Embedded newform 43.10.g.a.9.1

• Label: 43.10.g.a.9.1
• 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			9.1
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.1

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.1	✓	384	1.1	even	1	trivial
43.10.g.a.24.1	yes	384	43.24	even	21	inner