Embedded newform 187.2.r.a.9.9

• Label: 187.2.r.a.9.9
• 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			9.9
Character	\(\chi\)	\(=\)	187.9
Dual form			187.2.r.a.104.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0402376 + 0.0789707i) q^{2} +(2.17340 - 0.521788i) q^{3} +(1.17095 - 1.61168i) q^{4} +(0.213343 + 0.0167905i) q^{5} +(0.128658 + 0.150640i) q^{6} +(-0.980687 + 4.08486i) q^{7} +(0.349471 + 0.0553508i) q^{8} +(1.77840 - 0.906138i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{2} - 12 q^{3} - 20 q^{5} - 12 q^{6} - 12 q^{7} - 28 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0402376	+	0.0789707i	0.0284523	+	0.0558407i	0.904795	−	0.425847i	\(-0.140024\pi\)
							−0.876343	+	0.481688i	\(0.840024\pi\)
\(3\)	2.17340	−	0.521788i	1.25481	−	0.301254i	0.449059	−	0.893502i	\(-0.351759\pi\)
							0.805756	+	0.592248i	\(0.201759\pi\)
\(5\)	0.213343	+	0.0167905i	0.0954101	+	0.00750894i	0.126074	−	0.992021i	\(-0.459762\pi\)
							−0.0306642	+	0.999530i	\(0.509762\pi\)
\(7\)	−0.980687	+	4.08486i	−0.370665	+	1.54393i	0.405914	+	0.913911i	\(0.366953\pi\)
							−0.776579	+	0.630020i	\(0.783047\pi\)
\(11\)	−2.93687	−	1.54103i	−0.885500	−	0.464639i
							\(13\)	−4.73346	−	1.53800i	−1.31283	−	0.426563i	−0.432801	−	0.901489i	\(-0.642475\pi\)
−0.880026	+	0.474926i	\(0.842475\pi\)
\(17\)	4.11355	−	0.280619i	0.997681	−	0.0680601i
							\(19\)	−0.828375	+	5.23015i	−0.190042	+	1.19988i	0.689581	+	0.724209i	\(0.257795\pi\)
−0.879623	+	0.475671i	\(0.842205\pi\)
\(23\)	0.803940	−	0.333003i	0.167633	−	0.0694359i	−0.297289	−	0.954788i	\(-0.596082\pi\)
							0.464922	+	0.885352i	\(0.346082\pi\)
\(29\)	8.29093	−	5.08069i	1.53959	−	0.943460i	0.545126	−	0.838354i	\(-0.316482\pi\)
							0.994461	−	0.105106i	\(-0.0335180\pi\)
\(31\)	1.83042	−	2.14315i	0.328754	−	0.384921i	−0.571270	−	0.820762i	\(-0.693549\pi\)
							0.900024	+	0.435841i	\(0.143549\pi\)
\(37\)	2.99092	+	4.88075i	0.491705	+	0.802390i	0.998207	−	0.0598489i	\(-0.0190619\pi\)
							−0.506502	+	0.862239i	\(0.669062\pi\)
\(41\)	−4.01672	−	2.46145i	−0.627307	−	0.384414i	0.172208	−	0.985061i	\(-0.444910\pi\)
							−0.799515	+	0.600647i	\(0.794910\pi\)
\(43\)	−6.25236	−	6.25236i	−0.953476	−	0.953476i	0.0454890	−	0.998965i	\(-0.485515\pi\)
							−0.998965	+	0.0454890i	\(0.985515\pi\)
\(47\)	7.00227	+	9.63779i	1.02139	+	1.40582i	0.911228	+	0.411902i	\(0.135136\pi\)
							0.110157	+	0.993914i	\(0.464864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.r.a.9.9	✓	256
11.5	even	5	inner	187.2.r.a.60.8	yes	256
17.2	even	8	inner	187.2.r.a.53.8	yes	256
187.104	even	40	inner	187.2.r.a.104.9	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.r.a.9.9	✓	256	1.1	even	1	trivial
187.2.r.a.53.8	yes	256	17.2	even	8	inner
187.2.r.a.60.8	yes	256	11.5	even	5	inner
187.2.r.a.104.9	yes	256	187.104	even	40	inner