Embedded newform 43.4.g.a.13.2

• Label: 43.4.g.a.13.2
• Level: $43$
• Weight: $4$
• Character: 43.13
• 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			13.2
Character	\(\chi\)	\(=\)	43.13
Dual form			43.4.g.a.10.2

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.2	✓	120	43.10	even	21	inner
43.4.g.a.13.2	yes	120	1.1	even	1	trivial
1849.4.a.k.1.7		60	43.28	odd	42
1849.4.a.l.1.54		60	43.15	even	21