Embedded newform 731.2.bm.a.3.9

• Label: 731.2.bm.a.3.9
• Level: $731$
• Weight: $2$
• Character: 731.3
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $6144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bm (of order \(336\), degree \(96\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(6144\)
Relative dimension:	\(64\) over \(\Q(\zeta_{336})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{336}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	731.3
Dual form			731.2.bm.a.244.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.125640 + 2.23723i) q^{2} +(2.33558 + 0.649282i) q^{3} +(-3.00198 - 0.338241i) q^{4} +(0.799277 - 0.238383i) q^{5} +(-1.74603 + 5.14365i) q^{6} +(-0.0317705 + 0.484724i) q^{7} +(0.383211 - 2.25542i) q^{8} +(2.46380 + 1.48459i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6144 q - 112 q^{2} - 88 q^{3} - 80 q^{4} - 88 q^{5} - 48 q^{6} - 144 q^{7} - 112 q^{8} - 104 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{16}\right)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.125640	+	2.23723i	−0.0888409	+	1.58196i	0.564848	+	0.825195i	\(0.308935\pi\)
							−0.653689	+	0.756763i	\(0.726779\pi\)
\(3\)	2.33558	+	0.649282i	1.34845	+	0.374863i	0.865554	−	0.500815i	\(-0.166966\pi\)
							0.482895	+	0.875678i	\(0.339585\pi\)
\(5\)	0.799277	−	0.238383i	0.357447	−	0.106608i	−0.102069	−	0.994777i	\(-0.532546\pi\)
							0.459517	+	0.888169i	\(0.348023\pi\)
\(7\)	−0.0317705	+	0.484724i	−0.0120081	+	0.183208i	0.987641	+	0.156731i	\(0.0500956\pi\)
							−0.999649	+	0.0264772i	\(0.991571\pi\)
\(11\)	1.25739	+	0.649581i	0.379116	+	0.195856i	0.637971	−	0.770060i	\(-0.279774\pi\)
							−0.258855	+	0.965916i	\(0.583345\pi\)
\(13\)	4.27171	+	0.159836i	1.18476	+	0.0443306i	0.622493	−	0.782625i	\(-0.286120\pi\)
							0.562267	+	0.826956i	\(0.309929\pi\)
\(17\)	1.87237	+	3.67345i	0.454116	+	0.890943i
							\(19\)	2.38295	+	3.95472i	0.546687	+	0.907275i	0.999874	+	0.0158603i	\(0.00504870\pi\)
−0.453188	+	0.891415i	\(0.649713\pi\)
\(23\)	−3.88023	−	6.04906i	−0.809084	−	1.26132i	−0.961993	−	0.273075i	\(-0.911959\pi\)
							0.152908	−	0.988240i	\(-0.451136\pi\)
\(29\)	2.37678	−	1.85933i	0.441356	−	0.345269i	−0.370796	−	0.928714i	\(-0.620915\pi\)
							0.812152	+	0.583446i	\(0.198296\pi\)
\(31\)	−4.16558	−	9.79651i	−0.748160	−	1.75951i	−0.644109	−	0.764934i	\(-0.722772\pi\)
							−0.104051	−	0.994572i	\(-0.533181\pi\)
\(37\)	−2.24011	−	1.96453i	−0.368273	−	0.322966i	0.455129	−	0.890426i	\(-0.349593\pi\)
							−0.823402	+	0.567459i	\(0.807926\pi\)
\(41\)	−10.5508	+	6.22377i	−1.64776	+	0.971990i	−0.675728	+	0.737151i	\(0.736170\pi\)
							−0.972034	+	0.234839i	\(0.924544\pi\)
\(43\)	6.30079	+	1.81660i	0.960862	+	0.277029i
							\(47\)	−0.205470	+	1.82360i	−0.0299709	+	0.265999i	0.969786	+	0.243957i	\(0.0784455\pi\)
−0.999757	+	0.0220424i	\(0.992983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bm.a.3.9	✓	6144
17.6	odd	16	inner	731.2.bm.a.261.9	yes	6144
43.29	odd	42	inner	731.2.bm.a.717.9	yes	6144
731.244	even	336	inner	731.2.bm.a.244.9	yes	6144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bm.a.3.9	✓	6144	1.1	even	1	trivial
731.2.bm.a.244.9	yes	6144	731.244	even	336	inner
731.2.bm.a.261.9	yes	6144	17.6	odd	16	inner
731.2.bm.a.717.9	yes	6144	43.29	odd	42	inner