Embedded newform 187.2.p.a.4.6

• Label: 187.2.p.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	187.4
Dual form			187.2.p.a.47.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.830083 + 1.14251i) q^{2} +(0.831391 + 1.63170i) q^{3} +(0.00173994 + 0.00535500i) q^{4} +(0.0745546 - 0.470719i) q^{5} +(-2.55435 - 0.404570i) q^{6} +(-1.69465 + 3.32594i) q^{7} +(-2.69377 - 0.875258i) q^{8} +(-0.207866 + 0.286102i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 10 q^{3} + 16 q^{4} - 2 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.830083	+	1.14251i	−0.586957	+	0.807877i	−0.994437	−	0.105337i	\(-0.966408\pi\)
							0.407479	+	0.913214i	\(0.366408\pi\)
\(3\)	0.831391	+	1.63170i	0.480004	+	0.942060i	0.996324	+	0.0856646i	\(0.0273014\pi\)
							−0.516320	+	0.856395i	\(0.672699\pi\)
\(5\)	0.0745546	−	0.470719i	0.0333418	−	0.210512i	−0.965393	−	0.260800i	\(-0.916014\pi\)
							0.998735	+	0.0502876i	\(0.0160138\pi\)
\(7\)	−1.69465	+	3.32594i	−0.640518	+	1.25709i	0.311267	+	0.950323i	\(0.399247\pi\)
							−0.951785	+	0.306765i	\(0.900753\pi\)
\(11\)	2.94580	−	1.52390i	0.888191	−	0.459474i
							\(13\)	0.799572	+	0.580923i	0.221761	+	0.161119i	0.693119	−	0.720823i	\(-0.256236\pi\)
−0.471358	+	0.881942i	\(0.656236\pi\)
\(17\)	2.02545	+	3.59132i	0.491244	+	0.871022i
							\(19\)	−7.03198	−	2.28483i	−1.61325	−	0.524176i	−0.642912	−	0.765940i	\(-0.722274\pi\)
−0.970335	+	0.241765i	\(0.922274\pi\)
\(23\)	−3.76435	−	3.76435i	−0.784921	−	0.784921i	0.195736	−	0.980657i	\(-0.437290\pi\)
							−0.980657	+	0.195736i	\(0.937290\pi\)
\(29\)	6.98485	+	3.55896i	1.29705	+	0.660882i	0.959841	−	0.280545i	\(-0.0905154\pi\)
							0.337214	+	0.941428i	\(0.390515\pi\)
\(31\)	−5.74425	+	0.909800i	−1.03170	+	0.163405i	−0.649252	−	0.760573i	\(-0.724918\pi\)
							−0.382445	+	0.923978i	\(0.624918\pi\)
\(37\)	4.39010	+	2.23687i	0.721728	+	0.367739i	0.775954	−	0.630790i	\(-0.217269\pi\)
							−0.0542255	+	0.998529i	\(0.517269\pi\)
\(41\)	10.6521	−	5.42753i	1.66358	−	0.847637i	0.669048	−	0.743219i	\(-0.266702\pi\)
							0.994533	−	0.104418i	\(-0.0332980\pi\)
\(43\)			7.88436i			1.20235i	0.799116	+	0.601177i	\(0.205301\pi\)
							−0.799116	+	0.601177i	\(0.794699\pi\)
\(47\)	2.13749	−	6.57853i	0.311785	−	0.959577i	−0.665272	−	0.746601i	\(-0.731684\pi\)
							0.977057	−	0.212976i	\(-0.0683155\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.p.a.4.6	✓	128
11.3	even	5	inner	187.2.p.a.157.11	yes	128
17.13	even	4	inner	187.2.p.a.81.11	yes	128
187.47	even	20	inner	187.2.p.a.47.6	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.p.a.4.6	✓	128	1.1	even	1	trivial
187.2.p.a.47.6	yes	128	187.47	even	20	inner
187.2.p.a.81.11	yes	128	17.13	even	4	inner
187.2.p.a.157.11	yes	128	11.3	even	5	inner