Embedded newform 731.2.z.a.611.7

• Label: 731.2.z.a.611.7
• Level: $731$
• Weight: $2$
• Character: 731.611
• 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			611.7
Character	\(\chi\)	\(=\)	731.611
Dual form			731.2.z.a.67.7

$q$-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 \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.7	✓	768	731.67	even	42	inner
731.2.z.a.67.8	yes	768	43.24	even	21	inner
731.2.z.a.611.7	yes	768	1.1	even	1	trivial
731.2.z.a.611.8	yes	768	17.16	even	2	inner