Embedded newform 43.4.e.a.41.3

• Label: 43.4.e.a.41.3
• Level: $43$
• Weight: $4$
• Character: 43.41
• 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			41.3
Character	\(\chi\)	\(=\)	43.41
Dual form			43.4.e.a.21.3

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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	−0.908644	−	0.417572i	\(-0.862881\pi\)
							−0.240059	+	0.970758i	\(0.577167\pi\)
\(3\)	0.731791	+	0.352412i	0.140833	+	0.0678217i	0.502973	−	0.864302i	\(-0.332239\pi\)
							−0.362140	+	0.932124i	\(0.617954\pi\)
\(5\)	−1.33434	+	5.84613i	−0.119347	+	0.522894i	0.879544	+	0.475817i	\(0.157848\pi\)
							−0.998891	+	0.0470765i	\(0.985010\pi\)
\(7\)	−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	0.494421	−	0.869223i	\(-0.335380\pi\)
							−0.957448	+	0.288604i	\(0.906809\pi\)
\(13\)	−18.5331	+	81.1987i	−0.395396	+	1.73234i	0.249774	+	0.968304i	\(0.419644\pi\)
							−0.645170	+	0.764039i	\(0.723213\pi\)
\(17\)	−4.30954	−	18.8813i	−0.0614833	−	0.269376i	0.934838	−	0.355075i	\(-0.115545\pi\)
							−0.996321	+	0.0856991i	\(0.972688\pi\)
\(19\)	−0.907522	+	1.13800i	−0.0109579	+	0.0137408i	−0.787280	−	0.616595i	\(-0.788511\pi\)
							0.776322	+	0.630336i	\(0.217083\pi\)
\(23\)	35.4249	+	44.4214i	0.321157	+	0.402718i	0.916035	−	0.401098i	\(-0.131371\pi\)
							−0.594879	+	0.803816i	\(0.702800\pi\)
\(29\)	−168.826	+	81.3023i	−1.08104	+	0.520602i	−0.887650	−	0.460519i	\(-0.847663\pi\)
							−0.193392	+	0.981122i	\(0.561949\pi\)
\(31\)	−59.3522	+	28.5825i	−0.343870	+	0.165599i	−0.597845	−	0.801612i	\(-0.703976\pi\)
							0.253975	+	0.967211i	\(0.418262\pi\)
\(37\)	−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	0.0989553	−	0.995092i	\(-0.468450\pi\)
							0.839692	+	0.543063i	\(0.182736\pi\)
\(43\)	−137.283	+	246.293i	−0.486873	+	0.873473i
							\(47\)	69.5301	−	87.1879i	0.215787	−	0.270589i	−0.662143	−	0.749378i	\(-0.730353\pi\)
0.877930	+	0.478789i	\(0.158924\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.21.3	✓	60	43.21	even	7	inner
43.4.e.a.41.3	yes	60	1.1	even	1	trivial
1849.4.a.g.1.7		30	43.8	odd	14
1849.4.a.h.1.24		30	43.35	even	7