Embedded newform 147.4.o.a.101.10

• Label: 147.4.o.a.101.10
• 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			101.10
Character	\(\chi\)	\(=\)	147.101
Dual form			147.4.o.a.131.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.73197 - 1.46469i) q^{2} +(3.12695 - 4.14996i) q^{3} +(5.91786 + 5.49098i) q^{4} +(-2.72117 - 1.85526i) q^{5} +(-17.7481 + 10.9075i) q^{6} +(17.4082 - 6.32091i) q^{7} +(-0.126831 - 0.263367i) q^{8} +(-7.44434 - 25.9535i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 11 q^{3} - 234 q^{4} + 14 q^{6} + 28 q^{7} - 125 q^{9}+O(q^{10}) \)

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	−0.401979	−	0.915649i	\(-0.631678\pi\)
							−0.917471	+	0.397803i	\(0.869773\pi\)
\(3\)	3.12695	−	4.14996i	0.601782	−	0.798660i
							\(5\)	−2.72117	−	1.85526i	−0.243389	−	0.165940i	0.435482	−	0.900197i	\(-0.356578\pi\)
−0.678871	+	0.734258i	\(0.737530\pi\)
\(7\)	17.4082	−	6.32091i	0.939955	−	0.341297i
							\(11\)	−6.32797	−	41.9833i	−0.173450	−	1.15077i	−0.890914	−	0.454172i	\(-0.849935\pi\)
0.717464	−	0.696596i	\(-0.245303\pi\)
\(13\)	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	0.840251	−	0.542197i	\(-0.182407\pi\)
							0.388818	+	0.921315i	\(0.372884\pi\)
\(19\)	−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	0.970451	−	0.241297i	\(-0.0775728\pi\)
							−0.937752	+	0.347306i	\(0.887097\pi\)
\(29\)	−283.433	+	64.6918i	−1.81490	+	0.414240i	−0.988796	−	0.149272i	\(-0.952307\pi\)
							−0.826108	+	0.563512i	\(0.809450\pi\)
\(31\)	−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	−0.895222	−	0.445620i	\(-0.852983\pi\)
							−0.209762	+	0.977752i	\(0.567269\pi\)
\(43\)	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	−0.808582	−	0.588383i	\(-0.799765\pi\)
							0.992935	+	0.118660i	\(0.0378599\pi\)

(See \(a_n\) instead)

Twists

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

    

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