Embedded newform 43.10.g.a.9.5

• Label: 43.10.g.a.9.5
• Level: $43$
• Weight: $10$
• Character: 43.9
• 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			9.5
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-22.7164 - 28.4855i) q^{2} +(-193.287 - 29.1333i) q^{3} +(-181.456 + 795.011i) q^{4} +(1708.11 + 1164.57i) q^{5} +(3560.90 + 6167.67i) q^{6} +(-3657.76 + 6335.43i) q^{7} +(9961.31 - 4797.11i) q^{8} +(17702.5 + 5460.49i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−22.7164	−	28.4855i	−1.00393	−	1.25889i	−0.965710	−	0.259622i	\(-0.916402\pi\)
							−0.0382223	−	0.999269i	\(-0.512170\pi\)
\(3\)	−193.287	−	29.1333i	−1.37771	−	0.207656i	−0.581926	−	0.813242i	\(-0.697701\pi\)
							−0.795780	+	0.605586i	\(0.792939\pi\)
\(5\)	1708.11	+	1164.57i	1.22222	+	0.833297i	0.990233	−	0.139421i	\(-0.0445242\pi\)
							0.231989	+	0.972718i	\(0.425477\pi\)
\(7\)	−3657.76	+	6335.43i	−0.575803	+	0.997320i	0.420151	+	0.907454i	\(0.361977\pi\)
							−0.995954	+	0.0898660i	\(0.971356\pi\)
\(11\)	−5732.88	−	25117.4i	−0.118061	−	0.517259i	−0.999028	−	0.0440810i	\(-0.985964\pi\)
							0.880967	−	0.473178i	\(-0.156893\pi\)
\(13\)	−5739.84	+	76592.9i	−0.0557385	+	0.743778i	0.897395	+	0.441228i	\(0.145457\pi\)
							−0.953133	+	0.302550i	\(0.902162\pi\)
\(17\)	216590.	−	147668.i	0.628952	−	0.428812i	−0.206446	−	0.978458i	\(-0.566190\pi\)
							0.835398	+	0.549646i	\(0.185237\pi\)
\(19\)	−565889.	+	174554.i	−0.996185	+	0.307282i	−0.749638	−	0.661848i	\(-0.769772\pi\)
							−0.246547	+	0.969131i	\(0.579296\pi\)
\(23\)	−510146.	+	473346.i	−0.380118	+	0.352698i	−0.846973	−	0.531636i	\(-0.821578\pi\)
							0.466855	+	0.884334i	\(0.345387\pi\)
\(29\)	6.58775e6	−	992944.i	1.72960	−	0.260696i	0.792530	−	0.609832i	\(-0.208763\pi\)
							0.937072	+	0.349137i	\(0.113525\pi\)
\(31\)	−2.32410e6	+	5.92170e6i	−0.451988	+	1.15165i	0.505412	+	0.862878i	\(0.331341\pi\)
							−0.957399	+	0.288767i	\(0.906755\pi\)
\(37\)	−7.75487e6	−	1.34318e7i	−0.680247	−	1.17822i	−0.974905	−	0.222620i	\(-0.928539\pi\)
							0.294658	−	0.955603i	\(-0.404794\pi\)
\(41\)	−1.68090e7	−	2.10779e7i	−0.928999	−	1.16493i	−0.986032	−	0.166556i	\(-0.946735\pi\)
							0.0570325	−	0.998372i	\(-0.481836\pi\)
\(43\)	−1.04810e7	+	1.98177e7i	−0.467512	+	0.883987i
							\(47\)	−5.26159e6	+	2.30525e7i	−0.157281	+	0.689094i	0.833375	+	0.552708i	\(0.186406\pi\)
−0.990656	+	0.136385i	\(0.956451\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.5	✓	384
43.24	even	21	inner	43.10.g.a.24.5	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.5	✓	384	1.1	even	1	trivial
43.10.g.a.24.5	yes	384	43.24	even	21	inner