Embedded newform 43.6.g.a.9.6

• Label: 43.6.g.a.9.6
• 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			9.6
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.33890 - 2.93289i) q^{2} +(28.6156 + 4.31311i) q^{3} +(3.98929 - 17.4782i) q^{4} +(-77.5461 - 52.8700i) q^{5} +(-54.2792 - 94.0143i) q^{6} +(35.5619 - 61.5951i) q^{7} +(-168.746 + 81.2638i) q^{8} +(568.046 + 175.219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.33890	−	2.93289i	−0.413463	−	0.518466i	0.530872	−	0.847452i	\(-0.321865\pi\)
							−0.944335	+	0.328986i	\(0.893293\pi\)
\(3\)	28.6156	+	4.31311i	1.83569	+	0.276686i	0.973297	−	0.229550i	\(-0.0737254\pi\)
							0.862395	+	0.506236i	\(0.168964\pi\)
\(5\)	−77.5461	−	52.8700i	−1.38719	−	0.945768i	−0.999715	−	0.0238867i	\(-0.992396\pi\)
							−0.387472	−	0.921881i	\(-0.626652\pi\)
\(7\)	35.5619	−	61.5951i	0.274309	−	0.475117i	−0.695651	−	0.718379i	\(-0.744884\pi\)
							0.969961	+	0.243262i	\(0.0782175\pi\)
\(11\)	−113.232	−	496.104i	−0.282156	−	1.23621i	−0.895023	−	0.446020i	\(-0.852841\pi\)
							0.612867	−	0.790186i	\(-0.290016\pi\)
\(13\)	−55.2426	+	737.161i	−0.0906599	+	1.20977i	0.747467	+	0.664298i	\(0.231270\pi\)
							−0.838127	+	0.545475i	\(0.816349\pi\)
\(17\)	1081.85	−	737.593i	0.907914	−	0.619005i	−0.0167210	−	0.999860i	\(-0.505323\pi\)
							0.924635	+	0.380855i	\(0.124370\pi\)
\(19\)	432.455	−	133.395i	0.274825	−	0.0847724i	−0.154279	−	0.988027i	\(-0.549305\pi\)
							0.429104	+	0.903255i	\(0.358829\pi\)
\(23\)	1345.53	−	1248.47i	0.530362	−	0.492104i	−0.368927	−	0.929458i	\(-0.620275\pi\)
							0.899289	+	0.437354i	\(0.144084\pi\)
\(29\)	2316.92	−	349.219i	0.511582	−	0.0771086i	0.111822	−	0.993728i	\(-0.464331\pi\)
							0.399761	+	0.916620i	\(0.369093\pi\)
\(31\)	−941.518	+	2398.95i	−0.175964	+	0.448349i	−0.991460	−	0.130414i	\(-0.958369\pi\)
							0.815495	+	0.578763i	\(0.196465\pi\)
\(37\)	1233.33	+	2136.19i	0.148107	+	0.256529i	0.930528	−	0.366221i	\(-0.119349\pi\)
							−0.782421	+	0.622750i	\(0.786015\pi\)
\(41\)	−3620.71	−	4540.22i	−0.336383	−	0.421811i	0.584656	−	0.811281i	\(-0.301229\pi\)
							−0.921039	+	0.389471i	\(0.872658\pi\)
\(43\)	11876.7	+	2439.92i	0.979543	+	0.201236i
							\(47\)	−2629.85	+	11522.1i	−0.173655	+	0.760832i	0.810819	+	0.585298i	\(0.199022\pi\)
−0.984473	+	0.175534i	\(0.943835\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.6	✓	204
43.24	even	21	inner	43.6.g.a.24.6	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.6	✓	204	1.1	even	1	trivial
43.6.g.a.24.6	yes	204	43.24	even	21	inner