Embedded newform 43.4.e.a.21.3

• Label: 43.4.e.a.21.3
• Level: $43$
• Weight: $4$
• Character: 43.21
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			21.3
Character	\(\chi\)	\(=\)	43.21
Dual form			43.4.e.a.41.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.24902 - 1.56465i) q^{2} +(0.731791 - 0.352412i) q^{3} +(3.12011 + 3.91250i) q^{4} +(-1.33434 - 5.84613i) q^{5} -2.92901 q^{6} -26.3629 q^{7} +(2.40390 + 10.5322i) q^{8} +(-16.4229 + 20.5937i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.24902	−	1.56465i	−1.14870	−	0.553186i	−0.240059	−	0.970758i	\(-0.577167\pi\)
							−0.908644	+	0.417572i	\(0.862881\pi\)
\(3\)	0.731791	−	0.352412i	0.140833	−	0.0678217i	−0.362140	−	0.932124i	\(-0.617954\pi\)
							0.502973	+	0.864302i	\(0.332239\pi\)
\(5\)	−1.33434	−	5.84613i	−0.119347	−	0.522894i	−0.998891	−	0.0470765i	\(-0.985010\pi\)
							0.879544	−	0.475817i	\(-0.157848\pi\)
\(7\)	−26.3629			−1.42346			−0.711732	−	0.702451i	\(-0.752089\pi\)
							−0.711732	+	0.702451i	\(0.752089\pi\)
\(11\)	−16.8926	+	21.1826i	−0.463028	+	0.580618i	−0.957448	−	0.288604i	\(-0.906809\pi\)
							0.494421	+	0.869223i	\(0.335380\pi\)
\(13\)	−18.5331	−	81.1987i	−0.395396	−	1.73234i	−0.645170	−	0.764039i	\(-0.723213\pi\)
							0.249774	−	0.968304i	\(-0.419644\pi\)
\(17\)	−4.30954	+	18.8813i	−0.0614833	+	0.269376i	−0.996321	−	0.0856991i	\(-0.972688\pi\)
							0.934838	+	0.355075i	\(0.115545\pi\)
\(19\)	−0.907522	−	1.13800i	−0.0109579	−	0.0137408i	0.776322	−	0.630336i	\(-0.217083\pi\)
							−0.787280	+	0.616595i	\(0.788511\pi\)
\(23\)	35.4249	−	44.4214i	0.321157	−	0.402718i	−0.594879	−	0.803816i	\(-0.702800\pi\)
							0.916035	+	0.401098i	\(0.131371\pi\)
\(29\)	−168.826	−	81.3023i	−1.08104	−	0.520602i	−0.193392	−	0.981122i	\(-0.561949\pi\)
							−0.887650	+	0.460519i	\(0.847663\pi\)
\(31\)	−59.3522	−	28.5825i	−0.343870	−	0.165599i	0.253975	−	0.967211i	\(-0.418262\pi\)
							−0.597845	+	0.801612i	\(0.703976\pi\)
\(37\)	−164.226			−0.729691			−0.364845	−	0.931068i	\(-0.618878\pi\)
							−0.364845	+	0.931068i	\(0.618878\pi\)
\(41\)	246.421	+	118.670i	0.938647	+	0.452029i	0.839692	−	0.543063i	\(-0.182736\pi\)
							0.0989553	+	0.995092i	\(0.468450\pi\)
\(43\)	−137.283	−	246.293i	−0.486873	−	0.873473i
							\(47\)	69.5301	+	87.1879i	0.215787	+	0.270589i	0.877930	−	0.478789i	\(-0.158924\pi\)
−0.662143	+	0.749378i	\(0.730353\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.21.3	✓	60
43.16	even	7		1849.4.a.h.1.24		30
43.27	odd	14		1849.4.a.g.1.7		30
43.41	even	7	inner	43.4.e.a.41.3	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.21.3	✓	60	1.1	even	1	trivial
43.4.e.a.41.3	yes	60	43.41	even	7	inner
1849.4.a.g.1.7		30	43.27	odd	14
1849.4.a.h.1.24		30	43.16	even	7