Embedded newform 43.7.d.a.7.5

• Label: 43.7.d.a.7.5
• Level: $43$
• Weight: $7$
• Character: 43.7
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.5
Character	\(\chi\)	\(=\)	43.7
Dual form			43.7.d.a.37.17

$q$-expansion

\(f(q)\)	\(=\)	\(q-10.1559i q^{2} +(30.5813 - 17.6561i) q^{3} -39.1423 q^{4} +(101.658 - 58.6923i) q^{5} +(-179.314 - 310.581i) q^{6} +(191.878 + 110.781i) q^{7} -252.452i q^{8} +(258.977 - 448.562i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	10.1559i		−	1.26949i	−0.772723	−	0.634744i	\(-0.781106\pi\)
							0.772723	−	0.634744i	\(-0.218894\pi\)
\(3\)	30.5813	−	17.6561i	1.13264	−	0.653931i	0.188043	−	0.982161i	\(-0.439786\pi\)
							0.944598	+	0.328230i	\(0.106452\pi\)
\(5\)	101.658	−	58.6923i	0.813264	−	0.469538i	−0.0348238	−	0.999393i	\(-0.511087\pi\)
							0.848088	+	0.529855i	\(0.177754\pi\)
\(7\)	191.878	+	110.781i	0.559411	+	0.322976i	0.752909	−	0.658125i	\(-0.228650\pi\)
							−0.193498	+	0.981101i	\(0.561983\pi\)
\(11\)	1438.34			1.08065			0.540323	−	0.841457i	\(-0.318302\pi\)
							0.540323	+	0.841457i	\(0.318302\pi\)
\(13\)	−661.952	+	1146.53i	−0.301298	+	0.521864i	−0.976430	−	0.215833i	\(-0.930753\pi\)
							0.675132	+	0.737697i	\(0.264087\pi\)
\(17\)	−3957.19	+	6854.06i	−0.805454	+	1.39509i	0.110531	+	0.993873i	\(0.464745\pi\)
							−0.915984	+	0.401214i	\(0.868588\pi\)
\(19\)	−5836.08	+	3369.46i	−0.850864	+	0.491247i	−0.860942	−	0.508703i	\(-0.830125\pi\)
							0.0100783	+	0.999949i	\(0.496792\pi\)
\(23\)	−7712.75	−	13358.9i	−0.633907	−	1.09796i	−0.986746	−	0.162275i	\(-0.948117\pi\)
							0.352839	−	0.935684i	\(-0.385216\pi\)
\(29\)	11123.1	+	6421.92i	0.456070	+	0.263312i	0.710390	−	0.703808i	\(-0.248518\pi\)
							−0.254320	+	0.967120i	\(0.581852\pi\)
\(31\)	3864.08	+	6692.78i	0.129706	+	0.224658i	0.923563	−	0.383447i	\(-0.125263\pi\)
							−0.793857	+	0.608105i	\(0.791930\pi\)
\(37\)	12422.5	−	7172.14i	0.245247	−	0.141594i	−0.372339	−	0.928097i	\(-0.621444\pi\)
							0.617586	+	0.786503i	\(0.288111\pi\)
\(41\)	78223.5			1.13497			0.567487	−	0.823383i	\(-0.307916\pi\)
							0.567487	+	0.823383i	\(0.307916\pi\)
\(43\)	−53717.2	−	58615.9i	−0.675629	−	0.737242i
							\(47\)	104457.			1.00611			0.503055	−	0.864254i	\(-0.332209\pi\)
0.503055	+	0.864254i	\(0.332209\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.7.5	✓	42
43.37	odd	6	inner	43.7.d.a.37.17	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.5	✓	42	1.1	even	1	trivial
43.7.d.a.37.17	yes	42	43.37	odd	6	inner