Embedded newform 147.2.o.b.122.15

• Label: 147.2.o.b.122.15
• 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			122.15
Character	\(\chi\)	\(=\)	147.122
Dual form			147.2.o.b.47.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.749148 + 2.42868i) q^{2} +(-0.787851 + 1.54250i) q^{3} +(-3.68478 + 2.51224i) q^{4} +(2.16672 + 0.326581i) q^{5} +(-4.33644 - 0.757879i) q^{6} +(2.09729 - 1.61288i) q^{7} +(-4.88766 - 3.89778i) q^{8} +(-1.75858 - 2.43051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

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	\(0.260169\pi\)
							−0.154433	+	0.988003i	\(0.549355\pi\)
\(3\)	−0.787851	+	1.54250i	−0.454866	+	0.890560i
							\(5\)	2.16672	+	0.326581i	0.968987	+	0.146051i	0.614420	−	0.788979i	\(-0.289390\pi\)
0.354567	+	0.935030i	\(0.384628\pi\)
\(7\)	2.09729	−	1.61288i	0.792699	−	0.609613i
							\(11\)	2.76892	−	2.98419i	0.834862	−	0.899767i	−0.161335	−	0.986900i	\(-0.551580\pi\)
0.996197	+	0.0871325i	\(0.0277703\pi\)
\(13\)	−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	\(0.153105\pi\)
							−0.945590	+	0.325361i	\(0.894514\pi\)
\(19\)	−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	0.462922	−	0.886399i	\(-0.346801\pi\)
							0.589863	+	0.807504i	\(0.299182\pi\)
\(29\)	−0.510176	−	1.05939i	−0.0947374	−	0.196724i	0.848215	−	0.529652i	\(-0.177678\pi\)
							−0.942952	+	0.332928i	\(0.891963\pi\)
\(31\)	−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	−0.999929	−	0.0118877i	\(-0.996216\pi\)
							0.234095	+	0.972214i	\(0.424787\pi\)
\(43\)	−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	−0.100412	−	0.994946i	\(-0.532016\pi\)
							0.477509	+	0.878627i	\(0.341540\pi\)

(See \(a_n\) instead)

Twists

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

    

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