Embedded newform 43.6.g.a.10.4

• Label: 43.6.g.a.10.4
• Level: $43$
• Weight: $6$
• Character: 43.10
• 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			10.4
Character	\(\chi\)	\(=\)	43.10
Dual form			43.6.g.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.66421 - 7.29139i) q^{2} +(-15.5896 - 14.4651i) q^{3} +(-21.5637 + 10.3845i) q^{4} +(-29.9883 + 4.52001i) q^{5} +(-79.5260 + 137.743i) q^{6} +(-23.8750 - 41.3527i) q^{7} +(-37.6123 - 47.1643i) q^{8} +(15.6392 + 208.690i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 204 q - 468 q^{4} - 85 q^{5} - 22 q^{6} + 454 q^{7} + 698 q^{8} - 333 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.66421	−	7.29139i	−0.294194	−	1.28895i	−0.878628	−	0.477506i	\(-0.841541\pi\)
							0.584435	−	0.811441i	\(-0.301316\pi\)
\(3\)	−15.5896	−	14.4651i	−1.00008	−	0.927935i	−0.00267613	−	0.999996i	\(-0.500852\pi\)
							−0.997400	+	0.0720612i	\(0.977042\pi\)
\(5\)	−29.9883	+	4.52001i	−0.536447	+	0.0808564i	−0.411678	−	0.911329i	\(-0.635057\pi\)
							−0.124769	+	0.992186i	\(0.539819\pi\)
\(7\)	−23.8750	−	41.3527i	−0.184161	−	0.318976i	0.759132	−	0.650936i	\(-0.225623\pi\)
							−0.943293	+	0.331960i	\(0.892290\pi\)
\(11\)	456.305	+	219.745i	1.13703	+	0.547567i	0.905114	−	0.425168i	\(-0.139785\pi\)
							0.231919	+	0.972735i	\(0.425499\pi\)
\(13\)	146.712	−	373.816i	0.240773	−	0.613478i	−0.758498	−	0.651676i	\(-0.774066\pi\)
							0.999270	+	0.0381974i	\(0.0121616\pi\)
\(17\)	−229.910	−	34.6534i	−0.192946	−	0.0290819i	0.0518583	−	0.998654i	\(-0.483486\pi\)
							−0.244804	+	0.969573i	\(0.578724\pi\)
\(19\)	−4.64664	+	62.0051i	−0.00295294	+	0.0394043i	−0.998486	−	0.0550134i	\(-0.982480\pi\)
							0.995533	+	0.0944177i	\(0.0300989\pi\)
\(23\)	−1901.68	+	1296.55i	−0.749581	+	0.511055i	−0.876821	−	0.480816i	\(-0.840340\pi\)
							0.127241	+	0.991872i	\(0.459388\pi\)
\(29\)	−56.0053	+	51.9653i	−0.0123661	+	0.0114741i	−0.686332	−	0.727289i	\(-0.740780\pi\)
							0.673965	+	0.738763i	\(0.264590\pi\)
\(31\)	−5821.87	−	1795.81i	−1.08807	−	0.335626i	−0.301716	−	0.953398i	\(-0.597559\pi\)
							−0.786357	+	0.617772i	\(0.788036\pi\)
\(37\)	5183.75	−	8978.51i	0.622500	−	1.07820i	−0.366518	−	0.930411i	\(-0.619450\pi\)
							0.989019	−	0.147791i	\(-0.0472163\pi\)
\(41\)	1840.46	+	8063.60i	0.170989	+	0.749151i	0.985593	+	0.169134i	\(0.0540970\pi\)
							−0.814604	+	0.580017i	\(0.803046\pi\)
\(43\)	−11782.1	−	2861.99i	−0.971742	−	0.236046i
							\(47\)	2122.90	−	1022.34i	0.140180	−	0.0675070i	−0.362479	−	0.931992i	\(-0.618070\pi\)
0.502659	+	0.864485i	\(0.332355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.g.a.10.4	✓	204
43.13	even	21	inner	43.6.g.a.13.4	yes	204

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.g.a.10.4	✓	204	1.1	even	1	trivial
43.6.g.a.13.4	yes	204	43.13	even	21	inner