Embedded newform 731.2.bh.a.13.8

• Label: 731.2.bh.a.13.8
• 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			13.8
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22997 - 0.508977i) q^{2} +(-2.07186 - 0.0775236i) q^{3} +(2.91178 + 1.40224i) q^{4} +(-1.29034 - 1.74835i) q^{5} +(4.58074 + 1.22740i) q^{6} +(0.658840 + 2.45882i) q^{7} +(-2.20288 - 1.75674i) q^{8} +(1.29499 + 0.0970462i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22997	−	0.508977i	−1.57683	−	0.359901i	−0.657519	−	0.753438i	\(-0.728394\pi\)
							−0.919309	+	0.393537i	\(0.871251\pi\)
\(3\)	−2.07186	−	0.0775236i	−1.19619	−	0.0447583i	−0.568141	−	0.822932i	\(-0.692337\pi\)
							−0.628050	+	0.778173i	\(0.716147\pi\)
\(5\)	−1.29034	−	1.74835i	−0.577057	−	0.781884i	0.414708	−	0.909954i	\(-0.363884\pi\)
							−0.991765	+	0.128070i	\(0.959122\pi\)
\(7\)	0.658840	+	2.45882i	0.249018	+	0.929348i	0.971321	+	0.237771i	\(0.0764169\pi\)
							−0.722303	+	0.691577i	\(0.756916\pi\)
\(11\)	−0.138885	+	0.396910i	−0.0418753	+	0.119673i	−0.962922	−	0.269778i	\(-0.913050\pi\)
							0.921047	+	0.389451i	\(0.127335\pi\)
\(13\)	0.183906	+	0.468584i	0.0510062	+	0.129962i	0.954106	−	0.299469i	\(-0.0968095\pi\)
							−0.903100	+	0.429431i	\(0.858714\pi\)
\(17\)	3.11568	−	2.70047i	0.755663	−	0.654960i
							\(19\)	−3.38847	+	0.253931i	−0.777369	+	0.0582558i	−0.457504	−	0.889207i	\(-0.651257\pi\)
−0.319865	+	0.947463i	\(0.603638\pi\)
\(23\)	3.31285	−	0.626824i	0.690776	−	0.130702i	0.171341	−	0.985212i	\(-0.445190\pi\)
							0.519435	+	0.854510i	\(0.326142\pi\)
\(29\)	−0.299978	−	8.01707i	−0.0557044	−	1.48873i	−0.694576	−	0.719419i	\(-0.744408\pi\)
							0.638872	−	0.769313i	\(-0.279401\pi\)
\(31\)	2.50471	+	1.32378i	0.449859	+	0.237757i	0.676823	−	0.736146i	\(-0.263356\pi\)
							−0.226964	+	0.973903i	\(0.572880\pi\)
\(37\)	−2.16001	+	8.06125i	−0.355103	+	1.32526i	0.525252	+	0.850947i	\(0.323971\pi\)
							−0.880355	+	0.474315i	\(0.842696\pi\)
\(41\)	−3.41020	−	5.42731i	−0.532584	−	0.847603i	0.466738	−	0.884395i	\(-0.345429\pi\)
							−0.999323	+	0.0367920i	\(0.988286\pi\)
\(43\)	−5.45655	−	3.63677i	−0.832116	−	0.554602i
							\(47\)	2.27110	+	1.09370i	0.331274	+	0.159533i	0.592125	−	0.805846i	\(-0.298289\pi\)
−0.260852	+	0.965379i	\(0.584003\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.8	✓	1536
17.4	even	4	inner	731.2.bh.a.701.57	yes	1536
43.10	even	21	inner	731.2.bh.a.268.57	yes	1536
731.225	even	84	inner	731.2.bh.a.225.8	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.8	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.8	yes	1536	731.225	even	84	inner
731.2.bh.a.268.57	yes	1536	43.10	even	21	inner
731.2.bh.a.701.57	yes	1536	17.4	even	4	inner