Embedded newform 731.2.bd.a.7.1

• Label: 731.2.bd.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.52329 - 1.04518i) q^{2} +(0.167312 - 2.55269i) q^{3} +(3.86036 + 3.86036i) q^{4} +(1.64362 + 1.87419i) q^{5} +(-3.09020 + 6.26630i) q^{6} +(3.41978 - 3.89951i) q^{7} +(-3.61566 - 8.72898i) q^{8} +(-3.51390 - 0.462614i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.52329	−	1.04518i	−1.78423	−	0.739053i	−0.991598	−	0.129356i	\(-0.958709\pi\)
							−0.792634	−	0.609697i	\(-0.791291\pi\)
\(3\)	0.167312	−	2.55269i	0.0965977	−	1.47380i	−0.624534	−	0.780998i	\(-0.714711\pi\)
							0.721131	−	0.692798i	\(-0.243622\pi\)
\(5\)	1.64362	+	1.87419i	0.735050	+	0.838164i	0.991731	−	0.128332i	\(-0.0409623\pi\)
							−0.256681	+	0.966496i	\(0.582629\pi\)
\(7\)	3.41978	−	3.89951i	1.29255	−	1.47388i	0.497327	−	0.867563i	\(-0.334315\pi\)
							0.795227	−	0.606312i	\(-0.207352\pi\)
\(11\)	−1.18527	+	0.791975i	−0.357373	+	0.238789i	−0.721276	−	0.692648i	\(-0.756444\pi\)
							0.363903	+	0.931437i	\(0.381444\pi\)
\(13\)	1.61032	+	6.00978i	0.446621	+	1.66681i	0.711620	+	0.702565i	\(0.247962\pi\)
							−0.264998	+	0.964249i	\(0.585371\pi\)
\(17\)	−3.14021	−	2.67190i	−0.761614	−	0.648032i
							\(19\)	0.00731066	+	0.0555300i	0.00167718	+	0.0127395i	0.992263	−	0.124154i	\(-0.0396216\pi\)
−0.990586	+	0.136893i	\(0.956288\pi\)
\(23\)	1.62882	−	3.30293i	0.339633	−	0.688708i	−0.658315	−	0.752743i	\(-0.728730\pi\)
							0.997948	+	0.0640352i	\(0.0203970\pi\)
\(29\)	2.16093	+	0.733538i	0.401275	+	0.136215i	0.514779	−	0.857323i	\(-0.327874\pi\)
							−0.113504	+	0.993538i	\(0.536207\pi\)
\(31\)	0.152522	−	2.32703i	0.0273937	−	0.417947i	−0.962035	−	0.272926i	\(-0.912008\pi\)
							0.989429	−	0.145021i	\(-0.0463249\pi\)
\(37\)	0.482076	−	7.35505i	0.0792527	−	1.20916i	−0.754227	−	0.656613i	\(-0.771988\pi\)
							0.833480	−	0.552550i	\(-0.186345\pi\)
\(41\)	0.810375	−	4.07403i	0.126559	−	0.636256i	−0.864478	−	0.502671i	\(-0.832351\pi\)
							0.991037	−	0.133586i	\(-0.0426491\pi\)
\(43\)	6.36812	+	1.56429i	0.971130	+	0.238552i
							\(47\)	−1.34575	+	1.34575i	−0.196297	+	0.196297i	−0.798411	−	0.602113i	\(-0.794325\pi\)
0.602113	+	0.798411i	\(0.294325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.1	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.64	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.64	yes	1024
731.209	even	48	inner	731.2.bd.a.209.1	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.1	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.1	yes	1024	731.209	even	48	inner
731.2.bd.a.381.64	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.64	yes	1024	17.5	odd	16	inner