Embedded newform 731.2.bl.a.9.11

• Label: 731.2.bl.a.9.11
• Level: $731$
• Weight: $2$
• Character: 731.9
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $3072$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bl (of order \(168\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(3072\)
Relative dimension:	\(64\) over \(\Q(\zeta_{168})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{168}]$

Embedding invariants

Embedding label			9.11
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.06687 - 0.232880i) q^{2} +(-0.844141 - 0.508645i) q^{3} +(2.26786 + 0.517625i) q^{4} +(3.26327 - 2.13659i) q^{5} +(1.62628 + 1.24789i) q^{6} +(1.78259 + 0.234683i) q^{7} +(-0.640380 - 0.224079i) q^{8} +(-0.947951 - 1.79361i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.06687	−	0.232880i	−1.46150	−	0.164671i	−0.654875	−	0.755737i	\(-0.727279\pi\)
							−0.806623	+	0.591066i	\(0.798707\pi\)
\(3\)	−0.844141	−	0.508645i	−0.487365	−	0.293666i	0.251096	−	0.967962i	\(-0.419209\pi\)
							−0.738461	+	0.674296i	\(0.764447\pi\)
\(5\)	3.26327	−	2.13659i	1.45938	−	0.955511i	0.461728	−	0.887022i	\(-0.347230\pi\)
							0.997652	−	0.0684896i	\(-0.0218180\pi\)
\(7\)	1.78259	+	0.234683i	0.673757	+	0.0887017i	0.459639	−	0.888106i	\(-0.347979\pi\)
							0.214118	+	0.976808i	\(0.431312\pi\)
\(11\)	−2.41835	−	3.40834i	−0.729159	−	1.02765i	−0.997921	−	0.0644474i	\(-0.979472\pi\)
							0.268762	−	0.963207i	\(-0.413386\pi\)
\(13\)	−5.58346	−	0.418423i	−1.54857	−	0.116050i	−0.727194	−	0.686432i	\(-0.759176\pi\)
							−0.821379	+	0.570382i	\(0.806795\pi\)
\(17\)	3.74816	−	1.71793i	0.909063	−	0.416659i
							\(19\)	0.547225	−	1.03540i	0.125542	−	0.237537i	−0.813331	−	0.581802i	\(-0.802348\pi\)
0.938873	+	0.344265i	\(0.111872\pi\)
\(23\)	0.621353	−	1.35448i	0.129561	−	0.282428i	−0.833107	−	0.553112i	\(-0.813440\pi\)
							0.962668	+	0.270684i	\(0.0872498\pi\)
\(29\)	−2.22886	+	8.98718i	−0.413888	+	1.66888i	0.288808	+	0.957387i	\(0.406741\pi\)
							−0.702697	+	0.711490i	\(0.748021\pi\)
\(31\)	−0.748140	+	0.776657i	−0.134370	+	0.139492i	−0.784005	−	0.620754i	\(-0.786826\pi\)
							0.649635	+	0.760246i	\(0.274922\pi\)
\(37\)	−1.08937	−	8.27461i	−0.179092	−	1.36034i	−0.810887	−	0.585202i	\(-0.801015\pi\)
							0.631796	−	0.775135i	\(-0.282318\pi\)
\(41\)	0.480251	−	0.868949i	0.0750026	−	0.135707i	−0.835088	−	0.550117i	\(-0.814583\pi\)
							0.910091	+	0.414410i	\(0.136012\pi\)
\(43\)	2.75743	+	5.94950i	0.420504	+	0.907291i
							\(47\)	−3.03073	−	0.691745i	−0.442078	−	0.100901i	−0.00431253	−	0.999991i	\(-0.501373\pi\)
−0.437765	+	0.899089i	\(0.644230\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.11	✓	3072
17.2	even	8	inner	731.2.bl.a.138.54	yes	3072
43.24	even	21	inner	731.2.bl.a.196.54	yes	3072
731.325	even	168	inner	731.2.bl.a.325.11	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.11	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.54	yes	3072	17.2	even	8	inner
731.2.bl.a.196.54	yes	3072	43.24	even	21	inner
731.2.bl.a.325.11	yes	3072	731.325	even	168	inner