Embedded newform 147.2.o.b.131.7

• Label: 147.2.o.b.131.7
• Level: $147$
• Weight: $2$
• Character: 147.131
• 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			131.7
Character	\(\chi\)	\(=\)	147.131
Dual form			147.2.o.b.101.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.742405 + 0.291372i) q^{2} +(0.610083 - 1.62105i) q^{3} +(-0.999837 + 0.927713i) q^{4} +(-3.49434 + 2.38240i) q^{5} +(0.0194000 + 1.38124i) q^{6} +(-2.63459 + 0.242725i) q^{7} +(1.16405 - 2.41717i) q^{8} +(-2.25560 - 1.97795i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 14 q^{3} - 44 q^{4} - 28 q^{6} - 28 q^{7} - 26 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.742405	+	0.291372i	−0.524959	+	0.206031i	−0.612996	−	0.790086i	\(-0.710036\pi\)
							0.0880369	+	0.996117i	\(0.471941\pi\)
\(3\)	0.610083	−	1.62105i	0.352232	−	0.935913i
							\(5\)	−3.49434	+	2.38240i	−1.56271	+	1.06544i	−0.598222	+	0.801330i	\(0.704126\pi\)
−0.964492	+	0.264110i	\(0.914922\pi\)
\(7\)	−2.63459	+	0.242725i	−0.995783	+	0.0917415i
							\(11\)	−0.481626	+	3.19538i	−0.145216	+	0.963443i	0.790949	+	0.611882i	\(0.209587\pi\)
−0.936165	+	0.351561i	\(0.885651\pi\)
\(13\)	1.47939	+	1.17977i	0.410308	+	0.327210i	0.806796	−	0.590830i	\(-0.201200\pi\)
							−0.396488	+	0.918040i	\(0.629771\pi\)
\(17\)	0.330642	−	0.101990i	0.0801925	−	0.0247361i	−0.254399	−	0.967099i	\(-0.581878\pi\)
							0.334592	+	0.942363i	\(0.391402\pi\)
\(19\)	−3.79474	−	2.19089i	−0.870573	−	0.502626i	−0.00303461	−	0.999995i	\(-0.500966\pi\)
							−0.867539	+	0.497370i	\(0.834299\pi\)
\(23\)	−0.617691	+	2.00251i	−0.128798	+	0.417551i	−0.996886	−	0.0788585i	\(-0.974872\pi\)
							0.868088	+	0.496410i	\(0.165349\pi\)
\(29\)	3.02682	+	0.690852i	0.562066	+	0.128288i	0.494108	−	0.869400i	\(-0.335495\pi\)
							0.0679577	+	0.997688i	\(0.478352\pi\)
\(31\)	−6.61413	+	3.81867i	−1.18793	+	0.685853i	−0.957836	−	0.287314i	\(-0.907238\pi\)
							−0.230097	+	0.973168i	\(0.573904\pi\)
\(37\)	4.19171	+	3.88934i	0.689113	+	0.639404i	0.944911	−	0.327327i	\(-0.106148\pi\)
							−0.255798	+	0.966730i	\(0.582338\pi\)
\(41\)	−5.35893	−	2.58073i	−0.836925	−	0.403042i	−0.0342172	−	0.999414i	\(-0.510894\pi\)
							−0.802708	+	0.596373i	\(0.796608\pi\)
\(43\)	−1.13376	+	0.545989i	−0.172896	+	0.0832626i	−0.518330	−	0.855181i	\(-0.673446\pi\)
							0.345434	+	0.938443i	\(0.387732\pi\)
\(47\)	0.205644	+	0.523971i	0.0299962	+	0.0764290i	0.945087	−	0.326819i	\(-0.105977\pi\)
							−0.915091	+	0.403248i	\(0.867881\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.b.131.7	yes	192
3.2	odd	2	inner	147.2.o.b.131.10	yes	192
49.3	odd	42	inner	147.2.o.b.101.10	yes	192
147.101	even	42	inner	147.2.o.b.101.7	✓	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.b.101.7	✓	192	147.101	even	42	inner
147.2.o.b.101.10	yes	192	49.3	odd	42	inner
147.2.o.b.131.7	yes	192	1.1	even	1	trivial
147.2.o.b.131.10	yes	192	3.2	odd	2	inner