Embedded newform 43.6.g.a.24.1

• Label: 43.6.g.a.24.1
• Level: $43$
• Weight: $6$
• Character: 43.24
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			24.1
Character	\(\chi\)	\(=\)	43.24
Dual form			43.6.g.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.52161 + 8.17784i) q^{2} +(17.3296 - 2.61202i) q^{3} +(-17.2250 - 75.4676i) q^{4} +(2.46584 - 1.68118i) q^{5} +(-91.6562 + 158.753i) q^{6} +(94.8911 + 164.356i) q^{7} +(427.928 + 206.079i) q^{8} +(61.2883 - 18.9049i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.52161	+	8.17784i	−1.15287	+	1.44565i	−0.278468	+	0.960446i	\(0.589827\pi\)
							−0.874400	+	0.485205i	\(0.838745\pi\)
\(3\)	17.3296	−	2.61202i	1.11169	−	0.167561i	0.432582	−	0.901595i	\(-0.357603\pi\)
							0.679113	+	0.734034i	\(0.262365\pi\)
\(5\)	2.46584	−	1.68118i	0.0441102	−	0.0300738i	−0.541064	−	0.840981i	\(-0.681978\pi\)
							0.585174	+	0.810907i	\(0.301026\pi\)
\(7\)	94.8911	+	164.356i	0.731949	+	1.26777i	0.956049	+	0.293206i	\(0.0947222\pi\)
							−0.224101	+	0.974566i	\(0.571944\pi\)
\(11\)	−24.7212	+	108.311i	−0.0616009	+	0.269891i	−0.996344	−	0.0854333i	\(-0.972773\pi\)
							0.934743	+	0.355325i	\(0.115630\pi\)
\(13\)	58.7689	+	784.216i	0.0964470	+	1.28700i	0.810015	+	0.586410i	\(0.199459\pi\)
							−0.713567	+	0.700586i	\(0.752922\pi\)
\(17\)	−707.745	−	482.533i	−0.593957	−	0.404953i	0.228695	−	0.973498i	\(-0.426554\pi\)
							−0.822652	+	0.568545i	\(0.807506\pi\)
\(19\)	1767.36	+	545.157i	1.12316	+	0.346448i	0.800061	−	0.599919i	\(-0.204800\pi\)
							0.323095	+	0.946367i	\(0.395277\pi\)
\(23\)	−1916.48	−	1778.23i	−0.755412	−	0.700919i	0.205229	−	0.978714i	\(-0.434206\pi\)
							−0.960641	+	0.277794i	\(0.910397\pi\)
\(29\)	7555.17	+	1138.76i	1.66821	+	0.251441i	0.914152	−	0.405372i	\(-0.132858\pi\)
							0.754053	+	0.656813i	\(0.228096\pi\)
\(31\)	1231.87	+	3138.76i	0.230229	+	0.586615i	0.998572	−	0.0534238i	\(-0.0170134\pi\)
							−0.768342	+	0.640039i	\(0.778918\pi\)
\(37\)	2712.48	−	4698.15i	0.325733	−	0.564187i	−0.655927	−	0.754824i	\(-0.727722\pi\)
							0.981661	+	0.190637i	\(0.0610555\pi\)
\(41\)	−10847.0	+	13601.7i	−1.00774	+	1.26367i	−0.0433869	+	0.999058i	\(0.513815\pi\)
							−0.964355	+	0.264610i	\(0.914757\pi\)
\(43\)	8592.92	−	8553.96i	0.708712	−	0.705498i
							\(47\)	−6317.94	−	27680.7i	−0.417187	−	1.82782i	−0.548087	−	0.836421i	\(-0.684644\pi\)
0.130900	−	0.991396i	\(-0.458213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.24.1	yes	204
43.9	even	21	inner	43.6.g.a.9.1	✓	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.1	✓	204	43.9	even	21	inner
43.6.g.a.24.1	yes	204	1.1	even	1	trivial