Embedded newform 43.9.f.a.22.2

• Label: 43.9.f.a.22.2
• Level: $43$
• Weight: $9$
• Character: 43.22
• 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			22.2
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.1485 + 27.3031i) q^{2} +(64.2865 + 133.492i) q^{3} +(-412.962 - 517.838i) q^{4} +(-972.457 + 221.957i) q^{5} -4490.02 q^{6} +292.786i q^{7} +(12005.0 - 2740.07i) q^{8} +(-9596.72 + 12033.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.1485	+	27.3031i	−0.821779	+	1.70644i	−0.121479	+	0.992594i	\(0.538764\pi\)
							−0.700301	+	0.713848i	\(0.746951\pi\)
\(3\)	64.2865	+	133.492i	0.793660	+	1.64805i	0.761145	+	0.648581i	\(0.224637\pi\)
							0.0325150	+	0.999471i	\(0.489648\pi\)
\(5\)	−972.457	+	221.957i	−1.55593	+	0.355131i	−0.912077	−	0.410018i	\(-0.865522\pi\)
							−0.643853	+	0.765149i	\(0.722665\pi\)
\(7\)			292.786i			0.121944i	0.998139	+	0.0609718i	\(0.0194200\pi\)
							−0.998139	+	0.0609718i	\(0.980580\pi\)
\(11\)	2589.34	−	3246.93i	0.176855	−	0.221770i	−0.685501	−	0.728072i	\(-0.740417\pi\)
							0.862356	+	0.506302i	\(0.168988\pi\)
\(13\)	8875.90	+	38887.9i	0.310770	+	1.36157i	0.853249	+	0.521503i	\(0.174628\pi\)
							−0.542479	+	0.840069i	\(0.682514\pi\)
\(17\)	3387.48	−	14841.5i	0.0405584	−	0.177698i	−0.950592	−	0.310444i	\(-0.899522\pi\)
							0.991150	+	0.132746i	\(0.0423795\pi\)
\(19\)	−103922.	+	82875.2i	−0.797433	+	0.635931i	−0.935035	−	0.354556i	\(-0.884632\pi\)
							0.137602	+	0.990488i	\(0.456061\pi\)
\(23\)	173498.	−	217559.i	0.619987	−	0.777439i	−0.368356	−	0.929685i	\(-0.620079\pi\)
							0.988343	+	0.152246i	\(0.0486507\pi\)
\(29\)	−227500.	+	472408.i	−0.321654	+	0.667922i	−0.997616	−	0.0690154i	\(-0.978014\pi\)
							0.675962	+	0.736937i	\(0.263729\pi\)
\(31\)	358401.	+	172597.i	0.388082	+	0.186890i	0.617742	−	0.786380i	\(-0.288047\pi\)
							−0.229661	+	0.973271i	\(0.573762\pi\)
\(37\)			2.64788e6i			1.41283i	0.707796	+	0.706417i	\(0.249689\pi\)
							−0.707796	+	0.706417i	\(0.750311\pi\)
\(41\)	582380.	+	280459.i	0.206097	+	0.0992509i	0.534085	−	0.845431i	\(-0.320656\pi\)
							−0.327988	+	0.944682i	\(0.606371\pi\)
\(43\)	−2.87809e6	−	1.84521e6i	−0.841842	−	0.539724i
							\(47\)	−1.74737e6	−	2.19113e6i	−0.358091	−	0.449031i	0.569856	−	0.821745i	\(-0.306999\pi\)
−0.927947	+	0.372713i	\(0.878428\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.2	yes	174
43.2	odd	14	inner	43.9.f.a.2.2	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.2	✓	174	43.2	odd	14	inner
43.9.f.a.22.2	yes	174	1.1	even	1	trivial