Embedded newform 731.2.v.a.36.18

• Label: 731.2.v.a.36.18
• Level: $731$
• Weight: $2$
• Character: 731.36
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $512$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.v (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(64\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			36.18
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.07479 - 1.07479i) q^{2} +(-2.28358 - 1.75225i) q^{3} +0.310339i q^{4} +(-0.670369 + 0.873642i) q^{5} +(0.571063 + 4.33766i) q^{6} +(-1.27789 - 1.66538i) q^{7} +(-1.81603 + 1.81603i) q^{8} +(1.36788 + 5.10501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.07479	−	1.07479i	−0.759990	−	0.759990i	0.216330	−	0.976320i	\(-0.430591\pi\)
							−0.976320	+	0.216330i	\(0.930591\pi\)
\(3\)	−2.28358	−	1.75225i	−1.31842	−	1.01166i	−0.997923	−	0.0644126i	\(-0.979483\pi\)
							−0.320499	−	0.947249i	\(-0.603851\pi\)
\(5\)	−0.670369	+	0.873642i	−0.299798	+	0.390705i	−0.918825	−	0.394665i	\(-0.870861\pi\)
							0.619027	+	0.785370i	\(0.287527\pi\)
\(7\)	−1.27789	−	1.66538i	−0.482997	−	0.629454i	0.486757	−	0.873537i	\(-0.338180\pi\)
							−0.969754	+	0.244084i	\(0.921513\pi\)
\(11\)	−5.76900	−	2.38960i	−1.73942	−	0.720491i	−0.998821	−	0.0485412i	\(-0.984543\pi\)
							−0.740597	−	0.671949i	\(-0.765457\pi\)
\(13\)	0.794322	+	0.458602i	0.220305	+	0.127193i	0.606092	−	0.795395i	\(-0.292736\pi\)
							−0.385786	+	0.922588i	\(0.626070\pi\)
\(17\)	3.06754	+	2.75504i	0.743987	+	0.668195i
							\(19\)	−0.973307	+	3.63243i	−0.223292	+	0.833337i	0.759790	+	0.650169i	\(0.225302\pi\)
−0.983082	+	0.183168i	\(0.941365\pi\)
\(23\)	−1.10511	−	8.39414i	−0.230431	−	1.75030i	−0.581129	−	0.813811i	\(-0.697389\pi\)
							0.350698	−	0.936489i	\(-0.385944\pi\)
\(29\)	−7.00975	−	0.922852i	−1.30168	−	0.171369i	−0.552375	−	0.833596i	\(-0.686278\pi\)
							−0.749304	+	0.662227i	\(0.769612\pi\)
\(31\)	3.45787	+	2.65332i	0.621052	+	0.476550i	0.870718	−	0.491782i	\(-0.163654\pi\)
							−0.249666	+	0.968332i	\(0.580321\pi\)
\(37\)	−4.52822	−	3.47463i	−0.744435	−	0.571225i	0.165385	−	0.986229i	\(-0.447113\pi\)
							−0.909820	+	0.415004i	\(0.863780\pi\)
\(41\)	−1.62553	+	3.92438i	−0.253865	+	0.612885i	−0.998510	−	0.0545765i	\(-0.982619\pi\)
							0.744644	+	0.667461i	\(0.232619\pi\)
\(43\)	4.38581	+	4.87490i	0.668830	+	0.743415i
							\(47\)		−	5.69108i		−	0.830130i	−0.909792	−	0.415065i	\(-0.863759\pi\)
0.909792	−	0.415065i	\(-0.136241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.18	✓	512
17.9	even	8	inner	731.2.v.a.638.18	yes	512
43.6	even	3	inner	731.2.v.a.393.18	yes	512
731.264	even	24	inner	731.2.v.a.264.18	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.18	✓	512	1.1	even	1	trivial
731.2.v.a.264.18	yes	512	731.264	even	24	inner
731.2.v.a.393.18	yes	512	43.6	even	3	inner
731.2.v.a.638.18	yes	512	17.9	even	8	inner