Embedded newform 731.2.bl.a.9.1

• Label: 731.2.bl.a.9.1
• Level: $731$
• Weight: $2$
• Character: 731.9
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $3072$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bl (of order \(168\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(3072\)
Relative dimension:	\(64\) over \(\Q(\zeta_{168})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{168}]$

Embedding invariants

Embedding label			9.1
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.72251 - 0.306753i) q^{2} +(-0.620611 - 0.373954i) q^{3} +(5.36808 + 1.22523i) q^{4} +(0.404374 - 0.264759i) q^{5} +(1.57490 + 1.20847i) q^{6} +(-5.03855 - 0.663338i) q^{7} +(-9.06682 - 3.17262i) q^{8} +(-1.15649 - 2.18819i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.72251	−	0.306753i	−1.92510	−	0.216907i	−0.934732	−	0.355353i	\(-0.884361\pi\)
							−0.990370	+	0.138446i	\(0.955789\pi\)
\(3\)	−0.620611	−	0.373954i	−0.358310	−	0.215903i	0.325535	−	0.945530i	\(-0.394455\pi\)
							−0.683845	+	0.729627i	\(0.739694\pi\)
\(5\)	0.404374	−	0.264759i	0.180841	−	0.118404i	−0.454143	−	0.890929i	\(-0.650054\pi\)
							0.634984	+	0.772525i	\(0.281007\pi\)
\(7\)	−5.03855	−	0.663338i	−1.90439	−	0.250718i	−0.915836	−	0.401553i	\(-0.868470\pi\)
							−0.988559	+	0.150835i	\(0.951804\pi\)
\(11\)	−2.60702	−	3.67425i	−0.786046	−	1.10783i	−0.991620	−	0.129186i	\(-0.958763\pi\)
							0.205575	−	0.978641i	\(-0.434094\pi\)
\(13\)	−2.38567	−	0.178781i	−0.661664	−	0.0495849i	−0.260335	−	0.965518i	\(-0.583833\pi\)
							−0.401330	+	0.915934i	\(0.631452\pi\)
\(17\)	1.50303	+	3.83939i	0.364539	+	0.931188i
							\(19\)	0.596715	−	1.12904i	0.136896	−	0.259019i	−0.806130	−	0.591738i	\(-0.798442\pi\)
0.943026	+	0.332718i	\(0.107966\pi\)
\(23\)	2.85741	−	6.22880i	0.595810	−	1.29880i	−0.338040	−	0.941132i	\(-0.609764\pi\)
							0.933850	−	0.357664i	\(-0.116427\pi\)
\(29\)	0.866216	−	3.49275i	0.160852	−	0.648587i	−0.834392	−	0.551172i	\(-0.814181\pi\)
							0.995244	−	0.0974147i	\(-0.0310573\pi\)
\(31\)	0.484108	−	0.502561i	0.0869483	−	0.0902626i	−0.675316	−	0.737529i	\(-0.735993\pi\)
							0.762264	+	0.647266i	\(0.224088\pi\)
\(37\)	0.782153	+	5.94104i	0.128585	+	0.976701i	0.927123	+	0.374757i	\(0.122274\pi\)
							−0.798538	+	0.601944i	\(0.794393\pi\)
\(41\)	0.461524	−	0.835066i	0.0720780	−	0.130415i	−0.836752	−	0.547582i	\(-0.815548\pi\)
							0.908830	+	0.417167i	\(0.136977\pi\)
\(43\)	−2.50639	+	6.05954i	−0.382221	+	0.924071i
							\(47\)	−0.0475795	−	0.0108597i	−0.00694019	−	0.00158405i	0.219049	−	0.975714i	\(-0.429704\pi\)
−0.225990	+	0.974130i	\(0.572562\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.1	✓	3072
17.2	even	8	inner	731.2.bl.a.138.64	yes	3072
43.24	even	21	inner	731.2.bl.a.196.64	yes	3072
731.325	even	168	inner	731.2.bl.a.325.1	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.1	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.64	yes	3072	17.2	even	8	inner
731.2.bl.a.196.64	yes	3072	43.24	even	21	inner
731.2.bl.a.325.1	yes	3072	731.325	even	168	inner