Embedded newform 43.4.g.a.13.6

• Label: 43.4.g.a.13.6
• Level: $43$
• Weight: $4$
• Character: 43.13
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			13.6
Character	\(\chi\)	\(=\)	43.13
Dual form			43.4.g.a.10.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.113555 - 0.497516i) q^{2} +(4.85972 - 4.50917i) q^{3} +(6.97312 + 3.35808i) q^{4} +(-8.91725 - 1.34406i) q^{5} +(-1.69154 - 2.92983i) q^{6} +(2.21697 - 3.83991i) q^{7} +(5.00791 - 6.27973i) q^{8} +(1.26664 - 16.9021i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.113555	−	0.497516i	0.0401477	−	0.175898i	−0.950879	−	0.309563i	\(-0.899817\pi\)
							0.991027	+	0.133664i	\(0.0426744\pi\)
\(3\)	4.85972	−	4.50917i	0.935254	−	0.867789i	−0.0561765	−	0.998421i	\(-0.517891\pi\)
							0.991431	+	0.130632i	\(0.0417005\pi\)
\(5\)	−8.91725	−	1.34406i	−0.797584	−	0.120216i	−0.262408	−	0.964957i	\(-0.584517\pi\)
							−0.535176	+	0.844741i	\(0.679755\pi\)
\(7\)	2.21697	−	3.83991i	0.119705	−	0.207335i	−0.799946	−	0.600072i	\(-0.795138\pi\)
							0.919651	+	0.392737i	\(0.128472\pi\)
\(11\)	5.96279	−	2.87153i	0.163441	−	0.0787089i	−0.350375	−	0.936610i	\(-0.613946\pi\)
							0.513816	+	0.857901i	\(0.328232\pi\)
\(13\)	4.94815	+	12.6077i	0.105567	+	0.268980i	0.973891	−	0.227018i	\(-0.0728975\pi\)
							−0.868324	+	0.495998i	\(0.834802\pi\)
\(17\)	−125.638	+	18.9368i	−1.79245	+	0.270168i	−0.959199	−	0.282732i	\(-0.908759\pi\)
							−0.833248	+	0.552900i	\(0.813521\pi\)
\(19\)	6.32317	+	84.3768i	0.0763492	+	1.01881i	0.895451	+	0.445161i	\(0.146853\pi\)
							−0.819102	+	0.573648i	\(0.805528\pi\)
\(23\)	10.5279	+	7.17783i	0.0954446	+	0.0650731i	0.610102	−	0.792323i	\(-0.291128\pi\)
							−0.514657	+	0.857396i	\(0.672081\pi\)
\(29\)	60.8880	+	56.4958i	0.389884	+	0.361759i	0.850623	−	0.525777i	\(-0.176225\pi\)
							−0.460739	+	0.887536i	\(0.652416\pi\)
\(31\)	62.7379	−	19.3521i	0.363486	−	0.112121i	−0.107632	−	0.994191i	\(-0.534327\pi\)
							0.471118	+	0.882070i	\(0.343851\pi\)
\(37\)	−130.912	−	226.746i	−0.581669	−	1.00748i	−0.995282	−	0.0970272i	\(-0.969067\pi\)
							0.413613	−	0.910453i	\(-0.364267\pi\)
\(41\)	54.5075	−	238.813i	0.207625	−	0.909666i	−0.758516	−	0.651654i	\(-0.774075\pi\)
							0.966142	−	0.258012i	\(-0.0830674\pi\)
\(43\)	211.051	−	186.988i	0.748486	−	0.663150i
							\(47\)	−513.729	−	247.399i	−1.59436	−	0.767805i	−0.595009	−	0.803719i	\(-0.702852\pi\)
−0.999355	+	0.0359132i	\(0.988566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.13.6	yes	120
43.10	even	21	inner	43.4.g.a.10.6	✓	120
43.15	even	21		1849.4.a.l.1.27		60
43.28	odd	42		1849.4.a.k.1.34		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.10.6	✓	120	43.10	even	21	inner
43.4.g.a.13.6	yes	120	1.1	even	1	trivial
1849.4.a.k.1.34		60	43.28	odd	42
1849.4.a.l.1.27		60	43.15	even	21