Embedded newform 43.4.e.a.16.1

• Label: 43.4.e.a.16.1
• Level: $43$
• Weight: $4$
• Character: 43.16
• 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			16.1
Character	\(\chi\)	\(=\)	43.16
Dual form			43.4.e.a.35.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12822 - 4.94304i) q^{2} +(1.17167 - 5.13342i) q^{3} +(-15.9531 + 7.68259i) q^{4} +(-2.13250 - 2.67407i) q^{5} -26.6966 q^{6} +25.1802 q^{7} +(30.6843 + 38.4769i) q^{8} +(-0.653062 - 0.314498i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12822	−	4.94304i	−0.398885	−	1.74763i	−0.631799	−	0.775133i	\(-0.717683\pi\)
							0.232913	−	0.972497i	\(-0.425174\pi\)
\(3\)	1.17167	−	5.13342i	0.225488	−	0.987928i	−0.727782	−	0.685808i	\(-0.759449\pi\)
							0.953270	−	0.302119i	\(-0.0976941\pi\)
\(5\)	−2.13250	−	2.67407i	−0.190737	−	0.239176i	0.677263	−	0.735741i	\(-0.263166\pi\)
							−0.868000	+	0.496565i	\(0.834595\pi\)
\(7\)	25.1802			1.35960			0.679801	−	0.733397i	\(-0.262066\pi\)
							0.679801	+	0.733397i	\(0.262066\pi\)
\(11\)	−62.7822	−	30.2343i	−1.72087	−	0.828726i	−0.989113	−	0.147157i	\(-0.952988\pi\)
							−0.731754	−	0.681569i	\(-0.761298\pi\)
\(13\)	16.5220	+	20.7179i	0.352490	+	0.442008i	0.926190	−	0.377057i	\(-0.123064\pi\)
							−0.573700	+	0.819065i	\(0.694493\pi\)
\(17\)	54.8108	−	68.7305i	0.781974	−	0.980565i	−0.218015	−	0.975945i	\(-0.569958\pi\)
							0.999989	−	0.00461939i	\(-0.00147040\pi\)
\(19\)	−14.0614	+	6.77162i	−0.169785	+	0.0817640i	−0.516846	−	0.856079i	\(-0.672894\pi\)
							0.347061	+	0.937843i	\(0.387180\pi\)
\(23\)	0.387846	+	0.186777i	0.00351615	+	0.00169329i	0.435641	−	0.900121i	\(-0.356522\pi\)
							−0.432125	+	0.901814i	\(0.642236\pi\)
\(29\)	15.3481	+	67.2446i	0.0982786	+	0.430587i	0.999998	−	0.00173408i	\(-0.000551975\pi\)
							−0.901720	+	0.432321i	\(0.857695\pi\)
\(31\)	50.7834	+	222.497i	0.294225	+	1.28908i	0.878584	+	0.477588i	\(0.158489\pi\)
							−0.584359	+	0.811495i	\(0.698654\pi\)
\(37\)	276.114			1.22683			0.613417	−	0.789759i	\(-0.289794\pi\)
							0.613417	+	0.789759i	\(0.289794\pi\)
\(41\)	84.5473	+	370.426i	0.322051	+	1.41100i	0.833897	+	0.551919i	\(0.186104\pi\)
							−0.511847	+	0.859077i	\(0.671038\pi\)
\(43\)	−98.5697	−	264.180i	−0.349575	−	0.936908i
							\(47\)	53.3436	−	25.6889i	0.165552	−	0.0797259i	−0.349272	−	0.937021i	\(-0.613571\pi\)
0.514825	+	0.857295i	\(0.327857\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.16.1	✓	60
43.11	even	7		1849.4.a.h.1.29		30
43.32	odd	14		1849.4.a.g.1.2		30
43.35	even	7	inner	43.4.e.a.35.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.16.1	✓	60	1.1	even	1	trivial
43.4.e.a.35.1	yes	60	43.35	even	7	inner
1849.4.a.g.1.2		30	43.32	odd	14
1849.4.a.h.1.29		30	43.11	even	7