Embedded newform 731.2.y.a.4.3

• Label: 731.2.y.a.4.3
• Level: $731$
• Weight: $2$
• Character: 731.4
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.y (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			4.3
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.01642 - 1.60804i) q^{2} +(-0.987529 - 0.111268i) q^{3} +(1.03511 + 4.53513i) q^{4} +(3.97010 - 1.38920i) q^{5} +(1.81235 + 1.81235i) q^{6} +(2.87738 + 2.87738i) q^{7} +(2.96740 - 6.16187i) q^{8} +(-1.96195 - 0.447803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.01642	−	1.60804i	−1.42583	−	1.13706i	−0.968870	−	0.247569i	\(-0.920368\pi\)
							−0.456956	−	0.889489i	\(-0.651060\pi\)
\(3\)	−0.987529	−	0.111268i	−0.570150	−	0.0642405i	−0.177817	−	0.984064i	\(-0.556903\pi\)
							−0.392333	+	0.919823i	\(0.628332\pi\)
\(5\)	3.97010	−	1.38920i	1.77548	−	0.621269i	0.775535	−	0.631305i	\(-0.217480\pi\)
							0.999950	+	0.0100364i	\(0.00319473\pi\)
\(7\)	2.87738	+	2.87738i	1.08755	+	1.08755i	0.995780	+	0.0917676i	\(0.0292517\pi\)
							0.0917676	+	0.995780i	\(0.470748\pi\)
\(11\)	4.07083	−	2.55787i	1.22740	−	0.771227i	0.247148	−	0.968978i	\(-0.420507\pi\)
							0.980252	+	0.197751i	\(0.0633637\pi\)
\(13\)	−1.20337	−	0.579511i	−0.333754	−	0.160728i	0.259499	−	0.965743i	\(-0.416443\pi\)
							−0.593253	+	0.805016i	\(0.702157\pi\)
\(17\)	−0.296337	+	4.11244i	−0.0718724	+	0.997414i
							\(19\)	−2.09797	+	0.478847i	−0.481307	+	0.109855i	−0.456289	−	0.889832i	\(-0.650822\pi\)
−0.0250182	+	0.999687i	\(0.507964\pi\)
\(23\)	4.65625	−	2.92572i	0.970895	−	0.610054i	0.0495387	−	0.998772i	\(-0.484225\pi\)
							0.921357	+	0.388718i	\(0.127082\pi\)
\(29\)	0.388029	+	3.44385i	0.0720552	+	0.639507i	0.976890	+	0.213744i	\(0.0685658\pi\)
							−0.904835	+	0.425763i	\(0.860006\pi\)
\(31\)	1.09672	+	9.73364i	0.196976	+	1.74821i	0.571142	+	0.820852i	\(0.306501\pi\)
							−0.374165	+	0.927362i	\(0.622071\pi\)
\(37\)	1.07043	−	1.07043i	0.175977	−	0.175977i	−0.613622	−	0.789600i	\(-0.710288\pi\)
							0.789600	+	0.613622i	\(0.210288\pi\)
\(41\)	−3.82470	+	0.430940i	−0.597317	+	0.0673015i	−0.405447	−	0.914119i	\(-0.632884\pi\)
							−0.191871	+	0.981420i	\(0.561455\pi\)
\(43\)	−3.68316	−	5.42534i	−0.561677	−	0.827356i
							\(47\)	0.100197	+	0.438992i	0.0146152	+	0.0640336i	0.981710	−	0.190384i	\(-0.0609733\pi\)
−0.967094	+	0.254418i	\(0.918116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.3	✓	768
17.13	even	4	inner	731.2.y.a.47.62	yes	768
43.11	even	7	inner	731.2.y.a.140.62	yes	768
731.183	even	28	inner	731.2.y.a.183.3	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.3	✓	768	1.1	even	1	trivial
731.2.y.a.47.62	yes	768	17.13	even	4	inner
731.2.y.a.140.62	yes	768	43.11	even	7	inner
731.2.y.a.183.3	yes	768	731.183	even	28	inner