Embedded newform 147.2.o.b.101.3

• Label: 147.2.o.b.101.3
• Level: $147$
• Weight: $2$
• Character: 147.101
• 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			101.3
Character	\(\chi\)	\(=\)	147.101
Dual form			147.2.o.b.131.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71938 - 0.674808i) q^{2} +(1.70035 + 0.329871i) q^{3} +(1.03481 + 0.960164i) q^{4} +(2.08886 + 1.42416i) q^{5} +(-2.70095 - 1.71458i) q^{6} +(-1.82029 + 1.92004i) q^{7} +(0.471513 + 0.979106i) q^{8} +(2.78237 + 1.12179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71938	−	0.674808i	−1.21579	−	0.477162i	−0.331211	−	0.943557i	\(-0.607457\pi\)
							−0.884577	+	0.466395i	\(0.845553\pi\)
\(3\)	1.70035	+	0.329871i	0.981697	+	0.190451i
							\(5\)	2.08886	+	1.42416i	0.934165	+	0.636903i	0.931752	−	0.363094i	\(-0.118280\pi\)
0.00241237	+	0.999997i	\(0.499232\pi\)
\(7\)	−1.82029	+	1.92004i	−0.688005	+	0.725706i
							\(11\)	−0.349865	−	2.32120i	−0.105488	−	0.699869i	−0.977586	−	0.210537i	\(-0.932479\pi\)
0.872098	−	0.489332i	\(-0.162759\pi\)
\(13\)	−2.55243	+	2.03550i	−0.707917	+	0.564545i	−0.909892	−	0.414846i	\(-0.863836\pi\)
							0.201975	+	0.979391i	\(0.435264\pi\)
\(17\)	5.16345	+	1.59271i	1.25232	+	0.386290i	0.848817	−	0.528687i	\(-0.177316\pi\)
							0.403505	+	0.914977i	\(0.367792\pi\)
\(19\)	5.23611	−	3.02307i	1.20125	−	0.693540i	0.240414	−	0.970670i	\(-0.422717\pi\)
							0.960832	+	0.277130i	\(0.0893833\pi\)
\(23\)	−2.00874	−	6.51217i	−0.418851	−	1.35788i	−0.881853	−	0.471524i	\(-0.843704\pi\)
							0.463003	−	0.886357i	\(-0.346772\pi\)
\(29\)	−8.11264	+	1.85166i	−1.50648	+	0.343844i	−0.894514	−	0.447040i	\(-0.852478\pi\)
							−0.611966	+	0.790884i	\(0.709621\pi\)
\(31\)	−3.37206	−	1.94686i	−0.605640	−	0.349666i	0.165617	−	0.986190i	\(-0.447038\pi\)
							−0.771257	+	0.636524i	\(0.780372\pi\)
\(37\)	−1.70568	+	1.58264i	−0.280412	+	0.260185i	−0.807822	−	0.589426i	\(-0.799354\pi\)
							0.527410	+	0.849611i	\(0.323163\pi\)
\(41\)	−4.73887	+	2.28212i	−0.740088	+	0.356407i	−0.765643	−	0.643266i	\(-0.777579\pi\)
							0.0255551	+	0.999673i	\(0.491865\pi\)
\(43\)	−0.477066	−	0.229743i	−0.0727519	−	0.0350355i	0.397154	−	0.917752i	\(-0.369998\pi\)
							−0.469906	+	0.882717i	\(0.655712\pi\)
\(47\)	0.0175485	−	0.0447129i	0.00255971	−	0.00652204i	−0.929588	−	0.368600i	\(-0.879837\pi\)
							0.932148	+	0.362078i	\(0.117933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.101.3	✓	192
3.2	odd	2	inner	147.2.o.b.101.14	yes	192
49.33	odd	42	inner	147.2.o.b.131.14	yes	192
147.131	even	42	inner	147.2.o.b.131.3	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.101.3	✓	192	1.1	even	1	trivial
147.2.o.b.101.14	yes	192	3.2	odd	2	inner
147.2.o.b.131.3	yes	192	147.131	even	42	inner
147.2.o.b.131.14	yes	192	49.33	odd	42	inner