Embedded newform 43.5.h.a.5.4

• Label: 43.5.h.a.5.4
• Level: $43$
• Weight: $5$
• Character: 43.5
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	43.5
Dual form			43.5.h.a.26.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.91193 - 0.892874i) q^{2} +(-5.06803 + 16.4302i) q^{3} +(0.0905063 + 0.0435855i) q^{4} +(-31.2456 + 12.2630i) q^{5} +(34.4959 - 59.7486i) q^{6} +(70.0200 - 40.4261i) q^{7} +(49.8789 + 39.7771i) q^{8} +(-177.340 - 120.908i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.91193	−	0.892874i	−0.977984	−	0.223218i	−0.296485	−	0.955038i	\(-0.595814\pi\)
							−0.681499	+	0.731819i	\(0.738672\pi\)
\(3\)	−5.06803	+	16.4302i	−0.563115	+	1.82557i	0.00118053	+	0.999999i	\(0.499624\pi\)
							−0.564295	+	0.825573i	\(0.690852\pi\)
\(5\)	−31.2456	+	12.2630i	−1.24982	+	0.490520i	−0.895692	−	0.444676i	\(-0.853319\pi\)
							−0.354132	+	0.935195i	\(0.615224\pi\)
\(7\)	70.0200	−	40.4261i	1.42898	−	0.825022i	0.431939	−	0.901903i	\(-0.357830\pi\)
							0.997040	+	0.0768812i	\(0.0244962\pi\)
\(11\)	−25.6940	+	12.3736i	−0.212347	+	0.102261i	−0.537034	−	0.843561i	\(-0.680455\pi\)
							0.324687	+	0.945821i	\(0.394741\pi\)
\(13\)	−36.1794	−	5.45317i	−0.214080	−	0.0322673i	0.0411267	−	0.999154i	\(-0.486905\pi\)
							−0.255206	+	0.966887i	\(0.582143\pi\)
\(17\)	−36.6855	+	93.4731i	−0.126939	+	0.323436i	−0.980188	−	0.198067i	\(-0.936534\pi\)
							0.853249	+	0.521504i	\(0.174629\pi\)
\(19\)	−98.1119	−	143.904i	−0.271778	−	0.398625i	0.665949	−	0.745997i	\(-0.268027\pi\)
							−0.937728	+	0.347372i	\(0.887074\pi\)
\(23\)	10.9831	−	146.559i	0.0207619	−	0.277049i	−0.977155	−	0.212530i	\(-0.931830\pi\)
							0.997916	−	0.0645187i	\(-0.0205512\pi\)
\(29\)	−46.2674	−	149.995i	−0.0550148	−	0.178354i	0.923895	−	0.382646i	\(-0.124987\pi\)
							−0.978910	+	0.204292i	\(0.934511\pi\)
\(31\)	−553.725	−	513.782i	−0.576197	−	0.534632i	0.337343	−	0.941382i	\(-0.390472\pi\)
							−0.913540	+	0.406749i	\(0.866662\pi\)
\(37\)	−1925.04	−	1111.42i	−1.40617	−	0.811850i	−0.411150	−	0.911568i	\(-0.634873\pi\)
							−0.995016	+	0.0997172i	\(0.968206\pi\)
\(41\)	191.337	−	838.301i	0.113823	−	0.498692i	−0.885591	−	0.464466i	\(-0.846246\pi\)
							0.999414	−	0.0342257i	\(-0.0108965\pi\)
\(43\)	63.1739	+	1847.92i	0.0341665	+	0.999416i
							\(47\)	−1164.18	−	560.638i	−0.527015	−	0.253797i	0.151404	−	0.988472i	\(-0.451620\pi\)
−0.678419	+	0.734675i	\(0.737335\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.5.4	✓	168
43.26	odd	42	inner	43.5.h.a.26.4	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.5.4	✓	168	1.1	even	1	trivial
43.5.h.a.26.4	yes	168	43.26	odd	42	inner