Embedded newform 731.2.bd.a.7.9

• Label: 731.2.bd.a.7.9
• Level: $731$
• Weight: $2$
• Character: 731.7
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bd (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(64\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			7.9
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00010 - 0.828468i) q^{2} +(0.00800967 - 0.122204i) q^{3} +(1.89982 + 1.89982i) q^{4} +(0.138736 + 0.158198i) q^{5} +(-0.117262 + 0.237784i) q^{6} +(-2.95893 + 3.37401i) q^{7} +(-0.568950 - 1.37357i) q^{8} +(2.95946 + 0.389621i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00010	−	0.828468i	−1.41428	−	0.585815i	−0.460865	−	0.887470i	\(-0.652461\pi\)
							−0.953417	+	0.301655i	\(0.902461\pi\)
\(3\)	0.00800967	−	0.122204i	0.00462438	−	0.0705545i	−0.994923	−	0.100639i	\(-0.967911\pi\)
							0.999547	+	0.0300850i	\(0.00957780\pi\)
\(5\)	0.138736	+	0.158198i	0.0620447	+	0.0707484i	0.782030	−	0.623241i	\(-0.214185\pi\)
							−0.719985	+	0.693990i	\(0.755851\pi\)
\(7\)	−2.95893	+	3.37401i	−1.11837	+	1.27526i	−0.160821	+	0.986984i	\(0.551414\pi\)
							−0.957549	+	0.288272i	\(0.906919\pi\)
\(11\)	−4.61282	+	3.08219i	−1.39082	+	0.929314i	−0.390857	+	0.920452i	\(0.627821\pi\)
							−0.999961	+	0.00886288i	\(0.997179\pi\)
\(13\)	0.618082	+	2.30671i	0.171425	+	0.639767i	0.997133	+	0.0756691i	\(0.0241093\pi\)
							−0.825708	+	0.564098i	\(0.809224\pi\)
\(17\)	−3.77805	−	1.65117i	−0.916311	−	0.400468i
							\(19\)	−0.887231	−	6.73919i	−0.203545	−	1.54608i	−0.722558	−	0.691310i	\(-0.757034\pi\)
0.519013	−	0.854766i	\(-0.326300\pi\)
\(23\)	−1.79524	+	3.64039i	−0.374334	+	0.759075i	−0.999775	−	0.0211967i	\(-0.993252\pi\)
							0.625441	+	0.780271i	\(0.284919\pi\)
\(29\)	−1.16582	−	0.395743i	−0.216487	−	0.0734876i	0.211089	−	0.977467i	\(-0.432299\pi\)
							−0.427576	+	0.903979i	\(0.640632\pi\)
\(31\)	0.229855	−	3.50690i	0.0412831	−	0.629858i	−0.926217	−	0.376990i	\(-0.876959\pi\)
							0.967501	−	0.252869i	\(-0.0813741\pi\)
\(37\)	0.550805	−	8.40366i	0.0905518	−	1.38155i	−0.674828	−	0.737975i	\(-0.735782\pi\)
							0.765380	−	0.643578i	\(-0.222551\pi\)
\(41\)	1.96360	−	9.87168i	0.306663	−	1.54170i	−0.453077	−	0.891471i	\(-0.649674\pi\)
							0.759740	−	0.650227i	\(-0.225326\pi\)
\(43\)	3.75871	+	5.37327i	0.573198	+	0.819417i
							\(47\)	6.70764	−	6.70764i	0.978410	−	0.978410i	−0.0213614	−	0.999772i	\(-0.506800\pi\)
0.999772	+	0.0213614i	\(0.00680007\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.9	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.56	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.56	yes	1024
731.209	even	48	inner	731.2.bd.a.209.9	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.9	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.9	yes	1024	731.209	even	48	inner
731.2.bd.a.381.56	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.56	yes	1024	17.5	odd	16	inner