Embedded newform 731.2.z.a.611.6

• Label: 731.2.z.a.611.6
• 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			611.6
Character	\(\chi\)	\(=\)	731.611
Dual form			731.2.z.a.67.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53756 - 1.92804i) q^{2} +(0.193458 - 1.28351i) q^{3} +(-0.908197 + 3.97907i) q^{4} +(-0.441633 + 0.647756i) q^{5} +(-2.77211 + 1.60048i) q^{6} +(-4.06857 - 2.34899i) q^{7} +(4.62452 - 2.22705i) q^{8} +(1.25675 + 0.387655i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53756	−	1.92804i	−1.08722	−	1.36333i	−0.926486	−	0.376328i	\(-0.877186\pi\)
							−0.160730	−	0.986998i	\(-0.551385\pi\)
\(3\)	0.193458	−	1.28351i	0.111693	−	0.741035i	−0.860943	−	0.508701i	\(-0.830126\pi\)
							0.972636	−	0.232334i	\(-0.0746361\pi\)
\(5\)	−0.441633	+	0.647756i	−0.197504	+	0.289685i	−0.912196	−	0.409754i	\(-0.865615\pi\)
							0.714692	+	0.699439i	\(0.246567\pi\)
\(7\)	−4.06857	−	2.34899i	−1.53778	−	0.887835i	−0.998969	−	0.0454064i	\(-0.985542\pi\)
							−0.538807	−	0.842429i	\(-0.681125\pi\)
\(11\)	−1.00441	+	0.229251i	−0.302842	+	0.0691217i	−0.371242	−	0.928536i	\(-0.621068\pi\)
							0.0683996	+	0.997658i	\(0.478211\pi\)
\(13\)	−0.0853578	+	1.13902i	−0.0236740	+	0.315907i	0.972778	+	0.231740i	\(0.0744420\pi\)
							−0.996452	+	0.0841670i	\(0.973177\pi\)
\(17\)	−3.90555	+	1.32162i	−0.947235	+	0.320541i
							\(19\)	−2.48008	+	0.765003i	−0.568969	+	0.175504i	−0.565874	−	0.824491i	\(-0.691461\pi\)
−0.00309488	+	0.999995i	\(0.500985\pi\)
\(23\)	0.877607	+	0.945835i	0.182994	+	0.197220i	0.817886	−	0.575380i	\(-0.195146\pi\)
							−0.634892	+	0.772601i	\(0.718956\pi\)
\(29\)	0.280085	+	1.85824i	0.0520105	+	0.345067i	0.999668	+	0.0257776i	\(0.00820617\pi\)
							−0.947657	+	0.319290i	\(0.896556\pi\)
\(31\)	7.11867	+	2.79387i	1.27855	+	0.501794i	0.904840	−	0.425751i	\(-0.139990\pi\)
							0.373711	+	0.927545i	\(0.378085\pi\)
\(37\)	−0.902913	+	0.521297i	−0.148438	+	0.0857007i	−0.572379	−	0.819989i	\(-0.693980\pi\)
							0.423941	+	0.905690i	\(0.360646\pi\)
\(41\)	−0.962317	+	0.767422i	−0.150289	+	0.119851i	−0.695748	−	0.718286i	\(-0.744927\pi\)
							0.545459	+	0.838138i	\(0.316355\pi\)
\(43\)	3.97763	+	5.21330i	0.606583	+	0.795021i
							\(47\)	0.205362	−	0.899749i	0.0299551	−	0.131242i	−0.957739	−	0.287637i	\(-0.907130\pi\)
0.987695	+	0.156395i	\(0.0499874\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.611.6	yes	768
17.16	even	2	inner	731.2.z.a.611.5	yes	768
43.24	even	21	inner	731.2.z.a.67.5	✓	768
731.67	even	42	inner	731.2.z.a.67.6	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.5	✓	768	43.24	even	21	inner
731.2.z.a.67.6	yes	768	731.67	even	42	inner
731.2.z.a.611.5	yes	768	17.16	even	2	inner
731.2.z.a.611.6	yes	768	1.1	even	1	trivial