Embedded newform 147.4.o.a.101.52

• Label: 147.4.o.a.101.52
• 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			101.52
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.87159 + 1.91196i) q^{2} +(-5.14329 - 0.739270i) q^{3} +(14.2124 + 13.1872i) q^{4} +(-3.32241 - 2.26518i) q^{5} +(-23.6426 - 13.4352i) q^{6} +(1.07928 + 18.4888i) q^{7} +(25.8583 + 53.6954i) q^{8} +(25.9070 + 7.60457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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	0.999881	−	0.0154057i	\(-0.00490398\pi\)
							0.722486	+	0.691385i	\(0.242999\pi\)
\(3\)	−5.14329	−	0.739270i	−0.989828	−	0.142273i
							\(5\)	−3.32241	−	2.26518i	−0.297165	−	0.202604i	0.405554	−	0.914071i	\(-0.367079\pi\)
−0.702719	+	0.711467i	\(0.748031\pi\)
\(7\)	1.07928	+	18.4888i	0.0582756	+	0.998301i
							\(11\)	6.81829	+	45.2364i	0.186890	+	1.23994i	0.863787	+	0.503857i	\(0.168086\pi\)
−0.676897	+	0.736078i	\(0.736676\pi\)
\(13\)	−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	0.196543	−	0.980495i	\(-0.437029\pi\)
							−0.389941	+	0.920840i	\(0.627505\pi\)
\(19\)	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	\(-0.823844\pi\)
							0.406836	−	0.913501i	\(-0.366632\pi\)
\(29\)	220.122	−	50.2413i	1.40950	−	0.321710i	0.550994	−	0.834509i	\(-0.314249\pi\)
							0.858508	+	0.512799i	\(0.171392\pi\)
\(31\)	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	−0.0649137	−	0.997891i	\(-0.520677\pi\)
							0.739710	+	0.672926i	\(0.234963\pi\)
\(43\)	−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	\(0.244062\pi\)
							−0.999824	+	0.0187452i	\(0.994033\pi\)

(See \(a_n\) instead)

Twists

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

    

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