Embedded newform 731.2.y.a.4.7

• Label: 731.2.y.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.84201 - 1.46896i) q^{2} +(-3.11411 - 0.350875i) q^{3} +(0.790139 + 3.46182i) q^{4} +(-0.601633 + 0.210521i) q^{5} +(5.22081 + 5.22081i) q^{6} +(2.55945 + 2.55945i) q^{7} +(1.58534 - 3.29200i) q^{8} +(6.64976 + 1.51776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.84201	−	1.46896i	−1.30250	−	1.03871i	−0.996229	−	0.0867655i	\(-0.972347\pi\)
							−0.306272	−	0.951944i	\(-0.599082\pi\)
\(3\)	−3.11411	−	0.350875i	−1.79793	−	0.202578i	−0.851251	−	0.524758i	\(-0.824156\pi\)
							−0.946678	+	0.322180i	\(0.895584\pi\)
\(5\)	−0.601633	+	0.210521i	−0.269059	+	0.0941477i	−0.461436	−	0.887173i	\(-0.652666\pi\)
							0.192378	+	0.981321i	\(0.438380\pi\)
\(7\)	2.55945	+	2.55945i	0.967381	+	0.967381i	0.999485	−	0.0321040i	\(-0.0102208\pi\)
							−0.0321040	+	0.999485i	\(0.510221\pi\)
\(11\)	−1.22412	+	0.769166i	−0.369086	+	0.231912i	−0.703785	−	0.710413i	\(-0.748508\pi\)
							0.334699	+	0.942325i	\(0.391365\pi\)
\(13\)	−0.0896870	−	0.0431910i	−0.0248747	−	0.0119790i	0.421405	−	0.906873i	\(-0.361537\pi\)
							−0.446280	+	0.894894i	\(0.647251\pi\)
\(17\)	−2.65572	+	3.15391i	−0.644107	+	0.764935i
							\(19\)	4.78679	−	1.09255i	1.09816	−	0.250649i	0.365209	−	0.930926i	\(-0.380998\pi\)
0.732955	+	0.680277i	\(0.238140\pi\)
\(23\)	−2.80301	+	1.76125i	−0.584469	+	0.367246i	−0.791575	−	0.611072i	\(-0.790738\pi\)
							0.207106	+	0.978319i	\(0.433596\pi\)
\(29\)	−0.398536	−	3.53710i	−0.0740063	−	0.656824i	−0.974878	−	0.222739i	\(-0.928500\pi\)
							0.900872	−	0.434085i	\(-0.142928\pi\)
\(31\)	−0.425605	−	3.77735i	−0.0764409	−	0.678432i	−0.972242	−	0.233977i	\(-0.924826\pi\)
							0.895801	−	0.444455i	\(-0.146603\pi\)
\(37\)	4.05583	−	4.05583i	0.666774	−	0.666774i	−0.290194	−	0.956968i	\(-0.593720\pi\)
							0.956968	+	0.290194i	\(0.0937198\pi\)
\(41\)	1.92219	−	0.216578i	0.300195	−	0.0338238i	0.0394176	−	0.999223i	\(-0.487450\pi\)
							0.260777	+	0.965399i	\(0.416021\pi\)
\(43\)	4.51962	+	4.75111i	0.689236	+	0.724537i
							\(47\)	2.96860	+	13.0063i	0.433015	+	1.89716i	0.441623	+	0.897201i	\(0.354403\pi\)
−0.00860782	+	0.999963i	\(0.502740\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.7	✓	768
17.13	even	4	inner	731.2.y.a.47.58	yes	768
43.11	even	7	inner	731.2.y.a.140.58	yes	768
731.183	even	28	inner	731.2.y.a.183.7	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.7	✓	768	1.1	even	1	trivial
731.2.y.a.47.58	yes	768	17.13	even	4	inner
731.2.y.a.140.58	yes	768	43.11	even	7	inner
731.2.y.a.183.7	yes	768	731.183	even	28	inner