Embedded newform 731.2.bl.a.9.8

• Label: 731.2.bl.a.9.8
• 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			9.8
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29109 - 0.258143i) q^{2} +(-0.766885 - 0.462093i) q^{3} +(3.23258 + 0.737816i) q^{4} +(-3.34001 + 2.18683i) q^{5} +(1.63771 + 1.25666i) q^{6} +(-2.68123 - 0.352990i) q^{7} +(-2.86326 - 1.00190i) q^{8} +(-1.02722 - 1.94360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29109	−	0.258143i	−1.62004	−	0.182535i	−0.745165	−	0.666880i	\(-0.767629\pi\)
							−0.874877	+	0.484345i	\(0.839058\pi\)
\(3\)	−0.766885	−	0.462093i	−0.442762	−	0.266790i	0.277189	−	0.960816i	\(-0.410597\pi\)
							−0.719950	+	0.694026i	\(0.755835\pi\)
\(5\)	−3.34001	+	2.18683i	−1.49370	+	0.977979i	−0.499981	+	0.866036i	\(0.666660\pi\)
							−0.993715	+	0.111943i	\(0.964293\pi\)
\(7\)	−2.68123	−	0.352990i	−1.01341	−	0.133418i	−0.394512	−	0.918891i	\(-0.629086\pi\)
							−0.618896	+	0.785473i	\(0.712420\pi\)
\(11\)	3.48800	+	4.91587i	1.05167	+	1.48219i	0.866300	+	0.499523i	\(0.166491\pi\)
							0.185371	+	0.982669i	\(0.440651\pi\)
\(13\)	5.07210	+	0.380101i	1.40675	+	0.105421i	0.756347	−	0.654171i	\(-0.226982\pi\)
							0.650400	+	0.759592i	\(0.274601\pi\)
\(17\)	0.408983	+	4.10277i	0.0991929	+	0.995068i
							\(19\)	−2.19194	+	4.14735i	−0.502865	+	0.951466i	0.493778	+	0.869588i	\(0.335616\pi\)
−0.996642	+	0.0818781i	\(0.973908\pi\)
\(23\)	1.35588	−	2.95565i	0.282720	−	0.616295i	−0.713736	−	0.700415i	\(-0.752998\pi\)
							0.996456	+	0.0841196i	\(0.0268078\pi\)
\(29\)	−0.688953	+	2.77799i	−0.127935	+	0.515860i	0.871730	+	0.489987i	\(0.162998\pi\)
							−0.999665	+	0.0258730i	\(0.991763\pi\)
\(31\)	−6.33944	+	6.58108i	−1.13860	+	1.18200i	−0.157696	+	0.987488i	\(0.550407\pi\)
							−0.980901	+	0.194509i	\(0.937689\pi\)
\(37\)	−1.05638	−	8.02398i	−0.173667	−	1.31913i	−0.827060	−	0.562114i	\(-0.809988\pi\)
							0.653392	−	0.757020i	\(-0.273345\pi\)
\(41\)	−3.34689	+	6.05574i	−0.522696	+	0.945748i	0.475134	+	0.879913i	\(0.342400\pi\)
							−0.997831	+	0.0658346i	\(0.979029\pi\)
\(43\)	4.77561	−	4.49372i	0.728274	−	0.685286i
							\(47\)	−0.782524	−	0.178606i	−0.114143	−	0.0260523i	0.165068	−	0.986282i	\(-0.447216\pi\)
−0.279211	+	0.960230i	\(0.590073\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.8	✓	3072
17.2	even	8	inner	731.2.bl.a.138.57	yes	3072
43.24	even	21	inner	731.2.bl.a.196.57	yes	3072
731.325	even	168	inner	731.2.bl.a.325.8	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.8	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.57	yes	3072	17.2	even	8	inner
731.2.bl.a.196.57	yes	3072	43.24	even	21	inner
731.2.bl.a.325.8	yes	3072	731.325	even	168	inner