Embedded newform 147.2.o.b.122.2

• Label: 147.2.o.b.122.2
• 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			122.2
Character	\(\chi\)	\(=\)	147.122
Dual form			147.2.o.b.47.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.749148 - 2.42868i) q^{2} +(-1.62670 - 0.594854i) q^{3} +(-3.68478 + 2.51224i) q^{4} +(-2.16672 - 0.326581i) q^{5} +(-0.226072 + 4.39636i) q^{6} +(2.09729 - 1.61288i) q^{7} +(4.88766 + 3.89778i) q^{8} +(2.29230 + 1.93530i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.749148	−	2.42868i	−0.529728	−	1.71733i	−0.684161	−	0.729331i	\(-0.739831\pi\)
							0.154433	−	0.988003i	\(-0.450645\pi\)
\(3\)	−1.62670	−	0.594854i	−0.939175	−	0.343439i
							\(5\)	−2.16672	−	0.326581i	−0.968987	−	0.146051i	−0.354567	−	0.935030i	\(-0.615372\pi\)
−0.614420	+	0.788979i	\(0.710610\pi\)
\(7\)	2.09729	−	1.61288i	0.792699	−	0.609613i
							\(11\)	−2.76892	+	2.98419i	−0.834862	+	0.899767i	−0.996197	−	0.0871325i	\(-0.972230\pi\)
0.161335	+	0.986900i	\(0.448420\pi\)
\(13\)	−3.59737	−	0.821077i	−0.997732	−	0.227726i	−0.307683	−	0.951489i	\(-0.599554\pi\)
							−0.690048	+	0.723763i	\(0.742411\pi\)
\(17\)	0.243486	−	3.24909i	0.0590540	−	0.788021i	−0.886536	−	0.462660i	\(-0.846895\pi\)
							0.945590	−	0.325361i	\(-0.105486\pi\)
\(19\)	−2.07459	+	1.19777i	−0.475944	+	0.274787i	−0.718725	−	0.695295i	\(-0.755274\pi\)
							0.242780	+	0.970081i	\(0.421941\pi\)
\(23\)	−5.04898	+	0.378369i	−1.05278	+	0.0788953i	−0.589863	−	0.807504i	\(-0.700818\pi\)
							−0.462922	+	0.886399i	\(0.653199\pi\)
\(29\)	0.510176	+	1.05939i	0.0947374	+	0.196724i	0.942952	−	0.332928i	\(-0.108037\pi\)
							−0.848215	+	0.529652i	\(0.822322\pi\)
\(31\)	−5.91873	−	3.41718i	−1.06304	−	0.613744i	−0.136765	−	0.990603i	\(-0.543671\pi\)
							−0.926270	+	0.376860i	\(0.877004\pi\)
\(37\)	6.88075	+	4.69122i	1.13119	+	0.771231i	0.976240	−	0.216691i	\(-0.0695264\pi\)
							0.154949	+	0.987923i	\(0.450479\pi\)
\(41\)	4.90373	−	6.14909i	0.765834	−	0.960326i	−0.234095	−	0.972214i	\(-0.575213\pi\)
							0.999929	+	0.0118877i	\(0.00378407\pi\)
\(43\)	−5.60718	−	7.03118i	−0.855087	−	1.07225i	−0.996607	−	0.0823100i	\(-0.973770\pi\)
							0.141520	−	0.989935i	\(-0.454801\pi\)
\(47\)	−2.58525	+	0.797444i	−0.377097	+	0.116319i	−0.477509	−	0.878627i	\(-0.658460\pi\)
							0.100412	+	0.994946i	\(0.467984\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.122.2	yes	192
3.2	odd	2	inner	147.2.o.b.122.15	yes	192
49.47	odd	42	inner	147.2.o.b.47.15	yes	192
147.47	even	42	inner	147.2.o.b.47.2	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.47.2	✓	192	147.47	even	42	inner
147.2.o.b.47.15	yes	192	49.47	odd	42	inner
147.2.o.b.122.2	yes	192	1.1	even	1	trivial
147.2.o.b.122.15	yes	192	3.2	odd	2	inner