Embedded newform 43.9.d.a.7.14

• Label: 43.9.d.a.7.14
• Level: $43$
• Weight: $9$
• Character: 43.7
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.14
Character	\(\chi\)	\(=\)	43.7
Dual form			43.9.d.a.37.15

$q$-expansion

\(f(q)\)	\(=\)	\(q-4.04858i q^{2} +(-89.5347 + 51.6929i) q^{3} +239.609 q^{4} +(-50.9246 + 29.4013i) q^{5} +(209.283 + 362.488i) q^{6} +(839.413 + 484.635i) q^{7} -2006.51i q^{8} +(2063.81 - 3574.63i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 84 q^{3} - 6474 q^{4} - 3 q^{5} + 1791 q^{6} - 5031 q^{7} + 57858 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	4.04858i		−	0.253036i	−0.991964	−	0.126518i	\(-0.959620\pi\)
							0.991964	−	0.126518i	\(-0.0403802\pi\)
\(3\)	−89.5347	+	51.6929i	−1.10537	+	0.638184i	−0.937625	−	0.347647i	\(-0.886981\pi\)
							−0.167742	+	0.985831i	\(0.553647\pi\)
\(5\)	−50.9246	+	29.4013i	−0.0814793	+	0.0470421i	−0.540186	−	0.841546i	\(-0.681646\pi\)
							0.458707	+	0.888588i	\(0.348313\pi\)
\(7\)	839.413	+	484.635i	0.349610	+	0.201847i	0.664513	−	0.747276i	\(-0.268639\pi\)
							−0.314904	+	0.949124i	\(0.601972\pi\)
\(11\)	4021.73			0.274690			0.137345	−	0.990523i	\(-0.456143\pi\)
							0.137345	+	0.990523i	\(0.456143\pi\)
\(13\)	15521.4	−	26883.9i	0.543449	−	0.941281i	−0.455254	−	0.890362i	\(-0.650451\pi\)
							0.998703	−	0.0509195i	\(-0.0162152\pi\)
\(17\)	−72854.4	+	126188.i	−0.872289	+	1.51085i	−0.0126652	+	0.999920i	\(0.504032\pi\)
							−0.859623	+	0.510928i	\(0.829302\pi\)
\(19\)	−207832.	+	119992.i	−1.59477	+	0.920739i	−0.602294	+	0.798275i	\(0.705746\pi\)
							−0.992473	+	0.122464i	\(0.960920\pi\)
\(23\)	240574.	+	416687.i	0.859682	+	1.48901i	0.872233	+	0.489091i	\(0.162671\pi\)
							−0.0125509	+	0.999921i	\(0.503995\pi\)
\(29\)	453858.	+	262035.i	0.641694	+	0.370482i	0.785267	−	0.619158i	\(-0.212526\pi\)
							−0.143573	+	0.989640i	\(0.545859\pi\)
\(31\)	58207.0	+	100818.i	0.0630273	+	0.109167i	0.895817	−	0.444423i	\(-0.146591\pi\)
							−0.832790	+	0.553589i	\(0.813258\pi\)
\(37\)	−1.64540e6	+	949970.i	−0.877937	+	0.506877i	−0.869978	−	0.493091i	\(-0.835867\pi\)
							−0.00795945	+	0.999968i	\(0.502534\pi\)
\(41\)	−2.54720e6			−0.901422			−0.450711	−	0.892670i	\(-0.648830\pi\)
							−0.450711	+	0.892670i	\(0.648830\pi\)
\(43\)	−3.36570e6	+	600201.i	−0.984469	+	0.175559i
							\(47\)	4.13285e6			0.846950			0.423475	−	0.905908i	\(-0.360810\pi\)
0.423475	+	0.905908i	\(0.360810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.d.a.7.14	✓	56
43.37	odd	6	inner	43.9.d.a.37.15	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.d.a.7.14	✓	56	1.1	even	1	trivial
43.9.d.a.37.15	yes	56	43.37	odd	6	inner