Embedded newform 43.5.f.a.22.3

• Label: 43.5.f.a.22.3
• Level: $43$
• Weight: $5$
• Character: 43.22
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $78$
• 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			22.3
Character	\(\chi\)	\(=\)	43.22
Dual form			43.5.f.a.2.3

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.2.3	✓	78	43.2	odd	14	inner
43.5.f.a.22.3	yes	78	1.1	even	1	trivial