Embedded newform 731.2.bd.a.7.20

• Label: 731.2.bd.a.7.20
• 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			7.20
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20385 - 0.498652i) q^{2} +(0.0460112 - 0.701995i) q^{3} +(-0.213607 - 0.213607i) q^{4} +(-0.570187 - 0.650174i) q^{5} +(-0.405442 + 0.822155i) q^{6} +(2.09550 - 2.38946i) q^{7} +(1.14794 + 2.77137i) q^{8} +(2.48365 + 0.326979i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20385	−	0.498652i	−0.851252	−	0.352600i	−0.0859719	−	0.996298i	\(-0.527400\pi\)
							−0.765280	+	0.643697i	\(0.777400\pi\)
\(3\)	0.0460112	−	0.701995i	0.0265646	−	0.405297i	−0.963783	−	0.266689i	\(-0.914070\pi\)
							0.990347	−	0.138608i	\(-0.0442629\pi\)
\(5\)	−0.570187	−	0.650174i	−0.254995	−	0.290766i	0.610152	−	0.792285i	\(-0.291108\pi\)
							−0.865147	+	0.501518i	\(0.832775\pi\)
\(7\)	2.09550	−	2.38946i	0.792023	−	0.903129i	−0.205114	−	0.978738i	\(-0.565757\pi\)
							0.997138	+	0.0756087i	\(0.0240900\pi\)
\(11\)	4.14260	−	2.76800i	1.24904	−	0.834582i	0.257742	−	0.966214i	\(-0.417022\pi\)
							0.991299	+	0.131632i	\(0.0420217\pi\)
\(13\)	1.22499	+	4.57172i	0.339751	+	1.26797i	0.898626	+	0.438715i	\(0.144566\pi\)
							−0.558875	+	0.829252i	\(0.688767\pi\)
\(17\)	1.18238	+	3.94993i	0.286770	+	0.958000i
							\(19\)	0.870775	+	6.61419i	0.199769	+	1.51740i	0.738115	+	0.674675i	\(0.235716\pi\)
−0.538345	+	0.842724i	\(0.680950\pi\)
\(23\)	−0.354889	+	0.719643i	−0.0739994	+	0.150056i	−0.930729	−	0.365710i	\(-0.880826\pi\)
							0.856729	+	0.515766i	\(0.172493\pi\)
\(29\)	5.63440	+	1.91262i	1.04628	+	0.355165i	0.791074	−	0.611721i	\(-0.209522\pi\)
							0.255209	+	0.966886i	\(0.417856\pi\)
\(31\)	−0.560596	+	8.55304i	−0.100686	+	1.53617i	0.586655	+	0.809837i	\(0.300444\pi\)
							−0.687341	+	0.726335i	\(0.741222\pi\)
\(37\)	0.461599	−	7.04264i	0.0758864	−	1.15780i	−0.774827	−	0.632173i	\(-0.782163\pi\)
							0.850713	−	0.525630i	\(-0.176170\pi\)
\(41\)	1.56742	−	7.87994i	0.244790	−	1.23064i	−0.641360	−	0.767240i	\(-0.721629\pi\)
							0.886149	−	0.463400i	\(-0.153371\pi\)
\(43\)	−0.0197385	+	6.55741i	−0.00301010	+	0.999995i
							\(47\)	7.68379	−	7.68379i	1.12080	−	1.12080i	0.129174	−	0.991622i	\(-0.458767\pi\)
0.991622	−	0.129174i	\(-0.0412325\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.20	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.45	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.45	yes	1024
731.209	even	48	inner	731.2.bd.a.209.20	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.20	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.20	yes	1024	731.209	even	48	inner
731.2.bd.a.381.45	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.45	yes	1024	17.5	odd	16	inner