Embedded newform 731.2.v.a.36.11

• Label: 731.2.v.a.36.11
• 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			36.11
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.51428 - 1.51428i) q^{2} +(-0.591375 - 0.453778i) q^{3} +2.58612i q^{4} +(2.49753 - 3.25484i) q^{5} +(0.208361 + 1.58266i) q^{6} +(-1.77330 - 2.31101i) q^{7} +(0.887545 - 0.887545i) q^{8} +(-0.632647 - 2.36107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.51428	−	1.51428i	−1.07076	−	1.07076i	−0.997298	−	0.0734629i	\(-0.976595\pi\)
							−0.0734629	−	0.997298i	\(-0.523405\pi\)
\(3\)	−0.591375	−	0.453778i	−0.341431	−	0.261989i	0.423785	−	0.905763i	\(-0.360701\pi\)
							−0.765216	+	0.643774i	\(0.777368\pi\)
\(5\)	2.49753	−	3.25484i	1.11693	−	1.45561i	0.242697	−	0.970102i	\(-0.421968\pi\)
							0.874232	−	0.485509i	\(-0.161366\pi\)
\(7\)	−1.77330	−	2.31101i	−0.670246	−	0.873481i	0.327196	−	0.944956i	\(-0.393896\pi\)
							−0.997442	+	0.0714748i	\(0.977229\pi\)
\(11\)	−3.88226	−	1.60808i	−1.17054	−	0.484856i	−0.289172	−	0.957277i	\(-0.593380\pi\)
							−0.881373	+	0.472422i	\(0.843380\pi\)
\(13\)	3.79515	+	2.19113i	1.05258	+	0.607710i	0.923372	−	0.383907i	\(-0.125422\pi\)
							0.129212	+	0.991617i	\(0.458755\pi\)
\(17\)	1.22157	−	3.93799i	0.296274	−	0.955103i
							\(19\)	−0.315467	+	1.17734i	−0.0723731	+	0.270100i	−0.992625	−	0.121227i	\(-0.961317\pi\)
0.920252	+	0.391327i	\(0.127984\pi\)
\(23\)	−0.106905	−	0.812026i	−0.0222913	−	0.169319i	0.976678	−	0.214709i	\(-0.0688803\pi\)
							−0.998969	+	0.0453899i	\(0.985547\pi\)
\(29\)	−3.74767	−	0.493390i	−0.695925	−	0.0916202i	−0.225739	−	0.974188i	\(-0.572480\pi\)
							−0.470186	+	0.882568i	\(0.655813\pi\)
\(31\)	3.39229	+	2.60300i	0.609274	+	0.467512i	0.866730	−	0.498778i	\(-0.166218\pi\)
							−0.257456	+	0.966290i	\(0.582884\pi\)
\(37\)	0.851683	+	0.653519i	0.140016	+	0.107438i	0.676395	−	0.736540i	\(-0.263541\pi\)
							−0.536379	+	0.843977i	\(0.680208\pi\)
\(41\)	4.02781	−	9.72399i	0.629038	−	1.51863i	−0.211782	−	0.977317i	\(-0.567927\pi\)
							0.840820	−	0.541315i	\(-0.182073\pi\)
\(43\)	6.45709	+	1.14282i	0.984696	+	0.174278i
							\(47\)			1.09174i			0.159246i	0.996825	+	0.0796232i	\(0.0253717\pi\)
−0.996825	+	0.0796232i	\(0.974628\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.11	✓	512
17.9	even	8	inner	731.2.v.a.638.11	yes	512
43.6	even	3	inner	731.2.v.a.393.11	yes	512
731.264	even	24	inner	731.2.v.a.264.11	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.11	✓	512	1.1	even	1	trivial
731.2.v.a.264.11	yes	512	731.264	even	24	inner
731.2.v.a.393.11	yes	512	43.6	even	3	inner
731.2.v.a.638.11	yes	512	17.9	even	8	inner