Embedded newform 43.8.g.a.10.15

• Label: 43.8.g.a.10.15
• Level: $43$
• Weight: $8$
• Character: 43.10
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			10.15
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.620020 + 2.71649i) q^{2} +(14.7432 + 13.6797i) q^{3} +(108.329 - 52.1686i) q^{4} +(433.530 - 65.3441i) q^{5} +(-28.0197 + 48.5315i) q^{6} +(-202.241 - 350.291i) q^{7} +(431.251 + 540.771i) q^{8} +(-133.207 - 1777.52i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.620020	+	2.71649i	0.0548026	+	0.240106i	0.994909	−	0.100774i	\(-0.0321321\pi\)
							−0.940107	+	0.340880i	\(0.889275\pi\)
\(3\)	14.7432	+	13.6797i	0.315260	+	0.292518i	0.821891	−	0.569644i	\(-0.192919\pi\)
							−0.506632	+	0.862163i	\(0.669110\pi\)
\(5\)	433.530	−	65.3441i	1.55104	−	0.233782i	0.683055	−	0.730367i	\(-0.260651\pi\)
							0.867988	+	0.496585i	\(0.165413\pi\)
\(7\)	−202.241	−	350.291i	−0.222857	−	0.385999i	0.732818	−	0.680425i	\(-0.238205\pi\)
							−0.955674	+	0.294426i	\(0.904871\pi\)
\(11\)	−4827.60	−	2324.85i	−1.09360	−	0.526648i	−0.201957	−	0.979394i	\(-0.564730\pi\)
							−0.891639	+	0.452746i	\(0.850444\pi\)
\(13\)	710.304	−	1809.82i	0.0896690	−	0.228473i	−0.879003	−	0.476816i	\(-0.841791\pi\)
							0.968672	+	0.248343i	\(0.0798860\pi\)
\(17\)	24102.4	+	3632.85i	1.18984	+	0.179339i	0.713970	−	0.700177i	\(-0.246895\pi\)
							0.475869	+	0.879516i	\(0.342133\pi\)
\(19\)	−2065.28	+	27559.3i	−0.0690784	+	0.921787i	0.849764	+	0.527163i	\(0.176744\pi\)
							−0.918843	+	0.394624i	\(0.870875\pi\)
\(23\)	−68157.1	+	46468.7i	−1.16806	+	0.796367i	−0.982499	−	0.186267i	\(-0.940361\pi\)
							−0.185556	+	0.982634i	\(0.559409\pi\)
\(29\)	33921.6	−	31474.6i	0.258275	−	0.239645i	−0.540398	−	0.841410i	\(-0.681726\pi\)
							0.798673	+	0.601765i	\(0.205536\pi\)
\(31\)	171747.	+	52976.9i	1.03543	+	0.319389i	0.765464	−	0.643479i	\(-0.222510\pi\)
							0.269971	+	0.962868i	\(0.412986\pi\)
\(37\)	−269310.	+	466459.i	−0.874070	+	1.51393i	−0.0163208	+	0.999867i	\(0.505195\pi\)
							−0.857750	+	0.514068i	\(0.828138\pi\)
\(41\)	−69858.9	−	306072.i	−0.158299	−	0.693553i	−0.990319	−	0.138808i	\(-0.955673\pi\)
							0.832020	−	0.554745i	\(-0.187184\pi\)
\(43\)	−59399.9	+	517967.i	−0.113932	+	0.993489i
							\(47\)	−883900.	+	425664.i	−1.24183	+	0.598032i	−0.935309	−	0.353833i	\(-0.884878\pi\)
−0.306518	+	0.951865i	\(0.599164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.15	✓	300
43.13	even	21	inner	43.8.g.a.13.15	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.15	✓	300	1.1	even	1	trivial
43.8.g.a.13.15	yes	300	43.13	even	21	inner