Embedded newform 43.8.g.a.9.2

• Label: 43.8.g.a.9.2
• Level: $43$
• Weight: $8$
• Character: 43.9
• Analytic conductor: $13.433$
• Analytic rank: $0$
• Dimension: $300$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.4325560958\)
Analytic rank:	\(0\)
Dimension:	\(300\)
Relative dimension:	\(25\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	43.9
Dual form			43.8.g.a.24.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.0734 - 15.1396i) q^{2} +(-73.9419 - 11.1450i) q^{3} +(-54.9571 + 240.783i) q^{4} +(-26.8606 - 18.3132i) q^{5} +(724.002 + 1254.01i) q^{6} +(53.2365 - 92.2083i) q^{7} +(2075.71 - 999.609i) q^{8} +(3253.36 + 1003.53i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.0734	−	15.1396i	−1.06715	−	1.33816i	−0.938037	−	0.346534i	\(-0.887359\pi\)
							−0.129113	−	0.991630i	\(-0.541213\pi\)
\(3\)	−73.9419	−	11.1450i	−1.58113	−	0.238316i	−0.701037	−	0.713125i	\(-0.747279\pi\)
							−0.880089	+	0.474809i	\(0.842517\pi\)
\(5\)	−26.8606	−	18.3132i	−0.0960992	−	0.0655194i	0.514317	−	0.857600i	\(-0.328045\pi\)
							−0.610416	+	0.792081i	\(0.708998\pi\)
\(7\)	53.2365	−	92.2083i	0.0586632	−	0.101608i	−0.835202	−	0.549943i	\(-0.814650\pi\)
							0.893866	+	0.448335i	\(0.147983\pi\)
\(11\)	407.040	+	1783.36i	0.0922068	+	0.403984i	0.999877	−	0.0157003i	\(-0.00499778\pi\)
							−0.907670	+	0.419685i	\(0.862141\pi\)
\(13\)	−222.107	+	2963.82i	−0.0280389	+	0.374153i	0.965479	+	0.260480i	\(0.0838808\pi\)
							−0.993518	+	0.113673i	\(0.963738\pi\)
\(17\)	−22380.2	+	15258.6i	−1.10482	+	0.753256i	−0.971316	−	0.237792i	\(-0.923576\pi\)
							−0.133507	+	0.991048i	\(0.542624\pi\)
\(19\)	12680.4	−	3911.39i	0.424127	−	0.130826i	−0.0753415	−	0.997158i	\(-0.524005\pi\)
							0.499469	+	0.866332i	\(0.333528\pi\)
\(23\)	−79701.0	+	73951.7i	−1.36589	+	1.26736i	−0.435795	+	0.900046i	\(0.643533\pi\)
							−0.930096	+	0.367316i	\(0.880277\pi\)
\(29\)	119505.	−	18012.5i	0.909900	−	0.137145i	0.322619	−	0.946529i	\(-0.395437\pi\)
							0.587280	+	0.809384i	\(0.300199\pi\)
\(31\)	107987.	−	275147.i	0.651038	−	1.65882i	−0.0971004	−	0.995275i	\(-0.530957\pi\)
							0.748138	−	0.663543i	\(-0.230948\pi\)
\(37\)	35047.9	+	60704.8i	0.113751	+	0.197023i	0.917280	−	0.398243i	\(-0.130380\pi\)
							−0.803529	+	0.595266i	\(0.797047\pi\)
\(41\)	240046.	+	301009.i	0.543941	+	0.682080i	0.975499	−	0.220005i	\(-0.0706074\pi\)
							−0.431558	+	0.902085i	\(0.642036\pi\)
\(43\)	−446290.	−	269524.i	−0.856008	−	0.516962i
							\(47\)	61816.6	−	270836.i	0.0868486	−	0.380509i	−0.912760	−	0.408497i	\(-0.866053\pi\)
0.999608	+	0.0279885i	\(0.00891017\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.9.2	✓	300
43.24	even	21	inner	43.8.g.a.24.2	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.9.2	✓	300	1.1	even	1	trivial
43.8.g.a.24.2	yes	300	43.24	even	21	inner