Embedded newform 731.2.bl.a.9.19

• Label: 731.2.bl.a.9.19
• 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			9.19
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36186 - 0.153444i) q^{2} +(-0.627515 - 0.378115i) q^{3} +(-0.118749 - 0.0271036i) q^{4} +(-2.79353 + 1.82903i) q^{5} +(0.796566 + 0.611227i) q^{6} +(-4.13974 - 0.545007i) q^{7} +(2.74469 + 0.960410i) q^{8} +(-1.15100 - 2.17780i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36186	−	0.153444i	−0.962978	−	0.108502i	−0.383546	−	0.923522i	\(-0.625297\pi\)
							−0.579432	+	0.815020i	\(0.696726\pi\)
\(3\)	−0.627515	−	0.378115i	−0.362296	−	0.218305i	0.323284	−	0.946302i	\(-0.395213\pi\)
							−0.685580	+	0.727997i	\(0.740451\pi\)
\(5\)	−2.79353	+	1.82903i	−1.24930	+	0.817967i	−0.989051	−	0.147573i	\(-0.952854\pi\)
							−0.260254	+	0.965540i	\(0.583806\pi\)
\(7\)	−4.13974	−	0.545007i	−1.56468	−	0.205993i	−0.702158	−	0.712021i	\(-0.747780\pi\)
							−0.862517	+	0.506028i	\(0.831113\pi\)
\(11\)	−1.62904	−	2.29592i	−0.491174	−	0.692245i	0.492883	−	0.870095i	\(-0.335943\pi\)
							−0.984058	+	0.177850i	\(0.943086\pi\)
\(13\)	−5.83543	−	0.437305i	−1.61846	−	0.121287i	−0.765559	−	0.643365i	\(-0.777538\pi\)
							−0.852897	+	0.522079i	\(0.825157\pi\)
\(17\)	0.0150429	−	4.12308i	0.00364844	−	0.999993i
							\(19\)	−0.111056	+	0.210127i	−0.0254779	+	0.0482065i	−0.896526	−	0.442992i	\(-0.853917\pi\)
0.871048	+	0.491198i	\(0.163441\pi\)
\(23\)	−2.41230	+	5.25852i	−0.502999	+	1.09648i	0.473468	+	0.880811i	\(0.343002\pi\)
							−0.976467	+	0.215666i	\(0.930808\pi\)
\(29\)	−1.21334	+	4.89242i	−0.225312	+	0.908499i	0.745083	+	0.666972i	\(0.232410\pi\)
							−0.970394	+	0.241527i	\(0.922352\pi\)
\(31\)	0.405113	−	0.420555i	0.0727604	−	0.0755339i	−0.682834	−	0.730573i	\(-0.739253\pi\)
							0.755595	+	0.655040i	\(0.227348\pi\)
\(37\)	1.14925	+	8.72939i	0.188935	+	1.43510i	0.778633	+	0.627479i	\(0.215913\pi\)
							−0.589699	+	0.807623i	\(0.700753\pi\)
\(41\)	2.12585	−	3.84643i	0.332001	−	0.600711i	−0.656071	−	0.754699i	\(-0.727783\pi\)
							0.988073	+	0.153988i	\(0.0492116\pi\)
\(43\)	5.14467	−	4.06600i	0.784555	−	0.620059i
							\(47\)	0.491567	+	0.112197i	0.0717024	+	0.0163656i	0.258222	−	0.966086i	\(-0.416864\pi\)
−0.186519	+	0.982451i	\(0.559721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.19	✓	3072
17.2	even	8	inner	731.2.bl.a.138.46	yes	3072
43.24	even	21	inner	731.2.bl.a.196.46	yes	3072
731.325	even	168	inner	731.2.bl.a.325.19	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.19	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.46	yes	3072	17.2	even	8	inner
731.2.bl.a.196.46	yes	3072	43.24	even	21	inner
731.2.bl.a.325.19	yes	3072	731.325	even	168	inner