Embedded newform 731.2.bl.a.9.3

• Label: 731.2.bl.a.9.3
• 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			9.3
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.55439 - 0.287810i) q^{2} +(1.48660 + 0.895766i) q^{3} +(4.49220 + 1.02532i) q^{4} +(-2.99252 + 1.95932i) q^{5} +(-3.53955 - 2.71599i) q^{6} +(0.541511 + 0.0712913i) q^{7} +(-6.32713 - 2.21396i) q^{8} +(0.00578988 + 0.0109550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.55439	−	0.287810i	−1.80622	−	0.203513i	−0.856404	−	0.516307i	\(-0.827306\pi\)
							−0.949821	+	0.312794i	\(0.898735\pi\)
\(3\)	1.48660	+	0.895766i	0.858292	+	0.517171i	0.875799	−	0.482676i	\(-0.160335\pi\)
							−0.0175076	+	0.999847i	\(0.505573\pi\)
\(5\)	−2.99252	+	1.95932i	−1.33830	+	0.876233i	−0.997946	−	0.0640657i	\(-0.979593\pi\)
							−0.340350	+	0.940299i	\(0.610546\pi\)
\(7\)	0.541511	+	0.0712913i	0.204672	+	0.0269456i	0.232165	−	0.972676i	\(-0.425419\pi\)
							−0.0274930	+	0.999622i	\(0.508752\pi\)
\(11\)	−1.08606	−	1.53066i	−0.327459	−	0.461511i	0.618505	−	0.785781i	\(-0.287739\pi\)
							−0.945965	+	0.324270i	\(0.894881\pi\)
\(13\)	−0.560793	−	0.0420256i	−0.155536	−	0.0116558i	−0.00326552	−	0.999995i	\(-0.501039\pi\)
							−0.152271	+	0.988339i	\(0.548658\pi\)
\(17\)	−1.72284	−	3.74591i	−0.417851	−	0.908516i
							\(19\)	1.14067	−	2.15825i	0.261687	−	0.495137i	−0.717902	−	0.696144i	\(-0.754898\pi\)
0.979590	+	0.201007i	\(0.0644214\pi\)
\(23\)	1.02705	−	2.23884i	0.214154	−	0.466831i	−0.771365	−	0.636393i	\(-0.780426\pi\)
							0.985520	+	0.169562i	\(0.0542353\pi\)
\(29\)	0.818921	−	3.30204i	0.152070	−	0.613174i	−0.844868	−	0.534975i	\(-0.820321\pi\)
							0.996938	−	0.0781991i	\(-0.0249170\pi\)
\(31\)	2.72343	−	2.82724i	0.489142	−	0.507787i	−0.429397	−	0.903116i	\(-0.641274\pi\)
							0.918540	+	0.395328i	\(0.129369\pi\)
\(37\)	0.462011	+	3.50932i	0.0759542	+	0.576929i	0.986529	+	0.163585i	\(0.0523059\pi\)
							−0.910575	+	0.413344i	\(0.864361\pi\)
\(41\)	−1.81743	+	3.28840i	−0.283836	+	0.513562i	−0.978625	−	0.205651i	\(-0.934069\pi\)
							0.694790	+	0.719213i	\(0.255497\pi\)
\(43\)	3.55961	+	5.50719i	0.542835	+	0.839839i
							\(47\)	8.32722	+	1.90063i	1.21465	+	0.277236i	0.781396	−	0.624036i	\(-0.214508\pi\)
0.433254	+	0.901272i	\(0.357365\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.3	✓	3072
17.2	even	8	inner	731.2.bl.a.138.62	yes	3072
43.24	even	21	inner	731.2.bl.a.196.62	yes	3072
731.325	even	168	inner	731.2.bl.a.325.3	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.3	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.62	yes	3072	17.2	even	8	inner
731.2.bl.a.196.62	yes	3072	43.24	even	21	inner
731.2.bl.a.325.3	yes	3072	731.325	even	168	inner