Embedded newform 43.9.f.a.2.17

• Label: 43.9.f.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.22586 + 6.69857i) q^{2} +(-59.4708 + 123.492i) q^{3} +(125.149 - 156.932i) q^{4} +(267.347 + 61.0203i) q^{5} -1019.07 q^{6} +4439.62i q^{7} +(3310.53 + 755.607i) q^{8} +(-7622.86 - 9558.77i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.22586	+	6.69857i	0.201616	+	0.418661i	0.977122	−	0.212680i	\(-0.0682192\pi\)
							−0.775506	+	0.631341i	\(0.782505\pi\)
\(3\)	−59.4708	+	123.492i	−0.734207	+	1.52460i	0.113153	+	0.993578i	\(0.463905\pi\)
							−0.847360	+	0.531019i	\(0.821809\pi\)
\(5\)	267.347	+	61.0203i	0.427756	+	0.0976325i	0.430978	−	0.902362i	\(-0.358169\pi\)
							−0.00322237	+	0.999995i	\(0.501026\pi\)
\(7\)			4439.62i			1.84907i	0.381096	+	0.924535i	\(0.375547\pi\)
							−0.381096	+	0.924535i	\(0.624453\pi\)
\(11\)	9758.16	+	12236.3i	0.666496	+	0.835759i	0.994033	−	0.109079i	\(-0.0347902\pi\)
							−0.327538	+	0.944838i	\(0.606219\pi\)
\(13\)	−6426.39	+	28155.8i	−0.225006	+	0.985814i	0.728643	+	0.684893i	\(0.240151\pi\)
							−0.953649	+	0.300921i	\(0.902706\pi\)
\(17\)	−28076.8	−	123012.i	−0.336164	−	1.47283i	−0.806969	−	0.590594i	\(-0.798894\pi\)
							0.470805	−	0.882237i	\(-0.343964\pi\)
\(19\)	−32798.7	−	26156.1i	−0.251676	−	0.200705i	0.489523	−	0.871991i	\(-0.337171\pi\)
							−0.741199	+	0.671285i	\(0.765743\pi\)
\(23\)	79756.2	+	100011.i	0.285006	+	0.357386i	0.903640	−	0.428294i	\(-0.140885\pi\)
							−0.618634	+	0.785679i	\(0.712314\pi\)
\(29\)	−182316.	−	378582.i	−0.257770	−	0.535264i	0.731416	−	0.681931i	\(-0.238860\pi\)
							−0.989186	+	0.146667i	\(0.953145\pi\)
\(31\)	−447585.	+	215545.i	−0.484650	+	0.233395i	−0.660221	−	0.751072i	\(-0.729537\pi\)
							0.175571	+	0.984467i	\(0.443823\pi\)
\(37\)			3.20443e6i			1.70980i	0.518796	+	0.854898i	\(0.326380\pi\)
							−0.518796	+	0.854898i	\(0.673620\pi\)
\(41\)	1.45071e6	−	698625.i	0.513387	−	0.247234i	−0.159208	−	0.987245i	\(-0.550894\pi\)
							0.672595	+	0.740011i	\(0.265180\pi\)
\(43\)	2.84092e6	−	1.90194e6i	0.830970	−	0.556317i
							\(47\)	−2.10850e6	+	2.64398e6i	−0.432098	+	0.541834i	−0.949441	−	0.313944i	\(-0.898350\pi\)
0.517343	+	0.855778i	\(0.326921\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.17	✓	174
43.22	odd	14	inner	43.9.f.a.22.17	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.17	✓	174	1.1	even	1	trivial
43.9.f.a.22.17	yes	174	43.22	odd	14	inner