Embedded newform 43.6.g.a.10.3

• Label: 43.6.g.a.10.3
• Level: $43$
• Weight: $6$
• Character: 43.10
• 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			10.3
Character	\(\chi\)	\(=\)	43.10
Dual form			43.6.g.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91263 - 8.37977i) q^{2} +(7.67423 + 7.12065i) q^{3} +(-37.7314 + 18.1705i) q^{4} +(59.1521 - 8.91575i) q^{5} +(44.9914 - 77.9274i) q^{6} +(-90.2330 - 156.288i) q^{7} +(52.9405 + 66.3853i) q^{8} +(-9.96919 - 133.030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91263	−	8.37977i	−0.338108	−	1.48135i	−0.802999	−	0.595980i	\(-0.796764\pi\)
							0.464891	−	0.885368i	\(-0.346093\pi\)
\(3\)	7.67423	+	7.12065i	0.492302	+	0.456790i	0.886831	−	0.462094i	\(-0.152902\pi\)
							−0.394529	+	0.918884i	\(0.629092\pi\)
\(5\)	59.1521	−	8.91575i	1.05815	−	0.159490i	0.403159	−	0.915130i	\(-0.367912\pi\)
							0.654987	+	0.755640i	\(0.272674\pi\)
\(7\)	−90.2330	−	156.288i	−0.696017	−	1.20554i	−0.969836	−	0.243756i	\(-0.921620\pi\)
							0.273819	−	0.961781i	\(-0.411713\pi\)
\(11\)	115.836	+	55.7837i	0.288643	+	0.139003i	0.572603	−	0.819833i	\(-0.305934\pi\)
							−0.283959	+	0.958836i	\(0.591648\pi\)
\(13\)	−51.2529	+	130.590i	−0.0841124	+	0.214315i	−0.966742	−	0.255755i	\(-0.917676\pi\)
							0.882629	+	0.470070i	\(0.155771\pi\)
\(17\)	106.328	+	16.0263i	0.0892327	+	0.0134497i	0.193507	−	0.981099i	\(-0.438014\pi\)
							−0.104274	+	0.994549i	\(0.533252\pi\)
\(19\)	174.094	−	2323.12i	0.110637	−	1.47635i	−0.615141	−	0.788417i	\(-0.710901\pi\)
							0.725778	−	0.687929i	\(-0.241480\pi\)
\(23\)	−710.709	+	484.553i	−0.280138	+	0.190995i	−0.695235	−	0.718782i	\(-0.744700\pi\)
							0.415097	+	0.909777i	\(0.363748\pi\)
\(29\)	−418.056	+	387.900i	−0.0923081	+	0.0856494i	−0.724977	−	0.688773i	\(-0.758150\pi\)
							0.632669	+	0.774422i	\(0.281959\pi\)
\(31\)	9895.97	+	3052.50i	1.84950	+	0.570495i	0.999009	+	0.0445046i	\(0.0141709\pi\)
							0.850490	+	0.525991i	\(0.176305\pi\)
\(37\)	−2660.21	+	4607.62i	−0.319456	+	0.553315i	−0.980375	−	0.197143i	\(-0.936834\pi\)
							0.660918	+	0.750458i	\(0.270167\pi\)
\(41\)	−508.372	−	2227.32i	−0.0472304	−	0.206930i	0.945807	−	0.324729i	\(-0.105273\pi\)
							−0.993037	+	0.117799i	\(0.962416\pi\)
\(43\)	7054.05	+	9861.48i	0.581791	+	0.813338i
							\(47\)	20560.5	−	9901.40i	1.35765	−	0.653811i	0.393540	−	0.919307i	\(-0.371250\pi\)
0.964112	+	0.265497i	\(0.0855360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.10.3	✓	204
43.13	even	21	inner	43.6.g.a.13.3	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.10.3	✓	204	1.1	even	1	trivial
43.6.g.a.13.3	yes	204	43.13	even	21	inner