Embedded newform 43.5.f.a.8.5

• Label: 43.5.f.a.8.5
• 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			8.5
Character	\(\chi\)	\(=\)	43.8
Dual form			43.5.f.a.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.98221 - 0.680671i) q^{2} +(8.81268 - 2.01144i) q^{3} +(-5.98522 - 2.88233i) q^{4} +(26.0327 + 20.7604i) q^{5} -27.6504 q^{6} -60.4753i q^{7} +(54.1521 + 43.1848i) q^{8} +(0.638992 - 0.307722i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.98221	−	0.680671i	−0.745553	−	0.170168i	−0.167165	−	0.985929i	\(-0.553461\pi\)
							−0.578389	+	0.815761i	\(0.696318\pi\)
\(3\)	8.81268	−	2.01144i	0.979187	−	0.223493i	0.297166	−	0.954826i	\(-0.403959\pi\)
							0.682021	+	0.731333i	\(0.261101\pi\)
\(5\)	26.0327	+	20.7604i	1.04131	+	0.830415i	0.985780	−	0.168042i	\(-0.0537444\pi\)
							0.0555280	+	0.998457i	\(0.482316\pi\)
\(7\)		−	60.4753i		−	1.23419i	−0.786889	−	0.617095i	\(-0.788310\pi\)
							0.786889	−	0.617095i	\(-0.211690\pi\)
\(11\)	143.454	−	69.0838i	1.18557	−	0.570940i	0.266041	−	0.963962i	\(-0.414285\pi\)
							0.919529	+	0.393022i	\(0.128570\pi\)
\(13\)	188.486	−	236.354i	1.11530	−	1.39854i	0.207963	−	0.978137i	\(-0.433317\pi\)
							0.907337	−	0.420405i	\(-0.138112\pi\)
\(17\)	−23.5120	−	29.4831i	−0.0813563	−	0.102018i	0.739488	−	0.673170i	\(-0.235068\pi\)
							−0.820844	+	0.571153i	\(0.806496\pi\)
\(19\)	−201.183	+	417.760i	−0.557293	+	1.15723i	0.411969	+	0.911198i	\(0.364841\pi\)
							−0.969262	+	0.246033i	\(0.920873\pi\)
\(23\)	−402.470	+	193.819i	−0.760812	+	0.366388i	−0.773719	−	0.633529i	\(-0.781606\pi\)
							0.0129067	+	0.999917i	\(0.495892\pi\)
\(29\)	629.627	+	143.708i	0.748664	+	0.170878i	0.579797	−	0.814761i	\(-0.303132\pi\)
							0.168868	+	0.985639i	\(0.445989\pi\)
\(31\)	−327.497	+	1434.86i	−0.340788	+	1.49309i	0.456626	+	0.889659i	\(0.349058\pi\)
							−0.797414	+	0.603432i	\(0.793799\pi\)
\(37\)		−	1970.48i		−	1.43936i	−0.694306	−	0.719680i	\(-0.744288\pi\)
							0.694306	−	0.719680i	\(-0.255712\pi\)
\(41\)	−265.562	+	1163.50i	−0.157979	+	0.692150i	0.832447	+	0.554105i	\(0.186939\pi\)
							−0.990426	+	0.138046i	\(0.955918\pi\)
\(43\)	345.495	+	1816.43i	0.186855	+	0.982387i
							\(47\)	−1920.84	−	925.030i	−0.869554	−	0.418755i	−0.0547556	−	0.998500i	\(-0.517438\pi\)
−0.814798	+	0.579745i	\(0.803152\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.8.5	✓	78
43.27	odd	14	inner	43.5.f.a.27.5	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.8.5	✓	78	1.1	even	1	trivial
43.5.f.a.27.5	yes	78	43.27	odd	14	inner