Embedded newform 43.6.e.a.4.17

• Label: 43.6.e.a.4.17
• Level: $43$
• Weight: $6$
• Character: 43.4
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.17
Character	\(\chi\)	\(=\)	43.4
Dual form			43.6.e.a.11.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.33304 - 7.94138i) q^{2} +(4.01851 + 5.03906i) q^{3} +(-15.8374 - 69.3884i) q^{4} +(-53.4066 - 25.7193i) q^{5} +65.4665 q^{6} -5.04361 q^{7} +(-358.491 - 172.640i) q^{8} +(44.8289 - 196.408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 3 q^{2} - 9 q^{3} - 331 q^{4} + 67 q^{5} - 20 q^{6} - 624 q^{7} + 589 q^{8} - 579 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.33304	−	7.94138i	1.11953	−	1.40385i	0.215439	−	0.976517i	\(-0.430882\pi\)
							0.904094	−	0.427333i	\(-0.140547\pi\)
\(3\)	4.01851	+	5.03906i	0.257788	+	0.323256i	0.893836	−	0.448393i	\(-0.148004\pi\)
							−0.636049	+	0.771649i	\(0.719432\pi\)
\(5\)	−53.4066	−	25.7193i	−0.955366	−	0.460080i	−0.109802	−	0.993954i	\(-0.535022\pi\)
							−0.845564	+	0.533874i	\(0.820736\pi\)
\(7\)	−5.04361			−0.0389042			−0.0194521	−	0.999811i	\(-0.506192\pi\)
							−0.0194521	+	0.999811i	\(0.506192\pi\)
\(11\)	62.1446	−	272.273i	0.154854	−	0.678459i	−0.836580	−	0.547846i	\(-0.815448\pi\)
							0.991433	−	0.130614i	\(-0.0416947\pi\)
\(13\)	1004.45	+	483.716i	1.64842	+	0.793838i	0.999455	+	0.0330227i	\(0.0105134\pi\)
							0.648968	+	0.760816i	\(0.275201\pi\)
\(17\)	148.218	−	71.3780i	0.124388	−	0.0599021i	−0.370655	−	0.928771i	\(-0.620867\pi\)
							0.495043	+	0.868869i	\(0.335152\pi\)
\(19\)	355.367	+	1556.97i	0.225836	+	0.989453i	0.952995	+	0.302984i	\(0.0979830\pi\)
							−0.727159	+	0.686469i	\(0.759160\pi\)
\(23\)	−408.722	+	1790.73i	−0.161105	+	0.705846i	0.828254	+	0.560352i	\(0.189334\pi\)
							−0.989359	+	0.145494i	\(0.953523\pi\)
\(29\)	−929.436	+	1165.48i	−0.205222	+	0.257341i	−0.873782	−	0.486318i	\(-0.838340\pi\)
							0.668560	+	0.743658i	\(0.266911\pi\)
\(31\)	5257.60	−	6592.82i	0.982615	−	1.23216i	0.00994891	−	0.999951i	\(-0.496833\pi\)
							0.972666	−	0.232209i	\(-0.0745955\pi\)
\(37\)	−543.788			−0.0653018			−0.0326509	−	0.999467i	\(-0.510395\pi\)
							−0.0326509	+	0.999467i	\(0.510395\pi\)
\(41\)	−9978.85	+	12513.1i	−0.927087	+	1.16253i	0.0593243	+	0.998239i	\(0.481105\pi\)
							−0.986412	+	0.164292i	\(0.947466\pi\)
\(43\)	11158.6	−	4742.70i	0.920323	−	0.391160i
							\(47\)	−3292.01	−	14423.2i	−0.217378	−	0.952397i	−0.959406	−	0.282028i	\(-0.908993\pi\)
0.742028	−	0.670369i	\(-0.233864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.e.a.4.17	✓	108
43.11	even	7	inner	43.6.e.a.11.17	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.17	✓	108	1.1	even	1	trivial
43.6.e.a.11.17	yes	108	43.11	even	7	inner