Embedded newform 43.10.g.a.9.8

• Label: 43.10.g.a.9.8
• 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			9.8
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.6235 - 20.8452i) q^{2} +(172.616 + 26.0176i) q^{3} +(-44.2510 + 193.876i) q^{4} +(-1824.53 - 1243.94i) q^{5} +(-2327.13 - 4030.71i) q^{6} +(5716.05 - 9900.48i) q^{7} +(-7522.08 + 3622.44i) q^{8} +(10310.8 + 3180.45i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.6235	−	20.8452i	−0.734661	−	0.921236i	0.264406	−	0.964411i	\(-0.414824\pi\)
							−0.999068	+	0.0431755i	\(0.986253\pi\)
\(3\)	172.616	+	26.0176i	1.23037	+	0.185448i	0.731871	−	0.681443i	\(-0.238647\pi\)
							0.498497	+	0.866891i	\(0.333886\pi\)
\(5\)	−1824.53	−	1243.94i	−1.30553	−	0.890092i	−0.307417	−	0.951575i	\(-0.599465\pi\)
							−0.998108	+	0.0614830i	\(0.980417\pi\)
\(7\)	5716.05	−	9900.48i	0.899818	−	1.55853i	0.0720911	−	0.997398i	\(-0.477033\pi\)
							0.827727	−	0.561132i	\(-0.189634\pi\)
\(11\)	12668.8	+	55505.5i	0.260896	+	1.14306i	0.920282	+	0.391255i	\(0.127959\pi\)
							−0.659386	+	0.751804i	\(0.729184\pi\)
\(13\)	11329.9	−	151188.i	0.110023	−	1.46815i	−0.619932	−	0.784656i	\(-0.712840\pi\)
							0.729955	−	0.683496i	\(-0.239541\pi\)
\(17\)	213675.	−	145681.i	0.620488	−	0.423041i	−0.211856	−	0.977301i	\(-0.567951\pi\)
							0.832344	+	0.554259i	\(0.186998\pi\)
\(19\)	−741818.	+	228821.i	−1.30589	+	0.402813i	−0.868156	−	0.496291i	\(-0.834695\pi\)
							−0.437733	+	0.899105i	\(0.644219\pi\)
\(23\)	−661724.	+	613990.i	−0.493062	+	0.457495i	−0.887085	−	0.461606i	\(-0.847274\pi\)
							0.394023	+	0.919101i	\(0.371083\pi\)
\(29\)	−1.45061e6	+	218645.i	−0.380855	+	0.0574047i	−0.336680	−	0.941619i	\(-0.609304\pi\)
							−0.0441753	+	0.999024i	\(0.514066\pi\)
\(31\)	692928.	−	1.76555e6i	0.134760	−	0.343362i	−0.847525	−	0.530755i	\(-0.821908\pi\)
							0.982285	+	0.187393i	\(0.0600037\pi\)
\(37\)	5.99843e6	+	1.03896e7i	0.526175	+	0.911362i	0.999535	+	0.0304926i	\(0.00970760\pi\)
							−0.473360	+	0.880869i	\(0.656959\pi\)
\(41\)	4.20418e6	+	5.27187e6i	0.232356	+	0.291365i	0.884317	−	0.466888i	\(-0.154625\pi\)
							−0.651961	+	0.758253i	\(0.726053\pi\)
\(43\)	−1.48007e7	+	1.68384e7i	−0.660197	+	0.751093i
							\(47\)	−4.50949e6	+	1.97574e7i	−0.134799	+	0.590593i	0.861731	+	0.507365i	\(0.169380\pi\)
−0.996530	+	0.0832286i	\(0.973477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.8	✓	384
43.24	even	21	inner	43.10.g.a.24.8	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.8	✓	384	1.1	even	1	trivial
43.10.g.a.24.8	yes	384	43.24	even	21	inner