Embedded newform 43.9.f.a.2.13

• Label: 43.9.f.a.2.13
• Level: $43$
• Weight: $9$
• Character: 43.2
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) 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			43.9.f.a.22.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08134 - 2.24542i) q^{2} +(-24.6440 + 51.1739i) q^{3} +(155.741 - 195.293i) q^{4} +(-234.966 - 53.6295i) q^{5} +141.556 q^{6} -271.708i q^{7} +(-1228.94 - 280.497i) q^{8} +(2079.28 + 2607.33i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08134	−	2.24542i	−0.0675837	−	0.140339i	0.864440	−	0.502735i	\(-0.167673\pi\)
							−0.932024	+	0.362396i	\(0.881959\pi\)
\(3\)	−24.6440	+	51.1739i	−0.304247	+	0.631776i	−0.995901	−	0.0904534i	\(-0.971168\pi\)
							0.691653	+	0.722230i	\(0.256883\pi\)
\(5\)	−234.966	−	53.6295i	−0.375946	−	0.0858071i	0.0303729	−	0.999539i	\(-0.490331\pi\)
							−0.406319	+	0.913732i	\(0.633188\pi\)
\(7\)		−	271.708i		−	0.113164i	−0.998398	−	0.0565822i	\(-0.981980\pi\)
							0.998398	−	0.0565822i	\(-0.0180203\pi\)
\(11\)	−8442.85	−	10587.0i	−0.576658	−	0.723106i	0.404881	−	0.914370i	\(-0.367313\pi\)
							−0.981539	+	0.191263i	\(0.938742\pi\)
\(13\)	−5830.31	+	25544.2i	−0.204135	+	0.894375i	0.764251	+	0.644919i	\(0.223109\pi\)
							−0.968386	+	0.249456i	\(0.919748\pi\)
\(17\)	4856.65	+	21278.4i	0.0581488	+	0.254767i	0.995645	−	0.0932301i	\(-0.0297192\pi\)
							−0.937496	+	0.347997i	\(0.886862\pi\)
\(19\)	−139518.	−	111262.i	−1.07058	−	0.853755i	−0.0808485	−	0.996726i	\(-0.525763\pi\)
							−0.989727	+	0.142971i	\(0.954334\pi\)
\(23\)	−257839.	−	323320.i	−0.921378	−	1.15537i	−0.987509	−	0.157560i	\(-0.949637\pi\)
							0.0661317	−	0.997811i	\(-0.478934\pi\)
\(29\)	266264.	+	552903.i	0.376462	+	0.781731i	1.00000		6.21075e-5i	\(-1.97694e-5\pi\)
							−0.623538	+	0.781793i	\(0.714305\pi\)
\(31\)	−1.48961e6	+	717356.i	−1.61296	+	0.776762i	−0.999912	−	0.0132813i	\(-0.995772\pi\)
							−0.613051	+	0.790043i	\(0.710058\pi\)
\(37\)		−	2.79168e6i		−	1.48956i	−0.667308	−	0.744782i	\(-0.732554\pi\)
							0.667308	−	0.744782i	\(-0.267446\pi\)
\(41\)	−2.88103e6	+	1.38743e6i	−1.01956	+	0.490994i	−0.867532	−	0.497382i	\(-0.834295\pi\)
							−0.152028	+	0.988376i	\(0.548580\pi\)
\(43\)	2.30220e6	−	2.52746e6i	0.673395	−	0.739283i
							\(47\)	3.23468e6	−	4.05616e6i	0.662888	−	0.831235i	−0.330766	−	0.943713i	\(-0.607307\pi\)
0.993654	+	0.112477i	\(0.0358786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.13	✓	174
43.22	odd	14	inner	43.9.f.a.22.13	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.13	✓	174	1.1	even	1	trivial
43.9.f.a.22.13	yes	174	43.22	odd	14	inner