Embedded newform 731.2.z.a.67.7

• Label: 731.2.z.a.67.7
• Level: $731$
• Weight: $2$
• Character: 731.67
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			67.7
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.43695 + 1.80187i) q^{2} +(-0.348188 - 2.31008i) q^{3} +(-0.736893 - 3.22854i) q^{4} +(0.523790 + 0.768259i) q^{5} +(4.66280 + 2.69207i) q^{6} +(-2.81082 + 1.62283i) q^{7} +(2.72340 + 1.31152i) q^{8} +(-2.34851 + 0.724419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.43695	+	1.80187i	−1.01607	+	1.27412i	−0.0548084	+	0.998497i	\(0.517455\pi\)
							−0.961266	+	0.275621i	\(0.911117\pi\)
\(3\)	−0.348188	−	2.31008i	−0.201027	−	1.33372i	−0.830637	−	0.556814i	\(-0.812024\pi\)
							0.629611	−	0.776911i	\(-0.283214\pi\)
\(5\)	0.523790	+	0.768259i	0.234246	+	0.343576i	0.925306	−	0.379222i	\(-0.123808\pi\)
							−0.691060	+	0.722798i	\(0.742856\pi\)
\(7\)	−2.81082	+	1.62283i	−1.06239	+	0.613371i	−0.926092	−	0.377297i	\(-0.876853\pi\)
							−0.136297	+	0.990668i	\(0.543520\pi\)
\(11\)	3.36971	+	0.769115i	1.01601	+	0.231897i	0.697929	−	0.716167i	\(-0.254105\pi\)
							0.318078	+	0.948064i	\(0.396962\pi\)
\(13\)	−0.348764	−	4.65393i	−0.0967296	−	1.29077i	−0.808559	−	0.588415i	\(-0.799752\pi\)
							0.711830	−	0.702352i	\(-0.247867\pi\)
\(17\)	−2.10655	+	3.54435i	−0.510914	+	0.859632i
							\(19\)	−0.0838144	−	0.0258533i	−0.0192283	−	0.00593116i	0.285126	−	0.958490i	\(-0.407965\pi\)
−0.304354	+	0.952559i	\(0.598441\pi\)
\(23\)	2.02205	−	2.17925i	0.421626	−	0.454405i	−0.486140	−	0.873881i	\(-0.661596\pi\)
							0.907767	+	0.419476i	\(0.137786\pi\)
\(29\)	−0.633282	+	4.20155i	−0.117597	+	0.780208i	0.849829	+	0.527058i	\(0.176705\pi\)
							−0.967427	+	0.253150i	\(0.918533\pi\)
\(31\)	−7.51121	+	2.94793i	−1.34905	+	0.529465i	−0.926302	−	0.376781i	\(-0.877031\pi\)
							−0.422752	+	0.906246i	\(0.638936\pi\)
\(37\)	−3.78548	−	2.18555i	−0.622329	−	0.359302i	0.155446	−	0.987844i	\(-0.450319\pi\)
							−0.777775	+	0.628542i	\(0.783652\pi\)
\(41\)	−7.60574	−	6.06537i	−1.18782	−	0.947252i	−0.188424	−	0.982088i	\(-0.560338\pi\)
							−0.999393	+	0.0348355i	\(0.988909\pi\)
\(43\)	−5.93960	+	2.77870i	−0.905780	+	0.423747i
							\(47\)	−2.75961	−	12.0906i	−0.402530	−	1.76360i	−0.617095	−	0.786889i	\(-0.711690\pi\)
0.214565	−	0.976710i	\(-0.431167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.67.7	✓	768
17.16	even	2	inner	731.2.z.a.67.8	yes	768
43.9	even	21	inner	731.2.z.a.611.8	yes	768
731.611	even	42	inner	731.2.z.a.611.7	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.7	✓	768	1.1	even	1	trivial
731.2.z.a.67.8	yes	768	17.16	even	2	inner
731.2.z.a.611.7	yes	768	731.611	even	42	inner
731.2.z.a.611.8	yes	768	43.9	even	21	inner