Embedded newform 43.6.g.a.9.1

• Label: 43.6.g.a.9.1
• Level: $43$
• Weight: $6$
• Character: 43.9
• 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			9.1
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.1

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.1	✓	204	1.1	even	1	trivial
43.6.g.a.24.1	yes	204	43.24	even	21	inner