Embedded newform 43.6.g.a.9.15

• Label: 43.6.g.a.9.15
• 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			9.15
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.30682 + 6.65454i) q^{2} +(23.7101 + 3.57372i) q^{3} +(-8.99992 + 39.4312i) q^{4} +(-24.4442 - 16.6658i) q^{5} +(102.044 + 176.745i) q^{6} +(-48.7498 + 84.4371i) q^{7} +(-64.7633 + 31.1883i) q^{8} +(317.192 + 97.8407i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.30682	+	6.65454i	0.938122	+	1.17637i	0.984134	+	0.177427i	\(0.0567774\pi\)
							−0.0460118	+	0.998941i	\(0.514651\pi\)
\(3\)	23.7101	+	3.57372i	1.52100	+	0.229254i	0.855760	−	0.517374i	\(-0.173090\pi\)
							0.665242	+	0.746628i	\(0.268328\pi\)
\(5\)	−24.4442	−	16.6658i	−0.437272	−	0.298127i	0.324604	−	0.945850i	\(-0.394769\pi\)
							−0.761876	+	0.647723i	\(0.775721\pi\)
\(7\)	−48.7498	+	84.4371i	−0.376034	+	0.651310i	−0.990481	−	0.137648i	\(-0.956046\pi\)
							0.614447	+	0.788958i	\(0.289379\pi\)
\(11\)	0.448656	+	1.96569i	0.00111797	+	0.00489816i	0.975484	−	0.220071i	\(-0.0706290\pi\)
							−0.974366	+	0.224969i	\(0.927772\pi\)
\(13\)	62.1741	−	829.656i	0.102036	−	1.36157i	−0.677644	−	0.735390i	\(-0.736999\pi\)
							0.779679	−	0.626179i	\(-0.215382\pi\)
\(17\)	763.227	−	520.359i	0.640518	−	0.436698i	−0.199022	−	0.979995i	\(-0.563777\pi\)
							0.839541	+	0.543297i	\(0.182824\pi\)
\(19\)	−2527.76	+	779.712i	−1.60640	+	0.495507i	−0.962881	−	0.269926i	\(-0.913001\pi\)
							−0.643515	+	0.765434i	\(0.722524\pi\)
\(23\)	144.703	−	134.265i	0.0570371	−	0.0529227i	−0.651139	−	0.758958i	\(-0.725709\pi\)
							0.708176	+	0.706036i	\(0.249518\pi\)
\(29\)	2514.55	−	379.008i	0.555221	−	0.0836862i	0.134563	−	0.990905i	\(-0.457037\pi\)
							0.420659	+	0.907219i	\(0.361799\pi\)
\(31\)	2237.39	−	5700.79i	0.418156	−	1.06544i	−0.554292	−	0.832323i	\(-0.687011\pi\)
							0.972448	−	0.233122i	\(-0.0748940\pi\)
\(37\)	5029.16	+	8710.76i	0.603936	+	1.04605i	0.992219	+	0.124508i	\(0.0397353\pi\)
							−0.388282	+	0.921541i	\(0.626931\pi\)
\(41\)	−2641.03	−	3311.75i	−0.245366	−	0.307679i	0.643864	−	0.765140i	\(-0.277330\pi\)
							−0.889230	+	0.457461i	\(0.848759\pi\)
\(43\)	−11995.2	+	1767.18i	−0.989321	+	0.145750i
							\(47\)	−3524.83	+	15443.3i	−0.232752	+	1.01975i	0.714594	+	0.699540i	\(0.246612\pi\)
−0.947346	+	0.320213i	\(0.896246\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.15	✓	204
43.24	even	21	inner	43.6.g.a.24.15	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.15	✓	204	1.1	even	1	trivial
43.6.g.a.24.15	yes	204	43.24	even	21	inner