Embedded newform 43.10.g.a.24.1

• Label: 43.10.g.a.24.1
• 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			24.1
Character	\(\chi\)	\(=\)	43.24
Dual form			43.10.g.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-27.2199 + 34.1327i) q^{2} +(-145.141 + 21.8765i) q^{3} +(-310.186 - 1359.01i) q^{4} +(-589.424 + 401.863i) q^{5} +(3204.02 - 5549.53i) q^{6} +(4835.78 + 8375.81i) q^{7} +(34691.1 + 16706.4i) q^{8} +(1778.80 - 548.686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−27.2199	+	34.1327i	−1.20296	+	1.50847i	−0.395585	+	0.918429i	\(0.629458\pi\)
							−0.807376	+	0.590037i	\(0.799113\pi\)
\(3\)	−145.141	+	21.8765i	−1.03453	+	0.155931i	−0.644290	−	0.764781i	\(-0.722847\pi\)
							−0.390243	+	0.920712i	\(0.627609\pi\)
\(5\)	−589.424	+	401.863i	−0.421758	+	0.287550i	−0.755542	−	0.655100i	\(-0.772627\pi\)
							0.333785	+	0.942649i	\(0.391674\pi\)
\(7\)	4835.78	+	8375.81i	0.761246	+	1.31852i	0.942209	+	0.335027i	\(0.108745\pi\)
							−0.180962	+	0.983490i	\(0.557921\pi\)
\(11\)	−5251.26	+	23007.3i	−0.108142	+	0.473803i	0.891636	+	0.452753i	\(0.149558\pi\)
							−0.999778	+	0.0210501i	\(0.993299\pi\)
\(13\)	5177.58	+	69090.0i	0.0502784	+	0.670919i	0.964225	+	0.265085i	\(0.0853999\pi\)
							−0.913947	+	0.405834i	\(0.866981\pi\)
\(17\)	460200.	+	313759.i	1.33637	+	0.911121i	0.999536	−	0.0304436i	\(-0.00969199\pi\)
							0.336833	+	0.941565i	\(0.390644\pi\)
\(19\)	500379.	+	154347.i	0.880862	+	0.271710i	0.702011	−	0.712166i	\(-0.252286\pi\)
							0.178851	+	0.983876i	\(0.442762\pi\)
\(23\)	1.51715e6	+	1.40771e6i	1.13046	+	1.04891i	0.998335	+	0.0576760i	\(0.0183690\pi\)
							0.132120	+	0.991234i	\(0.457821\pi\)
\(29\)	1.08667e6	+	163789.i	0.285303	+	0.0430025i	0.290135	−	0.956986i	\(-0.406300\pi\)
							−0.00483158	+	0.999988i	\(0.501538\pi\)
\(31\)	760804.	+	1.93850e6i	0.147960	+	0.376997i	0.985569	−	0.169274i	\(-0.0541423\pi\)
							−0.837609	+	0.546271i	\(0.816047\pi\)
\(37\)	−1.19716e6	+	2.07354e6i	−0.105013	+	0.181888i	−0.913744	−	0.406291i	\(-0.866822\pi\)
							0.808730	+	0.588180i	\(0.200155\pi\)
\(41\)	1.25364e7	−	1.57202e7i	0.692862	−	0.868821i	−0.303606	−	0.952798i	\(-0.598191\pi\)
							0.996468	+	0.0839763i	\(0.0267620\pi\)
\(43\)	−1.68483e7	−	1.47894e7i	−0.751533	−	0.659696i
							\(47\)	1.23858e7	+	5.42656e7i	0.370240	+	1.62213i	0.726102	+	0.687587i	\(0.241330\pi\)
−0.355862	+	0.934538i	\(0.615813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.24.1	yes	384
43.9	even	21	inner	43.10.g.a.9.1	✓	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.1	✓	384	43.9	even	21	inner
43.10.g.a.24.1	yes	384	1.1	even	1	trivial