Embedded newform 147.2.o.b.122.4

• Label: 147.2.o.b.122.4
• Level: $147$
• Weight: $2$
• Character: 147.122
• 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			122.4
Character	\(\chi\)	\(=\)	147.122
Dual form			147.2.o.b.47.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.539030 - 1.74749i) q^{2} +(1.37089 + 1.05861i) q^{3} +(-1.11070 + 0.757259i) q^{4} +(-0.891238 - 0.134333i) q^{5} +(1.11096 - 2.96624i) q^{6} +(1.52731 - 2.16040i) q^{7} +(-0.937527 - 0.747653i) q^{8} +(0.758682 + 2.90248i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{37}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.539030	−	1.74749i	−0.381152	−	1.23566i	−0.919550	−	0.392973i	\(-0.871447\pi\)
							0.538399	−	0.842690i	\(-0.319029\pi\)
\(3\)	1.37089	+	1.05861i	0.791484	+	0.611190i
							\(5\)	−0.891238	−	0.134333i	−0.398574	−	0.0600753i	−0.0533029	−	0.998578i	\(-0.516975\pi\)
−0.345271	+	0.938503i	\(0.612213\pi\)
\(7\)	1.52731	−	2.16040i	0.577269	−	0.816554i
							\(11\)	3.88014	−	4.18179i	1.16991	−	1.26086i	0.210888	−	0.977510i	\(-0.432364\pi\)
0.959017	−	0.283348i	\(-0.0914451\pi\)
\(13\)	1.39947	+	0.319421i	0.388144	+	0.0885914i	0.412140	−	0.911121i	\(-0.364782\pi\)
							−0.0239954	+	0.999712i	\(0.507639\pi\)
\(17\)	−0.352296	+	4.70106i	−0.0854443	+	1.14017i	0.775456	+	0.631401i	\(0.217520\pi\)
							−0.860901	+	0.508773i	\(0.830099\pi\)
\(19\)	−5.39140	+	3.11273i	−1.23687	+	0.714108i	−0.968454	−	0.249194i	\(-0.919834\pi\)
							−0.268418	+	0.963302i	\(0.586501\pi\)
\(23\)	−3.46185	+	0.259430i	−0.721845	+	0.0540948i	−0.430586	−	0.902549i	\(-0.641693\pi\)
							−0.291259	+	0.956644i	\(0.594074\pi\)
\(29\)	2.44739	+	5.08206i	0.454469	+	0.943715i	0.994760	+	0.102236i	\(0.0325998\pi\)
							−0.540291	+	0.841478i	\(0.681686\pi\)
\(31\)	5.42328	+	3.13113i	0.974049	+	0.562368i	0.900468	−	0.434922i	\(-0.143224\pi\)
							0.0735810	+	0.997289i	\(0.476557\pi\)
\(37\)	−4.28219	−	2.91955i	−0.703987	−	0.479970i	0.157650	−	0.987495i	\(-0.449608\pi\)
							−0.861637	+	0.507525i	\(0.830561\pi\)
\(41\)	−0.873217	+	1.09498i	−0.136374	+	0.171007i	−0.845329	−	0.534246i	\(-0.820595\pi\)
							0.708955	+	0.705254i	\(0.249167\pi\)
\(43\)	2.24439	+	2.81438i	0.342266	+	0.429188i	0.922938	−	0.384950i	\(-0.125781\pi\)
							−0.580671	+	0.814138i	\(0.697210\pi\)
\(47\)	1.63076	−	0.503023i	0.237871	−	0.0733735i	−0.173528	−	0.984829i	\(-0.555517\pi\)
							0.411399	+	0.911455i	\(0.365040\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.122.4	yes	192
3.2	odd	2	inner	147.2.o.b.122.13	yes	192
49.47	odd	42	inner	147.2.o.b.47.13	yes	192
147.47	even	42	inner	147.2.o.b.47.4	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.47.4	✓	192	147.47	even	42	inner
147.2.o.b.47.13	yes	192	49.47	odd	42	inner
147.2.o.b.122.4	yes	192	1.1	even	1	trivial
147.2.o.b.122.13	yes	192	3.2	odd	2	inner