Embedded newform 43.5.f.a.2.13

• Label: 43.5.f.a.2.13
• Level: $43$
• Weight: $5$
• Character: 43.2
• 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			2.13
Character	\(\chi\)	\(=\)	43.2
Dual form			43.5.f.a.22.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.21422 + 6.67439i) q^{2} +(-3.25251 + 6.75392i) q^{3} +(-24.2405 + 30.3966i) q^{4} +(40.7370 + 9.29795i) q^{5} -55.5326 q^{6} -88.1833i q^{7} +(-165.237 - 37.7142i) q^{8} +(15.4662 + 19.3939i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.21422	+	6.67439i	0.803555	+	1.66860i	0.741909	+	0.670501i	\(0.233921\pi\)
							0.0616457	+	0.998098i	\(0.480365\pi\)
\(3\)	−3.25251	+	6.75392i	−0.361390	+	0.750435i	−0.999815	−	0.0192496i	\(-0.993872\pi\)
							0.638424	+	0.769685i	\(0.279587\pi\)
\(5\)	40.7370	+	9.29795i	1.62948	+	0.371918i	0.936948	−	0.349470i	\(-0.113638\pi\)
							0.692532	+	0.721388i	\(0.256495\pi\)
\(7\)		−	88.1833i		−	1.79966i	−0.436242	−	0.899829i	\(-0.643691\pi\)
							0.436242	−	0.899829i	\(-0.356309\pi\)
\(11\)	−22.9945	−	28.8342i	−0.190038	−	0.238300i	0.677680	−	0.735357i	\(-0.262985\pi\)
							−0.867718	+	0.497057i	\(0.834414\pi\)
\(13\)	−8.88087	+	38.9096i	−0.0525495	+	0.230235i	−0.994382	−	0.105847i	\(-0.966245\pi\)
							0.941833	+	0.336082i	\(0.109102\pi\)
\(17\)	−5.67166	−	24.8492i	−0.0196251	−	0.0859833i	0.964167	−	0.265296i	\(-0.0854697\pi\)
							−0.983792	+	0.179313i	\(0.942613\pi\)
\(19\)	351.764	+	280.523i	0.974416	+	0.777071i	0.974835	−	0.222929i	\(-0.0715618\pi\)
							−0.000418567		1.00000i	\(0.500133\pi\)
\(23\)	−432.544	−	542.393i	−0.817663	−	1.02532i	−0.999121	−	0.0419185i	\(-0.986653\pi\)
							0.181458	−	0.983399i	\(-0.441918\pi\)
\(29\)	−79.2283	−	164.519i	−0.0942073	−	0.195623i	0.848540	−	0.529131i	\(-0.177482\pi\)
							−0.942747	+	0.333508i	\(0.891768\pi\)
\(31\)	62.5334	−	30.1145i	0.0650712	−	0.0313366i	−0.401065	−	0.916050i	\(-0.631360\pi\)
							0.466136	+	0.884713i	\(0.345646\pi\)
\(37\)		−	545.619i		−	0.398553i	−0.979943	−	0.199276i	\(-0.936141\pi\)
							0.979943	−	0.199276i	\(-0.0638592\pi\)
\(41\)	314.714	−	151.559i	0.187219	−	0.0901597i	−0.337926	−	0.941173i	\(-0.609725\pi\)
							0.525145	+	0.851013i	\(0.324011\pi\)
\(43\)	1771.91	−	528.347i	0.958305	−	0.285747i
							\(47\)	14.3922	−	18.0472i	0.00651525	−	0.00816987i	−0.778563	−	0.627566i	\(-0.784051\pi\)
0.785078	+	0.619396i	\(0.212623\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.2.13	✓	78
43.22	odd	14	inner	43.5.f.a.22.13	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.2.13	✓	78	1.1	even	1	trivial
43.5.f.a.22.13	yes	78	43.22	odd	14	inner