363230i	\(0.118326\pi\)
\(23\)	−3.48829	−	3.48829i	−0.727358	−	0.727358i	0.242735	−	0.970093i	\(-0.421955\pi\)
							−0.970093	+	0.242735i	\(0.921955\pi\)
\(29\)	0.624758	−	3.94456i	0.116015	−	0.732487i	−0.859268	−	0.511526i	\(-0.829080\pi\)
							0.975283	−	0.220962i	\(-0.0709195\pi\)
\(31\)	−1.28551	−	0.655000i	−0.230885	−	0.117642i	0.334722	−	0.942317i	\(-0.391358\pi\)
							−0.565606	+	0.824676i	\(0.691358\pi\)
\(37\)	−1.86182	+	11.7551i	−0.306082	+	1.93252i	0.0515693	+	0.998669i	\(0.483578\pi\)
							−0.357651	+	0.933855i	\(0.616422\pi\)
\(41\)	−0.711076	−	4.48956i	−0.111051	−	0.701151i	−0.978903	−	0.204327i	\(-0.934499\pi\)
							0.867851	−	0.496824i	\(-0.165501\pi\)
\(43\)			11.3922i			1.73729i	0.495434	+	0.868645i	\(0.335009\pi\)
							−0.495434	+	0.868645i	\(0.664991\pi\)
\(47\)	7.66315	+	5.56761i	1.11779	+	0.812119i	0.983872	−	0.178874i	\(-0.0572454\pi\)
							0.133914	+	0.990993i	\(0.457245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.p.a.38.4	✓	128
11.9	even	5	inner	187.2.p.a.174.13	yes	128
17.13	even	4	inner	187.2.p.a.115.13	yes	128
187.64	even	20	inner	187.2.p.a.64.4	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.p.a.38.4	✓	128	1.1	even	1	trivial
187.2.p.a.64.4	yes	128	187.64	even	20	inner
187.2.p.a.115.13	yes	128	17.13	even	4	inner
187.2.p.a.174.13	yes	128	11.9	even	5	inner