Embedded newform 731.2.u.b.52.11

• Label: 731.2.u.b.52.11
• 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			52.11
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.614029 - 0.769968i) q^{2} +(1.49724 + 0.225673i) q^{3} +(0.229223 - 1.00429i) q^{4} +(0.606327 + 0.413386i) q^{5} +(-0.745588 - 1.29140i) q^{6} +(0.356455 - 0.617398i) q^{7} +(-2.68862 + 1.29477i) q^{8} +(-0.675919 - 0.208493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.614029	−	0.769968i	−0.434184	−	0.544450i	0.515816	−	0.856699i	\(-0.327489\pi\)
							−0.950000	+	0.312250i	\(0.898917\pi\)
\(3\)	1.49724	+	0.225673i	0.864432	+	0.130292i	0.566267	−	0.824222i	\(-0.308387\pi\)
							0.298165	+	0.954514i	\(0.403625\pi\)
\(5\)	0.606327	+	0.413386i	0.271157	+	0.184872i	0.691263	−	0.722603i	\(-0.257055\pi\)
							−0.420106	+	0.907475i	\(0.638007\pi\)
\(7\)	0.356455	−	0.617398i	0.134727	−	0.233354i	−0.790766	−	0.612119i	\(-0.790318\pi\)
							0.925493	+	0.378764i	\(0.123651\pi\)
\(11\)	−1.29884	−	5.69061i	−0.391616	−	1.71578i	−0.658956	−	0.752181i	\(-0.729002\pi\)
							0.267340	−	0.963602i	\(-0.413855\pi\)
\(13\)	0.0533787	−	0.712289i	0.0148046	−	0.197553i	−0.984897	−	0.173139i	\(-0.944609\pi\)
							0.999702	−	0.0244142i	\(-0.00777204\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	−4.97946	+	1.53596i	−1.14237	+	0.352374i	−0.807486	−	0.589887i	\(-0.799172\pi\)
−0.334881	+	0.942260i	\(0.608696\pi\)
\(23\)	2.38036	−	2.20865i	0.496339	−	0.460535i	−0.391840	−	0.920034i	\(-0.628161\pi\)
							0.888179	+	0.459498i	\(0.151971\pi\)
\(29\)	10.1236	−	1.52589i	1.87991	−	0.283351i	0.893680	−	0.448705i	\(-0.148115\pi\)
							0.986234	+	0.165353i	\(0.0528765\pi\)
\(31\)	1.85451	−	4.72522i	0.333080	−	0.848675i	−0.661991	−	0.749512i	\(-0.730288\pi\)
							0.995071	−	0.0991631i	\(-0.0316165\pi\)
\(37\)	−2.75043	−	4.76389i	−0.452168	−	0.783178i	0.546352	−	0.837555i	\(-0.316016\pi\)
							−0.998520	+	0.0543772i	\(0.982683\pi\)
\(41\)	5.92316	+	7.42741i	0.925042	+	1.15997i	0.986811	+	0.161876i	\(0.0517544\pi\)
							−0.0617690	+	0.998090i	\(0.519674\pi\)
\(43\)	6.54991	+	0.314154i	0.998852	+	0.0479081i
							\(47\)	−0.446903	+	1.95801i	−0.0651876	+	0.285605i	−0.997006	−	0.0773182i	\(-0.975364\pi\)
0.931819	+	0.362924i	\(0.118221\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.11	✓	348
43.24	even	21	inner	731.2.u.b.239.11	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.11	✓	348	1.1	even	1	trivial
731.2.u.b.239.11	yes	348	43.24	even	21	inner