Embedded newform 147.3.l.a.8.1

• Label: 147.3.l.a.8.1
• 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			8.1
Character	\(\chi\)	\(=\)	147.8
Dual form			147.3.l.a.92.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.00834 + 2.39907i) q^{2} +(-2.98897 + 0.256973i) q^{3} +(2.40448 - 10.5347i) q^{4} +(3.25362 + 6.75620i) q^{5} +(8.37534 - 7.94382i) q^{6} +(3.12235 + 6.26506i) q^{7} +(11.3620 + 23.5934i) q^{8} +(8.86793 - 1.53617i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 5 q^{3} + 62 q^{4} + 7 q^{6} - 14 q^{7} - 45 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00834	+	2.39907i	−1.50417	+	1.19953i	−0.581735	+	0.813379i	\(0.697626\pi\)
							−0.922434	+	0.386156i	\(0.873803\pi\)
\(3\)	−2.98897	+	0.256973i	−0.996325	+	0.0856578i
							\(5\)	3.25362	+	6.75620i	0.650723	+	1.35124i	0.921417	+	0.388575i	\(0.127033\pi\)
−0.270694	+	0.962666i	\(0.587253\pi\)
\(7\)	3.12235	+	6.26506i	0.446050	+	0.895008i
							\(11\)	−3.79209	+	3.02409i	−0.344736	+	0.274917i	−0.780516	−	0.625135i	\(-0.785044\pi\)
0.435781	+	0.900053i	\(0.356472\pi\)
\(13\)	6.23035	+	7.81261i	0.479258	+	0.600970i	0.961411	−	0.275117i	\(-0.0887166\pi\)
							−0.482153	+	0.876087i	\(0.660145\pi\)
\(17\)	5.74595	−	1.31148i	0.337997	−	0.0771456i	−0.0501537	−	0.998742i	\(-0.515971\pi\)
							0.388151	+	0.921596i	\(0.373114\pi\)
\(19\)	15.4107			0.811088			0.405544	−	0.914075i	\(-0.367082\pi\)
							0.405544	+	0.914075i	\(0.367082\pi\)
\(23\)	−30.2078	−	6.89473i	−1.31338	−	0.299771i	−0.492222	−	0.870470i	\(-0.663815\pi\)
							−0.821160	+	0.570699i	\(0.806672\pi\)
\(29\)	−16.3406	+	3.72964i	−0.563469	+	0.128608i	−0.494762	−	0.869029i	\(-0.664745\pi\)
							−0.0687075	+	0.997637i	\(0.521888\pi\)
\(31\)	−16.5671			−0.534423			−0.267212	−	0.963638i	\(-0.586102\pi\)
							−0.267212	+	0.963638i	\(0.586102\pi\)
\(37\)	11.1442	+	48.8261i	0.301196	+	1.31963i	0.868325	+	0.495996i	\(0.165197\pi\)
							−0.567129	+	0.823629i	\(0.691946\pi\)
\(41\)	−6.32752	−	13.1392i	−0.154330	−	0.320469i	0.809441	−	0.587201i	\(-0.199770\pi\)
							−0.963771	+	0.266732i	\(0.914056\pi\)
\(43\)	−18.7678	−	9.03808i	−0.436460	−	0.210188i	0.202734	−	0.979234i	\(-0.435017\pi\)
							−0.639194	+	0.769046i	\(0.720732\pi\)
\(47\)	5.87239	−	4.68308i	0.124945	−	0.0996400i	−0.559024	−	0.829152i	\(-0.688824\pi\)
							0.683968	+	0.729512i	\(0.260253\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.3.l.a.8.1	✓	216
3.2	odd	2	inner	147.3.l.a.8.36	yes	216
49.43	even	7	inner	147.3.l.a.92.36	yes	216
147.92	odd	14	inner	147.3.l.a.92.1	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.3.l.a.8.1	✓	216	1.1	even	1	trivial
147.3.l.a.8.36	yes	216	3.2	odd	2	inner
147.3.l.a.92.1	yes	216	147.92	odd	14	inner
147.3.l.a.92.36	yes	216	49.43	even	7	inner