Embedded newform 43.5.h.a.5.2

• Label: 43.5.h.a.5.2
• Level: $43$
• Weight: $5$
• Character: 43.5
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	43.5
Dual form			43.5.h.a.26.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.62097 - 1.51119i) q^{2} +(-2.79970 + 9.07640i) q^{3} +(27.1380 + 13.0690i) q^{4} +(20.2385 - 7.94303i) q^{5} +(32.2529 - 55.8637i) q^{6} +(-41.2058 + 23.7902i) q^{7} +(-74.9765 - 59.7917i) q^{8} +(-7.61736 - 5.19343i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.62097	−	1.51119i	−1.65524	−	0.377798i	−0.710008	−	0.704193i	\(-0.751309\pi\)
							−0.945233	+	0.326395i	\(0.894166\pi\)
\(3\)	−2.79970	+	9.07640i	−0.311078	+	1.00849i	0.656161	+	0.754621i	\(0.272179\pi\)
							−0.967239	+	0.253868i	\(0.918297\pi\)
\(5\)	20.2385	−	7.94303i	0.809540	−	0.317721i	0.0757757	−	0.997125i	\(-0.475857\pi\)
							0.733765	+	0.679404i	\(0.237761\pi\)
\(7\)	−41.2058	+	23.7902i	−0.840934	+	0.485513i	−0.857582	−	0.514348i	\(-0.828034\pi\)
							0.0166476	+	0.999861i	\(0.494701\pi\)
\(11\)	−69.9733	+	33.6974i	−0.578292	+	0.278491i	−0.700079	−	0.714066i	\(-0.746852\pi\)
							0.121787	+	0.992556i	\(0.461137\pi\)
\(13\)	−251.315	−	37.8796i	−1.48707	−	0.224139i	−0.645280	−	0.763947i	\(-0.723259\pi\)
							−0.841789	+	0.539807i	\(0.818497\pi\)
\(17\)	44.1311	−	112.444i	0.152703	−	0.389081i	−0.833967	−	0.551814i	\(-0.813936\pi\)
							0.986670	+	0.162733i	\(0.0520310\pi\)
\(19\)	−306.723	−	449.880i	−0.849647	−	1.24620i	−0.967424	−	0.253163i	\(-0.918529\pi\)
							0.117776	−	0.993040i	\(-0.462423\pi\)
\(23\)	−37.4804	+	500.141i	−0.0708515	+	0.945447i	0.842611	+	0.538523i	\(0.181018\pi\)
							−0.913462	+	0.406924i	\(0.866601\pi\)
\(29\)	−220.387	−	714.476i	−0.262053	−	0.849555i	−0.986996	−	0.160746i	\(-0.948610\pi\)
							0.724943	−	0.688809i	\(-0.241866\pi\)
\(31\)	998.602	+	926.567i	1.03913	+	0.964170i	0.999386	−	0.0350405i	\(-0.0111560\pi\)
							0.0397417	+	0.999210i	\(0.487346\pi\)
\(37\)	−1559.48	−	900.368i	−1.13914	−	0.657683i	−0.192923	−	0.981214i	\(-0.561797\pi\)
							−0.946218	+	0.323531i	\(0.895130\pi\)
\(41\)	116.774	−	511.619i	0.0694667	−	0.304354i	−0.928245	−	0.371970i	\(-0.878682\pi\)
							0.997711	+	0.0676166i	\(0.0215395\pi\)
\(43\)	−1821.35	+	318.555i	−0.985047	+	0.172285i
							\(47\)	−651.945	−	313.960i	−0.295131	−	0.142128i	0.280459	−	0.959866i	\(-0.409513\pi\)
−0.575590	+	0.817738i	\(0.695228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.5.2	✓	168
43.26	odd	42	inner	43.5.h.a.26.2	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.5.2	✓	168	1.1	even	1	trivial
43.5.h.a.26.2	yes	168	43.26	odd	42	inner