Embedded newform 43.4.g.a.10.2

• Label: 43.4.g.a.10.2
• Level: $43$
• Weight: $4$
• Character: 43.10
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.2
Character	\(\chi\)	\(=\)	43.10
Dual form			43.4.g.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08942 - 4.77305i) q^{2} +(-6.48010 - 6.01265i) q^{3} +(-14.3874 + 6.92863i) q^{4} +(20.4076 - 3.07594i) q^{5} +(-21.6392 + 37.4801i) q^{6} +(-5.26876 - 9.12575i) q^{7} +(24.3248 + 30.5023i) q^{8} +(3.82197 + 51.0006i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08942	−	4.77305i	−0.385167	−	1.68753i	−0.680995	−	0.732288i	\(-0.738453\pi\)
							0.295828	−	0.955241i	\(-0.404404\pi\)
\(3\)	−6.48010	−	6.01265i	−1.24710	−	1.15714i	−0.981171	−	0.193141i	\(-0.938132\pi\)
							−0.265924	−	0.963994i	\(-0.585677\pi\)
\(5\)	20.4076	−	3.07594i	1.82531	−	0.275121i	0.855289	−	0.518152i	\(-0.173380\pi\)
							0.970019	+	0.243031i	\(0.0781416\pi\)
\(7\)	−5.26876	−	9.12575i	−0.284486	−	0.492744i	0.687998	−	0.725712i	\(-0.258490\pi\)
							−0.972484	+	0.232968i	\(0.925156\pi\)
\(11\)	−7.67147	−	3.69438i	−0.210276	−	0.101264i	0.325781	−	0.945445i	\(-0.394373\pi\)
							−0.536057	+	0.844182i	\(0.680087\pi\)
\(13\)	4.91165	−	12.5147i	0.104788	−	0.266996i	−0.868857	−	0.495062i	\(-0.835145\pi\)
							0.973646	+	0.228067i	\(0.0732404\pi\)
\(17\)	−46.0361	−	6.93883i	−0.656788	−	0.0989949i	−0.187811	−	0.982205i	\(-0.560139\pi\)
							−0.468977	+	0.883210i	\(0.655377\pi\)
\(19\)	1.12481	−	15.0095i	0.0135815	−	0.181232i	−0.986287	−	0.165037i	\(-0.947226\pi\)
							0.999869	−	0.0161953i	\(-0.00515534\pi\)
\(23\)	125.014	−	85.2334i	1.13336	−	0.772712i	0.156731	−	0.987641i	\(-0.449904\pi\)
							0.976630	+	0.214929i	\(0.0689519\pi\)
\(29\)	−36.7826	+	34.1293i	−0.235530	+	0.218540i	−0.789103	−	0.614261i	\(-0.789454\pi\)
							0.553573	+	0.832800i	\(0.313264\pi\)
\(31\)	−114.166	−	35.2154i	−0.661444	−	0.204028i	−0.0541830	−	0.998531i	\(-0.517255\pi\)
							−0.607261	+	0.794503i	\(0.707732\pi\)
\(37\)	−77.8678	+	134.871i	−0.345983	+	0.599261i	−0.985532	−	0.169490i	\(-0.945788\pi\)
							0.639549	+	0.768751i	\(0.279121\pi\)
\(41\)	−56.9464	−	249.498i	−0.216915	−	0.950368i	−0.959742	−	0.280884i	\(-0.909372\pi\)
							0.742826	−	0.669484i	\(-0.233485\pi\)
\(43\)	184.251	−	213.445i	0.653441	−	0.756978i
							\(47\)	−42.2355	+	20.3395i	−0.131078	+	0.0631239i	−0.498273	−	0.867020i	\(-0.666032\pi\)
0.367195	+	0.930144i	\(0.380318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.10.2	✓	120
43.13	even	21	inner	43.4.g.a.13.2	yes	120
43.20	odd	42		1849.4.a.k.1.7		60
43.23	even	21		1849.4.a.l.1.54		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.2	✓	120	1.1	even	1	trivial
43.4.g.a.13.2	yes	120	43.13	even	21	inner
1849.4.a.k.1.7		60	43.20	odd	42
1849.4.a.l.1.54		60	43.23	even	21