Embedded newform 43.3.f.a.8.4

• Label: 43.3.f.a.8.4
• Level: $43$
• Weight: $3$
• Character: 43.8
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) 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			43.3.f.a.27.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.811544 - 0.185230i) q^{2} +(-3.07466 + 0.701771i) q^{3} +(-2.97958 - 1.43489i) q^{4} +(-3.71827 - 2.96522i) q^{5} +2.62521 q^{6} -0.589128i q^{7} +(4.75551 + 3.79239i) q^{8} +(0.852334 - 0.410462i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 7 q^{2} - 7 q^{3} + 5 q^{4} - 7 q^{5} - 20 q^{6} + 21 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.811544	−	0.185230i	−0.405772	−	0.0926148i	0.0147632	−	0.999891i	\(-0.495301\pi\)
							−0.420535	+	0.907276i	\(0.638158\pi\)
\(3\)	−3.07466	+	0.701771i	−1.02489	+	0.233924i	−0.701746	−	0.712427i	\(-0.747596\pi\)
							−0.323141	+	0.946351i	\(0.604739\pi\)
\(5\)	−3.71827	−	2.96522i	−0.743654	−	0.593044i	0.176638	−	0.984276i	\(-0.443478\pi\)
							−0.920292	+	0.391231i	\(0.872049\pi\)
\(7\)		−	0.589128i		−	0.0841612i	−0.999114	−	0.0420806i	\(-0.986601\pi\)
							0.999114	−	0.0420806i	\(-0.0133986\pi\)
\(11\)	−7.06877	+	3.40414i	−0.642615	+	0.309467i	−0.726662	−	0.686995i	\(-0.758930\pi\)
							0.0840474	+	0.996462i	\(0.473215\pi\)
\(13\)	5.70426	−	7.15292i	0.438790	−	0.550225i	−0.512434	−	0.858726i	\(-0.671256\pi\)
							0.951224	+	0.308502i	\(0.0998276\pi\)
\(17\)	−12.6426	−	15.8533i	−0.743683	−	0.932549i	0.255732	−	0.966748i	\(-0.417683\pi\)
							−0.999415	+	0.0341991i	\(0.989112\pi\)
\(19\)	4.75371	−	9.87118i	0.250195	−	0.519536i	−0.737611	−	0.675226i	\(-0.764046\pi\)
							0.987806	+	0.155690i	\(0.0497602\pi\)
\(23\)	−26.7597	+	12.8868i	−1.16346	+	0.560295i	−0.913051	−	0.407844i	\(-0.866281\pi\)
							−0.250413	+	0.968139i	\(0.580566\pi\)
\(29\)	−27.4545	−	6.26632i	−0.946709	−	0.216080i	−0.278822	−	0.960343i	\(-0.589944\pi\)
							−0.667887	+	0.744263i	\(0.732801\pi\)
\(31\)	−9.89971	+	43.3735i	−0.319346	+	1.39914i	0.519358	+	0.854557i	\(0.326171\pi\)
							−0.838704	+	0.544588i	\(0.816686\pi\)
\(37\)		−	58.9416i		−	1.59302i	−0.604628	−	0.796508i	\(-0.706678\pi\)
							0.604628	−	0.796508i	\(-0.293322\pi\)
\(41\)	−10.7433	+	47.0693i	−0.262031	+	1.14803i	0.657014	+	0.753878i	\(0.271819\pi\)
							−0.919045	+	0.394153i	\(0.871038\pi\)
\(43\)	39.4038	+	17.2144i	0.916369	+	0.400336i
							\(47\)	41.2786	+	19.8787i	0.878268	+	0.422952i	0.817992	−	0.575230i	\(-0.195088\pi\)
0.0602763	+	0.998182i	\(0.480802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.f.a.8.4	✓	42
3.2	odd	2		387.3.w.b.352.4		42
43.27	odd	14	inner	43.3.f.a.27.4	yes	42
129.113	even	14		387.3.w.b.199.4		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.f.a.8.4	✓	42	1.1	even	1	trivial
43.3.f.a.27.4	yes	42	43.27	odd	14	inner
387.3.w.b.199.4		42	129.113	even	14
387.3.w.b.352.4		42	3.2	odd	2