Embedded newform 731.2.u.a.52.15

• Label: 731.2.u.a.52.15
• 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			52.15
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0261406 + 0.0327793i) q^{2} +(-0.733041 - 0.110488i) q^{3} +(0.444651 - 1.94814i) q^{4} +(-0.752885 - 0.513308i) q^{5} +(-0.0155404 - 0.0269167i) q^{6} +(1.97546 - 3.42159i) q^{7} +(0.151031 - 0.0727325i) q^{8} +(-2.34158 - 0.722281i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0261406	+	0.0327793i	0.0184842	+	0.0231784i	0.790988	−	0.611832i	\(-0.209567\pi\)
							−0.772503	+	0.635011i	\(0.780996\pi\)
\(3\)	−0.733041	−	0.110488i	−0.423221	−	0.0637903i	−0.0660211	−	0.997818i	\(-0.521030\pi\)
							−0.357200	+	0.934028i	\(0.616269\pi\)
\(5\)	−0.752885	−	0.513308i	−0.336700	−	0.229558i	0.383152	−	0.923685i	\(-0.374838\pi\)
							−0.719853	+	0.694127i	\(0.755791\pi\)
\(7\)	1.97546	−	3.42159i	0.746653	−	1.29324i	−0.202765	−	0.979227i	\(-0.564993\pi\)
							0.949418	−	0.314014i	\(-0.101674\pi\)
\(11\)	−0.487441	−	2.13562i	−0.146969	−	0.643913i	−0.993717	−	0.111920i	\(-0.964300\pi\)
							0.846748	−	0.531994i	\(-0.178557\pi\)
\(13\)	−0.266592	+	3.55743i	−0.0739394	+	0.986653i	0.829658	+	0.558271i	\(0.188535\pi\)
							−0.903598	+	0.428382i	\(0.859084\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	2.37600	−	0.732899i	0.545092	−	0.168139i	−0.00996594	−	0.999950i	\(-0.503172\pi\)
0.555058	+	0.831812i	\(0.312696\pi\)
\(23\)	−6.88768	+	6.39084i	−1.43618	+	1.33258i	−0.581374	+	0.813637i	\(0.697484\pi\)
							−0.854808	+	0.518945i	\(0.826325\pi\)
\(29\)	6.25380	−	0.942609i	1.16130	−	0.175038i	0.460011	−	0.887913i	\(-0.347845\pi\)
							0.701291	+	0.712875i	\(0.252607\pi\)
\(31\)	−3.19301	+	8.13566i	−0.573482	+	1.46121i	0.290098	+	0.956997i	\(0.406312\pi\)
							−0.863580	+	0.504211i	\(0.831783\pi\)
\(37\)	−4.66611	−	8.08194i	−0.767103	−	1.32866i	−0.939127	−	0.343569i	\(-0.888364\pi\)
							0.172024	−	0.985093i	\(-0.444969\pi\)
\(41\)	3.89866	+	4.88877i	0.608869	+	0.763497i	0.986731	−	0.162365i	\(-0.0519121\pi\)
							−0.377862	+	0.925862i	\(0.623341\pi\)
\(43\)	−5.73389	−	3.18160i	−0.874409	−	0.485189i
							\(47\)	2.02093	−	8.85425i	0.294782	−	1.29153i	−0.583002	−	0.812470i	\(-0.698122\pi\)
0.877785	−	0.479055i	\(-0.159021\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.15	✓	348
43.24	even	21	inner	731.2.u.a.239.15	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.15	✓	348	1.1	even	1	trivial
731.2.u.a.239.15	yes	348	43.24	even	21	inner