Embedded newform 43.5.f.a.2.3

• Label: 43.5.f.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	43.2
Dual form			43.5.f.a.22.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.63624 - 3.39769i) q^{2} +(-3.03818 + 6.30885i) q^{3} +(1.10883 - 1.39043i) q^{4} +(-20.2554 - 4.62315i) q^{5} +26.4067 q^{6} +82.0298i q^{7} +(-65.3642 - 14.9189i) q^{8} +(19.9317 + 24.9935i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.63624	−	3.39769i	−0.409060	−	0.849422i	−0.999115	−	0.0420591i	\(-0.986608\pi\)
							0.590055	−	0.807363i	\(-0.299106\pi\)
\(3\)	−3.03818	+	6.30885i	−0.337576	+	0.700983i	−0.998788	−	0.0492156i	\(-0.984328\pi\)
							0.661213	+	0.750199i	\(0.270042\pi\)
\(5\)	−20.2554	−	4.62315i	−0.810215	−	0.184926i	−0.202706	−	0.979240i	\(-0.564974\pi\)
							−0.607508	+	0.794313i	\(0.707831\pi\)
\(7\)			82.0298i			1.67408i	0.547144	+	0.837039i	\(0.315715\pi\)
							−0.547144	+	0.837039i	\(0.684285\pi\)
\(11\)	4.13360	+	5.18337i	0.0341620	+	0.0428378i	0.798619	−	0.601837i	\(-0.205564\pi\)
							−0.764457	+	0.644675i	\(0.776993\pi\)
\(13\)	−41.4155	+	181.453i	−0.245062	+	1.07369i	0.691276	+	0.722590i	\(0.257049\pi\)
							−0.936339	+	0.351098i	\(0.885808\pi\)
\(17\)	29.1361	+	127.653i	0.100817	+	0.441707i	0.999992	+	0.00411755i	\(0.00131066\pi\)
							−0.899175	+	0.437590i	\(0.855832\pi\)
\(19\)	−244.758	−	195.188i	−0.678000	−	0.540687i	0.222831	−	0.974857i	\(-0.428470\pi\)
							−0.900831	+	0.434170i	\(0.857042\pi\)
\(23\)	−568.202	−	712.503i	−1.07411	−	1.34689i	−0.934211	−	0.356721i	\(-0.883895\pi\)
							−0.139896	−	0.990166i	\(-0.544677\pi\)
\(29\)	442.588	+	919.044i	0.526264	+	1.09280i	0.979507	+	0.201412i	\(0.0645530\pi\)
							−0.453242	+	0.891387i	\(0.649733\pi\)
\(31\)	923.104	−	444.543i	0.960566	−	0.462584i	0.113187	−	0.993574i	\(-0.463894\pi\)
							0.847378	+	0.530990i	\(0.178180\pi\)
\(37\)			789.147i			0.576440i	0.957564	+	0.288220i	\(0.0930635\pi\)
							−0.957564	+	0.288220i	\(0.906937\pi\)
\(41\)	2356.29	−	1134.73i	1.40172	−	0.675032i	0.428209	−	0.903680i	\(-0.359145\pi\)
							0.973509	+	0.228648i	\(0.0734306\pi\)
\(43\)	851.057	+	1641.49i	0.460280	+	0.887774i
							\(47\)	−788.331	+	988.536i	−0.356872	+	0.447504i	−0.927566	−	0.373659i	\(-0.878103\pi\)
0.570694	+	0.821163i	\(0.306674\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.2.3	✓	78
43.22	odd	14	inner	43.5.f.a.22.3	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.2.3	✓	78	1.1	even	1	trivial
43.5.f.a.22.3	yes	78	43.22	odd	14	inner