Embedded newform 43.9.d.a.7.6

• Label: 43.9.d.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	43.7
Dual form			43.9.d.a.37.23

$q$-expansion

\(f(q)\)	\(=\)	\(q-21.4690i q^{2} +(-45.9397 + 26.5233i) q^{3} -204.918 q^{4} +(-326.905 + 188.739i) q^{5} +(569.429 + 986.280i) q^{6} +(2218.06 + 1280.60i) q^{7} -1096.68i q^{8} +(-1873.53 + 3245.04i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 84 q^{3} - 6474 q^{4} - 3 q^{5} + 1791 q^{6} - 5031 q^{7} + 57858 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	21.4690i		−	1.34181i	−0.741542	−	0.670906i	\(-0.765905\pi\)
							0.741542	−	0.670906i	\(-0.234095\pi\)
\(3\)	−45.9397	+	26.5233i	−0.567157	+	0.327448i	−0.756013	−	0.654556i	\(-0.772855\pi\)
							0.188856	+	0.982005i	\(0.439522\pi\)
\(5\)	−326.905	+	188.739i	−0.523048	+	0.301982i	−0.738181	−	0.674603i	\(-0.764315\pi\)
							0.215133	+	0.976585i	\(0.430982\pi\)
\(7\)	2218.06	+	1280.60i	0.923805	+	0.533359i	0.884847	−	0.465882i	\(-0.154263\pi\)
							0.0389583	+	0.999241i	\(0.487596\pi\)
\(11\)	8625.40			0.589127			0.294563	−	0.955632i	\(-0.404826\pi\)
							0.294563	+	0.955632i	\(0.404826\pi\)
\(13\)	18295.9	−	31689.4i	0.640589	−	1.10953i	−0.344712	−	0.938708i	\(-0.612023\pi\)
							0.985301	−	0.170825i	\(-0.0546432\pi\)
\(17\)	−25995.9	+	45026.2i	−0.311250	+	0.539100i	−0.978633	−	0.205614i	\(-0.934081\pi\)
							0.667383	+	0.744714i	\(0.267414\pi\)
\(19\)	213487.	−	123257.i	1.63817	−	0.945796i	0.656702	−	0.754150i	\(-0.271951\pi\)
							0.981464	−	0.191646i	\(-0.0613825\pi\)
\(23\)	−66038.6	−	114382.i	−0.235986	−	0.408740i	0.723573	−	0.690248i	\(-0.242499\pi\)
							−0.959559	+	0.281508i	\(0.909165\pi\)
\(29\)	241268.	+	139296.i	0.341121	+	0.196946i	0.660768	−	0.750591i	\(-0.270231\pi\)
							−0.319647	+	0.947537i	\(0.603564\pi\)
\(31\)	−656646.	−	1.13734e6i	−0.711025	−	1.23153i	−0.964473	−	0.264182i	\(-0.914898\pi\)
							0.253448	−	0.967349i	\(-0.418435\pi\)
\(37\)	2.95734e6	−	1.70742e6i	1.57795	−	0.911032i	0.582809	−	0.812609i	\(-0.301954\pi\)
							0.995145	−	0.0984224i	\(-0.0313796\pi\)
\(41\)	2.63613e6			0.932893			0.466446	−	0.884549i	\(-0.345534\pi\)
							0.466446	+	0.884549i	\(0.345534\pi\)
\(43\)	3.36307e6	+	614800.i	0.983698	+	0.179829i
							\(47\)	5.67498e6			1.16298			0.581491	−	0.813553i	\(-0.302469\pi\)
0.581491	+	0.813553i	\(0.302469\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.d.a.7.6	✓	56
43.37	odd	6	inner	43.9.d.a.37.23	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.d.a.7.6	✓	56	1.1	even	1	trivial
43.9.d.a.37.23	yes	56	43.37	odd	6	inner