Embedded newform 731.2.j.a.509.10

• Label: 731.2.j.a.509.10
• 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			509.10
Character	\(\chi\)	\(=\)	731.509
Dual form			731.2.j.a.135.10

$q$-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	−0.375606	−	0.926779i	\(-0.622566\pi\)
							0.614811	+	0.788674i	\(0.289232\pi\)
\(5\)	−1.22310	+	0.706158i	−0.546988	+	0.315804i	−0.747906	−	0.663804i	\(-0.768941\pi\)
							0.200918	+	0.979608i	\(0.435607\pi\)
\(7\)	2.02420	+	1.16867i	0.765077	+	0.441717i	0.831116	−	0.556100i	\(-0.187703\pi\)
							−0.0660387	+	0.997817i	\(0.521036\pi\)
\(11\)		−	2.22029i		−	0.669444i	−0.942317	−	0.334722i	\(-0.891358\pi\)
							0.942317	−	0.334722i	\(-0.108642\pi\)
\(13\)	3.44285	−	5.96319i	0.954875	−	1.65389i	0.220222	−	0.975450i	\(-0.429322\pi\)
							0.734653	−	0.678443i	\(-0.237345\pi\)
\(17\)	3.14171	+	2.67015i	0.761976	+	0.647605i
							\(19\)	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	−0.794405	−	0.607388i	\(-0.792217\pi\)
							0.128811	+	0.991669i	\(0.458884\pi\)
\(29\)	−1.97503	−	1.14028i	−0.366754	−	0.211745i	0.305286	−	0.952261i	\(-0.401248\pi\)
							−0.672039	+	0.740516i	\(0.734581\pi\)
\(31\)	4.10408	−	2.36949i	0.737115	−	0.425574i	−0.0839043	−	0.996474i	\(-0.526739\pi\)
							0.821019	+	0.570900i	\(0.193406\pi\)
\(37\)	6.36747	−	3.67626i	1.04680	−	0.604373i	0.125052	−	0.992150i	\(-0.460090\pi\)
							0.921753	+	0.387777i	\(0.126757\pi\)
\(41\)			11.1475i			1.74094i	0.492221	+	0.870470i	\(0.336185\pi\)
							−0.492221	+	0.870470i	\(0.663815\pi\)
\(43\)	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.509.10	yes	128
17.16	even	2	inner	731.2.j.a.509.9	yes	128
43.6	even	3	inner	731.2.j.a.135.9	✓	128
731.135	even	6	inner	731.2.j.a.135.10	yes	128

    

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