Embedded newform 43.6.g.a.9.3

• Label: 43.6.g.a.9.3
• 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			9.3
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.47096 - 6.86037i) q^{2} +(-18.4151 - 2.77564i) q^{3} +(-10.0126 + 43.8679i) q^{4} +(64.4216 + 43.9219i) q^{5} +(81.7066 + 141.520i) q^{6} +(8.11387 - 14.0536i) q^{7} +(102.744 - 49.4787i) q^{8} +(99.2091 + 30.6020i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.47096	−	6.86037i	−0.967138	−	1.21275i	−0.977095	−	0.212803i	\(-0.931741\pi\)
							0.00995697	−	0.999950i	\(-0.496831\pi\)
\(3\)	−18.4151	−	2.77564i	−1.18133	−	0.178057i	−0.471136	−	0.882061i	\(-0.656156\pi\)
							−0.710196	+	0.704004i	\(0.751394\pi\)
\(5\)	64.4216	+	43.9219i	1.15241	+	0.785699i	0.979936	−	0.199315i	\(-0.0638716\pi\)
							0.172474	+	0.985014i	\(0.444824\pi\)
\(7\)	8.11387	−	14.0536i	0.0625868	−	0.108404i	−0.833034	−	0.553222i	\(-0.813398\pi\)
							0.895621	+	0.444818i	\(0.146732\pi\)
\(11\)	6.37310	+	27.9224i	0.0158807	+	0.0695778i	0.982247	−	0.187590i	\(-0.0600677\pi\)
							−0.966367	+	0.257168i	\(0.917211\pi\)
\(13\)	−2.03567	+	27.1642i	−0.00334079	+	0.0445798i	−0.998625	−	0.0524215i	\(-0.983306\pi\)
							0.995284	+	0.0970013i	\(0.0309251\pi\)
\(17\)	−9.20043	+	6.27275i	−0.00772122	+	0.00526424i	−0.567175	−	0.823598i	\(-0.691964\pi\)
							0.559453	+	0.828862i	\(0.311011\pi\)
\(19\)	2359.40	−	727.779i	1.49940	−	0.462504i	0.566936	−	0.823762i	\(-0.308129\pi\)
							0.932466	+	0.361258i	\(0.117653\pi\)
\(23\)	3242.52	−	3008.62i	1.27810	−	1.18590i	0.305814	−	0.952091i	\(-0.401071\pi\)
							0.972283	−	0.233809i	\(-0.0751190\pi\)
\(29\)	−3109.96	+	468.751i	−0.686689	+	0.103502i	−0.483114	−	0.875558i	\(-0.660494\pi\)
							−0.203575	+	0.979059i	\(0.565256\pi\)
\(31\)	−1819.66	+	4636.42i	−0.340084	+	0.866520i	0.653858	+	0.756617i	\(0.273149\pi\)
							−0.993942	+	0.109903i	\(0.964946\pi\)
\(37\)	3028.16	+	5244.92i	0.363642	+	0.629847i	0.988557	−	0.150846i	\(-0.0481998\pi\)
							−0.624915	+	0.780693i	\(0.714866\pi\)
\(41\)	7226.30	+	9061.49i	0.671361	+	0.841860i	0.994527	−	0.104482i	\(-0.0333185\pi\)
							−0.323166	+	0.946342i	\(0.604747\pi\)
\(43\)	10561.7	−	5954.77i	0.871088	−	0.491127i
							\(47\)	5154.19	−	22582.0i	0.340342	−	1.49114i	−0.458011	−	0.888946i	\(-0.651438\pi\)
0.798353	−	0.602189i	\(-0.205705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.3	✓	204
43.24	even	21	inner	43.6.g.a.24.3	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.3	✓	204	1.1	even	1	trivial
43.6.g.a.24.3	yes	204	43.24	even	21	inner