Embedded newform 731.2.n.a.208.20

• Label: 731.2.n.a.208.20
• Level: $731$
• Weight: $2$
• Character: 731.208
• 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			208.20
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.45

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.24128i q^{2} +(2.55087 + 0.683505i) q^{3} +0.459214 q^{4} +(1.75642 + 0.470632i) q^{5} +(0.848423 - 3.16636i) q^{6} +(-1.74616 + 0.467882i) q^{7} -3.05258i q^{8} +(3.44170 + 1.98707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.24128i		−	0.877720i	−0.898555	−	0.438860i	\(-0.855382\pi\)
							0.898555	−	0.438860i	\(-0.144618\pi\)
\(3\)	2.55087	+	0.683505i	1.47275	+	0.394622i	0.903872	−	0.427802i	\(-0.140712\pi\)
							0.568875	+	0.822424i	\(0.307379\pi\)
\(5\)	1.75642	+	0.470632i	0.785497	+	0.210473i	0.629207	−	0.777238i	\(-0.283380\pi\)
							0.156290	+	0.987711i	\(0.450047\pi\)
\(7\)	−1.74616	+	0.467882i	−0.659987	+	0.176843i	−0.573240	−	0.819388i	\(-0.694314\pi\)
							−0.0867466	+	0.996230i	\(0.527647\pi\)
\(11\)	1.36614	+	1.36614i	0.411908	+	0.411908i	0.882403	−	0.470495i	\(-0.155925\pi\)
							−0.470495	+	0.882403i	\(0.655925\pi\)
\(13\)	−1.50484	+	2.60646i	−0.417367	+	0.722901i	−0.995674	−	0.0929187i	\(-0.970380\pi\)
							0.578307	+	0.815819i	\(0.303714\pi\)
\(17\)	1.26930	−	3.92286i	0.307852	−	0.951434i
							\(19\)	0.780862	−	0.450831i	0.179142	−	0.103428i	−0.407747	−	0.913095i	\(-0.633686\pi\)
0.586890	+	0.809667i	\(0.300352\pi\)
\(23\)	−0.978224	+	3.65078i	−0.203974	+	0.761240i	0.785786	+	0.618498i	\(0.212259\pi\)
							−0.989760	+	0.142742i	\(0.954408\pi\)
\(29\)	0.0319814	+	0.119356i	0.00593880	+	0.0221639i	0.968832	−	0.247720i	\(-0.0796813\pi\)
							−0.962893	+	0.269884i	\(0.913015\pi\)
\(31\)	−9.88434	−	2.64850i	−1.77528	−	0.475685i	−0.785570	−	0.618773i	\(-0.787630\pi\)
							−0.989710	+	0.143088i	\(0.954297\pi\)
\(37\)	3.88326	+	1.04052i	0.638404	+	0.171060i	0.563480	−	0.826130i	\(-0.309462\pi\)
							0.0749234	+	0.997189i	\(0.476129\pi\)
\(41\)	2.90950	+	2.90950i	0.454388	+	0.454388i	0.896808	−	0.442420i	\(-0.145880\pi\)
							−0.442420	+	0.896808i	\(0.645880\pi\)
\(43\)	6.20731	−	2.11406i	0.946606	−	0.322392i
							\(47\)	0.665590			0.0970863			0.0485431	−	0.998821i	\(-0.484542\pi\)
0.0485431	+	0.998821i	\(0.484542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.20	✓	256
17.13	even	4	inner	731.2.n.a.251.45	yes	256
43.6	even	3	inner	731.2.n.a.565.20	yes	256
731.608	even	12	inner	731.2.n.a.608.45	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.20	✓	256	1.1	even	1	trivial
731.2.n.a.251.45	yes	256	17.13	even	4	inner
731.2.n.a.565.20	yes	256	43.6	even	3	inner
731.2.n.a.608.45	yes	256	731.608	even	12	inner