Embedded newform 147.4.o.a.101.21

• Label: 147.4.o.a.101.21
• Level: $147$
• Weight: $4$
• Character: 147.101
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			101.21
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56048 - 0.612445i) q^{2} +(-4.21730 + 3.03552i) q^{3} +(-3.80440 - 3.52996i) q^{4} +(5.95462 + 4.05979i) q^{5} +(8.44011 - 2.15402i) q^{6} +(8.80926 + 16.2910i) q^{7} +(9.59357 + 19.9212i) q^{8} +(8.57124 - 25.6034i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56048	−	0.612445i	−0.551714	−	0.216532i	0.0730735	−	0.997327i	\(-0.476719\pi\)
							−0.624788	+	0.780795i	\(0.714814\pi\)
\(3\)	−4.21730	+	3.03552i	−0.811620	+	0.584186i
							\(5\)	5.95462	+	4.05979i	0.532597	+	0.363119i	0.799558	−	0.600589i	\(-0.205067\pi\)
−0.266961	+	0.963707i	\(0.586019\pi\)
\(7\)	8.80926	+	16.2910i	0.475655	+	0.879632i
							\(11\)	−3.97701	−	26.3857i	−0.109010	−	0.723236i	−0.974841	−	0.222900i	\(-0.928448\pi\)
0.865831	−	0.500336i	\(-0.166790\pi\)
\(13\)	−18.0363	+	14.3835i	−0.384798	+	0.306867i	−0.796715	−	0.604355i	\(-0.793431\pi\)
							0.411917	+	0.911221i	\(0.364859\pi\)
\(17\)	−23.7012	−	7.31084i	−0.338140	−	0.104302i	0.121035	−	0.992648i	\(-0.461379\pi\)
							−0.459175	+	0.888346i	\(0.651855\pi\)
\(19\)	−113.090	+	65.2923i	−1.36550	+	0.788373i	−0.990350	−	0.138590i	\(-0.955743\pi\)
							−0.375153	+	0.926963i	\(0.622410\pi\)
\(23\)	−41.5785	−	134.794i	−0.376944	−	1.22202i	−0.923172	−	0.384386i	\(-0.874413\pi\)
							0.546228	−	0.837636i	\(-0.316063\pi\)
\(29\)	−28.6184	+	6.53196i	−0.183252	+	0.0418260i	−0.313162	−	0.949700i	\(-0.601388\pi\)
							0.129910	+	0.991526i	\(0.458531\pi\)
\(31\)	−234.740	−	135.527i	−1.36002	−	0.785207i	−0.370393	−	0.928875i	\(-0.620777\pi\)
							−0.989626	+	0.143668i	\(0.954110\pi\)
\(37\)	−43.1123	+	40.0023i	−0.191557	+	0.177739i	−0.770121	−	0.637898i	\(-0.779804\pi\)
							0.578563	+	0.815637i	\(0.303614\pi\)
\(41\)	6.52251	−	3.14108i	0.0248450	−	0.0119647i	−0.421420	−	0.906866i	\(-0.638468\pi\)
							0.446265	+	0.894901i	\(0.352754\pi\)
\(43\)	−89.1389	−	42.9270i	−0.316129	−	0.152240i	0.269090	−	0.963115i	\(-0.413277\pi\)
							−0.585219	+	0.810875i	\(0.698991\pi\)
\(47\)	−141.207	+	359.791i	−0.438239	+	1.11661i	0.525742	+	0.850644i	\(0.323788\pi\)
							−0.963981	+	0.265970i	\(0.914308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.21	✓	648
3.2	odd	2	inner	147.4.o.a.101.34	yes	648
49.33	odd	42	inner	147.4.o.a.131.34	yes	648
147.131	even	42	inner	147.4.o.a.131.21	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.21	✓	648	1.1	even	1	trivial
147.4.o.a.101.34	yes	648	3.2	odd	2	inner
147.4.o.a.131.21	yes	648	147.131	even	42	inner
147.4.o.a.131.34	yes	648	49.33	odd	42	inner