Embedded newform 43.8.g.a.10.23

• Label: 43.8.g.a.10.23
• Level: $43$
• Weight: $8$
• Character: 43.10
• 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			10.23
Character	\(\chi\)	\(=\)	43.10
Dual form			43.8.g.a.13.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.07723 + 17.8635i) q^{2} +(-15.0227 - 13.9390i) q^{3} +(-187.157 + 90.1302i) q^{4} +(240.957 - 36.3184i) q^{5} +(187.749 - 325.191i) q^{6} +(-782.008 - 1354.48i) q^{7} +(-910.836 - 1142.15i) q^{8} +(-132.050 - 1762.08i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 300 q - 40 q^{2} - 12 q^{3} - 3028 q^{4} + 235 q^{5} + 1274 q^{6} - 4122 q^{7} + 762 q^{8} + 8097 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.07723	+	17.8635i	0.360380	+	1.57893i	0.752232	+	0.658898i	\(0.228977\pi\)
							−0.391852	+	0.920028i	\(0.628166\pi\)
\(3\)	−15.0227	−	13.9390i	−0.321235	−	0.298063i	0.503027	−	0.864271i	\(-0.332220\pi\)
							−0.824263	+	0.566208i	\(0.808410\pi\)
\(5\)	240.957	−	36.3184i	0.862074	−	0.129937i	0.296900	−	0.954909i	\(-0.404047\pi\)
							0.565174	+	0.824972i	\(0.308809\pi\)
\(7\)	−782.008	−	1354.48i	−0.861724	−	1.49255i	−0.870264	−	0.492586i	\(-0.836052\pi\)
							0.00853980	−	0.999964i	\(-0.497282\pi\)
\(11\)	362.159	+	174.407i	0.0820398	+	0.0395083i	0.474454	−	0.880280i	\(-0.342646\pi\)
							−0.392414	+	0.919789i	\(0.628360\pi\)
\(13\)	3172.84	−	8084.26i	0.400540	−	1.02056i	−0.578280	−	0.815838i	\(-0.696276\pi\)
							0.978820	−	0.204721i	\(-0.0656288\pi\)
\(17\)	−1640.39	−	247.249i	−0.0809798	−	0.0122057i	0.108427	−	0.994104i	\(-0.465419\pi\)
							−0.189407	+	0.981899i	\(0.560657\pi\)
\(19\)	−905.865	+	12087.9i	−0.0302988	+	0.404310i	0.961349	+	0.275333i	\(0.0887882\pi\)
							−0.991648	+	0.128976i	\(0.958831\pi\)
\(23\)	37403.0	−	25500.9i	0.641001	−	0.437027i	−0.198712	−	0.980058i	\(-0.563676\pi\)
							0.839713	+	0.543031i	\(0.182723\pi\)
\(29\)	62935.5	−	58395.6i	0.479184	−	0.444618i	−0.403225	−	0.915101i	\(-0.632111\pi\)
							0.882409	+	0.470483i	\(0.155920\pi\)
\(31\)	−88093.9	−	27173.4i	−0.531104	−	0.163824i	0.0175927	−	0.999845i	\(-0.494400\pi\)
							−0.548697	+	0.836021i	\(0.684876\pi\)
\(37\)	271217.	−	469761.i	0.880260	−	1.52465i	0.0292072	−	0.999573i	\(-0.490702\pi\)
							0.851052	−	0.525081i	\(-0.175965\pi\)
\(41\)	−161476.	−	707472.i	−0.365901	−	1.60312i	−0.737917	−	0.674892i	\(-0.764190\pi\)
							0.372015	−	0.928227i	\(-0.378667\pi\)
\(43\)	−16544.9	+	521100.i	−0.0317339	+	0.999496i
							\(47\)	778119.	−	374722.i	1.09321	−	0.526462i	0.201692	−	0.979449i	\(-0.435356\pi\)
0.891517	+	0.452987i	\(0.149642\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.g.a.10.23	✓	300
43.13	even	21	inner	43.8.g.a.13.23	yes	300

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.g.a.10.23	✓	300	1.1	even	1	trivial
43.8.g.a.13.23	yes	300	43.13	even	21	inner