Embedded newform 43.6.g.a.9.5

• Label: 43.6.g.a.9.5
• 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			9.5
Character	\(\chi\)	\(=\)	43.9
Dual form			43.6.g.a.24.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.60873 - 4.52520i) q^{2} +(-0.0525050 - 0.00791386i) q^{3} +(-0.333852 + 1.46270i) q^{4} +(-19.3143 - 13.1682i) q^{5} +(0.153664 + 0.266155i) q^{6} +(53.2202 - 92.1801i) q^{7} +(-159.048 + 76.5937i) q^{8} +(-232.201 - 71.6247i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.60873	−	4.52520i	−0.637939	−	0.799950i	0.352805	−	0.935697i	\(-0.385228\pi\)
							−0.990743	+	0.135747i	\(0.956656\pi\)
\(3\)	−0.0525050	−	0.00791386i	−0.00336820	−	0.000507674i	0.147358	−	0.989083i	\(-0.452923\pi\)
							−0.150726	+	0.988576i	\(0.548161\pi\)
\(5\)	−19.3143	−	13.1682i	−0.345504	−	0.235561i	0.378117	−	0.925758i	\(-0.376572\pi\)
							−0.723622	+	0.690197i	\(0.757524\pi\)
\(7\)	53.2202	−	92.1801i	0.410517	−	0.711037i	−0.584429	−	0.811445i	\(-0.698681\pi\)
							0.994946	+	0.100408i	\(0.0320148\pi\)
\(11\)	72.6939	+	318.493i	0.181141	+	0.793629i	0.981089	+	0.193560i	\(0.0620033\pi\)
							−0.799948	+	0.600070i	\(0.795140\pi\)
\(13\)	−28.9290	+	386.031i	−0.0474762	+	0.633525i	0.921818	+	0.387622i	\(0.126704\pi\)
							−0.969294	+	0.245903i	\(0.920916\pi\)
\(17\)	−1252.61	+	854.014i	−1.05122	+	0.716709i	−0.960249	−	0.279145i	\(-0.909949\pi\)
							−0.0909700	+	0.995854i	\(0.528997\pi\)
\(19\)	−1795.13	+	553.723i	−1.14080	+	0.351891i	−0.806884	−	0.590710i	\(-0.798848\pi\)
							−0.333920	+	0.942601i	\(0.608372\pi\)
\(23\)	1876.56	−	1741.19i	0.739676	−	0.686319i	−0.217424	−	0.976077i	\(-0.569765\pi\)
							0.957100	+	0.289758i	\(0.0935748\pi\)
\(29\)	−3584.75	+	540.314i	−0.791524	+	0.119303i	−0.532346	−	0.846527i	\(-0.678690\pi\)
							−0.259178	+	0.965830i	\(0.583452\pi\)
\(31\)	3638.18	−	9269.93i	0.679954	−	1.73250i	−0.000817346	−	1.00000i	\(-0.500260\pi\)
							0.680772	−	0.732496i	\(-0.261645\pi\)
\(37\)	−5995.89	−	10385.2i	−0.720028	−	1.24713i	−0.960988	−	0.276590i	\(-0.910796\pi\)
							0.240960	−	0.970535i	\(-0.422538\pi\)
\(41\)	3849.10	+	4826.61i	0.357601	+	0.448418i	0.927794	−	0.373093i	\(-0.121703\pi\)
							−0.570193	+	0.821511i	\(0.693131\pi\)
\(43\)	−8519.04	+	8627.54i	−0.702618	+	0.711567i
							\(47\)	−232.404	+	1018.23i	−0.0153461	+	0.0672359i	−0.982021	−	0.188771i	\(-0.939550\pi\)
0.966675	+	0.256007i	\(0.0824069\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.9.5	✓	204
43.24	even	21	inner	43.6.g.a.24.5	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.9.5	✓	204	1.1	even	1	trivial
43.6.g.a.24.5	yes	204	43.24	even	21	inner