Embedded newform 43.6.g.a.9.17

• Label: 43.6.g.a.9.17
• 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			9.17
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.32555 + 7.93199i) q^{2} +(2.21950 + 0.334535i) q^{3} +(-15.7832 + 69.1506i) q^{4} +(53.6893 + 36.6047i) q^{5} +(11.3860 + 19.7211i) q^{6} +(51.6195 - 89.4076i) q^{7} +(-355.837 + 171.362i) q^{8} +(-227.390 - 70.1405i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.32555	+	7.93199i	1.11821	+	1.40219i	0.905119	+	0.425158i	\(0.139781\pi\)
							0.213090	+	0.977033i	\(0.431647\pi\)
\(3\)	2.21950	+	0.334535i	0.142381	+	0.0214604i	0.219846	−	0.975535i	\(-0.429445\pi\)
							−0.0774651	+	0.996995i	\(0.524683\pi\)
\(5\)	53.6893	+	36.6047i	0.960423	+	0.654805i	0.938618	−	0.344957i	\(-0.112107\pi\)
							0.0218044	+	0.999762i	\(0.493059\pi\)
\(7\)	51.6195	−	89.4076i	0.398170	−	0.689651i	−0.595330	−	0.803481i	\(-0.702979\pi\)
							0.993500	+	0.113830i	\(0.0363121\pi\)
\(11\)	20.3695	+	89.2448i	0.0507574	+	0.222383i	0.993945	−	0.109878i	\(-0.0350459\pi\)
							−0.943188	+	0.332260i	\(0.892189\pi\)
\(13\)	17.7788	−	237.241i	0.0291772	−	0.389343i	−0.963429	−	0.267964i	\(-0.913649\pi\)
							0.992606	−	0.121379i	\(-0.0387317\pi\)
\(17\)	−325.466	+	221.899i	−0.273139	+	0.186223i	−0.692141	−	0.721762i	\(-0.743332\pi\)
							0.419002	+	0.907985i	\(0.362380\pi\)
\(19\)	1621.60	−	500.198i	1.03053	−	0.317876i	0.267032	−	0.963688i	\(-0.413957\pi\)
							0.763497	+	0.645812i	\(0.223481\pi\)
\(23\)	1673.07	−	1552.38i	0.659467	−	0.611896i	−0.277745	−	0.960655i	\(-0.589587\pi\)
							0.937213	+	0.348758i	\(0.113397\pi\)
\(29\)	8257.74	−	1244.65i	1.82333	−	0.274823i	0.853947	−	0.520360i	\(-0.174202\pi\)
							0.969387	+	0.245536i	\(0.0789640\pi\)
\(31\)	−3376.13	+	8602.24i	−0.630979	+	1.60771i	0.153918	+	0.988084i	\(0.450811\pi\)
							−0.784898	+	0.619625i	\(0.787285\pi\)
\(37\)	−7700.94	−	13338.4i	−0.924782	−	1.60177i	−0.791912	−	0.610636i	\(-0.790914\pi\)
							−0.132870	−	0.991133i	\(-0.542419\pi\)
\(41\)	−3536.56	−	4434.70i	−0.328565	−	0.412007i	0.589921	−	0.807461i	\(-0.299159\pi\)
							−0.918486	+	0.395454i	\(0.870587\pi\)
\(43\)	−8795.47	+	8345.55i	−0.725417	+	0.688310i
							\(47\)	2559.75	−	11215.0i	0.169026	−	0.740549i	−0.817364	−	0.576122i	\(-0.804565\pi\)
0.986389	−	0.164427i	\(-0.0525775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.17	✓	204
43.24	even	21	inner	43.6.g.a.24.17	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.17	✓	204	1.1	even	1	trivial
43.6.g.a.24.17	yes	204	43.24	even	21	inner