Embedded newform 731.2.bd.a.7.17

• Label: 731.2.bd.a.7.17
• 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			7.17
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41151 - 0.584666i) q^{2} +(0.145113 - 2.21399i) q^{3} +(0.236307 + 0.236307i) q^{4} +(0.494155 + 0.563476i) q^{5} +(-1.49927 + 3.04022i) q^{6} +(-0.0148342 + 0.0169152i) q^{7} +(0.973943 + 2.35131i) q^{8} +(-1.90637 - 0.250978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41151	−	0.584666i	−0.998087	−	0.413421i	−0.176992	−	0.984212i	\(-0.556637\pi\)
							−0.821095	+	0.570791i	\(0.806637\pi\)
\(3\)	0.145113	−	2.21399i	0.0837808	−	1.27825i	−0.724427	−	0.689351i	\(-0.757896\pi\)
							0.808208	−	0.588897i	\(-0.200438\pi\)
\(5\)	0.494155	+	0.563476i	0.220993	+	0.251994i	0.851703	−	0.524024i	\(-0.175570\pi\)
							−0.630711	+	0.776018i	\(0.717236\pi\)
\(7\)	−0.0148342	+	0.0169152i	−0.00560682	+	0.00639335i	−0.754636	−	0.656143i	\(-0.772187\pi\)
							0.749030	+	0.662537i	\(0.230520\pi\)
\(11\)	4.13056	−	2.75995i	1.24541	−	0.832156i	0.254551	−	0.967059i	\(-0.418072\pi\)
							0.990859	+	0.134903i	\(0.0430724\pi\)
\(13\)	0.398723	+	1.48805i	0.110586	+	0.412712i	0.998919	−	0.0464878i	\(-0.0148029\pi\)
							−0.888333	+	0.459200i	\(0.848136\pi\)
\(17\)	−0.649262	−	4.07167i	−0.157469	−	0.987524i
							\(19\)	−0.235886	−	1.79173i	−0.0541160	−	0.411052i	−0.996966	−	0.0778352i	\(-0.975199\pi\)
0.942850	−	0.333217i	\(-0.108134\pi\)
\(23\)	−0.449787	+	0.912078i	−0.0937871	+	0.190181i	−0.938711	−	0.344705i	\(-0.887979\pi\)
							0.844924	+	0.534886i	\(0.179646\pi\)
\(29\)	−3.01199	−	1.02243i	−0.559312	−	0.189861i	0.0274459	−	0.999623i	\(-0.491263\pi\)
							−0.586758	+	0.809762i	\(0.699596\pi\)
\(31\)	0.329857	−	5.03264i	0.0592441	−	0.903890i	−0.860023	−	0.510255i	\(-0.829551\pi\)
							0.919267	−	0.393634i	\(-0.128782\pi\)
\(37\)	−0.622518	+	9.49779i	−0.102341	+	1.56143i	0.570287	+	0.821446i	\(0.306832\pi\)
							−0.672628	+	0.739981i	\(0.734835\pi\)
\(41\)	0.688614	−	3.46190i	0.107543	−	0.540657i	−0.889023	−	0.457862i	\(-0.848615\pi\)
							0.996567	−	0.0827951i	\(-0.0263847\pi\)
\(43\)	6.47438	−	1.04040i	0.987333	−	0.158660i
							\(47\)	−5.39392	+	5.39392i	−0.786784	+	0.786784i	−0.980966	−	0.194181i	\(-0.937795\pi\)
0.194181	+	0.980966i	\(0.437795\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.17	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.48	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.48	yes	1024
731.209	even	48	inner	731.2.bd.a.209.17	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.17	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.17	yes	1024	731.209	even	48	inner
731.2.bd.a.381.48	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.48	yes	1024	17.5	odd	16	inner