Embedded newform 731.2.bh.a.13.6

• Label: 731.2.bh.a.13.6
• 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			13.6
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.34214 - 0.534578i) q^{2} +(0.295194 + 0.0110454i) q^{3} +(3.39791 + 1.63635i) q^{4} +(0.0899681 + 0.121902i) q^{5} +(-0.685480 - 0.183674i) q^{6} +(-0.259001 - 0.966606i) q^{7} +(-3.32712 - 2.65329i) q^{8} +(-2.90459 - 0.217669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.34214	−	0.534578i	−1.65614	−	0.378004i	−0.710624	−	0.703572i	\(-0.751587\pi\)
							−0.945519	+	0.325568i	\(0.894445\pi\)
\(3\)	0.295194	+	0.0110454i	0.170430	+	0.00637704i	0.122470	−	0.992472i	\(-0.460918\pi\)
							0.0479600	+	0.998849i	\(0.484728\pi\)
\(5\)	0.0899681	+	0.121902i	0.0402349	+	0.0545164i	0.824253	−	0.566221i	\(-0.191595\pi\)
							−0.784018	+	0.620738i	\(0.786833\pi\)
\(7\)	−0.259001	−	0.966606i	−0.0978933	−	0.365343i	0.899549	−	0.436820i	\(-0.143895\pi\)
							−0.997442	+	0.0714772i	\(0.977229\pi\)
\(11\)	−0.728730	+	2.08259i	−0.219720	+	0.627925i	0.780276	+	0.625435i	\(0.215078\pi\)
							−0.999997	+	0.00249021i	\(0.999207\pi\)
\(13\)	0.646948	+	1.64840i	0.179431	+	0.457183i	0.992086	−	0.125563i	\(-0.0400736\pi\)
							−0.812655	+	0.582746i	\(0.801978\pi\)
\(17\)	3.39732	−	2.33628i	0.823971	−	0.566631i
							\(19\)	6.94095	−	0.520152i	1.59236	−	0.119331i	0.751185	−	0.660091i	\(-0.229482\pi\)
0.841177	+	0.540760i	\(0.181863\pi\)
\(23\)	−5.56502	+	1.05296i	−1.16039	+	0.219557i	−0.730233	−	0.683198i	\(-0.760589\pi\)
							−0.430154	+	0.902755i	\(0.641541\pi\)
\(29\)	0.108946	+	2.91165i	0.0202308	+	0.540679i	0.972154	+	0.234343i	\(0.0752938\pi\)
							−0.951923	+	0.306337i	\(0.900897\pi\)
\(31\)	2.60184	+	1.37511i	0.467304	+	0.246978i	0.684308	−	0.729193i	\(-0.260104\pi\)
							−0.217004	+	0.976171i	\(0.569628\pi\)
\(37\)	−2.51863	+	9.39964i	−0.414060	+	1.54529i	0.372652	+	0.927971i	\(0.378449\pi\)
							−0.786712	+	0.617321i	\(0.788218\pi\)
\(41\)	4.14565	+	6.59776i	0.647441	+	1.03040i	0.995667	+	0.0929860i	\(0.0296412\pi\)
							−0.348226	+	0.937410i	\(0.613216\pi\)
\(43\)	6.36337	−	1.58352i	0.970405	−	0.241485i
							\(47\)	−0.641867	−	0.309107i	−0.0936260	−	0.0450879i	0.386485	−	0.922296i	\(-0.373689\pi\)
−0.480111	+	0.877208i	\(0.659403\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.6	✓	1536
17.4	even	4	inner	731.2.bh.a.701.59	yes	1536
43.10	even	21	inner	731.2.bh.a.268.59	yes	1536
731.225	even	84	inner	731.2.bh.a.225.6	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.6	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.6	yes	1536	731.225	even	84	inner
731.2.bh.a.268.59	yes	1536	43.10	even	21	inner
731.2.bh.a.701.59	yes	1536	17.4	even	4	inner