Embedded newform 731.2.bl.a.9.9

• Label: 731.2.bl.a.9.9
• 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			9.9
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.25200 - 0.253740i) q^{2} +(2.54761 + 1.53509i) q^{3} +(3.05728 + 0.697804i) q^{4} +(0.169346 - 0.110877i) q^{5} +(-5.34772 - 4.10345i) q^{6} +(-3.68545 - 0.485199i) q^{7} +(-2.42979 - 0.850219i) q^{8} +(2.73203 + 5.16926i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.25200	−	0.253740i	−1.59241	−	0.179421i	−0.729211	−	0.684289i	\(-0.760113\pi\)
							−0.863196	+	0.504868i	\(0.831541\pi\)
\(3\)	2.54761	+	1.53509i	1.47086	+	0.886282i	0.999992	+	0.00408076i	\(0.00129895\pi\)
							0.470873	+	0.882201i	\(0.343939\pi\)
\(5\)	0.169346	−	0.110877i	0.0757338	−	0.0495858i	−0.509344	−	0.860563i	\(-0.670112\pi\)
							0.585077	+	0.810977i	\(0.301064\pi\)
\(7\)	−3.68545	−	0.485199i	−1.39297	−	0.183388i	−0.603591	−	0.797294i	\(-0.706264\pi\)
							−0.789379	+	0.613906i	\(0.789597\pi\)
\(11\)	−1.56408	−	2.20436i	−0.471588	−	0.664641i	0.508990	−	0.860773i	\(-0.330019\pi\)
							−0.980578	+	0.196132i	\(0.937162\pi\)
\(13\)	−2.07767	−	0.155700i	−0.576242	−	0.0431834i	−0.216578	−	0.976265i	\(-0.569490\pi\)
							−0.359664	+	0.933082i	\(0.617109\pi\)
\(17\)	−4.12160	−	0.111595i	−0.999634	−	0.0270659i
							\(19\)	−0.523550	+	0.990604i	−0.120111	+	0.227260i	−0.936838	−	0.349763i	\(-0.886262\pi\)
0.816728	+	0.577023i	\(0.195786\pi\)
\(23\)	−1.69049	+	3.68507i	−0.352492	+	0.768390i	0.647488	+	0.762075i	\(0.275819\pi\)
							−0.999980	+	0.00631430i	\(0.997990\pi\)
\(29\)	−2.41240	+	9.72723i	−0.447971	+	1.80630i	0.131508	+	0.991315i	\(0.458018\pi\)
							−0.579479	+	0.814987i	\(0.696744\pi\)
\(31\)	−7.27775	+	7.55516i	−1.30712	+	1.35695i	−0.411964	+	0.911200i	\(0.635157\pi\)
							−0.895159	+	0.445747i	\(0.852938\pi\)
\(37\)	−1.33045	−	10.1058i	−0.218725	−	1.66138i	−0.650581	−	0.759437i	\(-0.725474\pi\)
							0.431856	−	0.901943i	\(-0.357859\pi\)
\(41\)	1.15532	−	2.09040i	0.180431	−	0.326466i	−0.769651	−	0.638465i	\(-0.779570\pi\)
							0.950083	+	0.311999i	\(0.100998\pi\)
\(43\)	−1.41713	−	6.40248i	−0.216110	−	0.976369i
							\(47\)	9.26744	+	2.11523i	1.35180	+	0.308538i	0.836261	−	0.548331i	\(-0.184737\pi\)
0.515534	+	0.856869i	\(0.327594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.9	✓	3072
17.2	even	8	inner	731.2.bl.a.138.56	yes	3072
43.24	even	21	inner	731.2.bl.a.196.56	yes	3072
731.325	even	168	inner	731.2.bl.a.325.9	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.9	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.56	yes	3072	17.2	even	8	inner
731.2.bl.a.196.56	yes	3072	43.24	even	21	inner
731.2.bl.a.325.9	yes	3072	731.325	even	168	inner