Embedded newform 731.2.bh.a.13.7

• Label: 731.2.bh.a.13.7
• 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			13.7
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.27687 - 0.519681i) q^{2} +(1.99644 + 0.0747016i) q^{3} +(3.11213 + 1.49872i) q^{4} +(-0.407293 - 0.551863i) q^{5} +(-4.50682 - 1.20760i) q^{6} +(-0.238104 - 0.888617i) q^{7} +(-2.65525 - 2.11749i) q^{8} +(0.988594 + 0.0740848i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.27687	−	0.519681i	−1.60999	−	0.367470i	−0.679468	−	0.733705i	\(-0.737789\pi\)
							−0.930522	+	0.366235i	\(0.880647\pi\)
\(3\)	1.99644	+	0.0747016i	1.15265	+	0.0431290i	0.606870	−	0.794801i	\(-0.292425\pi\)
							0.545777	+	0.837930i	\(0.316234\pi\)
\(5\)	−0.407293	−	0.551863i	−0.182147	−	0.246801i	0.704005	−	0.710195i	\(-0.251393\pi\)
							−0.886152	+	0.463394i	\(0.846631\pi\)
\(7\)	−0.238104	−	0.888617i	−0.0899949	−	0.335866i	0.906218	−	0.422810i	\(-0.138956\pi\)
							−0.996213	+	0.0869447i	\(0.972290\pi\)
\(11\)	0.128268	−	0.366568i	0.0386742	−	0.110524i	−0.922927	−	0.384975i	\(-0.874210\pi\)
							0.961601	+	0.274450i	\(0.0884959\pi\)
\(13\)	−1.37613	−	3.50632i	−0.381670	−	0.972479i	−0.984621	−	0.174701i	\(-0.944104\pi\)
							0.602952	−	0.797778i	\(-0.293991\pi\)
\(17\)	4.11047	−	0.322577i	0.996935	−	0.0782365i
							\(19\)	−6.97321	+	0.522570i	−1.59976	+	0.119886i	−0.844507	−	0.535544i	\(-0.820107\pi\)
−0.755256	+	0.655430i	\(0.772488\pi\)
\(23\)	3.11054	−	0.588545i	0.648592	−	0.122720i	0.148802	−	0.988867i	\(-0.452458\pi\)
							0.499790	+	0.866147i	\(0.333411\pi\)
\(29\)	−0.197060	−	5.26653i	−0.0365931	−	0.977970i	−0.889787	−	0.456376i	\(-0.849147\pi\)
							0.853194	−	0.521594i	\(-0.174662\pi\)
\(31\)	−8.17713	−	4.32174i	−1.46866	−	0.776207i	−0.474106	−	0.880468i	\(-0.657228\pi\)
							−0.994550	+	0.104261i	\(0.966752\pi\)
\(37\)	1.95966	−	7.31355i	0.322166	−	1.20234i	−0.594964	−	0.803753i	\(-0.702834\pi\)
							0.917130	−	0.398588i	\(-0.130500\pi\)
\(41\)	5.50292	+	8.75785i	0.859412	+	1.36775i	0.928965	+	0.370169i	\(0.120700\pi\)
							−0.0695521	+	0.997578i	\(0.522157\pi\)
\(43\)	1.92960	+	6.26711i	0.294262	+	0.955725i
							\(47\)	5.90543	+	2.84391i	0.861396	+	0.414826i	0.811795	−	0.583943i	\(-0.198491\pi\)
0.0496008	+	0.998769i	\(0.484205\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.7	✓	1536
17.4	even	4	inner	731.2.bh.a.701.58	yes	1536
43.10	even	21	inner	731.2.bh.a.268.58	yes	1536
731.225	even	84	inner	731.2.bh.a.225.7	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.7	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.7	yes	1536	731.225	even	84	inner
731.2.bh.a.268.58	yes	1536	43.10	even	21	inner
731.2.bh.a.701.58	yes	1536	17.4	even	4	inner