Embedded newform 147.2.o.b.59.9

• Label: 147.2.o.b.59.9
• Level: $147$
• Weight: $2$
• Character: 147.59
• Analytic conductor: $1.174$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17380090971\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(16\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			59.9
Character	\(\chi\)	\(=\)	147.59
Dual form			147.2.o.b.5.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0569582 - 0.377893i) q^{2} +(-1.71778 - 0.221866i) q^{3} +(1.77159 + 0.546462i) q^{4} +(0.273303 + 3.64698i) q^{5} +(-0.181683 + 0.636501i) q^{6} +(-2.49430 + 0.882305i) q^{7} +(0.639038 - 1.32698i) q^{8} +(2.90155 + 0.762233i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0569582	−	0.377893i	0.0402755	−	0.267211i	−0.959625	−	0.281281i	\(-0.909241\pi\)
							0.999901	+	0.0140701i	\(0.00447882\pi\)
\(3\)	−1.71778	−	0.221866i	−0.991762	−	0.128094i
							\(5\)	0.273303	+	3.64698i	0.122225	+	1.63098i	0.635039	+	0.772480i	\(0.280984\pi\)
−0.512814	+	0.858500i	\(0.671397\pi\)
\(7\)	−2.49430	+	0.882305i	−0.942757	+	0.333480i
							\(11\)	0.795717	−	0.312296i	0.239918	−	0.0941607i	−0.242336	−	0.970192i	\(-0.577914\pi\)
0.482254	+	0.876032i	\(0.339818\pi\)
\(13\)	3.66163	+	2.92005i	1.01555	+	0.809877i	0.981870	−	0.189556i	\(-0.0607047\pi\)
							0.0336836	+	0.999433i	\(0.489276\pi\)
\(17\)	−3.49918	−	3.24677i	−0.848676	−	0.787456i	0.130291	−	0.991476i	\(-0.458409\pi\)
							−0.978967	+	0.204020i	\(0.934599\pi\)
\(19\)	2.80731	−	1.62080i	0.644041	−	0.371837i	−0.142128	−	0.989848i	\(-0.545395\pi\)
							0.786170	+	0.618011i	\(0.212061\pi\)
\(23\)	−2.30456	−	2.48372i	−0.480533	−	0.517892i	0.445526	−	0.895269i	\(-0.353017\pi\)
							−0.926060	+	0.377377i	\(0.876826\pi\)
\(29\)	0.0736869	+	0.0168186i	0.0136833	+	0.00312313i	0.229357	−	0.973342i	\(-0.426338\pi\)
							−0.215674	+	0.976465i	\(0.569195\pi\)
\(31\)	−4.42660	−	2.55570i	−0.795041	−	0.459017i	0.0466933	−	0.998909i	\(-0.485132\pi\)
							−0.841734	+	0.539892i	\(0.818465\pi\)
\(37\)	2.80340	−	0.864734i	0.460876	−	0.142161i	−0.0556183	−	0.998452i	\(-0.517713\pi\)
							0.516494	+	0.856291i	\(0.327237\pi\)
\(41\)	2.08912	+	1.00607i	0.326266	+	0.157122i	0.589845	−	0.807516i	\(-0.299189\pi\)
							−0.263579	+	0.964638i	\(0.584903\pi\)
\(43\)	−1.81967	+	0.876307i	−0.277497	+	0.133636i	−0.567457	−	0.823403i	\(-0.692073\pi\)
							0.289959	+	0.957039i	\(0.406358\pi\)
\(47\)	13.3590	+	2.01354i	1.94861	+	0.293705i	0.999757	−	0.0220259i	\(-0.00701162\pi\)
							0.948849	+	0.315731i	\(0.102250\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.59.9	yes	192
3.2	odd	2	inner	147.2.o.b.59.8	yes	192
49.5	odd	42	inner	147.2.o.b.5.8	✓	192
147.5	even	42	inner	147.2.o.b.5.9	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.5.8	✓	192	49.5	odd	42	inner
147.2.o.b.5.9	yes	192	147.5	even	42	inner
147.2.o.b.59.8	yes	192	3.2	odd	2	inner
147.2.o.b.59.9	yes	192	1.1	even	1	trivial