Embedded newform 43.6.g.a.24.12

• Label: 43.6.g.a.24.12
• Level: $43$
• Weight: $6$
• Character: 43.24
• 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			24.12
Character	\(\chi\)	\(=\)	43.24
Dual form			43.6.g.a.9.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.59848 - 3.25839i) q^{2} +(17.4103 - 2.62418i) q^{3} +(3.25566 + 14.2640i) q^{4} +(30.9820 - 21.1232i) q^{5} +(36.6897 - 63.5485i) q^{6} +(16.4293 + 28.4563i) q^{7} +(175.095 + 84.3211i) q^{8} +(64.0287 - 19.7502i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.59848	−	3.25839i	0.459351	−	0.576007i	−0.497177	−	0.867649i	\(-0.665630\pi\)
							0.956528	+	0.291642i	\(0.0942015\pi\)
\(3\)	17.4103	−	2.62418i	1.11687	−	0.168341i	0.435437	−	0.900219i	\(-0.356594\pi\)
							0.681436	+	0.731878i	\(0.261356\pi\)
\(5\)	30.9820	−	21.1232i	0.554224	−	0.377863i	−0.253580	−	0.967314i	\(-0.581608\pi\)
							0.807804	+	0.589451i	\(0.200656\pi\)
\(7\)	16.4293	+	28.4563i	0.126728	+	0.219500i	0.922407	−	0.386219i	\(-0.126219\pi\)
							−0.795679	+	0.605719i	\(0.792886\pi\)
\(11\)	−51.4133	+	225.256i	−0.128113	+	0.561300i	0.869603	+	0.493752i	\(0.164375\pi\)
							−0.997716	+	0.0675484i	\(0.978482\pi\)
\(13\)	−69.9136	−	932.932i	−0.114737	−	1.53106i	−0.696415	−	0.717639i	\(-0.745223\pi\)
							0.581679	−	0.813419i	\(-0.302396\pi\)
\(17\)	−1145.63	−	781.080i	−0.961444	−	0.655501i	−0.0225635	−	0.999745i	\(-0.507183\pi\)
							−0.938880	+	0.344244i	\(0.888135\pi\)
\(19\)	1320.70	+	407.383i	0.839309	+	0.258892i	0.684464	−	0.729046i	\(-0.260036\pi\)
							0.154844	+	0.987939i	\(0.450512\pi\)
\(23\)	−911.064	−	845.344i	−0.359112	−	0.333207i	0.479897	−	0.877325i	\(-0.340674\pi\)
							−0.839009	+	0.544118i	\(0.816864\pi\)
\(29\)	−4240.94	−	639.219i	−0.936413	−	0.141141i	−0.336930	−	0.941530i	\(-0.609389\pi\)
							−0.599482	+	0.800388i	\(0.704627\pi\)
\(31\)	1435.49	+	3657.58i	0.268285	+	0.683580i	0.999998	+	0.00196698i	\(0.000626110\pi\)
							−0.731713	+	0.681613i	\(0.761279\pi\)
\(37\)	−591.148	+	1023.90i	−0.0709891	+	0.122957i	−0.899335	−	0.437260i	\(-0.855949\pi\)
							0.828346	+	0.560217i	\(0.189282\pi\)
\(41\)	−2712.84	+	3401.79i	−0.252037	+	0.316044i	−0.891714	−	0.452600i	\(-0.850497\pi\)
							0.639677	+	0.768644i	\(0.279068\pi\)
\(43\)	−6884.11	+	9980.85i	−0.567776	+	0.823183i
							\(47\)	1258.61	+	5514.34i	0.0831089	+	0.364124i	0.999332	−	0.0365445i	\(-0.0116351\pi\)
−0.916223	+	0.400668i	\(0.868778\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.24.12	yes	204
43.9	even	21	inner	43.6.g.a.9.12	✓	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.12	✓	204	43.9	even	21	inner
43.6.g.a.24.12	yes	204	1.1	even	1	trivial