Embedded newform 731.2.bh.a.13.11

• Label: 731.2.bh.a.13.11
• 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			13.11
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91310 - 0.436653i) q^{2} +(1.16771 + 0.0436927i) q^{3} +(1.66735 + 0.802954i) q^{4} +(-2.60787 - 3.53354i) q^{5} +(-2.21487 - 0.593473i) q^{6} +(-0.817724 - 3.05179i) q^{7} +(0.229177 + 0.182763i) q^{8} +(-1.62997 - 0.122149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91310	−	0.436653i	−1.35277	−	0.308760i	−0.516126	−	0.856513i	\(-0.672626\pi\)
							−0.836640	+	0.547753i	\(0.815484\pi\)
\(3\)	1.16771	+	0.0436927i	0.674179	+	0.0252260i	0.372289	−	0.928117i	\(-0.378573\pi\)
							0.301890	+	0.953343i	\(0.402383\pi\)
\(5\)	−2.60787	−	3.53354i	−1.16627	−	1.58025i	−0.735069	−	0.677992i	\(-0.762850\pi\)
							−0.431206	−	0.902254i	\(-0.641912\pi\)
\(7\)	−0.817724	−	3.05179i	−0.309071	−	1.15347i	−0.929384	−	0.369113i	\(-0.879661\pi\)
							0.620314	−	0.784354i	\(-0.287005\pi\)
\(11\)	1.75251	−	5.00837i	0.528400	−	1.51008i	−0.302218	−	0.953239i	\(-0.597727\pi\)
							0.830618	−	0.556842i	\(-0.187987\pi\)
\(13\)	0.846531	+	2.15693i	0.234785	+	0.598223i	0.998902	−	0.0468550i	\(-0.0149199\pi\)
							−0.764116	+	0.645079i	\(0.776825\pi\)
\(17\)	−0.966440	−	4.00824i	−0.234396	−	0.972141i
							\(19\)	−0.357317	+	0.0267772i	−0.0819741	+	0.00614311i	−0.115654	−	0.993290i	\(-0.536896\pi\)
0.0336799	+	0.999433i	\(0.489277\pi\)
\(23\)	4.54323	−	0.859626i	0.947329	−	0.179244i	0.310781	−	0.950481i	\(-0.399409\pi\)
							0.636548	+	0.771237i	\(0.280362\pi\)
\(29\)	0.105310	+	2.81446i	0.0195555	+	0.522632i	0.974307	+	0.225226i	\(0.0723119\pi\)
							−0.954751	+	0.297406i	\(0.903879\pi\)
\(31\)	4.10706	+	2.17064i	0.737650	+	0.389859i	0.793492	−	0.608580i	\(-0.208261\pi\)
							−0.0558425	+	0.998440i	\(0.517784\pi\)
\(37\)	−0.251864	+	0.939969i	−0.0414062	+	0.154530i	−0.983534	−	0.180725i	\(-0.942156\pi\)
							0.942128	+	0.335255i	\(0.108822\pi\)
\(41\)	1.95288	+	3.10800i	0.304989	+	0.485388i	0.963692	−	0.267015i	\(-0.0860374\pi\)
							−0.658703	+	0.752403i	\(0.728895\pi\)
\(43\)	−0.00510714	−	6.55744i	−0.000778831	−	1.00000i
							\(47\)	−9.90444	−	4.76973i	−1.44471	−	0.695736i	−0.463043	−	0.886336i	\(-0.653243\pi\)
−0.981668	+	0.190599i	\(0.938957\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.11	✓	1536
17.4	even	4	inner	731.2.bh.a.701.54	yes	1536
43.10	even	21	inner	731.2.bh.a.268.54	yes	1536
731.225	even	84	inner	731.2.bh.a.225.11	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.11	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.11	yes	1536	731.225	even	84	inner
731.2.bh.a.268.54	yes	1536	43.10	even	21	inner
731.2.bh.a.701.54	yes	1536	17.4	even	4	inner