Embedded newform 43.9.d.a.7.16

• Label: 43.9.d.a.7.16
• 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			7.16
Character	\(\chi\)	\(=\)	43.7
Dual form			43.9.d.a.37.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+0.360163i q^{2} +(-80.5202 + 46.4884i) q^{3} +255.870 q^{4} +(539.177 - 311.294i) q^{5} +(-16.7434 - 29.0004i) q^{6} +(-1661.31 - 959.156i) q^{7} +184.357i q^{8} +(1041.83 - 1804.51i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 84 q^{3} - 6474 q^{4} - 3 q^{5} + 1791 q^{6} - 5031 q^{7} + 57858 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			0.360163i			0.0225102i	0.999937	+	0.0112551i	\(0.00358268\pi\)
							−0.999937	+	0.0112551i	\(0.996417\pi\)
\(3\)	−80.5202	+	46.4884i	−0.994076	+	0.573930i	−0.906490	−	0.422227i	\(-0.861249\pi\)
							−0.0875861	+	0.996157i	\(0.527915\pi\)
\(5\)	539.177	−	311.294i	0.862684	−	0.498071i	−0.00222626	−	0.999998i	\(-0.500709\pi\)
							0.864910	+	0.501927i	\(0.167375\pi\)
\(7\)	−1661.31	−	959.156i	−0.691923	−	0.399482i	0.112409	−	0.993662i	\(-0.464143\pi\)
							−0.804332	+	0.594180i	\(0.797477\pi\)
\(11\)	−10974.2			−0.749551			−0.374776	−	0.927116i	\(-0.622280\pi\)
							−0.374776	+	0.927116i	\(0.622280\pi\)
\(13\)	5789.23	−	10027.2i	0.202697	−	0.351081i	−0.746700	−	0.665161i	\(-0.768363\pi\)
							0.949396	+	0.314080i	\(0.101696\pi\)
\(17\)	67919.5	−	117640.i	0.813203	−	1.40851i	−0.0974078	−	0.995245i	\(-0.531055\pi\)
							0.910611	−	0.413265i	\(-0.135612\pi\)
\(19\)	129675.	−	74868.2i	0.995046	−	0.574490i	0.0882675	−	0.996097i	\(-0.471867\pi\)
							0.906779	+	0.421607i	\(0.138534\pi\)
\(23\)	−56547.5	−	97943.2i	−0.202070	−	0.349996i	0.747125	−	0.664683i	\(-0.231434\pi\)
							−0.949195	+	0.314688i	\(0.898100\pi\)
\(29\)	708962.	+	409319.i	1.00238	+	0.578722i	0.908950	−	0.416904i	\(-0.136885\pi\)
							0.0934256	+	0.995626i	\(0.470218\pi\)
\(31\)	−153624.	−	266085.i	−0.166346	−	0.288120i	0.770786	−	0.637094i	\(-0.219864\pi\)
							−0.937133	+	0.348974i	\(0.886530\pi\)
\(37\)	1.17477e6	−	678253.i	0.626824	−	0.361897i	−0.152697	−	0.988273i	\(-0.548796\pi\)
							0.779521	+	0.626376i	\(0.215463\pi\)
\(41\)	4.34586e6			1.53794			0.768972	−	0.639282i	\(-0.220769\pi\)
							0.768972	+	0.639282i	\(0.220769\pi\)
\(43\)	2.44668e6	−	2.38787e6i	0.715655	−	0.698454i
							\(47\)	−2.15285e6			−0.441187			−0.220594	−	0.975366i	\(-0.570799\pi\)
−0.220594	+	0.975366i	\(0.570799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.d.a.7.16	✓	56
43.37	odd	6	inner	43.9.d.a.37.13	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.d.a.7.16	✓	56	1.1	even	1	trivial
43.9.d.a.37.13	yes	56	43.37	odd	6	inner