Embedded newform 147.3.l.a.8.19

• Label: 147.3.l.a.8.19
• Level: $147$
• Weight: $3$
• Character: 147.8
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.l (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.00545988610\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(36\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			8.19
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.200107 - 0.159580i) q^{2} +(0.0939851 + 2.99853i) q^{3} +(-0.875507 + 3.83585i) q^{4} +(0.404578 + 0.840116i) q^{5} +(0.497313 + 0.585029i) q^{6} +(5.69424 - 4.07132i) q^{7} +(0.881135 + 1.82969i) q^{8} +(-8.98233 + 0.563634i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.200107	−	0.159580i	0.100054	−	0.0797901i	−0.572185	−	0.820124i	\(-0.693904\pi\)
							0.672239	+	0.740334i	\(0.265333\pi\)
\(3\)	0.0939851	+	2.99853i	0.0313284	+	0.999509i
							\(5\)	0.404578	+	0.840116i	0.0809157	+	0.168023i	0.937501	−	0.347982i	\(-0.113133\pi\)
−0.856586	+	0.516005i	\(0.827419\pi\)
\(7\)	5.69424	−	4.07132i	0.813462	−	0.581618i
							\(11\)	−10.2000	+	8.13426i	−0.927276	+	0.739478i	−0.965667	−	0.259785i	\(-0.916348\pi\)
0.0383904	+	0.999263i	\(0.487777\pi\)
\(13\)	6.91077	+	8.66583i	0.531597	+	0.666602i	0.973026	−	0.230694i	\(-0.0740997\pi\)
							−0.441429	+	0.897296i	\(0.645528\pi\)
\(17\)	−29.6457	+	6.76645i	−1.74387	+	0.398026i	−0.971479	−	0.237125i	\(-0.923795\pi\)
							−0.772388	+	0.635151i	\(0.780938\pi\)
\(19\)	21.9675			1.15618			0.578092	−	0.815972i	\(-0.303797\pi\)
							0.578092	+	0.815972i	\(0.303797\pi\)
\(23\)	16.9462	+	3.86786i	0.736792	+	0.168168i	0.574419	−	0.818561i	\(-0.305228\pi\)
							0.162373	+	0.986729i	\(0.448085\pi\)
\(29\)	46.1820	−	10.5407i	1.59248	−	0.363474i	0.667838	−	0.744307i	\(-0.267220\pi\)
							0.924644	+	0.380833i	\(0.124363\pi\)
\(31\)	−5.03379			−0.162380			−0.0811901	−	0.996699i	\(-0.525872\pi\)
							−0.0811901	+	0.996699i	\(0.525872\pi\)
\(37\)	15.4877	+	67.8559i	0.418585	+	1.83394i	0.540424	+	0.841393i	\(0.318264\pi\)
							−0.121839	+	0.992550i	\(0.538879\pi\)
\(41\)	−8.23439	−	17.0989i	−0.200839	−	0.417046i	0.776087	−	0.630626i	\(-0.217202\pi\)
							−0.976926	+	0.213580i	\(0.931488\pi\)
\(43\)	13.8561	+	6.67277i	0.322236	+	0.155181i	0.588008	−	0.808855i	\(-0.299912\pi\)
							−0.265772	+	0.964036i	\(0.585627\pi\)
\(47\)	13.8148	−	11.0170i	0.293933	−	0.234404i	−0.465409	−	0.885096i	\(-0.654093\pi\)
							0.759341	+	0.650692i	\(0.225521\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.19	yes	216
3.2	odd	2	inner	147.3.l.a.8.18	✓	216
49.43	even	7	inner	147.3.l.a.92.18	yes	216
147.92	odd	14	inner	147.3.l.a.92.19	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.18	✓	216	3.2	odd	2	inner
147.3.l.a.8.19	yes	216	1.1	even	1	trivial
147.3.l.a.92.18	yes	216	49.43	even	7	inner
147.3.l.a.92.19	yes	216	147.92	odd	14	inner