Embedded newform 43.8.c.a.6.9

• Label: 43.8.c.a.6.9
• Level: $43$
• Weight: $8$
• Character: 43.6
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(25\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			6.9
Character	\(\chi\)	\(=\)	43.6
Dual form			43.8.c.a.36.9

$q$-expansion

\(f(q)\)	\(=\)	\(q-6.76607 q^{2} +(25.9136 - 44.8837i) q^{3} -82.2203 q^{4} +(-60.8352 + 105.370i) q^{5} +(-175.333 + 303.686i) q^{6} +(496.526 + 860.008i) q^{7} +1422.37 q^{8} +(-249.529 - 432.197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.76607			−0.598042			−0.299021	−	0.954247i	\(-0.596660\pi\)
							−0.299021	+	0.954247i	\(0.596660\pi\)
\(3\)	25.9136	−	44.8837i	0.554119	−	0.959763i	−0.443852	−	0.896100i	\(-0.646388\pi\)
							0.997971	−	0.0636628i	\(-0.0202782\pi\)
\(5\)	−60.8352	+	105.370i	−0.217651	+	0.376982i	−0.954089	−	0.299522i	\(-0.903173\pi\)
							0.736439	+	0.676504i	\(0.236506\pi\)
\(7\)	496.526	+	860.008i	0.547141	+	0.947675i	0.998469	+	0.0553172i	\(0.0176170\pi\)
							−0.451328	+	0.892358i	\(0.649050\pi\)
\(11\)	3266.11			0.739873			0.369936	−	0.929057i	\(-0.379379\pi\)
							0.369936	+	0.929057i	\(0.379379\pi\)
\(13\)	−5369.52	−	9300.27i	−0.677850	−	1.17407i	−0.975627	−	0.219435i	\(-0.929579\pi\)
							0.297778	−	0.954635i	\(-0.403755\pi\)
\(17\)	−230.435	−	399.125i	−0.0113757	−	0.0197032i	0.860282	−	0.509819i	\(-0.170288\pi\)
							−0.871657	+	0.490116i	\(0.836954\pi\)
\(19\)	11342.6	−	19646.0i	0.379381	−	0.657108i	−0.611591	−	0.791174i	\(-0.709470\pi\)
							0.990972	+	0.134066i	\(0.0428035\pi\)
\(23\)	47151.6	−	81669.0i	0.808070	−	1.39962i	−0.106128	−	0.994352i	\(-0.533845\pi\)
							0.914199	−	0.405266i	\(-0.132821\pi\)
\(29\)	92418.3	+	160073.i	0.703664	+	1.21878i	0.967172	+	0.254124i	\(0.0817872\pi\)
							−0.263508	+	0.964657i	\(0.584879\pi\)
\(31\)	126965.	−	219910.i	0.765453	−	1.32580i	−0.174554	−	0.984648i	\(-0.555848\pi\)
							0.940007	−	0.341155i	\(-0.110818\pi\)
\(37\)	−203435.	+	352359.i	−0.660266	+	1.14361i	0.320280	+	0.947323i	\(0.396223\pi\)
							−0.980546	+	0.196291i	\(0.937110\pi\)
\(41\)	−244264.			−0.553499			−0.276749	−	0.960942i	\(-0.589257\pi\)
							−0.276749	+	0.960942i	\(0.589257\pi\)
\(43\)	518410.	−	55402.1i	0.994338	−	0.106264i
							\(47\)	914307.			1.28455			0.642273	−	0.766476i	\(-0.277992\pi\)
0.642273	+	0.766476i	\(0.277992\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.c.a.6.9	✓	50
43.36	even	3	inner	43.8.c.a.36.9	yes	50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.c.a.6.9	✓	50	1.1	even	1	trivial
43.8.c.a.36.9	yes	50	43.36	even	3	inner