Embedded newform 731.2.j.a.509.13

• Label: 731.2.j.a.509.13
• 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			509.13
Character	\(\chi\)	\(=\)	731.509
Dual form			731.2.j.a.135.13

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.04963 q^{2} +(-1.14468 + 0.660879i) q^{3} +2.20099 q^{4} +(-1.07261 + 0.619273i) q^{5} +(2.34616 - 1.35456i) q^{6} +(0.325723 + 0.188056i) q^{7} -0.411958 q^{8} +(-0.626478 + 1.08509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{2} + 116 q^{4} - 12 q^{8} + 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.04963			−1.44931			−0.724654	−	0.689113i	\(-0.758001\pi\)
							−0.724654	+	0.689113i	\(0.758001\pi\)
\(3\)	−1.14468	+	0.660879i	−0.660879	+	0.381559i	−0.792612	−	0.609727i	\(-0.791279\pi\)
							0.131733	+	0.991285i	\(0.457946\pi\)
\(5\)	−1.07261	+	0.619273i	−0.479687	+	0.276947i	−0.720286	−	0.693677i	\(-0.755989\pi\)
							0.240599	+	0.970625i	\(0.422656\pi\)
\(7\)	0.325723	+	0.188056i	0.123112	+	0.0710786i	0.560291	−	0.828296i	\(-0.310689\pi\)
							−0.437180	+	0.899374i	\(0.644023\pi\)
\(11\)		−	3.21595i		−	0.969645i	−0.874612	−	0.484823i	\(-0.838884\pi\)
							0.874612	−	0.484823i	\(-0.161116\pi\)
\(13\)	−1.51925	+	2.63143i	−0.421365	+	0.729826i	−0.996073	−	0.0885322i	\(-0.971782\pi\)
							0.574708	+	0.818359i	\(0.305116\pi\)
\(17\)	1.69471	+	3.75872i	0.411028	+	0.911623i
							\(19\)	2.17636	+	3.76957i	0.499292	+	0.864799i	1.00000		0.000817420i	\(-0.000260193\pi\)
−0.500708	+	0.865616i	\(0.666927\pi\)
\(23\)	6.19848	−	3.57870i	1.29247	−	0.746210i	0.313381	−	0.949627i	\(-0.398538\pi\)
							0.979092	+	0.203418i	\(0.0652050\pi\)
\(29\)	2.12867	+	1.22899i	0.395283	+	0.228217i	0.684447	−	0.729063i	\(-0.260044\pi\)
							−0.289163	+	0.957280i	\(0.593377\pi\)
\(31\)	−4.76401	+	2.75050i	−0.855641	+	0.494005i	−0.862550	−	0.505972i	\(-0.831134\pi\)
							0.00690913	+	0.999976i	\(0.497801\pi\)
\(37\)	−0.0633372	+	0.0365677i	−0.0104126	+	0.00601170i	−0.505197	−	0.863004i	\(-0.668580\pi\)
							0.494785	+	0.869016i	\(0.335247\pi\)
\(41\)			4.40087i			0.687301i	0.939098	+	0.343650i	\(0.111663\pi\)
							−0.939098	+	0.343650i	\(0.888337\pi\)
\(43\)	−6.23169	−	2.04110i	−0.950323	−	0.311264i
							\(47\)	1.26040			0.183848			0.0919240	−	0.995766i	\(-0.470698\pi\)
0.0919240	+	0.995766i	\(0.470698\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.j.a.509.13	yes	128
17.16	even	2	inner	731.2.j.a.509.14	yes	128
43.6	even	3	inner	731.2.j.a.135.14	yes	128
731.135	even	6	inner	731.2.j.a.135.13	✓	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.j.a.135.13	✓	128	731.135	even	6	inner
731.2.j.a.135.14	yes	128	43.6	even	3	inner
731.2.j.a.509.13	yes	128	1.1	even	1	trivial
731.2.j.a.509.14	yes	128	17.16	even	2	inner