Embedded newform 43.4.e.a.21.2

• Label: 43.4.e.a.21.2
• 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			21.2
Character	\(\chi\)	\(=\)	43.21
Dual form			43.4.e.a.41.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.96035 - 1.90720i) q^{2} +(-7.07654 + 3.40788i) q^{3} +(7.05904 + 8.85175i) q^{4} +(-1.30223 - 5.70543i) q^{5} +34.5251 q^{6} +12.3980 q^{7} +(-3.24916 - 14.2355i) q^{8} +(21.6295 - 27.1225i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 3 q^{3} - 31 q^{4} - 23 q^{5} + 16 q^{6} + 96 q^{7} + 61 q^{8} - 177 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.96035	−	1.90720i	−1.40020	−	0.674299i	−0.426995	−	0.904254i	\(-0.640428\pi\)
							−0.973201	+	0.229955i	\(0.926142\pi\)
\(3\)	−7.07654	+	3.40788i	−1.36188	+	0.655847i	−0.965055	−	0.262047i	\(-0.915602\pi\)
							−0.396825	+	0.917894i	\(0.629888\pi\)
\(5\)	−1.30223	−	5.70543i	−0.116475	−	0.510309i	−0.999184	−	0.0403897i	\(-0.987140\pi\)
							0.882709	−	0.469920i	\(-0.155717\pi\)
\(7\)	12.3980			0.669428			0.334714	−	0.942320i	\(-0.391360\pi\)
							0.334714	+	0.942320i	\(0.391360\pi\)
\(11\)	31.7363	−	39.7961i	0.869896	−	1.09081i	−0.125223	−	0.992129i	\(-0.539965\pi\)
							0.995119	−	0.0986860i	\(-0.0314640\pi\)
\(13\)	16.1060	+	70.5649i	0.343615	+	1.50548i	0.791380	+	0.611325i	\(0.209363\pi\)
							−0.447765	+	0.894152i	\(0.647780\pi\)
\(17\)	−16.0478	+	70.3098i	−0.228950	+	1.00310i	0.721548	+	0.692365i	\(0.243431\pi\)
							−0.950498	+	0.310731i	\(0.899426\pi\)
\(19\)	29.7720	+	37.3328i	0.359482	+	0.450776i	0.928380	−	0.371632i	\(-0.121202\pi\)
							−0.568898	+	0.822408i	\(0.692630\pi\)
\(23\)	121.240	−	152.030i	1.09914	−	1.37828i	0.180321	−	0.983608i	\(-0.442286\pi\)
							0.918822	−	0.394673i	\(-0.129142\pi\)
\(29\)	−103.224	−	49.7102i	−0.660975	−	0.318309i	0.0731524	−	0.997321i	\(-0.476694\pi\)
							−0.734127	+	0.679012i	\(0.762408\pi\)
\(31\)	97.5752	+	46.9897i	0.565323	+	0.272245i	0.694639	−	0.719358i	\(-0.255564\pi\)
							−0.129316	+	0.991603i	\(0.541278\pi\)
\(37\)	19.4210			0.0862918			0.0431459	−	0.999069i	\(-0.486262\pi\)
							0.0431459	+	0.999069i	\(0.486262\pi\)
\(41\)	437.142	+	210.516i	1.66512	+	0.801881i	0.998398	+	0.0565775i	\(0.0180188\pi\)
							0.666725	+	0.745304i	\(0.267695\pi\)
\(43\)	166.643	+	227.458i	0.590996	+	0.806674i
							\(47\)	−12.6269	−	15.8337i	−0.0391878	−	0.0491399i	0.761851	−	0.647752i	\(-0.224291\pi\)
−0.801039	+	0.598612i	\(0.795719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.e.a.21.2	✓	60
43.16	even	7		1849.4.a.h.1.26		30
43.27	odd	14		1849.4.a.g.1.5		30
43.41	even	7	inner	43.4.e.a.41.2	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.e.a.21.2	✓	60	1.1	even	1	trivial
43.4.e.a.41.2	yes	60	43.41	even	7	inner
1849.4.a.g.1.5		30	43.27	odd	14
1849.4.a.h.1.26		30	43.16	even	7