Embedded newform 43.4.e.a.4.1

• Label: 43.4.e.a.4.1
• Level: $43$
• Weight: $4$
• Character: 43.4
• 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			4.1
Character	\(\chi\)	\(=\)	43.4
Dual form			43.4.e.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.08835 + 3.87267i) q^{2} +(-0.198541 - 0.248963i) q^{3} +(-3.67950 - 16.1209i) q^{4} +(-11.1889 - 5.38829i) q^{5} +1.57731 q^{6} +3.53619 q^{7} +(38.0923 + 18.3443i) q^{8} +(5.98550 - 26.2242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.08835	+	3.87267i	−1.09190	+	1.36920i	−0.168344	+	0.985728i	\(0.553842\pi\)
							−0.923554	+	0.383468i	\(0.874730\pi\)
\(3\)	−0.198541	−	0.248963i	−0.0382092	−	0.0479129i	0.762360	−	0.647153i	\(-0.224041\pi\)
							−0.800569	+	0.599241i	\(0.795469\pi\)
\(5\)	−11.1889	−	5.38829i	−1.00077	−	0.481944i	−0.139568	−	0.990213i	\(-0.544571\pi\)
							−0.861198	+	0.508269i	\(0.830286\pi\)
\(7\)	3.53619			0.190936			0.0954681	−	0.995432i	\(-0.469565\pi\)
							0.0954681	+	0.995432i	\(0.469565\pi\)
\(11\)	−2.16594	+	9.48961i	−0.0593687	+	0.260111i	−0.995899	−	0.0904764i	\(-0.971161\pi\)
							0.936530	+	0.350588i	\(0.114018\pi\)
\(13\)	−52.4796	−	25.2729i	−1.11963	−	0.539187i	−0.219854	−	0.975533i	\(-0.570558\pi\)
							−0.899779	+	0.436346i	\(0.856272\pi\)
\(17\)	−86.9732	+	41.8841i	−1.24083	+	0.597552i	−0.935038	−	0.354547i	\(-0.884635\pi\)
							−0.305791	+	0.952099i	\(0.598921\pi\)
\(19\)	−29.0416	−	127.240i	−0.350663	−	1.53636i	−0.775654	−	0.631158i	\(-0.782580\pi\)
							0.424991	−	0.905198i	\(-0.360277\pi\)
\(23\)	−19.1809	+	84.0371i	−0.173891	+	0.761867i	0.810481	+	0.585765i	\(0.199206\pi\)
							−0.984372	+	0.176102i	\(0.943651\pi\)
\(29\)	48.6360	−	60.9876i	0.311430	−	0.390521i	−0.601341	−	0.798993i	\(-0.705367\pi\)
							0.912771	+	0.408471i	\(0.133938\pi\)
\(31\)	−64.7237	+	81.1609i	−0.374991	+	0.470224i	−0.933138	−	0.359518i	\(-0.882941\pi\)
							0.558147	+	0.829742i	\(0.311512\pi\)
\(37\)	433.942			1.92810			0.964050	−	0.265721i	\(-0.0856100\pi\)
							0.964050	+	0.265721i	\(0.0856100\pi\)
\(41\)	−165.335	+	207.323i	−0.629778	+	0.789717i	−0.989683	−	0.143272i	\(-0.954238\pi\)
							0.359905	+	0.932989i	\(0.382809\pi\)
\(43\)	41.8303	+	278.850i	0.148350	+	0.988935i
							\(47\)	−114.782	−	502.893i	−0.356227	−	1.56073i	−0.762504	−	0.646984i	\(-0.776030\pi\)
0.406277	−	0.913750i	\(-0.366827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.4.1	✓	60
43.11	even	7	inner	43.4.e.a.11.1	yes	60
43.21	even	7		1849.4.a.h.1.2		30
43.22	odd	14		1849.4.a.g.1.29		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.4.1	✓	60	1.1	even	1	trivial
43.4.e.a.11.1	yes	60	43.11	even	7	inner
1849.4.a.g.1.29		30	43.22	odd	14
1849.4.a.h.1.2		30	43.21	even	7