Embedded newform 731.2.u.a.52.20

• Label: 731.2.u.a.52.20
• 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			52.20
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.615504 + 0.771817i) q^{2} +(1.80469 + 0.272014i) q^{3} +(0.228185 - 0.999743i) q^{4} +(1.07898 + 0.735639i) q^{5} +(0.900850 + 1.56032i) q^{6} +(-2.48864 + 4.31044i) q^{7} +(2.69092 - 1.29588i) q^{8} +(0.316206 + 0.0975368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.615504	+	0.771817i	0.435227	+	0.545757i	0.950278	−	0.311402i	\(-0.100799\pi\)
							−0.515051	+	0.857159i	\(0.672227\pi\)
\(3\)	1.80469	+	0.272014i	1.04194	+	0.157047i	0.647649	−	0.761939i	\(-0.275752\pi\)
							0.394291	+	0.918986i	\(0.370990\pi\)
\(5\)	1.07898	+	0.735639i	0.482536	+	0.328988i	0.780036	−	0.625734i	\(-0.215201\pi\)
							−0.297500	+	0.954722i	\(0.596153\pi\)
\(7\)	−2.48864	+	4.31044i	−0.940616	+	1.62919i	−0.176316	+	0.984334i	\(0.556418\pi\)
							−0.764300	+	0.644861i	\(0.776915\pi\)
\(11\)	0.189415	+	0.829880i	0.0571106	+	0.250218i	0.995423	−	0.0955721i	\(-0.0304680\pi\)
							−0.938312	+	0.345790i	\(0.887611\pi\)
\(13\)	−0.375652	+	5.01273i	−0.104187	+	1.39028i	0.662886	+	0.748720i	\(0.269331\pi\)
							−0.767073	+	0.641560i	\(0.778288\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	7.10362	−	2.19118i	1.62968	−	0.502690i	0.660873	−	0.750498i	\(-0.270186\pi\)
0.968809	+	0.247807i	\(0.0797099\pi\)
\(23\)	−4.70090	+	4.36180i	−0.980206	+	0.909498i	−0.995907	−	0.0903785i	\(-0.971192\pi\)
							0.0157015	+	0.999877i	\(0.495002\pi\)
\(29\)	6.40067	−	0.964746i	1.18857	−	0.179149i	0.475165	−	0.879897i	\(-0.342388\pi\)
							0.713409	+	0.700748i	\(0.247150\pi\)
\(31\)	2.77386	−	7.06768i	0.498200	−	1.26939i	−0.431845	−	0.901948i	\(-0.642137\pi\)
							0.930045	−	0.367446i	\(-0.119768\pi\)
\(37\)	−0.937755	−	1.62424i	−0.154166	−	0.267023i	0.778589	−	0.627534i	\(-0.215936\pi\)
							−0.932755	+	0.360511i	\(0.882602\pi\)
\(41\)	0.0400439	+	0.0502135i	0.00625381	+	0.00784203i	0.784949	−	0.619561i	\(-0.212689\pi\)
							−0.778695	+	0.627403i	\(0.784118\pi\)
\(43\)	6.16593	−	2.23188i	0.940296	−	0.340359i
							\(47\)	1.41963	−	6.21981i	0.207074	−	0.907252i	−0.759427	−	0.650592i	\(-0.774521\pi\)
0.966502	−	0.256660i	\(-0.0826220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.20	✓	348
43.24	even	21	inner	731.2.u.a.239.20	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.20	✓	348	1.1	even	1	trivial
731.2.u.a.239.20	yes	348	43.24	even	21	inner