Embedded newform 43.5.h.a.3.12

• Label: 43.5.h.a.3.12
• Level: $43$
• Weight: $5$
• Character: 43.3
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.12
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.49210 - 5.17490i) q^{2} +(-7.43652 - 0.557291i) q^{3} +(-10.5932 - 13.2835i) q^{4} +(9.68683 - 31.4039i) q^{5} +(-21.4165 + 37.0945i) q^{6} +(-66.6650 + 38.4891i) q^{7} +(-5.54483 + 1.26557i) q^{8} +(-25.1040 - 3.78381i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.49210	−	5.17490i	0.623026	−	1.29373i	−0.315634	−	0.948881i	\(-0.602217\pi\)
							0.938660	−	0.344845i	\(-0.112069\pi\)
\(3\)	−7.43652	−	0.557291i	−0.826281	−	0.0619212i	−0.345125	−	0.938557i	\(-0.612164\pi\)
							−0.481155	+	0.876635i	\(0.659783\pi\)
\(5\)	9.68683	−	31.4039i	0.387473	−	1.25616i	−0.526428	−	0.850220i	\(-0.676469\pi\)
							0.913901	−	0.405937i	\(-0.133055\pi\)
\(7\)	−66.6650	+	38.4891i	−1.36051	+	0.785491i	−0.989692	−	0.143214i	\(-0.954256\pi\)
							−0.370819	+	0.928705i	\(0.620923\pi\)
\(11\)	42.0444	−	52.7220i	0.347474	−	0.435719i	−0.577128	−	0.816654i	\(-0.695826\pi\)
							0.924602	+	0.380935i	\(0.124398\pi\)
\(13\)	46.0248	−	42.7048i	0.272336	−	0.252691i	−0.532167	−	0.846639i	\(-0.678622\pi\)
							0.804503	+	0.593948i	\(0.202432\pi\)
\(17\)	364.016	−	112.284i	1.25957	−	0.388526i	0.408105	−	0.912935i	\(-0.366190\pi\)
							0.851467	+	0.524409i	\(0.175714\pi\)
\(19\)	−36.5253	−	242.329i	−0.101178	−	0.671272i	−0.980715	−	0.195442i	\(-0.937386\pi\)
							0.879537	−	0.475830i	\(-0.157852\pi\)
\(23\)	−190.472	−	485.315i	−0.360061	−	0.917420i	−0.990037	−	0.140806i	\(-0.955031\pi\)
							0.629976	−	0.776614i	\(-0.283065\pi\)
\(29\)	1040.08	−	77.9433i	1.23672	−	0.0926793i	0.559732	−	0.828674i	\(-0.310904\pi\)
							0.676987	+	0.735995i	\(0.263285\pi\)
\(31\)	−1412.03	+	962.705i	−1.46933	+	1.00177i	−0.476905	+	0.878955i	\(0.658241\pi\)
							−0.992429	+	0.122820i	\(0.960806\pi\)
\(37\)	814.342	+	470.160i	0.594844	+	0.343433i	0.767011	−	0.641634i	\(-0.221743\pi\)
							−0.172166	+	0.985068i	\(0.555077\pi\)
\(41\)	844.904	+	406.884i	0.502620	+	0.242049i	0.667973	−	0.744186i	\(-0.267162\pi\)
							−0.165353	+	0.986234i	\(0.552876\pi\)
\(43\)	538.910	+	1768.72i	0.291460	+	0.956583i
							\(47\)	−217.990	−	273.350i	−0.0986825	−	0.123744i	0.730038	−	0.683406i	\(-0.239502\pi\)
−0.828721	+	0.559662i	\(0.810931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.12	✓	168
43.29	odd	42	inner	43.5.h.a.29.12	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.12	✓	168	1.1	even	1	trivial
43.5.h.a.29.12	yes	168	43.29	odd	42	inner