Embedded newform 187.2.p.a.4.5

• Label: 187.2.p.a.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	187.4
Dual form			187.2.p.a.47.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.887944 + 1.22215i) q^{2} +(-0.0267877 - 0.0525737i) q^{3} +(-0.0871729 - 0.268291i) q^{4} +(0.507206 - 3.20238i) q^{5} +(0.0880390 + 0.0139440i) q^{6} +(1.63147 - 3.20194i) q^{7} +(-2.46815 - 0.801951i) q^{8} +(1.76131 - 2.42423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.887944	+	1.22215i	−0.627872	+	0.864191i	−0.997896	−	0.0648309i	\(-0.979349\pi\)
							0.370025	+	0.929022i	\(0.379349\pi\)
\(3\)	−0.0267877	−	0.0525737i	−0.0154659	−	0.0303535i	0.883144	−	0.469101i	\(-0.155422\pi\)
							−0.898610	+	0.438748i	\(0.855422\pi\)
\(5\)	0.507206	−	3.20238i	0.226830	−	1.43215i	−0.566856	−	0.823817i	\(-0.691840\pi\)
							0.793685	−	0.608329i	\(-0.208160\pi\)
\(7\)	1.63147	−	3.20194i	0.616637	−	1.21022i	−0.345696	−	0.938347i	\(-0.612357\pi\)
							0.962333	−	0.271872i	\(-0.0876428\pi\)
\(11\)	1.00289	+	3.16136i	0.302382	+	0.953187i
							\(13\)	−2.84858	−	2.06962i	−0.790054	−	0.574008i	0.117925	−	0.993022i	\(-0.462376\pi\)
−0.907980	+	0.419014i	\(0.862376\pi\)
\(17\)	−1.82825	+	3.69561i	−0.443415	+	0.896317i
							\(19\)	4.33784	+	1.40945i	0.995168	+	0.323350i	0.760933	−	0.648830i	\(-0.224741\pi\)
0.234235	+	0.972180i	\(0.424741\pi\)
\(23\)	0.118809	+	0.118809i	0.0247735	+	0.0247735i	0.719385	−	0.694612i	\(-0.244424\pi\)
							−0.694612	+	0.719385i	\(0.744424\pi\)
\(29\)	0.707515	+	0.360497i	0.131382	+	0.0669426i	0.518446	−	0.855110i	\(-0.326511\pi\)
							−0.387064	+	0.922053i	\(0.626511\pi\)
\(31\)	1.11975	−	0.177351i	0.201113	−	0.0318532i	−0.0550652	−	0.998483i	\(-0.517537\pi\)
							0.256178	+	0.966630i	\(0.417537\pi\)
\(37\)	7.07655	+	3.60568i	1.16338	+	0.592770i	0.925583	−	0.378545i	\(-0.123575\pi\)
							0.237795	+	0.971315i	\(0.423575\pi\)
\(41\)	−10.7035	+	5.45371i	−1.67161	+	0.851726i	−0.678488	+	0.734611i	\(0.737365\pi\)
							−0.993118	+	0.117115i	\(0.962635\pi\)
\(43\)			0.0466111i			0.00710813i	0.999994	+	0.00355407i	\(0.00113130\pi\)
							−0.999994	+	0.00355407i	\(0.998869\pi\)
\(47\)	3.53619	−	10.8833i	0.515806	−	1.58749i	−0.266005	−	0.963972i	\(-0.585704\pi\)
							0.781811	−	0.623515i	\(-0.214296\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.p.a.4.5	✓	128
11.3	even	5	inner	187.2.p.a.157.12	yes	128
17.13	even	4	inner	187.2.p.a.81.12	yes	128
187.47	even	20	inner	187.2.p.a.47.5	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.p.a.4.5	✓	128	1.1	even	1	trivial
187.2.p.a.47.5	yes	128	187.47	even	20	inner
187.2.p.a.81.12	yes	128	17.13	even	4	inner
187.2.p.a.157.12	yes	128	11.3	even	5	inner