Embedded newform 731.2.bl.a.9.6

• Label: 731.2.bl.a.9.6
• 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			9.6
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.50024 - 0.281710i) q^{2} +(0.980817 + 0.591000i) q^{3} +(4.22200 + 0.963643i) q^{4} +(2.79856 - 1.83232i) q^{5} +(-2.28579 - 1.75395i) q^{6} +(-0.0420865 - 0.00554079i) q^{7} +(-5.53481 - 1.93671i) q^{8} +(-0.789084 - 1.49302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.50024	−	0.281710i	−1.76794	−	0.199199i	−0.832822	−	0.553540i	\(-0.813276\pi\)
							−0.935116	+	0.354342i	\(0.884705\pi\)
\(3\)	0.980817	+	0.591000i	0.566275	+	0.341214i	0.770230	−	0.637766i	\(-0.220141\pi\)
							−0.203955	+	0.978980i	\(0.565380\pi\)
\(5\)	2.79856	−	1.83232i	1.25155	−	0.819440i	0.262206	−	0.965012i	\(-0.415550\pi\)
							0.989348	+	0.145572i	\(0.0465024\pi\)
\(7\)	−0.0420865	−	0.00554079i	−0.0159072	−	0.00209422i	0.122568	−	0.992460i	\(-0.460887\pi\)
							−0.138476	+	0.990366i	\(0.544220\pi\)
\(11\)	1.47507	+	2.07891i	0.444750	+	0.626816i	0.975312	−	0.220830i	\(-0.0708765\pi\)
							−0.530563	+	0.847646i	\(0.678019\pi\)
\(13\)	6.04283	+	0.452848i	1.67598	+	0.125597i	0.878498	−	0.477745i	\(-0.158546\pi\)
							0.797482	+	0.603343i	\(0.206165\pi\)
\(17\)	−4.11717	+	0.221109i	−0.998561	+	0.0536267i
							\(19\)	3.72206	−	7.04249i	0.853900	−	1.61566i	0.0676835	−	0.997707i	\(-0.478439\pi\)
0.786217	−	0.617951i	\(-0.212037\pi\)
\(23\)	0.426618	−	0.929976i	0.0889560	−	0.193913i	−0.859260	−	0.511538i	\(-0.829076\pi\)
							0.948216	+	0.317625i	\(0.102885\pi\)
\(29\)	−2.00056	+	8.06662i	−0.371494	+	1.49793i	0.431591	+	0.902069i	\(0.357952\pi\)
							−0.803085	+	0.595864i	\(0.796810\pi\)
\(31\)	−0.324392	+	0.336757i	−0.0582625	+	0.0604833i	−0.748701	−	0.662908i	\(-0.769322\pi\)
							0.690439	+	0.723391i	\(0.257417\pi\)
\(37\)	0.367678	+	2.79279i	0.0604459	+	0.459132i	0.994724	+	0.102587i	\(0.0327120\pi\)
							−0.934278	+	0.356545i	\(0.883955\pi\)
\(41\)	1.97002	−	3.56449i	0.307666	−	0.556680i	−0.675952	−	0.736946i	\(-0.736267\pi\)
							0.983618	+	0.180266i	\(0.0576959\pi\)
\(43\)	−1.89889	+	6.27648i	−0.289578	+	0.957155i
							\(47\)	−7.35451	−	1.67862i	−1.07277	−	0.244852i	−0.350567	−	0.936538i	\(-0.614011\pi\)
−0.722199	+	0.691686i	\(0.756868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.6	✓	3072
17.2	even	8	inner	731.2.bl.a.138.59	yes	3072
43.24	even	21	inner	731.2.bl.a.196.59	yes	3072
731.325	even	168	inner	731.2.bl.a.325.6	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.6	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.59	yes	3072	17.2	even	8	inner
731.2.bl.a.196.59	yes	3072	43.24	even	21	inner
731.2.bl.a.325.6	yes	3072	731.325	even	168	inner