Embedded newform 731.2.y.a.4.18

• Label: 731.2.y.a.4.18
• 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			4.18
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17476 - 0.936838i) q^{2} +(2.38910 + 0.269187i) q^{3} +(0.0573487 + 0.251261i) q^{4} +(-3.52574 + 1.23371i) q^{5} +(-2.55443 - 2.55443i) q^{6} +(-0.191598 - 0.191598i) q^{7} +(-1.13586 + 2.35864i) q^{8} +(2.71054 + 0.618664i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17476	−	0.936838i	−0.830679	−	0.662445i	0.112894	−	0.993607i	\(-0.463988\pi\)
							−0.943574	+	0.331162i	\(0.892559\pi\)
\(3\)	2.38910	+	0.269187i	1.37935	+	0.155415i	0.770281	−	0.637705i	\(-0.220116\pi\)
							0.609066	+	0.793120i	\(0.291545\pi\)
\(5\)	−3.52574	+	1.23371i	−1.57676	+	0.551731i	−0.969989	−	0.243148i	\(-0.921820\pi\)
							−0.606768	+	0.794879i	\(0.707534\pi\)
\(7\)	−0.191598	−	0.191598i	−0.0724171	−	0.0724171i	0.669971	−	0.742388i	\(-0.266307\pi\)
							−0.742388	+	0.669971i	\(0.766307\pi\)
\(11\)	4.24202	−	2.66543i	1.27902	−	0.803659i	0.290808	−	0.956781i	\(-0.406076\pi\)
							0.988207	+	0.153122i	\(0.0489329\pi\)
\(13\)	−1.40765	−	0.677886i	−0.390411	−	0.188012i	0.228372	−	0.973574i	\(-0.426660\pi\)
							−0.618783	+	0.785562i	\(0.712374\pi\)
\(17\)	3.64973	+	1.91820i	0.885189	+	0.465232i
							\(19\)	6.39825	−	1.46036i	1.46786	−	0.335030i	0.587457	−	0.809255i	\(-0.300129\pi\)
0.880403	+	0.474226i	\(0.157272\pi\)
\(23\)	1.31371	−	0.825462i	0.273928	−	0.172121i	−0.388061	−	0.921634i	\(-0.626855\pi\)
							0.661990	+	0.749513i	\(0.269712\pi\)
\(29\)	−0.618227	−	5.48692i	−0.114802	−	1.01890i	−0.910587	−	0.413317i	\(-0.864370\pi\)
							0.795785	−	0.605579i	\(-0.207058\pi\)
\(31\)	−1.15032	−	10.2094i	−0.206603	−	1.83365i	−0.486566	−	0.873644i	\(-0.661751\pi\)
							0.279962	−	0.960011i	\(-0.409678\pi\)
\(37\)	−1.94506	+	1.94506i	−0.319766	+	0.319766i	−0.848677	−	0.528911i	\(-0.822600\pi\)
							0.528911	+	0.848677i	\(0.322600\pi\)
\(41\)	0.967863	−	0.109052i	0.151155	−	0.0170311i	−0.0360628	−	0.999350i	\(-0.511482\pi\)
							0.187218	+	0.982318i	\(0.440053\pi\)
\(43\)	4.02521	−	5.17665i	0.613839	−	0.789432i
							\(47\)	1.95256	+	8.55471i	0.284810	+	1.24783i	0.891546	+	0.452930i	\(0.149621\pi\)
−0.606736	+	0.794903i	\(0.707522\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.18	✓	768
17.13	even	4	inner	731.2.y.a.47.47	yes	768
43.11	even	7	inner	731.2.y.a.140.47	yes	768
731.183	even	28	inner	731.2.y.a.183.18	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.18	✓	768	1.1	even	1	trivial
731.2.y.a.47.47	yes	768	17.13	even	4	inner
731.2.y.a.140.47	yes	768	43.11	even	7	inner
731.2.y.a.183.18	yes	768	731.183	even	28	inner