Embedded newform 731.2.j.a.135.11

• Label: 731.2.j.a.135.11
• 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			135.11
Character	\(\chi\)	\(=\)	731.135
Dual form			731.2.j.a.509.11

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.05109 q^{2} +(-2.62751 - 1.51699i) q^{3} +2.20698 q^{4} +(2.73044 + 1.57642i) q^{5} +(5.38926 + 3.11149i) q^{6} +(0.592018 - 0.341802i) q^{7} -0.424527 q^{8} +(3.10253 + 5.37374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 4 q^{2} + 116 q^{4} - 12 q^{8} + 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.05109			−1.45034			−0.725170	−	0.688570i	\(-0.758239\pi\)
							−0.725170	+	0.688570i	\(0.758239\pi\)
\(3\)	−2.62751	−	1.51699i	−1.51699	−	0.875836i	−0.999801	−	0.0199662i	\(-0.993644\pi\)
							−0.517192	−	0.855870i	\(-0.673023\pi\)
\(5\)	2.73044	+	1.57642i	1.22109	+	0.704996i	0.965150	−	0.261696i	\(-0.0842819\pi\)
							0.255939	+	0.966693i	\(0.417615\pi\)
\(7\)	0.592018	−	0.341802i	0.223762	−	0.129189i	−0.383929	−	0.923363i	\(-0.625429\pi\)
							0.607691	+	0.794174i	\(0.292096\pi\)
\(11\)			5.64679i			1.70257i	0.524703	+	0.851286i	\(0.324176\pi\)
							−0.524703	+	0.851286i	\(0.675824\pi\)
\(13\)	2.15005	+	3.72399i	0.596316	+	1.03285i	0.993360	+	0.115050i	\(0.0367028\pi\)
							−0.397044	+	0.917800i	\(0.629964\pi\)
\(17\)	3.49409	+	2.18891i	0.847441	+	0.530890i
							\(19\)	0.516837	−	0.895188i	0.118571	−	0.205370i	−0.800631	−	0.599158i	\(-0.795502\pi\)
0.919201	+	0.393788i	\(0.128835\pi\)
\(23\)	−5.57224	−	3.21713i	−1.16189	−	0.670819i	−0.210136	−	0.977672i	\(-0.567391\pi\)
							−0.951757	+	0.306853i	\(0.900724\pi\)
\(29\)	−5.65872	+	3.26706i	−1.05080	+	0.606678i	−0.922873	−	0.385105i	\(-0.874165\pi\)
							−0.127925	+	0.991784i	\(0.540832\pi\)
\(31\)	−1.62115	−	0.935974i	−0.291168	−	0.168106i	0.347301	−	0.937754i	\(-0.387098\pi\)
							−0.638468	+	0.769648i	\(0.720432\pi\)
\(37\)	−9.34986	−	5.39814i	−1.53711	−	0.887449i	−0.999006	−	0.0445691i	\(-0.985809\pi\)
							−0.538101	−	0.842880i	\(-0.680858\pi\)
\(41\)			2.21300i			0.345613i	0.984956	+	0.172806i	\(0.0552835\pi\)
							−0.984956	+	0.172806i	\(0.944716\pi\)
\(43\)	−3.47316	+	5.56212i	−0.529652	+	0.848215i
							\(47\)	5.35132			0.780571			0.390285	−	0.920694i	\(-0.372376\pi\)
0.390285	+	0.920694i	\(0.372376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.j.a.135.11	✓	128
17.16	even	2	inner	731.2.j.a.135.12	yes	128
43.36	even	3	inner	731.2.j.a.509.12	yes	128
731.509	even	6	inner	731.2.j.a.509.11	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.j.a.135.11	✓	128	1.1	even	1	trivial
731.2.j.a.135.12	yes	128	17.16	even	2	inner
731.2.j.a.509.11	yes	128	731.509	even	6	inner
731.2.j.a.509.12	yes	128	43.36	even	3	inner