Embedded newform 43.6.g.a.9.9

• Label: 43.6.g.a.9.9
• Level: $43$
• Weight: $6$
• Character: 43.9
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(17\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			9.9
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10270 - 1.38274i) q^{2} +(-5.76893 - 0.869526i) q^{3} +(6.42464 - 28.1482i) q^{4} +(43.6641 + 29.7697i) q^{5} +(5.15906 + 8.93576i) q^{6} +(20.6505 - 35.7677i) q^{7} +(-96.9964 + 46.7110i) q^{8} +(-199.680 - 61.5931i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10270	−	1.38274i	−0.194932	−	0.244436i	0.674754	−	0.738043i	\(-0.264250\pi\)
							−0.869685	+	0.493606i	\(0.835678\pi\)
\(3\)	−5.76893	−	0.869526i	−0.370077	−	0.0557801i	−0.0386289	−	0.999254i	\(-0.512299\pi\)
							−0.331448	+	0.943474i	\(0.607537\pi\)
\(5\)	43.6641	+	29.7697i	0.781086	+	0.532536i	0.886961	−	0.461844i	\(-0.152812\pi\)
							−0.105875	+	0.994379i	\(0.533764\pi\)
\(7\)	20.6505	−	35.7677i	0.159289	−	0.275897i	−0.775323	−	0.631564i	\(-0.782413\pi\)
							0.934612	+	0.355668i	\(0.115747\pi\)
\(11\)	−168.726	−	739.238i	−0.420437	−	1.84206i	−0.529894	−	0.848064i	\(-0.677768\pi\)
							0.109457	−	0.993992i	\(-0.465089\pi\)
\(13\)	11.5174	−	153.689i	0.0189015	−	0.252223i	−0.979747	−	0.200242i	\(-0.935827\pi\)
							0.998648	−	0.0519813i	\(-0.0165536\pi\)
\(17\)	920.466	−	627.563i	0.772477	−	0.526666i	−0.111743	−	0.993737i	\(-0.535643\pi\)
							0.884220	+	0.467072i	\(0.154691\pi\)
\(19\)	−1082.64	+	333.949i	−0.688017	+	0.212225i	−0.619001	−	0.785391i	\(-0.712462\pi\)
							−0.0690160	+	0.997616i	\(0.521986\pi\)
\(23\)	−2053.48	+	1905.35i	−0.809413	+	0.751025i	−0.971826	−	0.235698i	\(-0.924263\pi\)
							0.162414	+	0.986723i	\(0.448072\pi\)
\(29\)	8021.60	−	1209.06i	1.77119	−	0.266964i	0.819294	−	0.573374i	\(-0.194366\pi\)
							0.951900	+	0.306410i	\(0.0991277\pi\)
\(31\)	954.704	−	2432.55i	0.178429	−	0.454629i	−0.813478	−	0.581595i	\(-0.802429\pi\)
							0.991907	+	0.126966i	\(0.0405240\pi\)
\(37\)	3345.25	+	5794.14i	0.401720	+	0.695800i	0.993934	−	0.109981i	\(-0.0350790\pi\)
							−0.592213	+	0.805781i	\(0.701746\pi\)
\(41\)	−2434.70	−	3053.02i	−0.226197	−	0.283642i	0.655763	−	0.754967i	\(-0.272347\pi\)
							−0.881959	+	0.471325i	\(0.843776\pi\)
\(43\)	10628.7	+	5834.38i	0.876612	+	0.481198i
							\(47\)	5756.45	−	25220.6i	0.380110	−	1.66537i	−0.317013	−	0.948421i	\(-0.602680\pi\)
0.697123	−	0.716951i	\(-0.254463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.9	✓	204
43.24	even	21	inner	43.6.g.a.24.9	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.9	✓	204	1.1	even	1	trivial
43.6.g.a.24.9	yes	204	43.24	even	21	inner