Embedded newform 147.4.o.a.101.3

• Label: 147.4.o.a.101.3
• 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			101.3
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.87159 - 1.91196i) q^{2} +(-5.19603 - 0.0355553i) q^{3} +(14.2124 + 13.1872i) q^{4} +(3.32241 + 2.26518i) q^{5} +(25.2450 + 10.1078i) q^{6} +(1.07928 + 18.4888i) q^{7} +(-25.8583 - 53.6954i) q^{8} +(26.9975 + 0.369493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.87159	−	1.91196i	−1.72237	−	0.675979i	−0.722486	−	0.691385i	\(-0.757001\pi\)
							−0.999881	+	0.0154057i	\(0.995096\pi\)
\(3\)	−5.19603	−	0.0355553i	−0.999977	−	0.00684261i
							\(5\)	3.32241	+	2.26518i	0.297165	+	0.202604i	0.702719	−	0.711467i	\(-0.251969\pi\)
−0.405554	+	0.914071i	\(0.632921\pi\)
\(7\)	1.07928	+	18.4888i	0.0582756	+	0.998301i
							\(11\)	−6.81829	−	45.2364i	−0.186890	−	1.23994i	−0.863787	−	0.503857i	\(-0.831914\pi\)
0.676897	−	0.736078i	\(-0.263324\pi\)
\(13\)	−16.7221	+	13.3355i	−0.356760	+	0.284507i	−0.785425	−	0.618957i	\(-0.787556\pi\)
							0.428665	+	0.903464i	\(0.358984\pi\)
\(17\)	13.5558	+	4.18142i	0.193398	+	0.0596555i	0.389941	−	0.920840i	\(-0.372495\pi\)
							−0.196543	+	0.980495i	\(0.562971\pi\)
\(19\)	48.0199	−	27.7243i	0.579817	−	0.334757i	−0.181244	−	0.983438i	\(-0.558012\pi\)
							0.761060	+	0.648681i	\(0.224679\pi\)
\(23\)	48.9641	+	158.738i	0.443901	+	1.43909i	0.850737	+	0.525591i	\(0.176156\pi\)
							−0.406836	+	0.913501i	\(0.633368\pi\)
\(29\)	−220.122	+	50.2413i	−1.40950	+	0.321710i	−0.858508	−	0.512799i	\(-0.828608\pi\)
							−0.550994	+	0.834509i	\(0.685751\pi\)
\(31\)	262.183	+	151.371i	1.51901	+	0.877002i	0.999749	+	0.0223879i	\(0.00712687\pi\)
							0.519263	+	0.854614i	\(0.326206\pi\)
\(37\)	−106.009	+	98.3616i	−0.471019	+	0.437042i	−0.879624	−	0.475669i	\(-0.842206\pi\)
							0.408605	+	0.912711i	\(0.366015\pi\)
\(41\)	−177.153	+	85.3123i	−0.674796	+	0.324965i	−0.739710	−	0.672926i	\(-0.765037\pi\)
							0.0649137	+	0.997891i	\(0.479323\pi\)
\(43\)	−93.3354	−	44.9479i	−0.331012	−	0.159407i	0.260994	−	0.965340i	\(-0.415950\pi\)
							−0.592006	+	0.805933i	\(0.701664\pi\)
\(47\)	90.1080	−	229.592i	0.279651	−	0.712540i	−0.720173	−	0.693794i	\(-0.755938\pi\)
							0.999824	−	0.0187452i	\(-0.00596714\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.3	✓	648
3.2	odd	2	inner	147.4.o.a.101.52	yes	648
49.33	odd	42	inner	147.4.o.a.131.52	yes	648
147.131	even	42	inner	147.4.o.a.131.3	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.3	✓	648	1.1	even	1	trivial
147.4.o.a.101.52	yes	648	3.2	odd	2	inner
147.4.o.a.131.3	yes	648	147.131	even	42	inner
147.4.o.a.131.52	yes	648	49.33	odd	42	inner