Embedded newform 731.2.bd.a.7.6

• Label: 731.2.bd.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19786 - 0.910384i) q^{2} +(-0.118993 + 1.81548i) q^{3} +(2.58758 + 2.58758i) q^{4} +(-1.86406 - 2.12555i) q^{5} +(1.91431 - 3.88183i) q^{6} +(3.05317 - 3.48147i) q^{7} +(-1.51068 - 3.64712i) q^{8} +(-0.307458 - 0.0404776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19786	−	0.910384i	−1.55412	−	0.643739i	−0.570067	−	0.821598i	\(-0.693083\pi\)
							−0.984056	+	0.177860i	\(0.943083\pi\)
\(3\)	−0.118993	+	1.81548i	−0.0687004	+	1.04817i	0.814802	+	0.579739i	\(0.196846\pi\)
							−0.883503	+	0.468426i	\(0.844821\pi\)
\(5\)	−1.86406	−	2.12555i	−0.833633	−	0.950576i	0.165733	−	0.986171i	\(-0.447001\pi\)
							−0.999366	+	0.0355942i	\(0.988668\pi\)
\(7\)	3.05317	−	3.48147i	1.15399	−	1.31587i	0.213152	−	0.977019i	\(-0.431627\pi\)
							0.940839	−	0.338855i	\(-0.110040\pi\)
\(11\)	3.42237	−	2.28676i	1.03188	−	0.689483i	0.0802686	−	0.996773i	\(-0.474422\pi\)
							0.951616	+	0.307290i	\(0.0994222\pi\)
\(13\)	−0.120720	−	0.450532i	−0.0334816	−	0.124955i	0.947162	−	0.320755i	\(-0.103936\pi\)
							−0.980644	+	0.195800i	\(0.937270\pi\)
\(17\)	−2.37208	−	3.37242i	−0.575315	−	0.817932i
							\(19\)	0.415638	+	3.15708i	0.0953538	+	0.724284i	0.970616	+	0.240635i	\(0.0773556\pi\)
−0.875262	+	0.483649i	\(0.839311\pi\)
\(23\)	−3.19730	+	6.48348i	−0.666683	+	1.35190i	0.256127	+	0.966643i	\(0.417554\pi\)
							−0.922810	+	0.385256i	\(0.874113\pi\)
\(29\)	−9.33897	−	3.17015i	−1.73420	−	0.588683i	−0.737926	−	0.674882i	\(-0.764195\pi\)
							−0.996278	+	0.0861992i	\(0.972528\pi\)
\(31\)	0.522837	−	7.97695i	0.0939042	−	1.43270i	−0.647716	−	0.761882i	\(-0.724276\pi\)
							0.741620	−	0.670820i	\(-0.234058\pi\)
\(37\)	−0.199221	+	3.03952i	−0.0327517	+	0.499694i	0.949698	+	0.313167i	\(0.101390\pi\)
							−0.982450	+	0.186527i	\(0.940277\pi\)
\(41\)	0.836576	−	4.20575i	0.130651	−	0.656828i	−0.858841	−	0.512242i	\(-0.828815\pi\)
							0.989492	−	0.144586i	\(-0.0461850\pi\)
\(43\)	5.96297	−	2.72819i	0.909344	−	0.416045i
							\(47\)	8.17481	−	8.17481i	1.19242	−	1.19242i	0.216032	−	0.976386i	\(-0.430688\pi\)
0.976386	−	0.216032i	\(-0.0693116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.6	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.59	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.59	yes	1024
731.209	even	48	inner	731.2.bd.a.209.6	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.6	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.6	yes	1024	731.209	even	48	inner
731.2.bd.a.381.59	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.59	yes	1024	17.5	odd	16	inner