Embedded newform 731.2.j.a.509.9

• Label: 731.2.j.a.509.9
• Level: $731$
• Weight: $2$
• Character: 731.509
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			509.9
Character	\(\chi\)	\(=\)	731.509
Dual form			731.2.j.a.135.9

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.09187 q^{2} +(-0.414315 + 0.239205i) q^{3} +2.37591 q^{4} +(1.22310 - 0.706158i) q^{5} +(0.866692 - 0.500385i) q^{6} +(-2.02420 - 1.16867i) q^{7} -0.786343 q^{8} +(-1.38556 + 2.39986i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{2} + 116 q^{4} - 12 q^{8} + 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.09187			−1.47917			−0.739586	−	0.673062i	\(-0.764979\pi\)
							−0.739586	+	0.673062i	\(0.764979\pi\)
\(3\)	−0.414315	+	0.239205i	−0.239205	+	0.138105i	−0.614811	−	0.788674i	\(-0.710768\pi\)
							0.375606	+	0.926779i	\(0.377434\pi\)
\(5\)	1.22310	−	0.706158i	0.546988	−	0.315804i	−0.200918	−	0.979608i	\(-0.564393\pi\)
							0.747906	+	0.663804i	\(0.231059\pi\)
\(7\)	−2.02420	−	1.16867i	−0.765077	−	0.441717i	0.0660387	−	0.997817i	\(-0.478964\pi\)
							−0.831116	+	0.556100i	\(0.812297\pi\)
\(11\)			2.22029i			0.669444i	0.942317	+	0.334722i	\(0.108642\pi\)
							−0.942317	+	0.334722i	\(0.891358\pi\)
\(13\)	3.44285	−	5.96319i	0.954875	−	1.65389i	0.220222	−	0.975450i	\(-0.429322\pi\)
							0.734653	−	0.678443i	\(-0.237345\pi\)
\(17\)	−3.88327	−	1.38573i	−0.941831	−	0.336088i
							\(19\)	0.0342440	+	0.0593123i	0.00785611	+	0.0136072i	0.869927	−	0.493181i	\(-0.164166\pi\)
−0.862071	+	0.506788i	\(0.830833\pi\)
\(23\)	3.19208	−	1.84295i	0.665594	−	0.384281i	−0.128811	−	0.991669i	\(-0.541116\pi\)
							0.794405	+	0.607388i	\(0.207783\pi\)
\(29\)	1.97503	+	1.14028i	0.366754	+	0.211745i	0.672039	−	0.740516i	\(-0.265419\pi\)
							−0.305286	+	0.952261i	\(0.598752\pi\)
\(31\)	−4.10408	+	2.36949i	−0.737115	+	0.425574i	−0.821019	−	0.570900i	\(-0.806594\pi\)
							0.0839043	+	0.996474i	\(0.473261\pi\)
\(37\)	−6.36747	+	3.67626i	−1.04680	+	0.604373i	−0.921753	−	0.387777i	\(-0.873243\pi\)
							−0.125052	+	0.992150i	\(0.539910\pi\)
\(41\)		−	11.1475i		−	1.74094i	−0.492221	−	0.870470i	\(-0.663815\pi\)
							0.492221	−	0.870470i	\(-0.336185\pi\)
\(43\)	5.40537	+	3.71241i	0.824311	+	0.566138i
							\(47\)	−0.393738			−0.0574326			−0.0287163	−	0.999588i	\(-0.509142\pi\)
−0.0287163	+	0.999588i	\(0.509142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.j.a.509.9	yes	128
17.16	even	2	inner	731.2.j.a.509.10	yes	128
43.6	even	3	inner	731.2.j.a.135.10	yes	128
731.135	even	6	inner	731.2.j.a.135.9	✓	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.j.a.135.9	✓	128	731.135	even	6	inner
731.2.j.a.135.10	yes	128	43.6	even	3	inner
731.2.j.a.509.9	yes	128	1.1	even	1	trivial
731.2.j.a.509.10	yes	128	17.16	even	2	inner