Embedded newform 731.2.bl.a.9.14

• Label: 731.2.bl.a.9.14
• 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			9.14
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71083 - 0.192764i) q^{2} +(1.34544 + 0.810704i) q^{3} +(0.939930 + 0.214533i) q^{4} +(1.48366 - 0.971406i) q^{5} +(-2.14554 - 1.64633i) q^{6} +(-3.99522 - 0.525981i) q^{7} +(1.68338 + 0.589040i) q^{8} +(-0.248850 - 0.470847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71083	−	0.192764i	−1.20974	−	0.136305i	−0.516037	−	0.856566i	\(-0.672593\pi\)
							−0.693703	+	0.720261i	\(0.744022\pi\)
\(3\)	1.34544	+	0.810704i	0.776788	+	0.468060i	0.848389	−	0.529373i	\(-0.177573\pi\)
							−0.0716014	+	0.997433i	\(0.522811\pi\)
\(5\)	1.48366	−	0.971406i	0.663511	−	0.434426i	−0.171109	−	0.985252i	\(-0.554735\pi\)
							0.834620	+	0.550826i	\(0.185687\pi\)
\(7\)	−3.99522	−	0.525981i	−1.51005	−	0.198802i	−0.670430	−	0.741973i	\(-0.733890\pi\)
							−0.839622	+	0.543170i	\(0.817224\pi\)
\(11\)	1.09407	+	1.54195i	0.329876	+	0.464917i	0.946669	−	0.322209i	\(-0.104425\pi\)
							−0.616793	+	0.787126i	\(0.711568\pi\)
\(13\)	0.161425	+	0.0120971i	0.0447712	+	0.00335514i	0.0970969	−	0.995275i	\(-0.469044\pi\)
							−0.0523257	+	0.998630i	\(0.516663\pi\)
\(17\)	3.69906	−	1.82126i	0.897153	−	0.441720i
							\(19\)	−0.109649	+	0.207467i	−0.0251553	+	0.0475962i	−0.896373	−	0.443301i	\(-0.853807\pi\)
0.871217	+	0.490897i	\(0.163331\pi\)
\(23\)	2.89535	−	6.31153i	0.603723	−	1.31604i	−0.325156	−	0.945661i	\(-0.605417\pi\)
							0.928879	−	0.370384i	\(-0.120774\pi\)
\(29\)	0.954061	−	3.84696i	0.177165	−	0.714362i	−0.813903	−	0.581001i	\(-0.802661\pi\)
							0.991068	−	0.133361i	\(-0.0425769\pi\)
\(31\)	3.15484	−	3.27509i	0.566625	−	0.588223i	−0.374102	−	0.927388i	\(-0.622049\pi\)
							0.940727	+	0.339164i	\(0.110144\pi\)
\(37\)	−0.756795	−	5.74843i	−0.124416	−	0.945036i	−0.933810	−	0.357769i	\(-0.883537\pi\)
							0.809394	−	0.587266i	\(-0.199796\pi\)
\(41\)	−3.13773	+	5.67729i	−0.490031	+	0.886643i	0.509625	+	0.860397i	\(0.329784\pi\)
							−0.999655	+	0.0262468i	\(0.991644\pi\)
\(43\)	5.84466	−	2.97321i	0.891302	−	0.453410i
							\(47\)	−8.21620	−	1.87529i	−1.19846	−	0.273540i	−0.423701	−	0.905802i	\(-0.639269\pi\)
−0.774754	+	0.632262i	\(0.782126\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.14	✓	3072
17.2	even	8	inner	731.2.bl.a.138.51	yes	3072
43.24	even	21	inner	731.2.bl.a.196.51	yes	3072
731.325	even	168	inner	731.2.bl.a.325.14	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.14	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.51	yes	3072	17.2	even	8	inner
731.2.bl.a.196.51	yes	3072	43.24	even	21	inner
731.2.bl.a.325.14	yes	3072	731.325	even	168	inner