Embedded newform 147.3.l.a.8.3

• Label: 147.3.l.a.8.3
• Level: $147$
• Weight: $3$
• Character: 147.8
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• 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			8.3
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.88499 + 2.30070i) q^{2} +(1.88790 + 2.33149i) q^{3} +(2.13985 - 9.37531i) q^{4} +(-2.07438 - 4.30749i) q^{5} +(-10.8106 - 2.38284i) q^{6} +(6.99908 + 0.113215i) q^{7} +(8.99215 + 18.6724i) q^{8} +(-1.87169 + 8.80323i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.88499	+	2.30070i	−1.44250	+	1.15035i	−0.480598	+	0.876941i	\(0.659580\pi\)
							−0.961897	+	0.273411i	\(0.911848\pi\)
\(3\)	1.88790	+	2.33149i	0.629299	+	0.777163i
							\(5\)	−2.07438	−	4.30749i	−0.414876	−	0.861498i	−0.998766	−	0.0496654i	\(-0.984184\pi\)
0.583890	−	0.811833i	\(-0.301530\pi\)
\(7\)	6.99908	+	0.113215i	0.999869	+	0.0161736i
							\(11\)	13.8154	−	11.0174i	1.25594	−	1.00158i	0.256556	−	0.966529i	\(-0.417412\pi\)
0.999386	−	0.0350508i	\(-0.0111593\pi\)
\(13\)	7.90710	+	9.91519i	0.608238	+	0.762707i	0.986637	−	0.162936i	\(-0.0520965\pi\)
							−0.378398	+	0.925643i	\(0.623525\pi\)
\(17\)	−11.0157	+	2.51427i	−0.647983	+	0.147898i	−0.533866	−	0.845569i	\(-0.679261\pi\)
							−0.114118	+	0.993467i	\(0.536404\pi\)
\(19\)	−4.75673			−0.250354			−0.125177	−	0.992134i	\(-0.539950\pi\)
							−0.125177	+	0.992134i	\(0.539950\pi\)
\(23\)	44.3232	+	10.1165i	1.92710	+	0.439847i	0.997448	+	0.0714011i	\(0.0227470\pi\)
							0.929649	+	0.368446i	\(0.120110\pi\)
\(29\)	−36.8943	+	8.42089i	−1.27222	+	0.290376i	−0.804739	−	0.593629i	\(-0.797695\pi\)
							−0.467479	+	0.884004i	\(0.654838\pi\)
\(31\)	5.75431			0.185623			0.0928115	−	0.995684i	\(-0.470415\pi\)
							0.0928115	+	0.995684i	\(0.470415\pi\)
\(37\)	−1.33751	−	5.86002i	−0.0361490	−	0.158379i	0.953632	−	0.300975i	\(-0.0973121\pi\)
							−0.989781	+	0.142596i	\(0.954455\pi\)
\(41\)	15.0560	+	31.2640i	0.367219	+	0.762537i	0.999930	−	0.0118506i	\(-0.00377226\pi\)
							−0.632711	+	0.774388i	\(0.718058\pi\)
\(43\)	−1.33216	−	0.641533i	−0.0309804	−	0.0149194i	0.418329	−	0.908295i	\(-0.362616\pi\)
							−0.449310	+	0.893376i	\(0.648330\pi\)
\(47\)	36.8525	−	29.3889i	0.784095	−	0.625295i	−0.147388	−	0.989079i	\(-0.547087\pi\)
							0.931483	+	0.363784i	\(0.118515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.3	✓	216
3.2	odd	2	inner	147.3.l.a.8.34	yes	216
49.43	even	7	inner	147.3.l.a.92.34	yes	216
147.92	odd	14	inner	147.3.l.a.92.3	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.3	✓	216	1.1	even	1	trivial
147.3.l.a.8.34	yes	216	3.2	odd	2	inner
147.3.l.a.92.3	yes	216	147.92	odd	14	inner
147.3.l.a.92.34	yes	216	49.43	even	7	inner