Embedded newform 731.2.u.b.52.9

• Label: 731.2.u.b.52.9
• Level: $731$
• Weight: $2$
• Character: 731.52
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.u (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(29\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			52.9
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.916857 - 1.14970i) q^{2} +(-1.68959 - 0.254664i) q^{3} +(-0.0361471 + 0.158371i) q^{4} +(2.78996 + 1.90216i) q^{5} +(1.25632 + 2.17601i) q^{6} +(0.435372 - 0.754087i) q^{7} +(-2.43457 + 1.17243i) q^{8} +(-0.0768702 - 0.0237113i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.916857	−	1.14970i	−0.648316	−	0.812963i	0.343699	−	0.939080i	\(-0.388320\pi\)
							−0.992015	+	0.126117i	\(0.959748\pi\)
\(3\)	−1.68959	−	0.254664i	−0.975483	−	0.147030i	−0.358092	−	0.933686i	\(-0.616572\pi\)
							−0.617392	+	0.786656i	\(0.711811\pi\)
\(5\)	2.78996	+	1.90216i	1.24771	+	0.850673i	0.993203	−	0.116399i	\(-0.0371351\pi\)
							0.254505	+	0.967071i	\(0.418087\pi\)
\(7\)	0.435372	−	0.754087i	0.164555	−	0.285018i	−0.771942	−	0.635693i	\(-0.780714\pi\)
							0.936497	+	0.350675i	\(0.114048\pi\)
\(11\)	0.0420090	+	0.184053i	0.0126662	+	0.0554942i	0.980866	−	0.194684i	\(-0.0623680\pi\)
							−0.968200	+	0.250178i	\(0.919511\pi\)
\(13\)	0.0373828	−	0.498838i	0.0103681	−	0.138353i	−0.989617	−	0.143727i	\(-0.954091\pi\)
							0.999986	+	0.00537363i	\(0.00171049\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	6.51365	−	2.00919i	1.49433	−	0.460941i	0.563417	−	0.826173i	\(-0.309487\pi\)
0.930916	+	0.365232i	\(0.119010\pi\)
\(23\)	−3.29510	+	3.05740i	−0.687075	+	0.637512i	−0.944395	−	0.328814i	\(-0.893351\pi\)
							0.257320	+	0.966326i	\(0.417161\pi\)
\(29\)	8.42962	−	1.27056i	1.56534	−	0.235937i	0.691583	−	0.722297i	\(-0.256914\pi\)
							0.873759	+	0.486360i	\(0.161676\pi\)
\(31\)	2.44854	−	6.23877i	0.439770	−	1.12052i	−0.523511	−	0.852019i	\(-0.675378\pi\)
							0.963281	−	0.268496i	\(-0.0865267\pi\)
\(37\)	3.95932	+	6.85775i	0.650909	+	1.12741i	0.982903	+	0.184126i	\(0.0589453\pi\)
							−0.331994	+	0.943281i	\(0.607721\pi\)
\(41\)	−6.07203	−	7.61408i	−0.948292	−	1.18912i	−0.981845	−	0.189683i	\(-0.939254\pi\)
							0.0335538	−	0.999437i	\(-0.489317\pi\)
\(43\)	4.00705	−	5.19072i	0.611069	−	0.791577i
							\(47\)	2.15199	−	9.42850i	0.313900	−	1.37529i	−0.534158	−	0.845385i	\(-0.679371\pi\)
0.848058	−	0.529903i	\(-0.177772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.9	✓	348
43.24	even	21	inner	731.2.u.b.239.9	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.9	✓	348	1.1	even	1	trivial
731.2.u.b.239.9	yes	348	43.24	even	21	inner