Embedded newform 147.3.l.a.8.6

• Label: 147.3.l.a.8.6
• 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			8.6
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29766 + 1.83232i) q^{2} +(2.90685 + 0.741768i) q^{3} +(1.03175 - 4.52040i) q^{4} +(0.779870 + 1.61942i) q^{5} +(-8.03811 + 3.62196i) q^{6} +(-6.58273 + 2.38068i) q^{7} +(0.811795 + 1.68571i) q^{8} +(7.89956 + 4.31242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29766	+	1.83232i	−1.14883	+	0.916161i	−0.997383	−	0.0723039i	\(-0.976965\pi\)
							−0.151447	+	0.988465i	\(0.548393\pi\)
\(3\)	2.90685	+	0.741768i	0.968950	+	0.247256i
							\(5\)	0.779870	+	1.61942i	0.155974	+	0.323883i	0.964283	−	0.264876i	\(-0.0853310\pi\)
−0.808309	+	0.588759i	\(0.799617\pi\)
\(7\)	−6.58273	+	2.38068i	−0.940390	+	0.340097i
							\(11\)	−11.1904	+	8.92408i	−1.01731	+	0.811280i	−0.982150	−	0.188101i	\(-0.939767\pi\)
−0.0351638	+	0.999382i	\(0.511195\pi\)
\(13\)	13.9694	+	17.5171i	1.07457	+	1.34747i	0.933951	+	0.357402i	\(0.116337\pi\)
							0.140617	+	0.990064i	\(0.455091\pi\)
\(17\)	−8.47377	+	1.93408i	−0.498457	+	0.113770i	−0.464356	−	0.885649i	\(-0.653714\pi\)
							−0.0341012	+	0.999418i	\(0.510857\pi\)
\(19\)	−22.9346			−1.20708			−0.603541	−	0.797332i	\(-0.706244\pi\)
							−0.603541	+	0.797332i	\(0.706244\pi\)
\(23\)	−21.2655	−	4.85371i	−0.924587	−	0.211031i	−0.266380	−	0.963868i	\(-0.585828\pi\)
							−0.658207	+	0.752837i	\(0.728685\pi\)
\(29\)	34.8146	−	7.94621i	1.20050	−	0.274007i	0.424908	−	0.905236i	\(-0.360306\pi\)
							0.775596	+	0.631229i	\(0.217449\pi\)
\(31\)	−26.0458			−0.840189			−0.420094	−	0.907480i	\(-0.638003\pi\)
							−0.420094	+	0.907480i	\(0.638003\pi\)
\(37\)	8.17361	+	35.8109i	0.220908	+	0.967862i	0.956796	+	0.290759i	\(0.0939078\pi\)
							−0.735888	+	0.677103i	\(0.763235\pi\)
\(41\)	8.50159	+	17.6537i	0.207356	+	0.430579i	0.978547	−	0.206025i	\(-0.0660529\pi\)
							−0.771191	+	0.636604i	\(0.780339\pi\)
\(43\)	32.5867	+	15.6929i	0.757829	+	0.364951i	0.772561	−	0.634940i	\(-0.218975\pi\)
							−0.0147321	+	0.999891i	\(0.504690\pi\)
\(47\)	68.3608	−	54.5159i	1.45449	−	1.15991i	0.498355	−	0.866973i	\(-0.333938\pi\)
							0.956131	−	0.292941i	\(-0.0946339\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.6	✓	216
3.2	odd	2	inner	147.3.l.a.8.31	yes	216
49.43	even	7	inner	147.3.l.a.92.31	yes	216
147.92	odd	14	inner	147.3.l.a.92.6	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.6	✓	216	1.1	even	1	trivial
147.3.l.a.8.31	yes	216	3.2	odd	2	inner
147.3.l.a.92.6	yes	216	147.92	odd	14	inner
147.3.l.a.92.31	yes	216	49.43	even	7	inner