Embedded newform 43.6.g.a.9.14

• Label: 43.6.g.a.9.14
• 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			9.14
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.93041 + 4.92858i) q^{2} +(-23.5623 - 3.55144i) q^{3} +(-1.72210 + 7.54503i) q^{4} +(6.81244 + 4.64464i) q^{5} +(-75.1059 - 130.087i) q^{6} +(57.5727 - 99.7189i) q^{7} +(137.793 - 66.3574i) q^{8} +(310.364 + 95.7346i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.93041	+	4.92858i	0.694805	+	0.871258i	0.996624	−	0.0821071i	\(-0.0261649\pi\)
							−0.301818	+	0.953366i	\(0.597594\pi\)
\(3\)	−23.5623	−	3.55144i	−1.51152	−	0.227825i	−0.659648	−	0.751574i	\(-0.729295\pi\)
							−0.851872	+	0.523749i	\(0.824533\pi\)
\(5\)	6.81244	+	4.64464i	0.121865	+	0.0830859i	0.622718	−	0.782446i	\(-0.286028\pi\)
							−0.500854	+	0.865532i	\(0.666981\pi\)
\(7\)	57.5727	−	99.7189i	0.444091	−	0.769188i	−0.553898	−	0.832585i	\(-0.686860\pi\)
							0.997988	+	0.0633970i	\(0.0201934\pi\)
\(11\)	−91.0818	−	399.056i	−0.226960	−	0.994378i	−0.952103	−	0.305779i	\(-0.901083\pi\)
							0.725142	−	0.688599i	\(-0.241774\pi\)
\(13\)	59.3380	−	791.811i	0.0973811	−	1.29946i	−0.707792	−	0.706421i	\(-0.750309\pi\)
							0.805173	−	0.593040i	\(-0.202072\pi\)
\(17\)	−1375.34	+	937.693i	−1.15422	+	0.786934i	−0.980239	−	0.197815i	\(-0.936615\pi\)
							−0.173981	+	0.984749i	\(0.555663\pi\)
\(19\)	−485.693	+	149.816i	−0.308658	+	0.0952084i	−0.445215	−	0.895424i	\(-0.646873\pi\)
							0.136557	+	0.990632i	\(0.456396\pi\)
\(23\)	1972.01	−	1829.76i	0.777303	−	0.721231i	−0.188056	−	0.982158i	\(-0.560219\pi\)
							0.965359	+	0.260927i	\(0.0840281\pi\)
\(29\)	−3788.00	+	570.949i	−0.836402	+	0.126067i	−0.553258	−	0.833010i	\(-0.686616\pi\)
							−0.283144	+	0.959077i	\(0.591377\pi\)
\(31\)	1518.21	−	3868.33i	0.283744	−	0.722969i	−0.715948	−	0.698154i	\(-0.754005\pi\)
							0.999692	−	0.0248150i	\(-0.00789966\pi\)
\(37\)	3301.60	+	5718.54i	0.396479	+	0.686721i	0.993289	−	0.115661i	\(-0.0368987\pi\)
							−0.596810	+	0.802383i	\(0.703565\pi\)
\(41\)	3211.07	+	4026.56i	0.298326	+	0.374088i	0.908291	−	0.418340i	\(-0.137388\pi\)
							−0.609965	+	0.792428i	\(0.708817\pi\)
\(43\)	7967.55	−	9139.29i	0.657134	−	0.753774i
							\(47\)	−3639.13	+	15944.1i	−0.240300	+	1.05282i	0.700445	+	0.713706i	\(0.252985\pi\)
−0.940745	+	0.339115i	\(0.889872\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.14	✓	204
43.24	even	21	inner	43.6.g.a.24.14	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.14	✓	204	1.1	even	1	trivial
43.6.g.a.24.14	yes	204	43.24	even	21	inner