Embedded newform 43.5.f.a.2.1

• Label: 43.5.f.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	43.2
Dual form			43.5.f.a.22.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.00827 - 6.24674i) q^{2} +(-2.99008 + 6.20897i) q^{3} +(-19.9962 + 25.0745i) q^{4} +(23.3607 + 5.33193i) q^{5} +47.7808 q^{6} +14.3857i q^{7} +(108.636 + 24.7953i) q^{8} +(20.8920 + 26.1977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q - 7 q^{2} - 7 q^{3} + 69 q^{4} - 7 q^{5} - 140 q^{6} - 343 q^{8} + 486 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.00827	−	6.24674i	−0.752068	−	1.56169i	−0.825511	−	0.564386i	\(-0.809113\pi\)
							0.0734432	−	0.997299i	\(-0.476601\pi\)
\(3\)	−2.99008	+	6.20897i	−0.332231	+	0.689885i	−0.998437	−	0.0558823i	\(-0.982203\pi\)
							0.666206	+	0.745768i	\(0.267917\pi\)
\(5\)	23.3607	+	5.33193i	0.934428	+	0.213277i	0.662519	−	0.749046i	\(-0.269488\pi\)
							0.271910	+	0.962323i	\(0.412345\pi\)
\(7\)			14.3857i			0.293586i	0.989167	+	0.146793i	\(0.0468952\pi\)
							−0.989167	+	0.146793i	\(0.953105\pi\)
\(11\)	32.3093	+	40.5145i	0.267019	+	0.334831i	0.897206	−	0.441612i	\(-0.145593\pi\)
							−0.630187	+	0.776443i	\(0.717022\pi\)
\(13\)	46.3646	−	203.136i	0.274347	−	1.20199i	−0.630478	−	0.776207i	\(-0.717141\pi\)
							0.904825	−	0.425784i	\(-0.140002\pi\)
\(17\)	83.3506	+	365.183i	0.288410	+	1.26361i	0.886706	+	0.462333i	\(0.152987\pi\)
							−0.598296	+	0.801275i	\(0.704155\pi\)
\(19\)	482.671	+	384.918i	1.33704	+	1.06625i	0.991808	+	0.127736i	\(0.0407711\pi\)
							0.345232	+	0.938517i	\(0.387800\pi\)
\(23\)	261.923	+	328.442i	0.495129	+	0.620872i	0.965123	−	0.261797i	\(-0.0843152\pi\)
							−0.469994	+	0.882670i	\(0.655744\pi\)
\(29\)	−115.596	−	240.037i	−0.137451	−	0.285419i	0.820870	−	0.571116i	\(-0.193489\pi\)
							−0.958320	+	0.285697i	\(0.907775\pi\)
\(31\)	110.167	−	53.0538i	0.114638	−	0.0552068i	−0.375686	−	0.926747i	\(-0.622593\pi\)
							0.490324	+	0.871540i	\(0.336878\pi\)
\(37\)		−	675.037i		−	0.493088i	−0.969132	−	0.246544i	\(-0.920705\pi\)
							0.969132	−	0.246544i	\(-0.0792949\pi\)
\(41\)	−1715.52	+	826.149i	−1.02053	+	0.491463i	−0.867857	−	0.496814i	\(-0.834503\pi\)
							−0.152676	+	0.988276i	\(0.548789\pi\)
\(43\)	−1800.62	+	420.183i	−0.973837	+	0.227249i
							\(47\)	−232.750	+	291.859i	−0.105364	+	0.132123i	−0.831718	−	0.555199i	\(-0.812642\pi\)
0.726353	+	0.687322i	\(0.241214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.f.a.2.1	✓	78
43.22	odd	14	inner	43.5.f.a.22.1	yes	78

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.f.a.2.1	✓	78	1.1	even	1	trivial
43.5.f.a.22.1	yes	78	43.22	odd	14	inner