Embedded newform 731.2.j.a.135.14

• Label: 731.2.j.a.135.14
• Level: $731$
• Weight: $2$
• Character: 731.135
• 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			135.14
Character	\(\chi\)	\(=\)	731.135
Dual form			731.2.j.a.509.14

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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	\(-0.208721\pi\)
							−0.131733	+	0.991285i	\(0.542054\pi\)
\(5\)	1.07261	+	0.619273i	0.479687	+	0.276947i	0.720286	−	0.693677i	\(-0.244011\pi\)
							−0.240599	+	0.970625i	\(0.577344\pi\)
\(7\)	−0.325723	+	0.188056i	−0.123112	+	0.0710786i	−0.560291	−	0.828296i	\(-0.689311\pi\)
							0.437180	+	0.899374i	\(0.355977\pi\)
\(11\)		−	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	0.574708	−	0.818359i	\(-0.305116\pi\)
							−0.996073	+	0.0885322i	\(0.971782\pi\)
\(17\)	−4.10250	−	0.411696i	−0.995002	−	0.0998509i
							\(19\)	2.17636	−	3.76957i	0.499292	−	0.864799i	−0.500708	−	0.865616i	\(-0.666927\pi\)
1.00000		0.000817420i	\(0.000260193\pi\)
\(23\)	−6.19848	−	3.57870i	−1.29247	−	0.746210i	−0.313381	−	0.949627i	\(-0.601462\pi\)
							−0.979092	+	0.203418i	\(0.934795\pi\)
\(29\)	−2.12867	+	1.22899i	−0.395283	+	0.228217i	−0.684447	−	0.729063i	\(-0.739956\pi\)
							0.289163	+	0.957280i	\(0.406623\pi\)
\(31\)	4.76401	+	2.75050i	0.855641	+	0.494005i	0.862550	−	0.505972i	\(-0.168866\pi\)
							−0.00690913	+	0.999976i	\(0.502199\pi\)
\(37\)	0.0633372	+	0.0365677i	0.0104126	+	0.00601170i	0.505197	−	0.863004i	\(-0.331420\pi\)
							−0.494785	+	0.869016i	\(0.664753\pi\)
\(41\)			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.135.14	yes	128
17.16	even	2	inner	731.2.j.a.135.13	✓	128
43.36	even	3	inner	731.2.j.a.509.13	yes	128
731.509	even	6	inner	731.2.j.a.509.14	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.j.a.135.13	✓	128	17.16	even	2	inner
731.2.j.a.135.14	yes	128	1.1	even	1	trivial
731.2.j.a.509.13	yes	128	43.36	even	3	inner
731.2.j.a.509.14	yes	128	731.509	even	6	inner