Embedded newform 43.5.f.a.8.4

• Label: 43.5.f.a.8.4
• Level: $43$
• Weight: $5$
• Character: 43.8
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $78$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(78\)
Relative dimension:	\(13\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			8.4
Character	\(\chi\)	\(=\)	43.8
Dual form			43.5.f.a.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.61843 - 1.05413i) q^{2} +(-9.66855 + 2.20678i) q^{3} +(5.80324 + 2.79469i) q^{4} +(-23.4239 - 18.6799i) q^{5} +46.9798 q^{6} +27.2594i q^{7} +(35.4032 + 28.2331i) q^{8} +(15.6326 - 7.52825i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.61843	−	1.05413i	−1.15461	−	0.263532i	−0.397991	−	0.917389i	\(-0.630292\pi\)
							−0.756618	+	0.653858i	\(0.773150\pi\)
\(3\)	−9.66855	+	2.20678i	−1.07428	+	0.245198i	−0.722843	−	0.691012i	\(-0.757165\pi\)
							−0.351440	+	0.936210i	\(0.614308\pi\)
\(5\)	−23.4239	−	18.6799i	−0.936955	−	0.747196i	0.0306860	−	0.999529i	\(-0.490231\pi\)
							−0.967641	+	0.252333i	\(0.918802\pi\)
\(7\)			27.2594i			0.556314i	0.960536	+	0.278157i	\(0.0897235\pi\)
							−0.960536	+	0.278157i	\(0.910276\pi\)
\(11\)	186.057	−	89.6003i	1.53766	−	0.740498i	0.542621	−	0.839978i	\(-0.317432\pi\)
							0.995040	+	0.0994796i	\(0.0317178\pi\)
\(13\)	−10.0727	+	12.6307i	−0.0596015	+	0.0747379i	−0.810739	−	0.585408i	\(-0.800934\pi\)
							0.751137	+	0.660146i	\(0.229506\pi\)
\(17\)	57.6743	+	72.3213i	0.199565	+	0.250247i	0.871537	−	0.490330i	\(-0.163124\pi\)
							−0.671972	+	0.740577i	\(0.734552\pi\)
\(19\)	−186.098	+	386.436i	−0.515506	+	1.07046i	0.467004	+	0.884255i	\(0.345333\pi\)
							−0.982511	+	0.186205i	\(0.940381\pi\)
\(23\)	−144.505	+	69.5899i	−0.273166	+	0.131550i	−0.565452	−	0.824781i	\(-0.691298\pi\)
							0.292286	+	0.956331i	\(0.405584\pi\)
\(29\)	1095.51	+	250.043i	1.30263	+	0.297316i	0.816893	−	0.576790i	\(-0.195695\pi\)
							0.485735	+	0.874106i	\(0.338552\pi\)
\(31\)	191.171	−	837.575i	0.198929	−	0.871566i	−0.772646	−	0.634837i	\(-0.781067\pi\)
							0.971576	−	0.236729i	\(-0.0760755\pi\)
\(37\)			2390.89i			1.74645i	0.487320	+	0.873224i	\(0.337975\pi\)
							−0.487320	+	0.873224i	\(0.662025\pi\)
\(41\)	199.569	−	874.370i	0.118720	−	0.520148i	−0.880239	−	0.474532i	\(-0.842617\pi\)
							0.998959	−	0.0456169i	\(-0.0145254\pi\)
\(43\)	1626.19	+	879.944i	0.879498	+	0.475903i
							\(47\)	1202.97	+	579.319i	0.544576	+	0.262254i	0.685883	−	0.727712i	\(-0.259416\pi\)
−0.141307	+	0.989966i	\(0.545130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.8.4	✓	78
43.27	odd	14	inner	43.5.f.a.27.4	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.8.4	✓	78	1.1	even	1	trivial
43.5.f.a.27.4	yes	78	43.27	odd	14	inner