Embedded newform 147.3.l.a.8.18

• Label: 147.3.l.a.8.18
• 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			8.18
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.200107 + 0.159580i) q^{2} +(-2.28574 - 1.94303i) q^{3} +(-0.875507 + 3.83585i) q^{4} +(-0.404578 - 0.840116i) q^{5} +(0.767464 + 0.0240552i) q^{6} +(5.69424 - 4.07132i) q^{7} +(-0.881135 - 1.82969i) q^{8} +(1.44925 + 8.88255i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

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	−0.672239	−	0.740334i	\(-0.734667\pi\)
							0.572185	+	0.820124i	\(0.306096\pi\)
\(3\)	−2.28574	−	1.94303i	−0.761915	−	0.647677i
							\(5\)	−0.404578	−	0.840116i	−0.0809157	−	0.168023i	0.856586	−	0.516005i	\(-0.172581\pi\)
−0.937501	+	0.347982i	\(0.886867\pi\)
\(7\)	5.69424	−	4.07132i	0.813462	−	0.581618i
							\(11\)	10.2000	−	8.13426i	0.927276	−	0.739478i	−0.0383904	−	0.999263i	\(-0.512223\pi\)
0.965667	+	0.259785i	\(0.0836516\pi\)
\(13\)	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	0.772388	−	0.635151i	\(-0.219062\pi\)
							0.971479	+	0.237125i	\(0.0762051\pi\)
\(19\)	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	−0.162373	−	0.986729i	\(-0.551915\pi\)
							−0.574419	+	0.818561i	\(0.694772\pi\)
\(29\)	−46.1820	+	10.5407i	−1.59248	+	0.363474i	−0.924644	−	0.380833i	\(-0.875637\pi\)
							−0.667838	+	0.744307i	\(0.732780\pi\)
\(31\)	−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	0.976926	−	0.213580i	\(-0.0685124\pi\)
							−0.776087	+	0.630626i	\(0.782798\pi\)
\(43\)	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	−0.759341	−	0.650692i	\(-0.774479\pi\)
							0.465409	+	0.885096i	\(0.345907\pi\)

(See \(a_n\) instead)

Twists

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

    

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