Embedded newform 43.10.g.a.9.14

• Label: 43.10.g.a.9.14
• 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			9.14
Character	\(\chi\)	\(=\)	43.9
Dual form			43.10.g.a.24.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.98413 - 7.50386i) q^{2} +(-64.3608 - 9.70083i) q^{3} +(93.4326 - 409.355i) q^{4} +(-1644.31 - 1121.07i) q^{5} +(312.350 + 541.005i) q^{6} +(-2499.13 + 4328.62i) q^{7} +(-8058.28 + 3880.66i) q^{8} +(-14760.3 - 4552.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.98413	−	7.50386i	−0.264463	−	0.331627i	0.631814	−	0.775120i	\(-0.282311\pi\)
							−0.896278	+	0.443493i	\(0.853739\pi\)
\(3\)	−64.3608	−	9.70083i	−0.458750	−	0.0691454i	−0.0843979	−	0.996432i	\(-0.526897\pi\)
							−0.374352	+	0.927287i	\(0.622135\pi\)
\(5\)	−1644.31	−	1121.07i	−1.17657	−	0.802175i	−0.192739	−	0.981250i	\(-0.561737\pi\)
							−0.983835	+	0.179075i	\(0.942690\pi\)
\(7\)	−2499.13	+	4328.62i	−0.393412	+	0.681410i	−0.992897	−	0.118976i	\(-0.962039\pi\)
							0.599485	+	0.800386i	\(0.295372\pi\)
\(11\)	−4490.58	−	19674.5i	−0.0924774	−	0.405170i	0.907409	−	0.420249i	\(-0.138057\pi\)
							−0.999886	+	0.0150790i	\(0.995200\pi\)
\(13\)	5961.64	−	79552.5i	0.0578922	−	0.772518i	−0.890412	−	0.455156i	\(-0.849584\pi\)
							0.948304	−	0.317363i	\(-0.102797\pi\)
\(17\)	156423.	−	106647.i	0.454234	−	0.309691i	−0.314503	−	0.949256i	\(-0.601838\pi\)
							0.768737	+	0.639565i	\(0.220886\pi\)
\(19\)	−24850.6	+	7665.39i	−0.0437467	+	0.0134941i	−0.316551	−	0.948575i	\(-0.602525\pi\)
							0.272805	+	0.962069i	\(0.412049\pi\)
\(23\)	−494109.	+	458466.i	−0.368169	+	0.341611i	−0.842462	−	0.538756i	\(-0.818894\pi\)
							0.474292	+	0.880367i	\(0.342704\pi\)
\(29\)	6.85477e6	−	1.03319e6i	1.79971	−	0.271262i	0.838065	−	0.545571i	\(-0.183687\pi\)
							0.961642	+	0.274309i	\(0.0884490\pi\)
\(31\)	−1.30291e6	+	3.31977e6i	−0.253389	+	0.645625i	−0.999803	−	0.0198571i	\(-0.993679\pi\)
							0.746414	+	0.665482i	\(0.231774\pi\)
\(37\)	−7.80631e6	−	1.35209e7i	−0.684759	−	1.18604i	−0.973512	−	0.228634i	\(-0.926574\pi\)
							0.288753	−	0.957404i	\(-0.406759\pi\)
\(41\)	1.93053e7	+	2.42081e7i	1.06696	+	1.33793i	0.938138	+	0.346262i	\(0.112549\pi\)
							0.128825	+	0.991667i	\(0.458879\pi\)
\(43\)	−1.78909e7	−	1.35096e7i	−0.798039	−	0.602605i
							\(47\)	−3.43316e6	+	1.50417e7i	−0.102625	+	0.449631i	0.897340	+	0.441339i	\(0.145496\pi\)
−0.999966	+	0.00829146i	\(0.997361\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.10.g.a.9.14	✓	384
43.24	even	21	inner	43.10.g.a.24.14	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.10.g.a.9.14	✓	384	1.1	even	1	trivial
43.10.g.a.24.14	yes	384	43.24	even	21	inner