Embedded newform 187.2.r.a.15.4

• Label: 187.2.r.a.15.4
• Level: $187$
• Weight: $2$
• Character: 187.15
• Analytic conductor: $1.493$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.r (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.49320251780\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			15.4
Character	\(\chi\)	\(=\)	187.15
Dual form			187.2.r.a.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.270768 - 1.70956i) q^{2} +(-3.36059 + 0.264484i) q^{3} +(-0.947178 + 0.307757i) q^{4} +(-0.735432 + 1.20012i) q^{5} +(1.36209 + 5.67353i) q^{6} +(-0.137628 + 1.74873i) q^{7} +(-0.789004 - 1.54851i) q^{8} +(8.26056 - 1.30834i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(35\)	\(122\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.270768	−	1.70956i	−0.191462	−	1.20884i	−0.876886	−	0.480699i	\(-0.840383\pi\)
							0.685424	−	0.728145i	\(-0.259617\pi\)
\(3\)	−3.36059	+	0.264484i	−1.94024	+	0.152700i	−0.988189	−	0.153243i	\(-0.951028\pi\)
							−0.952050	+	0.305943i	\(0.901028\pi\)
\(5\)	−0.735432	+	1.20012i	−0.328895	+	0.536708i	−0.973757	−	0.227591i	\(-0.926915\pi\)
							0.644862	+	0.764299i	\(0.276915\pi\)
\(7\)	−0.137628	+	1.74873i	−0.0520186	+	0.660958i	0.914537	+	0.404502i	\(0.132555\pi\)
							−0.966556	+	0.256457i	\(0.917445\pi\)
\(11\)	2.23670	+	2.44892i	0.674389	+	0.738376i
							\(13\)	−1.59195	+	2.19113i	−0.441527	+	0.607710i	−0.970551	−	0.240897i	\(-0.922558\pi\)
0.529024	+	0.848607i	\(0.322558\pi\)
\(17\)	3.65414	+	1.90978i	0.886259	+	0.463191i
							\(19\)	−2.90573	+	1.48054i	−0.666619	+	0.339659i	−0.754338	−	0.656486i	\(-0.772042\pi\)
0.0877192	+	0.996145i	\(0.472042\pi\)
\(23\)	0.513189	−	1.23895i	0.107007	−	0.258339i	−0.861302	−	0.508094i	\(-0.830350\pi\)
							0.968309	+	0.249756i	\(0.0803502\pi\)
\(29\)	0.748007	+	0.875804i	0.138901	+	0.162633i	0.825496	−	0.564408i	\(-0.190895\pi\)
							−0.686595	+	0.727040i	\(0.740895\pi\)
\(31\)	−2.39104	+	9.95938i	−0.429443	+	1.78876i	0.166840	+	0.985984i	\(0.446644\pi\)
							−0.596283	+	0.802774i	\(0.703356\pi\)
\(37\)	−4.10977	+	3.51007i	−0.675641	+	0.577052i	−0.919659	−	0.392719i	\(-0.871535\pi\)
							0.244017	+	0.969771i	\(0.421535\pi\)
\(41\)	4.25329	−	4.97996i	0.664253	−	0.777740i	−0.321324	−	0.946969i	\(-0.604128\pi\)
							0.985577	+	0.169229i	\(0.0541279\pi\)
\(43\)	0.377603	−	0.377603i	0.0575840	−	0.0575840i	−0.677728	−	0.735312i	\(-0.737035\pi\)
							0.735312	+	0.677728i	\(0.237035\pi\)
\(47\)	−7.05246	−	2.29148i	−1.02871	−	0.334247i	−0.254428	−	0.967092i	\(-0.581887\pi\)
							−0.774279	+	0.632844i	\(0.781887\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.15.4	✓	256
11.3	even	5	inner	187.2.r.a.168.4	yes	256
17.8	even	8	inner	187.2.r.a.59.4	yes	256
187.25	even	40	inner	187.2.r.a.25.4	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.15.4	✓	256	1.1	even	1	trivial
187.2.r.a.25.4	yes	256	187.25	even	40	inner
187.2.r.a.59.4	yes	256	17.8	even	8	inner
187.2.r.a.168.4	yes	256	11.3	even	5	inner