Embedded newform 187.2.r.a.9.1

• Label: 187.2.r.a.9.1
• Level: $187$
• Weight: $2$
• Character: 187.9
• 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			9.1
Character	\(\chi\)	\(=\)	187.9
Dual form			187.2.r.a.104.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22225 - 2.39880i) q^{2} +(0.924642 - 0.221987i) q^{3} +(-3.08479 + 4.24585i) q^{4} +(3.50496 + 0.275847i) q^{5} +(-1.66265 - 1.94671i) q^{6} +(0.704149 - 2.93299i) q^{7} +(8.63717 + 1.36799i) q^{8} +(-1.86733 + 0.951454i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22225	−	2.39880i	−0.864263	−	1.69621i	−0.705279	−	0.708930i	\(-0.749178\pi\)
							−0.158983	−	0.987281i	\(-0.550822\pi\)
\(3\)	0.924642	−	0.221987i	0.533843	−	0.128164i	0.0424436	−	0.999099i	\(-0.486486\pi\)
							0.491399	+	0.870935i	\(0.336486\pi\)
\(5\)	3.50496	+	0.275847i	1.56747	+	0.123362i	0.832223	−	0.554441i	\(-0.187068\pi\)
							0.735244	+	0.677803i	\(0.237068\pi\)
\(7\)	0.704149	−	2.93299i	0.266143	−	1.10857i	−0.665849	−	0.746087i	\(-0.731930\pi\)
							0.931992	−	0.362479i	\(-0.118070\pi\)
\(11\)	0.447838	−	3.28625i	0.135028	−	0.990842i
							\(13\)	−0.680745	−	0.221187i	−0.188805	−	0.0613464i	0.213089	−	0.977033i	\(-0.431648\pi\)
−0.401893	+	0.915687i	\(0.631648\pi\)
\(17\)	3.39445	+	2.34045i	0.823275	+	0.567643i
							\(19\)	−0.130584	+	0.824478i	−0.0299581	+	0.189148i	−0.998129	−	0.0611416i	\(-0.980526\pi\)
0.968171	+	0.250290i	\(0.0805259\pi\)
\(23\)	−6.33596	+	2.62444i	−1.32114	+	0.547234i	−0.928115	−	0.372294i	\(-0.878571\pi\)
							−0.393025	+	0.919528i	\(0.628571\pi\)
\(29\)	−2.18265	+	1.33753i	−0.405308	+	0.248373i	−0.710180	−	0.704020i	\(-0.751386\pi\)
							0.304872	+	0.952393i	\(0.401386\pi\)
\(31\)	3.83941	−	4.49537i	0.689578	−	0.807392i	−0.299608	−	0.954063i	\(-0.596856\pi\)
							0.989185	+	0.146671i	\(0.0468558\pi\)
\(37\)	1.06310	+	1.73483i	0.174773	+	0.285204i	0.928017	−	0.372537i	\(-0.121512\pi\)
							−0.753244	+	0.657741i	\(0.771512\pi\)
\(41\)	−3.71603	−	2.27718i	−0.580346	−	0.355636i	0.201137	−	0.979563i	\(-0.435536\pi\)
							−0.781483	+	0.623927i	\(0.785536\pi\)
\(43\)	−2.22034	−	2.22034i	−0.338599	−	0.338599i	0.517241	−	0.855840i	\(-0.326959\pi\)
							−0.855840	+	0.517241i	\(0.826959\pi\)
\(47\)	3.49810	+	4.81473i	0.510251	+	0.702300i	0.983962	−	0.178380i	\(-0.0570858\pi\)
							−0.473711	+	0.880681i	\(0.657086\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.9.1	✓	256
11.5	even	5	inner	187.2.r.a.60.16	yes	256
17.2	even	8	inner	187.2.r.a.53.16	yes	256
187.104	even	40	inner	187.2.r.a.104.1	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.9.1	✓	256	1.1	even	1	trivial
187.2.r.a.53.16	yes	256	17.2	even	8	inner
187.2.r.a.60.16	yes	256	11.5	even	5	inner
187.2.r.a.104.1	yes	256	187.104	even	40	inner