Embedded newform 43.9.f.a.2.8

• Label: 43.9.f.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.10369 - 12.6745i) q^{2} +(64.4015 - 133.731i) q^{3} +(36.2267 - 45.4269i) q^{4} +(363.007 + 82.8539i) q^{5} -2088.06 q^{6} +1931.33i q^{7} +(-4307.89 - 983.247i) q^{8} +(-9645.74 - 12095.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.10369	−	12.6745i	−0.381481	−	0.792153i	−0.999980	−	0.00636757i	\(-0.997973\pi\)
							0.618499	−	0.785786i	\(-0.287741\pi\)
\(3\)	64.4015	−	133.731i	0.795080	−	1.65100i	0.0365784	−	0.999331i	\(-0.488354\pi\)
							0.758502	−	0.651671i	\(-0.225932\pi\)
\(5\)	363.007	+	82.8539i	0.580811	+	0.132566i	0.502825	−	0.864388i	\(-0.332294\pi\)
							0.0779857	+	0.996954i	\(0.475151\pi\)
\(7\)			1931.33i			0.804386i	0.915555	+	0.402193i	\(0.131752\pi\)
							−0.915555	+	0.402193i	\(0.868248\pi\)
\(11\)	−6279.53	−	7874.28i	−0.428900	−	0.537824i	0.519680	−	0.854361i	\(-0.326051\pi\)
							−0.948580	+	0.316537i	\(0.897480\pi\)
\(13\)	−8789.14	+	38507.7i	−0.307732	+	1.34826i	0.550429	+	0.834882i	\(0.314464\pi\)
							−0.858161	+	0.513381i	\(0.828393\pi\)
\(17\)	−31670.0	−	138755.i	−0.379186	−	1.66132i	−0.699971	−	0.714172i	\(-0.746804\pi\)
							0.320784	−	0.947152i	\(-0.396054\pi\)
\(19\)	−55624.5	−	44359.1i	−0.426827	−	0.340383i	0.386398	−	0.922332i	\(-0.373719\pi\)
							−0.813225	+	0.581949i	\(0.802290\pi\)
\(23\)	57783.9	+	72458.7i	0.206488	+	0.258928i	0.874282	−	0.485418i	\(-0.161333\pi\)
							−0.667794	+	0.744346i	\(0.732761\pi\)
\(29\)	20561.7	+	42696.9i	0.0290715	+	0.0603676i	0.914999	−	0.403456i	\(-0.132191\pi\)
							−0.885927	+	0.463824i	\(0.846477\pi\)
\(31\)	1.41806e6	−	682901.i	1.53549	−	0.739453i	0.540683	−	0.841226i	\(-0.318166\pi\)
							0.994808	+	0.101773i	\(0.0324515\pi\)
\(37\)		−	269704.i		−	0.143906i	−0.997408	−	0.0719532i	\(-0.977077\pi\)
							0.997408	−	0.0719532i	\(-0.0229232\pi\)
\(41\)	1.04465e6	−	503076.i	0.369688	−	0.178032i	−0.239810	−	0.970820i	\(-0.577085\pi\)
							0.609498	+	0.792788i	\(0.291371\pi\)
\(43\)	3.37626e6	+	537681.i	0.987555	+	0.157272i
							\(47\)	6.04210e6	−	7.57655e6i	1.23822	−	1.55267i	0.526580	−	0.850125i	\(-0.323474\pi\)
0.711636	−	0.702548i	\(-0.247955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.8	✓	174
43.22	odd	14	inner	43.9.f.a.22.8	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.8	✓	174	1.1	even	1	trivial
43.9.f.a.22.8	yes	174	43.22	odd	14	inner