Embedded newform 147.2.o.b.122.5

• Label: 147.2.o.b.122.5
• 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			122.5
Character	\(\chi\)	\(=\)	147.122
Dual form			147.2.o.b.47.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.459136 - 1.48848i) q^{2} +(-0.910820 + 1.47323i) q^{3} +(-0.352300 + 0.240194i) q^{4} +(-3.51468 - 0.529753i) q^{5} +(2.61107 + 0.679327i) q^{6} +(-2.23383 + 1.41775i) q^{7} +(-1.91642 - 1.52829i) q^{8} +(-1.34081 - 2.68369i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.459136	−	1.48848i	−0.324659	−	1.05252i	−0.959933	−	0.280230i	\(-0.909589\pi\)
							0.635274	−	0.772287i	\(-0.280887\pi\)
\(3\)	−0.910820	+	1.47323i	−0.525862	+	0.850570i
							\(5\)	−3.51468	−	0.529753i	−1.57181	−	0.236913i	−0.695454	−	0.718571i	\(-0.744797\pi\)
−0.876359	+	0.481658i	\(0.840035\pi\)
\(7\)	−2.23383	+	1.41775i	−0.844309	+	0.535857i
							\(11\)	0.609714	−	0.657116i	0.183836	−	0.198128i	−0.634409	−	0.772997i	\(-0.718757\pi\)
0.818245	+	0.574869i	\(0.194947\pi\)
\(13\)	−3.35936	−	0.766751i	−0.931718	−	0.212659i	−0.270386	−	0.962752i	\(-0.587151\pi\)
							−0.661332	+	0.750093i	\(0.730008\pi\)
\(17\)	−0.307383	+	4.10174i	−0.0745514	+	0.994819i	0.827015	+	0.562179i	\(0.190037\pi\)
							−0.901567	+	0.432640i	\(0.857582\pi\)
\(19\)	5.05861	−	2.92059i	1.16052	−	0.670029i	0.209094	−	0.977896i	\(-0.432948\pi\)
							0.951429	+	0.307867i	\(0.0996151\pi\)
\(23\)	−0.000984747	0	7.37966e-5i	−0.000205334	0	1.53877e-5i	−0.0748328	−	0.997196i	\(-0.523842\pi\)
							0.0746274	+	0.997211i	\(0.476223\pi\)
\(29\)	−0.494959	−	1.02779i	−0.0919115	−	0.190856i	0.849945	−	0.526871i	\(-0.176635\pi\)
							−0.941857	+	0.336015i	\(0.890921\pi\)
\(31\)	−8.65236	−	4.99544i	−1.55401	−	0.897208i	−0.997809	−	0.0661607i	\(-0.978925\pi\)
							−0.556201	−	0.831048i	\(-0.687742\pi\)
\(37\)	−5.03298	−	3.43143i	−0.827417	−	0.564123i	0.0739090	−	0.997265i	\(-0.476453\pi\)
							−0.901326	+	0.433142i	\(0.857405\pi\)
\(41\)	−2.38839	+	2.99495i	−0.373005	+	0.467733i	−0.932536	−	0.361076i	\(-0.882410\pi\)
							0.559532	+	0.828809i	\(0.310981\pi\)
\(43\)	−1.80566	−	2.26423i	−0.275361	−	0.345292i	0.624851	−	0.780744i	\(-0.285160\pi\)
							−0.900212	+	0.435452i	\(0.856588\pi\)
\(47\)	5.59735	−	1.72655i	0.816457	−	0.251844i	0.141725	−	0.989906i	\(-0.454735\pi\)
							0.674733	+	0.738062i	\(0.264259\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.122.5	yes	192
3.2	odd	2	inner	147.2.o.b.122.12	yes	192
49.47	odd	42	inner	147.2.o.b.47.12	yes	192
147.47	even	42	inner	147.2.o.b.47.5	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.47.5	✓	192	147.47	even	42	inner
147.2.o.b.47.12	yes	192	49.47	odd	42	inner
147.2.o.b.122.5	yes	192	1.1	even	1	trivial
147.2.o.b.122.12	yes	192	3.2	odd	2	inner