Embedded newform 147.4.o.a.101.45

• Label: 147.4.o.a.101.45
• 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			101.45
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.73197 + 1.46469i) q^{2} +(2.47351 + 4.56966i) q^{3} +(5.91786 + 5.49098i) q^{4} +(2.72117 + 1.85526i) q^{5} +(2.53792 + 20.6767i) q^{6} +(17.4082 - 6.32091i) q^{7} +(0.126831 + 0.263367i) q^{8} +(-14.7635 + 22.6062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.73197	+	1.46469i	1.31945	+	0.517846i	0.917471	−	0.397803i	\(-0.130227\pi\)
							0.401979	+	0.915649i	\(0.368322\pi\)
\(3\)	2.47351	+	4.56966i	0.476027	+	0.879431i
							\(5\)	2.72117	+	1.85526i	0.243389	+	0.165940i	0.678871	−	0.734258i	\(-0.262470\pi\)
−0.435482	+	0.900197i	\(0.643422\pi\)
\(7\)	17.4082	−	6.32091i	0.939955	−	0.341297i
							\(11\)	6.32797	+	41.9833i	0.173450	+	1.15077i	0.890914	+	0.454172i	\(0.150065\pi\)
−0.717464	+	0.696596i	\(0.754697\pi\)
\(13\)	20.3002	−	16.1888i	0.433096	−	0.345383i	−0.382549	−	0.923935i	\(-0.624954\pi\)
							0.815645	+	0.578553i	\(0.196382\pi\)
\(17\)	−86.1489	−	26.5734i	−1.22907	−	0.379118i	−0.388818	−	0.921315i	\(-0.627116\pi\)
							−0.840251	+	0.542197i	\(0.817593\pi\)
\(19\)	−40.5950	+	23.4375i	−0.490164	+	0.282997i	−0.724643	−	0.689125i	\(-0.757995\pi\)
							0.234478	+	0.972121i	\(0.424662\pi\)
\(23\)	−3.60688	−	11.6932i	−0.0326994	−	0.106009i	0.937752	−	0.347306i	\(-0.112903\pi\)
							−0.970451	+	0.241297i	\(0.922427\pi\)
\(29\)	283.433	−	64.6918i	1.81490	−	0.414240i	0.826108	−	0.563512i	\(-0.190550\pi\)
							0.988796	+	0.149272i	\(0.0476930\pi\)
\(31\)	−60.1859	−	34.7483i	−0.348700	−	0.201322i	0.315412	−	0.948955i	\(-0.397857\pi\)
							−0.664113	+	0.747633i	\(0.731190\pi\)
\(37\)	25.9091	−	24.0401i	0.115120	−	0.106816i	−0.620513	−	0.784196i	\(-0.713076\pi\)
							0.735633	+	0.677381i	\(0.236885\pi\)
\(41\)	290.090	−	139.700i	1.10498	−	0.532133i	0.209762	−	0.977752i	\(-0.432731\pi\)
							0.895222	+	0.445620i	\(0.147017\pi\)
\(43\)	52.6564	+	25.3580i	0.186745	+	0.0899316i	0.524920	−	0.851152i	\(-0.324095\pi\)
							−0.338175	+	0.941083i	\(0.609810\pi\)
\(47\)	−59.4013	+	151.352i	−0.184353	+	0.469723i	−0.992935	−	0.118660i	\(-0.962140\pi\)
							0.808582	+	0.588383i	\(0.200235\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.o.a.101.45	yes	648
3.2	odd	2	inner	147.4.o.a.101.10	✓	648
49.33	odd	42	inner	147.4.o.a.131.10	yes	648
147.131	even	42	inner	147.4.o.a.131.45	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.o.a.101.10	✓	648	3.2	odd	2	inner
147.4.o.a.101.45	yes	648	1.1	even	1	trivial
147.4.o.a.131.10	yes	648	49.33	odd	42	inner
147.4.o.a.131.45	yes	648	147.131	even	42	inner