Embedded newform 43.10.g.a.24.12

• Label: 43.10.g.a.24.12
• Level: $43$
• Weight: $10$
• Character: 43.24
• Analytic conductor: $22.147$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(22.1465409550\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(32\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			24.12
Character	\(\chi\)	\(=\)	43.24
Dual form			43.10.g.a.9.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.77399 + 12.2562i) q^{2} +(-129.957 + 19.5879i) q^{3} +(59.2473 + 259.579i) q^{4} +(828.781 - 565.053i) q^{5} +(1030.13 - 1784.23i) q^{6} +(3211.01 + 5561.63i) q^{7} +(-10991.9 - 5293.44i) q^{8} +(-2303.38 + 710.499i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.77399	+	12.2562i	−0.431953	+	0.541652i	−0.949403	−	0.314062i	\(-0.898310\pi\)
							0.517449	+	0.855714i	\(0.326882\pi\)
\(3\)	−129.957	+	19.5879i	−0.926306	+	0.139618i	−0.594835	−	0.803848i	\(-0.702783\pi\)
							−0.331470	+	0.943466i	\(0.607545\pi\)
\(5\)	828.781	−	565.053i	0.593027	−	0.404319i	−0.229282	−	0.973360i	\(-0.573638\pi\)
							0.822309	+	0.569041i	\(0.192685\pi\)
\(7\)	3211.01	+	5561.63i	0.505476	+	0.875509i	0.999980	+	0.00633416i	\(0.00201624\pi\)
							−0.494504	+	0.869175i	\(0.664650\pi\)
\(11\)	19692.6	−	86279.1i	0.405543	−	1.77680i	−0.198763	−	0.980047i	\(-0.563693\pi\)
							0.604306	−	0.796752i	\(-0.293450\pi\)
\(13\)	−10352.4	−	138143.i	−0.100530	−	1.34148i	−0.788184	−	0.615440i	\(-0.788978\pi\)
							0.687654	−	0.726038i	\(-0.258641\pi\)
\(17\)	433902.	+	295829.i	1.26000	+	0.859055i	0.994467	−	0.105046i	\(-0.0334989\pi\)
							0.265535	+	0.964101i	\(0.414451\pi\)
\(19\)	25458.7	+	7852.96i	0.0448172	+	0.0138243i	0.317083	−	0.948398i	\(-0.397297\pi\)
							−0.272266	+	0.962222i	\(0.587773\pi\)
\(23\)	−462657.	−	429283.i	−0.344734	−	0.319866i	0.488735	−	0.872432i	\(-0.337458\pi\)
							−0.833469	+	0.552566i	\(0.813649\pi\)
\(29\)	1.71103e6	+	257896.i	0.449228	+	0.0677103i	0.369760	−	0.929127i	\(-0.379440\pi\)
							0.0794679	+	0.996837i	\(0.474678\pi\)
\(31\)	2.23489e6	+	5.69442e6i	0.434640	+	1.10744i	0.965596	+	0.260045i	\(0.0837375\pi\)
							−0.530957	+	0.847399i	\(0.678167\pi\)
\(37\)	−2.27668e6	+	3.94333e6i	−0.199707	+	0.345904i	−0.948434	−	0.316976i	\(-0.897333\pi\)
							0.748726	+	0.662880i	\(0.230666\pi\)
\(41\)	2.08764e7	−	2.61782e7i	1.15379	−	1.44681i	0.280344	−	0.959900i	\(-0.409551\pi\)
							0.873450	−	0.486913i	\(-0.161877\pi\)
\(43\)	1.35359e7	+	1.78710e7i	0.603780	+	0.797151i
							\(47\)	−1.94419e6	−	8.51804e6i	−0.0581162	−	0.254624i	0.937521	−	0.347927i	\(-0.113114\pi\)
−0.995638	+	0.0933035i	\(0.970257\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.24.12	yes	384
43.9	even	21	inner	43.10.g.a.9.12	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.12	✓	384	43.9	even	21	inner
43.10.g.a.24.12	yes	384	1.1	even	1	trivial