Embedded newform 731.2.bh.a.13.12

• Label: 731.2.bh.a.13.12
• Level: $731$
• Weight: $2$
• Character: 731.13
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1536$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bh (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			13.12
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81482 - 0.414221i) q^{2} +(-1.61287 - 0.0603495i) q^{3} +(1.32006 + 0.635709i) q^{4} +(0.0618915 + 0.0838599i) q^{5} +(2.90208 + 0.777610i) q^{6} +(-1.28801 - 4.80691i) q^{7} +(0.778392 + 0.620747i) q^{8} +(-0.393892 - 0.0295181i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81482	−	0.414221i	−1.28327	−	0.292899i	−0.474101	−	0.880470i	\(-0.657227\pi\)
							−0.809172	+	0.587572i	\(0.800084\pi\)
\(3\)	−1.61287	−	0.0603495i	−0.931193	−	0.0348428i	−0.432512	−	0.901628i	\(-0.642373\pi\)
							−0.498681	+	0.866785i	\(0.666182\pi\)
\(5\)	0.0618915	+	0.0838599i	0.0276787	+	0.0375033i	0.818219	−	0.574907i	\(-0.194962\pi\)
							−0.790540	+	0.612411i	\(0.790200\pi\)
\(7\)	−1.28801	−	4.80691i	−0.486821	−	1.81684i	−0.571717	−	0.820451i	\(-0.693723\pi\)
							0.0848964	−	0.996390i	\(-0.472944\pi\)
\(11\)	1.21069	−	3.45994i	0.365036	−	1.04321i	−0.604148	−	0.796872i	\(-0.706486\pi\)
							0.969184	−	0.246340i	\(-0.0792279\pi\)
\(13\)	−0.707419	−	1.80247i	−0.196203	−	0.499916i	0.798585	−	0.601882i	\(-0.205582\pi\)
							−0.994788	+	0.101965i	\(0.967487\pi\)
\(17\)	0.887116	+	4.02654i	0.215157	+	0.976579i
							\(19\)	4.93486	−	0.369817i	1.13214	−	0.0848418i	0.504545	−	0.863386i	\(-0.331660\pi\)
0.627590	+	0.778544i	\(0.284041\pi\)
\(23\)	4.18278	−	0.791425i	0.872171	−	0.165024i	0.269487	−	0.963004i	\(-0.413146\pi\)
							0.602684	+	0.797980i	\(0.294098\pi\)
\(29\)	−0.336606	−	8.99599i	−0.0625062	−	1.67051i	−0.579841	−	0.814730i	\(-0.696885\pi\)
							0.517335	−	0.855783i	\(-0.326924\pi\)
\(31\)	−6.75930	−	3.57239i	−1.21401	−	0.641621i	−0.267267	−	0.963623i	\(-0.586120\pi\)
							−0.946739	+	0.322002i	\(0.895644\pi\)
\(37\)	−1.19892	+	4.47444i	−0.197102	+	0.735594i	0.794611	+	0.607119i	\(0.207675\pi\)
							−0.991713	+	0.128475i	\(0.958992\pi\)
\(41\)	−3.99194	−	6.35314i	−0.623437	−	0.992194i	−0.997889	−	0.0649474i	\(-0.979312\pi\)
							0.374452	−	0.927246i	\(-0.377831\pi\)
\(43\)	2.17288	+	6.18697i	0.331361	+	0.943504i
							\(47\)	−1.54429	−	0.743693i	−0.225258	−	0.108479i	0.317851	−	0.948141i	\(-0.397039\pi\)
−0.543109	+	0.839662i	\(0.682753\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.12	✓	1536
17.4	even	4	inner	731.2.bh.a.701.53	yes	1536
43.10	even	21	inner	731.2.bh.a.268.53	yes	1536
731.225	even	84	inner	731.2.bh.a.225.12	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.12	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.12	yes	1536	731.225	even	84	inner
731.2.bh.a.268.53	yes	1536	43.10	even	21	inner
731.2.bh.a.701.53	yes	1536	17.4	even	4	inner