Embedded newform 731.2.n.a.608.50

• Label: 731.2.n.a.608.50
• Level: $731$
• Weight: $2$
• Character: 731.608
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			608.50
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.79720i q^{2} +(-1.99454 + 0.534434i) q^{3} -1.22993 q^{4} +(0.473839 - 0.126965i) q^{5} +(-0.960485 - 3.58458i) q^{6} +(-3.30462 - 0.885471i) q^{7} +1.38398i q^{8} +(1.09448 - 0.631896i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.79720i			1.27081i	0.772178	+	0.635406i	\(0.219167\pi\)
							−0.772178	+	0.635406i	\(0.780833\pi\)
\(3\)	−1.99454	+	0.534434i	−1.15155	+	0.308556i	−0.783584	−	0.621285i	\(-0.786611\pi\)
							−0.367961	+	0.929841i	\(0.619944\pi\)
\(5\)	0.473839	−	0.126965i	0.211907	−	0.0567803i	−0.151304	−	0.988487i	\(-0.548347\pi\)
							0.363211	+	0.931707i	\(0.381680\pi\)
\(7\)	−3.30462	−	0.885471i	−1.24903	−	0.334677i	−0.427068	−	0.904219i	\(-0.640454\pi\)
							−0.821962	+	0.569543i	\(0.807120\pi\)
\(11\)	−1.62450	+	1.62450i	−0.489804	+	0.489804i	−0.908244	−	0.418440i	\(-0.862577\pi\)
							0.418440	+	0.908244i	\(0.362577\pi\)
\(13\)	−0.566387	−	0.981011i	−0.157087	−	0.272083i	0.776730	−	0.629834i	\(-0.216877\pi\)
							−0.933817	+	0.357751i	\(0.883544\pi\)
\(17\)	3.68792	−	1.84371i	0.894452	−	0.447165i
							\(19\)	0.798992	+	0.461298i	0.183301	+	0.105829i	0.588843	−	0.808248i	\(-0.299584\pi\)
−0.405542	+	0.914077i	\(0.632917\pi\)
\(23\)	−1.50527	−	5.61775i	−0.313871	−	1.17138i	−0.925036	−	0.379879i	\(-0.875966\pi\)
							0.611165	−	0.791503i	\(-0.290701\pi\)
\(29\)	1.39906	−	5.22138i	0.259800	−	0.969586i	−0.705557	−	0.708653i	\(-0.749303\pi\)
							0.965357	−	0.260933i	\(-0.0840302\pi\)
\(31\)	4.40471	−	1.18024i	0.791109	−	0.211977i	0.159432	−	0.987209i	\(-0.449034\pi\)
							0.631677	+	0.775232i	\(0.282367\pi\)
\(37\)	6.30282	−	1.68883i	1.03618	−	0.277643i	0.299648	−	0.954050i	\(-0.403131\pi\)
							0.736528	+	0.676407i	\(0.236464\pi\)
\(41\)	4.10891	−	4.10891i	0.641705	−	0.641705i	−0.309270	−	0.950974i	\(-0.600085\pi\)
							0.950974	+	0.309270i	\(0.100085\pi\)
\(43\)	−6.55585	−	0.144430i	−0.999757	−	0.0220253i
							\(47\)	−7.21339			−1.05218			−0.526091	−	0.850428i	\(-0.676343\pi\)
−0.526091	+	0.850428i	\(0.676343\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.50	yes	256
17.4	even	4	inner	731.2.n.a.565.15	yes	256
43.36	even	3	inner	731.2.n.a.251.50	yes	256
731.208	even	12	inner	731.2.n.a.208.15	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.15	✓	256	731.208	even	12	inner
731.2.n.a.251.50	yes	256	43.36	even	3	inner
731.2.n.a.565.15	yes	256	17.4	even	4	inner
731.2.n.a.608.50	yes	256	1.1	even	1	trivial